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+        "text": "Users often interact with large language models through black-box inference APIs, both for closed- and open-weight models (e.g., Llama models are popularly accessed via Amazon Bedrock and Azure AI Studios). In order to cut costs or add functionality, API providers may quantize, watermark, or finetune the underlying model, changing the output distribution — possibly without notifying users. We formalize detecting such distortions as Model Equality Testing, a two-sample testing problem, where the user collects samples from the API and a reference distribution, and conducts a statistical test to see if the two distributions are the same. We find that tests based on the Maximum Mean Discrepancy between distributions are powerful for this task: a test built on a simple string kernel achieves a median of  $77.4\\%$  power against a range of distortions, using an average of just 10 samples per prompt. We then apply this test to commercial inference APIs from Summer 2024 for four Llama models, finding that 11 out of 31 endpoints serve different distributions than reference weights released by Meta.",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Since running a large language model requires compute and technical expertise, many users rely on black-box APIs to handle inference. This applies to both closed-weight models, like GPT and Claude, and openweight ones: e.g., Amazon Bedrock, Microsoft Azure, and the seven other companies in Figure 1 all compete to offer Llama models as a service. While users can sample from black-box APIs, they have little to no insight into the underlying implementation of the model, including into questions like:",
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+            "1. How has the API modified the language model's distribution? To drive down costs, API providers may quantize or prune large model weights; they may also watermark outputs or incorrectly implement some decoding parameters. These changes distort the resulting distribution of completions. The problem is when such distortions are undisclosed: users assume that calling a third-party API is exactly equivalent to working with the original model. For example, benchmarks like HELM (Liang et al., 2022) evaluate models through third-party APIs, but quantized or watermarked models may be less capable than the intended model.",
+            "2. Is the API changing over time? Language model inference endpoints may also drift over time without notifying users (Chen et al., 2023; Eyuboglu et al., 2024), e.g., due to finetuning or updates to the inference stack. Unstable APIs affect research reproducibility (Pozzobon et al., 2023) and can disrupt user productivity in human-AI teams (Bansal et al., 2019)."
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+        "text": "Under the status quo, neither users nor regulators have a way to rigorously answer these questions for themselves. These concerns are important to address: tens of thousands of developers already rely on black-box inference APIs for applications (Amazon, 2024), and this dependence will increase as LLMs —",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
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+    {
+        "type": "page_footnote",
+        "text": "These problems are already experienced by users: e.g., see threads 1, 2 and 3.",
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+    {
+        "type": "page_number",
+        "text": "1",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/ffa5a3f202b1e62a6fb0b0357b2ce807d9547707475acdecfbb935239e47ceaa.jpg",
+        "image_caption": [
+            "Figure 1: (Left) We formalize auditing black-box language model inference APIs as Model Equality Testing. This enables us to assess an API's faithfulness to a reference distribution and its stability over time. (Right) We evaluate candidate tests and apply the most powerful one to Llama model APIs from Summer 2024, finding that 11 of 31 endpoints deviate from reference weights released by Meta."
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+    {
+        "type": "text",
+        "text": "and the corresponding infrastructure costs for hosting — grow larger. For example, most users must rely on third-party APIs to use Llama 3.1 405B because of its size.",
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+    {
+        "type": "text",
+        "text": "The current approach to this problem is for an outside auditor to monitor APIs' accuracies on multiple-choice or short-answer benchmarks (ArtificialAnalysis, 2024); these studies typically decode from the language model greedily. Such audits can be a poor match for user needs. Greedy decoding only checks that the modes of the next-token distributions match, rather than the overall distribution over completions, which is problematic because users often sample from models. Short-answer benchmarks cover only a small slice of possible prefixes, which may significantly differ from a particular user's task: e.g., common applications like code generation, dialogue, and summarization are longform tasks. Ideally, users could personally audit APIs on their custom tasks. Such a method should be sample-efficient, apply to tasks without automated verifiers, and assess with confidence whether the overall distribution of completions has shifted in a statistically significant way.",
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+        "type": "text",
+        "text": "We provide such a method. Suppose a user wishes to audit a an API on their task of interest. The user collects two samples: one from a reference distribution  $P$  and one from the test API's distribution  $Q$ . For example, to answer if an API has modified the distribution of an open-weight model,  $P$  might be from reference model weights released on Hugging Face. To answer if the API is changing over time,  $P$  might be from the API at an earlier point in time (Figure 1 left). The user then conducts a two-sample test for whether  $P = Q$  to examine if the API's distribution is statistically indistinguishable from the reference.",
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+    {
+        "type": "text",
+        "text": "Our setting is challenging because the distributions being compared are high-dimensional: they are defined over multi-token completions from large vocabularies. Two-sample kernel tests based on estimating the Maximum Mean Discrepancy (MMD) between  $P$  and  $Q$  (Gretton et al., 2012) are flexible tools for this setting, as they allow us to specify a featurization to reduce dimensionality. We find that a simple string kernel based on the Hamming distance between completions is particularly sample-efficient. In simulations (\\$4), this test achieves a median of  $77.4\\%$  power against a wide range of distortions — e.g., quantization, watermarking, and finetuning — using an average of just 10 samples per prompt for distributions over 20-25 prompts. We then apply this test to nine commercial inference API providers across four Llama models (\\$5, Figure 1 right). Our test flags 11 out of these 31 endpoints, with each audit costing less than $1.",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
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+    {
+        "type": "page_number",
+        "text": "2",
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+    {
+        "type": "text",
+        "text": "Because our test statistic is an estimate of a distance, we also explore how this machinery can quantify statistical distances between black-box endpoints. In §4.3, we estimate pairwise distances between the output distributions of 13 language models — without requiring log probability access — and find that models within the same family (e.g., the Llama family or GPT-3.5 family) output more similar distributions than models within the same size range (e.g., 7B or 70B models). In §5, we estimate the effect size of deviations between API endpoints and reference weights, finding that some implementations are further from the reference weights than if the provider had substituted in an entirely different language model.",
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+    {
+        "type": "text",
+        "text": "Summary of contributions. We unify several API auditing tasks under the formalization of Model Equality Testing, a two-sample distribution testing problem, and empirically validate kernel-based tests for this problem. We then apply this test to audit popular commercial inference APIs. To enable users to audit APIs for custom applications, we open-source a Python package. We also encourage future research in Model Equality Testing by releasing a dataset of 1 million LLM completions from five models.[2]",
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+    {
+        "type": "text",
+        "text": "2 THE MODEL EQUALITY TESTING PROBLEM",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Suppose an auditor is interested in a task parameterized by a distribution  $\\pi$  over  $m$  prompts and a maximum completion length of  $L$  tokens. The auditor has sample access to a reference distribution  $P$  and API distribution  $Q$ , both operating on the same vocabulary  $\\mathcal{V}$  with the same decoding parameters. The auditor samples  $N$  prompt-completion pairs:  $z := (x, y)$  from each distribution:",
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+        "type": "equation",
+        "text": "\n$$\n\\mathcal {D} _ {P} := \\left\\{z ^ {(i)}: x ^ {(i)} \\sim \\pi , y ^ {(i)} \\sim P (\\cdot \\mid x ^ {(i)}) \\right\\} _ {i = 1} ^ {N}, \\tag {1}\n$$\n",
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+        "type": "equation",
+        "text": "\n$$\n\\mathcal {D} _ {Q} := \\{z ^ {(i)}: x ^ {(i)} \\sim \\pi , y ^ {(i)} \\sim Q (\\cdot \\mid x ^ {(i)}) \\} _ {i = 1} ^ {N}.\n$$\n",
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+        "type": "text",
+        "text": "We wish to use these samples to test if  $P = Q$ , i.e., distinguish between the hypotheses",
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+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} H _ {0}: \\quad \\pi (x) P (y \\mid x) = \\pi (x) Q (y \\mid x), \\\\ H _ {1}: \\quad (\\cdot) P (\\cdot | \\cdot) \\cdot (\\cdot) Q (\\cdot | \\cdot). \\end{array} \\tag {2}\n$$\n",
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+    {
+        "type": "equation",
+        "text": "\n$$\nH _ {1}: \\pi (x) P (y \\mid x) \\neq \\pi (x) Q (y \\mid x).\n$$\n",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "We require that the Type-1 error rate is controlled at  $\\alpha$ . A good test will maximize power against unknown  $Q$  and generalize across several language models and prompt distributions  $\\pi$ . We are particularly interested in sample-efficient tests that are cheap to run: such tests are powerful when  $N$  is small compared to the size of the vocabulary  $|\\mathcal{V}|$  and the maximum completion length  $L$ . The latter parameters modulate the size of the space that the joint distributions are defined over: the set of all prompt-completion pairs has size  $m|\\mathcal{V}|^L$ , where  $m$  is the number of prompts captured in  $\\pi$ . Effective tests must navigate this high-dimensional space well. Fortunately, we expect the distributions in practice to be lower-dimensional, as language typically only places significant mass on a small number of tokens at each position.",
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+    {
+        "type": "text",
+        "text": "Throughout the paper, we will use  $\\mathbf{n}_s\\{S\\}$  to denote the count of object  $s$  in string or sample  $S$ ,  $\\pi P$  to denote the joint distribution of prompts and completions under  $P$ , and  $\\pi Q$  to denote the joint distribution under  $Q$ .",
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+    {
+        "type": "text",
+        "text": "3 METHOD",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "To tackle the problem, we employ a two-sample kernel test from Gretton et al. (2012). This test uses samples  $\\mathcal{D}_P$  and  $\\mathcal{D}_Q$  to estimate the Maximum Mean Discrepancy (MMD) between  $P$  and  $Q$ , which is a measure of the distance between the two distributions. Intuitively, if the estimated MMD is large, we reject the null hypothesis that  $P = Q$ .",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
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+    {
+        "type": "page_footnote",
+        "text": "2Package, experiment code, and dataset: https://github.com/i-gao/model-equality-testing.",
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+    {
+        "type": "page_footnote",
+        "text": "3In Appendix C.6, we discuss how to extend this setup to unequal sample sizes.",
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+    {
+        "type": "page_footnote",
+        "text": "As a concrete example, Llama-3 uses a vocabulary size of  $|\\mathcal{V}| = 128256$ , and users often sample  $L = 250$  tokens for longform generation tasks.",
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+        "type": "page_number",
+        "text": "3",
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+    {
+        "type": "text",
+        "text": "The MMD is defined with respect to a unit-norm kernel function  $k$  and its associated feature map  $\\phi$ . For our two joint distributions  $\\pi P$  and  $\\pi Q$ , the MMD is defined as the squared distance between the expected features from each distribution:",
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+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\mathrm {M M D} _ {k} (\\pi P, \\pi Q) = \\| \\mathbb {E} _ {z \\sim \\pi P} [ \\phi (z) ] - \\mathbb {E} _ {z \\sim \\pi Q} [ \\phi (z) ] \\| ^ {2}, \\\\ \\begin{array}{l} = \\| \\mathbb {E} _ {z} \\sim_ {\\pi P} [ \\phi (z) ] - \\mathbb {E} _ {z} \\sim_ {\\pi Q} [ \\phi (z) ] \\| ^ {2}, \\\\ = \\mathbb {E} _ {z, z ^ {\\prime} \\sim \\pi P} [ k (z, z ^ {\\prime}) ] + \\mathbb {E} _ {z, z ^ {\\prime} \\sim \\pi Q} [ k (z, z ^ {\\prime}) ] - 2 \\mathbb {E} _ {z \\sim \\pi P, z ^ {\\prime} \\sim \\pi Q} [ k (z, z ^ {\\prime}) ]. \\end{array} \\tag {3} \\\\ \\end{array}\n$$\n",
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+        "type": "text",
+        "text": "For simplicity, we select kernels of the form  $k(z,z^{\\prime}) = \\mathbf{1}\\{x = x^{\\prime}\\} \\tilde{k} (y,y^{\\prime})$ , where  $\\tilde{k}$  is a prompt-agnostic kernel over completions. Then  $\\mathrm{MMD}_k(\\pi P,\\pi Q) = \\mathbb{E}_{x\\sim \\pi}\\left[\\mathrm{MMD}_{\\tilde{k}}(P(y\\mid x),Q(y\\mid x))\\right]$ .",
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+    {
+        "type": "text",
+        "text": "To conduct a two-sample test with samples  $\\mathcal{D}_Q$  and  $\\mathcal{D}_P$ , the test statistic is the empirical estimator of (3):",
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+        "type": "equation",
+        "text": "\n$$\n\\widehat {\\operatorname {M M D}} \\left(\\mathcal {D} _ {Q}, \\mathcal {D} _ {P}\\right) = \\frac {1}{N (N - 1)} \\left[ \\sum_ {z, z ^ {\\prime} \\in \\mathcal {D} _ {P}} k \\left(z, z ^ {\\prime}\\right) + \\sum_ {z, z ^ {\\prime} \\in \\mathcal {D} _ {Q}} k \\left(z, z ^ {\\prime}\\right) \\right] - \\frac {2}{N ^ {2}} \\sum_ {z \\in \\mathcal {D} _ {P}} \\sum_ {z ^ {\\prime} \\in \\mathcal {D} _ {Q}} k \\left(z, z ^ {\\prime}\\right). \\tag {4}\n$$\n",
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+    {
+        "type": "text",
+        "text": "We can compute p-values by simulating the test statistic's distribution under the null, i.e., by repeatedly sampling both  $\\mathcal{D}_Q$  and  $\\mathcal{D}_P$  from  $P$  and computing (4). Alternatively, to avoid drawing extra samples from  $P$ , we can use the permutation procedure (Lehmann et al., 1986), at a potential cost to power. This procedure repeatedly shuffles samples between  $\\mathcal{D}_Q$  and  $\\mathcal{D}_P$  to recompute the test statistic (Appendix A.2).",
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+        "type": "text",
+        "text": "Kernel choice. The choice of kernel  $(k,\\phi)$  determines the test's semantics and power. For example, setting  $\\phi (z)$  to be an indicator of whether  $y$  passes an automated verifier for  $x$ , when one is available, leads to rejecting the null when  $P,Q$  result in different task accuracies. However, because  $\\phi$  is relatively coarse, the MMD may be zero even when  $P\\neq Q$ , limiting the test's power. At the other extreme are universal kernels, which guarantee that the MMD is zero if and only if  $P = Q$  (Gretton et al., 2012). One such universal kernel for strings is the computationally expensive all-substrings kernel (Borgwardt et al., 2006):",
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+    {
+        "type": "equation",
+        "text": "\n$$\n\\tilde {k} _ {\\text {a l l}} \\left(y, y ^ {\\prime}\\right) = \\sum_ {s \\in \\mathcal {V} ^ {\\leq L}} \\mathbf {n} _ {s} \\{y \\} \\cdot \\mathbf {n} _ {s} \\left\\{y ^ {\\prime} \\right\\}, \\tag {5}\n$$\n",
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+    {
+        "type": "text",
+        "text": "where  $\\mathbf{n}_s\\{y\\}$  is the number of times  $s$  appears in  $y$ . Another universal kernel is the one-hot kernel:",
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+    {
+        "type": "equation",
+        "text": "\n$$\n\\tilde {k} _ {\\text {o n e - h o t}} \\left(y, y ^ {\\prime}\\right) = \\mathbf {1} \\left\\{y = y ^ {\\prime} \\right\\}, \\tag {6}\n$$\n",
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+    {
+        "type": "text",
+        "text": "which results in a classical two-sample multinomial test between the joint distributions. While universal kernels can eventually detect differences between any  $P, Q$  with enough samples, they may have low power in the small-sample regime. For example, the one-hot MMD measures if there are more exact match collisions within  $\\mathcal{D}_P$  or  $\\mathcal{D}_Q$  than between them, but in small samples, we may see no duplicate completions at all.",
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+    {
+        "type": "text",
+        "text": "We posit that other string kernels, though not universal, provide more powerful features for testing with small samples. Specifically, we investigate a fast kernel related to the Hamming distance between completions:",
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+    {
+        "type": "equation",
+        "text": "\n$$\n\\tilde {k} _ {\\text {h a m m i n g}} \\left(y, y ^ {\\prime}\\right) = \\sum_ {i = 1} ^ {L} \\mathbf {1} \\left\\{y _ {i} = y _ {i} ^ {\\prime} \\right\\}, \\tag {7}\n$$\n",
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+    {
+        "type": "text",
+        "text": "where  $y$  shorter than  $L$  is right-padded with a special token. Intuitively, a test based on this kernel rejects if a significantly larger number of substitutions are needed to align completions between  $\\mathcal{D}_P$  and  $\\mathcal{D}_Q$  than within each sample. The associated Hamming MMD is a pseudo-metric, as it is zero when  $P = Q$  and obeys the triangle inequality, but may not separate all distributions (Appendix A.1). Despite this limitation, we find in the following sections that this kernel is empirically effective and well-suited to common distortions we encounter with language models: quantization, watermarking, finetuning, and related distortions tend to result in detectable inter-sample Hamming distances.",
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+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
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+        "text": "4",
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+    {
+        "type": "text",
+        "text": "4 EVALUATING TESTS IN SIMULATIONS",
+        "text_level": 1,
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+        "text": "In this section, we evaluate our test's power using different kernels at checking equivalence between known pairs of distributions. Specifically, we evaluate if tests can detect when a language model has been quantized or watermarked (§4.1), finetuned (§4.2), or swapped out for a different model altogether (§4.3).",
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+        "text": "All experiments in this section are run on a longform language modeling task. The prompt distribution  $\\pi$  is a uniform distribution over  $m = 25$  random 100-character strings sampled from English, German, Spanish, French, and Russian Wikipedia (Box 1). The maximum completion length is  $L = 50$ , and we sample using temperature 1. Power is computed from 100 Monte Carlo simulations. We estimate p-values by simulating the empirical distribution of the test statistic under the null 1000 times; in Appendix C, we validate that the permutation procedure results in the same trends.",
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+        "text": "Box 1: Sample prompt for the Wikipedia language modeling task",
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+        "type": "text",
+        "text": "Continue the paragraph. Do not output anything except the continuation to the paragraph. Start the continuation immediately.",
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+    {
+        "type": "text",
+        "text": "\"The British Arab Commercial Bank PLC (BACB) is an international wholesale bank incorporated in the U...\"",
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+        "text": "To evaluate tests' generalization across prompts, we repeat power experiments over ten different prompt distributions, where we resample 25 Wikipedia strings for each  $\\pi$ . All tests are conducted at a significance level of  $\\alpha = 0.05$ . Additional details can be found in Appendix B.",
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+    {
+        "type": "text",
+        "text": "4.1 DETECTING QUANTIZATION AND WATERMARKING",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In our first experiments, the reference distribution  $P$  represents full-precision weights published on Hugging Face. We evaluate if tests can distinguish  $P$  from alternative distributions  $Q$ :",
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+            "- Quantized models. These alternatives represent the model inferred at lower precisions: nf4 (Dettmers et al., 2024), int8 (Dettmers et al., 2022), and fp16. Some accounts suggest that quantization particularly degrades Llama-3 models on longform tasks (Reddit, 2024; Panda, 2024).",
+            "- Watermarked models. Some providers may watermark outputs so that they are later detectable as having been generated by the platform. We apply the watermarking algorithm from Kirchenbauer et al. (2023) with default bias of 2.5."
+        ],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We repeat evaluations for 5 instruction-tuned models: Mistral 7B Instruct (Jiang et al., 2023), Llama-3 8B and 70B Instruct, and Llama-3.1 8B and 70B Instruct (Meta, 2024). Additional models are in Appendix C.1.",
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+    {
+        "type": "text",
+        "text": "Tests. We compare three choices of kernels: the Hamming kernel (7), the all-substrings kernel (5), and the one-hot kernel (6). We also evaluate two tests from the multinomial testing literature (Balakrishnan & Wasserman, 2018; Bhattacharya & Valiant, 2015):",
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+        "type": "equation",
+        "text": "\n$$\n\\mathrm {L} 1 \\left(\\mathcal {D} _ {P}, \\mathcal {D} _ {Q}\\right) = \\sum_ {z \\in \\mathcal {D} _ {P} \\cup \\mathcal {D} _ {Q}} \\left| \\frac {\\mathbf {n} _ {z} \\left\\{\\mathcal {D} _ {P} \\right\\} - \\mathbf {n} _ {z} \\left\\{\\mathcal {D} _ {Q} \\right\\}}{N} \\right| \\tag {8}\n$$\n",
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+        "type": "equation",
+        "text": "\n$$\n\\chi^ {2} \\left(\\mathcal {D} _ {P}, \\mathcal {D} _ {Q}\\right) = N ^ {2} \\sum_ {z \\in \\mathcal {D} _ {P} \\cup \\mathcal {D} _ {Q}} \\frac {\\left(\\mathbf {n} _ {z} \\left\\{\\mathcal {D} _ {P} \\right\\} - \\mathbf {n} _ {z} \\left\\{\\mathcal {D} _ {Q} \\right\\}\\right) ^ {2} - \\mathbf {n} _ {z} \\left\\{\\mathcal {D} _ {P} \\right\\} - \\mathbf {n} _ {z} \\left\\{\\mathcal {D} _ {Q} \\right\\}}{\\mathbf {n} _ {z} \\left\\{\\mathcal {D} _ {P} \\right\\} + \\mathbf {n} _ {z} \\left\\{\\mathcal {D} _ {Q} \\right\\}}. \\tag {9}\n$$\n",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Results. Figure 2 (left) compares the empirical sample complexities of each test. To draw out a sample complexity curve, we vary the number of samples from  $N = 10m$  to  $N = 100m$ , where  $m = 25$  is the number of prompts in the prompt distribution. We observe that the Hamming MMD test attains the highest power with the fewest samples: at an average of 10 samples per prompt, this test has a median power",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
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+        "type": "page_number",
+        "text": "5",
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+    {
+        "type": "image",
+        "img_path": "images/21a9d4da6b41d6699b2dfc1c1cab8571e8a7e57d2c02e346097f51d65af0a123.jpg",
+        "image_caption": [
+            "Figure 2: (Left) Sample complexity of tests. At an average of just 10 samples per prompt, the Hamming MMD test is able to detect quantization and watermarking with nontrivial power. Curves first median power across alternative distributions  $Q$ , averaged over language models and prompt distributions, with shaded standard errors. Results stratified by language model and alternative are in Appendix C.1. (Middle) While other tests rapidly degrade in power when the user is interested in longer completions, the Hamming MMD test maintains power best across completion lengths. (Right) Power of the Hamming MMD test, stratified by alternative distribution. The test is significantly less powerful against the fp16 alternative."
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+        "type": "image",
+        "img_path": "images/acd615a60e86e2016818f639be216a4a53acb7bc9085e382dc7d3c59eba67b7c.jpg",
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+    {
+        "type": "text",
+        "text": "of  $77.4\\%$  across alternatives. In Figure 2 (right), we break down power by alternative distribution. The Hamming test is strong on all alternatives except fp16, where the initial power at  $N = 10m$  is much smaller. This suggests that fp16 and fp32 differ in ways that are not captured by the Hamming kernel.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To accommodate user tasks which require very long completions, it is important that tests retain power as the completion length  $L$  increases, even though the size of the sample space grows exponentially with  $L$ . In Figure 2 (middle), we fix  $N = 250$  (i.e.,  $N = 10m$ ) and vary the completion length  $L$  from 1 to 50 tokens. We observe that the Hamming MMD and all substring tests are more robust to increasing completion length than the other tests. This result is consistent with the intuition that a clever string kernel — as opposed to a one-hot kernel — can help MMD tests generalize to high-dimensional spaces.",
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+    {
+        "type": "text",
+        "text": "4.2 DETECTING FINETUNING",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Given their effectiveness in detecting quantized and watermarked models, we next ask if MMD-based tests can detect when a model has been finetuned. We finetune Llama-3 8B Instruct on two datasets: a disjoint, i.i.d. split of the testing Wikipedia task, and an out-of-distribution code dataset (Chaudhary, 2023). We use a small learning rate of  $1 \\times 10^{-6}$  with AdamW (Loshchilov, 2017). We then use the Hamming MMD test to compare finetuned checkpoints  $Q$  to the original model as the reference distribution  $P$ .",
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+    {
+        "type": "text",
+        "text": "Figure 3 (upper left) plots power against the checkpoint number of the finetuned model. The Hamming MMD test is always able to detect finetuning with nontrivial (greater than  $50\\%$ ) power, even after a single epoch (42 optimization steps). One might expect that finetuning on the out-of-distribution code dataset would not affect the model's distribution on the Wikipedia testing task, but we find this is not the case. Finetuning affects the model on other distributions enough to be detectable by statistical tests. These results suggest that it is challenging to isolate the effects of full finetuning to any single distribution, which may have implications for tasks such as unlearning or model editing (Hase et al., 2024).",
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+    {
+        "type": "text",
+        "text": "4.3 Distinguishing IF SAMPLES COME FROM DIFFERENT LANGUAGE MODELS",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We next explore if the Hamming MMD test can distinguish whether two bodies of text are generated from the same language model. In this setting,  $P$  and  $Q$  are two different language models, drawn from a pool of 13 instruction-tuned models (Figure 3 right), including eight open-weight models (Abdin et al., 2024; Groeneveld et al., 2024; Team et al., 2024) and five OpenAI closed-weight models. In order to compare",
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+        "page_idx": 5
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
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+    },
+    {
+        "type": "page_number",
+        "text": "6",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/94e69e505ff3f03c607f4b305ac5ebbfcb2d6189b2152bc8a05ba57072450e07.jpg",
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+    {
+        "type": "table",
+        "img_path": "images/d6809946c3f0f90427faa37f75d8130a0c93d4cb77ded96f2090af49857bef4a.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Alternative Q</td><td>Power (P = Llama-3.1 8B)</td></tr><tr><td>Llama-3 8B</td><td>0.76 (0.00)</td></tr><tr><td>Llama-3 70B</td><td>1.00 (0.00)</td></tr><tr><td>Llama-3.1 70B</td><td>0.75 (0.07)</td></tr><tr><td>OLMo 7B</td><td>1.00 (0.00)</td></tr><tr><td>gpt-4o-mini</td><td>1.00 (0.00)</td></tr><tr><td>Llama-3.1 8B (watermark)</td><td>0.32 (0.04)</td></tr><tr><td>Llama-3.1 8B (int8)</td><td>0.07 (0.01)</td></tr><tr><td>Llama-3.1 8B (nf4)</td><td>1.00 (0.00)</td></tr></table>",
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+    },
+    {
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+        "img_path": "images/095cbeaa7ae5be34560c123b1975d7386c811325ed8368921d2ba0e58c719847.jpg",
+        "image_caption": [
+            "Figure 3: (Upper left) The Hamming MMD test is able to detect when Llama-3 8B has been finetuned on datasets of 1000 samples, even after a single epoch. Power is higher, earlier, when the finetuning distribution is i.i.d. with the testing distribution. (Lower left) The Hamming MMD test can also detect when two models are different with near-perfect power. Standard errors are over prompt distributions. Full results are in Appendix C.2. (Right) The MMD framework allows us to estimate statistical distance between any models from which we can draw samples. The cells show average estimated MMDs over 10 bootstraps. Rows are sorted using spectral clustering with two components. Models within a family are typically clustered together, suggesting that factors like training data, rather than scale, determine model similarity."
+        ],
+        "image_footnote": [],
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+    },
+    {
+        "type": "text",
+        "text": "models with different tokenizers, samples must be compared in character space. We sample  $L = 50$ -token completions to the Wikipedia language modeling task as before, and then we decode these to characters, ignoring special tokens. The vocabulary of interest is now all Unicode characters (  $|\\mathcal{V}| = 155063$ ), the maximum completion length is  $L \\approx 1000$  characters, and we test with  $N = 10m$  samples.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Switching to character space makes the testing problem significantly more challenging, as the sample space is higher dimensional: Figure 3 (lower left) shows that power to detect quantization and watermarking is lower in this setting than in the token space setting ( $\\S 4.1$ ). However, even in this more challenging setting, we find that all model swaps are detectable with  $100\\%$  power, except for pairs within the Llama family, e.g., Llama-3.1 8B and Llama-3.1 70B (75% power) or Llama-3 8B and Llama-3.1 8B (76% power; Figure 3 lower left). Model swaps are significantly more detectable than watermarking or quantization. Appendix D.2 shows qualitative examples of completions from different models.",
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+    },
+    {
+        "type": "text",
+        "text": "Estimating distances between models. A useful feature of the MMD tests is that the test statistic is an estimator of a distance. As a result, we can reuse the machinery to quantify the degree to which two models differ by estimating  $\\mathbb{E}_{x\\sim \\pi}[\\mathrm{MMD}(P(\\cdot \\mid x),Q(\\cdot \\mid x))]$ .<sup>5</sup> Figure 3 (right) estimates the Hamming MMD between all pairs of models. To decrease estimator error, we increase the sample size to  $N = 100m$  samples from each model and report the average MMD over 10 simulations, along with standard errors. We observe that models within a family are typically clustered together, suggesting that training data, rather than",
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "page_footnote",
+        "text": "5Note that the Hamming MMD is a pseudometric:  $\\mathrm{MMD}_{\\mathrm{Hamming}}(P,Q) = 0$  does not imply  $P = Q$  (Appendix A.1).",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "page_number",
+        "text": "7",
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+    },
+    {
+        "type": "text",
+        "text": "scale, determines model similarity. Surprisingly, while generations of Llama models (3 and 3.1) are close in distance, some generations of GPT models (e.g., 4-preview and 3.5-turbo) are not. This result suggests the use of different training data or procedures between these models.",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 AUDITING INFERENCE API PROVIDERS",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "As a case study, we now apply our test to 31 commercial inference endpoints for four of Meta's Llama models (Figure 1). These endpoints are distributed across nine API providers from Summer 2024: Amazon Bedrock, Anyscale, $^{6}$  Azure AI Studio, Deepinfra, Fireworks AI, Groq, Perplexity, Replicate, and Together.ai. $^{7}$  We are interested in whether endpoints differ from weights published by Meta on Hugging Face inferred at commonly accepted precisions. Specifically, we consider two possible null distributions: the full-precision weights  $(P_{1})$  and the fp16-precision weights  $(P_{2})$ . The null and alternative hypotheses are",
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+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} H _ {0}: \\pi (x) P _ {1} (y \\mid x) = \\pi (x) Q (y \\mid x) \\quad \\text {O R} \\quad \\pi (x) P _ {2} (y \\mid x) = \\pi (x) Q (y \\mid x), \\\\ H _ {1}: \\pi (x) P _ {1} (y \\mid x) \\neq \\pi (x) Q (y \\mid x) \\quad \\text {A N D} \\quad \\pi (x) P _ {2} (y \\mid x) \\neq \\pi (x) Q (y \\mid x). \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "To test this composite hypothesis, we collect three samples:  $\\mathcal{D}_{P_1}$  and  $\\mathcal{D}_{P_2}$  from the two null distributions, and  $\\mathcal{D}_Q$  from the API. We then conduct 2 two-sample tests, one for  $(\\mathcal{D}_{P_1},\\mathcal{D}_Q)$  and another for  $(\\mathcal{D}_{P_2},\\mathcal{D}_Q)$ , and obtain p-values  $p_1$  and  $p_2$ . We set the overall rejection rule to",
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+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\delta \\left(\\mathcal {D} _ {P _ {1}}, \\mathcal {D} _ {P _ {2}}, \\mathcal {D} _ {Q}\\right) = \\mathbf {1} \\left\\{p _ {1} <   \\alpha \\wedge p _ {2} <   \\alpha \\right\\}. \\tag {11}\n$$\n",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "This rule continues to control the FPR at  $\\alpha$  under the composite null hypothesis: without loss of generality, suppose  $Q = P_{1}$ . Since  $P_{Q = P_1}(p_1 < \\alpha) \\leq \\alpha$ , we have  $P_{Q = P_1}(p_1 < \\alpha \\land p_2 < \\alpha) = P_{Q = P_1}(p_1 < \\alpha)P_{Q = P_1}(p_2 < \\alpha \\mid p_1 < \\alpha) \\leq \\alpha$ . Note that this rule may be pessimistic, reducing power.",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Experiment details. We consider testing with three prompt distributions  $\\pi$ . For all models, we test with one set of the Wikipedia completion task from §4, where  $\\pi$  is uniform over  $m = 25$  prompts, and  $L = 50$  tokens or around 1000 characters. For the smaller Llama-3 8B and Llama-3.1 8B models, we also test with the coding task HumanEval (Chen et al., 2021b) and instruction task UltraChat (Ding et al., 2023). Both  $\\pi$  are uniform over  $m = 20$  prompts, and  $L = 250$  tokens or 3000 characters. Because APIs often return decoded completions, rather than individual tokens, we conduct all tests in character space, as in §4.3. We explicitly requested all samples at temperature 1. Tests are conducted at level  $\\alpha = 0.01$  using  $N = 10m$  samples. To reduce variance, we repeat tests over ten samples and fail endpoints if the average rejection rate is  $\\geq 0.5$ . For the most expensive endpoint (Azure's Llama-3 70B), a Wikipedia audit costs $0.14, HumanEval $0.83, and UltraChat $0.93. For the cheapest endpoint (Fireworks' Llama-3 8B), all three audits cost less than $0.02. For additional details, including the dates we collect API samples, see Appendix B.",
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+        "type": "text",
+        "text": "Results. Despite power being generally reduced due to the composite decision rule, the test flags several endpoints (Table 1). Notably, Amazon Bedrock and Perplexity have the most endpoints flagged, with the latter failing all tests. In Box 2, we include an example comparing samples from the fp32 null and Perplexity; these samples suggest that Perplexity serves a lower entropy distribution than the full-precision model. Additional qualitative samples can be found in Appendix D.3. The HumanEval and UltraChat prompt distributions elicit more failures than the Wikipedia distribution; this may be because these knowledge-intensive distributions are more sensitive to changes such as quantization.",
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+        "type": "text",
+        "text": "To estimate the effect size of deviations, we estimate the MMD between providers and the nulls using ten bootstraps of  $N = 100m$  samples each. We find that some deviations are quite large: some APIs'",
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+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
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+    {
+        "type": "page_footnote",
+        "text": "<sup>6</sup>We collected samples from Anyscale's serverless endpoints from before they were deprecated in August 2024.",
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+    {
+        "type": "page_footnote",
+        "text": "At the time of writing, only two providers disclosed distribution-altering optimizations: Fireworks AI noted semantic caching, and Together.ai noted quantization.",
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+    {
+        "type": "page_footnote",
+        "text": "In Figure 1, we combine the results of the tests on the prompt distributions via a Bonferroni correction, setting the level of each test to  $\\alpha = 0.01 / 3$  and rejecting if the endpoint fails on any of the three prompt distributions.",
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+        "type": "page_number",
+        "text": "8",
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+    {
+        "type": "table",
+        "img_path": "images/04f49214416d29a431c0ab5eb2b409df8180ec2324865e6f4c793f98dddcf847.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"4\">Wikipedia</td><td colspan=\"2\">HumanEval</td><td colspan=\"2\">UltraChat</td></tr><tr><td></td><td>3 8B</td><td>3.1 8B</td><td>3 70B</td><td>3.1 70B</td><td>3 8B</td><td>3.1 8B</td><td>3 8B</td><td>3.1 8B</td></tr><tr><td>Amazon</td><td>✓</td><td>X</td><td>✓</td><td>X</td><td>✓</td><td>X</td><td>X</td><td>X</td></tr><tr><td>Anyscale</td><td>✓</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>Azure</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>Deepinfra</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>Fireworks</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>X</td><td>✓</td><td>X</td></tr><tr><td>Groq</td><td>✓</td><td>✓</td><td>✓</td><td>X</td><td>✓</td><td>X</td><td>✓</td><td>✓</td></tr><tr><td>Perplexity</td><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>Replicate</td><td>✓</td><td>—</td><td>X</td><td>—</td><td>✓</td><td>—</td><td>✓</td><td>—</td></tr><tr><td>Together</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr></table>",
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+        "type": "text",
+        "text": "Table 1: Audit results for 31 API endpoints across nine companies and four language models.  $\\times$  denotes an endpoint failure, i.e., the average rejection rate over ten samples is  $\\geq  {50}\\%$  .",
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+    {
+        "type": "image",
+        "img_path": "images/f626666dd9d33f77d7e6bf6685a239f41b44e9e520fafb58a6ee212508fb1a63.jpg",
+        "image_caption": [
+            "Figure 4: (Left) Average MMD (Hamming) between providers for each model. Amazon Bedrock's Llama-3 and -3.1 70B models are the most different from the other providers. (Right) Absolute difference in HumanEval average accuracy vs. the MMD (Hamming). There is a moderate positive correlation between MMD and task accuracy. Gray points indicate pairs where both distributions have accuracy  $< 10\\%$ . There are multiple ways to be wrong for a task, and the MMD captures these differences."
