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+# LEARNING ROBUST REPRESENTATIONS WITH LONG-TERM INFORMATION FOR GENERALIZATION IN VISUAL REINFORCEMENT LEARNING
+
+Rui Yang $^{1,2}$ , Jie Wang $^{1,2}$ , Qijie Peng $^{1,2}$ , Ruibo Guo $^{2}$ , Guoping Wu $^{2}$ , Bin Li $^{2}$ $^{1}$ MoE Key Laboratory of Brain-inspired Intelligent Perception and Cognition  
+ $^{2}$ University of Science and Technology of China  
+{yr0013, sa24006073, ruiboguo, guoping}@mail.ustc.edu.cn  
+{jiewangx, binli}@ustc.edu.cn
+
+# ABSTRACT
+
+Generalization in visual reinforcement learning (VRL) aims to learn agents that can adapt to test environments with unseen visual distractions. Despite advances in robust representations learning, many methods do not take into account the essential downstream task of sequential decision-making. This leads to representations that lack critical long-term information, impairing decision-making abilities in test environments. To tackle this problem, we propose a novel robust action-value representation learning (ROUSER) under the information bottleneck (IB) framework. ROUSER learns robust representations to capture long-term information from the decision-making objective (i.e., action values). Specifically, ROUSER uses IB to encode robust representations by maximizing their mutual information with action values for long-term information, while minimizing mutual information with state-action pairs to discard irrelevant features. As action values are unknown, ROUSER proposes to decompose robust representations of state-action pairs into one-step rewards and robust representations of subsequent pairs. Thus, it can use known rewards to compute the loss for robust representation learning. Moreover, we show that ROUSER accurately estimates action values using learned robust representations, making it applicable to various VRL algorithms. Experiments demonstrate that ROUSER outperforms several state-of-the-art methods in eleven out of twelve tasks, across both unseen background and color distractions.
+
+# 1 INTRODUCTION
+
+Generalization in visual reinforcement learning (VRL) has received considerable attention (Mnih et al., 2015; Yarats et al., 2021; Kirk et al., 2021; Zhu et al., 2023) due to its potential to learn agents that can address complex tasks across diverse environments in real-world applications, e.g., autonomous driving (Zhou et al., 2023b; Cao et al., 2023), robot control (Xing et al., 2021; Liu et al., 2023b), and chip design (Geng et al., 2024; Wang et al., 2024c; 2025b). It refers to agents' capability to directly use their learned skills to unknown environments, where visual distractions (e.g., dynamic backgrounds or object colors under control) may differ from those encountered during training (Li et al., 2021; Ali et al., 2023). Thus, the generalizable agents can execute tasks with high performance when encountering environments under unseen distractions without extensive retraining.
+
+To learn generalizable agents, one of promising approaches in VRL aims to develop robust representations against visual distractions in environments (Kemertas & Augmented-Armstrong, 2021; Mazoure et al., 2022). Specifically, some methods introduce data augmentations (Hansen & Wang, 2021; Huang et al., 2022) and contrastive learning (Laskin et al., 2020; Kim et al., 2021) to learn representations that are robust to irrelevant and spurious information. Other methods apply metric learning (Zhang et al., 2021; Lopez et al., 2022) to encode task-relevant information for robust representation learning. Then, based on the learned robust representations, these methods can directly use traditional VRL algorithms (Schulman et al., 2017; Fujimoto et al., 2018; Haarnoja et al., 2018)
+
+to optimize the objectives of the downstream task of sequential decision-making, i.e., maximizing the expected cumulative rewards (i.e., action values) over these environments.
+
+However, the aforementioned representation learning approaches often do not take into account the essential downstream decision-making. This results in the representations that cannot capture critical long-term robust information in sequential data, which is a key factor for generalization in VRL (Zhou et al., 2022; Qi et al., 2022; Wang et al., 2023a). Although such representations are robust against visual distractions, they cannot facilitate the generalization of sequential decision-making.
+
+In this paper, we propose a robust action-value representation learning (ROUSER) for VRL generalization under the information bottleneck (IB). ROUSER introduces IB to learn robust representations that capture long-term information from the sequential decision-making objective (i.e., action values).
+
+Specifically, ROUSER first applies IB to propose a robust action-value representation, which is correlated with the action value. This robust representation for each state-action pair maximizes mutual information with the action value, while minimizing mutual information with the associated state-action pair. Thus, this representation can capture long-term information without irrelevant features. Then, as the true action values are unknown, we cannot directly predict action values to learn the robust action-value representation. Inspired by task decomposition methods (Dietterich, 2000; Russell & Zimdars, 2003; Dayan, 1993; Barreto et al., 2017), ROUSER proposes to decompose the robust action-value representation into a compressed reward representation—which captures only information from a one-step reward—and a subsequent robust action-value representation. With this recursive form, ROUSER can use the known one-step rewards instead of unknown action values to compute the loss associated with robust action-value representations.
+
+Moreover, for compressed reward representations, ROUSER introduces a reward model in the IB framework. This model maximizes mutual information between one-step rewards and compressed reward representations, while minimizing it between such representations and associated state-action pairs. Therefore, ROUSER can use one-step rewards and state-action pairs to learn compressed reward representations that encode only information from rewards.
+
+This study proposes robust action-value representations in the IB framework, capturing long-term information during the robust representation learning process. Thus, it significantly enhances performance in sequential decision-making against unseen visual distractions. In addition, we provide a theoretical guarantee for ROUSER, establishing a bound between the true action-value function and the action-value function on top of learned robust action-value representations. This shows that ROUSER can accurately estimate true action values using learned robust action-value representations. Building upon this proof, we present an advantage of ROUSER, which is its applicability to various VRL algorithms to improve generalization. We can integrate ROUSER with the VRL critic for action-value estimation, using a critic's embedding as a robust action-value representation. Experiments in Section 5.2 demonstrate the applicability by combining ROUSER with traditional VRL algorithms, including policy gradient (Yarats et al., 2022) and value-based (Hosu & Rebedea, 2016) methods.
+
+It is worth noting that our study differs from task decomposition methods, which learn the representations from action values but do not take into account robustness against visual distractions. We use one-step rewards to extract long-term information from action values and leverage IB to discard irrelevant features in VRL. We summarize the major contributions below.
+
+- To the best of our knowledge, this study is the first to learn representations from action values against visual distractions. It encodes robust and long-term information rather than just robust features to facilitate downstream decision-making for generalization.  
+- We show that ROUSER can accurately estimate action values using the learned robust action-value representations. With this proof, we can integrate ROUSER with various VRL algorithms to estimate the action values, thereby enhancing robustness.  
+- Extensive experiments demonstrate that ROUSER outperforms several state-of-the-art VRL methods in eleven out of twelve tasks across both unseen background and color distractions.
+
+# 2 RELATED WORK
+
+Representation Learning for Generalization in VRL. Representation learning approaches for generalization in VRL aim to improve agents' performance on unseen test environments by extracting
+
+robust representations from training environments. Some approaches design an auxiliary contrastive task to maximize the mutual information between representations from similar observations while minimizing the mutual information between representations from dissimilar observations (Kim et al., 2019; Laskin et al., 2020; Fan & Li, 2022). To obtain similar observations, they always leverage data augmentations to generate observations (Hansen et al., 2021; Hansen & Wang, 2021; Huang et al., 2022; Ye et al., 2023; Wang et al., 2025a). However, they often do not take into account the downstream task of sequential decision-making, leading to representations that lack long-term information. Some approaches introduce the bisimulation metric with a recursive form (Ferns & Precup, 2014; Castro, 2020; Agarwal et al., 2021) to identify robust representations using long-term information, similar to our approach. However, it is difficult to directly compute the recursive metric. Thus, for implementation, they often simplify the computation to capture information from the immediate future. Other methods directly predict reward sequences to learn robust representations (Yang et al., 2022; Zhou et al., 2023a), which captures long-term task-relevant information for generalization. Instead of using reward sequences, our approach simply applies one-step rewards to encode the sequential information, and we further introduce the IB principle to improve robustness.
+
+Task Decomposition in RL. We use the task decomposition to represent methods that decompose a task into independent sub-tasks, so that each task has the same environment dynamics but a different reward function. Then, they propose to learn action-value representations invariant to different tasks/reward functions. Some of them, namely successor feature methods (Dayan, 1993; Barreto et al., 2017; 2018; 2020), use such representations to generalize to different tasks for transfer learning. Others, namely value decomposition methods (Sutton et al., 1999; Russell & Zimdars, 2003; Makino, 2023), apply such representations to improve sample efficiency in new tasks. See Appendix A for details. Motivated by their framework where action-value representations can be decomposed into an infinite sequence of reward representations, we (1) learn the action-value representations for long-term information using one-step rewards and (2) filter out irrelevant information from the one-step reward representations to improve robustness of overall action-value representations.
+
+It is worth noting that in this article, we aim to learn agents in training environments to generalize to test environments with unseen visual distractions, while remaining a consistent reward function. This differs from task decomposition methods, which primarily extract knowledge invariant to tasks between tasks with varying reward functions and no visual distractions. Moreover, task decomposition methods can easily encode irrelevant features in the setting of this article, as they learn representations from environment dynamics, which involve visual distractions in VRL (Yang et al., 2022).
+
+Information Bottleneck (IB). The IB principle in supervised learning (Tishby & Zaslavsky, 2015; Saxe et al., 2018) aims to learn compressed representations including minimal information relevant to downstream tasks, improving generalization performance. Specifically, it regularizes representations by minimizing the mutual information between inputs and representations, while maximizing the mutual information between representations and labels. In VRL, recent methods (Pei & Hou, 2019; Xiang et al., 2023) also focus on the IB principle that can learn compressed representations to improve generalization capabilities of agents. Different from supervised learning, in VRL, these methods only have one-step rewards but action values, i.e., the targets in VRL corresponding to labels in supervised learning. Thus, some of them (Federici et al., 2020; Fan & Li, 2022) follow the unsupervised learning paradigm, using the IB framework to regularize representations from similar inputs or multi-view image observations. Since the unsupervised learning in the IB framework is more challenging than supervised learning (Federici et al., 2020), we use one-step rewards—the known supervised signals in RL—to extract their compressed representations for robust action-value representation learning.
+
+# 3 PRELIMINARIES
+
+Visual RL. We consider a family of environments  $\mathcal{E}$ . Each environment  $e\in \mathcal{E}$  is a block Markov decision process (BMDP) (Zhang et al., 2020) denoted by  $\mathcal{M}^e = \langle S,\mathcal{O}^e,\mathcal{A},\mathcal{R},p,p^e,\gamma \rangle$ . Here,  $\mathcal{S}$  is the state space,  $\mathcal{O}^e$  is the observation space in  $e$ ,  $\mathcal{A}$  is the action space,  $\mathcal{R}$  is the reward space,  $p(s',r\mid s,a)$  is the state transition probability,  $p^e (o',r\mid o,a)$  is the observation transition probability (i.e., environment dynamics) varying with the environment  $e\in \mathcal{E}$ , and  $\gamma \in [0,1)$  is the discount factor. For simplicity, we use bold letters (e.g., o and a) to denote random variables, normal letters (e.g., o and a) to denote samples, and  $\mathcal{O}$  to denote the set of  $\mathcal{O}^e$  for all  $e\in \mathcal{E}$ .
+
+We suppose that an agent reaches an unseen latent state  $s$  and obtains an observation  $o$  on an environment  $e \in \mathcal{E}$ . In a BMDP, the observation is determined by a state and some task-irrelevant visual factors varying with environments, e.g., backgrounds or agent colors in decision-making process. Each state  $s$  does not involve irrelevant features in VRL and is invariant to the family of environments. Formally, let  $\mathcal{X}$  be the set of such visual factors. We introduce an observation function  $g: S \times \mathcal{X} \to \mathcal{O}$  (Zhang et al., 2020; Song et al., 2020) such that  $\mathbf{o} = g(\mathbf{s}, \mathbf{x})$ . Here,  $\mathbf{x}$  is a random variable in  $\mathcal{X}$ , which is independent of  $s$  and  $\mathbf{a}$  with a specific transition probability  $q^{e}(x' | x)$ .
+
+Moreover, we assume that the environments follow a generalized Block structure Zhang et al. (2020). That is, an observation  $o \in \mathcal{O}$  uniquely determines its generating state  $s$  and the visual factor  $x$ . This assumption implies that the observation function  $g(s,x)$  is invertible with respect to both  $s$  and  $x$ . Then, we have  $p^e (o',r|o,a) = p(s',r|s,a)q^e (x' | x)$ . Note that the expected reward  $r(o,a) = r(s,a) = \mathbb{E}_p[\mathbf{r}|s,a]$  depends on the corresponding state-action pair rather than the observation-action pair, where  $\mathbf{r}$  is the random variable of the one-step reward.
+
+We aim to learn an agent with a policy  $\pi : \mathcal{O} \to \Delta(\mathcal{A})$  that maximizes the expected cumulative reward  $\mathbb{E}^e\left[\sum_{t=0}^\infty \gamma^t r_t\right]$  simultaneously in all  $e \in \mathcal{E}$ , where  $\mathbb{E}^e[\cdot]$  means that the expectation is taken in  $e$ . We use  $Q^\pi(o_t, a_t) = Q^\pi(s_t, a_t) = \mathbb{E}_{\pi,p}\left[\sum_{i=0}^\infty \gamma^i r(o_{t+i}, a_{t+i})\right]$  to denote the action values.
+
+# 4 ALGORITHM
+
+In this paper, we propose to learn robust action-value representations that capture long-term information for decision-making. Then, we introduce robust and compressed reward representations and use temporal-difference (TD) learning (Sutton & Barto, 2018) paradigm to learn robust action-value representations, guided by an IB-based objective to ensure robustness throughout the learning process.
+
+# 4.1 ROBUST ACTION-VALUE REPRESENTATIONS IN THE IB FRAMEWORK
+
+In this subsection, we propose robust action-value representations in the IB framework.
+
+Firstly, motivated by task decomposition methods, we provide the action-value representation in Definition 4.1. This action-value representation is linearly correlated with the action-value function following a given policy, containing the long-term information from the action-value function. Therefore, we propose using the action-value representation to facilitate the agent's decision-making performance in VRL test environments.
+
+Definition 4.1. A representation in a  $D$ -dimensional space  $\mathcal{Z}$  is an action-value representation  $H^{\pi}(o_t, a_t)$  if there exists a linear mapping  $\Phi: \mathcal{Z} \to \mathbb{R}$  such that
+
+$$
+Q ^ {\pi} \left(o _ {t}, a _ {t}\right) = \Phi \left(H ^ {\pi} \left(o _ {t}, a _ {t}\right)\right) = \left\langle \omega , H ^ {\pi} \left(o _ {t}, a _ {t}\right) \right\rangle , \quad \forall o _ {t} \in \mathcal {O}, a _ {t} \in \mathcal {A}, \tag {1}
+$$
+
+where the dimension  $D > 1$ ,  $\langle \cdot, \cdot \rangle$  denotes the inner product, and  $w$  denotes the weight vector of  $\Phi$ .
+
+Then, for VRL generalization, due to visual factors  $x_{t} \in \mathcal{X}$ , the action-value representation may involve irrelevant information. This motivates us to introduce the IB principle, compressing the action-value representation to discard irrelevant information. Specifically, we employ the conditional IB (Chechik & Tishby, 2002) to provide Definition 4.2. In this definition, we propose a robust action-value representation, the action-value representation for each observation-action pair maintains maximum mutual information with the corresponding action value while minimizing mutual information with the observation-action pair given the action value.
+
+Definition 4.2. An action-value representation is a robust action-value representation  $Z^{\pi}(o_t, a_t) = \mathbb{E}[\mathbf{Z}_t]$  such that for all  $o_t \in \mathcal{O}$  and  $a_t \in \mathcal{A}$ ,
+
+$$
+\mathbf {Z} _ {t} = \underset {\mathbf {H} _ {t}} {\arg \min } \mathcal {I} \left(\mathbf {u} _ {t}; \mathbf {H} _ {t} \mid \mathbf {Q} _ {t}\right) - \beta \mathcal {I} \left(\mathbf {Q} _ {t}; \mathbf {H} _ {t}\right), \tag {2}
+$$
+
+where  $\mathbf{H}_t$  is the random variable of  $H^{\pi}(o_t, a_t)$ ,  $\mathbf{Q}_t$  is the random variable of  $Q^{\pi}(o_t, a_t)$ ,  $\mathbf{u}_t$  is the random variable of  $(o_t, a_t)$ ,  $\mathcal{I}(\mathbf{y}_1; \mathbf{y}_2)$  is the mutual information between random variables  $\mathbf{y}_1$  and  $\mathbf{y}_2$ ,  $\mathcal{I}(\mathbf{y}_1; \mathbf{y}_2 | \mathbf{y}_3)$  is the conditional mutual information to quantify the information between  $\mathbf{y}_1$  and  $\mathbf{y}_2$  given  $\mathbf{y}_3$ , and  $\beta$  is a hyperparameter.
+
+However, it is worth noting that we do not have the true action values during the training process of VRL, and we thus cannot directly use Equations 1 and 2 to learn robust action-value representations.
+
+# 4.2 COMPRESSED REWARD REPRESENTATIONS FROM ONE-STEP REWARDS
+
+To address the aforementioned challenge posed by unknown action values, we propose using one-step rewards instead. We first present how to learn simple action-value representations, and we then discuss how to learn our proposed robust action-value representations in this section.
+
+The Recursive Form of Action-Value Representations and the Reward Representations. To learn simple action-value representations, following task decomposition methods, we introduce a reward representation  $h \in \mathcal{Z}$  that is linearly related to the one-step reward, with a linear mapping  $f: \mathcal{Z} \to \mathbb{R}$  and the weight vector  $\omega'$ .
+
+$$
+r \left(o _ {t}, a _ {t}\right) = f \left(h \left(o _ {t}, a _ {t}\right)\right) = \left\langle \omega^ {\prime}, h \left(o _ {t}, a _ {t}\right) \right\rangle . \tag {3}
+$$
+
+Based on the reward representations, we can derive the action-value representations such that
+
+$$
+H ^ {\pi} \left(o _ {t}, a _ {t}\right) = \mathbb {E} _ {\pi , p} \left[ \sum_ {i = 0} ^ {\infty} \gamma^ {i} h \left(o _ {t + i}, a _ {t + i}\right) \right], \quad Q ^ {\pi} \left(o _ {t}, a _ {t}\right) = \left\langle \omega^ {\prime}, H ^ {\pi} \left(o _ {t}, a _ {t}\right) \right\rangle . \tag {4}
+$$
+
+Equation 4 shows that the action-value representation is linearly correlated with the infinite sequence of reward representations. We provide the details in Appendix C.1.
+
+With Equation 4, we can derive a recursive form, i.e., the action-value representation for each observation-action pair consists of a reward representation and the action-value representation of its subsequent observation-action pair.
+
+$$
+H ^ {\pi} \left(o _ {t}, a _ {t}\right) = h \left(o _ {t}, a _ {t}\right) + \gamma \mathbb {E} _ {\pi , p} \left[ H ^ {\pi} \left(o _ {t + 1}, a _ {t + 1}\right) \right]. \tag {5}
+$$
+
+With the linear relation in Equation 3 and the recursive form in Equation 5, we can simply compute the loss for learning such action-value representations. Moreover, we show that using Equation 5 can converge in tabular settings in Appendix C.2.
+
+Compressed Reward Representations for Robust Action-Value Representations. Based on Definition 4.2, Equations 4, and 5, as the action-value representations are decomposed into an infinite sequence of reward representations, which serve as their foundation, we can directly filter out task-irrelevant information from such reward representations to improve robustness of overall action-value representations, learning robust action-value representations without irrelevant features. Thus, we propose using the IB framework to regulate the reward representations.
+
+Since (1) the one-step reward is task-relevant without irrelevant features, and (2) the observation may involve irrelevant visual factors, we propose a compressed reward representation in Definition 4.3. This representation for each observation-action pair is linearly related to the corresponding one-step reward and does not involve any irrelevant information by preserving maximum mutual information with this one-step reward while minimizing mutual information with its observation-action pair.
+
+Definition 4.3. A reward representation is a compressed reward representation  $z(o_{t},a_{t}) = \mathbb{E}\left[\mathbf{z}_{t}\right]$  such that for all  $o_{t} \in \mathcal{O}$  and  $a_{t} \in \mathcal{A}$ ,
+
+$$
+\mathbf {z} _ {t} = \underset {\mathbf {h} _ {t}} {\arg \min } \mathcal {I} \left(\mathbf {u} _ {t}; \mathbf {h} _ {t} \mid \mathbf {r} _ {t}\right) - \beta \mathcal {I} \left(\mathbf {r} _ {t}; \mathbf {h} _ {t}\right), \tag {6}
+$$
+
+where  $\mathbf{h}_t$  is the random variable of  $h(o_t, a_t)$ , and  $\beta$  is a hyperparameter.
+
+In Equation 6, given the one-step reward  $\mathbf{r}_t$ , the conditional mutual information  $\mathcal{I}(\mathbf{u}_t; \mathbf{h}_t | \mathbf{r}_t)$  quantifies information in  $\mathbf{h}_t$  that is irrelevant. The mutual information  $\mathcal{I}(\mathbf{r}_t; \mathbf{h}_t)$  quantifies task-relevant information shared between the reward  $\mathbf{r}_t$  and the representations  $\mathbf{h}_t$ .  $\beta$  is a hyperparameter that determines the preference over the trade-off between task-relevant and task-irrelevant information.
+
+Based on Definition 4.3 and Equation 4, the robust action-value representation is linearly correlated with the infinite sequence of compressed reward representations. Thus, we can also use the recursive form of robust action-value representations, as shown in Equation 5, to learn these representations.
+
+Furthermore, in the following of this part, we discuss how to compute mutual information on Equation 6. We provide detail formulas of Equation 6 given the joint distribution and conditional joint distribution of  $\mathbf{u}_t$ ,  $\mathbf{z}_t$ , and  $\mathbf{r}_t$ . Note that for simplicity, we use  $\mathbf{z}$  instead of  $\mathbf{h}$ .
+
+$$
+\mathcal {I} (\mathbf {u} _ {t}; \mathbf {z} _ {t} | \mathbf {r} _ {t}) = \mathbb {E} _ {\mathbf {u} _ {t}, \mathbf {z} _ {t}, \mathbf {r} _ {t}} \left[ \log \frac {p (\mathbf {z} _ {t} | \mathbf {u} _ {t})}{p (\mathbf {z} _ {t} | \mathbf {r} _ {t})} \right], \quad \mathcal {I} (\mathbf {r} _ {t}; \mathbf {z} _ {t}) = \mathbb {E} _ {\mathbf {r} _ {t}, \mathbf {z} _ {t}} \left[ \log \frac {p (\mathbf {r} _ {t} | \mathbf {z} _ {t})}{p (\mathbf {r} _ {t})} \right]. (7)
+$$
+
+To estimate the probability density  $p(\mathbf{z}_t|\mathbf{u}_t)$  on Equation 7, we assume that a compressed reward representation  $z(\mathbf{o}_t,\mathbf{a}_t)$  for any observation-action pair  $(\mathbf{o}_t,\mathbf{a}_t)$  follows a Gaussian distribution  $\mathcal{N}_{\psi}$ , and we provide the detailed implementation in the next subsection.
+
+However, due to the modelling of  $\mathbf{z}_t$ , the approximation of  $p(\mathbf{z}_t|\mathbf{r}_t)$  is intractable. Inspired by the method that estimates conditional mutual information (Alemi et al., 2017; Fan & Li, 2022), we introduce a new representation  $\mathbf{k}_t \in \mathcal{Z}$  extracted directly from the one-step reward  $\mathbf{r}_t$  to estimate  $p(\mathbf{k}_t|\mathbf{r}_t)$  instead of  $p(\mathbf{z}_t|\mathbf{r}_t)$ . Thus, we can derive an upper bound of the conditional mutual information.
+
+$$
+\mathcal {I} \left(\mathbf {u} _ {t}; \mathbf {z} _ {t} \mid \mathbf {r} _ {t}\right) \leq \mathcal {D} _ {\mathcal {K L}} \left(p \left(\mathbf {z} _ {t} \mid \mathbf {u} _ {t}\right) \| p \left(\mathbf {k} _ {t} \mid \mathbf {r} _ {t}\right)\right). \tag {8}
+$$
+
+We provide details in Appendix C.3. We then can minimize the upper bound—i.e., the Kullback-Leibler (KL) divergence between  $p(\mathbf{z}_t|\mathbf{u}_t)$  and  $p(\mathbf{k}_t|\mathbf{r}_t)$ —to discard irrelevant and redundant information for compressed reward representation learning. Moreover, the representation  $\mathbf{k}_t$  is directly mapped from the one-step reward  $\mathbf{r}_t$ . To prevent this mapping from introducing potential redundant information into  $\mathbf{k}_t$ , we first use  $\mathcal{I}(\mathbf{r}_t;\mathbf{k}_t|\mathbf{u}_t)$  to quantify this information and then minimize  $\mathcal{I}(\mathbf{r}_t;\mathbf{k}_t|\mathbf{u}_t)$ . Therefore, we have the whole optimization objective to minimize for compressed reward representation learning.
+
+$$
+\mathcal {L} _ {\mathrm {c i b}} = \mathcal {I} \left(\mathbf {u} _ {t}; \mathbf {z} _ {t} \mid \mathbf {r} _ {t}\right) - \beta \mathcal {I} \left(\mathbf {r} _ {t}; \mathbf {z} _ {t}\right) + \mathcal {I} \left(\mathbf {r} _ {t}; \mathbf {k} _ {t} \mid \mathbf {u} _ {t}\right). \tag {9}
+$$
+
+# 4.3 ROUSER ALGORITHM
+
+Here, we present the detailed algorithm of ROUSER. Firstly, we provide the losses for compressed reward representations. Then, we introduce the loss with TD learning paradigm for robust action-value representation learning. Moreover, we provide the theoretical analysis of learned robust action-value representations. We provide the architecture and pseudocode of ROUSER in Appendix D.
+
+IB-based Losses for Compressed Reward Representation Learning. The optimization objective for learning compressed reward representations consists of two parts: a prediction loss based on Equation 3 and a conditional IB loss based on Equation 9.
+
+Firstly, we introduce a reward model  $\psi$  to encode the compressed reward representation from an observation-action pair and assume that a compressed reward representation  $z(o_{t},a_{t}) = \mathbb{E}[\mathbf{z}_{t}]$  for any observation-action pair  $(o_t,a_t)$  follows a Gaussian distribution  $\mathcal{N}_{\psi} = \mathcal{N}\big(z_{\psi}(o_t,a_t),\Sigma_{\psi}(o_t,a_t)\big)$  where  $z_{\psi}(o_t,a_t)$  is the mean and  $\Sigma_{\psi}(o_t,a_t)$  is the diagonal covariance. We also provide a parameterized model  $\eta$  to estimate the representation  $k(r_{t}) = \mathbb{E}[\mathbf{k}_{t}]$ .  $\mathbf{k}_t$  is drawn from the Gaussian distribution  $\mathcal{N}_{\eta} = \mathcal{N}\big(k_{\eta}(r_{t}),\Sigma_{\eta}(r_{t})\big)$ , where  $k_{\eta}(r_{t})$  is the mean of  $\mathbf{k}_t$ , and  $\Sigma_{\eta}(r)$  is the diagonal covariance.
+
+Secondly, we provide the prediction loss based on Equation 3 to maintain the linear mapping relationship between the compressed reward representations and corresponding one-step rewards. Building upon model-based RL methods (Janner et al., 2019; Wang et al., 2022; 2023c) that model one-step rewards as a Gaussian distribution, we predict the mean and the standard deviation of reward distribution to learn reward representations. We formulate the prediction loss based on KL divergence:
+
+$$
+\mathcal {L} _ {\text {p r e d}} (\psi , \omega^ {\prime}, \omega^ {\prime \prime}) = \mathbb {E} _ {(\mathbf {o} _ {t}, \mathbf {a} _ {t}, \mathbf {r} _ {t}) \sim \mathcal {B}} \left[ \frac {\left\| \left\langle \omega^ {\prime} , z _ {\psi} (\mathbf {o} _ {t} , \mathbf {a} _ {t}) \right\rangle - \mathbf {r} _ {t} \right\| _ {2} ^ {2}}{2 \cdot \left\langle \omega^ {\prime \prime} , z _ {\psi} (\mathbf {o} _ {t} , \mathbf {a} _ {t}) \right\rangle^ {2}} + \log \left\langle \omega^ {\prime \prime}, z _ {\psi} (\mathbf {o} _ {t}, \mathbf {a} _ {t}) \right\rangle \right], \tag {10}
+$$
+
+where  $\omega^{\prime}$  denote the weight vector mentioned in Equation 3 for the computation of the mean of one-step rewards,  $\omega''$  denote the weight vector to compute the standard deviation of one-step rewards, and  $\mathcal{B}$  denotes the replay buffer.
+
+Finally, we propose the conditional IB loss based on Equation 9 to learn compressed reward representations, discarding task-irrelevant information while preserving task-relevant information. With the models  $\psi$  and  $\eta$ , we can reformulate Equation 9 as:
+
+$$
+\mathcal {L} _ {\text {i n f o}} (\psi , \eta) = \mathbb {E} _ {\mathcal {B}} \left[ - \mathcal {I} \left(\mathbf {z} _ {t}; \mathbf {k} _ {t}\right) + \alpha \mathcal {D} _ {\mathcal {S K L}} \left(\mathcal {N} _ {\psi} | | \mathcal {N} _ {\eta}\right) \right], \tag {11}
+$$
+
+where  $\alpha$  derived from  $\beta$  is a hyperparameter that is tuned by an exponential scheduler during training, and  $\mathcal{D}_{\mathcal{SKL}}$  represents the symmetrized KL divergence (Jeffreys, 1998) obtained by averaging the KL divergences  $\mathcal{D}_{\mathcal{KL}}(\mathcal{N}_{\psi}||\mathcal{N}_{\eta})$  and  $\mathcal{D}_{\mathcal{KL}}(\mathcal{N}_{\eta}||\mathcal{N}_{\psi})$ . Please refer to Appendix C.3.
+
+The first term of  $\mathcal{L}_{\mathrm{info}}$  in Equation 11 maximizes mutual information between the representation  $\mathbf{z}_t$  and the one-step reward  $\mathbf{r}_t$ . This term in fact maximizes task-relevant information in the representation  $\mathbf{z}_t$ . The second term of  $\mathcal{L}_{\mathrm{info}}$  minimizes the conditional mutual information between the observation-action pair  $(\mathbf{o}_t, \mathbf{a}_t)$  and the representation  $\mathbf{z}_t$ . It represents that we can minimize irrelevant and redundant information in the representation  $\mathbf{z}_t$ . See Appendix C.3 for the derivation of Equation 11.
+
+Further, to estimate the mutual information  $\mathcal{I}(\mathbf{z}_t; \mathbf{k}_t)$  between  $\mathbf{z}_t$  and  $\mathbf{k}_t$  for a transition  $(\mathbf{o}, \mathbf{a}, \mathbf{r}) \sim \mathcal{B}$ , we can leverage the InfoNCE (van den Oord et al., 2018; Laskin et al., 2020) function. Thus, we have
+
+$$
+\mathcal {I} \left(\mathbf {z} _ {t}; \mathbf {k} _ {t}\right) = - \log \frac {\exp \left(\operatorname {s i m} \left(\mathbf {z} _ {t} , \mathbf {k} _ {t} ^ {+}\right)\right)}{\exp \left(\operatorname {s i m} \left(\mathbf {z} _ {t} , \mathbf {k} _ {t} ^ {+}\right)\right) + \sum_ {i = 1} ^ {N - 1} \exp \left(\operatorname {s i m} \left(\mathbf {z} _ {t} , \mathbf {k} _ {t , i}\right)\right)}, \tag {12}
+$$
+
+where  $\mathrm{sim}(\mathbf{x},\mathbf{y}) = \mathbf{x}^T\mathbf{y}$ $N$  is the batch size.
+
+TD-Based Loss for Robust Action-Value Representation Learning. As the robust action-value representation has a recursive nature, we employ the TD learning paradigm—the commonly used technique to learn an objective with a recursive form—to learn the representation.
+
+Based on Equations 2, 5, and 6, we derive a recursive form of the robust action-value representation.
+
+$$
+Z ^ {\pi} \left(\mathbf {o} _ {t}, \mathbf {a} _ {t}\right) = z \left(\mathbf {o} _ {t}, \mathbf {a} _ {t}\right) + \gamma \mathbb {E} _ {\pi , p} \left[ Z ^ {\pi} \left(\mathbf {o} _ {t + 1}, \mathbf {a} _ {t + 1}\right) \right], \tag {13}
+$$
+
+where  $Z^{\pi}$  and  $z$  subject to Equations 2 and 6, respectively.
+
+Then, we model the critic using the parameter  $\theta$  to estimate the action values. For robust action-value representation learning, we do not change critic's structure and directly use embeddings from critic's center layer as robust action-value representations (see Figure 1). Therefore, we also parameterize the robust action-value representations as  $Z_{\theta}$ . Based on Equation 13, we have the TD-based loss as:
+
+$$
+\mathcal {L} _ {\text {r o b u s t}} (\theta) = \mathbb {E} _ {\mathcal {B}} \left[ \left\| Z _ {\theta} \left(\mathbf {o} _ {t}, \mathbf {a} _ {t}\right) - \hat {Z} \left(\mathbf {o} _ {t}, \mathbf {a} _ {t}\right) \right\| _ {2} ^ {2} \right], \hat {Z} \left(\mathbf {o} _ {t}, \mathbf {a} _ {t}\right) = z _ {\psi} \left(\mathbf {o} _ {t}, \mathbf {a} _ {t}\right) + \gamma \mathbb {E} _ {\pi} \left[ Z _ {\bar {\theta}} \left(\mathbf {o} _ {t + 1}, \mathbf {a} _ {t + 1}\right) \right], \tag {14}
+$$
+
+where  $(\mathbf{o}_t,\mathbf{a}_t,\mathbf{r}_t,\mathbf{o}_{t + 1})\sim \mathcal{B}$ , and  $\bar{\theta}$  denotes the parameter of the target critic that are obtained as an exponentially moving average of the parameter  $\theta$  of the critic (Mnih et al., 2015). Moreover, if we apply the soft actor-critic framework (Haarnoja et al., 2018) that adds entropy of the policy to one-step rewards, we will also add this entropy into each dimension of compressed reward representations.
+
+According to Equation 14, we provide a theoretical guarantee for the robust action-value representation learning in the following theorem. It gives a bound between the true action-value function and the action-value function on top of learned robust action-value representations (see Appendix C.4).
+
+Theorem 4.4. Let  $Z$  be a learned robust action-value representation from any observation-action pair  $(o, a) \in \mathcal{O} \times \mathcal{A}$ ,  $Q_e^\pi : \mathcal{O} \times \mathcal{A} \to \mathbb{R}$  be the true action-value function of a policy  $\pi$  in the environment  $e \in \mathcal{E}$ ,  $f_e^* : \mathcal{Z} \to \mathbb{R}$  be the optimal linear mapping on the representation space, and  $\epsilon$  be a bound of estimation error for each compressed reward representation  $z$ , i.e.,  $|f_e^*(z) - r| \leq \epsilon$ . For any  $(o, a) \in \mathcal{O} \times \mathcal{A}$  and  $e \in \mathcal{E}$ , we have
+
+$$
+0 \leq \left| Q _ {e} ^ {\pi} (o, a) - f _ {e} ^ {*} (Z (o, a)) \right| \leq \frac {1}{1 - \gamma} \epsilon .
+$$
+
+This theoretical proof allows us to directly use embeddings of the critic's center layer as robust action-value representations for estimating action values. This shows that ROUSER can be integrated with various VRL methods. We combine ROUSER with traditional VRL methods in Section 5.2.
+
+In addition, to alleviate the overestimation bias of action values, Fujimoto et al. (2018) introduces double critics with  $\theta_{1}$  and  $\theta_{2}$ . We note that our representation learning based on Equation 14 also suffers from the overestimation bias as well as the action-value function approximation (Fujimoto et al., 2018). Thus, we propose to leverage the overestimation bias of robust action-value representations to alleviate the bias of estimated action values. We reformulate Equation 14 as:
+
+$$
+\mathcal {L} _ {\text {r o b u s t}} \left(\theta_ {1}, \theta_ {2}\right) = \mathbb {E} _ {\left(\mathbf {o} _ {t}, \mathbf {a} _ {t}, \mathbf {r} _ {t}, \mathbf {o} _ {t + 1}\right) \sim \mathcal {B}} \left[ \sum_ {i = 1, 2} \left\| Z _ {\theta_ {i}} \left(\mathbf {o} _ {t}, \mathbf {a} _ {t}\right) - \hat {Z} \left(\mathbf {o} _ {t}, \mathbf {a} _ {t}\right) \right\| _ {2} ^ {2} \right], \tag {15}
+$$
+
+$$
+\hat {Z} (\mathbf {o} _ {t}, \mathbf {a} _ {t}) = z _ {\psi} (\mathbf {o} _ {t}, \mathbf {a} _ {t}) + \gamma \mathbb {E} _ {\pi} \left[ \min  _ {i = 1, 2} \mathbb {I} Z _ {\bar {\theta} _ {i}} \left(\mathbf {o} _ {t + 1}, \mathbf {a} _ {t + 1}\right) \right],
+$$
+
+$$
+\min  _ {i = 1, 2} \mathbb {I} Z _ {\bar {\theta} _ {i}} \left(\mathbf {o} _ {t}, \mathbf {a} _ {t}\right) = \min  _ {i = 1, 2} \mathbb {I} _ {\omega^ {\prime} \geq 0} \cdot Z _ {\bar {\theta} _ {i}} \left(\mathbf {o} _ {t}, \mathbf {a} _ {t}\right) + \max  _ {i = 1, 2} \mathbb {I} _ {\omega^ {\prime} <   0} \cdot Z _ {\bar {\theta} _ {i}} \left(\mathbf {o} _ {t}, \mathbf {a} _ {t}\right), \tag {16}
+$$
+
+![](images/25dbc9599de9faa249e6c0303891ac6c557c8e3584132c316d764dc41f1b57ad.jpg)  
+Figure 1: The architecture of ROUSER. We first input  $(o_t, a_t, r_t)$  into the reward model to compute the reward loss, updating the reward model. Next, we input  $(o_t, a_t)$  into the critic model to compute the robust loss, which updates  $(\theta_1, \theta_2)$  to learn robust action-value representations. Moreover, we use traditional actor and critic losses to update the actor and the entire critic model, respectively.
+
+where  $\bar{\theta}_{1}$  and  $\bar{\theta}_{2}$  are parameters of target critics,  $\mathbb{I}_{\omega^{\prime} \geq 0}$  and  $\mathbb{I}_{\omega^{\prime} < 0}$  denote  $D$ -dimensional vectors of binary values from the indicator function  $\mathbb{I}$ , and  $\bar{\omega}^{\prime}$  is the parameter of the linear mapping  $f$  equivalent to  $\Phi$ . With the operator  $\min \mathbb{I}$ , we select minimums of robust action-value representations positively related to action values and maximums negatively related to action values. Thus, based on the linear relation between action values and robust action-value representations, we can alleviate the overestimation bias in the action-value estimation.
+
+To clearly overview our method, we provide the architecture in Figure 1. It illustrates our proposed two loss functions: the robust loss  $\mathcal{L}_{\mathrm{robust}}(\theta_1,\theta_2)$  and the reward loss  $\mathcal{L}_{\mathrm{pred}}(\psi ,\omega^{\prime},\omega^{\prime \prime}) + \mathcal{L}_{\mathrm{info}}(\psi ,\eta)$ . The detailed implementation of this architecture are presented in Appendix D.
+
+# 5 EXPERIMENTS
+
+We conduct extensive experiments to evaluate the generalization performance of ROUSER on testing environments with unseen visual distractions. Firstly, we demonstrate the effectiveness of ROUSER for solving the visual control tasks on DeepMind Control Suite (DMC, Tassa et al. (2018)) with task-irrelevant visual distractions, including background and color distractions. Then, we assess the applicability of ROUSER by combining it with traditional policy gradient and value-based VRL algorithms. Finally, we visualize the captured long-term information and evaluate the robustness of ROUSER. Moreover, we conduct careful ablation studies to show the effectiveness of ROUSER. We provide additional results of ROUSER in Appendix E.
+
+# 5.1 GENERALIZATION PERFORMANCE ON DMC BENCHMARK WITH CONTINUOUS ACTIONS
+
+We apply visual distractions in DMC benchmark, including color and background distractions (Stone et al., 2021), to investigate the generalization performance of VRL agents. For color distractions, we train the agents in a training environment with a weak color change and evaluate in various test environments with strong color changes. It is worth noting that the variance of the color distractions we use is larger than the hard setting proposed in Stone et al. (2021), which poses greater challenges for the agents. For background distractions, we follow the hard setting (Agarwal et al., 2021; Yang et al., 2022) to use 2 videos as backgrounds during training and evaluate the generalization performance on 30 unseen videos. See Appendix E.1 for more details. We conduct main experiments on six tasks of DMC. For each task, we train the agents with six random seeds and present means and standard errors of their cumulative rewards at 500,000 environment steps.
+
+We integrate ROUSER with DrQv2 (Yarats et al., 2022) and SRM (Huang et al., 2022) to evaluate the effectiveness of ROUSER on generalization. Note that in all experiments under both color and background distractions, ROUSER-DrQv2 consistently outperforms DrQv2, and ROUSER-SRM
+
+![](images/6b65542dba38832dea85d43b6dbd0a1f11f2576524a493cca5d282b896f5af18.jpg)
+
+![](images/0858681043cec2d699efb6af4ba331acc19db9c8dce489660fc9eb02c3789ffe.jpg)
+
+![](images/96cd07824f1c0d9b79715d03615c725a356db5423d45e44a6e7e24de91e20fd5.jpg)
+
+![](images/e5b246440bbf5ab73ea9888cc51501f94cfe3e2fc0b3230a90f8d21dc5bd7807.jpg)
+
+![](images/361190fb147a7ef5fa5b30bd852c257d4e3c2149c4991f4df56be5ec6ae1d7d1.jpg)
+
+![](images/317bd85c825d41edd944c30d35c565e43f317d971a8a3f2890f612f9aa5a3616.jpg)
+
+![](images/abd89c639674b6e5bbce3ea11a9aa7d5f1d845fcc4def2563036fb7178c0128f.jpg)
+
+![](images/612dd77e47d1d38d56b17688ce104461fa3fcd1901c9353e24c1571354bdb7b2.jpg)
+
+![](images/17b2115cea852a6db3f3be0630495b1d7a755df3bf57192b30fd0e618a3a1d17.jpg)
+
+![](images/299c7f0f9195c1bb57a04190f8c7224ecbb568015757a5dec3f4061b589082d8.jpg)
+
+![](images/40f09792c5c3617c597a3adfe0a153b4199b66bcba674f35eb70e8c9584edb5f.jpg)
+
+![](images/aa9fb7c7cdc621ea911c85a5df01f60db4fda43e074052f8628278ca40d416d7.jpg)
+
+![](images/a4fbed56ad064dbf8d923c8427a1d30eaf879bf7421cdba251d1dedc11e9a9c4.jpg)  
+Figure 2: We evaluate ROUSER on DMC with unseen color and video background distractions. Each result is averaged over 100 episodes at 500K environment steps using six random seeds.
+
+surpasses SRM. We also compare our approach ROUSER against several state-of-the-art (SOTA) methods: (1) CURL (Laskin et al., 2020), (2) RAP (Chen & Pan, 2022), (3) CRESP (Yang et al., 2022), (4) TACO (Zheng et al., 2024). See Appendix E.1 for detailed settings.
+
+Figure 2 illustrates results under color and background distractions, demonstrating that ROUSER outperforms other baselines in 11 out of 12 experiments. See Appendix E.1 for detailed results.
+
+It is worth noting that although ROUSER uses one-step rewards to learn robust action-value representations, it significantly improves generalization performance on tasks with sparse reward functions (e.g., ball_in_cup Catch, finger_spin, and reach easy tasks from DMC benchmark). Considering that the actor-critic algorithms (Fujimoto et al., 2018; Haarnoja et al., 2018)—which estimate action values using one-step rewards—also perform well on sparse reward tasks, we believe that our approach ROUSER is also effective in the contexts beyond sparse rewards.
+
+# 5.2 EXTENDING ROUSER TO PROCGEN WITH DISCRETE ACTIONS
+
+In this subsection, motivated by the result in Theorem 4.4, we extend ROUSER to decision tasks with discrete actions, assessing its applicability.
+
+Firstly, Figure 2 shows that ROUSER can enhance generalization when combined with policy gradient VRL algorithms, such as DrQv2 and SRM. Then, to demonstrate that ROUSER can be integrated with value-based methods, we conduct experiments in Procgen (Cobbe et al., 2020), a benchmark with image observations where each game has multiple levels to evaluate generalization. See Appendix E.2 for detailed settings. We integrate ROUSER with QR-DQN (Dabney et al., 2018), a prior SOTA value-based VRL method on Procgen for generaliz
+
+![](images/aa1fbc243041f38f1d22357a4abd36ea6d010f01f9c6c5a542b3d35dbcfe88a8.jpg)  
+Figure 3: Results on Procgen.
+
+Figure 3 shows Progen scores on test levels that are averaged over five random seeds with a batch size of 256 at  $25\mathrm{M}$  environment steps. The results demonstrate that ROUSER can also improve the generalization performance of agents when combined with the value-based VRL method for discrete control tasks. In addition, we provide the details of these results and the implementation for the combination of ROUSER and QR-DQN in Appendix E.2.
+
+![](images/de533274f62600304c49ae87bc6974ff9af0bc860adf12cc2e11a5bc944466da.jpg)  
+(a) Long-Term Information
+
+![](images/922e3feeb174e79c62106715a60879b8fabaf5ffd4223c7bafe0d2a6feb64283.jpg)  
+(b) Conditional IB Loss
+
+![](images/510d5c2fd1a0f4d7a7ae45111dced6fb7abd880272a7fdd8f7606aec355499aa.jpg)  
+(c) Noise in Rewards
+
+![](images/6c8eb609763929ec862f80947f2b05014e6b81929e8e5dd5448e3f38fdd4db34.jpg)  
+(d) Different Dimensions  
+Figure 4: We report four results on cheetah_run task with unseen color distractions.
+
+# 5.3 ANALYZING THE REPRESENTATIONS LEARNED BY ROUSER
+
+Long-Term Robust Information. We assess how ROUSER can capture long-term robust information. Specifically, we first collect 200K samples under unseen color distractions using a DrQv2's policy. Next, we compute the representations learned by each method from state-action pairs in the collected data. Such representations are extracted from the center layer of the critic in each method. Then, we fix these representations and input them into a 2-layer trainable MLP to predict the future reward sequences over a length of 300. During the prediction process, we use the Adam optimizer with a learning rate of 1e-4. As shown in Figure 4(a), we plot curves of the MSE loss on 1K evaluation samples across three seeds. The lower MSE losses of ROUSER indicate that its learned representations capture long-term information more effectively.
+
+![](images/1f82e5ae32c07a25e8b7bf2507defbb5d45cc0b896650d9eceb5484979607eab.jpg)  
+Figure 5: Results with different  $\alpha$  in different color scales.
+
+Effectiveness of IB. To assess the effectiveness of IB, we remove the conditional IB loss from the objectives of ROUSER. We then train agents to evaluate their generalization performance. Results (i.e., means and standard deviations) over three seeds in Figure 4(b) demonstrate the effectiveness of the conditional IB loss. Moreover, by tuning  $\alpha$  in the conditional IB loss from Equation 11, we find that larger values of  $\alpha$  result in smaller  $\mathcal{D}_{\mathcal{SK},\mathcal{L}}$ , potentially improving generalization performance. This observation aligns with our analysis in Section 4.3, demonstrating that minimizing  $\mathcal{D}_{\mathcal{SK},\mathcal{L}}$  (the second term of  $\mathcal{L}_{\mathrm{info}}$ ) effectively discards irrelevant and redundant information. We illustrate the results for different values of  $\alpha$  over three seeds in Figure 5. Each result is evaluated at  $200\mathrm{K}$  steps on cartpole-swingup task under color distractions (scale from 0.3 to 0.6, see Appendix E.5 for details).
+
+Noisy One-Step Rewards. We add noise drawn from  $\mathcal{N}(0,0.1)$  for each reward to show how ROUSER can be robust against noisy rewards. Figure 4(c) demonstrates that ROUSER-DrQv2 significantly outperforms DrQv2, remaining effective under the noisy reward setting.
+
+Ablation Study for Dimension. We illustrate results of ROUSER-DrQv2 for different dimensions of robust action-value representations in Figure 4(d). The results, averaged over three seeds on cheetah_run task with color distractions, show that ROUSER is not sensitive to the dimension  $D$ .
+
+# 6 CONCLUSION
+
+Many existing VRL approaches focus on learning robust representations against irrelevant visual distractions but often do not take into account the long-term decision-relevant information. This may degrade generalization performance of VRL agents. In this paper, we propose robust action-value representations in the IB framework, which encodes long-term information from action values while discarding irrelevant features from image observations during the robust representation learning process. Considering that action values are unknown, we propose to decompose robust action-value representations into infinite sequences associated with rewards. Thus, we can directly use known rewards to learn robust action-value representations. Experiments demonstrate that our approach significantly enhances generalization capabilities in the majority of tasks. We believe this method is general and applicable to other real-word applications, such as end-to-end autonomous driving.
+
+# REPRODUCIBILITY STATEMENT
+
+In this study, to ensure the reproducibility of our approach, we provide key information from the main text and Appendix as follows.
+
+1. Algorithm. We provide the architecture and pseudocode of our approach ROUSER in Appendix D. We also provide the detailed implementation of ROUSER in Appendices E.1 and E.2. See Appendix E.5 for the hyperparameters of ROUSER.  
+2. Source Code. According to the architecture and pseudocode in Appendix D, our approach ROUSER can directly build upon traditional VRL algorithms. Specifically, in Section 5.1, we follow Yuan et al. (2023) to use the provided code for DrQv2 and SRM, available at https://github.com/gemcollector/RL-ViGen. In Section 5.2, we apply the code from Wang et al. (2020) for QR-DQN, available at https://github.com/kaixin96/mixreg. Moreover, we implement our codes in Python version 3.7 and make the code available online<sup>1</sup>.  
+3. Experimental Details. We provide detailed experiment settings in Section 5.1, Appendices E.1, E.2, and E.3.  
+4. Theoretical Proofs. We provide all proofs in Appendix C.
+
+# ACKNOWLEDGMENTS
+
+The authors would like to thank all the anonymous reviewers for their valuable suggestions. This work was supported by the National Key R&D Program of China under contract 2022ZD0119801 and the National Nature Science Foundations of China grants U23A20388 and 62021001.
+
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+
+# APPENDIX
+
+# A RELATED WORK OF TASK DECOMPOSITION
+
+Value Decomposition. Hierarchical RL aims to solve complex tasks into layered sub-goals, enabling agents to tackle challenges at varying levels of abstraction for efficient and scalable learning (Wang et al., 2023b; 2024a; Ling et al., 2024; Kuang et al., 2024; Liu et al., 2024; 2025; 2023a). Some methods suggest that reusing learned skills from simple tasks can effectively improve the sample efficiency in solving the complex composite tasks (Sutton et al., 1999; Dietterich, 2000; Russell & Zimdars, 2003; Makino, 2023). Thus, for a complex sequential decision-making task that can be divided into a series of simple individual sub-tasks, these approaches estimate the action-value function for each sub-task to learn these reusable skills.
+
+Inspired by this hierarchical RL, some methods propose the reward decomposition for composite tasks, which decomposes the one-step rewards into a series of sub-rewards instead of breaking down the composite task (van Seijen et al., 2017; Lin et al., 2019; Grimm & Singh, 2019; Fatemi & Tavakoli, 2022). Specifically, under the assumption that one-step rewards of composite tasks are linear combinations of independent sub-rewards, these methods decompose each one-step reward either manually or automatically (i.e., leveraging deep learning techniques). Then, they combine the action values for each sub-task associated with each sub-reward additively to estimate the overall action-value function, thus efficiently addressing the complex composite task.
+
+In this article, although the reward representations extracted by ROUSER are similar to the sub-rewards from reward decomposition methods, they do not rely on the prior assumption that one-step rewards are linear combinations of independent sub-rewards.
+
+Successor Features. Successor features have garnered attention in the field of transfer RL, which allows flexible knowledge transfer across tasks with different reward functions. Barreto et al. (2017) assume a set of tasks with different reward functions but similar environment dynamics, and thus propose the successor features, which represent the features from environment dynamics involving the long-term sequential information. To learn such features, Barreto et al. (2017; 2018; 2020); Ma et al. (2020); Carvalho et al. (2023) first decouple rewards into the component associated with environment dynamics and the weight vector associated with different tasks. Then, they can use the reward component to learn successor features from the environment dynamics. By using the successor features and the weight vector, they can achieve the transfer learning. Moreover, based on successor features, Abdolshah et al. (2021) propose transferring knowledge across different environments by using a Gaussian distribution to capture the changes in environment dynamics.
+
+Our study differs from these methods, which do not take into account robustness against visual distractions. Also, the successor features learned by the aforementioned methods often involve irrelevant information from environment dynamics in the VRL generalization challenge. Instead, we use one-step rewards to extract long-term information from action values and leverage the IB framework to discard irrelevant features. It is worth NOTING that in all the aforementioned successor feature works, fine-tuning the successor features is essential when encountering unseen testing environments. In contrast, our approach focuses on zero-shot generalization without any fine-tuning.
+
+# B IMPACT STATEMENTS AND LIMITATIONS
+
+This paper proposes a novel approach called ROUSER to advance the field of generalization in VRL, enhancing the potential of agents in real-world applications. Although our primary focus is on technical innovation, we recognize the potential societal consequences of our work, as generalization in VRL has the potential to influence various domains such as autonomous driving and the financial field. We are committed to ethical research practices and attach great importance to the social implications and ethical considerations in the development of generalization research in VRL.
+
+It is essential for our approach to use the artificially designed rewards for robust action-value representation learning. We look forward to advancing the representation learning in VRL using inverse RL techniques that can derive the reward function from data. This will enable us to introduce a framework that supports autonomic iterative optimization. Moreover, our work may inspire some
+
+studies about the message invariance for message passing-based graph neural networks (Shi et al., 2023; 2025; 2024).
+
+# C PROOFS
+
+# C.1 PROOF OF THE MAPPING
+
+In this subsection, we prove that the action-value representation is linearly correlated with the infinite sequence of reward representations.
+
+Proof. Based on Equations 1, 3, and the definition of the action-value function, we have
+
+$$
+\begin{array}{l} Q ^ {\pi} \left(o _ {t}, a _ {t}\right) = \mathbb {E} _ {\pi , p} \left[ \sum_ {i = 0} ^ {\infty} \gamma^ {i} r \left(o _ {t + i}, a _ {t + i}\right) \right] (17) \\ = \mathbb {E} _ {\pi , p} \left[ \sum_ {i = 0} ^ {\infty} \gamma^ {i} \langle \omega^ {\prime}, h \left(o _ {t + i}, a _ {t + i}\right) \rangle \right] (18) \\ = \left\langle \omega^ {\prime}, \mathbb {E} _ {\pi , p} \left[ \sum_ {i = 0} ^ {\infty} \gamma^ {i} h \left(o _ {t + i}, a _ {t + i}\right) \right] \right\rangle (19) \\ = \left\langle \omega^ {\prime}, H ^ {\pi} \left(o _ {t}, a _ {t}\right) \right\rangle . (20) \\ \end{array}
+$$
+
+Therefore, we derive an action-value representation  $H^{\pi}(o_t, a_t)$ .
+
+![](images/0480db5d24f0965806f53bb789027a5edd1e39856d3a3f2823af2f9af4ed4a6e.jpg)
+
+# C.2 PROOF OF THE CONVERGENCE OF ACTION-VALUE REPRESENTATION LOSS
+
+In this section, we provide a detailed derivation of the convergence of action-value representation. ROUSER leverages the reward representations to introduce an auxiliary objective function with a recursive form. Similar to the TD-learning of action-value functions, we prove that Equations 4 and 5 can be reformulated as a contraction mapping  $\mathbf{T}$ . Here, we define  $Z^{\pi}$  as a generic function  $Z^{\pi}:\mathcal{O}\times \mathcal{A}\to \mathcal{Z}$ , and we define the mapping  $\mathbf{T}:\mathcal{Z}\rightarrow \mathcal{Z}$ .
+
+Theorem There exists a contraction mapping  $\mathbf{T}:\mathcal{Z}\to \mathcal{Z}$  defined as
+
+$$
+\left(\mathbf {T} Z ^ {\pi}\right) \left(o _ {t}, a _ {t}\right) = z \left(o _ {t}, a _ {t}\right) + \gamma \cdot \mathbb {E} _ {\pi , p} \left[ Z ^ {\pi} \left(o _ {t + 1}, a _ {t + 1}\right) \mid o _ {t}, a _ {t} \right].
+$$
+
+Proof. Let  $\left|Z^{\pi}(o_{t},a_{t})\right|_{d}$  denote the L1 value of the  $d$ th dimension of  $Z^{\pi}(o_{t},a_{t})$ , where  $1 \leq d \leq D$ . Due to the properties of contraction mappings (Sutton & Barto, 2018; Zhou et al., 2023b), we can apply the operator  $\mathbf{T}$  to compute the target auxiliary loss function until convergence in tabular settings.  $\mathbf{T}$  operator is a contraction in the sup-norm, which we provide as follows.
+
+$$
+\left| \left| \mathbf {T} Z _ {1} ^ {\pi} - \mathbf {T} Z _ {2} ^ {\pi} \right| \right| _ {\infty} \leq \gamma \cdot \left| \left| Z _ {1} ^ {\pi} - Z _ {2} ^ {\pi} \right| \right| _ {\infty}. \tag {21}
+$$
+
+We provide a detailed proof of Equation 21.
+
+$$
+\begin{array}{l} \left| \left| \mathbf {T} Z _ {1} ^ {\pi} - \mathbf {T} Z _ {2} ^ {\pi} \right| \right| _ {\infty} \\ = \max _ {d} | z + \gamma \cdot \mathbb {E} _ {\pi , p} \left[ Z _ {1} ^ {\pi} (o _ {t + 1}, a _ {t + 1}) | o _ {t}, a _ {t} \right] - z - \gamma \cdot \mathbb {E} _ {\pi , p} \left[ Z _ {2} ^ {\pi} (o _ {t + 1}, a _ {t + 1}) | o _ {t}, a _ {t} \right] | _ {d} \\ = \gamma \cdot \max _ {d} | \mathbb {E} _ {\pi , p} \left[ Z _ {1} ^ {\pi} (o _ {t + 1}, a _ {t + 1}) | o _ {t}, a _ {t} \right] - \mathbb {E} _ {\pi , p} \left[ Z _ {2} ^ {\pi} (o _ {t + 1}, a _ {t + 1}) | o _ {t}, a _ {t} \right] | _ {d} \\ = \gamma \cdot \max _ {d} | \mathbb {E} _ {\pi , p} \left[ Z _ {1} ^ {\pi} (o _ {t + 1}, a _ {t + 1}) - Z _ {2} ^ {\pi} (o _ {t + 1}, a _ {t + 1}) | o _ {t}, a _ {t} \right] | _ {d} \\ \leq \gamma \cdot \max  _ {d} \mathbb {E} _ {\pi , p} \left[ \left| Z _ {1} ^ {\pi} \left(o _ {t + 1}, a _ {t + 1}\right) - Z _ {2} ^ {\pi} \left(o _ {t + 1}, a _ {t + 1}\right) \right| _ {d} \mid o _ {t}, a _ {t} \right] \\ \leq \gamma \cdot \max_{\substack{d\\ o_{t + 1}\in \mathcal{O}\\ a_{t + 1}\in \mathcal{A}}}|Z_{1}^{\pi}(o_{t + 1},a_{t + 1}) - Z_{2}^{\pi}(o_{t + 1},a_{t + 1})|_{d} \\ = \gamma \cdot | | Z _ {1} ^ {\pi} - Z _ {2} ^ {\pi} | | _ {\infty}. \\ \end{array}
+$$
+
+![](images/f6c025e3f0e587550dc044eec3d079447e7a8cca8fb60dd19d88bc6d56a43413.jpg)
+
+Based on Equation 21, we can derive the convergence of auxiliary loss similar to action-value function estimation. So we can apply  $\mathbf{T}$  to compute the auxiliary loss until convergence in tabular settings.
+
+# C.3 INFORMATION BOTTLENECK
+
+Here, we provide a derivation of the formulas involved in the IB techniques in our ROUSER approach. We first list the mathematical formulas used in our proof.
+
+(P.1) KL Divergence
+
+$$
+\mathcal {D} _ {\mathcal {K L}} \left(p (\mathbf {x}) | | q (\mathbf {x})\right) = \mathbb {E} _ {\mathbf {x}} \left[ \log \frac {p (\mathbf {x})}{q (\mathbf {x})} \right]. \tag {22}
+$$
+
+(P.2) Mutual Information
+
+$$
+\mathcal {I} (\mathbf {x}; \mathbf {y}) = \mathbb {E} _ {\mathbf {x}, \mathbf {y}} \left[ \log \frac {p (\mathbf {x} , \mathbf {y})}{p (\mathbf {x}) p (\mathbf {y})} \right] = \mathcal {D} _ {\mathcal {K L}} \left(p (\mathbf {x}, \mathbf {y}) | | p (\mathbf {x}) p (\mathbf {y})\right). \tag {23}
+$$
+
+(P.3) Chain rule of Mutual Information
+
+$$
+\mathcal {I} (\mathbf {x}, \mathbf {y}; \mathbf {z}) = \mathcal {I} (\mathbf {y}; \mathbf {z}) + \mathcal {I} (\mathbf {x}; \mathbf {z} | \mathbf {y}). \tag {24}
+$$
+
+(P.4) Symmetry of Mutual Information
+
+$$
+\mathcal {I} (\mathbf {x}; \mathbf {y}) = \mathcal {I} (\mathbf {y}; \mathbf {x}). \tag {25}
+$$
+
+Then, for Equation 6, we can leverage (P.2) to expand the computation of mutual information and conditional mutual information as follows.
+
+$$
+\begin{array}{l} \mathcal {I} (\mathbf {r}; \mathbf {z}) = \mathbb {E} _ {\mathbf {r}, \mathbf {z}} \left[ \log \frac {p (\mathbf {r} , \mathbf {z})}{p (\mathbf {r}) p (\mathbf {z})} \right] \\ = \mathbb {E} _ {\mathbf {r}, \mathbf {z}} \left[ \log \frac {p (\mathbf {z}) p (\mathbf {r} | \mathbf {z})}{p (\mathbf {r}) p (\mathbf {z})} \right] \tag {26} \\ = \mathbb {E} _ {\mathbf {r}, \mathbf {z}} \left[ \log \frac {p (\mathbf {r} | \mathbf {z})}{p (\mathbf {r})} \right]. \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \mathcal {I} (\mathbf {u}; \mathbf {z} | \mathbf {r}) = \mathbb {E} _ {\mathbf {u}, \mathbf {z}, \mathbf {r}} \left[ \log \frac {p (\mathbf {u} , \mathbf {z} | \mathbf {r})}{p (\mathbf {u} | \mathbf {r}) p (\mathbf {z} | \mathbf {r})} \right] \\ = \mathbb {E} _ {\mathbf {u}, \mathbf {z}, \mathbf {r}} \left[ \log \frac {p (\mathbf {u} | \mathbf {r}) p (\mathbf {z} | \mathbf {u} , \mathbf {r})}{p (\mathbf {u} | \mathbf {r}) p (\mathbf {z} | \mathbf {r})} \right] \tag {27} \\ = \mathbb {E} _ {\mathbf {u}, \mathbf {z}, \mathbf {r}} \left[ \log \frac {p (\mathbf {z} | \mathbf {u} , \mathbf {r})}{p (\mathbf {z} | \mathbf {r})} \right] \\ \stackrel {*} {=} \mathbb {E} _ {\mathbf {u}, \mathbf {z}, \mathbf {r}} \left[ \log \frac {p (\mathbf {z} | \mathbf {u})}{p (\mathbf {z} | \mathbf {r})} \right]. \\ \end{array}
+$$
+
+Here in  $*$ , since  $\mathbf{z}$  is the representation of  $\mathbf{u}$ , we have  $p(\mathbf{z}|\mathbf{u},\mathbf{r}) = p(\mathbf{z}|\mathbf{u})$ . As discussed in Equation 8, we can introduce a new feature  $\mathbf{k}$  to provide an upper bound to estimate the conditional mutual information. The details of the proof in Equation 8 are as follows,
+
+$$
+\begin{array}{l} \mathcal {I} (\mathbf {u}; \mathbf {z} | \mathbf {r}) \\ = \mathbb {E} _ {\mathbf {u}, \mathbf {z}, \mathbf {r}} \left[ \log \frac {p (\mathbf {z} | \mathbf {u})}{p (\mathbf {z} | \mathbf {r})} \right] \\ = \mathbb {E} _ {\mathbf {u}, \mathbf {z}, \mathbf {r}} \left[ \log \frac {p (\mathbf {z} | \mathbf {u})}{p (\mathbf {k} | \mathbf {r})} \frac {p (\mathbf {k} | \mathbf {r})}{p (\mathbf {z} | \mathbf {r})} \right] \\ = \mathbb {E} _ {\mathbf {u}, \mathbf {z}, \mathbf {r}} \left[ \log \frac {p (\mathbf {z} \mid \mathbf {u})}{p (\mathbf {k} \mid \mathbf {r})} \right] - \mathbb {E} _ {\mathbf {u}, \mathbf {z}, \mathbf {r}} \left[ \log \frac {p (\mathbf {z} \mid \mathbf {r})}{p (\mathbf {k} \mid \mathbf {r})} \right] (28) \\ = \mathbb {E} _ {\mathbf {r} | \mathbf {u}, \mathbf {z}} \left(\mathbb {E} _ {\mathbf {u}, \mathbf {z}} \left[ \log \frac {p (\mathbf {z} | \mathbf {u})}{p (\mathbf {k} | \mathbf {r})} \right]\right) - \mathbb {E} _ {\mathbf {u} | \mathbf {z}, \mathbf {r}} \left(\mathbb {E} _ {\mathbf {z}, \mathbf {r}} \left[ \log \frac {p (\mathbf {z} | \mathbf {r})}{p (\mathbf {k} | \mathbf {r})} \right]\right) (26) \\ = \mathbb {E} _ {\mathbf {u}, \mathbf {z}} \left[ \log \frac {p (\mathbf {z} | \mathbf {u})}{p (\mathbf {k} | \mathbf {r})} \right] - \mathbb {E} _ {\mathbf {z}, \mathbf {r}} \left[ \log \frac {p (\mathbf {z} | \mathbf {r})}{p (\mathbf {k} | \mathbf {r})} \right] \\ = \mathcal {D} _ {\mathcal {K L}} \left(p (\mathbf {z} | \mathbf {u}) | | p (\mathbf {k} | \mathbf {r})\right) - \mathcal {D} _ {\mathcal {K L}} \left(p (\mathbf {z} | \mathbf {r}) | | p (\mathbf {k} | \mathbf {r})\right) \\ \leq \mathcal {D} _ {\mathcal {K L}} \Big (p (\mathbf {z} | \mathbf {u}) | | p (\mathbf {k} | \mathbf {r}) \Big) = \mathcal {D} _ {\mathcal {K L}} \Big (\mathcal {N} _ {\psi} | | \mathcal {N} _ {\eta} \Big). \\ \end{array}
+$$
+
+Since the features  $\mathbf{z} \sim \mathcal{N}_{\psi}$  and  $\mathbf{k} \sim \mathcal{N}_{\eta}$ , we can directly use the Gaussian distribution  $\mathcal{N}$  to replace the probability density function  $p$ . Note that the equality holds if the two distributions coincide. Similar to  $\mathcal{I}(\mathbf{u}; \mathbf{z}|\mathbf{r}), \mathcal{I}(\mathbf{r}; \mathbf{k}|\mathbf{u})$  is upper bounded by  $\mathcal{D}_{\mathcal{KL}}\big(\mathcal{N}_{\eta}||\mathcal{N}_{\psi}\big)$ .
+
+We can also leverage  $\mathbf{k}$  to provide a lower bound for the mutual information:
+
+$$
+\mathcal {I} (\mathbf {r}; \mathbf {z}) \stackrel {\text {(P . 3)}} {=} \mathcal {I} (\mathbf {k}, \mathbf {r}; \mathbf {z}) - \mathcal {I} (\mathbf {z}; \mathbf {k} | \mathbf {r}) \tag {29}
+$$
+
+$$
+\stackrel {*} {=} \mathcal {I} (\mathbf {k}, \mathbf {r}; \mathbf {z}) \tag {30}
+$$
+
+$$
+\stackrel {(\mathrm {P . 3})} {=} \mathcal {I} (\mathbf {k}; \mathbf {z}) + \mathcal {I} (\mathbf {z}; \mathbf {r} | \mathbf {k}) \tag {31}
+$$
+
+$$
+\geq \mathcal {I} (\mathbf {k}; \mathbf {z}). \tag {32}
+$$
+
+Since  $\mathbf{k}$  is the representation of  $\mathbf{r}$ , we have  $\mathcal{I}(\mathbf{z};\mathbf{k}|\mathbf{r}) = 0$ , and we thus derive the equality with  $*$  of Equation 30. Moreover, the equality in Equation 32 holds if  $\mathcal{I}(\mathbf{z};\mathbf{r}|\mathbf{k}) = 0$ , i.e., the representation  $\mathbf{k}$  captures all the information associated with the reward  $\mathbf{r}$ .
+
+Based on Equation 9, we can further derive the whole optimization objective  $\mathcal{L}_{\mathrm{cib}}$  as:
+
+$$
+\begin{array}{l} \mathcal {L} _ {\mathrm {c i b}} = \left[ \mathcal {I} (\mathbf {u}; \mathbf {z} | \mathbf {r}) + \mathcal {I} (\mathbf {r}; \mathbf {k} | \mathbf {u}) \right] - \beta \mathcal {I} (\mathbf {r}; \mathbf {z}) \\ \leq \mathcal {D} _ {\mathcal {K L}} \left(\mathcal {N} _ {\psi} | | \mathcal {N} _ {\eta}\right) + \mathcal {D} _ {\mathcal {K L}} \left(\mathcal {N} _ {\eta} | | \mathcal {N} _ {\psi}\right) - \beta \mathcal {I} (\mathbf {k}; \mathbf {z}) \tag {33} \\ = \mathcal {D} _ {\mathcal {S K L}} \left(\mathcal {N} _ {\psi} | | \mathcal {N} _ {\eta}\right) - \beta \mathcal {I} (\mathbf {k}; \mathbf {z}). \\ \end{array}
+$$
+
+By reparameterizing the objective, we can derive the final loss function:
+
+$$
+\mathcal {L} _ {\text {i n f o}} (\psi , \eta) = \mathbb {E} _ {\mathcal {B}} \left[ - \mathcal {I} (\mathbf {z}; \mathbf {k}) + \alpha \mathcal {D} _ {\mathcal {S K L}} \left(\mathcal {N} _ {\psi} | | \mathcal {N} _ {\eta}\right) \right]. \tag {34}
+$$
+
+# C.4 PROOF OF THE ACTION VALUE BOUND
+
+We provide the detailed proof of the Theorem 4.4.
+
+Theorem 4.4. Let  $Z$  be a learned robust action-value representation from any observation-action pair  $(o, a) \in \mathcal{O} \times \mathcal{A}$ ,  $Q_{e}^{\pi} : \mathcal{O} \times \mathcal{A} \to \mathbb{R}$  be the true action-value function of a policy  $\pi$  in the environment  $e \in \mathcal{E}$ ,  $f_{e}^{*} : \mathcal{Z} \to \mathbb{R}$  be the optimal linear mapping on the representation space, and  $\epsilon$  be a bound of estimation error for each compressed reward representation  $z$ , i.e.,  $|f_{e}^{*}(z) - r| \leq \epsilon$ . For any  $(o, a) \in \mathcal{O} \times \mathcal{A}$  and  $e \in \mathcal{E}$ , we have
+
+$$
+0 \leq \left| Q _ {e} ^ {\pi} (o, a) - f _ {e} ^ {*} (Z (o, a)) \right| \leq \frac {1}{1 - \gamma} \epsilon .
+$$
+
+Proof.
+
+$$
+\begin{array}{l} \left\| Q _ {e} ^ {\pi} (o, a) - f _ {e} ^ {*} \big (Z (o, a) \big) \right\| _ {1} = \left\| \mathbb {E} _ {\pi^ {*}} ^ {e} \left[ \sum_ {t = 0} ^ {\infty} \gamma^ {t} r _ {t} \right] - \mathbb {E} _ {\pi^ {*}} ^ {e} \left[ \sum_ {t = 0} ^ {\infty} \gamma^ {t} f _ {e} ^ {*} (z) \right] \right\| _ {1} \\ = \left\| \mathbb {E} _ {\pi^ {*}} ^ {e} \left[ \sum_ {t = 0} ^ {\infty} \gamma^ {t} \left(r _ {t} - f _ {e} ^ {*} (z)\right) \right] \right\| _ {1} \\ \leq \mathbb {E} _ {\pi^ {*}} ^ {e} \left[ \sum_ {t = 0} ^ {\infty} \gamma^ {t} \| r _ {t} - f _ {e} ^ {*} (z) \| _ {1} \right] \\ \leq \mathbb {E} _ {\pi^ {*}} ^ {e} \left[ \sum_ {t = 0} ^ {\infty} \gamma^ {t} \cdot \epsilon \right] \\ = \frac {1}{1 - \gamma} \epsilon . \\ \end{array}
+$$
+
+![](images/425a3bd9acb140428159ed9afbbacaf5bb8b37b5edc7fa04fddd7a271ec5d69c.jpg)
+
+# D ARCHITECTURE AND PSEUDOCODE
+
+As we directly use the embeddings from critic's center layer as robust action-value representations, we do not need to change the architecture of commonly used VRL algorithms. Specifically, we follow the traditional VRL actor-critic architecture to use a 3-layer feed-forward ConvNet with no residual connection as the encoder. Then, we apply a 3-layer MLP with hidden size 1024 as each critic, and we use Equation 15 to regulate the embeddings of the second layer of each critic. The actor uses the same architecture as the critic. Moreover, we use an additional 2-layer MLP with hidden size 1024 as the reward model  $\psi$  to output the compressed reward representations  $\mathbf{z}$ , as well as  $\eta$  to output the representations  $\mathbf{k}$  from one-step rewards. As the inputs are image observations, the reward model  $\psi$  is after the shared encoder for the actor and critic. We provide the architecture and the algorithm of ROUSER in Figure 1 and Algorithm 1, respectively.
+
+Algorithm 1 ROUSER in a general actor-critic framework  
+Initialize the critic network  $Q_{\theta}$  with parameters  $\theta$  actor network  $\pi_{\phi}$  with parameters  $\phi$  reward model  $\psi$  , and the linear mapping  $f$    
+Initialize target network  $Q_{\bar{\theta}}$  with weight  $\bar{\theta}\gets \theta$    
+Initialize the replay buffer  $\mathcal{B}$    
+for  $t = 1,\dots ,T$  do if  $t\leq T_0$  then Randomly select  $a$  under  $o$  Execute  $a$  to obtain  $r$  and  $o^{\prime}$  else  $a\sim \pi_{\phi}(\cdot |o)$ $o^\prime \sim p(o^\prime |o,a)$  Sample a batch  $\{o,a,r,o'\}$  from  $\mathcal{B}$  Update:  $f\gets f - \lambda_f\nabla_f\mathcal{L}_{\mathrm{pred}}$  linear network  $\psi \gets \psi -\lambda_{\psi}\nabla_{\psi}\bigl (\mathcal{L}_{\mathrm{pred}} + \mathcal{L}_{\mathrm{info}}\bigr)$  reward network  $\theta \leftarrow \theta -\lambda_{\theta}\nabla_{\theta}(\mathcal{L}_{\mathrm{robust}} + \mathcal{L}_{\mathrm{critic}})$  critic network  $\phi \gets \phi -\lambda_{\phi}\nabla_{\phi}\mathcal{L}_{\mathrm{actor}}$  actor network end if  $\mathcal{B}\gets \mathcal{B}\cup \{o,a,r,o'\}$    
+end for
+
+# EXPERIMENTS
+
+# E.1 EXPERIMENTS ON DMC
+
+**Environment Setting.** Color Distractions. On DMC benchmark, we apply the similar treatment of dynamic color distraction (Stone et al., 2021) to the objects of the environments. Specifically, we uniformly sample the color  $x_0 \sim \mathcal{U}(x - \beta, x + \beta)$  for each channel at the start of each episode, where  $x$  is the origin color in DCS, and  $\beta$  is a hyperparameter. We leverage a dynamic setting where the color  $x_t$  can change to  $x_{t+1} = \text{clip}(\widehat{x}_{t+1}, x_t - \beta, x_t + \beta)$ , where  $\widehat{x}_{t+1} \sim \mathcal{N}(x_t, 0.03 \cdot \beta)$ . We train the agents on the environment with weak dynamic color distractions ( $\beta = 0.2$ ). Then, we evaluate all agents in the test environments with strong dynamic color distractions ( $\beta = 0.5$ ).
+
+Background Distractions. We adopt the dynamic background settings from Stone et al. (2021). To establish different training environments, we utilize  $N$  videos from the DAVIS 2017 training set, where  $N$  represents the number of training environments. Each environment uses one video as the background and randomly samples a scene and a frame from the video at the start of every episode. Additionally, we set  $\beta_{\mathrm{bg}} = 1.0$ , indicating that we use the distracting background instead of the original skybox. For evaluation, we apply 30 videos from the DAVIS 2017 validation dataset as the unseen backgrounds. In each episode of the test environment, we randomly select one of the 30 dynamic backgrounds.
+
+![](images/03d6f2eaf447a2fe68c0e463726fb13a364148e22f94fcec4f906e8f252f1d8d.jpg)
+
+![](images/554db3558c11a1ff76cf75d6c9607a2ee1037cd0688d47daf86d855d1e5faa04.jpg)  
+Figure 6: The dynamics color and background distractions using in Section 5.1. The first row illustrates the dynamic color changes, and the second row shows one of dynamic backgrounds.
+
+![](images/0be7c033c085285cd3fbc24193c2b9a4e9a238e4c4342bf516ca8bd93e871c5d.jpg)
+
+![](images/a38fcc9b6b795a9551f99db2fb9be9ee470d79dff19c4fe43a67d0fa7e9b6f41.jpg)
+
+![](images/ad4ec49e0b6a69b72bb88c0d3b58f27093c84db3d12d1233d0deca60b8633a17.jpg)
+
+![](images/df8d4284460313fa874c8de341cc5218cee28c5325237b1571b6a33f67e9077a.jpg)
+
+![](images/0dc05fbd9b2942953dd4a3285b1125f31805d29335f937067e94312723ce217c.jpg)
+
+![](images/0393b37eb4b0e8612f57201d2361bc018548395ef9470e0f5f203dc985751dd6.jpg)
+
+![](images/e8bdc4b7f0c2e00cb258be49d15209bb148e9e9cb836ad7d283d5a03aa522604.jpg)  
+Figure 7: On cheetah_run task under color changes, the left of this figure shows the curves of MSE loss for capturing long-term information, and the right illustrates the performance of weight  $c$ .
+
+![](images/00b2a900c72e53d0768b62d7b8fa11c18060a8c352233a70c8c322f086e0e577.jpg)
+
+Network. For the shared encoder, we use the default setting of baselines combined with our approach. For example, in ROUSER-SRM, we use a shared pixel encoder with 11 convolutional layers to extract the image information, and employ  $3 \times 3$  kernels and 32 filters with a stride of 2 for the first convolutional layer and 1 for others. After each convolution, following the commonly used setting of VRL methods (e.g., DrQ, DrQv2, and SRM), we apply a fully connected layer to output 50-dimensional representations normalized by LayerNorm and a tanh activation. Note that the actor and critic each have a fully connected layer.
+
+After the fully connected layer, the actor network is parameterized with 3 fully connected layers using ReLU activations up until the last layer. The critic network employs Double Q-learning technique, where each Q-value is learned using a 3-layer fully connected network, similar to the one used in the actor network. The output dimension of these hidden layers in the actor and the critic network is 1024. Moreover, the gradients of the shared pixel encoder are computed through the critic's optimizer rather than the actor's.
+
+The reward model also has a fully connected layer to output 50-dimensional representations normalized by LayerNorm and a tanh activation from the outputs of the shared encoder. After the fully connected layer, the reward model has a 3-layer MLP. The output dimension of these hidden layers in the reward model is also 1024. The final outputs of the reward model is twice the dimension of compressed reward representations, half of which is the mean and half of which is the variance. Thus, we can use the outputs to model the Gaussian distribution of compressed reward representations. Then, we use a linear layer, mapping compressed reward representations into the corresponding one-step rewards. Moreover, we use a linear layer to get the representations  $\mathbf{k}$  from one-step rewards.
+
+![](images/e048382ba3034ad7df84d31151d56c76901ff13c72696d34bf79b1cd4e7343dd.jpg)
+
+![](images/394e45df2b705943a18a3f601acf455e192a7641a5f7be87ec876a4592c0e600.jpg)
+
+![](images/56a55d8adb667bc1295506e189a0b14a4e57c8e4964141e60d78db2f5bf566d1.jpg)
+
+![](images/0fbb6834325216a1e0c670140eb0f21a944088b1c61f037255eda1c294da4702.jpg)
+
+![](images/95ab82934183b445c7dda831be47ccc9c809d812e0cd87c92319768be6212cdf.jpg)
+
+![](images/314443e4146b07d3c18867d4e5d62e1314d0a05978c0e636aa94721b6bba118c.jpg)
+
+![](images/e759cd3bbf4682058b5d171bd1b3486802e4d4b25f0aa2b619cf5272424642a2.jpg)
+
+![](images/fd550f9b5eda38c63a2cf3a75de042c431d5a2c6ef9145ea58ec3b332acb2caf.jpg)
+
+![](images/2088768d642c32b15256bef9692fba0aa36825ccd7b917d58e1726bb487d127f.jpg)
+
+![](images/81fc1e1cbe12ac6e9146b3b401d1952881c06e7066f7ce91293c4a3d1344dfde.jpg)
+
+![](images/974b62473b95d8ff1b1fdd09d7c659e2f930cb0be7ac88cba394c491760fb2e8.jpg)
+
+![](images/589e1c267943bb5e52f2b45dcc5004bd7e5f388c88e8450438412f9a46f4a21b.jpg)
+
+![](images/27e61e3053a9ca816b75b2ba892bba6d95d74c8447b5c69c4f5d8b30899ee7ed.jpg)  
+Figure 8: Training curves over six seeds on DMC benchmark with color and video background distractions. Each point from the curves is evaluated over ten episodes on test environments with unseen distractions.
+
+**Implementations.** In the implementation of VRL, the multi-step reward is a commonly used trick without any theoretical guarantee:  $r(o_{t},a_{t}) = \sum_{i = 0}^{N}\gamma^{i}\hat{r} (o_{t + i},a_{t + i})$ , where  $\hat{r} (\cdot ,\cdot)$  is the raw reward function. Many VRL methods (Yarats et al., 2022; Huang et al., 2022; Hessel et al., 2018) use  $N = 3$ . In our approach, we do not change the default setting when combined with baselines. It is worth noting that we change the mode of our reward model. Specifically, we map the outputs of reward model into the  $N$  raw rewards instead of the multi-step reward, thus learning the compressed reward representations. Unlike CRESP, which predicts reward sequence distributions to update the shared encoder for representation learning, our reward model does not update the shared encoder with this mapping, although it establishes a mapping from the shared encoder's outputs to the multi-step rewards. This is because that we believe the multi-step rewards with length of  $N = 3$  do not involve much long-term robust information for sequential decision-making. We demonstrate this in the left of Figure 7. The experiment setting for this figure is the same as in Figure 4 (a). We use 200K collected data to learn a 2-layer MLP projector that maps the fixed representations from CRESP and ROUSER-DrQv2 to the long-term sequential rewards with the length of 300. The curves demonstrate that ROUSER can effectively capture more long-term robust information than CRESP.
+
+Compute Resource. We use NVIDIA GeForce RTX 3090 GPUs for six tasks on DMC benchmark under backgrounds or color changes. Trials of DrQv2, SRM, ROUSER-DrQv2, and ROUSER-SRM
+
+Table 1: Means and standard errors on six DMC environments with unseen dynamic color changes. Each result is averaged over 100 episodes at 500K environment steps using six random seeds.  
+
+<table><tr><td>ALGO</td><td>ball_in_cup Catch</td><td>cartpole_swingup</td><td>cheetah_run</td><td>finger_spin</td><td>walker_walk</td><td>reacher_easy</td></tr><tr><td>ROUSER-SRM</td><td>942.6 ± 12.3</td><td>683.5 ± 37.8</td><td>370.0 ± 25.1</td><td>853.4 ± 25.6</td><td>814.0 ± 19.2</td><td>754.8 ± 39.7</td></tr><tr><td>ROUSER-DrQv2</td><td>780.9 ± 60.3</td><td>461.4 ± 15.6</td><td>371.6 ± 23.7</td><td>424.2 ± 49.8</td><td>417.7 ± 22.7</td><td>154.6 ± 52.8</td></tr><tr><td>SRM</td><td>826.0 ± 125.6</td><td>640.2 ± 15.8</td><td>299.5 ± 23.4</td><td>779.7 ± 46.8</td><td>776.5 ± 27.8</td><td>659.4 ± 18.4</td></tr><tr><td>DrQv2</td><td>632.1 ± 49.5</td><td>377.0 ± 24.7</td><td>140.2 ± 10.2</td><td>234.0 ± 54.9</td><td>115.3 ± 41.8</td><td>93.7 ± 20.4</td></tr><tr><td>CRESP</td><td>584.8 ± 57.4</td><td>410.6 ± 15.2</td><td>241.5 ± 87.3</td><td>574.0 ± 204.2</td><td>394.1 ± 65.0</td><td>270.9 ± 17.6</td></tr><tr><td>CURL</td><td>339.2 ± 51.5</td><td>205.5 ± 31.5</td><td>173.3 ± 46.6</td><td>279.2 ± 27.6</td><td>278.2 ± 46.3</td><td>198.2 ± 37.4</td></tr><tr><td>TACO</td><td>360.5 ± 192.8</td><td>385.9 ± 26.3</td><td>138.7 ± 17.2</td><td>130.8 ± 45.7</td><td>177.8 ± 31.8</td><td>122.9 ± 28.2</td></tr><tr><td>RAP</td><td>149.2 ± 34.2</td><td>304.4 ± 44.4</td><td>90.8 ± 57.6</td><td>1.5 ± 0.5</td><td>86.5 ± 62.5</td><td>209.8 ± 59.7</td></tr></table>
+
+Table 2: Means and standard errors on six DMC environments with unseen dynamic backgrounds. Each result is averaged over 100 episodes at 500K environment steps using six random seeds.  
+
+<table><tr><td>ALGO</td><td>ball_in_cupCatch</td><td>cartpole_swingup</td><td>finger_spin</td><td>cheetah_run</td><td>walker_walk</td><td>reacher_easy</td></tr><tr><td>ROUSER-SRM</td><td>868.9 ± 47.0</td><td>822.4 ± 10.8</td><td>776.8 ± 24.0</td><td>372.9 ± 10.8</td><td>894.0 ± 12.6</td><td>901.5 ± 18.2</td></tr><tr><td>ROUSER-DrQv2</td><td>523.1 ± 67.5</td><td>731.9 ± 10.5</td><td>348.2 ± 174.1</td><td>408.0 ± 22.1</td><td>358.6 ± 74.7</td><td>440.0 ± 70.9</td></tr><tr><td>SRM</td><td>818.1 ± 47.6</td><td>807.7 ± 14.5</td><td>527.5 ± 27.2</td><td>333.8 ± 22.9</td><td>823.1 ± 17.5</td><td>889.2 ± 14.2</td></tr><tr><td>DrQv2</td><td>474.3 ± 72.6</td><td>717.8 ± 15.0</td><td>309.7 ± 113.8</td><td>359.2 ± 19.2</td><td>237.7 ± 106.9</td><td>404.5 ± 32.9</td></tr><tr><td>CRESP</td><td>665.3 ± 75.5</td><td>689.8 ± 20.0</td><td>778.1 ± 62.9</td><td>327.4 ± 20.1</td><td>794.1 ± 33.9</td><td>667.7 ± 33.5</td></tr><tr><td>CURL</td><td>167.0 ± 57.4</td><td>329.6 ± 18.4</td><td>745.9 ± 31.8</td><td>185.0 ± 15.9</td><td>746.1 ± 16.7</td><td>714.9 ± 33.1</td></tr><tr><td>TACO</td><td>816.1 ± 42.6</td><td>726.6 ± 67.4</td><td>737.2 ± 99.4</td><td>64.3 ± 59.5</td><td>505.4 ± 151.2</td><td>198.8 ± 51.2</td></tr><tr><td>RAP</td><td>76.1 ± 14.6</td><td>170.8 ± 21.7</td><td>6.9 ± 4.0</td><td>96.7 ± 12.4</td><td>115.1 ± 5.7</td><td>150.9 ± 16.4</td></tr></table>
+
+on `reacher_easy` task from DMC under dynamic backgrounds are trained for 12.24, 10.83, 16.93, and 15.83 hours on average. Moreover, the agents of ROUSER-DrQv2 and ROUSER-SRM require approximately 2800MB and 3600MB of memory using the batch size of 256, respectively.
+
+Results. We compare our approach ROUSER against several SOTA method: (1) CURL (Laskin et al., 2020), which leverages contrastive learning to maximize the mutual information between representations from observations and augmentations. (2) DrQv2 (Yarats et al., 2022), which is the prior state-of-the-art DRL algorithm for sample efficiency. (3) SRM (Huang et al., 2022), which adopts augmentation in the frequency domain to facilitate the learning of robust policies. (4) RAP (Chen & Pan, 2022), which effectively enhances the robustness of representations by leveraging the behavioral similarity. (5) CRESP (Yang et al., 2022), which predicts the characteristic function of reward sequences to learn task-relevant representations for generalization. (6) TACO (Zheng et al., 2024), which learns state and action representations that encompass sufficient information for control to improve sample efficiency. Moreover, we illustrate image observations from the environments with dynamic color and background distractions in Figure 6.
+
+We provide the means and standard errors at 500K environment steps in Tables 1 and 2. Note that Table 1 lists the detailed results of Figure 2 under unseen dynamic color changes, and Table 2 lists the detailed results of Figure 2 under unseen dynamic video backgrounds. As shown in Tables 1 and 2, ROUSER outperforms other baselines in 11 out of 12 experiments. Even in finger_spin task under dynamic backgrounds, our approach ROUSER is almost on par with the highest performance. In addition, we illustrate the training curves over six seeds in Figure 8.
+
+Analysis of learned Representations for Capturing Long-Term Robust Information. In Section 5.3, we present results from a task where learned representations are used to predict the average of future reward sequences over a length of 300. These results indicate the effectiveness of ROUSER in capturing long-term information.
+
+Moreover, we conduct additional experiments with varying reward sequence lengths, including 50, 100, 300, and 500. We report the results in Table 3, which are averaged over three seeds on cheetah_run task with unseen color distractions. These results, averaged over three seeds on cheetah_run task with unseen color distractions, demonstrate that the representations learned by ROUSER are able to accurately predict reward sequences of different lengths, with performance improving as the reward sequences become longer.
+
+<table><tr><td>Reward Sequence</td><td>ROUSER-SRM</td><td>SRM</td><td>ROUSER-DrQv2</td><td>DrQv2</td></tr><tr><td>Length=50</td><td>0.0169 ± 0.0051</td><td>0.0170 ± 0.0056</td><td>0.0169 ± 0.0053</td><td>0.0173 ± 0.0029</td></tr><tr><td>Length=100</td><td>0.0213 ± 0.0070</td><td>0.0210 ± 0.0069</td><td>0.0205 ± 0.0065</td><td>0.0211 ± 0.0004</td></tr><tr><td>Length=300</td><td>0.0244 ± 0.0122</td><td>0.0268 ± 0.0179</td><td>0.0239 ± 0.0182</td><td>0.0283 ± 0.0029</td></tr><tr><td>Length=500</td><td>0.0294 ± 0.0091</td><td>0.0383 ± 0.0004</td><td>0.0287 ± 0.0087</td><td>0.0319 ± 0.0011</td></tr></table>
+
+Table 3: MSE results and standard deviations with different reward sequence lengths over three seeds, including 50, 100, 300, and 500.  
+
+<table><tr><td>Game</td><td>PPO</td><td>MixReg</td><td>QR-DQN</td><td>ROUSER-QRDQN</td></tr><tr><td>bigfish</td><td>3.7 ± 1.3</td><td>7.1 ± 1.6</td><td>12.2 ± 1.1</td><td>13.6 ± 5.2</td></tr><tr><td>caveflyer</td><td>5.1 ± 0.4</td><td>6.1 ± 0.6</td><td>6.0 ± 0.7</td><td>6.4 ± 0.4</td></tr><tr><td>dodgeball</td><td>1.6 ± 0.1</td><td>1.7 ± 0.4</td><td>7.2 ± 0.2</td><td>8.0 ± 0.4</td></tr></table>
+
+Table 4: Average Progen scores on test levels after training on 25M environment steps. The mean and standard deviation are computed using five results with different random seeds. We boldface the results that have highest means. This table corresponds to the results in Figure 3.
+
+# E.2 EXPERIMENTS ON PROCGEN
+
+Settings. Procgen benchmark consists of 16 procedurally generated games. Each of these games has procedurally generated levels which present agents with meaningful generalization challenges (Raileanu et al., 2021). All environments of these games use a discrete 15-dimensional action space, and produce  $64 \times 64 \times 3$  RGB observations. Following Raileanu et al. (2021), we use the easy setting, where agents are learned on the training environments with 200 levels and tested on environments with unseen levels.
+
+**Implementation.** Value-based VRL methods mainly focus on non-continuous control tasks. They often use a critic to estimate the action-value function without a actor/policy network. Thus, their critic inputs an image observation and outputs a vector of action values for all possible discrete actions. Note that the last layer of their critic maps the representations of image observations into the action values of all possible discrete actions. Therefore, in the combination of ROUSER and QR-DQN, for the action value of each possible discrete action, we replace its last layer mapping into a 2-layer ensemble MLP, where the ensemble size is the number of discrete actions. The inputs of this 2-layer ensemble MLP are robust action-value representations, and this MLP outputs the quantile values of the action-value function for each discrete action.
+
+Moreover, we use the same design for the reward model  $\psi$  with a 3-layer MLP. Specifically, the reward model uses the image observations as inputs and outputs the compressed reward representations for each discrete action. It uses actions from a batch to select compressed representations for updating.
+
+Results. We compare our approach with: (1) PPO (Schulman et al., 2017), a popular policy gradient baseline upon which many competitive methods are developed; (2) MixReg (Wang et al., 2020), a regularization method applicable to both policy gradient and value-based RL algorithms to enhance generalization on Procogen; (3) QR-DQN, a prior SOTA value-based method on Procgen.
+
+Table 4 lists the Procgen scores on test levels after training on 25M environment steps using the batch size of 256. The mean and standard deviation are averaged over five runs with different random seeds. We adapt the tables from Raileanu & Fergus (2021). These results demonstrate that ROUSER can be also applicable to value-based RL algorithms, outperforming several previous methods.
+
+# E.3 EXPERIMENTS ON MUJoCo
+
+To demonstrate that ROUSER can not only improve robustness in VRL but also enhance sample efficiency in traditional state-based RL (i.e., RL using a vector as a state), we further conduct experiments on MuJoCo (Todorov et al., 2012) using vector states. We combine our approach with DDPG (Lillicrap et al., 2016) and SAC (Haarnoja et al., 2018), and further compare it against the
+
+![](images/c8d6030a478385678fa2c6ca7013280455eabc5b0f75ea2516d33a65ec0c40cf.jpg)
+
+![](images/a5154da385ea28b4b2e36bb00771d5ae926a5364a26d2048a5873fbabf42cf74.jpg)
+
+![](images/00b7cf3caf6769772089b5ab2fee7dd89a8624ff65dd9fd3fd92b68d1a29a9a9.jpg)
+
+![](images/2dc780cb1716653eb2e63ab2a5c4eb6120d73a04945fa8496211666343afdb6e.jpg)
+
+![](images/7eeaf138ba2afd6c203b4530cb2b64b41fa6f4d55c3bbbae6a34a0723b83c18e.jpg)
+
+![](images/972c44950fef37a55789cb5576e5bdb4fce639f7d8011426d21ee1d57c372737.jpg)
+
+![](images/002a45bd587dcff16247a31aa0190df0c1a917ca3a20451d085e6fa15cedb449.jpg)  
+Figure 9: Training curves during the training process with 1M steps over six random seeds.
+
+![](images/818b5d99fc25d9d02b2d6d0da33ecc0a8e88a778a4fb40c5ba82df6f0cf720a8.jpg)
+
+![](images/9bf2b44455da6b93c6b6c4e1a7e9b62c1c7a5b59918017a44fea965319440ee9.jpg)
+
+Table 5: Hyperparameters of ROUSER on DMC and Progen benchmarks.  
+
+<table><tr><td>Parameter</td><td>DMC Value</td><td>Progen Value</td></tr><tr><td>Optimizer</td><td>Adam</td><td>Adam</td></tr><tr><td>learning rate of critic/action-value network</td><td>1·10-4</td><td>2.5·10-4</td></tr><tr><td>learning rate of robust action-value representations</td><td>1·10-5</td><td>2.5·10-5</td></tr><tr><td>learning rate of actor network</td><td>1·10-4</td><td>None</td></tr><tr><td>learning rate of reward model</td><td>1·10-4</td><td>2.5·10-4</td></tr><tr><td>target smoothing coefficient</td><td>0.01</td><td>0.005</td></tr><tr><td>update frequency of target network</td><td>2</td><td>1</td></tr><tr><td>dimension of robust action-value representations D</td><td>512</td><td>512</td></tr><tr><td>batch size</td><td>256</td><td>512</td></tr><tr><td>total environment steps T</td><td>10^6</td><td>2.5·10^7</td></tr><tr><td>α scheduler start value</td><td>10-3</td><td>10-2</td></tr><tr><td>α scheduler end value</td><td>10-2</td><td>10-1</td></tr><tr><td>reward model update times for each step</td><td>5</td><td>5</td></tr><tr><td>nstep</td><td>3</td><td>3</td></tr></table>
+
+SOTA of task decomposition methods, VDFP (Tang et al., 2021). We illustrate training curves (mean and standard deviation) on six tasks from MoJoCo during the training process in Figure 9. Each result is run for 1 million time steps over six random seeds. Figure 9 shows that ROUSER outperforms baselines, achieving an average improvement of  $+18.7\%$ .
+
+# E.4 EXPERIMENTS ON DMC-GB
+
+Environment Setting. To comprehensively evaluate our approach, we have conducted additional experiments on six DMC-GB (Hansen & Wang, 2021) tasks across all their settings (i.e., coloreasy, color-hard, video_easy, and video-hard). All these experiment settings are the same as Hansen & Wang (2021) and Hansen et al. (2021). Moreover, we compare our approach against several SOTA generalization VRL methods, including PIE-G (Yuan et al., 2022), SVEA (Hansen et al., 2021), DrG (Ha et al., 2023), SECANT (Fan et al., 2021), SGQN (Bertoin et al., 2022), and MIIR (Wang et al., 2024b). Most of the baseline results we report are taken directly from their respective papers. For cases where results are not provided, we ran experiments using the available source code and hyperparameters under three random seeds.
+
+Table 6: Hyperparameters of ROUSER on MuJoCo benchmark for sample efficiency.  
+
+<table><tr><td>Parameter</td><td>Value</td></tr><tr><td>Optimizer</td><td>Adam</td></tr><tr><td>learning rate of critic network</td><td>1·10-3</td></tr><tr><td>learning rate of robust action-value representations</td><td>1·10-4</td></tr><tr><td>learning rate of actor network</td><td>1·10-4</td></tr><tr><td>learning rate of reward model</td><td>5·10-5</td></tr><tr><td>target smoothing coefficient</td><td>0.01</td></tr><tr><td>update frequency of target network</td><td>2</td></tr><tr><td>dimension of robust action-value representations D</td><td>512</td></tr><tr><td>batch size</td><td>256</td></tr><tr><td>total environment steps T</td><td>10^6</td></tr><tr><td>α scheduler start value</td><td>10-3</td></tr><tr><td>α scheduler end value</td><td>10-2</td></tr><tr><td>nstep</td><td>1</td></tr></table>
+
+Table 7: Results and standard deviations on DMC-GB under the color_easy and color-hard settings. Results of our approach are averaged over three random seeds.  
+
+<table><tr><td>colorEasy</td><td>ROUSER-SRM</td><td>PIE-G</td><td>DrG</td><td>SVEA</td><td>SGQN</td><td>TIA</td><td>SECANT</td></tr><tr><td>ball_in_cup Catch</td><td>963 ± 8</td><td>955 ± 4</td><td>831 ± 92</td><td>959 ± 2</td><td>907 ± 71</td><td>652 ± 402</td><td>-</td></tr><tr><td>cartpole_swingup</td><td>841 ± 20</td><td>624 ± 52</td><td>701 ± 43</td><td>826 ± 20</td><td>598 ± 92</td><td>483 ± 62</td><td>-</td></tr><tr><td>cheetah_run</td><td>616 ± 13</td><td>429 ± 12</td><td>375 ± 31</td><td>587 ± 39</td><td>304 ± 43</td><td>281 ± 2</td><td>-</td></tr><tr><td>finger_spin</td><td>933 ± 11</td><td>845 ± 5</td><td>876 ± 79</td><td>892 ± 59</td><td>628 ± 46</td><td>410 ± 49</td><td>-</td></tr><tr><td>walker_walk</td><td>942 ± 6</td><td>909 ± 40</td><td>812 ± 33</td><td>907 ± 23</td><td>845 ± 26</td><td>780 ± 140</td><td>-</td></tr><tr><td>walker_stand</td><td>971 ± 2</td><td>968 ± 4</td><td>910 ± 20</td><td>965 ± 10</td><td>963 ± 11</td><td>895 ± 39</td><td>-</td></tr><tr><td>color-hard</td><td>ROUSER-SRM</td><td>PIE-G</td><td>DrG</td><td>SVEA</td><td>SGQN</td><td>TIA</td><td>SECANT</td></tr><tr><td>ball_in_cup Catch</td><td>964 ± 5</td><td>960 ± 3</td><td>607 ± 46</td><td>961 ± 7</td><td>905 ± 71</td><td>575 ± 375</td><td>958 ± 7</td></tr><tr><td>cartpole_swingup</td><td>799 ± 24</td><td>520 ± 69</td><td>523 ± 38</td><td>837 ± 23</td><td>540 ± 76</td><td>407 ± 173</td><td>866 ± 15</td></tr><tr><td>cheetah_run</td><td>562 ± 10</td><td>376 ± 27</td><td>219 ± 8</td><td>456 ± 62</td><td>277 ± 43</td><td>249 ± 25</td><td>582 ± 64</td></tr><tr><td>finger_spin</td><td>979 ± 9</td><td>838 ± 10</td><td>758 ± 124</td><td>977 ± 5</td><td>461 ± 5</td><td>408 ± 79</td><td>910 ± 115</td></tr><tr><td>walker.walk</td><td>916 ± 9</td><td>824 ± 92</td><td>725 ± 134</td><td>760 ± 145</td><td>692 ± 153</td><td>616 ± 243</td><td>856 ± 31</td></tr><tr><td>walker_stand</td><td>962 ± 2</td><td>948 ± 15</td><td>731 ± 21</td><td>942 ± 26</td><td>905 ± 34</td><td>841 ± 37</td><td>939 ± 7</td></tr></table>
+
+1. We run PIE-G, DrG, and SGQN on all DMC-GB tasks across color_easy and color-hard settings.  
+2. We run DrG on finger_spin and walker_stand tasks across videoeasy and video-hard settings.  
+3. We run SVEA on all DMC-GB tasks in the color_easy setting and on cheetah_run task in the color-hard setting.  
+4. We run TIA  ${}^{2}$  on all DMC-GB tasks in all four settings.
+
+Note that since SECANT and MIIR did not release their source code, we only include the results reported in their respective papers.
+
+Results. We provide all results in Tables 7 and 8. These results indicate that ROUSER outperforms the aforementioned baselines in 17 out of 24 settings, demonstrating ROUSER's robustness and effectiveness in DMC-GB tasks.
+
+# E.5 HYPERPARAMETERS
+
+Hyperparameters. We provide the hyperparameters for DMC, Procgen, and MuJoCo in Tables 5 and 6. In DMC benchmark, the hyperparameter  $\alpha$  in the loss function  $\mathcal{L}_{\mathrm{info}}$  can be tuned from a small value  $10^{-3}$  to  $10^{-2}$  by an exponential scheduler.
+
+Table 8: Results and standard deviations on DMC-GB under the videoeasy and video-hard settings. Results of our approach are averaged over three random seeds.  
+
+<table><tr><td>videoeasy</td><td>ROUSER</td><td>PIE-G</td><td>DrG</td><td>SVEA</td><td>SGQN</td><td>MIIR</td><td>TIA</td><td>SECANT</td></tr><tr><td>ball_in_cupCatch</td><td>979 ± 21</td><td>922 ± 20</td><td>701 ± 36</td><td>871 ± 22</td><td>950 ± 24</td><td>973 ± 2</td><td>664 ± 416</td><td>903 ± 49</td></tr><tr><td>cartpole_swingup</td><td>830 ± 21</td><td>587 ± 61</td><td>572 ± 25</td><td>782 ± 27</td><td>761 ± 28</td><td>858 ± 16</td><td>326 ± 139</td><td>752 ± 38</td></tr><tr><td>cheetah_run</td><td>397 ± 30</td><td>287 ± 20</td><td>547 ± 21</td><td>249 ± 20</td><td>308 ± 34</td><td>393 ± 57</td><td>225 ± 41</td><td>428 ± 70</td></tr><tr><td>finger_spin</td><td>970 ± 6</td><td>837 ± 107</td><td>751 ± 43</td><td>808 ± 33</td><td>956 ± 26</td><td>978 ± 9</td><td>295 ± 16</td><td>861 ± 102</td></tr><tr><td>walker_walk</td><td>925 ± 17</td><td>871 ± 22</td><td>902 ± 23</td><td>819 ± 71</td><td>910 ± 24</td><td>919 ± 30</td><td>696 ± 245</td><td>842 ± 47</td></tr><tr><td>walker_stand</td><td>973 ± 3</td><td>957 ± 12</td><td>910 ± 17</td><td>961 ± 8</td><td>955 ± 9</td><td>971 ± 3</td><td>903 ± 18</td><td>932 ± 15</td></tr><tr><td>video-hard</td><td>ROUSER</td><td>PIE-G</td><td>DrG</td><td>SVEA</td><td>SGQN</td><td>MIIR</td><td>TIA</td><td>SECANT</td></tr><tr><td>ball_in_cupCatch</td><td>940 ± 5</td><td>786 ± 47</td><td>635 ± 26</td><td>403 ± 174</td><td>782 ± 57</td><td>929 ± 9</td><td>314 ± 36</td><td>-</td></tr><tr><td>cartpole_swingup</td><td>769 ± 37</td><td>401 ± 21</td><td>545 ± 23</td><td>393 ± 45</td><td>544 ± 43</td><td>765 ± 22</td><td>189 ± 29</td><td>-</td></tr><tr><td>cheetah_run</td><td>270 ± 12</td><td>154 ± 17</td><td>489 ± 11</td><td>105 ± 37</td><td>135 ± 44</td><td>268 ± 73</td><td>76 ± 25</td><td>-</td></tr><tr><td>finger_spin</td><td>937 ± 10</td><td>762 ± 59</td><td>437 ± 61</td><td>335 ± 58</td><td>822 ± 24</td><td>956 ± 18</td><td>182 ± 55</td><td>-</td></tr><tr><td>walker.walk</td><td>868 ± 59</td><td>600 ± 28</td><td>782 ± 37</td><td>377 ± 93</td><td>739 ± 21</td><td>821 ± 58</td><td>166 ± 60</td><td>-</td></tr><tr><td>walker_stand</td><td>968 ± 4</td><td>852 ± 56</td><td>819 ± 58</td><td>834 ± 46</td><td>851 ± 24</td><td>965 ± 2</td><td>673 ± 142</td><td>-</td></tr></table>
+
+Hyperparameter Search. In the right of Figure 7, we illustrate the results (mean and standard deviation) of ROUSER-DrQv2 on cheetah_run task from DMC benchmark under color changes for searching the learning rate  $l$  of robust action-value representations. Based on the learning rate  $\hat{l} = 0.0001$ , we introduce the weight  $c$  such that  $l = c \cdot \hat{l}$ . Each curve is averaged over three seeds.
+
+Moreover, in Figure 5 of the main text, we report the results without standard errors of different  $\alpha$  with three random seeds. Here, we provide the detailed results (refer to Table 9).
+
+Table 9: The mean and standard error of the results in Figure 5.  
+
+<table><tr><td>α</td><td>scale-0.3</td><td>scale-0.4</td><td>scale-0.5</td><td>scale-0.6</td></tr><tr><td>0.01-0.1</td><td>649.7 ± 9.1</td><td>488.0 ± 15.7</td><td>363.0 ± 17.3</td><td>316.1 ± 9.1</td></tr><tr><td>0.001-0.01</td><td>663.7 ± 7.5</td><td>524.4 ± 14.5</td><td>398.4 ± 19.4</td><td>283.3 ± 11.2</td></tr><tr><td>0.0001-0.001</td><td>683.1 ± 2.2</td><td>510.3 ± 16.8</td><td>367.9 ± 12.5</td><td>251.4 ± 11.5</td></tr></table>
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+# LEARNING SHAPE-INDEPENDENT TRANSFORMATION VIA SPHERICAL REPRESENTATIONS FOR CATEGORY-LEVEL OBJECT POSE ESTIMATION
+
+Huan Ren $^{1,2}$  Wenfei Yang $^{1,2,3}$  Xiang Liu $^{4}$  Shifeng Zhang $^{5}$  Tianzhu Zhang $^{1,2*}$
+
+<sup>1</sup>University of Science and Technology of China
+
+$^{2}$ National Key Laboratory of Deep Space Exploration, Deep Space Exploration Laboratory
+
+$^{3}$ Jianghuai Advance Technology Center  $^{4}$ Dongguan University of Technology  $^{5}$ Sangfor Technologies rh.hr_666@mail.ustc.edu.cn {yangwf, tzzhang}@ustc.edu.cn
+
+# ABSTRACT
+
+Category-level object pose estimation aims to determine the pose and size of novel objects in specific categories. Existing correspondence-based approaches typically adopt point-based representations to establish the correspondences between primitive observed points and normalized object coordinates. However, due to the inherent shape-dependence of canonical coordinates, these methods suffer from semantic incoherence across diverse object shapes. To resolve this issue, we innovatively leverage the sphere as a shared proxy shape of objects to learn shape-independent transformation via spherical representations. Based on this insight, we introduce a novel architecture called SpherePose, which yields precise correspondence prediction through three core designs. Firstly, We endow the pointwise feature extraction with SO(3)-invariance, which facilitates robust mapping between camera coordinate space and object coordinate space regardless of rotation transformation. Secondly, the spherical attention mechanism is designed to propagate and integrate features among spherical anchors from a comprehensive perspective, thus mitigating the interference of noise and incomplete point cloud. Lastly, a hyperbolic correspondence loss function is designed to distinguish subtle distinctions, which can promote the precision of correspondence prediction. Experimental results on CAMERA25, REAL275 and HouseCat6D benchmarks demonstrate the superior performance of our method, verifying the effectiveness of spherical representations and architectural innovations.
+
+# 1 INTRODUCTION
+
+Object pose estimation, which involves predicting the 3D rotation  $\mathbf{R} \in \mathrm{SO}(3)$  and 3D translation  $t \in \mathbb{R}^3$  of observed objects, has received considerable attention from the research community due to its crucial applications in augmented reality (Marchand et al., 2015; Su et al., 2019), robotic manipulation (Liu et al., 2023a; Wen et al., 2022), and hand-object interaction (Lin et al., 2023c), etc. While many prior instance-level object pose estimation methods (Peng et al., 2019; Wang et al., 2019a; 2021a) have achieved promising performance, their dependence on CAD models restricts the generalization ability. To mitigate this problem, category-level object pose estimation has been introduced in Wang et al. (2019b), which aims to reason about the 6D pose  $\{\mathbf{R}, t\} \in \mathrm{SE}(3)$  and 3D size  $s \in \mathbb{R}^3$  of unseen objects in specific categories, without the need for their CAD models.
+
+Existing category-level methods can be mainly divided into two groups, i.e., direct regression-based methods (Chen et al., 2021; Di et al., 2022; Lin et al., 2023b) and correspondence-based methods (Wang et al., 2019b; Tian et al., 2020; Liu et al., 2023b; Lin et al., 2024). The former approaches seek to directly regress object pose in an end-to-end manner. Although conceptually simple, they struggle with the pose-sensitive feature learning due to the non-linearity of the entire pose search space SE(3) (Lin et al., 2022a; 2023b). In contrast, the latter methods focus on establishing the correspondence between camera coordinate space and the Normalized Object Coordinate Space
+
+![](images/a81a359ac97d80946d64b2e049e617ae60a963bbd6bac3164ceb77183b6873c4.jpg)  
+Figure 1: We propose a novel approach for category-level object pose estimation based on spherical representations. (a) The classical pipeline of correspondence-based methods, where representations denote the organization form of observation data, e.g., point-based representations in  $\mathbb{R}^3$ , which is straightforward and commonly adopted in previous methods. (b) The overview of comparison between our method and previous methods. Note that each XYZ position of representations and NOCS coordinates is visualized as an RGB tuple. Our method employs spherical representations to learn shape-independent transformation, yielding smaller NOCS angle errors compared to DPDN (Lin et al., 2022b), which adopts point-based representations and suffers from the shape-dependence.
+
+(NOCS) (Wang et al., 2019b), and then solve the pose through the Umeyama algorithm (Umeyama, 1991) or deep estimators (Lin et al., 2022b), as illustrated in Figure 1(a). We follow this paradigm, where the correspondences in the linear coordinate search space  $\mathbb{R}^3$  are easier to learn.
+
+Nevertheless, existing correspondence-based methods typically adopt point-based representations, where discrete points back-projected from observed depth maps serve as the representations of observation data. Subsequently, the NOCS coordinates of these points are derived by geometric normalization and alignment of object shapes, which are inherently shape-dependent. As a result, observed points with similar semantics across diverse object shapes are mapped to distinct NOCS coordinates, also known as semantic incoherence in Wan et al. (2023). For example, points on the camera lens are mapped to different coordinates in NOCS if the camera length is different. Since the correspondence prediction need to take into account both pose information and large shape variation, it is hard to learn and generalize, which results in large NOCS angle errors as demonstrated in Figure 1(b)(Top). To resolve the above issue, we propose to innovatively leverage the sphere as a shared proxy shape of objects to learn shape-independent transformation via spherical representations. Specifically, we first uniformly divide the sphere with the Hierarchical Equal Area iso-Latitude Pixelation (HEALPix) grids (Gorski et al., 2005), each of which is represented by its center anchor. Then observed points are projected onto the spherical grids, with point-wise features assigned to the corresponding spherical anchors, which serves as the new representations of observation. Since these spherical anchors are coherent across various object shapes, the spherical NOCS coordinates are shape-independent, and thus more precise correspondence prediction can be achieved by focusing only on the pose information, as shown in Figure 1(b)(Bottom).
+
+Based on the sphere representations, we introduce a novel architecture for category-level object pose estimation, termed SpherePose. Given a centralized and scale-normalized observation<sup>1</sup>, point-wise features are first extracted and then assigned to spherical anchors with HEALPix spherical projection, yielding the spherical representations. The shape-independent transformation is subsequently
+
+learned between the spherical anchors and the spherical NOCS coordinates for rotation estimation. In order to boost the precision of correspondence prediction, we further present three core designs. The first design is the SO(3)-invariant point-wise feature extraction. It is intuitive that no matter how an object is rotated, the feature of a specific point on that object should remain invariant as it is mapped to a static coordinate in NOCS. Driven by this insight, for deep image features, we employ pretrained DINOv2 (Oquab et al., 2024) features with patch-wise consensus semantics, which are robust to rotations (Chen et al., 2024). As for deep point cloud features, we propose the ColorPointNet++ network by making minimal adjustments to PointNet++ (Qi et al., 2017), which derives input features from RGB values rather than absolute XYZ coordinates, inherently ensuring SO(3)-invariance by focusing on the color information and relative proximity of point cloud. The second design is the spherical feature interaction. Due to self-occlusion, depth cameras fail to perceive the occluded regions on the backside of objects, resulting in incomplete point cloud. Consequently, after projecting point-wise features onto the sphere, there exist numerous spherical anchors with no assigned features. To this end, we adopt the attention mechanism (Vaswani, 2017) with learnable position embedding to facilitate the interaction and propagation of features among anchors, where initially unassigned spherical features can be reasoned out. Moreover, by holistically integrating the relationship among spherical anchors, we acquire comprehensive spherical features which are robust to noise. The third design is a hyperbolic correspondence loss function. For the supervision of correspondences, we derive the gradients of several typical loss functions and realize that they exhibit minor gradients around zero, which is prone to yield ambiguous prediction. To achieve more precise prediction, we leverage the computation of correspondence distance in hyperbolic space (Lin et al., 2023a), which yields higher gradients near zero and can discern more subtle distinctions.
+
+To sum up, the main contributions of this paper are as follows:
+
+- We present an innovative approach that utilizes the sphere as a proxy shape of objects to learn shape-independent transformation via HEALPix spherical representations, which addresses the semantic inconsistency of shape-dependent canonical coordinates that plagues existing correspondence-based methods with point-based representations.  
+- We introduce an architecture for category-level object pose estimation that achieves precise correspondence prediction through three core designs: SO(3)-invariant point-wise feature extraction, spherical feature interaction, and a hyperbolic correspondence loss function.  
+- Extensive experiments on existing benchmarks demonstrate the superior performance of our method over state-of-the-art approaches, verifying the effectiveness of spherical representations and architectural innovations.
+
+# 2 RELATED WORK
+
+The task of category-level object pose estimation is first introduced in Wang et al. (2019b) to predict the 6D pose and size of unseen objects in a given category, and existing methods can be categorized into two groups, i.e., direct regression-based methods and correspondence-based methods.
+
+Direct Regression-based Methods. This category of approaches intends to directly regress the object pose in an end-to-end manner. FS-Net (Chen et al., 2021) proposes to decouple the rotation into two perpendicular vectors that simplifies the prediction, and utilizes a 3D graph convolution (3D-GC) autoencoder for feature extraction. GPV-Pose (Di et al., 2022) enhances the learning of pose-sensitive features with geometric consistency, and HS-Pose (Zheng et al., 2023) further extends 3D-GC to extract hybrid scope latent features that combine both global and local geometric information. VI-Net (Lin et al., 2023b) presents a novel rotation estimation network that decouples the rotation into a viewpoint rotation and an in-plane rotation, simplifying the task by leveraging spherical representations. Based on the decoupled rotation, SecondPose (Chen et al., 2024) extracts SE(3)-consistent semantic and geometric features to further enhance the pose estimation. However, these methods struggle with the pose-sensitive feature learning due to the non-linearity of SE(3).
+
+Correspondence-based Methods. This group of approaches aims to establish the correspondence between camera coordinate space and object coordinate space, and then acquire the pose through pose fitting algorithm, such as Umayama (Umayama, 1991). As a cornerstone, the Normalized Object Coordinate Space (NOCS) is introduced in Wang et al. (2019b), which serves as a shared canonical representation to align object instances within a given category. Later, SPD (Tian et al., 2020) proposes to handle the intra-class shape variation by reconstructing the 3D object model from
+
+a pre-learned categorical shape prior, and then match the observed point cloud to the reconstructed 3D model. Inspired by it, CR-Net (Wang et al., 2021b) and SGPA (Chen & Dou, 2021) are presented to improve the reconstruction quality. DPDN (Lin et al., 2022b) learns the shape prior deformation in the feature space, and replaces the traditional Umeyama algorithm with a deep estimator to directly regress the pose and size. Query6DoF (Wang et al., 2023) and IST-Net (Liu et al., 2023b) eliminate the dependence on explicit shape priors with implicit counterparts and implicit space transformation, respectively. More recently, AG-Pose (Lin et al., 2024) proposes to detect a set of sparse keypoints to represent the geometric structures of objects and establish robust keypoint-level correspondences for pose estimation. Although these methods have demonstrated strong performance, they all adopt point-based representations, where the NOCS coordinates are derived by geometric normalization and alignment of object shapes. When handling diverse object shapes, the NOCS coordinates become semantically incoherent, which are hard to learn and generalize. To tackle this issue, Semantically-aware Object Coordinate Space (SOCS) is presented in Wan et al. (2023), where semantically meaningful SOCS coordinates are built by keypoint-guided non-rigid object alignment to the categorical average shape. However, due to the non-rigid alignment process, there is a mismatch between the geometric structure of the observed point cloud and the SOCS coordinates. This discrepancy plagues the pose fitting, as the transformation may not be optimal for all parts of the object. In this work, we instead resort to spherical representations, where the sphere is leveraged as a shared proxy shape of objects to learn shape-independent transformation between spherical anchors and spherical NOCS coordinates. Thus precise correspondence prediction can be achieved without the need to account for large shape variation, thereby enhancing subsequent pose estimation.
+
+# 3 METHOD
+
+Target. Given an RGB-D image, we first acquire the instance segmentation masks with an offline Mask R-CNN (He et al., 2017), which then yield the cropped RGB image  $\pmb{I} \in \mathbb{R}^{H \times W \times 3}$  and segmented depth image for each instance. Partially observed point cloud  $P^O \in \mathbb{R}^{N \times 3}$  is derived by back-projecting and downsampling the segmented depth image, where  $N$  denotes the number of points. With  $I$  and  $P^O$  as inputs, our method predicts the rotation  $R \in \mathrm{SO}(3)$ , translation  $t \in \mathbb{R}^3$ , and size  $s \in \mathbb{R}^3$  of the observed instance.
+
+Overview. As illustrated in Figure 2, the proposed method mainly consists of two components. First, in the spherical representation projection phase (Section 3.1), point-wise features are extracted and then assigned to spherical anchors with HEALPix spherical projection, yielding the spherical representations. Subsequently, in the spherical rotation estimation phase (Section 3.2), spherical anchors perform feature interaction with each other through the attention mechanism, and then they are mapped to the spherical NOCS coordinates for rotation estimation. In Section 3.3, we elaborate on the training and testing workflow, including additional translation and size prediction for complete category-level object pose estimation. Finally, in Section 3.4, we emphasize the distinctions between our method and some relevant works on spherical representations and the proxy shape.
+
+# 3.1 SPHERICAL REPRESENTATION PROJECTION
+
+SO(3)-invariant Point-wise Feature Extraction. Taking image  $\pmb{I}$  and normalized points  $P^2$  as inputs, we start with point-wise feature extraction. Note that for point-wise image features, we first employ a 2D feature extractor to  $\pmb{I}$  and then leverage  $\pmb{P}$  to perform point-wise selection on the extracted features. To facilitate the transformation between camera coordinate space and object coordinate space, we recognize that the point-wise features  $\pmb{F}^{P} \in \mathbb{R}^{N \times C}$  should be SO(3)-invariant:
+
+$$
+\psi_ {\boldsymbol {G}} \left(\boldsymbol {F} ^ {P}\right) = \boldsymbol {F} ^ {P}, \quad \forall \boldsymbol {G} \in \mathrm {S O} (3), \tag {1}
+$$
+
+where  $\psi_G$  indicates the transformation of features when rotating points  $P$  by rotation  $G$ . Intuitively, the feature of a particular point on an object should remain invariant regardless of the object's rotation, given that the point is mapped to a fixed coordinate in NOCS. To this end, we synthesize the features of observation with SO(3)-invariance from a comprehensive perspective. Specifically, for image features, we employ RGB values with low-level texture information and the pretrained DINOv2 (Oquab et al., 2024) features with high-level semantic information. Owing to the pretraining on large-scale datasets, DINOv2 possesses robust generalization capabilities and can provide
+
+![](images/8e9c6fbc7ca41261da978f5f24c86011cfb34fb99ae29cff083d9e0a9684ca0a.jpg)  
+Figure 2: Overview of the proposed SpherePose. Given the observation  $\mathbf{I}$  and  $P$ , we first extract SO(3)-invariant point-wise features from four distinct perspectives and assign them to the spherical anchors  $\mathbf{A}$  with HEALPix spherical projection, yielding initial spherical features  $\mathbf{F}^{\mathbf{A}}$ . Then, the Transformer encoder module is employed for spherical feature interaction and integrates comprehensive spherical features  $\mathbf{F}$ . Finally, we predict the corresponding spherical NOCS coordinates  $\mathbf{O}$  via a NOCS predictor, which is applied to the estimation of rotation  $\mathbf{R}$ .
+
+semantically consistent patch-wise features that are robust to rotations (Chen et al., 2024). As for point cloud features, we adopt radius<sup>3</sup> values with low-level spatial information, along with features with high-level geometric information from the proposed ColorPointNet++, which is a variant of PointNet++ (Qi et al., 2017). The input features of the original PointNet++ are derived from absolute XYZ coordinates, which are inherently SO(3)-equivariant. To validate the importance of SO(3)-invariance for correspondence prediction, we make minimal adjustments to adapt the architecture for extracting SO(3)-invariant features, by replacing raw XYZ coordinates with RGB values. Given the SO(3)-invariance of color attributes and kNN operation in grouping and interpolating, our ColorPointNet++ yields SO(3)-invariant point-wise features, as detailed in Appendix A.1. The final point-wise features  $F^P \in \mathbb{R}^{N \times C}$  are the concatenation of all four point-wise features that capture distinct information from the observation, followed by feature dimension reduction to  $C$ .
+
+HEALPix Spherical Projection. Once point-wise features are extracted, previous methods adopt point-based representations to directly predict the NOCS coordinates corresponding to the observed points, which causes semantic incoherence across diverse object shapes, as introduced in Section 1. Instead, we propose to employ the sphere as a shared proxy shape of objects to learn shape-independent transformation via spherical representations. Specifically, we first uniformly divide the sphere with HEALPix grids (Gorski et al., 2005) as illustrated in Figure 4(b), each of which is equal in area and represented by its center anchor  $A_{m} \in \mathbb{R}^{3}, m \in \{1, \dots, M\}$ , where  $M$  denotes the number of grids. We then project the normalized points  $P$  onto the spherical grids, and assign point-wise features  $F^{P}$  to the corresponding spherical anchors, which serve as the new representations of observation with spherical features  $F^{A} \in \mathbb{R}^{M \times C}$ . In detail, consider a grid indexed by  $m$ , we follow VI-Net (Lin et al., 2023b) to search within this grid region for the point with the largest radius value, denoted as  $P_{n}$ , and assign its feature  $F_{n}^{P}$  to the anchor  $A_{m}$ . If there is no point projected onto the grid, we set  $F_{m}^{A} = 0$ . Since these spherical anchors are coherent across diverse objects, the corresponding spherical NOCS coordinates are shape-independent.
+
+# 3.2 SPHERICAL ROTATION ESTIMATION
+
+Spherical Feature Interaction. Due to self-occlusion, depth cameras fail to perceive the occluded regions on the backside of objects, causing incomplete point cloud. This phenomenon deteriorates when projected onto spherical grids, leaving almost half of the spherical anchors without assigned features. To this end, we employ a Transformer encoder with attention mechanism (Vaswani, 2017) to facilitate feature interaction and propagation among spherical anchors  $\mathbf{A} \in \mathbb{R}^{M \times 3}$  in a global perspective, yielding comprehensive spherical features  $\mathbf{F} \in \mathbb{R}^{M \times C}$  from the initial spherical features  $\mathbf{F}^A$  with a sequence of self-attention layers and MLP blocks, formulated as follows:
+
+$$
+\begin{array}{l} \boldsymbol {F} ^ {l + 1} = \operatorname {M L P} \left(\hat {\boldsymbol {F}} ^ {l + 1}\right) + \hat {\boldsymbol {F}} ^ {l + 1}, \\ \hat {\boldsymbol {F}} ^ {l + 1} = \operatorname {A t t e n t i o n} \left(\boldsymbol {F} ^ {l}\right) + \boldsymbol {F} ^ {l}, \end{array} \quad l \in \{0, \dots , L - 1 \}, \tag {2}
+$$
+
+where  $L$  is the number of layers and  $\pmb{F} = \pmb{F}^{L}$ ,  $\pmb{F}^{0} = \pmb{F}^{A}$ . Due to the disorder of spherical anchors, the learnable position embeddings  $\pmb{E}^{pos} \in \mathbb{R}^{M \times C}$  are assigned to corresponding anchors and contribute to the computation of attention for feature propagation. In formula, the self-attention in  $l$ -th layer is defined as:
+
+$$
+\operatorname {A t t e n t i o n} \left(\boldsymbol {F} ^ {l}\right) = \operatorname {S o f t m a x} \left(\frac {\left(\boldsymbol {Q} ^ {l} \boldsymbol {W} ^ {Q}\right) \left(\boldsymbol {K} ^ {l} \boldsymbol {W} ^ {K}\right) ^ {\top}}{\sqrt {C}}\right) \left(\boldsymbol {V} ^ {l} \boldsymbol {W} ^ {V}\right), \tag {3}
+$$
+
+$$
+\boldsymbol {Q} ^ {l} = \boldsymbol {F} ^ {l} + \boldsymbol {E} ^ {\text {p o s}}, \quad \boldsymbol {K} ^ {l} = \boldsymbol {F} ^ {l} + \boldsymbol {E} ^ {\text {p o s}}, \quad \boldsymbol {V} ^ {l} = \boldsymbol {F} ^ {l}, \tag {4}
+$$
+
+where  $W^{Q / K / V} \in \mathbb{R}^{C \times C}$  denotes learnable projection matrix for query, key and value, respectively, with independent parameters across layers. Through global feature interaction and propagation, those spherical features that are initially unassigned can be reasoned out. Moreover, by attending to the relationship among spherical anchors from a holistic perspective, we integrate pose-coherent spherical features which can alleviate the interference of noise.
+
+Correspondence-based Rotation Estimation. Given the spherical features  $\pmb{F}$ , we follow previous methods (Lin et al., 2022b; 2024) to first predict the spherical NOCS coordinates  $\pmb{O} \in \mathbb{R}^{M \times 3}$  corresponding to spherical anchors  $\pmb{A}$  with an MLP-based NOCS predictor. Since the spherical anchors and NOCS coordinates are on the unit sphere, we  $L^2$  normalize the output of MLP:
+
+$$
+\boldsymbol {O} _ {m} = \frac {\operatorname {M L P} \left(\boldsymbol {F} _ {m}\right)}{\| \operatorname {M L P} \left(\boldsymbol {F} _ {m}\right) \| _ {2}}, \quad m \in \{1, \dots , M \}. \tag {5}
+$$
+
+After acquiring the correspondences between spherical anchors and spherical NOCS coordinates, there are several approaches to acquire the rotation  $\mathbf{R} \in \mathrm{SO}(3)$ , such as the Umayama algorithm (Umayama, 1991) or deep estimators (Lin et al., 2022b). We adopt the Umayama algorithm with RANSAC (Fischler & Bolles, 1981) for outlier removal as it does not require extra training.
+
+# 3.3 CATEGORY-LEVEL OBJECT POSE ESTIMATION
+
+Training. For the estimation of translation  $t \in \mathbb{R}^3$  and size  $s \in \mathbb{R}^3$ , we follow VI-Net (Lin et al., 2023b) to employ a simple and lightweight PointNet++ (Qi et al., 2017). It takes the observed point cloud  $P^O$  and the point-wise selected RGB attributes from image  $I$  as input, and directly regresses  $t$  and  $s$  with  $L^1$  loss as follows:
+
+$$
+\mathcal {L} _ {\boldsymbol {t} \boldsymbol {s}} = \| \boldsymbol {t} - \boldsymbol {t} ^ {g t} \| _ {1} + \| \boldsymbol {s} - \boldsymbol {s} ^ {g t} \| _ {1}. \tag {6}
+$$
+
+As for the estimation of rotation  $\mathbf{R} \in \mathrm{SO}(3)$ , we adopt the correspondence-based paradigm to supervise the spherical NOCS coordinates  $\mathbf{O}$ . Specifically, we generate ground truth spherical NOCS coordinates  $\mathbf{O}^{gt}$  by mapping spherical anchors  $\mathbf{A}$  into NOCS using the ground truth rotation  $\mathbf{R}^{gt}$ :
+
+$$
+\boldsymbol {O} _ {m} ^ {g t} = \left(\boldsymbol {R} ^ {g t}\right) ^ {\top} \boldsymbol {A} _ {m}, \quad m \in \{1, \dots , M \}, \tag {7}
+$$
+
+and the prediction errors  $e$  can then be measured by the distance between them, defined as:
+
+$$
+\boldsymbol {e} _ {m} = \left\{ \begin{array}{l l} \| \boldsymbol {O} _ {m} - \boldsymbol {O} _ {m} ^ {g t} \| _ {1}, & \text {a s} L ^ {1} \text {d i s t a n c e} \\ \| \boldsymbol {O} _ {m} - \boldsymbol {O} _ {m} ^ {g t} \| _ {2}, & \text {a s} L ^ {2} \text {d i s t a n c e} \end{array} , \quad m \in \{1, \dots , M \}. \right. \tag {8}
+$$
+
+Previous methods employ either  $e$  directly as the loss function or the smooth  $L^1$  loss (Wang et al., 2019b), which uses a squared term  $e^2$  if  $e$  fall below a certain threshold and the term  $e$  otherwise. However, we examine the gradients of these loss functions in Figure 3 and discover that they exhibit minor gradients around zero, which is prone to generate ambiguous prediction. To promote the precision of correspondence prediction, we leverage the computation of distance in hyperbolic space (Lin et al., 2023a), which yields higher gradients near zero and can discern more subtle distinctions. In detail, the  $\operatorname{arccosh}(1 + x)$  function is employed to the errors  $e$  with  $L^2$  distance, and the final hyperbolic correspondence loss function for  $O$  is formulated as follows:
+
+$$
+\mathcal {L} _ {\text {c o r r}} = \frac {1}{M} \sum_ {m = 1} ^ {M} \operatorname {a r c o s h} \left(1 + \boldsymbol {e} _ {m}\right) = \frac {1}{M} \sum_ {m = 1} ^ {M} \operatorname {a r c o s h} \left(1 + \| \boldsymbol {O} _ {m} - \boldsymbol {O} _ {m} ^ {g t} \| _ {2}\right). \tag {9}
+$$
+
+Inference. Given the cropped RGB image  $\pmb{I}$  and point cloud  $P^O$ , we first predict the translation  $t$  and size  $s$  via PointNet++, and then normalize<sup>4</sup> the point cloud as  $P = (P^O - t) / \|s\|_2$ , which is fed into our SpherePose for the estimation of rotation  $R$ .
+
+![](images/528e4d07a705b169ed6cb67c0366c0254c3906fc6b44071b5c31b84d25bd04f6.jpg)  
+Figure 3: Illustration of several correspondence loss functions and their gradients.
+
+![](images/cf63c94fccb9a2932f4ff54ea64fb9b9f9ecc51b66ff3ffe0119fdc2a85d395e.jpg)  
+Figure 4: Overview of (a) equirectangular grids and (b) HEALPix grids.
+
+![](images/ad6e35db64d294b146c8c06f5754e40ade7d3ab965f26ef7c8acbba5c3c019db.jpg)
+
+# 3.4 DISCUSSION
+
+Spherical Representations. Although several previous methods (Lin et al., 2021; 2023b; Chen et al., 2024) have also adopted spherical representations for category-level object pose estimation, there are two fundamental distinctions. Firstly, they employ the equirectangular grids (Driscoll & Healy, 1994), which are evenly divided along latitude and longitude, as shown in Figure 4(a). However, this partition of sphere is non-uniform in area, with a higher sampling density near the poles, which is suboptimal for representing the point cloud data. This characteristic is particularly detrimental for objects with diverse poses, as non-uniform sampling of distinct semantic components results in a biased distribution of sampled data. Instead, we resort to the HEALPix grids (Gorski et al., 2005), which consistently yield the uniformly sampled spherical representations across various object poses, as illustrated in Figure 4(b). This uniform spherical partition is a more appropriate choice for the capture of point cloud structures, thereby enhancing the performance of pose estimation. Secondly, they intend to directly regress the pose in an end-to-end manner, which struggle with the pose-sensitive feature learning due to the non-linearity of  $\mathrm{SE}(3) / \mathrm{SO}(3)$ . DualPoseNet (Lin et al., 2021) employs spherical convolution in the spectral domain to learn  $\mathrm{SO}(3)$ -equivariant features. However, the Fourier transforms require tensors in the Fourier domain of  $\mathrm{SO}(3)$  that scale with the cube of the bandwidth, limiting the resolution. VI-Net (Lin et al., 2023b) utilizes spherical convolution in the spatial domain to learn viewpoint-equivariant features. Although it simplifies the regression process by decoupling the rotation into viewpoint rotation and in-plane rotation, the spatial convolution has a limited receptive field. Different from them, we follow with the correspondence-based paradigm to establish the correspondences between spherical anchors and spherical NOCS coordinates, which is easier to learn in the linear space  $\mathbb{R}^3$ . As for the spherical feature extraction, we adopt the attention mechanism to facilitate the feature interaction and propagation among spherical anchors from a holistic perspective, yielding the comprehensive spherical features.
+
+Proxy Shape. SAR-Net (Lin et al., 2022a) aims to deform the category-level proxy template point cloud to align with the observed point cloud for implicitly representing its rotational state. The object rotation is then solved from the categorical and deformed template point clouds by Umeyama algorithm. Although it learns shape-independent transformation by leveraging the category-shared template shape, the representations of the proxy shape are not well exploited to facilitate correspondence prediction. Specifically, the rotation information for guidance during the deformation process is derived from a single global feature vector of the observed point cloud, which leads to difficulties with rotation-sensitive feature learning. In contrast, we resort to the sphere as a shared proxy shape to learn shape-independent transformation via spherical representations. By projecting the observed point cloud onto the spherical grids, we preserve the intact information from the observation. The comprehensive spherical features are further extracted through the interaction and integration of features among spherical anchors from a holistic perspective, which can mitigate the interference of noise and promote the correspondence prediction. The proxy shape is also incorporated in MFOS (Lee et al., 2024) for image-aligned pose representation, which transforms the pose into a 2D image by rendering the 3D coordinates of a proxy shape (e.g., a cuboid or an ellipsoid) positioned according to the pose. Since MFOS operates with only 2D images as input, the proxy shape provides a mechanism for achieving 3D geometric perception, and the object pose is recovered by solving a PnP problem from the dense 2D-3D mapping. We differ by directly learning the 3D-3D transformation of the proxy sphere between the camera and object coordinate spaces.
+
+Table 1: Performance comparison with state-of-the-art methods on REAL275 dataset.  
+
+<table><tr><td></td><td>Method</td><td>Representation</td><td>IoU50</td><td>IoU75</td><td>5°2cm</td><td>5°5cm</td><td>10°2cm</td><td>10°5cm</td></tr><tr><td rowspan="6">Direct Regression</td><td>GPV-Pose (Di et al., 2022)</td><td>Point-based</td><td>-</td><td>64.4</td><td>32.0</td><td>42.9</td><td>-</td><td>73.3</td></tr><tr><td>HS-Pose (Zheng et al., 2023)</td><td>Point-based</td><td>82.1</td><td>74.7</td><td>46.5</td><td>55.2</td><td>68.6</td><td>82.7</td></tr><tr><td>GenPose (Zhang et al., 2023)</td><td>Point-based</td><td>-</td><td>-</td><td>52.1</td><td>60.9</td><td>72.4</td><td>84.0</td></tr><tr><td>DualPoseNet (Lin et al., 2021)</td><td>Spherical</td><td>79.8</td><td>62.2</td><td>29.3</td><td>35.9</td><td>50.0</td><td>66.8</td></tr><tr><td>VI-Net (Lin et al., 2023b)</td><td>Spherical</td><td>-</td><td>-</td><td>50.0</td><td>57.6</td><td>70.8</td><td>82.1</td></tr><tr><td>SecondPose (Chen et al., 2024)</td><td>Spherical</td><td>-</td><td>-</td><td>56.2</td><td>63.6</td><td>74.7</td><td>86.0</td></tr><tr><td rowspan="8">Correspondence</td><td>NOCS (Wang et al., 2019b)</td><td>Point-based</td><td>78.0</td><td>30.1</td><td>7.2</td><td>10.0</td><td>13.8</td><td>25.2</td></tr><tr><td>SPD (Tian et al., 2020)</td><td>Point-based</td><td>77.3</td><td>53.2</td><td>19.3</td><td>21.4</td><td>43.2</td><td>54.1</td></tr><tr><td>SGPA (Chen &amp; Dou, 2021)</td><td>Point-based</td><td>80.1</td><td>61.9</td><td>35.9</td><td>39.6</td><td>61.3</td><td>70.7</td></tr><tr><td>DPDN (Lin et al., 2022b)</td><td>Point-based</td><td>83.4</td><td>76.0</td><td>46.0</td><td>50.7</td><td>70.4</td><td>78.4</td></tr><tr><td>IST-Net (Liu et al., 2023b)</td><td>Point-based</td><td>82.5</td><td>76.6</td><td>47.5</td><td>53.4</td><td>72.1</td><td>80.5</td></tr><tr><td>Query6DoF (Wang et al., 2023)</td><td>Point-based</td><td>82.5</td><td>76.1</td><td>49.0</td><td>58.9</td><td>68.7</td><td>83.0</td></tr><tr><td>AG-Pose (Lin et al., 2024)</td><td>Point-based</td><td>83.7</td><td>79.5</td><td>54.7</td><td>61.7</td><td>74.7</td><td>83.1</td></tr><tr><td>SpherePose</td><td>Spherical</td><td>84.0</td><td>79.0</td><td>58.2</td><td>67.4</td><td>76.2</td><td>88.2</td></tr></table>
+
+# 4 EXPERIMENTS
+
+# 4.1 EXPERIMENTAL SETUP
+
+Datasets. We evaluate our methods on three benchmarks including CAMERA25, REAL275 (Wang et al., 2019b) and HouseCat6D (Jung et al., 2024). CAMERA25 is a synthetic dataset that contains 275K training images and 25K testing images from 6 object categories, which are generated by rendering foreground objects with real-world backgrounds using the mixed-reality technique. REAL275 is a real-world dataset that consists of 4.3K training images of 7 scenes and 2.75K testing images of 6 scenes, which shares the same categories with CAMERA25. HouseCat6D is an emerging real-world dataset that comprises 20K training frames of 34 scenes, 3K testing frames of 5 scenes and 1.4K validation frames of 2 scenes from 10 household categories. This collection encompasses a diverse range of photometrically challenging objects, including glass and cutlery, which are captured in a comprehensive manner across various viewpoints and occlusions.
+
+Evaluation Metrics. Following previous works (Wang et al., 2019b; Chen et al., 2024), we report the mean Average Precision (mAP) of  $n^{\circ}m$  cm for 6D pose estimation, which denotes the percentage of prediction with rotation error less than  $n^{\circ}$  and translation error less than  $m$  cm. We also report the mAP of Intersection over Union  $(\mathrm{IoU}_x)$  for 3D bounding boxes with thresholds of  $x\%$ .
+
+Implementation Details. For a fair comparison, we employ the same instance segmentation masks as DPDN (Lin et al., 2022b) from Mask R-CNN (He et al., 2017). The number of sampled points is  $N = 2,048$ . For DINOv2 features, images are first cropped and resized to  $224 \times 224$  and then fed into the frozen DINOv2, followed by bilinear interpolation upsampling to the original resolution for point-wise selection. After extracting four point-wise features, we concatenate them at the feature dimension, followed by a linear layer activated by GeLU (Hendrycks & Gimpel, 2016) that reduces the dimension to  $C = 128$ . The resolution of HEALPix grids is expressed by the parameter  $N_{side}$ , and we set it to 8, which results in  $M = 768$  grids. The number of the Transformer encoder layers is set to  $L = 6$  by default. All experiments are conducted on a single RTX3090Ti GPU with a batch size of 64 for 200K iterations. More details can be found in Appendix A.2.
+
+# 4.2 COMPARISON WITH STATE-OF-THE-ART METHODS
+
+Results on REAL275 and CAMERA25 Datasets. Table 1 shows the comparison of our method with existing direct regression-based methods and correspondence-based methods on REAL275 dataset. It should be noted that our approach represents a pioneering application of spherical representations within the correspondence-based paradigm. From the results we can see that SpherePose outperforms the prior state-of-the-art methods on the precision of 6D pose estimation. Specifically, when compared with direct regression-based methods, SpherePose surpasses SecondPose (Chen et al., 2024) by  $2.0\%$  on  $5^{\circ}2\mathrm{cm}$  and  $2.2\%$  on  $10^{\circ}5\mathrm{cm}$ , which also employs spherical representations and DINOv2 (Oquab et al., 2024) backbone. And when compared with correspondence-based methods, SpherePose outperforms AG-Pose (Lin et al., 2024) by  $3.5\%$  on  $5^{\circ}2\mathrm{cm}$  and  $5.1\%$  on  $10^{\circ}5\mathrm{cm}$ , which encounters the intra-class semantic incoherence arising from point-based representations. As
+
+Table 2: Performance comparison with state-of-the-art methods on HouseCat6D dataset.  
+
+<table><tr><td></td><td>Method</td><td>Representation</td><td>IoU25</td><td>IoU50</td><td>5°2cm</td><td>5°5cm</td><td>10°2cm</td><td>10°5cm</td></tr><tr><td rowspan="4">Direct Regression</td><td>FS-Net (Chen et al., 2021)</td><td>Point-based</td><td>74.9</td><td>48.0</td><td>3.3</td><td>4.2</td><td>17.1</td><td>21.6</td></tr><tr><td>GPV-Pose (Di et al., 2022)</td><td>Point-based</td><td>74.9</td><td>50.7</td><td>3.5</td><td>4.6</td><td>17.8</td><td>22.7</td></tr><tr><td>VI-Net (Lin et al., 2023b)</td><td>Spherical</td><td>80.7</td><td>56.4</td><td>8.4</td><td>10.3</td><td>20.5</td><td>29.1</td></tr><tr><td>SecondPose (Chen et al., 2024)</td><td>Spherical</td><td>83.7</td><td>66.1</td><td>11.0</td><td>13.4</td><td>25.3</td><td>35.7</td></tr><tr><td rowspan="2">Correspondence</td><td>AG-Pose (Lin et al., 2024)</td><td>Point-based</td><td>81.8</td><td>62.5</td><td>11.5</td><td>12.0</td><td>32.7</td><td>35.8</td></tr><tr><td>SpherePose</td><td>Spherical</td><td>88.8</td><td>72.2</td><td>19.3</td><td>25.9</td><td>40.9</td><td>55.3</td></tr></table>
+
+for the precision of 3D bounding boxes, we achieve comparable performance. Similar results on CAMERA25 dataset can be found in Table 8, please refer to Appendix A.3 for the details.
+
+Results on HouseCat6D Dataset. Table 2 provides the quantitative results of existing methods on HouseCat6D dataset. As can be seen from the table, SpherePose achieves the best performance against state-of-the-art approaches by a large margin under all metrics. Specifically, SpherePose surpasses SecondPose (Chen et al., 2024) by  $6.1\%$  on  $\mathrm{IoU}_{50}$  and  $8.3\%$  on  $5^{\circ}2\mathrm{cm}$ , and outperforms AG-Pose (Lin et al., 2024) by  $10.3\%$  on  $\mathrm{IoU}_{50}$  and  $7.8\%$  on  $5^{\circ}2\mathrm{cm}$ . When compared under the  $10^{\circ}5\mathrm{cm}$  metric, our method surpasses them by almost  $20\%$  (19.6% and  $19.5\%$ , respectively). It is notable that this dataset includes objects with photometric complexities and extensive viewpoint distributions, which is extremely challenging. The significant improvements on this benchmark further demonstrate the effectiveness of the proposed method.
+
+Results of Correspondence Errors. To validate the effectiveness of the proposed spherical representations compared to point-based representations in correspondence prediction, we calculate the mean NOCS errors on the REAL275 dataset. As shown in Table 3, the NOCS errors of Sphere-Pos are lower than DPDN (Lin et al., 2022b) and AG-Pose (Lin et al., 2024) on both the angle and
+
+Table 3: Comparison of correspondence errors.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Mean NOCS Error</td></tr><tr><td>Angle (°)</td><td>Distance (cm)</td></tr><tr><td>DPDN (Lin et al., 2022b)</td><td>16.62</td><td>10.50</td></tr><tr><td>AG-Pose (Lin et al., 2024)</td><td>10.36</td><td>5.88</td></tr><tr><td>SpherePose</td><td>4.52</td><td>3.89</td></tr></table>
+
+Euclidean distance, which indicates that the proposed method achieves more accurate correspondence prediction by learning shape-independent transformation on the proxy sphere.
+
+# 4.3 ABLATION STUDIES
+
+To shed more light on the superiority of our method, we conduct comprehensive ablation studies on REAL275 dataset, as detailed below.
+
+# Efficacy of SO(3)-invariant Point-wise Feature Extraction.
+
+Table 4 illustrates the impact of different deep point cloud backbones on the performance of object pose estimation. The original PointNet++ (Qi et al., 2017)
+
+Table 4: Ablation studies on the deep point cloud backbone.  
+
+<table><tr><td>Point Cloud Backbone</td><td>5°2cm</td><td>5°5cm</td><td>10°2cm</td><td>10°5cm</td><td>Parameters</td></tr><tr><td>None</td><td>55.9</td><td>64.6</td><td>74.3</td><td>86.7</td><td>0</td></tr><tr><td>PointNet++</td><td>56.3</td><td>64.7</td><td>73.2</td><td>84.4</td><td>1,391,104</td></tr><tr><td>ColorPointNet++</td><td>58.2</td><td>67.4</td><td>76.2</td><td>88.2</td><td>1,389,664</td></tr></table>
+
+inherently lacks SO(3)-invariance due to the injection of SO(3)-equivariant absolute XYZ coordinates. To validate the importance of SO(3)-invariant features for correspondence-based methods, we introduce minimal adaptations to PointNet++ to derive our ColorPointNet++, which enables the extraction of inherently SO(3)-invariant point cloud features while maintaining the architecture and the number of parameters almost unchanged. As shown in the table, with comparable number of parameters, our ColorPointNet++ outperforms the original PointNet++ by  $1.9\%$  on  $5^{\circ}2\mathrm{cm}$  and  $3.8\%$  on  $10^{\circ}5\mathrm{cm}$ . Notably, PointNet++ performs even worse than no point cloud backbone at all, showing a  $2.3\%$  drop on  $10^{\circ}5\mathrm{cm}$ . These results demonstrate the effectiveness of SO(3)-invariant point-wise features for correspondence-based object pose estimation methods.
+
+Efficacy of Spherical Feature Interaction. Table 5 illustrates the impact of varying numbers  $L$  of the Transformer encoder layers. When the Transformer encoder is not employed, there is a significant drop in performance to  $20.7\%$  on  $5^{\circ}2\mathrm{cm}$ , while when two layers of encoder are applied, the performance on  $5^{\circ}2\mathrm{cm}$  goes up to  $53.6\%$ . This result demonstrates the effectiveness of spherical
+
+Table 5: Impact of the number of encoder layers.  
+
+<table><tr><td>Encoder Layers</td><td>5°2cm</td><td>5°5cm</td><td>10°2cm</td><td>10°5cm</td></tr><tr><td>0</td><td>20.7</td><td>23.9</td><td>56.2</td><td>65.8</td></tr><tr><td>2</td><td>53.6</td><td>61.3</td><td>74.0</td><td>85.8</td></tr><tr><td>4</td><td>55.7</td><td>64.5</td><td>75.2</td><td>86.5</td></tr><tr><td>6</td><td>58.2</td><td>67.4</td><td>76.2</td><td>88.2</td></tr><tr><td>8</td><td>57.9</td><td>66.7</td><td>74.9</td><td>86.2</td></tr></table>
+
+Table 6: Impact of distinct loss functions.  
+
+<table><tr><td>Loss Function</td><td>5°2cm</td><td>5°5cm</td><td>10°2cm</td><td>10°5cm</td></tr><tr><td>L1</td><td>56.0</td><td>64.6</td><td>74.4</td><td>86.1</td></tr><tr><td>Smooth L1</td><td>54.6</td><td>63.2</td><td>72.6</td><td>83.3</td></tr><tr><td>Hyperbolic L1</td><td>58.0</td><td>66.5</td><td>73.5</td><td>85.0</td></tr><tr><td>L2</td><td>54.2</td><td>62.6</td><td>72.2</td><td>83.2</td></tr><tr><td>Hyperbolic L2</td><td>58.2</td><td>67.4</td><td>76.2</td><td>88.2</td></tr></table>
+
+feature interaction, which yields comprehensive spherical features from a holistic perspective. As the number of encoder layers increases, the precision of pose estimation is also improved, reaching a saturation at six layers. Further increasing the number of encoder layers does not lead to enhanced performance, but rather increases the inference overhead. Therefore, we set  $L = 6$  in our method.
+
+Efficacy of the Hyperbolic Correspondence Loss Function. Table 6 presents the impact of distinct loss functions on correspondence. As for the error with  $L^1$  distance in Equation 8, we conduct evaluations using  $L^1$  loss, smooth  $L^1$  loss and hyperbolic  $L^1$  loss, respectively. Since the smooth  $L^1$  loss has minor gradients near zero, it exhibits the worst performance of  $54.6\%$  on  $5^\circ 2\mathrm{cm}$ . Conversely, the hyperbolic  $L^1$  loss augments the gradients around zero to distinguish subtle distinctions, which improves the  $5^\circ 2\mathrm{cm}$  metric from  $56.0\%$  of the  $L^1$  loss to  $58.0\%$ . With regard to the error with  $L^2$  distance in Equation 8, we also derive the hyperbolic  $L^2$  loss, which results in an improvement on  $5^\circ 2\mathrm{cm}$  from  $54.2\%$  of the  $L^2$  loss to  $58.2\%$ . We assume that the  $L^2$  distance in Euclidean space is a better measure of the correspondence errors, so we adopt the hyperbolic  $L^2$  loss for correspondence prediction. For detailed formulas of these loss functions, please refer to Appendix A.2.
+
+Choice of Spherical Grids. Table 7 provides the performance comparison of adopting different spherical grids. To ensure a fair comparison, we use  $M = 768$  grids for both HEALPix (Gorski et al., 2005) and Super-Fibonacci Spirals (Alexa, 2022) grids. For Equirectangular (Driscoll
+
+& Healy, 1994) grids, the latitude and longitude resolution is set to  $28 \times 28$ , resulting in 784 grids, which is close to 768 in number. Since the HEALPix grids are area-uniform for the partition of sphere and can better capture the structure of point cloud, it achieves  $2.5\%$  and  $4.6\%$  higher performance on the  $5^{\circ}2\mathrm{cm}$  and  $10^{\circ}5\mathrm{cm}$  metrics, respectively, compared to the Equirectangular grids. In addition, we experiment with other nearly uniform spherical partitions, such as Super-Fibonacci Spirals, which also outperforms Equirectangular grids with a  $3.6\%$  gain on  $10^{\circ}5\mathrm{cm}$ , though still slightly below the performance of HEALPix.
+
+Table 7: Impact of the choice of spherical grids.  
+
+<table><tr><td>Spherical Grids</td><td>5°2cm</td><td>5°5cm</td><td>10°2cm</td><td>10°5cm</td></tr><tr><td>HEALPix</td><td>58.2</td><td>67.4</td><td>76.2</td><td>88.2</td></tr><tr><td>Equirectangular</td><td>55.7</td><td>64.7</td><td>72.1</td><td>83.6</td></tr><tr><td>Super-Fibonacci Spirals</td><td>56.5</td><td>65.0</td><td>75.8</td><td>87.2</td></tr></table>
+
+# 5 CONCLUSION
+
+In this work, we investigate the prevalent issue of intra-class semantic inconsistency of the canonical coordinates within existing correspondence-based approaches for category-level object pose estimation. We attribute this issue to shape-dependent point-based representations and propose to leverage the sphere as a shared proxy shape of objects to learn shape-independent transformation via spherical representations. In light of this insight, we introduce SpherePose, a novel architecture that achieves precise correspondence prediction through three core designs: SO(3)-invariant point-wise feature extraction for robust mapping between camera and object coordinate space, spherical feature interaction for holistic relationship integration, and a hyperbolic correspondence loss function for diagnosis of subtle correspondence errors. Experimental evaluations on existing benchmarks demonstrate the superiority of our method over state-of-the-art approaches.
+
+Limitations and Future Works. Although the employment of spherical representations equipped with spherical attention mechanism yields superior performance, the computational complexity of the attention mechanism, which scales quadratically with the number of tokens, limits the resolution of spherical grids. When the sphere is evenly partitioned into  $M = 768$  grids, each grid corresponds to an area of approximately 53.71 square degrees. For future work, we may explore linear attention mechanism (Wang et al., 2020) to facilitate spherical feature interaction. This could potentially enhance the resolution of spherical partitioning and improve the quality of spherical representations.
+
+# ACKNOWLEDGEMENT
+
+This work was partially supported by the National Nature Science Foundation of China (Grant 62306294, 62071122), Basic Strengthening Program Laboratory Fund (No.NKLDSE2023A009), Dreams Foundation of Jianghuai Advance Technology Center (NO.2023M743385), Youth Innovation Promotion Association of CAS.
+
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+
+# A APPENDIX
+
+# A.1 CLARIFICATIONS ON SO(3)-INVARIANCE
+
+In this section, we first provide a detailed explanation of SO(3)-invariance and equivariance. Subsequently, we elaborate on the SO(3)-invariance of DINOv2 and ColorPointNet++, respectively.
+
+SO(3)-Invariance/Equivariance. (1) Rotation estimation network should be SO(3)-equivariant. The rotation estimation task can be defined as  $\pmb{R} = \mathrm{Net}(\pmb{P})$ , where Net denotes the rotation estimation network. The SO(3)-equivariance between the input observed point cloud  $\pmb{P} \in \mathbb{R}^{N \times 3}$  and the output estimated rotation  $\pmb{R} \in \mathrm{SO}(3)$  is then formulated as:
+
+$$
+\operatorname {N e t} \left(\psi_ {\boldsymbol {G}} (\boldsymbol {P})\right) = \psi_ {\boldsymbol {G}} \left(\operatorname {N e t} (\boldsymbol {P})\right). \tag {10}
+$$
+
+(2) Correspondence predictor should ensure SO(3)-invariance. Correspondence-based methods can be expressed as  $\mathbf{R} = \operatorname{Net}(\mathbf{P}) = \Phi (\mathbf{P},\operatorname {Corr}(\mathbf{P}))$ , where  $\Phi$  is the rotation solver (e.g., the Umayama (Umayama, 1991) algorithm) and Corr is the correspondence (NOCS) predictor. Then, SO(3)-equivariance between the network input and output is formulated as:
+
+$$
+\Phi \left(\psi_ {\boldsymbol {G}} (\boldsymbol {P}), \operatorname {C o r r} \left(\psi_ {\boldsymbol {G}} (\boldsymbol {P})\right)\right) = \psi_ {\boldsymbol {G}} \left(\Phi (\boldsymbol {P}, \operatorname {C o r r} (\boldsymbol {P}))\right). \tag {11}
+$$
+
+Given fixed reference coordinates  $\mathrm{Corr}(\pmb {P})\in \mathbb{R}^{N\times 3}$ , the rotation solver  $\Phi$  is SO(3)-equivariant with respect to the other input  $\pmb{P}$ , which is written as:
+
+$$
+\Phi \left(\psi_ {G} (\boldsymbol {P}), \operatorname {C o r r} (\boldsymbol {P})\right) = \psi_ {G} \left(\Phi (\boldsymbol {P}, \operatorname {C o r r} (\boldsymbol {P}))\right). \tag {12}
+$$
+
+Consequently, the correspondence predictor Corr needs to be designed to ensure SO(3)-invariance:
+
+$$
+\operatorname {C o r r} \left(\psi_ {\boldsymbol {G}} (\boldsymbol {P})\right) = \operatorname {C o r r} (\boldsymbol {P}). \tag {13}
+$$
+
+It means that no matter how an object is rotated, a specific point on that object should be mapped to a static coordinate in NOCS.
+
+(3) Feature extractor in correspondence-based methods should be SO(3)-invariant. Since the correspondence predictor is typically defined as  $\mathrm{Corr}(\pmb {P}) = \mathrm{MLP}(f(\pmb {P}))$ , the point-wise feature extractor  $f(P)$  in correspondence-based approaches also needs to be SO(3)-invariant, formulated as:
+
+$$
+f \left(\psi_ {\mathbf {G}} (\mathbf {P})\right) = f (\mathbf {P}), \tag {14}
+$$
+
+which is consistent with Equation 1.
+
+SO(3)-invariance of DINOv2. To illustrate the SO(3)-invariance of DINOv2 (Oquab et al., 2024), Figure 5 presents a PCA visualization of DINOv2 features from the same instance under different rotations. The visualization shows that the DINOv2 features of the camera lens remain relatively consistent across various rotations, indicating their robustness to rotation variations. However, since DINOv2 does not strictly guarantee such semantic consistency, we conclude the features as approximately SO(3)-invariant, in line with the terminology stated in Second-Pose (Chen et al., 2024).
+
+![](images/e6755de8851afef5faed14279d9c43bd95028e0bddf3e6fc9b9564b35638b81b.jpg)  
+Figure 5: PCA visualization of DINOv2 features.
+
+SO(3)-invariance of ColorPointNet++. To further illustrate the SO(3)-invariance of the proposed ColorPointNet++ network, we show the pipeline comparison between ColorPointNet++ and PointNet++ (Qi et al., 2017) in Figure 6. Given the input points  $P \in \mathbb{R}^{N \times d}$ , where  $d$  denotes the coordinate dimension and is equal to 3 in Euclidean space, and associated point attributes  $F^{attr} \in \mathbb{R}^{N \times C_0}$  (e.g., RGB colors with  $C_0 = 3$ ), PointNet++ first concatenates them to yield the input features  $F^{in} \in \mathbb{R}^{N \times (d + C_0)}$ , and then aggregates the point-wise features  $F^{out} \in \mathbb{R}^{N \times C}$  by a hierarchical grouping and propagation strategy. It should be noted that in each set abstraction level and feature propagation level, PointNet++ concatenates the absolute XYZ coordinates onto the output features. The injection of SO(3)-equivariant XYZ coordinates on features results in the output features of PointNet++ lacking SO(3)-invariance. In contrast, the proposed ColorPointNet++ is structured to
+
+![](images/52105b38f4c038327a27a3decae7a609d9773e77f90cc32de8b1c9c6c88ff7af.jpg)  
+(a) PointNet++
+
+![](images/d2f0f0a2c304861d27d04c3526e9464482f9eccb12e7db1905d0a7f450b9f7ff.jpg)  
+(b) ColorPointNet++  
+Figure 6: Overview of comparison between the PointNet++ and our ColorPointNet++ network.
+
+achieve SO(3)-invariant feature extraction by avoiding the injection of absolute XYZ coordinates on the features. Specifically, XYZ coordinates are not used for input features and are not concatenated in each level, where the input features are derived exclusively from RGB values. Given that the k Nearest Neighbor (kNN) operation employed in the grouping and interpolation processes is SO(3)-invariant, the output features of our ColorPointNet++ network therefore exhibit SO(3)-invariance.
+
+# A.2 MORE IMPLEMENTATION DETAILS
+
+Training Details. For deep image feature extraction, we use the DINOv2 (Oquab et al., 2024) model with a ViT-S/14 (Dosovitskiy et al., 2020) architecture and feature dimension of 384. And for deep point cloud feature extraction, we implement the ColorPointNet++ network incorporating 4 set abstraction levels with multi-scale grouping, and the output feature dimension is 256. In terms of HEALPix grids (Gorski et al., 2005), the resolution is expressed by the parameter  $N_{side}$  and the number of grids is equal to  $N_{pix} = 12N_{side}^2$ , where we set  $N_{side} = 8$  and yield  $M = N_{pix} = 768$ . For data augmentation, we follow previous works (Lin et al., 2023b; 2024) to use perturbations with random translation  $\Delta t \sim U(-0.02, 0.02)$ , random scale  $\Delta s \sim U(0.8, 1.2)$ , and random rotational degree sampled from  $U(0, 20)$  for each axis. Regarding network optimization, we train SpherePose using the Adam (Kingma & Ba, 2015) optimizer for 200K iterations, with an initial learning rate of 0.001 and a cosine annealing schedule.
+
+Correspondence Loss Functions. The specific formulas corresponding to the loss functions in Table 6 within the ablation study section (Section 4.3) are provided below:
+
+$$
+\mathcal {L} _ {L ^ {1}} = \frac {1}{M} \sum_ {m = 1} ^ {M} \| O _ {m} - O _ {m} ^ {g t} \| _ {1}, \tag {15}
+$$
+
+$$
+\mathcal {L} _ {\text {S m o o t h} L ^ {1}} = \frac {1}{M} \sum_ {m = 1} ^ {M} \left\{ \begin{array}{l l} 5 \| \boldsymbol {O} _ {m} - \boldsymbol {O} _ {m} ^ {g t} \| _ {1} ^ {2}, & \| \boldsymbol {O} _ {m} - \boldsymbol {O} _ {m} ^ {g t} \| _ {1} \leq 0. 1 \\ \| \boldsymbol {O} _ {m} - \boldsymbol {O} _ {m} ^ {g t} \| _ {1} - 0. 0 5, & \| \boldsymbol {O} _ {m} - \boldsymbol {O} _ {m} ^ {g t} \| _ {1} > 0. 1 \end{array} , \right. \tag {16}
+$$
+
+$$
+\mathcal {L} _ {\text {H y p e r b o l i c} L ^ {1}} = \frac {1}{M} \sum_ {m = 1} ^ {M} \operatorname {a r c o s h} \left(1 + \| O _ {m} - O _ {m} ^ {g t} \| _ {1}\right), \tag {17}
+$$
+
+$$
+\mathcal {L} _ {L ^ {2}} = \frac {1}{M} \sum_ {m = 1} ^ {M} \| O _ {m} - O _ {m} ^ {g t} \| _ {2}, \tag {18}
+$$
+
+$$
+\mathcal {L} _ {\text {H y p e r b o l i c} L ^ {2}} = \frac {1}{M} \sum_ {m = 1} ^ {M} \operatorname {a r c o s h} \left(1 + \| O _ {m} - O _ {m} ^ {g t} \| _ {2}\right), \tag {19}
+$$
+
+Table 8: Performance comparison with state-of-the-art methods on CAMERA25 dataset.  
+
+<table><tr><td></td><td>Method</td><td>Representation</td><td>IoU50</td><td>IoU75</td><td>5°2cm</td><td>5°5cm</td><td>10°2cm</td><td>10°5cm</td></tr><tr><td rowspan="4">Direct Regression</td><td>GPV-Pose (Di et al., 2022)</td><td>Point-based</td><td>93.4</td><td>88.3</td><td>72.1</td><td>79.1</td><td>-</td><td>89.0</td></tr><tr><td>HS-Pose (Zheng et al., 2023)</td><td>Point-based</td><td>93.3</td><td>89.4</td><td>73.3</td><td>80.5</td><td>80.4</td><td>89.4</td></tr><tr><td>DualPoseNet (Lin et al., 2021)</td><td>Spherical</td><td>92.4</td><td>86.4</td><td>64.7</td><td>70.7</td><td>77.2</td><td>84.7</td></tr><tr><td>VI-Net (Lin et al., 2023b)</td><td>Spherical</td><td>-</td><td>-</td><td>74.1</td><td>81.4</td><td>79.3</td><td>87.3</td></tr><tr><td rowspan="7">Correspondence</td><td>NOCS (Wang et al., 2019b)</td><td>Point-based</td><td>83.9</td><td>69.5</td><td>32.3</td><td>40.9</td><td>48.2</td><td>64.4</td></tr><tr><td>SPD (Tian et al., 2020)</td><td>Point-based</td><td>93.2</td><td>83.1</td><td>54.3</td><td>59.0</td><td>73.3</td><td>81.5</td></tr><tr><td>SGPA (Chen &amp; Dou, 2021)</td><td>Point-based</td><td>93.2</td><td>88.1</td><td>70.7</td><td>74.5</td><td>82.7</td><td>88.4</td></tr><tr><td>IST-Net (Liu et al., 2023b)</td><td>Point-based</td><td>93.7</td><td>90.8</td><td>71.3</td><td>79.9</td><td>79.4</td><td>89.9</td></tr><tr><td>Query6DoF (Wang et al., 2023)</td><td>Point-based</td><td>91.9</td><td>88.1</td><td>78.0</td><td>83.1</td><td>83.9</td><td>90.0</td></tr><tr><td>AG-Pose (Lin et al., 2024)</td><td>Point-based</td><td>93.8</td><td>91.3</td><td>77.8</td><td>82.8</td><td>85.5</td><td>91.6</td></tr><tr><td>SpherePose</td><td>Spherical</td><td>94.8</td><td>92.4</td><td>78.3</td><td>84.3</td><td>84.8</td><td>92.3</td></tr></table>
+
+Table 9: IoU metric comparison with state-of-the-art methods on HouseCat6D dataset.  
+
+<table><tr><td></td><td colspan="2">Method</td><td>Representation</td><td>IoU25 / IoU50</td><td>Bottle</td><td>Box</td><td>Can</td><td>Cup</td><td>Remote</td><td>Teapot</td><td>Cutlery</td><td>Glass</td><td>Tube</td><td>Shoe</td></tr><tr><td rowspan="4">Direct Regression</td><td>FS-Net(Chen et al., 2021)</td><td>Point-based</td><td>74.9 / 48.0</td><td>65.3 / 45.0</td><td>31.7 / 1.2</td><td>98.3 / 73.8</td><td>96.4 / 68.1</td><td>65.6 / 46.8</td><td>69.9 / 59.8</td><td>71.0 / 51.6</td><td>99.4 / 32.4</td><td>79.7 / 46.0</td><td>71.4 / 55.4</td><td></td></tr><tr><td>GPV-Pose(Di et al., 2022)</td><td>Point-based</td><td>74.9 / 50.7</td><td>66.8 / 45.6</td><td>31.4 / 1.1</td><td>98.6 / 75.2</td><td>96.7 / 69.0</td><td>65.7 / 46.9</td><td>75.4 / 61.6</td><td>70.9 / 52.0</td><td>99.6 / 62.7</td><td>76.9 / 42.4</td><td>67.4 / 50.2</td><td></td></tr><tr><td>VI-Net(Lin et al., 2023b)</td><td>Spherical</td><td>80.7 / 56.4</td><td>90.6 / 79.6</td><td>44.8 / 12.7</td><td>99.0 / 67.0</td><td>96.7 / 72.1</td><td>54.9 / 17.1</td><td>52.6 / 47.3</td><td>89.2 / 76.4</td><td>99.1 / 93.7</td><td>94.9 / 36.0</td><td>85.2 / 62.4</td><td></td></tr><tr><td>SecondPose(Chen et al., 2024)</td><td>Spherical</td><td>83.7 / 66.1</td><td>94.5 / 79.8</td><td>54.5 / 23.7</td><td>98.5 / 93.2</td><td>99.8 / 82.9</td><td>53.6 / 35.4</td><td>81.0 / 71.0</td><td>93.5 / 74.4</td><td>99.3 / 92.5</td><td>75.6 / 35.6</td><td>86.9 / 73.0</td><td></td></tr><tr><td rowspan="3">Correspondence</td><td>NOCs(Wang et al., 2019b)</td><td>Point-based</td><td>50.0 / 21.2</td><td>41.9 / 5.0</td><td>43.3 / 6.5</td><td>81.9 / 62.4</td><td>68.8 / 2.0</td><td>81.8 / 59.8</td><td>24.3 / 0.1</td><td>14.7 / 6.0</td><td>95.4 / 49.6</td><td>21.0 / 4.6</td><td>26.4 / 16.5</td><td></td></tr><tr><td>AG-Pose(Lin et al., 2024)</td><td>Point-based</td><td>81.8 / 62.5</td><td>82.3 / 62.8</td><td>57.2 / 7.7</td><td>97.1 / 83.6</td><td>97.9 / 79.6</td><td>87.0 / 66.2</td><td>63.4 / 60.9</td><td>77.2 / 62.0</td><td>100.0 / 99.4</td><td>83.4 / 53.4</td><td>72.0 / 50.0</td><td></td></tr><tr><td>SpherePose</td><td>Spherical</td><td>88.8 / 72.2</td><td>98.6 / 87.4</td><td>58.9 / 5.6</td><td>97.4 / 87.2</td><td>100.0 / 97.7</td><td>79.0 / 63.5</td><td>94.8 / 87.3</td><td>85.5 / 74.5</td><td>99.6 / 98.0</td><td>74.5 / 30.0</td><td>99.8 / 90.8</td><td></td></tr></table>
+
+# A.3 ADDITIONAL EXPERIMENTAL RESULTS
+
+Results on CAMERA25 Dataset. In Table 8, we compare our method with the existing ones for category-level object pose estimation on CAMERA25 (Wang et al., 2019b) dataset. From the results we can see that SpherePose achieves the best performance. In detail, SpherePose outperforms the state-of-the-art correspondence-based method AG-Pose (Lin et al., 2024) by  $1.0\%$  on  $\mathrm{IoU}_{50}$ ,  $1.1\%$  on  $\mathrm{IoU}_{75}$ ,  $0.5\%$  on  $5^{\circ}2\mathrm{cm}$  and  $1.5\%$  on  $5^{\circ}5\mathrm{cm}$ , respectively.
+
+Results of IoU Metric on HouseCat6D Dataset. Table 9 presents the quantitative comparison of IoU metrics between our method and existing methods on HouseCat6D (Jung et al., 2024) dataset. Once again, our SpherePose attains the state-of-the-art performance, surpassing the second-best method SecondPose (Chen et al., 2024) by  $5.1\%$  on  $\mathrm{IoU}_{25}$  and  $6.1\%$  on  $\mathrm{IoU}_{50}$ .
+
+Ablation Studies on Point-wise Features. Table 10 presents the results of detailed ablation studies on point-wise features. For low-level image features, RGB values provide the low-level texture information, bringing a performance gain of  $1.4\%$  on  $5^{\circ}2\mathrm{cm}$ . For high-level image features, the frozen pretrained DINOv2 (Oquab et al., 2024) achieves a  $4.4\%$  higher performance on  $5^{\circ}2\mathrm{cm}$  compared to
+
+Table 10: Ablation studies on point-wise features.  
+
+<table><tr><td colspan="2">Point-wise Features</td><td>5°2cm</td><td>5°5cm</td><td>10°2cm</td><td>10°5cm</td></tr><tr><td rowspan="2">Low-level Image</td><td>RGB</td><td>58.2</td><td>67.4</td><td>76.2</td><td>88.2</td></tr><tr><td>None</td><td>56.8</td><td>65.9</td><td>75.9</td><td>87.7</td></tr><tr><td rowspan="3">High-level Image</td><td>DINOv2</td><td>58.2</td><td>67.4</td><td>76.2</td><td>88.2</td></tr><tr><td>None</td><td>48.7</td><td>57.1</td><td>69.2</td><td>80.3</td></tr><tr><td>ResNet18</td><td>53.8</td><td>62.5</td><td>72.8</td><td>83.8</td></tr><tr><td rowspan="2">Low-level Point Cloud</td><td>Radius</td><td>58.2</td><td>67.4</td><td>76.2</td><td>88.2</td></tr><tr><td>None</td><td>52.3</td><td>58.9</td><td>72.3</td><td>83.6</td></tr><tr><td rowspan="4">High-level Point Cloud</td><td>ColorPointNet++</td><td>58.2</td><td>67.4</td><td>76.2</td><td>88.2</td></tr><tr><td>None</td><td>55.9</td><td>64.6</td><td>74.3</td><td>86.7</td></tr><tr><td>PointNet++</td><td>56.3</td><td>64.7</td><td>73.2</td><td>84.4</td></tr><tr><td>HP-PPF</td><td>55.1</td><td>64.2</td><td>74.2</td><td>85.9</td></tr></table>
+
+the end-to-end trained ResNet18 (He et al., 2016). Due to the large-scale pretraining, DINOv2 produces semantically consistent patch-wise features that are robust to rotations, which facilitates correspondence prediction. For low-level point cloud features, removing the radius information results in a  $5.9\%$  drop on  $5^{\circ}2\mathrm{cm}$ , which highlights the importance of spatial structure awareness. Since the proposed ColorPointNet++ eliminates the injection of raw XYZ coordinates, its features lack implicit encoding of the underlying spatial structure, making explicit SO(3)-invariant radius information essential. For high-level point cloud features, ColorPointNet++ outperforms the original PointNet++ (Qi et al., 2017) by  $3.8\%$  on  $10^{\circ}5\mathrm{cm}$ , emphasizing the significance of SO(3)-invariant features. Additionally, We have tried HP-PPF (Chen et al., 2024), which is also SO(3)-invariant, but the performance is suboptimal compared to not using it. This might stem from the fact that HP-PPF samples 300 points from 2048 observed points to estimate point-wise normal vectors, making it prone to noise due to the sparse sampling. Moreover, as a parameter-free approach, HP-PPF lacks the deep feature extraction capabilities compared to the learnable architecture of ColorPointNet++.
+
+Table 11: Per-category results of our Sphere- Pose on REAL275 dataset.  
+
+<table><tr><td>Category</td><td>IoU50</td><td>IoU75</td><td>5°2cm</td><td>5°5cm</td><td>10°2cm</td><td>10°5cm</td></tr><tr><td>bottle</td><td>57.6</td><td>51.3</td><td>71.7</td><td>76.8</td><td>81.4</td><td>88.7</td></tr><tr><td>bowl</td><td>100.0</td><td>100.0</td><td>87.7</td><td>96.9</td><td>90.8</td><td>100.0</td></tr><tr><td>camera</td><td>90.9</td><td>77.5</td><td>5.4</td><td>5.7</td><td>43.9</td><td>50.1</td></tr><tr><td>can</td><td>71.4</td><td>70.6</td><td>76.5</td><td>81.6</td><td>91.9</td><td>98.4</td></tr><tr><td>laptop</td><td>84.8</td><td>75.5</td><td>59.5</td><td>92.7</td><td>60.0</td><td>97.0</td></tr><tr><td>mug</td><td>99.6</td><td>99.3</td><td>48.5</td><td>50.3</td><td>89.2</td><td>94.9</td></tr><tr><td>average</td><td>84.0</td><td>79.0</td><td>58.2</td><td>67.4</td><td>76.2</td><td>88.2</td></tr></table>
+
+Table 13: Per-category results of our SpherePose on CAMERA25 dataset.  
+
+<table><tr><td>Category</td><td>IoU50</td><td>IoU75</td><td>5°2cm</td><td>5°5cm</td><td>10°2cm</td><td>10°5cm</td></tr><tr><td>bottle</td><td>93.8</td><td>90.6</td><td>78.0</td><td>93.3</td><td>79.5</td><td>96.4</td></tr><tr><td>bowl</td><td>97.0</td><td>96.7</td><td>95.1</td><td>95.6</td><td>98.6</td><td>99.2</td></tr><tr><td>camera</td><td>94.5</td><td>91.6</td><td>69.2</td><td>72.8</td><td>78.9</td><td>86.9</td></tr><tr><td>can</td><td>92.2</td><td>91.7</td><td>96.7</td><td>97.9</td><td>97.2</td><td>98.6</td></tr><tr><td>laptop</td><td>97.9</td><td>91.4</td><td>73.1</td><td>87.4</td><td>75.6</td><td>92.1</td></tr><tr><td>mug</td><td>93.6</td><td>92.3</td><td>57.6</td><td>58.9</td><td>78.7</td><td>80.4</td></tr><tr><td>average</td><td>94.8</td><td>92.4</td><td>78.3</td><td>84.3</td><td>84.8</td><td>92.3</td></tr></table>
+
+Table 12: Per-category results of AG-Pose (Lin et al., 2024) on REAL275 dataset.  
+
+<table><tr><td>Category</td><td>IoU50</td><td>IoU75</td><td>5°2cm</td><td>5°5cm</td><td>10°2cm</td><td>10°5cm</td></tr><tr><td>bottle</td><td>57.7</td><td>50.3</td><td>62.0</td><td>64.9</td><td>83.4</td><td>88.0</td></tr><tr><td>bowl</td><td>100.0</td><td>100.0</td><td>88.7</td><td>94.3</td><td>94.1</td><td>99.7</td></tr><tr><td>camera</td><td>90.8</td><td>82.9</td><td>1.2</td><td>1.3</td><td>24.8</td><td>27.3</td></tr><tr><td>can</td><td>71.3</td><td>71.2</td><td>83.4</td><td>85.3</td><td>96.3</td><td>98.6</td></tr><tr><td>laptop</td><td>83.3</td><td>74.1</td><td>59.6</td><td>91.1</td><td>61.2</td><td>95.6</td></tr><tr><td>mug</td><td>99.4</td><td>98.5</td><td>32.9</td><td>33.4</td><td>88.3</td><td>89.3</td></tr><tr><td>average</td><td>83.7</td><td>79.5</td><td>54.7</td><td>61.7</td><td>74.7</td><td>83.1</td></tr></table>
+
+Table 14: Per-category results of AG-Pose (Lin et al., 2024) on CAMERA25 dataset.  
+
+<table><tr><td>Category</td><td>IoU50</td><td>IoU75</td><td>5°2cm</td><td>5°5cm</td><td>10°2cm</td><td>10°5cm</td></tr><tr><td>bottle</td><td>93.7</td><td>91.4</td><td>80.9</td><td>96.4</td><td>82.3</td><td>99.0</td></tr><tr><td>bowl</td><td>96.9</td><td>96.7</td><td>98.7</td><td>99.0</td><td>99.7</td><td>99.8</td></tr><tr><td>camera</td><td>89.2</td><td>84.3</td><td>57.0</td><td>60.9</td><td>73.6</td><td>81.1</td></tr><tr><td>can</td><td>92.1</td><td>92.0</td><td>99.7</td><td>99.8</td><td>99.7</td><td>99.9</td></tr><tr><td>laptop</td><td>97.5</td><td>90.8</td><td>76.1</td><td>85.9</td><td>80.6</td><td>92.4</td></tr><tr><td>mug</td><td>93.6</td><td>92.7</td><td>54.5</td><td>54.6</td><td>77.1</td><td>77.3</td></tr><tr><td>average</td><td>93.8</td><td>91.3</td><td>77.8</td><td>82.8</td><td>85.5</td><td>91.6</td></tr></table>
+
+Per-category Results. The REAL275 and CAMERA25 datasets share the same categories, including bottle, bowl, camera, can, laptop and mug. In order to conduct an in-depth analysis of the relative merits of our method, we show the per-category and average results of our Sphere-Pose and AG-Pose (Lin et al., 2024) on REAL275 dataset in Table 11 and Table 12, and on CAMERA dataset in Table 13 and Table 14, respectively. It can be observed that SpherePose outperforms AG-Pose mainly in the camera and mug categories, which have large intra-class shape variations. This result demonstrates the effective
+
+ness of our approach to learn shape-independent transformation via spherical representations, which can resolve the issue of intra-class semantic incoherence arising from point-based representations. We also provide the per-category results on HouseCat6D dataset in Table 15, which contains 10 household categories: bottle, box, can, cup, remote, teapot, cutlery, glass, tube and shoe. Note that we deal with all categories by a single model as in Lin et al. (2022b; 2023b; 2024).
+
+Convergence Speed of the Hyperbolic Correspondence Loss Function. Figure 7 visualizes the training convergence speed and accuracy comparison between our hyperbolic correspondence loss function and the traditional loss function. As discussed in Section 3.3, the hyperbolic  $L^2$  loss function is designed to improve the gradient of the traditional  $L^2$  loss function around zero, facilitating more fine-grained correspondence prediction. As shown in the figure, the hyperbolic  $L^2$  loss not only accelerates convergence during training but also achieves higher accuracy, demonstrating its effectiveness in improving correspondence prediction precision.
+
+Table 15: Per-category results of our SpherePose on HouseCat6D dataset.  
+
+<table><tr><td>Category</td><td>IoU25</td><td>IoU50</td><td>5°2cm</td><td>5°5cm</td><td>10°2cm</td><td>10°5cm</td></tr><tr><td>bottle</td><td>98.6</td><td>87.4</td><td>51.0</td><td>55.7</td><td>67.9</td><td>79.1</td></tr><tr><td>box</td><td>58.9</td><td>5.6</td><td>0.0</td><td>0.0</td><td>0.2</td><td>0.2</td></tr><tr><td>can</td><td>97.4</td><td>87.2</td><td>33.9</td><td>41.4</td><td>63.0</td><td>79.2</td></tr><tr><td>cup</td><td>100.0</td><td>97.7</td><td>2.8</td><td>3.0</td><td>60.6</td><td>68.4</td></tr><tr><td>remote</td><td>79.0</td><td>63.5</td><td>15.7</td><td>16.3</td><td>46.9</td><td>50.1</td></tr><tr><td>teapot</td><td>94.8</td><td>87.3</td><td>25.2</td><td>44.4</td><td>53.6</td><td>86.8</td></tr><tr><td>cutlery</td><td>85.5</td><td>74.5</td><td>1.4</td><td>1.6</td><td>8.2</td><td>13.0</td></tr><tr><td>glass</td><td>99.6</td><td>98.0</td><td>54.8</td><td>66.3</td><td>78.5</td><td>96.4</td></tr><tr><td>tube</td><td>74.5</td><td>30.0</td><td>1.0</td><td>1.2</td><td>5.3</td><td>6.7</td></tr><tr><td>shoe</td><td>99.8</td><td>90.8</td><td>7.3</td><td>29.3</td><td>24.5</td><td>73.2</td></tr><tr><td>average</td><td>88.8</td><td>72.2</td><td>19.3</td><td>25.9</td><td>40.9</td><td>55.3</td></tr></table>
+
+![](images/b453457419103fc015e6cf689034e92279f894368b866845bbd463ea0d5768ed.jpg)  
+Figure 7: Training comparison between the hyperbolic  $L^2$  and traditional  $L^2$  loss functions.
+
+Inference Speed and Overhead. Table 16 shows the comparison of the pose estimation performance, total number of parameters, and inference speed on REAL275 dataset. As demonstrated by the results, the increase in the number  $L$  of encoder layers leads to enhanced performance, reaching saturation at six layers. A further increase in the number of layers does not result in further perfor
+
+Table 16: Comparison of different methods in terms of performance on REAL275, inference speed and overhead. All the experiments for inference speed are conducted on a single RTX3090Ti GPU.  
+
+<table><tr><td>Method</td><td>5°2cm</td><td>5°5cm</td><td>10°2cm</td><td>10°5cm</td><td>Parameters (M)</td><td>Speed (FPS)</td></tr><tr><td>VI-Net (Lin et al., 2023b)</td><td>50.0</td><td>57.6</td><td>70.8</td><td>82.1</td><td>28.9</td><td>29.7</td></tr><tr><td>SecondPose (Chen et al., 2024)</td><td>56.2</td><td>63.6</td><td>74.7</td><td>86.0</td><td>60.2</td><td>14.3</td></tr><tr><td>DPDN (Lin et al., 2022b)</td><td>46.0</td><td>50.7</td><td>70.4</td><td>78.4</td><td>24.6</td><td>27.4</td></tr><tr><td>AG-Pose (Lin et al., 2024)</td><td>54.7</td><td>61.7</td><td>74.7</td><td>83.1</td><td>28.9</td><td>27.5</td></tr><tr><td>SpherePose with L = 2</td><td>53.6</td><td>61.3</td><td>74.0</td><td>85.8</td><td>25.7</td><td>26.4</td></tr><tr><td>SpherePose with L = 4</td><td>55.7</td><td>64.5</td><td>75.2</td><td>86.5</td><td>26.1</td><td>25.9</td></tr><tr><td>SpherePose with L = 6</td><td>58.2</td><td>67.4</td><td>76.2</td><td>88.2</td><td>26.5</td><td>25.3</td></tr><tr><td>SpherePose with L = 8</td><td>57.9</td><td>66.7</td><td>74.9</td><td>86.2</td><td>26.9</td><td>24.5</td></tr></table>
+
+mance gains, but rather increases the number of parameters and slows down the inference. Therefore, we set  $L = 6$  in our method. Moreover, our approach demonstrates superior performance over existing methods, retaining a comparable number of parameters and inference speed. Specifically, in comparison to the leading correspondence-based method AG-Pose (Lin et al., 2024), our SpherePose achieves a  $3.5\%$  improvement on  $5^{\circ}2\mathrm{cm}$ , with 2.4 million fewer parameters and only a slight decrease in FPS. When compared to the direct regression-based methods, SpherePose outperforms the advanced SecondPose (Chen et al., 2024) by  $2\%$  on  $5^{\circ}2\mathrm{cm}$ , with substantially fewer parameters (26.5 million and 60.2 million, respectively) and higher FPS (25.3 and 14.3, respectively).
+
+# A.4 CONCURRENT RELATED WORKS
+
+In this section, we discuss the distinctions between our approach and two concurrent related works.
+
+Correspondence-based work. Mariotti et al. (2024) propose a semantic correspondence estimation approach, which deal with the challenges of object symmetries and repeated parts by incorporating a weak geometric spherical prior to supplement leading self-supervised features with 3D perception capabilities. They project image features onto a spherical map and use viewpoint information to guide correspondence prediction. While both this work and ours utilize spherical representations, the primary focus and application are different. Mariotti et al. (2024) aim to improve semantic correspondence estimation, which involves finding local regions that correspond to the same semantic entities across images. In contrast, our SpherePose focuses on correspondence-based category-level object pose estimation, which involves establishing correspondences between observed points and normalized object coordinates, and then fitting the object pose. Mariotti et al. (2024) use spherical representations to learn 3D perception-driven semantic features, while we leverage spherical representations to learn shape-independent transformation, which is beneficial for handling large intra-class shape variations.
+
+Regression-based work. Lee & Cho (2024) propose a 3D rotation regression approach that directly predicts Wigner-D coefficients in the frequency domain, aligning with the operations of spherical CNNs. They use an SO(3)-equivariant pose harmonics predictor to ensure consistent pose estimation under arbitrary rotations, overcoming limitations of spatial parameterizations such as discontinuities and singularities. While both this work and ours aim to improve 3D rotation estimation, the operating domains and the way to estimate rotation are different. Lee & Cho (2024) focus on predicting Wigner-D coefficients in the frequency domain to capture 3D rotations, which is useful for ensuring consistent pose estimation under different rotations. Instead, our method operates in the spatial domain, leveraging spherical representations to learn shape-independent transformation rather than directly predicting rotations through the network.
+
+# A.5 VISUALIZATION
+
+Qualitative Comparison. In Figure 8, we provide the qualitative comparison of existing correspondence-based methods, i.e., DPDN (Lin et al., 2022b), AG-Pose (Lin et al., 2024) and our SpherePose on REAL275 dataset. The visualization results indicate that previous methods adopting point-based representations do not perform well when dealing with novel objects with large shape variations, such as the length variation in camera lenses and the curvature distinctions in mug han
+
+![](images/a61df54455cf56804d559f641f356b3b59f917522079d21337b0f447f63ec549.jpg)  
+Figure 8: Qualitative comparison of DPDN (Lin et al., 2022b), AG-Pose (Lin et al., 2024) and our SpherePose on REAL275 dataset. Red/Green indicates the predicted/GT results.
+
+dles. By learning shape-independent transformation on the sphere, the proposed method allows for enhanced pose estimation.
+
+More Visualization. We provide more visualization results of pose estimation from our SpherePose, including successful cases in Figure 9 and failed cases in Figure 10. We visualize all six previously unseen scenes within the REAL275 testing dataset, with three images for each scene. It can be observed that failures predominantly occur due to the omission of objects in the detection process and significant viewpoint variations that result in incomplete capture of some objects.
+
+![](images/bbcfaf67b23c749d63969fe2c667a7a8dcc3f2d0e3fd3ae5676747566e5490be.jpg)  
+Figure 9: Visualization of successful cases from our SpherePose on the 6 testing scenes of REAL275 dataset. Red/Green indicates the predicted/GT results.
+
+![](images/d6445e1d9fbc4422d6322c87c631a3a9b5dbec774ca68a01e4dbdada8ef82a2d.jpg)  
+Scene 1
+
+![](images/7a2feadb44f5b4a2c5289a7e654505bbd2028b22d47430b4a1519e699ac93e70.jpg)
+
+![](images/ecc815909fea91e2e3aec06ed7046c56abc9f871a07d098b9123a488f8b23293.jpg)
+
+![](images/c635bb4afde35f2393d619fb34f876d9af8f151f3fc753c038a93b8c5c18cc08.jpg)  
+Scene 2
+
+![](images/bb4058314e8c2af9fce88a9de77125020505bed5b3b7024e96dc8ab7488ecb0d.jpg)
+
+![](images/02b943e6f44a3ddba024f994560a862e1e8d86e660874cf14c44875613ba3bf3.jpg)
+
+![](images/5e3ace0a0eb310c3fdb024532fb807ece7dc7ccd5153e6f1accf88e4f09b2dbd.jpg)  
+Scene 3
+
+![](images/68154da035ef213f1a33513e3e7f79377c4aab23c5d36969df3f4f3658facfbf.jpg)
+
+![](images/c811f13dbc688aa96ca837abae0879314ce583d1f175cb3492d328713e169613.jpg)
+
+![](images/782780fa7b21d41838946b8a4e331640a91122ff0b36e357d7487d7148a50fed.jpg)  
+Scene 4
+
+![](images/216ab41a25a39d2c82f0a245b4372e5e9c76b2c686c48a0d63db8f17c4093abc.jpg)
+
+![](images/96b16ae9662f47bd87f5e7bf345b2581ad0076cd4848d9955c6959709bdaa9ac.jpg)
+
+![](images/475649f323c1c35877582e06bf8d2d98012e3493f3ded7595b9cf7c9dcf10e6b.jpg)  
+Scene 5
+
+![](images/1cf5686879ed8e93bfbeea21415dbccac18db69c4aa7005a429c7943bd4e47c0.jpg)
+
+![](images/585b0f39561db8e71206ccae36284d4c900f602eb5656c2b7887feed2eb6e573.jpg)
+
+![](images/e241a82a861211a1ddbc76f62c0bf191ba0a6a2f52a97792663d055e6cdde607.jpg)  
+Scene 6
+
+![](images/1a496c7b95573f5aea58f020426daba628338de0683987635db80e15f1a7f3eb.jpg)  
+Figure 10: Visualization of failed cases from our SpherePose on the 6 testing scenes of REAL275 dataset. Red/Green indicates the predicted/GT results.
+
+![](images/7944ffcd8eda56e1d476775852239428f6997ead8bd4d6ddd4dccbcb0e290b8d.jpg)
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+# LEARNING SPATIAL-SEMANTIC FEATURES FOR ROBUST VIDEO OBJECT SEGMENTATION
+
+Xin Li $^{1,2,3\dagger}$ , Deshui Miao $^{2\dagger}$ , Zhenyu He $^{2,1*}$ , Yaowei Wang $^{2,1*}$ , Huchuan Lu $^{4}$ , and Ming-Hsuan Yang $^{5,6}$
+
+$^{1}$ Pengcheng Laboratory  $^{2}$ Harbin Institute of Technology, Shenzhen  $^{3}$  Pazhou Lab (Huangpu)
+
+$^{4}$ Dalian University of Technology  $^{5}$ University of California at Merced  $^{6}$ Yonsei University
+
+# ABSTRACT
+
+Tracking and segmenting multiple similar objects with distinct or complex parts in long-term videos is particularly challenging due to the ambiguity in identifying target components and the confusion caused by occlusion, background clutter, and changes in appearance or environment over time. In this paper, we propose a robust video object segmentation framework that learns spatial-semantic features and discriminative object queries to address the above issues. Specifically, we construct a spatial-semantic block comprising a semantic embedding component and a spatial dependency modeling part for associating global semantic features and local spatial features, providing a comprehensive target representation. In addition, we develop a masked cross-attention module to generate object queries that focus on the most discriminative parts of target objects during query propagation, alleviating noise accumulation to ensure effective long-term query propagation. Extensive experimental results show that the proposed method achieves state-of-the-art performance on benchmark data sets, including the DAVIS2017 test (87.8%), YoutubeVOS 2019 (88.1%), MOSE val (74.0%), and LVOS test (73.0%), and demonstrate the effectiveness and generalization capacity of our model. The source code and trained models are released at https://github.com/yahooo-m/S3.
+
+# 1 INTRODUCTION
+
+Video Object Segmentation (VOS) aims to track and segment target objects specified in the initial frame with mask annotations in a video sequence (Xu et al., 2018; Pont-Tuset et al., 2017; Voigtlaender et al., 2019; Yang et al., 2021). This practice holds significant promise in various applications, particularly as the prevalence of video content increases in domains such as autonomous driving, augmented reality, and interactive video editing (Park et al., 2022; Bao et al., 2018). The main challenges faced by VOS include drastic target appearance change, occlusion, and identity confusion caused by similar objects and background clutter, which becomes more difficult in long-term videos.
+
+Existing VOS methods (Oh et al., 2019; Cheng et al., 2021a) typically perform video object segmentation by comparing the test frame with past frames. Specifically, they first use an association model to generate the correlated features of the test sample and target templates. Then, the target masks are predicted based on the correlated features. The pixel-wise correlated features (Wu et al., 2023) contribute to accurate mask predictions. To account for targets that appear differently over time, some methods (Yang et al., 2021; Cheng & Schwing, 2022) utilize a memory module to store these changing appearances. In addition, several recent approaches (Wang et al., 2023; Cheng et al., 2023a) introduce object queries to help distinguish different target objects to alleviate identity confusion.
+
+Although significant advances have been made in video object segmentation, existing methods do not perform well in challenging scenarios. First, when dealing with target objects with multiple complex or separate parts caused by occlusion, background clutter, and shape complexity, existing methods often generate incomplete prediction masks. This is because the pixel-wise correlation adopted by existing methods mainly focuses on detailed spatial information at the pixel level and neglects the semantic information at the instance level. Second, although object query improves the ID association
+
+![](images/3e8b95e356df50f6da371f96c9731fafdfe190c6a820dd8e3ff290a992e4d639.jpg)  
+Figure 1: Performance comparison in terms of the  $\mathcal{I}\&\mathcal{F}$  score on various datasets. The proposed method achieves favorable performance against state-of-the-art methods, including AOT_L (Yang et al., 2021), XMem (Cheng & Schwing, 2022), and Cutie (Cheng et al., 2023a) on all the datasets. All methods are trained using the YoutubeVOS and DAVIS datasets.
+
+accuracy, it does not work well in sequences with dramatic target appearance changes. Existing methods (Wang et al., 2023; Cheng et al., 2023a) update target queries based on the entire predicted sample, which can introduce noise and errors and further lead to tracking failures when targets are missed or their identities are swapped.
+
+To address the aforementioned issues, we propose a VOS algorithm, called S3, to learn spatial-semantic features and generate discriminative object queries for robust VOS. Specifically, we design a spatial semantic block with a semantic embedding module and a spatial dependencies modeling part. This block efficiently utilizes the semantic information and local details from pre-trained ViTs for VOS, avoiding the need to train all ViT backbone parameters. In addition, we propose a discriminative query that emphasizes representative target parts, enabling more reliable target representation and query updates. This approach mitigates noise accumulation during query propagation and enhances robustness in long-term videos. We evaluate the proposed method extensively on five benchmarks, including DAVIS 2017 (Pont-Tuset et al., 2017), Youtube VOS2018 (Xu et al., 2018), Youtube VOS2019, LVOS (Hong et al., 2023), and MOSE (Ding et al., 2023). Extensive experimental results show that our approach achieves state-of-the-art performance on all these datasets including the DAVIS2017 test  $(89.1\%)$ , YoutubeVOS 2019  $(88.5\%)$ , MOSE  $(75.1\%)$ , and LVOS test  $(73.0\%)$  datasets. In addition, the results achieved by only using the DAVIS and Youtube datasets for training show that our method generalizes well to different scenarios, as shown in Figure 1.
+
+In this work, we make the following contributions:
+
+- We propose a spatial-semantic network block to incorporate semantic information with spatial information for Video Object Segmentation. The spatial-semantic block integrates global semantic information from the CLS token of a pre-trained ViT backbone into the base features of the input sample and then models spatial dependencies using a spatial dependency modeling module, enhancing robust VOS performance when handling target objects with complex or separate parts.
+
+- We develop a discriminative query mechanism to capture the most representative region of the target for better target representation learning and updating. This significantly improves VOS datasets, especially in long-term VOS datasets.  
+- We demonstrate that the proposed method achieves state-of-the-art performance on five diverse datasets and evaluate the contribution of each proposed component with comprehensive ablation studies.
+
+# 2 RELATED WORK
+
+Video Object Segmentation. Video Object Segmentation (VOS) aims to separate objects from the background and identify each target object across frames based on the target mask annotations given in the initial frame. Early VOS methods (Caelles et al., 2017; Xiao et al., 2018; Voigtlaender & Leibe, 2017; Hu et al., 2017; Luiten et al., 2018) use test-time learning to adapt pre-trained segmentation models to separate the specified targets online. The test-time learning of a deep model is time-consuming and significantly reduces the run speed. To avoid online learning, propagation-based methods (Duke et al., 2021; Cheng et al., 2018; Tsai et al., 2016) use offline shift attention to model temporal correlations between adjacent frames, propagating target masks from the previous frame to the current one. Despite promising performance, these methods suffer from error accumulation caused by inaccurate predictions. Matching-based methods (Chen et al., 2018; Hu et al., 2018; Bhat et al., 2020; Yang et al., 2020) enable fast and efficient inference by comparing the target template with the test image and predicting target masks based on feature matching. For example, OML (Chen et al., 2018) and VideoMatch (Hu et al., 2018) perform pixel matching between adjacent frames for mask prediction. Several methods (Oh et al., 2019; Yang et al., 2021; Yang & Yang, 2022; Cheng & Schwing, 2022) improve the matching-based VOS framework by introducing target memory to enrich target templates and developing advanced matching strategies to generate more comprehensive correlated features. For example, SimVOS (Wu et al., 2023) and JointFormer (Mao et al., 2021) use ViT blocks (Dosovitskiy et al., 2021) to model features and patch correspondence for object modeling jointly. For better target association across frames, some recent methods (Wang et al., 2023; Cheng et al., 2023a) introduce object queries to enrich target representation. Despite good performance on simple target objects, existing methods lack modeling of object-level semantic information, so they do not work well on dealing with objects with complex structures and long-term deformation. Unlike existing methods, the proposed approach performs pixel- and object-level feature learning through a novel object-aware backbone and a query-level correlation mechanism, enabling robust VOS performance, particularly in complex and long-term video scenarios.
+
+Query-Based VOS Frameworks. DETR (Carion et al., 2020) introduces the transformer network (Dosovitskiy et al., 2021) to object detection tasks by using queries to represent target regions. In the segmentation domain, some methods (Cheng et al., 2022; Cavagnero et al., 2024) introduce masked cross-attention modules to use queries to represent targets for segmentation tasks. Motivated by the above methods, ISVOS (Wang et al., 2023) uses additional queries generated by Mask2Former (Cheng et al., 2022) to represent instance information for target modeling, enhancing target representation in the matching-based VOS framework. To enable end-to-end training, Cutie (Cheng et al., 2023a) learns object-level features and enhances the interaction between object-level and pixel-level information, leading to improved segmentation accuracy and efficiency. However, query-based VOS methods suffer from query drift due to ineffective propagation. These methods model the entire target sample in the query, incorporating redundant information that diminishes the saliency of the target object. Instead, the proposed method generates discriminative queries for target representation and propagation, which preserves key distinguishing information of target objects and offers greater robustness to variations.
+
+Adapters for Dense Tasks. The ViT-Adapter (Chen et al., 2022) efficiently leverages pre-trained models by introducing a dense adapter for enhanced information exchange within ViT blocks, improving detection and segmentation performance. To enhance spatial hierarchical features, ViT-CoMer (Xia et al., 2024) introduces a multi-receptive field feature pyramid module to provide multi-scale spatial features. However, both methods require a complete fine-tuning step, which is inefficient, particularly in the video domain. Meanwhile, they focus on spatial information enhancement while lacking semantic feature interaction between ViT blocks and multi-scale features.
+
+To better adapt to VOS tasks, the proposed method improves adapter-based VOS frameworks in the following aspects. First, we introduce a spatial-semantic module to integrate the global semantic features from ViT to multi-scale features. This module does not need extra modules to transmit information to ViT blocks, which is more efficient. Second, the proposed spatial-semantic integration block associates semantic priors with dense local features. Third, the proposed method freezes the ViT blocks during training, eliminating the need for full fine-tuning.
+
+# 3 PROPOSED ALGORITHM
+
+Our method aims to learn comprehensive target features that contain semantic, spatial, and discriminative information for video object segmentation. This helps us cope with complex target appearance variations and ID confusion between target objects with similar appearances in long-term videos. To this end, we propose a spatial-semantic feature learning network that first embeds semantic information from a trained ViT network with multi-scale features from a trainable CNN network. Then, we model spatial dependencies upon the fused feature, enabling spatial-semantic features for VOS. In addition, to ensure that the target query retains discriminative information during long-term appearance changes, we develop a discriminative query propagation module to capture and update local representations of the target object.
+
+# 3.1 OVERALL FRAMEWORK
+
+Figure 2(a) shows the overall framework of the proposed method, consisting of two core modules: a spatial-semantic feature generation module and a pixel-and-query dual-level target association module. Given a test frame  $\mathbf{X}_t \in \mathbb{R}^{3 \times H \times W}$ , the proposed method predicts the mask of all specified objects within it based on the initial frame  $\mathbf{X}_{init} \in \mathbb{R}^{3 \times H \times W}$  and its corresponding annotation mask  $\mathbf{M}_{init} \in \mathbb{R}^{H \times W}$ . First, the feature generation module takes the test frame  $\mathbf{X}_t$  as input and generates spatial-semantic features. Then, the target association module associates the spatial-semantic features with the reference samples pixel-wise and query-wise to better represent the correlated features. Finally, a decoder predicts the mask based on these correlated features. In addition, we develop a dual-level memory module consisting of target features and target queries to better represent video targets that vary over frames. Both kinds of memory are updated every few frames using the online predicted masks, enabling adaptive target representations for effective target association.
+
+# 3.2 SPATIAL-SEMANTIC FEATURE LEARNING
+
+Figure 2(b) shows the spatial-semantic network, which consists of a semantic embedding module that incorporates semantic information from a pre-trained ViT model into multi-scale features and a spatial dependency modeling module that learns spatial dependencies based on semantic features.
+
+Semantic embedding. As the VOS task is designed for generic objects and no class labels are given, it is challenging to learn semantic representations directly from the VOS dataset during training. Fortunately, the CLS token in a trained ViT model aggregates semantic information from the entire image and provides a rich, global representation of the image content. Thus, we incorporate the semantic features generated from the CLS token with the multi-scale features generated from CNN networks to obtain detailed semantic features at different scales. To be specific, we first concatenate the CLS token  $\mathcal{F}_{cls}^{i}$  with a global token representation  $\mathcal{F}_g$ , which is generated by averagely pooling the features from a ViT block, to generate global semantic features denoted as  $(\mathcal{F}_{cls}^{i},\mathcal{F}_{g})$ . Then, we use a cross-attention operation to generate the semantic-aware features  $\mathcal{F}_{sem}^{i}$  by taking  $(\mathcal{F}_{cls}^{i},\mathcal{F}_{g})$  as the Key and Value, and the spatial-temporal features  $(\mathcal{F}_{spsem}^{i - 1})$  from the  $(i - 1)$ -th spatial-semantic block as the Query, which is formulated as:
+
+$$
+\mathcal {F} _ {s e m} ^ {i} = f _ {s o f t m a x} \left(\frac {\left(W ^ {q} \left(\mathcal {F} _ {c l s} ^ {i} , \mathcal {F} _ {g} ^ {i}\right)\right) \left(W ^ {k} \mathcal {F} _ {s p s e m} ^ {i - 1}\right) ^ {\top}}{\sqrt {d}}\right) \left(W ^ {v} \mathcal {F} _ {s p s e m} ^ {i - 1}\right), \tag {1}
+$$
+
+where  $f_{\text{softmax}}$  is the SoftMax operation, and  $W^q$ ,  $W^k$ , and  $W^v$  are the projection matrices for Query, Key, and Value.
+
+Spatial dependency modeling. We note that learning spatial dependencies between semantic features is important for the VOS model to understand the relationships between different object parts, which
+
+![](images/6d21f91883e1d886438c6aefa5ed727c46f34340577554c6ce5eeff98df26b30.jpg)  
+Figure 2: Architecture of the proposed method. (a) shows the overall framework of the proposed method comprising a feature generation part, a target association component, and a decoder-based prediction part. The feature generation part contains multiple spatial-semantic blocks, as illustrated in (b), which learn spatial-semantic features by integrating semantic priors and spatial details. (c) illustrates the discriminative query propagation process, which learns to generate discriminative queries that represent high-level object information.
+
+helps handle objects with complex structures or separate parts. Given the features of the input image patches generated by a ViT backbone model  $\mathcal{F}_{vit}^{i}$ , we adopt the Multi-scale Deformable Cross-Attention operation (Carion et al., 2020) to capture the spatial information  $\mathcal{F}_{spa}^{i}$  as:
+
+$$
+\mathcal {F} _ {s p a} ^ {i} = \sum_ {j} f _ {\alpha} \left(\mathcal {F} _ {v i t} ^ {i}, f _ {g s} \left(\mathcal {F} _ {s e m} ^ {i}, \Delta p\right)\right) \cdot f _ {g s} \left(\mathcal {F} _ {s e m} ^ {i}, \Delta p\right), \tag {2}
+$$
+
+where  $\Delta p$  indicates the deformable offsets generated based on the query features  $\mathcal{F}_{vit}^{i}, f_{gs}$  performs grid sampling on  $\mathcal{F}_{sem}^{i}$  using the offsets  $\Delta p, f_{\alpha}$  denotes the attention weights generated based on the query features  $\mathcal{F}_{vit}^{i}$  and the grid sampled features, and  $\sum_{j}$  is the summation over all sampled points. To enable an effective long-range propagation through the cascade blocks, we use the residual connection (He et al., 2016) followed by a convolution-feed-forward to generate the final spatial-semantic feature of the current block:
+
+$$
+\mathcal {F} _ {s p s e m} ^ {i} = \mathcal {F} _ {s p a} ^ {i} + f _ {C o n F F N} \left(\mathcal {F} _ {s p a} ^ {i} + \mathcal {F} _ {s p s e m} ^ {i - 1}\right). \tag {3}
+$$
+
+The spatial-semantic block leverages the strengths of both a pre-trained ViT network for providing semantic information and a multi-scale CNN network for rich spatial information, enabling comprehensive spatial-semantic feature learning for robust VOS.
+
+Figure 3(a) shows how the feature changes during the processing of the proposed model by visualizing the feature maps in different processing steps. It shows that the semantic embedding module enhances the semantic information (boundaries) of the objects in the feature maps. In contrast, after spatial dependence modeling, the feature maps pay more attention to the details of the objects. Figure 3(b) compares the features generated by different networks. Compared to ViT and ResNet models, the proposed Spatial-Semantic model exhibits better target representation capabilities with significantly higher activations in target object areas.
+
+# 3.3 DISCRIMINATIVE QUERY PROPAGATION
+
+To propagate target queries effectively across frames, we update the target queries with the most distinctive feature of the target object.
+
+![](images/c3f7761de18cfee19e57219a56d34e1af1756bf46fb8c273a8d344f9167e7cfa.jpg)  
+Figure 3: Visualized feature maps from different backbones and stages. It shows that the proposed spatial-semantic model generates effective features for target representation.
+
+Discriminative feature generation. We select the discriminative feature of a target object by comparing the target query  $\mathcal{Q} \in \mathbb{R}^{N \times C}$  with every channel activation in the correlated feature map of the target and taking the most similar one. Formally, given the feature map  $\mathcal{F} \in \mathbb{R}^{H \times W \times C}$ , the salient channel activation feature of a target is computed as:
+
+$$
+\mathcal {Q} _ {s} = \mathcal {F} [ \underset {i \in (1, H * W)} {\arg \max } (\mathcal {F Q} ^ {T} \odot \mathcal {M}) ], \tag {4}
+$$
+
+where  $\odot$  denotes the dot product,  $\mathcal{M}$  represents a binary mask of the target, and  $\mathcal{F}[i]$  denotes the  $i$ -th channel activation of the feature map  $\mathcal{F}$ . We use the strategy in Cutie (Cheng et al., 2023a) to generate the target mask  $\mathcal{M}$ , which uses the predicted segmentation results in the previous frame.
+
+Query propagation. Based on the discriminative target feature  $\mathcal{Q}_s$  generated from the target sample and the input target query  $\mathcal{Q}_{\mathrm{in}}$  at the current stage, the output query  $\mathcal{Q}_{\mathrm{out}}$  is propagated as:
+
+$$
+\mathcal {Q} _ {\text {o u t}} = \left(\alpha \odot \frac {\mathcal {A}}{\| \mathcal {A} \| _ {2}} + \mathcal {Q} _ {s}\right) \mathcal {W} _ {\text {o u t}} + \mathcal {Q} _ {\text {i n}}, \tag {5}
+$$
+
+where  $\alpha$  is a learnable parameter to scale the matrix,  $\mathcal{W}_{\mathrm{out}}$  is a learnable projection matrix, and  $\mathcal{A}$  denotes the correspondence between salient query and salient pixel features, which is computed as.
+
+$$
+\mathcal {A} = \mathcal {Q} \odot \mathcal {Q} _ {s}. \tag {6}
+$$
+
+The proposed discriminative query propagation scheme adaptively refines target queries with the most representative features, which helps deal with the challenges of dramatic appearance variations in long-term videos. We also present the similarity map between the discriminative query and the feature in the last two rows, which indicates that our proposed discriminative query can represent and distinguish different objects.
+
+# 4 EXPERIMENTS
+
+# 4.1 EXPERIMENTAL SETTINGS
+
+Network. The full version of the proposed method uses a pre-trained ViT-base (Dosovitskiy et al., 2021) model as the ViT branch and constructs a simple CNN module for multi-scale feature generation.
+
+Table 1: Ablation study. All the variants are well aligned using the same settings and trained on the YoutubeVOS and DAVIS datasets. S3 denotes the proposed algorithm.  
+
+<table><tr><td>Dataset</td><td colspan="3">MOSE-val</td><td colspan="3">DAVIS 2017 test</td><td colspan="3">LVOS test</td><td colspan="3">LVOS val</td><td colspan="4">YouTube-VOS 2019 val</td><td></td></tr><tr><td>Method</td><td>J&amp;F</td><td>J</td><td>F</td><td>J&amp;F</td><td>J</td><td>F</td><td>J&amp;F</td><td>J</td><td>F</td><td>G</td><td>J</td><td>F</td><td>G</td><td>Js</td><td>Fs</td><td>Ju</td><td>Fu</td></tr><tr><td>XMem (Cheng &amp; Schwing, 2022) (Baseline)</td><td>53.3</td><td>62.0</td><td>57.6</td><td>81.0</td><td>77.4</td><td>84.5</td><td>50.0</td><td>45.5</td><td>54.4</td><td>49.5</td><td>45.2</td><td>53.7</td><td>85.5</td><td>84.3</td><td>88.6</td><td>80.3</td><td>88.6</td></tr><tr><td>Cutie (Cheng et al., 2023a)</td><td>64.0</td><td>60.0</td><td>67.9</td><td>84.2</td><td>80.6</td><td>87.7</td><td>56.2</td><td>51.8</td><td>60.5</td><td>60.1</td><td>55.9</td><td>64.2</td><td>86.1</td><td>85.8</td><td>90.5</td><td>80.0</td><td>88.0</td></tr><tr><td>+Discriminative Query</td><td>64.2</td><td>60.3</td><td>68.1</td><td>85.2</td><td>81.8</td><td>88.5</td><td>57.4</td><td>53.3</td><td>61.5</td><td>62.1</td><td>57.9</td><td>66.2</td><td>86.5</td><td>86.2</td><td>90.7</td><td>80.6</td><td>88.8</td></tr><tr><td>+ViT</td><td>64.2</td><td>60.2</td><td>68.3</td><td>85.6</td><td>82.0</td><td>89.2</td><td>58.3</td><td>53.7</td><td>62.8</td><td>58.9</td><td>54.1</td><td>63.6</td><td>86.7</td><td>86.4</td><td>90.3</td><td>81.0</td><td>88.7</td></tr><tr><td>+Spatial-semantic (S3)</td><td>68.5</td><td>64.5</td><td>72.6</td><td>86.7</td><td>82.7</td><td>90.8</td><td>66.5</td><td>62.1</td><td>70.8</td><td>67.7</td><td>63.1</td><td>72.2</td><td>87.5</td><td>86.8</td><td>91.8</td><td>81.3</td><td>89.9</td></tr></table>
+
+The Spatial-Semantic block performs feature modeling after every block of the ViT branch. Upon the target association part, the Decoder predicts target masks based on multi-scale features. We use a soft aggregation operation for the scenario containing multiple target objects to merge the predicted masks.
+
+In all experiments, we set the number of Spatial-Semantic Blocks to  $N = 4$ .
+
+Training. We adopt similar training settings as those of Cutie (Cheng et al., 2023a). Note that, apart from ResNet pre-training, we do not pre-train other models on static images and directly trained them on video datasets (YoutubeVOS (Xu et al., 2018) and DAVIS (Pont-Tuset et al., 2017)). To enhance the performance of our model, we use the MEGA dataset constructed by Cutie, which includes the YouTubeVOS (Xu et al., 2018), DAVIS (Pont-Tuset et al., 2017), OVIS (Qi et al., 2022), MOSE (Ding et al., 2023) and BURST (Athar et al., 2023) datasets. We sample 8 frames to train the model and randomly select 3 of them to train the matching process. For each sequence, we randomly choose at most 3 targets for training. Point supervision in loss is adopted to reduce the memory requirements.
+
+For optimization, the AdamW (Kingma & Ba, 2014) optimizer is used with a learning rate of 5e-5 and a weight decay of 0.5. We train the model for 125K with a batch size of 16 on the video dataset, and 195k on the MEGA dataset. For specific parameter settings, please refer to our supplementary material. All our models are trained on a machine with 8 x NVIDIA V100 GPUs and tested using one NVIDIA V100 GPU.
+
+Inference. Our feature and query memories are updated every 5th frame during the testing phase. For longer sequences, we employ a long-term fusion strategy (Cheng & Schwing, 2022) for updating. We skip frames without targets. We use two kinds of input sizes, 480 and 600/720, for more detailed comparisons. In this section, we compare all methods with input size 480p.
+
+# 4.2 ABLATION STUDY
+
+Effect of every component. Table 1 provides a comprehensive analysis of the components of our method. The first two rows describe the current methods upon which we build improvements. Cutie improves performance by using a query transformer within the model.
+
+Baseline. We adopt XMem as the baseline model, a representative VOS framework with a well-designed memory mechanism.
+
++ Discriminative Query, which performs discriminative query generation by adaptively refining target queries with the most representative features of the query transformer. To validate the effectiveness of our proposed discriminative query propagation, we first choose ResNet50 as the backbone for feature generation (simply comparing query propagation with Cutie (Cheng et al., 2023a)). Performance gains compared to Cutie demonstrate the advantages of target modeling of discriminative queries for VOS, especially in long-term benchmarks.  
++ ViT, which directly utilizes ViTDet (Li et al., 2022b) to incorporate pre-trained model features at multiple scales. The performance indicates that directly employing ViT to generate multi-scale features is minimally beneficial for VOS tasks.  
++ Spatial-semantic, which applies the spatial-semantic blocks to integrate semantic information and multi-scale features. It is clear that the proposed spatial dependency modeling significantly boosts model performance in terms of  $\mathcal{J}\&\mathcal{F}$  on all VOS benchmarks. These improvements validate the benefits of the spatial-semantic network, which improves the semantic understanding of targets.
+
+Table 2: Effect of different settings in every component. The ablation study examines the impact of pre-trained weights and the number of queries on training results for the YoutubeVOS and DAVIS datasets. Additionally, it explores the effects of different adapters on the MEGA dataset.  
+
+<table><tr><td rowspan="2">Dataset
+Method</td><td colspan="3">MOSE-val</td><td colspan="3">DAVIS 2017 test</td><td colspan="3">LVOS test</td><td colspan="5">YouTube-VOS 2018 val</td><td colspan="4">YouTube-VOS 2019 val</td><td></td></tr><tr><td>J&amp;F</td><td>J</td><td>F</td><td>J&amp;F</td><td>J</td><td>F</td><td>J&amp;F</td><td>J</td><td>F</td><td>G</td><td>Jn</td><td>Fn</td><td>Ju</td><td>Fu</td><td>G</td><td>Jn</td><td>Fn</td><td>Ju</td><td>Fu</td></tr><tr><td colspan="19">Effect of different pre-trained weights.</td><td></td></tr><tr><td>ResNet (He et al., 2016) (Baseline)</td><td>64.2</td><td>60.3</td><td>68.1</td><td>85.2</td><td>81.8</td><td>88.5</td><td>57.4</td><td>53.3</td><td>61.5</td><td>86.3</td><td>86.2</td><td>91.0</td><td>79.9</td><td>88.3</td><td>86.5</td><td>86.2</td><td>90.7</td><td>80.6</td><td>88.8</td></tr><tr><td>MAE (He et al., 2022)</td><td>67.0</td><td>62.6</td><td>71.3</td><td>85.3</td><td>81.3</td><td>89.3</td><td>64.4</td><td>60.2</td><td>68.7</td><td>86.4</td><td>85.6</td><td>90.2</td><td>80.6</td><td>89.0</td><td>86.4</td><td>85.4</td><td>89.7</td><td>81.2</td><td>89.3</td></tr><tr><td>DINov2 (Oquab et al., 2023)</td><td>67.5</td><td>63.5</td><td>71.6</td><td>86.1</td><td>82.7</td><td>89.6</td><td>63.2</td><td>58.7</td><td>67.6</td><td>87.3</td><td>86.6</td><td>91.7</td><td>80.9</td><td>89.9</td><td>86.7</td><td>86.2</td><td>91.0</td><td>80.6</td><td>89.1</td></tr><tr><td>ViT-small (Yang et al., 2024)</td><td>64.3</td><td>60.0</td><td>68.5</td><td>85.3</td><td>81.5</td><td>89.1</td><td>55.9</td><td>51.6</td><td>60.1</td><td>86.0</td><td>85.3</td><td>90.4</td><td>79.9</td><td>88.5</td><td>86.3</td><td>85.3</td><td>90.1</td><td>80.6</td><td>89.1</td></tr><tr><td>Depth_Anything</td><td>68.5</td><td>64.5</td><td>72.6</td><td>86.7</td><td>82.7</td><td>90.8</td><td>66.5</td><td>62.1</td><td>70.8</td><td>87.4</td><td>87.0</td><td>92.0</td><td>80.9</td><td>89.7</td><td>87.5</td><td>86.8</td><td>91.8</td><td>81.3</td><td>89.9</td></tr><tr><td colspan="19">Effect of different query numbers.</td><td></td></tr><tr><td>32-query</td><td>68.3</td><td>64.3</td><td>72.3</td><td>85.8</td><td>81.7</td><td>89.9</td><td>64.8</td><td>60.4</td><td>69.6</td><td>86.8</td><td>86.9</td><td>92.0</td><td>79.8</td><td>88.6</td><td>86.9</td><td>86.8</td><td>91.6</td><td>80.3</td><td>88.8</td></tr><tr><td>16-query</td><td>67.7</td><td>63.6</td><td>71.8</td><td>86.6</td><td>82.7</td><td>90.2</td><td>66.4</td><td>62.0</td><td>70.9</td><td>86.9</td><td>86.7</td><td>91.8</td><td>80.1</td><td>88.9</td><td>87.0</td><td>86.6</td><td>91.6</td><td>80.6</td><td>89.3</td></tr><tr><td>8-query</td><td>68.5</td><td>64.5</td><td>72.6</td><td>86.7</td><td>82.7</td><td>90.8</td><td>66.5</td><td>62.1</td><td>70.8</td><td>87.4</td><td>87.0</td><td>92.0</td><td>80.9</td><td>89.7</td><td>87.5</td><td>86.8</td><td>91.8</td><td>81.3</td><td>89.9</td></tr><tr><td colspan="19">Effect off different adapters.</td><td></td></tr><tr><td>Ours (Adapter (Chen et al., 2022))</td><td>72.2</td><td>67.8</td><td>76.4</td><td>87.2</td><td>84.1</td><td>90.4</td><td>71.1</td><td>66.7</td><td>75.6</td><td>87.1</td><td>86.1</td><td>91.0</td><td>81.6</td><td>89.8</td><td>87.1</td><td>85.9</td><td>90.9</td><td>81.4</td><td>90.1</td></tr><tr><td>Ours (CoMer (Xia et al., 2024))</td><td>73.2</td><td>69.1</td><td>77.3</td><td>87.0</td><td>83.2</td><td>90.8</td><td>73.4</td><td>68.8</td><td>77.9</td><td>87.3</td><td>87.1</td><td>92.2</td><td>80.7</td><td>89.4</td><td>87.4</td><td>87.0</td><td>91.7</td><td>81.3</td><td>89.6</td></tr><tr><td>Ours (Spatial-Semantic)</td><td>74.0</td><td>69.8</td><td>78.3</td><td>87.8</td><td>84.0</td><td>91.7</td><td>73.0</td><td>68.3</td><td>77.8</td><td>88.0</td><td>87.0</td><td>91.8</td><td>82.5</td><td>90.7</td><td>88.1</td><td>87.4</td><td>92.5</td><td>81.9</td><td>90.7</td></tr></table>
+
+Effect of different pre-trained weights. Table 2 also compares the performance of using different training parameters. We conduct experiments using three different pre-training parameters, and the results indicate that better pre-training parameters lead to more significant improvements in the model, especially those from Depth Anything (Yang et al., 2024) (8-query in Table 2). The Depth Anything model implicitly contains depth information about the scene, which contributes to its stronger representation capability, consistent with the findings of the Depth Anything paper.
+
+Effect of different query numbers. Table 2 shows that performance drops as the number of queries increases. This is because more queries introduce additional background noise, given the fewer targets in most VOS benchmarks, except MOSE (Ding et al., 2023).
+
+Effect of adapters. Table 2 presents a performance comparison between different adapters and our method. We replace the fusion module with various adapters and fully fine-tune them on the MEGA dataset, as indicated in related research that full fine-tuning achieves optimal performance (Chen et al., 2022; Xia et al., 2024). The experimental results demonstrate that our method is efficient in training and outperforms the other two methods in accuracy.
+
+# 4.3 COMPARISONS WITH STATE-OF-THE-ART METHODS
+
+DAVIS 2017 (Pont-Tuset et al., 2017) is a benchmark that offers densely annotated, high-quality, full-resolution videos with multiple objects of interest. Table 3 shows that our method achieves favorable performance  $(86.7\%)$  against state-of-the-art methods when trained on YoutubeVOS and DAVIS, without any pre-training. The gains are primarily due to the injection of semantic information, which helps handle sequences with object occlusion. When using the MEGA dataset as the training data, our method performs best on the DAVIS test set  $(87.8\%)$ .
+
+YouTubeVOS 2018 (Xu et al., 2018) contains 347 videos of 65 categories for training and videos for validation. For validation, there are 26 categories that the model has not seen in its training, allowing us to evaluate its generalization ability for 350 class-agnostic targets. Table 3 illustrates that our method performs better on seen and unseen categories than existing state-of-the-art models. The performance gain can be attributed to spatial-semantic feature representation and representative target correlation.
+
+YouTubeVOS 2019 is an extension of YouTube-VOS 2018, featuring a larger number of masked targets and more challenging sequences in its validation set. In Table 3, our approach exhibits competitive performance compared to the leading state-of-the-art methods. Remarkably, our model achieves the highest performance (87.5%) without relying on pre-training. This underscores the effectiveness of our method in leveraging inherent semantic-spatial information and discriminative query representation.
+
+MOSE (Ding et al., 2023) contains 2149 video clips and 5200 objects from 36 categories under more complex scenes. Compared to existing state-of-the-art methods, our model gains great improvement in this dataset. The benefits stem primarily from the discriminative query that captures more
+
+Table 3: Quantitative comparisons on the MOSE, LVOS, DAVIS 2017, YouTube-VOS 2018 & 2019 dataset. The best two results are shown in red and blue color. In the table, * denotes that the models are pre-trained using the static image datasets (Shi et al., 2016; Li et al., 2020; Zeng et al., 2019; Cheng et al., 2020; Wang et al., 2017). In the Appendix, we compare methods with external training data (e.g., large BL30K dataset for pre-training) and improved test size (600p/720p).  
+
+<table><tr><td>Dataset</td><td colspan="3">MOSE-val</td><td colspan="3">LVOS test</td><td colspan="3">DAVIS 2017 test</td><td colspan="5">YouTube-VOS 2018 val</td><td colspan="5">YouTube-VOS 2019 val</td><td rowspan="2">FPS</td></tr><tr><td>Method</td><td>J&amp;F</td><td>J</td><td>F</td><td>J&amp;F</td><td>J</td><td>F</td><td>J&amp;F</td><td>J</td><td>F</td><td>G</td><td>Js</td><td>Fs</td><td>Ju</td><td>Fu</td><td>G</td><td>Js</td><td>Fs</td><td>Ju</td><td>Fu</td></tr><tr><td colspan="21">Trained only on the YouTube VOS, and DAVIS datasets</td></tr><tr><td>MiVOS (Cheng et al., 2021b)*</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>78.6</td><td>74.9</td><td>82.2</td><td>82.6</td><td>81.1</td><td>85.6</td><td>77.7</td><td>86.2</td><td>82.4</td><td>80.6</td><td>84.7</td><td>78.1</td><td>86.4</td><td>-</td></tr><tr><td>STCN (Cheng et al., 2021a)*</td><td>52.5</td><td>48.5</td><td>56.6</td><td>45.8</td><td>41.1</td><td>50.5</td><td>77.8</td><td>74.3</td><td>81.3</td><td>84.3</td><td>83.2</td><td>87.9</td><td>79.0</td><td>87.3</td><td>84.2</td><td>82.6</td><td>87.0</td><td>79.4</td><td>87.7</td><td>13.2</td></tr><tr><td>Swin-B-AOT-L (Yang et al., 2021)*</td><td>59.4</td><td>53.6</td><td>65.2</td><td>54.4</td><td>49.3</td><td>59.4</td><td>81.2</td><td>77.3</td><td>85.1</td><td>85.1</td><td>90.1</td><td>78.4</td><td>86.9</td><td>85.3</td><td>84.6</td><td>89.5</td><td>79.3</td><td>87.7</td><td>12.1</td><td></td></tr><tr><td>DeAOT-R50 (Yang &amp; Yang, 2022)</td><td>59.0</td><td>54.6</td><td>63.4</td><td>-</td><td>-</td><td>-</td><td>80.7</td><td>76.9</td><td>84.5</td><td>86.0</td><td>84.9</td><td>89.9</td><td>80.4</td><td>88.7</td><td>85.6</td><td>84.2</td><td>89.2</td><td>80.2</td><td>88.8</td><td>11.7</td></tr><tr><td>XMem (Cheng &amp; Schwing, 2022)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>79.8</td><td>76.3</td><td>83.4</td><td>84.3</td><td>83.9</td><td>88.8</td><td>77.7</td><td>86.7</td><td>84.2</td><td>83.8</td><td>88.3</td><td>78.1</td><td>86.7</td><td></td></tr><tr><td>XMem (Cheng &amp; Schwing, 2022)*</td><td>53.3</td><td>62.0</td><td>57.6</td><td>50.0</td><td>45.5</td><td>54.4</td><td>81.0</td><td>77.4</td><td>84.5</td><td>85.7</td><td>84.6</td><td>89.3</td><td>80.2</td><td>88.7</td><td>85.5</td><td>84.3</td><td>88.6</td><td>80.3</td><td>88.6</td><td>22.6</td></tr><tr><td>ISVOS (Wang et al., 2023)*</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>82.8</td><td>79.3</td><td>86.2</td><td>86.3</td><td>85.5</td><td>90.2</td><td>80.5</td><td>88.8</td><td>86.1</td><td>85.2</td><td>89.7</td><td>80.7</td><td>88.9</td><td>5.8</td></tr><tr><td>SimVOS-B (Wu et al., 2023)</td><td>61.6</td><td>57.9</td><td>65.3</td><td>-</td><td>-</td><td>-</td><td>82.3</td><td>78.7</td><td>85.8</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>84.2</td><td>83.1</td><td>87.5</td><td>79.1</td><td>87.2</td><td>3.3</td></tr><tr><td>Cutie (Cheng et al., 2023a)*</td><td>64.0</td><td>60.0</td><td>67.9</td><td>56.2</td><td>51.8</td><td>60.5</td><td>84.2</td><td>80.6</td><td>87.7</td><td>86.1</td><td>85.5</td><td>90.0</td><td>80.6</td><td>88.3</td><td>86.1</td><td>85.8</td><td>90.5</td><td>80.0</td><td>88.0</td><td>36.4</td></tr><tr><td>JointFormer (Zhang et al., 2023)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>87.0</td><td>83.4</td><td>90.6</td><td>86.0</td><td>86.0</td><td>91.0</td><td>79.5</td><td>87.5</td><td>86.2</td><td>85.7</td><td>90.5</td><td>80.4</td><td>88.2</td><td>3.0</td></tr><tr><td>JointFormer (Zhang et al., 2023)*</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>87.6</td><td>84.2</td><td>91.1</td><td>87.0</td><td>86.2</td><td>91.0</td><td>81.4</td><td>89.3</td><td>87.0</td><td>86.1</td><td>90.6</td><td>82.0</td><td>89.5</td><td>3.0</td></tr><tr><td>S3 (Ours)</td><td>68.5</td><td>64.5</td><td>72.6</td><td>66.5</td><td>62.1</td><td>70.8</td><td>86.7</td><td>82.7</td><td>90.8</td><td>87.4</td><td>87.0</td><td>92.0</td><td>80.9</td><td>89.7</td><td>87.5</td><td>86.8</td><td>91.8</td><td>81.3</td><td>89.9</td><td>13.1</td></tr><tr><td colspan="21">Trained on the MEGA dataset</td></tr><tr><td>DEVA (Cheng et al., 2023b)</td><td>66.5</td><td>62.3</td><td>70.8</td><td>54.0</td><td>49.0</td><td>59.0</td><td>83.2</td><td>79.6</td><td>86.8</td><td>86.2</td><td>85.4</td><td>89.9</td><td>80.5</td><td>89.1</td><td>85.8</td><td>84.8</td><td>89.2</td><td>80.3</td><td>88.8</td><td>25.3</td></tr><tr><td>Cutie (Cheng et al., 2023a)*</td><td>69.9</td><td>65.8</td><td>74.1</td><td>66.7</td><td>62.4</td><td>71.0</td><td>86.1</td><td>82.4</td><td>89.9</td><td>87.0</td><td>86.4</td><td>91.1</td><td>81.4</td><td>89.2</td><td>87.0</td><td>86.0</td><td>90.5</td><td>82.0</td><td>89.6</td><td>36.4</td></tr><tr><td>S3 (Ours)</td><td>74.0</td><td>69.8</td><td>78.3</td><td>73.0</td><td>68.3</td><td>77.8</td><td>87.8</td><td>84.0</td><td>91.7</td><td>88.0</td><td>87.0</td><td>91.8</td><td>82.5</td><td>90.7</td><td>88.1</td><td>87.4</td><td>92.5</td><td>81.9</td><td>90.7</td><td>13.1</td></tr></table>
+
+Table 4: Comparison with SAM 2 (Ravi et al., 2024). It shows that our method performs favorably against SAM 2.  
+
+<table><tr><td>Dataset</td><td colspan="3">LVOSV2-val</td><td colspan="3">LVOS-val</td><td colspan="3">DAVIS17-test</td><td colspan="3">MOSE-val</td><td colspan="3">YouTube-VOS 2019 val</td></tr><tr><td>Method</td><td>J&amp;F</td><td>J</td><td>F</td><td>J&amp;F</td><td>J</td><td>F</td><td>J&amp;F</td><td>J</td><td>F</td><td>G</td><td>J</td><td>F</td><td>G</td><td>J</td><td>F</td></tr><tr><td>SAM 2 (Hiera-B+)</td><td>75.8</td><td>72.0</td><td>79.6</td><td>74.9</td><td>70.2</td><td>79.6</td><td>88.3</td><td>85.0</td><td>91.5</td><td>75.8</td><td>71.8</td><td>79.9</td><td>88.4</td><td>85.2</td><td>91.6</td></tr><tr><td>SAM 2 (Hiera-L)</td><td>78.1</td><td>74.3</td><td>81.9</td><td>76.1</td><td>71.6</td><td>80.6</td><td>89.0</td><td>85.8</td><td>92.2</td><td>77.2</td><td>73.3</td><td>81.2</td><td>89.1</td><td>86.0</td><td>92.2</td></tr><tr><td rowspan="3">S3 (ViTb+)</td><td rowspan="3">(+7.9)</td><td rowspan="3">83.7</td><td rowspan="3">80.1</td><td rowspan="3">(+1.1)</td><td rowspan="3">76.0</td><td rowspan="3">71.2</td><td rowspan="3">80.8</td><td rowspan="3">(+1.5)</td><td rowspan="3">89.8</td><td rowspan="3">86.4</td><td rowspan="3">93.1</td><td rowspan="3">(+0.1)</td><td rowspan="3">75.9</td><td>71.8</td><td>79.8</td></tr><tr><td>(+0.3)</td><td>88.7</td></tr><tr><td>85.5</td><td>91.9</td></tr></table>
+
+representative target information. Additionally, the feature extraction module provides matching with more semantic and spatially informative features.
+
+LVOS (Hong et al., 2023) is a new long-term dataset to validate the robustness of VOS methods when faced with these challenges. It contains videos 20 times longer than existing VOS datasets, typically featuring a single target per video. In this benchmark, our method achieves a performance gain  $(66.5\%)$  compared to  $56.2\%$ . This improvement is attributed to the quality of memorized features and queries, which enhances the robustness of our approach in long-term video scenarios. Our method demonstrates enhanced stability and effectiveness in long-duration contexts by incorporating more discriminative queries that capture crucial target information into the object memory.
+
+Comparison with Segment Anything 2 (Ravi et al., 2024). Table 4 shows the detailed comparison results between our method and SAM2. Specifically, SAM2 used a large-scale mixed training dataset (comprising 1.  $3\%$  DAVIS, approximately  $9.4\%$  MOSE, approximately  $9.2\%$  YouTubeVOS, approximately  $15.5\%$  SA-1B, approximately  $49.5\%$  SA-V, and approximately 15.  $1\%$  Internal, approximately  $120.6\mathbf{k}$  videos). In contrast, our model is trained solely on VOS datasets (DAVIS, MOSE, YouTubeVOS, OVIS, and BURST, about  $11.4\mathbf{K}$  videos). To ensure a relatively fair comparison, we use the same model size (ViTb) and the same input image size (1024p). In all benchmarks evaluated, our method outperforms SAM2 in all metrics, especially on the challenging LVOSV2 val dataset  $(+7.9\%)$ . Even when SAM2 (Ravi et al., 2024) uses Hiera-L as a feature extractor, our model demonstrates comparable results, especially on LVOS V2 (Hong et al., 2023) val dataset ( $83.7\%$  vs  $78.1\%$ ).
+
+Visualized Results. We visualize some challenging sequences, including scenarios involving part-to-whole-changing and faint objects and those where targets closely resemble the background. In Figure 4(a), our approach shows favorable segmentation performance against the state-of-the-art methods. For example, our model accurately segments targets and provides detailed results in a video that includes gorillas, necks, and a riding man. This can be attributed to our proposed discriminative query propagation, which enables our model to capture more detailed target information by focusing on the most representative features, thereby facilitating precise segmentation in these challenging scenarios. Additionally, incorporating semantic features in our model ensures the model understands
+
+![](images/83bfa699d0b38587ec6a96214e2e1c8493f6563fbc753c6947bc092d86eb3b91.jpg)  
+(b) Qualitative comparisons with target appearance variance.  
+Figure 4: Visualized results on challenging scenarios. The proposed method performs well on these challenging sequences, while other methods suffer from semantic understanding and thus cannot segment the whole objects. (b) Our method handles the challenging sequence with fast-motion objects better than Cutie (Cheng et al., 2023a).
+
+the semantic prior of the interested targets, such as in the case of the sequence of man riding from YoutubeVOS 2019.
+
+Figure 4(b) shows the segmentation and tracking performance on long-term sequences with fast motion and dramatic appearance variations. In this sequence, two similar dogs continuously change their position and posture, making it challenging for the model to identify and track them accurately. Our VOS method performs accurately in tracking and segmenting targets, whereas Cutie fails to identify and segment the targets in this scenario. Good performance in scenarios like this is due to the semantic-aware feature and discriminative query representation learned by our model. More visualization and videos can be found in the Supplementary Material.
+
+# 4.4 LIMITATIONS
+
+Our method enhances target representation by introducing spatial-semantic features and discriminative query propagation, significantly improving numerous challenging scenarios. However, in scenes with object parts as the target (some sequences in the VOT challenge, shown in the appendix), the proposed model finds it difficult to represent a part of the target and often extends the predicted mask to the whole target. For example, when the target is the head of a person, our model may gradually shift to segmenting the entire body. This tendency arises because our method tends to interpret the target as the entire object seen during the training phase. A potential direction to improve this issue could be to let the model learn more part-aware semantic features, which also needs a new benchmark. Moreover, the improvement of the inference speed will also be addressed in our future work.
+
+# 5 CONCLUSION
+
+In this paper, we propose a spatial-semantic network to learn comprehensive representations to improve the robustness of video object segmentation in handling objects with complex structures or separate parts. In addition, we develop a discriminative query propagation module that selects representative features from target objects for query update, enabling more reliable query propagation in long-term videos. Our approach achieves favorable performance compared with state-of-the-art methods on numerous datasets. Both quantitative and qualitative results demonstrate the effectiveness of the proposed method in tracking and segmenting video objects in complex and long-term scenes.
+
+Acknowledgments This work is supported by the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2024A1515011292), the National Natural Science Foundation of China (Grant No.62476148, 62172126), and the Shenzhen Research Council (No. JCYJ20210324120202006).
+
+# REFERENCES
+
+Ali Athar, Jonathon Luiten, Paul Voigtlaender, Tarasha Khurana, Achal Dave, Bastian Leibe, and Deva Ramanan. Burst: A benchmark for unifying object recognition, segmentation and tracking in video. In WACV, pp. 1674-1683, 2023. 7  
+L. Bao, B. Wu, and W. Liu. Cnn in mrf: Video object segmentation via inference in a cnn-based higher-order spatiotemporal mrf. In CVPR, 2018. 1  
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+
+# A APPENDIX
+
+The appendix is organized as follows:
+
+1. We provide a detailed implementation of training and inference.  
+2. We provide more experiments with different training settings and test hyperparameters.  
+3. More visualization and failure cases are shown.
+
+# A.1 IMPLEMENT DETAILS.
+
+# A.1.1 MULTI-OBJECT PROCESSING
+
+In this section, we provide additional implementation details for completeness. Our training and inference code will be released to ensure reproducibility.
+
+Following the previous work (Cheng & Schwing, 2022; Cheng et al., 2021a), we extend our methods to the multi-object setting. The predicted mask of each frame undergoes channel separation for different targets. The separated masks for each target are then concatenated with the RGB image and the aggregated mask of other targets for feature extraction. The extracted features are fused with RGB features to obtain pixel-level value features, which are then stored in memory. It is important to note that the pixel-level features stored in memory include the key (RGB feature) and the value.
+
+# A.1.2 TRAINING DETAILS
+
+As mentioned in the main paper, we train our network only on video-level datasets, which differs from other methods (Cheng et al., 2023a; Cheng & Schwing, 2022; Yang & Yang, 2022). The backbone weights are initialized from MAE pre-training, similar to the approaches used in SimVOS (Wu et al., 2023) and JointFormer (Mao et al., 2021). We perform spatial-semantic fusion every three ViT layers and five discriminative query transformer blocks for query propagation. We implement our network using PyTorch and train the model on 8 NVIDIA V100s, employing automatic mixed precision (AMP) during training.
+
+Two different training datasets are used for better improvements. The "DAVIS&YTVOS" settings combine the training sets of YouTubeVOS-2019 and DAVIS-2017, with the DAVIS-2017 dataset expanded five times to increase the data ratio. The "MEGA" set contains four different training sets, including DAVIS-2017, YouTubeVOS-2019, MOSE, and BURST datasets, with the DAVIS-2017 dataset expanded five times to increase the data ratio. To sample a training sequence, we randomly select a "seed" frame from all the frames and then randomly choose 7 other frames from the same video. We re-sample if any two consecutive frames have a temporal distance greater than the max-skip. Following the previous work (Cheng & Schwing, 2022; Cheng et al., 2023a), the max-skip is set to [5, 10, 15, 5] after  $[0\%, 10\%, 30\%, 80\%]$  of the training iterations, respectively. To avoid sampling frames without targets, we use the data statistics provided by Cutie (Cheng et al., 2023a).
+
+Table 5: Training parameters.  
+
+<table><tr><td>Config</td><td>DAVIS YTVOS</td><td>MEGA</td></tr><tr><td>optimizer</td><td>AdamW</td><td>AdamW</td></tr><tr><td>base learning rate</td><td>5e-5</td><td>5e-5</td></tr><tr><td>weight decay</td><td>0.05</td><td>0.05</td></tr><tr><td>droppath rate</td><td>0.15</td><td>0.15</td></tr><tr><td>batch size</td><td>16</td><td>16</td></tr><tr><td>num ref frames</td><td>3</td><td>3</td></tr><tr><td>num frames</td><td>8</td><td>8</td></tr><tr><td>max-skip</td><td>[5, 10, 15, 5]</td><td>[5, 10, 15, 5]</td></tr><tr><td>max-skip-itr</td><td>[0.1,0.3,0.8,1]</td><td>[0.1, 0.3, 0.8, 1]</td></tr><tr><td>Iterations</td><td>150,000</td><td>190,000</td></tr><tr><td>learning rate schedule</td><td>steplr</td><td>steplr</td></tr></table>
+
+Table 6: Quantitative comparisons with high-resolution input on the MOSE, LVOS, DAVIS 2017, YouTube-VOS 2018 & 2019 dataset. + means improve the input size.  
+
+<table><tr><td>Dataset</td><td colspan="3">MOSE-val</td><td colspan="3">DAVIS 2017 test</td><td colspan="6">YouTube-VOS 2018 val</td><td colspan="5">YouTube-VOS 2019 val</td></tr><tr><td>Method</td><td>J&amp;F</td><td>J</td><td>F</td><td>J&amp;F</td><td>J</td><td>F</td><td>G</td><td>Js</td><td>Fs</td><td>Ju</td><td>Fu</td><td>G</td><td>Js</td><td>Fs</td><td>Ju</td><td>Fu</td><td></td></tr><tr><td>Cutie-base (Cheng et al., 2023a)*</td><td>64.0</td><td>60.0</td><td>67.9</td><td>84.2</td><td>80.6</td><td>87.7</td><td>86.1</td><td>85.8</td><td>90.5</td><td>80.0</td><td>88.0</td><td>86.1</td><td>85.5</td><td>90.0</td><td>80.6</td><td>88.3</td><td></td></tr><tr><td>ISVOS (Wang et al., 2023)*+BL30K (Cheng et al., 2021b)</td><td>-</td><td>-</td><td>-</td><td>84.0</td><td>80.1</td><td>87.8</td><td>86.7</td><td>86.1</td><td>90.8</td><td>81.0</td><td>89.0</td><td>86.3</td><td>85.2</td><td>89.7</td><td>81.0</td><td>89.1</td><td></td></tr><tr><td>JointFormer (Zhang et al., 2023)*+BL30K (Cheng et al., 2021b)</td><td>-</td><td>-</td><td>-</td><td>88.1</td><td>84.7</td><td>91.6</td><td>87.6</td><td>86.4</td><td>91.0</td><td>82.2</td><td>90.7</td><td>87.4</td><td>86.5</td><td>90.9</td><td>82.0</td><td>90.3</td><td></td></tr><tr><td>Ours</td><td>68.5</td><td>64.5</td><td>72.6</td><td>87.1</td><td>83.1</td><td>91.1</td><td>87.4</td><td>87.0</td><td>92.0</td><td>80.9</td><td>89.7</td><td>87.5</td><td>86.8</td><td>91.8</td><td>81.3</td><td>89.9</td><td></td></tr><tr><td>Cutie-base (Cheng et al., 2023a)*+</td><td>66.2</td><td>62.3</td><td>70.1</td><td>85.9</td><td>82.6</td><td>89.2</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>86.9</td><td>86.2</td><td>90.7</td><td>81.6</td><td>89.2</td><td></td></tr><tr><td>Ours+</td><td>70.5</td><td>66.5</td><td>74.6</td><td>87.9</td><td>84.6</td><td>91.3</td><td>87.6</td><td>86.9</td><td>91.7</td><td>81.5</td><td>90.1</td><td>87.8</td><td>86.8</td><td>91.6</td><td>82.2</td><td>90.8</td><td></td></tr><tr><td>Cutie-base* (Cheng et al., 2023a) w/ MEGA</td><td>69.9</td><td>65.8</td><td>74.1</td><td>86.1</td><td>82.4</td><td>89.9</td><td>87.0</td><td>86.4</td><td>91.1</td><td>81.4</td><td>89.2</td><td>87.0</td><td>86.0</td><td>90.5</td><td>82.0</td><td>89.6</td><td></td></tr><tr><td>Ours w/MEGA</td><td>73.2</td><td>68.8</td><td>77.5</td><td>88.2</td><td>84.3</td><td>92.1</td><td>88.1</td><td>87.4</td><td>92.5</td><td>81.9</td><td>90.7</td><td>88.0</td><td>88.0</td><td>91.8</td><td>82.5</td><td>90.8</td><td></td></tr><tr><td>Cutie-base (Cheng et al., 2023a)*+ w/ MEGA</td><td>71.7</td><td>67.6</td><td>75.8</td><td>88.1</td><td>84.7</td><td>91.4</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>87.5</td><td>86.3</td><td>90.6</td><td>82.7</td><td>90.5</td><td></td></tr><tr><td>Ours+ w/MEGA</td><td>75.1</td><td>71.0</td><td>79.2</td><td>89.1</td><td>85.8</td><td>92.4</td><td>88.5</td><td>87.6</td><td>92.6</td><td>82.7</td><td>91.3</td><td>88.5</td><td>87.3</td><td>92.0</td><td>83.1</td><td>91.4</td><td></td></tr></table>
+
+We conduct data augmentation with methods of random horizontal mirroring, random affine transformations, and cut-and-paste. The training samples are also resized by crop (480 x 480). Then, random color jittering and random grayscaleing are performed in the training samples. The same augmentation is performed in the same video for stability. To ensure the images can be input into the patch embedding of ViT, we apply padding to both the images and masks so that their dimensions are multiples of 32.
+
+We initialize the model with the first frame and ground truth in the training phase. When the number of past frames is three or fewer, we use all of them as memory frames. If there are more than three past frames, we randomly sample three of them to be the memory frames. We compute the loss at all frames except the first one and perform back-propagation through time.
+
+# A.1.3 INFERENCE DETAILS
+
+During the inference phase, the model memorizes the feature every 5th frame like (Cheng & Schwing, 2022; Cheng et al., 2023a). The pixel-level memory always stores the first frame and its mask. The top-K filter is used to augment memory reading in pixel memory. We use the streaming average for object-level memory to accumulate the discriminative query representation like the operation in (Cheng et al., 2023a).
+
+# A.2 ADDITIONAL QUANTITATIVE RESULTS
+
+In this section, we provide additional ablation studies and SOTA comparisons with different settings.
+
+Table 6 shows detailed results about methods with further pre-training (e.g. JointFormer (Zhang et al., 2023) and ISVOS (Wang et al., 2023)) and improved test size (Cutie (Cheng et al., 2023a)) We note that our method achieves comparable results against SOTA methods (Cutie (Cheng et al., 2023a) and JointFormer (Zhang et al., 2023)) without any pre-training. When trained with MEGA datasets or improved test image size, our method achieves state-of-the-art results in all the benchmarks.
+
+In the main paper, we provide ablation studies for different components. To provide a broader comparison, we conduct tests using various combinations of training data and testing settings. The experimental results in Table 7 detail the impact of different components of our model on the performance, facilitating a better understanding of the effectiveness of our model.
+
+To compare with more methods on MOSE, we follow the training strategy from MOSE (Ding et al., 2023), and replace the YouTubeVOS with MOSE in the training process. Table 8 shows the results of the variants of our method. All the variants achieve great improvement in terms of  $\mathcal{J}\&\mathcal{F}$ , which further validates the effectiveness of the proposed spatial-semantic feature learning and discriminative query representation.
+
+Table 9 shows the results of our method on the more challenging VOTS2023 (Kristan et al., 2023) benchmark, which indicates that our method outperforms the winner (DMAOT) of last year on most metrics. Our method achieves the highest robustness thanks to our representation of target saliency information and the modeling of target semantic-spatial information.
+
+![](images/a2aebe0941ad5e12cea7e972355f38e956d0d12ecf9769d4cadd3d141a43d445.jpg)  
+Qualitative comparisons from various sequences.
+
+![](images/995a6821011bf51ece14e1c749abafdc4369a5d1e8607b3a8cb77be6cb7a7883.jpg)  
+Figure 5: Additional qualitative comparison against state-of-the-art methods including XMem, Swin-DeAOT, and Cutie. All the sequences are selected from YouTubeVOS 2019 and DAVIS 2017 datasets.  
+Qualitative comparisons from various sequences with appearance variance.  
+Figure 6: Additional qualitative comparison on sequence with giant appearance variance. All the sequences are picked from MOSE (Ding et al., 2023) dataset.
+
+Table 7: Model detailed ablation study.  
+
+<table><tr><td>Dataset</td><td colspan="3">MOSE-val</td><td colspan="3">DAVIS 2017 test</td><td colspan="3">LVOS test</td><td colspan="5">YouTube-VOS 2019 val</td></tr><tr><td>Method</td><td>J&amp;F</td><td>J</td><td>F</td><td>J&amp;F</td><td>J</td><td>F</td><td>G</td><td>J</td><td>F</td><td>G</td><td>Js</td><td>Fs</td><td>Ju</td><td>Fu</td></tr><tr><td colspan="15">Trained on the YouTubeVOS, and DAVIS datasets</td></tr><tr><td>XMem (Cheng &amp; Schwing, 2022) (Baseline)</td><td>53.3</td><td>62.0</td><td>57.6</td><td>81.0</td><td>77.4</td><td>84.5</td><td>50.0</td><td>45.5</td><td>54.4</td><td>85.5</td><td>84.3</td><td>88.6</td><td>80.3</td><td>88.6</td></tr><tr><td>Cutie (Cheng et al., 2023a)</td><td>64.0</td><td>60.0</td><td>67.9</td><td>84.2</td><td>80.6</td><td>87.7</td><td>56.2</td><td>51.8</td><td>60.5</td><td>86.1</td><td>85.8</td><td>90.5</td><td>80.0</td><td>88.0</td></tr><tr><td>+Discriminative Query</td><td>64.2</td><td>60.3</td><td>68.1</td><td>85.2</td><td>81.8</td><td>88.5</td><td>57.4</td><td>53.3</td><td>61.5</td><td>86.5</td><td>86.2</td><td>90.7</td><td>80.6</td><td>88.8</td></tr><tr><td>+ViT</td><td>64.2</td><td>60.2</td><td>68.3</td><td>85.6</td><td>82.0</td><td>89.2</td><td>58.3</td><td>53.7</td><td>62.8</td><td>86.7</td><td>86.4</td><td>90.3</td><td>81.0</td><td>88.7</td></tr><tr><td>+Spatial</td><td>68.2</td><td>64.0</td><td>72.4</td><td>86.2</td><td>82.4</td><td>90.1</td><td>67.4</td><td>62.9</td><td>71.9</td><td>87.3</td><td>86.4</td><td>91.3</td><td>81.5</td><td>90.3</td></tr><tr><td>+Semantic (Full)</td><td>68.5</td><td>64.5</td><td>72.6</td><td>86.7</td><td>82.7</td><td>90.8</td><td>66.5</td><td>62.1</td><td>70.8</td><td>87.5</td><td>86.8</td><td>91.8</td><td>81.3</td><td>89.9</td></tr><tr><td colspan="15">Improved test size (600/720)</td></tr><tr><td>Cutie (Cheng et al., 2023a)</td><td>66.2</td><td>62.3</td><td>70.1</td><td>85.9</td><td>82.6</td><td>89.2</td><td>56.2</td><td>51.8</td><td>60.5</td><td>86.9</td><td>86.2</td><td>90.7</td><td>81.6</td><td>89.2</td></tr><tr><td>+Discriminative Query</td><td>66.4</td><td>62.4</td><td>70.1</td><td>87.9</td><td>84.6</td><td>91.2</td><td>57.4</td><td>53.3</td><td>61.5</td><td>87.1</td><td>86.3</td><td>90.6</td><td>82.0</td><td>89.5</td></tr><tr><td>+Spatial</td><td>69.9</td><td>67.5</td><td>74.1</td><td>87.0</td><td>83.7</td><td>90.2</td><td>67.4</td><td>62.9</td><td>71.9</td><td>87.5</td><td>86.8</td><td>91.6</td><td>81.6</td><td>90.2</td></tr><tr><td>+Semantic ( full )</td><td>70.5</td><td>66.5</td><td>74.6</td><td>87.8</td><td>84.6</td><td>91.3</td><td>66.5</td><td>62.1</td><td>70.8</td><td>87.7</td><td>86.8</td><td>91.6</td><td>82.2</td><td>90.8</td></tr><tr><td colspan="15">Trained on the MEGA datasets</td></tr><tr><td>Cutie (Cheng et al., 2023a)</td><td>69.9</td><td>65.8</td><td>74.1</td><td>86.1</td><td>82.4</td><td>89.9</td><td>66.7</td><td>62.4</td><td>71.0</td><td>87.0</td><td>86.0</td><td>90.5</td><td>82.0</td><td>90.7</td></tr><tr><td>+Discriminative query</td><td>70.6</td><td>66.5</td><td>74.6</td><td>86.6</td><td>82.7</td><td>90.5</td><td>66.5</td><td>62.1</td><td>70.8</td><td>87.5</td><td>86.0</td><td>90.6</td><td>82.8</td><td>90.6</td></tr><tr><td>+Spatial</td><td>73.5</td><td>69.1</td><td>77.7</td><td>87.6</td><td>83.8</td><td>91.5</td><td>68.8</td><td>64.4</td><td>73.1</td><td>87.9</td><td>86.9</td><td>91.8</td><td>82.3</td><td>90.4</td></tr><tr><td>+Semantic ( Full )</td><td>74.0</td><td>69.8</td><td>78.3</td><td>87.8</td><td>84.0</td><td>91.7</td><td>73.0</td><td>68.3</td><td>77.8</td><td>88.1</td><td>87.4</td><td>92.5</td><td>81.9</td><td>90.7</td></tr><tr><td colspan="15">Improved test size (600/720)</td></tr><tr><td>Cutie (Cheng et al., 2023a)</td><td>71.7</td><td>67.6</td><td>75.8</td><td>88.1</td><td>84.7</td><td>91.4</td><td>66.7</td><td>62.4</td><td>71.0</td><td>87.5</td><td>86.3</td><td>90.6</td><td>82.7</td><td>90.5</td></tr><tr><td>+Discriminative query</td><td>71.6</td><td>67.7</td><td>75.5</td><td>88.1</td><td>84.6</td><td>91.4</td><td>66.5</td><td>62.1</td><td>70.8</td><td>88.0</td><td>86.2</td><td>90.6</td><td>83.6</td><td>91.5</td></tr><tr><td>+Spatial</td><td>75.3</td><td>71.3</td><td>79.2</td><td>89.0</td><td>85.8</td><td>92.3</td><td>68.8</td><td>64.4</td><td>73.1</td><td>88.3</td><td>87.0</td><td>91.9</td><td>83.0</td><td>91.1</td></tr><tr><td>+Semantic(Full)</td><td>75.1</td><td>71.0</td><td>79.2</td><td>89.1</td><td>85.8</td><td>92.4</td><td>73.0</td><td>68.3</td><td>77.8</td><td>88.5</td><td>87.3</td><td>92.0</td><td>83.1</td><td>91.4</td></tr></table>
+
+Table 8: Results only trained on MOSE datasets.  
+
+<table><tr><td>Dataset</td><td colspan="3">MOSE-val</td></tr><tr><td>Method</td><td>J&amp;F</td><td>J</td><td>F</td></tr><tr><td>RDE (Li et al., 2022a)</td><td>48.8</td><td>44.6</td><td>52.9</td></tr><tr><td>STCN (Cheng et al., 2021a)</td><td>50.8</td><td>46.6</td><td>55.0</td></tr><tr><td>AOT (Yang et al., 2021)</td><td>57.2</td><td>53.1</td><td>61.3</td></tr><tr><td>XMem (Cheng &amp; Schwing, 2022)</td><td>57.6</td><td>53.3</td><td>62.0</td></tr><tr><td>DeAOT (Yang &amp; Yang, 2022)</td><td>59.4</td><td>55.1</td><td>63.8</td></tr><tr><td>ResNet+Discriminative query</td><td>69.9</td><td>65.8</td><td>73.9</td></tr><tr><td>+Spatial</td><td>72.7</td><td>68.3</td><td>77.0</td></tr><tr><td>+Semantic</td><td>72.9</td><td>68.4</td><td>77.3</td></tr><tr><td colspan="4">Improved test size (720)</td></tr><tr><td>ResNet+Discriminative query</td><td>71.6</td><td>67.7</td><td>75.5</td></tr><tr><td>+Spatial</td><td>74.0</td><td>69.9</td><td>78.1</td></tr><tr><td>+Semantic</td><td>74.5</td><td>70.2</td><td>78.5</td></tr></table>
+
+Table 9: Results on VOTS2023 Challenge.  
+
+<table><tr><td>Dataset</td><td colspan="6">VOTS2023</td></tr><tr><td>Method</td><td>Q↑</td><td>A↑</td><td>R↑</td><td>NER↓</td><td>DRE↓</td><td>ADQ↓</td></tr><tr><td>Dynamic_DEAOT</td><td>0.59</td><td>0.69</td><td>0.84</td><td>0.07</td><td>0.1</td><td>0.57</td></tr><tr><td>M-VOSTrack</td><td>0.61</td><td>0.75</td><td>0.76</td><td>0.16</td><td>0.08</td><td>0.71</td></tr><tr><td>HQTrack</td><td>0.62</td><td>0.75</td><td>0.77</td><td>0.16</td><td>0.08</td><td>0.70</td></tr><tr><td>DMAOT</td><td>0.64</td><td>0.75</td><td>0.80</td><td>0.14</td><td>0.07</td><td>0.73</td></tr><tr><td>Ours</td><td>0.67 (+0.3)</td><td>0.74</td><td>0.86 (+0.6)</td><td>0.09 (+0.5)</td><td>0.05 (+0.02)</td><td>0.75 (+0.02)</td></tr></table>
+
+# A.3 ADDITIONAL QUALITATIVE COMPARISON
+
+We present a more qualitative comparison of most benchmarks in Figure 5 and Figure 6.
+
+In Figure 5, we categorize the scenarios into three main types: overall-to-part segmentation (man on bike and deer), small object segmentation (small monkey), blurry object segmentation (cat toy), and segmentation under varying lighting conditions (man-bike and golf). In these three cases, our method demonstrates precise segmentation compared to other methods. The semantic embedding allows our model to consider semantic information when parsing objects, spatial dependence modeling enhances the understanding of spatial positions and details of the object, and the discriminative query
+
+![](images/979b836a8248932fe294948298c79a2791f344a09bdc731b0045a91d14a80f0d.jpg)  
+(a) Visualization of the feature map output from different backbones
+
+![](images/d4443aee8890984e177cb5174346c29ff881abb26c0e07d5ea0429be2f8f6719.jpg)  
+(b) Visualization of the feature map after proposed semantic and spatial layers  
+Figure 7: Visualizations of feature maps from different architectural components.
+
+accurately represents the target. The integration of these three components enables the model to achieve excellent segmentation and tracking performance in complex and challenging scenarios.
+
+In Figure 6, we select more sequence results on MOSE, primarily divided into two categories: positional changes of similar targets and rapid movement of targets. In both cases, our method outperforms Cutie. This is attributed to our proposed spatial-semantic feature representation and discriminative query expression, which enable the model to distinguish appearance-similar targets and consistently track the target.
+
+Figure 9 compares sequences that require a semantic understanding of the target. Without the semantic embedding module, the method suffers from segmenting parts of the target. The addition of the semantic embedding module enables the model to fully understand the semantic information of the target, allowing for precise segmentation and tracking in these scenarios.
+
+Visualization of feature maps. Figure 7(a) shows that the proposed Spatial-Semantic block assigns higher activation weights to complex target objects compared to the CNN or ViT model, which indicates that the proposed model makes target representations more distinguishable than ViT and ResNet. Moreover, we show how the feature changes during the processing of the proposed model by visualizing the feature maps at different processing steps in 7(b). After passing through the semantic embedding module, the feature maps (semantic embedding) demonstrate that the module makes the contours of objects clearer, indicating richer semantic information. After passing through the spatial dependence module, the feature maps (spatial modeling) show that the details of the targets are more pronounced. The changes in the feature maps confirm that the proposed two modules significantly enhance the semantic and spatial information of the targets.
+
+We also visualize the similarity between the learned discriminative queries and the feature map using heat maps in Figure 8. The visualization shows that the learned discriminative queries distinctly express the target areas. We also mark the positions of the discriminative queries, which represent the locations of the pixels with the highest similarity.
+
+Failure cases are shown in Figure 10, in which our method tends to segment the whole instance rather than the part of it. The challenging sequences require the model to segment and track the head of the man, the flamingo, and the kangaroo. Our model interprets semantics as the entire instance during tracking and segmentation, such as a person, bird, or kangaroo, resulting in segmentation outcomes gradually favoring the whole instance.
+
+![](images/cbf909ee03de79c5d3f0d18ba6bcc59417e79e3ca221458679fc1a512b34eab1.jpg)  
+Figure 8: Visualization of the similarity map between query and features. The map indicates the discrimination of the target representation. The discriminative queries of different targets are plotted in the similarity map, which can be found in the black box.
+
+![](images/21df02a88f3efb4f11165e183bdbdb26e726d606bae8b3ecd9a1670da06c37c8.jpg)  
+Figure 9: Qualitative comparison between with and without semantic embedding block. All the sequences are picked from the YouTubeVOS 2019 dataset.
+
+![](images/452cbeb09672b207ef43906bd9e4c76abde0114a874631c8a78f12afd4a4295a.jpg)  
+Time line  
+Figure 10: Failure cases. All the sequences are picked from VOT Challenge (Kristan et al., 2023).
\ No newline at end of file
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+# LEARNING SPLITTING HEURISTICS IN DIVIDE-AND-CONQUER SAT SOLVERS WITH REINFORCEMENT LEARNING
+
+Shumao Zhai $^{1}$ , Ning Ge $^{1,2}$  *
+
+<sup>1</sup>Beihang University 2State Key Laboratory of Complex & Critical Software Environment (CCSE)
+
+# ABSTRACT
+
+We propose RDC-SAT, a novel approach to optimize splitting heuristics in Divide-and-Conquer SAT solvers using deep reinforcement learning. Our method dynamically extracts features from the current solving state whenever a split is required. These features, such as learned clauses, variable activity scores, and clause LBD (Literal Block Distance) values, are represented as a graph. A GNN integrated with an Actor-Critic model processes this graph to determine the optimal split variable. Unlike traditional linear state transitions characterized by Markov processes, divide-and-conquer challenges involve tree-like state transitions. To address this, we developed a reinforcement learning environment based on the Painless framework that efficiently handles these transitions. Additionally, we designed different discounted reward functions for satisfiable and unsatisfiable SAT problems, capable of handling tree-like state transitions. We trained our model using the Decentralized Proximal Policy Optimization (DPPO) algorithm on phase transition random 3-SAT problems and implemented the RDC-SAT solver, which operates in both GPU-accelerated and non-GPU modes. Evaluations show that RDC-SAT significantly improves the performance of D&C solvers on phase transition random 3-SAT datasets and generalizes well to the SAT Competition 2023 dataset, substantially outperforming traditional splitting heuristics.
+
+# 1 INTRODUCTION
+
+The propositional satisfiability problem (SAT) is one of the foundational problems in theoretical computer science. It involves determining the assignment of variables that satisfy a given Boolean formula. Due to its NP-complete nature, SAT plays a crucial role in theoretical computer science and has a wide range of applications, including software verification, hardware design verification, combinatorial optimization, and artificial intelligence, etc. Most modern complete SAT solvers primarily employ the conflict-driven clause learning (CDCL) algorithm (Marques-Silva et al., 2021), which has transformed the field by introducing clause learning mechanisms that effectively optimize the search space. CDCL solvers avoid revisiting previously explored paths and concentrate on promising areas of the search space, establishing them as the leading method for tackling complex and challenging SAT problems.
+
+In the realm of modern SAT solvers, parallelism has been embraced to fully harness the computational power of many-core machines (Hamadi & Sais, 2018). Two dominant parallelization strategies have emerged: Divide-and-Conquer (D&C) and Portfolio. The D&C strategy, often employing the guiding path technique, dynamically divides the problem into smaller, more manageable subspaces. Each subspace is independently tackled by individual sequential solvers, allowing the problem to be processed more efficiently in parallel. This method leverages problem structure to enable targeted sub-formula exploration, reducing overall solving time. The Portfolio approach adopts a competitive strategy in which multiple sequential solvers concurrently tackle the entire problem. This competition continues until one solver successfully solves the problem. Strategic diversity among solvers enhances solution coverage and success probability. Although D&C approaches seem to be the natural way to work in parallel, empirical results from annual SAT competitions indicate that Portfolio-based
+
+solvers currently lead in performance. However, D&C solvers still possess unique advantages in certain domains, such as better scalability in environments with large parallel computing power, like cloud platforms (Heisinger et al., 2020), and have been successfully employed to solve large-scale mathematical problems (Heule et al., 2016).
+
+In modern D&C SAT solvers, the problem is dynamically divided into smaller, more manageable subproblems. The splitting process entails selecting a variable and dividing the problem into subformulas by setting the variable to either True or False, which are subsequently solved using CDCL sequential solvers. This partitioning is guided by splitting heuristics, which largely determine the performance of D&C SAT solvers. An effective splitting heuristic minimizes total solving time while ensuring balanced subspaces for optimal load distribution. Splitting heuristics generally fall into two categories: look-ahead and look-back techniques. Look-ahead heuristics evaluate the potential impact of variables to minimize runtime, as demonstrated in the Cube-and-Conquer solver (Heule et al., 2011; Fleury & Heisinger, 2020), which creates smaller subspaces by choosing variables that maximize unit propagations. In contrast, look-back heuristics evaluate variables based on their historical effectiveness in the search, selecting split variables through methods like VSIDS-based variable activity (Audemard et al., 2016), the number of flips of variables (Le Frioux et al., 2019), and propagation rates (Nejati et al., 2017) of variables.
+
+Machine learning methods have already been widely applied to SAT problems (Holden et al., 2021; Guo et al., 2023; Yolcu & Poczos, 2019; Zhang et al., 2020). In recent years, GNNs and RL have shown potential in optimizing branch splitting, phase selection, clause deletion, etc., heuristics in CDCL solvers (Jaszczur et al., 2020; Kurin et al., 2020; Selsam & Bjørner, 2019; Cameron et al., 2020; Han, 2020; Wang et al., 2023; Shi et al., 2023; Liu et al., 2024). For example, Graph-Q-Sat (Kurin et al., 2019; 2020) uses GNNs and RL to optimize branch heuristics in CDCL solvers, significantly reducing the number of iterations. However, Graph-Q-Sat did not effectively reduce overall runtime due to CDCL's requirement for high-frequency variable selection calls. In contrast, splitting heuristics in D&C solvers are equally critical but invoked far less frequently, making RL and GNN-based optimization of these heuristics particularly promising. A more detailed discussion of related work is provided in Appendix A.
+
+In this paper, we present a technique that employs deep reinforcement learning to optimize the splitting euristics within parallel D&C SAT solvers, resulting in the development of the Reinforced Divide-and-Conquer SAT solver (RDC-SAT). When a split is required, the D&C solver parses information from the sequential solver, including learned clauses, variable activities, and clause LBD(Literal Block Distance), and represents this information as a graph. A GNN is then used to evaluate the importance of each variable and select the optimal split variable.
+
+To optimize this model using reinforcement learning, we developed an RL environment within the Painless (PArallel INstantiabLE Sat Solver) framework (Le Frioux et al., 2017; 2019) that efficiently manages the tree-like state transitions characteristic of D&C methods. Additionally, we designed distinct discounted reward functions for satisfiable and unsatisfiable SAT problems. We then trained a model optimized for splitting heuristics using the Distributed Proximal Policy Optimization(DPPO)(Heess et al., 2017) algorithm, resulting in the creation of RDC-SAT. Moreover, considering the computational overhead of the neural network in the SAT solver, we also implemented an approach similar to NeuroBack (Wang et al., 2023), which supports invoking the neural network only once at the initial stage to guide subsequent splits, making it suitable for GPU resource-poor computing environments. To validate our approach, we trained the model on a random dataset at the phase transition and evaluated its effectiveness and generalization on larger-scale problems. Our empirical results demonstrate that RDC-SAT outperforms baseline methods. On the random 3-SAT dataset, RDC-SAT achieved a  $25.08\%$  reduction in PAR2 score; on the SAT Competition 2023 dataset—which includes small to medium-sized SAT problems with hundreds of thousands of variables and clauses—it achieved a  $14.95\%$  reduction. Even without GPU acceleration, the insights from the initial neural network call effectively guide subsequent splits, ensuring efficient problem-solving with minimal reliance on neural network computations.
+
+# 2 PRELIMINARIES
+
+SAT problem. The Boolean Satisfiability Problem (SAT)(Biere et al., 2009) concerns the satisfiability determination of propositional logic formulas. A propositional logic formula is composed
+
+of variables and operators such as AND (conjunction,  $\land$ ), OR (disjunction,  $\lor$ ), NOT (negation,  $\neg$ ), and parentheses, typically expressed in Conjunctive Normal Form (CNF). In CNF, a formula is a conjunction of clauses, where a clause is a disjunction of literals, and a literal is either a variable (positive literal) or its negation (negative literal). For example,  $(x \lor \neg y) \land (\neg x \lor z)$  is a CNF formula, where  $x$  and  $z$  are positive literals of their respective variables, and  $\neg y$  is a negative literal. In the SAT problem, each variable can be assigned one of two values: True or False. If there is at least one assignment of variables that makes the entire formula True, then the formula is deemed satisfiable (SAT); otherwise, it is considered unsatisfiable (UNSAT). During the process of solving SAT problems, CDCL solvers generate numerous learned clauses to aid in resolving the problem efficiently. These learned clauses help reduce the search space by preventing the solver from revisiting conflicting assignments, thus facilitating quicker convergence towards a solution or proving unsatisfiability.
+
+Divide-and-Conquer SAT solvers. Modern D&C SAT solvers follow a hierarchical master-slave architecture.(Le Frioux et al., 2019) The master node manages the partitioning of the search space into smaller subspaces, which are tackled independently and in parallel by multiple sequential CDCL solvers. These subproblems are solved concurrently by worker nodes, which share learned clauses with each other to accelerate the solving process and reduce redundant computations. To ensure dynamic load balancing, D&C solvers utilize a work-stealing mechanism. When a worker node becomes idle, the master node identifies an ongoing subproblem currently being processed by another node, splits it into smaller tasks, and assigns one to the idle worker. This dynamic redistribution of tasks enhances computational efficiency. Incremental solvers further optimize this approach by enabling solvers to continue from previous states with retained learned clauses and variable activity scores, minimizing redundancy and improving overall solving speed.
+
+Graph Neural Networks. Graph Neural Networks (GNNs) (Wu et al., 2020; Zhou et al., 2020) are designed to handle data structured as graphs, allowing for the modeling of relationships and interactions within data. The GraphNets framework (Battaglia et al., 2018) utilizes an Encoder-Processor-Decoder architecture to effectively process graph-structured data. The encoder maps node and edge features into high-dimensional latent representations. The processor iteratively updates these representations through a message-passing mechanism, where each node aggregates information from its neighbors to refine its embedding. This mechanism enables the network to propagate and integrate information across the graph. Finally, the decoder reconstructs the desired outputs from the node embeddings for tasks such as node classification, graph classification, or link prediction, utilizing the embeddings either directly or through aggregation for graph-level predictions.
+
+Reinforcement Learning. Reinforcement Learning (RL) is a machine learning paradigm where an agent learns to make decisions by interacting with its environment, maximizing cumulative rewards. Central to RL is the classic Markov Decision Process (MDP), which models the decision-making problem through states  $s \in S$ , actions  $a \in A$ , state-transition probabilities  $P(s'|s,a)$ , and a reward function  $R(s,a)$ . This framework assumes the state transitions are generally linear, meaning the next state  $s'$  depends only on the current state  $s$  and action  $a$ . Advanced RL algorithms like Actor-Critic methods use a policy network (actor) to select actions and a value network (critic) to estimate the value function, improving both networks iteratively. Proximal Policy Optimization (PPO) (Schulman et al., 2017) introduces a clipped surrogate objective function to stabilize policy updates, preventing drastic changes and ensuring more reliable learning. Distributed Proximal Policy Optimization (DPPO) Heess et al. (2017) extends PPO by employing multiple agents across distributed environments, leveraging data parallelism to speed up learning while maintaining policy stability.
+
+# 3 INTERACTIVE DIVIDE-AND-CONQUER ENVIRONMENT
+
+We selected Painless (Le Frioux et al., 2017; 2019) as the foundational parallel divide-and-conquer framework. The Painless framework is a modular and high-performance parallel SAT-solving system, ideal for comparing different heuristics. It uses a master-slave architecture with a work-stealing strategy for dynamic load balancing and supports clause sharing to enhance solving efficiency. For our experiments, we used the best-performing settings identified, including MapleCOMSPS (Liang et al., 2016b;a) as the sequential solver and the "altoall" clause-sharing strategy. Building on this foundation, to construct a reinforcement learning environment suitable for optimizing splitting heuristics and handling tree-structured state transitions, we implemented features to extract SAT problem states from the SAT solver, store various information required for training using a binary
+
+tree, and retain only simple interfaces. To use Python for model training, we encapsulated the divide-and-conquer environment as a Python library using Pybind11 and created the PainlessEnv environment in Python. Specific implementation details can be found in Appendix B.
+
+# 3.1 ENVIRONMENT INTERFACE
+
+Our custom PainlessEnv environment is specifically designed for the needs of divide-and-conquer SAT problems, addressing the complexities of parallel and branching operations in SAT solving. It manages a binary tree structure to support the non-linear state transitions characteristic of these problems, unlike the linear transitions typical in traditional environments like Gym (Brockman et al., 2016). PainlessEnv primarily provides two interfaces for agent interaction: 'reset' and 'step'. The reset method initializes a new SAT problem and provides the initial state of the first subproblem that needs to be partitioned. Unlike Gym's 'step' method, which returns the next observation, reward, and done flag, our 'step' method only returns the next subproblem state and a 'done' flag. This is because, in the tree-structured state transitions, the next state is not always a direct successor of the previous state, and the reward can only be accurately determined from the tree structure. Therefore, our 'step' method does not return the reward directly.
+
+# 3.2 MANAGING STATE TRANSITIONS WITH BINARY TREES
+
+Due to the non-linear state transitions inherent in D&C problems, PainlessEnv cannot produce a continuous sequence of states and rewards for training. Instead, it utilizes a binary tree, as illustrated in Figure 1, to store all critical information during the solving process. The root node denotes the initial problem, internal nodes correspond to subproblems requiring splitting, and leaf nodes signify fully solved subproblems. When a worker requests a division or completes a solution, a new node is added to the tree with all pertinent data. After solving a SAT problem, the reinforcement learning agent retrieves training information from this binary tree. Additionally, In a D&C solver for SAT problems, once a subproblem is proven satisfiable, it indicates that the entire SAT problem is satisfiable. Typically, the D&C solver would interrupt all other active workers and return the result. However, for reinforcement learning training, the other unresolved subproblems are still meaningful. Interrupting the process leads to missing leaf nodes in the binary tree, which is suboptimal for training. To address this, even if a subproblem is confirmed as satisfiable, we continue processing until all subproblems are resolved before returning a result, ensuring a complete and informative dataset for training.
+
+# 3.3 SAT SUBPROBLEM STATE EXTRACTION
+
+Each time a split is required, the Painless environment interrupts a sequential solver and extracts basic information about the SAT problem to represent its state. MapleCOMSPS categorizes learned clauses into three levels: core, tier2, and local. Core learned clauses are considered the most crucial. They are generally derived from conflicts that involve decisions at deeper levels of the decision tree that are usually kept permanently. Tier2 learned clauses are of intermediate importance. They are often derived from conflicts at shallower depths and are kept as long as they are deemed useful. Local learned clauses are less critical and more temporary, typically generated from recent conflicts and are discarded periodically to manage memory and performance overhead. Additionally, it maintains variable activity values used in branch heuristics, the LBD values of clauses to evaluate their utility, and other statistical information such as the number of decisions and unit propagations. We extract these learned clauses at different levels, along with variable activity scores and other pertinent data, to comprehensively represent the current solving state of the SAT problem. This information is then represented as a graph, which our environment can be configured to utilize.
+
+# 4 REINFORCED DIVIDE-AND-CONQUER SAT
+
+We utilized a state representation and a GNN similar to Graph-Q-Sat (Kurin et al., 2020), with modifications to incorporate additional information that better represents the evolving state of the SAT problem. Additionally, we replaced DQN with an Actor-Critic network and trained it using DPPO.
+
+![](images/86b69ba1c06cb5939d8d46355e8b4435660aa51d14a19de110b175e54beaa9d8.jpg)  
+Figure 1: Binary tree maintained by PainlessEnv to store process information. Each node in the tree stores problem states, splitting times, solving times, split variables, satisfiability status, and other relevant information. Internal nodes represent subproblems that have been split, while leaf nodes represent solved subproblems. Dashed lines indicate satisfiable subproblems.
+
+![](images/b77bf69714d76e2b7f3da5493a686d179a86881e6472611ba0b6eb310c0ceaa0.jpg)  
+Figure 2: State representation of Boolean formula  $(x_{1}\lor x_{2})\land (\neg x_{2}\lor x_{3})$  . The first position of the node feature indicates node type: 1 for variable nodes and 0 for clause nodes, and vice versa for the second position.  $F_{x}$ $F_{c}$  ,and  $F_{g}$  represent variable activity features, clause LBD features, and global SAT problem statistics, respectively. Edge features represent the polarity of literals.
+
+# 4.1 STATE REPRESENTATION
+
+In RDC-SAT, the SAT problem is structured as a graph shown in Figure 2 where nodes represent variables and clauses, with node and edge features indicating variable polarity and type. We extract detailed information from the Maple solver, including variable data and learned clauses at various levels. Incorporating learned clauses provides a more dynamic reflection of the SAT problem's state. Variable activity levels and clause LBD values are normalized to enhance message passing efficiency. Additionally, global characteristics like restart counts, decision counts, conflict counts, and unit propagation counts are included as global features. This comprehensive representation captures the dynamic state changes in the SAT problem, enabling the GNN to process node and edge data more effectively.
+
+# 4.2 REWARD FUNCTION FOR TREE-STRUCTURED STATE TRANSITIONS
+
+In traditional reinforcement learning, rewards are typically calculated for linear state transitions, where the reward for each state is computed from the final state backward through the sequence, applying a discount factor at each step. This process involves calculating the reward for the final state first and then propagating it backward through the sequence. Correspondingly, for tree-structured state transitions, calculating the reward for each state requires first computing the discounted rewards for its child states. This necessitates using a post-order traversal to ensure that the discounted rewards of the child states are available before calculating the reward for the current state. Once the discounted rewards of the child states are obtained, they can be integrated into the discounted reward of current state using methods such as summation, averaging, taking the maximum, or the minimum of the child states' rewards. To illustrate this, consider a binary tree, where each state has two child states. Mathematically, if  $R_{L}$  and  $R_{R}$  are the discounted rewards of the left and right child states, and  $r_{s}$  is the immediate reward for the current state, the current state's reward  $R$  can be calculated as
+
+$$
+R = r _ {s} + \gamma \cdot f \left(R _ {L}, R _ {R}\right)
+$$
+
+The primary goal of this optimization is to minimize solving time. Thus, we simply use the negative solving time as the reward, making the maximization of this reward equivalent to minimizing the solving time. After solving SAT problems, we extract the splitting time and solving time from the binary tree in PainlessEnv to calculate the actual solving time for each state. For unsatisfiable problems, the total solving time is the sum of the splitting time and the longer solving time of the two subproblems. In contrast, for satisfiable problems, the total solving time is the sum of the splitting time and the shorter solving time of the satisfiable subproblems. The following formulas encapsulate
+
+the calculation of discounted reward:
+
+$$
+R _ {U} = - \left(T _ {s} + \max  \left(\Delta T _ {U c} + \gamma \cdot \left| R _ {U c} \right|\right)\right)
+$$
+
+$$
+R _ {S} = - \left(T _ {s} + \min \left(\Delta T _ {S c} + \gamma \cdot | R _ {S c} |\right)\right)
+$$
+
+Here,  $R_U$  and  $R_S$  are the discounted rewards for UNSAT and SAT problems, respectively;  $T_s$  is the splitting time, which is the time taken to analyze the problem state and decide the split variable;  $|R_{Uc}|$  and  $|R_{Sc}|$  are the absolute values of the discounted rewards for the child subproblems of UNSAT and SAT states, respectively. If a corresponding child state is a leaf node, its discounted reward is zero;  $\Delta T_{Uc}$  and  $\Delta T_{Sc}$  are the time intervals from the completion of the division to the return of the next child subproblem state in UNSAT and SAT scenarios, respectively; and  $\gamma$  is the discount factor applied to future rewards. Typically,  $\gamma$  is less than 1 to prioritize immediate rewards over future rewards. However, in our study, we set  $\gamma = 1$  because the solving process terminates within a definite time frame. This ensures that each action considers the entire solving process, and the reward directly reflects the total solving time under the current policy and action.
+
+![](images/46c348bfd39d81e86f5bc558db688b9aeed911111b0004dcf8fdfcae59bb6c4d.jpg)  
+Figure 3: Calculation of discounted reward in tree-structured state transitions.
+
+# 4.3 NEURAL NETWORK, ACTION DEFINITION, AND OBJECTIVE FUNCTION
+
+We use a Graph Neural Network (GNN) with an Encoder-Processor-Decoder architecture (Battaglia et al., 2018) to extract features, followed by an Actor-Critic network to make decisions. The Encoder transforms initial features into higher-dimensional representations. The Processor integrates local and global information through message passing. The Decoder outputs variable, clause, and global features. The RL action corresponds to selecting a variable as the split variable during the Divide-and-Conquer process. The Actor network assigns probabilities to valid variables, excluding those already chosen as split variables or eliminated during the solving process, ensuring the action space only contains currently selectable variables. The Critic network estimates the value of the current state based on clause and global features. We use the Proximal Policy Optimization (PPO)(Schulman et al., 2017) objective function. This objective combines policy loss, value loss, and entropy loss to optimize the learning process.
+
+# 4.4 TRAINING AND DEPLOYMENT
+
+Training data. We utilize randomly generated 3-SAT problems at the phase transition as our training data. The phase transition in SAT problems is characterized by a specific ratio of clauses  $(n)$  to variables  $(m)$ , given by  $n = 4.258m + 58.26m^{-\frac{2}{3}}$  (Crawford & Auton, 1996), where the likelihood of satisfiability sharply transitions from high to low as the number of clauses increases relative to the number of variables. This critical point presents the greatest computational challenge. Although these datasets may initially exhibit a uniform structure, CDCL solvers in D&C solvers can quickly generate a large number of learned clauses, making the subsequent SAT problems input to the neural network inherently rich in structural features. We selected two datasets: one with 300 variables and 1,278 clauses, and another with 350 variables and 1,491 clauses (hereafter referred to as the 300-1278 and 350-1491 datasets, respectively), each comprising 1,000 instances with a balanced ratio of satisfiable to unsatisfiable instances. These training datasets can be solved within seconds to tens of seconds, allowing for the generation of a large amount of training data. All these datasets can be solved within a reasonable timeframe, generating complete binary trees that are used for subsequent extraction of training data.
+
+Training. We implemented DPPO using a multi-agent architecture to enhance both exploration and learning efficiency. The system architecture includes five exploration agents continuously solving SAT problems and collecting training data, complemented by a single dedicated training agent. Each exploration agent runs 16 threads to solve SAT problems continuously. Upon completion, these agents parse the binary tree to extract training data, which is then transmitted to the training agent via
+
+a message queue. The training agent systematically retrieves this data from the queue, stores it in a replay buffer, and samples from this buffer to form batches for neural network training. We represent the state using a graph constructed from variables, original clauses, core learned clauses, and tier2 learned clauses, all extracted from the Sequential CDCL solver. We use the Adam optimizer to train the model. Post-training, the updated network parameters are synchronized across all exploration agents to maintain consistency and optimize policy performance. An  $\epsilon$ -greedy strategy is implemented to ensure a balance between exploration and exploitation, with the epsilon parameter decreasing progressively over time. Additionally, all rewards are normalized during training to maintain a consistent learning scale and enhance algorithmic stability. The exploration agents use only CPU resources, while the training agent uses only GPU resources. For exploration processes, we used LRB as the branch heuristics, as it performed better on the random dataset. Training took approximately 2-3 days. After completing this training, we switched the branch heuristics to VSIDS and trained a new model, which took about half a day. This training operation is hosted on a Linux server equipped with an Intel(R) Xeon(R) Platinum 8373C CPU, featuring 72 physical cores and 144 logical cores, supplemented by 256 GB of memory and a Tesla A100 GPU with 80 GB of memory. During training, we observed that even though random 3-SAT problems at the phase transition initially have just over a thousand clauses, the states used for training showed that the CDCL solver generated approximately ten to fifty thousand core and tier2 learned clauses, ensuring a rich structural representation.
+
+Deployment. After the training phase, we utilized TorchScript to export the model and integrate it with the Painless framework, thereby creating the RDC-SAT solver. This solver is a complete D&C SAT solver that no longer depends on Python, and it can dynamically generate variable probability rankings required for splitting decisions using the neural network. The solver can choose to run the neural network's forward propagation on either CPU or GPU. However, the relatively high overhead of neural networks often becomes a bottleneck in SAT solving with GNNs, especially when GPUs are unavailable and the forward propagation is done using CPUs. When multiple threads call the neural network simultaneously, resource contention may occur, slowing down the solving process. To address this and enable RDC-SAT to function effectively without GPU support, we implemented a new strategy: the neural network is used only once during the first required split to generate a ranked list of candidate split variables. For all subsequent splits, the solver directly selects split variables from this precomputed ranking. Additionally, when the problem size is very large, leading to excessive overhead from neural network calls or potential memory limitations, RDC-SAT will switch to the default splitting heuristic.
+
+# 5 EVALUATION
+
+In this chapter, we compare the performance of different splitting heuristics within the Painless framework across various datasets. We evaluate the performance of RDC-SAT against traditional splitting heuristics and analyze the cost of using GNNs.
+
+# 5.1 SETUP
+
+Solvers. We used the Painless framework to evaluate the performance of our RDC methods compared to baseline divide-and-conquer solvers. To ensure a fair comparison, we kept all configurations consistent except for the splitting heuristics. Specifically, we used FLIPS (Audemard et al., 2014b), VSIDS(Audemard et al., 2016), and PR (Nejati et al., 2017) splitting strategies from the work (Le Frioux et al., 2019), with P-CLONE-FLIPS showing the best overall performance, outperforming Treengeling and MapleAmpharos. Our RDC-SAT utilized core and tier2 learned clauses to construct the graph, employing two modes: RDC-GPU-always and RDC-CPU-once. RDC-GPU-always uses the GPU to accelerate neural network calls, selecting the split variable with GNN every time it splits. RDC-CPU-once, on the other hand, does not rely on GPU and only uses the CPU to call the GNN and generate variable sorting. For subsequent splitting, the worker uses the variable probability sorting to select split variables. Due to the poor performance of VSIDS on random datasets compared to LRB (as shown in Appendix E.2), we used Maple's LRB branch heuristics for the random dataset, and the RDC used the LRB model. In contrast, for the SAT Competition dataset, we used Maple's VSIDS branch heuristics, and the RDC used the VSIDS model.
+
+Dataset. We evaluated RDC-SAT on two datasets. (1) To investigate whether RDC-SAT has effectively learned the splitting heuristics, we randomly selected 200 SAT instances from the random
+
+dataset at the phase transition. This set includes 100 problems from the 450-1917 dataset and 100 from the 500-2129 dataset, with a 1:1 ratio of satisfiable to unsatisfiable instances. This dataset provides a suitable level of difficulty for the baseline solvers, enabling us to distinguish their performance. (2) To evaluate whether RDC-SAT, trained on random datasets, can generalize to more extensive, application-oriented datasets, we screened 305 small to medium-sized SAT problems from the 400 instances in the SAT Competition 2023. These problems, which make up the majority of the competition's instances, have a total number of variables and clauses below one million. The SAT Competition dataset includes a diverse range of difficult SAT instances, such as industrial applications and handmade problems. Since our model was trained solely on random datasets, without any knowledge of application-specific or handmade instances, this dataset serves as an ideal benchmark for evaluating the generalization capability of RDC-SAT.
+
+All experiments were conducted on the same machine used for training, with each solver allocated 16 threads. RDC-CPU-once was also limited to using 16 CPU cores for running the neural network's forward propagation process. The timeout for each instance was set to 5000 seconds. For instances that timed out, we used the same PAR2 score as in the SAT Competition, treating the timeouts as requiring double the timeout duration to solve. To analyze the performance of different splitting heuristics, we recorded the splitting and solving time. In our small-scale tests, we found that the PR method was not well-suited for solving random problems, and thus we did not evaluate PR on the random dataset.
+
+# 5.2 EVALUATION RESULT
+
+Table 1 presents all the results. Figure 4 illustrates the comparison of wall-clock time and problems solved across different splitting heuristics on two datasets. Instances that no solver could successfully solve within the time limit were excluded from the statistics.
+
+Table 1: Comparison of Splitting Heuristics across Datasets. #SAT, #UNSAT, and #ALL indicate the number of SAT problems, UNSAT problems, and total problems solved within the time limit. Clock Time is the average wall-clock time in seconds. PAR2 is the penalized average runtime. Splits are the average number of splits per problem. Splitting Time represents the average CPU time spent by all threads on splitting SAT subproblems. Solving Time represents the average CPU time spent by all threads using the Sequential CDCL solver to solve SAT subproblems. All time values are in seconds.
+
+<table><tr><td>Dataset</td><td>Splitting Heuristics</td><td>#SAT</td><td>#UNSAT</td><td>#All</td><td>Clock Time</td><td>PAR2</td><td>Splits</td><td>Splitting Time</td><td>Solving Time</td></tr><tr><td rowspan="4">Random (141)</td><td>RDC-GPU-always</td><td>76</td><td>53</td><td>129</td><td>1064.43</td><td>1489.96</td><td>60.42</td><td>20.78</td><td>16870.24</td></tr><tr><td>RDC-CPU-once</td><td>73</td><td>53</td><td>126</td><td>1238.04</td><td>1769.95</td><td>65.84</td><td>0.624</td><td>19316.30</td></tr><tr><td>FLIPS</td><td>74</td><td>50</td><td>124</td><td>1385.80</td><td>1988.64</td><td>49.21</td><td>0.00037</td><td>22090.20</td></tr><tr><td>LRB</td><td>70</td><td>50</td><td>120</td><td>1484.81</td><td>2229.49</td><td>74.77</td><td>0.00054</td><td>23617.79</td></tr><tr><td rowspan="5">SAT COMP 2023 (140)</td><td>RDC-GPU-always</td><td>50</td><td>85</td><td>135</td><td>764.00</td><td>942.57</td><td>162.19</td><td>142.47</td><td>10678.02</td></tr><tr><td>RDC-CPU-once</td><td>50</td><td>85</td><td>135</td><td>882.10</td><td>1060.67</td><td>191.95</td><td>8.77</td><td>11737.56</td></tr><tr><td>FLIPS</td><td>50</td><td>83</td><td>133</td><td>858.25</td><td>1108.25</td><td>80.02</td><td>0.0146</td><td>12296.52</td></tr><tr><td>VSIDS</td><td>46</td><td>84</td><td>130</td><td>901.12</td><td>1258.26</td><td>108.44</td><td>0.0013</td><td>12602.74</td></tr><tr><td>PR</td><td>49</td><td>72</td><td>121</td><td>1269.57</td><td>1948.14</td><td>104.19</td><td>0.0155</td><td>19232.39</td></tr></table>
+
+![](images/656112f5b57b2db70a74dcd94807ad75901a3405c867fb7cc617e9957590915c.jpg)  
+(a) Results on Random Dataset
+
+![](images/9edca4932d524f2d6f0f8a079983c63ffbdd4f7f961cc8f6798cf00910a75495.jpg)  
+(b) Results on SAT COMP 2023 Dataset  
+Figure 4: Comparison of wall-clock time and problems solved across splitting heuristics on two datasets.
+
+Evaluation on Random 3-SAT Dataset. For the Random 3-SAT dataset, the RDC-GPU-always and RDC-CPU-once modes were evaluated against traditional heuristics FLIPS and LRB. RDC-GPU-always successfully solved 129 out of 141 problems, while RDC-CPU-once solved 126, both significantly outperforming the traditional splitting heuristics in terms of average clock time and PAR2 score. Specifically, RDC-GPU-always achieved the best performance with an average clock time of 1064.43 seconds and a PAR2 score of 1489.96. Compared to the best baseline method, FLIPS, RDC-GPU-always reduced the PAR2 score by approximately  $25.08\%$ , demonstrating the model's capability to effectively learn and apply splitting heuristics from the random dataset, thus significantly enhancing solving efficiency. Even RDC-CPU-once, which only utilizes the initial SAT problem state for its analysis, reduced the PAR2 score by approximately  $10.99\%$ , showing substantial performance improvement over traditional heuristics.
+
+Generalization to SAT Competition Dataset. The RDC-SAT model, trained solely on random 3-SAT problems at the phase transition without any additional dataset information, was further evaluated on the SAT Competition dataset to test its generalization capability. This dataset includes a diverse range of problems, many of which are application-specific, providing a comprehensive test of RDC-SAT's robustness. Both RDC-GPU-always and RDC-CPU-once demonstrated competitive performance. Specifically, both methods successfully solved 135 problems, with RDC-GPU-always achieving the best average clock time of 764.00 seconds and a PAR2 score of 942.57. Compared to the best traditional method, FLIPS, which had a PAR2 score of 1108.25, RDC-GPU-always reduced the PAR2 score by approximately  $14.95\%$ , showcasing its effectiveness across varied problem types. RDC-CPU-once, while solving the same number of problems, recorded slightly higher average clock time and PAR2 score of 1060.67, yet still achieved a PAR2 reduction of approximately  $4.29\%$ , indicating substantial performance improvement even with limited reliance on neural network calls. These results underscore the ability of RDC-SAT to generalize well beyond its training conditions.
+
+Analysis of Splitting Time and Solving Time. To analyze the solving performance of our methods, we consider two key metrics: Splitting Time, which represents the total splitting time across all threads, and Solving Time, which represents the total solving time across all threads. In SAT solvers, incorporating a GNN is beneficial only if its performance improvements outweigh the overhead. In D&C solvers, the average number of splits per problem is relatively low; for example, in the random dataset, it ranges from 49.21 to 74.77 across various splitting heuristics (see Table 1). Therefore, even if the GNN is invoked at each split, it does not introduce significant performance overhead. Figures 5a and 5b illustrate the relationship between the splitting time per split and the total number of nodes for RDC-GPU-always and RDC-CPU-once on the two datasets. From the figures, it can be observed that the Splitting Time is approximately proportional to the total number of nodes. Notably, in RDC-GPU-always, the average time per split is less than 1 second. In RDC-CPU-once, although the initial GNN call takes slightly longer, subsequent splits are faster because they utilize precomputed variable rankings. When considering total CPU time, the Splitting Time of RDC-GPU-always accounts for only  $0.12\%$  and  $1.33\%$  of the total Solving Time on the random and SAT Competition datasets, respectively. Similarly, for RDC-CPU-once, it accounts for just  $0.003\%$  and  $0.075\%$ . Despite the slightly higher Splitting Time compared to baseline methods, the RDC methods significantly reduce the overall Solving Time. Specifically, RDC-GPU-always reduces the Solving Time by approximately  $23.63\%$  and  $13.17\%$  on the random and SAT Competition datasets, respectively, compared to the best-performing baseline, FLIPS. RDC-CPU-once achieves reductions of approximately  $12.56\%$  and  $4.54\%$ , respectively. Therefore, although the Splitting Time of both RDC-SAT methods is marginally higher than that of the baseline methods, they substantially decrease the Solving Time, leading to superior overall performance compared to the baselines.
+
+# 6 DISCUSSION
+
+In this paper, we propose a method to optimize splitting heuristics in D&C SAT solvers using reinforcement learning. We built a reinforcement learning environment tailored for Divide-and-Conquer tasks and trained a model, resulting in the RDC-SAT solver. Compared to baseline methods, RDC-SAT achieves significant performance improvements in wall-clock time. To the best of our knowledge, this is the first work addressing tree-structured state transitions induced by Divide-and-Conquer tasks using reinforcement learning. However, our focus was on exploring the potential of reinforcement learning and GNNs in optimizing splitting heuristics, rather than constructing a state-of-the-art (SOTA) parallel D&C SAT solver. Therefore, we did not modify any other components of
+
+Figure 5: Analysis of splitting Times for RDC-SAT on two dataset.  
+![](images/ce31fb0862bd6961aabca00358ba7d302b2d2fff649a39690b70567b76d2a18d.jpg)  
+(a) Splitting Time on Random Dataset. RDC-GPU-always: Avg. 72803.28 nodes, 0.344s per split. RDCCPU-once: Avg. 9043.16 nodes, 0.624s per split.
+
+![](images/cfe5c8e020675260a3a635bba34fe667413948a36717a7d8338efd8eece5659f.jpg)  
+(b) Splitting Time on SAT Competition 2023 Dataset. RDC-GPU-always: Avg. 248184.79 nodes, 0.878s per split. RDC-CPU-once: Avg. 191098.54 nodes, 8.77s per split.
+
+the Painless framework or train a general model on large-scale datasets containing multiple types of instances, leaving room for further optimization to reach better performance.
+
+Integration of GNNs with SAT Solvers. The integration of GNNs with SAT solving has become a hot research area in recent years. However, for SAT solvers, which are performance-critical tools, the overhead introduced by GNNs—including computational cost and GPU memory usage—is non-negligible, making it challenging to scale to very large SAT problems. Leading SAT solvers have not adopted GNNs. For example, attempts to swap CDCL solvers' branch heuristics with GNNs show that while GNNs can improve decision-making, their significant overhead might extend overall solving time. Our work expands the application of GNNs in SAT solving to a new direction. In a Divide-and-Conquer solver, the splitting heuristic significantly impacts performance, but its invocation frequency is relatively low, allowing RDC-SAT to improve wall-clock time. To extend GNNs to larger-scale SAT problems and reduce overhead, future efforts should prioritize reducing both the size and computational cost of the neural networks. Currently, the GNN employed in RDC-SAT is relatively large, resulting in substantial GPU memory consumption. We plan to reduce the size of the neural network to decrease memory usage and improve efficiency. For large SAT problems, sampling methods can be explored to construct the graph. Techniques such as sampled message passing could further reduce the overhead of GNNs. Minimizing GNN calls is another crucial focus. For example, using the RDC-once method proposed in this work, the GNN is called only initially or at specific points to generate information that guides subsequent decisions. Currently, RDC-once only calls the neural network once initially, without using any new information generated during the solving process. In the future, we will explore a hybrid approach that integrates initial GNN-generated information with default strategies for decision-making. Additionally, training specialized models for domain-specific problems(Li et al., 2022) may yield significant performance improvements for these applications.
+
+Further Improvements for RDC-SAT. The development of D&C solvers has lagged behind portfolio-based parallel solvers in recent years. Many D&C solvers, including the Painless framework we used, have not incorporated high-performance sequential solvers like Kissat and CaDiCal or recent optimizations. RDC-SAT currently uses default configurations of the Painless framework, such as MapleCOMSPS as the sequential solver. Replacing MapleCOMSPS with better-performing solvers, optimizing other components, and exploring different configurations could lead to substantial improvements. Additionally, since our experiments aimed to evaluate RDC-SAT's performance, we trained the model only on the random dataset without training a general model. Training on larger and more diverse datasets could enhance RDC-SAT's effectiveness and generalization. Furthermore, RDC-SAT uses limited information to represent states, focusing on key features of variables and clauses. Future work could explore incorporating additional features to achieve greater performance enhancements.
+
+In future work, we plan to implement these optimizations and evaluate the performance of RDC-SAT on larger datasets to further enhance its effectiveness.
+
+# ACKNOWLEDGMENTS
+
+This work was supported by National Key Research and Development Program of China Grant 2022YFB3305102, Natural Science Foundation of China Grant 61902011, and State Key Laboratory of Complex & Critical Software Environment (CCSE-2024ZX-16).
+
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+
+# A RELATED WORK
+
+# A.1 DIVIDE-AND-CONQUER SAT SOLVERS AND THEIR SPLITTING HEURISTICS
+
+Divide-and-Conquer (D&C) SAT solvers have gained prominence due to their ability to leverage parallel computing resources to tackle large and complex SAT problems efficiently. These solvers divide the problem into smaller subproblems, which are then solved independently. A crucial aspect of their performance is the heuristic used to select the best split variable, which significantly impacts the overall solving time and the balance of the resulting subspaces.
+
+Treengeling (Fleury & Heisinger, 2020) and Paracooba (Heisinger et al., 2020) are advanced SAT solvers that employ the Cube-and-Conquer method to harness parallelism. Treengeling utilizes a look-ahead heuristic during the Cube phase to divide the SAT problem into numerous subproblems, or cubes. This heuristic selects the split variable based on the number of unit propagations, aiming to create balanced subspaces. Treengeling generates more cubes than the available computational cores to ensure efficient utilization of all resources. Each cube is then solved independently by sequential CDCL solvers in the Conquer phase. Similarly, Paracooba extends the Cube-and-Conquer method to a distributed environment. It divides the SAT problem into many cubes using a look-ahead heuristic and distributes these cubes to different nodes across a network. Each node runs a CDCL solver to independently solve its assigned subproblems. By aggregating the computational power of multiple machines, Paracooba effectively handles significantly larger SAT problems, ensuring better load balancing and reducing overall solving time.
+
+In contrast to look-ahead techniques, some solvers use look-back methods that dynamically analyze the search performed by the solver, as well as formula statistics, to identify the best candidate at the current splitting point. For example, Ampharos (Audemard et al., 2016) picks the variable with the highest VSIDS activity, while MapleAmpharos (Nejati et al., 2017) uses a propagation-rate (PR) based splitting heuristic. This heuristic ranks variables by the ratio of the propagations they cause to the number of times they are branched on, selecting the highest-ranking variables for splitting. Another explored track is the number of flips of the variables (Audemard et al., 2014b;a). A flip occurs when a variable is propagated to the reverse of its last propagated value. Ranking the variables according to the number of their flips and choosing the highest one as a split variable helps to generate search subspaces with comparable computational time. This approach can also be used to limit the number of variables on which the look-ahead propagation is applied by preselecting a predefined percentage of variables with the highest number of flips. MaplePainlessDC (Nejati et al., 2020) uses a machine learning-based splitting heuristic to optimize divide-and-conquer parallel Boolean SAT solvers. The paper developed pairwise ranking models and minimum ranking models using a random forest classifier, comparing and selecting split variables based on structural features of the formula and dynamic probing statistics.
+
+The Painless Framework is a modular and high-performance parallel SAT solving framework that supports the implementation of various components, including multiple splitting heuristics. It allows developers to create parallel SAT solvers that can compete with state-of-the-art solvers. The framework implements several splitting heuristics, ensuring its solvers can match the performance of leading SAT solvers. Our reinforcement learning environment and the RDC-SAT solver are developed based on the Painless framework. This framework was also used for comparative experiments to evaluate the effectiveness of our approach.
+
+# A.2 OPTIMIZING SAT SOLVING WITH GNNS AND RL
+
+Graph Neural Networks (GNNs) and Reinforcement Learning (RL) have been increasingly applied to optimize SAT solving, showing promising results. Research like Graph-Q-Sat has utilized GNNs to learn branch heuristics in CDCL solvers, replacing VSIDS. Similarly, RL techniques have been used to train heuristics to enhance decision-making in SAT solvers. However, the overhead of calling GNNs for each variable decision in CDCL solvers is very high, making it difficult to apply to real-world problems. Consequently, recent research in this field has focused on reducing the overhead and frequency of GNN calls.
+
+Graph-Q-Sat(Kurin et al., 2020) is an approach that improves the heuristics of SAT solvers by employing GNNs and reinforcement learning. It uses value-based reinforcement learning to train a branching heuristic that significantly reduces the number of iterations compared to VSIDS. However,
+
+it requires calling GNNs at each decision point, which incurs high computational costs and does not reduce the actual wall-clock solving time. To mitigate this issue, other research, such as NeuroCore (Selsam & Bjørner, 2019), predicts the unsatisfiable core through periodic GNN inferences, reducing the frequency of GNN calls. While this approach enhances solver performance, it still demands substantial computational resources and relies heavily on GPUs. Neuroglue (Han, 2020) enhances SAT solvers by predicting glue variables, which are critical for efficient clause learning. This method periodically updates variable scores using GNN-based insights, significantly improving solver performance without incurring substantial computational costs. NeuroBack (Wang et al., 2023) builds upon these methods by making offline GNN predictions about variable phases before the SAT solving begins, thus eliminating the need for frequent online inferences and reducing computational costs. NeuroBack guides CDCL solvers like Kissat with a single GNN call for phase predictions, increasing performance on recent SAT competition benchmarks. SATformer (Shi et al., 2023) uses a GNN to extract clause representations only once, reducing computational costs, and applies a Transformer to predict unsatisfiability. Integrated into solvers like CaDiCaL and Kissat, SATformer improves solving efficiency over baseline methods. NeuroSelect (Liu et al., 2024) employs a GNN to guide the deletion of learned clauses, making only a single GNN call per instance. On instances with fewer than 400,000 nodes from the SAT Competition 2022, NeuroSelect improves performance over Kissat These methods primarily optimize branch heuristics and phase heuristics in CDCL solvers, but the overhead of GNNs remains a challenge.
+
+In contrast, our approach focuses on the importance of splitting heuristics in Divide-and-Conquer (D&C) solvers. The invocation frequency of these heuristics is relatively low, making it feasible to use GNNs without incurring high computational costs. Our RDC-GPU method leverages GNNs to significantly optimize D&C solver performance with minimal GNN calls. Even without GPUs, we can adopt a strategy similar to NeuroBack. By using RDC-CPU-once, we make a single GNN call to guide subsequent splitting decisions, thereby optimizing splitting heuristics while minimizing GNN overhead.
+
+# B IMPLEMENTATION DETAILS OF INTERACTIVE DIVIDE-AND-CONQUER ENVIRONMENT
+
+# B.1 DIVIDE-AND-CONQUER IN PAINLESS FRAMEWORK
+
+The Painless framework (Le Frioux et al., 2017; 2019) is a modular and high-performance parallel SAT-solving framework that is both flexible and ideal for comparing different algorithms. Within Painless, the D&C algorithm is implemented using a master-slave architecture that employs a workstealing strategy to achieve dynamic load balancing. Whenever a CPU core becomes idle, the master thread interrupts an active worker, splits its subproblem, and assigns it to two separate workers. Painless also supports clause sharing, allowing multiple workers to exchange learned clauses to speed up the solving process. This design makes the framework highly adaptable and efficient, empowering researchers to explore and compare various heuristics for parallel SAT solving.
+
+The Painless framework incorporates several D&C solvers, which can compete effectively with the leading solvers available. For the basic configuration in our experiment, we used the best-performing settings identified through experimentation. This includes utilizing MapleCOMSPS (Liang et al., 2016b) as the sequential solver, the "altoall" clause-sharing strategy, and the "clone" strategy. Specifically, MapleCOMSPS, which won the SAT Competition in 2016, serves as the sequential solver. The "altoall" clause-sharing strategy ensures that all solvers directly share clauses with an LBD below 4, which helps to improve the efficiency of the solving process. Additionally, the "clone" strategy allows a new worker, after requesting and dividing a partition, to duplicate the original solver and tackle the next problem. This helps to reuse learned clauses and internal solver information to accelerate problem-solving.
+
+Our Interactive Divide-and-Conquer Environment is primarily divided into two parts. First, we encapsulate the necessary components in the PainlessSolver within the Painless C++ environment, derived from painless-v2, The basic configuration is based on P-CLONE-FLIP (Le Frioux et al., 2019), and use Pybind11 to create a shared library. Then, we create a custom PainlessEnv in Python to facilitate interaction with the agent. PainlessSolver handles the solving process,
+
+![](images/b28ea337dacadb93bd9791abebdfdda73f76323c2ded29410c95652f4da16e68.jpg)  
+Figure 6: Interactive Divide-and-Conquer environment
+
+while PainlessEnv provides the interface for the agent to interact, allowing for dynamic state transitions and reward calculations.
+
+# B.2 PAINLESSSOLVER IN C++
+
+Algorithm 1: PainlessSolver::step  
+Input: action  
+Output: state, done  
+1 if there is a pending splitting request then  
+2 pass action as split variable to worker;  
+3 while not done do  
+4 if no globalEnding, no pending splitting requests, and no finished workers then  
+5 wait for signal;  
+6 parse finished workers and insert as leaf nodes in binary tree;  
+7 if there is a pending splitting request then  
+8 state = create graph from subproblem;  
+9 parse worker and insert as internal node in binary tree;  
+10 return state, done = false;  
+11 if globalEnding then  
+12 return root, done = true;
+
+We designed a new class, PainlessSolver, to handle all interactions and maintain statistics with the Python environment. When solving a new SAT problem, a PainlessSolver instance is created, initializing the Master thread, Worker threads, and Sharer thread, and setting the initial solving state. The Master thread manages the Worker threads, which use a CDCL solver to solve SAT problems. The Sharer thread handles the sharing of learned clauses among the Worker threads. Additionally, the PainlessSolver thread executes the logic described in the corresponding pseudocode. This class uses the solve() method to start the entire solving process. The interface built with Pybind11 enables the direct transmission of data generated by Painless to Python algorithms for processing and the reception of decision outputs from the Python environment to control the solving process. The core function step() of PainlessSolver is shown in Algorithm 1.