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+    {
+        "type": "text",
+        "text": "implementations are further from the reference weights than if the provider had substituted in an entirely different language model. For example, the deviation between Perplexity's Llama-3 8B and the fp32 null on the Wikipedia testing task is 0.03. This is comparable to the deviation between Llama-3 8B and GPT-3.5-Turbo (0.03; Figure 3 right), Phi-3 mini (0.04), or OLMo 7B (0.04).",
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+    {
+        "type": "text",
+        "text": "Correlating MMD with task accuracy. We now ask how well the Hamming MMD correlates with task accuracy when available. Automated verifiers exist for one of our prompt distributions, HumanEval. We find a moderate positive correlation between the absolute average accuracy difference and the Hamming MMD (Figure 4 right,  $R^2 = 0.392$ ). In several cases, the MMD is high but the accuracy difference is low. These are often when both task accuracies are low: the gray points in the figure highlight pairs where both distributions have accuracy  $< 10\\%$ . In these cases, the MMD captures that there are multiple ways to be wrong for a task. For example, the bottom-right-most point in the figure contrasts Llama-3.1 8B from Perplexity vs. the nf4 version; Perplexity answers with incorrect code snippets, while nf4 outputs nonsense completions (e.g., \"simulation!!!!!![...]\"). Samples from this comparison can be found in Appendix D.3.",
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+    },
+    {
+        "type": "text",
+        "text": "Which models do providers differ on most? Finally, we ask whether providers differ on some more models more than others. Figure 4 (left) estimates the MMD between providers for each model, and the large Llama-3.1 405B model. We find that deviations are larger on the earlier Llama models than the recent Llama-3.1 405B model.",
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "page_number",
+        "text": "9",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Box 2: Samples from fp32 vs. Perplexity for Llama-3 8B (Instruct)",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Prompt: Continue the paragraph. Do not output anything except the continuation to the paragraph. Start the continuation immediately.",
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "\"Weaver, Tom. I Talked with a Zombie: Interviews with 23 Veterans of Horror and Sci-Fi Films and Tele...\"",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "fp32",
+        "bbox": [
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+            209
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Perplexity",
+        "bbox": [
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+            210
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
+        "list_items": [
+            "\"...vision, Some of Whom Are Still Alive. New York: Pocket Books, 1981. According to Tom Weaver, Vincent Price was known for being a bit of a practical joker, and he would often play harmless pranks on the crew",
+            "\"...visions. New York: St. Martin's Press, 1988. This collection of interviews with horror and science fiction professionals is a fascinating look at the creative process and the people behind some of the most iconic films and TV shows of the",
+            "\"...vision, voodoo, and the Apocalypse. New York: St. Martin's Press, 1990. The book features interviews with notable horror and science fiction personalities, including Greg Cannom, makeup artist for The Lost Boys and RoboCop",
+            "\"...visions. New York: St. Martin's Press, 1988. In this collection of interviews, Weaver delves into the experiences of some of the most iconic figures in the horror and science fiction genres, including George Romero, John Carpenter,",
+            "\"...vision. Written by Tom Weaver, a renowned authority on classical horror cinema, this book brings together intimate and revealing conversations with some of the most legendary figures in the genre, including Vincent Price, Boris Karloff, and Bela Lugosi. Interviews",
+            "\"...vision. New York: St. Martin's Press, 1988. 224 pp. This collection of interviews with horror and science fiction professionals is a fascinating look at the people behind the genre's most iconic films. Weaver's conversational style"
+        ],
+        "bbox": [
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+            213,
+            812,
+            356
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 RELATED WORK",
+        "text_level": 1,
+        "bbox": [
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+            383,
+            315,
+            398
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Auditing ML APIs. Prior work either only applies to the classifier setting (Chen et al., 2021a; 2022) or audit by monitoring performance on low-temperature multiple-choice or short-answer benchmarks (ArtificialAnalysis, 2024; Chen et al., 2023; Eyuboglu et al., 2024). Concurrent work studies a general language model setting, but does not provide guarantees of audit correctness (Sam et al., 2025). In contrast, we formalize auditing as a hypothesis testing problem, allowing us to provide FPR guarantees, and we study the general setting of sampling completions from language models, including for tasks without automated verifiers. In cryptography, a body of work has aimed to verify ML API predictions through proof protocols (Ghodsi et al., 2017; Feng et al., 2021; Liu et al., 2021; Kang et al., 2022; Weng et al., 2023; Lee et al., 2024). These methods require APIs to cooperate and opt-in to providing proofs of valid inference alongside each prediction; public verifiers then check these proofs for correctness with perfect accuracy. Unfortunately, these methods scale too poorly to apply to large language models: in Kang et al. (2022), generating a proof takes 41 minutes per prediction for a 68M parameter MobileNet (Sandler et al., 2018).",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Two-sample testing. Testing whether two samples come from the same distribution is an established problem in statistics. A line of work focuses on testing multinomial distributions (Batu et al., 2013; Chan et al., 2014; Canonne, 2020), including when samples have unequal sizes (Bhattacharya & Valiant, 2015; Diakonikolas & Kane, 2016; Balakrishnan & Wasserman, 2018). MMD-based tests are a general approach that do not assume a specific distributional form (Gretton et al., 2012). These tests can be applied to structured data when paired with appropriate kernels (Lodhi et al., 2002; Borgwardt et al., 2006).",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "7 CONCLUSION",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "As the public grows more dependent on black-box APIs to interact with language models, tools for auditing these APIs are increasingly important. In this work, we unified several API auditing tasks under the formalization of Model Equality Testing, a two-sample distribution testing problem, and we extensively validated candidate tests for this problem. Future work could explore stronger tests than those we have presented, or explore how to adapt these tests to other modalities, such as image generation models. To help facilitate this research, we open-source the dataset of 1 million LLM completions used for this work at https://github.com/i-gao/model-equality-testing. This dataset contains completions from five models across quantized, watermarked, and API alternatives; also see Appendix B.1.",
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
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+            64
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "page_number",
+        "text": "10",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS",
+        "text_level": 1,
+        "bbox": [
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+            119,
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+            133
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The authors would like to thank Mert Yuksekgonul, Steven Cao, Chenchen Gu, Chenglei Si, Nicole Meister, Yifan Mai, Simon Guo, Teddi Worledge, Yuhui Zhang, and other members of the Guestrin and P-Lambda labs for feedback on this paper.",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "CONFLICTS OF INTEREST",
+        "text_level": 1,
+        "bbox": [
+            145,
+            196,
+            357,
+            212
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "PL is a co-founder of Together AI; this work was done in his Stanford capacity. The topic, research, and results of this work were not shared with Together, or any other API provider evaluated, until the public release of the paper.",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "REFERENCES",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2025",
+        "bbox": [
+            145,
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+        "page_idx": 11
+    },
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+    },
+    {
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+            "Reddit. Post by user: mo4gv9eywmpmw3xr. result: Llama 3 exl2 quant quality compared to gguf and llama 2. https://www.reddit.com/r/LocalLLaMA/comments/1cfbadc/result_llama_3_exl2_quant_quality_compared_to/, 2024.",
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2025/Model Equality Testing_ Which Model is this API Serving_/8d15512f-4850-44b9-9f62-11c1ca40138e_origin.pdf

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2025/Model Equality Testing_ Which Model is this API Serving_/full.md

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+# MODEL EQUALITY TESTING: WHICH MODEL IS THIS API SERVING?
+
+Irena Gao, Percy Liang, Carlos Guestrin
+
+irena@cs.stanford.edu, pliang@cs.stanford.edu, guestrin@stanford.edu
+
+Stanford University
+
+# ABSTRACT
+
+Users often interact with large language models through black-box inference APIs, both for closed- and open-weight models (e.g., Llama models are popularly accessed via Amazon Bedrock and Azure AI Studios). In order to cut costs or add functionality, API providers may quantize, watermark, or finetune the underlying model, changing the output distribution — possibly without notifying users. We formalize detecting such distortions as Model Equality Testing, a two-sample testing problem, where the user collects samples from the API and a reference distribution, and conducts a statistical test to see if the two distributions are the same. We find that tests based on the Maximum Mean Discrepancy between distributions are powerful for this task: a test built on a simple string kernel achieves a median of  $77.4\%$  power against a range of distortions, using an average of just 10 samples per prompt. We then apply this test to commercial inference APIs from Summer 2024 for four Llama models, finding that 11 out of 31 endpoints serve different distributions than reference weights released by Meta.
+
+# 1 INTRODUCTION
+
+Since running a large language model requires compute and technical expertise, many users rely on black-box APIs to handle inference. This applies to both closed-weight models, like GPT and Claude, and openweight ones: e.g., Amazon Bedrock, Microsoft Azure, and the seven other companies in Figure 1 all compete to offer Llama models as a service. While users can sample from black-box APIs, they have little to no insight into the underlying implementation of the model, including into questions like:
+
+1. How has the API modified the language model's distribution? To drive down costs, API providers may quantize or prune large model weights; they may also watermark outputs or incorrectly implement some decoding parameters. These changes distort the resulting distribution of completions. The problem is when such distortions are undisclosed: users assume that calling a third-party API is exactly equivalent to working with the original model. For example, benchmarks like HELM (Liang et al., 2022) evaluate models through third-party APIs, but quantized or watermarked models may be less capable than the intended model.  
+2. Is the API changing over time? Language model inference endpoints may also drift over time without notifying users (Chen et al., 2023; Eyuboglu et al., 2024), e.g., due to finetuning or updates to the inference stack. Unstable APIs affect research reproducibility (Pozzobon et al., 2023) and can disrupt user productivity in human-AI teams (Bansal et al., 2019).
+
+Under the status quo, neither users nor regulators have a way to rigorously answer these questions for themselves. These concerns are important to address: tens of thousands of developers already rely on black-box inference APIs for applications (Amazon, 2024), and this dependence will increase as LLMs —
+
+![](images/ffa5a3f202b1e62a6fb0b0357b2ce807d9547707475acdecfbb935239e47ceaa.jpg)  
+Figure 1: (Left) We formalize auditing black-box language model inference APIs as Model Equality Testing. This enables us to assess an API's faithfulness to a reference distribution and its stability over time. (Right) We evaluate candidate tests and apply the most powerful one to Llama model APIs from Summer 2024, finding that 11 of 31 endpoints deviate from reference weights released by Meta.
+
+![](images/71be53b4dcbcbf09ce6cacf1d5d4e223c674454353fb1534813ea550cd513673.jpg)
+
+and the corresponding infrastructure costs for hosting — grow larger. For example, most users must rely on third-party APIs to use Llama 3.1 405B because of its size.
+
+The current approach to this problem is for an outside auditor to monitor APIs' accuracies on multiple-choice or short-answer benchmarks (ArtificialAnalysis, 2024); these studies typically decode from the language model greedily. Such audits can be a poor match for user needs. Greedy decoding only checks that the modes of the next-token distributions match, rather than the overall distribution over completions, which is problematic because users often sample from models. Short-answer benchmarks cover only a small slice of possible prefixes, which may significantly differ from a particular user's task: e.g., common applications like code generation, dialogue, and summarization are longform tasks. Ideally, users could personally audit APIs on their custom tasks. Such a method should be sample-efficient, apply to tasks without automated verifiers, and assess with confidence whether the overall distribution of completions has shifted in a statistically significant way.
+
+We provide such a method. Suppose a user wishes to audit a an API on their task of interest. The user collects two samples: one from a reference distribution  $P$  and one from the test API's distribution  $Q$ . For example, to answer if an API has modified the distribution of an open-weight model,  $P$  might be from reference model weights released on Hugging Face. To answer if the API is changing over time,  $P$  might be from the API at an earlier point in time (Figure 1 left). The user then conducts a two-sample test for whether  $P = Q$  to examine if the API's distribution is statistically indistinguishable from the reference.
+
+Our setting is challenging because the distributions being compared are high-dimensional: they are defined over multi-token completions from large vocabularies. Two-sample kernel tests based on estimating the Maximum Mean Discrepancy (MMD) between  $P$  and  $Q$  (Gretton et al., 2012) are flexible tools for this setting, as they allow us to specify a featurization to reduce dimensionality. We find that a simple string kernel based on the Hamming distance between completions is particularly sample-efficient. In simulations (\$4), this test achieves a median of  $77.4\%$  power against a wide range of distortions — e.g., quantization, watermarking, and finetuning — using an average of just 10 samples per prompt for distributions over 20-25 prompts. We then apply this test to nine commercial inference API providers across four Llama models (\$5, Figure 1 right). Our test flags 11 out of these 31 endpoints, with each audit costing less than $1.
+
+Because our test statistic is an estimate of a distance, we also explore how this machinery can quantify statistical distances between black-box endpoints. In §4.3, we estimate pairwise distances between the output distributions of 13 language models — without requiring log probability access — and find that models within the same family (e.g., the Llama family or GPT-3.5 family) output more similar distributions than models within the same size range (e.g., 7B or 70B models). In §5, we estimate the effect size of deviations between API endpoints and reference weights, finding that some implementations are further from the reference weights than if the provider had substituted in an entirely different language model.
+
+Summary of contributions. We unify several API auditing tasks under the formalization of Model Equality Testing, a two-sample distribution testing problem, and empirically validate kernel-based tests for this problem. We then apply this test to audit popular commercial inference APIs. To enable users to audit APIs for custom applications, we open-source a Python package. We also encourage future research in Model Equality Testing by releasing a dataset of 1 million LLM completions from five models.[2]
+
+# 2 THE MODEL EQUALITY TESTING PROBLEM
+
+Suppose an auditor is interested in a task parameterized by a distribution  $\pi$  over  $m$  prompts and a maximum completion length of  $L$  tokens. The auditor has sample access to a reference distribution  $P$  and API distribution  $Q$ , both operating on the same vocabulary  $\mathcal{V}$  with the same decoding parameters. The auditor samples  $N$  prompt-completion pairs:  $z := (x, y)$  from each distribution:
+
+$$
+\mathcal {D} _ {P} := \left\{z ^ {(i)}: x ^ {(i)} \sim \pi , y ^ {(i)} \sim P (\cdot \mid x ^ {(i)}) \right\} _ {i = 1} ^ {N}, \tag {1}
+$$
+
+$$
+\mathcal {D} _ {Q} := \{z ^ {(i)}: x ^ {(i)} \sim \pi , y ^ {(i)} \sim Q (\cdot \mid x ^ {(i)}) \} _ {i = 1} ^ {N}.
+$$
+
+We wish to use these samples to test if  $P = Q$ , i.e., distinguish between the hypotheses
+
+$$
+\begin{array}{l} H _ {0}: \quad \pi (x) P (y \mid x) = \pi (x) Q (y \mid x), \\ H _ {1}: \quad (\cdot) P (\cdot | \cdot) \cdot (\cdot) Q (\cdot | \cdot). \end{array} \tag {2}
+$$
+
+$$
+H _ {1}: \pi (x) P (y \mid x) \neq \pi (x) Q (y \mid x).
+$$
+
+We require that the Type-1 error rate is controlled at  $\alpha$ . A good test will maximize power against unknown  $Q$  and generalize across several language models and prompt distributions  $\pi$ . We are particularly interested in sample-efficient tests that are cheap to run: such tests are powerful when  $N$  is small compared to the size of the vocabulary  $|\mathcal{V}|$  and the maximum completion length  $L$ . The latter parameters modulate the size of the space that the joint distributions are defined over: the set of all prompt-completion pairs has size  $m|\mathcal{V}|^L$ , where  $m$  is the number of prompts captured in  $\pi$ . Effective tests must navigate this high-dimensional space well. Fortunately, we expect the distributions in practice to be lower-dimensional, as language typically only places significant mass on a small number of tokens at each position.
+
+Throughout the paper, we will use  $\mathbf{n}_s\{S\}$  to denote the count of object  $s$  in string or sample  $S$ ,  $\pi P$  to denote the joint distribution of prompts and completions under  $P$ , and  $\pi Q$  to denote the joint distribution under  $Q$ .
+
+# 3 METHOD
+
+To tackle the problem, we employ a two-sample kernel test from Gretton et al. (2012). This test uses samples  $\mathcal{D}_P$  and  $\mathcal{D}_Q$  to estimate the Maximum Mean Discrepancy (MMD) between  $P$  and  $Q$ , which is a measure of the distance between the two distributions. Intuitively, if the estimated MMD is large, we reject the null hypothesis that  $P = Q$ .
+
+The MMD is defined with respect to a unit-norm kernel function  $k$  and its associated feature map  $\phi$ . For our two joint distributions  $\pi P$  and  $\pi Q$ , the MMD is defined as the squared distance between the expected features from each distribution:
+
+$$
+\begin{array}{l} \mathrm {M M D} _ {k} (\pi P, \pi Q) = \| \mathbb {E} _ {z \sim \pi P} [ \phi (z) ] - \mathbb {E} _ {z \sim \pi Q} [ \phi (z) ] \| ^ {2}, \\ \begin{array}{l} = \| \mathbb {E} _ {z} \sim_ {\pi P} [ \phi (z) ] - \mathbb {E} _ {z} \sim_ {\pi Q} [ \phi (z) ] \| ^ {2}, \\ = \mathbb {E} _ {z, z ^ {\prime} \sim \pi P} [ k (z, z ^ {\prime}) ] + \mathbb {E} _ {z, z ^ {\prime} \sim \pi Q} [ k (z, z ^ {\prime}) ] - 2 \mathbb {E} _ {z \sim \pi P, z ^ {\prime} \sim \pi Q} [ k (z, z ^ {\prime}) ]. \end{array} \tag {3} \\ \end{array}
+$$
+
+For simplicity, we select kernels of the form  $k(z,z^{\prime}) = \mathbf{1}\{x = x^{\prime}\} \tilde{k} (y,y^{\prime})$ , where  $\tilde{k}$  is a prompt-agnostic kernel over completions. Then  $\mathrm{MMD}_k(\pi P,\pi Q) = \mathbb{E}_{x\sim \pi}\left[\mathrm{MMD}_{\tilde{k}}(P(y\mid x),Q(y\mid x))\right]$ .
+
+To conduct a two-sample test with samples  $\mathcal{D}_Q$  and  $\mathcal{D}_P$ , the test statistic is the empirical estimator of (3):
+
+$$
+\widehat {\operatorname {M M D}} \left(\mathcal {D} _ {Q}, \mathcal {D} _ {P}\right) = \frac {1}{N (N - 1)} \left[ \sum_ {z, z ^ {\prime} \in \mathcal {D} _ {P}} k \left(z, z ^ {\prime}\right) + \sum_ {z, z ^ {\prime} \in \mathcal {D} _ {Q}} k \left(z, z ^ {\prime}\right) \right] - \frac {2}{N ^ {2}} \sum_ {z \in \mathcal {D} _ {P}} \sum_ {z ^ {\prime} \in \mathcal {D} _ {Q}} k \left(z, z ^ {\prime}\right). \tag {4}
+$$
+
+We can compute p-values by simulating the test statistic's distribution under the null, i.e., by repeatedly sampling both  $\mathcal{D}_Q$  and  $\mathcal{D}_P$  from  $P$  and computing (4). Alternatively, to avoid drawing extra samples from  $P$ , we can use the permutation procedure (Lehmann et al., 1986), at a potential cost to power. This procedure repeatedly shuffles samples between  $\mathcal{D}_Q$  and  $\mathcal{D}_P$  to recompute the test statistic (Appendix A.2).
+
+Kernel choice. The choice of kernel  $(k,\phi)$  determines the test's semantics and power. For example, setting  $\phi (z)$  to be an indicator of whether  $y$  passes an automated verifier for  $x$ , when one is available, leads to rejecting the null when  $P,Q$  result in different task accuracies. However, because  $\phi$  is relatively coarse, the MMD may be zero even when  $P\neq Q$ , limiting the test's power. At the other extreme are universal kernels, which guarantee that the MMD is zero if and only if  $P = Q$  (Gretton et al., 2012). One such universal kernel for strings is the computationally expensive all-substrings kernel (Borgwardt et al., 2006):
+
+$$
+\tilde {k} _ {\text {a l l}} \left(y, y ^ {\prime}\right) = \sum_ {s \in \mathcal {V} ^ {\leq L}} \mathbf {n} _ {s} \{y \} \cdot \mathbf {n} _ {s} \left\{y ^ {\prime} \right\}, \tag {5}
+$$
+
+where  $\mathbf{n}_s\{y\}$  is the number of times  $s$  appears in  $y$ . Another universal kernel is the one-hot kernel:
+
+$$
+\tilde {k} _ {\text {o n e - h o t}} \left(y, y ^ {\prime}\right) = \mathbf {1} \left\{y = y ^ {\prime} \right\}, \tag {6}
+$$
+
+which results in a classical two-sample multinomial test between the joint distributions. While universal kernels can eventually detect differences between any  $P, Q$  with enough samples, they may have low power in the small-sample regime. For example, the one-hot MMD measures if there are more exact match collisions within  $\mathcal{D}_P$  or  $\mathcal{D}_Q$  than between them, but in small samples, we may see no duplicate completions at all.
+
+We posit that other string kernels, though not universal, provide more powerful features for testing with small samples. Specifically, we investigate a fast kernel related to the Hamming distance between completions:
+
+$$
+\tilde {k} _ {\text {h a m m i n g}} \left(y, y ^ {\prime}\right) = \sum_ {i = 1} ^ {L} \mathbf {1} \left\{y _ {i} = y _ {i} ^ {\prime} \right\}, \tag {7}
+$$
+
+where  $y$  shorter than  $L$  is right-padded with a special token. Intuitively, a test based on this kernel rejects if a significantly larger number of substitutions are needed to align completions between  $\mathcal{D}_P$  and  $\mathcal{D}_Q$  than within each sample. The associated Hamming MMD is a pseudo-metric, as it is zero when  $P = Q$  and obeys the triangle inequality, but may not separate all distributions (Appendix A.1). Despite this limitation, we find in the following sections that this kernel is empirically effective and well-suited to common distortions we encounter with language models: quantization, watermarking, finetuning, and related distortions tend to result in detectable inter-sample Hamming distances.
+
+# 4 EVALUATING TESTS IN SIMULATIONS
+
+In this section, we evaluate our test's power using different kernels at checking equivalence between known pairs of distributions. Specifically, we evaluate if tests can detect when a language model has been quantized or watermarked (§4.1), finetuned (§4.2), or swapped out for a different model altogether (§4.3).
+
+All experiments in this section are run on a longform language modeling task. The prompt distribution  $\pi$  is a uniform distribution over  $m = 25$  random 100-character strings sampled from English, German, Spanish, French, and Russian Wikipedia (Box 1). The maximum completion length is  $L = 50$ , and we sample using temperature 1. Power is computed from 100 Monte Carlo simulations. We estimate p-values by simulating the empirical distribution of the test statistic under the null 1000 times; in Appendix C, we validate that the permutation procedure results in the same trends.
+
+# Box 1: Sample prompt for the Wikipedia language modeling task
+
+Continue the paragraph. Do not output anything except the continuation to the paragraph. Start the continuation immediately.
+
+"The British Arab Commercial Bank PLC (BACB) is an international wholesale bank incorporated in the U..."
+
+To evaluate tests' generalization across prompts, we repeat power experiments over ten different prompt distributions, where we resample 25 Wikipedia strings for each  $\pi$ . All tests are conducted at a significance level of  $\alpha = 0.05$ . Additional details can be found in Appendix B.
+
+# 4.1 DETECTING QUANTIZATION AND WATERMARKING
+
+In our first experiments, the reference distribution  $P$  represents full-precision weights published on Hugging Face. We evaluate if tests can distinguish  $P$  from alternative distributions  $Q$ :
+
+- Quantized models. These alternatives represent the model inferred at lower precisions: nf4 (Dettmers et al., 2024), int8 (Dettmers et al., 2022), and fp16. Some accounts suggest that quantization particularly degrades Llama-3 models on longform tasks (Reddit, 2024; Panda, 2024).  
+- Watermarked models. Some providers may watermark outputs so that they are later detectable as having been generated by the platform. We apply the watermarking algorithm from Kirchenbauer et al. (2023) with default bias of 2.5.
+
+We repeat evaluations for 5 instruction-tuned models: Mistral 7B Instruct (Jiang et al., 2023), Llama-3 8B and 70B Instruct, and Llama-3.1 8B and 70B Instruct (Meta, 2024). Additional models are in Appendix C.1.
+
+Tests. We compare three choices of kernels: the Hamming kernel (7), the all-substrings kernel (5), and the one-hot kernel (6). We also evaluate two tests from the multinomial testing literature (Balakrishnan & Wasserman, 2018; Bhattacharya & Valiant, 2015):
+
+$$
+\mathrm {L} 1 \left(\mathcal {D} _ {P}, \mathcal {D} _ {Q}\right) = \sum_ {z \in \mathcal {D} _ {P} \cup \mathcal {D} _ {Q}} \left| \frac {\mathbf {n} _ {z} \left\{\mathcal {D} _ {P} \right\} - \mathbf {n} _ {z} \left\{\mathcal {D} _ {Q} \right\}}{N} \right| \tag {8}
+$$
+
+$$
+\chi^ {2} \left(\mathcal {D} _ {P}, \mathcal {D} _ {Q}\right) = N ^ {2} \sum_ {z \in \mathcal {D} _ {P} \cup \mathcal {D} _ {Q}} \frac {\left(\mathbf {n} _ {z} \left\{\mathcal {D} _ {P} \right\} - \mathbf {n} _ {z} \left\{\mathcal {D} _ {Q} \right\}\right) ^ {2} - \mathbf {n} _ {z} \left\{\mathcal {D} _ {P} \right\} - \mathbf {n} _ {z} \left\{\mathcal {D} _ {Q} \right\}}{\mathbf {n} _ {z} \left\{\mathcal {D} _ {P} \right\} + \mathbf {n} _ {z} \left\{\mathcal {D} _ {Q} \right\}}. \tag {9}
+$$
+
+Results. Figure 2 (left) compares the empirical sample complexities of each test. To draw out a sample complexity curve, we vary the number of samples from  $N = 10m$  to  $N = 100m$ , where  $m = 25$  is the number of prompts in the prompt distribution. We observe that the Hamming MMD test attains the highest power with the fewest samples: at an average of 10 samples per prompt, this test has a median power
+
+![](images/21a9d4da6b41d6699b2dfc1c1cab8571e8a7e57d2c02e346097f51d65af0a123.jpg)  
+Figure 2: (Left) Sample complexity of tests. At an average of just 10 samples per prompt, the Hamming MMD test is able to detect quantization and watermarking with nontrivial power. Curves first median power across alternative distributions  $Q$ , averaged over language models and prompt distributions, with shaded standard errors. Results stratified by language model and alternative are in Appendix C.1. (Middle) While other tests rapidly degrade in power when the user is interested in longer completions, the Hamming MMD test maintains power best across completion lengths. (Right) Power of the Hamming MMD test, stratified by alternative distribution. The test is significantly less powerful against the fp16 alternative.
+
+![](images/156707823457cd2fdad9d9320c2c9630602a0da6ac9be28ea0bb18733c3b2387.jpg)
+
+![](images/9297a7d8405ed39b808b7aef766904a7242731ad37cdc3b2b717ac2bf1faa3ea.jpg)
+
+![](images/37c2999e6634af86627d73849646eef4322a4a9bc8ba93b6d4f0031f61743022.jpg)
+
+![](images/acd615a60e86e2016818f639be216a4a53acb7bc9085e382dc7d3c59eba67b7c.jpg)
+
+of  $77.4\%$  across alternatives. In Figure 2 (right), we break down power by alternative distribution. The Hamming test is strong on all alternatives except fp16, where the initial power at  $N = 10m$  is much smaller. This suggests that fp16 and fp32 differ in ways that are not captured by the Hamming kernel.
+
+To accommodate user tasks which require very long completions, it is important that tests retain power as the completion length  $L$  increases, even though the size of the sample space grows exponentially with  $L$ . In Figure 2 (middle), we fix  $N = 250$  (i.e.,  $N = 10m$ ) and vary the completion length  $L$  from 1 to 50 tokens. We observe that the Hamming MMD and all substring tests are more robust to increasing completion length than the other tests. This result is consistent with the intuition that a clever string kernel — as opposed to a one-hot kernel — can help MMD tests generalize to high-dimensional spaces.
+
+# 4.2 DETECTING FINETUNING
+
+Given their effectiveness in detecting quantized and watermarked models, we next ask if MMD-based tests can detect when a model has been finetuned. We finetune Llama-3 8B Instruct on two datasets: a disjoint, i.i.d. split of the testing Wikipedia task, and an out-of-distribution code dataset (Chaudhary, 2023). We use a small learning rate of  $1 \times 10^{-6}$  with AdamW (Loshchilov, 2017). We then use the Hamming MMD test to compare finetuned checkpoints  $Q$  to the original model as the reference distribution  $P$ .
+
+Figure 3 (upper left) plots power against the checkpoint number of the finetuned model. The Hamming MMD test is always able to detect finetuning with nontrivial (greater than  $50\%$ ) power, even after a single epoch (42 optimization steps). One might expect that finetuning on the out-of-distribution code dataset would not affect the model's distribution on the Wikipedia testing task, but we find this is not the case. Finetuning affects the model on other distributions enough to be detectable by statistical tests. These results suggest that it is challenging to isolate the effects of full finetuning to any single distribution, which may have implications for tasks such as unlearning or model editing (Hase et al., 2024).
+
+# 4.3 Distinguishing IF SAMPLES COME FROM DIFFERENT LANGUAGE MODELS
+
+We next explore if the Hamming MMD test can distinguish whether two bodies of text are generated from the same language model. In this setting,  $P$  and  $Q$  are two different language models, drawn from a pool of 13 instruction-tuned models (Figure 3 right), including eight open-weight models (Abdin et al., 2024; Groeneveld et al., 2024; Team et al., 2024) and five OpenAI closed-weight models. In order to compare
+
+![](images/94e69e505ff3f03c607f4b305ac5ebbfcb2d6189b2152bc8a05ba57072450e07.jpg)
+
+<table><tr><td>Alternative Q</td><td>Power (P = Llama-3.1 8B)</td></tr><tr><td>Llama-3 8B</td><td>0.76 (0.00)</td></tr><tr><td>Llama-3 70B</td><td>1.00 (0.00)</td></tr><tr><td>Llama-3.1 70B</td><td>0.75 (0.07)</td></tr><tr><td>OLMo 7B</td><td>1.00 (0.00)</td></tr><tr><td>gpt-4o-mini</td><td>1.00 (0.00)</td></tr><tr><td>Llama-3.1 8B (watermark)</td><td>0.32 (0.04)</td></tr><tr><td>Llama-3.1 8B (int8)</td><td>0.07 (0.01)</td></tr><tr><td>Llama-3.1 8B (nf4)</td><td>1.00 (0.00)</td></tr></table>
+
+![](images/095cbeaa7ae5be34560c123b1975d7386c811325ed8368921d2ba0e58c719847.jpg)  
+Figure 3: (Upper left) The Hamming MMD test is able to detect when Llama-3 8B has been finetuned on datasets of 1000 samples, even after a single epoch. Power is higher, earlier, when the finetuning distribution is i.i.d. with the testing distribution. (Lower left) The Hamming MMD test can also detect when two models are different with near-perfect power. Standard errors are over prompt distributions. Full results are in Appendix C.2. (Right) The MMD framework allows us to estimate statistical distance between any models from which we can draw samples. The cells show average estimated MMDs over 10 bootstraps. Rows are sorted using spectral clustering with two components. Models within a family are typically clustered together, suggesting that factors like training data, rather than scale, determine model similarity.
+
+models with different tokenizers, samples must be compared in character space. We sample  $L = 50$ -token completions to the Wikipedia language modeling task as before, and then we decode these to characters, ignoring special tokens. The vocabulary of interest is now all Unicode characters (  $|\mathcal{V}| = 155063$ ), the maximum completion length is  $L \approx 1000$  characters, and we test with  $N = 10m$  samples.
+
+Switching to character space makes the testing problem significantly more challenging, as the sample space is higher dimensional: Figure 3 (lower left) shows that power to detect quantization and watermarking is lower in this setting than in the token space setting ( $\S 4.1$ ). However, even in this more challenging setting, we find that all model swaps are detectable with  $100\%$  power, except for pairs within the Llama family, e.g., Llama-3.1 8B and Llama-3.1 70B (75% power) or Llama-3 8B and Llama-3.1 8B (76% power; Figure 3 lower left). Model swaps are significantly more detectable than watermarking or quantization. Appendix D.2 shows qualitative examples of completions from different models.
+
+Estimating distances between models. A useful feature of the MMD tests is that the test statistic is an estimator of a distance. As a result, we can reuse the machinery to quantify the degree to which two models differ by estimating  $\mathbb{E}_{x\sim \pi}[\mathrm{MMD}(P(\cdot \mid x),Q(\cdot \mid x))]$ .<sup>5</sup> Figure 3 (right) estimates the Hamming MMD between all pairs of models. To decrease estimator error, we increase the sample size to  $N = 100m$  samples from each model and report the average MMD over 10 simulations, along with standard errors. We observe that models within a family are typically clustered together, suggesting that training data, rather than
+
+scale, determines model similarity. Surprisingly, while generations of Llama models (3 and 3.1) are close in distance, some generations of GPT models (e.g., 4-preview and 3.5-turbo) are not. This result suggests the use of different training data or procedures between these models.
+
+# 5 AUDITING INFERENCE API PROVIDERS
+
+As a case study, we now apply our test to 31 commercial inference endpoints for four of Meta's Llama models (Figure 1). These endpoints are distributed across nine API providers from Summer 2024: Amazon Bedrock, Anyscale, $^{6}$  Azure AI Studio, Deepinfra, Fireworks AI, Groq, Perplexity, Replicate, and Together.ai. $^{7}$  We are interested in whether endpoints differ from weights published by Meta on Hugging Face inferred at commonly accepted precisions. Specifically, we consider two possible null distributions: the full-precision weights  $(P_{1})$  and the fp16-precision weights  $(P_{2})$ . The null and alternative hypotheses are
+
+$$
+\begin{array}{l} H _ {0}: \pi (x) P _ {1} (y \mid x) = \pi (x) Q (y \mid x) \quad \text {O R} \quad \pi (x) P _ {2} (y \mid x) = \pi (x) Q (y \mid x), \\ H _ {1}: \pi (x) P _ {1} (y \mid x) \neq \pi (x) Q (y \mid x) \quad \text {A N D} \quad \pi (x) P _ {2} (y \mid x) \neq \pi (x) Q (y \mid x). \\ \end{array}
+$$
+
+To test this composite hypothesis, we collect three samples:  $\mathcal{D}_{P_1}$  and  $\mathcal{D}_{P_2}$  from the two null distributions, and  $\mathcal{D}_Q$  from the API. We then conduct 2 two-sample tests, one for  $(\mathcal{D}_{P_1},\mathcal{D}_Q)$  and another for  $(\mathcal{D}_{P_2},\mathcal{D}_Q)$ , and obtain p-values  $p_1$  and  $p_2$ . We set the overall rejection rule to
+
+$$
+\delta \left(\mathcal {D} _ {P _ {1}}, \mathcal {D} _ {P _ {2}}, \mathcal {D} _ {Q}\right) = \mathbf {1} \left\{p _ {1} <   \alpha \wedge p _ {2} <   \alpha \right\}. \tag {11}
+$$
+
+This rule continues to control the FPR at  $\alpha$  under the composite null hypothesis: without loss of generality, suppose  $Q = P_{1}$ . Since  $P_{Q = P_1}(p_1 < \alpha) \leq \alpha$ , we have  $P_{Q = P_1}(p_1 < \alpha \land p_2 < \alpha) = P_{Q = P_1}(p_1 < \alpha)P_{Q = P_1}(p_2 < \alpha \mid p_1 < \alpha) \leq \alpha$ . Note that this rule may be pessimistic, reducing power.
+
+Experiment details. We consider testing with three prompt distributions  $\pi$ . For all models, we test with one set of the Wikipedia completion task from §4, where  $\pi$  is uniform over  $m = 25$  prompts, and  $L = 50$  tokens or around 1000 characters. For the smaller Llama-3 8B and Llama-3.1 8B models, we also test with the coding task HumanEval (Chen et al., 2021b) and instruction task UltraChat (Ding et al., 2023). Both  $\pi$  are uniform over  $m = 20$  prompts, and  $L = 250$  tokens or 3000 characters. Because APIs often return decoded completions, rather than individual tokens, we conduct all tests in character space, as in §4.3. We explicitly requested all samples at temperature 1. Tests are conducted at level  $\alpha = 0.01$  using  $N = 10m$  samples. To reduce variance, we repeat tests over ten samples and fail endpoints if the average rejection rate is  $\geq 0.5$ . For the most expensive endpoint (Azure's Llama-3 70B), a Wikipedia audit costs $0.14, HumanEval $0.83, and UltraChat $0.93. For the cheapest endpoint (Fireworks' Llama-3 8B), all three audits cost less than $0.02. For additional details, including the dates we collect API samples, see Appendix B.
+
+Results. Despite power being generally reduced due to the composite decision rule, the test flags several endpoints (Table 1). Notably, Amazon Bedrock and Perplexity have the most endpoints flagged, with the latter failing all tests. In Box 2, we include an example comparing samples from the fp32 null and Perplexity; these samples suggest that Perplexity serves a lower entropy distribution than the full-precision model. Additional qualitative samples can be found in Appendix D.3. The HumanEval and UltraChat prompt distributions elicit more failures than the Wikipedia distribution; this may be because these knowledge-intensive distributions are more sensitive to changes such as quantization.
+
+To estimate the effect size of deviations, we estimate the MMD between providers and the nulls using ten bootstraps of  $N = 100m$  samples each. We find that some deviations are quite large: some APIs'
+
+<table><tr><td></td><td colspan="4">Wikipedia</td><td colspan="2">HumanEval</td><td colspan="2">UltraChat</td></tr><tr><td></td><td>3 8B</td><td>3.1 8B</td><td>3 70B</td><td>3.1 70B</td><td>3 8B</td><td>3.1 8B</td><td>3 8B</td><td>3.1 8B</td></tr><tr><td>Amazon</td><td>✓</td><td>X</td><td>✓</td><td>X</td><td>✓</td><td>X</td><td>X</td><td>X</td></tr><tr><td>Anyscale</td><td>✓</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>Azure</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>Deepinfra</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>Fireworks</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>X</td><td>✓</td><td>X</td></tr><tr><td>Groq</td><td>✓</td><td>✓</td><td>✓</td><td>X</td><td>✓</td><td>X</td><td>✓</td><td>✓</td></tr><tr><td>Perplexity</td><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>Replicate</td><td>✓</td><td>—</td><td>X</td><td>—</td><td>✓</td><td>—</td><td>✓</td><td>—</td></tr><tr><td>Together</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr></table>
+
+Table 1: Audit results for 31 API endpoints across nine companies and four language models.  $\times$  denotes an endpoint failure, i.e., the average rejection rate over ten samples is  $\geq  {50}\%$  .
+
+![](images/f626666dd9d33f77d7e6bf6685a239f41b44e9e520fafb58a6ee212508fb1a63.jpg)  
+Figure 4: (Left) Average MMD (Hamming) between providers for each model. Amazon Bedrock's Llama-3 and -3.1 70B models are the most different from the other providers. (Right) Absolute difference in HumanEval average accuracy vs. the MMD (Hamming). There is a moderate positive correlation between MMD and task accuracy. Gray points indicate pairs where both distributions have accuracy  $< 10\%$ . There are multiple ways to be wrong for a task, and the MMD captures these differences.
+
+![](images/c38084b23298e453982973e89e5cc27b2c6f74e0b39f2f1402dc3be2caa5bae0.jpg)
+
+implementations are further from the reference weights than if the provider had substituted in an entirely different language model. For example, the deviation between Perplexity's Llama-3 8B and the fp32 null on the Wikipedia testing task is 0.03. This is comparable to the deviation between Llama-3 8B and GPT-3.5-Turbo (0.03; Figure 3 right), Phi-3 mini (0.04), or OLMo 7B (0.04).
+
+Correlating MMD with task accuracy. We now ask how well the Hamming MMD correlates with task accuracy when available. Automated verifiers exist for one of our prompt distributions, HumanEval. We find a moderate positive correlation between the absolute average accuracy difference and the Hamming MMD (Figure 4 right,  $R^2 = 0.392$ ). In several cases, the MMD is high but the accuracy difference is low. These are often when both task accuracies are low: the gray points in the figure highlight pairs where both distributions have accuracy  $< 10\%$ . In these cases, the MMD captures that there are multiple ways to be wrong for a task. For example, the bottom-right-most point in the figure contrasts Llama-3.1 8B from Perplexity vs. the nf4 version; Perplexity answers with incorrect code snippets, while nf4 outputs nonsense completions (e.g., "simulation!!!!!![...]"). Samples from this comparison can be found in Appendix D.3.
+
+Which models do providers differ on most? Finally, we ask whether providers differ on some more models more than others. Figure 4 (left) estimates the MMD between providers for each model, and the large Llama-3.1 405B model. We find that deviations are larger on the earlier Llama models than the recent Llama-3.1 405B model.
+
+# Box 2: Samples from fp32 vs. Perplexity for Llama-3 8B (Instruct)
+
+Prompt: Continue the paragraph. Do not output anything except the continuation to the paragraph. Start the continuation immediately.
+
+"Weaver, Tom. I Talked with a Zombie: Interviews with 23 Veterans of Horror and Sci-Fi Films and Tele..."
+
+fp32
+
+Perplexity
+
+"...vision, Some of Whom Are Still Alive. New York: Pocket Books, 1981. According to Tom Weaver, Vincent Price was known for being a bit of a practical joker, and he would often play harmless pranks on the crew  
+"...visions. New York: St. Martin's Press, 1988. This collection of interviews with horror and science fiction professionals is a fascinating look at the creative process and the people behind some of the most iconic films and TV shows of the  
+"...vision, voodoo, and the Apocalypse. New York: St. Martin's Press, 1990. The book features interviews with notable horror and science fiction personalities, including Greg Cannom, makeup artist for The Lost Boys and RoboCop  
+"...visions. New York: St. Martin's Press, 1988. In this collection of interviews, Weaver delves into the experiences of some of the most iconic figures in the horror and science fiction genres, including George Romero, John Carpenter,  
+"...vision. Written by Tom Weaver, a renowned authority on classical horror cinema, this book brings together intimate and revealing conversations with some of the most legendary figures in the genre, including Vincent Price, Boris Karloff, and Bela Lugosi. Interviews  
+"...vision. New York: St. Martin's Press, 1988. 224 pp. This collection of interviews with horror and science fiction professionals is a fascinating look at the people behind the genre's most iconic films. Weaver's conversational style
+
+# 6 RELATED WORK
+
+Auditing ML APIs. Prior work either only applies to the classifier setting (Chen et al., 2021a; 2022) or audit by monitoring performance on low-temperature multiple-choice or short-answer benchmarks (ArtificialAnalysis, 2024; Chen et al., 2023; Eyuboglu et al., 2024). Concurrent work studies a general language model setting, but does not provide guarantees of audit correctness (Sam et al., 2025). In contrast, we formalize auditing as a hypothesis testing problem, allowing us to provide FPR guarantees, and we study the general setting of sampling completions from language models, including for tasks without automated verifiers. In cryptography, a body of work has aimed to verify ML API predictions through proof protocols (Ghodsi et al., 2017; Feng et al., 2021; Liu et al., 2021; Kang et al., 2022; Weng et al., 2023; Lee et al., 2024). These methods require APIs to cooperate and opt-in to providing proofs of valid inference alongside each prediction; public verifiers then check these proofs for correctness with perfect accuracy. Unfortunately, these methods scale too poorly to apply to large language models: in Kang et al. (2022), generating a proof takes 41 minutes per prediction for a 68M parameter MobileNet (Sandler et al., 2018).
+
+Two-sample testing. Testing whether two samples come from the same distribution is an established problem in statistics. A line of work focuses on testing multinomial distributions (Batu et al., 2013; Chan et al., 2014; Canonne, 2020), including when samples have unequal sizes (Bhattacharya & Valiant, 2015; Diakonikolas & Kane, 2016; Balakrishnan & Wasserman, 2018). MMD-based tests are a general approach that do not assume a specific distributional form (Gretton et al., 2012). These tests can be applied to structured data when paired with appropriate kernels (Lodhi et al., 2002; Borgwardt et al., 2006).
+
+# 7 CONCLUSION
+
+As the public grows more dependent on black-box APIs to interact with language models, tools for auditing these APIs are increasingly important. In this work, we unified several API auditing tasks under the formalization of Model Equality Testing, a two-sample distribution testing problem, and we extensively validated candidate tests for this problem. Future work could explore stronger tests than those we have presented, or explore how to adapt these tests to other modalities, such as image generation models. To help facilitate this research, we open-source the dataset of 1 million LLM completions used for this work at https://github.com/i-gao/model-equality-testing. This dataset contains completions from five models across quantized, watermarked, and API alternatives; also see Appendix B.1.
+
+# ACKNOWLEDGEMENTS
+
+The authors would like to thank Mert Yuksekgonul, Steven Cao, Chenchen Gu, Chenglei Si, Nicole Meister, Yifan Mai, Simon Guo, Teddi Worledge, Yuhui Zhang, and other members of the Guestrin and P-Lambda labs for feedback on this paper.
+
+# CONFLICTS OF INTEREST
+
+PL is a co-founder of Together AI; this work was done in his Stanford capacity. The topic, research, and results of this work were not shared with Together, or any other API provider evaluated, until the public release of the paper.
+
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+Ingo Steinwart. On the influence of the kernel on the consistency of support vector machines. Journal of machine learning research, 2(Nov):67-93, 2001.
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+# MODEL RISK-SENSITIVE OFFLINE REINFORCEMENT LEARNING
+
+Gwangpyo Yoo, Honguk Woo*
+
+Department of Computer Science and Engineering, Sungkyunkwan University
+
+{necrocathy,hwoo}@skku.edu
+
+# ABSTRACT
+
+Offline reinforcement learning (RL) is becoming critical in risk-sensitive areas such as finance and autonomous driving, where incorrect decisions can lead to substantial financial loss or compromised safety. However, traditional risk-sensitive offline RL methods often struggle with accurately assessing risk, with minor errors in the estimated return potentially causing significant inaccuracies of risk estimation. These challenges are intensified by distribution shifts inherent in offline RL. To mitigate these issues, we propose a model risk-sensitive offline RL framework designed to minimize the worst-case of risks across a set of plausible alternative scenarios rather than solely focusing on minimizing estimated risk. We present a critic-ensemble criterion method that identifies the plausible alternative scenarios without introducing additional hyperparameters. We also incorporate the learned Fourier feature framework and the IQN framework to address spectral bias in neural networks, which can otherwise lead to severe errors in calculating model risk. Our experiments in finance and self-driving scenarios demonstrate that the proposed framework significantly reduces risk, by  $11.2\%$  to  $18.5\%$ , compared to the most outperforming risk-sensitive offline RL baseline, particularly in highly uncertain environments.
+
+# 1 INTRODUCTION
+
+Many application domains of offline reinforcement learning (RL), such as finance (Zhang et al., 2020; Wang & Ku, 2022), self-driving (Bernhard et al., 2019; Seres et al., 2023), and healthcare (Lu et al., 2020), are inherently risk-sensitive due to the potential for high costs or safety risks from interactions between the agent and the environment at deployment. In these scenarios, both premature and well-trained agents can make unsafe decisions, prompting active research efforts to integrate offline RL with risk-sensitive RL (Urpí et al., 2021; Ma et al., 2021; Rigter et al., 2024). These efforts aim to minimize risk instead of maximizing expected return in offline RL settings. Here, (aleatoric) risk is defined as a real number representing the severity of potential harm associated with the return distribution. The function that maps a return distribution to its corresponding risk is called risk measure. For example, the risk calculated by conditional value at risk  $(CV@R)$  risk measure with a confidence level  $\alpha \in [0,1]$  is the negation of the average from the most pessimistic to the  $\alpha$ -th quantile (Yoo et al., 2024).
+
+As these risk measures are designed to emphasize rare events, while underestimating frequent ones, the estimated risk becomes highly sensitive to changes in the underlying return distribution (Cont et al., 2010; Kou et al., 2013; Embrechts et al., 2015; Pesenti et al., 2016). Such errors can lead to incorrect risk management, especially when there is a distribution shift between the training and deployment environments. This situation is common in offline RL, where the agent cannot receive real-time feedback from the environment.
+
+In finance, the issue is addressed by minimizing the worst-case of risks across a set of plausible alternative scenarios rather than directly minimizing the risk estimated by the model (Bernard et al., 2023). This worst-case is referred to as model risk, which quantifies the worst-case consequence of
+
+![](images/f6d07a140258efe9d46f09a8b5c8a3ccca920e5e73dd5150983ce6da914d7969.jpg)  
+Figure 1: Difference between risk and model risk measured by  $CV@R_{\alpha}$ . The negation of the shaded area represents the risk.
+
+![](images/a5edb36884a9e42e7835fc12a18445e69c129980ccb741bc7e210719356215aa.jpg)
+
+using incorrect models (Breuer & Csiszár 2016; Bernard et al. 2023). Unlike traditional approaches that rely solely on the estimated risk and are vulnerable to model errors, the model risk acknowledges the potential for inaccuracies in the model, enabling more robust and reliable risk-sensitive decision-making. Inspired by this, our work in risk-sensitive offline RL shifts the focus from conventional risk minimization to model risk minimization. In doing so, we address the common issue of estimating return distribution statistics (mean and the standard deviation) in RL contexts, where the statistics are often unreliable or unknown. This contrasts with finance, where the mean and deviation are assumed to be known.
+
+Figure 1 illustrates the difference between traditional risk-sensitive offline RL approaches (left-side in the figure) and our model risk-sensitive offline RL (right-side in the figure). In the left-side, existing risk-sensitive offline RL approaches minimize the risk estimated by critic (green area). In the right-side, our model risk-sensitive RL minimizes risk (blue area) of the worst-case scenario (blue line) among the plausible alternative scenarios set (gray lines). By minimizing the model risk, the agent accounts for the possibility of being incorrect, aiming to achieve the best possible results despite potential errors.
+
+We present a model risk-sensitive offline RL framework that accounts for model errors, enhancing robustness in highly uncertain environments. To the best of our knowledge, this is the first approach that incorporates model risk into offline RL algorithms. Specifically, we develop a critic-ensemble criterion method that identifies plausible alternative scenarios, enabling the calculation of model risk without introducing additional hyperparameters. Achieving this requires accurate return distribution statistics, particularly the mean and deviation. To enhance their precision, which is essential for identifying plausible alternative scenarios, we adopt a Fourier feature network for quantile regression. Through case studies in finance and self-driving scenarios, we demonstrate that our model risk-sensitive offline RL framework is more robust than other approaches, achieving an  $11.2\%$  to  $18.5\%$  reduction in risk. Notably, our approach exhibits strong advantages in highly uncertain environments with a large possibility of model errors.
+
+Our contributions are as follows.
+
+- We present the model risk-sensitive offline framework that minimizes the model risk, enabling more robust decision-making compared to conventional risk-sensitive offline RL approaches.  
+- We introduce the critic-ensemble criterion, which adjusts a set of alternative scenarios by expanding or contracting it based on the level of uncertainty, thereby enabling the computation of model risk.  
+- We devise the Fourier feature quantile regression network to precisely estimate the statistics of distributions, which are required to calculate the model risk.  
+- Through several case studies, we demonstrate the robustness of our approach in highly uncertain environments.
+
+# 2 RELATED WORK
+
+Risk-sensitive RL. Traditional research on risk-sensitive RL is based on risk measures that can be computed without estimating whole return distributions (Howard & Matheson, 1972; Sato et al.,
+
+2001; Mihatsch & Neuneier, 2002; Tamar et al., 2015; Chow et al., 2017). With the rise of distributional RL, Dabney et al. (2018a) developed IQN framework, which enables the direct computation of risk via quantile regression, thereby unifying the risk-sensitive RL framework. Other approaches include the direct calculation of  $CV@R$  policy gradient without calculating the return distribution (Tamar et al., 2015; Chow et al., 2017), entropy-based methods (Mihatsch & Neuneier, 2002), dynamic  $CV@R$  (Du et al., 2022; Lim & Malik, 2022), and entropic value at risk ( $EV@R$ , Ni & Lai (2022)). Dynamic  $CV@R$  recursively accounts for risk at each timestep and is related to 1R2R (Rigter et al., 2024).  $EV@R$  is a risk measure that is out of our scope, but it is closely related to CODAC (Ma et al., 2021). Another interesting approach is integrating risk-sensitive RL with safe RL. For example, Kim & Oh (2022) integrate the risk-sensitive approach to the safe RL, constraining the  $CV@R$  of costs instead of expected costs. Meanwhile, Ying et al. (2021); Greenberg et al. (2022) constrain the risk using the safe RL method, while maximizing return. In an online RL setting, Jaimungal et al. (2022) have proposed a similar approach to ours. However, because their method is based on KDE and on-policy algorithm, therefore, it cannot be applied to offline RL.
+
+Risk-sensitive Offline RL. Research in risk-sensitive offline RL relies on integrating IQN and offline RL frameworks. Urpi et al. (2021) first proposed a risk-sensitive offline RL framework ORAAC by combining IQN and BCQ (Fujimoto et al., 2019). ORAAC is hindered by excessive constraints on the behavior policy, limiting its capability to generate better trajectories. To address the limitation, CODAC (Ma et al., 2021), which is based on CQL, was proposed. Yet, CODAC inherits the issues from CQL, such as hyperparameter sensitivity (Tarasov et al., 2024). Another approach, 1R2R (Rigter et al., 2024), is a model-based RL method that estimates risk with worst-case perturbations of the transition. However, its main limitation is that it is only guaranteed when transition probabilities follow a normal distribution.
+
+Model Risk. Our work is built on the concept of model risk, which has been extensively studied in the field of mathematical finance (Bernard et al., 2023; Breuer & Csiszár, 2016; Blanchet & Murthy, 2019; Glasserman & Xu, 2014). Focusing on the worst-case expectation among scenarios aligns with Breuer's approach to aleatoric risk using a coherent risk measure (Breuer & Csiszár, 2016; Glasserman & Xu, 2014). However, we focus on the worst-case of aleatoric risk among alternative scenarios, following Bernard et al. (2023), which leads to a different formulation. This approach allows comparison of value distributions with different support, which is crucial in offline RL, since the worst-case distribution may not match the model's support (Bernard et al., 2023).
+
+# 3 BACKGROUND AND PROBLEM FORMULATION
+
+# 3.1 BACKGROUND
+
+Risk-sensitive RL. We consider a Markov decision process  $(S, \mathcal{A}, R, \mathcal{P}, \gamma)$  where  $S$  is a set of states,  $\mathcal{A}$  is a set of actions,  $R: S \times \mathcal{A} \to \triangle(\mathbb{R})$  is an immediate reward which may be random, and  $\mathcal{P}: S \times \mathcal{A} \to \triangle(S)$  is a transition probability. Here  $\triangle(\mathcal{X})$  denotes a set of random variables whose supports are subsets of  $\mathcal{X}$ . Further, we assume that all random attributes' means and variances are finite. An optimal risk-sensitive policy is formulated as
+
+$$
+\pi^ {*} = \underset {\pi} {\arg \min } H _ {\phi} \left(Z ^ {\pi} (s, a)\right) \text {w h e r e} Z ^ {\pi} (s, a) = \sum_ {t = 0} ^ {T} R \left(s _ {t}, a _ {t}\right) \gamma^ {t} | s _ {0} = s, a _ {0} = a. \tag {1}
+$$
+
+Here  $H_{\phi}(Z^{\pi}(s,a))$  denotes the risk of policy  $\pi$  with respect to its return  $Z^{\pi}(s,a)$  and  $H_{\phi}$  denotes a risk measure given by the user or environment. As we are interested in spectral risk measures (Adam et al. (2008), in Def. ②), we use the quantile function of a random variable.
+
+Definition 1 (Koenker 2005) Let  $Z$  be a random variable. A quantile function of the random variable  $Z$ ,  $F_Z^{-1} : [0,1] \to \mathbb{R}$ , is defined as the left inverse of cumulative distribution of  $Z$ . Specifically,
+
+$$
+F _ {Z} ^ {- 1} (p) := \inf  \{x \in \mathbb {R} \mid p \leq F _ {Z} (x) \}, \tag {2}
+$$
+
+where  $F_Z: \mathbb{R} \to [0,1]$  is the cumulative distribution function of  $Z$ . For readability, we denote  $F_Z^{-1}(\cdot; s, a)$  as the quantile function of dependent random variable,  $Z|_s, a$ , rather than  $F_{Z|s,a}^{-1}(\cdot)$ .
+
+The spectral risk is the weighted integral of the quantile function.
+
+Definition 2 (Adam et al. 2008) A function  $H_{\phi}: L^{1} \to (\mathbb{R} \cup \{\infty\})$  is called spectral risk measure which has the following representation
+
+$$
+H _ {\phi} (Z) = \underbrace {- \int_ {0} ^ {1} F _ {Z} ^ {- 1} (p) \phi (p) d p} _ {u := F _ {\phi} (p) \Rightarrow d u = d F _ {\phi} (p) = \phi (p) d p w i t h p = F _ {\phi} ^ {- 1} (u)} = - \mathbb {E} _ {u \sim U [ 0, 1 ]} [ F _ {Z} ^ {- 1} (F _ {\phi} ^ {- 1} (u)) ], \quad (3)
+$$
+
+where  $\phi :[0,1]\to [0,\infty ]$  is a weight function; right-continuous, non-increasing with  $\int_0^1\phi (p)dp = 1$ ; i.e,  $\phi$  is a density function for some random variable. Further, the value of this function given  $Z$ ,  $H_{\phi}(Z)$ , is called the risk of  $Z$ . Here  $L^1$  is a set of random variables whose means are finite. The right most term is followed by and inverse transform sampling (Miller et al., 2010; Dabney et al., 2018a) and Monte-Carlo integral.
+
+Since  $\phi$  is non-increasing, the risk measure necessarily prioritizes worse outcomes, as intended. The negation is required in the definition, since we aim to minimize the risk. For example, when we want to optimize for the worst-case risk measure (i.e.,  $\phi$  is a Dirac-delta function), we minimize the negation of the worst-case outcome, the risk.
+
+To calculate the risk  $H_{\phi}(Z^{\pi}(s,a))$ , we use the quantile function of  $Z^{\pi}(s,a)$ ,  $F_{Z^{\pi}}^{-1}(\cdot ,s,a)$ . To learn the quantile function, we use the following distributional Bellman equation (Bellemare et al., 2017; Dabney et al., 2018b; a).
+
+$$
+Z ^ {\pi} \left(s _ {t}, a _ {t}\right) \stackrel {\text {d i s t r .}} {=} R \left(s _ {t}, a _ {t}\right) + \gamma Z ^ {\pi} \left(s _ {t + 1}, a\right), \tag {4}
+$$
+
+where  $a = \arg \min_{a'} H_{\phi}(Z(s_{t+1}, a'))$ . Specifically, we use the quantile regression method to estimate the quantile of return  $Z^{\pi}(s, a)$ . The quantile regression loss (Koenker, 2005; Dabney et al., 2018b) of the critic is
+
+$$
+\mathcal {L} _ {\text {c r i t .}} (\theta) = \frac {1}{N N ^ {\prime}} \sum_ {j = 1} ^ {N ^ {\prime}} \sum_ {i = 1} ^ {N} \underbrace {\left| p _ {i} - \mathbf {1} \left(\delta_ {i j} ^ {\theta} <   0\right) \right|} _ {\text {A s y m m e t r i c w e i g h t e d}} \cdot \underbrace {\left| \delta_ {i j} ^ {\theta} \right|} _ {L ^ {1} \text {l o s s}}, \tag {5}
+$$
+
+where  $\delta_{ij}^{\theta}$  is the distributional Bellman residual and  $\theta$  is a critic's parameter. The distributional Bellman residual is defined as a pairwise difference between the target quantile and the critic's quantile as
+
+$$
+\delta_ {i j} ^ {\theta} = R \left(s _ {t}, a _ {t}\right) + \gamma F _ {Z ^ {\pi}} ^ {- 1} \left(p _ {j}; s _ {t + 1}, a, \theta_ {\mathrm {t a r}}\right) - F _ {Z ^ {\pi}} ^ {- 1} \left(p _ {i}; s _ {t}, a _ {t}, \theta\right), \tag {6}
+$$
+
+where  $p_i, p_j$  denote  $N, N'$  numbers of independent random variables which follow  $U[0,1]$  and  $\theta_{\mathrm{tar}}$  denotes the parameter of target-critic.
+
+Risk-sensitive Offline RL. In offline settings, an agent is not allowed to interact directly with the environment and can only access a transition dataset  $\mathcal{D} := \{(s_i, a_i, r_i, s_i')\}_{i=1}^{|\mathcal{D}|}$ . Due to this constraint, naive RL approaches suffer from overestimation of  $Z^{\pi}(s, a)$  from distribution shifts (Fujimoto et al., 2019). Thus, an auxiliary regulation is commonly applied. The objective of offline RL is defined as
+
+$$
+\pi^ {*} = \underset {\pi} {\arg \max } \mathbb {E} \left[ Z ^ {\pi} (s, a) \right] \text {c o n s t r a i n t} D (\pi \| \pi_ {\mathcal {D}}) <   \epsilon , \tag {7}
+$$
+
+where  $D(\cdot \|\cdot)$  is some divergence (or metric),  $\pi_{\mathcal{D}}$  is the behavior policy for the dataset  $\mathcal{D}$ , and  $\epsilon > 0$  is a tolerance. Incorporating this regulation, the objective of risk-sensitive offline RL is extended from Eq.(1) as
+
+$$
+\pi^ {*} = \underset {\pi} {\arg \min } H _ {\phi} \left(Z ^ {\pi} (s, a)\right) \text {c o n s t r a i n t} D (\pi \| \pi_ {\mathcal {D}}) <   \epsilon . \tag {8}
+$$
+
+# 3.2 PROBLEM FORMULATION
+
+The primary concern of risk-sensitive offline RL is that the error of computing  $H_{\phi}(Z^{\pi}(s,a))$  can be significant, as the agent cannot receive direct feedback from the environment in offline settings. To address this, we instead compute the worst-case of risks among alternative scenarios (i.e., model risk) and focus on minimizing the model risk. The model risk is defined as the highest possible risk within a  $\sqrt{\varepsilon}$ -ball with mean  $\mu$  and standard deviation  $\sigma$ . The random variables in the ball specify the plausible alternative scenarios.
+
+![](images/af2a58bb6736e0d07974d304234cc2f6db6d39dc5edd05b4c4755fbd829669a4.jpg)  
+Figure 2: Overall framework of MR-IQN. Q.R. loss is abbreviation of quantile regression loss and s.g. trick is stop-gradient trick. Note that the stop-gradient trick does not change the value.
+
+Definition 3 (Bernard et al. (2023)) The model risk  $MR_{\phi}(Z^{\pi}(s,a);\mu ,\sigma ,\varepsilon)$  is defined as
+
+$$
+M R _ {\phi} \left(Z ^ {\pi} (s, a); \mu , \sigma , \varepsilon\right) := \sup  _ {X \in M \left(Z ^ {\pi} (s, a); \mu , \sigma , \varepsilon\right)} H _ {\phi} (X), \tag {9}
+$$
+
+where  $M(Z^{\pi}(s,a);\mu ,\sigma ,\varepsilon) = \{X\in L^{2}\mid \mathbb{E}[X] = \mu ,\mathbb{Var}[X] = \sigma^{2},D_{W;2}(X,Z^{\pi}(s,a)) < \sqrt{\varepsilon}\}$  and  $L^2$  is a set of random variables whose mean and variance are finite. Here,  $D_{W;2}$  is the Wasserstein 2-distance defined as  $D_{W;2}(X,Y) = \sqrt{\int_0^1|F_X^{-1}(p) - F_Y^{-1}(p)|^2dp}$ . The distribution of random variable  $Z^{\pi}(s,a)$  is called reference distribution.
+
+Our goal is to find a policy  $\pi^{*}$  such that
+
+$$
+\pi^ {*} = \underset {\pi} {\arg \min } \mathrm {M R} _ {\phi} \left(Z ^ {\pi} (s, a); \mu , \sigma , \varepsilon\right) \text {c o n s t r a i n t} D (\pi \| \pi_ {\mathcal {D}}) <   \epsilon . \tag {10}
+$$
+
+# 4 MODEL RISK-SENSITIVE OFFLINE RL ALGORITHM
+
+To achieve a model risk-sensitive offline RL agent satisfying the objective in Eq.(10), we introduce Model Risk IQN (MR-IQN), which integrates the TD3+BC framework (Fujimoto & Gu, 2021) with the model risk objective from Eq.(10). The MR-IQN consists of distributional critics and a deterministic actor. The critic is trained to minimize the loss function in Eq.(5), similar to other approaches (Urpi et al., 2021; Dabney et al., 2018a; Kuznetsov et al., 2020). Specifically, the critics are implemented using IQN (Dabney et al., 2018a) and TQC (Kuznetsov et al., 2020), following Yoo et al. (2024). Meanwhile the actor is trained to minimize the loss function,
+
+$$
+\mathcal {L} (\pi) = \underbrace {\lambda_ {\mathrm {q . l e a r n}} \mathrm {M R} _ {\phi} \left(Z ^ {\pi} (s , a) ; \mu , \sigma , \varepsilon\right)} _ {\text {M o d e l r i s k}} + \underbrace {(a - a _ {\mathcal {D}}) ^ {2}} _ {\text {B C l o s s}}. \tag {11}
+$$
+
+Here  $\lambda_{\mathrm{q,learn}} > 0$  is a scale parameter,  $a_{\mathcal{D}}$  is a batch action, and  $a = \pi (s)$
+
+Figure 2 describes the MR-IQN framework. We extract  $\mu$  and  $\sigma$  from the critic estimation. To precisely estimate the statistics,  $\mu$  and  $\sigma$ , we leverage the Fourier feature architecture (Tancik et al., 2020) for the critics, as in [Li & Pathak (2021)]. Next, we calculate  $\varepsilon$  as the distance between the ensemble of the critics obtained by quantile mixture of individual critics. Finally, we calculate  $\mathrm{MR}_{\phi}(Z^{\pi}(s,\pi (s)))$  and minimize the loss function in Eq.(11). To do so, we first introduce the theorem and corollary below for calculating model risk.
+
+Theorem 1 (Bernard et al. 2023) Let  $Z$  be a random variable. For  $\varepsilon > (\mu_Z - \mu)^2 + (\sigma_Z - \sigma)^2$ , there exists a unique quantile function  $h_\lambda : [0,1] \to \mathbb{R}$  which satisfies
+
+$$
+M R _ {\phi} (Z; \mu , \sigma , \varepsilon) = - \int_ {0} ^ {1} h _ {\lambda} (p) \phi (p) d p, \tag {12}
+$$
+
+where  $\mu_Z$  is the mean of  $Z$ ,  $\sigma_Z$  is the standard deviation of  $Z$ . We have
+
+$$
+h _ {\lambda} (p) = \mu - \sigma \left(\frac {\phi (p) + \lambda F _ {Z} ^ {- 1} (p) - a _ {\lambda}}{b _ {\lambda}}\right), \tag {13}
+$$
+
+where  $\lambda \geq 0$  is some constant, and  $a_{\lambda} = \mathbb{E}[\phi (U) + \lambda F_Z^{-1}(U)], b_{\lambda} = \text{std}[\phi (U) + \lambda F_Z^{-1}(U)]$ . Here  $U$  denotes the uniform random variable  $U[0,1]$  and std denotes the standard deviation operator. If  $\varepsilon \leq (\mu_Z - \mu)^2 + (\sigma_Z - \sigma)^2$  holds, the problem is ill-posed or it has only a trivial solution.
+
+Let  $c_{0} = \operatorname{corr}(F_{Z}^{-1}(U), \phi(U))$ . If  $\varepsilon < (\mu_{Z} - \mu)^{2} + (\sigma_{Z} - \sigma)^{2} + 2\sigma \sigma_{Z}(1 - c_{0})$  holds, we achieve
+
+$$
+\lambda = \frac {K}{\sigma_ {Z} ^ {2}} \sqrt {\frac {V \sigma_ {Z} ^ {2} - C _ {\phi , Z} ^ {2}}{\sigma^ {2} \sigma_ {Z} ^ {2} - K ^ {2}}} - \frac {C _ {\phi , Z} ^ {2}}{\sigma_ {Z}}, \tag {14}
+$$
+
+where  $V = \mathbb{Var}[\phi (U)]$ ,  $C_{\phi ,Z} = \mathbb{Cov}[F_Z^{-1}(U),\phi (U)]$ , and  $K = \frac{1}{2}\left((\mu_{Z} - \mu)^{2} + \sigma^{2} + \sigma_{Z}^{2} - \varepsilon\right)$ . Otherwise,  $\lambda = 0$ . Here, corr is the Pearson correlation.