+
+Due to Python's Global Interpreter Lock (GIL), only one thread can execute Python bytecode at a time. In a D&C solver, multiple workers may request splits simultaneously. To manage these requests,
+
+we use a queue. When the master thread detects idle threads, it interrupts a worker, adds its instance to the pending split queue, and signals the PainlessSolver thread(line 5). The worker then blocks, waiting for the split variable. The PainlessSolver thread dequeues the worker instance, parses the problem state, adds it as an internal node to the binary tree, and sends the state to the Python environment(line 7-10). After processing, the PainlessSolver thread returns the split variable, allowing the worker to continue solving(line 2). While one worker waits, others continue solving their subproblems. Additionally, PainlessSolver parses finished workers and inserts them as leaf nodes in the tree(line 6). Once solving is complete, it returns the root node and sets done to true. Locks and signals ensure thread-safe communication and protect data structures.
+
+Finally, we used Pybind11 to package PainlessSolver into a shared library. This allows the shared library to be directly imported into Python, enabling interaction with PainlessSolver through its provided interfaces.
+
+# B.3 PAINLESSENV IN PYTHON
+
+To facilitate interaction between the agent and the environment, we defined a custom environment similar to gym. Env. OpenAI's gym library provides a widely-used framework for custom reinforcement learning environments. However, gym. Env is designed for linear state transitions, where each state follows a sequential order, and rewards are computed based on this continuous sequence. In these environments, the step function returns the "next state," which is not suitable for our case. In a divide-and-conquer reinforcement learning environment, each split creates two subproblems, forming a binary tree structure rather than a direct "forward" relationship between states. Additionally, it is not possible to immediately obtain the reward in divide-and-conquer, as this information is embedded within the tree structure. To address these challenges, we created a custom class PainlessEnv, mimicking the basic structure of the gym library but without inheriting it. This allows us to define state transitions and reward logic flexibly, accurately handling parallel and tree-structured transitions.
+
+PainlessEnv accepts a set of SAT problem paths and related configuration parameters as initial inputs, such as solver timeout, CPU core count, and levels of learned clauses. It provides reset and step methods for interaction. The reset method fully reinitializes the solving environment, creating a new PainlessSolver instance, loading a new SAT problem, and starting the solving process. The agent returns a split variable, which the step method passes to the PainlessSolver environment, then waits for the next split state while recording necessary training data. Upon solving the problem or timeout, PainlessEnv retrieves the root node of the binary tree, analyzes the solving process, and calculates the reward for each action. It performs a post-order traversal from the root node to compute the discounted rewards for all internal node states. This tree structure details the solving paths from root to leaf nodes, storing all this data in a replay buffer for subsequent machine learning model training.
+
+# C DETAILS OF REINFORCED DIVIDE-AND-CONQUER
+
+# C.1 NEURAL NETWORK
+
+Our approach utilizes a modified Graph Neural Network (GNN) structure similar to Graph-Q-Sat, replacing the DQN structure with an Actor-Critic network. As shown in Figure 7, the neural network encodes the SAT problem information as a graph. The Encoder transforms initial features into higher-dimensional representations, the Processor performs iterative message passing between nodes, edges, and global attributes, and the Decoder outputs variable, clause, and global features. The Actor network evaluates the variable features, outputs the probabilities for selecting each variable after removing invalid split variables, and the Critic network uses clause and global features to assess the current state's value. We implemented our neural network using PyTorch(Paszke et al., 2019) and PyTorch Geometric(PyG)(Fey & Lenssen, 2019).
+
+We adopted Graph Networks (Battaglia et al., 2018) as the basis of our GNN architecture. This framework is known for its versatility in handling various graph structures. The GN framework consists of three main parts: Encoder, Processor, and Decoder. The Encoder independently converts node, edge, and global features into higher-dimensional representations using MLPs. The Processor, the core component, updates these features through multiple layers of message passing, aggregating information from connected nodes and edges, and applying batch normalization to ensure scale
+
+![](images/8442dc9d5d66d81d145a746cf7c4cc3c5a26137a9995f803d1f56c345b43f0f3.jpg)  
+Figure 7: Network of RDC-SAT.
+
+invariance across nodes. The Decoder simplifies the complex graph structure into actionable outputs, retaining node and global features for the subsequent PPO usage while discarding edge features. Layer normalization (Ba et al., 2016) is applied within MLPs to expedite training convergence and reduce sensitivity to input scale variations.
+
+The Actor-Critic network leverages features derived from the GNN to perform variable selection and value estimation. The Actor network, responsible for decision-making, processes the variable features through an MLP and applies a softmax function to produce a probability distribution over the variables, excluding already chosen or invalid ones via a masking strategy. The Critic network evaluates the cumulative rewards from the current state using clause and global features. It employs attention pooling networks to aggregate clause nodes and global features, which are then fed into an MLP to generate the state value. This architecture allows the system to separate decision-making from value assessment, facilitating more precise policy adjustments and long-term reward optimization.
+
+# C.2 SETTINGS AND HYPERPARAMETERS
+
+We implemented a Distributed Proximal Policy Optimization (DPPO) framework, as shown in Figure 8, to accelerate data collection and training using one training process and multiple exploration processes. This framework includes a message queue for transferring collected data from exploration processes to the training process and a parameter server for synchronizing updated network parameters from the training process to the exploration processes. The message queue allows real-time transmission of states, actions, rewards, and subsequent states to the training process, which then stores them in a replay buffer for training. After updating the network parameters, the training process synchronizes these updates to the parameter server. Exploration processes check the parameter server for updates before solving new problems and update their parameters accordingly.
+
+Table 2 shows hyperparameters were used in our experiments:
+
+# D DATASETS
+
+# D.1 TRAINING DATA
+
+For our task involving reinforcement learning and graph neural networks, the most critical factors are problem size, difficulty, and structural richness. The ideal dataset should have a moderate problem size, which demands less memory during training; moderate difficulty, allowing for the rapid collection of substantial training data; and rich structure, enabling the GNN to learn more information. Although datasets like SAT Competition possess rich structural features, their generally high difficulty and larger size significantly slow down the training process for reinforcement learning. In contrast, random datasets offer highly controllable size and difficulty, and the learning clauses generated during the process can introduce additional structural features. Therefore, we chose to use a random dataset for our experiments.
+
+Our training dataset comes from the dataset provided by the study(Cameron et al., 2020). This study provides 11 datasets ranging from 100 to 600 variables, each containing 5000 satisfiable (SAT) and
+
+![](images/7e90953df85b5034f6eb36e87121da043a4f5a6b4058fb2f7cb51ebe01455d47.jpg)  
+Figure 8: Architecture of. Exploration agents interact with the PainlessEnv on CPU, collect training data, and send it to the message queue. The training agent on GPU samples data from the replay buffer to update the model parameters, which are then synchronized back to the exploration agents.
+
+5000 unsatisfiable (UNSAT) problems. We randomly selected 500 SAT and 500 UNSAT problems from the datasets with 300 and 350 variables as our training dataset. These problems have solving times ranging from a few seconds to several tens of seconds, and during the solving process, they generate thousands to tens of thousands of core and tier2 learned clauses. Consequently, the resulting graphs have relatively rich structural features.
+
+# D.2 EVALUATION DATA
+
+To investigate whether RDC-SAT has effectively learned the splitting heuristics, we randomly selected 200 SAT instances from the random dataset at the phase transition. This set includes 100 problems from the 450-1917 dataset and 100 from the 500-2129 dataset, with a 1:1 ratio of satisfiable to unsatisfiable instances. This dataset provides a suitable level of difficulty for the baseline solvers, enabling us to distinguish their performance.
+
+To evaluate whether RDC-SAT, trained on random datasets, can generalize to other types of more extensive, application-oriented datasets, we focused on small to medium-sized SAT problems from the SAT Competitions dataset. Some problems in the competition are extremely large, with many SAT instances having tens of millions of clauses. Applying GNNs to these problems would consume significant memory and computational resources. For such large-scale problems, our RDC-SAT solver automatically switches to the default splitting heuristics. Therefore, we primarily focus on the performance of various methods on small to medium-sized SAT problems. We first screened 400 problems from the SAT Competition 2023 (Balyo et al., 2023) dataset and selected those with a total number of variables and clauses less than one million, resulting in 305 SAT problems. These small to medium-sized problems constitute the majority of the SAT Competition.
+
+# EXPERIMENT DETAILS
+
+# E.1 BASELINES SELECTION
+
+The Painless framework provides three splitting heuristics: based on Flips, PR, and using CDCL solver branch heuristics to select split variables. We used these three methods for our evaluations. However, in a small-scale test, we found that the PR method did not perform well on the random dataset, leading us to exclude it from our random dataset tests. Additionally, we chose not to use MaplePainlessDC as a baseline. MaplePainlessDC(Nejati et al., 2020) is also a D&C solver based on the Painless framework, which uses a machine learning method to optimize the splitting heuristic. While the performance reported in the original paper exceeds that of Treengeling, it has been observed
+
+Table 2: Hyperparameters used in our experiments  
+
+<table><tr><td>Parameter</td><td>Value</td><td>Description</td></tr><tr><td colspan="3">PainlessEnv Configuration</td></tr><tr><td>timeout</td><td>1000</td><td>Maximum time (in seconds) allowed for exploration agent</td></tr><tr><td>ncpus</td><td>16</td><td>The number of threads used by each exploration agent</td></tr><tr><td>use_learnt</td><td>2</td><td>In addition to the original clause, two levels of learning clauses, core and tire2, are used to build the graph</td></tr><tr><td>node feats</td><td>4</td><td>Number of node features</td></tr><tr><td>edge feats</td><td>2</td><td>Number of edge features</td></tr><tr><td>global feats</td><td>4</td><td>Number of global features</td></tr><tr><td>use_LRB</td><td>true</td><td>Use LRB as branch heuristic</td></tr><tr><td colspan="3">Network Configuration</td></tr><tr><td>encoder_out_dims</td><td>(32, 32, 32)</td><td>Dimensions for the encoder outputs (node, edge, global)</td></tr><tr><td>core_out_dims</td><td>(64, 64, 32)</td><td>Dimensions for the core outputs (node, edge, global)</td></tr><tr><td>decoder_out_dims</td><td>(32, 1, 32)</td><td>Dimensions for the decoder outputs (node, edge, global)</td></tr><tr><td>core_steps</td><td>4</td><td>Number of processing steps in the Processor</td></tr><tr><td>n Hidden</td><td>1</td><td>Number of hidden layers in Processor&#x27;s MLP</td></tr><tr><td>hidden_size</td><td>64</td><td>Size of hidden layers in GNN&#x27;s MLP</td></tr><tr><td>e2v_aggregator</td><td>sum + BN</td><td>Aggregation method for edge to vertex. After sum aggregation, we use batch normalization(Ioffe &amp; Szegedy, 2015) to normalize the features</td></tr><tr><td>independent_block_layers</td><td>0</td><td>Number of hidden layer of Encoder and Decoder</td></tr><tr><td>n HiddenActor</td><td>2</td><td>Number of hidden layer of actor&#x27;s MLP</td></tr><tr><td>n Hidden_critic</td><td>2</td><td>Number of hidden layer of critic&#x27;s MLP</td></tr><tr><td>activation_function</td><td>LeakyRelu(Maas et al., 2013)</td><td>Activation function unsed in MLP</td></tr><tr><td colspan="3">DPPO Configuration</td></tr><tr><td>optimizer</td><td>Adam(Kingma &amp; Ba, 2014)</td><td>Optimizer used</td></tr><tr><td>lr_gnn</td><td>1e-3</td><td>Learning rate for the GNN</td></tr><tr><td>lr_ACTor</td><td>1e-3</td><td>Learning rate for the actor network</td></tr><tr><td>lr_critic</td><td>1e-3</td><td>Learning rate for the critic network</td></tr><tr><td>policy_loss_weight</td><td>1.0</td><td>Weight of the policy loss</td></tr><tr><td>value_loss_weight</td><td>0.5</td><td>Weight of the value loss</td></tr><tr><td>entropy_loss_weight</td><td>0.1</td><td>Weight of the entropy loss</td></tr><tr><td>eps Clip</td><td>0.2</td><td>Clipping parameter for PPO</td></tr><tr><td>epsilon_start</td><td>1.0</td><td>Starting value for epsilon in epsilon-greedy exploration</td></tr><tr><td>epsilon_final</td><td>0.01</td><td>Final value for epsilon in epsilon-greedy exploration</td></tr><tr><td>epsilon Decay</td><td>200</td><td>Decay rate for epsilon</td></tr><tr><td>normalize Reward</td><td>True</td><td>Whether to normalize rewards</td></tr><tr><td>gamma</td><td>1.0</td><td>Discount factor for future rewards</td></tr><tr><td>train_epochs</td><td>1</td><td>Number of training epochs</td></tr><tr><td>train_max_steps</td><td>10000</td><td>Maximum number of training steps</td></tr><tr><td>sample_max_nodes</td><td>300000</td><td>Maximum number of nodes of one batch to sample</td></tr><tr><td>num_of_process</td><td>5</td><td>Number of exploration processes</td></tr></table>
+
+from the experimental results of SAT Competition 2020(Nejati & Ganesh) that MaplePainlessDC can produce incorrect results in a small number of SAT instances. This potential unreliability in solving performance led us to exclude MaplePainlessDC as a baseline in our evaluations.
+
+# E.2 EXPLANATION FOR USING LRB OVER VSIDS IN RANDOM DATASET EXPERIMENTS
+
+The decision to use different branching heuristics for random and SAT Competition benchmarks stems from the observed poor performance of VSIDS on random datasets, as commonly noted in empirical studies. We conducted a small experiment using VSIDS as the branching heuristic on 100 instances from the 400-1704 random dataset with a time limit of 1000 seconds. The results of this experiment are presented in Table 3. This performance is significantly worse compared to using LRB as the branching heuristic (see Table 4 for a comparison). Therefore, we opted for Maple's LRB branching heuristics for both training and evaluation on the random datasets. It is also worth noting that even with VSIDS, RDC-SAT still significantly outperforms the baseline.
+
+Table 3: Performance Comparison of Splitting Heuristics on 400-1704 random dataset with VSIDS as Branching Heuristics  
+
+<table><tr><td>Splitting Heuristics</td><td>Clock time (s)</td><td>PAR2</td><td>Solved</td></tr><tr><td>RDC-GPU-always</td><td>388.23</td><td>588.18</td><td>80</td></tr><tr><td>RDC-CPU-once</td><td>405.48</td><td>703.34</td><td>77</td></tr><tr><td>FLIPS</td><td>460.70</td><td>720.72</td><td>74</td></tr><tr><td>VSIDS</td><td>502.28</td><td>852.20</td><td>65</td></tr></table>
+
+# E.3 MORE EVALUATION RESULT ON RANDOM DATASET
+
+Table 4 shows a detailed comparison of various splitting heuristics across different datasets. This table includes all instances, including those that no solver could solve within the time limit.
+
+Firstly, for relatively easy SAT problems, such as those in the 350-1491 dataset, using the default splitting heuristics like FLIPS and LRB can yield quick results. In these cases, the overhead of using a GNN-based approach might actually slow down the process. For example, FLIPS and LRB both performed better than RDC-GPU-always and RDC-CPU-once in terms of average solving time. This is because the simpler problems do not require the sophisticated analysis that GNN provides, and the additional computation introduces unnecessary delay. However, as the problem difficulty increases, the advantages of RDC-SAT become more apparent. In the 450-1917 dataset, RDC-GPU-always significantly outperformed traditional methods, with a total average solving time of 526.21 seconds compared to FLIPS' 827.07 seconds and LRB's 780.34 seconds. On the most challenging dataset, 500-2129, RDC-GPU-always not only achieved a lower average solving time but also solved more instances than the baselines. Specifically, RDC-GPU-always solved 30 instances in this dataset, whereas the best baseline method solved only 26 instances, representing a  $15.38\%$  increase in the number of solved instances.
+
+Additionally, the table reveals that the PR splitting heuristic performed very poorly on the 350-1491 dataset, with an average solving time of 61.65 seconds. This performance suggests that the PR method may not be suitable for solving random SAT problems. Due to this, we did not test the PR method on larger datasets.
+
+Table 4: Comparison of splitting Heuristics across datasets  
+
+<table><tr><td>Dataset</td><td>Splitting Heuristics</td><td>SAT Avg (s)</td><td>#SAT</td><td>UNSAT Avg (s)</td><td>#UNSAT</td><td>Total Avg (s)</td><td>PAR2</td></tr><tr><td rowspan="5">random 350-1491</td><td>RDC-GPU-always</td><td>12.34</td><td>50</td><td>19.80</td><td>50</td><td>16.03</td><td>16.03</td></tr><tr><td>RDC-CPU-once</td><td>11.16</td><td>50</td><td>17.99</td><td>50</td><td>14.58</td><td>14.58</td></tr><tr><td>FLIPS</td><td>10.76</td><td>50</td><td>17.22</td><td>50</td><td>13.99</td><td>13.99</td></tr><tr><td>LRB</td><td>11.02</td><td>50</td><td>17.96</td><td>50</td><td>14.49</td><td>14.49</td></tr><tr><td>PR</td><td>22.36</td><td>50</td><td>103.83</td><td>50</td><td>61.65</td><td>61.65</td></tr><tr><td rowspan="4">random 400-1704</td><td>RDC-GPU-always</td><td>19.14</td><td>50</td><td>63.38</td><td>50</td><td>41.27</td><td>41.27</td></tr><tr><td>RDC-CPU-once</td><td>18.29</td><td>50</td><td>64.35</td><td>50</td><td>41.32</td><td>41.32</td></tr><tr><td>FLIPS</td><td>18.16</td><td>50</td><td>80.27</td><td>50</td><td>49.21</td><td>49.21</td></tr><tr><td>LRB</td><td>17.90</td><td>50</td><td>67.84</td><td>50</td><td>42.87</td><td>42.87</td></tr><tr><td rowspan="4">random 450-1917</td><td>RDC-GPU-always</td><td>291.48</td><td>50</td><td>760.94</td><td>49</td><td>526.21</td><td>576.21</td></tr><tr><td>RDC-CPU-once</td><td>353.63</td><td>50</td><td>835.91</td><td>49</td><td>594.75</td><td>644.77</td></tr><tr><td>FLIPS</td><td>390.83</td><td>50</td><td>1263.31</td><td>48</td><td>827.07</td><td>927.07</td></tr><tr><td>LRB</td><td>341.64</td><td>50</td><td>1219.04</td><td>47</td><td>780.34</td><td>930.34</td></tr><tr><td rowspan="4">random 500-2129</td><td>RDC-GPU-always</td><td>3072.05</td><td>26</td><td>4777.20</td><td>4</td><td>3924.63</td><td>7424.63</td></tr><tr><td>RDC-CPU-once</td><td>3397.68</td><td>23</td><td>4804.05</td><td>4</td><td>4100.86</td><td>7750.86</td></tr><tr><td>FLIPS</td><td>3255.54</td><td>24</td><td>4898.27</td><td>2</td><td>4076.91</td><td>7776.91</td></tr><tr><td>LRB</td><td>3627.84</td><td>20</td><td>4898.62</td><td>3</td><td>4263.23</td><td>8113.23</td></tr></table>
+
+# E.4 DOMAIN-SPECIFIC BENCHMARK COMPARISON OF RDC-SAT AND SOTA PORTFOLIO SOLVERS
+
+SAT Competition results indicate that D&C solvers generally lag behind portfolio approaches in average performance. However, D&C solvers demonstrate unique advantages in specific scenarios.
+
+For instance, we conducted a small-scale experiment on a dataset containing 100 randomly selected instances from the miter benchmarks, which are used to verify circuit equivalence. The timeout was set to 1000 seconds, with all methods using 16 threads and VSIDS as the branch heuristic for the Sequential Solver. Using the model trained in our work. Table 5 compares our RDC-SAT with the D&C solver FLIPS (Le Frioux et al., 2019) and PRS (Chen et al., 2023) – the SAT Competition 2023 Parallel Track champion portfolio solver.
+
+Table 5: Performance Comparison on Circuit Equivalence Verification  
+
+<table><tr><td>Method</td><td>Solved</td><td>Avg. Time (s)</td><td>PAR2</td></tr><tr><td>RDC-SAT (D&amp;C)</td><td>80</td><td>313.64</td><td>513.64</td></tr><tr><td>FLIPS (D&amp;C)</td><td>76</td><td>333.66</td><td>573.66</td></tr><tr><td>PRS (Portfolio)</td><td>79</td><td>385.44</td><td>595.44</td></tr></table>
+
+This result shows that D&C solvers can outperform Portfolio solvers in specific domains. In this experiment, RDC-SAT solved the most instances and achieved the lowest PAR2, reducing PAR2 by  $13.74\%$  compared to the SOTA Portfolio solver PRS. The RDC-SAT model, trained on a random dataset, generalized effectively to the circuit equivalence dataset.
+
+# F THREATS TO VALIDITY
+
+During the training phase of our D&C SAT solver using reinforcement learning, several potential threats to validity may impact the results. Firstly, due to Python's Global Interpreter Lock (GIL) enforcing single-threaded execution for Python bytecode, we use a message queue to manage all split requests. When multiple threads simultaneously request splits, only one worker can call the neural network to obtain the split point, while other threads enter a waiting state. This restriction may lead to inefficiencies and does not fully replicate the actual D&C solver's runtime environment. Additionally, during training, multiple threads simultaneously calling the neural network, along with the training process itself, can cause GPU resource contention, leading to delays in split decisions and affecting training performance. Consequently, we opted to use CPUs for neural network calls during training to avoid GPU contention. However, this results in overestimated split times during training, as it does not accurately reflect the performance of the D&C solver in a real-world environment. Furthermore, our current setup limits the efficient use of computational resources, particularly GPUs. If more computational resources were available, such as additional GPUs for running the DPPO framework, this issue could be significantly mitigated. Enhanced computational resources would allow for more realistic training conditions, better reflecting the actual operational environment of the D&C solver, and potentially leading to more effective learning outcomes. These factors may result in the model's performance not reaching its optimal potential. With more computational resources to train the model and better simulate actual environments, RDC-SAT could achieve better results.
+
+Another potential threat to validity pertains to the experimental evaluation of our solvers. All experiments were conducted on a machine equipped with a CPU featuring 144 logical cores, an A100 GPU, and 256 GB of RAM. Due to the extensive time required for the experiments, we executed multiple solvers in parallel, allocating 16 threads to each solver. For the RDC-CPU-once configuration, the neural network calls were restricted to using a maximum of 16 threads. Although the number of threads used per solver is less than the total logical cores, the total number of threads during the SAT Competition experiments exceeded the number of physical cores, potentially leading to resource contention. This contention could affect performance, but it should impact all solvers equally, thus maintaining fairness in the comparisons. We primarily used small to medium-sized SAT problems, so we did not observe significant memory contention.
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+# LEARNING STRUCTURED REPRESENTATIONS BY EMBEDDING CLASS HIERARCHY WITH FAST OPTIMAL TRANSPORT
+
+Siqi Zeng $^{1}$  Sixian Du $^{2}$  Makoto Yamada $^{3}$  Han Zhao $^{1}$
+
+<sup>1</sup>University of Illinois, Urbana-Champaign <sup>2</sup>Stanford University
+
+3Okinawa Institute of Science and Technology
+
+{siqi6, hanzhao}@illinois.edu; dusixian@stanford.edu, makoto.yamada{oist.jp
+
+# ABSTRACT
+
+To embed structured knowledge within labels into feature representations, prior work (Zeng et al., 2022) proposed to use the Cophenetic Correlation Coefficient (CPCC) as a regularizer during supervised learning. This regularizer calculates pairwise Euclidean distances of class means and aligns them with the corresponding shortest path distances derived from the label hierarchy tree. However, class means may not be good representatives of the class conditional distributions, especially when they are multi-mode in nature. To address this limitation, under the CPCC framework, we propose to use the Earth Mover's Distance (EMD) to measure the pairwise distances among classes in the feature space. We show that our exact EMD method generalizes previous work, and recovers the existing algorithm when class-conditional distributions are Gaussian. To further improve the computational efficiency of our method, we introduce the Optimal Transport-CPCC family by exploring four EMD approximation variants. Our most efficient OT-CPCC variant, the proposed Fast FlowTree algorithm, runs in linear time in the size of the dataset, while maintaining competitive performance across datasets and tasks. The code is available at https://github.com/uiuctml/OTCPCC.
+
+# 1 INTRODUCTION
+
+Supervised classification problems have been a cornerstone in machine learning. While traditional classification methods treat class labels as independent entities, there has been a growing interest in leveraging semantic similarity among classes to improve model performance while following existing label dependency. This is particularly important in fine-grained classification tasks where labels are inherently living on a hierarchy, such as in the CIFAR (Krizhevsky & Hinton, 2009) and ImageNet (Deng et al., 2009) datasets. To capture the hierarchical relationship between classes, Zeng et al. (2022) made a significant contribution by embedding the tree metric of a given class hierarchy into the feature space. They introduced a regularizer, the Cophenetic Correlation Coefficient (CPCC), to correlate the tree metric with the Euclidean distance in the class-conditioned representations. This approach of encoding hierarchical relationship in the latent space not only led to more interpretable features compared to other hierarchical classification methods (Shen et al., 2023), but also improved some generalization performance across multiple datasets.
+
+Despite the value of this framework for embedding class hierarchies, it presents a notable limitation. Specifically, Zeng et al. (2022) uses the  $\ell_2$  norm between two class Euclidean centroids, or equivalently, maximum mean discrepancy with the linear kernel, as the distance of two class distributions. However, this approach overlooks the difference of samples within the same class, and thus cannot fully capture the global structural information given pairs of class-conditional distributions, especially when they are multi-modal. On the other hand, the use of Optimal Transport (OT) as an alternative to Euclidean centroids has first gained attention in the context of clustering, where Euclidean centroids neglect the intrinsic geometry of data distributions (Cuturi & Doucet, 2014). For the same motivation, we propose addressing the limitation of  $\ell_2$ -CPCC (Zeng et al., 2022), by integrating the OT distance,
+
+also known as the Wasserstein distance, and its variants, as a more effective metric for measuring the distance between class distributions in the problem of embedding hierarchical representations.
+
+Our contributions are as follows: (i) We first generalize the  $\ell_2$ -CPCC with EMD, and propose a family of OT-CPCC algorithm to address the limitation of using  $\ell_2$  centroids that misrepresents class relationships. We provide a detailed comparative analysis of  $\ell_2$ -CPCC versus OT-CPCC about their forward and backward computation. (ii) We propose a novel linear time OT approximation algorithm, FastFT, and apply it in the CPCC framework to learn hierarchical representations that is significantly more efficient than the cubic time exact EMD solution. (iii) We empirically analyze the advantage of OT-CPCC over  $\ell_2$ -CPCC across a wide range real-world datasets and tasks.
+
+# 2 PRELIMINARIES
+
+Notation and Problem Setup Throughout the paper we focus on the supervised  $k$ -class classification problem. We denote a sample of input  $x$  with label  $y$  living in the space of  $\mathcal{X} \subseteq \mathbb{R}^p$  and  $\mathcal{Y} = [k] := \{1, \ldots, k\}$ . The dataset  $\mathcal{D}$  consists of  $N$  data pairs  $\{(x_i, y_i)\}_{i=1}^N$ .
+
+Given the label space  $\mathcal{V}$ , we have a corresponding label tree  $\mathcal{T}$  that characterizes the relationship between different subsets of  $\mathcal{V}$ . In particular, each node of  $\mathcal{T}$  is associated with a subset of  $\mathcal{V}$ : each leaf node  $v$  is associated with a particular class in  $\mathcal{V}$ ; any internal node corresponds to the subset that is the union of its leaf children classes; the root node of  $\mathcal{T}$  corresponds to the full  $\mathcal{V}$ . The metric function of  $\mathcal{V}$  is called tree metric  $t: \mathcal{V} \times \mathcal{V} \to \mathbb{R}_+$  which measures the similarity of two class labels  $u, v$  based on the shortest path distance of two leaf nodes on  $\mathcal{T}$ . Intuitively, the tree  $\mathcal{T}$  could be understood as a dendrogram of  $\mathcal{V}$  that characterizes the similarity of different classes.
+
+Let  $\mathcal{Z} \subseteq \mathbb{R}^d$  be the representation/feature space. A deep network  $h$  is the composition of two functions  $h_{w,\theta} = g_w \circ f_\theta : \mathcal{X} \to \Delta_k$  where  $\Delta_k$  is a  $k-1$  dimensional probability simplex. The feature encoder  $f_\theta$  contains all layers until the penultimate layer of a neural network. On top of that,  $g_w : \mathcal{Z} \to \Delta_k$  is a linear classifier with weights  $w$ . Let  $q$  be the one-hot encoding vector of  $y$ . The parameters  $\theta$  and  $w$  are learned by minimizing  $\mathcal{L}$  combining the cross entropy loss  $\ell_{\mathrm{CE}}$  with an optional regularization term  $\mathcal{R}$ :
+
+$$
+\mathcal {L} (x, y; w, \theta) = \sum_ {i = 1} ^ {k} - q _ {i} \log h _ {w, \theta} (x) _ {i} + \lambda \mathcal {R} (x, y, \theta). \tag {1}
+$$
+
+The hyperparameter  $\lambda$  is the regularization factor which controls the strength of  $\mathcal{R}$ . The goal of the problem is to embed the class hierarchical relationship in the feature space. At a high level, the distance of class-conditioned representations should approximate the tree metric of classes in  $\mathcal{T}$ . This extra constraint can be quantitatively characterized by CPCC as the regularizer term  $\mathcal{R}$ .
+
+# 2.1 COPHENETIC CORRELATION COEFFICIENT
+
+The Cophenetic Correlation Coefficient (CPCC) (Sokal & Rohlf, 1962) was introduced as a regularization term in Zeng et al. (2022) to embed the class hierarchy  $\mathcal{T}$  in the feature space. CPCC has the same formulation as Pearson's correlation coefficient that measures the linear dependence between two metrics  $t$  and  $\rho$ :
+
+$$
+\operatorname {C P C C} (t, \rho) = \frac {\sum_ {u <   v} (t (u , v) - \bar {t}) (\rho (u , v) - \bar {\rho})}{(\sum_ {u <   v} (t (u , v) - \bar {t}) ^ {2}) ^ {- 1 / 2} (\sum_ {u <   v} (\rho (u , v) - \bar {\rho}) ^ {2}) ^ {- 1 / 2}}, \tag {2}
+$$
+
+where  $\rho$  is a distance function between  $u, v$  on feature space  $\mathcal{Z}$ , and  $\bar{t}, \bar{\rho}$  are the average of each metric. Let  $\mathcal{D}_u = \{(x_i^u, u)\}_{i=1}^m$ ,  $\mathcal{D}_v = \{(x_i^v, v)\}_{i=1}^n \subseteq \mathcal{D}$  be datasets with the same class label  $u, v$ . When  $\ell_2$ -CPCC is used as a regularizer in Eq. 1,
+
+$$
+\rho (u, v) := \left\| \frac {1}{m} \sum_ {(x _ {i} ^ {u}, u) \in \mathcal {D} _ {u}} f (x _ {i} ^ {u}) - \frac {1}{n} \sum_ {(x _ {j} ^ {v}, v) \in \mathcal {D} _ {v}} f (x _ {j} ^ {v}) \right\| _ {2},
+$$
+
+which is the  $\ell_2$  distance of the mean of the two class-conditional features. We call  $z_i^u$  as the  $i$ -th feature vector with label  $u$ , when the context is clear, we drop the superscript  $u$  for convenience. By
+
+Algorithm 1 Greedy Flow Matching  
+Input: number of features  $m,n$  weights  $a = (a_{1},\dots ,a_{m})$ $b = (b_{1},\ldots ,b_{n})$  1:  $\pmb {P} = \operatorname {Array}[m\times n]$  3: for each node  $v$  at height h do 2:  $i = 0$  4: if  $v$  is a leaf then 5: Pass its flow to its parent 6: else 7:  $C_a(v) = \{z:a(z) > 0\} ,C_b(v) = \{z:$ $b(z) > 0\}$  .  $z$  is the leaf children of  $v$  8: while  $\exists z_{a}\in C_{a}(v)$  and  $z_{b}\in C_{b}(v)$  do 9:  $\eta = \min (a(z_a),b(z_b))$  10:  $P(z_{a},z_{b}) + = \eta ,a(z_{a}) - = \eta ,b(z_{b}) - = \eta$  11: Pass non-zero unmatched flows in either  $a$  or  $b$  to its parent Output:  $\pmb{P}$
+
+maximizing CPCC  $(\lambda < 0)$ ,  $\rho$  follows the relation in  $t$ : classes sharing a close common ancestor will be grouped together while classes with a higher common ancestor are pulled away with respect to Euclidean distance.
+
+# 2.2 EXACT EARTH MOVER'S DISTANCE
+
+Earth Mover's Distance (EMD), or Optimal Transport distances, aims to find the optimal transportation plan between two discrete measures. For  $\mathcal{D}_u$ , let  $a = (a_1,\ldots ,a_m)\in \Delta_m$  which is not necessarily uniform. Let  $b$  defined similarly for  $\mathcal{D}_v$ , and  $\delta_z$  is the Dirac delta at  $z$ . Define two discrete measures as  $\mu = \sum_{i = 1}^{m}a_{i}\delta_{z_{i}},\nu = \sum_{j = 1}^{n}b_{j}\delta_{z_{j}}$ . Then, EMD between probability distributions defined over  $\mathcal{Z}$ , is the objective value of the following linear program:
+
+$$
+\min  _ {\boldsymbol {P}} \langle \boldsymbol {P}, \boldsymbol {D} \rangle , \quad \text {s . t .} \boldsymbol {P} \mathbf {1} _ {n} = a, \boldsymbol {P} ^ {\top} \mathbf {1} _ {m} = b. \tag {3}
+$$
+
+In Eq.3,  $\langle \cdot ,\cdot \rangle$  is the Frobenius inner product,  $\mathbf{1}_N$  is the  $N$ -dimensional all ones vector,  $\pmb{D}$  is the  $m\times n$  pairwise distance matrix where  $D_{ij} = \| z_i - z_j\|$ , and  $\pmb{P}$  is called the flow matrix in the computer science literature, or optimal transport plan in the OT literature, the same size with  $\pmb{D}$  with nonnegative entries. Intuitively, this is a minimum cost flow problem. Both source and target flow sum to 1, and we want to find the optimal transportation plan  $\pmb{P}$  with the minimum cost with respect to ground similarity  $\pmb{D}$ . The norm  $\| \cdot \|$  depends on the ground metric of  $\pmb{D}$ . We use the  $\| \cdot \| _2$  norm as the ground metric throughout this paper.
+
+When the dimension of the feature is 1, fast computation of the EMD distance exists (Houry et al., 2024). Instead of solving a linear program in Eq.3, when  $m = n$ ,  $a_{i} = b_{i} = 1 / n$  for all  $i$ , the closed form solution for 1d EMD is:  $\mathrm{EMD}_{1\mathrm{d}}(\mu ,\nu) = \frac{1}{n}\sum_{i = 1}^{n}\| z_{(i)} - z_{(i)}^{\prime}\|$ , and  $z_{(i)}$  is the  $i$ -th order statistic of the sequence  $\{z_1,\dots z_n\}$  after sorting. When  $m\neq n$ ,  $\mathrm{EMD}_{1\mathrm{d}}$  generalizes to solving inverse CDF of  $\mu ,\nu$  (Peyre & Cuturi, 2019) which requires approximation. To avoid this approximation, after sorting, we can compute the flow matrix  $\pmb{P}$  (Hoffman, 1963) efficiently via a greedy algorithm in linear time (Alg.1) and use the inner product objective of EMD in Eq.3 to compute the result. This greedy flow matching algorithm that generates the optimal transport plan can be used for any  $m,n$ , including the special case when  $m = n$ , and fit any non-uniform weights of  $a,b$ .
+
+# 3 CPCC WITH OPTIMAL TRANSPORT
+
+# 3.1 OT-CPCC FAMILY
+
+Despite its simplicity, the use of class centroids  $\ell_2$  distance as a measure of class distances has two limitations. First, since there is no guarantee of the distribution of class conditional features, the class mean difference may not accurately reflect the average pairwise Euclidean distance between all the pairs of sample. For example, in Fig.1, class  $v$  is a non-unimodal (bi-modal in this example) distribution, which is very likely to happen when class  $v$  is an abstract concept, such as coarse classes like household furniture in CIFAR100 (Krizhevsky & Hinton, 2009) or the collection of diverse dog breeds in ImageNet Deng et al. (2009). The blue  $\ell_2$  line can be a biased estimation of the average of
+
+red lines. We will see later (c.f. Sec.3.3) that the direction of blue vector plays an important role in the feature geometry in  $\ell_2$ -CPCC, and therefore we misrepresent the class distribution in  $\ell_2$ -CPCC. Second, it is known (Linial et al., 1994) that, for certain trees such as a basic star graph with three edges, it is impossible to embed tree metric  $t$  into  $\ell_2$  exactly, no matter how large feature dimension  $d$  is. It implies if we use the Euclidean centroids to represent a class, we can never reach the optimal CPCC value 1 during training, which always cause some loss of hierarchical information in the learning of structured representation.
+
+How to overcome the above limitations? Instead of finding a one-to-one correspondence of a label and a point estimate of a class, we represent a label class with all samples that belong to the class, which leads to EMD-CPCC. We call the set of data-weight pairs containing  $m$  class-  $u$  conditioned features as  $\mu = \{(z_i,a_i)\}_{i = 1}^m$  and define  $\nu$  similarly for the dataset containing  $n$  class-  $v$  conditioned features. The weight  $a_{i}$  can be set as uniform  $(1 / m)$  by default, or non-uniform if external prior knowledge is available. We propose EMD-CPCC by implementing  $\rho (u,v)\coloneqq \mathrm{EMD}(\mu ,\nu)$ . In Eq.3,  $P$  depends on  $D$  derived from each sample in two datasets. Besides, EMD is actually a generalization of  $\ell_2$  method:
+
+Proposition 3.1 (EMD reduces to  $\ell_2$  between means of Gaussian with same covariance).  $\operatorname{EMD}(\mathcal{N}(\mu_z, \Sigma), \mathcal{N}(\mu_{z'}, \Sigma)) = \| \mu_z - \mu_{z'}\|$
+
+![](images/3cd8154a4facd7fae94e429d0bed3cc1ac3b935f985efe2a12952878ad584f2f.jpg)  
+Figure 1: Comparison between EMD (weighted sum of red lines) and the class mean  $\ell_2$  distance (blue line).
+
+Although EMD has many appealing properties, the computational cost of EMD is high, where the time complexity of linear programming is cubic time with respect to the data size. Many following work provides approximation methods to reduce the computational cost, such as Sinkhorn (Cuturi, 2013), Sliced Wasserstein Distance (SWD) (Bonneel et al., 2015), Tree Wasserstein Distance (TWD) (Yamada et al., 2022; Le et al., 2019; Takezawa et al., 2021), and FlowTree (FT) (Backurs et al., 2020) (detailed discussion of approximation methods in App.A). Similarly, we can replace  $\rho$  with all other EMD approximation methods, and we call all of them as the OT-CPCC family. As a side remark, since all approximation methods provide a  $P$  satisfying constraints in Eq.3 and since EMD is lower bounded by  $\ell_2$  (see proof of Prop.3.1 in App.C), all OT-CPCC methods are still lower bounded by  $\ell_2$ .
+
+# 3.2 FAST FLOWTREE
+
+FlowTree (Backurs et al., 2020) is a tree-based method where the structure of the tree  $\mathcal{T}$  is learned using QuadTree (Samet, 1984) to approximate the Euclidean space. Once the tree has been built, it firsts compute the optimal flow matrix  $P_{t}$  using  $t$  as the ground metric with a linear time bottom-up as shown in Alg.2 (Kalantari & Kalantari, 1995) (see App.Fig.8 in for visualization). Then, it estimates EMD with  $\langle P_t,D_{\ell_2}\rangle$ . Given the tree, the time complexity of computing the FT is  $O((m + n)d\log (d\Phi))$  where  $\Phi$  is the maximum Euclidean distance of any two feature vectors.
+
+Our Fast FlowTree The main computational bottleneck of FT is the construction of the tree. However, in our case, we have access to  $\mathcal{T}$  given as prior knowledge, and thus we can skip the tree construction step. To apply tree-based EMD approximation methods, instead of building a QuadTree in FT, we first construct an augmented tree by extending the leaves of the original label tree  $\mathcal{T}$  with samples. Specifically, since each leaf node in  $\mathcal{T}$  corresponds to a class, we extend each leaf node by adding all samples from the corresponding class, as demonstrated in Fig.2. We set all extended edges to have edge weight 1, and assign the instance weight  $a_{i}$  for each data point. The resulting algorithm that applies Alg.2 to the augmented label tree is called Fast FlowTree (FastFT).
+
+Key Insight that allows FastFT to be more efficient than FT is that the tree metric between any two data samples from different classes is always identical in the augmented tree  $\mathcal{T}$ , which allows us to reduce the computation of  $P_{t}$  to 1d OT greedy flow matching using Alg.1. Now we can prove that Alg.1 and Alg.2 are equivalent given our constructed augmented tree in Fig.2, and the time complexity of FastFT is  $O((m + n)d)$ , which is optimal among all OT variants.
+
+Theorem 3.2 (Correctness of Fast FlowTree). For any augmented label tree  $\mathcal{T}$ , Alg. 1 and Alg. 2 return the same EMD approximation. The Fast FlowTree can be computed in  $O((m + n)d)$ .
+
+<table><tr><td>Method</td><td>Time Complexity</td><td>∂L/∂θ exact</td><td>Use Tree</td></tr><tr><td>ℓ2</td><td>O((nk + k2)d)</td><td>✓</td><td>N/A</td></tr><tr><td>EMD</td><td>O(n(d + n2 log n))</td><td>✓</td><td></td></tr><tr><td>Sinkhorn</td><td>O(n2(d + I))</td><td>✓</td><td></td></tr><tr><td>TWD</td><td>O(n2dI)</td><td></td><td>✓</td></tr><tr><td>SWD</td><td>O(np(d + log n))</td><td></td><td></td></tr><tr><td>FT</td><td>O(nd log(dΦ))</td><td></td><td>✓</td></tr><tr><td>FastFT</td><td>Θ(nd)</td><td></td><td>✓</td></tr></table>
+
+Table 1: Time complexity comparison of different CPCC methods. Let  $I$  be the number of iterations for iterative methods,  $p$  be the number of projections for SWD,  $\Phi$  be the maximum Euclidean distance of any two feature vectors. We use  $\tilde{\cdot}$  to represent a factor of  $k^2$ . For simplicity we assume  $m = n$ . In the batch learning setting,  $d$  can be much larger than  $n$  particularly for the fine-grained classification problems. See App.I for detailed analysis.
+
+![](images/3bf86b288d267f2b7eba4b96a5b8d9920ad1381f563d67356287a4bf3a3f76f8.jpg)  
+Figure 2: An example of augmented  $\mathcal{T}$ . Leaves become samples of each class, and each sample is assigned some weight (ex., uniform) within a class. Whenever we call FastFT-CPCC, only a subtree rooted at the lowest common ancestor of a pair of class label will be used. For example, we use subtree rooted at  $F$  to compute optimal flow of samples with label  $A$ ,  $B$ , and subtree rooted at  $H$  for samples with label  $A$ ,  $C$ .
+
+Please refer to App.D for proof details of Theorem 3.2. Note that all OT-CPCC methods inherently need  $\Omega((m + n)d)$  because we access each of  $m + n$  data points once and look at each entry in  $d$ -dimensions. Additionally, FastFT can be applied to any label tree with no restriction on structure or edge weights. To control the importance of data points, the flexibility of FastFT can be achieved by tuning  $a_i$ .
+
+Comparison of OT-methods on Synthetic Dataset To further understand the behavior of different OT approximation methods, we present experiments on synthetic datasets in Fig.3 (see App.J for more details). We first measured the computation time for different OT methods across datasets of varying sizes. Second, we evaluated the error of different OT approximation methods, along with the  $\ell_2$  distance of the mean of two datasets, in approximating EMD. Two different scenarios were considered, either both distributions were Gaussian or non-Gaussian. Under Gaussian distribution, the OT methods' results were closely aligned with the  $\ell_2$  method, except for SWD, which supports Prop.3.1 and implies the similarity in performance between OT and  $\ell_2$ -CPCC under Gaussian distributions. In non-Gaussian distribution scenario, notable differences emerged between OT methods and  $\ell_2$ , where SWD is closer to  $\ell_2$  measurement. FastFT demonstrated exceptional advantage as an approximation method, outperforming Sinkhorn in both computational speed and precision.
+
+![](images/77d6fe5529679f80a1bf835abb6eb3f74cb399c48ce70d9da1e62f4cee02bbdb.jpg)  
+Figure 3: Efficiency and approximation error comparison of OT methods on synthetic datasets.
+
+![](images/6648eeddef9a437f8326c1d5996ce028596ab56aaffdaac38b8bb0ef2ab41a6d.jpg)
+
+![](images/31e605042d276522d92e5eb5f552a53492a64776e934d86118b02c0408234934.jpg)
+
+# 3.3 DIFFERENTIABILITY ANALYSIS OF OT-CPCC
+
+Since we use some non-differentiable algorithms during training (ex., linear programming in EMD, sorting in SWD, tree construction in TWD, greedy algorithm in FT and FastFT), it is unclear how does the backpropagation algorithm work exactly in this problem. Thanks to the Danksin's theorem (Danskin, 1966) and the structure of the underlying optimization problem, we can still
+
+efficiently compute the gradients of OT-CPCC. For simplicity, let us consider the case of EMD-CPCC, i.e.,  $\rho = \mathrm{EMD}$  exactly. The other approximate variants can be analyzed in a similar way.
+
+The underlying challenge of the problem is to take derivatives of the optimal value of a minimization problem. We state the differentiability section of Danskin's theorem formally and prove a lemma:
+
+Theorem 3.3 (Danskin (1966)). Let  $V(x) = \max_{z \in \mathcal{C}} f(x, z)$ , and  $Z_0(x)$  is the set of  $z$  that achieve the maximum of  $V$  at  $x$ . If  $f: \mathbb{R}^n \times \mathcal{C} \to \mathbb{R}$  is continuous and  $\mathcal{C}$  is compact, then the subgradient of  $V$  is the convex hull of set  $\left\{\frac{\partial f(x, z)}{\partial x} : z \in Z_0\right\}$ . Particularly, if  $Z_0 = \{z^*\}$ ,  $\frac{\partial V}{\partial x} = \frac{\partial f(x, z^*)}{\partial x}$ .
+
+Lemma 3.4. Denote  $P^*$  as the optimal flow of EMD problem, then  $\frac{\partial \rho_{EMD}}{\partial \theta} = P^* \cdot \frac{\partial D}{\partial \theta}$ .
+
+Proof. We rewrite the objective function as a maximization problem:  $\rho_{\mathrm{EMD}} = -\max_{P\in \mathcal{C}} - \langle P,D\rangle$  Roughly speaking, Danksin's theorem states that the derivative of maximization problems can be evaluated at the optimum. We can get (sub-)differential of  $\rho_{\mathrm{EMD}}$  with respect to  $\theta$  with one step of chain rule:
+
+$$
+\frac {\partial \rho_ {\mathrm {E M D}}}{\partial \theta} = \frac {\partial \rho_ {\mathrm {E M D}}}{\partial \boldsymbol {D}} \cdot \frac {\partial \boldsymbol {D}}{\partial \theta} = - \frac {\partial - \langle \boldsymbol {P} ^ {*} , \boldsymbol {D} \rangle}{\partial \boldsymbol {D}} \cdot \frac {\partial \boldsymbol {D}}{\partial \theta} = \boldsymbol {P} ^ {*} \cdot \frac {\partial \boldsymbol {D}}{\partial \theta},
+$$
+
+where the last term comes from the original flow of backpropagation containing the gradients with respect to network parameters. The key takeaway is the gradient of  $\rho_{\mathrm{EMD}}$  w.r.t. the network parameters does not require the gradient from  $P$  in the EMD setting, although  $P$  depends on  $\theta$ .
+
+Differentiability for Other OT-CPCC Methods The application of Danskin's relies on the optimality of  $P^*$ . How about the other approximation methods? The differentiability is not a problem in Sinkhorn as the iterative algorithm only involves matrix multiplication. But for the SWD, TWD, FT, the approximate solutions (the flow matrix  $P$ ) for these methods are not explicitly formulated as solutions to an optimization problem. Following EMD, except for the Sinkhorn variant, our method is to treat flow matrix  $P$  always as a constant for these approximation methods, and stop the gradient through  $P$ . For example, in Fast FlowTree, since the tree flow matrix  $P$  depends only on the structure of the tree based on external information and not on the model parameters  $\theta$ , we have  $\frac{\partial \rho_{\mathrm{FastFT}}}{\partial \theta} := P_{\mathrm{FastFT}} \cdot \frac{\partial D}{\partial \theta}$ .
+
+To provide a more fine-grained understanding of the proposed method, in what follows we analyze the gradients of the proposed CPCC variants and compare them with the  $\ell_2$ -CPCC.
+
+Gradient of  $\ell_2$ -CPCC If we want to know the difference between  $\ell_2$ -CPCC and EMD-CPCC, assume the same input and the same network parameters, ignoring the cross-entropy loss, the only difference of the gradient starts from the distance calculation between two classes. Assume representations from two classes are  $Z \in \mathbb{R}^{m \times d}$ ,  $Z' \in \mathbb{R}^{n \times d}$ .
+
+$$
+\frac {\partial \rho_ {l _ {2}}}{\partial Z} = \frac {\partial}{\partial Z} \| \mu_ {z} - \mu_ {z ^ {\prime}} \| = \frac {\mathbf {1} \left(\frac {1}{m} \mathbf {1} ^ {\top} Z - \frac {1}{n} \mathbf {1} ^ {\top} Z ^ {\prime}\right)}{m \left\| \frac {1}{m} Z ^ {\top} \mathbf {1} - \frac {1}{n} Z ^ {\prime} ^ {\top} \mathbf {1} \right\|}.
+$$
+
+Let  $i$ -th row of  $Z$  be  $Z_{i}$ . Each row of  $\frac{\partial\rho_{l_2}}{\partial Z}$  is the same, so
+
+$$
+\frac {\partial \rho_ {l _ {2}}}{\partial Z _ {i .}} = \frac {\mu_ {z} - \mu_ {z ^ {\prime}}}{m \| \mu_ {z} - \mu_ {z ^ {\prime}} \|}.
+$$
+
+Gradient of EMD-CPCC Let  $D \in \mathbb{R}^{m \times n}$  be the cost matrix and  $P$  is any constant flow matrix:
+
+$$
+\rho_ {\mathrm {E M D}} = \left\langle \boldsymbol {P} ^ {*}, \boldsymbol {D} \left(\boldsymbol {Z}, \boldsymbol {Z} ^ {\prime}\right) \right\rangle = \sum_ {i, j} \boldsymbol {P} _ {i j} ^ {*} \left\| Z _ {i \cdot} - Z _ {j \cdot} ^ {\prime} \right\|.
+$$
+
+Let's calculate the derivative by rows of  $Z$ . We only keep the terms related to  $Z_{i}$
+
+$$
+\frac {\partial \rho_ {\mathrm {E M D}}}{\partial Z _ {i .}} = \frac {\partial}{\partial Z _ {i .}} \sum_ {j} \boldsymbol {P} _ {i j} ^ {*} \| Z _ {i .} - Z _ {j .} ^ {\prime} \| = \sum_ {j = 1} ^ {n} \boldsymbol {P} _ {i j} ^ {*} \frac {Z _ {i .} - Z _ {j .} ^ {\prime}}{\| Z _ {i .} - Z _ {j .} ^ {\prime} \|}.
+$$
+
+Note that since  $P^{*}$  is the optimal solution of a linear program, it is a sparse matrix with at most  $m + n - 1$  non-zero entries (Peyre et al., 2019, Proposition 3.4). As a comparison to the gradient of  $\ell_{2}$ -CPCC, which assigns equal importance to all data points and only depends on the direction of class centroids, the gradient of EMD-CPCC assigns different importance to each pair of data points and depends on the pairwise transport probability between data points. From this analysis, we can see that the gradient provided by EMD-CPCC is more fine-grained and informative than that of  $\ell_{2}$ -CPCC.
+
+# 4 EXPERIMENTS
+
+For a fair comparison, we closely aligned our implementation with Zeng et al. (2022) to ensure consistency. This section will cover dataset benchmarks, the multi-modality of features, downstream hierarchical classification retrieval, and interpretability analysis. A detailed description of training hyperparameters is provided in App.F, along with additional experiments, including the effect of label hierarchy structure, label hierarchy tree weights, optimal transport flow settings (distribution of  $a$  in Eq.3), and comparisons with non-CPCC hierarchical methods, in App.H. Throughout this section, the Flat objective is the standard cross entropy loss in Eq.1 with  $\lambda = 0$  without tree information.
+
+Table 2: Dataset Statistics. Table 3: Preciseness of hierarchical information.  
+
+<table><tr><td>Dataset</td><td># train</td><td># test</td></tr><tr><td>CIFAR10</td><td>50K</td><td>10K</td></tr><tr><td>CIFAR100</td><td>50K</td><td>10K</td></tr><tr><td>INAT</td><td>500K</td><td>100K</td></tr><tr><td>L17</td><td>88K</td><td>3.4K</td></tr><tr><td>N26</td><td>132K</td><td>5.2K</td></tr><tr><td>E13</td><td>334K</td><td>13K</td></tr><tr><td>E30</td><td>307K</td><td>12K</td></tr></table>
+
+![](images/69caac0cb53e9f03afe20f36fe87200bf93a828b6c7821dbd2ca2cfc81cc0f59.jpg)
+
+![](images/89e6d6228057c33d655b01e098307bf493f2c79e908ea563f0bc41c14a974edc.jpg)  
+Figure 4: Hierarchical data split. Source and target dataset share the same coarse labels but different fine labels.
+
+<table><tr><td>Dataset</td><td>Objective</td><td>TestCPCC</td></tr><tr><td rowspan="6">CIFAR10</td><td>Flat</td><td>58.83 (8.80)</td></tr><tr><td>\( \ell_2 \)</td><td>99.94 (0.02)</td></tr><tr><td>FastFT</td><td>99.95 (0.00)</td></tr><tr><td>EMD</td><td>99.95 (0.01)</td></tr><tr><td>Sinkhorn</td><td>99.94 (0.01)</td></tr><tr><td>SWD</td><td>99.98 (0.00)</td></tr><tr><td rowspan="6">CIFAR100</td><td>Flat</td><td>22.13 (0.84)</td></tr><tr><td>\( \ell_2 \)</td><td>83.08 (0.11)</td></tr><tr><td>FastFT</td><td>93.85 (0.05)</td></tr><tr><td>EMD</td><td>89.60 (0.34)</td></tr><tr><td>Sinkhorn</td><td>93.81 (0.10)</td></tr><tr><td>SWD</td><td>73.17 (2.66)</td></tr><tr><td rowspan="6">INAT</td><td>Flat</td><td>31.97 (0.00)</td></tr><tr><td>\( \ell_2 \)</td><td>67.01 (0.04)</td></tr><tr><td>FastFT</td><td>73.20 (0.03)</td></tr><tr><td>EMD</td><td>65.80 (0.01)</td></tr><tr><td>Sinkhorn</td><td>73.20 (0.04)</td></tr><tr><td>SWD</td><td>50.88 (0.03)</td></tr></table>
+
+<table><tr><td>Dataset</td><td>Objective</td><td>TestCPCC</td></tr><tr><td rowspan="5">L17</td><td>Flat</td><td>47.63 (1.18)</td></tr><tr><td>\( \ell_2 \)</td><td>92.30 (0.35)</td></tr><tr><td>FastFT</td><td>91.71 (0.15)</td></tr><tr><td>Sinkhorn</td><td>91.77 (0.44)</td></tr><tr><td>SWD</td><td>95.10 (0.51)</td></tr><tr><td rowspan="5">N26</td><td>Flat</td><td>29.77 (0.53)</td></tr><tr><td>\( \ell_2 \)</td><td>93.03 (0.32)</td></tr><tr><td>FastFT</td><td>92.68 (0.21)</td></tr><tr><td>Sinkhorn</td><td>93.31 (0.24)</td></tr><tr><td>SWD</td><td>92.40 (0.05)</td></tr><tr><td rowspan="5">E13</td><td>Flat</td><td>49.20 (0.86)</td></tr><tr><td>\( \ell_2 \)</td><td>92.02 (0.08)</td></tr><tr><td>FastFT</td><td>91.97 (0.20)</td></tr><tr><td>Sinkhorn</td><td>91.30 (0.57)</td></tr><tr><td>SWD</td><td>88.32 (0.50)</td></tr><tr><td rowspan="5">E30</td><td>Flat</td><td>51.33 (0.09)</td></tr><tr><td>\( \ell_2 \)</td><td>93.37 (0.19)</td></tr><tr><td>FastFT</td><td>91.81 (0.44)</td></tr><tr><td>Sinkhorn</td><td>91.89 (0.61)</td></tr><tr><td>SWD</td><td>89.88 (0.68)</td></tr></table>
+
+# 4.1 DATASETS
+
+We conducted experiments across 7 diverse datasets, CIFAR10, CIFAR100 (Krizhevsky & Hinton, 2009), BREEDS benchmarks (Santurkar et al., 2021) including four settings LIVING17 (L17), ENTITY13 (E13), ENTITY30 (E30), NONLIVING26 (N26), and iNaturalist-mini (Van Horn et al., 2018) (INAT) to thoroughly evaluate the efficacy and influence of OT-CPCC methods. The dataset statistics is shown in Tab.2. Note that BREEDS and INAT are larger scale datasets with high resolution images. As subsets of ImageNet, BREEDS inherit a more complicated label hierarchy and more realistic, possibly multi-modal class distributions. The major difference from ImageNet is that BREEDS's class hierarchy is manually calibrated so that it reflects the visual hierarchy instead of the WordNet semantic hierarchy in the original ImageNet, which removes ill-defined hierarchical relationships unrelated to vision tasks.
+
+# 4.2 EXISTENCE OF MULTI-MODE CLASS-CONDITIONED FEATURE DISTRIBUTION
+
+To verify our hypothesis in Fig.1, we aim to determine whether class-conditioned clusters exhibit a multi-modal structure in practice. We track the class-conditioned features during training: at each epoch, for each class-conditioned feature cluster, we fit a Gaussian Mixture Model (GMM) with the number of components ranging from 1 to 10. We report the optimal number of components based on the lowest Akaike Information Criterion (AIC), averaged across all fine-level class labels. If the best number of Gaussian components exceeds 1, this may indicate the presence of a multi-modal structure in the feature distributions, assuming a sufficiently large batch size. From Fig.5, with default training hyperparameters in App.F, we observe that this phenomenon occurs across most datasets for both  $\ell_2$ -CPCC and OT-CPCC training, suggesting that using optimal transport distances in CPCC computation is more appropriate.
+
+![](images/8d065004e43dcc6b2c79d1d2bd47a255a9d721f5a70ef298e629a886e11af468.jpg)  
+Figure 5: The average best number of components of learned features (left) across training epochs on CIFAR100 for different CPCC methods (right) for the  $\ell_2$ -CPCC final trained features for all datasets.
+
+<table><tr><td>Dataset</td><td># Comp.</td></tr><tr><td>CIFAR10</td><td>9.90</td></tr><tr><td>CIFAR100</td><td>3.07</td></tr><tr><td>INAT</td><td>3.48</td></tr><tr><td>BREEDS</td><td>1.25</td></tr></table>
+
+![](images/3b86c6d975d3a2539bf2d3765ed877b6c41d11eeb2049181ffe1178e070f00d3.jpg)  
+Figure 6: CIFAR100 Euclidean UMAP for  $\ell_2$ -CPCC (left) vs. EMD-CPCC (right). Each cluster represents a fine label and we color data sample by coarse labels.
+
+# 4.3 HIERARCHICAL CLASSIFICATION AND RETRIEVAL
+
+Table 4: Generalization performance on coarse level. The prefix of metric represents the split of test set:  $\mathbf{s} =$  source,  $\mathbf{t} =$  target. The average number of three seeds are reported. See the detailed version including standard deviation in Tab.16,17 and performance on each BREEDS setting is in Tab.18,19. Due to the computation complexity, we do not report EMD results on training BREEDS from scratch.  
+
+<table><tr><td>Dataset</td><td>Objective</td><td>sAcc</td><td>tAcc</td><td>sMAP</td><td>tMAP</td><td>Dataset</td><td>sAcc</td><td>tAcc</td><td>sMAP</td><td>tMAP</td></tr><tr><td rowspan="5">CIFAR10</td><td>Flat</td><td>99.58</td><td>87.30</td><td>99.22</td><td>89.66</td><td rowspan="5">INAT</td><td>94.63</td><td>38.81</td><td>70.41</td><td>34.00</td></tr><tr><td>FastFT</td><td>99.61</td><td>87.79</td><td>99.91</td><td>93.04</td><td>94.66</td><td>39.43</td><td>72.90</td><td>35.63</td></tr><tr><td>EMD</td><td>99.61</td><td>87.45</td><td>99.88</td><td>93.07</td><td>94.64</td><td>41.01</td><td>73.87</td><td>35.58</td></tr><tr><td>Sinkhorn</td><td>99.61</td><td>87.41</td><td>99.87</td><td>92.60</td><td>94.30</td><td>36.94</td><td>68.75</td><td>34.92</td></tr><tr><td>SWD</td><td>99.56</td><td>87.63</td><td>99.36</td><td>90.12</td><td>94.52</td><td>39.38</td><td>75.13</td><td>38.11</td></tr><tr><td rowspan="5">CIFAR100</td><td>Flat</td><td>86.17</td><td>43.09</td><td>82.07</td><td>42.09</td><td rowspan="5">BREEDS</td><td>93.85</td><td>82.43</td><td>74.47</td><td>66.50</td></tr><tr><td>FastFT</td><td>86.96</td><td>44.29</td><td>91.71</td><td>42.20</td><td>94.25</td><td>82.89</td><td>92.84</td><td>79.69</td></tr><tr><td>EMD</td><td>86.93</td><td>44.03</td><td>91.82</td><td>42.32</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Sinkhorn</td><td>86.93</td><td>43.99</td><td>91.61</td><td>42.45</td><td>94.42</td><td>83.51</td><td>91.84</td><td>79.33</td></tr><tr><td>SWD</td><td>86.42</td><td>43.35</td><td>87.24</td><td>42.39</td><td>94.41</td><td>82.96</td><td>83.58</td><td>72.38</td></tr></table>
+
+Table 5: Generalization performance on fine level.  
+
+<table><tr><td>Dataset</td><td>Objective</td><td>sAcc</td><td>tAcc</td><td>sMAP</td><td>Dataset</td><td>sAcc</td><td>tAcc</td><td>sMAP</td></tr><tr><td rowspan="5">CIFAR10</td><td>l2</td><td>96.96</td><td>55.71</td><td>99.22</td><td rowspan="5">INAT</td><td>88.62</td><td>26.78</td><td>56.10</td></tr><tr><td>FastFT</td><td>96.90</td><td>55.99</td><td>99.24</td><td>88.49</td><td>27.10</td><td>56.21</td></tr><tr><td>EMD</td><td>97.05</td><td>56.12</td><td>99.24</td><td>88.68</td><td>26.78</td><td>56.83</td></tr><tr><td>Sinkhorn</td><td>96.95</td><td>54.89</td><td>99.27</td><td>88.08</td><td>26.77</td><td>51.56</td></tr><tr><td>SWD</td><td>96.96</td><td>59.21</td><td>99.45</td><td>88.46</td><td>26.78</td><td>54.19</td></tr><tr><td rowspan="5">CIFAR100</td><td>l2</td><td>80.98</td><td>23.76</td><td>77.52</td><td rowspan="5">BREEDS</td><td>82.66</td><td>45.95</td><td>62.86</td></tr><tr><td>FastFT</td><td>81.09</td><td>24.71</td><td>78.72</td><td>82.58</td><td>45.75</td><td>63.95</td></tr><tr><td>EMD</td><td>81.32</td><td>23.15</td><td>78.88</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Sinkhorn</td><td>80.99</td><td>23.53</td><td>78.51</td><td>82.99</td><td>46.87</td><td>61.54</td></tr><tr><td>SWD</td><td>80.50</td><td>26.18</td><td>86.82</td><td>82.90</td><td>46.33</td><td>76.24</td></tr></table>
+
+Hierarchical Dataset Split For downstream classification and retrieval, we split each dataset into source and target subsets, with both further divided into train and test sets, as illustrated in Fig.4. The source and target datasets share the same level of coarse labels but differ in fine labels, where the fine level refers to the more granular class labels that are exactly one level below the coarse labels. For example, in CIFAR10, the source set includes the fine labels (deer, dog, frog, horse) and (ship, truck), while the target set includes (bird, cat) and (airplane, automobile). Both sets share the same coarse labels: animal and transportation. We include details of hierarchical construction and splits for other datasets in App.E.
+
+Given the hierarchical splits, we define two evaluation protocols. First, we pretrain on the source-train set and evaluate on the source-test set, where the train and test distributions are i.i.d. Second, we pretrain on the source-train set and evaluate on the target-test set, introducing a subpopulation shift (Santurkar et al., 2021) between the train and test distributions. For each protocol, we conduct classification and retrieval experiments at two levels of granularity.
+
+Metrics For classification experiments, we measure performance using accuracy (Acc). At the fine level, source accuracy is computed directly using any pretrained model, while coarse source accuracy is calculated by summing the fine-level probabilities for predictions that share the same coarse parent label. Since the source and target sets share the same coarse labels, coarse target accuracy can be similarly computed using the model pretrained on the source split. For fine target accuracy, after pretraining on the source set, we train a linear classifier on top of the frozen pretrained encoder for one-shot learning on the target-train set and evaluate it on the target-test set. We include the visualization of all types of classification tasks evaluation in Fig.9 in the Appendix.
+
+For retrieval experiments, we use mean average precision (MAP) as the evaluation metric. Given a model pretrained on the source set, we compute the fine or coarse class prototypes by calculating the Euclidean centroid of the class-conditioned features. Using these class prototypes, we compute the cosine similarity score between the prototypes and test points, and evaluate whether test points are most similar to the class prototype with the same label via average precision. Since the source and target sets have different fine labels, we do not include target MAP in Tab.5.
+
+Discussion OT-CPCC outperforms  $\ell_2$ -CPCC consistently on fine level tasks: the oracle for the fine level tasks should be Flat, and Zeng et al. (2022) reported that  $\ell_2$ -CPCC does not perform well for several fine level tasks due to the constraint on coarse level. Tab.5 shows that OT-CPCC methods show better fine-level performance than  $\ell_2$ , while maintaining hierarchical information simultaneously. Besides, OT-CPCC outperforms the Flat baseline consistently on coarse level tasks as expected in Tab.4, which implies OT-CPCC methods successfully include the additional hierarchical information by optimizing the regularizer. In Fig.6, we visualize the learned hierarchical embeddings using UMAP (McInnes et al., 2018), with Euclidean distance as the similarity metric. We observe that  $\ell_2$ -CPCC displays a clearer coarse-level grouping pattern, likely due to the direct optimization of CPCC in Euclidean space. However, EMD-CPCC also retains a significant amount of hierarchical information and demonstrates better separation between fine-level classes, which aligns with the results observed in our classification and retrieval tasks.
+
+# 4.4 INTERPRETABILITY EVALUATION
+
+![](images/7d687947fc8c62b3dfecd953e39b28e4636fc5a4c51b58c2d707965f5ba1df13.jpg)  
+Figure 7: Test CPCC for various batch size, regularization strength, and architecture, from left to right. FastFT consistently performs the best across all settings.
+
+![](images/126708587b6f44f83d01ef4abb2f49e35d7ba57c1ead48168b80ab9a443f4872.jpg)
+
+![](images/fd7bbe3d461b48bad54f89e28d7d6f4dd132cc0d416bbb1a471c3176899e43f0.jpg)
+
+A model is more interpretable when its representations accurately capture hierarchical information. CPCC is a natural measure of the precision of the hierarchical structure embedded in corresponding metric spaces. Unlike rank-based metrics, where only the relative order of distances matters, CPCC not only requires the correct ranking of distances (i.e., fine classes sharing the same coarse class should be closer than fine classes with different coarse parents), but is also influenced by the absolute values of the node-to-node distances, because CPCC is affected by the given label tree weights. Therefore, we report CPCC, a stricter metric, on the test set. In Tab.3, we present  $\ell_2$ -CPCC for the Flat and  $\ell_2$  objectives, and use the corresponding OT distances for the OT-CPCC experiments.
+
+We observe that OT-CPCC preserves more hierarchical information, leading to better interpretability. As shown in Tab.3, in 5 out of 7 datasets, an OT-CPCC method achieves the best TestCPCC score. For E13 and E30, OT-CPCC methods slightly underperform compared to  $\ell_2$ , likely due to the BREEDS representations being less multi-modal than those of other datasets, as indicated in Tab.5.
+
+Ablation Study In our experiments with CIFAR100, shown in Fig.7, we investigated the impact of batch size, the regularization factor  $\lambda$  in Eq.1, and architecture on TestCPCC. Given CIFAR100's large number of fine classes, the average sample size per class is relatively small when applying CPCC with a standard batch size of 128. Consequently, there is minimal difference between using the centroid of a few data points, as in  $\ell_2$ -CPCC, and using pairwise distances for each data point in OT-CPCC. Therefore, larger batch sizes allow OT-CPCC to better capture class distributions, potentially leading to a more significant advantage in the quality of the learned hierarchy. We compared TestCPCC across various settings, including batch sizes of [64, 128, 512, 1024],  $\lambda$  values of [0.01, 0.05, 0.1, 0.5, 1, 2], and backbones such as ResNet18, ResNet34, ResNet50 (He et al., 2016), ViT-B/16, and ViT-L/16 (Dosovitskiy, 2020), using  $\ell_2$ , EMD, or FastFT CPCC. Across all comparisons, we observe that FastFT > EMD >  $\ell_2$  in terms of the quality of the learned hierarchy, highlighting the advantage of optimal transport-based methods.
+
+# 5 RELATED WORK
+
+Learning Hierarchical Representations We discuss methods that operate on feature representations, thereby directly influencing the geometry of the representation space when hierarchical information is provided. DeVise (Frome et al., 2013) is an early multimodal work to align image and text embedding evaluated on a hierarchical precision metric to reflect semantic knowledge in the representation. Ge et al. (2018); Zhang et al. (2016) modify triplet loss to capture class similarity for metric learning or image retrieval. Barz & Denzler (2019); Garg et al. (2022) embed classes on hypersphere based on hierarchical similarity, and similar ideas have been explored in the hyperbolic space (Kim et al., 2023) where tree can be embedded with low distortion. Yang et al. (2022) provides a multi-task proxy loss for deep metric learning. Nolasco & Stowell (2022) proposes a ranking loss to make Euclidean distance of feature close to a target  $d$ -dim sequence determined by the rank of tree metric. We work on the extension of Zeng et al. (2022) which is a special type of hierarchical methods where a regularizer is used. For a broader discussion on various hierarchical methods, we direct readers to the summary provided by Bertinetto et al. (2020).
+
+OT Methods and Hierarchical Learning The application of Optimal Transport (OT) in hierarchical learning contexts is explored in a limited number of studies. Ge et al. (2021), for instance, substitutes the conventional cross-entropy with a novel loss function that optimally transports the predictive distribution to the ground-truth one-hot vector, modifying the ground metric as tree metric and thus utilizing the hierarchical structure of the classes. Additionally, SEAL (Tan et al., 2023) employs the TWD to construct a latent hierarchy, enhancing traditional supervised and semi-supervised learning methodologies. It can be demonstrated that, given a tree  $\mathcal{T}$ , the optimization process for TWD aligns with  $\ell_1$  loss optimization and is upper bounded by the SumLoss baseline introduced by Zeng et al. (2022). Consequently, these methodologies can be categorized as hierarchical loss functions rather than regularization methods discussed in this work.
+
+# 6 CONCLUSION
+
+Building upon Zeng et al. (2022) using CPCC as a regularizer to learn structured representations, we have identified and targeted the limitations inherent in  $\ell_2$ -CPCC. Our proposed OT-CPCC family provides a more nuanced measurement of pairwise class distances in feature space, effectively capturing complex distributions theoretically and empirically. Furthermore, our novel linear FastFT algorithm itself is interesting in nature for OT communities. FastFT-CPCC can be viewed as a successful application in learning tasks, and we welcome future work to investigate more on the properties of our greedy approach.
+
+# ACKNOWLEDGMENT
+
+SZ and HZ are partially supported by an NSF IIS grant No. 2416897. HZ would like to thank the support from a Google Research Scholar Award. MY is partly supported by MEXT KAKENHI Grant Number 24K03004 and by JST ASPIRE JPMJAP2302. The views and conclusions expressed in this paper are solely those of the authors and do not necessarily reflect the official policies or positions of the supporting companies and government agencies.
+
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+
+# APPENDIX
+
+# A APPROXIMATION METHODS OF EARTH MOVER'S DISTANCE
+
+Most approximation methods do not rely on any tree structure. Sinkhorn (Cuturi, 2013) is a fully differentiable approximation method which leverages the entropic regularization setup of EMD and the convergence of doubly stochastic matrix. Sliced Wasserstein Distance (SWD) (Bonneel et al., 2015) projects data to 1 dimension through a random projection matrix and uses  $\mathrm{EMD}_{\mathrm{ld}}$  for fast calculation, but the quality of approximation highly relies on the number of random projections. Tree Wasserstein Distance (TWD) (Yamada et al., 2022; Le et al., 2019; Takezawa et al., 2021) is a tree based approximation algorithm that can be written in the closed matrix form  $\| \mathrm{diag}(\mathbf{w})\mathbf{B}(a - b)\| _1$ , where  $\mathbf{w}$  contains weights of  $\mathcal{T}$  and  $\pmb{B}$  is a 0-1 internal nodes by leaf nodes matrix that represents if a node is the ancestor of a leaf node. To make sure TWD is a good approximation, an extra step of lasso regression is performed to learn w such that the tree metric approximates Euclidean distance between features with iterative gradient based method. In Yamada et al. (2022), it requires us to learn the tree distance between all pairs of nodes, which leads to quadratic computational complexity with respect to sample size as shown in Tab.1.
+
+# B COMPARISON OF STRUCTURED REPRESENTATION LEARNING AND LEARNING LABEL REPRESENTATIONS
+
+Generally speaking, the problem of structured representation learning and the problem of learning label representations, or learning label embeddings, are two different topics. In label representation learning, each label is mapped to a high-dimensional vector for downstream tasks. For example, in standard multi-class classification, we typically use one-hot vector embeddings. Alternative approaches include using a soft label to assign probability values in (0,1) to each class (Nguyen et al., 2014), or learning label embeddings (Weston et al., 2011; Sun et al., 2017; Liu et al., 2024). Label representation learning methods have various downstream applications, such as text summarization (Sun et al., 2017), novelty detection (Barz & Denzler, 2019), and extreme classification (Jalan & Kar, 2019; Tagami, 2017; Bhatia et al., 2015; Yu et al., 2014; Evron et al., 2018; Gupta et al., 2019).
+
+While  $\ell_2$ -CPCC falls into this category, OT-CPCC methods are not label embedding methods. Instead of representing a class with a Euclidean centroid in  $\ell_2$ , OT-CPCC methods intentionally avoid this step by considering a more nuanced relationship between labels. OT-CPCC relies on pairwise relationships between data points from two classes to include more complex distributional geometry during learning. Since these methods depend on data point relations, there is no one-to-one correspondence between a label and a representation.
+
+# C EMD-CPCC GENERALIZES  $\ell_2$ -CPCC
+
+We can show EMD-CPCC is a generalization of previous work of  $\ell_2$ -CPCC through the following proposition.
+
+Proposition 3.1 (EMD reduces to  $\ell_2$  between means of Gaussian with same covariance).  $\mathrm{EMD}(\mathcal{N}(\mu_z,\Sigma),\mathcal{N}(\mu_{z'},\Sigma)) = \| \mu_z - \mu_{z'}\|$
+
+Proof. Since  $\mu, \nu$  in EMD definition are probability measures, the definition of EMD can be generalized to continuous distributions as well. Denote  $Z$  the random variable from  $\mu$ ,  $Z'$  from  $\nu$ , and  $q$  the coupling between  $\mu$  and  $\nu$ , then EMD can be also expressed in the following  $p$ -Wasserstein  $(\mathcal{W}_p)$  form when  $p = 1$ :
+
+$$
+\begin{array}{l} \mathcal {W} _ {p} = \left(\inf  _ {q} \int_ {\mathcal {Z} \times \mathcal {Z}} \| z _ {i} - z _ {j} \| ^ {p} q (z _ {i}, z _ {j}) d z _ {i} z _ {j}\right) ^ {\frac {1}{p}} \\ = \inf  _ {q} \left(\mathbb {E} \| Z - Z ^ {\prime} \| ^ {p}\right) ^ {\frac {1}{p}} \tag {4} \\ \end{array}
+$$
+
+With this definition, our statement can be proved in two steps. First, we want to show the lower bound of EMD is  $\| \mu_z - \mu_{z'}\|$  for any pairs of input random variables. By one step application of Jensen's
+
+inequality, since every norm is convex, we have the following result from the expectation form of  $\mathcal{W}_1$  in Eq.4:
+
+$$
+\begin{array}{l} \mathcal {W} _ {1} = \inf  \mathbb {E} \| Z - Z ^ {\prime} \| \geq \inf  \| \mathbb {E} (Z - Z ^ {\prime}) \| \\ = \| \mathbb {E} Z - \mathbb {E} Z ^ {\prime} \| = \| \mu_ {z} - \mu_ {z ^ {\prime}} \| \tag {5} \\ \end{array}
+$$
+
+Second, we need to show EMD  $\leq \| \mu_z - \mu_{z'}\|$  for the given Gaussian constraints. Following the convexity of quadratic function, apply Jensen's again, we have:
+
+$$
+\mathcal {W} _ {1} ^ {2} = \left(\inf  \mathbb {E} \| Z - Z ^ {\prime} \|\right) ^ {2} \leq \inf  \mathbb {E} \left(\| Z - Z ^ {\prime} \|\right) ^ {2} = \mathcal {W} _ {2} ^ {2} \tag {6}
+$$
+
+Dowson & Landau (1982) showed the following analytical form of
+
+$$
+\mathcal {W} _ {2} ^ {2} = \left\| \mu_ {z} - \mu_ {z ^ {\prime}} \right\| ^ {2} + \operatorname {t r} \left(\Sigma_ {z} + \Sigma_ {z ^ {\prime}} - 2 \left(\Sigma_ {z} \Sigma_ {z ^ {\prime}}\right) ^ {1 / 2}\right) \tag {7}
+$$
+
+This distance metric is also sometimes called Bures-Wasserstein (Bures) distance. The trace term becomes 0 if  $\Sigma_z = \Sigma_{z'} = \Sigma$ .
+
+# D DETAILS FOR FAST FLOW TREE CPCC ALGORITHM
+
+In this section, we first present the full pseudocode to compute FastFT-CPCC in Alg.3. Compared to FlowTree (Backurs et al., 2020) which uses the helper function in Alg.2 (Kalantari & Kalantari, 1995), we replace the greedy_flow_MATCHing algorithm with Alg.1 which aligns with algorithm that computes the 1 dimensional OT transportation plan.
+
+Algorithm 3 Train with Fast FlowTree CPCC  
+input epoch  $E$  , batch size  $B$  , train data  $\mathcal{D} = \{(x_i,y_i)\}_{i = 1}^N$  dictionary  $S$    
+1: for iteration  $= 0,1,\ldots ,E - 1$  do   
+2: for batch  $= 0,1,\dots ,\lfloor N / B\rfloor$  do   
+3: assume there are  $k$  classes in batch   
+4: for  $(a,b)$  in  $\binom{k}{2}$  pairs of classes do   
+5:  $z_{a} = (a_{1},\dots,a_{m})$    
+6:  $z_{b} = (b_{1},\dots,b_{n})$    
+7: if  $(m,n)$  in  $S$  or  $(n,m)$  in  $S$  then   
+8:  $P\gets S[(m,n)]$    
+9: else   
+10:  $P\gets$  greedy_flow_MATCHing  $(m,n,a=$  unif,b=unif) {Alg.1}   
+11:  $S[(m,n)]\gets P$    
+12: idx  $\leftarrow$  P.nonzero () {store nonzero index tuples of  $P$  }   
+13:  $\rho (a,b) = 0$    
+14: for  $(i,j)$  in idx do   
+15:  $\rho (a,b) + = P[i,j]\cdot \mathrm{norm}(z_a[i],z_b[j])$    
+16: Calculate Eq.1 with CPCC as  $\mathcal{R}$    
+17: Update network parameters with backpropagation
+
+Now we prove the correctness of Fast Flow Tree algorithm and its relation to 1 dimensional optimal transport distance.
+
+Theorem 3.2 (Correctness of Fast FlowTree). For any augmented label tree  $\mathcal{T}$ , Alg. 1 and Alg. 2 return the same EMD approximation. The Fast FlowTree can be computed in  $O((m + n)d)$ .
+
+Proof Sketch We can observe that the blue code blocks, which contain the update rule of the flow matrix, are almost identical between Alg.1 and Alg.2. According to Kalantari & Kalantari (1995), Alg.2 provides the optimal flow matrix for the optimal transport problem using the tree metric as the ground metric in  $D$ . From Fig.1, the tree distance between any two samples in the augmented tree is always the same, meaning the order of encountering any vertex within the source or target distribution does not matter. This effectively reduces the FlowTree problem in Alg.2 to a flat 1d EMD problem in Alg.1. The major difference between two algorithms is that, unlike Alg.2, Alg.1 does not require passing the flow from the leaf to the root node.
+
+Proof. Alg.2 will not start to match leaves until the current node has children from two different distributions, or in our context, from two different fine labels. This is because of the while statement in line 8: based on our construction of  $\mathcal{T}$  in Fig.2, if the current node  $v$  is either a leaf or a fine label node, then all of its children are from either  $C_a(v)$  or  $C_b(v)$ , so we will jump to line 11 to send all flows to  $v$ 's parent. Therefore, the matching starts at the lowest common ancestor of two label nodes.
+
+Now we have  $v$  as the lowest common ancestor of two fine class nodes (ex.,  $F$  or  $H$  in Fig.2). Before entering line 9, we always pass unmatched demands to the parent, and we can see the demands are always from one label until we reach the lowest common ancestor. From Fig.2, we also know at this point, we have collected all flows from both labels. Therefore, the sum of flow is 1 for both fine label nodes when the matching starts.
+
+Both Alg.2 and Alg.1 greedily remove one sample from either distribution. The greedy plan is always feasible and no unmatched flow is left after one scan of all  $m + n$  features, because we inherently remove the same amount of flow from both distributions every iteration in the while loop. Thus, on our  $\mathcal{T}$ , we can skip most steps in Alg.2: bottom up step in line 2, looping through all vertices at height  $h$ , the if-else check from line 4-7, and the last step of passing unmatched demands to parents are all unnecessary. Alg.2 terminates immediately once we run through all while loop steps, thus reducing to Alg.1.
+
+Alg.1, or the flow matrix solution for 1-d EMD problem, can be used when Monge Property or northwest-corner rules (Hoffman, 1963) hold. It can be shown by induction that this  $O(m + n)$  greedy algorithm can be applied whenever the following constraint is satisfied: if  $\{(z_i, a_i)\}$ ,  $\{(z_j, b_j)\}$  are indexed (1d feature, weight) sequences, for indices  $i_1 < i_2$  and  $j_1 < j_2$ :
+
+$$
+D _ {i _ {1}, j _ {1}} + D _ {i _ {2}, j _ {2}} \leq D _ {i _ {1}, j _ {2}} + D _ {i _ {2}, j _ {1}} \tag {8}
+$$
+
+If two sequences are ordered (in ascending order) by metric defining  $D$ , we have  $z_{i_1} \leq z_{i_2}$  and  $z_{j_1} \leq z_{j_2}$ . There are in total six cases for any  $i_1 < i_2, j_1 < j_2$ :
+
+1. If  $z_{j_1} \leq z_{j_2} \leq z_{i_1} \leq z_{i_2}$ , then Eq.8  $\Leftrightarrow z_{i_1} - z_{j_1} + z_{i_2} - z_{j_2} \leq z_{i_1} - z_{j_2} + z_{i_2} - z_{j_1}$  holds.  
+2. If  $z_{j_1} \leq \dot{z}_{i_1} \leq z_{j_2} \leq z_{i_2}$ , then Eq.8  $\Leftrightarrow \dot{z}_{i_1} - \dot{z}_{j_1} + \dot{z}_{i_2} - z_{j_2} \leq z_{j_2} - z_{i_1} + z_{i_2} - z_{j_1}$  holds because  $z_{i_1} \leq z_{j_2}$ .  
+3. If  $z_{j_1} \leq z_{i_1} \leq z_{i_2} \leq z_{j_2}$ , then Eq.8  $\Leftrightarrow$ $z_{i_1} - z_{j_1} + z_{j_2} - z_{i_2} \leq z_{j_2} - z_{i_1} + z_{i_2} - z_{j_1}$  holds because  $z_{i_1} \leq z_{i_2}$ .  
+4. If  $z_{i_1} \leq z_{j_1} \leq z_{j_2} \leq z_{i_2}$ , then Eq.8  $\Leftrightarrow$ $z_{j_1} - z_{i_1} + z_{j_2} - z_{i_2} \leq z_{j_2} - z_{i_1} + z_{i_2} - z_{j_1}$  holds because  $z_{j_1} \leq z_{i_2}$ .  
+5. If  $z_{i_1} \leq z_{j_1} \leq z_{i_2} \leq z_{j_2}$ , then Eq.8  $\Leftrightarrow$ $z_{j_1} - z_{i_1} + z_{j_2} - z_{i_2} \leq z_{j_2} - z_{i_1} + z_{i_2} - z_{j_1}$  holds because  $z_{j_1} \leq z_{i_2}$ .  
+6. If  $z_{i_1} \leq z_{i_2} \leq z_{j_1} \leq z_{j_2}$ , then Eq.8  $\Leftrightarrow$ $z_{j_1} - z_{i_1} + z_{j_2} - z_{i_2} \leq z_{j_2} - z_{i_1} + z_{j_1} - z_{i_2}$  holds.
+
+Therefore, if two sequences are ordered with respect to some metric, Eq.8 is satisfied.
+
+Recall we need the extra sorting step in the beginning of 1d OT plan to apply the greedy flow matching algorithm. The distance function in Eq.8 is decided by the ground metric of EMD, which does not make any difference for 1 dimensional features. In Flow Tree, a tree is constructed with QuadTree such that the tree distance of any two features is close to the  $\ell_2$  distance between them. In this case, in EMD computation, we can improve computational efficiency by Flow Tree. With extended label tree  $\mathcal{T}$ , because all same class sample leaves belong to one single parent, we have a special case here where the tree distance of any two samples from  $\mu, \nu$  are always the same, satisfying Eq.8 vacuously. This statement fails only if we assign different edge weights for each sample during the extension of label tree, and does not depend on the weights of original label tree. To avoid the limitation on the extended edge weights, we can instead assign different weights  $a_i$  for each sample to adjust the importance of each dat points. Therefore, FastFT is still very flexible.
+
+For the time complexity, Alg.1 requires at most  $O(m + n)$  steps from line 4. The most time-consuming aspect is computing the  $d$ -dimensional distance of  $O(m + n)$  pairs in  $D$ , since there will be at most  $m + n$  nonzero values in  $D$  after applying the greedy matching algorithm, which makes the final time complexity as  $O((m + n)d)$ .
+
+![](images/be9f39fa5023891cb62ab497cbd5ea2d2df74335611d041d44e237e40ad01d28.jpg)
+
+![](images/3345ba1a1785c7e191bb03b8bb8cba37c3c7450e0ed9d6f3b2e8d1de449b252e.jpg)  
+First skip leaf nodes.  
+Start from vertices on depth - 1.
+
+![](images/c0a4b1b4a4789e4149be32235beab691349ac16e0616c9dc25a749507773ecdc.jpg)  
+(a) Step 1
+
+![](images/25d7a6d3ae9005c44163ccbc5b0a35fe3bc2cce336a3fb143010baf7319170e2.jpg)  
+Greedy match pairs of demand and supply. Find minimum flow, then update. We only have one possible pair:  $\frac{1}{5} < \frac{1}{2}$ .  
+(b) Step 2
+
+![](images/1813a3fefc02fc64ae61a06dd9bc33973dc2882dd53a3265a6d109408538edf1.jpg)  
+(c) Step 3
+
+![](images/c443ba35800c7ef769af0344c59a5168e46a74f7d41d6d93343b8be080833542.jpg)  
+Pass the non-zero remaining flow to the parent of leaf nodes  $\mathbf{v}$  
+(d) Step 4
+
+![](images/ac01c93c6ce1a466c11bf75d8afc5841276505295dee642870e58bea14842e99.jpg)  
+(e) Step 5  
+(g) Step 7
+
+![](images/3682437f32558d17dc7bcffc5b46ae690072107decd3ea109ebda4c28895640f.jpg)  
+Repeat the same steps for  $\mathbf{v}'$  
+(f) Step 6
+
+![](images/a51beb19135a7f275dec88e662910b800a13cf7eeaaf3cdd5f84783d4b05ad63.jpg)  
+Repeat the same steps for  $\mathbf{v}'$ .
+
+![](images/9d64aeafcde02157aadf3d9279287d8959fdd27486a7375d273bc3aefa3c3c09.jpg)  
+(h) Step 8
+
+![](images/6669a05770bec5b156e8e36da6bf04927717d893a48768bfe8463231a3eda0f9.jpg)  
+(i) Step 9  
+(k) Step 11  
+Figure 8: Example run of Alg.2 Bottom-up Tree Flow Matching.
+
+![](images/e6e22d94fae7f1c1c069e4b6a17708f2fd4510924eaf93aea8099addecf3b147.jpg)  
+Go to one level above. Repeat all steps.  
+We get the full flow matrix.  $\mu$  
+(1) Step 12
+
+$$
+\begin{array}{c c} v & \left[ \begin{array}{c c} 1 / 5 & 0 \\ 0 & 1 / 5 \\ 0 & 1 / 5 \\ 1 / 1 0 & 1 / 1 0 \\ 1 / 5 & 0 \end{array} \right] \end{array}
+$$
+
+# E DATASET HIERARCHY CONSTRUCTION
+
+In this section, we discuss the details of dataset hierarchies and how we created the dataset splits. Throughout the main body of this paper, we consider a two-level hierarchy: coarse and fine. For each coarse node, except for BREEDS, we split the coarse node's fine children into approximately  $40\%$  for the target set and  $60\%$  for the source set. Following this procedure, the splits are detailed in Tables 6, 7, 8, 9, 10, 11, and 12. The original class labels for CIFAR10 and CIFAR100 are from https://www.cs.toronto.edu/~kriz/cifar.html, for iNaturalist from https://github.com/visipedia/inat_comp/tree/master/2021, and for BREEDS from WordNet (Miller, 1995).
+
+The CIFAR10 hierarchy was manually constructed by us, while the CIFAR100 hierarchy is provided in the original dataset. In INAT, fine and coarse classes correspond to the class and phylum labels from the original annotations. These higher-level labels are particularly suitable for evaluating multimodal settings, where OT-CPCC methods may have a more significant advantage over  $\ell_2$ -CPCC.
+
+In Zeng et al. (2022), fine classes in BREEDS are defined one level deeper than the original labels. However, due to the large number of classes and limited samples per class within a batch, the distinction between  $\ell_2$ -CPCC and OT-CPCC can become less apparent. To address this, we employ a three-level hierarchy, using the labels from Santurkar et al. (2021) as fine labels and setting their parent categories as mid and coarse labels. Consequently, due to BREEDS setting, the fine classes are the same for both source and target datasets, so there is no source-target split in Tables 9, 10, 11, and 12. Despite this hierarchical structure, as seen in Fig.5, BREEDS exhibits relatively low multi-modality. With this design, in Sec.4.3, for classification and retrieval experiments, one-shot fine-tuning is not required for BREEDS on the target split.
+
+Table 6: CIFAR10 hierarchy and data split.  
+
+<table><tr><td>Coarse</td><td>Source</td><td>Target</td></tr><tr><td>animal</td><td>deer, dog, frog</td><td>bird, cat</td></tr><tr><td>transportation</td><td>ship, truck</td><td>airplane, automobile</td></tr></table>
+
+Table 7: CIFAR100 hierarchy and data split.  
+
+<table><tr><td>Coarse</td><td>Source</td><td>Target</td></tr><tr><td>aquatic mammals</td><td>otter, seal, whale</td><td>beaver, dolphin</td></tr><tr><td>fish</td><td>ray, shark, trout</td><td>aquarium fish, flat fish</td></tr><tr><td>flowers</td><td>roses, sunflowers, tulips</td><td>orchids, poppies</td></tr><tr><td>food containers</td><td>cans, cups, plates</td><td>bottles, bowls</td></tr><tr><td>fruit and vegetables</td><td>oranges, pears, sweet pepper</td><td>apples, mushrooms</td></tr><tr><td>household electrical devices</td><td>lamp, telephone, television</td><td>clock, computer keyboard</td></tr><tr><td>household furniture</td><td>couch, table, wardrobe</td><td>bed, chair</td></tr><tr><td>insects</td><td>butterfly, caterpillar, cockroach</td><td>bee, beetle</td></tr><tr><td>large carnivores</td><td>lion, tiger, wolf</td><td>bear, leopard</td></tr><tr><td>large man-made outdoor things</td><td>house, road, skyscraper</td><td>bridge, castle</td></tr><tr><td>large natural outdoor scenes</td><td>mountain, plain, sea</td><td>cloud, forest</td></tr><tr><td>large omnivores and herbivores</td><td>chimpanzee, elephant, kangaroo</td><td>camel, cattle</td></tr><tr><td>medium-sized mammals</td><td>possum, raccoon, skunk</td><td>fox, porcupine</td></tr><tr><td>non-insect invertebrates</td><td>snail, spider, worm</td><td>crab, lobster</td></tr><tr><td>people</td><td>girl, man, woman</td><td>baby, boy</td></tr><tr><td>reptiles</td><td>lizard, snake, turtle</td><td>crocodile, dinosaur</td></tr><tr><td>small mammals</td><td>rabbit, shrew, squirrel</td><td>hamster, mouse</td></tr><tr><td>trees</td><td>palm, pine, willow</td><td>maple, oak</td></tr><tr><td>vehicles 1</td><td>motorcycle, pickup truck, train</td><td>bicycle, bus</td></tr><tr><td>vehicles 2</td><td>streetcar, tank, tractor</td><td>lawn-mower, rocket</td></tr></table>
+
+Table 8: INAT hierarchy and data split.  
+
+<table><tr><td>Coarse</td><td>Source</td><td>Target</td></tr><tr><td>Arthropoda</td><td>Diplopoda, Hexanauplia, Insecta, Malacostraca, Merostomata</td><td>Arachnida, Chilopoda</td></tr><tr><td>Chordata</td><td>Ascidiacea, Aves, Elasmobranchii, Mammalia, Reptilia</td><td>Actinopterygii, Amphibia</td></tr><tr><td>Cnidaria</td><td>Hydrozoa, Scyphozoa</td><td>Anthozoa</td></tr><tr><td>Echinodermata</td><td>Echinoidea, Holothuroidea, Ophiuroidea</td><td>Asteroidea</td></tr><tr><td>Mollusca</td><td>Cephalopoda, Gastropoda, Polyplacophora</td><td>Bivalvia</td></tr><tr><td>Ascomycota</td><td>Lecanoromycetes, Leotiomycetes, Pezizomycetes, Sordariomycetes</td><td>Athoniomycetes, Dothideomycetes</td></tr><tr><td>Basidiomycota</td><td>Dacrymycetes, Pucciniomycetes, Tremellomycetes</td><td>Agaricomycetes</td></tr><tr><td>Bryophyta</td><td>Polytrichopsida, Sphagnopsida</td><td>Bryopsida</td></tr><tr><td>Tracheophyta</td><td>Liliopsida, Lycopodiopsida, Magno-liopsida, Pinopsida, Polypodiopsida</td><td>Cycadopsida, Gnetopsida</td></tr></table>
+
+Table 9: LIVING17 dataset hierarchy.  
+
+<table><tr><td>Coarse</td><td>Fine</td></tr><tr><td>bird</td><td>grouse, parrot</td></tr><tr><td>amphibian</td><td>salamander</td></tr><tr><td>reptile, reptilian</td><td>turtle, lizard, snake, serpent, ophidian</td></tr><tr><td>arthropod</td><td>spider, crab, beetle, butterfly</td></tr><tr><td>mammal, mammalian</td><td>dog, domestic dog, Canis familiaris, wolf, fox, domestic cat, house cat, Felis domesticus, Felis catus, bear, ape, monkey</td></tr></table>
+
+Table 10: NONLIVING26 dataset hierarchy.  
+
+<table><tr><td>Coarse</td><td>Fine</td></tr><tr><td>structure, place</td><td>dwelling, home, domicile, abode, habitation, dwelling house, fence, fencing, mercantile establishment, retail store, sales outlet, outlet, out-building, roof</td></tr><tr><td>paraphernalia</td><td>ball, bottle, digital computer, keyboard instrument, percussion instru-ment, percussive instrument, pot, stringed instrument, timepiece, time-keeper, horologe, wind instrument, wind</td></tr><tr><td>food, nutrient</td><td>squash</td></tr><tr><td>apparel, toiletries</td><td>bag, body armor, body armour, suit of armor, suit of armour, coat of mail, cataphract, coat, hat, chapeau, lid, skirt</td></tr><tr><td>conveyance, transport</td><td>boat, bus,abus,coach, charabanc, double-decker, jitney,motorbus, motorcoach, omnibus, passenger vehicle, car, auto, automobile, machine, motorcar, ship, truck, motortruck</td></tr><tr><td>furnishing</td><td>chair</td></tr></table>
+
+# F TRAINING HYPERPARAMETERS
+
+We conducted all experiments with NVIDIA RTX A6000. The architectural and training setups for each dataset are as follows:
+
+Table 11: ENTITY13 dataset hierarchy.  
+
+<table><tr><td>Coarse</td><td>Fine</td></tr><tr><td>living thing, animate thing non-living thing</td><td>bird, reptile, reptilian, arthropod, mammal, mammalian garment, accessory, accoutrement, accouterment, craft, equipment, fur-niture, piece of furniture, article of furniture, instrument, man-made structure, construction, wheeled vehicle, produce, green goods, green groceries, garden truck</td></tr></table>
+
+Table 12: ENTITY30 dataset hierarchy.  
+
+<table><tr><td>Coarse</td><td>Fine</td></tr><tr><td>living thing, animate thing</td><td>serpentes, passerine, passeriform bird, saurian, arachnid, arachnoid, aquatic bird, crustacean, carnivore, insect, ungulate, hoofed mammal, primate, bony fish</td></tr><tr><td>non-living thing</td><td>barrier, building, edifice, electronic equipment, footwear, legwear, garment, headdress, headgear, home appliance, household appliance, kitchen utensil, measuring instrument, measuring system, measuring device, motor vehicle, automotive vehicle, musical instrument, instrument, neckwear, sports equipment, tableware, tool, vessel, watercraft, dish, vegetable, veggie, veg, fruit</td></tr></table>
+
+# F.1 CIFAR10, CIFAR100
+
+We conduct training from scratch using ResNet18, following the ResNet architecture design where the first convolution layer employs a 3x3 kernel size, while the max pooling layer is omitted. The images are preprocessed by normalizing them according to CIFAR10/CIFAR100's mean and standard deviation. We augment the training data by first zero-padding the images by 4 pixels, then cropping them back to their original dimensions, applying a  $50\%$  chance of horizontal flipping, and randomly rotating the images between -15 and 15 degrees.
+
+For CIFAR10, the pretraining process spans 100 epochs with a batch size of 64. The learning rate begins at 0.01 and is decreased by a factor of 10 every 60 steps, with momentum set at 0.9 and weight decay set to  $5 \times 10^{-4}$  in the SGD optimizer. For CIFAR100, the pretraining process spans 200 epochs with a batch size of 128. The learning rate begins at 0.1 and is decreased by a factor of 5 every 60 steps, with momentum set at 0.9 and weight decay set to  $5 \times 10^{-4}$  in the SGD optimizer. Additionally, for both datasets, in the fine-level one-shot transfer learning, we train for 100 epochs starting at a learning rate of 0.1, which is reduced by a factor of 10 every 60 steps. CPCC is applied to coarse, fine level and  $\lambda$  in Eq.1 is set to 1. The ratio between the distance between two fine classes sharing the same coarse parent and those sharing different parents is 1:2.
+
+# F.2 BREEDS
+
+We train a ResNet18 model from scratch originally designed for ImageNet, where the first convolution layer uses a 7x7 kernel. Adhering to the BREEDS training setup, we use a batch size of 128, an initial learning rate of 0.1, and SGD optimization with a weight decay of  $10^{-4}$ . For the  $ENTITY13$  and  $ENTITY30$  datasets, the model is trained for 300 epochs, reducing the learning rate by a factor of 10 every 100 steps, while for  $LIVING17$  and  $NONLIVING26$ , we extend training to 450 epochs, with the learning rate divided by 10 every 150 steps. Data augmentation for the training set includes a RandomResizedCrop to  $224 \times 224$ , horizontal flipping with a  $50\%$  probability, and color jittering that adjusts brightness, contrast, and saturation within a range of 0.9 to 1.1. The test set, on the other hand, is resized to  $256 \times 256$  and then center-cropped to  $224 \times 224$ .
+
+Since in our experiments with BREEDS, the source and target split share the same set of fine labels, to calculate fine target accuracy, it is unnecessary to use one-shot transfer learning. We directly use the source set pretrained models for target fine accuracy evaluation.
+
+To apply CPCC methods, we backtrack two levels up to find the mid and coarse labels from the original class labels in BREEDS. CPCC is applied to mid and coarse level and  $\lambda$  in Eq.1 is set to 1. The ratio between the distance between two fine classes sharing the same coarse parent and those sharing different parents is 1:2.
+
+# F.3 INAT
+
+We fine-tune an ImageNet-pretrained ResNet18 model where checkpoint can be downloaded in Pytorch (torchvision.models.resnet18(weights='IMAGENET1K_V1'), https://torch.org/vision/main/models/generated/torchvision.models.resnet18.html). The images are preprocessed by normalizing them with mean=[0.466, 0.471, 0.380] and standard deviation=[0.195, 0.194, 0.192]. We augment the training data by cropping a random portion (scale=(0.08, 1.0), ratio=(0.75, 1.33)) of image and resize it to 224x224, and apply a  $50\%$  chance of horizontal flipping. We use a batch size of 1024. The learning rate begins at 0.1 and is decreased by a factor of 5 every 60 steps, with momentum set at 0.9 and weight decay set to  $5 \times 10^{-4}$  in the SGD optimizer. In the downstream one-shot fine-tuning task, we train for 60 epochs starting at a learning rate of 0.05, which is reduced by a factor of 10 every 15 steps. CPCC is applied to coarse, fine level and  $\lambda$  in Eq.1 is set to 0.1. The ratio between the distance between two fine classes sharing the same coarse parent and those sharing different parents is 1:3.
+
+![](images/9d17da20eef8ecff3db29a90319642002ed3e3097653621835b36e1f111847e6.jpg)  
+Figure 9: Visualization of the evaluation of hierarchical classification.
+
+# G HIERARCHICAL BASELINE OBJECTIVES
+
+In this section, we introduce the details of hierarchical objectives we used in Fig.12.
+
+Flat is the naive cross entropy loss only on the fine level, so the expression is:
+
+$$
+\mathcal {L} _ {\text {F l a t}} (x, y _ {\text {f i n e}}) = \sum_ {(x, y) \in \mathcal {D}} \ell_ {\text {F l a t}} \left(y _ {\text {f i n e}}, h (x)\right). \tag {9}
+$$
+
+We adopt all hierarchical baselines from the comparative study in Zeng et al. (2022), including:
+
+Multi-Task Learning (MTL) (Caruana, 1997), where the backbone model is trained with two linear heads—one for coarse classes and one for fine classes—each with a cross-entropy loss. The total loss,  $\mathcal{L}_{\mathrm{MTL}}$ , is the sum of these two cross-entropy terms, with a scaling coefficient of 1 for both. Curriculum Learning (Curr) (Bengio et al., 2009) involves two training phases: first, training a coarse head for 50 epochs, followed by training the full network with a fine head for 200 epochs, starting from the coarse-trained features, assuming that coarse classification is an easier task. Sum Loss (Hu et al., 2018) uses only one fine classifier head, where coarse predictions are obtained by summing the fine class probabilities for the same coarse parent and applying a second cross-entropy loss. Hierarchical Cross Entropy (HXE) (Bertinetto et al., 2020) is the sum of weighted cross-entropy losses ( $\alpha = 0.4$ ), where the hierarchical structure is incorporated into the log-probabilities. The same paper also introduces Soft Label (Soft), which uses a single softmax loss based on the height of the least common ancestor between label pairs, with a hyperparameter  $\beta = 10$ . Quadruplet
+
+Loss (Quad) (Zhang et al., 2016) extends triplet loss by defining two types of positive labels based on the hierarchy. For detailed mathematical expressions of these objectives, refer to App.C in Zeng et al. (2022).
+
+We also include several recently published hierarchical losses:
+
+Soft Hierarchical Consistency (SHC) (Garg et al., 2022) is the combination of MTL and Sum Loss. For a two level hierarchy, they jointly train a two-headed network. Define the network as  $h: \mathcal{X} \to \Delta_{k_1} \times \Delta_{k_2}$  with a shared feature encoder  $f: \mathcal{X} \to \mathcal{Z}$  and two classifiers  $g_{\mathrm{coarse}}: \mathcal{Z} \to \Delta_{k_1}, g_{\mathrm{fine}}: \mathcal{Z} \to \Delta_{k_2}$ , with  $k_1$  as the number of coarse classes and  $k_2$  the number of fine classes. The ground truth coarse probability  $p_{\mathrm{coarse}} = g_{\mathrm{coarse}}(f(\cdot))$  is the output of coarse head after softmax normalization, and  $\hat{p}_{\mathrm{coarse}}$  is derived from the output of fine head. Let  $\mathbf{W}$  be a  $k_1$  by  $k_2$  matrix representing the relationships in the label tree: if a fine class  $i$  belongs to a coarse class  $j$ , then  $\mathbf{W}_{ji}$  is 1, else the entry is set to 0. Then  $\hat{p}_{\mathrm{coarse}} = \mathbf{W}g_{\mathrm{fine}}(f(\cdot))$ , so the probability of belonging to a given coarse class is the sum of the probabilities of its fine class descendants in the tree. Then, the SHC loss is defined as the Jensen-Shannon Divergence between these two probability distributions:
+
+$$
+\mathcal {L} _ {\mathrm {S H C}} \left(x, y _ {\text {c o a r s e}}, y _ {\text {f i n e}}\right) = \mathrm {J S D} \left(p _ {\text {c o a r s e}} \left(x, y _ {\text {c o a r s e}}\right) \| \hat {p} _ {\text {c o a r s e}} \left(x, y _ {\text {f i n e}}\right)\right). \tag {10}
+$$
+
+In Garg et al. (2022),  $\mathcal{L}_{\mathrm{SHC}}$  is used in the combination with Flat cross entropy loss. Therefore, we treat it as a counterpart of CPCC losses and use  $\mathcal{L}_{\mathrm{Flat}} + \lambda \cdot \mathcal{L}_{\mathrm{SHC}}$  as the training loss, with  $\lambda = 1$ .
+
+Geometric Consistency (GC) (Garg et al., 2022) is also computed under the MTL two-head pipeline. The coarse classifier can be decomposed to a linear layer and softmax function, i.e., let  $\mathbf{W}^c\in \mathbb{R}^{d\times k_1}$ ,  $g_{\mathrm{coarse}}(\cdot) = \mathrm{softmax}(\mathbf{W}^c (\cdot))$ , and we define  $\mathbf{W}^f$  similarly for the linear fine layer.
+
+$$
+\mathcal {L} _ {\mathrm {G C}} \left(x, y _ {\text {c o a r s e}}, y _ {\text {f i n e}}\right) = \sum_ {i = 1} ^ {k _ {1}} 1 - \cos \left(\mathbf {W} _ {i} ^ {c}, \hat {\mathbf {W}} _ {i} ^ {c}\right). \tag {11}
+$$
+
+As in SHC, GC aims to force the similarity between the predicted coarse weight and the true coarse weight. Here,  $\mathbf{W}_i^c$  is part of the linear classifier corresponds to each coarse class, and the predicted coarse weight  $\hat{\mathbf{W}}_i^c = \sum_{j\in \mathrm{children}(i)}\mathbf{W}_j^f /\left\| \sum_{j\in \mathrm{children}(i)}\mathbf{W}_j^f\right\|$  is the normalized sum of a part of fine classifiers based on the hierarchical relationship. We combine  $\mathcal{L}_{\mathrm{GC}}$  with Flat cross entropy during training and set  $\lambda$  in Eq.1 as 1.
+
+RankLoss (Rank) (Nolasco & Stowell, 2022) proposes a ranking loss to make Euclidean distance of feature  $\rho$  close to the target  $d$ -dim sequence determined by the rank of tree metric:
+
+$$
+\mathcal {L} _ {\text {r a n k}} (x, y _ {\text {c o a r s e}}, y _ {\text {f i n e}}) = \frac {1}{p} \sum_ {p} \left(1 - I _ {p}\right) (\rho - \text {T a r g e t R a n k} (\rho)). \tag {12}
+$$
+
+In this equation above,  $I_{p}$  is an indicator variable which shows whether the rank of pairwise  $\ell_2$  feature distance is the same as the rank of pairwise tree metric distance, and TargetRank is a function reflecting hierarchical similarity. Since this loss does not focus on discriminative tasks like classification, we combine it with Flat cross entropy, setting  $\lambda = 1$  in Eq.1 for downstream tasks evaluation.
+
+# H ADDITIONAL EXPERIMENTS
+
+# H.1 EFFECT OF TREE DEPTH
+
+Although in the main body we focus on a depth-3 label hierarchy consisting of a root node, a coarse level, and a fine level, we also test whether our method can be applied to more complex trees. In CIFAR100, beyond the standard fine-coarse levels, we introduce a mid level by grouping 5 fine classes from the same coarse class into two smaller groups  $(2 + 3)$ . Additionally, we create a coarser level by grouping two alphabetically neighboring coarse classes into a single coarser label. For example, for 10 fine labels, the hierarchical grouping is highlighted by the following color scheme: {{(orchids, poppies) (roses, sunflowers, tulips) [ (bottles, bowls) (cans, cups, plates) ]}}.
+
+In Fig.10, we aligned the axes to sort the fine labels based on the hierarchical structure for embeddings learned using trees of varying depth. We then computed the fine-to-fine class distances using either
+
+![](images/9fa3090faaf6ee9bd7698a918055baab9199f09523760c68a64f87e2a52c8494.jpg)  
+Figure 10: CIFAR100 class embedding pairwise distance matrix for different types of label tree with varying tree depth.
+
+the EMD or FastFT metric. On the left of Fig.10, we used the standard fine-coarse CIFAR100 label tree, where a clear pattern of twenty 5x5 coarse blocks appears on the diagonal for both optimal transport methods. In the middle of Fig.10, we applied CPCC to the fine-mid-coarse hierarchy, and within each 5x5 coarse block, we observed the expected fine-grained  $2 \times 2 + 3 \times 3$  patterns. Similarly, on the right of Fig.10, when we included the coarser grouping, we could observe the merging of neighboring 5x5 coarse blocks.
+
+Table 13: EMD and FastFT performance for tree with different depths on the full CIFAR100 dataset. The batch size is set to 128.  
+
+<table><tr><td>Objective</td><td>Setup</td><td>CPCC</td><td>FineAcc</td><td>CoarseAcc</td></tr><tr><td rowspan="3">EMD</td><td>depth = 3</td><td>88.80</td><td>71.91</td><td>82.21</td></tr><tr><td>depth = 4</td><td>85.09</td><td>72.59</td><td>82.93</td></tr><tr><td>depth = 5</td><td>88.64</td><td>72.20</td><td>82.50</td></tr><tr><td rowspan="3">FastFT</td><td>depth = 3</td><td>90.75</td><td>72.71</td><td>83.31</td></tr><tr><td>depth = 4</td><td>87.86</td><td>72.45</td><td>83.05</td></tr><tr><td>depth = 5</td><td>90.27</td><td>72.56</td><td>83.04</td></tr></table>
+
+We also include metrics for the preciseness of hierarchical information and hierarchical classification in Tab.13. As observed in Fig.10, the tree information is well-learned, leading us to expect high CPCC scores across all settings in Tab.13. For the hierarchical classification metrics, within the same OT method, the difference between FineAcc and CoarseAcc is small. It is worth noting that, because we train on the full CIFAR100 dataset for visualization purposes, the source-target hierarchical split shown in Fig.4 is not applied here. Both FineAcc and CoarseAcc are reported on the original CIFAR100 test set. This suggests that even when the tree used is not identical to the label hierarchy associated with the dataset, an approximate tree with a similar structure is sufficient for classification tasks.
+
+In summary, EMD-CPCC is a flexible structured representation learning method that adapts well to complex hierarchies, while FastFT-CPCC serves as a good approximation to EMD-CPCC. This
+
+observation is consistent with the results in Tab.3, where optimal transport methods achieve higher Test CPCC values.
+
+# H.2 EFFECT OF TREE WEIGHTS
+
+![](images/418b7f0b1b189701ab00c4d0ecf38f76aae3f4666c5c37a6ed73cf44c2d2568f.jpg)
+
+![](images/03df9c407ca25b710758bde7db6fe99bc4b3637433634a27ce49fedeebe92ae2.jpg)
+
+![](images/d938fe6a1a7afc2107cd6a6902cdb7fb26b10ae54c5805d176bc49d1aa74c754.jpg)
+
+![](images/0a73231158fc3604fe811992bbf051b076c504a1bd71346c1d1a7773679c6f90.jpg)  
+Figure 11: CIFAR100 class embedding pairwise distance matrix for different types of label tree with varying tree weights.
+
+![](images/d30a1f50f409961f005f1fbe711020c5cd1c292788ecab78f659798ec949d24f.jpg)
+
+![](images/db8e9d7d04c30e65597fb6c3097c1e75a5d5febbefd6c139a79647575abb63db.jpg)
+
+In this section, we demonstrate that all CPCC methods, including OT-CPCC, are influenced by input tree weights. In Fig.11, we begin with the original unweighted CIFAR100 tree (where all edges have a default weight of 1) and modify the edge weights of a subtree representing the coarse label aquatic_mammals, which includes five fine labels: beaver, dolphin, otter, seal, and whale. We visualize the learned representation through a distance matrix, as described in Sec.H.1. From left to right in Fig.11, the edge weights of this subtree are set to 1, 2, and 4, respectively. Consequently, we observe a color shift in the top-left coarse block of the matrix, with the block gradually darkening as the subtree weights increase. This demonstrates the flexibility of OT-CPCC methods, which can successfully embed weighted trees, and highlights why CPCC is a sensitive and effective metric for evaluating the interpretability and quality of the hierarchical information embedded in the learned representation, as discussed in Sec.4.4.
+
+Table 14: EMD and FastFT performance for tree with different edge weights within aquatic_mammals classes on the full CIFAR100 dataset. The batch size is set to 1024.  
+
+<table><tr><td>Objective</td><td>Setup</td><td>CPCC</td><td>FineAcc</td><td>CoarseAcc</td></tr><tr><td rowspan="3">EMD</td><td>weight 1x</td><td>89.13</td><td>75.76</td><td>85.50</td></tr><tr><td>weight 2x</td><td>88.85</td><td>75.57</td><td>85.22</td></tr><tr><td>weight 4x</td><td>79.86</td><td>75.56</td><td>85.35</td></tr><tr><td rowspan="3">FastFT</td><td>weight 1x</td><td>92.12</td><td>75.43</td><td>85.43</td></tr><tr><td>weight 2x</td><td>92.57</td><td>75.38</td><td>85.10</td></tr><tr><td>weight 4x</td><td>84.23</td><td>75.57</td><td>84.98</td></tr></table>
+
+We use the same experimental settings as those described in Sec.H.1. While the classification metrics remain comparable, the most significant trend emerges in the CPCC metric. As shown in
+
+Fig.11, we observe the expected pattern in the top-left block, which changes as the edge weights are multiplied. However, as the injected hierarchy deviates further from the given label hierarchy, it becomes increasingly unnatural. The increased weights make the tree embedding more challenging, resulting in a drop in CPCC values for larger weights.
+
+# H.3 EFFECT OF FLOW WEIGHT
+
+In Eq.3, the flow weight vectors  $a$  and  $b$  are set to uniform by default in all other experiments. However, in practice, these vectors can be customized by the user if prior information about the data distribution is available, which may significantly influence the learned embeddings. On CIFAR100, we tested three different flow weight settings:
+
+- Uniform (unif): default setting where we assume each data point is uniformly sampled from the class distribution, so  $a = (1/m, \dots, 1/m)$  and  $b = (1/n, \dots, 1/n)$ .  
+- Distance (dist): is an adaptive weighting method where during training, we compute the Euclidean centroid of each class and compute the distance of each sample to the corresponding class centroid. Then we can apply softmax function to normalize the distances into probabilities. Mathematically, if two class representations are  $Z$ ,  $Z'$ , then for each entry of flow weight vector,  $a_i = \frac{\exp(Z_i - \mu_z)}{\sum_{j=1}^m \exp(Z_j - \mu_z)}$ ,  $b_i = \frac{\exp(Z_i' - \mu_{z'})}{\sum_{j=1}^n \exp(Z_j' - \mu_{z'})}$ .  
+- Inverse-Distance (inv): is similar to dist, yet the only difference is we set the weight to be inversely scaled by the distance to the centroid, so the expressions of the flow weights are
+
+$$
+a _ {i} = \frac {\exp (\mu_ {z} - Z _ {i})}{\sum_ {j = 1} ^ {m} \exp (\mu_ {z} - Z _ {j})}, b _ {i} = \frac {\exp (\mu_ {z ^ {\prime}} - Z _ {i} ^ {\prime})}{\sum_ {j = 1} ^ {n} \exp (\mu_ {z ^ {\prime}} - Z _ {j} ^ {\prime})}.
+$$
+
+From Tab.15, we observe two key findings: first, tuning the flow weights can lead to improvements in out-of-distribution generalization metrics for the target set, but this comes at the expense of in-distribution source metrics. Second, the most significant differences are seen in the CPCC values, where it is nearly impossible to recover hierarchical information on the test set when using the Distance flow weight for both EMD and FastFT. The low interpretability scores suggest that not all flow weight settings are suitable for hierarchy learning.
+
+Table 15: Comparison of FastFT and EMD methods with different optimal transport flow weighting schemes across fine and coarse metrics, with cell colors representing the magnitude of values.  
+
+<table><tr><td rowspan="2">Objective</td><td rowspan="2">Weight</td><td rowspan="2">CPCC</td><td colspan="3">Fine Metrics</td><td colspan="4">Coarse Metrics</td></tr><tr><td>sAcc</td><td>sMAP</td><td>tAcc</td><td>sAcc</td><td>sMAP</td><td>tAcc</td><td>tMAP</td></tr><tr><td rowspan="3">FastFT</td><td>unif</td><td>93.85</td><td>81.32</td><td>78.88</td><td>23.15</td><td>86.96</td><td>91.71</td><td>44.29</td><td>42.20</td></tr><tr><td>dist</td><td>9.30</td><td>78.90</td><td>67.33</td><td>25.28</td><td>85.18</td><td>91.47</td><td>45.00</td><td>43.78</td></tr><tr><td>inv</td><td>66.52</td><td>79.73</td><td>68.31</td><td>25.45</td><td>85.81</td><td>92.03</td><td>43.70</td><td>43.70</td></tr><tr><td rowspan="3">EMD</td><td>unif</td><td>89.60</td><td>80.99</td><td>78.51</td><td>23.53</td><td>86.93</td><td>91.82</td><td>44.03</td><td>42.32</td></tr><tr><td>dist</td><td>19.87</td><td>79.23</td><td>73.79</td><td>26.05</td><td>85.91</td><td>91.30</td><td>43.55</td><td>42.53</td></tr><tr><td>inv</td><td>98.13</td><td>79.51</td><td>75.84</td><td>26.80</td><td>86.05</td><td>91.22</td><td>44.15</td><td>42.13</td></tr></table>
+
+# I ANALYSIS FOR THE TIME COMPLEXITY OF OT APPROXIMATION METHODS
+
+In this section, we provide the details and references for the reported time complexities in Tab.1.
+
+In the following time complexity analysis, denote  $b$  as batch size. We assume each class-conditioned cluster has the same size  $n$ , which implies  $b = nk$ , where  $k$  is the number of classes.
+
+For  $\ell_2$ , given one batch of batch size  $b$ , computing Euclidean centroids for all classes requires scanning through all  $d$ -dimensional data points once, which requires  $O bd) = O(nkd)$ . We have at most  $k$  classes in one batch, so computing the pairwise distances between class centroids will cost  $O(k^2 d)$ . The total cost for  $\ell_2$  is  $O((nk + k^2)d)$ .
+
+Table 16: Detailed generalization performance on fine level. Each value is the mean (std) of three seeds of experiments. Flat directly optimizes on the fine level classes, so we treat it as the upper bound of fine metrics and colored in grey. The bold values have the largest mean for a column metric within a dataset, ignoring the upper bound.  
+
+<table><tr><td>Dataset</td><td>Objective</td><td>sAcc</td><td>tAcc</td><td>sMAP</td><td>Dataset</td><td>sAcc</td><td>tAcc</td><td>sMAP</td></tr><tr><td rowspan="6">CIFAR10</td><td>Flat</td><td>96.92 (0.14)</td><td>54.23 (6.92)</td><td>99.43 (0.02)</td><td rowspan="6">INAT</td><td>88.70 (0.01)</td><td>26.77 (0.04)</td><td>63.97 (0.04)</td></tr><tr><td>\( \ell_2 \)</td><td>96.96 (0.16)</td><td>55.71 (7.38)</td><td>99.22 (0.06)</td><td>88.62 (0.01)</td><td>26.78 (0.04)</td><td>56.10 (0.05)</td></tr><tr><td>FastFT</td><td>96.90 (0.13)</td><td>55.99 (8.19)</td><td>99.24 (0.02)</td><td>88.49 (0.01)</td><td>27.10 (0.04)</td><td>56.21 (0.05)</td></tr><tr><td>EMD</td><td>97.05 (0.18)</td><td>56.12 (7.63)</td><td>99.24 (0.02)</td><td>88.68 (0.00)</td><td>26.78 (0.04)</td><td>56.83 (0.00)</td></tr><tr><td>Sinkhorn</td><td>96.95 (0.08)</td><td>54.89 (8.59)</td><td>99.27 (0.03)</td><td>88.08 (0.00)</td><td>26.77 (0.04)</td><td>51.56 (0.04)</td></tr><tr><td>SWD</td><td>96.96 (0.05)</td><td>59.21 (4.42)</td><td>99.45 (0.03)</td><td>88.46 (0.00)</td><td>26.78 (0.04)</td><td>54.19 (0.03)</td></tr><tr><td rowspan="6">CIFAR100</td><td>Flat</td><td>80.53 (0.31)</td><td>28.44 (1.89)</td><td>86.47 (0.21)</td><td rowspan="6">BREEDS</td><td>82.08 (0.26)</td><td>45.19 (0.38)</td><td>73.74 (0.46)</td></tr><tr><td>\( \ell_2 \)</td><td>80.98 (0.17)</td><td>23.76 (1.01)</td><td>77.52 (0.12)</td><td>82.66 (0.41)</td><td>45.95 (0.40)</td><td>62.86 (0.38)</td></tr><tr><td>FastFT</td><td>81.09 (0.18)</td><td>24.71 (0.90)</td><td>78.72 (0.19)</td><td>82.58 (0.34)</td><td>45.75 (0.45)</td><td>63.95 (0.36)</td></tr><tr><td>EMD</td><td>81.32 (0.19)</td><td>23.15 (0.40)</td><td>78.88 (0.18)</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Sinkhorn</td><td>80.99 (0.12)</td><td>23.53 (0.48)</td><td>78.51 (0.20)</td><td>82.99 (0.31)</td><td>46.87 (0.40)</td><td>61.54 (0.55)</td></tr><tr><td>SWD</td><td>80.50 (0.33)</td><td>26.18 (1.04)</td><td>86.82 (0.22)</td><td>82.90 (0.30)</td><td>46.33 (0.16)</td><td>76.24 (0.39)</td></tr></table>
+
+Table 17: Detailed generalization performance on coarse level. Each value is the mean (std) of three seeds of experiments.  $\ell_2$  directly optimizes on the coarse level classes, so we treat it as the upper bound of coarse metrics and colored in grey.  
+
+<table><tr><td>Dataset</td><td>Objective</td><td>sAcc</td><td>tAcc</td><td>sMAP</td><td>tMAP</td><td>Dataset</td><td>sAcc</td><td>tAcc</td><td>sMAP</td><td>tMAP</td></tr><tr><td rowspan="6">CIFAR10</td><td>ε2</td><td>99.60 (0.01)</td><td>87.51 (0.97)</td><td>99.88 (0.03)</td><td>92.84 (0.63)</td><td rowspan="6">INAT</td><td>94.52 (0.00)</td><td>40.86 (0.01)</td><td>74.34 (0.05)</td><td>37.35 (0.01)</td></tr><tr><td>Flat</td><td>99.58 (0.07)</td><td>87.30 (0.59)</td><td>99.22 (0.05)</td><td>89.66 (0.38)</td><td>94.63 (0.00)</td><td>38.81 (0.01)</td><td>70.41 (0.04)</td><td>34.00 (0.13)</td></tr><tr><td>FastFT</td><td>99.61 (0.06)</td><td>87.79 (0.41)</td><td>99.91 (0.03)</td><td>93.04 (0.57)</td><td>94.66 (0.00)</td><td>39.43 (0.02)</td><td>72.90 (0.05)</td><td>35.63 (0.01)</td></tr><tr><td>EMD</td><td>99.61 (0.03)</td><td>87.45 (0.60)</td><td>99.88 (0.03)</td><td>93.07 (0.63)</td><td>94.64 (0.00)</td><td>41.01 (0.01)</td><td>73.87 (0.01)</td><td>35.58 (0.01)</td></tr><tr><td>Sinkhorn</td><td>99.61 (0.05)</td><td>87.41 (0.53)</td><td>99.87 (0.03)</td><td>92.60 (0.13)</td><td>94.30 (0.00)</td><td>36.94 (0.02)</td><td>68.75 (0.02)</td><td>34.92 (0.00)</td></tr><tr><td>SWD</td><td>99.56 (0.04)</td><td>87.63 (0.62)</td><td>99.36 (0.09)</td><td>90.12 (0.43)</td><td>94.52 (0.00)</td><td>39.38 (0.01)</td><td>75.13 (0.03)</td><td>38.11 (0.01)</td></tr><tr><td rowspan="6">CIFAR100</td><td>ε2</td><td>87.00 (0.13)</td><td>44.55 (0.39)</td><td>92.13 (0.23)</td><td>43.03 (0.55)</td><td rowspan="6">BREEDS</td><td>94.24 (0.30)</td><td>83.13 (0.23)</td><td>95.00 (0.25)</td><td>81.73 (0.32)</td></tr><tr><td>Flat</td><td>86.17 (0.26)</td><td>43.09 (0.55)</td><td>82.07 (0.34)</td><td>42.09 (0.26)</td><td>93.85 (0.20)</td><td>82.43 (0.27)</td><td>74.47 (0.52)</td><td>66.50 (0.39)</td></tr><tr><td>FastFT</td><td>86.96 (0.13)</td><td>44.29 (0.33)</td><td>91.71 (0.11)</td><td>42.20 (0.44)</td><td>94.25 (0.27)</td><td>82.89 (0.31)</td><td>92.84 (0.27)</td><td>79.69 (0.34)</td></tr><tr><td>EMD</td><td>86.93 (0.28)</td><td>44.03 (0.31)</td><td>91.82 (0.14)</td><td>42.32 (0.31)</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Sinkhorn</td><td>86.93 (0.10)</td><td>43.99 (0.37)</td><td>91.61 (0.31)</td><td>42.45 (0.09)</td><td>94.42 (0.21)</td><td>83.51 (0.30)</td><td>91.84 (0.42)</td><td>79.33 (0.38)</td></tr><tr><td>SWD</td><td>86.42 (0.26)</td><td>43.35 (0.71)</td><td>87.24 (0.29)</td><td>42.39 (0.30)</td><td>94.41 (0.25)</td><td>82.96 (0.17)</td><td>83.58 (0.60)</td><td>72.38 (0.43)</td></tr></table>
+
+For OT methods, we have  $k$  class-conditioned clusters, resulting in  $O(k^2)$  pairs of matrices as input for all OT methods. For each pair, the computation time of OT methods can range from linear to cubic with respect to  $n$ . For instance, each FastFT computation takes  $O(nd)$  as shown in Theorem 3.2, so the total computation time becomes  $O(k^2 nd)$ . In general, the computation time for all OT methods can be expressed as  $O(k^2 \cdot \text{OT-time})$ . Next, we elaborate on the "OT-time" term.
+
+For EMD, the complexity is typically  $O(n^3 \log n)$  in the past literature (Cuturei, 2013), which primarily focuses on the influence of the number of samples  $n$ . However, in our setting, because  $d >> n$ , we aim to emphasize the effect of  $d$ . In the time complexity expression for EMD, the  $O(n^3 \log n)$  term arises from solving the linear programming problem, while the  $O(nd)$  term accounts for computing the objective function  $\langle P, D \rangle$ . Due to the sparse nature of the linear programming solution, there are only  $O(n)$  non-zero values in  $P$ . Consequently, we compute  $O(n)$ $d$ -dimensional pairwise distances in  $D$ , making the dot product computation require  $O(nd)$  in total. Therefore, the overall complexity of EMD is  $O(n(d + n^2 \log n))$ .
+
+To get the approximated flow matrix, Sinkhorn (Cuturi, 2013) mainly depends on the iterative matrix scaling algorithm. For each  $I$  iteration, we normalize the row and column of a matrix of size  $O(n^{2})$ , so the complexity for running this iterative algorithm is  $O(n^{2}I)$ . Unlike in EMD, the approximated flow matrix  $P$  derived from Sinkhorn is not sparse. Therefore, the dot product of the objective function will now take  $O(n^{2}d)$  in total. The total complexity of Sinkhorn is  $O(n^{2}(d + I))$ .
+
+To obtain the approximated flow matrix, Sinkhorn (Cuturi, 2013) relies on the iterative matrix scaling algorithm. During each of the  $I$  iterations, we normalize the rows and columns of a matrix of size  $O(n^{2})$ , resulting in a complexity of  $O(n^{2}I)$  for the iterative algorithm. Unlike EMD, the approximated flow matrix  $\pmb{P}$  derived from Sinkhorn is not sparse. Consequently, computing the dot product for the objective function requires  $O(n^{2}d)$  in total. Therefore, the overall complexity of Sinkhorn is  $O(n^{2}(d + I))$ .
+
+Table 18: Fine level generalization performance for each BREEDS setting.  
+
+<table><tr><td>Dataset</td><td>Objective</td><td>sAcc</td><td>tAcc</td><td>sMAP</td></tr><tr><td rowspan="5">L17</td><td>Flat</td><td>85.15 (0.30)</td><td>46.31 (0.31)</td><td>85.94 (0.18)</td></tr><tr><td>l2</td><td>85.74 (0.69)</td><td>47.50 (0.73)</td><td>74.07 (0.61)</td></tr><tr><td>FastFT</td><td>85.21 (0.38)</td><td>46.47 (0.75)</td><td>74.46 (0.16)</td></tr><tr><td>Sinkhorn</td><td>85.80 (0.49)</td><td>48.37 (0.69)</td><td>70.50 (0.53)</td></tr><tr><td>SWD</td><td>85.88 (0.26)</td><td>47.50 (0.09)</td><td>86.65 (0.51)</td></tr><tr><td rowspan="5">N26</td><td>Flat</td><td>79.76 (0.27)</td><td>36.06 (0.37)</td><td>70.75 (1.14)</td></tr><tr><td>l2</td><td>80.58 (0.54)</td><td>37.08 (0.26)</td><td>55.51 (0.50)</td></tr><tr><td>FastFT</td><td>80.78 (0.52)</td><td>36.69 (0.19)</td><td>58.87 (0.42)</td></tr><tr><td>Sinkhorn</td><td>81.74 (0.26)</td><td>38.16 (0.14)</td><td>57.54 (0.92)</td></tr><tr><td>SWD</td><td>81.79 (0.25)</td><td>37.71 (0.09)</td><td>76.03 (0.05)</td></tr><tr><td rowspan="5">E13</td><td>Flat</td><td>84.37 (0.24)</td><td>56.04 (0.27)</td><td>74.04 (0.34)</td></tr><tr><td>l2</td><td>84.77 (0.18)</td><td>55.69 (0.37)</td><td>66.92 (0.16)</td></tr><tr><td>FastFT</td><td>85.04 (0.15)</td><td>56.66 (0.44)</td><td>67.96 (0.64)</td></tr><tr><td>Sinkhorn</td><td>85.00 (0.15)</td><td>57.14 (0.40)</td><td>64.47 (0.20)</td></tr><tr><td>SWD</td><td>84.49 (0.18)</td><td>56.82 (0.12)</td><td>75.39 (0.54)</td></tr><tr><td rowspan="5">E30</td><td>Flat</td><td>79.05 (0.23)</td><td>42.35 (0.57)</td><td>64.23 (0.17)</td></tr><tr><td>l2</td><td>79.54 (0.21)</td><td>43.51 (0.23)</td><td>54.92 (0.23)</td></tr><tr><td>FastFT</td><td>79.29 (0.32)</td><td>43.16 (0.42)</td><td>54.49 (0.23)</td></tr><tr><td>Sinkhorn</td><td>79.40 (0.33)</td><td>43.82 (0.38)</td><td>53.66 (0.53)</td></tr><tr><td>SWD</td><td>79.42 (0.52)</td><td>43.28 (0.34)</td><td>66.87 (0.45)</td></tr></table>
+
+Table 19: Coarse level generalization performance for each BREEDS setting.  
+
+<table><tr><td>Dataset</td><td>Objective</td><td>sAcc</td><td>tAcc</td><td>sMAP</td><td>tMAP</td></tr><tr><td rowspan="5">L17</td><td>l2</td><td>93.13 (0.62)</td><td>79.03 (0.29)</td><td>93.93 (0.54)</td><td>69.13 (0.42)</td></tr><tr><td>Flat</td><td>92.68 (0.39)</td><td>78.35 (0.37)</td><td>76.66 (0.26)</td><td>56.13 (0.36)</td></tr><tr><td>FastFT</td><td>93.33 (0.26)</td><td>78.37 (0.47)</td><td>92.11 (0.23)</td><td>65.44 (0.24)</td></tr><tr><td>Sinkhorn</td><td>93.11 (0.20)</td><td>79.50 (0.43)</td><td>89.14 (0.36)</td><td>63.88 (0.32)</td></tr><tr><td>SWD</td><td>92.96 (0.41)</td><td>78.66 (0.26)</td><td>83.92 (0.38)</td><td>59.66 (0.21)</td></tr><tr><td rowspan="5">N26</td><td>l2</td><td>88.94 (0.42)</td><td>70.24 (0.50)</td><td>89.59 (0.26)</td><td>68.40 (0.52)</td></tr><tr><td>Flat</td><td>88.44 (0.12)</td><td>68.98 (0.41)</td><td>61.20 (1.01)</td><td>50.86 (0.16)</td></tr><tr><td>FastFT</td><td>88.83 (0.75)</td><td>69.43 (0.42)</td><td>83.89 (0.40)</td><td>65.22 (0.75)</td></tr><tr><td>Sinkhorn</td><td>89.94 (0.35)</td><td>70.70 (0.41)</td><td>83.14 (0.78)</td><td>66.07 (0.85)</td></tr><tr><td>SWD</td><td>89.88 (0.34)</td><td>69.93 (0.04)</td><td>74.06 (0.57)</td><td>58.50 (0.48)</td></tr><tr><td rowspan="5">E13</td><td>l2</td><td>97.80 (0.04)</td><td>91.51 (0.01)</td><td>98.25 (0.04)</td><td>94.15 (0.09)</td></tr><tr><td>Flat</td><td>97.53 (0.12)</td><td>91.30 (0.16)</td><td>83.89 (0.13)</td><td>82.37 (0.31)</td></tr><tr><td>FastFT</td><td>97.80 (0.01)</td><td>91.78 (0.13)</td><td>97.69 (0.13)</td><td>93.21 (0.10)</td></tr><tr><td>Sinkhorn</td><td>97.63 (0.02)</td><td>91.97 (0.14)</td><td>97.53 (0.23)</td><td>92.77 (0.18)</td></tr><tr><td>SWD</td><td>97.67 (0.13)</td><td>91.69 (0.06)</td><td>91.43 (0.72)</td><td>87.71 (0.16)</td></tr><tr><td rowspan="5">E30</td><td>l2</td><td>97.07 (0.10)</td><td>91.74 (0.10)</td><td>98.24 (0.14)</td><td>95.28 (0.24)</td></tr><tr><td>Flat</td><td>96.75 (0.18)</td><td>91.10 (0.15)</td><td>76.12 (0.66)</td><td>76.65 (0.74)</td></tr><tr><td>FastFT</td><td>97.02 (0.07)</td><td>91.98 (0.23)</td><td>97.65 (0.32)</td><td>94.90 (0.26)</td></tr><tr><td>Sinkhorn</td><td>96.99 (0.25)</td><td>91.87 (0.22)</td><td>97.54 (0.31)</td><td>94.59 (0.18)</td></tr><tr><td>SWD</td><td>97.12 (0.13)</td><td>91.55 (0.31)</td><td>84.89 (0.74)</td><td>83.66 (0.88)</td></tr></table>
+
+For FT (Backurs et al., 2020), the most computationally expensive step is the tree construction using the QuadTree algorithm. Backurs et al. (2020) demonstrate that this tree can be built in
+
+$O(nd\log (d\Phi))$ . Once the tree is constructed, as shown in Lemma 3.1 of Backurs et al. (2020), FT requires  $O(n(d + \log (d\Phi)))$  to run the bottom-up algorithm and compute the objective function, which is less expensive than the tree construction. Thus, the total complexity of FT is  $O(nd\log (d\Phi))$ .
+
+For Fast FlowTree, please refer to the proof of Thm. 3.2.
+
+For TWD, we follow Algorithm 1 in Yamada et al. (2022). The first step is to construct a tree. To save computation, instead of using QuadTree, we leverage the same technique as in FastFT to use our augmented tree, making this step  $O(1)$ . For each of the  $I$  iterations in supervised learning, we need to compute all entries in the pairwise distance matrix  $D$ , which requires  $O(n^{2}d)$ . Thus, even when the tree structure is known, the total complexity of TWD is  $O(n^{2}dI)$ .
+
+Finally, we want to highlight that Fig.3 is evaluated with  $k = 2$ , where we vary the number of samples in one source and one target distribution. In this scenario, although the theoretical computational complexities for  $\ell_2$  and Fast FlowTree are both  $O(nd)$ , we observe that  $\ell_2$  scales better in practice. This is due to the effect of standard vectorization optimizations in common matrix operation packages, which leverage parallelism along the dimension of  $n$ .
+
+# I.1 COMPARISON TO OTHER HIERARCHICAL BASELINES
+
+In Fig.12, we compared OT-CPCC methods with other hierarchical objectives described in Sec.G, and we report the Fine/Coarse Metric Avg., which averages all source/target classification and retrieval metrics. Based on CPCC, fine, and coarse metrics, we observe that OT-CPCC methods behave similarly to  $\ell_2$ -CPCC, exhibiting higher CPCC values and stronger performance on coarse metrics, but lower fine metrics (though OT methods still outperform  $\ell_2$  on fine metrics, as shown in Tab.5). In contrast, all other hierarchical baseline objectives perform more like flat cross-entropy methods, with the opposite trend—lower CPCC and coarse metrics but relatively higher fine metrics. Additionally, we excluded Rank loss from the correlation plot on the right side of Fig.12, as Tab.20 suggests that Rank loss is not suitable for any of the downstream tasks used in our evaluation, making it an outlier compared to the other objectives.
+
+<table><tr><td>Objective</td><td>CPCC</td></tr><tr><td>l2</td><td>83.08</td></tr><tr><td>FastFT</td><td>93.85</td></tr><tr><td>EMD</td><td>89.60</td></tr><tr><td>Sinkhorn</td><td>93.81</td></tr><tr><td>SWD</td><td>73.17</td></tr><tr><td>Flat</td><td>22.13</td></tr><tr><td>MTL</td><td>37.07</td></tr><tr><td>Curr</td><td>22.70</td></tr><tr><td>SumLoss</td><td>27.48</td></tr><tr><td>HXE</td><td>22.65</td></tr><tr><td>Soft</td><td>50.90</td></tr><tr><td>Quad</td><td>24.01</td></tr><tr><td>SHC</td><td>30.68</td></tr><tr><td>GC</td><td>21.88</td></tr><tr><td>Rank</td><td>24.75</td></tr></table>
+
+![](images/a17e8cf9de461a19ce16a70b3782aea3c7397affad638047841de8cdfe472cfe.jpg)
+
+![](images/dce083eaa3b5dff1898f619699fd51ad885c6a6f813f1dd96dc819a26c804314.jpg)  
+Figure 12: Left: Comparison of CPCC for hierarchical baselines on CIFAR100. Right: CPCC against coarse and fine metrics average. Rank is an outlier, so its value is removed from the visualization.
+
+# J IMPLEMENTATION DETAILS FOR SYNTHETIC DATASETS EXPERIMENTS
+
+Computational Efficiency We measured the computation time for different OT methods across datasets of varying sizes. This involved generating two random 128-dimensional datasets of identical shape and recording the wall-clock time for a single distance computation. This process was repeated
+
+Table 20: Comparison of fine and coarse metrics for Flat vs. Rank Loss.  
+
+<table><tr><td rowspan="2">Objective</td><td colspan="3">Fine Metrics</td><td colspan="4">Coarse Metrics</td></tr><tr><td>sAcc</td><td>tAcc</td><td>sMAP</td><td>sAcc</td><td>tAcc</td><td>sMAP</td><td>tMAP</td></tr><tr><td>Flat</td><td>80.53</td><td>28.44</td><td>86.47</td><td>86.17</td><td>43.09</td><td>82.07</td><td>42.09</td></tr><tr><td>Rank</td><td>77.89</td><td>20.48</td><td>74.39</td><td>83.41</td><td>39.25</td><td>46.60</td><td>24.31</td></tr></table>
+
+10 times, and the average time was reported. For illustrative purposes, we fixed the number of projections at 10 for SWD, and the regularization parameter at 10 for Sinkhorn, the same as in our experiment setting in Tab.4,5.
+
+Approximation Error We evaluated the error of different OT approximation methods, along with the  $\ell_2$  distance of the mean of two datasets, in approximating EMD. Two different distribution scenarios were considered:
+
+- Gaussian Distribution: Here, both distributions were created as Gaussian distributions with different means (1 and 4, respectively) and the same standard deviation of 1.  
+- Non-Gaussian Distribution: For this scenario, we generated the dataset from Gaussian mixtures. One dataset combined Gaussian distributions with means of 0 and 5, while the other merged Gaussian distributions with means of 2 and 3, all of them with a standard deviation of 1.
+
+Other experiment settings were consistent with those detailed in the Computational Efficiency paragraph above.
+
+# K COMPARISON OF RUNTIME OF OT-CPCC ON REAL-WORLD DATASETS
+
+Table 21: Comparison of the wall clock runtime of OT-methods on real datasets for training one epoch. The results are averaged for three random seeds.  
+
+<table><tr><td>Dataset</td><td>Objective</td><td>Per Epoch (s)</td><td>Dataset</td><td>Objective</td><td>Per Epoch (s)</td></tr><tr><td rowspan="6">CIFAR10(b=128)</td><td>Flat</td><td>85.88 (3.69)</td><td rowspan="6">INAT(b=1024)</td><td>Flat</td><td>147.67 (4.54)</td></tr><tr><td>l2</td><td>83.78 (9.80)</td><td>l2</td><td>149.39 (1.24)</td></tr><tr><td>FastFT</td><td>85.21 (4.26)</td><td>FastFT</td><td>132.01 (3.24)</td></tr><tr><td>EMD</td><td>87.70 (2.43)</td><td>EMD</td><td>234.68 (4.40)</td></tr><tr><td>Sinkhorn</td><td>85.65 (4.15)</td><td>Sinkhorn</td><td>165.69 (4.60)</td></tr><tr><td>SWD</td><td>92.90 (4.65)</td><td>SWD</td><td>152.52 (8.90)</td></tr></table>
+
+We include the wall clock run times for training different CPCC methods over one epoch in Tab.21, using two distinct settings: CIFAR10 with a batch size of 128 and model training from scratch, and INAT with a batch size of 1024 and model fine-tuning from an ImageNet checkpoint. Despite the differences in settings, we observe that FastFT consistently runs significantly faster than exact EMD computation and outperforms the other two approximation methods, SWD and Sinkhorn. This aligns with our theoretical analysis in Tab.1 and observations on synthetic data in Fig. 3. When comparing FastFT to  $\ell_2$ , we find that FastFT is not much slower, and in the case of INAT, it is even faster than  $\ell_2$ . Thus, we can confidently conclude that our proposed method, especially FastFT, is scalable.
+
+Similar to  $\ell_2$ , one could argue for using another closed-form Wasserstein expression, the Bures-Wasserstein distance (as defined in Eq.7), in CPCC optimization. However, Bures depends on the covariance term of the source and target distributions, which requires at least  $O(nd^2)$  computation. As shown in Tab.1, in  $\ell_2$ , calculating the mean of a class-conditioned cluster only requires  $O(nd)$ , and all other OT methods have a linear  $d$  term in their time complexity. In CIFAR100, even with a large batch size of 1024,  $n$  is approximately  $1024 / 100 \approx 10$ , assuming all classes are of equal size. Using ResNet18 as the backbone,  $d = 512$ , which is typically much larger than  $n$ . Therefore, the  $d^2$  factor from computing the covariance matrix in Bures can make it prohibitively expensive in our case.
+
+Table 22: Comparison of the wall clock runtime of Bures-CPCC vs. OT-CPCC on real datasets for training one batch. The results are averaged for three random seeds.  
+
+<table><tr><td>Dataset</td><td>Objective</td><td>Per Batch (s)</td></tr><tr><td rowspan="3">CIFAR100</td><td>l2</td><td>0.35 (0.18)</td></tr><tr><td>EMD</td><td>1.94 (0.12)</td></tr><tr><td>Bures</td><td>33.77 (4.73)</td></tr></table>
+
+To verify this, we computed the running time for CIFAR100 (batch size = 128) for different methods in Tab.22, considering one batch update. The results are averaged over three random seeds.
+
+As observed in Tab.22, Bures-Wasserstein is significantly slower than both  $\ell_{2}$  and EMD due to its quadratic dependence on  $d$ . Therefore, for scalability concerns, covariance-based Wasserstein metrics are not well-suited to our problem setup.
\ No newline at end of file
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+# LEARNING STRUCTURED UNIVERSE GRAPH WITH OUTLIER OOD DETECTION FOR PARTIAL MATCHING
+
+Zetian Jiang $^{1\dagger}$ , Jiaxin Lu $^{3\dagger}$ , Haozhao Fan $^{1}$ , Tianzhe Wang $^{1}$ , Junchi Yan $^{12*}$
+
+$^{1}$ Sch. of Computer Science & Sch. of Artificial Intelligence, Shanghai Jiao Tong University  
+$^{2}$ Shanghai Artificial Intelligence Laboratory  
+$^{3}$ Department of Computer Science, University of Texas at Austin
+
+{maple_jzt, usedtobe, yanjunchi}@sjtu.edu.cn, lujiaxin@utexas.edu
+
+# ABSTRACT
+
+Partial matching is a kind of graph matching where only part of two graphs can be aligned. This problem is particularly important in computer vision applications, where challenges like point occlusion or annotation errors often occur when labeling key points. Previous work has often conflated point occlusion and annotation errors, despite their distinct underlying causes. We propose two components to address these challenges: (1) a structured universe graph is learned to connect two input graphs  $\mathbf{X}_{ij} = \mathbf{X}_{iu}\mathbf{X}_{ju}^{\top}$ , effectively resolving the issue of point occlusion; (2) an energy-based out-of-distribution detection is designed to remove annotation errors from the input graphs before matching. We evaluated our method on the Pascal VOC and Willow Object datasets, focusing on scenarios involving point occlusion and random outliers. The experimental results demonstrate that our approach consistently outperforms state-of-the-art methods across all tested scenarios, highlighting the accuracy and robustness of our method.
+
+# 1 INTRODUCTION
+
+Graph matching has been a fundamental task in traditional computer vision for a long time, applied in areas such as structure from motion (Vijayanarasimhan et al., 2017), object tracking (Nam & Han, 2016; Iqbal et al., 2017), optical flow (Sun et al., 2018), stereo matching (Luo et al., 2016; Chang & Chen, 2018), and pose estimation (Gasse et al., 2019). With the rise of deep learning, integrating graph matching and neural networks has led to significant improvements in accuracy and efficiency (Zanfir & Sminchisescu, 2018; Wang et al., 2019), making it more suitable for downstream tasks. Beyond these, graph matching has also shown great potential in other fields, such as bioinformatics for protein structure comparison (Luo et al., 2021) and drug design (Kriege et al., 2019), as well as in social network analysis for community detection and network alignment (Chen et al., 2020).
+
+In particular, graph matching aims to find node correspondence between graph-structured data via both node and edge affinity (Gold & Rangarajan, 1996; Cho et al., 2010). It is a well-known NP-Hard problem and is often formulated as the quadratic assignment problem (QAP):
+
+$$
+\begin{array}{l} \max  _ {\mathbf {X}} \operatorname {v e c} (\mathbf {X}) ^ {\top} \mathbf {K} \operatorname {v e c} (\mathbf {X}) \\ s. t. \quad \mathbf {X} \in \{0, 1 \} ^ {n \times n}, \mathbf {X} \mathbf {1} _ {n} = \mathbf {1} _ {n}, \mathbf {X} ^ {\top} \mathbf {1} _ {n} = \mathbf {1} _ {n}. \\ \end{array}
+$$
+
+(1)
+
+Here,  $\mathbf{X}$  represents a binary permutation matrix, where  $\mathbf{X}_{ij} = 1$  indicates that node  $i$  in the first graph corresponds to node  $j$  in the second graph.  $\mathbf{K} \in \mathbf{R}^{n^2 \times n^2}$  is a hand-crafted or learned affinity matrix whose diagonal (off-diagonal) encodes node (edge) affinities.
+
+While theoretically elegant, the QAP's assumption of perfect one-to-one node correspondence between graphs rarely holds in real-world applications. Practical scenarios often involve partial matching, where only subsets of nodes between graphs correspond, as illustrated in Figure 1. Traditional approaches attempted to address this limitation by introducing dummy nodes with zero affinity. However, this solution proves inadequate for handling real-world challenges such as occlusions and annotation errors.
+
+![](images/667aa2463760c5aa4bf8c871a8ed9e08796a35ea3052b8a4f87372404355109f.jpg)  
+Figure 1: Overview of the problem setting and UGM. Partial Matching: Due to occlusion, viewpoint differences, and annotation errors, only a subset of key points on each graph can be matched. Left: Prior works have limited capability in detecting and filtering outliers. Right: UGM employs two key components to improve partial matching performance: (1) a universe graph that serves as a bridge to aggregate matches from different graphs, and (2) an outlier filter to remove spurious matches. This design ensures that only inliers are matched, significantly enhancing partial matching performance.
+
+Several recent works have explored ways to overcome this limitation. BBGM (Rolínek et al., 2020a), for instance, bypasses the constraint that each node must find a specific match by learning a negative affinity score. However, this approach heavily relies on the neural network's capacity, and the actual performance gains are often limited. GCAN (Jiang et al., 2022) reformulates graph matching as an integer linear programming (ILP), introducing a placeholder node for each graph to absorb outliers. While this helps, learning for ILP remains a challenging task. AFAT (Wang et al., 2023) predicts which node match to discard by analyzing matched points and the affinity matrix. However, as a post-processing step, its effectiveness is limited because outliers may have already disrupted the initial matching, weakening its ability to correct errors. Another approach, URL (Nurlanov et al., 2023), learns a universe point representation, which serves as a central reference for key point alignment across all images. Although it addresses the issue of point occlusion, it still suffers from redundant matches as the erroneous annotation outliers fail to find the corresponding universe point representation.
+
+A key insight often overlooked by these literature is the distinction between point occlusion and annotation errors: occlusion-caused outliers may find correspondences in other graphs, while annotation errors are inherently meaningless. Building on this observation, we propose Universe Graph Matching (UGM), which addresses these challenges through two key innovations: (1) A latent universe graph learning approach that utilizes the structured universe graph to serve as a bridge for matching graph pairs  $\mathbf{X}_{ij} = \mathbf{X}_{iu}\mathbf{X}_{ju}^{\top}$ . This method leverages the inherent structural information from input graphs, enhancing the accuracy and robustness of the learned universe graph for handling point occlusions. (2) An energy-based out-of-distribution detection method that effectively identifies and removes annotation errors at test time. Through training with a margin loss, we effectively differentiate between the energy scores of erroneous annotation outliers and valid in-distribution keypoints. By removing annotation errors, we significantly reduce the complexity of the partial matching problem (as we will discuss in Sec. 3.3), thereby improving overall matching accuracy.
+
+# Our primary contributions are:
+
+- We propose Latent Universe Graph Learning, which enhances universe matching robustness through combined node features and structural information.  
+- We are the first to introduce the Energy-based OOD Detection for annotation error identification, enabling pre-matching outlier filtering for improved accuracy.  
+- We evaluate the proposed UGM approach on keypoint matching tasks under occlusion and random outlier settings using both Pascal VOC and Willow datasets. Our experimental results demonstrate that UGM consistently outperforms state-of-the-art methods, highlighting the effectiveness and robustness of our method.
+
+# 2 RELATED WORKS
+
+# 2.1 CLASSICAL GRAPH MATCHING
+
+Two-graph matching has been widely studied, which involves constructing graphs for the objects to be matched and aligning the nodes and edges between the two graphs. (Gold & Rangarajan, 1996) solves graph matching via a graduated assignment (GA) algorithm, which computes the partial derivative
+
+of the objective function with Taylor expansion to turn the GM problem into a linear assignment problem. RRWM (Cho et al., 2010) constructs an association graph for matching graphs and applies the random walk algorithm with matching constraints. RRWHM (Lee et al., 2011) is the extended version of RRWM where the association graph is built with higher-order geometric information extracted from the graph. DS++, DS* (Dym et al., 2017; Bernard et al., 2018) design different ways to tightly relax GM to a convex problem and then project the results back to the original solution space. LPMP (Swoboda et al., 2017) proposes several additional Lagrangian relaxations of the graph matching problem and o leading solvers for this problem optimize the Lagrange decomposition duals with sub-gradient and dual ascent updates.
+
+In addition, numerous works have focused on matching multiple graphs, which is called Multi-Graph Matching. There are mainly two forms of methods. The first solves the problem in discrete space, which includes composition-based method CAO (Yan et al., 2015b), MGM-Floyd (Jiang et al., 2021), tree-structure-based approach MatchOpt (Yan et al., 2015a), DPMC (Wang et al., 2020b), and Majorize-Minimization framework based approach M3C (Lu et al., 2024). Another line of works (Pachauri et al., 2013; Huang & Guibas, 2013; Chen et al., 2014; Wang et al., 2018) etc. tries to relax the MGM problem into a continuous space and re-project the solution to obtain a discrete matching result.
+
+# 2.2 DEEP LEARNING OF GRAPH MATCHING
+
+Deep learning has recently been applied to graph matching on images (Zanfir & Sminchisescu, 2018), whereby CNN extracts node features from images followed by spectral matching and is learned using a regression-like node correspondence supervision. This work is improved via introducing GNN to encode structural (Wang et al., 2019; Wang et al., 2020) or geometric (Zhang & Lee, 2019) information, with a combinatorial loss based on cross-entropy loss, and Sinkhorn network (Adams & Zemel, 2011) as a differential matching solver. The work (Yu et al., 2020) extends PCA (Permutation loss and Cross-graph Affinity GM) (Wang et al., 2019) by edge embedding and Hungarian-based (Kuhn, 1955) attention mechanism to stabilize end-to-end training. BBGM (Blackbox Deep Graph Matching) (Rolinić et al., 2020b) proposes a better front-end feature extraction backbone with Spline Convolution (Fey et al., 2018), and the gradient is backpropagated by a fitting linear gradient of the discrete graph matching solver (Pogancić et al., 2019). NGM (Neural Graph Matching Nets) (Wang et al., 2021; 2020a) proposes to address the most general Lawler's QAP form, based on the novel feature extractors e.g. (Rolinić et al., 2020a) with the proposed learnable graph matching solver. DLGM (Deep Latent Graph Matching) (Yu et al., 2021), based on BBGM, predicts consistent graph topology utilizing deterministic and generative models to improve the matching quality. GCAN (Jiang et al., 2022) formulates the graph matching problem as an Integer Linear Programming problem and explores a novel node feature extraction framework with attention. AFAT (Wang et al., 2023) uses the attention model to predict which node match to discard by analyzing matched points and the affinity matrix. COMMON (Lin et al., 2023) adopts a contrastive learning approach, utilizing the infoNEC loss to supervise training similarity matrices for both edges and nodes. URL (Nurlanov et al., 2023) introduces a novel universe point representation learning method, applying universe key point matching to the graph matching field for the first time.
+
+# 2.3 OUT-OF-DISTRIBUTION DETECTION
+
+Out-of-distribution (OOD) detection is a growing field aimed at identifying data points that deviate from the distribution on which a model was trained. This is crucial for improving the robustness and safety of machine learning systems. Many OOD detection methods can be applied to pre-trained models without modifying the training process. These methods often rely on statistics from the model's output, such as maximum softmax probability, to distinguish between in-distribution (ID) and OOD data (Hendrycks & Gimpel, 2016; Liang et al., 2017). Energy-based OOD detection methods, on the other hand, assign an energy score to each input, where OOD data typically results in higher energy values (Liu et al., 2020; Bao et al., 2024). Generative models, such as VAEs and GANs, have also been used to estimate the likelihood that a given data point belongs to the training distribution (Nalisnick et al., 2018). More recently, self-supervised tasks have been employed to improve OOD detection by learning representations that are highly correlated with the training distribution, enabling the model to identify deviations (Sehwag et al., 2021). Contrastive learning has further advanced OOD detection by ensuring that OOD data points are positioned far from ID data in the learned representation space (Winkens et al., 2020).
+
+![](images/1c0c14abd129676c7ba68f9b9e04e1156b3a01436f2ce0106c6e5d9b3bfb2576.jpg)  
+Figure 2: Pipeline of Universe Graph Matching. The pipeline consists of two main components: 1) structured universe graph learning, which includes the training of node and edge affinities, and 2) outlier ood detection through energy score. The dark blue arrows indicate the training phase, while the light blue arrows represent the inference phase.
+
+# 3 METHODOLOGY
+
+In this section, we will introduce our method Universe Graph Matching (UGM) from two parts: structured universe graph learning and outlier ood detection in Section 3.1 and 3.2, respectively. The overall pipeline is shown in Figure 2, and more implementation and training details are provided in Appendix B. Meanwhile, we also discuss UGM in perspective of problem formulation and analyze its strength as well as drawbacks in Section 3.3.
+
+# 3.1 STRUCTURED UNIVERSE GRAPH LEARNING
+
+We begin by defining the structured universe graph, which serves as the foundation for our approach. The universe graph  $\mathbf{U} = \{\mathbf{N}_u\in \mathbf{R}^{n_u\times d},\mathbf{E}_u\in \mathbf{R}^{m_u\times d}\}$  is composed of both node embeddings  $\mathbf{N}_u$  and edge embeddings  $\mathbf{E}_u$ . The number of nodes  $n_u$  corresponds to the distinct types of key points present in the input graphs. We define this latent  $\mathbf{U}$  as a complete, directed graph, where every pair of nodes is connected by a directed edge, resulting in  $m_{u} = n_{u}\times (n_{u} - 1)$  edges. This ensures that any edge in the input graph will have a corresponding edge in the universe graph. Moreover, the directed edges capture the inherent asymmetry in real-world relationships between key points, such as varying geometric or contextual dependencies, offering greater flexibility in encoding complex structural information.
+
+We now introduce the pipeline to match the latent universe graph  $\mathbf{U}$  to the input graph  $\mathbf{G}_i$ . The input graph  $\mathbf{G}_i = \{\mathbf{N}_i \in \mathbf{R}^{n_i \times d}, \mathbf{E}_i \in \mathbf{R}^{m_i \times d}\}$  consists of node features  $\mathbf{N}_i$  and edge features  $\mathbf{E}_i$ . As in previous works, these features are initially extracted using a CNN-based backbone, followed by refinement through a GNN. Node affinity  $\mathbf{K}_n$  and Edge affinity  $\mathbf{K}_e$  are constructed by:
+
+$$
+\mathbf {K} _ {n} = \mathbf {N} _ {i} \mathbf {N} _ {u} ^ {\top}, \mathbf {K} _ {e} = \mathbf {E} _ {i} \mathbf {E} _ {u} ^ {\top}. \tag {2}
+$$
+
+The matching between the universe graph  $\mathbf{U}$  and the input graph  $\mathbf{G}_i$  is then obtained using a traditional graph matching solver, such as LPMP (Swoboda et al., 2017):
+
+$$
+\mathbf {X} _ {i u} = G M S o l v e r \left(\mathbf {K} _ {n}, \mathbf {K} _ {e}\right). \tag {3}
+$$
+
+In our training process, we use the ground truth universe matching  $\mathbf{X}_{iu}^{gt}$ , instead of ground truth pairwise matching  $\mathbf{X}_{ij}^{gt}$ , as the supervision signal.  $\mathbf{X}_{iu}^{gt} \in \{0,1\}^{n_i \times n_u}$  is a permutation matrix which denotes the ground truth correspondence between input graph  $\mathbf{G}_i$  and universe latent graph. In fact, the universe matching set  $\{\mathbf{X}_{iu}|1 \leq i \leq N\}$  and the pairwise matching set  $\{\mathbf{X}_{ij}|1 \leq i,j \leq N\}$  are fully equivalent and can be obtained via non-negative factorisation (Bernard et al., 2019):
+
+$$
+\mathcal {X} _ {p} = \left( \begin{array}{c c c c} \mathbf {X} _ {0 0} & \mathbf {X} _ {0 1} & \dots & \mathbf {X} _ {0 N} \\ \mathbf {X} _ {1 0} & \mathbf {X} _ {1 1} & \dots & \mathbf {X} _ {1 N} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf {X} _ {N 0} & \mathbf {X} _ {N 1} & \dots & \mathbf {X} _ {N N} \end{array} \right) = \left( \begin{array}{c} \mathbf {X} _ {0 u} \\ \mathbf {X} _ {1 u} \\ \vdots \\ \mathbf {X} _ {N u} \end{array} \right) \left(\mathbf {X} _ {0 u} ^ {\top} \quad \mathbf {X} _ {1 u} ^ {\top} \quad \dots \quad \mathbf {X} _ {N u} ^ {\top}\right) = \mathcal {X} _ {u} \mathcal {X} _ {u} ^ {T}. \tag {4}
+$$
+
+Even if some pairwise matchings are missing, we can still recover the universe matching by first applying spectral decomposition and then projecting the result back into the discrete space (Pachauri et al., 2013). It is worth noting that, in most computer vision tasks, the matching matrix is typically
+
+generated from keypoint annotations, and these annotations naturally serve as the ground truth universe matching.
+
+In practice, we cannot directly supervise  $\mathbf{X}_{iu}$  with  $\mathbf{X}_{iu}^{gt}$ , as most graph matching algorithms are non-differentiable. Even when using differentiable algorithms, such as RRWM (Cho et al., 2010), the high number of iterations would make gradient-based optimization difficult. Therefore, instead of supervising the solution, we directly act on the node and edge affinities. By treating the universe's nodes and edges indices as labels, we apply the cross-entropy loss to the node affinity  $\mathbf{K}_n$  and edge affinity  $\mathbf{K}_e$  for training:
+
+$$
+\mathcal {L} _ {\text {g r a p h}} = C E \left(\operatorname {s o f t m a x} \left(\mathbf {K} _ {n}\right), y _ {n}\right) + C E \left(\operatorname {s o f t m a x} \left(\mathbf {K} _ {e}\right), y _ {e}\right) \tag {5}
+$$
+
+where
+
+$$
+y _ {n} = \arg \max  \left(\mathbf {X} _ {i u} ^ {g t}\right), y _ {e} = y _ {n} (E d g e I D (0)) \times n _ {u} + y _ {n} (E d g e I D (1)). \tag {6}
+$$
+
+Note  $EdgeID \in \mathbb{N}^{2 \times m}$  denotes directed edges in the graph.  $EdgeID(0)$  denotes the indices of the source nodes and  $EdgeID(1)$  denotes the indices of the target nodes. Another important consideration in the matching process is the diversity of categories. The input may consist of pairs from different categories, each ideally requiring its own distinct latent universe graph. A straightforward approach would be to learn a separate graph representation for each category. However, this method would cause the latent universe graph to grow excessively as the number of categories increases. Consequently, the complexity of the graph matching solver could escalate quadratically or even worse, making it impractical for large-scale problems.
+
+To address this, we propose learning a single latent universe graph, where key points from different categories are mapped to the same universe node. For example, the left front wheel of a car and the left handlebar of a bicycle can share the same universe node embedding. In fact, a single universe node embedding is shared among all categories, allowing the graph's size to be limited by the maximum number of points across all categories. The same principle applies to edge embeddings, which further reduces space complexity.
+
+However, since the latent universe graph shares node and edge embeddings for all categories, the embeddings  $\mathbf{N}_u$  and  $\mathbf{E}_u$  inherently lack category-specific information. To reintroduce this information for affinity construction, we add a class embedding  $\mathbf{C}$  to each key point feature  $\mathbf{F}$  extracted by CNN-based backbone. This ensures that the nodes  $\mathbf{N}_i$  and edges  $\mathbf{E}_i$  of the input graph still carry the necessary class information. The class embedding is derived from the global features  $\mathbf{H}$  of the input images pair as follows:
+
+$$
+\mathbf {C} = \operatorname {E n c o d e r} \left(\mathbf {H} _ {i}, \mathbf {H} _ {j}\right), \mathbf {F} _ {i} = \operatorname {C o n c a t} \left(\mathbf {F} _ {i}, \mathbf {C}\right), \mathbf {F} _ {j} = \operatorname {C o n c a t} \left(\mathbf {F} _ {j}, \mathbf {C}\right). \tag {7}
+$$
+
+# 3.2 OUTLIER DETECTION
+
+Although the structured universe graph effectively handles point occlusion, it cannot address error annotation outliers, as these outliers do not have corresponding nodes in the universe graph. We frame the identification of these erroneous annotation outliers as an out-of-distribution (OOD) detection problem.
+
+Out-of-distribution detection typically refers to a model's ability to identify and appropriately handle data that deviates significantly from its training distribution. In the context of keypoint annotation, the correctly annotated keypoints adhere to a well-defined, coherent distribution that is consistent across all data. In contrast, erroneously annotated outliers are generated through random and unpredictable factors, causing them to deviate substantially from this established distribution. This inherent difference in data characteristics enables us to frame the outlier detection problem within an OOD detection framework, leveraging the disparity between the structured nature of valid annotations and the stochastic nature of annotation errors.
+
+Following Liu et al. (2020), we define an energy score function for each node  $v_{i}$  of input graph:
+
+$$
+\mathcal {E} \left(v _ {i}\right) = - T \cdot \ln \left(\sum_ {j} e ^ {K _ {n} (i, j) / T}\right), \tag {8}
+$$
+
+where  $K_{n}(i,j)$  denotes the node affinity between  $v_{i}$  on input graph and  $v_{j}$  on universe graph, and  $T$  is a temperature constant. When  $v_{i}$  is an inlier, it should have a corresponding point in the latent universe graph, resulting in a relatively low energy score. Conversely, when  $v_{i}$  is an outlier, all the values of  $K_{n}(i,j)$  will be relatively low, leading to a higher energy score.
+
+While the energy score can be used for a pre-trained neural network directly, the energy gap between in-distribution and out-of-distribution data may not always be optimal for differentiation. To address this, we also introduce a margin loss that fine-tunes the neural network to explicitly create a larger energy gap:
+
+$$
+\mathcal {L} _ {\text {e n e r g y}} = \mathbb {E} _ {v \sim \mathcal {D} _ {i n}} (\max  (0, \mathcal {E} (v) - m _ {i n})) + \mathbb {E} _ {v \sim \mathcal {D} _ {o u t}} (\max  (0, m _ {o u t} - \mathcal {E} (v))), \tag {9}
+$$
+
+where  $\mathcal{D}_{in}$  is the in-distribution training data,  $\mathcal{D}_{out}$  is the out-of-distribution training data,  $m_{in}$  and  $m_{out}$  are the margins for in- and out-of-distribution data, respectively. This loss assigns lower energy scores to in-distribution data and higher scores to OOD data. By contrastively shaping the energy landscape during training, the model becomes more effective at distinguishing between in- and out-of-distribution data.
+
+To apply the energy score in our network, we first compute the energy  $\mathcal{E}$  for each node in the input graph  $\mathbf{G}_i$  based on its affinity with nodes in the latent universe graph  $\mathbf{U}$ . Nodes with energy scores above a certain threshold  $\tau$ , indicating a high likelihood of being outliers, are removed before the graph matching process:
+
+$$
+\mathbf {G} ^ {\text {f i l t e r e d}} = \left\{\mathbf {V} = \left\{v _ {i} \mid \mathcal {E} \left(v _ {i}\right) \leq \tau \right\}, \mathbf {E} = \left\{e _ {i j} \mid \mathcal {E} \left(v _ {i}\right) \leq \tau , \mathcal {E} \left(v _ {j}\right) \leq \tau \right\} \right\} \tag {10}
+$$
+
+We use the newly obtained graph  $\mathbf{G}^{\mathrm{filtered}}$  to construct the filtered node affinity  $\mathbf{K}_n^{\mathrm{filtered}}$  and edge affinity  $\mathbf{K}_e^{\mathrm{filtered}}$ , and then feed it into the graph matching solver to obtain the matching result.
+
+$$
+\mathbf {X} _ {i u} ^ {\text {f i l t e r e d}} = G M S o l v e r \left(\mathbf {K} _ {n} ^ {\text {f i l t e r e d}}, \mathbf {K} _ {e} ^ {\text {f i l t e r e d}}\right). \tag {11}
+$$
+
+This preprocessing step improves the quality and accuracy of the subsequent matching between the input graph and the latent universe graph. After the matching process, the removed outlier points are reintroduced into the solution to maintain the integrity of the input data.
+
+$$
+\mathbf {X} _ {i u} \left(v _ {i}\right) = \left\{ \begin{array}{l l} \mathbf {0} ^ {\top} & \mathcal {E} \left(v _ {i}\right) > \tau , \\ \mathbf {X} _ {i u} ^ {\text {f i l t e r e d}} \left(v _ {i}\right) & \mathcal {E} \left(v _ {i}\right) \leq \tau . \end{array} \right. \tag {12}
+$$
+
+# 3.3 DISCUSSION
+
+Universe Graph Matching essentially decomposes the ambiguous partial matching problem into two well-defined subgraph matching problems. Partial matching, where only parts of the input graphs are aligned, is difficult to formalize with clear optimization objectives and constraints. Conventional approaches, such as adding dummy nodes or relaxing the condition  $\mathbf{X1}_n = \mathbf{1}_n$  to  $\mathbf{X1}_n \leq \mathbf{1}_n$ , have proven inadequate. Our key contribution lies in introducing latent universe graph, which enables us to decompose the partial matching problem between two graphs as  $\mathbf{X}_{ij} = \mathbf{X}_{iu}\mathbf{X}_{ju}^{\top}$ . This reformulation shifts the objective from maximizing the match  $\mathbf{X}_{ij}$  between the input graphs to maximizing the match  $\mathbf{X}_{iu}, \mathbf{X}_{ju}$  between each input graph and the latent universe graph. Since the latent universe graph contains all points and edges, the input graphs become its subgraphs, turning the solution of  $\mathbf{X}_{iu}$  into a well-defined subgraph matching problem. This more precise and structured problem definition in universe graph matching improves performance over traditional pairwise graph matching on partial matching tasks.
+
+However, the latent universe graph alone cannot fully address the error annotation points, which lack corresponding matches in the universe graph and violate the assumption that input graphs are subgraphs of the universe graph. While introducing dummy nodes in to the universe graph  $\mathbf{U}$  could potentially account for these outliers, this approach faces several challenges. Estimating the appropriate number of dummy nodes is difficult, and the randomness of error annotations makes it complicate to learning effective dummy node embeddings. Our solution adopts an OOD detection approach, treating error annotation outliers as out-of-distribution points and excluding them before solving the subgraph matching problem.
+
+Despite its advantages, our approach still faces several challenges: 1) Universe latent graph is inherently limited to closed-set problems, relying on class information and training data, which restricts generalization to unseen categories. However, few-shot learning can significantly alleviate this limitation, as shown in our experiments in Appendix ??, demonstrating the method's potential to adapt to new classes. 2) The universe graph is a complete graph; however, the distribution of edges in the input graphs is uneven. This mismatch leads to suboptimal learning of many edge embeddings, with some edges either absent or appearing only rarely in the training data. 3) While the universe graph is more stable and structured than universe point representation, the universe graph cannot fully prevent errors in  $\mathbf{X}_{iu}$  from propagating to all  $\mathbf{X}_{ij}$  matches. All in all, expanding graph partial matching to a more general form, both in terms of mathematical modeling and finer design, requires a long-term exploration.
+
+# 4 EXPERIMENT
+
+# 4.1 PROTOCOL
+
+Datasets We evaluate our method on Pascal VOC (Everingham et al., 2010) and Willow Object Class (Cho et al., 2013), two widely recognized datasets.
+
+The Pascal VOC (Everingham et al., 2010) with Berkeley annotations (Bourdev & Malik, 2009) contains images with bounding boxes surrounding objects of 20 classes. We follow the standard data preparation procedure of NGM and BBGM (Wang et al., 2021; Rolínek et al., 2020b). Each object is cropped to its bounding box and scaled to  $256 \times 256$  px. We follow the unfiltered setting (Rolínek et al., 2020b), which preserves all key points for graph pairs, to evaluate our method's performance under point occlusion. Moreover, we randomly sample the coordinates on images to add outliers, which serve as error annotation key points. We also follow prior works to split the train and test dataset, where training data includes 7,020 images and test data includes 1,682 images.
+
+The Willow Object Class (Cho et al., 2013) contains images from Caltech-256 (Griffin et al., 2007) and Pascal VOC 2007 (Everingham et al.), which consists of 256 images from 5 categories: 40 cars, 40 motorbikes, 50 ducks, 66 wine bottles, and 109 faces. Each image is annotated with the same ten key points. Following standard procedure, we also rescale the image to  $256 \times 256$  px. In experiments, we randomly drop 0 - 6 key points for each image for the 'occlusion' setting. We also add 1 - 10 random outliers as we do in PascalVOC for the 'random outlier' setting. We choose 20 images from each category as our training dataset and leave others for evaluation.
+
+Compared Method We compare our methods UGM with previous work of two kinds: traditional solver with pre-defined affinity and deep graph matching learning model. For the learning-free traditional solvers, we consider four works: GAGM (Gold & Rangarajan, 1996), RRWM (Cho et al., 2010), BPF (Wang et al., 2017), and ZAC (Wang et al., 2020). We generate node and edge affinities using the pre-trained BBGM learning framework and input these affinities into each solver to obtain their respective matching results. For learning-based models, we compare our method with NGMv2 (Wang et al., 2021), BBGM (Roliniak et al., 2020b), DLGM (Yu et al., 2021), GCAN (Jiang et al., 2022), AFAT (Wang et al., 2023), and URL (Lin et al., 2023). To report the performance of prior works, we adhere to the following principles: 1) If a prior work's experimental settings align with ours, we directly report the performance metrics provided in their paper. 2) If the experimental setting is unique to our work, we attempt to replicate the prior methods under their original experimental settings and then adapt them to our setting to obtain comparable performance metrics. However, we failed to replicate the methods of DLGM and URL because they did not release publicly available code. Therefore, we did not report their performance in some of the experiments (Table 2 and Table 3).
+
+Evaluation Metric We see the matching prediction as a binary classification task for each entry. Therefore, F1 score is applied as an evaluation metric, which is introduced by BBGM (Roliniak et al., 2020b). The F1 score is calculated as follows:
+
+$$
+\mathbf {F} _ {1} = 2 \cdot \frac {\text {P r e c i s i o n} \cdot \text {R e c a l l}}{\text {P r e c i s i o n} + \text {R e c a l l}} = \frac {2 \mathrm {T P}}{2 \mathrm {T P} + \mathrm {F P} + \mathrm {F N}} \tag {13}
+$$
+
+where TP denotes true positive, FP denotes false positive, and FN denotes false negative.
+
+# 4.2 PERFORMANCE ON PASCAL VOC
+
+To thoroughly evaluate the performance of our proposed UGM method, we conducted experiments on the Pascal VOC dataset under two distinct settings: the unfiltered setting, where all key points are preserved without the introduction of additional noise, and the random outlier setting, where two random outliers are further added to each input image. All the results are shown in Table 1 and 2.
+
+Our proposed UGM method demonstrates superior performance in both setting, surpassing the state-of-the-art (SOTA) models by a significant margin. Specifically, in the unfiltered setting, UGM outperforms the best-performing SOTA model by  $2.2\%$  in terms of average F1 score. Similarly, in the random outlier setting, UGM still leads the comparison, exceeding the next best method by  $4.8\%$ . This consistent improvement across both settings highlights the robustness and generalization ability of our approach.
+
+A comparison of the two tables reveals how much performance degrades between the unfiltered and random outlier settings, allowing us to evaluate the robustness of learning-based models in the presence of outliers. Our UGM method experiences a performance drop of  $8.8\%$ , ranking second
+
+Table 1: F1 Score performance on Pascal VOC under 'unfiltered' setting. All the key points are preserved, and no additional outliers are added. Top: learning-free graph matching solver with pre-learned affinity. Middle: deep learning graph matching model. Bottom: out method UGM.  
+
+<table><tr><td>Method</td><td>aero</td><td>bike</td><td>bird</td><td>boat</td><td>bottle</td><td>bus</td><td>car</td><td>cat</td><td>chair</td><td>cow</td><td>table</td><td>dog</td><td>horse</td><td>motor</td><td>person</td><td>plant</td><td>sheep</td><td>sofa</td><td>train</td><td>tv</td><td>avg</td></tr><tr><td>GAGM</td><td>37.3</td><td>60.3</td><td>47.7</td><td>39.1</td><td>75.0</td><td>53.7</td><td>36.1</td><td>59.2</td><td>34.8</td><td>56.2</td><td>40.2</td><td>54.4</td><td>51.9</td><td>54.3</td><td>39</td><td>83.8</td><td>48.8</td><td>19</td><td>58.3</td><td>68.7</td><td>50.9</td></tr><tr><td>RRWM</td><td>39.4</td><td>63.7</td><td>52.1</td><td>39.7</td><td>76.5</td><td>57.9</td><td>28.5</td><td>64.3</td><td>37.9</td><td>60.3</td><td>43</td><td>57.9</td><td>54.2</td><td>54</td><td>43.3</td><td>85</td><td>50.4</td><td>22</td><td>66.1</td><td>69.9</td><td>53.3</td></tr><tr><td>BPF</td><td>39.2</td><td>66.5</td><td>51.2</td><td>40.0</td><td>77.0</td><td>59.0</td><td>26.7</td><td>63.8</td><td>36.9</td><td>60.6</td><td>49.7</td><td>58.1</td><td>54.3</td><td>58.2</td><td>43.6</td><td>84.9</td><td>50.7</td><td>23</td><td>66.4</td><td>70.4</td><td>54.0</td></tr><tr><td>ZAC</td><td>43.5</td><td>62.8</td><td>57.5</td><td>43.9</td><td>72.1</td><td>61.2</td><td>33.6</td><td>69</td><td>38.3</td><td>64.1</td><td>46.4</td><td>66.2</td><td>61.8</td><td>61.4</td><td>46.4</td><td>83.2</td><td>58.1</td><td>27.9</td><td>69.8</td><td>72.6</td><td>57.0</td></tr><tr><td>NGMv2</td><td>45.5</td><td>65.3</td><td>55.3</td><td>45.8</td><td>88.4</td><td>64.3</td><td>45.9</td><td>58.6</td><td>43.3</td><td>59.1</td><td>39.2</td><td>55.7</td><td>58</td><td>65.3</td><td>44.4</td><td>95.4</td><td>50.3</td><td>41.2</td><td>72.4</td><td>81.8</td><td>58.8</td></tr><tr><td>BBGM</td><td>42.7</td><td>70.9</td><td>57.5</td><td>46.6</td><td>85.8</td><td>64.1</td><td>51</td><td>63.8</td><td>42.4</td><td>63.7</td><td>47.9</td><td>61.5</td><td>63.4</td><td>69</td><td>46.1</td><td>94.2</td><td>57.4</td><td>39</td><td>78</td><td>82.7</td><td>61.4</td></tr><tr><td>DLGM</td><td>43.8</td><td>72.9</td><td>58.5</td><td>47.4</td><td>86.4</td><td>71.2</td><td>53.1</td><td>66.9</td><td>54.6</td><td>67.8</td><td>64.9</td><td>65.7</td><td>66.9</td><td>70.8</td><td>47.4</td><td>96.5</td><td>61.4</td><td>48.4</td><td>77.5</td><td>83.9</td><td>64.8</td></tr><tr><td>GCAN</td><td>45</td><td>66.7</td><td>60.6</td><td>49.7</td><td>89.7</td><td>66.3</td><td>65.2</td><td>64.9</td><td>45.5</td><td>66.9</td><td>54.4</td><td>63.1</td><td>62.5</td><td>63.5</td><td>55</td><td>96.1</td><td>63.5</td><td>49.7</td><td>80.6</td><td>83.6</td><td>64.6</td></tr><tr><td>AFAT</td><td>47.1</td><td>70.8</td><td>58.1</td><td>45.8</td><td>90.8</td><td>66.5</td><td>49.6</td><td>58.8</td><td>50.6</td><td>64.6</td><td>47.2</td><td>60.5</td><td>62.3</td><td>65.7</td><td>46.3</td><td>95.4</td><td>52.7</td><td>47.4</td><td>74.2</td><td>83.8</td><td>62.0</td></tr><tr><td>URL</td><td>62.7</td><td>75.2</td><td>73</td><td>56.7</td><td>93.7</td><td>66.2</td><td>76.7</td><td>69.2</td><td>64.9</td><td>76.6</td><td>44.6</td><td>74.4</td><td>78.8</td><td>80.9</td><td>62.5</td><td>96.9</td><td>70.3</td><td>55.4</td><td>73.6</td><td>82.1</td><td>71.7</td></tr><tr><td>UGM</td><td>57.4</td><td>76.3</td><td>71.2</td><td>55.1</td><td>91.7</td><td>72.3</td><td>80.3</td><td>72.3</td><td>70.1</td><td>74.6</td><td>61.5</td><td>75.3</td><td>71.6</td><td>76.1</td><td>63.8</td><td>97.2</td><td>73.6</td><td>60.0</td><td>93.1</td><td>84.8</td><td>73.9</td></tr></table>
+
+Table 2: F1 Score performance on Pascal VOC under 'random outlier' setting. Two random outliers are further added to each input image. Top: learning-free graph matching solver with pre-learned affinity. Middle: deep learning graph matching model. Bottom: out method UGM.  
+
+<table><tr><td>Method</td><td>aero</td><td>bike</td><td>bird</td><td>boat</td><td>bottle</td><td>bus</td><td>car</td><td>cat</td><td>chair</td><td>cow</td><td>table</td><td>dog</td><td>horse</td><td>motor</td><td>person</td><td>plant</td><td>sheep</td><td>sofa</td><td>train</td><td>tv</td><td>avg</td></tr><tr><td>GAGM</td><td>26.7</td><td>49.3</td><td>36.6</td><td>25.1</td><td>61.9</td><td>40.7</td><td>9.0</td><td>45.4</td><td>22.5</td><td>39.3</td><td>13.1</td><td>41.0</td><td>37.8</td><td>39.1</td><td>27.9</td><td>59.8</td><td>29.1</td><td>19.7</td><td>26.0</td><td>54.9</td><td>35.2</td></tr><tr><td>RRWM</td><td>28.3</td><td>54.6</td><td>40.3</td><td>27.4</td><td>70.6</td><td>46.3</td><td>9.0</td><td>49.8</td><td>23.5</td><td>42.8</td><td>15.1</td><td>44.8</td><td>38.1</td><td>39.1</td><td>30.3</td><td>66.8</td><td>29.3</td><td>18.1</td><td>27.1</td><td>64</td><td>38.3</td></tr><tr><td>BPF</td><td>25.8</td><td>55.2</td><td>38.8</td><td>26.0</td><td>70.7</td><td>45.7</td><td>9.5</td><td>49.4</td><td>22.9</td><td>40.2</td><td>14.4</td><td>44.1</td><td>36.9</td><td>40.0</td><td>29.0</td><td>68.6</td><td>28.6</td><td>18.8</td><td>28.1</td><td>63.8</td><td>37.8</td></tr><tr><td>ZAC</td><td>37.2</td><td>62.3</td><td>50.8</td><td>34.0</td><td>76.0</td><td>55.9</td><td>23.8</td><td>63.7</td><td>33.2</td><td>58.8</td><td>37.7</td><td>59.4</td><td>55.8</td><td>55.4</td><td>39.1</td><td>76.8</td><td>49.3</td><td>29.0</td><td>49.1</td><td>68.9</td><td>50.8</td></tr><tr><td>NGMv2</td><td>35.6</td><td>60.2</td><td>45.0</td><td>34.3</td><td>72.1</td><td>51.3</td><td>40.2</td><td>50.2</td><td>29.5</td><td>49.4</td><td>23.7</td><td>47.6</td><td>49.8</td><td>51.5</td><td>38.7</td><td>69.3</td><td>42.9</td><td>40.8</td><td>45.0</td><td>62.2</td><td>47.0</td></tr><tr><td>BBGM</td><td>34.4</td><td>63.9</td><td>44.1</td><td>35.0</td><td>78.5</td><td>57.6</td><td>21.5</td><td>54.6</td><td>33.7</td><td>50.3</td><td>37.4</td><td>49.8</td><td>50.8</td><td>54.7</td><td>29.6</td><td>79.8</td><td>44.1</td><td>23.5</td><td>44.5</td><td>76.1</td><td>48.2</td></tr><tr><td>GCAN</td><td>44.4</td><td>61.7</td><td>51.2</td><td>47.5</td><td>78.8</td><td>72.3</td><td>48.9</td><td>58.0</td><td>50.2</td><td>62.8</td><td>53.0</td><td>54.6</td><td>60.3</td><td>61.2</td><td>51.5</td><td>83.2</td><td>62.6</td><td>48.8</td><td>70.0</td><td>84.8</td><td>60.3</td></tr><tr><td>AFAT</td><td>41.2</td><td>66.9</td><td>51.4</td><td>36.5</td><td>82.7</td><td>61.8</td><td>37.9</td><td>56.9</td><td>34.8</td><td>55.7</td><td>28.0</td><td>54.4</td><td>55.3</td><td>60.3</td><td>38.2</td><td>86.3</td><td>48.8</td><td>32.6</td><td>57.1</td><td>77.1</td><td>53.1</td></tr><tr><td>UGM</td><td>49.9</td><td>72.2</td><td>63.5</td><td>48.8</td><td>85.3</td><td>70.1</td><td>71.8</td><td>66.5</td><td>54.6</td><td>67.4</td><td>49.5</td><td>67.4</td><td>65.2</td><td>66.2</td><td>57.6</td><td>88.0</td><td>62.2</td><td>47.2</td><td>65.6</td><td>83.4</td><td>65.1</td></tr></table>
+
+among the learning-based models, just behind GCAN. In comparison, models like NGMv2 and BBGM exhibit larger performance declines of  $11.8\%$  and  $13.2\%$ , respectively. The relatively low decrease in UGM's performance demonstrates its robustness and adaptability, particularly in noisy and complex environments, making it a reliable option for real-world graph matching tasks.
+
+We also perform an random outlier pressure test on our method UGM. Results are shown in Figure 3. Classes like sofa, train, and table see a significant performance drop (32% - 61%) as the number of random outliers increases. This decline is primarily due to the limited amount of training data, particularly for classes like 'table', which has only 28 images, and 'sofa', which has 73 images in the train dataset. The fewer images available, the lower the quality of the constructed latent universe graph, making it more difficult to distinguish between inliers and error annotation outliers. On the other hand, classes like TV, boat, horse, and bus are more robust, with relatively smaller drops in performance (9.9% - 13.9%).
+
+# 4.3 PERFORMANCE ON WILLOW OBJECT
+
+In this experiment, we evaluate the performance of various learning-based graph matching models on the Willow dataset under three settings: occlusion, random outlier, and occlusion + random outlier. These settings simulate real-world challenges, where key points may be missing (occlusion), or noisy data (outliers) may be introduced. All the results are shown in Table 3.
+
+Our proposed UGM method demonstrates superior performance across all three settings. In the occlusion setting, UGM achieves the highest average F1 score of  $91.9\%$ , outperforming the next best model, GCAN, by a significant margin of  $9.7\%$ . In the random outlier setting, UGM again leads with an average score of  $85.4\%$ , surpassing AFAT, the second-best model, by  $1.7\%$ . Finally, in the occlusion + random outlier setting, UGM shows exceptional robustness, maintaining the highest average score of  $71.1\%$ , which is  $5.9\%$  higher than the second-place model.
+
+It is worth noting that the extent of our lead varies across the three settings.
+
+- In the occlusion setting, UGM demonstrates a significant advantage. This suggests that the latent universe graph structure offers a considerable benefit compared to pairwise matching approaches,
+
+![](images/887bc5aae116965dfe94650f2c4fb04c188e70fa3d50342050fa357ad089e018.jpg)  
+Figure 3: Random outlier pressure test of UGM on Pascal VOC. All the key points are preserved, and additional outliers are added to each input image.
+
+Table 3: F1 Score performance on Willow Object Class. Occlusion: randomly drop 0-6 key points for each input image. Random Outlier: randomly add 1-10 outliers for each input image. Occlusion + Random Outlier: both operations are applied. Only the learning model's performance is reported.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="6">Occlusion</td><td colspan="6">Random Outlier</td><td colspan="6">Occlusion + Random Outlier</td></tr><tr><td>car</td><td>duck</td><td>face</td><td>motor</td><td>bottle</td><td>avg</td><td>car</td><td>duck</td><td>face</td><td>motor</td><td>bottle</td><td>avg</td><td>car</td><td>duck</td><td>face</td><td>motor</td><td>bottle</td><td>avg</td></tr><tr><td>NGMv2</td><td>80.9</td><td>73.8</td><td>84.9</td><td>85.7</td><td>76.7</td><td>80.4</td><td>78.9</td><td>66.6</td><td>84.3</td><td>63.1</td><td>76</td><td>73.8</td><td>46.3</td><td>45.2</td><td>61.6</td><td>42.9</td><td>54.6</td><td>50.1</td></tr><tr><td>BBGM</td><td>77.5</td><td>78.8</td><td>92.7</td><td>83.1</td><td>82.3</td><td>82.9</td><td>65.1</td><td>60.7</td><td>85.5</td><td>71.6</td><td>65.5</td><td>69.7</td><td>36.2</td><td>35.5</td><td>57.2</td><td>35.0</td><td>52.4</td><td>43.3</td></tr><tr><td>GCAN</td><td>81.4</td><td>74.4</td><td>88.5</td><td>88.3</td><td>78.2</td><td>82.2</td><td>74.8</td><td>75.7</td><td>92.8</td><td>77.1</td><td>83.5</td><td>80.8</td><td>43.7</td><td>51.7</td><td>60.2</td><td>45.8</td><td>56.9</td><td>51.7</td></tr><tr><td>AFAT</td><td>77.1</td><td>73.7</td><td>88.0</td><td>76.6</td><td>86.4</td><td>80.4</td><td>82.2</td><td>77.7</td><td>92.7</td><td>77.2</td><td>88.6</td><td>83.7</td><td>70.8</td><td>54.7</td><td>79.1</td><td>53.8</td><td>67.9</td><td>65.2</td></tr><tr><td>UGM</td><td>87.7</td><td>86.2</td><td>99.5</td><td>90.2</td><td>96.0</td><td>91.9</td><td>88.1</td><td>72.9</td><td>96.8</td><td>82.5</td><td>86.5</td><td>85.4</td><td>64.4</td><td>62.1</td><td>89.6</td><td>65.1</td><td>74.5</td><td>71.1</td></tr></table>
+
+as Universe Graph Matching essentially decomposes the ambiguous pairwise partial matching problem into two well-defined subgraph matching problems, thereby improving its ability to handle missing key points.
+
+- In the random outlier setting, the performance gap between UGM and other models is narrower. While our OOD (out-of-distribution) detection technique offers a clear advantage, the improvement is less pronounced. This is likely due to the fact that our approach does not introduce additional training parameters, which limits the potential for further gains.  
+- Finally, in the occlusion + random outlier setting, UGM once again achieves a clear lead. This demonstrates the strong synergy between the latent universe graph learning and our energy based OOD detection technique. The two techniques work in harmony, with no conflicting or contradictory effects, allowing UGM to excel in the most challenging conditions.
+
+# 4.4 ABLATION STUDIES
+
+The ablation study results presented in Table 4 highlight the contribution of each key component—class embedding, universe graph, and outlier filter—to the overall performance. Without the outlier filter component, we matched all the points in the input graph to the latent universe graph. Without edge learning, we only learned the universe node representation and used a linear assignment solver to obtain the matching results between the input graph and the universe point representation. Without class embedding, we directly used the local key point features extracted by the backbone to learn the node embeddings.
+
+Table 4: Ablation study on Pascal VOC with 'random outlier' setting. All the key points are preserved, and two random outliers are added to each input image. F1 score is reported.  
+
+<table><tr><td>class embedding</td><td>edge learning</td><td>outlier filter</td><td>F1 score</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>65.12</td></tr><tr><td>✓</td><td>✓</td><td>x</td><td>56.88</td></tr><tr><td>✓</td><td>x</td><td>✓</td><td>63.49</td></tr><tr><td>x</td><td>✓</td><td>✓</td><td>58.71</td></tr></table>
+
+These results demonstrate that each component plays a unique role in enhancing the model's performance. Specifically, removing the outlier filter has the most significant negative impact, underscoring its importance in handling noise and maintaining robust graph matching.
+
+Furthermore, we also conduct a sensitivity test for hyperparameters on Pascal VOC under the 'random outlier' setting, as shown in Figure 4. In this sensitivity analysis, we investigate the effects of varying three hyperparameters—temperature  $T$ , threshold  $\tau$ , and the pair of values for  $m_{in}$  and  $m_{out}$ . Unless
+
+![](images/39c27db9dff28639018f7396e35835ccbd820142c0c9a8300a2ac06212d3ae44.jpg)  
+Figure 4: Sensitivity test for hyperparameters on Pascal VOC under 'random outlier' setting. Temperature  $T$  in energy function, OOD threshold  $\tau$ , and  $m_{in}, m_{out}$  in energy loss are tested.
+
+![](images/5f8d7c3858b646310c1a32346bdcdf45a8d28af9372c07193078ba1b0a529f8d.jpg)
+
+![](images/4ce9a751cee315f835ceb36ecff2643ebde62ae1ac9de63284ee68c375e387d0.jpg)
+
+![](images/f54e086d47df54d4e23df8b3811337135eac7d5266511d851894a76c6c53aa4e.jpg)
+
+![](images/50bcdd39d39e7a542e90a200f60ce60682413fe693db28eaf18937aba694ee84.jpg)  
+Figure 5: Visualization on Pascal VOC of UGM and peer methods. The green dashed lines represent correct matches, while the red dashed lines represent incorrect matches. Ground truth matching is shown at the bottom.
+
+specified otherwise, the default parameter values are set to  $T = 1.0$ ,  $m_{in} = -3$ ,  $m_{out} = -6$ , and  $\tau = (m_{in} + m_{out}) / 2$ .
+
+For the temperature experiment, we tested values ranging from 0.8 to 10. The results show that performance peaks at temp=1.0 with an average value of 65.1, and gradually decreases as the temperature increases. The threshold experiment explores a range from -1.5 to -6. The average performance is highest at  $\tau = -2.5$  with a value of 66.37, and the performance decreases consistently as the threshold becomes more negative. Although adjusting  $\tau$  can achieve a higher F1 score, we are concerned that this might lead to some overfitting in the model. Therefore, we still report the model results using the default value  $\tau = (m_{in} + m_{out}) / 2$  in other contexts. For the  $m_{\mathrm{in}}$  and  $m_{\mathrm{out}}$  experiments, we analyze both their margin (difference) and absolute offset. Performance peaks at (-3, -6), where the margin is 3, but the overall impact of varying the range (from 1 to 7) or shifting their absolute values is minimal.
+
+# 5 CONCLUSION
+
+In summary, this work presents a novel approach to addressing the challenges of partial matching in graph-based keypoint alignment tasks. We introduce Structured Universe Graph Learning to effectively resolve point occlusion by connecting input pairs via a learned latent graph. Additionally, by incorporating both node features and structural information, the robustness of the matching process is further enhanced. We also adopt Energy-based Out-of-Distribution Detection to filter out annotation errors before matching, improving the overall quality of the matching process. Through extensive evaluations of the Pascal VOC and Willow Object datasets, our method consistently outperforms state-of-the-art techniques, particularly in challenging scenarios involving both point occlusion and random outliers, demonstrating the effectiveness and robustness of our approach.
+
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+
+# APPENDIX
+
+# A NOTATION
+
+We first present all notations used in this paper for a better understanding of proposed algorithms and to facilitate the following discussion.
+
+Table 5: Main notations and description used in this paper.  
+
+<table><tr><td>Notations</td><td>Descriptions</td></tr><tr><td>N</td><td>Node embedding or node feature on the graph.</td></tr><tr><td>E</td><td>Edge embedding or node feature on the graph.</td></tr><tr><td>U</td><td>U = {Nu ∈ Rnun×d, Eu ∈ Rmu×d} denotes the latent universe graph, where nu and mu represents the node and edge number of the latent universe graph.</td></tr><tr><td>Gi</td><td>Gi = {Ni ∈ Rni×d, Ei ∈ Rmi×d} denotes the input graph, where ni and mi represents the node and edge number of the input graph.</td></tr><tr><td>EdgeID</td><td>EdgeID ∈ N2×m denotes directed edges in the graph. EdgeID(0) denote the indices of the source nodes and EdgeID(1) denote the indices of the target nodes.</td></tr><tr><td>Kn</td><td>Kn ∈ Rni×nu denotes the node affinity between the input graph and the universe graph.</td></tr><tr><td>Ke</td><td>Ke ∈ Rmi×mu denotes the edge affinity between the input graph and the universe graph.</td></tr><tr><td>Xij</td><td>Xij ∈ {0,1}ni×nu denotes the matching between the input graphs.</td></tr><tr><td>Xiu</td><td>Xi ∈ {0,1}ni×nu denotes the matching between the input graph and the universe graph.</td></tr><tr><td>C</td><td>Class embedding for image pair.</td></tr><tr><td>Hi</td><td>Global feature extracted by backbone for the input image.</td></tr><tr><td>Fi</td><td>Key point feature extracted by backbone for the input image.</td></tr><tr><td>E(vi)</td><td>Energy score for node vi.</td></tr><tr><td>τ</td><td>Energy score threshold for outlier filter.</td></tr><tr><td>Din, Dout</td><td>Din, Dout denotes in-distribution data and out-of-distribution data respectively.</td></tr></table>
+
+# B IMPLEMENTATION DETAIL
+
+We utilize the standard feature extractor pipeline with a few modifications.
+
+- Fisrt of all, we replace the VGG16 (Simonyan & Zisserman, 2014) backbone with ResNet50 (He et al., 2016). We compute the outputs of  $relu3\_5$ ,  $relu4\_1$  of the ResNet50 network pre-trained on ImageNet (Krizhevsky et al., 2012), to obtain feature  $\mathbf{F}_1$  and  $\mathbf{F}_2$ , respectively. These features are then concatenated to create the key point feature  $\mathbf{F}$ :
+
+$$
+\mathbf {F} = \text {C O N C A T} \left(\mathbf {F} _ {1}, \mathbf {F} _ {2}\right) \tag {14}
+$$
+
+Class embedding is added to  $\mathbf{F}$  as introduced in Equation 7, and we apply an MLP layer to reduce the feature dim to keep the same dimension with the output of the VGG16-based backbone.
+
+- Then we feed the obtained feature  $\mathbf{F}$  and the graph adjacency  $\mathcal{A}$  into the geometric feature refinement component. The graph adjacency  $\mathcal{A}$  is generated using Delaunay triangulation (Delaunay et al., 1934) based on keypoint locations. We apply SplineConv (Fey et al., 2018) to encode higher-order information and the geometric structure of the entire graph into node-wise features  $\mathbf{F}^n$ :
+
+$$
+\mathbf {N}, \mathbf {E} = S p l i n e C o n v (\mathbf {F}, \mathcal {A}) \tag {15}
+$$
+
+The Spline Conv operation is calculated as follows:
+
+$$
+\mathbf {F} _ {i} = \frac {1}{| \mathcal {N} (i) |} \sum_ {j \in \mathcal {N} (i)} \mathbf {F} _ {j} \cdot h _ {\Theta} \left(\mathbf {F} _ {i} - \mathbf {F} _ {j}\right) \tag {16}
+$$
+
+where  $\mathbf{F}_i$  represents the node feature of  $v_i$ ,  $\mathcal{N}(i)$  denotes the neighborhood of  $v_i$ , and  $h_\Theta$  denotes a kernel function defined over the weighted B-Spline tensor product basis.
+
+Algorithm 1 Universe Graph Matching  
+Require: Input images  $I$  input key points  $P$  , learnable parameters  $\theta$  , learning rate  $\eta$  , epoch number  $E$  , margins  $m_{in},m_{out}$  , Temperature  $T$  , OOD threshold  $\tau$    
+Ensure: Trained parameters  $\theta^{*}$  and universe latent graph  $\mathbf{U}^*$    
+1: Initialize  $\theta$  randomly   
+2: Initialize node and edge embedding of universe latent graph U randomly   
+3: for  $e = 1$  to  $E$  do   
+4: for image pair  $I_{i},I_{j}$  in train/test dataset do   
+5: # extract features   
+6: Extract key point feature F by CNN backbone via Eq. 14   
+7: Add class embedding to F via Eq. 7   
+8: Refine F with Spline Conv to obtain node N and edge feature E via Eq. 15 and Eq. 16   
+9:   
+10: # build affinity   
+11: Construct node affinity  $\mathbf{K}_n$  and edge affinity  $\mathbf{K}_e$  via Eq. 2   
+12: Calculate energy  $\mathcal{E}$  for each node via Eq. 8   
+13: Filter out random outlier with OOD threshold  $\tau$  via Eq. 10   
+14:   
+15: if training then   
+16: # loss and update   
+17: Calculate permutation loss with  $\mathbf{K}_n^{\mathrm{filtered}}$  and  $\mathbf{K}_e^{\mathrm{filtered}}$  for both  $I_{i}$  and  $I_{j}$    
+18: Calculate energy loss with  $\mathcal{E}$ $m_{in}$  ,and  $m_{out}$  via Eq. 9 for both  $I_{i}$  and  $I_{j}$    
+19: Final loss  $\mathcal{L} = \mathcal{L}_{\mathrm{permutation}} + \mathcal{L}_{\mathrm{energy}}$  , and compute gradient  $\nabla_{\theta}\mathcal{L},\nabla_{\mathbf{U}}\mathcal{L}$    
+20: Update parameters:  $\theta \leftarrow \theta -\eta \nabla_{\theta}\mathcal{L}$    
+21: Update universe embedding:  $\mathbf{U}\gets \mathbf{U} - \eta \nabla_{\mathbf{U}}\mathcal{L}$    
+22: else   
+23: # build pairwise matching   
+24: Use LPMP solver to obtain universe matching  $\mathbf{X}_{iu},\mathbf{X}_{ju}$  via Eq. 11 and Eq. 12   
+25: Build pairwise matching by  $\mathbf{X} = \mathbf{X}_{iu}\mathbf{X}_{ju}^{\top}$    
+26: end if   
+27: end for   
+28: end for   
+29: return  $\theta^{*}$  , U\*
+
+Node and edge affinity construction, universe matching, and outlier filter process have been introduced in Section 3.1 and 3.2. We finally build pairwise matching by  $\mathbf{X}_{ij} = \mathbf{X}_{iu}\mathbf{X}_{ju}^{\top}$ .
+
+All the models are trained on on a Linux workstation with i9-10920X CPU@3.50GHz CPU, one RTX3090, and 128GB RAM. The training and inference algorithm is shown in Algorithm 1. In default, we train our UGM with hyper parameters  $m_{in} = -6$ ,  $m_{out} = -3$ ,  $T = 1.0$ ,  $\tau = -4.5$ ,  $\eta = 1e - 3$ , and  $E = 15$ .
+
+# C BACKBONE COMPARISON
+
+To systematically evaluate the impact of network architecture on performance, we conducted comprehensive experiments comparing VGG and ResNet as backbone models across all methods. This analysis ensures fair comparison by providing results using consistent architectures while also quantifying how architectural choices influence each method's effectiveness.
+
+As demonstrated in Table 6, our method consistently outperforms existing approaches across multiple feature extractor configurations, including VGG+Spline2D, VGG+Attention, and ResNet+Spline2D. The sole exception occurs with VGG+Spline2D+3D, where URL achieves better results through its utilization of spline3D features. However, due to the unavailability of URL's implementation, we were unable to integrate this feature extractor into our UGM model. To facilitate a fair comparison, we adopted ResNet50 as the common backbone architecture.
+
+While evaluating architectural impacts, we observed that not all baseline methods exhibited performance improvements when transitioning to ResNet. To maintain scientific rigor and transparency, we have included these results in this appendix, allowing for a more proper discussion of architectural sensitivities across different approaches.
+
+Table 6: Backbone fairness comparison on Pascal VOC with unfiltered setting. The performance with the VGG backbone is reported by their paper, and the performance with the ResNet backbone is reproduced via publicly available code.  $-$  denotes that the method has not released its code.  
+
+<table><tr><td>Method</td><td>Backbone</td><td>GNN</td><td>F1 score</td><td>Backbone</td><td>GNN</td><td>F1 score</td></tr><tr><td>NGMv2</td><td>VGG</td><td>Spline2D</td><td>58.8</td><td>ResNet</td><td>Spline2D</td><td>57.46</td></tr><tr><td>BBGM</td><td>VGG</td><td>Spline2D</td><td>61.4</td><td>ResNet</td><td>Spline2D</td><td>64.45</td></tr><tr><td>DLGM</td><td>VGG</td><td>Spline2D</td><td>64.8</td><td>ResNet</td><td>Spline2D</td><td>-</td></tr><tr><td>GCAN</td><td>VGG</td><td>Attention</td><td>64.6</td><td>ResNet</td><td>Attention</td><td>64.67</td></tr><tr><td>AFAT</td><td>VGG</td><td>Attention</td><td>62</td><td>ResNet</td><td>Attention</td><td>59.55</td></tr><tr><td>URL</td><td>VGG</td><td>Spline2D</td><td>67.6</td><td>ResNet</td><td>Spline2D</td><td>-</td></tr><tr><td>URL</td><td>VGG</td><td>Spline2D+3D</td><td>71.7</td><td>ResNet</td><td>Spline2D+3D</td><td>-</td></tr><tr><td>UGM</td><td>VGG</td><td>Spline2D</td><td>70.5</td><td>ResNet</td><td>Spline2D</td><td>73.9</td></tr></table>
+
+# D EFFECT OF ENERGY MARGIN LOSS
+
+![](images/7744428b666dbf78cc0a51c248d85856f8b85e373c644a03a153604727902829.jpg)  
+Figure 6: Left: Energy histogram of ID and OOD samples without energy loss finetuning, Right: Energy histogram of ID and OOD samples with energy loss finetuning
+
+![](images/d0e11d3a7b41fd5cb87d98ff044c6cb636864f3dee88a10197861036defcb341.jpg)
+
+The effect of energy margin loss is shown in Figure 6. Left shows the energy distribution of indistribution (ID) and out-of-distribution (OOD) samples before applying the Energy Margin Loss, where the energy values for both ID and OOD samples overlap significantly, making it difficult to distinguish between them. After finetuning with the Energy Margin Loss, the energy values for ID samples shift toward lower energy levels, while OOD samples are pushed toward higher energy levels. This separation of energy distributions effectively enhances the model's ability to distinguish between ID and OOD samples, demonstrating the efficacy of the Energy Margin Loss in improving model robustness.
\ No newline at end of file
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+# LEARNING SUCCESSOR FEATURES WITH DISTRIBUTED HEBBIAN TEMPORAL MEMORY
+
+Evgenii Dzhivelikian
+
+MIPT, AIRI
+
+Moscow, Russia
+
+dzhivelikian.ea@phystech.edu
+
+Petr Kuderov
+
+AIRI, MIPT
+
+Moscow, Russia
+
+kuderov@airi.net
+
+Aleksandr Panov
+
+AIRI, MIPT
+
+Moscow, Russia
+
+panov@airi.net
+
+# ABSTRACT
+
+This paper presents a novel approach to address the challenge of online sequence learning for decision making under uncertainty in non-stationary, partially observable environments. The proposed algorithm, Distributed Hebbian Temporal Memory (DHTM), is based on the factor graph formalism and a multi-component neuron model. DHTM aims to capture sequential data relationships and make cumulative predictions about future observations, forming Successor Features (SFs). Inspired by neurophysiological models of the neocortex, the algorithm uses distributed representations, sparse transition matrices, and local Hebbian-like learning rules to overcome the instability and slow learning of traditional temporal memory algorithms such as RNN and HMM. Experimental results show that DHTM outperforms LSTM, RWKV and a biologically inspired HMM-like algorithm, CSCG, on non-stationary data sets. Our results suggest that DHTM is a promising approach to address the challenges of online sequence learning and planning in dynamic environments.
+
+# 1 INTRODUCTION
+
+Modelling sequential data is one of the essential tasks in artificial intelligence as it has many applications, including decision-making and learning world models (Moerland et al., 2023), natural language processing (Min et al., 2021), conversational AI (Dwivedi et al., 2023), time series analysis (Eraslan et al., 2019), and video and music generation (Ji et al., 2020). One of the classical approaches to modelling sequential data is to form a representation that stores and condenses the most relevant information about a sequence, and to find a general transformation rule of this information through the dimension of time (Lipton et al., 2015; Harshvardhan et al., 2020; Mathys et al., 2011). We refer to the class of algorithms that use this approach as temporal memory (TM) algorithms because they essentially model the cognitive ability of complex living organisms to remember the past experience and make future predictions based on that memory (Hochreiter & Schmidhuber, 1997; Friston et al., 2016; 2018; Parr & Friston, 2017).
+
+Temporal memories differ in their ability to capture data dependencies and the extent to which they can generalise their predictions to unseen sequences. There are two extremes known in psychology as cognitive maps Tolman (1948), which extract commonalities or schemas generalising experience, and episodic memory Tulving et al. (1972), which stores fragments of experience separately. They both can be viewed as temporal memories, however, the former is slower and require a lot of data to built up a useful schema, according to contemporary computational models based on HMM (George et al., 2021) and Transformers (Dedieu et al., 2024), while the latter is instant and is usually viewed as a data buffer (Zhang et al., 2021).
+
+In this paper, we explore advantages of an episodic-like memory in addressing the problem of online sequence learning for planning and decision-making under uncertainty, which can be formalized as reinforcement learning (RL) for a partially observable Markov decision process (POMDP) (Poupart, 2005; Singh et al., 2021). Inferring the hidden state in a partially observable environment is a sequence modelling problem, since it requires processing a sequence of observations to obtain enough information about the hidden state. One of the most efficient representations of hidden states for discrete POMDPs is the Successor Representation (SR), which disentangles hidden states and goals
+
+given by the reward function (Dayan, 1993; Samuel J. Gershman, 2018; Millidge & Buckley, 2022). The Successor Features framework is an extension of SR to continuous POMDP, using the same idea of value function decomposition, but instead for features of a hidden state (Barreto et al., 2017; Touati et al., 2022). Temporal memory (TM) algorithms can be used to make cumulative predictions about future states and their features, forming SR or SF. This work shows that the proposed algorithm, Distributed Hebbian Temporal Memory (DHTM), can dynamically form SFs for navigation tasks in Gridworld and AnimalAI (Fountas et al., 2020) environments.
+
+The most prominent TM algorithms, such as a Recurrent Neural Network (RNN) (Boden, 2002) or LSTM (Hochreiter & Schmidhuber, 1997), use backpropagation to capture data relationships, which is known to be unstable due to recurrent nonlinear derivatives. They also require complete data sequences to be available during training. Although the gradient vanishing problem can be partially circumvented in a way that Receptance Weighted Key Value (RWKV) (Peng et al., 2023) or Linear Recurrent Unit (LRU) (Orvieto et al., 2023) models do, the problem of online learning, i.e. learning on non-stationary data sets, is still a viable issue. Unlike graphical models such as HMM (Eddy, 2004), RNN models and their descendants also lack a probabilistic theoretical foundation, which is advantageous for modelling sequences captured from stochastic environments (Salaun et al., 2019; Zhao et al., 2020). There is little research on TM models that can be used in fully online adaptive systems interacting with partially observable stochastic environments with access to only one sequence data point at a time, a common case in reinforcement learning (Jahromi et al., 2022).
+
+We propose a Distributed Hebbian Temporal Memory (DHTM) algorithm based on factor graph formalism and a multi-compartment neuron model inspired by the HTM neuron (Hawkins & Ahmad, 2016). Factor graph is a flexible graphical representation of probabilistic distributions, which is commonly used in combination with sum-product algorithm (Loeliger, 2004). We found this representation particularly convenient in our attempt of probabilistic interpretation of the HTM framework. The resulting graphical structure of our model is similar to that of the factorial HMM (Ghahramani & Jordan, 1995). An important feature of our model is that the transition matrices for each factor are stored as different components (segments) of artificial neurons, which makes computation very efficient in the case of sparse transition matrices. Our TM forms sequence representations entirely online and uses only local Hebbian-like learning rules (Hebb, 2005; Churchland & Sejnowski, 1992; Lillicrap et al., 2020), which avoid the drawbacks of gradient methods and make the learning process much more sample efficient than gradient methods. This comes at the expense of hidden state generalization, since DHTM's hidden state uniquely encodes each observation sequence. Different trajectories leading to the same position in the partially observable Gridworld can result in different hidden states inferred by DHTM. In essence, this makes DHTM a model of episodic memory, or an adaptive trajectory buffer, with built-in machinery for making multistep future predictions. In this light, we also show that an agent forming SF with DHTM can be interpreted as a kind of episodic control (EC) (Lengyel & Dayan, 2007; Pritzel et al., 2017; Ritter et al., 2018; Emukpere et al., 2023).
+
+Though our approach is similar to EC, we are not aware of other works, demonstrating its application to partially observable environments. There are other temporal memories that, similarly to DHTM, use Hebbian rules and learn online, like Asymmetric Hopfield Network (AHN) (Chaudhry et al., 2023), Temporal Predictive Coding (tPC) (Tang et al., 2023) and Hierarchical Temporal Memory (Hawkins & Ahmad, 2016). Compared to DHTM, AHN is a first-order memory. That is, it is not able to disentangle identical observations that correspond to different underlying environmental states. Therefore, it is not suitable for partially observable environments. In contrast, tPC has a hidden layer that, potentially, makes it a high-order memory. Despite that it has been tested only in MDP environments (Tang et al., 2024) so far, this method paves a promising direction, which we aim to compare with in future work. Finally, the algorithm that inspired our work, HTM, is also a Hebbian high-order temporal memory, which, opposed to DHTM, doesn't allow for probabilistic predictions due to its all-or-nothing computations.
+
+The proposed TM is tested as an episodic memory for an RL agent architecture navigating in a Gridworld environment and a more challenging AnimalAI testbed (Crosby et al., 2020). The results demonstrate that our algorithm outperforms classical LSTM, RWKV and a biologically inspired HMM-like world model CSCG (George et al., 2021) in changing environments and is in order of magnitude more sample efficient than Dreamer V3 (Hafner et al., 2023). Another advantage of our algorithm is that it can be implemented on neuromorphic processors using only local learning rules.
+
+Our contribution to this work includes the following key elements:
+
+- A novel episodic memory model, called DHTM, which is based on a factor graph formalism and features an efficient, bio-inspired implementation.  
+- We demonstrate that the DHTM can operate fully online, using only local Hebbian-like learning rules.  
+- We show that DHTM learns at a speed comparable to model-based table episodic control while supporting distributed representations. This capability significantly broadens its applicability.
+
+# 2 BACKGROUND
+
+# 2.1 REINFORCEMENT LEARNING
+
+This paper considers decision-making in a partially observable environment (Poupart, 2005). The environment is defined by its state transition function  $P(s,a,s^{\prime}) = Pr(s^{\prime} \mid s,a)$ , determining its dynamics, where  $s^{\prime}, s \in S$  is called state space,  $a \in A$  — action space. Partial observability means that an agent has access to observations  $o \in O$ , which only partially inform about the environment's state  $s$  at each moment.
+
+In reinforcement learning (RL), the goal for the agent is set through the reward function  $R(s): S \to \mathbb{R}$ . The task of RL is to find the agent's policy  $\pi(a|s): S \times A \to [0,1]$  that maximizes the expected return  $G = \mathbb{E}[\sum_{t=0}^{T} \gamma^l R_t]$ , where  $T$  is an episode length and discount factor  $\gamma$  sets the importance of remote rewards. Value-based methods usually aim to estimate the Q-function given a policy  $\pi: Q^\pi(s_t, a_t) = \mathbb{E}[\sum_{l \geq t} \gamma^{l-t} R(s_{l+1}) | s_t, a_t, \pi]$ . For an optimal value function  $Q^*$ , an optimal policy can be defined as  $\pi(a|s) = \operatorname*{argmax}_{a} Q^*(s, a)$ .
+
+One of the drawbacks of value-based methods is that they require learning a new Q-function when the reward function changes, even if the environment remains the same. To get around this, the dynamics of the environment and the reward function can be decoupled, which leads us to the successor representation framework.
+
+# 2.2 SUCCESSOR REPRESENTATION
+
+Successor representations are such representations of hidden states from which we can linearly infer the state value given the reward function (Dayan, 1993). For a discrete state space, such representation can be derived just by rearranging sums in the equation for value function:
+
+$$
+V \left(h _ {t} = i\right) = \mathrm {E} \left[ \sum_ {l = 0} ^ {\infty} \gamma^ {l} R _ {t + l + 1} \mid h _ {t} = i \right] = \sum_ {j} \mathrm {S R} _ {i j} R _ {j}, \tag {1}
+$$
+
+where  $\gamma$  is a discount factor,  $i$ -th row in the matrix SR is the Successor Representation of a state  $i$ , and  $\mathrm{SR}_{ij} = \sum_{l=0}^{\infty} \gamma^{l} p(h_{t+l+1} = j \mid h_{t} = i)$ .  $R_{j}$  is a reward for observing the state  $j$ .
+
+An extension of SR for large or continuous state spaces that we use is Successor Features (Barreto et al., 2017), which is based on a similar idea of value function decomposition, but in a feature space instead (see details in Appendix A).
+
+It is easy to show that, similarly to the Q-function, both SR and SF can be obtained using dynamic programming. However, this limits its use to relatively stable environments, since changes in the transition function can't be taken into account immediately.
+
+The next step to improve an agent's adaptability is to use a world model to approximate the environment's transition function and use it directly to form SR according to its definition. And one of the most basic such world models is the Hidden Markov Model.
+
+# 2.3 HIDDEN MARKOV MODEL
+
+A partially observable environment can be approximated by a Hidden Markov Model (HMM) with hidden state space  $H$  and observation space  $O$ . Generally,  $H$  is not equal to environements state
+
+space  $S$  since the environemnt's true structure is usually unknown. For simplicity, we assume that actions are fully observable and that information about them is contained in hidden state. For the process of length  $T$  with state values  $h_{1:T} = (h_1,\dots ,h_T)$  and observations  $o_{1:T} = (o_1,\dots ,o_T)$ , the Markov property yields the following factorization of the generative model:
+
+$$
+p \left(o _ {1: T}, h _ {1: T}\right) = p \left(h _ {1}\right) \prod_ {t = 2} ^ {T} p \left(h _ {t} \mid h _ {t - 1}\right) \prod_ {t = 1} ^ {T} p \left(o _ {t} \mid h _ {t}\right). \tag {2}
+$$
+
+In the case of a discrete hidden state variable, a time-independent stochastic transition matrix can be learned with the Baum-Welch algorithm (Baum et al., 1970), a variant of the expectation maximization (EM) algorithm. To compute the statistics for the expectation step, it uses the forward-backward algorithm, which is a special case of the sum-product algorithm (Kschischang et al., 2001).
+
+Factor graph is a convenient graphical representation (Loeliger, 2004) of probabilistic models like HMM (see Fig.1A for an example). Formally, a factor graph is a bipartite graph, where one part consists of factors (squares), which, replacing conditional probabilities, define dependencies between random variables (circles) constituting the other part. Such representation also allows extension of forward-backward algorithm to sum-product algorithm. As shown in section 3.1, we base DHTM on ideas from the sum-product algorithm and the Factorial HMM, while adapting them for online learning.
+
+# 3 DISTRIBUTED HEBBIAN TEMPORAL MEMORY
+
+# 3.1 FACTOR GRAPH MODEL
+
+Distributed Hebbian Temporal Memory is based on the sum-product belief propagation algorithm in a factor graph (see Figure 1A). Analogous to Factorial HMM (Ghahramani & Jordan, 1997), we divide the hidden space  $H$  into subspaces  $H^k$ . There are four sets of random variables (RV) in the model:  $H_{t-1}^i$ -latent variables representing hidden states from the previous time step (context),  $H_t^k$ -latent variables for the current time step, and  $\Phi_t^k$ -feature variables. All random variables have categorical distributions. RV state values are denoted by corresponding lowercase letters:  $h_{t-1}^i$ ,  $h_t^k$ ,  $\varphi_t^k$ ,  $o_t^{lm}$ .
+
+In this work, each variable  $\Phi_t^k$  is considered independent and has a separate graphical model for increased computational efficiency. However, in practice, hidden variables of the same time step are statistically interdependent. We introduce their interdependence through a segment computation trick that goes beyond the standard sum-product algorithm (see Appendix B for details).
+
+The model also has three types of factors:  $M_{t-1}^i$ —messages from previous time steps,  $F_c^k$ —context factor (generalized transition matrix),  $F_e^k$ —emission factor. We assume that messages  $M_{t-1}^i$  contain posterior information from time step  $t-1$ , so we don't plot observable feature variables for previous time steps in Figure 1A.
+
+We will discuss only the upper block of the graph in Figure 1A, which is DHTM itself. The lower block—an encoder—is described in Appendix D. The only requirement for the encoder is that its output should be represented as states of categorical variables (features) for the current observation.
+
+The main routine of the DHTM is to estimate distributions of hidden state variables given by the equation 3, the computational flow of which is schematically shown in Figure 1B:
+
+$$
+p \left(h _ {t} ^ {k}\right) \propto \sum_ {\Omega_ {k}} \prod_ {i \in \omega_ {k}} M _ {t - 1} ^ {i} \left(h _ {t - 1} ^ {i}\right) F _ {c} ^ {k} \left(h _ {t} ^ {k}, \Omega_ {k}\right), \tag {3}
+$$
+
+where summation is over all possible hidden RV states in  $\Omega_{k} = \{h_{t - 1}^{i}:i\in \omega_{k}\}$ ,  $\omega_{k} = i_{1},\ldots ,i_{n}$  set of previous time step RV indexes included in  $F_{c}^{k}$  factor,  $(n + 1)$  —factor size.
+
+# 3.2 NEURAL IMPLEMENTATION
+
+For computational purposes, we translate the factor graph shown in Figure 1A to the Hebbian learning neural network architecture shown in Figure 1B (for a biological interpretation of the model,
+
+![](images/e97ae17619be265711aef579069328b7ed1c4cb50027fd827152b7f60621a08c.jpg)  
+Figure 1: A. Example of a factor graph for the DHTM. The input to the model is a sequence of observations  $o_{t}$ . The encoder block forms categorical features  $\Phi_t^k$ . Each feature  $\Phi$  has its own explaining hidden variable, which may depend on hidden variables of the other features and on itself from the previous time step.  $F_{c}^{k}$  and  $F_{e}^{k}$  are context and emission factors for the corresponding variables. Unary factors  $M_{t - 1}^{i}$  called messages represent accumulated information about previous time steps. B. Neural implementation. Random variables are represented by cell clusters (white circles), where each cell corresponds to a state and its spike frequency to the probability of the state  $p(h_{t}^{k})$ . The dendritic segments of the cell correspond to the context factor values  $f_{l}$  for a given combination of states (active presynaptic cells). Segments' excitations  $E_{l}$  are combined to determine the cell's spike frequency  $p(h_{t}^{k})$ . Segment synaptic weights reflect the specificity of the combination of presynaptic cells for the segment. Emission factors  $F_{e}^{k}$  are fixed and represented by minicolumns within a variable.
+
+![](images/7ba7d6259521186633103bdbe3aa2ad97dcb2b766ddb9b417513f0b2936b4cd3.jpg)
+
+see Appendix C). Each RV can be viewed as a set of neurons representing the states of the RV, i.e.  $p(h_t^k) = p(c_t^j = 1)$ , where  $j$ -is the index of a neuron corresponding to the state  $h_t^k$  and  $c_t^j \in \{0,1\}$  cell's activity. Since RVs are categorical, only one cell per variable can be active at a time and their activity is defined through sampling of RV's distribution.
+
+Since factors  $F_{c}^{k}$  represent dependencies between variables, in neural implementation we view them as collections of connections between cell groups corresponding to RVs that  $F_{c}^{k}$  takes as arguments. Each factor's entry maps a particular combination of states  $h_{t}^{k}, h_{t-1}^{i_{1}}, \ldots, h_{t-1}^{i_{n_{l}}}$  to a factor value  $f_{l}$ . We found it useful to represent each such entry as a separate computational unit, which we call segment (alluding to dendritic segments of a biological neuron). As shown in Figure 1B, segment  $l$  connects its presynaptic cells  $h_{t-1}^{i_{1}}, \ldots, h_{t-1}^{i_{n_{l}}}$  (one per each RV), constituting its receptive field, to the cell it excites  $h_{t}^{k}$ . The factor  $f_{l}$  value can be thought of as a segment's excitation efficiency, i.e. how strong its presynaptic pattern influences the cell's activation.
+
+Reformulation of belief propagation step in equation 3 in terms of segmental computations results in the following equations:
+
+$$
+p \left(h _ {t} ^ {k}\right) = \operatorname {s o f t m a x} _ {j \in \text {c e l l s} \left[ H _ {t} ^ {k} \right]} \left(\max  _ {l \in \operatorname {s e g} (j)} \left(E _ {l}\right)\right), \tag {4}
+$$
+
+$$
+E _ {l} = \log f _ {l} + w _ {l} \log \frac {1}{n _ {l}} \sum_ {u \in \operatorname {r e c} (l)} w _ {u l} m _ {u} + \sum_ {u \in \operatorname {r e c} (l)} \left(1 - w _ {u l}\right) \log m _ {u}, \tag {5}
+$$
+
+where cells  $[H_t^k ]$  --indexes of cells corresponding to states of the variable  $H_{t}^{k}$ $seg(j)$  all segments for cell  $j$ $w_{ul}$  synapse efficacy or neuron's specificity for segment, such that  $w_{ul} = p(s_l = 1|c_{t - 1}^u = 1)$ $w_{l} = \frac{1}{n_{l}}\sum_{u}w_{ul}$  average synapse efficacy for segment  $l$  , and  $n_l$  number of cells in segment's receptive field rec(l).  $m_{u}$  is presynaptic cell's activation probability  $p(c_{t - 1}^u = 1)$  , which carries messages from the previous time-step and corresponds to entry in  $M_{t - 1}^{i}$  For detailed derivation and rationale behind these equations, see Appendix B.
+
+Initially, cells have no segments. As learning progresses, new non-zero connections, grouped into segments, are created. In equation 5, we benefit from having sparse factors because their complexity depends linearly on the number of non-zero entries or segments. And that's usually the case in our
+
+model due to the one-step Monte Carlo learning and the special form of the emission factors  $F_{e}^{k}$ :
+
+$$
+F _ {e} ^ {k} \left(h _ {t} ^ {k}, \varphi_ {t} ^ {k}\right) = \mathbb {I} \left[ h _ {t} ^ {k} \in \operatorname {c o l} \left(\varphi_ {t} ^ {k}\right) \right], \tag {6}
+$$
+
+where  $\mathbb{I}$  indicator function,  $\mathrm{col}(\varphi_t^k)$  is a set of hidden states connected to the feature state  $\varphi_t^k$  forming a column. The shape of the emission factor is inspired by the presumed columnar structure of the neocortex and has been shown to induce a sparse transition matrix in HMM (George et al., 2021).
+
+The next step after computing the  $p(h_t^k)$  distribution is to incorporate information about the current feature states  $p(h_t^k \mid \varphi_t^k) \propto p(h_t^k) \mathbb{I}[h_t^k \in \mathrm{col}(\varphi_t^k)]$ . After that, the learning step is executed. The loop closing step of our TM algorithm is to assign the posterior of the current step  $p(h_t^k \mid \varphi_t^k)$  to  $M_{t-1}^i$ .
+
+# 3.3 LEARNING
+
+DHTM learns  $f_{l}$  and  $w_{ul}$  weights by Monte-Carlo Hebbian-like updates. First,  $h_{t - 1}^{i}$  and  $h_t^k$  are sampled from their posterior distributions:  $p(h_{t - 1}^{i}\mid \varphi_{t - 1}^{i})\propto M_{t - 1}^{i}$  and  $p(h_t^k\mid \varphi_t^k)$  respectively. Cells that correspond to sampled states become active. For cell  $j$  corresponding to  $h_t^k$  we check if there are segments matching the pattern of sampled states (active cells) from the previous time-step  $\{h_{t - 1}^i\}_i$ . If there are no such segments, the new segment connecting sampled cells is created. For all newly created segments and segments with receptive fields matching active cells from the previous time-step, its activity  $s_l$  is set to 1. Then  $f_{l}$  is updated according to the activity of the segment  $s_l$  and the activity of its cell  $c_t^j$ , so that  $f_{l}$  is proportional to several coincidences  $s_l = c_t^j = 1$  in the recent past (exponential moving average of coincidence):
+
+$$
+\Delta f _ {l} = \alpha \left(c _ {t} ^ {j} - f _ {l}\right) s _ {l}, \tag {7}
+$$
+
+where  $\alpha \in [0,1)$  is the segment's learning rate.
+
+That is,  $f_{l}$  is trained to be equal to the cell's average activation frequency when the segment is active. It's similar to Baum-Welch's update rule (Baum et al., 1970) for the transition matrix in HMM, which, in effect, counts transitions from one state to another, but in our case, the previous state (context) is represented by a group of RVs.
+
+Weights  $w_{ul}$  are also updated by the Hebbian rule to reflect the specificity of a presynaptic  $u$  for activating a segment  $l$ . That is, they are targeted to represent the probability  $p(s_l = 1 \mid c_{t-1}^u = 1)$  that segment  $s_l$  is active, given that cell  $u$  was active at the previous time step. In our algorithm, it is approximated as an exponential moving average of the frequency activation of segment  $s_l$ , given  $c_{t-1}^u = 1$ :
+
+$$
+\Delta w _ {u l} = \beta \cdot \mathbb {I} \left[ c _ {t - 1} ^ {u} = 1 \right] \cdot \left(\mathbb {I} \left[ s _ {l} = 1 \right] - w _ {u l}\right), \tag {8}
+$$
+
+where  $\beta \in [0,1)$  — learning rate.
+
+# 3.4 AGENT ARCHITECTURE
+
+We incorporate DHTM as a part of an RL agent. The same agent is used for other memories tested. The agent consists of a memory model and a feature reward function. The memory model generates SF by predicting cumulative future distributions of feature variables  $\Phi^k$ . We assume that the reward function can be decomposed linearly in feature space as  $R_{t} = \frac{1}{n}\sum_{k=1}^{n}r_{i}^{k}$ , where  $r_{i}^{k}$  is a reward associated with feature state  $\varphi_{t}^{k} = i$  and  $n$ -number of feature variables.  $r_{i}^{k}$  is learned during interaction with the environment and is used in combination with SF representations to estimate the action value function (see Appendix A for details).
+
+The agent training procedure is outlined in Algorithm 1. For each episode, the memory state is reset to a fixed initial message with RESET MEMORY() and the variable action is set to a fixed initial action. An observation returned by an environment (obs) is encoded as a set of categorical variables. In the OBSERVE() routine, memory learns to predict next feature states as described in section 3. An agent learns associations to map feature states and rewards in the REINFORCE() function:
+
+$$
+r _ {i} ^ {k} \leftarrow r _ {i} ^ {k} + \alpha \mathbb {I} \left[ \varphi_ {t} ^ {k} = i \right] \left(R _ {t} - r _ {i} ^ {k}\right) \tag {9}
+$$
+
+where  $\alpha$  is a learning rate,  $R_{t}$  —a reward for the current time step.
+
+Algorithm 1 General agent training procedure  
+1: for episode=1..n do  
+2: RESET MEMORY()  
+3: action  $\leftarrow$  initial_action  
+4: while (not terminal) and (steps  $<$  max_steps) do  
+5: obs, reward  $\leftarrow$  STEP()  
+6: features  $\leftarrow$  ENCODE(PREPROCESS(obs))  
+7: OBSERVE(features, action)  
+8: REINFORCE(reward, features)  
+9: action  $\leftarrow$  SAMPLE_ACTION()  
+10: ACT(action)  
+11: end while  
+12: end for
+
+We include actions in DHTM by forcing some of the hidden variables  $H_{t}^{k}$  to represent actions. That is, we assume that information about action is contained in the hidden state of the model. For example, if we have 4 actions, we set 4 states for one of the hidden variables and set its state from observing the action.
+
+In SAMPLE_ACION(), for each action and current state, TM predicts feature states up to time step  $T$  under policy  $\pi$ , and the predictions are combined to form SF (equation 10). For simplicity, we always build SF under a uniform policy, which doesn't guarantee optimality, but
+
+we found that it doesn't make a significant difference to iterative policy improvement in our experimental setups. The planning horizon  $T$  is determined dynamically from the distribution of  $\Phi_{t + l}^{k}$ . For each prediction step  $l$ , we check whether the distribution of  $\Phi_{t + l}^{k}$  is close to uniform (according to Kullback-Leibler distance) or whether any of states with positive reward has probability above a threshold; in that case, the prediction cycle is stopped. The action value is determined by combining the corresponding SF and feature state rewards as shown in equation 11. Finally, the action is sampled from the softmax distribution over the action values.
+
+$$
+\mathrm {S F} _ {t + T} ^ {\pi} \left(\varphi^ {k} = j \mid h _ {t}\right) = \sum_ {l = 0} ^ {T} \gamma^ {l} p _ {\pi} \left(\varphi_ {t + l + 1} ^ {k} = j \mid h _ {t}\right) \tag {10}
+$$
+
+$$
+Q ^ {\pi} \left(h _ {t}, a _ {t}\right) = \sum_ {j k} \sum_ {h _ {t + 1}} \mathrm {S F} _ {t + T} ^ {\pi} \left(\varphi^ {k} = j \mid h _ {t + 1}\right) \cdot p \left(h _ {t + 1} \mid h _ {t}, a _ {t}\right) \cdot r _ {j} ^ {k}, \tag {11}
+$$
+
+# 3.5 DHTM AND EPISODIC CONTROL
+
+Algorithm 2 Episodic memory learning  
+Input:  $\varphi_{t + 1},a_t$    
+1:  $h_{t + 1}\leftarrow \mathcal{D}_{a_t}(h_t)$    
+2:  $\varphi_{t + 1}^{*}\gets \operatorname {argmax}_{\varphi}F_{e}(h_{t + 1},\varphi)$    
+3: if  $h_{t + 1}$  is null or  $\varphi_{t + 1}^{*}$  is not  $\varphi_{t + 1}$  then   
+4:  $h_{t + 1}\gets \underset {h\notin K}{\operatorname{argmax}}F_{e}(h,\varphi_{t + 1})\# K$  is the set of keys for all actions   
+5:  $\mathcal{D}_{a_t}(h_t)\gets h_{t + 1}$    
+6: end if   
+7:  $h_t\gets h_{t + 1}$
+
+It can be shown that DHTM combined with the agent architecture described in the previous section is similar to episodic control.
+
+DTHM performs best when its hidden state space is large. Since the hidden state for unpredicted observations is formed randomly, it's important that there are no accidental state collisions when remembering transitions. That is, DHTM tries to remember new sequences without interfering with old memories. In effect, this makes DHTM a buffer of agent trajectories with a built-in recall engine. Episodic Control is based on a similar idea, and to investigate its difference from DHTM, we implemented an Episodic Control agent, a simpli
+
+fied counterpart of DHTM, reflecting its core property that presumably helps it solve RL tasks.
+
+Episodic Control (EC) is a general computational approach to storing and reusing an agent's experience, inspired by the notion of episodic memory from psychology (Tulving et al., 1972). The core of EC is a dictionary  $\mathcal{D}_a = (K_a, V_a)$  that stores key-value pairs for each discrete action  $a \in A$ . In our EC version,  $K_a$  is an array of hidden states  $h_t$  and  $V_a$  is an array of corresponding subsequent hidden states  $h_{t+1}$ . If the key state  $h_t$  already exists in the dictionary, its value is rewritten. That is,  $\mathcal{D}_a$  represents the transition matrix of a deterministic HMM for a feature variable  $\Phi_t$ . The hidden state  $h_t$  and the feature state  $\varphi_t$  have the same dependence as in DHTM (equation 6), but in EC we assume that there are infinite hidden states per feature state. To predict the next state, we just have to look up  $\mathcal{D}_a$  for the current state  $h_t$ , but if there is no entry for  $h_t$  or the prediction does not match the observed state  $\varphi_{t+1}$ , then the new state is formed. Unlike DHTM, the new state is not completely
+
+![](images/90caac5288a149f9b4d86c7f404e4557a43b3b7d51282698e8ca85d95d8e395b.jpg)  
+Figure 2: POMDP 5x5 Gridworld foraging task using different types of memory: DHTM, CSCG, LSTM, RWKV. All agents, except for DHTM, are trained in two modes. The first mode is with oracle, which tells the agent to reinitialize memory and clear the trajectory buffer during environmental change. The second mode is without oracle, so the agent continues using its current strategy to gather new trajectories and update memory. There is also an episodic control agent (e c) among baselines, which uses a simple dictionary to store trajectories and make predictions. A. Dynamic of steps required for the agent to reach the goal. In the 300th episode, the goal is blocked by a wall, so the agent needs to find a new optimal trajectory. B. Total steps taken during 1000 episodes.
+
+![](images/f4673f4cfcd5d45dcdbfbfd88e0d569b6356db976e4ac907f169f53de64b4c17.jpg)
+
+random, but to avoid collisions, the algorithm makes sure that this state hasn't been chosen before. The learning procedure of the EC agent's memory  $\mathcal{D}_a$  is summarized in the algorithm 2.
+
+Since  $\mathcal{D}_a$  can be transformed into a transition matrix, the process of forming SF can be reduced to one in Sec. 3.4. However, since the resulting transition matrix is sparse, it's more efficient to consider  $\mathcal{D}_a$  as a graph representation and to perform SF formation as a breadth-first search (see Appendix E.3 for details).
+
+# 4 EXPERIMENTS
+
+We test our model in a reinforcement learning task in the Gridworld environment and in a more challenging AnimalAI 3D environment. This section shows how different memory models affect the adaptability of an RL agent. For each setup, the results are averaged over five experiments with different seed values, and one standard deviation is shown everywhere, except for boxplots. See Appendix E for details on the baselines, their model parameters, and training regimes. See Appendix F for a more detailed description of the environments and setups<sup>1</sup>.
+
+# 4.1 GRIDWORLD
+
+The first test was conducted in partially observable Gridworld in a 5x5 alternating maze with a stationary end state and the agent's starting position. The agent can observe only the colour of the cell it is in and the colour of an obstacle it encounters, while being punished in addition to the constant step penalty. Each episode ends when the reward is collected or when the maximum number of action steps is reached. In each episode, we measure the number of action steps required for the agent to reach the goal.
+
+Each trial lasts 1000 episodes and consists of two phases. In the first phase, there are no obstacles on the shortest path to the terminal state. In the second phase, after 300 episodes, the previously optimal path is blocked by a wall, so the agent has to take a detour to get the reward. The experiment aims to show the ability of a memory to relearn the structure of the maze in the test time. If the agent is able to reach the end state in the second phase, it suggests that the model can forget, or at least suppress,
+
+![](images/541c334c1ae94e4613f5f18a5a79ec7f67e9a6a86c2762973286dbcbab023cb1.jpg)  
+Figure 3: Amount of action steps the agent performs to reach the goal in a  $10 \times 10$ -meter room in AnimalAI environment using DHTM, CSCG, LSTM, RWKV and EC dictionary. The agent and the goal have the same initial positions throughout the episodes. Each time-step, the agent gets an encoded image of a first-person view of the environment and produces one of three actions: turn left, turn right, or go forward. A. Dynamic of steps required for the agent to reach the goal. B. Total steps taken during 1000 episodes by the agent with DHTM with K-Means encoder and Categorical VAE. For the latter, DHTM's context factor size is varied (i.e., how many previous time-step hidden variables are considered for prediction).
+
+![](images/bb3b03e1a53386ec026e14c02e32f682fd10c8f58b36c2e55e3530f4576dd008.jpg)
+
+the memory of the old shortest path when it is blocked. Otherwise, the agent will continue to hit the wall and fail to reach the terminal state.
+
+In this experiment, we compare our memory model DHTM with CSCG (George et al., 2021), LSTM (Hochreiter & Schmidhuber, 1997) and RWKV (Peng et al., 2023) within the same agent architecture that uses memory to form successor features and evaluate actions (see Sec. 3.4). All memories, except DHTM, are trained in two modes. The first mode is with an oracle that tells the agent to reinitialize memory and clear the trajectory buffer just in time for the environmental change. The second mode is without an oracle, so the agent continues to use its current strategy to collect new trajectories and update memory in the second phase. There is also a simple episodic control agent (EC) among the baselines, which uses a dictionary to store trajectories and make predictions (see Section 3.5 for details).
+
+The results in Figure 2 show that CSCG effectively can't adapt to the environmental change in the online regime, but only if it is retrained from scratch on new trajectories. And, apparently, LSTM and RWKV are only able to learn a predictive model for a small horizon, since they perform significantly worse than other baselines in the second phase, even with oracle enabled. In contrast, DHTM and EC are able to adapt online even without explicit memory's trainable parameters reinitialisation. Another important observation is that the EC agent performs on par with DHTM, suggesting that its mechanism can explain most of DHTM's performance in this task. However, see Appendix H.2 for experiments that show their difference in scalability.
+
+# 4.2 ANIMALAI
+
+AnimalAI is a 3D environment capable of providing rich visual input to an agent in the form of RGB images. The goal of the experiment described here is to show that DHTM can handle distributed representations and is scalable to environments with larger state spaces.
+
+The agent's task is to reach a sphere of radius  $0.5\mathrm{m}$ , which is located in a square area of size 10 by 10 meters. The start position of the agent and the position of the ball are the same between episodes. In each episode, we measure the number of actions (steps), which are AnimalAI's standard left/right turn and forward walk, the agent takes to reach the goal, with the maximum number of actions limited to 40. Each action is repeated three times and three frames are skipped, so one agent action step corresponds to three steps in the environment.
+
+We test DHTM and EC agents using two different encoders: K-Means and Categorical VAE (Jang et al., 2017). Both encoders are pre-trained on RGB images collected with a uniform strategy during 2000 episodes in the same room used for agent testing. K-Means represents an image as a single categorical variable with 50 states, and Categorical VAE is configured for distributed representation with five variables with 50 states each (see Appendix D for details). In case of EC agent, since it doesn't support distributed representations, we map each unique VAE encoding to one feature state using a simple dictionary. Since the resulting number of states for such feature variable is large (typically more than 1000), it's infeasible to use the same mapping for CSCG. We also didn't find much difference for LSTM and RWKV using K-Means or VAE, so we only keep the results for VAE encoder.
+
+The results are shown in Figure 3A. It can be seen that after about 150 episodes, the agent with DHTM and the EC agent reach the goal stably, although the DHTM agent with distributed encoding (with a factor size of 4) converges to a slightly shorter path on average. The agent with CSCG slowly improves its strategy, but LSTM and RWKV agents don't show any learning on this time-step scale. To understand a common learning time-step scale for contemporary model-based algorithms, we also tested a small version of Dreamer V3 (18M parameters) (Hafner et al., 2023), which successfully solves the same task after about  $35 \times 10^{3}$  action steps (1000 episodes) versus  $3.5 \times 10^{3}$  steps (150 episodes) for DHTM (see Appendix H.1 for details).
+
+To show that DHTM can't simply be replaced by multiple parallel EC dictionaries or CSCGs in the case of distributed encoding, we also vary the context factor size of DHTM, that is, how many previous time-step variables are taken into account to predict the current state of a variable. The factor size of 1 corresponds to the graph of the Factorial HMM, with parallel HMMs. It can be seen from Figure 3B that in our case of categorical VAE, knowing only the previous state of the variable is not enough to reliably predict rewarding features, since the agent doesn't converge to the shortest path. We also tested the variant (factor 1*) where each feature variable is represented by three hidden states, to ensure that the result for factor size 1 is not due to DHTM's hidden state collisions. DHTM with a factor that takes into account more than one previous time-step hidden variable (excluding the action variable) performs significantly better. For diagrams depicting factor graphs used in these experiments, see Appendix E.4. For further experiments showing the importance of distributed encoding in DHTM, see Appendix H.3.
+
+# 5 CONCLUSION
+
+This paper introduces a novel episodic memory model, DHTM. It combines ideas from belief propagation, episodic control, and neurophysiological models of the neocortex, resulting in a fast probabilistic sequence memory.
+
+The experiments show that our memory model can quickly learn observation sequences in changing environments and effectively reuse them for RL tasks, on par with model-based table episodic control. We also show that, unlike CSCG and table EC, DHTM is able to handle distributed representations and account for statistical dependencies between variables of these representations. Potentially, it extends the application of our algorithm to higher state spaces, exploiting their factorizations for more efficient computations and to reduce the number of stored parameters.
+
+Our results suggest that DHTM can be viewed as an adaptive trajectory buffer for a slower generalizing algorithm, such as a Neural State Space Model (Jiang et al., 2023). Through prediction error learning, DHTM can selectively discard and replace obsolete transitions in observed trajectories without disrupting relevant experience. It can also provide an initial policy for an agent through episodic control until a slower but more accurate world model takes over. These features of a trajectory buffer are highly desirable in the case of online learning in non-stationary environments.
+
+Further improvements of DHTM can be directed towards the implementation of an automatic feature space factorization discovery to determine an optimal context factor size during learning. To avoid excessive segment growth in noisy environments, it's also possible to implement an adaptive segment activation threshold. One of the main limitations of this study is that the evaluation of the algorithms is limited to basic RL setups due to the simple agent architecture. More advanced use cases could be demonstrated using better exploration strategies and in combination with other memory models, which is also a matter of future research.
+
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+
+# A VALUE FUNCTION DECOMPOSITION
+
+In our agent model, we approximate the reward function  $\mathbf{R}(\mathbf{s})$  as a sum:
+
+$$
+R _ {t} = \frac {1}{n} \sum_ {k = 1} ^ {n} r \left(\varphi_ {t} ^ {k}\right), \tag {12}
+$$
+
+where  $r(\varphi_t^k)$  is a reward associated with state  $\varphi_t^k$ ,  $n$ -number of feature variables. Then, similarly to the Successor Representation idea (see Section 2.2), the value function can be represented as:
+
+$$
+\begin{array}{l} V \left(h _ {t}\right) = \mathrm {E} \left[ \sum_ {l = 0} ^ {\infty} \gamma^ {l} R _ {t + l + 1} \mid h _ {t} \right] = \sum_ {l = 0} ^ {\infty} \gamma^ {l} \mathrm {E} \left[ \frac {1}{n} \sum_ {k = 1} ^ {n} r \left(\varphi_ {t} ^ {k}\right) \mid h _ {t} \right] \\ = \frac {1}{n} \sum_ {k = 1} ^ {n} \sum_ {j} \sum_ {l = 0} ^ {\infty} \gamma^ {l} p (\varphi_ {t + l + 1} ^ {k} = j \mid h _ {t}) r _ {j} ^ {k} \\ = \frac {1}{n} \sum_ {k = 1} ^ {n} \sum_ {j} \mathrm {S F} _ {j} ^ {k} \left(h _ {t}\right) r _ {j} ^ {k}, \tag {13} \\ \end{array}
+$$
+
+where  $\mathrm{SF}_j^k (h_t) = \sum_{l = 0}^{\infty}\gamma^l p(\varphi_{t + l + 1}^k = j\mid h_t),h_t = (h_t^1,\dots,h_t^n)$  -hidden state vector of variables  $\{H_t^k\}_{k}$ .
+
+# B DETAILS OF NEURONAL IMPLEMENTATION
+
+As shown in Figure 1B, each RV can be viewed as a set of neurons representing the states of the RV, i.e.  $p(h_{t}^{k}) = p(c_{t}^{j} = 1)$ , where  $j$ -is the index of a neuron corresponding to the state  $h_{t}^{k}$ . Cell activity is binary  $c_{t}^{j} \in \{0,1\}$ , and the probability can be interpreted as a spike rate. Factors  $F_{c}^{k}$  and  $M_{t - 1}^{i}$  can be represented as vectors, where elements are factor values for all possible combinations of RV states included in the factor. Let's denote the elements of the vectors as  $f_{l}$  and  $m_{u}$  respectively, where  $l$  corresponds to a particular combination of state values  $h_{t}^{k}, h_{t - 1}^{i_{1}}, \ldots, h_{t - 1}^{i_{n_{l}}}$ , and  $u$  indexes all neurons representing states of RVs from previous time steps.
+
+Inspired by biological neural networks, we group the connections of a neuron into dendritic segments. A segment acts as an independent computational unit that detects a particular input pattern (a contextual state) defined by its receptive field. In our model, a segment connects the factor value  $f_{l}$  and the excitation  $E_{l}$  induced by the segment  $l$  to the cell to which it is connected. The segment is active, i.e.  $s_{l} = 1$  if all its presynaptic cells are active, otherwise  $s_{l} = 0$ . Computationally, a segment transmits its factor value  $f_{l}$  to the cell to which it is attached if the context matches the corresponding state combination.
+
+We can now rewrite equation 3 as the following:
+
+$$
+p \left(h _ {t} ^ {k}\right) \propto \sum_ {l \in \operatorname {s e g} (j)} L _ {l} f _ {l} ^ {k}, \tag {14}
+$$
+
+where  $L_{l} = \prod_{u\in \operatorname{rec}(l)}m_{u}$  is segment's likelihood as long as messages are normalized to probability distributions,  $\mathrm{seg}(j)$  — indexes of segments that are attached to cell  $j$ ,  $\operatorname{rec}(l)$  — indexes of presynaptic cells that constitute receptive field of a segment with index  $l$ .
+
+Initially, cells have no segments. As learning progresses, new non-zero connections, grouped into segments, are created. In equation 14, we benefit from having sparse factor value vectors because their complexity depends linearly on the number of non-zero components. And that's usually the case in our model due to the one-step Monte Carlo learning and the special form of the emission factors  $F_{e}^{k}$ :
+
+$$
+F _ {e} ^ {k} \left(h _ {t} ^ {k}, \varphi_ {t} ^ {k}\right) = \mathbb {I} \left[ h _ {t} ^ {k} \in \operatorname {c o l} \left(\varphi_ {t} ^ {k}\right) \right], \tag {15}
+$$
+
+where  $\mathbb{I}$  indicator function,  $\operatorname {col}(\varphi_t^k)$  is a set of hidden states connected to the feature state  $\varphi_t^k$  forming a column. The shape of the emission factor is inspired by the presumed columnar structure
+
+of the neocortex and has been shown to induce a sparse transition matrix in HMM (George et al., 2021).
+
+The segment likelihood  $L_{l}$  resulting from the sum-product algorithm is computed as if the presynaptic cells were independent. However, this isn't usually the case for sparse factors. To account for their interdependence, we substitute the following equation for segment log-likelihood:
+
+$$
+\log L _ {l} = w _ {l} \log \frac {1}{n _ {l}} \sum_ {u \in \operatorname {r e c} (l)} w _ {u l} m _ {u} + \sum_ {u \in \operatorname {r e c} (l)} \left(1 - w _ {u l}\right) \log m _ {u}, \tag {16}
+$$
+
+where  $w_{ul}$  — synapse efficacy or neuron specificity for segment, such that  $w_{ul} = p(s_l = 1|c_{t - 1}^u = 1)$ ,  $w_{l} = \frac{1}{n_{l}}\sum_{u}w_{ul}$  average synapse efficacy for segment  $l$ , and  $n_l$ -number of cells in segment's receptive field  $rec(l)$ .
+
+The idea behind the formula is to approximate between two extreme cases. The first is  $p(s_{l} = 1|c_{t - 1}^{u} = 1) \to 1$  for all  $u$ , which means that all cells in the receptive field are dependent and part of a cluster, i.e. they fire together. In this case,  $p(s_{l}) = m_{u}$  for all  $u$ , but we also reduce the prediction variance by averaging across different  $u$ . In the opposite case,  $p(s_{l} = 1|c_{t - 1}^{u} = 1) \to 0$  for all  $u$ , presynaptic cells don't cluster and the segment activation probability is just a product of the activation probability of each cell.
+
+The resulting equation for belief propagation in DHTM is the following:
+
+$$
+p \left(h _ {t} ^ {k}\right) = p \left(c _ {t} ^ {j} = 1\right) = \operatorname * {s o f t m a x} _ {j \in \text {c e l l s} \left[ H _ {t} ^ {k} \right]} \left(\max  _ {l \in \operatorname {s e g} (j)} \left(E _ {l}\right)\right), \tag {17}
+$$
+
+where  $E_{l} = \log f_{l} + \log L_{l}$ , cells  $[H_t^k ]$  —indexes of cells representing states for the variable  $H_{t}^{k}$ . Here we also approximate the logarithmic sum with the max operation, inspired by the neurophysiological model of segment aggregation by cells (Stuart & Spruston, 2015).
+
+The next step after computing the  $p(h_t^k)$  distribution parameters is to incorporate information about the current feature states  $p(h_t^k \mid \varphi_t^k) \propto p(h_t^k) \mathbb{I}[h_t^k \in \mathrm{col}(\varphi_t^k)]$ . After that, the learning step is executed. The loop closing step of our TM algorithm is to assign the posterior of the current step  $p(h_t^k \mid \varphi_t^k)$  to  $M_{t-1}^i$ .
+
+# C BIOLOGICAL INTERPRETATION
+
+Neural implementation of the DHTM is inspired by neocortical neural networks (see Fig. 4). Hidden variables  $H^{k}$  may be considered as populations of excitatory pyramidal neurons in cortical layer L2/3 of somatosensory areas, with lateral inhibition modelled as softmax function. Staiger & Petersen (2021) showed that neurons in this layer are responsible for temporal context formation.
+
+The neuronal activity at timestep  $t$  can be thought to carry messages  $M_{t-1}^{k}$ . Messages are propagated through synapses of dendritic segments, which correspond to factors  $F_{c}^{k}$ . Dendritic segments of biological neurons are known to be coincidence detectors of their synaptic input (Stuart & Spruston, 2015). That is, each segment can be thought of as a detector of a particular combination of active cells (states in our interpretation). If we associate a segment's parameter with factor value, a collection of dendritic segments represent a mapping from state space to factor values, representing context factor  $F_{c}$ .
+
+Our computational model of neuron and its dendrites is based on HTM neuron model (Hawkins & Ahmad, 2016). In this model, each segment has a receptive field and activates when a number of active cells in this receptive field reaches a threshold. The neuron is active if any of its segments are active. We use the same activation rule, but the threshold is fixed to the receptive field size. Our algorithm of segment's growth on demand is also similar to HTM's. In principle, DHTM can be considered as a probabilistic interpretation of HTM, allowing belief propagation.
+
+Similarly to HTM framework, our temporal memory can be interpreted as a model of cortical layers. Feature variables  $\Phi_t^k$  may be considered to represent cells of a granular layer (L4), as they are known to be the main hub for sensory excitation for L2/3. L2/3 cells that have common sensory input from the layer L4 are modelled as columns for particular feature states  $\operatorname{col}(\varphi_t^k)$  (Mountcastle, 1997).
+
+![](images/dff6c9ed50eb7cd707902252739bbc6ff8a2c2a7115ce3b67c07eb62d1da3246.jpg)  
+Figure 4: Biological view of the neural implementation of the DHTM. Variables  $H_{t-1}$  correspond to populations of neurons that have common sensory input and lateral inhibitory competition. Dendritic segments correspond to factor values  $f_l$ . Spike frequency of a neuron reflects state probability  $p(h_t^k)$  of a variable.
+
+# D ENCODING OBSERVATIONS
+
+Since DHTM is designed to process categorical random variables, we need a categorical encoder for RGB images to solve AnimalAI tasks.
+
+To compare our model to CSCG and EC in AnimalAI, we use the approach described in Guntupalli et al. (2023) to quantize observation space. K-Means (implementation from scikit-learn) with 50 clusters pretrained on  $4 \times 10^{4}$  64x64 RGB observations from trajectories sampled by uniform policy. To encode an image, the distance from each cluster is calculated and the index of the closest cluster is returned. In that way, AnimalAI's observation space is represented by one categorical variable.
+
+In order to test DHTM's ability to deal with distributed encoding, we also trained Categorical VAE (Jang et al., 2017) with the latent space represented by five categorical variables with 50 states each. We use a slightly modified implementation from Subramanian (2020). The only difference is that we modify the loss function by adding mean squared reward prediction error, which is linearly calculated from the latent codes to ensure the linear decomposition in feature space (equation 12). The model is trained on  $6 \times 10^{4}$  64x64 RGB images for 20 epochs with batch size 64. An image is encoded by first passing it through VAE's encoder block, then, to ensure stable encoding, instead of sampling from the latent distribution, the state with the highest logit is chosen for each variable.
+
+# E IMPLEMENTATION DETAILS
+
+As mentioned in Section 3.4, we incorporate DHTM as a part of an RL agent, which has a memory model and a feature reward function. All tested memory models share the same pipeline—conditioned on actions, they learn sequences of categorically encoded observations and are used to form SFs.
+
+In all experiments in Section 4.1, the agent uses softmax exploration with temperature equal to 0.04, except for EC agent we set softmax temperature to 0.01, since for higher temperatures its policy is unstable. For SF formation the agent uses  $\gamma = 0.8$  and  $\gamma = 0.9$  for Gridworld and AnimalAI experiments respectively. Maximum SF horizon is set to 50, but it is usually shorter since we employ early stop condition based on feature variables distribution. That is, if the prediction is close to uniform or the probability of a state with associated positive reward meets a threshold, the
+
+planning is interrupted. We use probability thresholds equal to 0.05 and 0.01 for Gridworld and AnimalAI respectively.
+
+In scaling Gridworld experiments (Appendix H.2), we increase maximum planning horizon up to 100 steps, and we also had to decrease agent's temperature to 0.02 for DHTM and to 0.002 for EC agent. Discount factor is 0.7 for DHTM and 0.9 for EC. We found that, in  $10 \times 10$  maze, DHTM's strategy becomes more stable if we increase the positive reward threshold up to 0.4. EC agent converges faster if we increase its softmax temperature, however, it converges to less optimal trajectory on average.
+
+In the noise tolerance AnimalAI experiment (Appendix H.3), for DHTM we use the same set of agent parameters, except we set the goal probability threshold to 0.5. We also set the goal detection threshold equal to the mean positive reward since we found that noise variable states mostly get reward assignment slightly above zero, in contrast to signal variables, in which only several states get positive reward assignment.
+
+# E.1 LSTM AND RWKV
+
+LSTM baseline was implemented with a single LSTMCell from PyTorch library (Paszke et al., 2019). It is supported by an additional symexp-layer to encode input before passing it to the LSTM cell and a symexp-layer to decode the LSTM cell's output from the LSTM's hidden state back to the input representation, where symexp activation function,  $\mathrm{symexp}(x) = \mathrm{sign}(x)e^{|x| - 1}$ , is a reverse of symlog function:  $\mathrm{symlog} = \mathrm{sign}(x)\log(|x| + 1)$ .
+
+The similar way we implemented RWKV baseline: a single RWKV layer supported by single-layer encoder and decoder. Current public RWKV implementation is a fast evolving framework (Bo, 2021), and for the increased performance it is tightly bound to the offline batch training common for the transformer architectures. In our case we needed a so-called sequential mode for online learning similar to LSTM. Thus, we adapted another public implementation mentioned in the official documentation (RWKV in 150 lines of code).
+
+For both Gridworld and AnimalAI tasks, LSTM and RWKV with a hidden state size of 100 are trained on a buffer of 1000 trajectories with learning rate 0.002 on 50 randomly sampled batches of size 50 every 10 episodes.
+
+We also incorporated some notion of random variables in hidden states by splitting the hidden state of the tested RNNs into groups. In all experiments the hidden state represents 10 categorical variables with 10 states. That is, RNN is forced to learn 10 categorical distributions with multi-cross-entropy loss to explain the observed sequences, which is a somewhat close to the multi-categorical hidden state representation used in Dreamer V2/V3 (Hafner et al., 2023). The idea of using symexp activation function, mentioned above, is inspired by Dreamer too, and is used to remedy the problem of learning extreme probability values. Without symexp the neural network has to represent zero probability with high negative logit values and one-probability with high positive logit values, which is hard to reach with low learning rate and may lead to instabilities. Thus, symexp function makes it faster to reach target values in log space.
+
+# E.2 CSCG
+
+CSCG baseline was implemented using code from the repository accompanying the paper (https: //github.com/vicariousinc/naturecomm_cscg). CSCG was trained on buffers of 500 action steps with 1000 EM steps until convergence. We iteratively calculated the exponential moving average of transition matrices obtained for different batches with a smoothing coefficient  $\alpha = 0.05$  as described in the paper (Dedieu et al., 2019). This smoothed transition matrix was used as an initialization for the next training batch and for inference. The first batch of observations is gathered using the uniform policy, and later batches are gathered using the current best policy.
+
+In so-called oracle mode, the transition matrix is trained from scratch every 500 steps. The buffer is discarded at the environmental change and the transition matrix is reinitialised.
+
+# E.3 EPISODIC CONTROL
+
+Here, we provide an algorithm for SF formation using EC dictionary  $\mathcal{D}_a$  (see Algorithm 3). In contrast to other memory algorithms, EC dictionary doesn't estimate state transition probabilities, but it stores connections between them in all-or-nothing fashion. Therefore, on each time step, all predicted observations are assumed equally probable. But it's not a limitation in our set of experiments, since we tested our algorithms in deterministic environments.
+
+Algorithm 3 SF formation for EC  
+Input:  $h_0,\gamma \in (0,1),T$    
+Output: SF   
+1: SF  $\leftarrow$  array of zeros   
+2: PS  $\leftarrow \{h_0\}$  # PS is the set of all states (nodes) on the current BFS depth   
+3: goal_found  $\leftarrow$  false   
+4: for  $l = 1..T$  do   
+5: PS  $\leftarrow \bigcup_{a\in A}\{\mathcal{D}_a(h)\}_{h\in \mathrm{PS}}$  # get next depth nodes assuming the policy is uniform   
+6: counts  $\leftarrow$  array of zeros   
+7: for all  $h\in \mathrm{PS}$  do   
+8:  $\varphi \gets \operatorname {argmax}F_e(h,\varphi)$    
+9: count  $\varphi =$  counts+1   
+10: goal_found  $\leftarrow r_{\varphi} > 0$  # check if the feature state is rewarding   
+11: end for   
+12: SF  $=$  SF  $^+\gamma^{l - 1}$  NORMALIZE(counts)   
+13: if goal_found is true then   
+14: break   
+15: end if   
+16: end for
+
+# E.4 DHTM
+
+In the Gridworld task and AnimalAI with K-Means encoder, the factor-graph for DHTM consists of three hidden variables  $H$  with 40 states per one feature state. Each hidden variable gets the same copied observation, and their predictions are averaged. Using several hidden variables here is essential, as it significantly increases the amount of trajectories memory is able to store by decreasing collision probability for hidden state representations of different trajectories.
+
+In AnimalAI experiments, hidden variables state spaces are of size 40 states per one feature state. For Categorical VAE encoder, we use one hidden variable per feature variable, except the case (factor  $1\star$ ) that tests that information from one feature variable is not sufficient for correct prediction, in which three hidden variables per observation variable are used and the factor of size three connecting to only hidden copies of the same feature variable. Examples of typical DHTM's graphical models used in our experiments are presented on Figure 5. It's important to note that edge colours denote separate graphical models, but the variables are shared (copied) between them. So there are no loops in these models. For simplicity, we define context factor size as the number of hidden variables from the previous time-step it has connections to, but it also connects to one hidden variable of the current step and the action variable.
+
+The most important DHTM's parameters, except the factor size, are factor learning rate  $\alpha$  and its initial value  $f_0$ . For better adaptability in changing and stochastic environments, the initial factor value should be set closer to zero, but in stationary environments, it can be safely set  $f_0 = 1$ , allowing faster convergence. In Gridworld we use  $\alpha = 0.1$  and  $f_0 = 0.05$ , and in AnimalAI we set  $\alpha = 0.001$  and  $f_0 = 1$ . We also set the limit on the number of segments  $5 \times 10^4$ , but practically never reach it in our experiments.
+
+![](images/f9a7e443692e243b7b875b3bd777815057c75c5f5a025cd013276eecc6ad5d26.jpg)  
+Figure 5: Examples of factor-graphs for two main setups. Edge colours show that edges correspond to separate graphical models with copied variables. All context factors are also connected to action variable  $A_{t-1}$ . A. Graphical model for one feature variable. In that case, feature value is copied for three hidden variables, to increase hidden state space and avoid memory collisions. B. Graphical model for Categorical VAE distributed encoding, example for factor of size 3 (i.e. takes into account tree hidden variables from the previous time-step).
+
+![](images/3f7b668fe7423b78d70fdf1c5a08472a7b3dc2c7a7b9a0114f854ba8b31e0850.jpg)
+
+# F EXPERIMENTAL SETUPS
+
+# F.1 GRIDWORLD
+
+We use a simple homebrew Gridworld implementation. Each Gridworld position maps to a colour, reward, and terminal state indicator. There are two special colours for the world border and obstacles. Each time-step, an agent gets the colour index of the current agent's position. An agent is able to see obstacles or border colour when trying to go through them, which just replace current position colour for one step without actually changing the agent's position. We also punish an agent trying to go through a wall or world border in addition to base step reward punishment to facilitate faster convergence to optimal trajectories.
+
+Samples of setups used in our experiments are depicted in Figure 6. Additional setups for scaling experiments are shown in Figure 7. Here, the red colour corresponds to obstacles (negative numbers) and the light gray colour to the reward (number 4). Letters A and G correspond to the agent's initial position and goal, correspondingly. Floor colours are resampled for each trial, but we ensure that there are no more than five floor colours, including the terminal state. In each episode, the agent and the terminal rewarding state have the same initial positions. The episode ends when the terminal state is entered or the limit of action steps is reached.
+
+# F.2 ANIMALAI
+
+AnimalAI is a testbed inspired by experiments with animals (Crosby et al., 2020). The environment consists of 3D area surrounded by a wall and many different objects that can be placed using a configuration file, including walls, food, ramps, trees, movable obstacles, and more. We tested our agent in a  $10 \times 10$ -meter room surrounded by walls (see Fig. 8). Each timestep, the agent gets an RGB image of a first-person view of the environment and is able to influence its behaviour by choosing one of three actions: turn left, turn right, or go forward. The agent and food item have the same fixed initial positions for each episode. The episode ends if the food item is collected or the maximum episode time is reached. To make an agent's actions more discretized, we repeat its choice three times and skip three frames.
+
+![](images/a6ee4366a898a2edff0a64ac5f7f9303cd04d0fc3bd11103faffda5e694b4d68.jpg)  
+Figure 6: Schematic view of POMDP Gridworld setups. The left picture shows the initial setup of the environment, with the reward delivered by the light gray square (number 4). The right picture represents the environment after the reward is blocked by obstacles (red squares or negative numbers). The letter A depicts the agent's initial position and the letter G—the goal.
+
+![](images/e23657e3bf04e049b7d8d637549272ada924b4e6a8e94d6dab5152d3c0f87f6f.jpg)
+
+![](images/a7b3ee5c1f04a0c993ec369272ca7a187a21df8e7979c4248d296800bebcd6b5.jpg)  
+Figure 7: Schematic view of POMDP Gridworld setups for scale experiments. From left to right:  $5 \times 5$ ,  $7 \times 7$ ,  $10 \times 10$ . The letter A depicts the agent's initial position and the letter G—the goal.
+
+![](images/ef233b8dab17e1a3b65ee40ce60fff01b99eb7f73c9e376839d8bb5ee23b5f8a.jpg)
+
+![](images/55585ae2a2fe373e7f2271cd1f8e4a4a654ecb31f7893d2c3dabb6a727e59059.jpg)
+
+![](images/1cf9064f71211905eaba375e7765a68927849cc64b864cd00a6ed70bca5ef7b7.jpg)  
+Figure 8: Sample of the agent's first-person view of our setup in AnimalAI environment. The green ball represents a goal, a food item to be collected.
+
+![](images/32613895544b1cb3d3993a329b63d2a363d92644035c9a2b9e9521b8f9a53445.jpg)  
+Figure 9: Dreamer V3 (18M) score in AnimalAI experiment described in this paper.
+
+# G CAPACITY
+
+In this section, we provide basic theoretical analysis of DHTM's capacity. Since the hidden state for unpredicted feature state transition is generated randomly, the main source of errors is accidental collision of hidden states. Due to special form of the emission factor (see equation 6), the collision is possible only for the same feature state. Here we analyse the capacity of DHTM for factor grpah depicted in Figure 5. Let's assume that the stream of feature states is sampled from uniformly distributed categorical random variable. Therefore, the probability to have a collision  $p_c = \frac{1}{n_\varphi n_v n_h}$  where  $n_v$  — number of hidden variables,  $n_\varphi, n_h$  are the number of states for feature and hidden variables correspondingly. Therefore, the probability of having no collisions for a  $l$  transitions is equal to  $(1 - p_c)^l$ . Let's set a threshold  $c \in (0,1)$ , which determines our confidence in that the sequence of transitions is collision free. Solving the equation  $1 - c = (1 - p_c)^l$  with respect to  $l$ , we get:
+
+$$
+l = \frac {\log (1 - c)}{\log \left(1 - \frac {1}{n _ {\varphi} n _ {v} n _ {h}}\right)}. \tag {18}
+$$
+
+Using the Taylor expansion for  $n_{\varphi}n_v n_h \to +\infty$ , we can see that  $l$  grows linearly with the total number of states  $n_{\varphi}n_v n_h$ . Therefore, DHTM's capacity scales linearly with the number of neurons. This, however, can be easily improved if we set additional prior on hidden state activation probability, which would take into account hidden state usage, but we leave it for future research.
+
+# H ADDITIONAL EXPERIMENTS
+
+# H.1 DREAMER
+
+In order to ground our experiments to state-of-the-art model based algorithms, as an additional baseline, we test the smallest version of Dreamer V3 with 18 million parameters. We use the implementation described in Voudouris et al. (2023), which is based on Hafner et al. (2023) and adapted for AnimalAI.
+
+We tested it with default parameters in the same setup as described in Section 4.2, except we don't apply frame skip. The agent successfully solves the task after  $105 \times 10^{3}$  action steps. Taking into account the frame skip used in our experiments with DHTM, it's approximately  $35 \times 10^{3}$  action steps versus  $3.5 \times 10^{3}$  for the DHTM agent to solve the task.
+
+# H.2 SCALABILITY
+
+In this section, we explore DHTM's scalability properties and compare them to the EC agent. In RL experiments, like presented in section 4, the number of segments largely depends on the RL algorithm used, its exploration strategy, and how fast it converges to a stable policy. We tested DHTM and EC agents in empty Gridworld of different sizes:  $5 \times 5$ ,  $7 \times 7$  and  $10 \times 10$ , where the agent and the goal are placed in opposite corners (see Fig. 7). As shown in Figure 10A, the DHTM-based
+
+agent requires fewer episodes to find the goal; however, the policy becomes less stable as the size of the Gridworld grows. As a consequence of faster convergence, DHTM produces fewer segments than the number of EC's dictionary entries throughout experiments. It also can be seen that for both DHTM and EC agents, the number of memory units grows non-linearly with the state space size (see Fig. 10B).
+
+![](images/b7dc866a1f980f64254c54eb6faff2024ae0141dbbbc1478a93c543e7a4282ef.jpg)  
+Figure 10: Comparison of EC and DHTM-based agents in foraging task in POMDP Gridworld of different sizes: 5x5, 7x7 and 10x10. A. Dynamic of steps required for the agent to reach the goal. B. Number of segments grown (or dictionary entries for EC agent) after 300 episodes with respect to state space size.
+
+![](images/42820ea17e67cfa028b22c7eae95b8583442e0317177958e67489d6fa9f45c0d.jpg)
+
+In principle, the number of segments reflects the number of transitions stored, which is proportional to the amount of experience provided to the memory. To show that, we tested DHTM and EC agent in 10x10 empty Gridworld using the same memory parameters as for experiments in section 4.1, but under uniform strategy. That is, agents freely explored the environment during 100 episodes and at the end the number of segments for DHTM and dictionary size for EC was recorded for each episode length. Figure 11 shows that DHTM's number of segments and EC's dictionary size grow linearly with the maximum episode length. In other words, since DHTM lacks generalisation abilities it largely serves as a trajectory buffer, in the same sense as a dictionary in EC.
+
+DHTM's computational complexity (time and memory) of the prediction step grows linearly with respect to the number of segments. Therefore, to use it for huge environments, there is need to prune unnecessary segments or remember only selected episodes. This is a common problem among episodic-like memories. The current approach to form SF also should be adjusted to be scalable, since it requires maximum planning horizon to be equal to the episode length. Among the improved approaches could be to employ n-step TD learning, hierarchical planning or SF caching, to recalculate it less frequently.
+
+# H.3 NOISE TOLERANCE
+
+In order to show the key difference between DHTM and its simplified EC version, we tested agents in the same AnimalAI setup as described in 4.2, but we add one noise variable to the VAE encoder. It's easy to see that the naive mapping from distributed VAE encoding to one categorical variable used for EC doesn't work in this case, since even identical observations will likely be encoded differently due to the noise variable. In contrast, since DHTM is able to handle distributed representations directly (as the name implies), it still can use non-noise variables independently to learn transitions.
+
+In this experiment, we vary DHTM's factor size (see Fig. 12). Since factors formed completely random, probability to include the noisy variable increases with factor size and this consequently worsen the performance of the agent. If the factor size is set to the total number of variables, DHTM becomes effectively unfactorised, similarly to its EC counterpart.
+
+![](images/f0dfbfbd6043498714d22f44c4dc462a709de084d890f71a4cd91c0efb533264.jpg)  
+Figure 11: Comparison of DHTM's number of segments with EC agent's dictionary size depending on episode length in Gridworld after uniform exploration during 100 episodes.
+
+![](images/f1243b4add336176d9c977179d4ed2e51d67a1f8b2ac4724177d37201eac4b3b.jpg)  
+Figure 12: Number of action steps the agent takes to reach the goal in a  $10 \times 10$ -meter room in the AnimalAI environment, using DHTM and EC models with a distributed Categorical VAE encoder that includes one additional noisy variable (six variables in total). DHTM's factor size is varied, determining the number of variables sampled for each factor.
+
+# I GLOSSARY
+
+Categorical Random Variable—a discrete random variable that can take on finite  $K$  possible states.
+
+Cortical Column or Minicolumn—a population of neurons in the neocortex that spans across layers and shares sensory input.
+
+Dendritic segment—a group of synapses (neuron's connections) that acts as an independent computational unit affecting the resulting neuron's activity.
+
+Factor Graph—bipartite graph representing the factorization of a probability distribution, with one part representing factor nodes and another—random variables.
+
+Factor size—number of variables connected to the factor. For simplicity, we refer only to the number of previous hidden states, used for prediction.
+
+Hidden Markov Model (HMM)—statistical model of a stochastic process where state probability depends only on previous state of the process.
+
+Multi-compartment neuron model—a model of neuron that divides neuron's connections into groups (segments) of different types (compartments), where each group may be considered as partly independent computational unit and groups of each compartment may affect the neuron's activity differently.
+
+Oracle—a mythical creature that gives up to our algorithms insightful information about the environment that is usually hidden.
+
+Successor Representations (SR)—a discounted sum of future [one-hot encoded] observations.
+
+Successor Features (SF)—a generalization of SR, a discounted sum of future latent states.
+
+Temporal Memory (TM)—in this work by this term we mean “memory for sequences”.
\ No newline at end of file
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+# LEARNING TASK BELIEF SIMILARITY WITH LATENT DYNAMICS FOR META-REINFORCEMENT LEARNING
+
+Menglong Zhang, Fuyuan Qian, Quanying Liu*
+
+Southern University of Science and Technology
+
+{zhangml2022, qianfy2023}@mail.sustech.edu.cn
+
+liuqy@sustech.edu.cn
+
+# ABSTRACT
+
+Meta-reinforcement learning requires utilizing prior task distribution information obtained during exploration to rapidly adapt to unknown tasks. The efficiency of an agent's exploration hinges on accurately identifying the current task. Recent Bayes-Adaptive Deep RL approaches often rely on reconstructing the environment's reward signal, which is challenging in sparse reward settings, leading to suboptimal exploitation. Inspired by bisimulation metrics, which robustly extracts behavioral similarity in continuous MDPs, we propose SimBelief—a novel metaRL framework via measuring similarity of task belief in Bayes-Adaptive MDP (BAMDP). SimBelief effectively extracts common features of similar task distributions, enabling efficient task identification and exploration in sparse reward environments. We introduce latent task belief metric to learn the common structure of similar tasks and incorporate it into the specific task belief. By learning the latent dynamics across task distributions, we connect shared latent task belief features with specific task features, facilitating rapid task identification and adaptation. Our method outperforms state-of-the-art baselines on sparse reward MuJoCo and panda-gym tasks.
+
+# 1 INTRODUCTION
+
+In meta-reinforcement learning, an agent is required to efficiently explore the environment to gather information relevant to the current task and use that information to adapt to new, unseen tasks (Duan et al., 2016; Finn et al., 2017; Gupta et al., 2018b; Humplik et al., 2019). However, in real-world scenarios, there are many distractions, and irrelevant information can cause the agent to engage in erroneous exploration behaviors, leading to the learning of a suboptimal policy. This challenge is exacerbated when the exploration space is vast, and rewards are sparse. Humans, on the other hand, have the ability to generalize across similar tasks by quickly identifying common patterns, such as recognizing that both opening a window and a drawer involve a pulling action. These shared structures can be abstracted and applied to new tasks through certain transformations, enabling rapid adaptation by leveraging prior knowledge. This process involves identifying similar features and relationships between tasks, with the underlying task belief facilitating effective knowledge transfer (Figure 1).
+
+Common approaches to task belief modeling include posterior sampling (Thompson, 1933; Osband et al., 2013; Rakelly et al., 2019)
+
+and Bayes-Adaptive Markov Decision Processes (BAMDPs) (Duff, 2002; Ghavamzadeh et al., 2015), where BAMDPs provide a structured framework for learning and adapting to new tasks by integrating prior information (Zintgraf et al., 2019). Previous works have addressed the issue of
+
+![](images/65bf7b36ee9e2d75ecd56421da7ecdea3b06a8bcb49728845655871e3f477b8d.jpg)  
+Figure 1: Learning shared structures among tasks in the latent space as task beliefs enables the agent to rapidly adapt to new tasks.
+
+uninformative rewards in meta-RL by using intrinsic rewards (Zhang et al., 2021c) to extract task-relevant information or exploration bonuses (Zintgraf et al., 2021) to enhance performance in hard exploration tasks. However, they often neglect the potential common structures shared across tasks.
+
+To efficiently learn the structural relationships embedded within task belief and address the challenges of effective task identification and rapid adaptation, it is necessary to develop methods that can efficiently recognize tasks with similar structures. This requires exploring strategies beyond BAMDP-based approaches, which may involve leveraging hierarchical reinforcement learning (Frans et al., 2018), or task embeddings (Liu et al., 2021; Lan et al., 2019) that can capture the underlying structure of tasks and facilitate transfer learning across related tasks.
+
+In this work, we propose SimBelief (Learning Similarity of Task Belief for Meta-RL) as a meta-RL framework under BAMDP to measure the difference of tasks in similar distributions by learning the dynamics and transferring capability in latent space. This results in a task belief similarity, which quantifies the representational relationships between any two random tasks in the learned latent space. However, directly reconstructing and exploring the environment in the latent space may lose crucial detailed information (Yarats et al., 2021; Kemertas & Aumentado-Armstrong, 2021; Hafner et al., 2019), which is important for accurately reconstructing the specific task and ensuring convergence to the Bayesian optimal policy(Zintgraf et al., 2019; Choshen & Tamar, 2023).
+
+Therefore, we combine the task belief similarity learned in the latent space, which provides an overall understanding of the task distribution and the relationships between tasks, with the belief of the specific task currently being learned, significantly enhancing the agent's adaptability to unknown environments. Compared to related meta-RL methods, our approach effectively learns dynamics in the latent space and leverages the learned latent representations to quickly discern relationships between tasks. This enables the agent to utilize prior knowledge from tasks, recognize the current environment, and accelerate exploration and adaptation to unseen tasks.
+
+Our main contributions are as follows:
+
+- We propose a task representation method for context-based meta-reinforcement learning algorithms in BAMDP, which enhances the agent's ability to recognize and adapt to unknown tasks by learning task belief similarity through latent reward model, transition model, and inverse dynamics model.  
+- In scenarios with sparse or uninformative reward signals, our algorithm more effectively extracts latent task space representations compared to other baselines and achieves state-of-the-art performance on sparse reward MuJoCo and panda-gym tasks.  
+- Our algorithm demonstrates stronger adaptability and generalization capabilities to out-of-distribution (OOD) tasks.  
+- We theoretically validate the effectiveness of the latent task belief metric in Bayes-Adaptive Markov Decision Processes (BAMDPs).
+
+# 2 BACKGROUND
+
+# 2.1 PROBLEM FORMULATION
+
+In meta-reinforcement learning (meta-RL), we consider a distribution  $p(\mathcal{T})$  over a space of tasks  $\mathcal{T}$ , where each task is modeled as a partially observable Markov decision process (POMDP; (Cassandra et al., 1994)) defined as  $\mathcal{T} = (\mathcal{S},\mathcal{A},\mathcal{O},P,R,\rho_0,\gamma ,H)$ . Here,  $\mathcal{S}$  and  $\mathcal{A}$  represent the shared state and action spaces across all tasks, while  $\mathcal{O}$  denotes the task-specific observation space. The transition function  $P(s^{\prime}\mid s,a)$ , reward function  $R(r\mid s,a)$ , and initial state distribution  $\rho_0$  define the task-specific dynamics.
+
+For context-based methods, the agent interacts with the environment over a horizon of  $H$  time steps in each episode of each task and utilizes a task-specific context  $\tau_{t} = (o_{1:t},a_{1:t - 1},r_{1:t - 1})$  to infer the hidden task dynamics. The objective is to learn a policy  $\pi (a_t\mid \tau_t)$  that can rapidly adapt to new tasks by leveraging the learned task belief encoded in the context. The full horizon comprises  $N$  task episodes, referred to as a meta-episode, each with its own set of  $T$  time steps, and the task context is updated as the agent gathers more observations. The objective in meta-RL is to maximize
+
+the expected return:
+
+$$
+J (\pi) = \mathbb {E} _ {\mathcal {T} \sim p (\mathcal {T})} \left[ \sum_ {t = 1} ^ {H} \mathbb {E} _ {\left(o _ {t}, a _ {t}, r _ {t}\right) \sim \pi} [ r _ {t} ] \right]. \tag {1}
+$$
+
+To address the multi-task adaptation problem in POMDPs, it is essential to avoid local optima and balance exploration with exploitation. Utilizing the framework of Bayes-Adaptive Markov Decision Processes (BAMDPs; (Duff, 2002)), we can represent uncertainty over task-relevant information in a belief space. By leveraging the current belief, we can reconstruct the dynamics model to better handle uncertainty, enabling more effective adaptation to the current task. The BAMDP is defined as  $M^{+} = (\mathcal{S}^{+},\mathcal{A},\rho_{0}^{+},P^{+},R^{+},H^{+})$ , where the hyper-state space  $S^{+} = \mathcal{S}\times \mathcal{B}$  combines the state space  $\mathcal{S}$  with the task belief space  $\mathcal{B}$ , representing the posterior over MDPs based on past experiences.  $P^{+}$  is the transition function,  $\rho_0^+$  is the initial hyper-state distribution, and  $R^{+}$  is the reward function. The total decision horizon is  $H^{+} = N\times H$ . The goal is to learn a meta-policy  $\pi^{+}(a_{t}\mid s_{t}^{+})$  that maximizes cumulative reward by balancing exploration of task uncertainty and exploitation of the current belief.
+
+# 2.2 BISIMULATION METRICS IN RL
+
+Bisimulation metrics provide a way to measure state similarity in reinforcement learning (RL) based on behavioral equivalence, focusing on transitions and rewards rather than raw state features. The concept originated in model checking (Givan et al., 2003). In RL, bisimulation aggregates states that lead to similar future outcomes. This allows for more efficient learning. Approximate bisimulation metrics (Ferns et al., 2004; Castro, 2020) extended this idea to high-dimensional environments, allowing RL agents to better generalize, reduce state space complexity, and improve exploration. These metrics are particularly useful in tasks with sparse rewards or partial observability (Guo et al., 2022).
+
+Definition 1 ( $\pi$ -Bisimulation Metric (Castro, 2020)). Given two states  $s_i$  and  $s_j$  from the state space  $S$ , the distance between these states can be evaluated using the  $\pi$ -bisimulation metric  $d_{\pi}(s_i, s_j)$  which measures behavioral similarity. We define the metric  $d_{\pi}(s_i, s_j)$  as:
+
+$$
+d _ {\pi} \left(s _ {i}, s _ {j}\right) = \left| R _ {s _ {i}} ^ {\pi} - R _ {s _ {j}} ^ {\pi} \right| + \gamma W _ {1} \left(d _ {\pi}\right) \left(P _ {s _ {i}} ^ {\pi}, P _ {s _ {j}} ^ {\pi}\right) \tag {2}
+$$
+
+where  $R_{s_i}^{\pi}$  and  $R_{s_j}^{\pi}$  represent the reward functions under policy,  $P^{\pi}$  denotes the transition distributions, and  $W_{1}$  is the Wasserstein distance between the distributions.
+
+# 2.3 UNINFORMATIVE REWARDS IN META-REINFORCEMENT LEARNING
+
+Uninformative rewards provide insufficient feedback for agents to quickly learn effective strategies (Andrychowicz et al., 2017; Dulac-Arnold et al., 2019). This issue can impede exploration, slow convergence, and lead to suboptimal policies. To address this challenge, methods such as intrinsic motivation and curiosity-driven exploration have been employed to encourage agents to explore state spaces even when extrinsic rewards are minimal (Pathak et al., 2017; Burda et al., 2018). Reward shaping techniques modify the reward function to offer more informative feedback, thereby accelerating learning (Ng et al., 1999). Balancing the exploration-exploitation trade-off is also crucial for efficient learning (Sutton, 2018). Enhancing meta-RL agents' ability to learn from uninformative rewards can significantly improve their performance across complex, real-world environments by incorporating advanced exploration strategies and reward augmentation methods (Gupta et al., 2018b).
+
+In this work, we learn the dynamics of multiple tasks in a latent space and use a task belief metric to measure the differences between similar tasks. By extracting similar structures within the latent space and representing these structures as task belief similarity, we map them to the current real and unknown environment to assist the agent in exploration. This approach aims to achieve rapid adaptation to multiple tasks in environments with uninformative rewards.
+
+# 3 METHOD
+
+In this section, we provide a detailed introduction to SimBelief (Figure 2), an effective off-policy meta-RL approach for online adaptation within the BAMDP framework. First, we propose a latent
+
+task belief metric to quantify the relationships between different tasks in the latent space in section 3.1. Then, we explain how task belief similarity is learned within the learned latent dynamics in section 3.2. We also describe the overall algorithmic process of SimBelief in detail in section 3.3. Finally, we theoretically prove the conditions under which the policy can transfer between tasks in the latent space in section 3.4.
+
+# 3.1 LATENT TASK BELIEF METRIC
+
+Previous work (Zhang et al., 2021b) learns an environment encoder to capture the relationships between Block MDPs (Du et al., 2019) for multi-task generalization. In (Zhang et al., 2021c), information gain is used as an intrinsic reward for exploration. However, in these approaches, the agent needs to know the current task information (i.e., task ID) during training. In contrast, in context-based meta-RL, these approaches limit adaptability to out-of-distribution tasks, as the agent must infer the task being executed from historical information and reason about the current task distribution during adaptation. Including task ID during training can partially constrain the agent's reasoning capabilities. In real-world scenarios, relying solely on the environment's inherent reward signal or manually designed intrinsic rewards often fails to effectively capture the common structure of tasks (Zheng et al., 2020). Therefore, learning the structural similarities between tasks and leveraging belief to facilitate rapid transfer across similar tasks is more advantageous in complex environments, especially those with sparse reward signals. To measure the relationships between tasks and make it applicable within the BAMDP, we define the latent task belief metric.
+
+Definition 2 (Latent Task Belief Metric). Given two latent task beliefs  $b_{i}$  and  $b_{j}$ , with corresponding samples  $z_{i}$  and  $z_{j}$  identified by their latent representations, let  $d_{\pi}$  denote the metric that evaluates the distance between these two task beliefs. We define  $d_{\pi}(z_i,z_j)$  as follows:
+
+$$
+\begin{array}{l} d _ {\pi} (z _ {i}, z _ {j}) = \left| R _ {i} ^ {\pi} (s _ {i} ^ {+}, a _ {i}) - R _ {j} ^ {\pi} (s _ {j} ^ {+}, a _ {j}) \right| + W _ {2} (d _ {\pi}) \left(T _ {i} ^ {\pi} (s _ {i} ^ {+}, a _ {i}), T _ {j} ^ {\pi} (s _ {j} ^ {+}, a _ {j})\right) \\ + \left\| I _ {i} ^ {\pi} \left(s _ {i} ^ {+}, s _ {i} ^ {\prime +}\right) - I _ {j} ^ {\pi} \left(s _ {j} ^ {+}, s _ {j} ^ {\prime +}\right) \right\| _ {1} \tag {3} \\ \end{array}
+$$
+
+where  $s_i^+ = (g(s_i), z_i)$  is the augmented state corresponding to the task  $i$ ,  $r_i = R_i^\pi(s_i^+, a_i)$  is the reward model,  $s'_i^+ = T_i^\pi(s_i^+, a_i)^1$  is the transition model, and  $a_i = I_i^\pi(s_i^+, s'_i^+)$  is the inverse dynamics model.
+
+Here, we map  $s$  to a latent compressed space as  $g(s)$ , and learn the task dynamics in the compressed space. Latent dynamic space can be defined as  $\mathcal{S}^{+} = \mathcal{G}(\mathcal{S})\times \mathcal{Z}$ . In Figure 2,  $z_{l}$  represents the sample from latent task belief  $b_{l}$  in the latent space and captures the similarity between different tasks. All task distributions share a common latent dynamic space, with different tasks distinguished by different  $z_{l}$ , where  $z_{l}$  contains the dynamic similarity information between any two tasks. We employ the inverse dynamics model to predict  $a$ , as it not only helps retain information relevant to what the agent can control but also enhances the agent's reasoning capability (see Appendix G.2). Unlike in Definition 1, here we use the dynamic model to measure task belief similarity to enhance exploration efficiency in unknown environments, while preserving the agent's control information in sparse reward settings. This approach is more effective for identifying similar tasks.
+
+# 3.2 LEARNING TASK BELIEF SIMILARITY IN LATENT SPACE
+
+Our goal is to use the latent task belief metric to capture the similarity information between any two tasks, which is embedded within the task belief. This enables SimBelief to achieve rapid task reasoning and adaptation. From the replay buffer  $\mathcal{D}$ , we randomly sample trajectories  $\tau$  from two tasks (i.e., task  $i$  and task  $j$ ) in the task set  $M$ . The trajectory is then fed into the context encoder  $q_{\phi}$  to obtain historical information  $h^i$  and  $h^j$ , which serve as prior knowledge for the tasks and are input into the latent space. The latent space primarily consists of three components: (1) a state encoder  $g_{\theta}$  that maps the original state space to the latent state space  $S^{+}$ , (2) a belief similarity learner  $\psi_l$  that maps  $h$  to the latent task belief  $b_l$ , and (3) a latent dynamics model  $p_{\theta}$  conditioned on the learned latent task belief (Figure 2).
+
+![](images/cfbc8f2103e9e531f1b1fe3e39ddfdcec4231edfd38340305c4af1d2d0f80103.jpg)  
+Figure 2: SimBelief architecture. Our framework consists of three components: learning latent task belief similarity, reconstructing the specific task, and the policy. In the compressed state space,  $\psi_l$  learns the latent task belief  $b_l^i$  through latent dynamics shared among all tasks in the distribution. In the real environment, we reconstruct the dynamics of the specific task using  $\psi_r$  to obtain the specific task belief  $b_r^i$ . The policy  $\pi$  is conditioned on the state and the combined belief  $b^i$ , which integrates  $b_r^i$  and  $b_l^i$ . In this paper, we use  $l$  and  $r$  to denote the latent space and real space, respectively, and  $i$  and  $j$  to represent task  $i$  and task  $j$ , respectively.
+
+To obtain the latent dynamic representation of all tasks conditioned on  $z_{l}$ , we reconstruct the latent dynamics  $p_{\theta}((s'^{+},r,a)\mid z_l)$ , which includes the reward model  $R_{l}$ , the transition model  $T_{l}$ , and the inverse dynamics model  $I_{l}$  through one-step predictions. The learning objective for the latent dynamics can be expressed as:
+
+$$
+\mathbb {E} _ {p (M)} \left[ \mathbb {E} _ {\psi_ {l} \left(z _ {l} \mid h\right)} \left[ \log p _ {\theta} \left(\left(s ^ {\prime +}, r, a\right) \mid z _ {l}\right) \right] \right]. \tag {4}
+$$
+
+This objective enables the model to capture task-specific latent dynamics by maximizing the likelihood of predicting the next augmented state  $s^{\prime +}$ , reward  $r$ , and action  $a$  given the latent task belief  $z_{l}$ .
+
+We aim to learn the similarity relationship between any two tasks using the latent task belief metric and the latent dynamics learned through Equation 4. This similarity relationship is obtained by optimizing the belief similarity learner  $\psi_{l}$ . The learning objective for task belief similarity using  $\psi_{l}$  is defined as follows:
+
+$$
+\begin{array}{l} \mathcal {L} _ {b i s i m} (\psi_ {l}) = \| \psi_ {l} (h _ {i}) - \psi_ {l} (h _ {j}) \| _ {1} - \left| \hat {R} \left(s _ {i} ^ {+}, a _ {i}\right) - \hat {R} \left(s _ {j} ^ {+}, a _ {j}\right) \right| - W _ {2} \left(\hat {T} (\cdot \mid s _ {i} ^ {+}, a _ {i}), \hat {T} (\cdot \mid s _ {j} ^ {+}, a _ {j})\right) \\ - \left\| \hat {I} (\cdot \mid s _ {i} ^ {+}, s _ {i} ^ {\prime +}) - \hat {I} (\cdot \mid s _ {j} ^ {+}, s _ {j} ^ {\prime +}) \right\| _ {1}, \tag {5} \\ \end{array}
+$$
+
+where the gradients of dynamics model are stopped, and  $b_{l}^{i} = \psi_{l}(h_{i})$ . We use the 2-Wasserstein metric because the  $W_{2}$  metric has a convenient closed-form solution (Zhang et al., 2021a). During training, through Equation 4 and 5,  $z_{l}^{i}$ , sampled from  $b_{l}^{i}$ , is optimized to encode both the latent dynamic information required to reconstruct task  $i$  and the similarity information between task  $i$  and any other task in the task set  $M$ . Through this approach, we can utilize  $\psi_{l}$  to identify contextual information at the current timestep and represent the similarity between any tasks through its output  $b_{l}$ .
+
+# 3.3 SIMBELIEF ALGORITHM
+
+We now proceed to describe how the learned task belief similarity (Section 3.2) can be leveraged to enhance the exploration capability and convergence efficiency of meta-RL. The algorithm pseudocode can be found in Appendix C.
+
+Reconstructing the specific task. In the BAMDP framework (Ghavamzadeh et al., 2015; Zintgraf et al., 2019), even with extremely sparse rewards, it is necessary to construct the specific task's reward model in the original state space using a variational auto-encoder (VAE, (Kingma, 2013))
+
+This is because reconstructing the specific task in the latent state space may result in the loss of crucial information, as the latent space is solely used for learning task belief similarity. Additionally,
+
+along the temporal dimension, we only reconstruct the past history, as we use a one-step prediction approach in the latent space. This approach effectively captures local information about the task distribution, enhancing the agent's reasoning ability for future predictions (see the upper part of Figure 2). The objective is to maximise
+
+$$
+\mathcal {L} _ {\mathrm {V A E}} (\phi) = \mathbb {E} _ {q _ {\phi} \left(z _ {r} \mid \tau : t\right)} \left[ \log p _ {\phi} \left(\tau : t \mid z _ {r}\right) \right] - D _ {\mathrm {K L}} \left(q _ {\phi} \left(z _ {r} \mid \tau : t\right) \| p \left(z _ {r}\right)\right), \tag {6}
+$$
+
+where  $q_{\phi}$  generates the specific task belief  $z_{r}$  for the current timestep using only the information from the historical trajectory  $\tau_{:t}$ . We consider  $q_{\phi}$  as the forward reasoning process to infer the task belief  $z_{r}$ , corresponding to  $z_{r} \sim \psi_{r}(b_{r} \mid h) q_{\phi}(h \mid \tau_{:t})$  in the Algorithm 1.
+
+Integrating latent task belief for effective online exploration. The learned latent task belief  $z_{l} \sim b_{l}$  enables the agent to quickly recall the similarities between different tasks during interactions with the environment, enhancing the efficiency of reasoning and exploration. This is the primary reason why SimBelief achieves robust adaptation to OOD tasks (Figure 4). Since the latent state space is an abstraction of the original task, the latent task belief may overlook the specific task information required by the agent during exploration in the real environment. Therefore, we choose to project the two distributions into a joint space  $\mathcal{B}$ , and then use a Gaussian mixture of the two distributions  $b_{r} = (\mu_{\psi_{r}}(h), \sigma_{\psi_{r}}(h))$  and  $b_{l} = (\mu_{\psi_{l}}(h), \sigma_{\psi_{l}}(h))$ , combined with the real environment's augmented state derived from  $s_{t}$ , as the input to the policy  $\pi$ . The formulation of the Gaussian mixture distribution  $b$  is given by:
+
+$$
+b = w _ {r} \cdot \mathcal {N} \left(z _ {r} \mid \mu_ {r}, \sigma_ {r} ^ {2}\right) + w _ {l} \cdot \mathcal {N} \left(z _ {l} \mid \mu_ {l}, \sigma_ {l} ^ {2}\right), \tag {7}
+$$
+
+where  $\mathcal{N}(z\mid \mu_r,\sigma_r^2)$  and  $\mathcal{N}(z\mid \mu_l,\sigma_l^2)$  are Gaussian distributions with means  $\mu_r$  and  $\mu_l$ , and variances  $\sigma_r^2$  and  $\sigma_l^2$ , respectively. The weights  $w_{r}$  and  $w_{l}$  satisfy  $w_{r} + w_{l} = 1$  and are non-negative (see Appendix G.3).
+
+In the early stages of training, the latent task belief enhances the agent's ability to identify tasks and explore efficiently in the real environment, providing a high-level understanding of the overall task distribution (see Appendix F.2). However, for the algorithm to converge stably, it need to incorporate finer-grained information about specific tasks, which is captured in the specific task belief  $b_{r}$ . Therefore, we minimize the discrepancy between the latent belief  $b_{l}$  and the specific task belief  $b_{r}$  when training  $\psi_{l}$ . This is achieved by introducing a KL divergence term between the two beliefs, serving as a regularization for the latent task belief. Combined with Equation 5, the overall optimization objective of  $\psi_{l}$  can be written as:
+
+$$
+\mathcal {J} (\psi_ {l}) = \mathbb {E} _ {p (M)} \left[ \mathcal {L} _ {\text {b i s i m}} (\psi_ {l}) + D _ {\mathrm {K L}} \left(\hat {q} _ {\phi} \left(z _ {r} \mid \tau_ {: t}\right) \| p (z _ {l})\right) \right], \tag {8}
+$$
+
+where the gradient of  $\hat{q}_{\phi}$  is stopped, and  $p(z_l)$  represents the distribution of  $z_{l} \sim \psi_{l}(b_{l} \mid h)$ .
+
+SimBelief utilizes Soft Actor-Critic (SAC) (Haarnoja et al., 2018) to optimize the policy, where the optimization of latent dynamics and latent task similarity is conducted jointly with policy optimization to ensure convergence to near Bayes-optimal policy. To enhance the consistency between latent dynamics and real environment exploration, without introducing perturbations that could disrupt belief similarity in the latent space, we use the loss of Q-function as the objective to train the offset  $(\Delta \mu, \Delta \sigma)$  of  $b_{l} = (\mu_{\psi_{l}}(h) + \Delta \mu, \sigma_{\psi_{l}}(h) + \Delta \sigma)$  (see Appendix G.4). The loss of SAC after integrating latent dynamic information is
+
+$$
+\mathcal {J} (\pi) = \mathbb {E} _ {(s, a) \sim \mathcal {D}} [ \alpha \log (\pi (a | (s, b))) - Q _ {\theta} ((s, b), a) ], \tag {9}
+$$
+
+where  $(s,b)$  is the augmented state in the real environment under the BAMDP framework.
+
+# 3.4 THEORETICAL ANALYSIS
+
+Based on the latent task belief metric  $d_{\pi}(z_i, z_j)$  defined in Definition 2, we can establish the following theorems regarding the difference in task dynamics between two tasks in the latent space within BAMDP.
+
+Theorem 1 (Value difference bound). Given two tasks  $\mathcal{M}_i$  and  $\mathcal{M}_j$  in the latent space with states  $s_i^+, s_j^+ \in S^+$ , and let  $V^\pi$  be the value function of policy  $\pi$ , the value difference bound between the tasks can be given by:
+
+$$
+\left| V ^ {\pi} \left(s _ {i} ^ {+}\right) - V ^ {\pi} \left(s _ {j} ^ {+}\right) \right| \leq d _ {\pi} \left(z _ {i}, z _ {j}\right), \tag {10}
+$$
+
+where  $d_{\pi}(z_i,z_j)$  is the latent task belief metric.
+
+![](images/1e3bc2e52a264df2d203aa68d51251f43347e905f8035777ff4e92ce4be2276f.jpg)
+
+![](images/7b1b38c4b6f3107006aa9f83f03d392f0a9c52ddc451a6407e305150c87b9300.jpg)
+
+![](images/1d4c20d6ead73264d7a079986c092a8e7546d8487842dfa267471aeacc06278b.jpg)
+
+![](images/a8fb221aaee252bd9bc25cbfdb86a7e96cf3f867e0778cbec1526717c8298dd1.jpg)  
+Figure 3: Meta-testing performance on sparse reward MuJoCo and panda-gym tasks over 3 random seeds. Our algorithm, SimBelief, demonstrates superior online adaptation capabilities compared to other algorithms.
+
+![](images/4aadc4613d918cdc61f911f6261b6bf75fb080b6f0d1d72302c6c067b940da5f.jpg)
+
+![](images/08d46c6e182f9a612d17003461cfc4c455876bb89761f73f29b578d8b0277b94.jpg)
+
+This theorem demonstrates that tasks with similar latent task beliefs  $z_{i}$  and  $z_{j}$  will have similar value function under the same policy  $\pi$ .
+
+Theorem 2 (Latent transfer bound). Let  $Q_{\mathcal{M}_j}^*$  be the optimal  $Q$ -function for task  $\mathcal{M}_j$ . The difference between  $Q_{\mathcal{M}_j}^*$  and the  $Q$ -function of the policy  $\pi$  learned from task  $\mathcal{M}_i$ , applied to task  $\mathcal{M}_j$ , is bounded as follows:
+
+$$
+\left\| Q _ {\mathcal {M} _ {j}} ^ {*} - \left[ Q _ {\mathcal {M} _ {i}} ^ {\pi} \right] _ {\mathcal {M} _ {j}} \right\| _ {\infty} \leq \epsilon_ {R} + \gamma \left(\epsilon_ {T} + \epsilon_ {I} + \| z _ {i} - z _ {j} \| _ {1}\right) \frac {R _ {\max }}{2 (1 - \gamma)}. \tag {11}
+$$
+
+This extended transfer bound theorem provides a formal bound on how well a policy  $\pi$  trained on task  $\mathcal{M}_i$  will transfer to task  $\mathcal{M}_j$ .  $\epsilon_R$ ,  $\epsilon_T$ , and  $\epsilon_I$  are approximation errors for rewards, transitions, and inverse dynamics. All proofs are provided in Appendix B.
+
+# 4 EXPERIMENTS
+
+In this section, we evaluate SimBelief on sparse-reward tasks in MuJoCo (Finn et al., 2017; Rakelly et al., 2019) and the more challenging panda-gym (Gallouédec et al., 2021) environment. We aim to address the following questions: 1. Can SimBelief achieve fast online adaptation in sparse reward tasks? 2. Can SimBelief leverage learned latent belief similarity representations to enhance out-of-distribution generalization? 3. What is the impact of latent task representations on rapid exploration? 4. How does the latent space correspond to the real environment?
+
+**Environments and baselines:** We conducted experiments on six complex sparse reward tasks, including Point-Robot-Sparse, Cheetah-Vel-Sparse, Walker-Rand-Params, Panda-Reach, Panda-Push, and Panda-Pick-And-Place (see Appendix E). While MuJoCo tasks are commonly used by current meta-learning algorithms, panda-gym simulates real-world robotic arm movements with extremely sparse rewards, making it a better benchmark for evaluating an algorithm's ability to handle real-world tasks. We compared SimBelief against PEARL (Rakelly et al., 2019), which is based on posterior sampling, and MetaCURE (Zhang et al., 2021c), which learns a separate policy, as well as VariBAD (Zintgraf et al., 2019), HyperX (Zintgraf et al., 2021), and RL² (Duan et al., 2016), which are based on the BAMDP framework (see Appendix D).
+
+Online Adaptation Performance. During the training phase, we performed meta-testing by calculating the meta-episode average return and success rate across different tasks to evaluate the algorithm's online performance. As shown in Figure 3, SimBelief consistently performed well across all tasks and exhibited superior adaptation capabilities compared to other algorithms. BAMDP-based algorithms often require more interactions with the environment to converge to a Bayes-optimal
+
+![](images/66895667903fc92748c8af7f4c5ac18c9bb6fea9f26130e7def1e631b2e340ba.jpg)
+
+![](images/32380b23d02687327b40556efa143214675995117fd4a89466b124ebc56b53aa.jpg)
+
+![](images/cd45b6ffdab5a36930c0356db849e0364d9dc974dccbaff86c767c884b2d4bc6.jpg)
+
+![](images/e9b875125402b094196b319e0b084231d542dab82a14c6bed00005240552d81d.jpg)  
+Figure 4: Average test performance for the first 5 rollouts on out-of-distribution tasks (Cheetah-Vel-Sparse and Point-Robot-Sparse) shows that as the task distribution gradually deviates from the training distribution, SimBelief demonstrates more efficient and robust adaptation compared to other algorithms, while still maintaining a high return.
+
+![](images/88943ff163895f85a4673c7b0c7eab21dad0df1bf8a2c6fc8ff5aab60f526e44.jpg)
+
+![](images/68f8c9cc6214762d32ad3050ca3452d4ee3896eee0f7dbb63eec3a4dc9745974.jpg)
+
+policy, whereas our algorithm, by integrating task belief similarity in the latent space, significantly improved the agent's ability to effectively extract task-relevant information in sparse reward environments, thus accelerating convergence. In sparse reward tasks like those in panda-gym, the performance of Varibad, HyperX, and  $\mathrm{RL}^2$  illustrates that overly relying on the environment's reward signal while neglecting task similarity information makes it difficult to achieve online adaptation in more complex scenarios. Additionally, SimBelief demonstrated more stable performance compared to posterior sampling-based algorithms like PEARL and MetaCURE, indicating that SimBelief accurately learned the common structure of the task distribution.
+
+Out-of-distribution Task Inference and Adaptation. We evaluated the algorithm's reasoning and generalization capabilities on out-of-distribution (OOD) tasks. In Figure 4, we show the agent's out-of-domain adaptation and generalization performance on the Cheetah-Vel-Sparse (top) and Point-Robot-Sparse (bottom) tasks. The leftmost column represents the task range used during the training phase, where the agent was trained by generating task goals within this range. During the testing phase, we adjusted the velocity range for Cheetah-Vel-Sparse and the semi-circle radius for Point-Robot-Sparse to assess the agent's robustness and task inference capabilities within five episodes. The results indicate that SimBelief exhibits stable out-of-distribution generalization, demonstrating that learned latent belief similarity representations can be leveraged to enhance out-of-distribution task inference ability. The latent task belief similarity we proposed also equips the agent with strong task transferability.
+
+Exploration Performance. We visualized the exploration and adaptation process of the agent in the Point-Robot-Sparse environment over two episodes, as shown in Figure 5. The test target appeared within a semicircle with a radius of 1.2, and five positions on the semicircle were randomly selected as the agent's targets. Compared to other algorithms, SimBelief successfully identified the randomly appearing target location within the semicircle and controlled the robot to reach the target point within two episodes. Moreover, in the out-of-distribution setting, SimBelief demonstrated more efficient learning of a near Bayes-optimal policy compared to BAMDP-based algorithms like HyperX. The latent task belief enabled the agent to efficiently analogize and identify tasks, leading to more effective exploration.
+
+Task Belief Visualization. We use t-SNE (Van der Maaten & Hinton, 2008) to visualize SimBelief's latent task belief and specific task belief (Figure 6), aim to illustrate how the task common structure learned in the latent space contributes to effective reconstruction of the real environment. This visualization reveals the underlying reason for SimBelief's ability to achieve rapid out-of-distribution generalization. The latent space, as an abstraction and holistic representation of the task distri
+
+![](images/20bba4239f651198a32ff6e85cd2949a5e4613dc8f69aa1d1e5e219d27277e4f.jpg)
+
+![](images/1b4dac1e2b5a012dfd873bc59c20a2e9af8ace90cccc2dae81478aba0a126a69.jpg)
+
+![](images/c6be0fb69eee078485e38094048911fd615768d3ac264607f9f2707fb9ce8065.jpg)
+
+![](images/2b7acd0c0a6f4458df6f8d0c1c800d91ebeb350b5f9e22ed9755b68ae701b1b4.jpg)  
+Figure 5: Exploration and adaptation performance in the 5 random out-of-distribution (radius = 1.2) tasks of Point-Robot-Sparse. SimBelief is capable of robustly learning a near Bayes-optimal policy.
+
+![](images/5bf349c9cdfb613791dc53ea6a53c3bca1445054f608a04c22ef02b0811397a5.jpg)
+
+![](images/ded09a0e23ccc2b5f064ec575a05b376a82e1419463aea001f146b081c55bc4d.jpg)
+
+![](images/bd7c924a862235c870b3f0dafd4c43d6dd9cd3ae476fa428fd7ddc5815f6e80d.jpg)
+
+![](images/bba4cfca5cd81f8452042a9d24ecdc2d198cd791f42f420b551b346ea59ae558.jpg)
+
+![](images/5e6663480f9c810e8e10fc52cdf5494629f6ff52f7eae8769f5aba6fb7a18933.jpg)
+
+![](images/2399c37e55f06fb979fe0f18badaf6613bf10c9b5b50ec52fb79a57bac2b70ec.jpg)
+
+![](images/be774e662ce9b11808015876a3f1bd0b5520f972748d4b13a3f813829c7b819a.jpg)  
+Figure 6: t-SNE visualization of task beliefs learned by the algorithm on 10 randomly sampled OOD tasks. The velocity range for Cheetah-Vel-Sparse is set to 3-4, and the semicircle radius for Point-Robot-Sparse is 1.1. SimBelief demonstrates a more distinct representation in the latent space, enhancing the specific task belief to represent similar tasks in the real environment.
+
+![](images/1383baa910c3e9f2a3bf98ce616300b487358a728702a3422b4f653ab9eb5539.jpg)
+
+![](images/9aff4a8c76dd5dd972e716b156f0407b1ffc6dfaceaba19187e2f0e534480b71.jpg)
+
+![](images/1c702aed6a4d5dd9bdfbff949a02ce786c08e3f961c35fdf27b95db76ac523d7.jpg)
+
+bution, enables efficient transfer of essential task information. This, in turn, allows the agent to leverage prior knowledge for identifying and reasoning quickly about unknown environments.
+
+The mechanism of SimBelief for rapid adaptation. SimBelief, through the latent task belief, essentially learns the transfer relationships between knowledge across different tasks. In extremely sparse reward scenarios, once the agent succeeds in one task, this prior knowledge of success can be quickly propagated to other similar tasks via the latent task belief, enabling the agent to rapidly adapt to similar types of tasks. The latent task belief acts as a bridge, facilitating the generalization of past successes to new, related tasks, significantly reducing exploration time and computational resources. A more detailed discussion is provided in Appendix F.
+
+# 5 RELATED WORKS
+
+Task representation in reinforcement learning focuses on how agents encode and utilize task information to enhance learning efficiency and generalization. Meta-reinforcement learning enables agents to quickly adapt to new tasks by learning task priors; for instance, (Finn et al., 2017) introduced Model-Agnostic Meta-Learning (MAML) for rapid adaptation with minimal updates, while
+
+(Rakelly et al., 2019) proposed PEARL to learn latent task representations for fast probabilistic adaptation. (Zintgraf et al., 2019; Dorfman et al., 2021) learn a latent variable model of task distribution for efficient Bayes-adaptive RL, and (Gupta et al., 2018a) explored unsupervised meta-learning without explicit task labels. Yuan & Lu (2022); Choshen & Tamar (2023) apply contrastive learning to enhance encoder representation. (Lee et al., 2023) decomposes complex tasks into subtasks to handle non-parametric task variability. In multi-task RL, (Teh et al., 2017) introduced Distral to learn shared policies across tasks with task-specific adaptations. Contextual task representations have also been explored: (Sodhani et al., 2021) use metadata to learn interpretable representations, and (Hausman et al., 2018) learned embedding spaces for transferable skills. These studies underscore the importance of efficient task representation to improve RL performance and generalization.