+
+Corollary 1 (Bernard et al. 2023) The solution of Eq.(9) exists and unique whenever  $h_\lambda$  exists as
+
+$$
+M R _ {\phi} \left(Z ^ {\pi} (s, a); \mu , \sigma , \varepsilon\right) = - \underbrace {(\mu - \sigma s t d [ \phi (U) ] c o r r [ \phi (U) , \phi (U) + \lambda F _ {Z \pi} ^ {- 1} (U ; s , a) ])} _ {\int_ {0} ^ {1} h _ {\lambda} (p) \phi (p) d p i n E q. \boxed {1 2}}), \tag {15}
+$$
+
+where  $\lambda \geq 0$  is the constant determined by Eq.(14) in Theorem I
+
+# 4.1 CRITIC-ENSEMBLE CRITERION
+
+To calculate the model risk for policy gradient using Eq.(15), it is required to determine  $\mu, \sigma$  and tolerance  $\varepsilon$ . For  $\varepsilon$ , we want to set  $\varepsilon$  large when there is a large inconsistency between an individual critic while setting  $\varepsilon$  small when the inconsistency is small. For conservatism, we choose  $\mu$  as the minimum value among the critics' estimation and  $\sigma$  as the maximum to reduce overestimation bias as discussed by Fujimoto et al. (2018). From now on, we use  $K$  to denote the number of critics.
+
+Step 1. Calculating  $\mu, \sigma$ . To ensure conservatism, we select  $\mu_0, \sigma$  as the smallest expectation and the largest deviation of  $Z^{\pi}$ , as follows.
+
+$$
+\mu_ {0} := \min  _ {i = 1, \dots , K} \mathbb {E} \left[ Z ^ {\pi} (s, a, \theta^ {(i)}) \right], \sigma := \max  _ {i = 1, \dots , K} \operatorname {s t d} \left[ Z ^ {\pi} (s, a, \theta^ {(i)}) \right] \tag {16}
+$$
+
+When  $\lambda = 0$ , the gradient of the policy in Eq.(15) solely depends on the mean-deviation risk measure, which often leads to local optima. To mitigate this, we apply the stop gradient trick to  $\mu$  as
+
+$$
+\mu := - H _ {\phi} \left(Z ^ {\pi} (s, a)\right) + \operatorname {s t o p - g r a d} \left(H _ {\phi} \left(Z ^ {\pi} (s, a)\right)\right) + \mu_ {0}, \tag {17}
+$$
+
+where  $H_{\phi}(Z^{\pi}(s,a)) = \max_{i = 1,\dots ,K}H_{\phi}(Z^{\pi}(s,a,\theta^{(i)}))$  , i.e., the most conservative critic.
+
+Step 2. Calculating  $\varepsilon$  through quantile mixture. We construct an ensemble of critics as a single quantile function by mixing their quantiles and calculate the distance between this ensemble and individual critics.
+
+We first form the critic ensemble by sorting the estimates from all critics and interpolating them over an equally divided space (i.e., quantile mixture). Next, we calculate the Wasserstein-2 distance between the ensemble of critics and individual critics. The largest of these distances is selected as  $\varepsilon$ .
+
+The first step involves calculating the sorted critic values  $y_{k}$  defined as
+
+$$
+\left\{y _ {1}, \dots , y _ {N K} \right\} = \cup_ {i = 1} ^ {K} \cup_ {j = 1} ^ {N} F _ {Z \pi} ^ {- 1} \left(p _ {j}; s, a, \theta^ {(i)}\right), y _ {1} \leq y _ {2} \leq \dots \leq y _ {N K}, \tag {18}
+$$
+
+where  $N$  represents the number of random variables  $p_j \sim U[0,1]$ . Let  $X$  be an equally divided set of  $[0,1]$  defined as  $X = \{x_1 = 0, x_2 = 1/(NK - 1), \ldots, x_{NK} = 1\}$ . By interpolating pairs  $\{(x_1, y_1), \ldots, (x_{NK}, y_{NK})\}$ , we construct the ensemble of critics,  $F_{\mathrm{ens.}}^{-1}$ . Figure  $\boxed{3}$  depicts the procedure of calculating  $F_{\mathrm{ens.}}^{-1}$ . Finally,  $\varepsilon$  is computed by
+
+$$
+\varepsilon = \max  _ {i = 1, \dots , K} \int_ {0} ^ {1} | F _ {\text {e n s .}} ^ {- 1} (x; s, a) - F _ {Z \pi} ^ {- 1} (x; s, a, \theta^ {(i)}) | ^ {2} d x = D _ {W; 2} ^ {2} \left(F _ {Z \pi} ^ {- 1} (\cdot ; s, a, \theta^ {i}), F _ {\text {e n s .}} ^ {- 1} (\cdot)\right). \tag {19}
+$$
+
+We use the trapezoid method to numerically calculate the integral in Eq. (19).
+
+Step 3. Calculate the model risk. We take the most conservative critic and take the reference distribution of the critic estimates in the place of  $F_{Z}^{-1}$  in Eq.(14) and Eq.(15).
+
+$$
+F _ {\text {r e f}} ^ {- 1} (\cdot) = F _ {Z \pi} ^ {- 1} (\cdot ; s, a, \theta^ {(*)}), \text {w h e r e} \theta^ {(*)} = \underset {\theta^ {(i)}} {\arg \max } H _ {\phi} \left(Z ^ {\pi} (s, a, \theta^ {(i)})\right). \tag {20}
+$$
+
+Algorithm 1 Critic-Ensemble Model Risk
+
+Inputs: Critic Parameters  $\theta^{(i)}$ , Risk Measure's Density  $\phi$ ,  $N$  the number of samples.
+
+Output: Model Risk  $\mathrm{MR}_{\phi}(Z^{\pi}(s,a),\mu ,\sigma ,\varepsilon)$
+
+Step1 Calculate  $\mu, \sigma$ :
+
+Sample  $p_j$  from  $\dot{U}[0,1]$  with  $j = 1,\dots ,N$ . To estimate risk and mean by inv. trans. sampling.
+
+$\mu_0 \gets \min_{i=1\dots K} \mathbb{E}_{p_j}[F_{Z^\pi}^{-1}(p_j;s,a,\theta^{(i)})]$ $\triangleright$  Get minimal mean over critics; mean over  $p_j$ .
+
+$p_j^{\prime}\gets F_{\phi}^{-1}(p_j),H_{\phi}(Z^{\pi}(s,a))\gets \max_{i = 1,\ldots ,K} - \mathbb{E}_{p_j'}[F_{Z\pi}^{-1}(p_j';s,a,\theta^{(i)})])$  Eq.(3)
+
+$\mu \leftarrow -H_{\phi}(Z^{\pi}(s,a)) + \mathrm{stop - grad}(H_{\phi}(Z^{\pi}(s,a))) + \mu_0$  Stop-Gradient Trick in Eq.(17)
+
+$\sigma \gets \max_{i = 1,\dots ,K}\mathrm{std}_{p_j}[F_{Z\pi}^{-1}(p_j,s,a,\theta^{(i)})]$ $\triangleright$  Get maximal standard deviation over critics.
+
+Step 2 Calculate  $\varepsilon$ :
+
+$\{y_{k}\}_{k = 1}^{NK}\gets \mathrm{sort}(\cup_{i = 1}^{K}\cup_{j = 1}^{N}\{F_{Z\pi}^{-1}(p_j;s,a,\theta^{(i)})\})$  ▷Sort
+
+$\{x_{1}\gets 0,x_{2}\gets \frac{1}{NK - 1},\ldots ,x_{NK - 1}\gets \frac{NK - 2}{NK - 1},x_{NK}\gets 1\}$ $\triangleright x\gets \mathrm{linspace}(0,1,NK)$
+
+$F_{\mathrm{ens.}}^{-1}\gets$  interpolation  $\left(\left\{\left(x_{k},y_{k}\right)\right\}_{k = 1}^{NK}\right)$  Rearrange and Interpolarate
+
+$\varepsilon = \max_{i = 1,\dots ,K}\int_0^1 |F_{\mathrm{ens}}^{-1}(x) - F_{Z\pi}^{-1}(x;s,a,\theta^{(i)})|^2 dx$ $\triangleright$  Calculate via numerical integral.
+
+Step 3 Calculate Model Risk:
+
+$\theta^{(*)}\leftarrow \arg \max_{\theta^{(i)}}H_{\phi}(Z^{\pi}(s,a,\theta^{(i)})),F_{\mathrm{ref}}^{-1}\leftarrow F_{Z^{\pi}}^{-1}(\cdot ;s,a,\theta^{(*)})\triangleright$  Take the most conservative.
+
+$\lambda \leftarrow$  calculate lambda  $(F_{\mathrm{ref}}^{-1},\mu ,\sigma ,\varepsilon ,\phi)$   using Eq.14
+
+$\mathbf{MR}_{\phi}(Z^{\pi}(s,a))\gets$  calculate model risk  $(F_{\mathrm{ref}}^{-1},\mu ,\sigma ,\varepsilon ,\phi)$  using Eq.(15)
+
+Return:  $\mathrm{MR}_{\phi}(Z^{\pi}(s,a))$
+
+![](images/60f452d539f13b297272d470df1cdfcaeb6e80052ae4a28b1e91e745a21a1f8d.jpg)  
+Quantile-Level
+
+![](images/71aa7d424b578b10dc37e7c2eb379617002e1b4fbbc20b89aa82a635fc898ce6.jpg)  
+$x_{1}$ $x_{2}$  ...  
+Figure 3: Procedure of calculating  $F_{\mathrm{ens.}}^{-1}$  with three critics.  $F_{\mathrm{ens.}}^{-1}$  is a quantile mixture of critics, therefore, it shows a mid-level pessimism similar to  $F_{Z^{\pi}}^{-1}(\cdot ,s,a,\theta^{(3)})$ .
+
+If a solution does not exist, (i.e.,  $\varepsilon < (\mu_Z - \mu)^2 + (\sigma_Z - \sigma)^2$ ), we substitute  $H_{\phi}(Z^{\pi}(s,a))$  for  $\mathrm{MR}_{\phi}(Z^{\pi}(s,a);\mu,\sigma,\epsilon)$  to preserve the gradient signal.
+
+Algorithm [1] outlines the procedure of computing model risk throughout the critic-ensemble criterion. First, the most conservative  $\mu$ ,  $\sigma$  among critics, are chosen. Next, we compute  $\varepsilon$  by measuring the  $D_{W;2}$  distance between each individual critic and the ensemble, which is obtained by quantile mixture via sort and rearrange. Finally, we compute the model risk using the parameters obtained from the previous steps.
+
+# 4.2 ADDRESSING SPECTRAL BIAS OF QUANTILE REGRESSION
+
+Although our framework accounts for model errors, it is essential to estimate the mean  $(\mu)$  and deviation  $(\sigma)$  as accurately as possible, since these statistics are assumed to approximate the ground truth. While quantile regression yields an unbiased estimator (Koenker, 2005), due to spectral bias (Rahaman et al., 2019), conventional neural networks often fail to capture these statistics effectively. Spectral bias refers to the phenomenon where neural networks struggle to learn high-frequency components of patterns. For risk-sensitive RL to be effective,  $Z^{\pi}$  must converge in the distributional sense to ensure proper signals can be back-propagated. According to the Lévy convergence theorem (Williams, 1991), this is equivalent to the pointwise convergence of the Fourier transform of  $Z^{\pi}$ , emphasizing the critical role of the high-frequency domain (and thus spectral bias) in convergence. The Fourier feature network (Tancik et al., 2020; Li & Pathak, 2021) is a known solution to the spectral bias problem.
+
+![](images/9eb5db2c43c152e11e92187629eaf4ecfb9dd0993107fc718c81073f191a24ff.jpg)  
+(a) Vanilla IQN  $D_{W;1}$ :  $2.23 \pm 0.05$  Crossing(%):  $6.07 \pm 2.08$
+
+![](images/8f46b681361e5eef291cc41e57db16ab7e975f3334ba2851802d4014721d1440.jpg)  
+(b) Fourier feature in level domain  $D_{W;1}$ :  $2.33 \pm 0.14$  Crossing  $(\%)$ :  $0.01 \pm 0.00$
+
+![](images/00c88e077aa3925ae6bbedf6df56bc7ef870722836704a15b06f4cbe8c4c0800.jpg)  
+(c) Full Fourier feature  $D_{W;1}$ :  $0.06 \pm 0.02$  Crossing(%):  $0.01 \pm 0.00$
+
+Figure 4 demonstrates the importance of the Fourier feature network in quantile regression. Here the networks are trained to estimate  $F_{\mathcal{N}(\mu, \sigma^2)}^{-1}$  given  $\mu \sim \mathcal{N}(0,1)$ ,  $\log \sigma \sim \mathcal{N}(0,1)$ . (a) is the result of a network trained by Vanilla IQN. (b) is the case when we handle the spectral bias in the level of quantile. (c) shows the case when we handle the spectral bias of all input domains. As shown, the spectral bias is the cause of inaccurate distribution estimation. Note that applying Fourier features in the
+
+level domain helps reduce the crossing quantile, but it does not improve overall accuracy. This is because the cosine embedding used in IQN already functions as a type of Fourier feature network. Figure 5 depicts the implementation of the Fourier feature network in (c). For parameters  $\xi$  initialized from  $\mathcal{N}(0,\sigma_{ff})$  and input  $X_0$ , the Fourier feature is
+
+![](images/9e376c37584df22d4b435ffaefd8c4431a4271c593ae080f68703054ec830228.jpg)  
+Figure 4: Results of quantile regression.  $D_{W;1}$  is the Wasserstein 1-distance between ground truth quantile, and Crossing is the ratio of the numbers of crossing quantiles (\%).  
+Figure 5: Fourier feature quantile regression network.  $\odot$  is pairwise multiplication operator.
+
+$$
+X _ {1} = \sin \left(2 \pi X _ {0} ^ {T} \xi\right) \# \cos \left(2 \pi X _ {0} ^ {T} \xi\right) \# X _ {0}, \tag {21}
+$$
+
+where  $\#$  denotes the concatenation operator, and  $X_{1}$  denotes the feature passed to the next layer. We also leverage using the Fourier feature IQN instead of cosine embedding IQN to reduce crossing quantile error. The remaining process is the same as conventional MLP.
+
+# 5 EXPERIMENTS
+
+# 5.1 BASELINES AND CRITERION
+
+Risk Measures. For the experiments, we are interested in following risk measures.
+
+(i)  $CV@R$ . considers average over below  $\alpha$ 's quantile. The risk measure's density  $\phi_{\alpha}^{\mathrm{CV@R}}$  parameterized by  $\alpha \in [0,1]$  of is
+
+$$
+\phi_ {\alpha} ^ {C V @ R} (p) = \frac {1}{\alpha} \mathbf {1} _ {[ 0, \alpha ]} (p). \tag {22}
+$$
+
+(ii) Wang. is a risk measure motivated by areas where one's cost becomes another's benefit, like an option trading [Wang 2002]. The density of Wang risk measure,  $\phi_{\alpha}^{\mathrm{wang}}$ , parameterized by  $\alpha < 0$  is
+
+$$
+\phi_ {\alpha} ^ {w a n g} (p) = \frac {\varphi \left(F _ {\mathcal {N} (0 , 1)} ^ {- 1} (1 - p) + \alpha\right)}{\varphi \left(F _ {\mathcal {N} (0 , 1)} ^ {- 1} (1 - p)\right)}, \tag {23}
+$$
+
+where  $F_{\mathcal{N}(0,1)}^{-1}$  and  $\varphi$  are the quantile and density of standard normal distribution, respectively.
+
+Baseline Algorithms. ORAAC (Urpí et al., 2021) is a risk-sensitive variant of BCQ (Fujimoto et al., 2019). It achieves the risk-sensitive offline RL objective by only having slight modifications in the dataset's actions. CODAC (Ma et al., 2021) is a risk-sensitive variant of CQL (Kumar et al., 2020). It achieves a dataset constraint by minimizing KL-divergence between the RL agent's actions and the dataset's actions using the Donsker-Varadhan representation. IQN-TD3+BC is a risk-sensitive variant of the TD3+BC (Fujimoto & Gu, 2021) algorithm, where the IQN-Critic is used. The optimal actor  $\pi_{\mathrm{TD3 + BC}}$  in IQN-TD3+BC is defined as
+
+$$
+\pi_ {\mathrm {T D} 3 + \mathrm {B C}} = \underset {\pi} {\arg \min } \lambda_ {\mathrm {q}, \text {l e a r n}} H _ {\phi} \left(Z ^ {\pi} (s, a)\right) + \left(a - a _ {\mathcal {D}}\right) ^ {2}. \tag {24}
+$$
+
+Here  $\lambda_{\mathrm{q. learn}} > 0$  is a scale hyperparameter,  $a_{\mathcal{D}}$  is a batch action, same as in Eq.(11).
+
+# 5.2 MAIN RESULTS
+
+We evaluate MR-IQN on finance and self-driving scenarios comparing with the baselines above. The label Mean is the average of the mean score over seeds and  $-H_{\phi}(Z^{\pi})$  is the mean negative risk over seeds. All reported scores are averaged across 5 seeds. Numerical descriptions provided without explicit criteria are the results of comparing the negative risk. We also present the D4RL results, comparing the baselines and 1R2R, in Appendix A.1. The results and analysis about  $CV@R(10\%)$  risks are in Appendix A.4, and the dataset details are provided in Appendix B.
+
+Table 1: Performance comparison in finance scenarios.  
+
+<table><tr><td colspan="2">Target Risk Measure</td><td colspan="2">CV@R (50%)</td><td colspan="2">Wang (-0.5)</td></tr><tr><td>Env.</td><td>Algorithm</td><td>Mean</td><td>-Hφ(Zπ)</td><td>Mean</td><td>-Hφ(Zπ)</td></tr><tr><td>Forex</td><td>CODAC</td><td>1577.47 ± 41.52</td><td>249.53 ± 28.33</td><td>1591.76 ± 8.12</td><td>892.34 ± 54.06</td></tr><tr><td>Env.</td><td>ORAAC</td><td>302.37 ± 19.27</td><td>-99.90 ± 0.01</td><td>304.89 ± 38.05</td><td>70.47 ± 17.22</td></tr><tr><td>(High)</td><td>IQN-TD3+BC</td><td>2103.16 ± 103.43</td><td>454.16 ± 11.76</td><td>2181.16 ± 123.41</td><td>1307.52 ± 90.05</td></tr><tr><td></td><td>MR-IQN (ours)</td><td>2537.52 ± 289.57</td><td>556.94 ± 172.92</td><td>2681.40 ± 114.00</td><td>1639.79 ± 57.88</td></tr><tr><td>Stock</td><td>CODAC</td><td>37.67 ± 7.54</td><td>13.98 ± 2.34</td><td>33.43 ± 7.01</td><td>21.89 ± 4.65</td></tr><tr><td>Env.</td><td>ORAAC</td><td>78.48 ± 3.27</td><td>26.07 ± 2.25</td><td>78.57 ± 6.09</td><td>50.97 ± 3.98</td></tr><tr><td>(Low)</td><td>IQN-TD3+BC</td><td>80.49 ± 1.87</td><td>30.92 ± 1.15</td><td>83.10 ± 3.05</td><td>55.55 ± 1.76</td></tr><tr><td></td><td>MR-IQN (ours)</td><td>74.78 ± 2.10</td><td>31.16 ± 1.02</td><td>87.03 ± 2.62</td><td>58.56 ± 1.61</td></tr></table>
+
+Finance. To investigate the real-world scenarios, we conduct experiments using the trading log data ranging from 2023-Feb-3rd to 2023-December-1st. The trading environment is based on the Meta-Trader Simulator (Amin, 2021). We evaluate 1000 episodes for each seed to calculate  $CV@R$  and Wang negative risk. For the details, see Appendix. Table II depicts the results.
+
+In the experiment, MR-IQN shows the highest performance in both Forex and Stock environments. In Forex environment, where uncertainty is high due to high leverage (High), the model risk approach effectively covers tail risk, maintaining high average returns and low risk, compared to the baseline approaches without model risk. IQN-TD3+BC, which is comparable to MR-IQN shows degraded performance  $(18.5\% - 11.2\%)$  in Forex, indicating that MR-IQN is particularly useful in handling higher uncertainty and errors. Meanwhile, MR-IQN shows a slightly degraded average compared to ORAAC and IQN-TD3+BC in Stock environment. This indicates that there is a trade-off between conservatism and average performance.
+
+Self-Driving. To simulate the self-driving scenario, we utilize the Airsim environment (Shah et al., 2017). In this scenario, RL agents are tasked with driving a quadcopter drone to a designated goal while avoiding obstacles, with wind disturbances affecting the drone's movement. The agents receive rewards based on the cosine of the angle between their movement direction and the goal. We evaluate performance in 100 episodes for each seed to measure negative risk. Table ② depicts the results.
+
+MR-IQN outperforms other baselines, showing the highest success rate (87.4%-88.8%) and the lowest collision rate (10.6%-11.6%). CODAC showed a relatively low average of rewards and a higher success rate compared to ORAAC. This implies that CODAC fails to generalize efficient actions and hovers over. The high collision rate of ORAAC implies that it fails to avoid obstacles because of the strict constraint to the batch action.
+
+Table 2: Performance comparison in self-driving scenarios  
+
+<table><tr><td>Risk Measure</td><td>Algorithm</td><td>Mean</td><td>-Hφ(Zπ)</td><td>Success(%)</td><td>Collision(%)</td></tr><tr><td rowspan="4">CV@R (50%)</td><td>CODAC</td><td>43.56 ± 89.45</td><td>-425.72 ± 97.73</td><td>65.0 ± 13.1</td><td>21.4 ± 5.5</td></tr><tr><td>ORAAC</td><td>92.37 ± 40.51</td><td>-88.37 ± 4.20</td><td>23.0 ± 15.7</td><td>68.0 ± 10.2</td></tr><tr><td>IQN-TD3+BC</td><td>411.78 ± 33.10</td><td>265.51 ± 45.25</td><td>81.8 ± 5.6</td><td>14.8 ± 5.6</td></tr><tr><td>MR-IQN(ours)</td><td>431.44 ± 30.41</td><td>290.96 ± 46.95</td><td>88.0 ± 4.6</td><td>11.6 ± 4.9</td></tr><tr><td rowspan="4">Wang (-0.5)</td><td>CODAC</td><td>6.67 ± 1.02</td><td>-2.15 ± 0.75</td><td>52.6 ± 19.3</td><td>31.2 ± 27.6</td></tr><tr><td>ORAAC</td><td>156.22 ± 32.09</td><td>51.73 ± 27.53</td><td>36.2 ± 6.4</td><td>58.4 ± 8.1</td></tr><tr><td>IQN-TD3+BC</td><td>412.98 ± 26.68</td><td>311.20 ± 31.03</td><td>83.8 ± 5.1</td><td>14.4 ± 3.9</td></tr><tr><td>MR-IQN(ours)</td><td>436.17 ± 19.16</td><td>341.26 ± 18.36</td><td>87.4 ± 1.8</td><td>10.6 ± 2.6</td></tr></table>
+
+# 5.3 ABLATION STUDY
+
+Table 3: Ablation results.  $\checkmark$  represents presence and  $X$  represents absence. M.R. and F.F. are the abbreviations for model risk and Fourier feature, respectively.  
+
+<table><tr><td colspan="3">Components</td><td colspan="2">CV@R (50%)</td><td colspan="2">Wang (-0.5)</td></tr><tr><td>M.R.</td><td>F.F.</td><td>TQC</td><td>Mean</td><td>-Hφ(Zπ)</td><td>Mean</td><td>-Hφ(Zπ)</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>2537.52 ± 289.57</td><td>556.94 ± 172.92</td><td>2681.40 ± 114.00</td><td>1639.79 ± 57.88</td></tr><tr><td>✓</td><td>✓</td><td>X</td><td>2003.37 ± 214.76</td><td>464.84 ± 57.01</td><td>1975.85 ± 104.49</td><td>1160.65 ± 87.82</td></tr><tr><td>✓</td><td>X</td><td>✓</td><td>2410.83 ± 86.94</td><td>513.02 ± 77.54</td><td>2324.31 ± 146.95</td><td>1376.27 ± 122.96</td></tr><tr><td>✓</td><td>X</td><td>X</td><td>1763.21 ± 278.16</td><td>278.16 ± 94.26</td><td>1607.13 ± 76.54</td><td>872.54 ± 56.23</td></tr><tr><td>X</td><td>✓</td><td>✓</td><td>2103.16 ± 103.43</td><td>454.16 ± 11.76</td><td>2181.16 ± 123.41</td><td>1307.52 ± 90.05</td></tr><tr><td>X</td><td>✓</td><td>X</td><td>1712.79 ± 75.40</td><td>295.10 ± 49.83</td><td>1650.28 ± 51.48</td><td>934.79 ± 30.00</td></tr><tr><td>X</td><td>X</td><td>✓</td><td>2170.68 ± 186.55</td><td>476.23 ± 75.94</td><td>2188.57 ± 134.40</td><td>1297.15 ± 88.15</td></tr><tr><td>X</td><td>X</td><td>X</td><td>1454.30 ± 76.87</td><td>153.84 ± 38.57</td><td>1414.44 ± 40.00</td><td>750.22 ± 37.45</td></tr></table>
+
+We also conduct an ablation study in the Forex environment, identifying model risk, Fourier features, and TQC as key contributors to the model's improvement. Table 3 presents the results. MR-IQN achieves an  $8.6\% -19.8\%$  performance gain from the Fourier feature and a  $19.8\% -41.3\%$  performance gain from TQC. The performance gain from model risk is reported in Section 5.2. Notably, model risk enhances performance when combined with other factors, due to the design of the critic ensemble criterion and the requirement for accurate estimation of  $\mu$  and  $\sigma$ , as discussed in Section 4.2.
+
+Our framework achieves approximately  $118.6\% - 262.0\%$  performance gains compared to the naive implementation, which omits the key components. The reason why TQC is introduced to our framework is to construct a target quantile which is similar to  $F_{\mathrm{ens}}^{-1}$ . As a result, the calculated model risk tolerance,  $\varepsilon$ , becomes more aligned with our intention, i.e., large  $\varepsilon$  for high inconsistency and vice versa. However, the performance improvement of TQC without other factors ( $72.9\% - 209.6\%$ ) implies that TQC itself is not a negligible factor.
+
+# 6 CONCLUSION
+
+We proposed a model risk-sensitive offline RL framework, devising the critic-ensemble criterion to capture model risk effectively. To ensure the precision of model risk calculation, we employed Fourier feature networks, which accurately estimate both the mean and standard deviation—critical components for calculating the model risk. The framework ensures more reliable decision-making in risk-sensitive applications by accounting for potential model errors and striving to make the best decisions despite them. While our framework is limited to spectral risk measures and cannot accommodate those outside this class, such as CMV (Vadori et al., 2020) or  $EV@R$  (Ni & Lai, 2022), the broad applicability of spectral risk measures covers many practical real-world problems. Our future work involves extending the framework to partially observable Markov decision processes, including embodied control and decision-making in mission-critical business applications.
+
+# ACKNOWLEDGEMENT
+
+This work was supported by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT), (RS-2022-II220043 (2022-0-00043), Adaptive Personality for Intelligent Agents, RS-2022-II221045 (2022-0-01045), Self-directed multi-modal Intelligence for solving unknown, open domain problems, RS-2025-02218768, Accelerated Insight Reasoning via Continual Learning, and RS-2019-II190421, Artificial Intelligence Graduate School Program (Sungkyunkwan University)), the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. RS-2023-00213118), IITP-ITRC (Information Technology Research Center) grant funded by the Korea government (MIST) (IITP-2025-RS-2024-00437633, 10%), IITP-ICT Creative Consilience Program grant funded by the Korea government (MSIT) (IITP-2025-RS-2020-II201821, 10%), and by Samsung Electronics.
+
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+
+# APPENDIX
+
+# A ADDITIONAL EXPERIMENTS
+
+# A.1 D4RL
+
+Since D4RL is a standard benchmark for offline RL training, we report the D4RL scores here. We trained MR-IQN using the Wang risk measure with a parameter of  $-0.5$ . MR-IQN shows comparable performance (68.59) to the state of the art risk-sensitive offline RL algorithm 1R2R (67.35-67.78) in overall scores and outperforms it when evaluating the average without random dataset (85.44), which is a metric often used in behavior clone based offline RL algorithms (Wang et al., 2023; Lu et al., 2023).
+
+Table A: D4RL Mujoco-v2 results (normalized score).  
+
+<table><tr><td></td><td>Environment</td><td>MR-IQN Wang-0.5</td><td>1R2RWang</td><td>1R2RCV@R</td><td>ORAAC</td><td>CODAC</td><td>TD3+BC</td></tr><tr><td rowspan="3">Random</td><td>HalfCheetah</td><td>20.48 ± 0.91</td><td>35.5</td><td>36.9</td><td>22.4</td><td>31.0</td><td>11.0</td></tr><tr><td>Hopper</td><td>26.47 ± 10.6</td><td>31.8</td><td>30.9</td><td>10.9</td><td>9.2</td><td>8.5</td></tr><tr><td>Walker2d</td><td>7.15 ± 1.43</td><td>8.1</td><td>7.6</td><td>2.4</td><td>12.8</td><td>1.6</td></tr><tr><td rowspan="3">Medium</td><td>HalfCheetah</td><td>61.14 ± 0.75</td><td>75.5</td><td>74.5</td><td>38.5</td><td>62.8</td><td>48.3</td></tr><tr><td>Hopper</td><td>98.84 ± 5.87</td><td>64.6</td><td>80.2</td><td>29.6</td><td>63.9</td><td>59.3</td></tr><tr><td>Walker2d</td><td>89.05 ± 1.24</td><td>83.4</td><td>63.9</td><td>45.5</td><td>84.0</td><td>83.7</td></tr><tr><td rowspan="3">Medium Replay</td><td>HalfCheetah</td><td>46.36 ± 2.19</td><td>65.6</td><td>65.7</td><td>39.3</td><td>53.4</td><td>44.6</td></tr><tr><td>Hopper</td><td>95.08 ± 7.14</td><td>93.2</td><td>92.9</td><td>22.6</td><td>68.8</td><td>60.94</td></tr><tr><td>Walker2d</td><td>81.07 ± 13.93</td><td>88.4</td><td>92.2</td><td>11.1</td><td>73.9</td><td>81.8</td></tr><tr><td rowspan="3">Medium Expert</td><td>HalfCheetah</td><td>78.54 ± 12.00</td><td>94.5</td><td>96.0</td><td>24.4</td><td>76.7</td><td>90.7</td></tr><tr><td>Hopper</td><td>105.62 ± 3.60</td><td>89.6</td><td>81.6</td><td>4.1</td><td>87.6</td><td>98.0</td></tr><tr><td>Walker2d</td><td>113.25 ± 0.36</td><td>78.1</td><td>90.9</td><td>42.8</td><td>112.0</td><td>110.1</td></tr><tr><td colspan="2">Average</td><td>68.59</td><td>67.35</td><td>67.78</td><td>31.83</td><td>61.34</td><td>58.86</td></tr><tr><td colspan="2">Average without Random</td><td>85.44</td><td>81.43</td><td>81.99</td><td>36.08</td><td>75.90</td><td>76.13</td></tr></table>
+
+# A.2 BEHAVIOR ANALYSIS
+
+![](images/5337f8f988be8564d0972b45f786e2cdc24ba48cbee679b1fea057c7567855b3.jpg)  
+(a) CODAC High
+
+![](images/527f31f3c8f6ae5516114bf6fbf324f638e1e8d784217f498e0680a0f753bb28.jpg)  
+(b) IQN TD3+BC High
+
+![](images/8da4f30da70f769cead3dc749f56f5d0ceed4e243abf51006833d8ba2c5966eb.jpg)  
+(c) ORAAC High
+
+![](images/eebe6eccd0a917fdd8f896c0593f3d05e4978580f9e8a1f2a1273d658b369f32.jpg)  
+(d) MR-IQN (ours) High
+
+![](images/ef7794fe662567b3fe48433ac1efc345a264d5738a0e4de64489d77f1a864857.jpg)  
+(e) CODAC Low
+
+![](images/ec1097d84e58f9dc46c82d9db0f4aeb222684c074a68d42a3e79398f586188b9.jpg)  
+(f) IQN TD3+BCLow
+
+![](images/2c3153cfe982f25a965a77d12a7cbf8baf91b9624a74a400aac573a5d5f83e17.jpg)  
+(g) ORAAC Low
+
+![](images/e6e26d0548f0975cbdfd001c18ade85b3a2f2e0687e16c5d48f2bd7ccd229650.jpg)  
+(h) MR-IQN (ours) Low  
+Figure A: Risky point mass environment. In the red zone, the agent gets a random reward. Top (a-d) figures depict the behavior of the policies when the variance of reward is high. Bottom (e-h) figures depict the behavior of the policies when the variance of reward is low. The blue lines are the trajectories of policies.
+
+To analyze the behavior of risk-sensitive offline RL algorithms, we compare the algorithms in a modified risky point mass environment (Ma et al., 2021). The agent is trained to reach the goal (green points in Figure A). In this environment, there is a red zone, where the agent gets normal white noise in the reward signal in 0.1 probability. We compare two situations: when the variance of the noise signal is high (100) and low (5). Figure A depicts the results.
+
+In this environment, MR-IQN (ours) shows the most risk-averse behavior, avoiding the red zone as much as possible, as desired. IQN TD3+BC sometimes fails to avoid the red zone, especially when the variance of the reward signal is low. This difference highlights the conservatism of model risk-sensitive offline RL compared to conventional risk-sensitive offline RL. In contrast, both CODAC and ORAAC consistently fail to avoid the red zone. CODAC's failure is due to the key assumption of strict monotonicity of quantile function (Ma et al., 2021), which does not hold in this environment, coupled with hyperparameter sensitivity inherited from CQL (Tarasov et al., 2024). ORAAC struggles because the strong constraints imposed by the behavior policy make it difficult for the policy to stitch together the desired trajectory.
+
+# A.3 NUMBER OF CRITICS
+
+![](images/3b2495bbd609954c0903f25f42647c915d09fc81fc748e172ace3a942e991315.jpg)  
+Figure B: Effect of the number of critics (i.e., ensemble size). The environment is Forex and the risk measure is Wang(-0.5).
+
+We also investigate how the number of critics affects performance. Figure B shows the results. As demonstrated in TQC Kuznetsov et al. (2020), the performance improvement begins to saturate between 5 and 7 critics for both the mean and risk performance. The observed performance degradation between 3 and 5 critics appears to be an illusion caused by the presence of outliers, as suggested by the large gaps in the error bars.
+
+# A.4 TAIL SCORE ANALYSIS
+
+Table B:  ${CV}@R\left( {{10}\% }\right)$  Performance in finance scenarios  
+
+<table><tr><td></td><td>Target Risk Measure</td><td>CODAC</td><td>ORAAC</td><td>IQN-TD3+BC</td><td>MR-IQN (ours)</td></tr><tr><td rowspan="3">Stock Env.</td><td>Wang(-0.5)</td><td>3.00 ± 0.91</td><td>7.92 ± 0.86</td><td>8.32 ± 0.71</td><td>10.14 ± 0.55</td></tr><tr><td>CV@R(50%)</td><td>2.71 ± 1.51</td><td>7.85 ± 0.82</td><td>8.33 ± 0.67</td><td>10.06 ± 0.53</td></tr><tr><td>CV@R(10%)</td><td>3.98 ± 0.86</td><td>7.91 ± 0.74</td><td>8.68 ± 0.43</td><td>10.35 ± 0.52</td></tr><tr><td rowspan="3">Forex Env.</td><td>Wang(-0.5)</td><td>-51.27 ± 22.79</td><td>-96.59 ± 1.04</td><td>-27.91 ± 48.02</td><td>9.44 ± 59.11</td></tr><tr><td>CV@R(50%)</td><td>-48.72 ± 25.25</td><td>-100 ± 0.00</td><td>-11.47 ± 28.97</td><td>25.71 ± 32.09</td></tr><tr><td>CV@R(10%)</td><td>-4.35 ± 2.06</td><td>-47.54 ± 20.94</td><td>0.25 ± 34.37</td><td>31.40 ± 36.02</td></tr></table>
+
+We further investigated the  $CV@R(10\%)$  (negative) risk in financial scenarios. The results are presented in Table B In Stock environment, only CODAC demonstrates a significant improvement in  $CV@R(10\%)$  risk, whereas the performance of other algorithms appears to have reached saturation.
+
+In Forex environment, all algorithms exhibit consistent behavior, with  $CV@R(10\%)$  optimization yielding the highest  $CV@R(10\%)$  performance, followed by  $CV@R(50\%)$  and  $Wang(-0.5)$  optimization criteria, respectively, except for ORRAC, which does not show meaningful  $CV@R(10\%)$  performance in either  $CV@R(50\%)$  or  $Wang(-0.5)$  risks.
+
+![](images/2f88e18b7ba5c723829d51943dbb0fbbc0407b5e0007a15e49ce2b672f41836e.jpg)  
+(a) Quantile of risk measures  
+Figure C: Visualizing risk measures
+
+![](images/4b8210d3c6c0cb36710c1ae9b3d4bf4d7d20ca90ebc06dec739027a9c07bc022.jpg)  
+(b) Projections of risk measures
+
+To examine this phenomenon, we investigate how risk measures are distinguishable. We leverage the Wasserstein-2 distance  $(D_{W;2})$  to measure distinguishability. The  $D_{W;2}$  between  $CV@R(50\%)$  and  $CV@R(10\%)$  is approximately 0.23, and  $D_{W;2}$  between  $Wang(-0.5)$  and  $CV@R(10\%)$  is approximately 0.40. Therefore,  $CV@R(50\%)$  is a more similar risk measure with  $CV@R(10\%)$  compared to  $Wang(-0.5)$ . This similarity explains the consistent behavior of algorithms. Figure  $\boxed{C}$  visualizes the risk measures. Figure  $\boxed{C}$  (a) describes the quantile function of risk measures, and Figure  $\boxed{C}$  (b) describes the vector projection of risk measures via hat function coefficient by finite element method approximation. We can verify that the previous argument.
+
+Note that spectral risk measure is defined as (Eq.3),
+
+$$
+H _ {\phi} (Z) = - \int_ {0} ^ {1} F _ {Z} ^ {- 1} (p) \phi (p) d p.
+$$
+
+The equation is governed by  $\phi$ , and  $\phi$  is a probability density function defined on  $[0,1]$ . Therefore, it is natural to use Wasserstein distance to measure distinguishability. Meanwhile, the model risk calculates the risk of some function whose  $D_{W;2}$  distance between reference distribution is constrained by  $\sqrt{\varepsilon}$  as defined in Eq.(9). Therefore, it shows similar behavior to other algorithms.
+
+# B EXPERIMENT DETAILS
+
+# B.1 DETAILED EXPLANATION OF BASELINES
+
+ORAAC policy  $\pi_{\mathrm{oraac}}$  is as follows:
+
+$$
+\pi_ {\text {o r a a c}} = \pi_ {\mathrm {b c}} (s) + \kappa \pi_ {\text {r i s k}} (s, \pi_ {\mathrm {b c}} (s)) \tag {A}
+$$
+
+where  $\pi_{\mathrm{bc}}$  is behavior cloning policy implemented using VAE,  $\pi_{\mathrm{risk}}$  is the risk-sensitive perturbation policy, and  $\kappa \geq 0$  is the modification parameter. We set  $\kappa = 0.25$  following Urpf's base implementation.
+
+CODAC's critic objective is as follows
+
+$$
+\begin{array}{l} Z^{*}_{\text{CODAC}} = \arg \min_{Z}\max_{\pi}\mathcal{L}_{\text{crit.}}(\theta) + \\ \lambda_ {p \sim U [ 0, 1 ]} \underbrace {\left(\underset {a \sim \mathcal {D}} {\mathbb {E}} \left[ F _ {Z} ^ {- 1} (p ; s , a , \theta) \right] - \log \underset {a \sim \pi} {\mathbb {E}} \left[ \exp \left(F _ {Z} ^ {- 1} (p ; s , a , \theta)\right) \right]\right)} _ {\text {K L - p o t e n t i a l b y D o n s k e r - V a r a d h a n R e p r e s e n t a t i o n}}. \tag {B} \\ \end{array}
+$$
+
+Table C: Performance of ORAAC and CODAC with/without observation normalization.  
+
+<table><tr><td colspan="2">Target Risk Measure</td><td colspan="2">CV@R (50%)</td><td colspan="2">Wang (-0.5)</td></tr><tr><td colspan="6">Without Observation Normalization</td></tr><tr><td>Env.</td><td>Algorithm</td><td>Mean</td><td>-Hφ(Zπ)</td><td>Mean</td><td>-Hφ(Zπ)</td></tr><tr><td rowspan="2">Forex</td><td>CODAC</td><td>-0.01 ± 0.04</td><td>-0.03 ± 0.09</td><td>3.63 ± 8.12</td><td>-1.74 ± 3.52</td></tr><tr><td>ORAAC</td><td>302.37 ± 19.27</td><td>-99.90 ± 0.01</td><td>304.89 ± 38.05</td><td>70.47 ± 17.22</td></tr><tr><td rowspan="2">Stock</td><td>CODAC</td><td>6.67 ± 1.02</td><td>-2.15 ± 0.75</td><td>5.89 ± 1.64</td><td>0.48 ± 1.40</td></tr><tr><td>ORAAC</td><td>68.6 ± 13.0</td><td>24.0 ± 3.59</td><td>73.43 ± 3.43</td><td>47.68 ± 2.21</td></tr><tr><td rowspan="2">Airsim</td><td>CODAC</td><td>-4190.24 ± 3073.29</td><td>-7460.17 ± 5227.62</td><td>6.67 ± 1.02</td><td>-2.15 ± 0.75</td></tr><tr><td>ORAAC</td><td>-149.58 ± 64.53</td><td>-431.80 ± 217.92</td><td>68.6 ± 13.0</td><td>24.0 ± 3.59</td></tr><tr><td colspan="6">With Observation Normalization</td></tr><tr><td rowspan="2">Forex</td><td>CODAC</td><td>1577.47 ± 41.52</td><td>249.53 ± 28.33</td><td>1591.76 ± 8.12</td><td>892.34 ± 54.06</td></tr><tr><td>ORAAC</td><td>0.00 ± 0.00</td><td>0.00 ± 0.00</td><td>0.00 ± 0.00</td><td>0.00 ± 0.00</td></tr><tr><td rowspan="2">Stock</td><td>CODAC</td><td>37.67 ± 7.54</td><td>13.98 ± 2.34</td><td>33.43 ± 7.01</td><td>21.89 ± 4.65</td></tr><tr><td>ORAAC</td><td>78.48 ± 3.27</td><td>26.07 ± 2.25</td><td>78.57 ± 6.09</td><td>50.97 ± 3.98</td></tr><tr><td rowspan="2">Airsim</td><td>CODAC</td><td>43.56 ± 89.45</td><td>-425.72 ± 97.73</td><td>-2.40 ± 17.59</td><td>-301.23 ± 109.30</td></tr><tr><td>ORAAC</td><td>92.37 ± 40.51</td><td>-88.37 ± 4.20</td><td>156.22 ± 32.09</td><td>51.73 ± 27.53</td></tr><tr><td colspan="6">Behavior Policy</td></tr><tr><td>Forex</td><td></td><td>191.42</td><td>-99.90</td><td>-</td><td>26.91</td></tr><tr><td>Stock</td><td></td><td>1.072</td><td>-11.67</td><td>-</td><td>-6.85</td></tr><tr><td>Airsim</td><td></td><td>378.93</td><td>78.57</td><td>-</td><td>188.29</td></tr></table>
+
+Here  $\mathcal{L}_{\mathrm{crit.}}$  is the quantile regression loss with respect to distributional Bellman residuals defined in Eq.(4), and  $\lambda > 0$  is constant or lagrangian coefficient. Note that the right most part is equal to learning f-GAN discriminator (Nowozin et al., 2016) with KL-divergence (Nguyen et al., 2010). Therefore, CODAC (and CQL) can be understood as  $\mathrm{GAN} + \mathrm{RL}$ .
+
+This leads the optimal actor  $\pi_{\mathrm{codac}}$  is defined as
+
+$$
+\pi_ {\mathrm {c o d a c}} = \underset {\pi} {\arg \min } H _ {\phi} \left(Z _ {\mathrm {C O D A C}} ^ {*}\right) = \underset {\pi} {\arg \min } H _ {\phi} \left(Z ^ {\pi} (s, a)\right), \text {c o n s t r a i n e d} D _ {K L} \left(\pi_ {\mathrm {c o d a c}} \| \pi_ {D}\right) <   \epsilon . \tag {C}
+$$
+
+for some  $\epsilon > 0$  and behavior policy  $\pi_{\mathcal{D}}$  in Eq.(7). See Eq.(8) as well.
+
+1R2R (Rigter et al., 2024) is a model-based risk-sensitive offline RL algorithm based on the dual formula for coherent risk measure as Eq. (D).
+
+$$
+H _ {\phi} \left(Z ^ {\pi} (s, a)\right) = - \inf  \mathbb {E} _ {\mathbb {Q} \in \mathcal {T} (\phi} \left[ Z ^ {\pi} (s, a) \right], \tag {D}
+$$
+
+where  $\mathcal{T}$  is a subset of probability measure which is absolutely continuous with respect to occupation measure of policy,  $\rho^{\pi}$ . The constrains of  $\mathcal{T}$  is  $\phi$  dependent. For example, for  $CV@R_{\alpha}$  risk measure,  $\mathcal{T} = \{\mathbb{P} \ll \rho^{\pi} | \frac{d\mathbb{P}}{d\rho^{\pi}} < \frac{1}{\alpha}\}$  where  $\ll$  denotes absolute continuity and  $\frac{d\mathbb{P}}{d\rho^{\pi}}$  denotes the Radon-Nikodym derivative between  $\mathbb{P}$  and  $\rho^{\pi}$ .
+
+To calculate risk based on Eq.(D), 1R2R samples next states,  $\{s_{t + 1}^{(i)}\}_{i = 0}^{N}$  given  $s_t,a_t$  , with Gaussian ensemble model. Then, construct empirical quantile function based on
+
+$$
+F _ {Z \pi} ^ {- 1} \left(p _ {i}; s, a\right) \approx R \left(s _ {t}, a _ {t}\right) + \gamma V ^ {\pi} \left(s _ {t + 1} ^ {(i)}\right) \tag {E}
+$$
+
+where  $V^{\pi}(s_{t + 1}^{(1)}) \leq V^{\pi}(s_{t + 1}^{(2)}) \leq \dots \leq V^{\pi}(s_{t + 1}^{(N)})$  and  $p_1 = 0, p_2 = \frac{1}{N - 1}, \ldots, p_N = 1$ .
+
+According to the risk measured by approximated quantile function in Eq.(E), the synthetic data  $(s_t, a_t, R(s, a), s_{t+1})$  is rejected or accepted; i.e.,  $s_{t+1}$  is resampled according to  $\phi$ . Therefore, 1R2R minimizes dynamic risk (time-dependent risk) instead of static risk (time-independent risk). However, 1R2R requires terminal function given observation a priori (Rigter et al. 2023). Therefore, we could not conduct the main experiments with 1R2R.
+
+ORAAC and CODAC Tuning: To improve the performance of the baselines (ORAAC and CODAC), we also apply observation normalization. The following tables present baseline performance both with and without observation normalization, and in the main text, we report the better score between them.
+
+# B.2 ENVIRONMENT DETAILS
+
+Finance. The trading environment is implemented using MT-sim (Amin, 2021). The data ranges from February 3rd to December 1st, 2023. In the Forex environment, the agent trades 'EURUSD' and 'USDJPY' with a leverage of 100, while in the Stock environment, it trades 'QQQ', 'SPY', 'IWM', and 'IYY' with a leverage of 10. The forex data is collected from MetaTrader 4 (Sajedi, 2024), and the stock data from Yahoo Finance (Perlin, 2023).
+
+The environment terminates when the agent's assets drop to  $0. To increase randomness, trades are initiated at a random interval, and the agent aims to maximize return within this interval. Observations include balance, equity, margin, orders, and features. The features consist of 15-day accumulated price and volume data, including open, close, low, and high prices. For numerical stability, the data is preprocessed using the arcsinh function, achieving logarithmic scaling. Action represents logits for buying, selling, or holding each symbol. The reward is the benefit ratio relative to the initial balance, fixed at$ 50,000.
+
+Self-driving. The self-driving environment is implemented using Airsim (Shah et al., 2017). The agent's goal is to pilot a quadcopter drone to the target while avoiding static obstacles with height constraints. Random walk noise distorts the agent's actions to simulate wind conditions, introducing randomness. The environment ends when the drone collides with an object or reaches the goal.
+
+The observations provided to the agent include lidar data, previous wind conditions, kinematic information (e.g., orientation, linear velocity), and the goal position. The lidar data has a fixed array shape as it is projected onto a grid in the spherical coordinate system. For each grid cell, the minimal value of radii is taken, capturing the closest detected object in that region. The agent's actions represent the pulse width modification (PWM) values of the motors which corresponds to the RPM of motors. The reward is determined by the cosine similarity between the drone's linear velocity and the goal position. Further, the agent gets 100 reward if it reaches the goal and  $-100$  if it collides. Further, the agent receives alive reward of 0.01 when the agent does not collide with anything. To avoid trivial solution, the agent receives height constrain reward when if flies to high;  $-\max(\text{current height} - \text{height constrain}, 0)$ .
+
+![](images/46b6676748d4c1c143f6488ddcf710ad242dc63e052fb87d5ef07f0d4016f5d2.jpg)  
+(a) Overview of map  
+Figure D: Airsim environment
+
+![](images/4c8a2bd630b1ccd55bd27547a254cc6a04496e233c32d3798225d8abd130c451.jpg)  
+(b) Example view of Airsim
+
+# B.3 HYPERPARAMETER DETAILS
+
+Table D lists up the detailed hyperparameters of the experiments.
+
+Table D: Hyperparameter settings  
+
+<table><tr><td>Hyperparameter (Common)</td><td>Finance</td><td>Self-Driving</td><td>D4RL</td></tr><tr><td>learning rate</td><td colspan="3">3 × 10-4</td></tr><tr><td>optimizer</td><td colspan="3">Adam (β1=0.9,β2=0.999)</td></tr><tr><td>discount factor (γ)</td><td>0.99</td><td>0.96</td><td>0.99</td></tr><tr><td>batch size</td><td></td><td>256</td><td></td></tr><tr><td>layer normalization</td><td></td><td>True</td><td></td></tr><tr><td>num. critics</td><td>5</td><td>5</td><td>3</td></tr><tr><td>num. quantiles for critic training</td><td></td><td>16</td><td></td></tr><tr><td>soft update ratio (τ)</td><td></td><td>5 × 10-3</td><td></td></tr><tr><td>IQN num. cosine (Fourier feature dim for p)</td><td></td><td>64</td><td></td></tr><tr><td>IQN embedding dim.</td><td></td><td>64</td><td></td></tr><tr><td>Model</td><td colspan="3">MR(ours) &amp; IQN-TD3+BC</td></tr><tr><td>q-learning scale (λq. learn)</td><td>2.5</td><td>1.0</td><td>5.0</td></tr><tr><td>Fourier feature init. (σff)</td><td></td><td>1 × 10-3</td><td></td></tr><tr><td>Fourier feature dimensions for s, a</td><td></td><td>250</td><td></td></tr><tr><td>TQC drop out per critic</td><td></td><td>3</td><td></td></tr><tr><td>num. quantiles for actor train (N)</td><td></td><td>100</td><td></td></tr><tr><td>policy delay</td><td></td><td>2</td><td></td></tr><tr><td>Model</td><td colspan="3">ORAAC</td></tr><tr><td>num. actions for inference</td><td></td><td>100</td><td></td></tr><tr><td>action distortion scale (κ)</td><td></td><td>0.25</td><td></td></tr><tr><td>VAE KL-divergence coef. (β)</td><td></td><td>0.5</td><td></td></tr><tr><td>Model</td><td colspan="3">CODAC</td></tr><tr><td>KL-Lagrangian</td><td></td><td>True</td><td></td></tr><tr><td>KL-Lagrangian target</td><td></td><td>10</td><td></td></tr><tr><td>entropy coef</td><td></td><td>0.2</td><td></td></tr></table>
+
+# C MINOR COMMENTS ABOUT TERMINOLOGY
+
+In the actuarial science literature, conditional value at risk  $(CV@R)$ , tail value at risk  $(TV@R)$ , expected shortfall  $(ES)$ , and conditional tail expectation  $(CTE)$  are more strictly distinguished. However, in the main text, we use  $CV@R$  to mean  $TV@R$ , following the ML literature. These terms must be distinguished when the quantile function of a random variable is discontinuous at the confidence level. Because the worst-case quantile function for calculating model risk might also be discontinuous at that level, we acknowledge that our usage actually refers to  $TV@R$  rather than  $CV@R$ , though we keep using  $CV@R$  for consistency with previous ML literature. For details on their differences, see Zhou et al. (2015).
+
+# REFERENCES
+
+Haghpanah Mohammad Amin. gym-mtsim. https://github.com/AminHP/gym-mtsim, 2021. Accessed: 2024-03-28, version 1.1.0.  
+Arsenii Kuznetsov, Pavel Shvechikov, Alexander Grishin, and Dmitry Vetrov. Controlling overestimation bias with truncated mixture of continuous distributional quantile critics. In International Conference on Machine Learning, pp. 5556-5566. PMLR, 2020.  
+Cheng Lu, Huayu Chen, Jianfei Chen, Hang Su, Chongxuan Li, and Jun Zhu. Contrastive energy prediction for exact energy-guided diffusion sampling in offline reinforcement learning. In International Conference on Machine Learning, pp. 22825-22855. PMLR, 2023.  
+Yecheng Ma, Dinesh Jayaraman, and Osbert Bastani. Conservative offline distributional reinforcement learning. Advances in neural information processing systems, 34:19235-19247, 2021.  
+XuanLong Nguyen, Martin J Wainwright, and Michael I Jordan. Estimating divergence functionals and the likelihood ratio by convex risk minimization. IEEE Transactions on Information Theory, 56(11):5847-5861, 2010.  
+Sebastian Nowozin, Botond Cseke, and Ryota Tomioka. f-gan: Training generative neural samplers using variational divergence minimization. Advances in Neural Information Processing Systems, 29, 2016.  
+Marcelo Perlin. Yahoo finance, 2023. URL https://github.com/ropensci/yfR. Python package version 0.2.38.  
+Marc Rigter, Bruno Lacerda, and Nick Hawes. One risk to rule them all: official code, 2023. URL https://github.com/marc-rigter/1R2R.  
+Marc Rigter, Bruno Lacerda, and Nick Hawes. One risk to rule them all: A risk-sensitive perspective on model-based offline reinforcement learning. Advances in Neural Information Processing Systems, 36, 2024.  
+Reza Sajedi. Meta trader5, 2024. URL https://www.metatrader5.com/en/trading-platform. Python package version 5.0.4200.  
+Shital Shah, Debadeepta Dey, Chris Lovett, and Ashish Kapoor. Airsim: High-fidelity visual and physical simulation for autonomous vehicles. In Field and Service Robotics, 2017. URL https://arxiv.org/abs/1705.05065.  
+Denis Tarasov, Alexander Nikulin, Dmitry Akimov, Vladislav Kurenkov, and Sergey Kolesnikov. Corl: Research-oriented deep offline reinforcement learning library. Advances in Neural Information Processing Systems, 36, 2024.  
+Zhendong Wang, Jonathan J Hunt, and Mingyuan Zhou. Diffusion policies as an expressive policy class for offline reinforcement learning. International Conference on Learning Representations, 2023.  
+Xianhua Zhou, Huadong Zhang, and Qingquan Fan. Optimal limited stop-loss reinsurance under var, tvar, and cte risk measures. Mathematical Problems in Engineering, 2015(1):143739, 2015.

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+# MODEL MERGING WITH SVD TO TIE THE KNOTS
+
+George Stoica $^{1*}$  Pratik Ramesh $^{1*}$  Boglarka Ecsedi $^{1}$  Leshem Choshen $^{2}$  Judy Hoffman $^{1}$
+
+<sup>1</sup>Georgia Tech 2IBM Research, MIT
+
+Correspondence emails: {gstoica3, pramesh39}@gatech.edu
+
+# ABSTRACT
+
+Recent model merging methods demonstrate that the parameters of fully-finetuned models specializing in distinct tasks can be combined into one model capable of solving all tasks without retraining. Yet, this success does not transfer well when merging LoRA finetuned models. We study this phenomenon and observe that the weights of LoRA finetuned models showcase a lower degree of alignment compared to their fully-finetuned counterparts. We hypothesize that improving this alignment is key to obtaining better LoRA model merges, and propose KnOTS to address this problem. KnOTS uses the SVD to jointly transform the weights of different LoRA models into an aligned space, where existing merging methods can be applied. In addition, we introduce a new benchmark that explicitly evaluates whether merged models are general models. Notably, KnOTS consistently improves LoRA merging by up to  $4.3\%$  across several vision and language benchmarks, including our new setting. We release our code at: https://github.com/gstoica27/KnOTS.
+
+# 1 INTRODUCTION
+
+Model merging (Garipov et al., 2018; Draxler et al., 2018; Wortzman et al., 2022a; Choshen et al., 2022) is an increasingly popular technique that can surprisingly create a single general model by combining weights of task-specific models. This allows creating multi-task models by accumulating skills (Stoica et al., 2024; Ilharco et al., 2023; Yadav et al., 2023; Ortiz-Jimenez et al., 2024), a desirable trait in various scenarios such as recycling models shared on hubs (Choshen et al., 2023), patching model weaknesses (Cai et al., 2023; Zaman et al., 2023) and collaborating to improve models (Don-Yehiya et al., 2023).
+
+Model merging approaches have found substantial success when merging models that are full-rank finetuned (i.e., all parameters are tuned with maximum rank, we denote it as FFT) to solve distinct tasks from the same pretrained checkpoint, into one model capable of solving all (Ilharco et al., 2023; Matena & Raffel, 2022; Yadav et al., 2023; Ortiz-Jimenez et al., 2024; Wortsman et al., 2022b). In many cases, performing a linear sum over the finetuned model weights without further training, can achieve strong performance(Wortsman et al., 2022b; Ilharco et al., 2023; Yadav et al., 2023).
+
+Interestingly, existing merging approaches do not always transfer well when applied to models finetuned with parameter efficient finetuning (PEFT) strategies (Tang et al., 2024), such as the extremely popular Low-Rank Adaptation (LoRA) (Hu et al., 2022). We probe into this phenomenon ( $\S 3$ ) by comparing the pairwise centered kernel alignment (CKA) (Kornblith et al., 2019) between FFT models and their equivalent LoRA finetuned counterparts. While we observe that FFT models have very high CKA—indicating that the finetuning-updates they apply to the respective pretrained weights (we call these "task-updates") have high alignment, LoRA models exhibit considerably lower CKA. This suggests that the task-updates between different LoRA models process inputs through misaligned subspaces. We hypothesize that aligning these task-updates is crucial to strengthening model merging operations, and propose a data and gradient free approach to do this.
+
+Our method, termed KnOTS (Knowledge Orientation Through SVD), builds upon singular value decomposition (SVD) to transform the task-updates of different LoRA models into a shared space, where merging methods can be applied (Fig. 1). Notably, KnOTS is simple and readily slots into
+
+![](images/f2039e0117cdc7626bc272c6237adfa37a6b76bced6fdbb9d730b24371e0f75a.jpg)  
+Figure 1: The KnOTS method for merging "task-updates" from an arbitrary layer- $j$  of different models. Each weight-update is denoted by  $\Delta W_{j}^{(i)}$ , where  $i$  is the task-update of the  $i^{th}$  model. KnOTS first concatenates the updates together and applies the SVD, to obtain  $U, \Sigma$  and a set of concatenated  $V^{(i)}$  matrices that each correspond to a particular task. KnOTS then merges the  $V$ 's into a single  $V^{(merged)}$  matrix. Finally, KnOTS multiplies the  $U, \Sigma$  and  $V^{(merged)}$  to obtain a merged-update to be added to the pretrained model.
+
+many existing merging methods. KnOTS first concatenates all the task-updates for a layer, and then decomposes the result with the SVD to obtain:  $U\Sigma V^T$  (see §4). We observe that in this construction,  $U$  consists of vectors that form the orthonormal basis for a shared representation space between all task updates, the non-negative diagonal values in  $\Sigma$  represent the scale associated with each basis vector in the aligned space, and  $V^T$  is a set of concatenated matrices (one for each task-update) that are all aligned to the common  $U$ . KnOTS then applies existing merging methods, such as those of Ilharco et al. (2023); Yadav et al. (2023), to the  $V^T$  task-matrices and obtains a  $V^{(merged)}$  matrix that is still aligned to the common  $U$ . After merging, KnOTS constructs the merged model by multiplying the resulting  $U\Sigma V^{(merged)}$  into one matrix at every layer, and adding it to the corresponding parameter matrix from the pretrained model. We validate KnOTS on several model merging settings spanning vision and natural language, demonstrating that KnOTS strengthens existing model merging approaches by up to 4.3% (§ 5).
+
+In addition, we introduce a new variant of the popular eight-vision task merging benchmark originally proposed by Ilharco et al. (2023), aimed at explicitly evaluating a merging method's ability to create general models. Specifically, we transform the benchmark to a "joint-evaluation" setting. This consists of evaluating the performance of arbitrary merged models on the union of all inputs and labels across all eight datasets, without providing the model with any information regarding the dataset a particular example comes from. It contrasts to the existing setting where only labels pertaining to the dataset an example stems from are given to the models to classify. We argue that the joint-evaluation setting is a step towards simulating the ability to create general merged models, capable of preserving the union of all skills captured by their base models (§ 5.3). Notably, KnOTS further improves merging performance in this challenging setting.
+
+# 2 RELATED WORK
+
+Landscape between models. The geometric landscape of non-convex loss functions used to train neural networks remains largely uncharted. However, Draxler et al. (2018); Garipov et al. (2018); Simsek et al. (2021); Frankle & Carbin (2019) reveal that the parameter values of independently trained neural networks can be interpolated without increasing the test loss, a phenomenon known as mode-connectivity. Neyshabur et al. (2020), shows that models are often mode-connected when they are finetuned on the same task from the same pretrained initialization. In parallel, it has become common practice to finetune models using pretrained weights (Dosovitskiy et al., 2021; Huang et al., 2023; Oquab et al., 2024; Hsu et al., 2021; Touvron et al., 2023; Radford et al., 2021). Together with recent discoveries Wortsman et al. (2022b); Matena & Raffel (2022) that even the parameters of models finetuned on different tasks from the same initialization may be merged to create strong multitask models, there has been a surge of model merging approaches for finetuned models.
+
+Model merging methods. Several approaches improve robustness or scores by averaging finetuned weights (Choshen et al., 2022; Wortsman et al., 2022b; Rame et al., 2022; 2024). Task arithmetic (TA) (Ilharco et al., 2023) first subtracts the parameter values of the pretrained model from those of the finetuned models, creating a set of "task-vectors." These are then linearly summed to create a merged model, without any gradient calculations or retraining. Ortiz-Jimenez et al. (2024) theoretically grounds TA, by showing that its success is dependent on the task-vectors governing disjoint regions in the pretrained model's function space—a concept known as "weight disentanglement." In practice, they find that the weights of different finetuned models heavily conflict, and propose a finetuning strategy that disentangles finetuned models to merge better with TA. Daheim et al. (2024); Shah et al. (2024) also find that TA-style merging can improve with finetuning and access to large amounts of training data. TIES (Yadav et al., 2023) improves TA by resolving the parameter interference between models when merging. Specifically, it prunes low-magnitude weights and then only averages weights which share the dominant sign. DARE (Yu et al., 2024) further explores this issue by randomly dropping fine-tuned weights and rescaling the remaining ones to create sparse task-vectors.
+
+Merging LoRA models. LoRA (Hu et al., 2022) has quickly become a very popular finetuning technique. However, existing merging methods (Ilharco et al., 2023; Yadav et al., 2023; Yu et al., 2024) do not transfer well to merging LoRA models (Tang et al., 2024). Tang et al. (2024) suggest this is due to increased weight-entanglement between the models, and propose to a finetuning method similar to Ortiz-Jimenez et al. (2024) which improve the performance of each merging method. To the best of our knowledge, we present the first framework capable of achieving strong merged LoRA models without finetuning.
+
+SVD-based LoRA Approaches. Employing the SVD on LoRA models is not a new idea for purposes other than merging and aligning weights. Meng et al. (2024) initialize the parameters of LoRA models using the SVD of the weights of the corresponding pretrained model, yielding improved finetuning performance. Zhang et al. (2023) use the SVD to improve the gradient updates in parameter efficient finetuning. The popular Hugging Face library (Mangrulkar et al., 2022) includes several merging methods for LoRA models that are suffixed by "-svd" (e.g., "TIES-svd"). These first merge the LoRA parameters of different models according solely to the prefix merging-method (e.g., "TIES"). Afterwards, they use the SVD to decompose the merged parameters back into LoRA weights. KnOTS is unrelated to these approaches. Instead, it significantly improves merging performance by aligning LoRA models with the SVD.
+
+# 3 BACKGROUND AND MOTIVATION
+
+Problem setting. We study gradient-free multitask model-merging (Ilharco et al., 2023; Stoica et al., 2024; Yadav et al., 2023; Jin et al., 2023). Suppose that we have a set of  $n$  models with the same architecture, that have each been finetuned with Low-Rank Adaptation (LoRA) (Hu et al., 2022) on a distinct task (e.g., different image classification problems) from the same pretrained model. Let  $\{f^{(1)}, f^{(2)}, \ldots, f^{(n)}\}$  denote the finetuned models and  $f^{(pt)}$  be the pretrained model. Model-merging methods fuse the parameters of these finetuned models into a single unified model capable of solving all tasks. We study gradient-free approaches as they are both data-efficient and training-free, making them lightweight tools for merging models on-the-fly (Wortsman et al., 2022a). We study models finetuned with LoRA due to its wide usage and because recent work observes notable challenges when applying existing gradient-free methods on such models (Tang et al., 2024).
+
+Model definitions. Let a pre-trained model  $f^{(pt)}$  have  $l$  layers, each containing weights and an optional bias. Assume all biases are concatenated in respective weights, and let weight of the  $j^{th}$  layer be represented as  $W_{j}^{(pt)}$ . Let  $\theta^{(pt)}$  be the collection of all model weights, such that  $\theta^{(pt)} = \{W_1^{(pt)},\dots ,W_j^{(pt)},\dots ,W_l^{(pt)}\}$ . Finetuned models are obtained by applying task-updates to each weight in  $\theta^{(pt)}$ . We denote these updates by  $\tau^{(i)} = \{\Delta W_1^{(i)},\dots ,\Delta W_j^{(i)},\dots ,\Delta W_l^{(i)}\}$ , where  $\Delta W_{j}^{(i)}$  is the update corresponding to the  $j^{th}$  layer in the  $i^{th}$  model. As LoRA constrains each update to be low-rank, we let all updates have a rank of  $r < < \min (I,O)$ , where  $I$  and  $O$  represent the input and output dimension. Every  $\Delta W_{j}^{(i)}\in \mathbb{R}^{O\times I}$  transforms input features  $x^{(i)}\in \mathbb{R}^I$  to output features  $y^{(i)}\in \mathbb{R}^O$  as follows:  $y^{(i)} = \Delta W_{j}^{(i)}x^{(i)}$ . Due to the low-rank constraint,  $\Delta W_{j}^{(i)}$  can only act on
+
+a subspace of size  $r < \min (I,O)$  over  $y^{(i)}$  (Hu et al., 2022). Finally, the parameters of  $f^{(i)}$  are given by  $\theta^{(pt)} + \tau^{(i)} = \{W_1^{(pt)} + \Delta W_1^{(i)},\dots ,W_j^{(pt)} + \Delta W_j^{(i)},\dots ,W_l^{(pt)} + \Delta W_l^{(i)}\}$
+
+Merging models. Existing methods merge models by first defining a function (e.g., identity in (Ilharco et al., 2023)) that transforms the task-updates  $\{\tau^{(1)},\dots \tau^{(n)}\}$ . Second, each update is uniformly scaled via a unique "scaling coefficient", denoted by  $\alpha^{(i)}\geq 0$ . Third, these updates are summed to create a single set of merged updates, which are then added to the pretrained model weights to obtain the final merged model. Optimal values for  $\alpha (i)\geq 0$  are selected according to the performance of the merged model on any available data Ilharco et al. (2023); Yadav et al. (2023); Yu et al. (2024). Although LoRA updates factorize into smaller matrices  $A$  and  $B$ , we utilize the full-matrix representation  $\Delta W_{j}^{(i)} = BA$  in all merge operations. We explain this decision in App. A.
+
+# 3.1 LORA MODELS ARE DIFFICULT TO MERGE
+
+Existing full-rank finetuned (FFT) model merging approaches perform well without additional data, even on different tasks (Ilharco et al., 2023; Yadav et al., 2023). However, Tang et al. (2024) observe that the same merging methods encounter significant challenges when merging LoRA finetuned models, even when these are finetuned on the same tasks. Previous works (Ilharco et al., 2023), Ilharco et al. (2023); Yadav et al. (2023); Tang et al. (2024) use "task-vector orthogonality" as a proxy for understanding when merging is difficult. This consists of flattening the updates  $\{\tau^{(1)},\dots \tau^{(n)}\}$  into a vector, and computing the pairwise cosine similarity between them. The intuition is that when the cosine similarity between vectors is near zero (i.e., they are orthogonal), their weights are disentangled and lie in distinct subspaces—enabling them to be merged without interference.
+
+Task-vector orthogonality may not reliably measure merge potential. To see why, consider a simple toy example. Let  $f^{(1)}(x) = W_2^{(1)}\mathrm{ReLU}(W_1^{(1)}x)$  and  $f^{(2)} = W_2^{(2)}\mathrm{ReLU}(W_1^{(2)}x)$  be two classification models that transform inputs  $x \in \mathbb{R}$  to some  $y^{(1)},y^{(2)} \in \mathbb{R}$  respectively, where  $W_{1}^{(1)},W_{2}^{(1)},W_{1}^{(2)},W_{2}^{(2)} \in \mathbb{R}$ . Let  $W_{2}^{(1)} = 1$ ,  $W_{1}^{(1)} = 1$ ,  $W_{2}^{(2)} = 1$ , and  $W_{1}^{(2)} = -1$ , and assume that each weight was initialized with 0. Here,  $f^{(1)}$  classifies all  $x \in \mathbb{R}^+$  as positive (i.e.,  $y^{(1)} > 0$ ) and all  $x \in \mathbb{R}^{-}$  as negative (i.e.,  $y^{(2)} \leq 0$ ), while  $f^{(2)}$  does the opposite. Calculating the task-vectors of  $f^{(1)}$  and  $f^{(2)}$  give [1, 1] and [-1, 1] respectively—making them orthogonal. However, it is trivial to see that merging their model weights catastrophically affects the performance of at least one model. Any scaled sum over their weights with an  $\alpha^{(i)}$  will preserve one model's predictions while flipping the other's. Despite having orthogonal task-vectors, merging the weights of these models is catastrophic. Thus, additional measures may be helpful to understand when models can be merged.