+
+Exploration with task inference. Integrating exploration with task inference in reinforcement learning enables agents to learn efficient policies by understanding task structures. Information-theoretic approaches encourage exploration by maximizing the entropy of state visitation distributions, promoting diverse behaviors (Liu & Abbeel, 2021). (Raileanu & Rocktäschel, 2020) propose a type of intrinsic reward which encourages the agent to take actions that lead to significant changes in its learned state representation. (Wan et al., 2023) introduce DEIR, a method that enhances exploration efficiency and robustness by using discriminative model-based episodic intrinsic rewards. (Yang et al., 2023) propose an unsupervised skill discovery method through contrastive learning among behaviors. (Rana et al., 2023) propose a low-level residual policy for skill adaptation enabling downstream RL agents to adapt to unseen tasks Planning-based methods enable agents to plan exploratory actions that maximize information gain by learning world models (Sekar et al., 2020). (Xie et al., 2020) leverage the idea of partial amortization for fast adaptation at test time.
+
+Bisimulation for control. Bisimulation has been effectively applied in various control tasks, demonstrating its value in reducing computational burdens without sacrificing policy performance (Zhang et al., 2021a). (Zhang et al., 2021b) propose a method for learning state abstractions that generalize across tasks, improving sample efficiency and performance in multi-task and meta-reinforcement learning by leveraging shared dynamics in complex environments. (Gelada et al., 2019) has leveraged bisimulation for efficient exploration by ensuring the agent treats bisimilar states similarly, reducing sample complexity. (Hansen-Estruch et al., 2022) propose a form of state abstraction that captures functional equivariance for goal-conditioned RL. (Kemertas & Aumentado-Armstrong, 2021) generalize value function approximation bounds for on-policy bisimulation metrics to non-optimal policies. These studies highlight the increasing applicability of bisimulation in reinforcement learning and control tasks. They offer a variety of solutions that reduce sample complexity, guide exploration, and enhance learning performance, particularly in goal-conditioned settings and beyond.
+
+# 6 CONCLUSION
+
+We present SimBelief, a meta-RL framework aimed at improving the agent's ability to quickly adapt and generalize in real-world sparse reward tasks. At the core of SimBelief is the latent task belief metric, which learns the similarity between the dynamics of various tasks in a shared latent space and represents this similarity as task belief similarity. This helps the agent efficiently capture the common structure of the task distribution, facilitating the recognition and reasoning about unknown tasks. By combining the latent task belief with the specific task belief learned through interactions in the real environment, SimBelief demonstrates strong adaptation and generalization capabilities during exploration, particularly on out-of-distribution tasks. It overcomes the slow convergence and inefficiency issues inherent in BAMDP. We believe that SimBelief will inspire future research on agent generalization, providing a powerful tool for enabling agents to more efficiently tackle the challenges and complexities of real-world environments.
+
+# ACKNOWLEDGMENTS
+
+This work was funded in part by National Natural Science Foundation of China (62472206), Shenzhen Excellent Youth Project (RCYX20231211090405003), Shenzhen Science and Technology Innovation Committee (2022410129, KJZD20230923115221044, KCXFZ20201221173400001).
+
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+Yuri Burda, Harrison Edwards, Amos Storkey, and Oleg Klimov. Exploration by random network distillation. arXiv preprint arXiv:1810.12894, 2018.  
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+
+# A THEORETICAL BACKGROUND AND ANALYSIS
+
+In this section, we introduce the theoretical background and the rationale behind the design of the latent task belief metric.
+
+Definition 3 (Bisimulation Relation (Givan et al., 2003)). Given an MDP  $\mathcal{M}$ , an equivalence relation  $E \subseteq S \times S$  is a bisimulation relation if whenever  $(s, t) \in E$ , the following properties hold, where  $S_E$  is the state space  $S$  partitioned into equivalence classes defined by  $E$ :
+
+$$
+\forall a \in A, \mathcal {R} (s, a) = \mathcal {R} (t, a) \tag {12}
+$$
+
+$$
+\forall a \in A, \forall c \in S _ {E}, \mathcal {P} (s, a) (c) = \mathcal {P} (t, a) (c), \text {w h e r e} \mathcal {P} (s, a) (c) = \sum_ {s ^ {\prime} \in c} \mathcal {P} (s, a) \left(s ^ {\prime}\right) \tag {13}
+$$
+
+Definition 4 (Diffuse Metric). A diffuse metric measures the distance between two points by considering not only the shortest path but also the distribution of multiple paths between the points. The diffuse metric satisfies the following properties:
+
+1. Non-negativity: For any two points  $x$  and  $y$ ,  $d(x, y) \geq 0$ , and  $d(x, y) = 0$  if and only if  $x = y$ .  
+2. Symmetry: For any two points  $x$  and  $y$ ,  $d(x, y) = d(y, x)$ .  
+3. Triangle inequality: For any three points  $x, y$ , and  $z$ ,  $d(x, z) \leq d(x, y) + d(y, z)$ .
+
+These properties ensure that the diffuse metric behaves as a valid distance measure across the space.
+
+To prove that the metric  $d_{\pi}(z_i,z_j)$  in Definition 2 is valid, we show it satisfies three key properties: non-negativity, symmetry, and the triangle inequality.
+
+Proof. Each term in the metric  $d_{\pi}(z_i, z_j)$  is non-negative:  $|R_i^\pi(s_i^+, a_i) - R_j^\pi(s_j^+, a_j)|$  is the absolute difference in rewards, which is non-negative.  $W_2(T_i^\pi(s_i^+, a_i), T_j^\pi(s_j^+, a_j))$  is the Wasserstein distance, which is non-negative.  $\| I_i^\pi(s_i^+, s_{i+1}^+) - I_j^\pi(s_j^+, s_{j+1}^+) \|_1$  is an  $L_1$ -norm, which is non-negative. Thus,  $d_{\pi}(z_i, z_j) \geq 0$ . Additionally, all three types of distances satisfy symmetry, hence  $d_{\pi}(z_i, z_j) = d_{\pi}(z_j, z_i)$ .
+
+To derive the triangle inequality for latent task belief metric, we need to verify that for any three task beliefs  $z_{i}, z_{j}$ , and  $z_{k}$ , the following holds:
+
+$$
+d _ {\pi} \left(z _ {i}, z _ {k}\right) \leq d _ {\pi} \left(z _ {i}, z _ {j}\right) + d _ {\pi} \left(z _ {j}, z _ {k}\right)
+$$
+
+Expand the distance  $d_{\pi}(z_i,z_j)$  and  $d_{\pi}(z_j,z_k)$
+
+$d_{\pi}(z_i,z_j) = |R_i^\pi (s_i^+,a_i) - R_j^\pi (s_j^+,a_j)| + W_2(d_\pi)(T_i^\pi (s_i^+,a_i),T_j^\pi (s_j^+,a_j)) + \| I_i^\pi (s_i^+,s_i'^+) - I_j^\pi (s_j^+,s_j'^+)\| _1$  and similarly,
+
+$$
+d _ {\pi} \left(z _ {j}, z _ {k}\right) = \left| R _ {j} ^ {\pi} \left(s _ {j} ^ {+}, a _ {j}\right) - R _ {k} ^ {\pi} \left(s _ {k} ^ {+}, a _ {k}\right) \right| + W _ {2} \left(d _ {\pi}\right) \left(T _ {j} ^ {\pi} \left(s _ {j} ^ {+}, a _ {j}\right), T _ {k} ^ {\pi} \left(s _ {k} ^ {+}, a _ {k}\right)\right) + \left\| I _ {j} ^ {\pi} \left(s _ {j} ^ {+}, s _ {j} ^ {\prime +}\right) - I _ {k} ^ {\pi} \left(s _ {k} ^ {+}, s _ {k} ^ {\prime +}\right) \right\| _ {1}
+$$
+
+Expand the distance  $d_{\pi}(z_i,z_k)$
+
+$$
+d _ {\pi} (z _ {i}, z _ {k}) = | R _ {i} ^ {\pi} (s _ {i} ^ {+}, a _ {i}) - R _ {k} ^ {\pi} (s _ {k} ^ {+}, a _ {k}) | + W _ {2} (d _ {\pi}) (T _ {i} ^ {\pi} (s _ {i} ^ {+}, a _ {i}), T _ {k} ^ {\pi} (s _ {k} ^ {+}, a _ {k})) + \| I _ {i} ^ {\pi} (s _ {i} ^ {+}, s _ {i} ^ {' +}) - I _ {k} ^ {\pi} (s _ {k} ^ {+}, s _ {k} ^ {' +}) \| _ {1}
+$$
+
+To prove the triangle inequality, we need to show that:
+
+$$
+\left| R _ {i} ^ {\pi} \left(s _ {i} ^ {+}, a _ {i}\right) - R _ {k} ^ {\pi} \left(s _ {k} ^ {+}, a _ {k}\right) \right| \leq \left| R _ {i} ^ {\pi} \left(s _ {i} ^ {+}, a _ {i}\right) - R _ {j} ^ {\pi} \left(s _ {j} ^ {+}, a _ {j}\right) \right| + \left| R _ {j} ^ {\pi} \left(s _ {j} ^ {+}, a _ {j}\right) - R _ {k} ^ {\pi} \left(s _ {k} ^ {+}, a _ {k}\right) \right|
+$$
+
+This follows directly from the standard triangle inequality for absolute values. Similarly, for the Wasserstein distance term:
+
+$$
+W _ {2} \left(d _ {\pi}\right) \left(T _ {i} ^ {\pi} \left(s _ {i} ^ {+}, a _ {i}\right), T _ {k} ^ {\pi} \left(s _ {k} ^ {+}, a _ {k}\right)\right) \leq W _ {2} \left(d _ {\pi}\right) \left(T _ {i} ^ {\pi} \left(s _ {i} ^ {+}, a _ {i}\right), T _ {j} ^ {\pi} \left(s _ {j} ^ {+}, a _ {j}\right)\right) + W _ {2} \left(d _ {\pi}\right) \left(T _ {j} ^ {\pi} \left(s _ {j} ^ {+}, a _ {j}\right), T _ {k} ^ {\pi} \left(s _ {k} ^ {+}, a _ {k}\right)\right)
+$$
+
+This holds due to the triangle inequality for the Wasserstein distance (Villani et al., 2009). Finally, for the inverse dynamics term:
+
+$$
+\| I _ {i} ^ {\pi} \left(s _ {i} ^ {+}, s _ {i} ^ {\prime +}\right) - I _ {k} ^ {\pi} \left(s _ {k} ^ {+}, s _ {k} ^ {\prime +}\right) \| _ {1} \leq \| I _ {i} ^ {\pi} \left(s _ {i} ^ {+}, s _ {i} ^ {\prime +}\right) - I _ {j} ^ {\pi} \left(s _ {j} ^ {+}, s _ {j} ^ {\prime +}\right) \| _ {1} + \| I _ {j} ^ {\pi} \left(s _ {j} ^ {+}, s _ {j} ^ {\prime +}\right) - I _ {k} ^ {\pi} \left(s _ {k} ^ {+}, s _ {k} ^ {\prime +}\right) \| _ {1}
+$$
+
+This also follows directly from the standard triangle inequality for norms. Since each of the three components satisfies the triangle inequality, we can conclude that the latent task belief metric  $d_{\pi}$  also satisfies the triangle inequality:
+
+$$
+d _ {\pi} \left(z _ {i}, z _ {k}\right) \leq d _ {\pi} \left(z _ {i}, z _ {j}\right) + d _ {\pi} \left(z _ {j}, z _ {k}\right)
+$$
+
+![](images/cc90db225ef48687f1860a1511f58a9e9d5d4b56e3f1c70077856f897c4daba5.jpg)
+
+# B THEOREMS AND PROOFS
+
+Lemma 1. Let  $S^{+} = \mathcal{G}(S)\times \mathcal{Z}$  be the latent dynamics space, and  $\pi$  is the improving policy. The  $\mathcal{F}$  function is defined as follows:
+
+$$
+\begin{array}{l} \mathcal {F} (d, \pi) \left(z _ {i}, z _ {j}\right) = \left| R _ {i} ^ {\pi} \left(s _ {i} ^ {+}, a _ {i}\right) - R _ {j} ^ {\pi} \left(s _ {j} ^ {+}, a _ {j}\right) \right| + W _ {2} \left(d _ {\pi}\right) \left(T _ {i} ^ {\pi} \left(s _ {i} ^ {+}, a _ {i}\right), T _ {j} ^ {\pi} \left(s _ {j} ^ {+}, a _ {j}\right)\right) \\ + \left\| I _ {i} ^ {\pi} \left(s _ {i} ^ {+}, s _ {i + 1} ^ {+}\right) - I _ {j} ^ {\pi} \left(s _ {j} ^ {+}, s _ {j + 1} ^ {+}\right) \right\| _ {1}, \tag {14} \\ \end{array}
+$$
+
+where  $s_i^+ = (g(s_i), z_i)$ ,  $s_j^+ = (g(s_j), z_j)$  are the augmented states corresponding to task  $i$  and task  $j$ , and  $z_i, z_j$  represent different task beliefs in latent space  $S^+$ . Then, there exists a unique least fixed point  $\tilde{d}$  such that  $\mathcal{F}(\tilde{d}) = \tilde{d}$ .
+
+Proof. To prove that  $\mathcal{F}$  has a unique least fixed point, we need to establish that  $\mathcal{F}$  is monotonic and continuous, ensuring the conditions for a least fixed-point result.
+
+We first show that  $\mathcal{F}$  is monotonic. Suppose  $d \leq d'$ , meaning  $d(z_i, z_j) \leq d'(z_i, z_j)$  for all  $z_i, z_j$  in a latent space. Then, by the definition of  $\mathcal{F}$ , we have:
+
+$$
+W _ {2} \left(d _ {\pi}\right) \left(T _ {i} ^ {\pi} \left(s _ {i} ^ {+}, a _ {i}\right), T _ {j} ^ {\pi} \left(s _ {j} ^ {+}, a _ {j}\right)\right) \leq W _ {2} \left(d _ {\pi} ^ {\prime}\right) \left(T _ {i} ^ {\pi} \left(s _ {i} ^ {+}, a _ {i}\right), T _ {j} ^ {\pi} \left(s _ {j} ^ {+}, a _ {j}\right)\right).
+$$
+
+Since the Wasserstein distance is non-decreasing in  $d$ , it follows that  $\mathcal{F}(d,\pi)\leq \mathcal{F}(d^{\prime},\pi)$ . Additionally, the reward difference and inverse dynamics term remain unchanged under ordering, confirming that  $\mathcal{F}$  is indeed monotonic.
+
+Next, we show that  $\mathcal{F}$  is continuous with respect to the pointwise supremum of an  $\omega$ -chain  $\{d_n\}$ . Consider the sequence  $\{d_n\}$  forming an increasing chain in  $\mathcal{M}$ , meaning  $d_{n}(z_{i},z_{j})$  converges to  $d_{\infty}(z_i,z_j)$ . By the properties of the Wasserstein metric and  $L1$  norm, limits commute with these operations, leading to:
+
+$$
+\begin{array}{l} \mathcal {F} ^ {\pi} (\bigsqcup_ {n \in \mathbb {N}} d _ {n}) (z _ {i}, z _ {j}) = | R _ {i} ^ {\pi} \left(s _ {i} ^ {+}, a _ {i}\right) - R _ {j} ^ {\pi} \left(s _ {j} ^ {+}, a _ {j}\right) | + W _ {2} (\bigsqcup_ {n \in \mathbb {N}} d _ {n}) \left(T _ {i} ^ {\pi} \left(s _ {i} ^ {+}, a _ {i}\right), T _ {j} ^ {\pi} \left(s _ {j} ^ {+}, a _ {j}\right)\right) \\ + \left\| I _ {i} ^ {\pi} \left(s _ {i} ^ {+}, s _ {i + 1} ^ {+}\right) - I _ {j} ^ {\pi} \left(s _ {j} ^ {+}, s _ {j + 1} ^ {+}\right) \right\| _ {1} \\ = \left| R _ {i} ^ {\pi} \left(s _ {i} ^ {+}, a _ {i}\right) - R _ {j} ^ {\pi} \left(s _ {j} ^ {+}, a _ {j}\right) \right| + \sup  _ {n \in \mathbb {N}} W _ {2} \left(d _ {n}\right) \left(T _ {i} ^ {\pi} \left(s _ {i} ^ {+}, a _ {i}\right), T _ {j} ^ {\pi} \left(s _ {j} ^ {+}, a _ {j}\right)\right) \\ + \left\| I _ {i} ^ {\pi} \left(s _ {i} ^ {+}, s _ {i + 1} ^ {+}\right) - I _ {j} ^ {\pi} \left(s _ {j} ^ {+}, s _ {j + 1} ^ {+}\right) \right\| _ {1} \\ = \sup  _ {n \in \mathbb {N}} \left(\left| R _ {i} ^ {\pi} \left(s _ {i} ^ {+}, a _ {i}\right) - R _ {j} ^ {\pi} \left(s _ {j} ^ {+}, a _ {j}\right) \right| + W _ {2} \left(d _ {n}\right) \left(T _ {i} ^ {\pi} \left(s _ {i} ^ {+}, a _ {i}\right), T _ {j} ^ {\pi} \left(s _ {j} ^ {+}, a _ {j}\right)\right) \right. \\ + \left\| I _ {i} ^ {\pi} \left(s _ {i} ^ {+}, s _ {i + 1} ^ {+}\right) - I _ {j} ^ {\pi} \left(s _ {j} ^ {+}, s _ {j + 1} ^ {+}\right) \right\| _ {1}) \\ = \left(\bigsqcup_ {n \in \mathbb {N}} \mathcal {F} ^ {\pi} \left(d _ {n}\right)\right) \left(z _ {i}, z _ {j}\right) \tag {15} \\ \end{array}
+$$
+
+This guarantees that  $\mathcal{F}$  is continuous. The policy  $\pi$  in the above equations is fixed. As the policy updates, it can converge to a fixed point.
+
+Since  $\mathcal{M}$  is a complete partial order (ω-CPO) and  $\mathcal{F}$  is a monotonic and continuous operator, the fixed-point theorem (Ferns et al., 2004) ensures the existence of a least fixed point  $\tilde{d}$  such that  $\mathcal{F}(\tilde{d}) = \tilde{d}$ .
+
+![](images/8ef6862a3800d7a7b72f55680ca1485792dc04a70ba67d64a88fac20cc65bb01.jpg)
+
+Theorem 1 (Value difference bound). Given two tasks  $\mathcal{M}_i$  and  $\mathcal{M}_j$  in the latent space with states  $s_i^+, s_j^+ \in S^+$ , and let  $V^\pi$  be the value function of policy  $\pi$ , the value difference bound between the tasks can be given by:
+
+$$
+\left| V ^ {\pi} \left(s _ {i} ^ {+}\right) - V ^ {\pi} \left(s _ {j} ^ {+}\right) \right| \leq d _ {\pi} \left(z _ {i}, z _ {j}\right), \tag {16}
+$$
+
+where  $d_{\pi}(z_i, z_j)$  is the latent task belief metric.
+
+Proof. We will use the standard value function update:
+
+$$
+V _ {n} ^ {\pi} (s) = R ^ {\pi} (s, a) + \gamma \sum_ {s ^ {\prime} \in S} P ^ {\pi} (s ^ {\prime} \mid s, a) V _ {n - 1} ^ {\pi} (s ^ {\prime})
+$$
+
+with  $V_0^\pi(s) = 0$ , and our task difference metric  $d_\pi(z_i, z_j)$ , and prove this by induction, showing that for all  $n \in \mathbb{N}$  and states  $s_i^+, s_j^+ \in S^+$ :
+
+$$
+\left| V _ {n} ^ {\pi} \left(s _ {i} ^ {+}\right) - V _ {n} ^ {\pi} \left(s _ {j} ^ {+}\right) \right| \leq d _ {\pi} ^ {n} \left(z _ {i}, z _ {j}\right).
+$$
+
+The base case holds trivially:
+
+$$
+0 = V _ {0} ^ {\pi} \left(s _ {i} ^ {+}\right) - V _ {0} ^ {\pi} \left(s _ {j} ^ {+}\right) = 0, \quad \text {s o a s s u m e t h i s h o l d s f o r} n.
+$$
+
+Now, for  $n + 1$ :
+
+$$
+| V _ {n + 1} ^ {\pi} (s _ {i} ^ {+}) - V _ {n + 1} ^ {\pi} (s _ {j} ^ {+}) | = | R _ {i} ^ {\pi} (s _ {i} ^ {+}, a _ {i}) + \gamma \sum_ {s ^ {\prime} \in S ^ {+}} T _ {i} ^ {\pi} (s _ {i} ^ {\prime +} \mid s _ {i} ^ {+}, a _ {i}) V _ {n} ^ {\pi} (s _ {i} ^ {\prime +}) - R _ {j} ^ {\pi} (s _ {j} ^ {+}, a _ {j}) - \gamma \sum_ {s ^ {\prime} \in S ^ {+}} T _ {j} ^ {\pi} (s _ {j} ^ {\prime +} \mid s _ {j} ^ {+}, a _ {j}) V _ {n} ^ {\pi} (s _ {j} ^ {\prime +}) |.
+$$
+
+We decompose this into rewards and transition dynamics:
+
+$$
+\leq | R _ {i} ^ {\pi} (s _ {i} ^ {+}, a _ {i}) - R _ {j} ^ {\pi} (s _ {j} ^ {+}, a _ {j}) | + \gamma \left| \sum_ {s ^ {\prime} \in S ^ {+}} \Big (T _ {i} ^ {\pi} (s _ {i} ^ {\prime +} \mid s _ {i} ^ {+}, a _ {i}) V _ {n} ^ {\pi} (s _ {i} ^ {\prime +}) - T _ {j} ^ {\pi} (s _ {j} ^ {\prime +} \mid s _ {j} ^ {+}, a _ {j}) V _ {n} ^ {\pi} (s _ {j} ^ {\prime +}) \Big) \right|.
+$$
+
+Using the Lipschitz property of the value function, we can bound the transition dynamics difference:
+
+$$
+\leq | R _ {i} ^ {\pi} (s _ {i} ^ {+}, a _ {i}) - R _ {j} ^ {\pi} (s _ {j} ^ {+}, a _ {j}) | + \gamma W _ {2} \left(T _ {i} ^ {\pi} (s _ {i} ^ {+}, a _ {i}), T _ {j} ^ {\pi} (s _ {j} ^ {+}, a _ {j})\right) + \left\| I _ {i} ^ {\pi} (s _ {i} ^ {+}, s _ {i} ^ {\prime +}) - I _ {j} ^ {\pi} (s _ {j} ^ {+}, s _ {j} ^ {\prime +}) \right\| _ {1}.
+$$
+
+By the definition of the latent task belief metric  $d_{\pi}(z_i,z_j)$ , we know:
+
+$$
+d _ {\pi} (z _ {i}, z _ {j}) = | R _ {i} ^ {\pi} (s _ {i} ^ {+}, a _ {i}) - R _ {j} ^ {\pi} (s _ {j} ^ {+}, a _ {j}) | + W _ {2} \left(T _ {i} ^ {\pi} (s _ {i} ^ {+}, a _ {i}), T _ {j} ^ {\pi} (s _ {j} ^ {+}, a _ {j})\right) + \left\| I _ {i} ^ {\pi} (s _ {i} ^ {+}, s _ {i} ^ {\prime +}) - I _ {j} ^ {\pi} (s _ {j} ^ {+}, s _ {j} ^ {\prime +}) \right\| _ {1}.
+$$
+
+We assume that  $z$  contains sufficient information about the task differences at the current time step, and we disregard the effect of the discount factor  $\gamma$  during the experiments. Thus, we have:
+
+$$
+| V _ {n + 1} ^ {\pi} (s _ {i} ^ {+}) - V _ {n + 1} ^ {\pi} (s _ {j} ^ {+}) | \leq d _ {\pi} ^ {n + 1} (z _ {i}, z _ {j}).
+$$
+
+Lemma 2. Let  $g$  be an  $(\epsilon_R, \epsilon_T, \epsilon_I)$ -approximate bisimulation abstraction of  $M$ . The states in the real space are mapped to the latent dynamics space  $S^+$  through  $g$ . For any two states in the real space,  $s_1$  and  $s_2$ , if their representations in the latent space are identical, i.e.,  $g(s_1) = g(s_2)$ , then
+
+$$
+\left| R \left(s _ {1}, a\right) - R \left(s _ {2}, a\right) \right| \leq \epsilon_ {R},
+$$
+
+$$
+\left\| T \left(s _ {1}, a\right) - T \left(s _ {2}, a\right) \right\| _ {1} \leq \epsilon_ {T},
+$$
+
+$$
+\| I (s _ {1}, s _ {1} ^ {\prime}) - I (s _ {2}, s _ {2} ^ {\prime}) \| _ {1} \leq \epsilon_ {I}.
+$$
+
+Proof. The proof follows a similar process to Lemma 3 in (Jiang, 2018). Here, we have added a representation constraint on the inverse dynamics.  $\square$
+
+Theorem 2 (Latent transfer bound). Let  $Q_{\mathcal{M}_j}^*$  be the optimal  $Q$ -function for task  $\mathcal{M}_j$ . The difference between  $Q_{\mathcal{M}_j}^*$  and the  $Q$ -function of the policy  $\pi$  learned from task  $\mathcal{M}_i$ , applied to task  $\mathcal{M}_j$ , is bounded as follows:
+
+$$
+\left\| Q _ {\mathcal {M} _ {j}} ^ {*} - \left[ Q _ {\mathcal {M} _ {i}} ^ {\pi} \right] \mathcal {M} _ {j} \right\| _ {\infty} \leq \epsilon_ {R} + \gamma \left(\epsilon_ {T} + \epsilon_ {I} + \| z _ {i} - z _ {j} \| _ {1}\right) \frac {R _ {\max }}{2 (1 - \gamma)}. \tag {17}
+$$
+
+Proof.  $\| z_{i} - z_{j}\|_{1}$  is the distance between the latent task beliefs of task  $\mathcal{M}_i$  and task  $\mathcal{M}_j$ . In the real state space, the augmented state is still conditioned on  $z_{l}$  (Section 3.3).
+
+We aim to bound the difference between the Q-function  $Q_{\mathcal{M}_j}^*$  (the optimal Q-function for task  $\mathcal{M}_j$ ) and  $[Q_{\mathcal{M}_i}^\pi ]_{\mathcal{M}_j}$  (the Q-function for policy  $\pi$  learned on task  $\mathcal{M}_i$ , but applied to task  $\mathcal{M}_j$ ). This difference is given by:
+
+$$
+\left\| Q _ {\mathcal {M} _ {j}} ^ {*} - \left[ Q _ {\mathcal {M} _ {i}} ^ {\pi} \right] \mathcal {M} _ {j} \right\| _ {\infty}.
+$$
+
+The Q-function for any task satisfies the Bellman equation:
+
+$$
+Q (s, a) = R (s, a) + \gamma \mathbb {E} _ {s ^ {\prime} \sim T (s, a)} \left[ \max  _ {a ^ {\prime}} Q \left(s ^ {\prime}, a ^ {\prime}\right) \right].
+$$
+
+Using this, we decompose the Q-function difference:
+
+$$
+\| Q _ {\mathcal {M} _ {j}} ^ {*} - [ Q _ {\mathcal {M} _ {i}} ^ {\pi} ] _ {\mathcal {M} _ {j}} \| _ {\infty} \leq \frac {1}{1 - \gamma} \| Q _ {\mathcal {M} _ {j}} ^ {*} - \mathcal {T} _ {\mathcal {M} _ {j}} ^ {\pi} Q _ {\mathcal {M} _ {i}} ^ {\pi} \| _ {\infty}.
+$$
+
+Here,  $\mathcal{T}_{\mathcal{M}_j}^\pi Q_{\mathcal{M}_i}^\pi = R_{\mathcal{M}_j}(s^+,a) + \gamma \mathbb{E}_{s'\sim T_{M_j}(s^+,a),a\sim I_{M_j}(s^+,s' + )}[Q_{\mathcal{M}_i}^\pi (s'^+,a')]$ .
+
+According to Lemma 2,
+
+$$
+\begin{array}{l} \sup  _ {s _ {i} ^ {+}, s _ {j} ^ {+} \in \mathcal {S} ^ {+}, g (s _ {i}) = g (s _ {j})} \left\| T _ {\mathcal {M} _ {i}} \left(s _ {i} ^ {+}, a\right) - T _ {\mathcal {M} _ {j}} \left(s _ {j} ^ {+}, a\right) \right\| _ {1} \\ \leq \sup _ {s _ {i} ^ {+}, s _ {j} ^ {+} \in \mathcal {S} ^ {+}, g (s _ {i}) = g (s _ {j})} \left(\left\| T _ {\mathcal {M} _ {i}} (s _ {i} ^ {+}, a) - T _ {\mathcal {M} _ {i}} (s _ {j} ^ {+}, a) \right\| _ {1} + \left\| T _ {\mathcal {M} _ {i}} (s _ {j} ^ {+}, a) - T _ {\mathcal {M} _ {j}} (s _ {j} ^ {+}, a) \right\| _ {1}\right) \\ \leq \sup_{s_{i}^{+},s_{j}^{+}\in \mathcal{S}^{+},g(s_{i}) = g(s_{j})}\left\| T_{\mathcal{M}_{i}}(s_{i}^{+},a) - T_{\mathcal{M}_{i}}(s_{j}^{+},a)\right\|_{1} + \sup_{s_{i}^{+},s_{j}^{+}\in \mathcal{S}^{+},g(s_{i}) = g(s_{j})}\left\| T_{\mathcal{M}_{i}}(s_{j}^{+},a) - T_{\mathcal{M}_{j}}(s_{j}^{+},a)\right\|_{1} \\ = \epsilon_ {T} + \left\| T _ {\mathcal {M} _ {i}} \left(s _ {j} ^ {+}, a\right) - T _ {\mathcal {M} _ {j}} \left(s _ {j} ^ {+}, a\right) \right\| _ {1}, \tag {18} \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \sup  \quad | R _ {\mathcal {M} _ {i}} \left(s _ {i} ^ {+}, a\right) - R _ {\mathcal {M} _ {j}} \left(s _ {j} ^ {+}, a\right) | \\ s _ {i} ^ {+}, s _ {j} ^ {+} \in \mathcal {S} ^ {+}, g (s _ {i}) = g (s _ {j}) \\ \leq \sup_{s_{i}^{+},s_{j}^{+}\in \mathcal{S}^{+},g(s_{i}) = g(s_{j})}\bigl(|R_{\mathcal{M}_{i}}(s_{i}^{+},a) - R_{\mathcal{M}_{i}}(s_{j}^{+},a)| + |R_{\mathcal{M}_{i}}(s_{j}^{+},a) - R_{\mathcal{M}_{j}}(s_{j}^{+},a)|\bigr) \\ \leq \sup_{s_{i}^{+},s_{j}^{+}\in \mathcal{S}^{+},g(s_{i}) = g(s_{j})}|R_{\mathcal{M}_{i}}(s_{i}^{+},a) - R_{\mathcal{M}_{i}}(s_{j}^{+},a)| + \sup_{s_{i}^{+},s_{j}^{+}\in \mathcal{S}^{+},g(s_{i}) = g(s_{j})}|R_{\mathcal{M}_{i}}(s_{j}^{+},a) - R_{\mathcal{M}_{j}}(s_{j}^{+},a)| \\ = \epsilon_ {R} + \left| R _ {\mathcal {M} _ {i}} \left(s _ {j} ^ {+}, a\right) - R _ {\mathcal {M} _ {j}} \left(s _ {j} ^ {+}, a\right) \right|, \tag {19} \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \sup  _ {s _ {i} ^ {+}, s _ {j} ^ {+} \in \mathcal {S} ^ {+}, g (s _ {i}) = g (s _ {j})} \left\| I _ {\mathcal {M} _ {i}} \left(s _ {i} ^ {+}, s _ {i} ^ {' +}\right) - I _ {\mathcal {M} _ {j}} \left(s _ {j} ^ {+}, s _ {j} ^ {' +}\right) \right\| _ {1} \\ \leq \sup _ {s _ {i} ^ {+}, s _ {j} ^ {+} \in \mathcal {S} ^ {+}, g (s _ {i}) = g (s _ {j})} \left(\left\| I _ {\mathcal {M} _ {i}} (s _ {i} ^ {+}, s _ {i} ^ {' +}) - I _ {\mathcal {M} _ {i}} (s _ {j} ^ {+}, s _ {j} ^ {' +}) \right\| _ {1} + \left\| I _ {\mathcal {M} _ {i}} (s _ {j} ^ {+}, s _ {j} ^ {' +}) - I _ {\mathcal {M} _ {j}} (s _ {j} ^ {+}, s _ {j} ^ {' +}) \right\| _ {1}\right) \\ \leq \sup_{s_{i}^{+},s_{j}^{+}\in \mathcal{S}^{+},g(s_{i}) = g(s_{j})}\left\| I_{\mathcal{M}_{i}}(s_{i}^{+},s_{i}^{' + }) - I_{\mathcal{M}_{i}}(s_{j}^{+},s_{j}^{' + })\right\|_{1} + \sup_{s_{i}^{+},s_{j}^{+}\in \mathcal{S}^{+},g(s_{i}) = g(s_{j})}\left\| I_{\mathcal{M}_{i}}(s_{j}^{+},s_{j}^{' + }) - I_{\mathcal{M}_{j}}(s_{j}^{+},s_{j}^{' + })\right\|_{1} \\ = \epsilon_ {I} + \left\| I _ {\mathcal {M} _ {i}} \left(s _ {j} ^ {+}, s _ {j} ^ {' +}\right) - I _ {\mathcal {M} _ {j}} \left(s _ {j} ^ {+}, s _ {j} ^ {' +}\right) \right\| _ {1}, \tag {20} \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \left\| T _ {\mathcal {M} _ {i}} \left(s _ {i} ^ {+}, a\right) - T _ {\mathcal {M} _ {j}} \left(s _ {j} ^ {+}, a\right) \right\| _ {1} + \left| R _ {\mathcal {M} _ {i}} \left(s _ {i} ^ {+}, a\right) - R _ {\mathcal {M} _ {j}} \left(s _ {j} ^ {+}, a\right) \right| + \left\| I _ {\mathcal {M} _ {i}} \left(s _ {i} ^ {+}, s _ {i} ^ {' +}\right) - I _ {\mathcal {M} _ {j}} \left(s _ {j} ^ {+}, s _ {j} ^ {' +}\right) \right\| _ {1} \\ \leq \epsilon_ {R} + \epsilon_ {T} + \epsilon_ {I} + \| z _ {i} - z _ {j} \| _ {1}, \tag {21} \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \left\| Q _ {\mathcal {M} _ {j}} ^ {*} - \mathcal {T} _ {\mathcal {M} _ {j}} ^ {\pi} Q _ {\mathcal {M} _ {i}} ^ {\pi} \right\| _ {\infty} \\ = \left| \left(\mathcal {T} _ {\mathcal {M} _ {j}} Q _ {\mathcal {M} _ {j}} ^ {*}\right) \left(s _ {j} ^ {' +}, a\right) - \left(\mathcal {T} \left[ Q _ {\mathcal {M} _ {j}} ^ {*} \right] _ {\mathcal {M} _ {i}}\right) \left(s _ {i} ^ {' +}, a\right) \right| \\ = \Big | R _ {\mathcal {M} _ {j}} (s _ {j} ^ {' +}, a) + \gamma \langle T _ {\mathcal {M} _ {j}} (s _ {j} ^ {' +}, a), V _ {\mathcal {M} _ {j}} ^ {*} \rangle - R _ {\mathcal {M} _ {i}} (s _ {i} ^ {' +}, a) - \gamma \langle T _ {\mathcal {M} _ {i}} (s _ {i} ^ {' +}, a), [ V _ {\mathcal {M} _ {j}} ^ {*} ] _ {\mathcal {M} _ {i}} \rangle \Big | \\ \leq \epsilon_ {R} + \gamma \left| \langle T _ {\mathcal {M} _ {j}} (s _ {j} ^ {' +}, a), V _ {\mathcal {M} _ {j}} ^ {*} \rangle - \langle T _ {\mathcal {M} _ {i}} (s _ {i} ^ {' +}, a), [ V _ {\mathcal {M} _ {j}} ^ {*} ] _ {\mathcal {M} _ {i}} \rangle \right| \\ \leq \epsilon_ {R} + \gamma \left| \langle T _ {\mathcal {M} _ {j}} (s _ {j} ^ {' +}, a), V _ {\mathcal {M} _ {j}} ^ {*} \rangle - \langle T _ {\mathcal {M} _ {i}} (s _ {i} ^ {' +}, a), [ V _ {\mathcal {M} _ {j}} ^ {*} ] _ {\mathcal {M} _ {i}} \rangle \right| + \gamma | I _ {\mathcal {M} _ {i}} (s _ {j} ^ {+}, s _ {j} ^ {' +}) - I _ {\mathcal {M} _ {j}} (s _ {j} ^ {+}, s _ {j} ^ {' +}) | \\ \leq \epsilon_ {R} + \gamma (\epsilon_ {T} + \epsilon_ {I} + \| z _ {i} - z _ {j} \| _ {1}) \| V _ {\mathcal {M} _ {j}} ^ {*} - \frac {R _ {\mathrm {m a x}}}{2 (1 - \gamma)} \mathbf {1} \| _ {\infty} \\ \leq \epsilon_ {R} + \gamma (\epsilon_ {T} + \epsilon_ {I} + \| z _ {i} - z _ {j} \| _ {1}) \frac {R _ {\mathrm {m a x}}}{2 (1 - \gamma)}. \\ \end{array}
+$$
+
+We get the final bound:
+
+$$
+\left\| Q _ {\mathcal {M} _ {j}} ^ {*} - \left[ Q _ {\mathcal {M} _ {i}} ^ {\pi} \right] _ {\mathcal {M} _ {j}} \right\| _ {\infty} \leq \epsilon_ {R} + \gamma \left(\epsilon_ {T} + \epsilon_ {I} + \| z _ {i} - z _ {j} \| _ {1}\right) \frac {R _ {\max }}{2 (1 - \gamma)}.
+$$
+
+![](images/f11de6843b2a81ebd6d07667baa2cfe97b3f5f5a0c15ba4df91518cd868fc56a.jpg)
+
+# C SIMBELIEF PSEUDO-CODE
+
+# Algorithm 1 SimBelief algorithm
+
+Input: Task distribution  $p(M)$
+
+Initialise: context encoder  $q_{\phi}$ , real envs dynamics  $p_{\phi}$ , actor  $\pi_{\theta}$ , critic  $Q_{\omega}$ , replay buffer  $\mathcal{D}$ , belief similarity learner  $\psi_l$ , latent dynamics  $p_{\theta}$ , state encoder  $g$ , distribution offset  $(\Delta \mu, \Delta \sigma)$
+
+1: while not done do  
+2: Sample tasks  $M_{train} = \{\mathcal{M}_i\}_{i=1}^N$  from  $p(M)$  
+3: Collect trajectories with  $\pi_{\theta}$  and add to buffer  $\mathcal{D}$  Data collection  
+4: for step in SAC training steps do  
+5: Sample training tasks  $M_{\mathrm{train}}$  from  $p(M)$  
+6: Sample batches from  $\mathcal{D}$  
+7: Infer specific task beliefs  $z_{r} \sim \psi_{r}(b_{r} \mid h) q_{\phi}(h \mid \tau_{:t})$  
+8: Infer latent task beliefs  $z_{l} \sim \psi_{l}(b_{l} \mid h) q_{\phi}(h \mid \tau_{t})$  
+9: Permute  $z_{l}^{i}$  to get  $z_{l}^{j}$  
+0:  $s_i^+ = (g(s),z_i),s_j^+ = (g(s),z_j)$  
+1: Update latent dynamics  $p_{\theta}$  and state encoder  $g$  using Eq. 4  
+2: Update belief similarity learner  $\psi_l$  using Eq. 8  
+3: Update distribution offset  $(\Delta \mu, \Delta \sigma)$  using Eq. 9  
+4: Integrate  $b_{l}^{i}$  with  $b_{r}^{i}$  to obtain  $b^{i}$  
+5: Update  $(\theta ,\omega)$  with SAC algorithm >SAC update  
+6: end for  
+7: for step in VAE training steps do  
+8: Sample  $\tau_{:T} \sim \mathcal{B}$  with trajectory length  $T$  
+19: Decode only the past trajectories and update  $q_{\phi}$ $\psi_r$  and  $p_{\phi}$  using Eq. 6 VAE update  
+20: end for  
+21: end while
+
+# D CONTEXT-BASED META-RL BASELINES
+
+In this section, we provide a detailed overview of the baseline methods compared in our experiments, highlighting their core methodologies and design principles.
+
+PEARL (Rakelly et al., 2019) (Probabilistic Embeddings for Actor-Critic RL) is an off-policy meta-reinforcement learning algorithm designed to enhance both meta-training sample efficiency and
+
+rapid adaptation to new tasks. By disentangling task inference from control, PEARL employs a probabilistic latent context variable that enables structured exploration and efficient posterior sampling. This design allows the policy to reason about task uncertainty and adapt quickly in sparse reward or dynamic environments. Built upon the Soft Actor-Critic framework, PEARL achieves better sample efficiency compared to on-policy meta-RL methods.
+
+MetaCURE (Zhang et al., 2021c) (Meta-RL with Efficient Uncertainty Reduction Exploration) is an off-policy meta-RL framework designed to address the challenges of sparse-reward environments. It explicitly separates exploration and exploitation by learning distinct policies for each, enhancing the efficiency of task inference and adaptation. The exploration policy is driven by an empowerment-based intrinsic reward that maximizes information gain about the task, enabling efficient collection of task-relevant experiences. MetaCURE leverages a shared probabilistic task inference mechanism, which improves sample efficiency by integrating exploration and exploitation processes. Compared to MetaCURE, SimBelief does not require knowledge of task IDs during the training phase. Instead, it learns the common structure of tasks to enable reasoning, resulting in stronger online adaptation capabilities.
+
+VariBAD (Zintgraf et al., 2019) (Variational Bayes-Adaptive Deep RL) is a meta-reinforcement learning framework that approximates Bayes-optimal policies by leveraging variational inference and latent task embeddings. The algorithm learns a posterior belief over tasks using a variational auto-encoder and conditions its policy on this belief to balance exploration and exploitation in uncertain environments. VariBAD is notable for its ability to perform structured online exploration by integrating task uncertainty directly into action selection. Unlike traditional methods that rely on computationally intractable posterior sampling or explicit planning, VariBAD offers a tractable and flexible approach to Bayes-adaptive policies.
+
+HyperX (Zintgraf et al., 2021) (Hyper-State Exploration) is a meta-reinforcement learning method that addresses sparse reward environments by leveraging exploration bonuses to meta-learn approximately Bayes-optimal task-adaptation strategies. It integrates two exploration bonuses during meta-training: (1) a bonus based on hyper-states (combining environment states and task beliefs) to encourage diverse task-exploration strategies, and (2) a reconstruction error bonus to incentivize the agent to collect data where task beliefs are inaccurate. By exploring hyper-states, HyperX efficiently gathers data for belief inference and optimally trades off exploration and exploitation in sparse or complex environments.
+
+$\mathbf{RL}^2$  (Duan et al., 2016) reformulates the reinforcement learning process as a meta-learning problem, embedding task-specific learning within the hidden state of a recurrent neural network (RNN). By leveraging a "slow" reinforcement learning algorithm to optimize the RNN's weights,  $\mathrm{RL}^2$  enables the network to store and process information about task dynamics across episodes. This design allows the agent to efficiently adapt to unseen tasks by utilizing its historical trajectory data, including observations, actions, rewards, and termination flags.  $\mathrm{RL}^2$  demonstrates competitive performance in solving multi-armed bandits and tabular MDPs, achieving results comparable to theoretically optimal algorithms. Additionally, its scalability to high-dimensional tasks, such as visual navigation in dynamic environments, highlights its potential as a versatile and efficient meta-RL framework.
+
+# E ENVIRONMENTS AND IMPLEMENTATION DETAILS
+
+Table 1: Adaptation length and goal settings for environments used for evaluation  
+
+<table><tr><td>Environment</td><td># of adaptation episodes</td><td>Max steps per episode</td><td>Goal type</td><td>Goal range</td><td>Goal radius</td></tr><tr><td>Cheetah-Vel-Sparse</td><td>2</td><td>200</td><td>Velocity</td><td>[0,3]</td><td>0.5</td></tr><tr><td>Point-Robot-Sparse</td><td>2</td><td>60</td><td>Position</td><td>Semicircle with radius 1</td><td>0.3</td></tr><tr><td>Walker-Rand-Params</td><td>2</td><td>200</td><td>Velocity</td><td>1.5</td><td>0.5</td></tr><tr><td>Panda-Reach</td><td>3</td><td>50</td><td>Position</td><td>/</td><td>0.05</td></tr><tr><td>Panda-Push</td><td>3</td><td>50</td><td>Position</td><td>/</td><td>0.05</td></tr><tr><td>Panda-Pick-And-Place</td><td>3</td><td>50</td><td>Position</td><td>/</td><td>0.05</td></tr></table>
+
+![](images/70ca98b5489b968d65855420669ae3f8df9090609ea24ca4cf2d8329e693c38d.jpg)  
+Panda-Reach
+
+![](images/1827563c3e3742031aedc39acea5c37f0944bf4279b31056072c35e34f2663e7.jpg)  
+Panda-Push  
+Figure 7: Panda-gym tasks. The Panda-Reach task involves controlling the Panda robotic arm to move its gripper to a randomly generated target position within a defined workspace, the Panda-Push task requires pushing a cube from its initial position to a randomly generated target position on a table, and the Panda-Pick-And-Place task focuses on picking up a cube and placing it at a randomly generated target location above the table.
+
+![](images/7f438edd61b56627a9d97812881b996c63f5e86d8092700c7cafb42308d19ea0.jpg)  
+Panda-Pick-And-Place
+
+Table 1 summarizes key settings for the environments used in this work, including the number of adaptation episodes, the maximum number of steps allowed per episode, the type of goal (whether velocity or position-based), the goal range, and the goal radius. The specific environments are described as follows:
+
+Cheetah-Vel-Sparse. The Cheetah-Vel-Sparse environment involves controlling a half-cheetah robot to achieve and maintain a target velocity, which is randomly sampled from a uniform distribution over the range [0, 3]. At each time step, the robot receives a reward based on how closely its current velocity matches the target velocity, with a sparse reward system in place. Specifically, if the difference between the current velocity and the target velocity is within a specified tolerance (goal radius of 0.5), the robot receives a reward; otherwise, the reward is zero. Additionally, a control cost is applied to penalize large actions. The observation space includes the robot's joint positions, velocities, and body orientation, while the action space consists of motor torques controlling the robot's movements. The sparse reward encourages the agent to achieve the target velocity while minimizing control effort.
+
+Point-Robot-Sparse. The Point-Robot-Sparse environment requires a point robot to navigate towards a goal randomly placed along a unit half-circle. The goal position is sampled at the beginning of each episode, and the robot receives rewards only when it reaches within a goal radius of 0.3 units from the target. The observation space consists of the robot's current position, and the action space allows movement in both x and y directions. The reward system is sparse, meaning that rewards are only given when the robot is within the specified goal radius, encouraging the robot to explore and move efficiently towards the target. The sparse reward structure makes the task more challenging, as the robot receives feedback only when it approaches the target, requiring it to learn how to navigate effectively with limited guidance.
+
+Walker-Rand-Params. The Walker-Rand-Params environment involves controlling a bipedal walker robot, with the added complexity of randomized physical parameters such as body mass, leg strength, and joint properties. These parameters are randomized at the start of each episode, requiring the robot to adapt to various physical configurations in order to move forward. The observation space includes the walker's joint angles, positions, and velocities, while the action space consists of motor commands applied to the walker's joints. The reward system is primarily based on how closely the walker's velocity matches a target velocity of 1.5 units per time step. If the deviation from the target velocity exceeds 0.5 units, the reward is set to 0. For smaller deviations, the reward is given as 0.8 - distance from the target velocity. Additionally, a small control cost proportional to the sum of the squared motor actions is subtracted from the reward to penalize large actions.
+
+Panda-Reach. The Panda-Reach task involves controlling the Panda robotic arm's gripper to reach a target position, randomly generated within a  $30\mathrm{cm} \times 30\mathrm{cm} \times 30\mathrm{cm}$  workspace. The task is considered successful when the distance between the gripper and the target is less than  $5\mathrm{cm}$ . The observation space for this task includes the position and velocity of the gripper (6 coordinates), augmented by two additional vectors representing the desired goal (target position) and the achieved goal (current gripper position). The action space comprises three movement coordinates  $(x, y, z)$  for controlling the gripper. The gripper remains closed throughout the task, and its movement along
+
+these axes determines the success of the task. A sparse reward function is used, with a reward of 0 if the target is reached, and -1 otherwise.
+
+Panda-Push. The Panda-Push task requires the robot to push a cube (side length of  $4\mathrm{cm}$ ) placed on a table to a target position. Both the target position and the initial position of the cube are randomly generated within a  $20\mathrm{cm} \times 20\mathrm{cm}$  area around the neutral position of the robot. The gripper remains closed during the task, and the objective is to move the cube to within  $5\mathrm{cm}$  of the target position. The observation space includes the gripper's position and velocity (6 coordinates), along with the cube's position, orientation, and velocity (12 coordinates). The action space consists of three movement coordinates  $(x, y, z)$  for controlling the gripper's movement. As in the Panda-Reach task, a sparse reward function is used, providing a reward of 0 if the task is completed successfully, and -1 otherwise.
+
+Panda-Pick-And-Place. The Panda-Pick-And-Place task involves the robot picking up a cube (side length of  $4\mathrm{cm}$ ) and placing it at a target location, which is randomly generated within a  $20\mathrm{cm} \times 20\mathrm{cm} \times 10\mathrm{cm}$  volume above the table. The task is completed when the cube is placed within  $5\mathrm{cm}$  of the target position. The observation space for this task includes the gripper's position and velocity (6 coordinates), the cube's position, orientation, and velocity (12 coordinates), and the opening state of the gripper (1 coordinate). The action space is expanded to include three movement coordinates (x, y, z) for controlling the gripper and an additional coordinate for controlling the gripper's opening and closing. A sparse reward function is employed, rewarding the robot with 0 for completing the task and -1 otherwise (Figure 7).
+
+All experiments were conducted using an Nvidia RTX 4090 GPU, the source code is available at: https://github.com/mlzhang-pr/SimBelief.
+
+Table 2: Hyperparameter settings for SimBelief in different environments  
+
+<table><tr><td>Parameter Name</td><td>Cheetah-Vel-Sparse Walker-Rand-Params</td><td>Point-Robot-Sparse</td><td>Panda-Reach</td><td>Panda-Push Panda-Pick-And-Place</td></tr><tr><td>Number of Tasks</td><td>120</td><td>100</td><td>100</td><td>60</td></tr><tr><td>Number of Training Tasks</td><td>100</td><td>80</td><td>80</td><td>50</td></tr><tr><td>Number of Evaluation Tasks</td><td>20</td><td>20</td><td>20</td><td>10</td></tr><tr><td>Number of Episodes</td><td>2</td><td>2</td><td>3</td><td>3</td></tr><tr><td>Number of Iterations</td><td>1000</td><td>2000</td><td>1000</td><td>4000</td></tr><tr><td>RL Updates per Iteration</td><td>2000</td><td>1000</td><td>1000</td><td>1000</td></tr><tr><td>Batch Size</td><td>256</td><td>256</td><td>256</td><td>256</td></tr><tr><td>Policy Buffer Size</td><td>1e6</td><td>1e6</td><td>1e6</td><td>1e6</td></tr><tr><td>VAE Buffer Size</td><td>1e5</td><td>5e4</td><td>5e4</td><td>5e4</td></tr><tr><td>Policy Layers</td><td>[128, 128, 128]</td><td>[128, 128]</td><td>[128, 128]</td><td>[128, 128, 128]</td></tr><tr><td>Actor Learning Rate</td><td>0.0003</td><td>0.00007</td><td>0.00007</td><td>0.00007</td></tr><tr><td>Critic Learning Rate</td><td>0.0003</td><td>0.00007</td><td>0.00007</td><td>0.00007</td></tr><tr><td>Discount Factor (γ)</td><td>0.99</td><td>0.9</td><td>0.9</td><td>0.9</td></tr><tr><td>Entropy Alpha</td><td>0.2</td><td>0.01</td><td>0.01</td><td>0.01</td></tr><tr><td>VAE Updates per Iteration</td><td>20</td><td>25</td><td>25</td><td>25</td></tr><tr><td>VAE Learning Rate</td><td>0.0003</td><td>0.001</td><td>0.001</td><td>0.001</td></tr><tr><td>KL Weight</td><td>1.0</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>Task Embedding Size</td><td>10</td><td>10</td><td>5</td><td>5</td></tr></table>
+
+# F THE MECHANISM OF SIMBELIEF FOR RAPID ADAPTATION
+
+In this section, we will explain, from the perspective of task belief representation, why SimBelief can quickly converge to a near Bayes-optimal policy during the training phase and how it enables adaptation to OOD tasks within a single episode in sparse reward environments.
+
+# F.1 THE EVOLUTION OF TASK BELIEF DURING THE TRAINING PHASE
+
+During the training phase, the agent needs to explore complex environments to acquire relevant information about the current task, which is represented by the specific task belief  $b_{r}$ . Other meta-RL algorithms are limited to task-specific information while ignoring the similarities between tasks. SimBelief's latent dynamics can capture the unique structure shared among tasks and distinguish these structures through the latent task belief  $b_{l}$ . Specifically, during the learning process, SimBelief
+
+![](images/138eb37fc5d75c694e6ce83320f33418945f07ecaec098b8f9d4a0c46c4182e0.jpg)
+
+![](images/a21b7733adda6df2d52bd97979770e884fc260df1e060aa075e340a8a8b463ad.jpg)
+
+![](images/3b0ea04260702a8455fb9587ffe30094e43a9fd4e41447d680eb9553ddfcadbb.jpg)
+
+![](images/539b48caaf54afc939db79ec7b886bee20f9e1b4b9dd1017d6cefa776a31c4ed.jpg)
+
+![](images/8cc04470a0199a8ead9b1586455dd7a89eada5fb1ac7cd60cff628c0a1848116.jpg)
+
+![](images/58cd179fd1f1d2c0ac80abe304d09bb407238c6ed388a7b450c55d6bc40dbc74.jpg)
+
+![](images/8e3578221d84f21fcde60306641ceb5df39b6756cb6b550f0644679dea4ea719.jpg)
+
+![](images/89a34d70b4fc02397218a0f7a5dc75c2d6b0e06e1d2daa58cafe22e3af91cb26.jpg)
+
+![](images/4715722f54b72f38c1804edaf1f4bab879cfa7dfee8ed021b7da4f92a29593dc.jpg)
+
+Figure 8: Visualization of different task beliefs during the training phase.  
+![](images/41797cad27206e7e9ebac85043649dcb4edecddf91b9ad692d2b1e6a5a9cc8e3.jpg)  
+Specific task belief
+
+![](images/db839ded10540cdc260ab5342d2a85059f704cb530629cf6abd9cb906074017a.jpg)  
+Latent task belief
+
+![](images/558d7a1a3e3b4e68970031741ecaf1783019b9cded6bc0fc128feecdc606883d.jpg)  
+Mixture distribution
+
+identifies the distribution of similar tasks via  $b_{l}$ , and then uses  $b_{r}$  to fit the agent to the exact specific task distribution, as illustrated in Figure 1. In Figure 8, we visualize the specific task belief, latent task belief, and their mixed Gaussian distribution. During the early and middle stages of training, the agent quickly identifies the distribution of the task being executed. After convergence, it can accurately fit the specific task distribution.
+
+# F.2 THE CORRELATION OF BELIEFS BETWEEN DIFFERENT TASKS DURING THE ADAPTATION PHASE
+
+We visualize the correlations of task beliefs across different tasks in Point-Robot-Sparse, PandaReach, Panda-Push, and Panda-Pick-Place environments. In each environment, 20 tasks are randomly generated, and the SimBelief agent's rollouts are used to extract the specific task beliefs and latent task beliefs. The cosine similarity between task beliefs across tasks is calculated, and the correlation matrices are visualized. As shown in Figure 9, specific task belief primarily captures the local information of task distributions (learning weaker correlations between tasks), while latent task belief emphasizes global information (capturing stronger inter-task correlations). This illustrates the principle behind SimBelief's ability to achieve rapid adaptation to OOD tasks in sparse reward environments: latent task belief enhances the agent's reasoning efficiency across tasks and
+
+improves the efficiency of knowledge transfer. Theorem 2 demonstrates the transferability of tasks in the latent space.
+
+# G ABLATION STUDY
+
+# G.1 LATENT DYNAMICS ABLATION
+
+To validate the effectiveness of our algorithm using the learned latent task belief, we froze the overall latent dynamics and, similar to VariBAD (Zintgraf et al., 2019), only used an external VAE for environment reconstruction. Our algorithm demonstrated significant improvements across all six tested tasks, with particularly notable gains in the more exploration-demanding tasks, Panda-Push and Panda-Pick-And-Place (Figure 10).
+
+# G.2 THE IMPACT OF INVERSE DYNAMICS ON ADAPTATION ABILITY
+
+To demonstrate the effectiveness of the inverse dynamics module in facilitating rapid learning of task similarity, we conducted ablation experiments by removing the inverse dynamics module from the latent space on Cheetah-Vel-Sparse (velocity range [4.0,5.0]) and Point-Robot-Sparse (radius=1.2) tasks. The results show that the inverse dynamics module plays a critical role in enabling the agent's reasoning and generalization on OOD tasks (Figure 11).
+
+# G.3 ABLATION STUDY ON  $w_{r}$  AND  $w_{l}$
+
+When combining  $b_{l}$  and  $b_{r}$ , we apply a mixture of Gaussians to the two beliefs. To evaluate the impact of different weights, we compared three weighting configurations on the Cheetah-Vel-Sparse task:  $[w_{r}, w_{l}] = [0.5, 0.5]$ ,  $[w_{r}, w_{l}] = [0.25, 0.75]$ , and  $[w_{r}, w_{l}] = [0.75, 0.25]$ . Among these,  $[w_{r}, w_{l}] = [0.5, 0.5]$  achieved higher exploration efficiency during training and demonstrated stronger adaptability to OOD tasks (Figure 12).
+
+# G.4 ABLATION STUDY ON OFFSET  $(\Delta \mu, \Delta \sigma)$
+
+During the training phase, we use the offset  $(\Delta \mu, \Delta \sigma)$  applied to the latent task belief primarily to improve the stability of the algorithm's convergence in continuous control tasks (e.g., Cheetah-Vel-Sparse). However, this does not have a significant impact on the overall performance of the algorithm. Even without using the offset during training, SimBelief still outperforms other baselines (Figure 13).
+
+# H ADDITIONAL VISUALIZATIONS
+
+In this section, we provide the task belief representations of SimBelief and other baselines on the Walker-Rand-Param tasks (Figure 14), the adaptation performance on in-distribution tasks PandaPick-And-Place and Walker-Rand-Params (Figure 15), and the exploration performance on the indistribution Point-Robot-Sparse task (Figure 16).
+
+![](images/238a76c4b525d278546841269bfabb39ee8eb95f57cf37ed6e80521bd94ccf5d.jpg)
+
+![](images/5f3c5553fd47e27ca991ee42bdfd653ce792e3a85a73e165e689bfc6f8cc373c.jpg)
+
+![](images/7e5ae9a0a9fc1005f4f628de8ca024dfdf53ec28e1a56fba3fef24da241affe9.jpg)
+
+![](images/f9871056dc02acebf6235e97eedaf5c57c7449a32d6a7d80ffae3345a1568ebc.jpg)
+
+![](images/4a5ea0e93c73fb06e814f255c8b86e3b5a5f29da2b6a144115f505eea20e9f15.jpg)
+
+![](images/eeff0d71a94b8b8b3258483b0148368d53a42121fe02cb6090cf9806e88a58f7.jpg)
+
+![](images/aa08163f1593ce39dec27a7aba779e06208ddd83a1c097f973d7da13d9af7696.jpg)  
+Figure 9: The correlation matrix of task beliefs across different tasks. Specific task belief focuses more on local information between tasks, while latent task belief captures the global characteristics of task distributions.
+
+![](images/94e976a063c77c41d13a1e147a68b121bdbd758e9641606ad7d5eb87f18c55f7.jpg)
+
+![](images/39ab8c0caa3da5ba6b93c75b7b45b0cf16c05bd0c7bb942cac7c1778f87dbf47.jpg)
+
+![](images/4af0718d2e6d2a4f2a221c751fc29c61c5443f3f052ad608787015470c424a5d.jpg)
+
+![](images/e199ccc5db26ffc1d1e3d419cca447197b2b97b68815400fdef357cc1a75c8b2.jpg)
+
+Figure 10: Ablation study on SimBlief's latent task belief.  
+![](images/469a0cc8ff830afdef8f60bfb4e68fa1a420089b50259519f9ac5baeb74fcb88.jpg)  
+SimBelief SimBelief w/o Latent Task Belief
+
+![](images/8193335933f94302e9431a5b3767827e13c9c23bb784f1ecc6343fb60e8c7a05.jpg)
+
+![](images/9c2ab0aac5d6cdbe5bfcd0d569954691c4ea9d0d6aaedec64cdc4da4fa60b88e.jpg)
+
+Figure 11: Ablation study on SimBlief's inverse dynamic module in latent space.  
+![](images/a5fb75fc71fa0b9901649009f8f9372e3e45dead2e20aa13126be13201bc4a35.jpg)  
+SimBelief
+
+![](images/e1a655558d5e6e9cbb4ec4ce8a5c8253d270352aa505be1d48498a501f32fbdb.jpg)  
+— SimBelief w/o inverse dynamic
+
+![](images/443d1945b5d6378940b28050bbe53d46835d5a272074aef82d5799484a8dd49b.jpg)  
+Figure 12: The performance of different Gaussian mixture weights on the Cheetah-Vel-Sparse task.
+
+![](images/24584c619f15f7e8db5792cbc4302307274c201319e658697042266ea1ed0046.jpg)
+
+![](images/f4724dea249e6a229627505e32da1b1e3fd4ecb9a198186ad767ab422d3f6f25.jpg)
+
+![](images/919f0f4a892e8b9c472edf93ae361a9d53fbebe4d3f0e1926d22e5e6aacb1b3b.jpg)
+
+![](images/f809792601119541e2bef24020a13f498073616afe9f35e8f6f75eeebf86da8c.jpg)
+
+![](images/5f664bc839439c85bd0cce1e5bf455bbfe4e95b00cf15a4418c1999d5fc60b7c.jpg)
+
+![](images/15ee85742bfb86547478b1f514fc23a2a73688063e5315469de9ac62f2596610.jpg)
+
+![](images/4afd72c3a890cec4f1830fb168acd55326e858eb961f9b25a2ce7f8cf43ab02e.jpg)
+
+![](images/2f9539d6bcb3260e62bd67d965aed536c851dd77048d86981c1724a9a703a90b.jpg)  
+Figure 13: Ablation study on offset  $(\Delta \mu, \Delta \sigma)$ .
+
+![](images/ec42cf70b636982753bc8ff39e755075cb60319de134b254411cd908f3f8d1f8.jpg)  
+—SimBelief
+
+![](images/eee889348d3dbc9f0e16946f481f8639068c30717929ed3575079677efb0b1db.jpg)  
+—SimBelief w/o offset
+
+![](images/c9a6bd7001c13eaf424e8481a30d5f19e161d9f08637dfb27f65a3d1de60df54.jpg)
+
+![](images/f3eeebe6aeb9ed70a4dd15bd9e3f715a224b7a1ee732d1d18025679c3b5299af.jpg)
+
+![](images/f8c83667ce4378b06e23e76ef620c2894d4cbe85ce056d0f59134f64b4c2bdad.jpg)
+
+![](images/f7b98d780c63d51821e68e7b5755c5a4cc20cef9c00011f357a4df258e7ab465.jpg)
+
+![](images/d2176a5f556365bbdc5aab828e8f91f7b35db0432bdfa66f9232f7415d899a70.jpg)
+
+![](images/ecf36e75149b481a281a6cde710a649634c567c233a6f6704e1217254ddd2b40.jpg)  
+Figure 14: The task belief representation of all experimental algorithms for 10 random indistribution Walker-Rand-Param tasks.
+
+![](images/7b5872868faafefd77991f7376f6678652c901997721a016f30e86779ed3e07d.jpg)
+
+![](images/82de3b0b5bedaa743d9171bb505fd12ae5cd504568917ece8e48914b912b10ae.jpg)
+
+![](images/3101abee638a5873393324dd3ffc02900eaab1064bc077d56e958f4c57da32d9.jpg)  
+Figure 15: Average test performance for the first 5 rollouts on randomly generated tasks of PandaPick-And-Place and Walker-Rand-Params.
+
+![](images/5f5c76ee9955dda75f670b1d884906ddd48cd4b4439d874a139646af779ebca4.jpg)
+
+![](images/f8315419292819740850967cfb0676569417da7610a9a4b46aa7f4e032df2f42.jpg)  
+Figure 16: Exploration and adaptation performance in the 5 random in-distribution (radius = 1.0) tasks of Point-Robot-Sparse.
+
+![](images/7e3dd138872237bf3860c2e9c612cc62c447ccd5239f334d958a9265be035d2f.jpg)
+
+![](images/00d1483e93c367b6af83ceebde2deac26d9d66781ff7f22538f21f03c2dea078.jpg)
\ No newline at end of file
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+# LEARNING VIDEO-CONDITIONED POLICY ON UNLABELLED DATA WITH JOINT EMBEDDING PREDICTIVE TRANSFORMER
+
+Hao Luo $^{1}$  and Zongqing Lu $^{1,2*}$
+
+$^{1}$ School of Computer Science, Peking University
+
+$^{2}$ Beijing Academy of Artificial Intelligence
+
+# ABSTRACT
+
+The video-conditioned policy takes prompt videos of the desired tasks as a condition and is regarded for its prospective generalizability. Despite its promise, training a video-conditioned policy is non-trivial due to the need for abundant demonstrations. In some tasks, the expert rollouts are merely available as videos, and costly and time-consuming efforts are required to annotate action labels. To address this, we explore training video-conditioned policy on a mixture of demonstrations and unlabeled expert videos to reduce reliance on extensive manual annotation. We introduce the Joint Embedding Predictive Transformer (JEPT) to learn a video-conditioned policy through sequence modeling. JEPT is designed to jointly learn visual transition prediction and inverse dynamics. The visual transition is captured from both demonstrations and expert videos, on the basis of which the inverse dynamics learned from demonstrations is generalizable to the tasks without action labels. Experiments on a series of simulated visual control tasks evaluate that JEPT can effectively leverage the mixture dataset to learn a generalizable policy. JEPT outperforms baselines in the tasks without action-labeled data and unseen tasks. We also experimentally reveal the potential of JEPT as a simple visual priors injection approach to enhance the video-conditioned policy.
+
+![](images/6c3dbe1a835ce42d56fae9ea2dbaca32f8245f006ebd7574c23e4396548abca5.jpg)  
+Figure 1: Illustration of the BC and JEPT for visual-conditioned policy learning. Video-conditioned Behavior Cloning (BC) (right top) directly models the policy from the demonstrations. JEPT (bottom) decomposes imitation learning into visual transition prediction (solid arrows) and inverse dynamics learning (dashed arrows). BC is constrained to the action-labeled dataset, whereas JEPT enables visual transition prediction on unlabeled data and leverages the mixture dataset.
+
+# 1 INTRODUCTION
+
+Learning generalizable policies through multi-task imitation learning remains a considerable challenge. A generalizable policy should adapt seamlessly to novel tasks whose demonstrations are absent from the training dataset. In response to this, the video-conditioned policy (Jiang et al., 2023; Bahl et al., 2023; Shah et al., 2023) has garnered attention for its superior potential to generalize across tasks. A video-conditioned policy takes a prompt video depicting the task as a condition and executes the desired task in the dynamics encountered. Utilizing a prompt video as a policy condition exhibits a better potential for generalization compared to other forms of task specifications, such as instructions or goals given in language (Brohan et al., 2023; Padalkar & et al., 2023; Jiang et al., 2019; Yenamandra et al., 2023) or image (Bousmalis et al., 2024; Lee et al., 2020; Du et al., 2024; Nair et al., 2018) formats. The abundant information conveyed in videos offers sufficient policy guidance for novel tasks, thus circumventing the misleading caused by language ambiguities or the absence of process depiction in static images (Jain et al., 2024).
+
+Although comprehensive information in prompt videos imbues video-conditioned policy with flexibility and generalization, learning such a policy is more than trivial. Imitating the tasks depicted in prompt videos requires models capable of both temporal reasoning to understand the task and fine-grained control to replicate it. To this end, previous methods (Shah et al., 2023; Jain et al., 2024) typically conduct behavior cloning on large datasets of paired prompt videos and expert demonstrations. However, procuring expert demonstrations can be prohibitively costly and time-consuming. In the tasks where expert policies are hard to obtain, we can acquire the videos of the expert rollouts through human intervention, but these videos are devoid of action labels. Conducting behavior cloning in this scenario requires additional efforts to annotate the action labels. This raises the question: Can these expert videos<sup>1</sup> be effectively harnessed without additional annotation, thus alleviating the data collection burden? Accordingly, we explore the possibility of training a generalizable video-conditioned policy on a dataset containing both expert demonstrations and unlabeled expert videos paired with prompt videos.
+
+Unlike methods that directly imitate expert demonstrations, behavior cloning is infeasible on the mixture dataset due to the absence of the action labels from the data samples with expert videos. Intuitively, we decompose the behavior cloning into two synergistic subtasks to fully exploit the mixture dataset of expert demonstrations and expert videos. Specifically, the process of video-conditioned imitation can be broken down into visual transition prediction and inverse dynamics learning. In the visual transition prediction, the model learns how the prompt videos should manifest in the dynamics of the tasks. In inverse dynamics learning, the model infers the actions required to realize the visual transition. By combining them, the model can learn to predict plausible future observations and then convert them into actions, enabling video-conditioned imitation.
+
+This design potentially suits the setting of mixture dataset. On the one hand, visual transition prediction is more task-specific in visual imitation learning task and can be more sufficiently learned from both expert videos and demonstrations. On the other hand, although the inverse dynamics is merely embedded in the demonstrations, it remains universally applicable across tasks. Consequently, the expert videos could enhance the learning of visual transitions, while the inverse dynamics derived from the demonstrations could generalize to guide the execution of the tasks without action labels.
+
+In light of these insights, we propose the Joint Embedding Predictive Transformer (JEPT) to encapsulate both visual transition and inverse dynamics for video-conditioned policy learning. Building on the framework of Decision Transformers (Chen et al., 2021; Lee et al., 2022; Furuta et al., 2022), we employ a video-conditioned Transformer-based architecture to perform sequence modeling, serving as the policy. As shown in Figure 1, we modify the typical behavior cloning sequence into a two-step process. First, JEPT predicts the embeddings of the next observations, thus capturing visual transition. Then, conditioned on these predicted embeddings, JEPT predicts the corresponding actions, thereby learning inverse dynamics. Additionally, we integrate the Joint Embedding Predictive Architecture (JEPA) (LeCun, 2022) into the video-conditioned Transformer. Prior works (Assran et al., 2023; Bardes et al., 2024) have demonstrated that JEPA is an effective visual representation learning approach, compressing visual inputs into predictive features. By incorporating JEPA, we aim to learn
+
+predictive visual representations of future observations, thereby improving the transfer of inverse dynamics between action-labeled and unlabeled data.
+
+To evaluate the effectiveness of JEPT, we conduct experiments on Meta-World (Yu et al., 2020a) and Robosuite (Zhu et al., 2020). In our Meta-World experiments, JEPT outperforms all the baselines regarding the average success rates of the tasks within the dataset and the unseen tasks. In the Robosuite experiments, we further validate the effectiveness of JEPT in handling tasks where there is a larger discrepancy between prompt videos and task dynamics. Additionally, we explore the potential of JEPT in injecting visual priors via adjusting the inputs of the joint embedding encoder. Our experiments on Meta-World reveal that the optical flow priors can effectively improve the performance. These results demonstrate the effectiveness of JEPT in jointly leveraging the mixture dataset of expert demonstrations and expert videos to train a generalizable video-conditioned policy.
+
+In summary, our contributions are as follows:
+
+- We propose a novel paradigm, JEPT, to learn video-conditioned policies using the dataset containing a mixture of expert videos and demonstrations.  
+- We design JEPT as a joint embedding predictive architecture to learn an abstract visual representation for learning a generalizable video-conditioned policy.  
+- We explore the JEPT as an approach of visual priors injection and experimentally find that JEPT can effectively leverage visual prior knowledge and achieve better performance.
+
+# 2 RELATED WORKS
+
+Video Prompt Policy Learning. As a densely informative data form, video inherently contains abundant information for task completion when used as a task description. The detailed guidance of the task completion process in the video makes it possible to learn the policy generalizable to novel tasks beyond the training dataset. Some works (Finn et al., 2017; Duan et al., 2017; Yu et al., 2018) employ meta-learning methods to adapt the policy to novel tasks depicted by prompt video, which requires the similarity between the tasks. More recent works focus on learning policy via behavior cloning on datasets of paired prompt videos and expert demonstrations. These works take the prompt videos as a condition input to the policy network and generalize to unseen tasks with corresponding prompt videos. Various auxiliary mechanisms have been explored to achieve generalizable imitation, such as inverse dynamics prediction (Dasari & Gupta, 2021), cross-painting (Chen et al., 2024), skill decomposition (Shin et al., 2024; 2023), hierarchical policy learning (Jain et al., 2023), text-aligned representation (Jang et al., 2022), contrastive video encoder (Chane-Sane et al., 2023) and observation-attentive representation (Jain et al., 2024). Some works (Sivakumar et al., 2022) design object-centric decomposition for some specific manipulation tasks and achieve generalization within the same task category. The core of these methods remains direct behavior cloning, necessitating access to action-labeled demonstration data. Recent works (Jain et al., 2024; Shah et al., 2023; Jiang et al., 2023) construct large-scale datasets for video-conditioned policy learning, evaluating that large-scale and well-aligned data can effectively improve the video-conditioned policy learning in terms of generalization and success rates. However, these works require extensive demonstration annotation efforts, which is costly and time-consuming. Our work basically follows the video-conditioned policy learning setting. Still, we explore how the video-conditioned policy can learn effectively when the dataset is a mixture of expert videos and demonstrations paired with prompt videos.
+
+Learning from Videos. Given the relative accessibility of video data compared to action-labeled demonstrations, some works explore leveraging unlabeled videos to aid in policy learning. Despite the absence of action labels, videos contain substantial decision-related knowledge due to their rich temporal information. A path for learning from videos involves self-supervised representation learning on videos, including masked autoencoder (Radosavovic et al., 2023; Xiao et al., 2022; Yang et al., 2024), temporal contrastive learning (Li et al., 2024; Nair et al., 2023), and video prediction (Seo et al., 2022; Luo et al., 2024). These methods aim to learn a compact and informative representation from videos that aid in policy learning. Other approaches focus on learning reward functions (Escontrela et al., 2023; Yu et al., 2020b; Bruce et al., 2023) or value functions (Bobrin et al., 2024; Chang et al., 2022) to expedite online policy learning. Due to the absence of action labels, these methods require fine-tuning on the downstream tasks for adaptation. Additionally, some research utilizes datasets combining videos and demonstrations, learning an inverse dynamics model (IDM) (Baker et al., 2022; Schmeckpeper et al., 2021; Zheng et al., 2023; Zhang et al., 2022; Kim
+
+![](images/137d420b3f06461e524646c3b41559e2df4654f3ddd49eb6a2ec05ccbacaa500.jpg)  
+Figure 2: The architecture of the Joint Embedding Predictive Transformer (JEPT). The model consists of a joint embedding encoder, a prompt video encoder, and a video-conditioned causal predictor. The demonstrations and the expert videos are processed as a sequence of tokens, and the causal predictor predicts the tokens causally. The visual embedding token of the next observation is predicted to learn a predictive joint embedding. We use  $t' \coloneqq t - k + 1$  to denote the start of the context.
+
+et al., 2023) or a latent inverse dynamics model (Schmidt & Jiang, 2024; Ye et al., 2022; Edwards et al., 2019) on the action-labeled data. The learned IDM can be used to annotate the video data, facilitating imitation learning to derive policies. In these works, the decoupling of inverse dynamics learning and visual imitation may limit the generalizability of the IDM. In our work, however, the learning of visual transitions and inverse dynamics is conducted jointly, offering better generalization.
+
+Joint Embedding Predictive Architecture. The Joint Embedding Predictive Architecture (JEPA) (LeCun, 2022) emerges as a promising representation learning framework. JEPA optimizes a predictive loss on the encoded embeddings to learn an embedding space where the embeddings are predictable to each other. Unlike the generative architectures, JEPA optimizes the predictive loss on embeddings rather than raw data, potentially discarding extraneous information to learn more compact and meaningful representations. In order to train a JEPA, it is a common practice to construct data pairs to predict and be predicted. JEPA is a versatile framework. Various works employ different data pair construction methods to enable JEPA to learn embedding spaces tailored to specific information, enhancing tasks such as image classification (Assran et al., 2023), mask classification (Kim et al., 2024), video understanding (Bardes et al., 2024), and motion and content learning (Bardes et al., 2023). Our work integrates JEPA with sequence modeling on trajectories, providing a novel data pair construction method for JEPA. This extends JEPA to architectures akin to the Decision Transformers, aiming to learn visual representations that facilitate inverse dynamics knowledge transfer.
+
+# 3 METHODOLOGY
+
+In this section, we describe how our Joint Embedding Predictive Transformer (JEPT) works on the mixture of expert demonstrations and expert videos paired with prompt videos.
+
+# 3.1 PRELIMINARIES
+
+We aim to learn a video-conditioned policy across a set of tasks. The environment is viewed as a collection of Partially Observable Markov Decision Process variants (POMDPs), with each task represented by a POMDP variant  $\mathcal{M}_i \coloneqq (\mathcal{S}, \mathcal{A}, \mathcal{P}, \mathcal{O}, \Omega, \mathcal{V}_i)$ . Here,  $\mathcal{S}, \mathcal{A}, \mathcal{P}, \mathcal{O}, \Omega$  denote the state space, action space, transition function, observation space, and observation function, respectively, shared across all tasks. Specifically,  $\mathcal{V}_i$  comprises a set of prompt videos depicting the task  $\mathcal{M}_i$ .
+
+Video-Conditioned Policy. During interaction with the environment, the agent receives a prompt video  $V \in \mathcal{V}_i$  illustrating the desired task at the start of each rollout. At each timestep  $t$ , the agent obtains a visual observation  $o_t \in \mathcal{O}$ , derived from the current state  $s_t \in S$  via the observation function  $\Omega \colon S \to \mathcal{O}$ . Upon executing an action  $a_t \in \mathcal{A}$ , the environment transitions to the subsequent
+
+state  $s_{t + 1}$  according to the transition function  $\mathcal{P}\colon \mathcal{S}\times \mathcal{A}\times \mathcal{S}\to [0,1]$ . A video-conditioned policy generates actions based on the visual observation context and the prompt video. Formally, a video-conditioned policy is denoted as  $\pi \left(a_{t}|o_{[t - k + 1:t]},V\right)$ , where  $o_{[t - k + 1:t]}$  represents the visual observation context over the last  $k$  timesteps.
+
+Mixture Dataset. We consider training the video-conditioned policy on a mixture dataset. Ideally, we collect three types of data for imitation learning: (1) the prompt video  $V \coloneqq (v_{1}, v_{2}, \ldots, v_{N})$  specifying the task with  $N$  frames, (2) the expert video  $O \coloneqq (o_{0}, o_{1}, \ldots, o_{T})$ , the visual observation sequence recorded alongside a  $T$ -step expert rollout of the task, and (3) the corresponding action sequence  $A \coloneqq (a_{0}, a_{1}, \ldots, a_{T-1})$  executed during the same rollout. In the tasks where expert demonstrations are available, we assume access to the dataset  $\mathcal{D}_{\mathrm{demo}}$ , and each data sample  $d_{\mathrm{demo}} \in \mathcal{D}_{\mathrm{demo}}$  is in the form:
+
+$$
+d _ {\mathrm {d e m o}} ^ {(i)} = \left(\left(v _ {1}, v _ {2}, \dots , v _ {N}\right), \left(o _ {0}, a _ {0}, o _ {1}, a _ {1}, \dots , o _ {T}\right) \sim \pi_ {i} ^ {\star}\right). \tag {1}
+$$
+
+Here,  $\pi^{\star}$  represents the expert policies,  $\sim$  denotes the rollout process, and  $i$  indexes the dataset sample. Conversely, in the tasks where merely videos of expert rollouts are available, we denote this dataset as  $\mathcal{D}_{\mathrm{vid}}$ , and each data sample  $d_{\mathrm{vid}} \in \mathcal{D}_{\mathrm{vid}}$  is in the form:
+
+$$
+d _ {\mathrm {v i d}} ^ {(i)} = \left(\left(v _ {1}, v _ {2}, \dots , v _ {N}\right), \left(o _ {0}, o _ {1}, \dots , o _ {T}\right) \sim \pi_ {i} ^ {\star}\right). \tag {2}
+$$
+
+Each data sample in  $\mathcal{D}_{\mathrm{demo}}$  consists of a prompt video  $V$  and a paired expert demonstration  $(O, A)$ , while each data sample in  $\mathcal{D}_{\mathrm{vid}}$  consists of a prompt video  $V$  and a paired expert video  $O$ . In summary, our objective is to train the video-conditioned policy  $\pi(a_{t} | o_{[t-k+1:t]}, V)$  on a dataset mixed of these two kinds of data, denoted as  $\mathcal{D}_{\mathrm{demo}} \cup \mathcal{D}_{\mathrm{vid}}$ .
+
+# 3.2 MODEL ARCHITECTURE
+
+Our JEPT employs a Transformer-based architecture to perform sequence modeling on the data, following the supervised learning paradigm of the Generalized Decision Transformers (Furuta et al., 2022). In this context, demonstrations are treated as sequences of observation and action tokens, where the sequence modeling serves as a video-conditioned policy via predicting the corresponding action tokens. To perform the sequence modeling, JEPT comprises two kinds of modules: (1) visual encoders that process the high-dimensional visual input and (2) a causal predictor aggregating the visual representations to causally predict tokens in the sequence. The comprehensive architecture of JEPT is illustrated in Figure 2, and we elucidate each component in detail below.
+
+# 3.2.1 VISUAL ENCODERS
+
+Joint Embedding Encoder. A Joint Embedding Encoder is employed to learn the visual representation for each observation. In our design, the Joint Embedding Encoder captures information from both single-timestep and contextual levels. Structurally, the Joint Embedding Encoder comprises a spatial encoder for single-timestep encoding and a bi-directional temporal encoder for contextual encoding. Formally, the Joint Embedding Encoder  $\Psi_{\mathrm{obs}}$  takes as input a visual observation context and outputs the visual representations:
+
+$$
+\left(h _ {t - k + 1}, \dots , h _ {t - 1}, h _ {t}\right) = \Psi_ {\text {o b s}} \left(o _ {t - k + 1}, \dots , o _ {t - 1}, o _ {t}\right). \tag {3}
+$$
+
+Each representation  $h_t$  consists of  $N_{\mathrm{obs}}$  tokens. The visual embeddings of the next observation is predicted by the causal predictor and thus a joint embedding will be learned. Via learning joint embeddings predictive of the previous observation tokens, the encoder can compress the high-dimensional visual space into a low-dimensional space and discard extraneous information for effective and generalizable policy learning.
+
+Prompt Video Encoder. A Prompt Video Encoder encodes a prompt video into an embedding that serves as a reference for the policy. To specify the desired task, the embedding of the prompt video is used for an implicit understanding of task execution. Specifically, the Prompt Video Encoder comprises a per-frame encoder and a Perceiver Resampler (Jaegle et al., 2022). The Perceiver Resampler aggregates frame embeddings from the per-frame encoder to produce an overall embedding  $E_{\mathrm{prt}}$  of the prompt video. Each representation  $E_{\mathrm{prt}}$  consists of  $N_{\mathrm{prt}}$  tokens.  $E_{\mathrm{prt}}$  abstracts task-relevant attributes from videos and serves as the policy condition. Formally, the Prompt Video Encoder outputs as:
+
+$$
+E _ {\mathrm {p r t}} = \Psi_ {\mathrm {p r t}} \left(v _ {1}, v _ {2}, \dots , v _ {N}\right). \tag {4}
+$$
+
+# 3.2.2 CAUSAL PREDICTOR
+
+Connecting to the visual encoders, a video-conditioned Causal Predictor integrates the visual representations of prompt videos and observations to conduct sequence modeling on the mixture dataset.
+
+To effectively leverage the mixture dataset, the predictor is designed to conduct two subtasks of behavior cloning, visual transition prediction and inverse dynamics learning. The visual transition delineates how observations should evolve to accomplish tasks depicted by the prompt videos. Formally, visual transition prediction approximates the distribution  $P\left( {{o}_{t + 1} \mid  {o}_{ \leq  t},V}\right)$  ,which is embedded in both  ${\mathcal{D}}_{\text{demo }}$  and  ${\mathcal{D}}_{\text{vid }}$  . Through visual transition prediction, the model learns to align the visual observation sequence with the prompt videos. Inverse dynamics learning approximates the distribution  $P\left( {{a}_{t} \mid  {o}_{t},{o}_{t + 1}}\right)$  ,reflecting the actions required to realize a given visual transition. Although the action labels are only available in  ${\mathcal{D}}_{\text{demo }},P\left( {{a}_{t} \mid  {o}_{t},{o}_{t + 1}}\right)$  is shared across tasks due to the shared transition function  $\mathcal{P}$  . Given the challenge of predicting in the raw visual observation space, we replace  ${o}_{t + 1}$  with the joint embedding of the next observation  ${h}_{t + 1}$  in seek of a compressed embedding space conducive to knowledge transfer. Accordingly, the causal predictor is tasked with capturing  $P\left( {{h}_{t + 1} \mid  {o}_{ \leq  t},V}\right)$  and  $P\left( {{a}_{t} \mid  {o}_{t},{h}_{t + 1}}\right)$  .
+
+The input and output sequences of the causal predictor are designed to concurrently capture the visual transition  $P(h_{t+1}|o_{\leq t}, V)$  and the inverse dynamics  $P(a_t|o_t, h_{t+1})$ . As shown in Figure 2, with the video representation  $E_{\mathrm{pr}}$  as the prompt, the causal predictor takes the trajectory sequence  $(o_{t-k+1}, h_{t-k+2}, \ldots, o_t, h_{t+1})$  as input to causally predict  $(\hat{h}_{t-k+2}, \hat{a}_{t-k+1}, \ldots, \hat{h}_{t+1}, \hat{a}_t)$  respectively. Structurally, the predictor comprises a causal Transformer encoder  $\Psi_{\mathrm{pred}}$  and two prediction heads,  $\Gamma_{\mathrm{obs}}$  and  $\Gamma_{\mathrm{act}}$ . Respectively,  $\Gamma_{\mathrm{obs}}$  predicts the joint embedding of the next observation  $h_{t+1}$  from the hidden states of observation tokens  $o_t$ , and  $\Gamma_{\mathrm{act}}$  predicts the action tokens  $a_t$  from the hidden states of  $h_{t+1}$ . Formally, the predictor encodes the input sequence:
+
+$$
+Z = \Psi_ {\text {p r e d}} \left(E _ {\text {p r t}}, o _ {t - k + 1}, h _ {t - k + 2}, \dots , o _ {t}, h _ {t + 1}\right) \tag {5}
+$$
+
+and predicts the tokens:
+
+$$
+\hat {h} _ {t + 1} = \Gamma_ {\text {o b s}} \left(Z _ {t} ^ {(o)}\right), \quad \hat {a} _ {t} = \Gamma_ {\text {a c t}} \left(Z _ {t + 1} ^ {(h)}\right). \tag {6}
+$$
+
+Here,  $Z_{t}^{[\cdot]}$  denotes the hidden states of the corresponding tokens in  $Z$ , while  $\hat{h}_{t+1}$  and  $\hat{a}_t$  represent the predicted tokens.
+
+In this predictive form, the causal predictor alternatively models  $P(h_{t + 1}|E_{\mathrm{prt}},h_{\leq t},o_{\leq t})$  to capture the visual transition and  $P(a_{t}|E_{\mathrm{prt}},o_{\leq t},h_{\leq t + 1})$  to capture the inverse dynamics. During inference, the predictor iteratively feeds the predicted joint embedding tokens back into the input sequence to predict tokens. By combining the two types of token prediction together, the causal predictor indeed works as a planning-based policy, which predicts the desired next observations first and then converts the plan into actions.
+
+# 3.3 TRAINING PROCEDURE
+
+With the designed sequence modeling form, JEPT can simultaneously capture visual transition and inverse dynamics. In this subsection, we describe how JEPT is optimized on the mixture dataset  $\mathcal{D}_{\mathrm{demo}} \cup \mathcal{D}_{\mathrm{vid}}$ . Specifically, two predictive losses, the visual transition loss and the inverse dynamics loss, are optimized to predict the joint embedding tokens and the action tokens, respectively.
+
+Visual Transition Loss. For the joint embedding prediction, we compute the average  $L_{2}$  distance between the predicted joint embedding and the encoded joint embedding as the visual transition loss  $\mathcal{L}_{\mathrm{obs}}$  to approximate the visual transition  $P(h_{t + 1}|E_{\mathrm{prt}},h_{\leq t},o_{\leq t})$ . For both  $\mathcal{D}_{\mathrm{demo}}$  and  $\mathcal{D}_{\mathrm{vid}}$ , the joint embedding of the next observations is available. The visual transition loss  $\mathcal{L}_{\mathrm{obs}}$  is defined as:
+
+$$
+\mathcal {L} _ {\mathrm {o b s}} = \mathbb {E} _ {(V, O) \sim \mathcal {D} _ {\mathrm {d e m o}} \cup \mathcal {D} _ {\mathrm {v i d}}} \left[ \frac {1}{k} \sum_ {i = t - k + 1} ^ {t} \left\| h _ {i + 1} - \hat {h} _ {i + 1} \right\| _ {2} \right]. \tag {7}
+$$
+
+Inverse Dynamics Loss. For the action prediction, we compute the average Cross-Entropy between the predicted actions and the discrete ground-truth actions as the inverse dynamics loss  $\mathcal{L}_{\mathrm{act}}$  to
+
+approximate the inverse dynamics  $P\left(a_{t}|E_{\mathrm{prt}},o_{\leq t},h_{\leq t + 1}\right)$ . The ground-truth action label is merely available in  $\mathcal{D}_{\mathrm{demo}}$ . The inverse dynamics loss  $\mathcal{L}_{\mathrm{act}}$  is defined as:
+
+$$
+\mathcal {L} _ {\mathrm {a c t}} = \mathbb {E} _ {(V, O, A) \sim \mathcal {D} _ {\mathrm {d e m o}}} \left[ \frac {1}{k} \sum_ {i = t - k + 1} ^ {t} a _ {i} \log \hat {a} _ {i} \right]. \tag {8}
+$$
+
+Via optimizing  $\mathcal{L}_{\mathrm{obs}}$ , JEPT learns to generate the visual transitions aligned with the prompt videos. By optimizing  $\mathcal{L}_{\mathrm{act}}$ , JEPT learns to convert the visual transitions into actions. Formally, all the components of JEPT are trained with the loss  $\mathcal{L}_{\mathrm{total}}$ :
+
+$$
+\mathcal {L} _ {\text {t o t a l}} = \mathcal {L} _ {\text {o b s}} + \mathbb {1} _ {d \in \mathcal {D} _ {\text {d e m o}}} \cdot c \cdot \mathcal {L} _ {\text {a c t}}, \tag {9}
+$$
+
+where  $c$  is a hyperparameter balancing the two losses, and  $\mathbb{1}_{d\in \mathcal{D}_{\mathrm{demo}}}$  is an indicator function that equals 1 when the data sample  $d$  is from  $\mathcal{D}_{\mathrm{demo}}$  and 0 otherwise. Considering that JEPT predicts the encoded joint embeddings instead of the raw visual observations, we adopt an alternative training procedure to separately optimize the Joint Embedding Encoder and other components in case of potential model collapse. Additionally, an exponential moving average (EMA) of the joint embedding is used when the Joint Embedding Encoder is a fixed target network to stabilize the training, which is widely adapted in the previous JEPAs (Assran et al., 2023; Bardes et al., 2024). The overall process is shown in Algorithm 1 in Appendix A.1.
+
+# 4 EXPERIEMNTS
+
+In this section, we evaluate the effectiveness of our proposed JEPT on the mixture dataset, where the model learns from action-labeled expert demonstrations and unlabeled expert videos. We evaluate JEPT and baselines on two simulated benchmarks, Meta-World (Yu et al., 2020a) and RoboSuite (Zhu et al., 2020). Both benchmarks provide a variety of robotic manipulation tasks and are widely used for evaluating visual control tasks. Via the experiments, we aim to figure out: (1) whether JEPT effectively leverages the additional  $\mathcal{D}_{\mathrm{vid}}$  to improve the performance on unlabeled tasks, (2) whether the learned policy can generalize to unseen tasks, and (3) whether the joint embedding predictive mechanism in JEPT is essential for knowledge transfer.
+
+# 4.1 EXPERIMENT SETUP
+
+We first introduce the experimental setup, including the environments, dataset, metrics, and baselines used in our experiments. More details are available in Appendix B.
+
+**Environments.** We replace the language task descriptions in the environments with prompt videos recorded in the environments as task specifications. In our Meta-World experiments, a Sawyer robot arm interacts with various objects. The prompt videos are recorded with the same robot arm performing identical manipulations, which means there is no visual gap between the prompt videos and the expert videos. In our Robosuite experiments, a Panda robot arm interacts with objects. The prompt videos are recorded by performing the same manipulation with various robot arms, including Panda, Sawyer, IIWA, and UR5e, introducing a visual gap between the prompt and expert videos.
+
+Dataset. We select 18 tasks for the Meta-World task set and 15 tasks for the Robosuite task set. In order to construct the mixture dataset  $\mathcal{D}_{\mathrm{demo}} \cup \mathcal{D}_{\mathrm{vid}}$  and evaluate on the unseen tasks, we split each task set into three subsets: (1)  $\mathcal{T}_{\mathrm{demo}}$ : the tasks with the prompt videos and the paired expert demonstrations, (2)  $\mathcal{T}_{\mathrm{vid}}$ : the tasks with the prompt videos and the paired expert videos, and (3)  $\mathcal{T}_{\mathrm{unseen}}$ : the tasks with merely the prompt videos. For each task, we construct an expert policy in the vector state space and collect demonstrations or videos by running this expert policy in the rendered environments.  $\mathcal{D}_{\mathrm{demo}}$  is collected from the tasks in  $\mathcal{T}_{\mathrm{demo}}$ , while  $\mathcal{D}_{\mathrm{vid}}$  is collected from the tasks in  $\mathcal{T}_{\mathrm{vid}}$ . We also collect prompt videos for the tasks from  $\mathcal{T}_{\mathrm{unseen}}$ , which are not included in the mixture dataset.
+
+Metrics. We evaluate the model performance in terms of success rate, which is the percentage of successful episodes within 50 trials with different random seeds. The success rates of the tasks from  $\mathcal{T}_{\mathrm{demo}}$ ,  $\mathcal{T}_{\mathrm{vid}}$  and  $\mathcal{T}_{\mathrm{unseen}}$  reflect the model's ability of behavior cloning, learning from videos, and one-shot imitation learning, respectively. We also calculate the average success rates of seen and unseen tasks to evaluate the overall performance.
+
+Baselines. The baselines chosen for comparison with JEPT in our setting are as follows:
+
+Table 1: Success Rates (\%) of JEPT and the baselines calculated from 50 trials for each task in Meta-World. The average success rate of the tasks in  $\mathcal{T}_{\mathrm{demo}}$  and  $\mathcal{T}_{\mathrm{vid}}$  are listed in the first two rows, while the individual success rate of the 4 tasks in  $\mathcal{T}_{\mathrm{unseen}}$  are listed in the following rows. The average success rates of the seen and unseen tasks are listed in the last two row.  
+
+<table><tr><td>Task</td><td>Vid2Robot-D</td><td>Vid2Robot-M</td><td>BC+IDM</td><td>JEPT+MWM</td><td>DT*</td><td>JEPT</td></tr><tr><td>Tdemo</td><td>51.8</td><td>46.3</td><td>49.5</td><td>44.5</td><td>61.3</td><td>51.3</td></tr><tr><td>Tvid</td><td>4.7</td><td>28.7</td><td>8.3</td><td>21.3</td><td>13.0</td><td>31.7</td></tr><tr><td>Handle Press</td><td>10.0</td><td>22.0</td><td>8.0</td><td>18.0</td><td>6.0</td><td>28.0</td></tr><tr><td>Lever Pull</td><td>0.0</td><td>4.0</td><td>0.0</td><td>4.0</td><td>0.0</td><td>10.0</td></tr><tr><td>Plate Slide Back</td><td>4.0</td><td>8.0</td><td>0.0</td><td>2.0</td><td>0.0</td><td>14.0</td></tr><tr><td>Faucet Open</td><td>4.0</td><td>14.0</td><td>4.0</td><td>14.0</td><td>0.0</td><td>22.0</td></tr><tr><td>Seen Average</td><td>28.2</td><td>37.5</td><td>28.9</td><td>32.9</td><td>37.1</td><td>41.5</td></tr><tr><td>Unseen Average</td><td>4.5</td><td>12.0</td><td>3.0</td><td>9.5</td><td>1.5</td><td>18.5</td></tr></table>
+
+Table 2: Success Rates (\%) of the ablations calculated from 50 trials for each task in Meta-World. We train JEPT on a dataset with various task numbers for  $\mathcal{D}_{\mathrm{vid}}$  collection. The average success rate of the tasks in  $\mathcal{T}_{\mathrm{demo}}$  is listed in the first row, while the individual success rate of the 4 tasks in  $\mathcal{T}_{\mathrm{unseen}}$  are listed in the following rows. The average success rate of the unseen tasks is calculated in the last row.  
+
+<table><tr><td>Task</td><td>None</td><td>2 Tasks</td><td>4 Tasks</td><td>6 Tasks (JEPT)</td></tr><tr><td>\( \mathcal{T}_{\text{demo}} \)</td><td>59.5</td><td>49.8</td><td>47.0</td><td>51.3</td></tr><tr><td>Handle Press</td><td>6.0</td><td>4.0</td><td>16.0</td><td>28.0</td></tr><tr><td>Lever Pull</td><td>0.0</td><td>0.0</td><td>4.0</td><td>10.0</td></tr><tr><td>Plate Slide Back</td><td>0.0</td><td>0.0</td><td>10.0</td><td>14.0</td></tr><tr><td>Faucet Open</td><td>0.0</td><td>2.0</td><td>20.0</td><td>22.0</td></tr><tr><td>Unseen Average</td><td>1.5</td><td>1.5</td><td>12.5</td><td>18.5</td></tr></table>
+
+- Vid2Robot (Jain et al., 2024): conducts behavior cloning and auxiliary representation learning on a large dataset where all the data is action-labeled. We train two variants of Vid2Robot: Vid2Robot-D refers to Vid2Robot trained merely on  $\mathcal{D}_{\mathrm{demo}}$ , while Vid2Robot-M refers to Vid2Robot trained on  $\mathcal{D}_{\mathrm{demo}} \cup \mathcal{D}_{\mathrm{vid}}$  with behavior cloning loss masked from  $\mathcal{D}_{\mathrm{vid}}$ .  
+- BC+IDM: Zheng et al. (2023); Kim et al. (2023) proposed a paradigm leveraging the mixture dataset for visual imitation learning. These methods separately learn an inverse dynamics model for action annotation and conduct behavior cloning on the mixture dataset. We accommodate these methods to video-conditioned imitation learning.  
+-  $\mathbf{DT}^{\star}$ : We remove the joint embedding predictive mechanism from JEPT, which indeed conducts behavior cloning with a DT variant on  $\mathcal{D}_{\mathrm{demo}}$ .  
+- JEPT+MMW (Seo et al., 2023): We replace the joint embedding in JEPT with the representation learned with MWM as a baseline. MWM provides a practical representation learning method for visual control tasks. Unlike the joint embedding in JEPT, MWM learns a generative Masked Auto-Encoder (MAE).
+
+# 4.2 META-WORLD EXPERIMENTS
+
+In the Meta-World experiments, we divide the 18 tasks into three subsets:  $\mathcal{T}_{\mathrm{demo}}$ ,  $\mathcal{T}_{\mathrm{vid}}$  and  $\mathcal{T}_{\mathrm{unseen}}$ , respectively containing 8, 6 and 4 tasks.
+
+Comparisons with Baselines. The task success rates of JEPT and the baselines are presented in Table 1, where JEPT surpasses all baselines in terms of average success rates. Regarding the success rates of tasks from  $\mathcal{T}_{\mathrm{demo}}$ , JEPT performs slightly worse than  $\mathrm{DT}^{\star}$ , similar to the gap between Vid2Robot-M and Vid2Robot-D. This may be a side-effect of the more tasks integrated into the policy, leading to a decrease in the performance of individual tasks. Notably, JEPT exhibits a significant advantage in tasks from both  $\mathcal{T}_{\mathrm{vid}}$  and  $\mathcal{T}_{\mathrm{unseen}}$ , indicating that JEPT effectively leverages the additional  $\mathcal{D}_{\mathrm{vid}}$  to learn a more generalizable policy. Compared to Vid2Robot-M, the sequence modeling design in JEPT, extracting visual transition and inverse dynamics, might suit the mixture dataset more. Although BC+IDM also captures the inverse dynamics, JEPT integrates transition-aware representation learning with inverse dynamics learning on the mixture dataset instead of an IDM learned merely on  $\mathcal{D}_{\mathrm{demo}}$ , which may capture more generalizable inverse dynamics knowledge and accounts for the better performance of JEPT. Additionally, the outperformance of JEPT over  $\mathrm{DT}^{\star}$
+
+Figure 3: Visualization of the observation embeddings learned in JEPT (right), MWM (mid), and Vid2Robot-M (left). The observation embeddings are calculated in a rollout video from 'Door Close' in  $\mathcal{T}_{\mathrm{demo}}$ , 'Drawer Close' in  $\mathcal{T}_{\mathrm{vid}}$ , and 'Facet Open' in  $\mathcal{T}_{\mathrm{unseen}}$ . We apply t-SNE on one whole embedding set and split the projected vectors according to the algorithms into three sub-figures.  
+![](images/84f060159c772e26851003fdd509bb9af65c5fddf444cae9aef0bf9ed7b54717.jpg)  
+Door Close
+
+![](images/6428e6c54b678985a08a6c3ba0c2dc6d58897cb89295cdbebf288eb4b3833b3e.jpg)  
+- Drawer Close
+
+![](images/045add1e667b2c3864e9571d553d1bff5deee57c8c3ba39c7acb3f9e88c5135a.jpg)  
+Faucet Open
+
+and JEPT+MWM indicates that the predictive joint embedding in JEPT is more suitable for learning on mixture datasets than non-specialized or alternative embedding mechanisms.
+
+Ablation Study. We conduct an ablation study to investigate the impact of the size of  $\mathcal{D}_{\mathrm{vid}}$ . As is listed in Table 2, we train JEPT on datasets where 0, 2, 4 and 6 tasks are covered in  $\mathcal{D}_{\mathrm{vid}}$ . The success rates of tasks increases as the size of  $\mathcal{T}_{\mathrm{vid}}$  grows. The results indicate that the additional video data is beneficial and JEPT can effectively leverage the data in  $\mathcal{D}_{\mathrm{vid}}$  to learn a more generalizable video-conditioned policy. Additionally, when the  $\mathcal{D}_{\mathrm{vid}}$  grows from 0 tasks to 6 tasks, the performance degradation on the seen tasks might result from a higher integration of the policy, which stands with our observation in Table 1.
+
+Visualizations. We also visualize the visual embeddings learned in JEPT, MWM, and Vid2Robot-M. Specifically, we select one task from each of  $\mathcal{T}_{\mathrm{demo}}$ ,  $\mathcal{T}_{\mathrm{vid}}$ , and  $\mathcal{T}_{\mathrm{unseen}}$  and calculate the observation embeddings of a rollout video from each task. We apply t-SNE to the whole set containing the observation embeddings of the three videos from JEPT, MWM, and Vid2Robot to project them into a shared 2D space. As shown in Figure 3, the observation embeddings from Vid2Robot-M and MWM demonstrate severe out-of-distribution phenomena among the action-labeled, unlabeled, and unseen tasks. The embeddings from JEPT exhibit greater similarity in the distribution, potentially contributing to its superior generalization.
+
+# 4.3 ROBOSUIT EXPERIMENTS
+
+We also conduct experiments on the Robosuite, which offers a variety of robotic arms for manipulation tasks. Thus we evaluate the performance of JEPT in the case where there is a visual gap between prompt videos and expert videos. We split the 15 tasks into three subsets:  $\mathcal{T}_{\mathrm{demo}}$ ,  $\mathcal{T}_{\mathrm{vid}}$ , and  $\mathcal{T}_{\mathrm{unseen}}$ , containing 6, 5, and 4 tasks, respectively. The results of these experiments are shown in Table 3. Although the success rates in these tasks are somewhat lower than those in the Meta-World experiments due to the increased difficulty, JEPT still outperforms the baselines across all tasks. Similarly to the Meta-world results, the success rate of JEPT in  $\mathcal{T}_{\mathrm{demo}}$  is lower than that of  $\mathrm{DT}^{\star}$ , possibly due to the integration of more tasks into a single policy. However, JEPT still exceeds all the baselines in  $\mathcal{T}_{\mathrm{vid}}$  and  $\mathcal{T}_{\mathrm{unseen}}$ , indicating that JEPT can effectively leverage  $\mathcal{D}_{\mathrm{vid}}$  to learn more generalizable policies despite visual gaps.
+
+# 4.4 VISUAL PRIOR INJECTION
+
+Considering that visual priors learned from external datasets may enhance the generalization of the model due to their universality, we additionally explore the possibility of injecting such priors from the pre-trained models into JEPT. A directly injection approach is adopted in our practice. We simply replace the output of the single-timestep spatial encoder in the Joint Embedding Encoder with the output from the visual encoders of some pre-trained visual models. Specifically, we experiment on Meta-World with FlowFormer (Huang et al., 2022), VideoMAE-v2 (Wang et al., 2023), Dino-v2 (Oquab et al., 2024), and SAM (Kirillov et al., 2023), representing visual priors related to optical flow, video reconstruction, depth estimation, and visual segmentation, respectively. As shown
+
+Table 3: Success Rates (\%) of JEPT and the baselines calculated from 50 trials for each task in Robosuite. The average success rate of the tasks in  $\mathcal{T}_{\mathrm{demo}}$  and  $\mathcal{T}_{\mathrm{vid}}$  are listed in the first two rows, while the individual success rate of the 4 tasks in  $\mathcal{T}_{\mathrm{unseen}}$  are listed in the following rows. 'X T' refers to the task where Panda robot arm performs the task T with the prompt video recorded with X robot arm. The average success rates of the seen and unseen tasks are listed in the last two rows.  
+
+<table><tr><td>Task</td><td>Vid2Robot-D</td><td>Vid2Robot-M</td><td>BC+IDM</td><td>JEPT+MWM</td><td>DT*</td><td>JEPT</td></tr><tr><td>Tdemo</td><td>26.8</td><td>20.7</td><td>26.7</td><td>24.3</td><td>36.3</td><td>27.2</td></tr><tr><td>Tvid</td><td>4.0</td><td>11.6</td><td>8.4</td><td>10.8</td><td>2.0</td><td>12.8</td></tr><tr><td>Panda Lift</td><td>4.0</td><td>16.0</td><td>6.0</td><td>22.0</td><td>8.0</td><td>38.0</td></tr><tr><td>Sawyer Lift</td><td>0.0</td><td>2.0</td><td>0.0</td><td>6.0</td><td>0.0</td><td>16.0</td></tr><tr><td>IIWA Lift</td><td>0.0</td><td>4.0</td><td>0.0</td><td>2.0</td><td>0.0</td><td>12.0</td></tr><tr><td>UR5e Lift</td><td>0.0</td><td>0.0</td><td>0.0</td><td>0.0</td><td>0.0</td><td>8.0</td></tr><tr><td>Seen Average</td><td>15.5</td><td>16.1</td><td>17.5</td><td>17.6</td><td>19.2</td><td>20.1</td></tr><tr><td>Unseen Average</td><td>1.0</td><td>5.5</td><td>1.5</td><td>7.5</td><td>2.0</td><td>18.5</td></tr></table>
+
+Table 4: Success Rates (\%) of JEPT injected with different visual priors calculated from 50 trials for each task in Meta-World tasks. The average success rate of the tasks in  $\mathcal{T}_{\mathrm{demo}}$  and  $\mathcal{T}_{\mathrm{vid}}$  are listed in the first two rows, while the individual success rate of the 4 tasks in  $\mathcal{T}_{\mathrm{unseen}}$  are listed in the following rows. The average success rate of the seen and unseen tasks are calculated in the last two row.  
+
+<table><tr><td>Task</td><td>JEPT+FlowFormer</td><td>JEPT+VideoMAE-v2</td><td>JEPT+Dino-v2</td><td>JEPT+SAM</td></tr><tr><td>Tdemo</td><td>59.8</td><td>33.3</td><td>38.5</td><td>41.3</td></tr><tr><td>Tvid</td><td>35.3</td><td>11.7</td><td>9.3</td><td>19.0</td></tr><tr><td>Handle Press</td><td>36.0</td><td>8.0</td><td>8.0</td><td>16.0</td></tr><tr><td>Lever Pull</td><td>10.0</td><td>2.0</td><td>2.0</td><td>4.0</td></tr><tr><td>Plate Slide Back</td><td>6.0</td><td>0.0</td><td>0.0</td><td>2.0</td></tr><tr><td>Faucet Open</td><td>26.0</td><td>4.0</td><td>4.0</td><td>10.0</td></tr><tr><td>Seen Average</td><td>47.5</td><td>22.5</td><td>23.9</td><td>30.1</td></tr><tr><td>Unseen Average</td><td>19.5</td><td>3.5</td><td>3.5</td><td>8.0</td></tr></table>
+
+in Table 4, JEPT+FlowFormer outperforms other visual priors and surpasses the original JEPT. This indicates that the optical flow priors are beneficial for visual control tasks and can generalize across tasks. Meanwhile, JEPT can effectively incorporate the optical flow priors to enhance model performance. The poor performance of other visual priors may result from the inherent unsuitability of these priors for visual control tasks or the inadequacy of the injection approach, requiring further exploration in future work.
+
+# 5 CONCLUSION
+
+In this paper, we propose the Joint Embedding Predictive Transformer (JEPT), a novel approach for video-conditioned policy learning. JEPT is designed to learn from a mixture of expert demonstrations and expert videos paired with prompt videos, aiming to reduce the burden of action label annotation. To suit the mixture dataset, we decompose the video-conditioned policy learning into two subtasks: visual transition prediction and inverse dynamics learning. By jointly learning the two subtasks in the sequence modeling, JEPT works as a planning-based policy. We implement JEPT as an extension of the joint embedding predictive architecture to learn an abstract representation of visual observation, which aids in generalizing video-conditioned policy. Experimentally, we evaluate the effectiveness of JEPT on a series of visual control tasks. Additionally, we explore the JEPT as a simple visual priors injection approach and find it valid in injecting optical flow knowledge.
+
+Limitations. A series of experimental results of JEPT indicate that joint learning visual transition and inverse dynamics allow it to effectively leverage the mixture dataset and derive a generalizable policy. Despite this, we have also identified some limitations. Since our work focuses on solving problems within datasets containing expert videos, our method does not include additional designs for the potential visual gap between the prompt videos and task dynamics. We have validated the effectiveness of JEPT in addressing a certain level of visual gap in Robosuite environments, but more design and validation are needed to apply JEPT to one-shot visual imitation with more discrepancy prompt videos such as human videos. In this regard, our attempt at injecting visual priors could be one beneficial approach.
+
+# ACKNOWLEDGMENTS
+
+This work was supported by NSFC under Grant 62450001 and 62476008. The authors would like to thank the anonymous reviewers for their valuable comments and advice. The authors express special thanks to Xinrun Xu (v.xinrun@gmail.com) for her assistance in the figures of this work.
+
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+Tianhe Yu, Deirdre Quillen, Zhanpeng He, Ryan Julian, Karol Hausman, Chelsea Finn, and Sergey Levine. Meta-world: A benchmark and evaluation for multi-task and meta reinforcement learning. In Proc. of Conference on Robot Learning, 2020a.  
+Xingrui Yu, Yueming Lyu, and Ivor Tsang. Intrinsic reward driven imitation learning via generative model. In Proc. of International Conference on Machine Learning, 2020b.  
+Qihang Zhang, Zhenghao Peng, and Bolei Zhou. Learning to drive by watching youtube videos: Action-conditioned contrastive policy pretraining. In Proc. of European Conference on Computer Vision, 2022.  
+Qinqing Zheng, Mikael Henaff, Brandon Amos, and Aditya Grover. Semi-supervised offline reinforcement learning with action-free trajectories. In Proc. of International Conference on Machine Learning, 2023.  
+Yuke Zhu, Josiah Wong, Ajay Mandlekar, Roberto Martin-Martin, Abhishek Joshi, Soroush Nasiriany, and Yifeng Zhu. robosuite: A modular simulation framework and benchmark for robot learning. arXiv preprint arXiv:2009.12293, 2020.
+
+# A IMPLEMENTATIONS
+
+We build our framework based on PyTorch (Paszke et al., 2019) and use the implementations of Transformer modules from the codebase x-transformers<sup>2</sup>. We use the 3090 Nvidia GPU and i9-12900K CPU for the JEPT training and testing. It takes around 10 hours to train JEPT on the Meta-World dataset and 16.5 hours to train JEPT on the Robosuite dataset. The details of the implementations are as follows.
+
+# A.1 ALGORITHM
+
+Algorithm 1 Joint Embedding Predictive Transformer  
+1: Input: Mixture dataset  $\mathcal{D}_{\mathrm{demo}} \cup \mathcal{D}_{\mathrm{vid}}$   
+2: Initialize: JEPT components  $\Psi_{\mathrm{obs}}, \Psi_{\mathrm{prt}}, \Psi_{\mathrm{pred}}, \Gamma_{\mathrm{obs}}, \Gamma_{\mathrm{act}}$  with random weights  
+3: for  $e = 1, 2, \ldots$  do  
+4: for  $d \in \mathcal{D}_{\mathrm{demo}} \cup \mathcal{D}_{\mathrm{vid}}$  do  
+5: Encode  $V$  and  $O$  into  $E_{\mathrm{prt}}$  and  $(h_{t-k+1}, \ldots, h_t)$  as Equations 3 and 4  
+6: Predict the joint embedding tokens and action tokens as Equations 5 and 6  
+7: Compute  $\mathcal{L}_{\mathrm{total}}$  as Equation 9  
+8: Update  $\Psi_{\mathrm{obs}}$  with  $\mathcal{L}_{\mathrm{total}}$   
+9: end for  
+10: for  $d \in \mathcal{D}_{\mathrm{demo}} \cup \mathcal{D}_{\mathrm{vid}}$  do  
+11: Repeat steps in lines 5-7  
+12: Update  $\Psi_{\mathrm{prt}}, \Psi_{\mathrm{pred}}, \Gamma_{\mathrm{obs}}, \Gamma_{\mathrm{act}}$  with  $\mathcal{L}_{\mathrm{total}}$   
+13: end for  
+14: end for
+
+# A.2 MODEL DETAILS
+
+Joint Embedding Encoder. To encode visual observations, we utilize a pre-trained ViT (Dosovitskiy et al., 2021) variant, 'vit_base_batch16_224', as the spatial visual encoder. To effectively compress the tokens of visual observations, a Perceiver-IO (Jaegle et al., 2022) with a fixed number of learnable queries is employed to compress the visual tokens of single-frame observations. The ViT weights remain fixed during training, allowing the Perceiver-IO to function as an adaptor network. The compressed visual tokens are concatenated with representations of the self-state vector produced by an MLP to form the observation tokens. A bi-directional Transformer temporal encoder is used to encode the observation tokens with context awareness. For visual prior injection approaches in Section 4.4, we simply replace the outputs of the fixed ViT with the embedding outputs of visual encoders pre-trained to capture visual priors. Notably, there are subtle distinctions in input processing among these four visual prior injections. For FlowFormer, observation tokens at each temporal step are computed from adjacent frames. VideoMAE v2 replicates all video frames excluding the first to preserve the original temporal length. Dino v2 and SAM v2 both utilize single-frame images as inputs. Moreover, these models diverge in architectural design, pre-training datasets, and methodological approaches. Despite these variations, our visual prior experiments do not aim to comparatively evaluate visual prior performance. Instead, our objective is to ascertain whether a joint embedding predictive of these representations with visual priors can further compress visual information and enhance generalization.
+
+Prompt Video Encoder. To encode the prompt videos, we utilize the same ViT variant as the Joint Embedding Encoder for encoding the visual tokens. Two cascaded Perceiver-IOs are employed to aggregate these tokens. The first Perceiver compresses the tokens from a single frame, while the second compresses the flattened tokens from the first layer to form the prompt video embeddings. During training, only the Perceiver-IOs are trained, the ViT weights remain fixed.
+
+Video-Conditioned Causal Predictor. To process observations in expert videos and demonstrations, we employ the same structure as the Joint Embedding Encoder to encode single-timestep observations. Notably, the single-timestep encoder shares the weights in our implementation, and the EMAs are
+
+calculated when producing the target joint embeddings. A causal Transformer is utilized, taking as input the sequence of all compressed tokens. A causal mask, where tokens attend only to preceding tokens and tokens from the same observations, ensures prediction causality. To predict the joint embedding tokens and action tokens, two prediction heads are applied to the causal Transformer's output. Specifically, an MLP predicts the joint embedding tokens, while a cross-attention Transformer with learnable action queries for each action dimension predicts the action tokens.
+
+# A.3 BASELINE ADAPTATION
+
+Vid2Robot We implement Vid2Robot following the original paper (Jain et al., 2024), as no codebase is released. Similar to our structure, a spatial visual encoder followed by a Perceiver processes the spatiotemporal information of videos and observation sequences. We use the same ViT variant as the Joint Embedding Encoder to encode the visual tokens of the prompt videos and observations, ensuring consistency in comparison. We align the Perceiver structure and the action prediction head in Vid2Robot with our framework. Since no text instruction is provided in our setting, we remove the auxiliary text-video contrastive loss in Vid2Robot.
+
+MWM We adapt the released codebase of MWM (Seo et al., 2023) to our setting. The only difference is the absence of a reward signal in our setting; thus, we remove the reward prediction loss.
+
+$\mathbf{DT}^*$  We adapt our implementation of JEPT to perform sequence modeling in a typical behavior cloning sequence  $P(a_{t}|E_{\mathrm{prt}},o_{1},a_{1},\ldots ,o_{t})$
+
+# A.4 HYPERPARAMETERS
+
+In our implementation, the Perceiver-IO structure is shared across the Joint Embedding Encoder, the Prompt Video Encoder, and the Video-Conditioned Causal Predictor. The hyperparameters for the Perceiver-IOs and other modules are detailed in Table 5. During training,  $E_{\mathrm{prt}}$  is initially excluded in the causal predictor as a warm-up strategy. Furthermore, the hyperparameters for the training process are listed in Table 6.
+
+# B ENVIRONMENT SETUP DETAILS
+
+# B.1 ENVIRONMENT
+
+Meta-World Environment. In the Meta-World environment, a Sawyer robot arm is directed to execute a variety of manipulation tasks. We employ 18 tasks originally from Meta-World to establish our video prompt visual control framework. As shown in Figure 4, for each task, a video of the identical manipulation in the same environment, albeit with a different random seed, is provided as the task prompt, while the original task descriptions are omitted. We utilize the customized camera perspective for visual observation and resize the image from the camera to  $224 \times 224$  pixels. In addition to the visual observation, we assume the agent is aware of its current self-state, which encompasses the joint angles, joint velocities, and end-effector position. A vector representing these self-state details is supplied alongside the visual camera image to form the observation. For action prediction using Cross-Entropy, the original continuous action space is discretized into 7 bins for each action dimension. During the rollout of each task, the episode is truncated after 250 steps, with an action repeat of 2.
+
+Robosuite Environment. In the Robosuite environment, we select tasks involving a Panda robot arm performing various manipulation tasks. We employ two tasks originally from the Robosuite simulator and four tasks implemented by MimicGen (Mandlekar et al., 2023) within the Robosuite framework to create our video prompt visual task. As illustrated in Figure 4, a video depicting a specific robot arm completing the same manipulation serves as the task prompt. The robot in these prompt videos may be a Panda, Sawyer, IIWA, or UR5e, indicating a potential visual gap between the prompt videos and the task dynamics. Two tasks are considered distinct if the robot arm in the prompt videos or the desired manipulation differs. Consequently, we construct 15 tasks from the 6 tasks of MimicGen. We utilize the 'agentview_image' attribute of the state as the visual image. Similar to the Meta-World environment, the visual image is resized to  $224 \times 224$  pixels and accompanied by the self-state vector. The continuous action space is discretized into 7 bins per dimension. During each
+
+Table 5: The hyperparameters of the JEPT Modules.  
+
+<table><tr><td colspan="2">Single-Frame Perceiver-IO</td></tr><tr><td>num latents</td><td>64</td></tr><tr><td>num queries</td><td>8</td></tr><tr><td>embedding channels</td><td>512</td></tr><tr><td>attention heads</td><td>8</td></tr><tr><td>encoder cross-attention layers</td><td>1</td></tr><tr><td>encoder self-attention layers</td><td>2</td></tr><tr><td>decoder cross-attention layers</td><td>1</td></tr><tr><td>attention feedforward dim</td><td>1024</td></tr><tr><td>attention dropout</td><td>0.1</td></tr><tr><td colspan="2">Temproal Perceiver-IO</td></tr><tr><td>num latents</td><td>128</td></tr><tr><td>num queries</td><td>32</td></tr><tr><td>embedding channels</td><td>512</td></tr><tr><td>attention heads</td><td>8</td></tr><tr><td>encoder cross-attention layers</td><td>1</td></tr><tr><td>encoder self-attention layers</td><td>2</td></tr><tr><td>decoder cross-attention layers</td><td>1</td></tr><tr><td>attention feedforward dim</td><td>892</td></tr><tr><td>attention dropout</td><td>0.08</td></tr><tr><td colspan="2">Bi-directional Temporal Encoder</td></tr><tr><td>joint embedding dim</td><td>256</td></tr><tr><td>attention layers</td><td>2</td></tr><tr><td>attention heads</td><td>8</td></tr><tr><td>attention dim</td><td>512</td></tr><tr><td>attention feedforward dim</td><td>1024</td></tr><tr><td>attention dropout</td><td>0.1</td></tr><tr><td>attention activation</td><td>GeLU</td></tr><tr><td colspan="2">Causal Transformer</td></tr><tr><td>Max prompt length</td><td>250</td></tr><tr><td>prompt tokens num Nprt</td><td>32</td></tr><tr><td>obs tokens num Nobs</td><td>9</td></tr><tr><td>context length k</td><td>8</td></tr><tr><td>embedding channels</td><td>584</td></tr><tr><td>attention heads</td><td>8</td></tr><tr><td>attention layers</td><td>4</td></tr><tr><td>attention feedforward dim</td><td>1168</td></tr><tr><td>attention dropout</td><td>0.05</td></tr><tr><td>Attention Activation</td><td>GeLU</td></tr><tr><td colspan="2">Action Prediction Head Γact</td></tr><tr><td>Cross-attention layers</td><td>2</td></tr><tr><td>Cross-attention heads</td><td>4</td></tr><tr><td>Cross-attention dim</td><td>256</td></tr><tr><td>Feedforward dim</td><td>512</td></tr><tr><td>activation</td><td>GeLU</td></tr><tr><td colspan="2">Joint Embedding Prediction Head Γobs</td></tr><tr><td>layers</td><td>3</td></tr><tr><td>hidden dim</td><td>256</td></tr><tr><td>activation</td><td>LeakyRelu</td></tr></table>
+
+Table 6: The hyperparameters of the training process.  
+
+<table><tr><td>Training Hyperparameters</td><td>Value</td></tr><tr><td>Minibatch size</td><td>90</td></tr><tr><td>Optimizer</td><td>AdamW</td></tr><tr><td>Joint Embedding Encoder Learning Rate</td><td>3e-5</td></tr><tr><td>Prompt Video Encoder Learning Rate</td><td>7e-5</td></tr><tr><td>Video-Conditioned Causal Predictor Learning Rate</td><td>7e-5</td></tr><tr><td>Weight Decay</td><td>1e-4</td></tr><tr><td>Max Gradient Clip</td><td>1.0</td></tr><tr><td>Warm Up Steps</td><td>200</td></tr><tr><td>Loss Weight c</td><td>8</td></tr></table>
+
+![](images/14e28641bb0fc00a537fe65bdc6c18740cc9e07c910d6e3209b3bf10d2a88c29.jpg)  
+Figure 4: The video prompt task setting of the Meta-World and Robosuite. In Meta-World, the task specification is a video of the same manipulation in the same environment with different random seed. In Robosuite, the task specification is a video of a certain robot arm executing the same manipulation, and the visual gap between the prompt videos and the task dynamics is likely.
+
+task rollout, the episode is truncated after 200 steps, although there is no restriction on episode length within the Robosuite environment.
+
+# B.2 DATASET
+
+Task Subset. In both Meta-World and Robosuite, tasks are manually categorized into three subsets: those with expert demonstrations and prompt videos,  $\mathcal{T}_{\mathrm{demo}}$ ; those with expert videos and prompt videos,  $\mathcal{T}_{\mathrm{vid}}$ ; and those with only prompt videos,  $\mathcal{T}_{\mathrm{unseen}}$ . The specific task divisions are detailed in Table 7 and 8. For Meta-World experiments, 500 expert videos/demonstrations are collected by executing expert policies on tasks from  $\mathcal{T}_{\mathrm{demo}} \cup \mathcal{T}_{\mathrm{vid}}$ . For Robosuite experiments, 1000 expert videos/demonstrations are gathered for each task in the dataset. For the tasks in  $\mathcal{T}_{\mathrm{unseen}}$ , 10 prompt videos are sampled for 5 trials with different random seeds on each to assess the one-shot imitation capability of the learned policy. In the ablation study of task numbers covered in  $\mathcal{T}_{\mathrm{vid}}$ , we use the 'Button Press Topdown Wall' and 'Door Open' tasks for the 2-task variant. Additionally, we include the 'Reach Wall' and 'Drawer Close' tasks for the 4-task variant.
+
+Expert Policy. To construct the data used for training and evaluation, we develop expert policies for individual tasks within the vector state space, collecting both prompt videos and corresponding expert demonstrations. Initially, expert policies are formulated in the continuous action space. In MetaWorld, policies are trained using PPO for each task, utilizing the vector state and reward function provided by the environment. We integrate rule-based policies from the GitHub repository<sup>3</sup> with PPO policies to form expert policies, executing rule-based policies with a probability of 0.85 and PPO policies with a probability of 0.15 to ensure diversity. For tasks in Robosuite, expert policies are learned through behavior cloning on the expert demonstrations from MimicGen. Additionally, expert policies for 'Stack' and 'Lift' tasks are trained using DrQ-v2 (Yarats et al., 2021) with no demonstrations available in Mimicgen. To collect expert rollouts in the discrete action space, continuous expert policies are used to generate actions, which are then replaced with the nearest discrete actions in the bins for interaction with the environment. A performance drop is noted in the discrete action space compared to the continuous action space. Specifically, the average success rate for expert policies in
+
+Table 7: The division of tasks in our Meta-World experiments.  
+
+<table><tr><td>Task Subset</td><td>Tasks</td></tr><tr><td>Tdemo</td><td>Button Press, Button Press Topdown, Handle Pull, Reach, Door Close, Window Open, Plate Slide, Faucet Close</td></tr><tr><td>Tvid</td><td>Button Press Topdown Wall, Door Open, Reach Wall,Drawer Close, Plate Slide Back Side, Window Close</td></tr><tr><td>Tunseen</td><td>Handle Press, Lever Pull, Plate Slide Back, Faucet Open</td></tr></table>
+
+Table 8: The division of tasks in our Robosuite experiments. 'X T' refers to the task where the Panda robot arm performs the task T with the prompt video recorded with X robot arm.  
+
+<table><tr><td>Task Subset</td><td>Tasks</td></tr><tr><td>Tdemo</td><td>Panda Stack, Sawyer Stack, IIWA Stack, UR5e Stack, Three Pieces Assembly, Coffee</td></tr><tr><td>Tvid</td><td>Panda Square, Sawyer Square, IIWA Square, UR5e Square, Stack Three</td></tr><tr><td>Tunseen</td><td>Panda Lift, Sawyer Lift, IIWA Lift, UR5e Lift</td></tr></table>
+
+Meta-World decreases from  $91.5\%$  to  $85.3\%$  post-discretization, and in Robosuite, it declines from  $78.3\%$  to  $69.7\%$ . For prompt video collection, we record visual observation sequences of the expert rollouts, while for expert videos or demonstrations, both visual observation sequences and self-state vector sequences are recorded.
+
+# C ADDITIONAL EXPERIMENTS
+
+# C.1 VISUAL PRIOR INJECTION ON ROBOSUITE
+
+In order to supplement the effects of various visual priors on JEPT in different environments, we conduct experiments on visual prior injection within the RoboSuite environment. The implementation of visual prior injection is consistent with the experiments conducted in Meta-World. The results, as shown in Table 9, indicate that the performance of JEPT can be slightly enhanced by utilizing priors based on optical flow to process the visual inputs. This suggests that optical flow might be a suitable prior for enhancing JEPT's performance across different environments. However, all of these visual encoders have been trained on large datasets across numerous visual tasks and possess certain generalizable priors for handling visual inputs. Our experiments with visual priors aim to ascertain whether a joint embedding predictive of these representations, when combined with visual priors, can further compress visual information and improve generalization. From the perspective of introducing more generalizable visual priors, these different priors for processing visual inputs should be logical. However, the success of learning predictive joint embeddings of these priors and effectively enhancing performance may depend on selecting more refined injection methods and better training designs. We merely highlight JEPT's potential in this domain, and further detailed research on other visual prior injections is left for future exploration.
+
+# C.2 ADDITIONAL ABLATIONS
+
+We conduct additional ablation studies to explore the influence of the hyperparameter  $c$  in JEPT. The value of  $c$  is varied within the set  $\{0.3, 1, 5, 8, 12\}$ , and the performance of JEPT is assessed in Meta-World. The results are presented in Table 10. When  $c$  remains within a reasonable range, JEPT exhibits similar performance. However, when  $c$  is excessively small or large, JEPT's performance declines significantly. This underscores the critical role of the hyperparameter  $c$  in JEPT's efficacy. In other experiments, we set  $c = 8$ , as it yields the optimal performance in Meta-World.
+
+To thoroughly investigate the impact of the ratio between  $D_{\mathrm{demo}}$  and  $D_{\mathrm{vid}}$  on model performance, we conduct additional ablation experiments. Specifically, beyond the original 14-task training dataset,
+
+we incorporate 6 additional tasks from Meta-World to collect training data (Handle Pull Side, Door Lock, Printer Open, Plate Slide Side, Dial Turn, Handle Press Side). We maintain a constant total amount of training data while varying the ratio of  $\mathcal{T}_{\mathrm{demo}}$  to  $\mathcal{T}_{\mathrm{vid}}$ . For various ratios, we experiment with different hyperparameters  $c$ , and the results are presented in Table 11. Given the variability of  $\mathcal{T}_{\mathrm{demo}}$  and  $\mathcal{T}_{\mathrm{vid}}$ , we report the average success rate on  $\mathcal{T}_{\mathrm{unseen}}$  under different settings. The results indicate that within a certain range,  $D_{\mathrm{vid}}$  effectively compensates for the lack of data with action labels. However, when  $D_{\mathrm{demo}}$  is insufficient, it may not adequately learn, making IDM difficult to generalize to unseen tasks. Moreover, the results suggest that when  $D_{\mathrm{vid}}$  is reduced, a larger  $c$  is often required to achieve better performance.
+
+Considering that employing a fixed video encoder mitigates the distribution shift in video representation between action-labeled and unlabeled data, thereby enhancing the generalization of the inverse dynamics model in BC+IDM, we conducted experiments on Meta-World using a pre-trained video encoder, denoted as  $\mathrm{BC + IDM}^{\star}$ . Most pre-trained video encoders utilize a kernel size of 2 for temporal compression, halving the representation length in the temporal dimension. Similar to our approach in visual prior injection with VideoMAE v2, we repeat all frames within the video except for the first frame to maintain consistent length and accurately decode the actions. Employing VideoMAE v2, the experimental results are presented in Table 12. There is a slight improvement in success rates on unlabeled tasks, but no gain on unseen tasks. Overall, the improvement from using a fixed video encoder is limited.
+
+# C.3 ADDITIONAL VISUALIZATION
+
+We conduct a series of visualizations within the Meta-World environment to reveal the distribution of some variants within the rollouts. Specifically, we process 150 videos for each of the 18 different tasks using a prompt video encoder to encode them into  $E_{prt}$ . We flatten these embeddings and apply t-SNE to reduce the dimensionality to two dimensions. As is shown in Figure 5, there are clear distinctions between different tasks, suggesting that JEPT effectively captures and represents the unique features of each task. Additionally, we observe a degree of similarity between related tasks, such as 'Faucet Close' and 'Faucet Open', as well as 'Window Open', 'Window Close', and 'Drawer Close'. This similarity reflects the inherent relationships between tasks, which may facilitate knowledge transfer during the learning process.
+
+Furthermore, we also apply t-SNE to visualize the actions and joint embeddings of the 150 trajectories for each task. Each point in the visualization is color-coded with varying transparency to denote its temporal position within the trajectory, with points closer to the end being less transparent. The visualizations are shown in Figures 7 and 6. From this analysis, we find that the overall distributions of actions and joint embeddings across different tasks are similar, which partially explains the generalization capability of JEPT. Additionally, there are specific concentrated regions in the distribution for each task, which likely arise from the unique nature of each task.
+
+Table 9: Success Rates (\%) of JEPT injected with different visual priors calculated from 50 trials for each task in RoboSuite tasks. The average success rate of the tasks in  $\mathcal{T}_{\mathrm{demo}}$  and  $\mathcal{T}_{\mathrm{vid}}$  are listed in the first two rows, while the individual success rate of the 4 tasks in  $\mathcal{T}_{\mathrm{unseen}}$  are listed in the following rows. The average success rate of the seen and unseen tasks are calculated in the last two row.  
+
+<table><tr><td>Task</td><td>JEPT+FlowFormer</td><td>JEPT+VideoMAE-v2</td><td>JEPT+Dino-v2</td><td>JEPT+SAM</td><td>JEPT</td></tr><tr><td>\( \mathcal{T}_{\text{demo}} \)</td><td>27.7</td><td>20.7</td><td>24.3</td><td>26.7</td><td>27.3</td></tr><tr><td>\( \mathcal{T}_{\text{vid}} \)</td><td>13.2</td><td>11.6</td><td>10.8</td><td>8.4</td><td>12.8</td></tr><tr><td>Panda Lift</td><td>34.0</td><td>14.0</td><td>16.0</td><td>12.0</td><td>38.0</td></tr><tr><td>Sawyer Lift</td><td>18.0</td><td>0.0</td><td>4.0</td><td>2.0</td><td>16.0</td></tr><tr><td>IIWA Lift</td><td>8.0</td><td>8.0</td><td>0.0</td><td>0.0</td><td>12.0</td></tr><tr><td>UR5e Lift</td><td>14.0</td><td>0.0</td><td>0.0</td><td>0.0</td><td>8.0</td></tr><tr><td>Seen Average</td><td>20.4</td><td>16.1</td><td>17.6</td><td>17.5</td><td>20.1</td></tr><tr><td>Unseen Average</td><td>18.5</td><td>5.5</td><td>5.0</td><td>3.5</td><td>18.5</td></tr></table>
+
+Table 10: Success Rates (\%) of JEPT and the ablations on  $c$  calculated from 50 trials for each task in Meta-World. The average success rates of the tasks in  $T_{\text{demo}}$  and  $T_{\text{vid}}$  are listed in the first two rows, while the individual success rates of the 4 tasks in  $T_{\text{unseen}}$  are listed in the following rows. The average success rates of the seen and unseen tasks are listed in the last two row.  
+
+<table><tr><td>Task</td><td>c = 0.3</td><td>c = 1</td><td>c = 5</td><td>c = 8(JEPT)</td><td>c = 10</td><td>c = 12</td></tr><tr><td>Tdemo</td><td>38.5</td><td>33.0</td><td>47.0</td><td>51.3</td><td>52.5</td><td>54.5</td></tr><tr><td>Tvid</td><td>19.0</td><td>17.7</td><td>24.3</td><td>31.7</td><td>27.7</td><td>12.3</td></tr><tr><td>Handle Press</td><td>4.0</td><td>12.0</td><td>24.0</td><td>28.0</td><td>14.0</td><td>6.0</td></tr><tr><td>Lever Pull</td><td>0.0</td><td>0.0</td><td>8.0</td><td>10.0</td><td>12.0</td><td>0.0</td></tr><tr><td>Plate Slide Back</td><td>2.0</td><td>8.0</td><td>6.0</td><td>14.0</td><td>16.0</td><td>2.0</td></tr><tr><td>Faucet Open</td><td>0.0</td><td>2.0</td><td>16.0</td><td>22.0</td><td>20.0</td><td>8.0</td></tr><tr><td>Seen Average</td><td>28.8</td><td>25.3</td><td>35.7</td><td>41.5</td><td>40.1</td><td>33.4</td></tr><tr><td>Unseen Average</td><td>1.5</td><td>5.5</td><td>13.5</td><td>18.5</td><td>15.5</td><td>4.0</td></tr></table>
+
+Table 11: Average Success Rates (\%) of JEPT variants on the 4 tasks in  $\mathcal{T}_{\text{unseen}}$  in Meta-World experiments. The columns represent the variants with different ratios of  $\mathcal{T}_{\text{demo}}$  and  $\mathcal{T}_{\text{vid}}$ , while the rows represent the variants with different values of hyperparameter  $c$ .  
+
+<table><tr><td>Tdemo + Tvid</td><td>(1) + (19)</td><td>(3) + (17)</td><td>(5) + (15)</td><td>(8) + (12)</td><td>(10) + (10)</td><td>(12) + (8)</td></tr><tr><td>c = 1</td><td>0.0</td><td>0.0</td><td>8.0</td><td>5.0</td><td>3.0</td><td>9.0</td></tr><tr><td>c = 5</td><td>0.0</td><td>1.0</td><td>5.5</td><td>7.5</td><td>10.0</td><td>8.5</td></tr><tr><td>c = 8</td><td>1.5</td><td>2.0</td><td>3.5</td><td>8.5</td><td>18.0</td><td>18.5</td></tr><tr><td>c = 10</td><td>0.0</td><td>0.0</td><td>6.0</td><td>10.0</td><td>16.0</td><td>10.0</td></tr><tr><td>c = 12</td><td>0.5</td><td>2.5</td><td>8.0</td><td>19.5</td><td>14.5</td><td>9.5</td></tr><tr><td>c = 16</td><td>0.0</td><td>5.0</td><td>12.5</td><td>12.0</td><td>10.0</td><td>10.5</td></tr><tr><td>c = 20</td><td>1.0</td><td>0.0</td><td>10.0</td><td>11.0</td><td>8.0</td><td>4.0</td></tr></table>
+
+Table 12: Success Rates (\%) of JEPT and two baselines calculated from 50 trials for each task in Meta-World. The average success rate of the tasks in  $\mathcal{T}_{\text{demo}}$  and  $\mathcal{T}_{\text{vid}}$  are listed in the first two rows, while the individual success rate of the 4 tasks in  $\mathcal{T}_{\text{unseen}}$  are listed in the following rows. The average success rates of the seen and unseen tasks are listed in the last two row.  
+
+<table><tr><td>Task</td><td>BC+IDM</td><td>BC+IDM*</td><td>JEPT</td></tr><tr><td>Tdemo</td><td>49.5</td><td>47.5</td><td>51.3</td></tr><tr><td>Tvid</td><td>8.3</td><td>14.7</td><td>31.7</td></tr><tr><td>Handle Press</td><td>8.0</td><td>7.0</td><td>28.0</td></tr><tr><td>Lever Pull</td><td>0.0</td><td>0.0</td><td>10.0</td></tr><tr><td>Plate Slide Back</td><td>0.0</td><td>0.0</td><td>14.0</td></tr><tr><td>Faucet Open</td><td>4.0</td><td>3.0</td><td>22.0</td></tr><tr><td>Seen Average</td><td>28.9</td><td>31.1</td><td>41.5</td></tr><tr><td>Unseen Average</td><td>3.0</td><td>2.5</td><td>18.5</td></tr></table>
+
+![](images/a91f3688bfe8908912ea76b3271e07156e0b86cd84d78f5c1bf0a7c6668cff65.jpg)  
+Figure 5: Visualization of the prompt video embeddings learned in JEPT. The prompt video embeddings are calculated from 150 videos of each task in Meta-World. We apply t-SNE on the whole embedding set.
+
+<table><tr><td>Window Open</td><td>Button Press Topdown</td><td>Reach Wall</td></tr><tr><td>Handle Pull</td><td>Door Close</td><td>Plate Slide Back</td></tr><tr><td>Button Press</td><td>Handle Press</td><td>Lever Pull</td></tr><tr><td>Window Close</td><td>Faucet Open</td><td>Plate Slide</td></tr><tr><td>Door Open</td><td>Reach</td><td>Drawer Close</td></tr><tr><td>Faucet Close</td><td>Plate Slide Back Side</td><td>Button Press Topdown Wall</td></tr></table>
+
+![](images/ad0dea6dc748facb256101ea08bc4947deac7f2b12a23dde76db8f944157dcc1.jpg)
+
+![](images/0cf9ec869bd1d77fa53829525832b56f0ed4b974cff2e7c6e682cf39e4869130.jpg)
+
+![](images/b64c056d81ec7c46f0d07abb63b2ed627573707d8ef975fc2a9bc74c1507c767.jpg)
+
+![](images/e4c1aa0d7882364601efa94b852c71068452de00e127b83dc802ea6ea19f97c6.jpg)
+
+![](images/3e792cf40cdbbecaaf52fda6d3fdef1df19503f216b484aae658943f819c2361.jpg)
+
+![](images/c8738227982cb0150879b93b0e1cb90a986d0fc262cbd30b38f1672630acb45b.jpg)
+
+![](images/08247793cf7473e10a75c56fa76ccf16d4d4c0826bec1dc24bdf66289af06bf3.jpg)
+
+![](images/cd2d565c512800586c475d77bb6a82f2df0647f07728d8bd2e7e6e7722837c15.jpg)
+
+![](images/9423759fdeeded9b45fa7bd1f981b26147b3e64a3461bce0cd8e9f207a4a61e5.jpg)
+
+![](images/2e65001e0579bd8835bf8faccd4d3518786126a0b6ff5832c5253bf0de78cd72.jpg)
+
+![](images/9d7dedcd2ad2863527c8a96cdf62237bf5d66ef93639e20d2109906fd3320521.jpg)
+
+![](images/63f1660103bc013b150d2c564415990a787436d886665b77db5a5accf52fd739.jpg)
+
+![](images/061b942b636ca5a1fb4f2a3f15693fbb45ef39a3fcfba7048dffd00fa2937aa2.jpg)
+
+![](images/9b90a997cb86c7f077c5870b4f54dd2627a0d2d2db337624a2d1633ed21e6a6c.jpg)
+
+![](images/2bdcfbb0e3a1d58584cfa0eee61ca2bed8a1dc0ba1d629b7ac04f839c2b257aa.jpg)
+
+![](images/3fab5970e280e71f1ef1de8969e0da70391cc19de3cc68ed8e392f77f2e1de82.jpg)  
+Figure 6: Visualization of the joint embeddings learned in JEPT. The joint embeddings are calculated from 150 trajectories of each task in Meta-World. We apply t-SNE on one whole embedding set and split the projected vectors according to the tasks into 18 sub-figures. The transparency of the points reflects the temporal positions of the corresponding embeddings, with points closer to the end being less transparent.
+
+![](images/3006f5d5365543094738e17f269a34f181e7d85154fc5ce59ee8bebf3563a88d.jpg)
+
+![](images/b0f59ca489391780053c2c3837c8aa8e6f6b0ecd65dfa2408dbf56ce85bf817e.jpg)
+
+![](images/cba20e75ff6a27dd55c94ef5a823f90a3a2c459c0eef2247ea6386454bab7c34.jpg)
+
+![](images/990bae704b367f5aa4cf8c6342a1999753a877dac9f9f7ebecc1e2ceaed931f0.jpg)
+
+![](images/e7b8f8972d69720f4b754d0a29782e89c7f540d8e063a532ba5da50010020c2a.jpg)
+
+![](images/8e97d915341db3a043b5581b4924dfb46af86624a6d69750cafebd57d519973b.jpg)
+
+![](images/3a48a36c147382c0b77736a063237489b16314b6e71c148edbef16b1ab51bf4d.jpg)
+
+![](images/c5c2e5439ba59d7b1846d352ea189d5eb62c82fd62c84dbc3294d97670038ee3.jpg)
+
+![](images/ded40506398cd9fe6e350278530f12a71f3cb51687667ee087165bb86b418dc6.jpg)
+
+![](images/fd3f80e7cd36e96fad8e5e927258a6e2be7260f04c2de9a89e1730ff17a76821.jpg)
+
+![](images/e5bfb1ec16c9244e40d367e095383a7ba3464d9f8dda7716fbd5c5a4aa361dcb.jpg)
+
+![](images/0ea140a578f79bb58bc2b23b8072d7e1bba8cbd3ef36a72c67c56ece77b000d2.jpg)
+
+![](images/172ae590cd274223b8bb19db24183a16ddde165b056235c323f9bc891cd4a9bb.jpg)
+
+![](images/7c5dbd825956ff7ee1602839abdc6845c7ad4c89bdb13b0041e58638508b1e37.jpg)
+
+![](images/5cdaf15c96c3d8c8a2c5dc5c068bf5cebaca9cfd1723b2aee3d8cb8dac54d187.jpg)
+
+![](images/c2fc245023a39f20cd2716bc1d67635babada69a0a27a3b9bc2739dce9f9cfe7.jpg)
+
+![](images/7523c10126290ce5165e6bda0ae146aa432dfc8a4fe5c29fa04932305f45f6c9.jpg)
+
+![](images/1e169aaaeeba28d26fb1b9a6076c76f93f74ac5aab49f0a5d53fdb586249324a.jpg)  
+Figure 7: Visualization of the actions learned in JEPT. The actions are collected from 150 trajectories of each task in Meta-World. We apply t-SNE on one whole embedding set and split the projected vectors according to the tasks into 18 sub-figures. The transparency of the points reflects the temporal positions of the corresponding embeddings, with points closer to the end being less transparent.
+
+![](images/957d83b4b6318a378a8d86d8715f3a23edc6be0fb114f8102a95f42edfe335d2.jpg)
+
+![](images/f75a19804efc12104e6394eca5242c55db49612e4611f7b193e6eb98df228d88.jpg)
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+# LEARNING VIEW-INVARIANT WORLD MODELS FOR VISUAL ROBOTIC MANIPULATION
+
+Jing-Cheng Pang $^{1,2,3,*}$ , Nan Tang $^{1,3,*}$ , Kaiyuan Li $^{1,3}$ , Yuting Tang $^{4,2}$ , Xin-Qiang Cai $^{2}$ , Zhen-Yu Zhang $^{2}$ , Gang Niu $^{2}$ , Masashi Sugiyama $^{2,4}$  & Yang Yu $^{1,3,\diamond}$
+
+$^{1}$  National Key Laboratory for Novel Software Technology, Nanjing University, China & School of Artificial Intelligence, Nanjing University, China;  
+2 RIKEN Center for Advanced Intelligence Project, Japan;  
+<sup>3</sup> Polixir.ai, China; <sup>4</sup> The University of Tokyo, Japan
+
+# ABSTRACT
+
+Robotic manipulation tasks often rely on visual inputs from cameras to perceive the environment. However, previous approaches still suffer from performance degradation when the camera's viewpoint changes during manipulation. In this paper, we propose ReViWo (Representation learning for View-invariant World model), leveraging multi-view data to learn robust representations for control under viewpoint disturbance. ReViWo utilizes an autoencoder framework to reconstruct target images by an architecture that combines view-invariant representation (VIR) and view-dependent representation. To train ReViWo, we collect multi-view data in simulators with known view labels. Meanwhile, ReViWo is simultaneously trained on Open X-Embodiment datasets without view labels. The VIR is then used to train a world model on pre-collected manipulation data and a policy through interaction with the world model. We evaluate the effectiveness of ReViWo in various viewpoint disturbance scenarios, including control under novel camera positions and frequent camera shaking, using the Meta-world & PandaGym environments. Besides, we also conduct experiments on real world ALOHA robot. The results demonstrate that ReViWo maintains robust performance under viewpoint disturbance, while baseline methods suffer from significant performance degradation. Furthermore, we show that the VIR captures task-relevant state information and remains stable for observations from novel viewpoints, validating the efficacy of the ReViWo approach.
+
+# 1 INTRODUCTION
+
+Developing robots capable of completing various manipulation tasks is a hallmark of machine intelligence. Previous approaches in this domain often rely on visual inputs from cameras to perceive the environment, with a policy trained on these visual inputs using reinforcement learning (RL) algorithms and pre-collected robotic manipulation data (Kalashnikov et al., 2018b; Quillen et al., 2018; Agarwal et al., 2020; Levine et al., 2020; Pang et al., 2024). However, the policy learned in this way often suffers from performance degradation when merely changing the camera position during deployment (Liu et al., 2024). This challenge, referred to as the viewpoint disturbance, arises from the policy's learned representations, which fail to separate the view-invariant information from the observation: they encode the entirety of the observations, including both the view-invariant task state and the view-dependent information like light. As a result, the changes in the viewpoint can lead to a substantial transformation in the learned representations, which can compromise the policy's effectiveness when exposed to novel viewpoints (Liu et al., 2024).
+
+There are existing works focusing on training robust robotic manipulation policies under viewpoint disturbances. For example, MVWM (Seo et al., 2023) tries to train a robust image representation using a multi-view masked autoencoder, by encoding a masked image and reconstructing target images in different viewpoints. The learned representation is then utilized for policy decision-making.
+
+However, the learned representation can still be sensitive to variations in viewpoint because it relies on a single encoder model to extract information, and then reconstruct complex, viewpoint-dependent visual information which can be highly challenging. RT-X series works (Brohan et al., 2023; Zitkovich et al., 2023; O'Neill et al., 2024) attempt to address viewpoint disturbance by training policies on extensive data. RoboUniView (Liu et al., 2024) learns a unified representation from 3D multi-view images using an autoencoder (Bank et al., 2023), but it requires all viewpoints to contain similar information, and relies on 3D data with camera calibration. Although these approaches have demonstrated a degree of effectiveness in addressing viewpoint disturbances, they have yet to achieve the complete and effective decoupling of view-invariant state information from the viewpoint, which is important for robust robotic control.
+
+In this paper, we investigate robust robotic manipulation under viewpoint disturbance. Motivated by human's ability to adapt to changes in viewpoint, a skill that is supported by cognitive science research in human psychology (Souto & Kerzel, 2021), we propose to decompose the visual observation into a view-invariant representation (VIR) for robot control, as shown in Fig. 1. More specifically, we propose Representation learning for the View-invariant World model (ReViWo), which employs two encoders to separately extract view-invariant and view-dependent
+
+![](images/81837784a6f7ce82242de1a2341ce19367bd11491fe6a5f1320e7f332293e2ee.jpg)  
+Figure 1: An illustration of the robotic manipulation with view-invariant representation.
+
+representations from the image, and a decoder to reconstruct a target image that combines the view-invariant and viewpoint information from two encoders' inputs, respectively. This training process requires the view labels, which can be obtained when collecting multi-view data from fixed-position cameras. Leveraging the learned VIR, ReViWo trains a world model to learn the environment dynamics from the offline manipulation data. Finally, ReViWo trains policy with RL algorithm and using the world model.
+
+Our contributions are as follows: First, we introduce an observation decomposition idea for enhancing visual robotic manipulation under viewpoint disturbances, which enables the robot to focus on invariant information across viewpoints. This decomposition distinguishes this work from previous works that process the observation into a single representation. Second, we implement this idea by proposing ReViWo, comprising a world model and an autoencoder framework that learns the view-invariant representation from pre-collected multi-view data with view labels. Besides, we also train ReViWo with Open X-Embodiment (O'Neill et al., 2024), to incorporate the knowledge from more diverse robotic manipulation data. Lastly, we conduct extensive experiments to verify the efficacy of the proposed method: Meta-world (Yu et al., 2019) and PandaGym (Gallouédec et al., 2021) simulation environments and real world ALOHA robotics arm. The experiment results demonstrate the robustness of ReViWo in the face of two types of viewpoint disturbance: a novel camera installation that results in  $10 \rightarrow 90$  degrees of viewpoint offset from the training setting, and a scenario with continuous camera shaking.
+
+# 2 BACKGROUND
+
+RL and Offline RL. We consider an RL problem aiming at learning a policy that maximizes the expected cumulative discounted reward in a Markov Decision Process (MDP) (Puterman, 1994; Sutton & Barto, 1998; Yuan et al., 2023; Cai et al., 2023b), which is represented by the tuple  $\mathcal{M} = (\mathcal{S},\mathcal{A},\mathcal{P},\mathcal{R},\gamma)$ . In this tuple,  $\mathcal{S}$  denotes the state space,  $\mathcal{A}$  the action space,  $\mathcal{P}: S \times \mathcal{A} \to \mathcal{S}$  the transition function of the environment,  $\mathcal{R}: S \times \mathcal{A} \to \mathbb{R}$  the reward function that evaluates the quality of the agent's action, and  $\gamma \in (0,1)$  the discount factor which balances the immediate and future rewards. A policy  $\pi: \mathcal{S} \to \mathcal{A}$  defines the agent's strategy, mapping states to a distribution over possible actions. The RL agent interacts with the environment as follows: at each timestep  $t$ , the agent observes a state  $s_t$  from the environment. It then selects an action  $a_t$  based on the policy  $\pi(\cdot | s_t)$  and executes it in the environment. Next, the agent receives a reward  $r_t$  and the environment transitions to a new state  $s_{t+1}$  according to the transition function  $\mathcal{P}(\cdot | s_t, a_t)$ . The objective of RL is to find a policy that maximizes the expected sum of rewards over time:  $\mathbb{E}_{\pi}[\sum_{t \geq 0} \gamma^t r_t]$ . In offline RL, we only have access to pre-collected interaction data:  $\{(s_t^i, a_t^i, r_t^i, s_{t+1}^i)_i\}$  and no further
+
+![](images/e7d3a5cc3dd3df1d33caf85fe13af56336d4a489825c94d98cf9ec061431010e.jpg)  
+Figure 2: Overall training framework of ReViWo method. (A) View-invariant representation learning process trains two encoders to decompose view-invariant and view-dependent information within the images, and a decoder to reconstruct the target image. (B) The World modeling process builds a world model upon the learned VIR. (C) Policy learning by interacting with the world model.
+
+![](images/1080ea93efc64af6d30804c4cb6b33e35d21484263ac6ed274e0e72be20d6751.jpg)
+
+environment interacting is allowed. The objective of offline RL is to find a policy  $\pi (a_t|s_t)$  that, when deployed, maximizes the cumulative reward within the confines of the MDP as characterized by the dataset.
+
+VAE and VQ-VAE. Variational Autoencoder (VAE) (Kingma & Welling, 2014) is an effective tool to extract information from the image, which learns to encode an image into a continuous latent space and subsequently reconstruct the image from this space. VAE represents an image with a latent vector that captures the essential information required to reconstruct the image. We use  $o \in \mathbb{R}^{H \times W \times C}$  to denote the image, where  $H$ ,  $W$ , and  $C$  represent the height, the width, and the number of channels of image  $o$ . The encoder network  $q_{\phi}(z|o)$  maps the input image to a distribution over the latent space, and the latent vector  $z$  is sampled from this probability density:  $z \sim q_{\phi}(z|o)$ . The decoder network  $p_{\theta}(o|z)$  aims to reconstruct an image from the latent vector. The objective of a VAE is to minimize the negative log-likelihood as below:
+
+$$
+\mathcal {L} _ {\mathrm {V A E}} (\theta , \phi) = \mathbb {E} _ {q _ {\phi} (z | o)} [ - \log p _ {\theta} (o | z) ] + D _ {K L} \left(q _ {\phi} (z | o) \| p (z)\right), \tag {1}
+$$
+
+where the Kullback-Leibler divergence term regularizes the latent space by enforcing similarity to a prior distribution  $p(z)$ , typically chosen to be a standard normal distribution.
+
+Vector Quantized VAE (VQ-VAE) (van den Oord et al., 2017) extends VAE by introducing a discrete latent space, which can be more suitable for information extraction. VQ-VAE introduces two main modifications: (a) The continuous latent vector is discretized by mapping it to the nearest vector in a learnable codebook  $e = \{e_1, e_2, \dots, e_K\}$ . This mapping results in a discrete latent variable  $z_q$ , which is used for reconstruction; (b) The training objective includes a codebook loss. The codebook loss consists of two terms: a commitment loss that encourages the encoder outputs to commit to a codebook vector, and a quantization loss that moves the codebook vectors towards the encoder outputs. The overall loss function is:
+
+$$
+\mathcal {L} _ {\mathrm {V Q - V A E}} (\theta , \phi ; o) = \mathbb {E} _ {q _ {\phi} (z | o)} [ - \log p _ {\theta} (o | z) ] + \| \operatorname {s g} [ q _ {\phi} (z | o) ] - e \| _ {2} ^ {2} + \| q _ {\phi} (z | o) - \operatorname {s g} [ e ] \| _ {2} ^ {2}, \tag {2}
+$$
+
+where  $\mathrm{sg}[\cdot ]$  denotes the stop-gradient operator.
+
+# 3 METHOD
+
+This section presents the proposed ReViWo method. The main idea of ReViWo is to decompose the visual observation into view-invariant representation (VIR) and view-dependent representation (VDR) separately. In the next, we give a formal problem definition and then elaborate on the two key steps of ReViWo method: (1) view-invariant representation learning and (2) world modeling and behavior learning.
+
+# 3.1 PROBLEM FORMULATION
+
+This study focuses on training a policy for visual robotic control, which is robust to viewpoint disturbances. Consider we have the following two datasets:
+
+1. A multi-view dataset  $\mathcal{O} = \{(o_{s_i}^{v_1}, o_{s_i}^{v_2}, \ldots, o_{s_i}^{v_N})_i\}$ , where  $v_i \in \mathcal{V}$  is a specific viewpoint within the set of viewpoints  $\mathcal{V}$ ,  $o$  denotes a visual observation,  $N$  the number of viewpoints, and  $s_i$  the state presented in the image. Here we have multiple trajectories from  $N$  fixed viewpoints. In practical, we can collect this dataset by recording robotic manipulation with multiple cameras.  
+2. A robotic manipulation dataset  $\mathcal{D} = \{(s_t^i, a_t^i, r_t^i, s_{t+1}^i)_i\}$ , where the state  $s_t^i$  is visual observation from a fixed position camera,  $a_t^i$  is robot behaviors and  $r_t^i$  denotes the observed reward.
+
+The primary objective is to derive a policy that maximizes the rewards on visual observations of novel viewpoints that are not present in the manipulation dataset, thus ensuring the policy's decision-making capability is not limited to the viewpoints within the training data. We will provide a detailed discussion in Sec. 4.1 on how to construct viewpoint disturbance in the experiments.
+
+# 3.2 VIEW-INVARIANT REPRESENTATION LEARNING
+
+The first step of ReViWo is to learn VIR using multi-view images. The main idea is to separate the view-invariant and view-dependent information within the image and find a training objective to enable both two representations to be meaningful. Then the policy can make decisions based on the view-invariant information, which is robust to the viewpoint changes. ReViWo learns the representation using an autoencoder architecture, as shown in Fig. 2 (A). The framework comprises two encoders and a decoder, where each encoder accounts for capturing specific information within an image, and the decoder reconstructs the images from the encoders' output.
+
+Encoders for information extraction. ReViWo utilizes two encoders to extract view-invariant and view-dependent from visual images separately. Specifically, we use  $q_{\phi}^{S}(o)$  to represent the view-invariant encoder (VIE) and  $q_{\phi}^{V}(o)$  for view-dependent encoder (VDE), where  $\phi$  denotes the parameters of the autoencoder model. The encoders are implemented upon the vision transformer (ViT) architecture (Dosovitskiy et al., 2021), which processes an image by partitioning it into a sequence of patches. Each patch is converted into a token embedding via a learnable convolutional neural network (CNN). These token embeddings are then concatenated with their corresponding positional embeddings, which are computed using the function (1-timestep), to form the inputs for the transformer at each timestep. We present our implementation of ViT architecture in Fig. 9. For the VIE, the transformation can be formalized as  $z_{s} = q_{\phi}^{S}(o_{s}^{v})$ , where  $o_{s}^{v} \in \mathbb{R}^{M \times H \times W \times 3}$  denotes the input image patches and  $z_{s} \in \mathbb{R}^{M \times C}$  represents the outputted VIR. Here, the dimensions  $W \times H$  denote the size of each image patch,  $M$  the patch number and  $C$  denotes the dimensions of the output feature space. Thus the output can be regarded as a sequence of features corresponding to each input patch. The VDE employs the same architecture as the VIE and yields an output  $z_{v} = q_{\phi}^{V}(o_{s}^{v})$ , capturing the view-dependent features.
+
+Decoder for image reconstruction. The decoder is also a transformer-based architecture, which inputs both view-invariant and view-dependent features and outputs a reconstructed image:  $\tilde{o}_s^v = p_\phi (z_s;z_v)$ . The features  $z_{s}$  and  $z_{v}$  could be extracted from different images within the same domain, and the goal for the decoder is to generate a novel image that integrates the features of the two inputs. The decoder's input is the concatenated outputs from the two encoders. Instead of generating the image patch by patch in an auto-regressive manner, we follow the Genie approach (Bruce et al., 2024) to output all patches simultaneously to enhance efficiency. Each output feature is then processed by a CNN to form an image patch, with the collective patches constituting the entire image.
+
+Training objective. The autoencoder is trained to reconstruct pixels following a VAE-like objective (Kingma & Welling, 2014), which involves optimizing the parameters  $\phi$  by minimizing the negative log-likelihood as in Eq. (3):
+
+$$
+\mathcal {L} _ {\mathrm {A E}} (\phi) = \underset {o _ {s _ {i}} ^ {v _ {j}}, o _ {s _ {m}} ^ {v _ {n}}, o _ {s _ {i}} ^ {v _ {n}} \sim \mathcal {O}} {\mathbb {E}} [ - \log p _ {\phi} \left(o _ {s _ {i}} ^ {v _ {n}} \mid q _ {\phi} ^ {S} \left(o _ {s _ {i}} ^ {v _ {j}}\right); q _ {\phi} ^ {V} \left(o _ {s _ {m}} ^ {v _ {n}}\right) \right] + \lambda_ {1} \mathcal {L} _ {\mathrm {V Q}} (\phi) + \lambda_ {2} \mathcal {L} _ {\text {C o n t r a s t i v e}} (\phi). \tag {3}
+$$
+
+The first term, the image reconstruction loss, measures the model's ability to predict an image of a given state from a novel viewpoint, by using state information from one image and viewpoint information from a different image. Minimizing this term enables the model to disentangle state and view information, as it must accurately predict the target image by fusing these two distinct information types. The reconstruction of the image,  $o_{s_i}^{v_n}$ , conditioned on the state-related features from  $o_{s_i}^{v_j}$ , trains
+
+the model to concentrate on certain aspects in the image that are invariant to viewpoint changes, such as the state of the object. The inclusion of  $z_{v}$  in the reconstruction encourages the model to learn viewpoint-specific features, such as lighting, angle, background, and perspective, which are independent of the object's state. The second term,  $\mathcal{L}_{\mathrm{VQ}}$ , is from VQ-VAE (van den Oord et al., 2017), which is known for its efficacy in image reconstruction. It comprises a commitment loss that ensures the encoder's output vectors are proximate to their nearest vectors in the quantization dictionary and a quantization loss that ensures the quantized vectors accurately represent the encoder's outputs. We refer readers to Eq. (2) for VQ-VAE loss. The term  $\mathcal{L}_{\mathrm{Contrastive}}$  is designed to optimize the variance of the latent representations  $z_{s}$  and  $z_{v}$  for multi-view images under a variety of conditions. This term encourages  $z_{s}$  remain consistent across identical states and varies across different states. Similarly, it ensures that  $z_{v}$  is consistent for identical viewpoints and varies for different viewpoints, thereby enhancing the model's ability to capture the nuances of multi-view image representations. Refer to Appendix A for the mathematical justification that Eq. (3) can effectively separate view-invariant and view-independent representations.
+
+Integration of Open X-Embodiment data without view labels. In addition to the data with view labels, we also involve multi-view data without view labels from the Open X-Embodiment dataset (O'Neill et al., 2024), which are readily available on the internet. This dataset encompasses a wide array of robotic manipulation tasks performed by various robots, thereby enriching the training of the autoencoder with a broader spectrum of information and diversity. Note that in the absence of labels, the inputs to the VIE and the VDE correspond to identical states observed from different viewpoints. To prevent the VDE from encoding the entirety of the image information for image reconstruction, we introduce a weighting factor in the loss calculation for these unlabeled data.
+
+# 3.3 WORLD MODELLING AND POLICY LEARNING FROM OFFLINE MANIPULATION DATA
+
+View-invariant representation as the state for visual control. The previous section introduces how ReViWo learns the VIRs, which capture the invariant features of the environment that are essential for decision-making and are not affected by viewpoint changes. This is similar to how humans focus on the relevant aspects of a task, regardless of how they are viewing the scene. By focusing on these invariant features, the policy can make decisions based on the true state of the environment rather than on extraneous visual details that may change with the camera angle. Therefore, we naturally regard the VIR as the task state for control. Building on this foundation, ReViWo proceeds to train a world model and policy via offline reinforcement learning.
+
+World modeling and behavior learning. We employ the COMBO algorithm (Yu et al., 2021) for world modeling and behavior learning, which is an offline model-based reinforcement learning methodology. COMBO algorithm first trains an ensemble of world models, utilizing ensemble methods (Ganaie et al., 2022) to improve the prediction accuracy and robustness of the world model. Then, the policy interacts with the world model, from an initial state that is sampled from the offline dataset, for a rollout. The policy is optimized by utilizing a combination of the real offline dataset and the simulated rollouts generated by the world model. The world model  $\mathcal{M}_{\theta}(q_{\phi}^{S}(s_{t}),a_{t})$ , parameterized by  $\theta$ , is trained by minimizing the difference between the predicted next state with the target state, as shown in Eq. (4):
+
+$$
+\mathcal {L} _ {\mathrm {W M}} (\theta) = \underset {(s _ {t}, a _ {t}, s _ {t + 1}) \sim \mathcal {D}} {\mathbb {E}} [ - \log \mathcal {M} _ {\theta} \left(q _ {\phi} ^ {S} \left(s _ {t + 1}\right) \mid q _ {\phi} ^ {S} \left(s _ {t}\right), a _ {t}\right) ]. \tag {4}
+$$
+
+In addition to the world model, we also train a reward model by supervised learning on the offline control data, which is capable of predicting the rewards for given state-action pairs. This reward model provides the necessary reward signals for the policy's learning process.
+
+# 4 EXPERIMENT
+
+In this section, we conduct extensive experiments to evaluate our proposed method for robust control under viewpoint disturbance. The goal of the experiments is to answer the following key questions: (1) How does ReViWo perform compared to existing baseline methods under viewpoint disturbance (Sec. 4.2)? (2) How is the algorithm's robustness on viewpoints variations, and how much data is required to train ReViWo(Sec. 4.2)? (3) Does ReViWo learn meaningful view-invariant and view-dependent representation (Sec. 4.3)? (4) What is the impact of each component in ReViWo on the overall performance of the algorithm (Sec. 4.4)? We first introduce the experiment setup.
+
+![](images/ebcab2dfeb228ec401691c63002f7fc2bdb960d09f96b3bfed2c02be133fda2b.jpg)  
+(A)Meta-world
+
+![](images/b3a1cfc843a971b45845ab3631af438d60395a2d1ac4ab4015760027bf9db59e.jpg)  
+Figure 3: A visualization of the environments in our experiments. (A) In the Meta-world, the agent controls a Sawyer robot to manipulate various objects such as door, drawer and window. (B) In the PandaGym environment, the agent controls a Franka robot with 7-DoF. (C) ALOHA robot that manipulates a bottle to the plate.
+
+![](images/e0e11e530fa7ed98c5e3c2f29f3e3a04a4d15e08ba8664907aefbf84633e05d9.jpg)  
+(B) PandaGym
+
+![](images/10c9ebbd091439636bc0f66a9b5a5ac52207870e82a9e92b7b2432300c708ae2.jpg)  
+(C) ALOHA
+
+# 4.1 EXPERIMENT SETUP
+
+Evaluation environments. We conduct experiments on two robotics manipulation environments: Meta-world (Yu et al., 2019) and PandaGym (Gallouédec et al., 2021), as shown in Fig. 3. (1) Meta-World: This environment requires the agent to control a Sawyer robotics arm with 7 degrees of freedom (DoF) and a parallel finger gripper. The action space is a 2-tuple. The first element of the tuple is the displacement in 3D space of the end-effector, and the second is a normalized torque to be applied by the gripper fingers. Meta-World offers a suite of 50 distinct manipulation tasks, covering a wide array of scenarios, such as interactions with doors, windows, drawers, and balls. For our experiments, we assess the performance of our methods on a subset of tasks: door opening, drawer opening, and window closing. (2) PandaGym: This environment involves a Panda robotic arm by Franka Emika, which has 7 DoF and a parallel finger gripper. PandaGym is primarily focused on block manipulation tasks, which are designed to test the robot's foundational skills, such as reaching a goal point. The action space is defined by the gripper's movement command, which includes three coordinates for spatial movement, and the finger movement, which is a single coordinate reflecting the gripper's aperture. For certain tasks, the gripper is fixed in a closed position, which reduces the action space to only the gripper's spatial movement command. (3) ALOHA: We include a real world ALOHA robot to manipulate a bottle to a plate. The task involves three stages: reaching for a bottle (stage 1), grasping it (stage 2), and then placing it on a plate (stage 3).
+
+We mainly use Door Open,Drawer Open, and Window Close from Meta-world, and Reach from PandaGym for evaluation, and we also conduct experiments on Coffee Button, Faucet Open, Dial Turn (Appendix F.1). The observations on all tasks are images with  $128 \times 128$  pixels, which are captured by the fixed-position third-person camera. To introduce viewpoint disturbance, we can modify camera parameters, such as its azimuth, pitch, and height, to alter the viewpoint. For our experiments, we specifically alter the azimuth value, as this adjustment alone can introduce considerable visual perturbations and is sufficient to evaluate the robustness of the proposed methods.
+
+Dataset for training. Multi-view dataset for training autoencoder includes three parts: (1) Meta-world data. We collect 51 trajectories of 17 tasks from Meta-world and record them utilize 20 cameras from different viewpoints, resulting in 1020 observation sequences (112040 observations) in total. The trajectories are sampled by pre-trained expert policies with Gaussian noise. (2) PandaGym. In this environment, we collect 30 trajectories of 5 tasks with 20 viewpoints, resulting in 600 observation sequences (30k observations) in total. (3) Open X-Embodiment (Routing Primitive Dataset (Luo et al., 2023)), including 4 viewpoints for manipulating multi-stage cable routing tasks. This dataset does not provide a viewpoint label. We utilize 101 trajectories in this dataset to keep a data balance, with 10064 observations in total. We refer readers to Appendix C.1 for the detailed setup of camera parameters for collecting data. For offline control data, we collect 400 trajectories on Meta-world, with a single viewpoint that doesn't exist in the multi-view data and 100 trajectories on PandaGym. Then, we train the world model and RL policy on two tasks separately. We investigate the algorithm's performance when collecting multi-view images on a narrower range of available viewpoints, and present the result in Sec. 4.4. For real world data collection, we collect a dataset of 128 trajectories using three cameras (two third-person cameras and one gripper camera). The dataset includes 10 different types of bottles and plates, which are placed randomly within the operational area during data collection. We use this real-world dataset to train both VIE and policy.
+
+![](images/ec3334543c0f5331237bfe4d8419d56dd6fc580cccd43f6a07f239eaab740987.jpg)  
+(a) Door Open
+
+![](images/643954fcd76c60763379da7e9f7a9bc95e6f2a7fcbbeac93284c2dd8f59a3e93.jpg)  
+(b)Drawer Open  
+Figure 4: Performance of different methods under various viewpoint disturbances. The x-axis denotes the disturbance type, and the y-axis denotes the average success rate of the last two checkpoints, by evaluation for 30 episodes. The error bars stand for the half standard deviation over three random seeds. Results on more tasks are shown in Appendix F.1.
+
+![](images/03f527bd86b7b183966d84eebd4b469d831e54929d5067e536ecfb5a44ee75bf.jpg)  
+(c) Window Close
+
+![](images/85a102a877cf8025b4dee543754792db92167826f2c804c2f2465b0f5160f113.jpg)  
+(d) Reach
+
+Viewpoint disturbance setup. In our experiments, we investigate the policy robustness under two distinct types of viewpoint disturbances: (1) Camera installation position (CIP): This category of viewpoint disturbance is designed to simulate the situation where the camera's installation position deviates from that of the policy training dataset. To this end, we adjust the camera's azimuth angle. Specifically, for the Meta-world experiments, we set the azimuth to a 10-degree offset, while for the PandaGym experiments, we apply a 90-degree offset. This alteration is intended to evaluate the robustness of policies to variations in the camera's installation angle. We also conduct experiments to investigate ReViWo's performance on various azimuth offsets (Fig. 5). (2) Camera shaking (CSH): This disturbance is relevant for scenarios with a hand-held camera, where the viewpoint can be continuously changing. To mimic this, we introduce a dynamic disturbance where the camera's azimuth angle is continuously adjusted during the robot's manipulation task. It is challenging because the model learns with a fixed viewpoint during training while being evaluated on a variable one. We present the visualization of the viewpoint disturbance in Appendix C.1.
+
+Implementation Details. For autoencoder training, For world modelling and behavior learning for both ReViWo and baseline methods, we utilize OfflineRL-kit (Sun, 2023), a well-verified offline RL codebase. For all methods, the model is trained with an offline RL algorithm for 25000 gradient steps, and evaluated for 40 episodes. We conduct all experiments with four random seeds, and the shaded area or the error bars in the figures represent the standard deviation across four trials. We use 64 CPU cores (AMD EPYC 9654 @ 2.4GHz) and 4 GPUs (NVIDIA GeForce RTX 4090) for our experiments. More implementation details can be found in Appendix C.
+
+# 4.2 PERFORMANCE UNDER VIEWPOINT DISTURBANCE
+
+Baselines for comparison. We choose the following representative approaches in the domain of multi-view robotics manipulation or offline RL for comparison. (1) MVWM (Seo et al., 2023) handles the view disturbance by learning a multi-view masked autoencoder which reconstructs pixels of randomly masked viewpoints. It then learns a world model operating on the representations from the autoencoder. In MVWM's original implementation, the world model is learned in an online setting. We align it with our setting by training only with offline data and utilizing COMBO to train both world model and policy. (2) COMBO (Yu et al., 2021) is a model-based offline RL algorithm that trains a value function using both the offline dataset and the world model generated data while also additionally regularizing the value function on out-of-support state-action tuples generated via model rollouts. In the experiments, the COMBO method first pre-trains the vision representation on the same multi-view data as ReViWo using a standard VAE and then trains a policy that makes decisions based on the learned vision representation. (3) Behavior Cloning (BC) adopts a supervised learning approach to mimic the actions within the offline dataset. Similar to COMBO, we also use VAE to process the visual observation when implementing the BC baseline. All these baseline methods use the same multi-view dataset to pre-train vision representation with VAE. (4) Conservative Q-Learning (CQL) (Kumar et al., 2020) is an offline RL method. In our experiment in Sec. 4.4, we utilize CQL to train policy with the learned VIR, to verify the effect of the world model in ReViWo.
+
+Main results. Fig. 4 presents the success rate of different methods under various levels of viewpoint disturbance. In general, our proposed ReViWo method outperforms the baseline methods across two types of viewpoint disturbances. While the baselines demonstrate considerable performance from the training viewpoint, they suffer from a clear decline when subjected to viewpoint variations.
+
+This conclusion is supported by COMBO performance  $(23.3\to 0$  on Door Open,  $94.4\to 0$  on Drawer Open, and  $98.9\rightarrow 22.8$  on Window Close), which demonstrates the influence of viewpoint disturbance on these visual-based policies. BC shows promising performance on certain training viewpoints, yet its success rate substantially decreases under viewpoint disturbances. This performance degradation is due to the limitations of supervised learning on offline control data, which fails to preserve its robust performance when encountering novel observations. In contrast, ReViWo not only performs comparably to the training viewpoints and the CIP on tasks such as Door Open, but also demonstrates its robustness to viewpoint disturbances. This robustness is further evidenced by ReViWo's ability to handle tasks with a camera shaking, which is a significant challenge for policies to perform robustly (Seo et al., 2023). Interestingly, ReViWo achieves the highest performance on the training viewpoint for certain tasks. We hypothesize this is due to that VIR focuses on task-related domains within the observation and masks some unrelated information, thereby enhancing the policy learning for specific tasks.
+
+Impact of training data variations & Performance under various viewpoint offsets. In this experiment, we study two important questions: the quantity of data necessary for training the autoencoder using ReViWo, and the degree of its robustness to changes in viewpoint. We train the autoencoder using images collected from Meta-world, employing two configurations: one with 10 random viewpoints across a 90-degree azimuth range (10V+90D), and another with 20 random viewpoints over a 180-degree azimuth range (20V+180D). For evaluation, we evaluate the algorithm's performance across various degrees of azimuth offset, ranging from 0 to 15 degrees. Fig. 5 shows the comparative performance of ReViWo trained with different data against various levels of camera viewpoint offset. Key findings include: (1)  $10\mathrm{V} + 90\mathrm{D}$  setting is enough for ReViWo to achieve a good performance, while additional data can further improve the algorithm's efficacy. For instance, the performance of ReViWo with  $10\mathrm{V} + 90\mathrm{D}$  is comparable to that with  $20\mathrm{V} + 180\mathrm{D}$  when the azimuth offset is  $\leq 10$ , and only marginally inferior when the offset  $>10$ . In contrast, the COMBO method has a noticeable decline under even slight disturbance (e.g.,  $98.9 \rightarrow 55.6$  when the offset is 2.5); (2) ReViWo can effectively maintain the policy performance even when offset reaches 15 degrees, which is a considerable range during the practical robotic deployment. These insights serve as a guideline for applying ReViWo, and highlight its adaptability to viewpoint variations.
+
+Real world evaluation. Tab. 1 presents the real world experiment results. We implement ReViWo-BC as follow: training VIE on the real world images and conducting behavior cloning on the control data with VIE representation. In this experiments, the policy inputs with all three camera images. To simulate viewpoint disturbances, we adjusted the azimuth by +15 degrees and the elevation by -15 degrees on two third-person cameras. We evaluate the method on both training and CIP viewpoint for 10 trajectories with random initialization. We also consider a baseline, ACT (Action Chunking Transformer) (Zhao et al., 2023), for comparison. ACT performs end-to-end imitation learning directly from real demonstrations, with pre-trained ResNet-50 as the vision encoder. The results indicate that the proposed method maintains robust control when subjected to novel viewpoints in a real-world setting, while imitation learning-based method suffers from a performance degradation Cai et al. (2023c;a). This result serves as a preliminary evidence of the applicability of ReViWo method. In contrast, ACT performs comparably to ReViWo-BC on training viewpoint, while exhibiting a significant decline in performance when viewpoint changes. These real world results further substantiate the efficacy of the ReViWo in handling viewpoint disturbances.
+
+<table><tr><td>Success Rate</td><td>ReViWo-BC (Training)</td><td>ReViWo-BC (CIP)</td><td>ACT (Training)</td><td>ACT (CIP)</td></tr><tr><td>Stage 1</td><td>100%</td><td>100%</td><td>100%</td><td>100%</td></tr><tr><td>Stage 2</td><td>80%</td><td>60%</td><td>60%</td><td>0%</td></tr><tr><td>Stage 3</td><td>60%</td><td>50%</td><td>60%</td><td>0%</td></tr></table>
+
+Table 1: Results on real world ALOHA robot.
+
+# 4.3 ANALYSIS ON VIEW-INVARIANT REPRESENTATION LEARNING
+
+Previous results demonstrate ReViWo can effectively handle common viewpoint disturbance compared to baseline methods. In this section, we investigate the source of the robustness and analyze the quality of the learned view-invariant representation.
+
+Analysis on the generated VIR. We analyze the effectiveness of the learned view-invariant representation by projecting the resulting representations onto a two-dimensional plane using t-SNE (der
+
+![](images/9d929bcb6863a268d77e9705ecd907941658bbd7207c0197b36a314fb98680c9.jpg)  
+Figure 5: Performance across different levels of viewpoint offsets.
+
+<table><tr><td>Method Task</td><td colspan="2">w/ Open-X</td><td>w/o Open-X</td></tr><tr><td rowspan="2">Door Open</td><td>CIP</td><td>42.2</td><td>22.8</td></tr><tr><td>CSH</td><td>19.4</td><td>25.0</td></tr><tr><td rowspan="2">Drawer Open</td><td>CIP</td><td>55.0</td><td>30.6</td></tr><tr><td>CSH</td><td>37.8</td><td>25.6</td></tr></table>
+
+Table 2: Ablation study of ReViWo that trains w/ and w/o Open X-Embodiment dataset.
+
+![](images/4183e7e540484d8dfce09880787458adb3e79afc3ecfdd4fe64fe7c960c971dc.jpg)  
+(a) ReViWo
+
+![](images/cb37154ca76769ac32631777b051ba83690a1b731c6c0a7a230c23e8ab892aed.jpg)  
+(b) MVWM  
+Figure 6: The t-SNE projections of representations generated by various methods. Points with the same color encode six images for the same state but with six different novel viewpoints that do not exist in the multi-view datasets. The presented ReViWo representation is generated by VIE.
+
+![](images/973fc9cad84e3407178c0d8816c7c026f8294d8010a0d7ca4ac56b897fc10568.jpg)  
+(c) VAE
+
+Maaten et al., 2008). We first sample the state from the Meta-World environment, from which we render a series of images from multiple viewpoints. Subsequently, we encode these images using the VIE, as well as several baseline models for comparison. The t-SNE projections of these encoded images are illustrated in Fig. 6, where each color of points correspond to the representations of the same state from various viewpoints. We observe a notable clustering of VIR points, which are in closer proximity to each other as compared to those of the baseline models. In contrast, the MVWM and VAE representations exhibit greater diversity compared to ReViWo. These results suggest their representations are significantly different across viewpoints, potentially distracting the robust policy execution. This finding indicates the learned VIR is more adept at capturing the essential state information of a task, even in the presence of novel viewpoints.
+
+Examples of the decoder output. To investigate whether the trained autoencoder learns meaningful representation, we present the decoded images when VIE and VDE are from different images, as shown in Fig. 7. We have several conclusions from the results. First, VIR correctly focuses on the task state, which is consistent despite changes in viewpoint. For example in each row in Fig. 7, the decoder can generate the images for the same state from two viewpoints. Second, the view-dependent representation indeed captures the property of viewpoint, thus facilitating the generation of the image with the same viewpoint as the viewpoint reference image. We present more examples of decoder output in Appendix F.3. Besides, the trained autoencoder can be applied in different domains (i.e., Meta-world and PandaGym).
+
+# 4.4 ABLATION STUDY AND APPLICABILITY ANALYSIS
+
+Effect of the Open X-Embodiment data. To evaluate the impact of Open X-Embodiment data on the efficacy of our proposed method, ReViWo, we conducted a comparative analysis of the model's performance with and without the integration of this data. The results of this analysis are detailed in Tab. 2. The integration of Open X-Embodiment data into ReViWo significantly enhances the model's robustness to variations in viewpoint, as shown in the experiment results. This could be attributed to that the diverse visual representations in the Open X-Embodiment data contribute to a more comprehensive understanding of the world by the model, as well as a better understanding of different viewpoints. These results imply a potential to scale up the ReViWo training for more challenging robotic manipulation. Note that the performance decline on Door Open (CSH) can be attributed to the diverse and unstructured nature of the Open X-Embodiment dataset, which introduces variability that the model struggles to generalize under dynamic conditions.
+
+Effect of the world model. We verify the effect of the world model by training a policy with the learned VIR and offline RL method, CQL (Kumar et al., 2020), on the offline RL dataset. Tab.
+
+![](images/46c65527482d7a70255e0a627152f8565975631574559e0b8178e38ad4d0c6a3.jpg)  
+Figure 7: Examples of the decoder output the corresponding ground truth image. The decoder can generate the images that combine the task state in VIE inputs, and the viewpoint in VDE inputs.
+
+![](images/294b3ca0fc666adb810f2af309733b9fa6cce49f5c24f647ed88628f59abb5cb.jpg)
+
+3 presents the performance of ReViWo method with and without world model (WM). The results present a consistent performance enhancement for ReViWo w/ WM, suggesting that the imaginary rollouts generated by the WM contribute to the enhancement of both the robustness and the overall learning efficiency of the ReViWo framework. The reduced performance on Painter Open (CSH) is likely due to the world model's difficulty in accurately predicting the next state in highly variable and unstable visual conditions, leading to less reliable policy execution. Notably, the ReViWo w/o WM still surpasses the performance of baselines like COMBO (result in Sec. 4). This finding reveals the efficacy of the VIR in facilitating robust control, independent of the world model.
+
+<table><tr><td>Task Method</td><td colspan="3">Drawer Open</td><td colspan="3">Window Close</td></tr><tr><td rowspan="2">w/ WM</td><td>Training</td><td>CIP</td><td>CSH</td><td>Training</td><td>CIP</td><td>CSH</td></tr><tr><td>92.8</td><td>30.6</td><td>25.6</td><td>98.9</td><td>71.4</td><td>23.3</td></tr><tr><td>w/o WM</td><td>60.0</td><td>28.3</td><td>27.2</td><td>97.2</td><td>68.3</td><td>17.8</td></tr></table>
+
+Table 3: Performance of ReViWo w/ and w/o world model (WM). We implement ReViWo w/o WM by training policy with CQL and using the learned view-invariant representation.
+
+# 5 CONCLUSION AND LIMITATION
+
+This study explores robust robotic manipulation under viewpoint disturbances. We propose a novel approach, ReViWo, which learns viewpoint-invariant representations that are subsequently leveraged for robotic control. We conduct extensive experiments and demonstrate that ReViWo is capable of being applied to a variety of robotic platforms, enabling robust control under various types of viewpoint disturbance. Despite the promising results, there are still limitations. One limitation of our work is the reliance on the labeled viewpoints. In a practical setting, these labels can be generated by sampling multiple trajectories using fixed-position cameras. However, it is worth exploring to eliminate this dependency and to leverage a broader spectrum of multi-view data from the real world. A potential solution is to train with label-free multi-view data and constrain the information contained within VDR, by information bottleneck (Tishby & Slonim, 2000) technique or by masking portions of the image input to the VDR, thereby preventing the VDR contains too much information. Besides, the experiment scale is limited, in terms of the dataset scale and model size. In future works, we hope to scale up the framework to solve more challenging tasks. For instance, employing a pre-trained model such as CLIP (Radford et al., 2021), and training with more data from Open X-Embodiment and other open multi-view data (Yu et al., 2023). Lastly, currently we implement the world model based on a simple multi-layer perceptron architecture. To deal with robot manipulation tasks with more complex dynamics, it would be beneficial to use more powerful structures to build the world model, e.g., recurrent state-space model (Hafner et al., 2019). We believe these interesting directions are worth further exploration for developing smarter and more robust robots with the support of world models and reinforcement learning.
+
+# AUTHORSHIP AND CREDIT Attribution
+
+The contributions of the authors are as follows:
+
+- Y. Y.: led the direction, oversaw the project;  
+J.-C. P.: designed the method;  
+N. T.: proved the theory;  
+- N. T.: implemented the codes;  
+- J.-C. P.: designed the experiments;  
+K. L.: processed the data;  
+N. T., K. L.: conducted experiments;  
+J.-C. P., K. L., Y. T.: analyzed the results;  
+J.-C. P., X.-Q. C., Y. T., Z.-Y. Z.: wrote the paper;  
+Y.Y.,M.S.,G.N.: revised the paper;  
+J.-C. P., G. N.: discussed and wrote the response;  
+- Y. Y., M. S., G. N.: provided guidance.
+
+All authors discussed and contributed to the final manuscript.
+
+# ACKNOWLEDGMENT
+
+This work was supported by the National Science Foundation of China (62495093) and Jiangsu Science Foundation (BK20243039). The authors would like to thank Zhilong Zhang, Junyin Ye and Haoxiang Ren for their assistance in real world deployment, and anonymous reviewers & chairs for their valuable comments.
+
+Masashi Sugiyama was supported by JST ASPIRE Grant Number JPMJAP2405 and by a grant from Apple, Inc. Any views, opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and should not be interpreted as reflecting the views, policies or position, either expressed or implied, of Apple Inc.
+
+# ETHICS STATEMENT
+
+In the development and evaluation of ReViWo for robotic manipulation, we have carefully considered the ethical implications of this research, particularly as they pertain to the use of robotic manipulation tasks and artificial intelligence. The proposed involves the collection and use of multiview data in simulators and Open X-Embodiment datasets and is designed to respect privacy and ensure the security of data. The datasets used do not contain any personal or sensitive information, and all data collection processes comply with relevant legal standards and best practices in research ethics. The potential for bias and discrimination has been addressed by ensuring that the VIR does not inadvertently encode any biased representations of the environment. This is particularly important in maintaining fairness and avoiding any form of discrimination that could arise from biased training data. The research has been conducted with a commitment to research integrity, including thorough documentation and adherence to IRB guidelines where applicable. We recognize that the insights and methods presented in this paper must be applied responsibly, avoiding any potentially harmful applications. The technology developed is intended for beneficial purposes and should not be used in ways that could cause harm or diminish the safety of individuals. The experiment results are reported with the most transparency and accuracy, reflecting our commitment to advancing knowledge in the field of robotic manipulation while upholding the highest ethical standards.
+
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+
+# Appendix
+
+# Table of Contents
+
+A Mathematical Justification for Representation Learning 16  
+B Related Work 17
+
+C More Implementation Details & Experiment Setup 18
+
+C.1 Viewpoints for Autoencoder and RL Training 18  
+C.2 Architecture and Optimization Details 18  
+C.3 Computation 18  
+C.4 Hyperparameters 19  
+C.5 Detailed Descriptions of Baselines 19
+
+D Algorithm Description 21  
+E Discussion on the Usage of Dataset with View Labels 21  
+F Additional Results 22
+
+F.1 Results on More Tasks 22  
+F.2 Imitation Learning with View-invariant Representation 22  
+F.3 More Examples of Decoder Output 24
+
+# A MATHEMATICAL JUSTIFICATION FOR REPRESENTATION LEARNING
+
+Here we provide a mathematical explanation to justify why ReViWo could separately learn the view-invariant and view-dependent representations from the learning objective in Eq. (3). The intuition is that the disentanglement can be achieved by minimizing the InfoNCE losses, which equals to the contrastive term in Eq. (3).
+
+Denote  $V, L$  as the random variable of the viewpoint and the scene occupancy,  $Z_v, Z_l$  as the VDE and VIE. It is obvious that the viewpoint is independent of the occupancy, which means that  $I(V; L) = 0$ , where  $I$  denotes the mutual information. Each RGB image  $X$  corresponds to a unique pair  $(V, L)$ , i.e.  $X = X(V, L)$ .
+
+During training, we sample a training image batch with size  $(N_{l},N_{v})$  at each step. In this sample matrix, the samples of the same row are sampled from the same state and the samples of the same column are sampled from the same viewpoint.
+
+We use  $f(\cdot, \cdot)$  as the similarity metric, and for a certain VDE  $z_v$ , denote  $z_v^P$  its positive sample and  $z_v^k$  its negative samples, the InfoNCE loss for VDE is:
+
+$$
+\begin{array}{l} L _ {\mathrm {n c e (V D E)}} = \sum_ {i = 1} ^ {N _ {l}} \sum_ {j = 1} ^ {N _ {v}} \left[ - \frac {1}{N _ {l} \cdot N _ {v}} \cdot \log \frac {\exp (f (z _ {v} ^ {i , j} , z _ {v} ^ {P _ {i , j}}))}{\sum_ {k = 1} ^ {N} \exp (f (z _ {v} ^ {i , j} , z _ {v} ^ {k}))} \right] \\ \approx \mathbb {E} _ {v \sim P _ {v}, l \sim P _ {l}, z _ {v} \sim q _ {v} (\cdot | x (v, l))} \left[ - \log \frac {\exp (f (z _ {v} , z _ {v} ^ {P}))}{\sum_ {k = 1} ^ {N} \exp (f (z _ {v} , z _ {v} ^ {k}))} \right] \\ \approx \mathbb {E} _ {v \sim P _ {v}, l \sim P _ {l}, z _ {v} \sim q _ {v} (\cdot | x (v, l))} \left[ - \log \frac {\exp (f (z _ {v} , z _ {v} ^ {P}))}{N \cdot \mathbb {E} [ \exp (f (z _ {v} , z _ {v} ^ {\prime})) ]} \right] \\ \approx \mathbb {E} _ {v \sim P _ {v}, l \sim P _ {l}, z _ {v} \sim q _ {v} (\cdot | x (v, l)), l ^ {P} \sim P _ {l}, z _ {v} ^ {P} \sim q _ {v} (\cdot | x (v, l ^ {p}))} \left[ - f (z _ {v}, z _ {v} ^ {P}) \right] + \\ \mathbb {E} _ {v \sim P _ {v}, l \sim P _ {l}, z _ {v} \sim q _ {v} (\cdot | x (v, l))} \left[ \log \left(\mathbb {E} _ {v ^ {\prime} \sim P _ {v}, l ^ {\prime} \sim P _ {l}, z _ {v} ^ {\prime} \sim q _ {v} (\cdot | x (v ^ {\prime}, l ^ {\prime}))} \left[ \exp \left(f \left(z _ {v}, z _ {v} ^ {\prime}\right)\right) \right]\right) \right] + \log N \\ \geq \mathbb {E} _ {v \sim P _ {v}, l \sim P _ {l}, z _ {v} \sim q _ {v} (\cdot | x (v, l)), l ^ {P} \sim P _ {l}, z _ {v} ^ {P} \sim q _ {v} (\cdot | x (v, l ^ {p}))} \left[ - f (z _ {v}, z _ {v} ^ {P}) \right] + \\ \mathbb {E} _ {v \sim P _ {v}, l \sim P _ {l}, z _ {v} \sim q _ {v} (\cdot | x (v, l)), v ^ {\prime} \sim P _ {v}, l ^ {\prime} \sim P _ {l}, z _ {v} ^ {\prime} \sim q _ {v} (\cdot | x (v ^ {\prime}, l ^ {\prime}))} \left[ f \left(z _ {v}, z _ {v} ^ {\prime}\right) \right] + \log N. \tag {5} \\ \end{array}
+$$
+
+The last inequality in Eq. (5) is derived from Jenson inequality. We set  $f(z_v, z_v') = \log P(z_v, z_v') = \log P(z_v) + \log P(z_v'|z_v)$ , i.e. the energy function, we have:
+
+$$
+L _ {n c e (V D E)} \geq \mathbb {E} _ {v \sim P _ {v}, l \sim P _ {l}, z _ {v} \sim q _ {v} (\cdot | x (v, l)), l ^ {P} \sim P _ {l}, z _ {v} ^ {P} \sim q _ {v} (\cdot | x (v, l ^ {p}))} \left[ - f (z _ {v}, z _ {v} ^ {P}) \right] +
+$$
+
+$$
+\begin{array}{l} \mathbb {E} _ {v \sim P _ {v}, l \sim P _ {l}, z _ {v} \sim q _ {v} (\cdot | x (v, l)), v ^ {\prime} \sim P _ {v}, l ^ {\prime} \sim P _ {l}, z _ {v} ^ {\prime} \sim q _ {v} (\cdot | x (v ^ {\prime}, l ^ {\prime}))} \left[ f \left(z _ {v}, z _ {v} ^ {\prime}\right) \right] + \log N \\ = \mathbb {E} _ {v \sim P _ {v}, l \sim P _ {l}, z _ {v} \sim q _ {v} (\cdot | x (v, l)), l ^ {P} \sim P _ {l}, z _ {v} ^ {P} \sim q _ {v} (\cdot | x (v, l ^ {p}))} \left[ - \log P \left(z _ {v}\right) - \log P \left(z _ {v} ^ {P} | v\right) \right] + \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \mathbb {E} _ {v \sim P _ {v}, l \sim P _ {l}, z _ {v} \sim q _ {v} (\cdot | x (v, l)), v ^ {\prime} \sim P _ {v}, l ^ {\prime} \sim P _ {l}, z _ {v} ^ {\prime} \sim q _ {v} (\cdot | x (v ^ {\prime}, l ^ {\prime}))} \left[ \log P (z _ {v}) + \log P (z _ {v} ^ {\prime}) \right] + \log N \\ = H \left(Z _ {v} | v\right) - H \left(Z _ {v}\right) + \log N \\ = - I \left(Z _ {v}; V\right) + \log N. \tag {6} \\ \end{array}
+$$
+
+We justify that  $f(z_v, z_v')$  can be represented as  $\log P(z_v, z_v')$ , which allows us to relate the similarity metric to probabilistic terms. This relationship is crucial for interpreting the InfoNCE loss in terms of entropy and mutual information. Similarly, we obtain:
+
+$$
+L _ {\mathrm {n c e} (\mathrm {V I E})} \geq - I \left(Z _ {l}; L\right) + \log N, \tag {7}
+$$
+
+We then prove that the distance metric  $d(X, Y)$ , defined in terms of mutual information, satisfies the triangle inequality. This metric is used to measure the disentanglement between  $Z_{v}$  and  $Z_{l}$ . We denote
+
+$$
+d (X, Y) = 1 - \frac {I (X ; Y)}{\operatorname* {m a x} (H (X) , H (Y))},
+$$
+
+and
+
+$$
+A \left(X _ {1}, X _ {2}, \dots , X _ {n}\right) = \max  \left(H \left(X _ {1}\right), \dots , H \left(X _ {n}\right)\right).
+$$
+
+According to triangle inequality, we have:
+
+$$
+d (V, L) \leq d \left(Z _ {v}, V\right) + d \left(Z _ {l}, L\right) + d \left(Z _ {v}, Z _ {l}\right), \tag {8}
+$$
+
+which can be simplified as:
+
+$$
+\begin{array}{l} I \left(Z _ {v}; Z _ {l}\right) \leq A \left(Z _ {v}, Z _ {l}\right) \cdot \left[ 2 - \frac {I (V ; Z _ {v})}{A (V , Z _ {v})} - \frac {I (L ; Z _ {L})}{A (L , Z _ {L})} \right] \\ \leq A \left(Z _ {v}, Z _ {l}, V, L\right) \cdot \left[ 2 - \frac {I \left(V ; Z _ {v}\right)}{A \left(V , Z _ {v} , L , Z _ {l}\right)} - \frac {I \left(L ; Z _ {L}\right)}{A \left(L , Z _ {L} , V , Z _ {v}\right)} \right] \tag {9} \\ = 2 \cdot A \left(Z _ {v}, Z _ {l}, V, L\right) - I (V; Z _ {v}) - I (L; Z _ {l}) \\ \leq 2 \cdot A \left(Z _ {v}, Z _ {l}, V, L\right) + L _ {\mathrm {n c e} (\mathrm {V D E})} + L _ {\mathrm {n c e} (\mathrm {V I E})} - 2 \log N. \\ \end{array}
+$$
+
+From Eq. (9), we can see that minimizing  $A(Z_{v},Z_{l},V,L)$ ,  $L_{\mathrm{nce(VDE)}}$  and  $L_{\mathrm{nce(VDE)}}$  equals to minimizing the upper bound of the mutual information between the view-invariant and view-dependent representations.
+
+Based on the Data Processing Inequality, we show that  $A(Z_v,Z_l,V,L)$  is a constant  $C$ , which represents the maximum entropy among the variables. This conclusion is based on the fact that processing data cannot increase mutual information:
+
+$$
+\begin{array}{l} A \left(Z _ {v}, Z _ {l}, V, L\right) = \max  \left(H \left(Z _ {v}\right), H \left(Z _ {l}\right), H (V), H (L)\right) \\ = \max  (H (V), H (L)) \tag {10} \\ = C. \\ \end{array}
+$$
+
+Finally, we establish a clear connection between the InfoNCE loss and mutual information. Minimizing the InfoNCE losses for  $Z_{v}$  and  $Z_{l}$  corresponds to minimizing the mutual information between these representations, thus facilitating their disentanglement.
+
+# B RELATED WORK
+
+Robotic manipulation with visual inputs has become a prominent field of study, leveraging cameras to capture environmental data and employing learning-based methods, such as reinforcement learning, to train control policies (Zeng et al., 2018; Kalashnikov et al., 2018b; Quillen et al., 2018; Ebert et al., 2018; Pang et al., 2023a;b; 2021). For instance, the work of VPG (Zeng et al., 2018) utilizes the Deep Q-Network algorithm (Mnih et al., 2015) to train a policy for a block grasping task. Similarly, QT-Opt (Kalashnikov et al., 2018a) has demonstrated the feasibility of real-world grasping by introducing a self-supervised, vision-based reinforcement learning framework.
+
+A critical challenge in training policies with visual inputs is the issue of viewpoint disturbance, the camera position changes during deployment. Prior research has investigated a variety of methods to mitigate the effects of viewpoint changes. A prevalent technique is domain randomization (Tobin et al., 2017), which augments the training data with synthetic variations to emulate potential environmental disturbances. Another approach is the adoption of unified view representations (Liu et al., 2024; Sermanet et al., 2018; Yin et al., 2022). For instance, RoboUniView (Liu et al., 2024) and TCN (Sermanet et al., 2018) aim to learn a unified representation for images captured from disparate viewpoints. However, the enforcement of viewpoint invariance, which presumes the same information across different viewpoints, necessitates the careful selection of positive and negative data pairs. This is akin to the challenges faced by contrastive learning methods, which often rely on complicated choices with regard to the sampling of such pairs (Saunshi et al., 2019; Seo et al., 2023). Imitation learning methods, such as RT-1 (Brohan et al., 2023), RT-2 (Zitkovich et al., 2023), and RT-X (O'Neill et al., 2024), seek to learn robust control by leveraging extensive manipulation datasets, but this approach inevitably increases data processing and computational demands. While these methods have shown a certain level of success in handling viewpoint disturbances, they fail to isolate view-invariant information, which is essential for robust robotic control Kang (2024), from the viewpoint information. In contrast, this work proposes a novel approach to this challenge.
+
+We introduce an autoencoder framework with dual encoders, designed to decompose view-invariant information, which is shown to be effective in robust robotic control.
+
+World Models for Robotics Manipulation. Model-based RL trains a world model that learns the environment dynamics through supervised learning. In robotics, world models are crucial for enabling robots to predict the outcomes of their actions. There are two common ways to utilize the world model in robotics: planning (Alterovitz et al., 2016) and policy learning (Ha & Schmidhuber, 2018). For example, Schmerling et al. (2018) proposes model-based planning for human-machine interaction. Ha & Schmidhuber (2018) introduces the concept of world models in the context of reinforcement learning, demonstrating that compact and efficient representations of the environment can be learned, which in turn can be used to train agents with fewer interactions with the real world.
+
+Due to the challenge in modeling the environment with image as input/output, recent works propose to separate the representation learning and world modeling (Hafner et al., 2020; 2021; 2023). However, these world models seldom consider the viewpoint disturbance. In this work, we follow the representation learning  $^+$  world modeling framework but attempt to learn a view-invariant world model that is robust to the viewpoint disturbances.
+
+# C MORE IMPLEMENTATION DETAILS & EXPERIMENT SETUP
+
+# C.1 VIEWPOINTS FOR AUTOENCODER AND RL TRAINING
+
+To construct the dataset to train the autoencoder, we use a camera configuration that encompasses a diverse set of azimuth angles. Specifically, the azimuth angles employed in Meta-world are  $\{8, 11, 17, 20, 28, 35, 136, 144, 148, 152, 159, 180, 194, 195, 198, 219, 324, 333, 339, 351\}$ . Note that these angles are selected to span a 180-degree range, encompassing two distinct 90-degree intervals: [-45, 45] and [135, 225] degrees, respectively. For PandaGym, the azimuth angles are  $\{6, 21, 39, 60, 69, 85, 114, 125, 153, 159, 162, 170, 179, 205, 209, 213, 217, 224, 238, 244\}$  selected from an interval of [0, 270] degrees.
+
+For RL training, we select a distinct azimuth angle of 22.5 degrees for the viewpoint of MetaWorld and 45 degrees for the viewpoint of PandaGym. This angle is set to be different from any of the azimuth angles used for training the autoencoder to ensure diverse and comprehensive learning. This choice of viewpoint is designed to evaluate the generalization capabilities of the RL algorithm by exposing it to a novel viewpoint that is not encountered during the autoencoder training phase. We present the visualization of different viewpoints in Fig. 8.
+
+# C.2 ARCHITECTURE AND OPTIMIZATION DETAILS
+
+The autoencoder architecture is based on the ViT (Dosovitskiy et al., 2021). The architecture of the encoder and decoder are presented in Fig. 9. We first utilize a one-layer CNN to project the input image with size  $128 \times 128$  into patch embeddings where the patch size is  $16 \times 16$ . We employ a bidirectional transformer structure for both encoder and decoder block, consisting of 8-head self-attention mechanisms and feed-forward networks with layer normalization and residual connections included. The transformer's intermediate embedding dimension is 256 and the dropout rate is set 0.1. For VIE, we utilize a vector quantizer with a codebook size of 512 and an embedding dimension of 64 to obtain the view-invariant encoding. To prevent codebook collapse, we utilized k-means initialization for our vector quantizer. As for VDE, we introduce a simple linear layer to get the view-dependent encoding. The view-invariant encoding and view-dependent encoding are concatenated and fed into the decoder block, after which the decoder output is projected into the image space through a one-layer deconvolution.
+
+# C.3 COMPUTATION
+
+We use 64 CPU cores (AMD EPYC 9654 @ 2.4GHz) and 4 GPUs (NVIDIA GeForce RTX 4090) for our experiments. The software stack employed for our experiments includes Python 3.11 and PyTorch 2.1.0. On average, the training of the autoencoder to the point of convergence on our dataset
+
+![](images/6d24ce1683f4abc69f43fa6bc2a64b7afd0437accd29e788b3ee40eeb7fc7b9e.jpg)  
+(a) Meta-world (training)
+
+![](images/14505e82a9bfdcca94dbe535b8246aaaab92215b80d34b9980e471f8bba77e91.jpg)  
+(b) Meta-world (offset=10)
+
+![](images/b9976db5f825689e2adad0750df53d8bbae8469deab44d5552b17c49625a8b64.jpg)  
+(c) Meta-world (offset=20)
+
+![](images/6eb93dc4480e91d9cd64f9f002b97e4095aab2cb16dcb4cfc5680b6074f9d61e.jpg)  
+(d) PandaGym (training)
+
+![](images/c357feaa3ca264a25b25f4b3939a818bfa48b348b943d60fe317fc8db2ae1aab.jpg)  
+(e) PandaGym (offset=45)
+
+![](images/029cf25b68ddb1fb8d34506fed71c37896eb306017b7304f6b1d05d0758245c2.jpg)  
+(f) PandaGym (offset=90)
+
+![](images/a3dfca729e6f2163cdd544b7ef506ab75b15631b262930eafafc4a7334e3860a.jpg)  
+Figure 8: Visualization of the training viewpoint and novel viewpoint for robotics control. In our experiments, Meta-world (offset=10) and PandaGym (offset=90) are utilized for evaluating policy performance under CIP.  
+(1) Encoder  
+Figure 9: Network architecture of encoder and decoder of ReViWo.
+
+![](images/006a34440d513900145934ffc26afc40962e4b6f789767945b4727dca6ec4d6a.jpg)  
+(2) Decoder
+
+requires approximately 12 hours. Subsequently, the training of the world model and the policy are completed in an average of 2 hours.
+
+# C.4 HYPERPARAMETERS
+
+The hyper-parameters for implementing ReViWo are presented in Table 4. When implementing baseline methods, we use the same hyper-parameters of offline RL for training the world model and policy.
+
+# C.5 DETAILED DESCRIPTIONS OF BASELINES
+
+MVWM (Seo et al., 2023): MVWM is a reinforcement learning framework that trains a multi-view masked autoencoder for representation learning and a world model to solve visual manipulation tasks. The autoencoder consists of a synergistic combination of view-masking: which masks viewpoints at random, and video autoencoding: which reconstructs video frames of both masked and unmasked viewpoints. In Seo et al. (2023), the authors find that the autoencoder effectively learns representations that capture useful information from the current viewpoint but also the cross-view information from different viewpoints. For behavior learning, MVWM learns a world model on
+
+Table 4: Hyper-parameters for training ReViWo and baselines.  
+
+<table><tr><td>Hyper-parameters</td><td>Value</td></tr><tr><td>Patch Num.</td><td>8 × 8 = 64</td></tr><tr><td>Embedding dimension for each patch</td><td>256</td></tr><tr><td>Encoder output dimension</td><td>64</td></tr><tr><td>Enocoder block Num.</td><td>8</td></tr><tr><td>Decoder block Num.</td><td>8</td></tr><tr><td>Coef. for VQ loss</td><td>0.25</td></tr><tr><td>Coef. for VDE contrastive loss</td><td>0.1</td></tr><tr><td>Coef. for VIE contrastive loss</td><td>1</td></tr><tr><td>Learning rate of authencoder</td><td>3e-5</td></tr><tr><td>Optimizer for autoencoder</td><td>Adam</td></tr><tr><td>Learning rate of world model</td><td>1e-3</td></tr><tr><td>Optimizer for world model</td><td>Adam</td></tr><tr><td>Learning rate for policy</td><td>1e-4</td></tr><tr><td>Optimizer for policy</td><td>Adam</td></tr><tr><td>World model structure</td><td>[input dim, 2048, 512, 256, 256, output dim]</td></tr><tr><td>World model ensemble Num.</td><td>7</td></tr><tr><td>Policy structure</td><td>[input dim, 2048, 512, 256, 256, output dim]</td></tr></table>
+
+frozen representations from either single-view or multi-view data, which is particularly feasible as the autoencoder consists of vision transformer layers that take inputs of variable sizes. Then the actor and critic are trained with imaginary trajectories from the world model. Although MVWM is originally implemented for online RL, it can be easily adapted to offline settings by combining its representation learning technique with offline model-based RL (Agarwal et al., 2020).
+
+COMBO (Yu et al., 2021): COMBO is a novel approach in the field of reinforcement learning, designed to effectively learn policies from offline data without further interaction with the environment. It addresses the challenge of distributional shift, where the policy encounters states that are not well-represented in the offline dataset, which can lead to poor performance or even catastrophic failures when deployed. COMBO integrates conservative policy evaluation with model-based planning, using a learned dynamics model to simulate future states and rewards. By combining conservative value estimation and model-based rollouts, COMBO aims to safely optimize policies while mitigating the risks associated with out-of-distribution states. This approach allows for more robust policy learning from static datasets, expanding the applicability of reinforcement learning to scenarios where active data collection is impractical or impossible.
+
+CQL (Kumar et al., 2020): CQL is an algorithm developed for offline reinforcement learning, where the goal is to learn effective policies from a fixed dataset without further interaction with the environment. The key challenge in offline reinforcement learning is to avoid overestimation of the action values (Q-values) for state-action pairs not well-represented in the dataset, which can lead to suboptimal or dangerous policies when executed in the real environment. CQL addresses this issue by introducing a conservative estimation of Q-values during training. It does so by penalizing the Q-values of actions that are not supported by the data and by ensuring that the learned policy does not deviate too much from the behavior policy that generated the dataset. This conservative approach reduces the likelihood of overestimating the Q-values of unseen actions, leading to more reliable policy performance. CQL is particularly useful in scenarios where data collection is expensive or risky, such as robotics, healthcare, or finance, where it is critical to learn from limited data without the opportunity for trial-and-error learning in the actual environment. By focusing on the safe and robust optimization of policies from offline data, CQL represents a significant step forward in the practical application of reinforcement learning.
+
+BC: Behavior cloning (BC) is a straightforward method in the realm of robotic learning, where the goal is to mimic expert behavior. BC trains a policy (typically represented by a neural network) to replicate the actions taken by an expert in various states. This is achieved by collecting a dataset of state-action pairs from an expert's demonstrations and then using this dataset to train the policy via supervised learning, treating the problem as a simple function approximation task where the input
+
+is the state and the output is the action. The appeal of BC lies in its simplicity and efficiency, as it does not require reinforcement signals or interaction with the environment during training. Thus, BC has emerged as a popular method employed in the realm of robots, where learning a capable policy by online interaction is challenging and expensive. However, it also has limitations, such as the tendency to accumulate errors due to covariate shift, where the policy encounters states that are not well-represented in the training data, leading to actions that deviate from the experts', and thus to states that are even less well-represented, in a potentially compounding fashion. Despite its limitations, BC can be quite effective for tasks where the expert's policy is easy to capture with supervised learning, and it serves as a foundation for more complex imitation learning algorithms that seek to address its shortcomings.
+
+# D ALGORITHM DESCRIPTION
+
+The practical implementation of ReViWo method is presented in the form of pseudo-code in Algorithm 1.
+
+Algorithm 1 Representation learning for View-invariant World model (ReViWo)  
+Required: a multi-view image dataset  $\mathcal{O}$  offline control data  $\mathcal{D}$  and a empty replay buffer for world model rollout  $\tilde{\mathcal{D}}$    
+Output: the optimized robotic control policy  $\pi$  . 1: Initialize the autoencoder  $p_{\phi},q_{\phi}$  , world model  $\mathcal{M}_{\theta}$  , reward model  $\mathcal{R}_{\Theta}$  and policy  $\pi_{\Phi}$  , where the subscript denotes their parameters. 2: // Training autoencoder 3: while training not converge do 4: Sample  $(o_{s_1}^{v_1},o_{s_2}^{v_2},o_{s_1}^{v_2})$  data from  $\mathcal{O}$  5: Update  $\phi$  based on AE training objective in Eq. (3). 6: end while 7: Pre-process visual observation in  $\mathcal{D}$  from  $s_t$  to  $z_{s_t}$  8: Update the offline control dataset with  $(z_{s_t},a_t,z_{s_{t + 1}},r_t)$  9: // Training world model 10: while AE training not converge do 11: Sample  $(s_t,a_t,s_{t + 1},r_t)$  from  $\mathcal{D}$  12: Update world model parameters  $\theta$  with world model training objective in Eq. (4). 13: Update reward model parameters  $\Theta$  with supervised learning on reward data. 14: end while 15: // Training policy 16: while policy training not converge do 17: Sample in the world model with  $\pi_{\Phi}$  and  $\mathcal{R}_{\Theta}$  to collect  $(\tilde{z}_{s_t},a_t,\tilde{z}_{s_{t + 1}},\tilde{r}_t)$  18: Update replay buffer:  $\tilde{\mathcal{D}} = \tilde{\mathcal{D}}\cup (\tilde{z}_{s_t},a_t,\tilde{z}_{s_{t + 1}},\tilde{r}_t)$  19: Sample from both  $\mathcal{D}$  and  $\tilde{\mathcal{D}}$  to update  $\Phi$  with COMBO algorithm. 20: end while 21: return the optimized policy  $\pi$
+
+# E DISCUSSION ON THE USAGE OF DATASET WITH VIEW LABELS
+
+In this research, we present a method for robust robotic manipulation that relies on a multi-view dataset that has implicit view labels to train the autoencoder. We would like to discuss the rationality of employing such labels for robust robotic control. The view information is important in visual robotics control. The ability to discriminate between features that are relevant under different viewing conditions is paramount. View labels facilitate this process by guiding the model to associate specific features with the viewpoints they are most relevant to, thereby enhancing the model's ability to generalize these features across a range of similar viewpoints. Some previous works (Liu et al., 2024) also attempt to train with the view label. Different from Liu et al. (2024), we do not explicitly use the view labels like camera parameters, but the multi-view images that are naturally aligned in states or viewpoints.
+
+Besides, in practice, it is not expensive to obtain the viewpoint labels, as we can record multiple manipulation trajectories from some fixed-position cameras. Such view information is not just a facilitator for quicker learning, but also a critical component that imbues the model with a deeper understanding of its operational context. It empowers the model to develop a robust, interpretable, and adaptable framework for robotic manipulation that is essential for real-world applications. Moreover, we demonstrate that ReViWo can be integrated with multi-view data with that is without a view label. Our experiments in Tab. 2 demonstrate the efficacy of such integration.
+
+# F ADDITIONAL RESULTS
+
+# F.1 RESULTS ON MORE TASKS
+
+We conduct experiments on more tasks in the Meta-world environment, including Coffee Button, Faucet Open, and Dial Turn. The experiment results are presented in Fig. F.1.
+
+![](images/d0963d8e3e4d151ef4010fa7a3e2a59e0137b6dc3de981bd3186b101c1b67ae9.jpg)  
+(a) Coffee Button Open
+
+![](images/74aef7024e558697e38f0d973486104d7b2162aa68d023376b194496a31f44a0.jpg)  
+(b)Faucet Open  
+Figure 10: Performance of Imitation Learning with View-invariant Representation. The x-axis denotes the disturbance type, and the y-axis denotes the average success rate of the last two checkpoints, by evaluation for 30 episodes.
+
+![](images/068eae5b2ea88593bcf728dcd4364a4345dbcf0113c795288f24badc09840352.jpg)  
+(c) Dial Turn
+
+# F.2 IMITATION LEARNING WITH VIEW-INVARIANT REPRESENTATION
+
+Imitation learning is a widely-considered setup in the field of robotics. We further investigate whether the learned VIR can be utilized for imitation learning. To this end, we apply BC with the learned VIR using the offline control data. ReViWo's capability to learn view-invariant representations enables the application of Imitation Learning as an alternative to training a world model for robotic control policy acquisition. Fig. 11 illustrates the success rate of Behavior Cloning utilizing view-invariant representations (RiVi-BC). While RiVi-BC does not surpass ReViWo, it outperforms conventional Behavior Cloning with a VAE encoder. The performance drop of BC is primarily attributed to distributional shift and the accumulation of errors during sequential decision-making. BC is inherently susceptible to these issues, as it relies on imitating actions from a fixed dataset. However, it is noteworthy that when BC is paired with VIR, it outperforms BC with VAE representations. This outcome underscores the robustness of VIR in novel viewpoints and improving overall performance.
+
+![](images/ae2c98d1652c3ec4870f1d144df57bb124f17e32b860e3563c85a46091e6ab04.jpg)  
+(a) Door Open
+
+![](images/feeac3a6bd23c9e6c29e54f422d34065c56242574636c3140f20ac281ad0601a.jpg)  
+(b)Drawer Open  
+Figure 11: Performance of Imitation Learning with View-invariant Representation. The x-axis denotes the disturbance type, and the y-axis denotes the average success rate of the last two checkpoints, by evaluation for 30 episodes.
+
+![](images/99e11dd1864991f51c7e3a538cdc206929f425d676bc65374a4181451036b3d7.jpg)  
+(c) Window Close
+
+# F.3 MORE EXAMPLES OF DECODER OUTPUT
+
+![](images/d42fcd2b0860825224729e8f0362c3ae06ce3720d9abebb8ac96a03e40b028f7.jpg)  
+Fig. 12 presents additional examples of the decoder output.  
+VIE  
+Input  
+VDE  
+Input  
+Decoder  
+Output  
+Ground  
+Truth  
+Figure 12: Additional examples of the decoder output the corresponding ground truth image. The decoder can generate the images that combine the task state in VIE inputs, and the viewpoint in VDE inputs.
+
+![](images/decc6f68614156e9179c1a04396ad22d81a1049e29e32eb220a642715d15651f.jpg)  
+VIE  
+Input  
+VDE  
+Input  
+Decoder  
+Output  
+Ground  
+Truth
\ No newline at end of file
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+# LEARNING A FAST MIXING EXOGENOUS BLOCK MDP USING A SINGLE TRAJECTORY
+
+Alexander Levine*
+
+The University of Texas at Austin
+
+Peter Stone
+
+The University of Texas at Austin  
+Sony AI
+
+Amy Zhang
+
+The University of Texas at Austin
+
+# ABSTRACT
+
+In order to train agents that can quickly adapt to new objectives or reward functions, efficient unsupervised representation learning in sequential decision-making environments can be important. Frameworks such as the Exogenous Block Markov Decision Process (Ex-BMDP) have been proposed to formalize this representation-learning problem (Efroni et al., 2022b). In the Ex-BMDP framework, the agent's high-dimensional observations of the environment have two latent factors: a controllable factor, which evolves deterministically within a small state space according to the agent's actions, and an exogenous factor, which represents time-correlated noise, and can be highly complex. The goal of the representation learning problem is to learn an encoder that maps from observations into the controllable latent space, as well as the dynamics of this space. Efroni et al. (2022b) has shown that this is possible with a sample complexity that depends only on the size of the controllable latent space, and not on the size of the noise factor. However, this prior work has focused on the episodic setting, where the controllable latent state resets to a specific start state after a finite horizon.
+
+By contrast, if the agent can only interact with the environment in a single continuous trajectory, prior works have not established sample-complexity bounds. We propose STEEL, the first provably sample-efficient algorithm for learning the controllable dynamics of an Ex-BMDP from a single trajectory, in the function approximation setting. STEEL has a sample complexity that depends only on the sizes of the controllable latent space and the encoder function class, and (at worst linearly) on the mixing time of the exogenous noise factor. We prove that STEEL is correct and sample-efficient, and demonstrate STEEL on two toy problems. Code is available at: https://github.com/midi-lab/steel.
+
+# 1 INTRODUCTION
+
+This work considers the unsupervised representation learning problem in sequential control environments. Suppose an agent (e.g., a robot) is able to make observations and take actions in an environment for some period of time, but does not yet have an externally-defined task to accomplish. We want the agent to learn a model of the environment that may be useful for many downstream tasks: the question is then how to efficiently explore the environment to learn such a model.
+
+Sequential decision-making tasks are often modeled as Markov Decision Processes (MDPs). In the unsupervised setting, an MDP consists of a set of possible observations  $\mathcal{X}$ , a set of possible actions  $\mathcal{A}$ , a distribution over initial observations  $\pi_0 \in \mathcal{P}(\mathcal{X})$ , and a transition function  $\mathcal{T}: \mathcal{X} \times \mathcal{A} \to \mathcal{P}(\mathcal{X})$ . The agent does not have direct access to  $\mathcal{T}$ . Instead, at each timestep  $t$ , the agent observes  $x_{t} \in \mathcal{X}$  and selects action  $a_{t}$ . The next observation  $x_{t+1}$  is then sampled as  $x_{t+1} \sim \mathcal{T}(x_{t}, a_{t})$ .
+
+In the totally generic MDP setting, the only model learning possible is to directly learn the transition function  $\mathcal{T}$ . However, if the space of possible observations is large, this task becomes intractable. Therefore, prior works have attempted to simplify the problem by assuming that the MDP has some underlying structure, which a learning algorithm can exploit. One such structural assumption is the Ex-BMDP (Exogenous Block MDP) framework, introduced by Efroni et al. (2022b). The Ex-
+
+BMDP framework captures situations where the space of observations  $\mathcal{X}$  is very large, but the parts of the environment that the agent has control over can be represented by a much smaller latent state.
+
+An Ex-BMDP has an observation space  $\mathcal{X}$ , a controllable (or endogenous) latent state space  $\mathcal{S}$ , and an exogenous state space  $\mathcal{E}$ . In the version of this setting that we consider, the controllable state  $s_t \in \mathcal{S}$  evolves deterministically according to a latent transition function  $T: \mathcal{S} \times \mathcal{A} \to \mathcal{S}$ . That is:  $s_{t+1} = T(s_t, a_t)$ . The exogenous state  $e_t \in \mathcal{E}$  represents time correlated noise: it evolves stochastically, independently of actions, according to a transition function  $\mathcal{T}_e: \mathcal{E} \to \mathcal{P}(\mathcal{E})$ . That is:  $e_{t+1} \sim \mathcal{T}_e(e_t)$ . Neither  $s$  nor  $e$  is directly observed. Instead, the observation  $x_t$  is sampled as  $x_t \sim \mathcal{Q}(s_t, e_t)$ , where  $\mathcal{Q} \in \mathcal{S} \times \mathcal{E} \to \mathcal{P}(\mathcal{X})$  is the emission function. We make a block assumption on  $\mathcal{Q}$  with respect to  $\mathcal{S}$ : that is, we assume that for distinct latent states  $s, s' \in \mathcal{S}$ , the sets of possible observations that can be sampled from  $\mathcal{Q}(s, \cdot)$  and  $\mathcal{Q}(s', \cdot)$  are disjoint. In other words, there exists a deterministic partial inverse of  $\mathcal{Q}$ , which is  $\phi^*: \mathcal{X} \to \mathcal{S}$ , such that if  $x \sim \mathcal{Q}(s, e)$ , then  $\phi^*(x) = s$ . Hence, it is always possible in principle to infer  $s$  from  $x$ .<sup>1</sup>
+
+As in the general MDP setting, the agent only directly observes  $x_{t} \in \mathcal{X}$ , and chooses actions  $a_{t}$  in response. However, rather than attempting to learn the full transition dynamics  $\mathcal{T}$  of the system (which is determined by  $T$ ,  $\mathcal{T}_e$ , and  $\mathcal{Q}$  together), the objective of the agent is instead to efficiently model only the latent encoder  $\phi^{*}$  and the latent transition function  $T$ . Together, these models allow the agent to plan or learn in downstream tasks using the encoded representations  $\phi^{*}(x)$ , modeling only the parts of the environment that the agent can actually control (the latent state  $s \in S$ ) while ignoring the potentially-complex dynamics of time-correlated noise.
+
+Specifically, the aim of efficient representation learning in this setting is to learn  $\phi^{*}$  and  $T$ , using a number of environment steps of exploration that is dependent only on  $|\mathcal{S}|$  and the size of the function class  $\mathcal{F}$  that the encoder  $\phi^{*}$  belongs to, and is not dependent on the size of  $\mathcal{X}$  or  $\mathcal{E}$ . This allows  $\mathcal{X}$  and  $\mathcal{E}$  to be very large or potentially even infinite, but still allows for representation learning to be tractable. Efroni et al. (2022b) proposes an algorithm, PPE, with this property. However, PPE only works in a finite-horizon setting, where the agent interacts with the environment in episodes of fixed length  $H$ . After each episode, the controllable state (almost) always resets to a deterministic start state  $s_0 \in S$  at the beginning of each episode. In this work, we instead consider the single-trajectory, no-reset setting, where the agent interacts with the environment in a single episode of unbounded length, with no ability to reset the state. This better models real-world cases, where, for example, expensive human intervention would be required to "reset" the environment that a robot trains in: we would rather not require this intervention. This no-reset Ex-BMDP setting was previously considered by Lamb et al. (2023) and Levine et al. (2024); however, the algorithms presented in those works do not have sample-complexity guarantees.
+
+By contrast, the algorithm presented in this work is guaranteed to learn  $\phi^{*}$  and  $T$  using samples polynomial in  $|\mathcal{S}|$  and  $\log |\mathcal{F}|$ , with no dependence on  $|\mathcal{E}|$  and  $|\mathcal{X}|$ . We only require that the mixing time of the exogenous noise is bounded. In other words, the requirement is that  $t_{\mathrm{mix}}$ , the mixing time of the Markov chain on  $\mathcal{E}$  induced by  $\mathcal{T}_{e}$ , is at most some known quantity  $\hat{t}_{\mathrm{mix}}$ . Note that we do not require that the endogenous state  $s$  mixes quickly under any particular policy (although we do require - as do Lamb et al. (2023) and Levine et al. (2024) - that all states in  $\mathcal{S}$  are eventually reachable from one another). In this setting, we derive an algorithm with the following asymptotic sample complexity (where  $\mathcal{O}^*(f(x)) \coloneqq \mathcal{O}(f(x)\log(f(x))))$ :
+
+$$
+\mathcal {O} ^ {*} \left(N D | \mathcal {S} | ^ {2} | \mathcal {A} | \cdot \log \frac {| \mathcal {F} |}{\delta} + | \mathcal {S} | | \mathcal {A} | \hat {t} _ {\text {m i x}} \cdot \log \frac {N | \mathcal {F} |}{\delta} + \frac {| \mathcal {S} | ^ {2} D}{\epsilon} \cdot \log \frac {| \mathcal {F} |}{\delta} + \frac {| \mathcal {S} | \hat {t} _ {\text {m i x}}}{\epsilon} \cdot \log \frac {| \mathcal {F} |}{\delta}\right), \tag {1}
+$$
+
+where  $N$  is a predetermined upper-bound on  $|\mathcal{S}|$ ,  $\delta$  is the failure rate of the algorithm,  $D$  is the maximum distance between any two latent states in  $\mathcal{S}$  (at most  $|\mathcal{S}|$ ), and  $\epsilon$  is the minimum accuracy of the output learned encoder  $\phi$  on any latent state class  $s \in \mathcal{S}$ . Note that this expression is at worst polynomial in  $|\mathcal{S}|$ , and linear in  $\hat{t}_{\mathrm{mix}}$  and  $\log |\mathcal{F}|$ .
+
+Our algorithm proceeds iteratively, at each iteration taking a certain sequence of actions repeatedly in a loop. Because the latent state dynamics  $T$  are deterministic, this process is guaranteed to (after some transient period) enter a cycle of latent states, of bounded length. Because the latent states in
+
+this cycle are repeatedly re-visited, the algorithm is then able to predictably collect many samples of the same latent state, without the need to "re-set" the environment. Furthermore, because this looping can be continued indefinitely, the algorithm can "wait out" the mixing time of the exogenous dynamics, in order to collect near-i.i.d. samples of each latent state. We call our algorithm Single-Trajectory Exploration for Ex-BMDPs via Looping, or STEEL. In summary, we: (1) introduce STEEL, the first provably sample-efficient algorithm for learning Ex-BMDPs in a general functionapproximation setting from a single trajectory, (2) prove the correctness and sample complexity of STEEL, and (3) empirically test STEEL on two toy problems to demonstrate its efficacy.
+
+# 2 RELATED WORKS
+
+# 2.1 REPRESENTATION LEARNING FOR EX-BMDP AND EXO-MDPS
+
+The Ex-BMDP model was originally introduced by Efroni et al. (2022b), who propose the PPE algorithm to learn the endogenous state encoder  $\phi(\cdot)$  and latent transition dynamics  $T$  of an Ex-BMDP. PPE has explicit sample-complexity guarantees that are polynomial in  $|\mathcal{S}|$  and  $\log |\mathcal{F}|$ : crucially, the sample complexity does not depend explicitly on  $|\mathcal{E}|$  or  $|\mathcal{X}|$ . However, unlike the method proposed in this work, PPE is restricted to the episodic, finite horizon setting with (nearly) deterministic resets. After each episode, the endogenous state is (nearly) always reset to the same starting latent state  $s_0 \in S$ , and the exogenous state  $e_0 \in \mathcal{E}$  is i.i.d. resampled from a fixed starting distribution. Similarly to this work, PPE assumes that the latent transition dynamics  $T$  are (close to) deterministic. Because both  $s_0$  and  $T$  are nearly deterministic, PPE can collect i.i.d. samples of observations  $x$  associated with any latent state  $s \in S$  simply by executing the same sequence of actions starting from  $s_0$  after each reset. By contrast, in our setting, we cannot reset the Ex-BMDP state, so it is more challenging to collect samples of a given latent state  $s$ .
+
+Other works have considered the Ex-BMDP setting without latent state resets. Lamb et al. (2023) and Levine et al. (2024) consider a setting similar to ours, where the agent interacts with the environment in a single trajectory. However, these methods do not provide sample-complexity guarantees, and instead are only guaranteed to converge to the correct encoder in the limit of infinite samples.
+
+Efroni et al. (2022a) considers a related "Exo-MDP" setting, and proposes the ExoRL algorithm. In this setting, while the environment is episodic, the latent state  $s_0$  resets to a starting value sampled randomly from a fixed distribution after each episode. Additionally, the latent transition dynamics may be non-deterministic. However, unlike our work, Efroni et al. (2022a) does not consider the general function-approximation setting for state encoders. Instead, the observation  $x$  is explicitly factorized into  $d$  factors, and the controllable state  $s$  consists of some unknown subset of  $k$  of these factors: the representation learning problem is reduced to identifying which  $k$  of the  $d$  factors are action-dependent. ExoRL guarantees a sample-complexity polynomial in  $2^k = |\mathcal{S}|$  and  $\log(d)^2$ .
+
+Recently, Mhammedi et al. (2024) have proposed an algorithm for provably sample-efficient policy optimization in the episodic Ex-BMDP setting with rewards, that can handle nondeterministic latent dynamics. However, the algorithm requires simulator access to the environment: this means that the agent is able to reset the environment to any observation  $x \in \mathcal{X}$  that has been previously observed. This requirement is considerably stronger than even the requirement of deterministic resets to a single latent state found in Efroni et al. (2022b).
+
+In addition to our main claim that our proposed algorithm is the first provably sample-efficient algorithm for representation learning in the single-trajectory Ex-BMDP setting, another property of our method is that we do not make a "block" assumption on the exogenous state  $e$ . For a fixed  $s \in S$ , in our setting, the same observation  $x \in \mathcal{X}$  may be emitted by multiple distinct exogenous states  $e, e' \in \mathcal{E}$ . Prior works (Efroni et al., 2022b;a; Lamb et al., 2023; Levine et al., 2024) have stated assumptions that require that  $e$  may be uniquely inferred from  $x$ .<sup>3</sup> Removing this restriction allows one to model a greater range of phenomena. For example, suppose an agent can turn on or turn off
+
+<table><tr><td></td><td>No-Reset Setting</td><td>Sample-Complexity Guarantees</td><td>Function Approximation</td><td>Nondeterministic Reset State</td><td>Partially-Observed Exogenous State</td><td>Nondeterministic Latent Transitions</td></tr><tr><td>STEEL</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✗</td></tr><tr><td>(Efroni&#x27;22b)</td><td>✗</td><td>✓</td><td>✓</td><td>✗</td><td>?</td><td>✗</td></tr><tr><td>(Lamb&#x27;23)</td><td>✓</td><td>✗</td><td>✓</td><td>✓</td><td>?</td><td>✗</td></tr><tr><td>(Levine&#x27;24)</td><td>✓</td><td>✗</td><td>✓</td><td>✓</td><td>?</td><td>✗</td></tr><tr><td>(Efroni&#x27;22a)</td><td>✗</td><td>✓</td><td>✗</td><td>✓</td><td>✗</td><td>✓</td></tr></table>
+
+Table 1: Comparison to Prior Works for learning Ex-BMDP Latent Dynamics
+
+a "noisy TV": i.e., the agent can control whether or not some source of temporally-correlated noise is observable. This is allowed in our version of the Ex-BMDP formulation, but is not allowed if  $e$  must be fully inferable from  $x$ . One prior work, Wu et al. (2024), also (implicitly) removes the block restriction on the exogenous state  $e$ . That work extends Ex-BMDP representation learning to the partially-observed state setting, with the assumption that the observation history within some known window is sufficient to infer the latent state  $s$ . However, Wu et al. (2024) does not provide any sample-complexity guarantees. Wang et al. (2022) and Kooi et al. (2023) consider similar settings with continuous controllable latent state. However, the proposed methods require explicitly modeling the exogenous noise state  $e$ , and there are no sample complexity guarantees. Trimponias & Dietterich (2023) considers the sample-efficiency of reward-based reinforcement learning in Ex-BMDPs assuming known endogenous and exogenous state encoders; however, it does not address the sample complexity of the representation learning problem.
+
+# 2.2 REPRESENTATION LEARNING FOR BLOCK MDPS AND LOW-RANK MDPS
+
+The Ex-BMDP framework can be considered as a generalization of the Block MDP framework (Dann et al., 2018; Du et al., 2019). Like the Ex-BMDP setting, the Block MDP setting models environments where the observed state space  $\mathcal{X}$  of the overall MDP is much larger than an action-dependent latent state space  $S$ . Some works in the Block MDP framework (Mhammedi et al., 2023; Misra et al., 2020) also allow for nondeterministic latent state transitions: that is  $s_{t+1} \sim \mathcal{T}_s(s_t, a_t)$ . However, unlike the Ex-BMDP setting, there is no exogenous latent state  $e \in \mathcal{E}$  or exogenous dynamics  $\mathcal{T}_e$ : the observation is simply sampled as  $x_t \sim \mathcal{Q}(s_t)$ . In other words, the Block MDP setting does not allow for time correlated noise outside of the modelled latent state  $s$ . Therefore, even when stochastic latent-state transition are allowed, any time-correlated noise must be captured in  $S$ , and so impacts the sample complexity (which is typically polynomial in  $|\mathcal{S}|$ ).
+
+The Low-Rank MDP framework can also be considered as an extension the Block MDP framework, but is an orthogonal extension to the Ex-BMDP framework. In Low Rank MDPs, there exist functions  $\phi : \mathcal{X} \times \mathcal{A} \to \mathbb{R}^d$  and  $\mu : \mathcal{X} \to \mathbb{R}^d$ , such that  $\operatorname*{Pr}(x_{t+1} = x'|x_t = x, a_t = a) = \phi(x, a)^T\mu(x')$ . The sample complexity depends only polynomially on  $d$  and logarithmically on the size of the function classes for the state encoders  $\phi$  and  $\mu$ ; it should not depend explicitly on  $|\mathcal{X}|$ . Works under this framework include Agarwal et al. (2020); Uehara et al. (2022) and Cheng et al. (2023). Other works in the Low Rank MDP framework use a reward signal and only explicitly learn part of the representation (the encoder  $\phi$ ), including Mhammedi et al. (2023) and Jiang et al. (2017); see Mhammedi et al. (2023) for a recent, thorough comparison of these works. Note that while BMDPs can be formulated as low-rank MDPs with  $d = |\mathcal{S}|$ , this does not hold for Ex-BMDPs: the rank of the transition probabilities on  $\mathcal{X}$  depends on  $|\mathcal{E}|$  - as noted by Efroni et al. (2022b).
+
+# 3 NOTATION AND ASSUMPTIONS
+
+- The Ex-BMDP,  $M$ , has observation space  $\mathcal{X}$ , with discrete endogenous states  $\mathcal{S}$  that have deterministic, controllable dynamics, and possibly continuous exogenous states  $\mathcal{E}$  with nondeterministic dynamics that do not depend on actions. Concretely, we have that  $s_{t+1} = T(s_t, a_t)$ , where  $T$  is a deterministic function, and  $e_{t+1} \sim \mathcal{T}_e(e_t)$ . Let  $x_t \sim \mathcal{Q}(s_t, e_t)$ , for  $x_t \in \mathcal{X}$ ,  $s_t \in \mathcal{S}$ ,  $e_t \in \mathcal{E}$ , with the block assumption on  $\mathcal{S}$ . That is, a given  $x \in \mathcal{X}$  can be emitted only by one particular  $s \in \mathcal{S}$ , which we define as  $\phi^*(x_t) = s_t$ . We assume that  $M$  is accessed in one continuous trajectory. The initial endogenous state is an arbitrary  $s_{\mathrm{init}} \in \mathcal{S}$ , and the initial exogenous state  $e_{\mathrm{init}} \sim \pi_{\mathcal{E}}^{\mathrm{init}}$ , where  $\pi_{\mathcal{E}}^{\mathrm{init}} \in \mathcal{P}(\mathcal{E})$ .
+
+- The exogenous dynamics on  $\mathcal{E}$  are irreducible and aperiodic, with stationary distribution  $\pi_{\mathcal{E}}$ . There is a known upper bound  $\hat{t}_{\mathrm{mix}}$  on the mixing time  $t_{\mathrm{mix}}$ , where (as defined in
+
+Levin & Peres (2017) and elsewhere)  $t_{\mathrm{mix}} \coloneqq t_{\mathrm{mix}}(1/4)$ , where  $t_{\mathrm{mix}}(\epsilon)$  is defined such that:
+
+$$
+\forall e \in \mathcal {E}, \| \Pr \left(e _ {t + t _ {\operatorname* {m i x}} (\epsilon)} = e ^ {\prime} \mid e _ {t} = e\right) - \pi_ {\mathcal {E}} \left(e ^ {\prime}\right) \| _ {\mathrm {T V}} \leq \epsilon . \tag {2}
+$$
+
+This assumption bounds how "temporally correlated" the noise in the Ex-BMDP is: it ensures that the exogenous noise state  $e_{t}$  at time  $t$  is relatively unlikely to affect  $e_{t + \hat{t}_{\mathrm{mix}}}$ .
+
+- We have a known upper bound on the number of endogenous latent states,  $N \geq |S|$ . Additionally, we assume that all endogenous latent states can be reached from one another in at most  $D$  steps, for some finite  $D$  (note that we do not assume that all pairs of states in  $S$  can be reached from one another in exactly  $D$  steps). We assume that there is a known upper bound on this diameter:  $\hat{D} \geq D$ . Trivially, if all endogenous latent states are reachable from one another then  $D \leq N - 1$ , so if a tighter bound is not available then we can use  $\hat{D} \coloneqq N - 1$ . (In fact, it is not very important to use a tight bound here:  $\hat{D}$  does not appear in the asymptotic sample complexity.)
+
+- There is an encoder hypothesis class  $\mathcal{F} : \mathcal{X} \to \{0,1\}$ , with realizability for one-vs-rest classification of endogenous states. That is,
+
+$$
+\forall s \in \mathcal {S}, \exists f \in \mathcal {F}: \forall x \in \mathcal {X}, f (x) = \mathbb {1} _ {\phi^ {*} (x) = s}. \tag {3}
+$$
+
+In other words, for every latent state  $s \in S$ , there is some function  $f \in \mathcal{F}$  such that  $f(x) = 1$  if and only if  $\phi^{*}(x) = s$ .
+
+- The algorithm has access to a training oracle for  $\mathcal{F}$ , which, given two finite multi-sets  $D_0$  and  $D_1$  each with elements from  $\mathcal{X}$ , returns a classifier  $f \in \mathcal{F}$ . The only requirement that we have for this oracle is that, if there exist any classifiers  $\mathcal{F}^* \subset \mathcal{F}$ , such that, for  $f^* \in \mathcal{F}^*$ ,  $\forall x \in D_0$ ,  $f^*(x) = 0$  and  $\forall x \in D_1$ ,  $f^*(x) = 1$ , then the oracle will return some member of  $\mathcal{F}^*$ . Note that an optimizer that minimizes the 0-1 loss on  $D_0 \cup D_1$  will satisfy this requirement. However, it is not strictly necessary to minimize the 0-1 loss in particular.  
+- General notation: Let  $\mathcal{M}(\mathcal{A})$  be the set of all multisets of the set  $\mathcal{A}$ . Let  $\perp$  represent an undefined value. For lists  $x, y$ , let  $x \cdot y$  represent their concatenation: that is,  $[a, b] \cdot [c, d] = [a, b, c, d]$ . For multisets  $A$  and  $B$ , let  $A \oplus B$  be their union, where their multiplicities are additive. Let  $\%$  be the modulo operator, so that  $a\% b \equiv a \pmod{b}$  and  $0 \leq a\% b \leq b - 1$ .
+
+# 4 ALGORITHM
+
+The STEEL algorithm is presented in full in Appendix A; in this section, we give a high-level overview of the algorithm, and present bounds on its sample-complexity. We state the sample-complexity and correctness of STEEL in the following theorem, which is proved in Appendix B.
+
+Theorem 1. For an Ex-BMDP  $M = \langle \mathcal{X},\mathcal{A},\mathcal{S},\mathcal{E},\mathcal{Q},T,\mathcal{T}_e,\pi_{\mathcal{E}}^{init}\rangle$  starting at an arbitrary endogenous latent state  $s_{init} \in S$ , with  $|\mathcal{S}| \leq N$ , where the exogenous Markov chain  $\mathcal{T}_{e}$  has mixing time at most  $\hat{t}_{mix}$ , and where all states in  $S$  are reachable from one another in at most  $\hat{D}$  steps; and corresponding encoder function class  $\mathcal{F}$  such that Equation 3 holds, the algorithm STEEL(  $M,\mathcal{F},N,\hat{D},\hat{t}_{mix},\delta ,\epsilon$  ) will output a learned endogenous state space  $S'$ , transition model  $T'$ , and encoder  $\phi'$ , such that, with probability at least  $1 - \delta$ ,
+
+-  $|\mathcal{S}'| = |\mathcal{S}|$ , and under some bijective function  $\sigma : \mathcal{S} \to \mathcal{S}'$ , it holds that
+
+$$
+\forall s \in \mathcal {S}, a \in \mathcal {A}: \sigma (T (s, a)) = T ^ {\prime} (\sigma (s), a), a n d, \tag {4}
+$$
+
+Under the same bijection  $\sigma$
+
+$$
+\forall s \in \mathcal {S}, \Pr_ {\substack {x \sim \mathcal {Q} (s, e), \\ e \sim \pi_ {\mathcal {E}}}} \left(\phi^ {\prime} (x) = \sigma \left(\phi^ {*} (x)\right)\right) \geq 1 - \epsilon , \tag{5}
+$$
+
+where  $\pi_{\varepsilon}$  is the stationary distribution of  $\mathcal{T}_e$
+
+Furthermore, the number of steps that STEEL executes on  $M$  scales as:
+
+$$
+\mathcal {O} ^ {*} \Big (N D | \mathcal {S} | ^ {2} | \mathcal {A} | \cdot \log \frac {| \mathcal {F} |}{\delta} + | \mathcal {S} | | \mathcal {A} | \hat {t} _ {m i x} \cdot \log \frac {N | \mathcal {F} |}{\delta} + \frac {| \mathcal {S} | ^ {2} D}{\epsilon} \cdot \log \frac {| \mathcal {F} |}{\delta} + \frac {| \mathcal {S} | \hat {t} _ {m i x}}{\epsilon} \cdot \log \frac {| \mathcal {F} |}{\delta} \Big),
+$$
+
+where  $\mathcal{O}^*(f(x)) \coloneqq \mathcal{O}(f(x) \log(f(x)))$ .
+
+![](images/66dfc98daf4cbcb3ae7172b7908ccbe35c1b32fc7a4fb032e3c088fd8fddc68f.jpg)  
+Figure 1: STEEL discovers the latent dynamics  $S$  and  $T$  by iteratively adding cycles to the learned dynamics graph. In this simple example, the initially-unknown "true" dynamics consist of 6 states arranged in a grid, where the agent can move (U)p, (D)own, (L)eft, or (R)ight. STEEL takes 12 iterations to discover the full dynamics: each pane corresponds to an iteration, and shows the still-unknown parts of the dynamics graph in grey, the already-known parts of the dynamics graph in black, and the cycle being explored in red. States are represented as circles and transitions as arrows.
+
+We now present a high-level overview of STEEL. The algorithm proceeds in three phases. In the first phase, the algorithm learns the transition dynamics; in the second phase, it collects additional samples of observations of each latent state in  $S$ ; in the final phase, the encoder  $\phi'$  is learned.
+
+STEEL Phase 1: Learning latent dynamics. In the first phase, STEEL constructs a learned latent state space  $S'$  and learned transition dynamics  $T'$  by iteratively adding cycles to the known transition graph. At each iteration, a sequence of actions  $\hat{a}$  is chosen such that, starting anywhere in the known  $T'$ , taking the actions in  $\hat{a}$  is guaranteed to traverse a transition not already in  $T'$ . (We explain the process of constructing  $\hat{a}$  below.)
+
+STEEL then takes the actions in  $\hat{a}$  repeatedly, collecting a sequence of observations  $x_{CF}$ . Because the transitions in  $T$  are deterministic, this sequence of transitions must eventually (after at most  $|\mathcal{S}||\hat{a}|$  steps) enter a cycle of latent states, of length  $n_{\mathrm{cyc}}|\hat{a}|$ , for some  $n_{\mathrm{cyc}} \leq |\mathcal{S}|$ . Because  $\hat{a}$  was chosen to always escape the known transitions in  $T'$ , this cycle cannot be contained in  $T'$ , so adding the states and transitions of the new cycle to  $\mathcal{S}'$  and  $T'$  is guaranteed to expand the known dynamics graph by at least one edge: this process will discover the full transition dynamics after at most  $|\mathcal{S}||\mathcal{A}|$  iterations. See Figure 1 for an example of how STEEL constructs  $\mathcal{S}'$  and  $T'$  by adding cycles to the dynamics. Throughout this process, STEEL also collects a dataset  $\mathcal{D}(s) \in \mathcal{M}(\mathcal{X})$  for each newly-discovered latent state  $s \in \mathcal{S}'$ , so that each observation in  $\mathcal{D}(s)$  has latent state  $s$ . The observations within each dataset  $\mathcal{D}(s)$  are collected at least  $\hat{t}_{\mathrm{mix}}$  steps apart from one another, so they are near-i.i.d.
+
+Constructing  $\hat{a}$ . To ensure that  $\hat{a}$  always escapes the known transition graph given by  $T'$ , at each iteration, STEEL uses a recurrent procedure to construct  $\hat{a}$ . At the beginning of this procedure,  $\hat{a}$  is initialized as empty, and a set of latent states  $\mathcal{B}$  is initialized with all of the learned states in  $S'$ . At each step of the procedure,  $\mathcal{B}$  represents the set of known states  $s \in S'$  that can be reached by starting at any arbitrary state  $s' \in S'$  and taking the action sequence given by the partially-constructed action list  $\hat{a}$ , following the known dynamics in  $T'$ . At each step, STEEL chooses a state  $s \in \mathcal{B}$ , and plans the shortest path from  $s$  to any unknown transition in  $T'$ . The corresponding sequence of actions,  $\hat{a}'$ , is then appended to  $\hat{a}$ , and  $\mathcal{B}$  is updated by replacing each state  $b \in \mathcal{B}$  with the state that results from starting at  $b$  and taking the sequence of actions  $\hat{a}'$ , according to the learned partial transition function  $T'$ . If this path reaches an unknown transition in  $T'$ , then  $b$  is removed from  $\mathcal{B}$  and not replaced. By construction, we know that the state  $s$  will be removed, so  $\mathcal{B}$  will shrink by at least one state at each step of the process. By the end,  $\mathcal{B}$  is empty, and we are guaranteed that taking action sequence  $\hat{a}$  from any state in  $S'$  will lead to an unknown transition in  $T'$ , as desired. Note that at each step, the shortest distance to an unknown transition has length at most  $D + 1$ , and the procedure continues for at most  $|S|$  steps, so  $|\hat{a}| \leq |S'| \cdot (D + 1)$ .
+
+Identifying latent states. At each iteration, to identify the distinct latent states in the cycle, STEEL uses a subroutine called CycleFind. CycleFind additionally collects the datasets  $\mathcal{D}(s)$  for each newly-discovered latent state. CycleFind itself has two phases:
+
+![](images/a1a613d3a8159743b771ba38eec1d80f5b33c1d22d8337119b3ceb7102328edc.jpg)  
+Figure 2: CycleFind determines the period of the cycle in  $x_{CF}$ . See Section 4 under "CycleFind Phase 1." We show the sequence  $x_{CF}$  sampled from  $M$ : specifically, we show every  $|\hat{a}|$ 'th observation, where the same actions  $\hat{a}$  are taken between each one. The observations' latent states are color-coded as red, blue, and green: a pattern repeats every  $3|\hat{a}|$  steps, so  $n_{cyc} = 3$ .  $D_1$  consists of the first observation in each  $(n_{cyc}'|\hat{a}|)$ -cycle, and  $D_0$  the other observations taken between executions of  $\hat{a}$ . (Spans of length  $\geq \hat{t}_{\mathrm{mix}}$  are skipped to ensure certain subsets of the datasets are near-i.i.d.)
+
+CycleFind Phase 1: Finding the cycle's periodicity. To identify the latent states in the cycle in  $x_{CF}$ , CycleFind first determines cycle's period,  $n_{\mathrm{cyc}}|\hat{a}|$ . To find  $n_{\mathrm{cyc}}$ , CycleFind tests all possible values of  $n_{\mathrm{cyc}}$  from  $N$  to 1, in decreasing order. To check whether some candidate value,  $n_{\mathrm{cyc}}'$ , is in fact  $n_{\mathrm{cyc}}$ , CycleFind constructs two datasets,  $D_0$  and  $D_1$  from  $x_{CF} = [x_1, \ldots]$ . These datasets are constructed so that if  $n_{\mathrm{cyc}}' = n_{\mathrm{cyc}}$ , then  $D_1$  contains observations of only one controllable latent state  $s$ , and  $D_0$  contains no observations of  $s$ . Therefore, if one attempts to train a classifier  $f$  to perfectly distinguish all observations in  $D_0$  from those in  $D_1$ , then such a classifier is unlikely to exist in  $\mathcal{F}$  if  $n_{\mathrm{cyc}}' > n_{\mathrm{cyc}}$ , but is guaranteed to exist if  $n_{\mathrm{cyc}}' = n_{\mathrm{cyc}}$ , by the realizability assumption. In this way, CycleFind uses the training oracle to determine  $n_{\mathrm{cyc}}$ .
+
+Specifically, the datasets used are (see Figure 2 for illustration.):
+
+$$
+D _ {0} := \left\{x _ {\bar {t} _ {\mathrm {m i x}} + \left(2 \bar {t} _ {\mathrm {m i x}} + n _ {\mathrm {c y c}} ^ {\prime} | \hat {a} |\right) i + j \cdot | \hat {a} | + \text {o f f s .}} \mid i \in \{0, \dots , k - 1 \}, j \in \{1, \dots , n _ {\mathrm {c y c}} ^ {\prime} - 1 \} \right\} \tag {6}
+$$
+
+$$
+D _ {1} := \left\{x _ {\left(2 \bar {t} _ {\text {m i x}} + n _ {\text {c y c}} ^ {\prime} \cdot | \hat {a} |\right) i + \text {o f f s .}} \mid i \in \{0, \dots , k - 1 \} \right\} \tag {7}
+$$
+
+where  $\bar{t}_{\mathrm{mix}} := \lceil \hat{t}_{\mathrm{mix}} / (n_{\mathrm{cyc}}' \cdot |\hat{a}|) \rceil \cdot n_{\mathrm{cyc}}' \cdot |\hat{a}|$  is  $\hat{t}_{\mathrm{mix}}$  rounded up to the nearest multiple of  $n_{\mathrm{cyc}}' \cdot |\hat{a}|$ ; the number of samples needed for  $D_1$  is  $k$  (which depends on  $\log(|F|)$ ,  $\log(\delta)$ , and other parameters); and offs. :=  $\max((N - 1) \cdot |\hat{a}|, \hat{t}_{\mathrm{mix}})$  is a constant offset to ensure that the Ex-BMDP endogenous state has entered the terminal cycle induced by  $\hat{a}$ , and that the exogenous state has mixed.
+
+To see why  $D_{1}$  and  $D_{0}$  are perfectly distinguishable only if  $n_{\mathrm{cyc}}^{\prime} = n_{\mathrm{cyc}}$ , consider the sequence of repeated latent states that compose the cycle,  $[s_0^{\mathrm{cyc}}, \ldots, s_{n_{\mathrm{cyc}} \cdot |\hat{a}| - 1}^{\mathrm{cyc}}]$ . If we only consider every  $|\hat{a}|$ 'th state in the cycle, then the resulting sequence,  $[s_0^{\mathrm{cyc}}, s_{|\hat{a}|}^{\mathrm{cyc}}, \ldots, s_{(n_{\mathrm{cyc}} - 1) \cdot |\hat{a}|}^{\mathrm{cyc}}]$ , cannot contain any repeated states. Therefore,
+
+$$
+\forall n, m, \quad n \equiv m \quad (\mathrm {m o d} n _ {\mathrm {c y c}}) \Leftrightarrow \phi^ {*} (x _ {| \hat {a} | n + \text {o f f s .}}) = \phi^ {*} (x _ {| \hat {a} | m + \text {o f f s .}}), \tag {8}
+$$
+
+and in particular, if  $n_{\mathrm{cyc}}' = n_{\mathrm{cyc}}$ , then  $n \equiv m$  (mod  $n_{\mathrm{cyc}}'$ )  $\Leftrightarrow \phi^{*}(x_{|\hat{a}|n + \mathrm{offs}.}) = \phi^{*}(x_{|\hat{a} |m + \mathrm{offs}.})$ . Then, through modular arithmetic, we can conclude that  $D_{1}$  contains observations of only one latent state of the cycle, while  $D_0$  contains observations of all of the other latent states.
+
+Meanwhile, if  $n_{\mathrm{cyc}}' > n_{\mathrm{cyc}}$ , then for each observation in  $D_1$ , there is a corresponding observation in  $D_0$  with the same latent state, that is nearly-identically and independently distributed (i.e., they are collected  $\geq \hat{t}_{\mathrm{mix}}$  steps apart). In particular, consider the (unknown) subset  $D_0^{(j')} \subseteq D_0$ , defined as:
+
+$$
+D _ {0} ^ {(j ^ {\prime})} := \left\{x _ {\bar {t} _ {\operatorname* {m i x}} + \left(2 \bar {t} _ {\operatorname* {m i x}} + n _ {\mathrm {c y c}} ^ {\prime} | \hat {a} |\right) i + j ^ {\prime} \cdot | \hat {a} | + \text {o f f s .} \right| i \in \{0, \dots , k - 1 \} \} \tag {9}
+$$
+
+where  $j^{\prime}:= (-\lceil \hat{t}_{\mathrm{mix}} / (n_{\mathrm{cyc}}^{\prime}\cdot |\hat{a}|)\rceil \cdot n_{\mathrm{cyc}}^{\prime} - 1)\% n_{\mathrm{cyc}} + 1.$  From arithmetic and applying Equation 8, we see that all observations in  $D_0^{(j')}$  and all observations in  $D_{1}$  have the same endogenous latent state. Additionally, all observations in  $D_0^{(j')}$  and  $D_{1}$  are collected at least  $\hat{t}_{\mathrm{mix}}$  steps apart.
+
+CycleFind Phase 2: Identifying latent states in the cycle. Once  $n_{\mathrm{cyc}}$  is known, CycleFind can identify the latent states which re-occur every  $n_{\mathrm{cyc}}|\hat{a}|$  steps in  $x_{CF}$ . These latent states are not necessarily distinct from each other, and may also have been discovered already in a previous iteration of CycleFind. Therefore, CycleFind extracts from  $x_{CF}$  datasets  $\mathcal{D}_i'$  for each position in the cycle:  $i \in \{0, \dots, n_{\mathrm{cyc}}|\hat{a}| - 1\}$ . CycleFind also uses datasets  $\mathcal{D}(s)$  collected in previous iterations representing the already-discovered states in  $S'$ . CycleFind determines whether two datasets (either collected in this call to CycleFind, or collected in previous calls) represent the same latent state by attempting to learn a classifier  $f \in \mathcal{F}$  that distinguishes them: if they both consist of near-i.i.d. samples of the same latent state, then it is highly unlikely that such a classifier exists.
+
+To ensure the near-i.i.d. property "well enough," we only need that samples are separated by  $\hat{t}_{\mathrm{mix}}$  steps within each individual  $\mathcal{D}_i^\prime$ ; and that, when trying to distinguish two datasets  $\mathcal{D}_i^\prime$ ,  $\mathcal{D}_j^\prime$  which were both collected during this round of CyceFind, there are two (sufficiently large) subsets of  $\mathcal{D}_i^\prime$  and  $\mathcal{D}_j^\prime$  respectively such that all samples in the two subsets are collected at least  $\hat{t}_{\mathrm{mix}}$  steps apart - ensuring this second condition only doubles the number of samples we must collect. Thus, we do not need to "wait"  $\hat{t}_{\mathrm{mix}}$  steps between collecting each usable sample from  $x_{CF}$ ; rather, we collect a usable sample for each latent state once for every roughly  $2\max (\hat{t}_{\mathrm{mix}},n_{\mathrm{cyc}}|\hat{a}|)$  steps. This is why  $\hat{t}_{\mathrm{mix}}$  does not appear in the largest term (in  $|S|$ ) of our asymptotic sample complexity.
+
+If it is determined that some  $\mathcal{D}_i$  represents a newly-discovered latent state, then a new state  $s^{\prime}$  is inserted into  $\mathcal{S}^{\prime}$  and  $\mathcal{D}'(s')$  is initialized as  $\mathcal{D}_i$ . Once all states in  $[s_0^{\mathrm{cyc}}, \dots, s_{n_{\mathrm{cyc}} \cdot |\hat{a}| - 1}^{\mathrm{cyc}}]$  have been identified, the action sequence  $\hat{a}$  can be used to add them to the learned transition dynamics  $T'$ .
+
+STEEL Phase 2: Collecting additional samples to train encoder. Once we have the complete latent dynamics graph, the determinism of the latent dynamics allows us to use open-loop planning to efficiently re-visit each latent state, in order to collect enough samples to learn a highly-accurate encoder. Note that we can navigate to any arbitrary latent state in  $D$  steps, so we can visit every latent state in  $|S|D$  steps. STEEL collects datasets  $\mathcal{D}(s)$  for each latent state  $s$  where, within each  $\mathcal{D}(s)$ , the samples are collected at least  $\hat{t}_{\mathrm{mix}}$  steps apart: therefore, it can add one sample to each dataset  $\mathcal{D}(s)$  at worst roughly every  $\max(|S|D, \hat{t}_{\mathrm{mix}})$  steps.
+
+STEEL Phase 3: Training the encoder. Finally, STEEL trains the encoder. Specifically, for each latent state  $s \in S'$ , it trains a binary classifier  $f_s \in \mathcal{F}$  to distinguish  $\mathcal{D}(s)$  from  $\uplus_{s' \in S' \setminus \{s\}} \mathcal{D}(s')$ . To ensure that only the correct binary classifier,  $f_{\sigma(\phi^*(x))}(x)$ , returns 1, we ensure that each  $f_s$  has an accuracy of  $1 - \epsilon / |S|$  on each latent state. We guarantee the accuracy of each classifier on each latent state separately and apply a union bound: note that because we use a union bound here, we do not need the samples in different datasets  $\mathcal{D}(s)$ ,  $\mathcal{D}(s')$  to be independent, which is why we are able to collect samples more frequently that every  $\hat{t}_{\mathrm{mix}}$  steps. Finally, we define  $\phi'(x) \coloneqq \arg \max_s f_s(x)$ .
+
+# 5 SIMULATION EXPERIMENTS
+
+We test the STEEL algorithm on two toy problems: an infinite-horizon environment inspired by the "combination lock" environment from Efroni et al. (2022b), and a version of the "multi-maze"
+
+![](images/0e3a56877a325793ef42d626043c7a8ea9510460be18ae3a9fa72a324e505df1.jpg)  
+Figure 3: Visualisations of the simulation experiment environments. For both environments, we show the ground-truth latent dynamics  $T$  (in the case of the combination lock, we show an arbitrary instance of  $T$ , for some  $[a_0^*, \dots a_{K-1}^*]$ ), and an example transition in the observed space  $\mathcal{X}$ .
+
+![](images/3260999414e4e58cb61781d1728dbb46215b109ad91facaafaaff9bfe575f65b.jpg)
+
+environment from Lamb et al. (2023). In our combination lock environment,  $\mathcal{A} = \{0,1\}$ ,  $\mathcal{S} = \{0,..,K - 1\}$ , and there is some sequence of "correct" actions  $[a_0^*,\dots a_{K - 1}^*]$ , such that  $T(i,a_i^*) = i + 1$ , but  $T(i,1 - a_i^*) = 0$ . In other words, in order to progress through the states, the agent must select the correct next action from the (arbitrary) sequence  $[a_0^*,\dots a_{K - 1}^*]$ ; otherwise, the latent state is reset to 0. The observation space  $\mathcal{X} = \{0,1\}^{L}$ , where  $L\gg K$ . Some arbitrary subset of size  $K$  of the components in  $\mathcal{X}$  are indicators for each latent state in  $\mathcal{S}$ : that is,  $\forall i\in S,\exists j\in \{0,\ldots ,L - 1\} : (x_t)_j = 1\leftrightarrow s_t = i$ . The other  $L - K$  components in  $\mathcal{X}$  are independent two-state Markov chains with states  $\{0,1\}$ , each with different arbitrary transition probabilities (bounded such that no transition probability for any of the two-state chains is less than 0.1). Because each component is time-correlated, they all must be contained in  $\mathcal{E}$ , so  $|\mathcal{E}| = 2^{L - K}$ . In the multi-maze environment, the agent learns to navigate a four-room maze (similar to the one in Sutton et al. (1999)) using actions  $\mathcal{A} = \{\mathrm{Up},\mathrm{Down},\mathrm{Left},\mathrm{Right}\}$ . The latent state space has size  $|S| = 68$ . However, the observation  $x\in \mathcal{X}$  in fact consists of nine copies of this maze, each containing a different apparent "agent." Eight of these "agents" move according to random actions: the true controllable agent is only present in one of the mazes. Because the eight distractor mazes can be in any configuration and have temporally-persistent state, we have that  $|\mathcal{E}| = 68^8$ . For both environments, we use the hypothesis class  $\mathcal{F} := \{(x\to (x)_i|i\in \{0,\dim (\mathcal{X}) - 1\}\}$ . In other words, the hypothesis class assumes that for each latent state  $s$ , there is some component of  $i$  of the observations such that  $\phi^{*}(x) = s$  if and only if  $(x)_i = 1$ . The two environments are visualized in Figure 3.
+
+There are four sources of potential variability in these simulation experiments: (1) the random elements of the environments' dynamics,  $\mathcal{T}_e$ ,  $\mathcal{Q}$ , and  $e_{\mathrm{init}}$ ; (2) the starting latent state  $s_{\mathrm{init}}$ ; (3) steps in Algorithm 1 that allow for arbitrary choices (e.g., the choice of action  $\hat{a} = [a]$  in the first invocation of CycleFind); and (4) the parameters of the environment, such as the "correct" action sequence  $[a_0^*, \dots a_{K-1}^*]$  in the combination lock. STEEL is designed in such a way that, with high probability (i.e., if the algorithm succeeds), no choice that the algorithm makes in terms of control flow or ac
+
+<table><tr><td></td><td>Combo. Lock (K = 20)</td><td>Combo. Lock (K = 30)</td><td>Combo. Lock (K = 40)</td><td>Multi-Maze</td></tr><tr><td>Fixed Env. Accuracy</td><td>20/20</td><td>20/20</td><td>20/20</td><td>20/20</td></tr><tr><td>Fixed Env. Steps</td><td>1886582±0</td><td>4286241±0</td><td>7914856±0</td><td>41003875±0</td></tr><tr><td>Variable Env. Accuracy</td><td>20/20</td><td>20/20</td><td>20/20</td><td>20/20</td></tr><tr><td>Variable Env. Steps</td><td>2.00·106±1.28·105</td><td>4.78·106±4.36·105</td><td>9.59·106±1.13·106</td><td>4.13·107±1.11·106</td></tr></table>
+
+Table 2: Success rate and number of steps taken for STEEL on both simulation environments. For all experiments, we set  $\delta = \epsilon = .05$ . For the combination lock experiments, we set  $L = 512$ , and use the (intentionally loose) upper bounds  $N = \hat{D} = K + 10$  ( $|S| + 10$ ) and  $\hat{t}_{\mathrm{mix}} = 40$ . For the multi-maze environment, we use  $N = \hat{D} = 80$  ( $|S| = 68$ ), and  $\hat{t}_{\mathrm{mix}} = 300$ . See Appendix D for how we chose the (loose) bounds  $\hat{t}_{\mathrm{mix}} \geq t_{\mathrm{mix}}$ .
+
+tions will depend on exogenous noise. Therefore, if we hold (2-4) constant, we expect the number of environment steps taken to be constant, regardless of the exogenous noise. To verify this, we test both environments for 20 simulations, in both a "fixed environment" setting with (2-4) held constant, and a "variable environment" setting with (2-4) set randomly. We test the combination lock environment with latent states  $K \in \{20,30,40\}$ . We measure the success rate in exactly learning  $\phi^{*}(x)$  and  $T$  (up to permutation) and the number of steps taken. Results are shown in Table 2.
+
+STEEL correctly learned the latent dynamics  $T$  and optimal encoder  $\phi^*$  in every simulation run; and we verify that the step counts do not depend on exogenous noise. In the combination lock experiments for large  $K$ , which are hard exploration problems, the total step counts were many orders of magnitude smaller than either the size of the observation space  $(\approx 10^{154})$ ; or the reciprocal-probability of a uniformly-random policy navigating from state 0 to state  $K - 1$  ( $\approx 10^{12}$  for  $K = 40$ ). This shows that STEEL is effective at learning latent dynamics for hard exploration problems under high-dimensional, time-correlated noise. For the multi-maze experiment (which is not a hard exploration task), STEEL took a few orders of magnitude greater steps than reported in Lamb et al. (2023) or Levine et al. (2024) for the same environment ( $\approx 10^3 - 10^4$  steps). However, unlike these prior methods, STEEL is guaranteed to discover the correct encoder with high probability; this requires the use of conservative bounds when defining sample counts  $d$ ,  $n_{\mathrm{samp,cyc}}$  and  $n_{\mathrm{samp}}$  in Algorithms 1 and 2, and in making other adversarial assumptions in the design of the algorithm that ensure that it is correct and sample-efficient even in pathological cases. Additionally, note that the encoder hypothesis class  $\mathcal{F}$  used in this experiment has no spatial priors. By contrast, Lamb et al. (2023) choose a neural-network encoder for this environment with strong spatial priors that favor focusing attention on a single maze, using sparsely-gated patch encodings (and Levine et al. (2024) use this same network architecture in order to compare to Lamb et al. (2023)) – this difference in priors over representations may also account for some of the gap in apparent sample efficiency.
+
+In Appendix F, we present an additional set of experiments on a family of tabular Ex-BMDPs which are known to be particularly challenging to "multistep inverse" methods, such as those proposed by Lamb et al. (2023) and Levine et al. (2024). We find that, for sufficiently large instances of these environments, STEEL can in fact empirically outperform these prior "practical" methods.
+
+# 6 LIMITATIONS AND CONCLUSION
+
+A major limitation of STEEL that may constrain its real-world applicability is its strict determinism assumption on  $T$ . In the episodic setting, Efroni et al. (2022b) can get away with allowing rare deviations from deterministic latent transitions (no more often on average than once every  $4|S|$  episodes) because the environment resets "erase" these deviations before they can propagate for too long. By contrast, in the single-trajectory setting, the STEEL algorithm is fragile to even rare deviations in latent dynamics; ameliorating this issue may require significant changes to the algorithm.
+
+A second barrier to practical applicability of STEEL is the need for an optimal training oracle for  $\mathcal{F}$ . While this is tractable for, e.g., linear models (with the realizability assumption ensuring linear separability), it becomes computationally intractable for anything much more complicated. However, this kind of assumption is common in sample-complexity results; and could be worked around in adapting STEEL to practical settings.[7]
+
+Two additional limitations to our work are the assumption of reachability of all endogenous latent states  $s \in S$ , and the requirement that an upper-bound on the mixing time of the exogenous noise be known a priori. However, in Appendix E, we argue that these assumptions are in fact necessary, for any algorithm in the single-trajectory, no-resets Ex-BMDP setting.
+
+Finally, the core assumption that  $S$  is finite and small is of course a major limitation: sample-efficient reinforcement learning in combinatorial and continuous state spaces is a broad area of ongoing and future work. Despite these limitations, STEEL represents what we hope is an important contribution to representation learning in scenarios where resetting the environment during training is not possible, and observations are impacted by high-dimensional, time-correlated noise.
+
+# ACKNOWLEDGEMENTS
+
+A portion of this research has taken place in the Learning Agents Research Group (LARG) at the Artificial Intelligence Laboratory, The University of Texas at Austin. LARG research is supported in part by the National Science Foundation (FAIN-2019844, NRT-2125858), the Office of Naval Research (N00014-18-2243), Army Research Office (W911NF-23-2-0004, W911NF-17-2-0181), Lockheed Martin, and Good Systems, a research grand challenge at the University of Texas at Austin. The views and conclusions contained in this document are those of the authors alone. Peter Stone serves as the Executive Director of Sony AI America and receives financial compensation for this work. The terms of this arrangement have been reviewed and approved by the University of Texas at Austin in accordance with its policy on objectivity in research. Alexander Levine is supported by the NSF Institute for Foundations of Machine Learning (FAIN-2019844). Amy Zhang and Alexander Levine are supported by National Science Foundation (2340651) and Army Research Office (W911NF-24-1-0193).
+
+# REFERENCES
+
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+Christoph Dann, Nan Jiang, Akshay Krishnamurthy, Alekh Agarwal, John Langford, and Robert E Schapire. On oracle-efficient pacrl with rich observations. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett (eds.), Advances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018.  
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+
+# A FULL ALGORITHM
+
+The STEEL algorithm is presented here in full as Algorithm 1, with a major subroutine, CycleFind, split out as Algorithm 2.
+
+Algorithm 1: STEEL  
+Input: Access to Ex-BMDP  $M$  , access to training oracle for function class  $\mathcal{F}$  , knowledge of upper bounds  $N,\hat{D},\hat{t}_{\mathrm{mix}}$  parameters  $\delta ,\epsilon$    
+Initialize learned latent state set  $S^{\prime}$  , initially empty;   
+Initialize table of collected datasets for each latent state:  $\mathcal{D}:S^{\prime}\to \mathcal{M}(\mathcal{X})$  .   
+Initialize learned latent dynamics:  $T^{\prime}:(S^{\prime}\cup \{\bot \})\times A\rightarrow (S^{\prime}\cup \{\bot \})$  . (When a state  $s$  is added to  $S^{\prime}$  we initially set  $\forall a\in \mathcal{A}$ $T(s,a)\coloneqq \bot$  . Also, we set  $\forall a\in \mathcal{A}$ $T(\bot ,a)\coloneqq \bot$  as a permanent definition.); // Phase 1: Discover latent dynamics  $T$  -   
+Chose arbitrary  $a\in \mathcal{A}$  ..   
+ $S^{\prime},D,T^{\prime},s_{\mathrm{corr.}}\gets$  CycleFind([a],  $S^{\prime},D,T^{\prime}$  );// Special case for first iteration   
+while  $\exists s\in S^{\prime},a\in \mathcal{A}:T^{\prime}(s,a)\coloneqq \bot$  do Initialize  $\mathcal{B}\leftarrow S^{\prime}$  , and initialize action list  $\hat{a}\gets []$  while  $\mathcal{B}$  non-empty do Chose arbitrary  $s\in \mathcal{B}$  ..  $\mathcal{B}\gets \mathcal{B}\setminus \{s\}$  . Let  $\hat{a}^\prime :=$  a minimum-length sequence of actions such that  $T^{\prime}(T^{\prime}(.T^{\prime}(.T^{\prime}(s,\hat{a}_0^{\prime}),\hat{a}_1^{\prime}),\hat{a}_2^{\prime}),\dots,\hat{a}_{|\hat{a}^{\prime}| - 1}^{\prime}) = \bot$  . (This can be found using Dijkstra's algorithm.);  $\hat{a}\gets \hat{a}\cdot \hat{a}^{\prime}$  ..  $\mathcal{B}\gets \{s''\in S'|\exists s'\in \mathcal{B}:T'(T'(..T'(s',\hat{a}_0'),\hat{a}_1'),\hat{a}_2'),\dots,\hat{a}_{|\hat{a} '| - 1}^{\prime}) = s''\} ;$ $S^{\prime},D,T^{\prime},s_{\mathrm{corr.}}\gets$  CycleFind(  $\hat{a},S^{\prime},D,T^{\prime}$  . // Phase 2: Collect additional latent samples to train encoder.   
+Let  $d:= [\frac{3|S'| \ln(16|S'|^{2}|F| / \delta)}{\epsilon} ]$    
+while  $\exists s\in S':|\mathcal{D}(s)| <   d$  do Let  $\mathcal{C}:= \{s\in S'||\mathcal{D}(s)| <   d\land s\neq s_{\mathrm{corr.}}\}$  Use  $T^{\prime}$  to plan a cycle of actions  $\bar{a}$  starting at  $s_{\mathrm{corr.}}$  that visits all states C and then returns to  $s_{\mathrm{corr.}}$  by greedily applying Dijkstra's algorithm repeatedly; If  $|\bar{a} | <   t_{\mathrm{mix}}$  , use  $T^{\prime}$  to extend  $\bar{a}$  by repeatedly inserting the shortest-length self-loop of some state visited in  $\bar{a}$  into  $\bar{a}$  until  $|\bar{a} |\geq t_{\mathrm{mix}}$  ; Execute all actions in  $\bar{a}$  once without collecting data; while  $\forall s\in \mathcal{C}:|\mathcal{D}(s)| <   d$  do for a in  $\bar{a}_0,\ldots ,\bar{a}_{|\bar{a} | - 1}$  do Take action a on M;  $s_{\mathrm{corr.}}\gets T'(s_{\mathrm{corr.}},a)$  if  $s_{\mathrm{corr.}}$  is being visited for the first time in this execution through a then Let  $x_{\mathrm{corr.}}:=$  the observed state of M;  $\mathcal{D}(s_{\mathrm{corr.}})\gets \mathcal{D}(s_{\mathrm{corr.}})\uplus \{x_{\mathrm{corr.}}\} ;$  // Phase 3: Train latent state encoder  $\phi '$    
+for  $s\in S'$  do Let  $D_0:= \oplus_{s'\in S'\backslash \{s\}}D(s')$  ..  $D_1:= D(s)$  Apply training oracle to distinguish  $D_0$  from  $D_{1}$  yielding  $f_{s}\in F$  return  $S^{\prime},T^{\prime}$  ,and  $\phi '(x):= arg\max_s f_s(x)$
+
+# B PROOFS
+
+# B.1 STEEL
+
+Here, we explain the STEEL algorithm (Algorithm 1), and prove the correctness of Theorem 1. STEEL proceeds in three phases: in the first phase, we learn a tabular representation of the endogenous latent states  $\mathcal{S}$  and associated dynamics  $T$  of the Ex-BMDP. For each  $s \in S$ , we also begin to collect a dataset  $\mathcal{D}(s)$ , where for each  $x \in \mathcal{D}(s)$ , we have that  $\phi^{*}(x) = s$ , and additionally where all samples in  $\mathcal{D} := \bigcup_{s \in S} \mathcal{D}(s)$  were collected from the Ex-BMDP  $M$  at least  $\hat{t}_{\mathrm{mix}}$  time steps apart.
+
+Algorithm 2: CycleFind Subroutine  
+Input: Action list  $\hat{a}$ ; current learned state set  $S'$ , datasets  $\mathcal{D}$ , and transition dynamics  $T'$ . Also, access to Ex-BMDP  $M$ , access to training oracle for function class  $\mathcal{F}$ , knowledge of upper bounds  $N$ ,  $\hat{t}_{\mathrm{mix}}$ , and  $\hat{D}$ , and parameters  $\delta$ ,  $\epsilon$ .  
+// Phase 1: find length of cycle,  $n_{\mathrm{cyc}} \cdot |\hat{a}|$ .  
+Let  $n_{\mathrm{samp.cyc.}} := \left[\ln \left(\frac{\delta}{4|\mathcal{A}| \cdot N \cdot (N-1) \cdot |\mathcal{F}|}\right) / \ln \left(\frac{9}{16}\right)\right]$ ;  
+Let  $c_{\mathrm{init}} := (2\hat{t}_{\mathrm{mix}} + 3N \cdot |\hat{a}| - 2) \cdot n_{\mathrm{samp.cyc.}} - \hat{t}_{\mathrm{mix}} - N \cdot |\hat{a}| + 1 + \max((N-1) \cdot |\hat{a}|, \hat{t}_{\mathrm{mix}})$ ;  
+Collect a sequence of observation  $x_{CF} := [x_1, \ldots x_{c_{\mathrm{init}}}]$  from  $M$  by taking the actions in  $\hat{a}$  repeatedly in a loop, for a total of  $c_{\mathrm{init}}$  actions. (Action  $\hat{a}_i \% |\hat{a}|$  is taken after observing  $x_i$ );  
+Let  $\bar{x}_i := x_i \cdot |\hat{a}| + \max((N-1) \cdot |\hat{a}|, \hat{t}_{\mathrm{mix}})$ ;  
+Initialize  $n_{\mathrm{cyc}} \gets 1$ ; // Default value if no other  $n_{\mathrm{cyc}}'$  is  $n_{\mathrm{cyc}}$   
+for  $n_{\mathrm{cyc}}' \in [N, N-1, ..., 3, 2]$  do  
+    Let  $q := [\hat{t}_{\mathrm{mix}} / (n_{\mathrm{cyc}}' \cdot |\hat{a}|)]$ ,  $r := q \cdot n_{\mathrm{cyc}}'$ ,  $k := \lfloor \frac{c_{\mathrm{init}} + r \cdot |\hat{a}| - \max((N-1) \cdot |\hat{a}|, \hat{t}_{\mathrm{mix}})}{2r \cdot |\hat{a}| + n_{\mathrm{cyc}}' | \hat{a}|} \rfloor$ ;  
+    Let  $D_0 := \{\bar{x}_{r+(2r+n_{\mathrm{cyc}}')i+j} | i \in \{0, \ldots, k-1\}, j \in \{1, \ldots, n_{\mathrm{cyc}}' - 1\}\}$ ;  
+    Let  $D_1 := \{\bar{x}_{(2r+n_{\mathrm{cyc}}')i} | i \in \{0, \ldots, k-1\}\}$ ;  
+    Apply training oracle to distinguish  $D_0$  from  $D_1$ , yielding  $f \in \mathcal{F}$ ;  
+    if ( $\forall x \in D_0, f(x) = 0$  and  $\forall x \in D_0, f(x) = 1$ ) then  
+         $n_{\mathrm{cyc}} \gets n_{\mathrm{cyc}}'$ ;  
+    break;  
+// Phase 2: Assemble datasets for observations from cycle, identify new latent states, and update  $S'$ ,  $D$ , and  $T'$ .  
+Let  $n_{\mathrm{samp.}} := \left[\ln \left(\frac{\delta}{4|\mathcal{A}| \cdot N^4 (\hat{D} + 1) \cdot |\mathcal{F}|}\right) / \ln \left(\frac{9}{16}\right)\right]$ ;  
+Let  $c := 2 \cdot n_{\mathrm{cyc}} \cdot |\hat{a}| \cdot ((n_{\mathrm{samp.}} - 1) \cdot \left[\frac{\hat{t}_{\mathrm{mix}}}{|\hat{a}| \cdot n_{\mathrm{cyc}}} + 1\right) + \hat{t}_{\mathrm{mix}} + \max((N - n_{\mathrm{cyc}}) \cdot |\hat{a}|, \hat{t}_{\mathrm{mix}})$ ;  
+Extend the sequence of observation  $x_{CF}$  to length at least  $c$  by taking the actions  $\hat{a}$  repeatedly in a loop on  $M$ , for max(0, c - cinit) additional steps, so that  $x_{CF} = [x_1, \ldots x_c]$ ;  
+Let  $n_0 := \max((N - n_{\mathrm{cyc}}) \cdot |\hat{a}|, \hat{t}_{\mathrm{mix}})$ ,  $n_0' := n_0 + (n_{\mathrm{samp.}} - 1) \cdot |\hat{a}| \cdot n_{\mathrm{cyc}} \cdot \left[\frac{\hat{t}_{\mathrm{mix}}}{|\hat{a}| \cdot n_{\mathrm{cyc}}}\right] + |\hat{a}| \cdot n_{\mathrm{cyc}} + \hat{t}_{\mathrm{mix}}$ $\forall i \in \{0, \ldots, n_{\mathrm{cyc}} \cdot |\hat{a}| - 1\}$ , Let:  
+\[ D_i' = \left\{x_j | x_k \in \{0, \ldots, n_{\mathrm{samp.}} - 1\}, \exists offset \in \{n_0, n_0'\} : j = k \cdot |\hat{a}| \cdot n_{\mathrm{cyc}} \cdot \left[\frac{\hat{t}_{\mathrm{mix}}}{|\hat{a}| \cdot n_{\mathrm{cyc}}} + offset + (i - offset)\% (n_{\mathrm{cyc}} \cdot |\hat{a}|)\right]; j = k \cdot |\hat{a}| \cdot n_{\mathrm{cyc}} \cdot s_i' < s_i; (Optionally; and only if  $D(s)$  was not initialized or modified already during this call to CycleFind.)  
+\[ D(s) = D(s) + D_i'; (StateAlreadyFound? \gets True; break; if not StateAlreadyFound? then Insert new state  $s'$  into  $S'$ ;  
+\[ D(s') = D_i'; s_i' < s_i; (s_i' < s_i; s_i' < s_i; s_i' < s_i; s_i' < s_i; s_i' < s_i; s_i' < s_i; s_i' < s_i; s_i' < s_i; s_i' < s_i; s_i' < s_i; s_i' < s_i; s_i' < s_i; s_i' < s_i; s_i' < s_i; s_i' < s_i; s_i', t', s_i';
+
+Then, in the second phase, we use the learned dynamics model  $T'$  to efficiently collect additional samples for each learned latent state  $s \in S'$  and add them to  $\mathcal{D}(s)$ , until  $|\mathcal{D}(s)| \geq d$ , where:
+
+$$
+d := \lceil \frac {3 | \mathcal {S} ^ {\prime} | \ln (1 6 | \mathcal {S} ^ {\prime} | ^ {2} | \mathcal {F} | / \delta)}{\epsilon} \rceil . \tag {10}
+$$
+
+Finally, in the third phase, we use  $\mathcal{D}$  to learn an encoder  $\phi^{\prime}$ , which approximates  $\phi^{*}$  with high probability when the exogenous state  $e$  of the Ex-BMDP is sampled from its stationary distribution  $\pi_{e}$ .
+
+STEEL relies on the CycleFind subroutine, which is described in detail and proven correct in Section B.1.1. This subroutine is given a list of actions  $\hat{a}$  and the previously-learned states  $S^{\prime}$ , datasets  $\mathcal{D}$ , and dynamics  $T^{\prime}$ . It identifies and collects samples of all latent states in some state cycle which is traversed by taking the actions  $\hat{a}$  repeatedly, and also identifies the latent state transitions in this cycle.
+
+# We restate Theorem 1 here:
+
+Theorem 1. For an  $Ex-BMDP M = \langle \mathcal{X}, \mathcal{A}, \mathcal{S}, \mathcal{E}, \mathcal{Q}, T, \mathcal{T}_e, \pi_{\mathcal{E}}^{init} \rangle$  starting at an arbitrary endogenous latent state  $s_{init} \in S$ , with  $|\mathcal{S}| \leq N$ , where the exogenous Markov chain  $\mathcal{T}_e$  has mixing time at most  $\hat{t}_{mix}$ , and where all states in  $\mathcal{S}$  are reachable from one another in at most  $\hat{D}$  steps; and corresponding encoder function class  $\mathcal{F}$  such that Equation 3 holds, the algorithm STEEL $(M, \mathcal{F}, N, \hat{D}, \hat{t}_{mix}, \delta, \epsilon)$  will output a learned endogenous state space  $\mathcal{S}'$ , transition model  $T'$ , and encoder  $\phi'$ , such that, with probability at least  $1 - \delta$ ,
+
+-  $|\mathcal{S}'| = |\mathcal{S}|$ , and under some bijective function  $\sigma : \mathcal{S} \to \mathcal{S}'$ , it holds that
+
+$$
+\forall s \in \mathcal {S}, a \in \mathcal {A}: \sigma (T (s, a)) = T ^ {\prime} (\sigma (s), a), a n d, \tag {4}
+$$
+
+Under the same bijection  $\sigma$
+
+$$
+\forall s \in \mathcal {S}, \Pr_ {\substack {x \sim \mathcal {Q} (s, e), \\ e \sim \pi_ {\mathcal {E}}}} (\phi^ {\prime} (x) = \sigma (\phi^ {*} (x))) \geq 1 - \epsilon , \tag{5}
+$$
+
+where  $\pi_{\mathcal{E}}$  is the stationary distribution of  $\mathcal{T}_e$ .
+
+Furthermore, the number of steps that STEEL executes on  $M$  scales as:
+
+$$
+\mathcal {O} ^ {*} \Big (N D | \mathcal {S} | ^ {2} | \mathcal {A} | \cdot \log \frac {| \mathcal {F} |}{\delta} + | \mathcal {S} | | \mathcal {A} | \hat {t} _ {m i x} \cdot \log \frac {N | \mathcal {F} |}{\delta} + \frac {| \mathcal {S} | ^ {2} D}{\epsilon} \cdot \log \frac {| \mathcal {F} |}{\delta} + \frac {| \mathcal {S} | \hat {t} _ {m i x}}{\epsilon} \cdot \log \frac {| \mathcal {F} |}{\delta} \Big),
+$$
+
+where  $\mathcal{O}^*(f(x)) \coloneqq \mathcal{O}(f(x) \log(f(x)))$ .
+
+For the sake of our proof, we will treat the ground-truth properties of the Ex-BMDP, such as the latent dynamics  $T$ , as (unknown, arbitrary) fixed quantities, not as random variables. Similarly, we will treat the initial latent state  $s_{init}$  as an arbitrary but fixed quantity, rather than a random variable. Furthermore, in the proof, we will treat decisions that are specified (implicitly or explicitly) as "arbitrary" in Algorithms 1 and 2 (such as the choice of the action  $\hat{a} = [a]$  in the first invocation of CycleFind, or the choice of "shortest" paths in cases of ties when Dijkstra's algorithm is used) as being made deterministically, such as by a pseudorandom process - crucially, we require that these choices are made in a way that does not depend on the observations of the Ex-BMDP.
+
+This leaves the exogenous noise transitions  $\mathcal{T}_e$ , the emission function  $\mathcal{Q}$ , and the initial exogenous latent state  $e_{init}$  as the only sources of randomness in the algorithm. We note that the sample space over these three processes together as  $\Omega$ . Throughout the algorithm, we will ensure that decisions such as control flow choices, choices of actions, and choices of how to assemble datasets, are made deterministically, independently of  $\Omega$ , with high probability. That is, unless the algorithm fails, these decisions will be fully determined by  $s_0$ ,  $T$ , and algorithm parameters. While whether or not the algorithm fails will depend (solely) on  $\Omega$ , we will ultimately bound the total probability of failure as less than  $\delta$  by union bound, so that at each step in the proof, we can treat the algorithm's choices as statistically independent of  $\Omega$ .
+
+STEEL begins by repeatedly applying the CycleFind subroutine. CycleFind identifies a cycle in the latent dynamics  $T$  of the Ex-BMDP, and collects observations of the states in that cycle. In Phase 1
+
+of STEEL, throughout these application of CycleFind, the algorithm maintains a representation of the learned Ex-BMDP "so far", in the form of an incomplete set of learned states  $S'$ , learned latent dynamics  $T': (S' \cup \{\bot\}) \times \mathcal{A} \rightarrow (S' \cup \{\bot\})$ , and a table of datasets corresponding to each latent state  $\mathcal{D}: S' \rightarrow \mathcal{M}(\mathcal{X})$ .
+
+We first describe the CycleFind subroutine in detail:
+
+# B.1.1 CYCLEFIND SUBROUTINE
+
+Here, we describe CycleFind, and prove its correctness. CycleFind accepts as input a list of actions  $\hat{a} := [\hat{a}_0, \dots, \hat{a}_{|\hat{a}| - 1}]$ , and the Ex-BMDP  $M$  starting at an arbitrary state  $x_0$ . CycleFind also takes the representation of the learned Ex-BMDP "so far", in the form  $\mathcal{S}'$ ,  $\mathcal{D}$  and  $T'$ . We assume that, for each  $s \in S'$  all of the previously-observed observations in  $\mathcal{D}(s)$  were collected with gaps of at least  $\hat{t}_{\mathrm{mix}}$  steps between them. Also, we assume that  $\forall s \in S'$ ,  $|\mathcal{D}(s)| \geq n_{\mathrm{samp}}$ , where, in terms of the upper bounds  $N \geq |S|$  and  $\hat{t}_{\mathrm{mix}} \geq t_{\mathrm{mix}}$ , and total failure probability  $\delta$ ,
+
+$$
+n _ {\text {s a m p .}} := \left[ \ln \left(\frac {\delta}{4 | \mathcal {A} | \cdot N ^ {4} \cdot (\hat {D} + 1) \cdot | \mathcal {F} |}\right) / \ln \left(\frac {9}{1 6}\right) \right]. \tag {11}
+$$
+
+CycleFind first proceeds to take the actions  $\hat{a}_0, \dots, \hat{a}_{|\hat{a}| - 1}$  repeatedly in a loop. CycleFind then uses this collected sequence of observations (which may need to be extended by cycling through  $\hat{a}$  for additional iterations) to learn new states and update  $\mathcal{S}'$ ,  $\mathcal{D}$ , and  $T'$ .
+
+We will show that sequence of latent states visited by CycleFind is eventually periodic; that is, it guaranteed to eventually get stuck in a cycle of latent states, with a period in the form  $n_{\mathrm{cyc}} \cdot |\hat{a}|$ , for some  $n_{\mathrm{cyc}} \leq N$ . The goal of CycleFind is to:
+
+1. Identify the period of this cycle. (That is, determine  $n_{\mathrm{cyc}}$ .)  
+2. Use this period to extract from the sequence of observed states some new multiset of observations  $\mathcal{D}_i' \in \mathcal{M}(\mathcal{X})$  for  $i \in \{0, \dots, n_{\mathrm{cyc}} \cdot |\hat{a}| - 1\}$ , which each contain only one unique latent state, corresponding to the position  $i$  in the cycle. These multiset will only contain observations collected at least  $\hat{t}_{\mathrm{mix}}$  timesteps apart, so will be close-to-i.i.d. samples. (Depending on  $n_{\mathrm{cyc}}$ , we may need to perform additional cycles of data collection at this step.)  
+3. Identify which of these multisets  $\mathcal{D}_i^\prime$  have the same latent states that have been previously identified in  $S^{\prime}$ , and which have a new latent state, and determine among the new multisets which ones have the same latent states to each other, and which are distinct. This allows us to update  $\mathcal{S}'$  and  $\mathcal{D}$  with the new samples from  $\mathcal{D}_i^\prime$ , while maintaining the property that  $\forall s \in S', \forall x, x' \in \mathcal{D}(s)$ ,  $\phi^*(x) = \phi^*(x')$ , and  $\forall s, s' \in S', \forall x \in \mathcal{D}(s)$ ,  $x' \in \mathcal{D}(s')$ ,  $\phi^*(x) \neq \phi^*(x')$ . We also update the learned transitions  $T'$ .  
+4. Return the updated learned state set  $S'$ , datasets  $\mathcal{D}$ , transition function  $T'$ , and the current latent state of  $M$ ,  $s_{\mathrm{curr.}} \in S'$ .
+
+Specifically, CycleFind has the following property:
+
+Proposition 1. For any action sequence  $\hat{a}$  of length at most  $(D + 1)N$ , there exists at least one sequence of ground-truth states in  $\mathcal{S}$ ,  $[s_0^{cyc*}, s_1^{cyc*}, \ldots, s_{|\hat{a}| \cdot n_{cyc} - 1}^{cyc*}]$ , for some  $n_{cyc} \leq N$ , such that  $\forall i \in \{0, 1, \ldots, |\hat{a}| \cdot n_{cyc} - 1\}$ ,  $T(s_i^{cyc*}, \hat{a}_{i\%|\hat{a}|}) = s_{(i+1)\%(|\hat{a}| \cdot n_{cyc})}^{cyc*}$ . Given a sequence of actions  $\hat{a}$ , learned partial state set  $\mathcal{S}'$ , transition dynamics  $T'$ , and datasets  $\mathcal{D}$  which meet the following inductive assumptions:
+
+- There exists an injective mapping  $\sigma^{-1}: S' \to S$  such that
+
+$$
+\forall s \in \mathcal {S} ^ {\prime}, a \in \mathcal {A}, T ^ {\prime} (s, a) = \bot \vee \sigma^ {- 1} \left(T ^ {\prime} (s, a)\right) = T \left(\sigma^ {- 1} (s), a\right) \tag {12}
+$$
+
+and additionally,
+
+$$
+\forall s \in \mathcal {S} ^ {\prime}, \forall x \in \mathcal {D} (s), \phi^ {*} (x) = \sigma^ {- 1} (s). \tag {13}
+$$
+
+-  $\forall s \in S'$ ,  $|\mathcal{D}(s)| \geq n_{\text{samp.}}$ ; and for each  $s$ , the samples in  $\mathcal{D}(s)$  were all sampled from  $M$  at least  $\hat{t}_{\text{mix}}$  steps apart. Additionally, the choice to add any sample  $x$  to  $\mathcal{D}(s)$  was made
+
+fully deterministically (as a function of  $T$ ,  $s_{init}$ , the timestep  $t$  at which  $x$  was collected, and algorithm parameters), and independently of the random processes captured by  $\Omega$ .
+
+- The choice of action sequence  $\hat{a}$  is similarly fully deterministic and independent of  $\Omega$ .
+
+then, with probability at least:
+
+$$
+1 - \frac {\delta}{2 \cdot | \mathcal {A} | \cdot N} \tag {14}
+$$
+
+CycleFind will return updated  $S'$ ,  $D$ ,  $T'$ , and  $s_{\text{curr}}$  which meet the same inductive assumptions, and for which additionally:
+
+- The image of the updated  $\mathcal{S}_{(new)}'$ ,  $\sigma^{-1}(\mathcal{S}_{(new)}')$  is a (non-strict) superset of  $\sigma^{-1}(\mathcal{S}')$ , which additionally includes all unique states in some  $[s_0^{cyc*}, s_1^{cyc*}, \dots, s_{|\hat{a}| \cdot n_{cyc} - 1}^{cyc*}]$ .  
+- The transition matrix  $T_{(new)}'$  is a (non-strict) superset of the old transition matrix  $T'$  (in the sense that its domain is now  $S_{(new)}' \supseteq S_{(old)}'$ , and if  $T_{(old)}'(s, a) \neq \bot$  then  $T_{(new)}'(s, a) \neq \bot$ ), and  $T_{(new)}'$  additionally includes the transitions corresponding to the cycle; that is:
+
+$$
+\begin{array}{l} \forall i \in \{0, \dots, | \hat {a} | \cdot n _ {c y c} - 1 \}, \exists s, s ^ {\prime} \in \mathcal {S} ^ {\prime}: \sigma^ {- 1} (s) = s _ {i} ^ {c y c *} \wedge \sigma^ {- 1} (s ^ {\prime}) = s _ {(i + 1) \% (\hat {a} | \cdot n _ {c y c})} ^ {c y c *} \\ T _ {(n e w)} ^ {\prime} \left(s, \hat {a} _ {i \% | \hat {a} _ {|}}\right) = s ^ {\prime}. \tag{15} \\ \end{array}
+$$
+
+- The final observation  $x$  sampled by CycleFind from  $M$  is such that  $\sigma^{-1}(s_{\text{curr.}}) = \phi^{*}(x)$ .
+
+Additionally, CycleFind will take at most:
+
+$$
+\begin{array}{l} \max  \left(\left(2 \hat {t} _ {\text {m i x}} + 3 N \cdot | \hat {a} | - 2\right) \cdot n _ {\text {s a m p . c y c .}} - N \cdot | \hat {a} |, 2 \cdot \left(\hat {t} _ {\text {m i x}} + | \mathcal {S} | \cdot | \hat {a} | - 1\right) \cdot n _ {\text {s a m p .}} + 1\right) \tag {16} \\ + \max  (N \cdot | \hat {a} | - | \hat {a} | - \hat {t} _ {m i x}, 0) + 1 a c t i o n s, \\ \end{array}
+$$
+
+where  $n_{\text{samp.}}$  is defined in Equation 11 and  $n_{\text{samp. cyc.}}$  is defined in Equation 18.
+
+# Proof. Determining  $n_{\mathrm{cyc}}$ :
+
+CycleFind initially takes  $c_{\mathrm{init}}$  actions, where, in terms of the upper bounds  $N \geq |S|$  and  $\hat{t}_{\mathrm{mix}} \geq t_{\mathrm{mix}}$ ,
+
+$$
+c _ {\text {i n i t}} := \left(2 \hat {t} _ {\text {m i x}} + 3 N \cdot | \hat {a} | - 2\right) \cdot n _ {\text {s a m p . c y c .}} - \hat {t} _ {\text {m i x}} - N \cdot | \hat {a} | + 1 + \max  \left(\left(N - 1\right) \cdot | \hat {a} |, \hat {t} _ {\text {m i x}}\right), \tag {17}
+$$
+
+where
+
+$$
+n _ {\text {s a m p . c y c .}} := \left[ \ln \left(\frac {\delta}{4 | \mathcal {A} | \cdot N \cdot (N - 1) \cdot | \mathcal {F} |}\right) / \ln \left(\frac {9}{1 6}\right) \right]. \tag {18}
+$$
+
+CycleFind first takes action  $\hat{a}_0$ , then  $\hat{a}_1$ , then  $\hat{a}_2$ , etc, until taking action  $\hat{a}_{|\hat{a}| - 1}$ , at which point it repeats the process starting at  $\hat{a}_0$ , for a total of  $c_{\mathrm{init}}$  steps. The observation after each of these actions is recorded as  $x_{CF} \coloneqq [x_1, \dots, x_{c_{\mathrm{init}}}]$ . Let  $s_{CF} \coloneqq [s_1, \dots, s_{c_{\mathrm{init}}}]$  be the (initially unknown) latent states corresponding to these observations; that is,  $\phi^*(x)$  for each  $x$  in  $x_{CF}$ . (For indexing purposes,  $x_0$  and  $s_0$  will refer to the observation and latent state, respectively, of the Ex-BMDP before the first action was taken by CycleFind. However, these will not be used by the algorithm.)
+
+First, we show that  $s_{CF}$  must in fact end in a cycle of period  $n_{\mathrm{cyc}} \cdot |\hat{a}|$ , for some  $n_{\mathrm{cyc}} \leq N$ . Let  $s_{per}$  consist of every  $|\hat{a}|$ 'th element in  $s_{CF}$  starting at an offset  $m := \max(0, \hat{t}_{\mathrm{mix}} - (N-1) \cdot |\hat{a}|)$ ; that is,  $s_{per.} := [s_m, s_{m+|\hat{a}|}, s_{m+2|\hat{a}|}, \dots, s_{m+\lfloor(c_{\mathrm{init}}-m)/|\hat{a}|\rfloor|\hat{a}|}]$ . Note that the evolution from one state to the next in  $s_{per.}$  is deterministic, because it is caused by the same sequence of actions,  $[\hat{a}_m\%|\hat{a}|, \hat{a}_{(m+1)\%|\hat{a}|}, \dots, \hat{a}_{(m+|\hat{a}|-1)\%|\hat{a}|}]$ , being taken after each state. That is, if  $s_{m+i\cdot|\hat{a}|} = s$  and  $s_{m+j\cdot|\hat{a}|} = s$  and  $s_{m+(i+1)\cdot|\hat{a}|} = s'$ , then  $s_{m+(j+1)\cdot|\hat{a}|} = s'$ . As a consequence, if  $s_{m+i\cdot|\hat{a}|} = s$  and the next occurrence of the latent state  $s$  in the sequence  $s_{per.}$  is  $s_{m+(i+t)\cdot|\hat{a}|} = s$ , then all subsequent states in the sequence  $s_{per.}$  will consist of repetitions of the sequence of  $s_{m+(i+1)\cdot|\hat{a}|}$  through  $s_{m+(i+t)\cdot|\hat{a}|}$ .
+
+Then  $s_{per}$  must consist of some sequence of 'transient' latent states which occur only once at the beginning of the sequence, followed by a repeated cycle. Because these states never re-occur, the transient part lasts length  $n_{\mathrm{trn}} \leq N - 1$ .
+
+Then, the period of the cycle is  $n_{\mathrm{cyc}}$ , where  $n_{\mathrm{trn}} + n_{\mathrm{cyc}} \leq N$ . Note that the cycle in  $s_{per}$  contains no repeated states. (Otherwise, the span between the first two repetitions of a state in the cyclic sequence in  $s_{per}$  will itself repeat indefinitely, so we can analyse this smaller cycle as the cycle of length  $n_{\mathrm{cyc}}$ .)
+
+There is a corresponding cycle in  $s_{CF}$ , of length  $n_{\mathrm{cyc}} \cdot |\hat{a}|$ . To see this, note that for all  $i \geq n_{\mathrm{trn}}$  we have that  $s_{m+i|\hat{a}|} = s_{m+i|\hat{a}|+n_{\mathrm{cyc}} \cdot |\hat{a}|}$ . Furthermore, for all  $j$  in  $\{0, \dots, |\hat{a}| - 1\}$ , the sequence of actions taken between  $s_{m+i|\hat{a}|}$  and  $s_{m+i|\hat{a}|+j}$  is the same as the sequence of actions taken between  $s_{m+i|\hat{a}|+j}$  and  $s_{m+i|\hat{a}|+j+n_{\mathrm{cyc}} \cdot |\hat{a}|}$ . Therefore  $s_{m+i|\hat{a}|+j} = s_{m+i|\hat{a}|+j+n_{\mathrm{cyc}} \cdot |\hat{a}|}$ . Thus, for any general  $i' \geq n_{\mathrm{trn}}|\hat{a}|$  (which always can be written as  $i' = i|\hat{a}| + j$ ) we have that  $s_{m+i'} = s_{m+i' + n_{\mathrm{cyc}} \cdot |\hat{a}|}$ . However, this cycle may contain repeated states.
+
+In order to avoid the transient part of  $s_{CF}$ , and to prevent sampling observations with exogenous noise that is correlated to samples taken in previous iterations of CycleFind, we skip the first  $\max((N - 1) \cdot |\hat{a}|, \hat{t}_{\mathrm{mix}})$  transitions in  $s_{CF}$ . For convenience, we will let  $\bar{s}_i := s_{i \cdot |\hat{a}| + \max((N - 1) \cdot |\hat{a}|, \hat{t}_{\mathrm{mix}})}$ , and similarly  $\bar{x}_i := x_{i \cdot |\hat{a}| + \max((N - 1) \cdot |\hat{a}|, \hat{t}_{\mathrm{mix}})}$ . Note that the sequence  $[\bar{s}_0, \bar{s}_1, \dots]$  is equivalent to  $s_{per}$ . After skipping the first  $N - 1 \geq n_{trn}$  elements of the sequence.
+
+Because the cycle in  $s_{per}$  contains no repeated states, we have that
+
+$$
+\forall i, j \in \mathbb {N}, \bar {s} _ {i} = \bar {s} _ {j} \Leftrightarrow i \equiv j \quad (\mathrm {m o d} n _ {\mathrm {c y c}}) \tag {19}
+$$
+
+In order to find  $n_{\mathrm{cyc}}$ , we test the hypothesis that  $n_{\mathrm{cyc}} = n_{\mathrm{cyc}}'$ , for each  $n_{\mathrm{cyc}}' \in \{N, \dots, 2\}$ , in order, until we identify  $n_{\mathrm{cyc}}$ . If none of the tests pass, then we know that  $n_{\mathrm{cyc}} = 1$ . The test for each hypothesis  $n_{\mathrm{cyc}} = n_{\mathrm{cyc}}'$  has a zero false-negative rate. Consequently, the loop will always end before  $n_{\mathrm{cyc}} > n_{\mathrm{cyc}}'$ , so at each iteration, it must always be the case that  $n_{\mathrm{cyc}} \leq n_{\mathrm{cyc}}'$ . A failure can only occur if the test that  $n_{\mathrm{cyc}} = n_{\mathrm{cyc}}'$  has a false positive, when in fact  $n_{\mathrm{cyc}} < n_{\mathrm{cyc}}'$ .
+
+Each test proceeds as follows:
+
+- Let  $q \coloneqq \lceil \hat{t}_{\mathrm{mix}} / (n_{\mathrm{cyc}}' \cdot |\hat{a}|) \rceil$  and  $r \coloneqq q \cdot n_{\mathrm{cyc}}'$ .  
+Let  $k := \left\lfloor \frac{c_{\mathrm{init}} + r \cdot |\hat{a}| - \max((N - 1) \cdot |\hat{a}|, \hat{t}_{\mathrm{mix}})}{2r \cdot |\hat{a}| + n_{\mathrm{cyc}}' \cdot |\hat{a}|} \right\rfloor$  
+- Let  $D_0 \coloneqq \{\bar{x}_{r + (2r + n_{\mathrm{cyc}}')i + j} \mid i \in \{0, \dots, k - 1\}, j \in \{1, \dots, n_{\mathrm{cyc}}' - 1\}\}$ .  
+- Let  $D_{1} \coloneqq \{\bar{x}_{(2r + n_{\mathrm{cyc}}^{\prime})i}|i \in \{0,\dots,k - 1\}\}$ .  
+- Use the training oracle to try to learn to distinguish  $D_0$  from  $D_{1}$ , yielding  $f \in \mathcal{F}$ .  
+- If  $n_{\mathrm{cyc}} = n_{\mathrm{cyc}}'$ , then,
+
+Note that, because  $n_{\mathrm{cyc}}|r$
+
+$$
+\forall i \in \mathbb {N}, (2 r + n _ {\text {c y c}} ^ {\prime}) i \equiv 0 \quad (\mathrm {m o d} n _ {\text {c y c}}), \tag {20}
+$$
+
+but
+
+$$
+\forall i \in \mathbb {N}, j \in \{1, \dots , n _ {\text {c y c}} ^ {\prime} - 1 \}, \tag {21}
+$$
+
+$$
+r + (2 r + n _ {\text {c y c}} ^ {\prime}) i + j \equiv j \not \equiv 0 \pmod {n _ {\text {c y c}}},
+$$
+
+- Consequently, by Equation 19, all elements of  $D_{1}$  will have the same latent state, and none of the elements of  $D_{0}$  have this latent state. By realizability,  $f$  will have  $100\%$  accuracy on the training set. (This is the "true positive" case of the test.)
+
+- Conversely, if  $n_{\mathrm{cyc}} < n_{\mathrm{cyc}}'$ , there is only a small chance that any classifier  $f$  will have  $100\%$  accuracy on the training set.
+
+- Define  $j'$  as the (unknown) value  $j' := (-r - 1)\% n_{\mathrm{cyc}} + 1$ . Noting, by assumption, that  $n_{\mathrm{cyc}} < n_{\mathrm{cyc}}'$ , we have that  $j' \in \{1,..,n_{\mathrm{cyc}}' - 1\}$ . Then,  $\forall i \in \{0,..,k - 1\}$ , we have that  $\bar{x}_{r + (2r + n_{\mathrm{cyc}}')i + j'} \in D_0$ , while  $\bar{x}_{(2r + n_{\mathrm{cyc}}')i} \in D_1$ . However, we also have that:
+
+$$
+r + \left(2 r + n _ {\text {c y c}} ^ {\prime}\right) i + j ^ {\prime} \equiv \left(2 r + n _ {\text {c y c}} ^ {\prime}\right) i \quad (\mathrm {m o d} n _ {\text {c y c}}) \tag {22}
+$$
+
+which by Equation 19 implies that
+
+$$
+\bar {s} _ {r + \left(2 r + n _ {\text {c y c}} ^ {\prime}\right) i + j ^ {\prime}} = \bar {s} _ {\left(2 r + n _ {\text {c y c}} ^ {\prime}\right) i}. \tag {23}
+$$
+
+- Now, we can define  $D_0^{(j')} \subseteq D_0$  as
+
+$$
+D _ {0} ^ {(j ^ {\prime})} := \left\{\bar {x} _ {r + \left(2 r + n _ {\text {c y c}} ^ {\prime}\right) i + j ^ {\prime}} \mid i \in \{0, \dots , k - 1 \} \right\}. \tag {24}
+$$
+
+Fix any arbitrary classifier  $f^{\prime}\in \mathcal{F}$
+
+- In order for  $f'$  to have  $100\%$  accuracy on the training set, we must have  $f'(x) = 1$  for all  $x \in D_1$ , and  $f'(x) = 0$  for all  $x \in D_0^{(j')}$ .
+
+- Note that all observations in  $D_{1} \oplus D_{0}^{(j')}$  are collected at least  $t_{\mathrm{mix}}$  steps apart from one another. (Specifically,  $\bar{x}_{r + (2r + n_{\mathrm{cyc}}')i + j'}$  is collected  $(r + j') \cdot |\hat{a}| \geq r \cdot |\hat{a}| \geq t_{mix}$  steps after  $\bar{x}_{(2r + n_{\mathrm{cyc}}')i}$ , and  $(r + n_{\mathrm{cyc}}' - j') \cdot |\hat{a}| \geq r \cdot |\hat{a}| \geq t_{mix}$  steps before  $\bar{x}_{(2r + n_{\mathrm{cyc}}')(i + 1)}$ .)