+
+Merging is easier when model layers share similar activations. Entezari et al. (2022); Ainsworth et al. (2023); Jordan et al. (2023); Stoica et al. (2024) extensively show that models whose layers extract similar intermediate activations for the same inputs, are easier to merge—even when trained on distinct tasks (Stoica et al., 2024). This often occurs when the weights across models extract the same kinds of meanings in a similar order from the inputs (Entezari et al., 2022; Stoica et al., 2024), and are thus aligned. We posit that understanding model alignment from this perspective can serve as a helpful tool for understanding the challenges behind merging. One way to measure the possibility of this is using the centered kernel alignment (CKA) measure Kornblith et al. (2019). CKA measures the similarity between the intermediate activations of two models, at each layer. CKA values close to one indicate that model parameters can be transformed to process information similarly—and thus transform inputs in a functionally aligned manner. When CKA is low, the reverse is true.
+
+FFT models have very aligned CKA representations. Fig. 2a shows the average pairwise CKAs between the task-updates of the FFT models presented in Ilharco et al. (2023). We observe very high CKA, indicating that the updates between models process information in a way that ensures the intermediate feature activations between their layers are well aligned. This suggests that their updates extract similar information from inputs at each layer, and that merging them is less likely to cause significant disruption—as seen by (Ilharco et al., 2023).
+
+LoRA models have considerably less aligned CKA representations. Conducting the same CKA analysis on LoRA variants of the same models from (Ilharco et al., 2023) shows that the models
+
+![](images/832163117c0f3ad82d194a9327039327e558a13814601faefb58fdd356a18e16.jpg)  
+(a) Full-rank finetuned model alignments  
+(b) LoRA finetuned model alignments  
+(c) LoRA finetuned model alignments with KnOTS  
+Figure 2: Finetuning strategy impacts representation alignments between models trained on different tasks. The figure shows the average pairwise centered kernel alignment (CKA—Kornblith et al. (2019)) between the outputs solely given by finetuning updates (e.g., a  $\Delta W_{j}^{(i)}$ ) across every attention layer, from models finetuned on different tasks with different strategies (defined in §5.2). High CKA indicates that the task-updates of different models are aligned. (a) Full-rank finetuned models exhibit high CKA and alignment.(b) LoRA finetuned models are drastically less aligned. (c) However, they are dramatically more aligned with KnOTS.
+
+exhibit dramatically lower average CKA between task-updates (Fig. 2b). This suggests that they are functionally misaligned: from the same inputs each LoRA extracts unrelated features. Entezari et al. (2022); Jordan et al. (2023); Stoica et al. (2024) observe that merging models in these circumstances can yield poor performance. We speculate that this phenomenon may be due to the low-rank constraint imposed on LoRA models. Specifically, each task-update can only act on a subspace of the input and output activation space. Since each LoRA model is finetuned on a different dataset, it must carefully pick the subspace it uses to extract information. Thus, it is unlikely that the task-updates across different models extract the same kind of information, leading to low activation alignment between updates and poor merges.
+
+Our hypothesis. Coupling the CKAs of FFT model task-updates with those of LoRA models, we hypothesize that aligning LoRA task-updates is a crucial first step to improving their mergeability, and the performance of existing merging methods. We seek to design such an alignment method without assuming access to data or gradients.
+
+# 4 METHOD:KNOTS
+
+We propose a data-free and gradient-free method for aligning LoRA models, to improve the performance of existing model merging methods. Our method, termed KnOTS (Knowledge Orientation Through SVD), uses singular value decomposition (SVD) to jointly align the representation spaces between LoRA models (see illustration in Fig. 1).
+
+Subspace alignment with the SVD. KnOTS aligns the updates across different LoRA models layer-wise. Thus, it suffices to consider how we align the updates for an arbitrary layer  $j$ . Recall that these updates are denoted by:  $\Delta W_{j}^{(1)},\ldots ,\Delta W_{j}^{(i)},\ldots ,\Delta W_{j}^{(n)}$ , each extracting different kinds of information. For notational ease, we drop the subscript  $j$  for the remainder of this section. Most often SVD is used with square matrices, aligning vectors in them. Instead, we propose to align these updates into a shared space where they can extract similar information, by concatenating the updates across tasks and deconstructing the result with SVD:
+
+$$
+\begin{array}{l} S V D \left[ \Delta W ^ {(1)}; \Delta W ^ {(2)}; \dots ; \Delta W ^ {(n)} \right] = U \Sigma V ^ {T} (1) \\ = U \Sigma \left[ V ^ {(1)}; V ^ {(2)}; \dots ; V ^ {(n)} \right] ^ {T} (2) \\ \end{array}
+$$
+
+where,  $U\in \mathbb{R}^{O\times k}$ $\Sigma \in \mathrm{Diag}(\mathbb{R}^k)$ $\{[V^{(1)}]^T,\ldots ,[V^{(n)}]^T\} \in \mathbb{R}^{k\times I}$ $1\leq k\leq \min (I,O,nr)$  Here,  $\Delta W^{(1)} = U\Sigma [V^{(1)}]^T$  .  $\Delta W^{(2)} = U\Sigma [V^{(2)}]^T;\dots ;\Delta W^{(n)} = U\Sigma [V^{(n)}]^T$  . With the SVD,
+
+each task-update is decomposed into a shared  $U\Sigma$  term—defining the orthogonal basis for the activations from all task-updates—along with task specific components,  $[V^{(i)}]^T$ . By construction, each  $[V^{(i)}]^T$ 's from any task-update acts to transform the same information (given by  $U\Sigma$ ) with any input, thereby increasing their alignment. We visualize this effect in Fig. 2c. Specifically, we compute the CKA between the intermediate activations obtained from passing the same inputs through each  $[V^{(1)}]^T, \ldots [V^{(n)}]^T$  of our LoRA models, and averaging over all layers to be merged. We observe that the CKAs are significantly higher compared to Fig. 2b, indicating that the task-updates between LoRA models are substantially more aligned under KnOTS.
+
+Applying merge methods. In many cases, we can directly apply existing merging methods on the aligned  $[V^{(1)}]^T,\ldots [V^{(n)}]^T$  without modifications. Specifically, we can apply some merging function (e.g., weighted sum a-la-(Ilharco et al., 2023)) over these parameters to obtain a single merged  $[V^{(merged)}]^T$ . We can then construct the merged update with the remaining SVD components:  $\Delta W_{j}^{(merged)} = U\Sigma [V^{(merged)}]^{T}$ , which is finally added to the pre-trained layer weights  $W_{j}$ .
+
+# 5 EXPERIMENTS AND RESULTS
+
+We validate KnOTS across diverse benchmarks spanning both vision and language domains. We first evaluate KnOTS on the popular "per-task" setting across both vision and language tasks (§ 5.2). Here, models finetuned on distinct datasets are merged and then evaluated on each dataset separately. KnOTS consistently improves the capabilities of existing merging methods across all experiments. Second, we study the capabilities of merging methods building general models by introducing a new benchmark (§ 5.3). Despite being very challenging, KnOTS is still capable of uplifting existing merging approaches. Third, we conduct extensive analysis on different facets of KnOTS (§ 5.4).
+
+# 5.1 EXPERIMENTAL DETAILS
+
+Models. We use ViT-B/32 or ViT-L/16 (Dosovitskiy et al., 2021) for all our vision experiments. These are two variants of the CLIP vision encoder (Radford et al., 2021) that are finetuned separately on a variety of tasks. Our natural language experiments are conducted with LLaMa3-8B model (AI, 2024) across different natural language inference (NLI) tasks. As in (Hu et al., 2022), we only apply LoRA on the weight matrices in the attention layers: namely the key, query, value and output layers. Unless otherwise specified, each LoRA has a rank of 16. Further training details are found in App. D.
+
+Merging methods. As discussed in § 3, we restrict our scope to gradient-free merging methods in this work. To the best of our knowledge, there are only four such merging methods for our setting: RegMean (Jin et al., 2023), Task-Arithmetic (TA) (Ilharco et al., 2023), TIES (Yadav et al., 2023), and DARE (Yu et al., 2024). RegMean (Jin et al., 2023) is a merging approach that also aligns the weights of each model. It does this by solving a closed-form locally linear regression problem at every model layer. This results in obtaining a transformation matrix for each models' task-updates, that when applied aligns the updates between models. In contrast, TA, TIES and DARE assume that the task updates between models stemming from a shared pre-trained checkpoint are functionally aligned and directly merge them.
+
+TA (Ilharco et al., 2023) merges models finetuned on different tasks with the same pretrained checkpoint by linearly summing their parameters. They apply a summation-weight (termed "scaling coefficient") to the parameters of each model that achieves the best merged model performance over a held-out validation set. Instead of tuning a unique scaling co-efficient for each task-vector which may be expensive and does not scale well, we only tune a single scaling co-efficient for all models as recommended by Ilharco et al. (2023); Yadav et al. (2023). TIES (Yadav et al., 2023) extends TA by reducing noise and conflicts between parameters when merged. It reduces noise by pruning  $k$  parameters with the lowest magnitudes, and reduces parameter conflicts by also pruning those that lie in outlying directions—a step called "sign-resolution". TIES then linearly sums the model parameters using the scaling coefficients and  $k$  for pruning, that achieve the best merged model over the held-out set. DARE (Yu et al., 2024) reduces noise by randomly pruning parameters following a Bernoulli distribution with probability  $p$ . The remaining parameters are then rescaled by  $\frac{1}{(1 - p)}$  to account for scale lost from pruned parameters. Afterwards, DARE typically employs sign-resolution of TIES (specified by DARE-TIES). To account for its random pruning step, we run
+
+Table 1: Eight models per-task results. We merge eight ViT-B/32 models finetuned with LoRA on different image classification datasets. "Finetuned" refers to the accuracy of each finetuned model on the dataset it was trained on. We report the per-task (including the average) normalized-accuracies across other merging baselines. These describe how close they get to the "Finetuned" accuracy. KnOTS-TIES improves over baselines by up to  $4.3\%$  average accuracy.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="9">Datasets</td></tr><tr><td>Cars</td><td>DTD</td><td>EuroSAT</td><td>GTSRB</td><td>MNIST</td><td>RESISC45</td><td>SUN397</td><td>SVHN</td><td>Avg</td></tr><tr><td colspan="10">Per-Task Absolute Accuracies (%)</td></tr><tr><td>Finetuned</td><td>74.0</td><td>58.3</td><td>99.0</td><td>92.7</td><td>99.3</td><td>88.4</td><td>64.5</td><td>96.2</td><td>84.1</td></tr><tr><td colspan="10">Per-Task Accuracies of Merged Models Normalized Against Finetuned Models (%)</td></tr><tr><td>RegMean</td><td>80.2</td><td>71.3</td><td>37.9</td><td>47.3</td><td>43.1</td><td>70.5</td><td>93.9</td><td>43.0</td><td>60.9</td></tr><tr><td>TA</td><td>82.0</td><td>73.6</td><td>48.8</td><td>42.1</td><td>53.1</td><td>71.5</td><td>97.5</td><td>41.2</td><td>63.7</td></tr><tr><td>TIES</td><td>82.2</td><td>72.8</td><td>50.0</td><td>36.8</td><td>56.8</td><td>69.4</td><td>96.9</td><td>44.6</td><td>63.7</td></tr><tr><td>DARE-TIES</td><td>81.4</td><td>74.5</td><td>50.8</td><td>39.2</td><td>55.0</td><td>70.7</td><td>97.6</td><td>40.1</td><td>63.7</td></tr><tr><td>KnOTS-TIES</td><td>82.7</td><td>73.7</td><td>49.3</td><td>48.9</td><td>68.9</td><td>70.9</td><td>95.5</td><td>53.8</td><td>68.0</td></tr><tr><td>KnOTS-DARE-TIES</td><td>81.8</td><td>75.9</td><td>50.7</td><td>40.3</td><td>53.2</td><td>70.2</td><td>97.9</td><td>41.0</td><td>63.9</td></tr></table>
+
+all DARE experiments across three separate seeds, and report the best performance. KnOTS can be coupled with merging methods such as these latter three by applying them directly on aligned  $[V^{(i)}]^T$  parameters. Note that we make an exception for methods like TIES (Yadav et al., 2023) that involve magnitude-based model pruning. While model-scale is absent in  $[V^{(i)}]^T$  (its row-wise magnitudes are  $\leq 1$  by definition), it is retained in  $\Sigma$ . Thus, we may apply such operations on  $\Sigma [V^{(i)}]^T$ , but merge using the newly pruned  $[V^{(i)}]^T$  s. Additionally, we observe that applying KnOTS on TA reduces to performing TA:  $\sum_{i=1}^{n} \alpha^{(i)} \Delta W_j^{(i)} = \sum_{i=1}^{n} \alpha_i U \Sigma [V_j^{(i)}]^T = U \Sigma \sum_{i=1}^{n} \alpha_i [V_j^{(i)}]^T$ . Thus, KnOTS on TA can be considered a generalization of TA: KnOTS reduces to the standard merging method when its alignment is not leveraged. We refer to all methods we apply KnOTS to by prefixing their names with KnOTS (e.g., KnOTS on TIES is denoted by KnOTS-TIES). Inspired by the recommendations made by each merging baseline, we tune hyperparameters such as scaling-coefficient, top-K and DARE-pruning-co-efficient as described in C. Note: KnOTS does not require any new hyperparameter to be tuned apart from the ones used by the original merging methods.
+
+In some experiments, we add the "Ensemble" as a baseline, which consists of all models used in a particular merging evaluation. The ensemble passes an input in parallel to all models, and predicts its class according to the highest confidence prediction across all models.
+
+Metrics. We report the absolute accuracy of all individual finetuned models on their respective datasets, and utilize these for merging. Similar to Ilharco et al. (2023); Yadav et al. (2023), we compare our merging methods via "normalized-accuracy" wherever applicable. This is obtained by dividing the performance of a merged model on a task (e.g., Cars (Krause et al., 2013)) by the performance of the original model finetuned on the task. For instance, the normalized accuracy for a given task-i is computed as  $\frac{\text{Accuracy of merged model on task-i}}{\text{Accuracy of finetuned model on task-i}}$ . This metric shows how close the merged model gets to the original finetuned model for each task. Certain experiment settings study different generalization properties and thus have different metrics. For the remainder of this paper, we assume all accuracy measurements are normalized, and define other metrics in their relevant sections.
+
+# 5.2 PER-TASK EVALUATIONS ON VISION AND NLP SETTINGS
+
+We first compare KnOTS's ability to improve existing merging methods in the popular "per-task" experiment settings (Ilharco et al., 2023; Yadav et al., 2023; Tang et al., 2024). These settings comprise of merging a set of models independently finetuned on different datasets into a single model, and then evaluating it on each dataset independently—using only its examples and labels. This way, the merged model is judged on its ability to preserve the individual skills of each original model.
+
+Merging eight ViT-B/32 models finetuned on image classification datasets. We follow the image classification benchmark from Ilharco et al. (2023) and merge models finetuned on eight different datasets: Cars (Krause et al., 2013), DTD (Cimpoi et al., 2014), EuroSAT (Helber et al., 2019),
+
+GTSRB Stallkamp et al. (2011), MNIST (LeCun, 1998), RESISC45 (Cheng et al., 2017), SUN397 (Xiao et al., 2016) and SVHN (Netzer et al., 2011). Like Ilharco et al. (2023); Yadav et al. (2023), we report normalized per-task accuracies and their average. Tab. 1 reports merging performances. Interestingly, nearly all merging methods achieve comparable performance. However, applying KnOTS on both TIES and DARE-TIES elevates their respective performances, with KnOTS-TIES strengthening TIES by  $4.3\%$  average normalized accuracy. Note, we also compare against a popular gradient-based approach—Fisher weight averaging (Matena & Raffel, 2022) in App. E and achieve considerable gains, though gradient-based merging is not our focus.
+
+KnOTS scales to ViT-L/14 models. We also evaluate how KnOTS is affected by merging larger vision models. Specifically, we merge eight ViT-L/14 models finetuned with LoRA on the same datasets as our ViT-B/32 counterparts. We then evaluate the merged model's normalized per-task accuracies over each dataset. Results for several merging methods are summarized in Tab. 2.
+
+While performance of all merging methods improve with the larger models, KnOTS notably con
+
+sistently outperforms the baselines in both KnOTS-TIES and KnOTS-DARE-TIES. Additionally, KnOTS-TIES is still capable of improving TIES by  $3\%$ , showing its ability to maintain performance with scale.
+
+Table 2: Normalized per-task avg. image classification results. We merge eight ViT-L/14 models finetuned with LoRA on eight vision datasets. We report the average normalized accuracies against average absolute accuracy of the finetuned models:  $92.3\%$ . KnOTS is best.
+
+<table><tr><td rowspan="2">TA</td><td rowspan="2">TIES</td><td rowspan="2">DARE-TIES</td><td colspan="2">KnOTS</td></tr><tr><td>TIES</td><td>DARE-TIES</td></tr><tr><td>74.4</td><td>75.2</td><td>74.7</td><td>78.2</td><td>75.6</td></tr></table>
+
+KnOTS performs well on LLMs. We also evaluate KnOTS in the NLI setting, by merging six PEFT llama3-8B (AI, 2024) models finetuned on SNLI (Bowman et al., 2015), MNLI (Williams et al., 2018), SICK (Marelli et al., 2014), QNLI, RTE (Wang et al., 2019), and SCITAIL (Khot et al., 2018). Each of these tasks involves performing a 3-way classification to determine whether a given "hypothesis" is true (entailment), false (contradiction) or inconclusive (neutral) compared to a given "premise." QNLI, RTE and SCITAIL
+
+only employ two of the three classes, so we simply mask the missing label when finetuning and evaluating on these datasets. Tab. 3 shows the results of merging these models using TA, TIEs, DARE-TIES, KnOTS-TIES and KnOTS-DARE-TIES. Overall, we observe that KnOTS-TIES significantly outperforms baseline merging methods by up to  $2.9\%$  average normalized accuracy, and KnOTS-DARE-TIES further improves DARE-TIES by  $2\%$ . This demonstrates the power of utilizing KnOTS to align LoRA models even on dramatically larger models (8B parameters), and its robustness across modalities.
+
+Table 3: Normalized per-task avg. NLI results. We merge six LLama3-8B models finetuned with LoRA on different NLI datasets. All numbers are normalized against the absolute average per-task accuracy of the individual finetuned models:  $92.9\%$ . KnOTS performs best.
+
+<table><tr><td rowspan="2">TA</td><td rowspan="2">TIES</td><td rowspan="2">DARE-TIES</td><td colspan="2">KnOTS</td></tr><tr><td>TIES</td><td>DARE-TIES</td></tr><tr><td>90.2</td><td>90.0</td><td>90.9</td><td>92.9</td><td>91.1</td></tr></table>
+
+# 5.3 A NEW BENCHMARK TOWARDS BUILDING GENERAL MODELS
+
+We further introduce a new experimental setting to the eight vision per-task benchmark in Tab. 1, that we call the "joint-task." This task differs from the "per-task" regime in that it evaluates merged models over the union of all inputs and labels across every vision-task. The joint-task is significantly more challenging than the per-task counterpart as it explicitly examines whether a merged model is a general model: whether it can classify any input to any label.
+
+Table 4: Eight models joint-task results. We merge eight ViT-B/32 models finetuned with LoRA on different datasets. We report the joint-task "Union" performances for several merging methods. KnOTS-TIES performs the best.
+
+<table><tr><td rowspan="3">Metric</td><td colspan="6">Method</td></tr><tr><td rowspan="2">Ensemble</td><td rowspan="2">TA</td><td rowspan="2">TIES</td><td rowspan="2">DARE-TIES</td><td colspan="2">KnOTS</td></tr><tr><td>TIES</td><td>DARE-TIES</td></tr><tr><td>Hits@1</td><td>40.7</td><td>43.6</td><td>43.6</td><td>44.0</td><td>46.8</td><td>45.2</td></tr><tr><td>Hits@3</td><td>63.1</td><td>65.3</td><td>65.3</td><td>66.4</td><td>68.1</td><td>66.9</td></tr><tr><td>Hits@5</td><td>72.6</td><td>74.0</td><td>73.9</td><td>75.1</td><td>76.3</td><td>75.3</td></tr></table>
+
+After aggregating the labels across all eight tasks and removing duplicates (e.g., MNIST (LeCun, 1998) and SVHN (Netzer et al., 2011) have the same labels), we obtain 748 unique labels over which to classify all the images across all datasets. As some labels in one task are hyponyms of those in another (e.g., "islet" in SUN397 (Xiao et al., 2016) and
+
+"island" in RESISC45 (Cheng et al., 2017)) making it challenging to distinguish labels, we report performance using "Hits@k." This refers to the number of times the expected label is within the top-  $k$  predictions of a model (e.g., Hits@1 is accuracy). Tab. 4 shows the results over the "Union" of all images and labels across all tasks, when merging our ViT-B/32 LoRA models. Please see App. F for performances broken down by dataset. Overall, KnOTS-TIES significantly outperforms every baseline at all Hits@k levels on the extremely challenging "Union" evaluation—by up to  $3.2\%$  on Hits@1. Interestingly, we also observe that the ensemble performs particularly poorly in the joint setting. We posit this is due to certain models making over-confidently incorrect predictions on data from tasks they are not finetuned on, a notable issue in ensemble models (Kardan et al., 2021). We argue that the joint-task is an important benchmark for assessing a merged model's generality, and we hope future work expands upon it.
+
+# 5.4 ADDITIONAL ANALYSIS & ABLATIONS
+
+We also conduct three analysis experiments to understand different facets of KnOTS. We conduct our analysis with KnOTS-TIES as it consistently achieves the best performance across all settings.
+
+KnOTS scales better with the number of models. Following Ilharco et al. (2023); Yadav et al. (2023), we evaluate the performance of KnOTS on the eight per-task vision benchmark with our ViT-B/32 models as we increase the number of tasks being merged. Fig. 3 illustrates the same measuring performance with average normalized accuracies of the merged model, where the dots represent the performance of a model evaluated only on the tasks included in the merge, while the bars indicate the  $95\%$  confidence intervals based on random subselection of 28 task combinations.
+
+Notably, KnOTS-TIES achieves significantly better merging results compared to the baselines, with a performance gap that remains consistently  $>4\%$  (for  $>2$  tasks being merged), underscoring its robustness as the number of tasks increases.
+
+KnOTS is robust to different LoRA ranks. Fig. 4 shows the performance of KnOTS-TIES against TIES (Yadav et al., 2023) in merging our ViT-B/32 models trained at various LoRA ranks {4, 16, 64, 256, 768} on the eight per-task vision benchmark. Note that these ViTs each have a feature dimension of 768, making our LoRA rank 768 setting "full rank". The results consistently demonstrate that KnOTS-TIES outperforms TIES across all rank settings. In the very low rank setting (e.g., rank four), KnOTS improves TIES by  $6\%$  (64.6% normalized accuracy vs.  $58.6\%$ ). The performance uplift continues with increasing LoRA rank, with KnOTS improving TIES by  $4\%$  in the full rank setting. This experiment highlights how KnOTS is both robust and scalable to merging models finetuned on diverse tasks with arbitrary LoRA rank.
+
+![](images/696730e9d9089d8a9fbeb8efa53be1f4654459b0048a3e385bb50245878073bf.jpg)  
+Figure 3: KnOTS boosts performance with scale. KnOTS-TIES continues to see gains, outperforming original TIES (Yadav et al., 2023) and Task Arithmetic (TA) (Ilharco et al., 2023) when merging an increasing number of tasks in the per-task evaluation vision setting § 5.2. Performance is the average normalized accuracy with  $95\%$  confidence intervals over merging different combinations of the tasks
+
+![](images/0e3e21d7754b599ca14bed46107d3cb16e71598f7bfc94b95f125fb22d9bbffc.jpg)  
+Figure 4: KnOTS boosts performance across LoRA ranks. KnOTS-TIES consistently outperforms TIES (Yadav et al., 2023) across varying LoRA ranks in the per-task evaluation vision setting. Performance is reported in terms of average normalized accuracy across eight vision tasks.
+
+The way task-updates are concatenated matters. KnOTS concatenates task-updates column-wise before computing the SVD. In this experiment, we investigate the effects of concatenating them
+
+row-wise instead. We conduct our analysis using the same vision models from Tab. 1 and merge them with KnOTS-TIES. We observe that applying KnOTS-TIES after concatenating the rows of the task-updates achieves an average normalized accuracy of  $65.4\%$ . It performs  $2.6\%$  worse than KnOTS-TIES with column-wise task-update concatenation. We posit this may be due to the difference in the deconstructed task-update terms when concatenating them row-wise. Specifically, each update would be denoted by  $\Delta W_{j}^{(i)} = U^{(i)}\Sigma V$ , where  $\Sigma V$  are shared across all updates with different  $U^{(i)}$ . In this way, the shared  $\Sigma V$  would act on distinct  $U^{(i)}$  containing different information—which in turn may lower alignment. These results suggest that concatenating the updates column-wise is crucial to obtaining strong performance with KnOTS.
+
+# 6 CONCLUSION
+
+In this paper, we study merging LoRA models sharing the same pretrained checkpoint and finetuned on different tasks without additional finetuning. We find that prior work does not transfer well in this setting, and observe that this is due to parameter misalignment between LoRA models. We introduce KnOTS to tackle this problem by using the singular value decomposition (SVD) to map the parameters of each LoRA model into a shared representation space with aligned elements, over which prior work can be applied to merge the parameters. In addition, we introduce a novel benchmark designed to evaluate whether merged models can be general. Notably, KnOTS improves prior work by up to  $4.3\%$  across both vision and language settings, including our new setting.
+
+# 7 REPRODUCIBILITY STATEMENT
+
+We describe the procedure for finetuning all our models, including compute resources, datasets, and hyperparameter searches in App. D. App. C describes our hyperparameter searches for all merging methods. App. B explains how we calculated our centered kernel alignment (CKA) visualizations in Fig. 2. In addition, App. F provides extensive details on how we created our novel benchmark.
+
+# 8 ACKNOWLEDGEMENTS
+
+This work was partially sponsored by NSF awards #2403297 and #2144194, an NSF-GRFP, and Google. All views and conclusions expressed in this work are those of the authors and not a reflection of these sources.
+
+We would like to thank Simar Kareer for his valuable insights and discussions that contributed to the brainstorming process of this work.
+
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+
+# A WHICH LORA REPRESENTATION IS BEST FOR MERGING?
+
+LoRA is a popular Parameter Efficient Finetuning (PEFT) method that finetunes models by applying low-rank updates to the pretrained parameters. For instance, let  $W_{j}^{(i)}, \Delta W_{j}^{(i)} \in \mathbb{R}^{O \times I}$ . With LoRA, each  $\Delta W_{j}^{(i)}$  is rank  $r << \min(O, I)$ , and can be factorized into two low-rank matrices  $A_{j}^{(i)} \in \mathbb{R}^{r \times I}, B_{j}^{(i)} \in \mathbb{R}^{O \times r}$ , with  $\Delta W_{j}^{(i)} = B_{j}^{(i)}A_{j}^{(i)}$ . The existence of  $A_{j}^{(i)}$ , and  $B_{j}^{(i)}$  for each  $\Delta W_{j}^{(i)}$  may suggest that merging should be done on them separately, and the results multiplied to achieve the merged-update  $\Delta W_{j}^{(m)}$ . However, this immediately leads to issues when applying existing merging methods. For instance, let us apply task-arithmetic (TA) Ilharco et al. (2023) on a single arbitrary layer (e.g., the  $j^{th}$  layer) across  $f^{(1)}, \ldots, f^{(n)}$ . First, TA merges the factorizations of  $\Delta W_{j}^{(1)}, \ldots, \Delta W_{j}^{(n)}$  by computing  $\sum_{i=1}^{n}\lambda^{(i)}A_{j}^{(i)}, \sum_{i=1}^{n}\lambda^{(i)}B_{j}^{(i)}$ . Multiplying each to obtain  $\Delta W_{j}^{(m)}$  yields,
+
+$$
+\begin{array}{l} \Delta W _ {j} ^ {(m)} = \left(\sum_ {i = 1} ^ {n} \lambda^ {(i)} B _ {j} ^ {(i)}\right) \left(\sum_ {i = 1} ^ {n} \lambda^ {(i)} A _ {j} ^ {(i)}\right) (3) \\ = \underbrace {\left(\sum_ {i = 1} ^ {l} \lambda^ {(i)} \lambda^ {(i)} B _ {j} ^ {(i)} A _ {j} ^ {(i)}\right)} _ {\text {P r o d u c t s o f a l i n g f a c t o r i z a t i o n s}} + \underbrace {\left(\sum_ {i \neq k} ^ {l} \lambda^ {(i)} \lambda^ {(k)} B _ {j} ^ {(i)} A _ {j} ^ {(k)}\right)} _ {\text {P r o d u c t s o f m i s a l i n g f a c t o r i z a t i o n s}}. (4) \\ \end{array}
+$$
+
+Notably,  $\Delta W_{j}^{(m)}$  consists of two terms: one composed of  $B_{j}^{(i)}$  and  $A_{j}^{(i)}$  matrices from the updates of the same model, and one composed of  $B_{j}^{(i)}$  and  $A_{j}^{(k)}$  matrices from updates of different models. Unfortunately, there is no guarantee that  $B_{j}^{(i)}$  expects the same input representation as the output of  $A_{j}^{(k)}$  when  $i \neq k$ , and including these terms in the merged model can incur significant drop in model performance Stoica et al. (2024). Similarly, this same issue persists as well when applying Yadav et al. (2023). Thus, we only conduct merging on the original update representations:  $\Delta W_{j}^{(i)}$  of each model. This trivially avoids mismatches when applying merging methods.
+
+# B COMPUTING CENTERED KERNEL ALIGNMENTS ACROSS TASKS
+
+We create the pairwise centered kernel alignment (CKA) (Kornblith et al., 2019) matrices presented in Fig. 2 using models that are finetuned from the same pretrained checkpoint on the different tasks shown in Tab. 1. Each matrix is generated using a set of unlabeled heldout data from all eight datasets. Specifically, this heldout set consists of the validation data of the respective dataset when it exists and otherwise randomly samples  $20\%$  of the test set. Note that in situations where we sample  $20\%$  of a dataset's test split, we always evaluate any merged model on the remaining  $80\%$  of examples. For fair comparison, all CKA plots are created using the same data.
+
+Pairwise CKA on full-rank finetuned models. We compute Fig. 2a) as follows. We first take the set of eight full-rank finetuned (FFT) ViT-B/32 (Dosovitskiy et al., 2021) models released by Ilharco et al. (2023), along with their shared pretrained model. We then collect the intermediate outputs of every key, query, value and projection layer across all nine models over all the data from our heldout set. This yields nine different intermediate activations for each layer across all models. We then subtract the activations of the pretrained model from the eight activations of the finetuned models, at every layer. We then compute the CKA (following (Kornblith et al., 2019)) between activations of two different models at every layer, across all model pairs. Finally, we average the results over all layers.
+
+Pairwise CKA on LoRA finetuned models. We compute Fig. 2b) as follows. We first take the set of eight LoRA models we merge in in Tab. 1, and extract the intermediate activations over all LoRA layers of each model and over the same data used in our FFT CKA plots. Specifically, these layers are every key, query, value and projection layer of each model. Since LoRA finetuning adds new parameters to a pretrained model, there are no layers from the pretrained model to subtract. Thus, we
+
+directly compute CKA over the activations at the same layer across LoRA model pairs, and average over all layers.
+
+Pairwise CKA on LoRA finetuned models aligned with KnOTS. We compute Fig. 2c) as follows. This procedure is extremely similar to computing the CKAs for LoRA finetuned models, with one small difference. Specifically, rather than computing the CKAs from the output activations of LoRA layers, we instead calculate the CKA over the outputs of different  $[V^{(i)}]^T$  for the same layer of different LoRA models (using the same data again). Afterwards, we average the results across all layers.
+
+# C TUNING THE HYPERPARAMETERS FOR DIFFERENT MERGING BASELINES
+
+Scaling co-efficient: A single scalar scaling co-efficient is tuned across the range [0.1, 0.2, 0.3,..., 1.0]
+
+top- $k$ : Like Yadav et al. (2023), we define top- $k$  the percentage of task-update elements retained when merging. It is tuned across the range [10, 20, 30,..., 100]
+
+DARE-pruning co-efficient  $(p)$ : [0.99, 0.9, 0.8, ..., 0.1]
+
+DARE-Seeds: We evaluated DARE over the following five seeds: [420, 421, 422, 423, 424].
+
+For methods such as TIES (Yadav et al., 2023) and DARE (Yu et al., 2024) which involve tuning more than one scaling co-efficient we tune by performing a linear-search by first tuning the scaling coefficient and then the corresponding pruning hyperparameter i.e. top- $k$  for TIES and DARE-pruning co-efficient for DARE. The default value used for TIES top- $k$  is 30 and for the DARE-pruning co-efficient is 0.9 as recommended by their respective baselines.
+
+# D TRAINING DETAILS
+
+Training Vision models. We make use of the CLIP (Radford et al., 2021) based ViT-B/32 and ViT-L/14 models from Hugging Face (HF). These models are then LoRA fine-tuned using HF's PEFT LoRA library. Across all our experiments we initialize the LoRA layers across the query, key, value and output layer. Note that these are the only learnable layers. We set the LoRA rank to be 16, LoRA alpha to be 16, LoRA dropout to be 0.1 and disable the use of bias parameters. All models are trained using the AdamW (Loshchilov & Hutter, 2019) optimizer, with a cosine learning rate scheduler (Loshchilov & Hutter, 2017) using Cross-Entropy loss.
+
+The ViT-B/32 models were fine-tuned on the 8 vision tasks using a standard learning rate of 1e-5, weight decay of 1e-1 and label smoothing set to 0.
+
+The ViT-L/14 models were fine-tuned on the 8 vision tasks using a standard learning rate of 3e-4, weight decay of 1e-4 and label smoothing set to 0.
+
+Across all our experiments the text encoder in the CLIP-model remains frozen and the text embeddings are obtained by passing the class labels through the text encoder.
+
+Training LLMs on NLI tasks. For the setting in § 5.2 we use the LLama-3 8B parameter model. We initialize the model using the "AutoModelForSequenceClassification" model class from Hugging Face's "AutoModel" to classify across 3 classes. The models are LoRA fine-tuned using the PEFT LoRA library. We initialize rank-16 LoRA weights across the weights of the attention layer only i.e. the query, key, value and output layer. The models were trained using AdamW (Loshchilov & Hutter, 2019) optimizer using a linear learning rate scheduler, with a learning rate of 3e-5 and warm steps set to 6% of the total number of training steps. For datasets like QNLI (Wang et al., 2019), RTE (Wang et al., 2019) and SCITAIL (Khot et al., 2018) which only employ two of the three classes, we simply mask the missing label when finetuning and evaluating on these datasets.
+
+During merging we only merge the Llama back-bone with the LoRA layers and use task-specific head when evaluating across each dataset. Even for the merged model we use the merged backbone and task-specific head during evaluation.