+
+- Because, additionally,  $D_0^{(j')}$  and  $D_{1}$  are defined independently of  $\Omega$ , by Lemma 1 we have:
+
+$$
+\forall t \in \left\{t ^ {\prime} \mid \bar {x} _ {t ^ {\prime}} \in D _ {1} \uplus D _ {0} ^ {(j ^ {\prime})} \right\}, \tag {25}
+$$
+
+$$
+p _ {s} - 1 / 4 \leq \Pr \left(f ^ {\prime} (\bar {x} _ {t}) = 1 \mid \left(D _ {1} \uplus D _ {0} ^ {(j ^ {\prime})}\right) _ {<   t}, \phi^ {*} (\bar {x} _ {t}) = s\right) \leq p _ {s} + 1 / 4
+$$
+
+where  $(D_{1} \oplus D_{0}^{(j')})_{<t}$  refers to the samples in  $D_{1} \oplus D_{0}^{(j')}$  collected before  $\bar{x}_{t}$  and:
+
+$$
+\forall s \in \mathcal {S}, p _ {s} := \Pr (f ^ {\prime} (x) = 1 | x \sim \mathcal {Q} (s, e), e \sim \pi_ {\varepsilon}). \tag {26}
+$$
+
+Then, by Equation 26, the probability that  $f'$  returns 1 on all samples in  $D_{1}$ , and 0 on all samples in  $D_{0}^{(j')}$  is at most:
+
+$$
+\Pi_ {i = 0} ^ {k - 1} \left(p _ {\bar {s} _ {(2 r + n _ {\mathrm {c y c}} ^ {\prime}) i}} + 1 / 4\right) \cdot \Pi_ {i = 0} ^ {k - 1} \left(1 - \left(p _ {\bar {s} _ {r + (2 r + n _ {\mathrm {c y c}} ^ {\prime}) i + j ^ {\prime}}} - 1 / 4\right)\right). \tag {27}
+$$
+
+By Equation 23, this is:
+
+$$
+\prod_ {i = 0} ^ {k - 1} \left(p _ {\bar {s} _ {(2 r + n _ {\mathrm {c y c}} ^ {\prime}) i}} + 1 / 4\right) \cdot \prod_ {i = 0} ^ {k - 1} \left(1 - \left(p _ {\bar {s} _ {(2 r + n _ {\mathrm {c y c}} ^ {\prime}) i}} - 1 / 4\right)\right). \tag {28}
+$$
+
+Rearranging gives us:
+
+$$
+\operatorname {F P R} \left(f ^ {\prime}\right) \leq \Pi_ {i = 0} ^ {k - 1} \left(- p _ {\bar {s} (2 r + n _ {\text {c y c}} ^ {\prime}) i} ^ {2} + p _ {\bar {s} (2 r + n _ {\text {c y c}} ^ {\prime}) i} + 5 / 1 6\right). \tag {29}
+$$
+
+Because  $\forall p, -p^2 + p + 5/16 \leq 9/16$ , we can upper-bound this as:
+
+$$
+\operatorname {F P R} \left(f ^ {\prime}\right) \leq \prod_ {i = 0} ^ {k - 1} \left(- p _ {\bar {s} _ {(2 r + n _ {\text {c y c}} ^ {\prime}) i}} ^ {2} + p _ {\bar {s} _ {(2 r + n _ {\text {c y c}} ^ {\prime}) i}} + 5 / 1 6\right) \leq \left(\frac {9}{1 6}\right) ^ {k} \tag {30}
+$$
+
+- As a uniform convergence bound:
+
+$$
+F P R (f) \leq | \mathcal {F} | \left(\frac {9}{1 6}\right) ^ {k} \tag {31}
+$$
+
+- Finally, we take a union bound over all values of  $n_{\mathrm{cyc}}'$ . To do this, we must lower bound  $k$  for all values of  $n_{\mathrm{cyc}}'$ . First, note that
+
+$$
+\hat {t} _ {\text {m i x}} + N | \hat {a} | - 1 \geq \hat {t} _ {\text {m i x}} + n _ {\text {c y c}} ^ {\prime} | \hat {a} | - 1 \geq r | \hat {a} | \tag {32}
+$$
+
+Then:
+
+$$
+\begin{array}{l} k = \left\lfloor \frac {\left(\hat {t} _ {\text {m i x}} + N \cdot | \hat {a} | - 1\right) \cdot \left(2 \cdot n _ {\text {s a m p . c y c .}} - 1\right) + N \cdot | \hat {a} | \cdot n _ {\text {s a m p . c y c .}} + r \cdot | \hat {a} |}{2 r \cdot | \hat {a} | + n _ {\text {c y c .}} ^ {\prime} \cdot | \hat {a} |} \right\rfloor \\ \geq \left\lfloor \frac {r \cdot | \hat {a} | \cdot \left(2 \cdot n _ {\text {s a m p . c y c .}} - 1\right) + N \cdot | \hat {a} | \cdot n _ {\text {s a m p . c y c .}} + r \cdot | \hat {a} |}{2 r \cdot | \hat {a} | + N \cdot | \hat {a} |} \right\rfloor \tag {33} \\ \geq \left\lfloor n _ {\text {s a m p . c y c .}} \right\rfloor \\ \geq n _ {\text {s a m p . c y c .}}. \\ \end{array}
+$$
+
+So we have that, by union bound over all values of  $n_{\mathrm{cyc}}'$ :
+
+$$
+F P R (f) \leq (N - 1) | \mathcal {F} | \left(\frac {9}{1 6}\right) ^ {n _ {\text {s a m p . c y c .}}} \tag {34}
+$$
+
+# Collecting  $\mathcal{D}_i^t$ :
+
+We now know that  $s_{CF}$  eventually enters a cycle of length  $n_{\mathrm{cyc}} \cdot |\hat{a}|$ , where  $n_{\mathrm{cyc}}$  is known. This is the latent-state cycle  $[s_0^{cyc*}, s_1^{cyc*}, \dots, s_{|\hat{a}| \cdot n_{\mathrm{cyc}} - 1}^{cyc*}]$  mentioned in Proposition 1.
+
+Depending on the value of  $n_{\mathrm{cyc}}$ , we might now need to extend  $x_{CF}$  (and, respectively  $s_{CF}$ ) by making additional loops through  $\hat{a}$ , until the length of  $x_{CF}$  is at least  $c$ , where:
+
+$$
+c := 2 \cdot n _ {\mathrm {c y c}} \cdot | \hat {a} | \cdot \left((n _ {\mathrm {s a m p .}} - 1) \cdot \left\lceil \frac {\hat {t} _ {\mathrm {m i x}}}{| \hat {a} | \cdot n _ {\mathrm {c y c}}} \right\rceil + 1\right) + \hat {t} _ {\mathrm {m i x}} + \max  ((N - n _ {\mathrm {c y c}}) \cdot | \hat {a} |, \hat {t} _ {\mathrm {m i x}}). \tag {35}
+$$
+
+This will entail taking an additional  $\max(c - c_{\mathrm{init}}, 0)$  steps on  $M$ . Note that in the worst case, this means that CycleFind takes a total of at most:
+
+$$
+\begin{array}{l} \max  (c _ {\text {i n i t}}, c) \leq \max  \left((2 \hat {t} _ {\text {m i x}} + 3 N \cdot | \hat {a} | - 2) \cdot n _ {\text {s a m p . c y c .}} - N \cdot | \hat {a} |, \right. \\ \left. 2 \cdot \left(\hat {t} _ {\text {m i x}} + | \mathcal {S} | \cdot | \hat {a} | - 1\right) \cdot n _ {\text {s a m p .}} + 1\right) \tag {36} \\ + \max  (N \cdot | \hat {a} | - | \hat {a} | - \hat {t} _ {\text {m i x}}, 0) + 1 \text {a c t i o n s}. \\ \end{array}
+$$
+
+We now define how to collect two datasets for each position in the cycle in  $s_{CF}$ ,  $\mathcal{D}_i^A$  and  $\mathcal{D}_i^B$  for each  $i \in \{0, \dots, n_{\mathrm{cyc}} \cdot |\hat{a}| - 1\}$ . Specifically we take:
+
+$$
+\begin{array}{l} \mathcal {D} _ {i} ^ {A} = \left\{x _ {j} | \exists k \in \{0, \dots , n _ {\text {s a m p .}} - 1 \}: \right. \\ j = k \cdot \left(|\hat{a} | \cdot n _ {\mathrm {cyc}} \cdot \left\lceil \frac {\hat {t} _ {\mathrm {mix}}}{|\hat{a} | \cdot n _ {\mathrm {cyc}}} \right\rceil\right) + n _ {0} + (i - n _ {0}) \% \left(n _ {\mathrm {cyc}} \cdot |\hat{a} |\right) \Bigg \} \tag{37} \\ \end{array}
+$$
+
+where we let
+
+$$
+n _ {0} := \max  \left(\left(N - n _ {\text {c y c}}\right) \cdot | \hat {a} |, \hat {t} _ {\text {m i x}}\right), \tag {38}
+$$
+
+and
+
+$$
+\begin{array}{l} \mathcal {D} _ {i} ^ {B} = \left\{x _ {j} | \exists k \in \{0, \dots , n _ {\text {s a m p .}} - 1 \}: \right. \\ j = k \cdot \left(|\hat{a} | \cdot n_{\mathrm{cyc}} \cdot \left\lceil \frac{\hat{t}_{\mathrm{mix}}}{|\hat{a} | \cdot n_{\mathrm{cyc}}} \right\rceil\right) + n _ {0} ^ {\prime} + (i - n _ {0} ^ {\prime}) \% \left(n_{\mathrm{cyc}} \cdot |\hat{a} |\right) \Bigg\} \tag{59} \\ \end{array}
+$$
+
+where
+
+$$
+n _ {0} ^ {\prime} := n _ {0} + \left(n _ {\text {s a m p .}} - 1\right) \cdot \left(\left| \hat {a} \right| \cdot n _ {\text {c y c}} \cdot \left[ \frac {\hat {t} _ {\text {m i x}}}{\left| \hat {a} \right| \cdot n _ {\text {c y c}}} \right]\right) + \left| \hat {a} \right| \cdot n _ {\text {c y c}} + \hat {t} _ {\text {m i x}}. \tag {40}
+$$
+
+Note that, because we know that  $s_{CF}$  enters a cycle of length  $n_{\mathrm{cyc}} \cdot |\hat{a}|$  after at most  $(N - n_{\mathrm{cyc}}) \cdot |\hat{a}| \leq n_0$  transitions, we have that,
+
+$$
+\forall i, j \geq n _ {0}, i \equiv j \quad (\mathrm {m o d} n _ {\mathrm {c y c}} \cdot | \hat {a} |) \rightarrow s _ {i} = s _ {j}. \tag {41}
+$$
+
+![](images/43d61f2fe4026207194d36c33e7b2a5d622d36d6729c23634205e8893228b8c9.jpg)  
+Figure 4: Illustration of the sampling procedure for datasets  $\mathcal{D}_i^\prime$ . The first goal is to ensure that for each cycle position  $i$ , the samples in  $\mathcal{D}_i^\prime$  are sampled  $t_\mathrm{mix}$  steps apart from each other, and are therefore nearly i.i.d. We also want to ensure that for any pair of cycle positions  $i,j$ , there is a large subset of  $\mathcal{D}_i^\prime$  that only contains samples retrieved at least  $t_\mathrm{mix}$  steps apart from some large subset of  $\mathcal{D}_j^\prime$ . (This second goal is meant to guarantee that if  $\mathcal{D}_i^\prime$  and  $\mathcal{D}_j^\prime$  represent the same latent state, then it is unlikely that any classifier exists than can separate the two subsets perfectly, which would be strictly necessary to perfectly separate  $\mathcal{D}_i^\prime$  and  $\mathcal{D}_j^\prime$ ). However, it is not necessary for all samples in  $\cup_i\mathcal{D}_i$  to be collected  $t_\mathrm{mix}$  steps apart. Therefore, we collect an observation of each cycle position  $i\in \{0,n_{\mathrm{cyc}}\cdot |\hat{a} |\}$ , all together, every  $\hat{t}_{\mathrm{mix}}$  steps (rounded up to the cycle period). We continue until  $n_{\mathrm{samp}}$  observations of each position are collected; then wait  $\hat{t}_{\mathrm{mix}}$  steps and collect  $n_{\mathrm{samp}}$  additional observations of each cycle position. This process ensures that  $\mathcal{D}_i^\prime$  and  $\mathcal{D}_j^\prime$  contain complementary subsets  $\mathcal{D}_i^A$  and  $\mathcal{D}_j^B$ , each with at least  $n_{\mathrm{samp}}$  samples, such all samples in  $\mathcal{D}_i^A\cup \mathcal{D}_j^B$  are near-i.i.d.
+
+Therefore, because:
+
+$$
+k \cdot \left(|\hat{a} | \cdot n _ {\mathrm {cyc}} \cdot \left\lceil \frac {\hat {t} _ {\mathrm {mix}}}{|\hat{a} | \cdot n _ {\mathrm {cyc}}} \right\rceil\right) + n _ {0} + (i - n _ {0}) \% \left(n _ {\mathrm {cyc}} \cdot |\hat{a} |\right) \equiv i \quad (\text {mod } n _ {\mathrm {cyc}} \cdot |\hat{a} |) \tag{42}
+$$
+
+(and a similar equivalence holds for  $n_0'$ ), we have that, for any fixed  $i$ , all observations in  $\mathcal{D}_i^A \uplus \mathcal{D}_i^B$  must share the same latent state. Also, for any fixed  $i$ , all observations in  $\mathcal{D}_i^A$  and  $\mathcal{D}_i^B$  are collected at least  $\hat{t}$  steps apart. Additionally,  $\mathcal{D}_i^A$  and  $\mathcal{D}_i^B$  are defined solely in terms of  $n_{\mathrm{cyc}}$  and  $\hat{a}$ , and so the selection of samples to put in these sets only depends on the sequence of latent states  $s$  that the Ex-BMDP traverses, and is therefore defined deterministically and independently of  $\Omega$  (assuming  $n_{\mathrm{cyc}}$  is correctly determined). We therefore define
+
+$$
+\mathcal {D} _ {i} ^ {\prime} := \mathcal {D} _ {i} ^ {A} \uplus \mathcal {D} _ {i} ^ {B}, \tag {43}
+$$
+
+and note that all elements in this set both share the same latent state and were collected at least  $\hat{t}$  steps apart from one another.
+
+Additionally, for any fixed pair  $i,j$ , all observations in  $\mathcal{D}_i^A \oplus \mathcal{D}_j^B$  are collected at least  $\hat{t}$  steps apart from one another.
+
+Using the  $c$  samples in  $x_{CF}$ , this allows us to construct  $\mathcal{D}_i^A$  and  $\mathcal{D}_i^B$ , for each  $i \in \{0, n_{\mathrm{cyc}} \cdot |\hat{a}| - 1\}$  where  $|\mathcal{D}_i^A| = |\mathcal{D}_i^B| = n_{\mathrm{samp.}}$ . See Figure 4 for an illustration of the sampling procedure.
+
+# Identifying new latent states from  $\mathcal{D}'$ :
+
+At this point, each set  $\mathcal{D}_i^\prime$  consists of observations of a single latent state  $s$ , but two such sets  $\mathcal{D}_i^\prime$  and  $\mathcal{D}_j^\prime$  may represent the same latent state, and  $\mathcal{D}_i^\prime$  may contain the same latent state as some previously-collected  $\mathcal{D}(s)$  for some  $s \in S'$ .
+
+In order to identify the newly-discovered latent states to add to  $S^{\prime}$ , and appropriately update  $\mathcal{D}(\cdot)$  and  $T^{\prime}$ , we proceed as follows:
+
+- For  $i \in \{0, \dots, n_{\mathrm{cyc}} \cdot |\hat{a}| - 1\}$ :
+
+- For each  $s \in S'$ , use the training oracle to learn a classifier  $f \in \mathcal{F}$ , with  $D_0 := \mathcal{D}(s)$  and  $D_1 := \mathcal{D}_i'$ . If  $f$  can distinguish  $D_0$  from  $D_1$  with 100% training set accuracy, then we conclude (with high probability) that  $\mathcal{D}(s)$  and  $\mathcal{D}_i'$  represent two different latent states. Otherwise, we conclude that  $\mathcal{D}(s)$  and  $\mathcal{D}_i'$  both represent the same latent state.  
+- If  $\mathcal{D}_i^\prime$  is identified as representing some already-discovered latent state  $s\in S^{\prime}$  then discard  $\mathcal{D}_i^\prime$ . (Or, we can update  $\mathcal{D}(s)$  by merging the samples in  $\mathcal{D}_i^\prime$  into it; this choice does not affect our analysis - however, we should avoid doing this if  $\mathcal{D}(s)$  was either defined, or already updated, during this call of CycleFind: this is because two datasets  $\mathcal{D}_i^\prime$  and  $\mathcal{D}_j^\prime$  from the same iteration of CycleFind may contain observations that were sampled fewer than  $\hat{t}_{\mathrm{mix}}$  steps apart from each other, which would break the inductive assumption on  $\mathcal{D}(s)$  if they are both merged into  $\mathcal{D}(s)$ .) Record this latent state  $s$  as:
+
+$$
+s _ {i} ^ {\text {c y c}} := s \tag {44}
+$$
+
+- Otherwise, if  $\mathcal{D}_i^\prime$  does not represent the any latent state  $s\in S^{\prime}$ , then  $\mathcal{D}_i^\prime$  (and  $\mathcal{D}_i^{\prime \prime}$ ) represents a newly-discovered state. We update  $\mathcal{S}^{\prime}$  by inserting a new state  $s^\prime$  into it, and update  $\mathcal{D}(s)$  by associating  $s^\prime$  with  $\mathcal{D}_i^\prime$ :
+
+$$
+\mathcal {S} ^ {\prime} \leftarrow \mathcal {S} ^ {\prime} \cup \{s ^ {\prime} \} \tag {45}
+$$
+
+$$
+\mathcal {D} (s ^ {\prime}) := \mathcal {D} _ {i} ^ {\prime}
+$$
+
+Finally, we also record this new latent state as:
+
+$$
+s _ {i} ^ {\text {c y c}} := s ^ {\prime} \tag {46}
+$$
+
+- To analyse the success rate of using the training oracle to determine if a given  $\mathcal{D}(s)$  and  $\mathcal{D}_i^\prime$  represent the same latent state, consider the following:
+
+- If  $\mathcal{D}(s)$  and  $\mathcal{D}_i^\prime$  contain different latent states, then  $f$  will be able to distinguish  $D_0$  from  $D_{1}$ , deterministically, with  $100\%$  accuracy on the training set (due to our realizability assumption.)  
+- Otherwise,  $\mathcal{D}(s)$  and  $\mathcal{D}_i^\prime$  both contain samples entirely of the same latent state,  $s$ . Then, either:
+
+* The latent state  $s$  was identified before the current run of the CycleFind subroutine. Therefore, some subset of samples  $D_0' \subseteq D_0 = \mathcal{D}(s)$  were added to  $\mathcal{D}(s)$  before the current run of CycleFind, such that  $|D_0'| \geq n_{\mathrm{samp}}$ . Let  $D_1' := D_1 = \mathcal{D}_i'$ , and note also that all samples in  $D_0' \oplus D_1'$  were collected at least  $\hat{t}_{\mathrm{mix}}$  steps apart from one another. (This is by inductive hypothesis for  $\mathcal{D}(s)$ , by construction for  $\mathcal{D}_i'$ , and by the fact that each run of CycleFind starts by "wasting" at least  $\hat{t}_{\mathrm{mix}}$  steps.)  
+* The latent state  $s$  was identified during the current run of CycleFind, such that  $\mathcal{D}(s) = \mathcal{D}_j'$  for some  $j < i$ . Then let  $D_0' := \mathcal{D}_j^A \subseteq D_0$  and  $D_1' := \mathcal{D}_i^B \subseteq D_1$ . Note that  $|D_0'|, |D_1'| \geq n_{\mathrm{samp}}$ , and all observations in  $D_0' \oplus D_1'$  were collected at least  $\hat{t}_{\mathrm{mix}}$  steps apart from one another.  
+* In either case, the choice of samples to include in  $D_0' \oplus D_1'$  was made deterministically and independently of  $\Omega$  (by construction and/or assumption).
+
+We define  $p_s$  as in Equation 72. Note that the samples in  $D_0'$  and  $D_1'$  were observed at least  $\hat{t}_{\mathrm{mix}}$  steps apart, at deterministically-chosen timesteps. Then Lemma 1 is applicable, and we have that the probability that an arbitrary  $f' \in \mathcal{F}$  returns 1 on all samples from  $D_1 \supseteq D_1'$ ; and also returns 0 on all samples from  $D_0 \supseteq D_0'$  is at most:
+
+$$
+\begin{array}{l} \left(p _ {s} + 1 / 4\right) ^ {n _ {\text {s a m p .}}} \cdot \left(1 - \left(p _ {s} - 1 / 4\right)\right) ^ {n _ {\text {s a m p .}}} = \\ \left(- p _ {s} ^ {2} + p _ {s} + 5 / 1 6\right) ^ {n _ {\text {s a m p .}}} \leq \left(\frac {9}{1 6}\right) ^ {n _ {\text {s a m p .}}} \tag {47} \\ \end{array}
+$$
+
+As a uniform convergence bound, we then have that:
+
+$$
+F P R (f) \leq | \mathcal {F} | \left(\frac {9}{1 6}\right) ^ {n _ {\text {s a m p .}}} \tag {48}
+$$
+
+- Note that at all iterations,  $|\mathcal{S}'| \leq |\mathcal{S}|$ , so we train at most  $|\mathcal{S}| \cdot n_{\mathrm{cyc}} \cdot |\hat{a}|$  classifiers. Therefore, by union bound, the total failure rate is bounded by:
+
+$$
+\Pr (\text {f a i l}) \leq | \mathcal {S} | \cdot n _ {\text {c y c}} \cdot | \hat {a} | \cdot | \mathcal {F} | \left(\frac {9}{1 6}\right) ^ {n _ {\text {s a m p .}}} \tag {49}
+$$
+
+- Note that the states  $s_i^{\mathrm{cyc}}$  now represent the latent states associated with the cyclic part of  $x_{CF}$ . Because we know the actions in the cycle, we can use this information to update  $T'$ . Specifically,  $\forall i \in \{0,1,\dots,|\hat{a}| \cdot n_{\mathrm{cyc}} - 1\}$ , the action taken after  $s_i^{\mathrm{cyc}}$  and before  $s_{(i+1)\%(|\hat{a}| \cdot n_{\mathrm{cyc}})}^{\mathrm{cyc}}$  is  $\hat{a}_{i\% |\hat{a}|}$ . We can then update:
+
+$$
+T ^ {\prime} \left(s _ {i} ^ {\text {cyc}}, a _ {i \% | \hat {a} |}\right) \leftarrow s _ {(i + 1) \% (| \hat {a} | \cdot n _ {\text {cyc}})} ^ {\text {cyc}} \tag{50}
+$$
+
+Return the updated  $S^{\prime},\mathcal{D}$  , and  $T^{\prime}$  , as well as  $s_{\mathrm{curr.}}$  ..
+
+Returning the updated  $S', \mathcal{D}$ , and  $T'$  is straightforward. Note that the choice to assign or merge a given  $\mathcal{D}_i'$  in to a given  $\mathcal{D}(s)$  depends only on the latent states  $s$  in the datasets, and so is independent of  $\Omega$ .
+
+We have then shown that, if CycleFind succeeds, then states  $[s_0^{\mathrm{cyc}},\dots,s_{n_{\mathrm{cyc}}\cdot |\hat{a} | - 1}^{\mathrm{cyc}}]$ , have been added to  $S^{\prime}$ , if they were not present already. These states correspond to the states in the cycle  $[s_0^{cyc*},\dots,s_{n_{\mathrm{cyc}}\cdot |\hat{a} | - 1}^{cyc*}]$ , and the corresponding transitions have been added to  $T^{\prime}$ ; furthermore, the datasets  $\mathcal{D}(s)$  have been updated appropriately.
+
+To determine the learned latent state of the Ex-BMDP  $M$  after CycleFind is run, simply note that this is equivalently the state corresponding to the observation  $x_{c}$ , which we know belongs to dataset  $\mathcal{D}_{c\% (n_{\mathrm{cyc}}|\hat{a}|)}'$ . We then know that this observation must have the same latent state as the rest of  $\mathcal{D}_{c\% (n_{\mathrm{cyc}}|\hat{a}|)}'$ ; that is, the observation  $s_{c\% (n_{\mathrm{cyc}}|\hat{a}|)}^{\mathrm{cyc}}$ .
+
+The total failure rate for the CycleFind algorithm can be bounded by union bound from the failure rates of Parts 1 and 3 of the algorithm; that is, Equations 34 and 49. That is:
+
+$$
+\begin{array}{l} \Pr (\text {f a i l}) \leq (N - 1) \cdot | \mathcal {F} | \left(\frac {9}{1 6}\right) ^ {n _ {\text {s a m p . c y c .}}} + | \mathcal {S} | \cdot n _ {\text {c y c}} \cdot | \hat {a} | \cdot | \mathcal {F} | \left(\frac {9}{1 6}\right) ^ {n _ {\text {s a m p .}}} \\ \leq (N - 1) \cdot | \mathcal {F} | \frac {\delta}{4 | \mathcal {A} | N (N - 1) | \mathcal {F} |} + N ^ {3} \cdot (D + 1) \cdot | \mathcal {F} | \frac {\delta}{4 | \mathcal {A} | N ^ {4} (\hat {D} + 1) | \mathcal {F} |} \tag {51} \\ \leq \frac {\delta}{2 | \mathcal {A} | \cdot N}. \\ \end{array}
+$$
+
+All claims of Proposition 1 have therefore been proven.
+
+![](images/5568e97fbc73b27014bed41dcc85f6bfbb58b5bd52280921333d2d0d119bfebe.jpg)
+
+# B.1.2 STEEL PHASE 1
+
+Note that given a fixed  $\hat{a}$ , there might be multiple different state cycles that could be discovered by CycleFind. However, only one will actually be discovered, depending on the state that the Ex-BMDP starts in as well as the not-yet-discovered parts of the state dynamics. For example, consider an Ex-BMDP  $\mathcal{A} \coloneqq \{L, R\}$ ,  $\mathcal{S} \coloneqq \{\hat{1}, \hat{2}\}$  with the following latent dynamics:
+
+![](images/4fcd00cd2aba3251a6b9eb8de6545c353595535df0184fa09be2b2e49f463f71.jpg)
+
+If we set  $\hat{a} \coloneqq [L]$ , then, depending on the initial state, CycleFind will either collect samples of  $\hat{1}$  and discover its self-loop transition, or collect samples of  $\hat{2}$  and discover its self-loop transition.
+
+In order to learn the complete latent dynamics of the Ex-BMDP, we maintain a representation  $T'$  of the partial transition graph that has been discovered so far, and iteratively apply CycleFind using, at each step, an action sequence  $\hat{a}$  that is guaranteed to produce a cycle that is not entirely contained in the partial graph discovered so far.
+
+Note that this is not as simple as choosing a sequence of actions that leads to an unknown state transition from the final latent state of the Ex-BMDP reached in the previous iteration of CycleFind. For example, consider the following partially-learned latent state dynamics (with  $\mathcal{A} \coloneqq \{L, R\}$ ):
+
+![](images/637fe9bb58721abf844d409bf2ef1878a437d82e4f7e8b37fdef94fb61695944.jpg)
+
+Here, the only unknown transition from an known state is the effect of the 'R' action from  $\hat{4}$ . Suppose we know from the previous iteration of CycleFind that the current latent state of the Ex-BMDP is  $s_{\mathrm{curr.}} = \hat{3}$ . Naively, it might seem as if running CycleFind with  $\hat{a} = [R,R]$  would learn some new transition dynamics or states, because it would navigate through the unknown transition. However, this might not be the case in fact. In particular, the 'R'-transition from  $\hat{4}$  might only be visited transiently. For example, suppose the full latent dynamics of the Ex-BMDP are as follows (with the currently unknown parts shown in gray):
+
+![](images/25800c91a8dd6aba7513c2aa3bddf41bf3fc9aef8ecc439a0ed822ca5a5eaab0.jpg)
+
+Then, if we run CycleFind with  $\hat{a} = [R,R]$ , it will converge to a cycle between the nodes  $\hat{1}$  and  $\hat{2}$ :
+
+![](images/923c07e9e835a9dd24a45a6f45e206cabc7f41065821db23d6003be3d170643d.jpg)
+
+Note that the states  $(\hat{1}$  and  $\hat{2})$  and associated transitions that CycleFind converges on were already explored, so we learn no new information from this application of CycleFind.
+
+Instead, at each iteration, we design  $\hat{a}$  so that no cycle of the actions  $\hat{a}$  can be entirely contained within the currently-known partial transition graph. We show that the length of the resulting  $\hat{a}$  is at most  $(D + 1)|\mathcal{S}|$ .
+
+We proceed as follows. Note that in the first iteration, before any latent states are known, we can simply use  $\hat{a} = [a]$  for some arbitrary  $a\in \mathcal{A}$ . Otherwise, we use the following algorithm:
+
+- Initialize  $\mathcal{B}$  with all of the previously-learned latent states (that is,  $\mathcal{B} \gets S'$ .)  
+- While  $\mathcal{B}$  is non-empty:
+
+- Remove some latent state  $s$  from  $\mathcal{B}$ .
+
+- Use Dijkstra's algorithm to compute a shortest path in the partial transition graph that starts at  $s$  and ends at any not-yet-defined transition. (that, is, any transition for which  $T^{\prime}(\cdot ,\cdot) = \bot$ ). (Note this must be possible. Otherwise, because all states can reach each other in the full latent dynamics, if there were no such undiscovered edge in the
+
+same connected component as  $s$ , then we would know that we have already found the complete latent dynamics.) Also note that the shortest path through such an edge can have length at most  $D + 1$ , simply because:
+
+* The length of the shortest path from  $s$  to the state with the undefined edge in the full transition graph  $T$  is at most  $D$ .  
+* Suppose that some transition on this shortest path is missing in the partial transition graph  $T'$ . Concretely, let  $d$  be the first state along this path such that the transition out of it is missing. Then, we have a path from  $s$  to  $d$  of length less than  $D$ , and we know that  $d$  is itself missing a transition. Then  $d$  can be used in place of the original state with the undefined edge: it has a missing transition, and it is at most  $D$  steps, in  $T'$ , from  $s$ .
+
+- Let  $\hat{a}'$  be the list of actions on the path we have found from  $s$  through an undefined edge. Note that taking actions  $\hat{a}'$  from  $s$  will result in taking an unknown transition, and that  $|\hat{a}'| \leq D + 1$ .  
+- Replace  $\mathcal{B}$  with the set of states that can result from starting at any state  $s' \in \mathcal{B}$  and then taking actions  $\hat{a}'$ , according to the learned partial transition graph  $T'$ . For a given  $s' \in \mathcal{B}$ , if this path leads to an unknown transition, do not insert any state corresponding to  $s'$  into the new  $\mathcal{B}$ .  
+- Concatenate  $\hat{a}^\prime$  to the end of  $\hat{a}$
+
+Note that at every iteration,  $|\mathcal{B}|$  decreases by at least 1, so the algorithm runs for at most  $|S'|$  iterations, so the final length of  $\hat{a}$  is at most  $(D + 1)|S'|$ . Also note that, by construction, taking all actions in  $\hat{a}$  will traverse an unknown transition at some point, starting at any latent state that has been learned so far. As a consequence, any cycle traversed by taking  $\hat{a}$  repeatedly must involve at least one transition (and possibly some states) that are not yet included in the partial transition graph. Therefore, applying CycleFind using an  $\hat{a}$  constructed in this way is guaranteed to learn at least one new transition. Therefore, to fully learn the transition dynamics, we must apply CycleFind at most  $|\mathcal{A}| \cdot |S|$  times.
+
+Also, note that the process of constructing  $\hat{a}$  at each iteration depends only on the partial latent dynamics model  $T'$ , which in turn depends only on the choices of  $\hat{a}$  in previous invocations of CycleFind, and ultimately these depend only on the starting latent state  $s_{\mathrm{init}}$  and the ground-truth dynamics  $T$ . Therefore  $\hat{a}$  is at every iteration independent of  $\Omega$ , as required by Proposition 1.
+
+Then, assuming CycleFind succeeds at each invocation, by the end of Phase 1, STEEL will have discovered the complete state set  $S$  and transition function  $T$ , up to permutation.
+
+# B.1.3 STEEL PHASE 2
+
+In the next phase, once we have completely learned  $T'$  (that is, once there are no state-action pairs  $s \in S', a \in \mathcal{A}$  for which  $T'(s, a)$  is undefined), we collect additional samples of each latent state, until the total number of samples collected for each is at least  $d$ , where:
+
+$$
+d := \lceil \frac {3 | \mathcal {S} ^ {\prime} | \ln (1 6 | \mathcal {S} ^ {\prime} | ^ {2} | \mathcal {F} | / \delta)}{\epsilon} \rceil . \tag {52}
+$$
+
+We can leverage the fact that we now have a complete latent transition graph as well as knowledge of the current latent state  $s_{\mathrm{curr}}$  from the last iteration of CycleFind.
+
+To do this, we proceed as follows:
+
+- Use  $T'$  to plan a sequence of actions  $\bar{a}$  such that:
+
+-  $\hat{t}_{\mathrm{mix}} \leq |\bar{a}| \leq \max(|\mathcal{S}| \cdot D, \hat{t}_{\mathrm{mix}} + D)$ , and  
+- Taking the actions in  $\bar{a}$  starting at  $s_{\mathrm{curr}}$  traverses a cycle. That is,
+
+$$
+T ^ {\prime} \left(T ^ {\prime} \left(T ^ {\prime} \left(\dots T ^ {\prime} \left(s _ {\text {c u r r .}}, \bar {a} _ {0}\right), \bar {a} _ {1}\right), \bar {a} _ {2}\right), \dots , \bar {a} _ {| \bar {a} | - 1}\right)\left. \right) = s _ {\text {c u r r .}} \tag {53}
+$$
+
+and,
+
+- Taking the actions in  $\bar{a}$  starting at  $s_{\mathrm{curr.}}$  visits all latent states in  $s \in S' \setminus \{s_{\mathrm{curr.}}\}$  such that  $|\mathcal{D}(s)| < d$  at least once.
+
+Note that planning such a sequence  $\bar{a}$  always must be possible. For example, starting at  $s_{\mathrm{curr.}}$ , we can greedily plan a route to the nearest as-of-yet-unvisited latent state  $s \in S' \setminus \{s_{\mathrm{curr.}}\}$  such that  $|\mathcal{D}(s)| < d$  and repeat until all such states have been visited, and then navigate back to  $s_{\mathrm{curr.}}$ . This takes at most  $|\mathcal{S}'| \cdot D$  steps. If this sequence has length less than  $\hat{t}_{\mathrm{mix}}$ , then we can insert a self-loop at any state in the sequence (such as the state with the shortest self-loop) and repeat this self-loop as many times as necessary until  $|\bar{a}| \geq \hat{t}_{\mathrm{mix}}$ . Because all self-loops are of length at most  $D + 1$ , this can "overshoot" by at most  $D$ , so we have that  $|\bar{a}| \leq \max(|S| \cdot D, \hat{t}_{\mathrm{mix}} + D)$ .
+
+- Execute the actions in  $\bar{a}$  on  $M$  once without collecting data, in order to ensure that within each set  $\mathcal{D}(s)$ , the newly-collected observations are collected at least  $\hat{t}_{\mathrm{mix}}$  steps after observations added in previous phases of STEEL.  
+- Repeatedly take the actions  $\bar{a}$  on  $M$ , collecting the observation of each latent state  $s$  the first time in the cycle that it is visited and inserting the observation into  $\mathcal{D}(s)$ , until  $\forall s \in S'$ ,  $|\mathcal{D}(s)| \geq d$ . Note that for a given latent state  $s$ , we collect observations of  $s$  exactly  $|\bar{a}|$  steps apart. Because  $|\hat{a}| \geq \hat{t}_{\mathrm{mix}}$ , this ensures that the observation added to  $\mathcal{D}(s)$  are collected at least  $\hat{t}_{\mathrm{mix}}$  steps apart. Because each  $\mathcal{D}(s)$  will already contain at least one sample (from CycleFind), this process will take at most  $d - 1$  iterations.  
+- Note that if for some state  $s \in S'$ ,  $|\mathcal{D}(s)|$  reaches  $d$  during some iteration of taking the actions  $\bar{a}$ , then for the next iteration, we can re-plan a shorter  $\bar{a}$  that does not necessarily visit  $s$ . However, when we do this, we must execute the newly-planned cycle  $\bar{a}$  once without collecting data, in order to ensure that all observation added to any particular  $\mathcal{D}(s)$  are collected at least  $\hat{t}_{\mathrm{mix}}$  steps apart. This could require at most  $|S|$  additional iterations through some  $\bar{a}$ .
+
+This process will ensure that  $\forall s\in S^{\prime}$ $|\mathcal{D}(s)|\geq d$  , in at most
+
+$$
+\left(d - 1 + | \mathcal {S} |\right) \cdot \max  \left(D + \hat {t} _ {\text {m i x}}, | \mathcal {S} | \cdot D\right) \text {s t e p s .} \tag {54}
+$$
+
+Also, note that all samples collected during this phase are sorted into the appropriate dataset  $\mathcal{D}(s)$  entirely by open-loop planning on  $T^{\prime}$ , so the choice of samples in each  $\mathcal{D}(s)$  remains independent of  $\Omega$ , and, in principle, can be a deterministic function of  $s_{\mathrm{init}}$ .
+
+# B.1.4 STEEL PHASE 3
+
+Finally, for each learned latent state  $s \in S'$ , we train a classifier  $f_s$  to distinguish  $D_0 := \oplus_{s' \in S' \setminus \{s\}} \mathcal{D}(s')$  from  $D_1 := \mathcal{D}(s)$ . This set of classifiers allows us to perform one-versus-rest classification to identify the latent state of any observation  $x$ , by defining:
+
+$$
+\phi^ {\prime} (x) := \arg \max  _ {s} f _ {s} (x). \tag {55}
+$$
+
+Along with the learned transition dynamics  $T'$ , this should be a sufficient representation of the latent dynamics.
+
+We want to guarantee that when the exogenous state  $e$  of the Ex-BMDP is at equilibrium (that is, is sampled from its stationary distribution), for any latent state  $s$ , if  $x \sim \mathcal{Q}(s, e)$ , then the probability that  $f_s(x) = 1$  and,  $\forall s' \neq s$ ,  $f(s') = 0$  is at least  $1 - \epsilon$ . By union bound, we can do this by ensuring that the accuracy of each classifier  $f_s$ , on each latent state  $s' \in S$ , is at least  $1 - \epsilon / |S|$ . By realizability, we know that  $\forall s$ , there exists some classifier  $f_s^* \in \mathcal{F}$  for which  $f_s^*(s) = 1$  iff  $\phi^*(x) = s$ . Therefore, we need to upper-bound the probability that  $\exists f' \in \mathcal{F}$ , for which  $\forall x \in D_1$ ,  $f(x) = 1$  and  $\forall x \in D_0$ ,  $f(x) = 0$ , but for which either
+
+$$
+\Pr_ {x \sim \mathcal {Q} (s, e); e \sim \pi} \left(f ^ {\prime} (x) = 0\right) \geq \frac {\epsilon}{| \mathcal {S} |} \tag {56}
+$$
+
+Or, for any  $s' \neq s$ ,
+
+$$
+\Pr_ {x \sim \mathcal {Q} \left(s ^ {\prime}, e\right); e \sim \pi} \left(f ^ {\prime} (x) = 1\right) \geq \frac {\epsilon}{| \mathcal {S} |}. \tag {57}
+$$
+
+For all  $s$ , all samples in  $\mathcal{D}(s)$  are collected at least  $t_{\mathrm{mix}}$  samples apart at timesteps chosen deterministically and independently of  $\Omega$ . Therefore, for any single fixed classifier  $f$ , we can use Lemma 2 and the fact that  $\forall s, |\mathcal{D}(s)| \geq d$  to bound the false-positive rates in Equations 56 and 57 as:
+
+$$
+\Pr \left(\forall x \in \mathcal {D} (s), f ^ {\prime} (x) = 1 \bigwedge_ {x \sim \mathcal {Q} (s, e); e \sim \pi} \Pr f ^ {\prime} (x) = 0 \geq \frac {\epsilon}{| \mathcal {S} |}\right) \leq 8 e ^ {- \frac {\epsilon \cdot d}{3 | \mathcal {S} |}}. \tag {58}
+$$
+
+and,  $\forall s^{\prime}\in S^{\prime}\setminus \{s\}$
+
+$$
+\Pr \left(\forall x \in \mathcal {D} \left(s ^ {\prime}\right), f ^ {\prime} (x) = 0 \bigwedge_ {x \sim \mathcal {Q} \left(s ^ {\prime}, e\right); e \sim \pi} \Pr f ^ {\prime} (x) = 1 \geq \frac {\epsilon}{| S |}\right) \leq 8 e ^ {- \frac {\epsilon \cdot d}{3 | S |}}. \tag {59}
+$$
+
+Taking the union bound bound over  $s$  and all latent states  $s' \in S' \setminus \{s\}$  gives a total false positive rate for learning  $f'$  as  $f_s$  as:
+
+$$
+\operatorname {F P R} \left(f ^ {\prime}, s\right) \leq 8 | \mathcal {S} | e ^ {- \frac {e \cdot d}{3 | \mathcal {S} |}}. \tag {60}
+$$
+
+Taking the union bound over all  $f \in \mathcal{F}$  gives:
+
+$$
+\operatorname {F P R} \left(f _ {s}\right) \leq 8 | \mathcal {S} | | \mathcal {F} | e ^ {- \frac {\epsilon \cdot d}{3 | \mathcal {S} |}}. \tag {61}
+$$
+
+Finally, taking the union bound over each classifier  $f_{s}$  gives:
+
+$$
+\operatorname {F P R} \leq 8 | S | ^ {2} | \mathcal {F} | e ^ {- \frac {\epsilon \cdot d}{3 | S |}}. \tag {62}
+$$
+
+# B.1.5 BOUNDING THE OVERALL FAILURE RATE AND SAMPLE COMPLEXITY
+
+Here, we bound the overall failure rate of the STEEL algorithm. We do this by separately bounding the failure rate of the first phase of the algorithm (the repeated applications of CycleFind) and the final phase of the algorithm, the learning of classifiers  $f_{s}$ . We let each of these failure rates be at most  $\delta / 2$ . Therefore, we must have, over the at most  $|\mathcal{S}| \cdot |\mathcal{A}|$  iterations of CycleFind, a failure rate of at most
+
+$$
+\frac {\delta}{2} \geq | \mathcal {S} | \cdot | \mathcal {A} | \cdot \Pr (\text {C y c l e F i n d F a i l s}). \tag {63}
+$$
+
+This is satisfied by Proposition 1 (noting that  $N \geq |\mathcal{S}|$ ). The number of samples needed for these  $|\mathcal{S}| \cdot |\mathcal{A}|$  iterations of CycleFind, each with  $|\hat{a}| \leq |\mathcal{S}| \cdot (D + 1)$ , is (by Equation 36) at most:
+
+$$
+\begin{array}{l} | \mathcal {S} | \cdot | \mathcal {A} | \cdot \left(\max  \left((2 \hat {t} _ {\operatorname {m i x}} + 3 N \cdot | \mathcal {S} | \cdot (D + 1) - 2\right) \cdot n _ {\text {s a m p . c y c .}} - N \cdot | \mathcal {S} | \cdot (D + 1), \right. \\ \left. 2 \cdot \left(\hat {t} _ {\text {m i x}} + | \mathcal {S} | ^ {2} \cdot (D + 1) - 1\right) \cdot n _ {\text {s a m p .}} + 1\right) \tag {64} \\ \left. + \max  \left((N - 1) \cdot | \mathcal {S} | \cdot (D + 1) - \hat {t} _ {\operatorname {m i x}}, 0\right) + 1\right) \\ \end{array}
+$$
+
+Which is upper-bounded by:
+
+$$
+\begin{array}{l} | \mathcal {S} | \cdot | \mathcal {A} | \cdot \left(\max  \left((2 \hat {t} _ {\operatorname {m i x}} + 3 N \cdot | \mathcal {S} | \cdot (D + 1) - 2\right) \cdot n _ {\text {s a m p . c y c .}}, \right. \\ 2 \cdot \left(\hat {t} _ {\text {m i x}} + | \mathcal {S} | ^ {2} \cdot (D + 1) - 1\right) \cdot n _ {\text {s a m p .}} \tag {65} \\ \left. + 2 + \max  \left(\left(N - 1\right) \cdot | \mathcal {S} | \cdot \left(D + 1\right) - \hat {t} _ {\operatorname {m i x}}, 0\right)\right) \\ \end{array}
+$$
+
+where  $n_{\mathrm{samp.}}$  is given by Equation 11 and  $n_{\mathrm{samp. cyc.}}$  is given by Equation 18. Meanwhile, the overall failure rate of the second phase is at most
+
+$$
+\frac {\delta}{2} \geq 8 | \mathcal {S} | ^ {2} | \mathcal {F} | e ^ {- \frac {e \cdot d}{3 | \mathcal {S} |}}. \tag {66}
+$$
+
+Solving for  $d$  in Equation 66 gives:
+
+$$
+\frac {3 | \mathcal {S} | \ln (1 6 | \mathcal {S} | ^ {2} | \mathcal {F} | / \delta)}{\epsilon} \leq d. \tag {67}
+$$
+
+Which is indeed satisfied by Equation 52, given that the structure of the latent dynamics were correctly learned using CycleFind in the first phase of the algorithm, so that  $|\mathcal{S}'| = |\mathcal{S}|$ .
+
+By Equation 54, we then know that the number of samples needed for this phase is at most:
+
+$$
+\left. \max  \left(D + \hat {t} _ {\operatorname {m i x}}, | \mathcal {S} | \cdot D\right) \cdot \left(\lceil \frac {3 | \mathcal {S} | \ln (1 6 | \mathcal {S} | ^ {2} | \mathcal {F} | / \delta)}{\epsilon} \rceil - 1 + | \mathcal {S} |\right). \right. \tag {68}
+$$
+
+Combining the number of samples over both phases and simplifying gives us an overall upper-bound of the number of required samples of:
+
+$$
+\begin{array}{l} \max (D + \hat {t} _ {\mathrm {m i x}}, | \mathcal {S} | \cdot D) \cdot \left(\frac {3 | \mathcal {S} | \ln (1 6 | \mathcal {S} | ^ {2} | \mathcal {F} | / \delta)}{\epsilon} + | \mathcal {S} |\right) + \\ | \mathcal {S} | \cdot | \mathcal {A} | \cdot \left(\max  \left((2 \hat {t} _ {\text {m i x}} + 3 N \cdot | \mathcal {S} | \cdot (D + 1) - 2\right) \cdot \right. \\ \left. \right.\left(\ln \left(4 | \mathcal {A} | \cdot N \cdot (N - 1) \cdot | \mathcal {F} | / \delta\right) / \ln (1 6 / 9) + 1\right), \tag {69} \\ 2 \cdot (\hat {t} _ {\text {m i x}} + | \mathcal {S} | ^ {2} \cdot (D + 1) - 1) \cdot \\ \left. \left(\ln \left(4 | \mathcal {A} | \cdot N ^ {4} \cdot (\hat {D} + 1) \cdot | \mathcal {F} | / \delta\right) / \ln (1 6 / 9) + 1\right) \right. \\ \left. + 2 + \max  \left(\left(N - 1\right) \cdot | \mathcal {S} | \cdot \left(D + 1\right) - \hat {t} _ {\operatorname {m i x}}, 0\right)\right) \\ \end{array}
+$$
+
+This gives us a big-O sample complexity of (using that  $\hat{D} \leq N$  and  $|\mathcal{S}| \leq N$ ):
+
+$$
+\begin{array}{l} \mathcal {O} \Big (| \mathcal {S} | ^ {2} \cdot N \cdot D \cdot | \mathcal {A} | \cdot (\log | \mathcal {A} | + \log (N) + \log | \mathcal {F} | + \log (1 / \delta)) + \\ \left| \mathcal {S} \right| \cdot \left| \mathcal {A} \right| \cdot \hat {t} _ {\text {m i x}} \cdot (\log | \mathcal {A} | + \log (N) + \log | \mathcal {F} | + \log (1 / \delta)) + \tag {70} \\ \left| \mathcal {S} \right| ^ {2} \cdot D \cdot (1 / \epsilon) \cdot \left(\log \left(\left| \mathcal {S} \right|\right) + \log \left| \mathcal {F} \right| + \log (1 / \delta)\right) + \\ \left. \left| \mathcal {S} \right| \cdot \hat {t} _ {\operatorname {m i x}} \cdot (1 / \epsilon) \cdot \left(\log \left(| \mathcal {S} |\right) + \log | \mathcal {F} | + \log (1 / \delta)\right)\right) \\ \end{array}
+$$
+
+Using the notation  $\mathcal{O}^* (f(x))\coloneqq \mathcal{O}(f(x)\log (f(x)))$  , we can write this as:
+
+$$
+\mathcal {O} ^ {*} \left(N D | \mathcal {S} | ^ {2} | \mathcal {A} | \cdot \log \frac {| \mathcal {F} |}{\delta} + | \mathcal {S} | | \mathcal {A} | \hat {t} _ {\text {m i x}} \cdot \log \frac {N | \mathcal {F} |}{\delta} + \frac {| \mathcal {S} | ^ {2} D}{\epsilon} \cdot \log \frac {| \mathcal {F} |}{\delta} + \frac {| \mathcal {S} | \hat {t} _ {\text {m i x}}}{\epsilon} \cdot \log \frac {| \mathcal {F} |}{\delta}\right). \tag {71}
+$$
+
+Therefore, we have shown that, with high probability, STEEL returns (up to permutation) the correct latent dynamics for the Ex-BMDP, and a high-accuracy latent-state encoder  $\phi'$ , within the sample-complexity bound stated in Theorem 1. This completes the proof.
+
+# C USEFUL LEMMATA
+
+Lemma 1. Consider an Ex-BMDP  $M = \langle \mathcal{X}, \mathcal{A}, \mathcal{S}, \mathcal{E}, \mathcal{Q}, T, \mathcal{T}_e, \pi_{\mathcal{E}}^{\text{init}} \rangle$  starting at an arbitrary latent endogenous state  $s_{\text{init}} \in \mathcal{S}$ . Let  $\Omega$  represent the sample space of the three sources of randomness in  $M$ : that is,  $\mathcal{T}_e$ ,  $\mathcal{Q}$ , and the initial exogenous latent state  $e_{\text{init}}$ . Assume that all actions on  $M$  are taken deterministically and independently of  $\Omega$ . Let  $f \in \mathcal{X} \to \{0,1\}$  be a fixed arbitrary function, and for each  $s \in \mathcal{S}$  let
+
+$$
+p _ {s} := \Pr (f (x) = 1 | x \sim \mathcal {Q} (s, e), e \sim \pi_ {\mathcal {E}}). \tag {72}
+$$
+
+where  $\pi_{\mathcal{E}}$  is the stationary distribution of  $\mathcal{T}_e$ . Consider a trajectory sampled from this Ex-BMDP denoted as  $x_{traj} \coloneqq x_0', x_1', x_2'$ , with endogenous latent states  $s_{traj} \coloneqq s_0', s_1', s_2'$  (so that  $s_0' = s_{init}$ ) and exogenous states  $e_{traj} \coloneqq e_0', e_1', e_2'$ . Then, for any fixed  $t_1, t_2 \in \mathbb{N}$ , selected independently of  $\Omega$ , where  $t_2 - t_1 \geq t_{mix}$ , we have that:
+
+$$
+p _ {s} - 1 / 4 \leq \Pr \left(f \left(x _ {t _ {2}} ^ {\prime}\right) = 1 \mid x _ {\leq t _ {1}} ^ {\prime}, s _ {t _ {2}} ^ {\prime} = s\right) \leq p _ {s} + 1 / 4, \tag {73}
+$$
+
+where  $x_{\leq t_1}^{\prime}$  denotes the observations in the trajectory  $x_{\text{traj}}$  up to and including  $x_{t_1}^{\prime}$ .
+
+Note that this does not necessarily hold if  $t_1, t_2$  depend on  $\Omega$ .
+
+Proof. From the definition of mixing time:
+
+$$
+\forall e \in \mathcal {E}, \| \Pr \left(e _ {t _ {2}} ^ {\prime} = \cdot \mid e _ {t _ {1}} ^ {\prime} = e\right) - \pi_ {\mathcal {E}} \| _ {\mathrm {T V}} \leq \frac {1}{4}. \tag {74}
+$$
+
+Then,
+
+$$
+\forall e \in \mathcal {E}, \left| \Pr (f (x) = 1 | x \sim \mathcal {Q} (s _ {t _ {2}} ^ {\prime}, e ^ {\prime}), e ^ {\prime} \sim \pi_ {\mathcal {E}}) - \underset {x \sim \mathcal {Q} (s _ {t _ {2}} ^ {\prime}, e _ {t _ {2}} ^ {\prime})} {\Pr} (f (x) = 1 | e _ {t _ {1}} ^ {\prime} = e) \right| \leq \frac {1}{4}. \tag {75}
+$$
+
+Because  $e_{t_2}'$  depends on  $x_{\leq t_1}'$  only through  $e_{t_1}'$ , we have:
+
+$$
+\left| \Pr (f (x) = 1 | x \sim \mathcal {Q} (s _ {t _ {2}} ^ {\prime}, e ^ {\prime}), e ^ {\prime} \sim \pi_ {\mathcal {E}}) - \underset {x \sim \mathcal {Q} (s _ {t _ {2}} ^ {\prime}, e _ {t _ {2}} ^ {\prime})} {\operatorname * {P r}} (f (x) = 1 | x _ {\leq t _ {1}} ^ {\prime}) \right| \leq \frac {1}{4}. \tag {76}
+$$
+
+Then by Equation 72,
+
+$$
+\left| p _ {s _ {t _ {2}} ^ {\prime}} - \Pr_ {x \sim \mathcal {Q} \left(s _ {t _ {2}} ^ {\prime}, e _ {t _ {2}} ^ {\prime}\right)} \left(f (x) = 1 \mid x _ {\leq t _ {1}} ^ {\prime}\right) \right| \leq \frac {1}{4}, \tag {77}
+$$
+
+which directly implies Equation 73. Note that this is does not hold if  $t_1$  and  $t_2$  can depend on  $\Omega$ . For example, if we define  $t_2 \coloneqq (\min t$  such that  $f(x_t') = 0$  and  $t \geq t_1 + t_{\mathrm{mix}}$ ), then trivially Equation 73 may not apply.
+
+Lemma 2. Consider an irreducible, aperiodic Markov chain  $\mathcal{T}_e$  on states  $\mathcal{E}$  with mixing time  $t_{mix}$  and stationary distribution  $\pi_{\mathcal{E}}$ , and an arbitrary function  $f:\mathcal{E}\to \{0,1\}$ . Suppose  $\operatorname*{Pr}_{e\sim \pi_{\mathcal{E}}} (f(e) = 1)\leq 1 - \epsilon$ . Consider a fixed sequence of  $N$  timesteps  $t_1,\ldots ,t_N$ , where  $\forall i$ ,  $t_i - t_{i - 1}\geq t_{mix}$ . Now, for a trajectory  $e_0,\dots,e_{t_N}$  sampled from the Markov chain, starting at an arbitrary  $e_0$ , we have that:
+
+$$
+\Pr \left(\bigcap_ {i = 1} ^ {N} f \left(e _ {t _ {i}}\right) = 1\right) \leq 8 e ^ {- \frac {\epsilon - N}{3}}. \tag {78}
+$$
+
+Proof. Define  $\epsilon'$  as:
+
+$$
+\epsilon^ {\prime} := \Pr_ {e \sim \pi_ {\varepsilon}} (f (e) = 0). \tag {79}
+$$
+
+Note that we know that  $\epsilon' \geq \epsilon$ . Now, fix any  $i \geq t_{\mathrm{mix}}$ . Let  $M^i$  denote the linear operator on state distributions corresponding to taking  $i$  steps of the Markov chain: that is,  $M^i\pi$  gives the distribution after  $i$  time steps. Also, let  $\Pi$  be the linear operator defined as:
+
+$$
+\Pi \pi := \left(\int_ {e \in \mathcal {E}} \pi (e) d e\right) \pi_ {e}. \tag {80}
+$$
+
+Define the linear operator  $\Delta^i$  as:
+
+$$
+\Delta^ {i} := M ^ {i} - \Pi . \tag {81}
+$$
+
+By linearity and noting that both  $M^i$  and  $\Pi$  are stochastic operators, we have that, for any  $\pi$ ,
+
+$$
+\int_ {e \in \mathcal {E}} \left(\Delta^ {i} \pi\right) (e) d e = 0. \tag {82}
+$$
+
+Also, from the definition of mixing time, we have, for any function  $\pi$ :
+
+$$
+\left\| \Delta^ {i} \pi \right\| _ {1} \leq \frac {1}{2} \| \pi \| _ {1}. \tag {83}
+$$
+
+(To see this, for any  $e \in \mathcal{E}$  consider the unit vector  $\vec{e}$ . Then, note that:
+
+$$
+\left\| \Delta^ {i} \vec {e} \right\| _ {1} = 2 \cdot \left\| \pi_ {e} - M ^ {i} \vec {e} \right\| _ {T V} \leq \frac {1}{2}. \tag {84}
+$$
+
+Then, for any  $\pi$ , we have:
+
+$$
+\left\| \Delta^ {i} \pi \right\| _ {1} = \left\| \int_ {e \in \mathcal {E}} \pi (e) \Delta^ {i} \vec {e} d e \right\| _ {1} \leq \int_ {e \in \mathcal {E}} | \pi (e) | \| \Delta^ {i} \vec {e} \| _ {1} d e \leq \frac {1}{2} \| \pi \| _ {1}. \tag {85}
+$$
+
+Because  $\pi_e$  is a stationary distribution of  $M^i$ , we also have that:
+
+$$
+\Delta^ {i} \pi_ {e} = (| M ^ {i} - \Pi) \pi_ {e} = \pi_ {e} - \pi_ {e} = 0. \tag {86}
+$$
+
+Additionally, let  $\Gamma$  be the linear operator defined as:
+
+$$
+\Gamma \pi := \int_ {e \in \mathcal {E}} \pi (e) f (e) \vec {e} d e \tag {87}
+$$
+
+In other words,  $(\Gamma \pi)(e) \coloneqq f(e) \cdot \pi(e)$ . One useful fact about this operator is that, for any function  $\pi_0$  such that  $\int_{e \in \mathcal{E}} \pi_0(e) de = 0$ , we have that
+
+$$
+\int_ {e \in \mathcal {E}} (\Gamma \pi_ {0}) (e) d e \leq \frac {1}{2} \| \pi_ {0} \| _ {1}. \tag {88}
+$$
+
+(To see this, note that we have:
+
+$$
+\int_ {e \in \mathcal {E}} (\Gamma \pi_ {0}) (e) d e + \int_ {e \in \mathcal {E}} ((I - \Gamma) \pi_ {0}) (e) d e = 0, \tag {89}
+$$
+
+and also that, because  $\Gamma \pi_0$  and  $(I - \Gamma)\pi_0$  are nonzero for disjoint  $e$ 's:
+
+$$
+\left\| \Gamma \pi_ {0} \right\| _ {1} + \left\| (I - \Gamma) \pi_ {0}) \right\| _ {1} = \left\| \pi_ {0} \right\| _ {1}. \tag {90}
+$$
+
+Then, we also have:
+
+$$
+\left| \int_ {e \in \mathcal {E}} \left(\Gamma \pi_ {0}\right) (e) d e \right| \leq \| \Gamma \pi_ {0} \| _ {1}, \tag {91}
+$$
+
+and
+
+$$
+\left| \int_ {e \in \mathcal {E}} (\Gamma \pi_ {0}) (e) d e \right| = \left| \int_ {e \in \mathcal {E}} ((I - \Gamma) \pi_ {0}) (e) d e \right| \leq \| (I - \Gamma) \pi_ {0} \| _ {1}. \tag {92}
+$$
+
+Combining these equations and inequalities yields Equation 88.)
+
+Now, consider the operator  $M^i\Gamma$ . This operator, when applied to a probability distribution  $\pi$ , yields the (unnormized) probability density that results from applying  $\mathcal{T}_e$  to  $e'$  times, where  $e'$  is sampled from  $\pi$  conditioned on  $f(e') = 1$ . More precisely, it is the density of  $e \sim \mathcal{T}_e^i(e')$ , scaled down by the probability that  $f(e') = 1$ . In other words, it is given by:
+
+$$
+(M ^ {i} \Gamma \pi) (e) = p \left(e _ {i} = e \mid e \sim \mathcal {T} _ {e} ^ {i} \left(e ^ {\prime}\right) \wedge f \left(e ^ {\prime}\right) = 1 \wedge e ^ {\prime} \sim \pi\right) \cdot \Pr_ {e ^ {\prime} \sim \pi} \left(f \left(e ^ {\prime}\right) = 1\right). \tag {93}
+$$
+
+Now, consider any vector  $v$ . Note that  $v$  always can be uniquely decomposed as follows:
+
+$$
+v := a \pi_ {e} + b \bar {v} \text {w h e r e} \int_ {e \in \mathcal {E}} \bar {v} (e) d e = 0 \text {a n d} \| \bar {v} \| _ {1} = 1 \text {a n d} b \geq 0. \tag {94}
+$$
+
+(Specifically, we must set  $a \coloneqq \int_{e \in \mathcal{E}} v(e) \, d\epsilon$  and  $b \coloneqq \|v - a\pi_e\|_1$  and  $\bar{v} \coloneqq (v - a\pi_e)/b$ .) Assume that  $v$  is such that  $a \geq 0$ . Now, consider the equation:
+
+$$
+v ^ {\prime} = M ^ {i} \Gamma v \tag {95}
+$$
+
+If we consider the above decomposition, we have:
+
+$$
+a ^ {\prime} \pi_ {e} + b ^ {\prime} \bar {v} ^ {\prime} = M ^ {i} \Gamma \left(a \pi_ {e} + b \bar {v}\right) \tag {96}
+$$
+
+Note that this can be re-written as:
+
+$$
+a ^ {\prime} \pi_ {e} + b ^ {\prime} \bar {v} ^ {\prime} = \Pi \Gamma \left(a \pi_ {e} + b \bar {v}\right) + \Delta^ {i} \Gamma \left(a \pi_ {e} + b \bar {v}\right). \tag {97}
+$$
+
+Note that by Equation 80, the image of  $\Pi$  can be written in the form  $a^\prime \pi_e$ , while, by Equation 82, the image of  $\Delta^i$  can be written as  $b^{\prime}\bar{v}$ , where  $b^{\prime}$  and  $\bar{v}$  are constrained as in Equation 97. Then, we have (using Equation 80):
+
+$$
+a ^ {\prime} = \int_ {e \in \mathcal {E}} (\Gamma (a \pi_ {e} + b \bar {v})) (e) d e = a \int_ {e \in \mathcal {E}} (\Gamma \pi_ {e})) (e) d e + b \int_ {e \in \mathcal {E}} (\Gamma \bar {v})) (e) d e \tag {98}
+$$
+
+and:
+
+$$
+b ^ {\prime} = \| \Delta^ {i} \Gamma (a \pi_ {e} + b \bar {v}) \| _ {1} \leq a \| \Delta^ {i} \Gamma \pi_ {e} \| _ {1} + b \| \Delta^ {i} \Gamma \bar {v} \| _ {1}. \tag {99}
+$$
+
+Now, from Equation 98, we have:
+
+$$
+\begin{array}{l} a ^ {\prime} = a \int_ {e \in \mathcal {E}} (\Gamma \pi_ {e}) (e) d e + b \int_ {e \in \mathcal {E}} (\Gamma \bar {v}) (e) d e \\ \leq a \int_ {e \in \mathcal {E}} (\Gamma \pi_ {e}) (e) d e + \frac {b}{2} \| \bar {v} \| _ {1} \quad \text {(b y E q u a t i o n 8 8)} \tag {100} \\ \leq a \left(1 - \epsilon^ {\prime}\right) + \frac {b}{2} \| \bar {v} \| _ {1} \quad \text {(b y E q u a t i o n 7 9)} \\ \leq a \left(1 - \epsilon^ {\prime}\right) + \frac {b}{2} \quad (\text {b y}) \\ \end{array}
+$$
+
+And, from Equation 99, we have:
+
+$$
+\begin{array}{l} b ^ {\prime} \leq a \| \Delta^ {i} \Gamma \pi_ {e} \| _ {1} + b \| \Delta^ {i} \Gamma \bar {v} \| _ {1} \\ \leq a \| \Delta^ {i} (I - (I - \Gamma)) \pi_ {e} \| _ {1} + b \| \Delta^ {i} \Gamma \bar {v} \| _ {1} \\ \leq a \| \Delta^ {i} \pi_ {e} \| _ {1} + a \| \Delta^ {i} (I - \Gamma) \pi_ {e} \| _ {1} + b \| \Delta^ {i} \Gamma \bar {v} \| _ {1} \\ \leq a \| \Delta^ {i} (I - \Gamma) \pi_ {e} \| _ {1} + b \| \Delta^ {i} \Gamma \bar {v} \| _ {1} \quad (byEquation86) \\ \leq \frac {a}{2} \| (I - \Gamma) \pi_ {e} \| _ {1} + \frac {b}{2} \| \Gamma \bar {v} \| _ {1} \quad (\text {b y E q u a t i o n 8 3}) (101) \\ \leq \frac {a \epsilon^ {\prime}}{2} + \frac {b}{2} \| \Gamma \bar {v} \| _ {1} \quad \text {(b y E q u a t i o n 7 9)} \\ \leq \frac {a \epsilon^ {\prime}}{2} + \frac {b}{2} \| \bar {v} \| _ {1} \quad \text {(b y E q u a t i o n 9 0)} \\ \leq \frac {a \epsilon^ {\prime}}{2} + \frac {b}{2} \quad (\text {b y d e f i n i t i o n o f} \bar {v}.) \\ \end{array}
+$$
+
+We can summarize these results as:
+
+$$
+\left[ \begin{array}{l} a ^ {\prime} \\ b ^ {\prime} \end{array} \right] \leq \left[ \begin{array}{c c} 1 - \epsilon^ {\prime} & \frac {1}{2} \\ \frac {\epsilon^ {\prime}}{2} & \frac {1}{2} \end{array} \right] \left[ \begin{array}{l} a \\ b \end{array} \right]. \tag {102}
+$$
+
+where there “ $\leq$ ” sign applies elementwise. Also, because the elements of this matrix are all nonnegative, we have:
+
+$$
+\left[ \begin{array}{c} x ^ {\prime} \\ y ^ {\prime} \end{array} \right] \leq \left[ \begin{array}{l} x \\ y \end{array} \right] \Rightarrow \left[ \begin{array}{c c} 1 - \epsilon^ {\prime} & \frac {1}{2} \\ \frac {\epsilon^ {\prime}}{2} & \frac {1}{2} \end{array} \right] \left[ \begin{array}{c} x ^ {\prime} \\ y ^ {\prime} \end{array} \right] \leq \left[ \begin{array}{c c} 1 - \epsilon^ {\prime} & \frac {1}{2} \\ \frac {\epsilon^ {\prime}}{2} & \frac {1}{2} \end{array} \right] \left[ \begin{array}{l} x \\ y \end{array} \right]. \tag {103}
+$$
+
+Now, let  $\pi_{t_1}$  represent the probability distribution of the Markov chain at timestep  $t_1$ , and  $\pi_{t_N + t_{\mathrm{mix}}}$  be the probability distribution at timestep  $t_N + t_{\mathrm{mix}}$ . Applying Equation 93 repeatedly gives us that:
+
+$$
+M ^ {t _ {\text {m i x}}} \Gamma M ^ {t _ {N} - t _ {N - 1}} \Gamma M ^ {t _ {N - 1} - t _ {N - 2}} \Gamma \dots M ^ {t _ {2} - t _ {1}} \Gamma \pi_ {t _ {1}} =
+$$
+
+$$
+\pi_ {t _ {N} + t _ {\operatorname {m i x}}} \cdot \Pr (f \left(e _ {t _ {1}}\right) = 1) \cdot \Pr (f \left(e _ {t _ {2}}\right) = 1 | f \left(e _ {t _ {1}}\right) = 1) \dots \tag {104}
+$$
+
+$$
+\Pr (f \left(e _ {t _ {N - 1}}\right) = 1 | \cap_ {i = 1} ^ {N - 2} f \left(e _ {t _ {i}}\right) = 1) \cdot \Pr (f \left(e _ {t _ {N}}\right) = 1 | \cap_ {i = 1} ^ {N - 1} f \left(e _ {t _ {i}}\right) = 1)
+$$
+
+This gives us that:
+
+$$
+\int_ {e \in \mathcal {E}} \left(M ^ {t _ {\operatorname* {m i x}}} \Gamma M ^ {t _ {N} - t _ {N - 1}} \Gamma \dots M ^ {t _ {2} - t _ {1}} \Gamma \pi_ {t _ {1}}\right) (e) d e = \Pr \left(\cap_ {i = 1} ^ {N} f \left(e _ {t _ {i}}\right) = 1\right) \tag {105}
+$$
+
+where the right-hand side is the probability that we are ultimately trying to bound.
+
+Now, let  $v_{0} \coloneqq \pi_{t_{1}}$ ; for  $1 \leq i \leq N - 1$ , let  $v_{i} \coloneqq M^{t_{i + 1} - t_{i}}\Gamma M^{t_{i} - t_{i - 1}}\Gamma \dots \Gamma \pi_{t_{1}}$ ; and finally let  $v_{N} \coloneqq M^{t_{\mathrm{mix}}}\Gamma M^{t_{N} - t_{N - 1}}\Gamma \dots \Gamma \pi_{t_{1}}$ . Let  $a_{i}$  and  $b_{i}$  represent the components  $a$  and  $b$  in the decomposition given in Equation 94 of  $v_{i}$ . Note that:
+
+$$
+a _ {N} = \int_ {e \in \mathcal {E}} \left(M ^ {t _ {\operatorname {m i x}}} \Gamma M ^ {t _ {N} - t _ {N - 1}} \Gamma \dots M ^ {t _ {2} - t _ {1}} \Gamma \pi_ {t _ {1}}\right) (e) d e = \Pr \left(\cap_ {i = 1} ^ {N} f \left(e _ {t _ {i}}\right) = 1\right). \tag {106}
+$$
+
+Also, note that  $\forall j\in [1,N]$ , we have that  $v_{j} = M^{i}\Gamma v_{j - 1}$ , for some  $i\geq t_{\mathrm{mix}}$ . Therefore, by Equation 102,
+
+$$
+\left[ \begin{array}{l} a _ {i} \\ b _ {i} \end{array} \right] \leq \left[ \begin{array}{c c} 1 - \epsilon^ {\prime} & \frac {1}{2} \\ \frac {\epsilon^ {\prime}}{2} & \frac {1}{2} \end{array} \right] \left[ \begin{array}{l} a _ {i - 1} \\ b _ {i - 1} \end{array} \right]. \tag {107}
+$$
+
+(Additionally, each  $a_i$  represents a probability, and so  $a_i \geq 0$ , so Equation 102 is applicable.) Now, due to the relation shown in Equation 103, we can apply this inequality recursively:
+
+$$
+\left[ \begin{array}{c} a _ {i} \\ b _ {i} \end{array} \right] \leq \left[ \begin{array}{c c} 1 - \epsilon^ {\prime} & \frac {1}{2} \\ \frac {\epsilon^ {\prime}}{2} & \frac {1}{2} \end{array} \right] \left[ \begin{array}{c} a _ {i - 1} \\ b _ {i - 1} \end{array} \right] \bigwedge \left[ \begin{array}{c} a _ {i - 1} \\ b _ {i - 1} \end{array} \right] \leq \left[ \begin{array}{c c} 1 - \epsilon^ {\prime} & \frac {1}{2} \\ \frac {\epsilon^ {\prime}}{2} & \frac {1}{2} \end{array} \right] \left[ \begin{array}{c} a _ {i - 2} \\ b _ {i - 2} \end{array} \right] \Longrightarrow
+$$
+
+$$
+\left[ \begin{array}{c} a _ {i} \\ b _ {i} \end{array} \right] \leq \left[ \begin{array}{c c} 1 - \epsilon^ {\prime} & \frac {1}{2} \\ \frac {\epsilon^ {\prime}}{2} & \frac {1}{2} \end{array} \right] \left[ \begin{array}{c} a _ {i - 1} \\ b _ {i - 1} \end{array} \right] \bigwedge
+$$
+
+$$
+\left[ \begin{array}{l l} 1 - \epsilon^ {\prime} & \frac {1}{2} \\ \frac {\epsilon^ {\prime}}{2} & \frac {1}{2} \end{array} \right] \left[ \begin{array}{l} a _ {i - 1} \\ b _ {i - 1} \end{array} \right] \leq \left[ \begin{array}{l l} 1 - \epsilon^ {\prime} & \frac {1}{2} \\ \frac {\epsilon^ {\prime}}{2} & \frac {1}{2} \end{array} \right] \left[ \begin{array}{l l} 1 - \epsilon^ {\prime} & \frac {1}{2} \\ \frac {\epsilon^ {\prime}}{2} & \frac {1}{2} \end{array} \right] \left[ \begin{array}{l} a _ {i - 2} \\ b _ {i - 2} \end{array} \right] \Longrightarrow \tag {108}
+$$
+
+$$
+\left[ \begin{array}{c} a _ {i} \\ b _ {i} \end{array} \right] \leq \left[ \begin{array}{c c} 1 - \epsilon^ {\prime} & \frac {1}{2} \\ \frac {\epsilon^ {\prime}}{2} & \frac {1}{2} \end{array} \right] \left[ \begin{array}{c} a _ {i - 1} \\ b _ {i - 1} \end{array} \right] \leq \left[ \begin{array}{c c} 1 - \epsilon^ {\prime} & \frac {1}{2} \\ \frac {\epsilon^ {\prime}}{2} & \frac {1}{2} \end{array} \right] \left[ \begin{array}{c c} 1 - \epsilon^ {\prime} & \frac {1}{2} \\ \frac {\epsilon^ {\prime}}{2} & \frac {1}{2} \end{array} \right] \left[ \begin{array}{c} a _ {i - 2} \\ b _ {i - 2} \end{array} \right]
+$$
+
+So that we have:
+
+$$
+\left[ \begin{array}{c} a _ {N} \\ b _ {N} \end{array} \right] \leq \left(\left[ \begin{array}{l l} 1 - \epsilon^ {\prime} & \frac {1}{2} \\ \frac {\epsilon^ {\prime}}{2} & \frac {1}{2} \end{array} \right]\right) ^ {N} \left[ \begin{array}{c} a _ {0} \\ b _ {0} \end{array} \right]. \tag {109}
+$$
+
+The matrix  $\left[ \begin{array}{cc}1 - \epsilon^{\prime} & \frac{1}{2}\\ \frac{\epsilon^{\prime}}{2} & \frac{1}{2} \end{array} \right]$  has eigenvalues  $\frac{\left(\frac{3}{2} - \epsilon^{\prime}\right)\pm\sqrt{\epsilon^{\prime 2} + \frac{1}{4}}}{2}$ ; we can use the closed-form solution to the Nth power of an arbitrary  $2\times 2$  matrix given by Williams (1992) to exactly write this upper-bound on  $a_{N}$ :
+
+$$
+\begin{array}{l} a _ {N} \leq \left[ \left(\frac {\left(\frac {3}{2} - \epsilon^ {\prime}\right) + \sqrt {\epsilon^ {\prime 2} + \frac {1}{4}}}{2}\right) ^ {N} \frac {1}{\sqrt {\epsilon^ {\prime 2} + \frac {1}{4}}} \left[ 1 - \epsilon^ {\prime} - \frac {\left(\frac {3}{2} - \epsilon^ {\prime}\right) - \sqrt {\epsilon^ {\prime 2} + \frac {1}{4}}}{2} \right] ^ {T} \right. \\ - \left(\frac {\left(\frac {3}{2} - \epsilon^ {\prime}\right) - \sqrt {\epsilon^ {\prime 2} + \frac {1}{4}}}{2}\right) ^ {N} \frac {1}{\sqrt {\epsilon^ {\prime 2} + \frac {1}{4}}} \left[ 1 - \epsilon^ {\prime} - \frac {\left(\frac {3}{2} - \epsilon^ {\prime}\right) + \sqrt {\epsilon^ {\prime 2} + \frac {1}{4}}}{2} \right] ^ {T} \left. \right]\left[\begin{array}{c}1\\b _ {0}\end{array}\right]. \\ \end{array}
+$$
+
+Where we are also using that  $\pi_{t_1}$  is a normalized probability distribution, so  $a_0 = 1$ . Simplifying:
+
+$$
+\begin{array}{l} a _ {N} \leq \left[ \left(\frac {\left(\frac {3}{2} - \epsilon^ {\prime}\right) + \sqrt {\epsilon^ {\prime 2} + \frac {1}{4}}}{2}\right) ^ {N} \frac {1}{\sqrt {\epsilon^ {\prime 2} + \frac {1}{4}}} \left(\frac {1}{4} - \frac {3 \epsilon^ {\prime}}{2} + \frac {\sqrt {\epsilon^ {\prime 2} + \frac {1}{4}}}{2} + \frac {b _ {0}}{2}\right) \right. \\ \left. - \left(\frac {\left(\frac {3}{2} - \epsilon^ {\prime}\right) - \sqrt {\epsilon^ {\prime 2} + \frac {1}{4}}}{2}\right) ^ {N} \frac {1}{\sqrt {\epsilon^ {\prime 2} + \frac {1}{4}}} \left(\frac {1}{4} - \frac {3 \epsilon^ {\prime}}{2} - \frac {\sqrt {\epsilon^ {\prime 2} + \frac {1}{4}}}{2} + \frac {b _ {0}}{2}\right) \right] \\ \end{array}
+$$
+
+This gives us:
+
+$$
+\begin{array}{l} a _ {N} \leq \left[ \left| \frac {\left(\frac {3}{2} - \epsilon^ {\prime}\right) + \sqrt {\epsilon^ {\prime 2} + \frac {1}{4}}}{2} \right| ^ {N} \frac {1}{\sqrt {\epsilon^ {\prime 2} + \frac {1}{4}}} \left| \frac {1}{4} - \frac {3 \epsilon^ {\prime}}{2} + \frac {\sqrt {\epsilon^ {\prime 2} + \frac {1}{4}}}{2} + \frac {b _ {0}}{2} \right| \right. \tag {110} \\ \left. + \left| \frac {\left(\frac {3}{2} - \epsilon^ {\prime}\right) - \sqrt {\epsilon^ {\prime 2} + \frac {1}{4}}}{2} \right| ^ {N} \frac {1}{\sqrt {\epsilon^ {\prime 2} + \frac {1}{4}}} \left| \frac {1}{4} - \frac {3 \epsilon^ {\prime}}{2} - \frac {\sqrt {\epsilon^ {\prime 2} + \frac {1}{4}}}{2} + \frac {b _ {0}}{2} \right| \right] \\ \end{array}
+$$
+
+Note that because  $\pi_{t_1}$  is a normalized probability distribution, we have that  $b_{0}$  is at most 2 (with this maximum achieved when  $\pi_{t_1}$  has disjoint support from  $\pi_e$ .) Numerically one can see that, for  $\epsilon' \in [0,1]$ ,
+
+$$
+\begin{array}{l} - . 7 <   \frac {1}{4} - \frac {3 \epsilon^ {\prime}}{2} + \frac {\sqrt {\epsilon^ {\prime 2} + \frac {1}{4}}}{2} \leq . 5 (111) \\ - 1. 9 <   \frac {1}{4} - \frac {3 \epsilon^ {\prime}}{2} - \frac {\sqrt {\epsilon^ {\prime 2} + \frac {1}{4}}}{2} \leq 0 (112) \\ \end{array}
+$$
+
+Additionally, we can bound  $\frac{1}{\sqrt{\epsilon'^2 + \frac{1}{4}}}\leq 2$ . Then this gives us:
+
+$$
+a _ {N} \leq \left[ 4 \left| \frac {\left(\frac {3}{2} - \epsilon^ {\prime}\right) + \sqrt {\epsilon^ {\prime 2} + \frac {1}{4}}}{2} \right| ^ {N} + 4 \left| \frac {\left(\frac {3}{2} - \epsilon^ {\prime}\right) - \sqrt {\epsilon^ {\prime 2} + \frac {1}{4}}}{2} \right| ^ {N} \right] \tag {113}
+$$
+
+Because, for  $\epsilon' \in [0,1]$ ,  $\left|\frac{\left(\frac{3}{2} - \epsilon'\right) - \sqrt{\epsilon'^2 + \frac{1}{4}}}{2}\right| < \left|\frac{\left(\frac{3}{2} - \epsilon'\right) + \sqrt{\epsilon'^2 + \frac{1}{4}}}{2}\right|$ , and using Equation 106, we can bound:
+
+$$
+\Pr \left(\bigcap_ {i = 1} ^ {N} f \left(e _ {t _ {i}}\right) = 1\right) \leq 8 \left(\frac {\left(\frac {3}{2} - \epsilon^ {\prime}\right) + \sqrt {\epsilon^ {\prime 2} + \frac {1}{4}}}{2}\right) ^ {N} \tag {114}
+$$
+
+This bound is clearly somewhat unwieldy; to obtain a more manageable bound, it is helpful to consider the asymptotic behavior near  $\epsilon' = 0$ . Letting  $\delta \coloneqq \operatorname*{Pr}(\bigcap_{i=1}^{N} f(e_{t_i}) = 1)$ , we have:
+
+$$
+\ln \delta \leq \ln 8 + N \ln \frac {\left(\frac {3}{2} - \epsilon^ {\prime}\right) + \sqrt {\epsilon^ {\prime 2} + \frac {1}{4}}}{2} \tag {115}
+$$
+
+For  $\epsilon^{\prime}$  small, we have  $\epsilon^{\prime 2} \ll \frac{1}{4}$ , so  $\sqrt{\epsilon^{\prime 2} + \frac{1}{4}} \approx \frac{1}{2}$ . Then:
+
+$$
+\ln \delta \lesssim \ln 8 + N \ln \left(1 - \frac {\epsilon^ {\prime}}{2}\right) \tag {116}
+$$
+
+Then, using the standard approximation  $\ln (1 - x)\approx -x$  gives us:
+
+$$
+\ln \delta \lesssim \ln 8 - \frac {N \epsilon^ {\prime}}{2}. \tag {117}
+$$
+
+This approximation would give us  $\delta \lesssim 8e^{-\frac{N\epsilon'}{2}}$ . However, while this holds approximately for small  $\epsilon'$ , it does not hold exactly. Despite this, it does suggest a form for our final bound. If we try  $8e^{-\frac{N\epsilon'}{3}}$ , we find that it holds that:
+
+$$
+\frac {\left(\frac {3}{2} - \epsilon^ {\prime}\right) + \sqrt {\epsilon^ {\prime 2} + \frac {1}{4}}}{2} \leq e ^ {- \frac {\epsilon^ {\prime}}{3}} \text {o n t h e i n t e r v a l} 0 \leq \epsilon^ {\prime} \leq 0. 4 4. \tag {118}
+$$
+
+Combining with Equation 114, this implies that
+
+$$
+\delta \leq 8 e ^ {- \frac {N \epsilon^ {\prime}}{3}} \text {o n t h e i n t e r v a l} 0 \leq \epsilon^ {\prime} \leq 0. 4 4. \tag {119}
+$$
+
+For very large  $\epsilon'$ , we can use a much simpler bound on  $\operatorname*{Pr}(\bigcap_{i=1}^{N} f(e_{t_i}) = 1)$ . Recall that:
+
+$$
+\begin{array}{l} \delta = \Pr \left(\bigcap_ {i = 1} ^ {N} f \left(e _ {t _ {i}}\right) = 1\right) = \Pr \left(f \left(e _ {t _ {N}}\right) = 1 \middle | \bigcap_ {i = 1} ^ {N - 1} f \left(e _ {t _ {i}}\right) = 1\right) \\ \cdot \Pr \left(f \left(e _ {t _ {N - 1}}\right) = 1 \mid \bigcap_ {i = 1} ^ {N - 2} f \left(e _ {t _ {i}}\right) = 1\right) \tag {120} \\ \end{array}
+$$
+
+.
+
+$$
+\cdot \Pr (f \left(e _ {t _ {2}}\right) = 1 | f \left(e _ {t _ {1}}\right) = 1) \cdot \Pr (f \left(e _ {t _ {1}}\right) = 1)
+$$
+
+However, because  $\forall i$ ,  $t_i - t_{i-1} \geq t_{\mathrm{mix}}$ , we have that:
+
+$$
+\forall i > 1, \Pr \left(f \left(e _ {t _ {i}}\right) = 1 \mid \bigcap_ {j = 1} ^ {i - 1} f \left(e _ {t _ {j}}\right) = 1\right) \leq \Pr_ {e \sim \pi_ {e}} (f (e) = 1) + \frac {1}{4} = \frac {5}{4} - \epsilon^ {\prime} \tag {121}
+$$
+
+Combining these equations gives us:
+
+$$
+\begin{array}{l} \delta \leq \left(\frac {5}{4} - \epsilon^ {\prime}\right) ^ {N - 1} \cdot \Pr \left(f \left(e _ {t _ {1}}\right) = 1\right) \tag {122} \\ \leq \left(\frac {5}{4} - \epsilon^ {\prime}\right) ^ {N - 1} = \left(\frac {5}{4} - \epsilon^ {\prime}\right) ^ {- 1} \left(\frac {5}{4} - \epsilon^ {\prime}\right) ^ {N} \leq 8 \left(\frac {5}{4} - \epsilon^ {\prime}\right) ^ {N} \\ \end{array}
+$$
+
+Where we used that  $\epsilon' \leq 1$  in the last step. Finally, it holds that:
+
+$$
+\left(\frac {5}{4} - \epsilon^ {\prime}\right) \leq e ^ {- \frac {\epsilon^ {\prime}}{3}} \text {o n t h e i n t e r v a l} 0. 3 7 \leq \epsilon^ {\prime} \leq 1. \tag {123}
+$$
+
+This then implies that:
+
+$$
+\delta \leq 8 e ^ {- \frac {N \epsilon^ {\prime}}{3}} \text {o n} 0. 3 7 \leq \epsilon^ {\prime} \leq 1. \tag {124}
+$$
+
+Combining with Equation 119, we have that
+
+$$
+\forall \epsilon^ {\prime} \in [ 0, 1 ], \Pr \left(\bigcap_ {i = 1} ^ {N} f \left(e _ {t _ {i}}\right)\right) \leq 8 e ^ {- \frac {N _ {\epsilon} ^ {\prime}}{3}} \tag {125}
+$$
+
+Because  $\epsilon' \geq \epsilon$ , we have:
+
+$$
+\Pr \left(\bigcap_ {i = 1} ^ {N} f \left(e _ {t _ {i}}\right)\right) \leq 8 e ^ {- \frac {N \epsilon^ {\prime}}{3}} \leq 8 e ^ {- \frac {N \epsilon}{3}}. \tag {126}
+$$
+
+which was to be proven.
+
+![](images/7fbe8ccd0f91508677aa140d6960ffaec37618233f61bcccd46a2c86ef1f6cd4.jpg)
+
+# D UPPER-BOUNDING MIXING TIMES FOR EXAMPLES
+
+Here, we prove that the values of  $\hat{t}_{\mathrm{mix}}$  used in the simulation experiments are in fact (somewhat loose) upper bounds on the true mixing times of  $\mathcal{T}_e$  for these environments. While in practice, the true mixing times would not be known a priori, it is important for the validity of our examples that the true  $t_{\mathrm{mix}}$  is in fact  $\leq \hat{t}_{\mathrm{mix}}$ .
+
+We use the following following well-known fact:
+
+For distributions  $\mathcal{A} \coloneqq \mathcal{A}_1 \otimes \mathcal{A}_2 \otimes \ldots \otimes \mathcal{A}_n$  and  $\mathcal{B} \coloneqq \mathcal{B}_1 \otimes \mathcal{B}_2 \otimes \ldots \otimes \mathcal{B}_n$ :
+
+$$
+\left\| \mathcal {A} - \mathcal {B} \right\| _ {T V} \leq \sum_ {i = 1} ^ {n} \left\| \mathcal {A} _ {i} - \mathcal {B} _ {i} \right\| _ {T V} \tag {127}
+$$
+
+First, we deal with the combination lock experiment. We can write the transition matrix for any arbitrary two-state Markov chain as
+
+$$
+\left[ \begin{array}{c c} 1 - \epsilon_ {0} & \epsilon_ {1} \\ \epsilon_ {0} & 1 - \epsilon_ {1} \end{array} \right] \tag {128}
+$$
+
+where  $0 \leq \{\epsilon_0, \epsilon_1\} \leq 1$ . Note that in our particular example, we have  $0.1 \leq \{\epsilon_0, \epsilon_1\} \leq 0.9$ .
+
+This matrix has eigenvalues 1 and  $1 - \epsilon_0 - \epsilon_1$ , and the stationary distribution (the eigenvector corresponding to the eigenvalue 1) is
+
+$$
+\pi_ {\infty} := \left[ \epsilon_ {1} / \left(\epsilon_ {0} + \epsilon_ {1}\right), \epsilon_ {0} / \left(\epsilon_ {0} + \epsilon_ {1}\right) \right] ^ {T}. \tag {129}
+$$
+
+Using the closed-form formula for the  $n$ 'th power of a two-state Markov Chain given by Williams (1992), we have:
+
+$$
+\left(\left[ \begin{array}{c c} 1 - \epsilon_ {0} & \epsilon_ {1} \\ \epsilon_ {0} & 1 - \epsilon_ {1} \end{array} \right]\right) ^ {n} = \frac {1}{\epsilon_ {0} + \epsilon_ {1}} \left[ \left[ \begin{array}{l l} \epsilon_ {1} & \epsilon_ {1} \\ \epsilon_ {0} & \epsilon_ {0} \end{array} \right] - (1 - \epsilon_ {0} - \epsilon_ {1}) ^ {n} \left[ \begin{array}{c c} - \epsilon_ {0} & \epsilon_ {1} \\ \epsilon_ {0} & - \epsilon_ {1} \end{array} \right] \right] \tag {130}
+$$
+
+To compute the mixing time, we compute the state distribution  $\pi_{n}$ ,  $n$  timesteps after each starting state:
+
+$$
+\left(\left[ \begin{array}{c c} 1 - \epsilon_ {0} & \epsilon_ {1} \\ \epsilon_ {0} & 1 - \epsilon_ {1} \end{array} \right]\right) ^ {n} \left[ \begin{array}{l} 1 \\ 0 \end{array} \right] = \frac {1}{\epsilon_ {0} + \epsilon_ {1}} \left[ \begin{array}{c} \epsilon_ {1} + (1 - \epsilon_ {0} - \epsilon_ {1}) ^ {n} \epsilon_ {0} \\ \epsilon_ {0} - (1 - \epsilon_ {0} - \epsilon_ {1}) ^ {n} \epsilon_ {0} \end{array} \right] \tag {131}
+$$
+
+and,
+
+$$
+\left(\left[ \begin{array}{c c} 1 - \epsilon_ {0} & \epsilon_ {1} \\ \epsilon_ {0} & 1 - \epsilon_ {1} \end{array} \right]\right) ^ {n} \left[ \begin{array}{l} 0 \\ 1 \end{array} \right] = \frac {1}{\epsilon_ {0} + \epsilon_ {1}} \left[ \begin{array}{l} \epsilon_ {1} - (1 - \epsilon_ {0} - \epsilon_ {1}) ^ {n} \epsilon_ {1} \\ \epsilon_ {0} + (1 - \epsilon_ {0} - \epsilon_ {1}) ^ {n} \epsilon_ {1} \end{array} \right] \tag {132}
+$$
+
+Note that the TV distance between either of these distributions and the stationary distribution  $\pi$  is at most
+
+$$
+\left\| \pi_ {n} - \pi_ {\infty} \right\| _ {T V} \leq \frac {\left| \left(1 - \epsilon_ {0} - \epsilon_ {1}\right) \right| ^ {n} \max  \left(\epsilon_ {0} , \epsilon_ {1}\right)}{\left(\epsilon_ {0} + \epsilon_ {1}\right)} \leq \left| \left(1 - \epsilon_ {0} - \epsilon_ {1}\right) \right| ^ {n}. \tag {133}
+$$
+
+The parameters  $\{\epsilon_0,\epsilon_1\}$  for each two-state Markov chain are chosen uniformly at random, such that  $0.1\leq \{\epsilon_0,\epsilon_1\} \leq 0.9$  . Therefore,  $0\leq |(1 - \epsilon_0 - \epsilon_1)|\leq 0.8$  . Then, for any individual chain, we have:
+
+$$
+\left\| \pi_ {n} - \pi_ {\infty} \right\| _ {T V} \leq \left| \left(1 - \epsilon_ {0} - \epsilon_ {1}\right) \right| ^ {n} \leq 0. 8 ^ {n} \tag {134}
+$$
+
+In the combination lock experiments, there are up to  $L = 512$  of these noise Markov chains; the probability distribution over the exogenous noise  $\mathcal{E}$  is the product distribution over these chains. Then we use Equation 127 to bound the total TV distance for the chain  $\mathcal{T}_e$  to its stationary distribution  $\pi_{\mathcal{E}}$ ; that is:
+
+$$
+\left\| \pi_ {n} ^ {\text {t o t a l}} - \pi_ {\mathcal {E}} \right\| _ {T V} \leq 5 1 2 \cdot 0. 8 ^ {n}. \tag {135}
+$$
+
+Then, by the definition of mixing time, we have:
+
+$$
+t _ {\text {m i x}} \leq \min  n, \text {s u c h} 5 1 2 \cdot 0. 8 ^ {n} \leq \frac {1}{4} \tag {136}
+$$
+
+Which gives us:
+
+$$
+t _ {\text {m i x}} \leq \left\lceil \frac {- \log (2 0 4 8)}{\log (0 . 8)} \right\rceil = 3 5. \tag {137}
+$$
+
+So the value that we use in the experiment,  $\hat{t}_{\mathrm{mix}} = 40$  , is a valid upper bound.
+
+For the multi-maze experiment, the exogenous noise state consists of eight identical mazes, with agents moving uniformly at random in each of them. Unlike the "combination lock" example, where the individual components of the exogenous noise are conditioned on parameters  $\epsilon_0$ ,  $\epsilon_1$ , which can vary, in the multi-maze example the individual mazes always represent instances of exactly the same, specific Markov chain. Let the transition matrix of this chain be  $M$ , with stationary distribution  $\pi_M$ . Then, by Equation 127, for the whole exogenous state  $\mathcal{T}_e$ , we have:
+
+$$
+\begin{array}{l} t _ {\text {m i x}} = \min  n, \text {s u c h} \forall s _ {0}, \| \left(\mathcal {T} _ {e}\right) ^ {n} s _ {0} - \pi_ {\mathcal {E}} \| \leq \frac {1}{4} \\ = \min  n, \text {s u c h} \forall s _ {0} ^ {(1)}, s _ {0} ^ {(2)}, \dots s _ {0} ^ {(8)}, \| M ^ {n} s _ {0} ^ {(1)} \otimes M ^ {n} s _ {0} ^ {(2)} \otimes \dots \otimes M ^ {n} s _ {0} ^ {(8)} - \\ \left. \pi_ {M} \otimes \pi_ {M} \otimes \dots \otimes \pi_ {M} \right\| \leq \frac {1}{4} \tag {138} \\ \leq \min  n, \text {s u c h t h a t} \forall s _ {0}, \sum_ {i = 1} ^ {8} \| M ^ {n} s _ {0} - \pi_ {M} \| \leq \frac {1}{4} \\ = \min  n, \text {s u c h} \forall s _ {0}, \| M ^ {n} s _ {0} - \pi_ {M} \| \leq \frac {1}{3 2} \\ \end{array}
+$$
+
+Which is to say that  $t_{mix}$  for the entire exogenous noise chain is upper-bounded by  $t_{mix}(1/32)$  for the individual maze chain  $M$ . Furthermore, while the state space for the entire chain is of size  $|\mathcal{E}| = 68^8$ , the individual maze chain  $M$  operates on a state of size 68. This is small enough that it is tractable to exactly compute  $t_{mix}(1/32)$  for  $M$  using numerical techniques. We performed the computation (source code is provided with the supplementary materials), and found that
+
+$$
+\min  n, \text {s u c h} \forall s _ {0}, \| M ^ {n} s _ {0} - \pi_ {M} \| \leq \frac {1}{3 2} = 2 9 3. \tag {139}
+$$
+
+Therefore, for the full exogenous noise chain  $\mathcal{T}_e$ , we have that  $t_{mix} \leq 293$ . Then the value that we use in the experiment,  $\hat{t}_{\mathrm{mix}} = 300$ , is a valid upper bound.
+
+# E DISCUSSION OF ASSUMPTIONS
+
+In this section, we argue that two major assumptions of the STEEL algorithm are necessary assumptions. In other words, if we remove these assumptions, then, we argue, achieving similar guarantees to our algorithm would become effectively impossible for any algorithm to accomplish in the single-trajectory, no-rews Ex-BMDP setting (at least without making additional assumptions to compensate). These assumptions are (1) reachability: the assumption that all endogenous states  $s \in S$  are reachable from one another in  $T$ ; and (2) the availability of a known upper bound on the exogenous state mixing time  $t_{\mathrm{mix}}$ .
+
+# E.1 REACHABILITY
+
+Here, we argue that any algorithm for single-trajectory, no-rews representation learning of Ex-BMDPs must necessarily make a reachability assumption: that all endogenous latent states must be reachable from one another. At a high level, we argue that, if all latent states are not reachable from one another, then an algorithm must visit states in a particular order in order to explore the entire dynamics with a single trajectory. However, because the latent dynamics are not known a priori, it is impossible for an algorithm to guarantee that it visits states in the appropriate order.
+
+More formally, recall that the reachability assumption is that for any pair of latent states  $s_1, s_2$ , there exists both a path from  $s_1$  to  $s_2$  and a path from  $s_2$  to  $s_1$ . Consider any Ex-BMDP  $M$  where this condition does not hold, such that  $|\mathcal{A}| \geq 2$ , and any learning algorithm  $A$ . Firstly, if there is any pair of states  $s_1$  and  $s_2$  where neither can reach the other, then clearly a single trajectory is insufficient to learn the Ex-BMDP, because it cannot visit both  $s_1$  and  $s_2$ . Therefore, we restrict to the case where, for each pair of states, either both are reachable from each other, or (without loss of generality)  $s_2$  is reachable from  $s_1$  but  $s_1$  is not reachable from  $s_2$ .
+
+Consider every edge  $(s, a, s')$  of the state transition graph defined by the state transition function  $T$ . If, for all such edges,  $s$  is reachable from  $s'$ , then the reachability assumption holds on the entire dynamics (because all single "steps" are invertible by some sequence of actions, so for any pair  $s_1, s_2$ , the path from  $s_1$  to  $s_2$  implies the existence of a path from  $s_2$  to  $s_1$ ). Therefore, if reachability does not hold, then there exists some edge  $(s, a, s')$  such that  $s$  is not reachable from  $s'$ . Now, consider the first time that the algorithm  $A$  encounters the state  $s$ . Regardless of the details of  $A$ , there must exist some action  $a'$  such that, on this first encounter, the probability of  $A$  taking action  $a'$  is at least  $1 / |\mathcal{A}|$ .
+
+Therefore, if  $a' = a$ , then probability that the algorithm  $A$  never revisits  $s$ , and so never takes any other action from  $s$  apart from  $a'$ , is at least  $1 / |\mathcal{A}|$ . Then, with substantial probability, the algorithm  $A$  never explores the  $|\mathcal{A}| - 1$  other possible transitions from  $s$ , and cannot possibly learn the full dynamics of the Ex-BMDP. (To be more precise, the algorithm's output will not depend at all on the ground-truth value of  $T(s, a'')$  for  $a'' \in \mathcal{A} \setminus \{a\}$ , and so is highly unlikely to return the correct values for these transitions on arbitrary Ex-BMDPs.)
+
+Alternatively, if  $a' \neq a$ , then consider the alternative Ex-BMDP  $M'$ , which is identical to  $M$  in every way (in terms of dynamics, exogenous state, emission function, etc.), except that the effects of actions  $a$  and  $a'$  on the latent state  $s$  are swapped. Note that before first encountering the latent state  $s$ , the MDPs  $M$  and  $M'$  will produce identically-distributed sequences of observations, so the algorithm  $A$  will behave identically on them, and have identical internal memory/state. Then, when first encountering  $s$ , the algorithm  $M$  on  $A$  will take action  $a'$  with substantial probability, and then transition to  $s'$  and be unable to revisit  $s$ .
+
+Therefore, for any Ex-BMDP  $M$  that violates reachability, any algorithm  $A$  is either likely to fail on  $M$ , or to fail on a slightly-modified version of  $M$ . In any case, no such algorithm will be able to succeed with high probability on any general class of Ex-BMDPs that does not require reachability.
+
+# E.2 KNOWN UPPER BOUND ON THE MIXING TIME  $t_{\mathrm{MIX}}$
+
+Here, we argue that the assumption that an upper bound on the mixing time is provided is also necessary. We argue that:
+
+1. Any general, provably sample-efficient algorithm for learning, with high probability, an Ex-BMDP from a single trajectory must necessarily require an upper bound on the mixing time of the exogenous noise to be provided a priori (unless some other information about the exogenous noise is provided.)  
+2. The runtime of any algorithm for learning an Ex-BMDP must, in the worst case, include some term that scales at least linearly with the mixing time of the exogenous noise.
+
+At a high level, we argue that, with a single trajectory, it can be impossible to distinguish between a static exogenous dynamics (i.e., where only one exogenous state exists) and an extremely slow-mixing exogenous dynamics, in time sublinear in the mixing time. For any proposed algorithm that does not have access to a bound on the mixing time, and any Ex-BMDP with a single exogenous state (that is, a Block MDP), we can construct an extremely slow-mixing Ex-BMDP that behaves identically to the Block MDP with substantial probability during the entire duration of the runtime of the learning algorithm, but which at equilibrium has a quite different distribution of observations. This will make the learned encoder fail on the Ex-BMDP at equilibrium.
+
+More precisely, to show (1): suppose the converse is true: that there exists some algorithm  $A$  that learns Ex-BMDP latent dynamics and state encoders, which is not provided any upper-bound on the mixing time of the exogenous noise of the Ex-BMDP, or any other information about the exogenous noise distribution. We assume that  $A$  can learn the correct endogenous dynamics, as well as an encoder with accuracy (for every endogenous state, under the stationary exogenous distribution) of at least  $1 - \epsilon$ , with probability at least  $1 - \delta$ , for small values of  $\epsilon$  and  $\delta$ . Then, consider any Ex-BMDP  $M_1$  and related parameterized family of Ex-BMDPs  $M_2(\gamma), M_3(\gamma)$  with the following properties:
+
+-  $M_{1}$  has  $\mathbf{N}$  endogenous latent states  $s_1, \ldots, s_N$  with some transition function  $T$ , a single exogenous latent state  $e_1$ , and an emission function  $\mathcal{Q}(s, e_1)$ . (In other words,  $M_{1}$  is any Block MDP).  
+-  $M_2(\gamma)$  has the same endogenous states and transition probabilities as  $M_1$ , but has two exogenous latent states  $e_1, e_2$ . The state  $e_1$  transitions to  $e_2$ , and  $e_2$  transitions to  $e_1$ , each with probability  $\gamma$ . Note that the stationary distribution of the exogenous state is uniform over  $e_1$  and  $e_2$ . We also assume that the initial exogenous state distribution is uniform over  $e_1$  and  $e_2$ . Regardless of  $\gamma$ , the emission function of  $M_2(\gamma)$  is defined such that  $\forall s_i, Q_{M_2}(s_i, e_1) = Q_{M_1}(s_i, e_1)$ .  
+-  $M_3(\gamma)$  is identical to  $M_2(\gamma)$ , except that its emission function is defined as  $\forall s_i, \mathcal{Q}_{M_3}(s_i, e_1) := \mathcal{Q}_{M_1}(s_i, e_1)$ , but  $\forall s_i, \mathcal{Q}_{M_3}(s_i, e_2) := \mathcal{Q}_{M_2}(s_{(i+1)\% N}, e_2)$ . In other words, when the exogenous state is equal to  $e_2$ , the emission distributions for the endogenous states are permuted in  $M_3$  compared to  $M_2$ .
+
+We also assume that the encoder hypothesis class can represent the inverses of  $\mathcal{Q}_{M_1}$ ,  $\mathcal{Q}_{M_2}$ , and  $\mathcal{Q}_{M_3}$ . However, note that by construction, any fixed encoder  $\phi(\cdot)$  which has accuracy at least  $1 - \epsilon$  on  $M_2(\gamma)$  (for every endogenous state, under the stationary exogenous distribution) can only have accuracy at most  $0.5 + \epsilon$  on  $M_3(\gamma)$ , because, when the exogenous state is  $e_2$ , any time that the encoder returns the correct latent state for  $M_2(\gamma)$ , it will return the incorrect latent state for  $M_3(\gamma)$ .<sup>8</sup>
+
+Now, consider what happens when we run the algorithm  $A$  on  $M_1$ . The number of environment steps that  $A$  takes on  $M_1$  forms some distribution; let  $t$  be the 90'th percentile of this distribution, so that with probability 0.9,  $A$  stops sampling before step  $t$ .
+
+Now, we can set  $\gamma \coloneqq 1 - 0.9^{1 / t}$ , so that, with probability 0.9, the endogenous state of  $M_2(\gamma)$  or  $M_3(\gamma)$  do not change before step  $t$ . Therefore, by union bound, on  $M_2(\gamma)$  or  $M_3(\gamma)$ , if the initial exogenous state is  $e_1$ , then with probability at least 0.8, the exogenous state will stay constant at  $e_1$
+
+up to timestep  $t$ , and the algorithm  $A$ , seeing exactly the same distribution of observations as if the MDP was  $M_{1}$ , will halt before timestep  $t$ .
+
+Considering the  $50\%$  probability that the exogenous state starts at  $e_1$ , this gives at least a  $40\%$  chance that  $A$  applied to  $M_2(\gamma)$  or  $M_3(\gamma)$  never encounter the exogenous state  $e_2$ . In this case, the distribution of encoders output by  $A$  should be the same for the two Ex-BMDPs: let this distribution be  $\mathcal{G}$ . (That is,  $\mathcal{G}$  is the conditional distribution of encoders output by  $A$  when applied to  $M_2(\gamma)$  or  $M_3(\gamma)$ , given that  $A$  never encounters  $e_2$ .) However, because  $A$  fails on  $M_2(\gamma)$  with probability at most  $\delta$ , we can conclude that at least  $(1 - 0.4^{-1}\delta)$  of the encoders from the distribution  $\mathcal{G}$  represent successes of  $A$  on  $M_2(\gamma)$ .
+
+Because  $A$  on  $M_3(\gamma)$  also draws from  $\mathcal{G}$  with probability at least 0.4, we can conclude that at least  $0.4 - \delta$  of the time,  $A$  on  $M_3(\gamma)$  produces an encoder that is highly accurate on  $M_2(\gamma)$ . As discussed above, any encoder that is highly-accurate on  $M_2(\gamma)$  cannot be highly accurate on  $M_3(\gamma)$ , so  $A$  must fail on  $M_3$  with substantial probability.
+
+Therefore, we can conclude that for any algorithm  $A$  that has no "hint" about the exogenous dynamics (such as knowing the mixing time), there exists some  $\gamma$  such that  $A$  cannot possibly succeed with high probability on both  $M_2(\gamma)$  and  $M_3(\gamma)$ .
+
+To show point (2), simply note that in this construction, in order to ensure that  $e_2$  is observed with probability  $(1 - \delta)$ , an algorithm must observe at least the first  $\lceil \log(2\delta) / \log(1 - \gamma) \rceil$  timesteps. Meanwhile, the mixing time of the two-state Markov chain exogenous state is given by  $\lceil -1 / \log_2(1 - 2\gamma) \rceil$ . For a fixed  $\delta$  and as  $\gamma$  approaches 0, the ratio between these quantities approaches a constant: therefore, the number of steps required to ensure that  $e_2$  is observed with high probability is linear in the mixing time.
+
+# F DOUBLE-PRIME LOOP EXPERIMENTS
+
+In this section, we present additional experiments which demonstrate that, on some types on Ex-BMDPs, the STEEL algorithm can empirically outperform the algorithms proposed in Lamb et al. (2023) and Levine et al. (2024), which do not have sample-complexity guarantees. We show that there are certain structures of Ex-BMDP latent dynamics which are difficult for these prior methods to learn efficiently, but on which STEEL performs well. Specifically, we look at a family of "double-prime loop" tabular Ex-BMDPs which are discussed by Levine et al. (2024). First, though, we describe the algorithms proposed by Lamb et al. (2023) and Levine et al. (2024), and motivate why certain dynamics structures such as "double-prime loops" present a challenge to these methods.
+
+# F.1 BACKGROUND
+
+# F.1.1 AC-STATE AND ACDF ALGORITHMS
+
+Lamb et al. (2023) and Levine et al. (2024) both propose algorithms to learn endogenous state encoders in the Ex-BMDP framework.
+
+Lamb et al. (2023) first proposed the AC-State algorithm, which aims to learn an encoder  $\phi : \mathcal{X} \to \mathbb{N}$ , such that, under some one-to-one permutation  $\sigma$ ,  $\sigma(\phi(x)) = \phi^*(x)$ . To accomplish this task, AC-State optimizes  $\phi$  using a multistep inverse dynamics objective.
+
+Specifically, in the theoretical treatment, AC-State tries to find the encoder  $\phi$  which optimizes the following objective:
+
+$$
+\begin{array}{l} \mathcal {L} (\phi) := \min  _ {g \in \mathbb {N} \times \mathbb {N} \times \mathbb {N} \rightarrow \mathcal {P} (\mathcal {A})} \underset {k \sim \{1, \dots , K \} (x _ {t}, a _ {t}, x _ {t + k}) \sim \mathcal {D}} {\mathbb {E}} - \log \left(g \left(\phi \left(x _ {t}\right), \phi \left(x _ {t + k}\right), k\right) [ a _ {t} ]\right) \tag {140} \\ \phi_ {\mathrm {A C - S t a t e}} := \operatorname * {a r g   m i n} _ {\mathcal {L} (\phi) = \min  _ {\phi^ {\prime}} \mathcal {L} (\phi^ {\prime})} \\ \end{array}
+$$
+
+In other words, the loss  $\mathcal{L}(\phi)$  is minimized on tuples  $(x_{t},a_{t},x_{t + k})$  sampled from a trajectory, consisting of an observation  $x_{t}$ , the following action  $a_{t}$ , and the observation  $k$  steps in the future,  $x_{t + k}$ .
+
+The encoder  $\phi (x)$  is trained to retain any information about the observations  $\phi (x_{t})$  and  $\phi (x_{t + k})$  that is useful for predicting  $a_{t}$ . Specifically, an inverse-dynamics model  $g$  is simultaneously trained with
+
+$\phi$  to predict  $a_{t}$  given  $\phi (x_{t}),\phi (x_{t + k})$ , and  $k$ . The loss  $\mathcal{L}(\phi)$  is then taken as the minimum loss for  $\phi$  over all such functions  $g$ .
+
+However, note that, based on this loss function alone, the identity encoder  $\phi(x) := x$  achieves the minimum possible value of  $\mathcal{L}(\phi)$ . Therefore, in order to filter exogenous information, the final encoder returned by AC-State is the minimal range encoder that minimizes the loss function.
+
+Lamb et al. (2023) claims that, if the data  $\mathcal{D}$  is collected by a policy  $\pi(x)$  that only depends on  $x$  through  $\phi^{*}(x)$  (that is, the policy ignores exogenous noise); and the maximum segment length  $K$  is at least the endogenous dynamics diameter  $D$ ; and the endogenous dynamics are deterministic, then the final encoder  $\phi_{\mathrm{AC-State}}$  returned will correctly encode the endogenous latent state (in the limit of infinite data; and with sufficient function approximation).
+
+When applied in practice,  $\phi$  and  $g$  are learned neural networks, and a vector quantization bottleneck is used to restrict the range of  $\phi$  (so the output of  $\phi$  is a quantized vector, rather than an integer.)
+
+Levine et al. (2024) subsequently show that, in some cases, the AC-State objective in Equation 140 is not sufficient to learn a correct latent state encoder of an Ex-BMDP, even with unbounded amounts of data. Levine et al. (2024) identify specific flaws in the proofs of Lamb et al. (2023), and propose a modified loss function to address these flaws. In particular,
+
+1. The maximum segment length  $K$  required to correctly learn the endogenous encoder must in some cases be larger than  $D$ ; a corrected upper bound on the necessary segment length of  $2D^2 + D$  is given.  
+2. If the endogenous dynamics are periodic, then the multistep inverse objective can be insufficient, for any choice of  $K$ . Levine et al. (2024) then propose to add an auxiliary loss function to the loss in Equation 140, namely a latent forward dynamics loss:
+
+$$
+\mathcal {L} _ {\text {f o r w a r d}} (\phi) := \min  _ {h \in \mathbb {N} \times \mathcal {A} \rightarrow \mathcal {P} (\mathbb {N})} \underset {(x _ {t}, a _ {t}, x _ {t + 1}) \sim \mathcal {D}} {\mathbb {E}} - \log \left(h \left(\phi (x _ {t}), a _ {t}\right) [ \phi (x _ {t + 1}) ]\right). \tag {141}
+$$
+
+This loss term enforces that the transitions in the learned endogenous latent state space are deterministic; Levine et al. (2024) prove that enforcing this constraint is sufficient, when combined with the multistep inverse loss, to ensure that a correct endogenous representations are learned even when the endogenous latent dynamics are periodic. Levine et al. (2024) name the AC-State algorithm with this modified loss function ACDF.
+
+Note that because  $D$  is in general unknown a priori,  $K$  must be treated as a hyperparameter of the algorithm, so point (1) above is primarily of theoretical interest, but may aid in setting this hyperparameter if there is some prior knowledge of  $D$ .
+
+So far, we have only described the learning objectives of the two proposed algorithms: now, we address how these algorithms collect the trajectory from which the tuples  $(x_{t},a_{t},x_{t + k})$  are sampled. Recall that a condition on the correctness of AC-State (and ACDF) is that the data-collection policy does not depend on the noise in the observation, but only on the endogenous latent state. This condition naturally raises the question of how exactly an algorithm intended to discover  $\phi^{*}$  can take actions that depend only on  $\phi^{*}(x)$ , without already knowing  $\phi^{*}$  in the first place.
+
+Lamb et al. (2023) performs experiments using two data-collection policies: (1). a uniformly random policy, which meets the condition simply by ignoring the observation  $x$  entirely; and (2) a goal-seeking policy, which aims to maximize latent state coverage. For the latter policy, the data is collected simultaneously with training  $\phi$ , and the partially-trained encoder  $\phi_t$  is used to (imperfectly) filter out exogenous noise and plan in (an approximation of) the endogenous latent space. Lamb et al. (2023) explicitly acknowledge that this bootstrapping approach breaks the condition that the action  $a_t$  depends only on the true endogenous state  $\phi^*(x_t)$ , which is necessary in their proof of the correctness of their algorithm. Despite this, they observe that the approach seems to work well empirically. Meanwhile, Levine et al. (2024) only use uniformly random actions in their experiments.
+
+For now, we will assume that data is collected under a uniformly-random policy. We will return to discussing the latent-state-coverage maximizing approach proposed by Lamb et al. (2023) later, in Section F.3.
+
+![](images/6254c00d634e31d1aa60ead168655370096ab6d2121da5586ef10f37d6bb5351.jpg)  
+Figure 5: Latent dynamics of DoublePrime(11, 13).
+
+# F.1.2 DOUBLE-PRIME LOOPS
+
+To specifically demonstrate that  $K \approx \mathcal{O}(D^2)$  can be necessary to learn a correct encoder, even when a forward dynamics loss is also being used, Levine et al. (2024) give a concrete example. This example is the family of tabular "double-prime loop" Ex-BMDPs. For any two primes  $p, q$ , with  $p < q$ , let the Ex-BMDP DoublePrime  $(p, q)$  be defined as follows:
+
+-  $\mathcal{S} = \{0,1,\dots,(q - 1),0',1',\dots,(q - 1)'\}$ ;  $\mathcal{A} = \{0,1\}$ ;  $\mathcal{E} = \{e_0\}$ .  
+-  $\mathcal{X} = \mathcal{S};\mathcal{Q}(s,e_0) = s;\mathcal{T}_e(e_0) = e_0.$  
+- The endogenous latent state transition function  $T$  is defined as:
+
+$$
+T (s, a) = \left\{ \begin{array}{l l} (s + 1) \% q & \text {if } s \in \{1, \dots, q - 1 \} \text {or} (s = 0 \text {and} a = 0) \\ ((s + 1) \% q) ^ {\prime} & \text {if } s \in \{1 ^ {\prime}, \dots, (q - 1) ^ {\prime} \} \text {or} (s = 0 ^ {\prime} \text {and} a = 0) \\ (q - p + 1) ^ {\prime} & \text {if} (s = 0 \text {and} a = 1) \\ (q - p + 1) & \text {if} (s = 0 ^ {\prime} \text {and} a = 1). \end{array} \right. \tag{142}
+$$
+
+In other words, DoublePrime  $(p,q)$  is a noise-free Ex-BMDP, where all of the dynamics are deterministic, and the latent state  $s$  is directly observed as  $x$ . The dynamics consist of two connected loops, each of  $q$  states. The actions are generally ignored and the agent continues to progress through a loop, except for in states 0 and  $0'$ , where taking the action 1 transports the agent to the other loop, at position  $q - p + 1$ . As an example, the dynamics of DoublePrime(11, 13) are shown in Figure 5.
+
+There is also a related family of "single-prime loop" Ex-BMDPs. Let  $\mathrm{SinglePrime}(p,q)$  be defined as (borrowing notation from Levine et al. (2024)):
+
+-  $\mathcal{S} = \{0^*, 1^*, \dots, (q - 1)^*\}$ ;  $\mathcal{A} = \{0, 1\}$ ;  $\mathcal{E} = \{e_0\}$ ;  $\mathcal{T}_e(e_0) = e_0$ .  
+-  $\mathcal{X} = \{0,1,\dots,(q - 1),0',1',\dots,(q - 1)'\}$ .  
+$\mathcal{Q}(s^{*},e_{0}) = \left\{ \begin{array}{ll}s & \mathrm{with~probability~}1 / 2\\ s^{\prime} & \mathrm{with~probability~}1 / 2. \end{array} \right.$  
+- The endogenous latent state transition function  $T$  is defined as:
+
+$$
+T \left(s ^ {*}, a\right) = \left\{ \begin{array}{l l} \left(\left(s + 1\right) \% q\right) ^ {*} & \text {if} s \in \left\{1 ^ {*}, \dots , (q - 1) ^ {*} \right\} \text {or} (s = 0 ^ {*} \text {and} a = 0) \\ \left(q - p + 1\right) ^ {*} & \text {if} (s = 0 ^ {*} \text {and} a = 1). \end{array} \right. \tag{143}
+$$
+
+In other words,  $\mathrm{SinglePrime}(p,q)$  has the same observation space  $\mathcal{X}$  and action space  $\mathcal{A}$  as  $\mathrm{DoublePrime}(p,q)$ , but fewer controllable latent states. In  $\mathrm{SinglePrime}(p,q)$ , when the agent is at latent state  $s_t = i^*$ , either the observation  $x_{t} = i$  or the observation  $x_{t} = i'$  is emitted, each with probability 0.5. The agent progresses through the loop of latent states regardless of actions, except
+
+![](images/f2c7be9ceb6cd838772c795e536f64d0be1bbfc9154979676f54c13ba42c1a08.jpg)  
+Figure 6: Latent dynamics of SinglePrime(11, 13).
+
+in latent state  $0^{*}$ , where taking action 1 transitions the latent state to  $(q - p + 1)^{*}$ . (See Figure 6 for an example.) Note that if we ignore the distinction between the observations  $i$  and  $i'$ , then DoublePrime  $(p, q)$  and SinglePrime  $(p, q)$  appear to have identical observed dynamics.
+
+Let  $\phi_{DP}(x) \coloneqq x$  be the optimal encoder for DoublePrime  $(p, q)$ , and
+
+$$
+\phi_ {S P} (x) := i ^ {*} \text {i f} x = i \text {o r} x = i ^ {\prime} \tag {144}
+$$
+
+be the optimal encoder for SinglePrime  $(p,q)$
+
+Levine et al. (2024) show that, on the Ex-BMDP DoublePrime  $(p,q)$  under a uniformly random behavioral policy, in the limit of infinite training data, the encoders  $\phi_{DP}$  and  $\phi_{SP}$  have exactly the same loss under the loss function in Equation 140, with  $K\leq (q - 1)p$ . Note that if  $p$  and  $q$  are close, then  $(q - 1)p\approx D^2 /4$ . Because Equation 140 prefers the smallest-range encoder among encoders with identical losses, this means that AC-State will incorrectly return  $\phi_{SP}$  (which only has  $q$  distinct outputs, rather than  $2q$ ). Furthermore, the forward-dynamics loss suggested by Levine et al. (2024) does not help in this case: it will be zero for both encoders under data generated by DoublePrime  $(p,q)$ .
+
+It is important to note that  $\mathrm{SinglePrime}(p,q)$  and  $\mathrm{DoublePrime}(p,q)$  have truly distinct controllable latent dynamics: in  $\mathrm{DoublePrime}(p,q)$ , given sufficient lead-time, the agent can (eventually) control whether it is in state  $1'$  or 1 at a future time step, while an agent in  $\mathrm{SinglePrime}(p,q)$  can never control this. Therefore, AC-State is making an error if it returns  $\phi_{SP}$  rather than  $\phi_{DP}$ .
+
+Levine et al. (2024) only considers the case of unlimited data, and only introduces the double-prime loop Ex-BMDPs to make the point that the hyperparameter  $K$  must be set very high in some cases. Here, however, we will also show empirically that, even when using a carefully-tuned and very large value of  $K$ , AC-State and ACDF can take many samples to correctly learn these dynamics.
+
+# F.1.3 TABULAR EXPERIMENTS IN LEVINE ET AL. (2024)
+
+Levine et al. (2024) empirically compare AC-State (Lamb et al., 2023) to their proposed ACDF method, and empirically explore the effect of the hyperparameter  $K$ . To do so from a purely statistical perspective (i.e., controlling for differences in optimization), in one set of experiments, Levine et al. (2024) perform tests on small tabular Ex-BMDPs, including SinglePrime(3,5) and DoublePrime(3,5). $^{10}$  For these experiments, the final learned  $\phi_{\mathrm{AC-State}}$  (or  $\phi_{\mathrm{ACDF}}$ ) is found by exhaustively computing the losses in Equations 140 and 141 on all possible encoders  $\phi \in \mathcal{X} \rightarrow \{0,\dots,|\mathcal{X}| - 1\}$  (up to a relabeling perturbation of the output). For each encoder  $\phi$ , the multistep inverse model  $g$  (and, for ACDF, the forward model  $h$ ) is estimated as a tabular function based on
+
+the collected data:
+
+$$
+g \left(s, s ^ {\prime}, k\right) [ a ] := \frac {\# \text {o f i n s t a n c e s o f} \left(x _ {t} , a _ {t} , x _ {t + k}\right) \text {s u c h t h a t} \phi \left(x _ {t}\right) = s ; a _ {t} = a ; \text {a n d} \phi \left(x _ {t + k}\right) = s ^ {\prime}}{\# \text {o f i n s t a n c e s o f} \left(x _ {t} , a _ {t} , x _ {t + k}\right) \text {s u c h t h a t} \phi \left(x _ {t}\right) = s \text {a n d} \phi \left(x _ {t + k}\right) = s ^ {\prime}}. \tag {145}
+$$
+
+However, due to overfitting, this tabular definition of  $g$ , if used naively, would always lead to the identity encoder  $\phi(x) \coloneqq x$  having the lowest empirical loss, regardless of the true latent encoder. Therefore, Levine et al. (2024) collect two trajectories for each experiment, and uses one to fit  $g$  and  $h$  for each possible encoder  $\phi$ , and the other to evaluate the loss on each  $\phi$  in order to determine the final output encoder  $\phi_{\mathrm{AC-State}}$  (or  $\phi_{\mathrm{ACDF}}$ ). We will call these sets of tuples the fitting set and optimization set, respectively.
+
+Because the number of possible encoders grows very quickly with  $|\mathcal{X}|$ , Levine et al. (2024) limit these experiments to very small tabular Ex-BMDPs, with  $|\mathcal{X}| \leq 10$ .
+
+With this background out of the way, we describe our experiments:
+
+# F.2 EXPERIMENTS
+
+# F.2.1 SETUP
+
+Given that Levine et al. (2024) describe how learning the correct latent state encoder for DoublePrime  $(p, q)$  using a multistep-inverse loss requires taking into account specific long-duration dependencies between states, we hypothesized that learning such an encoder using multistep-inverse methods may be considerably less sample-efficient than doing so using STEEL, particularly for large  $p$  and  $q$ . We therefore adapted the tabular Ex-BMDP experimental setup from the released code of Levine et al. (2024) and tested the algorithms head-to-head.
+
+However, for large  $p$  and  $q$ , the process of exhaustively searching all possible encoders (that is, all partitions of  $2q$  elements) quickly becomes intractable. Therefore, we restricted the hypothesis space of encoders to only include the two hypotheses  $\phi_{SP}$  and  $\phi_{DP}$ . Note that this modification makes the learning task strictly "easier." For a fair comparison, we also restricted the hypothesis class for STEEL to the minimum-possible set of "one-versus-rest" classifiers that would ensure realizability for both DoublePrime  $(p,q)$  and SinglePrime  $(p,q)$ . In particular, this is  $3q$  classifiers  $f \in \mathcal{F}$ : for each  $i \in \{0,\dots,1 - q\}$ ,  $\mathcal{F}$  includes a hypothesis which distinguishes  $i$  from all other observations, a hypothesis which distinguishes  $i'$  from all other observations, and a hypothesis which distinguishes  $i$  or  $i'$  from all other observations.
+
+We make the following further modifications to the experimental protocol from Levine et al. (2024):
+
+- When assessing the "minimum-range" minimal loss encoder as in Equation 140, Levine et al. (2024) include an empirical "fudge factor": their protocol returns the minimum-range encoder that achieves a loss within  $0.1\%$  of the true minimum loss over all possible encoders. We found that even without this fudge factor, the multistep-inverse methods were still heavily biased towards returning  $\phi_{SR}$ , when applied either to SinglePrime  $(p,q)$ ; or to DoublePrime  $(p,q)$  with too-small  $K$  or too-few samples. This observation has a simple statistical explanation: if, in the "infinite sample" limit,  $\operatorname{Pr}(a_t|s_t = i, s_{t+1} = j) = \operatorname{Pr}(a_t|s_t = i', s_{t+1} = j')$ , then fitting  $g(i,j,k)$  to both samples of  $(x_t = i, a_t, x_{t+k} = j)$  and samples of  $(x_t = i', a_t, x_{t+k} = j')$  from the "fitting set" will lead to a higher-quality multistep inverse model  $g$ , and therefore lower loss on the "optimization set", compared to using only the (smaller number of) samples of  $(x_t = i, a_t, x_{t+k} = j)$  alone. We therefore remove the fudge factor entirely, and in fact return  $\phi_{GR}$  in the case of exact numerical ties.  
+- Because the forward-model loss in Equation 141 is zero for both  $\phi_{SR}$  and  $\phi_{GR}$  for any dataset collected from DoublePrime $(p, q)$ , we do not use it in our experiments: that is, we regard AC-State and ACDF as equivalent in this setting, and refer to AC-State alone from here onward. (Technically, the forward model loss would help identify the correct encoder  $\phi_{SR}$  in data generated from SinglePrime $(p, q)$ . However, as mentioned above,  $\phi_{SR}$  is already returned essentially "by default" by AC-State, so this loss term turns out to be unnecessary.)  
+- To better match the spirit of "learning an Ex-BMDP from a single trajectory", we collect the "fitting" and "optimization" sets from one single trajectory, one after the other, with each corresponding to half of the trajectory: we regard the total sample complexity as the
+
+length of the entire trajectory. Within of these two sub-trajectories, we collect all available tuples  $(x_{t},a_{t},x_{t + k})$  for all  $k\leq K$ . There are therefore slightly fewer tuples with  $k = K$  than with  $k = 1$ : we weight all tuples equally in the loss (rather than weighting all values of  $k$  equally; Lamb et al. (2023) is unclear about the "correct" behavior here, and it is not theoretically important).
+
+- Based on Equation 145,  $g(s,s^{\prime},k)[a]$  can be zero, which, by Equation 140, can lead to an infinite loss on a sample in the "optimization" set. In order to avoid this, Levine et al. (2024) set  $g(s,s^{\prime},k)[a]$  to an arbitrary floor value of  $10^{-7}$ . As a more principled and scale-invariant solution, we use standard Laplace ("add-one") smoothing when fitting  $g$ .
+
+We empirically compare the sample-efficiency of STEEL to AC-State on these problems with  $(p,q) = (3,5),(5,7),(11,13),(17,19),(29,31)$ , and (41,43).
+
+In our comparison, we run each algorithm under the "best" choice of hyperparameters. For AC-State, the conduct a large hyperparameter search to find the optimal  $K$ , while for STEEL, we simply choose a single set of hyperparameters that will succeed on both DoublePrime $(p,q)$  and SinglePrime $(p,q)$  based on our prior knowledge of the problems. Specifically:
+
+- For AC-State, for each tested value of  $(p,q)$ , we first collect 20 validation trajectories of DoublePrime $(p,q)$  (under a uniformly random policy), for increasing dataset sizes starting at  $T = 100$  steps. At each tested dataset size  $T$ , we compute the optimal  $\phi_{\mathrm{AC-State}}$  for each of the 20 trajectories, with each possible value of  $K$  from  $K = 1$  to  $K = 50,000$ . We first repeatedly increase the dataset size  $T$  by factors of 10. Once we first identify a  $T = 10^{m}$  such that, for at least some  $K$ , AC-State correctly returns  $\phi_{\mathrm{AC-State}} = \phi_{DP}$  for all 20 trajectories, we then test with  $T = \{2 \cdot 10^{m-1}, 3 \cdot 10^{m-1}, \dots, 9 \cdot 10^{m-1}\}$ . At the earliest of these timesteps at which, for some  $K$ , a correct encoder is learned for all 20 trajectories, we record the median such  $K$  as the "tuned" value  $K_{\mathrm{opt}}$  of this hyperparameter. (See Figure 7 for the results of the hyperparameter search.) At this point, we perform the actual test: we evaluate the success rate of AC-State on DoublePrime $(p,q)$ , using only  $K = K_{\mathrm{opt}}$  for values of  $T$  starting at  $T = 10^{m-1}$  and increasing by increments of  $10^{m-1}$ , on 20 new trajectories for each tested dataset size. We stop when this test-time accuracy reaches 20/20 on trajectories from DoublePrime $(p,q)$ , at some  $T_{\mathrm{max}}$ . Finally, we verify that AC-State can also consistently, correctly learn  $\phi_{SP}$  on data from SinglePrime $(p,q)$  with  $K = K_{\mathrm{opt}}$  and a trajectory length of  $T_{\mathrm{max}}$ , for 20 trajectories. (This always held in practice. $^{11}$ )
+
+- For STEEL, we set  $N = \hat{D} = 2q$  (to match the maximum number of states in DoublePrime(p,q) or SinglePrime(p,q));  $\hat{t}_{\text{mix}} = 1$  (because neither DoublePrime(p,q) nor SinglePrime(p,q) have time-correlated noise);  $\epsilon = 0.49$  (because, due to the simple emission distributions  $\mathcal{Q}$  of SinglePrime and DoublePrime, a  $51\%$  encoder accuracy on each latent state implies a  $100\%$  encoder accuracy); and  $\delta = 0.05$ . We test for 20 trials each of SinglePrime(p,q) and DoublePrime(p,q). All of these tests returned the correct encoders, so the only metric we needed to consider was the number of environment steps taken for each run.
+
+# F.2.2 RESULTS
+
+Our top-line results are reported in Table 3. Our first reported statistic is the "Max Steps": the number of environment steps at which all 20 out of 20 tested trajectories for each of DoublePrime  $(p, q)$  and SinglePrime  $(p, q)$  returned the correct encoder. (For STEEL, this is simply the maximum steps taken over all 40 trajectories; for AC-State, this is the  $T_{\mathrm{max}}$  discovered in the test-time search described in the previous section.) We see that, while for small instances  $(|S| \leq 14)$  of DoublePrime  $(p, q)$ , AC-State is more sample-efficient than STEEL, STEEL scales efficiently to larger instances of the problem.
+
+![](images/d6f796a99337de46df3d698c362df8b08e2f829151080b81c0dd0975658a834a.jpg)
+
+![](images/c83d5cd62bea833961ffee9319e6953b57079f7edd4a6e1addfb477fd0ce6b02.jpg)
+
+![](images/25ca088f5b270d07cd2efc8264e7d2de6cfab4c0f0cfc30cb77d409e5d0b98f0.jpg)
+
+![](images/b2b6b8f73d8af6fb9d5d42854a28c73f0353088f37de02b0e582f646654d69d7.jpg)
+
+![](images/c24542e788cf72e5d206423887c008419b19e811db700c77dc91697276acf535.jpg)  
+Figure 7: Results from the hyperparameter search for the maximum-step-count parameter  $K$  in Equation 140. The objective is to find a  $K$  which leads to a high success rate in learning the correct latent state encoder for DoublePrime( $p, q$ ), for the lowest possible number of environment steps.
+
+![](images/2c509e0537b4da4d0e9ab46035046e988b3921d206eb945f267b20440477bfcb.jpg)
+
+<table><tr><td>(p,q)</td><td>(3,5)</td><td>(5,7)</td><td>(11,13)</td><td>(17,19)</td><td>(29,31)</td><td>(41,43)</td></tr><tr><td>Max Steps for STEEL</td><td>5.97·104</td><td>1.49·105</td><td>9.07·105</td><td>2.60·106</td><td>1.10·107</td><td>2.83·107</td></tr><tr><td>Max Steps for AC-State</td><td>4·103</td><td>4·104</td><td>1·106</td><td>8·106</td><td>1·108</td><td>5·108</td></tr><tr><td>Max Steps: (AC-State / STEEL)</td><td>≈0.07</td><td>≈0.3</td><td>≈1</td><td>≈3</td><td>≈9</td><td>≈18</td></tr><tr><td>Median Steps (STEEL, DoublePrime)</td><td>2.50·104</td><td>7.48·104</td><td>5.23·105</td><td>1.08·106</td><td>1.74·106</td><td>3.29·106</td></tr><tr><td>Median Steps (AC-State, DoublePrime)</td><td>3·103</td><td>3·104</td><td>8·105</td><td>7·106</td><td>9·107</td><td>5·108</td></tr><tr><td>Median Steps: (AC-State / STEEL)</td><td>≈0.1</td><td>≈0.4</td><td>≈1.5</td><td>≈6</td><td>≈50</td><td>≈150</td></tr></table>
+
+Table 3: Comparison of the empirical sample complexity of STEEL and AC-State on prime-loop MDPs. See text of Section F.2.2.
+
+Further, we noticed an interesting trend in the data. AC-State tended to transition very abruptly from learning the wrong encoder on all tests of DoublePrime to learning the correct encoder on all tests as  $T$  increased (as can be observed in the hyperparameter search in Figure 7). By contrast, the distribution of steps taken for STEEL on DoublePrime( $p, q$ ) was highly skewed: most trials took significantly fewer than the maximum observed number of steps. We therefore also compare the median number of steps taken to learn a correct encoder for DoublePrime( $p, q$ ). For STEEL, this is computed as simply the median length of the 20 DoublePrime( $p, q$ ) trajectories tested. For AC-State, it is the first timestep  $T$  in the test-time search where AC-State with  $K = K_{\mathrm{opt}}$  learns a correct encoder for at least 10 of the 20 trajectories. Here, we find that, especially for large ( $p, q$ ), STEEL's advantage over AC-State becomes even more pronounced.
+
+# F.3 EXPLORATION POLICIES WITH AC-STATE
+
+So far, we have only compared STEEL to AC-State, where AC-State uses a uniformly random exploration policy. This may seem unfair: STEEL takes decidedly nonrandom actions, and Lamb et al. (2023) proposes an active exploration method for AC-State (albeit, as mentioned in Section F.1.1 above, a method without strict theoretical grounding).
+
+However, there is reason to believe that exploration policies similar to the one proposed by Lamb et al. (2023) would be unlikely to help correctly identify the dynamics of DoublePrime  $(p,q)$ .
+
+To start with, the stated goal of the exploration policy in Lamb et al. (2023) is to "achieve high coverage of the control-endogenous state space." A natural question to ask is: how poorly do uniformly random policies perform at achieving this goal? To quantify this, we examined the state coverage for a single  $10^{6}$ -step trajectory on DoublePrime(41, 43), and computed the ratio of the state visitation of the most-visited state to the state visitation of the least-visited state. Surprisingly, this was  $\approx 2$ . In other words, DoublePrime is not a hard exploration problem, and so techniques designed to improve state coverage would seem to be unlikely to provide much of a benefit over a uniformly random policy.
+
+Still, we will now examine the specific exploration technique proposed by Lamb et al. (2023). At a high level, the technique proceeds as follows:
+
+- Throughout data collection, the agent maintains an approximate version of the encoder  $\phi'$  obtained by optimizing the AC-State loss (Equation 140), and an approximate transition function  $T'$ , obtained through counting. The algorithm also maintains state visitation counts for each of the learned latent states encoded by  $\phi'$ .  
+- At the start of each round of exploration (time  $t$ ), the agent selects a goal learned-latent state  $s_g$ , with probability inversely proportional to the visitation count. The agent then takes a single random action  $a_t$ , and then, starting at time  $t + 1$ , plans shortest-path to  $s_g$  and computes the number of steps  $k'$  it will take to get to  $s_g$ . Then, the agent proceeds to navigate to  $s_g$  in a closed-loop manner, using both  $\phi'$  and  $T'$ . After  $k'$  steps, regardless of whether  $s_g$  is reached, a new goal is set, and the process starts over.  
+- The encoder  $\phi$  is only trained on the tuples  $(x_{t},a_{t},x_{t+k'+1})$  which begin with a random action and end with a goal state.
+
+We performed experiments of a version of this algorithm adapted for our tabular setting, on the DoublePrime environments. Here, we explain the modifications we made to this exploration algorithm from Lamb et al. (2023) for our tests. Examining the original algorithm, we immediately
+
+<table><tr><td>(p,q)</td><td>(3,5)</td><td>(5,7)</td><td>(11,13)</td><td>(17,19)</td><td>(29,31)</td></tr><tr><td>Max Steps (AC-State, DoublePrime, Uniform Policy)</td><td>4·103</td><td>4·104</td><td>1·106</td><td>8·106</td><td>1·108</td></tr><tr><td>Max Steps (AC-State, DoublePrime, Exploration)</td><td>8·103</td><td>6·104</td><td>2·106</td><td>2·107</td><td>2·108</td></tr><tr><td>Median Steps (AC-State, DoublePrime, Uniform Policy)</td><td>3·103</td><td>3·104</td><td>8·105</td><td>7·106</td><td>9·107</td></tr><tr><td>Median Steps (AC-State, DoublePrime, Exploration)</td><td>6·103</td><td>4·104</td><td>2·106</td><td>2·107</td><td>2·108</td></tr></table>
+
+Table 4: Comparison of the empirical sample complexity of AC-State on the DoublePrime  $(p,q)$  environments, with and without active exploration. (Note that we did not perform a final comparison for  $p = 41$ ,  $q = 43$ , because the hyperparameter search for the active exploration policy (see Figure 8) with a maximum  $K$  of 50000 did not rule out a  $K_{\mathrm{opt}} > 50000$ , so we could not fairly tune this hyperparameter for the active exploration policy. We were unable to perform a hyperparameter search with  $K > 50000$  due to technical limitations.)
+
+notice an issue: the encoder is only trained with segments of length  $k = k' + 1$ , which (assuming  $\phi'$  and  $T'$  are anywhere close to accurate) will only scale linearly with  $D$ . However, properly learning the encoder for the DoublePrime environments requires examining longer segments of trajectories than the diameter of the dynamics graph. Therefore, in order to adapt this exploration method to the setting we consider, we eliminate this feature of the algorithm and train  $\phi$  on the entire collected trajectory, as in Equation 140.
+
+Furthermore, we give the exploration method what should be a large, unnatural advantage, by planning, choosing goals, and assessing state visitation on the true ground truth dynamics  $T$ , with the ground-truth state encoder  $\phi$ . This represents what should be the "best possible" case for the algorithm. It also enables us to test in this setting without having to optimize  $\phi'$  continuously during data collection, and to assess many values of the hyperparameter  $K$  (which would impact the intermediate encoder  $\phi'$ ) simultaneously on a single collected trajectory.
+
+In our implementation, we reset the state visitation counts between collecting the "fitting" and "optimization" portions of the trajectory, in order to ensure that they are distributed in the same way. We also take uniformly random actions in cases where both actions produce a path of the same length to the goal. Otherwise, we simply take the shortest available path to the goal state, and, when it is reached, pick a new goal and take one random action.
+
+Hyperparameter search results are shown in Figure 8, and top-line results are shown in Table 4. We see in Table 4 that AC-State with the active exploration method seems to slightly underperform random exploration on the DoublePrime  $(p,q)$  environments, requiring slightly more samples to find the correct encoder, for all values of  $(p,q)$  that were tested. $^{12}$  While it is not immediately obvious why we see this gap, these results support our initial hypothesis that, because the DoublePrime environments are well-explored by uniformly random policies, techniques designed to improve latent state coverage are not helpful in improving AC-State's performance on these environments.
+
+# F.4 DISCUSSION
+
+It is important to note that the DoublePrime environments are not hard-exploration environments, nor do they have any time-correlated noise, nor do they have rich observations. In fact, adding time-correlated noise or rich observations would likely only scale the sample-complexity of STEEL on these environments only modestly (linearly in  $\hat{t}_{\text{mix}}$  and  $\log(|\mathcal{F}|)$ ). Rather, STEEL outperforms AC-State on large instances of these environments because the environments' transition dynamics are arranged in a way that is in a sense "adversarial" to multistep-inverse methods. By contrast, because STEEL has well-understood sample-complexity that applies for any transition graph structure (as long as the reachability assumption holds), we do not expect that any such "adversarial" environments to exist for STEEL.
+
+![](images/825e1962aec4b6b07b6201cee0dd9db505acc91c05e2f917b1b23f4fa8d790a3.jpg)
+
+![](images/ea72a7519719aad54c758ebcc2a72839c50f341574f4d5f6bba85ce8a1dae75c.jpg)
+
+![](images/7dbb3a802a464ea5d1b8868c0a409b6a422f5ee41c61ab2ccc88b15be1517066.jpg)
+
+![](images/cd4eb8f31acb88f9da71a393d9f4a75761549d809fa398774122b915d6c17c08.jpg)
+
+![](images/e7c2df9cfecfa8639dcfb3f3613647348d8831fdf382696c55bddad41521de04.jpg)  
+Figure 8: Results from the hyperparameter search for the maximum-step-count parameter  $K$  for DoublePrime $(p, q)$ , using the active exploration policy described in Section F.3.
+
+![](images/57fc97ac15d72e53a6dcf128bb365e7fd6d4cdee4503c49e5ff8854f5c9ce94a.jpg)
\ No newline at end of file
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+# LEARNING A NEURAL SOLVER FOR PARAMETRIC PDESTO ENHANCE PHYSICS-INFORMED METHODS
+
+Lise Le Boudec\*
+
+Emmanuel de Bezenac 2
+
+Louis Serrano
+
+Ramon Daniel Regueiro-Espino
+
+Yuan Yin  $^{3}$  †
+
+Patrick Gallinari  $^{1,4}$
+
+$^{1}$  Sorbonne Université, CNRS, ISIR, F-75005 Paris, France  
+2 INRIA Paris, France  
+$^{3}$  Valeo.ai, Paris, France  
+$^{4}$  Criteo AI Lab, Paris, France
+
+# ABSTRACT
+
+Physics-informed deep learning often faces optimization challenges due to the complexity of solving partial differential equations (PDEs), which involve exploring large solution spaces, require numerous iterations, and can lead to unstable training. These challenges arise particularly from the ill-conditioning of the optimization problem caused by the differential terms in the loss function. To address these issues, we propose learning a solver, i.e., solving PDEs using a physics-informed iterative algorithm trained on data. Our method learns to condition a gradient descent algorithm that automatically adapts to each PDE instance, significantly accelerating and stabilizing the optimization process and enabling faster convergence of physics-aware models. Furthermore, while traditional physics-informed methods solve for a single PDE instance, our approach extends to parametric PDEs. Specifically, we integrate the physical loss gradient with PDE parameters, allowing our method to solve over a distribution of PDE parameters, including coefficients, initial conditions, and boundary conditions. We demonstrate the effectiveness of our approach through empirical experiments on multiple datasets, comparing both training and test-time optimization performance. The code is available at https://github.com/2ailesB/neural-parametric-solver.
+
+# 1 INTRODUCTION
+
+Partial Differential Equations (PDEs) are ubiquitous as mathematical models of dynamical phenomena in science and engineering. Solving PDEs is of crucial interest to researchers and engineers, leading to a huge literature on this subject (Evans, 2010; Salsa, 2015). Traditional approaches to solving PDEs such as finite difference, finite element analysis, or spectral methods (Zienkiewicz et al., 2005; LeVeque, 2007) often come with stability and convergence guarantees but suffer from a high computational cost. Improving numerical PDE solvers through faster and more accurate algorithms remains an active research topic (Zienkiewicz et al., 2005).
+
+PDE solvers usually rely on discretization and/or linearization of the problem through various techniques to simplify the computations. Iterative methods such as Jacobi, Gauss-Seidel, Conjugate Gradient, and Krylov subspace methods can then be used to solve the resulting systems. Unfortunately, many PDEs have an ill-conditioned nature, and these iterative processes can demand extensive computational resources. Preconditioning techniques are often essential to mitigate this, though they require precise customization to the specific PDE problem, making the development of effective solvers a significant research endeavor in itself. Yet, the computational demands, time, and expertise required to develop these algorithms sometimes make them impractical or sub-optimal for specific
+
+classes of problems. Instead of relying on hand-designed algorithms, researchers have investigated, as an alternative, the use of machine learning to train iterative PDE solvers Hsieh et al. (2019); Li et al. (2023a); Rudikov et al. (2024); Kopaničáková et al. (2023). These approaches usually parallel the classical numerical methods by solving a linear system resulting from the discretization of a PDE, for example, using finite differences or finite elements. A preconditioner is learned from data by optimizing a residual loss computed w.r.t. a ground truth solution obtained with a PDE solver. This preconditioner is used on top of a baseline iterative solver and aims at accelerating its convergence. Examples of baseline solvers are the conjugate gradient (Li et al., 2023a; Rudikov et al., 2024) or the Jacobi method Hsieh et al. (2019).
+
+Another recent research direction investigates the use of neural networks for building surrogate models in order to accelerate the computations traditionally handled by numerical techniques. These methods fall into two main categories: supervised and unsupervised. The supervised methodology consists of first solving the PDE using numerical methods to generate input and target data and then regressing the solution using neural networks in the hope that this surrogate could solve new instances of the PDE. Many models, such as Neural Operators, lie within this class (Li et al., 2020; Raonic et al., 2023; Bartolucci et al., 2023) and focus on learning the solution operator directly through a single neural network pass. Unsupervised approaches, involve considering a neural network as a solution of the PDE. The neural network parameters are found by minimizing the PDE residual with gradient descent. Methods such as Physics-Informed Neural Networks (PINNs) (Raissi et al., 2019), or DeepRitz (E & Yu, 2018) fall under this category. This family of methods is attractive as it does not rely on any form of data, but only on information from the PDE residual. However, they exhibit severe difficulties during training (Krishnapriyan et al., 2021; Ryck et al., 2023), often requiring many optimization steps and sophisticated training schemes (Krishnapriyan et al., 2021; Rathore et al., 2024). The ill-conditioned nature of PDE residual loss appears again in this context, making standard optimizers such as Adam inappropriate (see appendix E.6 and appendix E.7 for a visualization of the ill conditioning of this loss landscape). A detailed review of the existing literature is described in appendix A.
+
+In this work, we consider having access to the PDE as in unsupervised approaches and also to some data for training our neural solver. Our objective is to solve the optimization issues mentioned above by learning an iterative algorithm that solves the PDE from its residual, defined as in the PINNs framework (see fig. 1). This neural solver is trained from data, either simulations or observations. Different from the classical ML training problem which aims at learning the parameters of a statistical model from examples, the problem we handle is learning to learn, Andrychowicz et al. (2016), i.e. learning an iterative algorithm that will allow us to solve a learning problem. When vanilla PINNs handle a single PDE instance, requiring retraining for each new instance, we consider the more complex setting of solving parametric PDEs, the parameters may include boundary or initial conditions, forcing terms, and PDE coefficients. Each specific instance of the PDE, sampled from the PDE parameter distribution, will then be considered as a training example. The objective is then to learn a solver from a sample of the parametric PDE distribution in order to accelerate inference on new instances of the equation. With respect to unsupervised approaches, our model implements an improved optimization algorithm, tailored to the parametric PDE problem at hand instead of using a hand-defined method such as stochastic gradient descent (SGD) or Adam. As demonstrated in the experimental section, this approach proves highly effective for the ill-posed problem of optimizing the PINNs objective, enabling convergence in just a few steps. This is further illustrated through gradient trajectory visualizations in appendix E.6. In the proposed methodology, the neural solver will make use of the gradient information computed by a baseline gradient method to accelerate its convergence. In our instantiation, we will use SGD as our baseline algorithm, but the method could be easily extended to other baselines. Our model deviates from the traditional preconditioning methods by directly optimizing the non-linear PDE residual loss (Raissi et al., 2019) without going through the discretization steps. Our contribution includes :
+
+- Setting an optimization framework for learning to iteratively solve parametric PDEs from physics-informed loss functions. We develop an instantiation of this idea using an SGD baseline formulation. We detail the different components of the framework as well as training and inference procedures.  
+- Evaluating this method on challenging PDEs for physics-informed method, including failure cases of classical PINNs and showing that it solves the associated optimization issues and accelerates the convergence.
+
+- Extending the comparison to several parametric PDEs with varying parameters from 1d static to 2d+time problems. We perform a comparison with baselines demonstrating a significant acceleration of the convergence w.r.t. baselines.
+
+![](images/80b19b1a362f06d6227c87400612273dd74278917d2fbcc586ee51c7da29dd90.jpg)  
+Figure 1: Optimization scheme of a physics-informed method with our framework.
+
+# 2 MOTIVATION
+
+Let us first motivate our objective with a simple example. Physics-informed neural networks (PINNs) are a promising tool for optimizing neural networks in an unsupervised way to solve partial differential equations (PDEs). However, these methods are notoriously difficult to train (Krishnapriyan et al., 2021; Ryck et al., 2023). As an illustrative example of this challenge, let us solve the zero-boundary Poisson equation in 1d on  $\Omega = [-\pi, \pi]$ . Note that this section is intentionally informal, we provide rigorous statements and proofs in Appendix B.
+
+Poisson equation, 1d. The solution is given by  $u(x) = \sin (kx)$ .
+
+$$
+u ^ {\prime \prime} (x) = - k ^ {2} \sin (k x),
+$$
+
+$$
+u (- \pi) = 0, u (\pi) = 0. \tag {1}
+$$
+
+Physics-informed machine learning relies on an ansatz space of parametric functions, e.g. neural networks  $u_{\Theta}:\Omega \mapsto \mathbb{R}$ , minimizing the following loss in order to satisfy the constraints in equation 1:
+
+$$
+\mathcal {L} _ {\mathrm {P D E}} = \mathcal {L} _ {\mathrm {R e s}} + \lambda \mathcal {L} _ {\mathrm {B C}}, \quad \mathcal {L} _ {\mathrm {R e s}} = \int_ {\Omega} | u _ {\Theta} ^ {\prime \prime} (x) - f (x) | ^ {2} d x,
+$$
+
+$$
+\mathcal {L} _ {\mathrm {B C}} = \frac {1}{2} \left[ u _ {\Theta} (- \pi) ^ {2} + u _ {\Theta} (\pi) ^ {2} \right].
+$$
+
+As a simple example, consider the parametrization given by considering a linear combination of Fourier features widely used (Tancik et al., 2020),  $u_{\Theta}(x) = \sum_{k = -K}^{K}\theta_{k}\phi_{k}(x)$ , with  $\phi_0(x) = \frac{1}{\sqrt{2\pi}}$ ,  $\phi_{-k}(x) = \frac{1}{\sqrt{\pi}}\cos (kx)$  and  $\phi_k(x) = \frac{1}{\sqrt{\pi}}\sin (kx)$  for  $1\leq k\leq K$ .
+
+This simple but informative example yields a tractable gradient descent algorithm, as the associated updates are linear in the parameters, governed by a matrix  $A$  and constant  $b$ :
+
+$$
+\begin{array}{l} \Theta_ {l + 1} = \Theta_ {l} - \eta \nabla \mathcal {L} _ {\mathrm {P D E}} (\Theta_ {l}) \\ = (I - \eta A) \Theta_ {l} + b \tag {2} \\ \end{array}
+$$
+
+with  $A$  whose condition number is  $\kappa(A) \coloneqq \lambda_{\max}(A) / \lambda_{\min}(A) \geq K^4$ :
+
+$$
+A = \left[ \begin{array}{c c c c} 0 ^ {4} & 0 & \dots & 0 \\ 0 & 1 ^ {4} & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & K ^ {4} \end{array} \right] + \lambda \left[ \begin{array}{c} \phi_ {1} (\pi) \\ \phi_ {2} (\pi) \\ \vdots \\ \phi_ {K} (\pi) \end{array} \right] \left[ \begin{array}{c c c c} \phi_ {1} (\pi) & \phi_ {2} (\pi) & \dots & \phi_ {K} (\pi) \end{array} \right]. \tag {3}
+$$
+
+This implies that the condition number of  $A$  increases extremely rapidly in the ratio between the highest and lowest frequencies of the network. Given that the rate of convergence to the optimum  $\Theta^{*} = \Theta_{0} + A^{-1}b$  can be bounded as
+
+$$
+\left\| \Theta_ {l} - \Theta^ {*} \right\| _ {2} \leq \left(1 - c / \kappa (A)\right) ^ {l} \| \Theta_ {0} - \Theta^ {*} \| _ {2}, \tag {4}
+$$
+
+the number of steps  $N(\varepsilon)$  required to obtain an error of size at most  $\varepsilon$ , i.e.,  $\| \Theta_l - \Theta^* \|_2 \leq \varepsilon$  increases linearly in the condition number, i.e. as the fourth power of the maximal frequency  $K$ :
+
+$$
+N (\varepsilon) = O \left(\kappa (A) \ln \frac {1}{\epsilon}\right) = O \left(K ^ {4} \ln \frac {1}{\epsilon}\right). \tag {5}
+$$
+
+We believe that this simple example clearly illustrates and highlights the fact that PINNs—even when considering a linear basis, and when the PDE is linear-suffer heavily from ill-conditioning: if 500 steps are required in order to achieve a given error when  $K = 5$ , roughly speaking, 312 500 steps are required for only  $K = 25$ . This result extends to more general linear systems of equations and linear ansatz, as explained in appendix B.
+
+Our objective in the following will be to accelerate the convergence of such systems in this context—as well as extend them to the non-linear setting. To do so, in the following section, we will learn how to transform the optimization problem in such a way that the number of gradient descent iterations is small. The resulting method can be seen as a standalone, iterative solver as it is not only applicable to different PDEs but can handle a wide range of initial/boundary conditions and parameters.
+
+# 3 APPROACH
+
+In order to optimize PDE-based losses, we propose to learn a physics-based optimizer that will fulfill two objectives: (i) allowing a fast test-time optimization given a new PDE and (ii) solving without retraining parametric PDEs, with varying PDE coefficients  $\gamma^2$ , forcing terms  $f$ , and initial/boundary conditions  $g$  using the same model. We present the general framework below and propose an instantiation that leverages a linear combination of basis functions as the ansatz.
+
+# 3.1 PROBLEM STATEMENT
+
+Let us consider the following family of boundary value problems parameterized by  $\gamma$  with domain  $\Omega$ , representing both space and time, with  $\mathcal{N}$  a potentially nonlinear differential operator,  $\mathcal{B}$  the boundary operator,  $g$  the initial/boundary conditions, and source term  $f$ :
+
+$$
+\mathcal {N} (u; \gamma) = f \quad \text {i n} \Omega , \tag {6}
+$$
+
+$$
+\mathcal {B} (u) = g \quad \text {o n} \partial \Omega . \tag {7}
+$$
+
+Note that different PDEs can be represented in this form, amounting to changing the parameters  $\gamma$ . The goal here is to develop a generic algorithm that is able to solve the above problem, yielding an approximate solution  $u$  given the PDE and different sets of inputs  $(\gamma, f, g)$ .
+
+For training, we assume access to a dataset of  $M$  problem instances, represented by the PDE parameters  $(\gamma_{i},f_{i},g_{i})_{i = 1}^{M}$  and to associated target solution  $(u_i)_{i = 1}^M$  given on a  $m$  point grid  $(x_j)_{j = 1}^m$ . The solutions  $(u_{i})_{i = 1}^{M}$  will be used to train the neural solver. At inference, for a new PDE instance, only the PDE parameters are provided and we do not have access to solution points  $(u_{i})_{i = 1}^{M}$ .
+
+# 3.2 METHODOLOGY
+
+Physics-informed neural networks consider an ansatz  $u_{\Theta}$  parametrized by some finite-dimensional  $\Theta$ . The parameters  $\Theta$  are iteratively updated by minimizing a criterion  $\mathcal{L}_{\mathrm{PDE}}$  (e.g. the PDE residual), which assesses how well the ansatz  $u_{\Theta}$  meets the conditions specified in equations 6 and 7. As introduced for example for PINNs (Raissi et al., 2019) or Deep Galerkin method (Sirignano & Spiliopoulos, 2018), we consider  $\mathcal{L}_{\mathrm{PDE}}$  to be given by the strong formulation of the residual  $\mathcal{L}_{\mathrm{Res}}$ , plus a boundary discrepancy term<sup>3</sup>  $\mathcal{L}_{\mathrm{BC}}$ :  $\mathcal{L}_{\mathrm{PDE}} = \mathcal{L}_{\mathrm{Res}} + \lambda \mathcal{L}_{\mathrm{BC}}, \quad \lambda > 0$ .
+
+$$
+\mathcal {L} _ {\mathrm {R e s}} = \sum_ {x _ {j} \in \Omega} | \mathcal {N} (u _ {\Theta}; \gamma) (x _ {j}) - f (x _ {j}) | ^ {2}, \quad \mathcal {L} _ {\mathrm {B C}} = \sum_ {x _ {j} \in \partial \Omega} | \mathcal {B} (u _ {\Theta}) (x _ {j}) - g (x _ {j}) | ^ {2} \tag {8}
+$$
+
+As illustrated in section 2, performing gradient descent, or alternatives such as Adam and L-BFGS on such a highly ill-conditioned loss  $\mathcal{L}_{\mathrm{PDE}}$  leads to severe training difficulties (Krishnapriyan et al., 2021). The key idea in our work is to improve a baseline gradient descent algorithms with the neural solver. More precisely starting from a baseline gradient algorithm, SGD in our instantiation, instead of considering the classical update, we first transform the gradient using a neural network  $\mathcal{F}_{\varrho}$  with parameters  $\varrho$ , depending on the values of the PDE parameters as well as on other inputs such as the residual gradient provided by SGD:  $\nabla_{\Theta} \mathcal{L}_{\mathrm{PDE}}$ . The objective is to transform, through the neural solver  $\mathcal{F}_{\varrho}$ , the ill-conditioned problem into a new, simpler problem that requires fewer steps in order to achieve a given error.
+
+Once the neural solver  $\mathcal{F}_{\varrho}$  has been learned, inference can be performed on any new PDE as follows (see inference algorithm 1). Starting from an initial ansatz parameter  $\Theta_0$ , it is iteratively updated by this solver. At iteration  $l$ , the steepest direction of the loss  $\mathcal{L}_{\mathrm{PDE}}$  is first computed with autograd. Then, the gradient is transformed, in a PDE parameter dependant way, with  $\mathcal{F}_{\varrho}$ :
+
+$$
+\Theta_ {l + 1} = \Theta_ {l} - \eta \mathcal {F} _ {\varrho} (\nabla_ {\Theta} \mathcal {L} _ {\mathrm {P D E}} (\Theta_ {l}), \gamma , f, g) \tag {9}
+$$
+
+The objective is to iteratively refine the ansatz to closely approximate the true solution after a series of  $L$  iterations, ideally small for efficiency.
+
+This approach can be seen as learning the iterates in a PDE solver to achieve a low loss, similar to the residual minimization methods in PDEs (Elman et al., 2014). By design, this solver is intended to be applicable to different PDEs, as well as various sources, boundary conditions, and initial conditions. This flexibility allows for a broad range of applications, making it a versatile tool in solving complex PDEs with varying characteristics.
+
+Designed as a parametric PDE solver,  $\mathcal{F}_{\varrho}$  is trained with input target data from different sets of PDE parameters, as outlined in section 3.1. Once trained, it will be used without retraining on new instances of the PDE, i.e. with new values of the PDE parameters. The underlying hypothesis is that even though the solutions may be different for different inputs and parameters, the solution methodology remains relatively consistent. This consistency is expected to enhance the algorithm's ability to generalize across novel scenarios effectively.
+
+# 3.3 TRAINING OF A PHYSICS-INFORMED SOLVER
+
+Choice of Ansatz  $u_{\Theta}$ . A very common choice (Shen et al., 2011) is to consider a family of basis functions  $\Psi(x) = \{\psi_i(x)\}_{i=1}^N$  and consider the ansatz to be given by its linear span  $u_{\Theta}(x) = \sum_{i=0}^{N} \theta_i \psi_i(x)$ . In the following, we consider this linear reconstruction, although our formulation is generic in the sense that it can also accommodate nonlinear variants.
+
+As indicated in eq. (10), the solver  $\mathcal{F}_{\varrho}$  will be trained from samples of the PDE parameter distribution  $(\gamma, f, g)$  and from the associated samples of the solution  $u$ . We first describe below the inference step aiming at iteratively updating the parameters  $\Theta$  of the solution function  $u_{\Theta}$  while  $\mathcal{F}_{\varrho}$  is held fixed. We then describe how the  $\varrho$  parameters of the solver are trained. Please refer to fig. 1 that illustrates the interaction between the two steps.
+
+Note that other formulations of the loss may also be considered in a straightforward manner.  
+4 In the following we use "solver" with  $\mathcal{F}_{\varrho}$  as a short-hand to refer to our proposed method.  
+Although we have found this may further complicate training.
+
+Inference The inference step is performed at fixed values of the solver parameters  $\mathcal{F}_{\varrho}$ . It consists, for a given instance of the PDE with parameters  $(\gamma_{i},f_{i},g_{i})$ , in finding the best  $\Theta$  with a few steps of the solver using eq. (9). It is illustrated in fig. 1 - grey box and formalized in algorithm 1: starting from initial parameters  $\Theta_0$ , we compute the Physical loss  $\mathcal{L}_{\mathrm{PDE}}$  using the ansatz  $u_{\Theta_0}$ . The PDE derivatives in  $\mathcal{L}_{\mathrm{PDE}}$ , can be computed by hand or automatic differentiation depending on the application  $^6$ . Then, eq. (9) is used to update the parameters  $\Theta$  for a given number of steps  $L$ . The final solution is reconstructed using the linear combination,  $u_{\Theta}(x) = \sum_{i = 0}^{N}\theta_{i}\psi_{i}(x)$ , introduced in section 3.2 with the computed coefficients  $\Theta_L$ . Note that inference does not make use of the sampled target solutions  $(u_i)_{i = 1}^M$  computed on the grid points  $(x_j)_{j = 1}^m$ . These targets are used exclusively for training the neural solver.
+
+Training the neural solver Training amounts to learning the parameters of the solver  $\mathcal{F}_{\varrho}$  and is performed with a training set of PDE parameters and simulation data considered as ground truth  $(\gamma_{i},f_{i},g_{i},u_{i})_{i = 1}^{M}$ , corresponding to PDE instances (i.e. with different parameters
+
+# Algorithm 1: Inference using the neural PDE solver.
+
+Data:  $\Theta_0\in \mathbb{R}^n$  PDE  $(\gamma ,f,g)$
+
+Result:  $\Theta_L\in \mathbb{R}^n$
+
+for  $1 = 0\dots \mathrm{L - 1}$  do
+
+$$
+\left. \right| \Theta_ {l + 1} = \Theta_ {l} - \eta \mathcal {F} _ {\varrho} \left(\nabla \mathcal {L} _ {\mathrm {P D E}} \left(\Theta_ {l}\right), \gamma , f, g\right)
+$$
+
+end
+
+return  $\Theta_L$
+
+# Algorithm 2: Training algorithm for learning to optimize physics-informed losses.
+
+Data:  $\Theta_0\in \mathbb{R}^n$  ,PDE  $(\gamma ,f,g)$  ,sample values  $u(x)$
+
+Result:  $\mathcal{F}_{\varrho}$
+
+for  $e = 1\ldots$  epochs do
+
+for (PDE, x, u) in dataset do
+
+Initialize  $\Theta_0$
+
+Estimate  $\Theta_L$  from  $\Theta_0, (\gamma, f, g)$  using algorithm 1
+
+Reconstruct  $u_{\Theta_L}(x)$
+
+Update  $\varrho$  with gradient descent from the data loss in eq. (10)
+
+end
+
+end
+
+return  $\mathcal{F}_{\varrho}$
+
+$\gamma$  and/or forcing terms  $f$  and/or initial/boundary conditions  $g$ ). See fig. 1 - white box and algorithm 2. The objective is to learn a solver  $\mathcal{F}_{\varrho}$  able, at inference, to converge to a target solution in a small (2 to 5 in our experiments) number of steps. For that, an optimizer (Adam in our experiments) is used to update the  $\mathcal{F}_{\varrho}$  parameters. The training algorithm makes use of the data associated with the different PDE instances by sampling PDEs in batches and running algorithm 1 on several PDE instances. For each PDE instance, one starts from an initial parameter value  $\Theta_0$  and then performs two optimization steps (see algorithm 2): (i) one consists in solving in the ansatz parameters  $\Theta$  using the neural solver using algorithm 1, leading to  $\Theta_L$ ; (ii) the second one is the optimization of the solver parameters  $\varrho$ . We train the outputs directly to match the associated ground truth  $(u_i(x_j))_{j=1}^m$  using the data loss:
+
+$$
+\mathcal {L} _ {\mathrm {D A T A}} = \mathbb {E} _ {\gamma , f, g} \left[ \left| \left| u _ {\Theta_ {L}} - u _ {\gamma , f, g} \right| \right| \right]. \tag {10}
+$$
+
+The expectation is computed on the distribution of the PDE parameters  $(\gamma, f, g)$ . The solution  $u$  is entirely determined by these parameters as indicated by the notation  $u_{\gamma, f, g}$ .  $||u_{\Theta_L} - u||$  denotes a distance between the target  $(u(x_j))_{j=1}^m$  and the forecast  $(u_{\Theta_L}(x_j))_{j=1}^m$  with  $m$  the trajectory size<sup>7</sup>. In practice, one samples a set of PDE instances  $(\gamma_i, f_i, g_i)$  and for each instance a corresponding sample  $u_i$ .
+
+Theoretical analysis and relation to preconditioning Analyzing the behavior of the inference algorithm is challenging due to the non linear nature of the solver. We however could get some intuition using simplifying assumptions. We build on the ideas introduced in section 2 for the simple case of the Poisson equation, for which an explicit analytical solution could be derived. We provide in appendix B, a proof for a more general case and give below in theorem 1 our main result. This shows that the number of steps induced by  $\mathcal{F}$  for the proposed algorithm is significantly less than the number of steps required by the baseline PINNs algorithm. This results is obtained under two main assumptions: (i)  $\mathcal{F}$  behaves like its linearization and (ii) the descent operator  $\mathcal{F}$  used in our algorithm, allows us to reach the optimum of  $\mathcal{L}_{\mathrm{DATA}}$ .
+
+Theorem 1. (Convergence rate in the linear case). Given a linear ansatz  $u_{\Theta}(x) = \sum_{i=1}^{N} \theta_i \phi_i(x)$ , assume the conditioner  $\mathcal{F}$  behaves like its linearization  $P = \text{Jacobian}(\mathcal{F})$ , meaning that  $\mathcal{F}$  can be replaced by  $P$  at any point. Let  $A$  be the matrix derived from the PDE loss as eq. (3) for the Poisson equation or eq. (15) in the more general case. Denote by  $\kappa(A)$  the condition number of the matrix  $A$ . The number of steps  $N'(\varepsilon)$  required to achieve an error  $\| \Theta_l - \Theta^* \|_2 \leq \varepsilon$  satisfies:
+
+$$
+N ^ {\prime} (\varepsilon) = O \left(\kappa (P A) \ln \left(\frac {1}{\varepsilon}\right)\right), \tag {11}
+$$
+
+Moreover, if  $\mathcal{F}$  minimizes  $\mathcal{L}_{\text{DATA}}$  this necessarily implies  $\kappa(PA) = 1 \leq \kappa(A)$ . Consequently, the number of steps is effectively reduced, i.e.,  $N'(\varepsilon) \ll N(\varepsilon)$  with  $N(\varepsilon)$  the number of steps of the vanilla PINNs.
+
+Proof. We sketch the main insights here and refer to appendix B for the proof and a detailed analysis.
+
+- Using a linearization of the neural solver, it can be shown that the solver performs as a pre-conditioner on the linear system.  
+- Assuming that solution  $u_{L}$  provided by the solver reaches the optimum  $u^{*}$ , and that the training set is such that the learned parameter  $\Theta$  vectors span the whole parameter space of the model, then the convergence of the solver is guaranteed at an optimal rate.  
+- In practice, and as shown in the experiments (section 4), the convergence rate is significantly improved w.r.t. the reference baseline gradient algorithm.
+
+# 4 EXPERIMENTS
+
+![](images/dd1793b6d1cb0a5445ee6ce80c06f68418c74562a0f10d2645f8dabf087914d9.jpg)
+
+We present the datasets used in the experiments in section 4.1, a comparison with selected baselines in section 4.2, and a test-time comparison with different optimizers demonstrating the remarkable effectiveness of the proposed method in section 4.3. Finally, we make a comparison of the training and inference time in section 4.4. Experimental details and additional experiments can be found in the appendices: ablations are in appendix E and additional results and visualization are in appendix E.6, appendix E.7, and appendix F.
+
+# 4.1 DATASETS
+
+We consider several representative parametric equations for our evaluation. More details about the data generation are presented in appendix C. Our objective is to learn a neural solver able to solve quickly and accurately a new instance of a PDE, given its parametric form, and the values of the parameters  $\gamma$ , forcing terms  $f$  and initial/boundary conditions  $g$ , i.e.  $(\gamma, f, g) \mapsto u$ . Solving is performed with a few iterations of the neural solver (algorithm 1). For that, one trains the neural solver on a sample of the PDE parameter instances, see table 1 for the parameter distributions used for each parametric PDE. Evaluation is performed on unseen sets of parameters within the same PDE family. Helmholtz: We generate a dataset following the  $1d$  static Helmholtz equation  $u''(x) + \omega^2 u(x) = 0$  with
+
+Table 1: Parameters changed between each trajectory in the considered datasets.  
+
+<table><tr><td>Dataset</td><td>Parameters</td><td>Distribution</td></tr><tr><td rowspan="3">Helmholtz</td><td>ω</td><td>U[0.5, 50]</td></tr><tr><td>u0</td><td>N(0,1)</td></tr><tr><td>v0</td><td>N(0,1)</td></tr><tr><td rowspan="3">Poisson</td><td>Ai</td><td>U[-100, 100]</td></tr><tr><td>u0</td><td>N(0,1)</td></tr><tr><td>v0</td><td>N(0,1)</td></tr><tr><td rowspan="2">NLRD</td><td>ν</td><td>U[1, 5]</td></tr><tr><td>ρ</td><td>U[-5, 5]</td></tr><tr><td>Darcy</td><td>a(x)</td><td>ψ#N(0, (-Δ + 9I)-2) with ψ = 12 * 1R+ + 3 * 1R+</td></tr><tr><td rowspan="5">Heat</td><td>ν</td><td>U[2 × 10-3, 2 × 10-2]</td></tr><tr><td>Jmax</td><td>{1, 2, 3, 4, 5}</td></tr><tr><td>A</td><td>U[0.5, -0.5]</td></tr><tr><td>Kx, Ky</td><td>{1, 2, 3}</td></tr><tr><td>φ</td><td>U[0, 2π]</td></tr></table>
+
+boundary conditions  $u(0) = u_0$  and  $u'(0) = v_0$ . We generate 1, 024 trajectories with varying  $\omega$ ,  $u_0$ ,
+
+and  $v_{0}$  with a spatial resolution of 256. Poisson: We generate a dataset following the  $1d$  static Poisson equation with forcing term:  $-u''(x) = f(x)$  with  $u(0) = u_{0}$  and  $u'(0) = v_{0}$ . The forcing term  $f$  is a periodic function,  $f(x) = \frac{\pi}{K}\sum_{i=1}^{K}a_{i}i^{2r}\sin(\pi x)$ , with  $K = 16$  and  $r = -0.5$ . We generate 1,000 trajectories with varying  $u_{0}, v_{0}$ , and  $f$  (through changing  $a_{i}$ ) with a spatial resolution of 64. Reaction-Diffusion: In Krishnapriyan et al. (2021); Toloubidokhti et al. (2024), the authors propose a non-linear reaction-diffusion (NLRD). This PDE has been shown to be a failure case for PINNs (Krishnapriyan et al., 2021). We generate 1,000 trajectories by varying the parameters of the PDE:  $\nu$  and  $\rho$  (see table 1). Spatial resolution is 256 and temporal resolution is 100. The PDE is solved on  $[0,1]^{2}$ . Darcy Flow: The 2d Darcy Flow dataset is taken from (Li et al., 2020) and is commonly used in the operator learning literature (Li et al., 2023b; Goswami et al., 2022). For this dataset, the forcing term  $f$  is kept constant  $f = 1$ , and  $a(x)$  is a piece-wise constant diffusion coefficient taken from (Li et al., 2020). We kept 1,000 trajectories (on the 5,000 available) with a spatial resolution is  $64 \times 64$ . Heat: The  $2d + t$  Heat equation is simulated as proposed in (Zhou & Farimani, 2024). For this dataset, the parameter  $\nu$  is sampled from  $U[2 \times 10^{-3}, 2 \times 10^{-2}]$  and initial conditions are a combination of sine functions with a varying number of terms, amplitude, and phase. A summary of the datasets and the varying parameters for each PDE are presented in table 1 and more details on the dataset are provided in appendix C. Experiments have been conducted on NVIDIA TITAN V (12 Go) for 1d datasets to NVIDIA RTX A6000 GPU with 49Go for  $1d +$  time or  $2d$  datasets. For all datasets, 800 PDEs are considered during training and 200 for testing. All metrics reported are evaluated on test samples (i.e. PDEs not seen during training. Coefficients as well as initial and/or boundary conditions can vary from training).
+
+# 4.2 COMPARISON WITH BASELINES
+
+We performed comparisons with several baselines including fully data-driven supervised approaches trained from a data-loss only, unsupervised methods relying only on a PDE loss, and hybrid techniques trained from PDE + DATA losses. Network size and training details are described in appendix D. In this experiment, we considered training the models using the training sets (physical losses or MSE when possible) unless stated otherwise.
+
+Fully supervised We train a standard MLP to learn the mapping  $(\gamma, f, g) \mapsto \Theta$ , using as loss function  $\mathcal{L}_{\mathrm{DATA}} = \mathbb{E}_{\gamma, f, g, u} [||u_{\Theta_L} - u_{\gamma, f, g}||]$  with  $u_{\Theta}(x) = \sum_{i=0}^{N} \theta_i \psi_i(x)$ , the  $\psi_i(.)$  being fixed B-Spline basis functions (see appendix D). We denote this baseline as  $MLP+$  basis. Unsupervised We compare our approach with unsupervised physics-informed models (Raissi et al., 2019). While the initial version of PINNs solves only one PDE instance at a time and requires retraining for each new instance, we developed here a parametric version of PINNs (PPINNs) where the PDE parameters are fed to the network (similarly to (Zhang et al., 2023)). Finally, we used (Cho et al., 2024)'s (P2INNs) method as a physics-informed baseline specifically designed for parametric PDEs. In addition to PINNs-methods, we also compare our solver to the Physic-informed DeepONet (PO-DeepONet for Physics-Only DeepONet) (Wang et al., 2021b), which is designed to learn an operator for function-to-function mappings from physical losses and handles parametric PDEs. The mapping learned for the two unsupervised baselines is  $(x, \gamma, f, g) \mapsto u_{\gamma, f, g}(x)$ . In order to provide a fair comparison with our optimization method, we fine-tuned the unsupervised baselines for each specific PDE instance for a few steps (10 or 20). Comparison to preconditioning We compare our approach with vanilla PINNs (Raissi et al., 2019), i.e. by fitting one PINN per PDE in the test set and averaging the final errors. We optimize the PDE losses using L-BFGS (Liu & Nocedal, 1989) and refer to this baseline as  $PINNs + L-BFGS$ . As discussed in (Rathore et al., 2024), L-BFGS can be considered as a nonlinear preconditioning method for Physics-Informed methods and fastens convergence. Finally, we use the training strategy proposed by (Rathore et al., 2024) i.e. trained PINNs using successive optimizer (Adam + L-BFGS). This baseline is denoted as PINNs-multi-opt. For these baselines, one model is trained and evaluated for each PDE in the test set. We report the reader to appendix D for more details on the training procedure. Hybrid Finally, we compare our proposed method with neural operators, i.e., models trained to learn mappings  $(x, \gamma, f, g) \mapsto u_{\gamma, f, g}(x)$  using a combination of physical and data loss:  $\mathcal{L}_{\mathrm{DATA}} + \mathcal{L}_{\mathrm{PDE}}$ . We use as baselines Physics-Informed Neural Operator (PINO) (Li et al., 2023b) and Physics-Informed DeepONet (PI-DeepONet) (Goswami et al., 2022). As already indicated, for a fair comparison, the Unsupervised and Hybrid baselines are fine-tuned on each specific PDE instance for a few steps (10 on all datasets except for Heat for which 20 steps are made). Ours We represent the solution  $u_{\Theta}$  with a linear combination of B-Spline functions for  $\Psi$  (Piegl & Tiller, 1996). This was motivated by the nature of B-Splines which allows to capture local
+
+phenomena. However, other bases could be used such as Fourier, Wavelet or Chebychev Polynomials. The neural solver  $\mathcal{F}_{\varrho}$  is composed of Fourier Layers (FNO) (Li et al., 2020) that allow us to capture the range of frequencies present in the phenomenon. We refer the reader to appendix D for more details about the construction of the B-Spline basis and the training hyper-parameters.
+
+Table 2: Results of trained models - metrics in Relative MSE on the test set. Best performances are highlighted in bold, and second best are underlined.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Baseline</td><td colspan="2">1d</td><td>1d+time</td><td>2d</td><td>2d+time</td></tr><tr><td>Helmholtz</td><td>Poisson</td><td>NLRD</td><td>Darcy-Flow</td><td>Heat</td></tr><tr><td>Supervised</td><td>MLP + basis</td><td>4.66e-2</td><td>1.50e-1</td><td>2.85e-4</td><td>3.56e-2</td><td>6.00e-1</td></tr><tr><td rowspan="5">Unsupervised</td><td>PINNs+L-BFGS</td><td>9.86e-1</td><td>8.83e-1</td><td>6.13e-1</td><td>9.99e-1</td><td>9.56e-1</td></tr><tr><td>PINNS-multi-opt</td><td>8.47e-1</td><td>1.18e-1</td><td>7.57e-1</td><td>8.38e-1</td><td>6.10e-1</td></tr><tr><td>PPINNs</td><td>9.89e-1</td><td>4.30e-2</td><td>3.94e-1</td><td>8.47e-1</td><td>1.27e-1</td></tr><tr><td>P2INNs</td><td>9.90e-1</td><td>1.50e-1</td><td>5.69e-1</td><td>8,38e-1</td><td>1.78e-1</td></tr><tr><td>PO-DeepONet</td><td>9.83e-1</td><td>1.43e-1</td><td>4.10e-1</td><td>8.33e-1</td><td>4.43e-1</td></tr><tr><td rowspan="2">Hybrid</td><td>PI-DeepONet</td><td>9.79e-1</td><td>1.20e-1</td><td>7.90e-2</td><td>2.76e-1</td><td>9.18e-1</td></tr><tr><td>PINO</td><td>9.99e-1</td><td>2.80e-3</td><td>4.21e-4</td><td>1.01e-1</td><td>9.09e-3</td></tr><tr><td>Neural Solver</td><td>Ours</td><td>2.41e-2</td><td>5.56e-5</td><td>2.91e-4</td><td>1.87e-2</td><td>2.31e-3</td></tr></table>
+
+Results: Table 2 presents the comparison with the baselines. We recall that the evaluation set is composed of several PDE instances sampled from unseen PDE parameters  $(\gamma, f, g)$ . The proposed method is ranked first or second on all the evaluations. The most comparable baselines are the unsupervised methods, since at inference they leverage only the PDE residual loss, as our method does. Therefore our method should be primarily compared to these baselines. Supervised and hybrid methods both incorporate data loss and make different assumptions while solving a different optimization problem.
+
+Table 2 clearly illustrates that unsupervised Physics-informed baselines all suffer from ill-conditioning and do not capture the dynamics. Compared to these baselines, the proposed method improves at least by one order of magnitude in all cases. PINNs baseline performs poorly on these datasets because of the ill-conditioning nature of the PDE, requiring numerous optimization steps to achieve accurate solutions (appendix D). This is observed on PINNs models for parametric PDEs (PPINNs and P2INNs) as well as on PINNs fitted on one equation only (PINNs+L-BFGS and PINNs-multi-opt). We observe that our neural solver has better convergence properties than other Physics-Informed methods. As will be seen later it also converges much faster.
+
+The supervised baseline performs well on all the PDEs except Poisson and Heat. The data loss used for training this model is the mean square error which is well-behaved and does not suffer from optimization problems as the PDE loss does. We note that our method reaches similar or better performances on every datasets, while relying only on physical information at inference (algorithm 1) and solving a more complex optimization problem.
+
+The hybrid approaches, do not perform well despite taking benefits from the PDE+DATA loss and from adaptation steps at test time. Again, the proposed method is often one order of magnitude better than the hybrids except on NLRD, where it has comparable performances. This shows that the combination of physics and data losses is also hard to optimize, and suffers from ill-conditioning.
+
+# 4.3 OPTIMIZATION FOR SOLVING NEW EQUATIONS
+
+The main motivation for our learned PDE solver is to accelerate the convergence to a solution, w.r.t. predefined solvers, for a new equation. In order to assess this property, we compare the convergence speed at test time inference with classical solvers, PINNs, and pre-trained PINO as detailed below. Results are presented in fig. 2 for the Poisson equation with performance averaged on 20 new instances of the Poisson equation. This experiment is also performed on the other datasets in appendix F.
+
+Baseline optimizers As for the classical optimizer baselines, we used SGD, Adam (Kingma & Ba, 2015), and L-BFGS (Krishnapriyan et al., 2021). These optimizers are used to learn the coefficient of
+
+the B-Spline basis expansion in the model  $u_{\Theta}(x) = \sum_{i=0}^{N} \theta_i \psi_i(x)$ . This provides a direct comparison to our iterative neural solver. PINNs - We also compare to the standard PINNs (Raissi et al., 2019), i.e. by fitting one Neural Network (NN) per equation. Note that this requires full training from scratch for each new equation instance and this is considerably more computationally demanding than solving directly the parametric setting. This baseline is similar to the Adam optimizer mentioned above, except that the ansatz for this experience is a multilayer perceptron instead of a linear combination of a B-Spline basis. Hybrid pre-training strategies - Finally we compare against the hybrid PINO pretrained on a set of several parametric PDE instances and then fine-tuned on a new PDE instance using only the PDE loss associated to this instance. Ours - We train our model as explained in algorithm 2 and show here the optimization process at test time. In order to perform its optimization, our model leverages the gradient of the physical loss and the PDE parameters (coefficients, initial/boundary conditions). In this experiment, we use  $L = 5$  steps for a better visualization (whereas, we used  $L = 2$  in table 2).
+
+Result: fig. 2 compares the number of optimization steps required for the different methods. Our neural solver converges very fast in only a few steps (5 here) to a good approximation of the solution, while all the other methods require thousands of iterations - we stopped here at 10,000 steps. The classical optimizers (SGD, Adam, L-BFGS) do not converge for a new equation. The baseline PINNs trained here from scratch on each new equation show an erratic convergence behavior. Pretrained PINO behaves better than the other baselines but still did not converge after 10,000 steps. This clearly demonstrates the potential of our learned solver to deal with physical losses w.r.t. alternative pre-defined solvers.
+
+![](images/063f6a55a9d916db9a383e3e0de16a5bcdb00be981a0a5aee8d9376e65359297.jpg)  
+Figure 2: Test-time optimization based on the physical residual loss  $\mathcal{L}_{\mathrm{PDE}}$  for new PDE on Poisson.
+
+# 4.4 COMPUTATIONAL TIME
+
+An important aspect of solving PDEs is the computational time required for each solution. Methods should find a trade-off between achieving high performance and maintaining reasonable computational costs. In appendix E.3, we provide the training (table 16) and inference (table 17) times for our method compared to various baselines. Our results show that while our method has comparable training and inference times to other approaches, it demonstrates substantially better precision (table 2).
+
+# 5 CONCLUSION
+
+We have presented a PDE solver learned from data that allows fast test-time optimization of physical losses. Our method succeeds to considerably accelerate the optimization process for the complex problem of minimizing physical losses and is several orders of magnitude faster than classical hand-defined optimization methods such as Adam or L-BFGS. Limitations and Future Work While efficient, the proposed method can be further improved. First, training our iterative algorithms requires more memory than standard machine learning models due to the complexity of backpropagation through iterations, which becomes challenging in higher-dimensional bases. Second, we have focused on solution approximations expressed as a linear expansion in predefined bases. More expressive representations, such as neural networks, could be explored; however, our preliminary experiments indicate increased ill-conditioning due to the compositional nature of neural networks. More sophisticated training schemes could enhance the optimization process. Future work will investigate these directions to improve scalability and broaden the applicability of the proposed method. Reproducibility Statement Hyper-parameters, baselines configurations and training details are detailed in appendix D, tables 4 and 5, and algorithms 1 and 2. The creation of the datasets is explained in appendix C and tables 1 and 3. Finally, we provide a theoretical analysis of the model, under an ideal scenario in appendix B. Code is available at https://github.com/2ailesB/neural-parametric-solver.
+
+# ETHICS STATEMENT
+
+Solving PDE is of crucial interest in many applications of science and engineering. While we do not directly target such real-world applications in this paper, one should acknowledge that solvers can be used in various ranges of scenarios including weather, climate, medical, aerodynamics, industry, and military applications.
+
+# ACKNOWLEDGMENTS
+
+We acknowledge the financial support provided by DL4CLIM (ANR-19-CHIA-0018-01), DEEPNUM (ANR-21-CE23-0017-02), PHLUSIM (ANR-23-CE23-0025-02), and PEPR Sharp (ANR-23-PEIA-0008), "ANR", "FRANCE 2030").
+
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+
+# A DETAILED RELATED WORK
+
+PDE solvers: Many tools for numerically solving PDEs have been developed for years. The standard methods for PDE include Finite Differences (FD), Finite Volume (FV), Finite Element Method (FEM), spectral and multigrid methods, and many others (S. H, 2012; Liu, 2009). While these methods are widely used, they often suffer from a high computational cost for complex problems or high-precision simulations. To address these challenges, integrating deep learning (DL) into solvers has emerged as a promising approach. Current solutions include incorporating correction terms into mathematical solvers to reduce numerical errors (Um et al., 2021). Some work such as Hsieh et al. (2019); Li et al. (2023a); Rudikov et al. (2024); Kopaničáková et al. (2023) build a method to directly enhance the convergence of numerical solvers through preconditioner learning. As an example, Rudikov et al. (2024); Li et al. (2023a) use a neural operator to approximate conditioner for the flexible conjugate gradient method or Hsieh et al. (2019) for the Jacobi method. Another example of preconditioner learning lies in Li et al. (2023a), where the author uses GNN to assess symmetry and positive definiteness.
+
+Unsupervised training: Physics-Informed Neural Networks (PINNs) (Raissi et al., 2019) have been a pioneering work in the development of DL method for physics. In these models, the solution is a neural network that is optimized using the residual loss of the PDE being solved. However, this method suffers from several drawbacks. First, as formulated in Raissi et al. (2019), PINNs can solve one instance of an equation at a time. Any small change in the parameters of the PDE involves a full retraining of the network. Efforts such as Beltran-Pulido et al. (2022); Zhang et al. (2023); Cho et al. (2024) have attempted to address this limitation by introducing parametric versions of PINNs capable of handling parametric equations, while Huang et al. (2022); Qin et al. (2022) explores meta-learning approaches. Other approaches to improve PINNs generalization include using neural operators (Wang et al., 2021b), or hyper-network (de Avila Belbute-Peres et al., 2021). Moreover, PINNs have shown convergence difficulties: Krishnapriyan et al. (2021) show that PINNs' losses have complex optimization landscapes, complicating training despite adequate neural network expressiveness. Approaches like those detailed in Wang et al. (2022) adopt a Neural Tangent Kernel (NTK) perspective to identify reasons for failure and suggest using adaptive weights during training to enhance performance. Additionally, studies such as Ryck et al. (2023) demonstrate that PINNs suffer from ill-conditioned losses, resulting in slow convergence of gradient descent algorithms. Recently, (Rathore et al., 2024) shows how training strategies can improve the convergence of Physics-Informed Neural Networks and show that specific optimizers such as L-BFGS act as conditioners on the physical losses.
+
+Supervised training: In contrast to the unsupervised training of Physics-informed Neural Networks, purely data-driven models have demonstrated remarkable capabilities for PDE simulation and forecasting. In most of the existing literature, the entire solver is replaced by a DL architecture and focuses on directly computing the solution from a given input data. A widely studied setting is operator learning which learns mappings between function spaces (Li et al., 2020; Kovachki et al., 2023; Lu et al., 2021). This method is very efficient, with the downside of relying on quite large quantities of data for training in order to ensure adequate generalization. Additionally, the neural network does not have access to the PDE in itself, only indirectly through the data. To ensure physical constraints in purely data-driven training, hybrid models have been proposed. The latter relies on both the available physical knowledge and some data. Some examples include the Aphinity model (Yin et al., 2021) (where the authors assume partial knowledge of the physics and learned the remaining dynamics from data), Physics-informed Deep Operator Networks (PIDON) (Wang et al., 2021b; Goswami et al., 2022), Physics-informed Neural Operator (PINO) (Li et al., 2023b) (DeepONet architecture (Lu et al., 2021) or Neural Operator models (Kovachki et al., 2023; Li et al., 2020) respectively with a combination of data and physical losses).
+
+Learning to solve: Improving the learning scheme and optimizers through data-driven training has been studied since Li & Malik (2016) and Andrychowicz et al. (2016). These works propose to learn the optimizer of neural networks, which are classically optimized through gradient-based algorithms such as Adam. They focus on improving training strategies for neural networks, which do not suffer from the optimization issues and ill-conditioning properties of physics-informed losses. We refer the reader to the survey of Chen et al. (2021) for a complete overview. The closer work to ours is the very
+
+recent work of Bihlo (2024) in which the author assesses the capabilities of learned optimizers for physics-informed neural networks. The main difference with our work relies on the problem setting. Bihlo (2024) considers learning an optimizer on a single equation, and for different neural networks initialization, while we focus on efficiently solving several instances of parametric PDE with varying PDE parameters  $\gamma, f, g$ .
+
+# B THEORETICAL ANALYSIS OF OUR METHOD AND PINNS
+
+Setting. We consider the following linear PDE:
+
+$$
+\mathcal {D} u (x) = f (x), \quad x \in \Omega , \tag {12}
+$$
+
+$$
+u (x) = g (x), \quad x \in \partial \Omega ,
+$$
+
+where  $\Omega \subset \mathbb{R}^d$  is an open bounded domain,  $\mathcal{D}$  is a linear differential operator,  $f(x)$  is a given function in  $\Omega$ , and  $g(x)$  is a given function on the boundary  $\partial \Omega$ .
+
+# B.1 THEORETICAL ANALYSIS OF PINNS
+
+Our aim is to find an approximate solution  $u_{\Theta}(x)$ , parameterized by  $\Theta \in \mathbb{R}^N$ ,  $\Theta = \{\theta_i\}_{i=1}^N$  that minimizes the loss function:
+
+$$
+\mathcal {L} _ {\mathrm {P D E}} (\Theta) = \mathcal {L} _ {\mathrm {R e s}} (\Theta) + \lambda \mathcal {L} _ {\mathrm {B C}} (\Theta), \tag {13}
+$$
+
+where:
+
+$$
+\mathcal {L} _ {\mathrm {R e s}} (\Theta) = \frac {1}{2} \int_ {\Omega} \left(\mathcal {D} u _ {\Theta} (x) - f (x)\right) ^ {2} d x, \quad \mathcal {L} _ {\mathrm {B C}} (\Theta) = \frac {1}{2} \int_ {\partial \Omega} \left(u _ {\Theta} (x) - g (x)\right) ^ {2} d x,
+$$
+
+and  $\lambda > 0$  is a regularization parameter balancing the PDE residual and boundary conditions.
+
+We perform gradient descent updates with step size  $\eta$ . At step  $k > 0$ , updates write as:
+
+$$
+\Theta_ {k + 1} = \Theta_ {k} - \eta \nabla_ {\Theta} \mathcal {L} _ {\mathrm {P D E}} (\Theta_ {k}).
+$$
+
+We establish the following theorem regarding the convergence rate of gradient descent.
+
+Theorem 2 (Convergence rate of PINNs). Given a linear ansatz  $u_{\Theta}(x) = \sum_{i=1}^{N} \theta_i \phi_i(x)$ , the number of steps  $N(\varepsilon)$  required to achieve an error  $\| \Theta_k - \Theta^* \|_2 \leq \varepsilon$  satisfies:
+
+$$
+N (\varepsilon) = O \left(\kappa (A) \ln \left(\frac {1}{\varepsilon}\right)\right), \tag {14}
+$$
+
+where  $\kappa(A)$  is the condition number of the matrix  $A \in \mathbb{R}^{n \times n}$  defined by:
+
+$$
+A _ {i, j} = \int_ {\Omega} \left(\mathcal {D} \phi_ {i} (x)\right) \left(\mathcal {D} \phi_ {j} (x)\right) d x + \lambda \int_ {\partial \Omega} \phi_ {i} (x) \phi_ {j} (x) d x. \tag {15}
+$$
+
+Proof. Since  $u_{\Theta}(x) = \sum_{i=1}^{N} \theta_i \phi_i(x)$ , we have:
+
+$$
+\frac {\partial u _ {\Theta} (x)}{\partial \theta_ {i}} = \phi_ {i} (x), \quad \frac {\partial (\mathcal {D} u _ {\Theta} (x))}{\partial \theta_ {i}} = \mathcal {D} \phi_ {i} (x).
+$$
+
+The gradient of the residual loss is:
+
+$$
+\nabla_ {\Theta} \mathcal {L} _ {\mathrm {R e s}} (\Theta) = \int_ {\Omega} \left(\mathcal {D} u _ {\Theta} (x) - f (x)\right) \mathcal {D} \phi (x) d x,
+$$
+
+where  $\mathcal{D}\phi (x)$  is the vector with components  $\mathcal{D}\phi_{i}(x)$ . Similarly, the gradient of the boundary loss is:
+
+$$
+\nabla_ {\Theta} \mathcal {L} _ {\mathrm {B C}} (\Theta) = \int_ {\partial \Omega} (u _ {\Theta} (x) - g (x)) \phi (x) d x,
+$$
+
+where  $\phi (x)$  is the vector of basis functions evaluated at  $x$ . Therefore, the total gradient is:
+
+$$
+\nabla_ {\Theta} \mathcal {L} _ {\mathrm {P D E}} (\Theta) = A \Theta - b,
+$$
+
+where the (positive, semi-definite) matrix  $A$  and vector  $b$  are defined as:
+
+$$
+A _ {i, j} = \int_ {\Omega} \left(\mathcal {D} \phi_ {i} (x)\right) \left(\mathcal {D} \phi_ {j} (x)\right) d x + \lambda \int_ {\partial \Omega} \phi_ {i} (x) \phi_ {j} (x) d x, \tag {16}
+$$
+
+$$
+b _ {i} = \int_ {\Omega} f (x) \mathcal {D} \phi_ {i} (x) d x + \lambda \int_ {\partial \Omega} g (x) \phi_ {i} (x) d x.
+$$
+
+Thus, the gradient descent update becomes:
+
+$$
+\Theta_ {k + 1} = \Theta_ {k} - \eta (A \Theta_ {k} - b).
+$$
+
+Subtracting  $\Theta^{*}$  (the optimal parameter vector satisfying  $A\Theta^{*} = b$ ) from both sides:
+
+$$
+\Theta_ {k + 1} - \Theta^ {*} = \Theta_ {k} - \Theta^ {*} - \eta A (\Theta_ {k} - \Theta^ {*}).
+$$
+
+Simplifying:
+
+$$
+\Theta_ {k + 1} - \Theta^ {*} = (\operatorname {I d} - \eta A) (\Theta_ {k} - \Theta^ {*}).
+$$
+
+By recursively applying the update rule, we obtain:
+
+$$
+\Theta_ {k} - \Theta^ {*} = (\operatorname {I d} - \eta A) ^ {k} (\Theta_ {0} - \Theta^ {*}).
+$$
+
+Since  $A$  is symmetric positive definite, it has eigenvalues  $\lambda_1 \leq \lambda_2 \leq \dots \leq \lambda_n$  with  $\lambda_i > 0$ . To ensure convergence, we require  $0 < \eta < \frac{2}{\lambda_{\max}(A)}$ . Choosing  $\eta = \frac{c}{\lambda_{\max}(A)}$  with  $0 < c < 2$ , we have:
+
+$$
+1 - \eta \lambda_ {i} = 1 - c \frac {\lambda_ {i}}{\lambda_ {\max} (A)}.
+$$
+
+The spectral radius  $\rho$  of  $\operatorname{Id} - \eta A$  is:
+
+$$
+\rho = \max \left\{\left| 1 - c \frac {\lambda_ {\min} (A)}{\lambda_ {\max} (A)} \right|, | 1 - c | \right\} = \max \left\{1 - \frac {c}{\kappa (A)}, | 1 - c | \right\},
+$$
+
+where  $\kappa(A) = \frac{\lambda_{\max}(A)}{\lambda_{\min}(A)}$  is the condition number of  $A$ . By choosing  $0 < c < 1$ , we ensure  $|1 - c| < 1$ , and since  $\kappa(A) \geq 1$ , we have  $1 - \frac{c}{\kappa(A)} < 1$ . Thus, the convergence factor is:
+
+$$
+\rho = 1 - \frac {c}{\kappa (A)}.
+$$
+
+Therefore:
+
+$$
+\left\| \Theta_ {k} - \Theta^ {*} \right\| _ {2} \leq \left(1 - \frac {c}{\kappa (A)}\right) ^ {k} \left\| \Theta_ {0} - \Theta^ {*} \right\| _ {2}.
+$$
+
+To achieve  $\| \Theta_k - \Theta^* \|_2 \leq \varepsilon$ , the number of iterations  $N(\varepsilon)$  satisfies:
+
+$$
+N (\varepsilon) \geq \frac {\ln \left(\varepsilon / \| \Theta_ {0} - \Theta^ {*} \| _ {2}\right)}{\ln \left(1 - \frac {c}{\kappa (A)}\right)}.
+$$
+
+Using the inequality  $\ln (1 - x) \leq -x$  for  $0 < x < 1$ , we get:
+
+$$
+N (\varepsilon) \leq \frac {\kappa (A)}{c} \ln \left(\frac {\| \Theta_ {0} - \Theta^ {*} \| _ {2}}{\varepsilon}\right).
+$$
+
+Thus:
+
+$$
+N (\varepsilon) = O \left(\kappa (A) \ln \left(\frac {1}{\varepsilon}\right)\right).
+$$
+
+![](images/6875b882d5d5e304811ce015c6f8fff67c31a4327df46decde0648189ffb95e5.jpg)
+
+We have shown that for a linear ansatz  $u_{\Theta}(x) = \sum_{i=1}^{N} \theta_i \phi_i(x)$ , the convergence rate of gradient descent depends linearly on the condition number  $\kappa(A)$  of the system matrix  $A$ . A large condition number impedes convergence, requiring more iterations to achieve a desired accuracy  $\varepsilon$ .
+
+![](images/d4ef158cc8e801013206136bb16aa4e0ae998c49ab16b1f6347fd05883ec95fd.jpg)
+
+# B.2 THEORETICAL ANALYSIS OF OUR METHOD
+
+In practice, we often work with multiple data points. For each data point, there is an associated parameter vector  $\Theta \in \mathbb{R}^N$ . We are interested in the iterative update where the gradient is transformed by a neural network  $\mathcal{F}$ :
+
+$$
+\Theta_ {l + 1} = \Theta_ {l} - \eta \mathcal {F} \left(\nabla_ {\Theta} \mathcal {L} _ {\mathrm {P D E}} \left(\Theta_ {l}\right)\right), \tag {17}
+$$
+
+where  $\Theta_{l}$  represents the parameter vector at iteration  $l$ . Recall that  $\mathcal{F}$  is trained to minimize the loss after  $L$  iteration steps for  $M$  data points:
+
+$$
+\mathcal {L} _ {\mathrm {D A T A}} = \frac {1}{m} \sum_ {k = 1} ^ {M} \left\| u _ {\Theta_ {L} ^ {(k)}} - u _ {k} ^ {*} \right\| _ {2} ^ {2}, \tag {18}
+$$
+
+Theorem 3. (Convergence rate of our method). Given a linear ansatz  $u_{\Theta}(x) = \sum_{i=1}^{N} \theta_i \phi_i(x)$ , assume  $\mathcal{F}$  behaves like its linearization  $P = \left. \frac{\partial \mathcal{F}}{\partial v} \right|_{v=0}$ . The number of steps  $N'(\varepsilon)$  required to achieve an error  $\| \Theta_l - \Theta^* \|_2 \leq \varepsilon$  satisfies:
+
+$$
+N ^ {\prime} (\varepsilon) = O \left(\kappa (P A) \ln \left(\frac {1}{\varepsilon}\right)\right), \tag {19}
+$$
+
+Moreover, if  $\mathcal{F}$  minimizes  $\mathcal{L}_{DATA}$  this necessarily implies  $\kappa(PA) = 1 \leq \kappa(A)$ . Consequently, the number of steps is effectively reduced, i.e.,  $N'(\varepsilon) \ll N(\varepsilon)$ .
+
+Proof. Since  $\mathcal{F}$  behaves like its linearization  $P$ , the gradient descent update becomes (refer to proof of Theorem 2 for steps):
+
+$$
+\Theta_ {l + 1} = \Theta_ {l} - \eta P (A \Theta_ {l} - b).
+$$
+
+Let  $\Theta^{*}$  be the optimal parameter vector minimizing  $\mathcal{L}_{\mathrm{PDE}}$ . Then, the difference between the parameter vector at iteration  $l$  and the optimal parameter vector is:
+
+$$
+\Theta_ {l + 1} - \Theta^ {*} = \Theta_ {l} - \Theta^ {*} - \eta P A (\Theta_ {l} - \Theta^ {*}) = (\operatorname {I d} - \eta P A) (\Theta_ {l} - \Theta^ {*}).
+$$
+
+By recursively applying this update until the final step  $L$ , we obtain:
+
+$$
+\Theta_ {L} - \Theta^ {*} = (\operatorname {I d} - \eta P A) ^ {L} (\Theta_ {0} - \Theta^ {*}).
+$$
+
+Since we have multiple data points, each with its own parameter vector, we consider the concatenation when necessary. Let's introduce  $\Xi_{l}$  as the matrix whose columns are the parameter vectors:
+
+$$
+\Xi_ {L} = \left[ \Theta_ {L} ^ {(1)}, \Theta_ {L} ^ {(2)}, \dots , \Theta_ {L} ^ {(m)} \right].
+$$
+
+Similarly,  $\Xi^{*}$  contains the optimal parameter vectors for each data point. The update for all data points can be written collectively:
+
+$$
+\Xi_ {L} - \Xi^ {*} = (\operatorname {I d} - \eta P A) ^ {L} (\Xi_ {0} - \Xi^ {*}).
+$$
+
+Since  $\mathcal{F}$  minimizes  $\mathcal{L}_{\mathrm{DATA}}$ , we have  $\Xi_L = \Xi^*$ , implying:
+
+$$
+(\operatorname {I d} - \eta P A) ^ {L} (\Xi_ {0} - \Xi^ {*}) = 0.
+$$
+
+Given that the values of  $\Xi_0$  are iid and sampled randomly from a continuous distribution, because the set of singular matrices has measure zero, the square matrix  $(\Xi_0 - \Xi^*)(\Xi_0 - \Xi^*)^\top$  is full rank (i.e., invertible), with probability 1. Thus, the only way for the above equality to hold is if:
+
+$$
+(\operatorname {I d} - \eta P A) ^ {L} = 0.
+$$
+
+This means  $\operatorname{Id} - \eta PA$  is nilpotent of index  $L$ . Consequently, all eigenvalues of  $\operatorname{Id} - \eta PA$  are zero, implying that all eigenvalues of  $PA$  are equal to  $\frac{1}{\eta}$ , leading to  $\kappa(PA) = \lambda_{\max}(PA) / \lambda_{\min}(PA) = 1$ , which is the optimal condition number. Referring to the convergence analysis in Theorem 2, we have:
+
+$$
+N ^ {\prime} (\varepsilon) \leq \frac {\kappa (P A)}{c} \ln \left(\frac {\| \Xi_ {0} - \Xi^ {*} \| _ {2}}{\varepsilon}\right).
+$$
+
+Which directly implies
+
+$$
+N ^ {\prime} (\varepsilon) = O \left(\kappa (P A) \ln \left(\frac {1}{\varepsilon}\right)\right), \tag {20}
+$$
+
+With  $\kappa(PA) = 1$ , this leads us to the desired result:
+
+$$
+N ^ {\prime} (\varepsilon) = O \left(\ln \left(\frac {1}{\varepsilon}\right)\right) \ll O \left(\kappa (A) \ln \left(\frac {1}{\varepsilon}\right)\right) =: N (\epsilon),
+$$
+
+Thus, the number of iterations required is significantly reduced compared to the case without the neural network preconditioner.
+
+![](images/24ccfc44063a1e1b0e818bdfdaab92d34980f72a15f528b9b027eeaa473afc6f.jpg)
+
+Discussion The convergence proofs for our method fundamentally rely on the assumption of linearity in the underlying problems. It is important to note that the theoretical analysis does not extend to non-linear cases. Consequently, for non-linear scenarios, the theory should be viewed primarily as a tool for building intuition or providing motivation, rather than a definitive proof. This is due to the lack of established methods for rigorously studying the non-linear regime, as no known results currently address such cases.
+
+Under these conditions:
+
+- This optimal condition number implies that convergence is not only guaranteed but also optimal, requiring fewer iterations.  
+- Guaranteed Convergence: The method reliably achieves convergence to the optimal solution due to the reduced condition number.  
+- Optimal Convergence Speed: With  $\kappa(PA) = 1$ , the neural network provides an enhanced convergence rate, resulting in fewer required iterations compared to the original system without the neural network.
+
+![](images/f36896708739914ec577bba14cadb05e11daf22c8a96e22af2a7e024b991a91b.jpg)
+
+# C DATASET DETAILS
+
+For all datasets, we kept 800 samples for training and 200 as testing examples (except otherwise stated in the experiments).
+
+# C.1 HELMHOLTZ
+
+We generate a dataset following the  $1d$  static Helmholtz equation eq. (21). For  $x \in [0, 1[$ ,
+
+$$
+\left\{ \begin{array}{l l} u (x) ^ {\prime \prime} + \omega^ {2} u (x) & = 0, \\ u (0) & = u _ {0}, \\ u ^ {\prime} (0) & = v _ {0}. \end{array} \right. \tag {21}
+$$
+
+The solution can be analytically derived:  $u(x) = \alpha \cos(\omega x + \beta)$ , with  $\beta = \arctan\left(\frac{-v_0}{\omega u_0}\right)$ ,  $\alpha = \frac{u_0}{\cos(\beta)}$  and directly computed from the PDE data. We generate 1,024 trajectories for training and 256 for testing with  $u_0, v_0 \sim \mathcal{N}(0,1)$ , and  $\omega \sim \mathcal{U}(0.5,50)$  and compute the solution on [0,1] with a spatial resolution of 256. For training, we keep 800 samples and use the complete dataset for the additional experiments presented in figs. 12a and 12b. Moreover, we sub-sample the spatial resolution by 4 and keep 64 points for training.
+
+![](images/8127550a930df51efa770cffb6363c38afee88c179ead6f75f794d70bdc8ac9d.jpg)  
+(a)  
+Figure 3: Samples from the Helmholtz Dataset.
+
+![](images/ef16f9f5f3ab75441c53b1851eae97557a8830177dad80e513315fe67c8ed6b1.jpg)  
+(b)
+
+# C.2 POISSON
+
+We generate a dataset following the  $1d$  static Poisson equation eq. (22) with forcing term. For  $x \in [0, 1]$ ,
+
+$$
+\left\{ \begin{array}{l l} - u ^ {\prime \prime} (x) & = f (x), \\ u (0) & = u _ {0}, \\ u ^ {\prime} (0) & = v _ {0}. \end{array} \right. \tag {22}
+$$
+
+We chose  $f$  to be a non-linear forcing term:  $f(x) = \frac{\pi}{K}\sum_{i=1}^{K}a_{i}i^{2r}\sin(\pi x)$ , with  $a_{i} \sim \mathcal{U}(-100,100)$ , we used  $K = 16$ ,  $r = -0.5$ , and solve the equation using a backward finite difference scheme. We generate 1,000 trajectories with  $u_{0}, v_{0} \sim \mathcal{N}(0,1)$  and compute the solution on [0, 1] with a spatial resolution of 64.
+
+Reaction-Diffusion We use a non-linear reaction-diffusion used in (Krishnapriyan et al., 2021; Toloubidokhti et al., 2024). This PDE has been shown to be a failure case for PINNs (Krishnapriyan et al., 2021). The PDE states as follows:
+
+$$
+\frac {\partial u (t , x)}{\partial t} - \nu \frac {\partial^ {2} u (t , x)}{\partial x ^ {2}} - \rho u (t, x) (1 - u (t, x) = 0, \tag {23}
+$$
+
+$$
+u (0, x) = e ^ {- 3 2 (x - 1 / 2) ^ {2}}. \tag {24}
+$$
+
+![](images/423ee546fe28a3a945dd25dafff93f934ee5822d33ea61023a3f80cb2816db76.jpg)  
+(a)
+
+![](images/759d83f3fc5a5b060d15dbec7da24f70ce11edc9b88d6eafe049e2a40911ecb5.jpg)  
+Figure 4: Samples from the Poisson Dataset.
+
+![](images/5cb58353d311d7cd7a5754ad1d24d45b779553655e6aa5f935e501f0333d730e.jpg)  
+(b)
+
+![](images/34837b7658674612338d3454fd3cf4c3d769b1fa6bf0390041f7b014df00fbc9.jpg)
+
+We generate 800 trajectories by varying  $\nu$  in [1, 5] and  $\rho$  in  $[-5, 5]$ . Spatial resolution is 256 and temporal resolution is 100, which we sub-sample by 4 for training, leading to a spatial resolution of  $64 \times 25$ . The PDE is solved on  $[0, 1]^2$  as in (Toloubidokhti et al., 2024).
+
+![](images/f95bf3b81a0d098060ab9f91eadf1e78c715d7d0f9a9f6ffc6342eac9a1b5598.jpg)  
+Figure 5: Samples from the Reaction-Diffusion Dataset.
+
+![](images/78554ba4c3a98a28b99d288804fa268f64d1c34d1950cca6fd71660e8c828c26.jpg)
+
+Reaction-Diffusion with initial conditions: To complexify the setting, we also change the initial condition of the problem (NLRDIC in the following). The initial condition is expressed as follows:
+
+$$
+u (x, 0) = \sum_ {i = 1} ^ {3} a _ {i} e ^ {- \frac {\left(\frac {x - h / 4}{h}\right) ^ {2}}{4}}. \tag {25}
+$$
+
+Where  $a_i$  are randomly chosen in [0, 1] and  $h = 1$  is the spatial resolution.
+
+![](images/6f00d98116d7d9ebbc450bdcc52ab3ea4a5315b21505556fd43dca1b43c73931.jpg)  
+Figure 6: Samples from the Reaction-Diffusion Dataset with initial conditions changed.
+
+![](images/e7dd61b26a22f078eebe58e56cc640d72977cae7a4a72873c7712369d22f8c49.jpg)
+
+# C.3 DARCY FLOW
+
+The 2d Darcy Flow dataset is taken from (Li et al., 2020) and is commonly used in the operator learning literature (Li et al., 2023b; Goswami et al., 2022).
+
+$$
+- \nabla . (a (x) \nabla u (x)) = f (x) \quad x \in (0, 1) ^ {2}, \tag {26}
+$$
+
+$$
+u (x) = 0 \quad x \in \partial (0, 1) ^ {2}. \tag {27}
+$$
+
+For this dataset, the forcing term  $f$  is kept constant  $f = 1$ , and  $a(x)$  is a piece-wise constant diffusion coefficient taken from (Li et al., 2020). We kept 1,000 trajectories (on the 5,000 available) with a spatial resolution of  $64 \times 64$ .
+
+![](images/94a96a9c152d05720fe21675b6e8496243930542fa5e1c13e0a033d36be80023.jpg)  
+(a)  
+Figure 7: Samples from the Darcy Dataset.
+
+![](images/21d8cdabcbf168081c606b97dc866d01591d02665e26bce7bb090ad6ea13045e.jpg)  
+(b)
+
+# C.4 HEAT
+
+As proof that our method can handle  $2\mathrm{d} +$  time, we consider the dataset proposed by (Zhou & Farimani, 2024).
+
+$$
+\frac {\partial u (x , y , t)}{\partial t} - \nu \nabla^ {2} u (x, y, t) = 0, \tag {28}
+$$
+
+$$
+u (x, y, 0) = \sum_ {j = 1} ^ {J} A _ {j} \sin \left(\frac {2 \pi l _ {x j} x}{L} + \frac {2 \pi l _ {y j} y}{L} + \phi_ {i}\right). \tag {29}
+$$
+
+Where  $L = 2$ ,  $\nu$  is randomly chosen between  $[2 \times 10^{-3}, 2 \times 10^{-2}]$ ,  $A_{j}$  in  $[-0.5, 0.5]$ ,  $l_{xj}$ ,  $l_{xy}$  are integers in  $\{1, 2, 3\}$  and  $\phi$  is in  $[0, 2\pi]$ . As a difference with (Zhou & Farimani, 2024), we randomly
+
+chose  $J$  between 1 and  $J_{max} = 5$  to have more diversity in the represented frequencies in the data. The PDEs are sampled with spatial resolution 64 in x and y and temporal resolution 100. However, during training the spatial resolution are subsampled by 4 and the coordinates are re scaled between 0 and 1. As for other PDEs, we use 800 trajectories for training and 200 for testing.
+
+![](images/70643fe87b520fcbc186520d9d714f592b89e47d7268d1cbd38fdec68540ba03.jpg)  
+(a)
+
+![](images/96c8ec1a68af00e7913ea4d03b49a2cc722d70d4fabc58db76463dc34e5ddce9.jpg)  
+(b)  
+Figure 8: Samples from the Heat dataset.
+
+# C.5 ADDITIONAL DATASET: ADVECTION
+
+The dataset is taken from (Takamoto et al., 2023).
+
+$$
+\frac {\partial u (t , x)}{\partial t} + \beta \frac {\partial u (t , x)}{\partial x} = 0, \quad x \in (0, 1), t \in (0, 2 ], \tag {30}
+$$
+
+$$
+u (0, x) = u _ {0} (x), \quad x \in (0, 1). \tag {31}
+$$
+
+Where  $\beta$  is a constant advection speed, and the initial condition is  $u_0(x) = \sum_{k_i = k_1\dots k_N}A_i\sin (k_ix + \phi_i)$ , with  $k_{i} = \frac{2\pi n_{i}}{L_{x}}$  and  $n_i$  are randomly selected in [1, 8]. The author used  $N = 2$  for this PDE. Moreover,  $A_{i}$  and  $\phi_{i}$  are randomly selected in [0, 1] and  $(0,2\pi)$  respectively. Finally,  $L_{x}$  is the size of the domain (Takamoto et al., 2023).
+
+The PDEBench's Advection dataset is composed of several configurations of the parameter  $\beta$  ( $\{0.1, 0.2, 0.4, 0.7, 1, 2, 4, 7\}$ ), each of them is composed of 10,000 trajectories with varying initial conditions. From these datasets, we sampled a total of 1,000 trajectories for  $\beta \in \{0.2, 0.4, 0.7, 1, 2, 4\}$  (which gives about 130 trajectories for each  $\beta$ ). This gives a dataset with different initial conditions and parameters. Moreover, during training, we sub-sampled the trajectories by 4, leading to a grid of resolution 25 for the t-coordinate and 256 for the x-coordinate.
+
+![](images/8d2aed89ffbac66384f0c4582302a19f71a917e0754d86a3b1826454fcb53112.jpg)  
+Trajectory sample for Advection
+
+![](images/fffefbf34d4ce20a27f259fa451e9b3c771e4d7a23e54bf66f9ee70cab4e7481.jpg)  
+(a)  
+Figure 9: Samples from the Advections Dataset.
+
+![](images/c12aa0d16ac753b098b9cfc073f6abf52b883b00e4a46f8ce703596998224172.jpg)  
+Trajectory sample for Advection
+
+![](images/60cecd66f6e800682d1d4444b5987c327c3b72bc10c6fc025387bd90349d9cd3.jpg)  
+(b)
+
+# C.6 SUMMARY OF PROBLEM SETTINGS CONSIDERED
+
+A summary of the datasets and parameters changing between 2 trajectories is presented in table 3.
+
+Table 3: Parameters changed between each trajectory in the considered datasets in the main part of the paper as well as additional datasets (Advections and NLRDIC).  
+
+<table><tr><td>Dataset</td><td>Changing PDE data</td><td>Range / Generation</td></tr><tr><td rowspan="3">Helmholtz</td><td>ω</td><td>[0.5, 50]</td></tr><tr><td>u0</td><td>N(0,1)</td></tr><tr><td>v0</td><td>N(0,1)</td></tr><tr><td rowspan="3">Poisson</td><td>Ai</td><td>[-100, 100]</td></tr><tr><td>u0</td><td>N(0,1)</td></tr><tr><td>v0</td><td>N(0,1)</td></tr><tr><td rowspan="2">Reaction-diffusion</td><td>ν</td><td>[1, 5]</td></tr><tr><td>ρ</td><td>[-5, 5]</td></tr><tr><td>Darcy</td><td>a(x)</td><td>ψ#N(0, (-Δ + 9I)-2) with ψ = 12 * 1R+ + 3 * 1R+</td></tr><tr><td rowspan="5">Heat</td><td>ν</td><td>[2 × 10-3, 2 × 10-2]</td></tr><tr><td>Jmax</td><td>{1, 2, 3, 4, 5}</td></tr><tr><td>A</td><td>[0.5, -0.5]</td></tr><tr><td>Kx, Ky</td><td>{1, 2, 3}</td></tr><tr><td>φ</td><td>[0, 2π]</td></tr><tr><td rowspan="4">Advection</td><td>β</td><td>{0.2, 0.4, 0.7, 1, 2, 4}</td></tr><tr><td>Ai</td><td>[0, 1]</td></tr><tr><td>φi</td><td>[0, 2π]</td></tr><tr><td>ki</td><td>{2kπ}8k=1</td></tr><tr><td rowspan="3">NLRDIC</td><td>ν</td><td>[1, 5]</td></tr><tr><td>ρ</td><td>[-5, 5]</td></tr><tr><td>ai</td><td>[0, 1]</td></tr></table>
+
+# D IMPLEMENTATION DETAILS
+
+We add here more details about the implementation and experiments presented in section 4.
+
+We implemented all experiments with PyTorch (Ansel et al., 2024). We estimate the computation time needed for development and the different experiments to be approximately 300 days.
+
+# D.1 B-SPLINE BASIS
+
+We chose to use a B-Spline basis to construct the solution. We manually build the spline and compute its derivatives thanks to the formulation and algorithms proposed in (Piegl & Tiller, 1996). We used Splines of degree  $d = 3$  and constructed the Splines with 2 different configurations:
+
+- Take  $N + d + 1$  equispaced nodes of multiplicity 1 from  $\frac{d}{N}$  to  $1 + \frac{d}{N}$ . This gives a smooth local basis with no discontinuities (see fig. 10a) represented by a shifted spline along the x-axis (denoted as shifted in the following).  
+- Use  $N + 1 - d$  nodes of multiplicity 1 and 2 nodes of multiplicity  $d$  (typically on the boundary nodes: 0 and 1). This means that nodes 0 and 1 are not differentiable (see fig. 10b). We call this set-up equispaced.
+
+![](images/c7c26e0c97ff47af317be733b08ea3cb12a768c51df5463d1a6232a43e989b5d.jpg)  
+(a)
+
+![](images/b69ed1d6634d3d3dd1ca20db6cc74f38d53b3cdf175397e5725aba52ee76f43f.jpg)  
+(b)  
+Figure 10: B-spline basis with  $N = 10$  terms with shifted spline (Left) and higher multiplicity nodes (Right). Dashed lines represent nodes' position with color the darker, the higher the multiplicity.
+
+Higer-dimension basis For  $1\mathrm{d} +$  time, 2d dataset, and  $2\mathrm{d} +$  time we build a 2 (or 3) dimensional B-spline basis, i.e., we treat the time coordinates as spatial ones. To build such bases, we compute the Cartesian product between the 2 (or 3) 1d-bases, 1 per dimension. This means that for a 2d dataset, for which we chose to use bases of size  $N1$  and  $N2$  for each coordinate, the resulting basis will have  $N1 + N2 + N1*N2$  terms. This method makes the training more costly and several techniques to improve its scalability could be used. For 3d datasets, the number of terms in the basis is cubic.
+
+# D.2 TRAINING DETAILS
+
+In our experiments, neural networks are trained using the Adam optimizer. For network optimization, we employ a smooth  $l1$ -loss for our solver while for other baselines, we use MSE loss and/or physical losses. All models are trained for at least 1,500 epochs on datasets composed of some sampling of  $\gamma$  and/or  $f$  and/or  $g$ . If not stated otherwise, we train our proposed method for 750 epochs and baselines for 1,500 epochs. We use the Adam optimizer and an initial learning rate of 0.001. We use an exponential learning rate scheduler that lowers the learning rate by 0.995 every epoch. Experiments are conducted on NVIDIA Titan V (12 Go) for 1d datasets and NVIDIA RTX A6000 GPU with 49Go for  $1d +$  time, 2d datasets or  $2d +$  time. We recall in algorithm 3 the pseudo-code for training our proposed method.
+
+Algorithm 3: Training algorithm for learning to optimize physics-informed losses.  
+```latex
+Data:  $\Theta_0\in \mathbb{R}^n$  ,PDE  $(\gamma ,f,g)$  ,sample values  $u(x)$    
+Result:  $\mathcal{F}_{\varrho}$    
+for  $e = 1\dots$  epochs do for (PDE, x, u) in dataset do Initialize  $\Theta_0$  Estimate  $\Theta_L$  from  $\Theta_0$ $(\gamma ,f,g)$  using Algorithm 1 Reconstruct  $u_{\Theta_L}(x)$  Update the parameters  $\varrho$  of  $\mathcal{F}$  with gradient descent from the data loss in Equation 10. end   
+end   
+return  $\mathcal{F}_{\varrho}$
+```
+
+We make use of two nested components: the solver for providing the approximate solution to the PDE and the optimizer that conditions the training of the solver. Both are using gradient descent but with different inputs and objectives. The former optimizes the PDE loss (inner loop), while the latter optimizes the gradient steps of the solver through conditioning (outer loop). Optimization of the two components proceeds with an alternate optimization scheme. In particular, this implies that the map  $F_{\varrho}$  is kept fixed during the inner optimization process.
+
+# D.3 MODELS
+
+We present here the training details for our model and the baseline. The results can be found in table 2. In all our experiments, we use a GeLU (Hendrycks & Gimpel, 2016) activation function. The details of the model architecture on each dataset are presented in table 4.
+
+Hyperparameters for Helmholtz: For the baselines, we empirically searched hyperparameters to allow the network to handle the high frequencies involved in the Helmholtz dataset. Unfortunately, other network architectures did not improve the results. We proceed similarly for other datasets.
+
+Training of PPINNs: For the Helmholtz dataset, we trained our model for 2000 epochs with a plateau learning rate scheduler with patience of 400 epochs. For the Poisson dataset, we trained our parametric PINN model for 5000 epochs with a cosine annealing scheduler with a maximum number of iterations of 1000. For the Reaction-diffusion dataset, we consider an initial learning rate of 0.0001 instead of 0.001. Finally, P2INNs is trained for 5,000 epochs using the Adam optimizer and an exponential scheduler with patience 50.
+
+PINNs baselines (PINNs + L-BFGS and PINNs-multi-opt): The conditioning of the problem highly depends on the parameters of the PDE and initial/boundary conditions. This can lead to unstable training when optimizing PINNs, requiring specific parameter configurations for each PDE. To avoid extensive research of the best training strategy, we found a configuration that allowed a good fitting of most of the testing dataset. This means that in the values reported in table 2, we removed trainings for which the losses exploded (only a few hard PDEs were removed, typically between none to 20). This prevents us from extensive hyper-parameter tuning on each PDE. Please note that this lowers the reported relative MSE, thus advantaging the baseline. We detail in tables 5 and 6 the hyper-parameters for each dataset.
+
+Basis configuration: For all datasets, we use Splines of degree 3, built with shifted nodes. We change the number of terms in the basis depending on the problem considered. For the  $1d$ -problem, we use 32 terms. For the  $2d$ -problem, 40 elements are in the basis of each dimension, except for Reaction-Diffusion where the variable  $t$  has 20 terms. Moreover, during training, we use the projection of the initial conditions and/or parameters and/or forcing terms function in the basis as input to the networks. Finally, for experiments using the Heat dataset (i.e. 3d basis), we used 15 terms for the  $x$  and  $y$  spatial coordinates and 10 terms for the  $t$ -coordinate.
+
+Table 4: Architecture details of our model and baselines  
+
+<table><tr><td rowspan="3">Model</td><td rowspan="3">Architecture</td><td colspan="5">Dataset</td></tr><tr><td colspan="2">1d</td><td>1d + time</td><td>2d</td><td>2d + time</td></tr><tr><td>Helmholtz</td><td>Poisson</td><td>NLRD</td><td>Darcy</td><td>Heat</td></tr><tr><td rowspan="2">PINNs</td><td>MLP depth</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td></tr><tr><td>MLP width</td><td>256</td><td>256</td><td>256</td><td>256</td><td>256</td></tr><tr><td rowspan="2">PPINNs</td><td>MLP depth</td><td>8</td><td>3</td><td>5</td><td>3</td><td>3</td></tr><tr><td>MLP width</td><td>64</td><td>256</td><td>256</td><td>256</td><td>256</td></tr><tr><td rowspan="4">PO-DeepONet</td><td>Branch Net depth</td><td>3</td><td>2</td><td>5</td><td>5</td><td>5</td></tr><tr><td>Branch Net width</td><td>256</td><td>256</td><td>256</td><td>256</td><td>256</td></tr><tr><td>Trunk Net depth</td><td>3</td><td>2</td><td>5</td><td>5</td><td>5</td></tr><tr><td>Trunk Net width</td><td>256</td><td>256</td><td>256</td><td>256</td><td>256</td></tr><tr><td rowspan="9">P2INNs</td><td>Enc params depth</td><td>4</td><td>4</td><td>4</td><td>4</td><td>4</td></tr><tr><td>Enc params width</td><td>256</td><td>256</td><td>256</td><td>256</td><td>256</td></tr><tr><td>Emb params</td><td>128</td><td>128</td><td>128</td><td>128</td><td>128</td></tr><tr><td>Enc coord depth</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td></tr><tr><td>Enc coord width</td><td>256</td><td>256</td><td>256</td><td>256</td><td>256</td></tr><tr><td>Emb coord</td><td>128</td><td>128</td><td>128</td><td>128</td><td>128</td></tr><tr><td>Dec depth</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td></tr><tr><td>Dec width</td><td>256</td><td>256</td><td>256</td><td>256</td><td>256</td></tr><tr><td>Activations</td><td>GeLU</td><td>GeLU</td><td>GeLU</td><td>GeLU</td><td>GeLU</td></tr><tr><td rowspan="2">MLP + basis</td><td>MLP depth</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td></tr><tr><td>MLP width</td><td>256</td><td>256</td><td>1,024</td><td>1,024</td><td>1,024</td></tr><tr><td rowspan="4">PI-DeepONet</td><td>Branch Net depth</td><td>3</td><td>2</td><td>5</td><td>5</td><td>5</td></tr><tr><td>Branch Net width</td><td>256</td><td>256</td><td>256</td><td>256</td><td>256</td></tr><tr><td>Trunk Net depth</td><td>3</td><td>2</td><td>5</td><td>5</td><td>5</td></tr><tr><td>Trunk Net width</td><td>256</td><td>256</td><td>256</td><td>256</td><td>256</td></tr><tr><td rowspan="6">PINO</td><td>FNO depth</td><td>3</td><td>3</td><td>4</td><td>3</td><td>3</td></tr><tr><td>FNO width</td><td>64</td><td>64</td><td>64</td><td>64</td><td>64</td></tr><tr><td>FNO modes 1</td><td>16</td><td>16</td><td>10</td><td>20</td><td>7</td></tr><tr><td>FNO modes 2</td><td>-</td><td>-</td><td>5</td><td>20</td><td>7</td></tr><tr><td>FNO modes 3</td><td>-</td><td>-</td><td>-</td><td>-</td><td>5</td></tr><tr><td>FNO fc dim</td><td>64</td><td>64</td><td>64</td><td>64</td><td>64</td></tr><tr><td rowspan="6">Ours</td><td>FNO depth</td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td></tr><tr><td>FNO width</td><td>64</td><td>64</td><td>64</td><td>64</td><td>64</td></tr><tr><td>FNO modes 1</td><td>16</td><td>16</td><td>10</td><td>20</td><td>7</td></tr><tr><td>FNO modes 2</td><td>-</td><td>-</td><td>5</td><td>20</td><td>7</td></tr><tr><td>FNO modes 3</td><td>-</td><td>-</td><td>-</td><td>-</td><td>5</td></tr><tr><td>FNO fc dim</td><td>64</td><td></td><td>64</td><td>64</td><td>64</td></tr></table>
+
+Table 5: Hyperparameters for PINNs+L-BFGS baseline.  
+
+<table><tr><td rowspan="3">Hyper-parameter</td><td colspan="5">Dataset</td></tr><tr><td colspan="2">1d</td><td>1d + time</td><td>2d</td><td>2d + time</td></tr><tr><td>Helmholtz</td><td>Poisson</td><td>NLRD</td><td>Darcy</td><td>Heat</td></tr><tr><td>epochs</td><td>1,000</td><td>1,000</td><td>1,500</td><td>1,500</td><td>1,000</td></tr><tr><td>learning rate</td><td>1e-4</td><td>1e-5</td><td>1e-4</td><td>1e-3</td><td>1e-3</td></tr></table>
+
+Table 6: Hyperparameters for PINNs-multi-opt baseline.  
+
+<table><tr><td rowspan="3">Hyper-parameter</td><td colspan="5">Dataset</td></tr><tr><td colspan="2">1d</td><td>1d + time</td><td>2d</td><td>2d + time</td></tr><tr><td>Helmholtz</td><td>Poisson</td><td>NLRD</td><td>Darcy</td><td>Heat</td></tr><tr><td>epochs Adams</td><td>800</td><td>800</td><td>800</td><td>1,200</td><td>1,200</td></tr><tr><td>epochs L-BFGS</td><td>200</td><td>200</td><td>200</td><td>300</td><td>300</td></tr><tr><td>epochs total</td><td>1,000</td><td>1,000</td><td>1,000</td><td>1,500</td><td>1,500</td></tr><tr><td>learning rate Adam</td><td>1e-4</td><td>1e-5</td><td>1e-4</td><td>1e-3</td><td>1e-3</td></tr><tr><td>learning rate L-BFGS</td><td>1</td><td>1</td><td>1e-3</td><td>1</td><td>1</td></tr></table>
+
+# E ADDITIONAL EXPERIMENTS
+
+# E.1 ABLATION
+
+We experimentally show some properties of our iterative method. These evaluations are made on a test set composed of several instances of PDE with varying configurations  $(\gamma, f, g)$  that are unseen during training. These experiments are performed on the Helmholtz equation which appeared as one of the most complex to optimize in our evaluation. We used  $L = 5$  in our method for a better visualization, unless stated otherwise.
+
+Error w.r.t the number of steps L On fig. 11, we show that having more optimizer steps allows for a better generalization. However, we observed in our experiments that the generalization error stabilizes or even increases after the proposed 5 steps. This limitation of the solver should be investigated in future work in order to allow the model to make more iterations. Note that for this experiment, we lowered batch size (and adapted the learning rate accordingly) when the number of steps increased (fig. 11).
+
+![](images/77cb18eaa3a77f9f0bfe20ff4dcc3346fbfd728a3292bce59633955fb697ed0a.jpg)  
+Figure 11: Error on the test set (Helmholtz equation) w.r.t the number of iterations
+
+Error w.r.t the number of training samples On figs. 12a and 12b, we show that compared to other physics-informed baselines, our solver requires less data to learn to solve PDE. Note that in this simple example, the  $MLP + \text{basis}$  baseline also performed well. However, as shown in table 2, this is not the case for all the datasets. The contribution of the iterative procedure clearly appears since with  $2 \times$  less data, our model performs better than this baseline.
+
+![](images/235b5b005f908806fc359b93fb9e9ac67afc800f9eb46e0edee97560abb472be.jpg)  
+(a)
+
+![](images/1bc42c26f27939e8e0ee59f9caa9a15248c3741d6b7a2d9aaf7ac6d46f558f98.jpg)  
+(b)  
+Figure 12: Relative MSE on the test set w.r.t the number of training samples on Helmholtz (left) and Darcy (right) datasets.
+
+Optimization with  $\mathcal{L}_{\mathrm{PDE}}$  Table 7 shows that optimizing our network with physical loss greatly complicates training. Indeed, adding an ill-conditioned loss to the standard MSE makes training ill-conditioned.
+
+Table 7: How the physical loss complexifies training and lowers performances. Metrics in Relative MSE on the test set of Helmholtz equation.
+
+<table><tr><td>Input</td><td>Relative MSE</td></tr><tr><td>LPDE + L_DATA</td><td>1.11</td></tr><tr><td>L_DATA</td><td>2.19e-2</td></tr></table>
+
+Iterative update & SGD-based update In table 8, we compare two optimizer configurations. "Direct" means that the network directly predicts the parameters for the next step (i.e.  $\Theta_{k+1} = \mathcal{F}_{\varrho}(\nabla \mathcal{L}_{\mathrm{PDE}}(\Theta_k), \gamma, f, g)$ ), while "GD", corresponds to the update rule described before and using SGD as the base algorithm  $\Theta_{k+1} = \Theta_k - \eta \mathcal{F}_{\varrho}(\nabla \mathcal{L}_{\mathrm{PDE}}(\Theta_k), \gamma, f, g)$ . The latter clearly outperforms the direct approach and shows that learning increments is more efficient than learning a direct mapping between two updates. As shown in table 8 increasing the number of steps improves the performance (shown here for 1 and 5 update steps). However, the performance does not improve anymore after a few steps (not shown here).
+
+Table 8: Comparison of different optimizer configurations for solving the Helmholtz equation. Metrics in Relative MSE on the test set.
+
+<table><tr><td rowspan="2">N-steps</td><td colspan="2">Relative MSE</td></tr><tr><td>Direct</td><td>GD</td></tr><tr><td>1-step</td><td>1.08e-1</td><td>8.5e-2</td></tr><tr><td>5-step</td><td>9.07e-2</td><td>2.19e-2</td></tr></table>
+
+Optimization with different inner learning rates In table 9, we study the performance of our proposed method with different inner learning rates  $\eta$ . As expected, a higher learning rate leads to better performances since the optimization is taking bigger steps.
+
+Table 9: Ablation on the inner learning rate. Metrics in Relative MSE on the test set of Helmholtz equation.
+
+<table><tr><td>learning rate</td><td>Relative MSE</td></tr><tr><td>0.01</td><td>7.32e-2</td></tr><tr><td>0.1</td><td>4.93e-2</td></tr><tr><td>1</td><td>2.19e-2</td></tr></table>
+
+Quantifying the importance of input feature for the learned solver As indicated in eq. (9) the inputs of our learned solver are  $(\gamma ,f,g,\nabla \mathcal{L}_{\mathrm{PDE}}(\theta_L))$  . We performed experiments by removing either  $\gamma$  or  $\nabla \mathcal{L}_{\mathrm{PDE}}(\theta_l)$  from the input (For the Helmholtz equation, there is no forcing term  $f$  ). The BC  $g$  are kept since they are part of the PDE specification and are required to ensure the uniqueness of the solution. This experiment (table 10) illustrates that conditioning on the PDE parameters  $\gamma$  is indeed required to solve the parametric setting. Without  $\gamma$  , the solver has no hint on which instance should be used. The addition of the gradient information,  $\nabla$  , which is at the core of our method, is also crucial for improving the convergence and validates our setting.
+
+Table 10: Effect of using the gradient as input w.r.t the PDE parameters. Metrics in Relative MSE on the test set of Helmholtz equation.  
+
+<table><tr><td>Input</td><td>Relative MSE</td></tr><tr><td>γ + g</td><td>3.75e-1</td></tr><tr><td>∇ + g</td><td>1.07e-1</td></tr><tr><td>∇ + γ + g</td><td>2.19e-2</td></tr></table>
+
+We showed by a simple experiment that our model can handle nonlinear cases (see table 11). We propose to model the solution  $u$  using a non-linear combination of the basis terms  $\phi_i$ . The relation between the  $\phi_i$  is modeled using a simple NN with one hidden layer and a tanh activation function. This experiment is performed on the Poisson PDE.
+
+Table 11: Nonlinear combination of the basis. Relative MSE on the test set for our proposed method and comparison with other non linear models and optimizers.  
+
+<table><tr><td>bases</td><td>Relative MSE</td></tr><tr><td>PINNs+L-BFGS</td><td>8.83e-1</td></tr><tr><td>PPINNs</td><td>4.30e-2</td></tr><tr><td>Ours</td><td>3.44e-3</td></tr></table>
+
+Network architecture We show in table 12, an ablation on the layer type used for  $\mathcal{F}_{\varrho}$  in our experiments: MLP, Residual Network (ResNet), FNO and a modified version of a MLP (ModMLP) taken from (Wang et al., 2021a), inspired from attention mechanism. We conducted this experiment on the Helmholtz dataset.
+
+Table 12: Ablation on different layer types. Metric on the test set of the Helmholtz equation.  
+
+<table><tr><td>Layer type</td><td>Relative MSE</td></tr><tr><td>MLP</td><td>8.25e-1</td></tr><tr><td>ResNet</td><td>6.87e-1</td></tr><tr><td>ModMLP</td><td>5.55e-2</td></tr><tr><td>FNO</td><td>2.41e-2</td></tr></table>
+
+Irregular grids We show in table 13, an ablation on different types of grids: regular as in table 2, and irregular. The latter were created by sampling uniformly  $75\%$  of the points in the original grid. The sampled grids are different between each trajectory both during the training and testing phases. We conducted this experiment on the Helmholtz dataset. Finally, we show some reconstruction examples in figs. 13a and 13b.
+
+Table 13: Comparison of the performances when training our solver using regular or irregular and different grids for each PDE. Metrics on the test set.  
+
+<table><tr><td>Grid</td><td>Relative MSE</td></tr><tr><td>regular</td><td>2.41e-2</td></tr><tr><td>irregular</td><td>3.38e-1</td></tr></table>
+
+This experiment shows that our method is capable of handling irregular grids. We observe a decrease in the performances (table 13) and in the reconstruction quality figs. 13a and 13b. However, we also note that our method is still capable of reconstructing the dynamic of the PDE, where most of the considered baselines failed to capture the oscillation in the Helmholtz PDE solution.
+
+![](images/982f4074b6b67fcad709de4af7aef54b8ca92c008548f3bc1bf55fce8c8899de.jpg)  
+(a)
+
+![](images/d777b0533142171d39d89a5e0820dda9d8e74c514cc42ef52df82bdc3a4acbfb.jpg)  
+(b)  
+Figure 13: Comparison of the reconstruction of the solution when models are trained on regular (left) and irregular grid (right).
+
+Error as a function of the PDE parameters values We illustrate in fig. 14 the behavior of the reconstruction of the MSE varying the PDE parameter values. We conducted this experiment on the Helmholtz PDE and varied  $\omega$  from  $-5$  to  $55$  i.e. extrapolation of  $10\%$  beyond the parameter distribution (and kept fixed boundary conditions).
+
+![](images/7995166a6cdbe6295a21f55c47a75029657b867cb7d0203380d143b4370630bc.jpg)  
+Figure 14: MSE comparison with PDE parameters  $\omega$  of the Helmholtz dataset and extrapolation outside of the training distribution.
+
+We observe in fig. 14 that, as expected, solving Helmholtz PDE outside the training distribution fails for our model as well as for all baselines. The proposed method performs well inside the training distribution (as already observed in table 2) and behaves similarly to fully supervised methods on Out-Of-Distribution (OOD) examples. Finally, other baselines, involving physical only or hybrid training, only predict a mean solution.
+
+# E.2 STATISTICAL ANALYSIS
+
+We provide a statistical analysis on some datasets and baselines. These experiments are conducted using  $L = 5$  (instead of  $L = 2$  in table 2). For computational cost reasons, we did not make this experiment on all datasets but kept 2 datasets in 1d (table 14), 1 dataset in 1d+time, and 1 dataset in 2d (table 15) so that several configurations and data sizes are represented.
+
+This analysis shows the robustness of our proposed method w.r.t. initial seed.
+
+Table 14: Results of trained models with error bars (std errors) on 1d datasets - metrics in Relative MSE on test set. Best performances are highlighted in bold, and second best are underlined.  
+
+<table><tr><td></td><td colspan="3">1d</td></tr><tr><td></td><td>Baseline</td><td>Helmholtz</td><td>Poisson</td></tr><tr><td>Supervised</td><td>MLP + basis</td><td>5.26e-2 ± 7.56e-3</td><td>1.58e-1 ± 7.98e-3</td></tr><tr><td rowspan="2">Unsupervised</td><td>PPINNs</td><td>8.33e-1±5.61e-3</td><td>3.59e-2±2.11e-2</td></tr><tr><td>PO-DeepONet</td><td>9.84e-1±6.93e-4</td><td>1.79e-1±3.09e-2</td></tr><tr><td rowspan="3">Hybrid</td><td>PI-DeepONet</td><td>9.81e-1 ± 2.25e-3</td><td>1.25e-1 ±1.04e-2</td></tr><tr><td>PINO</td><td>9.95e-1 ± 3.30e-3</td><td>3.27e-3 ± 1.38e-3</td></tr><tr><td>Ours</td><td>2.17e-2 ± 1.12e-3</td><td>4.07e-5 ± 2.65e-5</td></tr></table>
+
+Table 15: Results of trained models with error bars (std errors) on 1d + time, and 2d datasets - metrics in Relative MSE on test set. Best performances are highlighted in bold, and second best are underlined.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Baseline</td><td>1d + time</td><td>2d</td></tr><tr><td>1dnldr</td><td>Darcy-Flow</td></tr><tr><td>Supervised</td><td>MLP + basis</td><td>2.83e-5 ± 6.83e-7</td><td>3.78e-2±2.09e-3</td></tr><tr><td rowspan="2">Unsupervised</td><td>PPINNs</td><td>4.64e-1 ± 1.92e-2</td><td>9.99e-1±2.63e-2</td></tr><tr><td>PO-DeepONet</td><td>4.18e-1 ± 1.04e-2</td><td>8.32e-1±2.51e-4</td></tr><tr><td rowspan="3">Hybrid</td><td>PI-DeepONet</td><td>7.88e-2 ± 1.96e-4</td><td>2.72e-1 ± 4.44e-3</td></tr><tr><td>PINO</td><td>8.00e-5 ± 1.00e-5</td><td>1.17e-1 ± 1.42e-2</td></tr><tr><td>Ours</td><td>2.61e-5 ± 2.53e-6</td><td>1.62e-2 ± 3.06e-4</td></tr></table>
+
+# E.3 COMPUTATIONAL COST
+
+Finally, we detail here the training and inference times of our method as well as baselines.
+
+As we can see in table 16, our model takes longer to train due to the iterative process occurring at each epoch. However, note that this is training time; inference time is similar to other methods (see table 17). We detail a justification for this additional training time compared to PINNs variants below:
+
+- Comparison with vanilla PINNs: Consider the following experiment. Suppose we train a classical PINN on a single instance of the Darcy PDE. Based on the training times shown in table 16, using this method, we performed 1,500 steps, which took 420 minutes for training (please note that the performances were less accurate than our model's performance). If we wanted to train a PINNs on each PDE of our entire test dataset for 15000 epochs (sometimes even more steps are required), this would take 4200 minutes or stated otherwise approximately 3 days. Suppose now, that one wants to solve an additional equation. This will require an average of 0.226 seconds (see table 17) with our method, while PINNs would require an entirely new training session of approximately 20 minutes for 15,000 steps. This makes our method 5,000 times faster than traditional PINNs for solving any new equation.
+
+- Comparison with PINNs parametric variants: Now let us consider two parametric variants of PINNs designed to handle multiple PDEs (PPINNs for parametric PINNs, P2INNs for the model proposed by (Cho et al., 2024)). In table 2, models was trained for 5,000 epochs only. Let us consider training it further as suggested for vanilla PINNs, until 15,000 epochs. First, this training would require approximately  $19\mathrm{h}30\mathrm{m}$ . Then, the optimization problem is still ill-conditioned, training further would probably not significantly improve the performance. We can extend this reasoning to the P2INNs baseline, for which we observed similar performance and behaviors.
+
+Table 16: Training time of the experiments shown in Table 2. of the paper on a single NVIDIA Titan RTX (25 Go) GPU. d stands for days, h for hours, m for minutes.  
+
+<table><tr><td>Dataset</td><td>Helmholtz</td><td>Poisson</td><td>1dnlrd</td><td>Darcy</td><td>Heat</td></tr><tr><td>MLP + basis</td><td>30m</td><td>20m</td><td>1h10m</td><td>2h</td><td>4h45</td></tr><tr><td>PPINNs</td><td>15m</td><td>20m</td><td>4h15m</td><td>6h30m</td><td>1d2h</td></tr><tr><td>P2INNs</td><td>2h</td><td>3h</td><td>11h</td><td>1d7h</td><td>1d8h</td></tr><tr><td>PODON</td><td>10m</td><td>10m</td><td>3h30m</td><td>1d9h</td><td>22h</td></tr><tr><td>PIDON</td><td>10m</td><td>10m</td><td>3h30m</td><td>1d10h</td><td>22h</td></tr><tr><td>PINO</td><td>15m</td><td>10m</td><td>1h10m</td><td>45m</td><td>2h40</td></tr><tr><td>Ours</td><td>30m</td><td>30m</td><td>4h30</td><td>10h15</td><td>1d 13h</td></tr></table>
+
+Table 17: Inference time, averaged (in seconds) on the test set. All experiments are conducted on a single NVIDIA RTX A6000 (48Go). We report the mean time computed on the test set to evaluate the baselines as performed in table 2 in the paper (i.e. with 10 test-time optimization steps when applicable and 20 steps on the Heat dataset). We consider as inference the solving of a PDE given its parameters and/or initial/boundary conditions.  
+
+<table><tr><td>Dataset</td><td>Helmholtz</td><td>Poisson</td><td>1dnlrd</td><td>Darcy</td><td>Heat</td></tr><tr><td>MLP + basis</td><td>1.12e-2</td><td>1.18e-2</td><td>1.25e-2</td><td>1.19e-2</td><td>1.66e-2</td></tr><tr><td>PINNs+L-BFGS</td><td>274</td><td>136</td><td>369</td><td>126</td><td>234</td></tr><tr><td>PINNs-multi-opt</td><td>15.5</td><td>25.5</td><td>16.5</td><td>105</td><td>90</td></tr><tr><td>PPINNs</td><td>3.09e-1</td><td>2.03e-1</td><td>2.91e-1</td><td>3.22e-1</td><td>5.45e-1</td></tr><tr><td>P2INNs</td><td>2.84e-1</td><td>3.09e-1</td><td>6.76e-1</td><td>1.29</td><td>1.23</td></tr><tr><td>PODON</td><td>3.27e-1</td><td>2.71e-1</td><td>4.38e-1</td><td>6.32e-1</td><td>8.85e-1</td></tr><tr><td>PIDON</td><td>3.32e-1</td><td>2.96e-1</td><td>4.43e-1</td><td>6.35e-1</td><td>8.80e-1</td></tr><tr><td>PINO</td><td>3.14e-1</td><td>1.24e-1</td><td>5.19e-1</td><td>2.21e-1</td><td>8.08e-1</td></tr><tr><td>Ours</td><td>2.58e-1</td><td>2.16e-1</td><td>2.84e-1</td><td>2.26e-1</td><td>2.90e-1</td></tr></table>
+
+# E.4 ADDITIONAL DATASETS
+
+We provide additional experiments on 2 new datasets: Non-linear Reaction-Diffusion in a more complex setting and Advections. We refer the reader to appendix C for the details about the PDE setting. These datasets were not included in the main part of the paper due to a lack of space in the results table.
+
+Table 18: Results of trained models on additional datasets - metrics in Relative MSE on the test set. Best performances are highlighted in bold, and second best are underlined.  
+
+<table><tr><td></td><td>Baseline</td><td>NLRDIC</td><td>Advections</td></tr><tr><td>Supervised</td><td>MLP + basis</td><td>9.88e-4</td><td>6.90e-2</td></tr><tr><td rowspan="2">Unsupervised</td><td>PPINNs</td><td>3.71e-1</td><td>4.50e-1</td></tr><tr><td>PO-DeepONet</td><td>4.36e-1</td><td>5.65e-1</td></tr><tr><td rowspan="3">Hybrid</td><td>PI-DeepONet</td><td>5.39e-2</td><td>4.26e-1</td></tr><tr><td>PINO</td><td>3.79e-3</td><td>6.51e-4</td></tr><tr><td>Ours</td><td>1.41e-3</td><td>5.39e-3</td></tr></table>
+
+These additional datasets show cases where baselines are performing very well on the considered PDE. For Advections, PINO reached very good performances. We believe that the Fourier layers used in the model fit well to the phenomenon. Indeed, the solution is represented using a combination of sine moving with time. This simple dynamics is easily captured using Fourier transformations. Our B-Splines basis can be sub-optimal for this dataset. For the complexified version of NLRD, NLRDIC, it is the supervised baseline that performs best. This dataset does not present high frequencies, and the MLP looks sufficiently expressive to find the coefficient in the basis. However, this baseline had difficulties in reconstructing the higher frequencies in Advections. Even if our model does not
+
+perform best on these datasets, it is ranked second. We believe that these additional results show the robustness of our method across different physical phenomena.
+
+# E.5 TRAINING BEHAVIOR
+
+We show in this section the evolution of the MSE as the training progresses (see fig. 15).
+
+![](images/fba4af27c1985215c2549eb713945ac72df16e438f39e7b3e4f6c472ab0e7dbe.jpg)  
+Figure 15: MSE during training on the training and testing sets. Example shown on the Helmholtz equation for results as presented in table 2.
+
+![](images/25a20285aea550982d37d1a0bf5afadbb58ad37189e224cd30313f528ff7ae22.jpg)  
+(a)
+
+![](images/84434f0b2faa471224b1cfc069e6996667aa5f8316b96dc6cc9b90d7d1e51c6a.jpg)  
+(b)  
+Figure 16: Reconstruction of the solution PDEs during training. Example drawn from the test set.
+
+In fig. 15, we show the MSE evolution on both the train and test sets (evaluation every 50 epochs) and on figs. 16a and 16b some reconstruction (from the test set), with respect to the training epochs. The training set (800 PDE trajectories) corresponds to PDEs used to update the network parameters, while the test set (200 trajectories) are unseen PDEs for the network. This means that the model has not been trained or optimized on these PDEs. The test set is composed of PDEs with varying PDE parameter values  $(\omega)$  and boundary conditions  $(u_0, v_0)$ . This illustrates that the generalization performance on new PDEs within the training distribution is rapidly achieved by the network.
+
+# E.6 LOSS LANDSCAPES
+
+In this section, we propose a visual representation of the optimization paths in the loss landscape. Figures 17 and 18 illustrates the behavior of the vanilla PINNs algorithm and of our "learning to optimize" method. The plots represent respectively a 2D visualization of the physical loss landscape  $\mathcal{L}_{PDE}$  (fig. 17) and of the data loss landscape  $\mathcal{L}_{\mathrm{DATA}}$  (fig. 18), around the approximate solution (in our basis) of a given Helmholtz equation. The function basis used in this experiment has a size of 32. We used the technique in (Garipov et al., 2018) to visualize the solution coordinates in the basis  $\Theta$  in the loss landscape (more details in appendix E.7.1). Superimposed on the loss background, we plot three trajectories obtained by starting from an initial random vector  $\Theta_0$ .
+
+Figure 17 visualizes the sharp and ill conditioned landscape of the physical loss  $\mathcal{L}_{PDE}$ , while Figure 18 shows the better conditioned landscape of the MSE data loss  $\mathcal{L}_{\mathrm{DATA}}$ . The figures provide intuition on how the proposed algorithm operates and improves the convergence. The optimizer  $\mathcal{F}$  modifies the direction and magnitude of the stochastic gradient descent (SGD) updates in the physical loss landscape (Figure 17, left plot). It achieves this by utilizing solution values at various sample points to adjust the gradient, steering it toward the corresponding minimum in the mean squared error (MSE) landscape (Figure 18, central plot). While the descent remains within the residual physics error space, the learned optimizer provides an improved gradient direction, enhancing convergence efficiency (Figure 17, right plot). We describe below the different figures.
+
+- Left Columns in Figures 17 and 18: these figures show the gradient path (100 steps) obtained by directly optimizing the physical loss  $\mathcal{L}_{PDE}$ , similar to the vanilla PINNs algorithm. This trajectory highlights the ill-conditioning of the optimization problem associated to  $\mathcal{L}_{PDE}$ .  
+- Center Column in Figures 17 and 18: these figures plot the trajectory of a gradient-based optimization algorithm trained with a mean squared error  $\mathcal{L}_{\mathrm{DATA}}$  data loss (100 steps), under the assumption that the solution values are known at collocation points. While these quantities are not available in our case, this visualization is included to illustrate the differences in convergence behavior between physics-based and MSE-based loss functions when both are accessible.  
+- Right Column in Figures 17 and 18: these plots illustrate the behavior of our algorithm (the solver is trained with 2 gradient steps for this example). It demonstrates the effect of our learned optimizer and the significant improvements achieved compared to a standard gradient descent algorithm on the physical loss.
+
+. We describe in appendix E.7 the construction of the figures figs. 17 and 18.
+
+![](images/eaf40229ef8a1b5db1fa3e854905d7cd7774532f749ba4f4239dacc0d14687b1.jpg)  
+Figure 17: Loss landscapes and optimization trajectories on an instance of the Helmholtz PDE. The figure represents the PDE loss  $\mathcal{L}_{\mathrm{PDE}}$  landscape and superimposed are examples of optimization paths computed using SGD on the physical loss (left), the data loss (center), and our method (right).
+
+![](images/8f3da2222ce730aef22f56d8a58e52d9be6cd666d31ccaddcaa3dff6f8e80619.jpg)  
+Figure 18: Loss landscapes and optimization trajectories on an instance of the Helmholtz PDE. The figures represent the DATA loss  $\mathcal{L}_{\mathrm{DATA}}$  and superimposed are examples of optimization paths computed using SGD on the physical loss (left), the data loss (center), and our method (right).
+
+# E.7 PROJECTION OF THE BASIS FOR VISUALIZATION
+
+Following the technique described in (Garipov et al., 2018), we plot a 2D slice of the loss function around the minimum solution. We explain below the method used for creating figs. 17 and 18.
+
+# E.7.1 CREATE A BASIS FOR PROJECTION
+
+We want to visualize the loss landscapes around a parameter solution  $\Theta^{\star}$  of a given PDE. Due to the high dimensionality of the parameter space, we use the method proposed in (Garipov et al., 2018), that involves creating a 2d slice of the loss landscape for visualization. This methods projects the multidimensional landscape onto a 2D basis  $(u,v)$  so that any solution vector  $\Theta$  could be written as  $\Theta = \alpha u + \beta v$ , with  $\alpha, \beta \in \mathbb{R}$  and  $u,v \in \mathbb{R}^N$ , 2 orthonormal vectors. Let  $\Omega_{u,v} := \mathrm{Vect}\{u,v\}$  denote this projection space. For the visualization, the basis must include the solution vectors found by our algorithm. For that:
+
+1. We start by running our optimization algorithm from a random vector parameters  $\Theta_0$  to find  $\Theta_L \in \mathbb{R}^N$ , our approximate PDE solution. We set  $w_1 = \Theta_L$  and set a second vector  $w_2 = \Theta_0$ , chosen here as the initial vector of our optimization process.  
+2. We generate a third vector  $w_{3}\in \mathbb{R}^{N}$  randomly.  
+3. We then set:  $u = w_{2} - w_{1}$  and  $v = (w_{3} - w_{1}) - \frac{<w_{3} - w_{1},w_{2} - w_{1}>}{||w_{2} - w_{1}||^{2}}(w_{2} - w_{1})$  
+4. The resulting normalized vectors  $\hat{u} = \frac{u}{||u||}$  and  $\hat{v} = \frac{v}{||v||}$  form an orthogonal basis containing  $w_{1}, w_{2}$  and  $w_{3}$ .  
+5. Using the solution parameters  $\Theta_L$  as the origin for our newly created basis, we can compute a 2D slice of the loss landscape where each solution  $\Theta$  could be represented in the  $(u,v)$  basis with its projection coordinates  $(\alpha ,\beta)$  as  $\Theta_{projection} = w_1 + \alpha \hat{u} +\beta \hat{v}$ .
+
+# E.7.2 PLOT TRAJECTORIES
+
+Finally, we run a baseline optimization procedures and get trajectories  $\{\Theta_{i}\}_{i = 0}^{L}$ , with L being the number of optimization steps, e.g. a gradient descent, on the physical loss, the MSE loss or our proposed method. We project the  $\Theta_{i}$ 's in the created basis (see appendix E.7.1) and plot this projection of the optimization trajectory on the loss landscape  $\Omega_{u,v}$  as visualized on figs. 17 and 18.
+
+# E.7.3 ILLUSTRATION OF THE ILL-CONDITIONING OF THE PDE LOSS
+
+As an illustration of the ill-conditioning of the PDE loss, we replicate fig. 17 using a basis that further emphasizes this aspect. To build this basis, we use the procedure described in appendix E.7.1 by using 2 eigenvectors of the hessian of the PDE loss. First, we compute  $\mathrm{Hess}(\mathcal{L}_{\mathrm{PDE}})$  and its
+
+eigenvectors decomposition. We select the vectors associated to the highest and lowest eigenvalues, and respectively set them to  $w_{2}$  and  $w_{3}$ . The resulting landscape visualization is shown in fig. 19.
+
+![](images/77c421314b5dce7fdce0dcefb53bb8956466060c8731dbb89fbf73b30f391573.jpg)  
+Optimization path in the  $\mathcal{L}_{PDE}$  loss landscape.  
+Figure 19: Loss landscapes and optimization trajectories on an instance of the Helmholtz PDE. These optimization trajectories are computed using SGD on the physical loss (left), the data loss (center), and our method (right). The background represents the PDE loss  $\mathcal{L}_{\mathrm{PDE}}$ .
+
+Figure 19, clearly illustrates the characteristic shape of the ill-conditioned function  $\mathcal{L}_{\mathrm{PDE}}$ . The two directions, extracted from the highest and lowest eigenvalues, are clearly visible on this PDE loss landscape. This highlights the difficulty of this optimization problem for standard descent methods.
+
+# F QUALITATIVE RESULTS
+
+This section is dedicated to visualization of the results of our model, baselines and optimizers, presented in section 4. For each PDE considered, we chose 2 samples in the test sets and compute the solutions with our model and the different baselines. We provide visualization samples with  $L = 5$  i.e. results proposed in tables 14 and 15 to detail more precisely the evolution of the solution at several steps of optimization. 3 datasets are shown with  $L = 2$ : Heat for computational reasons (training with  $L = 5$  is much more expensive when the dimension of the problem increases) and the 2 additional datasets trained only with  $L = 2$  (Advection and NLRDIC). Then, we show the evolution of the reconstruction of the solution with our method i.e. we plot the solution at each step of the optimization (figs. 20a, 20b, 23a, 23b, 26a, 26b, 29a, 29b, 32a, 32b, 35a, 35b, 38a and 38b) and we compare the final prediction with baselines' (figs. 21a, 21b, 24a, 24b, 27a, 27b, 30a, 30b, 33a, 33b, 36a, 36b, 39a and 39b). Finally, we chose 20 (6 for Heat) PDEs and we reproduce fig. 2 for every dataset. More precisely, we optimize one PINN per PDE using Adam, we fit our basis using several optimizers (GD, ADAM, L-BFGS and our learned optimization process) and we fine-tune the learned PINO for 10,000 steps and visualize the evolution of the MSE (averaged at each step on the selected PDE). These figures show the relevance of learning the optimizer when using physics-informed losses (figs. 22, 25, 28, 31, 34, 37 and 40).
+
+# F.1 HELMHOLTZ
+
+![](images/648aef56260534778d247771aabb217f5dcde991bd2482cd5fbdaf96b2bd6870.jpg)  
+(a)
+
+![](images/3f2f6db486ef35a8e43277135e793f0c5c34885d74751b3a28b2bfd412993d23.jpg)  
+(b)  
+Figure 20: Reconstruction of the solution using our optimizer on the Helmholtz dataset.
+
+![](images/ede0c87f809b00968a29d468a01cf948362ba8541d6af4edb8ae8004d57f7e7f.jpg)  
+(a)  
+Figure 21: Visual comparison of the solutions for the Helmholtz equation.
+
+![](images/6938e712b3867857b7ab1aa1161d66f2f0eadde23a37bd1a08d3a560db3ef6f7.jpg)  
+(b)
+
+![](images/27777d6145987d495becc96497dbdb6b0bcdbae928129e84f50109b961c13df8.jpg)  
+Figure 22: Test-time optimization based on the physical residual loss  $\mathcal{L}_{\mathrm{PDE}}$  on Helmholtz. Note that, even though hardly visible on this figure, the optimization is running very slowly and the PINN MSE (orange) decreases for the last steps. This dataset will probably need even more steps before convergence.
+
+# F.2 POISSON
+
+![](images/4d0da8d8ee0112f122b551a20d10401f099a55518a9b5c227f56ca328a244aec.jpg)  
+(a)
+
+![](images/79545dc875239fd38be4a46dd59d5704598d4d89de43840f6373da368b9ab0b0.jpg)  
+(b)  
+Figure 23: Reconstruction of the solution using our optimizer on the Poisson dataset.
+
+![](images/6dedb2a1e5091d8bf43ef61331d82f2072d84b83147b93a2e531717644cabc03.jpg)  
+(a)  
+Figure 24: Visual comparison of the solutions for the Poisson equation.
+
+![](images/3324a3653a26f8a827b024804a0a611338dff57994a2f6c17b0a07759701425e.jpg)  
+(b)
+
+![](images/db8f4e3c9d3000c83bb5dbc9022e939446d107d52d487f4c346b898991712e39.jpg)  
+Figure 25: Test-time optimization based on the physical residual loss  $\mathcal{L}_{\mathrm{PDE}}$  on Poisson.
+
+# F.3 REACTION-DIFFUSION
+
+Evolution of the reconstruction of the solution with optimization steps.
+
+![](images/17f938bf3839a78b00a2adace930b17abcef848a5295731a9f941bc57a10f45a.jpg)
+
+![](images/b9ef79981a075bc9dbcf8c8457a469ad78d74d4f22e7c8853a815e4d3336134e.jpg)
+
+![](images/43f3502523a3380c0c9bc8db466ed4dca5233b2de8dfb9239ed12bca20bc5505.jpg)
+
+![](images/679531529dfa62ecf787279c89da79d7c566780d2de6ee9db3040d91e9f506d9.jpg)  
+(a)
+
+![](images/d3049e61d9d6ce38e601278b06d72de1ce123324890eea12f2473295027bdb4e.jpg)
+
+![](images/65bdf2ac1bf5b85c884d9d8b3ee3993f049f249ba272b87e192248f12338d5d8.jpg)
+
+![](images/8a9063d8f4d853ea7b4eed1aabcd0f56279d6ab86cc4ea7750d3a0b03858de7a.jpg)
+
+Evolution of the reconstruction of the solution with optimization steps.
+
+![](images/385dc35c6d0d7111c423c958ad7112b067b00839f675af2ef09cb4be986ba897.jpg)  
+Figure 26: Reconstruction of the solution using our optimizer on the Reaction-Diffusion dataset.
+
+![](images/d11da6d22036190d1f93084c17713c1716dfb6abcea2a130cf578546e868ff9e.jpg)
+
+![](images/a03007e58c8ae08a9c34c37825378aaf145527dbe8437ef1bed2232741b5e769.jpg)
+
+![](images/c63afa1396f25334d5cf0b9963df697137ffe52355463a89139cef3dd2756a72.jpg)  
+(b)
+
+![](images/ceb46e2e07f1f601bb6c422e7f40ef0f9a0c59ce598ba8f215f989f0a45fc618.jpg)
+
+![](images/4722795d2b3d336dc261d6cb4063d3b343461cd810e8b947b613ef774b4645d6.jpg)
+
+![](images/c39692002df0d72ad3956221896879229b77b43094a31dd37224f305a89e1ba2.jpg)
+
+![](images/e8d6c76927a812bce92338f9668597a9518413ed534a66bc723e9d9c71419dc6.jpg)  
+Figure 27: Visual comparison of the solutions for the Reaction-Diffusion equation.
+
+![](images/bb0a494f388ddc0b0572a6bf9667fb8b10b40d898633c328971827d81a659c47.jpg)  
+Figure 28: Test-time optimization based on the physical residual loss  $\mathcal{L}_{\mathrm{PDE}}$  on NLRD.
+
+# F.4 DARCY
+
+Evolution of the reconstruction of the solution with optimization steps.
+
+![](images/1636bf7b9583e097bf6c1f480a5e39935b9cc7bc8743cef4ed3027aae7fb4abd.jpg)  
+(a)
+
+Evolution of the reconstruction of the solution with optimization steps.
+
+![](images/0499d9da3f45aa933322c3a15815352c5491ad419496025b61305afc1b11d3ff.jpg)  
+(b)  
+Figure 29: Reconstruction of the solution using our optimizer on the Darcy dataset.
+
+Solution for Darcy flow
+
+![](images/6151dfa817aaa9247bf99df6adaea7bc9085846d9fed9e4972858471640d8a70.jpg)  
+(a)
+
+Solution for Darcy flow
+
+![](images/4ac344e2bd7863e561720aba07a50828ca9898a8e024f02eb4d5cecd9564d294.jpg)  
+(b)  
+Figure 30: Visual comparison of the solutions for the Darcy equation.
+
+![](images/511d077e4f21c9d86bb867773c968632f7e6496d46175ec2f6559dbcdb0f5a20.jpg)  
+Figure 31: Test-time optimization based on the physical residual loss  $\mathcal{L}_{\mathrm{PDE}}$  on Darcy.
+
+# F.5 HEAT
+
+![](images/fea5e7220ebf76684f317bb479d4cd69e3a43c10678b567d1adc7ff3c00e8ff1.jpg)  
+Evolution of the reconstruction of the solution with optimization steps.
+
+![](images/e7960727b52ebcd94cd147ac29f1ebe3d4671bc42d1fb8fc20d4a0b264e2bed1.jpg)  
+(a)  
+Evolution of the reconstruction of the solution with optimization steps.  
+(b)  
+Figure 32: Reconstruction of the solution using our optimizer on the Heat dataset.
+
+![](images/3251e6fb88cf54e39535bb5962a5f684482a1a4fe3b85c180d4ba210ba9a8251.jpg)
+
+![](images/09af595b37a2f70dd715ab0b80d42dad258ccb932693a7f2ae44b6a436aced3b.jpg)
+
+![](images/b54d489b45abed6e3dbf9fefde0571d80311ce1717965e27d47771c50a04ef87.jpg)
+
+![](images/ef2e29cfd6cd01758ac8ed3ebfe1a0a242eb9796d2f9668e177acf32daa4048c.jpg)  
+(a)
+
+![](images/08f9500321dfd619dbea68087ef036f5ab400e662b801a5b233a7e4c07266fb5.jpg)
+
+![](images/10bdccbb8f86c57e62be0669becd320fe4e070778bdfc4046e9280691db4299a.jpg)
+
+![](images/c7cd1faea07ba5f5f34e94170a479c9a547c933729b7a3cc8f4a1e487d4b6f15.jpg)
+
+![](images/234e2c9fc5a249172119e9190b2760291075888fc4174d5a95ef1f33372071c7.jpg)  
+Figure 33: Visual comparison of the solutions for the Heat equation.
+
+![](images/a5875cc070e044ce0da595b004d8c832177b36442691b93c31f9ddcce9e0ebad.jpg)
+
+![](images/371fd62876643163a0f86c5311983ff9ac955241807445881846910c2ba4cc25.jpg)
+
+![](images/186652e1fcf587ac27cc0cce9aefb99892faf906e410183466cba19817bfb11d.jpg)  
+(b)
+
+![](images/7c16c3a6e1c59b29388d5bffd653ff99ef9e343df7f7ff82b721c8befae0d6be.jpg)
+
+![](images/f816e5bd402794d000ff3d924316b897c6ddb8fe231b9f294b561761e574f6c7.jpg)
+
+![](images/f67afc6e035e5fe7534f42b60ee90ada32c0d57d9b33bf87ad7b624d349e8dab.jpg)
+
+![](images/f4715f41b529dfc486f41dd33ec65b66cf78651ea0e8c510c3389b84d9893bfc.jpg)  
+Figure 34: Test-time optimization based on the physical residual loss  $\mathcal{L}_{\mathrm{PDE}}$  on Heat. For computational reasons, this experiment has been conducted on 1,000 steps only.
+
+# F.6 ADVECTION
+
+Evolution of the reconstruction of the solution with optimization steps.
+
+![](images/e1bf877838ebc4a7c2b03fa0fa5bee611d5dceb88526b946ef015ee621969a83.jpg)  
+(a)
+
+Evolution of the reconstruction of the solution with optimization steps.
+
+![](images/105eda4b877d77261ff815525e6fd00156aecfe0de4b8ea5526d6dffc41be79c.jpg)  
+(b)  
+Figure 35: Reconstruction of the solution using our optimizer on the Advection dataset.
+
+![](images/bab420918b2b15d65e0d0609fa0a0826d6e76cb2c1d430bbb46f00aad14ab22c.jpg)
+
+![](images/3cc65b996e122db6595394bc0c34aabebbdd253875c9b19182217aced7b0a42f.jpg)  
+(a)  
+(b)
+
+![](images/f920a2e16e9469614d19c8fcc060f392828f985f3ef844b958db27e7243d36ed.jpg)  
+Figure 36: Visual comparison of the solutions for the Advection equation.  
+Figure 37: Test-time optimization based on the physical residual loss  $\mathcal{L}_{\mathrm{PDE}}$  on Advection.
+
+# F.7 NON-LINEAR REACTION-DIFFUSION WITH INITIAL CONDITIONS
+
+Evolution of the reconstruction of the solution with optimization steps.
+
+![](images/d6bc7653eb8b9509ad84fe0747f87e4e0131331e4d3bf814846b22e0c83cdbbb.jpg)
+
+![](images/78bae9c73d69d53d99aa252079df84fecf1614d29ca12c00519a51096c81f4fa.jpg)  
+(a)
+
+![](images/6505f8d0a938431b5dd0261d026852d87d6ec3bd75d4ccffef0fe56a9a5b561d.jpg)
+
+![](images/73f099d9e1580db54a23509ab8299eeacd51034dac5c64236c2ced808e735033.jpg)
+
+Evolution of the reconstruction of the solution with optimization steps.
+
+![](images/c9d6b5bd236c29ad4d92e3919023536cd4d7536939933fce22c5822a20d48961.jpg)  
+Figure 38: Reconstruction of the solution using our optimizer on the NLRDIC dataset.
+
+![](images/b575950370e6c2f7650600ee99ce279ff9dee15f878775f429df00fd110b7c56.jpg)  
+(b)
+
+![](images/c7d93e87b4e0ac0c078090b343ddb8bdf413d7097624a3a52fdecc10718e6cb5.jpg)
+
+![](images/9f911a798c4e8657d2731f8ec408ffe290c53f9c9aaa541e814a4d01a57ff635.jpg)
+
+![](images/ed259b063c152d3e03bae2e41a53fac62b8efbea0f867a3b634d5284ff98baf2.jpg)  
+Figure 39: Visual comparison of the solutions for the NLRD with varying IC equation.
+
+![](images/38314105654ccc223cf1843e914db5cae78348d4de7dda23135e07258a4b44be.jpg)  
+Figure 40: Test-time optimization based on the physical residual loss  $\mathcal{L}_{\mathrm{PDE}}$  on NLRD with varying IC.
\ No newline at end of file
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+# LEARNING FROM IMPERFECT HUMAN FEEDBACK: A TALE FROM CORRUPTION-ROBUST DUELING
+
+# Yuwei Cheng
+
+Department of Statistics
+
+University of Chicago
+
+Chicago, IL 60637, USA
+
+yuweicheng@uchicago.edu
+
+# Xuefeng Liu*
+
+Department of Computer Science
+
+University of Chicago
+
+Chicago, IL 60637, USA
+
+xuefeng@uchicago.edu
+
+# Fan Yao*
+
+Department of Computer Science
+
+University of Virginia
+
+Charlottesville, VA 22904, USA
+
+fy4bc@virginia.edu
+
+# Haifeng Xu
+
+Department of Computer Science
+
+University of Chicago
+
+Chicago, IL 60637, USA
+
+haifengxu@uchicago.edu
+
+# ABSTRACT
+
+This paper studies Learning from Imperfect Human Feedback (LIHF), addressing the potential irrationality or imperfect perception when learning from comparative human feedback. Building on evidences that human's imperfection decays over time (i.e., humans learn to improve), we cast this problem as a concave-utility continuous-action dueling bandit but under a restricted form of corruption: i.e., the corruption scale is decaying over time as  $t^{\rho - 1}$  for some "imperfection rate"  $\rho \in [0, 1]$ . With  $T$  as the total number of iterations, we establish a regret lower bound of  $\Omega(\max\{\sqrt{T}, T^{\rho}\})$  for LIHF, even when  $\rho$  is known. For the same setting, we develop the Robustified Stochastic Mirror Descent for Imperfect Dueling (RoSMID) algorithm, which achieves nearly optimal regret  $\tilde{O}(\max\{\sqrt{T}, T^{\rho}\})$ . Core to our analysis is a novel framework for analyzing gradient-based algorithms for dueling bandit under corruption, and we demonstrate its general applicability by showing how this framework can be easily applied to obtain corruption-robust guarantees for other popular gradient-based dueling bandit algorithms. Our theoretical results are validated by extensive experiments.
+
+# 1 INTRODUCTION
+
+Many real-world problems, such as personalized recommendation (Yue & Joachims, 2009; Immorlica et al., 2020; Yao et al., 2022) and fine-tuning generative models (Bai et al., 2022; Casper et al., 2023; Han et al., 2024), require learning from human feedback. Expressing preferences as numerical values or concrete functions is generally challenging for humans. Therefore, an approach that has achieved significant real-world success is to learn humans' preferences by eliciting their comparative feedback between two options (Bai et al., 2022). A natural theoretical framework capturing such learning task is the seminal dueling bandits framework introduced by Yue & Joachims (2009). The dueling bandit problem features a sequential online learning problem, during which a pair of actions are selected at each round and only their comparison result is revealed to the learner. The comparison outcome between two actions is modeled using a utility function of actions, alongside a link function that determines the probability of each action winning based on their utility difference. This modeling approach is termed as utility-based dueling bandit and achieves great success in generating summaries closely aligned with human preferences (Stiennon et al., 2020). However, human feedback is often imperfect, a crucial factor that has been largely overlooked in previous studies of dueling bandit. As we are all aware of, humans are not always rational (Posner, 1997) neither perfectly know our preferences (Pu et al., 2012). A powerful framework that can capture this problem is robust bandit learning under adversarial corruption (Bogunovic et al., 2020; Saha &
+
+Gaillard, 2022; Di et al., 2024). However, assuming fully adversarial feedback from a regular human seems over-pessimistic at the first glance, hence might make our algorithm development overly conservative. Indeed, ample behavioral studies show that humans often refine their preferences through interactions with the system, navigating a complex tradeoff between exploration and exploitation (Cohen et al., 2007; Wilson et al., 2014). Furthermore, their response errors may exhibit systematic patterns that depend on their past consumption history, rather than merely random noise. Though the sources of imperfections vary a lot in human feedback, one common finding in these studies is that the feedback's imperfection tends to decrease over time, as humans interact more with the system (Cohen et al., 2007; Wilson et al., 2014; Immorlica et al., 2020; Yao et al., 2022). This premise is the core motivation of this work, which studiesueling bandit learning under corruption that can be arbitrary but have decaying scales.
+
+Our Model and Contributions. We cast Learning from Imperfect Human Feedback (LIHF) as a concave-utility continuous-action dueling bandit learning problem under adversarial utility corruption yet with decaying scale, upper bounded by  $\mathcal{O}(t^{\rho -1})$  at round  $t$ , for some  $\rho \in [0,1]$  (hence the total amount of attack within time  $T$  is  $\mathcal{O}(T^{\rho})$ ). A fundamental research question that motivates our study is whether LIHF is fundamentally easier than learning under arbitrarily corrupted feedback.
+
+Our first main technical result hints on a potentially negative answer to the above question. We prove a strong regret lower bound of  $\Omega(d\max\{\sqrt{T}, T^{\rho}\})$  under LIHF, where  $d$  is the action's dimension. This bound holds even when the decaying rate of imperfection index  $\rho$  is known. This lower bound has the same order as recent lower bounds ofueling bandits under (the harder) arbitrary corruption (Agarwal et al., 2021; Di et al., 2024), thus hinting that learning from imperfect human feedback may be no easier than learning from arbitrary adversarial corruption. Notably, however, we use the phrase "hint", while not an affirmative claim, because our setting with continuous action space and concave utility function is different from the  $K$ -armed bandits setting of Agarwal et al. (2021) and linear utility function case of Di et al. (2024). Therefore, neither their lower bounds nor their proof techniques are directly applicable to our setting with corruption to strictly concave utilities; see more discussions and comparisons below.
+
+Our second main result is an efficient algorithm with regret upper bound  $\tilde{\mathcal{O}}(d\max\{\sqrt{T}, T^{\rho}\})$  that matches the above lower bound, up to logarithmic terms, in the same setting (i.e., decaying corruption level  $t^{\rho-1}$  with known  $\rho$ ). Unlike previous corruption-robust dueling bandit algorithms based on robust  $K$ -armed bandits (Agarwal et al., 2021) or robustified LinUCB (Di et al., 2024), our algorithm is a gradient-based algorithm that is built upon the recent Noisy Comparison-based Stochastic Mirror Descent (NC-SMD) of Kumagai (2017). This is natural for our setup with continuous actions and strictly concave utility. While our algorithm itself can be viewed as a natural generalization of Kumagai (2017), its analysis is highly non-trivial and much more complex, due to the necessity of accounting for corruption.
+
+Our main technical contribution is a new framework for analyzing the regret of gradient-based algorithms under arbitrary corruption. This framework introduces a novel regret decomposition lemma for dueling bandits, which upper bounds total regret by combining decision regret and feedback error due to corruption, based on quantifying bias in gradient estimation. It yields a tight analysis of our new algorithm and implies regret bounds for popular existing algorithms in (even fully) adversarial corruption settings, such as the dueling bandit gradient descent (DBGD, Yue & Joachims (2009)).
+
+These upper bounds are looser than our lower bound for the LIHF setting. It is an intriguing open question to understand whether these upper bounds under arbitrary corruption are tight, or the lower bound under arbitrary corruption may be stronger than our lower bound under LIHF. Finally, we conduct simulations which validate our theory.
+
+Comparisons with Related Works. Since multiple recent works (Agarwal et al., 2021; Saha & Gaillard, 2022; Di et al., 2024) study dueling bandits under arbitrarily adversarial corruption, it is worthwhile to discuss the key difference between these works and ours. The first key difference is the setting, which leads to fundamentally different algorithms hence analysis. Specifically, our setting with continuous action space and a completely unknown concave utility function is motivated by addressing problems from modern recommender systems where the action (i.e., contents) space is extremely large, not enumerable, thus often embedded as high-dimensional continuous vectors (Chen et al., 2019; 2021). This setting and our techniques are crucially different from earlier works. Agarwal et al. (2021); Saha & Gaillard (2022) consider the  $K$ -arm dueling bandit setting and utilizes
+
+the reduction of Ailon et al. (2014) from dueling bandits to multi- $K$ -armed bandits (MAB); (Di et al., 2024) considers linear utility setting where the unknowns is a parameter vector  $\theta$ , and utilizes techniques from linear contextual bandits based on the optimism principle. The methods all rely on parameter estimation which are not applicable to our setting with arbitrary concave utilities. Moreover, they often need to enumerate all actions to identify the best one under optimism, which are not computationally efficient for our setting with continuous action space. This is why our algorithm is gradient-based and falls within the family of stochastic mirror descent. To the best of our knowledge, this is the first time a corruption-robust algorithm is developed for dueling bandits with continuous actions and strictly concave utilities, which is also why it is necessary for us to develop a new analysis framework. Additionally, we also note that dueling bandits with strictly concave utility and linear utility are generally not comparable, both in terms of their problem difficulties and methodologies. A strictly more general situation is the concave utility case (no need to be strongly concave), however the best upper bound so far for general concave utility is  $\mathcal{O}(T^{3/4})$  (Flaxman et al., 2005; Yue & Joachims, 2009), which are considerably worse than the  $\mathcal{O}(T^{1/2})$  for linear and for strongly concave utility.
+
+The second main difference between our work and previous is the more restricted class of adversarial environments. Our model is motivated by imperfect human feedback whereas previous studies are motivated by arbitrarily adversarial adversaries. Finally, a more subtle difference is that the corruption in our model is on utility with motivations from learning from imperfect human feedback, whereas corruption in all previous settings directly flip the comparison outcome. These two differences lead to a very different proof techniques of our lower bound. Our proof is more involved because the adversary has limited capability to alter the comparison outcomes. Specifically, the corruption are decaying; moreover, a constant amount of utility corruption only slightly shifts the outcome probability whereas in previous settings each corruption completely changes the outcome.
+
+# 2 THE PROBLEM OF LEARNING FROM IMPERFECT HUMAN FEEDBACK
+
+Notation. For a positive integer  $T$ , we use  $[T]$  to denote  $\{1, 2, \dots, T\}$ . We use standard asymptotic notations including  $\mathcal{O}(\cdot), \Omega(\cdot), \Theta(\cdot)$ . We use  $\tilde{\mathcal{O}}(\cdot), \tilde{\Omega}(\cdot), \tilde{\Theta}(\cdot)$  to hide logarithmic factors. We use  $\| \cdot \|_2$  to define  $L_2$  norm,  $\| \cdot \|_\infty$  to define infinity norm, and  $\lambda_{\max}(\cdot)$  to denote maximum eigenvalue.
+
+Motivated by alignment of machine learning models with human preferences (Bai et al., 2022), we study learning from comparative human feedback within the seminal dueling bandit framework (Yue & Joachims, 2009) but account for "imperfections" in human feedback, as described below.
+
+Basics of Continuous Dueling Bandits. We consider theueling bandit framework with continuous action set  $\mathcal{A}\subset \mathbb{R}^d$  (Yue & Joachims, 2009). At each round  $t\in [T]$ , the learner chooses two actions  $a_{t},a_{t}^{\prime}$  to present to some human agent, henceforth denoted as the "user". The user receives utility  $\mu (a_t),\mu (a_t^{\prime})$  from the two actions respectively, and will pick one of these actions following a link function  $\sigma (\cdot)$ . In the absence of corruption, the user selects action  $a_{t}$  with probability  $\sigma (\mu (a_t) - \mu (a_t'))$  and chooses action  $a_{t}^{\prime}$  otherwise. Since the comparison outcome has less errors compared to the exact utility value while still carries useful information about the underlying utility function  $\mu$ , this form of human feedback has been widely used for learning human preferences (Ailon et al., 2014; Maystre & Grossglauser, 2017; Bengs et al., 2021; Bai et al., 2022). Following standard assumptions in this field (Yue & Joachims, 2009; Kumagai, 2017), we also assume
+
+1. the action set  $\mathcal{A}$  is a convex compact set that contains the origin, has non-empty interior, and is contained in a  $d$ -dimensional ball of radius  $R$ ;  
+2. the utility function  $\mu : \mathcal{A} \to \mathbb{R}$  is strongly concave and twice-differentiable. The following constants are useful for our algorithm analysis:  $\mu$  is  $L^{\mu}$ -Lipschitz,  $\alpha$ -strongly concave, $^1$ $\kappa$ -smooth, bounded,  $R^{\mu} \coloneqq \sup_{a,a' \in \mathcal{A}} \mu(a) - \mu(a')$ , and  $a^* \coloneqq \arg \max_{a \in \mathcal{A}} \mu(a)$  is the unique optimal within  $\mathcal{A}$ ;  
+3. the link function  $\sigma : \mathbb{R} \to [0,1]$  is smooth, rotation-symmetric (i.e.  $\sigma(x) = 1 - \sigma(-x)$ ) and concave for  $\mathbb{R}^+$ . Since  $R^\mu$  is finite, there exists positive  $l_1^\sigma$ ,  $L_1^\sigma$  and  $R_2^\sigma$  such that  $l_1^\sigma \leq \sigma' \leq L_1^\sigma$  on  $[-R^\mu, R^\mu]$ , and the second derivative is bounded above by  $R_2^\sigma$  and  $L_2^\sigma$ . Lipschitz on  $[-R^\mu, R^\mu]$ . For a more detailed discussion on the generality of the modeling assumptions, please refer to Appendix A.
+
+A broad range of natural link functions satisfy our assumptions, including the standard logistic distribution function, the cumulative standard Gaussian distribution function, and the linear function. This distinguishes our work from some prior studies, working with concrete link function formats such as logistic function (Maystre & Grossglauer, 2017; Saha, 2021; Xu et al., 2024) or the linear function (Ailon et al., 2014; Chen & Frazier, 2017; Zimmert & Seldin, 2018; Lin & Lu, 2018).
+
+Modeling Imperfect Human Feedback as a Restricted Form of Adversarial Corruption. To incorporate imperfect human feedback, we study a generalization of the dueling bandit framework above, by introducing a corruption term,  $c(a, a')$ , to the utility difference. Formally, let  $a \succ a'$  represent the event where the user chooses  $a$  over  $a'$ . The probability of this event in the presence of corruption  $c(a, a')$  is denoted by  $\hat{\mathbb{P}}(a \succ a')$ , expressed as  $\hat{\mathbb{P}}(a \succ a') := \sigma(\mu(a) - \mu(a') + c(a, a'))$ . We denote the user's preferential feedback under corruption as  $\hat{\mathcal{F}}(a, a')$ , referred to as imperfect dueling feedback. Mathematically,  $\hat{\mathcal{F}}(a, a')$  follows a binomial distribution with mean  $\hat{\mathbb{P}}(a \succ a')$ , as expressed by the following equation.
+
+$$
+\hat {\mathcal {F}} (a, a ^ {\prime}) := \left\{ \begin{array}{l} 1 \text {w . p .} \hat {\mathbb {P}} (a \succ a ^ {\prime}) \\ 0 \text {w . p .} 1 - \hat {\mathbb {P}} (a \succ a ^ {\prime}) \end{array} \right.
+$$
+
+The "hat" notation in  $\hat{\mathcal{F}}$  and  $\hat{\mathbb{P}}$  is to emphasize the presence of corruption. When there is no corruption, we use  $\mathbb{P}(a\succ a^{\prime})\coloneqq \sigma (\mu (a) - \mu (a^{\prime}))$  to represent the probability of the event  $a\succ a^{\prime}$ , and  $\mathcal{F}(a,a^{\prime})$  to denote the correspondingueling feedback.
+
+We remark that the term  $c(a, a')$  above is often called "strong corruption" in the literature of adversarial corruption because it can be introduced after observing actions  $a$  and  $a'$  and the corruption function,  $c(\cdot)$ , can depend on  $\mu$  and past actions (Bogunovic et al., 2020; He et al., 2022; Di et al., 2024). The only difference between our model and the above works on adversarial corruption is a natural restriction to the scale of the corrupted term  $c(a, a')$ . Our motivation is that human feedback, while imperfect, should not be arbitrarily adversarial and tends to improve as human agent interact with the learner. This motivates our following definition of imperfect human feedback.
+
+Definition 2.1 ( $\rho$ -Imperfect Human Feedback). The human feedback is said to be  $\rho$ -imperfect for some  $\rho \in [0,1]$  if there exists a constant  $C_{\kappa}$  such that corruption  $c_{t}(a_{t},a_{t}^{\prime})$  satisfies  $|c_{t}(a_{t},a_{t}^{\prime})|\leq C_{\kappa}t^{\rho -1},\forall t\in [T]^{2}$
+
+The definition shows that  $\rho$ -imperfect human feedback is mathematically equivalent to a restricted form of strong adversarial corruption to user utility functions. The total corruption is bounded by  $\sum_{t=1}^{T}|c_t(a_t,a_t')| \leq C_\kappa T^\rho$ , which is a key parameter influencing the intrinsic difficulty of the learning problem. We thus cast the Learning from Imperfect Human Feedback (LIHF) problem asueling bandits under strong corruption but with decaying scales. For clarity, we use "arbitrary adversarial corruption" as corruption without decaying constraints and "  $\rho$ -Imperfect Human Feedback" as corruption with decaying constraints, as described in Definition 2.1. Through this modeling, we aim to: first create a more optimistic framework for LIHF compared to arbitrary adversarial corruption, and second, explore LIHF's intrinsic difficulty to achieve more efficient learning guarantees than those possible under arbitrary adversarial corruption. We direct readers to Appendix B for a more comprehensive discussion on challenges in developing robust algorithms for continuousueling bandits with generally concave utility under arbitrary adversarial corruption.
+
+Several points are worth noting about Definition 2.1. While we are not the first to model imperfect human feedback, we are the first to apply it toueling bandits with strictly concave utilities. Recent works, motivated by recommendation systems, have explored learning user preferences from imperfect feedback (Immorlica et al., 2020; Yao et al., 2022; Wang et al., 2023). These studies assume that users do not know their true expected reward  $\theta_{i}$  for each choice  $i$  (i.e., arm) and instead behave based on an estimated reward  $\hat{\theta}_{t,i}$  at time  $t$ . The utility difference  $|\theta_{i} - \hat{\theta}_{t,i}| = c_{t}$  is modeled as a decreasing function of time,  $t^{\rho -1}$ , indicating that users improve their preference estimates over time. By viewing the human utility  $\mu (a)$  as the average reward, our modeling of  $\rho$ -Imperfect Human Feedback shares the same spirit; it captures imperfect yet gradually improving human feedback (e.g., due to inaccuracies in utility perception or initial ignorance of true preferences). As one of our motivations, when generative models like ChatGPT learn to create personalized content, the user
+
+could undergo a process of preference refinement during interactions with the model. Additionally, our model differs from recent corruption-robust dueling bandit studies (Komiyama et al., 2015; Saha & Gaillard, 2022; Di et al., 2024), which corrupt realized outcomes (e.g., flipping  $a^\prime \succ a$  to  $a\succ a^{\prime}$ ) and are motivated by adversaries like malicious users or fraudulent clicks (Deshpande & Montanari, 2013; Lykouris et al., 2018). In contrast, our model focuses on utility corruption caused by imperfect perception of true utility, not outcome manipulation.
+
+Learning goal: regret minimization by Learning from  $\rho$ -Imperfect Human Feedback ( $\rho$ -LIHF). The goal of the learner is to optimize her sequential actions to minimize the following dueling regret in the  $\rho$ -LIHF problem:
+
+$$
+\operatorname {R e g} _ {T} := \mathbb {E} \left(\sum_ {t = 1} ^ {T} \sigma \left(\mu \left(a ^ {*}\right) - \mu \left(a _ {t}\right)\right) + \sigma \left(\mu \left(a ^ {*}\right) - \mu \left(a _ {t} ^ {\prime}\right)\right) - 1\right). \tag {1}
+$$
+
+This regret measure has been extensively studied in prior literature (Yue & Joachims, 2009; Komiyama et al., 2015; Kumagai, 2017; Saha & Gaillard, 2022). Other regret measures, such as the cumulative difference between optimal and obtained utility, are also considered. We discuss the equivalence of various regret measures in Lemma D.1.
+
+# 3 THE INTRINSIC LIMIT OF LIHF
+
+In this section, we study the intrinsic limit of learning from  $\rho$ -Imperfect Human Feedback. We are particularly interested in understanding whether learning from this restricted version of corrupted feedback is fundamentally "easier" than learning from arbitrarily adversarial corruption. Somewhat surprisingly, the answer seems to be no, as illustrated by the following lower bound result.
+
+Theorem 3.1. There exists a  $\rho$ -Imperfect Human Feedback (see Definition 2.1), strongly concave utility function  $\mu$ , and link function  $\sigma$  such that any learner has to suffer  $Reg_{T} \geq \Omega\left(d\max \{\sqrt{T}, T^{\rho}\}\right)$ , even with the knowledge of  $\rho$ .
+
+Notably, similar lower bound results have appeared in recent studies on adversarial corruption in bandits (Bogunovic et al., 2022), includingueling bandits (Agarwal et al., 2021; Di et al., 2024), showing the same lower bound  $\Omega(d\max\{\sqrt{T}, T^{\rho}\})$  (though they often use  $C \coloneqq \Theta(T^{\rho})$  to denote total corruption budget). However, a few key distinctions are worth noting. First, Theorem 3.1 presents a stronger result, showing that even when corruption decays over time, this structure (which defines our LIHF problem) does not make learning easier. Second, our corruption targets utility values, which is quite different from the corruption of comparison outcomes studied in (Agarwal et al., 2021; Di et al., 2024). In their case, corruption can fully flip comparison outcomes, which is critical for promoting sub-optimal arms in their lower bound proofs. However, such full control of the outcome is infeasible in our corruption of only the utility, which only slightly shifts the comparison probabilities since the scale of the corruption is also upper bounded by  $\mathcal{O}(t^{\rho - 1})$ .
+
+The aforementioned difference from previous works (Agarwal et al., 2021; Di et al., 2024) also render our proof of Theorem 3.1 fundamentally different from (and much more involved than) their proofs. Specifically, due to limited and diminishing corruption power, the corruption in our LIHF problem are insufficient to completely "mask" the instance as one with a different sub-optimal arm, which is the key strategy in previous lower bound proofs. Thus, our proof has to leverage information-theoretic lower bounds of statistical distributions to understand how small-scale corruption at each round could influence the overall function estimation errors. Core of our proof is the follow lower bound result for a different, and intuitively easier, problem: the standard bandit setup with direct reward feedback shown in Lemma 3.1 below. We then convert this lower bound for direct reward feedback toueling feedback through a linear link function, i.e.  $\sigma (x) = \frac{1 + x}{2}$ .
+
+Lemma 3.1 (Lower Bound under Direct Reward Feedback). Consider bandit learning with direct reward feedback, where reward  $r(a) \coloneqq \mu(a) + \epsilon$ ,  $\epsilon$  follows standard normal distribution. The action space  $\mathcal{A}$  is contained in a  $d$ -dimensional unit ball, and  $\mu$  is  $\mu_{\theta}(a) \coloneqq \theta^{\top}a - \frac{1}{2}\|a\|_2^2$ , with  $\theta \in \mathbb{R}^d$ ,  $\| \theta \|_2 \leq 1$ . Under  $\rho$ -Imperfect Human Feedback (Definition 2.1), for any  $T, d \leq \frac{1}{C_{\kappa}} T^{1 - \rho}$ , and for any learner, there exists a  $\theta$  such that under direct reward feedback, even with the knowledge of  $\rho^3$ , has to suffer regret  $Reg_T \coloneqq \mathbb{E}\{\sum_{t=1}^{T} \mu_{\theta}(a^*) - \mu_{\theta}(a_t)\} \geq \frac{d}{4} C_{\kappa} T^\rho$ .
+
+Proof Sketch of Lemma 3.1. We aim to show that there exists a corruption strategy such that the regret incurred by any learner with direct reward feedback is no less than  $\Omega(dT^{\rho})$ . The formulation of such a corruption strategy hinges on the observation that the regret incurred by the learner is bounded below by the sum of the Kullback-Leibler (KL) divergence between the distribution of the corrupted reward feedback  $\hat{v}_t$  conditioned on different values of  $\theta$ . Specifically, let  $\theta$  be uniformly drawn from  $\{-\beta, \beta\}^d$ , for some constant  $\beta > 0$ . Given an action  $a_t$  and index  $i$ , conditioned on the value of the  $i$ -th coordinate of  $\theta$ ,  $\theta_i$ , if  $\theta_i > 0$ , then  $\hat{v}_t$  is
+
+$$
+\hat {v} _ {t} = \mu_ {\theta} (a _ {t}) + c _ {t} (a _ {t} | \theta_ {i} > 0) + \xi_ {a _ {t}} = \left(- \frac {1}{2} \| a _ {t} \| ^ {2} + \sum_ {j \neq i} \theta_ {j} a _ {t, j}\right) + \beta a _ {t, i} + c _ {t} (a _ {t} | \theta_ {i} > 0) + \xi .
+$$
+
+Conditioned on  $\theta_{i} < 0$ , the corrupted reward feedback  $\hat{v}_{t}^{\prime}$  is
+
+$$
+\hat {v} _ {t} ^ {\prime} = \mu_ {\theta} (a _ {t}) + c _ {t} (a _ {t} | \theta_ {i} <   0) + \xi_ {a _ {t}} = \left(- \frac {1}{2} \| a _ {t} \| ^ {2} + \sum_ {j \neq i} \theta_ {j} a _ {t, j}\right) - \beta a _ {t, i} + c _ {t} (a _ {t} | \theta_ {i} <   0) + \xi .
+$$
+
+The noise  $\xi$  follows standard normal distribution. Consider the following corruption strategy which sets  $c_{t}(a_{t}|\theta_{i} > 0) := -\beta a_{t,i}$  and  $c_{t}(a_{t}|\theta_{i} < 0) := \beta a_{t,i}$  to minimize the KL divergence between  $\hat{v}_{t}$  and  $\hat{v}_{t}^{\prime}$ . Together with 1-strongly concavity of  $\mu_{\theta}$ , we can establish  $\mathbb{E}\left\{\sum_{t=1}^{T}\mu_{\theta}(a^{*}) - \mu_{\theta}(a_{t})\right\} \geq \frac{1}{2}\max\left\{\sum_{t=1}^{T}\left(\frac{C_{\kappa}T^{\rho-1}}{\beta} - \beta\right)^{2}, \frac{dT\beta^{2}}{2}\right\}$ . Since  $d \leq \frac{1}{C_{\kappa}} T^{1-\rho}$ , we can set  $\beta := \sqrt{C_{\kappa}} T^{\frac{\rho-1}{2}}$  to satisfy  $\| \theta \|_2 \leq 1$  and to optimize the aforementioned lower bound. Then we obtain  $\mathbb{E}\left\{\sum_{t=1}^{T}\mu_{\theta}(a^{*}) - \mu_{\theta}(a_{t})\right\} \geq dC_{\kappa}T^{\rho}/4$ . Notice that the corruption level each round is bounded by  $\beta \| a_{t}\|_{\infty}$ . In the scenario when the radius of  $\mathcal{A}$  is upper bounded by  $\beta$ , the corruption budget each round is bounded by  $C_{\kappa}t^{\rho-1}$ . It implies that the corruption is  $\rho$ -Imperfect Human Feedback, completing the proof.
+
+Theorem 3.1 can be proved by converting the lower bound in Lemma 3.1 for direct reward feedback to dueling feedback (see Lemma C.1). Before concluding this section, we note that the strongly concave utility function in the lower bound of Theorem 3.1 is not necessary. In particular, Proposition 1 below shows that similar lower bound applies to linear utilities as well. This result happens to resolve an open problem posed by Yao et al. (2022) which studies a similar learning from imperfect user problem with linear utility and dueling feedback (they termed it the "learning from a learning user" problem), develops a no-regret learning algorithm for that setting, but left the lower bound as an open problem. Proposition 1 implies that their upper bound's dependence on  $T$  is tight. We direct readers to Appendix C.1 and C.2 for proof details of Lemma 3.1 and Proposition 1.
+
+Proposition 1. There exists an LIHF instance with  $\rho$ -Imperfect Human Feedback (Definition 2.1), linear user utility  $\mu$ , and link function  $\sigma$ , such that any learner has to suffer  $Reg_{T} \geq \Omega(d \max \{\sqrt{T}, T^{\rho}\})$ , even with the knowledge of  $\rho$ .
+
+# 4 AN EFFICIENT AND TIGHT  $\rho$ -LIHF ALGORITHM
+
+In this section, we present a learning algorithm with a regret upper bound that matches the  $\Omega(d\max\{\sqrt{T}, T^{\rho}\})$  lower bound, up to a logarithmic term, for the  $\rho$ -LIHF setting discussed in Section 3. Our algorithm, Robustified Stochastic Mirror Descent for Imperfect Dueling (RoSMID), outlined in Algorithm 1, generalizes the Noisy Comparison-Based Stochastic Mirror Descent (NC-SMD) from Kumagai (2017). RoSMID employs a corruption-aware learning rate,  $\eta_{\rho} := \sqrt{\log T} / (dT^{\max\{0.5,\rho\}})$ , to slow down the learning when the imperfection level  $\rho$  is large, with NC-SMD being the special case with  $\rho = 0$ . Although RoSMID may initially seem like a pretty natural extension of NC-SMD, the choice of optimal  $\eta_{\rho}$  is in fact through careful derivation from a novel framework for analyzing continuous bandits under utility corruption. We conclude by demonstrating the applicability of this new framework to analyzing other gradient-based algorithms.
+
+# Theorem 4.1. RoSMID satisfies  $Reg_{T} \leq \tilde{\mathcal{O}}(d \max\{\sqrt{T}, T^{\rho}\})$  for any  $\rho$ -LIHF problem.
+
+The main technical contribution underlying this theorem is a novel framework for analyzing continuous dueling bandits with arbitrary strongly concave, yet possibly imperfect or corrupted utilities, which we believe has independent value. This framework not only enables a tight regret upper bound analysis, as shown in Theorem 4.1, but also allows us to easily derive regret guarantees for other dueling bandit algorithms under adversarial corruption, as demonstrated in Subsection 4.1. While recent works have explored corruption-robust regret analysis for dueling bandits with  $K$  arms
+
+Algorithm 1: Robustified Stochastic Mirror Descent for Imperfect Dueling (RoSMID)  
+Input: Learning rate  $\eta_{\rho}:= \sqrt{\log T} / (dT^{\max\{0.5,\rho\}})$ ,  $\nu$ -self-concordant function  $\mathcal{R}^4$ , time horizon  $T$ , tuning parameters  $\lambda \leq \frac{l_1^\sigma \alpha}{2}$ ,  $\phi \geq ((L_1^\sigma)^3 L_2^\sigma / \lambda)^2$   
+1 Initialize  $a_1 = \arg \min_{a \in A} \mathcal{R}(a)$   
+2 for  $t \in [T]$  do  
+3 Update concordant function  $\mathcal{R}_t(a) = \mathcal{R}(a) + \frac{\lambda \eta_{\rho}}{2} \sum_{i=1}^{t} \|a - a_i\|^2 + \phi \|a\|^2$   
+4 Sample direction  $u_t$  uniformly at random from a unit sphere  $\mathbb{S}^d$   
+5 Obtain corrupted feedback  $\hat{\mathcal{F}}(a_t', a_t) \in \{0,1\}$ , for  $a_t$  and  $a_t' = a_t + \nabla^2 \mathcal{R}_t(a_t)^{-1/2} u_t$   
+6 Compute the corrupted gradient:  $\hat{g}_t = \hat{\mathcal{F}}(a_t', a_t) d \cdot \nabla^2 \mathcal{R}_t(a_t)^{1/2} \cdot u_t$   
+7 Set  $a_{t+1} = \nabla \mathcal{R}_t^{-1} (\nabla \mathcal{R}_t(a_t) - \eta_{\rho} \hat{g}_t)$   
+Output:  $a_{T+1}$
+
+(Saha & Gaillard, 2022) or linear utilities (Di et al., 2024), neither their algorithms nor analysis can be applied to our setting with arbitrary concave utilities in continuous action space.
+
+The full proof of Theorem 4.1 is quite involved and is deferred to Appendix D. Here, we outline the main ideas, divided into four key steps. Step 1-3 are applicable for analyzing continuous dueling bandits under any form of corruption, whereas last Step 4 is the only step that hinges on the decaying corruption level assumptions of  $\rho$ -LIHF and is also the most involved and novel step.
+
+Proof Sketch. The proof of Theorem 4.1 is divided into four major steps.
+
+# Step 1: quantifying bias of gradient estimation in  $\rho$ -LIHF
+
+Our proof starts from quantifying the bias of the gradient  $\hat{g}_t$  caused by  $\rho$ -Imperfect Human Feedback, as shown in the following Lemma 4.1.
+
+Lemma 4.1 (Corrupted Gradients Estimation). The gradient  $\hat{g}_t$  in Line 6 of Algorithm 1 satisfies
+
+$$
+\mathbb {E} \left(\hat {g} _ {t} \mid a _ {t}\right) = \nabla \mid_ {a = a _ {t}} \bar {P} _ {t} (a) - \mathbb {E} \left(b _ {t} \mid a _ {t}\right).
+$$
+
+where  $\bar{P}_t(a) := \mathbb{E}_{x \in \mathbb{B}}\left[\mathbb{P}(a_t \succ a + \nabla^2 R_t(a_t)^{-\frac{1}{2}}x)\right]$  (with  $\mathbb{B}$  as the unit ball) is the smoothed probability that  $a_t$  is preferred over any action  $a$ , and  $b_t = d\left(\mathbb{P}(a_t \succ a_t') - \hat{\mathbb{P}}(a_t \succ a_t')\right)\nabla^2\mathcal{R}_t(a_t)^{\frac{1}{2}}u_t$ .
+
+# Step 2: decomposing regret for dueling bandits into regret of decision and feedback error
+
+Core to our proof is a "regret decomposition" lemma below that upper bounds the regret by two parts: regret of sub-optimal decision making under uncertainty, referred as regret of decision (first term in equation 2), and regret due to corruption, referred as feedback error (second term in equation 2).
+
+Lemma 4.2 (Regret Decomposition for Dueling Bandits). The  $Reg_{T}$  of Algorithm 1 satisfies:
+
+$$
+R e g _ {T} \leq \underbrace {\frac {C _ {0}}{\eta_ {\rho}} \log (T) + 8 d ^ {2} \eta_ {\rho} T + 2 L ^ {\mu} L _ {1} ^ {\sigma} R} _ {\text {R e g r e t o f D e c i s i o n}} + \underbrace {2 \mathbb {E} \left\{\sum_ {t = 1} ^ {T} b _ {t} ^ {\top} \left(a _ {t} - a _ {T} ^ {*}\right) \right\}} _ {\text {F e e d b a c k E r r o r}}, \tag {2}
+$$
+
+where  $C_0 \coloneqq (2\nu + 4L_1^\sigma \kappa + 4R_2^\sigma (L^\mu)^2 + L^\mu L_1^\sigma \kappa) / \lambda$ , and  $a_T^* \coloneqq \frac{1}{T} a_1 + (1 - \frac{1}{T})a^*$ .
+
+Similar regret decomposition has been shown to be useful for reinforcement learning which has direct reward feedback (Foster et al., 2023). However, to the best of our knowledge, Lemma 4.2 is the first to exhibit such a decomposition forueling feedback, which contains much sparser information than direct reward feedback as in classic RL or online learning. Hence the lemma may be of independent interest for future research onueling bandits under corruption or erroneous feedback.
+
+# Step 3: upper bounding the regret from feedback error
+
+Bounding the regret of decision term in equation 2 is standard. Thus, this step develops a novel upper bound for bounding the regret from feedback error, as shown below.
+
+Lemma 4.3 (Feedback Error). The feedback error term in equation 2 can be bounded as follows:
+
+$$
+\mathbb {E} \left\{\sum_ {t = 1} ^ {T} b _ {t} ^ {\top} (a _ {t} - a _ {T} ^ {*}) \right\} \leq C _ {1} d \sqrt {\eta_ {\rho}} \mathbb {E} \left\{\sum_ {t = 1} ^ {T} \sqrt {t} | c _ {t} (a _ {t}, a _ {t} ^ {\prime}) | \sqrt {\mu (a ^ {*}) - \mu (a _ {t})} \right\} + C _ {2} d T ^ {\rho} + 2 d \sqrt {\lambda} \sqrt {\eta_ {\rho} T},
+$$
+
+where  $C_1 \coloneqq \sqrt{\frac{2\lambda(L^\sigma)^2}{\alpha}}$ , and  $C_2 \coloneqq 2L^\sigma R\sqrt{\sup_{a\in\mathcal{A}}\lambda_{\max}(\nabla^2\mathcal{R}(a)) + 2\phi}$ .
+
+To our knowledge, such a bound is new for dueling bandits. The less common term in the bound is  $\sqrt{\mu(a^{*}) - \mu(a_{t})}$ . It comes from the strong concavity of the utility function  $\mu$ , which implies
+
+$$
+\langle b _ {t}, a _ {t} - a ^ {*} \rangle \leq \| b _ {t} \| _ {2} \| a _ {t} - a ^ {*} \| _ {2} \leq \| b _ {t} \| _ {2} \sqrt {\frac {2}{\alpha} (\mu (a ^ {*}) - \mu (a _ {t}))}.
+$$
+
+# Step 4: The iterative regret refinement analysis and resultant optimal learning rate
+
+Our last and also the most intriguing step is to pin down the regret upper bound. Given the bounds from previous steps, the only remaining term to upper bound is  $\mathbb{E}\left\{\sum_{t=1}^{T} \sqrt{t} |c_t(a_t, a_t')| \sqrt{\mu(a^*) - \mu(a_t)}\right\}$  in Lemma 4.3. We can upper bound  $\sqrt{t} |c_t(a_t, a_t')|$  by  $C_\kappa t^{\rho - \frac{1}{2}}$  under  $\rho$ -Imperfect Human Feedback assumption (this is the point where we start to need the decaying structure in Definition 2.1). This is a tricky question, since this term involving optimal action  $a^*$  closely connects to the regret  $\operatorname{Reg}_T$  itself (see equation 1). This is precisely the motivation for our analysis approach of "iterative regret refinement". Concretely, it turns out that a good upper bound for the term  $A := \mathbb{E}\left(\sum_{t=1}^{T} \mu(a^*) - \mu(a_t)\right)$  can be leveraged to derive a good upper bound for term  $B := \mathbb{E}\left\{\sum_{t=1}^{T} t^{\rho - \frac{1}{2}} \sqrt{\mu(a^*) - \mu(a_t)}\right\}$ , which can then be leveraged to derive a good upper bound for the regret  $\operatorname{Reg}_T$ , which can be used to refine the original upper bound of term  $A$ , at which point we can re-apply the above refinement process.
+
+Two key factors are essential for the iterative regret refinement mentioned above to result in a tight bound. The first step is to derive a good upper bound for term  $B$  using term  $A$ , which is shown in Lemma 4.4. The second step is to prove that this iterative refinement consistently improves the regret bound, eventually reaching a limit. This is demonstrated by Lemma 4.5. The choice of learning rate  $\eta_{\rho} = \frac{\sqrt{\log T}}{dT^{\max\{1/2,\rho\}}}$  is crucial in the second step; in a sense, it is the only choice that converges to the tight regret upper bound  $\tilde{\mathcal{O}}\left(d\max\{\sqrt{T},T^{\rho}\}\right)$  in the limit, rather than exploding the bound.
+
+Lemma 4.4. If  $A = \mathbb{E}\left(\sum_{t=1}^{T}[\mu(a^{*}) - \mu(a_{t})]\right) \leq \mathcal{O}(T^{\frac{1}{2} + \psi})$  for some  $\psi \in [0, \frac{1}{2})$ , then we must have  $B = \mathbb{E}\left(\sum_{t=1}^{T}t^{\rho - \frac{1}{2}}\sqrt{\mu(a^{*}) - \mu(a_{t})}\right) \leq \mathcal{O}(T^{-\frac{1}{4} + \frac{\psi}{2} + \rho})$ .
+
+To prove Lemma 4.4, we first bound term  $B$  by  $\sum_{t=1}^{T} t^{\rho-1/2} \sqrt{\mathbb{E}[\mu(a^{*}) - \mu(a_{t})]}$  using the concavity of the square root function. The key novelty of our proof is to use the Abel's equation (Williams, 1963) to re-arrange the term  $\sum_{t=1}^{T} t^{\rho-1/2} \sqrt{\mathbb{E}[\mu(a^{*}) - \mu(a_{t})]}$  and establish its connection with another term  $C = \sum_{t=1}^{T} \sqrt{\mathbb{E}[\mu(a^{*}) - \mu(a_{t})]}$ . We finally connect term  $C$  with the term  $A$  in the lemma statement, by proving  $\sum_{t=1}^{T} \sqrt{\mathbb{E}[\mu(a^{*}) - \mu(a_{t})]} \leq \mathcal{O}(T^{\frac{3}{4}+\frac{\psi}{2}})$  if  $\mathbb{E}\left(\sum_{t=1}^{T} \mu(a^{*}) - \mu(a_{t})\right) \leq \mathcal{O}(T^{\frac{1}{2}+\psi})$  via concavity of square root. Together with the decay scale  $t^{\rho-1}$ , which allows  $t^{\rho-1} - (t+1)^{\rho-1} \leq t^{\rho-2}$ , we obtain the tight bound  $\mathcal{O}(T^{-\frac{1}{4}+\frac{\psi}{2}+\rho})$ .
+
+Armed with Lemma 4.4, we can use it to prove the induction claim (Lemma 4.5). The challenge in this proof lies in identifying the constant  $K$ , which must hold across all iterations,  $k$ , of the analysis to ensure the bound remains stable and does not explode.
+
+Lemma 4.5 (Induction Claim). Consider  $T > \sqrt{2L^{\mu}L_{1}^{\sigma}R}$  and  $\eta_{\rho} = \frac{\sqrt{\log T}}{dT^{\max\{1/2,\rho\}}}$ . If  $Reg_{T} \leq 144dK\sqrt{\log T} T^{\rho + \frac{3}{2^{k}}}$  for some integer  $k$ , then we must also have
+
+\[
+\begin{array}{rcl}
+Reg_T & \leq & 144dK\sqrt{\log T} T^{\rho + \frac{\frac{3}{2} - \rho}{2^{k + 1}}}.\\
+& \max \left\{C_0, \frac{C_2C_\kappa}{\sqrt{\log T}}, (L^\sigma C_\kappa)^2, 2\sqrt{\lambda} RC_\kappa L^\sigma \right\}, & C_0 \text{ and } C_2 \text{ defined in Lemma 4.2 and 4.3 respectively,}
+\end{array}
+\]
+
+# 4.1 ADDITIONAL APPLICATIONS OF OUR REGRET ANALYSIS FRAMEWORK
+
+To showcase useful applications of the new regret analysis framework above for proving Theorem 4.1, in this subsection we employ the framework to study continuous dueling bandits under arbitrary and agnostic corruption — i.e., remove decaying constraints on  $c_t$  and  $\rho$  is unknown to the learner. We analyze natural variants of RoSIMD, which apply to strongly concave utilities, and the well-known algorithm Dueling Bandit Gradient Descent (DBGD) (Yue & Joachims, 2009), which apply to general concave utilities. The proofs of these results are direct applications of the analysis from Steps 1-3 outlined above (without Step 4, since it is the only one requiring knowledge of  $\rho$  and decaying corruption). Additionally, we present these results to offer a set of principled benchmark algorithms for the experiments in Section 5. One thing to highlight is that our robustified algorithms retain the same computational complexity as the originals, ensuring added robustness without compromising efficiency. For further details, see Appendix G.1.
+
+Proposition 2 (Efficiency-Robustness Tradeoff in RoSMID). If the total corruption level satisfies  $\sum_{t=1}^{T}|c_t(a_t,a_t')|\leq\mathcal{O}(T^\rho)$ , then for any  $\alpha\in[\frac{1}{2},1)$ , if the utility function is strongly concave, for a sufficiently large round  $T$ , by choosing learning rate  $\eta=\frac{\sqrt{\log T}}{2d}T^{-\alpha}$
+
+1. (Robustness) RoSMID incurs  $Reg_{T} \leq \tilde{\mathcal{O}}(dT^{\alpha} + \sqrt{d} T^{\frac{1}{2}(1 - \alpha) + \rho} + dT^{\rho})$ .  
+2. (Efficiency) There exists a strongly concave utility function  $\mu$  and a link function  $\sigma$  such that  $R e g_{T}$  suffered by RoSMID is at least  $\Omega (T^{\alpha})$  in scenario without corruption.
+
+Proposition 3 (Efficiency-Robustness Tradeoff in DBGD). If the total corruption level satisfies  $\sum_{t=1}^{T}|c_t(a_t,a_t')|\leq\mathcal{O}(T^\rho)$ , then for any  $\alpha\in(0,\frac{1}{4}]$ , if utility function  $\mu$  is generally concave, for a sufficiently large round  $T$ , choosing  $\gamma=\frac{R}{\sqrt{T}}$  and  $\delta=\frac{\sqrt{2Rd}}{\sqrt{13L^{\sigma}L^{\mu}T^{\alpha}}}$  for DBGD
+
+1. (Robustness)  $Reg_{T}$  incurred by DBGD satisfies  $Reg_{T} \leq \mathcal{O}(\sqrt{d} T^{1 - \alpha} + \sqrt{d} T^{\alpha + \rho})$ .  
+2. (Efficiency) There exists a linear utility function  $\mu$  and a link function  $\sigma$  such that  $Reg_{T}$  suffered by DBGD is at least  $\Omega\left(T^{1 - \alpha}\right)$  in scenario without corruption.
+
+Both propositions illustrate the tradeoff between learning efficiency and robustness to adversarial corruption, achieved through their tunable learning rate  $\eta_{\rho}, \gamma, \delta$  (learning rate in DBGD is  $\frac{\gamma\delta}{d}$ ; we direct reader to Algorithm 3 for detail). Specifically, greater tolerance for agnostic corruption (i.e., larger  $\alpha$  in Proposition 2) results in worse regret in non-corrupt settings. A similar robustness-accuracy trade-off has been studied in classification problems with adversarial examples, both empirically (Tsipras et al., 2018) and theoretically (Zhang et al., 2019). However, to our knowledge, this is the first formal analysis of such a trade-off in online learning. While expanding the confidence region for UCB-type algorithms to handle agnostic corruption in linear contextual dueling bandits increases tolerance for corruption but worsens regret—sharing a similar idea of a trade-off (Di et al., 2023)—it's unclear if this approach necessarily reduces efficiency. In contrast, our work demonstrates that adjusting learning rates in gradient-based algorithms to enhance robustness against agnostic corruption inevitably decreases efficiency when no corruption is present. In particular, the efficiency statements in both propositions can be interpreted as algorithm-dependent lower bounds, concretely demonstrating that for certain problem instances, increasing robustness necessarily decreases the learning efficiency of the algorithm. In fact, the "efficiency-robustness tradeoff" is inherent under unknown adversarial attacks. Lemma E.1 in the appendix shows that algorithms that learn faster tend to be more vulnerable to unknown corruption. However, the nature of this tradeoff varies across different algorithms. It is a very intriguing and fundamental open question to study what the optimal tradeoff is under unknown adversarial corruption and how to achieve it.
+
+# 5 EXPERIMENTS
+
+In this section, we validate our theoretical analysis of RoSMID and DBGD under  $\rho$ -Imperfect Human Feedback through simulations. We also compare their performance against two baseline algo
+
+rithms: Doubler and the heuristic algorithm, Sparring, proposed by Ailon et al. (2014). For Sparring, we use bandit gradient descent (Flaxman et al., 2005) as the black-box bandit algorithm. We direct readers to Appendix G for implementation details.
+
+Experiment Setup. We consider a standard experiment setup, which adopts a strongly concave utility  $\mu_{\theta}(a) := \theta^{\top}a - \frac{1}{2}\|a\|_2^2$ , and a logistic link function  $\sigma(x) = \frac{1}{1 + \exp(-x)}$ . We choose  $d = 5$  and  $T = 10^5$ . Our action space  $\mathcal{A}$  is a  $d$ -dimensional ball with radius  $R = 10$ . The preference parameter  $\theta$  is randomly sampled from the surface of  $\mathcal{A}$ . In our problem setting, the optimal action  $a^* = \theta$  and  $\mu_{\theta}(a^*) = 50$ . We simulate  $\rho$ -Imperfect Human Feedback for  $\rho \in [0.5, 1]$ .
+
+Results and Discussion. Figure 1 presents a log-log plot of regret versus iteration,  $t$ , which the order of regret in  $T$  is represented by the slope of the line. Each line represents the average over five trials with different random seeds, and each line corresponds to a distinct value of  $\rho$ . The shaded region indicates  $\pm$  one standard deviation. In the legend, "o" denotes the estimated order of regret. We estimate it using least squares on the last  $1\%$  of the data. Figure 1 supports Theorem 4.1, as for  $\rho \in [0.5, 1)$ , the estimated slope is less than  $\rho$ , suggesting that the choice of  $\eta_{\rho} = \frac{\sqrt{\log T}}{dT^{\max\{1/2,\rho\}}}$  for RoSMID results in a regret bound of at most  $\tilde{\mathcal{O}}(T^{\rho})$ .
+
+![](images/378a089ea056e40f67bcc19aa96fb4639d67b36ec59902858bb40408fc86f315.jpg)  
+Figure 1: Robustness of RoS-MID for  $\rho$ -LIHF
+
+Figure 2 compares the performance of our proposed algorithms, variants of DBGD and RoSMID, with two baseline algorithms, Sparring and Doubler, in the setting when  $c_t$  is arbitrary and agnostic. We set the parameter  $\alpha = 1/4$  for DBGD and  $\alpha = 1/2$
+
+for RoSMID. When  $\rho$  is unknown to the algorithms, these parameter choices enable DBGD and RoSMID to tolerate  $\mathcal{O}(T^{3/4})$  levels of unknown corruption, since the slope estimates are less than 1, implying sublinear regret, aligning with our theoretical predictions. Experimentally, RoSMID demonstrates the best performance under strongly concave utility functions. Sparring performs comparably to RoSMID, despite being a heuristic approach without theoretical guarantees. In contrast, Doubler exhibits the worst performance, likely because its theoretical guarantees are only applicable to linear link functions, which differs from our experimental setup. We direct readers to Appendix G for experiment results with other choice of  $\alpha$  for efficiency-robustness tradeoff.
+
+![](images/420d642c12b5f0e271a7118bed89e777256a69fb5d0318d6c96872fd6087725a.jpg)  
+Figure 2: For each algorithm, we tested its performance under  $\rho$ -Imperfect Human feedback with  $\rho = 0.5, 0.6, 0.75$ . For each  $\rho$ , we presented a line plot of the average regret over five simulations, accompanied by  $\pm$  one standard deviation shown by the shaded region. In the legend,  $o$  denotes the estimated line slope, calculated using least squares on the last  $1\%$  of the data.
+
+# 6 CONCLUSION
+
+In conclusion, this paper enhances the understanding of continuous dueling bandits by addressing  $\rho$ -Imperfect Human Feedback with concave utilities. We established fundamental regret lower bounds, introduced the RoSMID algorithm with nearly optimal performance, and developed a novel regret analysis framework that highlights the inherent efficiency-robustness trade-off in gradient-based algorithms. Our experimental results corroborate the theoretical predictions. A promising future direction is the development of an algorithm that achieves  $\tilde{O}(d\sqrt{T} +dT^{\rho})$  regret for continuous dueling bandits with strongly concave utilities under arbitrary adversarial corruption. Additionally, incorporating contextual information to better model real-world scenarios could further enhance the applicability and robustness of these algorithms. We left these directions for future research.
+
+Acknowledgment. This work is supported by the AI2050 program at Schmidt Sciences (Grant G-24-66104), Army Research Office Award W911NF23-1-0030, and NSF Award CCF-2303372. Fan Yao conducted this research while he is visiting the University of Chicago.
+
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+
+# Appendix to Learning from Imperfect Human Feedback: A Tale from Corruption-Robust Dueling
+
+A DISCUSSION ON THE GENERALITY OF MODELING ASSUMPTIONS
+
+The assumptions of a continuous action space are standard in the online convex optimization literature (Flaxman et al., 2004; Yue & Joachims, 2009; Shamir, 2013; Kumagai, 2017). Moreover, they are particularly natural for today's preference learning for recommendation systems since languages, texts, videos are mostly processed as embedding vectors which are continuous by nature. We assumed strong concave utility functions. This is standard due to at least two reasons. First, there is already a rich literature studying online optimization of strongly convex or concave functions, and our work subscribes to that rich literature (Shamir, 2013; Kumagai, 2017; Wan et al., 2022; Saha et al., 2022). Second, strongly concave utilities functions capture essential properties like diminishing marginal returns and risk-averse behavior. These characteristics align closely with real-world scenarios in economics, behavioral studies, and decision-making processes. Strongly concave functions are frequently utilized in economic literature for modeling utility functions such as the "indirect utility function" (Montrucchio, 1987; Sorger, 1995; Venditti, 1997), which represents the maximum utility a consumer can achieve based on a specific income level and set of prices. Our assumption on the link function aligns with those in Kumagai (2017) but is slightly stronger than those in Yue & Joachims (2009). Nevertheless, it encompasses many natural link functions, including the sigmoid function, the cumulative Gaussian distribution function, and the linear function, more general than merely sigmoid link functions (Maystre & Grossglauer, 2017; Saha, 2021; Xu et al., 2024) or linear link function (Ailon et al., 2014; Chen & Frazier, 2017; Zimmert & Seldin, 2018; Lin & Lu, 2018), as assumed in some earlier works.
+
+# B CHALLENGES IN DEVELOPING ROBUST ALGORITHMS FOR CONTINUOUS DUELING BANDITS WITH GENEALLY CONCAVE UTILITY UNDER CORRUPTION INDUCED BY STRONG ADVERSARY
+
+There is a line of work that studies bandits under corruption by a weak adversary (Lykouris et al., 2018; Gupta et al., 2019; Agarwal et al., 2021; Zimmert & Seldin, 2021; Saha & Gaillard, 2022; Wei et al., 2022), where the corruption is introduced before observing the agent's actions. This research develops efficient algorithms that achieve sublinear regret without requiring prior knowledge of the total corruption. Specifically, for corruption induced by a weak adversary, Wei et al. (2022) adopts the idea of model selection and introduces a robust framework called Corruption-Robustness through Balancing and Elimination (COBE). This framework allows an algorithm designed for a known corruption level to be adapted for scenarios with an unknown corruption level. Notably, in the case of weak adversaries, Zimmert & Seldin (2021); Saha & Gaillard (2022) develop "best-of both-world" algorithms, which achieve optimal performance under both begin and corrupted settings.
+
+However, for corruption induced by a strong adversary—where the corruption occurs after observing the agent's actions (the scenario considered in our setting)—it may not be possible to develop a best-of both-world algorithm. This limitation arises because, for any algorithm that does not know the total corruption, there exists a problem instance where the algorithm suffers linear regret (Bogunovic et al., 2020; Kang et al., 2024). To achieve sublinear regret in such settings, it is crucial for the algorithm to have precise knowledge of the total corruption or at least an accurate estimation of its upper bound.
+
+For cases with known corruption, nearly optimal algorithms are proposed for various settings, including stochastic linear bandits in discrete action spaces (Bogunovic et al., 2020), Gaussian process bandits (Bogunovic et al., 2022), linear contextual bandits (He et al., 2022), Lipschitz bandits (Kang et al., 2024), and linear contextual dueling bandits (Di et al., 2024). These approaches typically rely on phase elimination techniques to eliminate suboptimal arms or Upper Confidence Bound (UCB)-based algorithms to pull optimal arms with high probability. The success of these methods heavily depends on constructing precise utility estimates to guide decision-making.
+
+However, for learning from nonlinear utility, constructing utility estimates underueling feedback is a very challenging problem, as only preferential (directional) information is available. This limited
+
+information motivates the exploration of a new approach: robust gradient-based algorithms. Zimmert & Seldin (2021) and Saha & Gaillard (2022) utilize robust stochastic mirror descent algorithms to update the probability distribution for sampling  $K$  arms. However, it is difficult to efficiently generalize this design to continuous action spaces, as it requires transforming the probability vector into a probability density function and efficiently managing its updates and approximations. While Yue & Joachims (2009) and Kumagai (2017) exploitsueling bandit gradient descent and stochastic mirror descent and achieves sublinear regret for continuousueling bandits, their performance under corruption remains unexplored. It is a nontrivial question and still remains an interesting open problem of developing robust and computationally efficient algorithms for continuousueling bandits with generally concave utility functions under corruption induced by strong adversaries.
+
+# C PROOFS FOR THEOREM 3.1
+
+Theorem (Theorem 3.1 restated). There exists a  $\rho$ -Imperfect Human Feedback (see Definition 2.1), strongly concave utility function  $\mu$ , and link function  $\sigma$  such that any learner has to suffer  $Reg_{T} \geq \Omega\left(d \max \{\sqrt{T}, T^{\rho}\}\right)$ , even with the knowledge of  $\rho$ .
+
+Proof. Before showing the regret lower bound proof for Theorem 3.1, let's first establish the following Lemma C.1, which builds the connection between regret incurred under bandit reward feedback andueling feedback. This proof is inspired by Theorem 6 in Yao et al. (2022). To prove Lemma C.1, in addition to  $\mathrm{Reg}_T$ , we introduce a new metric to measure the performance of algorithm underueling feedback, which we coin functional regret  $\mathrm{Reg}_T^{\mathrm{FO}}$ , defined as follows.
+
+$$
+\operatorname {R e g} _ {T} ^ {\mathrm {F O}} := \mathbb {E} \left\{\sum_ {t = 1} ^ {T} \mu \left(a ^ {*}\right) - \mu \left(a _ {t}\right) + \mu \left(a ^ {*}\right) - \mu \left(a _ {t} ^ {\prime}\right) \right\}. \tag {3}
+$$
+
+Lemma C.1. Given a utility function  $\mu$ , if the regret lower bound for algorithm with bandit reward feedback is  $\overline{Reg}$ , then any learner with dueling feedback has to suffer regret  $Reg_T^{FO} \geq 2\overline{Reg}$ .
+
+Proof. We prove our claim by contradiction. Let  $(a_{0,t}, a_{1,t})$  be the pair of recommendations at round  $t$ . Suppose  $\mathrm{Reg}_T^{\mathrm{FO}} < 2\overline{\mathrm{Reg}}$ . As a result, at least one of the following inequality must hold:
+
+$$
+\mathbb {E} \left\{\sum_ {t = 1} ^ {T} \mu \left(a ^ {*}\right) - \mu \left(a _ {0, t}\right) \right\} \leq \overline {{\operatorname {R e g}}} \tag {4}
+$$
+
+$$
+\mathbb {E} \left\{\sum_ {t = 1} ^ {T} \mu \left(a ^ {*}\right) - \mu \left(a _ {1, t}\right) \right\} \leq \overline {{\operatorname {R e g}}}
+$$
+
+Consider a principal who can observe the interaction between a user and the learner  $\mathcal{L}$ , then we can construct two algorithms  $\mathcal{L}_0$  and  $\mathcal{L}_1$  as follows.
+
+# Algorithm 2: Algorithm  $\mathcal{L}_i$
+
+Input: the time horizon  $T$    
+1 for  $t\leq T$  do   
+2 Call the learner  $\mathcal{L}$  to generate two candidates  $(a_{0,t},a_{1,t})$  . Present  $(a_{0,t},a_{1,t})$  to user and received the feedback.   
+4 Return user feedback to the learner  $\mathcal{L}$  and update  $\mathcal{L}$    
+Output: the sequential decision  $\{a_{i,t}\}_{t = 1}^{T}$
+
+From equation 4, we know that at least one of  $\{\mathcal{L}_0,\mathcal{L}_1\}$  achieves an expected regret lower than Reg. However, we know all algorithms with bandit feedback with utility function  $\mu$  has to occur regret at least Reg, contradiction. Therefore, Lemma C.1 must hold, which completes the proof.
+
+Next, we prove Theorem 3.1. First, consider the case  $\rho \leq 0.5$ . Given that the utility function is strongly concave, by applying Lemma C.2, we have  $\overline{\mathrm{Reg}} = 0.02\min \{T,d\sqrt{T}\}$ . From Lemma C.1,
+
+we also obtain  $\mathrm{Reg}_T^{\mathrm{FO}}\geq 0.04\min \{T,d\sqrt{T}\}$ . Using the linear link function  $\sigma (x) = \frac{1}{2} +\frac{1}{2} x$ , it follows that  $\mathrm{Reg}_T\geq 0.02\min \{T,d\sqrt{T}\} \geq \Omega (d\max \{\sqrt{T},T^\rho \})$ , completing the proof.
+
+Lemma C.2 (Theorem 6 in (Shamir, 2013)). Let the number of rounds  $T$  be fixed. Then for any learner, there exists a quadratic function of the form  $\mu_{\theta}(a) \coloneqq \theta^{\top}a - \frac{1}{2}\| a\|^{2}$  which is minimized at  $\theta$  and  $\| \theta \|_2 \leq 0.5$  such that
+
+$$
+\mathbb {E} \left(\sum_ {t = 1} ^ {T} \mu_ {\theta} \left(a ^ {*}\right) - \mu_ {\theta} \left(a _ {t}\right)\right) \geq 0. 0 2 \min  \left\{T, d \sqrt {T} \right\}.
+$$
+
+Now, let's focus on the scenario when  $\rho > 0.5$ . In essence, we want to extend Lemma C.2 to the scenario in presence of adversarial corruption induced by  $\rho$ -Imperfect Human Feedback. If we can show the regret lower bound can be generalized to  $\Omega(dT^{\rho})$ , then applying the same technique which uses Lemma C.1 to connect regret suffered under bandit reward feedback and regret suffered under dueling feedback and choose linear link function to connect  $\mathrm{Reg}_T^{\mathrm{FO}}$  and  $\mathrm{Reg}_T$ , we will get the desired regret lower bound. The extension of Lemma C.2 is proven at section C.1.
+
+# C.1 PROOF FOR LEMMA 3.1
+
+Lemma (Lemma 3.1 restated). Consider bandit learning with direct reward feedback, where reward  $r(a) \coloneqq \mu(a) + \epsilon$ ,  $\epsilon$  follows standard normal distribution. The action space  $\mathcal{A}$  is contained in a  $d$ -dimensional unit ball, and  $\mu$  is  $\mu_{\theta}(a) \coloneqq \theta^{\top}a - \frac{1}{2}\|a\|_2^2$ , with  $\theta \in \mathbb{R}^d$ ,  $\|\theta\|_2 \leq 1$ . Under  $\rho$ -Imperfect Human Feedback (Definition 2.1), for any  $T$ ,  $d \leq \frac{1}{C_{\kappa}} T^{1-\rho}$ , and for any learner, there exists a  $\theta$  such that under direct reward feedback, even with the knowledge of  $\rho$ , has to suffer regret  $Reg_T \coloneqq \mathbb{E}\{\sum_{t=1}^T \mu_{\theta}(a^*) - \mu_{\theta}(a_t)\} \geq \frac{d}{4} C_{\kappa} T^\rho$ .
+
+Proof. To prove Lemma 3.1, we aim to prove that there exists a corruption strategy such that the regret incurred by any learner with imperfect dueling feedback is not less than  $\Omega(dT^{\rho})$ . The formulation of such a corruption strategy hinges on the observation that when the utility function  $\mu_{\theta}$  adopts the quadratic form parameterized by  $\theta$ , specifically  $\mu_{\theta}(a) = \theta^{\top}a - \frac{1}{2}\|a\|_2^2$ , which is both smooth and strongly concave, then the regret incurred by the learner is bounded below by the sum of the Kullback-Leibler (KL) divergence between the distribution of the corrupted reward feedback obtained at round  $t$ , denoted  $\hat{v}_t$ ,  $t \in [T]$ , conditioned on different possible values of  $\theta$ . (see Lemma C.4). Assume that  $\theta$  is uniformly drawn from  $\{-\beta, \beta\}^d$ , given an action  $a_t$ , conditioned on  $\theta_i > 0$ , the corrupted reward feedback  $\hat{v}_t$  is
+
+$$
+\hat {v} _ {t} = \mu_ {\theta} \left(a _ {t}\right) + c _ {t} \left(a _ {t} \mid \theta_ {i} > 0\right) + \xi = \underbrace {\left(- \frac {1}{2} \left\| a _ {t} \right\| ^ {2} + \sum_ {j \neq i} \theta_ {j} a _ {t , j}\right) + \beta a _ {t , i} + c _ {t} \left(a _ {t} \mid \theta_ {i} > 0\right)} _ {\mu_ {1}} + \xi . \tag {5}
+$$
+
+Conditioned on  $\theta_{i} < 0$ , the corrupted reward feedback  $\hat{v}_{t}$  is
+
+$$
+\hat {v} _ {t} = \mu_ {\theta} \left(a _ {t}\right) + c _ {t} \left(a _ {t} \mid \theta_ {i} <   0\right) + \xi = \underbrace {\left(- \frac {1}{2} \| a _ {t} \| ^ {2} + \sum_ {j \neq i} \theta_ {j} a _ {t , j}\right) - \beta a _ {t , i} + c _ {t} \left(a _ {t} \mid \theta_ {i} <   0\right)} _ {\mu_ {2}} + \xi . \tag {6}
+$$
+
+We use  $c_{t}(a_{t}|\theta_{i} > 0)$  and  $c_{t}(a_{t}|\theta_{i} < 0)$  to empathize the fact that the magnitude and sign of corruption can be dependent on  $\theta$ .  $\xi$  follows a standard Gaussian distribution. Therefore,  $\hat{v}_{t}$  in Equation 5 follows  $\mathrm{N}(\mu_1,1)$ .  $\hat{v}_{t}$  in Equation 6 follows  $\mathrm{N}(\mu_2,1)$ . By using Lemma C.3, we have
+
+$$
+\mathrm {D} _ {\mathrm {K L}} (\mathrm {N} (\mu_ {1}, 1) | | \mathrm {N} (\mu_ {2}, 1)) = (\mu_ {1} - \mu_ {2}) ^ {2}.
+$$
+
+Lemma C.3 (KL Divergence for Normal Distribution). Let  $N(\mu, \sigma^2)$  represent a Gaussian distribution variable with mean  $\mu$  and variance  $\sigma^2$ . Then
+
+$$
+D _ {K L} (N (\mu_ {1}, \sigma^ {2}) | | N (\mu_ {2}, \sigma^ {2})) = \frac {(\mu_ {1} - \mu_ {2}) ^ {2}}{2 \sigma^ {2}}.
+$$
+
+To optimize the lower bound in Lemma C.4, the adversary selects  $c_{t}(a_{t}|\theta_{i} > 0)$  and  $c_{t}(a_{t}|\theta_{i} < 0)$  to minimize the difference between  $\mu_{1}$  and  $\mu_{2}$ . Consider the following corruption strategy: when  $\theta_{i} > 0$ , set  $c_{t}(a_{t}|\theta_{i} > 0) = -\beta a_{t,i}$ ; when  $\theta_{i} < 0$ , set  $c_{t}(a_{t}|\theta_{i} < 0) = \beta a_{t,i}$ . This strategy ensures that  $\mu_{1} = \mu_{2}$ .
+
+Next step is to determine the value of  $\beta$ . Notice that the corruption budget is  $C_{\kappa}T^{\rho}$ . To execute the corruption strategy described above, it requires
+
+$$
+\sum_ {t = 1} ^ {T} \left| c _ {t} \left(a _ {t}\right) \right| \leq \beta \sum_ {t = 1} ^ {T} \left\| a _ {t} \right\| _ {\infty} \leq C _ {\kappa} T ^ {\rho}. \tag {7}
+$$
+
+Therefore, by exploiting the fact that  $\mu_{\theta}$  is 1-strongly concave, we establish another lower bound for the regret by
+
+$$
+\begin{array}{l} \mathbb {E} \left\{\sum_ {t = 1} ^ {T} \mu_ {\theta} \left(a ^ {*}\right) - \mu_ {\theta} \left(a _ {t}\right) \right\} \geq \frac {1}{2} \mathbb {E} \left\{\sum_ {t = 1} ^ {T} \| a _ {t} - \theta \| _ {2} ^ {2} \right\} \\ \geq \frac {1}{2} \mathbb {E} \left\{\sum_ {t = 1} ^ {T} \left(\left\| a _ {t} \right\| _ {\infty} - \beta\right) ^ {2} \right\} \\ \geq \frac {1}{2} \sum_ {t = 1} ^ {T} \left(C _ {\kappa} T ^ {\rho - 1} / \beta - \beta\right) ^ {2}. \\ \end{array}
+$$
+
+Since  $|a_{t,i} - \theta_i| \geq ||a_{t,i}| - \beta|$  for all  $i$ , we obtain the second inequality. By utilizing the corruption constraint equation 7, we derive the last inequality, which is minimized when  $\| a_t\|_\infty = \frac{C_\kappa T^{\rho - 1}}{\beta}$  for all  $t$ . Together with Lemma C.4, we have:
+
+$$
+\mathbb {E} \left\{\sum_ {t = 1} ^ {T} \mu_ {\theta} \left(a ^ {*}\right) - \mu_ {\theta} \left(a _ {t}\right) \right\} \geq \frac {1}{2} \max  \left\{\sum_ {t = 1} ^ {T} \left(C _ {\kappa} T ^ {\rho - 1} / \beta - \beta\right) ^ {2}, \frac {d T \beta^ {2}}{2} \right\}. \tag {8}
+$$
+
+We select  $\beta$  to optimize the lower bound in Equation equation 8. Since  $d \leq \frac{1}{C_{\kappa}} T^{1 - \rho}$ , we can set  $\beta = \sqrt{C_{\kappa}} T^{\frac{\rho}{2} - \frac{1}{2}}$  to satisfy  $\| \theta \|_2 \leq 1$ . Then we obtain  $\mathbb{E}\left\{\sum_{t=1}^{T} \mu_\theta(a^*) - \mu_\theta(a_t)\right\} \geq dC_{\kappa}T^\rho / 4$ . Notice that the corruption level each round is bounded by  $\beta \| a_t \|_\infty$ . Consider the scenario when the radius of the action space  $\mathcal{A}$  is upper bounded by  $\beta$ , then the corruption budget each round is bounded by  $C_{\kappa} t^{-1 + \rho}$ , which satisfies the definition of  $\rho$ -Imperfect Human Feedback, which completes the proof.
+
+Lemma C.4. Let's consider the utility function  $\mu_{\theta}(a) := \theta^{\top}a - \frac{1}{2}\| a\|_2^2$ . Let  $\hat{v}_1, \hat{v}_2, \dots, \hat{v}_T$  be a sequence of corrupted reward feedback obtained by a learner. Then there exists a  $\theta \in \mathbb{R}^d$ ,  $\|\theta\|_2 \leq 1$ , uniformly drawn from  $\{-\beta, \beta\}^d$ , such that the regret suffered by the learner is
+
+$$
+\mathbb {E} \left\{\sum_ {t = 1} ^ {T} \mu_ {\theta} (a ^ {*}) - \mu_ {\theta} (a _ {t}) \right\} \geq \frac {d T \beta^ {2}}{4} \left(1 - \sqrt {\frac {1}{d} \sum_ {i = 1} ^ {d} \sum_ {t = 1} ^ {T} \mathbb {D} _ {t , i}}\right).
+$$
+
+$\mathbb{D}_{t,i} \coloneqq \sup_{\{\theta_j\}_{j \neq i}} D_{KL}\left(\mathbb{P}(\hat{v}_t|\theta_i > 0, \{\theta_j\}_{j \neq i}, \{\hat{v}_l\}_{l=1}^{t-l})||\mathbb{P}(\hat{v}_t|\theta_i < 0, \{\theta_j\}_{j \neq i}, \{\hat{v}_l\}_{l=1}^{t-l})\right)$ .  $D_{KL}$  is the KL divergence between two distributions.
+
+Proof. Using the similar argument in Shamir (2013), assume that the learner is deterministic:  $a_{t}$  is a deterministic function of the realized corrupted reward feedback  $\hat{v}_{1}, \hat{v}_{2}, \dots, \hat{v}_{t-1}$  at  $a_{1}, a_{2}, \dots, a_{t-1}$ . This assumption is without loss of generality, since any random learners can be seen as a randomization over deterministic learning algorithms. Thus a lower bound which holds uniformly for any deterministic learner would also hold over a randomization. To lower bound equation 9, we use Lemma C.5, which relates this to the question of how informative are the query values (as measured by Kullback-Leibler divergence) for determining the sign of  $\theta$ 's coordinates. Intuitively, the more similar the query values are, the smaller is the KL divergence and the harder it is to distinguish the true sign of  $\theta_{i}$ , leading to a larger lower bound. In addition, we are facing a powerful adversary who
+
+has the complete knowledge of the problem and is able to add corruption on the query value to make they are even more similar, which resulting a even smaller KL divergence, consequently, an even larger lower bound. Let  $\bar{a}_T \coloneqq \frac{1}{T} \sum_{t=1}^{T} a_t$  represent the average action, we have
+
+$$
+\begin{array}{l} \mathbb {E} \left\{\sum_ {t = 1} ^ {T} \mu_ {\theta} \left(a ^ {*}\right) - \mu_ {\theta} \left(a _ {t}\right) \right\} = T \mathbb {E} \left\{\frac {1}{T} \sum_ {t = 1} ^ {T} \mu_ {\theta} \left(a ^ {*}\right) - \mu_ {\theta} \left(a _ {t}\right) \right\} \\ \geq T \mathbb {E} \left(\mu_ {\theta} \left(a ^ {*}\right) - \mu_ {\theta} (\bar {a} _ {T})\right) \\ \geq T \mathbb {E} \left(\frac {1}{2} \| \bar {a} _ {T} - \theta \| ^ {2}\right) \\ = T \mathbb {E} \left(\frac {1}{2} \sum_ {i = 1} ^ {d} \left(\bar {a} _ {i} - \theta_ {i}\right) ^ {2}\right) \\ \geq \mathbb {E} \left(\frac {\beta^ {2} T}{2} \sum_ {i = 1} ^ {d} \mathbb {I} _ {\bar {a} _ {i} \theta_ {i} <   0}\right) \tag {9} \\ \geq \frac {d T \beta^ {2}}{4} \left(1 - \sqrt {\frac {1}{d} \sum_ {i = 1} ^ {d} \sum_ {t = 1} ^ {T} \mathbb {D} _ {t , i}}\right). \\ \end{array}
+$$
+
+We get the second inequality by using the fact that  $\mu_{\theta}$  is 1-strongly concave. We get the last inequality by using Lemma C.5, which completes the proof.
+
+Lemma C.5 (Lemma 4 in (Shamir, 2013)). Let  $\theta$  be a random vector, none of those coordinates is supported on  $0$ . Let  $\hat{v}_1, \hat{v}_2, \ldots, \hat{v}_T$  be a sequence of values obtained by a deterministic learner returning a point  $\bar{a}_T$  (so that the action  $a_t$  is a deterministic function of  $\hat{v}_1, \ldots, \hat{v}_{t-1}$  and  $\bar{a}_T$  is a deterministic function of  $\hat{v}_1, \ldots, \hat{v}_T$ ). Then we have
+
+$$
+\mathbb {E} \left(\sum_ {i = 1} ^ {d} \mathbb {I} _ {\bar {a} _ {i} \theta_ {i}}\right) \geq \frac {d}{2} \left(1 - \sqrt {\frac {1}{d} \sum_ {i = 1} ^ {d} \sum_ {t = 1} ^ {T} \mathbb {D} _ {t , i}}\right),
+$$
+
+where  $\mathbb{D}_{t,i} = \sup_{\{\theta_j\}_{j\neq i}}D_{KL}\left(\mathbb{P}(\hat{v}_t|\theta_i > 0,\{\theta_j\}_{j\neq i},\{\hat{v}_l\}_{l = 1}^{t - l})||\mathbb{P}(\hat{v}_t|\theta_i < 0,\{\theta_j\}_{j\neq i},\{\hat{v}_l\}_{l = 1}^{t - l})\right)$ ,  $D_{KL}$  represents the KL divergence between two distributions.
+
+![](images/bac73c4d65468fde2d89d004737a246075f964220cec55b8e407337472138ab7.jpg)
+
+![](images/8264d61f69075a9bbb6fd9a5c176154427dc9232e0b6fa64ead6124d407529ae.jpg)
+
+# C.2 PROOF FOR PROPOSITION 1:
+
+Proposition (Proposition 1 restated). There exists an LIHF instance with  $\rho$ -Imperfect Human Feedback (Definition 2.1), linear user utility  $\mu$ , and link function  $\sigma$ , such that any learner has to suffer  $Reg_{T} \geq \Omega(d \max\{\sqrt{T}, T^{\rho}\})$ , even with the knowledge of  $\rho$ .
+
+Proof. The proof structure is very similar to the lower bound proof in Theorem 3.1. We start by discussing the value of  $\rho$ . Consider the scenario when  $\rho \leq 0.5$ . Applying Lemma C.6 and C.1 together with linear link function, we have  $\mathrm{Reg}_T \geq \Omega(d \max \{\sqrt{T}, T^\rho\})$ , which completes the proof.
+
+Lemma C.6 ((Dani et al., 2008)). Let  $\mathcal{A} = [-1,1]^d$  and  $\Theta = [-T^{-\frac{1}{2}}, T^{-\frac{1}{2}}]^d$ . Consider the linear reward function  $r_t = \theta^\top A_t + \epsilon_t$ ,  $\epsilon_t$  is independent standard Gaussian noise. Then for any learner, there exists a vector  $\theta \in \Theta$  such that
+
+$$
+R e g _ {T} (\mathcal {A}, \theta) \geq \frac {\exp (- 2)}{8} d \sqrt {T}.
+$$
+
+Now, let's focus on the scenario when  $\rho > 0.5$  and the essence is to extend Lemma C.6 to the scenario in presence of  $\rho$ -Imperfect Human Feedback, which is shown in Lemma C.7
+
+Lemma C.7. Assume that the action space  $\mathcal{A} \subset [-1,1]^d$ . Given  $\rho$ -Imperfect Human Feedback (Definition 2.1), for stochastic linear bandit, there exists a  $\theta \in [-C_{\kappa}T^{\rho - 1}, C_{\kappa}T^{\rho - 1}]^d$  such that any learner even with the knowledge of  $\rho$  has to suffer regret no less than  $\frac{d}{8} C_{\kappa}T^{\rho}$ .
+
+Proof. The proof extends Lemma C.6 to a scenario in presence of corruption. We want to construct a parameter family and a corruption strategy such that for all algorithms, it will occur at least  $\Omega(dT^{\rho})$  regret. Consider the action set  $\mathcal{A} \in [-1, 1]^d$  and  $\Theta = \{-\beta, \beta\}^d$ . For any learner, we can lower bound its regret by
+
+$$
+\begin{array}{l} \operatorname {R e g} _ {T} (\mathcal {A}, \theta) = \mathbb {E} _ {\theta} \left[ \sum_ {t = 1} ^ {T} \sum_ {i = 1} ^ {d} (\operatorname {s i g n} (\theta_ {i}) - a _ {t i}) \theta_ {i} \right] \\ \geq \beta \sum_ {i = 1} ^ {d} \mathbb {E} _ {\theta} \left[ \sum_ {t = 1} ^ {T} \mathbb {I} \{\operatorname {s i g n} \left(a _ {t i}\right) \neq \operatorname {s i g n} \left(\theta_ {i}\right) \} \right] \\ \geq \frac {T \beta}{2} \sum_ {i = 1} ^ {d} \mathbb {P} _ {\theta} \left(\sum_ {t = 1} ^ {T} \mathbb {I} \{\operatorname {s i g n} \left(a _ {t i}\right) \neq \operatorname {s i g n} \left(\theta_ {i}\right) \} \geq \frac {T}{2}\right) \\ \end{array}
+$$
+
+Let's denote
+
+$$
+p _ {\theta_ {i}} = \mathbb {P} _ {\theta} \left(\sum_ {t = 1} ^ {T} \mathbb {I} \{\operatorname {s i g n} \left(a _ {t i}\right) \neq \operatorname {s i g n} \left(\theta_ {i}\right) \} \geq \frac {T}{2}\right)
+$$
+
+Let  $i \in [d]$  and  $\theta \in \Theta$  be fixed, and let  $\theta_j' = \theta_j$ , for  $j \neq i$  and  $\theta_i' = -\theta_i$ . Then using Lemma C.8, we have
+
+$$
+p _ {\theta_ {i}} + p _ {\theta_ {i} ^ {\prime}} \geq \frac {1}{2} \exp \left(- \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \mathrm {D} _ {\mathrm {K L}} \left(P _ {a _ {t}}, P _ {a _ {t}} ^ {\prime}\right) \right]\right).
+$$
+
+$P_{a_t}$  is the distribution of corrupted reward observed by the learner after playing action  $a_t$  when reward parameter is  $\theta$ . Similarly,  $P_{a_t}'$  is the distribution of corrupted observed by the learner after playing action  $a_t$  when the reward parameter is  $\theta'$ .
+
+Lemma C.8 (Bretagnolle-Huber inequality). Let  $\mathbb{P}$  and  $\mathbb{Q}$  be probability measures on the same measurable space  $(\Omega, \mathcal{F})$ . Let  $A \in \mathcal{F}$  be any arbitrary event and  $A^c$  is the complement of  $A$ . Then we have
+
+$$
+\mathbb {P} (A) + \mathbb {Q} \left(A ^ {c}\right) \geq \frac {1}{2} \exp \left(- D _ {K L} (\mathbb {P}, \mathbb {Q})\right). \tag {10}
+$$
+
+In presence of adversarial corruption, for  $\theta$ , the corrupted reward is
+
+$$
+\hat {r} _ {t} = \sum_ {j \neq i} a _ {t j} \theta_ {j} + a _ {t i} \theta_ {i} + c _ {t} \left(a _ {t} | \theta\right) + \epsilon_ {t}. \tag {11}
+$$
+
+For  $\theta^{\prime}$  , we have
+
+$$
+\hat {r} _ {t} = \sum_ {j \neq i} a _ {t j} \theta_ {j} - a _ {t i} \theta_ {i} + c _ {t} \left(a _ {t} \mid \theta^ {\prime}\right) + \epsilon_ {t}. \tag {12}
+$$
+
+Consider the corruption strategy such that  $c_{t}(a_{t}|\theta) = -a_{t,i}\theta_{i}$  in equation 11 and  $c_{t}(a_{t}|\theta^{\prime}) = a_{t,i}\theta_{i}$  in equation 12. This is achievable since we assume the adversary has complete knowledge of the problem instance. By doing so, the corrupted reward  $\hat{r}_t$  are from the same distribution regardless whether the preference parameter is  $\theta$  or  $\theta^{\prime}$ . This is because  $\theta$  and  $\theta^{\prime}$  only differs in the  $i$ -th coordinate with the magnitude  $\beta$ , and this difference could be masked by the corruption, resulting in  $\mathbb{E}\left[\sum_{t=1}^{T}D_{\mathrm{KL}}(P_{a_t},P_{a_t}')\right] = 0$ , which makes it indistinguishable by the learner. Notice that executing such a corruption strategy requires total corruption budget
+
+$$
+\sum_ {t = 1} ^ {T} \left| c _ {t} \left(a _ {t}\right) \right| \leq \beta \sum_ {t = 1} ^ {T} \left\| a _ {t} \right\| _ {\infty} \leq C _ {\kappa} T ^ {\rho}. \tag {13}
+$$
+
+Choose  $\beta = C_{\kappa}T^{\rho -1}$ , Equation equation 13 holds. Therefore, we have
+
+$$
+p _ {\theta_ {i}} + p _ {\theta_ {i} ^ {\prime}} \geq \frac {1}{2}.
+$$
+
+Applying an "averaging hammer" over all  $\theta \in \Theta$ , which satisfies  $|\Theta| = 2^d$ , we get
+
+$$
+\sum_ {\theta \in \Theta} \frac {1}{| \Theta |} \sum_ {i = 1} ^ {d} p _ {\theta_ {i}} = \frac {1}{| \Theta |} \sum_ {i = 1} ^ {d} \sum_ {\theta \in \Theta} p _ {\theta_ {i}} \geq \frac {d}{4}.
+$$
+
+This implies that there exists a  $\theta \in \Theta$  such that  $\sum_{i=1}^{d} p_{\theta_i} \geq \frac{d}{4}$ . Therefore we have
+
+$$
+\operatorname {R e g} _ {T} (\mathcal {A}, \theta) \geq \frac {d C _ {\kappa}}{8} T ^ {\rho},
+$$
+
+which completes the proof. We want to highlight that the corruption level each round  $|c_{t}(a_{t})|$  is bounded by  $\beta \| a_{t}\|_{\infty} \leq C_{\kappa}T^{\rho -1} \leq C_{\kappa}t^{\rho -1}, \forall t$ , satisfying definition of  $\rho$ -Imperfect Human Feedback.
+
+![](images/951838e87277a5cfc4dee4c05f3985875fe0fd71aa28b1ee3b22d5e7ff99d87f.jpg)
+
+# D PROOF FOR THEOREM 4.1
+
+Theorem (Theorem 4.1 restated). RoSMID satisfies  $Reg_{T} \leq \tilde{\mathcal{O}}(d\max\{\sqrt{T}, T^{\rho}\})$  for any  $\rho$ -LIHF problem.
+
+Proof. At the beginning of the proof, we formally define self-concordance (see Definition D.1). The input,  $\nu$ -self-concordant function  $\mathcal{R}$ , serves as a regularizer in RoSMID (see Algorithm 1).
+
+Definition D.1. (Self-concordance.) A function  $\mathcal{R} : \operatorname{int}(\mathcal{A}) \to \mathbb{R}$  is self-concordant if it satisfies
+
+1.  $\mathcal{R}$  is three times continuously differentiable, convex, and approaches infinity along any sequence of points approaching the boundary of int(A).  
+2. For every  $h \in \mathbb{R}^d$  and  $x \in \text{int}(\mathcal{A})$ ,  $|\nabla^3\mathcal{R}(x)[h,h,h]| \leq 2(h^\top\nabla^2\mathcal{R}(x)h)^{\frac{3}{2}}$  holds, where  $|\nabla^3\mathcal{R}(x)[h,h,h] := \frac{\partial^3\mathcal{R}}{\partial t_1\partial t_2\partial t_3}(x + t_1h + t_2h + t_3h)|_{t_1 = t_2 = t_3 = 0}$ .
+
+In addition to these two conditions, if for every  $h \in \mathbb{R}^d$  and  $x \in \operatorname{int}(\mathcal{A})$ ,  $|\nabla \mathcal{R}(x)^\top h| \leq \nu^{\frac{1}{2}}(h^\top \nabla^2 \mathcal{R}(x)h)^{\frac{1}{2}}$  for a positive real number  $\nu$ ,  $\mathcal{R}$  is  $\nu$ -self-concordant.
+
+Additionally, we remind the reader of the standard assumptions (Yue & Joachims, 2009; Kumagai, 2017) used in our analysis.
+
+1. the action set  $\mathcal{A}$  is a convex compact set that contains the origin, has non-empty interior, and is contained in a  $d$ -dimensional ball of radius  $R$ ;  
+2. the utility function  $\mu : \mathcal{A} \to \mathbb{R}$  is strongly concave and twice-differentiable. The following constants are useful for our algorithm analysis:  $\mu$  is  $L^{\mu}$ -Lipschitz,  $\alpha$ -strongly concave,  $\kappa$ -smooth, bounded  $R^{\mu} \coloneqq \sup_{a,a'} \mu(a) - \mu(a')$ , and  $a^{*} \coloneqq \arg \max_{a \in \mathcal{A}} \mu(a)$  is the unique optimal within  $\mathcal{A}$ ;  
+3. the link function  $\sigma : \mathbb{R} \to [0,1]$  is smooth, rotation-symmetric (i.e.  $\sigma(x) = 1 - \sigma(-x)$ ) and concave for any  $x \geq 0$ . For the ease of analysis: let  $l_1^\sigma$  [resp.  $L_1^\sigma$ ] denote the lower [resp. upper] bound of the first-order derivative of  $\sigma$ .  $\sigma$  is  $L^{\sigma}$ -Lipschitz and its second-order derivative  $\sigma''$  is  $L_2^\sigma$ -Lipschitz and upper bounded by  $R_2^\sigma$ .
+
+Under these assumption, Lemma D.1 implies  $\mathrm{Reg}_T$  defined in equation 1 and  $\mathrm{Reg}_T^{\mathrm{FO}}$  defined in equation 3 has the same order.
+
+Lemma D.1 (Lemma 12 in Kumagai (2017)). With  $Reg_T^{FO}$  defined in equation 3, we have
+
+$$
+\frac {R e g _ {T}}{L _ {1} ^ {\sigma}} \leq R e g _ {T} ^ {F O} \leq \frac {R e g _ {T}}{l _ {1} ^ {\sigma}}. \tag {14}
+$$
+
+In the following, we show the proof. The main proof can be divided into four key steps. Step 1 is natural, which quantifies the bias in gradient estimation by RoSMID due imperfect feedback from corrupted utilities. Building on this, Step 2 develops a regret decomposition lemma forueling bandits under corruption, decomposing the regret into two components: regret from sub-optimal decisions and regret from feedback error. Techniques from previous analyses of continuousueling bandits handle the sub-optimal decision regret, thus in Step 3, we focus on deriving new techniques to bound the regret from feedback error, a unique challenge in our setting. We identify a tight upper bound for this error. Finally, Step 4, the only step relying on the decaying corruption level in  $\rho$ -imperfect human feedback, uses this assumption, along with the carefully chosen learning rate  $\eta_{\rho}$  and an iterative regret refinement process, to ensure the tightness of the analysis in Step 3, thus completing the proof.
+
+# Step 1: quantifying bias of gradient estimation in  $\rho$ -LIHF
+
+Our proof starts from quantifying the bias of the gradient  $\hat{g}_t$  caused by  $\rho$ -imperfect human feedback, as shown in the following Lemma 4.1.
+
+Lemma (Lemma 4.1 restated). The gradient  $\hat{g}_t$  in Line 6 of Algorithm 1 satisfies
+
+$$
+\mathbb {E} \left(\hat {g} _ {t} \mid a _ {t}\right) = \nabla_ {a = a _ {t}} \bar {P} _ {t} (a) - \mathbb {E} \left(b _ {t} \mid a _ {t}\right).
+$$
+
+where  $\bar{P}_t(a) := \mathbb{E}_{x \in \mathbb{B}}\left[\mathbb{P}(a_t \succ a + \nabla^2 R_t(a_t)^{-\frac{1}{2}} x)\right]$  (with  $\mathbb{B}$  as the unit ball) is the smoothed probability that  $a_t$  is preferred over any action  $a$ , and  $b_t = d\left(\mathbb{P}(a_t \succ a_t') - \hat{\mathbb{P}}(a_t \succ a_t')\right) \nabla^2 \mathcal{R}_t(a_t)^{\frac{1}{2}} u_t$ .
+
+Proof. If we do not have corruption (i.e. the probability for observing noisy comparative feedback  $\mathcal{F}(a, a') = 1$  is based on true cost difference  $\mu(a) - \mu(a')$ ), then the uncorrupted gradient  $g_t \coloneqq \mathcal{F}(a_t', a_t) d\nabla^2 \mathcal{R}_t(a_t)^{1/2} u_t$  should be an unbiased estimate of  $\nabla \bar{P}_t(a_t)$ , i.e.
+
+$$
+\mathbb {E} \left(g _ {t} \mid a _ {t}\right) = \nabla \bar {P} _ {t} (a _ {t}).
+$$
+
+We use  $\nabla \bar{P}_t(a_t)$  as a shorthand of  $\nabla|_{a=a_t} \bar{P}_t(a)$ . We use  $P_t(a)$  as a shorthand of  $P_t(a) := \sigma(\mu(a_t) - \mu(a))$  equivalent to  $\mathbb{P}(a_t \succ a)$ , and  $\hat{P}_t(a) := \sigma(\mu(a_t) - \mu(a) + c_t(a_t, a_t'))$ , equivalent to  $\hat{\mathbb{P}}(a_t \succ a)$  under our modelling assumption. The proof is similar to Lemma 2.1 in Flaxman et al. (2005). Using the Law of total expectation, we have
+
+$$
+\begin{array}{l} \mathbb {E} \left(g _ {t} \mid a _ {t}\right) = \mathbb {E} _ {u _ {t}} \left[ \mathbb {E} \left(g _ {t} \mid a _ {t}, u _ {t}\right) \right] \\ = \mathbb {E} _ {u _ {t}} \left(d \mathbb {E} \left(P _ {t} \left(a _ {t} + \nabla^ {2} \mathcal {R} _ {t} \left(a _ {t}\right) ^ {- \frac {1}{2}} u _ {t}\right) \nabla^ {2} \mathcal {R} _ {t} \left(a _ {t}\right) ^ {\frac {1}{2}} u _ {t} \mid a _ {t}, u _ {t}\right)\right) \\ = d \mathbb {E} \left(P _ {t} \left(a _ {t} + \nabla^ {2} \mathcal {R} _ {t} \left(a _ {t}\right) ^ {- \frac {1}{2}} u _ {t}\right) \nabla^ {2} \mathcal {R} _ {t} \left(a _ {t}\right) ^ {\frac {1}{2}} u _ {t} \mid a _ {t}\right) \\ = \nabla \mathbb {E} _ {x \in \mathbb {B}} \left(P _ {t} \left(a _ {t} + \nabla^ {2} \mathcal {R} _ {t} \left(a _ {t}\right) ^ {- \frac {1}{2}} x \mid a _ {t}\right)\right) \\ = \nabla \bar {P} _ {t} (a _ {t}). \\ \end{array}
+$$
+
+We get the second inequality by using the definition of  $\mathcal{F}(a_t', a_t)$ . We use the Stroke's Theorem to get the second last equality. The gradient which we get to perform gradient descent  $\hat{g}_t$  is corrupted. If we let  $a_t' = a_t + \nabla^2 \mathcal{R}_t(a_t)^{-\frac{1}{2}} u_t$ , we have
+
+$$
+\begin{array}{l} \mathbb {E} \left(\hat {g} _ {t} \mid a _ {t}\right) = \mathbb {E} _ {u _ {t}} \left(\mathbb {E} \left(\hat {g} _ {t} \mid a _ {t}, u _ {t}\right)\right) \\ = \mathbb {E} _ {u _ {t}} \left(d \mathbb {E} (\hat {P} _ {t} (a _ {t} ^ {\prime}) \nabla^ {2} \mathcal {R} _ {t} (a _ {t}) ^ {\frac {1}{2}} u _ {t} | a _ {t}, u _ {t})\right) \\ = d \mathbb {E} \left(\hat {P} _ {t} \left(a _ {t} ^ {\prime}\right) \nabla^ {2} \mathcal {R} _ {t} \left(a _ {t}\right) ^ {\frac {1}{2}} u _ {t} \mid a _ {t}\right) \\ = d \mathbb {E} \left(\left[ P _ {t} \left(a _ {t} ^ {\prime}\right) + \hat {P} _ {t} \left(a _ {t} ^ {\prime}\right) - P _ {t} \left(a _ {t} ^ {\prime}\right) \right] \nabla^ {2} R _ {t} \left(a _ {t}\right) ^ {\frac {1}{2}} u _ {t} \mid a _ {t}\right) \\ = \nabla \mathbb {E} _ {x \in \mathbb {B}} \left(P _ {t} \left(a _ {t} + \nabla^ {2} \mathcal {R} _ {t} \left(a _ {t}\right) ^ {- \frac {1}{2}} x \mid a _ {t}\right)\right) + d \mathbb {E} \left[ \left(\hat {P} _ {t} \left(a _ {t} ^ {\prime}\right) - P _ {t} \left(a _ {t} ^ {\prime}\right)\right) \nabla^ {2} \mathcal {R} _ {t} \left(a _ {t}\right) ^ {\frac {1}{2}} u _ {t} \mid a _ {t} \right] \\ = \nabla \bar {P} _ {t} (a _ {t}) + d \mathbb {E} \left[ \left(\hat {P} _ {t} \left(a _ {t} ^ {\prime}\right) - P _ {t} \left(a _ {t} ^ {\prime}\right)\right) \nabla^ {2} \mathcal {R} _ {t} \left(a _ {t}\right) ^ {\frac {1}{2}} u _ {t} | a _ {t} \right] \\ = \mathbb {E} (g _ {t} | a _ {t}) + d \mathbb {E} \left[ \left(\hat {P} _ {t} (a _ {t} ^ {\prime}) - P _ {t} (a _ {t} ^ {\prime})\right) \nabla^ {2} \mathcal {R} _ {t} (a _ {t}) ^ {\frac {1}{2}} u _ {t} | a _ {t} \right]. \\ \end{array}
+$$
+
+We get the second last equality by using the fact that  $g_{t}$  is unbiased, which we proved earlier. If we defined  $b_{t}$  as
+
+$$
+b _ {t} := d \Big (P _ {t} (a _ {t} + \nabla^ {2} \mathcal {R} _ {t} (a _ {t}) ^ {- \frac {1}{2}} u _ {t}) - \hat {P} _ {t} (a _ {t} + \nabla^ {2} \mathcal {R} _ {t} (a _ {t}) ^ {- \frac {1}{2}} u _ {t}) \Big) \nabla^ {2} \mathcal {R} _ {t} (a _ {t}) ^ {\frac {1}{2}} u _ {t},
+$$
+
+we get
+
+$$
+\mathbb {E} \left(g _ {t} \mid a _ {t}\right) = \mathbb {E} \left(\hat {g} _ {t} \mid a _ {t}\right) + \mathbb {E} \left[ b _ {t} \mid a _ {t} \right].
+$$
+
+![](images/31435c112ef4e212a5b665a45cff6bd9b81c6dcceffee332864dfdf5f079583f.jpg)
+
+# Step 2: decomposing regret for dueling bandits into regret of decision and feedback error
+
+Core to our proof is a "regret decomposition" lemma below that upper bounds the regret by two parts: regret of sub-optimal decision making under uncertainty, referred as regret of decision (first term in equation 2), and feedback error (second term in equation 2).
+
+Lemma (Lemma 4.2 restated). The Reg $_T$  of Algorithm 1 satisfies:
+
+$$
+R e g _ {T} \leq \underbrace {\frac {C _ {0}}{\eta_ {\rho}} \log (T) + 8 d ^ {2} \eta_ {\rho} T + 2 L ^ {\mu} L _ {1} ^ {\sigma} R} _ {R e g r e t o f D e c i s i o n} + \underbrace {2 \mathbb {E} \left\{\sum_ {t = 1} ^ {T} b _ {t} ^ {\top} (a _ {t} - a _ {T} ^ {*}) \right\}} _ {F e e d b a c k E r r o r},
+$$
+
+where  $C_0 \coloneqq (2\nu + 4L_1^\sigma \kappa + 4R_2^\sigma (L^\mu)^2 + L^\mu L_1^\sigma \kappa) / \lambda$ , and  $a_T^* \coloneqq \frac{1}{T} a_1 + (1 - \frac{1}{T})a^*$ .
+
+Proof. The cornerstone of the analysis below is to separate the impact of  $b_{t}$  from the total regret.
+
+$$
+\begin{array}{l} \operatorname {R e g} _ {T} \leq 2 \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \left(P _ {t} \left(a _ {t}\right) - P _ {t} \left(a _ {T} ^ {*}\right)\right) \right] + \frac {L ^ {\mu} L _ {1} ^ {\sigma} \kappa}{\lambda \eta_ {\rho}} + 2 L ^ {\mu} L _ {1} ^ {\sigma} R \\ \leq 2 \left(\mathbb {E} \left\{\sum_ {t = 1} ^ {T} \left(\bar {P} _ {t} \left(a _ {t}\right) - \bar {P} _ {t} \left(a _ {T} ^ {*}\right)\right) \right\} + \mathbb {E} \left\{\sum_ {t = 1} ^ {T} \left(P _ {t} \left(a _ {t}\right) - \bar {P} _ {t} \left(a _ {t}\right)\right) \right\} + \mathbb {E} \left\{\sum_ {t = 1} ^ {T} \left(\bar {P} _ {t} \left(a _ {T} ^ {*}\right) - P _ {t} \left(a _ {T} ^ {*}\right)\right) \right\}\right) + \frac {L ^ {\mu} L _ {1} ^ {\sigma} \kappa}{\lambda \eta_ {\rho}} + 2 L ^ {\mu} L _ {1} ^ {\sigma} R \\ \leq 2 \mathbb {E} \left\{\sum_ {t = 1} ^ {T} \left(\bar {P} _ {t} \left(a _ {t}\right) - \bar {P} _ {t} \left(a _ {T} ^ {*}\right)\right) \right\} + \frac {4 L _ {1} ^ {\sigma} \kappa + 4 R _ {2} ^ {\sigma} \left(L ^ {\mu}\right) ^ {2} + L ^ {\mu} L _ {1} ^ {\sigma} \kappa}{\lambda \eta_ {\rho}} \log T + 2 L ^ {\mu} L _ {1} ^ {\sigma} R. \\ \end{array}
+$$
+
+We get the first inequality by using Lemma D.2. We get the last inequality by because
+
+$$
+\bar {P} _ {t} (a) - P _ {t} (a) \leq \frac {L _ {1} ^ {\sigma} \kappa + R _ {2} ^ {\sigma} (L ^ {\mu}) ^ {2}}{2} \| \nabla^ {2} \mathcal {R} _ {t} (a _ {t}) ^ {- \frac {1}{2}} u _ {t} \| ^ {2} \leq \frac {L _ {1} ^ {\sigma} \kappa + R _ {2} ^ {\sigma} (L ^ {\mu}) ^ {2}}{\lambda \eta_ {\rho} t},
+$$
+
+and
+
+$$
+\mathbb {E} \left\{\sum_ {t = 1} ^ {T} \left(P _ {t} \left(a _ {t}\right) - P _ {t} \left(a _ {T} ^ {*}\right)\right) \right\} \leq \mathbb {E} \left\{\sum_ {t = 1} ^ {T} \left(\bar {P} _ {t} \left(a _ {t}\right) - \bar {P} _ {t} \left(a _ {T} ^ {*}\right)\right) \right\} + 2 \frac {L _ {1} ^ {\sigma} \kappa + R _ {2} ^ {\sigma} \left(L ^ {\mu}\right) ^ {2}}{\lambda \eta_ {\rho}} \log T.
+$$
+
+Lemma D.2 (Lemma 5 in Kumagai (2017)).
+
+$$
+R e g _ {T} \leq 2 \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \left(P _ {t} \left(a _ {t}\right) - P _ {t} \left(a _ {T} ^ {*}\right)\right) \right] + \frac {L ^ {\mu} L _ {1} ^ {\sigma} \kappa}{\lambda \eta_ {\rho}} + 2 L ^ {\mu} L _ {1} ^ {\sigma} R.
+$$
+
+Then it remains to bound  $\mathbb{E}\{\sum_{t=1}^{T} \bar{P}_t(a_t) - \bar{P}_t(a_T^*)\}$ . And we have the following.
+
+$$
+\begin{array}{l} \mathbb {E} \left\{\bar {P} _ {t} \left(a _ {t}\right) - \bar {P} _ {t} \left(a _ {T} ^ {*}\right) \right\} \leq \mathbb {E} \left[ \nabla \hat {P} _ {t} ^ {\top} \left(a _ {t}\right) \left(a _ {t} - a _ {T} ^ {*}\right) - \frac {L _ {1} ^ {\sigma} \alpha}{4} \| a _ {t} - a _ {T} ^ {*} \| ^ {2} \right] \\ = \mathbb {E} \left[ g _ {t} ^ {\top} \left(a _ {t} - a _ {T} ^ {*}\right) - \frac {L _ {1} ^ {\sigma} \alpha}{4} \| a _ {t} - a _ {T} ^ {*} \| ^ {2} \right] \\ = \mathbb {E} \left[ \left(\hat {g} _ {t} + b _ {t}\right) ^ {\top} \left(a _ {t} - a _ {T} ^ {*}\right) - \frac {L _ {1} ^ {\sigma} \alpha}{4} \| a _ {t} - a _ {T} ^ {*} \| ^ {2} \right] \\ = \mathbb {E} \left[ \hat {g} _ {t} ^ {\top} (a _ {t} - a _ {T} ^ {*}) - \frac {L _ {1} ^ {\sigma} \alpha}{4} \| a _ {t} - a _ {T} ^ {*} \| ^ {2} \right] + \mathbb {E} \{b _ {t} ^ {\top} (a _ {t} - a _ {T} ^ {*}) \}, \\ \end{array}
+$$
+
+where the first and second equality is resulted from Lemma 4.1. Using the definition of  $a_{t+1}$  in Algorithm 1, we have
+
+$$
+\nabla \mathcal {R} _ {t} (a _ {t + 1}) - \nabla \mathcal {R} _ {t} (a _ {t}) = \eta_ {\rho} \hat {g} _ {t}.
+$$
+
+Therefore we have
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \hat {g} _ {t} ^ {\top} (a _ {t} - a _ {T} ^ {*}) - \frac {L _ {1} ^ {\sigma} \alpha}{4} \| a _ {t} - a _ {T} ^ {*} \| ^ {2} \right] = \frac {1}{\eta_ {\rho}} \mathbb {E} \left[ (\nabla \mathcal {R} _ {t} (a _ {t + 1}) - \nabla \mathcal {R} _ {t} (a _ {t})) ^ {\top} (a _ {t} - a _ {T} ^ {*}) - \frac {L _ {1} ^ {\sigma} \alpha \eta_ {\rho}}{4} \| a _ {t} - a _ {T} ^ {*} \| ^ {2} \right] \\ = \frac {1}{\eta_ {\rho}} \mathbb {E} \left[ D _ {\mathcal {R} _ {t}} (a _ {T} ^ {*}, a _ {t}) + D _ {\mathcal {R}} (a _ {t}, a _ {t + 1}) - D _ {\mathcal {R} _ {t}} (a _ {T} ^ {*}, a _ {t + 1}) - \frac {L _ {1} ^ {\sigma} \alpha \eta_ {\rho}}{4} \| a _ {t} - a _ {T} ^ {*} \| ^ {2} \right]. \\ \end{array}
+$$
+
+$D_{\mathcal{R}}(a,b)$  is the Bregman divergence associated with  $\mathcal{R}$ , defined by
+
+$$
+D _ {\mathcal {R}} (a, b) = \mathcal {R} (a) - \mathcal {R} (b) - \nabla \mathcal {R} (b) ^ {\top} (a - b).
+$$
+
+Then we can get equation 15 by using Lemma D.3, D.4, D.5.
+
+$$
+\mathbb {E} \left\{\sum_ {t = 1} ^ {T} \left(\bar {P} _ {t} \left(a _ {t}\right) - \bar {P} _ {t} \left(a _ {T} ^ {*}\right)\right) \right\} \leq \frac {\nu \log (T)}{\eta_ {\rho}} + 4 d ^ {2} \eta_ {\rho} T + \mathbb {E} \left\{\sum_ {t = 1} ^ {T} b _ {t} ^ {\top} \left(a _ {t} - a _ {T} ^ {*}\right) \right\}. \tag {15}
+$$
+
+Lemma D.3 (Lemma 10 in Kumagai (2017)). Let  $\mathcal{R}_t^* (a) = \sup_{x\in \mathbb{R}^d}x^\top a - \mathcal{R}_t(a)$  denote the Frenchel dual of  $\mathcal{R}_t$ . Then we have
+
+$$
+\sum_ {t = 1} ^ {T} \mathbb {E} \left[ \hat {g} _ {t} ^ {\top} (a _ {t} - a _ {T} ^ {*}) - \frac {L _ {1} ^ {\sigma} \alpha}{4} \| a _ {t} - a _ {T} ^ {*} \| ^ {2} \right] \leq \frac {1}{\eta_ {\rho}} \left(\mathcal {R} (a _ {T} ^ {*}) - \mathcal {R} (a _ {1}) + \mathbb {E} \left[ \sum_ {t = 1} ^ {T} D _ {\mathcal {R} _ {t} ^ {*}} (\nabla \mathcal {R} _ {t} (a _ {t}) - \eta_ {\rho} \hat {g} _ {t}, \nabla \mathcal {R} _ {t} (a _ {t})) \right]\right).
+$$
+
+Lemma D.4 (Lemma 11 in Kumagai (2017)). When  $\eta_{\rho} \leq \frac{1}{2d}$ , we have
+
+$$
+D _ {\mathcal {R} _ {t} ^ {*}} \left(\nabla \mathcal {R} _ {t} (a _ {t}) - \eta_ {\rho} \hat {g} _ {t}, \nabla \mathcal {R} _ {t} (a _ {t})\right) \leq 4 d ^ {2} \eta_ {\rho} ^ {2}.
+$$
+
+Lemma D.5 (Lemma 4 in Hazan & Levy (2014)).  $\mathcal{R}(a_T^*) - \mathcal{R}(a_1) \leq \nu \log(T)$ .
+
+If we let  $C_0 \coloneqq (2\nu + 4L_1^\sigma \kappa + 4R_2^\sigma (L^\mu)^2 + L^\mu L_1^\sigma \kappa) / \lambda$ , we have
+
+$$
+\begin{array}{l} \operatorname {R e g} _ {T} \leq \frac {2 \nu \log (T)}{\eta_ {\rho}} + 8 d ^ {2} \eta_ {\rho} T + 2 \mathbb {E} \left\{\sum_ {t = 1} ^ {T} b _ {t} ^ {\top} \left(a _ {t} - a _ {T} ^ {*}\right) \right\} + \frac {4 L _ {1} ^ {\sigma} \kappa + 4 R _ {2} ^ {\sigma} \left(L ^ {\mu}\right) ^ {2} + L ^ {\mu} L _ {1} ^ {\sigma} \kappa}{\lambda \eta_ {\rho}} \log T + 2 L ^ {\mu} L _ {1} ^ {\sigma} R \\ = \underbrace {\frac {C _ {0}}{\eta_ {\rho}} \log (T) + 8 d ^ {2} \eta_ {\rho} T + 2 L ^ {\mu} L _ {1} ^ {\sigma} R} _ {R e g r e t o f D e c i s i o n} + \underbrace {2 \mathbb {E} \left\{\sum_ {t = 1} ^ {T} b _ {t} ^ {\top} (a _ {t} - a _ {T} ^ {*}) \right\}} _ {F e e d b a c k E r r o r}, \\ \end{array}
+$$
+
+which completes the proof.
+
+![](images/3114d8ca6ce409db7a0c18e2eb6a435b882a7691acf946fa7b33d9090bef0309.jpg)
+
+# Step 3: upper bounding the regret from feedback error
+
+Bounding the regret of decision term in equation 2 is standard. Thus, this step develops a novel upper bound for bounding the regret from feedback error, as shown below.
+
+Lemma (Lemma 4.3 restated). The feedback error term in equation 2 can be bounded as follows:
+
+$$
+\mathbb {E} \left\{\sum_ {t = 1} ^ {T} b _ {t} ^ {\top} (a _ {t} - a _ {T} ^ {*}) \right\} \leq C _ {1} d \sqrt {\eta_ {\rho}} \mathbb {E} \left\{\sum_ {t = 1} ^ {T} \sqrt {t} | c _ {t} (a _ {t}, a _ {t} ^ {\prime}) | \sqrt {\mu (a ^ {*}) - \mu (a _ {t})} \right\} + C _ {2} d T ^ {\rho} + 2 d \sqrt {\lambda} \sqrt {\eta_ {\rho}} T,
+$$
+
+where  $C_1 \coloneqq \sqrt{\frac{2\lambda(L^\sigma)^2}{\alpha}}$ , and  $C_2 \coloneqq 2L^\sigma R\sqrt{\sup_{a \in \mathcal{A}} \lambda_{\max}(\nabla^2\mathcal{R}(a)) + 2\phi}$ .
+
+Proof. In the following, we will analyze the expected cumulative sum of the bias  $b_{t}$  on the total regret, which is  $\mathbb{E}\left\{\sum_{t = 1}^{T}b_{t}^{\top}(a_{t} - a_{T}^{*})\right\}$ . By Cauchy-Schwartz inequality,  $b_{t}^{\top}(a_{t} - a_{T}^{*}) \leq \| b_{t}\|_{2}\| (a_{t} - a_{T}^{*})\|_{2}$ . We can bound the  $\| b_{t}\|_{2}$  by the following.
+
+$$
+\begin{array}{l} b _ {t} ^ {\top} b _ {t} = d ^ {2} \left(P _ {t} (a _ {t} + \nabla^ {2} \mathcal {R} _ {t} (a _ {t}) ^ {- \frac {1}{2}} u _ {t}) - \hat {P} _ {t} (a _ {t} + \nabla^ {2} \mathcal {R} _ {t} (a _ {t}) ^ {- \frac {1}{2}} u _ {t})\right) ^ {2} u _ {t} ^ {\top} \nabla^ {2} \mathcal {R} _ {t} (a _ {t}) u _ {t} \\ \leq \Big (P _ {t} (a _ {t} + \nabla^ {2} \mathcal {R} _ {t} (a _ {t}) ^ {- \frac {1}{2}} u _ {t}) - \hat {P} _ {t} (a _ {t} + \nabla^ {2} \mathcal {R} _ {t} (a _ {t}) ^ {- \frac {1}{2}} u _ {t}) \Big) ^ {2} d ^ {2} \lambda_ {\max} (\nabla^ {2} \mathcal {R} _ {t} (a _ {t})). \\ \end{array}
+$$
+
+We get the first equality by Lemma 4.1. We get the inequality by using the fact  $\| u_t\| _2 = 1$ . If we let  $a_{t}^{\prime} = a_{t} + \nabla^{2}\mathcal{R}_{t}(a_{t})^{-\frac{1}{2}}u_{t}$ , then we have
+
+$$
+\begin{array}{l} \left| P _ {t} \left(a _ {t} ^ {\prime}\right) - \hat {P} _ {t} \left(a _ {t} ^ {\prime}\right) \right| = \left| \sigma \left(f \left(a _ {t}\right) - f \left(a _ {t} ^ {\prime}\right)\right) - \sigma \left(f \left(a _ {t}\right) - f \left(a _ {t} ^ {\prime}\right) + c _ {t} \left(a _ {t}, a _ {t} ^ {\prime}\right)\right) \right| \\ \leq \min  \left(2, L ^ {\sigma} \left| c _ {t} \left(a _ {t}, a _ {t} ^ {\prime}\right) \right|\right). \\ \end{array}
+$$
+
+We get the first equality by using the definition of  $P_{t}(a_{t})$  and  $\hat{P}_{t}(a_{t})$ . We get the first inequality by using the Lipschitz property of  $\sigma$ . Therefore, we have
+
+$$
+\left\| b _ {t} \right\| _ {2} \leq d \sqrt {\lambda_ {\max } \left(\nabla^ {2} \mathcal {R} _ {t} \left(a _ {t}\right)\right)} \min  \left\{2, L ^ {\sigma} \left| c _ {t} \left(a _ {t}, a _ {t} ^ {\prime}\right) \right| \right\}.
+$$
+
+If we use  $\lambda_R^* \coloneqq \sup_{a \in \mathcal{A}} \lambda_{\max}(\nabla^2 \mathcal{R}(a))$ , then we have
+
+$$
+\lambda_ {\max } \left(\nabla^ {2} \mathcal {R} _ {t} \left(a _ {t}\right)\right) = \lambda_ {\max } \left(\nabla^ {2} \mathcal {R} \left(a _ {t}\right)\right) + \lambda \eta_ {\rho} t + 2 \phi \leq \lambda_ {R} ^ {*} + 2 \phi + \lambda \eta_ {\rho} t.
+$$
+
+Therefore we have
+
+$$
+\begin{array}{l} \mathbb {E} \left\{\sum_ {t = 1} ^ {T} b _ {t} ^ {\top} \left(a _ {t} - a _ {T} ^ {*}\right) \right\} \leq \mathbb {E} \left\{\sum_ {t = 1} ^ {T} \| b _ {t} \| _ {2} \| a _ {t} - a _ {T} ^ {*} \| _ {2} \right\} \\ \leq \mathbb {E} \left\{\sum_ {t = 1} ^ {T} (d \sqrt {\lambda_ {\max } (\nabla^ {2} \mathcal {R} _ {t} (a _ {t}))} \min  (2, L ^ {\sigma} | c _ {t} (a _ {t}, a _ {t} ^ {\prime}) |)) \| a _ {t} - a _ {T} ^ {*} \| _ {2} \right\} \\ \leq \mathbb {E} \left\{\sum_ {t = 1} ^ {T} \left(d \sqrt {\lambda_ {R} ^ {*} + 2 \phi + \lambda \eta_ {\rho} t} \min  \left(2, L ^ {\sigma} | c _ {t} \left(a _ {t}, a _ {t} ^ {\prime}\right) |\right)\right) \| a _ {t} - a _ {T} ^ {*} \| _ {2} \right\} \\ \leq \mathbb {E} \left\{ \right.\sum_ {t = 1} ^ {T} \left( \right.d \sqrt {\lambda_ {R} ^ {*} + 2 \phi} \min  \left(2, L ^ {\sigma} | c _ {t} \left(a _ {t}, a _ {t} ^ {\prime}\right) |\right) \| a _ {t} - a _ {T} ^ {*} \| _ {2} \left. \right\} + \mathbb {E} \left\{ \right.\sum_ {t = 1} ^ {T} \left( \right.d \sqrt {\lambda \eta_ {\rho} t} \min  \left(2, L ^ {\sigma} | c _ {t} \left(a _ {t}, a _ {t} ^ {\prime}\right) |\right) \| a _ {t} - a _ {T} ^ {*} \| _ {2} \left. \right\}\left. \right. \\ \leq 2 R d L ^ {\sigma} \sqrt {\lambda_ {R} ^ {*} + 2 \phi} T ^ {\rho} + d \sqrt {\lambda \eta_ {\rho}} \mathbb {E} \left\{\sum_ {t = 1} ^ {T} \sqrt {t} \min  \left(2, L ^ {\sigma} \left| c _ {t} \left(a _ {t}, a _ {t} ^ {\prime}\right) \right|\right) \| a _ {t} - a _ {T} ^ {*} \| _ {2} \right\} (16) \\ \leq 2 R d L ^ {\sigma} \sqrt {\lambda_ {R} ^ {*} + 2 \phi} T ^ {\rho} + 2 d \sqrt {\lambda \eta_ {\rho} T} + d \sqrt {\lambda \eta_ {\rho}} \mathbb {E} \left\{\sum_ {t = 1} ^ {T} \sqrt {t} \min  \left(2, L ^ {\sigma} | c _ {t} \left(a _ {t}, a _ {t} ^ {\prime}\right) |\right) \| a _ {t} - a ^ {*} \| _ {2} \right\} (17) \\ \leq 2 R d L ^ {\sigma} \sqrt {\lambda_ {R} ^ {*} + 2 \phi} T ^ {\rho} + 2 d \sqrt {\lambda \eta_ {\rho} T} + d \sqrt {\lambda \eta_ {\rho}} \mathbb {E} \left\{\sum_ {t = 1} ^ {T} \sqrt {t} \min  \left(2, L ^ {\sigma} | c _ {t} \left(a _ {t}, a _ {t} ^ {\prime}\right) |\right) \min  \left(2 R, \sqrt {\frac {2}{\alpha} \left(\mu \left(a ^ {*}\right) - \mu \left(a _ {t}\right)\right)}\right) \right\}. (18) \\ \end{array}
+$$
+
+We get equation 16 by using the fact that  $\| a_t - a_T^* \|_2 \leq 2R$ ,  $\sum_{t=1}^{T} |c_t(a_t, a_t')| \leq T^\rho$ . We get equation 17 the definition of  $a_T^*$ , and the fact  $\sqrt{a + b} \leq \sqrt{a} + \sqrt{b}$ . Using the  $\alpha$ -strong convexity of  $\mu$ , we have  $\| a_t - a_* \|_2 \leq \min \left( 2R, \sqrt{\frac{2}{\alpha} (\mu(a^*) - \mu(a_t))} \right)$  we get the last inequality, which completes the proof.
+
+# Step 4: The iterative regret refinement analysis and resultant optimal learning rate
+
+Lemma (Lemma 4.4 restated). If  $A = \mathbb{E}\left(\sum_{t=1}^{T}[\mu(a^{*}) - \mu(a_{t})]\right) \leq \mathcal{O}(T^{\frac{1}{2} + \psi})$  for some  $\psi \in [0, \frac{1}{2})$ , then we must have  $B = \mathbb{E}\left(\sum_{t=1}^{T}t^{\rho - \frac{1}{2}}\sqrt{\mu(a^{*}) - \mu(a_{t})}\right) \leq \mathcal{O}(T^{-\frac{1}{4} + \frac{\psi}{2} + \rho}).$
+
+Proof. To prove Lemma 4.4, it is equivalent to prove the following Lemma D.6. If Lemma D.6 holds, let  $n_t = \sqrt{\mathbb{E}(\mu(a^*) - \mu(a_t))}$ ,  $m_t = \min (2,L^\sigma |c_t(a_t,a_t'))|$ , proves Lemma 4.4.
+
+Lemma D.6. Consider  $\sum_{t=1}^{T} n_t^2 \leq C' T^{\frac{1}{2} + \alpha}$ , with constants  $\alpha \geq 0$ ,  $C' > 0$ . In addition,  $n_t$  is uniformly upper bounded by a constant  $K$ . Specifically, we have  $0 \leq n_t \leq K, \forall t$ .  $m_t \leq c_k t^{\rho - 1}$  with constant  $c_k \geq 0$ . Then we have  $\sum_{t=1}^{T} \sqrt{t} m_t n_t \leq 5 \sqrt{C'} c_k T^{-\frac{1}{4} + \frac{\alpha}{2} + \rho}$ .
+
+Proof. Using Abel's equality (Lemma D.7), we have
+
+$$
+\sum_ {t = 1} ^ {T} \sqrt {t} m _ {t} n _ {t} \leq \sqrt {T} \left(\sum_ {t = 1} ^ {T} m _ {t} n _ {t}\right)
+$$
+
+Since we have  $\sum_{t=1}^{T} n_t^2 \leq C' T^{\frac{1}{2} + \alpha}$  and  $0 \leq n_t \leq K$ , we have  $\sum_{t=1}^{T} n_t \leq \sqrt{C'} T^{\frac{3}{4} + \frac{\alpha}{2}}, \forall t$ . Moreover, for  $t \geq 1$ ,  $t^{-1 + \rho} - (t + 1)^{-1 + \rho} \leq t^{\rho - 2}$  holds for all  $t \geq 1$ . This is because
+
+$$
+f (t) = t ^ {- 1 + \rho} \left((1 + \frac {1}{t}) ^ {- 1 + \rho} + \frac {1}{t} - 1\right).
+$$
+
+is decreasing over  $t \geq 1$  and  $\lim_{t \to \infty} f(t) = 0$ . Consequently, applying Abel's Summation Equation again, we have
+
+$$
+\begin{array}{l} \sum_ {t = 1} ^ {T} m _ {t} n _ {t} = \left(\sum_ {t = 1} ^ {T} n _ {t}\right) m _ {T} + \sum_ {t = 1} ^ {T - 1} \left(\sum_ {i = 1} ^ {t} n _ {i}\right) (m _ {t} - m _ {t + 1}) \\ \leq \sqrt {C ^ {\prime}} c _ {k} T ^ {\frac {3}{4} + \frac {\alpha}{2}} T ^ {- 1 + \rho} + \sqrt {C ^ {\prime}} c _ {k} \sum_ {t = 1} ^ {T - 1} \left(\sum_ {i = 1} ^ {t} n _ {i}\right) \left(t ^ {- 1 + \rho} - (t + 1) ^ {- 1 + \rho}\right) \\ \leq \sqrt {C ^ {\prime}} c _ {k} T ^ {\frac {3}{4} + \frac {\alpha}{2}} T ^ {- 1 + \rho} + \sqrt {C ^ {\prime}} c _ {k} \sum_ {t = 1} ^ {T - 1} \left(\sum_ {i = 1} ^ {t} n _ {i}\right) t ^ {- 2 + \rho} \\ \leq \sqrt {C ^ {\prime}} c _ {k} T ^ {\frac {3}{4} + \frac {\alpha}{2}} T ^ {- 1 + \rho} + \sqrt {C ^ {\prime}} c _ {k} \sum_ {t = 1} ^ {T - 1} t ^ {\frac {3}{4} + \frac {\alpha}{2}} t ^ {- 2 + \rho} \\ \leq 5 \sqrt {C ^ {\prime}} c _ {k} T ^ {- \frac {1}{4} + \frac {\alpha}{2} + \rho}. \\ \end{array}
+$$
+
+Since  $0 \leq n_{t} \leq K$ ,  $\sum_{i=1}^{t} n_{i}$  is increasing and the increasing rate is upper bound by  $t$ . Together with the constraint  $\sum_{t=1}^{T} n_{t} \leq \sqrt{C'} T^{\frac{3}{4} + \frac{\alpha}{2}}$ ,  $\sum_{t=1}^{T-1} \left( \sum_{i=1}^{t} n_{i} \right) t^{-\frac{1}{2} + \rho}$  is optimized when  $\sum_{i=1}^{t} n_{i}$  increasing in the speed of  $\sqrt{C'} t^{\frac{3}{4} + \frac{\alpha}{2}}$ . Because of this, the second last inequality holds.
+
+Lemma D.7. (Abel's Summation Equation (Williams, 1963)). For any numbers  $a_{k}, b_{k}$ , we have
+
+$$
+\sum_ {k = 1} ^ {n} a _ {k} b _ {k} = \left(\sum_ {k = 1} ^ {n} b _ {k}\right) a _ {n} + \sum_ {k = 1} ^ {n - 1} \left(\sum_ {i = 1} ^ {k} b _ {i}\right) (a _ {k} - a _ {k + 1}).
+$$
+
+![](images/95f56d18de93847775cfe88dbd283d16435c0193561c7fb14864a944e284aafc.jpg)
+
+![](images/916ab256f7507b9af35ae67e8e851e411c0c540eea43a52ad40b8473951fb543.jpg)
+
+After establishing Lemma 4.4, we can use it to prove the induction claim (Lemma 4.5). The challenge in this proof lies in identifying the constant  $K$ , which must hold across all iterations of the analysis to ensure the bound remains stable and does not diverge. This requires careful refinement of the analysis and calculations.
+
+Lemma (Lemma 4.5 restated). Consider  $T > \sqrt{2L^{\mu}L_{1}^{\sigma}R}$  and  $\eta_{\rho} = \frac{\sqrt{\log T}}{dT^{\max\{1/2,\rho\}}}$ . If  $Reg_{T} \leq 144dK\sqrt{\log T}T^{\rho + \frac{3}{2} - \rho}$  for some integer  $k$ , then we must also have  $Reg_{T} \leq 144dK\sqrt{\log T}T^{\rho + \frac{3}{2} - \rho}$ .  $K$  is a instance-dependent constant, where  $K := \max \left\{C_{0}, \frac{C_{2}C_{\kappa}}{\sqrt{\log T}}, (L^{\sigma}T_{\kappa}^{\rho})^{2}, 2\sqrt{\lambda}RC_{\kappa}L^{\sigma}\right\}$ ,  $C_{0}$  and  $C_{2}$  defined in Lemma 4.2 and 4.3 respectively,
+
+Proof. In the following, we denote
+
+$$
+m _ {t} := \min  \left(2, L ^ {\sigma} | c _ {t} \left(a _ {t}, a _ {t} ^ {\prime}\right) |\right), n _ {t} := \min  \left(2 R, \sqrt {\frac {2}{\alpha} \left(\mu \left(a ^ {*}\right) - \mu \left(a _ {t}\right)\right)}\right)
+$$
+
+Continue from equation 18, we have the following.
+
+The Base Case: When  $k = 1$ , let  $C_3 = C_2 dT^\rho + 2d \sqrt{\lambda \eta_\rho T}$ , we have
+
+$$
+\begin{array}{l} \operatorname {R e g} _ {T} \leq \frac {C _ {0}}{\eta_ {\rho}} \log (T) + 8 d ^ {2} \eta_ {\rho} T + 2 L ^ {\mu} L _ {1} ^ {\sigma} R + 2 C _ {3} + 2 d \sqrt {\lambda \eta_ {\rho}} \mathbb {E} \left(\sum_ {t = 1} ^ {T} \sqrt {t m _ {t}} n _ {t}\right). (19) \\ \leq \frac {C _ {0}}{\eta_ {\rho}} \log (T) + 8 d ^ {2} \eta_ {\rho} T + 2 L ^ {\mu} L _ {1} ^ {\sigma} R + 2 C _ {3} + 4 d R \sqrt {\lambda \eta_ {\rho}} \mathbb {E} \left\{\sum_ {t = 1} ^ {T} \sqrt {t} m _ {t} \right\} (20) \\ \leq \frac {C _ {0}}{\eta_ {\rho}} \log (T) + 8 d ^ {2} \eta_ {\rho} T + 2 L ^ {\mu} L _ {1} ^ {\sigma} R + 2 C _ {3} + 4 d R \sqrt {\lambda \eta_ {\rho} T} \left(\sum_ {t = 1} ^ {T} m _ {t}\right) (21) \\ \leq \frac {C _ {0}}{\eta_ {\rho}} \log (T) + 8 d ^ {2} \eta_ {\rho} T + 2 L ^ {\mu} L _ {1} ^ {\sigma} R + 2 C _ {3} + 4 d L ^ {\sigma} R \sqrt {\lambda \eta_ {\rho}} C _ {\kappa} T ^ {\rho}. (22) \\ \end{array}
+$$
+
+We get equation 19 according to Lemma 4.2 and 4.3. We get equation 20 because of  $\mathbb{E}\left(n_t\right)\leq 2R$ . We get equation 21 by Abel's Summation Equation (see Lemma D.7). We get equation 22 because of corruption budget. Choosing the learning rate  $\eta_{\rho}$  according to Theorem 4.1, we obtain  $\mathrm{Reg}_T\leq 144\sqrt{\log T} KdT^{\frac{\rho + 1}{2}}$ . This confirms the validity of the claim when  $k = 1$ .
+
+The Induction Argument: Let's assume that the claim holds true for a general step  $k$ . In the following, we will demonstrate that the claim also holds true for step  $k + 1$ . Because of Equation equation 14 and the induction claim, we obtain  $\mathrm{Reg}_T^{\mathrm{FO}} \leq \frac{144}{l_1^\sigma} dK\sqrt{\log T} T^{\rho + \frac{3/2 - \rho}{2^k}}$ . This suggests that
+
+$$
+\sum_ {t = 1} ^ {T} \mathbb {E} \left(\mu (a ^ {*}) - \mu (a _ {t})\right) \leq \frac {1 4 4}{l _ {1} ^ {\sigma}} d K \sqrt {\log T} T ^ {\rho + \frac {3 / 2 - \rho}{2 ^ {k}}}.
+$$
+
+By Jensen's inequality, we have
+
+$$
+\mathbb {E} (n _ {t}) = \min  \left\{2, \sqrt {\frac {2}{\alpha}} \mathbb {E} \left(\sqrt {\mu (a ^ {*}) - \mu (a _ {t})}\right) \right\} \leq \sqrt {\frac {2}{\alpha}} \sqrt {\mathbb {E} \left(\mu (a ^ {*}) - \mu (a _ {t})\right)}.
+$$
+
+Therefore we have
+
+$$
+\mathbb {E} \left(\sum_ {t = 1} ^ {T} \sqrt {t} m _ {t} n _ {t}\right) \leq \sqrt {\frac {2}{\alpha}} \sum_ {t = 1} ^ {T} \sqrt {t} m _ {t} \sqrt {\mathbb {E} \left(\mu \left(a ^ {*}\right) - \mu \left(a _ {t}\right)\right)}.
+$$
+
+By the definition of  $\rho$ -Imperfect Human Feedback, we have  $|c_{t}(a_{t},a_{t}^{\prime})|$  is upper bounded by  $C_k t^{-1 + \rho}$ .
+
+Therefore, using Lemma 4.4 we can upper bound  $\sqrt{\lambda}\sum_{t=1}^{T}\sqrt{tm}_tn_t$  by
+
+$$
+\sqrt {\lambda} \sum_ {t = 1} ^ {T} \sqrt {t} m _ {t} n _ {t} \leq 6 0 L ^ {\sigma} \sqrt {\frac {2 \lambda}{\alpha l _ {1} ^ {\sigma}}} (\log T) ^ {0. 2 5} C _ {\kappa} \sqrt {d K} T ^ {\frac {3}{2} \rho + \frac {3 / 2 - \rho}{2 k + 1}}
+$$
+
+Since  $\lambda \leq \alpha l_1^\sigma / 2$ , choosing  $\eta_\rho = \frac{1}{d} \frac{\sqrt{\log T}}{T^\rho}$ , we have
+
+$$
+\begin{array}{l} \operatorname {R e g} _ {T} \leq 2 d \left(C _ {0} + \frac {C _ {2}}{\sqrt {\log T}} + 6 0 C _ {\kappa} \sqrt {K} T ^ {\frac {3 / 2 - \rho}{2 k + 1}}\right) \sqrt {\log T} T ^ {\rho} + 1 2 K d \sqrt {\log T} T ^ {\rho} \\ \leq 1 4 4 d K \sqrt {\log T} T ^ {\rho + \frac {3 / 2 - \rho}{2 ^ {k + 1}}}. \\ \end{array}
+$$
+
+The claim holds true for step  $k + 1$ . Therefore, it implies that the claim holds true for all  $k$ . Repeating this process infinitely many times, we obtain
+
+$$
+\operatorname {R e g} _ {T} \leq \mathcal {O} \left(d \sqrt {\log T} T ^ {\rho}\right),
+$$
+
+which completes the proof when  $\rho \in [0.5,1]$ .
+
+□
+
+□
+
+# E PROOF FOR PROPOSITION 2
+
+Proposition (Proposition 2 restated). If the total corruption level satisfies  $\sum_{t=1}^{T}|c_t(a_t,a_t')| \leq \mathcal{O}(T^\rho)$ , then for any  $\alpha \in [\frac{1}{2},1)$ , if the utility function is strongly concave, for sufficiently large round  $T$ , by choosing learning rate  $\eta_\rho \coloneqq \frac{\sqrt{\log T}}{2d} T^{-\alpha}$
+
+1. (Robustness) RoSMID incurs  $Reg_{T} \leq \tilde{\mathcal{O}}(dT^{\alpha} + \sqrt{d} T^{\frac{1}{2}(1 - \alpha) + \rho} + dT^{\rho})$ .  
+2. (Efficiency) There exists a strongly concave utility function  $\mu$  and a link function  $\sigma$  such that  $R e g_{T}$  suffered by RoSMID is at least  $\Omega (T^{\alpha})$  in scenario without corruption.
+
+# E.1 PROOF FOR ROBUSTNESS STATEMENT IN PROPOSITION 2
+
+Proof. From Lemma 4.3, we know
+
+$$
+\begin{array}{l} \mathbb {E} \left\{\sum_ {t = 1} ^ {T} b _ {t} ^ {\top} \left(a _ {t} - a _ {T} ^ {*}\right) \right\} \leq 2 R d L ^ {\sigma} \sqrt {\lambda_ {R} ^ {*} + 2 \phi} T ^ {\rho} + d \sqrt {\lambda \eta_ {\rho}} \mathbb {E} \left\{\sum_ {t = 1} ^ {T} \sqrt {t} \min  \left(2, L ^ {\sigma} \left| c _ {t} \left(a _ {t}, a _ {t} ^ {\prime}\right) \right|\right) \| a _ {t} - a _ {T} ^ {*} \| _ {2} \right\} \\ \leq 2 R d L ^ {\sigma} \sqrt {\lambda_ {R} ^ {*} + 2 \phi} T ^ {\rho} + 2 R d \sqrt {\lambda \eta_ {\rho}} L ^ {\sigma} \sum_ {t = 1} ^ {T} \sqrt {t} \left| c _ {t} \left(a _ {t}, a _ {t} ^ {\prime}\right) \right| \\ \leq 2 R d L ^ {\sigma} \sqrt {\lambda_ {R} ^ {*} + 2 \phi} T ^ {\rho} + 2 R d \sqrt {\lambda \eta_ {\rho}} L ^ {\sigma} T ^ {\rho}. \\ \end{array}
+$$
+
+We get the second inequality by using the fact that the diameter of the action space of  $R$ . We get the last inequality by using Abel Summation Equation (see Lemma D.7). Therefore, by Lemma 4.2, we have
+
+$$
+\operatorname {R e g} _ {T} \leq \frac {C _ {0}}{\eta_ {\rho}} \log (T) + 8 d ^ {2} \eta_ {\rho} T + 2 L ^ {\mu} L _ {1} ^ {\sigma} R + 4 R d L ^ {\sigma} \sqrt {\lambda_ {R} ^ {*} + 2 \phi} T ^ {\rho} + 4 R d \sqrt {\lambda \eta_ {\rho} T} L ^ {\sigma} T ^ {\rho}.
+$$
+
+Choosing  $\eta_{\rho} = \frac{\sqrt{\log T}}{2d} T^{-\alpha}, \alpha \in [0.5, 1)$ , we have
+
+$$
+\begin{array}{l} \mathrm {R e g} _ {T} \leq 2 d C _ {0} \sqrt {\log T} T ^ {\alpha} + 4 d \sqrt {\log T} T ^ {1 - \alpha} + 4 R d L ^ {\sigma} \sqrt {\lambda_ {R} ^ {*} + 2 \phi} T ^ {\rho} + 4 R \sqrt {d} L ^ {\sigma} (\log T) ^ {0. 2 5} T ^ {\frac {1}{2} (1 - \alpha)} T ^ {\rho} + 2 L ^ {\mu} L _ {1} ^ {\sigma} R \\ \leq O \left(d \sqrt {\log T} T ^ {\alpha} + \sqrt {d} (\log T) ^ {\frac {1}{4}} T ^ {\frac {1}{2} (1 - \alpha)} T ^ {\rho} + d T ^ {\rho}\right), \\ \end{array}
+$$
+
+which completes the proof of robustness statement.
+
+![](images/4eee4cc1c6dde4bf2179025ee2c1f4b2338afb3086101c75161594175d96cd7e.jpg)
+
+# E.2 PROOF FOR EFFICIENCY STATEMENT IN PROPOSITION 2
+
+Proof. The robustness statement in Proposition 2 implies that RoSMID could afford agnostic corruption level  $\mathcal{O}(T^{\frac{1 + \alpha}{2}})$ . If Lemma E.1 holds, this implies that there exists a hard instance which makes RoSMID incur regret in order of  $\Omega(T^{\alpha})$  in non-corrupt setting. We can prove this by contradiction. Assume the efficiency statement in Proposition 2 is not true, which implies that for all  $\mu$ , and  $\sigma$ , there exists an  $\epsilon > 0$  such that  $\mathrm{Reg}_T = c_0 T^{\alpha - \epsilon}$ , for some constant  $c_0 > 0$ , we assume  $\epsilon$  to be the least possible. In this scenario, when we consider a linear link function  $\sigma(x) = \frac{1 + x}{2}$ , Lemma E.1 implies that an agnostic corruption budget  $2\sqrt{c_0 T^{\frac{1 + \alpha - \epsilon}{2}}}$  suffices to induce linear regret, which contradicts the robustness statement in Proposition 2.
+
+Lemma E.1. For any algorithm  $\mathcal{G}$  incurs regret  $Reg_{T} \leq c_{0} T^{\alpha}$ , for some constant  $c_{0} > 0$ , on learning from bandit with reward feedback for strongly concave utilities  $\mu$  in non-corrupted setting, then there exists an instance  $\mu$  such that  $\mathcal{G}$  will suffer linear regret under agnostic corruption with budget  $\sqrt{c_{0}} T^{\frac{1 + \alpha}{2}}$ .
+
+Proof. Consider the scenario when  $d = 2$ , the action space  $\mathcal{A} \coloneqq \{(a_1, a_2) : a_1 \geq 0, a_2 \geq 0, a_1 + a_2 \leq 1\}$ , the utility function  $\mu_{\theta}(a) = \langle \theta, a \rangle - \frac{1}{2} \| a \|_2^2$  with  $\theta \coloneqq [1, 0]$ . In this scenario, the optimal action is  $a^* \coloneqq [1, 0]$ . Assume that the adversary wants to "fool" the algorithm  $\mathcal{G}$  to make it believe the  $\theta \coloneqq [1, 1]$ , with optimal arm  $a^* = [\frac{1}{2}, \frac{1}{2}]$ , each round, the corruption it should introduce is  $c_t(a_t) = a_{t2}$ , where  $a_{ti}$  is the  $i$ -th coordinate value of action  $a_t$ . To make reward
+
+indistinguishable from these two environments, the total corruption budget which the adversary has to pay is  $\sum_{t=1}^{T}|a_{t2}|$ . We know that  $\mathcal{G}$  incurs regret  $\mathrm{Reg}_T := \sum_{t=1}^{T}\langle\theta,a^*-a_t\rangle-\frac{1}{2}\|a^*\|_2^2+\frac{1}{2}\|a_t\|_2^2=\frac{1}{2}-a_{t1}+\frac{1}{2}a_{t1}^2+\frac{1}{2}a_{t2}^2\leq c_0T^\alpha$ . Given this constraint, the maximum possible corruption paid by the adversary to make  $\theta=[1,1]$  and  $\theta=[1,0]$  indistinguishable is  $\max \sum_{t=1}^{T}|a_{t2}|=\sqrt{c_0T^{\frac{\alpha+1}{2}}}$ . Therefore, for any agnostic corruption no less than  $\sqrt{c_0T^{\frac{\alpha+1}{2}}}$ ,  $\mathcal{G}$  can not distinguish  $\theta=[1,0]$  and  $\theta=[1,1]$  and has to incur linear regret in either of these environments.
+
+![](images/d33ec097c7c95aeaca67376770d757efdb60a5f2a25f62d0c22e03b38fdb7434.jpg)
+
+# F PROOF FOR PROPOSITION 3:
+
+In this section, we present the regret upper for dueling bandit gradient descent Yue & Joachims (2009) in presence of arbitrary and agnostic corruption. Before presenting the proof, we make several remarks. First, in Yue & Joachims (2009), the utility function  $\mu$  is assumed to be strictly concave to ensure the uniqueness of the maximizer. However, in scenarios where the maximizer is unique within the action set for a concave utility function  $\mu$ , the requirement for "strictness" can be relaxed. Second, we highlight the notation differences. We use  $a$  to denote actions while Yue & Joachims (2009) uses  $w$ . In our proof, we define  $P_{t}(a) \coloneqq \sigma(\mu(a_{t}) - \mu(a))$  to represent the probability of the event  $a_{t} \succ a$ . While in Yue & Joachims (2009), it is denoted as  $c_{t}(w)$ . Just to highlight, its corrupted version  $\hat{P}_{t}(a) \coloneqq \sigma(\mu(a_{t}) - \mu(a) + c_{t}(a_{t}, a))$  is newly introduced in our paper. Moreover,  $\bar{P}_{t}(a) \coloneqq \mathbb{E}_{x \in \mathbb{B}}[P_{t}(\mathcal{P}_{A}(a + \delta x))]$  is denoted as  $\hat{\epsilon}_{t}(w) + \frac{1}{2}$  in Yue & Joachims (2009). The definition of  $\mathcal{F}(\mathcal{P}_{A}(a_{t} + \delta u_{t}), a_{t})$  is same as  $X_{t}(\mathcal{P}_{W}(w_{t} + \delta u_{t}))$  defined in Yue & Joachims (2009), except different notation. We remark  $\mathcal{P}_{A}(a)$  represents the projection of action  $a$  on  $\mathcal{A}$ .
+
+Algorithm 3: Dueling Bandit Gradient Descent (DBGD) (Yue & Joachims, 2009)  
+Input: Exploration rate  $\delta$  exploitation rate  $\gamma$  , initial action  $a_1 = 0\in \mathcal{A}$    
+for  $t\in [T]$  do Sample unit vector  $u_{t}$  uniformly and set  $a_t^\prime = \mathcal{P}_A(a_t + \delta u_t)$  Present action pair  $(a_{t},a_{t}^{\prime})$  to user and receive corrupted dueling feedback  $\hat{\mathcal{F}} (a_t',a_t)$  Compute gradient  $\hat{g}_t = -\frac{d}{\delta}\hat{\mathcal{F}} (a_t',a_t)u_t$  Set learning rate  $\eta = \frac{\gamma\delta}{d}$  and update  $a_{t + 1} = \mathcal{P}_{\mathcal{A}}(a_{t} - \eta \hat{g}_{t})$
+
+Proposition (Proposition 3 restated). If the total corruption level satisfies  $\sum_{t=1}^{T}|c_t(a_t,a_t')| \leq \mathcal{O}(T^\rho)$ , then for any  $\alpha \in (0,\frac{1}{4}]$ , if utility function  $\mu$  is generally concave, for a sufficiently large round  $T$ , choosing  $\gamma := \frac{R}{\sqrt{T}}$  and  $\delta := \frac{\sqrt{2}Rd}{\sqrt{13L^{\sigma}L^{\mu}T^{\alpha}}}$  for Algorithm 3
+
+1. (Robustness)  $Reg_T$  incurred by Algorithm 3 satisfies  $Reg_T \leq \mathcal{O}(\sqrt{d} T^{1 - \alpha} + \sqrt{d} T^{\alpha + \rho})$ .  
+2. (Efficiency) There exists a linear utility function  $\mu$  and a link function  $\sigma$  such that  $Reg_{T}$  suffered by Algorithm 3 is at least  $\Omega\left(T^{1 - \alpha}\right)$  in scenario without corruption.
+
+# F.1 PROOF FOR ROBUSTNESS STATEMENT IN PROPOSITION 3
+
+Proof. The proof of Theorem 3 builds on the proof of Yue & Joachims (2009), where we extend it to a setting in presence of agnostic corruption. The proof follows Step 1-3 introduced in Theorem 4.1. Similarly, in Step 1, we decompose bias in the gradient caused by corruption. Notice that without corruption, DBGD performs expected gradient ascent, where the gradient  $g_{t}$  is defined as follows
+
+$$
+g _ {t} = - \frac {d}{\delta} \mathcal {F} \left(\mathcal {P} _ {\mathcal {A}} \left(a _ {t} + \delta u _ {t}\right), a _ {t}\right) u _ {t}.
+$$
+
+In the presence of adversarial corruption, the corrupted gradient is
+
+$$
+\hat {g} _ {t} = - \frac {d}{\delta} \hat {\mathcal {F}} \left(\mathcal {P} _ {\mathcal {A}} \left(a _ {t} + \delta u _ {t}\right), a _ {t}\right) u _ {t}.
+$$
+
+Therefore, we quantify the bias  $\mathbb{E}(b_t|a_t)$  in the following lemma.
+
+Lemma F.1. Let  $P_{t}(a)$  represent the likelihood of action  $a_{t}$  being preferred over  $a$ , where  $P_{t}(a) \coloneqq \sigma(\mu(a_{t}) - \mu(a))$ . Likewise, the corrupted version is defined as  $\hat{P}_{t}(a) = \sigma(\mu(a_{t}) - \mu(a) + c_{t}(a_{t}, a))$ . The smoothed version of  $P_{t}$  over  $\mathcal{A}$  is  $\bar{P}_{t}(a) \coloneqq \mathbb{E}_{x \in \mathbb{B}}[P_{t}(\mathcal{P}_{\mathcal{A}}(a + \delta x))]$ . Let  $a_{t}' = \mathcal{P}_{\mathcal{A}}(a_{t} + \delta u_{t})$  where  $u_{t}$  is uniformly sampled from  $\mathbb{S}$  and  $\hat{g}_{t} = -\frac{d}{\delta} \hat{\mathcal{F}}(a_{t}', a_{t}) u_{t}$ . Let  $b_{t} \coloneqq \frac{d}{\delta}(\hat{P}_{t}(a_{t}') - P_{t}(a_{t}')') u_{t}$ . We have  $\mathbb{E}(\hat{g}_{t}|a_{t}) = \mathbb{E}(\nabla \bar{P}_{t}(a_{t})|a_{t}) - \mathbb{E}(b_{t}|a_{t})$ .
+
+Proof. (Proof for Lemma F.1). Since  $a_{t}^{\prime} \coloneqq \mathcal{P}_{\mathcal{A}}(a_{t} + \delta u_{t})$ , we have
+
+$$
+\begin{array}{l} \mathbb {E} \left(\hat {g} _ {t} \mid a _ {t}\right) = \mathbb {E} _ {u _ {t}} \left(\mathbb {E} \left(\hat {g} _ {t} \mid a _ {t}, u _ {t}\right)\right) \\ = - \frac {d}{\delta} \mathbb {E} _ {u _ {t}} \left(\mathbb {E} \left(\hat {\mathcal {F}} \left(a _ {t} ^ {\prime}, a _ {t}\right) u _ {t} \mid a _ {t}, u _ {t}\right)\right) \\ = - \frac {d}{\delta} \mathbb {E} \left(\hat {P} _ {t} \left(a _ {t} ^ {\prime}\right) u _ {t} \mid a _ {t}\right) \\ = - \frac {d}{\delta} \mathbb {E} \left(\left[ P _ {t} \left(a _ {t} ^ {\prime}\right) + \hat {P} _ {t} \left(a _ {t} ^ {\prime}\right) - P _ {t} \left(a _ {t} ^ {\prime}\right) \right] u _ {t} \mid a _ {t}\right) \\ = \nabla \mathbb {E} _ {x \in \mathbb {B}} \left(P _ {t} \left(\mathcal {P} _ {\mathcal {A}} \left(a _ {t} + \delta x\right)\right) \mid a _ {t}\right) - \frac {d}{\delta} \mathbb {E} \left[ \left(\hat {P} _ {t} \left(a _ {t} ^ {\prime}\right) - P _ {t} \left(a _ {t} ^ {\prime}\right)\right) u _ {t} \mid a _ {t} \right] (23) \\ = \nabla \bar {P} _ {t} \left(a _ {t}\right) - \frac {d}{\delta} \mathbb {E} \left[ \left(\hat {P} _ {t} \left(a _ {t} ^ {\prime}\right) - P _ {t} \left(a _ {t} ^ {\prime}\right)\right) u _ {t} \mid a _ {t} \right] \\ = \mathbb {E} \left(g _ {t} \mid a _ {t}\right) - \frac {d}{\delta} \mathbb {E} \left[ \left(\hat {P} _ {t} \left(a _ {t} ^ {\prime}\right) - P _ {t} \left(a _ {t} ^ {\prime}\right)\right) u _ {t} \mid a _ {t} \right]. (24) \\ \end{array}
+$$
+
+We get the equation equation 23 by using Lemma F.2. We get equation equation 24 by using Lemma F.3.
+
+Lemma F.2 (Lemma 2 in Yue & Joachims (2009)). Fix  $\delta > 0$ , over random unit vector  $u$ , we have
+
+$$
+\mathbb {E} [ P _ {t} (\mathcal {P} _ {\mathcal {A}} (a + \delta u)) u ] = \frac {\delta}{d} \nabla \bar {P} _ {t} (a).
+$$
+
+Lemma F.3 (Lemma 1 in Yue & Joachims (2009)).  $\mathbb{E}_{\mathcal{F},u}[\mathcal{F}(a_t',a_t)u] = -\mathbb{E}_u[P_t(a_t')u]$ .
+
+![](images/45d9a39e615f5c0e4148ef13ae809cb5dcdd45189df5a032bcf2223173dd31ef.jpg)
+
+In Step 2, we decompose regret under dueling bandits into two parts, regret of decision and feedback error.
+
+Lemma F.4 (Regret Decomposition for DBGD). Define  $\lambda := \frac{L^{\sigma}}{L^{\sigma} - \delta L^{\mu}L_{2}}$ ,  $L_{2}$  is the Lipschitz constant for  $\sigma'$ ,  $\gamma = R / \sqrt{T}$ , for  $b_{t}$  defined in Lemma F.1, we have
+
+$$
+R e g _ {T} \leq \underbrace {\lambda \left(\frac {2 R d \sqrt {T}}{\delta} + 1 3 \delta L ^ {\sigma} L ^ {\mu} T\right)} _ {R e g r e t o f D e c i s i o n} + \underbrace {2 \lambda \sum_ {t = 1} ^ {T} \mathbb {E} \left(b _ {t} ^ {\top} (a _ {t} - a ^ {*})\right)} _ {F e e d b a c k E r r o r}.
+$$
+
+Proof. Because of Lemma F.5, we have
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \bar {P} _ {t} (a _ {t}) - \bar {P} _ {t} (a ^ {*}) \right] \leq \sum_ {t = 1} ^ {T} \mathbb {E} \left(\lambda \nabla \bar {P} _ {t} (a _ {t}) (a _ {t} - a ^ {*}) + (3 + \lambda) \delta L ^ {\sigma} L ^ {\mu}\right) \\ = \lambda \sum_ {t = 1} ^ {T} \mathbb {E} \left(\mathbb {E} \left(g _ {t} \mid a _ {t}\right) ^ {\top} \left(a _ {t} - a ^ {*}\right)\right) + (3 + \lambda) \delta L ^ {\sigma} L ^ {\mu} T \\ = \lambda \sum_ {t = 1} ^ {T} \mathbb {E} \left(\mathbb {E} \left(\hat {g} _ {t} + b _ {t} \mid a _ {t}\right) ^ {\top} \left(a _ {t} - a ^ {*}\right)\right) + (3 + \lambda) \delta L ^ {\sigma} L ^ {\mu} T \\ = \lambda \sum_ {t = 1} ^ {T} \mathbb {E} \left(\hat {g} _ {t} ^ {\top} \left(a _ {t} - a ^ {*}\right)\right) + \lambda \sum_ {t = 1} ^ {T} \mathbb {E} \left(b _ {t} ^ {\top} \left(a _ {t} - a ^ {*}\right)\right) + (3 + \lambda) \delta L ^ {\sigma} L ^ {\mu} T \\ \end{array}
+$$
+
+Lemma F.5 (Lemma 4 in Yue & Joachims (2009)). Fix  $\delta \in (0, \frac{L^{\sigma}}{L^{\mu}L_{2}})$ , for  $\lambda$  defined in Lemma F.4, we have
+
+$$
+\mathbb {E} \left[ \sum_ {t = 1} ^ {T} \bar {P} _ {t} (a _ {t}) - \bar {P} _ {t} (a ^ {*}) \right] \leq \sum_ {t = 1} ^ {T} \mathbb {E} \left(\lambda \nabla \bar {P} _ {t} (a _ {t}) ^ {\top} (a _ {t} - a ^ {*}) + (3 + \lambda) \delta L ^ {\sigma} L ^ {\mu}\right).
+$$
+
+Since  $a_{t + 1} = \mathcal{P}_{\mathcal{A}}(a_t - \eta \hat{g}_t)$ , and  $\eta = \frac{\gamma\delta}{d}$ , by applying the telescoping sum and because  $a_1 = 0$ ,  $\| g_t\| _2\leq G,\forall t$ , we have
+
+$$
+\lambda \sum_ {t = 1} ^ {T} \mathbb {E} \left(\hat {g} _ {t} ^ {\top} \left(a _ {t} - a ^ {*}\right)\right) \leq \lambda \left(\frac {R ^ {2}}{2 \eta} + \frac {T \eta G ^ {2}}{2}\right).
+$$
+
+![](images/08449c9b7af9e9360ac8459e369168a99c9376d5a3090590c4e14fcaca77d3e3.jpg)
+
+Noticing that  $\| b_t\| _2\leq \frac{d}{\delta}\min \left(2,L^\sigma |c_t(a_t,a_t')|\right)$ . Moreover,  $\| a_{t} - a^{*}\|_{2}\leq 2R$ . Therefore, in Step 3, we can control the feedback error by the following.
+
+Lemma F.6.  $\sum_{t=1}^{T} \mathbb{E}\left(b_t^\top(a_t - a^*)\right) \leq 2RdL^\sigma T^\rho/\delta.$
+
+Proof. (Proof for Lemma F.6).
+
+$$
+\begin{array}{l} \sum_ {t = 1} ^ {T} \mathbb {E} \left(b _ {t} \left(a _ {t} - a ^ {*}\right)\right) \leq 2 R \frac {d}{\delta} \sum_ {t = 1} ^ {T} \min  \left(2, L ^ {\sigma} \left| c _ {t} \left(a _ {t}, a _ {t} ^ {\prime}\right) \right|\right) \\ \leq 2 R \frac {d L ^ {\sigma}}{\delta} \sum_ {t = 1} ^ {T} \left| c _ {t} \left(a _ {t}, a _ {t} ^ {\prime}\right) \right| \\ = 2 R \frac {d L ^ {\sigma} T ^ {\rho}}{\delta}. \\ \end{array}
+$$
+
+![](images/63057175b310b08f721180fc193c3154a66b884518241ed88ebee05ac2757403.jpg)
+
+Lemma F.7 (Lemma 3 in Yue & Joachims (2009)).  $Reg_{T} \leq 2\mathbb{E}\left[\sum_{t=1}^{T}\bar{P}_{t}(a_{t}) - \bar{P}_{t}(a^{*})\right] + 5\delta L^{\sigma}L^{\mu}T$
+
+Set  $G = \frac{d}{\delta}$ , and by Lemma F.7, we have
+
+$$
+\operatorname {R e g} _ {T} \leq \lambda \left(\frac {2 R d \sqrt {T}}{\delta} + 1 3 \delta L ^ {\sigma} L ^ {\mu} T\right) + 4 R \frac {\lambda d L ^ {\sigma} T ^ {\rho}}{\delta}.
+$$
+
+Therefore, by choosing  $\delta := \frac{\sqrt{2Rd}}{\sqrt{13L^{\sigma}L^{\mu}}T^{\alpha}}$ ,  $\gamma := \frac{R}{\sqrt{T}}$ , and  $T > \left(\frac{\sqrt{2Rd}L_{\mu}L_{2}}{\sqrt{13L^{\sigma}L^{\mu}}L^{\sigma}}\right)^{4}$ , we have
+
+$$
+\operatorname {R e g} _ {T} \leq 2 \lambda_ {T} \sqrt {2 6 R d L ^ {\sigma} L ^ {\mu}} T ^ {1 - \alpha} + 2 L ^ {\sigma} \sqrt {2 6 R d L ^ {\sigma} L ^ {\mu}} T ^ {\alpha + \rho}.
+$$
+
+$\lambda_{T} = \frac{L^{\sigma}\sqrt{13L^{\sigma}L^{\mu}}T^{\alpha}}{L^{\sigma}\sqrt{13L^{\sigma}L^{\mu}}T^{\alpha} - L^{\mu}L_{2}\sqrt{2Rd}}$  completes the proof.
+
+![](images/ba2d4c9cc437f1bb3dbcb52f28e187ef75a98279ea3dca26befb33820b2d81df.jpg)
+
+# F.2 PROOF FOR EFFICIENCY STATEMENT IN PROPOSITION 3
+
+Proof. The robustness statement in Proposition 3 implies that DBGD could afford agnostic corruption level  $\mathcal{O}(T^{1 - \alpha})$ . This implies that there exists a hard instance which makes DBGD attain regret in order of  $\Omega (T^{1 - \alpha})$  in non-corrupt setting, formally described as follows. To start a proof by contradiction, we assume that the efficiency statement in Proposition 3 is false. Specifically, for all problem instance  $\mu ,\sigma$ , there exist a  $\epsilon >0$ , such that  $\mathrm{Reg}_T\leq \mathcal{O}(T^{1 - \alpha -\epsilon})$ . We assume  $\epsilon$  to be the least possible. In particular, there exists a pair of instance  $\mu ,\sigma$  such that  $\mathrm{Reg}_T = c_0T^{1 - \alpha -\epsilon}$ ,  $c_{0}$  is a positive constant. In other words, it says that there exists a pair of instance  $\mu ,\sigma$  such that DBGD suffers regret in  $\Theta (T^{1 - \alpha -\epsilon})$ .
+
+Consider the following problem instance.  $\mu (a) = \theta^{\top}a$ , specifically  $\mu$  is linear function. Let  $d = 2$  with action set  $\mathcal{A}_1\coloneqq \{(a_1,a_2):a_1\geq 0,a_2\geq 0,\frac{1}{2} a_1 + a_2 - \frac{1}{4}\leq 0\}$  with  $\theta = [\frac{1}{2},\frac{1}{2} ]$ . The optimal arm is  $a_1 = [\frac{1}{2},0]$ , which is unique. Let the link function  $\sigma (x) = \frac{1}{2} +\frac{1}{2} x$ . It is easy to verify that it is rotational symmetric and Lipschitz. Denote  $\mathrm{Reg}_T$  as the regret occurred by DBGD on this problem instance. We know  $\mathrm{Reg}_T\leq \mathcal{O}(T^{1 - \alpha -\epsilon})$ . Consequently, it implies that  $a_2$  is at most proposed  $8c_{0}T^{1 - \alpha -\epsilon}$  times. This is because  $\mathrm{Reg}_T\leq c_0T^{1 - \alpha -\epsilon}$  and if the proposed action pair  $(a_t,a_t')$  including  $a_2$  at round  $t$ , it occurs regret at least  $\frac{1}{8}$ .
+
+Now consider a different problem instance with the same  $\mu, \sigma$  but different action set, which is  $\mathcal{A}_2 := \{(a_1, a_2) : a_1 \geq 0, a_2 \geq 0, \frac{3}{2} a_1 + a_2 - \frac{3}{4} \leq 0\}$ . The optimal arm in this action set  $a_2 = [0, \frac{3}{4}]$ , which is also unique. Proposing arm  $a_1 = [\frac{1}{2}, 0]$  occurs at least  $\frac{1}{8}$  regret. Consider an adversary that pulls the utility of  $a_2$  from  $\frac{3}{8}$  down to  $\frac{1}{8}$  whenever it is pulled at a cost of  $c_t(A_t) = \frac{1}{4}$ . It is equivalent to say that every time when the proposed arm  $w$  falls in the region  $\mathcal{W} = \{a_1 \geq 0, a_2 \geq 0, \frac{1}{2} a_1 + a_2 - \frac{1}{4} \geq 0, \frac{3}{2} a_1 + a_2 - \frac{3}{4} \leq 0\}$ , we assume that the utility is sampled from  $\theta^\top \mathcal{P}_{\mathcal{A}_1}(w)$  and  $|c_t(w)| = |\theta^\top \mathcal{P}_{\mathcal{A}_1}(w) - \theta^\top w| \leq \frac{1}{4}$ . Choosing the total corruption budget as  $2c_0 T^{1-\alpha-\epsilon}$ , the adversary can afford to corrupt  $8c_0 T^{1-\alpha-\epsilon}$  times.
+
+This means that as long as the adversary is corrupting, the utility observed in the first problem instance is exactly the same as the utility observed in the second problem instance, in which we pull  $a_2$  as most  $8c_0T^{1 - \alpha -\epsilon}$ . However,  $a_2$  is the optimal arm in the second problem instance, which implies that the regret occurred on the second problem instance is at least  $T - 8c_0T^{1 - \alpha -\epsilon}$ , which is linear in  $T$ . This contradicts the robustness statement in Proposition 3, which says that when agnostic corruption is  $2c_0T^{1 - \alpha -\epsilon}$ , the regret upper bound is  $O(T^{1 - \epsilon})$ , which is sublinear. To reconcile the conflicts, we must have the efficiency statement in Proposition 3 true to make the agnostic corruption budget at least  $\Omega (T^{1 - \alpha})$  to make the parallel world argument valid.
+
+# G ADDITIONAL EXPERIMENTS
+
+In this section, we list all the experiment details. At the beginning, we introduce the regret order fitting method, and baseline algorithms for comparison.
+
+Fitted Order of Regret. We introduce the methodology that we use to compute the order of  $\mathrm{Reg}_T$  in terms of the number of iteration,  $T$ . To fit the order of the regret, we first convert it into log scale. Then we input the last  $1\%$  of data, run linear regression, and use ordinary least squares to estimate the slope of the line, which is the fitted order of  $\mathrm{Reg}_T$ .
+
+Baseline Algorithms. We consider three baseline algorithms for comparison, Doubler (Ailon et al., 2014) and Sparring (Ailon et al., 2014)
+
+**Doubler** is the first approach that transforms aueling bandit problem into a standard multi-armed bandit (MAB) problem. It operates in epochs of exponentially increasing length: in each epoch, the left arm is sampled from a fixed distribution, and the right arm is selected using a MAB algorithm to minimize regret against the left arm. The feedback received by the MAB algorithm is the number of wins the right arm achieves compared to the left arm. Under linear link assumption, Doubler has been proven to experience regret as the same order as underlying MAB algorithm. For continuous action space and general concave utility, we choose Bandit Gradient Descent (BGD, (Flaxman et al., 2004)), with regret  $\mathcal{O}(T^{3/4})$ , as the underlying MAB algorithm.
+
+Sparring initializes two MAB instances and lets them compete against each other. It is a heuristic improvement over Doubler. Although it does not come with a regret upper bound guarantee, it is reported to enjoy better performance compared to Doubler (Ailon et al., 2014). We also choose BGD as the underlying MAB algorithm.
+
+To validate efficiency-robustness trade-off in Proposition 2 and Proposition 3, we try different  $\alpha$  values. In the first row of Figure 3, we consider  $\alpha = 0.05$  for DBGD and  $\alpha = 0.9$  for RoSMID. We consider  $\rho \in [0.5,0.95]$ . According to the theoretical prediction, both DBGD and RoSMID can tolerate at most  $\mathcal{O}(T^{0.95})$  agnostic corruption, which aligns with experiment results. This is because the estimated order of regret is less than 1. In the second row of Figure 3, we consider  $\alpha = 0.1$  for DBGD and  $\alpha = 0.8$  for RoSMID. According to the theoretical prediction, both DBGD and RoSMID can tolerate at most  $\mathcal{O}(T^{0.9})$  agnostic corruption. From the figure, we can see they
+
+have smaller fitted order of regret when  $\rho = 0.5$  while at a cost of tolerating smaller magnitude of agnostic corruption (it has linear regret when  $\rho = 0.95$ ), which reveals intrinsic tradeoff between efficiency and robustness.
+
+![](images/793e80753eb2ee02e084d3356cbc49550f1e5b0560d31501871d88281a7833ad.jpg)  
+Figure 3: In the first row, we consider  $\alpha = 0.05$  for DBGD and  $\alpha = 0.9$  for RoSMID. In the second row, we consider  $\alpha = 0.1$  for DBGD and  $\alpha = 0.8$  for RoSMID. Given the  $\alpha$ , for each algorithm, we tested its performance under  $\rho$ -Imperfect Human feedback with  $\rho = 0.5, 0.8, 0.95$ . For each  $\rho$ , we presented a line plot of the average regret over five simulations, accompanied by  $\pm$  one standard deviation shown by the shaded region. In the legend,  $o$  denotes the estimated line slope, calculated using least squares on the last  $1\%$  of the data.
+
+# G.1 COMPUTATIONAL COMPLEXITY
+
+Firstly, we emphasize that our robustified algorithm retains the same computational complexity as the originals. This indicates that incorporating robustness does not compromise computational efficiency, ensuring the practicality and scalability of the enhanced algorithm for real-world applications. Specifically, RoSMID has a running time of  $O(d^3 T)$ , where the  $d^3$  term arises from the singular value decomposition. Similarly, the robustified DBGD algorithm runs in  $O(dT)$ , assuming that  $P_A(a_t + \delta u_t)$  can be efficiently computed through matrix transformations.
+
+As our work is the first to address corruption-robust learning from a continuous-action strictly concave utility, there is no directly comparable literature for this exact setting. However, we provide relevant comparisons for reference. RCDB (Di et al., 2024), designed for corruption-robust learning in a linear contextual dueling bandit framework, has a running time of  $O(K^2 d^2 T)$ , where  $K$  is the number of actions. Similarly, Versatile-DB (Saha & Gaillard, 2022), which addresses dueling bandit problems in a multi-armed bandit setting under adversarial attacks, has a running time of  $O(K^2 T)$ , where  $K$  represents the number of arms.  $K^2$  comes from solving  $\arg \max_{\boldsymbol{x} \in S} (\langle \boldsymbol{x}, \boldsymbol{y} \rangle - f(\boldsymbol{x}))$  using second order Newton methods and  $f$  is a convex function.
\ No newline at end of file
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+# LEARNING FROM WEAK LABELERS AS CONSTRAINTS
+
+Vishwajeet Agrawal*, Rattana Pukdee*, Maria-Florina Balcan, Pradeep Ravikumar Carnegie Mellon University
+
+{vishwaja, rpukdee, ninamf, pkr}@cs.cmu.edu
+
+# ABSTRACT
+
+We study programmatic weak supervision, where, in contrast to labeled data, we have access to weak labelers, each of which either abstains or provides noisy labels corresponding to any input. Most previous approaches typically employ latent generative models that model the joint distribution of the weak labels and the latent "true" label. The caveats are that this relies on assumptions that may not always hold in practice, such as conditional independence assumptions over the joint distribution of the weak labelers and the latent true label, and more general implicit inductive biases in the latent generative models. In this work, we consider a more explicit form of side information that can be leveraged to denoise the weak labeler, namely the bounds on the average error of the weak labelers. We then propose a novel but natural weak supervision objective that minimizes a regularization functional subject to satisfying these bounds. This turns out to be a difficult constrained optimization problem due to discontinuous accuracy bound constraints. We provide a continuous optimization formulation for this objective through an alternating minimization algorithm that iteratively computes soft pseudo labels on the unlabeled data satisfying the constraints while being close to the model, and then updates the model on these labels until all the constraints are satisfied. We follow this with a theoretical analysis of this approach and provide insights into its denoising effects in training discriminative models given multiple weak labelers. Finally, we demonstrate the superior performance and robustness of our method on a popular weak supervision benchmark.
+
+# 1 INTRODUCTION
+
+Acquiring high-quality labeled data, which is critical for supervised learning, is often very expensive. The burgeoning field of programmatic weak supervision addresses this caveat by training models from "weak labelers," which are functions that either abstain or provide noisy labels corresponding to any input (Zhang et al., 2022). Such weak labelers could range from domain-expert-specified functions to rule-based systems, heuristics, or even pre-trained models. There are two main challenges in learning a predictor using only weak labelers: each weak labeler covers only part of the input space (due to abstention) and it is typically noisy. Thus, one has to generalize beyond the coverage regions, as well as denoise within them.
+
+To learn from weak labelers, most prior works draw from developments in related fields such as crowdsourcing (Estellés-Arolas & González-Ladrón-de Guevara, 2012; Wazny, 2017), employing latent generative models that model the joint distribution of the weak labels and the latent "true" label. Conditional inference on the true label then provides a natural approach to aggregate multiple noisy labels into less noisy "pseudo-labels" which provide an estimate of the true label. However, this relies on the implicit inductive bias since non-parametric mixture models are known to not be identifiable without any further assumptions (Kasahara & Shimotsu, 2014; Aragam et al., 2020). For example, Snorkel (Ratner et al., 2017; 2018) imposes both (a) conditional independence assumptions among the weak labelers and the true label (b) parametric assumptions on the latent generative model. However, these assumptions might not be applicable and are, in general, difficult to validate with domain experts. In addition, the denoising step does not take advantage of modern hypothesis classes such as deep neural networks. Instead, it is common to follow a two-staged approach: denoising step to estimate pseudo-labels and then train a neural network using these pseudo-labels.
+
+We also note that this learning from weak labelers is closely related to a subfield of learning with noisy labels that also faces a denoising problem (Angluin & Laird, 1988; Blum et al., 1998; Natarajan et al., 2013). Here, we assume that a noisy label is obtained by passing the unknown true label through a noise channel that is usually assumed to be independent of the input (Natarajan et al., 2013). There is also a general noise model that could depend on the input called Massart noise (Massart & Nédélec, 2006) which is known to lead to a computationally hard learning problem (Diakonikolas et al., 2019a; 2022a). A recent line of work shows that we can make headway in this difficult setting, provided we have additional side information such as the marginal distribution is structured (Awasthi et al., 2015; 2016; Diakonikolas et al., 2022b). There have been considerable recent advances in learning under such noise models and other difficult noise models such as the malicious noise model (Awasthi et al., 2014; 2017; Diakonikolas et al., 2018; 2019b; Prasad et al., 2018). However, we remark that results from learning with noisy labels are not directly applicable to our setting since the levels of noise induced by our weak labelers do not satisfy the amount of structure needed by these works.
+
+Regardless, we believe that programmatic weak supervision can considerably benefit from connections with the field of learning from noisy labels and that it forms a rich area of research. As our first step towards leveraging such a connection, we consider the following natural and explicit inductive bias that domain experts might provide, a bound on the accuracy of each weak labeler (similar to a bound on the fraction of corrupted data in the learning from noisy label setting), where accuracy can either be with respect to the target Bayes optimal classifier or with respect to the underlying stochastic label distribution. Our first contribution is to develop a constraint-based approach to aggregate weak labelers by viewing the accuracy bound as a constraint, and accordingly impose this constraint on the model to be learned. We pose the problem as learning the simplest model (with respect to some regularization functional) that satisfies all of these constraints (Section 2). This objective is however difficult along multiple facets: it is a constrained optimization problem, with discontinuous accuracy bound constraints. While there is a rich literature on continuous convex surrogates of classification accuracy (or error), they are not immediately applicable in our setting since a bound on the accuracy does not translate to a bound on any particular surrogate loss, and recall that it is these accuracy bounds that form our domain-expert provided side information.
+
+Our second contribution is a scalable optimization algorithm. This is based on a projection function that projects the model onto the set of all functions that satisfy the constraints. First, we show that the projection objective can be reduced to a linear program when the error bounds are given with respect to the target Bayes optimal classifier (Section 3.1), and a convex optimization problem when they are given with respect to the target label distribution (Section 3.2). The convex optimization problem for the latter case results in an efficient and robust algorithm that can also be used as an approximation when the constraints are given with respect to the target classifier.
+
+Our third contribution is a rigorous analysis of our constrained estimator (Section 4). We note that the traditional statistical learning theoretic tools are not applicable since we have no labeled data. We provide two tools to bound the error of the learned classifier: i) an agreement region based analysis ii) corrected triangle inequality. Both approaches involve disparate denoising coefficients that track the benefit of having multiple rather than a single weak labeler. We further refine this to provide a tighter extension of the bounds from the limited support to the overall space using a subtle Lipschitz notion that might be of independent interest even outside the context of this paper. Lastly, we demonstrate the robustness and competitive performance of our method on a popular weak supervision benchmark (Zhang et al., 2021) in Section 5.
+
+# 2 LEARNING FROM WEAK LABELER CONSTRAINTS
+
+Setup: We consider the general setting of multi-class classification, with input space  $\mathcal{X}$ , output space  $\mathcal{Y} = \{1,\dots ,K\}$ , joint distribution  $P$  over  $\mathcal{X}\times \mathcal{Y}$ , and marginal distribution over  $\mathcal{X}$  denoted by  $P_{X}$  which we assume to be continuous. We use uppercase letters (e.g.,  $X$ ) to represent random variables and lowercase letters (e.g.,  $x$ ) for deterministic variables. For a vector  $x\in \mathbb{R}^d$ , we denote  $x_{k}$  as the  $k$ th entry of  $x$ . Let  $p^{*}(x) = p(y|x)\in \mathbb{R}^{K}$  be the conditional distribution of  $y$  given  $x$ , and let  $f^{*}(x) = \arg \max_{k}p^{*}(x)_{k}$  denote the Bayes optimal classifier. Our goal is to learn a classifier  $f:\mathcal{X}\to \mathcal{Y}$  that minimizes the expected misclassification error with respect to the Bayes optimal classifier  $f^{*}$ , i.e. minimizes  $\operatorname *{Pr}(f(X)\neq f^{*}(X))$ , or the error with respect to the stochastic label
+
+itself:  $\operatorname*{Pr}(f(X)\neq Y)$ . In our setting, we assume that we only have access to unlabeled data drawn i.i.d. from  $P_{X}$  and weak labelers  $\{g_j\}_{j\in [m]}$ , where each weak labeler  $g_{j}:\mathcal{X}\to \mathcal{Y}\cup \{\emptyset \}$  maps any point in the instance space to the label space or abstains from prediction (denoted by  $\varnothing$ ). We denote the coverage set of the weak labeler by  $S_{j}\coloneqq \{x\in \mathcal{X}\mid g_{j}(x)\neq \emptyset \}$ . We consider formulating a classifier from a smooth real-valued function  $h:\mathcal{X}\rightarrow \mathbb{R}^{K}$  from some hypothesis class  $\mathcal{F}$  e.g. neural networks. This function induces a conditional distribution denoted by  $p_h$ , which we obtain by applying the softmax function,  $p_h(x)_k = \exp (h(x)_k) / \sum_{i = 1}^K\exp (h(x)_i)$ . For any conditional distribution  $p$  such that  $\arg \max_y p(x)_y$  is unique for each  $x$ , we denote the corresponding classifier as  $\operatorname {clf}(p)$  where  $\operatorname {clf}(p)(x) = \arg \max_k p(x)_k$ .
+
+Accuracy bounds and the corresponding constraints: We assume we have additional side information about the average error of each weak labeler  $g_{j}$  on its coverage set  $S_{j}$ . First, this can be an error bound with respect to the Bayes optimal classifier  $f^{*}$  given by
+
+$$
+\rho_ {\operatorname {c l f}} \left(g _ {j}, f ^ {*}\right) := \Pr_ {X \sim P _ {X}} \left(g _ {j} (X) \neq f ^ {*} (X) \mid X \in S _ {j}\right) \leq \eta_ {j}. \tag {1}
+$$
+
+Since our target Bayes optimal classifier  $f^{*}$  only differs from  $g_{j}$  by a fraction of at most  $\eta_{j}$  on  $S_{j}$ , a natural learning strategy is to encourage our hypothesis to also have the same behavior (differ from  $g_{j}$  at most  $\eta_{j}$ ). This leads to a constraint  $\rho_{\mathrm{clf}}(g_j,\operatorname {clf}(p_f))\leq \eta_j$ . We propose to learn the simplest hypothesis  $f$  (with respect to a regularization functional  $\mathcal{R}(f)$ ) that satisfies all of these constraints which can be formalized as the following objective with constraints on classifier outputs:
+
+$$
+\min  _ {f \in \mathcal {F}} \mathcal {R} (f) \text {s . t .} \rho_ {\operatorname {c l f}} \left(g _ {j}, \operatorname {c l f} \left(p _ {f}\right)\right) \leq \eta_ {j}, \forall j \in [ m ]. \tag {2}
+$$
+
+On the other hand, if we may instead have access to an error with respect to the stochastic labels sampled from  $p^*$ , the bound is given by
+
+$$
+\rho_ {\text {d i s t}} \left(g _ {j}, p ^ {*}\right) := \Pr_ {X \sim P _ {X}, Y \sim p ^ {*} (X)} \left(g _ {j} (X) \neq Y \mid X \in S _ {j}\right) \leq \eta_ {j}. \tag {3}
+$$
+
+Here  $g_{j}$  is a classifier but  $Y$  is a random variable distributed according to  $p^{*}(X)$ , so the probability that  $g_{j}(x)$  is not equal to  $y$  for each  $x$  is given by  $\sum_{k \neq g_{j}(x)} p^{*}(x)_{k}$ , thus  $\rho_{\mathrm{dist}}(g, p^{*}) \equiv \mathbb{E}_{X \sim P_{X}}[\sum_{k \neq g_{j}(x)} p^{*}(x)_{k}]$ . Similar to before, we have the following objective with constraints on classifier probabilistic labels:
+
+$$
+\min  _ {f \in \mathcal {F}} \mathcal {R} (f) \text {s . t .} \rho_ {\text {d i s t}} \left(g _ {j}, p _ {f}\right) \leq \eta_ {j}, \forall j \in [ m ] \tag {4}
+$$
+
+# 3 ALGORITHMS
+
+In this section, we will first examine an algorithm for solving the learning objective with constraints on classifiers (equation 2) through an alternating minimization algorithm. This requires a projection step that projects a conditional distribution into the set of conditional distributions that satisfy the constraints. We will describe a method for estimating this projection and show that it can be reduced to a linear program. Finally, we will extend our algorithm to solve the objective with constraints on the distribution (equation 4). We provide proofs for results in this section in Appendix B.
+
+# 3.1 LEARNING OBJECTIVE WITH CONSTRAINTS ON CLASSIFIER
+
+The key challenge in solving equation 2 is that it is a constrained optimization problem and these constraints are not continuous. As a key initial step, we define a projection loss of a conditional distribution  $p$  to a class of classifiers that satisfies the constraint. We denote a set of all conditional distributions that satisfy the constraints as
+
+$$
+Q _ {\mathrm {c l f}} = \bigcap_ {j} Q _ {j, \mathrm {c l f}}, \text {w h e r e} Q _ {j, \mathrm {c l f}} = \left\{q: \mathcal {X} \rightarrow \Delta^ {K} \mid \rho_ {\mathrm {c l f}} \left(g _ {j}, \operatorname {c l f} (q)\right) \leq \eta_ {j} \right\}. \tag {5}
+$$
+
+We consider a projection loss from any conditional distribution  $p$  to a set of conditional distributions  $Q$  given by,
+
+$$
+d (p, Q) = \inf  _ {q \in Q} \mathbb {E} _ {X} \left[ D _ {\mathrm {K L}} (q (X) | | p (X)) \right] \tag {6}
+$$
+
+where  $D_{\mathrm{KL}}$  is the KL-divergence. We can show that  $d(p, Q_{\mathrm{clf}})$  is zero if and only if  $p \in Q_{\mathrm{clf}}$ .
+
+Lemma 3.1. For any  $p$  such that  $\operatorname{clf}(p)$  is well defined, i.e.  $\arg \max_y p(x)_y$  is unique for each  $x$ , we have  $d(p, Q_{\mathrm{clf}}) = 0$  if and only if  $p \in Q_{\mathrm{clf}}$ .
+
+From Lemma 3.1, we may replace the constraints in equation 2 with the projection loss instead. This would still lead to the same optimal solution. We have an objective
+
+$$
+\min  _ {f \in \mathcal {F}} \mathcal {R} (f) \text {s . t .} d \left(p _ {f}, Q _ {\mathrm {c l f}}\right) = 0 \tag {7}
+$$
+
+We consider a relaxation of this objective into an unconstrained optimization problem,
+
+$$
+\min  _ {f \in \mathcal {F}} \mathcal {R} (f) + \alpha \cdot d \left(p _ {f}, Q _ {\mathrm {c l f}}\right) \tag {8}
+$$
+
+where  $\alpha$  trades off satisfying the constraints with a notion of simplicity of distribution specified through the regularization. This relaxed objective is now more suitable for a gradient-based optimization since it is continuous. We can rewrite this objective as
+
+$$
+\min  _ {q \in Q _ {\mathrm {c l f}}, f \in \mathcal {F}} \mathcal {L} (f, q) := \mathcal {R} (f) + \alpha \mathbb {E} _ {X} [ D _ {\mathrm {K L}} (q (X) | | p _ {f} (X)) ] \tag {9}
+$$
+
+This is a minimization problem in  $f \in \mathcal{F}, q \in Q_{\mathrm{clf}}$  for which we propose to use an alternating minimization algorithm where we keep track of a sequence of  $(f_t, q_t)$  and update as follows:
+
+1.  $f_{t + 1} = \arg \min_{f\in \mathcal{F}}\mathcal{R}(f) + \alpha \mathbb{E}_X[D_{\mathrm{KL}}(q_t(X)||p_f(X))]$  (10)  
+2.  $q_{t + 1} = \arg \min_{q\in Q_{\mathrm{clf}}}\mathbb{E}_X[D_{\mathrm{KL}}(q(X)||p_{f_{t + 1}}(X))]$  (11)
+
+We can show that this algorithm leads to a sequence of  $(f_t, q_t)$  that is decreasing in  $\mathcal{L}(f_t, q_t)$ :
+
+Lemma 3.2. A sequence  $(f_t, q_t)$  with an update in equation 10 and equation 11 satisfies  $\mathcal{L}(f_{t+1}, q_{t+1}) \leq \mathcal{L}(f_t, q_t)$ .
+
+Proof.  $\mathcal{L}(f_{t + 1},q_{t + 1})\leq \mathcal{L}(f_{t + 1},q_t)$  from minimization in equation 11 and  $\mathcal{L}(f_{t + 1},q_t)\leq \mathcal{L}(f_t,q_t)$  from minimization in equation 10.
+
+Solving step 1 (equation 10). It can be seen that the KL term in equation 10 is simply the cross-entropy loss with respect to the pseudo soft labels given by  $q$ . Thus this first step of the alternating minimization algorithm simply fits a regularized classifier to these pseudo labels. Further, one does not need to estimate the exact minima at each step, as long as the next iterate reduces the objective, i.e.  $\mathcal{L}(f_{t + 1},q_{t + 1}) < \mathcal{L}(f_t,q_t)$ . Thus one can take a single or few steps of gradient descent in equation 10 to get  $f_{t + 1}$  from  $f_{t}$ .
+
+Solving step 2 (equation 11). This step of the alternating minimization algorithm computes the soft pseudo labels on the training data so that the constraints are satisfied. This is a much more involved step, which we detail in the following section, where we show that estimating the projection  $q_{t}$  can be reduced to a linear program. Before doing so, we first remark that the constraints on classifiers may not be robust to a small perturbation. For example, if one has a uniform conditional distribution  $p_{\mathrm{unif}} = (1 / K,\dots ,1 / K)$ , we can make a small perturbation  $\Delta p$  of size at most  $\epsilon$  such that  $\operatorname {clf}(p_{\mathrm{unif}} + \Delta p) = k$  for any label  $k$ . This implies that there exists a conditional distribution  $p$  such that  $d(p,Q)$  is small but  $\operatorname {clf}(p)$  can be arbitrary.
+
+Lemma 3.3. Let  $Q = \{q\in \mathcal{X}\to \Delta^K\mid \operatorname {clf}(q)\in \mathcal{H}\}$  for some set of classifiers  $\mathcal{H}$ . For any  $\epsilon >0$  and any classifier  $f$ , there exists a conditional distribution  $p$  such that  $\operatorname {clf}(p) = f$  and  $d(p,Q) < \epsilon$ .
+
+To alleviate this problem, we enforce an additional constraint by making sure that  $q$  puts a higher probability on one label than the rest,
+
+$$
+\Lambda_ {\epsilon} := \{q \in \mathcal {X} \rightarrow \Delta^ {K} \mid \text {f o r e a c h} x, \exists y: q (x) _ {y} \geq (1 + \epsilon) q (x) _ {k} \text {f o r} k \neq y \} \tag {12}
+$$
+
+With this additional constraint, we can show that  $d(p, Q)$  bounds the error in satisfaction of the classifier constraints, and further show that the projection problem can be solved efficiently.
+
+Proposition 3.4. For any  $\epsilon, \nu > 0$  there exists  $\delta(\epsilon, \nu) > 0$  such that if  $d(p, \Lambda_{\epsilon} \cap Q_{\mathrm{clf}}) < \delta(\epsilon, \nu)$  then  $\rho_{\mathrm{clf}}(g_j, \operatorname{clf}(p)) \leq \eta_j + \nu / |S_j|$ , where  $|S_j| = P_X(X \in S_j)$  is the size of coverage set  $S_j$  of weak labeler  $g_j$  with respect to marginal distribution  $P_X$ .
+
+Proposition 3.5. Given  $Q = \Lambda_{\epsilon} \cap Q_{\mathrm{clf}}$  for  $\Lambda_{\epsilon}$  defined in equation 12 and  $Q_{\mathrm{clf}}$  defined in equation 5, for any conditional distribution  $p$ , the optimal projection  $q^{*} = \arg \min_{q \in Q} \mathbb{E}[D_{KL}(q(X)||p(X))]$  can be found as follows,
+
+1. Solve for optimal  $\operatorname{clf}(q^*)$  as
+
+$$
+\operatorname {c l f} \left(q ^ {*}\right) = \underset {h \in \mathcal {X} \rightarrow \mathcal {Y}} {\arg \min } \mathbb {E} _ {X} \left[ \operatorname {P r o j} _ {K L} (p (X); \Delta_ {h (X), \epsilon} ^ {K}) ]: \rho_ {\operatorname {c l f}} \left(g _ {j}, h\right) \leq \eta_ {j} \right. \tag {13}
+$$
+
+where
+
+$$
+\Delta_ {y, \epsilon} ^ {K} = \left\{r \in \Delta^ {K} \mid r _ {y} \geq (1 + \epsilon) r _ {k}, \forall k \neq y \right\},
+$$
+
+$$
+\operatorname {P r o j} _ {K L} (u; S) = \min  _ {r \in S} D _ {K L} (r \| u). \tag {14}
+$$
+
+2. Solve for an optimal distribution  $q^{*}$  given the optimal  $\operatorname {clf}(q^*)$  . For each  $x$
+
+$$
+q ^ {*} (x) = \arg \min  _ {r} D _ {K L} (r \| p (x)): r \in \Delta_ {\operatorname {c l f} \left(q ^ {*}\right) (x)} ^ {K, \epsilon} \tag {15}
+$$
+
+A key ingredient is the subset of the simplex  $\Delta_{y,\epsilon}^{K}$ , comprising probabilities that are more confident about the label  $y$  with some margin  $\epsilon$ . For any  $u \in \Delta^{K}$ , and label  $y \in [K]$ ,  $\operatorname{Proj}_{\mathrm{KL}}(u; \Delta_{y,\epsilon}^{K})$  is a convex optimization problem since the objective  $D_{\mathrm{KL}}(r||u)$  is convex in  $r$  and the constraints in  $\Delta_{y,\epsilon}^{K}$  are linear. While we can use a black box convex optimization solver, we solve this more efficiently by reparameterizing into an unconstrained objective and using a fast heuristic approximation (see Appendix D.2 for more details).
+
+Given  $c(x,y) = \mathrm{Proj}_{\mathrm{KL}}(p(x);\Delta_{y,\epsilon}^{K})$  for each  $x,y$ , the optimization in step 1 can be framed as an integer linear programming (ILP) problem by using a one-hot representation of the optimizing variable  $h$  as  $\mu : \mathcal{X} \to \{0,1\}^K$ , with  $\mu(x)_y = 1$  for  $y = h(x)$  and  $\mu(x) = 0$  otherwise. The objective then becomes  $\mathbb{E}_X[\sum_{y=1}^K \mu(X)_yc(X,y)]$  and the constraint  $\rho_{\mathrm{clf}}(g_j,h) \leq \eta_j$  becomes  $\mathbb{E}[\sum_{y \neq g_j(X)} \mu(X)_y] \leq \eta_j$ . For each  $x$ , we have an additional constraint  $\sum_{y=1}^K \mu(x)_y = 1$ . Both the objective and constraints are linear in  $\mu$ , resulting in an ILP. ILPs are usually inefficient to solve, however, we find empirically that the LP relaxation works very well for this class of ILPs (see Appendix D.1 for more details). One issue with solving the ILP is that it can still be slow even with LP relaxation since it involves one constraint for each unlabeled data  $x$ . We also can't take a small batch size when using a stochastic gradient descent since the constraints involve the population means of the weak labeler accuracies. Finally, this is not robust to misspecification of the error bounds, as the constraints might become infeasible. In the next section, we propose our method for the learning objective with constraints on distribution (equation 4). Interestingly, we found that it can also be viewed as an approximation of the algorithm presented in this section, yet it overcomes issues of efficiency and robustness, resulting in a more practical solution that is effective in both settings.
+
+# 3.2 LEARNING OBJECTIVE WITH CONSTRAINTS ON DISTRIBUTION
+
+Let  $Q_{\mathrm{dist}}$  be the set of conditional distributions that satisfy the constraints on distribution,
+
+$$
+Q _ {\text {d i s t}} = \bigcap_ {j} Q _ {j, \text {d i s t}}, \text {w h e r e} Q _ {j, \text {d i s t}} = \left\{q: \mathcal {X} \rightarrow \Delta^ {K} \mid \rho_ {\text {d i s t}} \left(g _ {j}, q\right) \leq \eta_ {j} \right\}. \tag {16}
+$$
+
+Now, with a similar proof, we would also have a version of Lemma 3.1 for the constraints on distribution, and further  $d(p, Q_{\mathrm{dist}})$  bounds the error in satisfying the constraints without the need of imposing any additional constraint.
+
+Proposition 3.6. For  $Q_{\mathrm{dist}}$  defined in equation 16, we have  $d(p, Q_{\mathrm{dist}}) = 0$  if and only if  $p \in Q_{\mathrm{dist}}$ . Furthermore, for any  $\nu > 0$  there exists  $\delta(\nu) > 0$  such that if  $d(p, Q_{\mathrm{dist}}) < \delta(\nu)$  then  $\rho_{\mathrm{dist}}(g_j, p) \leq \eta_j + \nu / |S_j|$ , where  $|S_j| = P_X(S_j)$  is the size of coverage set  $S_j$  of weak labeler  $g_j$  with respect to marginal distribution  $P_X$ .
+
+This allows us to derive the same alternating minimization algorithm as in equation 10 and equation 11. While the first step is identical, the main benefit of working with the constraints on distribution is that the second projection step can be directly posed as a convex optimization problem.
+
+$$
+\min  _ {q: \mathcal {X} \mapsto \Delta^ {| K |}} \mathbb {E} [ D _ {\mathrm {K L}} (q (X) | | p (X)) ] \quad s. t. \quad \mathbb {E} [ q (X) _ {g _ {j} (X)} | x \in S _ {j} ] \geq 1 - \eta_ {j}, \forall j \in [ m ]. \tag {17}
+$$
+
+We can also derive the optimal solution of equation 17.
+
+Proposition 3.7. Given  $Q_{\mathrm{dist}}$  as defined in equation 16, for any conditional distribution  $p$ , the optimal projection  $q^{*} = \arg \min_{q\in Q_{\mathrm{dist}}}\mathbb{E}[D_{KL}(q(X)||p(X))]$  is given by  $q^{*} = p^{\lambda^{*}}$  where
+
+$$
+p ^ {\boldsymbol {\lambda}} (x) _ {y} := Z _ {\boldsymbol {\lambda}} (x) \exp \left(\sum_ {j: g _ {j} (x) = y} \boldsymbol {\lambda}\right) p (x) _ {y} \tag {18}
+$$
+
+$Z_{\lambda}(x)$  is the normalization constant. The optimal  $\lambda^{*}$  is given by the following optimization:
+
+$$
+\boldsymbol {\lambda} ^ {*} = \min  _ {\boldsymbol {\lambda} \geq \mathbf {0}} \sum_ {j = 1} ^ {m} \boldsymbol {\lambda} _ {j} \quad s. t. \quad \rho_ {\text {d i s t}} \left(g _ {j}, p ^ {\boldsymbol {\lambda}}\right) \leq \eta_ {j} \forall j \in [ m ] \tag {19}
+$$
+
+The optimization for  $\lambda^{*}$  can be solved by a coordinate-wise alternating minimization. Given the current iterate  $\lambda$ , pick some  $j$ , and update  $\lambda_{j}$  as
+
+$$
+\lambda_ {j} = \min  _ {\lambda \geq 0} \lambda \text {s . t .} \rho_ {\operatorname {d i s t}} \left(g _ {j}, p ^ {\lambda}\right) = 1 - \mathbb {E} \left[ Z _ {\lambda} (x) \exp (\lambda) p ^ {\lambda - j} (X) _ {g _ {j} (X)} \right] \leq \eta_ {i} \tag {20}
+$$
+
+where  $\lambda_{-j} = (\lambda_1, \dots, \lambda_{j-1}, 0, \lambda_{j+1}, \dots, \lambda_m)$ . Note that while optimizing for a particular coordinate  $\lambda_j$ , we only consider the constraint associated with  $g_j$ . This can be seen as a relaxation of the true constraint, which is an intersection of  $m$  constraints. While this does not ensure satisfaction of all of the constraints as the alternating minimization proceeds, it lends us a simple method to solve this by a Newton-Raphson root-finding method as follows:
+
+Reset:  $\lambda_{i} = 0$
+
+Repeat until convergence:  $\lambda_{i}\gets \lambda_{i} + \log (1 + (1 - \eta_{i} - \tau_{i}(h_{i}))_{+} / \tau_{i}(h_{i}(1 - h_{i})))$
+
+$$
+h _ {i} (x) = p ^ {\lambda} (x) _ {g _ {i} (x)} \tag {21}
+$$
+
+$$
+\tau_ {i} (h) = \mathbb {E} [ h (X) | X \in S _ {i} ]
+$$
+
+For derivations and proofs, see appendix B. For computational efficiency, we also propose a single parallel update for all  $\lambda_{i}$  instead of performing the alternating minimization sequentially by using a single step of the above Newton-Raphson method parallelly across all  $i$  given the starting point  $\lambda = 0$  as:
+
+$$
+\boldsymbol {\lambda} _ {i} = \log (1 + (1 - \eta_ {i} - \tau_ {i} (p _ {i})) _ {+} / \tau_ {i} (p _ {i} (1 - p _ {i}))), (p _ {i} (x) = p (x) _ {g _ {i} (x)}). \tag {22}
+$$
+
+This does not give an exact projection of  $p$  on  $Q_{\mathrm{dist}}$ , but can be thought of as an approximate projection that finds a  $q$  that is closer to  $Q_{\mathrm{dist}}$  than  $p$  is to  $Q$ , on average. Overall, we summarize our learning algorithm (equation 10, equation 11) with our proposed projection step below,
+
+Summary of our learning algorithm. We have an alternating minimization algorithm where we track the pair  $(f_t, q_t)$ . Here,  $f_t: \mathcal{X} \to \mathbb{R}^K$  is a real-valued function from our hypothesis class (e.g., neural networks), and  $q_t$  represents its projection onto the set of distributions that satisfy our constraints. At each iteration  $t$ , we perform the following steps
+
+# 1. Estimate  $q_{t + 1}$  given  $f_t$ .
+
+(a) Calculate a conditional distribution given by  $f_{t}$  with a softmax function  $\sigma$ ,
+
+$$
+p _ {f _ {t}} (x) = \operatorname {s o f t m a x} \left(f _ {t} (x)\right) \in \Delta^ {K} \tag {23}
+$$
+
+(b) We use a one-step parallel Newton-Raphson update to calculate  $\lambda_{j}$  for each weak labeler  $j$  (equation 22). For simplicity, for each unlabeled data  $x_{i}$  for  $i = 1,\dots ,n$ , we denote the  $g_{j}(x_{i})^{th}$  coordinate of  $p_{f_t}(x_i)$  as  $z_{ij} \coloneqq p_{f_t}(x_i)_{g_j(x_i)}$  then
+
+$$
+\boldsymbol {\lambda} _ {j} = \log (1 + (1 - \eta_ {j} - \frac {1}{c _ {j}} \sum_ {i: g _ {j} (x _ {i}) \neq \emptyset} z _ {i j}) _ {+} / (\frac {1}{c _ {j}} \sum_ {i: g _ {j} (x _ {i}) \neq \emptyset} (z _ {i j}) (1 - z _ {i j}))), \tag {24}
+$$
+
+$c_{j}$  is the number of data points that  $g_{j}$  does not abstain,  $c_{j} = \sum_{i = 1}^{n}1[g_{j}(x_{i})\neq \emptyset ]$
+
+(c) Finally, we use  $\lambda$  to calculate  $q_{t + 1}(x)$ . We first compute the unnormalized version of  $q_{t + 1}$ ,
+
+$$
+\tilde {q} _ {t + 1} \left(x _ {i}\right) _ {k} = p _ {f _ {t}} \left(x _ {i}\right) _ {k} \exp \left(\sum_ {j: g _ {j} \left(x _ {i}\right) = k} \boldsymbol {\lambda} _ {j}\right). \tag {25}
+$$
+
+Then  $q_{t + 1}(x_i) = \tilde{q}_{t + 1}(x_i) / \sum_{k = 1}^{K}\tilde{q}_{t + 1}(x_i)_k$ .
+
+2. Estimate  $f_{t + 1}$  given  $q_{t + 1}$ . This step is equivalent to fitting a  $f_{t + 1}$  with respect to the soft pseudo labels given by  $q_{t + 1}$ . Our loss of  $f_{t}$  is given by
+
+$$
+\mathcal {L} \left(f _ {t}; q _ {t + 1}\right) = \mathcal {R} \left(f _ {t}\right) + \alpha \cdot \frac {1}{n} \sum_ {i = 1} ^ {n} \ell_ {\mathrm {C E}} \left(f _ {t} \left(x _ {i}\right), q _ {t + 1} \left(x _ {i}\right)\right)
+$$
+
+where  $\ell_{\mathrm{CE}}$  is a cross-entropy loss. We then update  $f_{t + 1}$  with a gradient descent;  $f_{t + 1} = f_t - \nu \cdot \nabla \mathcal{L}$  where  $\nu$  is a learning rate.
+
+We also provide a compact version of our algorithm in Algorithm 1 in Appendix C. Finally, we remark that the sets  $Q_{j,\mathrm{dist}}$  can also be viewed as a convex approximation of the constraint sets  $Q_{j,\mathrm{clf}}$  (see Appendix E for more details). Therefore, the method developed in this section is also applicable (as an approximation) for the setting of constraints on classifiers but it is more efficient and also robust to misspecification in error bounds  $\eta_j$  compared to the algorithm in Section 3.1.
+
+# 4 THEORETICAL ANALYSIS OF THE OBJECTIVE
+
+In this section, we provide some theoretical insights into the denoising effect of weak labeler constraints. We focus on the setting of constraints on classifiers (estimation objective equation 2). We learn a conditional distribution  $p_f$  and thus correspondingly a classifier  $\operatorname{clf}(p_f)$  that minimizes a regularization functional  $\mathcal{R}(f)$  subjected to weak labeler constraints. Minimizing  $\mathcal{R}(f)$  implicitly defines a hypothesis class of classifiers as  $\mathcal{F} = \{\operatorname{clf}(p_f) : \mathcal{R}(f) < \delta\}$  for some  $\delta > 0$ . Let  $\mathcal{H}_j = \{h : \mathcal{X} \to [K] \mid \rho_{\mathrm{clf}}(g_j, h) \leq \eta_j\}$  (here  $\mathcal{H}_j$  is a set of classifiers that differ from  $Q_j$  which is a set of conditional distributions). We learn a classifier  $f \in \widetilde{\mathcal{F}} := \mathcal{F} \cap \bigcap_j \mathcal{H}_j$ . The constraints lead to a smaller class  $\widetilde{\mathcal{F}} \subseteq \mathcal{F}'$ , which implies that a typical measure of complexity  $\mathcal{V}$  of the function class  $\widetilde{\mathcal{F}}$  such as Rademacher complexity is also smaller than  $\mathcal{F}$ . Thus, we can quantify the benefit of the weak labeler constraints via the ratio  $\mathcal{V}(\widetilde{\mathcal{F}}) / \mathcal{V}(\mathcal{F})$ . However, such a function class statistical complexity analysis merely makes the natural qualitative point that we require less labeled data to learn from  $\widetilde{\mathcal{F}}$  than  $\mathcal{F}$ . It does not provide a bound per se on the accuracy of our learned classifier, because we are not using any labeled data to select  $f$ .
+
+# 4.1 ERROR GUARANTEE ON AGREEMENT REGION
+
+Recall that we minimize  $\mathcal{R}(f)$  subject to satisfying constraints. We start with the observation that the weak labeler constraint directly restricts the output of a classifier to be similar to that of the weak labeler in its coverage set. For example, given a weak labeler that makes a constant prediction in a small region, any linear separator that satisfies the accuracy bound constraint can only slice off the coverage set by a fraction of at most  $\eta$  (Figure 1 (left)). This leads to an area in the middle where the decision boundary can't pass through: otherwise, the classifier would differ too much from the weak labeler. This resembles the classical notion of agreement regions (Cohn et al., 1994; Balcan et al., 2006), which we thus use to characterize the impact of weak labelers on  $\mathcal{F}$ .
+
+Definition 4.1. For a set of classifiers  $F$ , the agreement region is the set of all points where any classifier in  $F$  agrees,  $\mathrm{Agree}(F) = \{x \in \mathcal{X} \mid \forall f_1, f_2 \in F, f_1(x) = f_2(x)\}$ .
+
+Under realizability, we can show that the error of any  $f \in \widetilde{\mathcal{F}}$  in the agreement region is zero. Furthermore, we can show that as multiple constraints are used, the resulting agreement region will always be larger than or equal to the union of agreement regions for individual constraints. In Figure 1, we provide an example with linear classifiers where the agreement regions from multiple constraints are larger than the union of the agreement region from each individual constraint. This is a possible explanation of the implicit denoising effect of multiple weak labelers when learning from a function class  $\mathcal{F}$ . Formally, we denote an error of a classifier  $f$  within the set  $S$  as
+
+$$
+\operatorname {e r r} _ {S} (f) := \Pr (f (X) \neq f ^ {*} (X) \mid S). \tag {26}
+$$
+
+Lemma 4.2. For any function class  $\mathcal{F}$ , and corresponding constrained class  $\widetilde{\mathcal{F}}$ , so long as  $f^{*} \in \widetilde{\mathcal{F}}$ , for any  $f \in \widetilde{\mathcal{F}}$ ,  $\mathrm{err}_{\mathrm{Agree}(\widetilde{\mathcal{F}})}(f) = 0$ .
+
+Lemma 4.3. For a function class  $\mathcal{F}$  and constraints  $\mathcal{H}_1, \mathcal{H}_2$ , we have
+
+$$
+\operatorname {A g r e e} (F \cap \mathcal {H} _ {1}) \cup \operatorname {A g r e e} (F \cap \mathcal {H} _ {2}) \subseteq \operatorname {A g r e e} (F \cap \mathcal {H} _ {1} \cap \mathcal {H} _ {2}).
+$$
+
+![](images/fd08c5492c8ada2d5bc1a30b3795f89ca2b3104472dfb82a8ca91c4464396d9a.jpg)  
+Figure 1: The figure shows the agreement region of the class of linear separators given two weak labeler constraints: one weak labeler labels red in the red circle, and the other labels blue in the blue circle. A linear separator can only differ from each weak labeler by a fraction of  $\eta$  of its support, and therefore the agreement region is given by the middle region of each coverage set (left). The agreement regions from multiple constraints are larger than the union of the agreement region from each individual constraint (right).
+
+# 4.2 ERROR GUARANTEE ON THE COVERAGE UNION
+
+In the previous section, we used the geometry of constrained function classes in general. Here, we note that the specific form of our constraints (weak labelers error bound) naturally leads to an error guarantee via triangle inequality.
+
+Lemma 4.4. Let  $\mathcal{H}_j = \{h : \mathcal{X} \to [K] \mid \rho_{\mathrm{clf}}(g_j, h) \leq \eta_j\}$  then for any  $f \in \mathcal{H}_j$
+
+$$
+\operatorname {e r r} _ {S _ {j}} (f) \leq \eta_ {j} + \rho_ {\mathrm {c l f}} \left(g _ {j}, f ^ {*}\right) \tag {27}
+$$
+
+We show that it is possible to have a better guarantee than the combination of the error bound from the triangle inequality when we have multiple constraints with conflicting regions.
+
+Theorem 4.5. Let  $\mathcal{H}_1, \ldots, \mathcal{H}_k$  be constraints of the form  $\mathcal{H}_j = \{h : \mathcal{X} \to [K] \mid \rho_{\mathrm{clf}}(g_j, h) \leq \eta_j\}$ , each with different constant weak labels,  $g_j(X) = j - 1$  for all  $j = 1, \ldots, k$ , then for any  $f \in \bigcap_j \mathcal{H}_j$ , we have
+
+$$
+\operatorname {e r r} _ {S} (f) \leq \sum_ {j = 1} ^ {k} \left(\eta_ {j} + \rho_ {\mathrm {c l f}} \left(g _ {j}, f ^ {*}\right)\right) \frac {\Pr \left(S _ {j}\right)}{\Pr (S)} - \sum_ {\substack {\sigma \subseteq \{1, \dots , k \}, \\ |\sigma | \geq 2}} (2 | \sigma | - 3) \frac {\Pr \left(B _ {\sigma}\right)}{\Pr (S)}. \tag{28}
+$$
+
+when  $S = \bigcup_{j=1}^{k} S_{j}$ ,  $B_{\sigma} = \left(\bigcap_{i \in \sigma} S_{i}\right) \cap \left(\bigcap_{i \in \{1, \ldots, k\} \setminus \sigma} S_{i}^{c}\right)$  are minterms.
+
+The first term on the RHS is the error guarantee from naively combining multiple error guarantees based on the triangle inequality. The improvement is in the second term which is the region where weak labelers have a conflict  $(B_{\sigma})$ , more conflict leads to a better error bound. The main idea is that whenever there is a conflict, a classifier can match with only one weak labeler. Therefore, to be approximately close to all weak labelers, it must closely match the other weak labelers in areas outside of the conflict. We note that while our result is applicable to when each weak labeler has a different label if we have multiple weak labelers that predict the same label, we can merge them into a super weak labeler by combining the coverage region e.g.  $S_{1} \cup S_{2}$  and then derive the new  $\eta$  as a linear combination of  $\eta_{1}, \eta_{2}$  e.g.  $\eta = \frac{\eta_{1} \operatorname*{Pr}(S_{1}) + \eta_{2} \operatorname*{Pr}(S_{2})}{\operatorname*{Pr}(S_{1}) + \operatorname*{Pr}(S_{2})}$ , before applying Theorem 4.5.
+
+# 4.3 EXTENDING ERROR GUARANTEES TO ENTIRE INPUT SPACE
+
+The previous two sections provided error guarantees in subsets of the input space: the first over the agreement region of the constrained function class, and the second on the union of the coverage sets of the weak labelers. In this section, we provided a more nuanced analysis based on a smoothness argument which can be used to generalize beyond the coverage set or the agreement region. Intuitively, if  $f$  performs well on one region, and the underlying probability distribution is smooth then we would expect it to perform well in a neighbor region as well.
+
+Theorem 4.6. Let  $P$  be a joint distribution over  $\mathcal{X} \times \mathcal{Y}$  such that  $\operatorname*{Pr}(Y = y \mid X = x)$  is  $L$ -Lipschitz w.r.t. a metric  $d$  where for all  $k = 1, \ldots, K$  and  $x_1, x_2 \in \mathcal{X}$ ,
+
+$$
+\left| \Pr (Y = k \mid X = x _ {1}) - \Pr (Y = k \mid X = x _ {2}) \right| \leq L d \left(x _ {1}, x _ {2}\right), \tag {29}
+$$
+
+then for any classifier  $f$  and a region  $S$ , we have
+
+$$
+\Pr (f (X) \neq Y) \leq \Pr (f (X) \neq Y \mid X \in S) (1 + t _ {f} (S)) + L u _ {f} (S) \tag {30}
+$$
+
+The terms  $t_f(S), u_f(S)$  are distance between  $P_X$  and  $P_X$  condition on  $X \in S$ ,
+
+$$
+t _ {f} (S) = \sup  _ {k} \left| \frac {\Pr (f (X) = k \mid X \in S) - \Pr (f (X) = k)}{\Pr (f (X) = k \mid X \in S)} \right| \tag {31}
+$$
+
+$$
+u _ {f} (S) = \sum_ {k} \Pr (f (X) = k) W _ {d} \left(P _ {X \mid S \cap \{x \mid f (x) = k \}}, P _ {X \mid \{x \mid f (x) = k \}}\right) \tag {32}
+$$
+
+and  $W_{d}$  is a Wasserstien distance and we denote  $P_{X|A}$  for a distribution  $P_{X}$  condition on  $X \in A$ .
+
+In this theorem, we analyze an error w.r.t. to  $y$  instead of  $f^{*}$  since we require the smoothness of the underlying data distribution. The generalization bound is small whenever  $P_{X}$  is close to  $P_{X}$  condition on  $X\in S$ . First,  $t_f(S)$  is zero whenever  $\operatorname *{Pr}(f(X) = k\mid X\in S) = \operatorname *{Pr}(f(X) = k)$  for all  $k$ , that is, the proportion of label  $k$  predicted by  $f$  is the same in both original distribution and the distribution conditional on  $S$ . Second,  $u_{f}(S)$  is a bit stronger where it is small when the Wasserstein distance is small which is when  $S$  covers areas with high probability mass in  $\mathcal{X}$ . The smoothness of  $f$  is implicit in the term  $u_{f}(S)$  in terms of the regions  $\{x\mid f(x) = k\}$ . We provide full proof in Appendix I. We remark that  $t_f(S),u_f(S)$  only depends on the marginal distribution and, therefore, can be calculated solely from unlabeled data for any classifier  $f$ .
+
+# 5 EXPERIMENTAL EVALUATION
+
+<table><tr><td></td><td>Bio</td><td>CDR</td><td>Chem</td><td>IMDB</td><td>Semeval</td><td>TREC</td><td>Yelp</td><td>Youtube</td><td>Avg. rank</td></tr><tr><td>Sup (V)</td><td>64.21.1</td><td>59.90.6</td><td>35.91.4</td><td>64.31.3</td><td>59.63.0</td><td>52.13.7</td><td>73.41.4</td><td>80.91.2</td><td>6.8</td></tr><tr><td>MV</td><td>56.40.4</td><td>68.40.4</td><td>52.60.5</td><td>75.70.7</td><td>65.32.5</td><td>37.31.1</td><td>69.10.5</td><td>82.20.6</td><td>5.8</td></tr><tr><td>Snorkel</td><td>63.40.5</td><td>61.20.4</td><td>41.81.0</td><td>71.30.6</td><td>59.21.0</td><td>33.41.7</td><td>71.80.9</td><td>76.81.3</td><td>7.8</td></tr><tr><td>LoL(S)</td><td>55.50.8</td><td>67.40.4</td><td>51.60.4</td><td>75.30.6</td><td>53.42.0</td><td>38.72.8</td><td>70.61.4</td><td>79.10.9</td><td>7</td></tr><tr><td>Ours (C)</td><td>63.50.5</td><td>68.50.7</td><td>52.20.8</td><td>74.31.3</td><td>67.52.1</td><td>46.72.7</td><td>78.60.5</td><td>80.71.3</td><td>4.6</td></tr><tr><td>Ours (V)</td><td>62.60.8</td><td>67.90.5</td><td>53.30.5</td><td>72.51.1</td><td>79.42.4</td><td>63.12.1</td><td>75.30.7</td><td>90.21.4</td><td>4.1</td></tr><tr><td>Sup (T)</td><td>75.40.8</td><td>80.70.5</td><td>80.10.2</td><td>81.40.6</td><td>82.91.9</td><td>66.92.3</td><td>87.00.3</td><td>89.70.6</td><td>1.1</td></tr><tr><td>Ours (T)</td><td>64.80.7</td><td>69.10.2</td><td>55.60.3</td><td>74.81.1</td><td>77.31.4</td><td>66.32.9</td><td>78.30.3</td><td>88.00.4</td><td>2.8</td></tr><tr><td>Oursclf(T)</td><td>60.90.9</td><td>66.00.4</td><td>49.80.7</td><td>73.31.0</td><td>81.01.9</td><td>57.21.2</td><td>75.80.5</td><td>87.30.3</td><td>5.1</td></tr></table>
+
+Table 1: Average test accuracy and the corresponding standard error (over 5 random train-val-test split of the data) of our proposed algorithm and the baselines. We bold methods that are within a standard error of the best-performing method. Sup (T) and Sup (V) are supervised learning on the training set and the validation set respectively. MV, Snorkel, and Lol(S) are Majority vote, Snorkel, and Losses over labels (Simple) baselines. Ours is our proposed method (Algorithm 1) with different ways to estimate the error bound  $\eta_{j}$ . Ours (V) is when we estimate  $\eta_{j}$  from the labels from the validation set, given a Beta(1,2) prior. Ours (C) is when we use a constant  $\eta_{j}$  and tune this on the validation set. Ours (T) is when  $\eta_{j}$  is estimated from the labels from the training set. Finally, Ours<sub>clf</sub> is our algorithm with respect to constraints on the classifier (Section 3.1).
+
+Experiment details: We show a comparison of our proposed method and baselines on 8 text classification datasets from the WRENCH benchmark Zhang et al. (2021). We provide the dataset statistics in Appendix G.1. For all methods and datasets, we use a neural network with a single hidden layer and a hidden size of 16 on top of the pre-trained BERT text embeddings. The neural network is trained with a full batch gradient descent with an Adam optimizer with a learning rate in [0.001, 0.003, 0.01], weight decay in [0.001, 0.003, 0.01] and a number of epochs in range(1, 500, 5). We tune these hyperparameters on the validation set of size 100 for each dataset. For our proposed method, we implement Algorithm 1 with an L2 regularization. Since the WRENCH benchmark does not provide error bounds  $\eta_{j}$  for each weak labeler, we consider 3 strategies to estimate  $\eta_{j}$ ; i) we estimate  $\eta_{j}$  from the labels from the validation set, given a  $\mathrm{Beta}(1,2)$  prior, ii) we treat  $\eta_{j}$  as a fixed constant  $\eta$  for all weak labelers and tune this  $\eta$  on the validation set, iii) we estimate  $\eta_{j}$  from the labels from the training set which we treat as a ground truth error of each weak labeler.
+
+See Appendix F for more details on estimating error bound from the validation set. Finally, we also implement our algorithm with respect to constraints on the classifier (Section 3.1) with a linear programming relaxation of the integer linear program (see Appendix G.3 for more details).
+
+Baselines: We compare with three baselines: Majority Vote (MV), Snorkel Metal (Ratner et al., 2017) and Losses over Labels simple (LoL (S)) (Sam & Kolter, 2023). The majority vote and Snorkel are two-staged approaches where we first aggregate weak labels and then train a neural network on the aggregated labels. The Majority vote simply takes the majority vote between the non-abstain weak labels for each data point while Snorkel deploys a latent generative model to estimate soft labels from the weak labels. On the other hand, Losses over Labels (simple) is a one-staged approach that directly minimizes a weighted linear combination of the loss with respect to each weak labeler. We also compare to a baseline of simply performing supervised learning on labels from the validation set, to avoid concerns regarding possibly large validation sets as raised by Zhu et al. (2023).
+
+Results: Our proposed method (Ours (V)) is the best performing approach compared to other baselines (Table 1). This does not solely come from the fact that we have a large number of labels in the validation set since supervised learning on the validation set (with the best hyperparameters) still performs worse than ours on almost every dataset. Surprisingly, our method with a constant  $\eta_{j}$  (Ours (C)), also performs competitively and is the second best performing approach overall. On the other hand, having access to a more accurate error bound can lead to a performance increase across the board (Ours (T)). In addition, we found that most constraints are satisfied by the models trained by our algorithms (see Table G.2).
+
+Ablation on noisy weak labeler error bounds: We assume that the error bound  $\eta_{j}$  is given as a form of domain knowledge (or estimated from a small validation set). As such, we may have  $\eta_{j}$  that can be far from the ground truth error. In this section, we explore this impact on the performance of our algorithm and we do so by adding noise to the "ground truth"  $\eta_{j}$  (estimated from the training labels), given by  $\tilde{\eta}_{j} = \eta_{j} + \epsilon, \epsilon \sim \mathrm{Uniform}(-a, a)$  and use  $\tilde{\eta}_{j}$  as inputs of our algorithm. We see that our algorithm is quite robust and the performance drops graciously as the noise level increases (Figure 2).
+
+![](images/b8a9848dbf97af74cd51f69248cd47c0f07d9edb6510735f319ba437e9165c8d.jpg)  
+Figure 2: Impact of using noisy error bounds  $\tilde{\eta}_j = \eta_j + \epsilon, \epsilon \sim \mathrm{Uniform}(-a,a)$  for  $a \in [0,0.5]$ . The accuracy is averaged over 5 random splits of data and the shaded area is the standard error.
+
+# 6 CONCLUSION
+
+In this paper we provided a principled approach to learn from weak labelers - weak classifiers that label a subset of the input space, without making stringent assumptions about dependencies between them, or about their noise model. Instead, we consider the expected error / accuracy as the only source of information that characterizes the weakness of labelers. This information can either be given by domain experts or can be estimated from a small amount of labeled data given only a weak prior on the accuracy. We finally proposed a robust and scalable algorithm that can learn from multiple noisy weak labelers even when only noisy estimates of expected errors are provided. An interesting future direction could be to combine information about dependencies between weak labelers when such information is given along with the information about the expected error, as well as developing alternative techniques based on mixed integer linear programming for solving our constraint-based approach to aggregate weak labelers by viewing the accuracy bound as a constraint.
+
+# ACKNOWLEDGMENTS
+
+We acknowledge the support of NSF via IIS-2211907 and IIS-1901403, ONR via N00014-23-1-2368 and Bloomberg Data Science Fellowship.
+
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+
+# A ADDITIONAL RELATED WORK
+
+# A.1 PROGRAMMATIC WEAK SUPERVISION.
+
+Programmatic weak supervision Ratner et al. (2016), is a learning paradigm where subject matter expert encodes their domain knowledge in terms of weak labelers (labeling functions). These are functions that either provide labels (weak labels) or abstain from prediction on any input. The key question is how to learn from such weak labels that are potentially noisy. A general approach involves two steps of aggregating weak labels into less noisy labels and then training a discriminative classifier on the inferred labels. Various statistical label models have been developed for label aggregation, and most of these only depend on the weak labels Ratner et al. (2016; 2018; 2019); Bach et al. (2017); Varma et al. (2017); Fu et al. (2020); Kuang et al. (2022). Recent methods incorporate the input information into this weakly supervised learning pipeline Chen et al. (2022); Pukdee et al. (2022); Zhang et al. (2023) e.g. smoothness of the label with respect to the input instance or pretrained-embedding. Alternatively, there is a line of works that learns from weak labels in an end-to-end fashion. The label model and the discriminative model are parameterized and are trained jointly Cachay et al. (2021); Karamanolakis et al. (2021); Awasthi et al. (2020); Sam & Kolter (2023). A key prior work is constrained label learning Arachie & Huang (2021), which involves learning pseudo labels under constraints before training a discriminative model. Unlike our method, CLL's initial step is independent of the function class and remains a two-step process.
+
+# A.2 LEARNING WITH NOISY LABELS
+
+Learning from data with noisy labels has been a prominent research problem that has received substantial attention from the machine learning community. The main assumption is that the observed labels are flipped with some probability. There is a line of works in theoretical machine learning that focuses on the learnability and polynomial-time algorithms for various noise models Kearns & Vazirani (1994); Cesa-Bianchi et al. (1999); Angluin & Laird (1988); Awasthi et al. (2014; 2015); Diakonikolas et al. (2019a); Balcan & Haghtalab (2020). On the other hand, in the statistical learning theory community, various works are concerned with different surrogate losses or unbiased estimators Natarajan et al. (2013); Prasad et al. (2020); Natarajan et al.; Charoenphakdee et al. (2019); Xia et al. (2019; 2020). Recently, numerous studies have focused on learning noisy labels on deep neural networks Song et al. (2022); Karimi et al. (2020); Yu et al. (2019); Han et al. (2018; 2020). Despite rich theoretical and empirical research in this area, these approaches cannot be employed in our context since the noisy labels in our setting are deterministic given the input and label.
+
+# A.3 LEARNING FROM CONSTRAINTS
+
+Constrained optimization focuses on finding the optimal solution to a problem within given constraints Bertsekas (2014); Gill et al. (2019); Chong et al. (2023). The principal method involves converting the constrained optimization problem into an unconstrained one. This can be accomplished by adding auxiliary variables, as seen in the Lagrange multiplier method Rockafellar (1993), or by incorporating a penalty term that penalizes the objective function when constraints are violated Coit et al. (1996); Fortin & Glowinski (2000); Yeniay (2005). Additionally, gradient-based approaches like PGD include a projection step to ensure the solution stays within a feasible region after each update Chen & Wainwright (2015); Madry et al. (2018). For specific types of constraints such as linear or quadratic, there are dedicated algorithms designed to address these constrained scenarios Heady et al. (1963); Dantzig (2002); Frank et al. (1956); Boggs & Tolle (1995). In the field of machine learning, the optimization goal is the empirical loss, and constraints can be used to enforce desirable features in a machine learning model, such as fairness Zemel et al. (2013); Mehrabi et al. (2021), robustness Cohen et al. (2019); Kurakin et al. (2016), or privacy Dwork et al. (2006). Recent studies have begun incorporating constraints derived from domain knowledge into models, such as physics constraints Zhao et al. (2019); Zhu et al. (2019); Karniadakis et al. (2021), explanation constraints Pukdee et al. (2023), and weak supervision Ganchev et al. (2010); Hu et al. (2016), with the aim of improving model performance.
+
+# B PROOFS OF PROPERTIES OF PROJECTION LOSS AND COMPUTING THE PROJECTION
+
+Lemma B.1. For  $Q_{\mathrm{clf}}$  defined in equation 5, i.e.
+
+$$
+Q _ {\mathrm {c l f}} = \bigcap_ {j} Q _ {j, \mathrm {c l f}}, \text {w h e r e} Q _ {j, \mathrm {c l f}} = \left\{q: \mathcal {X} \rightarrow \Delta^ {K} \mid \rho_ {\mathrm {c l f}} \left(g _ {j}, \operatorname {c l f} (q)\right) \leq \eta_ {j} \right\}. \tag {33}
+$$
+
+where
+
+$$
+\rho_ {\operatorname {c l f}} \left(g _ {j}, \operatorname {c l f} (q)\right) = \mathbb {E} \left[ g _ {j} (X) \neq \operatorname {c l f} (q) (X) \mid g _ {j} (X) \neq \phi \right] \tag {34}
+$$
+
+and  $\operatorname{clf}(q)$  is defined as
+
+$$
+\operatorname {c l f} (q) (x) = \arg \max  _ {y} q (x) _ {y} \tag {35}
+$$
+
+(assuming  $\arg \max_y q(x)_y$  is unique. If  $\arg \max_y q(x)_y$  is not unique for some  $x$  then  $\operatorname{clf}(q)$  is not well defined.)
+
+then for any  $p$  such that  $\operatorname{clf}(p)$  is well defined, we have  $d(p, Q_{\mathrm{clf}}) = \inf_{q \in Q_{\mathrm{clf}}} \mathbb{E}[D_{KL}(p(X) || q(X))] = 0$  if and only if  $p \in Q_{\mathrm{clf}}$ .
+
+Proof. If  $p \in Q_{\mathrm{clf}}$ , we would have  $d(p, Q_{\mathrm{clf}}) \leq \mathbb{E}[D_{\mathrm{KL}}(p(X)||p(X))] = 0$ . Conversely when  $d(p, Q_{\mathrm{clf}}) = 0$  we prove that  $p \in Q_{\mathrm{clf}}$  i.e.  $\rho_{\mathrm{clf}}(g_j, p) \leq \eta_j$  for all  $j \in [m]$ .
+
+First, we prove that
+
+$$
+\inf  _ {q \in Q _ {\mathrm {c l f}}} \mathbb {E} [ d _ {T V} (p (X), q (X)) ] = 0 \tag {36}
+$$
+
+where  $d_{TV}$  is defined as
+
+$$
+d _ {T V} (u, v) = 1 / 2 \cdot \sum_ {k = 1} ^ {K} \left| u _ {k} - v _ {k} \right| \tag {37}
+$$
+
+Using Pinsker's inequality,
+
+$$
+d _ {T V} (u, v) \leq \sqrt {D _ {\mathrm {K L}} (u \| v) / 2} \tag {38}
+$$
+
+we have
+
+$$
+\begin{array}{l} \mathbb {E} \left[ d _ {T V} (p (X), q (X)) \right] = \mathbb {E} \left[ d _ {T V} (p (X), q (X)) \right] (39) \\ \leq \mathbb {E} \left[ \sqrt {D _ {\mathrm {K L}} (p (X) | | q (X)) / 2} \right] (40) \\ \leq \sqrt {\mathbb {E} \left[ D _ {\mathrm {K L}} (p (X) | | q (X)) \right] / 2} (\text {f r o m J e n s o n ' s}) (41) \\ \end{array}
+$$
+
+Thus
+
+$$
+\begin{array}{l} \inf  _ {q \in Q _ {\mathrm {c l f}}} \mathbb {E} [ d _ {T V} (p (X), q (X)) ] \leq \inf  _ {q \in Q _ {\mathrm {c l f}}} \sqrt {\mathbb {E} [ D _ {\mathrm {K L}} (p (X) | | q (X)) ] / 2} (42) \\ \leq \sqrt {\inf  _ {q \in Q _ {\mathrm {c l f}}} \mathbb {E} [ D _ {\mathrm {K L}} (p (X) | | q (X)) ] / 2} (43) \\ \leq 0 (44) \\ \end{array}
+$$
+
+Let us say  $p \notin Q_{\mathrm{clf}}$ , there exists some  $\epsilon > 0$  such that  $\rho_{\mathrm{clf}}(g_j, p) \geq \eta_j + \epsilon$  for some  $j \in [m]$ .
+
+$$
+\Pr \left(g _ {j} (X) \neq \operatorname {c l f} (p) (X) \mid X \in S _ {j}\right) \geq \eta_ {j} + \epsilon \tag {45}
+$$
+
+where  $S_{j} = \{x:g_{j}(x)\neq \phi \}$  is the coverage set of  $g_{j}$ . Let  $S^{\prime}\subseteq S_{j}$  defined as  $S^{\prime} = \{x:g_{j}(x)\neq \operatorname {clf}(p)(x)\}$ . Denoting  $|S| = \operatorname *{Pr}(X\in S)$ , since  $|S^{\prime}| / |S_{j}|\geq \eta_{j} + \epsilon$ , and assuming  $|S_{j}| > 0$ , we have  $|S^{\prime}| > 0$ .
+
+Now for any  $q \in Q_{\mathrm{clf}}$ , we have
+
+$$
+\Pr (\operatorname {c l f} (q) (X) \neq g _ {j} (X) | X \in S _ {j}) \leq \eta_ {j} \tag {46}
+$$
+
+Then
+
+$$
+\begin{array}{l} \Pr (\operatorname {c l f} (q) (X) \neq g _ {j} (X) | X \in S _ {j}) = \Pr (\operatorname {c l f} (q) (X) \neq g _ {j} (X) | X \in S ^ {\prime}, X \in S _ {j}) P (X \in S ^ {\prime} | X \in S _ {j}) (47) \\ = \Pr (\operatorname {c l f} (q) (X) \neq g _ {j} (X) | X \in S ^ {\prime}) P (X \in S ^ {\prime}, X \in S _ {j}) / P (X \in S _ {j}) (48) \\ = \Pr (\operatorname {c l f} (q) (X) \neq g _ {j} (X) | X \in S ^ {\prime}) P \left(X \in S ^ {\prime}\right) / P \left(X \in S _ {j}\right) (49) \\ = \Pr (\operatorname {c l f} (q) (X) \neq g _ {j} (X) | X \in S ^ {\prime}) | S ^ {\prime} | / | S _ {j} | (50) \\ \end{array}
+$$
+
+Thus,
+
+$$
+\begin{array}{l} \Pr (\operatorname {c l f} (q) (X) \neq g _ {j} (X) | X \in S ^ {\prime}) = \Pr (\operatorname {c l f} (q) (X) \neq g _ {j} (X) | X \in S _ {j}) | S _ {j} | / | S ^ {\prime} | (51) \\ \leq \eta_ {j} | S _ {j} | / | S ^ {\prime} | (52) \\ \leq \eta_ {j} / (\eta_ {j} + \epsilon) (53) \\ \end{array}
+$$
+
+Hence,
+
+$$
+\begin{array}{l} \Pr (\operatorname {c l f} (q) (X) = g _ {j} (X) \mid X \in S ^ {\prime}) = 1 - \Pr (\operatorname {c l f} (q) (X) \neq g _ {j} (X) \mid X \in S ^ {\prime}) (54) \\ \geq 1 - \eta_ {j} / (\eta_ {j} + \epsilon) (55) \\ \geq \epsilon / (\eta_ {j} + \epsilon) (56) \\ \end{array}
+$$
+
+Since  $\operatorname{clf}(p)(x) \neq g_j(x)$  for  $x \in S'$ ,  $\operatorname{clf}(q)(x) = g_j(x)$  implies  $\operatorname{clf}(p)(x) \neq \operatorname{clf}(q)(x)$ . Thus
+
+$$
+\Pr (\operatorname {c l f} (q) (X) \neq \operatorname {c l f} (p) (X) | X \in S ^ {\prime}) \geq \epsilon / \left(\eta_ {j} + \epsilon\right) \tag {57}
+$$
+
+Let  $S'' = \{x \in S' : \operatorname{clf}(q)(x) \neq \operatorname{clf}(p)(x)\}$ , we have
+
+$$
+\Pr \left(X \in S ^ {\prime \prime} \mid X \in S ^ {\prime}\right) \geq \epsilon / \left(\eta_ {j} + \epsilon\right) \tag {58}
+$$
+
+And thus,
+
+$$
+\Pr \left(X \in S ^ {\prime \prime} \mid X \in S ^ {\prime}\right) P \left(X \in S ^ {\prime}\right) = P \left(X \in S ^ {\prime \prime}, X \in S ^ {\prime}\right) = P \left(X \in S ^ {\prime \prime}\right) \tag {59}
+$$
+
+$$
+\Pr (X \in S ^ {\prime \prime} | X \in S ^ {\prime}) = P (X \in S ^ {\prime \prime}) / P (X \in S ^ {\prime}) \tag {60}
+$$
+
+$$
+P \left(X \in S ^ {\prime \prime}\right) / P \left(X \in S ^ {\prime}\right) \geq \epsilon / \left(\eta_ {j} + \epsilon\right) \tag {61}
+$$
+
+$$
+\left| S ^ {\prime \prime} \right| \geq \left| S ^ {\prime} \right| \epsilon / \left(\eta_ {j} + \epsilon\right) > 0 \tag {62}
+$$
+
+Now let
+
+$$
+\mu (x) = \max  _ {k \neq \operatorname {c l f} (p) (x)} p (x) _ {\operatorname {c l f} (p) (x)} - p (x) _ {k} \tag {63}
+$$
+
+For  $\operatorname{clf}(p)$  to be well defined,  $\mu(x)$  must be greater than 0, i.e.  $\mu(x) > 0$  for all  $x$ .
+
+Since  $\operatorname{clf}(p)(x) \neq \operatorname{clf}(q)(x)$  for  $x \in S''$ , we have for any  $x \in S''$
+
+$$
+p (x) _ {\operatorname {c l f} (p) (x)} \geq p (x) _ {\operatorname {c l f} (q) (x)} + \mu (x) \tag {64}
+$$
+
+$$
+q (x) _ {\operatorname {c l f} (q) (x)} \geq q (x) _ {\operatorname {c l f} (p) (x)} \tag {65}
+$$
+
+Or
+
+$$
+p (x) _ {\operatorname {c l f} (p) (x)} \geq p (x) _ {\operatorname {c l f} (q) (x)} + \mu (x) \tag {66}
+$$
+
+$$
+- q (x) _ {\operatorname {c l f} (p) (x)} \geq - q (x) _ {\operatorname {c l f} (q) (x)} \tag {67}
+$$
+
+$$
+p (x) _ {\operatorname {c l f} (p) (x)} - q (x) _ {\operatorname {c l f} (p) (x)} \geq p (x) _ {\operatorname {c l f} (q) (x)} - q (x) _ {\operatorname {c l f} (q) (x)} + \mu (x) \tag {68}
+$$
+
+$$
+(p (x) _ {\operatorname {c l f} (p) (x)} - q (x) _ {\operatorname {c l f} (p) (x)}) + (q (x) _ {\operatorname {c l f} (q) (x)} - p (x) _ {\operatorname {c l f} (q) (x)}) \geq \mu (x) \tag {69}
+$$
+
+$$
+\left| p (x) _ {\operatorname {c l f} (p) (x)} - q (x) _ {\operatorname {c l f} (p) (x)} \right| + \left| p (x) _ {\operatorname {c l f} (q) (x)} - q (x) _ {\operatorname {c l f} (q) (x)} \right| \geq \mu (x) \tag {70}
+$$
+
+$$
+d _ {T V} (p (x), q (x)) \geq \mu (x) / 2 \tag {71}
+$$
+
+$$
+\mathbb {E} \left[ d _ {T V} (p (X), q (X)) | X \in S ^ {\prime \prime} \right] \geq \mathbb {E} \left[ \mu (X) / 2 | X \in S ^ {\prime \prime} \right] \tag {72}
+$$
+
+Since  $\inf_{q\in Q_{\mathrm{clf}}}\mathbb{E}[d_{TV}(p(X),q(X))] = 0$  , there exists some  $q\in Q_{\mathrm{clf}}$  such that
+
+$$
+\mathbb {E} \left[ d _ {T V} (p (X), q (X)) \right] <   | S | \mathbb {E} [ \mu (X) / 2 | X \in S ] \tag {73}
+$$
+
+for any set  $S$  with  $|S| > 0$ . In particular this holds for  $S = S''$  since  $|S''| > 0$ . Then
+
+$$
+\begin{array}{l} \mathbb {E} \left[ d _ {T V} (p (X), q (X)) \right] \geq \mathbb {E} \left[ d _ {T V} (p (X), q (X)) \mathbb {I} \left(X \in S ^ {\prime \prime}\right) \right] (74) \\ = \mathbb {E} \left[ d _ {T V} (p (X), q (X)) | X \in S ^ {\prime \prime} \right] \Pr (X \in S ^ {\prime \prime}) (75) \\ = \mathbb {E} \left[ d _ {T V} (p (X), q (X)) \mid X \in S ^ {\prime \prime} \right] \left| S ^ {\prime \prime} \right| (76) \\ \end{array}
+$$
+
+Thus
+
+$$
+\mathbb {E} \left[ d _ {T V} (p (X), q (X) | X \in S ^ {\prime \prime}) ] \mid S ^ {\prime \prime} \right| <   \left| S ^ {\prime \prime} \right| \mathbb {E} [ \mu (X) / 2 | X \in S ^ {\prime \prime} ] \tag {77}
+$$
+
+$$
+\mathbb {E} \left[ d _ {T V} (p (X), q (X) | X \in S ^ {\prime \prime}) \right] <   \mathbb {E} \left[ \mu (X) / 2 | X \in S ^ {\prime \prime} \right] \tag {78}
+$$
+
+which leads to a contradiction.
+
+Lemma B.2. Let  $Q = \{q : \operatorname{clf}(q) \in \mathcal{H}\}$  for some set  $\mathcal{H}$  of classifiers. For any classifier  $f$  and any  $\epsilon > 0$ , there exists a conditional distribution  $p$  such that  $\operatorname{clf}(p) = f$  and  $d(p, Q) < \epsilon$ .
+
+Proof. We claim that for any classifiers  $f_{1}, f_{2}$  there exists sequences of conditional distributions  $\{q_1^t\}_{t=1}^\infty, \{q_2^t\}_{t=1}^\infty$  such that  $\operatorname{clf}(q_1^t) = f_1, \operatorname{clf}(q_2^t) = f_2$  for all  $t$ , and  $\lim_{t \to \infty} q_1^t = \lim_{t \to \infty} q_2^t$ . The main idea is that we can slightly perturb a uniform distribution to achieve any classifier. Formally, we define  $q_1^t(x)_k = 1 / K - \delta / ((K - 1) \cdot t)$  for  $k \neq f_1(x)$  and  $q_1^t(x)_k = 1 / K + \delta / t$  for  $k = f_1(x)$ , where  $K = |\mathcal{Y}|$  is the number of classes. We can define  $q_2^t$  in the same manner. It's clear that both sequence converges to a uniform distribution, that is,  $\lim_{t \to \infty} q_1^t(x)_y = \lim_{t \to \infty} q_2^t(x)_y = 1 / K$  for all  $x, y$  even when  $f_1, f_2$  are completely different.
+
+Now for any classifier  $h \in \mathcal{H}$  and any classifier  $f$ , there exists a sequence  $\{q_h^t\}$ ,  $\{q_f^t\}$  such that  $\operatorname{clf}(q_h^t) = h$ ,  $\operatorname{clf}(q_f^t) = f$  for all  $t$  and that  $\lim_{t \to \infty} q_h^t = \lim_{t \to \infty} q_f^t$ . Since  $\operatorname{clf}(q_h^t) = h$ , we know that  $q_h^t \in Q$  and this implies that
+
+$$
+\begin{array}{l} \lim  _ {t \rightarrow \infty} d \left(q _ {f} ^ {t}, Q\right) = \lim  _ {t \rightarrow \infty} \min  _ {q \in Q} \mathbb {E} \left[ D _ {\mathrm {K L}} (q (X) | | q _ {f} ^ {t} (X)) \right] (79) \\ \leq \lim  _ {t \rightarrow \infty} \mathbb {E} \left[ D _ {\mathrm {K L}} \left(q _ {h} ^ {t} (X) \mid \mid q _ {f} ^ {t} (X)\right)\right] = 0. (80) \\ \end{array}
+$$
+
+The last line comes from the fact that  $\lim_{t\to \infty}q_h^t = \lim_{t\to \infty}q_f^t$ . As a result, we can conclude that for any  $\epsilon$ , there exists some  $t$  for which  $p = q_h^t$  satisfies  $d(p,Q) < \epsilon$  and  $\operatorname{clf}(p) = f$ .
+
+Proposition B.3. For any  $\epsilon, \nu > 0$  there exists  $\delta(\epsilon, \nu) > 0$  such that if  $d(p, \Lambda_{\epsilon} \cap Q_{\mathrm{clf}}) < \delta(\epsilon, \nu)$  then  $\rho_{\mathrm{clf}}(g_j, \mathrm{clf}(p)) \leq \eta_j + \nu / |S_j|$ , where  $|S_j| = P_X(X \in S_j)$  is the size of coverage set  $S_j$  of weak labeler  $g_j$  with respect to marginal distribution  $P_X$ . In particular  $\delta(\epsilon, \nu) = \frac{\nu L(\epsilon)}{(1 + \nu)}$  where  $L(\epsilon)$  is a strictly increasing function in  $\epsilon$  given by
+
+$$
+L (\epsilon) = \frac {(1 + \epsilon) \log (1 + \epsilon) - (2 + \epsilon) \log (1 + \epsilon / 2)}{K + \epsilon} \tag {81}
+$$
+
+where  $K$  is the number of classes. The bound can be rephrased as the following inequality: For all  $p$  such that  $d(p, Q_{\mathrm{clf}}^{\epsilon}) < L(\epsilon)$ , we have
+
+$$
+\rho_ {\mathrm {c l f}} \left(g _ {j}, \operatorname {c l f} (p)\right) \leq \eta_ {j} + \frac {d \left(p , Q _ {\mathrm {c l f}} ^ {\epsilon}\right)}{\left(L (\epsilon) - d \left(p , Q _ {\mathrm {c l f}} ^ {\epsilon}\right)\right) \left| S _ {j} \right|} \tag {82}
+$$
+
+where  $Q_{\mathrm{clf}}^{\epsilon} = \Lambda_{\epsilon}\cap Q_{\mathrm{clf}}$
+
+Proof. First, we prove the following
+
+$$
+\rho_ {\mathrm {c l f}} \left(g _ {j}, \operatorname {c l f} (p)\right) \leq \eta_ {j} + \min  _ {q \in Q _ {\mathrm {c l f}}} d _ {\mathrm {c l f}} (p, q) / \left| S _ {j} \right| \tag {83}
+$$
+
+where  $d_{\mathrm{clf}}(p,q)$  is defined as
+
+$$
+d _ {\operatorname {c l f}} (p, q) = \Pr_ {X} (\operatorname {c l f} (p) (X) \neq \operatorname {c l f} (q) (X)) \tag {84}
+$$
+
+$$
+\begin{array}{l} \rho_ {\operatorname {c l f}} \left(g _ {j}, \operatorname {c l f} (p)\right) = \Pr_ {X} (\operatorname {c l f} (p) (X) \neq g _ {j} (X) | X \in S _ {j}) \\ \leq \Pr_ {X} (\operatorname {c l f} (p) (X) \neq \operatorname {c l f} (q) (X) | X \in S _ {j}) + \Pr_ {X} (\operatorname {c l f} (q) (X) \neq g _ {j} (X) | X \in S _ {j}) \text {f o r a l l} q \\ \end{array}
+$$
+
+Thus
+
+$$
+\begin{array}{l} \rho_ {\operatorname {c l f}} (g _ {j}, \operatorname {c l f} (p)) \leq \min  _ {q \in Q _ {\operatorname {c l f}}} \Pr_ {X} (\operatorname {c l f} (p) (X) \neq \operatorname {c l f} (q) (X) | X \in S _ {j}) + \Pr_ {X} (\operatorname {c l f} (q) (X) \neq g _ {j} (X) | X \in S _ {j}) \\ \leq \min  _ {q \in Q _ {\mathrm {c l f}}} \Pr_ {X} (\operatorname {c l f} (p) (X) \neq \operatorname {c l f} (q) (X) | X \in S _ {j}) + \max  _ {q \in Q _ {\mathrm {c l f}}} \Pr_ {X} (\operatorname {c l f} (q) (X) \neq g _ {j} (X) | X \in S _ {j}) \\ \leq \min  _ {q \in Q _ {\mathrm {c l f}}} \operatorname * {P r} _ {X} (\operatorname {c l f} (p) (X) \neq \operatorname {c l f} (q) (X) | X \in S _ {j}) + \eta_ {j} \\ \leq \min  _ {q \in Q _ {\mathrm {c l f}}} \operatorname * {P r} _ {X} (\operatorname {c l f} (p) (X) \neq \operatorname {c l f} (q) (X), X \in S _ {j}) / | S _ {j} | + \eta_ {j} \\ \leq \min  _ {q \in Q _ {\mathrm {c l f}}} \Pr_ {X} (\operatorname {c l f} (p) (X) \neq \operatorname {c l f} (q) (X)) / | S _ {j} | + \eta_ {j} \\ = \eta_ {j} + \min  _ {q \in Q _ {\mathrm {c l f}}} d _ {\mathrm {c l f}} (p, q) / | S _ {j} | \\ \end{array}
+$$
+
+Next, we will prove the bound on  $\inf_{q\in Q_{\mathrm{clf}}}d_{\mathrm{clf}}(p,q)$ , i.e.
+
+$$
+\inf  _ {q \in Q _ {\mathrm {c l f}}} d _ {\mathrm {c l f}} (p, q) \leq \frac {d \left(p , Q _ {\mathrm {c l f}} ^ {\epsilon}\right)}{\left(L (\epsilon) - d \left(p , Q _ {\mathrm {c l f}} ^ {\epsilon}\right)\right)} \tag {85}
+$$
+
+For any conditional distributions  $p, q \in \mathcal{X} \to \Delta^K$  define the following
+
+$$
+d _ {\mathrm {k l}} (p, q) = \mathbb {E} \left[ D _ {\mathrm {K L}} (q (X) | | p (X)) \right] \tag {86}
+$$
+
+$$
+d _ {\mathrm {k l}} ^ {+} (p, q) = \mathbb {E} \left[ D _ {\mathrm {K L}} (q (X) | | p (X)) | \operatorname {c l f} (p) (X) = \operatorname {c l f} (q) (X) \right] \tag {87}
+$$
+
+$$
+d _ {\mathrm {k l}} ^ {-} (p, q) = \mathbb {E} \left[ D _ {\mathrm {K L}} (q (X) | | p (X)) | \operatorname {c l f} (p) (X) \neq \operatorname {c l f} (q) (X) \right] \tag {88}
+$$
+
+Consider the following sublemma.
+
+Sublemma B.3.1. For any  $q \in Q_{\mathrm{clf}}^{\epsilon}$ , if  $d_{\mathrm{kl}}(p,q) < L(\epsilon)$  then
+
+$$
+d _ {\mathrm {c l f}} (p, q) \leq \frac {d _ {\mathrm {k l}} (p , q)}{L (\epsilon) - d _ {\mathrm {k l}} (p , q)} \tag {89}
+$$
+
+where
+
+$$
+L (\epsilon) = \inf  _ {q \in Q _ {\mathrm {c l f}} ^ {\epsilon}, p} d _ {\mathrm {k l}} ^ {-} (p, q) \tag {90}
+$$
+
+The bound in equation 85 directly follows from the above sublemma. This is proved as follows.
+
+$$
+\begin{array}{l} \inf  _ {q \in Q _ {\mathrm {c l f}}} d _ {\mathrm {c l f}} (p, q) = \inf  _ {q \in Q _ {\mathrm {c l f}} ^ {\epsilon}} d _ {\mathrm {c l f}} (p, q) \\ \leq \inf  _ {q \in Q _ {\mathrm {c l f}} ^ {\epsilon}} \frac {d _ {\mathrm {k l}} (p , q)}{L (\epsilon) - d _ {\mathrm {k l}} (p , q)} \\ \leq \frac {\operatorname* {i n f} _ {q \in Q _ {\mathrm {c l f}} ^ {\epsilon}} d _ {\mathrm {k l}} (p , q)}{\sup  _ {q \in Q _ {\mathrm {c l f}} ^ {\epsilon}} L (\epsilon) - d _ {\mathrm {k l}} (p , q)} \\ = \frac {\inf  _ {q \in Q _ {\mathrm {c l f}} ^ {\epsilon}} d _ {\mathrm {k l}} (p , q)}{L (\epsilon) - \inf  _ {q \in Q _ {\mathrm {c l f}} ^ {\epsilon}} d _ {\mathrm {k l}} (p , q)} \\ = \frac {d (p , Q _ {\mathrm {c l f}} ^ {\epsilon})}{L (\epsilon) - d (p , Q _ {\mathrm {c l f}} ^ {\epsilon})} \\ \end{array}
+$$
+
+Finally  $d(p, Q_{\mathrm{clf}}^{\epsilon}) \leq \frac{\nu L(\epsilon)}{(1 + \nu)}$  gives us  $\inf_{q \in Q_{\mathrm{clf}}} d_{\mathrm{clf}}(p, q) \leq \nu$ , and thus  $\rho_{\mathrm{clf}}(g_j, \operatorname{clf}(p)) \leq \nu / |S_j| + \eta_j$ .
+
+Now we proof sublemma B.3.1. First note that,
+
+$$
+d _ {\mathrm {k l}} (p, q) = d _ {\mathrm {c l f}} (p, q) d _ {\mathrm {k l}} ^ {-} (p, q) + \left(1 - d _ {\mathrm {c l f}} (p, q)\right) d _ {\mathrm {k l}} ^ {+} (p, q) \tag {91}
+$$
+
+Now
+
+$$
+\begin{array}{l} d _ {\mathrm {k l}} ^ {+} (p, q) = \frac {d _ {\mathrm {k l}} (p , q) - d _ {\mathrm {c l f}} (p , q) d _ {\mathrm {k l}} ^ {-} (p , q)}{1 - d _ {\mathrm {c l f}} (p , q)} \\ \leq \frac {d _ {\mathrm {k l}} (p , q) - d _ {\mathrm {c l f}} (p , q) L (\epsilon)}{1 - d _ {\mathrm {c l f}} (p , q)} \\ \leq d _ {\mathrm {k l}} (p, q) + \left(d _ {\mathrm {k l}} (p, q) - L (\epsilon)\right) \frac {d _ {\mathrm {c l f}} (p , q)}{1 - d _ {\mathrm {c l f}} (p , q)} \\ <   d _ {\mathrm {k l}} (p, q) \\ \end{array}
+$$
+
+Further,
+
+$$
+\begin{array}{l} d _ {\mathrm {c l f}} (p, q) = \frac {d _ {\mathrm {k l}} (p , q) - d _ {\mathrm {k l}} ^ {+} (p , q)}{d _ {\mathrm {k l}} ^ {-} (p , q) - d _ {\mathrm {k l}} ^ {+} (p , q)} \\ \leq \frac {d _ {\mathrm {k l}} (p , q)}{d _ {\mathrm {k l}} ^ {-} (p , q) - d _ {\mathrm {k l}} ^ {+} (p , q)} \\ <   \frac {d _ {\mathrm {k l}} (p , q)}{L (\epsilon) - d _ {\mathrm {k l}} (p , q)} \\ \end{array}
+$$
+
+Now we are only left to prove the form of  $L(\epsilon)$  given in equation 81.
+
+Recall  $\Lambda_{\epsilon}$  was defined in equation 12 as
+
+$$
+\Lambda_ {\epsilon} := \left\{q: \mathcal {X} \rightarrow \Delta^ {k} \mid \text {f o r e a c h} x, \exists y: q (x) _ {y} \geq (1 + \epsilon) q (x) _ {k} \text {f o r} k \neq y \right\} \tag {92}
+$$
+
+For a distribution  $u \in \Delta^K$ , let  $c_j(u)$  refer to the top  $j$ 'th class, i.e.  $c_1(u), \ldots, c_K(u)$  is a permutation of  $[K]$  s.t.  $u_{c_1(u)} \geq \ldots \geq u_{c_K(u)}$ . Let  $t_j(u)$  refer to the corresponding probability i.e.  $t_j(u) = u_{c_j(u)}$ .
+
+Let  $\lambda_{\epsilon} = \{u\in \Delta^{K}:t_{1}(u)\geq (1 + \epsilon)t_{j}(u)\forall j > 1\}$ . This is connected to  $\Lambda_{\epsilon}$  as
+
+$$
+\Lambda_ {\epsilon} = \left\{q: \mathcal {X} \rightarrow \Delta^ {K}: \forall x, q (x) \in \lambda_ {\epsilon} \right\}
+$$
+
+Now
+
+$$
+\begin{array}{l} L (\epsilon) = \inf  _ {q \in Q _ {\mathrm {c l f}} ^ {\epsilon}, p} d _ {\mathrm {k l}} ^ {-} (p, q) (93) \\ = \inf  _ {q \in Q _ {\operatorname {c l f}} ^ {\epsilon}, p} \mathbb {E} \left[ D _ {\mathrm {K L}} (q (X) | | p (X)) | \operatorname {c l f} (p) (X) \neq \operatorname {c l f} (q) (X) \right] (94) \\ = \inf  _ {h \in \mathcal {H}, p} \mathbb {E} \left[ \inf  _ {u \in \lambda_ {\epsilon}, c _ {1} (u) = h (X)} D _ {\mathrm {K L}} (u \| p (X)) | \operatorname {c l f} (p) (X) \neq c _ {1} (u) \right] (95) \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \left(\text {w h e r e} \mathcal {H} = \{h: \rho_ {\operatorname {c l f}} \left(g _ {j}, h\right) \leq \eta_ {j} \}\right) (96) \\ = \inf  _ {h \in \mathcal {H}} \mathbb {E} \left[ \inf  _ {u \in \lambda_ {\epsilon}: c _ {1} (u) = h (X), v \in \Delta^ {K}: c _ {1} (v) \neq c _ {1} (u)} D _ {\mathrm {K L}} (u | | v) \right] (97) \\ = \inf  _ {u \in \lambda_ {\epsilon}, v \in \Delta^ {K}: c _ {1} (v) \neq c _ {1} (u)} D _ {\mathrm {K L}} (u \| | v) (98) \\ = \inf  _ {u, v \in \Delta^ {K}} D _ {\mathrm {K L}} (u | | v): c _ {1} (v) \neq c _ {1} (u), t _ {1} (u) \geq (1 + \epsilon) t _ {j} (u) \forall j > 1 (99) \\ \end{array}
+$$
+
+Now without loss of generality, let  $c_{j}(u) = j$ . With some abuse of notation, we also use  $u, v$  to refer to the minimizer of objective in equation 99. The minimizer exists since the constraints induce closed sets. Then first we will prove that
+
+$$
+c _ {j} (v) = c _ {j} (u) = j \text {f o r} j \geq 3 \tag {100}
+$$
+
+Assume the contrary. Then there exists  $k \geq 3$  such that  $v_{k} > v_{2}$ . This implies
+
+$$
+\left(u _ {2} - u _ {k}\right) \log \left(v _ {k} / v _ {2}\right) > 0 \tag {101}
+$$
+
+$$
+u _ {2} \log \left(1 / v _ {2}\right) + u _ {k} \log \left(1 / v _ {k}\right) > u _ {2} \log \left(1 / v _ {k}\right) + u _ {k} \log \left(1 / v _ {2}\right) \tag {102}
+$$
+
+$$
+u _ {2} \log \left(u _ {2} / v _ {2}\right) + u _ {k} \log \left(u _ {k} / v _ {k}\right) > u _ {2} \log \left(u _ {2} / v _ {k}\right) + u _ {k} \log \left(u _ {k} / v _ {2}\right) \tag {103}
+$$
+
+$$
+D _ {\mathrm {K L}} (u | | v) > D _ {\mathrm {K L}} (u | | v ^ {\prime}) \tag {104}
+$$
+
+where  $v^{\prime}$  is defined as
+
+$$
+v _ {2} ^ {\prime} = v _ {k}, v _ {k} ^ {\prime} = v _ {2}, v _ {j} ^ {\prime} = v _ {j} \text {f o r} j \neq k, 2 \tag {105}
+$$
+
+But  $u, v$  was the minimizer of the objective in equation 99, which contradicts the above implication in equation 104.
+
+Since  $c_{1}(u) = 1 \neq c_{1}(v)$ , we get  $c_{1}(v) = 2$ , and  $c_{2}(v) = 1$ . To recap we have
+
+$$
+c _ {j} (u) = j \tag {106}
+$$
+
+$$
+c _ {1} (v) = 2, c _ {2} (v) = 1, c _ {j} (v) = j \text {f o r} j \geq 3 \tag {107}
+$$
+
+Next we will prove that
+
+$$
+v _ {1} = v _ {2} \tag {108}
+$$
+
+Let
+
+$$
+x = \left(v _ {2} - v _ {1}\right) / 2 \tag {109}
+$$
+
+$$
+y = \left(v _ {2} + v _ {1}\right) / 2 \tag {110}
+$$
+
+Since  $c_{1}(v) = 2, v_{2} \geq v_{1}$  and so  $x, y \geq 0$ . We have
+
+$$
+v _ {1} = y - x \tag {111}
+$$
+
+$$
+v _ {2} = y + x \tag {112}
+$$
+
+$$
+\sum_ {j = 3} ^ {K} v _ {j} = 1 - 2 y \tag {113}
+$$
+
+Now,
+
+$$
+L (\epsilon) = u _ {1} \log (u _ {1} / y - x) + u _ {2} \log (u _ {2} / y + x) + \sum_ {j = 3} ^ {K} u _ {j} \log (u _ {j} / v _ {j})
+$$
+
+$$
+\begin{array}{l} \nabla_ {x} L (\epsilon) = u _ {1} / (y - x) - u _ {2} (y + x) \\ = \frac {u _ {1} (y + x) - u _ {2} (y - x)}{y ^ {2} - x ^ {2}} \\ = \frac {x (u _ {1} + u _ {2}) + y (u _ {1} - u _ {2})}{y ^ {2} - x ^ {2}} \\ > 0 \\ \end{array}
+$$
+
+Thus  $L(\epsilon)$  is minimized for  $x = 0$
+
+Next, we have the following inequality
+
+$$
+\sum_ {i = 1} ^ {n} a _ {i} \log \left(a _ {i} / b _ {i}\right) \geq \left(\sum_ {i = 1} ^ {n} a _ {i}\right) \log \left(\sum_ {i = 1} ^ {n} a _ {i} / \sum_ {i = 1} ^ {n} b _ {i}\right) \tag {114}
+$$
+
+This follows Jenson's inequality, given a discrete distribution  $f: \sum_{i=1}^{n} f_i = 1$ , we have
+
+$$
+\sum_ {i = 1} ^ {n} f _ {i} \log (t _ {i}) \geq \log (\sum_ {i = 1} ^ {n} f _ {i} t _ {i})
+$$
+
+Substituting  $f_{i} = a_{i} / \sum_{i=1}^{n} a_{i}$ , and  $t_{i} = b_{i} / a_{i}$ , we get the required inequality. We use this inequality to have
+
+$$
+\sum_ {j = 3} ^ {K} u _ {j} \log \left(u _ {j} / v _ {j}\right) \geq \left(\sum_ {j = 3} ^ {K} u _ {j}\right) \log \left(\sum_ {j = 3} ^ {K} u _ {j} / \sum_ {j = 3} ^ {K} v _ {j}\right) \tag {115}
+$$
+
+This gives us
+
+$$
+L (\epsilon) \geq u _ {1} \log \left(u _ {1} / v _ {1}\right) + u _ {2} \log \left(u _ {2} / v _ {2}\right) + \left(1 - u _ {1} - u _ {2}\right) \log \left(\frac {1 - u _ {1} - u _ {2}}{1 - v _ {1} - v _ {2}}\right) \tag {116}
+$$
+
+We had  $v_{1} = v_{2}$ . So, substituting  $v_{1} = v_{2} = y$ , we have
+
+$$
+L (\epsilon) \geq f (y) = u _ {1} \log u _ {1} / y + u _ {2} \log u _ {2} / y + (1 - u _ {1} - u _ {2}) \log \left(\frac {1 - u _ {1} - u _ {2}}{1 - 2 y}\right)
+$$
+
+$$
+\begin{array}{l} \nabla_ {y} f (y) = - u _ {1} / y - u _ {2} / y + \frac {2 (1 - u _ {1} - u _ {2})}{1 - 2 y} \\ = \frac {2 y - u _ {1} - u _ {2}}{1 - 2 y} \\ \end{array}
+$$
+
+$$
+\nabla_ {y} ^ {2} f (y) = \frac {2 (1 - u _ {1} - u _ {2})}{(1 - 2 y) ^ {2}} > 0
+$$
+
+Thus  $f(y)$  is minimized for  $\nabla_y f(y) = 0$ , giving  $y = v_1 = v_2 = (u_1 + u_2) / 2$ . Substituting in equation 116, we get
+
+$$
+L (\epsilon) \geq u _ {1} \log \left(2 u _ {1} / \left(u _ {1} + u _ {2}\right)\right) + u _ {2} \log \left(2 u _ {2} / \left(u _ {1} + u _ {2}\right)\right) \tag {117}
+$$
+
+Let  $x = (u_{1} - u_{2}) / 2$  (and thus  $u_{1} = y + x, u_{2} = y - x$ ) we have
+
+$$
+\begin{array}{l} L (\epsilon) \geq f (x) = (y + x) \log (1 + x / y) + (y - x) \log (1 - x / y) \\ \nabla_ {x} f (x) = \log (1 + x / y) - \log (1 - x / y) > 0 \\ \end{array}
+$$
+
+Since  $u_{1} \geq (1 + \epsilon)u_{2}$ , we have  $y + x \geq (1 + \epsilon)(y - x)$ , thus  $f(x)$  is minimized for  $x = y\epsilon /(2 + \epsilon)$ . Using  $y = (u_{1} + u_{2}) / 2$ , we have
+
+$$
+\begin{array}{l} L (\epsilon) \geq (u _ {1} + u _ {2}) ((1 + \epsilon) / (2 + \epsilon) \log (2 (1 + \epsilon) / (2 + \epsilon)) + 2 y / (2 + \epsilon) \log (2 / (2 + \epsilon))) \\ = \left(\frac {u _ {1} + u _ {2}}{2 + \epsilon}\right) ((1 + \epsilon) \log (1 + \epsilon) - (2 + \epsilon) \log (1 + \epsilon / 2)) \\ \end{array}
+$$
+
+Now to bound  $u_{1} + u_{2}$ , consider the following.
+
+Since  $u_{j}\leq u_{2}$  for  $j\geq 3$  , we have
+
+$$
+(K - 1) u _ {2} \geq \sum_ {j = 2} ^ {K} u _ {j} = 1 - u _ {1}
+$$
+
+$$
+u _ {2} \geq \frac {1 - u _ {1}}{K - 1}
+$$
+
+$$
+u _ {1} + u _ {2} \geq \frac {1 + u _ {1} (K - 2)}{K - 1}
+$$
+
+Further since  $u_{1}\geq (1 + \epsilon)u_{j}$  for  $j\geq 2$  , we have,
+
+$$
+(K - 1) u _ {1} \geq (1 + \epsilon) \sum_ {j = 2} ^ {K} u _ {j}
+$$
+
+$$
+u _ {1} \geq \frac {(1 + \epsilon) (1 - u _ {1})}{K - 1}
+$$
+
+$$
+u _ {1} \geq \frac {1 + \epsilon}{K + \epsilon}
+$$
+
+Substituting back we get
+
+$$
+u _ {1} + u _ {2} \geq \frac {2 + \epsilon}{K + \epsilon}
+$$
+
+Substituting in the bound for  $L(\epsilon)$  we finally get
+
+$$
+L (\epsilon) \geq \frac {(1 + \epsilon) \log (1 + \epsilon) - (2 + \epsilon) \log (1 + \epsilon / 2)}{K + \epsilon} \tag {118}
+$$
+
+Now to prove  $L(\epsilon)$  is strictly increasing and positive for  $\epsilon > 0$ , we have  $L(0) = 0$  and
+
+$$
+\begin{array}{l} \nabla_ {\epsilon} L (\epsilon) = \nabla_ {\epsilon} f (\epsilon) / (K + \epsilon) - f (\epsilon) / (K + \epsilon) ^ {2} \\ = \frac {(K - 1) \log (1 + \epsilon) - (K - 2) \log (1 + \epsilon / 2)}{(K + \epsilon) ^ {2}} \\ > 0 \\ \end{array}
+$$
+
+![](images/5ace87219cc02fab2062e1bfebc1aa9ffeef463ef27db4a5743740c1ff4fa0a3.jpg)
+
+Proposition B.4. Given  $Q = \Lambda_{\epsilon} \cap Q_{\mathrm{clf}}$  for  $\Lambda_{\epsilon}$  defined in equation 12 and  $Q_{\mathrm{clf}}$  defined in equation 5, for any conditional distribution  $p$ , the optimal projection  $q^{*} = \arg \inf_{q \in Q} \mathbb{E}[D_{KL}(q(X)||p(X))]$  can be found as follows,
+
+1. Solve for optimal  $\operatorname{clf}(q^{*})$  as
+
+$$
+\operatorname {c l f} \left(q ^ {*}\right) = \underset {h \in \mathcal {X} \rightarrow \mathcal {Y}} {\arg \min } \mathbb {E} _ {X} \left[ \operatorname {P r o j} _ {K L} (p (X); \Delta_ {h (X), \epsilon} ^ {K}) ]: \rho_ {\operatorname {c l f}} \left(g _ {j}, h\right) \leq \eta_ {j} \right. \tag {119}
+$$
+
+where
+
+$$
+\Delta_ {y, \epsilon} ^ {K} = \left\{r \in \Delta^ {K} \mid r _ {y} \geq (1 + \epsilon) r _ {k}, \forall k \neq y \right\},
+$$
+
+$$
+\operatorname {P r o j} _ {K L} (u; S) = \inf  _ {r \in S} D _ {K L} (r | | u). \tag {120}
+$$
+
+2. Solve for an optimal distribution  $q^{*}$  given the optimal  $\operatorname{clf}(q^{*})$ . For each  $x$ ,
+
+$$
+q ^ {*} (x) = \arg \min  _ {r} D _ {K L} (r \| p (x)): r \in \Delta_ {\operatorname {c l f} \left(q ^ {*}\right) (x)} ^ {K, \epsilon} \tag {121}
+$$
+
+Proof. Given  $\Lambda_{\epsilon} = \{q \in \mathcal{X} \to \Delta^{K} \mid \forall x, q(x) \in \bigcup_{k=1}^{K} \Delta_{k,\epsilon}^{K}\}$ . We have
+
+$$
+\begin{array}{l} q^{*} = \operatorname *{arg  min}_{q\in Q}\mathbb{E}[D_{\mathrm{KL}}(q(X)||p(X))] \\ = \operatorname *{arg  min}_{q\in \Lambda_{\epsilon}}\mathbb{E}\left[D_{\mathrm{KL}}(q(X)||p(X))]:q\in Q_{\mathrm{cf}}\right. \\ = \operatorname * {a r g   m i n} _ {q \in \Lambda_ {\epsilon}} \mathbb {E} [ D _ {\mathrm {K L}} (q (X) | | p (X)) ]: \rho (g _ {j}, \operatorname {c l f} (q)) \leq \eta_ {j} \forall j \in [ m ] \\ = \operatorname * {a r g   m i n} _ {q \in \Lambda_ {\epsilon}} \min  _ {h \in \mathcal {X} \to \mathcal {Y}} \mathbb {E} [ D _ {\mathrm {K L}} (q (X) | | p (X)) ]: \rho (g _ {j}, h) \leq \eta_ {j}, h = \operatorname {c l f} (q) \\ = \operatorname *{arg  min}_{q:q = q^{\prime}}\min_{q^{\prime}\in \Lambda_{\epsilon}}\min_{h\in \mathcal{X}\to \mathcal{Y}}\mathbb{E}[D_{\mathrm{KL}}(q^{\prime}(X)||p(X))]:\rho (g_{j},h)\leq \eta_{j}, h = \operatorname {clf}(q^{\prime}) \\ = \operatorname *{arg  min}_{q:q = q^{\prime}}\min_{h\in \mathcal{X}\to \mathcal{Y}}\min_{q^{\prime}\in \Lambda_{\epsilon}}\mathbb{E}[D_{\mathrm{KL}}(q^{\prime}(X)||p(X))]:\rho (g_{j},h)\leq \eta_{j}, h = \operatorname {clf}(q^{\prime}) \\ = \operatorname * {a r g   m i n} _ {q: q = q ^ {\prime}} \min  _ {h \in \mathcal {X} \to \mathcal {Y}} \left(\min  _ {q ^ {\prime} \in \Lambda_ {\epsilon}} \mathbb {E} [ D _ {\mathrm {K L}} (q ^ {\prime} (X) | | p (X)) ]: \operatorname {c l f} (q ^ {\prime}) = h\right): \rho (g _ {j}, h) \leq \eta_ {j} \\ = \operatorname * {a r g   m i n} _ {q: q (x) = u _ {x}} \operatorname * {m i n} _ {h \in \mathcal {X} \rightarrow \mathcal {Y}} \mathbb {E} \left[ \operatorname * {m i n} _ {u _ {X} \in \Delta^ {K}} D _ {\mathrm {K L}} (u _ {X} | | p (X)): u _ {X} \in \Delta_ {h (X), \epsilon} ^ {K}) \right]: \rho (g _ {j}, h) \leq \eta_ {j} \\ \end{array}
+$$
+
+Thus
+
+$$
+q ^ {*} (x) = \underset {u \in \Delta^ {K}} {\arg \min } D _ {\mathrm {K L}} (u \| p (X)): u \in \Delta_ {h ^ {*} (X), \epsilon} ^ {K} \tag {122}
+$$
+
+where
+
+$$
+h ^ {*} = \underset {h \in \mathcal {X} \rightarrow \mathcal {Y}} {\arg \min } \mathbb {E} \left[ \right. \underset {u _ {X} \in \Delta^ {K}} {\min } D _ {\mathrm {K L}} \left(u _ {X} \| p (X)\right): u _ {X} \in \Delta_ {h (X), \epsilon} ^ {K}\left. \right): \rho \left(g _ {j}, h\right) \leq \eta_ {j} \tag {123}
+$$
+
+Since  $u_{x}^{*} \in \Delta_{h^{*}(x),\epsilon}^{K}$ , it implies  $\operatorname{clf}(u^{*}) = h^{*}$ , and since  $q^{*}(x) = u_{x}^{*}$ ,  $\operatorname{clf}(q^{*}) = \operatorname{clf}(u^{*}) = h^{*}$ . Thus
+
+$$
+\operatorname {c l f} \left(q ^ {*}\right) = \min  _ {h \in \mathcal {X} \rightarrow \mathcal {Y}} \mathbb {E} \left[ \operatorname {P r o j} _ {\mathrm {K L}} \left(p (X); \Delta_ {h (X), \epsilon} ^ {K}\right)\right]: \rho \left(g _ {j}, h\right) \leq \eta_ {j} \tag {124}
+$$
+
+and
+
+$$
+q ^ {*} (x) = \underset {u \in \Delta^ {K}} {\arg \min } D _ {\mathrm {K L}} (u \| p (X)): u \in \Delta_ {\operatorname {c l f} \left(q ^ {*}\right) (X), \epsilon} ^ {K} \tag {125}
+$$
+
+![](images/cb6e3b2e907cc9448a3009108a1b2795cfafba54ed5bf7a0e032eeeb6f1b705f.jpg)
+
+Proposition B.5. For  $Q_{\mathrm{dist}}$  defined in equation 16, we have  $d(p, Q_{\mathrm{dist}}) = 0$  if and only if  $p \in Q_{\mathrm{dist}}$ . Furthermore, for any  $\nu > 0$  there exists  $\delta(\nu) > 0$  such that if  $d(p, Q_{\mathrm{dist}}) < \delta(\nu)$  then  $\rho_{\mathrm{dist}}(g_j, p) \leq \eta_j + \nu / |S_j|$ , where  $|S_j| = P_X(S_j)$  is the size of coverage set  $S_j$  of weak labeler  $g_j$  with respect to marginal distribution  $P_X$ . In particular  $\delta(\nu) = \nu^2 / 2$ .
+
+Rephrasing we get the following bound:
+
+$$
+\rho_ {\mathrm {d i s t}} (g _ {j}, p) \leq \eta_ {j} + \sqrt {2 d (p , Q _ {\mathrm {d i s t}})} / | S _ {j} |
+$$
+
+Proof. Define  $E(p) = \min_{q \in Q_{\mathrm{dist}}} \mathbb{E}_X[d_{TV}(p(X), q(X))]$ , where  $d_{TV}$  is the TV distance:
+
+$$
+d _ {T V} (u, v) = 1 / 2 \cdot \sum_ {k = 1} ^ {K} | u _ {k} - v _ {k} | \tag {126}
+$$
+
+First, we prove the following
+
+$$
+\rho_ {\text {d i s t}} \left(g _ {j}, p\right) \leq \eta_ {j} + 2 \min  _ {q \in Q _ {\text {d i s t}}} \mathbb {E} _ {X} \left[ d _ {T V} (p (X), q (X)) \right] / | S _ {j} | \tag {127}
+$$
+
+$$
+\begin{array}{l} \rho_ {\text {d i s t}} \left(g _ {j}, p\right) = \mathbb {E} \left[ \sum_ {k \neq g _ {j} (X)} p (X) _ {g _ {j} (X)} \mid X \in S _ {j} \right] (128) \\ = \mathbb {E} [ \sum_ {k \neq g _ {j} (X)} p (X) _ {g _ {j} (X)} - q (X) _ {g _ {j} (X)} | X \in S _ {j} ] + \mathbb {E} [ \sum_ {k \neq g _ {j} (X)} q (X) _ {g _ {j} (X)} | X \in S _ {j} ] \text {f o r a l l} q (129) \\ \leq \mathbb {E} [ \sum_ {k} | p (X) _ {k} - q (X) _ {k} | X \in S _ {j} ] + \mathbb {E} [ \sum_ {k \neq g _ {j} (X)} q (X) _ {g _ {j} (X)} | X \in S _ {j} ] (130) \\ \leq \mathbb {E} \left[ \sum_ {k} | p (X) _ {k} - q (X) _ {k} | \mathbb {I} (X \in S _ {j}) \right] / \left| S _ {j} \right| + \rho_ {\text {d i s t}} \left(g _ {j}, q\right) (131) \\ \leq \mathbb {E} \left[ \sum_ {k} | p (X) _ {k} - q (X) _ {k} | \right] / | S _ {j} | + \rho_ {\text {d i s t}} \left(g _ {j}, q\right) (132) \\ \end{array}
+$$
+
+$$
+\rho_ {\text {d i s t}} \left(g _ {j}, p\right) \leq 2 \mathbb {E} \left[ d _ {T V} \left(p (X), q (X)\right) \right] / \left| S _ {j} \right| + \rho_ {\text {d i s t}} \left(g _ {j}, q\right) \tag {133}
+$$
+
+The above holds for all  $q$ . Thus
+
+$$
+\rho_ {\text {d i s t}} \left(g _ {j}, p\right) \leq \min  _ {q \in Q _ {\text {d i s t}}} 2 \mathbb {E} \left[ d _ {T V} (p (X), q (X)) \right] / \left| S _ {j} \right| + \rho_ {\text {d i s t}} \left(g _ {j}, q\right) \tag {134}
+$$
+
+$$
+\leq \min  _ {q \in Q _ {\text {d i s t}}} 2 \mathbb {E} [ d _ {T V} (p (X), q (X)) ] / | S _ {j} | + \max  _ {q \in Q _ {\text {d i s t}}} \rho_ {\text {d i s t}} \left(g _ {j}, q\right) \tag {135}
+$$
+
+$$
+\leq 2 \min  _ {q \in Q _ {\text {d i s t}}} \mathbb {E} [ d _ {T V} (p (X), q (X)) ] / | S _ {j} | + \eta_ {j} \tag {136}
+$$
+
+Further using Pinsker's Inequality
+
+$$
+d _ {T V} (u, v) \leq \sqrt {D _ {\mathrm {K L}} (u | | v) / 2}
+$$
+
+we have
+
+$$
+\begin{array}{l} \mathbb {E} \left[ d _ {T V} (p (X), q (X)) \right] = \mathbb {E} \left[ d _ {T V} (q (X), p (X)) \right] (138) \\ \leq \mathbb {E} \left[ \sqrt {D _ {\mathrm {K L}} (q (X) | | p (X)) / 2} \right] (139) \\ \leq \sqrt {\mathbb {E} \left[ D _ {\mathrm {K L}} (q (X) | | p (X)) \right] / 2} (\text {f r o m J e n s o n ' s}) (140) \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \min  _ {q \in Q _ {\text {d i s t}}} \mathbb {E} [ d _ {T V} (p (X), q (X)) ] \leq \min  _ {q \in Q _ {\text {d i s t}}} \sqrt {\mathbb {E} [ D _ {\mathrm {K L}} (q (X) | | p (X)) ] / 2} (141) \\ \leq \sqrt {\min  _ {q \in Q _ {\text {d i s t}}} \mathbb {E} \left[ D _ {\mathrm {K L}} (q (X) | | p (X)) \right] / 2} (142) \\ \leq \sqrt {d (p , Q _ {\text {d i s t}}) / 2} (143) \\ \end{array}
+$$
+
+Finally  $d(p, Q_{\mathrm{dist}}) \leq \nu^2 / 2$  implies  $\min_{q \in Q_{\mathrm{dist}}} \mathbb{E}[d_{TV}(p(X), q(X))] \leq \nu / 2$ , and thus  $\rho_{\mathrm{dist}}(g_j, p) \leq \nu / |S_j| + \eta_j$ .
+
+Proposition B.6. Given  $Q_{\mathrm{dist}}$  as defined in equation 16, for any conditional distribution  $p$ , the optimal projection  $q^{*} = \arg \min_{q\in Q_{\mathrm{dist}}}\mathbb{E}[D_{KL}(q(X)||p(X))]$  is given by  $q^{*} = p^{\lambda^{*}}$  where
+
+$$
+p ^ {\lambda} (x) _ {y} := Z _ {\lambda} (x) \exp \left(\sum_ {j: g _ {j} (x) = y} \lambda\right) p (x) _ {y} \tag {144}
+$$
+
+$Z_{\lambda}(x)$  is the normalization constant. The optimal  $\lambda^{*}$  is given by the following optimization:
+
+$$
+\boldsymbol {\lambda} ^ {*} = \min  _ {\boldsymbol {\lambda} \geq \mathbf {0}} \sum_ {j = 1} ^ {m} \boldsymbol {\lambda} _ {j} \quad s. t. \quad \rho_ {\text {d i s t}} \left(g _ {j}, p ^ {\boldsymbol {\lambda}}\right) \leq \eta_ {j} \forall j \in [ m ] \tag {145}
+$$
+
+Proof. The problem is a convex optimization with affine constraints since  $\rho_{\mathrm{dist}}(g_j,q) = \mathbb{E}[\sum_{y\neq g_j(X)}q(X)_y]$ . To restate it is,
+
+$$
+\min  _ {q \in \mathcal {X} \rightarrow \Delta^ {K}} \mathbb {E} \left[ D _ {\mathrm {K L}} (q (X) | | p (X)) \right] \tag {146}
+$$
+
+$$
+s. t. \mathbb {E} [ \sum_ {y \neq g _ {j} (X)} q (X) _ {y} ] \leq \eta_ {j} \forall j \in [ m ] \tag {147}
+$$
+
+In terms of unconstrained variable  $q \in \mathcal{X} \to \mathbb{R}^K$ , this becomes
+
+$$
+\min  _ {q \in \mathcal {X} \rightarrow \mathbb {R} ^ {K}} \mathbb {E} \left[ \sum_ {k = 1} ^ {K} q (X) _ {k} \log \left(q (X) _ {k} / p (X) _ {k}\right)\right] \tag {148}
+$$
+
+$$
+s. t. \mathbb {E} [ \sum_ {y \neq g _ {j} (X)} q (X) _ {y} ] \leq \eta_ {j} \forall j \in [ m ] \tag {149}
+$$
+
+$$
+\sum_ {y = 1} ^ {K} q (x) _ {y} = 1 \text {f o r a l l} x \tag {150}
+$$
+
+$$
+q (x) _ {y} \in [ 0, 1 ] \text {f o r a l l} x, \text {a n d} y \in [ K ] \tag {151}
+$$
+
+Since  $\log$  is undefined for nonpositive values and assuming  $p(x)_y > 0$  for all  $x,y$ , we only need to consider  $q(x)_y \in \mathbb{R}_+$ , thus we can ignore the constraint  $q(x)_y \geq 0$ .  $q(x)_y \leq 1$  also follows from the constraint  $\sum_{y=1}^{K} q(x)_y = 1$ . Hence the constraint  $q(x)_y \in [0,1]$  can be ignored. Rephrasing we have
+
+$$
+\min  _ {q \in \mathcal {X} \rightarrow \mathbb {R} _ {+} ^ {K}} \mathbb {E} \left[ \sum_ {k = 1} ^ {K} q (X) _ {k} \log \left(q (X) _ {k} / p (X) _ {k}\right)\right] \tag {152}
+$$
+
+$$
+s. t. \mathbb {E} [ q (X) _ {g _ {j} (X)} ] \geq 1 - \eta_ {j} \forall j \in [ m ] \tag {153}
+$$
+
+$$
+\sum_ {k = 1} ^ {K} q (x) _ {k} = 1 \text {f o r a l l} x \tag {154}
+$$
+
+Here we subtracted from 1, both sides of the inequality constraint.
+
+Now we form the lagrangian  $\mathcal{L}(q,\lambda ,\mu)$  where  $\lambda \in \mathbb{R}_+^m$ , a Lagrange multiplier for each inequality constraint, and  $\mu \in \mathcal{X}\to \mathbb{R}$ , the Lagrange multiplier for equality constraints.
+
+$$
+\begin{array}{l} \mathcal {L} (q, \boldsymbol {\lambda}, \mu) = \mathbb {E} \left[ \sum_ {k = 1} ^ {K} q (X) _ {k} \log (q (X) _ {k} / p (X) _ {k}) \right] \\ + \mathbb {E} [ \mu (X) (1 - \sum_ {k = 1} ^ {K} q (X) _ {k}) ] + \sum_ {j = 1} ^ {m} \lambda_ {j} (1 - \eta_ {j} - \mathbb {E} [ q (X) _ {g _ {j} (X)} | X \in S _ {j} ]) \\ \end{array}
+$$
+
+We have
+
+$$
+\min  _ {q \in \mathcal {X} \to \mathbb {R} _ {+} ^ {K}} \max  _ {\boldsymbol {\lambda} \in \mathbb {R} _ {+} ^ {m}, \mu \in \mathcal {X} \to \mathbb {R}} \mathcal {L} (q, \boldsymbol {\lambda}, \mu) = \min  _ {q \in Q _ {\text {d i s t}}} \mathbb {E} [ D _ {\mathrm {K L}} (q (X) | | p (X)) ]
+$$
+
+Since the optimization is convex, a point  $(q^{*},\lambda^{*},\mu^{*})$  that satisfies the KKT conditions is optimal. KKT stationarity condition implies that
+
+$$
+\nabla_ {q} \mathcal {L} \left(q ^ {*}, \lambda^ {*}, \mu^ {*}\right) = 0 \tag {155}
+$$
+
+where derivative of an objective  $L(g) = \mathbb{E}[\sum_{k=1}^{K} f(g(X, k))]$  where  $f: \mathbb{R} \to \mathbb{R}$  is a scalar function, with respect to a function  $g$  in  $\mathcal{X} \times [K] \to \mathbb{R}$  is defined using methods in calculus of variation as a function in  $\mathcal{X} \times [K] \to \mathbb{R}$
+
+$$
+\nabla_ {g} L (g) (x, y) = \mathbb {E} [ \delta (X = x) f ^ {\prime} (g (X, y)) ] = d (x) f ^ {\prime} (g (x, y)) \tag {156}
+$$
+
+where  $d$  is the probability density associated with marginal distribution  $P$  on  $\mathcal{X}$ . Thus we have
+
+$$
+\nabla_ {q} \mathcal {L} \left(q ^ {*}, \lambda^ {*}, \mu^ {*}\right) (x, y) = d (x) \left(1 + \log q ^ {*} (x) _ {y} / p (x) _ {y} - \mu^ {*} (x) - \sum_ {j} \lambda_ {j} ^ {*} \mathbb {I} (y = g _ {j} (x))\right) = 0 \tag {157}
+$$
+
+Using  $\sum_{y}q^{*}(x)_{y} = 1$ , we can eliminate  $\mu^{*}(x)$  to get  $q^{*}(x)_{y}$  for  $x$  in support of  $P(d(x) > 0)$
+
+$$
+q ^ {*} (x) _ {y} = Z _ {\boldsymbol {\lambda} ^ {*}} (x) \exp \left(\sum_ {j: g _ {j} (x) = y} \boldsymbol {\lambda} _ {j} ^ {*}\right) p (x) _ {y} \tag {158}
+$$
+
+where  $Z_{\lambda^*}(x)$  is the normalization constant
+
+$$
+Z _ {\boldsymbol {\lambda} ^ {*}} (x) = 1 / \left(\sum_ {y = 1} ^ {K} \exp \left(\sum_ {j: g _ {j} (x) = y} \boldsymbol {\lambda} _ {j} ^ {*}\right) p (x) _ {y}\right) \tag {159}
+$$
+
+Complementary slackness condition on the inequality constraints imply
+
+$$
+\boldsymbol {\lambda} _ {i} ^ {*} \left(1 - \eta_ {i} - \mathbb {E} \left[ q ^ {*} (X) _ {g _ {i} (X)} \mid X \in S _ {i} \right]\right) = 0; \forall i \in [ m ] \tag {160}
+$$
+
+Singling out  $\lambda_i^*$  from expression of  $q^{*}(x)_{y}$  in equation 158,  $q^{*}(x)_{g_{i}(x)}$  becomes
+
+$$
+q ^ {*} (x) _ {g _ {i} (x)} = \frac {\exp \left(\boldsymbol {\lambda} _ {i} ^ {*} + \sum_ {j \neq i : g _ {j} (x) = g _ {i} (x)} \boldsymbol {\lambda} _ {j} ^ {*}\right) p (x) _ {g _ {i} (x)}}{\exp \left(\boldsymbol {\lambda} _ {i} ^ {*} + \sum_ {j \neq i : g _ {j} (x) = g _ {i} (x)} \boldsymbol {\lambda} _ {j} ^ {*}\right) p (x) _ {g _ {i} (x)} + \sum_ {y \neq g _ {i} (x)} \exp \left(\sum_ {j : g _ {j} (x) = y} \boldsymbol {\lambda} _ {j} ^ {*}\right) p (x) _ {y}} \tag {161}
+$$
+
+or
+
+$$
+q ^ {*} (x) _ {g _ {i} (x)} = \exp \left(\boldsymbol {\lambda} _ {i} ^ {*}\right) f _ {i} (x) / \left(\exp \left(\boldsymbol {\lambda} _ {i} ^ {*}\right) f _ {i} (x) + 1 - f _ {i} (x)\right) \tag {162}
+$$
+
+where
+
+$$
+f _ {i} (x) = \exp \left(\sum_ {j \neq i: g _ {j} (x) = g _ {i} (x)} \lambda_ {j} ^ {*}\right) p (x) _ {g _ {i} (x)} / \left(\sum_ {y = 1} ^ {K} \exp \left(\sum_ {j \neq i: g _ {j} (x) = y} \lambda_ {j} ^ {*}\right) p (x) _ {y}\right) \tag {163}
+$$
+
+$f_{i}$  depends on  $\lambda_j^*$  for  $j\neq i$  but dependence is dropped for convenience.
+
+The complementary slackness condition in equation 160 now becomes
+
+$$
+\left. \boldsymbol {\lambda} _ {i} ^ {*} \left(1 - \eta_ {i} - \mathbb {E} \left[ \frac {\exp \left(\boldsymbol {\lambda} _ {i} ^ {*}\right) f _ {i} (X)}{\exp \left(\boldsymbol {\lambda} _ {i} ^ {*}\right) f _ {i} (X) + 1 - f _ {i} (X)} \mid X \in S _ {i} \right]\right) = 0 \right. \tag {164}
+$$
+
+Let  $\ell(t) = \mathbb{E}\left[\frac{t \cdot f_i(X)}{t \cdot f_i(X) + 1 - f_i(X)} | X \in S_i\right]$ . We have
+
+$$
+\frac {\delta}{\delta t} \ell (t) = \mathbb {E} \left[ \frac {f _ {i} (X) (1 - f _ {i} (X))}{(t f _ {i} (X) + 1 - f _ {i} (X)) ^ {2}} | X \in S _ {i} \right] > 0 \tag {165}
+$$
+
+and
+
+$$
+\frac {\delta^ {2}}{\delta t ^ {2}} \ell (t) = \mathbb {E} \left[ \frac {- 2 \left(f _ {i} (X)\right) ^ {2} \left(1 - f _ {i} (X)\right)}{\left(t f _ {i} (X) + 1 - f _ {i} (X)\right) ^ {3}} | X \in S _ {i} \right] <   0 \tag {166}
+$$
+
+(We assume  $p(x)_y \in (0,1)$  and thus  $f_i(x) \in (0,1)$  to get the above inequalities.)
+
+Therefore,  $\ell$  is a strictly increasing and concave function in  $t$ , and thus  $\ell(\exp(\lambda))$  is strictly increasing in  $\lambda$ . Thus from complementary slackness condition in equation 164, if  $\ell(\exp(0)) \geq 1 - \eta_i$  then  $\pmb{\lambda}_i^* = 0$ , else  $\pmb{\lambda}_i^* = \lambda : \ell(\exp(\lambda)) = 1 - \eta_i$ . Together this can be written as
+
+$$
+\lambda_ {i} ^ {*} = \min  _ {\lambda \geq 0} \lambda : \ell (\exp (\lambda)) \geq 1 - \eta_ {i} \tag {167}
+$$
+
+or
+
+$$
+\boldsymbol {\lambda} _ {i} ^ {*} = \min  _ {\lambda \geq 0} \lambda : \mathbb {E} \left[ \frac {\exp (\lambda) f _ {i} (X)}{\exp (\lambda) f _ {i} (X) + 1 - f _ {i} (X)} | X \in S _ {i} \right] \geq 1 - \eta_ {i} \tag {168}
+$$
+
+Plugging back  $f_{i}$  defined in equation 163, and using definition of  $p^{\lambda}$ , this is
+
+$$
+\boldsymbol {\lambda} _ {i} ^ {*} = \min  _ {\lambda \geq 0} \lambda : \mathbb {E} \left[ p ^ {\lambda} (X) _ {g _ {i} (X)} \right] \geq 1 - \eta_ {i} \tag {169}
+$$
+
+where  $\lambda_{i} = \lambda$ , and  $\lambda_{j} = \lambda_{j}^{*}$  for  $j \neq i$ . Or equivalently
+
+$$
+\boldsymbol {\lambda} _ {i} ^ {*} = \underset {\boldsymbol {\lambda} _ {i} \geq 0} {\arg \min } \sum_ {j = 1} ^ {m} \boldsymbol {\lambda} _ {j}: \rho \left(g _ {i}, p ^ {\boldsymbol {\lambda}}\right) \leq \eta_ {i} \tag {170}
+$$
+
+where  $\lambda_{j} = \lambda_{j}^{*}$  for  $j\neq i$
+
+And thus combining for all  $i$  we get,
+
+$$
+\boldsymbol {\lambda} ^ {*} = \underset {\boldsymbol {\lambda} \geq \mathbf {0}} {\arg \min } \sum_ {j = 1} ^ {m} \boldsymbol {\lambda} _ {j}: \rho \left(g _ {j}, p ^ {\boldsymbol {\lambda}}\right) \leq \eta_ {j} \forall j \in [ m ] \tag {171}
+$$
+
+![](images/1de6c1535a444e86e1b133c0c92b3e935eb60dd683e1de9bb52584cf11504c83.jpg)
+
+Proposition B.7. Given the optimization problem
+
+$$
+\boldsymbol {\lambda} ^ {*} = \underset {\boldsymbol {\lambda} \geq \mathbf {0}} {\arg \min } \sum_ {j = 1} ^ {m} \boldsymbol {\lambda} _ {j}: \rho \left(g _ {j}, p ^ {\boldsymbol {\lambda}}\right) \leq \eta_ {j} \forall j \in [ m ] \tag {172}
+$$
+
+The solution found by the following alternating minimization procedure is optimal if the outer loop converges (The inner loop (equation 174) always converges).
+
+Start with  $\lambda = 0$ . Then repeat until convergence:
+
+1. Pick  $j \in [m]$ , either randomly or in a round robin fashion.
+
+2. Solve for
+
+$$
+\lambda_ {j} = \arg \min  _ {\lambda} \lambda : \rho \left(g _ {j}, p ^ {\lambda_ {- j}}\right) \leq \eta_ {j} \tag {173}
+$$
+
+where  $\lambda_{-j} = [\lambda_1\ldots \lambda_{j - 1},0,\lambda_{j + 1},\lambda_m]$  as follows:
+
+$$
+\text {S t a r t :} \lambda_ {j} = 0
+$$
+
+Repeat until convergence:  $\lambda_{j}\gets \lambda_{j} + \log (1 + (1 - \eta_{j} - \tau_{j}(h_{j}))_{+} / \tau_{j}(h_{j}(1 - h_{j})))$
+
+$$
+\tau_ {j} (h) = \mathbb {E} [ h (X) | X \in S _ {j} ] \tag {174}
+$$
+
+where  $h_j(x) = p^\lambda (x)_{g_j(x)}$
+
+Proof. First, we show that the solution to the constrained optimization in equation 173 with respect to  $\lambda$  for a fixed  $\lambda_{-j}$  can be found by the iterative algorithm described in equation 174. The LHS of the constraint in equation 173,  $\rho(g_j, p^{\lambda_{-j}})$  is a function of the form
+
+$$
+\mathbb {E} \left[ \exp (\lambda) f _ {j} (X) / \left(\exp (\lambda) f _ {j} (X) + 1 - f _ {j} (X)\right) \mid X \in S _ {j} \right] \tag {175}
+$$
+
+where
+
+$$
+f _ {j} (x) = \exp (\sum_ {k \neq j: g _ {k} (x) = g _ {j} (x)} \lambda_ {k}) p (x) _ {g _ {j} (x)} / (\sum_ {y} \exp (\sum_ {k \neq j: g _ {k} (x) = y} \lambda_ {k}) p (x) _ {y})
+$$
+
+which is strictly increasing and concave in  $\exp (\lambda)$  as proved in equation 165. Suppose  $\mathbb{E}[h_j(X;\lambda^t,\pmb{\lambda}_{-j})|X\in S_j] < 1 - \eta_j$ . Then  $\pmb{\lambda}_j$  can also be written as
+
+$$
+\boldsymbol {\lambda} _ {j} = \min  _ {\lambda \geq 0} \lambda : \mathbb {E} \left[ h _ {j} \left(X; \lambda^ {t} + \lambda , \boldsymbol {\lambda} _ {- j}\right) | X \in S _ {j} \right] \geq 1 - \eta_ {j}
+$$
+
+We have
+
+$$
+h _ {j} (x; \lambda^ {t} + \lambda , \pmb {\lambda} _ {- j}) = \exp (\lambda) h _ {j} (x; \lambda^ {t}, \pmb {\lambda} _ {- j}) / (\exp (\lambda) h _ {j} (x; \lambda^ {t}, \pmb {\lambda} _ {- j}) + 1 - h _ {j} (x; \lambda^ {t}, \pmb {\lambda} _ {- j}))
+$$
+
+Letting  $\mu(s) = \mathbb{E}[h_j(X; \log s + \lambda^t, \lambda_{-j}) | X \in S_j]$ . This is of the form in 175 and is strictly increasing and concave in  $s$  as proved in equation 165, and equation 166 respectively.  $\lambda_j$  is thus given as  $\exp(s)$ ,  $s > 1$  s.t.  $\mu(s) = 1 - \eta_j$ . If a differentiable function  $\mu$  is strictly increasing and concave we have for any  $s_1, s_2$
+
+$$
+\mu \left(s _ {1}\right) <   \mu \left(s _ {1} + \left(\mu \left(s _ {2}\right) - \mu \left(s _ {1}\right)\right) / \mu^ {\prime} \left(s _ {1}\right)\right) <   \mu \left(s _ {2}\right) \tag {176}
+$$
+
+Observe that  $s = s_1 + (\mu(s_2) - \mu(s_1)) / \mu'(s_1)$  is also one iteration of the Newton-Raphson method to solve for  $s' : \mu(s') = \mu(s_2)$  starting with  $s = s_1$ .
+
+Using  $s_1 = 1$  and  $s_2: \mu(s_2) = 1 - \eta_i$ , we get
+
+$$
+\mu (1) <   \mu (1 + (1 - \eta_ {i} - \mu (1)) / \mu^ {\prime} (1)) <   1 - \eta_ {i} \tag {177}
+$$
+
+We have  $\mu(1) = \mathbb{E}[h_j(X; \lambda^t, \pmb{\lambda}_{-j}) | X \in S_j]$  and  $\mu'(1) = \mathbb{E}[h_j(X; \lambda^t, \pmb{\lambda}_{-j})(1 - h_j(X; \lambda^t, \pmb{\lambda}_{-j})) | X \in S_j]$ . Thus
+
+$$
+\lambda^ {t + 1} = \lambda^ {t} + \log \left(1 + \frac {\left(1 - \eta_ {i} - \mathbb {E} \left[ h _ {j} \left(X ; \lambda^ {t} , \boldsymbol {\lambda} _ {- j}\right) \mid X \in S _ {j} \right]\right)}{\mathbb {E} \left[ h _ {j} \left(X ; \lambda^ {t} , \boldsymbol {\lambda} _ {- j}\right) \left(1 - h _ {j} \left(X ; \lambda^ {t} , \boldsymbol {\lambda} _ {- j}\right)\right) \mid X \in S _ {j} \right]}\right) \tag {178}
+$$
+
+satisfies
+
+$$
+\mathbb {E} \left[ h _ {j} \left(X; \lambda^ {t}, \boldsymbol {\lambda} _ {- j}\right) \mid X \in S _ {j} \right]) <   \mathbb {E} \left[ h _ {j} \left(X; \lambda^ {t + 1}, \boldsymbol {\lambda} _ {- j}\right) \mid X \in S _ {j} \right]) <   1 - \eta_ {i} \tag {179}
+$$
+
+In particular given  $\lambda^t$ , we apply one step update of Newton-Raphson method on  $\mathbb{E}[h_j(X;\lambda^t,\pmb{\lambda}_{-j})|X\in S_j]$  to get  $\lambda^{t + 1}$  and thus subsequently use  $h_j(\cdot ;\lambda^{t + 1},\pmb{\lambda}_{-j})$  to apply the next update. Convergence of these updates is faster than directly applying Newton's method on the starting function  $\mathbb{E}[h_j(X;\lambda^0 = 0,\pmb{\lambda}_{-j})|X\in S_j]$  which is
+
+$$
+\exp \left(\lambda^ {t + 1}\right) = \exp \left(\lambda^ {t}\right) + \left(1 + \frac {\left(1 - \eta_ {i} - \mathbb {E} \left[ h _ {j} \left(X ; \lambda^ {t} , \boldsymbol {\lambda} _ {- j}\right) \mid X \in S _ {j} \right]\right)}{\mathbb {E} \left[ h _ {j} \left(X ; \lambda^ {t} , \boldsymbol {\lambda} _ {- j}\right) \left(1 - h _ {j} \left(X ; \lambda^ {t} , \boldsymbol {\lambda} _ {- j}\right)\right) \mid X \in S _ {j} \right]}\right) \tag {180}
+$$
+
+or
+
+$$
+\lambda^ {t + 1} = \lambda^ {t} + \log \left(1 + \frac {\exp (- \lambda^ {t}) (1 - \eta_ {i} - \mathbb {E} [ h _ {j} (X ; \lambda^ {t} , \boldsymbol {\lambda} _ {- j}) | X \in S _ {j} ])}{\mathbb {E} [ h _ {j} (X ; \lambda^ {t} , \boldsymbol {\lambda} _ {- j}) (1 - h _ {j} (X ; \lambda^ {t} , \boldsymbol {\lambda} _ {- j})) | X \in S _ {j} ]}\right) \tag {181}
+$$
+
+This results in a lower update since  $\lambda^t > 0$  or  $\exp(-\lambda^t) < 1$ .
+
+Now we show that if the alternating minimization converges it leads to the optimal solution  $\lambda^{*}$ .
+
+Suppose the algorithm converges but the solution  $\lambda$  found is not optimal. This implies that  $\lambda$  satisfies all of the constraints, but the KKT complementary slackness condition given in equation 164 is violated. Since  $\mathbb{E}[h_j(X;\lambda ,\lambda_{-j})|X\in S_j]$  is a strictly increasing function in  $\lambda$ , and  $\lambda_{j}$  solves for  $\lambda$  in equation 173 given  $\lambda_{-j}$ , if  $\lambda_{j} > 0$ , then the equality must be satisfied, otherwise  $\lambda_{j}$  can be reduced (since  $\lambda_{j} > 0$ ) without violating the inequality constraint. Thus either  $\mathbb{E}[h_j(X;\lambda ,\lambda_{-j})|X\in S_j] = 1 - \eta_j$  or  $\lambda_{j} = 0$  which satisfies the KKT complementary slackness condition.
+
+# C COMPACT VERSION OF OUR ALGORITHM
+
+Algorithm 1 Learning from weak labeler constraints  
+Input: Unlabeled data  $\{x_i\}_{i = 1}^n$  , weak labelers  $\{g_j\}_{j = 1}^m$  and the corresponding error bounds  $\{\eta_j\}_{j = 1}^m$  . A smooth real-valued function  $f:\mathcal{X}\rightarrow \mathbb{R}^{K}$ $w_{ij}\coloneqq g_j(x_i)\in [K]\cup \{\emptyset \}$  // Weak labels   
+ $c_{j}\coloneqq \sum_{i}s_{ij}$  // Coverage   
+for each epoch do   
+ $z_{i}\gets f(x_{i})$  // Logits   
+ $p_i\gets \sigma (z_i)$  // Probabilities   
+ $e_j\gets (1 - \eta_j - \sum_{i:w_{ij}\neq \emptyset}p_i / c_j)_+$  // Error in constraints   
+ $d_j\leftarrow \sum_{i:w_{ij}\neq \emptyset}p_i(1 - p_i) / c_j$ $\lambda_j\gets \log (1 + e_j / d_j)$  // One step parallel newton update to find  $\lambda$  (equation 22)   
+ $(h_i)_k\gets (z_i)_k + \sum_j\lambda_j\mathbb{I}(w_{ij} = k)$  // Logits of projected distribution (equation 22)   
+loss  $\leftarrow \frac{1}{n}\sum_{i = 1}^{n}\ell_{ce}(z_i,\sigma (h_i))$  // Cross entropy loss   
+ $f\gets update(f,\mathcal{R}(f) + \alpha \cdot \mathrm{loss})$  // Gradient descent update (equation 10)   
+end for
+
+![](images/11192ddfdeae7df7b80afc0119d545b4e97e13c06ffd4ed8a67675d551937186.jpg)  
+Figure 3: Max error in satisfying convex approximation of constraints  $(\max_j\rho_{\mathrm{dist}}(g_j,q) - \eta_j)$ , (left) vs max error in satisfying classifier constraints  $(\max_j\rho (g_j,\operatorname {clf}(q)) - \eta_j)$ , (right) averaged over 100 runs of random draws of  $p$  and weak labelers, where  $q = \pi (p,Q_{\mathrm{dist}})$ , i.e. the projection  $q$  is found by projecting  $p$  on constraints on distribution as described in section 3.2.
+
+# D ADDITIONAL RESULTS ON SOLVING THE ILP
+
+# D.1 SOLVING THE ILP THROUGH LP
+
+For the particular ILP in equation 13, we have empirically found that the optimal solution of the relaxed LP is mostly integral, and approximating the non-integer by its integral solution almost always satisfies the constraints (the error in constraint satisfaction  $\rho_{\mathrm{clf}}(g_j,\operatorname {clf}(q)) - \eta_j)$
+
+In Figure 4 and 5, we show the performance of solving the ILP after LP relaxation. We used 'scipy.linprog' in Python to solve the LP. In particular, Figure 4 shows that the constraints are satisfied well (the error  $\rho_{\mathrm{clf}}(g_j,\operatorname {clf}(q)) - \eta_j$  is small), and 5 shows that only a small fraction of the variables (one variable for each unlabeled data  $x$ ) are non-integral.
+
+![](images/3a156c29381d90aa9bb9048e37ee16b0562a543ac49a2597f157895039dbefd2.jpg)  
+Figure 4: Max error in satisfaction of classifier constraints  $(\max_j\rho (g_j,\operatorname {clf}(q)) - \eta_j)$ , averaged over 100 runs of random draws of  $p$  and weak labelers, where  $q = \pi (p,Q_{\mathrm{clf}})$ , i.e. the projection  $q$  is found by projecting  $p$  on the constraints on classifier through LP relaxation of the ILP described in 3.5.
+
+![](images/c8ed312a46b2b26af613704eec3c146b2c38c563bcc1d430b5988a18691adcb8.jpg)  
+Figure 5: Fraction of non-integral points after solving the LP relaxed version of ILP averaged over 100 runs of random draws of  $p$  and weak labelers.
+
+![](images/c33245b4e814040bcd6b1ac0f7be559c2ad97c1a3abcf7c1d811724eab469284.jpg)
+
+![](images/d4a0bca7cc1bb91b610037d907297a95928fe9df959dee0bbd096be949b374ba.jpg)  
+Figure 6: Max of  $D_{\mathrm{KL}}(h||p) / D_{\mathrm{KL}}(h^* ||p)$  over 1000 uniformly random draws of  $p$ .  $h$  denotes the solution of 182 found by approximation (equation 184) or gradient descent (20 steps after initial approximation) and  $h^*$  denotes the optimal solution as found by "cvxpy.CLARABEL" convex solver.
+
+# D.2 ESTIMATION OF THE COST VECTOR
+
+$\operatorname{Proj}_{\mathrm{KL}}(p(x);\Delta_{y,\epsilon}^{K})$  and  $q^{*}(x)$  from equation 14, and equation 15 requires solving an optimization problem. We expand the  $D_{\mathrm{KL}}$  term to have
+
+$$
+\min  _ {h \in \Delta^ {K}} \sum_ {i = k} ^ {K} h _ {i} \log \left(h _ {k} / p (x) _ {k}\right) \text {s . t .} h _ {y} \geq (1 + \epsilon) h _ {k}, \forall k \neq y \tag {182}
+$$
+
+which can be solved by a black box convex solver. However, it can also be reparameterized in  $\lambda \in \mathbb{R}_+^{K - 1}$  to get an unconstrained optimization as,
+
+$$
+h _ {k} = \exp (- \lambda_ {k}) / Z \text {f o r} k \neq y, h _ {y} = (1 + \epsilon) / Z; Z = \sum_ {k} h _ {k} \tag {183}
+$$
+
+The objective is no longer convex in  $\lambda$  but projected gradient descent (projection to  $\mathbb{R}_+^{K-1}$ ) can still be used to find a local minima. In particular, we find the approximation
+
+$$
+\lambda_ {i} = \log \left(\max  _ {j} p (x) _ {j} / p (x) _ {i}\right) \tag {184}
+$$
+
+yields  $h$  for which  $D_{\mathrm{KL}}(h||p(x))$  is empirically found to be within a small constant factor from the optimal. This can further be used as a starting point for gradient descent, which then results in a solution close to optimal. In Figure 6, we show empirically that our approximation is within a constant factor from the optimal as found by 'cvxpy.CLARABEL' convex solver in Python, and a few iterations of projected gradient descent after this approximation find a solution very close to the optimal.
+
+# E CONVEX APPROXIMATION OF THE CONSTRAINT ON CLASSIFIER WITH THE CONSTRAINT ON DISTRIBUTION
+
+We remarked that the constraint sets  $Q_{j,\mathrm{dist}}$  that impose constraints on distribution, i.e.
+
+$$
+Q _ {j, \text {d i s t}} = \left\{q \in \mathcal {X} \rightarrow \Delta^ {K} \mid \rho_ {\text {d i s t}} \left(g _ {j}, q\right) \leq \eta_ {j} \right\} \tag {185}
+$$
+
+can be considered as a convex approximation to the constraint sets  $Q_{j,\mathrm{clf}}$  that impose constraints on classifiers, i.e.
+
+$$
+Q _ {j, \operatorname {c l f}} = \left\{q \in \mathcal {X} \rightarrow \Delta^ {K} \mid \rho_ {\operatorname {c l f}} \left(g _ {j}, \operatorname {c l f} (q)\right) \leq \eta_ {j} \right\} \tag {186}
+$$
+
+First, we note that for distributions that are very confident on one label, i.e. for any  $x$ , there is a  $y$  such that  $p(x)_y > 1 - \epsilon$  for a small  $\epsilon$ , we can quantify the extent of approximation.
+
+Lemma E.1. If for any  $x$  there is a  $y$ , such that  $p(x)_y > 1 - \epsilon$ , then
+
+$$
+\rho_ {\operatorname {c l f}} \left(g _ {j}, \operatorname {c l f} (p)\right) \leq \rho_ {\operatorname {d i s t}} \left(g _ {j}, p\right) / (1 - \epsilon)
+$$
+
+and
+
+$$
+\rho_ {\mathrm {d i s t}} (g _ {j}, p) \leq \rho_ {\mathrm {c l f}} (g _ {j}, \operatorname {c l f} (p)) (1 - \epsilon) + \epsilon
+$$
+
+Thus any  $p$  that satisfies  $\rho_{\mathrm{clf}}(g_j, \operatorname{clf}(p)) \leq (\eta_j - \epsilon) / (1 - \epsilon)$  lies in  $Q_{j,\mathrm{dist}}$  and any  $p \in Q_{j,\mathrm{dist}}$  satisfies  $\rho_{\mathrm{clf}}(g_j, \operatorname{clf}(p)) \leq \eta_j / (1 - \epsilon)$ .
+
+Proof.
+
+$$
+\begin{array}{l} \rho_ {\text {d i s t}} (g, p) = \mathbb {E} \left[ \sum_ {k \neq g (X)} p (X) _ {k} | X \in S, \operatorname {c l f} (p) (X) = g (X) \right] \Pr (\operatorname {c l f} (p) (X) = g (X) | X \in S) \\ + \mathbb {E} \left[ \sum_ {k \neq g (X)} p (X) _ {k} | X \in S, \operatorname {c l f} (p) (X) \neq g (X) \right] \Pr (\operatorname {c l f} (p) (X) \neq g (X) | X \in S) \\ \leq \epsilon \left(1 - \rho_ {\operatorname {c l f}} (g, \operatorname {c l f} (p))\right) + \rho_ {\operatorname {c l f}} (g, \operatorname {c l f} (p)) \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \rho_ {\text {d i s t}} (g, p) \leq \rho_ {\text {c l f}} (g, \operatorname {c l f} (p)) (1 - \epsilon) + \epsilon \\ \rho_ {\text {d i s t}} (g, p) \geq 0 + (1 - \epsilon) \rho_ {\text {c l f}} (g, \operatorname {c l f} (p)) \\ \rho_ {\operatorname {c l f}} (g, \operatorname {c l f} (p)) \leq \rho_ {\operatorname {d i s t}} (g, p) / (1 - \epsilon) \\ \end{array}
+$$
+
+We dropped subscript  $j$  for convenience. Substituting  $\rho_{\mathrm{clf}}(g_j,\operatorname {clf}(p))\leq (\eta_j - \epsilon) / (1 - \epsilon)$  in the bound for  $\rho_{\mathrm{clf}}$ , we get  $\rho_{\mathrm{dist}}(g_j,p)\leq \eta_j$ . Similarly, substituting  $\rho_{\mathrm{dist}}(g_j,p)\leq \eta_j$  in the bound for  $\rho_{\mathrm{clf}}$ , we get  $\rho_{\mathrm{clf}}(g_j,\operatorname {clf}(p))\leq \eta_j / (1 - \epsilon)$ .
+
+The above lemma suggests that if the true distribution makes confident predictions, (which is often the case for many real-world classification setting), then imposing a constraint on the distribution is close to imposing a constraint on classifier. However, this is only a sufficient condition and the convex approximation can be good without such a condition as well.
+
+In Figure, 3 we show the maximum error in satisfaction of any classifier constraint  $\max_j\rho_{\mathrm{clf}}(g_j,\mathrm{clf}(q)) - \eta_j$  for  $p$  sampled uniformly at random when we use the method for projecting on constraints on distribution, i.e.  $q = \pi (p,Q_{\mathrm{dist}})$ . The data for weak labelers was generated randomly by first drawing  $S_{j}$  such that  $|S_{j}|\sim \mathcal{N}(0.6,0.3)$  and error bound  $\eta_j\sim \mathcal{N}(0.3,0.3)$  and then choosing weak labels uniformly at random such that their expected error in  $S_{j}$  is  $\eta_{j}$ . To estimate the projection  $\pi (p,Q_{\mathrm{dist}})$ , 20 steps of alternating minimization in a round-robin fashion (iterating over all constraints one by one) is performed to find  $\lambda$  (proposition B.7), where each step for each constraint  $i$  is optimized for 10 steps to find  $\lambda_{i}$  given current the iterate  $\lambda_{-i}$ .
+
+# F ESTIMATING ACCURACY BOUND FROM THE VALIDATION SET
+
+Our algorithm needs some estimate of the error bounds  $\eta_{j}$ . Often they are not given but we have a small validation set with true labels, which can be leveraged to get bounds.
+
+One can be tempted to estimate  $\eta_{j}$  for each labeler  $g_{j}$  separately from available labels in its coverage region  $S_{j}$ . This approach is problematic since many weak labelers have a small coverage set  $S_{j}$  such
+
+that it is unlikely to have coverage in the validation set. Indeed if the validation set was large enough to have good coverage for all weak labelers, supervised learning may perform well enough rendering any additional information provided by weak labeler constraints uninformative. We consider two ways to alleviate this problem of low coverage.
+
+First, we consider a prior distribution on weak labeler errors  $\eta_{j}$  and then compute a Bayesian posterior from the labeled set (and use its mean as the estimate of the bound  $\eta_{j}$ ). When there are no labels in the coverage, the posterior remains the same as the prior. One example of a prior is a Beta prior  $\beta (\alpha ,\beta)$  under which observing  $k$  incorrect out of  $n$  samples induces the posterior with mean  $(k + \alpha) / (n + \alpha +\beta)$ . This can also be viewed as domain experts providing a general prior belief about errors instead of individual error estimates/ bounds for each weak labeler.
+
+Second, we consider a shared parameterization of errors as  $\eta_{j} = h_{\phi}(g_{j},S_{j})$  and then treat  $\phi$  as a hyperparameter to be tuned on the validation set. A simple instance of this shared parameterization is a common bound  $\eta_{j} = \phi$ , i.e.  $h_\phi (\cdot) = \phi$ . This can also be viewed as domain experts providing relative information about errors, such as the average errors of different weak labelers are similar/correlated.
+
+# G ADDITIONAL EXPERIMENT DETAILS
+
+# G.1 DATASET STATISTICS
+
+<table><tr><td>Dataset</td><td>Num classes</td><td>Num labelers</td><td>Mean Accuracy</td><td>Mean Coverage</td></tr><tr><td>Youtube</td><td>2</td><td>10</td><td>0.83 ± 0.14</td><td>0.17 ± 0.09</td></tr><tr><td>IMDB</td><td>2</td><td>5</td><td>0.71 ± 0.15</td><td>0.24 ± 0.31</td></tr><tr><td>Bioresponse</td><td>2</td><td>20</td><td>0.64 ± 0.09</td><td>0.17 ± 0.20</td></tr><tr><td>Yelp</td><td>2</td><td>8</td><td>0.74 ± 0.09</td><td>0.18 ± 0.20</td></tr><tr><td>CDR</td><td>2</td><td>33</td><td>0.72 ± 0.13</td><td>0.06 ± 0.12</td></tr><tr><td>Chemprot</td><td>10</td><td>26</td><td>0.47 ± 0.17</td><td>0.06 ± 0.07</td></tr><tr><td>Semeval</td><td>9</td><td>164</td><td>0.77 ± 0.40</td><td>0.01 ± 0.03</td></tr><tr><td>Trec</td><td>6</td><td>68</td><td>0.73 ± 0.37</td><td>0.03 ± 0.08</td></tr></table>
+
+Table 2: Dataset statistics includes the number of classes, the number of weak labelers, and the mean accuracy and coverage of the weak labelers.
+
+# G.2 ERROR IN SATISFACTION OF CONSTRAINTS
+
+Table G.2 shows the errors in satisfaction of classifier constraints for our trained classifier  $f_{\mathrm{clf}}^{*}$ , when projecting on  $Q_{\mathrm{clf}}$  (row Ourclf(T) in table 1) as  $\Delta_{\mathrm{clf}}^{j}$ , and  $f_{\mathrm{dist}}^{*}$  when projecting on  $Q_{\mathrm{dist}}$  (row Our(T) in table 1) as  $\Delta_{\mathrm{dist}}^{j}$ .
+
+$$
+\Delta_ {\mathrm {c l f}} ^ {j} = \left(\rho_ {\mathrm {c l f}} \left(g _ {j}, f _ {\mathrm {c l f}} ^ {*}\right)\right) - \eta_ {j}) _ {+} \tag {187}
+$$
+
+$$
+\Delta_ {\text {d i s t}} ^ {j} = \left(\rho_ {\text {c l f}} \left(g _ {j}, f _ {\text {d i s t}} ^ {*}\right)\right) - \eta_ {j}) _ {+} \tag {188}
+$$
+
+$$
+\operatorname {m e a n} _ {j} \Delta^ {j} = \sum_ {j = 1} ^ {m} w _ {j} \Delta^ {j} \tag {189}
+$$
+
+$$
+w _ {j} = \left| S _ {j} \right| / \sum_ {j = 1} ^ {m} \left| S _ {j} \right| \tag {190}
+$$
+
+For most datasets, we see that almost all constraints are satisfied. Despite projecting on  $Q_{\mathrm{dist}}$ ,  
+
+<table><tr><td></td><td>Bio</td><td>CDR</td><td>Chem</td><td>IMDB</td><td>Semeval</td><td>Trec</td><td>Yelp</td><td>Youtube</td></tr><tr><td># Classes</td><td>2</td><td>2</td><td>10</td><td>2</td><td>9</td><td>6</td><td>2</td><td>2</td></tr><tr><td># Labelers</td><td>20</td><td>33</td><td>26</td><td>5</td><td>164</td><td>68</td><td>8</td><td>10</td></tr><tr><td>meanjΔjdist</td><td>0.00.0</td><td>0.00.0</td><td>0.00.0</td><td>0.020.02</td><td>0.00.0</td><td>0.00.0</td><td>0.00.0</td><td>0.00.0</td></tr><tr><td>meanjΔjclf</td><td>0.00.0</td><td>0.00.0</td><td>0.020.01</td><td>0.050.01</td><td>0.00.0</td><td>0.020.01</td><td>0.010.0</td><td>0.010.01</td></tr><tr><td>maxjΔjdist</td><td>0.080.15</td><td>0.070.07</td><td>0.190.29</td><td>0.110.05</td><td>0.00.01</td><td>0.120.06</td><td>0.030.01</td><td>0.010.01</td></tr><tr><td>maxjΔjclf</td><td>0.030.01</td><td>0.050.05</td><td>0.280.22</td><td>0.120.03</td><td>0.050.03</td><td>0.550.29</td><td>0.060.01</td><td>0.040.03</td></tr></table>
+
+Table 3: Mean and Max error in satisfaction of constraints
+
+$\Delta_{\mathrm{dist}}^j$  which measures error in satisfying classifier constraint is small, indicating that projecting on distribution works well empirically for satisfying constraints on classifier.
+
+# G.3 EXPERIMENT DETAILS OF SOLVING THE ILP
+
+In table 1, we also reported our results for the method of projecting on the constraint on classifiers through solving the ILP in 3.5, as  $\mathrm{Our}_{\mathrm{clf}}(T)$ . We did not have access to the true conditional distribution, and so we could not estimate accuracy bounds with respect to the target Bayes optimal classifier equation 1. Instead, we use the same bounds as estimated from training labels by interpreting them as noisy bounds with respect to the target classifier.
+
+We use the python function "scipy.linprog" to solve the ILP and estimate the projection.  $\epsilon$  in 12 was set to 0.2. The cost vector  $\mathrm{Proj}_{\mathrm{KL}}(p(X);\Delta_{y,\epsilon}^{K})$  of the ILP is estimated as described in D.2, in particular we use the approximation  $\lambda_{k} = \log (\max_{j}p(x)_{j} / p(x)_{k})$  for  $\lambda_{k}$  in equation 183. To limit the number of constraints in the linear program, we use stochastic gradient descent with a batch size of 1000. We take a single gradient step in each iteration of the alternating minimization in 3.2.
+
+# H ERROR GUARANTEE INSIDE THE COVERAGE OF WEAK LABELERS
+
+Theorem H.1. Let  $\mathcal{H}_1, \ldots, \mathcal{H}_k$  be constraints of the form  $\mathcal{H}_j = \{h : \mathcal{X} \to [K] \mid \rho_{\mathrm{clf}}(g_j, h) \leq \eta_j\}$ , each with different constant weak labels,  $g_j(X) = j - 1$  for all  $j = 1, \ldots, K$ , then for any  $f \in \bigcap_j \mathcal{H}_j$ , we have
+
+$$
+\operatorname {e r r} _ {S} (f) \leq \sum_ {j = 1} ^ {k} \left(\eta_ {j} + \operatorname {e r r} _ {S _ {j}} \left(g _ {j}\right)\right) \frac {\Pr \left(S _ {j}\right)}{\Pr (S)} - \sum_ {\substack {\sigma \subseteq \{1, \dots , k \}, \\ |\sigma | \geq 2}} (2 | \sigma | - 3) \frac {\Pr \left(B _ {\sigma}\right)}{\Pr (S)}. \tag{191}
+$$
+
+when  $S = \bigcup_{j=1}^{k} S_{j}$ ,  $B_{\sigma} = \left(\bigcap_{i \in \sigma} S_{i}\right) \cap \left(\bigcap_{i \in \{1, \ldots, k\} \setminus \sigma} S_{i}^{c}\right)$  are minterms.
+
+Proof. (Theorem 4.5) Let  $B_{\sigma} = (\bigcap_{i \in \sigma} S_i) \cap (\bigcap_{i \in \{1, \ldots, k\} \setminus \sigma} S_i^c)$  be minterms. For simplicity, we denote  $B_j = B_{\{j\}}$  and write  $\operatorname*{Pr}(f(X) \neq f^*(X); B) = \operatorname*{Pr}(X \in \{x \in \mathcal{X} \mid f(x) \neq f^*(x)\} \cap B)$ . Since the weak labelers  $g_j(x)$  are constant and different for each  $g_j$ , without loss of generality, we assume that  $g_j(x) = j - 1$  for any  $x \in S_j$  and for all  $j = 1, \ldots, k$ . Intuitively,  $B_j$  are regions in  $S_j$  with no other conflicting weak labels. We observe that
+
+$$
+\begin{array}{l} \operatorname {e r r} _ {S} (f) \Pr (X \in S) (192) \\ = \Pr (f (X) \neq f ^ {*} (X); S) (193) \\ = \sum_ {j = 1} ^ {k} \Pr (f (X) \neq f ^ {*} (X); B _ {j}) + \Pr (f (X) \neq f ^ {*} (X); S - \bigcup_ {j = 1} ^ {k} B _ {j}) \quad \left(B _ {i} \cap B _ {j} = \emptyset , i \neq j\right) (194) \\ \leq \sum_ {j = 1} ^ {k} \Pr (f (X) \neq j - 1; B _ {j}) + \Pr (f ^ {*} (X) \neq j - 1; B _ {j}) + \Pr (X \in S - \bigcup_ {j = 1} ^ {k} B _ {j}) \quad (\triangle \text {i n e q u a l i t y} (195) \\ \end{array}
+$$
+
+Since  $f\in \bigcap_{j}\mathcal{H}_{j}$  , for  $j = 1,\ldots ,k$  , we have
+
+$$
+\Pr (f (X) \neq j - 1; S _ {j}) \leq \eta_ {j} \Pr (X \in S _ {j}) \tag {196}
+$$
+
+$$
+\Pr (f (X) \neq j - 1; B _ {j}) + \Pr (f (X) \neq j - 1; S _ {j} - B _ {j}) \leq \eta_ {j} \Pr (X \in S _ {j}) \tag {197}
+$$
+
+$$
+\Pr (f (X) \neq j - 1; B _ {j}) \leq \eta_ {j} \Pr (X \in S _ {j}) - \Pr (f (X) \neq j - 1; S _ {j} - B _ {j}). \tag {198}
+$$
+
+Similarly, since  $f^{*} \in \bigcap_{j} \mathcal{H}_{j}$ , for  $j = 1, \ldots, k$ , we also have
+
+$$
+\Pr \left(f ^ {*} (X) \neq j - 1; B _ {j}\right) \leq \operatorname {e r r} _ {S _ {j}} \left(g _ {j}\right) \Pr \left(S _ {j}\right) - \Pr \left(f ^ {*} (X) \neq j - 1; S _ {j} - B _ {j}\right). \tag {199}
+$$
+
+Substitute these inequalities to the above, we have
+
+$$
+\begin{array}{l} \operatorname {e r r} _ {S} (f) \Pr (X \in S) \leq \sum_ {j = 1} ^ {k} \left(\eta_ {j} + \operatorname {e r r} _ {S _ {j}} \left(g _ {j}\right)\right) \Pr (X \in S _ {j}) - \Pr (f (X) \neq j - 1; S _ {j} - B _ {j}) (200) \\ - \Pr (f ^ {*} (X) \neq j - 1; S _ {j} - B _ {j}) + \Pr (X \in S - \bigcup_ {j = 1} ^ {k} B _ {j}) (201) \\ \end{array}
+$$
+
+The region  $S_{j} - B_{j}$  is the area in  $S_{j}$  with at least one other conflicting region, we can write this as a combination of minterms that represent each weak label in each region,
+
+$$
+S _ {j} - B _ {j} = \bigcup_ {\substack {\sigma \subseteq \{1, \dots , k \}, \\ j \in \sigma , | \sigma | \geq 2}} B _ {\sigma} \tag{202}
+$$
+
+Therefore,
+
+$$
+\begin{array}{l} \sum_ {j = 1} ^ {k} \Pr (f (X) \neq j - 1; S _ {j} - B _ {j}) (203) \\ = \sum_ {j = 1} ^ {k} \Pr (X \in S _ {j} - B _ {j}) - \Pr (f (X) = j - 1; S _ {j} - B _ {j}) (204) \\ = \sum_ {j = 1} ^ {k} \Pr (X \in S _ {j} - B _ {j}) - \sum_ {j = 1} ^ {k} \sum_ {\substack {\sigma \subseteq \{1, \dots , k \}, \\ j \in \sigma , | \sigma | \geq 2}} \Pr (f (X) = j - 1; B _ {\sigma}) \quad (\text {from equation 202}) (205) \\ = \sum_ {j = 1} ^ {k} \Pr (X \in S _ {j} - B _ {j}) - \sum_ {\substack {\sigma \subseteq \{1, \dots , k \}, \\ | \sigma | \geq 2}} \sum_ {j \in \sigma} \Pr (f (X) = j - 1; B _ {\sigma}) (206) \\ = \sum_ {j = 1} ^ {k} \Pr (X \in S _ {j} - B _ {j}) - \sum_ {\substack {\sigma \subseteq \{1, \dots , k \}, \\ | \sigma | \geq 2}} \Pr (f (X) + 1 \in \sigma ; B _ {\sigma}) \quad (\text {l a b e l s t a r t s f r o m 0}) (207) \\ \geq \sum_ {j = 1} ^ {k} \Pr (X \in S _ {j} - B _ {j}) - \sum_ {\substack {\sigma \subseteq \{1, \dots , k \}, \\ | \sigma | \geq 2}} \Pr (X \in B _ {\sigma}) (208) \\ \end{array}
+$$
+
+The second to last step holds from the fact that for each  $\sigma$ , we will see  $B_{\sigma}$ ,  $|\sigma|$  times for each element  $j \in \sigma$  in terms of  $\operatorname{Pr}(f(X) = j - 1; B_{\sigma})$ . The same argument also holds for  $f^{*}$ . Substitute this back to the equation equation 200
+
+$$
+\begin{array}{l} \operatorname {e r r} _ {S} (f) \Pr (X \in S) \leq \sum_ {j = 1} ^ {k} \left(\eta_ {j} + \operatorname {e r r} _ {S _ {j}} \left(g _ {j}\right)\right) \Pr (X \in S _ {j}) - 2 \sum_ {j = 1} ^ {k} \Pr (X \in S _ {j} - B _ {j}) (209) \\ + 2 \sum_ {\sigma \subseteq \{1, \dots , k \}, | \sigma | \geq 2} \Pr (X \in B _ {\sigma}) + \Pr (X \in S - \bigcup_ {j = 1} ^ {k} B _ {j}). (210) \\ \end{array}
+$$
+
+Finally, we can write each term as a combination of  $B_{\sigma}$
+
+$$
+\Pr (X \in S - \bigcup_ {j = 1} ^ {k} B _ {j}) = \sum_ {\substack {\sigma \subseteq \{1, \dots , k \}, \\ | \sigma | \geq 2}} \Pr (X \in B _ {\sigma}), \tag{211}
+$$
+
+$$
+\sum_ {j = 1} ^ {k} \Pr (X \in S _ {j} - B _ {j}) = \sum_ {\substack {\sigma \subseteq \{1, \dots , k \}, \\ | \sigma | \geq 2}} | \sigma | \Pr (X \in B _ {\sigma}) \tag{212}
+$$
+
+Substitute in, we have the result,
+
+$$
+\operatorname {e r r} _ {S} (f) \Pr (X \in S) \leq \sum_ {j = 1} ^ {k} \left(\eta_ {j} + \operatorname {e r r} _ {S _ {j}} \left(g _ {j}\right)\right) \Pr (X \in S _ {j}) - \sum_ {\substack {\sigma \subseteq \{1, \dots , k \}, \\ | \sigma | \geq 2}} (2 | \sigma | - 3) \Pr (X \in B _ {\sigma}). \tag{213}
+$$
+
+or in short, we write
+
+$$
+\operatorname {e r r} _ {S} (f) \leq \sum_ {j = 1} ^ {k} \left(\eta_ {j} + \operatorname {e r r} _ {S _ {j}} \left(g _ {j}\right)\right) \frac {\Pr \left(S _ {j}\right)}{\Pr (S)} - \sum_ {\substack {\sigma \subseteq \{1, \dots , k \}, \\ | \sigma | \geq 2}} (2 | \sigma | - 3) \frac {\Pr \left(B _ {\sigma}\right)}{\Pr (S)}. \tag{214}
+$$
+
+![](images/65ece10f6dc4650cd90cacf9fa09c812e855c55b52f33e7cf98e9e9f94c16f8c.jpg)
+
+# I EXTENDING THE ERROR GUARANTEE TO THE AREA OUTSIDE OF THE WEAK LABELERS COVERAGE WITH A SMOOTHNESS PROPERTY
+
+To prove Theorem 4.6, we rely on the following observation.
+
+Lemma I.1. Let  $f$  be an L-Lipschitz function w.r.t. a metric  $d$ . For any distribution  $P, Q$  we have
+
+$$
+\mathbb {E} _ {X \sim P} [ f (X) ] - \mathbb {E} _ {Y \sim Q} [ f (Y) ] \leq W _ {d} (P, Q) \tag {215}
+$$
+
+where  $W_{d}(P,Q)$  is a Wasserstein distance between  $P, Q$  where the cost function is given by  $d$ .
+
+Proof. For any distribution  $P, Q$  and a transport plan  $T: \mathcal{X} \times \mathcal{X}$  between  $P, Q$ , we have
+
+$$
+\begin{array}{l} \mathbb {E} _ {X \sim P} [ f (X) ] - \mathbb {E} _ {Y \sim Q} [ f (Y) ] = \int f (x) p (x) d x - \int f (y) q (y) d y (216) \\ = \iint f (x) T (x, y) d y d x - \iint f (y) T (x, y) d x d y (217) \\ = \iint | f (x) - f (y) | T (x, y) d y d x \leq L \iint d (x, y) T (x, y) d y d x. (218) \\ \end{array}
+$$
+
+When  $T$  is the optimal transport plan, the right-hand side term is the smallest, which is the definition of the Wasserstein distance  $W_{d}(P,Q)$ .
+
+The following Theorem is a generalization of Theorem 4.6.
+
+Theorem 1.2. Let  $P$  be a joint distribution over  $\mathcal{X} \times \mathcal{Y}$  such that  $\operatorname*{Pr}(Y = y \mid X = x)$  is  $L$ -Lipschitz w.r.t. a metric  $d$  where for all  $k = 1, \ldots, K$  and  $x_1, x_2 \in \mathcal{X}$ ,
+
+$$
+\left| \Pr (Y = k \mid X = x _ {1}) - \Pr (Y = k \mid X = x _ {2}) \right| \leq L d \left(x _ {1}, x _ {2}\right), \tag {219}
+$$
+
+For any set  $S$  and  $f$  we write an error of  $f$  with respect to  $P$  as  $\widetilde{\mathrm{err}}_S(f) = \operatorname*{Pr}(f(X) \neq Y \mid X \in S)$ , then for any subsets  $C, D \subseteq \mathcal{X}$ , we have
+
+$$
+\widetilde {\operatorname {e r r} _ {D}} (f) \leq \widetilde {\operatorname {e r r} _ {C}} (f) \left(1 + t _ {f} (C, D)\right) + L u _ {f} (C, D) \tag {220}
+$$
+
+when
+
+$$
+t _ {f} (C, D) = \sup  _ {k} \left| \frac {\Pr_ {C} (f (X) = k \mid X \in C) - \Pr (f (X) = k \mid X \in D)}{\Pr (f (X) = k \mid X \in C)} \right| \tag {221}
+$$
+
+and
+
+$$
+u _ {f} (C, D) = \sum_ {j} \Pr_ {D} (f (X) = k) W _ {d} \left(P _ {X | C \cap \{x \mid f (x) = k \}}, P _ {X | D \cap \{x \mid f (x) = k \}}\right), \tag {222}
+$$
+
+$W_{d}$  is a Wasserstein distance and we denote  $P_{X|A}$  for a distribution  $P_{X}$  condition on  $X \in A$ .
+
+Proof. First, we write an error term as a combination of errors for each prediction of  $f(x)$ ,
+
+$$
+\begin{array}{l} \widetilde {\operatorname {e r r}} _ {D} (f) = \Pr (f (X) \neq Y \mid X \in D) (223) \\ = \sum_ {k = 1} ^ {K} \Pr (f (X) \neq Y \mid f (X) = k, X \in D) \Pr (f (X) = k \mid X \in D) (224) \\ = \sum_ {k = 1} ^ {K} \Pr (Y \neq k \mid f (X) = k, X \in D) \Pr (f (X) = k \mid X \in D). (225) \\ \end{array}
+$$
+
+We observe that
+
+$$
+\begin{array}{l} | \Pr (Y \neq k \mid f (X) = k, X \in D) - \Pr (Y \neq k \mid f (X) = k, X \in C) | (226) \\ = | \Pr (Y = k \mid f (X) = k, X \in D) - \Pr (Y = k \mid f (X) = k, X \in C) | (227) \\ = \left| \mathbb {E} _ {X \sim P _ {X \mid D \cap \{x \mid f (x) = k \}}} [ \Pr (Y = k \mid X) ] - \mathbb {E} _ {X \sim P _ {X \mid C \cap \{x \mid f (x) = k \}}} [ \Pr (Y = k \mid X) ] \right| (228) \\ \leq L W _ {d} \left(P _ {X | D \cap \{x \mid f (x) = k \}}, P _ {X | C \cap \{x \mid f (x) = k \}}\right) (229) \\ \end{array}
+$$
+
+In the second to last line, we can think of  $\operatorname{Pr}(Y = k|X)$  as a function of  $X$  for which we can then apply Lemma I.1 to achieve the last line. For compactness of our notation, we denote  $\operatorname{Pr}_C(\cdot) = \operatorname{Pr}_X(\cdot \mid X \in C)$ . Substitute this back to the equation above, we have
+
+$$
+\begin{array}{l} \widetilde {\operatorname {e r r} _ {D}} (f) \leq \sum_ {k = 1} ^ {K} \Pr_ {C} (Y \neq k \mid f (X) = k) \Pr_ {D} (f (X) = k) (230) \\ + \sum_ {k = 1} ^ {K} \left| \Pr_ {D} (Y \neq k \mid f (X) = k) - \Pr_ {C} (Y \neq k \mid f (X) = k) \right| \Pr_ {D} (f (X) = k) (231) \\ \leq \sum_ {k = 1} ^ {K} \Pr_ {C} (Y \neq k \mid f (X) = k) \Pr_ {D} (f (X) = k) (232) \\ + \sum_ {k = 1} ^ {K} L W _ {d} \left(P _ {X \mid D \cap \{x \mid f (x) = k \}}, P _ {X \mid C \cap \{x \mid f (X) = k \}}\right) \Pr_ {D} (f (x) = k) (233) \\ = \sum_ {k = 1} ^ {K} \Pr_ {C} (Y \neq k \mid f (X) = k) \Pr_ {D} (f (X) = k) + L u _ {f} (C, D). (234) \\ \end{array}
+$$
+
+Finally, we also observe that
+
+$$
+\begin{array}{l} \sum_ {k = 1} ^ {K} \Pr_ {C} (Y \neq k \mid f (X) = k) \Pr_ {D} (f (X) = j) (235) \\ = \sum_ {k = 1} ^ {K} \Pr_ {C} (Y \neq k \mid f (X) = k) \Pr_ {C} (f (X) = k) \left(1 + \frac {\Pr_ {D} (f (X) = k) - \Pr_ {C} (f (X) = k)}{\Pr_ {C} (f (X) = k)}\right) (236) \\ \leq \sum_ {k = 1} ^ {K} \Pr_ {C} (Y \neq k \mid f (X) = k) \Pr_ {C} (f (X) = k) | 1 + \frac {\Pr_ {D} (f (X) = k) - \Pr_ {C} (f (X) = k)}{\Pr_ {C} (f (X) = k)} | (237) \\ \leq \left(\sum_ {k = 1} ^ {K} \Pr_ {C} (Y \neq k \mid f (X) = k) \Pr_ {C} (f (X) = k)\right) \left(1 + \sup  _ {k} \left| \frac {\Pr_ {D} (f (X) = k) - \Pr_ {C} (f (X) = k)}{\Pr_ {C} (f (X) = k)} \right|\right) (238) \\ = \widetilde {\operatorname {e r r}} _ {C} (f) \left(1 + t _ {f} (C, D)\right) (239) \\ \end{array}
+$$
+
+Combine with the above, we have
+
+$$
+\widetilde {\operatorname {e r r} _ {D}} (f) \leq \widetilde {\operatorname {e r r} _ {C}} (f) \left(1 + t _ {f} (C, D)\right) + L u _ {f} (C, D) \tag {240}
+$$
+
+![](images/9d2778a09c2e3be466971ba2c1dfc62df909aa55a6370e5a4990a227608c5eed.jpg)
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+# LEARNING MIRROR MAPS IN POLICY MIRROR DESCENT
+
+Carlo Alfano*
+
+Department of Statistics
+
+University of Oxford
+
+Sebastian Towers†
+
+FLAIR
+
+University of Oxford
+
+Silvia Sapora†
+
+FLAIR, Department of Statistics
+
+University of Oxford
+
+Chris Lu
+
+FLAIR
+
+University of Oxford
+
+Patrick Rebeschini
+
+Department of Statistics
+
+University of Oxford
+
+# ABSTRACT
+
+Policy Mirror Descent (PMD) is a popular framework in reinforcement learning, serving as a unifying perspective that encompasses numerous algorithms. These algorithms are derived through the selection of a mirror map and enjoy finite-time convergence guarantees. Despite its popularity, the exploration of PMD's full potential is limited, with the majority of research focusing on a particular mirror map—namely, the negative entropy—which gives rise to the renowned Natural Policy Gradient (NPG) method. It remains uncertain from existing theoretical studies whether the choice of mirror map significantly influences PMD's efficacy. In our work, we conduct empirical investigations to show that the conventional mirror map choice (NPG) often yields less-than-optimal outcomes across several standard benchmark environments. Using evolutionary strategies, we identify more efficient mirror maps that enhance the performance of PMD. We first focus on a tabular environment, i.e. Grid-World, where we relate existing theoretical bounds with the performance of PMD for a few standard mirror maps and the learned one. We then show that it is possible to learn a mirror map that outperforms the negative entropy in more complex environments, such as the MinAtar suite. Additionally, we demonstrate that the learned mirror maps generalize effectively to different tasks by testing each map across various other environments.
+
+# 1 INTRODUCTION
+
+Policy gradient (PG) methods (Williams & Peng, 1991; Sutton et al., 1999; Konda & Tsitsiklis, 2000; Baxter & Bartlett, 2001) are some of the most widely-used mechanisms for policy optimization in reinforcement learning (RL). These algorithms are gradient-based methods that optimize over a class of parameterized policies and have become a popular choice for RL problems, both in theory (Kakade, 2002; Peters & Schaal, 2008; Bhatnagar et al., 2009; Schulman et al., 2015; Mnih et al., 2016; Schulman et al., 2017; Lan, 2022a) and in practice (Shalev-Shwartz et al., 2016; Berner et al., 2019; Ouyang et al., 2022).
+
+Among PG methods, some of the most successful algorithms are those that employ some form of regularization in their updates, ensuring that the newly updated policy retains some degree of similarity to its predecessor. This principle has been implemented in different ways. For instance, trust region policy optimization (TRPO) Schulman et al. (2015) imposes a Kullback-Leibler divergence Kullback & Leibler (1951) hard constraint for its updates, while proximal policy optimization (PPO) (Schulman et al., 2017) uses a clipped objective to penalize large updates. A framework that has recently attracted attention and belongs to this heuristic is that of policy mirror descent (PMD) (Tomar et al., 2022; Lan, 2022a; Xiao, 2022; Kuba et al., 2022; Vaswani et al., 2022; Alfano et al., 2023), which applies mirror descent Nemirovski & Yudin (1983) to RL to regularize the policy updates.
+
+PMD consists of a wide class of algorithms, each derived by selecting a mirror map that introduces distinct regularization characteristics. In recent years, PMD has been investigated through numerical
+
+experiments (Tomar et al., 2022), but it has mainly been analysed from a theoretical perspective. To the best of our knowledge, research has been mostly focused either on the particular case of the negative entropy mirror map, which generates the natural policy gradient (NPG) algorithm (Kakade, 2002; Agarwal et al., 2021), or on finding theoretical guarantees for a generic mirror map.
+
+NPG has been proven to converge to the optimal policy, up to a difference in expected return (or error floor), in several settings, e.g. using tabular, linear, general parameterization (Agarwal et al., 2021) or regularizing rewards (Cen et al., 2021). It has been shown that NPG benefits from implicit regularization (Hu et al., 2022), that it can exploit optimism (Zanette et al., 2021; Liu et al., 2023), and some of its variants have been evaluated in simulations (Vaswani et al., 2022; 2023). When considering other specific mirror maps investigated in the RL literature, the Tsallis entropy has been noted for enhancing performance in offline settings (Tomar et al., 2022) and for offering improved sample efficiency in online settings (Li & Lan, 2023), when compared to the negative entropy.
+
+There is a substantial body of theoretical research focused on the general case of mirror maps. This research demonstrates that PMD achieves convergence to the optimal policy under the same conditions as NPG, as evidenced by various studies (Xiao, 2022; Lan, 2022b; Yuan et al., 2023; Alfano et al., 2023). Except for the setting where we have access to the true value of the policy, convergence guarantees in stochastic settings are subject to an error floor due to the inherent randomness or bias within the algorithm. In the majority of PMD analyses involving generic mirror maps, both the convergence rate and the error floor show mild dependence on the specific choice of mirror map. Typically, the effect of the mirror map appears explicitly as a multiplicative factor in the convergence rate and it appears implicitly in the error floor. These analyses often rely on upper bounds, meaning they may not accurately reflect the algorithms' actual performance in applications.
+
+In this work, we contribute to the literature with an empirical investigation of PMD, with the objectives of finding a mirror map that consistently outperforms the negative entropy mirror map and of understanding how the theoretical guarantees of PMD relate to simulations. We first consider a set of tabular environments, i.e. Grid-World (Oh et al., 2020), where we compare the empirical results to the theoretical guarantees given by Xiao (2022), which we can compute as we have full control over the environment. In this setting, we provide a learned mirror map which outperforms the negative entropy in all tested environments. Our experiments suggest that the error floor appearing in prototypical PMD convergence guarantees is not a good performance indicator: our learned mirror map achieves the best value while presenting the worst theoretical error floor, implying that the upper bounds typically considered in the PMD literature are loose with respect to the choice of the mirror map. Additionally, our experiments indicate that having small policy updates leads to smoother value improvements over time with less instances of performance degradation, as suggested by the monotonic policy improvement property given by Xiao (2022). We then consider two non-tabular settings, i.e. the Basic Control Suite and the MinAtar Suite, which are more realistic but also more complex, and therefore prevent us from computing the exact theoretical guarantees. We learn a mirror map for each of these environments and, also in this case, show that the learned mirror maps lead to a higher performance of PMD than the negative entropy. Moreover, we show that the learned mirror maps generalize well to other tasks, by testing each of the learned mirror maps on all the other environments. Lastly, we tackle continuous control tasks in MuJoCo (Todorov et al., 2012), where we show that the mirror map learned on one environment surpasses the negative entropy across several environments.
+
+To establish our findings, we employ the standard formulation of PMD (Xiao, 2022) for the tabular setting and the continuous control tasks, and we used a generalized version, Approximate Mirror Policy Optimization (AMPO) (Alfano et al., 2023), for the non-tabular setting with discrete action spaces. To allow optimization over the space of mirror maps, we introduce parameterization schemes for mirror maps, one for PMD and one for AMPO. Specifically, we propose a parameterization for  $\omega$ -potentials, which have been shown to induce a wide class of mirror maps (Krichene et al., 2015). We use evolution strategies (ES) to search for the mirror map that maximizes the performance of PMD and AMPO over an environment.
+
+AMPO (Alfano et al., 2023) is a recently-proposed PMD framework designed to integrate general parameterization schemes, in our case neural networks, and arbitrary mirror maps. It benefits from theoretical guarantees, as Alfano et al. (2023) show that AMPO has quasi-monotonic updates as well as sub-linear and linear convergence rates, depending on the step-size schedule. These desirable properties make AMPO particularly suitable for our numerical investigation.
+
+ES are a type of population-based stochastic optimization algorithm that leverages random noise to generate a diverse pool of candidate solutions, and have been successfully applied to a variety of tasks (Real et al., 2019; Salimans et al., 2017; Such et al., 2018). The main idea consists in iteratively selecting higher-performing individuals, w.r.t. a fitness function, resulting in a gradual convergence towards the optimal solution. ES algorithms are gradient-free and have been shown to be well-suited for optimisation problems where the objective function is noisy or non-differentiable and the search space is large or complex (Beyer, 2000; Lu et al., 2022; 2023). We use ES to search over the parameterized classes of mirror maps we introduce, by defining the fitness of a particular mirror map as the value of the last policy outputted by PMD and AMPO, for fixed hyper-parameters.
+
+The rest of the paper is organized as follows. In Section 2, we introduce the setting of RL as well as the PMD and AMPO algorithms. We describe the methodology behind our numerical experiments in Section 3, which are then discussed in Section 4. Finally, we discuss related works from the literature on automatic discovery of machine learning algorithms in Appendix A and give our conclusions in Section 5.
+
+# 2 PRELIMINARIES
+
+# 2.1 REINFORCEMENT LEARNING
+
+Define a discounted Markov Decision Process (MDP) as the tuple  $\mathcal{M} = (\mathcal{S},\mathcal{A},P,r,\gamma ,\mu)$ , where  $\mathcal{S}$  and  $\mathcal{A}$  are respectively the state and action spaces,  $P(s^{\prime}\mid s,a)$  is the transition probability from state  $s$  to  $s^\prime$  when taking action  $a$ ,  $r(s,a)\in [0,1]$  is the reward function,  $\gamma$  is a discount factor, and  $\mu$  is a starting state distribution. A policy  $\pi \in (\Delta (\mathcal{A}))^{S}$ , where  $\Delta (\mathcal{A})$  is the probability simplex over  $\mathcal{A}$ , represents the behavior of an agent on an MDP, whereby at state  $s\in S$  the agents take actions according to the probability distribution  $\pi (\cdot \mid s)$ .
+
+Our objective is for the agent to find a policy that maximizes the expected discounted cumulative reward for the starting state distribution  $\mu$ . That is, we want to find
+
+$$
+\pi^ {\star} \in \underset {\pi \in (\Delta (\mathcal {A})) ^ {S}} {\operatorname {a r g m a x}} \mathbb {E} _ {s \sim \mu} \left[ V ^ {\pi} (s) \right]. \tag {1}
+$$
+
+Here  $V^{\pi}:\mathcal{S}\to \mathbb{R}$  denotes the value function associated with policy  $\pi$  and is defined as
+
+$$
+V ^ {\pi} (s) := \mathbb {E} \Big [ \sum_ {t = 0} ^ {\infty} \gamma^ {t} r (s _ {t}, a _ {t}) \mid \pi , s _ {0} = s \Big ],
+$$
+
+where  $s_t$  and  $a_t$  are the current state and action at time  $t$  and the expectation is taken over the trajectories generated by  $a_t \sim \pi(\cdot | s_t)$  and  $s_{t+1} \sim P(\cdot | s_t, a_t)$ .
+
+Similarly to the value function, we define the  $Q$ -function associated with a policy  $\pi$  as
+
+$$
+Q ^ {\pi} (s, a) := \mathbb {E} \Big [ \sum_ {t = 0} ^ {\infty} \gamma^ {t} r (s _ {t}, a _ {t}) \mid \pi , s _ {0} = s, a _ {0} = a \Big ],
+$$
+
+where the expectation is once again taken over the trajectories generated by the policy  $\pi$ . When the state and action spaces are finite, the  $Q$ -function can be expressed as  $Q^{\pi} = (I - \gamma P^{\pi})^{-1}r$ , where  $P^{\pi}$  is a square matrix where the position  $((s,a),(s',a'))$  is occupied by  $\pi(a'|s')P(s'|s,a)$ . We also define the discounted state visitation distribution as
+
+$$
+d _ {\mu} ^ {\pi} (s) := (1 - \gamma) \mathbb {E} _ {s _ {0} \sim \mu} \Big [ \sum_ {t = 0} ^ {\infty} \gamma^ {t} P (s _ {t} = s \mid \pi , s _ {0}) \Big ],
+$$
+
+where  $P(s_{t} = s \mid \pi, s_{0})$  represents the probability of the agent being in state  $s$  at time  $t$  when following policy  $\pi$  and starting from  $s_{0}$ . The probability distribution over states  $d_{\mu}^{\pi}(s)$  represents the proportion of time spent on state  $s$  when following policy  $\pi$ .
+
+# 2.2 POLICY MIRROR DESCENT
+
+We review the PMD framework, starting from mirror maps (Bubeck, 2015). Let  $\mathcal{X} \subseteq \mathbb{R}^{\mathcal{A}}$  be a convex set. A mirror map  $h: \mathcal{X} \to \mathbb{R}$  is a strictly convex, continuously differentiable and essentially smooth function that satisfies  $\nabla h(\mathcal{X}) = \mathbb{R}^{\mathcal{A}}$ . In particular, we consider mirror maps belonging to the  $\omega$ -potential mirror map class, which contains most mirror maps used in the literature.
+
+Definition 2.1 ( $\omega$ -potential mirror map (Krichene et al., 2015)). For  $u \in (-\infty, +\infty]$ ,  $\omega \leq 0$ , an  $\omega$ -potential is defined as an increasing  $C^1$ -diffeomorphism  $\phi : (-\infty, u) \to (\omega, +\infty)$  such that
+
+$$
+\lim  _ {x \to - \infty} \phi (x) = \omega , \lim  _ {x \to u} \phi (x) = + \infty , \int_ {0} ^ {1} \phi^ {- 1} (x) d x \leq \infty .
+$$
+
+For any  $\omega$ -potential  $\phi$ , we define the associated mirror map  $h_{\phi}$  as
+
+$$
+h _ {\phi} (\pi_ {s}) = \sum_ {a \in \mathcal {A}} \int_ {1} ^ {\pi (a | s)} \phi^ {- 1} (x) d x.
+$$
+
+When  $\phi(x) = e^x$  we recover the negative entropy mirror map, which is the standard choice of mirror map in the RL literature (Agarwal et al., 2021; Tomar et al., 2022; Hu et al., 2022; Vaswani et al., 2022; 2023), while we recover the  $\ell_2$ -norm when  $\phi(x) = x$  (see Appendix B.1). Mirror maps belonging to this class are particularly advantageous as they can be defined by a single scalar function  $\phi$ , without having to account for the dimension of the action space  $\mathcal{A}$ . The Bregman divergence (Bregman, 1967; Censor & Zenios, 1997) induced by the mirror map  $h$  is defined as
+
+$$
+\mathcal {D} _ {h} (x, y) := h (x) - h (y) - \langle \nabla h (y), x - y \rangle ,
+$$
+
+where  $\mathcal{D}_h(x,y)\geq 0$  for all  $x,y\in \mathcal{X}$ . As we further discuss in Appendix C, the Bregman divergence measures how far two points are in a geometry induced by the mirror map. Given a starting policy  $\pi^0$ , a learning rate  $\eta$  and a mirror map  $h$ , PMD can be formalized as an iterative algorithm: for all iterations  $t\geq 0$
+
+$$
+\pi^ {t + 1} \in \operatorname {a r g m a x} _ {\pi \in (\Delta (\mathcal {A})) ^ {s}} \mathbb {E} _ {s \sim d _ {\mu} ^ {t}} [ \eta_ {t} \langle Q _ {s} ^ {t}, \pi_ {s} \rangle - \mathcal {D} _ {h} (\pi_ {s}, \pi_ {s} ^ {t}) ], \tag {2}
+$$
+
+where we used the shorthand:  $V^{t} \coloneqq V^{\pi^{t}}$ ,  $Q^{t} \coloneqq Q^{\pi^{t}}$ ,  $d_{\mu}^{t} \coloneqq d_{\mu}^{\pi^{t}}$  and  $y_{s} \coloneqq y(s, \cdot) \in \mathbb{R}^{\mathcal{A}}$ , for any function  $y : S \times \mathcal{A} \to \mathbb{R}$ . PMD benefits from several theoretical guarantees and, in particular, Xiao (2022) shows that PMD enjoys quasi-monotonic updates and convergence to the optimal policy. We give here a slight modification of the statements of these results. Specifically, we do not upper-bound a term regarding the distance between subsequent policies in the result on quasi-monotonic updates, which we use to draw connections between theory and practice. Additionally, to obtain the convergence rate for PMD with constant step-size in the setting where we do not have access to the true Q-function, we combine the analyses on the sublinear convergence of PMD and linear convergence of inexact PMD given by Xiao (2022). We provide a proof in Appendix D.
+
+Theorem 2.2 (Xiao (2022)). Following update (2), we have that, for all  $t \geq 0$
+
+$$
+V ^ {t + 1} (\mu) - V ^ {t} (\mu) \geq - \frac {1}{1 - \gamma} \max  _ {s \in \mathcal {S}} \left\| \widehat {Q} _ {s} ^ {t} - Q _ {s} ^ {t} \right\| _ {\infty} \max  _ {s \in \mathcal {S}} \left\| \pi_ {s} ^ {t + 1} - \pi_ {s} ^ {t} \right\| _ {1}, \tag {3}
+$$
+
+where  $\| \cdot \|_{\infty}$  and  $\| \cdot \|_1$  represent the  $\ell_{\infty}$  and the  $\ell_1$  norms, respectively, and  $\widehat{Q}$  is an estimate of the true  $Q$ -function. Additionally, at each iteration  $T > 0$ , we have
+
+$$
+V ^ {\star} (\mu) - \sum_ {t <   T} \frac {\mathbb {E} [ V ^ {t} (\mu) ]}{T} \leq \frac {1}{T} \left(\frac {\mathbb {E} _ {s \sim d _ {\mu} ^ {\star}} [ \mathcal {D} _ {h} \left(\pi_ {s} ^ {\star} , \pi_ {s} ^ {0}\right) ]}{\eta_ {t} (1 - \gamma)} + \frac {1}{(1 - \gamma) ^ {2}}\right) + 4 \frac {\max _ {t <   T , s \in \mathcal {S}} \left\| \widehat {Q} _ {s} ^ {t} - Q _ {s} ^ {t} \right\| _ {\infty}}{(1 - \gamma) ^ {2}}. \tag {4}
+$$
+
+The statement of Theorem 2.2 is similar to many results in the literature on PMD and NPG (Agarwal et al., 2021; Xiao, 2022; Lan, 2022b; Hu et al., 2022). That is, the convergence guarantee involves two terms, i.e. a convergence rate, which involves the Bregman divergence between the optimal policy and the starting policy, and an error floor, which involves the estimation error  $\max_{s\in S}\| \widehat{Q}_s^t -Q_s^t\|_\infty$ . Given that by setting  $\eta_0 = \mathbb{E}_{s\sim d_\mu^*}[\mathcal{D}_h(\pi_s^\star ,\pi_s^0)](1 - \gamma)$  we obtain the convergence rate  $2(T(1 - \gamma)^{2})^{-1}$ , and that the error floor has no explicit dependence on the mirror map, Equation (4) suggests that the mirror map has a mild influence on the performance of PMD. The only way the mirror map seems to affect the convergence guarantee is, implicitly, by changing the path of the algorithm and therefore influencing the estimation error. Similar observations can be made for several results in the PMD literature that share the same structure of convergence guarantees. On the other hand, Equation (3) suggests that mirror maps that prevent large updates of the policy cause the PMD algorithm to be less prone to performance degradation, as the lower bound is close to 0 when the policy update distance  $\max_{s\in S}\| \pi_s^{t + 1} - \pi_s^t\| _1$  is small. One of the contributions of
+
+our work is to challenge and refute the conclusion that the mirror map has little influence on the convergence of PMD, highlighting a gap between theoretical guarantees, which are based on upper bounds, and the actual performance observed in PMD-based methodologies. Our empirical studies reveal that the choice of mirror map significantly influences both the speed of convergence and the minimum achievable error floor in PMD. Additionally, we provide evidence that a mirror map that prevents large updates throughout training leads to a better performance, as suggested by (3).