+
+Table A1: Eight models per-task results including Fisher weight averaging. We merge eight ViT-B/32 models finetuned with LoRA on different image classification datasets. "Finetuned" refers to the accuracy of each finetuned model on the dataset it was trained on. We report the per-task (including the average) normalized-accuracies across other merging baselines. These describe how close they get to the "Finetuned" accuracy.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="9">Datasets</td></tr><tr><td>Cars</td><td>DTD</td><td>EuroSAT</td><td>GTSRB</td><td>MNIST</td><td>RESISC45</td><td>SUN397</td><td>SVHN</td><td>Avg</td></tr><tr><td colspan="10">Per-Task Absolute Accuracies (%)</td></tr><tr><td>Finetuned</td><td>74.0</td><td>58.3</td><td>99.0</td><td>92.7</td><td>99.3</td><td>88.4</td><td>64.5</td><td>96.2</td><td>84.1</td></tr><tr><td colspan="10">Per-Task Accuracies of Merged Models Normalized Against Finetuned Models (%)</td></tr><tr><td>RegMean</td><td>80.2</td><td>71.3</td><td>37.9</td><td>47.3</td><td>43.1</td><td>70.5</td><td>93.9</td><td>43.0</td><td>60.9</td></tr><tr><td>Fisher</td><td>84.5</td><td>72.4</td><td>44.4</td><td>55.8</td><td>47.8</td><td>70.9</td><td>96.1</td><td>39.2</td><td>63.9</td></tr><tr><td>TA</td><td>82.0</td><td>73.6</td><td>48.8</td><td>42.1</td><td>53.1</td><td>71.5</td><td>97.5</td><td>41.2</td><td>63.7</td></tr><tr><td>TIES</td><td>82.2</td><td>72.8</td><td>50.0</td><td>36.8</td><td>56.8</td><td>69.4</td><td>96.9</td><td>44.6</td><td>63.7</td></tr><tr><td>DARE-TIES</td><td>81.4</td><td>74.5</td><td>50.8</td><td>39.2</td><td>55.0</td><td>70.7</td><td>97.6</td><td>40.1</td><td>63.7</td></tr><tr><td>KnOTS-TIES</td><td>82.7</td><td>73.7</td><td>49.3</td><td>48.9</td><td>68.9</td><td>70.9</td><td>95.5</td><td>53.8</td><td>68.0</td></tr><tr><td>KnOTS-DARE-TIES</td><td>81.8</td><td>75.9</td><td>50.7</td><td>40.3</td><td>53.2</td><td>70.2</td><td>97.9</td><td>41.0</td><td>63.9</td></tr></table>
+
+Compute resources. All of our experiments were conducted on machines with one Nvidia A40 with 48GB of VRAM, and a CPU that has 8 workers. We trained all our models across these machines, and also applied every merging algorithm in this environment. KnOTS is capable of running entirely on the CPU. We compute the SVD using the Pytorch (Paszke et al., 2019) "torch.linalg.svd" solver. However, we note that more efficient SVD algorithms can easily be employed with our approach, such as the recent Fast SVD (Xu et al., 2023) which is designed for low-rank matrices.
+
+Datasets and licenses. This paper uses the following datasets and associated licenses. Cars (Krause et al., 2013) and GTSRB (Stallkamp et al., 2011) both use the Creative Commons License. EuroSAT (Helber et al., 2019) is under the MIT license and MNIST (LeCun, 1998) is under the GNU General Public License. We could not find the licenses of DTD (Cimpoi et al., 2014), RESISC45 (Cheng et al., 2017), SVHN (Netzer et al., 2011) and SUN397 (Xiao et al., 2016).
+
+# E COMPARISON WITH A FINETUNING BENCHMARK
+
+We also compare KnOTS against the popular finetuning benchmark Fisher Weight Averaging (Matena & Raffel, 2022) in the same setting as Tab. 1. We classify this method as finetuning because it adds learnable parameters to every model that are optimized via gradient descent to obtain the best merged model. We select the best reported Fisher model according to the hyperparameter configuration which achieved the best performance on the same held-out validation data as all methods in this setting. Our hyperparameter search consisted of two ranges. First, the number of examples used to compute the Fisher information matrices: [256, 512, 1024, 2048] (selected from the training data of each dataset). Second, the scaling term to merge model parameters: [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9]. The best performing configuration on our held-out validation dataset used a scaling coefficient of 1.0 and required 256 examples to compute the Fisher weights. Tab. A1 summarizes the results. Overall, we observe that while Fisher matches TIES, it is substantially outperformed by KnOTS—a method that doesn't require any training.
+
+# F JOINT TASK FULL PERFORMANCES
+
+Tab. A2 shows performances for each merging method across all datasets on the Joint-task. Each dataset column shows results on images only from the dataset, but with using the joint labels-set of 748 labels.
+
+Finding synonymous labels. After removing duplicates from the joint label space, we are left with 748 total labels. However, a manual search over these labels reveals that some may be synonyms/hyponyms of each other. To get a better idea of how many there are, we encode each label using a
+
+Table A2: Eight models joint-task results. We merge eight ViT-B/32 models finetuned with LoRA on different datasets. We report the joint-task performances for several merging methods. KnOTS-TIES improves over baselines by significant margins across across nearly every evaluation, often only trading best performances with the "Ensemble."  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Metric</td><td colspan="9">Joint-Task Performances (%)</td></tr><tr><td>Cars</td><td>DTD</td><td>EuroSAT</td><td>GTSRB</td><td>MNIST</td><td>RESISC45</td><td>SUN397</td><td>SVHN</td><td>Union</td></tr><tr><td rowspan="3">Ensemble</td><td>Hits@1</td><td>58.5</td><td>41.3</td><td>16.9</td><td>29.5</td><td>35.8</td><td>54.2</td><td>62.4</td><td>25.1</td><td>40.7</td></tr><tr><td>Hits@3</td><td>83.6</td><td>61.4</td><td>31.4</td><td>59.5</td><td>58.0</td><td>78.2</td><td>84.0</td><td>44.3</td><td>63.1</td></tr><tr><td>Hits@5</td><td>91.0</td><td>72.4</td><td>40.0</td><td>73.0</td><td>69.1</td><td>85.9</td><td>89.5</td><td>55.2</td><td>72.6</td></tr><tr><td rowspan="3">TA</td><td>Hits@1</td><td>60.7</td><td>40.7</td><td>15.3</td><td>38.8</td><td>31.8</td><td>59.7</td><td>61.9</td><td>29.2</td><td>43.5</td></tr><tr><td>Hits@3</td><td>84.9</td><td>63.7</td><td>23.0</td><td>66.1</td><td>48.4</td><td>83.6</td><td>83.9</td><td>50.1</td><td>65.2</td></tr><tr><td>Hits@5</td><td>92.0</td><td>74.0</td><td>31.0</td><td>77.9</td><td>55.7</td><td>90.2</td><td>89.9</td><td>61.6</td><td>74.0</td></tr><tr><td rowspan="3">TIES</td><td>Hits@1</td><td>60.4</td><td>39.7</td><td>13.0</td><td>35.0</td><td>33.4</td><td>58.6</td><td>61.3</td><td>32.9</td><td>43.6</td></tr><tr><td>Hits@3</td><td>84.9</td><td>61.9</td><td>21.9</td><td>63.3</td><td>48.2</td><td>82.8</td><td>83.7</td><td>53.6</td><td>65.3</td></tr><tr><td>Hits@5</td><td>92.1</td><td>72.4</td><td>29.0</td><td>75.3</td><td>54.0</td><td>89.4</td><td>89.9</td><td>64.1</td><td>73.9</td></tr><tr><td rowspan="3">DARE-TIES</td><td>Hits@1</td><td>60.8</td><td>39.3</td><td>12.8</td><td>33.7</td><td>34.3</td><td>57.5</td><td>60.4</td><td>35.5</td><td>44.0</td></tr><tr><td>Hits@3</td><td>85.3</td><td>61.4</td><td>18.1</td><td>63.6</td><td>50.2</td><td>82.3</td><td>82.6</td><td>57.8</td><td>66.4</td></tr><tr><td>Hits@5</td><td>92.6</td><td>73.0</td><td>20.9</td><td>74.6</td><td>55.7</td><td>89.0</td><td>89.1</td><td>69.5</td><td>75.1</td></tr><tr><td rowspan="3">KnOTS-TIES</td><td>Hits@1</td><td>61.7</td><td>40.5</td><td>16.2</td><td>44.2</td><td>39.1</td><td>59.0</td><td>60.6</td><td>36.7</td><td>46.8</td></tr><tr><td>Hits@3</td><td>85.8</td><td>63.8</td><td>22.3</td><td>69.0</td><td>52.6</td><td>83.9</td><td>82.8</td><td>58.4</td><td>68.1</td></tr><tr><td>Hits@5</td><td>92.6</td><td>74.5</td><td>31.1</td><td>79.8</td><td>58.4</td><td>90.4</td><td>89.1</td><td>68.5</td><td>76.3</td></tr><tr><td rowspan="3">KnOTS-DARE-TIES</td><td>Hits@1</td><td>60.4</td><td>40.3</td><td>15.9</td><td>41.9</td><td>34.6</td><td>58.4</td><td>60.4</td><td>34.8</td><td>45.2</td></tr><tr><td>Hits@3</td><td>85.0</td><td>63.5</td><td>21.1</td><td>68.4</td><td>50.0</td><td>83.8</td><td>82.4</td><td>56.5</td><td>66.9</td></tr><tr><td>Hits@5</td><td>92.2</td><td>74.5</td><td>26.6</td><td>78.9</td><td>55.9</td><td>90.4</td><td>88.9</td><td>67.4</td><td>75.3</td></tr></table>
+
+pretrained "distilbert-base-nli-mean-tokens" model taken from the Sentence Transformers library (Reimers & Gurevych, 2019), and compute the pairwise cosine-similarities between each label. We then take all label pairs with cosine-similarity  $\geq 0.8$  to be "synonyms," and reproduce them at the bottom of this section in dictionary form where keys and values are tuples containing "(label, dataset origin)." In total, we find 111 synonyms, leading to an average of 0.15 synonyms per-label and 1.95 synonyms per-labels that have synonyms. Thus, measuring performance on the joint-task using "Hits@k" with  $k = \{1,3,5\}$  appears to be a suitable choice.
+
+Below we print all synonyms found by our automatic search.
+
+```javascript
+found synonyms  $=$  { (lake natural', 'sun397'): [(lake', 'resisc45')], ('forest', 'resisc45'): [ ('tree house', 'sun397'), ('rainforest', 'sun397'), ('forest broadleaf', 'sun397'), ('forest road', 'sun397'), ('forest needleleaf', 'sun397'), ('forest path', 'sun397') ], (athletic field outdoor', 'sun397'): [(ground track field', 'resisc45')], ('lake or sea', 'eurosat'): [(lake', 'resisc45')], ('rainforest', 'sun397'): [(forest', 'resisc45')], ('iceberg', 'sun397'): [(snowberg', 'resisc45')], ('meadow', 'resisc45'): [ ('field wild', 'sun397'), ('park', 'sun397'), ('field cultivated', 'sun397'), ('pasture land', 'eurosat'), ('yard', 'sun397'), ('pasture', 'sun397') ], (pond', 'sun397'): [(lake', 'resisc45')], ('ground track field', 'resisc45'): [ ('athletic field outdoor', 'sun397'), ('track outdoor', 'sun397') ], (desert vegetation', 'sun397'): [(desert', 'resisc45')], ('marsh', 'sun397'): [(wetland, 'resisc45')], ('freeway', 'resisc45'): [(highway', 'sun397')], ('islet', 'sun397'): [(island, 'resisc45')], ('permanent crop land', 'eurosat'): [ ('rectangular farmland', 'resisc45'), ('field cultivated', 'sun397') ], (track outdoor', 'sun397'): [(ground track field', 'resisc45')], ('swamp', 'sun397'): [(wetland, 'resisc45')],
+```
+
+```javascript
+('sea ice', 'resisc45'): [(ice shelf', 'sun397'), ('ice floe', 'sun397')], ('desert', 'resisc45'): [(desert vegetation', 'sun397'), ('desert sand', 'sun397')], ('snowberg', 'resisc45'): [(iceberg', 'sun397'), ('mountain snowy', 'sun397'), ('snowfield', 'sun397')], ('railway station', 'resisc45'): [ ('train railway', 'sun397'), ('railroad track', 'sun397'), ('train station platform', 'sun397') ], ('street', 'sun397'): [(commercial area', 'resisc45'), ('intersection', 'resisc45') ], ('rectangular farmland', 'resisc45'): [(permanent crop land', 'eurosat}], ('field wild', 'sun397'): [(meadow', 'resisc45')], ('thermal power station', 'resisc45'): [(electrical substation', 'sun397')], ('lake', 'resisc45'): [(lake natural', 'sun397'), ('lake or sea', 'eurosat'), ('pond', 'sun397') ], ('runway', 'sun397'): [(airplane', 'resisc45'), ('airport', 'resisc45')], ('baseball field', 'sun397'): [(baseball diamond', 'resisc45')], ('industrial area', 'sun397'): [(industrial buildings or commercial buildings', 'eurosat') ], ('ice shelf', 'sun397'): [(sea ice', 'resisc45')], ('airplane', 'resisc45'): [(runway', 'sun397'), ('airplane cabin', 'sun397')], ('park', 'sun397'): [(forest', 'resisc45'), (meadow', 'resisc45')], ('thermal power station', 'resisc45'): [(electrical substation', 'sun397')], ('electrical substation', 'sun397'): [(thermal power station', 'resisc45')], ('field cultivated', 'sun397'): [(meadow', 'resisc45'), ('permanent crop land', 'eurosat'), ('circular farmland', 'resisc45'), ('pasture land', 'eurosat') ], ('patio', 'sun397'): [(terrace', 'resisc45')], ('shopfront', 'sun397'): [(commercial area', 'resisc45')], ('highway', 'sun397'): [(freeway', 'resisc45'), (highway or road, 'eurosat')], ('circular farmland', 'resisc45'): [(field cultivated', sun397'), (pasture land, 'eurosat'), ('pasture', sun397') ], ('residential buildings or homes or apartments', 'eurosat'): [(residential neighborhood', sun397) ], ('pasture land', 'eurosat'): [(meadow', resisc45'), (field cultivated', sun397'), (circular farmland, resisc45'), (yard, sun397'), (pasture, sun397) ], ('desert sand', sun397): [(desert, resisc45)], ('train railway', sun397): [(railway, resisc45)], ('wetland', resisc45): [(marsh, sun397), (swamp, sun397)], ('commercial area', resisc45): [(street, sun397), (shopfront, sun397), (industrial buildings or commercial buildings, 'eurosat') ], ('industrial buildings or commercial buildings', 'eurosat'): [(industrial area, sun397), (commercial area, 'resisc45') ],
+```
+
+```txt
+('airport', 'resisc45'): [(airport terminal', 'sun397'), ('runway', 'sun397')],  
+('tennis court outdoor', 'sun397'): [(tennis court', 'resisc45')],  
+('baseball diamond', 'resisc45'): [  
+    ('baseball field', 'sun397'), ('stadium baseball', 'sun397'),  
+    ('batters box', 'sun397')  
+],  
+('railway', 'resisc45'): [  
+    ('train railway', 'sun397'), ('railroad track', 'sun397'),  
+    ('train station platform', 'sun397')  
+],  
+('highway or road', 'eurosat'): [(highway, 'sun397')],  
+('yard', 'sun397'): [(meadow, 'resisc45'), ('pasture land', 'eurosat'),  
+('railroad track', 'sun397'): [  
+    ('railway station', 'resisc45'), ('railway', 'resisc45')  
+],  
+('tennis court', 'resisc45'): [  
+    ('tennis court outdoor', 'sun397'), ('tennis court indoor', 'sun397')  
+],  
+('residential neighborhood', 'sun397'): [  
+    ('residential buildings or homes or apartments', 'eurosat')  
+],  
+('island', 'resisc45'): [(islet, 'sun397')],  
+('train station platform', 'sun397'): [(railway station, 'resisc45'),  
+('pasture', 'sun397'): [  
+    ('meadow', 'resisc45'), ('circular farmland', 'resisc45'),  
+    ('pasture land', 'eurosat')  
+],  
+('terrace', 'resisc45'): [  
+    ('courtyard', 'sun397'), ('promenade deck', 'sun397'),  
+    ('pavilion', 'sun397'), ('patio', 'sun397'),  
+    ('carrousel', 'sun397'), ('balcony exterior', 'sun397'),  
+    ('veranda', 'sun397')  
+],
+```

2025/Model merging with SVD to tie the Knots/images.zip

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2025/Model-Agnostic Knowledge Guided Correction for Improved Neural Surrogate Rollout/ad4c2f01-4f37-441e-a45e-bb12f3d79389_origin.pdf

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2025/Model-Agnostic Knowledge Guided Correction for Improved Neural Surrogate Rollout/full.md

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+# MODEL-AGNOSTIC KNOWLEDGE GUIDED CORRECTION FOR IMPROVED NEURAL SURROGATE ROLLOUT
+
+Bharat Srikishan $^{1*}$ , Daniel O'Malley $^{2}$ , Mohamed Mehana $^{2}$ , Nicholas Lubbers $^{2}$ , Nikhil Muralidhar $^{1}$
+
+<sup>1</sup>Tevens Institute of Technology, <sup>2</sup>Los Alamos National Laboratory {bsrikish, nmurali1}@stevens.edu, {omalled, mzm, nlabbers}@lanl.gov
+
+# ABSTRACT
+
+Modeling the evolution of physical systems is critical to many applications in science and engineering. As the evolution of these systems is governed by partial differential equations (PDEs), there are a number of computational simulations which resolve these systems with high accuracy. However, as these simulations incur high computational costs, they are infeasible to be employed for large-scale analysis. A popular alternative to simulators are neural network surrogates which are trained in a data-driven manner and are much more computationally efficient. However, these surrogate models suffer from high rollout error when used autoregressively, especially when confronted with training data paucity. Existing work proposes to improve surrogate rollout error by either including physical loss terms directly in the optimization of the model or incorporating computational simulators as 'differentiable layers' in the neural network. Both of these approaches have their challenges, with physical loss functions suffering from slow convergence for stiff PDEs and simulator layers requiring gradients which are not always available, especially in legacy simulators. We propose the Hybrid PDE Predictor with Reinforcement Learning (HyPER) model: a model-agnostic, RL based, cost-aware model which combines a neural surrogate, RL decision model, and a physics simulator (with or without gradients) to reduce surrogate rollout error significantly. In addition to reducing in-distribution rollout error by  $47\% - 78\%$ , HyPER learns an intelligent policy that is adaptable to changing physical conditions and resistant to noise corruption. Code available at https://github.com/scailab/HyPER.
+
+# 1 INTRODUCTION
+
+Scientific simulations have been the workhorse enabling novel discoveries across many scientific disciplines. However, executing fine-grained simulations of a scientific process of interest is a costly undertaking requiring large computational resources and long execution times. In the past decade, the advent of low-cost, efficient GPU architectures has enabled the re-emergence of a powerful function approximation paradigm called deep learning (DL). These powerful DL models, with the ability to represent highly non-linear functions can be leveraged as surrogates to costly scientific simulations. Recently, the rapid progress of DL has greatly impacted scientific machine learning (SciML) with the development of neural surrogates in numerous application domains. Some highly-impactful applications include protein structure prediction, molecular discovery Schauerl & Denny (2022); Smith et al. (2018) and domains governed by partial differential equations (PDE) Brunton & Kutz (2024); Raissi et al. (2019); Lu et al. (2021b). Neural surrogates have also been successfully employed for modeling fluid dynamics in laminar regimes like modeling blood flow in cardiovascular systems Kissas et al. (2020) and for modeling turbulent Duraisamy et al. (2019) and multi-phase flows Muralidhar et al. (2021); Raj et al. (2023); Siddani et al. (2021).
+
+Neural Surrogates are Data Hungry. Although neural surrogates are effective at modeling complex functions, this ability is usually conditioned upon learning from a large trove of representative data. The data-hungry nature of popular neural surrogates like neural operators is well known in
+
+existing work (Li et al., 2020; Lu et al., 2021a; Tripura et al., 2024; Howard et al., 2023; Lu et al., 2022). However, many scientific applications suffer from data paucity due to the high cost of the data collection process (i.e., primarily due to high cost of scientific simulations). Hence, neural surrogates employed to model a scientific process of interest, need to address the data paucity bottleneck by learning effectively with a low volume of training data.
+
+Rollout Errors in Neural Surrogates. Although computational simulations have been designed for modeling various types of physical systems, those exhibiting transient dynamics are especially challenging to model. Solutions to systems exhibiting transient dynamics are usually
+
+obtained by discrete-time evolution of the dynamics. Simulators used to model such systems are invoked autoregressively and thereby encounter numerical instability and error buildup over long estimation horizons. Such error buildup during autoregressive invocation is termed rollout error. Effective techniques have been developed to reduce rollout error of computational simulations and increase their numerical stability over long rollouts. Although autoregressive rollout of neural surrogates is also affected by rollout error, solutions to minimize this error buildup have not been widely investigated. Recently, (Margazoglou & Magri, 2023) has inspected the stability of echo-state networks during autoregressive rollout and List et al. (2024); Carey et al. (2024); Lippe et al. (2024) have characterized rollout errors in more general neural surrogates. However, a systematic solution to alleviate rollout error in neural surrogates for modeling transient dynamics is still lacking.
+
+![](images/aea3df3c903fc690d99d88ad281fef512838851e666ef337d910d29eaf0e6ea8.jpg)  
+Figure 1: Cumulative MSE depicting rollout error for a single trajectory of HyPER vs surrogate only methods. x's mark the timesteps during the trajectory where our RL policy calls the simulator.
+
+Knowledge-Guided Neural Surrogates. One popular method of addressing errors due to data paucity in neural surrogates is to leverage knowledge of the theoretical model governing the scientific process. Previous efforts have incorporated domain knowledge (as ODEs, PDEs) while training the DL surrogate to develop knowledge-guided learning pipelines (Raissi et al., 2019; Karpatne et al., 2022; Rackauckas et al., 2020; Gao et al., 2021). Most of these approaches incorporate the PDE governing the system dynamics as soft regularizers while training the neural surrogates. A majority of such approaches exhibit slow convergence and catastrophic failures in challenging, stiff PDE conditions (Krishnapriyan et al., 2021; Wang et al., 2022).
+
+Hybrid-Modeling All approaches discussed thus far are so-called surrogate-only (SUG) approaches. Here, the computational simulator is employed only as a means of generating data to train the neural surrogate and discarded post the training. SUG approaches employ only the pre-trained neural surrogate during inference. Although SUG provide instantaneous responses relative to computational simulations, they generally have limited generalization ability outside the domain of the training data. An effective complement to SUG approaches are hybrid-modeling approaches (Kurz et al., 2022; Karpatne et al., 2022), that jointly resolve a query by incorporating surrogates in conjunction with computational solvers. Otherwise stated, hybrid-modeling pipelines employ a 'solver-in-the-loop' (Um et al., 2020) approach. In addition to neural surrogates, there exist a number of classical hybrid modeling techniques which combine a full order model (FOM) with a reduced order model (ROM) such as proper orthogonal decomposition (Willcox & Peraire, 2002), dynamic model decomposition (Kutz et al., 2016), and multi-scale methods. While these methods are used to accelerate scientific simulations, they are often limited in expressivity, especially in modeling complex non-linear dynamics. Recent hybrid models (Suh et al., 2023) generally have a static coupling between the components in the model and require a static interaction/transition between the FOM and the ROM, while our method proposes an adaptable and learnable interaction between the neural surrogate and the simulator. Our proposed method allows dynamic integration of an FOM (fine-grained simulator) with ROM (neural surrogate) using reinforcement learning.
+
+Knowledge-Guidance with Hybrid-Modeling. Hybrid-modeling approaches are inherently knowledge-guided. A majority of the recent hybrid-modeling approaches are based on directly incorporating PDE solvers as additional layers in the deep learning architecture of neural surro
+
+![](images/66b69cf106f2cabbfb972c8b7b7dd811933904be1d5bef626db9aa0838f4901b.jpg)  
+Figure 2: Overview of Hybrid PDE Predictor with RL (HyPER) with example rollout. Here  $\pi_{\theta}$  is the decision model,  $f_{\phi}$  is the surrogate, and  $S$  is the simulator. At  $t = 9.0$  in the above trajectory, the decision policy invokes the simulator, correcting the trajectory to reduce rollout error. The effect of this knowledge-guided correction can be observed by a reduction in absolute error (dotted green circle) in the figure.
+
+gates (Chen et al., 2018; Belbute-Peres et al., 2020; Donti et al., 2021; Pachalieva et al., 2022). While such approaches address the issues of large errors under data paucity and during rollout, inherent in SUG approaches, they impose the crippling restriction of differentiability on the computational solvers to be incorporated as part of the DL pipeline. Most solvers and computational simulations are NOT differentiable out-of-the-box and hence imposing such differentiability constraints drastically curtails the applicability of current hybrid-modeling approaches.
+
+To address the existing challenges with surrogate-only and hybrid modeling approaches, we propose the Hybrid PDE Predictor with RL (HyPER) framework. HyPER is a model-agnostic, simulator-agnostic framework that learns to invoke the costly computational simulator (in a cost-aware manner) as knowledge-guided correction to alleviate the effects of rollout errors in surrogates trained with low volumes of training data. Fig. 2 depicts the proposed HyPER framework. Our contributions are as follows.
+
+- HyPER is a first of its kind knowledge-guided correction mechanism that incorporates simulators in the loop without the requirement of the simulators to be differentiable.  
+- HyPER is model agnostic (i.e., functions with any neural surrogate, scientific simulator) and trained in a cost-aware manner, to intelligently invoke the simulator to correct the rollout error of the neural surrogates.  
+- We demonstrate through rigorous experimentation on in-distribution, out-of-distribution and noisy data that HyPER significantly reduces rollout error relative to SUG approaches by comparing with state-of-the-art neural surrogates.
+
+# 2 METHOD
+
+# 2.1 PDE PREDICTION
+
+We aim to solve PDEs involving spatial dimensions  $\pmb{x} = [x_{1}, x_{2}, \dots, x_{m}] \in \mathbb{R}^{m}$  and scalar time  $t \in [0, T]$ . These PDEs relate solution function  $\pmb{u}(\pmb{x}, t): \mathbb{R}^{m} \times [0, T] \to \mathbb{R}^{n}$  to its partial derivatives over the domain. We assume we have initial conditions  $\pmb{u}(\pmb{x}, t = 0)$  and boundary conditions  $\pmb{u}_{B}(\pmb{x} = \pmb{x}_{B}, t)$  which define the field values at time 0 and at the boundaries of the domain respectively. These time-dependent PDEs can be generally defined as:
+
+$$
+\frac {\partial \boldsymbol {u}}{\partial t} = \mathcal {F} (\boldsymbol {x}, t, \boldsymbol {u}, \frac {\partial \boldsymbol {u}}{\partial \boldsymbol {x}}, \frac {\partial^ {2} \boldsymbol {u}}{\partial \boldsymbol {x} ^ {2}}, \dots) \tag {1}
+$$
+
+We focus on autoregressive rollout of these PDEs over time, which can be defined with an function  $g$  taking current state and time as inputs and producing the next state:
+
+$$
+g (\boldsymbol {u} (\boldsymbol {x}, t), \Delta t) = \boldsymbol {u} (\boldsymbol {x}, t + \Delta t) \tag {2}
+$$
+
+Note that the equation above can be applied autoregressively over any number of timesteps to unroll a PDE trajectory. The cumulative error of this autoregressive process is defined as the rollout error, which we aim to minimize.
+
+# 2.2 HYPER COMPONENTS
+
+Surrogate ML Model For the sake of conciseness, we shorten  $\pmb{u}(\pmb{x},t)$  to  $\pmb{u}_t$ . We begin with a machine learning model  $f_{\phi}(\pmb{u}_t)$ , which we denote as the surrogate. This model can be any deep learning model with parameters  $\phi$  that predicts next state  $\pmb{u}_{t + 1}$  given starting state and time  $\pmb{u}_t$ .
+
+**Simulator** We also define a PDE simulator  $S(\boldsymbol{u}_t)$  which numerically solves the PDE to find the next state  $\boldsymbol{u}_{t+1}$ . Crucially, this simulator is only required to return the next state without any gradient information during training and inference.
+
+Decision Model Finally, we have a decision model  $d_{\theta}(\pmb{u}_t)$  which takes current state at time  $t$  and outputs a next action: either call the surrogate  $f_{\phi}$  or the simulator  $S$ . In HyPER we implement our decision model as a learned policy  $\pi_{\theta}$  which we train using reinforcement learning (RL).
+
+We formalize our decision model using a Markov Decision Process (MDP) which is a tuple  $(\mathbb{S},\mathbb{A},P,r)$ . Our states  $\mathbb{S}$  consist of current state  $\pmb{u}$  and current time  $t$ . Our action space is binary at each timestep and defined as  $a:\{0 = \text{call surrogate}, 1 = \text{call simulator}\}$ . Our reward function for a trajectory of length  $T$  is:
+
+$$
+R (\boldsymbol {a}) = \sum_ {t = 0} ^ {T - 1} - \mathcal {L} \left(f _ {\phi}, S, \boldsymbol {u} _ {t}, \boldsymbol {u} _ {t + 1}, a _ {t}\right) + b \left(f _ {\phi}, S, \boldsymbol {u} _ {t}\right) - \mathcal {C} (\boldsymbol {a}, \lambda , T) \tag {3}
+$$
+
+Here  $\mathcal{L}$  represents an error function,  $b$  is a baseline function, and  $\mathcal{C}$  is a cost function. The baseline function stabilizes the reward values and leads to better policy learning. The baseline function  $b$  we use is the mean squared error of randomly calling the simulator the same number of times as our current HyPER policy. Our error function is defined as the mean squared error of either calling the surrogate or simulator according to the policy action:
+
+$$
+\mathcal {L} \left(f _ {\phi}, S, \boldsymbol {u} _ {t}, \boldsymbol {u} _ {t + 1}, a _ {t}\right) = \left(\mathcal {G} \left(\boldsymbol {u} _ {t}\right) - \boldsymbol {u} _ {t + 1}\right) ^ {2} \tag {4}
+$$
+
+$$
+\mathcal {G} \left(\boldsymbol {u} _ {t}, a _ {t}\right) = \left\{ \begin{array}{l l} f _ {\phi} (\boldsymbol {u} _ {t}) & \text {w h e n} a _ {t} = 0 \\ S (\boldsymbol {u} _ {t}) & \text {w h e n} a _ {t} = 1 \end{array} \right. \tag {5}
+$$
+
+$T$  is the length of the trajectory and  $\lambda$  is a hyperparameter set by the user to specify what percentage of the trajectory to call the simulator. For example, if  $\lambda = 0.5$  then the cost function will penalize the reward if the simulator is not called  $50\%$  of the time. Our cost function is defined as the absolute difference between  $\lambda$  and the percent of the trajectory that our policy called the simulator:
+
+$$
+\mathcal {C} (\boldsymbol {a}, \lambda , T) = \left| \frac {\| \boldsymbol {a} \| _ {1}}{T} - \lambda \right| \tag {6}
+$$
+
+Intuitively, reward function 3 optimizes the RL decision model to minimize mean squared error while calling the simulator for  $\lambda$  proportion of the trajectory. To learn a policy  $\pi_{\theta}(a|\pmb{u}_t)$  we use the REINFORCE policy gradient algorithm (Sutton et al., 1999) training with the update:
+
+$$
+\nabla_ {\theta} J (\theta) = \mathbb {E} _ {\pi_ {\theta}} \left[ \sum_ {t = 0} ^ {T - 1} \nabla_ {\theta} \log \pi_ {\theta} \left(a _ {t} \mid \boldsymbol {u} _ {t}\right) \cdot R (\boldsymbol {a}) \right] \tag {7}
+$$
+
+For the full training algorithm and details of HyPER see Appendix A.1.
+
+# 2.3 EXPERIMENTS
+
+2D Navier Stokes Dataset. We create a 2D incompressible Navier Stokes fluid flow dataset using  $\Phi$  Flow (Holl & Thuerey, 2024) and follow the example of Gupta & Brandstetter (2022). We use the Navier-Stokes equations in vector velocity form along with an additional scalar field representing particle concentration:
+
+$$
+\frac {\partial \boldsymbol {v}}{\partial t} = - \boldsymbol {v} \cdot \nabla \boldsymbol {v} + \mu \nabla^ {2} \boldsymbol {v} - \nabla p + f \tag {8}
+$$
+
+$$
+f = \{0, 0. 5 c \} \tag {9}
+$$
+
+$$
+\nabla \cdot \boldsymbol {v} = 0 \tag {10}
+$$
+
+$$
+\frac {\partial c}{\partial t} = - \boldsymbol {v} \cdot \nabla c \tag {11}
+$$
+
+Eq. 8 comprises a convection term  $-\boldsymbol{v} \cdot \nabla \boldsymbol{v}$ , diffusion term  $\mu \nabla^2 \boldsymbol{v}$  where  $\mu$  indicates kinematic viscosity, a pressure term  $\nabla p$ , and external force term  $f$ . Eq. 9 defines the buoyant force which is applied in the y-direction. The velocity divergence term Eq. 10 enforces conservation of mass. Eq. 11 models the particle concentration field  $c$  that is advected by the velocity vector field  $\boldsymbol{v}$ . Note that  $c$  influences  $\boldsymbol{v}$  through force  $f$  and  $\boldsymbol{v}$  affects  $c$  through Eq. 11, creating complex dynamics between fields. For our experiments, we generate 1,000 trajectories of 20 timesteps with  $\Delta t = 1.5s$  at a grid size of 64x64, and diffusion coefficient  $\mu = 0.01$ . We set our velocity boundary condition to  $\boldsymbol{v} = 0$  (Dirichlet) and our concentration boundary condition to  $\partial c / \partial x = 0$  (Neumann). While we simulate all the fields above, our experiments focus on predicting the particle concentration field  $c$ . We split our 1000 trajectories into 3 sets: 400 for surrogate training, 400 for RL training, and 200 for testing. Our RL model selects 4 actions at a time, however HyPER puts no restrictions on action selection.
+
+Subsurface Flow Dataset. To evaluate HyPER's ability to work with different PDEs and problem scales, we generate a subsurface flow dataset using the Julia-based DPFEHM (Pachalieva et al., 2022) simulator. We use DPFEHM to generate a 2D dataset of subsurface fluid flows modeled by the Richards equation:
+
+$$
+\frac {\partial \theta}{\partial t} = \nabla \cdot \mathbf {K} (h) (\nabla h + \nabla z) - T ^ {- 1} \tag {12}
+$$
+
+Eq. 12 models fluid flow underground in an unsaturated medium.  $\theta$  represents the volumetric fluid content,  $\mathbf{K}(h)$  is the unsaturated hydraulic conductivity,  $h$  is the pressure,  $\nabla z$  is the geodetic head gradient, and  $T^{-1}$  is the fluid sink term. We generate 500 trajectories of 100 timesteps each with a grid size of 50x50 and timestep size of 10 seconds. Out of the 500 trajectories, 200 are used for surrogate training, 200 are used for RL training, and 100 are used for testing. We use Dirchlet boundary conditions (0) for volumetric flux on all sides of our domain but inject fluid at the top center of the domain at a rate of  $0.01m / s$ . In our experiments, we predict the pressure field  $h$ .