+
+When applying PMD to continuous control tasks, we replace the tabular policy in (2) with a parametrized one. That is, for all iterations  $t \geq 0$ , we obtain the updated policy as
+
+$$
+\pi_ {\theta^ {t + 1}} \in \operatorname {a r g m a x} _ {\pi_ {\theta}: \theta \in \Theta} \mathbb {E} _ {s \sim d _ {\mu} ^ {t}} \left[ \eta_ {t} \mathbb {E} _ {a \sim \pi_ {\theta} (\cdot | s)} \left(Q ^ {t} (s, a)\right) - \mathcal {D} _ {h} \left(\pi_ {\theta} (\cdot | s), \pi^ {t} (\cdot | s)\right) \right], \tag {5}
+$$
+
+where  $\{\pi_{\theta}:\theta \in \Theta \}$  is a class of parametrized policies.
+
+# 2.3 APPROXIMATE MIRROR POLICY OPTIMIZATION
+
+AMPO is a theoretically sound framework for deep reinforcement learning based on mirror descent, as it inherits the quasi-monotonic updates and convergence guarantees from the tabular case (Alfano et al., 2023). Given a parameterized function class  $\mathcal{F}^{\Theta} = \{f^{\theta}:S\times \mathcal{A}\to \mathbb{R},\theta \in \Theta \}$ , an initial scoring function  $f^{\theta^0}$ , a step-size  $\eta$  and an  $\omega$ -potential mirror map  $h_\phi$ , AMPO can be described, for all iterations  $t$ , by the two-step update
+
+$$
+\pi^ {t} (a \mid s) = \sigma \left(\phi \left(\eta f ^ {t} (s, a) + \lambda_ {s} ^ {t}\right)\right) \quad \forall s \in \mathcal {S}, a \in \mathcal {A}, \tag {6}
+$$
+
+$$
+\theta^ {t + 1} \in \underset {\theta \in \Theta} {\operatorname {a r g m i n}} \mathbb {E} _ {s \sim d _ {\mu} ^ {t}, a \sim \pi (\cdot | s)} \left[ \left(f ^ {\theta} (s, a) - Q ^ {t} (s, a) - \eta^ {- 1} \max  \left(\eta f ^ {t} (s, a) + \lambda_ {s} ^ {t}, \phi^ {- 1} (0)\right)\right) ^ {2} \right] \tag {7}
+$$
+
+where  $\lambda_s^t\in \mathbb{R}$  is a normalization factor to ensure  $\pi_s^t\in \Delta (\mathcal{A})$  for all  $s\in S$ ,  $f^{t}\coloneqq f^{\theta^{t}}$ , and  $\sigma (z) = \max (z,0)$  for  $z\in \mathbb{R}$ . Theorem 1 by Krichene et al. (2015) ensures that there always exists a normalization constant  $\lambda_s^t\in \mathbb{R}$ . As shown by Alfano et al. (2023), AMPO recovers the standard formulation of PMD in (2) in the tabular setting. Assuming for simplicity that  $\phi (x) > 0$  for all  $x\in \mathbb{R}$ , the minimization problem in (7) implies that, at each iteration  $t$ ,  $f^{t}$  is an approximation of the sum of the  $Q$ -functions up to that point, that is  $f^{t}\simeq \sum_{i = 0}^{t - 1}Q^{i}$ . Therefore, the scoring function  $f^{t}$  serves as an estimator of the value of an action.
+
+# 3 METHODOLOGY
+
+Denote by  $\mathcal{H}$  the class of  $\omega$ -potentials mirror maps. Our objective is to search for the mirror map that maximizes the value of the last policy outputted by our mirror descent based algorithms, for a fixed time horizon  $T$ . That is, we want to find
+
+$$
+h ^ {\star} \in \underset {h \in \mathcal {H}} {\operatorname {a r g m a x}} \mathbb {E} \left[ V ^ {T} (\mu) \right], \tag {8}
+$$
+
+where the expectation is taken over the randomness of the policy optimization algorithm and the policy updates are based on the mirror map  $h$ . Depending on the setting, we parameterize the mirror map by parameterizing either  $\phi^{-1}$  or  $\phi$  as monotonically increasing functions. One of the primary objectives of this work is to motivate further research into the choice of mirror maps, as influenced by the choice of the function  $\phi$ , moving beyond the conventional use of  $\phi(x) = e^x$ . This is achieved by examining how different choices of  $\phi$  influence the trajectory of training and demonstrating that, in many cases, there is a mirror map that outperforms the negative entropy by a large margin.
+
+# 3.1 POLICY MIRROR DESCENT
+
+We parameterize  $\phi^{-1}$  as a one layer neural network with 126 hidden units, where all kernels are non-negative and the activation functions are equally split among the following convex and concave monotonic non-linearities:  $x^{3}, (x)_{+}^{2}, (x)_{+}^{1/2}, (x)_{+}^{1/3}, \log((x)_{+} + 10^{-3})$  and  $e^{x}$ , where  $(x)_{+} = \max(x, 0)$ . To ensure that we are able to recover the negative entropy and the  $\ell_2$ -norm, we add  $ax + b\log(x)$  to the final output, where  $a, b \geq 0$
+
+To search for the best mirror map within this class, we employ a slight variation of the OpenAI-ES strategy (Salimans et al., 2017), adapted to the multi-task setting (Jackson et al., 2024). Denote by  $\psi$  the parameters of the mirror map and by  $F(\psi)$  the objective function in (8). Given a distribution of tasks  $\mathcal{E}$ , we estimate the gradient  $\nabla_{\psi} F(\psi)$  as
+
+$$
+\left. E _ {\epsilon \sim \mathcal {N} \left(0, I _ {d}\right)} \left[ E _ {e \sim \varepsilon} \left[ \frac {\epsilon}{2 \sigma} \left(F _ {e} (\psi + \sigma \epsilon) - F _ {e} (\psi - \sigma \epsilon)\right) \right] \right], \right. \tag {9}
+$$
+
+where  $\mathcal{N}(0, I_d)$  is the multivariate normal distribution,  $d$  is the number of parameters, and  $\sigma > 0$  is a hyperparameter regulating the variance of the perturbations. To account for different reward scales across tasks, we perform a rank transformation of the objective functions, whereby, for each sampled task  $e$ , we return 1 for the higher performing member between  $\psi + \sigma \epsilon$  and  $\psi - \sigma \epsilon$ , and 0 for the other. We note that, when the distribution of tasks  $\mathcal{E}$  covers a single task, we recover the standard OpenAI-ES strategy.
+
+# 3.2 APPROXIMATE MIRROR POLICY OPTIMIZATION
+
+As non-tabular environments with discrete action space, we consider the Basic Control Suite (BCS) and the MinAtar suite. Given the higher computational cost of simulations on these environments w.r.t. to the tabular setting, we replace the neural network parameterization for the mirror map with one with fewer parameters, in order to reduce the dimension of the mirror map class we search over. We define the parameterized class  $\Phi = \{\phi_{\psi}:\mathbb{R}\to [0,1],\psi \in \mathbb{R}_{+}^{n}\}$ , with
+
+$$
+\phi_ {\psi} (x) = \left\{ \begin{array}{l l} 0 & \text {i f} x \leq 0, \\ \frac {x}{\psi_ {1} n} & \text {i f} 0 <   x \leq \psi_ {1}, \\ \frac {j}{n} + \frac {x - \sum_ {i = 1} ^ {j} \psi_ {i}}{n \psi_ {j + 1}} & \text {i f} \sum_ {i = 1} ^ {j} \psi_ {i} <   x \leq \sum_ {i = 1} ^ {j + 1} \psi_ {i}, \\ 1 & \text {i f} x > 1, \end{array} \right.
+$$
+
+where  $1 \leq j \leq n - 1$ . In other words,  $\phi$  is defined as a piece-wise linear function with  $n$  steps where, for all  $j \leq n$ ,  $\phi \left( \sum_{i=1}^{j} \psi_i \right) = j / n$  and subsequent points are interpolated with a straight segment. This is illustrated in Figure 1. We note that the  $\omega$ -potentials within  $\Phi$  violate some of the constraints in Definition 2.1, as they are non-decreasing instead of increasing and  $\lim_{x \to \infty} \phi(x) = 1$  for all  $\phi \in \Phi$ . In Appendix B.2, we show that, if  $\phi \in \Phi$ , we can construct an  $\omega$ -potential  $\phi'$  that satisfies the constraints in Definition 2.1 such that  $\phi$  and  $\phi'$  induce the same policies along the path of AMPO.
+
+To effectively learn the hyperparameters, we employ the Separable Covariance Matrix Adaptation Evolution Strategy (sep-CMA) (Ros & Hansen, 2008), a variant of the popular algorithm CMA-ES (Hansen & Ostermeier,
+
+![](images/99f409f748be7b43a5758977402c42ede20c868c5ea00cd74e28c034c28a08c2.jpg)  
+Figure 1: A plot visually demonstrating the parameterization for  $\phi$ .
+
+2001). CMA-ES is, essentially, a second-order method adapted for gradient free optimization. At every generation, it samples  $n$  new points from a normal distribution, parameterized by a mean vector  $m_{k}$  and a covariance matrix  $C_k$ . That is, the samples for generation  $k$ ,  $x_1^k, \ldots, x_n^k$ , are distributed i.i.d. according to  $x_i^k \sim \mathcal{N}(m_k, C_k)$ . For each generation  $k$ , denote the  $m \leq n$  best performing samples as  $x_1^{*k}, \ldots, x_m^{*k}$ . Then update the mean vector as  $m_{k+1} = \sum_{i=1}^{m} w_i x_i^{*k}$ , where  $\sum_{i=1}^{m} w_i = 1$ , so that the next generation is distributed around the weighted mean of the best performing samples.  $C_{k+1}$  is also updated to reflect the covariance structure of  $x_1^{*k}, \ldots, x_m^{*k}$ , in a complex way beyond the scope of this text. Due to the need to update  $C_k$  based on covariance information, CMA-ES exhibits a quadratic scaling behavior with respect to the dimensionality of the search space, potentially hindering its efficiency in high-dimensional settings. To improve computational efficiency, we adopt sep-CMA, which introduces a diagonal constraint on the covariance matrix and reduces the computational complexity of the algorithm.
+
+![](images/d141602b7e07354ea6b6f2f0a4b142e5ba31b2fe02b021137fb1c41fef515736.jpg)  
+Figure 2: Comparison between the learned map and the negative entropy and  $\ell_2$ -norm mirror maps across a range of held-out configurations of Grid-World. We display the average over 256 runs and report the standard error as a shaded region. The column "Random tasks", reports the averaged metrics for 256 randomly sampled configurations of Grid-World.
+
+# 4 EXPERIMENTS
+
+In this section, we discuss the results of our numerical experiments. We start by presenting the tabular setting, where we track errors in order to understand what properties are desirable in a mirror map, and proceed by showing our results in the non-tabular setting.
+
+# 4.1 TABULAR SETTING: GRID-WORLD
+
+As tabular setting we adopt a discounted and infinite horizon version of Grid-World (Oh et al., 2020), which is a large class of tabular MDPs.
+
+Model architecture and training We define the tabular policy as a one-layer neural network with a softmax head, which takes as input a one-hot encoding of the environment state and outputs a distribution over the action space. We train the policy using the PMD update in (2), where we solve the minimization problem through stochastic gradient descent and estimate the  $Q$ -function through generalized advantage estimation (GAE) (Schulman et al., 2016). We perform a simple grid-search over the hyperparameters to maximize the performance for the negative entropy and the  $\ell_2$ -norm mirror maps, in order to have a fair comparison. We report the chosen hyperparameters in Appendix E. The training procedure is implemented in Jax, using evosax (Lange, 2022a) for the evolution. We run on four A40 GPUs, and the optimization process takes roughly 12 hours.
+
+Environment We adopt the version of Grid-World implemented by Jackson et al. (2024), and adapt it to the discounted and infinite horizon setting. We learn a single mirror map by training PMD on a continuous distribution of Grid-World environments, and test PMD with the learned mirror map on five held-out configurations from previous publications (Oh et al., 2020; Chevalier-Boisvert et al., 2024) and on 256 randomly sampled configurations. For all PMD iterations  $t$ , we track two quantities that appear in Theorem 2.2, that is the estimation error  $\max_{s\in S}\| \widehat{Q}_s^t -Q_s^t\|_\infty$ , and the distance between policy updates  $\max_{s\in S}\| \pi_s^{t + 1} - \pi_s^t\|_1$ . To obtain the true  $Q$ -function, we compute the transition matrix  $P^{\pi^t}$  and use the formula specified in Section 2, that is  $Q^{t} = (I - \gamma P^{\pi^{t}})^{-1}r$ . PMD is run for 128 iterations and  $2^{18}\simeq 250k$  total environment steps for all Grid-World configurations.
+
+We show the results of our simulations in Figure 2. As shown by the first row, the learned mirror map, the  $\ell_2$ -norm, and the negative entropy consistently rank first, second and third, respectively, in all tested configurations, in terms of final value. These results advocate the effectiveness of our methodology, as we are able to find a mirror map that outperforms the benchmark mirror maps in all tested environments. We highlight that the first five environments in Figure 2 are not part of the task
+
+![](images/1a713531b17838e2ac4cff1ee27cbf2a92c3ed98bc7c08c8c45e579c551b46a8.jpg)  
+Figure 3: Comparison between the learned mirror map, the  $\ell_2$ -norm and the negative entropy across a range of standard environments. The top plots present the performance of AMPO for all mirror maps, reporting the average over 100 realizations and a shaded region denoting the standard error around the average. The middle plots report the  $\omega$ -potentials that induce the mirror maps. The bottom plots report the policy distribution according to (6), for each mirror map and score scales. The score scales are obtained by multiplying the vector  $[1, \dots, |\mathcal{A}|]$  by a variable  $c \in [0,4]$ .
+
+distribution used during the evolution, demonstrating the generalizability of our approach. Moreover, we have that the  $\ell_2$ -norm consistently outperforms the negative entropy, further proving that the negative entropy is not always the best choice of mirror map. Another shared property among all training curves, is that the learned mirror map presents a slower convergence in the initial iterations w.r.t. the  $\ell_2$ -norm and to the negative entropy, as testified by a lower value, but convergence to a higher value in the long run. Lastly, we note that in all environments most of the value improvement happens in the first 50k environment steps.
+
+To gain a better understanding of why the learned mirror maps outperforms the negative entropy and the  $\ell_2$ -norm, and to draw connections with the theoretical results outlined in Section 2.2, we report the estimation error and the distance between policy updates for all mirror maps. The first conclusion that we draw is that a smaller estimation error does not seem to be related to a higher performance. On the contrary, the second row of Figure 2 shows how the three mirror maps consistently have the same ranking in both value and estimation error, which is exactly the opposite of what Theorem 2.2 would suggest. On the other hand, the first and third rows of Figure 2 show how the lower bound on the performance improvement in (3) brings some valid insight on the behaviour of PMD during its first iterations, which are the ones that bring the largest improvement. In all configurations, we have that in the initial iterations of PMD the learned mirror map induces the smallest policy update distances as well as the performance curve with fewer dips in value, while both the negative entropy and the  $\ell_2$ -norm induce larger policy update distances and performance curves with several dips in value. This observation confirms the behaviour described by (3), whereby a small distance between policy updates prevents large performance degradation in policy updates.
+
+# 4.2 NON-TABULAR SETTING: BASIC CONTROL AND MINATAR SUITES
+
+Model architecture and training We define the scoring function as a deep neural network, which we train using the AMPO update in (7) and (6), where (7) is solved through Adam and the  $Q$ -function is estimated through GAE. We optimize the hyper-parameters of AMPO for the negative entropy mirror map for each suite, using the hyper-parameter tuning framework Optuna (Akiba et al., 2019). This is done to ensure we are looking at a fair benchmark of performance when using the negative entropy mirror map. We report the chosen hyper-parameters in Appendix E. We then initialize the
+
+Table 1: The table contains, for each entry, the value of the final policy outputted by AMPO trained on the environment corresponding to the column with the mirror map learned on the environment corresponding to the row. The last row represents the performance of AMPO with the negative entropy for the corresponding column environments. The value is averaged over 100 runs. Green cells correspond to a value higher than that associated to the negative entropy.  
+
+<table><tr><td></td><td>Acrobot</td><td>CartPole</td><td>Asterix</td><td>Freeway</td><td>SpaceInvaders</td></tr><tr><td>Acrobot</td><td>-88.49</td><td>476.41</td><td>24.03</td><td>56.00</td><td>144.01</td></tr><tr><td>CartPole</td><td>-83.76</td><td>499.93</td><td>27.26</td><td>52.26</td><td>100.07</td></tr><tr><td>Asterix</td><td>-103.55</td><td>490.86</td><td>30.22</td><td>58.56</td><td>122.00</td></tr><tr><td>Freeway</td><td>-82.51</td><td>457.47</td><td>3.20</td><td>58.21</td><td>143.93</td></tr><tr><td>SpaceInvaders</td><td>-78.29</td><td>489.56</td><td>4.36</td><td>22.27</td><td>170.24</td></tr><tr><td>Negative entropy</td><td>-105.63</td><td>359.14</td><td>17.80</td><td>53.69</td><td>81.77</td></tr></table>
+
+parameterized mirror map to be an approximation of the negative entropy $^2$  and run Sep-CMA-ES on each environment separately. The whole training procedure is implemented in JAX, using gymnas environments (Lange, 2022b) and evosax (Lange, 2022a) for the evolution. We run on 8 GTX 1080Ti GPUs, and the optimization process takes roughly 48 hours for a single environment.
+
+**Environments** We test AMPO on the Basic Controle Suite (BCS) and the MinAtar Suite. For BCS, we run the evolution for 600 generation, each with 500k timesteps. For MinAtar, we run the evolution for 500 generation with 1M timesteps, then run 100 more generations with 10M timesteps.
+
+Our empirical results are illustrated in Figure 3, where we show the performance of AMPO for the learned mirror map, for the negative entropy and for the  $\ell_2$ -norm. For all environments and mirror maps, we use the hyper-parameters returned by Optuna for AMPO with the negative entropy. Our learned mirror map leads to a better overall performance in all environments, apart from Acrobot, where it ties with the  $\ell_2$ -norm. We observe the largest improvement in performance on Asterix and SpaceInvaders, where the average return for the learned mirror map is more than double the one for the negative entropy. Figure 3 also suggests that different mirror maps may result in different error floors, as shown by the performance curves in Acrobot, Asterix and SpaceInvaders, where the negative entropy converges to a lower point than the learned mirror map.
+
+The second and third row of Figure 3 illustrate the properties of the learned mirror map, in comparison to the negative entropy and the  $\ell_2$ -norm. In particular, the second row shows the corresponding  $\omega$ -potential, while the third row shows the policy distribution induced by the mirror map according to (6), depending on the scores assigned by the scoring function to each action. A shared property among all the learned mirror maps is that they all lead to assigning 0 probability to the worst actions for relatively small score scales, whilst the negative entropy always assigns positive weights to all actions. In more complex environments, where the evaluation of a certain action may be strongly affected by noise or where the optimal state may be combination locked (Misra et al., 2020), this behaviour may lead to a critical lack of exploration. However, it appears that this is not the case in these environments, and we hypothesise that by setting the probability of the worst actions to 0 the learned mirror maps avoid wasting samples and hence can converge to the optimal policy more rapidly.
+
+Our last result consists in testing each learned mirror map across the other environments we consider. In Table 1, we report the value of the final policy outputted by AMPO for all learned maps, plus the negative entropy, and for all environments, averaged over 100 runs. The table shows that the learned mirror maps generalize well to different environments and to different sets of hyper-parameters, which are shared within BCS and MinAtar but not between them. In particular, we have that the mirror maps learned in Acrobot and Asterix outperform the negative entropy in all environments, those learned in CartPole and Freeway outperform the negative entropy in 4 out of 5 environments, and that learned in SpaceInvaders outperforms the negative entropy in 3 out of 5 environments. These results show that our methodology can be useful in practice, as it benefits from good generalization across tasks.
+
+# 4.3 CONTINUOUS CONTROL: MUJOCO
+
+Model architecture and training We define the policy as a deep neural network and optimize it using the PMD update in (5), which is computed using Adam and the estimate of the  $Q$ -function obtained through GAE. As previously discussed, we ensure a fair comparison by tuning the hyper-parameters to maximize the performance of the negative entropy. We report the chosen hyper-parameters in Appendix E. The mirror map is initialized to be close to the negative entropy and is trained for 256 generations using OpenAI-ES on a single MuJoCo environment. We run on eight A40 GPUs, and the optimization process takes roughly 24 hours.
+
+**Environments** We use the brax library (Freeman et al., 2021) to simulate three MuJoCo environments, i.e. Hopper, Halfcheetah, and Ant. The mirror map is learned on Hopper and is then tested on all environments, using  $10^{7}$  environment steps and 488 PMD update steps.
+
+![](images/ad90e918b694be51420d0fee0c02f7093725b3977677c5d6467ec469c3a0719c.jpg)  
+Figure 4: Comparison between the mirror map learned on Hopper and the negative entropy on three MuJoCo environments. The plots present the performance of PMD for both mirror maps, reporting the average over 8 realizations and a shaded region denoting the standard error around the average.
+
+Figure 4 shows that the mirror map learned on Hopper significantly improves in terms of final performance upon the negative entropy in both train and test environments. This result confirms the ability of our methodology to learn generalizable mirror maps, even in complex continuous control tasks.  
+![](images/b606c2dda616db18f9e237f2a9c346125842ad1e6a381e8281f0d92d9df232f8.jpg)  
+Learned Negative entropy
+
+![](images/be512d73eeb010147dc34b957644e2d0458c6d9e14b3d3f3b17597bf2cea89c7.jpg)
+
+# 5 CONCLUSION
+
+Our study presents an empirical examination of PMD, where we successfully test the possibility of learning a mirror map that outperforms the negative entropy in both the tabular and non-tabular settings. In particular, we have shown that the learned mirror maps perform well on a set of configurations in Grid-World and that they can generalize to different tasks in BCS, in MinAtar, and in MuJoCo. Additionally, we have compared the theoretical findings established in the literature and the actual performance of PMD methods in the tabular setting, highlighting how the estimation error is not a good indicator of performance and validating the intuition that small policy updates lead to less instances of performance degradation. Our findings indicate that the choice of mirror map significantly impacts PMD's effectiveness, an aspect not adequately reflected by existing convergence guarantees.
+
+Our research introduces several new directions for inquiry. From a theoretical perspective, obtaining convergence guarantees that reflect the impact of the mirror map is an area for future exploration. On the practical side, investigating how temporal awareness (Jackson et al., 2024) or specific environmental challenges, such as exploration, robustness to noise, or credit assignment (Osband et al., 2019), can inform the choice of a mirror map to improve performance represents another research focus.
+
+# ACKNOWLEDGMENTS
+
+Carlo Alfano is funded by the Engineering and Physical Sciences Research Council. Patrick Rebeschini was funded by UK Research and Innovation (UKRI) under the UK government's Horizon Europe funding guarantee [grant number EP/Y028333/1]. For the purpose of Open Access, the authors have applied a CC BY public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission.
+
+# REPRODUCIBILITY STATEMENT
+
+The implementation of our experiments can be found at https://github.com/c-alfano/Learning-mirror-maps.
+
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+
+# A RELATED WORKS
+
+Discovering reinforcement learning algorithm has been an active area of research in the last years, where researchers have shown that handcrafted algorithms are not always optimal and can be outperformed by automatically discovered algorithms. Oh et al. (2020) obtained an algorithm capable of solving GridWorld and Atari tasks without relying on common notions in the field, such as Q-value functions. Houthooft et al. (2018) used ES to meta-train a policy loss network that outperforms PPO and Kirsch et al. (2020) discovered a new loss function for deterministic policies. Most closely related to our work are Lu et al. (2022) and Jackson et al. (2024), who employ ES to discover RL algorithms within the Mirror Learning class (Kuba et al., 2022). Given a function  $f$ , the class of algorithms they consider follows the policy update step
+
+$$
+\pi_ {\theta^ {t + 1}} \in \operatorname * {a r g m a x} _ {\pi_ {\theta}: \theta \in \Theta} \mathbb {E} _ {s \sim d _ {\mu} ^ {t}} \left[ \eta_ {t} \mathbb {E} _ {a \sim \pi_ {\theta} (\cdot | s)} (Q ^ {t} (s, a)) - \mathbb {E} _ {a \sim \pi^ {t} (\cdot | s)} \left[ \left(\frac {\pi_ {\theta} (\cdot | s)}{\pi^ {t} (\cdot | s)}\right) \right] \right],
+$$
+
+This update is similar to the one in (5) but replaces the Bregman divergence with a different penalty term called drift, which recovers the  $f$ -divergences when  $f$  is a convex function. To the best of our knowledge, this class of algorithms has fewer theoretical guarantees than PMD, in particular we are not aware of finite-time convergence guarantees or convergence guarantees involving approximation or estimation errors (Belousov & Peters, 2017; Kuba et al., 2022).
+
+# B FURTHER DISCUSSION ON  $\omega$ -POTENTIALS
+
+# B.1 NEGATIVE ENTROPY AND  $\ell_2$ -NORM
+
+If  $\phi (x) = e^{x - 1}$ , then the associated mirror map  $h_\phi$  is the negative entropy. We have that
+
+$$
+\int_ {0} ^ {1} \phi^ {- 1} (x) d x = \int_ {0} ^ {1} \log (x) d x = [ x \log (x) - x ] _ {0} ^ {1} = - 1 \leq + \infty .
+$$
+
+The mirror map  $h_{\phi}$  becomes the negative entropy, up to a constant, as
+
+$$
+h _ {\phi} (\pi_ {s}) = \sum_ {a \in \mathcal {A}} \int_ {1} ^ {\pi (a | s)} \log (x) d x = | \mathcal {A} | - 1 + \sum_ {a \in \mathcal {A}} \pi (a \mid s) \log (\pi (a \mid s)).
+$$
+
+If  $\phi (x) = x$ , then the associated mirror map  $h_\phi$  is the  $\ell_2$ -norm. We have that
+
+$$
+\int_ {0} ^ {1} \phi^ {- 1} (x) d x = \int_ {0} ^ {1} x d x = \left[ \frac {x ^ {2}}{2} \right] _ {0} ^ {1} = \frac {1}{2} \leq + \infty .
+$$
+
+The mirror map  $h_{\phi}$  becomes the  $\ell_2$ -norm, up to a constant, as
+
+$$
+h _ {\phi} (\pi_ {s}) = \sum_ {a \in \mathcal {A}} \int_ {1} ^ {\pi (a | s)} x d x = \frac {1}{2} \sum_ {a \in \mathcal {A}} \pi (a | s) ^ {2} - 1.
+$$
+
+# B.2 PARAMETRIC  $\phi$
+
+We show here that the parametric class of  $\omega$ -potentials we introduce in Section 3 results in a well defined algorithm when used for AMPO, even if it breaks some of the constraints in Definition 2.1. In particular,  $\omega$ -potentials within  $\Phi$  are not  $C^1$ -diffeomorphisms and are only non-decreasing. We can afford not meeting the first constraint as the proof for Theorem 1 by Krichene et al. (2015), which establishes the existence of the normalization constant in Equation (6), only requires Lipschitz-continuity. As to the second constraint, we build an augmented class  $\Phi'$  that contains increasing  $\omega$ -potentials and show that it results in the same updates for AMPO as  $\Phi$ .
+
+Let  $\Delta_{n}$  be the  $n$ -dimensional probability simplex. We define the parameterized class  $\Phi' = \{\phi_{\psi}': \mathbb{R} \to [0,1], \psi \in \Delta_{n}\}$ , with
+
+$$
+\phi^{\prime}_{\psi}(x) = \left\{ \begin{array}{ll}e^{x} - 1 & \text{if} x\leq 0,\\ \frac{x}{\psi_{1}n} & \text{if} 0 <   x\leq \psi_{1},\\ \frac{j}{n} +\frac{x - \sum_{i = 1}^{j}\psi_{i}}{n\psi_{j + 1}} & \text{if}\sum_{i = 1}^{j}\psi_{i} <   x\leq \sum_{i = 1}^{j + 1}\psi_{i},\\ x & \text{if} x > 1, \end{array} \right.
+$$
+
+where  $1 \leq j \leq n - 1$ . The functions within  $\Phi'$  are increasing and continuous. We start by showing that  $\phi_{\psi}^{\prime} \in \Phi^{\prime}$  and  $\phi_{\psi} \in \Phi$  are equivalent for Equation (6), for the same parameters  $\psi$ . At each iteration  $t$ , we have that
+
+$$
+\begin{array}{l} \pi^ {t + 1} (a \mid s) = \sigma \left(\phi^ {\prime} \left(\eta f ^ {t + 1} (s, a) + \left(\lambda^ {\prime}\right) _ {s} ^ {t + 1}\right)\right) \\ = \sigma \left(\phi \left(\eta f ^ {t + 1} (s, a) + \left(\lambda^ {\prime}\right) _ {s} ^ {t + 1}\right)\right) \\ = \sigma \left(\phi \left(\eta f ^ {t + 1} (s, a) + \lambda_ {s} ^ {t + 1}\right)\right). \\ \end{array}
+$$
+
+This due to the following two facts. For  $x \leq 0$ ,  $\sigma(\phi'(x))$  and  $\sigma(\phi(x))$  are the same function. Also,  $\sigma(\phi'(\eta f^{t+1}(s, a) + (\lambda')_s^{t+1}))$  has to be less or equal to 1, as a result of the projection, meaning that  $\eta f^{t+1}(s, a) + (\lambda')_s^{t+1} \leq 1$ , where  $\sigma(\phi'(\cdot))$  and  $\sigma(\phi(\cdot))$  are equivalent.
+
+Since  $\phi$  and  $\phi'$  induce the same policy, share that same normalization constant, and have the same value at 0, they also induce the same expression for Equation (7).
+
+# C FURTHER DISCUSSION ON BREGMAN DIVERGENCE
+
+To provide an intuition on the Bregman divergence, we report here some discussion from Orabona (2020). Given a mirror map  $h$ , the associated Bregman divergence between two points  $x, y \in \mathcal{X}$  is defined as
+
+$$
+\mathcal {D} _ {h} (x, y) := h (x) - h (y) - \langle \nabla h (y), x - y \rangle .
+$$
+
+Since  $h$  is convex, the Bregman divergence is always non-negative for  $x, y \in \mathcal{X}$ . We can build more intuition for the Bregman divergence using Taylor's theorem. Assume that  $h$  is twice differentiable in an open ball  $B$  around  $y$  and  $x \in B$ . Then there exists  $0 \leq \alpha \leq 1$  such that
+
+$$
+B _ {\psi} (x; y) = h (x) - h (y) - \nabla h (y) ^ {\top} (x - y) = \frac {1}{2} (x - y) ^ {\top} \nabla^ {2} h (z) (x - y),
+$$
+
+where  $z = \alpha x + (1 - \alpha)y$ . Therefore, the Bregman divergence can be approximated by squared local norms that depend on the Hessian of the mirror map  $h$ . This means that the Bregman divergence will behave differently on different areas of  $\mathcal{X}$ , depending on the value of Hessian of the mirror map.
+
+# D PROOF OF THEOREM 2.2
+
+In this section we outline how we use the proofs given by Xiao (2022) in order to obtain Theorem 2.2.
+
+# D.1 PRELIMINARY LEMMAS
+
+We start by presenting two preliminary lemmas, which are ubiquitous in the literature on PMD. The first characterizes the difference in value between two policies, while the second characterizes the PMD update.
+
+Lemma D.1 (Performance difference lemma, Lemma 1 in Xiao (2022)). For any policy  $\pi, \pi' \in \Delta(\mathcal{A})^S$  and  $\mu \in \Delta(S)$ ,
+
+$$
+V ^ {\pi} (\mu) - V ^ {\pi^ {\prime}} (\mu) = \frac {1}{1 - \gamma} \mathbb {E} _ {s \sim d _ {\mu} ^ {\pi}} \left[ \langle Q _ {s} ^ {\pi^ {\prime}}, \pi_ {s} - \pi_ {s} ^ {\prime} \rangle \right].
+$$
+
+Lemma D.2 (Three-point decent lemma, Lemma 6 in Xiao (2022)). Suppose that  $\mathcal{C} \subset \mathbb{R}^m$  is a closed convex set,  $f: \mathcal{C} \to \mathbb{R}$  is a proper, closed convex function,  $\mathcal{D}_h(\cdot, \cdot)$  is the Bregman divergence generated by a mirror map  $h$ . Denote  $\operatorname{rint} \operatorname{dom} h$  as the relative interior of  $\operatorname{dom} h$ . For any  $x \in \operatorname{rint} \operatorname{dom} h$ , let
+
+$$
+x ^ {+} \in \arg \min  _ {u \in \operatorname {d o m} h \cap \mathcal {C}} \{f (u) + \mathcal {D} _ {h} (u, x) \}.
+$$
+
+Then  $x^{+}\in \mathrm{rint}$  dom  $h\cap \mathcal{C}$  and for any  $u\in \mathrm{dom}h\cap \mathcal{C}$
+
+$$
+f (x ^ {+}) + \mathcal {D} _ {h} (x ^ {+}, x) \leq f (u) + \mathcal {D} _ {h} (u, x) - \mathcal {D} _ {h} (u, x ^ {+}).
+$$
+
+# D.2 QUASI-MONOTONIC UPDATES
+
+We first prove that PMD enjoys quasi-monotonic updates (Equation (3)), that is PMD updates have an upper bound on how much they can deteriorate performance.
+
+Proposition D.3 (Lemma 11 in Xiao (2022)). At each time  $t \geq 0$ , we have
+
+$$
+\langle \eta_ {t} \widehat {Q} _ {s} ^ {t}, \pi_ {s} ^ {t + 1} - \pi_ {s} ^ {t} \rangle \geq 0.
+$$
+
+Additionally, we have that
+
+$$
+V ^ {t + 1} (\mu) - V ^ {t} (\mu) \geq - \frac {1}{1 - \gamma} \max  _ {s \in \mathcal {S}} \left\| \widehat {Q} _ {s} ^ {t} - Q _ {s} ^ {t} \right\| _ {\infty} \max  _ {s \in \mathcal {S}} \left\| \pi_ {s} ^ {t + 1} - \pi_ {s} ^ {t} \right\| _ {1}. \tag {10}
+$$
+
+Proof. Using Lemma D.2 with  $x^{+} = \pi_{s}^{t + 1}$ ,  $\mathcal{C} = \Delta (\mathcal{A})$ ,  $f(u) = \langle \widehat{Q}_s^t,u\rangle$ ,  $x = \pi^t$  and  $u = \pi^{t + 1}$ , we obtain
+
+$$
+\left\langle \eta_ {t} \widehat {Q} _ {s} ^ {t}, \pi_ {s} ^ {t} - \pi_ {s} ^ {t + 1} \right\rangle \leq \mathcal {D} _ {h} \left(\pi_ {s} ^ {t}, \pi_ {s} ^ {t}\right) - \mathcal {D} _ {h} \left(\pi_ {s} ^ {t + 1}, \pi_ {s} ^ {t}\right) - \mathcal {D} _ {h} \left(\pi_ {s} ^ {t}, \pi_ {s} ^ {t + 1}\right). \tag {11}
+$$
+
+By rearranging terms and noticing  $\mathcal{D}_h(\pi_s^t,\pi_s^t) = 0$ , we have
+
+$$
+\left\langle \eta_ {t} \widehat {Q} _ {s} ^ {t}, \pi_ {s} ^ {t + 1} - \pi_ {s} ^ {t} \right\rangle \geq \mathcal {D} _ {h} \left(\pi_ {s} ^ {t + 1}, \pi_ {s} ^ {t}\right) + \mathcal {D} _ {h} \left(\pi_ {s} ^ {t}, \pi_ {s} ^ {t + 1}\right) \geq 0. \tag {12}
+$$
+
+Equation (10) can be obtained using the performance difference lemma and (12):
+
+$$
+\begin{array}{l} V ^ {t + 1} (\mu) - V ^ {t} (\mu) = \frac {1}{1 - \gamma} \mathbb {E} _ {s \sim d _ {\mu} ^ {t + 1}} \left[ \langle Q _ {s} ^ {t}, \pi_ {s} ^ {t + 1} - \pi_ {s} ^ {t} \rangle \right] \\ = \frac {1}{1 - \gamma} \mathbb {E} _ {s \sim d _ {\mu} ^ {t + 1}} \left[ \langle \widehat {Q} _ {s} ^ {t}, \pi_ {s} ^ {t + 1} - \pi_ {s} ^ {t} \rangle \right] \\ + \frac {1}{1 - \gamma} \mathbb {E} _ {s \sim d _ {\mu} ^ {t + 1}} \left[ \langle \widehat {Q} _ {s} ^ {t} - Q _ {s} ^ {t}, \pi_ {s} ^ {t + 1} - \pi_ {s} ^ {t} \rangle \right] \\ \geq - \frac {1}{1 - \gamma} \mathbb {E} _ {s \sim d _ {\mu} ^ {t + 1}} \left[ \left\| \widehat {Q} _ {s} ^ {t} - Q _ {s} ^ {t} \right\| _ {\infty} \left\| \pi_ {s} ^ {t + 1} - \pi_ {s} ^ {t} \right\| _ {1} \right]. \\ \end{array}
+$$
+
+![](images/77e28188153a291bf3551166f2fab8d76b29c227df5a5d9cf78c86a545e6a601.jpg)
+
+# D.3 CONVERGENCE GUARANTEE
+
+We can now prove the convergence guarantee reported in Equation (4). The proof of our result is obtained combining the analysis from the sublinear convergence guarantee with the analysis for the inexact linear convergence guarantee given by (Xiao, 2022, Theorems 8 and 13). For two different time  $t, t' \geq 0$ , denote the expected Bregman divergence between the policy  $\pi^t$  and policy  $\pi^{t'}$ , where the expectation is taken over the discounted state visitation distribution of the optimal policy  $d_{\mu}^{\star}$ , by
+
+$$
+\mathcal {D} _ {t ^ {\prime}} ^ {t} := \mathbb {E} _ {s \sim d _ {\mu} ^ {*}} \left[ \mathcal {D} _ {h} \left(\pi_ {s} ^ {t}, \pi_ {s} ^ {t ^ {\prime}}\right) \right].
+$$
+
+Similarly, denote the expected Bregman divergence between the optimal policy  $\pi^{\star}$  and  $\pi^t$  by
+
+$$
+\mathcal {D} _ {t} ^ {\star} := \mathbb {E} _ {s \sim d _ {\mu} ^ {\star}} \left[ \mathcal {D} _ {h} (\pi_ {s} ^ {\star}, \pi_ {s} ^ {t}) \right].
+$$
+
+Theorem D.4 (Theorems 8 and 13 in Xiao (2022)). Consider the PMD update in (2), at each iteration  $T \geq 0$ , we have
+
+$$
+V ^ {\star} (\mu) - \sum_ {t <   T} \mathbb {E} [ V ^ {t} (\mu) ] \leq \frac {1}{T} \left(\frac {\mathbb {E} [ \mathcal {D} _ {0} ^ {\star} ]}{\eta_ {t} (1 - \gamma)} + \frac {1}{(1 - \gamma) ^ {2}}\right) + \frac {4}{(1 - \gamma) ^ {2}} \mathbb {E} \left[ \max _ {s \in \mathcal {S}} \left\| \widehat {Q} _ {s} ^ {t} - Q _ {s} ^ {t} \right\| _ {\infty} \right].
+$$
+
+Proof. Using Lemma D.2 with  $x^{+} = \pi_{s}^{t + 1}$ ,  $\mathcal{C} = \Delta (\mathcal{A})$ ,  $f(u) = \langle \widehat{Q}_s^t,u\rangle$ ,  $x = \pi^t$  and  $u = \pi^{t + 1}$ , we have that
+
+$$
+\langle \eta_ {t} \widehat {Q} _ {s} ^ {t}, \pi_ {s} ^ {\star} - \pi_ {s} ^ {t + 1} \rangle \leq \mathcal {D} _ {h} (\pi^ {\star}, \pi^ {t}) - \mathcal {D} _ {h} (\pi^ {\star}, \pi^ {t + 1}) - \mathcal {D} _ {h} (\pi^ {t + 1}, \pi^ {t}),
+$$
+
+which can be decomposed as
+
+$$
+\langle \eta_ {t} \widehat {Q} _ {s} ^ {t}, \pi_ {s} ^ {t} - \pi_ {s} ^ {t + 1} \rangle + \langle \eta_ {t} \widehat {Q} _ {s} ^ {t}, \pi_ {s} ^ {\star} - \pi_ {s} ^ {t} \rangle \leq \mathcal {D} _ {h} (\pi^ {\star}, \pi^ {t}) - \mathcal {D} _ {h} (\pi^ {\star}, \pi^ {t + 1}) - \mathcal {D} _ {h} (\pi^ {t + 1}, \pi^ {t}).
+$$
+
+Taking expectation with respect to the distribution  $d_{\mu}^{\star}$  over states and with respect to the randomness of PMD and dividing both sides by  $\eta_t$ , we have
+
+$$
+\mathbb {E} \left[ \mathbb {E} _ {s \sim d _ {\mu} ^ {\star}} \left[ \langle \widehat {Q} _ {s} ^ {t}, \pi_ {s} ^ {t} - \pi_ {s} ^ {t + 1} \rangle \right] \right] + \mathbb {E} \left[ \mathbb {E} _ {s \sim d _ {\mu} ^ {\star}} \left[ \langle \widehat {Q} _ {s} ^ {t}, \pi_ {s} ^ {\star} - \pi_ {s} ^ {t} \rangle \right] \right] \leq \frac {1}{\eta_ {t}} \mathbb {E} [ \mathcal {D} _ {t} ^ {\star} - \mathcal {D} _ {t + 1} ^ {\star} - \mathcal {D} _ {t} ^ {t + 1} ]. \tag {13}
+$$
+
+We lower bound the two terms on the left hand side of (13) separately. For the first term, we have that
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \mathbb {E} _ {s \sim d _ {\mu} ^ {\star}} \left[ \langle \widehat {Q} _ {s} ^ {t}, \pi_ {s} ^ {t} - \pi_ {s} ^ {t + 1} \rangle \right] \right] \stackrel {(a)} {\geq} \frac {1}{1 - \gamma} \mathbb {E} \left[ \mathbb {E} _ {s \sim d _ {d _ {\mu} ^ {\star}} ^ {t + 1}} \left[ \langle \widehat {Q} _ {s} ^ {t}, \pi_ {s} ^ {t} - \pi_ {s} ^ {t + 1} \rangle \right] \right] \\ = \frac {1}{1 - \gamma} \mathbb {E} \left[ \mathbb {E} _ {s \sim d _ {d _ {\mu} ^ {\star}} ^ {t + 1}} \left[ \langle Q _ {s} ^ {t}, \pi_ {s} ^ {t} - \pi_ {s} ^ {t + 1} \rangle \right] \right] + \frac {1}{1 - \gamma} \mathbb {E} \left[ \mathbb {E} _ {s \sim d _ {d _ {\mu} ^ {\star}} ^ {t + 1}} \left[ \langle \widehat {Q} _ {s} ^ {t} - Q _ {s} ^ {t}, \pi_ {s} ^ {t} - \pi_ {s} ^ {t + 1} \rangle \right] \right] \\ \stackrel {(b)} {=} \mathbb {E} \left[ V ^ {t} (d _ {\mu} ^ {\star}) - V ^ {t + 1} (d _ {\mu} ^ {\star}) \right] + \frac {1}{1 - \gamma} \mathbb {E} \left[ \mathbb {E} _ {s \sim d _ {d _ {\mu} ^ {\star}} ^ {t + 1}} \left[ \langle \widehat {Q} _ {s} ^ {t} - Q _ {s} ^ {t}, \pi_ {s} ^ {t} - \pi_ {s} ^ {t + 1} \rangle \right] \right] \\ \geq \mathbb {E} \left[ V ^ {t} (d _ {\mu} ^ {\star}) - V ^ {t + 1} (d _ {\mu} ^ {\star}) \right] - \frac {1}{1 - \gamma} \mathbb {E} \left[ \mathbb {E} _ {s \sim d _ {d _ {\mu} ^ {\star}} ^ {t + 1}} \left[ \left\| \widehat {Q} _ {s} ^ {t} - Q _ {s} ^ {t} \right\| _ {\infty} \left\| \pi_ {s} ^ {t + 1} - \pi_ {s} ^ {t} \right\| _ {1} \right] \right] \\ \geq \mathbb {E} \left[ V ^ {t} (d _ {\mu} ^ {\star}) - V ^ {t + 1} (d _ {\mu} ^ {\star}) \right] - \frac {2}{1 - \gamma} \mathbb {E} \left[ \max _ {s \in \mathcal {S}} \left\| \widehat {Q} _ {s} ^ {t} - Q _ {s} ^ {t} \right\| _ {\infty} \right] \\ \end{array}
+$$
+
+where  $(a)$  follows from Lemmas D.3 and the fact that  $d_{d_{\mu}^{\star}}^{t + 1}(s) \geq (1 - \gamma)d_{\mu}^{\star}(s) \forall s \in S$ , and  $(b)$  follows from D.1. For the second term, we have that
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \mathbb {E} _ {s \sim d _ {\mu} ^ {\star}} \left[ \langle \widehat {Q} _ {s} ^ {t}, \pi_ {s} ^ {\star} - \pi_ {s} ^ {t} \rangle \right] \right] = \mathbb {E} \left[ \mathbb {E} _ {s \sim d _ {\mu} ^ {\star}} \left[ \langle Q _ {s} ^ {t}, \pi_ {s} ^ {\star} - \pi_ {s} ^ {t} \rangle \right] + \mathbb {E} \left[ \mathbb {E} _ {s \sim d _ {\mu} ^ {\star}} \left[ \langle \widehat {Q} _ {s} ^ {t} - Q _ {s} ^ {t}, \pi_ {s} ^ {\star} - \pi_ {s} ^ {t} \rangle \right] \right] \right. \\ \stackrel {(b)} {=} \mathbb {E} [ V ^ {\star} (\mu) - V ^ {t} (\mu) ] (1 - \gamma) + \mathbb {E} \left[ \mathbb {E} _ {s \sim d _ {\mu} ^ {\star}} \left[ \langle \widehat {Q} _ {s} ^ {t} - Q _ {s} ^ {t}, \pi_ {s} ^ {\star} - \pi_ {s} ^ {t} \rangle \right] \right], \\ \geq \mathbb {E} [ V ^ {\star} (\mu) - V ^ {t} (\mu) ] (1 - \gamma) - \frac {2}{1 - \gamma} \mathbb {E} \left[ \max _ {s \in \mathcal {S}} \left\| \widehat {Q} _ {s} ^ {t} - Q _ {s} ^ {t} \right\| _ {\infty} \right], \\ \end{array}
+$$
+
+where  $(b)$  follows from Lemma D.1.
+
+Plugging the two bounds in (13), dividing both sides by  $(1 - \gamma)$  and rearranging, we obtain
+
+$$
+\begin{array}{l} \frac {\mathbb {E} \left[ \mathcal {D} _ {t} ^ {t + 1} \right]}{\eta_ {t} (1 - \gamma)} + \mathbb {E} \left[ V ^ {\star} (\mu) - V ^ {t} (\mu) \right] \leq \frac {\mathbb {E} \left[ \mathcal {D} _ {t} ^ {\star} - \mathcal {D} _ {t + 1} ^ {\star} \right]}{\eta_ {t} (1 - \gamma)} + \frac {\mathbb {E} \left[ V ^ {t} \left(d _ {\mu} ^ {\star}\right) - V ^ {t + 1} \left(d _ {\mu} ^ {\star}\right) \right]}{1 - \gamma} \\ + \frac {4}{(1 - \gamma) ^ {2}} \mathbb {E} \left[ \max _ {s \in \mathcal {S}} \left\| \widehat {Q} _ {s} ^ {t} - Q _ {s} ^ {t} \right\| _ {\infty} \right]. \\ \end{array}
+$$
+
+Summing up from 0 to  $T - 1$  and dropping some positive terms on the left hand side and some negative terms on the right hand side, we have
+
+$$
+\begin{array}{l} T V ^ {\star} (\mu) - \sum_ {t <   T} \mathbb {E} [ V ^ {t} (\mu) ] \leq \frac {\mathbb {E} [ \mathcal {D} _ {0} ^ {\star} ]}{\eta_ {t} (1 - \gamma)} - \frac {\mathbb {E} \left[ V ^ {0} \left(d _ {\mu} ^ {\star}\right) - V ^ {T} \left(d _ {\mu} ^ {\star}\right) \right]}{1 - \gamma} \\ + \frac {4}{(1 - \gamma) ^ {2}} \mathbb {E} \left[ \max  _ {s \in \mathcal {S}} \left\| \widehat {Q} _ {s} ^ {t} - Q _ {s} ^ {t} \right\| _ {\infty} \right]. \\ \end{array}
+$$
+
+Notice that  $\mathbb{E}\left[V^0 (d_\mu^\star) - V^T (d_\mu^\star)\right]\leq \frac{1}{1 - \gamma}$  as  $r(s,a)\in [0,1]$ . By dividing  $T$  on both side, we yield the statement.
+
+# E TRAINING DETAILS
+
+We give the hyper-parameters we use for training in Tables 2, 3 and 4. Hyper-parameter tuning was performed differently for each method. For Gridworld, we performed a grid search over the hyper-parameters and selected those that maximized the averaged performance of the negative entropy
+
+and  $\ell_2$ -norm over 256 randomly sampled Gridworld environments. For Basic Control Suite and MinAtar, we used the optuna library to search over the hyper-parameters to maximize the average performance of the negative entropy over all environments. We used "CmaEsSampler" as option for the sampler. Lastly, we used Weights and Biases to optimize the hyperparameters for MuJoCo in order to maximize the average performance of the negative entropy over all environments.
+
+Table 2: Hyper-parameter settings of PMD for different sets of environments  
+
+<table><tr><td>Parameter</td><td>Grid-World</td><td>MuJoCo</td></tr><tr><td>Number of environment steps</td><td>218 ≈ 2.5e5</td><td>1e7</td></tr><tr><td>Number of environments</td><td>64</td><td>2048</td></tr><tr><td>Unroll length</td><td>32</td><td>10</td></tr><tr><td>Number of minibatches</td><td>1</td><td>128</td></tr><tr><td>Number of update epochs</td><td>32</td><td>8</td></tr><tr><td>Adam learning rate</td><td>-</td><td>1e-4</td></tr><tr><td>Sgd learning rate</td><td>40</td><td>-</td></tr><tr><td>Gamma</td><td>0.99</td><td>0.99</td></tr><tr><td>Max grad norm</td><td>-</td><td>1.0</td></tr><tr><td>η</td><td>0.1</td><td>0.5</td></tr></table>
+
+Table 3: Hyper-parameter settings of AMPO for different sets of environments  
+
+<table><tr><td>Parameter</td><td>BCS</td><td>MinAtar</td></tr><tr><td>Number of environment steps</td><td>5e5</td><td>1e7</td></tr><tr><td>Number of environments</td><td>4</td><td>256</td></tr><tr><td>Unroll length</td><td>128</td><td>128</td></tr><tr><td>Number of minibatches</td><td>4</td><td>8</td></tr><tr><td>Number of update epochs</td><td>16</td><td>8</td></tr><tr><td>Adam learning rate</td><td>4e-3</td><td>7e-4</td></tr><tr><td>Gamma</td><td>0.99</td><td>0.99</td></tr><tr><td>Max grad norm</td><td>1.4</td><td>1</td></tr><tr><td>AMPO learning rate</td><td>0.9</td><td>0.9</td></tr></table>
+
+Table 4: Hyper-parameter settings of OpenAI-ES, which we used in the tabular setting, and of Sep-CMA-ES, which we used in the non-tabular case.  
+
+<table><tr><td></td><td>OpenAI-ES</td><td>Sep-CMA-ES</td></tr><tr><td>Population Size</td><td>512</td><td>128</td></tr><tr><td>Number of generations</td><td>512 (256 for MuJoCo)</td><td>600</td></tr><tr><td>Sigma init</td><td>0.5</td><td>2</td></tr><tr><td>Sigma Decay</td><td>0.995</td><td>-</td></tr><tr><td>Learning rate</td><td>0.01</td><td>-</td></tr></table>
+
+# F COMPUTATIONAL COSTS OF ES
+
+The computational costs associated with ES are inevitable in this field of research that focuses on automatically discovering optimizers. In the literature, the most popular methods to discover optimizers are meta-gradients (Oh et al., 2020; Jackson et al., 2023) and ES (Lu et al., 2022). Discovering algorithms using meta-gradients consists in introducing some meta-parameters that influence the agent training procedure, training a batch of agents, and differentiating through the training procedure w.r.t. the meta-parameters to maximize the final performance of the agents. ES consists in running several training procedures in parallel and estimating the meta-gradient as in (9), therefore avoiding the need for differentiation. This property is particularly desirable in settings, like ours, where the agent is updated many times and differentiating through the whole training procedure is unfeasible due to memory constraints, meaning that it is necessary to limit the differentiation to the final updates. We decided to employ ES due to this property and because it has been found to be more successful in discovering optimizers (Jackson et al., 2024). Additionally, our experiments are implemented in JAX, which through parallelism and just in time compilation renders the computational costs of ES feasible.
\ No newline at end of file
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+# LEARNING ON ONE MODE: ADDRESSING MULTI-MODALITY IN OFFLINE REINFORCEMENT LEARNING
+
+Mianchu Wang* Yue Jin* Giovanni Montana\*†
+
+*University of Warwick †The Alan Turing Institute
+
+{mianchu.wang, yue.jin.3, g.montana}@warwick.ac.uk
+
+# ABSTRACT
+
+Offline reinforcement learning (RL) seeks to learn optimal policies from static datasets without interacting with the environment. A common challenge is handling multi-modal action distributions, where multiple behaviours are represented in the data. Existing methods often assume unimodal behaviour policies, leading to suboptimal performance when this assumption is violated. We propose weighted imitation Learning on One Mode (LOM), a novel approach that focuses on learning from a single, promising mode of the behaviour policy. By using a Gaussian mixture model to identify modes and selecting the best mode based on expected returns, LOM avoids the pitfalls of averaging over conflicting actions. Theoretically, we show that LOM improves performance while maintaining simplicity in policy learning. Empirically, LOM outperforms existing methods on standard D4RL benchmarks and demonstrates its effectiveness in complex, multi-modal scenarios.
+
+# 1 INTRODUCTION
+
+Offline reinforcement learning (RL) enables policy learning from static datasets, without active environment interaction, making it ideal for high-stakes applications like autonomous driving and robot manipulation (Levine et al., 2020; Ma et al., 2022; Wang et al., 2024a). A key challenge in offline RL is managing the discrepancy between the learned policy and the behaviour policy that generated the dataset. Small discrepancies can hinder policy improvement, while large discrepancies push the learned policy into uncharted areas, causing significant extrapolation errors and poor generalisation (Fujimoto et al., 2019; Yang et al., 2023). Addressing these challenges, existing research has proposed various solutions. Conservative approaches penalise actions that stray into out-of-distribution (OOD) regions (Yu et al., 2020; Kumar et al., 2020), while others regularise the policy by minimising its divergence from the behaviour policy, ensuring better fidelity to the dataset (Fujimoto & Gu, 2021; Wu et al., 2019). Another solution is weighted imitation learning, which aims to replicate actions from the dataset through supervised learning techniques (Mao et al., 2023; Peng et al., 2019).
+
+Many real-world datasets introduce an additional challenge: multi-modal action distributions (Wang et al., 2024b; Chen et al., 2022; Zhou et al., 2020). These datasets are common in practice because they often integrate data from diverse sources, such as multiple policies, human demonstrations, or distinct exploration strategies. This diversity arises naturally in domains where the same task can be approached in various ways, leading to states with multiple valid but potentially conflicting actions. For instance, in autonomous driving, different driving styles — conservative versus aggressive — may lead to different but equally valid ways of navigating a road. Similarly, in robotic manipulation, an object can be grasped in various ways depending on the robot's approach, the object's position, and environmental constraints. This multi-modality is not an exception but rather a frequent occurrence in complex, real-world decision-making tasks, as systems often integrate experience from various sources to handle different scenarios. Thus, it becomes crucial to model and manage these multimodal action distributions effectively.
+
+Most offline RL approaches implicitly assume a unimodal action distribution, which can force policies to converge toward an average action that may not exist in the dataset, leading to degraded performance. This limitation is particularly evident in scenarios where policies fail to capture complex multi-modal distributions, instead collapsing into suboptimal or invalid averages, as studied by Cai et al. (2023); Wang et al. (2024b); Yang et al. (2022b) and illustrated in Appendix A. Recent work has attempted to address this by modelling the full multi-modal action distribution using expressive
+
+generative models such as GANs, VAEs, and diffusion models (Zhou et al., 2020; Chen et al., 2022; Wang et al., 2023). However, these models often overcomplicate the learning process by capturing the entire action distribution, which is unnecessary when only a subset of the modes is relevant for optimal decision-making.
+
+We propose a simpler yet effective approach: Weighted Imitation Learning on One Mode (LOM). Our insight is that learning from a single mode — the one with the highest expected return — is sufficient to generate optimal actions. Instead of modelling the full multi-modal distribution, LOM identifies and focuses on the most promising mode for each state.
+
+LOM operates through three key steps (illustrated in Figure 1). First, it models the behaviour policy as a Gaussian mixture model (GMM), capturing the inherent multi-modality in the action space. Each mode in the GMM represents a distinct cluster of actions associated with a state. Second, a novel hyper Q-function is introduced to evaluate the expected return of each mode, enabling the dynamic selection of the most advantageous one. Finally, LOM performs weighted imitation learning on the actions from the selected mode, ensuring that the learned policy focuses on the most beneficial actions while retaining the simplicity of unimodal policies. This targeted, mode-specific learning strategy simplifies the policy learning process while maintaining or even enhancing performance in multi-modal scenarios. By dynamically selecting the optimal mode for each state, LOM achieves robust results with reduced complexity.
+
+This paper has four key contributions: (1) we propose LOM, a novel weighted imitation learning method designed to address the multi-modality problem in offline RL; (2) we introduce hyper Q-functions and hyper-policies for evaluating and selecting action modes; (3) we provide theoretical guarantees of consistent performance improvements over both the behaviour policy and the optimal action mode; and (4) we empirically demonstrate that LOM outperforms state-of-the-art (SOTA) offline RL methods across various benchmarks, particularly in multimodal datasets.
+
+![](images/a75094262944d9ae72361b734e87e3076e41813b482b95d377a4df54a4aef959.jpg)
+
+![](images/1455980cc36ea134fb2f3b0bb56ebae0a0412060f40452a4d4ec3789a20e084c.jpg)
+
+![](images/29fbdec2d79e78f7ee6d29f2824a08f0014bf1a3794641742606daec50afb7ce.jpg)  
+Figure 1: The three steps of LOM. (1) Learn a network producing the parameters of a GMM to model the behaviour policy. (2) Evaluate each mode via the expected return of its actions and then select the optimal mode  $\phi^1$ . (3) Sample actions from  $\phi^1$  for weighted imitation learning.
+
+# 2 RELATED WORK
+
+Offline RL The primary challenge in offline RL lies in managing the distribution shift between the behaviour policy, which generated the offline dataset, and the learned policy. This shift can cause the learned policy to produce actions that are not well-represented in the offline dataset, leading to inaccurate value function estimations and degraded performance (Levine et al., 2020; Fujimoto et al., 2019). To mitigate the risks associated with OOD actions, one approach involves using conservative value functions, which penalise actions that deviate from the behaviour policy (Fujimoto et al., 2019; Kumar et al., 2020; Bai et al., 2022; Sun et al., 2022; An et al., 2021). Another strategy focuses on regularising the learned policy to ensure its proximity to the behaviour policy. This regularisation can be measured using metrics such as mean squared error (MSE) (Beeson & Montana, 2024; Fujimoto & Gu, 2021) or more sophisticated metrics like the Wasserstein distance, Jensen-Shannon divergence (Yang et al., 2022b; Wu et al., 2019), and weighted MSE (Ma et al., 2024). These methods aim to keep the learned policy within a safer, well-understood operational space, reducing the likelihood of selecting OOD actions. For a comprehensive review of value penalties and policy regularisation techniques, we refer readers to (Wu et al., 2019). In contrast to these approaches, LOM tackles the distribution shift problem by focusing on a single, optimal mode of the behaviour policy, thereby reducing the risk of OOD actions while still allowing for significant policy improvement.
+
+Weighted Imitation Learning Our proposed method contributes to the growing field of weighted imitation learning, which revolves around two fundamental questions: which actions should be imitated and how should these actions be weighted (Brandfonbrener et al., 2021). Most existing methods imitate actions directly from the dataset (Peng et al., 2019; Wang et al., 2018; Nair et al., 2021; Wang et al., 2020), while some approaches (Siegel et al., 2020; Mao et al., 2023) suggest imitating actions from the most recent policies. These methods typically weigh actions based on their advantage, conditioned either on the behaviour policy (Peng et al., 2019; Wang et al., 2018) or on the most recently updated policy (Nair et al., 2021; Wang et al., 2020; Siegel et al., 2020; Mao et al., 2023). However, a key limitation of these approaches is their underlying assumption that the imitated actions are drawn from a unimodal Gaussian distribution, an oversimplification that often fails in complex, real-world datasets where multi-modal distributions are common (Lynch et al., 2020; Yang et al., 2022b). LOM addresses this limitation by forgoing the unimodal assumption, instead extracting a highly-rewarded Gaussian mode from the behaviour policy.
+
+Multi-modality in Offline RL Offline RL datasets are often collected from multiple, unknown policies, leading to states with several valid action labels, which may conflict with one another. Most existing methods employ a unimodal Gaussian policy, which inadequately captures the inherent multimodality in such datasets. To address this limitation, PLAS (Zhou et al., 2020) decodes variables sampled from a VAE latent space, while LAPO (Chen et al., 2022) leverages advantage-weighted latent spaces for further policy optimisation. Additionally, Yang et al. (2022b) demonstrate that GANs can model multiple action modes, and GOPlan (Wang et al., 2024b) extends this by applying exponentiated advantage weighting to highlight highly-rewarded modes. DAWOG (Wang et al., 2024a) and DMPO (Osa et al., 2023) separate the state-action space and learn a conditioned Gaussian policy or a mixture of deterministic policies to solve the regions individually. Techniques such as normalising flows and diffusion models can also model multi-modal distributions by progressively transforming an initial distribution into the target distribution (Wang et al., 2023; Akimov et al., 2022). However, these techniques are computationally intensive, making their inference processes inefficient. Beyond direct policy improvements, approaches like TD3+RKL (Cai et al., 2023) exploit the mode-seeking property of reverse KL-divergence to narrow action coverage, while BAW (Yang et al., 2022a) filters actions with values in the leftmost quantile. Despite these advances, most methods attempt to capture or model the entire multi-modal distribution, leading to increased complexity. In contrast, LOM simplifies the learning process by extracting and focusing on a single, optimal mode from the behaviour policy, effectively addressing the multi-modality challenge without the need to model the entire distribution.
+
+# 3 PRELIMINARIES
+
+To lay the foundation for our method, we first introduce the Markov Decision Process (MDP) and the concept of weighted imitation learning. An MDP is defined by the tuple  $\mathcal{M} = \langle S, \mathcal{A}, \mathcal{P}, r, \gamma \rangle$ , where  $S$  is the state space,  $\mathcal{A}$  the action space, and  $\mathcal{P}$  the transition dynamics. The function  $r$  represents the reward, and  $\gamma$  is the discount factor. The objective in RL is to learn a policy  $\pi$  that maximises the expected discounted return:
+
+$$
+J (\pi) = \mathbb {E} _ {\tau \sim p _ {\pi}} \left[ \sum_ {t = 0} ^ {T} \gamma^ {t} r (s _ {t}, a _ {t}) \right],
+$$
+
+where  $\tau = \{s_0, a_0, \dots, s_T, a_T\}$  is a trajectory sampled under policy  $\pi$ . The state-action value function  $Q^{\pi}(s_t, a_t)$  represents the expected return starting from state  $s_t$ , taking action  $a_t$ , and following  $\pi$  thereafter:
+
+$$
+Q ^ {\pi} \left(s _ {t}, a _ {t}\right) = \mathbb {E} _ {s _ {t + 1}, a _ {t + 1}, \dots \sim \pi} \left[ \sum_ {l = t} ^ {T} \gamma^ {l} r \left(s _ {l}, a _ {l}\right) \right]. \tag {1}
+$$
+
+The value function is defined as  $V^{\pi}(s_t) = \mathbb{E}_{a_t\sim \pi (\cdot |s_t)}[Q^{\pi}(s_t,a_t)]$ , and the advantage function is  $A^{\pi}(s_t,a_t) = Q^{\pi}(s_t,a_t) - V^{\pi}(s_t)$ .
+
+In offline RL, the agent learns exclusively from a fixed dataset  $\mathcal{D}$  collected by one or more behaviour policies, denoted as  $\pi_{b}$ , without further interaction with the environment (Levine et al., 2020). Our approach is based on weighted imitation learning, where the objective is to optimize
+
+$$
+J (\pi) = \mathbb {E} _ {s \sim d _ {\pi_ {b}}, a \sim \pi_ {b} (\cdot | s)} \left[ \exp \left(\frac {1}{\beta} A ^ {\pi_ {b}} (s, a)\right) \log \pi (a \mid s) \right],
+$$
+
+where  $\beta$  is a hyper-parameter. This formulation encourages learning a policy  $\pi$  that imitates actions from  $\mathcal{D}$ , weighted by the exponentiated advantage of the behaviour policy  $\pi_{b}$ , thereby implicitly constraining the KL-divergence between  $\pi$  and  $\pi_{b}$  (Wang et al., 2018). Recent works have extended this approach by imitating actions from the current learned policy (Siegel et al., 2020; Mao et al., 2023), or weighting actions based on the advantage conditioned on the current policy (Nair et al., 2021; Wang et al., 2020).
+
+# 4 METHOD
+
+In this section, we present the LOM method, specifically designed for offline RL with heterogeneous datasets. We begin by modelling the behaviour policy using a Gaussian Mixture Model (GMM) to capture the inherent multi-modality. Following this, we formalise the problem using a hyperMarkov decision process (H-MDP), where a hyper-policy dynamically selects the most promising mode for each state based on expected returns. Next, we propose a hyper Q-function to evaluate the hyper-policy and introduce a greedy hyper-policy for mode selection at each step for weighted imitation learning. Finally, we prove that the resulting policy of our LOM method outperforms both the behaviour policy and a greedy policy.
+
+# 4.1 MULTI-MODAL BEHAVIOURAL POLICY
+
+To address the challenge of multi-modal action distributions in offline RL, we model the behaviour policy as a Gaussian Mixture Model (GMM). GMMs are flexible in representing complex distributions and can approximate any density function with sufficient components (McLachlan & Basford, 1988), making them well-suited for capturing the heterogeneous nature of offline datasets that often arise from multiple policies or exploration strategies. The behaviour policy  $\pi_b(a\mid s)$  is expressed as:
+
+$$
+\pi_ {b} (a \mid s) = \sum_ {i = 1} ^ {M} \alpha^ {i} (s) \phi^ {i} (a \mid s), \tag {2}
+$$
+
+where  $\alpha^i (s)$  represents the mixing coefficients for each mode  $i$ , and  $\phi^i (a\mid s)$  denotes the Gaussian component for mode  $i$  at state  $s$ . Each mode captures a distinct cluster of actions associated with the state. By using a GMM, our model captures the diverse modes present in real-world offline RL datasets, where multiple valid strategies may coexist for the same task. This approach preserves the richness of the original data while avoiding the limitations of unimodal policy approximations.
+
+# 4.2 HYPER-MARKOV DECISION PROCESS
+
+To extend the traditional MDP framework introduced in Section 3 and formulate the offline RL problem with a multi-modal behaviour policy, we introduce the Hyper-Markov Decision Process (H-MDP). The H-MDP accounts for multiple policy modes, extending the standard MDP to a higher level of abstraction. Formally, the H-MDP is defined as  $\mathcal{M}_H = \langle S, \mathcal{A}, \mathcal{P}, r, \gamma, \Omega, \mathcal{A}_H, \mathcal{P}_H, r_H \rangle$ , where  $S$ ,  $\mathcal{A}$ ,  $\mathcal{P}$ ,  $r$ , and  $\gamma$  retain their standard meanings from the MDP. The novel components introduced are:  $\Omega = \{\phi^1, \dots, \phi^M\}$ , representing the Gaussian modes, and the hyper-action space  $\mathcal{A}_H = \{1, \dots, M\}$ , which indexes these modes.
+
+The transition dynamics  $\mathcal{P}_H$  conditioned on the hyper-action  $u_{t}\in \mathcal{A}_{H}$  and the reward function  $r_H$  for  $u_{t}$  are defined as:
+
+$$
+\mathcal {P} _ {H} (s _ {t + 1} \mid s _ {t}, u _ {t}) = \mathbb {E} _ {a _ {t} \sim \phi^ {u _ {t}} (a | s _ {t})} \left[ \mathcal {P} (s _ {t + 1} \mid s _ {t}, a _ {t}) \right],
+$$
+
+$$
+r _ {H} \left(s _ {t}, u _ {t}\right) = \mathbb {E} _ {a _ {t} \sim \phi^ {u _ {t}} \left(a \mid s _ {t}\right)} \left[ r \left(s _ {t}, a _ {t}\right) \right].
+$$
+
+At each time step, the agent selects a Gaussian mode  $u$  based on a hyper-policy  $\zeta$ , which maps states to a probability distribution over modes,  $\zeta(u \mid s): S \times \mathcal{A}_H \to [0,1]$ . The hyper-policy determines the likelihood of selecting each mode in a given state. The agent then selects an action  $a$  according to the Gaussian distribution  $\phi^u(a \mid s)$ . Consequently, the action follows a composite policy  $\pi_{\zeta}$ , which corresponds to the hyper-policy  $\zeta$  and can be expressed as:
+
+$$
+\pi_ {\zeta} (a \mid s) = \sum_ {u \in \mathcal {A} _ {H}} \zeta (u \mid s) \phi^ {u} (a \mid s), \tag {3}
+$$
+
+In the H-MDP, the agent's objective is to learn a hyper-policy  $\zeta$  that maximises the expected discounted return:
+
+$$
+J (\zeta) = \mathbb {E} _ {\tau \sim p _ {\zeta}} \left[ \sum_ {t = 0} ^ {T} \gamma^ {t} r _ {H} \left(s _ {t}, u _ {t}\right) \right], \tag {4}
+$$
+
+where  $p_{\zeta}$  is the distribution of state-mode trajectories  $\tau = (s_t, u_t, s_{t+1}, u_{t+1}, \ldots)$  induced by following the hyper-policy  $\zeta$ .
+
+# 4.3 HYPER Q-FUNCTION
+
+To evaluate the quality of selecting a mode  $u$  in a given state  $s$ , we define the hyper  $Q$ -function  $Q_H^\zeta(s, u)$ . This function quantifies the expected return when choosing mode  $u$  at state  $s$  and subsequently following the hyper-policy  $\zeta$ . Formally, the hyper  $Q$ -function is defined as:
+
+$$
+Q _ {H} ^ {\zeta} (s, u) = \mathbb {E} _ {s _ {t + 1}, u _ {t + 1}, \dots \sim \zeta} \left[ \sum_ {t = 0} ^ {T} \gamma^ {t} r _ {H} \left(s _ {t}, u _ {t}\right) \mid s _ {0} = s, u _ {0} = u \right]. \tag {5}
+$$
+
+This expectation is taken over future states and modes encountered while following the hyper-policy  $\zeta$  after selecting mode  $u$  in state  $s$ . The term  $r_{H}(s_{t}, u_{t})$  denotes the expected reward for selecting mode  $u_{t}$  in state  $s_{t}$ , with  $\gamma$  being the discount factor.
+
+Proposition 1. The hyper  $Q$ -function can be linked to the standard value function  $Q^{\pi}(s,a)$  via:
+
+$$
+Q _ {H} ^ {\zeta} (s, u) = \mathbb {E} _ {a \sim \phi^ {u} (\cdot | s)} \left[ Q ^ {\pi_ {\zeta}} (s, a) \right]. \tag {6}
+$$
+
+The proof can be found in Appendix B.1.
+
+This proposition establishes a critical relationship between the hyper Q-function and the standard Q-function. It shows that the expected return for selecting mode  $u$  (i.e., the hyper-action) is equivalent to the expected value of actions sampled from the Gaussian component  $\phi^u(a \mid s)$ , evaluated under the composite policy  $\pi_{\zeta}$ . This result is particularly important when dealing with multi-modal behaviour policies, as it allows the agent to effectively compare and evaluate different modes  $u$  based on their associated action distributions  $\phi^u(a \mid s)$ .
+
+# 4.4 MODE SELECTION
+
+Since the hyper Q-function evaluates the expected return after selecting mode  $u$  at a state, we now propose a greedy hyper-policy that improves upon the behaviour policy by selecting the most advantageous mode at each state based on the hyper Q-function.
+
+In correspondence with the multi-modal behaviour policy  $\pi_b$  defined in Eq. 2, we introduce the behavioural hyper-policy, denoted as  $\zeta_b$ , which reflects the mode-selection strategy implicit in the behaviour policy that generated the offline dataset. Specifically,  $\zeta_b(u|s) = \alpha^u (s)$ , where  $\alpha^u (s)$  is the mixing coefficient for the  $u$ -th Gaussian component in the GMM at state  $s$ . This formulation establishes the connection between the original behaviour policy  $\pi_b$  and the H-MDP framework, allowing us to represent the behaviour policy in terms of mode selection.
+
+To improve upon the behavioural hyper-policy  $\zeta_{b}$ , we define a greedy hyper-policy  $\zeta_{g}$ , which selects the mode that maximises the hyper Q-function  $Q_{H}^{\zeta_{b}}(s,u)$  for each state:
+
+$$
+\zeta_ {g} (s) = \arg \max  _ {u \in \mathcal {A} _ {H}} Q _ {H} ^ {\zeta_ {b}} (s, u). \tag {7}
+$$
+
+This greedy hyper-policy ensures that the agent selects the mode with the highest expected return at each state, thereby improving the overall policy performance relative to the behaviour policy.
+
+Theorem 1. The composite policy induced by the greedy hyper-policy improves upon the behaviour policy:
+
+$$
+V ^ {\pi_ {\zeta_ {g}}} (s) \geq V ^ {\pi_ {b}} (s), \quad \forall s \in \mathcal {S}. \tag {8}
+$$
+
+The proof can be found in Appendix B.2.
+
+This improvement property guaranteed by Theorem 1 forms the foundation of our LOM algorithm. It ensures that by systematically selecting modes with the highest expected return at each state, the agent's composite policy will, over time, perform at least as well as the original behaviour policy.
+
+# 4.5 WEIGHTED IMITATION LEARNING ON ONE MODE
+
+Based on the greedy hyper-policy  $\zeta_{g}$ , we obtain the corresponding greedy policy  $\pi_{\zeta_g}$ , which, according to Theorem 1, is not worse than  $\pi_{b}$ . However, rather than simply imitating  $\pi_{\zeta_g}$ , we apply weighted imitation learning to further improve the policy. Specifically, we aim to find a policy  $\pi$  that maximises the expected improvement  $\eta (\pi) = J(\pi) - J(\pi_{\zeta_g})$ . The expected improvement  $\eta (\pi)$  can be expressed in terms of the advantage  $A^{\pi_{\zeta_g}}(s,a)$  (Kakade & Langford, 2002; Schulman et al., 2015):  $\eta (\pi) = \mathbb{E}_{s\sim d_{\pi}(\cdot),a\sim \pi (\cdot |s)}[A^{\pi_{\zeta_g}}(s,a)]$ . However, in offline RL, state samples from the distribution  $d_{\pi}(s)$  are unavailable. Instead, we approximate the expected improvement by using an estimate  $\hat{\eta}$  (Kakade & Langford, 2002):  $\hat{\eta} (\pi) = \mathbb{E}_{s\sim d_{\pi_{\zeta_g}}(\cdot),a\sim \pi (\cdot |s)}[A^{\pi_{\zeta_g}}(s,a)]$ . This approximation provides a good estimate of  $\eta (\pi)$  if  $\pi$  and  $\pi_{\zeta_g}$  are close in terms of KL-divergence (Schulman et al., 2015). As a result, we can formulate the following constrained policy optimisation problem:
+
+$$
+\arg \max _ {\pi} \sum_ {s} d _ {\pi_ {\zeta_ {g}}} (s) \sum_ {a} \pi (a \mid s) A ^ {\pi_ {\zeta_ {g}}} (s, a),
+$$
+
+$$
+\sum_ {s} d _ {\pi_ {\zeta_ {g}}} (s) D _ {K L} (\pi (\cdot \mid s) | | \pi_ {\zeta_ {g}} (\cdot \mid s)) \leq \epsilon . \tag {9}
+$$
+
+Solving the corresponding Lagrangian leads to the optimal policy  $\pi^{*}\colon \pi^{*}(a\mid s) = \frac{1}{Z(s)}\pi_{\zeta_{g}}(a\mid s)\exp \left(\frac{1}{\beta} A^{\pi_{\zeta_{g}}} (s,a)\right)$ . We then learn a policy by minimising the KL-divergence to this optimal policy, resulting in the weighted imitation learning objective:
+
+$$
+\arg \min  _ {\pi} \mathbb {E} _ {s \sim d _ {\pi_ {b}} (\cdot)} [ D _ {K L} \left(\pi^ {*} (\cdot | s) \mid \mid \pi (\cdot | s)\right) ] \tag {10}
+$$
+
+$$
+= \arg \max  _ {\pi} \mathbb {E} _ {s \sim d _ {\pi_ {b}} (\cdot), a \sim \pi_ {\zeta_ {g}} (\cdot | s)} \left[ \exp \left(\frac {1}{\beta} A ^ {\pi_ {\zeta_ {g}}} (s, a)\right) \log \pi (a \mid s) \right]. \tag {11}
+$$
+
+In this framework, the actions to be imitated are sampled from  $\pi_{\zeta_g}$ , a unimodal Gaussian policy, thereby addressing the multi-modality issue. Furthermore, actions within this mode are weighted according to their advantage, encouraging the policy to concentrate on the highly-rewarded actions. As shown in Theorem 2, the policy learned by LOM has a theoretical advantage over both  $\pi_{\zeta_g}$  and the original behaviour policy  $\pi_b$ .
+
+Theorem 2. The LOM algorithm learns a policy  $\pi_L$  that is at least as good as both the composite policy induced by the greedy hyper-policy  $\pi_{\zeta_g}$  and the behaviour policy  $\pi_b$ . Specifically, for all states  $s \in S$ :
+
+$$
+V ^ {\pi_ {L}} (s) \geq V ^ {\pi_ {\zeta_ {g}}} (s) \geq V ^ {\pi_ {b}} (s). \tag {12}
+$$
+
+The proof is provided in Appendix B.3.
+
+This theorem implies that our method introduces a novel two-step improvement process. LOM achieves further policy improvement compared to one-step algorithms while requiring less off-policy evaluation than multi-step algorithms (Brandfonbrener et al., 2021).
+
+Theorem 3. The LOM algorithm learns a policy  $\pi_L$ , whose improvement upon the composite policy induced by the greedy hyper-policy  $\pi_{\zeta_g}$  is bounded. For all  $s \in S$ , we have
+
+$$
+V ^ {\pi_ {L}} (s) - V ^ {\pi_ {\zeta_ {g}}} (s) \geq \frac {1}{1 - \gamma} \hat {\eta} (\pi_ {L}) - \frac {A _ {\max }}{1 - \gamma} \sqrt {\frac {1}{2} D _ {K L} \left(d _ {\pi_ {L}} \mid \mid d _ {\pi_ {\zeta_ {g}}}\right)}, \tag {13}
+$$
+
+where  $A_{\mathrm{max}} = \max_{s,a}|A^{\pi \zeta_g}(s,a)|$ . The proof is provided in Appendix B.4.
+
+The bound shows that the learned policy  $\pi_L$  is guaranteed to perform at least as well as the baseline policy  $\pi_{\zeta_g}$ , with the performance improvement quantified by the maximal advantage and reduced by a penalty term proportional to the divergence in their state visitation distributions — emphasizing that maximising advantage while minimising divergence leads to better performance.
+
+# 5 PRACTICAL ALGORITHM
+
+In this section, we present the practical implementation of the LOM algorithm. First, we model the multi-modal behaviour policy using a mixture density network, as described in Section 5.1. In Section 5.2, we outline the mode selection process, which involves learning the hyper Q-function for the behavioural hyper-policy. This hyper Q-function is then used to derive the greedy hyper-policy  $\zeta_{g}$  and the corresponding greedy policy  $\pi_{\zeta_g}$ , which selects a single mode. Finally, in Section 5.3, we describe how to learn a policy through weighted imitation learning on the selected mode.
+
+Algorithm 1 Weighted imitation learning on one mode (LOM).  
+Initialise: A MDN behaviour policy  $\pi_{\rho}$  with parameter  $\rho$  , a target policy  $\pi_{\theta}$  with parameter  $\theta$  , a value function  $Q_{\psi}$  with parameter  $\psi$  and its slowly-updated copy  $Q_{\psi -}$  , a behaviour hyper Q-function  $Q_{\phi}$  with parameter  $\phi$  ; an offline dataset  $\mathcal{D}$    
+1: # Learn the MDN behaviour policy.   
+2: for  $i = 1,\dots,I_M$  do   
+3: Update  $\rho$  by minimising  $\mathcal{L}(\rho) = \mathbb{E}_{(s_t,a_t)\sim \mathcal{D}}[\log \pi_\rho (a_t\mid s_t)]$  .  $\triangleright$  Eq. 15.   
+4: end for   
+5: for  $i = 1,\ldots ,I_G$  do   
+6: Sample transitions  $\pmb {\tau} = \{s_t,a_t,r_t,s_{t + 1},a_{t + 1}\} \sim \mathcal{D}$    
+7: # Learn the value function.   
+8: Update  $\psi$  by minimising  $\mathcal{L}(\psi) = \mathbb{E}_{\tau}[(Q_{\psi}(s_t,a_t) - (r_t + \gamma Q_{\psi^{-}}(s_{t + 1},a_{t + 1}))^2 ]$  .  $\triangleright$  Eq. 16   
+9: # Learn behavioural hyper Q-function.   
+10: Update  $\phi$  by minimising  $\mathcal{L}(\phi) = \mathbb{E}_{s_t\sim \mathcal{D},u\sim \mathrm{Uniform}(\mathcal{A}_H)}[(Q_\phi (s_t,u) - \mathbb{E}_{a'_t\sim \phi^u (\cdot |s)}[Q_\psi (s_t,a'_t)])^2 ]$    
+11: # Learn the target policy.   
+12: Select the optimal mode  $u = \arg \max_{u\in \{1,\dots,M\}}Q_{\phi}(s_t,u)$    
+13: Sample action from the optimal mode  $\hat{a}_t\sim \phi^u (s_t)$    
+14: Get the mean of the mode  $\bar{a}_t = \phi_\mu^u (s_t)$ $\triangleright$  The mean of the mode   
+15: Estimate the advantage  $A(s_{t},\hat{a}_{t}) = Q_{\psi}(s_{t},\hat{a}_{t}) - Q_{\psi}(s_{t},\bar{a}_{t})$    
+16: Update  $\theta$  by minimising  $\mathcal{L}(\theta) = -\mathbb{E}_{s_t,\hat{a}_t}[\exp (A(s_t,\hat{a}_t))\log \pi_\theta (\hat{a}_t\mid s_t)]$  .  $\triangleright$  Eq. 18   
+17: # Update the target network.   
+18: if i mode update_delay  $= 0$  then   
+19:  $\psi^{-}\gets \rho \psi^{-} + (1 - \rho)\psi$    
+20: end if   
+21: end for
+
+# 5.1 BEHAVIOUR POLICY MODELLING
+
+At the start of the LOM algorithm, we model the behaviour policy using a mixture density network (MDN) (Bishop, 1994). The MDN receives a state  $s$  and represents the resulting action distribution with a GMM, including its mixing coefficients, locations, and scales. We index the Gaussian components as the hyper-actions, ranging from 1 to  $M$ .
+
+In the algorithm, we learn a neural network  $\pi_{\rho}(a|s)$  parameterised by  $\rho$  to estimate the behaviour policy. The network produces the parameters  $\{z_i^\mu ,z_i^\sigma ,z^{\alpha^i}\}_{i = 1}^M$ , which are used to estimate the probability density function for each policy mode:
+
+$$
+\phi^ {i} (a \mid s) = \frac {1}{\sqrt {2 \pi} \sigma_ {i} (s)} \exp \left(- \frac {\left| \left| a - \mu_ {i} (s) \right| \right| ^ {2}}{2 \sigma_ {i} (s) ^ {2}}\right), \tag {14}
+$$
+
+where  $\mu_i(s) = z_i^\mu$ , and  $\sigma_i(s) = \exp(z_i^\sigma)$ . The mixing coefficient  $\alpha^i(s)$  is processed through a softmax function:  $\alpha^i(s) = \exp(z^{\alpha^i}) / \sum_{j=1}^{M} \exp(z^{\alpha^j})$ . The parameters of the network are updated by minimising the negative log-likelihood:
+
+$$
+\mathcal {L} (\rho) = - \mathbb {E} _ {(s, a) \sim \mathcal {D}} [ \log \pi_ {\rho} (a \mid s) ]. \tag {15}
+$$
+
+LOM trains the MDN until convergence, after which the parameter  $\rho$  is fixed for the remainder of the algorithm.
+
+# 5.2 HYPER Q-FUNCTION LEARNING
+
+We learn the hyper Q-function of the behavioural hyper-policy based on Proposition 1. Specifically, we first learn a Q-function,  $Q^{\pi_b}$ , of the behaviour policy, and then estimate its expectation over actions sampled from mode  $u$ . We instantiate  $Q^{\pi_b}$  with a neural network  $Q_{\psi}$  parameterised by  $\psi$  and updated to minimising the TD error:
+
+$$
+\mathcal {L} (\psi) = \mathbb {E} _ {\left(s _ {t}, a _ {t}, r _ {t}, s _ {t + 1}, a _ {t + 1}\right) \sim \mathcal {D}} \left[ \left(Q _ {\psi} \left(s _ {t}, a _ {t}\right) - \left(r _ {t} + \gamma Q _ {\psi^ {-}} \left(s _ {t + 1}, a _ {t + 1}\right)\right)\right) ^ {2} \right]. \tag {16}
+$$
+
+where  $Q_{\psi^{-}}$  is a slowly updated copy of  $Q_{\psi}$ . Following Proposition 1, we learn  $Q_H^{c_b}$ , a hyper Q-function of the behavioural hyper-policy, which is denoted as  $Q_{\phi}$  parameterised by  $\phi$ . The optimisation objective of learning  $Q_{\phi}$  is:
+
+$$
+\mathcal {L} (\phi) = \mathbb {E} _ {s _ {t} \sim \mathcal {D}, u \sim \operatorname {U n i f o r m} \left(\mathcal {A} _ {H}\right)} \left[ \left(Q _ {\phi} \left(s _ {t}, u\right) - \mathbb {E} _ {a _ {t} \sim \phi^ {u} (\cdot | s)} \left[ Q _ {\psi} \left(s _ {t}, a _ {t}\right) \right]\right) ^ {2} \right]. \tag {17}
+$$
+
+With this hyper Q-function, we get a greedy hyper-policy  $\zeta_g(s) = \arg \max_{u\in \mathcal{A}_H}Q_\phi (s,u)$ . Corresponding to  $\zeta_g(s)$ , we get the greedy policy with one mode,  $\pi_{\zeta_g}(a\mid s) = \phi^{\zeta_g(s)}(a\mid s)$ .
+
+Table 1: Averaged normalised scores on D4RL benchmarks over 4 random seeds. 0 represents the performance of a random policy and 100 represents the performance of an expert policy. Standard deviations are provided as subscripts for multi-modal handling algorithms. The highest mean is highlighted in bold.  
+
+<table><tr><td>Dataset</td><td>Env</td><td>OneStep</td><td>AWAC</td><td>CQL</td><td>TD3BC</td><td>LAPO</td><td>DMPO</td><td>WCGAN</td><td>WCVAE</td><td>LOM</td></tr><tr><td rowspan="3">medium</td><td>halfcheetah</td><td>50.4</td><td>47.9</td><td>47.0</td><td>48.3</td><td>45.9±0.3</td><td>47.5±0.4</td><td>48.2±1.3</td><td>50.5±1.1</td><td>51.0±0.7</td></tr><tr><td>hopper</td><td>87.5</td><td>59.8</td><td>53.0</td><td>59.3</td><td>51.6±3.2</td><td>71.2±6.5</td><td>78.6±2.4</td><td>89.0±2.0</td><td>100.8±1.4</td></tr><tr><td>walker2d</td><td>84.8</td><td>83.1</td><td>73.3</td><td>83.7</td><td>80.7±0.8</td><td>79.4±4.7</td><td>82.4±1.7</td><td>85.0±1.5</td><td>85.1±0.7</td></tr><tr><td rowspan="3">medium-replay</td><td>halfcheetah</td><td>42.7</td><td>44.8</td><td>45.5</td><td>44.6</td><td>44.7±0.3</td><td>45.2±0.8</td><td>42.3±4.2</td><td>45.0±0.9</td><td>48.8±0.7</td></tr><tr><td>hopper</td><td>98.5</td><td>69.8</td><td>88.7</td><td>60.9</td><td>58.6±3.8</td><td>89.2±8.1</td><td>90.3±3.1</td><td>99.0±2.1</td><td>99.2±1.1</td></tr><tr><td>walker2d</td><td>61.7</td><td>78.1</td><td>81.8</td><td>81.8</td><td>71.7±5.2</td><td>82.1±3.8</td><td>72.6±4.0</td><td>74.0±3.1</td><td>84.8±1.0</td></tr><tr><td rowspan="3">medium-expert</td><td>halfcheetah</td><td>75.1</td><td>64.9</td><td>75.6</td><td>90.7</td><td>93.0±1.0</td><td>91.1±3.4</td><td>76.5±3.1</td><td>80.0±1.7</td><td>92.7±1.3</td></tr><tr><td>hopper</td><td>108.6</td><td>100.1</td><td>105.6</td><td>98.0</td><td>105.2±4.7</td><td>78.4±19.0</td><td>110.0±2.4</td><td>109.0±1.6</td><td>110.1±1.4</td></tr><tr><td>walker2d</td><td>111.3</td><td>110.0</td><td>107.9</td><td>110.1</td><td>111.1±0.2</td><td>109.9±0.4</td><td>99.7±1.0</td><td>111.0±0.7</td><td>111.3±0.8</td></tr><tr><td rowspan="3">expert</td><td>halfcheetah</td><td>88.2</td><td>81.7</td><td>96.3</td><td>96.7</td><td>95.9±0.2</td><td>97.0±1.0</td><td>90.7±1.7</td><td>89.0±0.8</td><td>95.2±0.4</td></tr><tr><td>hopper</td><td>106.9</td><td>109.5</td><td>96.5</td><td>107.8</td><td>106.7±3.6</td><td>93.6±15.1</td><td>107.3±1.8</td><td>108.0±1.3</td><td>111.0±0.8</td></tr><tr><td>walker2d</td><td>110.7</td><td>110.1</td><td>108.5</td><td>110.2</td><td>112.2±0.1</td><td>111.4±0.3</td><td>109.3±1.4</td><td>111.0±1.4</td><td>109.3±1.1</td></tr><tr><td rowspan="3">full-replay</td><td>halfcheetah</td><td>64.7</td><td>66.6</td><td>73.6</td><td>71.7</td><td>74.2±1.3</td><td>71.4±1.2</td><td>64.2±3.1</td><td>66.0±3.4</td><td>76.6±1.2</td></tr><tr><td>hopper</td><td>69.8</td><td>100.1</td><td>98.2</td><td>76.8</td><td>100.0±3.3</td><td>101.5±5.9</td><td>80.2±5.3</td><td>86.5±4.4</td><td>102.0±2.7</td></tr><tr><td>walker2d</td><td>67.1</td><td>78.3</td><td>92.7</td><td>90.2</td><td>96.2±2.8</td><td>95.4±1.8</td><td>53.0±5.4</td><td>72.0±3.7</td><td>97.9±0.9</td></tr></table>
+
+# 5.3 POLICY LEARNING
+
+Finally, we learn the policy  $\pi_{\theta}$  by maximising the objective:
+
+$$
+J (\theta) = \mathbb {E} _ {s \sim \mathcal {D}, a \sim \pi_ {\zeta_ {g}} (\cdot | s)} \left[ \exp_ {\operatorname {c l i p}} \left(\frac {1}{\beta} A ^ {\pi_ {\zeta_ {g}}} (s, a)\right) \log \pi_ {\theta} (a \mid s) \right], \tag {18}
+$$
+
+where  $\beta$  is a hyper-parameter,  $\exp_{clip}(\cdot)$  is the exponential function with a clipped range  $(0,C]$ , where  $C$  is a positive number for numerical stability. We estimate the advantage using:  $A^{\pi_{\zeta_g}}(s,a)\approx Q_\psi (s,a) - \mathbb{E}_{a\sim \pi_{\zeta_g}(\cdot |s)}[Q_\psi (s,a)]$ , where we use the Q-function conditioned on the behaviour policy  $\pi_b$  rather than  $\pi_{\zeta_g}$  in order to reduce extrapolation error and computational complexity. Further, we employ the approach used in (Mao et al., 2023) to replace the expectation over  $a\sim \pi_{\zeta_g}(\cdot |s)$  with the Q-value of the mean action of the Gaussian policy  $\pi_{\zeta_g}$  to reduce computation.
+
+Implementation details and the pseudocode can be found in Algorithm 1. The algorithm starts with training a MDN to model the behaviour policy, and then iteratively update the hyper Q-function  $Q_{\phi}$ , the value function  $Q_{\psi}$  and the target policy  $\pi_{\theta}$ . The code has been open sourced<sup>1</sup>.
+
+# 6 EXPERIMENTAL RESULTS
+
+We aim to answer the following questions: (1) How does LOM compare to SOTA offline RL algorithms, particularly those handling multi-modality? (2) Can performance improvements be attributed to learning on one mode? (3) How does the number of Gaussian components affect LOM's performance?
+
+# 6.1 LOM ACHIEVES SOTA PERFORMANCE IN D4RL BENCHMARK
+
+We evaluate LOM on three MuJoCo locomotion tasks from the D4RL benchmark (Fu et al., 2020): halfcheetah, hopper, and walker2d. Each environment contains five dataset types: (i) medium — 1M samples from a policy trained to approximately one-third of expert performance; (ii) medium-replay — the replay buffer of a policy trained to match the performance of the medium agent (0.2M for halfcheetah, 0.4M for hopper, 0.3M for walker2d); (iii) medium-expert — a 50-50 split of medium and expert data (just under 2M samples); (iv) expert — 1M samples from a fully trained SAC policy (Haarnoja et al., 2018); and (v) full-replay — 1M samples from the final replay buffer of an expert policy. Notably, the medium-replay and full-replay datasets are highly multi-modal, as states may have multiple action labels from different policies.
+
+Table 1 shows the benchmark results of our method against SOTA offline algorithms: OneStep (Brandfonbrener et al., 2021), AWAC (Nair et al., 2021), CQL (Kumar et al., 2020), and TD3BC (Fujimoto & Gu, 2021). We also include four algorithms designed for handling multi-modality: LAPO (Chen et al., 2022), deterministic mixture policy optimisation (DMPO) (Osa et al., 2023),
+
+![](images/b592ed301039a700db88b54d8ccc6801d36b455263f64e589f55542bcad2d520.jpg)  
+(a) Environment
+
+![](images/7ad67de3601874e1e1b49a0ff1ec45fe9379a881bba524db4f4653e1ea660ff7.jpg)  
+(b) BC
+
+![](images/4548c38858e61c2391bfd7185c839a2580675dc2d62a387d122af711a3c482e9.jpg)  
+(c) MDN  $(\mathbf{M} = 4)$  
+Figure 2: Comparative study in the FetchReach task with highly multi-modal datasets. (a) The FetchReach robot is tasked with reaching one of four specified goals using an expert dataset. The robot arm receives a reward of 2 for reaching the goal in the first quadrant and 1 for reaching any of the other three goals. The dataset contains actions directed toward all four goals, with conflicting directions. (b) Action distribution learned by behaviour cloning using a unimodal Gaussian policy model. (c) Action distribution learned by MDN using a GMM policy model. (d) Action distribution learned by AWAC, which applies weighted imitation learning over the entire action distribution using a unimodal Gaussian policy. (e) Action distribution learned by LOM.
+
+![](images/76194a2b43d131abba815062f0dd3a9d6561c76393e9abcf262de1858e8d008f.jpg)  
+(d) AWAC
+
+![](images/efb2192e6ca3d24dba08f68007797947389650ca4185ff0a3c5d087e97cb8507.jpg)  
+(e) LOM  $(\mathbf{M} = 4)$
+
+weighted conditioned GAN (WCGAN), and weighted conditioned VAE (WCVAE) (Wang et al., 2024b). The experimental results are shown in Table 1, where the baselines' results are mainly sourced from Mao et al. (2023) or reproduced using their official implementations. The results demonstrate that LOM outperforms the baseline algorithms in 12 out of 15 tasks. In tasks with multi-modal datasets, LOM surpasses all the baselines, and achieves a performance improvement ranging from  $0.2\%$  to  $7.9\%$  over the SOTA results.
+
+# 6.2 LEARNING ON ONE MODE DRIVES IMPROVEMENTS
+
+To explore the source of LOM's improvements, we design three tasks with highly multi-modal datasets: FetchReach, FetchPush, and FetchPickAndPlace. In FetchReach, as shown in Figure 2a, the objective is to control a robotic arm to reach one of four specified positions, symmetrically distributed on the  $xy$ -plane. The robot receives a reward of 2 for reaching the position in the first quadrant, 1 for reaching any of the other positions, and 0 otherwise. In FetchPush and FetchPickAndPlace, the robot is tasked with moving a cube to one of four positions on a desk, with the same reward structure. Each dataset contains  $4 \times 10^6$  transitions collected by an expert policy. Given a state and four goals in different quadrants, the dataset contains four trajectories corresponding to each goal, making the datasets highly multi-modal.
+
+We compare LOM with behaviour cloning (BC), AWAC, and MDN, introduced in Section 5.1. Specifically, BC and MDN imitate the behaviour policy using a unimodal Gaussian policy and a multi-modal Gaussian policy, respectively. AWAC applies weighted imitation learning to imitate the behaviour policy across all modes (Nair et al., 2021). LOM, however, distinguishes multiple behaviour modes using MDN and focuses on learning from the best mode. Table 2 shows that LOM outperforms the baseline algorithms by an average of  $53\%$ .
+
+In Figure 2, we further investigate the behaviour of different policies. Given a state, the multi-modal dataset contains four actions, each targeting distinct goals in separate quadrants. Figure 2b shows that BC learns an averaged action distribution. Figure 2c demonstrates that MDN effectively reconstructs the action distribution, allowing LOM to select the highest-rewarding mode. Figure 2d shows that while AWAC can learn the best mode (in the first quadrant), it tends to generate OOD actions. In contrast, Figure 2e illustrates that LOM imitates actions from the best single mode, filtering out suboptimal actions and focusing on highly-rewarding ones. These experiments demonstrate that learning on one mode effectively improves policy performance by concentrating on the most rewarding actions.
+
+Table 2: Average scores on the tasks with extremely multi-modal datasets.  
+
+<table><tr><td></td><td>BC</td><td>MDN</td><td>AWAC</td><td>WCGAN</td><td>WCVAE</td><td>LOM</td></tr><tr><td>FetchReach</td><td>14.3±3.4</td><td>17.1±2.3</td><td>32.4±0.2</td><td>42.3±3.1</td><td>40.8±2.1</td><td>47.2±3.1</td></tr><tr><td>FetchPush</td><td>11.8±0.9</td><td>18.9±4.7</td><td>26.8±1.4</td><td>41.2±3.7</td><td>41.8±2.4</td><td>44.3±2.7</td></tr><tr><td>FetchPick</td><td>3.6±0.8</td><td>8.1±1.3</td><td>22.7±2.3</td><td>29.2±2.8</td><td>30.5±2.1</td><td>34.2±3.6</td></tr></table>
+
+![](images/6ce24d47b2dd71b5672b0be44faff55b3ad0121ff50c7663c2b080e9a7f7caea.jpg)  
+(a)  $\mathbf{M} = 2$
+
+![](images/814b1b212a8d6639175c498749e52143fc1a717ce67e642fe922a4c7ee791414.jpg)  
+(b)  $\mathrm{M} = 4$
+
+![](images/1b953cfe02e05c1eff0c95193694825d5de747b45a6dcd9b2b52de625fea42a9.jpg)  
+(c)  $\mathrm{M} = {10}$  
+Figure 4: Action modes learned by MDN with varying numbers of mixtures. Red dots represent samples from the highest-reward mode. The original actions are clustered around  $(1,1)$ ,  $(1,-1)$ ,  $(-1,-1)$ , and  $(-1,1)$  but do not extend beyond these points. In (d) and (e), the red dots collapse into a single point in the first quadrant due to the small standard deviation of the mode.
+
+![](images/c4f0d32254b2e672c4fde536e7039df1a7740f2031cd6617e7d840171010539c.jpg)  
+(d)  $\mathrm{M} = {20}$
+
+![](images/60f86da6f36d871a22af36828544ec99855da6d4197625d75b93ab947ab8be25.jpg)  
+(e)  $\mathrm{M} = {50}$
+
+# 6.3 EFFECT OF THE NUMBER OF GAUSSIAN COMPONENTS
+
+We analyse the influence of the number  $M$  of Gaussian components in the LOM algorithm. The results, shown in Figure 4, are based on the multi-modal FetchReach environment. The figure illustrates the decomposition of action modes using blue contours, while red dots represent the actions imitated by LOM. When  $M$  is too large (Figure 4d and 4e), the standard deviation of the imitated Gaussian mode becomes excessively small, leading to overly narrow the domains of action learning. Conversely, smaller values of  $M$  results in modes with a large standard deviation (Figure 4a), causing LOM to capture actions outside the dataset and imitate OOD actions. This highlights the need to investigate LOM's sensitivity to this hyper-parameter.
+
+Figure 3 compares LOM across  $M = \{1,5,10,15,20\}$  in the medium-replay and full-replay datasets from the D4RL benchmark. The performance generally improves as the number of mixture components increases, particularly in the medium-replay dataset, where larger  $M$  values lead to continuous performance gains. However, in the full-replay datasets, performance increases only up to  $M = 10$ . This indicates that while increasing the number of components benefits performance, there
+
+may be diminishing returns beyond a certain threshold. Our experiments suggest that  $M$  primarily influences the domain of the imitated actions, and improvements can be achieved with a moderate, though not highly specific, value of  $M$ .
+
+![](images/ba2b95dd3f7f077a2198bbd2a53cf76f1b3d677d2812f0552271f69d1f23849c.jpg)  
+Figure 3: Normalised scores for varying numbers of Gaussian components in the medium replay and full replay datasets.
+
+![](images/5be2e71b5d3612c51c9b7581cac9cf683f18f46f2930f2865108574800e5cbe3.jpg)
+
+# 7 CONCLUSIONS AND DISCUSSIONS
+
+In this paper, we introduced LOM, a novel offline RL method specifically designed to tackle the challenge of multi-modality in offline RL datasets. Unlike existing methods that aim to model the entire action distribution, LOM focuses on imitating the highest-rewarded action mode. Through extensive experiments, we demonstrated that LOM achieves SOTA performance, and we attributed these improvements to the central idea of learning on one mode, which simplifies the learning process while maintaining robust performance.
+
+We observed that the hyper Q-function satisfies a Bellman-like equation, opening the possibility for learning it through Bellman updates. However, one challenge that emerges is the difficulty in accurately estimating the Bellman target for the hyper Q-function, as it requires computing the expectation over all possible hyper-actions, which may lead to extrapolation errors. Exploring a deterministic hyper-policy is a promising future direction to mitigate this issue, potentially simplifying the estimation process and further improving performance. Furthermore, the hyperparameter  $M$  in LOM is intrinsically linked to the multi-modality present in the dataset. This parameter can be estimated using statistical techniques like bump hunting (Friedman & Fisher, 1999) and peak finding (Du et al., 2006), eliminating the need for fine-tuning. However, a key challenge lies in the fact that each state corresponds to a distinct multi-modal action distribution, making it computationally intensive to identify and count the modes for every individual state.
+
+Acknowledgments We acknowledge support from a UKRI AI Turing Acceleration Fellowship (EPSRC EP/V024868/1).
+
+# REFERENCES
+
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+Gaon An, Seungyong Moon, Jang-Hyun Kim, and Hyun Oh Song. Uncertainty-based offline reinforcement learning with diversified Q-ensemble. In Advances in Neural Information Processing Systems, 2021.  
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+Alex Beeson and Giovanni Montana. Balancing policy constraint and ensemble size in uncertainty-based offline reinforcement learning. Machine Learning, 113(1):443-488, Jan 2024. ISSN 1573-0565.  
+Christopher M. Bishop. Mixture density networks. Technical report, Aston University, Birmingham, 1994.  
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+Xi Chen, Ali Ghadirzadeh, Tianhe Yu, Jianhao Wang, Alex Yuan Gao, Wenzhe Li, Liang Bin, Chelsea Finn, and Chongjie Zhang. LAPO: Latent-variable advantage-weighted policy optimization for offline reinforcement learning. In Advances in Neural Information Processing Systems, 2022.  
+Pan Du, Warren A. Kibbe, and Simon M. Lin. Improved peak detection in mass spectrum by incorporating continuous wavelet transform-based pattern matching. Bioinformatics, 22(17): 2059-2065, 07 2006. ISSN 1367-4803.  
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+Justin Fu, Aviral Kumar, Ofir Nachum, George Tucker, and Sergey Levine. D4RL: Datasets for deep data-driven reinforcement learning, 2020.  
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+Sergey Levine, Aviral Kumar, George Tucker, and Justin Fu. Offline reinforcement learning: Tutorial, review, and perspectives on open problems, 2020.
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+Yecheng Jason Ma, Jason Yan, Dinesh Jayaraman, and Osbert Bastani. Offline goal-conditioned reinforcement learning via f-advantage regression. In Advances in Neural Information Processing Systems, 2022.  
+Yixiu Mao, Hongchang Zhang, Chen Chen, Yi Xu, and Xiangyang Ji. Supported trust region optimization for offline reinforcement learning. In International Conference on Machine Learning, 2023.  
+Geoffrey J McLachlan and Kaye E Basford. Mixture models: Inference and applications to clustering, volume 38. M. Dekker New York, 1988.  
+Ashvin Nair, Abhishek Gupta, Murtaza Dalal, and Sergey Levine. AWAC: Accelerating online reinforcement learning with offline datasets, 2021.  
+Takayuki Osa, Akinobu Hayashi, Pranav Deo, Naoki Morihira, and Takahide Yoshiike. Offline reinforcement learning with mixture of deterministic policies. Transactions on Machine Learning Research, 2023. ISSN 2835-8856.  
+Xue Bin Peng, Aviral Kumar, Grace Zhang, and Sergey Levine. Advantage-weighted regression: Simple and scalable off-policy reinforcement learning, 2019.  
+John Schulman, Sergey Levine, Pieter Abbeel, Michael Jordan, and Philipp Moritz. Trust region policy optimization. In International Conference on Machine Learning, 2015.  
+Noah Siegel, Jost Tobias Springenberg, Felix Berkenkamp, Abbas Abdelmaleki, Michael Neunert, Thomas Lampe, Roland Hafner, Nicolas Heess, and Martin Riedmiller. Keep doing what worked: Behavior modelling priors for offline reinforcement learning. In International Conference on Learning Representations, 2020.  
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+Mianchu Wang, Rui Yang, Xi Chen, Hao Sun, Meng Fang, and Giovanni Montana. GOPlan: Goal-conditioned offline reinforcement learning by planning with learned models. Transactions on Machine Learning Research, 2024b. ISSN 2835-8856.  
+Qing Wang, Jiechao Xiong, Lei Han, peng sun, Han Liu, and Tong Zhang. Exponentially weighted imitation learning for batched historical data. In Advances in Neural Information Processing Systems, 2018.  
+Zhendong Wang, Jonathan J Hunt, and Mingyuan Zhou. Diffusion policies as an expressive policy class for offline reinforcement learning. In International Conference on Learning Representations, 2023.  
+Ziyu Wang, Alexander Novikov, Konrad Zolna, Josh S Merel, Jost Tobias Springenberg, Scott E Reed, Bobak Shahriari, Noah Siegel, Caglar Gulcehre, Nicolas Heess, and Nando de Freitas. Critic regularized regression. In Advances in Neural Information Processing Systems, 2020.  
+Yifan Wu, George Tucker, and Ofir Nachum. Behavior regularized offline reinforcement learning, 2019.
+
+Rui Yang, Yiming Lu, Wenzhe Li, Hao Sun, Meng Fang, Yali Du, Xiu Li, Lei Han, and Chongjie Zhang. Rethinking goal-conditioned supervised learning and its connection to offline RL. In International Conference on Learning Representations, 2022a.  
+Rui Yang, Lin Yong, Xiaoteng Ma, Hao Hu, Chongjie Zhang, and Tong Zhang. What is essential for unseen goal generalization of offline goal-conditioned RL? In International Conference on Machine Learning, 2023.  
+Shentao Yang, Zhendong Wang, Huangjie Zheng, Yihao Feng, and Mingyuan Zhou. A behavior regularized implicit policy for offline reinforcement learning, 2022b.  
+Tianhe Yu, Garrett Thomas, Lantao Yu, Stefano Ermon, James Y Zou, Sergey Levine, Chelsea Finn, and Tengyu Ma. MOPO: Model-based offline policy optimization. In Advances in Neural Information Processing Systems, 2020.  
+Wenxuan Zhou, Sujay Bajracharya, and David Held. PLAS: Latent action space for offline reinforcement learning. In Conference on Robot Learning, 2020.
+
+# A MOTIVATING EXAMPLE
+
+To illustrate the challenge of capturing complex multi-modal action distributions, we present a toy example in Fig. 5. In this example, we generate a dataset where each state is associated with multiple valid actions, simulating the multi-modal nature of real-world offline RL datasets. We evaluate four models — Gaussian, Conditional Variational Auto-Encoder (CVAE) (Zhou et al., 2020; Chen et al., 2022), Conditional Generative Adversarial Networks (CGAN) (Yang et al., 2022b; Wang et al., 2024b), and Mixture Density Network (MDN) (Bishop, 1994) — on their ability to model the multi-modal action distribution.
+
+Subfigures 5 (a-e) show the action distributions learned by each model. Specifically, we visualise how well each model captures distinct modes in the action space. The Gaussian model fails to separate modes, while the CVAE, CGAN, and MDN demonstrate stronger mode separation, with the MDN showing minimal mode overlap and the clearest boundaries between modes.
+
+Additionally, we assess each model's ability to capture high-reward actions. Subfigures 5 (f-j) illustrate the models' performance in learning positively-rewarded multi-modal action distributions. Our results reveal that the MDN with ranked components closely adheres to in-distribution actions and achieves superior outcomes in terms of reward, compared to the other models.
+
+![](images/d8034baf7f56cf8bff2e496ac60963f345450b9a83bbaccd96ae7be5932a6c07.jpg)  
+(a)
+
+![](images/a82dbc0af0a6b095799bd96d9f528e52358d92e5dfdbbaa24d5027a0867a6459.jpg)  
+(b)
+
+![](images/c4450d2926b2ef7b0b86c22db1e14a0b1c733a84a22a735b4f38d4d8e1e45f15.jpg)  
+(c)
+
+![](images/a24fb229e8ff56bd4a3a261d79c0c84fe998cae6914bd44786ffe5aa85ef854a.jpg)  
+(d)
+
+![](images/aad7b0c90819aa2b0be1312a5c3bb6155fc93ef8f9ba18f48956cf5624413570.jpg)  
+(e)
+
+![](images/ff146dd97281ce6c84ff7a01f5b13b961a0f41c06cecfe0bfd08863b46436575.jpg)  
+(f)
+
+![](images/81601e8cb7a26dd1e708cd8bd4d7746a8eb324a9dd672710e01c88697fcdd77b.jpg)  
+(g)
+
+![](images/25a664c1ed88529393b68404a9ebed1fd5a323a6d0351c1427b011d1f5d411a5.jpg)  
+(h)
+
+![](images/473fa6e382679a38ad387a9310f92d279c49d99c8e3ccff8a906c96529bfbb5c.jpg)  
+(i)  
+Figure 5: An example of modelling the multi-modal behaviour policy in a one-step MDP. The  $x$ -axis represents the state, and the  $y$ -axis represents the corresponding multi-modal actions. (a) shows the action distribution from the offline dataset. (b)-(e) illustrate the action distributions learned by a Gaussian model, a conditional VAE, a conditional GAN, and an MDN, respectively. (f) shows the action distribution of the offline dataset with rewards, where actions in the first and third quadrants receive a reward of 1, and others receive 0. (g)-(i) illustrate the action distributions learned by weighted Gaussian, weighted conditional VAE, and weighted conditional GAN models, respectively. (j) shows the action distribution learned by the top 10 of the 20 MDN components, ranked by a hyper Q-function.
+
+![](images/f95bb57b5e2e1b1e2b50af4d3bb478c3bb426411370f3c09b1a857fcc421b58b.jpg)  
+(j)
+
+# B THEORETICAL RESULTS
+
+# B.1 PROOF OF PROPOSITION 1
+
+Proposition 1 The hyper Q-function can be linked to the standard value function  $Q^{\pi}(s,a)$  via:
+
+$$
+Q _ {H} ^ {\zeta} (s, u) = \mathbb {E} _ {a \sim \phi^ {u} (\cdot | s)} \left[ Q ^ {\pi_ {\zeta}} (s, a) \right]. \tag {19}
+$$
+
+Proof:
+
+$$
+\begin{array}{l} Q _ {H} ^ {\zeta} (s, u) = \mathbb {E} _ {s _ {1}, u _ {1}, \dots \sim \zeta} \left[ \sum_ {t = 0} ^ {T} \gamma^ {t} r _ {H} \left(s _ {t}, u _ {t}\right) \mid s _ {0} = s, u _ {0} = u \right] \\ = r _ {H} (s, u) + \sum_ {t = 0} ^ {T - 1} \sum_ {s, u} p \left(s _ {t + 1} = s \mid \zeta\right) \zeta (u \mid s) \gamma^ {t + 1} r _ {H} (s, u) \\ = \sum_ {a} \phi^ {u} (a \mid s) r (s, a) + \sum_ {t = 0} ^ {T - 1} \sum_ {s} p \left(s _ {t + 1} = s \mid \zeta\right) \sum_ {u} \zeta (u \mid s) \gamma^ {t + 1} \sum_ {a} \phi^ {u} (a \mid s) r (s, a) \\ = \sum_ {a} \phi^ {u} (a \mid s) r (s, a) + \sum_ {t = 0} ^ {T - 1} \sum_ {s} p \left(s _ {t + 1} = s \mid \zeta\right) \sum_ {u, a} \zeta (u \mid s) \phi^ {u} (a \mid s) \gamma^ {t + 1} r (s, a) \\ = \sum_ {a} \phi^ {u} (a \mid s) r (s, a) + \sum_ {t = 0} ^ {T - 1} \sum_ {s, a} p (s _ {t + 1} = s \mid \zeta) \pi_ {\zeta} (a \mid s) \gamma^ {t + 1} r (s, a) \\ = \sum_ {a} \phi^ {u} (a \mid s) \left[ r (s, a) + \sum_ {t = 0} ^ {T - 1} \sum_ {s, a} p \left(s _ {t + 1} = s \mid \zeta\right) \pi_ {\zeta} (a \mid s) \gamma^ {t + 1} r (s, a) \right] \\ = \sum_ {a} \phi^ {u} (a \mid s) \left[ r (s, a) + \sum_ {t = 0} ^ {T - 1} \sum_ {s, a} p \left(s _ {t + 1} = s \mid \pi_ {\zeta}\right) \pi_ {\zeta} (a \mid s) \gamma^ {t + 1} r (s, a) \right] \\ = \mathbb {E} _ {a \sim \phi^ {u} (\cdot | s)} \left[ Q ^ {\pi_ {\zeta}} (s, a) \right]. \tag {20} \\ \end{array}
+$$
+
+# B.2 PROOF OF THEOREM 1
+
+Theorem 1. The composite policy induced by the greedy hyper-policy improves upon the behaviour policy:
+
+$$
+V ^ {\pi_ {\mathcal {C} _ {g}}} (s) \geq V ^ {\pi_ {b}} (s), \quad \forall s \in \mathcal {S}.
+$$
+
+Proof. Now, we prove that the improved policy  $\pi_{\zeta_g}$  is uniformly as good as or better than the behaviour policy  $\pi_b$ .
+
+$$
+\begin{array}{l} V ^ {\pi_ {\zeta_ {g}}} (s _ {t}) = \mathbb {E} _ {a _ {t} \sim \pi_ {\zeta_ {g}}} [ r (s _ {t}, a _ {t}) + \gamma \mathbb {E} _ {s _ {t + 1}, a _ {t + 1} \sim \pi_ {\zeta_ {g}}} [ Q (s _ {t + 1}, a _ {t + 1}) ] ] \\ = \mathbb {E} _ {a _ {t} \sim \pi_ {\zeta_ {g}}} [ r (s _ {t}, a _ {t}) + \gamma \mathbb {E} _ {s _ {t + 1}, a _ {t + 1} \sim \pi_ {\zeta_ {g}}} [ r (s _ {t + 1}, a _ {t + 1}) + \\ \gamma \mathbb {E} _ {s _ {t + 2}, a _ {t + 2} \sim \pi_ {\zeta_ {g}}} [ Q ^ {\pi_ {\zeta_ {g}}} (s _ {t + 2}, a _ {t + 2}) ] ]) ] \\ = \mathbb {E} _ {a _ {t} \sim \pi_ {\zeta g}} [ r (s _ {t}, a _ {t}) + \gamma \mathbb {E} _ {s _ {t + 1}, a _ {t + 1} \sim \pi_ {\zeta g}} [ r (s _ {t + 1}, a _ {t + 1}) + \dots + \\ \gamma \mathbb {E} _ {s _ {t + H - 1}, a _ {t + H - 1} \sim \pi_ {\zeta_ {g}}} [ r (s _ {t + H - 1}, a _ {t + H - 1}) + \gamma \mathbb {E} _ {s _ {t + H}, a _ {t + H} \sim \pi_ {\zeta_ {g}}} [ r (s _ {t + H}, a _ {t + H}) ]) ] \\ \geq \mathbb {E} _ {a _ {t} \sim \pi_ {\zeta_ {g}}} [ r (s _ {t}, a _ {t}) + \gamma \mathbb {E} _ {s _ {t + 1}, a _ {t + 1} \sim \pi_ {\zeta_ {g}}} [ r (s _ {t + 1}, a _ {t + 1}) + \dots + \\ \gamma \mathbb {E} _ {s _ {t + H - 1}, a _ {t + H - 1} \sim \pi_ {\zeta_ {g}}} [ r (s _ {t + H - 1}, a _ {t + H - 1}) + \gamma \mathbb {E} _ {s _ {t + H}, a _ {t + H} \sim \pi_ {\boldsymbol {b}}} [ r (s _ {t + H}, a _ {t + H}) ]) ] \\ = \mathbb {E} _ {a _ {t} \sim \pi_ {\zeta g}} [ r (s _ {t}, a _ {t}) + \gamma \mathbb {E} _ {s _ {t + 1}, a _ {t + 1} \sim \pi_ {\zeta g}} [ r (s _ {t + 1}, a _ {t + 1}) + \dots + \\ \gamma \mathbb {E} _ {s _ {t + H - 1}, a _ {t + H - 1} \sim \pi_ {\zeta g}} \left[ Q ^ {\pi_ {b}} \left(s _ {t + H - 1}, a _ {t + H - 1}\right) \right] ] ] (21) \\ \geq \mathbb {E} _ {a _ {t} \sim \pi_ {\zeta g}} [ r (s _ {t}, a _ {t}) + \gamma \mathbb {E} _ {s _ {t + 1}, a _ {t + 1} \sim \pi_ {\zeta g}} [ r (s _ {t + 1}, a _ {t + 1}) + \dots + \\ \left. \gamma \mathbb {E} _ {s _ {t + H - 1}, a _ {t + H - 1} \sim \pi_ {\boldsymbol {b}}} \left[ Q ^ {\pi_ {\boldsymbol {b}}} \left(s _ {t + H - 1}, a _ {t + H - 1}\right) \right] \right] (22) \\ > \dots \\ \geq \mathbb {E} _ {a _ {t} \sim \pi_ {\zeta g}} \left[ r \left(s _ {t}, a _ {t}\right) + \gamma \mathbb {E} _ {s _ {t + 1}, a _ {t + 1} \sim \tilde {\pi} _ {b}} \left[ Q ^ {\pi_ {b}} \left(s _ {t + 1}, a _ {t + 1}\right) \right] \right] \\ \geq \mathbb {E} _ {a _ {t} \sim \pi_ {b}} [ r (s _ {t}, a _ {t}) + \gamma \mathbb {E} _ {s _ {t + 1}, a _ {t + 1} \sim \tilde {\pi} _ {b}} [ Q ^ {\pi_ {b}} (s _ {t + 1}, a _ {t + 1}) ] ] ] \\ = V ^ {\pi_ {b}} \left(s _ {t}\right) \\ \end{array}
+$$
+
+The derivation from Eq. 21 to Eq. 22 is based on:
+
+$$
+\begin{array}{l} \mathbb {E} _ {s, a \sim \pi_ {\mathcal {C} g}} \left[ Q ^ {\pi_ {b}} (s, a) \right] \\ = \mathbb {E} _ {s, u \sim \zeta_ {g}} \mathbb {E} _ {a \sim \phi^ {u} (\cdot | s)} \left[ Q ^ {\pi_ {b}} (s, a) \right] \\ = \mathbb {E} _ {s} \mathbb {E} _ {u \sim \zeta_ {g}} \left[ Q _ {H} ^ {\zeta_ {b}} (s, u) \right] \tag {23} \\ \geq \mathbb {E} _ {s} \mathbb {E} _ {u \sim \zeta_ {b}} \left[ Q _ {H} ^ {\zeta_ {b}} (s, u) \right] \\ = \mathbb {E} _ {s, u \sim \zeta_ {b}} \mathbb {E} _ {a \sim \phi^ {u} (\cdot | s)} \left[ Q ^ {\pi_ {b}} (s, a) \right] \\ = \mathbb {E} _ {s, a \sim \pi_ {b}} \left[ Q ^ {\pi_ {b}} (s, a) \right] \\ \end{array}
+$$
+
+![](images/494484890acfd00e128fd53a7776485b0a30da37253b47f6808376895256683d.jpg)
+
+# B.3 PROOF OF THEOREM 2
+
+Theorem 2 LOM learns a policy  $\pi_L$ , which is uniformly as good as or better than the improved policy  $\pi_{\zeta_g}$  and the behaviour policy  $\pi_b$ . That is,
+
+$$
+\forall s \in \mathcal {S}, V ^ {\pi_ {L}} (s) \geq V ^ {\pi_ {\zeta_ {g}}} (s) \geq V ^ {\pi_ {b}} (s). \tag {24}
+$$
+
+Proof. The proof is structured in two parts. First, we show that the LOM-learned policy  $\pi_L$  is at least as good as the improved policy  $\pi_{\zeta_g}$ ; specifically, for all states  $s \in S$ , we establish that  $V^{\pi_L}(s) \geq V^{\pi_{\zeta_g}}(s)$ . Second, we prove that the improved policy  $\pi_{\zeta_g}$  performs no worse than the behaviour policy  $\pi_b$ , i.e.,  $V^{\pi_{\zeta_g}}(s) \geq V^{\pi_b}(s)$  for all  $s \in S$ .
+
+As established in Wang et al. (2018), the following lemma provides a sufficient condition for proving that one policy is no worse than another. We restate this result as the following lemma:
+
+Lemma (from Wang et al. (2018)): Suppose two policies  $\pi$  and  $\tilde{\pi}$  satisfy:
+
+$$
+g \left(\tilde {\pi} (a \mid s)\right) = g \left(\pi (a \mid s)\right) + h (s, A ^ {\pi} (s, a)), \tag {25}
+$$
+
+where  $g(\cdot)$  is a monotonically increasing function, and  $h(s,\cdot)$  is monotonically increasing for any fixed  $s$ . Then we have:
+
+$$
+V ^ {\bar {\pi}} (s) \geq V ^ {\pi} (s), \quad \forall s \in \mathcal {S}. \tag {26}
+$$
+
+This means that  $\tilde{\pi}$  is uniformly as good as or better than  $\pi$ .
+
+Since the LOM-learned policy imitates the optimal policy  $\pi^{*}$ , we have:
+
+$$
+\pi^ {*} (a \mid s) = \frac {1}{Z (s)} \pi_ {\zeta_ {g}} (a \mid s) \exp \left(\frac {1}{\beta} A ^ {\pi_ {\zeta_ {g}}} (s, a)\right), \tag {27}
+$$
+
+which gives:
+
+$$
+\log \pi_ {L} (a \mid s) = \log \frac {1}{Z (s)} + \log \pi_ {\zeta_ {g}} (a \mid s) + \frac {1}{\beta} A ^ {\pi_ {\zeta_ {g}}} (s, a), \tag {28}
+$$
+
+where  $\log (\cdot)$  is a monotonically increasing function and  $\log \frac{1}{Z(s)}$  is a constant. Therefore,  $\pi_L$  is uniformly as good as or better than the improved policy  $\pi_{\zeta_g}$ .
+
+# B.4 PROOF OF THEOREM 3
+
+Theorem 3 The LOM algorithm learns a policy  $\pi_L$ , whose improvement upon the composite policy induced by the greedy hyper-policy  $\pi_{\zeta_g}$  is bounded: for all  $s \in S$ :
+
+$$
+V ^ {\pi_ {L}} (s) - V ^ {\pi_ {\zeta_ {g}}} (s) \geq \frac {1}{1 - \gamma} \hat {\eta} (\pi_ {L}) - \frac {A _ {\max }}{1 - \gamma} \sqrt {\frac {1}{2} D _ {K L} \left(d _ {\pi_ {L}} \mid \mid d _ {\pi_ {\zeta_ {g}}}\right)}, \tag {29}
+$$
+
+where  $A_{\mathrm{max}} = \max_{s,a}|A^{\pi_{\zeta_g}}(s,a)|$ . The proof is provided in Appendix B.4.
+
+Proof. Following performance difference lemma (Kakade & Langford, 2002), we have
+
+$$
+V ^ {\pi_ {L}} (s) - V ^ {\pi_ {\zeta_ {g}}} (s) = \frac {1}{1 - \gamma} \mathbb {E} _ {s \sim d _ {\pi_ {L}}, a \sim \pi_ {L}} \left[ A ^ {\pi_ {\zeta_ {g}}} (s, a) \right]. \tag {30}
+$$
+
+Our goal is to express the right-hand side in terms of  $\hat{\eta} (\pi_L)$ , which uses  $d_{\pi_g}$  rather than  $d_{\pi_L}$ . To do this, we define the discrepancy between  $d_{\pi_L}$  and  $d_{\pi_{\zeta g}}$  as:
+
+$$
+\Delta (s) = d _ {\pi_ {L}} (s) - d _ {\pi_ {\zeta_ {g}}} (s). \tag {31}
+$$
+
+We can decompose the expectation over  $d_{\pi_L}$  as:
+
+$$
+\mathbb {E} _ {s \sim d _ {\pi_ {L}}} [ f (s) ] = \mathbb {E} _ {s \sim d _ {\pi_ {\zeta_ {g}}}} [ f (s) ] + \mathbb {E} _ {s \sim \Delta} [ f (s) ], \tag {32}
+$$
+
+where  $\mathbb{E}_{s\sim \Delta}[f(s)] = \sum_s\Delta (s)f(s)$
+
+Applying this to the expression:
+
+$$
+V ^ {\pi_ {L}} (s) - V ^ {\pi_ {\zeta_ {g}}} (s) = \frac {1}{1 - \gamma} \left(\mathbb {E} _ {s \sim d _ {\pi_ {\zeta_ {g}}}, a \sim \pi_ {L}} \left[ A ^ {\pi_ {\zeta_ {g}}} (s, a) \right] + \mathbb {E} _ {s \sim \Delta , a \sim \pi_ {L}} \left[ A ^ {\pi_ {\zeta_ {g}}} (s, a) \right]\right). \tag {33}
+$$
+
+The first term is exactly  $\hat{\eta} (\pi_L)$  scaled by  $\frac{1}{1 - \gamma}$ . We need to bound the error term:
+
+$$
+\epsilon = \frac {1}{1 - \gamma} \mathbb {E} _ {s \sim \Delta , a \sim \pi_ {L}} \left[ A ^ {\pi_ {\zeta_ {g}}} (s, a) \right] \tag {34}
+$$
+
+Assuming the advantage function  $A^{\pi \zeta_g}(s,a)$  is bounded:
+
+$$
+\left| A ^ {\pi \zeta_ {g}} (s, a) \right| \leq A _ {\max } \tag {35}
+$$
+
+We can then bound  $\epsilon$  by
+
+$$
+| \epsilon | \leq \frac {A _ {\operatorname* {m a x}}}{1 - \gamma} \sum_ {s} | \Delta (s) | = \frac {A _ {\operatorname* {m a x}}}{1 - \gamma} \| d _ {\pi_ {L}} - d _ {\pi_ {\zeta_ {g}}} \| _ {1} = \frac {A _ {\operatorname* {m a x}}}{1 - \gamma} D _ {T V} \left(d _ {\pi_ {L}} \| d _ {\pi_ {\zeta_ {g}}}\right) \tag {36}
+$$
+
+By applying Pinsker's inequity, we have
+
+$$
+D _ {T V} \left(d _ {\pi_ {L}} \mid \mid d _ {\pi_ {\zeta_ {g}}} \right. \leq \sqrt {\frac {1}{2} D _ {K L} \left(d _ {\pi_ {L}} \mid \mid d _ {\pi_ {\zeta_ {g}}}\right)} \tag {37}
+$$
+
+Finally, we have
+
+$$
+V ^ {\pi_ {L}} (s) \geq V ^ {\pi_ {\zeta_ {g}}} (s) + \frac {1}{1 - \gamma} \hat {\eta} \left(\pi_ {L}\right) - \frac {A _ {\max }}{1 - \gamma} \sqrt {\frac {1}{2} D _ {K L} \left(d _ {\pi_ {L}} \mid \mid d _ {\pi_ {\zeta_ {g}}}\right)} \tag {38}
+$$
+
+![](images/8689b5d3b1fdec9c7bdba838a5b5a5b859bf73ea1c641d795adeb8b11bf3c75a.jpg)
+
+# C FURTHER EXPERIMENTAL DETAILS
+
+# C.1 BASELINES
+
+In the experiments discussed in Section 6, we compare LOM with other multi-modal modelling approaches, such as WCGAN and WCVAE, in capturing complex action distributions in offline reinforcement learning. These models are evaluated on their ability to reconstruct and learn from the multi-modal behaviour policy in the FetchReach, FetchPush, and FetchPickAndPlace tasks, and across various D4RL benchmarks.
+
+Weighted Conditional GAN (WCGAN): The WCGAN learns a discriminator  $D$  and a generator  $\pi$  to optimise the following adversarial objectives:
+
+$$
+\max  _ {D} \min  _ {\pi} \quad \mathbb {E} _ {(s, a) \sim \mathcal {D}} [ w (s, a) \log D (s, a) ] + \mathbb {E} _ {s \sim \mathcal {D}, a ^ {\prime} \sim \pi (\cdot | s)} [ \log (1 - D (s, a ^ {\prime})) ], \tag {39}
+$$
+
+where  $w(s, a) = \exp_{clip}\left(\frac{1}{\beta} A^{\pi_b}(s, a)\right)$ . Here,  $A^{\pi_b}$  represents the advantage of the behaviour policy  $\pi_b$ . The WCGAN aims to generate action distributions that are weighted by this advantage, prioritising actions that yield higher returns.
+
+Weighted Conditional VAE (WCVAE): The WCVAE learns a conditional variational autoencoder with an encoder  $E$  and decoder  $D$ . The objective is to optimize:
+
+$$
+\max  _ {E, D} \mathbb {E} _ {(s, a) \sim \mathcal {D}, z \sim E (s, a)} [ w (s, a) \log D (a \mid z, s) - w (s, a) D _ {K L} (E (z \mid s, a) \| p (z \mid s)) ], \tag {40}
+$$
+
+where  $p(z \mid s)$  is the unit Gaussian prior distribution of the latent variable conditioned on  $s$ . The weighting function  $w(s, a)$  encourages the WCVAE to focus on high-reward actions.
+
+# C.2 HYPERPARAMETERS
+
+The hyperparameters used in LOM were largely the same for all of the experiments reported. In Table 3, we give a list and description of them, as well as their default values. The hyperparameters  $M$  used in Table 1 are selected from  $\{2,5,10,15,20\}$ .
+
+<table><tr><td>Symbol in paper</td><td>Description</td><td>Default values</td></tr><tr><td>M</td><td>Number of Gaussian components</td><td>Please check Figure 3</td></tr><tr><td>β</td><td>Advantage weights</td><td>5</td></tr><tr><td>C</td><td>Weight clips</td><td>50</td></tr><tr><td>ρ</td><td>Ployak averaging coefficient</td><td>0.995</td></tr><tr><td>update_delay</td><td>Target update delay</td><td>2</td></tr><tr><td>πρ</td><td>Gaussian mixture model</td><td>[state_dim, 512, 512, 2 * ac_dim + M]</td></tr><tr><td>πθ</td><td>Policy learned by LOM</td><td>[state_dim, 512, 512, ac_dim]</td></tr><tr><td>Qψ</td><td>Q-network</td><td>[state_dim + ac_dim, 512, 512, 1]</td></tr><tr><td>Qφ</td><td>Behaviour hyper Q-function</td><td>[state_dim + M, 512, 512, M]</td></tr></table>
+
+Table 3: Hyperparameters used in the experiments.
\ No newline at end of file
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+# LEARNING SYSTEM DYNAMICS WITHOUT FORGETTING
+
+# Xikun Zhang
+
+Generative AI Lab
+
+College of Computing and Data Science
+
+Nanyang Technological University
+
+Singapore 639798
+
+xikun.zhang@ntu.edu.sg
+
+# Dongjin Song & Yushan Jiang
+
+Department of Computer Science and Engineering
+
+University of Connecticut
+
+Storrs, CT
+
+USA
+
+{dongjin.song, yushan.jiang}@uconn.edu
+
+# Yixin Chen
+
+Washington University in St. Louis
+
+St. Louis, MO
+
+USA
+
+chen@cse.wustl.edu
+
+# Dacheng Tao
+
+Generative AI Lab
+
+College of Computing and Data Science
+
+Nanyang Technological University
+
+Singapore 639798
+
+dacheng.tao@gmail.com
+
+# ABSTRACT
+
+Observation-based trajectory prediction for systems with unknown dynamics is essential in fields such as physics and biology. Most existing approaches are limited to learning within a single system with fixed dynamics patterns. However, many real-world applications require learning across systems with evolving dynamics patterns, a challenge that has been largely overlooked. To address this, we systematically investigate the problem of Continual Dynamics Learning (CDL), examining task configurations and evaluating the applicability of existing techniques, while identifying key challenges. In response, we propose the Mode-switching Graph ODE (MS-GODE) model, which integrates the strengths LG-ODE and sub-network learning with a mode-switching module, enabling efficient learning over varying dynamics. Moreover, we construct a novel benchmark of biological dynamic systems for CDL, Bio-CDL, featuring diverse systems with disparate dynamics and significantly enriching the research field of machine learning for dynamic systems.
+
+Our code available at https://github.com/QueuQ/MS-GODE.
+
+# 1 INTRODUCTION
+
+Scientific research often involves systems composed of interacting objects, such as multi-body systems in physics and cellular systems in biology, with their evolution governed by underlying dynamic rules. However, due to potentially unknown or incomplete dynamic rules or incomplete observations, deriving explicit equations to simulate system evolution can be extremely challenging. As a result, data-driven approaches based on machine learning have emerged as a promising solution for predicting the future trajectories of system states purely from ob-
+
+![](images/00fad754250cb84c218c2b802aabf34dfa61d8f51643db69748225309e129db6.jpg)
+
+Exportin: The cargo to be translocated across the membrane.
+
+RAN: A type of protein that facilitates the translocation of cargo.
+
+RAN*: RAN protein activated with the assistance of RCC1.
+
+RCC1: A type of protein activates RAN through phosphorylation.
+
+Cytoplasm: The substance in a cell between the cell membrane and the nucleus.
+
+Nuclear Membrane: The layer that encloses the nucleus in a cell
+
+Nucleus: The structure within a cell containing the genetic information
+
+Figure 1: Illustration of the key components of one biological cellular system studied in our work: the RAN-regulated nucleocytoplasmic transport (Moore, 2013). Briefly speaking, this model depicts the translocation of cargo proteins (Exportin) via nuclear pores with the assistance of RAN proteins. RAN is first activated  $(RAN^{*})$  and then binds to cargo forming a complex. Next, the complex is translocated across the nuclear membrane into the cytoplasm with the assistance of RAN. Finally, RAN and Exportin are dissociated.
+
+servational data. For instance, the Interaction Network (IN) model (Battaglia et al., 2016) explicitly
+
+learns interactions between pairs of objects and has demonstrated superior performance in simulated physics systems, showcasing the potential of machine learning in studying physical system dynamics. IN has inspired many subsequent works (Kipf et al., 2018; Sanchez-Gonzalez et al., 2019; Huang et al., 2022; Liu et al., 2024). Later, to enable the modeling of incomplete and temporarily irregular system observations, ODE-based models (Huang et al., 2020; 2021) were proposed to learn the continuous dynamics of the systems.
+
+Despite the success of these methods, existing approaches are often limited to learning within a single system with a fixed type of dynamics. However, in many real-world scenarios, system dynamics are subject to change over time. For example, in cellular systems (Figure 1), the dynamics of a set of variables are subject to change when the kinetic factors are altered. Besides, as a widely adopted task in the field, predicting the trajectories of n-body physical systems (Huang et al., 2020; Battaglia et al., 2016) may also require a model to learn over systems governed by different dynamics factors including interaction types (e.g., elastic force or electrostatic force) and interaction strengths (e.g., different amounts of charges on charged objects), as illustrated in Figure 7. In this work, we term these learning scenarios as continual dynamics learning (CDL), and for the first time formally formulate the setting of CDL. In CDL, continually learning from systems with varying dynamics may overwrite a model's knowledge encoded in the model weights and trigger the catastrophic forgetting problem. In other words, the model may only accurately predict the most recently observed system dynamics while failing on earlier systems. This phenomenon is empirically verified and reported in Section 4.6. Moreover, many real-world systems exhibit repeated dynamics, and mitigating the forgetting is actually crucial in a broad range of scenarios. For example, dynamics of many physics systems are controlled by environmental factors, e.g. temperatures (Huang et al., 2023). The values of these factors like temperatures typically oscillate within a certain range, therefore the dynamics of the systems will also repeat. This is also true in biological systems. Moreover, biological cellular systems also go through different phases of cell cycle. Catastrophic forgetting phenomenon has also been observed in other fields like computer vision (Van de Ven & Tolias, 2019; Wang et al., 2024) and graph learning (Zhang et al., 2024b), and different approaches have been proposed to facilitate the models in the continual learning setting. However, as demonstrated in our experiments (Section 4.3), most existing continual techniques, which are based on regularization or memory-replay (Yoon et al., 2017; Kirkpatrick et al., 2017; Lopez-Paz & Ranzato, 2017), are designed to accommodate the patterns of different tasks<sup>1</sup> within one set of model weights and fail to effectively alleviate the forgetting issue in the context of dynamics learning, especially when the consecutive systems contain different number of objects and exhibit significantly different dynamics patterns.
+
+Targeting the challenge, we turn to the parameter-isolation based continual learning techniques, and propose a novel mode-switching graph ODE (MS-GODE) model, which can continually learn over varying system dynamics and automatically switch to the optimal mode during the test stage. MS-GODE consists of three major components: a prediction network serving as the backbone, a sub-network learning module for encoding the observed dynamics into masks, and a mode-switching module for switching the sub-network mode based on the observation. To support irregular and incomplete observational data in the practical scenario studied in our work, our backbone network is built based on LG-ODE model (Huang et al., 2020), which has a Variational AutoEncoder (VAE) (Kingma & Welling, 2013) structure facilitated by ODE-based prediction (Rubanova et al., 2019). The basic workflow of MS-GODE is as follow: Given the observational data, an encoder network first encodes the data into latent states, upon which an ODE-based generator predicts the future system trajectories within the latent space. Finally, a decoder network maps the predicted latent states back into the data space. Upon this framework, we adopt the sub-network learning strategy (Wortsman et al., 2020; Ramanujan et al., 2020; Zhou et al., 2019), which fixes the model weights after initialization and optimize a unique binary mask over the parameters for each system during training. In this way, unlike standard training strategy that encodes data patterns solely in one set of model weights, MS-GODE encodes different types of dynamics in different sub-networks by the collaboration between the binary masks and the fixed-weight backbone network. In the test stage, the model will be switched to the optimal mode by the switching module via selecting the most suitable mask that can most accurately reconstruct the given observation. Moreover, targeting that the existing entropy-based mask selection technique is only applicable to classification tasks, we further develop the novel observation reconstruction based mask selection strategy in the mode-switching
+
+module. In this way, despite the significant difference between consecutive systems, catastrophic forgetting is avoided.
+
+Besides innovatively formulating the CDL setting and the technical contribution of an effective model in CDL scenario, we have also created a novel dynamic system benchmark, Bio-CDL consisting of biological cellular systems based on the VCell platform (Schaff et al., 1997; Cowan et al., 2012; Blinov et al., 2017). Compared to the widely adopted simulated physics systems, cellular systems contain heterogeneous objects and interactions, offering richer and more challenging patterns of system dynamics. Therefore, Bio-CDL will significantly enhance the research of machine learning-based system dynamics prediction. In experiments, we thoroughly investigate the influence of the system sequence configuration on model performance in CDL with both the widely adopted physics systems and our newly constructed BioCDL, which demonstrates the advantage of MS-GODE over existing state-of-the-art techniques.
+
+# 2 RELATED WORKS
+
+# 2.1 LEARNING BASED DYNAMIC SYSTEM PREDICTION
+
+In recent years, graph neural networks (GNNs) have been proven to be promising in modeling and predicting the complex evolution of systems consisting of interacting objects (Battaglia et al., 2016; Kipf et al., 2018; Sanchez-Gonzalez et al., 2019; Huang et al., 2022; Liu et al., 2024; Luo et al., 2024b; Zhang et al., 2019; Luo et al., 2024a; Wu et al., 2021; Luo et al., 2025; Jiang et al., 2024; Guo et al., 2024; Zhang et al., 2020b; Shen et al., 2024; Yin et al., 2024; Chen et al., 2025). This was firstly demonstrated by (Battaglia et al., 2016) with Interaction Network (IN), which iteratively infers the effects of the pair-wise interactions within a system and predicts the changes of the system states. Following IN, (Kipf et al., 2018) proposed neural relational inference (NRI) to predict systems consisting of objects with unknown relationships. (Mrowca et al., 2018) proposed the Hierarchical Relation Network (HRN) that extends the predictions to systems consisting of deformable objects. (Sanchez-Gonzalez et al., 2019)
+
+![](images/1c4a455176187aefdfb4876be645d69121172dd8c2fdbe397f319dc9bcf5779c.jpg)  
+Figure 2: A molecule system may enter different phases and exhibit different dynamics as environmental factors (e.g. temperature) change. Molecules in solid state can only vibrate at fixed locations because of the strong interaction between them. Upon entering the liquid state, the interaction strength decreases and molecules can move around. In gas state, molecules move more freely with little molecule-wise interaction.
+
+proposed the Hamiltonian ODE graph network (HOGN), which injects Hamiltonian mechanics into the model as a physically informed inductive bias. Later, to better consider the intrinsic symmetry of the target systems, GNNs with different invariance and equivariance are proposed (Satorras et al., 2021; Huang et al., 2022; Han et al., 2022; Brandstetter et al., 2021; Wu et al., 2023; Liu et al., 2024). To better capture the complex system interactions, High-order graph ODE (HOPE) (Luo et al., 2023) innovatively incorporates information from high-order spatial neighborhood and high-order derivatives into dynamical system modeling. Considering that the observation of real-world systems may be incomplete and irregular samples, (Huang et al., 2020) proposed LG-ODE, which is capable of generating continuous system dynamics based on the latent ordinary differential equations. Later, Coupled Graph ODE (CG-ODE) (Huang et al., 2021) was proposed to apply ODE-based modeling to both node features and interactions. Similar ideas are also adopted in other time series research (Rubanova et al., 2019). By separating the commonalities intrinsic to the systems and the environmental factors causing the dynamics shift, Generalized Graph Ordinary Differential Equations (GG-ODE) (Huang et al., 2023) improves the generalization across systems in different environments. Similarly, Prototypical Graph ODE (PGODE) (Luo et al.) disentangles object states and system states to independently model their influence and improve the generalization capability. Disentangled Intervention-based Dynamic graph Attention networks (DIDA) (Zhang et al., 2022c) disentangles the invariant and variant patterns in dynamic graphs, and leverages the invariant patterns to ensure a stable prediction performance under spatio-temporal distribution shift. Context-attended Graph ODE (CARE) (Luo et al., 2024c) models the continuously varying environmental factors with a context variable, which is leveraged to better predict the system evolution with temporal environmental variation. Online Relational Inference (ORI) (Kang et al., 2024) models the relationship as trainable parameters, which is accompanied by AdaRelation for the online setting. Despite the substantial contributions these methods have made to dynamic system prediction, they have been limited to learning a single system with fixed dynamics. As one of the two major components of our MS-GODE,
+
+the backbone model for dynamics system prediction is mainly built upon the LG-ODE framework (Huang et al., 2020). Different from the other approaches that are typically limited to data with regular intervals or complete observation at each time stamp, LG-ODE supports irregular and incomplete observations, which is essential to the applications studied in our work.
+
+# 2.2 CONTINUAL LEARNING & MASKED NETWORKS
+
+Existing continual learning methods can be categorized into three types (Parisi et al., 2019; Van de Ven & Tolias, 2019; De Lange et al., 2021; Zhang et al., 2022a; Konishi et al., 2023; Zhang et al., 2024a; 2023b; 2022b; Fionda et al., 2023). Regularization-based methods slow down the adaption of important model parameters via regularization terms, so that the forgetting problem is alleviated (Wu et al., 2024; Goswami et al., 2023). For example, Elastic Weight Consolidation (EWC) (Kirkpatrick et al., 2017) and Memory Aware Synapses (MAS) (Aljundi et al., 2018) estimate the importance of the model parameters to the learned tasks, and add penalty terms to slow down the update rate of the parameters that are important to the previously learned tasks. Second, experience replay-based methods replay the representative data stored from previous tasks to the model when learning new tasks to prevent forgetting (Liang & Li, 2023; Rolnick et al., 2019; Rebuffi et al., 2017; Prabhu et al., 2020). For example, Gradient Episodic Memory (GEM) (Lopez-Paz & Ranzato, 2017) leverages the gradients computed based on the buffered data to modify the gradients for learning the current task and avoid the negative interference between learning different tasks. Finally, parameter isolation-based methods gradually introduce new parameters to the model for new tasks to prevent the parameters that are important to previous tasks (Qiao et al., 2023; Yoon et al., 2017). For example, Progressive Neural Network (PNN) (Rusu et al., 2016) allocates new network branches for new tasks, such that the learning on new tasks does not modify the parameters encoding knowledge of the old tasks. Our proposed MS-GODE also belongs to the parameter isolation-based methods, and is related to subnetwork-based ones (Wortsman et al., 2020; Kang et al., 2022; Zhou et al., 2019) and the edge-popup algorithm (Ramanujan et al., 2020). SupSup studies continual learning for classification tasks with an output entropy-based mask selection, which is not applicable to our task. Edge-popup algorithm provides a simple yet efficient strategy to select a sub-network, and is adopted by us to optimize the binary masks over the model parameters.
+
+# 3 LEARNING SYSTEM DYNAMICS WITHOUT FORGETTING
+
+# 3.1 PRELIMINARIES
+
+In CDL, a model is required to sequentially learn on multiple systems. A system is composed of multiple interacting objects, and is naturally represented as a graph  $\mathcal{G} = \{\mathbb{V},\mathbb{E}\}$ .  $\{\mathbb{V}$  is the node set denoting the objects of the system, and  $\mathbb{E}$  is the edge set containing the information of the relation and interaction between the objects. Based on  $\mathbb{E}$ , the spatial neighbors of a node  $v$  is defined as  $\mathcal{N}_s(v) = \{u|e_{u,v}\in \mathbb{E}\}$ . Each object node  $v$  is accompanied by observational data containing the observed states at certain time steps  $\mathbb{X}_v = \{\mathbf{x}_v^t |t\in \mathbb{T}_v\}$ , i.e. the trajectory of the system evolution. With a system structured as a graph, its trajectory is naturally a spatial-temporal graph, in which each node is an observed state  $\mathbf{x}_v^t$ . In the following, we will refer to  $\mathbf{x}_v^t$  as a 'state'. In a multi-body system, the trajectory records the 3D locations of the particles over time. While in other systems, e.g. cellular systems, a state could be the amount of a certain substance instead of locations, and a trajectory records the states at different time stamps. The set  $\mathbb{T}_v$  contains the time steps (real numbers) when the states of  $v$  are observed and can vary across different objects. For the prediction task, the observations lie within a certain period, i.e.  $\bigcup_{v\in \mathbb{V}}\mathbb{T}_v\in [t_0,t_1]$ , and the task is to predict the system states at future time steps beyond  $t_1$ . We denote the future time steps to predict for an object  $v$  as  $\mathbb{T}_v^{pred}$ . In CDL, different systems in a sequence may exhibit different dynamics and contain different objects (i.e.  $\mathbb{V}$ ).
+
+# 3.2 FRAMEWORK OVERVIEW
+
+In this subsection, we provide a high-level introduction to the workflow of MS-GODE (Figure 3), while the details of each component are provided in the following subsections.
+
+Overall, MS-GODE consists of three core components: 1) The backbone network; 2) The subnetwork learning module; 3) The mode switching module. The backbone network consists of: 1) an encoder network  $\mathrm{Enc}(\cdot;\theta_E)$  parameterized by the parameters  $\theta_E$  for encoding the trajectories into the latent space; 2) An ODE-based generator  $\mathrm{Gen}(\cdot;\theta_G)$  parameterized by  $\theta_G$  for predicting the future trajectories within the latent space; 3) A decoder network  $\mathrm{Dec}(\cdot;\theta_D)$  parameterized by  $\theta_D$  for transforming the predicted latent states back into the data space. Within a standard learning
+
+![](images/77d5da1d3921bb213956d965f65e27081eade0070f9a3be5062e4e637dc8efd0.jpg)
+
+![](images/977f5aca33d261fe05e9e9960b9a3631791d0dae258fa662e6499642ce130fe5.jpg)  
+Figure 3: The upper part illustrates the workflow of MS-GODE during mask selection and inference, which are denoted by dashed and solid lines. During mask selection, the observation is split into two parts, and the first part is fed into the model for selecting the mask that can best predict the second part. During inference, the entire observation is fed into the model for prediction of future states. The lower part demonstrates the structure of the masked encoder, masked generator, and masked decoder. Different components fetch their corresponding mask from the mask pool and apply the mask onto the parameters.
+
+scheme, the model will be trained by updating the parameters  $\{\theta_E,\theta_G,\theta_D\}$ . However, as introduced above, when continually training the model on multiple systems with different dynamics, this learning scheme will bias the model towards the most recently observed system, causing the catastrophic forgetting problem. In this work, inspired by the recent advances in sub-network learning (Ramanujan et al., 2020; Wortsman et al., 2020), we propose to avoid direct optimization of the parameters  $\{\theta_E,\theta_G,\theta_D\}$ . Instead, we train the model by optimizing the connection topology of the backbone model and encode the dynamics of each system into a sub-network. This is equivalent to encoding each system-specific dynamics pattern into a binary mask overlaying the shared parameters with fixed values. In this way, the interference between learning on different systems with different dynamics, i.e. the forgetting problem, can be avoided. Moreover, this approach is also memory efficient since the space for storing the binary masks is negligible. With the system-specific masks, the three model components are formulated as:
+
+$$
+\operatorname {E n c} \left(\because ; \theta_ {E} \odot \mathbf {M} _ {E} ^ {s}\right); \quad \operatorname {G e n} \left(\because ; \theta_ {G} \odot \mathbf {M} _ {G} ^ {s}\right); \quad \operatorname {D e c} \left(\because ; \theta_ {D} \odot \mathbf {M} _ {D} ^ {s}\right), \tag {1}
+$$
+
+where the superscript  $s$  is the system index. The details of the binary mask optimization during training and mask selection during testing are provided in Section 3.5. In the following subsections, all parameters are subsets of  $\theta_{E}$ ,  $\theta_{G}$ , or  $\theta_{D}$  and are controlled by the corresponding masks. For example,  $\mathbf{W}_{msg}$  in Section 3.3 is part of the encoder parameters  $\theta_{E}$  and is under the control of  $\mathbf{M}_E^s$ .
+
+# 3.3 MASKED ENCODER NETWORK
+
+As the first component of the model, the masked encoder network serves to encode the dynamics pattern in the observational data into the latent space. As introduced in Section 3.1, the trajectory of a system is a spatial-temporal graph. Therefore, the encoder is constructed as an attention-based spatial-temporal graph neural network framework (ST-GNN) (Huang et al., 2020; Hu et al., 2022; Huang et al., 2021; Zhang et al., 2020a). Originally, the graph attention network (GAT) (Veličković et al., 2017) is designed to aggregate information over the spatially neighboring nodes. On spatial-temporal graphs, the information aggregation is extended to both spatial and temporal neighboring nodes. Such a spatial-temporal neighborhood of a state  $\mathbf{x}_v^t$  is defined as the states of the spatially connected nodes within a specified temporal window  $\delta_{window}$ ,
+
+$$
+\mathcal {N} _ {s t} \left(\mathbf {x} _ {v} ^ {t}\right) := \left\{\mathbf {x} _ {w} ^ {q} \mid e _ {u, v} \in \mathbb {E} \quad a n d \quad | q - t | <   \delta_ {\text {w i n d o w}} \right\}. \tag {2}
+$$
+
+Then, iterative message passing is conducted over the spatial-temporal edges to update the representation of each state. The update of the hidden representation of a state  $\mathbf{x}_v^t$  at the  $l$ -the layer of the ST-GNN is formulated as,
+
+$$
+\mathbf {h} _ {v, t} ^ {l} = \mathbf {h} _ {v, t} ^ {l - 1} + \sigma \left(\sum_ {x _ {u} ^ {q} \in \mathcal {N} _ {s t} \left(\mathbf {x} _ {v} ^ {t}\right)} \mathrm {a} \left(\mathbf {h} _ {v, t} ^ {l - 1}, \mathbf {h} _ {u, q} ^ {l - 1}, q - t\right) \cdot \mathbf {W} _ {m s g} \cdot \mathrm {m s g} \left(\mathbf {h} _ {u, t} ^ {l - 1}, q - t\right)\right), \tag {3}
+$$
+
+where  $\mathrm{a}(\cdot, \cdot)$  calculates the attention score between a pair of states, and  $\mathrm{msg}(\cdot, \cdot)$  is the message function commonly adopted in GNNs (Gilmer et al., 2017). However, different from the message function in typical GNNs, the temporal relationship between the states is a crucial part to understand the system dynamics. Therefore, we follow the strategy in LG-ODE (Huang et al., 2020) to incorporate the temporal distance between the states in the message function and attention function,
+
+$$
+\operatorname {m s g} \left(\mathbf {h} _ {u, t} ^ {l - 1}, q - t)\right) := \sigma \left(\mathbf {W} _ {t m p} \cdot \operatorname {c o n c a t} \left(\mathbf {h} _ {u, t} ^ {l - 1}, q - t\right)\right) + \operatorname {T E} (q - t), \tag {4}
+$$
+
+$$
+\mathrm {a} \left(\mathbf {h} _ {v, t} ^ {l - 1}, \mathbf {h} _ {u, q} ^ {l - 1}, q - t\right) := \frac {\exp \left(\operatorname {m s g} \left(\mathbf {h} _ {u , q} ^ {l - 1} , q - t\right)\right) ^ {T} \cdot \mathbf {h} _ {v , t} ^ {l - 1}\left. \right)}{\sum_ {x _ {w} ^ {p} \in \mathcal {N} _ {s t} \left(\mathbf {x} _ {v} ^ {t}\right)} \exp \left(\operatorname {m s g} \left(\mathbf {h} _ {w , p} ^ {l - 1} , p - t\right)\right) ^ {T} \cdot \mathbf {h} _ {v , t} ^ {l - 1})}, \tag {5}
+$$
+
+where  $\mathrm{TE}(\cdot)$  is a temporal position encoding developed based on the position encoding in Transformer (Vaswani et al., 2017) for incorporating the temporal information into the representation. Finally, for subsequent state prediction, the state representations are averaged over the temporal dimension,
+
+$$
+\mathbf {h} _ {\text {f i n a l}} ^ {v} = \frac {1}{| \mathbb {T} _ {v} |} \sum_ {t \in \mathbb {T} _ {v}} \sigma \left(\bar {\mathbf {h}} _ {v} ^ {T} \cdot \operatorname {m s g} \left(\mathbf {h} _ {v, t} ^ {L}\right)\right) \cdot \operatorname {m s g} \left(\mathbf {h} _ {v, t} ^ {L}, t - t _ {0}\right), \tag {6}
+$$
+
+where  $t_0$  denotes the starting time of all states in the observational data (Section 3.1), and the average term  $\overline{\mathbf{h}}_v$  of each node  $v$  is a weighted summation over the representations at all time steps,
+
+$$
+\bar {\mathbf {h}} _ {v} = \sigma \left(\frac {1}{| \mathbb {T} _ {v} |} \sum_ {t \in \mathbb {T} _ {v}} \operatorname {m s g} \left(\mathbf {h} _ {v, t} ^ {L}, t - t _ {0}\right) \cdot \mathbf {W} _ {a v g}\right). \tag {7}
+$$
+
+# 3.4 MASKED ODE-BASED GENERATOR
+
+ODE-based generator (Huang et al., 2020; Chen et al., 2018; Rubanova et al., 2019) ensures that the model can handle observations with irregular temporal intervals and incomplete states, as well as predict future states at any time denoted by real numbers. Specifically, the trajectory prediction is formulated as solving an ODE initial value problem (IVP), where the initial values of the objects  $\{z_v^{t_1}|v\in \mathbb{V}\}$  are generated from the final representation of the states  $(\{\mathbf{h}_{final}^v |v\in \mathbb{V}\}$ , Section 3.3). Mathematically, the procedure of predicting the future trajectory of a system  $s$  is formulated as,
+
+$$
+z _ {v} ^ {t _ {1}} \sim \mathrm {p} \left(z _ {v} ^ {t _ {1}}\right), v \in \mathbb {V} \tag {8}
+$$
+
+$$
+\left\{z _ {v} ^ {\tau} \mid v \in \mathbb {V}, \tau \in \mathbb {T} _ {v} ^ {p r e d} \right\} = \operatorname {G e n} \left(\left\{z _ {v} ^ {t _ {1}} \mid v \in \mathbb {V} \right\}, \left\{\mathbb {T} _ {v} ^ {p r e d} \mid v \in \mathbb {V} \right\}; \theta_ {G} \odot \mathbf {M} _ {G} ^ {s}\right)), \tag {9}
+$$
+
+To estimate the posterior distribution  $q(\{z_v^{t_1} | v \in \mathbb{V}\} | \{\mathbb{X}_v | v \in \mathbb{V}\})$  based on the observation (i.e.  $\{\mathbb{X}_v | v \in \mathbb{V}\}$ ), the distribution is assumed to be Gaussian. Then the mean  $\mu_v$  and standard deviation  $\sigma_v$  are generated from  $\{\mathbf{h}_{final}^v | v \in \mathbb{V}\}$  with a multi-layer perceptron (MLP),
+
+$$
+q \left(z _ {v} ^ {t _ {1}} \mid \left\{\mathbb {X} _ {v} \mid v \in \mathbb {V} \right\}\right) = \mathcal {N} \left(\mu_ {v}, \sigma_ {v}\right) = \mathcal {N} \left(\operatorname {m l p} \left(\mathbf {h} _ {\text {f i n a l}} ^ {v}; \theta_ {G} \odot \mathbf {M} _ {G} ^ {s}\right)\right), v \in \mathbb {V}. \tag {10}
+$$
+
+As noted in Section 3.2, the parameters of  $\mathrm{mlp}(\cdot)$  and  $\mathrm{F}_{int}(\cdot)$  are part of  $\theta_{G}$  and controlled by  $\mathbf{M}_G^s$ .
+
+Based on the approximate posterior distribution  $q(\{z_v^{t_1} | v \in \mathbb{V}\} | \{\mathbb{X}_v | v \in \mathbb{V}\})$ , we sample an initial state for each object, upon which the ODE solver will be applied for generating the predicted states in the latent space. The dynamics of each object in the system are governed by its interaction with all the other objects. Therefore, the core part of the ODE-based generator is a trainable interaction network that encodes the dynamics in the form of the derivative of each  $\mathbf{z}_v$
+
+$$
+\left. \frac {\mathrm {d} \mathbf {z} _ {v}}{\mathrm {d} t} \right| _ {t = t ^ {\prime}} = \operatorname {F} _ {i n t} \left(\left\{\mathbf {z} _ {u} ^ {t ^ {\prime}} \mid u \in \mathcal {N} _ {s} (v) \right\}; \theta_ {G} \odot \mathbf {M} _ {G}\right), \tag {11}
+$$
+
+where the function  $\mathrm{F}_{int}(\cdot)$  parameterized by  $\theta_I$  predicts the dynamics (derivative) of each object  $v$  based on all the other objects interacting with  $v$  (i.e.  $\mathcal{N}_s(v)$  defined in Section 3.1), and  $t'$  denotes any possible future time. Note that  $\mathcal{N}_s(v)$  only contains the spatial neighbors and is different from
+
+$\mathcal{N}_{st}(\cdot)$ , because the object-wise interaction at a certain time  $t'$  is not dependent on system states at other times. For example, in a charged particle system governed by electrostatic force, the force between a pair of particles at  $t'$  is solely determined by the relative positions (i.e. the states) of the particles (Figure 7) at  $t'$ . In our work, we adopt the Neural Relational Inference (NRI) (Kipf et al., 2018) as the function  $\mathrm{F}_{int}(\cdot)$ . Based on  $\mathrm{F}_{int}(\cdot)$ , the future state of the system at an arbitrary future time  $t_2$  can be obtained via an integration
+
+$$
+\mathbf {z} _ {v} ^ {t _ {2}} = \mathbf {z} _ {v} ^ {t _ {1}} + \int_ {t _ {1}} ^ {t _ {2}} \frac {\mathrm {d} \mathbf {z} _ {v}}{\mathrm {d} t} \mathrm {d} t = \mathbf {z} _ {v} ^ {t _ {1}} + \int_ {t _ {1}} ^ {t _ {2}} \operatorname {F} _ {i n t} \left(\left\{\mathbf {z} _ {u} ^ {t} \mid u \in \mathcal {N} _ {s} (v) \right\}; \theta_ {G} \odot \mathbf {M} _ {G}\right) \mathrm {d} t, \tag {12}
+$$
+
+which can be solved numerically by mature ODE solvers, e.g. Runge-Kutta method. After obtaining the latent representations of the states at the future time steps ( $\{\mathbf{z}_v^t | t \in \mathbb{T}_{future}, v \in \mathbb{V}\}$ ), the predictions are generated via a masked decoder network that projects the latent representations back into the data space
+
+$$
+\mathbf {y} _ {v} ^ {t} = \operatorname {D e c} \left(\mathbf {z} _ {v} ^ {t}; \theta_ {D} \odot \mathbf {M} _ {D}\right), t \in \mathbb {T} _ {\text {f u t u r e}}, v \in \mathbb {V}. \tag {13}
+$$
+
+In our work,  $\mathrm{Dec}(\cdot ;\theta_D)$  is instantiated as a multi-layer perceptron (MLP).
+
+# 3.5 SUB-NETWORK LEARNING
+
+MS-GODE is trained by maximizing the Evidence Lower Bound (ELBO). Denoting the concatenation of all the initial latent states  $(\{\mathbf{z}_v^{t_1} | v \in \mathbb{V}\})$  as  $\mathbf{Z}_{\mathbb{V}}^{t_1}$ , the ELBO is formulated as
+
+$$
+\begin{array}{l} \operatorname {E L B O} \left(\mathbf {M} ^ {s}\right) = \mathbb {E} _ {\mathbf {Z} _ {\mathbb {V}} ^ {t _ {1}} \sim q \left(z _ {v} ^ {t _ {1}} \mid \{\mathbb {X} _ {v} | v \in \mathbb {V} \}\right)} \left[ \log \left(p \left(\left\{\mathbf {x} _ {v} ^ {t} \mid t \in \mathbb {T} _ {\text {f u t u r e}}, v \in \mathbb {V} \right\}\right)\right) \right] (14) \\ - \mathbf {K L} [ q (\mathbf {Z} _ {\mathbb {V}} ^ {t _ {1}} | \{\mathbb {X} _ {v} | v \in \mathbb {V} \}) ] | | p (\mathbf {Z} _ {\mathbb {V}} ^ {t _ {1}}) ]. (15) \\ \end{array}
+$$
+
+where  $p(\mathbf{Z}_{\mathbb{V}}^{t_1})$  denotes the prior distribution of  $\mathbf{Z}_{\mathbb{V}}^{t_1}$ , which is typically chosen as standard Gaussian.  $\mathbf{M}^s$  denotes the union of all masks over different modules of the framework (i.e.  $\mathbf{M}^s = \{\mathbf{M}_E^s, \mathbf{M}_I^s, \mathbf{M}_P^s\}$ ). In our model, the parameters  $(\theta_E, \theta_I, \text{and } \theta_P)$  will be fixed, and the ELBO will be maximized by optimizing the binary masks  $\mathbf{M}^s$  overlaying the parameters via the Edge-popup algorithm (Ramanujan et al., 2020). After learning each system, the obtained mask is added into a mask pool  $\mathbb{M}$  to be used in testing.
+
+# 3.6 MODE-SWITCHING
+
+During testing, MS-GODE will be evaluated on a sequence of systems exhibiting diverse dynamics patterns, and it will automatically switch to the optimal mode that ensures the highest accuracy. This is achieved by applying the most suitable mask based on the performance of reconstructing part of the given observations. To obtain the most suitable mask, a given observation from  $[t_0,t_1]$  is first split it into two periods  $[t_0,\frac{t_0 + t_1}{2} ]$  and  $\left[\frac{t_0 + t_1}{2},t_1\right]$ . Then, the first half is fed into the model and the correct mask can be chosen by selecting the one that can reconstruct the second half with the lowest error.
+
+# 4 EXPERIMENTS
+
+In this section, we aim to answer the following questions. 1. How to properly configure MS-GODE for optimal performance? 2. How would the configuration of the system sequence influence the performance? 3. How is the performance of the existing CL techniques? 4. Can MS-GODE outperform the baselines?
+
+# 4.1 EXPERIMENTAL SYSTEMS
+
+In experiments, we adopt physics and biological cellular systems. A detailed introduction to the system sequence construction is provided in Appendix A.1.
+
+Simulated physics systems are commonly adopted to evaluate the machine learning models in the task of learning system dynamics (Liu et al., 2024; Battaglia et al., 2016; Huang et al., 2020). The physics systems adopted in this work include spring-connected particles and charged particles (Figure 7). We carefully adjust the system configuration and construct 3 system sequences with different levels of dynamics shift (Appendix A.1).
+
+Biological cellular systems are innovatively introduced in this work based on Virtual Cell platform (Schaff et al., 1997; Cowan et al., 2012; Blinov et al., 2017). Currently, Bio-CDL includes rule-based model of EGFR and compartmental rule based model of translocation based on the Ran protein
+
+![](images/08c521274e0ef45738eafd25c39cbe52d593b75ef8a780d37c3af05f8aff0878.jpg)  
+(a)
+
+![](images/777975c1c66ea4607b89255a451b37eb12b8e1a0a570ad1b807bec420e5e91c0.jpg)  
+(b)  
+Figure 4: Performance comparison among different strategies to binarize the mask values. (a) Comparison over the cellular system sequence  $\mathrm{EGFR}_1\rightarrow \mathrm{Ran}_1\rightarrow \mathrm{EGFR}_2\rightarrow \mathrm{Ran}_2$  . (b)(c)(d) Comparison over different physics system sequences. Blue line denotes the performance of top-k selection with different thresholds. Red line demonstrates the performance of using fast selection.
+
+![](images/674a4e27a9dc57ae9b7f0f9de2e176303da4de868b78fe0b0140a1564de28b69.jpg)  
+(c)
+
+![](images/c307f7ff66274a55b7a57ec76f8e98f1cf457b22203c98155d7229e867520ee7.jpg)  
+(d)
+
+(GTPase). In experiments, we adjust the coefficients of the models to construct a sequence containing 2 EGFR and 2 Ran systems interleaved with each other (EGFR $_1$  → Ran $_1$  → EGFR $_2$  → Ran $_2$ ). Full details are provided in Appendix A.1.2.
+
+# 4.2 EXPERIMENTAL SETUPS & EVALUATION
+
+Model evaluation in CDL. The models in this work learn sequentially on multiple systems under the continual learning setting, thus the evaluation is significantly different from standard learning settings. After learning each new task, the model is tested on all learned tasks and the results form a performance matrix  $\mathbf{M}^p\in \mathbb{R}^{N\times N}$ , where  $\mathbf{M}_{i,j}^p$  denotes the performance on the  $j$ -system after learning from the 1-st to the  $i$ -th system, and  $N$  is the number of systems in the sequence. In our experiments, each entry of  $\mathbf{M}^p$  is a mean square error (MSE) evaluating the performance on a single system. To evaluate the performance over all systems, average performance (AP) can be calculated. For example,  $\frac{\sum_{j = 1}^{N}\mathrm{M}_{N,j}^{p}}{N}$  is the average performance after learning the entire sequence with  $N$  tasks. Similarly, average forgetting (AF) can be calculated as  $\frac{\sum_{j = 1}^{N - 1}\mathrm{M}_{N,j}^{p} - \mathrm{M}_{j,j}^{p}}{N - 1}$ . More details on model evaluation can be found in Appendix A.4. The baseline are configured by combining existing continual learning techniques with the LG-ODE (Huang et al., 2020) model. Other dynamics learning models were excluded because they typically don't support the practical setting with irregular and incomplete observation studied in this work. All experiments are repeated 5 times on a Nvidia Titan Xp GPU. The results are reported with average and standard deviations.
+
+Baselines & model settings. We adopt state-of-the-art baselines including the performance upper (joint training) and lower bounds (fine-tune) in experiments. The state-of-the-art baselines adopted in this work include Elastic Weight Consolidation (EWC) (Kirkpatrick et al., 2017) based on regularization, Learning without Forgetting (LwF) (Li & Hoiem, 2017) based on knowledge distillation, Gradient Episodic Memory (GEM) (Lopez-Paz & Ranzato, 2017) based on both memory replay and regularization, Bias-Correction (Chrysakis & Moens, 2023) based Memory Replay (BCMR) that integrates the idea of data sampling bias correction into memory replay, and Scheduled Data Prior (SDP) (Koh et al., 2023) that adopts a data-driven approach to balance the contribution of past and current data. Besides, joint training and fine-tuning are commonly adopted in continual learning works. Joint training trains the model jointly over all systems, which does not follow the continual learning setting. Fine-tune directly trains the model incrementally on new systems without any continual learning technique. As revealed by Ramanujan et al. (2020), for learning subnetworks, pre-trained models are the most suitable for serving as the backbone. However, unlike the Computer Vision (CV) tasks studied by Ramanujan et al. (2020), pre-trained models are not readily available in the context of dynamics system modeling. For example, Seifner et al. (2024) provides a promising pre-trained model with valuable insights into pre-training for dynamical systems, but the model is still limited to interpolation tasks. However, we will investigate how to adapt the provided model to extrapolation tasks once their code and model are released. Therefore, we adopt the random initialization strategy, which is shown by Ramanujan et al. (2020) to be less effective but could be comparable to pre-trained models with a proper ratio of remaining parameters, which is thoroughly investigated in our experiments reported in Section 4.3 (Figure 4). More details of experimental settings and baselines and experimental settings are provided in Appendix A.1 A.3.
+
+# 4.3 MODEL CONFIGURATION AND PERFORMANCE (RQ1)
+
+Table 1: Performance comparisons on physics system sequences (  $\downarrow$  lower means better).  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Seq1: Low-level dynamics shift</td><td colspan="2">Seq2: Mid-level dynamics shift</td><td colspan="2">Seq3: High-level dynamics shift</td></tr><tr><td>AP ↓</td><td>AF ↓</td><td>AP ↓</td><td>AF /% ↓</td><td>AP ↓</td><td>AF /% ↓</td></tr><tr><td>Fine-tune</td><td>0.369±0.027</td><td>0.115±0.016</td><td>0.391±0.044</td><td>0.314±0.030</td><td>0.258±0.025</td><td>0.086±0.037</td></tr><tr><td>EWC 2017</td><td>0.208±0.015</td><td>-0.007±0.019</td><td>0.227±0.038</td><td>0.008±0.022</td><td>0.148±0.011</td><td>0.008±0.017</td></tr><tr><td>GEM 2017</td><td>0.251±0.037</td><td>0.079±0.020</td><td>0.379±0.023</td><td>0.302±0.030</td><td>0.163±0.037</td><td>-0.091±0.033</td></tr><tr><td>LwF 2017</td><td>0.258±0.011</td><td>0.104±0.042</td><td>0.363±0.039</td><td>0.312±0.042</td><td>0.130±0.025</td><td>0.016±0.032</td></tr><tr><td>BCMR 2023</td><td>0.284±0.017</td><td>-0.006±0.031</td><td>0.298±0.028</td><td>-0.001±0.033</td><td>0.233±0.017</td><td>0.027±0.023</td></tr><tr><td>SDP 2023</td><td>0.352±0.021</td><td>0.121±0.018</td><td>0.303±0.026</td><td>0.352±0.058</td><td>0.213±0.035</td><td>0.024±0.045</td></tr><tr><td>Joint</td><td>0.194±0.006</td><td>-</td><td>0.186±0.015</td><td>-</td><td>0.116±0.009</td><td>-</td></tr><tr><td>Ours</td><td>0.200±0.003</td><td>0.002±0.004</td><td>0.204±0.005</td><td>-0.001±0.001</td><td>0.113±0.001</td><td>-0.000±0.000</td></tr></table>
+
+When optimizing the system-specific masks using the edge-popup algorithm (Ramanujan et al., 2020) (Appendix A.2), each entry of the mask is assigned with a continuous value for gradient descent after backpropagation. During inference, different strategies can be adopted to transform the continuous scores into binary values. In our experiments, we tested both 'fast selection' and 'top-k selection' with different thresholds. 'Fast selec
+
+tion' sets all entries with positive score values into '1's and the other entries into '0's. 'Top-k selection' first ranks the score values and sets a specified ratio of entries with the largest values into '1's. In Figure 4, we show the performance of different strategies. Overall, 'top-k selection' is inferior to 'fast selection'. This is potentially because 'fast selection' does not limit the number of selected entries, therefore allowing more flexibility for optimization. We also observe that the performance of 'top-k' selection is more sensitive on the cellular systems compared to the physics systems, indicating that the cellular systems have higher optimization difficulty for sub-network (binary mask) learning over system sequences.
+
+Second, sub-network learning will deactivate some neurons in the model, which resembles the dropout mechanism widely adopted in machine learning models and may cause the model to be over-sparsified. Therefore, we investigate the influence of dropout rate in MS-GODE. From Figure 5, we can see that a smaller dropout rate results in lower error (better performance). This corroborates our hypothesis that the mask selection mechanism and dropout complement each other, and the dropout rate should be decreased when the masking strategy is adopted.
+
+![](images/856d58e37337f94cd82d288301d7ea1537024dedce9fab684fb1efd16758f255.jpg)  
+(a)
+
+![](images/43ed3c4c811787f13e0734c5a4ea8675de558d9bc5790c305c1744e434869a58.jpg)  
+(b)  
+Figure 5: AP (a) and AF (b) of MS-GODE with different dropout rate.
+
+# 4.4 SEQUENCE CONFIGURATION AND PERFORMANCE (RQ2)
+
+In this subsection, we investigate the influence of system sequence configuration on the learning difficulty and performance. Specifically, we construct 3 physics system sequences with increasing level of dynamics shift (Appendix A.1.1). Sequence 1 (low-level dynamics shift) consists of 8 spring connected particle systems, in which consecutive systems are only different in one system coefficient. Sequence 2 (mid-level dynamics shift) is constructed to have a higher level of dynamics shift by simultaneously varying 2 system coefficients. Finally, sequence 3 (high-level dynamics shift) is constructed by interleaving spring-connected particle systems and charged particle systems with disparate dynamics. As shown in Table 1, in terms of both AP and AF, most methods, including MS-GODE, obtain similar performance over Sequence 1 and 2, and obtain better performance on Sequence 3. For MS-GODE, since the systems with more diverse dynamics are easier to distinguish for selecting the masks during inference. For all the methods in general, two possible cases are: 1. The model is not well trained on any system (it has not encoded much information), therefore it has nothing to forget. 2. The model is well trained and can maintain enough information from each system. However, the lower AP (low MSE) obtained on the system sequence with high-level dynamics shift indicates that the model has well adapted to each system, i.e., case 1 is not true. When case 2 holds, it indicates that the model's parameters are sufficiently updated to fit each system. When learning on one system, two possible cases are: 1. All parameters are modified uniformly. 2. A subset of parameters are modified more than the others. When the baseline is Fine-tune, if all the parameters are uniformly modified, it would be almost impossible to maintain the performance on previous systems. Therefore, the potential case is that each system relies more on a subset of the
+
+![](images/771fd1df7ad6748fd8822d783f7dc608d4826409098dd837c3cc57a96381b457.jpg)  
+Figure 6: Visualization of performance matrices. (a), (b), (c), (d) corresponds to MS-GODE, fine-tune, EWC, and LwF on the cellular system sequence. (e), (f), (g), (h) are the performance of MS-GODE, EWC, fine-tune, and GEM, on the physics systems. Lower values indicate better performance.
+
+![](images/2c1c066d6ee6d8dbb9396af469747d3fe04195e0d832e47a218516b6015a4588.jpg)
+
+![](images/926292e0731c2d77bc08d5f28c9d36a228f49e5e4102ed9a46e63eebc95b140d.jpg)
+
+![](images/51def80757d792629259b0579b21f216a3a30bcc8bdcc05212d4458723a4a509.jpg)
+
+![](images/e9d5e04d8776394fe942181c85f736bfdaa478cdb25cabca6e2a845cbfb5785c.jpg)
+
+parameters, and systems with larger difference in dynamics may lead to less overlap in the subsets of parameters they rely on.
+
+# 4.5 COMPARISONS WITH STATE-OF-THE-ARTS (RQ3,4)
+
+In this subsection, we compare MS-GODE with state-of-the-arts methods including the joint training. The experiments are conducted on both physics systems (Table 1) and cellular systems (Table 2). Besides, by comparing the results across different sequences in the two tables, we find that the sequences with gradual dynamics shift (Sequence 1,2 of physics systems) are more difficult to learn than the sequences with abrupt dynamics shift (Sequence 3 of the physics systems and the cellular system). The advantages of MS-GODE over the baselines mainly come from two perspectives. First, since the learning of the subnetworks are learned in a completely
+
+Table 2: Performance comparisons on biological cellular systems (  $\downarrow$  lower means better).  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">EGFR1→Ran1→EGFR2→Ran2</td></tr><tr><td>AP ↓</td><td>AF ↓</td></tr><tr><td>Fine-tune</td><td>0.355±0.089</td><td>0.226±0.037</td></tr><tr><td>EWC 2017</td><td>0.312±0.028</td><td>-0.013±0.019</td></tr><tr><td>GEM 2017</td><td>0.316±0.083</td><td>0.352±0.109</td></tr><tr><td>LwF 2017</td><td>0.330±0.036</td><td>0.349±0.046</td></tr><tr><td>BCMR 2023</td><td>0.149±0.025</td><td>0.013±0.044</td></tr><tr><td>SDP 2023</td><td>0.197±0.025</td><td>0.167±0.045</td></tr><tr><td>Joint</td><td>0.055±&lt;0.001</td><td>-</td></tr><tr><td>Ours</td><td>0.144±0.012</td><td>-0.003±0.036</td></tr></table>
+
+independent manner, the plasticity when learning new systems is guaranteed. Second, the independency of the subnetworks also eliminates the forgetting issue and protect the performance stability on previously learned systems.
+
+# 4.6 IN-DEPTH INVESTIGATION ON THE LEARNING DYNAMICS (RQ3,4)
+
+Table 1 and 2 provide the overall performance. However, as introduced in Section 4.2, to obtain an in-depth understanding of the performance of different methods, we have to seek help from the performance matrix. In Figure 6, we visualize the performance matrices of different methods. The  $i$ -th column demonstrates the performance of the  $i$ -th task when learning sequentially over the systems. Comparing MS-GODE ((a) and (e)) and the other methods, we find that MS-GODE maintains a much more stable performance. EWC ((c) and (f)) also maintains a relatively stable performance of each system based on its regularization strategy. However, compared to MS-GODE, the model becomes less and less adaptive to new systems, because the regularization is applied to more parameters when proceeding to each new task. Fine-tune ((b) and (g)) is not limited by regularization, therefore is more adaptive on new systems but less capable of preserving the performance on previous systems compared to EWC. LwF (d), although based on knowledge distillation, does not directly limit the adaptation of the parameters. Finally, based on memory and gradient modification, GEM (h) maintains the performance better than fine-tune (g), and is more adaptive to new tasks than EWC (f). More details on the performance matrices are provided in Appendix A.5.
+
+# 5 CONCLUSION
+
+In this paper, we systematically study the problem of continual dynamics learning (CDL) from different perspectives, including investigating the influence of task configuration and evaluating the performance of existing continual learning techniques. Based on the findings, we propose Mode-switching Graph ODE (MS-GODE). Additionally, we also construct Bio-CDL, consisting of biological cellular systems and significantly enriching the research field of machine learning on system dynamics. Finally, we conduct comprehensive experiments on both physics and cellular system sequences, which not only demonstrate the effectiveness of MS-GODE, but also provide insights into the problem of machine learning over system sequences with dynamics shift.
+
+# ACKNOWLEDGMENTS
+
+This project is supported by the National Research Foundation, Singapore, under its NRF Professorship Award No. NRF-P2024-001.
+
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+
+# A APPENDIX / SUPPLEMENTAL MATERIAL
+
+# A.1 SYSTEM SEQUENCE CONFIGURATION
+
+# A.1.1 PHYSICS SYSTEM SEQUENCE
+
+Simulated physics systems are commonly adopted to evaluate the machine learning models in the task of learning system dynamics (Liu et al., 2024; Battaglia et al., 2016; Huang et al., 2020). The physics systems adopted in this work include spring connected particles and charged particles (Figure 7) with disparate dynamics, therefore are ideal for constructing system sequences to evaluate a model's continual learning capability under severe significant dynamics shift. Besides, the configuration of each system type is also adjustable. For the spring connected particles, the number of particles, strength of the springs, and the size of the box containing the particles are adjustable. For the charged particles, the number of particles, charge sign, and the size of box are adjustable. In our experiments, we constructed multiple systems with different configurations, which are aligned into sequences for the model to learn.
+
+The physics system sequences are constructed to have different types of dynamics changes. System sequence 1 is composed of 8 spring connected particle systems. Each system contains 5 particles, and some pairs of particles are connected by springs. An illustration is given in Figure 7. For each system, besides the number of particles, the size of the box containing the particles and the strength of spring are adjustable. In Sequence 1, the first 4 systems have constant spring strength and decreasing box size. From the 5-th system, the box size is fixed, and the spring strength is gradually increased. Sequence 2 also contains 8 systems of spring connected particles and is designed to possess more severe dynamics shift. Specifically, both the box size and spring strength vary from the first to the last system, and the values are randomly aligned instead of monotonically increasing or decreasing. Sequence 3 is designed to possess more significant dynamics shift than Sequence 2 by incorporating the charged particle systems in the sequence. Specifically, Sequence 3 contains 4 spring connected particle systems and 4 charged particle systems, which are aligned alternatively. The box size gradually decreases and the interaction strength (spring strength or amount of charge on the particles) gradually increases. In a charged particle system, the particles could carry either positive or negative charge, and the system dynamics is governed by electrostatic force, which is significantly different from the spring connected particle system.
+
+<table><tr><td>System</td><td>S1</td><td>S2</td><td>S3</td><td>S4</td><td>S5</td><td>S6</td><td>S7</td><td>S8</td><td>S9</td><td>S10</td></tr><tr><td>Type</td><td>Spring</td><td>Spring</td><td>Spring</td><td>Spring</td><td>Spring</td><td>Spring</td><td>Spring</td><td>Spring</td><td>Spring</td><td>Spring</td></tr><tr><td># particles</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td></tr><tr><td>Box size</td><td>10.0</td><td>5.0</td><td>3.0</td><td>1.0</td><td>0.5</td><td>0.5</td><td>0.5</td><td>0.5</td><td>3.0</td><td>1.0</td></tr><tr><td>Interaction strength</td><td>0.01</td><td>0.01</td><td>0.01</td><td>0.01</td><td>0.01</td><td>0.1</td><td>0.5</td><td>1.0</td><td>0.1</td><td>0.5</td></tr></table>
+
+Table 3: Spring connected system configurations.  
+
+<table><tr><td>System</td><td>C1</td><td>C2</td><td>C3</td><td>C4</td></tr><tr><td>Type</td><td>Charge</td><td>Charge</td><td>Charge</td><td>Charge</td></tr><tr><td># particles</td><td>5</td><td>5</td><td>5</td><td>5</td></tr><tr><td>Box size</td><td>10.0</td><td>3.0</td><td>1.0</td><td>0.5</td></tr><tr><td>Interaction strength</td><td>0.01</td><td>0.1</td><td>0.5</td><td>1.0</td></tr></table>
+
+Table 4: Charged particle system configurations.
+
+Specifically, we list the configurations of the systems in Table 3 and 4. Then the three sequences can be precisely represented as:
+
+1. Sequence 1:  $S_{1} \rightarrow S_{2} \rightarrow S_{3} \rightarrow S_{4} \rightarrow S_{5} \rightarrow S_{6} \rightarrow S_{7} \rightarrow S_{8}$  
+2. Sequence 2:  $S_{1} \rightarrow S_{8} \rightarrow S_{2} \rightarrow S_{7} \rightarrow S_{3} \rightarrow S_{6} \rightarrow S_{5} \rightarrow S_{5}$  
+3. Sequence 3:  $S_{1} \to C_{1} \to S_{9} \to C_{2} \to S_{10} \to C_{3} \to S_{8} \to C_{4}$
+
+For each system, the simulation runs for 6,000 steps, and the observation is sampled every 100 steps, resulting in a 60-step series. During training, the first  $60\%$  part of the trajectory of each
+
+system is fed to the model to generate prediction for the remaining  $40\%$ . For each system sequence, 1,000 sequences are used for training, and another 1,000 sequences are used for testing.
+
+![](images/cb780deb3dc16bf8329d73c58ba5f5ef936c1663a8728744c4f11d9b5a7bda42.jpg)  
+Figure 7: Illustration of the continual learning over different physics systems with different dynamics. The factors determining the dynamics shown in this figure include the type and strength of the interactions.
+
+# A.1.2 BIOLOGICAL CELLULAR SYSTEM
+
+In this paper, we build a novel benchmark, Bio-CDL, containing biological cellular dynamic systems based on Virtual Cell (Schaff et al., 1997; Cowan et al., 2012; Blinov et al., 2017) with different system configurations and variable selection. Currently, Bio-CDL is built based on two types of cellular models. The first one is rule-based model of EGFR receptor interaction with two adapter proteins Grb2 and Shc. The second is a compartmental rule based model of translocation through the nuclear pore of a cargo protein based on the Ran protein (GTPase). An illustration of the Ran system is provided in Figure 8.
+
+Detailed description of these two systems can be found via https://vcell.org/webstart/VCell_Tutorials/VCell6.1_Rule-Based_Tutorial.pdf and https://vcell.org/webstart/VCell_Tutorials/VCell6.1_Rule-Based_Ran_Transport_Tutorial.pdf. Based on these 2 types of models, we construct multiple systems, which are aligned into different system sequences. Details are provided below.
+
+![](images/492373c814243d3f4e1efd487c40d3c9196b72777804aa37e386432bd619f895.jpg)  
+Figure 9
+
+![](images/f6040ff4f49c26d9d2533de8a3dc5cbd83f828ac58286092fdb75da025d57f4b.jpg)  
+Figure 10
+
+Our new benchmark is constructed based on the EGFR receptor interaction model and Ran transportation model. The length of the observe trajectories ranges from 250 to 550 time steps for EGFR systems and 40 to 150 time steps for Ran systems.
+
+We provide below the details of data generation, and will release our generated data later. The code for generating the configuration files for the simulations is contained in
+
+VCell_config_gen.py.
+
+We have restricted the model coefficients to a specific range to ensure the simulations remain stable. After generating the configuration files, they should be uploaded into the VCell platform for generating the simulations. The specific steps are listed below.
+
+1. Download the VCell client through https://vcell.org/run-vcell-software, create a free account and log in. 2. Select the 'BioModel' tab, in the 'Search' box, locate and load the models 'Rule-based_egfr_tutorial' or 'rule-based_Ran_TRANport' under the 'Tutorials' directory for the EGFR model or Ran model, respectively (Figure 9).
+
+![](images/c91363168b20cf9af6f61cfb007a9362fe70fff437e934387aa2956958631eef.jpg)  
+Figure 8: Illustration of the RAN-regulated nucleocytoplasmic transport (Moore, 2013). Briefly speaking, this model depicts the translocation of cargo proteins (Exportin 1) via nuclear pores with the assistance of RAN proteins. RAN-GDP is first activated into RAN-GTP and then binds to cargo molecules (Exportin 1) forming a complex containing RAN-GTP and Exportin 1. Next, the complex is translocated across the nuclear membrane into the cytoplasm with the assistance of RAN. Finally, RAN and Exportin 1 are dissociated after the translocation.
+
+4. In the upper left part of the window, click the 'Applications', then click the 'Simulations' under the 'network_determ' tab (Figure 10).  
+4. In the main window on the upper right, select the 'Simulations' tab and click the first icon under the 'Simulations' tab to create a new simulation (Figure 11).  
+5. Click to select the created simulation, then click the icon with a blue arrow and a gear to load the previously generated simulations. 6. After the simulations are done, select all the simulations and click the icon with a green arrow and a gear to export all the simulation results into a specified folder. 7. Specify the
+
+datastore_path
+
+![](images/bdb09279f2651ea3dd291ea55a2a26b0c90894fcae33ed5fa8bf764b13855950.jpg)  
+Figure 11
+
+and run
+
+generate_dataset.py
+
+to obtain the data files that can be loaded and used by the MS-GODE model.
+
+This implementation of MS-GODE is based on [Pytorch Geometric]https://github.com/rustyls/pytorch/geometric API.
+
+Example command for running the experiments with MS-GODE:
+
+```shell
+python run_models.py --cut_num 20 \
+--nepos 20 \
+--device 5 \
+--n Operators_to_viz 10 \
+--thresholding 'fast' \
+--mask True \
+--dropout_mask '0.0' \
+--batch-size 10 \
+--mode "ex" \
+--normalizeVCellfeat 'universal' \
+--system 'VCell' \
+--repeats 5 \
+--overwrite_results 'False' \
+--fix_random(seed 'False' \
+--save_results True
+```
+
+Example scripts for data generation of cellular/physics systems:
+
+```shell
+python ./data/generate_dataset.py --simulation VCell \
+--num-train 3000 \
+--num-test 3000 \
+--n-balls 5
+```
+
+The generation of cellular simulation data requires first obtaining simulation data from VCell platform, which is introduced in the Appendix A.1.2 of the submission.
+
+```shell
+python ./data/generate_dataset.py --simulation simulation \
+--num-train 3000 \
+--num-test 3000 \
+--n-balls 5
+```
+
+# Setup
+
+The mdoel implementation is based on the following packages:
+
+- [Python 3.6.10](https://www.python.org/)  
+- [Pytorch 1.4.0](https://pytorch.org/)  
+- [pytorch_physical 1.4.3](https://pytorch-beigeometric.readthedocs.io/)  
+- torch-cluster==1.5.3 - torch-scatter==2.0.4 - torch-sparse==0.6.1  
+- [torchdiffreq](https://github.com/rtqichen/torchdiffreq)  
+- [numpy 1.16.1](https://numpy.org/)
+
+In our experiments, we adjust the parameter configurations of these two types of models to construct multiple systems with different dynamics, which are aligned into a sequence for the experiments. We generate simulated data using the Virtual Cell platform (VCell) and pre-process the data. In experiments, we adopt a 4-system sequence using the simulated data by alternatively align 2 EGFR systems and 2 Ran systems. EGFR system is rule-based model of EGFR receptor interaction with two adapter proteins Grb2 and Shc. Ran system is a compartmental rule based model of translocation through the nuclear pore of a cargo protein based on the Ran protein (GTPase). For constructing systems with different dynamics, different EGFR and Ran systems are constructed by selecting a subset of the observed variables from the complete systems. This not only enrich the dynamics diversity across different systems in the learning sequence, but also enhance the learning difficulty over the systems. The IDs (inherited from VCell platform) of the selected variables in  $\mathrm{EGFR}_1$ ,  $\mathrm{EGFR}_2$ ,  $\mathrm{Ran}_1$ ,  $\mathrm{Ran}_2$  are {s15,s16,s21,s22,s29,s33,s34,s38,s45,s47,s59,s74,s9,s96}, {s24,s25,s26,s27,s30,s36,s43,s44,s58,s68}, {Ran_C_nuc,RCC1,s10,s11,s12,s13,s14,s15,s16,s17,s18,s19,s2,s20,s21,s22,s23,s24,s25,s26,s27,s28,s3,s30,s31,s32,s33,s34,s35,s4,s5,s6,s7,s8,s9}, {s14,s15,s16,s23,s24,s25,s31,s5}, respectively.
+
+Using the code for generating cellular system data requires the following steps.
+
+1. Using the script contained in our provided code 'MS-GODE/VCell_config_gen' to generate the configuration files for VCell simulation. Users can freely adjust the configurations in the script.  
+2. Download the VCell application from https://vcell.org/.  
+3. Load the generated configuration files into the VCell platform.  
+4. Generate simulation results. Instruction of using this script is also provided in the ReadMe file of our code.  
+5. Using the script 'MS-GODE/data/generate_dataset.py' for processing the simulation results. After this step, the data can be used for experiments.
+
+Since the entire dataset is much larger than the  $100\mathrm{MB}$  limit, we cannot provide the data in the supplementary materials. But we have provided all code and instruction for generating the data, and will release the complete Bio-CDL benchmark to the public later. Besides the sequence  $\mathrm{EGFR}_1\rightarrow \mathrm{Ran}_1\rightarrow \mathrm{EGFR}_2\rightarrow \mathrm{Ran}_2$  adopted in the experiments, the complete Bio-CDL also contain more different system configurations and sequence configurations. Moreover, we are also working on enriching Bio-CDL with more diverse system types and system sequences.
+
+For each cellular system, we generate 100 iterations of simulation using VCell. Different from the physics systems, the temporal interval between two time steps of cellular system is not constant. Similar to the physics systems, the first  $60\%$  part of the observation of each system simulation is fed into the model to reconstruct the remaining  $40\%$  during training. For each system, we use 20 trajectories for training and 20 trajectories for testing.
+
+# A.1.3 ADDITIONAL DETAILS OF EXPERIMENTAL SETTINGS
+
+For the encoder network, the number of layers is 2, and the hidden dimension is 64. 1 head is used for the attention mechanism. The interaction network of the generator is configured as 1-layer network, and the number of hidden dimensions is 128. Finally, the decoder network is a fully-connected
+
+layer. The model is trained for 20 epochs over each system in the given sequence. We adopt the AdamW optimizer (Loshchilov & Hutter, 2017) and set the learning rate as 0.0005. In our work, task boundaries are provided to the models during training.
+
+# A.1.4 ADDITIONAL DISCUSSION ON PERFORMANCE AND LOW-LEVEL DYNAMICS SHIFT
+
+The performance of MS-GODE is mainly determined on two factors, including the performance on each system and the forgetting issue. Therefore, to further improve the performance under gradual dynamics shift, there are two promising approaches:
+
+1. Enhancing the performance on each single system. As revealed in Ramanujan et al. (2020), the larger the backbone network, the more probable the masked sub-network can reach the capacity of the full backbone network. In other words, increasing the size of the model can improve the performance.  
+2. Reducing the performance decrease after learning new systems. Since MS-GODE completely separate the masks for different dynamics, the forgetting issue has been eliminated. Therefore, the performance decrease after learning new systems mainly comes from the incorrect mask selection. The strategy above to increase model size can help improve the mask selection, because better fitting to each system increases the difference between the masks. Furthermore, the mask selection can also be improved by incorporating a mixture of selection criteria. Specifically, in our experiments, the mode-switching module of MS-GODE selects the mask based on the error of reconstructing the second  $50\%$  of the observation. This can be enhanced by incorporating different splitting ratios and mixing the result. Since the correct mask tend to exhibit lower error with most splitting ratios, selecting the one that succeeds in more splitting ratios could increase the possibility of finding the correct one.
+
+# A.2 EDGE-POPUP ALGORITHM
+
+In this work, we adopt the edge-popup algorithm (Ramanujan et al., 2020) for optimizing the binary masks. The main idea is to optimize a continuous score value for each entry of the mask during the backpropagation, and binarize the values into discrete binary values during forward propagation (Appendix A.2). Accordingly, the strategy for binarizing the mask entry values is a crucial factor influencing the performance. For convenience of the readers, we provide the details about this algorithm in this subsection.
+
+Given a fully connected layer, the input to a neuron  $v$  in the  $l$ -th layer can be formulated as a weighted summation of the output of the neurons in the previous later,
+
+$$
+I _ {v} = \sum_ {u \mathcal {V} ^ {l - 1}} w _ {u v} z _ {u}, \tag {16}
+$$
+
+where  $\mathcal{V}^{l-1}$  denotes the nodes in the previous layer and  $z_u$  refers to the output of neuron  $u$ .
+
+With the edge-popup strategy, the output is reformulated as
+
+$$
+I _ {v} = \sum_ {u \mathcal {V} ^ {l - 1}} w _ {u v} z _ {u} h \left(s _ {u v}\right), \tag {17}
+$$
+
+where  $h(s_{uv})$  is the binary value over the weight  $w_{uv}$  denoting whether this weight is selected and  $s_{uv}$  the continuous score used in gradient descent based optimization. During backpropagation, the gradient will ignore  $h(\cdot)$  and goes through it, therefore the gradient over  $s_{uv}$  is
+
+$$
+g _ {s _ {u v}} = \frac {\partial \mathcal {L}}{\partial I _ {v}} \frac {\partial I _ {v}}{\partial s _ {u v}} = \frac {\partial \mathcal {L}}{\partial I _ {v}} w _ {u v} z _ {u}, \tag {18}
+$$
+
+where  $\mathcal{L}$  denotes the loss function.
+
+During forward propagation,  $h(\cdot)$  can take different options as we mentioned and studied in Section 4.3. In our experiments 4.3, we tested both 'fast selection' and 'top-k selection' with different thresholds. 'Fast selection' set all entries with positive values into '1's and the other entries into '0's. 'Top-k selection' will rank the entry values and set a specified ratio of entries with the largest values into '1's.
+
+# A.3 BASELINES
+
+1. Fine-tune denotes using the backbone model without any continual learning technique.  
+2. Elastic Weight Consolidation (EWC) (Kirkpatrick et al., 2017) applies a quadratic penalty over the parameters of a model based on their importance to maintain its performance on previous tasks.  
+3. Gradient Episodic Memory (GEM) (Lopez-Paz & Ranzato, 2017) selects and stores representative data in an episodic memory buffer. During training, GEM will modify the gradients calculated based on the current task with the gradient calculated based on the stored data to avoid updating the model into a direction that is detrimental to the performance on previous tasks.  
+4. Learning without Forgetting (LwF) (Li & Hoiem, 2017) is a knowledge distillation based method, which minimizes the discrepancy between the old model output and the new model output to preserve the knowledge learned from the old tasks.  
+5. Bias Correction based Memory Replay (BCMR) (Chrysakis & Moens, 2023). This baseline is constructed by integrating the navie memory replay with the data sampling bias correction strategy (Chrysakis & Moens, 2023). In other words, the method does not train the data immediately after observing the data. Instead, it stores the observed data into a memory buffer. Whenever testing is required, the model will be trained over all buffered data.  
+6. Scheduled Data Prior (SDP) (Koh et al., 2023) considers that the importance of new and old data is dependent on the specific characteristics of the given data, therefore balance the contribution of new and old data based on a data-driven approach. Since the condition assumed for  $c^2$  in SDP (Koh et al., 2023) does not align with the actual situation in our experiments, the parameter  $\alpha$  and  $\beta$  is Eq. (9) of SDP (Koh et al., 2023) are manually selected in our experiments.  
+7. Joint Training (Joint) jointly trains a given model on all data instead of following the sequential continual learning setting.
+
+# A.4 MODEL EVALUATION
+
+Different from standard learning setting with only one task to learn and evaluate, in our setting, the model will continually learn on a sequence of systems, therefore the setting and evaluation are significantly different. In the model training stage, the model is trained over a system sequence. In the testing stage, the model will be tested on all learned tasks. Therefore the model will have multiple performance corresponding to different tasks, and the most thorough evaluation metric is the performance matrix  $\mathbf{M}^p\in \mathbb{R}^{N\times N}$ , where  $\mathbf{M}_{i,j}^p$  denotes the performance on the  $j$ -system after learning from the 1-st to the  $i$ -th system, and  $N$  is the number of systems in the sequence. In our experiments, each entry of  $\mathbf{M}^p$  is a mean square error (MSE) value. To evaluate the overall performance on a sequence, the average performance (AP) over all learnt tasks after learning multiple tasks could be calculated. For example, i.e.,  $\frac{\sum_{j = 1}^{i}\mathbf{M}_{N,j}^{p}}{N}$  corresponds to the average model performance after learning the entire sequence with  $N$  tasks. Similarly, the average forgetting (AF) after  $N$  tasks can be formulated as  $\frac{\sum_{j = 1}^{N - 1}\mathbf{M}_{N,j}^{p} - \mathbf{M}_{j,j}^{p}}{N - 1}$ . These metrics are widely adopted in continual learning works (Chaudhry et al., 2018; Lopez-Paz & Ranzato, 2017; Liu et al., 2021; Zhang et al., 2023a; Zhou & Cao, 2021), although the names are different in different works.
+
+For convenience, the performance matrix can be visualized as a color map (Figure 6). For example, they are named as Average Accuracy (ACC) and Backwarde Transfer (BWT) in (Chaudhry et al., 2018; Lopez-Paz & Ranzato, 2017), Average Performance (AP) and Average Forgetting (AF) in (Liu et al., 2021), Accuracy Mean (AM) and Forgetting Mean (FM) in (Zhang et al., 2023a), and performance mean (PM) and forgetting mean (FM) in (Zhou & Cao, 2021).
+
+# A.5 PERFORMANCE MATRIX ANALYSIS
+
+Given a visualized performance matrix, we should approach it from two different dimensions. First, the  $i$ -th row of the matrix denotes the performance on each previously learned system after the
+
+model has learned from the 1-st system to the  $i$ -th system. Second, to check the performance of a specific system over the entire learning process, we check the corresponding column. For example, in Figure 6 (e), each column maintains a stable color from the top to the bottom. This indicates that the performance of each system is perfectly maintained with little forgetting. But if the color becomes darker and darker from top to bottom (increasing MSE), it indicates that the corresponding method is exhibiting obvious forgetting problem.
\ No newline at end of file
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+# LEARNING THE COMPLEXITY OF WEAKLY NOISY QUANTUM STATES
+
+Yusen Wu $^{1,2}$ , Bujiao Wu $^{*3,4,5,6}$ , Yanqi Song $^{7}$ , Xiao Yuan $^{6}$  & Jingbo B. Wang $^{\dagger 2}$
+
+1 School of Artificial Intelligence, Beijing Normal University, Beijing, 100875, China  
+2 Department of Physics, The University of Western Australia, Perth, WA 6009, Australia  
+3 Shenzhen Institute for Quantum Science and Engineering,  
+Southern University of Science and Technology, Shenzhen 518055, China  
+4 International Quantum Academy, Shenzhen 518048, China  
+5 Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany  
+6 Center on Frontiers of Computing Studies, Peking University, Beijing 100871, China  
+7 China Academy of Information and Communications Technology, Beijing, 100191, China  
+$^{*}$ bujiaowu@gmail.com, †jingbo.wang@uwa.edu.au
+
+# ABSTRACT
+
+Quantifying the complexity of quantum states is a longstanding key problem in various subfields of science, ranging from quantum computing to the black-hole theory. The lower bound on quantum pure state complexity has been shown to grow linearly with system size (Haferkamp et al., 2022). However, extending this result to noisy circuit environments, which better reflect real quantum devices, remains an open challenge. In this paper, we explore the complexity of weakly noisy quantum states via the quantum learning method. We present an efficient learning algorithm, that leverages the classical shadow representation of target quantum states, to predict the circuit complexity of weakly noisy quantum states. Our algorithm is proved to be optimal in terms of sample complexity accompanied with polynomial classical processing time. Our result builds a bridge between the learning algorithm and quantum state complexity, meanwhile highlighting the power of learning algorithm in characterizing intrinsic properties of quantum states.
+
+# 1 INTRODUCTION
+
+The concept of quantum complexity has deep connections to high-energy physics, quantum many-body systems, and black-hole physics (Bouland et al., 2019b; Brown et al., 2016a,b; Stanford & Susskind, 2014; Susskind, 2016). A problem is deemed "easy" if it is soluble with a (quantum) circuit whose size grows polynomially with the size of the inputs, while the problem is deemed "hard" if the size of the (quantum) circuit scales exponentially. In the context of quantum computation, the complexity of an  $n$ -qubit quantum state corresponds to the minimum number of gates required to prepare it from the initial state  $|0^n\rangle$  (Haferkamp et al., 2022). Brown and Susskind's conjecture suggests that the complexity of quantum states generated by random quantum circuits grows linearly before it saturates after reaching an exponential size (Brown & Susskind, 2018; Susskind, 2018), which is supported by the complexity geometry theory proposed by Nielsen et al. (2006). Recent works theoretically proved this conjecture by connecting quantum state complexity to unitary  $t$ -designs (Brandão et al., 2021; Jian et al., 2022) and the dimension of semi-algebraic sets (Haferkamp et al., 2022), demonstrating a rigorous computational ability separation between shallow and deep quantum circuits.
+
+However, in the practical world, the quantum system may interact with the surrounding environment, which inevitably introduces a noise signal, making the pure state noisy. For example, in the current noisy-intermediate-scale-quantum device (Boixo et al., 2018; Arute et al., 2019; Zhong et al., 2020; Wu et al., 2021), a certain level (constant) of noise exists in each quantum gate, resulting in noisy states prepared by  $\Omega (\log n)$ -depth noisy circuits is classically simulable in both mean value computation and random circuit sampling problems (Stilck Franca & Garcia-Patron, 2021; Aharonov et al., 2022). These facts exhibit a trend that is completely different from the case of pure quantum
+
+states: increasing the depth of the noisy circuit reduces the quantum state complexity. However, it is still unclear how to extend previous arts (Brandão et al., 2021; Jian et al., 2022; Haferkamp et al., 2022) to the noisy environment, and how to characterize the noisy quantum state complexity is still an open problem.
+
+On the other hand, learning algorithms have recently been considered a powerful means to study and understand many-particle quantum systems and the associated quantum processes (Carleo & Troyer, 2017; Carrasquilla & Melko, 2017; Glasser et al., 2018; Torlai et al., 2018; Moreno et al., 2020; Torlai & Melko, 2016; Schindler et al., 2017; Greplova et al., 2020; Wetzel, 2017; Huang et al., 2022; Wu et al., 2023b; Du et al., 2022). Specifically, Huang et al. (2022) explored the power of the learning algorithm in classifying quantum phases of matter by learning through the classical shadow (Huang et al., 2020), an efficient classical representation of the quantum state. The strong connection between quantum state complexity and quantum phase transition phenomena Huang et al. (2015) motivates us to explore the complexity of noisy quantum states using a learning algorithm.
+
+Here, we focus primarily on the complexity of weakly noisy quantum states to avoid the anti-concentration property (Deshpande et al., 2022), an undesirable distribution in quantum simulation and quantum approximate optimization algorithms. The  $n$ -qubit weakly noisy quantum states are generated by circuits with a depth of  $\tilde{R} \leq O(\mathrm{poly}\log n)$  and a noise strength of  $\mathcal{O}(1 / n)$ .<sup>1</sup> We focus on an essential and natural problem: Given  $\mathrm{poly}(n)$  copies of  $n$ -qubit unknown weakly noisy quantum state, how to predict its complexity?
+
+To answer this question, we develop a learning approach to solve this problem, which is illustrated in Fig. 1. Following the strong quantum state complexity proposed by Brandão et al. (2021), the learning algorithm pertains to determine whether an unknown weakly noisy state  $\rho^2$  can be  $\epsilon$ -distinguished from the maximally mixed state by a measurement operator induced by a specific quantum circuit architecture (QCA) (shown in Fig. 1(b)). The size of the measurement operator provides a state complexity upper bound (Brandão et al., 2021). More specifically, the problem asks whether there exists a measurement operator  $M$  induced by a specific QCA, such that
+
+$$
+\max  _ {\substack {M = U \left|0^{n}\right\rangle \left\langle 0^{n} \mid U^{\dagger} \right. \\ U \in \mathcal{U}_{A}(R)}} \left| \operatorname{Tr}\left(M\left(\rho - I_{n} / 2^{n}\right)\right) \right| \geq \Omega (1 - \epsilon), \tag{1}
+$$
+
+where  $\mathcal{A}$  denotes a QCA architecture and  $\mathcal{U}_A(R)$  contains all  $R$ -depth QCA circuits induced by  $\mathcal{A}$ . The quantum state complexity can be bounded by  $\mathcal{O}(nR)$ , provided that Eq. 1 is satisfied. In general, the circuit set  $\mathcal{U}_A$  contains an exponential number of candidate circuits  $U$ , making direct enumeration impractical. We address this challenge by revealing a specific property of QCA circuits, as outlined in Theorem 1. This property allows the behavior of an observable  $M$ , generated by any circuit in  $\mathcal{U}_A$ , to be approximated by a linear combination of  $\mathrm{poly}(n)$  random QCA circuits from  $\mathcal{U}_A$ . Leveraging this, we reformulate the complexity prediction problem into the optimization of a linear function over a compact set, which is both sample-efficient and computationally efficient, as demonstrated in Theorem 2 and illustrated in Fig. 1 (c). We finally show the sample complexity of our learning algorithm is optimal with respect to the noisy circuit depth  $\tilde{R}$ , as demonstrated in Theorem 3.
+
+We note that learning the quantum state complexity has broad applications. Firstly, the predicted complexity enables the classification of the target unknown state  $\rho$  into one of the following scenarios: (i)  $\rho$  can be approximated by an estimator generated by constant-depth noiseless circuits, which is amenable to classical simulation (Napp et al., 2022; Bravyi et al., 2021), (ii)  $\rho$  can be approximated using noiseless circuits with sub-logarithmic depth, a task that may present classical computational challenges, as discussed in Ref. (Deshpande et al., 2022), and (iii)  $\rho$  cannot be approximated by any linear combination of circuits constrained to a specific architecture with a depth of at most  $\log n$ . Besides benchmarking the NISQ computational power (whether its output can be classically simulated), the proposed quantum algorithm and predicted weakly noisy state complexity may have
+
+practical applications in various fields, including understanding the black hole theory and the quantum phase transition. For example, in the context of the Anti-de-Sitter-space/Conformal Field Theory (AdS/CFT) correspondence, the "complexity equals volume" conjecture (Stanford & Susskind, 2014) suggests that the boundary state of the correspondence has a complexity proportional to the volume behind the event horizon of a black hole in the bulk geometry. However, when measuring the boundary state, the interaction with the surrounding environment inevitably introduces a noise signal, making the pure state noisy. Furthermore, in quantum many-body systems, quantum phase transitions occur when the external parameters varies (Sachdev, 1999), and the ability to correctly predict the quantum phase transition boundary can help us understand many strong-correlated systems (Zheng et al., 2017). It is known that quantum topological phases can be distinguished by their ground state complexity (Huang et al., 2015), and the shadow tomography (Huang et al., 2020; 2022) method utilized quantum channels to provide a noisy state approximation to the ground state. Predicting complexity of such noisy ground state approximations may recognize the topological phases of matter.
+
+![](images/dfe31a8c24fa5b8ce276495e93c31afeb40852477d4dd7aabbf97b340e1ce245.jpg)  
+(a)  
+(b)
+
+![](images/ab2e4d6ad83a4b8e08e6eecc27c8c4a99f0653d0b37805b6948b4dd6ac77730a.jpg)  
+Figure 1: (a) An efficient quantum-to-classical representation conversion method that leverages the classical shadow of a noisy state (Huang et al., 2020). By measuring  $\mathrm{poly}(n)$  copies of the state, it can construct a classical representation  $\hat{\rho}$  that enables the prediction of the state's properties with a rigorous performance guarantee. Here,  $\rho_{\mathrm{un}}$  represents a noisy state generated by a  $\bar{R}$ -depth noisy quantum circuit model which is defined as Def. 2, and  $\hat{\rho}$  represents the classical shadow of  $\rho_{\mathrm{un}}$ . (b) A QCA model with  $R$  blocks, where each block corresponds to a causal slice, and the overall architecture follows the design of  $\mathcal{A}$ . (c) Visualization of the complexity prediction process via a quantum learning algorithm (Alg. 1), where  $\mathcal{L}_R(\vec{\beta},\hat{\rho}_{\mathrm{QCA}},\hat{\rho})$  represents a metric in measuring the distance between  $\hat{\rho}$  and  $\hat{\rho}_{\mathrm{QCA}}$  states, which is defined in Eq. 11. The blue Bloch sphere illustrates the relationship between a  $\bar{R}$ -block weakly noisy quantum state  $\hat{\rho}$  and its nearest pure state. All pure states reside on the surface of the  $n$ -qubit Bloch sphere, with the maximum mixed state  $I_n / 2^n$  located at the center of the sphere. In the regime where  $\tilde{R} < \mathcal{O}(\log n)$  and for a small noise strength  $p < 1 / n$ , the weakly noisy state  $\hat{\rho} = \rho_1$  is located near the surface of the sphere, while  $\hat{\rho} = \rho_2$  locates near the maximum mixed state.
+
+![](images/e72d054b5bfef11318cac3b242226244903e81f822be971c80a4d25378fc055a.jpg)  
+(c)
+
+# 2 THEORETICAL BACKGROUND
+
+To clearly demonstrate the motivation and contribution of this work, we review the related theoretical backgrounds in terms of quantum state complexity, noisy quantum states, and the architecture of quantum circuits.
+
+Here, we consider the quantum state complexity of an  $n$ -qubit quantum pure state  $|\psi \rangle$ . The complexity of a quantum state is the minimal circuit size required to implement a measurement operator that
+
+suffices to distinguish  $|\psi \rangle \langle \psi |$  from the maximally mixed state  $I_{n} / 2^{n}$ . Since any pure state  $|\psi \rangle \langle \psi |$  satisfies  $\frac{1}{2}\| |\psi \rangle \langle \psi | - I_n / 2^n\| _1 = 1 - 2^{-n}$ , which is achieved by the optimal measurement strategy  $M = |\psi \rangle \langle \psi |$ , such trace distance can be used to quantify the quantum state complexity. Let  $\mathbb{H}_{2^n}$  denote the space of  $2^{n}\times 2^{n}$  Hermitian matrices, and for fixed  $c\in \mathbb{N}$ , we consider a class of measurement operators  $M_{c}(2^{n})\subset \mathbb{H}_{2^{n}}$  that can be constructed by at most  $c$  2-local gates. The maximal bias achievable for quantum states with such a restricted set of measurements of the solution is defined as:
+
+$$
+\xi_ {Q S} (c, | \psi \rangle) = \max  _ {M \in M _ {c} \left(2 ^ {n}\right)} \left| \operatorname {T r} \left(M \left(| \psi \rangle \langle \psi | - I _ {n} / 2 ^ {n}\right)\right) \right|, \tag {2}
+$$
+
+where  $|\psi \rangle = V|0^n\rangle$  for some  $V\in \mathrm{SU}(2^{n})$ . Noting that the above metric  $\xi_{QS}(c,|\psi \rangle)$  degenerates to  $1 - 2^{-n}$  when  $c\to \infty$ . For example, if the quantum state  $|\psi \rangle$  can be easily prepared by a quantum computer (such as computational basis states),  $\xi_{QS}(c,|\psi \rangle)$  converges to  $1 - 2^{-n}$  rapidly as  $c$  increases. In contrast, for a general quantum state  $|\psi \rangle$ ,  $\xi_{QS}$  requires an exponentially large  $c$  to approach  $1 - 2^{-n}$ . Using this property, the quantum state complexity can be defined as follows:
+
+Definition 1 (Approximate State Complexity (Brandão et al., 2021)) Given an integer  $c$  and  $\epsilon \in (0,1)$ , we say a pure quantum state  $|\psi\rangle$  has  $\epsilon$ -strong state complexity at most  $c$  if and only if
+
+$$
+\xi_ {Q S} (c, | \psi \rangle) \geq 1 - \frac {1}{2 ^ {n}} - \epsilon , \tag {3}
+$$
+
+which is denoted as  $C_{\epsilon}(|\psi \rangle) \leq c$ .
+
+Due to imperfections in quantum hardware devices, two causal slices  $^3$  are separated by a quantum channel  $\mathcal{E}$ . In this paper, we consider  $\mathcal{E}$  to represent a general noise channel which is both gate-independent and time-invariant, such as the local-depolarizing channel, the global-depolarizing channel, the bit-flip channel and other common noise models.
+
+Definition 2 (Weakly Noisy Quantum State) We assume that the noise in the quantum device is modeled by a gate-independent Pauli noise channel  $\mathcal{E}$  with error rate  $p$ . Let  $\mathcal{U}$  be a causal slice, and let  $\mathcal{E} \circ \mathcal{U}$  be the representation of a noisy gate. We define the  $\tilde{R}$ -depth noisy quantum state with noise strength  $p$  as
+
+$$
+\rho_ {p, \tilde {R}} = \mathcal {E} \circ \mathcal {U} _ {\tilde {R}} \circ \mathcal {E} \circ \mathcal {U} _ {\tilde {R} - 1} \circ \dots \circ \mathcal {E} \circ \mathcal {U} _ {1} (| 0 ^ {n} \rangle \langle 0 ^ {n} |). \tag {4}
+$$
+
+We use the term "weakly noisy quantum states" to refer to noisy states  $\rho_{p,\tilde{R}}$  with  $\tilde{R} \leq \mathcal{O}(\mathrm{poly}\log n)$  and small error rate  $p$  such that  $p < \mathcal{O}(n^{-1})$ .
+
+# 3 PROBLEM STATEMENT
+
+# 3.1 WEAKLY NOISY STATE COMPLEXITY
+
+In this paper, we assume multiple copies of a weakly noisy quantum state  $\rho$  are provided, and we utilize a learning approach to predict its quantum state complexity, accessing only one copy at a time. In the learning phase, we don't have any information on  $\rho$ , as a result, it is generally difficult to exclude shortcuts that could improve the efficiency of a computation. As a result, deriving quantum complexity measures for weakly noisy states may be challenging without additional assumptions. Here, we supplement limitations to the operator architecture, which leads to the limited-structured quantum state complexity.
+
+Definition 3 (Limited-Structured (LS) Complexity of Weakly Noisy State) Given an integer  $c$  and  $\epsilon \in (0,1)$ , we say a weakly noisy state  $\rho$  has  $\epsilon$ -LS complexity at most  $c$  if and only if
+
+$$
+\max  _ { \begin{array}{c} M _ {c} = U | 0 ^ {n} \rangle \langle 0 ^ {n} | U ^ {\dagger} \\ U \in \mathcal {U} _ {A} ([ c / L ]) \end{array} } \left| \operatorname {T r} \left(M _ {c} \left(\rho - I _ {n} / 2 ^ {n}\right)\right) \right| \geq 1 - \frac {1}{2 ^ {n}} - \epsilon \tag {5}
+$$
+
+which is denoted as  $C_{\epsilon}^{\mathrm{lim}, \mathcal{A}}(\rho) \leq c$ . The notation  $L$  represents the number of gates in each layer of  $U \in \mathcal{U}_{\mathcal{A}}^{4}$ , and  $\epsilon$  is termed as the LS error.
+
+The measurement operator  $M_{c}$  (in Def 3) is limited by the architecture  $\mathcal{U}_{A}$ . It is interesting to note that the LS complexity provides an upper bound for  $C_{\epsilon}(\rho_{\mathrm{un}})$ , that is,
+
+$$
+C _ {\epsilon} (\rho) \leq C _ {\epsilon} ^ {\operatorname * {l i m}, \mathcal {A}} (\rho) \leq c.
+$$
+
+Now, we provide an insight on why Def. 1 can be modified to Def. 3 which characterized the weakly noisy quantum state complexity.
+
+Fact 1 Suppose we are given a general noise channel  $\mathcal{E}(\cdot) = \sum_{l=1}^{K} K_l(\cdot) K_l^\dagger$  and a  $\tilde{R}$ -depth noisy state  $\rho_{\tilde{R}} = \bigcirc_{r=1}^{\tilde{R}} \mathcal{E} \circ \mathcal{U}_r(|0^n\rangle \langle 0^n|)$ , where the  $K_l$  represent  $n$ -qubit Kraus operators and the  $\mathcal{U}_r$  are drawn independently from a unitary 2-design set  $\mathbb{U}$ . If the following relationship holds:
+
+$$
+\tilde {R} \leq \frac {\log \left(\frac {F - 1}{d (d + 1)} (\eta - 1 / d) ^ {- 1}\right)}{\log \left(d ^ {2} - 1\right) - \log (F - 1)}, \tag {6}
+$$
+
+where  $d = 2^n$  and  $F = \sum_{l=1}^{K} |\mathrm{Tr}(K_l)|^2$ , then for each noisy state  $\rho_{\tilde{R}}$ , their corresponding  $\tilde{R}$ -depth pure state  $|\Psi_{\mathcal{U}_1, \ldots, \mathcal{U}_{\tilde{R}}}\rangle = \bigcirc_{r=1}^{\tilde{R}} \mathcal{U}_r(|0^n\rangle \langle 0^n|)$  satisfies
+
+$$
+\mathbb {E} _ {\mathcal {U} _ {1}, \dots , \mathcal {U} _ {\bar {R}} \sim \mathbb {U}} \left[ \langle \Psi_ {\mathcal {U} _ {1}, \dots , \mathcal {U} _ {\bar {R}}} | \left(\bigcirc_ {r = 1} ^ {\tilde {R}} \mathcal {E} \circ \mathcal {U} _ {r} (| 0 ^ {n} \rangle \langle 0 ^ {n} |\right) | \Psi_ {\mathcal {U} _ {1}, \dots , \mathcal {U} _ {\bar {R}}} \rangle \right] \geq \eta . \tag {7}
+$$
+
+Specifically,  $\tilde{R}$  saturates to the bound given in Eq. 6 implies
+
+$$
+\eta (\tilde {R}) = \left(\frac {F - 1}{d ^ {2} - 1}\right) ^ {\tilde {R} - 1} \frac {F - 1}{d (d + 1)} + \frac {1}{d}. \tag {8}
+$$
+
+We leave proof details to Appendix F. A particularly noteworthy aspect of the presented fact is its applicability to general noise models. This allows us to establish a precise relationship between the depth of a quantum circuit  $\tilde{R}$ , the noise model  $\mathcal{E}$ , and the quality of approximation (purity)  $\eta(\tilde{R}) = 1 - \epsilon$  that can be achieved. This result further implies using measurement operators prepared by pure states suffice to distinguish a weakly noisy quantum state to the maximally mixed state, as defined by Def. 3. In light of this, we consider a noisy quantum state with depth  $\tilde{R} = \mathcal{O}(\mathrm{poly~log~}n)$ , where the noise is modeled as a local depolarizing channel  $\mathcal{E}_i(\cdot) = (1 - p)(\cdot) + p\mathrm{Tr}_i(\cdot)I_2 / 2$  and we have  $\eta(\tilde{R}) \approx (1 - p\tilde{R})$  by Eq. 8. Furthermore, we note that Eq. 7 implies that there exists a quantum weakly noisy state  $\bigcirc_{r=1}^{\tilde{R}}\mathcal{E} \circ \mathcal{U}_r(|0^n\rangle\langle 0^n|)$  and a quantum pure state  $|\Psi_{\mathcal{U}_1,\ldots,\mathcal{U}_{\tilde{R}}}\rangle = \mathcal{U}_{\tilde{R}} \cdots \mathcal{U}_1|0^n\rangle$  such that
+
+$$
+\left. \right. \operatorname {T r} \left[ M _ {\mathcal {U} _ {\tilde {R}} \dots \mathcal {U} _ {1}} \left(\bigcirc_ {r = 1} ^ {\tilde {R}} \mathcal {E} \circ \mathcal {U} _ {r} \left(| 0 ^ {n} \rangle \langle 0 ^ {n} |\right) - I _ {n} / 2 ^ {n}\right)\right] \approx 1 - p \tilde {R} - 2 ^ {- n} \tag {9}
+$$
+
+in the worst-case scenario over the choice of  $\mathcal{U}_1,\dots,\mathcal{U}_{\tilde{R}}\sim \mathbb{U}$  (otherwise Eq. 7 may be violated), where the measurement operator  $M_{\mathcal{U}_{\tilde{R}}\ldots \mathcal{U}_1} = |\Psi_{\mathcal{U}_1,\ldots ,\mathcal{U}_{\tilde{R}}}\rangle \langle \Psi_{\mathcal{U}_1,\ldots ,\mathcal{U}_{\tilde{R}}}|.$  As a consequence, Definition 1 can be naturally transferred into Definition 3 which characterizes the complexity of quantum weakly noisy states.
+
+# 4 QUANTUM LEARNING TASK AND ALGORITHM
+
+We start by introducing the intrinsic structure of  $\mathcal{U}_{\mathcal{A}}$  to devise a parameterized observable  $M$  for distinguishing  $\rho$  from the maximal entangled state. In general, optimizing an observable  $M = U|0^{n}\rangle \langle 0^{n}|U^{\dagger}$  with  $\mathcal{U}_A(R)$  is challenging, however, the intrinsic structure of a specific QCA allows for efficient optimization of the observable  $M$ . Specifically, we randomly generate a set of  $N = \mathrm{poly}(R,n)$  quantum neural network states  $\hat{\rho}_{\mathrm{QCA}}(R,\mathcal{A},N) = \{\hat{\rho}_i\}_{i = 1}^N$  with  $\hat{\rho}_i$  represents the classical shadow representation of  $U_{i}|0^{n}\rangle$  for  $U_{i}\in \mathcal{U}_{A}(R)$  (Huang et al., 2020) (Details refer to Appendix K). We then design a parameterized operator that takes the form of  $M(\vec{\beta}) = \sum_{i = 1}^{N}\beta_{i}\hat{\rho}_{i}$ .
+
+Theorem 1 (Intrinsic Structure of specific QCA) Randomly select  $N$  unitaries from the QCA model  $\mathcal{U}_A(R)$  to generate  $\hat{\rho}_{\mathrm{QCA}}(R,\mathcal{A},N) = \{\hat{\rho}_i\}_{i = 1}^N$ , where each layer in  $U_{i}$  contains  $L$  gates. Then for any  $n$ -qubit quantum state  $\rho$  and projector  $M_{\vec{x}} = U(\vec{x})|0^{n}\rangle \langle 0^{n}|U^{\dagger}(\vec{x})$  with
+
+$U(\vec{\pmb{x}})\in \mathcal{U}_{\mathcal{A}}(R)^5$  , there exists a vector  $\vec{\beta} (\vec{x})$  belongs to an  $N$  -dimensional compact set  $\mathcal{D}_{\beta}^{6}$  and  $\sum_{j = 1}^{N}\vec{\beta}_{j}(\vec{x}) = 1$  , such that
+
+$$
+\mathbb {E} _ {\vec {\boldsymbol {x}}} \left| \operatorname {T r} \left[ \sum_ {j = 1} ^ {N} \vec {\beta} _ {j} (\vec {\boldsymbol {x}}) \hat {\rho} _ {i} \rho \right] - \operatorname {T r} \left[ M _ {\vec {\boldsymbol {x}}} \rho \right] \right| \leq \sqrt {\frac {L R n ^ {2} \log n}{N}}. \tag {10}
+$$
+
+If  $N = LRn^{2}\log n / \epsilon^{2}$ , the above approximation error is upper bounded by  $\epsilon$  (We leave proof details to the Appendix  $G$ ).
+
+# 4.1 METRIC CONSTRUCTION
+
+Here, we assume QCA states  $\hat{\rho}_i$  are sampled from  $\mathcal{U}_A(R)$  following the probability distribution  $\vec{q}$ , as a result, the metric
+
+$$
+\mathcal {L} _ {R} (\vec {\boldsymbol {\beta}}) = \left| \mathbb {E} _ {\hat {\rho} _ {i} \sim \vec {\boldsymbol {q}}} \operatorname {T r} \left[ M (\vec {\boldsymbol {\beta}}) \left(\hat {\rho} _ {i} - \rho\right) \right] \right| \tag {11}
+$$
+
+is defined over the compact set  $\vec{\beta} \in D_{\vec{\beta}}$ . Given a specific parameter  $\vec{\beta}$ , we can efficiently calculate the corresponding value of  $\mathcal{L}_R(\vec{\beta})$  using classical shadow techniques (Huang et al., 2020; Wu et al., 2023a; Nguyen et al., 2022; Akhtar et al., 2022; Bertoni et al., 2022).
+
+To clarify our algorithm, we define the upper decision interval  $\mathrm{UDI}(\epsilon)$  and the lower decision interval  $\mathrm{LDI}(\epsilon)$  as follows.
+
+Definition 4 (Decision interval)  $\mathrm{UDI}(\epsilon)$  is defined as the integer interval  $[u,\infty)$  such that for any  $R\in \mathrm{UDI}(\epsilon)$ , it holds that  $\max_{\vec{\beta}}\mathcal{L}_R\left(\vec{\beta}\right)\leq \epsilon$ . Similarly,  $\mathrm{LDI}(\epsilon)$  is defined as the integer interval  $[0,l]$  such that for any  $R\in \mathrm{LDI}(\epsilon)$ , we have  $\min_{\vec{\beta}}\mathcal{L}_R\left(\vec{\beta}\right)\geq 2\epsilon$ .
+
+Before proposing the quantum learning algorithm, we need the following lemmas to support our method. We elaborate proof details in Appendices H.1 and H.2.
+
+Lemma 1 Consider the metric function  $\mathcal{L}_R(\vec{\beta})$ . If the relationship  $\max_{\vec{\beta}} \mathcal{L}_R(\vec{\beta}) \leq \epsilon$  holds for any distribution  $\vec{q}$ , then  $\rho_{\mathrm{un}}$  has the state complexity  $C_{\epsilon}^{\lim, \mathcal{A}}(\rho_{\mathrm{un}}) \leq LR$ , where  $C_{\epsilon}^{\lim, \mathcal{A}}(\cdot)$  is defined in Def. 3.
+
+Lemma 2 If there exists a distribution  $\vec{q}$  such that  $\min_{\vec{\beta}} \mathcal{L}_R(\vec{\beta}) > 2\epsilon$ , then with nearly unit probability,  $\rho_{\mathrm{un}}$  follows the quantum state complexity lower bound  $C_{\epsilon}^{\lim, \mathcal{A}}(\rho_{\mathrm{un}}) > LR$ .
+
+# 4.2 LEARNING TASK STATEMENT
+
+Given the preliminary background above, we now formally define the learning task.
+
+Task 1 (Structured Complexity Prediction  $(\mathrm{SCP}(\mathcal{A},\epsilon)))$  Given an architecture  $\mathcal{A}$ , an  $n$ -qubit weakly noisy quantum state  $\rho_{\mathrm{un}}$  (as defined in Eq. 4), and an approximation error  $\epsilon$ , design a learning algorithm  $\mathrm{QL}_n$  that runs on an ideal quantum device with polynomially many qubits in  $n$ , and learns from the unknown quantum state  $\rho_{\mathrm{un}}$ , such that the following conditions hold:
+
+1. (Completeness) If there exists an integer  $x < \log n$  such that  $x \in \mathrm{UDI}(\epsilon)$ , then  $\mathrm{QL}_n$  returns True.  
+2. (Soundness) If  $\log n \in \mathrm{LDI}(\epsilon)$ , then  $\mathrm{QL}_n$  returns False.  
+3. (Indeterminate case) If all integers  $\pmb{x} \in [0, \log n]$  lie in  $\mathrm{UDI}^c(\epsilon) \cup \mathrm{LDI}^c(\epsilon)$ , then  $\mathrm{QL}_n$  may return either True or False arbitrarily.
+
+To provide a clearer analysis of this task, we introduce the following definition.
+
+Definition 5 (Distinguishable Property,  $\mathrm{DP}(\mathcal{A},\epsilon)$ ) We say an  $n$ -qubit weakly noisy quantum state  $\rho_{\mathrm{un}}$  satisfies  $\mathrm{DP}(\mathcal{A},\epsilon)$  if  $\rho_{\mathrm{un}}$  can be distinguished from  $I_n / 2^n$  with the bias at least  $1 - 2^{-n} - \epsilon$  by using a  $R$ -depth ( $R < \log n$ ) QCA measurement  $M_{\mathrm{QCA}} = U|0^n\rangle \langle 0^n|U^\dagger$ , where  $U \in \mathcal{U}_A$ .
+
+We note that the learning algorithm  $\mathrm{QL}_n$  returns True (respectively, False) to indicate that  $\rho_{\mathrm{un}}$  does (does not) satisfy the  $\mathrm{DP}(\mathcal{A},\epsilon)$  property.
+
+# 4.3 QUANTUM LEARNING ALGORITHM FOR LIMITED-STRUCTURED COMPLEXITY ANALYSIS
+
+Based on Theorem 1, optimizing the metric function  $\mathcal{L}_R(\vec{\beta})$  can be limited into a compact set. Taking the maximization task as an example, we show how to utilize the Bayesian optimization on a compact set as Alg. 2 in Appendix I, which is termed as the BMaxS. Specifically, the subroutine BMaxS, which is essentially a Bayesian optimization, takes unknown state  $\rho_{\mathrm{un}}$ , QCA state set, and LS-error  $\epsilon$  as inputs, after  $T$  steps, it outputs True if  $R \in \mathrm{UDI}(\epsilon)$  holds, outputs False otherwise. This subroutine can be used as a building block to construct a quantum learning algorithm that verifies whether the completeness condition of Task 1 holds. Likewise, BMaxS can also be applied to verify the soundness condition. For the remaining (inconclusive) case, we return a random outcome (True or False), thereby providing an efficient solution to Task 1.
+
+Given the fact that a structured unitary set that  $\mathcal{U}_A(R - 1)$  is strictly contained in  $\mathcal{U}_A(R)$  (Haferkamp et al., 2022), the boolean function
+
+$$
+\mathcal {P} (R, \epsilon) = \operatorname {B M a x S} \left(\rho_ {\mathrm {u n}}, \hat {\rho} _ {\mathrm {Q C A}} (R, \mathcal {A}, N), T, \epsilon\right) \tag {12}
+$$
+
+is a monotone predicate in the interval  $R \in [1, \log n]$ . Therefore a quantum learning algorithm can be designed by a binary search program where  $\mathcal{P}(R, \epsilon)$  is packaged as an oracle. A monotone predicate  $\mathcal{P}$  is a boolean function defined on a totally ordered set with the property: if  $\mathcal{P}(R, \epsilon) = \text{True}$ , then  $\mathcal{P}(R', \epsilon) = \text{True}$  for all  $R' \geq R$  in the domain. In our case,  $\mathcal{P}$  returns True for the input  $R$  when  $R \in \mathrm{UDI}(\epsilon)$  holds. As a result, if the noisy state  $\rho_{\mathrm{un}}$  satisfies the  $\mathrm{DP}(\mathcal{A}, \epsilon)$  property, the  $\mathrm{QL}_n$  outputs the minimum  $R_{\mathrm{min}} \in [1, \log n]$  enabling  $C_{\epsilon}^{\lim, \mathcal{A}}(\rho_{\mathrm{un}}) \leq LR_{\mathrm{min}}$  (True). Otherwise, we test whether  $\log n \in \mathrm{LDI}(\epsilon)$ . If this holds, the  $\mathrm{QL}_n$  outputs  $C_{\epsilon}^{\lim, \mathcal{A}}(\rho_{\mathrm{un}}) > L \log n$  (False). Otherwise, the studied  $\rho_{\mathrm{un}}$  and threshold  $\epsilon$  would be an invalid case. Details are provided in Alg. 1.
+
+Algorithm 1: Quantum Learning Algorithm for Limited-Structured Complexity Prediction  
+Input: Noisy quantum state  $\rho_{\mathrm{un}},\epsilon$  Output: The minimum depth  $R$ $(R <   \log n)$  such that  $C_\epsilon^{\mathrm{lim},A}(\rho_{\mathrm{un}})\leq LR$  (True); False if such  $R$  does not exist; Or return True/False arbitrarily for invalid cases; 1 Initialize  $R\gets 1,s\gets \log (n)$  .   
+2 while  $s - R > 1$  do   
+3 Set  $N = LRn^{2}\log n / \epsilon^{2}$  and  $T = N^{2}n^{k}$  such that  $k\log (n) <   n^{k / 2 - 1}\epsilon$  for large  $n$  .   
+4 if  $\mathcal{P}((R + s) / 2,\epsilon) = \text{True do}$    
+5  $s\gets \lceil (R + s) / 2\rceil$    
+6 else do   
+7  $R\gets \lceil (R + s) / 2\rceil$    
+8 if  $\mathcal{P}(R,\epsilon) = \text{True do}$    
+9 return  $C_\epsilon^{\mathrm{lim},A}(\rho_{\mathrm{un}})\leq LR$  (True)   
+10 else do   
+11 if  $\mathcal{P}(R,2\epsilon) = \text{False,return} C_\epsilon^{\mathrm{lim},A}(\rho_{\mathrm{un}}) > L\log n$  (False)   
+12 else do   
+13 return True or False arbitrarily
+
+# 5 THEORETICAL PERFORMANCE GUARANTEE
+
+We will demonstrate that the required number of samplings and processing time for Alg. 1 are both efficient, and the sample complexity is optimal with respect to the noisy circuit depth. Using Theorem 1, we can state the following result.
+
+Theorem 2 Given  $\mathrm{poly}(n)$  copies of an  $n$ -qubit unknown weakly noisy state  $\rho_{\mathrm{un}}$  that is generated by a noisy quantum device with depth  $\tilde{R} = \mathcal{O}(\log n)$  (Def. 2) and a particular architecture  $\mathcal{A}$ , there exists a  $\mathrm{poly}(n, \tilde{R}, 1/\epsilon)$  quantum and classical cost learning algorithm which can efficiently solve the SCP  $(\mathcal{A}, \epsilon)$  problem.
+
+The proof of Theorem 2 depends on evaluating the sample complexity of QCA states and unknown noisy state, as well as the related iteration complexity during training the quantum learning algorithm. We analyze the sample and classical computational complexity in the next two subsections as part of the proof outline for this theorem.
+
+# 5.1 SAMPLE COMPLEXITY
+
+The sample complexity of QCA states is promised by Theorem 1, where the number of copies of QCA states  $N = LRn^{2}\log n / \epsilon^{2}$ . To evaluate  $\mathcal{L}_R(\vec{\beta})$  for  $1\leq R\leq \tilde{R}\leq \mathcal{O}(\log n)$  within  $\epsilon$ -additive error, the classical shadow representation of  $\rho_{\mathrm{un}}$  is shared by all QCA states, by leveraging the classical shadow method (Huang et al., 2020). As a result, the sample complexity of an unknown weakly noisy state is at most  $m\leq \mathcal{O}\left(\frac{n^2LR\log n\log(1 / \delta)}{\epsilon^4}\right)$ , where the failure probability  $\delta \in (0,1)$ .
+
+Meanwhile, we also give a sample complexity lower bound to demonstrate our learning algorithm is optimal in terms of the circuit depth  $\bar{R}$ .
+
+Theorem 3 Given an unknown weakly noisy quantum state  $\rho$  prepared by a  $R$ -depth quantum circuit affected by  $p$ -strength local Pauli noise channels, then any algorithm designed to learn  $\rho$  requires at least  $m$  samplings in the worst-case scenario, where  $m = \frac{(1 - p)^{-2c\tilde{R}}(1 - \delta)^2}{2n}$ ,  $c = 1/(2\ln 2)$  and constant  $\delta \in \mathcal{O}(1)$ .
+
+From the above result and the definition of the quantum weakly noisy state, we know that  $p = \mathcal{O}(n^{-1})$  which results in the sample complexity lower bound  $\Omega((1 + 2cp\tilde{R})(1 - \delta)^2/(2n))$ . This implies the sample complexity lower bound may linearly grow with the increasing of the circuit layer  $\tilde{R}$ , and our learning algorithm is nearly-optimal in terms of the circuit depth  $\tilde{R}$ . We leave proof details to Appendix J.
+
+# 5.2 RUNNING TIME ANALYSIS
+
+Here, we provide an analysis of the computational complexity for Alg. 1, including the number of iterations in Alg. 1 and the subroutine BMaxS (Alg. 2) which is a Bayesian optimization algorithm. Denote the global optimum  $\vec{\beta}^{*} = \max_{\vec{\beta}\in \mathcal{D}_{\vec{\beta}}}\mathcal{L}_{R}(\vec{\beta})$ , and a natural performance metric for optimizing  $\mathcal{P}(R)$  is the simple regret  $s_T = \mathcal{L}_R(\vec{\beta}^*) - \mathcal{L}_R(\vec{\beta}^{(T)})$ , which is the difference between the global maximum  $\mathcal{L}_R(\vec{\beta}^*)$  and  $\mathcal{L}_R(\vec{\beta}^{(T)})$ . Here,  $\vec{\beta}^{(T)}$  represents the updated parameter in the  $T$ -th step. Obviously, simple regret is non-negative and asymptotically decreases with the increasing iteration complexity  $T$ . To build up an explicit connection between  $s_T$  and  $T$ , the average regret  $\mathrm{avr}_T$  is introduced. Specifically,  $\mathrm{avr}_T = 1 / T\sum_{t = 1}^{T}\left[\mathcal{L}_R(\vec{\beta}^*) - \mathcal{L}_R(\vec{\beta}^{(T)})\right]$ . Noting that the relationship  $s_T\leq \mathrm{avr}_T$  holds for any  $T\geq 1$ . In the following, we show that  $\mathrm{avr}_T$  is upper bounded by  $\mathcal{O}(N\log T / \sqrt{T})$ , and the simple regret  $s_T\leq \mathrm{avr}_T\to 0$  with the increase of  $T$ . The following theorem derives the average regret bounds for  $\mathcal{P}(R) = \mathrm{BMaxS}(\rho_{\mathrm{un}},\hat{\rho}_{\mathrm{QCA}}(R,\mathcal{A},N),T,\epsilon)$ .
+
+Theorem 4 Take the target weakly noisy state  $\rho_{\mathrm{un}}$  and QCA state set  $\hat{\rho}_{\mathrm{QCA}}(R,\mathcal{A},N)$  into the subroutine  $\mathcal{P}(R)$  (Eq. 12). Pick the failure probability  $\delta \in (0,1)$ , then there exists a Bayesian approach (details refer to Alg. 2) such that the average regret  $\mathrm{avr}_T$  can be upper bounded by
+
+$$
+\operatorname {a v r} _ {T} \leq \mathcal {O} \left(\sqrt {\frac {4 N ^ {2} \log^ {2} T + 2 N \log T \log \left(\pi^ {2} / (6 \delta)\right)}{T}}\right) \tag {13}
+$$
+
+after  $T$  optimization steps with  $1 - \delta$  success probability, where  $N$  represents the number of samples in the QCA state set.
+
+![](images/2222fd878fa2b4808641963db0614125a2e6bc20e76c2ac42a236899f2439bb0.jpg)  
+(a)
+
+![](images/bf7172551098c90a096ec44e6766f67e5d78a958fa2c0ada5c1a8e2988302db0.jpg)  
+(b)
+
+![](images/d31dfd1f090b72507c24a2e583c45a18f8f11086cd7a24ab57e6f089b073c019.jpg)  
+(c)  
+(f)
+
+![](images/37cb413672dd6cfeda0b2c357cc7623c627eca12c4f57b7c69a569dcfecfcc92.jpg)  
+(d)  
+(e)  
+Figure 2: (a) Visualization of the 2D transverse field Ising model. (b)-(d) illustrate the trend of the function  $\min_{\vec{\beta}}\mathcal{L}_R$  as it varies with the circuit depth  $R$  of QCA set. To extract the complexity lower bound of the target quantum state from the plot, we begin by drawing a horizontal line representing the approximation error of the quantum state. For example, in (b), we set  $\epsilon' = 0.25$ . Next, we identify the last point where the curve remains above this horizontal line before crossing it, and record the corresponding horizontal coordinate  $R$ . This  $R$  represents the complexity lower bound of the target quantum state. For instance, in Fig. (b), the blue curve remains above  $\epsilon' = 0.25$  until  $R = 4$ , indicating a complexity lower bound of  $C_{0.125}^{\mathrm{lim},A}(\rho_{\mathrm{TI}}(2,p)) > 4L$ . Similarly, the green curve last remains above  $\epsilon' = 0.25$  at  $R = 2$ , yielding a complexity lower bound of  $C_{0.125}^{\mathrm{lim},A}(\rho_{\mathrm{TI}}(10,p)) > 2L$ , where  $L$  represents the number of two-qubit gates in each layer of the architecture  $\mathcal{A}$ .
+
+![](images/77ea17f5856826be657d941a539b437cfd4f5c46f874e4a857d9604a41d6bc20.jpg)
+
+![](images/7de29c012cdff874f0c605e7631b5010ebda473f41ff494a7b27b964baef41ac.jpg)
+
+Specifically, select an integer  $k$  such that  $k\log (n) < n^{k / 2}\epsilon$  for all  $n > n_0$ , where  $n_0$  represents a fixed integer. Then  $T = N^2 n^k$  enables the simple regret  $s_T$  can be upper bounded by  $\epsilon$ , where  $N = LRn^2\log n\epsilon^{-2}$  (Theorem 1). The proof details refer to Appendix I.
+
+Finally, noting that Alg. 1 is essentially a binary search program on the interval  $[1, \tilde{R}]$  by using the oracle  $\mathcal{P}(R,\epsilon) = \mathrm{BMaxS}(\rho_{\mathrm{un}},\hat{\rho}_{\mathrm{QCA}}(R,\mathcal{A},N),T,\epsilon)$ . Therefore, Alg. 1 requires
+
+$$
+\mathcal {O} \left(\frac {1}{\epsilon^ {4}}, n ^ {4 + k}, L ^ {2}, \tilde {R} ^ {2} \log (\tilde {R}), \log (1 / \delta)\right)
+$$
+
+classical time complexity to answer the SCP problem. This thus completes the proof of Theorem 2.
+
+# 6 NUMERICAL SIMULATIONS
+
+Here, we demonstrate how to use the proposed learning method to benchmark the capabilities of noisy state computation, providing numerical evidence to support our theoretical findings. Specifically, we address the fundamental question: Does the complexity of weakly noisy quantum states grow linearly with circuit depth?
+
+We consider to simulate the time dynamics of the Hamiltonian  $H = -J\sum_{\langle i,j\rangle}Z_iZ_j + h\sum_iX_i$  on a two-dimensional grid with  $(a\times b)$  size, where  $J > 0$  represents the coupling of nearest-neighbour spins and  $h$  represents the global transverse field strength (see Fig. 2 (a)). To simulate the time evolution circuit  $e^{-iH\tau}$ , the first-order Trotter decomposition is utilized (Kim et al., 2023), that is  $e^{-iH\tau} = \left[e^{-iH_{\mathrm{ZZ}}\delta t}e^{-iH_{\mathrm{X}}\delta t}\right]^{\tau /\delta t} + \mathcal{O}((\delta t)^2)$ , where  $H_{\mathrm{ZZ}}$  represents the spin term,  $H_{\mathrm{X}}$  represents the transverse-field term and the evolution time  $\tau$  is discretized into  $(\tau /\delta t)$  time slices. Then its quantum circuit implementation can be decomposed by  $e^{-iH_{\mathrm{ZZ}}\delta t} = \prod_{\langle i,j\rangle}R_{Z_iZ_j}(-2J\delta t)$  and
+
+$e^{-iH_X\delta t} = \prod_iR_{X_i}(2h\delta t)$ . We focus on a small-scale scenario studied in Ref. (Kim et al., 2023), where the system size is  $3\times 4$ , angle rotations  $-2J\delta t = -\pi /2$ ,  $h\delta t\in \{\pi /8,\pi /4,\pi /2,3\pi /4,\pi \}$  and each quantum gate is affected by a local depolarizing channel  $\mathcal{E}_i$  with strength  $p = 10^{-3}$ . Let  $\tilde{R} = \lceil \tau /\delta t\rceil$ , then we denote the output quantum weakly noisy state as
+
+$$
+\rho_ {\mathrm {T I}} (\tilde {R}, p) = \bigcirc_ {r = 1} ^ {\tilde {R}} \left[ \mathcal {E} _ {p} \circ \mathcal {U} _ {Z Z} \circ \mathcal {E} _ {p} \circ \mathcal {U} _ {X} \right] (| 0 ^ {n} \rangle \langle 0 ^ {n} |), \tag {14}
+$$
+
+where  $\mathcal{U}_{ZZ}(\cdot) = e^{-iH_{ZZ}\delta t}(\cdot)e^{iH_{ZZ}\delta t},\mathcal{U}_X(\cdot) = e^{-iH_X\delta t}(\cdot)e^{iH_X\delta t}$  and the quantum noise channel  $\mathcal{E}_p = \otimes_{i = 1}^n\mathcal{E}_i$  . In the numerical simulation, we demonstrated our algorithm on a server with 64 vCPUs and 128 GiB of memory, where the density matrix  $\rho_{\mathrm{TI}}(\tilde{R},p)$  and classical shadow set are prepared by the Pennylane package (Bergholm et al., 2018).
+
+In our analysis, we mainly focus on the circuit complexity lower bound of noisy quantum states  $\rho_{\mathrm{TI}}(\tilde{R},p)$  with  $\tilde{R}\in \{2,10\}$  by using Alg. 1, considering a standard 2D lattice quantum circuit architecture  $\mathcal{A}$  to prepare  $\hat{\rho}_{\mathrm{QCA}}$  , where each layer has  $L = \mathcal{O}(n)$  random two-qubit gates. Visualization of the architecture  $\mathcal{A}$  in 1D scenario is given by Fig. 1 (b). To estimate the lower bound, we thus set a small error  $(\epsilon = 10^{-2})^7$  and randomly generate  $N = n^{2}R$  quantum circuits with varying circuit depths  $R\in \{2,3,4,5\}$  based on the architecture  $\mathcal{A}$  . Precisely, we tune the linear coefficient  $\vec{\beta}$  to minimize the metric functions outlined in Lemma 2, as depicted in Figure 2, where each point represents the mean value of  $\min_{\vec{\beta}}\mathcal{L}_R$  by repeating 10 independent experiments, and the error bar represents the standard variance. This strategic adjustment allows us to derive an estimate for the quantum circuit lower bound. More specifically, in Figure 2 (b), we showcase a clear circuit complexity separation between weakly noisy quantum states  $\rho_{\mathrm{TI}}(2,p)$  and  $\rho_{\mathrm{TI}}(10,p)$  . To see this relationship, Figure 2 (b) demonstrated that  $\min_{\vec{\beta}}\mathcal{L}_2(\rho_{\mathrm{TI}}(10,p)) > 0.25$  , meanwhile  $\min_{\vec{\beta}}\mathcal{L}_4(\rho_{\mathrm{TI}}(2,p)) > 0.25$  which demonstrate that the circuit complexity  $C_{0.125}^{\lim ,\mathcal{A}}(\rho_{\mathrm{TI}}(\tilde{R} = 10,p)) > 2L$  and  $C_{0.125}^{\lim ,\mathcal{A}}(\rho_{\mathrm{TI}}(\tilde{R} = 2,p)) > 4L$  (according to Lemma 2), where  $L = 3n$  represents the number of random two qubit gates in each layer. This result highlights weakly noisy quantum state complexity lower bound may not grow with the circuit depth, which is dramatically different to that of pure states. Similar phenomenons are witnessed in subfigures. 2 (c)-(f), where a shallower circuit depth weakly noisy states possess higher state complexity lower bound.
+
+# 7 CONCLUSION
+
+The quantum state complexity serves as a measure of inherent properties within quantum states, thereby facilitating a deeper understanding of quantum entanglement information, quantum topological phases, and computational capabilities. In practical applications, collected quantum states are often subject to noise originating from state preparation and quantum measurement (SPAM), as well as limitations imposed by the quantum hardware. Consequently, original pure states are transformed to noisy states through quantum channels. Thus, investigating the quantum state complexity of noisy states holds significant importance in studying information scrambling, the spread of local noise and entanglement throughout the entire system, which is expected to illuminate studies in the field of black-hole theory and condensed-matter physics. In this paper, we investigate the complexity of weakly noisy quantum states through a quantum learning algorithm, which connects two significant concepts in the quantum computational theory. The proposed quantum learning algorithm exploits the intrinsic structure of QCA to build a learning model  $\mathcal{L}_R(\beta)$ , whose extreme points reveal the limited-structured complexity. Meanwhile, when considering the sample complexity of target noisy quantum state, our algorithm achieves optimal in terms of the circuit depth  $\tilde{R}$ . Moreover, we emphasize that the Bayesian optimizer (given by Alg. 2) is not the unique option, and other optimization algorithms may also work with a similar iteration steps. This highlights the universality of the intrinsic structure of QCA in combination with optimization subroutines.
+
+# ACKNOWLEDGMENTS
+
+We thank Jens Eisert, Philippe Faist, Haihan Wu, Tavis Bennett, John Tanner, and all anonymous reviewers for their helpful comments on this paper. Y. Wu acknowledges the support from the Natural Science Foundation of China Grant (No. 62371050). B. Wu acknowledges the support from the National Natural Science Foundation of China Grant (No. 12405014) and the Research fund of the post-doctor who came to Shenzhen. X. Yuan is supported by the Innovation Program for Quantum Science and Technology Grant (No. 2023ZD0300200), the National Natural Science Foundation of China Grant (No. 12175003 and No. 12361161602), NSAF Grant (No. U2330201). J. B. Wang acknowledges continued support from Pawsey Supercomputing Research Centre.
+
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+
+# A COMPARISON WITH RELATED WORKS
+
+In the context of condensed matter physics, a topological phase transition occurs when the ground states of a family of Hamiltonians exhibit different circuit complexities. This suggests that topological phase classification may help distinguish between low-complexity and high-complexity ground states. The use of learning algorithms to classify quantum phases of matter has been widely studied. Proposals include quantum neural networks (Cong et al., 2019), classical neural networks (Beach et al., 2018; Carrasquilla & Melko, 2017; Greplova et al., 2020; Schindler et al., 2017; Van Nieuwenburg et al., 2017), and other classical machine learning models (Huang et al., 2022; Rodriguez-Nieva & Scheurer, 2019; Wetzel, 2017). However, most previous works lack rigorous theoretical guarantees. Among these studies, Huang et al. (Huang et al., 2022) utilized shadow tomography of the concerned ground states to design an unsupervised machine learning approach guaranteed to classify accurately under certain conditions. The proposed "shadow kernel" can generate quantum entropies by tuning hyperparameters, enabling the approximation of various topological phase order parameters. Compared with Ref. Huang et al. (2022), our learning algorithm can be applied to noisy states and predict complexity, whereas Ref. Huang et al. (2022) mainly focuses on pure ground state classification without providing an explicit metric for characterizing complexity.
+
+On the other hand, learning quantum states and circuits is a long-standing task in the field of quantum machine learning. Previous works generally utilized parameterized circuits to learn approximations of target states and circuits (Mitarai et al., 2018; Yang et al., 2020; Huang et al., 2021; Jerbi et al., 2021; Cerezo et al., 2022). However, variational-based methods generally lack theoretical guarantees and may suffer from the barren plateau phenomenon even in low-depth parameterized circuits (McClean et al., 2018; Cerezo et al., 2021; Anschuetz & Kiani, 2022). Very recently, classical shadow-based learning algorithms have been proposed for reconstructing shallow quantum circuits  $U$  (Huang et al., 2024; Landau & Liu, 2024). Nevertheless, these methods require querying  $U^{\dagger}$ , which introduces intrinsic limitations in noisy environments since most noisy channels (CPTP maps) may not be invertible. Compared with related works, our method bypasses these limitations by introducing the "Intrinsic Structure property" (as given by Theorem 1), which enables an efficient quantum learning algorithm to predict state complexity in noisy environments.
+
+we would like to point out that Ref. Schuster et al. (2024) does not rule out the possibility that the pure state complexity can be predicted when the target states exhibit certain physical structures, such as the ground state of XXZ models and the toric code, as studied in Ref. Huang et al. (2022). Furthermore, recent works (Huang et al., 2024; Landau & Liu, 2024) have also demonstrated that fixed shallow quantum circuits and quantum pure states (including 1D- $\mathcal{O}(\log n)$ -depth and all-to-all  $\mathcal{O}(\log \log n)$ -depth) can be efficiently learned. These results demonstrate significant differences between scenarios with and without 'randomness'. Compared with Ref. (Schuster et al., 2024), our work is more similar to Refs. (Huang et al., 2022; 2024; Landau & Liu, 2024), which focus on a specific quantum state rather than an ensemble.
+
+Furthermore, we would like to highlight the differences between weakly noisy quantum states and pseudorandom quantum states. Specifically, we consider the second-moment statistic property of a set of pseudorandom quantum circuits  $U$  whose circuit depth  $\tilde{R} \leq \mathrm{poly}\log n$ . In Ref. (Schuster et al., 2024), they claimed that even  $\mathcal{O}(\log(n))$ -depth Clifford circuits may approximate the unitary 2-design property given by Haar measure. Then, we may suppose  $U = U_{1} \cdots U_{\tilde{R}}$  representing a random Clifford circuit, then for the initial state  $|0^{n}\rangle \langle 0^{n}|$  and arbitrary observable  $\tilde{O}$ , we have
+
+$$
+M _ {2} (U) = \int_ {U \sim \operatorname {C l} \left(2 ^ {n}\right)} \operatorname {T r} \left[ U | 0 ^ {n} \rangle \langle 0 ^ {n} | U ^ {\dagger} O \right] \operatorname {T r} \left[ U | 0 ^ {n} \rangle \langle 0 ^ {n} | U ^ {\dagger} O \right] = \frac {1}{d \left(d ^ {2} - 1\right)} \left[ \operatorname {T r} ^ {2} [ O ] + \operatorname {T r} [ O ^ {2} ] \right]. \tag {15}
+$$
+
+On the other hand, in the context of weakly noisy environment, the Clifford circuit  $U$  may transform to the channel representation  $\mathcal{U} = \bigcirc_{r=1}^{\tilde{R}}\mathcal{E} \circ \mathcal{U}_r$  where  $\mathcal{E}$  represents a  $n$ -qubit Pauli channel and  $\mathcal{U}_r = U_r(\cdot)U_r^\dagger$  represents a layer of Clifford gate. According to the "Channel Pushing" lemma given by Ref Quek et al. (2024) (Lemma 10), we may rewrite the noisy channel by  $\mathcal{U} = \mathcal{E}' \circ \bigcirc_{r=1}^{\tilde{R}}\mathcal{U}_r$ , where  $\mathcal{E}'$  also represents an  $n$ -qubit Pauli channel. Let the channel  $\mathcal{E}' = \sum_l K_l(\cdot)K_l^\dagger$ , where  $K_l$
+
+represents the Kraus operator satisfying  $\sum_{l}K_{l}^{\dagger}K_{l} = I$ , we have
+
+$$
+\begin{array}{l} M _ {2} (\mathcal {U}) = \int_ {U \sim \mathrm {C l} (2 ^ {n})} \operatorname {T r} [ \mathcal {U} (| 0 ^ {n} \rangle \langle 0 ^ {n} |) O ] \operatorname {T r} [ \mathcal {U} (| 0 ^ {n} \rangle \langle 0 ^ {n} |) O ] \\ = \sum_ {l _ {1}, l _ {2}} \int_ {U \sim \operatorname {C l} \left(2 ^ {n}\right)} \operatorname {T r} \left[ K _ {l _ {1}} U \left(\left| 0 ^ {n} \right\rangle \langle 0 ^ {n} |\right) U ^ {\dagger} K _ {l _ {1}} ^ {\dagger} O \right] \operatorname {T r} \left[ K _ {l _ {2}} U \left(\left| 0 ^ {n} \right\rangle \langle 0 ^ {n} |\right) U ^ {\dagger} K _ {l _ {2}} ^ {\dagger} O \right] \tag {16} \\ = \sum_ {l _ {1}, l _ {2}} \left(\frac {1}{d (d ^ {2} - 1)} \left(\operatorname {T r} \left(K _ {l _ {1}} O K _ {l _ {1}} ^ {\dagger}\right) \operatorname {T r} \left(K _ {l _ {2}} O K _ {l _ {2}} ^ {\dagger}\right) + \operatorname {T r} \left(K _ {l _ {1}} O K _ {l _ {1}} ^ {\dagger} K _ {l _ {2}} O K _ {l _ {2}} ^ {\dagger}\right)\right)\right). \\ \end{array}
+$$
+
+We note that the interaction term  $\sum_{l_1,l_2}\mathrm{Tr}(K_{l_1}OK_{l_1}^\dagger K_{l_2}OK_{l_2}^\dagger)$  may not equal to  $\mathrm{Tr}(O^2)$ , and this leads to  $M_2(U)\neq M_2(\mathcal{U})$ . This comparison indicates that even if an algorithm can learn the difference between noisy ensembles and Haar-random states, this does not imply that the algorithm can be used to distinguish between pseudorandom ensembles and Haar-random ensembles.
+
+The above discussions demonstrate the fundamental difference between our work and Ref. (Schuster et al., 2024). Such difference implies that the efficiency of our learning algorithm may not contradict the findings presented in Ref. (Schuster et al., 2024).
+
+# B NOISE MODELS ASSUMPTION IN THIS WORK
+
+In the proposed learning approach (Alg 1), the algorithm requires (i) the target weakly noisy quantum state  $\rho_{\mathrm{un}}$ , and (ii) the QCA state set  $\hat{\rho}_{\mathrm{QCA}}$ . The target weakly noisy state naturally contains noise signals, while the QCA state set  $\hat{\rho}_{\mathrm{QCA}}$  is provided by the classical shadow representation (Huang et al., 2020). When preparing the classical shadow, recent result successfully embedded the quantum error mitigation method into the classical shadow protocol, which provides rigorous theoretical guarantees even in the noisy environment (Jnane et al., 2024).
+
+In quantum computing research, the gate-independent noise model assumes that the noise affecting quantum operations is uniform across all gates, regardless of their type or implementation. This simplification is widely adopted for several reasons:
+
+Theoretical Simplification: Assuming gate-independent noise allows researchers to develop and analyze error correction protocols and fault-tolerant methods without delving into the complexities introduced by gate-specific noise characteristics. This uniformity facilitates the derivation of general results and theoretical bounds (Nielsen & Chuang, 2001).
+
+Practical Approximations: In certain quantum systems, particularly those with well-calibrated gates operating on the same number of qubits and utilizing uniform control mechanisms, noise variations across different gates can be negligible (Shor, 1996; Arute et al., 2019). In such cases, the gate-independent noise model serves as a reasonable approximation, streamlining analysis without significantly compromising accuracy.
+
+Alignment with Twirled Noise Models: Techniques like Pauli twirling are employed to transform complex noise channels into diagonal forms on the Pauli basis (Chen et al., 2023). While twirling simplifies the noise structure, it does not inherently eliminate gate dependence. However, in many scenarios, the resulting noise can be approximated as gate-independent, aligning with the assumptions of the model.
+
+The gate-independent noise model provides a foundational framework for understanding error propagation and developing correction strategies, which is a useful abstraction for theoretical exploration and the initial development of error correction methods. We leave an open question of how to depict the gate-dependent noise, which usually happens in larger, more complex quantum architectures.
+
+# C QUANTUM HARDWARE REQUIREMENT
+
+We note that our learning algorithm has very few limitations to the practical quantum hardware. In the Alg. 1, we only require (i) the target weakly noisy quantum state  $\rho_{\mathrm{un}}$ , and (ii) the QCA state set  $\hat{\rho}_{\mathrm{QCA}}$ , which is essentially the classical shadow representation Huang et al. (2020). When preparing
+
+the classical shadow, recent work has successfully integrated quantum error mitigation methods into the classical shadow protocol, providing rigorous theoretical guarantees in preparing shadows even in noisy environments Jnane et al. (2024). As a result, our Alg 1 only require a quantum computer which supports the quantum error mitigation function, which should have  $50 - 100$  noisy qubits with  $\geq 0.99$  quantum gate fidelity. Some popular quantum computation platforms may satisfy these requirements and can be used to prepare the required classical shadow, such as Eagle used in Ref. Kim et al. (2023) and Sycamore Arute et al. (2019). This work is essentially a theoretical research, and the algorithm has not yet been explicitly tested on these practical platforms.
+
+# D COMPARISON BETWEEN LOCAL AND GLOBAL NOISE MODELS
+
+We note that the global depolarizing channel is essentially a special case of contracting  $n$  local noise models. Specifically, given the global depolarizing channel  $\mathcal{E}$  with noisy strength  $p_g$  and local depolarizing channel  $\otimes_j \mathcal{E}_j$  with strength  $p_l$ , according to the relative entropy inequality, for any input state  $\rho$ , we can observe the relationships
+
+$$
+D \left(\mathcal {E} (\rho) \| I _ {n} / 2 ^ {n}\right) \leq \left(1 - p _ {g}\right) D \left(\rho \| I _ {n} / 2 ^ {n}\right), \tag {17}
+$$
+
+and
+
+$$
+D \left(\otimes_ {j = 1} ^ {n} \mathcal {E} _ {j} (\rho) \| I _ {n} / 2 ^ {n}\right) \leq (1 - p _ {l}) ^ {n} D \left(\rho \| I _ {n} / 2 ^ {n}\right) \tag {18}
+$$
+
+hold. These inequalities indicate that the global depolarizing channel is much weaker than local noise, given the same noise strength. Consequently, for the global depolarizing channel, our algorithm is capable of handling scenarios where  $p_{g} \leq \mathcal{O}(1)$ , while maintaining comparable prediction accuracy. This suggests that the algorithm can effectively handle both types of noise, with only a minor difference in performance under different noise strengths.
+
+# E RELATED DEFINITIONS
+
+Definition 6 (Architecture Haferkamp et al. (2022); Bouland et al. (2019a)) An architecture  $\mathcal{A}$  is a directed acyclic graph that contains  $|\mathcal{V}| \in \mathbb{Z}_{>0}$  vertices (gates), and two edges (qubits) enter each vertex, and two edges exit. A quantum circuit induced by the architecture  $\mathcal{A}$  is denoted as  $U_{\mathcal{A}}$ . The circuit set which contains all quantum circuits induced by the architecture  $\mathcal{A}$  is denoted by  $\mathcal{U}_{\mathcal{A}}$ . The circuit set  $\mathcal{U}_{\mathcal{A}}(R)$  contains all  $R$ -depth quantum circuits induced by the architecture  $\mathcal{A}$ , and we have the relationship
+
+$$
+\mathcal {U} _ {\mathcal {A}} = \cup_ {R > 1} \mathcal {U} _ {\mathcal {A}} (R). \tag {19}
+$$
+
+Definition 7 (Causal Slice) The circuit  $U_{\mathcal{A}}$  is a causal slice if there exists a qubit-reachable path between any two qubit-pairs, where the path only passes through vertices (gates) in the architecture  $\mathcal{A}$ .
+
+# F PROOF OF FACT 1
+
+Here, we are interested in measuring the quantity  $\mathbb{E}_{\mathcal{U}_1,\dots,\mathcal{U}_{\tilde{R}}}\left[\mathrm{Tr}(\rho_{1,\tilde{R}}\rho_{2,\tilde{R}})\right]$ , where  $\rho_{1,\tilde{R}} = \bigcirc_{r=1}^{\tilde{R}}\mathcal{E} \circ \mathcal{U}_r(\rho_1), \rho_{2,\tilde{R}} = \bigcirc_{r=1}^{\tilde{R}}\mathcal{U}_r(\rho_2)$  and  $\rho_1, \rho_2$  represent initial pure states. Suppose  $\mathcal{E}$  can be decomposed by Kraus operators, that is  $\mathcal{E}(\cdot) = \sum_{l=1}^{r}K_{l}(\cdot)K_{l}^{\dagger}$ . Then we first consider the scenario  $\tilde{R} = 1$ :
+
+$$
+\begin{array}{l} \operatorname {T r} \left(\rho_ {1, 1} \rho_ {2, 1}\right) = \operatorname {T r} \left[ \left(\mathcal {E} \circ \mathcal {U} _ {1} (\rho_ {1}) \mathcal {U} _ {1} (\rho_ {2})\right) \right] \\ = \operatorname {T r} \left[ \operatorname {S w a p} \left(\mathcal {E} \circ \mathcal {U} _ {1} \left(\rho_ {1}\right) \otimes \mathcal {U} _ {1} \left(\rho_ {2}\right)\right) \right] \tag {20} \\ = \sum_ {l = 1} ^ {r} \operatorname {T r} \left[ \operatorname {S w a p} \left(K _ {l} \mathcal {U} _ {1} (\rho_ {1}) K _ {l} ^ {\dagger} \otimes \mathcal {U} _ {1} (\rho_ {2})\right) \right]. \\ \end{array}
+$$
+
+Then taking the average value on a unitary 2 design set, we obtain
+
+$$
+\mathbb {E} _ {\mathcal {U} _ {1}} \operatorname {T r} \left(\rho_ {1, 1} \rho_ {2, 1}\right) = \sum_ {l = 1} ^ {r} \mathbb {E} _ {\mathcal {U} _ {1}} \operatorname {T r} \left[ \operatorname {S w a p} \left((K _ {l} \otimes I _ {n}) (U _ {1} \otimes U _ {1}) (\rho_ {1} \otimes \rho_ {2}) (U _ {1} ^ {\dagger} \otimes U _ {1} ^ {\dagger}) (K _ {l} ^ {\dagger} \otimes I _ {n})\right) \right]. \tag {21}
+$$
+
+Considering the relationship
+
+$$
+\begin{array}{l} \mathbb {E} _ {\mathcal {U} \sim \mathbb {U}} \left[ (U \otimes U) A (U ^ {\dagger} \otimes U ^ {\dagger}) \right] = \left(\frac {\operatorname {T r} (A)}{d ^ {2} - 1} - \frac {\operatorname {T r} (\operatorname {S w a p} A)}{d (d ^ {2} - 1)}\right) I _ {n} \otimes I _ {n} + \left(\frac {\operatorname {T r} (\operatorname {S w a p} A)}{d ^ {2} - 1} - \frac {\operatorname {T r} (A)}{d (d ^ {2} - 1)}\right) \text {S w a p} \tag {22} \\ = \alpha I _ {n} \otimes I _ {n} + \beta \text {S w a p} \\ \end{array}
+$$
+
+where  $\mathbb{U}$  is unitary 2-design, then Eq. 21 can be further calculated by
+
+$$
+\begin{array}{l} \mathbb {E} _ {\mathcal {U} _ {1}} \operatorname {T r} \left(\rho_ {1, 1} \rho_ {2, 1}\right) = \sum_ {l = 1} ^ {r} \mathbb {E} _ {\mathcal {U} _ {1}} \operatorname {T r} \left[ \operatorname {S w a p} \left((K _ {l} \otimes I _ {n}) (\alpha I _ {n} \otimes I _ {n} + \beta \operatorname {S w a p}) (K _ {l} ^ {\dagger} \otimes I _ {n})\right) \right] \\ = \alpha \sum_ {l = 1} ^ {r} \operatorname {T r} \left[ \operatorname {S w a p} \left(K _ {l} K _ {l} ^ {\dagger} \otimes I _ {n}\right) \right] + \beta \sum_ {l = 1} ^ {r} \operatorname {T r} \left[ \operatorname {S w a p} \left(K _ {l} \otimes I _ {n}\right) \operatorname {S w a p} \left(K _ {l} ^ {\dagger} \otimes I _ {n}\right) \right] \tag {23} \\ = \alpha \sum_ {l = 1} ^ {r} \operatorname {T r} \left[ K _ {l} K _ {l} ^ {\dagger} \right] + \beta \sum_ {l = 1} ^ {r} | \operatorname {T r} [ K _ {l} ] | ^ {2}. \\ \end{array}
+$$
+
+The last equality comes from
+
+$$
+\begin{array}{l} \operatorname {T r} \left[ \operatorname {S w a p} \left(K _ {l} \otimes I _ {n}\right) \operatorname {S w a p} \left(K _ {l} ^ {\dagger} \otimes I _ {n}\right) \right] = \operatorname {T r} \left[ \operatorname {S w a p} \left(K _ {l} \otimes I _ {n}\right) \left(I _ {n} \otimes K _ {l} ^ {\dagger}\right) \right] \\ = \operatorname {T r} \left[ \operatorname {S w a p} \left(K _ {l} \otimes K _ {l} ^ {\dagger}\right) \right] \tag {24} \\ = \left| \operatorname {T r} \left[ K _ {l} \right] \right| ^ {2}. \\ \end{array}
+$$
+
+Consider  $\sum_{l=1}^{r} \operatorname{Tr}\left[K_l K_l^\dagger\right] = \operatorname{Tr}[\mathcal{E}(I_n)] = d$  and denote  $F = \sum_{l=1}^{r} |\operatorname{Tr}[K_l]|^2$ , Eq. 21 can be finally expressed as
+
+$$
+\begin{array}{l} \mathbb {E} _ {\mathcal {U} _ {1}} \operatorname {T r} \left(\rho_ {1, 1} \rho_ {2, 1}\right) = \left(\frac {1}{d ^ {2} - 1} - \frac {\operatorname {T r} \left(\rho_ {1} \rho_ {2}\right)}{d \left(d ^ {2} - 1\right)}\right) \operatorname {T r} [ \mathcal {E} (I _ {n}) ] + \left(\frac {\operatorname {T r} \left(\rho_ {1} \rho_ {2}\right)}{d ^ {2} - 1} - \frac {1}{d \left(d ^ {2} - 1\right)}\right) F \tag {25} \\ = \frac {F - 1}{d ^ {2} - 1} \operatorname {T r} \left(\rho_ {1} \rho_ {2}\right) + \frac {1}{d ^ {2} - 1} (d - F / d), \\ \end{array}
+$$
+
+where  $d = 2^n$ . Therefore, we obtain a recursive formula for the overlap of the output states, as defined in 25. We have
+
+$$
+\mathbb {E} _ {\mathcal {U} _ {1}, \mathcal {U} _ {1}, \dots , \mathcal {U} _ {\tilde {R}}} \operatorname {T r} \left(\rho_ {1, \tilde {R}} \rho_ {2, \tilde {R}}\right) = \frac {F - 1}{d ^ {2} - 1} \mathbb {E} _ {\mathcal {U} _ {1}, \mathcal {U} _ {1}, \dots , \mathcal {U} _ {\tilde {R} - 1}} \operatorname {T r} \left(\rho_ {1, \tilde {R} - 1} \rho_ {2, \tilde {R} - 1}\right) + \frac {1}{d ^ {2} - 1} (d - F / d). \tag {26}
+$$
+
+Then, we can use this iteration relationship to construct a geometric sequence, that is
+
+$$
+\begin{array}{l} \mathbb {E} _ {\mathcal {U} _ {1}, \mathcal {U} _ {1}, \dots , \mathcal {U} _ {\tilde {R}}} \mathrm {T r} \left(\rho_ {1, \tilde {R}} \rho_ {2, \tilde {R}}\right) - \frac {1}{d} = \left(\frac {F - 1}{d ^ {2} - 1}\right) \left(\mathbb {E} _ {\mathcal {U} _ {1}, \mathcal {U} _ {1}, \dots , \mathcal {U} _ {\tilde {R} - 1}} \mathrm {T r} (\rho_ {1, \tilde {R} - 1} \rho_ {2, \tilde {R} - 1}) - \frac {1}{d}\right) \\ = \left(\frac {F - 1}{d ^ {2} - 1}\right) ^ {\tilde {R} - 1} \left(\mathbb {E} _ {\mathcal {U} _ {1}} \operatorname {T r} \left(\rho_ {1, 1} \rho_ {2, 1}\right) - \frac {1}{d}\right) \\ = \left(\frac {F - 1}{d ^ {2} - 1}\right) ^ {\tilde {R} - 1} \left(\frac {F - 1}{d ^ {2} - 1} + \frac {d ^ {2} - F}{d (d ^ {2} - 1)} - \frac {1}{d}\right) \tag {27} \\ = \left(\frac {F - 1}{d ^ {2} - 1}\right) ^ {\tilde {R} - 1} \frac {F - 1}{d (d + 1)}, \\ \end{array}
+$$
+
+where the last equality comes from initial states  $\rho_{1} = \rho_{2} = |0^{n}\rangle \langle 0^{n}|$ . Generally,  $F\leq d^{2}$ , then if
+
+$$
+\tilde {R} \leq \frac {\log \left(\frac {F - 1}{d (d + 1)} (\eta - 1 / d) ^ {- 1}\right)}{\log \left(d ^ {2} - 1\right) - \log (F - 1)}, \tag {28}
+$$
+
+we have  $\mathbb{E}_{\mathcal{U}_1,\mathcal{U}_1,\ldots ,\mathcal{U}_{\tilde{R}}}\mathrm{Tr}\left(\rho_{1,\tilde{R}}\rho_{2,\tilde{R}}\right)\geq \eta$
+
+# G PROOF OF THEOREM 1
+
+Consider  $U(\pmb{\alpha}) = \prod_{r=1}^{LR} U(\pmb{\alpha}_r)$  is composed of  $LR$  two-qubit gates
+
+$$
+U \left(\boldsymbol {\alpha} _ {r}\right) = \exp \left(- i \sum_ {j _ {1}, j _ {2} = 0} ^ {4} \alpha_ {r} \left(j _ {1}, j _ {2}\right) \left(P _ {j _ {1}} \otimes P _ {j _ {2}}\right)\right) = \exp \left(- i \langle \boldsymbol {\alpha} _ {r}, \boldsymbol {P} _ {r} \rangle\right), \tag {29}
+$$
+
+where  $P_{j}\in \{I,X,Y,Z\}$  and each  $\alpha_{r}(j_{1},j_{2})\in [-1,1]^{\mathrm{8}}$  Using Taylor series, one obtains
+
+$$
+U (\boldsymbol {\alpha}) = \prod_ {r = 1} ^ {L R} \sum_ {k = 0} ^ {\infty} \frac {\left(- i \langle \boldsymbol {\alpha} _ {r} , \boldsymbol {P} _ {r} \rangle\right) ^ {k}}{k !}. \tag {30}
+$$
+
+Denote
+
+$$
+U \left(\boldsymbol {\alpha} _ {r}\right) _ {\mathrm {t r}} = \sum_ {k = 0} ^ {K} \frac {\left(- i \langle \boldsymbol {\alpha} _ {r} , \boldsymbol {P} _ {r} \rangle\right) ^ {k}}{k !}, \tag {31}
+$$
+
+therefore  $U(\pmb{\alpha}_r) - U(\pmb{\alpha}_r)_{\mathrm{tr}} = \sum_{k = K + 1}^{\infty}\frac{(-i\langle\pmb{\alpha}_r,\pmb{P}_r\rangle)^k}{k!}$ . For arbitrary bit-string  $x,y$ , we can apply standard bound on Taylor series to bound
+
+$$
+\left\| \langle x | (U \left(\boldsymbol {\alpha} _ {r}\right) - U \left(\boldsymbol {\alpha} _ {r}\right) _ {\mathrm {t r}}) | y \rangle \right\| _ {1} \leq \kappa / K! \tag {32}
+$$
+
+for some constant  $\kappa$ . Therefore we have
+
+$$
+\begin{array}{l} \langle 0 ^ {n} | U ^ {\dagger} (\boldsymbol {\alpha}) \rho U (\boldsymbol {\alpha}) | 0 ^ {n} \rangle = \sum_ {i, j = 0} ^ {2 ^ {n} - 1} \rho_ {i j} \langle 0 ^ {n} | U ^ {\dagger} (\boldsymbol {\alpha}) | i \rangle \langle j | U (\boldsymbol {\alpha}) | 0 ^ {n} \rangle \\ = \sum_{i,j = 0}^{2^{n} - 1}\rho_{ij}\left(\sum_{\substack{y_{1},y_{2},\ldots y_{LR - 1}\in \{0,1\}^{n}\\ y_{LR} = i}}\prod_{r = 1}^{LR}\langle 0^{n}|U(\boldsymbol{\alpha}_{r})|y_{r}\rangle\right)\left(\sum_{\substack{y_{1},y_{2},\ldots y_{LR - 1}\in \{0,1\}^{n}\\ y_{LR} = j}}\prod_{r = 1}^{LR}\langle y_{r}|U(\boldsymbol{\alpha}_{r})|0^{n}\rangle\right) \\ = \sum_ {i, j = 0} ^ {2 ^ {n} - 1} \rho_ {i j} \left(\sum_ {\substack {y _ {1}, y _ {2}, \dots y _ {L R - 1} \in \{0, 1 \} ^ {n} \\ y _ {L R} = i}} \prod_ {r = 1} ^ {L R} \langle 0 ^ {n} | \sum_ {k = 0} ^ {\infty} \frac {(- i \langle \boldsymbol {\alpha} _ {r} , \boldsymbol {P} _ {r} \rangle) ^ {k}}{k !} | y _ {r} \rangle\right). \tag{33} \\ \end{array}
+$$
+
+$$
+\left(\sum_{\substack{y_{1},y_{2},\ldots y_{LR - 1}\in \{0,1\}^{n}\\ y_{LR} = j}}\prod_{r = 1}^{LR}\langle y_{r}|\sum_{k = 0}^{\infty}\frac{(-i\langle\boldsymbol{\alpha}_{r},\boldsymbol{P}_{r}\rangle)^{k}}{k!} |0^{n}\rangle \right),
+$$
+
+where the  $y_{1}, y_{2}, \ldots$  represent the Feymann integration path. According to inequality 32,  $\langle y_r | U(\pmb{\alpha}_r) | 0^n \rangle$  can be approximated by a polynomial of degree  $K$  based on Taylor truncated method, the above expression can be rewritten by
+
+$$
+\sum_ {r, s = 0} ^ {2 ^ {n} - 1} \rho_ {r s} \left(f _ {r} \left(\boldsymbol {\alpha} _ {1}, \dots \boldsymbol {\alpha} _ {L R}\right) + \mathcal {O} \left(\frac {2 ^ {L R n}}{(K !) ^ {L R}}\right)\right) \left(f _ {s} \left(\boldsymbol {\alpha} _ {1}, \dots \boldsymbol {\alpha} _ {L R}\right) + \mathcal {O} \left(\frac {2 ^ {L R n}}{(K !) ^ {L R}}\right)\right), \tag {34}
+$$
+
+where  $f_{r}(\alpha_{1},\dots \alpha_{LR})$  represents a multi-variable polynomial of degree  $LRK$ .
+
+Furthermore, we will show that Eq. 34 can be approximated by a low-degree function with at most  $\binom{LR}{q}(K)^q$  terms, where  $q = \mathcal{O}(1)$ . To show this fact, we rewrite Eq. 34 by
+
+$$
+\sum_ {r, s = 0} ^ {2 ^ {n} - 1} \rho_ {r s} f _ {r} \left(\boldsymbol {\alpha} _ {1}, \dots \boldsymbol {\alpha} _ {L R}\right) f _ {s} \left(\boldsymbol {\alpha} _ {1}, \dots \boldsymbol {\alpha} _ {L R}\right). \tag {35}
+$$
+
+For the convenience of the proof, we denote  $f_{r}(\alpha_{1},\dots \alpha_{LR}) = f_{r}$ . Let
+
+$$
+f _ {r} f _ {s} = 4 ^ {L R n} \sum_ {\vec {\imath}} \sum_ {j _ {1}, j _ {2}} \boldsymbol {a} _ {\vec {\imath}} \alpha_ {1} ^ {i _ {1}} (j _ {1}, j _ {2}) \dots \alpha_ {L R} ^ {i _ {L R}} (j _ {1}, j _ {2}),
+$$
+
+where the factor  $4^{LRn}$  aims to 'normalize' each term in the summation,  $\vec{i} = (i_1,\dots,i_{LR})$ ,  $\vec{\alpha} = (\alpha_{1}(j_{1},j_{2}),\ldots ,\alpha_{LR}(j_{1},j_{2}))$ ,  $j_{1},j_{2}\in [15]$  and each  $0\leq i_l\leq 2K,l\in [LR]$ . For the convenience of the proof, we omit the index  $(j_{1},j_{2})$  in  $f_{r}f_{s}$  in the following procedure.
+
+Given a constant value  $q \leq \mathcal{O}(1)$ , for every term like  $a_{\vec{i}}^{\alpha_1^{i_1}} \cdots \alpha_v^{i_v}$  with  $i_1 = \cdots = i_v \geq K$  and  $v > q$ , its corresponding parameter  $|a_{\vec{i}}| \leq 1 / (K!)^q$  (based on Taylor series). Truncate above high-degree terms, and denote
+
+$$
+\tilde {f} _ {r, s} = \sum_ {\substack {j _ {1}, \dots , j _ {q} \leq K - 1 \\ s _ {1} \leq s _ {2} \dots \leq s _ {q} \leq L R}} \boldsymbol {a} _ {\vec {j}, \vec {s}} ^ {j _ {1}} \alpha_ {s _ {1}} ^ {j _ {1}} \dots \alpha_ {s _ {q}} ^ {j _ {q}}, \tag{36}
+$$
+
+therefore, the relationship
+
+$$
+\left| f _ {r} f _ {s} - \tilde {f} _ {r, s} \right| = 4 ^ {L R n} \left| \sum_ {\vec {i}} \boldsymbol {a} _ {\vec {i}} \alpha_ {1} ^ {i _ {1}} \dots \alpha_ {L R} ^ {i _ {L R}} - \sum_ {\substack {j _ {1}, \dots , j _ {q} \leq K - 1 \\ s _ {1} \leq s _ {2} \dots \leq s _ {q} \leq L R}} \boldsymbol {a} _ {\vec {j}, \vec {s}} \alpha_ {s _ {1}} ^ {j _ {1}} \dots \alpha_ {s _ {q}} ^ {j _ {q}} \right| \leq 4 ^ {L R n} \left(\frac {K ^ {L R}}{(K !) ^ {q}}\right) \tag{37}
+$$
+
+holds. Above inequality comes from the fact that there are  $\mathcal{O}(K^{LR})$  terms in  $f_{r}$  where the norm of each truncated term is upper bounded by  $\leq \mathcal{O}(1 / (K!)^q)$ .
+
+Then let  $q = \mathcal{O}(1)$ ,  $LR = \mathcal{O}(n\log (n))$ ,  $\tilde{f}_{r,s}$  can provide an estimation to  $f_{r}f_{s}$  within  $2^{-\mathrm{poly}(n)}$  additive error according to the Stirling's formula. Specifically, let  $q = 1$  and use Stirling's formula, the error
+
+$$
+4 ^ {L R n} \left(\frac {K ^ {L R}}{(K !) ^ {q}}\right) = \frac {4 ^ {n ^ {2} \log n} K ^ {n \log n}}{K !} \approx \frac {4 ^ {n ^ {2} \log n} K ^ {n \log n}}{\sqrt {2 \pi K} (K / e) ^ {K}}. \tag {38}
+$$
+
+Let  $K = n^{2}\log n$ , one obtains
+
+$$
+\begin{array}{l} \frac {4 ^ {n ^ {2} \log n} (n ^ {2} \log n) ^ {n \log n}}{\sqrt {2 \pi n ^ {2} \log n} (n ^ {2} \log n / e) ^ {n ^ {2} \log n}} \leq \frac {(4 ^ {\log n}) ^ {n ^ {2} \log n} (n ^ {2} \log n) ^ {n \log n}}{(n ^ {2} \log n / e) ^ {n ^ {2} \log n} \sqrt {2 \pi n ^ {2} \log n}} \\ = \frac {\left(n ^ {2} \log n\right) ^ {n \log n}}{\sqrt {2 \pi n ^ {2} \log n}} \left(\frac {e}{\log n}\right) ^ {n ^ {2} \log n} \tag {39} \\ = \frac {1}{\sqrt {2 \pi n ^ {2} \log n}} \left(\frac {\left(e ^ {n} n ^ {2} \log n\right)}{(\log n) ^ {n}}\right) ^ {n \log n} \\ \leq \frac {1}{\sqrt {2 \pi n ^ {2}}} 2 ^ {- n ^ {2}}, \\ \end{array}
+$$
+
+where the last inequality holds for large  $n$ . Then Eq. 34 can be represented by a multi-variable polynomial function  $f(\vec{\alpha},\rho)$  with  $LR$  variables and at most  $\mathrm{poly}(n)$  terms, and the relationship
+
+$$
+\left| \langle 0 ^ {n} | U ^ {\dagger} (\vec {\alpha}) \rho U (\vec {\alpha}) | 0 ^ {n} \rangle - f (\vec {\alpha}, \rho) \right| \leq 2 ^ {- n ^ {2}} \tag {40}
+$$
+
+holds. Suppose the target observable  $M = U(\pmb{\alpha}^{*})|0^{n}\rangle \langle 0^{n}|U^{\dagger}(\pmb{\alpha}^{*})$ . Then we may write
+
+$$
+\operatorname {T r} (M \rho) \approx f (\vec {\boldsymbol {\alpha}} ^ {*}, \rho) = \sum_ {\substack {j _ {1}, \dots , j _ {q} \leq (K - 1) \\ s _ {1} \leq s _ {2} \dots \leq s _ {q} \leq L R}} \boldsymbol {b} _ {\vec {\boldsymbol {\jmath}}, \vec {\boldsymbol {\mathfrak {s}}}} (\rho) \alpha_ {s _ {1}} ^ {j _ {1}, *} \dots \alpha_ {s _ {q}} ^ {j _ {q}, *}, \tag{41}
+$$
+
+where  $b_{\vec{j},\vec{s}}(\rho) = \sum_{s,r=0}^{2^n-1} \rho_{r,s} \pmb{a}_{\vec{j},\vec{s}}$ . On the other hand, consider a machine learning procedure with data points  $\{(x_i = \vec{\alpha}_i, y_i = f(\vec{\alpha}_i, \rho))\}$ . Let the feature map  $\Psi(\vec{\alpha}) = (\alpha_{s_1}^{j_1} \cdots \alpha_{s_q}^{j_q})_{\vec{s},\vec{j}}$ , and the target is
+
+to synthesis the function  $f(\vec{\alpha}, \rho) = \langle \vec{b}(\rho), \Psi(\vec{\alpha}) \rangle$ , where  $\vec{b}(\rho) \in \mathbb{R}^{LRn^2}$ . Consider the loss function
+
+$$
+\left. \min  _ {\vec {b} (\rho)} \lambda \langle \vec {b} (\rho), \vec {b} (\rho) \rangle + \sum_ {i = 1} ^ {N} \left(\langle \vec {b} (\rho), \Psi (\vec {\alpha} _ {i}) \rangle - y _ {i}\right) ^ {2}, \right. \tag {42}
+$$
+
+where  $\lambda > 0$  is a hyper-parameter. Define the feature matrix  $\Psi = (\Psi(\vec{\alpha}_1), \ldots, \Psi(\vec{\alpha}_N))$  and the kernel matrix
+
+$$
+\mathrm {K} = \Psi^ {\dagger} \Psi = \left[ \mathrm {K} \left(\vec {\alpha} _ {i}, \vec {\alpha} _ {j}\right) \right] _ {i, j = 1} ^ {N}, \tag {43}
+$$
+
+where the kernel function
+
+$$
+\mathrm {K} \left(\boldsymbol {\alpha}, \boldsymbol {\alpha} ^ {\prime}\right) = \sum_ {l = 0} ^ {K} \sum_ {1 \leq i _ {1} <   \dots <   i _ {q} \leq L R} \left(\alpha_ {i _ {1}} \alpha_ {i _ {1}} ^ {\prime} + \dots + \alpha_ {i _ {q}} \alpha_ {i _ {q}} ^ {\prime}\right) ^ {l}. \tag {44}
+$$
+
+Without loss of generality, we can normalize  $\mathrm{K}(\alpha, \alpha')$  enabling  $\mathrm{Tr}(\mathrm{K}) = \mathrm{N}$ . Therefore, the optimal solution
+
+$$
+\vec {\boldsymbol {b}} (\rho) _ {\text {o p t}} = \sum_ {i = 1} ^ {N} \sum_ {j = 1} ^ {N} \Psi \left(\vec {\boldsymbol {\alpha}} _ {i}\right) (\mathrm {K} + \lambda \mathrm {I}) _ {i j} ^ {- 1} f \left(\vec {\boldsymbol {\alpha}} _ {j}\right). \tag {45}
+$$
+
+As a result, the trained machine learning model
+
+$$
+\begin{array}{l} g (\vec {\boldsymbol {x}}) = \langle \vec {\boldsymbol {b}} (\rho) _ {\mathrm {o p t}}, \Psi (\vec {\boldsymbol {x}}) \rangle = \sum_ {i = 1} ^ {N} \sum_ {j = 1} ^ {N} \left(\mathrm {K} + \lambda \mathrm {I}\right) _ {i j} ^ {- 1} \mathrm {K} (\vec {\boldsymbol {\alpha}} _ {i}, \vec {\boldsymbol {x}}) f (\vec {\boldsymbol {\alpha}} _ {j}) \\ = \sum_ {j = 1} ^ {N} \left(\sum_ {i = 1} ^ {N} (\mathrm {K} + \lambda I) _ {i j} ^ {- 1} \mathrm {K} \left(\vec {\alpha} _ {i}, \vec {\mathbf {x}}\right)\right) f \left(\vec {\alpha} _ {j}\right) \tag {46} \\ = \sum_ {j = 1} ^ {N} \vec {\beta} _ {j} (\boldsymbol {x}) f (\vec {\alpha} _ {j}) \\ = \sum_ {j = 1} ^ {N} \vec {\beta} _ {j} (\boldsymbol {x}) \langle 0 ^ {n} | U ^ {\dagger} (\vec {\alpha} _ {j}) \rho U (\vec {\alpha} _ {j}) | 0 ^ {n} \rangle + \mathcal {O} \left(\frac {2 ^ {L R n}}{(K !) ^ {L R}}\right). \\ \end{array}
+$$
+
+Now we analyze the prediction error of  $g(\vec{x})$  on the domain  $[0, 2\pi]^{LR}$ . Denote
+
+$$
+\begin{array}{l} \tilde {\epsilon} (\vec {x}) = \epsilon (\vec {x}) + | \operatorname {T r} (M (\vec {x}) \rho) - f (\vec {x}, \rho) | = | g (\vec {x}) - f (\vec {x}, \rho) | + | \operatorname {T r} (M (\vec {x}) \rho)) - f (\vec {x}, \rho) | \tag {47} \\ \leq \left| g (\vec {x}) - f (\vec {x}, \rho) \right| + 2 ^ {- \operatorname {p o l y} (n)}, \\ \end{array}
+$$
+
+and the expected prediction error
+
+$$
+\mathbb {E} _ {\vec {\boldsymbol {x}}} \left[ \epsilon (\vec {\boldsymbol {x}}) \right] = \frac {1}{N} \sum_ {i = 1} ^ {N} \epsilon (\vec {\boldsymbol {x}} _ {i}) + \left(\mathbb {E} _ {\vec {\boldsymbol {x}}} \left[ \epsilon (\vec {\boldsymbol {x}}) \right] - \frac {1}{N} \sum_ {i = 1} ^ {N} \epsilon (\vec {\boldsymbol {x}} _ {i})\right). \tag {48}
+$$
+
+Using the Cauchy-Schwartz inequality, the above first term can be upper bounded by
+
+$$
+\frac {1}{N} \sum_ {i = 1} ^ {N} \epsilon (\vec {\boldsymbol {x}} _ {i}) \leq \sqrt {\frac {\lambda^ {2} \sum_ {i = 1} ^ {N} \sum_ {j = 1} ^ {N} \left(\mathrm {K} + \lambda I\right) _ {i j} ^ {- 1} y _ {i} y _ {j}}{N}}. \tag {49}
+$$
+
+Therefore, if the matrix  $\mathrm{K}$  is invertable and hyper-parameter  $\lambda = 0$ , the training error is zero. Without loss of generality, we set  $\lambda = 1 / \mathrm{poly}(n)$ .
+
+The generalized error can be characterized by the Rademacher complexity Mohri et al. (2018), that is
+
+$$
+\mathbb {E} _ {\vec {\boldsymbol {x}}} \epsilon (\vec {\boldsymbol {x}}) - \frac {1}{N} \sum_ {i = 1} ^ {N} \epsilon (\vec {\boldsymbol {x}} _ {i}) \leq \sqrt {\frac {\operatorname* {s u p} \left(\mathrm {K} (x , x)\right) \| \vec {\boldsymbol {a}} _ {\text {o p t}} \| _ {\Psi}}{N}} = \sqrt {\frac {\| \vec {\boldsymbol {b}} (\rho) _ {\text {o p t}} \| _ {\Psi}}{N}}, \tag {50}
+$$
+
+where  $\| \vec{b} (\rho)\|_{\Psi} = \langle \vec{b} (\rho)_{\mathrm{opt}},\vec{b} (\rho)_{\mathrm{opt}}\rangle$  . Ideally,  $\vec{b} (\rho)_{\mathrm{opt}} = (b_{\vec{j},\vec{s}}(\rho))$  for  $j_{1},\ldots ,j_{q}\le K - 1,s_{1}\le s_{2}\ldots \le s_{q}\le LR$  and  $\vec{b} (\rho)$  is induced by the polynomial kernel function, therefore the 2-norm of the vector  $(\pmb {b}_{\vec{j},\vec{s}}(\rho))_{\vec{j},\vec{s}}$  can be used to estimate  $\| \vec{b} (\rho)\|_{\Psi}$  . Noting that the multi-variable polynomial function  $\sum_{\vec{j},\vec{s}}\pmb {b}_{\vec{j},\vec{s}}(\rho)\alpha_{s_1}^{j_1}\dots \alpha_{s_q}^{j_q}$  belongs to [0, 1] for all  $\vec{\alpha}\in [0,2\pi ]^{LR}$  . Let  $\vec{\alpha}$  takes value from the bitstring  $\{0,1\}^{LR}$ , we know that all  $\left|\pmb {b}_{\vec{j},\vec{s}}(\rho)\right|\in [0,1]$ . Therefore, we can upper bound  $\| \vec{b} (\rho)\|_{\Psi}$  by  $(LR)K = (LR)n^{2}\log n$
+
+Combine all together, for any  $M(\vec{\pmb{x}}) = U(\vec{\pmb{x}})|0^n\rangle \langle 0^n|U^\dagger (\vec{\pmb{x}})$  for  $U\in \mathcal{U}_A(R)$  and arbitrary density matrix  $\rho$ , we have the relationship
+
+$$
+\mathbb {E} _ {\vec {x}} \left| \sum_ {j = 1} ^ {N} \vec {\beta} _ {j} (\boldsymbol {x}) \langle 0 ^ {n} | U ^ {\dagger} (\vec {\alpha} _ {j}) \rho U (\vec {\alpha} _ {j}) | 0 ^ {n} \rangle - \operatorname {T r} \left(M (\vec {x}) \rho\right) \right| \leq \sqrt {\frac {\lambda^ {2} \sum_ {i = 1} ^ {N} \sum_ {j = 1} ^ {N} \left(\mathrm {K} + \lambda I\right) _ {i j} ^ {- 1} y _ {i} y _ {j}}{N}} + \sqrt {\frac {L R n ^ {2} \log n}{N}} \tag {51}
+$$
+
+In the above first term, hyper-parameter  $\lambda$  can take arbitrary value, and  $\lambda = \sqrt{\lambda_{\mathrm{min}}(\mathrm{K})} / (nN)$  enables that the first term is upper bounded by  $1/n$ , where  $\lambda_{\mathrm{min}}(\mathrm{K})$  represents the minimum eigenvalue of the kernel matrix  $\mathrm{K}$ . In the second term, let  $N = \tilde{\mathcal{O}}\left((LR)n^{2}\epsilon^{-2}\right)$ , and the above error can be upper bounded by  $\epsilon$ .
+
+# H PROOF OF QUANTUM LEARNING PRINCIPLES
+
+# H.1 PROOF OF LEMMA 2
+
+The proof of Lemma 2 depends on the intrinsic structure of the QCA model. Let  $M_{\mathrm{opt}}$  represent an observable
+
+$$
+M _ {\text {o p t}} = \arg \max  _ {M = V | 0 ^ {n} \rangle \langle 0 ^ {n} | V ^ {\dagger}, V \in \mathcal {U} _ {\mathcal {A}} (R)} \left| \operatorname {T r} \left(M \left(\rho_ {p, \tilde {R}} - I _ {n} / 2 ^ {n}\right)\right) \right|. \tag {52}
+$$
+
+Given the QCA circuit set  $\hat{\rho}_{\mathrm{QCA}}(R,\mathcal{A},N) = \{U(\vec{\alpha}_j)\}_{j = 1}^N$  with  $N = LRn^{2}\epsilon^{-2}$ , the intrinsic structure of QCA promises that there exists a vector  $\vec{\beta}^{\mathrm{opt}}$  such that
+
+$$
+\left| \sum_ {j = 1} ^ {N} \vec {\beta} _ {j} ^ {\mathrm {o p t}} (\vec {\boldsymbol {x}}) \operatorname {T r} \left(U (\vec {\boldsymbol {\alpha}} _ {j}) P _ {0} U ^ {\dagger} (\vec {\boldsymbol {\alpha}} _ {j}) \rho\right) - \operatorname {T r} \left(M _ {\mathrm {o p t}} \rho\right) \right| \leq \epsilon \tag {53}
+$$
+
+for any  $n$ -qubit density matrix  $\rho$  and projector  $P_0 = |0^n\rangle \langle 0^n|$ . Denote
+
+$$
+M _ {R} (\vec {\boldsymbol {\beta}}) = \sum_ {i = 1} ^ {N} \beta_ {i} U (\vec {\boldsymbol {\alpha}} _ {i}) P _ {0} U ^ {\dagger} (\vec {\boldsymbol {\alpha}} _ {i}),
+$$
+
+according to the assumption in Lemma 2, we have
+
+$$
+\begin{array}{l} \epsilon + \tilde {\epsilon} <   \min _ {\vec {\boldsymbol {\beta}}} \left| \mathbb {E} _ {| \Psi_ {i} \rangle \sim (\hat {\rho} _ {\mathrm {Q C A}}, \vec {\boldsymbol {q}})} \left[ \mathrm {T r} (M (\vec {\boldsymbol {\beta}}) (| \Psi_ {i} \rangle \langle \Psi_ {i} | - \rho_ {p, \tilde {R}})) \right] \right| \\ \leq \left| \mathbb {E} _ {| \Psi_ {i} \rangle \sim (\hat {\rho} _ {\mathrm {Q C A}}, \vec {q})} \left[ \operatorname {T r} (M (\vec {\boldsymbol {\beta}} _ {\mathrm {o p t}}) (| \Psi_ {i} \rangle \langle \Psi_ {i} | - \rho_ {p, \vec {R}})) \right] \right| \\ \leq \left| \mathbb {E} _ {| \Psi_ {i} \rangle \sim \left(\hat {\rho} _ {\mathrm {Q C A}}, \vec {\boldsymbol {q}}\right)} \left[ \operatorname {T r} \left(M _ {\text {o p t}} \left(| \Psi_ {i} \rangle \langle \Psi_ {i} | - \rho_ {p, \tilde {R}}\right)\right) \right] + \epsilon \right| \tag {54} \\ = \left| \mathbb {E} _ {| \Psi_ {i} \rangle \sim (\hat {\rho} _ {\mathrm {Q C A}}, \vec {q})} \left[ \mathrm {T r} (M _ {\mathrm {o p t}} (| \Psi_ {i} \rangle \langle \Psi_ {i} | - I _ {n} / 2 ^ {n})) \right] - \mathrm {T r} (M _ {\mathrm {o p t}} (\rho_ {p, \tilde {R}} - I _ {n} / 2 ^ {n})) + \epsilon \right| \\ \leq 1 - \frac {1}{2 ^ {n}} - \mathrm {T r} (M _ {\mathrm {o p t}} (\rho_ {p, \tilde {R}} - I _ {n} / 2 ^ {n})) + \epsilon , \\ \end{array}
+$$
+
+where the third line comes from the intrinsic structure of specific quantum circuit architecture, the last inequality comes from  $\sum_{i}q_{i} = 1$  and  $\langle \Psi_i|M_{\mathrm{opt}}|\Psi_i\rangle \leq 1$ . As a result, we have
+
+$$
+\operatorname {T r} \left(M _ {\text {o p t}} \left(\rho_ {p, \tilde {R}} - I _ {n} / 2 ^ {n}\right)\right) <   1 - \frac {1}{2 ^ {n}} - \tilde {\epsilon}. \tag {55}
+$$
+
+# H.2 PROOF OF LEMMA 1
+
+According to the assumption, the relationship
+
+$$
+\begin{array}{l} \epsilon \geq \max  _ {\vec {q}, M (\vec {\beta})} \left| \mathbb {E} _ {| \Psi_ {i} \rangle \sim \left(\hat {\rho} _ {\mathrm {Q C A}}, \vec {q}\right)} \operatorname {T r} \left(M (\vec {\beta}) \left(| \Psi_ {i} \rangle \langle \Psi_ {i} | - \rho_ {p, \tilde {R}}\right)\right) \right| \tag {56} \\ = \max _ {\vec {\boldsymbol {q}}, M (\vec {\boldsymbol {\beta}})} \left| \mathbb {E} _ {| \Psi_ {i} \rangle \sim (\hat {\rho} _ {\mathrm {Q C A}}, \vec {\boldsymbol {q}})} \left[ \operatorname {T r} \left(M (\vec {\boldsymbol {\beta}}) (| \Psi_ {i} \rangle \langle \Psi_ {i} | - \sigma)\right) - \operatorname {T r} \left(M (\vec {\boldsymbol {\beta}}) (\rho_ {p, \tilde {R}} - \sigma)\right) \right] \right| \\ \end{array}
+$$
+
+holds, where maximal entangled state  $\sigma = I / d$ ,  $d = 2^n$  and  $M(\vec{\beta})$  is defined as Eq. 11. Randomly choose an index  $t \in [N]$ , and let
+
+$$
+M _ {t} = \arg \max  _ {M = V | 0 \rangle \langle 0 | V ^ {\dagger}} \operatorname {T r} \left(M \left(\left| \Psi_ {t} \right\rangle \left\langle \Psi_ {t} \right| - \sigma\right)\right), \tag {57}
+$$
+
+where  $V\in \mathcal{U}_{\mathcal{A}}(R)$ . Based on the intrinsic structure in QCA, there exists a unit vector  $\vec{\beta}^{(t)}$  such that
+
+$$
+\left| \operatorname {T r} \left(M _ {t} \left(\left| \Psi_ {t} \right\rangle \left\langle \Psi_ {t} \right| - \sigma\right)\right) - \operatorname {T r} \left(M \left(\vec {\beta} ^ {(t)}\right) \left(\left| \Psi_ {t} \right\rangle \left\langle \Psi_ {t} \right| - \sigma\right)\right) \right| \leq \epsilon . \tag {58}
+$$
+
+Then assign the probability distribution  $q_{t} = 1 - (N - 1) / d$  and  $q_{j} = 1 / d$  for  $j \neq t$ . As a result, Eq. 56 can be further lower bounded by
+
+$$
+\left| \left(1 - \frac {N - 1}{d}\right) \operatorname {T r} \left(M \left(\vec {\boldsymbol {\beta}} ^ {(t)}\right) \left(\left| \Psi_ {t} \right\rangle \left\langle \Psi_ {t} \right| - \sigma\right)\right) + \frac {1}{d} \sum_ {j \neq t} \operatorname {T r} \left(M \left(\vec {\boldsymbol {\beta}} ^ {(t)}\right) \left(\left| \Psi_ {j} \right\rangle \left\langle \Psi_ {j} \right| - \sigma\right)\right) - \operatorname {T r} \left(M \left(\vec {\boldsymbol {\beta}} ^ {(t)}\right) \left(\rho_ {p, \tilde {R}} - \sigma\right)\right) \right|. \tag {59}
+$$
+
+Since  $|\Psi_t\rangle$  is generated by a  $R$ -depth quantum circuit  $U_{t}\in \mathcal{U}_{A}(R)$ , then  $C_{\epsilon}(|\Psi_t\rangle)\leq LR$  which implies
+
+$$
+\operatorname {T r} \left[ M _ {t} \left(\left| \Psi_ {t} \right\rangle \left\langle \Psi_ {t} \right| - \frac {I}{d}\right) \right] \geq 1 - \frac {1}{d} - \epsilon , \tag {60}
+$$
+
+then the relationship
+
+$$
+\operatorname {T r} \left[ M \left(\vec {\boldsymbol {\beta}} ^ {(t)}\right) \left(\left| \Psi_ {t} \right\rangle \left\langle \Psi_ {t} \right| - \frac {I}{d}\right) \right] \geq \operatorname {T r} \left[ M _ {t} \left(\left| \Psi_ {t} \right\rangle \left\langle \Psi_ {t} \right| - \frac {I}{d}\right) \right] - \epsilon \geq 1 - \frac {1}{d} - 2 \epsilon \tag {61}
+$$
+
+holds. Therefore,  $\left(1 - \frac{N - 1}{d}\right)\operatorname {Tr}\left(M(\vec{\beta}^{(t)})(|\Psi_t\rangle \langle \Psi_t| - \sigma)\right)\geq (1 - \frac{N - 1}{d})(1 - \frac{1}{d} -2\epsilon)$ . Combining the result  $\frac{1}{d}\sum_{j\neq t}\mathrm{Tr}\left(M(\vec{\beta}^{(t)})(|\Psi_j\rangle \langle \Psi_j| - \sigma)\right)\geq \frac{-(N - 1)}{d^2}$ , where  $\mathrm{Tr}\left(M(\vec{\beta}^{(t)})(|\Psi_j\rangle \langle \Psi_j| - \sigma)\right) > - 1 / d$ , we thus have
+
+$$
+\begin{array}{l} \operatorname {T r} \left(M \left(\vec {\boldsymbol {\beta}} ^ {(t)}\right) \left(\rho_ {p, \tilde {R}} - \sigma\right)\right) \geq \left(1 - \frac {N - 1}{d}\right) \left(1 - \frac {1}{d} - 2 \epsilon\right) + \frac {(N - 1)}{d ^ {2}} - \epsilon \tag {62} \\ = 1 - \frac {1}{d} - 3 \epsilon - \frac {N - 1}{d} \left(1 - \frac {2}{d} - 2 \epsilon\right). \\ \end{array}
+$$
+
+Note that  $N = LRn^{2}\log (n)\epsilon^{-2},d = 2^{n}$  , therefore
+
+$$
+\frac {N - 1}{d} \left(1 - \frac {2}{d} - 2 \epsilon\right) = \frac {\operatorname {p o l y} (n)}{2 ^ {n}} \left(1 - \frac {1}{2 ^ {n}} - 2 \epsilon\right) <   \mathcal {O} (\epsilon) \tag {63}
+$$
+
+for large  $n\in \mathbb{Z}_{>0}$  and  $\epsilon = 1 / \mathrm{poly}(n)$ . Finally,
+
+$$
+\begin{array}{l} \max  _ {M = V | 0 ^ {n} \rangle \langle 0 ^ {n} | V ^ {\dagger}} \left| \operatorname {T r} \left(M \left(\rho_ {p, \tilde {R}} - I _ {n} / 2 ^ {n}\right)\right) \right| \geq \operatorname {T r} \left(M _ {t} \left(\rho_ {p, \tilde {R}} - I _ {n} / 2 ^ {n}\right)\right) \\ \geq \operatorname {T r} \left(M \left(\vec {\boldsymbol {\beta}} ^ {(t)}\right) \left(\rho_ {p, \tilde {R}} - I _ {n} / 2 ^ {n}\right)\right) - \epsilon \tag {64} \\ \geq 1 - \frac {1}{d} - \mathcal {O} (\epsilon), \\ \end{array}
+$$
+
+where the first inequality is valid since  $M_{t}$  (defined by Eq. 57) is one of the instances in the set  $\{V|0^n\rangle \langle 0^n |V^\dagger \}$ , the second line comes from the intrinsic structure of QCA, and the third line comes from inequality 62. This implies  $C_\epsilon^{\mathrm{lim},\mathcal{A}}(\rho_{p,\tilde{R}})\leq LR$ .
+
+# I BAYESIAN OPTIMIZATION
+
+# I.1 OPTIMIZATION SUBROUTINE
+
+In the following, we show how to maximize the loss function  $\mathcal{L}_R(\vec{\beta})$  via Bayesian optimization on a compact set. Bayesian optimization is composed by two significant components:  $(i)$  a statistical model, in general Gaussian process, provides a posterior distribution conditioned on a prior distribution and a set of observations over  $\mathcal{L}_R(\vec{\beta})$ .  $(ii)$  an acquisition function determines the position of the next sample point, based on the current posterior distribution over  $\mathcal{L}_R(\vec{\beta})$ .
+
+Gaussian process is a set of random variables, where any subset forms a multivariate Gaussian distribution. For the optimization task considered in the main file, the random variables represent the value of the objective function  $\mathcal{L}_R(\vec{\beta})$  at the point  $\vec{\beta}$ . As a distribution over  $\mathcal{L}_R(\vec{\beta})$ , a Gaussian process is completely specified by mean function and covariance function
+
+$$
+\mu (\vec {\beta}) = \mathbb {E} _ {\vec {\beta}} [ \mathcal {L} _ {R} (\vec {\beta}) ]
+$$
+
+$$
+k (\vec {\beta}, \vec {\beta} ^ {\prime}) = \mathbb {E} _ {\vec {\beta}} [ (\mathcal {L} _ {R} (\vec {\beta}) - \mu (\vec {\beta})) (\mathcal {L} _ {R} (\vec {\beta} ^ {\prime}) - \mu (\vec {\beta} ^ {\prime})) ], \tag {65}
+$$
+
+and the Gaussian process is denoted as  $\mathcal{L}_R(\vec{\beta})\sim \mathcal{GP}(\mu (\vec{\beta}),k(\vec{\beta},\vec{\beta}'))$ . Without loss of generality, we assume that the prior mean function  $\mu (\vec{\beta}) = 0$ . In the  $t$ -th iteration step, assuming observations  $\mathrm{Acc}(t) = \{(\vec{\beta}^{(1)},y(\vec{\beta}^{(1)})),\ldots ,(\vec{\beta}^{(t)},y(\vec{\beta}^{(t)}))\}$  are accumulated, where  $y(\vec{\beta}^{(i)}) = \mathcal{L}_R(\vec{\beta}^{(i)}) + \epsilon_i$ , with the quantum measurement error  $\epsilon_{i}\sim \mathcal{N}(0,1 / 4\hat{M})$  for  $i\in [t]$ , where  $\hat{M}$  represents the measurement complexity in our algorithm. Conditioned on the accumulated observations  $\mathrm{Acc}(t)$ , the posterior distribution of  $\mathcal{L}_R(\vec{\beta})$  is a Gaussian process with mean function  $\mu_t(\vec{\beta}) = \mathbb{E}_{\vec{\beta}}[\mathcal{L}_R(\vec{\beta})|\mathrm{Acc}(t)]$  and covariance function  $k_{t}(\vec{\beta},\vec{\beta}^{\prime}) = \mathbb{E}_{\vec{\beta}}[(\mathcal{L}_{R}(\vec{\beta}) - \mu (\vec{\beta}))(\mathcal{L}_{R}(\vec{\beta}^{\prime})-$ $\mu (\vec{\beta}^{\prime}))|\mathrm{Acc}(t)]$ , specified by
+
+$$
+\mu_ {t} (\vec {\beta}) = \boldsymbol {k} _ {t} ^ {\mathrm {T}} \left[ \boldsymbol {K} _ {t} + \boldsymbol {I} _ {t} / 4 \hat {M} \right] ^ {- 1} \boldsymbol {y} _ {1: t}
+$$
+
+$$
+\begin{array}{l} \boldsymbol {\mu} _ {t} (\boldsymbol {\beta}) = \boldsymbol {\kappa} _ {t} \left[ \boldsymbol {I} _ {t} + \boldsymbol {I} _ {t} / 4 \hat {M} \right] - \boldsymbol {y} _ {1, t} \\ k _ {t} (\vec {\boldsymbol {\beta}}, \vec {\boldsymbol {\beta}} ^ {\prime}) = k (\vec {\boldsymbol {\beta}}, \vec {\boldsymbol {\beta}} ^ {\prime}) - \boldsymbol {k} _ {t} ^ {\mathrm {T}} [ \boldsymbol {K} _ {t} + \boldsymbol {I} _ {t} / 4 \hat {M} ] ^ {- 1} \boldsymbol {k} _ {t}, \end{array} \tag {66}
+$$
+
+where  $\pmb{k}_t = [k(\vec{\beta},\vec{\beta}^{(1)})\dots k(\vec{\beta},\vec{\beta}^{(t)})]^{\top}$ , the positive definite covariance matrix  $\pmb{K}_{t} = [k(\vec{\beta},\vec{\beta}^{\prime})]_{\vec{\beta},\vec{\beta}^{\prime}\in \vec{\beta}_{1:t}}$  with  $\vec{\beta}_{1:t} = \{\vec{\beta}^{(1)},\ldots ,\vec{\beta}^{(t)}\}$  and  $y_{1:t} = [y(\vec{\beta}^{(1)}),\ldots ,y(\vec{\beta}^{(t)})]^{\top}$ . The posterior variance of  $\mathcal{L}_R(\vec{\beta})$  is denoted as  $\sigma_t^2 (\vec{\beta}) = k_t(\vec{\beta},\vec{\beta})$ . The mean function  $\mu_t(\vec{\beta})$  is related to the expected value of  $\mathcal{L}_R(\vec{\beta})$ , while the covariance  $k_{t}$  estimates the deviations of  $\mu_t(\vec{\beta})$  from the value of  $\mathcal{L}_R(\vec{\beta})$ . Then the prediction is obtained by conditioning the prior Gaussian process on the observations and returns a posterior distribution described by a Gaussian process multivariate distribution. Using the Sherman-Morrison-Woodbury formula (Seeger, 2004), the predictive distribution can be explicitly expressed as Eq. 66.
+
+In the  $t$ -th iteration of Bayesian optimization, the acquisition function  $\mathcal{A}(\vec{\beta})$  learns from the accumulated observations  $\mathrm{Acc}(t - 1)$  and leads the search to the next point  $\vec{\beta}^{(t)}$  which is expected to gradually converge to the optimal parameters of  $\mathcal{L}_R(\vec{\beta})$ . This procedure is achieved via maximizing  $\mathcal{A}(\vec{\beta})$ . In detail, the design of acquisition function should consider exploration (exploring domains where  $\mathcal{L}_R(\vec{\beta})$  has high uncertainty) and exploitation (exploring domains where  $\mathcal{L}_R(\vec{\beta})$  is expected to have large image value). The upper confidence bound is a widely used acquisition function, which is defined as
+
+$$
+\mathcal {A} _ {\mathrm {U C B}} (\vec {\beta}) = \mu_ {t - 1} (\vec {\beta}) + \sqrt {\kappa_ {t}} \sigma_ {t - 1} (\vec {\beta}), \tag {67}
+$$
+
+and the next point  $\vec{\beta}^{(t)}$  is decided by  $\vec{\beta}^{(t)} = \arg \max_{\vec{\beta}\in \mathcal{D}_{\mathrm{domain}}}\mathcal{A}_{\mathrm{UCB}}(\vec{\beta})$ . Here,  $\kappa_{t}$  is a significant hyper-parameter, and a suitable  $\kappa_{t}$  may lead  $\vec{\beta}^{(t)}$  rapidly convergence to  $\vec{\beta}_{opt}$ . In Theorem 4, a specific  $\kappa_{t} = 2N\log (t^{2}N) + 2\log (t^{2} / \delta)$  is used. Details for maximizing  $\mathcal{L}_R(\vec{\beta})$  are shown in Alg 2.
+
+Algorithm 2: Bayesian Maximize Subroutine, BMaxS  $(\rho_{\mathrm{un}},\hat{\rho}_{\mathrm{QCA}}(R,\mathcal{A},N),T,\epsilon)$  
+Input: Noisy quantum state  $\rho_{\mathrm{un}}$  , a quantum state set  $\hat{\rho}_{\mathrm{QCA}}(R,\mathcal{A},N)$  , failure probability  $\delta \in (0,1)$  , iteration steps  $T$  , approximation error  $\epsilon$    
+Output: True/False;   
+Initialize  $\mu_0(\vec{\beta}) = 0$ $\sigma_0$  , the covariance function  $k(\cdot ,\cdot)$  .   
+for  $t = 1,2,\ldots ,T$  do Select  $\kappa_{t} = 2N\log (t^{2}N) + 2\log (t^{2} / \delta)$  Choose  $\vec{\beta}^{(t)} = \arg \max_{\vec{\beta}\in \mathcal{D}_{\beta}}\mu_{t - 1}(\vec{\beta}) + \sqrt{\kappa_t}\sigma_{t - 1}(\vec{\beta})$  Estimate  $\mathcal{L}_R(\vec{\beta}^{(t)})$  with shadow tomography of  $\rho_{\mathrm{un}}$  and  $\hat{\rho}_{\mathrm{QCA}}$  (obtained from  $\hat{M}$  -snapshot measurements), that is  $|y(\vec{\beta}^{(t)}) - \mathcal{L}_R(\vec{\beta}^{(t)})|\leq \epsilon_t$  where  $\epsilon_t\sim \mathcal{N}(0,1 / 4\hat{M})$  Update  $\mu_t,\sigma_t^2$  as Eq.66;   
+if  $y(\vec{\beta}^{(T)})\leq \epsilon$  do return True   
+else do return False
+
+# I.2 PROOF OF THEOREM 4
+
+Theorem 5 (Formal version of Theorem 4) Take the weakly noisy state  $\rho_{\mathrm{un}}$  and  $\hat{\rho}_{\mathrm{QCA}}(R,\mathcal{A},N)$  into Alg. 2. Pick the failure probability  $\delta \in (0,1)$  and let
+
+$$
+\kappa_ {t} = 2 N \log \left(t ^ {2} N\right) + 2 \log \left(t ^ {2} / \delta\right) \tag {68}
+$$
+
+in the  $t$ -th iteration step, then the average regret  $\mathrm{avr}_T$  can be upper bounded by
+
+$$
+\operatorname {a v r} _ {T} \leq \mathcal {O} \left(\sqrt {\frac {4 N ^ {2} \log^ {2} T + 2 N \log T \log \left(\pi^ {2} / (6 \delta)\right)}{T}}\right) \tag {69}
+$$
+
+with  $1 - \delta$  success probability.
+
+We need following two lemmas to support our proof.
+
+Lemma 3 (Lemma 5.1 in Srinivas et al. (2012)) Pick failure probability  $\delta \in (0,1)$  and set  $\kappa_{t} = 2\log(|\mathcal{D}_{\mathrm{domain}}| \pi_{t} / \delta)$ , where  $\sum_{t \geq 1} \pi_{t}^{-1} = 1$  and  $\pi_{t} > 0$ . Then
+
+$$
+\left| \mathcal {L} (\vec {\boldsymbol {\beta}}) - \mu_ {t - 1} (\vec {\boldsymbol {\beta}}) \right| \leq \kappa_ {t} ^ {1 / 2} \sigma_ {t - 1} (\vec {\boldsymbol {\beta}}) \tag {70}
+$$
+
+holds for any  $t \geq 1$  and  $\vec{\beta} \in \mathcal{D}_{\mathrm{domain}}$ .
+
+Lemma 4 (Lemma 5.4 in Srinivas et al. (2012)) Pick failure probability  $\delta \in (0,1)$  and let  $\kappa_{t}$  be defined as in Lemma 3. Then the following holds with probability  $\geq 1 - \delta$ :
+
+$$
+\sum_ {t = 1} ^ {T} 4 \kappa_ {t} \sigma_ {t - 1} ^ {2} (\vec {\boldsymbol {\beta}} _ {t}) \leq \kappa_ {T} \gamma_ {T}, \tag {71}
+$$
+
+where  $\gamma_{T} = \max_{A\in \mathcal{D}_{\mathrm{domain}}}\frac{1}{2}\log \left|I + \sigma^{-2}K_{A}\right|$ , and  $K_{A}$  represents the used covariance matrix in Bayesian optimization.
+
+We first consider the continuity of the loss function  $\mathcal{L}(\vec{\beta})$ . Considering  $\sum_{i=1}^{N} \beta_i = 1$ , the gradient function can be upper bounded by
+
+$$
+\left| \frac {\partial \mathcal {L} (\vec {\beta})}{\partial \beta_ {j}} \right| = \left| \mathbb {E} _ {\hat {\rho} _ {i} \sim \vec {\mathbf {q}}} \operatorname {T r} \left[ \hat {\rho} _ {j} \left(\hat {\rho} _ {i} - \rho_ {\mathrm {u n}}\right) \right] \right| \leq 1, \tag {72}
+$$
+
+where the inequality comes from  $|\mathrm{Tr}[\hat{\rho}_j(\hat{\rho}_i - \rho_{\mathrm{un}})]| \leq 1$ . Therefore, the relationship
+
+$$
+\left| \mathcal {L} (\vec {\beta}) - \mathcal {L} \left(\vec {\beta} ^ {\prime}\right) \right| \leq \left| \vec {\beta} - \vec {\beta} ^ {\prime} \right| _ {1} \tag {73}
+$$
+
+holds for any  $\vec{\beta},\vec{\beta}^{\prime}\in \mathcal{D}_{\mathrm{domain}}$ . Now let us choose a discretization  $\mathcal{D}_{\mathrm{domain}}^{t}$  of size  $(\tau_{t})^{N}$  such that for all  $\vec{\beta}\in \mathcal{D}_{\mathrm{domain}}$
+
+$$
+\left\| \vec {\boldsymbol {\beta}} - [ \vec {\boldsymbol {\beta}} ] _ {t} \right\| _ {1} \leq N / \tau_ {t}, \tag {74}
+$$
+
+where  $[\vec{\beta}]_t$  represents the closest point in discretization on  $\mathcal{D}_{\mathrm{domain}}^t$  to  $\vec{\beta}$ . Combine Eqs. 73-74, one obtains
+
+$$
+\left| \mathcal {L} (\vec {\boldsymbol {\beta}}) - \mathcal {L} ([ \vec {\boldsymbol {\beta}} ] _ {t}) \right| \leq N \tau_ {t} ^ {- 1} = t ^ {- 2}, \tag {75}
+$$
+
+where the last inequality comes from  $\tau_t = t^2 N$ , furthermore,  $|\mathcal{D}_{\mathrm{domain}}^t| = (t^2 N)^N$ .
+
+Using Lemma 3, we know that
+
+$$
+\kappa_ {t} = 2 \log \left(\left| \mathcal {D} _ {\text {d o m a i n}} \right| a _ {t} / \delta\right) \tag {76}
+$$
+
+enables the relationship
+
+$$
+\left| \mathcal {L} (\vec {\boldsymbol {\beta}}) - \mu_ {t - 1} (\vec {\boldsymbol {\beta}}) \right| \leq \sqrt {\kappa_ {t}} \sigma_ {t - 1} (\vec {\boldsymbol {\beta}}) \tag {77}
+$$
+
+holds with probability at least  $1 - \delta$ , where  $a_{t} > 0$  and  $\sum_{t\geq 1}a_t^{-1} = 1$ . A common selection is  $a_{t} = \pi t^{2} / 6$ . Taking  $|\mathcal{D}_{\mathrm{domain}}^t| = (t^2 N)^N$  into  $\kappa_{t}$  (Eq. 76), one obtains
+
+$$
+\kappa_ {t} = 2 \log \left(\left(t ^ {2} N\right) ^ {N} a _ {t} / \delta\right). \tag {78}
+$$
+
+Combine Eq. 75 and 77, we have
+
+$$
+\left| \mathcal {L} \left(\vec {\boldsymbol {\beta}} ^ {*}\right) - \mu_ {t - 1} \left(\left[ \vec {\boldsymbol {\beta}} ^ {*} \right] _ {t}\right) \right| \leq \left| \mathcal {L} \left(\vec {\boldsymbol {\beta}} ^ {*}\right) - \mathcal {L} \left(\left[ \vec {\boldsymbol {\beta}} ^ {*} \right] _ {t}\right) \right| + \left| \mathcal {L} \left(\left[ \vec {\boldsymbol {\beta}} ^ {*} \right] _ {t}\right) - \mu_ {t - 1} \left(\left[ \vec {\boldsymbol {\beta}} ^ {*} \right] _ {t}\right) \right| \leq t ^ {- 2} + \sqrt {\kappa_ {t}} \sigma_ {t - 1} \left(\left[ \vec {\boldsymbol {\beta}} ^ {*} \right] _ {t}\right) \tag {79}
+$$
+
+for any  $t \geq 1$ , where  $\vec{\beta}^* = \arg \max_{\vec{\beta} \in \mathcal{D}_{\beta}} \mathcal{L}(\vec{\beta})$ . Now we connect the relationship between above inequality to the regret bound.
+
+By the definition of  $\vec{\beta}^{(t)}$  (maximizing the  $\mathcal{A}_{\mathrm{UCB}}(\vec{\beta})$  in the  $t$ -th step):  $\mu_{t-1}(\vec{\beta}^{(t)}) + \sqrt{\kappa_t}\sigma_{t-1}(\vec{\beta}^{(t)}) \geq \mu_{t-1}([\vec{\beta}^*]_t) + \sqrt{\kappa_t}\sigma_{t-1}([\vec{\beta}^*]_t)$ . Also, by Eq. 79, we have  $\mathcal{L}(\vec{\beta}^*) \leq \mu_{t-1}([\vec{\beta}^*]_t) + \sqrt{\kappa_t}\sigma_{t-1}([\vec{\beta}^*]_t) + 1/t^2$ . Therefore, instantaneous regret
+
+$$
+\begin{array}{l} r _ {t} = \mathcal {L} (\vec {\boldsymbol {\beta}} ^ {*}) - \mathcal {L} (\vec {\boldsymbol {\beta}} ^ {(t)}) \\ \leq \sqrt {\kappa_ {t}} \sigma_ {t - 1} (\vec {\beta} ^ {(t)}) + 1 / t ^ {2} + \mu_ {t - 1} (\vec {\beta} ^ {(t)}) - \mathcal {L} (\vec {\beta} ^ {(t)}) \\ \leq 2 \sqrt {\kappa_ {t}} \sigma_ {t - 1} (\vec {\beta} ^ {(t)}) + 1 / t ^ {2}, \\ \end{array}
+$$
+
+where the last inequality comes from Eq. 77. Using Lemma 4,  $\sum_{t=1}^{T} 4\kappa_t \sigma_{t-1}(\vec{\beta}_t) \leq \kappa_T \gamma_T$ , where
+
+$$
+\gamma_ {T} = \max  _ {A \in \mathcal {D} _ {\text {d o m a i n}}} \frac {1}{2} \log \left| I + 2 \hat {M} K _ {A} \right|,
+$$
+
+$K_{A}$  represents the used covariance matrix in  $\mathrm{BMaxS}(\rho_{p,\tilde{R}},\hat{\rho}_{\mathrm{QCA}}(R,\mathcal{A},N),T)$  and  $\hat{M}$  represents the number of measurement in generating the shadow tomography. In the linear function case,  $K_{A}$  can be selected as the polynomial kernel function, and  $\gamma_T = \mathcal{O}(N\log (T))$  Vakili et al. (2021). Furthermore, using the Cauchy-Schwartz inequality to  $\sum_{t = 1}^{T}4\kappa_{t}\sigma_{t - 1}^{2}(\vec{\beta}_{t})\leq \kappa_{T}\gamma_{T}$ , one obtains  $\sum_{t = 1}^{T}2\kappa_{t}^{1 / 2}\sigma_{t - 1}(\vec{\beta}_{t})\leq \sqrt{T\kappa_{T}\gamma_{T}}$
+
+Finally, we have the result
+
+$$
+\begin{array}{l} \operatorname {a v r} _ {T} = \frac {1}{T} \sum_ {t = 1} ^ {T} r _ {t} \\ \leq \frac {1}{T} \sum_ {t = 1} ^ {T} 2 \kappa_ {t} ^ {1 / 2} \sigma_ {t - 1} (\vec {\boldsymbol {\beta}} _ {t}) + \frac {\pi^ {2}}{6 T} \\ \leq \sqrt {\frac {\kappa_ {T} \gamma_ {T}}{T}} + \frac {\pi^ {2}}{6 T} \tag {80} \\ = \mathcal {O} \left(\sqrt {\frac {2 N ^ {2} \log (T ^ {2} N) \log (T) + 2 N \log (T ^ {2} \pi^ {2} / (6 \delta)) \log (T)}{T}}\right) \\ \leq \mathcal {O} \left(\sqrt {\frac {4 N ^ {2} \log^ {2} T + 2 N \log T \log (\pi^ {2} / (6 \delta))}{T}}\right), \\ \end{array}
+$$
+
+where the first inequality results from  $\sum_{t\geq 1}t^{-2} = \pi^2 /6$  and the last inequality comes from  $N\leq T$
+
+# J PROOF OF SAMPLE COMPLEXITY LOWER BOUND
+
+we require the following lemmas to support our proof.
+
+Lemma 5 (Lemma 6 in Wang et al. (2021)) Consider a single instanoise channel  $\mathcal{N} = \mathcal{N}_1 \otimes \dots \otimes \mathcal{N}_n$  where each local noise channel  $\{\mathcal{N}_j\}_{j=1}^n$  is a Pauli noise channel that satisfies  $\mathcal{N}_j(\sigma) = q_\sigma \sigma$  for  $\sigma \in \{X, Y, Z\}$  and  $q_\sigma$  be the Pauli strength. Then we have
+
+$$
+D _ {2} \left(\mathcal {N} (\rho) \| \frac {I ^ {\otimes n}}{2 ^ {n}}\right) \leq q ^ {2 c} D _ {2} \left(\rho \| \frac {I ^ {\otimes n}}{2 ^ {n}}\right), \tag {81}
+$$
+
+where  $D_2(\cdot \|\cdot)$  represents the 2-Renyi relative entropy,  $q = \max_{\sigma} q_{\sigma}$  and  $c = 1/(2\ln 2)$ .
+
+Lemma 6 Given an arbitrary  $n$ -qubit density matrix and maximally mixed state  $I^{\otimes n} / 2^n$ , we have
+
+$$
+D \left(\rho \| I ^ {\otimes n} / 2 ^ {n}\right) \leq D _ {2} \left(\rho \| I ^ {\otimes n} / 2 ^ {n}\right), \tag {82}
+$$
+
+where  $D(\cdot \| \cdot)$  denotes the relative entropy and  $D_{2}(\cdot \| \cdot)$  denotes the 2-Renyi relative entropy.
+
+Proof: Given quantum states  $\rho$  and  $\sigma$ , the quantum 2-Renyi entropy
+
+$$
+D _ {2} (\rho \| \sigma) = \log \operatorname {T r} \left[ \left(\sigma^ {- 1 / 4} \rho \sigma^ {- 1 / 4}\right) ^ {2} \right]. \tag {83}
+$$
+
+When  $\sigma = I^{\otimes n} / 2^n$ , we have  $D_{2}(\rho \| I^{\otimes n} / 2^{n}) = \log \operatorname{Tr}\left[\left((I^{\otimes n} / 2^{n})^{-1}\rho^{2}\right)\right] = n + \log \operatorname{Tr}[\rho^{2}]$ . Noting that the function  $y = x^{2} - x\log x\geq 0$  when  $x\in [0,1]$ , and this implies  $\mathrm{Tr}(\rho^2)\geq \mathrm{Tr}(\rho \log \rho)$ . Finally, we have
+
+$$
+D \left(\rho \| I ^ {\otimes n} / 2 ^ {n}\right) = n + \operatorname {T r} \left[ \rho \log \rho \right] + n \leq \operatorname {T r} \left[ \rho^ {2} \right] + n = D _ {2} \left(\rho \| I ^ {\otimes n} / 2 ^ {n}\right). \tag {84}
+$$
+
+Task 2 Consider a pure quantum state  $\rho_0$  and two quantum circuit  $\mathcal{C}_1$  and  $\mathcal{C}_2$  with  $R_{1}$  and  $R_{2}$  depth  $(R_{1} < R_{2})$ , respectively. Each quantum circuit is affected by  $p$ -strength Pauli channel in each layer. Suppose that a distinguisher is given access to copies of the quantum states  $\Phi_{\mathcal{C}_1}(\rho_0)$  and  $\Phi_{\mathcal{C}_2}(\rho_0)$ , then what is the fewest number of copies sufficing to identify quantum states  $\Phi_{\mathcal{C}_1}(\rho_0)$  and  $\Phi_{\mathcal{C}_2}(\rho_0)$  with high probability?
+
+Obviously, if a quantum state complexity prediction problem can predict the complexity  $R_{1}$  and  $R_{2}$ , then we can classify quantum states  $\Phi_{\mathcal{C}_1}(\rho_0)$  and  $\Phi_{\mathcal{C}_2}(\rho_0)$  easily. The sample complexity of Task 2
+
+thus can be used to benchmark the sample complexity lower bound of the quantum state complexity prediction problem.
+
+Now we prove the sample complexity lower bound to Task 2. We consider the sample complexity  $m$  in distinguishing quantum states  $\Phi_{\mathcal{C}_1}(\rho_0)$  and  $\Phi_{\mathcal{C}_2}(\rho_0)$ . When their trace distance is quite large, let  $\delta \in (0,1)$  and we have
+
+$$
+\begin{array}{l} 1 - \delta \leq \frac {1}{2} \left\| \Phi_ {\mathcal {C} _ {1}} (\rho_ {0}) ^ {\otimes m} - \Phi_ {\mathcal {C} _ {2}} (\rho_ {0}) ^ {\otimes m} \right\| _ {1} \\ \leq \frac {1}{2} \left(\left\| \Phi_ {\mathcal {C} _ {1}} \left(\rho_ {0}\right) ^ {\otimes m} - \left(I _ {n} / 2 ^ {n}\right) ^ {\otimes m} \right\| _ {1} + \left\| \Phi_ {\mathcal {C} _ {2}} \left(\rho_ {0}\right) ^ {\otimes m} - \left(I _ {n} / 2 ^ {n}\right) ^ {\otimes m} \right\| _ {1}\right) \tag {85} \\ \leq \frac {1}{\sqrt {2}} \left(D ^ {1 / 2} \left(\Phi_ {\mathcal {C} _ {1}} ^ {\otimes m} (\rho_ {0}) \| (I _ {n} / 2 ^ {n}) ^ {\otimes m}\right) + D ^ {1 / 2} \left(\Phi_ {\mathcal {C} _ {2}} ^ {\otimes m} (\rho_ {0}) \| (I _ {n} / 2 ^ {n}) ^ {\otimes m}\right)\right), \\ \end{array}
+$$
+
+where the second line comes from the triangle inequality and the third line comes from the Pinsker's inequality. Using Lemmas 5 and 6, we have
+
+$$
+\begin{array}{l} 1 - \delta \leq \frac {1}{\sqrt {2}} \left(D _ {2} ^ {1 / 2} \left(\Phi_ {\mathcal {C} _ {1}} ^ {\otimes m} (\rho_ {0}) \| (I _ {n} / 2 ^ {n}) ^ {\otimes m}\right) + D _ {2} ^ {1 / 2} \left(\Phi_ {\mathcal {C} _ {2}} ^ {\otimes m} (\rho_ {0}) \| (I _ {n} / 2 ^ {n}) ^ {\otimes m}\right)\right) \\ \leq \frac {\sqrt {n m}}{\sqrt {2}} \left(\left(1 - p\right) ^ {c R _ {1}} + \left(1 - p\right) ^ {c R _ {2}}\right) \tag {86} \\ \leq \sqrt {2 n m} (1 - p) ^ {c R _ {1}}, \\ \end{array}
+$$
+
+As a result we have
+
+$$
+m \geq \frac {(1 - p) ^ {- 2 c R _ {1}} (1 - \delta) ^ {2}}{2 n}. \tag {87}
+$$
+
+# K REVIEW OF SHADOW TOMOGRAPHY
+
+In quantum computation, the basic operators are the Pauli operators  $\{I,\sigma^x,\sigma^y,\sigma^z\}$  which provide a basis for the density operators of a single qubit as well as for the unitaries that can be applied to them. For an  $n$  -qubit case, one can construct the Pauli group according to
+
+$$
+\mathbf {P} _ {n} = \left\{e ^ {i \alpha \pi / 2} \sigma^ {j _ {1}} \otimes \dots \otimes \sigma^ {j _ {n}} \mid j _ {k} \in \{I, x, y, z \}, 1 \leq k \leq n \right\},
+$$
+
+where  $e^{i\alpha \pi /2} \in \{0,\pm 1,\pm i\}$  is a global phase. Then the Clifford group  $\mathbf{Cl}(2^n)$  is defined as the group of unitaries that normalize the Pauli group:
+
+$$
+\mathbf {C l} (2 ^ {n}) = \left\{U | U \mathbf {P} ^ {(1)} U ^ {\dagger} = \mathbf {P} ^ {(2)}, \left(\mathbf {P} ^ {(1)}, \mathbf {P} ^ {(2)} \in \mathbf {P} _ {n}\right) \right\}.
+$$
+
+The  $n$ -qubit Clifford gates are then defined as elements in the Clifford group  $\mathbf{Cl}(2^n)$ , and these Clifford gates compose the Clifford circuit.
+
+Randomly sampling Clifford circuit  $U$  can reproduce the first 3 moments of the full Clifford group endowed with the Haar measure  $\mathrm{d}\mu (U)$  which is the unique left- and right- invariant measure such that
+
+$$
+\int_ {\mathbf {C l} (2 ^ {n})} \mathbf {d} \mu (U) f (U) = \int \mathbf {d} \mu (U) f (V U) = \int \mathbf {d} \mu (U) f (U V)
+$$
+
+for any  $f(U)$  and  $V \in \mathbf{Cl}(2^n)$ . Using this property, one can sample Clifford circuits  $U \in \mathbf{Cl}(2^n)$  with the probability  $\operatorname*{Pr}(U)$ , and the corresponding expectation  $\mathrm{E}_{U \in \mathbf{Cl}(2^n)}\left[\left(U\rho U^\dagger\right)^{\otimes t}\right]$  can be expressed as
+
+$$
+\sum_ {U \in \mathbf {C l} (2 ^ {n})} \Pr (U) \left(U \rho U ^ {\dagger}\right) ^ {\otimes t} = \int_ {\mathbf {C l} (2 ^ {n})} \mathbf {d} \mu (U) \left(U \rho U ^ {\dagger}\right) ^ {\otimes t},
+$$
+
+for any  $n$ -qubit density matrix  $\rho$  and  $t = 1,2,3$ , where  $\operatorname{Pr}(U)$  indicates the probability on sampling  $U$ . The right hand side of the above equation can be evaluated explicitly by representation theory, this thus yields a closed-form expression for sampling from a Clifford group.
+
+To extract meaningful information from an  $n$ -qubit unknown quantum state  $\rho$ , the shadow tomography technique was proposed by Huang et al. (2020). The Clifford sampling is implemented by
+
+repeatedly performing a simple measurement procedure: apply a random unitary  $U \in \mathbf{Cl}(2^n)$  to rotate the state  $\rho$  and perform a  $\sigma^z$ -basis measurement. The number of repeating times of this procedure is defined as the Clifford sampling complexity. On receiving the  $n$ -bit measurement outcome  $|b\rangle \in \{0,1\}^n$ , according to the Gottesman-Knill theorem (Gottesman, 1997), we can efficiently store an classical description of  $U^\dagger |b\rangle \langle b|U$  in the classical memory. This classical description encodes meaningful information of the state  $\rho$  from a particular angle, and it is thus instructive to view the average mapping from  $\rho$  to its classical snapshot  $U^\dagger |b\rangle \langle b|U$  as a quantum channel:
+
+$$
+\mathcal {M} (\rho) = \mathrm {E} _ {U \in \mathbf {C l} (2 ^ {n})} \left(\mathrm {E} _ {b \in \{0, 1 \} ^ {n}} [ U ^ {\dagger} | b \rangle \langle b | U ]\right), \tag {88}
+$$
+
+where the quantum channel  $\mathcal{M}$  depends on the ensemble of unitary transformation, and the quantum channel  $\mathcal{M}$  can be further expressed as
+
+$$
+\mathcal {M} (\rho) = \mathrm {E} _ {U \in \mathbf {C l} (2 ^ {n})} \sum_ {\widehat {b} \in \{0, 1 \} ^ {n}} \langle \widehat {b} | U \rho U ^ {\dagger} | \widehat {b} \rangle U ^ {\dagger} | \widehat {b} \rangle \langle \widehat {b} | U = \frac {\rho + \operatorname {T r} (\rho) I}{(2 ^ {n} + 1) 2 ^ {n}}, \tag {89}
+$$
+
+where  $I$  indicates the  $2^{n} \times 2^{n}$  identity matrix. Therefore the inverse of quantum channel  $\mathcal{M}^{-1}(\rho) = (2^{n} + 1)\rho - I$ , and an estimation of  $\rho$  is defined as
+
+$$
+\widehat {\rho} = \mathcal {M} ^ {- 1} \left(U ^ {\dagger} | b \rangle \langle b | U\right).
+$$
+
+Repeat this procedure  $M$  times results in an array of Clifford samples of  $\rho$ :
+
+$$
+S (\rho ; M) = \left\{\widehat {\rho_ {1}} = \mathcal {M} ^ {- 1} \left(U _ {1} ^ {\dagger} | b _ {1} \rangle \langle b _ {1} | U _ {1}\right), \dots , \widehat {\rho_ {M}} = \mathcal {M} ^ {- 1} \left(U _ {M} ^ {\dagger} | b _ {M} \rangle \langle b _ {M} | U _ {M}\right) \right\}, \tag {90}
+$$
+
+and  $\frac{1}{M}\sum_{m = 1}^{M}\hat{\rho}_{m}$  is defined as the classical shadow of the quantum state  $\rho$ .
+
+# L NUMERICAL SIMULATIONS FOR STRESS-TESTING
+
+Here, we conducted stress testing of the quantum learning algorithm in this study. Specifically, we applied the algorithm to estimate the complexity lower bound of quantum states under noisy environments for one-dimensional dynamical evolution. We tested the transverse-field Ising model with system sizes of  $1 \times 6$ ,  $1 \times 8$ , and  $1 \times 10$ , and the numerical results are presented in Figure 3. Our findings indicate that the complexity lower bound of noisy quantum circuits does not grow linearly with circuit depth, which is consistent to the results given by Figure 2 in the main file.
+
+Furthermore, we observed that as the system size and complexity increase, the curves corresponding to two different quantum states,  $\tilde{R} = 2$  and  $\tilde{R} = 10$ , exhibit similar overall trends for the system size  $n \in \{6,8,10\}$ . Moreover, the variance of the algorithm does not increase with the growing system size but remains consistently stable. At the same time, the gap between the curves of quantum states with different complexities does not show a closing trend as the system size grows. These characteristics demonstrate the robustness of our learning algorithm against increases in system size and complexity. Therefore, our algorithm remains reliable for predicting the complexity of large-scale weakly noisy quantum states.
+
+![](images/0ef6f280e8c8e9a3a77481104ba5015f9fce51cb68df145bc2baa4d3a7a3b3b3.jpg)  
+Figure 3: These three subfigures illustrate the trend of the function  $\min_{\vec{\beta}}\mathcal{L}_R$  as it varies with the circuit depth  $R$  of QCA set. In the stress-testing experiments, we focus on the Hamiltonian dynamics of 1D transverse field Ising models, with system sizes ranging from 6 to 10. In all cases, we observe that the weakly noisy state with  $\tilde{R} = 2$  exhibits a larger quantum state complexity lower bound compared to the weakly noisy state with  $\tilde{R} = 10$ .
+
+![](images/5f685cb4d4c458e3ec365ccdc3c3c88ae42778e0f4a1fae2e57924030850c2f6.jpg)
+
+![](images/c27b086ebba9f6249348910ff281233f951f59597eb249be1351ccc9e052603e.jpg)
\ No newline at end of file
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+# LEARNING THE OPTIMAL STOPPING FOR EARLY CLASSIFICATION WITHIN FINITE HORIZONS VIASEQUENTIAL PROBABILITY RATIO TEST
+
+Akinori F. Ebihara Taiki Miyagawa Kazuyuki Sakurai Hitoshi Imaoka NEC Corporation aebihara@nec.com
+
+# ABSTRACT
+
+Time-sensitive machine learning benefits from Sequential Probability Ratio Test (SPRT), which provides an optimal stopping time for early classification of time series. However, in finite horizon scenarios, where input lengths are finite, determining the optimal stopping rule becomes computationally intensive due to the need for backward induction, limiting practical applicability. We thus introduce FIRMBOUND, an SPRT-based framework that efficiently estimates the solution to backward induction from training data, bridging the gap between optimal stopping theory and real-world deployment. It employs density ratio estimation and convex function learning to provide statistically consistent estimators for sufficient statistic and conditional expectation, both essential for solving backward induction; consequently, FIRMBOUND minimizes Bayes risk to reach optimality. Additionally, we present a faster alternative using Gaussian process regression, which significantly reduces training time while retaining low deployment overhead, albeit with potential compromise in statistical consistency. Experiments across independent and identically distributed (i.i.d.), non-i.i.d., binary, multiclass, synthetic, and real-world datasets show that FIRMBOUND achieves optimalities in the sense of Bayes risk and speed-accuracy tradeoff. Furthermore, it advances the tradeoff boundary toward optimality when possible and reduces decision-time variance, ensuring reliable decision-making. Code is included in the supplementary materials.
+
+# 1 INTRODUCTION
+
+Sequential Probability Ratio Test (SPRT) (Wald, 1945) offers a theoretically optimal framework for early classification of time series (ECTS) (Xing et al., 2009). ECTS is a task to sequentially observe an input time series and classify it as early and accurately as possible, balancing speed and accuracy (Gupta et al., 2020; Mori et al., 2016). This is vital in real-world scenarios with high sampling costs or where delays can have severe implications: e.g., medical diagnosis (Evans et al., 2015; Griffin & Moorman, 2001; Vats & Chan, 2016), stock crisis identification (Ghalwash et al., 2014), and autonomous driving (Doná et al., 2019). While the multi-objective nature of ECTS presents challenges, SPRT, with log class-likelihood ratios (LLRs), is optimal for binary i.i.d. samples and asymptotically optimal for multi-class, non-i.i.d. time series. SPRT's optimality ensures decisions within the shortest possible time with a controlled error rate (Tartakovsky, 1998; 1999).
+
+A key limitation of SPRT in real-world applications is the finite horizon (Grinold, 1977; Xiong et al., 2022): the deadline for classification. While the original SPRT assumes an indefinite sampling period to reach its decision threshold (Tartakovsky et al., 2014), practical scenarios often demand earlier decisions. For instance, detecting a face spoofing attack at a biometric checkpoint requires classification before the subject passes through (Labati et al., 2016). This constraint frequently results in suboptimal performance, as early thresholds may cause either delayed or rushed decisions (Fig. 1a).
+
+Fortunately, the optimal decision boundary for the finite horizon can be derived by solving backward induction, a recursive formula progressing from the horizon to the start of the time series (Chow et al., 1991; Peskir & Shiryaev, 2006). It optimizes the boundary by minimizing Bayes risk, or average  $a$
+
+![](images/da26a150bc96cee15f00891da63b21aae581f055fa9e5b8ed2b5c42c45a51f72.jpg)  
+Figure 1: Visual guide to the optimal stopping under finite horizon. (a) Finite horizon SPRT. Prematurely set decision boundaries lead to suboptimal results. Starbursts mark the stopping times of three decision boundaries for class 1: (Right) a static boundary (upper gray line) leads to delayed decision making; (Center) an optimal decision boundary within a finite horizon (yellow curve) achieves a faster stopping time; and (Left) a lower static boundary (lower gray line) can achieve the same hitting time (center starburst) but increases the risk of classifying another sequence (blue trajectory) to a wrong class. (b) FIRMBOUND & Pareto front. FIRMBOUND's goal is to delineate the Pareto-optimal point (meaning "optimal in the speed-accuracy multi-objective optimization problem") on the speed-accuracy tradeoff (SAT) curve. It achieves the Pareto-optimal point within the existing front (blue star) or discovers a new Pareto-optimal point (red star) if possible.
+
+![](images/3eff59e3c071191d5e42007c39ed4437724dc07380481f73ff7af89dc94c4943.jpg)
+
+posteriori risk (AAPR), which accounts for both classification accuracy and sampling costs. The resulting optimal boundary typically tapers monotonically as it nears the finite horizon (Fig. 1a).
+
+Unfortunately, applying the backward induction under real-world conditions is impractical due to its prohibitively high computational costs and the lack of true LLRs (Tartakovsky et al., 2014). To solve the backward induction, numerically calculating the conditional expectation of future risks (Eq. 5) is required because no analytical solution has been identified. This calculation necessitates intensive computational resources when applied to large-scale, high-dimensional real-world datasets. For instance, a naive use of sampling-based methods, such as Monte Carlo integration, is ineffective because ECTS demands instantaneous evaluation of the conditional expectation on the fly (Wang & Scott, 2019). Moreover, the lack of true LLRs, which are the sufficient statistic required for the backward induction and SPRT, further complicates their practical application within finite horizons.
+
+Thus, we propose the FInite-horizon average a posteriori Risk Minimizer for optimal BOUNDary (FIRMBOUND), a framework designed to estimate the solution to (i) the backward induction and (ii) the sufficient statistic, with theoretical guarantees. For estimating the backward induction, we offer two approaches: first, recognizing the concave nature of the conditional expectation, we formulate its estimation as convex function learning (CFL) to provide a statistically consistent estimator. Second, due to the high training costs of CFL (Siahkamari et al., 2022), we explore a faster alternative using Gaussian process (GP) regression (Hensman et al., 2013). Although it can compromise on statistical consistency, GP regression is trained 30 times faster than CFL with comparable performance. Both CFL and GP regression models offer low deployment overhead during the test phase, making them suitable for real-time ECTS. To address the absence of the sufficient statistic (i.e., true LLRs), FIRMBOUND integrates a sequential density ratio estimation (DRE) algorithm (Ebihara et al., 2021) to handle both i.i.d. and non-i.i.d. time series of any class size, producing statistically consistent LLR estimates, on which the optimal decision is learned (Fig. 2b,c).
+
+Our extensive experiments demonstrate that FIRMBOUND effectively approaches Bayes optimality (i.e., minimizes the AAPR) and delineates the Pareto-optimal points (meaning "optimal in the speed-accuracy multi-objective optimization problem") of the speed-accuracy tradeoff (SAT). In contrast to most existing ECTS methods, which lack theoretical guarantees (Gupta et al., 2020), FIRMBOUND significantly outperforms these baselines with less parameter sensitivity, substantiating the theoretical predictions across a wide range of synthetic and real-world datasets: two- and three-class sequential i.i.d. Gaussian datasets, non-i.i.d. damped oscillating LLR (DOL) datasets, and real-world datasets such as Spoofing in the Wild (SiW) (Liu et al., 2018b), the human motion database HMDB51 (Kuehne et al., 2011), the action recognition dataset UCF101 (Soomro et al., 2012), and FordA from UCR time series classification archive (Dau et al., 2018). Moreover, FIRMBOUND often achieves a lower
+
+(a) Estimated a posteriori risk at time step=48 (b) Learned optimal boundary for two-class test (c) Learned optimal boundary for three-class test
+
+![](images/cc75ed080af78e2f9b78965a3bae883c0e26d9d16b9373525b94e811e11630e0.jpg)  
+Figure 2: Learning Decision Boundaries. (a) Estimation of the continuation risk function  $\tilde{G}$ . Convex Function Learning (CFL) and Gaussian Process (GP) regression on a two-class sequential Gaussian dataset are used. The decision boundary at the current time step  $(= 48)$  is defined by the intersection of  $\tilde{G}$  and the stopping risk function  $G^{\mathrm{st}}$  (Thm. 2.1) (b, c) Decision boundaries (thresholds) derived from a two-class (b) and three-class(c) sequential Gaussian dataset.
+
+error rate than SPRT with static thresholds, illustrating its ability to advance the tradeoff boundary to discover new Pareto fronts (see Fig. 1b). Importantly, we empirically show that FIRMBOUND reduces decision-time variance even when it does not advance the Pareto front, contributing to reliable decision making, which is crucial for practical applications. In summary, our contribution is threefold:
+
+1. Statistically consistent and computationally efficient estimation alternatives of the optimal decision boundaries of SPRT for ECTS within finite horizons.  
+2. Comprehensive data handling under real-world scenarios, being capable of processing both i.i.d. and non-i.i.d. data series, and both binary to multiple, large class datasets.  
+3. Pareto-optimal decision making with an ability to identify potential new Pareto fronts and reduce variance of decision making time.
+
+A comprehensive literature review can be found in App. E.
+
+# 2 PRELIMINARIES: NOTATIONS AND SPRT
+
+We provide informal definitions here due to page limitations. Detailed mathematical foundations are provided in App. A and Tartakovsky et al. (2014). Let  $X^{(1,t)} \coloneqq \{x^{(t')}\}_{t' = 1}^t$  and  $y \in [K] \coloneqq \{1, \ldots, K\}$  be random variables that represent an input sequence with length  $t \in [T]$  and its class label, respectively, where  $x^{(t')} \in \mathbb{R}^{d_{\mathrm{feat}}}$  is a feature vector, and  $T \in \mathbb{N}$  is the fixed maximum length of sequences, or the finite horizon.  $X^{(1,t)}$  and  $y$  follow the joint density  $p(X^{(1,t)}, y)$ . Their samples denoted by  $X_m^{(1,t)} \coloneqq \{x_m^{(t')}\}_{t' = 1}^t$  and  $y_m \in [K] \coloneqq \{1, \ldots, K\}$  consist of a dataset, where  $m \in [M]$ , and  $M \in \mathbb{N}$  is the dataset size. The log-likelihood ratio (LLR) contrasting class  $k \in [K]$  and  $l \in [K]$  is defined as  $\lambda_{kl}(T) \coloneqq \lambda_{kl}(X^{(1,T)}) \coloneqq \log(p(X^{(1,T)}|y = k)/p(X^{(1,T)}|y = l))$ . The posterior of class  $k \in [K]$  given  $X^{(1,t)}$  is denoted by  $\pi_k(X^{(1,t)}) \coloneqq p(y = k|X^{(1,t)})$ . Let  $d_t : X^{(1,t)} \mapsto d_t(X^{(1,t)}) \in [K]$  and  $\tau : X^{(1,T)} \mapsto \tau(X^{(1,T)}) \in [T]$  denote the terminal decision rule (i.e., a class predictor) and stopping time (i.e., decision time or hitting time) of the input sequence, respectively. The terminal decision rule may not depend on  $t$ , in which case we omit the subscript  $t$ . The stopping time may not require the whole sequence  $X^{(1,T)}$  and may be able to calculate from the first  $t$  samples  $X^{(1,t)}$ , depending its algorithm. Our task is to construct the terminal decision rule  $\{d_t\}_{t \in [T]}$ , which will turn out to be time-independent, and the stopping time  $\tau$  that are "optimal" and can be computed efficiently.
+
+Sequential probability ratio test (SPRT). Our model is based on SPRT:
+
+Definition 2.1 (SPRT). Given the thresholds  $a_{k}^{(t)} \in \mathbb{R}$  ( $k \in [K]$  and  $t \in [T]$ ) for LLRs of input sequences, SPRT is defined as a tuple of a time-independent terminal decision rule and stopping time,
+
+![](images/c41b4d4cc664aaf6a260da9fe717ae9d347c821da288a316c12b830b63d6bb79.jpg)  
+Figure 3: Conceptual figure of FIRMBOUND. (a) The intersections of  $\tilde{G}$  and  $G^{\mathrm{st}}$  delineates the decision boundary. (b) FIRMBOUND estimates conditional expectations using either GP or CFL, based on available sufficient statistic such as (estimated) posterior probabilities  $\pi$  or LLRs  $\lambda$ .
+
+denoted by  $\delta^{*} \coloneqq (d^{*}, \tau^{*})$ , such that
+
+$$
+d ^ {*} \left(X ^ {(1, T)}\right) = d ^ {*} \left(X ^ {\left(1, \tau^ {*}\right)}\right) \in \underset {k \in [ K ]} {\arg \max } \left\{\underset {l (\neq k) \in [ K ]} {\min } \lambda_ {k l} \left(X ^ {(1, t)}\right) - a _ {k} ^ {(t)} \mid t = \tau^ {*} \left(X ^ {(1, T)}\right) \right\}, \tag {1}
+$$
+
+$$
+\tau^ {*} (X ^ {(1, T)}) = \tau^ {*} (X ^ {(1, \tau^ {*})}) := \min  \{t \in [ T ] \mid \max  _ {k \in [ K ]} \left\{\min  _ {l (\neq k) \in [ K ]} \lambda_ {k l} (X ^ {(1, t)}) - a _ {k} ^ {(t)} \right\} \geq 0 \}. \tag {2}
+$$
+
+This algebraic definition may seem complex, but the graphical descriptions are given in Figs. 1a & 2b. Note that the terminal decision rule (Eq. 1) is equivalent to choosing the argmax of the gaps between the class posteriors and the thresholds w.r.t.  $k \in [K]$  at the stopping time, i.e., choosing the most likely class.
+
+A key feature of SPRT is its various optimalities—asymptotic, non-asymptotic, and Bayes—which theoretically establish SPRT as the best model for ECTS. In this paper, we exploit the Bayes optimality, with the other optimalities summarized in App. B.
+
+To define the Bayes optimality of SPRT, we introduce the sufficient statistic, a posterior risk (APR), and average APR (AAPR). The sufficient statistic for sequential tests here means  $\mathcal{S}_t \coloneqq (\lambda_{kl}(X^{(1,t)}))_{k,l \in [K]}$  (or equivalently, we can use  $\mathcal{S}_t \coloneqq (\pi_k(X^{(1,t)}))_{k \in [K]}$  interchangeably), providing all necessary information for decision at  $t$  and serving as the fundamental variable in SPRT's optimality instead of  $X^{(1,t)}$  (see also App. C for the formal definition). Then, for a given  $d_t$ , APR at  $t \in [T]$  is defined as
+
+$$
+\operatorname {A P R} _ {t} \left(\mathscr {S} _ {t}, d _ {t} \left(X ^ {(1, t)}\right) = k\right) := \bar {L} _ {k} \left(1 - \pi_ {k} \left(X ^ {(1, t)}\right)\right) + c t, \tag {3}
+$$
+
+where  $c \in \mathbb{R}_{\geq 0}$  is a sampling cost, and  $\bar{L}_k$  is the  $k$ -th element of a penalty vector  $\bar{L} \in \mathbb{R}_{\geq 0}^K$  with  $k \in [K]$ , which penalizes incorrect classifications of  $d_t$ . The average APR (AAPR), or the Bayes risk, for  $\{d_t\}_{t \in [T]}$  and  $\tau$  is defined as
+
+$$
+\operatorname {A A P R} \left(\left\{d _ {t} \right\} _ {t \in [ T ]}, \tau\right) := \mathbb {E} \left[ \operatorname {A P R} _ {\tau} \left(\mathscr {S} _ {\tau}, d _ {\tau}\right) \right]. \tag {4}
+$$
+
+SPRT is Bayes optimal in the sense that it can provide a terminal decision rule and stopping time that minimize AAPR if the thresholds for LLRs are properly chosen. The following theorem provides how to compute the optimal thresholds to achieve the Bayes optimality (Arrow et al., 1949; Tartakovsky et al., 2014).
+
+Theorem 2.1 (Backward induction equation). Let  $\mathcal{S}_t$  be  $(\pi_1(X^{(1,t)},\ldots ,\pi_K(X^{(1,t)}))$  w.l.o.g.  $SPRT\delta^{*}$  is Bayes optimal if time-dependent thresholds  $a_{k}^{(t)}$  in Eqs. 1 & 2 are given by the intersections of the continuation risk function  $\tilde{G}_t(\mathcal{S}_t)$  and the stopping risk function  $G_{t}^{\mathrm{st}}(\mathcal{S}_{t})$ , which are defined for a pre-defined density  $p$ , not for each sample of  $X^{(1,T)}$ , and satisfy the following backward induction equation:
+
+$$
+\tilde {G} _ {t} \left(\mathcal {S} _ {t}\right) = \mathbb {E} \left[ G _ {t + 1} ^ {\min } \left(\mathcal {S} _ {t + 1}\right) \mid \mathcal {S} _ {t} \right] + c \tag {5}
+$$
+
+$$
+G _ {t} ^ {\mathrm {s t}} \left(\mathscr {S} _ {t}\right) = \min  _ {k} \left\{\bar {L} _ {k} \left(1 - \pi_ {k} \left(X ^ {(1, t)}\right)\right) \right\}, \tag {6}
+$$
+
+where  $G_{t}^{\min}(\mathcal{S}_{t})$  is referred to as the minimum risk function:
+
+$$
+G _ {t} ^ {\min } \left(\mathscr {S} _ {t}\right) := \left\{ \begin{array}{l l} G ^ {\mathrm {s t}} \left(\mathscr {S} _ {t}\right) & (t = T) \\ \min  \left\{G ^ {\mathrm {s t}} \left(\mathscr {S} _ {t}\right), \tilde {G} _ {t} \left(\mathscr {S} _ {t}\right) \right\} & (1 \leq t <   T). \end{array} \right. \tag {7}
+$$
+
+Therefore, the optimal stopping region is  $\{(\pi_1(X^{(1,t)}),\ldots ,\pi_K(X^{(1,t)}))\mid G^{\mathrm{st}}(\mathcal{S}_t) = \tilde{G}_t(\mathcal{S}_t)\}_{t\in [T]}\subset \mathbb{R}^{K\times T}$ . A similar theorem holds for  $\mathcal{S}_t = (\lambda_{kl}(X^{(1,t)}))_{k,l\in [K]}$ , rewriting  $\{\pi_k(X^{(1,t)})\}_{k\in [K]}$  by  $\{\lambda_{kl}(X^{(1,t)})\}_{k,l\in [K]}$ .
+
+The formal proof is provided in Tartakovsky et al. (2014). For an intuitive explanation, see App. F.
+
+This theorem indicates that the optimal stopping time is given by  $\tau^{*}(X^{(1,T)}) = \tau^{*}(X^{(1,\tau^{*})}) = \min \{t\in [T]\mid G_t^{\mathrm{st}}(\mathcal{S}_t)\leq \tilde{G}_t(\mathcal{S}_t)\}$  and that the optimal terminal decision rule simplifies to  $d^{*}(X^{(1,T)}) = d^{*}(X^{(1,\tau^{*})})\in \arg \min_{k\in [K]}\{\bar{L}_k(1 - \pi_k(X^{(1,t)}))\mid t = \tau^* (X^{(1,T)})\}$ . Note that an explicit formula of the dynamic threshold  $a_{k}^{(t)}$  as a function of the sufficient statistic  $\mathcal{S}_t$  is unnecessary to compute  $d^{*}$  and  $\tau^{*}$  (Figs. 2a, b, and 3a serve only for visualization). Note also that once the optimal stopping region is determined, calculating  $\tau^{*}$  no longer require a backward computation each time a new sequence arrives because  $G_{t}^{\mathrm{st}}$  and  $\tilde{G}_{t}$  are defined for the underlying density, not for individual sample sequences.
+
+# 3 FIRMBOUND
+
+Unfortunately, solving and deploying Eqs.5-7 in real-world scenarios presents significant challenges. First, these equations lack closed-form solutions. A naive numerical computation, such as Monte Carlo integration, would be possible, but it suffers from the curse of dimensionality ( $K$  can be  $>100$ , requiring an exponentially large number of samples for convergence) (see also App. D). Second, obtaining a well-calibrated sufficient statistic  $\mathcal{S}_t$  is challenging. Although computing softmax logits as class posteriors is common in classification problems (He et al., 2016a;b; Krizhevsky et al., 2012; LeCun et al., 1998), high-dimensional classifiers often produce overconfident or miscalibrated outputs (Guo et al., 2017; Melotti et al., 2022; Müller et al., 2019; Mukhoti et al., 2020).
+
+We address these challenges by transforming the backward induction into a pair of estimation problems and providing statistically consistent estimators (Secs. 3.1 & 3.2). Our proposed model, FIRMBOUND, is then proved to be statistically consistent with the Bayes optimal solution (Thm. 3.2).
+
+# 3.1 ESTIMATING THE CONDITIONAL EXPECTATION
+
+The first key idea is to transform the computation of the conditional expectation in the backward induction equation into a regression problem. An important observation is that the conditional expectation function in Eq. 5 is concave (Jarrett & van der Schaar, 2020; Tartakovsky et al., 2014):
+
+Theorem 3.1.  $\tilde{G}_t$  and  $G_t^{\mathrm{min}}$  are concave functions of vector  $(\pi_1(X^{(1,t)}),\ldots ,\pi_K(X^{(1,t)}))$  for all  $t\in [T]$
+
+Equipped with the concavity, we propose to build a consistent estimator of the Eq. 5 though convex function learning (CFL).
+
+CFL. CFL aims to build a statistically consistent estimator of a convex function from noisy data points, assuming the target function is inherently convex (Argyriou et al., 2008; Bach, 2010; Bartlett et al., 2005; Boyd & Vandenberghe, 2010; Mendelson, 2004). Assume that we have estimates of the sufficient statistic  $\mathcal{S}_t$  for all  $t\in [T]$  estimated on a given training dataset  $\{(X_m^{(1,T)},y_m)\}_{m = 1}^M$  via the algorithm given in Sec. 3.2. Then, our task toward solving the backward induction equation (Eq. 5-7) is to estimate  $G_{t}^{\mathrm{st}}(\mathcal{S}_{t})$  and  $\tilde{G}_t(\mathcal{S}_t)$  for all  $t\in [T]$ .  $G_{t}^{\mathrm{st}}$  can be computed from the estimated sufficient statistic via Eq. 6. Thus, we focus on the continuation risk  $\tilde{G}_t(\mathcal{S}_t) = \mathbb{E}\left[G_{t + 1}^{\min}(\mathcal{S}_{t + 1})|\mathcal{S}_t\right] + c$ . To
+
+estimate it from the estimated sufficient statistic, we first rewrite  $\tilde{G}_t$  as
+
+$$
+\begin{array}{l} \tilde {G} _ {t} \left(\mathcal {S} _ {t} \left(X _ {m} ^ {(1, t)}\right)\right) = \mathbb {E} _ {X ^ {(t + 1)}} \left[ G _ {t + 1} ^ {\min } \left(\mathcal {S} _ {t + 1} \left(X ^ {(1, t + 1)}\right)\right) \mid \mathcal {S} _ {t} \left(X ^ {(1, t)} = X _ {m} ^ {(1, t)}\right) \right] + c (8) \\ = \int d P \left(X ^ {(t + 1)} \mid X ^ {(1, t)} = X _ {m} ^ {(1, t)}\right) G _ {t + 1} ^ {\min } \left(\mathscr {S} _ {t + 1} \left(X ^ {(1, t + 1)}\right)\right) + c (9) \\ = G _ {t + 1} ^ {\min } \left(\mathcal {S} _ {t + 1} \left(X _ {m} ^ {(1, t + 1)}\right)\right) - \epsilon_ {m} ^ {(t)} + c =: \mathcal {G} _ {m} ^ {(t + 1)} - \epsilon_ {m} ^ {(t)} + c, (10) \\ \end{array}
+$$
+
+where  $\mathcal{G}_m^{(t + 1)}\coloneqq G_{t + 1}^{\min}(\mathcal{S}_{t + 1}(X_m^{(1,t + 1)}))$ $P$  is a properly defined probability measure, and  $\epsilon_{m}^{(t)}$  is a random variable representing the deviation of  $\mathcal{G}_m^{(t + 1)}$  from the expectation integral in Eq. 9. Suppose that the backward induction equation is solved for  $T,T - 1,\ldots ,t + 1$  ; i.e.,  $G_{t + 1}^{\mathrm{min}}$  is given. Then,  $\mathcal{G}_m^{(t + 1)}$  is computable from the estimated sufficient statistic by definition. Therefore, we regard  $\{\mathcal{G}_m^{(t)}\}_{m,t}$  as a given dataset henceforth, leading to the idea that the dataset  $\{\mathcal{G}_m^{(t)}\}_{m,t}$  can be regarded as a set of noisy observations of the ground truth continuation risk  $\tilde{G}_t(\mathcal{S}_t(X_m^{(1,t)}))$  of  $X_{m}^{(1,t)}$  (up to a constant  $c$ ) because  $\tilde{G}_t(\mathcal{S}_t(X_m^{(1,t)})) = \mathcal{G}_m^{(t + 1)} - \epsilon_m^{(t)} + c$  (Eq.10)  $\Leftrightarrow$
+
+$$
+\mathcal {G} _ {m} ^ {(t + 1)} = \tilde {G} _ {t} \left(\mathscr {S} _ {t} \left(X _ {m} ^ {(1, t)}\right)\right) + \epsilon_ {m} ^ {(t)} - c. \tag {11}
+$$
+
+This change of view, together with the fact that  $\tilde{G}_t(\mathcal{S}_t)$  is concave w.r.t.  $\mathcal{S}_t = (\pi_1,\dots ,\pi_K)$ , leads to the following noisy convex regression problem:
+
+$$
+\hat {\tilde {G}} _ {t} \left(\left\{X _ {m} ^ {(1, T)} \right\} _ {m = 1} ^ {M}\right) \in \underset {f: \text {c o n c a v e}} {\arg \min } \left\{\frac {1}{M} \sum_ {m = 1} ^ {M} \left(f \left(\mathscr {S} _ {t} \left(X _ {m} ^ {(1, t)}\right)\right) - \mathscr {G} _ {m} ^ {(t)}\right) ^ {2} + \lambda \| f \| \right\} + c, \tag {12}
+$$
+
+where  $\| f\|$  is a regularizer,  $\lambda$  is a hyperparameter, and  $\hat{G}_t(\{\mathcal{G}_m^{(t)}\}_{m,t})$  denotes the continuation risk estimated on  $\{\mathcal{G}_m^{(t)}\}_{m,t}$  or, equivalently,  $\{X_m^{(1,T)}\}_{m = 1}^M$ . With this novel reformulation, we employ an efficient solver, the 2-block Alternating Direction Method of Multipliers (ADMM) algorithm integrated with the augmented Lagrangian method with a concavity constraint (Siahkamari et al., 2022). Specifically,  $f$  in Eq. 12 is represented as a piecewise linear function, and  $\| f\|$  is defined as the  $L^1$  penalty terms (see App. G for the complete algorithm). This algorithm is known to converge to the ground truth function as  $M\to \infty$ ; i.e., it is consistent (Siahkamari et al., 2022).
+
+Consequently, given the estimates of the sufficient statistic, we now have the estimates of the continuation risks  $\tilde{G}_t$  and the stopping risks  $G_t^{\mathrm{st}}$  as functions of  $\mathcal{S}_t$  for all  $t\in [T]$ . Therefore, in the test phase, we can compute the optimal stopping region given in Thm. 2.1 for any  $\mathcal{S}_t$  and  $t\in [T]$  without re-solving the backward induction equation.
+
+Gaussian process (GP) regression. Although the aforementioned CFL algorithm is theoretically sound and computationally tractable, we further propose a more computationally efficient estimator using GP regression. GP regression is a Bayesian approach to regression used for probabilistic predictions, assuming the objective function values follow a Gaussian distribution defined by a covariance kernel (Wang, 2020).
+
+Evaluating the conditional expectation, or the continuation risk,  $\mathbb{E}[G_{t + 1}^{\min}(\mathcal{S}_{t + 1})|\mathcal{S}_t]$  at any  $\mathcal{S}_t$  and  $t\in [T]$  is formulated below (we omit inducing points here for brevity). Suppose that a set of estimated sufficient statistics at any  $t\in [T]$ , denoted by  $\{\mathcal{S}_{t,m}\coloneqq \mathcal{S}_t(X_m^{(1,t)})\}_{m\in [M]}^t$ , is given. We begin with our reformulation discussed above (Eq. 11):
+
+$$
+\mathcal {G} _ {m} ^ {(t + 1)} + c = \tilde {G} _ {t} \left(\mathscr {S} _ {t, m}\right) + \epsilon_ {m} ^ {(t)}. \tag {11}
+$$
+
+We make the following fundamental assumptions of GP regression. First, the observation noise  $\epsilon_{m}^{(t)}$  for any  $t\in [T]$  and  $m\in [M]$  follows a Gaussian distribution. Second,  $\{\tilde{G}_t(\mathcal{S}_{t,m})\}_{m\in [M]}$  for any  $t\in [T]$  forms a Gaussian process. Under these assumptions, Eq. 11 can be regarded as a GP regression problem with the latent function  $\tilde{G}_t$ , the explanatory variable  $\mathcal{S}_{t,m}$ , and the response variable  $\mathcal{G}_m^{(t + 1)} + c$ . Therefore, the predictive distribution of the continuation risk can be calculated, using the standard methods for the evidence lower bound (ELBO) maximization. Specifically, we use the variational GP with an inducing point method (Hensman et al., 2015; Matthews, 2017) via minibatch training to maximize the ELBO. This algorithm uses standard functions from GPyTorch
+
+![](images/84b9effc910e5fc213fbfd7009070d56f6246f6b2e6e07f8b1428385234ffdca.jpg)  
+Figure 4: Training and Testing. (Top) In the training phase, the sequential DRE algorithm SPRT-TANDEM is trained, followed by the training of CFL or GP models using the backward induction. (Bottom) In the testing phase, the trained DRE model is loaded to sequentially update the LLRs, with which the trained CFL/GP model calculates  $\tilde{G}_t$  and compares it with  $G_t^{\mathrm{st}}$  to make decisions at time  $t$ .
+
+(Gardner et al., 2018). For implementation details, see our code. For further detailed mathematical foundations of GP regression, see App. H.
+
+Together with the estimated sufficient statistics, the predictive distribution thus obtained provide  $G_{t}^{\mathrm{st}}$  and  $\tilde{G}_{t}$  for any  $\mathcal{S}_t$  and  $t\in [T]$ . Therefore, in the test phase, we can compute the optimal stopping region given in Thm. 2.1 for any  $\mathcal{S}_t$  and  $t\in [T]$  without re-solving the backward induction equation. We empirically validate that the training (sometimes referred to as inference in the Bayesian context) with GP regression is 30 times faster than the CFL training.
+
+# 3.2 DENSITY RATIO ESTIMATION (DRE) FOR ECTS
+
+Our remaining task is to estimate the sufficient statistic  $\mathcal{S}_t$  for all  $t\in [T]$ , the second estimation problem mentioned at the beginning of Sec. 3. A simple approach to this end is to estimate LLRs via a sequential density ratio estimation algorithm. It enhances precision by estimating the ratio of probabilities directly, rather than estimating each probability independently, thus reducing degrees of freedom (Belghazi et al., 2018; Gutmann & Hyvarinen, 2012; Hjelm et al., 2019; Liu et al., 2018a; Moustakides & Basioti, 2019; Oord et al., 2018; Sugiyama et al., 2010; 2008; 2012). Specifically, we employ SPRT-TANDEM algorithm (Ebihara et al., 2021; Miyagawa & Ebihara, 2021; Ebihara et al., 2023), which involves a consistent loss function, named Log-Sum-Exp Loss (LSEL):
+
+$$
+\hat {L} _ {\text {L S E L}} \left(\boldsymbol {w}; \left\{\left(X _ {m} ^ {(1, T)}\right), y _ {m} \right\} _ {m \in [ M ]}\right) := \frac {1}{K M} \sum_ {k \in [ K ]} \sum_ {t \in T} \frac {1}{M _ {k}} \sum_ {i \in I _ {k}} \log \left(1 + \sum_ {l (\neq k) \in [ K ]} e ^ {- \hat {\lambda} _ {k l} \left(\boldsymbol {w}, X ^ {(1, t)}\right)}\right) \tag {13}
+$$
+
+where  $\pmb{w} \in \mathbb{R}^d$  is the trainable parameters, e.g., the weights of a neural network,  $I_k := \{i \in [M] \mid y_i = k\}$  is the index set of class  $k$ ,  $M_k := |I_k|$  is the size of  $I_k$ , and  $\hat{\lambda}_{kl}(\pmb{w}, X^{(1,t)})$  is the estimated LLR parameterized by  $\pmb{w}$ . By minimizing LSEL, the estimated LLRs approaches the true LLRs as  $M \to \infty$  (Miyagawa & Ebihara, 2021); i.e., LSEL is consistent. We provide further details of SPRT-TANDEM in App. I.
+
+Finally, integrating CFL and LSEL and solving the backward induction equation, we establish that FIRMBOUND is statistically consistent.
+
+Theorem 3.2 (Informal). Under several technical assumptions, FIRMBOUND with CFL is statistically consistent with the Bayes optimal algorithm for ECTS; i.e., it minimizes AAPR as  $M \to \infty$ .
+
+Main assumptions are (i) a sufficiently large dataset size, (ii) a sufficiently large number of iterations of the ADMM algorithm in CFL, and (iii) a sufficiently large neural network for LSEL. The complete set of assumptions, the formal statement, and the proof are provided in App. J, as they are technical and lengthy. In the following, we empirically validate Thm. 3.2, demonstrating that FIRMBOUND minimizes AAPR, and highlight its practical strengths of FIRMBOUND across various datasets.
+
+# 4 EXPERIMENTS AND RESULTS
+
+These experiments are designed for a fair comparison with baseline models without exploring all possible configurations, as such variations would not alter our study's conclusion. To ensure fairness, the same feature extractor and feature vector size  $d_{\mathrm{feat}}$  are used across all models. All
+
+![](images/9d94f90e92bd67875a103a9e198c06144489dfc7f2182aa80c409d4c136ce599.jpg)
+
+![](images/6e07e8f5424339757a81993fc686ec1380d3d2457da4aec3d5c15b1bda53ef06.jpg)
+
+![](images/1f7d05fc941c6a3cc496ec65c2cec5980f4e96f3b54504b25a57d51779fc515d.jpg)
+
+![](images/6cd2e0b7234ef848bf7930b95f464a49177bd11bbcde7e62d7328e1ed40888ef.jpg)  
+Figure 5: Averaged a posteriori risk (AAPR) curves. AAPRs of FIRMBOUND are compared with static-threshold SPRTs. Horizontal and vertical axes are mean hitting time and AAPR, respectively. Note that we only show models with well-calibrated sufficient statistic here, as ill-calibrated statistic does not necessarily correlate with ECTS performance by definition and thus not meaningful discussing its minima (but see App. M for AAPR of other baseline models). Error bars represent the standard error of the mean.
+
+![](images/d85c3d7a8e3ba007a49db94a50d0b9b9432f62c346030263c6256d7ed4dbae18.jpg)
+
+![](images/85f5c34d619374d3a694a100bcb20dcf817de412de05d6e16ec249bef5a94fad.jpg)
+
+hyperparameters, including those for the baseline models, are optimized using Optuna (Akiba et al., 2019) with the Tree-structured Parzen Estimator (Bergstra et al., 2011). Details on parameter selection can be found in App. K. Additional parameter sensitivity test on GP models can be found in App. L, showing robustness against kernel choice. Fig. 4 shows The training and testing pipeline.
+
+Baselines. We evaluate the performance of FIRMBOUND by comparing it against SPRT with static thresholds and four ECTS models. To conduct SPRT on real-world datasets lacking true LLRs, we utilize SPRT-TANDEM (Ebihara et al., 2021; Miyagawa & Ebihara, 2021; Ebihara et al., 2023) to estimate LLRs. ECTS baseline models include: LSTMms, which enhances monotonic score growth (Ma et al., 2016), the reinforcement learning algorithm EARLIEST (with two fixed hyperparameters lambda  $= 10^{-1}$  and  $10^{-10}$ ) (Hartvigsen et al., 2019), the convolutional neural network-transformer hybrid, TCN-Transformer (TCNT, with two fixed hyperparameters  $\alpha = 0.3$  and 0.5) (Chen et al., 2022), and Calibrated eArLy tIME sERies cIAsifier (CALIMERA, with fixed hyperparameters, delay penalty  $= 0.1$ , 0.5.1.0) (Bilski & Jastrzewska, 2023).
+
+Evaluation criteria. Our evaluation metrics are AAPR and SAT curve. We compute APR at the decision time using softmax probabilities as class posteriors, with a fixed  $\bar{L}_k = L = 10$  for all  $k\in [K]$ , and up to three variations of  $c\in \{L / T,2L / T,0.1L / T\}$  (Fig. 5), where  $c = L / T$  is set such that the two terms in APR (Eq. 3) are of comparable magnitude. We do not vary  $L$  because decision boundaries are invariant to the scaling of  $L$  and  $c$  (see App. N for the proof). The SAT curve is derived from the averaged per-class error rate (i.e., macro-averaged recall) measured at the stopping time (Fig. 6).
+
+CFL. CFL model is trained at each time step to estimate the conditional expectation (Eq. 7), utilizing a custom training routine adapted from (Siahkamari et al., 2022). We optimize hyperparameters by randomly sampling 1,000 sequential data points. This process is repeated 30 times to identify the hyperparameters that minimize mean squared error using Optuna (Akiba et al., 2019). Once hyperparameter is set, we sample 5,000 sequential data points to model the conditional expectation curve with 5 epochs of training. The requirement of a convex function over a finite input space lets us use posteriors  $\pi_k(X_m^{(1,t)})$  as the sufficient statistic  $\mathcal{S}_t$ . Training on a two-class sequential Gaussian dataset (details provided below) takes approximately 10 hours on NVIDIA RTX 2080Ti.
+
+![](images/39ef145ccacdd0874f48b401f784c187c9905573fd6448ec224497177d0dfdea.jpg)  
+Figure 6: Speed-Accuracy Tradeoff (SAT) Curves. The performance of ECTS is evaluated through SAT curves. The horizontal axis represents the mean hitting time, while the vertical axis shows the averaged per-class error rate, equivalent to macro-averaged recall. Thus, models closest to the bottom-left corner perform best. Error bars represent the standard error of the mean.
+
+![](images/817847bb73cf601b8e7a06dcda1e2ff483b7aa8374994f851d5922fe2dda0749.jpg)  
+Figure 7: Additional ablation tests. Random hitting time and artificial tapering thresholds evaluating (a) AAPR curve and (b) SAT curve. The artificial thresholds start with three different magnitude at  $t = 1$ , gradually tapering to zero as approaching the horizon  $t = T$ . See App. O for details.
+
+GP regression. GP model is trained at each time step same as CFL. Models are trained for 30 epochs with a batch size of 2,000 with 200 randomly selected inducing points. Our empirical comparisons indicate that either LLRs or posteriors can serve as the sufficient statistic  $\mathcal{S}_t$ , yielding similar results (App. C). In subsequent analyses, we use LLRs for synthetic data and opt for posteriors for real-world data because of their lower dimensionality. Training on a two-class sequential Gaussian dataset (details provided below) typically requires approximately 20 minutes on NVIDIA RTX 2080Ti.
+
+Ablation tests. To assess the impact of different stopping rules, we conducted ablation tests using LLRs. The primary baseline is SPRT with static thresholds on estimated LLRs, as shown in Figs. 5 and 6. Additional tests are random stopping times to establish a chance level and monotonically descending decision boundaries generated using a power function (see App. O for details). Neither variant surpassed FIRMBOUND in terms of AAPR or SAT (Fig. 7). The datasets used in these experiments are detailed below.
+
+Dataset: sequential i.i.d. Gaussian datasets with known LLRs. Preliminary assessments are conducted on sequential Gaussian datasets to find that FIRMBOUND can minimize APR to achieve Pareto-optimal both with ground-truth and estimated LLRs. Let  $p_0(x), p_1(x)$ , and  $p_2(x)$  be the 128-dimensional Gaussian densities with an identity covariance matrix. The mean vectors are defined as  $(0.5, 0, 0, \dots, 0)$ ,  $(0, 0.5, 0, \dots, 0)$ , and  $(0, 0, 0.5, \dots, 0)$  for  $p_0(x), p_1(x)$ , and  $p_2(x)$ , respectively. Only  $p_0(x)$  and  $p_1(x)$  are used for the two-class dataset. We randomly sampled sequences of length  $T = 50$  from these Gaussian distributions to form the datasets. The sizes of the training, validation, and test datasets are as follows: for the two-class dataset, 80,000, 2,000, and 80,000 samples respectively; for
+
+the three-class dataset, 60,000, 6,000, and 120,000 samples respectively. SPRT-TANDEM is trained with the sampled vectors to provide estimated LLRs. Fig. 5a, b and 6a, b shows that FIRMBOUND effectively minimize AAPR to reach the best speed-accuracy tradeoff given a sampling cost.
+
+Dataset: sequential non-i.i.d. Damped-Oscillating LLRs (DOLs). We explore the potential for mitigating early inadvertent error to find a new Pareto-front (Fig. 1a). Nonlinear two-class LLRs  $\Lambda(t)$  of length  $T = 50$  are generated as  $\Lambda(t) = \gamma(1 - (1 - t/T)^{\exp(\kappa)}) + A\exp(-\beta t)\sin(\omega t) + \mathcal{N}(0, \sigma)$ , where  $\gamma \in \{-1,1\}$  denotes the class label value towards which the first term converges, the second term introduces a damped oscillation, and the third term represents Gaussian noise. The dataset is generated using parameters  $\kappa, A, \beta, \omega$ , and  $\sigma$ , chosen from a predefined parameter space. This results in 20,000 training samples, 2,000 validation samples, and 80,000 test samples. For additional details on LLR trajectories and dataset parameters, see App. P. Fig. 6c demonstrate that FIRMBOUND achieves notably low errors, effectively advancing the Pareto-front to a new optimal level.
+
+Dataset: real-world datasets for ECTS. Four datasets are used: SiW (two-class) (Liu et al., 2018b), HMDB51 (51-class) (Kuehne et al., 2011), UCF101 (101-class) (Soomro et al., 2012), and FordA dataset (two-class) (Dau et al., 2018). For the SiW dataset, a ResNet-152 (He et al., 2016a;b) is trained as a feature extractor to generate 512-dimensional feature vectors for each frame. The pretrained Microsoft Vision Model ResNet  $50^{1}$ , without fine-tuning, is used to extract 2048-dimensional feature vectors from the HMDB51 datasets. The pretrained vision transformer DINOv2 (the largest model without distillation) with registers (Dosovitskiy et al., 2021; Oquab et al., 2024; Darcet et al., 2024) is used to extract 1,538-dimensional feature vectors from UCF101 datasets. The dataset sizes and sequence lengths are as follows: SiW comprises 46,729 training, 4,968 validation, and 43,878 test samples, all with a sequence length of  $T = 50$ ; HMDB51 includes 5,277 training, 519 validation, and 2,434 test samples with  $T = 79$ ; UCF101 consists of 35,996 training, 4,454 validation, and 15,807 test samples, each with  $T = 20$ . Figs. 5d-f and 6d-f show that FIRMBOUND reaches Pareto-front by minimizing AAPR, and extend the frontier in a few datasets. FordA is used as an experiment on a non-vision modality dataset, presented in App. Q, showing the same trend.
+
+Reducing the variance of hitting time. Although FIRMBOUND consistently identifies the Pareto-optimal point, it does not always show performance gains compared with the vanilla SPRT. However, the variance of the hitting time is statistically significantly smaller across the database (Tab. 1, Wilcoxon signed-rank test,  $p = 2.00 \times 10^{-8} \ll 0.001$ ), demonstrating FIRMBOUND's advantage in reducing the variance of hitting time to enable reliable decision making across data.
+
+Table 1: Mean variance of hitting time (MVHT). FIRMBOUND with CFL provides smaller variance of hitting times than vanilla SPRT when evaluated at the same mean hitting time and corresponding macro averaged recall. The variance reduction is statistically significant (see main text).  
+
+<table><tr><td>Dataset</td><td>Gauss2est.</td><td>Gauss3est.</td><td>DOL</td><td>SiW</td><td>HMDB</td><td>UCF101</td><td>FordA</td></tr><tr><td>Trial repeats</td><td>5</td><td>3</td><td>3</td><td>5</td><td>6</td><td>10</td><td>2</td></tr><tr><td>↓MVHT, vanilla SPRT with static threshold</td><td>10.47</td><td>44.02</td><td>489.89</td><td>2.87</td><td>199.31</td><td>0.55</td><td>32.15</td></tr><tr><td>↓MVHT, FIRMBOUND with CFL</td><td>9.01</td><td>42.78</td><td>405.78</td><td>1.97</td><td>195.35</td><td>0.53</td><td>23.39</td></tr><tr><td>↑Difference in MVHT (positive is better)</td><td>1.45</td><td>1.24</td><td>84.11</td><td>0.90</td><td>3.97</td><td>0.017</td><td>8.76</td></tr></table>
+
+# 5 CONCLUSION
+
+With two statistically consistent estimators for backward induction and the sufficient statistic, FIRMBOUND delineates stable Pareto fronts across diverse datasets. Unlike existing ECTS models, which lack theoretical guarantees and are sensitive to hyperparameters and datasets (Fig.6d-f), FIRMBOUND consistently achieves optimal performance with reduced hitting time variance, approaching optimal performance for ECTS in real-world scenarios. For further discussion, see App.F.
+
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+
+# ACKNOWLEDGEMENTS
+
+We sincerely thank the anonymous reviewers for their constructive comments and the area chair for a thorough, fair, and objective evaluation of our work. The authors also extend their gratitude to Yukai Fujii for reading an early draft of the manuscript and providing invaluable feedback.
+
+# AUTHOR CONTRIBUTIONS
+
+A.F.E. and T.M. conceived the overall research direction, developed the theoretical framework, and wrote the manuscript. A.F.E. formulated the methodology, ran a pilot study to establish feasibility, conducted the experiments, and organized the code for release. K.S. and H.I. supervised the project.
+
+# APPENDIX
+
+# CONTENTS
+
+1 Introduction 1  
+2 Preliminaries: Notations and SPRT 3  
+3 FIRMBOUND 5
+
+3.1 Estimating the Conditional Expectation 5  
+3.2 Density Ratio Estimation (DRE) for ECTS 7
+
+4 Experiments and Results 7  
+5 Conclusion 10
+
+References 11
+
+A Mathematical Foundations 19
+
+A.1 Probability Measure and Data Randomness 19  
+A.2 Decision Rule, Terminal Decision, and Stopping Rule 19
+
+B Sequential Probability Ratio Test and Its Optimality 20  
+C LLRs and Posteriors as Sufficient Statistic 21
+
+D Computational Complexity of FIRMOUND and sampling method 22
+
+D.1 Direct Estimation Approach (FIRMBOUND) 22  
+D.2 Monte Carlo Integration with KDE Approach 22  
+D.3 Comparison 23
+
+E Supplementary Related Work 24
+
+E.1SPRT and Its Optimality 24  
+E.2 Optimal Stopping Theory 24  
+E.3SPRT's Backward Induction and Conditional Expectation 24  
+E.4 Sequential Design 24  
+E.5 Reinforcement Learning (RL) 25  
+E.6 Active Learning 25  
+E.7 Convex Function Learning (CFL) 25  
+E.8 Gaussian Process (GP) Regression 25  
+E.9 Other ECTS Algorithms 26  
+E.10 Neurophysiological underpinnings of SPRT. 26
+
+F Supplementary Discussion 27
+
+F.1 Intuitive Understanding of FIRMBOUND 27  
+F.2 Challenges in Estimating Conditional Expectations Using Monte Carlo and Kernel Density Estimation 27
+
+F.2.1 Curse of Dimensionality 27  
+F.2.2 Computational Complexity 28  
+F.2.3 Bandwidth Selection 28  
+F.2.4 Sampling Efficiency 28  
+F.2.5 Mitigation Strategies 28  
+F.2.6 Advantage of Having a Direct Estimator FIRMBOUND 28
+
+F.3 Performance under Small Datasets 29  
+F.4 Broader impact 29  
+F.5 Limitations and future work 30  
+F.6 Frequently Asked Questions 31
+
+# G Lagrangian Function for Convex Function Learning 33
+
+# H Stochastic Variational ELBO Maximization 35
+
+# I SPRT-TANDEM 36
+
+# J FIRMBOUND Is Statistically Consistent 37
+
+J.1 Assumptions 37  
+J.2 Formal Statement 38  
+J.3 Proof 38
+
+# K Experimental Details and Supplementary Results 40
+
+K.1 FIRMBOUND with CFL 40  
+K.2 FIRMBOUND with GP Regression 40  
+K.3 Dataset Preparation 40  
+K.4 Training ECTS Models 41
+
+K.4.1 Two-class Gaussian dataset 42  
+K.4.2 Three-class Gaussian dataset 42  
+K.4.3 SiW 43  
+K.4.4 HMDB51 45  
+K.4.5 UCF101 47
+
+K.5 Computing Infrastructure 49
+
+# L On Hyperparameter Sensitivity of FIRMBOUND 50
+
+# M AAPR on Baseline Models 52
+
+# N Parameter Space of  $L$  and  $c$  53
+
+O Ablation Study Details 54  
+P Dumped Oscillating Log-likelihood Ratio Function 55  
+Q Supplementary experiment on the UCR FordA Dataset 56  
+Supplementary References 57
+
+# A MATHEMATICAL FOUNDATIONS
+
+In the main text, we introduced concise notations to avoid delving into unnecessarily technical details. Here, we provide more rigorous definitions. See Tartakovsky et al. (2014) for details.
+
+# A.1 PROBABILITY MEASURE AND DATA RANDOMNESS
+
+We consider a standard probability space  $(\Omega, \mathcal{F}, P)$ , where  $\Omega$  is a sample space,  $\mathcal{F} \subset \mathcal{P}(\Omega)$  is a  $\sigma$ -algebra of  $\Omega$ , where  $\mathcal{P}(\Omega)$  denotes the power set of  $\Omega$ , and  $P$  is a probability measure satisfying Kolmogorov's axioms:
+
+$P(\Omega) = 1$  
+-  $P(A) \geq 0$  for any  $A \in \mathcal{F}$ ,  
+-  $P\left(\bigcup_{i=1}^{\infty} A_i\right) = \sum_{i=1}^{\infty} P(A_i)$  for any countable collection  $\{A_i\}_{i=1}^{\infty} \subset \mathcal{F}$  of pairwise disjoint sets (i.e.,  $A_i \cap A_j = \emptyset$  for  $i \neq j$ ).
+
+A function  $X = X(\omega)$  defined on the space  $(\Omega, \mathcal{F})$  (with values in  $\mathbb{R}^{d_{\text{feat}}}$  ( $d_{\text{feat}} \in \mathbb{N}$ ) in our paper) is called random variable if it is  $\mathcal{F}$ -measurable. The probability that a random variable  $X$  takes values in a set  $B \subset \mathbb{R}^{d_{\text{feat}}}$  is defined as  $P(X \in B) := P(X^{-1}(B))$ , where  $X^{-1}$  is the preimage of  $X$ .
+
+Let  $\{\mathcal{F}_t\}_{t\geq 0}$  be a filtration, which is a non-decreasing sequence of sub- $\sigma$ -algebras of  $\mathcal{F}$ ; i.e.,  $\mathcal{F}_s \subset \mathcal{F}_t \subset \mathcal{F}$  for all  $0 \leq s \leq t$ . Each element of the filtration can be interpreted as the available information at a given point  $t$ . The tuple  $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t\geq 0}, P)$  is called a filtered probability space.
+
+In our problem setting,  $X_{m}^{(1,T)}$  in the dataset  $S = \{X_{m}^{(1,T)}\}_{m=1}^{M}$  represents a sequence of observations for the  $m$ -th sample, which is treated as a stochastic process or as a realization of the stochastic process  $X^{(1,T)}$  interchangeably in our paper.  $y_{m}$  is the fixed class label associated with  $X_{m}^{(1,T)}$ .
+
+# A.2 DECISION RULE, TERMINAL DECISION, AND STOPPING RULE
+
+The decision rule  $\delta$  is defined as the pair  $(d_t, \tau)$ , where  $d_t$  is the terminal decision rule at time  $\tau = t$  ( $t \in \{1, \dots, T\}$ ) and  $\tau \in \{1, \dots, T\}$  is the stopping time. We provide their definitions below.
+
+The task of hypothesis testing as a time series classification involves identifying which one of the densities  $p_1, \ldots, p_K$  the sequence  $X^{(1,T)}$  is sampled from. Formally, this tests the hypotheses  $H_1: y = 1, \ldots, H_K: y = K$ .
+
+The decision function or test for a stochastic process  $X^{(1,T)}$  is denoted by  $d_t(X^{(1,T)}): \Omega \to \{1, \ldots, K\}$ . For each realization of  $X^{(1,T)}$ , we identify  $d_T: \mathbb{R}^{d_{feat} \times T} \to \{1, \ldots, K\}$ , i.e.,  $X^{(1,T)}(\omega) \mapsto y$ , where  $y \in \{1, \ldots, K\}$ . For simplicity, we write  $d_t$  instead of  $d_t(X^{(1,T)})$ .
+
+The stopping time  $\tau$  of  $X^{(1,T)}$  with respect to a filtration  $\{\mathcal{F}_t\}_{t\geq 1}$  is defined as  $\tau \coloneqq \tau (X^{(1,T)}):\Omega \to \mathbb{R}_{\geq 0}$  such that  $\{\omega \in \Omega |\tau (\omega)\leq t\} \in \mathcal{F}_t$
+
+Accordingly, for a fixed  $T \in \mathbb{N}$  and  $y \in \{1, \ldots, K\}$ , the set  $\{d_t = y\}$  represents the time-series data for which the decision function accepts the hypothesis  $H_i (i \in \{1, \ldots, K\})$  with a finite stopping time. Specifically,  $\{d_t = y\} = \{\omega \in \Omega | d_t(X^{(1,T)})(\omega) = y, \tau(X^{(1,T)})(\omega) < \infty\}$ .
+
+# B SEQUENTIAL PROBABILITY RATIO TEST AND ITS OPTIMALITY
+
+Our work centers around the optimality of Wald's SPRT. Below, we briefly review the optimality statements for both i.i.d. and non-i.i.d., multiclass classification scenarios. Note that the assumption of increasing LLRs is not applicable under the finite horizon setting discussed in the main manuscript.
+
+# SPRT's optimality with i.i.d., binary class data series.
+
+Theorem B.1. I.I.D. Optimality Let the time-series data points  $x^{(t)}$ ,  $t = 1,2,\ldots$  be i.i.d. with density  $f_{0}$  under  $H_{0}$  and with density  $f_{1}$  under  $H_{1}$ , where  $f_{0} \neq f_{1}$ . Let  $\alpha_{0} > 0$  and  $\alpha_{1} > 0$  be fixed constants such that  $\alpha_{0} + \alpha_{1} < 1$ . If the thresholds  $-a_{o}$  and  $a_{1}$  satisfies  $\alpha_{0}^{*}(a_{0},a_{1}) = \alpha_{0}$  and  $\alpha_{1}^{*}(a_{0},a_{1}) = \alpha_{1}$ , then  $SPRT\delta^{*} = (d^{*},\tau^{*})$  satisfies
+
+$$
+\inf  _ {\delta = (d, \tau) \in C \left(\alpha_ {0}, \alpha_ {1}\right)} \left\{\mathbb {E} [ \tau | H _ {0} ] \right\} = \mathbb {E} [ \tau^ {*} | H _ {0} ] \quad a n d \quad \inf  _ {\delta = (d, \tau) \in C \left(\alpha_ {0}, \alpha_ {1}\right)} \left\{\mathbb {E} [ \tau | H _ {1} ] \right\} = \mathbb {E} [ \tau^ {*} | H _ {1} ] \tag {14}
+$$
+
+A similar optimality also holds for continuous-time processes (Irle & Schmitz, 1984). Thus, SPRT terminates at the earliest expected stopping time compared to any other decision rule achieving the same or lower error rates—establishing the optimality of SPRT.
+
+Thm. B.1 demonstrates that, given user-defined thresholds, SPRT achieves the optimal mean hitting time. Additionally, these thresholds determine the error rates (Wald, 1947). Therefore, SPRT can minimize the required number of samples while maintaining desired upper bounds on false positive and false negative rates.
+
+SPRT's Asymptotic Optimality with Non-I.i.d., Multiclass Data Series. Intuitively, Thm. B.2 (Tartakovsky et al., 2014) suggests that if the LLRs  $\lambda_{kl}$  increase as samples accumulate, SPRT algorithm achieves asymptotic optimality. In this condition, the moments of the stopping time are minimized up to order  $r$  for a specified classification error rate.
+
+Theorem B.2 (Asymptotic optimality of SPRT under a multiclass, non-i.i.d. case). Assume that a non-negative increasing function  $\psi(t)$  ( $\psi(t) \xrightarrow{t \to \infty} \infty$ ) and positive finite constants  $I_{kl}$  ( $k, l \in [K]$ ,  $k \neq l$ ) exist, such that for some  $r > 0$ ,  $\lambda_{kl}(t) / \psi(t) \xrightarrow{P_{k-r-quickly}} I_{kl}$ . Then for all  $m \in (0, r]$  and  $k \in [K]$ ,  $\inf_{\delta} \mathbb{E}_k[\tau]^m \approx \mathbb{E}_k[\tau^*]^m$  as  $\max_{k, l} a_{kl} \to \infty$ .
+
+The precise definition of  $r$ -quick convergence and a more detailed discussion can be found, e.g., in Tartakovsky et al. (2014). Fig. 8 shows a graphical guide to the multiclass-SPRT decision rule.
+
+![](images/d4e365a5aec076708ba420370b98914715acddaa1c2c422552974f3995a9fce1.jpg)  
+Figure 8: Procedure of multiclass SPRT with static thresholds. (a) Example LLR trajectories for a three-class sequential Gaussian dataset, represented as an LLR matrix. (b, c) The two minimum operations defined in Def. 2.1 to determine the stopping time  $\tau^{*}$ .
+
+# C LLRS AND POSTERIORS AS SUFFICIENT STATISTIC
+
+The backward induction equation (Eq.7) depends on a sufficient statistic, which encapsulates all necessary information for decision-making. In hypothesis testing, true LLRs or posterior probabilities suffice to make decisions with a predefined error rate (Wald, 1947), thus both LLRs and posteriors qualify as sufficient statistics. The conversion is expressed by  $\pi_k(X^{(1,t)}) = 1 / (1 + \sum_{i\neq k}\chi_{ik}\exp (\lambda_{ik}(X^{(1,t)}))$  , where  $\chi_{kl}\coloneqq p(y = k) / p(y = l)$  represents the prior ratio. A formal definition of a sufficient statistic is available in Tartakovsky et al. (2014), as follows:
+
+Definition C.1 (Sufficient Statistic). A sequence  $\{\mathcal{S}_t\}_{t\geq 1}$  is defined to be sufficient statistic for the sequential decision problem if it satisfies the following conditions:
+
+1. Transitivity: The sequence is transitive, meaning there exists a function  $\phi_n(\cdot)$  such that
+
+$$
+\mathcal {S} _ {t + 1} = \phi_ {t} \left(\mathcal {S} _ {t}, x ^ {(t + 1)}\right), \quad a l m o s t s u r e l y, f o r n \geq 1.
+$$
+
+2. Equality of Conditional Probability Density Function (pdf): The conditional pdf of  $X_{t+1}$  given the past observations  $X^t$  can be expressed solely in terms of  $\mathcal{S}_t$ :
+
+$$
+p _ {t + 1} \left(x ^ {(t + 1)} \mid X ^ {(1, t)}\right) = p _ {t + 1} \left(x ^ {(t + 1)} \mid \mathscr {S} _ {t}\right), \quad \text {a l m o s t s u r e l y , f o r} t \geq 1.
+$$
+
+3. Equality of Risks: The A Posteriori Risk (APR) when using the sufficient statistic  $\mathcal{S}_t$  equals the APR calculated directly from the observations:
+
+$$
+\operatorname {A P R} \left(X ^ {(1, t)}\right) = \operatorname {A P R} \left(\mathscr {S} _ {t}\right), \quad \text {a l m o s t s u r e l y , f o r} n \geq 1.
+$$
+
+Note that the online DRE algorithm SPRT-TANDEM is transitive, providing consistent estimation of the sufficient statistic.
+
+Which statistic to use, LLRs or posteriors? In principle, the CFL algorithm can handle either LLRs or posteriors as the sufficient statistic for calculating the conditional expectation. Our experiments confirm that both LLRs and posteriors yield equivalent results; however, we opt to use posteriors to reduce input dimensionality.
+
+Conversely, our use of GP regression is predicated on the assumption that the risk distribution is jointly Gaussian, which motivates us to use LLRs as the sufficient statistic. Nonetheless, an experiment with the two-class Gaussian dataset confirms that GP regression provides equivalent results regardless of the type of statistic used (Fig. 9).
+
+![](images/e37467b6c7f0dc20ac27966d3f44323aa8201d633524c1930960f86bd2eee5c4.jpg)  
+Figure 9: Comparing LLRs and posteriors as sufficient statistics for GP Regression. LLRs and posteriors are used as sufficient statistics to evaluate (a) the AAPR curve and (b) the SAT curve. The two-class Gaussian dataset provides the ground-truth LLRs and the corresponding converted posteriors.
+
+# D COMPUTATIONAL COMPLEXITY OF FIRMBOUND AND SAMPLING METHOD
+
+Here, we provide a detailed comparison of the computational complexity for the inference stage of both the direct estimation approach and the Monte Carlo Integration with Kernel Density Estimation (KDE) approach.
+
+# D.1 DIRECT ESTIMATION APPROACH (FIRMBOUND)
+
+The direct estimation approach uses the following function to evaluate the conditional expectation:
+
+```txt
+@torch.no_grad()  
+def predict(self, X: Tensor, *args, **kwargs) -> Tensor:  
+    pred, _ = torch.max(  
+        torch/matmul(X, self.a.T) + self.y_hat.reshape(1, -1), dim=1)  
+    return pred
+```
+
+In this function:
+
+-  $X$  is the input tensor of size  $[B, K]$ , where  $B$  is the batch size and  $K$  is the number of classes.  
+- self.a and self.y_hat are parameter tensors of size  $[I, K]$  and  $[I]$ , respectively, where  $I \ll M$  is the subset data number.
+
+The computational complexity for each step in the inference stage is as follows:
+
+1. Matrix Multiplication: The operation torch/matmul(X, self.a.T) has a complexity of  $\mathcal{O}(B \cdot K \cdot I)$ . Broadcasting and Addition: The operation torch/matmul(X, self.a.T) + self.y_hat.reshape(1, -1) involves broadcasting and addition, which has a complexity of  $\mathcal{O}(B \cdot I)$ .  
+2. Maximum Value Selection: The operation torch.max(., dim=1) finds the maximum value along the specified dimension, which has a complexity of  $\mathcal{O}(B \cdot I)$ .
+
+Thus, the total computational complexity for the inference stage of the direct estimation approach is dominated by the matrix multiplication step, resulting in:
+
+$$
+\mathcal {O} (B \cdot K \cdot I)
+$$
+
+# D.2 MONTE CARLO INTEGRATION WITH KDE APPROACH
+
+The Monte Carlo Integration with KDE approach involves the following steps for the inference stage:
+
+1. Generate  $S$  samples from the conditional density  $p(\mathcal{S}_{t + 1} \mid \mathcal{S}_t)$  using KDE.  
+2. Evaluate the function  $G_{t + 1}(\mathcal{S}_{t + 1})$  for each sample.  
+3. Compute the average to estimate the conditional expectation.
+
+Assuming:
+
+-  $B$  is the batch size (number of input samples in  $\mathcal{S}_t$ ).  
+-  $K$  is the dimensionality (number of classes).  
+-  $M$  is the total number of data points.  
+-  $S$  is the number of Monte Carlo samples.
+
+In the Monte Carlo Integration with KDE approach, the dimensionality  $K$  affects the number of Monte Carlo samples  $S$  required for convergence. Let  $S(K)$  denote the number of samples as a function of  $K$ , typically increasing with  $K$ . The computational complexity for each step is as follows:
+
+1. Sampling from KDE: Generating  $S(K)$  samples for each of the  $B$  input samples, each requiring  $\mathcal{O}(M \cdot K)$  operations, resulting in a complexity of  $\mathcal{O}(B \cdot S(K) \cdot M \cdot K)$ .  
+2. Function Evaluation: Evaluating  $G_{t + 1}(\mathcal{S}_{t + 1})$  for each sample with complexity  $\mathcal{O}(K)$ , resulting in a total complexity of  $\mathcal{O}(B\cdot S(K)\cdot K)$ .  
+3. Monte Carlo Integration: The averaging step has a complexity of  $\mathcal{O}(B\cdot S(K)\cdot K)$
+
+Thus, the total computational complexity for the inference stage of the Monte Carlo Integration with KDE approach is dominated by the sampling step, resulting in:
+
+$$
+\mathcal {O} (B \cdot S (K) \cdot M \cdot K)
+$$
+
+# D.3 COMPARISON
+
+The direct estimation approach has a computational complexity of  $\mathcal{O}(B\cdot K\cdot I)$  for the inference stage, while the Monte Carlo Integration with KDE approach has a complexity of  $\mathcal{O}(B\cdot S(K)\cdot M\cdot K)$ . Given that  $I\ll M$  and considering that higher dimensionality  $(K)$  increases the number of samples required for convergence  $(S(K))$ , the direct estimation approach is significantly more efficient in terms of computational complexity during inference. This efficiency is particularly advantageous for real-time applications and large-scale datasets.
+
+Table 2: Comparison of Inference Stage Computational Complexity  
+
+<table><tr><td>Approach</td><td>Inference Stage Complexity</td></tr><tr><td>Direct Estimation with FIRMBOUND</td><td>O(B·K·I)</td></tr><tr><td>Monte Carlo Integration with KDE</td><td>O(B·S(K)·M·K)</td></tr></table>
+
+# E SUPPLEMENTARY RELATED WORK
+
+# E.1 SPRT AND ITS OPTIMALITY
+
+SPRT is Bayes optimal in binary classification with i.i.d. samples and is also known to require the minimal sample size to achieve a predefined error rate (Wald & Wolfowitz, 1948; 1950). The properties of SPRT under multiclass scenarios (Armitage, 1950; Baum & Veeravalli, 1994; Chernoff, 1959; Dragalin, 1987; Dragalin & Novikov, 1999; Kiefer & Sacks, 1963; Lorden, 1977; Paulson, 1963; Pavlov, 1991; 1984; Simons, 1967; Sobel & Wald, 1949), and with non-i.i.d. samples (Dragalin et al., 1999; Lai, 1981; Tartakovsky, 1998), have also been studied (App. B). Several algorithms employ SPRT with estimated density ratio with kernel method (Teng & Ertin, 2016) or boosting (Sochman & Matas, 2005) approach. However, they often assume i.i.d. samples and limited to binary classification, without considering the finite horizon.
+
+# E.2 OPTIMAL STOPPING THEORY
+
+Optimal stopping theory helps decide the best time to act, minimizing expected cost. It applies to various settings like the secretary problem, parking problem, one-armed bandit, change-point detection, and sequential statistical decision problems.
+
+Among these, finite-horizon problems are particularly relevant to our study, which involve a known upper bound on the length of the sequence. Discrete-time, finite-horizon problems are typically solved using dynamic programming techniques like backward induction, a type of Bellman equations. However, backward induction poses significant computational challenges. It requires storing and computing all possible histories, leading to high computational costs and analytical intractability unless the underlying distribution is known and simple (Ferguson, 2006; Tec et al., 2023). Several approximation methods, such as  $k$ -step and  $k$ -time look-ahead rules, have been proposed, but they fall short of optimality unless the problem is monotonic, which is often not the case with real-world data.
+
+# E.3 SPRT'S BACKWARD INDUCTION AND CONDITIONAL EXPECTATION
+
+Applying the backward induction under real-world conditions is impractical (Tartakovsky et al., 2014). No analytical solution has been identified, and although numerical computation on simulated datasets is feasible (Jarrett & van der Schaar, 2020), calculating the conditional expectation of future risks—a critical component of backward induction—is computationally intensive. This often necessitates approximations such as assuming conditional independence of temporal evidence (Ahmad & Yu, 2013; Naghshvar & Javidi, 2013), discretizing continuous variables (Frazier & Yu, 2007), or adopting a one-step look-ahead approach (Kleinegesse et al., 2020; Najemnik & Geisler, 2005). Moreover, ECTS demands instantaneous evaluation of the conditional expectation, precluding the use of sampling-based methods of the conditional expectation on the fly (Wang & Scott, 2019). The lack of true LLRs, which are the sufficient statistic required for SPRT and backward induction, further complicates their practical application within finite horizons.
+
+# E.4 SEQUENTIAL DESIGN
+
+Sequential design, particularly simulation-based Bayesian sequential design, offers a practical approach to these challenges. This method, grounded in statistical decision theory, approximates the objective function (e.g., minimum risk) using simulated trajectories on finite grid points rather than exhaustive computation of all possible histories (Brockwell & Kadane, 2003; Müller et al., 2007; Kadane & Vlachos, 2002). Notable approaches within this framework include constrained backward induction and sequential design with optimizing decision boundaries. Constrained backward induction iteratively approximates expected utility using simulated trajectories, while sequential design with optimizing decision boundaries transforms the sequential decision problem into a non-sequential optimization of parametric decision boundaries (Rossell et al., 2007). Both methods rely on simulated trajectories, unlike our model, which utilizes real-world data trajectories and avoids the tradeoff between precision and computational cost associated with grid-based methods.
+
+# E.5 REINFORCEMENT LEARNING (RL)
+
+RL is another domain where backward induction, often referred to as the Bellman equation, is extensively applied. In RL, algorithms like Q-learning and policy gradient methods can be viewed as constrained backward induction and sequential decision-making with optimizing boundaries, respectively. However, RL faces significant challenges, including poor sample efficiency and training instability, often leading to catastrophic forgetting and high variance in policy gradient estimates (Atkinson et al., 2018; Bjorck et al., 2021; Cetin et al., 2022; Kumar et al., 2020; Nikishin et al., 2018; Sullivan et al., 2022). RL approach is often combined with the sequential design (Asano, 2022; Blau et al., 2022).
+
+# E.6 ACTIVE LEARNING
+
+Active learning is a machine learning paradigm aimed at achieving high accuracy with minimal labeled data by strategically querying the most informative samples. It encompasses several strategies, including active sensing and active hypothesis testing. In active sensing, the system optimizes sensor placements and parameters to gather the most relevant data, while in active hypothesis testing, the goal is to identify the correct hypothesis as efficiently as possible. One foundational work by (Cohn et al., 1996) demonstrated the effectiveness of active learners over passive learners by querying the most informative data points. (Lewis & Gale, 1994) introduced uncertainty sampling, where instances with the highest uncertainty are selected for labeling. Another key method, query-by-committee (QBC) by (Seung et al., 1992), selects instances based on the disagreement among multiple models. More recently, approaches like Bayesian active learning by disagreement (Houlsby et al., 2011) and core-set approaches (Sener & Savarese, 2017) have been developed to handle the complexity of neural networks. Jarrett & van der Schaar (2020) developed a framework for timely decision-making under context-dependent time pressure.
+
+# E.7 CONVEX FUNCTION LEARNING (CFL)
+
+CFL aims to infer a convex function from data points, assuming the target function is inherently convex. This assumption ensures that any local minimum is also a global minimum (Argyriou et al., 2008; Bach, 2010; Bartlett et al., 2005; Boyd & Vandenberghe, 2010), thereby simplifying the optimization landscape and enhancing the efficiency of solving optimization problems (Mendelson, 2004). Within this framework, the Alternating Direction Method of Multipliers (ADMM) has proven to be particularly effective (Amos et al., 2016), allowing for the decomposition of complex optimization tasks into smaller, more manageable subproblems that are solved iteratively (Eckstein, 2012; Gabay & Mercier, 1976; Glowinski & Marroco, 1975). ADMM's capability extends to solving the augmented Lagrangean equation on a piecewise linear function, optimizing each segment effectively (Siahkamari et al., 2020). However, the standard ADMM can be slow to converge. To address this, enhancements such as the 2-block ADMM have been developed to accelerate convergence, thus improving the overall performance of CFL applications (Siahkamari et al., 2022).
+
+# E.8 GAUSSIAN PROCESS (GP) REGRESSION
+
+GP regression is a Bayesian approach that makes probabilistic predictions (Wang, 2020). Unlike traditional regression methods that presuppose a specific form for the regression function, GP regression treats observed function values as jointly Gaussian, with a mean function and a covariance defined by a kernel function. The kernel encapsulates assumptions about the function's smoothness and the nature of correlation between function values at different points in the input space. The inherent flexibility of GP regression, which does not require the explicit specification of the function form, renders GP regression widely applicable in diverse fields including geostatistics—often referred to as Kriging (Huang, 2020; Tao et al., 2022; Richter & Toledano-Ayala, 2015), financial modeling (Gonzalvez et al., 2019; Herfurth, 2020; Petelin et al., 2011), to robotics (Cheng et al., 2022; Jakkala & Akella, 2023; Xu et al., 2022).
+
+Traditional GP model, however, faces significant computational challenges when applied to large datasets due to the  $\mathcal{O}(M^3)$  scaling with respect to the number of data points  $M$ . To making it infeasible for large-scale applications. To mitigate this, inducing point methods have been developed to approximate the full GP, substantially reducing the computational load while largely retaining
+
+the model's expressive power (Candela & Rasmussen, 2005). By summarizing the dataset with a smaller set of  $m$  inducing points, the complexity is reduced to  $\mathcal{O}(m^2 M)$ . Additionally, stochastic variational inference (SVI) optimizes variational parameters using minibatches of data, which significantly decreases the computational demands to  $\mathcal{O}(m^3)$  per update, independent of the full dataset size (Hensman et al., 2014). This approach not only makes GP regression scalable but also adapts well to modern computational infrastructures, such as GPUs, enabling the handling of extensive datasets within constrained resource settings (Deisenroth & Ng, 2015; Wilson & Nickisch, 2015).
+
+# E.9 OTHER ECTS ALGORITHMS
+
+ECTS is pivotal in scenarios requiring prompt and accurate classification decisions from incomplete data streams. Applications of ECTS include, but not limited to, medical diagnosis (Evans et al., 2015; Griffin & Moorman, 2001; Vats & Chan, 2016), stock crisis identification (Ghalwash et al., 2014), autonomous driving (Doná et al., 2019), action recognition (Weng et al., 2020), and e-commerce user profiling (Duan et al., 2024). Delays in classification can have critical consequences, positioning ECTS as a key area of research within time series analysis. This field inherently presents a multi-objective optimization challenge aimed at maximizing classification accuracy while minimizing decision time (Mori et al., 2018; Mori et al., 2015; Xing et al., 2012). Recent advancements have integrated deep learning techniques due to their robust representational capacities (Dennis et al., 2018; Ismail Fawaz et al., 2019; Lv et al., 2023; Sun et al., 2023; Suzuki et al., 2018; Hartvigsen et al., 2021). For example, LSTM-s and LSTM-m, have been developed to impose monotonicity on classification scores and enhance inter-class margins, respectively, thereby accelerating action detection (Ma et al., 2016). The Early and Adaptive Recurrent Label ESTimator (EARLIEST) leverages a combination of reinforcement learning and recurrent neural networks to dynamically decide the timing and classification of data (Hartvigsen et al., 2019). Moreover, the incorporation of transformer technologies, as seen in TCN-Transformer, merges temporal convolution with transformer architecture to prioritize early classification through specialized loss functions (Chen et al., 2022). Several algorithms empirically predict future risk to decide when to halt the sampling (Martinez et al., 2020; Wang et al., 2024; Zafar et al., 2021). For example, Calibrated eArLy tIMe sERies clAsifier (CALIMERA) (Bilski & Jastrzejbska, 2023) predicts the minima of risk function where the decision making should be made.
+
+# E.10 NEUROPHYSIOLOGICAL UNDERPINNINGS OF SPRT.
+
+SPRT has been identified as a neural decision-making algorithm within the primate brain's lateral intraparietal cortex (LIP, (Roitman & Shadlen, 2002)). During alternative-choice tasks, LIP neurons gradually accumulate sensory evidence, represented by an increasing firing rate of single neurons (Latimer et al., 2015), the average population activity (Shadlen et al., 2016), or a high-dimensional manifold of neural populations (Okazawa et al., 2021). Since neural activities are proportional to LLRs, the behavior of LIP neurons and primates' decision strategies can be best explained by SPRT Kira et al. (2015). For more information, readers are directed to review articles such as (Doya, 2008; Gallivan et al., 2018; Gold & Shadlen, 2007).
+
+Multiple studies investigate decision-making under time pressure (Churchland et al., 2008; Drugowitsch et al., 2012; Hanks et al., 2014). Some papers report closing boundaries under such conditions (Kira et al., 2024), reminiscent of the optimal decision boundary computed with the backward induction (Fig. 2b), while others identify an urgency signal, a linearly increasing offset added to the ramping neural activity (i.e., corresponding to the sufficient statistic in optimal stopping theory), which accelerates decisions as the deadline approaches. Both the closing boundary and the urgency signal have psychophysically equivalent effects, compelling quicker decisions with less confidence as the finite horizon approaches.
+
+# F SUPPLEMENTARY DISCUSSION
+
+# F.1 INTUITIVE UNDERSTANDING OF FIRMBOUND
+
+APR. The first term of Eq. 3 imposes a heavy penalty if the terminal decision  $d_{t} = k$  corresponds to a low posterior probability  $\pi_{k}$ . The second term accumulates the sampling costs up to the current time step  $t$ .
+
+Theorem 2.1 (backward induction). Theorem 2.1 can be interpreted as a recursive decision-making process that minimizes Bayes risk at each time step. The Sequential Probability Ratio Test (SPRT) at each step must either (i) continue sampling to refine the sufficient statistic, or (ii) make a final classification decision with the current sufficient statistic. Each choice—(i) continuing or (ii) stopping—incurs a form of risk: the continuation risk,  $\tilde{G}_t$  (Eq. (5)), and the stopping risk,  $G_t^{\mathrm{st}}$  (Eq. (6)). At each time step, the lower of these two values defines the minimum risk,  $G_t^{\mathrm{min}}$  (Eq. (7)), up to the classification deadline or finite horizon.
+
+Initiation of backward induction. At the finite horizon  $(t = T)$ , the minimum risk  $G_{T}^{\mathrm{min}}$  is always equal to the stopping risk  $G^{\mathrm{st}}$ , setting the initial condition for the risk distribution at time  $T$ . Subsequently,  $G_{t}^{\mathrm{min}}$  for earlier time steps is recursively calculated using the risk distribution of the subsequent time step (Fig. 3b).
+
+Risk comparison during deployment. Initially, the sufficient statistic is small, leading to a higher  $G^{\mathrm{st}}$  than  $\tilde{G}$ . As more samples are collected, the statistic increases, enhancing decision confidence and reducing  $G^{\mathrm{st}}$  below  $\tilde{G}$ . The decision is made when  $G^{\mathrm{st}}$  is less than or equal to  $\tilde{G}$ , with the decision boundary at  $\tau^{*}$  being the intersection of  $\tilde{G}$  and  $G^{\mathrm{st}}$  (Fig. 3a).
+
+# F.2 CHALLENGES IN ESTIMATING CONDITIONAL EXPECTATIONS USING MONTE CARLO AND KERNEL DENSITY ESTIMATION
+
+Estimating conditional expectations such as  $\mathbb{E}\left[G_{t + 1}(\mathcal{S}_{t + 1})\mid \mathcal{S}_t\right]$  is a common problem in various scientific and engineering disciplines. One approach to achieve this estimation is by employing Monte Carlo integration techniques (Kroese et al., 2011; Robert & Casella, 2004) in conjunction with Kernel Density Estimation (KDE, (Scott, 1992; Silverman, 1986)) and its application to conditional density, Kernel Conditional Density Estimation (Rosenblatt, 1969) to approximate  $p(\mathcal{S}_{t + 1}\mid \mathcal{S}_t)$ . While this method is theoretically sound and flexible, it comes with several significant challenges, particularly in high-dimensional settings. This section outlines the primary difficulties associated with this estimation method, including issues related to the curse of dimensionality, computational complexity, bandwidth selection, and sampling efficiency.
+
+# F.2.1 CURSE OF DIMENSIONALITY
+
+Sparsity of data In high-dimensional spaces, data points tend to become sparse (Botev et al., 2010). The volume of the space increases exponentially with the number of dimensions, which means that even large datasets may not provide sufficient coverage of the space. This sparsity makes it difficult to accurately estimate the conditional density  $p(\mathcal{S}_{t + 1} \mid \mathcal{S}_t)$  using KCDE because the kernel functions may have to cover large regions with very few data points, leading to high variance in the density estimates. Indeed, Wang & Scott (2019) defines high-dimensional data for the kernel density method at most 50-dimensional, indicating the difficulty of modeling probability distributions over sufficient statistics  $\mathcal{S}$ , given that  $\mathcal{S}$  is at least  $K$ -dimensional where  $K$  is the class number.
+
+Bandwidth selection. Selecting an appropriate bandwidth for the kernel is critical for accurate density estimation (Bashtannyk & Hyndman, 2001; Wang & Wang, 2007). In high-dimensional settings, a single bandwidth parameter is often insufficient, and a multidimensional bandwidth matrix is required. However, selecting and optimizing such a bandwidth matrix is computationally intensive and challenging. If the bandwidth is too large, the estimate will be overly smooth, missing important details. Conversely, if it is too small, the estimate will be too noisy, capturing random fluctuations rather than the true underlying structure.
+
+# F.2.2 COMPUTATIONAL COMPLEXITY
+
+High computational cost. Kernel density estimation involves computing distances between data points and evaluating kernel functions. In high dimensions, these computations become increasingly expensive. The number of operations required grows with both the number of data points  $M$  and the dimensionality of the space. For KCDE, which requires estimating the joint and marginal densities, the computational cost is even higher. Given that our application is online ECTS, waiting for the estimation to converge at each time step is very impractical. Rather, FIRMBOUND provides a function that is readily be evaluated with convergence guarantee, enabling deployment under real-world scenarios.
+
+# F.2.3 BANDWIDTH SELECTION
+
+Data-dependent bandwidth. Adaptive methods, where the bandwidth varies locally depending on the density of data points, can provide better estimates but add another layer of complexity. These methods require careful tuning and can be computationally demanding, especially in high-dimensional spaces.
+
+# F.2.4 SAMPLING EFFICIENCY
+
+Efficient sampling techniques. Even with an accurate estimate of the conditional density  $p(\mathcal{S}_{t + 1} \mid \mathcal{S}_t)$ , efficiently sampling from this distribution can be challenging. Techniques such as rejection sampling or Metropolis-Hastings may be necessary, but these can be computationally intensive and may not scale well with dimensionality.
+
+# F.2.5 MITIGATION STRATEGIES
+
+To address these challenges, several strategies can be employed, each with its own assumptions and potential sources of error:
+
+Dimensionality Reduction: Techniques such as Principal Component Analysis (PCA) or autoencoders can be used to reduce the dimensionality of  $\mathcal{S}$  while preserving important structures in the data. This can help alleviate the curse of dimensionality and improve the efficiency of density estimation. However, this assumes that the reduced dimensions adequately capture the necessary information, which may not always be true.
+
+Sparse Kernel Methods: Utilizing a subset of the data points (e.g., random sampling or clustering-based methods) can reduce the computational burden. The assumption here is that the subset is representative of the full dataset, which might not hold in all cases, potentially leading to biased estimates.
+
+Localized Methods: Adaptive kernel methods, where the bandwidth varies depending on the local density of data points, can provide more accurate estimates in high-dimensional spaces. These methods assume that local adaptation can adequately capture the density variations, but improper tuning can introduce significant errors.
+
+Grid-Based Methods: For moderate-dimensional cases, grid-based methods can approximate the density on a discretized grid, reducing computational complexity. The main assumption is that the grid resolution is fine enough to capture the density details, but this can lead to high memory and computation costs if the dimensionality is still relatively high.
+
+# F.2.6 ADVANTAGE OF HAVING A DIRECT ESTIMATOR FIRMBOUND
+
+The development of a direct estimator for the conditional expectation  $\mathbb{E}\left[G_{t + 1}(\mathcal{S}_{t + 1})\mid \mathcal{S}_t\right]$  presents significant advantages over the traditional Monte Carlo Integration approach combined with Kernel Density Estimation. Firstly, a direct estimator offers computational efficiency by providing instantaneous evaluations, which is crucial for real-time applications and large-scale datasets. This efficiency eliminates the need for extensive sampling and repeated function evaluations inherent in Monte Carlo methods, thus reducing computational overhead. Additionally, the direct estimator ensures statistical consistency, guaranteeing that as the sample size increases, the estimator converges to the true conditional expectation, thereby enhancing the reliability and accuracy of the estimates. In contrast,
+
+![](images/de5e831891b5db21dd8b87ebc9b6a7fbd7aefd057e6b6c7d7979830861f8071e.jpg)  
+Figure 10: Performance comparison across datasets of different sizes. The three-class Gaussian dataset is reduced from the original size  $(M = 60000)$  to  $M = 6000$  and  $M = 600$  for training FIRMBOUND, while using the same test dataset. Hyperparameter settings remain fixed, and experiments are repeated five times to compute error bars. (a) The AAPR curve and (b) the SAT curve demonstrate that FIRMBOUND maintains competitive performance with datasets one order of magnitude smaller ( $M = 6000$ ).
+
+Monte Carlo methods are prone to sampling errors and require careful tuning of kernel functions and bandwidth parameters, adding complexity and potential sources of error. Furthermore, the direct estimator simplifies the implementation process by obviating the need for density estimation, which can be particularly challenging in high-dimensional spaces due to the curse of dimensionality. This simplicity, coupled with reduced memory requirements, makes the direct estimator more robust and scalable, offering a clear advantage in handling high-dimensional data and real-time decision-making scenarios.
+
+# F.3 PERFORMANCE UNDER SMALL DATASETS
+
+While FIRMBOUND ensures statistical consistency, its practicality could be questioned if performance degrades significantly with reduced dataset sizes. To address this concern, we conduct an additional experiment. Using the three-class Gaussian dataset, we train FIRMBOUND with datasets up to two orders of magnitude smaller than the original size while keeping the same test dataset. Specifically, dataset sizes of  $M = 600$  and  $M = 6000$  are compared to the original size of  $M = 60000$ . Hyperparameter settings are kept fixed, and experiments are repeated five times to compute error bars.
+
+The results show that even with a dataset one order of magnitude smaller ( $M = 6000$ ), FIRMBOUND demonstrates almost negligible differences in performance compared to the original dataset in terms of mean hitting time, AAPR, and mean per-class error rate. Notably, the real-world HMDB51 dataset has a similar order of magnitude ( $M = 5277$ ), further showcasing FIRMBOUND's robustness in real-world scenarios. However, with the smallest dataset ( $M = 600$ ), AAPR and mean per-class error rates increase significantly, indicating that the dataset size is insufficient to accurately estimate the sufficient statistics. Despite this, the mean hitting time remains close to the original, highlighting FIRMBOUND's stability/ and its ability to make reliable best-effort decisions even under highly limited data conditions.
+
+# F.4 BROADER IMPACT
+
+FIRMBOUND enhances the performance of ETCS in real-world settings and prompts further research across both machine learning and neuroscience. It facilitates the backward induction on real-world datasets, effectively removing constraints associated with i.i.d. or non-i.i.d. data, thereby expanding its utility for time-sensitive tasks. Furthermore, backward induction is instrumental in fields like active sensing and sequential design, where unlike ECTS, an agent proactively selects actions to gather informative evidence. FIRMBOUND is ideally suited for such applications, enabling its deployment in dynamic environments. Additionally, the tapering optimal threshold is reminiscent of decision-making processes observed in humans, providing a potential bridge to understanding neural thresholding mechanisms within finite horizon, which, despite extensive study (Churchland et al., 2008; Drugowitsch et al., 2012; Gold & Shadlen, 2007; Latimer et al., 2015; Okazawa et al., 2021), remain elusive.
+
+Our method is designed to optimize the speed-accuracy tradeoff in real-world applications, which is expected to lead to positive societal impacts. The potential for negative effects is minimal and mainly confined to instances where models are intentionally trained to prioritize speed or accuracy by using an extreme value of sampling cost, which could compromise decision speed or quality.
+
+# F.5 LIMITATIONS AND FUTURE WORK
+
+Domain gap. While FIRMBOUND provides a theoretical guarantee to minimize the AAPR, it is important to acknowledge that it may not achieve the global minimum on test data when there is a domain gap between the training and test sets. A domain gap occurs when the distribution of the test data differs from that of the training data, which can lead to suboptimal performance of the model, even with the optimal stopping rule. FIRMBOUND, as proposed, assumes that the test data follows the same distribution as the training data, and thus, its effectiveness may be compromised in the presence of such domain discrepancies.
+
+We recognize the importance of addressing domain gaps in machine learning research. However, it is important to note that handling domain gaps is beyond the current scope of our paper, which focuses on developing an optimal stopping rule for early classification within finite horizons. Employing a domain adaptation algorithm or foundation models would require a different methodological approach and additional research efforts. Potential directions include incorporating domain adaptation techniques and robustifying FIRMBOUND against such discrepancies.
+
+Future theoretical directions. While FIRMBOUND is "doubly consistent" estimator of both the backward induction and log-likelihood ratio, several theoretical directions are yet to be investigated. One example is the convergence rate of the algorithm. While it is presumably given by the sum of LSEL's and CFL's. The latter is given in the paper, but the former requires an additional extensive analysis because Lemma J.4 (consistency of LSEL), on which our consistency proof relies, is an asymptotics of the probability that the estimated parameters deviate from the optimal parameter set. Similarly, minimax bound cannot be derived straightforwardly, warrant a separate, focused study.
+
+Potential density chasm problem. We observe that Fig. 5a (for the two-class Gaussian dataset) shows a discrepancy in the minimal averaged posterior risk (AAPR) locations between true and estimated LLRs. Interestingly, this trend is negligible in Fig. 5b (for the three-class Gaussian dataset), where AAPR locations for true and estimated LLRs align more closely. One possible explanation is the density chasm problem, a known issue specific to density ratio estimation on "easy" problems, which increases the absolute value of the density ratio and could contribute to these errors (Rhodes et al., 2020). A countermeasure to the density chasm problem, telescoping density ratio estimation, was proposed in (Rhodes et al., 2020), and employing this approach may help mitigate the error on simple datasets.
+
+These issues are likely specific to simple datasets and are less prevalent in complex real-world datasets with arbitrary class counts. It is important to note that while statistical consistency guarantees minimization of estimation error, it does not address approximation or optimization errors, which may be the main contributing factors here.
+
+Performance gain under dynamic environments. FIRMBOUND often shows limited performance on datasets with monotonic trajectory of sufficient statistics (see also App. F.6 for more detailed discussion). This minimal performance gain does not diminish FIRMBOUND's practical value. The real-world datasets examined (SiW, HMDB51, UCF101, and FordA) involve relatively small domain gaps and fewer fluctuations, producing stable, monotonic trajectories that limit opportunities for improvement over static thresholds. However, in more adversarial real-world scenarios, such as those with dim or variable lighting, we would expect the trajectory to fluctuate similarly to the DOL dataset, where FIRMBOUND shows robust performance gains. Furthermore, as noted in Section 4, FIRMBOUND consistently reduces the variance of hitting times across all datasets. Along with its capability to handle i.i.d., non-i.i.d., and multiclass data, this variance reduction demonstrates FIRMBOUND's potential for reliable decision making. In future work, we aim to explore FIRMBOUND's performance under more dynamically adversarial conditions, which we anticipate will reveal greater gains similar to those seen in our DOL experiments.
+
+Retraining requirement at cost change. FIRMBOUND effectively delineates the Pareto front on the SAT. A potential limitation arises if a user is unsatisfied with the resulting speed or accuracy and wants to select a different point on the Pareto front; in such cases, retraining FIRMBOUND with a new cost parameter  $c$  is required. The training process can be computationally intensive, especially if the CFL algorithm is used to estimate the conditional expectation. However, there is a remedy for this issue. Without incurring additional computational costs, users can re-evaluate the AAPR curve on the current sufficient statistic  $\mathcal{S}$  using different values of  $c$ . This allows them to efficiently identify the optimal  $c$  that yields the desired mean hitting time and corresponding error rate on the SAT curve by finding the AAPR curve whose minimum is closest to the desired mean hitting time. This strategy effectively avoids the need for retraining FIRMBOUND when a specific error rate  $\alpha$  must be achieved on the SAT curve, which is often crucial in high-security applications.
+
+Extremely large LLRs' magnitude. Large LLRs, which can occur when the classification task is relatively easy, can significantly hinder the training of FIRMBOUND in the following ways. When LLRs are extremely large, the corresponding posterior probabilities derived from them often degenerate to either zero or one, making the training data less informative. This forces FIRMBOUND to learn from a dataset with extreme and non-informative posterior probabilities, potentially leading to overfitting and reduced generalization performance. An alternative approach is to train FIRMBOUND directly on the LLRs instead of the posteriors. However, in multiclass classification with  $K$  classes, the number of pairwise LLRs required is at least  $K(K - 1)/2$ , which can be extremely large. This approach can be prohibitively memory-intensive, especially when  $K$  is large, making training on standard devices challenging. Additionally, the unbounded nature of LLRs can introduce instability into the training process. However, it is important to note that in cases where posterior probabilities degenerate, time-series analysis may not be necessary. The fact that classification can be resolved entirely within the first (or first few) steps suggests that FIRMBOUND may not need to be employed in such scenarios, indicating that this limitation is not a direct weakness of the method.
+
+# F.6 FREQUENTLY ASKED QUESTIONS
+
+How would you justify FIRMBOUND's two-component framework, given that it introduces additional complexity and potential error propagation? The doubly consistent estimation of both the conditional expectation in backward induction and the LLR—required by FIRMBOUND necessitates a multi-component framework. While this design may introduce additional complexity and the potential for error propagation across components, FIRMBOUND guarantees the minimization of estimation errors, particularly in large datasets.
+
+Consistent estimation itself represents a significant advancement in ECTS. Most existing ECTS methods rely on empirical heuristics, whether they follow a two-component approach (e.g., CALIMERA) or a one-component, end-to-end framework (e.g., LSTMms, TCN-Transformer, EARLIEST). Our experiments demonstrate that these heuristic-based approaches are consistently outperformed by FIRMBOUND, emphasizing the practical benefits derived from our theoretically grounded, multi-component framework.
+
+When is a new Pareto-front available? FIRMBOUND extends the Pareto front on some datasets, depending on LLR monotonicity. In Gaussian datasets with monotonically increasing LLRs, it expedites decisions without increasing error rates (Fig. 1a, 6a, b). Importantly, even without a new Pareto-front, FIRMBOUND ensures reliability by reducing hitting time variance (Tab. 1), crucial for safe deployment in diverse scenarios. Conversely, in non-monotonic DOL datasets with initial noise followed by stabilization, it effectively achieves new Pareto-fronts (Fig. 6c).
+
+Is it possible to establish consistency with GP? When should I use GP? Yes, it is possible to construct a consistent estimator using Gaussian Processes (GP) under certain conditions. The GP regressor can be a consistent estimator if, with an appropriate choice of the kernel,  $\epsilon_m^{(t)}$  follows a Gaussian distribution for all  $t\in [T]$  and  $m\in [M]$ , and  $\{\tilde{G}_t(\mathcal{S}_{t,m})\}_{m = 1}^M$  is a Gaussian process for all  $t\in [T]$ . However, these conditions are difficult to guarantee under arbitrary circumstances. Thus, we introduced CFL as a more general and robust solution.
+
+When should I use GP? What's your recommendation? Albeit the consistency loss, our experiments show that the GP regressor performs competitively with CFL, which is the theoretically optimal approach. This makes GP a practical choice in scenarios with limited computational resources. It is worth noting that GP regression is particularly efficient during the *training* stage. During testing or deployment, both CFL and GP regression are sufficiently fast to support real-time decision-making. Therefore, GP regression is an effective option in environments with constrained training resources, such as edge computing settings.
+
+Why some models are excluded from Fig. 5? Effective risk minimization relies on well-calibrated statistics. As shown in Fig. 14 in App. M shows that model rankings based on AAPR don't always align with their SAT performance due to overconfidence or miscalibration, where predicted confidence levels don't match actual accuracy Guo et al. (2017); Melotti et al. (2022); Müller et al. (2019); Mukhoti et al. (2020). While it is feasible to minimize AAPR on the miscalibrated outputs, it can lead to suboptimal decision-making. By employing well-calibrated Log-Likelihood Ratios (LLRs), FIRMBOUND effectively minimizes risk, achieving Pareto-optimality at a given sampling cost. This calibration ensures that the model's confidence levels are more aligned with the true probabilities, thereby enhancing the reliability of the decision boundaries used in the stopping rule.
+
+What is the tackled problem? Finite horizon Early Classification of Time Series (ECTS) aims to minimize the decision time  $\tau < T$  while maintaining a desired error rate  $\alpha$ , where  $T$  is the maximum possible time step. As mentioned in Sec. 2, SPRT optimally solves ECTS under an infinite horizon, detailed in App. B. For finite horizons (including infinite as a special case, 1. 108), the average a posteriori risk (AAPR) must be defined and minimized via backward induction to find the optimal stopping boundary (Sec. 3).
+
+Why does computing the backward induction equation yield minimal AAPR? The minimal AAPR is defined as the expected optimal risk at the initial time step, computed recursively from the finite horizon back to the start using the backward equation. For a detailed explanation, see (Tartakovsky et al., 2014).
+
+Why is FIRMBOUND necessary, given the availability of simulation studies? FIRMBOUND accommodates a broad range of time series data, including i.i.d., non-i.i.d., binary, and multiclass series. In contrast, many existing simulation studies focus on artificial datasets with limited classes and do not reflect real-world complexities. Additionally, optimal threshold searching through numerical simulation might require intensive grid sampling, which becomes computationally impractical with large classes—for example, assigning unique posterior probabilities to 101 classes with 0.1 steps could result in up to 47 trillion combinations. FIRMBOUND facilitates the application of backward induction to real-world datasets without restrictions related to the data distribution, thereby extending its applicability to time-sensitive tasks.
+
+Why are some experimental results not state-of-the-art (SOTA)? As discussed in the Experiments and Results section, we do not claim to achieve state-of-the-art results. For example, the pretrained feature extractor ResNet50 was not fine-tuned for HMDB51 and UCF101 datasets. Our focus is on conducting a fair comparison rather than achieving the highest performance, as reaching state-of-the-art would not alter the conclusions of our study.
+
+Why don't you include a classification penalty in APR? While it is feasible to incorporate a classification penalty into the APR to potentially reduce errors, we aim to maintain a simple APR definition as stated in Sec. 3.2. Adding a classification penalty is redundant since the estimation of sufficient statistics inherently addresses error reduction. We utilize a density ratio estimation (DRE) algorithm to ensure statistically consistent estimation of LLRs, thereby reducing errors without the need for an additional penalty term in the APR.
+
+# G LAGRANGIAN FUNCTION FOR CONVEX FUNCTION LEARNING
+
+In this section, we review the 2-block ADMM algorithm (Siahkamari et al., 2022) used for solving the convex regression problem (Eq. 12). We solve the following noisy convex regression problem with regularization:
+
+$$
+\hat {f} \triangleq \arg \min  _ {f} \frac {1}{n} \sum_ {i = 1} ^ {n} \left(y _ {i} - f \left(\boldsymbol {x} _ {i}\right)\right) ^ {2} + \lambda \| f \|, \tag {15}
+$$
+
+where  $\pmb{x}_i \in \mathbb{R}^d$ , and  $\lambda$  is a hyperparameter affecting convergence. Note that regression labels  $y_i$  are noisy; i.e., they have bounded random discrepancies from the true label.
+
+The 2-block ADMM solves this problem by using piecewise linear functions:
+
+$$
+\min  _ {\hat {y} _ {i}, a _ {i}} \frac {1}{n} \sum_ {i = 1} ^ {n} \left(\hat {y} _ {i} - y _ {i}\right) ^ {2} + \lambda \sum_ {l = 1} ^ {d} \max  _ {i = 1} ^ {n} \left| a _ {i, l} \right| \tag {16}
+$$
+
+$$
+\text {s . t .} \hat {y} _ {i} - \hat {y} _ {j} - \left\langle a _ {i}, \boldsymbol {x} _ {i} - \boldsymbol {x} _ {j} \right\rangle \leq 0 \quad i, j \in [ n ] \times [ n ]. \tag {17}
+$$
+
+Then, we estimate  $f(\pmb{x})$  via
+
+$$
+\hat {f} (\boldsymbol {x}) \triangleq \max  _ {i} \left\langle a _ {i}, \boldsymbol {x} - \boldsymbol {x} _ {i} \right\rangle + \hat {y} _ {i}. \tag {18}
+$$
+
+The 2-block ADMM is summarized in Algorithms 1 & 2 and Updates 1-4 below. The algorithm uses the augmented Lagrange method and leverages the decomposition of the optimization problem into two blocks that are updated iteratively, focusing on the primal and dual variables. In Siahkamari et al. (2022), it is proven that the 2-block ADMM converges to the ground truth function  $f$  when  $\mathcal{T} \to \infty$  and  $n \to \infty$  if  $\lambda$  is in an appropriate region (e.g.,  $\bar{\lambda} \geq \frac{3}{\sqrt{2nd}}$  is necessary). The convergence rate is also derived. The algorithm is implemented in Python class ConvexRegressionModel in our code.
+
+# Update 1.
+
+$$
+\boldsymbol {a} _ {i} = \boldsymbol {\lambda} _ {i} \left(\boldsymbol {\theta} _ {i} + \hat {y} _ {i} \boldsymbol {x} _ {i} + \frac {1}{n} \sum_ {k} \hat {y} _ {k} \boldsymbol {x} _ {k}\right), \tag {19}
+$$
+
+where
+
+$$
+\boldsymbol {\lambda} _ {i} \triangleq \left(\boldsymbol {x} _ {i} \boldsymbol {x} _ {i} ^ {\mathcal {T}} + \frac {1}{n} I + \frac {1}{n} \sum_ {j} \boldsymbol {x} _ {j} \boldsymbol {x} _ {j} ^ {\mathcal {T}}\right) ^ {- 1},
+$$
+
+$$
+\boldsymbol {\theta} _ {i} \triangleq \frac {1}{n} \left(\boldsymbol {p} _ {i} ^ {+} - \boldsymbol {p} _ {i} ^ {-} - \boldsymbol {\eta} _ {i} + \sum_ {j} \left(\alpha_ {i, j} + s _ {i, j}\right) \left(\boldsymbol {x} _ {i} - \boldsymbol {x} _ {j}\right)\right).
+$$
+
+# Update 2.
+
+$$
+\hat {\boldsymbol {y}} = \boldsymbol {\Omega} ^ {- 1} \left(\frac {2 \boldsymbol {y}}{n ^ {2} \rho} + \boldsymbol {v} - \boldsymbol {\beta}\right) \tag {20}
+$$
+
+where  $\pmb {y} = [y_1,\dots ,y_n]^\mathcal{T}$ $\hat{\pmb{y}} = [\hat{\pmb{y}}_1,\dots ,\hat{\pmb{y}}_n]^\mathcal{T}$  , and
+
+$$
+\beta_ {i} \triangleq \frac {1}{n} \sum_ {j} \alpha_ {i, j} - \alpha_ {j, i} + s _ {i, j} - s _ {j, i},
+$$
+
+$$
+v _ {i} \triangleq \boldsymbol {x} _ {i} ^ {\mathcal {T}} \boldsymbol {\lambda} _ {i} \theta_ {i} + \boldsymbol {x} _ {i} ^ {\mathcal {T}} \frac {1}{n} \sum_ {j} \boldsymbol {\lambda} _ {j} \theta_ {j} - \frac {1}{n} \sum_ {j} \boldsymbol {x} _ {j} ^ {\mathcal {T}} \boldsymbol {\lambda} _ {j} \theta_ {j},
+$$
+
+$$
+\boldsymbol {\Omega} _ {i, j} \triangleq \left(\frac {2}{n ^ {2} \rho} + 2 - \boldsymbol {x} _ {i} ^ {\mathcal {T}} \boldsymbol {\lambda} _ {i} \boldsymbol {x} _ {i}\right) \mathbf {1} (i = j) - \frac {1}{n} D _ {i, j},
+$$
+
+$$
+D _ {i, j} \triangleq \boldsymbol {x} _ {i} ^ {\mathcal {T}} \left(\boldsymbol {\lambda} _ {i} + \boldsymbol {\lambda} _ {j} + \frac {1}{n} \sum_ {k} \boldsymbol {\lambda} _ {k}\right) \boldsymbol {x} _ {j} - \boldsymbol {x} _ {j} ^ {\mathcal {T}} \boldsymbol {\lambda} _ {j} \boldsymbol {x} _ {j} - \frac {1}{n} \sum_ {k} \boldsymbol {x} _ {k} \boldsymbol {\lambda} _ {k} \boldsymbol {x} _ {j}.
+$$
+
+Algorithm 1 L-update  
+Require:  $\{\gamma_i,c_i\}_{i = 1}^n$  ,and  $\rho /\lambda$    
+1: knot2n,..., knot1  $\leftarrow$  sort{  $\gamma_{i} + c_{i},\gamma_{i} - c_{i}\right]_{i = 1}^{n}$    
+2:  $f\gets \lambda /\rho$    
+3:  $f^{\prime}\gets 0$    
+4: for  $j = 2$  to  $2n$  do   
+5:  $f^{\prime}\gets f^{\prime} + \frac{1}{2}$    
+6:  $f\gets f + f^{\prime}\cdot (knot_{j} - knot_{j - 1})$    
+7: if  $f\leq 0$  then   
+8: return  $(knot_{j} - \frac{f}{f^{\prime}})^{+}$    
+9: end if   
+10: end for   
+11: return  $(knot_{2n} - \frac{f}{n})^{+}$
+
+Algorithm 2 Convex regression  
+Update 3.  
+Update 4.  
+Require:  $\{(x_i,y_i)\}_{i = 1}^n,\rho ,\lambda$  and  $\mathcal{T}$    
+1:  $\hat{y}_i = s_{i,j} = \alpha_{i,j}\gets 0$    
+2:  $L = a_{i} = p_{i} = u_{i} = \eta_{i} = \gamma_{i}\gets 0_{d\times 1}$    
+3: for  $t = 1$  to  $\mathcal{T}$  do   
+4: Update  $\hat{y}$  by Eq.20   
+5: Update  $a_i$  by Eq.19   
+6:  $L_{l}\leftarrow \mathbf{L}_{-}\mathbf{u}\mathbf{p}\mathbf{d}(|\gamma_{i,l},|\eta_{i,l} + a_{i,l}|]_{i\in [n]},\lambda /\rho)$    
+7: Update  $u_{i,l},p_{i,l}^{+},p_{i,l}^{-},s_{i,j}$  by Eq.21   
+8: Update  $\alpha_{i,j},\gamma_{i,l},\eta_{i,l}$  by Eq.22   
+9: end for   
+10: return  $f(\cdot)\triangleq \max_{i = 1}^{n}(\langle a_i,\cdot -x_i\rangle +\hat{y}_i)$
+
+$$
+s _ {i, j} = \left(- \alpha_ {i, j} - \hat {y} _ {i} + \hat {y} _ {j} + \left\langle a _ {i}, x _ {i} - x _ {j} \right\rangle\right) ^ {+}, \tag {21}
+$$
+
+$$
+u _ {i, l} = \left(L _ {l} - \gamma_ {i, l} - | \eta_ {i, l} + a _ {i, l} |\right) ^ {+},
+$$
+
+$$
+p _ {i, l} ^ {+} = \frac {1}{2} \left(L _ {l} - \gamma_ {i, l} - u _ {i, l} + \eta_ {i, l} + a _ {i, l}\right) ^ {+},
+$$
+
+$$
+p _ {i, l} ^ {-} = \frac {1}{2} \left(L _ {l} - \gamma_ {i, l} - u _ {i, l} - \eta_ {i, l} - a _ {i, l}\right) ^ {-}.
+$$
+
+$$
+\alpha_ {i, j} = \alpha_ {i, j} + s _ {i, j} \tag {22}
+$$
+
+$$
+\hat {y} _ {i} - \hat {y} _ {j} - \left\langle \boldsymbol {a} _ {i}, \boldsymbol {x} _ {i} - \boldsymbol {x} _ {j} \right\rangle \quad i, j \in [ n ] \times [ n ]
+$$
+
+$$
+\gamma_ {i, l} = \gamma_ {i, l} + u _ {i, l} + p _ {i, l} ^ {+} + p _ {i, l} ^ {-} - L _ {l} \quad i, l \in [ n ] \times [ d ]
+$$
+
+$$
+\eta_ {i, l} = \eta_ {i, l} + a _ {i, l} - p _ {i, l} ^ {+} + p _ {i, l} ^ {-} \quad i, l \in [ n ] \times [ d ]
+$$
+
+# H STOCHASTIC VARIATIONAL ELBO MAXIMIZATION
+
+The problem of evaluating  $\mathbb{E}[G_{t + 1}^{\min}|\mathcal{S}_t]$  at each time step is formulated below. Given any set of  $M$  values at time  $t\{\mathcal{S}_t(X_m^{(t)})\}_{m = 1}^M$ , we assume that the joint distribution of the random variables  $\{f(\mathcal{S}_t(X_m^{(t)}))\}_{m = 1}^M$  are multivariate Gaussian distributions. Inducing points  $Z = \{z_i\}_{i = 1}^I$  with  $I\ll M$  are randomly sampled from the training dataset  $\{\mathcal{S}_t(X_m^{(t)})\}_{m = 1}^M$ . The prior distribution of the function is defined as:
+
+$$
+\left[ \begin{array}{l} f _ {\mathcal {S}} \\ f _ {Z} \end{array} \right] = \mathcal {N} \left(0 \quad \left[ \begin{array}{l l} K _ {\mathcal {S S}} & K _ {\mathcal {S Z}} \\ K _ {\mathcal {S Z}} ^ {T} & K _ {Z Z} \end{array} \right]\right), \tag {23}
+$$
+
+where  $f_{\mathcal{S}}$  and  $f_{Z}$  are latent functions of and  $z$ , and  $K$  is the covariance matrix defined with the Redial Basis Function (RBF) kernel:  $K = k(s, s') = \sigma^2 \exp \left\{\frac{(s - s')^2}{2l^2}\right\}$ . Note that  $\sigma$  and  $l$  are trainable model parameters.
+
+To approximate the posterior distribution  $p(f_{\mathcal{S}},f_z|G_{t + 1}^{\min})$ , We define a variational distribution  $q(f_{\mathcal{S}},f_z)\coloneqq p(f_{\mathcal{S}}|f_z)q(f_z)$ , where the marginal variational distribution is also defined with a Gaussian:  $q(f_{z}) = \mathcal{N}(f_{z}|\mu ,\Sigma)$ . The observed  $G_{t + 1}^{\mathrm{min}} = \{g_{t + 1,1}^{\mathrm{min}},g_{t + 1,2}^{\mathrm{min}},\ldots ,g_{t + 1,M}^{\mathrm{min}}\}$  are modeled with a Gaussian likelihood that assumes a homoskedastic noise is used:  $p(G_{t + 1}^{\mathrm{min}}|f)\sim N(G_{t + 1}^{\mathrm{min}}|f,\eta^{2}I)$ , we compute the marginal log likelihood and its variational evidence lower bound (ELBO):
+
+$$
+\begin{array}{l} \log (p \left(G _ {t + 1} ^ {\min }\right)) = \log \int \int p \left(G _ {t + 1} ^ {\min } \mid f _ {\mathcal {S}}, f _ {z}\right) p \left(f _ {\mathcal {S}}, f _ {z}\right) d f _ {\mathcal {S}}, d f _ {z} \\ \geq \mathbb {E} _ {q (f _ {\mathcal {S}})} \left[ \log p \left(G _ {t + 1} ^ {\min } \mid f _ {\mathcal {S}}\right) \right] - D _ {K L} \left[ p \left(f _ {z} \right\lvert   | q \left(f _ {z}\right) \right], \tag {24} \\ \end{array}
+$$
+
+where  $D_{KL}$  is the Kullback-Leibler divergence (KLD). The r.h.s. of Eq. 24 is defined as  $\mathcal{L}_{\mathrm{ELBO}}$ . Given that the second term of  $\mathcal{L}_{\mathrm{ELBO}}$  is independent of training data, an empirical approximation of  $\mathcal{L}_{\mathrm{ELBO}}$  for minibatch computation can be found as:
+
+$$
+\mathcal {L} _ {\mathrm {E L B O}} \sim \frac {1}{M ^ {\prime}} \sum_ {i = 1} ^ {M ^ {\prime}} \mathbb {E} _ {q \left(f _ {s i t}\right)} \left[ \log p \left(G _ {t + 1, i} ^ {\min } \mid f _ {s t i}\right) \right] - D _ {K L} \left[ p \left(f _ {z} \mid \mid q \left(f _ {z}\right) \right], \right. \tag {25}
+$$
+
+where  $M^{\prime}\leq M$  is a minibatch size.
+
+After the model training, we can predict the distribution of  $f_{new}$  given a new set of  $\mathcal{S}_{new}$  by computing the predictive distribution:
+
+$$
+p \left(f _ {n e w} \mid G _ {t + 1} ^ {\min }\right) = \int p \left(f _ {n e w} \mid f _ {z}\right) q \left(f _ {z}\right) d f _ {z}. \tag {26}
+$$
+
+Given that  $p(f_{new}, f_z)$  is a multivariate Gaussian distribution, the solution of Eq. 26 is analytical and also a Gaussian. Using  $\mathcal{S}_{new} = \mathcal{S}_t$  as inputs we thus approximate  $\mathbb{E}[G_{t+1}^{\min}|\mathcal{S}_t]$  with the mean of the predictive distribution,  $K_{newZ}K_{ZZ}^{-1}\mu$ .
+
+# I SPRT-TANDEM
+
+SPRT-TANDEM is a sequential DRE algorithm specifically designed for conducting SPRT on real-world sequential datasets (Ebihara et al., 2021). It employs a feature vector extractor followed by a temporal integrator (TI, Fig. 11), utilizing either recurrent networks or transformers as TIs. In our experiments, both LSTM (Hochreiter & Schmidhuber, 1997) and Transformer (Vaswani et al., 2017) are implemented. The TI outputs class posteriors, which are converted to LLRs using the TANDEM formula (Thm. I.1) in a transitive manner. Initially developed for binary-class, SPRT-TANDEM has been adapted for multiclass classification (Miyagawa & Ebihara, 2021), incorporating a statistically consistent LLR estimator, LSEL (Eq. 13). The LLR saturation problem, notably significant when the absolute value of the ground-truth LLR exceeds 100 nats, has also been addressed (Ebihara et al., 2023).
+
+A distinctive feature of SPRT-TANDEM is its absence of a dedicated loss function for promoting earliness, despite its design for ECTS. This is because the precision in estimating the sufficient statistic (i.e., LLR) ensures the minimum required data sampling to achieve a predefined error rate. Thus, SPRT-TANDEM is trained using LSEL (Eq. 13) and multiplet cross-entropy loss (MCE, Def. I.1), without a specific loss function for earliness.
+
+Theorem I.1 (TANDEM formula). Assuming that  $X^{(1,T)}$  are  $N$ -th order Markov series,  $\lambda_{kl}(X^{(1,t)})$  can be approximated as:
+
+$$
+\lambda_ {k l} \left(X ^ {(1, t)}\right) = \sum_ {s = N + 1} ^ {t} \log \frac {\pi_ {k} \left(X ^ {(s - N , s)}\right)}{\pi_ {l} \left(X ^ {(s - N , s)}\right)} - \sum_ {s = N + 2} ^ {t} \log \frac {\pi_ {k} \left(X ^ {(s - N , s - 1)}\right)}{\pi_ {k} \left(X ^ {(s - N , s - 1)}\right)} - \log \chi_ {k l}, \tag {27}
+$$
+
+where  $\chi_{kl} = \log (p(y = k) / p(y = l))$  is a log class prior probability.
+
+Definition I.1 (MCE).
+
+$$
+L _ {\mathrm {M C E}} := \frac {1}{M (T - N)} \sum_ {i = 1} ^ {M} \sum_ {k = 1} ^ {N + 1} \sum_ {t = k} ^ {T - (N + 1 - k)} \left(- \log \pi_ {y _ {i}} \left(X _ {i} ^ {(t, t - k + 1)}\right)\right). \tag {28}
+$$
+
+![](images/a355f9e0f6088b1f8e919490184b1d366951a4a110fc7f4b71688cbd546e4a60.jpg)  
+Figure 11: LLR estimation with SPRT-TANDEM. Feature vectors  $x$ , typically extracted by a feature vector extractor network, are sequentially fed into the temporal integrator (TI) network to output class posterior probabilities,  $\pi \coloneqq (\pi_1, \dots, \pi_K)$ . The TANDEM formula (Eq. I.1, denoted as "TANDEM" in the figure) is used to convert these posteriors to LLRs, enabling an online sequential update of the estimation.
+
+# J FIRMBOUND IS STATISTICALLY CONSISTENT
+
+We provide the full assumptions, the formal statement, and the proof of Thm. 3.2.
+
+# J.1 ASSUMPTIONS
+
+Most of the necessary assumptions are given in the following Lems. J.1-J.4 and Thm. 2.1.
+
+Lemma J.1 (CFL is statistically consistent (Prop. 1 in (Siahkamari et al., 2022))). See App. G for notations. With the appropriate choice of  $\lambda$  which requires knowledge of the bound on  $f$  and  $n \geq d$ , it holds that with probability at least  $1 - \delta$  over the data, the estimator  $\hat{f}$  of Eq. 18 has excess risk upper bounded by
+
+$$
+\mathbb {E} \left[ | f (x) - \hat {f} (x) | ^ {2} \right] \leq O \left(\left(\frac {n}{d}\right) ^ {\frac {- 2}{d + 4}} \log \left(\frac {n}{d}\right) + \sqrt {\frac {\log (1 / \delta)}{n}}\right). \tag {29}
+$$
+
+See (Siahkamari et al., 2022) for the proof. Note that the bound limits to zero as the dataset size  $n$  (denoted by  $M$  in the main text) limits to infinity. Note also that  $d$  used in this lemma corresponds to the number of classes  $K$  in the main text.
+
+Lemma J.2 (CFL converges (Thm. 2 in (Siahkamari et al., 2022))). See App.  $G$  for notations. Let  $\{\hat{y}_i^t, \pmb{a}_i^t\}$  be the output of Alg. 2 at the  $t^{\text{th}}$  iteration,  $\tilde{y}_i := \frac{1}{\mathcal{T}} \sum_{t=1}^{\mathcal{T}} \hat{y}_i^t$  and  $\tilde{\pmb{a}}_i := \frac{1}{\mathcal{T}} \sum_{t=1}^{\mathcal{T}} \pmb{a}_i^t$ . Denote  $\tilde{f}_{\mathcal{T}}(\pmb{x}) := \max_i \langle \tilde{\pmb{a}}_i, \pmb{x} - \pmb{x}_i \rangle + \tilde{y}_i$ . Assume  $\max_{i,l} |x_{i,l}| \leq 1$  and  $\operatorname{Var}(\{y_i\}_{i=1}^n) \leq 1$ . If we choose
+
+$$
+\rho = \frac {\sqrt {d} \lambda^ {2}}{n}, \quad \lambda \geq \frac {3}{\sqrt {2 n d}}, \quad a n d \quad \mathcal {T} \geq n \sqrt {d},
+$$
+
+we have:
+
+$$
+\frac {1}{n} \sum_ {i = 1} ^ {n} \left(\tilde {f} _ {\mathcal {T}} \left(\boldsymbol {x} _ {i}\right) - y _ {i}\right) ^ {2} + \lambda \| \tilde {f} _ {\mathcal {T}} \| \leq \min  _ {\hat {f} \in \mathcal {F}} \left(\frac {1}{n} \sum_ {i = 1} ^ {n} \left(\hat {f} \left(x _ {i}\right) - y _ {i}\right) ^ {2} + \lambda \| \hat {f} \|\right) + \frac {6 n \sqrt {d}}{\mathcal {T} + 1}, \tag {30}
+$$
+
+where  $\mathcal{F} \coloneqq \{f : \mathbb{R}^d \to \mathbb{R} \mid f \text{ is convex}\}$ .
+
+See (Siahkamari et al., 2022) for the proof. The inputs  $\pmb{x}$  and outputs  $y$  in the lemma correspond to  $(\pi_1(X^{(1,t)}),\ldots ,\pi_K(X^{(1,t)}))$ , and to  $\tilde{G}_t(\mathcal{S}_t(X^{(1,t)}))$ , respectively.  $\pi_k(X^{(1,t)})$  for all  $k\in [K]$  are obviously bounded by one, and thus, the assumption  $\max_{i,l}|x_{i,l}|\leq 1$  is satisfied. Also, the assumption  $\mathrm{Var}(\{y_i\}_{i = 1}^n)\leq 1$  is satisfied because we only consider integrable functions, and the continuation risk  $\tilde{G}_t$  is bounded. As a corollary of Lem. J.2, we have:
+
+Lemma J.3 (Convergence rate of CFL (Cor. 1 in (Siahkamari et al., 2022))). See App.  $G$  for notations. The CFL algorithm used in FIRMBOUND, outlined in App.  $G$ , needs  $\frac{6n\sqrt{d}}{\epsilon}$  iterations to achieve  $\epsilon$  error. Each iteration requires  $\mathcal{O}(n^2 d + nd^2)$  flops operations. Preprocessing costs  $\mathcal{O}(nd^3)$ . Therefore the total computational complexity is  $\mathcal{O}\left(\frac{n^3d^{1.5} + n^2d^{2.5} + nd^3}{\epsilon}\right)$ .
+
+Note that  $n$  and  $d$  used in this lemma correspond to  $M$  and  $K$  in the main text. See (Siahkamari et al., 2022) for the proof.
+
+Next, let us define
+
+$$
+L _ {\text {L S E L}} [ \lambda ] := \frac {1}{K T} \sum_ {k \in [ K ]} \sum_ {t \in [ T ]} \int d X ^ {(1, t)} p \left(X ^ {(1, t)} | k\right) \log \left(1 + \sum_ {l (\neq k)} e ^ {- \lambda_ {k l} \left(X ^ {(1, t)}\right)}\right). \tag {31}
+$$
+
+Let  $S \coloneqq \{(X_i^{(1,T)}, y_i)\}_{i=1}^M \sim p(X^{(1,T)}, y)^M$  be a training dataset, where  $M \in \mathbb{N}$  is the sample size. The empirical approximation of Eq. 31 is
+
+$$
+\hat {L} _ {\mathrm {L S E L}} (\boldsymbol {w}; S) := \frac {1}{K T} \sum_ {k \in [ K ]} \sum_ {t \in [ T ]} \frac {1}{M _ {k}} \sum_ {i \in I _ {k}} \log \left(1 + \sum_ {l (\neq k)} e ^ {- \hat {\lambda} _ {k l} \left(X _ {i} ^ {(1, t)}; \boldsymbol {w}\right)}\right). \tag {13}
+$$
+
+$M_{k}$  and  $I_{k}$  denote the sample size and index set of class  $k$ , respectively; i.e.,  $M_{k} = |I_{k}| = |\{i \in [M] | y_{i} = k\}$  and  $\sum_{k} M_{k} = M$ . Let  $L(\boldsymbol{w})$  and  $\hat{L}_S(\boldsymbol{w})$  denote  $L_{\mathrm{LSEL}}[\hat{\lambda}(\cdot; \boldsymbol{w})]$  and  $\hat{L}_{\mathrm{LSEL}}(\boldsymbol{w}; S)$ , respectively. Let  $\hat{\boldsymbol{w}}_S$  be the empirical risk minimizer of  $\hat{L}_S$ ; namely,  $\hat{\boldsymbol{w}}_S \in \arg \min_{\boldsymbol{w}} \hat{L}_S(\boldsymbol{w})$ .
+
+Lemma J.4 (LSEL is statistically consistent (Thm. 3.1 in (Miyagawa & Ebihara, 2021))). Let  $\pmb{W}^{*} \coloneqq \left\{\pmb{w}^{*} \in \mathbb{R}^{d} \mid \hat{\lambda}(X^{(1,t)}; \pmb{w}^{*}) = \lambda(X^{(1,t)}) (\forall t \in [T])\right\}$  be the target parameter set. Assume, for simplicity of the proof, that each  $\pmb{w}^{*}$  is separated in  $\pmb{w}^{*}$ ; i.e.,  $\exists \delta > 0$  such that  $\mathcal{B}(\pmb{w}^{*}; \delta) \cap \mathcal{B}(\pmb{w}^{*'}; \delta) = \emptyset$  for arbitrary  $\pmb{w}^{*}$  and  $\pmb{w}^{*'}$ , where  $\mathcal{B}(\pmb{w}; \delta)$  denotes an open ball at center  $\pmb{w}$  with radius  $\delta$ . Assume the following three conditions:
+
+(a)  $\forall k,l\in [K],\forall t\in [T],p(X^{(1,t)}\mid k) = 0\iff p(X^{(1,t)}\mid l) = 0.$  
+(b)  $\sup_{\boldsymbol{w}} |\hat{L}_S(\boldsymbol{w}) - L(\boldsymbol{w})| \xrightarrow{\mathbb{P}} 0$  as  $M \to \infty$ ; i.e.,  $\hat{L}_S(\boldsymbol{w})$  converges in probability uniformly over  $\boldsymbol{w}$  to  $L(\boldsymbol{w})$ .  
+(c) For all  $\boldsymbol{w}^{*}\in \mathbf{W}^{*}$ , there exist  $t\in [T]$ ,  $k\in [K]$ , and  $l\in [K]$ , such that the following  $d\times d$  matrix is full-rank:
+
+$$
+\int d X ^ {(1, t)} p (X ^ {(1, t)} \mid k) \nabla_ {\boldsymbol {w} ^ {*}} \hat {\lambda} _ {k l} (X ^ {(1, t)}; \boldsymbol {w} ^ {*}) \nabla_ {\boldsymbol {w} ^ {*}} \hat {\lambda} _ {k l} (X ^ {(1, t)}; \boldsymbol {w} ^ {*}) ^ {\top}.
+$$
+
+Then,  $\mathbb{P}(\hat{\boldsymbol{w}}_S \notin W^*) \xrightarrow{M \to \infty} 0$ ; i.e.,  $\hat{\boldsymbol{w}}_S$  converges in probability into  $\boldsymbol{W}^*$ .
+
+See (Miyagawa & Ebihara, 2021) for the proof. Assumption (a) ensures that LLRs  $\lambda(X^{(1,t)}) \coloneqq \{\lambda_{kl}(X^{(1,t)})\}_{k,l \in [K]}$  exists and is finite. Assumption (b) can be satisfied under the standard assumptions of the uniform law of large numbers (compactness, continuity, measurability, and dominance) (Jennrich, 1969; Newey & McFadden, 1986). Assumption (c) is a technical requirement, often assumed in the literature (Gutmann & Hyvarinen, 2012). We additionally assume that the neural network represented by  $w$  is so large that it can represent target LLRs, which can be satisfied according to the universal approximation theorem of neural networks.
+
+# J.2 FORMAL STATEMENT
+
+Now, we provide the formal statement of Thm. 3.2:
+
+Theorem J.1 (FIRMBOUND is statistically consistent). Suppose that all the assumptions mentioned in App. J.1 are satisfied. Suppose that we have the sufficient statistics estimated with LSEL on a dataset with size  $M$ . Suppose also that we have the continuation risk estimated on with CFL the same dataset. Then, with arbitrary precision, we can solve the backward induction equation in Thm. 2.1, which yields the Bayes optimal terminal decision rule  $d^{*}$  and stopping time  $\tau^{*}$ , with high probability over the data and as  $M \to \infty$ .
+
+# J.3 PROOF
+
+We provide the proof of Thm. J.1.
+
+Proof. We first show that the CFL combined with the density ratio estimation (DRE) with LSEL yields a consistent estimate of function  $\tilde{G}_t$ .
+
+Observation 1. According to Lems. J.3 & J.2, for any dataset, the output function of the CFL algorithm can be arbitrarily close to any convex function if  $\mathcal{T}, M \to \infty$  with  $\mathcal{T} > \Omega(M\sqrt{K})$ , where  $\Omega(\cdot)$  here denotes a Landau symbol. Therefore, according to Lem. J.1, with high probability over the data, the output function of the CFL algorithm can be arbitrarily close to any convex function if  $\mathcal{T}, M \to \infty$  with  $\mathcal{T} > \Omega(M\sqrt{K})$ .
+
+Observation 2. According to Lem. J.4, the estimated LLRs  $\hat{\lambda}_{kl}$  can be arbitrarily close to the true LLRs  $\lambda_{kl}$  as  $M\to \infty$
+
+CFL with DRE is consistent. Therefore, according to Observation 1 & 2, with high probability over the data, as  $M \to \infty$ , CFL with DRE can estimate any continuation risk  $\tilde{G}_t$  at any  $\mathcal{S}_t (= (\pi_1, \dots, \pi_K))$  because  $\tilde{G}_t$  is a continuous function of  $\mathcal{S}_t$ . That is, CFL with DRE is a statistically consistent estimator of  $\tilde{G}_t$ .
+
+FIRMOUND is consistent. Using the estimated continuation risk, we can solve the backward induction equation in Thm. 2.1 with arbitrary precision, which yields the Bayes optimal terminal decision rule  $d^{*}$  and stopping time  $\tau^{*}$ , with high probability over the data and as  $M \to \infty$ . This means that FIRMOUND (= CFL + DRE + backward induction) yields a statistically consistent estimator of the Bayes optimal algorithm in the sense of Thm. 2.1, minimizing AAPR.
+
+# K EXPERIMENTAL DETAILS AND SUPPLEMENTARY RESULTS
+
+Throughout the experiments, Optuna (Akiba et al., 2019) with the default algorithm, Tree-structured Parzen Estimator (TPE) (Bergstra et al., 2011), is used to find the best hyperparameter combinations from the predefined search space. TPE is a Bayesian optimization algorithm that models beliefs about the optimal hyperparameters using Parzen Estimation and optimizes the search process using a tree-like graph. The training procedure described below is common across all datasets unless specified otherwise.
+
+# K.1 FIRMBOUND WITH CFL
+
+Our custom code enables hyperparameter tuning at each time step, determining the lambda parameter (not to be confused with LLR  $\lambda$ ; we maintain the original notation from Siahkamari et al. (2022) for consistency) used in the augmented Lagrangian algorithm. The concave conditional expectation is negated for optimizing CFL models. Adam (Kingma & Ba, 2014) is employed as the optimizer.
+
+Tuning. A total of 1000 data points (i.e., posteriors  $\pi$  as the sufficient statistic) are randomly selected from the training dataset. Using Optuna, we search for the optimal lambda at each time step as follows: the initial value of lambda is log-uniformly selected from the range  $[1e - 3, 1e1]$ . A 5-fold cross-validation, consisting of 3 epochs each, is conducted to evaluate the mean squared error between the predictions and observed data points. This tuning trial is repeated 30 times to ensure comprehensive parameter exploration.
+
+Fitting. A subset of 5000 data points is randomly selected from the training dataset. The optimal lambda parameter, identified from the tuning trials, is used to train the final CFL models over 3 epochs on training data, which will be used for future online ECTS. The evaluation of AAPR and SAT curves is conducted 5 times on test data to validate performance.
+
+# K.2 FIRMBOUND WITH GP REGRESSION
+
+Similar to CFL, GP regression models are trained at each time step  $t$ . Adam (Kingma & Ba, 2014) is utilized as the optimizer.
+
+Initialization. A Cholesky Variational Distribution is used to estimate the true posterior, initialized with 200 inducing points (i.e., sufficient statistics, either LLRs or posteriors) that are randomly selected from the training dataset. The GP model is initialized with a constant mean and a covariance module, the latter employing a Radial Basis Function (RBF) kernel. A Gaussian likelihood module is also initialized to evaluate the Evidence Lower Bound (ELBO).
+
+Fitting. The negative variational ELBO is computed and minimized across minibatches of size 2000. After 30 epochs of training on training data, the predictive distribution is evaluated on all sufficient statistics in the training data to assess the conditional expectation. The evaluation of AAPR and SAT curves is repeated 30 times on test data.
+
+Supplementary Results. Fig. 12 shows representative fitting results on the two-class sequential Gaussian dataset.
+
+# K.3 DATASET PREPARATION
+
+Following the methodologies described in Ebihara et al. (2021) and Miyagawa & Ebihara (2021), we prepare feature vectors for the SiW and action recognition datasets UCF101/HMDB51, respectively. All pixel values are divided by 127.5 and then subtracted by 1 before feeding into the feature extractor. For the SiW videos, we use ResNet152 version 2 (He et al., 2016a;b) to produce a 512-dimensional feature vector (trainable parameters: 3.7M). For the UCF101 and HMDB51 videos, we employ the Pretrained Microsoft Vision Model ResNet50, which is used without fine-tuning to extract 2048-dimensional vector elements (trainable parameters: 23.5M). The train/test split for UCF101 and HMDB51 adheres to official splitting pattern #1. A validation set is derived from the training dataset
+
+![](images/956c02c14ed785fbe083391b6f11bd35b44a7fc2c1adf6a0500c4ed950209ac4.jpg)
+
+![](images/7fa09766aeade71c9828734cad7cef0e18f245df1ad9354f46a95434160c6cac.jpg)
+
+![](images/6afaeb16424f44574c8fdf613f2ac5efca0f16610b51c535d90cc21b979dece3.jpg)
+
+![](images/832e110b218ee130af0e0cbc76e3b159f5395ab4ca54eb902c2d40fb0ab75a26.jpg)
+
+![](images/6ed7aa0f19f6f04862e9ef683d0bacd91c3274c569f63e20aa1388fbba0457fc.jpg)
+
+![](images/bc87b630c019367cbc01eb47bdb885e4357f44cabd9e28f802ae3db9e83ac29b.jpg)
+
+![](images/b31d9036845dd8faf993eeab12f43c589803a0ec47983a82b894cd77651fa879.jpg)  
+Figure 12: Typical risk curves estimated with GP regression. Two-class Gaussian distribution dataset is used to generate observed continuation risk  $\tilde{G}$ , on which GP models are trained to provide estimations of the conditional expectation.
+
+![](images/6d0ed5e7b8d51c892410c52ead115d4d1077c2520721df4ceb6c39bfd005b2cc.jpg)
+
+![](images/64760c6e0ca33508c509b4f4073bd195d3fa223390d037433db3e3eb753ea63f.jpg)
+
+while maintaining the original class frequency. All videos are clipped or repeated to standardize the time steps to 50 and 79, respectively.
+
+Table 3: Extracted datasets.  
+
+<table><tr><td>Orig. dataset</td><td>Train set</td><td>Val. set</td><td>Test set</td><td>Feat. dim</td><td>Time steps</td></tr><tr><td>SiW (Liu et al., 2018b)</td><td>46,729</td><td>4,968</td><td>43,878</td><td>512</td><td>50</td></tr><tr><td>HMDB51 (Kuehne et al., 2011)</td><td>1,026</td><td>106</td><td>105</td><td>2,048</td><td>79</td></tr><tr><td>UCF101 (Soomro et al., 2012)</td><td>35,996</td><td>4,454</td><td>15,807</td><td>2,048</td><td>50</td></tr><tr><td>FordA (Soomro et al., 2012)</td><td>6,600</td><td>6,005</td><td>12,000</td><td>24</td><td>20</td></tr></table>
+
+# K.4 TRAINING ECTS MODELS
+
+For real-world datasets lacking ground-truth LLRs, we train the DRE model SPRT-TANDEM (Ebihara et al., 2021) to provide a statistically consistent estimator of LLRs. Additionally, ECTS baseline models, including LSTMMs (Ma et al., 2016), EARLIEST (Hartvigsen et al., 2019), TCNTransformer (Chen et al., 2022), and CALIMERA (Bilski & Jastrzewska, 2023) are trained.
+
+Similar to F I R M B O u N D, we utilize Optuna for hyperparameter optimization. The evaluation criterion is the averaged per-class error rate, or macro-averaged recall. A conservative  $40\%$  percentile pruner is used for early stopping of unpromising parameter combinations. The training settings common across models and databases, along with detailed pruner settings, are provided in Tab. 4. The choice of optimizer includes Adam (Kingma & Ba, 2014), RMSprop (Graves, 2013), and Lion (Chen et al., 2023). An exception is CALIMERA, which are trained with fixed parameters. CALIMERA employs linear and ridge classifiers, which leverage closed-form solutions for parameter estimation, ensuring a deterministic and efficient optimization process that is less prone to the hyperparameter sensitivities often associated with deep learning models.
+
+In subsequent analyses, the coefficient  $\gamma$  demonstrates that batch size and learning rate can be scaled equivalently to maintain consistent training dynamics, as per the linear scaling law (Goyal et al., 2017).
+
+Table 4: Common Hyperparameter Tuning Setup  
+
+<table><tr><td>Number of iterations</td><td>Number of training data * Number of epochs / Batch size</td></tr><tr><td>Pruner type</td><td>40% percentile</td></tr><tr><td>Pruner startup trials</td><td>Number of trials / 2</td></tr><tr><td>Pruner warmup steps</td><td>Number of iterations / 2</td></tr><tr><td>Pruner interval steps</td><td>Number of iterations / Number of epochs</td></tr></table>
+
+# K.4.1 TWO-CLASS GAUSSIAN DATSET
+
+Table 5: SPRT-TANDEM on two-class Gaussian dataset: parameter space.  
+
+<table><tr><td></td><td>Hyperparameter</td><td>Space</td><td>Optimal value</td></tr><tr><td rowspan="4">Fixed parameters</td><td>Batch size</td><td>200 × γ</td><td>N.A. (fixed)</td></tr><tr><td>Epochs</td><td>15 × γ</td><td>N.A. (fixed)</td></tr><tr><td># Tuning trials</td><td>200</td><td>N.A. (fixed)</td></tr><tr><td># Repeated test trials</td><td>60</td><td>N.A. (fixed)</td></tr><tr><td rowspan="8">Searched hyperparameters</td><td>Learning rate</td><td>[10-6, 10-3] × γ</td><td>0.0001</td></tr><tr><td>Weight decay</td><td>[0.0, 10-5]</td><td>0.0005</td></tr><tr><td>Optimizer</td><td>{Adam, RMSprop, Lion}</td><td>Adam</td></tr><tr><td>Order SPRT</td><td>{0, 1, ..., 10}</td><td>5</td></tr><tr><td>MCE weight</td><td>[0.0, 1.0]</td><td>1.0</td></tr><tr><td>LLR estim. loss weight</td><td>[0.0, 1.0]</td><td>0.8</td></tr><tr><td>FC activation</td><td>{B2Bsqrt, tanh, ReLU, GeLU}</td><td>ReLU</td></tr><tr><td>Temporal integrator</td><td>{LSTM, Transformer}</td><td>Transformer</td></tr><tr><td rowspan="5">Backbone-specific parameters</td><td>num blocks</td><td>[1, 3]</td><td>1</td></tr><tr><td>num heads</td><td>[2, 4]</td><td>4</td></tr><tr><td>Dropout</td><td>[0.0, 0.5]</td><td>0.4</td></tr><tr><td>MLP.units</td><td>[32, 64]</td><td>64</td></tr><tr><td>FF_dim</td><td>[32, 64]</td><td>64</td></tr></table>
+
+# K.4.2 THREE-CLASS GAUSSIAN DATASET
+
+Table 6: SPRT-TANDEM on three-class Gaussian dataset: parameter space.  
+
+<table><tr><td></td><td>Hyperparameter</td><td>Space</td><td>Optimal value</td></tr><tr><td rowspan="4">Fixed parameters</td><td>Batch size</td><td>200 × γ</td><td>N.A. (fixed)</td></tr><tr><td>Epochs</td><td>15 × γ</td><td>N.A. (fixed)</td></tr><tr><td># Tuning trials</td><td>200</td><td>N.A. (fixed)</td></tr><tr><td># Repeated test trials</td><td>30</td><td>N.A. (fixed)</td></tr><tr><td rowspan="8">Searched hyperparameters</td><td>Learning rate</td><td>[10-6, 10-3] × γ</td><td>0.0001</td></tr><tr><td>Weight decay</td><td>[0.0, 10-5]</td><td>0.00025</td></tr><tr><td>Optimizer</td><td>{Adam, RMSprop, Lion}</td><td>Lion</td></tr><tr><td>Order SPRT</td><td>{0, 1, ..., 10}</td><td>0</td></tr><tr><td>MCE weight</td><td>[0.0, 1.0]</td><td>0.7</td></tr><tr><td>LLR estim. loss weight</td><td>[0.0, 1.0]</td><td>0.1</td></tr><tr><td>FC activation</td><td>{B2Bsqrt, tanh, ReLU, GeLU}</td><td>ReLU</td></tr><tr><td>Temporal integrator</td><td>{LSTM, Transformer}</td><td>Transformer</td></tr><tr><td rowspan="5">Backbone-specific parameters</td><td>num blocks</td><td>[1, 3]</td><td>1</td></tr><tr><td>num heads</td><td>[2, 4]</td><td>2</td></tr><tr><td>Dropout</td><td>[0.0, 0.5]</td><td>0.3</td></tr><tr><td>MLP.units</td><td>[32, 64]</td><td>64</td></tr><tr><td>FF_dim</td><td>[32, 64]</td><td>32</td></tr></table>
+
+# K.4.3 SIW
+
+Table 7:SPRT-TANDEM on SiW:parameter space.  
+
+<table><tr><td></td><td>Hyperparameter</td><td>Space</td><td>Optimal value</td></tr><tr><td rowspan="4">Fixed parameters</td><td>Batch size</td><td>83 × γ</td><td>N.A. (fixed)</td></tr><tr><td>Epochs</td><td>18</td><td>N.A. (fixed)</td></tr><tr><td># Tuning trials</td><td>200</td><td>N.A. (fixed)</td></tr><tr><td># Repeated test trials</td><td>200</td><td>N.A. (fixed)</td></tr><tr><td rowspan="8">Searched hyperparameters</td><td>Learning rate</td><td>[10-6, 10-3] × γ</td><td>0.0001</td></tr><tr><td>Weight decay</td><td>[0.0, 10-5]</td><td>0.0</td></tr><tr><td>Optimizer</td><td>{Adam, RMSprop, Lion}</td><td>Adam</td></tr><tr><td>Order SPRT</td><td>{0, 1, ..., 10}</td><td>9</td></tr><tr><td>MCE weight</td><td>[0.0, 1.0]</td><td>1.0</td></tr><tr><td>LLR estim. loss weight</td><td>[0.0, 1.0]</td><td>1.0</td></tr><tr><td>FC activation</td><td>{B2Bsqrt, tanh, ReLU, GeLU}</td><td>ReLU</td></tr><tr><td>Temporal integrator</td><td>{LSTM, Transformer}</td><td>LSTM</td></tr><tr><td rowspan="2">Backbone-specific parameters</td><td>LSTM output activation</td><td>{B2Bsqrt, tanh, GeLU}</td><td>B2Bsqrt</td></tr><tr><td>LSTM hidden dim.</td><td>[32, 256]</td><td>256</td></tr></table>
+
+Table 8: LSTMms on SiW: parameter space.  
+
+<table><tr><td></td><td>Hyperparameter</td><td>Space</td><td>Optimal value</td></tr><tr><td rowspan="4">Fixed parameters</td><td>Batch size</td><td>100 × γ</td><td>N.A. (fixed)</td></tr><tr><td>Epochs</td><td>15</td><td>N.A. (fixed)</td></tr><tr><td># Tuning trials</td><td>100</td><td>N.A. (fixed)</td></tr><tr><td># Repeated test trials</td><td>26</td><td>N.A. (fixed)</td></tr><tr><td rowspan="7">Searched hyperparameters</td><td>Learning rate</td><td>[10-6, 10-3] × γ</td><td>0.0011</td></tr><tr><td>Weight decay</td><td>[0.0, 10-5]</td><td>0.001</td></tr><tr><td>Optimizer</td><td>{Adam, RMSprop, Lion}</td><td>Lion</td></tr><tr><td>Cross entropy weight</td><td>[0.0, 1.0]</td><td>1.0</td></tr><tr><td>Loss type</td><td>{LSTMm, LSTMs}</td><td>LSTMs</td></tr><tr><td>Loss weight</td><td>[0.0, 1.0]</td><td>1.0</td></tr><tr><td>LSTM hidden dim.</td><td>[32, 512]</td><td>76</td></tr></table>
+
+Table 9: EARLIEST (lambda=1e-1) on SiW: parameter space.  
+
+<table><tr><td></td><td>Hyperparameter</td><td>Space</td><td>Optimal value</td></tr><tr><td rowspan="5">Fixed parameters</td><td>EARLIEST param. lambda</td><td>1e-1</td><td>N.A. (fixed)</td></tr><tr><td>Batch size</td><td>256 × γ</td><td>N.A. (fixed)</td></tr><tr><td>Epochs</td><td>50</td><td>N.A. (fixed)</td></tr><tr><td># Tuning trials</td><td>200</td><td>N.A. (fixed)</td></tr><tr><td># Repeated test trials</td><td>30</td><td>N.A. (fixed)</td></tr><tr><td rowspan="4">Searched hyperparameters</td><td>Learning rate</td><td>[10-6, 10-3] × γ</td><td>0.000951</td></tr><tr><td>Weight decay</td><td>[0.0, 10-5]</td><td>0.0006</td></tr><tr><td>Optimizer</td><td>{Adam, RMSprop, Lion}</td><td>Lion</td></tr><tr><td>LSTM hidden dim.</td><td>[32, 256]</td><td>16</td></tr></table>
+
+Table 10: EARLIEST (lambda=1e-10) on SiW: parameter space.  
+
+<table><tr><td></td><td>Hyperparameter</td><td>Space</td><td>Optimal value</td></tr><tr><td rowspan="5">Fixed parameters</td><td>EARLIEST param. lambda</td><td>1e-10</td><td>N.A. (fixed)</td></tr><tr><td>Batch size</td><td>256 × γ</td><td>N.A. (fixed)</td></tr><tr><td>Epochs</td><td>50</td><td>N.A. (fixed)</td></tr><tr><td># Tuning trials</td><td>200</td><td>N.A. (fixed)</td></tr><tr><td># Repeated test trials</td><td>30</td><td>N.A. (fixed)</td></tr><tr><td rowspan="4">Searched hyperparameters</td><td>Learning rate</td><td>[10-6, 10-3] × γ</td><td>0.000441</td></tr><tr><td>Weight decay</td><td>[0.0, 10-5]</td><td>0.001</td></tr><tr><td>Optimizer</td><td>{Adam, RMSprop, Lion}</td><td>RMSprop</td></tr><tr><td>LSTM hidden dim.</td><td>[32, 256]</td><td>16</td></tr></table>
+
+Table 11: TCNT (alpha=0.3) on SiW: parameter space.  
+
+<table><tr><td></td><td>Hyperparameter</td><td>Space</td><td>Optimal value</td></tr><tr><td rowspan="5">Fixed parameters</td><td>TCNT param. alpha</td><td>0.3</td><td>N.A. (fixed)</td></tr><tr><td>Batch size</td><td>256 × γ</td><td>N.A. (fixed)</td></tr><tr><td>Epochs</td><td>20</td><td>N.A. (fixed)</td></tr><tr><td># Tuning trials</td><td>100</td><td>N.A. (fixed)</td></tr><tr><td># Repeated test trials</td><td>30</td><td>N.A. (fixed)</td></tr><tr><td rowspan="7">Searched hyperparameters</td><td>Learning rate</td><td>[10-6, 10-3] × γ</td><td>0.00425</td></tr><tr><td>Weight decay</td><td>[0.0, 10-5]</td><td>0.0003</td></tr><tr><td>Dropout</td><td>[0.0, 0.5]</td><td>0.3</td></tr><tr><td>Optimizer</td><td>{Adam, RMSprop, Lion}</td><td>Lion</td></tr><tr><td># Blocks</td><td>[1, 3]</td><td>1</td></tr><tr><td># Num heads</td><td>[2, 4]</td><td>4</td></tr><tr><td>TCN channels</td><td>[256, 1024]</td><td>256</td></tr></table>
+
+Table 12: TCNT (alpha=0.5) on SiW: parameter space.  
+
+<table><tr><td></td><td>Hyperparameter</td><td>Space</td><td>Optimal value</td></tr><tr><td rowspan="5">Fixed parameters</td><td>TCNT param. alpha</td><td>0.5</td><td>N.A. (fixed)</td></tr><tr><td>Batch size</td><td>256 × γ</td><td>N.A. (fixed)</td></tr><tr><td>Epochs</td><td>20</td><td>N.A. (fixed)</td></tr><tr><td># Tuning trials</td><td>100</td><td>N.A. (fixed)</td></tr><tr><td># Repeated test trials</td><td>30</td><td>N.A. (fixed)</td></tr><tr><td rowspan="7">Searched hyperparameters</td><td>Learning rate</td><td>[10-6, 10-3] ×γ</td><td>0.000002</td></tr><tr><td>Weight decay</td><td>[0.0, 10-5]</td><td>0.000</td></tr><tr><td>Dropout</td><td>[0.0, 0.5]</td><td>0.4</td></tr><tr><td>Optimizer</td><td>{Adam, RMSprop, Lion}</td><td>Lion</td></tr><tr><td># Blocks</td><td>[1, 3]</td><td>1</td></tr><tr><td># Num heads</td><td>[2, 4]</td><td>4</td></tr><tr><td>TCN channels</td><td>[256, 512]</td><td>32</td></tr></table>
+
+Table 13: CALIMERA on SiW: parameter space.  
+
+<table><tr><td></td><td>Hyperparameter</td><td>Space</td><td>Optimal value</td></tr><tr><td rowspan="2">Fixed parameters</td><td>Delay penalty</td><td>{0.1, 0.5, 1.0}</td><td>N.A. (fixed)</td></tr><tr><td># Repeated test trials</td><td>5</td><td>N.A. (fixed)</td></tr></table>
+
+# K.4.4 HMDB51
+
+Table 14:SPRT-TANDEM on HMDB51:parameter space.  
+
+<table><tr><td></td><td>Hyperparameter</td><td>Space</td><td>Optimal value</td></tr><tr><td rowspan="4">Fixed parameters</td><td>Batch size</td><td>128 × γ</td><td>N.A. (fixed)</td></tr><tr><td>Epochs</td><td>24</td><td>N.A. (fixed)</td></tr><tr><td># Tuning trials</td><td>200</td><td>N.A. (fixed)</td></tr><tr><td># Repeated test trials</td><td>5</td><td>N.A. (fixed)</td></tr><tr><td rowspan="9">Searched hyperparameters</td><td>Learning rate</td><td>[10-6, 10-3] × γ</td><td>10-4</td></tr><tr><td>Weight decay</td><td>[0.0, 10-5]</td><td>10-4</td></tr><tr><td>Optimizer</td><td>{Adam, RMSprop, Lion}</td><td>Adam</td></tr><tr><td>Order SPRT</td><td>{0, 1, ..., 10}</td><td>4</td></tr><tr><td>MCE weight</td><td>[0.0, 1.0]</td><td>0.1</td></tr><tr><td>LLR estim. loss type</td><td>{LLLRR, LSEL}</td><td>LSEL</td></tr><tr><td>LLR estim. loss weight</td><td>[0.0, 1.0]</td><td>1.0</td></tr><tr><td>FC activation</td><td>{B2Bsqrt, tanh, ReLU, GeLU}</td><td>tanh</td></tr><tr><td>Temporal integrator</td><td>{LSTM, Transformer}</td><td>LSTM</td></tr><tr><td rowspan="2">Backbone-specific parameters</td><td>LSTM output activation</td><td>{B2Bsqrt, tanh, GeLU}</td><td>B2Bsqrt</td></tr><tr><td>LSTM hidden dim.</td><td>[32, 256]</td><td>256</td></tr></table>
+
+Table 15: LSTMms on HMDB: parameter space.  
+
+<table><tr><td></td><td>Hyperparameter</td><td>Space</td><td>Optimal value</td></tr><tr><td rowspan="4">Fixed parameters</td><td>Batch size</td><td>100 × γ</td><td>N.A. (fixed)</td></tr><tr><td>Epochs</td><td>15</td><td>N.A. (fixed)</td></tr><tr><td># Tuning trials</td><td>100</td><td>N.A. (fixed)</td></tr><tr><td># Repeated test trials</td><td>20</td><td>N.A. (fixed)</td></tr><tr><td rowspan="7">Searched hyperparameters</td><td>Learning rate</td><td>[10-6, 10-3] × γ</td><td>0.000594</td></tr><tr><td>Weight decay</td><td>[0.0, 10-5]</td><td>0.0009</td></tr><tr><td>Optimizer</td><td>{Adam, RMSprop, Lion}</td><td>Lion</td></tr><tr><td>Cross entropy weight</td><td>[0.0, 1.0]</td><td>0.7</td></tr><tr><td>Loss type</td><td>{LSTMm, LSTMs}</td><td>LSTMm</td></tr><tr><td>Loss weight</td><td>[0.0, 1.0]</td><td>0.3</td></tr><tr><td>LSTM hidden dim.</td><td>[32, 512]</td><td>282</td></tr></table>
+
+Table 16: EARLIEST (lambda=1e-1) on HMDB51: parameter space.  
+
+<table><tr><td></td><td>Hyperparameter</td><td>Space</td><td>Optimal value</td></tr><tr><td rowspan="5">Fixed parameters</td><td>EARLIEST param. lambda</td><td>1e-1</td><td>N.A. (fixed)</td></tr><tr><td>Batch size</td><td>256 × γ</td><td>N.A. (fixed)</td></tr><tr><td>Epochs</td><td>50</td><td>N.A. (fixed)</td></tr><tr><td># Tuning trials</td><td>200</td><td>N.A. (fixed)</td></tr><tr><td># Repeated test trials</td><td>30</td><td>N.A. (fixed)</td></tr><tr><td rowspan="4">Searched hyperparameters</td><td>Learning rate</td><td>[10-6, 10-3] × γ</td><td>0.000273</td></tr><tr><td>Weight decay</td><td>[0.0, 10-5]</td><td>0.0009</td></tr><tr><td>Optimizer</td><td>{Adam, RMSprop, Lion}</td><td>RMSprop</td></tr><tr><td>LSTM hidden dim.</td><td>[32, 256]</td><td>159</td></tr></table>
+
+Table 17: EARLIEST (lambda=1e-10) on HMDB51: parameter space.  
+
+<table><tr><td></td><td>Hyperparameter</td><td>Space</td><td>Optimal value</td></tr><tr><td rowspan="5">Fixed parameters</td><td>EARLIEST param. lambda</td><td>1e-10</td><td>N.A. (fixed)</td></tr><tr><td>Batch size</td><td>256 × γ</td><td>N.A. (fixed)</td></tr><tr><td>Epochs</td><td>50</td><td>N.A. (fixed)</td></tr><tr><td># Tuning trials</td><td>200</td><td>N.A. (fixed)</td></tr><tr><td># Repeated test trials</td><td>30</td><td>N.A. (fixed)</td></tr><tr><td rowspan="4">Searched hyperparameters</td><td>Learning rate</td><td>[10-6, 10-3] × γ</td><td>0.000148</td></tr><tr><td>Weight decay</td><td>[0.0, 10-5]</td><td>0.000</td></tr><tr><td>Optimizer</td><td>{Adam, RMSprop, Lion}</td><td>Lion</td></tr><tr><td>LSTM hidden dim.</td><td>[32, 256]</td><td>147</td></tr></table>
+
+Table 18: TCNT (alpha=0.3) on HMDB51: parameter space.  
+
+<table><tr><td></td><td>Hyperparameter</td><td>Space</td><td>Optimal value</td></tr><tr><td rowspan="5">Fixed parameters</td><td>TCNT param. alpha</td><td>0.3</td><td>N.A. (fixed)</td></tr><tr><td>Batch size</td><td>256 × γ</td><td>N.A. (fixed)</td></tr><tr><td>Epochs</td><td>15</td><td>N.A. (fixed)</td></tr><tr><td># Tuning trials</td><td>300</td><td>N.A. (fixed)</td></tr><tr><td># Repeated test trials</td><td>10</td><td>N.A. (fixed)</td></tr><tr><td rowspan="7">Searched hyperparameters</td><td>Learning rate</td><td>[10-6, 10-3] × γ</td><td>0.000776</td></tr><tr><td>Weight decay</td><td>[0.0, 10-5]</td><td>0.000</td></tr><tr><td>Dropout</td><td>[0.0, 0.5]</td><td>0.1</td></tr><tr><td>Optimizer</td><td>{Adam, RMSprop, Lion}</td><td>Adam</td></tr><tr><td># Blocks</td><td>[1, 3]</td><td>1</td></tr><tr><td># Num heads</td><td>[2, 4]</td><td>2</td></tr><tr><td>TCN channels</td><td>[256, 1024]</td><td>1024</td></tr></table>
+
+Table 19: TCNT (alpha=0.5) on HMDB51: parameter space.  
+
+<table><tr><td></td><td>Hyperparameter</td><td>Space</td><td>Optimal value</td></tr><tr><td rowspan="5">Fixed parameters</td><td>TCNT param. alpha</td><td>0.5</td><td>N.A. (fixed)</td></tr><tr><td>Batch size</td><td>256 × γ</td><td>N.A. (fixed)</td></tr><tr><td>Epochs</td><td>15</td><td>N.A. (fixed)</td></tr><tr><td># Tuning trials</td><td>300</td><td>N.A. (fixed)</td></tr><tr><td># Repeated test trials</td><td>10</td><td>N.A. (fixed)</td></tr><tr><td rowspan="7">Searched hyperparameters</td><td>Learning rate</td><td>[10-6, 10-3] ×γ</td><td>0.000453</td></tr><tr><td>Weight decay</td><td>[0.0, 10-5]</td><td>0.010</td></tr><tr><td>Dropout</td><td>[0.0, 0.5]</td><td>0.4</td></tr><tr><td>Optimizer</td><td>{Adam, RMSprop, Lion}</td><td>Adam</td></tr><tr><td># Blocks</td><td>[1, 3]</td><td>1</td></tr><tr><td># Num heads</td><td>[2, 4]</td><td>4</td></tr><tr><td>TCN channels</td><td>[256, 1024]</td><td>1024</td></tr></table>
+
+Table 20: CALIMERA on HMDB51: parameter space.  
+
+<table><tr><td></td><td>Hyperparameter</td><td>Space</td><td>Optimal value</td></tr><tr><td rowspan="2">Fixed parameters</td><td>Delay penalty</td><td>{0.1, 0.5, 1.0}</td><td>N.A. (fixed)</td></tr><tr><td># Repeated test trials</td><td>5</td><td>N.A. (fixed)</td></tr></table>
+
+# K.4.5 UCF101
+
+Table 21: SPRT-TANDEM on UCF101: parameter space.  
+
+<table><tr><td></td><td>Hyperparameter</td><td>Space</td><td>Optimal value</td></tr><tr><td rowspan="4">Fixed parameters</td><td>Batch size</td><td>16 × γ</td><td>N.A. (fixed)</td></tr><tr><td>Epochs</td><td>25</td><td>N.A. (fixed)</td></tr><tr><td># Tuning trials</td><td>200</td><td>N.A. (fixed)</td></tr><tr><td># Repeated test trials</td><td>14</td><td>N.A. (fixed)</td></tr><tr><td rowspan="9">Searched hyperparameters</td><td>Learning rate</td><td>[10-6, 10-3] × γ</td><td>0.000027</td></tr><tr><td>Weight decay</td><td>[0.0, 10-5]</td><td>0.0006</td></tr><tr><td>Optimizer</td><td>{Adam, RMSprop, Lion}</td><td>Lion</td></tr><tr><td>Order SPRT</td><td>{0, 1, ..., 10}</td><td>6</td></tr><tr><td>MCE weight</td><td>[0.0, 1.0]</td><td>0.2</td></tr><tr><td>LLR estim. loss type</td><td>{LSEL, LLLR}</td><td>LSEL</td></tr><tr><td>LLR estim. loss weight</td><td>[0.0, 1.0]</td><td>0.4</td></tr><tr><td>FC activation</td><td>{B2Bsqrt, tanh, ReLU, GeLU}</td><td>tanh</td></tr><tr><td>Temporal integrator</td><td>{LSTM, Transformer}</td><td>Transformer</td></tr><tr><td rowspan="5">Backbone-specific parameters</td><td># Blocks</td><td>[1, 3]</td><td>1</td></tr><tr><td># Heads</td><td>[2, 4]</td><td>4</td></tr><tr><td>Dropout</td><td>[0.0, 0.5]</td><td>0.1</td></tr><tr><td>MLP.units</td><td>[256, 416]</td><td>288</td></tr><tr><td>FF_dim</td><td>[256, 416]</td><td>256</td></tr></table>
+
+Table 22: LSTMms on UCF101: parameter space.  
+
+<table><tr><td></td><td>Hyperparameter</td><td>Space</td><td>Optimal value</td></tr><tr><td rowspan="4">Fixed parameters</td><td>Batch size</td><td>100 × γ</td><td>N.A. (fixed)</td></tr><tr><td>Epochs</td><td>15</td><td>N.A. (fixed)</td></tr><tr><td># Tuning trials</td><td>100</td><td>N.A. (fixed)</td></tr><tr><td># Repeated test trials</td><td>11</td><td>N.A. (fixed)</td></tr><tr><td rowspan="7">Searched hyperparameters</td><td>Learning rate</td><td>[10-6, 10-3] × γ</td><td>0.000184</td></tr><tr><td>Weight decay</td><td>[0.0, 10-5]</td><td>0.008</td></tr><tr><td>Optimizer</td><td>{Adam, RMSprop, Lion}</td><td>RMSprop</td></tr><tr><td>Cross entropy weight</td><td>[0.0, 1.0]</td><td>0.5</td></tr><tr><td>Loss type</td><td>{LSTMm, LSTMs}</td><td>LSTMs</td></tr><tr><td>Loss weight</td><td>[0.0, 1.0]</td><td>0.4</td></tr><tr><td>LSTM hidden dim.</td><td>[32, 512]</td><td>362</td></tr></table>
+
+Table 23: EARLIEST (lambda=1e-1) on UCF101: parameter space.  
+
+<table><tr><td></td><td>Hyperparameter</td><td>Space</td><td>Optimal value</td></tr><tr><td rowspan="5">Fixed parameters</td><td>EARLIEST param. lambda</td><td>1e-1</td><td>N.A. (fixed)</td></tr><tr><td>Batch size</td><td>256 × γ</td><td>N.A. (fixed)</td></tr><tr><td>Epochs</td><td>50</td><td>N.A. (fixed)</td></tr><tr><td># Tuning trials</td><td>200</td><td>N.A. (fixed)</td></tr><tr><td># Repeated test trials</td><td>30</td><td>N.A. (fixed)</td></tr><tr><td rowspan="4">Searched hyperparameters</td><td>Learning rate</td><td>[10-6, 10-3] × γ</td><td>0.000026</td></tr><tr><td>Weight decay</td><td>[0.0, 10-5]</td><td>0.0005</td></tr><tr><td>Optimizer</td><td>{Adam, RMSprop, Lion}</td><td>Lion</td></tr><tr><td>LSTM hidden dim.</td><td>[32, 256]</td><td>238</td></tr></table>
+
+Table 24: EARLIEST (lambda=1e-10) on UCF101: parameter space.  
+
+<table><tr><td></td><td>Hyperparameter</td><td>Space</td><td>Optimal value</td></tr><tr><td rowspan="5">Fixed parameters</td><td>EARLIEST param. lambda</td><td>1e-10</td><td>N.A. (fixed)</td></tr><tr><td>Batch size</td><td>256 × γ</td><td>N.A. (fixed)</td></tr><tr><td>Epochs</td><td>50</td><td>N.A. (fixed)</td></tr><tr><td># Tuning trials</td><td>200</td><td>N.A. (fixed)</td></tr><tr><td># Repeated test trials</td><td>30</td><td>N.A. (fixed)</td></tr><tr><td rowspan="4">Searched hyperparameters</td><td>Learning rate</td><td>[10-6, 10-3] × γ</td><td>0.000758</td></tr><tr><td>Weight decay</td><td>[0.0, 10-5]</td><td>0.0006</td></tr><tr><td>Optimizer</td><td>{Adam, RMSprop, Lion}</td><td>RMSprop</td></tr><tr><td>LSTM hidden dim.</td><td>[32, 256]</td><td>196</td></tr></table>
+
+Table 25: TCNT (alpha=0.3) on UCF101: parameter space.  
+
+<table><tr><td></td><td>Hyperparameter</td><td>Space</td><td>Optimal value</td></tr><tr><td rowspan="5">Fixed parameters</td><td>TCNT param. alpha</td><td>0.3</td><td>N.A. (fixed)</td></tr><tr><td>Batch size</td><td>256 × γ</td><td>N.A. (fixed)</td></tr><tr><td>Epochs</td><td>15</td><td>N.A. (fixed)</td></tr><tr><td># Tuning trials</td><td>300</td><td>N.A. (fixed)</td></tr><tr><td># Repeated test trials</td><td>15</td><td>N.A. (fixed)</td></tr><tr><td rowspan="7">Searched hyperparameters</td><td>Learning rate</td><td>[10-6, 10-3] × γ</td><td>0.000585</td></tr><tr><td>Weight decay</td><td>[0.0, 10-5]</td><td>0.001</td></tr><tr><td>Dropout</td><td>[0.0, 0.5]</td><td>0.1</td></tr><tr><td>Optimizer</td><td>{Adam, RMSprop, Lion}</td><td>Adam</td></tr><tr><td># Blocks</td><td>[1, 3]</td><td>1</td></tr><tr><td># Num heads</td><td>[2, 4]</td><td>4</td></tr><tr><td>TCN channels</td><td>[256, 1024]</td><td>512</td></tr></table>
+
+Table 26: TCNT (alpha=0.5) on UCF101: parameter space.  
+
+<table><tr><td></td><td>Hyperparameter</td><td>Space</td><td>Optimal value</td></tr><tr><td rowspan="5">Fixed parameters</td><td>TCNT param. alpha</td><td>0.5</td><td>N.A. (fixed)</td></tr><tr><td>Batch size</td><td>256 × γ</td><td>N.A. (fixed)</td></tr><tr><td>Epochs</td><td>15</td><td>N.A. (fixed)</td></tr><tr><td># Tuning trials</td><td>300</td><td>N.A. (fixed)</td></tr><tr><td># Repeated test trials</td><td>15</td><td>N.A. (fixed)</td></tr><tr><td rowspan="7">Searched hyperparameters</td><td>Learning rate</td><td>[10-6, 10-3] ×γ</td><td>0.001106</td></tr><tr><td>Weight decay</td><td>[0.0, 10-5]</td><td>0.001</td></tr><tr><td>Dropout</td><td>[0.0, 0.5]</td><td>0.2</td></tr><tr><td>Optimizer</td><td>{Adam, RMSprop, Lion}</td><td>Adam</td></tr><tr><td># Blocks</td><td>[1, 3]</td><td>1</td></tr><tr><td># Heads</td><td>[2, 4]</td><td>4</td></tr><tr><td>TCN channels</td><td>[256, 1024]</td><td>512</td></tr></table>
+
+Table 27: CALIMERA on UCF: parameter space.  
+
+<table><tr><td></td><td>Hyperparameter</td><td>Space</td><td>Optimal value</td></tr><tr><td rowspan="2">Fixed parameters</td><td>Delay penalty</td><td>{0.1, 0.5, 1.0}</td><td>N.A. (fixed)</td></tr><tr><td># Repeated test trials</td><td>5</td><td>N.A. (fixed)</td></tr></table>
+
+# K.5 COMPUTING INFRASTRUCTURE
+
+All experiments were carried out using custom Python scripts on NVIDIA GeForce RTX 2080 Ti graphics cards. For mathematical computations, Numpy (Harris et al., 2020) and Scipy (Virtanen et al., 2020) were employed. Machine learning frameworks used include PyTorch 2.0.0 (Paszke et al., 2019) and TensorFlow 2.8.0 (Abadi et al., 2015). Gaussian process regression was performed using stochastic variational inference in GPyTorch (Gardner et al., 2018).
+
+# L ON HYPERPARAMETER SENSITIVITY OF FIRMBOUND
+
+Our algorithm either requires minimal hyperparameter tuning or can be easily tuned on the training dataset. Below is a list of major hyperparameters for the two approaches:
+
+# Convex Function Learning (CFL)
+
+- Lambda  
+Number of training data for tuning  
+Number of training data for fitting  
+Tuning trials  
+- Epochs
+
+# Gaussian Processes (GP)
+
+- Kernel type  
+Number of inducing points  
+- Batch size  
+- Learning rate  
+- Optimizer  
+- Epochs
+
+The most critical hyperparameter is the lambda parameter (do not confuse with the LLR or the baseline model EARLIEST's hyperparameter) in CFL, which controls the flexibility of the fitting curves. As described in the Sec. 4, we keep the other hyperparameter settings consistent across all datasets, including i.i.d., non-i.i.d., artificial, and real-world. Our experiments show that FIRMBOUND reliably minimizes the average a posteriori risk (AAPR) to delineate the Pareto front.
+
+As an additional experiment for hyperparameter sensitivity, we tested GP approach with varying kernel and number of inducing points using the two-class Gaussian dataset (Fig. 13). The number of inducing points is varied from the default 200 to 50 and 1000, while keeping the original Radial Basis Function (RBF) kernel. Alternatively, the number of inducing points is fixed at 200, and the Matérn kernel is used instead of RBF. Matérn kernel is a generalization of RBF kernel with a parameter  $\nu$  controls its smoothness. As  $\nu$  approaches to infinity, Matérn kernel converges to the RBF kernel. In Fig. 13, two values, 1.5 and 2.5, are used as the parameter  $\nu$ . Cost parameter is  $c = \bar{L} / T$ . The results are robust against any of the above hyperparameters, while Matérn kernel slightly off from the optimality (also see Fig. 5a and 6a of the main manuscript).
+
+![](images/2d69eae71a12005d0c980974fdbfba7d952efe7c5f053195004438c6ff9a499f.jpg)  
+Figure 13: GP's hyperparameter sensitivity test on the two-class Gaussian dataset. The number of inducing points is varied from 50 to 1000, with the Radial Basis Function (RBF) kernel, or fixed at 200 using the Matérn kernel with  $\nu$  values of 1.5 and 2.5. Note that  $\nu \rightarrow \infty$  converges to the RBF kernel.  $c = \bar{L} /T$ .
+
+# M AAPR ON BASELINE MODELS
+
+In this section, we demonstrate that ill-calibrated ECTS models can misleadingly exhibit small AAPRs. LSTMms, EARLIEST, and TCN-Transformer models were trained on the two-class Gaussian dataset and evaluated using two performance criteria: AAPR and the SAT curve. As shown in Fig 14a, while LSTMms achieves a lower AAPR than FIRMBOUND and SPRT-TANDEM, which maintain well-calibrated statistics, it is outperformed by them in terms of the SAT curve (i.e., ECTS results). This discrepancy arises from overconfidence, which inflates the statistic beyond the calibrated level. Consequently, AAPR alone does not reliably predict SAT performance when using an ill-calibrated statistic. Notably, FIRMBOUND records the minimal AAPR across all models.
+
+![](images/7179721e2d15b5e1d8a28ed4918eee238e1d6299df3f78f3ad89d75ae258ed0f.jpg)  
+Figure 14: AAPR and SAT curve with other ECTS algorithms. ECTS algorithms, LSTMms, EARLIEST, and TCN-Transformer are used to compare the two evaluation criteria on the two-class sequential Gaussian dataset.
+
+![](images/eb59df6c8ac25e66496d34e50e43af4ec3fd79c688decc0e8025faac455578e5.jpg)
+
+# N PARAMETER SPACE OF  $L$  AND  $c$
+
+Here, we prove that the possible parameter search space of coefficients  $\bar{L}_k = L$  (for all  $k\in [K]$ ) and  $c$  of APR is confined, and thus, we only need to consider a ratio of  $L$  and  $c$ . First, given that the continuation risk  $\tilde{G}_t(\pi)$  is concave and  $\tilde{G}_t(\pi = 0) = \tilde{G}_t(\pi = 1) = c$ , the maximum value of the stopping risk  $G_{t}^{\mathrm{st}}(\pi)$  needs to be larger than  $c$  in order to have more than one intersection (i.e., threshold):
+
+$$
+\begin{array}{l} \max  \left\{G _ {t} ^ {\mathrm {s t}} (\pi) \right\} = L \left(1 - \frac {1}{K}\right) \\ > c, \tag {32} \\ \end{array}
+$$
+
+where  $K$  is the number of classes.
+
+Second, the intersections of the two risk functions  $\tilde{G}_t(\pi)$  and  $G_t^{\mathrm{st}}(\pi)$  remain invariant under the scaling transformation of  $L$  and  $c$  by a factor  $\alpha \in \mathbb{R}_{\geq 0}$ . Specifically, at  $t = T$ ,
+
+$$
+\begin{array}{l} G _ {T} ^ {\mathrm {s t}} (\pi ; \alpha L) = \min  _ {k} \left\{\alpha L \left(1 - \pi_ {k} \left(X _ {m} ^ {(1, T)}\right)\right) \right\} \\ = \alpha \min _ {k} \left\{L \left(1 - \pi_ {k} \left(X _ {m} ^ {(1, T)}\right)\right) \right\} \\ = \alpha G _ {T} ^ {\mathrm {s t}} (\pi ; L, c) \tag {33} \\ \end{array}
+$$
+
+thus,
+
+$$
+\begin{array}{l} G _ {T} ^ {\mathrm {m i n}} (\pi ; \alpha L, \alpha c) = G _ {T} ^ {\mathrm {s t}} (\pi ; \alpha L) \\ = \alpha G _ {T} ^ {\min } (\pi ; L, c) \tag {34} \\ \end{array}
+$$
+
+then at  $t = T - 1$
+
+$$
+\begin{array}{l} \tilde {G} _ {T - 1} (\pi ; \alpha L, \alpha c) = \mathbb {E} \left[ G _ {T} ^ {\min } (\pi ; \alpha L, \alpha c) \mid \pi_ {k} \left(X _ {m} ^ {(1, T)}\right) \right] + \alpha c \\ = \mathbb {E} \left[ \alpha G _ {T} ^ {\min } (\pi ; L, c) \mid \pi_ {k} \left(X _ {m} ^ {(1, T)}\right) \right] + \alpha c \\ = \alpha \left(\mathbb {E} \left[ G _ {T} ^ {\min } (\pi ; L, c) \mid \pi_ {k} \left(X _ {m} ^ {(1, T)}\right) \right] + c\right) \\ = \alpha \tilde {G} _ {T - 1} (\pi ; L, c) \tag {35} \\ \end{array}
+$$
+
+$$
+\begin{array}{l} G _ {T - 1} ^ {\min } (\pi ; \alpha L, \alpha c) = \min  \left\{G _ {T - 1} ^ {\mathrm {s t}} (\pi ; \alpha L), \tilde {G} _ {T - 1} (\pi ; \alpha L, \alpha c) \right\} \\ = \min  \left\{G _ {T - 1} ^ {\alpha \mathrm {s t}} (\pi ; L), \alpha \tilde {G} _ {T - 1} (\pi ; L, c) \right\} \\ = \alpha G _ {T - 1} ^ {\min } (\pi ; L, c) \tag {36} \\ \end{array}
+$$
+
+holds true for a scaling factor  $\alpha \in \mathbb{R}$ . By induction, the above linearity holds true for general  $t \leq T$ . Given that the threshold is defined as the intersection of  $G^{\mathrm{st}}$  and  $\tilde{G}$ , scaling  $L$  and  $c$  by a constant  $\alpha$  does not alter the threshold.
+
+To equalize the magnitudes of the terms in APR (Eq. 3), we set  $c_{\mathrm{def}} = L / T$  as the default in our experiments. This choice consistently balances the speed-accuracy tradeoff and minimizes APR, as elaborated in Sec. 4.
+
+# O ABLATION STUDY DETAILS
+
+As stated in the main manuscript, vanilla SPRT with static threshold, whether applied to true LLRs or estimated LLRs, is crucial for our ablation studies. Figures 5 and 6 demonstrate that SPRT with a static threshold can lead to either a larger APR or a suboptimal speed-accuracy tradeoff. In this supplementary section on ablation studies, we detail the other two conditions tested: random stopping times and artificial tapering thresholds.
+
+Random stopping times. To establish a chance-level baseline, we randomly generate integers of size  $M$  within the range  $[1,\dots ,T]$  to use as stopping times. This experiment is repeated five times, with the computed AAPR and SAT points plotted in Fig. 7. These points typically fall in the middle of the figures, delineating the chance levels.
+
+Artificial tapering thresholds. Optimal stopping theory suggests that the optimal threshold computed with backward induction typically descends monotonically as it approaches the finite horizon (Tartakovsky et al., 2014). This insight motivates us to create artificial decision thresholds as economical alternatives. The following power function with  $\kappa \in -1.5, 0, 1.5$  is used to generate concave, linear, and convex curves, respectively (Fig. 15):
+
+$$
+f (t; A, T, \kappa) = A \left(1 - \frac {t}{T}\right) ^ {e ^ {\kappa}} \tag {37}
+$$
+
+Resulting AAPR and SAT are plotted in Fig. 7. The magnitude  $A$  is set to  $a, 2/a, 0$  where  $a = \max_{m} \{ \lambda(X_m^{(1,T)}) \}$ , whose result corresponding to the three points in Fig. 7.
+
+![](images/b0d0490ac8adb09fe58ed58921f2c396280bd6e3fbf6ee761daea034dd73fba6.jpg)  
+Figure 15: Tapering thresholds generated with the power function. According to Eq. 37, concave, linear, and convex tapering threshold are generated with  $\kappa \in \{-1.5,0,1.5\}$ , respectively.
+
+# P DUMPED OSCILLATING LOG-LIKELIHOOD RATIO FUNCTION
+
+To simulate non-i.i.d., non-monotonic LLR trajectories, we generate binary class LLRs according to Eq. 38:
+
+$$
+\Lambda (t) = \gamma \left(1 - \left(1 - \frac {t}{T}\right) ^ {\exp (\kappa)}\right) + A \exp (- \beta t) \sin (\omega t) + \mathcal {N} (0, \sigma), \tag {38}
+$$
+
+where  $\gamma \in \{-1,1\}$  corresponds to class targets that the trajectories asymptotically approach.  $A$ $\beta$  , and  $\omega$  denote the oscillation amplitude, damping coefficient, and angular frequency of the wave, respectively.  $\sigma$  indicates the noise level. Example trajectories are depicted in Fig. 16.
+
+The parameters and their respective prior distributions used in this study are detailed in Tab. 28. The notation  $\mathcal{N}(\mu, \sigma)$  represents a Gaussian (normal) distribution with mean  $\mu$  and standard deviation  $\sigma$ . The notation  $\mathcal{U}(a, b)$  denotes a uniform distribution sampled within the interval  $[a, b]$ .
+
+Table 28: Parameter space of DOL dataset.  
+
+<table><tr><td>Parameter</td><td>Distribution</td></tr><tr><td>A</td><td>Gaussian N(μ = 2, σ = 2)</td></tr><tr><td>β</td><td>Uniform U(0.02, 0.2)</td></tr><tr><td>ω</td><td>Uniform U(-2, 3)</td></tr><tr><td>κ</td><td>Uniform U(-2.5, 0)</td></tr><tr><td>σ</td><td>Gaussian N(μ = 0.0, σ = 1.0)</td></tr></table>
+
+![](images/49c82df949b78a7fb64fcfc552477665a6ceb3d8b75e5d9f27d7f082a0bbab19.jpg)  
+Figure 16: Tapering thresholds generated with the power function (Eq. 38). Note that the trajectories are generated at a higher sampling rate just for visualization purposes. In the experiment, we sample points at each time step  $t \in 1, \dots, T$ .
+
+# Q SUPPLEMENTARY EXPERIMENT ON THE UCR FORDA DATASET
+
+To test FIRMBOUND's risk minimization capability on continuous signals, we conduct additional experiments on the UCR FordA dataset. FordA is a time series binary classification dataset with 500 samples. Each time series is sliced into non-overlapping segments of 100 time steps. Then each 100-step segment was further processed using a sliding window approach with a window size of 24 and a stride of 4. This resulted in multiple windows per segment:
+
+- The number of windows  $W$  generated from each segment is calculated as:
+
+$$
+W = 1 + \left\lfloor \frac {1 0 0 - 2 4}{4} \right\rfloor = 2 0
+$$
+
+- Therefore, each 100-step segment was transformed into 20 windows, each of length 24.
+
+The resulting data are reshaped into a 3-dimensional array with dimensions  $(M\times T\times 24)$ , where  $M$  is the number of original time series (6,600 and 18,005 for training and test dataset, respectively),  $T = 20$  is time steps, or the number of windows per segment, and 24 is the feature dimension, or the window length.
+
+The result shows that FIRMBOUND effectively find minima of AAPR to optimize the speed-accuracy tradeoff, as shown in Fig. 17.
+
+![](images/25c0b287bda0206dde5ce5dd42e3801c52048635ce3889c184a90596ade2d0ec.jpg)  
+Figure 17: AAPR and SAT curves on UCR FordA dataset. FIRMBOUND effectively find minima of AAPR to optimize the speed-accuracy tradeoff.
+
+![](images/0448dc51bade121aa13100de906db4fedbc1a1611c5951329699f55a8035ce31.jpg)
+
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+# LEARNING TO ADAPT FROZEN CLIP FOR FEW-SHOT TEST-TIME DOMAIN ADAPTATION
+
+Zhixiang Chi $^{1}$ , Li Gu $^{2}$ , Huan Liu $^{3}$ , Ziqiang Wang $^{2}$ , Yanan Wu $^{2\dagger}$ , Yang Wang $^{2\dagger}$ , Konstantinos N Plataniotis $^{1\dagger}$
+
+<sup>1</sup> University of Toronto, <sup>2</sup> Concordia University, <sup>3</sup> McMaster University
+
+$\boxed{ \begin{array}{r l} \end{array} }$  zhixiang.chi@mail.utoronto.ca
+
+# ABSTRACT
+
+Few-shot Test-Time Domain Adaptation focuses on adapting a model at test time to a specific domain using only a few unlabeled examples, addressing domain shift. Prior methods leverage CLIP's strong out-of-distribution (OOD) abilities by generating domain-specific prompts to guide its generalized, frozen features. However, since downstream datasets are not explicitly seen by CLIP, solely depending on the feature space knowledge is constrained by CLIP's prior knowledge. Notably, when using a less robust backbone like ViT-B/16, performance significantly drops on challenging real-world benchmarks. Departing from the state-of-the-art of inheriting the intrinsic OOD capability of CLIP, this work introduces learning directly on the input space to complement the dataset-specific knowledge for frozen CLIP. Specifically, an independent side branch is attached in parallel with CLIP and enforced to learn exclusive knowledge via revert attention. To better capture the dataset-specific label semantics for downstream adaptation, we propose to enhance the inter-dispersion among text features via greedy text ensemble and refinement. The text and visual features are then progressively fused in a domain-aware manner by a generated domain prompt to adapt toward a specific domain. Extensive experiments show our method's superiority on 5 large-scale benchmarks (WILDS and DomainNet), notably improving over smaller networks like ViT-B/16 with gains of +5.1 in F1 for iWildCam and +3.1% in WC Acc for FMoW. Our Code: L2C
+
+# 1 INTRODUCTION
+
+Deep models excel when test and training data distributions align, but real-world scenarios often involve domain shifts (Gulrajani & Lopez-Paz, 2020; Taori et al., 2020; Su et al., 2024b; Yang et al., 2024a; 2023), leading to performance drop. Few-shot Test-Time Domain Adaptation (FSTT-DA) (Chi et al., 2024; Zhong et al., 2022) addresses this by introducing a test-time learning phase to adapt generic models to unseen target domains using a few unlabeled samples. It faces several challenges: i) limited domain-specific information due to a few unlabeled data (unseen target domains), ii) one-time adaptation for each target domain, iii) strict source-free environment during test-time on unseen target domains, and iv) handling diverse target domains with varying complexities and domain shifts.
+
+Therefore, developing an adaptive learning system using source domain data is crucial (Ahmed et al., 2021), as it embodies dataset-specific knowledge—including labels, semantics, and domains. MetaDMoE (Zhong et al., 2022) adapts to unseen target domains by querying relevant knowledge from source expert models and then updating an adaptive student model through knowledge distillation (Ye et al., 2024; Yang & Ye, 2024; Hamidi et al., 2024). MABN (Wu et al., 2024b) learns source distributions during offline training and pinpoints domain-specific parameters for updates during test-time. However, both MetaDMoE and MABN involve model fine-tuning, which can compromise the inherent OOD generalization of vision foundation models like CLIP (Wortsman et al., 2022b).
+
+VDPG (Chi et al., 2024) leverages CLIP's inherent OOD capabilities and robustness (Zhang et al., 2023) by operating solely on its visual features. It compacts source domain knowledge into a learnable knowledge bank. A generator then creates domain-specific prompts from this bank, conditioned on
+
+![](images/afa7ba638080e0a1429cb22d9017341a396c5c14da872311778fb4ab62a3ed78.jpg)  
+(a) High-level comparison.
+
+![](images/7a6928004521fb6527ef80a0821c158ba745ca70adfbfa2794abf4f645fbfd39.jpg)  
+(b) Comparison on ViT-B/16 and ViT-L/14.
+
+![](images/b7d4dc8392cf30005ba1810b7cb59e64340c95044e600a0bfa04831ac3a62e32.jpg)  
+Figure 1: VDPG preserves CLIP's OOD capabilities by operating in the feature space and relying on pretrained CLIP. However, with a weaker backbone like ViT-B/16, performance drops on benchmarks like iWildCam and FMoW. Learning directly from the image space complements CLIP, improving ViT-B/16 as shown in (b).
+
+the features of unlabeled data, to steer the frozen CLIP features toward the target domain. While effective in generating diverse prompts across domains, VDPG has notable drawbacks: its ability to produce domain-specific prompts and utilize source knowledge is limited by CLIP's general, non-dataset-specific knowledge. As shown in Fig. 1b, with a much less robust backbone, ViT-B/16, VDPG suffers significant performance drop. Moreover, VDPG overlooks class semantic cues (Yoon et al., 2024), which are vital for downstream datasets but not addressed by its vision-only encoder.
+
+In this work, we adopt VDPG's black-box approach to retain CLIP's OOD capabilities and aim to improve it by adapting both frozen image and text features. On the image side, we attach a parallel module named CPNet to learn directly from the input space and ComPlement the frozen visual feature of CLIP at its output. It contradicts the methods that involve feature interactions among intermediate layers (Yin et al., 2023; Xu et al., 2023; Chi et al., 2018), and allows CPNet to remain lightweight and applicable for black-box settings. CPNet is encouraged via revert attention to focus on learning only the necessary dataset-specific information, both semantic and domain, that may be absent from CLIP's generalized knowledge. Fig. 1a demonstrates high-level comparison with VDPG.
+
+Noting the significant diversity in image features across domains, even for the same group of classes, we deduce that text features must also adapt accordingly. To benefit downstream adaptation, we aim to enhance the discrimination among text features (termed inter-dispersion) (Cho et al., 2023a), reducing domain bias prior to adaptation. We propose a greedy text ensemble strategy to select prompt templates that improve this discrimination, combined with a lightweight refinement module that uses an inter-dispersion loss to further enhance class differentiation. Importantly, because the greedy ensemble is executed as a pre-processing step, the CLIP text encoder can be discarded when training starts, minimizing its impact on overall training costs (less than  $0.01\%$  of total cost).
+
+To adapt the complemented visual and enhanced text features towards the unseen target domain, we take advantage of CPNet which extracts the unique domain knowledge that CLIP may exclude. Specifically, we reshape the batched unlabeled data so that the inter-attention is computed among the batch instances. Within the same domain, the domain information is typically consistent across data instances (Zhong et al., 2022; Chi et al., 2024). It allows us to treat the propagated batch information as domain-specific knowledge. We then integrate it with a learnable domain cache to form a domain-specific prompt. This prompt guides the fusion of text and image features, enhancing the coherence of domain-specific outputs, and thus adapting to a particular target domain.
+
+We name our framework as Learning to Complement (L2C) and our contributions are: 1) We propose a parallel CPNet to learn dataset-specific knowledge to complement the generalized frozen CLIP visual feature; 2) We propose effortless greedy ensemble and lightweight refinement to enhance the class-wise inter-dispersion for text features to benefit adaptation; 3) We improve the domain knowledge extraction process to adapt both text and visual features in a domain-aware manner; 4) We evaluate L2C on 5 benchmarks, especially on challenging real-world WILDS dataset with smaller backbones (i.e., +5.1 in F1 for iWildCam and +3.1% in WC Acc for FMoW with ViT-B/16).
+
+# 2 RELATED WORK
+
+Distribution shifts often degrade the learning-based methods (Zhang et al., 2021a). To address this, Domain Generalization (DG) (Zhou et al., 2020; Lv et al., 2022) and Unsupervised Domain Adaptation (UDA) (Zhang, 2021; Peng et al., 2019; Pei et al., 2018) have been explored. DG extracts
+
+domain-invariant features for multiple domains (Li et al., 2018; Long et al., 2018), but a single model often falls short. UDA adapts source knowledge to unlabeled target data through extensive target-specific training, but its scale and resource demands limit practicality. PØDA (Fahes et al., 2023) and ULDA (Yang et al., 2024b) achieve zero-shot adaptation by leveraging natural language descriptions of target domains without accessing data. In contrast, FSTT-DA uses domain cues from target domain images, making it suitable for scenarios where descriptions or labels are unavailable.
+
+Test-time adaptation (TTA) (Su et al., 2022; 2024c;a) is an emerging learning paradigm that incorporates an additional learning phase at test time before inference, to mitigate distribution shifts. This phase often utilizes unsupervised objectives like entropy minimization (Wang et al., 2021; Niu et al., 2022; Zhang et al., 2022a; Gong et al., 2022; Zhao et al., 2023; Wang et al., 2024), teacher-student self-training (Yuan et al., 2023; Marsden et al., 2022; 2023; Liu et al., 2022a; Zhang et al., 2024), auxiliary tasks (Sun et al., 2020; Liu et al., 2023; Chi et al., 2021; Liu et al., 2022b), and contrastive learning (Chen et al., 2022a; Wu et al., 2023; Liang et al., 2022) for supervision. Although effective, these approaches often require model fine-tuning or a complex design of learnable parameters. It challenges their scalability and intrinsic OOD capabilities in larger foundation models (Wortsman et al., 2022a). Recent developments include the use of vision prompts (Han et al., 2023), which adapt by modifying only a minimal number of parameters to leverage the existing knowledge within large models (Zhang et al., 2021b; Gan et al., 2023). HybridPrompt (Wu et al., 2024a) and ProD (Wu et al., 2024a) introduce prompt-based algorithms to extract domain-specific knowledge to address domain shifts in cross-domain few-shot learning (CD-FSL) where labelled support set is availale (Guo et al., 2020; Wang & Deng, 2021; Fu et al., 2023). However, these prompts are inserted into various layers and require access to the weights of the base model. Therefore, they incur additional computational costs and pose challenges in scenarios where privacy concerns or proprietary models limit flexibility (An et al., 2022). Our work introduces a practical, gradient-free adaptation method, enabling model deployment in black-box environments.
+
+Few-shot test-time domain adaptation (FSTT-DA). FSTT-DA utilizes a few unlabeled data samples for domain adaptation, providing a practical edge over instance-level methods. MetaDMoE (Zhong et al., 2022) separately trains a pool of domain-specific experts, a process that creates boundaries in knowledge transfer among source domains. MABN (Wu et al., 2024b) focuses on identifying and updating domain-specific parameters via a self-supervised auxiliary branch. It makes its effectiveness dependent on the auxiliary task. VDPG (Chi et al., 2024) harnesses the inherent OOD generalization capabilities of VFMs (Zhang et al., 2023) to create a domain prompt generator that aligns VFM features to specific domains, yet it is limited by a lack of dataset-specific knowledge. In contrast, our method directly learns from the input space, effectively integrating dataset-specific knowledge with the robust OOD capabilities of foundation models.
+
+Efficient tuning with side network. Recent trends favor employing a smaller, parallel side network over inserting learnable parameters into the main backbone. This approach has proven effective in dense prediction (Chen et al., 2022b; Xu et al., 2023) and recognition tasks (Fu et al., 2024; Wang et al., 2023; Sung et al., 2022). However, these methods typically require accessing or modifying the main backbone's intermediate features for efficient adaptation. Our proposed framework diverges by integrating a revert attention mechanism that learns dataset-specific knowledge, aiming to enhance the output of pre-trained foundation models without intervening in their internal processes.
+
+# 3 PRELIMINARIES
+
+Problem setting. In this study, we address Few-Shot Test-time Domain Adaptation (FSTT-DA) (Zhong et al., 2022; Wu et al., 2024b; Chi et al., 2024). In this setting, a model is trained on  $N$  labeled source domains  $\mathcal{D}_s = \{\mathcal{D}_s^n = (x_s, y_s)^n\}_{n=1}^N$ , and then is tested on  $M$  target domains with only input images:  $\mathcal{D}_t = \{\mathcal{D}_t^m = (x_t)^m\}_{m=1}^M$ . We assume distribution shifts occur between any source and target domain pairs and they share the same label space  $\mathcal{Y}_s = \mathcal{Y}_t$ . At test-time, for each target domain  $\mathcal{D}_t^m$ , a few-shot of unlabeled data  $\mathbf{x}$  is used to adapt the model which is used for inference on  $\mathcal{D}_t^m$ . This adaptation stage is source-free, as source data are not used post-training. Appendix B depicts the setting of FSTT-DA.
+
+Motivations. Foundation models like CLIP, trained on web-scale datasets (Oquab et al., 2023; Radford et al., 2021), have markedly improved downstream tasks (Cho et al., 2023b; Goyal et al., 2023). However, it remains a significant challenge to adapt these models to unseen domains using
+
+![](images/f167dcfcdf5a6e320121edd70b4e69dfdf79b0badd294460389652d3bd19817b.jpg)  
+Figure 2: Training process of L2C on source domains. (Top) For a dataset, our greedy strategy selects text prompts with larger inter-dispersion which will be refined subsequently ( $\tilde{\mathbf{T}}^{gre}$ ). (Bottom) CPNet is proposed in parallel with CLIP image encoder to learn dataset-specific knowledge to complement the generalized knowledge in CLIP. To adapt to a domain, a few unlabeled data samples (support set  $S$ ) are used to first generate a domain prompt  $\mathbf{DP}$  via CPNet and domain cache.  $\mathbf{DP}$  is then used to adapt all the data (query set  $\mathcal{Q}$  with image feature:  $\tilde{\mathbf{I}}(\mathbf{x}_{\mathcal{Q}}^{in})$  and text feature:  $\tilde{\mathbf{T}}^{gre}$ ) in that domain via domain-aware fusion (DAF).
+
+minimal unlabeled data as in FSTT-DA. A key approach (VDPG) has been proposed to harness their inherent OOD generalization capabilities (Chi et al., 2024). This involves using domain-specific prompts based on a few data features to adapt CLIP's broad features to particular domains. Nevertheless, CLIP has not specifically seen the downstream datasets. The method of deriving domain-specific knowledge strictly from a generalized feature space has inherent limitations. Consequently, VDPG's reliance on CLIP's pre-trained knowledge restricts its performance. As shown in Fig. 1b and Table 1, using a weaker model like ViT-B/16 yields poor results on the challenging WILDS benchmarks. These shortcomings have motivated us to develop an efficient framework that learns directly from the input space. Our approach not only taps into dataset-specific knowledge including semantics and distribution/domain cues to complement generalized CLIP features, but also leverages text features to enrich label semantics, thus significantly enhancing adaptation capability.
+
+# 4 METHOD
+
+Overview. We aim to adapt both image and text features to unseen domains, as illustrated in Fig. 2. In Sec.4.1, we introduce CPNet to acquire dataset-specific knowledge, complementing CLIP's visual features. Sec.4.2 covers our greedy ensemble approach and lightweight refinement for enhancing text features. In Sec.4.3, we first demonstrate the generation of domain-specific prompts and then adapt the features using domain-aware fusion. Sec.4.4 outlines the training and inference process.
+
+# 4.1 LEARNING DATASET-SPECIFIC VISUAL KNOWLEDGE TO COMPLEMENT CLIP
+
+Parallel CPNet. Freezing CLIP is effective in retaining its OOD capability (Wortsman et al., 2022b). We propose an independent CPNet in parallel with the CLIP image encoder to learn dataset-specific knowledge to complement CLIP. We use boldface  $\mathbf{x}$  to refer to a batch of images and use  $x$  to indicate one image. Given an image  $x\in \mathbb{R}^{1\times H\times W\times C}$ , it is first split into  $l$  patches and encoded into embeddings with dimension  $d$ . A class token [CLS] is pre-pended to form the input  $(in)$  tokens as  $x^{in}\in \mathbb{R}^{1\times (l + 1)\times d}$ . Let  $\mathbf{I}$  represent the CLIP image encoder, we denote its output as  $\mathbf{I}(x^{in})\in \mathbb{R}^{1\times (l + 1)\times d}$ . We impose minimal architectural constraints on CPNet but only match its output dimension with that of the CLIP encoder, thus we express CPNet as  $\mathbf{CP}(x^{in})\in \mathbb{R}^{1\times (l + 1)\times d}$ .
+
+Given that CLIP has mastered extensive generalized knowledge, it is strategically beneficial for CPNet to only acquire necessary dataset-specific semantic and domain information not encompassed by CLIP. Consequently, we introduce a parameter-free Revert Attention (RT) mechanism (Chen et al., 2018) to specifically target the learning of CPNet. We employ the Scaled Dot-Product Attention method (Vaswani et al., 2017) to calculate the attention between their outputs and then compute its complement with respect to  $\mathbf{1}$ . The resulting reverted attention map  $\mathbf{A}$  is reapplied to  $\mathbf{CP}(x^{in})$  using a dot product:
+
+$$
+\mathbf {C P} ^ {R T} \left(x ^ {i n}\right) = \mathbf {A} \cdot \mathbf {C P} \left(x ^ {i n}\right), \quad \text {w h e r e} \mathbf {A} = \mathbf {1} - \operatorname {s o f t m a x} \left(\mathbf {C P} \left(x ^ {i n}\right) \cdot \mathbf {I} \left(x ^ {i n}\right)\right), \tag {1}
+$$
+
+where  $\cdot$  represents the dot-product. This approach ensures that CPNet is focused solely on learning information distinctive from CLIP, rendering efficiency and compactness (e.g., requiring only 3 transformer blocks to complement 12-layer ViT-B/16 on the DomainNet dataset). The dataset-specific information is added back to complement the CLIP visual feature:  $\tilde{\mathbf{I}}(x^{in}) = \mathbf{I}(x^{in}) + \mathbf{C}\mathbf{P}^{RT}(x^{in})$ .
+
+# 4.2 ENHANCING THE LEARNING ON DATASET-SPECIFIC LABEL SEMANTICS
+
+For classification with CLIP, text features act as class prototypes, generated by pairing class names
+
+with a template (e.g., A photo of a [CLASS]). We freeze CLIP to leverage its OOD generalization, so the same set of text features is shared across domains. In Fig. 3, we calculate the average difference in image embeddings for two selected class pairs across 6 domains in DomainNet. The significant semantic variation, even within the same class pairs, highlights the need for domain-specific text features. Since CLIP relies on the unified space of text and image, we focus on adapting text features alongside image features.
+
+To facilitate the adaptation process, we propose reducing domain biases in the text features by increasing their discrimination (inter-dispersion). This ensures that text features remain neutral across all domains before adaptation. To achieve this, we introduce a greedy text ensemble strategy as a preprocessing step, selecting text templates that enhance
+
+![](images/09c426b46412128a3de07ccc25019ba3c06ef0fe5c84e124c476640eae1e1520.jpg)  
+Figure 3: Image embedding differences are calculated as 1 minus cosine similarity for the displayed classes and domains, highlighting that semantic variations differ notably across domains, which requires customized text features for each domain.
+
+inter-dispersion. Let  $\mathbf{T}(\mathbf{P}_{\mathbf{C}})$  represent the text features from a prompt template  $\mathrm{P}$  and class labels  $\mathbf{C}$  using text encoder  $\mathbf{T}$ . The inter-dispersion of text features is quantified by their uniformity in a hypersphere (Cho et al., 2023a; Wang & Isola, 2020), described as:
+
+$$
+\mathcal {L} _ {u n i} \left(\mathbf {T} \left(\mathrm {P} _ {\mathbf {C}}\right)\right) = \sum_ {i, j \in | \mathbf {C} |, i \neq j} \exp (- t \| \mathbf {T} _ {i} \left(\mathrm {P} _ {\mathbf {C}}\right) - \mathbf {T} _ {j} \left(\mathrm {P} _ {\mathbf {C}}\right) \| _ {2} ^ {2}), \tag {2}
+$$
+
+where  $i, j$  represent  $i^{th}$  or  $j^{th}$  class,  $t = 2$  by default. Assuming  $P$  prompt templates, we sort them by increasing  $\mathcal{L}_{uni}$  values:  $\{\mathbf{T}(\mathbf{P}_{\mathbf{C}}^p)\}_{p=1}^P$ . We begin our ensemble with the most uniform embedding,  $\mathbf{T}(\mathbf{P}_{\mathbf{C}}^1)$ , and incrementally add others. The  $p^{th}$  prompt is retained in the ensemble list  $\mathbf{E}$  if it reduces the overall uniformity metric:
+
+$$
+\mathcal {L} _ {u n i} \left(A v e \left(\left[ \mathbf {E}, \mathbf {T} \left(\mathbf {P} _ {\mathrm {C}} ^ {p}\right) \right]\right)\right) <   \mathcal {L} _ {u n i} \left(A v e (\mathbf {E})\right). \tag {3}
+$$
+
+where  $\text{Ave}$  represents averaging ensemble. After selecting prompts greedily, we ensemble  $\mathbf{E}$  into  $\mathbf{T}^{gre} \in \mathbb{R}^{|C| \times d}$  via averaging, a well-informed initialization that encapsulates CLIP's text knowledge. Pseudocode for the greedy ensemble is provided in Appendix E. In Appendix F.2, we show another alternative to quantify inter-dispersion. At this stage, the text encoder  $\mathbf{T}$  can be discarded making the ensemble process effortless, occupying less than 0.01% of the total cost as in Appendix D.2. To further improve the inter-dispersion, we introduce a lightweight module to refine  $\mathbf{T}^{gre}$  when training on the source data:
+
+$$
+\tilde {\mathbf {T}} ^ {g r e} = M _ {c} \mathbf {T} ^ {g r e} M _ {d} + \mathbf {T} ^ {g r e}, \tag {4}
+$$
+
+where  $M_{c}\in \mathbb{R}^{|\mathbf{C}|\times |\mathbf{C}|}$  and  $M_{d}\in \mathbb{R}^{d\times d}$  adjust along label and feature dimensions, respectively. Additionally, we apply the uniformity loss at the output as  $\mathcal{L}_{uni}(\tilde{\mathbf{T}}^{gre})$ .
+
+# 4.3 ADAPTING TO A DOMAIN VIA DOMAIN-AWARE FUSION
+
+Domain prompt computation. We aim to compute a domain-specific prompt to adapt both image  $(\tilde{\mathbf{I}}(x^{in}))$  and text  $(\tilde{\mathbf{T}}^{gre})$  features towards unseen target domains. The source domain knowledge is critical in helping the computation of domain prompt (Chi et al., 2024). We follow the Cache-based learning methods (Zhang et al., 2022b; Zhu et al., 2023) to build a learnable key  $(\mathbf{K} \in \mathbb{R}^{L \times d})$  - value  $(\mathbf{V} \in \mathbb{R}^{L \times d})$  domain cache to store and query such learned source knowledge, where  $L$  is the cache size. Given a batch of  $b$  unlabeled images  $\mathbf{x}$  from a domain in FSTT-DA,  $\mathbf{K}$  is used to compute the similarity between that domain and the source domains, which will be used to query the source knowledge from  $\mathbf{V}$ .
+
+To process  $\mathbf{x}$ , we first transform it into an embedding  $\mathbf{x}^e \in \mathbb{R}^{b \times l \times d}$ . VDPG directly feeds  $\mathbf{x}^e$  into a transformer. Since the attention mechanism operates along the  $l$  dimension, excluding the batch dimension  $b$ , this results in separate attention for each image in  $\mathbf{x}^e$ . This approach is non-intuitive, as domain knowledge should be instance-agnostic. Instead, we propose computing interrelations within the batch (Blattmann et al., 2023) using the dataset-specific CPNet. To achieve this, we reshape  $\mathbf{x}^e$  by combining the first two dimensions into  $\tilde{\mathbf{x}}^e \in \mathbb{R}^{1 \times (b \times l) \times d}$ . This allows the attention mechanism to operate along the  $(b \times l)$  dimension, interleaving all the images in  $\mathbf{x}$ .
+
+Analogous to classification, where a CLS token aggregates global information for an image, we prepend a learnable domain token  $(\mathbb{D})$  so that all information in  $\tilde{\mathbf{x}}^e$  is propagated to  $\mathbb{D}$  through attention (Dosovitskiy et al., 2020). The prepended token  $[\mathbb{D},\tilde{\mathbf{x}}^e ]$  is fed to CP. We then retrieve  $\tilde{\mathbf{D}}$  from  $\mathbf{CP}([\mathbb{D},\tilde{\mathbf{x}}^e ])$  to query the source domain information from  $\mathbf{K - V}$  cache by computing their similarity as softmax  $(\mathbf{K}\tilde{\mathbf{D}}^{T})\cdot \mathbf{V}$ . The domain prompt (DP) is the concatenation of the queried source knowledge and  $\tilde{\mathbb{D}}$  which represents the domain-specific knowledge of that domain:
+
+$$
+\mathbf {D P} \in \mathbb {R} ^ {(L + 1) \times d} = \left[ \operatorname {s o f t m a x} \left(\mathbf {K} \tilde {\mathsf {D}} ^ {T}\right) \cdot \mathbf {V}, \tilde {\mathsf {D}} \right]. \tag {5}
+$$
+
+Domain-aware fusion. Once the domain prompt  $\mathbf{D}\mathbf{P}$  is obtained by Eq. 5 with unlabeled data, we aim to adapt all the data  $x^{in}$  in that domain. To this end, we propose cross-attentions among  $\mathbf{D}\mathbf{P}$ ,  $\tilde{\mathbf{I}}(x^{in})$  and  $\tilde{\mathbf{T}}^{gre}$  to progressively fuse them with a domain-aware fusion module: DAF(DP,  $\tilde{\mathbf{I}}(x^{in})$ ,  $\tilde{\mathbf{T}}^{gre}$ ). Specifically, we first separately project  $\tilde{\mathbf{I}}(x^{in})$  and  $\tilde{\mathbf{T}}^{gre}$  into that domain by conditioning on  $\mathbf{D}\mathbf{P}$  using cross-attention (CA) (Jaegle et al., 2021):
+
+$$
+\mathbf {D P} ^ {\mathbf {T}} = \mathbf {C A} (K = \tilde {\mathbf {T}} ^ {g r e}, V = \tilde {\mathbf {T}} ^ {g r e}, Q = \mathbf {D P}), \tag {6}
+$$
+
+$$
+\mathbf {D P} ^ {\mathbf {I}} = \mathbf {C A} (K = \tilde {\mathbf {I}} (x ^ {i n}), V = \tilde {\mathbf {I}} (x ^ {i n}), Q = \mathbf {D P}),
+$$
+
+Note, that we omit the  $QKV$  weight matrices and the FFN layer for simplicity. Now,  $\mathbf{D}\mathbf{P}^{\mathrm{T}}$  and  $\mathbf{D}\mathbf{P}^{\mathrm{I}}$  contain their modality information in the same domain, we then cross fuse them into other modality:
+
+$$
+\mathbf {T} ^ {d m} = \mathbf {C A} (K = \mathbf {D P} ^ {\mathbf {T}}, V = \mathbf {D P} ^ {\mathbf {T}}, Q = \tilde {\mathbf {I}} (x ^ {i n})), \tag {7}
+$$
+
+$$
+\mathbf {I} ^ {d m} = \mathbf {C A} (K = \mathbf {D P} ^ {\mathbf {I}}, V = \mathbf {D P} ^ {\mathbf {I}}, Q = \tilde {\mathbf {T}} ^ {g r e}),
+$$
+
+where  $dm$  represents domain. We then obtain their class token  $I^{dm}$  from  $\mathbf{I}^{dm}$  and its corresponding text feature  $T^{dm}$  from  $\mathbf{T}^{dm}$  as  $[(I_1^{dm}, T_1^{dm}), \dots, (I_B^{dm}, T_B^{dm})]$ , where  $B$  is the batch size. We finally follow the original CLIP loss (Goyal et al., 2023; Radford et al., 2021) on the adapted text and image features as:
+
+$$
+\mathcal {L} _ {c l i p} = \sum_ {i = 1} ^ {B} - \log \frac {\exp \left(\left(I _ {i} ^ {d m}\right) \cdot \left(T _ {i} ^ {d m}\right)\right)}{\sum_ {j = 1} ^ {B} \exp \left(\left(I _ {i} ^ {d m}\right) \cdot \left(T _ {j} ^ {d m}\right)\right)} + \sum_ {i = 1} ^ {B} - \log \frac {\exp \left(\left(I _ {i} ^ {d m}\right) \cdot \left(T _ {i} ^ {d m}\right)\right)}{\sum_ {j = 1} ^ {B} \exp \left(\left(I _ {j} ^ {d m}\right) \cdot \left(T _ {i} ^ {d m}\right)\right)}. \tag {8}
+$$
+
+The final loss is defined as:  $\mathcal{L}_{total} = \mathcal{L}_{clip} + \lambda \mathcal{L}_{uni}(\tilde{\mathbf{T}}^{gre})$ , where  $\lambda$  balances two losses.
+
+# 4.4 DOMAIN-CENTRIC LEARNING TO ADAPT
+
+Training on source domains. Our ultimate goal is to adapt to unseen target domains using only a few unlabeled data samples. It is essential to align the training objective directly with the evaluation protocol, embodying the system learning to adapt. Therefore, instead of uniformly sampling the data across domains, we follow VDPG to learn at the domain level and mimic the adaptation at test-time as in meta-learning (Finn et al., 2017; Chi et al., 2022; Gu et al., 2022).
+
+Algorithm 1 Domain-centric learning to adapt  
+Require:  $\mathbf{I} / \mathbf{T}$  : CLIP image/text encoders;  $\{\mathrm{P}^p\}_{p = 1}^P$ $P$  text prompt templates; C: C classes with names;  $\mathcal{D}_s$  : source domains;  $\alpha$  : learning rate; CP: CPNet; K/V: K-V domain cache; DAF: domain-aware fusion module;  $M_{c} / M_{d}$  : text refinement;   
+1: //Greedy text feature ensemble   
+2:  $\{\mathbf{T}(\mathbf{P}_{\mathbf{C}}^{p})\}_{p = 1}^{P}$ $\triangleright$  Compute and sort text features for all text prompt templates   
+3: Obtain  $\mathbf{T}^{gre}$  via greedy ensemble using Eq. 3, then discard the text encoder.   
+4: // Learning to complement CLIP and adapt to a particular domain   
+5: for itr=1 to Max_iteration do   
+6:  $(\mathbf{x}_S),(\mathbf{x}_Q,\mathbf{y}_Q)\sim \mathcal{D}_s^n\triangleright$  Sample a source domain and support and query sets   
+7:  $\tilde{\mathbf{x}}_S^e\gets \mathbf{x}_S$ $\tilde{\mathbf{x}}_Q^n\gets \mathbf{x}_Q$ $\triangleright$  Form input embeddings   
+8:  $\tilde{\mathbb{D}}\gets \mathbf{CP}([\tilde{\mathbb{D}},\tilde{\mathbf{x}}_{\tilde{S}}^{e}])$ $\triangleright$  Aggregate domain information from support set   
+9:  $\mathbf{DP} = [\mathrm{softmax}(\mathbf{K}\tilde{\mathbb{D}}^T)\cdot \mathbf{V},\tilde{\mathbb{D}}]$ $\triangleright$  Form a domain prompt for domain  $\mathcal{D}_s^n$    
+10:  $\tilde{\mathbf{I}} (\mathbf{x}_{\mathcal{Q}}^{in})\gets \mathbf{I}(\mathbf{x}_{\mathcal{Q}}^{in}) + \mathbf{C P}^{RT}(\mathbf{x}_{\mathcal{Q}}^{in})\triangleright$  Compute complemented visual feature   
+11:  $\tilde{\mathbf{T}}^{gre}\gets M_c\mathbf{T}^{gre}M_d + \mathbf{T}^{gre},$ $\triangleright$  Refine ensembled text feature   
+12:  $\tilde{\mathbf{T}}_{\mathcal{Q}}^{dm},\tilde{\mathbf{I}}_{\mathcal{Q}}^{dm}\gets \mathrm{DAF}(\mathbf{DP},\tilde{\mathbf{I}} (\mathbf{x}_{\mathcal{Q}}^{in}),\tilde{\mathbf{T}}^{gre})\triangleright$  Adapt query towards domain  $\mathcal{D}_s^n$    
+13: (CP, K/V, DAF,  $M_1 / M_2)\gets (\mathbf{CP},\mathbf{K} / \mathbf{V},\mathrm{DAF},M_1 / M_2) - \alpha \nabla \mathcal{L}_{total}$    
+14: end for
+
+Algo. 1 and Fig. 2 show our training scheme. For each dataset, the text process is only executed once to obtain the ensembled text feature (L2-3). The entire text encoder can be discarded before official training. For each iteration, we consider it as an adaptation task on a randomly sampled source domain  $\mathcal{D}_s^n$ . Two disjoint support set  $(\mathbf{x}_S)$  and query set  $(\mathbf{x}_{\mathcal{Q}}, \mathbf{y}_{\mathcal{Q}})$  are sampled.  $(\mathbf{x}_S)$  is used to generate the domain prompt (L8-9). Then the complemented visual feature is computed for  $\mathbf{x}_{\mathcal{Q}}$  and adapted by the domain prompt (L10-12). The
+
+whole system is evaluated by the loss on the query set (L13).
+
+Adapting to unseen target domain at inference. After iterations of the adaptation task trained on source domains, L2C is ready to adapt to unseen domains. Algo.2 in Appendix C outlines the inference process, which consists of two phases: 1) For each target domain, a few unlabeled data samples are first drawn, and the domain prompt is obtained. Afterward, the K-V cache can be discarded. 2) The domain prompt is then used to adapt every data sample in that domain. Fig. 8a & 8b in Appendix C demonstrate the two phases.
+
+# 5 EXPERIMENTS
+
+Datasets and evaluation. We follow VDPG to evaluate on DomainNet (Peng et al., 2019), which comprises 569K images across 345 classes in 6 domains. We follow the official leave-one-domain-out protocol to train 6 models and report accuracy. We also evaluate on 4 WILDS (Koh et al., 2021) benchmarks, known for their real-world challenges and notably low CLIP zero-shot accuracy (Chi et al., 2024). This includes classification benchmarks such as iWildCam (Beery et al., 2021), Camelyon17 (Bandi et al., 2018), and FMoW (Christie et al., 2018). Although CLIP is primarily designed for classification, we also adapt our framework for regression (PovertyMap (Yeh et al., 2020)), detailed in Appendix F.3. Evaluation metrics include accuracy, Macro F1, worst-case accuracy, Pearson correlation (r), and its worst-case.
+
+Architecture and training details. We use official CLIP pre-trained ViT-B/16 and ViT-L/14 as the foundation models. Their feature dimensions  $(d)$  are 768 and 1024 respectively. Therefore, our CPNet is stacked by regular transformer modules as in ViT with the same feature dimensions. The model is trained for 20 epochs with SGD using cosine decay with initial learning rates of  $2.5e^{-3}$  and  $1e^{-3}$  for WILDS and DomainNet.  $\lambda$  is set to 0.1 to balance the losses. We use 16 images for adaptation at inference. Appendix G&H lists additional hyperparameters and the text prompts.
+
+# 5.1 MAIN RESULTS
+
+Evaluation on WILDS. The WILDS benchmarks reveal complex real-world domain shifts, like wild-camera setups, remote sensing, and medical imaging. It is characterized by significant data imbalances at domain and class levels. CLIP demonstrates notably low zero-shot accuracies in these scenarios. However, as Table 1 indicates, our method substantially exceeds previous approaches. It surpasses VDPG with improvements of 2.1 and 5.1 in Macro-F1 for iWildCam, and enhances WC Acc by  $0.3\%$  and  $3.1\%$  for FMoW with ViT-L/14 and ViT-B/16, respectively. ViT-B/16 shows notably weaker learning capabilities compared to ViT-L/14. Our method, which learns directly from the input space, effectively harnesses domain-specific and data-specific knowledge, thus outperforming VDPG, particularly in models with lower capacities (i.e., ViT-B/16) across diverse WILDS datasets.
+
+Evaluation on DomainNet. Table 2 presents the accuracy across various domains and their overall averages. Our approach significantly surpasses VDPG, in 4/6 and 5/6 domains with average accuracy
+
+Table 1: Evaluation on challenging WILDS image testbeds under OOD conditions. It reveals that our method excels in both classification and regression tasks, significantly outperforming SOTA methods. Notably, with a smaller network (ViT-B/16), our method surpasses VDPG due to independently learned data-specific knowledge. (*: results obtained using official code; †: main evaluation metrics in WILDS; ◆: 3/8 channels utilized in PovertyMap as in VDPG.)  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Backbone</td><td colspan="2">iWildCam</td><td>Camelyon17</td><td colspan="2">FMoW</td><td colspan="2">PovertyMap (Regression)</td></tr><tr><td>Acc</td><td>Macro F1†</td><td>Acc†</td><td>WC Acc†</td><td>Avg Acc</td><td>WC Pearson r†</td><td>Pearson r</td></tr><tr><td>ERM</td><td></td><td>71.6 (2.5)</td><td>31.0 (1.3)</td><td>70.3 (6.4)</td><td>32.3 (1.25)</td><td>53.0 (0.55)</td><td>0.45 (0.06)</td><td>0.78 (0.04)</td></tr><tr><td>CORAL</td><td></td><td>73.3 (4.3)</td><td>32.8 (0.1)</td><td>59.5 (7.7)</td><td>31.7 (1.24)</td><td>50.5 (0.36)</td><td>0.44 (0.06)</td><td>0.78 (0.05)</td></tr><tr><td>IRM</td><td></td><td>59.8 (3.7)</td><td>15.1 (4.9)</td><td>64.2 (8.1)</td><td>30.0 (1.37)</td><td>50.8 (0.13)</td><td>0.43 (0.07)</td><td>0.77 (0.05)</td></tr><tr><td>ARM-CML</td><td>CNNs</td><td>70.5 (0.6)</td><td>28.6 (0.1)</td><td>84.2 (1.4)</td><td>27.2 (0.38)</td><td>45.7 (0.28)</td><td>0.37 (0.08)</td><td>0.75 (0.04)</td></tr><tr><td>ARM-BN</td><td></td><td>70.3 (2.4)</td><td>23.7 (2.7)</td><td>87.2 (0.9)</td><td>24.6 (0.04)</td><td>42.0 (0.21)</td><td>0.49 (0.21)</td><td>0.84 (0.05)</td></tr><tr><td>Meta-DMoE</td><td></td><td>77.2 (0.3)</td><td>34.0 (0.6)</td><td>91.4 (1.5)</td><td>35.4 (0.58)</td><td>52.5 (0.18)</td><td>0.51 (0.04)</td><td>0.80 (0.03)</td></tr><tr><td>MABN</td><td></td><td>78.4(0.6)</td><td>38.3(1.2)</td><td>92.4(1.9)</td><td>36.6(0.41)</td><td>53.2(0.52)</td><td>0.56 (0.05)</td><td>0.84 (0.04)</td></tr><tr><td>Zero-shot (ZS)</td><td>ViT-B/16</td><td>14.9</td><td>9.7</td><td>50.1</td><td>14.5</td><td>16.3</td><td>0.27</td><td>0.58</td></tr><tr><td>VDPG*</td><td>CLIP</td><td>71.4 (0.2)</td><td>30.1 (0.3)</td><td>93.2 (0.3)</td><td>37.8 (0.5)</td><td>52.7 (0.3)</td><td>0.38 (0.02)</td><td>0.77 (0.02)</td></tr><tr><td>L2C (ours)</td><td></td><td>73.4 (0.4)</td><td>35.2 (0.3)</td><td>94.2 (0.2)</td><td>40.9 (0.4)</td><td>54.8 (0.1)</td><td>0.50 (0.02)</td><td>0.80 (0.03)</td></tr><tr><td>Zero-shot (ZS)</td><td></td><td>28.7</td><td>1.0</td><td>64.2</td><td>13.3</td><td>21.1</td><td>0.35</td><td>0.62</td></tr><tr><td>FLYP</td><td>ViT-L/14</td><td>72.2 (0.4)</td><td>41.9 (0.3)</td><td>-</td><td>46.0 (0.3)</td><td>63.3 (0.4)</td><td>-</td><td>-</td></tr><tr><td>VDPG</td><td>CLIP</td><td>78.8 (0.2)</td><td>46.5 (0.3)</td><td>96.0 (0.4)</td><td>46.4 (0.5)</td><td>61.9 (0.4)</td><td>0.51 (0.03)</td><td>0.83 (0.04)</td></tr><tr><td>L2C (ours)</td><td></td><td>77.3 (0.1)</td><td>48.6 (0.4)</td><td>96.1 (0.3)</td><td>46.7 (0.3)</td><td>61.4 (0.3)</td><td>0.56 (0.02)</td><td>0.84 (0.03)</td></tr></table>
+
+Table 2: Evaluation on DomainNet. Our method surpasses SOTA, achieving top accuracy in 4/6 and 5/6 domains, with average gains of  $+1.4\%$  and  $+2.2\%$  using ViT-B/16 and ViT-L/14, respectively.  
+
+<table><tr><td>Method</td><td>Backbone</td><td>Clip</td><td>Info</td><td>Paint</td><td>Quick</td><td>Real</td><td>Sketch</td><td>Avg.</td></tr><tr><td>ERM</td><td></td><td>58.1 (0.3)</td><td>18.8 (0.3)</td><td>46.7 (0.3)</td><td>12.2 (0.4)</td><td>59.6 (0.1)</td><td>49.8 (0.4)</td><td>40.9</td></tr><tr><td>Mixup</td><td></td><td>55.7 (0.3)</td><td>18.5 (0.5)</td><td>44.3 (0.5)</td><td>12.5 (0.4)</td><td>55.8 (0.3)</td><td>48.2 (0.5)</td><td>39.2</td></tr><tr><td>CORAL</td><td></td><td>59.2 (0.1)</td><td>19.7 (0.2)</td><td>46.6 (0.3)</td><td>13.4 (0.4)</td><td>59.8 (0.2)</td><td>50.1 (0.6)</td><td>41.5</td></tr><tr><td>MTL</td><td rowspan="2">CNNs</td><td>57.9 (0.5)</td><td>18.5 (0.4)</td><td>46.0 (0.1)</td><td>12.5 (0.1)</td><td>59.5 (0.3)</td><td>49.2 (0.1)</td><td>40.6</td></tr><tr><td>SegNet</td><td>57.7 (0.3)</td><td>19.0 (0.2)</td><td>45.3 (0.3)</td><td>12.7 (0.5)</td><td>58.1 (0.5)</td><td>48.8 (0.2)</td><td>40.3</td></tr><tr><td>ARM</td><td></td><td>49.7 (0.3)</td><td>16.3 (0.5)</td><td>40.9 (1.1)</td><td>9.4 (0.1)</td><td>53.4 (0.4)</td><td>43.5 (0.4)</td><td>35.5</td></tr><tr><td>Meta-DMoE</td><td></td><td>63.5 (0.2)</td><td>21.4 (0.3)</td><td>51.3 (0.4)</td><td>14.3 (0.3)</td><td>62.3 (1.0)</td><td>52.4 (0.2)</td><td>44.2</td></tr><tr><td>MABN</td><td></td><td>64.2</td><td>23.6</td><td>51.5</td><td>15.2</td><td>64.6</td><td>54.1</td><td>45.5</td></tr><tr><td>DoPrompt</td><td>ViT-B/16 IMN</td><td>67.6 (0.2)</td><td>24.6 (0.1)</td><td>54.9 (0.1)</td><td>17.5 (0.2)</td><td>69.6 (0.3)</td><td>55.2 (0.5)</td><td>48.3</td></tr><tr><td>Zero-shot (ZS)</td><td></td><td>69.9</td><td>48.2</td><td>65.4</td><td>14.5</td><td>82.3</td><td>62.5</td><td>57.1</td></tr><tr><td>ERM</td><td rowspan="3">ViT-B/16 CLIP</td><td>68.0 (0.1)</td><td>22.5 (0.6)</td><td>46.5 (4.2)</td><td>18.5 (0.9)</td><td>58.7 (2.7)</td><td>52.5 (1.2)</td><td>44.4</td></tr><tr><td>MIRO</td><td>74.9 (0.2)</td><td>37.1 (0.4)</td><td>59.8 (0.6)</td><td>18.7 (1.2)</td><td>72.2 (0.2)</td><td>61.2 (0.9)</td><td>54.0</td></tr><tr><td>VDPG</td><td>76.3 (0.2)</td><td>49.3 (0.1)</td><td>67.8 (0.1)</td><td>17.4 (0.2)</td><td>81.5 (0.3)</td><td>66.6 (0.2)</td><td>59.8</td></tr><tr><td>L2C (ours)</td><td></td><td>75.6 (0.1)</td><td>52.1 (0.1)</td><td>69.4 (0.1)</td><td>17.3 (0.2)</td><td>85.5 (0.1)</td><td>67.1 (0.2)</td><td>61.2</td></tr><tr><td>Zero-shot (ZS)</td><td rowspan="2">ViT-L/14 CLIP</td><td>78.1</td><td>54.0</td><td>71.6</td><td>21.8</td><td>86.0</td><td>71.2</td><td>63.8</td></tr><tr><td>VDPG</td><td>82.4</td><td>54.9</td><td>73.1</td><td>22.7</td><td>85.0</td><td>73.2</td><td>65.2</td></tr><tr><td>L2C (ours)</td><td></td><td>82.3</td><td>58.7</td><td>75.2</td><td>24.0</td><td>88.6</td><td>75.4</td><td>67.4</td></tr></table>
+
+improvements of +1.4 and +2.2 using ViT-B/16 and ViT-L/14, respectively. These improvements underscore the benefits of directly learning dataset-specific knowledge. In Appendix F.1, we further compare with prompt-based methods: CoOp (Zhou et al., 2022b), CoCoOp (Zhou et al., 2022a) and side branch-based DTL (Fu et al., 2024)
+
+# 5.2 ABLATION STUDIES
+
+We conduct ablation studies on DomainNet-Info, iWildcam and FMoW using CLIP ViT-B/16 on various components, including CPNet, Revert Attention (RT), text refinement (Text ref.), greedy ensemble (Greedy), uniformity loss  $(\mathcal{L}_{uni})$ , DAF module and training schemes in Table 3. Note, if DAF is not incorporated, the domain branch is also omitted, which will be discussed in Table 4.
+
+CPNet and revert attention. As highlighted in Index 1 vs. 2 of Table 3, incorporating CPNet alongside a frozen CLIP markedly enhances performance across all datasets. On particularly challenging iWildCam and FMoW, there is exceptionally low zero-shot performance. However, effective learning of dataset-specific knowledge from the input space results in substantial performance gains, specifically, +13.1 on F1 for iWildCam and 13.8% WC Acc for FMoW. Additionally, integrating reverted attention directs CPNet to assimilate knowledge overlooked by CLIP, sharpening its focus on essential dataset-specific insights. This strategy leads to further enhancements (Index 4 vs. 5).
+
+Table 3: Ablation on various components of our work on DomainNet-Info, iWildCam and FMoW.  
+
+<table><tr><td rowspan="2">Index</td><td rowspan="2">CPNet</td><td rowspan="2">Text ref.</td><td rowspan="2">Greedy</td><td rowspan="2">Luni</td><td rowspan="2">DAF</td><td rowspan="2">Training</td><td>Info</td><td colspan="2">iWildCam</td><td colspan="2">FMoW</td></tr><tr><td>Acc</td><td>Acc</td><td>F1</td><td>WC Acc</td><td>Acc</td></tr><tr><td>1 (ZS)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>48.2</td><td>14.9</td><td>9.7</td><td>14.5</td><td>16.3</td></tr><tr><td>2</td><td>✓</td><td>-</td><td>-</td><td>-</td><td>-</td><td>ERM</td><td>49.6</td><td>65.8</td><td>22.8</td><td>28.3</td><td>49.0</td></tr><tr><td>3</td><td>✓</td><td>✓</td><td>-</td><td>-</td><td>-</td><td>ERM</td><td>50.3</td><td>68.7</td><td>24.0</td><td>31.4</td><td>50.9</td></tr><tr><td>4</td><td>✓</td><td>✓</td><td>✓</td><td>-</td><td>-</td><td>ERM</td><td>51.0</td><td>69.1</td><td>26.9</td><td>33.3</td><td>51.6</td></tr><tr><td>5</td><td>RT</td><td>✓</td><td>✓</td><td>-</td><td>-</td><td>ERM</td><td>51.5</td><td>71.5</td><td>32.9</td><td>36.1</td><td>53.1</td></tr><tr><td>6</td><td>RT</td><td>✓</td><td>✓</td><td>✓</td><td>-</td><td>ERM</td><td>51.7</td><td>72.2</td><td>33.2</td><td>36.0</td><td>53.8</td></tr><tr><td>7</td><td>RT</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>ERM</td><td>51.5</td><td>71.9</td><td>32.8</td><td>36.0</td><td>52.8</td></tr><tr><td>8</td><td>RT</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>Domain-centric</td><td>52.1</td><td>73.4</td><td>35.2</td><td>40.9</td><td>54.8</td></tr></table>
+
+![](images/7e68a47477e7e9bf0e3c10ec38190d55d07be8cc5be6399194dd8d160be29be0.jpg)  
+Figure 4: Analysis on a different number of transformer layers of CPNet.
+
+![](images/9cca7052882d5cd2eea718777a4ccccc01c7789346120fe4e1a6b6609354fec6.jpg)
+
+![](images/daf3996d842bfb2699d17c83b23bba6e1818658efd0465ab29fe35fa635101b5.jpg)
+
+![](images/112d76ab585f1751802f11ab279a2c32b99168ec22c04c5c1e15eccd33d8d344.jpg)
+
+![](images/ac1b7729219f8ed9a0b3b4531fe6c6357859f56a2b2cb6ee12a4cb6216afb9ec.jpg)
+
+Text refinement and greedy ensemble. Text features from frozen CLIP serve as initialization and refining them as per Eq. 4 proves beneficial for subsequent adaptation (Index 2 VS. 3). Since text features act as class prototypes, enhancing class discrimination is crucial. The goal is to make these features more discriminative by increasing the inter-dispersion among them, reducing domain biases. This results in more neutral features for unseen target domains. Therefore, using our greedy ensemble (Index 3 VS. 4) and enforcing the uniformity loss (Index 5 VS. 6) leads to positive gains. Appendix F.4 provides visualization using t-SNE (Van der Maaten & Hinton, 2008), and Appendix F.6 reports sensitivity on  $\lambda$ .
+
+Domain-aware adaptation and fusion (DAF). The text and image features  $\tilde{\mathbf{I}}(x^{in})$  and  $\tilde{\mathbf{T}}^{gre}$  are not inherently tailored to a specific domain. By integrating a domain-aware fusion model, both features are adapted to that domain, facilitating the fusion of text and image modalities within that domain. Hence, Index 8 demonstrates substantial improvement over domain-agnostic predictions (Index 6).
+
+Training schemes. ERM samples batches uniformly without considering domain labels, misaligning it with the protocol of adapting to a specific domain using limited unlabeled data. In contrast, our domain-centric learning to adapt optimizes at the domain level, treating each iteration as a task of FSTT-DA. Thus, a domain-centric approach yields further improvements (Index 7 VS. 8).
+
+Effect on number of transformer layers of CPNet. Fig. 4 illustrates the effect of the number of transformer layers in CPNet. Different downstream datasets require varying learning capacities depending on their complexity. For instance, while DomainNet is more stable, the more challenging remote sensing scenario in FMoW requires additional learnable blocks to achieve reasonable performance (Wang et al., 2022). Nevertheless, even with just 1 layer in CPNet, substantial gains have been observed over zero-shot performance, thanks to the complement of dataset-specific knowledge.
+
+Analysis on domain prompt DP. Our domain prompt DP aims to adapt the domain-agnostic features  $\tilde{\mathbf{I}}(x^{in})$  and  $\tilde{\mathbf{T}}^{gre}$  towards a particular domain. It consists of two components: knowledge queried from K-V cache, representing source domain knowledge and current domain knowledge  $\tilde{\mathbf{D}}$  aggregated from its unlabeled data. We omit some parts of DP, as reported in Table 4, equipping with both domain knowledge is essential to better adapt the features towards a domain.
+
+Analysis on domain information aggregation. VDPG independently processes all the unlabeled data and then aggregates their domain information via averaging. However, we perform simple reshaping and allow the attention to be performed on every pair of the tokens in that data batch which
+
+Table 4: Ablation on domain prompt. Table 5: Ablation on domain information aggregation.  
+
+<table><tr><td rowspan="2">Domain prompt</td><td colspan="2">iWildCam</td><td colspan="2">FMoW</td><td rowspan="2">Aggregation</td><td colspan="2">iWildCam</td><td colspan="2">FMoW</td></tr><tr><td>Acc</td><td>F1</td><td>WC Acc</td><td>Acc</td><td>Acc</td><td>F1</td><td>WC Acc</td><td>Acc</td></tr><tr><td>K-V cache only</td><td>73.0</td><td>34.1</td><td>37.8</td><td>50.6</td><td>Mean</td><td>73.2</td><td>34.4</td><td>37.8</td><td>52.8</td></tr><tr><td>D only</td><td>71.3</td><td>32.1</td><td>35.4</td><td>50.4</td><td>Max</td><td>71.9</td><td>34.5</td><td>36.7</td><td>52.6</td></tr><tr><td>DP</td><td>73.4</td><td>35.2</td><td>40.9</td><td>54.8</td><td>Reshape → D</td><td>73.4</td><td>35.2</td><td>40.9</td><td>54.8</td></tr></table>
+
+![](images/954e1937055c015585cc7d554812fc8a9a3c36c17036b8fbacd11be7ca2b1482.jpg)  
+Figure 5: Correlation between every pair of  $\mathbf{V}$ -vectors in  $\mathbf{K} - \mathbf{V}$  domain cache.
+
+has shown superiority as reported in Table 5. For Mean and Max, we do not reshape the tensor but take the mean or max over the batch dimension.
+
+Analysis on K-V domain cache.  $L$  is the size of the domain cache as the number of learnable vectors. Ideally, each can condense the distinct domain specificity from the source domains. Such property is exhibited by computing the correlations between every pair of the V-vectors, as illustrated in Fig. 5. Please note, that we did not apply external constraints on the cache (e.g., correlation loss (Chi et al., 2024)). Appendix F.5 reports the sensitivity on the size of the cache.
+
+Additional analysis on greedy ensemble and text feature uniformity. Table 6 reports the comparison between the average ensemble (CLIP) and our greedy approach using ViT-B/16 while holding off other components the same. Greater gains on more challenging WILDS benchmarks are observed compared to more structured common objects as in DomainNet. Fig. 6. illustrate that a lower uniformity among text features is potentially beneficial for the final performance.
+
+<table><tr><td>Ensemble method</td><td>DomainNet (Acc.)</td><td>iWildCam (F1)</td><td>FMoW (WC Acc)</td></tr><tr><td>Ensemble (CLIP)</td><td>60.9</td><td>33.6</td><td>40.1</td></tr><tr><td>Greedy ensemble</td><td>61.2</td><td>35.2</td><td>40.9</td></tr></table>
+
+Table 6: Comparison between average ensemble (CLIP) and our greedy approach using ViT-B/16. The greedy ensemble improves across benchmarks. However, DomainNet contains more structured, common objects. Greater gains on more challenging WILDS benchmarks are observed.
+
+![](images/5ac1f20fae96f0ae192e75ac3ae5e8621cbb20bf720fe8b7b819eb132b11e61a.jpg)  
+Figure 6: Final accuracy VS. text uniformity on DomainNet with ViT-B/16.
+
+# 6 CONLUSION
+
+In this work, we introduce L2C to address FSTT-DA. L2C adapts a model trained on source domains to unseen target domains using just a few unlabeled data points. Our method builds on the inherent OOD capability of CLIP, complementing it with a parallel network that learns data-specific knowledge at the input space through revert attention. Additionally, we propose a greedy text feature ensemble to effectively integrate data-specific label semantics. To facilitate domain adaptation, we generate a domain prompt that guides the integration of enhanced text and visual features through domain-aware fusion. Our extensive experiments validate L2C's effectiveness, showcasing its superior performance across five large-scale benchmarks in DomainNet and WILDS.
+
+# REPRODUCIBILITY STATEMENT
+
+For a fair comparison, we use the VDPG codebase, with data processing following the official WILDS code. The pre-trained CLIP models are directly sourced from the OpenAI CLIP repository. Sample code for the greedy ensemble is provided in Appendix E. Other components, such as CPNet, DAF, text refinement, and K-V cache, utilize standard PyTorch functions. Pre-trained models and the full code will be released upon publication of the paper.
+
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+Kaiyang Zhou, Jingkang Yang, Chen Change Loy, and Ziwei Liu. Conditional prompt learning for vision-language models. In IEEE Conference on Computer Vision and Pattern Recognition, 2022a.  
+Kaiyang Zhou, Jingkang Yang, Chen Change Loy, and Ziwei Liu. Learning to prompt for vision-language models. International Journal of Computer Vision, 2022b.  
+Xiangyang Zhu, Renrui Zhang, Bowei He, Aojun Zhou, Dong Wang, Bin Zhao, and Peng Gao. Not all features matter: Enhancing few-shot clip with adaptive prior refinement. In Proceedings of the IEEE/CVF International Conference on Computer Vision, 2023.
+
+# APPENDIX
+
+# A LIMITATION
+
+Our method enhances the generalized knowledge of robust CLIP, which serve as the primary contributor. However, when the downstream dataset significantly diverges from the pre-training, the load on our CPNet increases, necessitating a larger network. Despite this, our approach focuses on acquiring the excluded knowledge directly from the input space. Consequently, the trade-off between computational demand and performance is more favourable compared to previous methods (VDPG).
+
+# B ILLUSTRATION FOR FSTT-DA SETTING
+
+![](images/96ddfbd563a1aa60002ebadc0429e0f80052445b620b9ac27e624dc4ad4b4fea.jpg)  
+Figure 7: Illustration of FSTT-DA setting. After training on the source domains, the model adapts to each of the target domains using a few unlabeled data samples. Each target domain has a tailored model which will be used to infer all of the data in that domain.
+
+# C INFERENCE PROCESS (ADAPTING TO A TARGET DOMAIN)
+
+```latex
+Algorithm 2 Inference: adapting to an unseen target domain  
+Require: I: CLIP image encoder;  $\mathcal{D}_t^m$ : an unseen target domain; CP: CPNet; K/V: K-V domain cache; DAF: domain-aware fusion module;  $\tilde{\mathbf{T}}^{gre}$ : trained text feature;  
+1: // Compute the domain prompt  
+2:  $(\mathbf{x}_{S}) \sim \mathcal{D}_{t}^{m}$  ▷ Sample a few unlabeled data samples from the target domain  
+3:  $\tilde{\mathbf{x}}_{S}^{e} \gets \mathbf{x}_{S}$  ▷ Form input embeddings  
+4:  $\tilde{\mathbf{D}} \gets \mathbf{CP}([\mathbb{D}, \tilde{\mathbf{x}}_{S}^{e}])$  ▷ Aggregate domain information from unlabeled data  
+5:  $\mathbf{DP} = [\text{softmax}(\mathbf{K}\tilde{\mathbf{D}}^T) \cdot \mathbf{V}, \tilde{\mathbf{D}}]$  ▷ Compute the domain prompt for domain  $\mathcal{D}_t^m$   
+6: Discard K-V domain cache  
+7: //Adapting every data in the target domain  
+8: for every image  $\mathbf{x}_{Q}$  in  $\mathcal{D}_t^m$  do  
+9:  $\mathbf{x}_{Q}^{in} \gets \mathbf{x}_{Q}$  ▷ Form input embeddings  
+10:  $\tilde{\mathbf{I}}(\mathbf{x}_{Q}^{in}) \gets \mathbf{I}(\mathbf{x}_{Q}^{in}) + \mathbf{CP}^{RT}(\mathbf{x}_{Q}^{in})$  ▷ Compute complemented visual feature  
+11:  $\mathbf{T}_{Q}^{dm}$ ,  $\mathbf{I}_{Q}^{dm} \gets \text{DAF}(\mathbf{DP}, \tilde{\mathbf{I}}(\mathbf{x}_{Q}^{in}), \tilde{\mathbf{T}}^{gre})$  ▷ Adapt the image  $\mathbf{x}_{Q}$  towards domain  $\mathcal{D}_t^m$   
+12: Logits=Cosine_similarity( $\mathbf{T}_{Q}^{dm}$ ,  $\mathbf{I}_{Q}^{dm}$ ) ▷ Compute predictions  
+13: end for
+```
+
+Algo. 2 shows the process of inference for adapting to a particular unseen target domain. It contains two phases:
+
+Domain prompt computation. For an unseen target domain, we first collect a few unlabeled data samples and compute the domain prompt using the CPNet and the domain cache. Such a step is illustrated in Fig. 8a and L1-5 of Algo. 2. Please note, that after this step, the domain cache can be ignored.
+
+Adapting the data using the domain prompt. Once the domain prompt is computed, it is utilized to adapt all data samples in that domain. This stage follows the process as in L7-12 of Algo. 2. Also as illustrated in Fig. 8b, this step only involved CPNet, CLIP image encoder and proposed DAF.
+
+![](images/578f0a53f0b3769d54966b8d20b8eca3eb26a32c3078f076376297f20506ee7b.jpg)  
+(a) Computing the domain prompt (DP) for the target domain using a few unlabeled data samples.
+
+![](images/70e4de7dcdcd4b82c93de186d5c9e9849b175cb7e60f92d64c97f2390c15780c.jpg)  
+(b) Adapting every data in the target domain.  
+Figure 8: Inference process for adapting to an unseen domain. Adapting to an unseen target domain involved two phases. The first one is the domain prompt computation using a few unlabeled data sample as in (a). The second phase is to utilize the domain prompt for inferencing all the data in that domain as in (b).
+
+# D COMPUTATIONAL RESOURCES
+
+# D.1 COMPUTATIONAL COMPARISON OVER THE ENTIRE FRAMEWORKS
+
+Table 7: Comparison on speed and memory (batch size of 64).  
+
+<table><tr><td></td><td>CPNet(1 layer)</td><td>CPNet(3 layer)</td><td>CPNet(6 layer)</td><td>VDPG</td></tr><tr><td>Datasets</td><td>Camelyon17,PovertyMap</td><td>DomainNet,iWildCam</td><td>FMoW</td><td>All datasets</td></tr><tr><td># Learnable params (M)</td><td>13.8</td><td>27.9</td><td>49.2</td><td>32.1</td></tr><tr><td>Train memory(MB)</td><td>3872</td><td>5554</td><td>8106</td><td>3672</td></tr><tr><td>Train speed(s/batch)</td><td>0.84</td><td>0.88</td><td>0.97</td><td>0.87</td></tr><tr><td>Inference memory(MB)</td><td>2270</td><td>2330</td><td>2430</td><td>2798</td></tr><tr><td>Inference speed(s/batch)</td><td>0.64</td><td>0.67</td><td>0.73</td><td>0.69</td></tr></table>
+
+Table 7 reports the memory usage and speed during both training and inference. Despite introducing a parallel CPNet, the resource consumption remains comparable to VDPG. While VDPG relies on a heavy guidance module, our framework primarily allocates parameters within CPNet. The increase in memory during training is largely due to tensor reshaping, allowing attention to be applied across the entire batch. However, the domain prompt computation is performed only once per target domain and is gradient-free. As shown in Fig. 8b, the main computation during inference is streamlined—modules such as the text encoder, text feature refinement module, and K-V cache are all eliminated, making our framework highly efficient.
+
+# D.2 EFFICIENCY ON GREEDY TEXT ENSEMBLE.
+
+Greedy Ensemble is executed as a pre-processing step before large-scale training (L3 of Algo. 1). Once the text features for all templates are obtained, the entire text encoder can be discarded, making assembling highly efficient. For instance, in DomainNet-real with a batch size of 64, the image encoder requires 120K forward passes over 20 epochs, while the text encoder only needs 80 forward passes to compute the text features, resulting in minimal resource usage ( $< 0.01\%$ ).
+
+E PYTORCH-LIKE SAMPLE CODE FOR GREEDY ENSEMBLE  
+```python
+#P :Number of text prompt templates   
+#C :Number of classes   
+#d :feature dimension   
+#TE : Tensor, shape  $= [P,C,d]$    
+# P sets of text embeddings   
+#Score: Tensor, shape  $= [P]$    
+# corresponding uniformity loss for TE   
+def uniformity_loss(text_embedding, t=2): # text_embedding: shape  $= [C,d]$  return torch.pdist(text_embedding, p=2).pow(2.0).mul(-t).exp().mean()   
+def sort_uniformity (TE, Score): sort_index = torch.argsort (Score).cpu().numpy() return TE[sortidx]   
+def ensemble (TE_list): return torch.stack(TE_list, dim=0).mean(dim=0)   
+def greedy_ensemble (TE, Score): final_TE = [] TE_sorted  $=$  sort_uniformity(TE, Score) # take the text prompt embedding with least uniformity loss as base final_TE.append(TE_sorted[0]) for i in range(1,P): temp_TE  $=$  final_TE + TE_sorted[i] if uniformity_loss(ensemble(temp_TE))< uniformity_loss(ensemble(final_TE)) final_TE  $=$  temp_TE return ensemble(final_TE)
+```
+
+# F ADDITIONAL EXPERIMENTS
+
+# F.1 COMPARISON WITH PROMPT-BASED METHODS AND SIDE BRANCH-BASED METHODS
+
+We further provide a comparison with prompt-based methods: CoOp (Zhou et al., 2022b), CoCoOp (Zhou et al., 2022a) and side branch-based DTL (Fu et al., 2024). Since these methods are non-adaptive, we applied test-time optimization as in TPT to minimize the entropy (Shu et al., 2022).
+
+Table 8 outlines the architectural differences. TPT relies on gradient updates, making gradient flow crucial and thus limited to white-box settings. In contrast, VDPG and our proposed L2C generate domain-specific prompts for adaptation, enabling them to operate in the more challenging black-box setting. Despite these constraints, L2C still surpasses all other methods, as shown in Table 9.
+
+Table 8: Architectural comparison.  
+
+<table><tr><td>Method</td><td>Black/white box</td><td>Gradient at test-time</td></tr><tr><td>CoOp + TPT</td><td>white box</td><td>required</td></tr><tr><td>CoCoOp + TPT</td><td>white box</td><td>required</td></tr><tr><td>DTL + TPT</td><td>white box</td><td>required</td></tr><tr><td>VDPG</td><td>black box</td><td>gradient-free</td></tr><tr><td>L2C (Ours)</td><td>black box</td><td>gradient-free</td></tr></table>
+
+Table 9: Performance comparison.  
+
+<table><tr><td>Method</td><td>Ave Acc. (DomainNet, ViT-B16)</td></tr><tr><td>CoOp</td><td>58.8</td></tr><tr><td>CoOp + TPT</td><td>60.6</td></tr><tr><td>CoCoOp</td><td>59.4</td></tr><tr><td>CoCoOp + TPT</td><td>60.4</td></tr><tr><td>DTL</td><td>57.6</td></tr><tr><td>DTL + TPT</td><td>59.2</td></tr><tr><td>VDPG</td><td>59.8</td></tr><tr><td>L2C (Ours)</td><td>61.2</td></tr></table>
+
+# F.2 ALTERNATIVE CRITERIA FOR TEXT EMBEDDING UNIFORMITY
+
+The criteria we used for text feature inter-dispersion among classes is determined by the uniformity in a hypersphere (i.e., Eq. 2). It can also be determined by measuring the Average Text Feature
+
+Dispersion (ATFD) which calculates the distance of all class embedding to their centroid (Yoon et al., 2024). For a text features  $\mathbf{T}(\mathbf{P}_{\mathbf{C}})$  with a prompt template  $\mathrm{P}$  and class labels  $\mathbf{C}$ , ATFD is computed as:
+
+$$
+\operatorname {A T F D} = \frac {1}{| \mathbf {C} |} \sum_ {i \in | \mathbf {C} |} \| \mathbf {T} _ {\text {c e n t r o i d}} - \mathbf {T} _ {i} \left(\mathrm {P} _ {\mathrm {C}}\right) \| _ {2}, \quad \text {w h e r e} \quad \mathbf {T} _ {\text {c e n t r o i d}} = \frac {1}{| \mathbf {C} |} \sum_ {i = 1} ^ {| \mathbf {C} |} \mathbf {T} _ {i} \left(\mathrm {P} _ {\mathrm {C}}\right), \tag {9}
+$$
+
+where  $\| \cdot \|_2$  measures the L2 distance. Please note, a smaller ATFD indicates the class text features are closer to the centroid, therefore, the class features are more closely clustered. In contrast, a larger ATFD indicates a more dispersed distribution of the text features. Therefore, to integrate ATFD into our greedy ensemble pipeline, we need to sort the features of text in reverse order based on ATFD and select the prompts that can maximize ATFD after ensemble. The loss function then becomes  $\mathcal{L}_{total} = \mathcal{L}_{clip} - \lambda \cdot \mathrm{ATFD}$ .
+
+Table 10 reports selected text prompt templates for DomainNet with both  $\mathcal{L}_{uni}$  and ATFD. The selected templates match with each other and the average is also very close to each other. Table 11 reports a performance comparison of using  $\mathcal{L}_{uni}$  and ATFD on additional benchmarks, showing close results.
+
+Table 10: Comparison between metrics of using two different criteria for text feature inter-dispersion.
+
+<table><tr><td></td><td colspan="2">DomainNet (ViT-B/16)</td></tr><tr><td>Inter-dispersion criteria</td><td>Luni</td><td>ATFD</td></tr><tr><td>Selected prompt templates</td><td>a blurry photo of a {}. 
+a embroidered {}. 
+itap of the {}. 
+itap of my {}. 
+itap of a {}. 
+a black and white photo of a {}.</td><td>a blurry photo of a {}. 
+a embroidered {}. 
+itap of the {}. 
+itap of my {}. 
+itap of a {}. 
+a black and white photo of a {}.</td></tr><tr><td>Average accuracy on 6 domains</td><td>61.2</td><td>61.1</td></tr></table>
+
+Table 11: Additional performance comparison of using two different criteria for text feature inter-dispersion.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">iWildCam</td><td colspan="2">FMoW</td><td>DomainNet</td></tr><tr><td>Acc</td><td>Macro F1</td><td>WC Acc</td><td>Avg Acc</td><td>Acc</td></tr><tr><td>Luni</td><td>73.4</td><td>35.2</td><td>40.9</td><td>54.8</td><td>61.2</td></tr><tr><td>ATFD</td><td>73.5</td><td>35.3</td><td>40.8</td><td>54.8</td><td>61.1</td></tr></table>
+
+# F.3 ADOPT CLIP FOR REGRESSION AS IN POVERTYMAP DATASET
+
+Regression task on CLIP: Regression task requires a single number output, therefore, it is equivalent to setting the number of output class as 1 (Chi et al., 2024). Since the PovertyMap aims to estimate the wealth index for a region, therefore, we simply use a sentence prompt as the text input of the CLIP text encoder: A satellite image showing the wealth_index, yielding a text embedding with output dimension as  $\mathbf{T}^{dm} \in \mathbb{R}^{1 \times d}$ . With the adapted image feature  $\mathbf{I}^{dm} \in \mathbb{R}^{B \times d}$ , the logit is obtained by matrix multiplication between them, with shape  $\in \mathbb{R}^{B \times 1}$ . Therefore, each image has a single output number in our framework. For the regression task, we train the model using MSE loss, and omit the text uniformity loss as there is only one "class".
+
+Surprisingly, without any training, the original CLIP model is able to predict zero-shot regression (e.g., the regional wealth index) just like zero-shot classification (as shown in Table 1, the zero-shot regression shows positive correlation on the wealth index). Our method can significantly boost performance by learning the complement knowledge and adapting to those unseen target domains.
+
+# F.4 T-SNE VISUALIZATION OF COMPARISON ON TEXT FEATURES
+
+![](images/596e384886d38fb2e332acfb9b9fe5d5f4495ce62e936188601dbffdd01513fc.jpg)  
+Figure 9: t-SNE (Van der Maaten & Hinton, 2008) visualization of comparison on text features with prompt [CLASS], greedy ensemble and refined output (ViT-B/16 on DomainNet).
+
+Fig. 9 shows the visualization of the text features for different text prompting methods and also our simple yet effective refinement output. It clearly shows that the features with prompt [CLASS] have more clustered features which are less discriminative. Our greedy ensemble greatly increases the distance among the class features. With simple refinement using  $M_{c}$ ,  $M_{d}$  and the uniformity loss, the features are further separated.
+
+Table 12: Sensitivity on domain cache size.  
+
+<table><tr><td rowspan="2">L size</td><td colspan="2">iWildCam</td><td colspan="2">FMoW</td></tr><tr><td>Acc</td><td>F1</td><td>WC Acc</td><td>Acc</td></tr><tr><td>1</td><td>69.2</td><td>32.8</td><td>33.2</td><td>50.6</td></tr><tr><td>5</td><td>73.2</td><td>35.0</td><td>40.9</td><td>54.8</td></tr><tr><td>10</td><td>73.4</td><td>35.2</td><td>40.7</td><td>54.5</td></tr></table>
+
+Table 13: Sensitivity on loss balancing weight  $\lambda$  .  
+
+<table><tr><td rowspan="2">λ</td><td colspan="2">iWildCam</td><td colspan="2">FMoW</td></tr><tr><td>Acc</td><td>F1</td><td>WC Acc</td><td>Acc</td></tr><tr><td>0.01</td><td>73.2</td><td>35.1</td><td>40.8</td><td>54.7</td></tr><tr><td>0.1</td><td>73.4</td><td>35.2</td><td>40.9</td><td>54.8</td></tr><tr><td>1.0</td><td>73.5</td><td>34.9</td><td>40.4</td><td>54.8</td></tr></table>
+
+# F.5 SENSITIVITY ON DOMAIN Cache SIZE:
+
+Table 12 reports the performance with different size of domain cache. When size=1, the learning capability is too small. Increasing to 5 or 10 makes it more stable.
+
+# F.6 SENSITIVITY ON LOSS BALANCING WEIGHT  $\lambda$ :
+
+Table 13 reports the sensitivity on  $\lambda$ . Our framework is less sensitive to  $\lambda$  as it is only applied to the text features at the beginning, with only  $M_{c}$  and  $M_{d}$  to optimize. The effect is the convergence speed, but the ultimate performance is quite stable.
+
+# G ADDITIONAL HYPER-PARAMETERS
+
+We utilize the same configurations of transformer blocks as of ViT-B/16 or ViT-L/14, therefore, the feature size can be matched between the main backbone and the CPNet. The only hyper-parameter we tune is the number of transformer blocks. We show the number of layers for different datasets and the total number of learnable parameters and compare them with FLYP and VDPG when ViT-B/16 is utilized in Table14. The size of the domain cache is reported in Table 15. All the experiments can be conducted with a single NVIDIA V100 GPU. We set the batch size as 64 (12 images for support and 52 images for query set). Each iteration runs 0.4 seconds.
+
+Table 14: Configuration of CPNet and the total number of learnable parameters.  
+
+<table><tr><td></td><td colspan="5">L2C (ours)</td><td>FLYP</td><td>VDPG</td></tr><tr><td></td><td>DomainNet</td><td>iWildCam</td><td>Camelyon17</td><td>FMoW</td><td>PovertyMap</td><td>-</td><td>-</td></tr><tr><td># of transformer blocks</td><td>3</td><td>3</td><td>1</td><td>6</td><td>1</td><td>-</td><td>-</td></tr><tr><td>Learnable parameters</td><td>27.9M</td><td>27.9M</td><td>13.8M</td><td>49.2M</td><td>13.8M</td><td>149M</td><td>32.1M</td></tr></table>
+
+Table 15: Size of domain cache  $L$  .  
+
+<table><tr><td></td><td>DomainNet</td><td>iWildCam</td><td>Camelyon17</td><td>FMoW</td><td>PovertyMap</td></tr><tr><td>L</td><td>10</td><td>10</td><td>5</td><td>5</td><td>10</td></tr></table>
+
+# H DETAILS ON TEXT PROMPTSTEMPLATES:
+
+In this section, we describe the candidate test prompts used to perform the greedy ensemble, the full list of attached in supplementary material (MS Excel spreadsheet (TextPromptTemplates)). Table 16 shows the selected text prompts among the candidate text prompt as follows:
+
+DomainNet & iWildCam: We use 80 text prompt templates that used for ImageNet (Deng et al., 2009) from CLIP (Radford et al., 2021) and FLYP (Goyal et al., 2023).
+
+FMoW: We use 14 text prompt templates from FLYP (Goyal et al., 2023). The prompts describe the photos in remote-sensing scenarios.
+
+Camelyon17: We filter some text prompts from the 80 ImageNet text prompts that can be used to describe the medical issue and generate some using ChatGPT. In total, there are 56 text prompts.
+
+Please note that PovertyMap is a regression task, the prompts for this task and the experimental setting for regression are reported in Sec. F.3.
+
+Table 16: Selected text prompts by our proposed greedy ensemble.  $\{\}$  is replaced by class name.  
+
+<table><tr><td>iWildCam</td><td>DomainNet</td><td>FMoW</td><td>Camelyon17</td></tr><tr><td>a blurry photo of the {}. 
+a dark photo of the {}. 
+a cropped photo of the {}. 
+a black and white photo of the {}. 
+a black and white photo of a {}. 
+a close-up photo of the {}. 
+a photo of many {}. 
+itap of my {}. 
+a bright photo of the {}. 
+itap of a {}. 
+a good photo of a {}.</td><td>a blurry photo of a {}. 
+a embroidered {}. 
+itap of the {}. 
+itap of my {}. 
+itap of a {}. 
+a black and white photo of a {}.</td><td>aerial photo of a {} in oceania. 
+{}.
+a dark photo of the {}. 
+a black and white photo of the {}. 
+a good photo of the {}.</td><td></td></tr></table>
\ No newline at end of file
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+# LEARNING TO CLARIFY:  
+MULTI-TURN CONVERSATIONS WITH ACTION-BASED  
+CONTRASTIVE SELF-TRAINING
+
+Maximillian Chen\*1,2, Ruoxi Sun1, Tomas Pfister1, Sercan O. Ark1
+
+<sup>1</sup>Google <sup>2</sup>Columbia University
+
+maxchen@cs.columbia.edu, {ruoxis,tpfister,soarik}@google.com
+
+# ABSTRACT
+
+Large language models (LLMs), optimized through human feedback, have rapidly emerged as a leading paradigm for developing intelligent conversational assistants. However, despite their strong performance across many benchmarks, LLM-based agents might still lack conversational skills such as disambiguation – when they are faced with ambiguity, they often overhedge or implicitly guess users' true intents rather than asking clarification questions. Under task-specific settings, high-quality conversation samples are often limited, constituting a bottleneck for LLMs' ability to learn optimal dialogue action policies. We propose Action-Based Contrastive Self-Training (ACT), a quasi-online preference optimization algorithm based on Direct Preference Optimization (DPO), that enables data-efficient dialogue policy learning in multi-turn conversation modeling. We demonstrate ACT's efficacy under in data-efficient tuning scenarios, even when there is no action label available, using multiple real-world conversational tasks: tabular-grounded question-answering, machine reading comprehension, and AmbigSQL, a novel task for disambiguating information-seeking requests for complex SQL generation towards data analysis agents. Additionally, we propose evaluating LLMs' ability to function as conversational agents by examining whether they can implicitly recognize and reason about ambiguity in conversation. ACT demonstrates substantial conversation modeling improvements over standard tuning approaches like supervised fine-tuning and DPO.
+
+# 1 INTRODUCTION
+
+Conversations offer a natural and effective way for humans and intelligent systems to collaborate (Amershi et al., 2019; Lemon, 2012). The impressive capabilities of large language models (LLMs) have powered the rapid development of many generalist conversational assistants such as ChatGPT<sup>1</sup> and Gemini (Gemini Team et al., 2023), which present an opportunity for users to verbalize their need for assistance on complex tasks. However, the promises of a conversational interfaces also come with the complexities of language. Human conversation is riddled with ambiguity, whether it be due to humans' tendency to underspecify (Zipf, 1949) or even due to syntactic errors (Messer, 1980). Moreover, disambiguation becomes even more important in complex domains where it can be a difficult multi-turn process to achieve common ground (Beers et al., 2006). As it stands, existing LLM-powered conversational agents continue to struggle with modeling ambiguity (Liu et al., 2023), and tend to exhibit unwanted behavior such as overhedging (Ouyang et al., 2022) or generating responses which represent a "guess" of the user's intent (Deng et al., 2023a) (see Figure 1).
+
+One of the primary reasons that LLMs may exhibit unwanted conversational behaviors is that their language modeling objective during pre-training or supervised fine-tuning (SFT) is not directly aligned with this goal (Ouyang et al., 2022). While approaches like Ouyang et al. (2022) propose LLM "alignment" using post-training approaches like reinforcement learning from human feedback (RLHF) (Christiano et al., 2017), existing models still struggle with conversational tasks spanning
+
+![](images/7d2cde29ad4e2b845166d7767f9f63db2d70805049cb46722613dc42fec1a874.jpg)  
+Figure 1: Simplified example of ambiguity present at tabular-grounded conversational question answering based on Deng et al. (2022). A conversational agent should recognize when there is ambiguity and ask a clarifying question towards a more accurate final answer.
+
+multiple turns (Wang et al., 2023). This is partly due to the fact that existing approaches do not directly optimize for pragmatic skills (e.g. Bai et al. (2022)). Moreover, there is often high variance in the target distribution of a particular use case, so it is imperative that downstream adaptation approaches can effectively steer LLM policies. Given large-scale in-distribution training data, this may be feasible with standard SFT or RLHF. But, dialogue policy learning can be particularly challenging given limited data (Chen et al., 2022a; Dong et al., 2023) and collecting high-quality conversational datasets can be difficult for reasons such as annotation costs and privacy concerns (Chen et al., 2023a).
+
+This motivates the design of a conversational adaptation approach for LLMs which is more closely aligned with the goal of modeling actions in multi-turn conversation. We focus on improving LLMs' abilities to implicitly select conversational strategies in ambiguous contexts, and propose an approach called Action-Based Contrastive Self-Training (ACT). ACT is a sample-efficient, quasi-online Direct Preference Optimization algorithm (Rafailov et al., 2024) which focuses on contrasting the differences between an agent's possible pragmatic conversational actions. We demonstrate ACT's sample-efficient performance on a diverse range of mixed-initiative conversational tasks: (i) tabular-grounded question answering, (ii) machine reading comprehension, and (iii) text-to
+
+![](images/9c2b9e216043f525b877021c2f58f8a22445bdb32dc9b3916c7cfb1f88754cfe.jpg)  
+Figure 2: ACT greatly outperforms standard tuning approaches in data-efficient settings for conversational modeling, as exemplified here on PACIFIC.
+
+SQL generation, demonstrating large improvements compared to standard adaptation approaches (see Figure 2). Our work highlights the necessity of considering action-based preferences for conversational tasks, and we propose a workflow for evaluating LLMs' ability to recognize and reason about ambiguity in conversation.
+
+# 2 RELATED WORK
+
+# 2.1 MIXED-INITIATIVE CONVERSATIONAL AGENTS
+
+Neural approaches to building mixed-initiative conversational agents typically consist of two core components: an understanding and planning module (e.g., a binary prediction task to determine whether to ask a clarifying question or provide an answer), and a generation module which can be controlled at a pragmatic level using the output of the planning module (Chen et al., 2017; 2022b; Qian et al., 2022; Yu, 2017) (e.g., forming an utterance which follows the predicted action).
+
+Generation Many existing works focus on novel training methodologies to improve conditional generation as a complement to planning, with approaches such as multi-objective SFT (Chen et al., 2022b; Wen et al., 2016) or introducing specialized embeddings for control codes (Keskar et al., 2019). LLMs have vastly improved performance in pragmatically-controlled generation (Chen et al., 2023b), but all of these approaches still depend on conversational planning. Planning remains a difficult task – natural interaction is not deterministic and often requires long-horizon planning.
+
+Planning The planning task can be viewed as a stochastic Markov Decision Process (Wang et al., 2020b; Yang et al., 2021; Yu et al., 2023) in which some dialogue state is drawn from a potentially unknown distribution, given the previous dialogue state and an imposed action. However, the action itself is not literally presented to the interacting parties; rather, an action is a low-dimensional representation of the pragmatic intent carried by a given dialogue utterance (i.e., a dialogue act Sadek (1991); Stolcke et al. (2000); Wu et al. (2023); Yu & Yu (2021)). As such, training planning modules often requires complex long-horizon reasoning and simulation to model the responses and intents of each interacting party. Such efforts have examined combining neural models with search algorithms (Cheng et al., 2022; Väth et al., 2023; Yu et al., 2023) and simulation (Deng et al., 2023c; Wang et al., 2020a; Yu et al., 2023). However, such modular approaches can incur high computational overhead (Yu et al., 2023) and may result in error propagation while not directly optimizing for response quality itself. We propose directly optimizing dialogue action planning as an implicit subtask of response generation in mixed-initiative conversation contexts, as we discuss in Sec. 3.2.
+
+# 2.2 LEARNING FOR LLM ALIGNMENT
+
+The current paradigm of LLM training for downstream use cases consists of three phases: pretraining, supervised fine-tuning (SFT) for instruction-following, and tuning for alignment with human preferences (Tunstall et al., 2023; Rafailov et al., 2024; Lee et al., 2023; Ouyang et al., 2022). We primarily focus on the phase of tuning for alignment. These approaches typically start with an initial policy model obtained by conducting SFT on a target task  $(\pi_{SFT})$ , before performing tuning (often with RL), using contrastive preference examples (most commonly collected through human feedback (Ouyang et al., 2022) or a similar proxy like LLM-generated feedback (Lee et al., 2023)). In the case of online algorithms like PPO, a reward model is first fit over the preference examples so that it could be used for RL optimization (Ouyang et al., 2022). Such algorithms have certain advantages which may benefit the Markov Decision Process-like nature of conversations — namely, a diverse search space as opposed to a fixed dataset, flexible reward functions, and broader policy exploration. However, PPO is notoriously difficult to tune, and offline algorithms such as DPO (Rafailov et al., 2024), SLiC (Zhao et al., 2023), and IPO (Azar et al., 2024) have become widely adopted as an LLM adaptation approach because they bypasses explicit reward modeling and thus only require one set of hyperparameters to optimize (Huang et al., 2024; Rafailov et al., 2024; Zhao et al., 2023; Zheng et al., 2023) while still achieving similar empirical results given a fixed preference dataset.
+
+On-Policy DPO Many of our contemporaries also question the limits of fully offline preference learning algorithms and have examined "online" variants of them (Guo et al., 2024; Xu et al., 2023; 2024b). Yuan et al. (2024) proposes iterative DPO, and Chen et al. (2024) proposes a variant where ground-truth responses are considered winning, and responses sampled from the previous iteration of a policy model are considered losing. Pang et al. (2024) applies a variant of iterative DPO to optimize externalized reasoning chains. Our work differs from these in that we are proposing a novel approach to customize LLMs for specific conversational settings, in particular, multi-turn conversational settings. While other works look at applying DPO to conversations in general (e.g. Sun et al. (2024)), their focus is still on single-turn response optimization. ACT considers multi-turn trajectories for preference optimization, and to our knowledge, our work is the first paper to consider contrastive learning on the basis of conversational actions.
+
+# 3 METHODS
+
+# 3.1 PROBLEM SETUP
+
+We consider the task of tuning an LLM to function as a mixed-initiative conversational agent. Through a series of dialogue interactions with a user, the LLM is expected to assist the user by eventually providing a correct response to their request. Unlike the common agent interaction setting where users completely control the flow of interaction with the expectation that agents may autonomously complete tasks such as online shopping (Liu et al., 2024), mixed-initiative agents should understand how to redirect the flow of the interaction (Allen et al., 1999) through the execution of conversational actions or strategies such as clarifying questions (Chu-Carroll, 2000; Peng et al., 2018).
+
+Notation Consider a goal-oriented conversational environment. Let  $\pi_{\theta_i}$  be an LLM's policy parameterized by  $\theta$  at timestep  $i \geq 0$ , with  $\pi_{ref}$  being the reference policy model (i.e.,  $\pi_{ref} \gets \pi_{\theta_0}$ ). Let  $D$  be a dataset consisting of conversations. Let each conversation  $c$  in  $D$  contain  $n$  dialogue turns, through which a user is requesting one or more pieces of information from an agent. The turn state of a conversation (the observed utterances and actions given by each interacting party) at timestep
+
+![](images/bf216c7f429a809456e2f3f748f1962a7539deea7d6f8461664a36ec6a9b5721.jpg)  
+Figure 3: Overview of the tuning phase of  $ACT$ . For each initial contrastive pairing from  $D_{pref}$  (constructed as in Sec. 3.2.1), we sample an on-policy response from the model being tuned. After evaluating the sampled response's trajectory, we update the contrastive pairing by either replacing the existing winning or losing response. The model policy is updated using the objective in Eq. 1.
+
+$i$  can be represented by  $t_i$ . Implicitly, each  $t_i$  is part of a trajectory which ends when the user's question expressed at an earlier timestep  $j \leq i$  is answered. Any turn  $t_i$  can first be broken down into two primary components:  $p_i$  and  $r_i$ , where  $p_i$  can be a prompt at  $i$ , consisting of any task-specific information (e.g. a SQL database schema, tabular data or retrieved passages) combined with any existing dialogue context, and  $r_i$  is the ground truth system-side response at  $i$ . Next, we can let  $g_i$  be the goal response which resolves  $t_i$ 's implicit trajectory, i.e., the answer to the user's original question after any possible clarification turns. In the single-turn trajectory case,  $g_i \gets r_i$ . Each  $r_i$  implicitly expresses an action,  $a_i$ , where  $a_i$  exists in the latent Action Space  $S$  of a particular task and  $a_i$  can be inferred by some Action Annotation Agent  $G^2$ . Thus, we can formally represent turn state  $t_i$  using the tuple  $(p_i, r_i, g_i, a_i)$ . For the datasets considered in our experiments,  $S = [\text{CLARIFY}, \text{ANSWER}]$  (although the method can be extended to a broader action space). We assume access to a controllable generation model  $(M)$ , Action Classifier  $(A)$  and model which can be controlled to function as a User Simulator  $(U)$ . As we discuss in Sec. 3.2,  $M$  is used for preference data creation whereas  $A$  and  $U$  are used during tuning and evaluation. We illustrate this notation in Fig. A4.
+
+User Simulators Works such as Deng et al. (2023c); Yu et al. (2023) directly prompt LLMs for goal-oriented tasks conditioned on dialogue context and task objectives. Our implementation of  $U$  is inspired by this setup. We first prompt an LLM to summarize the user's information-seeking goal. Then, we form another prompt using this summary along with the current dialogue context to simulate a user response. Prompting with this goal summary allows for more flexibility than directly providing the user simulator with the ground truth information objective. We provide details on our implementation of  $U$  in Appendix H.
+
+Action Classifiers In the datasets we considered, the possible actions are to either "clarify" or "directly answer" a question. We directly use few-shot in-context learning as action classifier  $A$ . We provide details on our in-context examples for  $A$  in Appendix G.
+
+# 3.2 ACT: ACTION-BASED CONTRASTIVE SELF-TRAINING
+
+One of the north stars in developing intelligent conversational models is the ability to automatically produce responses which take actions which lead to the highest probability of conversational success (Wu et al., 2023; Zhao et al., 2019). We propose ACT, an approach that adapts generic LLMs for dialogue generation and models action planning as an implicit subtask. ACT is a quasi-online extension of the DPO algorithm which maintains its ease of use of offline method while incorporating the flexible exploration found during online learning. ACT relies on a few intuitions. 1) Contrastive preferences are an intuitive medium for demonstrating the pragmatic differences between the implicit actions of "winning" and "losing" dialogue responses. 2) Conversational improvements require multi-turn optimization, which are difficult to express using only single-turn contrast pairings. 3) The
+
+gradient of the objective of DPO-like algorithms (see Eq. 2) is weighted based on the log probabilities assigned to the winning and losing responses, and by construction, on-policy response sampling yields high-probability token sequences.  $ACT$  is summarized in Fig. 3.  $ACT$  consists of two phases: action-based contrast dataset construction (Alg. 1) and contrastive self-training (Alg. 2).
+
+# 3.2.1 CONSTRUCTION OF PREFERENCE DATA
+
+The preference dataset primarily consists of contrastive winning-losing action pairs, as shown in Alg. 1. That is, for each conversation turn  $t_i$  in a dataset  $D$ , we can construct  $D_{pref}$  consisting of augmented  $t_i'$  tuples. We add rejected action  $a_i'$  which is sampled from  $S \setminus a_i$ , winning response  $y_{wi} \gets r_i$ , and  $y_{l_i}$  which is a losing response sampled using  $M$ . Given that each  $a_i'$  is predefined when constructing  $D_{pref}$ , we use a high capacity LLM (Chen et al., 2023b) rather than tuning a smaller one or asking crowdworkers for losing response construction (more details in Appendix F).
+
+# Algorithm 1 Building Contrastive Action Pairs
+
+input Dataset  $D$ , Conditional generation model  $M$ , Action Space  $S$ , Action Annotation Agent  $G$
+
+1: Initialize empty dataset  $D_{\text{pref}}$  
+2: for conversation turn  $t_i \in D$  do  
+3: Let  $a_i = G(p_i, r_i)$  Infer Contextual Action  
+4: Let  $a_{i}^{\prime} = S \setminus a_{i}$  Determine Rejected Action  
+5: Let  $y_{wi} = r_i$  
+6: Sample  $y_{li} \sim P_M(\cdot | p_i, a_i')$ .  
+7: Let  $t_i' = (p_i, r_i, g_i, a_i, a_i', y_{wi}, y_{li})$ .  
+8: Add  $t_i'$  to  $D_{\text{pref}}$
+
+output  $D_{\text{pref}}$
+
+Action optimization for unlabeled conversations "in-the-wild" Obtaining gold-standard ambiguity annotations may not always be possible. In such settings, one can obtain pseudo-label supervision using a classifier as the Action Annotation Agent  $G$  rather than human annotation. We discuss details and analyze performance in Sec. 5.4. Depending on the data, it may be appropriate to introduce an initial preprocessing step which involves inferring user satisfaction similarly to Shi et al. (2024).
+
+# 3.2.2 SELF-TRAINING USING ON-POLICY CONVERSATION TRAJECTORY SIMULATION
+
+As in DPO training, we continuously sample batches from  $D_{pref}$ . Although each conversation turns  $t_i$  in each batch  $j$  has a default winning  $(y_{wi})$  and losing  $(y_{li})$  response, we also sample an on-policy response  $y_i$  from  $\pi_{\theta_j}$ . We use  $A$  to determine whether the implicit action of  $y_i^3$  matches the inferred action  $a_i$  of the ground truth response. If the implicit action of  $y_i$  is incorrect, we set  $y_{li} = y_i$ . If it does match  $a_i$ , then we simulate the outcome  $g_i'$  of the trajectory resulting from  $y_i$  using  $U^4$  and  $\pi_{\theta_j}$ . If the trajectory outcome  $g_i'$  fails to meet task-specific heuristics (e.g., low semantic similarity or an incorrect execution), we set  $y_{li}$  to the entire simulated trajectory resulting from  $y_i$  (e.g., "Are you looking for..." + "What is the average total number..." + "SELECT max..." in Figure 3). Otherwise, we set  $y_{wi}$  to the simulated trajectory (e.g. "Are you looking for..." + "What is the average total number..." + "SELECT avg..." in Figure 3).
+
+# 3.2.3 CONTRASTIVE RL TUNING FOR ALIGNMENT
+
+After constructing the up-to-date winning  $y_{wi}$  and losing  $y_{li}$  pairing at turn  $i$  through simulation (Sec. 3.2.2), we update the policy model  $(\pi_{\theta})$  using the DPO training objective (Rafailov et al., 2024), which is as follows (we ignore the  $i$  iterator for simplicity):
+
+$$
+\mathcal {L} _ {\mathrm {D P O}} \left(\pi_ {\theta}; \pi_ {r e f}\right) = - \mathbb {E} _ {\left(p, y _ {w}, y _ {l}\right) \sim \mathcal {D}} \left[ \log \sigma \left(\beta \log \frac {\pi_ {\theta} \left(y _ {w} \mid p\right)}{\pi_ {r e f} \left(y _ {w} \mid p\right)} - \beta \log \frac {\pi_ {\theta} \left(y _ {l} \mid p\right)}{\pi_ {r e f} \left(y _ {l} \mid p\right)}\right) \right], \tag {1}
+$$
+
+where  $p$  is a prompt consisting of a concatenation between task info and conversation history  $= \{x_{1},y_{1},\dots,x_{i - 1},y_{i - 1},x_{i}\}$  with each  $x_{i}$  and  $y_{i}$  representing observed user-side and system-side utterances at turn  $i$ ;  $y_{w}$  and  $y_{l}$  are the designated "winning" and "losing" responses or trajectories as set in Sec. 3.2.2;  $\pi_{ref}$  is the initial reference policy model; and  $\beta$  is a hyperparameter that regularizes the ratio between  $\pi_{\theta}$  and  $\pi_{ref}$ . The gradient of this objective is given as follows:
+
+$$
+\begin{array}{l} \nabla_ {\theta} \mathcal {L} _ {\mathrm {D P O}} (\pi_ {\theta}; \pi_ {r e f}) = \\ \left. - \beta \mathbb {E} _ {(p, y _ {w}, y _ {l}) \sim \mathcal {D}} \left[ \sigma \left(\hat {R} _ {\theta} (p, y _ {l}) - \hat {R} _ {\theta} (p, y _ {w})\right) \left[ \nabla_ {\theta} \log \pi \left(y _ {w} \mid p\right) - \nabla_ {\theta} \log \pi \left(y _ {l} \mid p\right) \right] \right], \right. \tag {2} \\ \end{array}
+$$
+
+Algorithm 2 ACT: Action-Based Contrastive Self-Training  
+input Initial Policy Model  $\pi_{\theta_0}$ , Action Contrast Dataset  $D_{pref}$ , Number of Batches  $B$ , Action Classifier  $A$ , User Simulator  $U$ , Task Heuristic  $H$ , Heuristic Tolerance  $\epsilon$   
+1: for conversation turn  $t_i$  in batch  $b_j$  sampled from  $D_{pref}$  where  $0 \leq j \leq B$  do  
+2: Sample  $y_i \sim P_{\theta_j}(|p_i|)$  ▷ Sample a response from the current model policy  
+3: if Action  $A(y_i) \neq$  Action  $a_i$  then  
+4: Set  $y_{li} = y_i$  ▷ Implicit pragmatic action does not match ground truth  
+5: else  
+6: Initialize Trajectory  
+7: Add  $y_i$  to Trajectory  
+8: while  $A(y_i) \neq$  ANSWER do  
+9: Clarification Answer  $= P_U(p; y_i)$  ▷ Simulate User Clarification  
+10: Add Clarification Answer to Trajectory  
+11:  $y_{i+1}' = P_{\pi_\theta}(P; y_i)$  ▷ Simulate next policy response  
+12: Add  $y_{i+1}'$  to Trajectory  
+13: if  $H(Trajectory$  outcome, Ground Truth Outcome  $g_i) > \epsilon$  then  
+14: Let  $y_{wi} =$  Trajectory ▷ Reward acceptable trajectory outcome  
+15: else  
+16: Let  $y_{li} =$  Trajectory ▷ Penalize bad trajectory outcome  
+17:  $\theta \gets$  Update  $(\theta)$  until convergence (eq 2)  
+output  $\pi_{\theta_B}$
+
+where  $R(p,y) = \beta \log \frac{\pi(y|p)}{\pi_{ref}(y|p)}$  for a given policy model  $\pi$  and reference model  $\pi_{ref}$ , as proven under the assumptions in Rafailov et al. (2024). The intuition behind the objective is that the gradient of the loss function would increase the likelihood of winning responses  $y_{w} \in Y_{w}$  and would decrease the likelihood of losing responses  $y_{l} \in Y_{l}$ , with each example being weighed by the magnitude of how incorrectly the implicitly defined reward model ranks the paired responses.
+
+# 4 EXPERIMENTAL SETUP
+
+$ACT$  is a sample-efficient approach to adapt an LLM to a conversational action policy. We are primarily concerned with learning optimized implicit selection for agent-side clarification question asking, and we thus evaluate  $ACT$  as a tuning approach for three complex conversational information-seeking tasks. As a base model for our tuning experiments, we use Zephyr  $\beta$ , a version of Mistral 7B (Jiang et al., 2023) which has been instruction tuned on UltraChat and aligned to human preferences on UltraFeedback (Cui et al., 2023; Ding et al., 2023; Tunstall et al., 2023).