+
+# 3 RESULTS AND DISCUSSION
+
+In this section, we investigate the effectiveness of HyPER to model transient PDE systems efficiently and with minimal rollout error compared to surrogate-only (SUG) rollout. We compare with state-of-the-art neural surrogates including UNet and Fourier Neural Operator (FNO), MPP, and PDE-Refiner. Specifically, we investigate the effectiveness of HyPER rollouts under changing PDE dynamics and in noisy data settings. Further, we also demonstrate the surrogate-agnostic and simulator-agnostic nature of HyPER. Our experiments seek to answer the following research questions:
+
+- RQ1: How effective are HyPER rollouts compared to SUG rollouts for transient PDE systems?  
+- RQ2: Are HyPER rollouts effective under changed physical conditions?  
+- RQ3: Are HyPER rollouts effective under noisy data conditions?  
+- RQ4: How crucial is the intelligent decision model for effective HyPER rollouts?  
+- RQ5: What is the error/efficiency trade-off between SUG, HyPER, and simulator-only paradigms?
+
+# 3.1 RQ1: HYPER ROLLOUTS VS SUG
+
+We begin by comparing HyPER to six SUG baselines: UNet, FNO, UNet-Multistep, MPP-ZS, MPP, and PDE-Refiner. All baselines except UNet-Multistep are trained in a one-step-ahead manner i.e., given the current state of the resolved field they are trained to predict the next state. Mean squared error (MSE) loss is used to train all surrogates over the full resolution trajectory. Note that all these baselines except MPP-ZS are trained with the same dataset that HyPER is trained with. For full training details see Appendix A.1 and for baseline details see Appendix A.6.
+
+The UNet model is based on the modern UNet architecture in PDEArena (Gupta & Brandstetter, 2022) while the FNO model is built with the neuraloperator library (Kovachki et al., 2021; Li et al.,
+
+Table 1: 2D Navier-Stokes results with best results in bold. Notice that HyPER rollout incurs significantly lower rollout (cumulative) error compared to SUG models.  
+
+<table><tr><td>2D Navier-Stokes</td><td>UNet</td><td>FNO</td><td>UNet-Multistep</td><td>MPP-ZS</td><td>MPP</td><td>PDE-Refiner</td><td>HyPER</td></tr><tr><td>Final MSE</td><td>0.019</td><td>0.053</td><td>0.016</td><td>0.22</td><td>0.036</td><td>0.024</td><td>0.011</td></tr><tr><td>Cumulative MSE</td><td>0.312</td><td>0.882</td><td>0.311</td><td>4.495</td><td>0.747</td><td>0.405</td><td>0.164</td></tr></table>
+
+![](images/fec3ee6b4bd5d640041e5146aa23b82640bafdba3cc8300fba74a54015fb27c3.jpg)  
+Figure 3: Predictions and absolute error snapshots of HyPER rollout for two distinct trajectories.
+
+![](images/a511c6910091cabfedf63a9ce731448a2bbe6f31b6e86a22d1641256914f2067.jpg)
+
+2020). UNet-Multistep is trained to specialize for rollout trajectory prediction, i.e. it minimizes MSE for 20-step autoregressive prediction. MPP-ZS (zero-shot) and MPP (McCabe et al., 2023) are large transformer based multi-task models which have been pretrained on a variety of fluid prediction tasks. MPP-ZS is the off-the-shelf pretrained model, while MPP is finetuned on our data. PDE-Refiner (Lippe et al., 2024) is a modern diffusion based model which both predicts the next timestep while adding a multi-step noisig/denoising process and has been shown to reduce rollout error. HyPER utilizes a UNet model that is pretrained on 400 trajectories while the RL model of HyPER is trained on a separate set of 400 trajectories with  $\lambda = 0.3$ . All SUG baselines are trained using 800 trajectories (same as HyPER RL plus surrogate dataset) for 200 epochs. Table 1 shows the MSE of the final trajectory state (Final MSE) as well as the aggregated MSE (Cumulative MSE) over all trajectory states in a rollout. Both metrics are calculated as an average across 200 test trajectories for both SUG and HyPER rollouts. HyPER outperforms all baselines significantly, with an average improvement in cumulative rollout error of  $68.30\%$ . While UNet-Multistep seems to be the best performing SUG method, because it trains across the full trajectory it learns a smoothed prediction that has poor high frequency detail (suffering from spectral bias) in comparison to UNet (see A.4 for examples). By incorporating the simulator, HyPER yields significantly lower cumulative rollout error while only invoking the simulator during  $\sim 30\%$  of the trajectory for knowledge-guided correction of surrogate rollout.
+
+In Figure 1 we plot the rollout MSE performance of HyPER versus the surrogate only (SUG) methods for a single sample trajectory. Note that HyPER outperforms both SUG models by a large margin while only invoking the simulator six times in the 20-step trajectory. Also see Figures 3(a) and 3(b) which depict qualitative snapshots from two distinct HyPER rollouts.
+
+# 3.2 RQ2: HYPER ROLLOUTS WITH CHANGING PHYSICAL CONDITIONS
+
+Pretrained neural surrogates are often confronted with modeling contexts with similar PDE dynamics but varied initial / boundary conditions. Effective rollouts under changing physical conditions is hence a crucial requirement for the effective use of neural surrogates in computational science. To this end, we inspect rollout errors of SUG and HyPER, under changing boundary conditions. Specifically, to test the adaptability of HyPER, we investigate HyPER rollouts with changing boundary conditions in our Navier-Stokes experiment. We generate a separate Navier-Stokes dataset which follows the same initial conditions of our previous experiment, but comprises a different velocity
+
+boundary condition at the top boundary of the domain. Specifically, the velocity at the top boundary of the domain is increased from 0.0 to  $0.5\mathrm{m / s}$  (imposing an intermittent external forcing effect causing the fluid to escape from the top of the domain) for four intermediate time-steps of the trajectory (timesteps 12-16). Following this, the boundary condition is reverted back to 0.0 for the rest of the trajectory. This results in the fluid escaping out of the top during these time steps changing the PDE dynamics significantly.
+
+Table 2: Results depict rollout error for 2D Navier-Stokes with changing boundary conditions. Notice that HyPER accumulates lower rollout error compared to SUG approaches.  
+
+<table><tr><td>Changing Boundary</td><td>UNet-P</td><td>FNO-P</td><td>UNet</td><td>FNO</td><td>HyPER</td></tr><tr><td>Final MSE</td><td>0.362</td><td>0.412</td><td>0.014</td><td>0.027</td><td>0.027</td></tr><tr><td>Cumulative MSE</td><td>1.84</td><td>2.538</td><td>0.92</td><td>0.969</td><td>0.527</td></tr></table>
+
+In this case, surrogates UNet-P and FNO-P are not re-trained with the changed boundary condition and hence show poor performance accumulating significant rollout error. UNet and FNO are trained on the changing boundary data (the same dataset as HyPER), but still suffer from poor rollout error. We train our HyPER RL policy with a pretrained surrogate (not trained on to the changed boundary setting) and a simulator (fully aware of the changed boundary condition). The RL policy of HyPER is trained with 400 of these changed trajectories to learn the optimal policy of (frugally) invoking the simulator to apply knowledge-guided correction to reduce surrogate rollout errors. As shown in Table 2, HyPER reduces cumulative error in this scenario by  $71.36\%$  and  $79.20\%$  relative to UNet-P and FNO-P rollout errors respectively. Even when compared to surrogates trained on the changing boundary data UNet and FNO, HyPER reduces rollout error by  $42.72\%$  and  $45.61\%$  respectively. In Figure 4a we demonstrate that HyPER has much lower rollout error than the UNet-P and FNO-P models. Figure 4b shows sample qualitative predictions of HyPER and SUG rollouts under the changing boundary scenario of interest, to further reinforce our point. As illustrated, our model prediction is much closer to the ground truth after the boundary condition has changed and fluid has escaped the box owing to the appropriately invoked knowledge-guided correction by the learned RL policy in response to increasing surrogate rollout error. Fig. 4a shows the significantly lower rollout error for HyPER rollout relative to UNet-P and FNO-P models.
+
+![](images/c261ddf599481c639dc2f18defd418bdc16100b0139b9acbc800946a788f91b2.jpg)  
+Figure 4: Predictions and absolute error of HyPER vs UNet and FNO for a single trajectory and timestep. Fig. 4a shows the rollout error accumulation over the trajectory with 'x' marking the times the simulator is called. Fig. 4b shows the resolved system state for a the same trajectory at a single timestep. Fig. 4b(i) shows the ground truth field while Fig. 4b(ii)-(iv) indicate the result of UNet, FNO and HyPER rollouts respectively. Fig. 4b(v)-(vii) depict the corresponding absolute errors. We notice that only HyPER rollouts capture the correct characteristics relative to the ground truth owing to the knowledge-guided correction while SUG models are unable to faithfully resolve the trajectory under changing physical conditions.
+
+# 3.3 RQ3: HYPER ROLLOUTS WITH NOISY DATA
+
+Neural surrogates, in addition to being data hungry and accumulating rollout error, have also been known to exhibit catastrophic failures in challenging PDE conditions (e.g., stiff PDEs) Krishnapriyan et al. (2021). As neural surrogates are a crucial part of HyPER rollouts, it is imperative
+
+Table 3: Results depicting rollout error for the 2D-Navier-Stokes experiment, with random Gaussian noise added to inputs, at timesteps 12-16.  $\sigma^2$  is the scale of the noise. The percentage reduction of cumulative MSE rollout error by HyPER, over the best performing SUG model is in parentheses.  
+
+<table><tr><td rowspan="2">Experiment</td><td colspan="5">Cumulative MSE</td></tr><tr><td>UNet-P</td><td>FNO-P</td><td>UNet</td><td>FNO</td><td>HyPER</td></tr><tr><td>Unimodal σ2= 1.0</td><td>2.931</td><td>2.523</td><td>1.61</td><td>1.713</td><td>1.291 (19.81%)</td></tr><tr><td>Unimodal σ2= 0.75</td><td>1.746</td><td>1.916</td><td>1.013</td><td>1.291</td><td>0.8 (21.03%)</td></tr><tr><td>Unimodal σ2= 0.5</td><td>1.009</td><td>1.487</td><td>0.642</td><td>1.137</td><td>0.49 (23.68%)</td></tr><tr><td>Unimodal σ2= 0.25</td><td>0.612</td><td>1.246</td><td>0.519</td><td>0.862</td><td>0.26 (49.90%)</td></tr></table>
+
+to investigate whether HyPER is capable of adapting to such failures by invoking the knowledge-guided correction (i.e., the simulator) to minimize the propagation (and accumulation) of such local failures over the remaining trajectory rollout. Further, another crucial property to investigate is the robustness of the intelligent decision mechanism in HyPER, to noisy, low-quality surrogate predictions. To jointly investigate both goals, we consider a simple (contrived) context with noisy inputs supplied to the HyPER RL policy. This additive noise, injected at specific steps to corrupt the surrogate output in the trajectory, serves to mimic low-quality surrogate predictions. Hence, experiments with such noisy inputs help characterize the ability of HyPER to adapt to sudden local changes during rollout (like catastrophic surrogate failure) in addition to demonstrating its ability for robust decision-making under noisy data conditions.
+
+To carry out this investigation, we add random Gaussian noise with mean 0 at four separate variance  $(\sigma^2)$  scales at fixed timesteps of our trajectory. The UNet-P and FNO-P surrogate models are never trained with this noisy data and therefore perform poorly when encountering it. The UNet and FNO models are trained with the noisy data and show improvement. We train the RL policy of HyPER with a small set (400) of these noisy trajectories while keeping our surrogate model static (using the UNet-P model). We test two different noise corruption scenarios added to a 20 step trajectory rollout. (i) unimodal noise: a case where noise is added at timesteps [12-16] and (ii) bimodal noise: a more sophisticated case where noise is added at two different time windows of [2-4] and [15-16]. The unimodal noise results are presented in Table 3, where we see that HyPER rollouts outperform state-of-the-art SUG approaches. In this case we notice a reduction in cumulative MSE by  $19.81\% - 49.90\%$  across the four noise scales. Note that HyPER achieves this significant reduction in rollout error while using the un-specialized UNet-P surrogate, demonstrating that HyPER succeeds in adapting pretrained models to noisy data.
+
+# 3.4 RQ4: INVESTIGATING HYPER ROLLOUTS VS. RANDOM POLICY ROLLOUTS
+
+To evaluate whether HyPER learns an effective RL policy, we compare it to a Random Policy baseline. This baseline is designed to invoke the simulator the same number of times as the HyPER RL policy for a particular rollout, but with uniform random probability over each timestep. By comparing the HyPER policy rollout to a Random Policy rollout with the same budget, we show that HyPER learns a superior performing policy in our experimental scenarios.
+
+Table 4: HyPER versus a random policy which calls the simulator the same number of times.  
+
+<table><tr><td rowspan="2">Experiment</td><td colspan="2">Final MSE</td><td colspan="2">Cumulative MSE</td><td>Cumulative Wins %</td></tr><tr><td>Random Policy</td><td>HyPER</td><td>Random Policy</td><td>HyPER</td><td>HyPER</td></tr><tr><td>Noise Free</td><td>0.011</td><td>0.011</td><td>0.186</td><td>0.164</td><td>57.00%</td></tr><tr><td>Bimodal σ2= 1.0</td><td>0.121</td><td>0.07</td><td>2.443</td><td>1.706</td><td>73.00%</td></tr><tr><td>Bimodal σ2= 0.75</td><td>0.055</td><td>0.025</td><td>1.305</td><td>0.739</td><td>84.50%</td></tr><tr><td>Bimodal σ2= 0.5</td><td>0.037</td><td>0.027</td><td>0.891</td><td>0.683</td><td>67.00%</td></tr><tr><td>Bimodal σ2= 0.25</td><td>0.022</td><td>0.019</td><td>0.495</td><td>0.424</td><td>61.00%</td></tr><tr><td>Changing Boundary</td><td>0.149</td><td>0.027</td><td>0.865</td><td>0.527</td><td>80.50%</td></tr></table>
+
+We summarize these results in Table 4, which also shows the percentage of trajectories over which HyPER reduces MSE compared to Random Policy (Cumulative Wins%). A 'win' is characterized by a HyPER rollout that yields a lower cumulative MSE than the corresponding Random Policy rollout. In the 'Noise Free' trajectories, HyPER only wins for  $57\%$  of the trajectories because the MSEs at each timestep in the Noise Free case have low variance, so a uniform Random Policy performs fairly well. However, in the bimodal noise scales (with fast and large rollout error accumulation), precise invocation of the simulator for knowledge-guided correction is imperative to prevent error accumulation. Hence, we see HyPER owing to its intelligent (RL-based) decision policy, has a higher percentage of wins  $(\approx 70\%)$  and significantly lower cumulative MSE at higher noise scales. We also see the strength of HyPER's learned policy when considering the changing boundary condition experiment which has a  $80.50\%$  win rate over the Random Policy and reduces cumulative MSE by  $39.08\%$ . This result demonstrates the effectiveness of HyPER in realistic physical scenarios and noisy conditions.
+
+# 3.5 RQ5: COST VERSUS ACCURACY TRADE-OFF
+
+![](images/916ab9a1618d970e079c8a7fbde433be64f4e2044bc9683ebc08e14d41f5bc59.jpg)  
+(a)
+
+![](images/505b64e5d7729b6c02d25e90f7b22386046646211eb5c214292dbbb5f56e7f47.jpg)  
+(b)  
+Figure 5: Figure 5(a) shows the average time of PDE prediction of a full trajectory for each method. Figure 5(b) illustrates the error per unit time (lower is better) for each method. We do not show the Sim-Only case here as we assume error is effectively zero for the simulator.
+
+A natural question is how much of a cost we pay in wall clock time when utilizing HyPER compared to baselines? We evaluate this by measuring the average time of each method on our 200 test trajectories and show results in Figure 5(a). While the wall clock time of SUG methods is lower than HyPER, our method has much lower rollout error, which we illustrate in Figure 5(b). Here we show the error per unit time (lower is better) and we see that HyPER has lower rollout error per second by a large margin in comparison to the baselines (note this plot is on a log scale). While calling the simulator incurs a time cost, the user can specify the  $\lambda$  parameter to adjust to their requirements allowing HyPER to be flexible in varying settings.
+
+# 3.6 SURROGATE-AGNOSTIC AND SIMULATOR-AGNOSTIC DESIGN OF HYPER
+
+To demonstrate the model-agnostic capability of HyPER, we train it on the subsurface flow task (dataset details in Sec. 2.3), comprising longer (i.e., 100 step) trajectories. In this case, we train HyPER using the Julia simulator DPFEHM, which simulates underground fluid flow in porous media. As we see in Table 5, HyPER outperforms both SUG baselines by a cumulative MSE per timestep
+
+Table 5: 2D Subsurface flow experiment results for a 100 timestep trajectory.  
+
+<table><tr><td rowspan="2">Experiment</td><td colspan="2">Cumulative MSE</td><td>per timestep</td></tr><tr><td>UNet-P</td><td>FNO-P</td><td>HyPER</td></tr><tr><td>Subsurface</td><td>4.345</td><td>10.938</td><td>0.271</td></tr></table>
+
+reduction of  $93.76\%$  and  $97.52\%$ . The integration of a distinct Julia-based simulator in a scenario with larger physical scales and times, shows that HyPER is surrogate and simulator agnostic and can perform well and reduce rollout error in multiple problem settings.
+
+# 4 RELATED WORK
+
+SUG approaches circumvent the use of computational simulations during inference and only employ simulations to generate training data. The U-Net (Ronneberger et al., 2015) model is a popular approach owing to its ability to capture spatial and temporal dynamics at multiple scales. This model has recently (Gupta & Brandstetter, 2022) demonstrated state of the art performance on various fluid dynamics benchmarks. Operator learning (Kovachki et al., 2021) approaches that learn function families of PDEs rather than single PDE instances have also been investigated to be effective neural surrogates. Two notable operator learning models are the deep operator network (Lu et al., 2021a) and the Fourier neural operator (FNO) (Li et al., 2020) models. Multiple investigations have been carried out employing operator learning techniques including extending them to multi-resolution (Howard et al., 2023; Lu et al., 2022) settings. Recently Takamoto et al. (2022) have also demonstrated that FNOs yield state-of-the-art results on benchmark tasks.
+
+Knowledge-guided SUG approaches like the popular Physics-informed neural network (PINN) (Raissi et al., 2019; Jagtap & Karniadakis, 2020; Cuomo et al., 2022) have also been effectively employed for improved generalization. A related paradigm of Universal Differential Equations (Rackauckas et al., 2020) utilizes data-driven surrogates to estimate focused sub-components of governing equations for better process modeling. Such approaches assume end-to-end gradient based training in a physics-informed manner and are known to converge slowly and to trivial solutions under data paucity and stiff PDE conditions Krishnapriyan et al. (2021) owing to catastrophic gradient imbalance Wang et al. (2021) between data-driven and physics-guided loss terms.
+
+SUG for Transient PDE Dynamics. All SUG approaches struggle to model transient PDE systems in an autoregressive manner and incur rollout error (Carey et al., 2024). In the work of Lippe et al. (2024), it is demonstrated how the spectral bias of traditional neural surrogates leads to significant error accumulation and they propose an initial refinement solution inspired by the process in diffusion modeling. Separately, List et al. (2024) have conducted a characterization of SUG rollout error and comment about the significant improvement obtainable by incorporating simulators in-the-loop.
+
+Hybrid-modeling for Transient PDE Dynamics. Hybrid-modeling approaches retain the simulation and resolve each query employing the neural surrogate and the simulator 'in-the-loop'. One recent example is Zhang et al. (2022) which selects between a simulator and a surrogate (in a predefined rule-based manner) to resolve a PDE trajectory. In their work, the simulator and surrogate are each invoked a pre-fixed number of times. In contract, HyPER learns a dynamic policy, capable of adapting to scenarios such as changing boundary conditions. Another major drawback with many such approaches (Chen et al., 2018; Belbute-Peres et al., 2020; Um et al., 2020; Donti et al., 2021; Pachalieva et al., 2022) is the requirement of simulators to be differentiable as they are mostly employed as additional layers in the neural network architecture, to be trained end-to-end with the neural surrogates.
+
+# 5 CONCLUSION AND FUTURE WORK
+
+This work presents a first of its kind knowledge-guided correction mechanism to reduce rollout errors in neural surrogates that model transient PDE systems. Our proposed method Hybrid PDE Predictor with RL (HyPER) learns a reinforcement learning based cost-aware control policy to parsimoniously invoke (costly) simulation steps to correct erroneous surrogate predictions. In contrast to existing approaches that employ simulators 'in-the-loop' with neural surrogates, HyPER does not impose any differentiability restrictions on the computational simulations. Further, HyPER is surrogate and simulator agnostic and is applicable to any neural surrogate and off-the-shelf simulator capable of resolving transient PDE systems.
+
+We have demonstrated the effectiveness of our proposed HyPER model in traditional in-distribution rollouts, under changing physical conditions and under noisy data conditions. Overall HyPER yields significant improvements of cumulative rollout error over state-of-the-art surrogate-only approaches with an average  $68.30\%$  improvement for in-distribution rollouts,  $75.28\%$  improvement for rollouts under changing physical conditions and  $28.61\%$  improvement for rollouts under noisy data conditions. In the future, we will investigate more sophisticated actor-critic based RL policies based to further improve the sample efficiency of HyPER. We will also explore extensions of HyPER to more challenging problems in multi-physics contexts as well as multi-phase flows.
+
+# ACKNOWLEDGMENTS
+
+DO gratefully acknowledges support from the Department of Energy, Office of Science, Office of Basic Energy Sciences, Geoscience Research program under Award Number LANLECA1.
+
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+
+# A APPENDIX
+
+# A.1 HYPER TRAINING PROCEDURE
+
+Below we detail the RL training procedure of HyPER.
+
+# Algorithm 1 HyPER Training Algorithm
+
+<table><tr><td colspan="3">Require: Dataset: D, Pretrained Surrogate Model: fφ, Simulator: S, Decision Model: dθ, Trajectory length: τ, Simulator proportion hyperparameter: λ, Learning rate: η, Error function (MSE in our case): ℓ, Cost function: c</td></tr><tr><td>1: for u, z ∈ D do</td><td colspan="2">For every trajectory in dataset, get field u and conditional features z</td></tr><tr><td>2: k ← 0</td><td></td><td>▷ Initialize number of simulator calls</td></tr><tr><td>3: Rd ← []</td><td></td><td>▷ Initialize decision model rewards list</td></tr><tr><td>4: Rb ← []</td><td></td><td>▷ Initialize baseline rewards list</td></tr><tr><td>5: L ← []</td><td></td><td>▷ List to store log probabilities of each action</td></tr><tr><td>6: u(x, -1) ← 0</td><td></td><td>▷ Set initial field prediction to 0</td></tr><tr><td>7: for t ∈ [0, τ] do</td><td></td><td>▷ For every time-step in trajectory</td></tr><tr><td>8: p ← dθ(ˆu(x, t-1), z, t)</td><td></td><td>▷ Get RL model probabilities for next action</td></tr><tr><td>9: a ~ Bernoulli(p)</td><td></td><td>▷ Sample next action</td></tr><tr><td>10: L += a log(p) + (1-a) log(1-p)</td><td></td><td>▷ Store log probability of action</td></tr><tr><td>11: if a = 0 then</td><td></td><td></td></tr><tr><td>12: u(x, t) ← fφ(ˆu(x, t-1), t)</td><td></td><td>▷ Call surrogate for next step prediction</td></tr><tr><td>13: else if a = 1 then</td><td></td><td></td></tr><tr><td>14: u(x, t) ← S(ˆu(x, t-1), t)</td><td></td><td>▷ Call simulator for next step prediction</td></tr><tr><td>15: k += 1</td><td></td><td>▷ Track number of times simulator called</td></tr><tr><td>16: end if</td><td></td><td></td></tr><tr><td>17: Rd += -ℓ(ˆu(x, t), u(x, t)) - c(λ, k, τ)</td><td></td><td>▷ Store policy reward based on MSE and cost function</td></tr><tr><td>18: end for</td><td></td><td></td></tr><tr><td>19: I ~ UniformWithoutReplacement([0, τ], k)</td><td></td><td>▷ Sample k times between [0, τ] without replacement, this list will contain the time-steps at which the random baseline will call the simulator</td></tr><tr><td>20: for t ∈ [0, τ] do</td><td></td><td>▷ Run random baseline for trajectory</td></tr><tr><td>21: if t∉ I then</td><td></td><td></td></tr><tr><td>22: u(x, t) ← fφ(ˆu(x, t-1), t)</td><td></td><td>▷ Call surrogate for next step prediction</td></tr><tr><td>23: else if t ∈ I then</td><td></td><td></td></tr><tr><td>24: u(x, t) ← S(ˆu(x, t-1), t)</td><td></td><td>▷ Call simulator for next step prediction</td></tr><tr><td>25: end if</td><td></td><td></td></tr><tr><td>26: Rb += -ℓ(ˆu(x, t), u(x, t)) - c(λ, k, τ)</td><td></td><td>▷ Store baseline reward using MSE and cost function</td></tr><tr><td>27: end for</td><td></td><td></td></tr><tr><td>28: ∇θJ ← -∇θL · StopGradient(Rd - Rb)</td><td></td><td>▷ Calculate policy gradient, this is done element-wise and then summed</td></tr><tr><td>29: θ ← θ - η∇θJ</td><td></td><td>▷ Update RL decision model parameters</td></tr><tr><td>30: end for</td><td></td><td></td></tr></table>
+
+# A.2 RQ3: HYPER ROLLOUTS WITH NOISY DATA: BIMODAL NOISE EXPERIMENTS
+
+Table 6 depicts the performance of HyPER in the bimodal noise case. We notice results similar to the unimodal noise case, with a reduction in cumulative MSE error of  $41.46\% - 65.24\%$ . We believe this improved performance in a more complex noise distribution is a result of our RL policy learning the more complex error distribution while the SUG methods accumulate larger error over two different noise windows.
+
+Table 6: Results depicting rollout error for the 2D-Navier-Stokes experiment with random Gaussian noise added at timesteps 2-4, 15-16.  $\sigma^2$  is the scale of the noise. The percentage reduction of cumulative MSE rollout error by HyPER, over the best performing SUG model is in parentheses.  
+
+<table><tr><td rowspan="2">Experiment</td><td colspan="3">Final MSE</td><td colspan="3">Cumulative MSE</td></tr><tr><td>UNet-P</td><td>FNO-P</td><td>HyPER</td><td>UNet-P</td><td>FNO-P</td><td>HyPER</td></tr><tr><td>Bimodal σ2= 1.0</td><td>0.844</td><td>0.097</td><td>0.07</td><td>6.648</td><td>2.914</td><td>1.706 (41.46%)</td></tr><tr><td>Bimodal σ2= 0.75</td><td>0.25</td><td>0.079</td><td>0.025</td><td>3.436</td><td>2.126</td><td>0.739 (65.24%)</td></tr><tr><td>Bimodal σ2= 0.5</td><td>0.087</td><td>0.07</td><td>0.027</td><td>1.942</td><td>1.576</td><td>0.683 (56.67%)</td></tr><tr><td>Bimodal σ2= 0.25</td><td>0.046</td><td>0.068</td><td>0.019</td><td>1.095</td><td>1.266</td><td>0.424 (61.28%)</td></tr></table>
+
+![](images/efa41e0c55282d04607e8adb0cda5fc6f41fd39e424761585dbbeb26514b05d1.jpg)  
+Figure 6: Average Cumulative MSE over all test trajectories of HyPER vs. UNet and FNO.
+
+# A.3 HYPER AVERAGE PERFORMANCE
+
+Figure 6 shows the average rollout error of HyPER versus UNet and FNO methods for all 200 Navier-Stokes test trajectories. This demonstrates that HyPER is effective in reducing rollout error significantly over a large set of unseen test trajectories.
+
+# A.4 UNET-MULTISTEP PREDICTION QUALITY
+
+In Figure 7 below we show five different sample predictions of UNet versus UNet-Multistep. Our investigations revealed that in comparison to UNet, all UNet-Multistep predictions look "smoothed". Essentially we found that while UNet-Multistep had reasonably low MSE, it failed to capture high frequency detail and suffered from spectral bias. Thus, UNet was chosen as our surrogate model in HyPER across all experiments because it showed comparable cumulative MSE to UNet-Multistep (Table 1).
+
+# A.5 HYPER VERSUS UNET PREDICTION
+
+We contrast the predictions and errors of a HyPER rollout and UNet rollout in Figure 8. Notice that at  $t = \{16.5s, 21.0s\}$ , UNet suffers from spectral bias and has high error on the plume edges where high frequency detail is required. We see that HyPER successfully mitigates this spectral bias error by incorporating the simulator, thereby significantly reducing rollout (cumulative) error over the full trajectory.
+
+![](images/0d2aa0db678113e06da7c8b1b98762be89a8e33ddcf97150c7a904ebed520ef2.jpg)  
+Figure 7: Sample predictions of UNet vs UNet-Multistep on five different noise free trajectories. Note that UNet-Multistep predictions suffer from spectral bias and have poor high frequency detail.
+
+# A.6 MODEL ARCHITECTURES AND PARAMETERS
+
+# A.6.1 MODEL TRAINING DETAILS
+
+We build and train all our neural surrogate models using the PyTorch library on a single Nvidia RTX A6000 GPU. The  $\Phi$  Flow simulator used in the Navier-Stokes experiment runs on the same GPU. The DPFEHM Julia simulator runs using multi-threading on an Intel(R) Xeon(R) Platinum 8358 CPU @ 2.60GHz.
+
+The UNet, FNO, UNet-Multistep, MPP, and PDE-Refiner baselines are trained with same number of samples as HyPER for the sake of fair comparison. The UNet-P and FNO-P baseline models are trained with a smaller dataset of 400 trajectories which are separate from the 400 trajectories HyPER's RL policy is trained with. All baselines are trained for 200 epochs.
+
+![](images/c71c13a354f13520bc8bf0f8ae848471a02c95331716ac0f5732754d5ff826e0.jpg)  
+Figure 8: Sample trajectory predictions and absolute errors of HyPER vs UNet. At various timesteps including  $t = 21.0s$ , we see that UNet has high error on the edges of the plume where high frequency behavior is present while HyPER successfully reduces this error.
+
+We only train the HyPER RL ResNet model for 30 epochs to demonstrate that HyPER does not require extensive training to outperform the baselines.
+
+Table 7: Model training times and datasets for SUG methods and HyPER.  
+
+<table><tr><td>Model</td><td>Training Time (Hours)</td><td>Epochs</td><td>Dataset Size (# Trajectories)</td></tr><tr><td>UNet-P</td><td>1.46</td><td>200</td><td>400</td></tr><tr><td>FNO-P</td><td>0.55</td><td>200</td><td>400</td></tr><tr><td>UNet</td><td>3.29</td><td>200</td><td>800</td></tr><tr><td>FNO</td><td>2.02</td><td>200.</td><td>800</td></tr><tr><td>UNet-Multistep</td><td>4.10</td><td>200</td><td>800</td></tr><tr><td>MPP</td><td>9.58</td><td>200</td><td>800</td></tr><tr><td>PDE-Refiner</td><td>3.66</td><td>200</td><td>800</td></tr><tr><td>HyPER</td><td>5.21</td><td>30</td><td>400</td></tr></table>
+
+# A.6.2 MODEL PARAMETERS
+
+Our UNet model is built on top of the PDEArena modern UNet architecture with wide residual blocks and training parameters in Table 8.
+
+Table 8: UNet parameters.  
+
+<table><tr><td>Parameter</td><td>Value</td></tr><tr><td>Model Size (# parameters)</td><td>12,295,233</td></tr><tr><td>Hidden Channels</td><td>64</td></tr><tr><td>Activation Function</td><td>GELU</td></tr><tr><td>Channel Multipliers</td><td>[1, 2, 2]</td></tr><tr><td>Num Residual Blocks Per Channel</td><td>2</td></tr><tr><td>Sinusoidal Time Embedding</td><td>Yes</td></tr><tr><td>Learning Rate</td><td>1e-4</td></tr><tr><td>Optimizer</td><td>Adam</td></tr><tr><td>Loss Function</td><td>MSE</td></tr><tr><td>Epochs</td><td>200</td></tr></table>
+
+Our FNO model is built using the neuraloperator library and is constructed to have a similar number of parameters as our UNet for a fair comparison. See Table 9 for details.
+
+Table 9: FNO parameters.  
+
+<table><tr><td>Parameter</td><td>Value</td></tr><tr><td>Model Size (# parameters)</td><td>12,437,057</td></tr><tr><td>Number of Fourier Modes</td><td>27</td></tr><tr><td>Hidden Channels</td><td>64</td></tr><tr><td>Lifting Channels</td><td>256</td></tr><tr><td>Projection Channels</td><td>256</td></tr><tr><td>Learning Rate</td><td>1e-5</td></tr><tr><td>Optimizer</td><td>Adam</td></tr><tr><td>Loss Function</td><td>MSE</td></tr><tr><td>Epochs</td><td>200</td></tr></table>
+
+The HyPER RL model is a lightly modified version of the ResNet34 model from the Torchvision library. See Table 10 for parameters.
+
+The Multiple Physics Pretrained model (MPP) is adapted from MPP github. Note that we train this model for 200 epochs while UNet and FNO are only trained for 50. See Table 11 for details.
+
+Table 10: RL ResNet parameters.  
+
+<table><tr><td>Parameter</td><td>Value</td></tr><tr><td>Model Size (# parameters)</td><td>11,751,300</td></tr><tr><td>Number of Layers</td><td>[3, 4, 6, 3]</td></tr><tr><td>Hidden Channels</td><td>64</td></tr><tr><td>Sinusoidal. Time Embedding</td><td>Yes</td></tr><tr><td>Activation Function</td><td>ReLU</td></tr><tr><td>Learning Rate</td><td>1e-5</td></tr><tr><td>Optimizer</td><td>Adam</td></tr><tr><td>Reward Function</td><td>Equation 3</td></tr><tr><td>Epochs</td><td>30</td></tr></table>
+
+Table 11: Multiple Physics Pretraining parameters.  
+
+<table><tr><td>Parameter</td><td>Value</td></tr><tr><td>Model Size (# parameters)</td><td>28,979,436</td></tr><tr><td>Patch Size</td><td>16x16</td></tr><tr><td>Embedding Dimension</td><td>384</td></tr><tr><td>Number of Axial Attention Heads</td><td>6</td></tr><tr><td>Number of Transformer Blocks</td><td>12</td></tr><tr><td>Epochs</td><td>200</td></tr></table>
+
+The PDE-Refiner model was adapted from PDE-Refiner to work with our UNet model. Note that we train this model for 200 epochs while UNet and FNO are only trained for 50. See Table 12 for details.
+
+Table 12: PDE-Refiner parameters.  
+
+<table><tr><td>Parameter</td><td>Value</td></tr><tr><td>Model Size (# parameters)</td><td>12,378,241</td></tr><tr><td>Number of Refinement/Denoising Steps</td><td>3</td></tr><tr><td>Minimum Noise Scale</td><td>4e-7</td></tr><tr><td>Hidden Channels</td><td>64</td></tr><tr><td>Activation Function</td><td>GELU</td></tr><tr><td>Channel Multipliers</td><td>[1, 2, 2]</td></tr><tr><td>Num Residual Blocks Per Channel</td><td>2</td></tr><tr><td>Sinusoidal Time Embedding</td><td>Yes</td></tr><tr><td>Learning Rate</td><td>1e-4</td></tr><tr><td>Optimizer</td><td>Adam</td></tr><tr><td>Loss Function</td><td>MSE</td></tr><tr><td>Epochs</td><td>200</td></tr></table>

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