+
+Table 1: Experimental results on PACIFIC's public evaluation set.  $ACT$  achieves the strongest performance compared to all tuning approaches across every condition in every metric. Tuning-based adaptation strategies are not given any in-context examples at inference time, whereas ICL baselines include 10 in-context conversation examples using the same strategy.  
+
+<table><tr><td colspan="3">Adapton Setting</td><td>Action-level</td><td colspan="3">Content-level</td></tr><tr><td>Base Model</td><td>Approach</td><td>Conversations</td><td>Macro F1 ↑</td><td>Turn F1 ↑</td><td>Traj. F1 ↑</td><td>Post-Clarify F1 ↑</td></tr><tr><td>Gemini Pro</td><td>Standard ICL</td><td>10</td><td>81.4</td><td>59.7</td><td>58.7</td><td>49.7</td></tr><tr><td>Claude Sonnet</td><td>Standard ICL</td><td>10</td><td>71.9</td><td>43.7</td><td>42.0</td><td>28.5</td></tr><tr><td>Gemini Pro</td><td>SFT</td><td>50</td><td>71.2</td><td>51.8</td><td>45.7</td><td>9.9</td></tr><tr><td>Gemini Pro</td><td>SFT</td><td>100</td><td>75.2</td><td>64.3</td><td>54.6</td><td>8.5</td></tr><tr><td>Gemini Pro</td><td>SFT</td><td>250</td><td>88.0</td><td>67.4</td><td>59.3</td><td>10.2</td></tr><tr><td>Zephyr 7B-β</td><td>SFT</td><td>50</td><td>69.0</td><td>57.8</td><td>61.3</td><td>43.5</td></tr><tr><td>Zephyr 7B-β</td><td>IRPO</td><td>50</td><td>67.7</td><td>59.1</td><td>56.7</td><td>34.4</td></tr><tr><td>Zephyr 7B-β</td><td>ACT (ours)</td><td>50</td><td>82.2</td><td>62.8</td><td>61.9</td><td>57.2</td></tr><tr><td>Zephyr 7B-β</td><td>SFT</td><td>100</td><td>82.3</td><td>58.6</td><td>60.3</td><td>49.9</td></tr><tr><td>Zephyr 7B-β</td><td>IRPO</td><td>100</td><td>84.5</td><td>60.4</td><td>55.2</td><td>38.2</td></tr><tr><td>Zephyr 7B-β</td><td>ACT (ours)</td><td>100</td><td>86.0</td><td>65.0</td><td>62.0</td><td>57.4</td></tr><tr><td>Zephyr 7B-β</td><td>SFT</td><td>250</td><td>86.9</td><td>65.1</td><td>63.3</td><td>56.7</td></tr><tr><td>Zephyr 7B-β</td><td>IRPO</td><td>250</td><td>85.4</td><td>64.9</td><td>58.4</td><td>40.3</td></tr><tr><td>Zephyr 7B-β</td><td>ACT (ours)</td><td>250</td><td>89.6</td><td>68.1</td><td>65.7</td><td>62.0</td></tr></table>
+
+Table 2: Abg-CoQA test set evaluation results. Although Claude Sonnet achieves the highest Action-level performance when prompted with in-distribution in-context conversation examples, it does not result in improved multi-turn goal completion.  $ACT$  combines on-policy sampling with multi-turn simulation to achieve the best multi-turn goal completion ability on all data settings.  
+
+<table><tr><td colspan="3">Adaptation Setting</td><td>Action-level</td><td colspan="2">Content-level</td></tr><tr><td>Base Model</td><td>Approach</td><td>Conversations</td><td>Macro F1 ↑</td><td>Turn Similarity ↑</td><td>Traj. Similarity ↑</td></tr><tr><td>Gemini Pro</td><td>Standard ICL</td><td>10</td><td>55.5</td><td>67.0</td><td>72.2</td></tr><tr><td>Claude Sonnet</td><td>Standard ICL</td><td>10</td><td>66.0</td><td>50.1</td><td>54.3</td></tr><tr><td>Zephyr 7B-β</td><td>SFT</td><td>50</td><td>44.6</td><td>53.3</td><td>64.2</td></tr><tr><td>Zephyr 7B-β</td><td>ACT (ours)</td><td>50</td><td>52.3</td><td>66.2</td><td>68.8</td></tr><tr><td>Zephyr 7B-β</td><td>SFT</td><td>100</td><td>52.6</td><td>63.1</td><td>69.4</td></tr><tr><td>Zephyr 7B-β</td><td>ACT (ours)</td><td>100</td><td>51.1</td><td>69.5</td><td>71.4</td></tr><tr><td>Zephyr 7B-β</td><td>SFT</td><td>250</td><td>53.5</td><td>64.0</td><td>66.2</td></tr><tr><td>Zephyr 7B-β</td><td>ACT (ours)</td><td>250</td><td>53.3</td><td>72.5</td><td>75.1</td></tr></table>
+
+# 4.1 DATASETS
+
+We investigate three mixed-initiative conversation tasks in which a user interacts with an assistant to retrieve some information. In our setup of each task, a user asks a query which may or may not be underspecified. The assistant is tasked with providing a response which may either be a clarifying question or an attempt to directly answer the user's query. For each task, we synthesize the initial rejected responses by prompting Gemini Ultra as the conditional generation model,  $M$ . ACT is evaluated on a diverse set of datasets spanning various domains: tabular conversational QA, conversational QA for machine reading comprehension, and conversational text-to-SQL generation.
+
+# 4.1.1 PACIFIC: CONVERSATIONAL QA FOR TABULAR DATA
+
+PACIFIC is a task for proactive conversational question answering grounded on a mixture of tabular and textual financial data (Deng et al., 2022). This may involve generating the correct words from a given span, from multiple spans, or providing a correct arithmetic expression. The official evaluation for PACIFIC uses a numeracy-focused token overlap metric called DROP F1.
+
+# 4.1.2 ABG-COQA: CONVERSATIONAL QA FOR MACHINE READING COMPREHENSION
+
+Abg-CoQA is a conversational question answering dataset for disambiguation in machine reading comprehension (Guo et al., 2021). As there are no arithmetic expressions, we use embedding-based semantic distance with SentenceBERT (Reimers & Gurevych, 2019) as an evaluation metric, which has been used to more flexibly measure question-answering performance (Risch et al., 2021).
+
+# 4.1.3 AMBIGSQL: AMBIGUOUS CONVERSATIONAL TEXT-TO-SQL GENERATION
+
+AmbigSQL is our new task for SQL-grounded conversational disambiguation. We systematically perturbed unambiguous queries from Spider, a popular single-turn text-to-SQL benchmark (Yu et al., 2018), resulting in paired training examples which can be easily incorporated into contrastive RL tuning. Each trajectory is evaluated by whether the final proposed SQL query matches the ground truth query's execution result.
+
+Our motivation in constructing AmbigSQL stems from the idea that disambiguation can lead to improved task performance. We prompt an LLM to introduce three types of ambiguous information requests. Those in which the requested information is ambiguous (e.g., "Show details about singers ordered by age from the oldest to the youngest"), those in which the requested population is ambiguous (e.g., "Which ones who live in the state of Indiana?"; see Table A12), and finally, those in which the requested presentation of results is ambiguous (e.g. "Show name, country, age for all singers ordered by age"; see Table A13). We found that constructing a SQL query for an underspecified request with and without clarifications can result in a performance gap of up to  $45.8\%$  (see Table A14), demonstrating the necessity of clarifying questions. We provide additional details in Appendix C.
+
+# 4.2 EVALUATION SETUP
+
+We conduct evaluations of  $ACT$ 's ability to reason about ambiguity in conversation to better accomplish conversational goals along two dimensions.
+
+Table 3: AmbigSQL test set evaluation. Zephyr tuned with  $ACT$  is able to achieve the strongest task performance within each data setting. There are especially large performance improvements in post-clarification SQL execution match when data resources are more scarce.  
+
+<table><tr><td colspan="3">Adaptation Setting</td><td colspan="2">Action-level</td><td colspan="2">Content-level</td></tr><tr><td>Base Model</td><td>Approach</td><td>Conversations</td><td>Accuracy ↑</td><td>Macro F1 ↑</td><td>Execution Match ↑</td><td>PC Execution Match ↑</td></tr><tr><td>Gemini Pro</td><td>Standard ICL</td><td>10</td><td>72.1</td><td>70.9</td><td>63.5</td><td>75.2</td></tr><tr><td>Claude Sonnet</td><td>Standard ICL</td><td>10</td><td>68.5</td><td>63.8</td><td>66.5</td><td>72.4</td></tr><tr><td>Zephyr 7B-β</td><td>SFT</td><td>50</td><td>77.4</td><td>77.4</td><td>21.9</td><td>13.9</td></tr><tr><td>Zephyr 7B-β</td><td>IRPO</td><td>50</td><td>91.0</td><td>91.0</td><td>27.8</td><td>30.8</td></tr><tr><td>Zephyr 7B-β</td><td>ACT (ours)</td><td>50</td><td>80.8</td><td>80.7</td><td>43.6</td><td>38.1</td></tr><tr><td>Zephyr 7B-β</td><td>SFT</td><td>100</td><td>97.2</td><td>97.2</td><td>43.3</td><td>34.3</td></tr><tr><td>Zephyr 7B-β</td><td>IRPO</td><td>100</td><td>96.2</td><td>96.1</td><td>45.0</td><td>37.0</td></tr><tr><td>Zephyr 7B-β</td><td>ACT (ours)</td><td>100</td><td>99.2</td><td>99.3</td><td>48.0</td><td>49.6</td></tr><tr><td>Zephyr 7B-β</td><td>SFT</td><td>250</td><td>99.8</td><td>99.7</td><td>51.0</td><td>50.7</td></tr><tr><td>Zephyr 7B-β</td><td>IRPO</td><td>250</td><td>97.0</td><td>97.1</td><td>49.7</td><td>45.6</td></tr><tr><td>Zephyr 7B-β</td><td>ACT (ours)</td><td>250</td><td>99.9</td><td>99.8</td><td>52.3</td><td>53.0</td></tr><tr><td>Zephyr 7B-β</td><td>SFT</td><td>14,000 (All)</td><td>99.8</td><td>99.8</td><td>63.1</td><td>60.4</td></tr></table>
+
+Agent task performance: We evaluate whether  $ACT$  improves multi-turn task completion capabilities. PACIFIC and Abg-CoQA are initially proposed only with static single-turn evaluations. We mirror this by conducting a turn-level evaluation where we compare the model's response to the ground truth utterance given in response to the user's query, using the task-specific heuristics given in Sec. 4.1. Since we are specifically concerned with improving LLMs' multi-turn capabilities, we additionally propose a multi-turn evaluation scheme for the trajectory outcomes in all three tasks considered. While the sampled response from an LLM is a clarifying question, we simulate a user response and re-sample another response from the evaluated LLM until it attempts to answer the original query. We evaluate this outcome against the user's ground truth information-seeking goal. We use  $A$  and  $U$  for simulation as described in Sec. 3.2.2 for  $ACT$ , and use the heuristics defined in Sec. 4.1. An example is illustrated in Fig A5. In PACIFIC and AmbigSQL, we also compute task performance on the simulated responses in which the model has previously asked any clarifying questions, in order to get a more fine-grained measure of the model's ability to reason about its own clarification questions. Details of each evaluation metric for each task are provided in Appendix D.
+
+Implicit ambiguity recognition: To help further understand an agent's multi-turn task completion ability, we consider "dialogue act accuracy" (Chen et al., 2023b). Assuming access to ground-truth ambiguity labels, given a contextually-ambiguous user request, a model should generate a clarifying question, otherwise, it should attempt to provide the requested information. We primarily consider Macro F1 since PACIFIC and Abg-CoQA have highly imbalanced classes.
+
+# 4.3 BASELINES
+
+Prompting baselines We compare our tuning approaches with smaller models against various prompt-based approaches for multiple frontier LLMs: Gemini 1.5 Pro, Gemini 1.5 Flash, Claude 3.5 Sonnet, and Claude 3.0 Haiku<sup>5</sup>. The results for Gemini Flash and Claude Haiku are included in Appendix B due to space constraints. We use 10 conversations as in-context examples, with three different prompting frameworks: i.) "Standard" which uses the same instruction formatting used for tuning; ii.) chain-of-thought reasoning (Wei et al., 2022); and iii.) "Proactive MIPrompt", the prompting baseline in Deng et al. (2023c), which is a combination of the mixed-initiative prompting approach used in Chen et al. (2023b) and Proactive Prompting (Deng et al., 2023b). We provide a detailed description of each style with examples in Appendix E.
+
+Tuning baselines We compare ACT with supervised fine-tuning (SFT) as well as other off-policy and on-policy approaches to DPO-based alignment. For SFT, we use the ground truth responses for each dataset's training split. As for DPO-based alignment, an on-policy variant called Iterative Reasoning Preference Optimization (IRPO) was recently proposed and has gained traction for improving model performance in reasoning tasks such as arithmetic. We have thus evaluated IRPO on our two quantitative reasoning tasks, PACIFIC and AmbigSQL. A popular off-policy approach is to sample responses from two high capacity models, with  $Y_{w}$  coming from whichever model is
+
+Table 4: Examining ACT on PACIFIC with unlabeled conversational data. We assume no access to action labels and instead use 0-shot Gemini Pro as the source of action label supervision.  
+
+<table><tr><td colspan="4">Task Adaptation Environment</td><td>Action-level</td><td colspan="3">Content-level</td></tr><tr><td>Base Model</td><td>Framework</td><td>Action Supervision</td><td>Tuning Ex.</td><td>Macro F1 ↑</td><td>Turn F1 ↑</td><td>Traj. F1 ↑</td><td>Post-Clarify F1 ↑</td></tr><tr><td>Zephyr 7B-β</td><td>SFT</td><td>NA</td><td>50</td><td>69.0</td><td>57.8</td><td>61.3</td><td>43.5</td></tr><tr><td>Zephyr 7B-β</td><td>ACT</td><td>Crowdsourced</td><td>50</td><td>82.2</td><td>62.8</td><td>61.9</td><td>57.2</td></tr><tr><td>Zephyr 7B-β</td><td>ACT</td><td>Pseudo-labeled</td><td>50</td><td>80.1</td><td>62.4</td><td>61.1</td><td>54.7</td></tr><tr><td>Zephyr 7B-β</td><td>SFT</td><td>NA</td><td>100</td><td>82.3</td><td>58.6</td><td>60.3</td><td>49.9</td></tr><tr><td>Zephyr 7B-β</td><td>ACT</td><td>Crowdsourced</td><td>100</td><td>86.0</td><td>65.0</td><td>62.0</td><td>57.4</td></tr><tr><td>Zephyr 7B-β</td><td>ACT</td><td>Pseudo-labeled</td><td>100</td><td>84.8</td><td>63.5</td><td>61.5</td><td>56.1</td></tr><tr><td>Zephyr 7B-β</td><td>SFT</td><td>NA</td><td>250</td><td>86.9</td><td>65.1</td><td>63.3</td><td>56.7</td></tr><tr><td>Zephyr 7B-β</td><td>ACT</td><td>Crowdsourced</td><td>250</td><td>89.6</td><td>68.1</td><td>65.7</td><td>62.0</td></tr><tr><td>Zephyr 7B-β</td><td>ACT</td><td>Pseudo-labeled</td><td>250</td><td>89.0</td><td>68.1</td><td>64.9</td><td>61.0</td></tr></table>
+
+of higher capacity (henceforth DPO-Dist; see Mitra et al. (2023); Mukherjee et al. (2023); Xu et al. (2024a)). We present DPO-Dist results in Appendix B.
+
+# 5 EXPERIMENTAL RESULTS
+
+To emulate real-world scenarios with limited data, we evaluate  $ACT$  as a tuning approach in different scenarios with limited conversation samples across a set of diverse tasks.
+
+# 5.1 CONVERSATIONAL QA WITH TABULAR GROUNDING
+
+In Table 1, we see that across all three data-efficient settings considered,  $ACT$  achieves the strongest performance across all metrics compared to both SFT and IRPO, which has the advantage of additional test-time computation (Snell et al., 2024; Pang et al., 2024). In particular,  $ACT$  achieves up to a  $19.1\%$  relative improvement over SFT when measuring the tuned model's ability to implicitly recognize ambiguity (from 69.0 to 82.2 Macro F1) given only 50 conversations as tuning data. We also observe that  $ACT$  has greatly improved data efficiency compared to adapter-based SFT with Gemini Pro, with a relative improvement of as high as  $35.7\%$  in multi-turn task performance (from 45.6 to 61.9 in terms of trajectory-level DROP F1). Additionally, tuning with  $ACT$  in these limited data settings grants the model the ability to match or outperform frontier LLMs used with in-context learning despite having zero in-context examples during inference. Overall, we find that on-policy learning and multi-turn trajectory simulation are crucial for improved multi-turn goal completion.
+
+# 5.2 CONVERSATIONAL QA FOR MACHINE READING COMPREHENSION
+
+Our results for Abg-CoQA are presented in Table 2. In all three data settings, we observe that  $ACT$  achieved the strongest performance in terms of task-specific metrics (notably, in terms of trajectory-level embedding similarity). However, in the setting with 100 and 250 conversations, Zephyr tuned with SFT slightly outperforms  $ACT$  in terms of implicit action recognition, although action-level performance primarily helps to contextualize clarification reasoning ability. We discuss this point further in Appendix A. Our approach leads to the strongest turn-level and trajectory-level task performance in all conditions, indicating improved multi-turn reasoning.
+
+# 5.3 CONVERSATIONAL TEXT-TO-SQL GENERATION
+
+We find that although the prompting baselines do not achieve as high Action Accuracy, the benchmarked frontier LLMs can achieve relatively strong downstream performance in terms of execution match. In contrast, tuning Zephyr with both SFT and  $ACT$  results in rather high Action Accuracy but lower text-to-SQL performance compared to the frontier LLMs. We observe that holistically,  $ACT$  achieves the largest relative performance improvements in multi-turn task performance compared to other tuning approaches, although the downstream SQL generation ability of larger models is much greater than that of smaller models. This is primarily due to the SQL generation benefiting greatly from scale (Sun et al., 2023b). It is possible that multi-turn performance on larger models can be improved further if  $ACT$  is applied, as it is even able to yield larger performance improvements than baseline approaches for quantitative reasoning such as IRPO.
+
+# 5.4 ACT IN-THE-WILD: LEARNING WITHOUT DIALOGUE ACTION SUPERVISION
+
+Although we have ambiguity labels in the tasks considered here and use them for supervision in Tables 1-3, we also demonstrate that it is possible to perform action-based tuning in the absence
+
+Table 5: Ablation study of various conditions using PACIFIC's 50 conversation setting.  
+
+<table><tr><td></td><td>Macro F1 ↑</td><td>Turn F1 ↑</td><td>Traj. F1 ↑</td><td>Post-Clarify F1 ↑</td></tr><tr><td colspan="5">Action Importance</td></tr><tr><td>ACT</td><td></td><td></td><td></td><td></td></tr><tr><td>w/ Random Actions</td><td>63.2</td><td>55.3</td><td>58.7</td><td>32.8</td></tr><tr><td colspan="5">Ablation of ACT subcomponents</td></tr><tr><td>ACT</td><td></td><td></td><td></td><td></td></tr><tr><td>w/o on-policy sampling</td><td>74.8</td><td>61.5</td><td>59.1</td><td>40.5</td></tr><tr><td>ACT</td><td></td><td></td><td></td><td></td></tr><tr><td>w/ sampling but w/o simulation</td><td>81.4</td><td>60.8</td><td>60.2</td><td>50.1</td></tr><tr><td>ACT (full)</td><td>82.2</td><td>62.8</td><td>61.9</td><td>57.2</td></tr><tr><td colspan="5">ACT with unaligned foundation models</td></tr><tr><td>Gemma 2B SFT</td><td>57.7</td><td>38.0</td><td>40.5</td><td>17.0</td></tr><tr><td>Gemma 2B ACT</td><td>62.7</td><td>42.6</td><td>44.0</td><td>24.8</td></tr><tr><td>Mistral 7B SFT</td><td>57.7</td><td>53.8</td><td>51.4</td><td>27.7</td></tr><tr><td>Mistral 7B ACT</td><td>75.7</td><td>58.1</td><td>57.6</td><td>31.9</td></tr></table>
+
+of action-label supervision. We use a pre-existing LLM, Gemini 1.5 Pro, as a zero-shot action annotator to re-label the ground truth Assistant-side turns on the PACIFIC corpus. We find that there is astonishingly high agreement (98.5%) with the ground truth action labels. Our results in terms of both Action-level and Content-level metrics reflect that there is nearly no empirical difference in performance. This highlights the potential of  $ACT$  being highly effective for adaptation to "in-the-wild" settings with small amounts of unlabeled conversational data.
+
+# 5.5 ABLATION STUDIES
+
+Are action-based preferences necessary? One of the key factors of  $ACT$  is that the contrastive pairs highlight differences between conversational actions. In Table 5 (" $ACT$  w/ Random Actions"), we additionally examine the importance of action selection by randomly sampling both the winning and losing action when constructing the preference pair, and observe this underperforms normal  $ACT$ .
+
+Do we need on-policy sampling? In Table 5 ("ACT w/o on-policy sampling"), we examine the importance of on-policy sampling by evaluating normal off-policy DPO on the dataset as constructed in Sec. 3.2.1. While we do observe some improvements over SFT (e.g., from 69.0 to 74.8 Macro F1), the overall improvements are much larger when using on-policy sampling as with full ACT. This may be due to the fact that the off-policy negative responses are not guaranteed to lie in the language manifold of the policy model, and distribution shift may be too difficult to overcome with off-policy learning (Guo et al., 2024).
+
+Is trajectory simulation necessary?  $ACT$  is better-aligned with multi-turn conversations due to its on-policy trajectory simulation. Without multi-turn simulation, our approach can be viewed similarly to on-policy DPO variants like Pang et al. (2024), but with a conversation-specific reward signal which accounts for conversation actions and task heuristics. In Table 5 (" $ACT$  w/ sampling w/o simulation"), we find that this trajectory-level simulation is critical to improving multi-turn performance, especially the policy model's ability to reason about its own clarification questions.
+
+Is ACT model agnostic? The base model in our main experiments, Zephyr, is obtained by aligning Mistral. In Table 5 ("ACT with unaligned foundation models") we observe a performance gap of 6.5 Action F1 and 4.3 Trajectory F1 after ACT tuning for the two models. However, our results demonstrate ACT can improve performance regardless of pre-existing alignment with human feedback, although it can help as an improved model initialization. Overall, we find that improving base model performance with ACT is model agnostic.
+
+# 6 CONCLUSION
+
+We propose  $ACT$ , a model agnostic quasi-online contrastive tuning approach for sample-efficient conversational task adaptation, along with a workflow for evaluation of conversational agents. We demonstrate encouraging evidence that  $ACT$  is highly effective for task adaptation in the limited data regime, even when there are no action labels available. Future work may also consider combining  $ACT$  with existing sophisticated tuning approaches for complex tasks like text-to-SQL generation, as well as generalization to large-scale data and multi-task environments.
+
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+George KingsleyZipf.Human behavior and the principle of least effort.1949.
+
+# Prompt  $(\mathbf{p_i})$  ..
+
+You are a helpful Assistant answering some questions from a User about a Story. Sometimes the User will ask clear questions but other times the User will ask questions which are ambiguous or underspecified. [Story]
+
+Materialism is a form of philosophical monism which holds that matter is the fundamental substance in nature, and that all phenomena, including mental phenomena and consciousness, are identical with material interactions.
+
+Materialism is closely related to physicalism, the view that all that exists is ultimately physical. Philosophical physicalism has evolved from materialism with the discoveries of the physical sciences to incorporate more sophisticated notions of physicality than mere ordinary matter, such as: spacetime, physical energies and forces, dark matter, and so on. Thus the term "physicalism" is preferred over "materialism" by some, while others use the terms as if they are synonymous.
+
+Materialism belongs to the class of monist ontology. As such, it is different from ontological theories based on dualism or pluralism. For singular explanations of the phenomenal reality, materialism would be in contrast to idealism, neutral monism, and spiritualism.
+
+Despite the large number of philosophical schools and subtle nuances between many, all philosophies are said to fall into one of two primary categories, which are defined in contrast to each other: Idealism, and materialism.[a] The basic proposition of these two categories pertains to the nature of reality, and the primary distinction between them is the way they answer two fundamental questions: "what does reality consist of?" and "how does it originate?" To idealists, spirit or mind or the objects of mind (ideas) are primary, and matter secondary. To materialists, matter is primary, and mind or spirit or ideas are secondary, the product of matter acting upon matter. [Conversation]
+
+User: What would be secondary to them?
+
+Assistant: matter is primary, and mind or spirit or ideas are secondar
+
+User: What does the basic proposition of these two categories pertain to?
+
+Assistant: nature of reality
+
+User: What would materialism be in contrast to Assistant:
+
+Chosen Response  $(\mathbf{y}_{\mathbf{wi}})$ : Do you want to know what materialism would be in contrast to for singular explanations of the phenomenal reality or for the categories that all philosophies are said to fall into?
+
+# Chosen Action  $(\mathbf{a_j})$  :CLARIFY
+
+Rejected Response  $(\mathbf{y}_{\mathrm{II}})$ : For singular explanations of phenomenal reality, materialism contrasts with idealism, neutral monism, and spiritualism.
+
+# Rejected Action  $(\mathbf{a_i}^{\prime})$  : Answer
+
+Information Goal  $(\mathbf{g}_i)$ : In the context of the two primary philosophical categories, materialism is contrasted with idealism.
+
+# Gold Trajectory:
+
+Assistant: Do you want to know what materialism would be in contrast to for singular explanations of the phenomena reality or for the categories that all philosophies are said to fall into?
+
+User: For the categories that all philosophies are said to fall into.
+
+Assistant: In the context of the two primary philosophical categories, materialism is contrasted with idealism."
+
+Figure A4: Example of a contrastive pairing constructed for RL tuning with Abg-CoQA (Guo et al., 2021). The notation used is as described in Section 3.1.
+
+# A LIMITATIONS, ETHICAL CONSIDERATIONS, AND BROADER IMPACTS
+
+# A.1 DISCUSSION OF LIMITATIONS
+
+We assume that the clarification questions are appropriately timed. However, crowdsourced conversation datasets are often noisy (Chen et al., 2023b), and relying on noisy annotations or token sequences may result in suboptimal learned policies (from the perspective of asking unnecessary clarifying questions, as well as generating disfluent language). Depending on the source of data, it may be necessary to do an additional preprocessing stage in which one infers whether an action is useful or not. Shi et al. (2024) infers user satisfaction given model responses in-the-wild, whereas Yu et al. (2023) ranks dialogue actions using Monte-Carlo Tree Search.
+
+Label noise can also affect the implicit action recognition evaluation, which assumes that an action in a benchmark task is "optimal." In a corpus like PACIFIC with high inter-annotator agreement (0.62), this is a reasonable assumption. However, we observe inconsistency in Abg-CoQA which may be a result of the low inter-annotator agreement (0.26) reported in Guo et al. (2021). Recent work has demonstrated the promise of many-shot in-context learning Agarwal et al. (2024) compared to supervised fine-tuning at the trade-off of inference-time latency. Yet, Table A6 indicates that even with a greatly increased number of in-context conversation examples (e.g. 50, 100, and 250), the
+
+![](images/d21553465d76834880a23b6d3d64ae7992f6e4a8c4ab11724af560d444f526b8.jpg)  
+Figure A5: Trajectory-level content evaluation using the example scenario from Figure 1. Trajectory-level evaluation seeks to measure the extent to which a candidate LLM can interact with a "User" to reach a target information goal. The "interactive" evaluation of a given instance continues until the candidate LLM attempts to resolve the User's request by providing a direct answer. The candidate trajectory resolution is scored using downstream task metrics. In this example, DROP F1 is used following the task metrics for PACIFIC.
+
+Table A6: Analysis of the impact of additional data on Abg-CoQA. Additionally many-shot examples do not necessarily improve implicit action recognition performance. ACT tuning with Zephyr 7B greatly outperforms many-shot Gemini performance.  
+
+<table><tr><td colspan="3">Adaptation Setting</td><td>Action-level</td><td colspan="2">Content-level</td></tr><tr><td>Base Model</td><td>Approach</td><td>Conversations</td><td>Macro F1 ↑</td><td>Turn Similarity ↑</td><td>Traj. Similarity ↑</td></tr><tr><td>Gemini Pro</td><td>ICL</td><td>50</td><td>56.4</td><td>64.5</td><td>68.9</td></tr><tr><td>Zephyr 7B-β</td><td>ACT (ours)</td><td>50</td><td>52.3</td><td>66.2</td><td>68.8</td></tr><tr><td>Gemini Pro</td><td>ICL</td><td>100</td><td>59.2</td><td>67.0</td><td>72.0</td></tr><tr><td>Zephyr 7B-β</td><td>ACT (ours)</td><td>100</td><td>51.1</td><td>69.5</td><td>71.4</td></tr><tr><td>Gemini Pro</td><td>ICL</td><td>250</td><td>58.8</td><td>66.0</td><td>71.1</td></tr><tr><td>Zephyr 7B-β</td><td>ACT (ours)</td><td>250</td><td>53.3</td><td>72.5</td><td>75.1</td></tr></table>
+
+downstream disambiguation ability does not improve uniformly. We thus posit that in such scenarios, it is more important that for such corpora, multi-turn task completion is a more important measure of a model. We do find that even with 250 in-context examples, tuning a smaller model with  $ACT$  and the same conversation samples has the potential to outperform frontier models with many-shot examples.
+
+$ACT$  also makes use of task-specific heuristics. While this was intentional since success criteria can vary greatly across domains, this may also require more customization and engineering expertise/effort. Our overall approach to tuning and evaluation also makes heavy use of existing LLMs. We prompt Gemini for purposes such as Action Classification or User Simulation, but such approaches are not perfect and may occasionally result in unwanted behavior. These prompting approaches similarly may require substantial customization efforts. We also realize that not all researchers may have access to commercial LLMs due to researchers for financial or privacy reasons.
+
+Our study also focuses specifically on the limited data regime. We believe that such contexts (e.g., when the target user population is unknown; when conversational data cannot be collected due to privacy concerns; when a conversational system is in its early stages and collecting abundant data for development iteration is not feasible; etc.) would benefit the most from focused adaptation designed to fundamentally teach conversational skills approaches such as  $ACT$ . As such, in our paper, this was the focus of all of our experiments. It is not clear how much our findings would generalize in settings in which there is an abundance of training data whose distribution closely matches the target
+
+distribution. Intuitively, if much more in-distribution data is made available, even the performance of unaligned objectives like SFT would start to catch up to the performance of focused approaches.
+
+Is ACT online learning? Levine et al. (2020) defines offline reinforcement learning as using a fixed dataset of experiences, whereas online reinforcement learning relies on interacting with an environment in real-time. Additionally, Guo et al. (2024) defines on-policy sampling in contrastive RL tuning as settings where both the winning and losing responses are sampled from the policy model. In our case, during tuning, we sample a single response from the policy model.
+
+As such, we define  $ACT$  as a quasi-online contrastive RL tuning approach.  $ACT$  does rely on action-based preference dataset, as is common in fully-offline reinforcement learning. However,  $ACT$  continuously samples responses from the policy model in order to update the contrast pairing with good or bad trajectories. Overall, it has both dynamic and static components, so we refer to it as quasi-online.  $ACT$  also specifically is different from fully online DPO where both the winning and losing responses are sampled (i.e. in Guo et al. (2024)) because our focus is on conversational actions. There is no guarantee that sampling a response from the policy model twice will result in differing actions, unless you change the prompt. However, in that case, you would no longer be computing the DPO objective.
+
+By nature of the domains considered, the extent to which  $ACT$  allows for online exploration is also limited. As previously mentioned, our experiments are constrained by the fact that there is an objective right/wrong target answer. For instance, if the target answer is an arithmetic expression as is common in PACIFIC, there are a fairly limited number of unique trajectories (when inspected in terms of the number of tokens) that will arrive at that particular expression. In such cases, the trajectory sampled from the policy model during  $ACT$  tuning may not be any different from the offline trajectory found in the training data.
+
+# A.2 ETHICAL CONSIDERATIONS
+
+Usage and Safety We do not condone the use of our work for any unethical, unlawful, and/or harmful applications. In our work, we do not release any new model artifacts or web-scraped data, but we do not specifically introduce any model guardrails as a part of  $ACT$ . However, our implementation of  $ACT$  relies on other LLMs such as Gemini to produce an initial preference dataset, and to perform user simulation. Gemini is released with safety guardrails in place, but these inference-time guardrails may not be available when using open-source LLMs instead. We advise that any deployments of models tuned with  $ACT$  should consider adding safety guardrails at inference-time.
+
+Hallucinations One commonly documented concern with using LLMs is their tendency to hallucinate answers in Assistant QA contexts. A solution is to provide an LLM with information from a retriever (i.e., retrieval-augmented generation). Two of the datasets we use, PACIFIC and Abg-CoQA, mirror this setting by performing grounded QA using a mixture of long-context textual passages and tabular data. It follows that ACT could be further studied in combination with approaches for improved retrieval-augmented generation.
+
+Our evaluation criteria in this paper are also rather restrictive towards hallucinations. In PACIFIC, we use a token-level metric (DROP F1); in Abg-CoQA, we evaluate a candidate response's semantic similarity with a ground truth answer; in AmbigSQL we use execution match, which is a fully objective metric. As such, it is difficult to perform well on PACIFIC and Abg-CoQA if a model consistently hallucinates answers, and in AmbigSQL, a "hallucinated" response would not consist of the appropriate SQL code.
+
+# A.3 BROADER IMPACTS
+
+There is an abundance of modern conversational assistants.  $ACT$  seeks to improve multi-turn conversational experiences, and thus, it can be used to improve many applications used by potentially millions of users around the world. However, the popularity of conversational assistants also creates an increased risk of misuse. Some people may develop conversational for unethical applications such as misinformation generation, or gray areas such as optimizing dialogue generation for content which is not suitable for the workplace. As discussed above, we do not condone the use of  $ACT$  for any
+
+Table A7: Experimental results on PACIFIC's public evaluation set with additional results using Gemini Flash and Claude Haiku. ACT achieves the strongest performance compared to all tuning approaches across every condition in every metric. Tuning-based adaptation strategies are not given any in-context examples at inference time, whereas inference-time adaptation strategies are prompted with 10 in-context conversation examples using the same strategy.  
+
+<table><tr><td colspan="3">Adaption Setting</td><td>Action-level</td><td></td><td colspan="2">Content-level</td></tr><tr><td>Base Model</td><td>Approach</td><td>Conversations</td><td>Macro F1 ↑</td><td>Turn F1 ↑</td><td>Traj. F1 ↑</td><td>Post-Clarify F1 ↑</td></tr><tr><td>Gemini Pro</td><td>Standard Prompt</td><td>10</td><td>81.4</td><td>59.7</td><td>58.7</td><td>49.7</td></tr><tr><td>Gemini Pro</td><td>Chain-of-Thought</td><td>10</td><td>86.3</td><td>66.3</td><td>17.1</td><td>19.2</td></tr><tr><td>Gemini Pro</td><td>Proactive MIPrompt</td><td>10</td><td>78.9</td><td>63.4</td><td>61.1</td><td>18.9</td></tr><tr><td>Gemini Flash</td><td>Standard Prompt</td><td>10</td><td>67.4</td><td>58.8</td><td>58.7</td><td>17.9</td></tr><tr><td>Gemini Flash</td><td>Chain-of-Thought</td><td>10</td><td>77.1</td><td>62.0</td><td>16.9</td><td>20.0</td></tr><tr><td>Gemini Flash</td><td>Proactive MIPrompt</td><td>10</td><td>76.8</td><td>64.0</td><td>62.0</td><td>24.4</td></tr><tr><td>Claude Sonnet</td><td>Standard Prompt</td><td>10</td><td>71.9</td><td>43.7</td><td>42.0</td><td>28.5</td></tr><tr><td>Claude Sonnet</td><td>Chain-of-Thought</td><td>10</td><td>80.0</td><td>37.2</td><td>13.0</td><td>6.8</td></tr><tr><td>Claude Sonnet</td><td>Proactive MIPrompt</td><td>10</td><td>74.9</td><td>47.2</td><td>45.9</td><td>7.6</td></tr><tr><td>Claude Haiku</td><td>Standard Prompt</td><td>10</td><td>46.9</td><td>26.4</td><td>26.2</td><td>—</td></tr><tr><td>Claude Haiku</td><td>Chain-of-Thought</td><td>10</td><td>48.6</td><td>23.7</td><td>12.0</td><td>2.9</td></tr><tr><td>Claude Haiku</td><td>Proactive MIPrompt</td><td>10</td><td>48.3</td><td>18.6</td><td>18.2</td><td>7.3</td></tr><tr><td>Gemini Pro</td><td>SFT</td><td>50</td><td>71.2</td><td>51.8</td><td>45.7</td><td>9.9</td></tr><tr><td>Gemini Pro</td><td>SFT</td><td>100</td><td>75.2</td><td>64.3</td><td>54.6</td><td>8.5</td></tr><tr><td>Gemini Pro</td><td>SFT</td><td>250</td><td>88.0</td><td>67.4</td><td>59.3</td><td>10.2</td></tr><tr><td>Zephyr 7B-β</td><td>SFT</td><td>50</td><td>69.0</td><td>57.8</td><td>61.3</td><td>43.5</td></tr><tr><td>Zephyr 7B-β</td><td>DPO-Dist (Pro v. Flash)</td><td>50</td><td>75.5</td><td>61.7</td><td>55.7</td><td>30.8</td></tr><tr><td>Zephyr 7B-β</td><td>DPO-Dist (Sonnet v. Haiku)</td><td>50</td><td>74.8</td><td>62.0</td><td>56.3</td><td>31.9</td></tr><tr><td>Zephyr 7B-β</td><td>IRPO</td><td>50</td><td>67.7</td><td>59.1</td><td>56.7</td><td>34.4</td></tr><tr><td>Zephyr 7B-β</td><td>ACT (ours)</td><td>50</td><td>82.2</td><td>62.8</td><td>61.9</td><td>57.2</td></tr><tr><td>Zephyr 7B-β</td><td>SFT</td><td>100</td><td>82.3</td><td>58.6</td><td>60.3</td><td>49.9</td></tr><tr><td>Zephyr 7B-β</td><td>DPO-Dist (Pro v. Flash)</td><td>100</td><td>68.8</td><td>53.3</td><td>53.3</td><td>31.7</td></tr><tr><td>Zephyr 7B-β</td><td>DPO-Dist (Sonnet v. Haiku)</td><td>100</td><td>83.0</td><td>59.0</td><td>53.7</td><td>29.3</td></tr><tr><td>Zephyr 7B-β</td><td>IRPO</td><td>100</td><td>84.5</td><td>60.4</td><td>55.2</td><td>38.2</td></tr><tr><td>Zephyr 7B-β</td><td>ACT (ours)</td><td>100</td><td>86.0</td><td>65.0</td><td>62.0</td><td>57.4</td></tr><tr><td>Zephyr 7B-β</td><td>SFT</td><td>250</td><td>86.9</td><td>65.1</td><td>63.3</td><td>56.7</td></tr><tr><td>Zephyr 7B-β</td><td>DPO-Dist (Pro v. Flash)</td><td>250</td><td>65.6</td><td>53.6</td><td>54.1</td><td>30.9</td></tr><tr><td>Zephyr 7B-β</td><td>DPO-Dist (Sonnet v. Haiku)</td><td>250</td><td>82.8</td><td>43.3</td><td>38.6</td><td>19.6</td></tr><tr><td>Zephyr 7B-β</td><td>IRPO</td><td>250</td><td>85.4</td><td>64.9</td><td>58.4</td><td>40.3</td></tr><tr><td>Zephyr 7B-β</td><td>ACT (ours)</td><td>250</td><td>89.6</td><td>68.1</td><td>65.7</td><td>62.0</td></tr></table>
+
+unethical, harmful, or unlawful applications, and we suggest the usage of safety guardrails for any deployments.
+
+# B ADDITIONAL EXPERIMENTAL RESULTS
+
+As described in Section 4.3, we include additional prompting experiments using Gemini 1.5 Flash and Claude 3.0 Haiku using Standard Prompting, Chain-of-Thought Prompting, and Proactive Mixed-Initiative Prompting. We also include additional comparisons against an off-policy DPO baseline, DPO-Dist, which involves distilling from two LLMs of differing sizes. The full results for PACIFIC are displayed in Table A7, the full results for Abg-CoQA are in Table A8, and the full results for AmbigSQL are in Table A9.
+
+Table A8: Abg-CoQA test set evaluation results with additional results using Gemini Flash and Claude Haiku.  $ACT$  outperforms SFT across all evaluations in all three data settings. However, Gemini Ultra achieves the strongest downstream task performance when prompted with in-distribution in-context conversation examples.  
+
+<table><tr><td colspan="3">Adaptation Setting</td><td>Action-level</td><td colspan="2">Content-level</td></tr><tr><td>Base Model</td><td>Approach</td><td>Conversations</td><td>Macro F1 ↑</td><td>Turn Similarity ↑</td><td>Traj. Similarity ↑</td></tr><tr><td>Gemini Pro</td><td>Standard Prompt</td><td>10</td><td>55.5</td><td>67.0</td><td>72.2</td></tr><tr><td>Gemini Pro</td><td>Chain-of-Thought</td><td>10</td><td>61.2</td><td>63.4</td><td>39.1</td></tr><tr><td>Gemini Pro</td><td>Proactive MIPrompt</td><td>10</td><td>55.5</td><td>63.3</td><td>33.3</td></tr><tr><td>Gemini Flash</td><td>Standard Prompt</td><td>10</td><td>52.6</td><td>62.5</td><td>67.4</td></tr><tr><td>Gemini Flash</td><td>Chain-of-Thought</td><td>10</td><td>61.2</td><td>56.5</td><td>36.6</td></tr><tr><td>Gemini Flash</td><td>Proactive MIPrompt</td><td>10</td><td>58.1</td><td>61.7</td><td>36.1</td></tr><tr><td>Claude Sonnet</td><td>Standard Prompt</td><td>10</td><td>66.0</td><td>50.1</td><td>54.3</td></tr><tr><td>Claude Sonnet</td><td>Chain-of-Thought</td><td>10</td><td>63.7</td><td>46.2</td><td>36.8</td></tr><tr><td>Claude Sonnet</td><td>Proactive MIPrompt</td><td>10</td><td>57.2</td><td>60.8</td><td>32.9</td></tr><tr><td>Claude Haiku</td><td>Standard Prompt</td><td>10</td><td>49.3</td><td>40.9</td><td>41.7</td></tr><tr><td>Claude Haiku</td><td>Chain-of-Thought</td><td>10</td><td>46.2</td><td>30.7</td><td>28.0</td></tr><tr><td>Claude Haiku</td><td>Proactive MIPrompt</td><td>10</td><td>45.2</td><td>34.5</td><td>31.4</td></tr><tr><td>Zephyr 7B-β</td><td>SFT</td><td>50</td><td>44.6</td><td>53.3</td><td>64.2</td></tr><tr><td>Zephyr 7B-β</td><td>DPO-Dist (Pro v. Flash)</td><td>50</td><td>46.9</td><td>57.2</td><td>61.2</td></tr><tr><td>Zephyr 7B-β</td><td>DPO-Dist (Sonnet v. Haiku)</td><td>50</td><td>44.7</td><td>57.9</td><td>61.5</td></tr><tr><td>Zephyr 7B-β</td><td>ACT (ours)</td><td>50</td><td>52.3</td><td>66.2</td><td>68.8</td></tr><tr><td>Zephyr 7B-β</td><td>SFT</td><td>100</td><td>52.6</td><td>63.1</td><td>69.4</td></tr><tr><td>Zephyr 7B-β</td><td>DPO-Dist (Pro v. Flash)</td><td>100</td><td>47.8</td><td>61.9</td><td>67.1</td></tr><tr><td>Zephyr 7B-β</td><td>DPO-Dist (Sonnet v. Haiku)</td><td>100</td><td>44.8</td><td>62.0</td><td>66.4</td></tr><tr><td>Zephyr 7B-β</td><td>ACT (ours)</td><td>100</td><td>51.1</td><td>69.5</td><td>71.4</td></tr><tr><td>Zephyr 7B-β</td><td>SFT</td><td>250</td><td>53.5</td><td>64.0</td><td>66.2</td></tr><tr><td>Zephyr 7B-β</td><td>DPO-Dist (Pro v. Flash)</td><td>250</td><td>46.0</td><td>61.9</td><td>66.3</td></tr><tr><td>Zephyr 7B-β</td><td>DPO-Dist (Sonnet v. Haiku)</td><td>250</td><td>46.3</td><td>62.6</td><td>67.0</td></tr><tr><td>Zephyr 7B-β</td><td>ACT (ours)</td><td>250</td><td>53.3</td><td>72.5</td><td>75.1</td></tr></table>
+
+Table A9: AmbigSQL test set evaluation with additional results using Gemini Flash and Claude Haiku. Zephyr tuned with  $ACT$  is able to achieve the strongest task performance within each data setting. There are especially large performance improvements in post-clarification SQL execution match when data resources are more scarce.  
+
+<table><tr><td colspan="3">Adaptation Setting</td><td>Action-level</td><td colspan="2">Content-level</td></tr><tr><td>Base Model</td><td>Approach</td><td>Conversations</td><td>Accuracy ↑</td><td>Execution Match ↑</td><td>PC Execution Match ↑</td></tr><tr><td>Gemini Pro</td><td>Standard Prompt</td><td>10</td><td>72.1</td><td>63.5</td><td>75.2</td></tr><tr><td>Gemini Flash</td><td>Standard Prompt</td><td>10</td><td>75.6</td><td>64.2</td><td>66.2</td></tr><tr><td>Claude Sonnet</td><td>Standard Prompt</td><td>10</td><td>68.5</td><td>66.5</td><td>72.4</td></tr><tr><td>Claude Haiku</td><td>Standard Prompt</td><td>10</td><td>73.8</td><td>57.3</td><td>65.3</td></tr><tr><td>Zephyr 7B-β</td><td>SFT</td><td>50</td><td>77.4</td><td>21.9</td><td>13.9</td></tr><tr><td>Zephyr 7B-β</td><td>DPO-Dist (Pro v. Flash)</td><td>50</td><td>77.7</td><td>42.6</td><td>31.5</td></tr><tr><td>Zephyr 7B-β</td><td>DPO-Dist (Sonnet v. Haiku)</td><td>50</td><td>78.0</td><td>40.9</td><td>41.2</td></tr><tr><td>Zephyr 7B-β</td><td>IRPO</td><td>50</td><td>91.0</td><td>27.8</td><td>30.8</td></tr><tr><td>Zephyr 7B-β</td><td>ACT (ours)</td><td>50</td><td>80.8</td><td>43.6</td><td>38.1</td></tr><tr><td>Zephyr 7B-β</td><td>SFT</td><td>100</td><td>97.2</td><td>43.3</td><td>34.3</td></tr><tr><td>Zephyr 7B-β</td><td>DPO-Dist (Pro v. Flash)</td><td>100</td><td>98.7</td><td>45.1</td><td>45.3</td></tr><tr><td>Zephyr 7B-β</td><td>DPO-Dist (Sonnet v. Haiku)</td><td>100</td><td>99.8</td><td>47.8</td><td>44.8</td></tr><tr><td>Zephyr 7B-β</td><td>IRPO</td><td>100</td><td>96.2</td><td>45.0</td><td>37.0</td></tr><tr><td>Zephyr 7B-β</td><td>ACT (ours)</td><td>100</td><td>99.2</td><td>48.0</td><td>49.6</td></tr><tr><td>Zephyr 7B-β</td><td>SFT</td><td>250</td><td>99.8</td><td>51.0</td><td>50.7</td></tr><tr><td>Zephyr 7B-β</td><td>DPO-Dist (Pro v. Flash)</td><td>250</td><td>97.3</td><td>49.7</td><td>44.2</td></tr><tr><td>Zephyr 7B-β</td><td>DPO-Dist (Sonnet v. Haiku)</td><td>250</td><td>99.7</td><td>50.7</td><td>50.3</td></tr><tr><td>Zephyr 7B-β</td><td>IRPO</td><td>250</td><td>97.0</td><td>49.7</td><td>45.6</td></tr><tr><td>Zephyr 7B-β</td><td>ACT (ours)</td><td>250</td><td>99.9</td><td>52.3</td><td>53.0</td></tr><tr><td>Zephyr 7B-β</td><td>SFT</td><td>14,000 (All)</td><td>99.8</td><td>63.1</td><td>60.4</td></tr></table>
+
+# C AmbigSQL: MODELING Ambiguity IN CONVERSATIONAL TEXT-TO-SQL
+
+Table A10: Overview of AmbigSQL, an ambiguous Text-to-SQL dataset synthesized from Spider.  
+
+<table><tr><td></td><td>Train</td><td>Dev</td><td>Test</td></tr><tr><td>Num. Unambiguous Requests</td><td>7,000</td><td>1,034</td><td>1,034</td></tr><tr><td>Num. Ambiguous Requests</td><td>7,000</td><td>1,034</td><td>1,034</td></tr><tr><td>Num. Unique Schemas</td><td>1,056</td><td>145</td><td>145</td></tr><tr><td>Types of Ambiguity</td><td>3</td><td>3</td><td>3</td></tr></table>
+
+There is growing interest in using LLM-based agents for coding tasks (Liu et al., 2024). Particularly, due to the complexity of such tasks, multi-turn interactions in which an agent is able to clarify assumptions and user intents should intuitively help with goal completion (Nijkamp et al., 2023). Despite this, there are few existing resources for multi-turn code generation tasks. One example is CoSQL, a conversational text-to-SQL task which also includes linguistic ambiguities (Yu et al., 2019), but the proposed task does not include agent-style interaction in which a model must learn to ask clarifying questions. Upon our inspection of the dataset, there are also various inconsistencies related to the "system-side" clarification questions given in the dataset's conversational contexts, which we highlight in Table A11. As a result, we propose AmbigSQL, our own synthetically constructed resource for ambiguous conversational text-to-SQL.
+
+Table A11: Conversations in CoSQL with noisy "clarification questions" (highlighted in red). Example 1) is a remnant of crowdsourcing in which the system-side party makes mention of the task guideline. Example 2) demonstrates a system-side clarification question being asked prior to the user making any information requests. Example 3) The system-side clarification question makes reference to some prior database search result, but the execution feedback is not made accessible to the system during inference.  
+
+<table><tr><td>No.</td><td>Interacting Party</td><td>Utterance</td></tr><tr><td></td><td>User</td><td>Can you list all the singer ids that aren’t present in the song table?</td></tr><tr><td></td><td>Assistant</td><td>SELECT Name FROM singer WHERE Singer_ID NOT IN ...</td></tr><tr><td></td><td>User</td><td>Thanks!</td></tr><tr><td>1</td><td>Assistant</td><td>You should ask at least 3 questions</td></tr><tr><td>2</td><td>Assistant</td><td>Did you want the full name of makers and the number?</td></tr><tr><td>3</td><td>Assistant</td><td>Do you mean the address of the customer with first name Luis?</td></tr></table>
+
+# C.1 AMBIGSQL CONSTRUCTION
+
+We start from Spider, a popular single-turn text-to-SQL dataset and benchmark. An overview of AmbigSQL is given in Table A10. Due to the nature of the single-turn requests in Spider, each instance can be viewed as a conversation consisting of a single state  $t_1$  (see notation defined in Section 3). In  $t_1$ ,  $p_1$  contains any instructions, the database schema, and the user's request.  $r_1$  is the correct SQL query which yields the information requested by the user.  $g_1 = r_1$  because the trajectory ends on the same turn due to the system yielding the correct query.  $a_1 = ANSWER$  because the only action possible is to provide a SQL query.
+
+Our desired result in constructing AmbigSQL is to have a corpus which can be used to demonstrate to an LLM the linguistic differences between ambiguous and clear user requests. That is, given a fixed database schema and a target SQL query, we want a pair of requests such that one requires asking a clarification question and one does not. This would also result in a balanced dataset in which half of the requests require asking clarification questions, and half do not.
+
+We focus on three fundamental types of ambiguous information requests. Those in which the requested information is ambiguous (e.g., "Show details about singers ordered by age from the oldest to the youngest"), those in which the requested population is ambiguous (e.g., "Which ones who live
+
+Table A12: In-context example given as part of a prompt for creating information requests in which the target population is ambiguous. The format of the black text represents how a ground-truth request would be used to form the prompt for a target example. The blue text represents the content that would be synthesized from an LLM. We omit the database schema from the paper.  
+```txt
+[Database Schema Omitted]   
+The target SQL query is the following:   
+SELECT professional_id, last_name, cell_number FROM Professionals   
+WHERE state  $=$  'Indiana' UNION SELECT T1.professional_id, T1.last_name, T1.cell_number FROM Professionals AS T1 JOIN Treatments AS T2 ON T1.professional_id = T2.professional_id   
+GROUP BY T1.professional_id HAVING count  $(\star)$  >2   
+Here is a clear request that would correspond to this SQL query: "Which professionals live in the state of Indiana or have done treatment on more than 2 treatments? List his or her id, last name and cell phone."   
+Here is the same request converted into an ambiguous format by underspecifying the target columns: "Which ones live in the state of Indiana or have done treatment on more than 2 treatments?"   
+Here is an appropriate clarifying question to recover the clear request from the ambiguous request: "Are you asking about the Professionals?"
+```
+
+in the state of Indiana?"; see Table A12), and finally, those in which the requested presentation of results is ambiguous (e.g. "Show name, country, age for all singers ordered by age"; see Table A13).
+
+We iterate through each of the examples in Spider, and use an LLM (Gemini Ultra 1.0) to synthesize a perturbed version of each unambiguous query, along with an appropriate clarifying question. For queries which require some manipulation of results presentation, we prompt an LLM to perturb the query such that the requested presentation style becomes ambiguous. Otherwise, we randomly select a perturbation strategy: either masking the requested information (Table A13) or masking the requested population (Table A12). For each of these three strategies, we use five in-context examples demonstrating the appropriate masking process. The exact in-context examples are given in the attached code. Each of these ambiguous queries are thus associated with conversations containing "ground truth" states at two timesteps,  $t_1$  and  $t_2$ .  $p_1$  contains an ambiguous user request,  $r_1$  is the synthesized clarification question, and accordingly,  $a_1 = \text{CLARIFY}$ .  $p_2$  contains the disambiguated user request,  $r_2$  is the correct ground truth SQL query, and accordingly,  $a_2 = \text{ANSWER}$ .
+
+The code to create AmbigSQL will be released publicly. Each instance includes the database metadata included in Spider, but for all experiments used in this paper, the database schema is linearized into the format used for prompting in Sun et al. (2023a;b).
+
+Table A13: In-context example given as part of a prompt for creating information requests in which the target columns are ambiguous. The format of the black text represents how a ground-truth request would be used to form the prompt for a target example. The blue text represents the content that would be synthesized from an LLM. We omit the database schema from the paper.  
+```txt
+[Database Schema Omitted]   
+The target SQL query is the following:   
+SELECT professional_id, last_name, cell_number FROM Professionals WHERE state  $=$  'Indiana' UNION SELECT T1.professional_id, T1.last_name, T1.cell_number FROM Professionals AS T1 JOIN Treatments AS T2 ON T1.professional_id = T2.professional_id GROUP BY T1.professional_id HAVING count  $(\star)$  > 2   
+Here is a clear request that would correspond to this SQL query: "Which professionals live in the state of Indiana or have done treatment on more than 2 treatments? List his or her id, last name and cell phone."   
+Here is the same request converted into an ambiguous format by underspecifying the target columns: "Which professionals live in the state of Indiana or have done treatment on more than 2 treatments?"   
+Here is an appropriate clarifying question to recover the clear request from the ambiguous request: "Which information of the professionals do you want to know?"
+```
+
+Table A14: Examination of ambiguity in AmbigSQL. Competitive high capacity LLMs struggle with producing "correct" SQL queries given only ambiguous user requests. Including disambiguation turns in the prompts greatly improves execution match.  
+
+<table><tr><td>Model</td><td>Ambiguous Request Execution Match</td><td>Execution Match with Clarification Turns</td></tr><tr><td>Gemini Pro</td><td>28.5</td><td>68.7</td></tr><tr><td>Gemini Ultra</td><td>31.2</td><td>77.0</td></tr></table>
+
+# C.2 EXAMINING CLARIFICATION NEED IN AMBIGSQL
+
+We primarily are concerned with examining the extent to which clarification questions are necessary in providing the requested SQL queries. Inspired by the notion of "recovery" presented in Toles et al. (2023), we examine the performance in constructing each of the unique SQL queries with and without the gold clarification turns. Concretely, we first evaluated the Execution Match performance achieved by prompting LLMs with only the ambiguous versions of each user request, with each instruction including the instruction that the LLM must construct a SQL query. Then, we prompted LLMs to construct the same SQL queries, but given the disambiguation turns as conversation history (i.e., with context consisting of the original ambiguous request, the clarification question, and then non-ambiguous request).
+
+We conducted this analysis on the test set using two competitive LLMs, Gemini Pro and Gemini Ultra, with the Execution Match tool from Spider and CoSQL (Yu et al., 2018; 2019). Our results are shown in Table A14. Given only an ambiguous request, both Gemini Pro and Gemini Ultra struggle to consistently construct the correct SQL query. However, given disambiguation turns, Execution Match improves dramatically, approximating the performance on the validation set given in Sun et al. (2023a;b).
+
+# C.3 EXAMPLES
+
+Table A15 and Table A16 each contain a pair of examples from AmbigSQL's test set. Each example contains the prompt which is provided to an LLM, the immediate ground truth response to the user request provided as part of the prompt, and the resulting ground truth trajectory (for examples which include clarifying questions). The examples in each table are paired — the top example is directly taken from Spider and converted into a conversational format, and the bottom example is the result of introducing ambiguity into the first example.
+
+Table A15: Examples from the AmbigSQL test set using the "concert_singer" database. Black text: context provided to an LLM as a prompt. Blue text: ground-truth single-turn response. Purple text: target trajectory which should be achieved through simulation.  
+
+<table><tr><td>[Instruction]
+You are an Assistant having a conversation with a User who would like to retrieve some information from a SQL database with the schema provided below. If you are confident in the User&#x27;s intent, you should provide the appropriate SQL query. Otherwise, you should ask the User a clarifying question.
+[Database Schema Omitted]
+[Conversation]
+User: How many singers do we have?
+Assistant: SELECT count(*) FROM singer</td></tr><tr><td>[Instruction]
+You are an Assistant having a conversation with a User who would like to retrieve some information from a SQL database with the schema provided below. If you are confident in the User&#x27;s intent, you should provide the appropriate SQL query. Otherwise, you should ask the User a clarifying question.
+[Database Schema Omitted]
+[Conversation]
+User: Tell me about the singers.
+Assistant: What specifically would you like to know about the singers? For example, would you like to know how many there are, or see a list of their names?
+User: How many singers do we have?
+Assistant: SELECT count(*) FROM singer</td></tr></table>
+
+Table A16: Examples from the AmbigSQL test set using the "flight_2" database. Black text: context provided to an LLM as a prompt. Blue text: ground-truth single-turn response. Purple text: target trajectory which should be achieved through simulation.  
+
+<table><tr><td>[Instruction]
+You are an Assistant having a conversation with a User who would like to retrieve some information from a SQL database with the schema provided below. If you are confident in the User&#x27;s intent, you should provide the appropriate SQL query. Otherwise, you should ask the User a clarifying question.
+[Database Schema Omitted]
+[Conversation]
+User: Return the number of airports.
+Assistant: SELECT count(*) FROM AIRPORTS</td></tr><tr><td>[Instruction]
+You are an Assistant having a conversation with a User who would like to retrieve some information from a SQL database with the schema provided below. If you are confident in the User&#x27;s intent, you should provide the appropriate SQL query. Otherwise, you should ask the User a clarifying question.
+[Database Schema Omitted]
+[Conversation]
+User: How many are there?
+Assistant: Could you please specify which table you are referring to when you ask &#x27;How many are there?&#x27; 
+User: Return the number of airports.
+Assistant: SELECT count(*) FROM AIRPORTS</td></tr></table>
+
+# D ADDITIONAL DETAILS ON EVALUATION
+
+We define the following action-level evaluation metrics:
+
+Accuracy: Percentage of correct implicit actions  
+- Macro F1: Unweighted Average of F1 for each action
+
+We define the following content-level evaluation metrics:
+
+- Turn-level DROP F1: Average immediate response DROP F1 Deng et al. (2022); Dua et al. (2019)  
+- Trajectory-level DROP F1: Average trajectory-outcome DROP F1 Deng et al. (2022); Dua et al. (2019)  
+- Post-Clarification DROP F1: Average DROP F1 Deng et al. (2022); Dua et al. (2019) of responses which follow agent clarification turns  
+- Turn-level Similarity: Immediate response embedding similarity  
+- Trajectory-level Similarity: Trajectory outcome embedding similarity  
+- Trajectory-level Execution Match: Percentage of trajectory outcomes with correct execution results  
+- Post-Clarification Execution Match: Percentage of trajectory outcomes with correct execution results out of those that which contain clarification turns
+
+PACIFIC As described in Section 4, PACIFIC is a conversational question-answering dataset in which the final answers may involve generating the correct words from a given span, from multiple spans, or providing a correct arithmetic expression. As such, the authors propose using DROP F1 as the official evaluation metric. The way DROP F1 is used in the original paper Deng et al. (2022) is analogous to our aforementioned "Turn-level DROP F1." However, as this does not fully represent a model's conversational reasoning abilities, we additionally evaluate LLMs in the PACIFIC environment using Macro F1, Trajectory-level DROP F1, and Post-Clarification DROP F1. Concretely, the evaluation for some LLM  $\pi$  is as follows. Assume we have some example with prompt  $p$ , winning action  $a$ , ground truth response  $r$ , and trajectory-level information goal  $g$ . We sample a candidate response from the LLM:  $y \sim P_{\theta}(\cdot | p)$ . We then simulate the trajectory resulting from each response  $y$  according to Lines 6-12 in Algorithm 2 and obtain trajectory outcome  $g'$ . The aforementioned action-level metrics are computed using the implicit actions of each  $y$  with each ground truth implicit action  $a$ . Turn-level DROP F1 is computed between all sampled responses  $y$  and all ground truth responses  $r$ , and Trajectory-level DROP F1 is computed over all simulated trajectory outcomes  $g'$  and all ground truth information goals  $g$ . Post-Clarification F1 is defined as Trajectory-level F1 for only the subset of trajectories which include clarification turns.
+
+Abg-CoQA As previously mentioned in Section 4, Abg-CoQA is a conversational question-answering dataset for machine reading comprehension. Thus, we use embedding similarity Risch et al. (2021) as it allows for producing more coherent and diverse responses which may be scored lowly by criteria such as token-level F1 score. In the original paper, language models are only evaluated in terms of QA performance Guo et al. (2021), rather than their ability to disambiguate requests. Thus, for our evaluation, we remove all clarification turns from the prompt and require the LLM to produce clarifying questions on its own. However, unlike the other tasks considered in this paper, each ambiguous request is paired with all of the possible trajectories (i.e., the reasons why the request is considered ambiguous). We thus perform an evaluation for every ground truth trajectory, so that it is impossible for an LLM to achieve a high cumulative trajectory-level score simply by getting lucky at guessing the user's intent.
+
+Concretely, the evaluation for some LLM  $\pi$  is as follows. Assume we have some example with prompt  $p$ , winning action  $a$ , ground truth response  $r$ , and the set of trajectory-level information goals  $G$ . For every individual trajectory-level goal  $g \in G$ , we sample a candidate response from the LLM:  $y \sim P_{\theta}(\cdot | p)$  then simulate the trajectory resulting from each response  $y$  according to Lines 6-12 in Algorithm 2 and obtain trajectory outcome  $g'$ . As with PACIFIC, we compute Macro F1 using the implicit actions of each  $y$  with each ground truth implicit action  $a$ . We compute Turn-level similarity for each unique  $p$  between sampled responses  $y$  and ground truth responses  $r$ . We compute
+
+Table A17: "Standard" in-context example given as part of a prompt to an LLM for response generation in PACIFIC. The purple text is the overall system instruction. The blue text is what an LLM would have to produce at inference time.  
+
+<table><tr><td>You are an Assistant answering questions from a User. You should either attempt to answer the question or ask a clarifying question if there is any ambiguity. 
+[Table and Passage Omitted] 
+User: How much does the company expect to make contributions to the defined plans during the next fiscal year? 
+Assistant: [&#x27;What kind of defined plans are you asking about?] 
+User: The defined benefit plans and the defined contribution plan respectively. 
+Assistant: [&#x27;$5.1 million&#x27;, &#x27;$0.6 million&#x27;] 
+User: What is the value of the expected defined contribution plan as a percentage of the defined benefit plan during the next fiscal year? 
+Assistant: 0.6/5.1 = 11.76 
+User: How much would change if there is a 1% increase and a 1% decrease respectively in the discount rate? 
+Assistant: [&#x27;What kind of change are you asking about?] 
+User: The pension. 
+Assistant: [&#x27;$(39,145)&#x27;, &#x27;$49,361&#x27;] 
+User: What is the difference in that amount if the discount rate had a 1% increase and a 1% decrease? 
+Assistant: 49,361 - (39,145) = 88506 
+User: How much would the pension change if there is a 1% increase and a 1% decrease respectively? 
+Assistant: [&#x27;Which rate of increase and decrease are you talking about?] 
+User: The future salary growth. 
+Assistant: [&#x27;$7,572&#x27;, &quot;$(6,919)] 
+User: What is the difference between them? 
+Assistant: 7,572 - (6,919) = 14491</td></tr></table>
+
+Trajectory-level similarity over all simulated trajectory outcomes  $g'$  and all ground truth information goals  $g$ .
+
+AmbigSQL is a ambiguous text-to-SQL generation task. By design, the labels are balanced, so we only use Accuracy for action-level metrics. The primary objective is to produce SQL queries which result in correct executions, so we evaluate LLMs in terms of Trajectory-level and Post-Clarification Execution Match. For each instance in the test set, we sample a response from an LLM and compute the trajectory as specified in Lines 6-12 in Algorithm 2. Trajectory-level Execution Match includes SQL queries which are constructed with or without clarification turns, whereas Post-Clarification Execution Match only looks at SQL queries which are constructed following clarification turns.
+
+# E DESCRIPTION OF IN-CONTEXT LEARNING BASELINES
+
+We use several in-context learning baselines with frontier LLMs in Section 4. For each condition, we randomly sample 10 conversation examples from each task's 250-instance data pool, and apply one of the following prompting frameworks.
+
+Standard Prompting We simply provide in-context examples that are structurally identical to the inputs used for model tuning. Our format is similar to the formats used in Chen et al. (2023b); Deng et al. (2023b) and we provide an example in Table A17.
+
+Chain-of-Thought Prompting We integrate the popular reasoning framework, chain-of-thought prompting (Wei et al., 2022) into our aforementioned "Standard" conversational prompt format. Effectively, we ask an LLM to do end-to-end dialogue generation by having it first produce a reasoning chain which states whether the current context is ambiguous or not. We provide an example in Table A18.
+
+Proactive Mixed-Initiative Prompting Following the baselines given in Deng et al. (2023c), we apply the Proactive Prompting framework Deng et al. (2023b) mixed with the Mixed-Initiative Prompting style Chen et al. (2023b). Ultimately, the LLM conditions on the possible set of actions,
+
+Table A18: "Chain-of-Thought" in-context example given as part of a prompt to an LLM for response generation in PACIFIC. The purple text is the overall system instruction. The blue text is what an LLM would have to produce at inference time.  
+
+<table><tr><td>You are an Assistant answering questions from a User. You should either attempt to answer the question or ask a clarifying question if there is any ambiguity. 
+[Table and Passage Omitted] 
+User: What is the value of the expected defined contribution plan as a percentage of the defined benefit plan during the next fiscal year? 
+Instruction: If the user&#x27;s question is ambiguous, ask an appropriate clarifying question. Otherwise, directly answer the user&#x27;s question using the information from the passage context and the table. Let&#x27;s think step by step. 
+Reasoning: The user&#x27;s question is not ambiguous. Assistant: 0.6/5.1 = 11.76 
+User: How much would change if there is a 1% increase and a 1% decrease respectively in the discount rate? 
+Instruction: If the user&#x27;s question is ambiguous, ask an appropriate clarifying question. Otherwise, directly answer the user&#x27;s question using the information from the passage context and the table. Let&#x27;s think step by step. 
+Reasoning: The user&#x27;s question was ambiguous. Assistant: [’What kind of change are you asking about?’] 
+User: The pension. 
+Instruction: If the user&#x27;s question is ambiguous, ask an appropriate clarifying question. Otherwise, directly answer the user&#x27;s question using the information from the passage context and the table. Let&#x27;s think step by step. 
+Reasoning: The user&#x27;s question is not ambiguous. Assistant: [’$(39,145)’, $49,361] 
+User: What is the difference in that amount if the discount rate had a 1% increase and a 1% decrease? 
+Instruction: If the user&#x27;s question is ambiguous, ask an appropriate clarifying question. Otherwise, directly answer the user&#x27;s question using the information from the passage context and the table. Let&#x27;s think step by step. 
+Reasoning: The user&#x27;s question is not ambiguous. Assistant: 49,361 - (39,145) = 88506 User: How much would the pension change if there is a 1% increase and a 1% decrease respectively? 
+Instruction: If the user&#x27;s question is ambiguous, ask an appropriate clarifying question. Otherwise, directly answer the user&#x27;s question using the information from the passage context and the table. Let&#x27;s think step by step. 
+Reasoning: The user&#x27;s question was ambiguous. 
+Assistant: [’Which rate of increase and decrease are you talking about?’]</td></tr></table>
+
+along with interweaved natural language instructions that describe which actions correspond to existing dialogue turns. We provide an example in Table A19. We use this framework for each of the long-context dialogue corpora (PACIFIC and Abg-CoQA).
+
+Table A19: “Proactive Mixed-Initiative” in-context example given as part of a prompt to an LLM for response generation in PACIFIC. The purple text is the overall system instruction. The blue text is what an LLM would have to produce at inference time.  
+
+<table><tr><td>You are an Assistant answering questions from a User. You should either attempt to answer the question or ask a clarifying question if there is any ambiguity.
+[Table and Passage Omitted]
+User: How much does the company expect to make contributions to the defined plans during the next fiscal year?
+The user&#x27;s last question was ambiguous. The Assistant asks a clarifying question.
+Assistant: [&#x27;What kind of defined plans are you asking about?] 
+User: The defined benefit plans and the defined contribution plan respectively.
+The user&#x27;s last question was unambiguous. The Assistant directly answers the question.
+Assistant: [&#x27;5.1 million&#x27;, &#x27;0.6 million&#x27;]
+User: What is the value of the expected defined contribution plan as a percentage of the defined benefit plan during the next fiscal year?
+The user&#x27;s last question was unambiguous. The Assistant directly answers the question.
+Assistant: 0.6/5.1 = 11.76
+User: How much would change if there is a 1% increase and a 1% decrease respectively in the discount rate?
+The user&#x27;s last question was ambiguous. The Assistant asks a clarifying question.
+Assistant: [&#x27;What kind of change are you asking about?] 
+User: The pension.
+The user&#x27;s last question was unambiguous. The Assistant directly answers the question.
+Assistant: [&#x27;(39, 145)&#x27;, &#x27;49,361&#x27;]
+User: What is the difference in that amount if the discount rate had a 1% increase and a 1% decrease?
+The user&#x27;s last question was unambiguous. The Assistant directly answers the question.
+Assistant: 49,361 - (39,145) = 88506
+User: How much would the pension change if there is a 1% increase and a 1% decrease respectively?
+Actions: [&quot;Directly Answer&quot;, &quot;Ask a Clarification Question&quot;]
+Prompt: Given the task background and the conversation history, please use appropriate actions to generate the response.
+Response: The user&#x27;s last question was ambiguous. The Assistant asks a clarifying question.
+Assistant: [&#x27;Which rate of increase and decrease are you talking about?]</td></tr></table>
+
+Table A20: In-context example given as part of a prompt to an LLM for action classification with PACIFIC. The blue text represents the action for this example.  
+
+<table><tr><td>[Table and Passage Omitted]</td></tr><tr><td>User: What was the total number of acquisitions in FY19?</td></tr><tr><td>Assistant: 35 acquisitions</td></tr><tr><td>User: What are the benefits of active growth strategy?</td></tr><tr><td>Assistant: strengthens and scales the National Storage operating platform which drives efficiencies across the business.</td></tr><tr><td>User: What was the total NLA?</td></tr><tr><td>Assistant: Which region are you asking about?</td></tr><tr><td>The last Assistant utterance is a clarifying question.</td></tr></table>
+
+Table A21: In-context example given as part of a prompt to an LLM for action classification with Abg-CoQA. The blue text represents the action for this example.  
+
+<table><tr><td>[Passage Omitted]</td></tr><tr><td>User: And who in the group was anxious?</td></tr><tr><td>Assistant: Peppe</td></tr><tr><td>User: Was she well-rested?</td></tr><tr><td>Assistant: no</td></tr><tr><td>The last Assistant utterance is a direct answer.</td></tr></table>
+
+# F CONDITIONAL GENERATION MODEL DETAILS
+
+As mentioned Section 3.1, we make use of a high capacity LLM as a conditional generation model,  $M$ . For all experiments considered, we use Gemini Ultra to construct the initial action-based contrastive preference dataset. We follow the format of "mixed-initiative prompting" described in Chen et al. (2023b), rather than performing zero-shot inference, we use ten in-context examples to adapt the LLM to the prompting structure. Concretely, for a given input/output pair to be used as an in-context example, we interweave each System-side utterance with a narrative instruction that states that the next utterance is either a clarifying question or a direct answer. This yields "control" over the pragmatic action of the generated utterance.
+
+# G ACTION CLASSIFIER DETAILS
+
+Compared to recognizing whether a request is ambiguous or not, action classification (i.e., recognizing whether an existing utterance is a question or answer attempt) is considerably simpler. We directly prompt Gemini Ultra 1.0 with 10 in-context examples to serve as  $A$  (as per the notation given in Section 3.1).
+
+Table A20 is an example of an in-context example used to demonstrate a system-side response with the "Clarify" action in PACIFIC ("Assistant: Which region are you asking about?"). All of the conversation history is made available in addition to the relevant table and passage in the prompt. We include all contextual information available as part of the input to the Action Classifier, along with the last Assistant response which is an example of the classification target.
+
+Table A21 is an example of an in-context example used to demonstrate a system-side response with the "Answer" action in Abg-CoQA ("Assistant: no"). As is common in this corpus, a subset of the conversation is made available in addition to the relevant passage in the prompt. We include all contextual information available as part of the input to the Action Classifier, along with the last Assistant response which is an example of the classification target.
+
+Table A22 is an example of an in-context example used to demonstrate a system-side response with the "Answer" action in AmbigSQL. In AmbigSQL, the information provided as input is the database schema, and the user's initial information request. This information is provided as part of the input to the Action Classifier, along with the last Assistant response which is an example of the classification target.
+
+Table A22: In-context example given as part of a prompt to an LLM for action classification with AmbigSQL. The blue text represents the action for this example.  
+```txt
+[Database Schema Omitted]   
+User: "Which professionals live in the state of Indiana or have done treatment on more than 2 treatments?   
+List his or her id, last name and cell phone."   
+Here is the same request converted into an ambiguous format by underspecifying the target columns: Assistant:   
+SELECT professional_id, last_name, cell_number FROM Professionals WHERE state  $=$  'Indiana' UNION SELECT T1.professional_id, T1.last_name, T1.cell_number FROM Professionals AS T1 JOIN Treatments AS T2 ON T1.professional_id = T2.professional_id   
+GROUP BY T1.professional_id HAVING count  $(\star)$  >2   
+The last Assistant utterance is a direct answer.
+```
+
+# H USER SIMULATOR DETAILS
+
+In order to iterate on system-side dialogue models, many works investigate the use of a user simulator,  $U$  (as per the notation given in Section 3.1) (e.g. Chandramohan et al. (2011); Kreyssig et al. (2018); Shah et al. (2018)). At its core, it requires a response generation model which can robustly represent user-side intents. Traditionally, this involved training a model specifically on user-side responses but due to the aforementioned control capabilities of modern LLMs Chen et al. (2023b), we directly perform few-shot prompting for user simulation, following Deng et al. (2023c); Yu et al. (2023). For a given conversation during training, we again prompt a high-capacity LLM with in-context examples to identify the information-seeking user intents. Then, conditioned on the generated information-seeking intents, we prompt the same LLM with the responses from the policy model to simulate the sub-goal trajectory. This allows  $U$  to ground on the real intents of the user while retaining enough flexibility to respond coherently to some system-side response. In our work, we use Gemini Ultra 1.0 with three hand-crafted in-context examples for both intent summarization and user response generation.
+
+Table A23: In-context example given as part of a prompt to an LLM for intent summarization with PACIFIC. The purple text represents the system prompt used for this task. The blue text represents the summarized information-seeking intents.  
+```txt
+The following is a conversation between a User and an Assistant. The User is asking some questions. Summarize what information the User is looking for. [Table and Passage Omitted]   
+User: What does Walletron deliver?   
+Assistant: patented mobile wallet technology.   
+User: How much did it cost the company to acquire certain technology assets of RevChip, LLC ("RevChip") and TranSend Integrated Technologies Inc. ("TranSend")?   
+Assistant: $7.0 million   
+User: What was the pro forma revenue in 2019?   
+Assistant: $1,382,957   
+User: What was the change in its amount between 2018 and 2019?   
+Assistant: 21228   
+User: How about that in pro forma net income?   
+Assistant: -6425   
+[Information] The user wants to know: 1. What technology Walletron delivers, 2. How much it cost Walletron to acquire certain technology assets of RevChip, LLC and TranSend Integrated Technologies Inc., 3. What the pro forma revenue was in 2019, 4. What the change in pro forma revenue was between 2018 and 2019, and 5. What the change in pro forma net income was between 2018 and 2019.
+```
+
+PACIFIC We provide one of our three handcrafted in-context examples of summarizing information-seeking intent in Table A23. While the information summary is handcrafted, the conversation and grounding data are taken directly from the PACIFIC. After intent summarization,  $U$  is given the existing conversation history along with an instruction that includes the information
+
+seeking intents and asked to provide the next user response. This format is structurally the same as the three simulation-specific in-context examples — an example is provided in Table A24.
+
+Table A24: In-context example given as part of a prompt to an LLM for user simulation with PACIFIC. The purple text represents the system prompt used for this task, which contains the information-seeking intents summarized from Table A23. The entire conversation is used for this example to demonstrate grounding between a conversation and intents.  
+
+<table><tr><td>The following is a conversation between a User and an Assistant. The User is asking some questions. The user wants to know: 1. What technology Walletron delivers, 2. How much it cost Walletron to acquire certain technology assets of RevChip, LLC and TranSend Integrated Technologies Inc., 3. What the pro forma revenue was in 2019, 4. What the change in pro forma revenue was between 2018 and 2019, and 5. What the change in pro forma net income was between 2018 and 2019. 
+[Table and Passage Omitted] 
+Assistant: $7.0 million 
+User: What was the pro forma revenue in 2019? 
+Assistant: $1,382,957 
+User: What was the change in its amount between 2018 and 2019? 
+Assistant: 21228 
+User: How about that in pro forma net income? 
+Assistant: -6425</td></tr></table>
+
+Abg-CoQA We conduct user simulation for Abg-CoQA similarly to PACIFIC. Table A25 is one of three hand-crafted examples of intent summarization, and Table A26 is one of three hand-crafted examples used for simulating the final user response.
+
+Table A25: In-context example given as part of a prompt to an LLM for intent summarization with Abg-CoQA. The purple text represents the system prompt used for this task. The blue text represents the summarized information-seeking intents.  
+
+<table><tr><td>[Passage Omitted]</td></tr><tr><td>User: What was his ranking?</td></tr><tr><td>Assistant: General</td></tr><tr><td>User: Did someone else have horse fighters?</td></tr><tr><td>Assistant: yes</td></tr><tr><td>User: Who?</td></tr><tr><td>Assistant: Do you want to know who had 2,500 horse fighters initially?</td></tr><tr><td>User: No, I want to know who had a considerable force of horse fighters west of him.</td></tr><tr><td>Assistant: General (Bishop) Polk.</td></tr><tr><td>Instruction: Summarize what information the user is looking for.</td></tr><tr><td>Summary: The user wants to know: 1. What his ranking was. 2. Whether someone else had horse fighters. 3. Who had a considerable force of horse fighters west of him.</td></tr></table>
+
+AmbigSQL We conduct user simulation slightly differently for AmbigSQL. Rather than having a separate step for knowledge-seeking intent summarization, we directly condition  $U$  on the target SQL query. As demonstrated in Table A27, the prompt includes the target SQL query, the original user request, and the clarification question sampled from the Assistant. In the example given in Table A27, the last line is the ground truth disambiguated query, and in practice, this would be the generation target of  $U$ .
+
+# I QUALITATIVE EXAMPLES
+
+In Table A28 we compare two trajectories produced by Zephyr 7B tuned on 50 conversations from PACIFIC using SFT and  $ACT$ , respectively, given the same conversation context and tabular information. The SFT model provided an incorrect answer to the user request, matching the "Incorrect Assumption" case in Figure 1. In contrast, the model tuned using  $ACT$  asked a clarifying question before eventually producing the correct answer.
+
+Table A26: In-context example given as part of a prompt to an LLM for user simulation with Abg-CoQA. The purple text represents the system prompt used for this task, which contains the information-seeking intents summarized from Table A25. The entire conversation is used for this example to demonstrate grounding between a conversation and intents.  
+
+<table><tr><td>The following is a snippet from a conversation between an Assistant and a User who is asking some questions about the Story below. The user only wants to know the following information: 1. What his ranking was. 2. Whether someone else had horse fighters. 3. Who had a considerable force of horse fighters west of him.</td></tr><tr><td>[Passage Omitted]</td></tr><tr><td>User: What was his ranking?</td></tr><tr><td>Assistant: General</td></tr><tr><td>User: Did someone else have horse fighters?</td></tr><tr><td>Assistant: yes</td></tr><tr><td>User: Who?</td></tr><tr><td>Assistant: Do you want to know who had 2,500 horse fighters initially?</td></tr><tr><td>User: No, I want to know who had a considerable force of horse fighters west of him.</td></tr><tr><td>Assistant: General (Bishop) Polk.</td></tr></table>
+
+Table A27: In-context example given as part of a prompt to an LLM for user simulation with AmbigSQL. The purple text represents the system prompt used for this task, which contains ground truth SQL query goal. The entire conversation is used for this example to demonstrate SQL query-grounded request disambiguation.  
+
+<table><tr><td>A user is asking an assistant to retrieve some information from a SQL database. The command that the assistant should ultimately return is as follows:</td></tr><tr><td>SELECT county FROM campuses where campus = &#x27;California State University-Chico&#x27;</td></tr><tr><td>The assistant will ask some questions to clarify the user&#x27;s intent. The user should respond with a rephrased request that reflects their desired query.</td></tr><tr><td>User: what is the county?</td></tr><tr><td>Assistant: Are you asking for a list of all of the counties in the database?</td></tr><tr><td>User: I&#x27;m looking for the county of the campus &#x27;California State University-Chico&#x27;</td></tr></table>
+
+Table A28: Comparison of trajectories using different tuning approaches. Left: snippet of tabular data from the PACIFIC evaluation set. Middle: trajectory resulting from the SFT model in response to the user query. Right: trajectory resulting from the ACT model. Red text: sampled from  $U$ .  
+
+<table><tr><td>Relevant Table Data</td><td>SFT</td><td>ACT</td></tr><tr><td>Year: 2019 || 2018</td><td>User: What were the total liabilities of IMFT?</td><td>User: What were the total liabilities of IMFT?</td></tr><tr><td>Total current liabilities: 851 || 167</td><td>Assistant: $909 ×</td><td>Assistant: Which year are you asking about?</td></tr><tr><td>Other noncurrent liabilities: 5 || 74</td><td></td><td>User: 2018</td></tr><tr><td>Total Liabilities: $909 || $1,305</td><td></td><td>Assistant: $1,305 ✓</td></tr></table>
+
+Table A29: Comparison of PACIFIC trajectories using different tuning approaches. Left: snippet of from a passage from the PACIFIC test set. Middle: trajectory resulting from the SFT model in response to the user query. Right: trajectory resulting from the ACT model. Red text: sampled from  $U$ . Both the SFT and ACT model answered the question incorrectly.  
+
+<table><tr><td></td><td>SFT</td><td>ACT</td></tr><tr><td>Relevant Data</td><td>User: What was the net profit/(loss) after tax?</td><td>User: What was the net profit/(loss) after tax?</td></tr><tr><td>Net profit/(loss) after tax was $(9.8) million</td><td>Assistant: $(9,819) ×</td><td>Assistant: Which year are you asking the amount of?</td></tr><tr><td>Net profit/(loss) after tax : (9,819) || 6,639 || (248%)</td><td></td><td>User: FY19
+Assistant: $(9,819) ×</td></tr></table>
+
+Table A30: Comparison of PACIFIC trajectories using different tuning approaches. Left: snippet of from a passage from the PACIFIC test set. Middle: trajectory resulting from the SFT model in response to the user query. Right: trajectory resulting from the ACT model. Red text: sampled from  $U$ . Both the SFT and ACT model answered the question incorrectly, and neither asked a clarification question.  
+
+<table><tr><td></td><td>SFT</td><td>ACT</td></tr><tr><td colspan="3">Relevant Table Data</td></tr><tr><td>Year: 2019 || 2018</td><td>User: In which year was the amount of Investments higher?</td><td>User: In which year was the amount of Investments higher?</td></tr><tr><td>Investments: 1,216.0 || 1,212.9</td><td>Assistant: [2018] ×</td><td>Assistant: [2018, 2019] ×</td></tr></table>
+
+Table A29 includes an erroneous trajectory produced by the ACT model in the 50 conversation setting of PACIFIC. In this particular example, both the provided passage and the provided table provided similar information regarding net profit/loss after tax. In the dataset, the ground truth answer selected a span from the passage, whereas the ACT model produced a clarification question and then eventually an answer using tabular information. The SFT model directly assumed the intent of the user and similarly produced an answer from the tabular information.
+
+Table A30 includes another pair of erroneous trajectories in the 50 conversation setting of PACIFIC. The requested information was accessible through the provided table data, but both the SFT model and the  $ACT$  model failed to produce the correct answer. This example was not annotated as requiring a clarifying question, so it simply indicates that there is much room for improvement in terms of downstream task performance.
+
+Table A31 includes a pair of trajectories resulting from SFT and  $ACT$  in the 50 conversation setting of Abg-CoQA. The requested information is given in the portion of the passage in the left panel. The SFT model asked an off-topic clarification question and failed to bring the questioning on track. The  $ACT$  model successfully asked an on-track clarification question and eventually produced the correct answer.
+
+Table A31: Comparison of Abg-CoQA trajectories using different tuning approaches. Left: snippet of a relevant passage from the Abg-CoQA test set. Middle: trajectory resulting from the SFT model in response to the user query. Right: trajectory resulting from the ACT model. Red text: sampled from  $U$ . Ground truth outcome: Meghan asked Lizzie, "Are you awake?"  
+
+<table><tr><td></td><td>SFT</td><td>ACT</td></tr><tr><td>Relevant Passage</td><td>User: What did Meghan ask?</td><td>User: What did Meghan ask?</td></tr><tr><td>She yawned, stretching out like a lazy cat.</td><td>Assistant: Do you mean where did Lizzie see her father?</td><td>Assistant: Do you mean that morning or the night before?</td></tr><tr><td>Her sister was also awake. &quot;Lizzie&quot;,</td><td>User: No</td><td>User: The night before.</td></tr><tr><td>Meghan whispered, &quot;are you awake?&quot;</td><td>Assistant: Lizzie had only met her father in person once, according to the story. ×</td><td>Assistant: Meghan asked Lizzie if she was awake that morning. ✓</td></tr></table>
+
+# J ADDITIONAL EXPERIMENTAL DETAILS
+
+# J.1 COMPUTE RESOURCES
+
+We conduct all experiments using one Google Compute Engine Virtual Machine with 8x 80GB A100 GPUs. All of our tuning experiments use full-model multi-GPU tuning with Deepspeed. Full ACT with 250 conversations (1,923 training examples) from PACIFIC took the longest amount of time and required approximately 16 hours to run for 12 epochs.
+
+Beyond the experiments reported in the paper, there were many preliminary experiments that took place as a part of this overall research project that were ultimately directionally different from our final contribution.
+
+# J.2 TRAINING HYPERPARAMETERS
+
+For all of our SFT experiments with Zephyr, Mistral, and Gemma, we tune the model for up to 8 epochs. We choose the best-performing model with learning rates from  $\{1e - 4,2e - 5,1e - 5\}$  with the AdamW optimizer.
+
+For our SFT experiments with Gemini Pro, we use the Vertex AI API and tune for up to 4 epochs with an Adapter size of 4.
+
+For all of our RL tuning experiments, we allow the model to train for up to 12 epochs, and select the checkpoint that results in the highest reward margin on the validation set (which is an action-based preference dataset constructed as described in Section 3.2 using each task's original validation set). For all experiments, we use a batch size of 4, and a maximum sequence length of 1, 280.
+
+Hyperparameters for Equation 2 For experiments with Zephyr 7B on PACIFIC, we achieve our strongest results using  $\beta = 0.01$  and a learning rate of  $5e - 7$ . On AmbigSQL, we use  $\beta = 0.01$  and a learning rate of  $5e - 7$ . On AmbigSQL, we use  $\beta = 0.5$  and a learning rate of  $5e - 7$ .
+
+# K ASSETS USED
+
+All resources used have been cited appropriately in the paper. In this section, we enumerate each of the existing artifacts used in our work along with their license.
+
+# Existing Models
+
+- Gemma Gemma Team et al. (2024): Gemma Open-Source License. https://ai.google.dev/gemma/terms  
+- Gemini 1.0 Ultra (gemini-1.0-ultra), Gemini 1.5 Pro (gemini-1.5-pro-001), Gemini 1.5 Flash (gemini-1.5-flash-001) (Gemini Team et al., 2023): Accessed through the Google Cloud Vertex AI Platform. https://cloud.google.com/products/gemini?hl=en  
+- Claude 3.5 Sonnet, Claude 3.0 Haiku (Anthropic AI, 2024): Accessed through the Google Cloud Vertex AI Platform. https://cloud.google.com/products/gemini?hl=en  
+- MiniLM-L6-v2 (Reimers & Gurevych, 2019): Apache 2.0. https://huggingface.co/sentence-transformers/all-MiniLM-L6-v2  
+- Mistral 7B-v0.1 (Jiang et al., 2023): Apache 2.0. https://huggingface.co/mistralai/Mistral-7B-v0.1  
+- Zephyr 7B- $\beta$  (with Mistral 7B as a Base Model) (Tunstall et al., 2023): MIT Open-Source License. https://huggingface.co/HuggingFaceH4/zephyr-7b-beta
+
+# Existing Datasets
+
+- Abg-CoQA (Guo et al., 2021): MIT Open-Source License. https://github.com/MeiqiGuo/AKBC2021-Abg-CoQA  
+- PACIFIC (Deng et al., 2022): MIT Open-Source License. https://github.com/dengyang17/PACIFIC/tree/main  
+- Spider (Yu et al., 2018): CC BY-SA 4.0. https://yale-lily.github.io/spider
+
+# Existing Algorithms and Software
+
+- Direct Preference Optimization (Rafailov et al., 2024): CC BY 4.0.  
+- Google Cloud Pipeline Components: Apache 2.0. https://cloud.google.com/vertex-ai/docs/pipelines/components-introduction  
+- HuggingFace Transformers (Wolf et al., 2020): Apache 2.0. https://github.com/huggingface/transformers/tree/main  
+- PyTorch (Paszke et al., 2019): PyTorch Open Source License. https://github.com/pytorch/pytorch/tree/main  
+- Vertex AI SDK: Apache 2.0. https://cloud.google.com/vertex-ai/docs/ python-sdk/use-vertex-aipython-sdk
\ No newline at end of file
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+# LEARNING TO COMMUNICATE THROUGH IMPLICIT COMMUNICATION CHANNELS
+
+Han Wang
+
+The Chinese University of Hong Kong, Shenzhen xwanghan@gmail.com
+
+Binbin Chen
+
+ByteDance Inc.  
+chenbinbin.1996@bytedance.com
+
+Tieying Zhang
+
+ByteDance Inc.
+
+tieying.zhang@bytedance.com
+
+Baoxiang Wang*
+
+The Chinese University of Hong Kong, Shenzhen
+
+Vector Institute
+
+bxiangwang@cuhk.edu.cn
+
+# ABSTRACT
+
+Effective communication is an essential component in collaborative multi-agent systems. Situations where explicit messaging is not feasible have been common in human society throughout history, which motivate the study of implicit communication. Previous works on learning implicit communication mostly rely on theory of mind (ToM), where agents infer the mental states and intentions of others by interpreting their actions. However, ToM-based methods become less effective in making accurate inferences in complex tasks. In this work, we propose the Implicit Channel Protocol (ICP) framework, which allows agents to communicate through implicit communication channels similar to the explicit ones. ICP leverages a subset of actions, denoted as the scouting actions, and a mapping between information and these scouting actions that encodes and decodes the messages. We propose training algorithms for agents to message and act, including learning with a randomly initialized information map and with a delayed information map. The efficacy of ICP has been tested on the tasks of Guessing Numbers, Revealing Goals, and Hanabi, where ICP significantly outperforms baseline methods through more efficient information transmission.
+
+# 1 INTRODUCTION
+
+Effective communication is pivotal in collaborative multi-agent systems, especially in environments characterized by incomplete information (Panait & Luke, 2005; Busoniu et al., 2008; Tuyls & Weiss, 2012). Communication acts as a vital conduit, enabling agents to exchange private information, coordinate joint actions, and infer real-world states (Wang et al., 2021). These processes synergistically foster tighter cooperation and enhance collective performance (Li et al., 2002; Cao et al., 2012). We focus on multi-agent reinforcement learning (MARL) methods for communication, where communication is broadly categorized into explicit and implicit strategies (Dafoe et al., 2020).
+
+Explicit communication uses direct channels independent of the environment dynamics (Sukhbaatar et al., 2016; Foerster et al., 2016; Jiang & Lu, 2018), allowing agents to transmit observations, intentions, and advice to facilitate decision-making and coordination (Zhu et al., 2022; Qu et al., 2021). This approach, analogous to human language or verbal exchanges (Havrylov & Titov, 2017; Baker et al., 1999), has been widely employed in MARL to enhance collaboration. However, dependence on direct channels introduces significant computational and memory overheads (Roth et al., 2006),
+
+which makes it challenging to implement in certain scenarios. Examples are tasks without communication channels or decentralized frameworks (Oliehoek et al., 2008; Kraemer & Banerjee, 2016).
+
+Situations where explicit messaging is not feasible have been common in human society throughout history. From early humans engaging in hunting and gathering through silent cooperation (Klein, 2009; Tomasello & Vaish, 2013), to modern military operations using gestures and codes for covert communication (Tzu, 2008), and even in everyday social interactions where intentions are conveyed through expressions, tone, and body language (Pease, 1984; Duncan Jr, 1969). Implicit communication has established an effective mechanism for information sharing without explicit language.
+
+Learning methods for implicit communication have been investigated by the MARL community. A prominent method is the theory of mind (ToM) (Premack & Woodruff, 1978), where agents infer the mental states and intentions of others by interpreting their actions (Heider & Simmel, 1944). By modeling the beliefs, desires, and intentions of other agents, ToM enables agents to coordinate in a variety of simple tasks (Baker et al., 2017; Zhao et al., 2023; Nguyen et al., 2020). However, ToM-based approaches face significant challenges, including the difficulty of making accurate inferences and the high computational complexity involved in modeling other agents. These issues become particularly pronounced in dynamic environments where agents must constantly update their models based on limited or ambiguous information.
+
+To address these challenges associated with ToM methods, we introduce Implicit Channel Protocol (ICP), a novel framework that allows agents to communicate through communication protocols in implicit communication likewise how it was done in explicit communication. ICP leverages a subset of actions, denoted as the scouting actions, which have no or uniform effects on environment dynamics. A centralized mapping between information and these scouting actions is established to encode and decode the messages. Agents exchange information by deliberately taking scouting actions, forming an implicit communication channel. We further demonstrate how agents' strategies are trained on this channel, including training with a randomly initialized information map and training with a delayed information map.
+
+We validate the effectiveness of ICP through comprehensive experiments on the tasks of Guessing Numbers, Revealing Goals, and Hanabi (Bard et al., 2020). These environments share a common characteristic: they lack direct communication but agents must collaboratively make decisions to achieve shared rewards. This setting introduces significant challenges, including sparse and delayed reward feedback, along with difficulty in credit assignment both temporally and among agents. Despite these hurdles, our experiments on Guessing Numbers and Revealing Goals demonstrate that ICP significantly enhances performance, through more efficient information transmission, compared to baseline methods. In Hanabi, which is a popular card game played by humans, our approach achieved an average score of 24.91 out of 25, which surpasses the best available learning algorithm which obtains 23.81.
+
+# 2 BACKGROUND AND RELATED WORK
+
+Decentralized Partially Observable Markov Decision Process The cooperative multi-agent problem can be formulated as a Dec-POMDP game (Bernstein et al., 2002), which is described by the tuple  $G = (S, U, N, \mathcal{T}, O, R, \gamma)$ , where  $S$  represents the true global state of the environment. At each discrete time step  $t$ , every agent  $i \in N := \{1, \dots, n\}$  selects an action  $u_i$  from its action space  $U_i$ . The state transition function  $\mathcal{T}(s'|s, u): S \times U \times S \to P(S)$  governs the transition of states, where  $u = (u_1, \dots, u_n)$  denotes the joint action. In POMDP, the global state remains inaccessible, and each agent  $i$  can only perceive its individual observation  $o_i$  through the observation function  $O_i(s): S \times N \to O$ .  $R_i(s, u_i): S \times U_i \to R$  represents the reward function for each agent  $i$ . In the cooperative scenarios, all agents share a common reward function  $R(s, u): S \times U \to R$ , known as team reward. The objective for each agent is to maximize the expected return, which makes effective cooperation among all agents necessary.
+
+Learning to Communicate Communication among agents is critical for effective collaboration in MARL. Most researches in this field focused on explicit communication protocols (Tucker et al., 2022; Peng et al., 2017; Kong et al., 2017; Pesce & Montana, 2020; Kim et al., 2019; Wang et al., 2020; Freed et al., 2020b;a; Gupta et al., 2023; Foerster et al., 2016), where agents exchange messages containing critical information. Additionally, techniques like attention mechanisms or graph-
+
+based methods that used to build more effective communication connection (Jiang & Lu, 2018; Das et al., 2019; Jiang et al., 2018; Sukhbaatar et al., 2016; Chen et al., 2024; Tucker et al., 2022) and intention sharing or theory of mind (ToM) reasoning to transmit more useful information (Wang et al., 2021; Kim et al., 2020; Qu et al., 2021), further enhancing the effectiveness of MARL systems.
+
+Explicit communication on direct channels often incurs significant communication overhead and may not be available in environments with limited bandwidth and high communication costs. Implicit communication, where agents convey information through their behaviors, has been explored as an alternative approach (Li et al., 2024; Tian et al., 2023; Li et al., 2021; Shaw et al., 2022; Gruben et al., 2022). ToM-based methods represent one main research strand for implicit communication, where agents infer the intentions and beliefs of others to anticipate actions and adjust strategies accordingly. Methods in Rabinowitz et al. (2018); Tian et al. (2020); Nguyen et al. (2020); Zhao et al. (2023) aim to enhance coordination by modeling and predicting others' behavior based on observed actions. However, ToM-based approaches can be computationally intensive and may struggle to accurately infer intentions in complex environments.
+
+# 3 SETTING
+
+Consider in a fully cooperative Dec-POMDPs, where no direct communication channel is available. Each agent maintains a joint action-observation history  $\tau_{t,i} = \{o_{0,i},u_0,r_1,\dots,r_t,o_{t,i}\}$  and makes decisions and executes actions based on it. Within this setting, we define a subset of actions as scouting actions  $U^s$  which have no or uniform effects on the state  $s_t$  or reward  $r_t$  and mainly affect the observation function  $O_{i}(s_{t})$ . The remaining actions are defined as regular actions  $U^{r}$ .
+
+In an idealized situation, scouting actions serve to influence the observation mapping function, which allows the agents to gather critical information from the environment to improve their decision-making and coordination. A scouting action might reveal information about an agent's surroundings or the state of other agents without changing the environment unpredictably. For instance, in the game of Hanabi, Hint actions reveal information to other players while consuming an information token, thereby uniformly altering the game state. Similarly, in StarCraft II, scouting the map with Scan from Terrans requires depleting energy, which impacts the environment and provides new information in the agent's observations.
+
+We are interested in this setting because agents generally need to employ scouting actions to collect sufficient information before taking other actions to obtain rewards. Both the information collection phase and the reward acquisition phase may require cooperation among agents. For example, agents will only receive positive rewards if one agent scouts the correct information for another agent, and the latter correctly utilizes this information. In this context, sparse and delayed reward feedback, along with the credit assignment problem both temporally and among agents, poses significant challenges for agents to learn optimal strategies.
+
+# 4 INFORMATION CARRIED BY SCOUTING ACTIONS
+
+Scouting actions carry two types of information.
+
+1. Information reflected through the environment. When an agent performs a scouting action, it influences the observation function, and the new observations provide information. This type of information depends on the observation function, the states, and the joint actions of all agents.  
+2. Information reflected through the choice of the scouting action. The specific scouting action chosen by an agent can carry intentional meaning. This type of information depends only on other agents' scouting action policy. If the receiver agent cannot understand the sender agent's intention, no information will be transmitted.
+
+General multi-agent reinforcement learning (MARL) methods tend to focus more on the first type of information. In Dec-POMDPs without modeling other agents' strategies, both partial observation of states and other agents' policies introduce uncertainty. In the exploration phase of online MARL training, the second type of information cannot be utilized due to the unknown policies of others. Therefore, MARL methods mostly learn to use the information reflected through the environment.
+
+For the second type of information which is not limited by observation function and states, when agents' intention is understood correctly, it is more fixable and could be more useful for decision-making and cooperation. Theory of mind (ToM) methods utilize the second type of information by modeling strategies or inferring the intention of other agents. By estimating the internal states of other agents, an agent can interpret the intentional choices behind their scouting actions. One approach is to build an explicit belief model that infers the state from observed actions to achieve communication. However, this method incurs a computational cost that grows exponentially with the size of the game and the number of agents. Moreover, the precise state distribution information is often difficult to fully utilize. Another approach involves training a neural network to map actions to states or intentions, which requires exponentially more data compared to action policy training. Since this method does not produce explicit inferences, the resulting information is inherently biased, which could negatively impact the agent's decision-making process.
+
+We propose a simpler and more effective solution to take advantage of the second type of information conveyed by scouting actions while also mitigating the computational overhead and inaccuracies of ToM methods. By establishing a shared communication protocol where all agents follow the same strategy for selecting scouting actions, the intended information can be easily decoded from the chosen action. This approach eliminates the need for state estimation and intention inference required in ToM methods. In our setting, this communication protocol has the following benefits.
+
+1. Low Computational Overhead and Accurate Information Transmission. By having all agents follow the same strategy for selecting scouting actions, there is no need for additional computational inference or training of a separate inference model to deduce intentions. The inverse of the strategy can be used to directly interpret the information, resulting in low computational overhead and accurate transmission of intentions.  
+2. Efficient Communication Enabled by Broadcasted Information. Scouting actions are observable by all agents in the system, allowing any information conveyed through these actions to be inherently broadcasted. This characteristic facilitates the use of efficient embedding techniques, such as the hat mapping method, to achieve more effective communication.  
+3. Independence from Observation-Based Information. When different embeddings are used, the information conveyed by altering the observation function may differ, making the information carried by scouting actions not necessarily correlated with the environment-based information. Both types of information can be optimized concurrently to enhance cooperation and decision-making.
+
+# 5 STRATEGY TRAINING ON CONSTRUCTED IMPLICIT CHANNEL
+
+To establish a shared communication protocol, we propose Implicit Channel Protocol (ICP), a new framework for agents' implicit communication. In this framework, at each step, agents choose whether to send information. If an agent chooses not to send information, a regular action  $u^r$  will be taken according to the action policy. If otherwise an agent wishes to convey message  $m_i \in M$ , it will output a scouting action  $u^s = \mathcal{P}(m_i)$ , which maps  $m_i$  through the centralized mapping mechanism  $\mathcal{P}$ . This mechanism  $\mathcal{P}$  needs to be decodable, which provides output messages from scouting actions. By allowing agents to send information through encoded scouting actions and other agents to decode the messages from the globally observed scouting actions, an implicit communication channel is constructed.
+
+As agents need to decide whether to send information, the new action space becomes  $U' = \{U - U^s, \text{send_info}\}$ , then agents sample actions  $u_i \in U'$  from new action policy  $u_i \sim \pi_i(\cdot \mid \tau_{t,i}, m_{-i})$  where  $m_{-i}$  are decoded messages transmitted from other agents. For message  $m_i$ , it is selected by the message strategy  $m_i \sim \phi_i(\cdot \mid \tau_{t,i}, m_{-i})$ . Based on this process, we can formulate agents' value functions as  $V^{\pi_i, \phi_i, \mathcal{P}}(s) := \mathbb{E}[\sum_{t=1}^{T} \gamma^{t-1} r_t(s_t, u_t, s_{t+1}) \mid s_1 = s]$ , for  $s_{t+1} \sim \mathcal{T}(\cdot \mid s_t, u_t)$  and  $u_t = (u_{t,1}, \dots, u_{t,n})$ ,
+
+$$
+u _ {t, i} = \left\{ \begin{array}{l l} u ^ {s} = \mathcal {P} \left(m _ {i} \sim \phi_ {i} (\cdot \mid \tau_ {t, i}, m _ {- i})\right) & \text {i f s e n d - i n f o ,} \\ u ^ {r} \sim \pi_ {i} (\cdot \mid \tau_ {t, i}, m _ {- i}) & \text {e l s e .} \end{array} \right. \tag {1}
+$$
+
+Equation (1) highlights three essential components of the framework: the agents' action policy  $\pi$ , the message strategy  $\phi$ , and the centralized mapping mechanism  $\mathcal{P}$ . To develop these components, we investigate two different methodologies. The first one is training with a random initial information map and the second one is training with a delayed information map. Both approaches construct an implicit communication channel within the environment, with which the agents exchange information and achieve implicit coordination. In the rest of the section, we will discuss the details of these approaches.
+
+# 5.1 STRATEGY TRAINING WITH RANDOM INITIAL MAP
+
+In this approach, we first establish a one-to-one mapping between information and actions using a randomly initialized embedding, that makes the size of the message the same as the size of the scouting action space  $M = \{1,\dots,|U^s |\}$ . Once this one-to-one mapping is set, the inverse of  $\mathcal{P}$  will obtain the original message  $m_{i} = \mathcal{P}^{-1}(u_{i}^{s})$ .
+
+In our implementation, we use a value-based method for training the action policy, incorporating parameter sharing and value decomposition techniques to facilitate effective coordination among agents. For the information strategy, we use the communication gradient method (Foerster et al., 2016), which is designed to optimize communication protocols within a predefined channel. Both the action policy and the information strategy are parameterized as Q-networks, for action policy  $\pi_i(\tau_{t,i}) = \mathrm{argmax}Q^{\theta_{1,i}}(\tau_{t,i},u_i)$ , and message strategy  $\phi_i(\tau_{t,i}) = \mathrm{argmax}Q^{\theta_{2,i}}(\tau_{t,i},m_i)$ . Given the value function  $V^{\pi_i,\phi_i,\mathcal{P}}(s)$ , the Q-functions are defined by  $Q^{\pi_i}(\tau_{t,i},u_i) := \mathbb{E}_{m\sim \phi ,\mathcal{P}}[r_t(s_t,u_t,s_{t + 1}) + \gamma V^{\pi_i,\phi_i,\mathcal{P}}]$ , and  $Q^{\phi_i}(\tau_{t,i},m_i) := \mathbb{E}_{u^r\sim \pi ,\mathcal{P}}[r_t(s_t,u_t,s_{t + 1}) + \gamma V^{\pi_i,\phi_i,\mathcal{P}}]$ , where  $\pi = (\pi_1,\dots,\pi_n)$ ,  $\phi = (\phi_1,\dots,\phi_n)$  and  $m = (m_1,\dots,m_n)$ .
+
+Parameter Sharing and Value Decomposition Parameter sharing enables different agents with the same observation and action spaces to learn from a shared network. Despite using the same network, agents evolve different hidden states and receive distinct observations, allowing them to behave differently. We incorporate the agent index  $i$  into the network input, allowing for specialization through rich representations in deep Q-networks.
+
+To further enhance learning efficiency, we apply value decomposition (Sunehag et al., 2018) for the action policies  $\pi$ . The joint action-value function  $Q_{\mathrm{tot}}^{\pi}(\tau ,u)$  is expressed as the sum of individual value functions:  $Q_{\mathrm{tot}}^{\pi}(\tau ,u) = \sum_{i = 1}^{N}Q^{\pi}(\tau_{t,i},u_{i};\theta_{1,i})$ , where  $\tau = (\tau_{i},\dots,\tau_{N})$ . This decomposition simplifies learning by allowing the gradients to propagate through individual agents' policies while maintaining a shared reward structure, which accelerates the overall training process.
+
+Communication Gradient Since our message space is  $M = \{1, \dots, |U^s|\}$ , which is discrete and non-binary, we introduce the Gumbel-Softmax technique (Havrylov & Titov, 2017; Jang et al., 2016) to sample one-hot vector messages  $\vec{m}$ . The Gumbel-Softmax method enables us to generate discrete samples that are differentiable, facilitating end-to-end training with gradient descent. Specifically, we sample the one-hot vector message  $\vec{m}$  such that  $\vec{m}_i = 1$  if  $i = \arg \max_{j \in M} (\log(\phi(\cdot | \tau_{t,i}, m_{-i}) + g_j))$ , and  $\vec{m}_i = 0$  otherwise, where  $g_j = -\log(-\log(u_j))$  and  $u_j \sim U(0,1)$ .
+
+This method ensures a continuous relaxation between  $\phi$  and  $\vec{m}$ . As a result of this relaxation, the game becomes fully differentiable and can be trained using the backpropagation algorithm. By enabling gradients to flow smoothly through the communication process, this approach facilitates more effective and efficient training of agents. Comparing the communication gradient implementation in DIAL (Foerster et al., 2016), we adopt this Gumbel-Softmax technique to replace the Discretize/Regularize Unit (DRU). This substitution allows agents to send discrete and non-binary messages during the learning phase, thereby enhancing the richness and accuracy of the information exchanged.
+
+# 5.2 STRATEGY TRAINING WITH DELAYED MAP
+
+In the training process with a delayed map, we begin by learning the information strategies and action policies using a direct communication channel. This initial phase provides agents with a flexible environment that allows for direct communication and enable them to develop their strategies more
+
+effectively. Once these strategies are established, we determine the mapping mechanism that maps the messages to actions. Finally, we fine-tune both the action policies and information strategies in the ICP framework to ensure optimal performance within the constraints of the final environment.
+
+Given that the information strategies were initially trained using a direct communication channel, there may be discrepancies between this channel and the implicit communication channel constructed by scouting actions, particularly in terms of capacity and whether the channel is discrete or noiseless. To minimize any potential performance loss resulting from these differences, we can limit the capacity of the direct channel or apply efficient embedding techniques (such as the hat mapping method) during the message-to-action mapping process. These steps help to align information strategies developed in the direct channel with the constraints of the implicit channel and ensure messages are decodable.
+
+During implementation, we utilize standard explicit communication algorithms designed for direct channels, such as RGMComm, to pre-train the information strategy. Then we use two kinds of mapping methods, including simple one-to-one mapping and the hat mapping method (Brown & Tanton, 2009; Bushi, 2012; Butler et al., 2009; Feige, 2004; Havil, 2011; Winkler, 2002) to obtain new information strategies. The one-to-one mapping ensures that each message corresponds directly to an action, while the hat mapping allows an agent to communicate efficiently with multiple receivers. Finally, we fine-tune both the information strategy and the action policy to optimize performance within the implicit communication framework.
+
+Pre-training with Direct Channel In the pre-training phase, we utilize a direct communication channel to facilitate the learning of the communication strategies. During this phase, the message size can be larger than the scouting action space, allowing for more complex and detailed communication between agents. This flexibility enables the agents to explore a wider range of communication strategies and coordination behaviors, which are crucial for solving complex tasks in partially observable environments. However, after pre-training, the message strategy needs to transition to the implicit communication framework and it is necessary to compress these messages without loss of information. For one-to-one mapping, this mapping requires that the message size matches the scouting action space exactly. It ensures a direct and consistent mapping between the learned messages and the corresponding actions.
+
+Hat Mapping Implementation When the local observations of different agents' have large overlaps, the hat principle-based mapping method utilizes broadcast channels to achieve a single transmission of messages to multiple receivers. The method is inspired by the multi-color hat guessing game, where players use logical deduction based on limited communication to maximize team success. In this game, players can observe the hat colors of others but not their own, and they must guess their own hat color using a pre-defined strategy. A well-known strategy involves each player leveraging the sum of the hat colors they observe, modulo the total number of possible colors. For example, in an 8-color version of the game, each player can compute the sum of the colors they observe (assigned numerical values) and use a pre-determined rule, such as calculating the sum modulo 8, to make a logical deduction about their own hat color.
+
+In our implementation, at each step, each agent computes local messages intended for every other agent. These local messages are derived from the shared information strategy and the observations that the target agent cannot see but are overlapped with other agents—much like the "hat" in the hat guessing game. When an agent sends information, it transmits a structured public message that combines all these local messages using a sum modulo operation. As a result, each receiving agent can also use the sum modulo operation to combine the received public message with the local messages of all agents except themselves and the sender. This allows them to infer their own specific local message from the combined public message. This method enables a single broadcast to convey individualized information to multiple agents simultaneously, enhancing communication efficiency even when explicit communication is not possible.
+
+# 6 EXPERIMENTS
+
+In this section, we explore the application of the ICP across various environments where no direct communication is available. ICP plays a vital role in these settings by constructing an implicit
+
+![](images/1ef6f576a19c4fce68d64b9bbd550db08c7bd77cfab7a8cea4674ad93d310a26.jpg)  
+Figure 1: Left: Guessing Numbers Environment. Agents cannot see their own digits, but can reveal others' segments' states by collaboratively giving hints. They can deduce their own digit and obtain shared rewards. Right: Revealing Goals Environment. The agents are positioned in a random spots in a grid world and are assigned a unique target. However, they can only observe others' targets which do not include themselves. By revealing each other's targets in the nearby grid, they can eventually find their own targets and reach them.
+
+![](images/32dce9d1b6c85b4154a0773e444002eb3d5369a3a256955d11cf4638856661b2.jpg)
+
+communication channel that enhances the agents' ability to share and interpret information. We will demonstrate how ICP facilitates effective collaboration and decision-making in 3 tasks, namely Guessing Numbers, the Revealing Goals, and the Hanabi card game. The first two tasks are designed by us and can be reused by future works as testing environments. The Hanabi game is a popular card game played by humans, and is described as The Hanabi Challenge (Bard et al., 2020) by the community. Each of these environments presents unique challenges, and we will show how ICP optimizes the use of scouting actions to improve overall performance and strategy.
+
+# 6.1 GUESSING NUMBERS
+
+In the Guessing Numbers game, we present a collaborative, turn-based multi-agent reinforcement learning environment involving  $N$  agents. Each agent  $i$  is uniquely assigned a digit  $d_{i} \in \{0,1,\dots ,9\}$ , which is visible to all other agents  $j \neq i$  but remains unknown to agent  $i$  themselves. The primary objective for each agent is to deduce and correctly guess their own digit  $d_{i}$ . During each turn, an agent can choose to either guess their own digit by selecting  $\hat{d}_i \in \{0,1,\ldots ,9\}$  or provide a hint about another agent's digit using the Radioland Slim font—a seven-segment digital display representation. Specifically, hints involve revealing the state ("on" or "off") of one of the seven segments of another agent's digit, and these hints are public and observable by all agents. Each agent's action space comprises  $10 + 7 \times (N - 1)$  actions: 10 options for guessing their own digit and 7 hint options for each of the  $N - 1$  other agents. The game encourages strategic collaboration, as agents must balance between gathering information to deduce their own digit and assisting others through informative hints, aiming for the collective success of all agents correctly guessing their digits.
+
+In this game, successfully guessing their digit rewards an agent with 10, while performing a hint action incurs a small penalty of  $-0.1$  to encourage efficient communication and collaboration. The game imposes a limit  $l$  on the number of hint actions, and each agent is allowed only 1 guess. The game concludes when all agents have made their guesses, with the goal being to accurately guess all digits within these constraints.
+
+In the Guessing Numbers experiment, we evaluate the performance of 5 approaches: VDN-on-policy, VDN-off-policy, ICP with the random initial map approach (ICN-DIAL-RM), ICP with the delayed map approach (ICN-DIAL-DM), and a cheating approach where a direct communication channel is available (DIAL-Cheat). Each approach is evaluated over 1k episodes with 6 random seeds, and running on a Linux metal machine with 256 GB RAM and 3090Ti GPU for 36 hours.
+
+For VDN-off-policy, we begin by warming up the replay buffer until its size exceeds the batch size. During each training step, we add 10 episodes to the replay buffer and randomly sample a batch of episodes from the buffer for training. In contrast, for VDN-on-policy and our proposed method, we utilize a vectorized environment to sample a batch of episodes at each training step and use these samples for training.
+
+Despite differences in network architecture, these approaches share similar hyperparameters. Specifically, we set the hidden size of the MLP and GRU to 256, use 2 layers in the GRU, and set the
+
+![](images/90471b87c719483cce5e291ac03eaa6d0d927c74402e21693cf2afe88efcde96.jpg)  
+(a)
+
+![](images/eb204fb4d990ce070a9d13ecb37b00b7b33b1497e43ffca9c79a0f021099372d.jpg)  
+(b)  
+Figure 2: (a): The training curves of Guessing Numbers over a total of 100k steps with  $N = 3$ ,  $l = 11$ . (b): Average episode length running by different algorithms in Guessing Numbers.
+
+learning rate to  $5 \times 10^{-4}$ , batch size to 256. The target network update rate is set to 10,  $\gamma$  is set to 0.99,  $\epsilon$  is set to 0.1, and we apply gradient clipping with a threshold of 10.
+
+The results are presented in Figure 2. All algorithms are capable of eventually guessing their own digit correctly, but the differences of number of hit actions needed is significant among the algorithms. By examining the saved models and averaging over  $10\mathrm{k}$  episodes, we find that ICN-DIALRM has an average episode length of approximately 8.63, while VDN-on-policy and VDN-off-policy have average episode lengths of around 12.4. This observation further demonstrates that ICN achieves more efficient information transfer by using scouting actions in implicit channels.
+
+# 6.2 REVEALING GOALS
+
+In the Revealing Goals task, we present a non-sequential collaborative that emphasizes information sharing among agents. The environment consists of  $N$  agents operating within an  $H \times H$  grid world. Each agent  $i$  starts at a random position and is assigned a unique goal location  $g_{i}$  that is at least two grid units away from its starting position. Critically, agents cannot perceive their own goal locations; they can only observe the goal locations of other agents  $j \neq i$ . At each time step, agents select from a fixed action space of eight actions: moving up, down, left, or right, and reveal information about the adjacent grid cells in these directions. When an agent performs a reveal action, the adjacent grid cell in the specified direction becomes revealed, and all agents gain information about whether their own goals are located in that cell. The grid world features wrap-around edges, creating a toroidal topology where, for example, moving left from cell  $(0,y)$  leads to cell  $(H - 1,y)$ .
+
+An agent observes the locations of all agents, the goal points of other agents, and any goals on the revealed grids. Since they cannot directly perceive their own goals, agents must rely on information revealed by themselves and others to infer the locations of their own goals. The objective is for each agent to navigate to its designated goal location  $g_{i}$ , upon which all agents receive a shared reward of 1. When an agent reaches its goal, a new goal is randomly assigned to it but is at least two grid units away from its current position. The game proceeds for a total of  $T$  time steps. This environment encourages strategic collaboration, as agents must balance between exploring the grid to find their own goals and assisting others by revealing grid cells that may contain teammates' goals. The agents will need to communicate implicitly, using the fixed action space of size 8.
+
+In the experiments for the Revealing Goals task, we use the same training hyperparameters and model architecture as in the Guessing Numbers experiments. We also compare the performance of 5 approaches: DIAL-Cheat, ICN-DIAL-DM, ICN-DIAL-RM, VDN-on-policy, and VDN-off-policy. The experimental results are shown in Figure 3(a). ICN-DIAL-RM significantly outperforms the baseline methods by achieving an average score 2.17 times that of the baselines. One reason for its effectiveness is that, in each round, each of the  $N$  agents can engage in  $N$  broadcast communications through implicit channels, which significantly enhances information efficiency compared to relying solely on environmental feedback. This result further validates the effectiveness of ICP in more complicated grid world tasks.
+
+![](images/e92af8272ead06a3633725fc8a5bc31e2adcceb1ed3b561ccfcbbb5c35b71063.jpg)  
+Figure 3: (a): The training curves of Revealing Goals over total 100k train steps with  $N = 4$ ,  $H = 5$ ,  $T = 50$ . (b): The training curves of 4-players Hanabi with on-policy algorithms take around 150 hours and off-policy algorithms take around 20 hours.
+
+![](images/3841b4f26bdbefce63888e2a66010627ecd9ff6f50097f1756069c8c5d453df2.jpg)
+
+# 6.3 HANABI
+
+To demonstrate ICP's broader applicability, we also evaluate it in the game of Hanabi. Hanabi is a fully cooperative card game where players work together to play cards in a specific order to form sequences, aiming to achieve the highest possible score. When it is played by humans, people will mute themselves and coordinate only through taking and observing actions. This game requires effective implicit communication and strategic information sharing among the agents.
+
+Because the game rule of Hanabi is quite involved, we refer it to a popular website https://hanabi.github.io/ that discusses both rules and human-play strategies. The action space for each agent includes three options: playing a card out of their 5-card hand, discarding a card, or giving a hint towards another player's 5-card hand. The hint could reveal all cards in a player's hand of a specific color or rank. Since they cannot see their own cards, agents must rely on hints from others to infer what cards they hold. The success of the game heavily depends on the agents' capacity to use these hints effectively to coordinate their actions and play the correct cards in order.
+
+In our experiments, we focus on the 4-player full version of Hanabi. Since hint actions in Hanabi are natural scouting actions, we select them to serve as information carriers for our ICP implementation. We implement several baseline algorithms, including VDN-off-policy and the original version of SAD (Hu & Foerster, 2019), as well as a variant of SAD with the hint action masks removed. For the ICP algorithm, we explore several variations, including ICN-DIAL-RM-on-policy, which involves strategy training with a random initial map using DIAL's communication gradient technique resulting in an on-policy learning method. We also implement ICN-DIAL-RM-cut-gradient-off-policy, similar to ICN-DIAL-RM-on-policy but with the communication gradient chain cut off to enable off-policy learning. Additionally, we use ICN-RGMComm-DM, which is strategy training with a delayed map using RGMComm (Chen et al., 2024), an off-policy MARL communication algorithm, and ICN-RGMComm-DM-hat-mapping, an extension of ICN-RGMComm-DM that utilizes the hat mapping method for improved communication.
+
+Our presented results are based on the best runs selected from more than three random seeds. From the results shown in Figure 3(b), we observe that the ICP method using the hat mapping approach, ICN-RGMComm-DM-hat-mapping, achieved the best performance. The score achieved by ICP is 24.91, which surpasses the best available learning algorithm (SAD, 23.81 points) and vastly outperforms theory of mind-based methods (Fuchs et al. (2021) 19.13 points) and average human players (17 points, reported in Kantack (2021)). It is worth remarking that the state-of-the-art strategy in the Hanabi game is a search-based algorithm, WTFWThat+search, which reports an averaged score of 24.96 points (Lerer et al., 2020). With the maximum score of the game being 25, and there are certain deals that prevent the players from winning 25 points in any way, both ICP and WTFWThat+search are close to fully solving the game.
+
+For additional details on our implementation, please refer to Appendix A. A comparison between ICP and Hanabi human conventions is provided in Appendix B, along with a detailed analysis of the Hanabi results in Appendix C.
+
+# 7 DISCUSSIONS
+
+# 7.1 COMPATIBILITY OF ICP WITH OTHER TRAINING METHODS
+
+In the ICP framework, action policy allows for the use of various algorithms for training. For instance, the action policy can be trained using value-based methods like VDN (Sunehag et al., 2018) or policy-based methods such as MADDPG (Lowe et al., 2017). This compatibility ensures that ICP can be tailored to various problem settings and agent dynamics. One could choose the most suitable algorithm depending on the specific characteristics of the environment and the agents.
+
+The communication strategy within the ICP framework is also versatile. It can be implemented using a variety of direct channel communication algorithms that support discrete channels. These include well-established methods like DIAL (Foerster et al., 2016) and RGMComm (Chen et al., 2024), which are designed to optimize communication under different constraints and requirements. The flexibility in choosing both the action policy and the communication algorithm highlights the comparability and broad applicability of the ICP framework, making it a general approach for multi-agent reinforcement learning scenarios.
+
+# 7.2 POTENTIAL OF FURTHER UTILIZING THE ENVIRONMENT INFORMATION
+
+As described in Section 4, scouting actions carry two types of information. The ICP framework utilizes the information reflected through the choice of the scouting action (choice information) to establish an implicit communication channel. However, even when we leverage this information, the information reflected through the environment (environment information) remains useful. Although in most cases the constructed implicit channel transmits more stable and useful information, in some situations the environment information is also significant. For example, in the Revealing Goals environment, a revealing action intended to transmit embedded information might also reveal other agents' goals, which helps agents to make better decisions.
+
+To verify the impact of this environmental information on performance, we randomly shuffle 6 different embeddings on the ICN-RGMComm-hat-mapping policy saved from the previous Hanabi experiment. We then fine-tune the action policy while keeping the information strategy fixed. The results are shown in Figure 4 in the appendix, where, under a fixed information strategy, some randomly shuffled embeddings enjoy improved performance. This suggests that one could further utilize the environment information for more effective communication in future works.
+
+# 8 CONCLUSION AND FUTURE WORK
+
+The Implicit Channel Protocol (ICP) introduced in this paper is an advancement in implicit communication through multi-agent reinforcement learning (MARL). By mapping information to scouting actions to construct implicit channels and optimizing both action and communication strategies, ICP facilitates communication protocols to be established even without explicit channels. The experimental results validate the protocol's effectiveness across various MARL scenarios and demonstrate its capacity to significantly improve both coordination among agents and performance in the tasks.
+
+A possible direction for future work is extending ICP to environments that lack inherent scouting actions. In such settings, identifying appropriate actions to serve as information carriers becomes challenging, as these actions must both convey information effectively and interact with the environment beneficially. Future research could focus on developing algorithms that dynamically identify or design such actions, balancing effective communication with minimal impact on the environment. These advancements would enhance the applicability of ICP across a wider range of scenarios, making it a more versatile tool in multi-agent communication.
+
+# REPRODUCIBILITY STATEMENT
+
+The code and our designed environments are freely available in the supplementary material. The details of the implementation can be found in Appendix A. The hyper-parameters used in experiments are also mentioned in Section 6.
+
+# ACKNOWLEDGEMENT
+
+Han Wang and Baoxiang Wang are partially supported by the National Natural Science Foundation of China (62106213, 72394361), an extended support project from the Shenzhen Science and Technology Program, Longgang District Key Laboratory of Intelligent Digital Economy Security, and ByteDance.
+
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+Peter Sunehag, Guy Lever, Audrunas Gruslys, Wojciech Marian Czarnecki, Vinicius Zambaldi, Max Jaderberg, Marc Lanctot, Nicolas Sonnerat, Joel Z Leibo, Karl Tuyls, et al. Value-decomposition networks for cooperative multi-agent learning based on team reward. In Proceedings of the 17th International Conference on Autonomous Agents and MultiAgent Systems, pp. 2085-2087, 2018.  
+Kai Tian, Athanasios Tzigieras, Chongfeng Wei, Yee Mun Lee, Christopher Holmes, Matteo Leonetti, Natasha Merat, Richard Romano, and Gustav Markkula. Deceleration parameters as implicit communication signals for pedestrians' crossing decisions and estimations of automated vehicle behaviour. Accident Analysis & Prevention, 190:107173, 2023.  
+Zheng Tian, Shihao Zou, Ian Davies, Tim Warr, Lisheng Wu, Haitham Bou Ammar, and Jun Wang. Learning to communicate implicitly by actions. In Proceedings of the AAAI Conference on Artificial Intelligence, 2020.  
+Michael Tomasello and Amrisha Vaish. Origins of human cooperation and morality. Annual review of psychology, 64(1):231-255, 2013.  
+Mycal Tucker, Roger Levy, Julie A Shah, and Noga Zaslavsky. Trading off utility, informativeness, and complexity in emergent communication. Advances in neural information processing systems, 35:22214-22228, 2022.  
+Karl Tuyls and Gerhard Weiss. Multiagent learning: Basics, challenges, and prospects. *Ai Magazine*, 33(3):41-41, 2012.  
+Sun Tzu. The art of war. In Strategic Studies, pp. 63-91. Routledge, 2008.  
+Rundong Wang, Xu He, Runsheng Yu, Wei Qiu, Bo An, and Zinovi Rabinovich. Learning efficient multi-agent communication: An information bottleneck approach. In International Conference on Machine Learning, pp. 9908-9918. PMLR, 2020.  
+Yuanfei Wang, Jing Xu, Yizhou Wang, et al. Tom2c: Target-oriented multi-agent communication and cooperation with theory of mind. In International Conference on Learning Representations, 2021.  
+Peter Winkler. Games people don't play. Puzzer's Tribute: a Feast for the Mind, pp. 301-313, 2002.  
+Zhuoya Zhao, Feifei Zhao, Yuxuan Zhao, Yi Zeng, and Yinqian Sun. A brain-inspired theory of mind spiking neural network improves multi-agent cooperation and competition. *Patterns*, 4(8), 2023.  
+Changxi Zhu, Mehdi Dastani, and Shihan Wang. A survey of multi-agent reinforcement learning with communication. arXiv preprint arXiv:2203.08975, 2022.
+
+# A TRAINING DETAILS
+
+# A.1 ICP IMPLEMENTATION WITH DIAL AND VDN
+
+In our implementation, we employ a 2-head input and 2-head output Implicit Channel Net (ICN) to parameterize all Q functions. Specifically, at each time step  $t$ , for agent  $i$ , observation-action pairs  $(o_{t,i}, u_{t-1})$  are input to one Multilayer Perceptron (MLP) and Rectified Linear Unit (ReLU) layer. Additionally, the last message vectors from other agents  $(\vec{m_{k,l}}$ , where  $k = \max(j \mid \vec{m_{j,l}} \in \vec{M}))$  for  $l \in (1, \dots, N)$  are stacked together along with each message sender's ID, then input to another MLP and ReLU layer. The outputs of these two layers are summed and input to a Gated Recurrent Unit (GRU) network (Chung et al., 2014). This process approximately achieves the Q function input with action-observation history  $\tau$  and other agents' messages  $m_{-i}$  by utilizing hidden states and inputs at each time step.
+
+Subsequently, the output of the GRU is directed to two distinct heads: an action head and a message head. The action head generates logits for actions, with an output dimension of  $|U - U^s| + 1$ , while the message head is specialized to handle the agent's communication behavior, producing logits for messages. Specifically, the output dimension of the message head is set to  $|U^s| + 1$ , where the additional dimension represents the NOOP (no operation) action. This design ensures that the agent refrains from sending a message when the action sampled from the action logits does not correspond to send_info. To enforce this behavior, a mask is applied to the output of the message head. When the action sampled from the action logits is send_info, the mask restricts Gumbel-Softmax sampling to select the NOOP action. If the agent decides to send a message, the sampled one-hot vector of messages  $\vec{m}$  directly maps to the  $j$ -th information action  $u^{\mathrm{info}}$ , where  $j = \operatorname*{argmax}_{l\in (1,\dots,|U^s|)}\vec{m}_l$ .
+
+During centralized training sessions, action sampling follows an  $\epsilon$ -greedy policy to ensure exploration, while we maintain the gradient of the one-hot vector of messages  $\vec{m}$  to ensure gradients pass through the communication process. However, since gradient information becomes inaccurate for updated policies, we can only use sampled episodes once, making our method on-policy. In decentralized execution sessions, we simply utilize the shared parameter network to manage greedy actions and one-hot vector messages sampled from Gumbel-Softmax.
+
+# A.2 ICP IMPLEMENTATION WITH RGMCOMM AND VDN
+
+In the pre-training phase of this approach, we used RGMComm, which utilized Regularized Information Maximization loss (Chen et al., 2024) to generate discrete messages. This method represents the latest state-of-the-art explicit communication algorithm in multi-agent reinforcement learning (MARL).
+
+RGMComm itself is based on the MADDPG (Multi-Agent Deep Deterministic Policy Gradient) framework (Lowe et al., 2017). Initially, a full observability centralized actor-critic model (Konda & Tsitsiklis, 1999) is trained to develop a policy with access to global information. This centralized policy serves as a reference to guide the communication strategy for each agent. During this stage, the agents use a centralized critic with a complete view of the environment to assess the action-value functions and optimize the communication strategy. After training this full-observability policy, RGMComm samples different information sets based on local observations. It then clusters these sampled information sets (Harsanyi, 1967) to assign specific messages to each cluster. By doing so, RGMComm derives a discrete communication protocol tailored to each agent's local observations, optimizing their joint policy under partial observability.
+
+In our implementation, instead of using MADDPG, we employed an RDQN (Recurrent Deep Q-Network) (Hausknecht & Stone, 2015) to train the policy with full observability. For example, in environments like Hanabi, the difference between the full observation (global view) and the local observation is minimal, with only additional knowledge of the agent's own hand being considered. Using this full-observation policy, we performed clustering on the information sets, similarly to the RGMComm approach, to obtain the message strategy.
+
+Following the pre-training phase, we proceed to the fine-tuning phase. This phase involves using VDN (Value Decomposition Network) to train the action policy. During this process, the communication strategy is embedded into the scouting action space and is fine-tuned to adapt to the environ-
+
+Algorithm 1: ICP implementation with DIAL and VDN  
+Input: environment env, max-train-step, max-episode-length, target-update-rate,  $\gamma$  , exploration constant  $\epsilon$  , learning rate  $\alpha$  , selection of information action space  $U_{\mathrm{info}}$    
+Output: Trained model parameters   
+1 Initialize shared RNN network weights  $(\theta_{1},\theta_{2})$  for each agent  $i$  and centralized Mapping Mechanism  $\mathcal{P}$  , target net weights  $(\theta_1^{\prime},\theta_2^{\prime})\gets (\theta_1,\theta_2)$  .   
+for Train step  $N = 1,2,\ldots$  , max-train-step do   
+3 Reset environment,  $o_0 = \mathsf{env}$  reset();   
+4 Reset buffer  $B$  .   
+5 Initialize empty one-hot messages  $m_0$  and target empty one-hot messages  $m_0^\prime$  , 2 RNNs' hidden states  $(h_1,h_2)$  and 2 target RNNs' hidden states  $(h_1',h_2')$  .   
+6 for time step  $t = 1,2,\dots$  , max-episode-length do   
+7 Initialize empty one-hot messages  $m_t$  .   
+8 for each agent i do Sample message and update hidden state:  $m_{\mathrm{sample}},h_2 = GumbelSoftmax(\varphi_{\theta_2}(o_{t,i},m_{t - 1, - i},h_2));$  Sample a random action  $u_{\mathrm{random}}\in U_{\mathrm{non - info}}$  . Compute greedy action and update hidden state:  $u_{\mathrm{greedy}},h_1 = \arg \max_u(\pi_{\theta_1}(o_{t,i},m_{t - 1, - i},h_1))$  .  $\epsilon$  greedy sample action  $u_{t,i}\leftarrow \epsilon \cdot u_{\mathrm{random}} + (1 - \epsilon)\cdot u_{\mathrm{greedy}}$  Get action  $u_{t,i}$  s q:  $q_{t,i} = Q_{\theta_1}(u_{t,i}\mid o_{t,i},m_{t - 1, - i},h_1)$  if  $u_{t,i} = |U_{\mathrm{non - info}} - 1|$  (send_info) then Update action  $u_{t,i}\gets u_{info} = \mathcal{P}(m_{\mathrm{sample}},\bigcap_{i\in N}\tau_{t,i})$  Update message  $m_{t,i} = m_{\mathrm{sample}}$  Sample target message and update hidden state:  $m_{\mathrm{sample}}^{\prime},h_2^{\prime} = GumbelSoftmax(\varphi_{\theta_2}^{\prime}(o_{t,i},m_{t - 1, - i}^{\prime},h_2^{\prime}))$  Compute target greedy action's  $q$  and update hidden state:  $q_{t,i}^{\prime},h_1^{\prime} = \max_u(\pi_{\theta_1}^{\prime}(o_{t,i},m_{t - 1, - i}^{\prime},h_1^{\prime}))$  if arg max  $\boldsymbol {u}_{(}\pi_{\theta_{1}^{\prime}}(o_{t,i},m_{t - 1, - i}^{\prime},h_{1}^{\prime})) = |U_{\mathrm{non - info}} - 1|$  (send_info) then Update message  $m_{t,i}^{\prime} = m_{\mathrm{sample}}^{\prime}$  Step environment,  $o_{t + 1},r_t = \mathrm{env}.step(u_t)$  Store  $(q_t,q_t',rt)$  into buffer B; Use samples in buffer to compute sum-  $q$  's TD-error:  $\delta_t = r_t + \gamma \cdot \sum_i(q_{t + 1,i}' - \sum_i(q_t,i);$  Update network weights  $\theta_{1},\theta_{2}$  using Adam Optimizer with loss  $\delta_{t,i}^{2}$  if Train step N mod target-update-rate  $\equiv 0$  then Update target network weights  $(\theta_1',\theta_2')\gets (\theta_1,\theta_2)$
+
+ment. By freezing the RGMComm-generated communication strategy during the initial stages of fine-tuning, we ensure stability and later adjust the embedding to better align with the action policy.
+
+# A.3 IMPLEMENTATION OF DIAL-CHEAT AND ICN-DIAL-DM
+
+DIAL with a Direct Communication Channel (DIAL-Cheat): In this approach, we added a discrete direct channel to the original environment, with the message size matching the size of the scouting action space. This channel allows agents to broadcast information while simultaneously taking actions.
+
+ICP Implemented with the Delayed Map Approach (ICN-DIAL-DM): For this method, we first removed scouting actions from the environment and introduced a discrete direct channel (with the same message size as the scouting action space). Using this modified environment, we pre-trained the agents with DIAL to learn the information strategy. The learned information strategy was then transferred back to the original environment for fine-tuning, where both the information strategy and action policy were optimized.
+
+# B HUMAN CONVENTION IN HANABI
+
+Hanabi serves as a compelling platform for studying cooperation and ad-hoc teamwork, particularly within the MARL community, while also being an engaging game in its own right. Beyond its academic appeal, many players have developed highly sophisticated strategies, as detailed in the referenced H-Group conventions<sup>1</sup>. These conventions outline a framework that enables players to achieve high scores by adhering to agreed-upon strategies.
+
+![](images/75e8cdb8924c99439c40c7c4146e635a0cc5c32d580ad0c9a1a35cf3f80d7662.jpg)  
+Figure 4: By shuffling the embedding of information into scouting actions, even if the information strategy stays the same, environment information will also change. After fine-tuning, the performance of implementation with the same information strategy but different embedding varies.
+
+Interestingly, some of these strategies embed elements of ToM reasoning into the conventions themselves. For example, techniques like Delayed Play Clues explicitly define the information that certain actions are intended to convey when teammates perform specific moves. Even the Basic and Advanced Strategies, such as misleading bluffs, leverage second-order ToM reasoning to facilitate more effective collaboration.
+
+In a manner similar to ICP, the H-Group conventions construct a stable implicit communication channel by explicitly codifying the ToM reasoning into the actions. When all players follow these conventions, actions serve as reliable carriers of information, ensuring a consistent and effective communication channel within the game.
+
+Compared with H-Group conventions, our ICP framework differs in that the communication strategy itself is learned as part of the training process, rather than being predefined. The communication through informative actions becomes akin to following a learned convention where agents develop their own patterns for encoding and transmitting information during the training phase. This allows agents to use scouting or informative actions as explicit communication mechanisms, while regular actions are selected through an online decision-making policy during gameplay. Notably, our method achieves a score of approximately 24.91 in the Hanabi game, which significantly surpasses the average human score of around 17 (Kantack, 2021). Thus, ICP offers a dynamic communication framework where the convention of communication emerges from the agents' learning process, enabling more flexible and adaptive strategies depending on the environment and task at hand.
+
+# C ANALYSIS OF HANABI RESULTS
+
+In Figure 3(b), we present a comparison between our proposed method and SAD, focusing on differences in average scores. While the difference in average score appears minimal in the figure, a detailed analysis provided below reveals that our method achieves a substantial improvement in overall performance.
+
+According to the results reported in the original SAD paper, SAD achieved an average score of 23.81/25 with a win rate of  $41.45\%$ . In contrast, our proposed method achieved an average score of 24.91/25 with a win rate of  $91\%$ . This represents a more than twofold increase in the win rate. The relatively small difference in average scores, despite the significant improvement in win rates, can be attributed to the score ceiling of 25 in Hanabi.
+
+Hanabi imposes inherent constraints on scoring due to its game structure. Certain starting hands, such as those with multiple high-value cards (e.g., 4s and 5s), can make achieving the maximum score of 25 impossible. Considering these limitations, achieving a win rate of  $91\%$  and scores close to 24 in the remaining  $9\%$  of games—demonstrates that our method performs near the theoretical maximum.
\ No newline at end of file
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+# LEARNING TO CONTEXTUALIZE WEB PAGES FOR ENHANCED DECISION MAKING BY LLM AGENTS
+
+Dongjun Lee\*, Juyong Lee\*, Kyuyoung Kim, Jihoon Tack
+
+Jinwoo Shin<sup>1</sup>, Yee Whye Teh<sup>2</sup>, Kimin Lee<sup>1</sup>
+
+{dgjun32，agi.is}@kaist.ac.kr
+
+KAIST AI<sup>1</sup>, University of Oxford<sup>2</sup>
+
+# ABSTRACT
+
+Recent advances in large language models (LLMs) have led to a growing interest in developing LLM-based agents for automating web tasks. However, these agents often struggle with even simple tasks on real-world websites due to their limited capability to understand and process complex web page structures. In this work, we introduce LCoW, a framework for Learning language models to Contextualize complex Web pages into a more comprehensible form, thereby enhancing decision making by LLM agents. LCoW decouples web page understanding from decision making by training a separate contextualization module to transform complex web pages into comprehensible format, which are then utilized by the decision-making agent. We demonstrate that our contextualization module effectively integrates with LLM agents of various scales to significantly enhance their decision-making capabilities in web automation tasks. Notably, LCoW improves the success rates of closed-source LLMs (e.g., Gemini-1.5-flash, GPT-4o, Claude-3.5-Sonnet) by an average of  $15.6\%$ , and demonstrates a  $23.7\%$  average improvement in success rates for open-source LMs (e.g., Llama-3.1-8B, Llama-3.1-70B) on the WorkArena benchmark. Moreover, the Gemini-1.5-flash agent with LCoW achieves state-of-the-art results on the WebShop benchmark, outperforming human experts. The relevant code materials are available at our project page: https://lcowiclr2025.github.io.
+
+# 1 INTRODUCTION
+
+Large language models (LLMs) have demonstrated strong potential in automating web tasks by treating web browsing as a sequential decision-making process, where web pages serve as observations and user interactions, such as clicking and typing, function as actions (Yao et al., 2022a;b). Various approaches have been developed to enhance the performance of LLM agents in these tasks. One such method involves fine-tuning open-source LLMs using demonstration data from web browsing tasks (Furuta et al., 2023; Lai et al., 2024). While promising, this approach requires extensive data collection and significant computational resources for effective fine-tuning. Alternatively, several studies have utilized advanced closed-source LLMs, such as GPT-4o (OpenAI, 2024), with carefully designed prompting techniques (Drouin et al., 2024; Zhou et al., 2023a; Sodhi et al., 2024; Pan
+
+et al., 2024). By leveraging the general world knowledge and reasoning capabilities of these models, the methods enhance web automation but at the cost of reduced controllability.
+
+![](images/0d7272f3687c9e8a6e5a4b255ab1d41c448a9c678708130e12e302868f0b84d6.jpg)  
+Figure 1: Success rate of the Gemini-1.5 flash agent on 40 WorkArena tasks. We selected a subset of 40 tasks by choosing the first 40 tasks based on the task indices. When the agent leverages observations contextualized by GPT-4o (yellow), its success rate improves by  $31\%$ , with further improvements achieved with our method (green).
+
+However, despite the advancements, state-of-the-art LLM agents often struggle to process complex raw web content, such as HTML and accessibility trees, posing significant challenges for their effec
+
+![](images/61ed5e9069ac8640bc406f5faf425ddf285eac61e271c89a95f64785210c0930.jpg)  
+Figure 2: (Top) In the conventional pipeline, LLM agents decide on the next action based on raw, complex web page observations (e.g., HTML, accessibility trees), which often hinder accurate decision making. (Bottom) In our proposed pipeline, a contextualization module transforms these complex web page observations into a more comprehensible format, thereby enabling LLM agents to make more accurate decisions by enhancing their understanding of the web page.
+
+tive use in web task automation. While LLMs excel in tasks that require detailed reasoning, such as solving mathematical problems or coding, we hypothesize that their underperformance in seemingly simple decision-making tasks like web browsing is not due to a lack of decision-making capabilities but rather to difficulties in understanding and processing complex web page observations.
+
+To validate our hypothesis, we conducted an initial experiment that demonstrated an LLM agent based on Gemini-1.5-flash can achieve substantial improvements in web browsing tasks when equipped with a module designed to contextualize complex web page observations (i.e., contextualization module). This module enhances task performance by removing irrelevant UI elements and highlighting key components with explanations, thereby simplifying the decision-making process. We evaluated the performance of this agent on 40 tasks from WorkArena (Drouin et al., 2024), a benchmark designed to assess web agents on real-world, enterprise-related websites. As shown in Figure 1, utilizing GPT-4o as the contextualization module (yellow) resulted in a  $31\%$  absolute improvement in the agent's success rate compared to direct processing of raw observations (red). These results support our hypothesis that the difficulty in understanding web pages is a major bottleneck for LLM-based web agents.
+
+In this work, we propose LCoW, a framework that includes a contextualization module and a training algorithm to fine-tune this module to enhance the decision-making capabilities of LLM agents in web automation. As illustrated in Figure 2, the contextualization module transforms complex web page observations into a comprehensible format, enabling LLM agents to make more accurate decisions. Furthermore, to enable the contextualization module to provide context more grounded in real websites, we propose an iterative algorithm designed to train the contextualization module. Our training algorithm consists of three phases: (i) trajectory collection, (ii) sampling contextualized observations, and (iii) updating the contextualization module. For each observation from the collected trajectories, we generate multiple contextualized observations using the current contextualization module. Each observation is then assigned a reward based on whether a set of LLM agents can accurately predict the correct action given the contextualized observation. Finally, we select the one with the highest reward as the target and train the contextualization module to maximize the likelihood of the target given the original raw observation.
+
+As demonstrated in our initial experiment using the Gemini-1.5-flash (see Figure 1), LCoW significantly enhances the decision-making capabilities of LLM agents, even beyond the improvements seen with state-of-the-art LLMs like GPT-4o used as a contextualization module. In our experiments, we conduct comprehensive evaluations of our proposed approach on three well-established benchmarks for evaluating agent performance in web environments: WebShop (Yao et al., 2022a), WorkArena (Drouin et al., 2024), and WebArena (Zhou et al., 2023a). In detail, we first demonstrate that LCoW significantly enhances the overall performance of LLM agents with varying scales in the main experiment. Furthermore, we analyze the behavior of the contextualization module in
+
+detail, including qualitative examination of the observation refined by the contextualization module, comparison with training LLM agents using behavior cloning, and generalization ability.
+
+# 2 BACKGROUND
+
+In this section, we describe the formulation of web browsing as a sequential decision-making problem and the use of LLMs as decision-making agents.
+
+Web browsing can be formulated as a Partially Observable Markov Decision Process (POMDP), defined by  $\langle S, \mathcal{O}, \mathcal{A}, T, \mathcal{R} \rangle$ . The state  $s_t \in S$  represents the internal configuration of the web browser at time step  $t$ , which is only partially observable. The observation  $o_t \in \mathcal{O}$  corresponds to the web page rendered by the web browser given  $s_t$ , which can take various forms (e.g., screenshot, HTML, accessibility trees). The action space  $\mathcal{A}$  is the set of all possible interactions with the UI elements (e.g., clicking, typing). The state transition function  $T$  defines the probability of transitioning from state  $s$  to state  $s'$  after performing an action  $a_t \in \mathcal{A}$ , such as clicking a link or scrolling. While transitions are typically deterministic, occasional stochastic events (e.g., pop-ups, network errors) can occur. The reward function  $\mathcal{R}$  assesses the functional correctness, evaluating whether the resulting state  $s_t$  satisfies pre-defined criteria for successful task completion.
+
+Leveraging their ability to interpret web pages and generate actions in text form, LLMs are increasingly employed as agents for automating web-based tasks. In this framework, an LLM agent  $\pi$  generates an action  $a_{t}$  to interact with UI elements at each time step  $t$ , based on the user instruction [TASK], the current web page observation  $o_{t}$ , and the history of previous actions  $a_{< t}$ . The objective of the agent is to complete the given task in order to maximize the reward.
+
+# 3 METHOD
+
+In this section, we present LCoW, a framework for enhancing the capability of LLM agents by contextualizing complex web pages. Section 3.1 outlines the concept of the contextualization module and its integration with LLM agents for web task automation. Section 3.2 introduces an iterative algorithm for training the contextualization modules to improve decision making of LLM agents.
+
+# 3.1 CONTEXTUALIZATION MODULE
+
+In this work, we decouple web page understanding from decision making of LLM agents. Our hypothesis is that while LLM agents possess strong decision-making capabilities, their performance can significantly degrade when they rely on lengthy, non-contextualized observations, such as HTML and accessibility trees. To address this limitation, we introduce a contextualization module, a separate language model designed to enhance LLM agents by contextualizing complex web page observations into a form that is easier to process and comprehensible. Intuitively, a proper contextualization of observations can enhance the agent's understanding of web content and its decision making based on the improved understanding (see Figure 3 for an example of the input and output of the module). Formally, given a web page observation  $o_{t}$  at time step  $t$ , the objective of our contextualization module  $f_{\theta}$  is to generate a contextualized observation  $o_{t}^{\circ}$  that serves as input to the LLM agent  $\pi$  to enhance its decision making. Specifically,  $f_{\theta}$  uses the task instruction [TASK], the previous actions of the agent  $a_{<t}$ , and the current web page observation  $o_{t}$  to generate a contextualized observation:
+
+$$
+o _ {t} ^ {\circ \circ} = f _ {\theta} \left(\left[ \text {T A S K} \right], a _ {<   t}, o _ {t}\right).
+$$
+
+The LLM agent  $\pi$  then predicts the next action based on the contextualized observation:
+
+$$
+a _ {t} = \pi \big ([ T A S K ], a _ {<   t}, o _ {t} ^ {\circledast} \big).
+$$
+
+While an arbitrary language model can serve as a contextualization module  $f_{\theta}$ , it is important for the module to learn from experience in the web environment to provide more grounded context for decision making, such as role of a particular button or interaction with specific UI elements.
+
+# 3.2 ALGORITHM FOR TRAINING THE CONTEXTUALIZATION MODULE
+
+We now describe an iterative algorithm for training the contextualization module  $f_{\theta}$  to enhance decision making of LLM agents. In a nutshell, the algorithm involves an iterative process of col
+
+```txt
+Role: Your task is to generate a "Reasoning" and a "Refined observation" based on the provided information.  
+Task: "Purchase me an iPad pro with silver color and ..."  
+Action history:  
+1. click [a32] 'Hardware'  
+2. click [a45] 'iPad pro'  
+Web page observation:  
+RootWebArea 'Catalog | ServiceNow'  
+generic ', live=' assertive', ...  
+generic ', live=' polite', ...  
+[55] navigation 'Global skip links'  
+...  
+[a157] rowgroup ': [a158] row ': [a159] gridcell ': ... [a401] LayoutRow ': [a402] LayoutTableCell 'Shopping Cart' [a403] LayoutTableCell 'Empty'
+```
+
+```txt
+Reasoning: We have navigated to the Hardware store and clicked on the iPad pro link. The actions needed to accomplish the user instruction are to choose color option and disk option.  
+Refined observation: I can focus on following elements:  
+[a192] combobox 'Quantity' value='1', checked=true, It confirms that 1 is already selected as a quantity.  
+[a195] radiogroup 'Mandatory' [a196] radio '\uf Silver', checked=True [a197] radio '\uf Space Gray'  
+This group contains the color options.  
+[a198] radiogroup 'Mandatory' [a200] radio '\uf137 128 GB', checked=True [a203] radio '\uf137 256 GB [add $100.00]'  
+This group contains the storage options.  
+[a206] button 'Order Now'  
+This button allows to finalize the process.
+```
+
+Figure 3: An example of a input of contextualization module including lengthy web page observation (left) and an observation contextualized by the contextualization module trained using LCoW (right). The module converts raw observations into a more concise form to enhance decision making in agents. The prompt used is provided in Appendix C.
+
+lecting paired input-output data for training the contextualization module and subsequently updating the module based on the collected data. For data collection, we begin by gathering trajectories of successfully completed tasks from the web browsing environment. For each observation  $o_{t}$  in the collected trajectories, we sample multiple candidate contextualized observations, and select the one that best provides the relevant context for multiple LLM agents to accurately predict the next action  $a_{t}$ . Based on the chosen target observations, we update  $f_{\theta}$  via supervised fine-tuning. We now outline a single iteration of LCoW, followed by a detailed explanation of the design of the reward used for evaluating the candidate contextualized observations.
+
+Single iteration A single iteration of LCoW starts with the contextualization module  $f_{\theta^{(i)}}$  and aims to update this module to  $f_{\theta^{(i + 1)}}$ . This process consists of three steps:
+
+Step 1 (Trajectory collection). Given a set of training tasks, we roll out the LLM agent  $\pi$  in the web environment to collect trajectory data. Specifically, the agent determines the next action based on the contextualized observation produced by  $f_{\theta^{(i)}}$  until the episode terminates. We collect only those trajectories that end in the successful completion of the tasks.
+
+Step 2 (Sampling optimal contextualization). As illustrated in Figure 4, we start by sampling multiple candidates  $o_{t}^{\circ \circ}$  from the current contextualization module  $f_{\theta^{(i)}}$  (i.e.,  $o_{t}^{\circ \circ} \sim f_{\theta^{(i)}}(\cdot | [\text{TASK}], a_{<t}, o_{t})$ ) for each web page observation  $o_{t}$  in the collected trajectories. Each candidate is then assigned a reward based on whether a set of LLM agents can accurately predict the ground-truth action  $a_{t}$  given  $o_{t}$ , with the candidate receiving the maximum reward selected as the optimal contextualized observation. If all candidates receive a zero reward, we retry the sampling process with the ground-truth action  $a_{t}$  provided as additional context to  $f_{\theta^{(i)}}$  (i.e.,  $o_{t}^{\circ \circ} \sim f_{\theta^{(i)}}(\cdot | [\text{TASK}], a_{<t}, o_{t}, a_{t})$ ) to guide the generation of valid contextualized observations.
+
+Step 3 (Model update). We update the current module  $f_{\theta^{(i)}}$  by fine-tuning it with the optimal contextualized observations collected in Step 2. With the updated module  $f_{\theta^{(i + 1)}}$ , we return to Step 1 and repeat the process.
+
+Algorithm 1 outlines the steps for a single iteration of the training algorithm. Starting with an initial contextualization module  $f_{\theta^{(0)}}$ , we iteratively train the module through  $M$  iterations. After each iteration, the module updates from  $f_{\theta^{(i)}}$  to  $f_{\theta^{(i + 1)}}$  until reaching the final  $f_{\theta^{(M)}}$ . Additionally,  $\mathcal{T}$  can be initialized as a set of human demonstrations in order to facilitate the training process.
+
+Reward for contextualized observations The reward for the contextualized observation  $o_{t}^{\circ \circ}$  is defined as the sum of the action-matching scores computed using multiple LLM agents. Each action matching score evaluates whether an LLM agent  $\pi$  correctly predicts the ground-truth action  $a_{t}$  given the contextualized observation  $o_{t}^{\circ \circ}$ . By leveraging multiple LLM agents to compute the reward, we
+
+![](images/35020fb6be5f64053765e8d754336430e857ebf1a7140c29851ec23e4a4b201e.jpg)  
+Figure 4: Illustration of sampling optimal contextualization. First, we sample multiple candidates of contextualized observations, given user instruction [TASK], previous actions  $a_{<t}$ , and observation  $o_t$ . Subsequently, multiple LLM agents predict the next action based on each candidate, and the reward for each candidate is computed according to how many LLM agents correctly predict the ground-truth action  $a_t$ .
+
+Algorithm 1 One iteration of LCoW  
+Require: a contextualization module  $f_{\theta(i)}$ , an LLM agent  $\pi$ , a set of LLM agents for computing the reward  $\Pi = \{\pi_i\}_{i=1}^K$ , a trajectory buffer  $\mathcal{T}$ , an empty data buffer  $\mathcal{D}$ , and a set of training tasks  $\mathcal{G}_{tr}$   
+1: // Trajectory collection  
+2: for [TASK]  $\in \mathcal{G}_{tr}$  do ▷ Collect successful trajectories from training environment  
+3:  $\tau, R \sim \pi(\cdot | [\text{TASK}], f_{\theta(i)})$   
+4: if  $R = 1.0$  then  
+5:  $\mathcal{T}$ .append( $\tau$ )  
+6: // Sampling optimal contextualizations  
+7: for  $(o_t, a_t)$  in  $\mathcal{T}$  do  
+8: for  $n \gets 1$  to  $N$  do ▷ Sample  $N$  candidate contextualized observations and assign rewards  
+9:  $o_{t,n}^{\circ} \sim f_{\theta(i)}(\cdot | [\text{TASK}], a_{<t}, o_t)$   
+10:  $r_{t,n} = \sum_{\pi \in \Pi} \text{ActionMatchingScore}(\pi([\text{TASK}], a_{<t}, o_{t,n}^{\circ}), a_t)$   
+11: if  $\max_n(r_{t,n}) = 0$  then  
+12: for  $n \gets 1$  to  $N$  do ▷ Retry the sampling if rewards are zero for all candidates  
+13:  $o_{t,n}^{\circ} \sim f_{\theta(i)}(\cdot | [\text{TASK}], a_{<t}, o_t)$   
+14:  $r_{t,n} = \sum_{\pi \in \Pi} \text{ActionMatchingScore}(\pi([\text{TASK}], a_{<t}, o_{t,n}^{\circ}), a_t)$   
+15:  $o_{t,*}^{\circ} = \arg \max_n(r_{t,n})$   
+16:  $\mathcal{D}$ .append([[[TASK], a_{<t}, o_t], o_{t,*}^{\circ}])  
+17: // Parameter update  
+18:  $\theta^{(i+1)} := \arg \max_{\theta(i)} \mathbb{E}_{([TASK], o_{t,*}^{\circ}, a_{<t}, o_t)} \sim \mathcal{D}[f_{\theta(i)}(o_{t,*}^{\circ}][\text{TASK}], a_{<t}, o_t]$   
+19: return  $f_{\theta(i+1)}$
+
+ensure that the module produces contextualized observations that generalize across a diverse set of agents, preventing overfitting to the behavior of any single agent and enhancing the module's adaptability to arbitrary LLM agents.
+
+# 4 EXPERIMENTS
+
+We design our experiments to investigate the following questions:
+
+- How effective is LCoW in training a contextualization module for improving decision making of LLM agents? (Section 4.2)  
+- Can the contextualization module trained with LCoW generalize to arbitrary LLMs with varying scales? (Section 4.2)  
+- What form do the web page observations take after contextualization and how the contextualized observation aid the decision making of LLM agents? (Section 4.3)  
+- Can the contextualization module trained with LCoW generalize to unseen tasks? (Section 4.3)
+
+![](images/44c82451df0e277ff7c3b88cd80879ed368d5c2a2d1bbac71895299643aaa5b8.jpg)  
+Figure 5: Success rate on 500 evaluation tasks from WebShop. Average human performance and expert human performance are  $50\%$  and  $59.6\%$ , respectively (Yao et al., 2022a). The Gemini-1.5-flash agent with the contextualization module trained for three iterations achieves a state-of-the-art success rate of  $62.8\%$ , outperforming the human expert performance, as well as previous baselines (Yao et al., 2022b; Furuta et al., 2023; Putta et al., 2024; Sridhar et al., 2023; Ma et al., 2024; Gur et al., 2023).
+
+# 4.1 BENCHMARKS & EVALUATION SETUP
+
+In this section, we describe three benchmarks used for our experiments. We consider the three benchmarks, WebShop (Yao et al., 2022a), WorkArena (Drouin et al., 2024), and WebArena (Zhou et al., 2023a), which are well-established benchmarks designed to evaluate the capabilities of web agents in completing various web tasks. For the main experiment, we exploit Webshop and WorkArena. For the additional analysis, we also conduct experiments in WebArena. Additional details about hyperparameters and prompts are described in Appendix B and Appendix C, respectively.
+
+WebShop WebShop provides a simulated online shopping environment with real-world product data, consisting of 500 evaluation tasks and 5,500 training tasks, each defined by natural language instructions to purchase products that meet specific criteria. During each decision-making step, the agent receives accessibility tree formatted web page observation to predict the next action, and at the end of an episode, it receives a reward ranging from 0 to 1 based on how well the attributes of the purchased item align with the intended product criteria. When the reward is equal to 1, the episode is considered to be a success. In our experiment, we train the contextualization module using LCoW on environments associated with 500 of the 5,500 training tasks in WebShop. After training, we evaluate the module on 500 evaluation tasks, measuring both the success rate and average reward.
+
+WorkArena WorkArena focuses on assessing web agents' ability to complete enterprise-related tasks, such as creating user accounts and ordering products from a service catalog, on realistic websites. This benchmark comprises 33 task types, with each task type containing up to 1,000 individual task instances, where agents have to navigate over web pages by following natural language instructions and receive rewards based on successful task completion. The agent receives binary rewards (i.e., 0 or 1) based on whether the instruction is achieved via web navigation. In our experiment, we train the contextualization module on 15 task instances for each of 33 task types (i.e., 495 tasks as a total) and evaluate the success rate on 5 task instances for each of 33 task types (i.e., 165 evaluation tasks).
+
+WebArena WebArena consists of 812 tasks spanning over 6 websites (Shopping, Gitlab, Reddit, Map, Wikipedia, and content management systems), where 812 individual tasks are created based on 190 task templates. Similar to WorkArena, WebArena is also based on a realistic web environment, which incorporates extremely long and complex web page observation. For each task, defined as a natural language instruction, the agent receives binary rewards based on whether the instruction is achieved by web navigation. Additionally, Liu et al. (2024) proposed 165 evaluation tasks filtered from the original 812 tasks in WebArena, for efficient evaluation. For the experiment, we evaluate the success rate on these 165 tasks while utilizing the remaining 647 tasks in the original WebArena benchmark as training tasks.
+
+<table><tr><td rowspan="2"></td><td colspan="2">GPT-4o</td><td colspan="2">Gemini-1.5-flash</td><td colspan="2">Claude-3.5-Sonnet</td><td colspan="2">Llama-3.1-70B (Unseen)</td></tr><tr><td>Success</td><td>Reward</td><td>Success</td><td>Reward</td><td>Success</td><td>Reward</td><td>Success</td><td>Reward</td></tr><tr><td>Raw observation</td><td>34.8%</td><td>0.496</td><td>43.6%</td><td>0.693</td><td>26.6%</td><td>0.336</td><td>34.2%</td><td>0.590</td></tr><tr><td>Self-ctx</td><td>26.2%</td><td>0.459</td><td>46.4%</td><td>0.608</td><td>12.4%</td><td>0.146</td><td>40.2%</td><td>0.547</td></tr><tr><td>LCoW (iter 1)</td><td>27.8%</td><td>0.545</td><td>46.4%</td><td>0.705</td><td>39.4%</td><td>0.600</td><td>39.2%</td><td>0.666</td></tr><tr><td>LCoW (iter 2)</td><td>46.0%</td><td>0.647</td><td>58.2%</td><td>0.796</td><td>58.8%</td><td>0.780</td><td>55.0%</td><td>0.781</td></tr><tr><td>LCoW (iter 3)</td><td>50.6%</td><td>0.666</td><td>62.8%</td><td>0.803</td><td>59.8%</td><td>0.771</td><td>59.6%</td><td>0.803</td></tr></table>
+
+Table 1: We investigate the efficacy of LCoW across multiple LLM agents in WebShop. For all LLM agents, LCoW consistently improves both success rate and reward over iterations, surpassing self-contextualization (self-ctx) and even human expert-level success rate by the third iteration. Additionally, LCoW is also effective when combined with Llama-3.1-70B, which was not used for computing the action-matching reward (i.e., unseen) during training the contextualization module.  
+
+<table><tr><td></td><td>GPT-4o</td><td>Gemini-1.5-flash</td><td>Claude-3.5-Sonnet</td><td>Llama-3.1-70B (Unseen)</td><td>Llama-3.1-8B (Unseen)</td></tr><tr><td>Raw observation</td><td>38.2%</td><td>11.5%</td><td>44.8%</td><td>26.1%</td><td>1.2%</td></tr><tr><td>Self-ctx</td><td>43.0%</td><td>12.7%</td><td>50.3%</td><td>29.1%</td><td>7.3%</td></tr><tr><td>LCoW (iter 1)</td><td>44.2%</td><td>41.2%</td><td>55.8%</td><td>40.0%</td><td>37.0%</td></tr></table>
+
+Table 2: We evaluate the success rate of five LLM agents with varying scales on 165 tasks in the WorkArena benchmark. Single iteration of LCoW shows improvement of success rate over all LLMs, even generalized to Llama-3.1-70B and Llama-3.1-8B, which were not used for computing the action matching reward.
+
+# 4.2 MAIN RESULTS
+
+We first demonstrate the effectiveness of LCoW on WebShop and WorkArena benchmarks and show that the contextualization module trained via LCoW enhances performance even for an LLM agent that was not involved in the LCoW training process, such as Llama-3.1-70B. We evaluate against two baselines: (i) decision making based solely on raw observations without the contextualization module (i.e., Raw observation), and (ii) decision making based on the observation contextualized by the LLM agent itself (i.e., self-contextualization).
+
+Effectiveness of LCoW As shown in Table 1, in WebShop benchmark, both the success rates and the average rewards achieved by all three LLM agents improve substantially when integrated with the contextualization module trained with LCoW. Particularly, both the Gemini-1.5-flash and Claude-3.5-Sonnet agents surpass the average human performance of  $50.5\%$  when combined with the contextualization module trained for 2 iterations. When the contextualization module is trained for 3 iterations, the agents exceed the expert human-level performance of  $59.6\%$ . As illustrated in Figure 5, the Gemini-1.5-flash agent combined with LCoW achieves state-of-art performance on the WebShop benchmark, outperforming prior methods in success rate by more than  $12\%$ . Notably, the Claude-3.5-Sonnet agent, lower performing than the other agents on WebShop, achieves  $59.8\%$  of success rate when integrated with our contextualization module, demonstrating that LCoW can effectively enhance the decision-making capabilities of LLM agents.
+
+As shown in Table 2, on WorkArena, the Claude-3.5-Sonnet agent outperforms the GPT-4o agent, achieving a success rate of  $44.8\%$  in our evaluation setup. When integrated with LCoW, Claude-3.5-Sonnet agent achieves an even higher success rate of  $55.8\%$ , which is higher than all baselines evaluated. GPT-4o and Gemini-1.5-flash also improve  $5\%$  and  $30\%$ , respectively, when combined with LCoW.
+
+Generalization to arbitrary LLM agents We evaluate the Llama-3.1-8B agent and Llama-3.1-70B agent integrated with LCoW to assess its effectiveness with LLM agents not involved in the training process (i.e., those not used for computing action-matching rewards). As shown in Table 1, the Llama-3.1-70B agent combined with the contextualization module trained for two iterations outperforms average human performance  $(50.0\%)$  on WebShop and achieves the success rate of human experts  $(59.6\%)$  when trained for three iterations. Furthermore, Table 2 shows that the contextualization module also enhances Llama-3.1-8B and Llama-3.1-70B agent on WorkArena, improving
+
+# Contextualized observation:
+
+The AXTree observation shows we are on the Shipping Report page. The page contains a filter section with date range inputs, order status options, and other report parameters. There's also an export function and a table to display the report results.
+
+···
+
+Following elements are crucial for completing the user instruction:
+
+```txt
+[753] LabelText StaticText ' StaticText'
+```
+
+[756] textbox 'From * value='08/05/2022'  
+[757] button 'undefined\ue627'
+
+[760] LabelText 'StaticText'To StaticText
+
+[763] textbox 'To *' value='03/01/2023', focused [764] button 'undefined\se627'
+
+This section contains the filter parameters for the shipping report. The date range inputs (From and To) have already been filled with the specified dates (08/05/2022 and 03/01/2023, respectively).
+
+# Contextualized observation:
+
+The AXTree observation shows a list of forums on a website called Postmill. Each forum is presented as an article with its name, number of subscribers, and number of submissions. The forums are sorted by the number of submissions in descending order.
+
+··
+
+Additionally, for context, we can include the first forum in the list to show where we are currently: [141] article ''
+
+[143] heading 'AskReddit - AskReddit' [144] link 'AskReddit - AskReddit'
+
+[149] paragraph ' ' [152] Section '
+
+[154] button 'Subscribe No subscribers' StaticText '0'
+
+[160] paragraph '' StaticText '10,041 submissions'
+
+This extraction shows the first forum in the list, which is "AskReddit" with 10,041 submissions. It confirms that we are currently on the first page of forums and need to navigate to the next page to continue our search for the UpliftingNews subreddit.
+
+Figure 6: Examples of how the contextualization module refines complicated web pages into comprehensible format from WebArena benchmark (Left) and WorkArena benchmark (Right). As indicated by blue color, the contextualization module provides comprehensible context by verbally explaining the web page and UI elements relevant to the given task.
+
+the success rate by approximately  $36\%$  and  $13\%$ , respectively. It is noteworthy that a relatively small LLM agent (i.e., Llama 3.1-8B) struggles to perform tasks when given raw observations, but when combined with LCoW, its success rate rises on significant margin. It demonstrates that LCoW can elicit a significant level of decision-making capability even from smaller models at the 8B scale.
+
+# 4.3 ANALYSIS
+
+How web pages are contextualized? We qualitatively analyze how the contextualization module, trained via LCoW, processes real-world web page observations and how the contextualization enhance decision making of LLM agents on web tasks. As shown in Figure 6, we observe that the contextualization module not only simplifies the web page by extracting UI elements relevant to the given task, but also verbally explains functionalities of the UI elements in web pages. These explanations aid LLM agents significantly in making more accurate decisions, while reducing the selection of inadmissible or redundant actions, which is the main failure mode of the LLM agent for web browsing. This knowledge of the functionality and interaction of UI elements grounded to the real website appears to have been learned during the LCoW training process, where the contextualization module explores various observation contextualizations and learns to generate contextualized observations that lead to more accurate decision making by LLM agents. More detailed examples and analyses are provided in the Appendix A.
+
+Comparison to directly training LLM agents One might argue that, given available demonstrations, it would be more straightforward to train the LLM agent directly rather than using the demonstrations to train a contextualization module. To demonstrate the superiority of training the contextualization module with an equivalent amount of demonstration data, we conduct comparative experiments against directly training LLM agent. In this analysis, we fine-tune Llama-3.1-8B with 264 seed demonstrations, the same demonstrations we used for training the contextualization module, using behavior cloning (BC) as a baseline. To ensure a fair comparison in terms of model scale, we define a Llama-3.1-8B agent that performs tasks based on the output of a contextualization module trained using LCoW as the direct comparison group. As shown in Figure 7, both smaller LLM agents, such as Llama-3.1-8B, and larger LLM agents achieve higher success rates when using the LCoW-trained contextualization module compared to the BC baseline.
+
+Generalization to tasks with unseen type Furthermore, we evaluate LCoW's generalization capabilities in WebArena benchmark. Specifically, we assess whether the contextualization module trained on specific task types generalizes to tasks belonging to task types unseen during the train
+
+![](images/ea2244a16ceac00a4135affbbe536ffa0f00202fd6b75763e48c63292b80acad.jpg)  
+Figure 7: Comparison to training LLM agent via behavior cloning (BC) in WorkArena. We fine-tune Llama-3.1-8B with demonstrations as a BC baseline.
+
+![](images/040cd2e88d0303f200b160598475f011abc1eda66f117e160eec17c6fa83d424.jpg)  
+Figure 8: Comparison of GPT-4o agent navigating web pages based on raw observation and observation contextualized by LCoW in WebArena.
+
+ing. Details on the definition of unseen task types are provided in the Appendix B.2. Among 165 evaluation tasks in WebArena, 117 tasks belong to the seen task types, and the remaining 48 tasks belong to the unseen task types. As shown in Figure 8, LCoW consistently shows  $6\%$  improvement in the success rate on both seen task types and unseen task types, compared to the GPT-4o agent without LCoW. We expect that generalization to unseen task types occurs due to the fact that the contextualization module learns to explain the UI elements in the web pages (e.g., the role of UI elements in complex web page observation) via LCoW-training, which enhances accurate decision making of LLM agents even in the unseen-type tasks. We also provide additional studies on the generalization ability of LCoW with the WorkArena benchmark in Appendix B.3.
+
+Can LCoW optimize the contextualization module for specific LLM agents? We demonstrate that LCoW effectively optimizes the contextualization module to generate context that leads to more accurate decision-making of LLM agents. In the WebShop benchmark, we used the contextualization module checkpoint from each LCoW training iteration to generate contextualized observations for the 1,372 raw observations present in the 397 seed demonstrations. We then calculated the action-matching reward by comparing the actions predicted from these contextualized observations with the ground-truth actions derived from the demonstrations. Figure 9 shows the average action-matching reward across three iterations, demonstrating that with each round of LCoW-training, the contextualization module increasingly produce contextualized observations that enhance decision making of the LLM agents used in data collection (i.e., GPT-4o, Gemini-1.5-flash, Claude-3.5-Sonnet). This shows the effectiveness of LCoW in optimizing the contextualization module for specific LLM agents. Furthermore, it suggests the potential to extend LCoW
+
+as a method for indirectly optimizing the behavior of closed-source LLM agents, where direct optimization is not feasible, across a wide range of text-based decision-making tasks.
+
+![](images/39f5270b5c64fc3f3add94d7cc3a1e082735bf6121b51720e326a4cda5df4fdf.jpg)  
+Figure 9: Average action-matching rewards across three iterations on WebShop show a consistent increase, suggesting that the contextualization module is optimized to generate observations that enhance decision-making in LLM agents.
+
+# 5 RELATED WORK
+
+LLM agents for web automation As LLMs continue to improve and demonstrate remarkable performance across diverse domains, developing web automation agents based on LLMs is gaining growing interest (Zhou et al., 2023a; Drouin et al., 2024). Recent works have explored various methods to utilize LLMs as agents for automating real-world web tasks. Pan et al. (2024) propose to apply a self-refine mechanism (Madaan et al., 2023) to improve decision making of agents through
+
+self-generated feedback. Sodhi et al. (2024) enhance web automation by using LLMs to manage low-level workflows handcrafted by humans, while Wang et al. (2024) extend this approach by introducing agent workflow memory, which extracts reusable routines from past experiences.
+
+Similar to our work, a promising direction for enhancing web automation involves integrating a summarization module that condenses web page observations, enabling agents to predict actions based on these summarized inputs. For example, Deng et al. (2024) propose MindAct, which consists of an HTML extraction module and an action prediction module. The HTML extraction module ranks individual HTML elements using a ranking language model trained to assess relevance based on user instructions and past actions. Additionally, Gur et al. (2024) introduce HTML-T5, a specialized language model for web page understanding pre-trained on a large corpus of HTML documents and fine-tuned for extractive summarization.
+
+Automated prompting for closed-source models The quality of outputs from closed-source LLMs heavily depends on the prompts used, leading to a substantial body of research dedicated to prompt engineering to elicit more effective responses from these models. (Kojima et al., 2022; Yao et al., 2022b; Lightman et al., 2024). For example, several recent studies have explored automating the discovery of more effective prompting formats (Shin et al., 2020; Zhou et al., 2023b). Manas et al. (2024) demonstrate that prompt refinement can yield more consistent responses from closed-source models, especially with text-to-image models. Also, Xu et al. (2024) have suggested compressing the input content for LLMs, focusing on the task of retrieval-augmented generation.
+
+# 6 LIMITATIONS AND FUTURE DIRECTIONS
+
+Dataset collection procedure We leveraged successful trajectories during training when using LCoW in our experiment. We acknowledge that this can be a bottleneck in learning completely new tasks that are not covered by the collected successful trajectories. To overcome this, incorporating an LLM-based search algorithm in the trajectory collection phase to gather successful trajectories for more challenging tasks can be an interesting future direction.
+
+Contextualization cost LCoW incurs latency in the decision-making process due to the computational cost required to generate the contextualized observation. Considering that most tokens of contextualized observation correspond to extraction from the raw observation, we expect LCoW can be integrated with an efficient decoding strategy (e.g., speculative decoding) for resolving the latency problem.
+
+Generalization ability Although we find that LCoW demonstrates generalization in several cases (e.g., unseen task types or unseen website), we acknowledge that we did not observe generalization to a broader range of tasks incorporating various unseen UI elements (see Appendix B.3 for more details), due to limited scale of experiments. We believe that scaling the dataset for the training contextualization module can be a promising direction for better generalization. For example, we expect leveraging large-scale web browsing demonstrations covering a wide range of websites as seed demonstrations would improve generalization to a broader range of tasks and websites.
+
+# 7 CONCLUSION
+
+In this work, we introduce LCoW, a novel approach to enhancing LLM agents in completing web tasks by leveraging language models to contextualize complex web pages into a more comprehensible form. Our approach separates the understanding of web content from the decision-making process by training a specialized module that generates contextualized representations of complex web pages, which are utilized by LLM agents for enhanced decision making. Through extensive experiments, we demonstrate that this contextualization module significantly improves the decision-making capabilities of LLM agents of varying scales.
+
+# ETHICS STATEMENT
+
+We introduce LCoW, a framework for improving decision-making capability of LLM agents for web automation. We caution that LLM agents may cause safety issues such as cybersecurity or risks regarding private information, while we believe that LCoW can be valuable for guiding the agents from potential mistakes by providing more contextualized information on the UI elements. Additionally, we believe the improved efficiency and capability of LLM agents with our methods can provide social opportunities to improve user interactions by using digital devices for those with disabilities.
+
+# REPRODUCIBILITY STATEMENT
+
+For the reproducibility of our results, we have provided a detailed description of our experimental setups and prompts in Section 4.1, Appendix B, and Appendix C. Across the entire experiment, we set the temperature hyperparameter of backbone LLM as 0.0 to enable reproduction. Additionally, to further facilitate reproduction, we open-source our work.
+
+# ACKNOWLEDGMENTS
+
+This research was partly supported by the MSIT(Ministry of Science, ICT), Korea, under the Global Research Support Program in the Digital Field program)(RS-2024-00436680,  $70\%$  ) supervised by the IITP(Institute for Information & Communications Technology Planning & Evaluation). This work was partly supported by Institute for Information & communications Technology Planning & Evaluation(IITP) grant funded by the Korea government(MSIT) (RS-2019-II190075, Artificial Intelligence Graduate School Program(KAIST),  $10\%$ ). This work was partly supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. RS-2024-00414822,  $10\%$ ). This work was partly supported by the Institute of Information & Communications Technology Planning & Evaluation(IITP)-ITRC(Information Technology Research Center) grant funded by the Korea government(MSIT)(IITP-2025-RS-2024-00436857,  $10\%$ )
+
+# REFERENCES
+
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+
+# A QUALITATIVE ANALYSIS
+
+In this section, we first analyze (1) examples of contextualized real-world web pages and (2) the decision-making process of LLM agents with and without LCoW.
+
+# A.1 EXAMPLES OF CONTEXTUALIZED WEB PAGES
+
+We provide 5 examples of contextualized web page observations, along with task instructions. As demonstrated in the following examples, the contextualization model trained via LCoW not only summarizes lengthy web pages but also provides contexts (e.g., what happens if a certain UI element is clicked) that enable more accurate decision-making of LLM-based web agents.
+
+- Example 1. The contextualized observation provides the necessary information required for sorting the list, including several column headers of the spreadsheet. Especially, it explains the "Personalize List" UI element, which initiates sorting. In many cases, LLM agents without contextualization do not click the "Personalize List" button due to a lack of knowledge about this UI element.  
+- Example 2. This example is drawn from the unseen-type tasks in WebArena, where the GPT-4o agent with LCoW succeeded, while the GPT-4o agent with raw observation failed. In this example, the contextualization module correctly extracts the rows corresponding to the latest order from the raw observation, thereby informing the GPT-4o agent of the status of the latest order. However, without LCoW, due to complex raw observation corresponding to this contextualized version, GPT-4o agent fails to identify the latest order. Although this task belongs to an unseen-type task, as table format frequently appears in web page observations in the training tasks, the contextualization module correctly extracts content from complex web pages, thereby improving the decision-making of LLM agents.  
+- Example 3. This example is also drawn from WebArena, where GPT-4o agent with LCoW succeeded, while GPT-4o with raw observation failed. In this example, the contextualization module explains that the agent need to click "Profile" link in order to edit posts or submission, thereby leading to correct decision making of LLM agent. However, given raw observation, the GPT-4o agent consecutively scrolls down the page to find the corresponding post.  
+- Example 4. In this example, the contextualized observation contains an explanation about the UI for keyword search, thereby guiding the GPT-4o agent to find a white desk based on search functionality. However, the GPT-4o agent without LCoW repeatedly scrolls down the page and moves on to the next page to find the white desk, thereby failing to add a white desk to the wish list within the maximum step limit.  
+- Example 5. In the task of filling out a form to create a new user, it is crucial to check for detailed requirements and entries before submitting the form. LCoW- contextualized web page includes, thereby guiding LLM agents to submit the complete form without mistakes (e.g., missing required entries).
+
+# Example 1 (WorkArena)
+
+Instruction: Sort the "users" list by the following fields:- Updates (descending)
+
+# Contextualized web page observation:
+
+```ini
+[a183] grid 'Users.', multiselectable=False  
+This is the main table containing the list of users.  
+We need to focus on this element to find ways to interact with the column header.  
+[a186] rowgroup ''  
+[a187] row ''  
+[a188] columnheader 'Select All'  
+[a193] columnheader '\ufle4 Show column search row'  
+[a197] columnheader 'UserID User ID column options'  
+[a204] columnheader 'Name Name column options'  
+[a211] columnheader 'Email Email column options'  
+[a218] columnheader 'Avatar Avatar column options'  
+[a225] columnheader 'Title Title column options'
+```
+
+```txt
+[a232] columnheader 'Company Company column options'  
+[a239] columnheader 'Department Department column options'  
+[a246] columnheader 'Location Location column options'  
+[a253] columnheader 'Time zone Time zone column options'  
+[a260] columnheader 'Business phone Business phone column options'  
+These are the column headers in the users table.  
+We need to focus on one of these elements to trigger the sorting action.  
+However, the AXTree does not show any direct options to sort these columns.  
+[a77] button '\uf13e Personalize List'  
+This button might provide options to customize the list view, potentially including sorting options.
+```
+
+# Example 2 (WebArena)
+
+Instruction: Tell me the status of my latest order and when will it arrive.
+
+# Contextualized web page observation:
+
+```txt
+[1415] strong''
+StaticText 'Recent Orders'
+[1416] link 'View All'
+[1420] table 'Recent Orders'
+[1421] caption''
+[1422] rowgroup''
+[1423] row''
+[1424] columnheader 'Order #
+[1425] columnheader 'Date'
+[1426] columnheader 'Ship To'
+[1427] columnheader 'Order Total'
+[1428] columnheader 'Status'
+[1429] columnheader 'Action'
+[1430] rowgroup''
+[1431] row''
+[1432] gridcell '000000170'
+[1433] gridcell '5/17/23'
+[1434] gridcell 'Emma Lopez'
+[1435] gridcell '$365.42'
+[1437] gridcell 'Canceled'
+[1438] gridcell 'View OrderReorder'
+[1439] link 'View Order'
+[1441] link 'Reorder'
+```
+
+```txt
+This extraction shows the "Recent Orders" section, which contains a table with information about recent orders. The table includes columns for Order #, Date, Ship To, Order Total, Status, and Action. The "View All" link [1416] likely leads to a page with more detailed information about all orders.
+```
+
+```yaml
+The first row in the table [1431] contains information about the latest order:
+- Order number: 000000170
+- Date: 5/17/23
+- Ship To: Emma Lopez
+- Order Total: $365.42
+- Status: Canceled
+- Action: View Order [1439] or Reorder [1441]
+```
+
+# Example 3 (WebArena)
+
+Instruction: Edit my post on Star Trek Starfleet Academy series by adding a line to the body that says "Every watch makes me feel like a kid again.
+
+# Contextualized web page observation:
+
+```txt
+[68] button 'MarcelsGrantMan136', focused, hasPopup='menu', expanded=True  
+This is the currently focused and expanded user account menu button.  
+It contains the user's username and is likely to lead to profile settings.
+```
+
+```txt
+[74] list 'This list contains the submenu items under the user account menu.
+```
+
+```txt
+[80] listing
+[81] link 'Profile'
+This is the "Profile" link we need to click to access the user's profile settings.
+It's nested under the user account menu.
+```
+
+These elements are crucial for navigating to the user's profile settings, where we can then look for options to edit posts or submissions. The "Profile" link [81] is particularly important as it's the next element we need to interact with to progress towards our goal of editing the post.
+
+# Example 4 (WebArena)
+
+Instruction: Add a white desk to my wish list.
+
+# Contextualized web page observation:
+
+```powershell
+[1850] heading 'My Wish List'  
+StaticText 'You have no items in your wish list.'
+```
+
+```txt
+This element shows the "My Wish List" heading and indicates that the wish list is currently empty. It's important to note this because it confirms that we haven't added any items to the wish list yet, including the white desk we want to add.
+```
+
+```txt
+[1430] button 'Add to Wish List' StaticText '\ue601'
+```
+
+```txt
+This button is crucial as it directly allows us to add items to the wish list. The button with ID 1430 is likely the one we need to click to add the white desk to the wish list.
+```
+
+```javascript
+[723] menuitem '\ue622 Office Products', hasPopup='menu'
+```
+
+```txt
+This menuitem is relevant because it's the currently selected category (Office Products). It confirms that we are in the correct category to find office-related items, including desks.
+```
+
+```txt
+[269] Section
+[271] LabelText
+StaticText '\ue615'
+StaticText 'Search'
+```
+
+[274] combobox '\ue615 Search',  
+autocomplete='both', hasPopup='listbox', expanded=False  
+[277] link 'Advanced Search'  
+[279] button 'Search', disabled=True
+
+This section contains the search functionality, which could be useful if we need to search for "white desk" to find the specific item we want to add to the wish list.
+
+These elements provide the necessary information and functionality to complete the user's instruction of adding a white desk to the wish list.
+
+# Example 5 (WorkArena)
+
+Instruction: Create a new user with a value of "8273378363827782819" for field "UserID", a value of "Webster-Richard" for field "Last name", a value of "Lynn-David" for field "First name", a value of "lynn-david.webster-richard@workarena.com" for field "Email", a value of "false" for field "Internal Integration User", a value of "Outlook" for field "Calendar integration", a value of "(514) 357-8997" for field "Mobile phone", a value of "false" for field "Password needs reset", a value of "English" for field "Language", and a value of "false" for field "Web service access only".
+
+# Contextualized web page observation:
+
+we should focus on the following elements:
+
+```ini
+[a298] LabelText  
+[a299] note  
+StaticText 'Internal Integration User'  
+[a305] checkbox 'Internal Integration User', checked='false'
+```
+
+This element represents the "Internal Integration User" field. It's a checkbox that is currently unchecked (false), which aligns with the user instruction. No action is needed for this field.
+
+```ini
+[a353] LabelText''
+[a354] note''
+StaticText 'Calendar integration'
+[a359] combobox 'Calendar integration' value='Outlook',
+hasPopup='menu', expanded=False
+[a360] menuitem -- None --', selected=False
+[a361] menuitem 'Outlook', selected=True
+```
+
+This element represents the "Calendar integration" field. It's a combobox that is already set to "Outlook", which aligns with the user instruction. No action is needed for this field.
+
+```ini
+[a429] LabelText
+[a430] note
+StaticText 'Mobile phone'
+[a435] textbox 'Mobile phone'
+```
+
+This element represents the "Mobile phone" field.  
+It's a textbox where we need to enter "(514) 357-8997" as per the user instruction.
+
+```ini
+[a327] LabelText''
+[a328] note''
+StaticText 'Language'
+[a333] combobox 'Language' value='-- None --',
+hasPopup='menu', expanded=False
+[a334] menuitem '-- None --', selected=True
+```
+
+```txt
+[a335]餐具 'English', selected=False  
+This element represents the "Language" field.  
+It's a combobox where we need to select "English" as per the user instruction.  
+[a285] LabelText ''  
+[a286] note ''  
+StaticText 'Web service access only'  
+[a292] checkbox 'Web service access only', checked='false'
+```
+
+# A.2 ANALYSIS ON DECISION MAKING PROCESS
+
+We provide the action sequences obtained from rollouts in evaluation tasks. For a given instruction, we compare the rollouts with and without the application of LCoW, highlighting the differences in decision-making behavior and task completion efficiency. Specifically, LCoW helps the LLM agent to accomplish task more efficiently without repeating redundant actions (see Figure 10 and Figure 12), and also helps to avoid getting stuck by informing proper interaction method regarding UI elements (see Figure 11 and Figure 13). Furthermore, as shown in Figure 14, with the contextualization module, the average number of decision-making steps by the agent decreases in both Claude-3.5-Sonnet and Llama-3.1-70B agent, allowing tasks to be completed more efficiently.
+
+Instruction: Go to the hardware store and order 9 "iPad pro" with configuration {'Choose the colour': 'Silver', 'Choose the storage': '128'}
+
+# Llama-3.1-70B
+
+Action 1: click [a252] link 'Hardware'
+
+Action 2: click [a268] link 'iPad pro'
+
+Action 3: select 'Silver' from [a198] radiogroup 'Mandatory'
+
+Action 4: click [a184] radio 'uf137 Silver'
+
+Action 5: hover the cursor on [a184] radio 'uf137 Silver'
+
+Action 6: click [a185] LabelText "
+
+Action 7: select '9' from [a244] combobox 'Quantity'
+
+Action 8: click [a285] button 'Order Now'
+
+# Llama-3.1-70B + LCoW
+
+Action 1: click [a252] link 'Hardware'
+
+Action 2: click [a268] link 'iPad pro'
+
+Action 3: click [a184] radio 'uf137 Silver'
+
+Action 4: click [a185] LabelText
+
+Action 5: select '9' from [a244] combobox 'Quantity'
+
+Action 6: click [a285] button 'Order Now'
+
+Figure 10: Llama-3.1-70B agent repeats redundant actions while manipulating UI element for selecting silver color, while LCoW helps to decide on actions more efficiently.
+
+Instruction: Add a white desk to my wish list.
+
+# GPT-40
+
+Action 1: click [716] menuitem 'ue622 Office Products'
+
+Action 2: click [1809] link 'Office Furniture&Lighting(372 item)'
+
+Action 3: click [1789] link 'ue608 Page Next'
+
+Action 4: click [1815] link \`ue608 Page Next'
+
+Action 5: scroll_down
+
+Action 6: scroll_down
+
+Action 7: scroll_down
+
+# GPT-4o + LCoW
+
+Action 1: click [716] menuitem 'ue622 Office Products'
+
+Action 2: fill 'white desk' in [274] combobox 'ue615 Search'
+
+Action 3: click [279] button 'Search'
+
+Action 4: click [1434] button 'Add to Wish List'
+
+Figure 11: Given complex web page observation, GPT-4o agent repeats moving to the next page and scrolling down the page for finding out white desk, resulting in task failure. In contrast, LCoW enables LLMs to utilize search functionality for finding out the white desk efficiently.
+
+Instruction: I recently moved, my address is 111 Magnolia Path, Atlanta, GA, 30303, update my information on OneStopShopping accordingly.
+
+# GPT-40
+
+Action 1: click [181] link 'My Account'
+
+Action 2: click [1398] link 'Edit Address'
+
+Action 3: clear contents in [1400] textbox 'Street Address* Street Address: Line 1'
+
+Action 4: fill '111 Magnolia Path' in [1400] textbox 'Street Address* Street Address: Line 1'
+
+Action 5: clear contents in [1737] textbox 'City *'
+
+Action 6: fill 'Atlanta' in [1737] textbox 'City *'
+
+Action 7: select 'Georgia' from [1665] combobox 'State/Province *'
+
+Action 8: clear contents in [1742] textbox 'Zip/Postal Code *
+
+Action 9: fill '30303' in [1742] textbox 'Zip/Postal Code *
+
+Action 10: click [1752] button 'Save Address'
+
+# GPT-4o + LCoW
+
+Action 1: click [219] link 'My Account'
+
+Action 2: click [1384] link 'Manage Addresses'
+
+Action 3: click [1374] link 'Change Billing Address'
+
+Action 4: fill '111 Magnolia Path' in [1400] textbox 'Street Address'  
+Street Address: Line 1
+
+Action 5: fill 'Atlanta' in [1737] textbox 'City *'
+
+Action 6: select 'Georgia' from [1665] combobox 'State/Province **
+
+Action 7: fill '30303' in [1742] textbox 'Zip/Postal Code'
+
+Action 8: click [1752] button 'Save Address'
+
+Figure 12: GPT-4o agent meaninglessly clear the empty textbox before filling out the address form. In contrast, LCoW explains the status of UI element (e.g., textbox is empty), thereby enabling more efficient decision making.
+
+Instruction: Post a notice on a virtual/meup for Harry Potter enthusiasts on July 8th in the books subreddit.
+
+# GPT-40
+
+Action 1: click [63] link 'Submit'
+
+Action 2: fill 'Virtual Meetup for Harry Potter Enthusiasts on July 8th' in [137] textbox 'Title This field is required.'
+
+Action 3: fill 'Join us for a virtual meetup for Harry Potter enthusiasts on July 8th! We will discuss our favorite books in [142] textbox 'Body'
+
+Action 4: click [378] combobox 'Choose one...'
+
+Action 5: select 'books' from [378] combobox 'Choose one...
+
+Action 6: press 'ArrowDown' keys while focusing on [378] combobox 'Choose one...'
+
+Action 7: press 'ArrowDown' keys while focusing on [378] combobox 'Choose one...'
+
+Action 8: press 'ArrowDown' keys while focusing on [378] combobox 'Choose one...'
+
+# GPT-4o + LCoW
+
+Action 1: click [62] link 'Submit'
+
+Action 2: fill 'Join Our Harry Potter Virtual Meetup on July 8th! in [137] textbox 'Title This field is required.'
+
+Action 3: fill 'Join us for a magical virtual meetup where Harry Potter enthusiasts from around the world come together to discuss their favorite books in [142] textbox 'Body'
+
+Action 4: fill 'books' in [379] textbox 'Choose one...'
+
+Action 5: click [378] combobox 'Choose one...'
+
+Action 6: click [453] option 'books'
+
+Action 7: click [395] button 'Create submission'
+
+Figure 13: GPT-4o agent struggles to interact with UI element for selecting subreddit, resulting in task failure. In contrast, LCoW verbally explains the interaction method of each UI element in the web page, enabling task success.
+
+![](images/c9de8e1a2ccb241c0e227245a4281c61c7f50d1587cf03cb49b21e4f59635f78.jpg)
+
+Figure 14: In order to compare the efficiency of task execution between LLM agent with LCoW and without LCoW, we select the tasks that LLM agent with LCoW and without LCoW both succeed, and visualize the distribution of action steps required for accomplishing the task.
+
+# B EXPERIMENTAL DETAILS & ADDITIONAL EXPERIMENTS
+
+# B.1 MAIN EXPERIMENT SETUP
+
+WebShop For the WebShop benchmark, we fine-tune Phi-3-mini-Instruct as a contextualization module, setting the learning rate, warmup ratio, and batch size to 1e-5, 1e-2, and 32, respectively. The module is trained for a single epoch over the collected data for each LCoW iteration, and we utilize demonstrations corresponding to 397 individual tasks provided in the WebShop benchmark as seed demonstrations. In the WebShop environment, we employ Gemini-1.5-flash as the LLM agent, combined with the contextualization module during the trajectory collection phase. It is also utilized for sampling optimal contextualization at the initial iteration of LCoW.
+
+WorkArena & WebArena For WorkArena and WebArena, we fine-tune Llama-3.1-8B-Instrcut as an observation contextualization module, and we set the learning rate, warm-up ratio, and batch size as 1e-5, 1e-1, and 128. Additionally, we set the training epochs as 4 and 3 for WorkArena and WebArena, respectively. Since no successful trajectories are provided in WorkArena, we collected 264 seed demonstrations (i.e., successful trajectories) among 495 training tasks across 33 task types using Claude-3.5-Sonnet and GPT-4o, but we could not collect any successful trajectories corresponding to 10 task types. Summary statistics of the collected seed demonstrations across task types can be found in Table 5. In WebArena, we collected 363 seed demonstrations (i.e., successful trajectories) among 647 training tasks (excluding 165 evaluation tasks in WebArena-Lite from 812 WebArena tasks) using GPT-4o and trajectories open-sourced by AgentOccam (Yang et al., 2024). Additionally, as the determination of action matching based on parsing is infeasible due to open-ended actions (e.g., sending a message to the user or filling out a form), we exploit GPT-4o as an action-matching evaluator. The prompt used for the action-matching evaluator is provided in Appendix C. For WorkArena and WebArena, we use Claude-3.5-Sonnet for sampling optimal contextualization at the initial iteration.
+
+# B.2 GENERALIZATION EXPERIMENT SETUP
+
+In this section, we first provide details of the generalization experiment in WebArena and WorkArena.
+
+WebArena In WebArena, we define unseen-type tasks based on the task template used to create 812 individual tasks in the WebArena benchmark. Specifically, 812 tasks in WebArena are all different but created based on 190 task templates. Therefore, tasks created from the same task template are different but similar. For example, "What is the top-1 best-selling product in 2022" and "What is the top-3 best-selling product in 2023" are from the same task template (i.e., same task type), while "Tell me the email address, name, phone number of the customer who has the most cancellations in the history" and "How many reviews our shop received in Apr 2023?" are from the different task template (i.e., different task type). Among the 165 evaluation tasks, we define the 117 tasks whose task templates overlap with tasks used for training as seen-type tasks, while the 48 tasks whose task templates do not overlap with any training tasks as unseen-type tasks.
+
+WorkArena WorkArena features a two-level task hierarchy: task categories at the top level, and task types within each category. WorkArena provides 7 categories of tasks (i.e., " Dashboard", "Menus", "Service catalog", "Knowledge base", "Forms", "Sort list", and "Filter list"). For instance, "Form" is a task category, and within it, "creating and submitting an incident report" and "creating new user information" are task types. We consider two levels of generalization: 1) Unseen-type tasks, tasks of a different type within the same category (i.e., medium-level generalization), and 2) Unseen-category tasks, tasks of a different type and category (i.e., hard-level generalization). As specified in Table 6, among 33 task types, we utilize 13 task types as seen task types, 14 task types as unseen task types, and the remaining 6 task types corresponding to "Filter List" task category as unseen-category tasks. For evaluation, we utilize 5 individual tasks from each task type.
+
+# B.3 ADDITIONAL GENERALIZATION EXPERIMENT RESULTS
+
+In this section, we provide the result of additional generalization experiments in the WorkArena and WebArena benchmarks.
+
+Generalization to tasks with unseen type in WorkArena We first evaluate generalization to unseen-type tasks in the WorkArena benchmark, where 70 tasks corresponding to 14 task types are considered unseen-type tasks. As shown in Table 3, LCoW improves GPT-4o and Gemini-1.5-flash agents by  $7.2\%$  and  $22.6\%$  success rate in 70 unseen-type tasks, respectively. This is an observation consistent with the WebArena experiment in Section 4.3.
+
+Generalization to tasks with unseen category in WorkArena Furthermore, we verify whether the contextualization module generalizes to tasks from entirely different categories (i.e., unseen-category tasks) in the WorkArena benchmark. This requires an understanding of unseen UI elements and webpage configurations, making it more challenging than generalization to unseen-type tasks. Specifically, we set the 30 tasks included in "Filter-List" task category as unseen-category tasks. As shown in Table 3, LCoW fails to improve both GPT-4o and Gemini-1.5-flash agents in unseen-category tasks. The main reason for this failure is the unseen-category tasks is that the contextualization module does not extract the necessary UI element required to dropdown the hidden menu when manipulating filters in Filter-List-related tasks. Since the training tasks do not involve any UI elements related to filter functionality, the contextualization module does not learn any knowledge about the UI element during training, resulting in no performance improvement in unseen-category tasks.
+
+<table><tr><td></td><td colspan="2">GPT-4o</td><td colspan="2">Gemini-1.5-flash</td></tr><tr><td></td><td>Unseen-type</td><td>Unseen-category</td><td>Unseen-type</td><td>Unseen-category</td></tr><tr><td>Raw observation</td><td>35.7%</td><td>0.0%</td><td>14.5%</td><td>0.0%</td></tr><tr><td>LCoW</td><td>42.9%</td><td>0.0%</td><td>37.1%</td><td>0.0%</td></tr></table>
+
+Generalization to tasks with unseen websites We additionally study the feasibility of LCoW for generalization to unseen websites in the WebArena benchmark, where tasks are distributed over 6 websites (Shopping, GitLab, Reddit, Map, Content Management System, and Wikipedia). We evaluate the contextualization module on tasks defined in the Shopping website, where the contextualization module is trained based on tasks in the remaining 5 websites. In this setting, we found that LCoW improves the performance of the GPT-4o agent by  $4.3\%$  even in the unseen website, as shown in Table 4. We conjecture that this is feasible because, even though the websites may differ in overall design and layout, common UI elements (e.g., search query input fields and calendar date selectors) are often shared across websites. The presence of shared UI components allows the knowledge gained from one website's task to be transferred effectively to another website, enabling the contextualization to generalize to tasks in unseen websites. We believe scaling LCoW for generalization to unseen websites where unseen UI elements are prevalent is a highly interesting future direction.
+
+Table 3: LCoW is shown to be generalized to unseen task types within the same task category on both GPT-4o and Gemini-1.5-flash backbone, but it struggles to generalize to tasks corresponding to unseen category.  
+
+<table><tr><td>(Unseen website)</td><td>GPT-4o</td></tr><tr><td>Raw observation</td><td>17.4%</td></tr><tr><td>LCoW</td><td>21.7%</td></tr></table>
+
+Table 4: LCoW improves the performance of GPT-4o agent on tasks corresponding to the unseen website. Although websites are different, there are UI elements commonly used across websites (e.g., UI elements for searching in Map or UI elements for filtering results based on data in the content management system), which enables LCoW to generalize across different websites.
+
+<table><tr><td>Task type</td><td>Number of seed demonstrations</td></tr><tr><td>Multi-chart-value retrieval</td><td>8</td></tr><tr><td>Multi-chart-minmax retrieval</td><td>12</td></tr><tr><td>Single-chart-value-retrieval</td><td>12</td></tr><tr><td>Single-chart-minmax retrieval</td><td>11</td></tr><tr><td>Create change request</td><td>8</td></tr><tr><td>Create incident</td><td>0</td></tr><tr><td>Create hardward asset</td><td>0</td></tr><tr><td>Create problem</td><td>14</td></tr><tr><td>Create user</td><td>11</td></tr><tr><td>Knowledge base search</td><td>12</td></tr><tr><td>Filter asset list</td><td>0</td></tr><tr><td>Filter change request list</td><td>0</td></tr><tr><td>Filter hardware list</td><td>0</td></tr><tr><td>Filter incident list</td><td>0</td></tr><tr><td>Filter service catalog item list</td><td>0</td></tr><tr><td>Filter user list</td><td>0</td></tr><tr><td>Sort asset list</td><td>0</td></tr><tr><td>Sort change request list</td><td>0</td></tr><tr><td>Sort hardware list</td><td>3</td></tr><tr><td>Sort incident list</td><td>3</td></tr><tr><td>Sort service-catalog list</td><td>5</td></tr><tr><td>Sort user list</td><td>4</td></tr><tr><td>All menu</td><td>15</td></tr><tr><td>Impersonation</td><td>12</td></tr><tr><td>Order developer laptop</td><td>15</td></tr><tr><td>Order iPad mini</td><td>14</td></tr><tr><td>Order iPad pro</td><td>15</td></tr><tr><td>Order sales laptop</td><td>15</td></tr><tr><td>Order standard laptop</td><td>15</td></tr><tr><td>Order apple watch</td><td>15</td></tr><tr><td>Order apple MacBook pro</td><td>15</td></tr><tr><td>Order development laptop PC</td><td>15</td></tr><tr><td>Order loander laptop</td><td>15</td></tr><tr><td>Total</td><td>264</td></tr></table>
+
+Table 5: We collected trajectories from 15 individual tasks for each of 33 task types in the WorkArena benchmark using GPT-4o-0806 and Claude-3.5-sonnet agent, thereby collecting 264 successful trajectories. We utilized them as seed demonstrations for LCoW.
+
+<table><tr><td>Task type</td><td>Seen / Unseen</td></tr><tr><td>Single-chart-value-retrieval</td><td>seen</td></tr><tr><td>Single-chart-minmax retrieval</td><td>seen</td></tr><tr><td>Multi-chart-value retrieval</td><td>unseen-type</td></tr><tr><td>Multi-chart-minmax retrieval</td><td>unseen-type</td></tr><tr><td>Create change request</td><td>seen</td></tr><tr><td>Create problem</td><td>seen</td></tr><tr><td>Create incident</td><td>unseen-type</td></tr><tr><td>Create hardward asset</td><td>unseen-type</td></tr><tr><td>Create user</td><td>unseen-type</td></tr><tr><td>Knowledge base search</td><td>seen</td></tr><tr><td>Sort user list</td><td>seen</td></tr><tr><td>Sort hardware list</td><td>seen</td></tr><tr><td>Sort asset list</td><td>unseen-type</td></tr><tr><td>Sort change request list</td><td>unseen-type</td></tr><tr><td>Sort incident list</td><td>unseen-type</td></tr><tr><td>Sort service-catalog list</td><td>unseen-type</td></tr><tr><td>All menu</td><td>seen</td></tr><tr><td>Impersonation</td><td>unseen-type</td></tr><tr><td>Order developer laptop</td><td>seen</td></tr><tr><td>Order iPad mini</td><td>seen</td></tr><tr><td>Order iPad pro</td><td>seen</td></tr><tr><td>Order apple watch</td><td>seen</td></tr><tr><td>Order standard laptop</td><td>seen</td></tr><tr><td>Order sales laptop</td><td>unseen-type</td></tr><tr><td>Order apple Macbook pro</td><td>unseen-type</td></tr><tr><td>Order development laptop PC</td><td>unseen-type</td></tr><tr><td>Order loander laptop</td><td>unseen-type</td></tr><tr><td>Filter asset list</td><td>unseen-category</td></tr><tr><td>Filter change request list</td><td>unseen-category</td></tr><tr><td>Filter hardware list</td><td>unseen-category</td></tr><tr><td>Filter incident list</td><td>unseen-category</td></tr><tr><td>Filter service catalog item list</td><td>unseen-category</td></tr><tr><td>Filter user list</td><td>unseen-category</td></tr></table>
+
+Table 6: Task type split for evaluation of generalization at different levels in WorkArena benchmark.
+
+# C PROMPTS
+
+In this section, we provide prompts used across experiments in WebShop, WorkArena, and WebArena. In WorkArena and WebArena experiments, we utilize identical prompts, where the prompt for the LLM agent is provided by BrowserGym (Drouin et al., 2024). In the WebShop experiment, we utilize the prompt for the LLM agent proposed by ReAct (Yao et al., 2022b).
+
+# C.1 WORKAREA & WEBAREA
+
+We present the prompt formats for the experiment in the WorkArena and WebArena benchmark.
+
+# Prompt for contextualization module
+
+```txt
+<system prompt>  
+You are an agent tasked with extracting and refining a subset of the webpage's observations based on the content of the page and user instructions.
+```
+
+```txt
+<main prompt>  
+You are currently on the {domain_info} website.  
+Your task is to generate a "Reasoning" and a "Refined observation" based on the provided inputs.
+```
+
+```txt
+First, review the "User instruction" and "History of interactions" and, then, generate the "Reasoning". Analyze the progress made so far, and provide a rationale for the next steps needed to efficiently accomplish the user instruction on the {domain_info} website.
+```
+
+```txt
+Second, refine the "AXTree observation at the current time step" into a "Refined observation".
+```
+
+```txt
+Select a subset of the AXTree observation that is essential for completing the user instruction and provide explanations for the corresponding elements in the selected subset.
+```
+
+```txt
+[Information source]   
+#User instruction   
+{goal}   
+#History of interactions   
+{history}   
+#AXTree observation at the current time step   
+{observation}
+```
+
+# Prompt for contextualization module in retry phase
+
+```txt
+<system prompt>  
+You are an agent tasked with extracting and refining a subset of the webpage's observations based on the content of the page and user instructions.
+```
+
+```txt
+<main prompt>  
+You are currently on the {domain_info} website.  
+Your task is to generate a "Reasoning" and a "Refined observation" based on the provided inputs.
+```
+
+```txt
+First, review the "User instruction" and "History of interactions" and, then, generate the "Reasoning". Analyze the progress made so far, and provide a rationale for the next steps needed to efficiently accomplish the user instruction on the {domain_info} website.
+```
+
+Second, refine the "AXTree observation at the current time step" into a "Refined observation".
+
+Select a subset of the AXTree observation that is necessary for completing the user instruction.
+
+You may refer to the Hints, which consists of the ground truth next action, but do not explicitly mention these hints in your output.
+
+[Information source]
+
+#User instruction
+
+{goal}
+
+History of interactions
+
+{history}
+
+AXTree observation at the current time step {observation}
+
+Hint
+
+Ground-truth next action: {action}
+
+# Prompt for self-contextualization
+
+<system prompt>
+
+You are an agent tasked with extracting and refining a subset of the webpage's observations based on the content of the page and user instructions.
+
+<main prompt>
+
+[General instructions]
+
+You are currently on the {domain_info} website.
+
+Your task is to generate a "Reasoning" and a "Refined observation" based on the provided inputs.
+
+First, review the "User instruction" and "History of interactions" and, then, generate the "Reasoning".
+
+Analyze the progress made so far, and provide a rationale for the next steps needed to efficiently accomplish the user instruction on the {domain_info} website.
+
+Second, refine the "AXTree observation at the current time step" into a "Refined observation".
+
+Extract a subset of the AXTree observation (e.g., chart, table, menu items) that contains necessary information for completing the user instruction, and explain the extracted elements.
+
+Ensure that the information on the elements (e.g., numeric element ID) are correctly included.
+
+Please follow the format in the [Reasoning & Refinement example] carefully.
+
+[Information source]
+
+User instruction {goal}
+
+History of interactions {history}
+
+AXTree observation at the current time step {observation}
+
+[Reasoning & Refinement example]
+
+Abstract example
+
+Here is an abstract version of the answer, describing the content of each tag.
+
+Make sure you follow this structure, but replace the content with your own answer:
+
+<reasoning>
+
+Think step by step. Based on the "User instruction,"
+
+"History of interaction," and "AXTree observation at the current time step":
+
+1. Provide a high-level description of the "AXTree observation at the current time step."
+
+2. Based on the "User instruction" and "History of interaction," track your progress and provide your reasoning on the next action needed to accomplish the "User instruction."
+
+</reasoning>
+
+<extraction>
+
+Based on your reasoning, identify the elements
+
+(e.g., links, buttons, static text, table row, chart) to focus on. Then, explain the semantics and functionalities of each extracted element.
+
+Ensure that:
+
+You do not alter the structure of the AXTree observation.
+
+You extract the element ID (id in [ ]) accurately without any errors.
+
+When extracting chart or table, you must extract the entire chart or table to avoid any confusion or loss of information.
+
+</extraction>
+
+# Prompt for LLM agent
+
+<system prompt>
+
+You are an agent trying to solve a web task based on the content of the page and a user instructions.
+
+You can interact with the page and explore.
+
+Each time you submit an action it will be sent to the browser and you will receive a new page.
+
+<main prompt>
+
+Instructions
+
+Review the current state of the page and all other information to find the best possible next action to accomplish your goal.
+
+Your answer will be interpreted and executed by a program, make sure to follow the formatting instructions.
+
+Goal:
+
+{goal}
+
+{history}
+
+Refined observation of current step:
+
+{refined observation}
+
+Action space:
+
+13 different types of actions are available.
+
+noop(wait_ms: float = 1000)
+
+Description: Do nothing, and optionally wait for the given time (in milliseconds).
+
+Examples:
+
+noop()
+
+noop(500)
+
+send msg_to_user(text: str)
+
+Description: Send a message to the user.
+
+You should send a short answer as a message and
+
+```txt
+do not ask questions through message. Examples: send msg_to_user(\'the city was built in 1751.\') send msg_to_user(\'Yes\') send msg_to_user(\'No\') send msg_to_user(\'31112\') send msg_to_user(\'Yoshua Bengio\')   
+scroll (delta_x: float, delta_y: float) Description: Scroll horizontally and vertically. Amounts in pixels, positive for right or down scrolling, negative for left or up scrolling. Dispatches a wheel event. Examples: scroll(0, 200) scroll(-50.2, -100.5)   
+fill (bid: str, value: str) Description: Fill out a form field. It focuses the element and triggers an input event with the entered text. It works for <input>, <textarea> and [contenteditable] elements. Examples: fill('237', 'example value') fill('45', 'multi-line\nexample') fill('a12', 'example with "quotes"')   
+select_option (bid: str, options: str | list [str]) Description: Select one or multiple options in a <select> element. You can specify option value or label to select. Multiple options can be selected. Examples: select_option('48', 'blue') select_option('48', ['red', 'green', 'blue'])   
+click (bid: str, button: Literal['left', 'middle', 'right'] = 'left', modifiers: list[typing.Literal['Alt', 'Control', 'Meta', 'Shift']) =[]) Description: Click an element. Examples: click('51') click('b22', button='right') click('48', button='middle', modifiers=['Shift'])   
+dblclick (bid: str, button: Literal['left', 'middle', 'right'] = 'left', modifiers: list[typing.Literal['Alt', 'Control', 'Meta', 'Shift']) = ]) Description: Double click an element. Examples: dblclick('12') dblclick('ca42', button='right') dblclick('178', button='middle', modifiers=['Shift'])   
+hover (bid: str) Description: Hover over an element. Examples: hover('b8')   
+press (bid: str, key_comb: str) Description: Focus the matching element and press a combination of keys. It accepts the logical key names that are emitted in the keyboardEvent. key property of the keyboard events: Backquote, Minus, Equal,
+```
+
+```txt
+Backslash, Backspace, Tab, Delete, Escape, ArrowDown, End, Enter, Home, Insert, PageDown, PageUp, ArrowRight, ArrowUp, F1 - F12, Digit0 - Digit9, KeyA - KeyZ, etc. You can alternatively specify a single character you'd like to produce such as "a" or "#". Following modification shortcuts are also supported: Shift, Control, Alt, Meta. Examples: press('88', 'Backspace') press('a26', 'Control+a') press('a61', 'Meta+Shift+t')   
+focus(bid: str) Description: Focus the matching element. Examples: focus('b455')   
+clear(bid: str) Description: Clear the input field. Examples: clear('996')   
+drag_and_drop(from_bid: str, to_bid: str) Description: Perform a drag & drop. Hover the element that will be dragged. Press left mouse button. Move mouse to the element that will receive the drop. Release left mouse button. Examples: drag_and_drop('56', '498')   
+upload_file(bid: str, file: str | list[str]) Description: Click an element and wait for a "filechooser" event, then select one or multiple input files for upload. Relative file paths are resolved relative to the current working directory. An empty list clears the selected files. Examples: upload_file('572', 'my_receipt.pdf') upload_file('63', ['/home/bob/Documents/image.jpg', '/home/bob/Documents/file.zip'])   
+Only a single action can be provided at once. Example: fill('a12', 'example with "quotes"') Multiple actions are meant to be executed sequentially without any feedback from the page. Don't execute multiple actions at once if you need feedback from the page.   
+# Abstract Example Here is an abstract version of the answer with description of the content of each tag. Make sure you follow this structure, but replace the content with your answer: Think step by step. If you need to make calculations such as coordinates, write them here. Describe the effect that your previous action had on the current content of the page. </think>   
+<action> One single action to be executed. You can only use one action at a time.
+```
+
+```twig
+</action>   
+#Concrete Example   
+Here is a concrete example of how to format your answer.   
+Make sure to follow the template with proper tags:   
+<think> My memory says that I filled the first name and last name, but I can't see any content in the form. I need to explore different ways to fill the form. Perhaps the form is not visible yet or some fields are disabled. I need to replan. </think>   
+<action> fill('a12'，'example with"quotes")   
+</action>
+```
+
+# C.2 WEBSHOP
+
+We present the prompt formats for the experiment in the WebShop benchmark.
+
+# Prompt for contextualization module
+
+<system prompt>  
+You are an agent tasked with extracting and rephrasing a subset of the webpage's observations based on the content of the page and user instructions.
+
+<main prompt>  
+You are currently on the online shopping website.  
+Your task is to generate a "Reasoning" and a "Refined observation" based on the provided inputs.
+
+First, review the "User instruction" and "History of interactions" and, then, generate the "Reasoning". Analyze the progress made so far, and provide a rationale for the next steps needed to efficiently accomplish the user instruction on the online shopping website.
+
+Second, rephrase the "AXTree observation at the current time step" into a "Rephrased observation". Select a subset of the AXTree observation that is essential for completing the user instruction and provide explanations for the corresponding elements in the selected subset.
+
+```txt
+[Information source]   
+#User instruction   
+{goal}   
+#History of interactions   
+{previous_actions}   
+#AXTree observation at the current time step   
+{obs}
+```
+
+# Prompt for self-contextualization
+
+<system prompt>  
+You are an agent tasked with extracting and rephrasing a subset of the webpage's observations based on the content of the page and user instructions.
+
+<main prompt>
+
+The current webpage on the web shopping site is described in the observation.
+
+Evaluate the current progress based on previous actions and current observation.
+
+Determine the next action by reasoning based on goal and progress.
+
+Condense the observation into a concise format, highlighting clickable buttons indicated by [].
+
+Ensure the summary includes only elements relevant to the goal and not already covered in previous actions.
+
+Focus on clickable buttons indicated as [].
+
+Here are a few examples.
+
+$\star \star$  goal\*\*: i would like a 3 ounce bottle of bright citrus deodorant for sensitive skin, and price lower than 50.00 dollars
+
+\*\*previous actions\*\*:
+
+1. search[3 ounce bright citrus deodorant sensitive skin]
+
+\*\*current observation\*\*:
+
+[Back to Search]
+
+Page 1 (Total results: 50)
+
+[Next  $>$  ]
+
+[B078GWRC1J]
+
+Bright Citrus Deodorant by Earth Mama | Natural and Safe for Sensitive Skin, Pregnancy and Breastfeeding,
+
+Contains Organic Calendula 3-Ounce $10.99
+
+[B078GTKVXY]
+
+Ginger Fresh Deodorant by Earth Mama | Natural and Safe for Sensitive Skin, Pregnancy and Breastfeeding, Contains Organ Calendula 3-Ounce $10.99
+
+[B08KBVJ4XN]
+
+Barrel and Oak - Aluminum-Free Deodorant, Deodorant for Men, Essential Oil-Based Scent, 24-Hour Odor Protection, Cedar & Patchouli Blend, Gentle on Sensitive Skin (Mountain Sage, 2.7 oz, 2-Pack)  
+$15.95
+
+$\star \star$  rephrased observation\*\*:
+
+Progress: I searched the keyword '3 ounce bright citrus deodorant sensitive skin' to see the relevant items, And now I am looking at the item list.
+
+Reasoning: the next step is to choose an item satisfying the specification of bright citrus deodorant.
+
+I can focus on:
+
+[B078GWRC1J]
+
+Bright Citrus Deodorant by Earth Mama | Natural and Safe for Sensitive Skin, Pregnancy and Breastfeeding, Contains Organic Calendula 3-Ounce $10.99
+
+$\star \star$  goal\*\*: i would like a 3 ounce bottle of bright citrus deodorant for sensitive skin, and price lower than 50.00 dollars
+
+\*\*\*previous actions\*\*:
+
+1. search[3 ounce bright citrus deodorant sensitive skin]
+
+2. click[B078GWRC1J]
+
+\*\*current observation\*\*:
+
+[Back to Search]
+
+[<Prev]
+
+size
+
+```ini
+[ travel set (4-pack) ]
+[ 3 ounce (pack of 1) ]
+[ 3-ounce (2-pack) ]
+scent
+[ assorted scents ]
+[ bright citrus ]
+[ calming lavender ]
+[ ginger fresh ]
+[ simply non-scents ]
+Bright Citrus Deodorant by Earth Mama | Natural and Safe for 
+Sensitive Skin, Pregnancy and Breastfeeding, Contains Organic 
+Calendula 3-Ounce
+Price: $10.99
+Rating: N.A.
+[ Description ]
+[ Features ]
+[ Reviews ]
+[ Buy Now ]
+**rephrased observation**: 
+Progress: I searched and and clicked the item seems to be most 
+relevant to the goal specification. I am looking at the option list. 
+Reasoning: As the goal requires 3-ounce bottle, I can focus 
+on the size option. 
+I can focus on:
+size
+[ travel set (4-pack) ]
+[ 3 ounce (pack of 1) ]
+[ 3-ounce (2-pack) ]
+**goal**: i would like a 3 ounce bottle of bright citrus deodorant 
+for sensitive skin, and price lower than 50.00 dollars
+**previous actions**: 
+1. search[3 ounce bright citrus deodorant sensitive skin]
+2. click[B078GWRC1J]
+3. click[3 ounce (pack of 1)]
+**current observation**: 
+You have clicked 3 ounce (pack of 1). 
+[ Back to Search ]
+[ < Prev ]
+size
+[ travel set (4-pack) ]
+[ 3 ounce (pack of 1) ]
+[ 3-ounce (2-pack) ]
+scent
+[ assorted scents ]
+[ bright citrus ]
+[ calming lavender ]
+[ ginger fresh ]
+[ simply non-scents ]
+Bright Citrus Deodorant by Earth Mama | Natural and Safe for 
+Sensitive Skin, Pregnancy and Breastfeeding, Contains Organic 
+Calendula 3-Ounce
+Price: $10.99
+Rating:N.A.
+[ Description ]
+[ Features ]
+[ Reviews ]
+[ Buy Now ]
+**rephrased observation**: 
+Progress: I searched and and clicked the item id. 
+Among the option list, and I clicked size option.
+```
+
+```txt
+Reasoning: According to the progress, I have to focus on the scent option as a next step.  
+I can focus on:  
+scent  
+[ assorted scents ]  
+[ bright citrus ]  
+[ calming lavender ]  
+[ ginger fresh ]  
+[ simply non-scents ]
+```
+
+```ini
+**goal**: i would like a 3 ounce bottle of bright citrus deodorant for sensitive skin, and price lower than 50.00 dollars
+**previous actions**: 
+1. search[3 ounce bright citrus deodorant sensitive skin]
+2. click[B078GWRC1J]
+3. click[3 ounce (pack of 1)]
+4. click[bright citrus]
+**current observation**: 
+You have clicked 3 ounce (pack of 1). 
+You have clicked bright citrus.
+[ Back to Search ]
+[ < Prev ]
+size
+[ travel set (4-pack) ]
+[ 3 ounce (pack of 1) ]
+[ 3-ounce (2-pack) ]
+scent
+[ assorted scents ]
+[ bright citrus ]
+[ calming lavender ]
+[ ginger fresh ]
+[ simply non-scents ]
+Bright Citrus Deodorant by Earth Mama | Natural and Safe for Sensitive Skin, Pregnancy and Breastfeeding,
+Contains Organic Calendula 3-Ounce
+Price: $10.99
+Rating: N.A.
+[ Description ]
+[ Features ]
+[ Reviews ]
+[ Buy Now ]
+```
+
+```txt
+\*\*rephrased observation\*\*:   
+Progress: Based on \*\*observation\*\* and \*\*previous actions\*\*, I clicked size option and scent option.   
+Reasoning: As there is no more options to select and I met all requirements specified in the goal, next step is to buy the item. I can focus on: [Buy Now]
+```
+
+Here is the task.
+
+```txt
+**goal**
+{goal}
+**previous actions**
+{previous_actions}
+**current observation**
+{obs}
+```
+
+```latex
+$\star \star$  rephrased observation\*\*:
+```
+
+# Prompt for LLM agent
+
+```txt
+Webshop   
+Instruction:   
+i would like a 3 ounce bottle of bright citrus deodorant for sensitive skin, and price lower than 50.00 dollars [ Search ]   
+Action: search[3 ounce bright citrus deodorant sensitive skin]   
+Observation:   
+Progress: I searched the keyword '3 ounce bright citrus deodorant sensitive skin' to see the relevant items, And now I am looking at the item list.   
+Reasoning: Based on the Progress and current observation, the next step is to choose an item satisfying the specification. I can focus on: [B078GWRC1J] Bright Citrus Deodorant by Earth Mama | Natural and Safe for Sensitive Skin, Pregnancy and Breastfeeding, Contains Organic Calendula 3-Ounce \ $10.99 [B078GTKVXY] Ginger Fresh Deodorant by Earth Mama | Natural and Safe for Sensitive Skin, Pregnancy and Breastfeeding, Contains Organic Calendula 3-Ounce \$ 10.99   
+Action: click[B078GWRC1J]   
+Observation:   
+Progress: I searched and and clicked the item seems to be most relevant to the goal specification. I am looking at the option list. Reasoning: As the goal requires 3-ounce bottle, I can focus on the size option. I can focus on: size [ travel set (4-pack) ] [ 3 ounce (pack of 1) ] [ 3-ounce (2-pack) ]   
+Action: click[3-ounce (pack of 1)]   
+Observation:   
+Progress: I searched and and clicked the item id. Among the option list, and I clicked size option. Reasoning: According to the progress, I have to focus on the scent option as a next step. I can focus on: scent [ assorted scents ] [ bright citrus ] [ calming lavender ] [ ginger fresh ] [ simply non-scents ]   
+Action: click[bright citrus]   
+Observation:   
+Progress: Based on \*\*observation\*\* and \*\*previous actions\*\*, I clicked size option and scent option.
+```
+
+```txt
+Reasoning: As there is no more options to select and I met all requirements specified in the goal, next step is to buy the item. I can focus on: [Buy Now]   
+Action: click[Buy Now]   
+Now Here is the task..   
+Instruction: {instruction}   
+{History of observations and actions}   
+Observation: {observation}   
+Action:
+```
+
+# C.3 ACTION MATCHING EVALUATION PROMPT
+
+We present the prompt formats for the action matching described in Algorithm 1.
+
+# Prompt for model-based evaluation of action matching
+
+```txt
+<system prompt>   
+Your task is to evaluate whether the given two action commands are semantically aligned.   
+<main prompt>   
+You will be given   
+1). \*\*reference action\*\* which indicates an correct action.   
+2). \*\*predicted action\*\* which is predicted by assistant agent   
+Your task is to assess whether the message in \*\*predicted action\*\* is semantically aligned with message in the \*\*reference action\*. Please make sure you read and understand these instructions carefully.   
+Please keep this document open while reviewing, and refer to it as needed.
+```
+
+```python
+Evaluation Criteria:   
+Alignment  $= 1$  : the predicted action is semantically aligned with the reference action. send msg_to_user('30') and send msg_to_user('The percentage of amount of pending orders among entire orders is  $30\%$  ) are semantically aligned. click('a34') and click('a34', button  $\equiv$  left') is semantically aligned.
+```
+
+```python
+Alignment = 0: the predicted action is semantically not aligned with the reference action. send msg_to_user('\ $25') and send msg_to_user('The requested value is \$ 29') are not semantically aligned. click('a34') and click('a34', button='left') are semantically aligned.
+```
+
+# Evaluation Steps:
+
+1. Write a simple feedback that assess whether the predicted action is semantically aligned with the reference action.  
+2. After writing a feedback, write a score that is 0 or 1. You should refer to the Evaluation Criteria.  
+3. The output format should look as follows: "Feedback: (write a
+
+feedback for criteria) [RESULT] (an integer number among 0 or 1)" 4. Please do not generate any other opening, closing, and explanations.
+
+$\star \star$  reference action: {ref_action}  
+ $\star \star$  predicted action: {pred_action}
+
+Feedback:
\ No newline at end of file
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+# LEARNING TO EXPLORE AND EXPLOIT WITH GNNS FOR UNSUPERVISED COMBINATORIAL OPTIMIZATION
+
+Utku Umur Acikalin
+
+ua45@cornell.edu
+
+Aaron Ferber
+
+amf272@cornell.edu
+
+Carla P. Gomes<sup>1</sup>
+
+gomes@cs.cornell.edu
+
+<sup>1</sup>Cornell University, Department of Computer Science
+
+# ABSTRACT
+
+Combinatorial optimization (CO) problems are pervasive across various domains, but their NP-hard nature often necessitates problem-specific heuristic algorithms. Recent advancements in deep learning have led to the development of learning-based heuristics, yet these approaches often struggle with limited search capabilities. We introduce Explore-and-Exploit GNN ( $X^2$ GNN, pronounced x-squared GNN), a novel unsupervised neural framework that combines exploration and exploitation for combinatorial search optimization: i) Exploration -  $X^2$ GNN generates multiple solutions simultaneously, promoting diversity in the search space; (ii) Exploitation -  $X^2$ GNN employs neural stochastic iterative refinement to exploit partial existing solutions, guiding the search toward promising regions and helping escape local optima. By balancing exploration and exploitation,  $X^2$ GNN achieves superior performance and generalization on several graph CO problems including Max Cut, Max Independent Set, and Max Clique. Notably, for large Max Clique problems,  $X^2$ GNN consistently generates solutions within 1.2% of optimality, while other state-of-the-art learning-based approaches struggle to reach within 22% of optimal. Moreover,  $X^2$ GNN consistently generates better solutions than Gurobi on large graphs for all three problems under reasonable time budgets. Furthermore,  $X^2$ GNN exhibits exceptional generalization capabilities. For the Maximum Independent Set problem,  $X^2$ GNN outperforms state-of-the-art methods even when trained on smaller or out-of-distribution graphs compared to the test set. Our framework offers a more effective and flexible approach to neural combinatorial optimization, addressing a key challenge in the field and providing a promising direction for future research in learning-based heuristics for combinatorial optimization. $^1$
+
+# 1 INTRODUCTION
+
+Combinatorial optimization (CO) problems aim to find a discrete solution that optimizes an objective function from a discrete set of feasible solutions constrained by specific problem parameters. These optimization problems frequently emerge in commercial, governmental, and scientific contexts, prompting extensive study in fields such as mathematics, computer science, and operations research. Many combinatorial optimization problems are NP-hard, indicating no polynomial-time exact algorithm exists unless  $\mathrm{P} = \mathrm{NP}$ .
+
+Exact algorithms, which achieve optimality by implicitly or explicitly considering all possible solutions, are typically only tractable for small instances due to their exponential worst-case time complexity. Consequently, another body of work focuses on heuristic algorithms that quickly attain high-quality solutions without optimality guarantees. Work here seeks to balance computational time and solution quality, often exploring the search space through multiple different solutions, and exploiting promising ones (Eiben and Schippers, 1998).
+
+Heuristics are often hand-crafted for a specific problem, exploring problem intricacies to achieve peak performance for a given problem distribution, requiring time and domain expertise. The rise of deep
+
+learning has enabled new learning-based heuristics that can be automatically tuned for performance using data.
+
+However, current approaches are often limited in their search capabilities, often only iteratively improving a single solution, or naively restarting and forgetting previously generated solutions.
+
+In this paper, we introduce Explore-and-Exploit GNN ( $X^2$ GNN, pronounced X-squared GNN), a novel unsupervised neural combinatorial optimization framework.  $X^2$ GNN combines effective exploration of the solution space with intelligent exploitation of partial solutions.
+
+# Our Main Contributions are:
+
+1. We propose  $X^{2}$ GNN, which combines exploration and exploitation for combinatorial search optimization: (i) Exploration -  $X^{2}$ GNN simultaneously generates multiple coupled solutions, promoting diversity in the search space; (ii) Exploitation -  $X^{2}$ GNN employs neural stochastic iterative refinement to exploit partial existing solutions, guiding the search toward promising regions and helping escape local optima.  
+2. State-of-the-art performance:  $X^{2}$  GNN outperforms existing learning-based approaches on benchmark datasets for the maximum cut, maximum independent set, and maximum clique problems. Additionally,  $X^{2}$  GNN is competitive with general OR approaches like Gurobi, and problem specific heuristics like KaMIS, offering improved or comparable solution quality at similar time cutoffs.  
+3. Strong generalization capabilities:  $X^{2}$  GNN generalizes to graphs that are out-of-distribution or up to 4 times larger than those seen during training, while still significantly outperforming other learning-based methods trained on the same distribution as the test set.  
+4. Rigorous Evaluation: We enhance existing benchmark datasets commonly used in the ML for combinatorial optimization community by including strong traditional baselines and evaluating solvers at comparable runtimes. We additionally allow solvers a 30-minute time limit, which is at least 24 times longer than our longest-running model.
+
+# 2 RELATED WORK
+
+The broad intersection between machine learning (ML) and combinatorial optimization (CO) has seen much work with different facets explored in various surveys (Bengio et al., 2021; Kotary et al., 2021; Cappart et al., 2023). State-of-the-art learning-based primal heuristics specifically can be broadly categorized by their training supervision and solution construction methods. Supervised learning approaches use training data composed of problem instances and corresponding solutions derived from existing solvers (Khalil et al., 2016; Selsam et al., 2019; Nair et al., 2020; Sun and Yang, 2023). However, these may face challenges such as the unavailability of high-quality solvers for all problems and poor generalization capabilities across different problem instances (Yehuda et al., 2020). Despite these challenges, recent studies have shown that diffusion-based training can enhance generalization in supervised learning (Sun and Yang (2023)).
+
+Unsupervised learning approaches have also been explored, differing primarily in whether solutions are constructed autoregressively or not. Earlier non-autoregressive models generate a 'soft' solution in a single step, which is then decoded into a final solution using methods ranging from simple greedy (Karalias and Loukas, 2020) decoding to more sophisticated techniques (Min et al., 2022). As Sanokowski et al. (2024) noted, these approaches can be classified as single-step diffusion methods. These models are notably faster and more scalable than their autoregressive counterparts. Sanokowski et al. (2023) suggest that non-autoregressive solution construction may fail to capture essential dependencies among problem variables and they refer to these types of methods as mean-field approximations.
+
+The earlier single-step non-autoregressive methods are outperformed by autoregressive construction governed by MDPs (Sanokowski et al., 2023; Zhang et al., 2023). However, these models are trained using reinforcement learning (RL) and face high computational needs and poor generalization (Sun and Yang, 2023). Additionally, autoregressive construction does not allow modification of fixed decisions, unlike diffusion-based construction where all variables can be altered at each step.
+
+The success of generative diffusion models (Sohl-Dickstein et al., 2015) made it appealing for CO. For diffusion-based CO approaches, noise is sequentially added to the optimal solution obtained from other solvers in the forward process, and the model learns to iteratively remove this noise in the reverse process. Sun and Yang (2023) models the CO problems as a discrete diffusion problem using Bernoulli and Categorical noise. For MIS, they outperform non-autoregressive models but have similar performance to autoregressive models, suffering from long diffusion schedules. Li et al. (2023) follow the same training procedure as Sun and Yang (2023) but employ gradient-guided noising-denosing rounds during inference. This improves upon Sun and Yang (2023); however, long diffusion schedules and reliance on the gradients hinder effectiveness. Sanokowski et al. (2024) trains a diffusion model to sample from the Boltzmann distribution with the probability of sampling a solution being positively correlated to its objective value. They derive an unsupervised loss function using a continuous Lagrangian relaxation, showing that longer generation schedules increase quality.
+
+# 3 PRELIMINARIES
+
+We consider the broad class of combinatorial optimization problems on graphs and instantiate  $X^{2}$  GNN for three NP-hard problems on undirected unweighted graphs  $G = (V, E)$ . As in many real-world scenarios, we consider distributional versions of these problems, where we are asked to train an algorithm on a dataset of instances and then deploy the algorithm on unseen instances.
+
+Maximum Clique (MC): A clique in a graph  $G$  is a subset of vertices where every two vertices are adjacent. The Maximum Clique problem involves finding the largest clique in  $G$ .
+
+Maximum Independent Set (MIS): An independent set in a graph  $G$  is a subset of vertices, none of which are adjacent. The Maximum Independent Set problem aims to find the largest independent set.
+
+Maximum Cut (MCut): For a graph  $G = (V, E)$ , the Maximum Cut problem seeks to partition the set of nodes  $V$  into two subsets  $S$  and  $V \setminus S$ , maximizing the number of edges between  $S$  and  $V \setminus S$ .
+
+Solutions for these problems can be represented by a binary decision for each node,  $Y \in \{0,1\}^{|V|}$ , indicating solution inclusion,  $Y_{i} = 1$  if  $v_{i} \in S$  otherwise 0. Additionally, the maximum clique and maximum independent set problems are closely related; a clique  $S$  in graph corresponds to an independent set  $S$  in its complementary graph (Cormen et al. (2001)). We use this relationship for all approaches and solve MC problems by solving MIS on the corresponding complementary graph.
+
+# 4  $X^2$  GNN FRAMEWORK
+
+$X^{2}$  GNN, illustrated in Figure 1, is an iterative framework that explores the search space by simultaneously generating a pool of  $K$ -Coupled solutions and exploiting promising ones via stochastic refinement.
+
+$K$ -Coupled solutions form a group of  $K$  solutions that are built collectively, we refer to each group as a  $K$ -Couple. We model the  $K$ -Couple using a multilayer graph, copying the original graph  $K$  times to represent the  $K$  solutions, and adding auxiliary edges between corresponding nodes in different layers. We model the solution values themselves as node features. We then iteratively feed the  $K$ -Couple into a graph neural network (GNN) which makes a prediction on each node corresponding to a new  $K$ -Couple. We train the GNN's outputs at each iteration using a combination of an unsupervised optimization loss, a constraint satisfaction loss, and a diversity loss on the  $K$ -Couple. Importantly, before feeding a  $K$ -Couple as input to the GNN, we randomly perturb the solution to help escape local optima. Furthermore, we randomly initialize the  $K$ -Couple.
+
+Formally, given a problem represented by graph  $G = (V,E)$ , we represent the GNN input of  $K$  solutions at iteration  $t$  using  ${}^t X\in [0,1]^{K\times |V|}$ , with  ${}^t X_u^k$  denoting the feature of node  $u\in V$  for solution  $k\in 1,\ldots ,K$  in iteration  $t$ . We use  ${}^t\hat{Y}$  to denote the  $K$ -Coupled solution generated at iteration  $t$ . That is,  ${}^t\hat{Y} = g_\theta (G,tX)$ . Similarly, we use the notation  ${}^t\hat{Y}_u^k$  to denote the probability of node  $u\in V$  being in solution  $k\in 1,\dots ,K$  generated at iteration  $t$ .
+
+Solution Generation and Stochastic Iterative Refinement: We refer to the first iteration  $(t = 1)$  of  $X^2$  GNN as construction and the subsequent iterations  $(t\geq 2)$  as refinement. During construction, we randomly initialize node features  $^{1}X_{u}^{k}$  to 0.5 with probability  $p$  and 0 with probability  $1 - p$
+
+![](images/54642a30638337aa18f4320148d0fdbca9fb64d01767a6e9f54a23946f0d4dd9.jpg)  
+Figure 1: Illustration of  $X^{2}$  GNN for a Minimum Independent Set instance. First, a multilayer graph is created from  $K = 2$  copies of the original graph, with cross edges to couple solutions. Copies of the original edges  $(E_{O}^{\prime})$  are drawn in blue, and cross edges  $(E_{C}^{\prime})$  are drawn in red. Node features correspond to the probability of being in the solution, representing soft solutions. Initially, when generating the coupled solutions, the features are random. These features are fed into a GNN to obtain  $K$  soft solutions. During stochastic refinement, the GNN iteratively takes solutions from the previous time step, randomly perturbs them, and generates new solutions. Stochastic refinement can be repeatedly applied at inference and is done once during training. Finally, the training loss is calculated for all generated solutions using the objective value, Lagrangian term, and diversity.
+
+with  $p$  being 0.95 in practice. Essentially, we initialize the  $K$ -Couple with unbiased solutions while introducing diversity to break symmetries for nodes with identical degrees.
+
+During refinement, we randomly set the previous iteration's output  $t^{-1}\hat{Y}$  to 0 with probability  $\phi$  to generate new node features  ${}^t X$ .
+
+The parameter  $\phi$  offers a natural way to control exploration and exploitation. If  $\phi = 1$ , no information from the previous iteration is used, maximizing exploration. If  $\phi = 0$ , the previously generated solutions are maintained, and the method will deterministically refine the  $K$ -Couple, maximizing exploitation. Thus, high values of  $\phi$  lead to exploration whereas low values lead to exploitation. We refer to one step of this approach as Stochastic Refinement and the general application of multiple iterations as Stochastic Iterative Refinement.
+
+Aligning the model's input and output enables repeated use of the recurrent model. Recurrent models trained on short iterations can be deployed for longer iterations to solve more complex problems (Schwarzschild et al., 2021); however, overuse can result in 'overthinking' (Bansal et al., 2022).  $X^{2}$  GNN mitigates this by not iterating over the hidden representations, but rather iterating stochastically over the output space. The stochastic sampling process significantly impacts performance by facilitating exploration around the current solution, allowing exploitation, and escaping local optima. Moreover, it helps generalization to bigger and out-of-distribution datasets.
+
+Converting Soft Solutions to Hard Solutions: Since the GNN outputs soft solutions  $\hat{Y}^k$ , we convert them to discrete feasible solutions  $S$  depending on the problem. For MCut, we select a node  $u$  into  $S$  iff  $\hat{Y}^k \geq 0.5$ , yielding a feasible solution due to the absence of constraints. For MIS and MC, we add node  $u$  into  $S$  in order of decreasing probability  $\hat{Y}_u^k$  as long as  $S \cup \{u\}$  satisfies problem constraints.
+
+$\pmb{K}$ -Coupled Solutions: To couple solutions, we construct a multilayer graph  $G_{M} = (V^{\prime},E^{\prime})$  from the original graph  $G = (V,E)$ .  $G_{M}$  contains  $K$  "layers" each containing a copy of the original graph.
+
+Additional edges (cross edges) connect nodes in different layers corresponding to the same node in  $G$ . We denote  $E_O'$  as the edges corresponding to original edges  $E$ , and  $E_C'$  as the cross edges, such that  $E' = E_O' \cup E_C'$ . We construct  $G_M$  as follows: For each node  $v_i \in V$ , we create node  $v_i^k$  for all layers  $k = 1, \ldots, K$ . For each edge  $(v_i, v_j) \in E$ , we create an original edge between  $v_i^k$  and  $v_j^k$  for all  $k = 1, \ldots, K$ , forming  $E_O'$ . We then add cross edges  $E_C'$ , such that for each node  $i \in V$  all copies of  $i$  have a pairwise edge between them in  $G_M$ . In practice, we select  $K = 2$ .
+
+$X^{2}$ GNN Neural Network Architecture: The GNN used by  $X^{2}$ GNN to construct and refine solutions consists of  $2L$  layers combining Graph Isomorphism Networks (GIN) (Xu et al., 2019) and Graph Attention Networks (GAT) (Velickovic et al., 2018; Brody et al., 2022). The layers alternate between GIN layers operating on  $(V', E_O')$  to work on individual solutions and GAT layers operating on  $(V', E_C')$  to enable information sharing between solutions. This alternating design enables the simultaneous generation of  $K$ -coupled solutions. Note that the same model parameters are used for both construction and refinement.
+
+Training and Loss Functions: We train  $X^{2}$  GNN using unsupervised combinatorial optimization losses which take the form of Lagrangian relaxations of the original problem. We adopt nonlinear programming formulations for MCut, MIS, and MC problems from Sanokowski et al. (2024). For MIS and MC, the objective is to include as many nodes in  $S$  as possible, penalizing constraint violation, whereas in MCut the objective is to include as many edges in the cut as possible. We additionally propose adding a loss function to promote diversity among the  $K$ -coupled solutions. In the MIS and MC settings, we want solutions to contain different nodes, whereas we are interested in having different cut edges for MCut. We write the optimization problems, continuous relaxations, and Lagrangian terms for the constraints in Table 1.
+
+Overall,  $X^{2}$  GNN trains the GNN parameters  $\theta$  to jointly optimize objective quality  $(\mathcal{L}_o)$ , constraint satisfaction  $(\mathcal{L}_c)$ , and solution diversity  $(\mathcal{L}_d)$  over the training set  $\mathcal{G}$ :
+
+$$
+\min _ {\theta} \mathbb {E} _ {G \in \mathcal {G}} \left[ \sum_ {t} \left[ \sum_ {k} \mathcal {L} _ {o} (G, ^ {t} \hat {Y} ^ {k}) + \lambda_ {1} \mathcal {L} _ {c} (G, ^ {t} \hat {Y} ^ {k}) \right] + \lambda_ {2} \mathcal {L} _ {d} (G, ^ {t} \hat {Y}) \right]
+$$
+
+For MC and MIS, we impose node diversity:
+
+$$
+\mathcal{L}_{d}(G,\hat{Y}) = \frac{1}{K(K - 1)}\sum_{\substack{1\leq k_{1},k_{2}\leq K\\ k_{1}\neq k_{2}}}\sum_{u\in V}\hat{Y}_{u}^{k_{1}}\hat{Y}_{u}^{k_{2}}
+$$
+
+For MCut, we impose cut edge diversity:
+
+$$
+\mathcal{L}_{d}(G,\mathcal{Y}) = \frac{1}{K(K - 1)}\sum_{\substack{1\leq k_{1},k_{2}\leq K\\ k_{1}\neq k_{2}}}\sum_{(u,v)\in E}\frac{1 - (2\hat{Y}_{u}^{k_{1}} - 1)(2\hat{Y}_{v}^{k_{1}} - 1)}{2}\frac{1 - (2\hat{Y}_{u}^{k_{2}} - 1)(2\hat{Y}_{v}^{k_{2}} - 1)}{2}
+$$
+
+<table><tr><td>Problem</td><td colspan="2">Formulation</td><td>Objective</td><td>Constraint loss</td></tr><tr><td rowspan="2">MC</td><td>max Y∈{0,1}|V|</td><td>∑u∈Y Yu</td><td>∑Yu</td><td>∑YukYk</td></tr><tr><td>s.t.</td><td>YuYv=0, ∀(u,v)∉E</td><td>∑u∈V</td><td>∑(u,v)∉E</td></tr><tr><td rowspan="2">MIS</td><td>max Y∈{0,1}|V|</td><td>∑u∈Yu</td><td>∑Yu</td><td>∑YukYk</td></tr><tr><td>s.t.</td><td>YuYv=0, ∀(u,v)∈E</td><td>∑u∈V</td><td>∑(u,v)∈E</td></tr><tr><td>MCut</td><td>max Y∈{0,1}|V|</td><td>1-(2Yu-1)(2Yv-1)/2</td><td>1-(2Yuk-1)(2Yv-1)/2</td><td>-</td></tr></table>
+
+Table 1: Mathematical formulation, objective loss  $\mathcal{L}_o$ , and constraint loss  $\mathcal{L}_c$  for our problems.
+
+We train  $X^{2}$  GNN using a two-stage training procedure. In the first stage, the model learns to construct solutions, and in the second stage, the model learns to stochastically refine constructed solutions for one step. The proposed two-stage training procedure leads to better initial solutions and more stable solution refinement for  $X^{2}$  GNN.
+
+Inference: Unlike during training, during inference we use the stochastic refinement step multiple times leading to stochastic iterative refinement. Training the model on a single stochastic refinement iteration is enough to teach the model to generally improve solutions. We show that using the stochastic iterative refinement longer leads to a significant increase in solution quality. Additionally, instead of generating just one  $K$ -coupled solution, we generate  $C$ $K$ -coupled solutions to increase exploration. These  $K$ -coupled solutions are independent and effectively run  $X^2$  GNN simultaneously with different random seeds.
+
+$X^{2}$  GNN offers time-quality trade-offs by selecting  $C$ , the number of  $K$ -Coupled solutions, and  $T$  the number of iterations. By increasing  $C$  and/or  $T$ , we can consider more solutions to improve solution quality by using more time. For each choice of  $C$  and  $T$ ,  $X^{2}$  GNN generates  $C \times K$  solutions at the first iteration that are then refined for  $T - 1$  iterations to generate  $C \times K \times T$  solutions in total. Hence, the computational impact of  $C$  and  $T$  is very similar. A natural question is how to select  $C$  and  $T$  for a fixed computational budget. Under a fixed budget, increasing C promotes exploration by maintaining more solutions, whereas increasing T enhances exploitation by allowing more refinement iterations on existing solutions. This mechanism controls exploration and exploitation in our optimization framework, which is crucial for effectively navigating the search landscape of complex problems.
+
+# 5 EXPERIMENTS
+
+Datasets: Previous literature has identified that some problem instances for MIS and MC are relatively easy (Dai et al., 2020). To ensure rigorous evaluation, we use synthetically generated hard instances, following previous work (Karalias and Loukas, 2020; Zhang et al., 2023; Sanokowski et al., 2024). Our datasets include RB graphs (Xu and Li, 2000), a revision to model B graphs (Gent et al., 2001; Smith and Dyer, 1996), known to generate challenging instances for MC and MIS. Specifically, we use RB graphs with 200-300 nodes(RB250) and 800-1200 nodes (RB1000). For MIS, we also evaluate on Erdős-Rényi (ER) graphs (Erdős and Rényi, 1959) with 700-800 nodes and edge probability 0.15 (ER750), as well as regular graphs where each node has either 3 (d=3) or 5 (d=5) neighbors. Following Schuetz et al. (2022), we generate 20 regular graphs for each degree  $d \in \{3,5\}$  and each size  $n \in \left[10^2,10^3,10^4,10^5,10^6\right]$  for testing. To train learning-based methods, we generate 4,000 additional graphs with  $n = 10^3$ , enabling evaluation of both generalization and scalability. For MCut, we use Barabási-Albert (BA) graphs (Barabási and Albert, 1999) with 250 nodes (BA250) and 1,000 nodes (BA1000). We train on 4,000 graphs and test on 500 graphs except for ER (128 test graphs) and regular graphs (20 test graphs for each size and d).
+
+Baselines: We compare  $X^{2}$  GNN against both Operations Research (OR) and Machine Learning (ML) techniques. We compare against Gurobi (Gurobi Optimization, LLC, 2024) on all tasks as it is a general-purpose exact solver that is highly performant on many CO tasks due to years of development. For MIS, we compare against KAMIS (Lamm et al., 2016), a highly specialized MIS solver, as well as learning-based approaches such as PPO (Ahn et al., 2020), Gflow (Zhang et al., 2023), DIFFUSCO (Sun and Yang, 2023), T2T (Li et al., 2023), and DiffUCO (Sanokowski et al., 2024). For MC, we benchmark against KAMIS used on the complement graph, greedy algorithms, mean-field annealing (MFA), and learning-based methods including ERDOS and its annealed version ANNEAL (Karalias and Loukas, 2020; Sun et al., 2022), DiffUCO, and Gflow. MCut comparisons include semi-definite-programming (SDP) based approximation algorithm (Goemans and Williamson, 1995), Tabu Search (TS) (Nath and Kuhnle, 2024), and learning-based methods RUN-CSP (Tonshoff et al., 2020), ANYCSP (Tonshoff et al., 2023), ERDOS, ANNEAL, DiffUCO, and Gflow. When given, we use Fast, Quality, and 30min to denote that we set time limits around the twice the fastest version of  $X^{2}$  GNN, twice the slowest version of  $X^{2}$  GNN, and 30 minutes respectively.
+
+Evaluation Metrics: We employ three metrics: the mean objective value (Size), the mean drop in quality relative to the best-known solution (Drop), and the mean runtime in seconds (Time). Overall, better methods find solutions with lower solution quality drop at smaller runtimes. Since all problems are maximization problems, larger size is better. For instance, a  $10\%$  drop means the method generates solutions with a mean objective value of 90, while the best method achieves 100. In all tables, learning-based methods are shaded. Bold entries denote the best learning-based method, and italics indicate the best method, learning or traditional. Additionally, we denote the method type categorizing methods into operations research (OR), heuristic (H), supervised learning (SL), and unsupervised learning (UL).
+
+Table 2: Results for Max Clique on small and large RB graphs, presenting the mean clique size, drop in quality compared to the optimal, and runtime in seconds. Learning-based methods are shaded, and the best learning-based result is bolded. The best global result is in italics.  $X^{2}$ GNN generates solutions at least 14% to 23% better than all learning-based methods.  $X^{2}$ GNN solves RB250 optimally, with a similar run time as Gurobi and KaMIS.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Type</td><td colspan="3">RB250</td><td colspan="3">RB1000</td></tr><tr><td>Size ↑</td><td>Drop ↓</td><td>Time ↓</td><td>Size ↑</td><td>Drop ↓</td><td>Time ↓</td></tr><tr><td>KaMIS</td><td>OR</td><td>19.074</td><td>0%</td><td>10</td><td>40.652</td><td>0%</td><td>51</td></tr><tr><td>Gurobi (30min)</td><td>OR</td><td>19.074</td><td>0%</td><td>0.73</td><td>40.652</td><td>0%</td><td>287</td></tr><tr><td>Gurobi (Quality)</td><td>OR</td><td>19.068</td><td>0.03%</td><td>0.61</td><td>36.23</td><td>10.88%</td><td>47</td></tr><tr><td>Gurobi (Fast)</td><td>OR</td><td>14.62</td><td>23.35%</td><td>0.17</td><td>25.36</td><td>37.62%</td><td>3</td></tr><tr><td>Greedy</td><td>H</td><td>13.53</td><td>29.07%</td><td>0.03</td><td>26.71</td><td>34.30%</td><td>0.04</td></tr><tr><td>MFA</td><td>H</td><td>14.82</td><td>22.30%</td><td>0.04</td><td>27.94</td><td>31.27%</td><td>0.21</td></tr><tr><td>Erdos</td><td>UL</td><td>12.02</td><td>36.98%</td><td>0.06</td><td>25.43</td><td>37.44%</td><td>0.2</td></tr><tr><td>Anneal</td><td>UL</td><td>14.1</td><td>26.08%</td><td>0.06</td><td>27.46</td><td>32.45%</td><td>0.2</td></tr><tr><td>Gflow</td><td>UL</td><td>16.24</td><td>14.86%</td><td>0.06</td><td>31.42</td><td>22.71%</td><td>0.44</td></tr><tr><td>DiffUCO</td><td>UL</td><td>16.3</td><td>14.54%</td><td>4.13</td><td>30.5</td><td>24.97%</td><td>7.92</td></tr><tr><td>X2GNN (RB250)(2x64)</td><td>UL</td><td>19.04</td><td>0.18%</td><td>0.09</td><td>39.83</td><td>2.02%</td><td>1.5</td></tr><tr><td>X2GNN (RB250)(8x64)</td><td>UL</td><td>19.072</td><td>0.01%</td><td>0.37</td><td>40.09</td><td>1.38%</td><td>5.8</td></tr><tr><td>X2GNN (RB250)(32x64)</td><td>UL</td><td>19.074</td><td>0%</td><td>1.41</td><td>40.17</td><td>1.19%</td><td>23.5</td></tr></table>
+
+For generalization, we denote the training dataset with (RB250) or (BA250) in the model name. We denote variants of  $X^{2}$  GNN that generate  $C$  2-Coupled solutions and use  $T - 1$  stochastic refinement steps with (2CxT) in the model name.
+
+# 5.1 RESULTS ON MAXIMUM CLIQUE
+
+Results for MC on small and large RB datasets are shown in Table 2. For MC, we train  $X^{2}$  GNN on RB250 and showcase generalization to larger instances.  $X^{2}$  GNN generates solutions of at least  $14\%$  and  $23\%$  higher objective value than the second-best learning-based methods on RB250 and RB1000, respectively.
+
+Compared to traditional algorithms,  $X^{2}$ GNN solves every RB250 instance optimally, with a similar run time as Gurobi and KaMIS. On the larger dataset, on which  $X^{2}$ GNN wasn't trained,  $X^{2}$ GNN generates solutions that are within 2% of optimality while almost being 50 and 190 times faster than KaMIS and Gurobi, respectively. Additionally, at a similar runtime,  $X^{2}$ GNN has substantially better solution quality than Gurobi.
+
+# 5.2 RESULTS ON MAXIMUM INDEPENDENT SET
+
+Table 3 presents results for Maximum Independent Set (MIS) on small RB, large RB, and ER graphs. For all datasets except regular graphs, the metaheuristic KaMIS achieves the best solution quality. Again,  $X^{2}$ GNN outperforms all learning-based methods by a large margin, especially for the largest RB1000 dataset, where  $X^{2}$ GNN generates solutions that are  $9\%$  better than the second best. Even the model trained on RB250 dataset is able to outperform the other learning-based methods on both RB1000 and ER750 datasets, showing that  $X^{2}$ GNN can successfully generalize to harder and different graph distributions.
+
+Comparison with the traditional algorithms is more nuanced. On ER750,  $X^{2}$  GNN generates better solutions than KaMIS and Gurobi when they are given either 15 or 60 seconds. On RB1000,  $X^{2}$  GNN generates better solutions than Gurobi but slightly worse solutions than KaMIS at around 20 seconds.
+
+Figure 2 presents the solution quality relative to the theoretical lower bound and the running time for regular graphs of varying sizes. Given the large graph sizes, obtaining optimal solutions is intractable. Following Schuetz et al. (2022), we use analytical upper bounds for random d-regular graphs, with the best-known ratios  $\alpha_{3} / n = 0.45537$  and  $\alpha_{5} / n = 0.38443$  for  $d = 3$  and  $d = 5$ , respectively (Duckworth and Zito, 2009). However, these bounds may not be tight. For instances with  $n = 100$  and  $d = 3$ , both  $X^{2}$  GNN and KaMIS find optimal solutions, as verified by Gurobi. Notably, the analytical upper bound remains  $2.6\%$  above these solutions, indicating that actual performance may be better than suggested.
+
+Table 3: Results for Max Independent Set on small and large RB graphs and ER graphs, presenting the mean independent set size, drop in quality compared to the virtual best (KaMIS), and runtime in seconds.  $X^{2}$ GNN substantially outperforms learning-based approaches on all datasets. When generalizing from small RB250 instances,  $X^{2}$ GNN outperforms learning-based methods trained on the larger and in-distribution problems. On ER instances,  $X^{2}$ GNN outperforms traditional OR approaches given similar time limits.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Type</td><td colspan="3">RB250</td><td colspan="3">RB1000</td><td colspan="3">ER750</td></tr><tr><td>Size ↑</td><td>Drop ↓</td><td>Time ↓</td><td>Size ↑</td><td>Drop ↓</td><td>Time ↓</td><td>Size ↑</td><td>Drop ↓</td><td>Time ↓</td></tr><tr><td>KaMIS (30min)</td><td>OR</td><td>20.106</td><td>0%</td><td>3.92</td><td>43.218</td><td>0%</td><td>381</td><td>45.234</td><td>0%</td><td>382</td></tr><tr><td>Gurobi (30min)</td><td>OR</td><td>20.106</td><td>0%</td><td>0.31</td><td>42.96</td><td>0.60%</td><td>550</td><td>43.62</td><td>3.57%</td><td>1800</td></tr><tr><td>KaMIS (Quality)</td><td>OR</td><td>20.106</td><td>0%</td><td>3.92</td><td>42.98</td><td>0.55%</td><td>18</td><td>44.84</td><td>0.87%</td><td>61</td></tr><tr><td>Gurobi (Quality)</td><td>OR</td><td>20.106</td><td>0%</td><td>0.42</td><td>42.25</td><td>2.24%</td><td>22.27</td><td>43.5</td><td>3.83%</td><td>120</td></tr><tr><td>KaMIS (Fast)</td><td>OR</td><td>20.032</td><td>0.37%</td><td>1.16</td><td>42.66</td><td>1.29%</td><td>6.5</td><td>43.46</td><td>3.92%</td><td>16</td></tr><tr><td>Gurobi (Fast)</td><td>OR</td><td>19.16</td><td>4.71%</td><td>0.1</td><td>38.81</td><td>10.20%</td><td>1.23</td><td>41.31</td><td>8.67%</td><td>3.62</td></tr><tr><td>PPO</td><td>UL</td><td>19.01</td><td>5.45%</td><td>0.15</td><td>32.32</td><td>25.22%</td><td>0.91</td><td>41.11</td><td>9.12%</td><td>2.11</td></tr><tr><td>GFlow</td><td>UL</td><td>19.18</td><td>4.61%</td><td>0.05</td><td>37.48</td><td>13.28%</td><td>0.4</td><td>41.14</td><td>9.05%</td><td>1.03</td></tr><tr><td>DIFUSCO</td><td>SL</td><td>17.68</td><td>12.07%</td><td>0.87</td><td>35.82</td><td>17.12%</td><td>41.11</td><td>40.35</td><td>10.80%</td><td>15.46</td></tr><tr><td>T2T</td><td>SL</td><td>18.35</td><td>8.73%</td><td>2.32</td><td>35.822</td><td>17.11%</td><td>26.55</td><td>41.37</td><td>8.54%</td><td>13.92</td></tr><tr><td>DiffUCO</td><td>UL</td><td>19.24</td><td>4.31%</td><td>0.42</td><td>38.87</td><td>10.06%</td><td>5</td><td>43.63</td><td>3.55%</td><td>0.71</td></tr><tr><td>X2GNN(16x8)</td><td>UL</td><td>19.51</td><td>2.96%</td><td>0.034</td><td>40.53</td><td>6.22%</td><td>0.3</td><td>42.05</td><td>7.04%</td><td>0.31</td></tr><tr><td>X2GNN(64x8)</td><td>UL</td><td>19.82</td><td>1.42%</td><td>0.128</td><td>41.54</td><td>3.88%</td><td>1.18</td><td>43.06</td><td>4.81%</td><td>1.07</td></tr><tr><td>X2GNN(256x8)</td><td>UL</td><td>19.98</td><td>0.63%</td><td>0.5</td><td>42.19</td><td>2.38%</td><td>4.66</td><td>43.82</td><td>3.13%</td><td>3.91</td></tr><tr><td>X2GNN(256x32)</td><td>UL</td><td>20.072</td><td>0.17%</td><td>1.94</td><td>42.48</td><td>1.71%</td><td>18.36</td><td>44.43</td><td>1.78%</td><td>15.26</td></tr><tr><td>X2GNN(1024x32)</td><td>UL</td><td>20.098</td><td>0.04%</td><td>7.11</td><td>42.81</td><td>0.94%</td><td>74.4</td><td>44.91</td><td>0.72%</td><td>57.18</td></tr><tr><td>X2GNN(RB250)(256x32)</td><td>UL</td><td>20.072</td><td>0.17%</td><td>1.94</td><td>39.28</td><td>9.1%</td><td>9.43</td><td>44.15</td><td>2.40%</td><td>7.96</td></tr></table>
+
+![](images/cf41f3e90125f596abfc8e218c6f3f097f7fa57157a3d9c9a1906450aab91b83.jpg)  
+Figure 2: The left plot shows the relative solution quality compared to the theoretical upper bound for  $X^{2}$  GNN, DiffUCO, and KaMIS (higher is better). The right plot shows the average running time for each method (lower is better). Lines with squares and triangles show the results for regular graphs where each node has 3 and 5 neighbors, respectively.  $X^{2}$  GNN outperforms DiffUCO on all graph sizes and families in terms of solution quality while being faster on larger graphs. It also surpasses KaMIS in larger graphs with a better run time.
+
+![](images/1bb9cab330d1119df4d095aaa898b7383ba8c3908ea498772f8e3eec7b4cd650.jpg)
+
+$X^{2}$  GNN surpasses DiffUCO in both solution quality and computational efficiency on larger graphs. DiffUCO fails to solve  $n = 10^6$  graphs within two hours, while  $X^{2}$  GNN efficiently produces high-quality solutions in about 600 seconds. Although KaMIS slightly outperforms  $X^{2}$  GNN on smaller instances,  $X^{2}$  GNN excels on larger graphs with lower computational time, scaling effectively to instances three orders of magnitude larger than those in training.
+
+# 5.3 RESULTS ON MAXIMUM CUT
+
+Results for Maximum Cut on small and large BA datasets are shown in Table 4.  $X^{2}$  GNN outperforms all state-of-the-art learning-based methods on both datasets. However, ANYCSP and DiffUCO notably have only slightly worse performance than  $X^{2}$  GNN both in terms of speed and quality. For the large dataset,  $X^{2}$  GNN outperforms Gurobi with a time limit of 30 minutes per instance while only using 0.2 seconds. Similarly,  $X^{2}$  GNN outperforms Tabu Search, finding better solutions faster.
+
+Table 4: Results for Max Cut on small and large BA graphs, presenting the mean cut size, drop in quality to the virtual best, and runtime in seconds.  $X^{2}$ GNN outperforms learning-based methods, slightly outperforming ANYCSP. Additionally, on BA1000,  $X^{2}$ GNN outperforms Gurobi given 30 minutes by generating better solutions in 0.2s.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Type</td><td colspan="3">BA250</td><td colspan="3">BA1000</td></tr><tr><td>Size ↑</td><td>Drop ↓</td><td>Time ↓</td><td>Size ↑</td><td>Drop ↓</td><td>Time ↓</td></tr><tr><td>Gurobi (30min)</td><td>OR</td><td>735.32</td><td>0%</td><td>759</td><td>2966.58</td><td>0.76%</td><td>1800</td></tr><tr><td>Gurobi (Quality)</td><td>OR</td><td>731.99</td><td>0.45%</td><td>2.5</td><td>2931.02</td><td>1.95%</td><td>4.24</td></tr><tr><td>Gurobi (Fast)</td><td>OR</td><td>731.83</td><td>0.47%</td><td>0.16</td><td>2930.99</td><td>1.95%</td><td>0.66</td></tr><tr><td>SDP</td><td>OR</td><td>700.04</td><td>4.8%</td><td>4.2</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Tabu Search</td><td>H</td><td>733.79</td><td>0.21%</td><td>3</td><td>2926.6</td><td>2.10%</td><td>11.5</td></tr><tr><td>Greedy</td><td>H</td><td>688.31</td><td>6.39%</td><td>0.02</td><td>2761.06</td><td>7.64%</td><td>0.29</td></tr><tr><td>MFA</td><td>H</td><td>704.03</td><td>4.26%</td><td>0.15</td><td>2833.86</td><td>5.20%</td><td>0.66</td></tr><tr><td>Erdos</td><td>UL</td><td>693.45</td><td>5.69%</td><td>0.07</td><td>2870.34</td><td>3.98%</td><td>0.25</td></tr><tr><td>Anneal</td><td>UL</td><td>696.73</td><td>5.25%</td><td>0.07</td><td>2863.23</td><td>4.22%</td><td>0.24</td></tr><tr><td>GFlow</td><td>UL</td><td>704.3</td><td>4.22%</td><td>0.27</td><td>2864.61</td><td>4.17%</td><td>1.95</td></tr><tr><td>RUN-CSP</td><td>UL</td><td>726.96</td><td>1.14%</td><td>0.78</td><td>2925.8</td><td>2.12%</td><td>10.8</td></tr><tr><td>ANYCSP</td><td>UL</td><td>735.12</td><td>0.03%</td><td>2.5</td><td>2988.6</td><td>0.02%</td><td>7.37</td></tr><tr><td>DiffUCO</td><td>UL</td><td>733.5</td><td>0.25%</td><td>2.77</td><td>2981.1</td><td>0.27%</td><td>6.6</td></tr><tr><td>X2GNN(16x8)</td><td>UL</td><td>734.21</td><td>0.15%</td><td>0.08</td><td>2985.2</td><td>0.14%</td><td>0.2</td></tr><tr><td>X2GNN(64x8)</td><td>UL</td><td>734.92</td><td>0.05%</td><td>0.15</td><td>2987.7</td><td>0.05%</td><td>0.48</td></tr><tr><td>X2GNN(256x8)</td><td>UL</td><td>735.17</td><td>0.02%</td><td>1.2</td><td>2988.7</td><td>0.02%</td><td>1.95</td></tr><tr><td>X2GNN(256x32)</td><td>UL</td><td>735.26</td><td>0.01%</td><td>4.1</td><td>2989.3</td><td>0%</td><td>7.3</td></tr><tr><td>X2GNN (BA250)(256x32)</td><td>UL</td><td>735.26</td><td>0.01%</td><td>4.1</td><td>2985.5</td><td>0.13%</td><td>5.4</td></tr></table>
+
+These results indicate that on 3 CO problems,  $X^{2}$  GNN outperforms neural baselines, is competitive with specialized metaheuristics like KaMIS, and improves over general solvers like Gurobi.
+
+# 5.4 ABLATION
+
+In this section, we analyze the impact of  $K$ -coupled solutions for different values of  $K$ . We also measure the impact of stochastic refinement, two-stage training, and encouraging diversity.
+
+Table 5: Effects of the parameter  $K$  on  $K$ -coupled solutions, showing  $K = 2$  gives the lowest drop.  
+
+<table><tr><td>Problem</td><td>K=1</td><td>K=2</td><td>K=4</td><td>K=8</td></tr><tr><td>MIS</td><td>4.01%</td><td>0.43%</td><td>3.81%</td><td>9.47%</td></tr><tr><td>MC</td><td>0.14%</td><td>0.01%</td><td>0.21%</td><td>1.79%</td></tr><tr><td>MCut</td><td>0.030%</td><td>0.015%</td><td>0.021%</td><td>0.157%</td></tr></table>
+
+Table 5 demonstrates that  $K = 2$  is the optimal choice for each problem. Its impact is particularly significant for MIS, while more subtle for MC and MCut.
+
+We evaluate the impact of ablating aspects of  $X^{2}$  GNN by comparing to our standard version which achieves a drop value of  $0.43\%$  on MIS. Using deterministic refinement instead of stochastic refinement significantly increases the drop value to  $4.97\%$ . Training the full framework in a single stage increased the drop value to  $1.44\%$ . Similarly, ignoring the diversity loss raises the drop value to  $1.81\%$ . These findings highlight the cumulative benefits of our proposed techniques. The combination of  $K = 2$  coupled solutions, stochastic refinement, two-stage training, and diversity loss is crucial for the superior performance of  $X^{2}$  GNN.
+
+# 5.5 NEURAL SEARCH DYNAMICS
+
+For a fixed budget,  $X^{2}$  GNN can controllably balance exploration and exploitation by trading off the number of solution couples generated at each iteration  $C$ , with the number of iterations  $T$  taken.
+
+MC and MIS problems benefit from different exploration-exploitation trade-offs due to their distinct feasible region structures: In MC, selecting a node restricts subsequent additions to only its neighbors, causing the search space to contract rapidly with each decision. Early choices have high impact, potentially eliminating optimal solutions if poor selections are made. Consequently, exploring diverse starting points (higher  $C$ ) becomes crucial. In MIS, selecting a node only eliminates its neighbors
+
+from consideration, typically leaving a substantial portion of nodes available. This more gradual search space reduction allows solutions to recover from suboptimal early choices through additional refinement iterations.
+
+As shown in Figure 3, our experiments confirm these insights. Figure 3a demonstrates that MC benefits from prioritizing exploration (higher C), with  $C = 64$  providing optimal performance. Conversely, Figure 3b shows that MIS performs better with emphasis on exploitation (higher T, lower C), with  $C = 4$  yielding the best results.
+
+These findings highlight that while all combinatorial optimization problems require both strategies, their relative importance varies based on how solution construction progressively constrains the search space.
+
+![](images/ae568910326bc920a88d2f18d58844d9013f3a7f5e94008f5a2deb246592fea3.jpg)  
+(a)
+
+![](images/815dbf4f64c3adeb2b9a33ae8e8627b2d9ab95ecf3cc1d24f1350dc9365cd3bc.jpg)  
+(b)  
+Figure 3: Search dynamics on RB250: Each line corresponds to one setting of  $C$ , where higher values increase exploration. We present the drop in solution quality (lower is better) over various computational budgets. For MC, higher values of  $C$  yield better performance, suggesting the search benefits from broader exploration of the solution space. Conversely, for MIS, lower values of  $C$  lead to better solutions, indicating the search benefits from deeper exploitation of promising regions.
+
+# 6 CONCLUSION
+
+In this work, we introduce Explore-and-Exploit GNN ( $X^{2}$ GNN), a novel unsupervised neural framework that addresses a key challenge in learning-based combinatorial optimization (CO). Unlike most existing approaches that focus on constructing a limited number of solutions,  $X^{2}$ GNN effectively explores the vast search space of NP-hard CO problems through two key mechanisms:
+
+(i) Exploration:  $X^{2}$  GNN generates multiple solutions simultaneously, and promotes solution diversity.  
+(ii) Exploitation:  $X^{2}$  GNN employs neural stochastic iterative refinement, using sampled partial solutions to guide the search toward promising regions and escape local optima.
+
+Our experiments on three canonical NP-Hard CO problems - Maximum Clique (MC), Maximum Independent Set (MIS), and Maximum Cut (MCut) - demonstrate that  $X^{2}$  GNN significantly outperforms state-of-the-art learning-based approaches. Notably, for large MC problems,  $X^{2}$  GNN consistently generates solutions within 1.2% of optimality, while other learning-based methods struggle to reach within 22% of optimal. Moreover,  $X^{2}$  GNN exhibits exceptional generalization capabilities, outperforming existing methods even when trained on smaller or out-of-distribution graphs. The iterative nature of  $X^{2}$  GNN allows users to trade off runtime and solution quality, as the model can be applied indefinitely to refine solutions. This feature, combined with its strong performance and generalization performance, positions  $X^{2}$  GNN as a competitive framework with promising directions for future research in learning-based heuristics for combinatorial optimization. By balancing exploration and exploitation,  $X^{2}$  GNN offers a more effective and adaptable approach to neural combinatorial optimization, addressing the limitations of existing methods and paving the way for more robust solutions to complex CO problems across various domains.
+
+# ACKNOWLEDGMENTS
+
+This project is partially supported by Schmidt Sciences programs, an AI2050 Senior Fellowship and two Eric and Wendy Schmidt AI in Science Postdoctoral Fellowships; the National Science Foundation (NSF); the National Institute of Food and Agriculture (USDA/NIFA); the Air Force Office of Scientific Research (AFOSR). Additionally, Utku Umur Acikalin is supported by the Turkish Ministry of National Education.
+
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+
+# A EXPERIMENTAL DETAILS
+
+# A.1 NETWORK ARCHITECTURE
+
+We use a GNN architecture where Graph Isomorphism Network (GIN) (Xu et al. (2019)) and Graph Attention Network (GAT) (Velickovic et al. (2018); Brody et al. (2022)) layers are used in an interleaved manner. We use  $n_L$  GIN and GAT layers. The initial node features (node probabilities) are transformed to a higher dimension by a linear layer before they are passed to the first GIN layer.
+
+The GIN layers work on the original edges and GAT layers work on the crossed edges. GIN layers are composed of two-layer MLPs with RELU activations after the MLP layers. We apply Layer Normalization after each MLP layer in GINs and after each GAT layer. We add skip connections to both GIN and GAT layers. Finally, the output of each GAT layer is concatenated and fed into a final two-layer MLP. This architecture is executed recurrently starting from the last hidden representation from MLP  $n_R$  times. The final output is transformed into logits and the probability that each node belongs to S is computed via Softmax. For all problems, the architecture hidden representation size is 64.
+
+# A.2 HYPERPARAMETERS
+
+All the hyperparameters on each dataset is given in Table 6.  $n_{L}$  and  $n_{R}$  represents the number of layers and number of recurrent steps.  $K$  and  $C$  means  $C$ $K$ -Coupled solutions are generated during training. On all settings, we select  $K = 2$  and  $C = 4$  during training.  $K$  and  $C$  values used during inference are shown right next to the model.  $\lambda_{1}$  and  $\lambda_{2}$  are the weights of the constraint and diversity loss, respectively.
+
+<table><tr><td>Dataset</td><td>ep</td><td>lr</td><td>bs</td><td>nL</td><td>nR</td><td>h</td><td>K</td><td>C</td><td>φ</td><td>λ1</td><td>λ2</td></tr><tr><td>RB-250 MIS</td><td>50</td><td>0.001</td><td>64</td><td>4</td><td>2</td><td>64</td><td>2</td><td>4</td><td>0.5</td><td>1</td><td>0.75</td></tr><tr><td>RB-1000 MIS</td><td>50</td><td>0.001</td><td>64</td><td>4</td><td>4</td><td>64</td><td>2</td><td>4</td><td>0.5</td><td>1</td><td>0.75</td></tr><tr><td>ER-750 MIS</td><td>50</td><td>0.001</td><td>64</td><td>4</td><td>4</td><td>64</td><td>2</td><td>4</td><td>0.5</td><td>1</td><td>0.75</td></tr><tr><td>RB-250 MC</td><td>50</td><td>0.001</td><td>64</td><td>4</td><td>2</td><td>64</td><td>2</td><td>4</td><td>0.8</td><td>1</td><td>0.75</td></tr><tr><td>BA-250 MCut</td><td>50</td><td>0.001</td><td>64</td><td>4</td><td>2</td><td>64</td><td>2</td><td>4</td><td>0.8</td><td>-</td><td>0.75</td></tr><tr><td>BA-1000 MCut</td><td>50</td><td>0.001</td><td>64</td><td>4</td><td>4</td><td>64</td><td>2</td><td>4</td><td>0.8</td><td>-</td><td>0.75</td></tr></table>
+
+Table 6: Hyperparameters used in training.
+
+# A.3 COMPUTATIONAL RESOURCES AND TIME MEASUREMENTS
+
+All experiments are conducted on a machine with 2 Intel Xeon 6348 processors with 28 cores, 1TB of memory, and 4 A100 80GB GPUs. All traditional heuristics are run using a single core and Gurobi is run with 4 cores, allowing 40GB of memory for each instance.
+
+We only use 1 A100 during training or inference for all datasets. The training time for MIS is roughly 1 hour and 9 hours for RB250 and RB1000 datasets, respectively. The training time for MCut is roughly 1 hour and 3 hours for BA250 and BA1000 datasets, respectively. The training time for MC is roughly 2 hours for the RB250 dataset.
+
+For baselines, we run each method in our machine and report the time we obtain on our machine to have a fair comparison. However, we use the original size when available on the same dataset split.
+
+# B ADDITIONAL EXPERIMENTS
+
+# B.1 ADDITIONAL EXPERIMENTS FOR MAXIMUM CLIQUE
+
+For MC, we consider a challenging dataset from DIMACS implementation challenges related Maximum Clique (DIMACS). This dataset contains 37 challenging graphs. We use the  $X^{2}$  GNN model trained on the RB-1000 dataset as well as its fine-tuned version. We report the performance of  $X^{2}$  GNN 256x32 with and without fine-tuning as well as the best traditional heuristic KaMIS.
+
+Table 7: Table shows the clique size and the running time for each instance for  $X^{2}$  GNN and KaMIS, considering that  $X^{2}$  GNN FT fine-tunes  $X^{2}$  GNN on these instances, and  $X^{2}$  GNN is trained on only RB1000 instances. The clique sizes matching the best-known solutions are shown in bold.  
+
+<table><tr><td rowspan="2">Instance</td><td rowspan="2">n</td><td rowspan="2">m</td><td rowspan="2">Best known</td><td colspan="3">Size</td><td>Kamis</td><td colspan="3">Runnin Time</td></tr><tr><td>KaMIS</td><td>\(X^2\)GNN FT</td><td>\(X^2\)GNN</td><td>KaMIS</td><td>\(X^2\)GNN FT</td><td>\(X^2\)GNN</td><td></td></tr><tr><td>brock200_2</td><td>200</td><td>9,876</td><td>12</td><td>12</td><td>11</td><td>11</td><td>7.14</td><td>4.95</td><td>4.95</td><td></td></tr><tr><td>brock200_4</td><td>200</td><td>13,089</td><td>17</td><td>16</td><td>16</td><td>16</td><td>5.45</td><td>3.7</td><td>3.67</td><td></td></tr><tr><td>brock400_2</td><td>400</td><td>59,786</td><td>29</td><td>24</td><td>24</td><td>24</td><td>9.17</td><td>4.32</td><td>4.35</td><td></td></tr><tr><td>brock400_4</td><td>400</td><td>59,765</td><td>33</td><td>25</td><td>25</td><td>25</td><td>8.77</td><td>4.28</td><td>4.29</td><td></td></tr><tr><td>brock800_2</td><td>800</td><td>208,166</td><td>24</td><td>20</td><td>20</td><td>20</td><td>33.91</td><td>10.84</td><td>10.79</td><td></td></tr><tr><td>brock800_4</td><td>800</td><td>207,643</td><td>26</td><td>20</td><td>21</td><td>21</td><td>36.02</td><td>10.87</td><td>10.8</td><td></td></tr><tr><td>C1000.9</td><td>1,000</td><td>450,079</td><td>68</td><td>66</td><td>66</td><td>64</td><td>19.06</td><td>6.68</td><td>6.59</td><td></td></tr><tr><td>C125.9</td><td>125</td><td>6,963</td><td>34</td><td>34</td><td>34</td><td>34</td><td>5.26</td><td>3.72</td><td>3.6</td><td></td></tr><tr><td>C2000.5</td><td>2,000</td><td>999,836</td><td>16</td><td>15</td><td>14</td><td>0</td><td>188.80</td><td>72.63</td><td>72.4</td><td></td></tr><tr><td>C2000.9</td><td>2,000</td><td>1,799,532</td><td>80</td><td>75</td><td>75</td><td>73</td><td>45.28</td><td>17.67</td><td>17.59</td><td></td></tr><tr><td>C250.9</td><td>250</td><td>27,984</td><td>44</td><td>44</td><td>44</td><td>44</td><td>5.15</td><td>3.72</td><td>3.65</td><td></td></tr><tr><td>C4000.5</td><td>4,000</td><td>4,000,268</td><td>18</td><td>16</td><td>13</td><td>0</td><td>596.32</td><td>367.1</td><td>369.75</td><td></td></tr><tr><td>C500.9</td><td>500</td><td>112,332</td><td>57</td><td>56</td><td>56</td><td>55</td><td>6.88</td><td>3.8</td><td>3.74</td><td></td></tr><tr><td>DSJC1000_5</td><td>1,000</td><td>499,652</td><td>15</td><td>15</td><td>12</td><td>12</td><td>70.18</td><td>20.1</td><td>19.96</td><td></td></tr><tr><td>DSJC500_5</td><td>500</td><td>125,248</td><td>13</td><td>13</td><td>13</td><td>13</td><td>26.61</td><td>7.29</td><td>7.24</td><td></td></tr><tr><td>gen200_p0.9_44</td><td>200</td><td>17,910</td><td>44</td><td>44</td><td>44</td><td>44</td><td>5.20</td><td>3.71</td><td>3.59</td><td></td></tr><tr><td>gen200_p0.9_55</td><td>200</td><td>17,910</td><td>55</td><td>55</td><td>55</td><td>55</td><td>5.20</td><td>3.74</td><td>3.6</td><td></td></tr><tr><td>gen400_p0.9_55</td><td>400</td><td>71,820</td><td>55</td><td>53</td><td>55</td><td>55</td><td>5.35</td><td>3.71</td><td>3.58</td><td></td></tr><tr><td>gen400_p0.9_65</td><td>400</td><td>71,820</td><td>65</td><td>65</td><td>65</td><td>65</td><td>5.22</td><td>3.71</td><td>3.58</td><td></td></tr><tr><td>gen400_p0.9_75</td><td>400</td><td>71,820</td><td>75</td><td>75</td><td>75</td><td>75</td><td>5.65</td><td>3.75</td><td>3.58</td><td></td></tr><tr><td>hamming10-4</td><td>1,024</td><td>434,176</td><td>40</td><td>38</td><td>40</td><td>38</td><td>12.24</td><td>9.53</td><td>9.4</td><td></td></tr><tr><td>hamming8-4</td><td>256</td><td>20,864</td><td>16</td><td>16</td><td>16</td><td>16</td><td>5.76</td><td>3.66</td><td>3.55</td><td></td></tr><tr><td>keller4</td><td>171</td><td>9,435</td><td>11</td><td>11</td><td>11</td><td>11</td><td>5.40</td><td>3.71</td><td>3.58</td><td></td></tr><tr><td>keller5</td><td>776</td><td>225,990</td><td>27</td><td>26</td><td>23</td><td>27</td><td>10.97</td><td>8.32</td><td>8.26</td><td></td></tr><tr><td>keller6</td><td>3,361</td><td>4,619,898</td><td>59</td><td>55</td><td>39</td><td>43</td><td>61.87</td><td>75.01</td><td>75.2</td><td></td></tr><tr><td>MANN_a27</td><td>378</td><td>70,551</td><td>126</td><td>126</td><td>126</td><td>92</td><td>5.05</td><td>3.69</td><td>3.57</td><td></td></tr><tr><td>MANN_a45</td><td>1,035</td><td>533,115</td><td>345</td><td>344</td><td>343</td><td>244</td><td>5.09</td><td>3.7</td><td>3.59</td><td></td></tr><tr><td>MANN_a81</td><td>3,321</td><td>5,506,380</td><td>1100</td><td>1100</td><td>1097</td><td>664</td><td>5.06</td><td>4.25</td><td>4.14</td><td></td></tr><tr><td>p_hat1500-1</td><td>1,500</td><td>284,923</td><td>12</td><td>11</td><td>10</td><td>0</td><td>88.97</td><td>61.61</td><td>60.96</td><td></td></tr><tr><td>p_hat1500-2</td><td>1,500</td><td>568,960</td><td>65</td><td>65</td><td>65</td><td>62</td><td>85.44</td><td>39.96</td><td>40.47</td><td></td></tr><tr><td>p_hat1500-3</td><td>1,500</td><td>847,244</td><td>94</td><td>94</td><td>94</td><td>94</td><td>44.08</td><td>22.49</td><td>22.45</td><td></td></tr><tr><td>p_hat300-1</td><td>300</td><td>10,933</td><td>8</td><td>8</td><td>8</td><td>8</td><td>18.69</td><td>5.27</td><td>5.19</td><td></td></tr><tr><td>p_hat300-2</td><td>300</td><td>21,928</td><td>25</td><td>25</td><td>25</td><td>25</td><td>12.02</td><td>4.48</td><td>4.38</td><td></td></tr><tr><td>p_hat300-3</td><td>300</td><td>33,390</td><td>36</td><td>36</td><td>36</td><td>36</td><td>7.06</td><td>3.75</td><td>3.57</td><td></td></tr><tr><td>p_hat700-1</td><td>700</td><td>60,999</td><td>11</td><td>11</td><td>9</td><td>2</td><td>42.16</td><td>15.36</td><td>15.27</td><td></td></tr><tr><td>p_hat700-2</td><td>700</td><td>121,728</td><td>44</td><td>44</td><td>44</td><td>44</td><td>25.06</td><td>11.49</td><td>11.42</td><td></td></tr><tr><td>p_hat700-3</td><td>700</td><td>183,010</td><td>62</td><td>62</td><td>62</td><td>62</td><td>13.62</td><td>7.38</td><td>7.31</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td>3.87%</td><td>6.68%</td><td>17.41%</td><td>15.60</td><td>8.46</td><td>8.33</td><td></td></tr></table>
+
+Table 7 shows that  $X^{2}$ GNN is able to generalize and generate optimal or near-optimal solutions on many instances even though it is trained on RB1000 instances, a set of much sparser instances. However, it does fail to generate good solutions on a few instances. With fine-tuning, solution quality improves quite a lot both for these failing cases and also in general, leading to an average gap of  $6.68\%$  from  $17.41\%$ . Even the most successful traditional heuristic KaMIS achieves an average gap of  $3.87\%$ . Considering this dataset is designed to be a challenging dataset, learning a general rule based on RB1000 instances that can solve many problems, and achieving a gap of  $6.68\%$  after fine-tuning is noteworthy.
+
+# B.2 ADDITIONAL EXPERIMENTS FOR MAXIMUM INDEPENDENT SET
+
+For MIS, we evaluate  $X^{2}$  GNN's performance on an additional dataset that comprises instances from Coding Theory applications, specifically error correction codes, with graph sizes ranging from 64 to 4,096 nodes. For this dataset, we utilize the  $X^{2}$  GNN model trained on the RB1000 dataset. We compare against KaMIS, the state-of-the-art MIS solver.
+
+On the Coding Theory dataset (Table 8),  $X^{2}$ GNN finds the best-known solution in 20 of 32 instances, compared to KaMIS's 28 instances.  $X^{2}$ GNN achieves an average gap of  $3.37\%$  from best-known solutions (versus  $0.34\%$  for KaMIS). Excluding one outlier instance where  $X^{2}$ GNN finds a solution less than half the best-known value,  $X^{2}$ GNN's average gap improves to  $1.74\%$ . This performance is notable given no domain-specific tuning was performed.
+
+To demonstrate  $X^{2}$  GNN's adaptability to weighted problems, we extend it to weighted maximum independent set problems with minimal modifications: adding a weight embedding layer combined
+
+Table 8: The table shows the size of the independent sets found by  $X^2$  GNN and KaMIS and the running time in seconds for each instance. The last row shows the average gap in percentages from the best-known solution. The instances where a method found the best-known solutions are shown in bold.  
+
+<table><tr><td>Graph</td><td>Best Known</td><td>X2GNN 256x32</td><td>KaMIS</td><td>X2GNN 256x32 Time</td><td>KaMIS Time</td></tr><tr><td>1dc.64</td><td>10</td><td>10</td><td>10</td><td>5.06</td><td>5.14</td></tr><tr><td>1dc.128</td><td>16</td><td>16</td><td>16</td><td>3.87</td><td>4.04</td></tr><tr><td>1dc.256</td><td>30</td><td>30</td><td>30</td><td>4.37</td><td>4.50</td></tr><tr><td>1dc.512</td><td>52</td><td>52</td><td>52</td><td>6.33</td><td>6.49</td></tr><tr><td>1dc.1024</td><td>94</td><td>94</td><td>93</td><td>11.29</td><td>11.40</td></tr><tr><td>1dc.2048</td><td>172</td><td>172</td><td>172</td><td>24.25</td><td>24.40</td></tr><tr><td>1et.64</td><td>18</td><td>18</td><td>18</td><td>3.81</td><td>1.31</td></tr><tr><td>1et.128</td><td>28</td><td>28</td><td>28</td><td>3.81</td><td>3.85</td></tr><tr><td>1et.256</td><td>50</td><td>50</td><td>50</td><td>3.82</td><td>3.90</td></tr><tr><td>1et.512</td><td>100</td><td>98</td><td>100</td><td>4.68</td><td>4.74</td></tr><tr><td>1et.1024</td><td>171</td><td>165</td><td>171</td><td>7.12</td><td>7.23</td></tr><tr><td>1et.2048</td><td>316</td><td>300</td><td>316</td><td>13.95</td><td>14.02</td></tr><tr><td>1tc.8</td><td>4</td><td>4</td><td>4</td><td>3.71</td><td>0.48</td></tr><tr><td>1tc.16</td><td>8</td><td>8</td><td>8</td><td>3.76</td><td>0.81</td></tr><tr><td>1tc.32</td><td>12</td><td>12</td><td>12</td><td>3.76</td><td>0.30</td></tr><tr><td>1tc.64</td><td>20</td><td>20</td><td>20</td><td>3.79</td><td>1.10</td></tr><tr><td>1tc.128</td><td>38</td><td>38</td><td>38</td><td>3.75</td><td>1.15</td></tr><tr><td>1tc.256</td><td>64</td><td>63</td><td>63</td><td>3.78</td><td>3.88</td></tr><tr><td>1tc.512</td><td>110</td><td>109</td><td>110</td><td>4.39</td><td>4.48</td></tr><tr><td>1tc.1024</td><td>196</td><td>189</td><td>196</td><td>6.64</td><td>6.73</td></tr><tr><td>1tc.2048</td><td>352</td><td>332</td><td>352</td><td>12.91</td><td>13.00</td></tr><tr><td>1zc.128</td><td>18</td><td>18</td><td>18</td><td>3.77</td><td>3.87</td></tr><tr><td>1zc.256</td><td>36</td><td>36</td><td>36</td><td>3.99</td><td>4.10</td></tr><tr><td>1zc.512</td><td>62</td><td>62</td><td>62</td><td>5.46</td><td>5.55</td></tr><tr><td>1zc.1024</td><td>112</td><td>109</td><td>112</td><td>9.13</td><td>9.32</td></tr><tr><td>1zc.2048</td><td>198</td><td>181</td><td>195</td><td>18.78</td><td>18.89</td></tr><tr><td>1zc.4096</td><td>379</td><td>326</td><td>353</td><td>40.19</td><td>40.41</td></tr><tr><td>2dc.128</td><td>5</td><td>5</td><td>5</td><td>4.52</td><td>1.46</td></tr><tr><td>2dc.256</td><td>7</td><td>7</td><td>7</td><td>8.22</td><td>10.06</td></tr><tr><td>2dc.512</td><td>11</td><td>11</td><td>11</td><td>19.07</td><td>22.31</td></tr><tr><td>2dc.1024</td><td>16</td><td>15</td><td>16</td><td>52.72</td><td>57.81</td></tr><tr><td>2dc.2048</td><td>24</td><td>11</td><td>24</td><td>152.61</td><td>162.73</td></tr><tr><td colspan="2">Average Gap</td><td>3.37%</td><td>0.34%</td><td></td><td></td></tr></table>
+
+with node representations via summation, and incorporating weights into the loss function calculation. Using the RB250 dataset with uniform random integer weights between 1 and 5, we compare against optimal solutions from Gurobi. Table 9 shows  $X^2$ GNN maintains high solution quality with a  $0.8\%$  optimality gap.
+
+Table 9: Results for Weighted Maximum Independent Set on small RB graphs, presenting the mean independent set size, drop in quality compared to the optimal, and run time in seconds.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Type</td><td colspan="3">RB250</td></tr><tr><td>Size ↑</td><td>Drop ↓</td><td>Time ↓</td></tr><tr><td>Gurobi</td><td>OR</td><td>82.94</td><td>0%</td><td>0.21</td></tr><tr><td>X²GNN(16x8)</td><td>UL</td><td>78.52</td><td>5.33%</td><td>0.03</td></tr><tr><td>X²GNN(64x8)</td><td>UL</td><td>80.26</td><td>3.23%</td><td>0.09</td></tr><tr><td>X²GNN(256x8)</td><td>UL</td><td>81.31</td><td>1.97%</td><td>0.35</td></tr><tr><td>X²GNN(256x32)</td><td>UL</td><td>81.83</td><td>1.34%</td><td>1.25</td></tr><tr><td>X²GNN(1024x32)</td><td>UL</td><td>82.28</td><td>0.8%</td><td>4.92</td></tr></table>
+
+In an ablation study, we replaced the GAT layer with a simple MLP for processing cross-edges. Table 10 shows this variant still outperforms other learning-based approaches but underperforms compared to the GAT version, indicating GAT's superior capability in aggregating information across different solutions.
+
+Table 10: The results for replacing GAT layers with a simple MLP layer.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Type</td><td colspan="3">RB250</td></tr><tr><td>Size ↑</td><td>Drop ↓</td><td>Time ↓</td></tr><tr><td>KaMIS (30min)</td><td>OR</td><td>20.106</td><td>0%</td><td>3.92</td></tr><tr><td>Gurobi (30min)</td><td>OR</td><td>20.106</td><td>0%</td><td>0.31</td></tr><tr><td>KaMIS (Quality)</td><td>OR</td><td>20.106</td><td>0%</td><td>3.92</td></tr><tr><td>Gurobi (Quality)</td><td>OR</td><td>20.106</td><td>0%</td><td>0.42</td></tr><tr><td>KaMIS (Fast)</td><td>OR</td><td>20.032</td><td>0.37%</td><td>1.16</td></tr><tr><td>Gurobi (Fast)</td><td>OR</td><td>19.16</td><td>4.71%</td><td>0.1</td></tr><tr><td>PPO</td><td>UL</td><td>19.01</td><td>5.45%</td><td>0.15</td></tr><tr><td>GFlow</td><td>UL</td><td>19.18</td><td>4.61%</td><td>0.05</td></tr><tr><td>DIFUSCO</td><td>SL</td><td>17.68</td><td>12.07%</td><td>0.87</td></tr><tr><td>T2T</td><td>SL</td><td>18.35</td><td>8.73%</td><td>2.32</td></tr><tr><td>DiffUCO</td><td>UL</td><td>19.24</td><td>4.31%</td><td>0.42</td></tr><tr><td>X²GNN-GAT(16x8)</td><td>UL</td><td>19.51</td><td>2.96%</td><td>0.034</td></tr><tr><td>X²GNN-GAT(64x8)</td><td>UL</td><td>19.82</td><td>1.42%</td><td>0.128</td></tr><tr><td>X²GNN-GAT(256x8)</td><td>UL</td><td>19.98</td><td>0.63%</td><td>0.5</td></tr><tr><td>X²GNN-GAT(256x32)</td><td>UL</td><td>20.072</td><td>0.17%</td><td>1.94</td></tr><tr><td>X²GNN-GAT(1024x32)</td><td>UL</td><td>20.098</td><td>0.04%</td><td>7.11</td></tr><tr><td>X²GNN-MLP(16x8)</td><td>UL</td><td>18.95</td><td>5.75%</td><td>0.027</td></tr><tr><td>X²GNN-MLP(64x8)</td><td>UL</td><td>19.41</td><td>3.46%</td><td>0.096</td></tr><tr><td>X²GNN-MLP(256x8)</td><td>UL</td><td>19.698</td><td>2.03%</td><td>0.38</td></tr><tr><td>X²GNN-MLP(256x32)</td><td>UL</td><td>19.886</td><td>1.09%</td><td>1.35</td></tr></table>
+
+# B.3 ADDITIONAL EXPERIMENTS FOR MCUT
+
+For the Maximum Cut (MCut) problem, we evaluate  $X^{2}$  GNN's out-of-distribution performance on two additional benchmark sets. The first dataset, introduced in Mirka and Williamson (2023), comprises diverse graphs from the SNAP Networks repository (referred to as SNAP dataset). This dataset is particularly suitable for assessing generalization capabilities due to its heterogeneous graph distributions. The second dataset, known as Gset  $^{2}$ , is a well-established benchmark collection traditionally used to evaluate MCut algorithms.
+
+To train  $X^2$  GNN, we generate 4,000 Erdős-Rényi (ER) graphs with sizes uniformly sampled from between 200 and 500 and edge probabilities from [0.1, 0.75]. We then evaluate the trained model on both SNAP and Gset datasets.
+
+Table 11 presents results for the SNAP dataset, comparing  $X^{2}$ GNN against ANYCSP (trained on the same dataset), Tabu Search (TS), Semidefinite Programming (SDP) relaxation, and BMZ heuristic from Mirka and Williamson (2023).  $X^{2}$ GNN discovers the best solutions among all compared methods for all but two instances, demonstrating superior performance over both ANYCSP and traditional heuristics.
+
+For the Gset evaluation, we use on all unweighted GSET instances, demonstrating that  $X^{2}$ GNN can generalize to larger instances. Table 12 shows the cut size achieved by the traditional heuristics BLS, DSDP, KHLWG and learning based heuristics X2GNN and ANYCSP for each instance. To aggregate the results, we first compute the virtual best solution by taking the maximum value found by any of the compared algorithms. Then, for each instance we compute the average gap from the best as 1 - solution/best and take the mean to calculate the average gap from the virtual best solution.
+
+Table 11: Table shows the cut sizes for instance and method. The cut sizes matching the best are shown in bold.  
+
+<table><tr><td>Graph</td><td>n</td><td>m</td><td>X2GNN (256x32)</td><td>ANYCSP</td><td>TS</td><td>BMZ</td><td>SDP</td></tr><tr><td>ENZYMES8</td><td>88</td><td>133</td><td>126</td><td>126</td><td>126</td><td>126</td><td>126</td></tr><tr><td>johnson16-2-4</td><td>120</td><td>5460</td><td>3036</td><td>2941</td><td>3036</td><td>3036</td><td>3036</td></tr><tr><td>hamming6-2</td><td>64</td><td>1824</td><td>992</td><td>946</td><td>992</td><td>992</td><td>992</td></tr><tr><td>ia-infect-hyper</td><td>113</td><td>2196</td><td>1279</td><td>1208</td><td>1279</td><td>1278</td><td>1275</td></tr><tr><td>soc-dolphins</td><td>62</td><td>159</td><td>122</td><td>122</td><td>122</td><td>122</td><td>122</td></tr><tr><td>email-enron-only</td><td>143</td><td>623</td><td>427</td><td>427</td><td>427</td><td>426</td><td>422</td></tr><tr><td>dwt_209</td><td>209</td><td>976</td><td>557</td><td>557</td><td>557</td><td>557</td><td>551</td></tr><tr><td>ca-netscience</td><td>379</td><td>914</td><td>620</td><td>580</td><td>627</td><td>634</td><td>634</td></tr><tr><td>ia-infect-dublin</td><td>410</td><td>2765</td><td>1771</td><td>1709</td><td>1758</td><td>1767</td><td>1750</td></tr><tr><td>road-chesapeake</td><td>39</td><td>170</td><td>126</td><td>126</td><td>126</td><td>126</td><td>125</td></tr><tr><td>Erdos991</td><td>492</td><td>1417</td><td>1036</td><td>1036</td><td>1012</td><td>1031</td><td>1019</td></tr><tr><td>dwt_503</td><td>503</td><td>3265</td><td>1938</td><td>1938</td><td>1937</td><td>1931</td><td>1934</td></tr><tr><td>p-hat700-1</td><td>700</td><td>60999</td><td>33413</td><td>31856</td><td>33426</td><td>33440</td><td>33450</td></tr><tr><td>email-univ</td><td>1133</td><td>5451</td><td>3775</td><td>3764</td><td>3657</td><td>3765</td><td>3736</td></tr></table>
+
+The results indicate that X2GNN achieves an average gap of  $0.12\%$  from the best solutions among the compared ones under the 64x256 setting, which has a maximum running time of 11 seconds on the largest instance with  $10^{4}$  nodes. ANYCSP under the 64x256 setting has a similar running time and achieves an average gap of  $0.69\%$  from the best solution. When we increase the number of refinement iterations by one order of magnitude, the average gaps achieved by X2GNN and ANYCSP are  $0.12\%$  and  $0.56\%$ , respectively. Despite having an order of magnitude longer runtime, ANYCSP achieves a higher average gap than X2GNN at even a short runtime. These results show that both methods are scalable and can generate high-quality solutions to larger and different distributions than those seen during training; however, X2GNN performs better than ANYCSP.
+
+Additionally, X2GNN generates solutions with a smaller gap on average than the second best traditional heuristic KHLWG, whereas ANYCSP consistently generates solutions with higher gap than KHLWG.
+
+Table 12: Results for Max Cut on Gset dataset, presenting the cut size for each instance. The last row shows the average gap in percentages from the best solution.  
+
+<table><tr><td>Graph</td><td>BLS</td><td>DSDP</td><td>KHLWG</td><td>X2GNN 64x256</td><td>ANYCSP 64x256</td><td>X2GNN 64x2560</td><td>ANYCSP 64x2560</td></tr><tr><td>G1</td><td>11624</td><td>-</td><td>11624</td><td>11583</td><td>11582</td><td>11598</td><td>11616</td></tr><tr><td>G14</td><td>3064</td><td>2922</td><td>3061</td><td>3054</td><td>3021</td><td>3058</td><td>3035</td></tr><tr><td>G15</td><td>3050</td><td>2938</td><td>3050</td><td>3043</td><td>3011</td><td>3036</td><td>3024</td></tr><tr><td>G16</td><td>3052</td><td>-</td><td>3052</td><td>3040</td><td>3011</td><td>3045</td><td>3024</td></tr><tr><td>G17</td><td>3047</td><td>-</td><td>3046</td><td>3037</td><td>3002</td><td>3039</td><td>3018</td></tr><tr><td>G2</td><td>11620</td><td>-</td><td>11620</td><td>11575</td><td>11585</td><td>11588</td><td>11601</td></tr><tr><td>G22</td><td>13359</td><td>12960</td><td>13359</td><td>13339</td><td>13263</td><td>13347</td><td>13285</td></tr><tr><td>G23</td><td>13344</td><td>13006</td><td>13342</td><td>13319</td><td>13247</td><td>13329</td><td>13309</td></tr><tr><td>G24</td><td>13377</td><td>12933</td><td>13337</td><td>13310</td><td>13273</td><td>13319</td><td>13276</td></tr><tr><td>G25</td><td>13340</td><td>-</td><td>13332</td><td>13311</td><td>13248</td><td>13327</td><td>13287</td></tr><tr><td>G26</td><td>13328</td><td>-</td><td>13328</td><td>13296</td><td>13236</td><td>13306</td><td>13272</td></tr><tr><td>G3</td><td>11622</td><td>-</td><td>11620</td><td>11583</td><td>11581</td><td>11591</td><td>11614</td></tr><tr><td>G35</td><td>7684</td><td>-</td><td>7678</td><td>7648</td><td>7543</td><td>7646</td><td>7575</td></tr><tr><td>G36</td><td>7678</td><td>-</td><td>7670</td><td>7646</td><td>7529</td><td>7654</td><td>7566</td></tr><tr><td>G37</td><td>7689</td><td>-</td><td>7682</td><td>7665</td><td>7546</td><td>7666</td><td>7593</td></tr><tr><td>G38</td><td>7687</td><td>-</td><td>7683</td><td>7649</td><td>7547</td><td>7658</td><td>7570</td></tr><tr><td>G4</td><td>11646</td><td>-</td><td>11646</td><td>11597</td><td>11612</td><td>11617</td><td>11633</td></tr><tr><td>G43</td><td>6660</td><td>-</td><td>6660</td><td>6655</td><td>6622</td><td>6659</td><td>6651</td></tr><tr><td>G44</td><td>6650</td><td>-</td><td>6639</td><td>6645</td><td>6622</td><td>6646</td><td>6631</td></tr><tr><td>G45</td><td>6654</td><td>-</td><td>6652</td><td>6640</td><td>6625</td><td>6652</td><td>6641</td></tr><tr><td>G46</td><td>6649</td><td>-</td><td>6649</td><td>6639</td><td>6622</td><td>6644</td><td>6637</td></tr><tr><td>G47</td><td>6657</td><td>-</td><td>6665</td><td>6651</td><td>6626</td><td>6654</td><td>6647</td></tr><tr><td>G48</td><td>6000</td><td>6000</td><td>6000</td><td>6000</td><td>6000</td><td>6000</td><td>6000</td></tr><tr><td>G49</td><td>6000</td><td>6000</td><td>6000</td><td>6000</td><td>6000</td><td>6000</td><td>6000</td></tr><tr><td>G5</td><td>11631</td><td>-</td><td>11631</td><td>11596</td><td>11583</td><td>11594</td><td>11623</td></tr><tr><td>G50</td><td>5880</td><td>5880</td><td>5880</td><td>5872</td><td>5846</td><td>5880</td><td>5876</td></tr><tr><td>G51</td><td>3848</td><td>-</td><td>3847</td><td>3838</td><td>3791</td><td>3837</td><td>3810</td></tr><tr><td>G52</td><td>3851</td><td>-</td><td>3849</td><td>3835</td><td>3792</td><td>3840</td><td>3805</td></tr><tr><td>G53</td><td>3850</td><td>-</td><td>3848</td><td>3841</td><td>3796</td><td>3844</td><td>3812</td></tr><tr><td>G54</td><td>3852</td><td>-</td><td>3851</td><td>3842</td><td>3791</td><td>3841</td><td>3802</td></tr><tr><td>G55</td><td>10294</td><td>9960</td><td>10236</td><td>10241</td><td>10066</td><td>10283</td><td>10141</td></tr><tr><td>G58</td><td>19263</td><td>-</td><td>19248</td><td>19167</td><td>18843</td><td>19186</td><td>18937</td></tr><tr><td>G60</td><td>14176</td><td>13610</td><td>14057</td><td>14121</td><td>13855</td><td>14146</td><td>13952</td></tr><tr><td>G63</td><td>26997</td><td>8017</td><td>26963</td><td>26863</td><td>26405</td><td>26905</td><td>26512</td></tr><tr><td>G70</td><td>9541</td><td>9456</td><td>9458</td><td>9492</td><td>9189</td><td>9525</td><td>9290</td></tr><tr><td>GeoMean</td><td>7726.5</td><td>-</td><td>7718.6</td><td>7703.5</td><td>7638.6</td><td>7710.3</td><td>7667.0</td></tr><tr><td>Mean Gap</td><td>0.01%</td><td>-</td><td>0.13%</td><td>0.12%</td><td>0.69%</td><td>0.11%</td><td>0.56%</td></tr></table>
\ No newline at end of file
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+# LEARNING TO GENERATE DIVERSE PEDESTRIAN MOVEMENTS FROM WEB VIDEOS WITH NOISY LABELS
+
+Zhizheng Liu, Joe Lin, Wayne Wu, Bolei Zhou
+
+Department of Computer Science, University of California, Los Angeles
+
+![](images/528bce62114c5beeb55d2138e75799ecc12d6e831bd293bcacff3db4db003221.jpg)  
+Figure 1: Pedestrian Movement Generation. Our method can generate diverse pedestrian movements in real-world (top row) and simulated (bottom row) urban environments.
+
+# ABSTRACT
+
+Understanding and modeling pedestrian movements in the real world is crucial for applications like motion forecasting and scene simulation. Many factors influence pedestrian movements, such as scene context, individual characteristics, and goals, which are often ignored by the existing human generation methods. Web videos contain natural pedestrian behavior and rich motion context, but annotating them with pre-trained predictors leads to noisy labels. In this work, we propose learning diverse pedestrian movements from web videos. We first curate a large-scale dataset called CityWalkers that captures diverse real-world pedestrian movements in urban scenes. Then, based on CityWalkers, we propose a generative model called PedGen for diverse pedestrian movement generation. PedGen introduces automatic label filtering to remove the low-quality labels and a mask embedding to train with partial labels. It also contains a novel context encoder that lifts the 2D scene context to 3D and can incorporate various context factors in generating realistic pedestrian movements in urban scenes. Experiments show that PedGen outperforms existing baseline methods for pedestrian movement generation by learning from noisy labels and incorporating the context factors. In addition, PedGen achieves zero-shot generalization in both real-world and simulated environments. The code, model, and data are available at https://genforce.github.io/PedGen/.
+
+# 1 INTRODUCTION
+
+In a bustling city, office workers rush through the crosswalk to nearby buildings, tourists wander around along storefronts looking at items on display, and friends sit and enjoy coffee on the outdoor patio. Pedestrians are essential participants in urban spaces. Their movements represent the social lives and interactions with the surrounding environments. Understanding and modeling pedestrian movements is critical to many applications. For example, city designers simulate pedestrian movements to optimize public areas and transportation systems (Mehta, 2008); forecasting the future pedestrian path is crucial for the safe deployment of autonomous vehicles (Lyssenko et al., 2021).
+
+Generating diverse and realistic pedestrian movements remains challenging. Multiple context factors affect pedestrian movements in the scenes. The first factor is the surrounding environment. As
+
+people constantly interact with the environment, it is important to model the scene context where the interaction happens. For example, objects like trash bins, plants, and other pedestrians in the scene can influence the walking behavior of a pedestrian (Daamen & Hoogendoorn, 2003), while the space types like crosswalks and waiting zones decide the overall movement pattern (Sime, 1995). The second factor is the individual characteristics. Studies have shown that walking speed and gait are influenced by age (Ostrosky et al., 1994), gender (Yamasaki et al., 1991), and body weight (Heglund & Taylor, 1988) and can also reflect the pedestrian's fitness level (Dridi, 2015). The last factor is the goal of the pedestrian, which decides the walking route in the scene (Hoogendoorn & Bovy, 2004).
+
+Existing human motion generation research mainly focuses on the breadth of activities (Mahmood et al., 2019), but few have studied generating natural real-world pedestrian movements in diverse scene contexts. For example, existing outdoor human motion datasets have limited scenes and human subjects with unnatural motion performed by actors with specific instructions (Kaufmann et al., 2023; Dai et al., 2023). Moreover, current motion generation methods (Guo et al., 2020; Tevet et al., 2022) lean toward generating complex motions from clean MoCap data, where many motion categories are rare in daily scenes. Few have considered learning the diverse motion contexts from noisy labels.
+
+Web videos, captured by many people walking and touring in different cities worldwide and shared on the YouTube website, contain diverse scene contexts and pedestrian movements in the most natural forms. However, labeling pedestrian movements in these web videos with pretrained predictors leads to inevitable label noise. How to harness the web data with large noise yet rich context becomes the pivot of modeling and generating diverse pedestrian movements. To this end, we first collect CityWalkers, a large-scale real-world dataset containing pedestrians in urban scenes, annotated with pseudo-labels by off-the-shelf 4D human motion estimation models. CityWalkers captures diverse real-world pedestrian movements regarding various moving speeds, gaits, headings, and local motions. Each movement is also paired with labels of context factors, such as the pedestrian's body shape, route destinations, and the environment's semantics and geometry.
+
+We then develop a new diffusion-based generative model PedGen for learning context-aware pedestrian movements with the noisy pseudo-labels from the CityWalkers dataset. PedGen has two key designs: 1) To mitigate the anomaly and incomplete labels from pseudo-labeling techniques, PedGen adopts a data iteration strategy to identify and remove low-quality labels from the dataset automatically and a motion mask embedding to train with partial labels; 2) To model the important context factors, PedGen considers the surrounding environment, the individual characteristics, and the goal points as input conditions to generate realistic and long-term pedestrian movements in urban scenes. As web videos only contain scene context labels in 2D, we propose a novel Context Encoder that can lift the environment context from 2D images into a 3D local scene representation with geometry and semantic information and also encode the other context factors to help generate realistic and long-term 3D pedestrian movements. We show some randomly sampled results of PedGen in Fig. 1.
+
+Experiment results on the CityWalkers validation set, the real-world Waymo open dataset (Sun et al., 2020) and CARLA simulator (Dosovitskiy et al., 2017) show that PedGen can predict more realistic and accurate future pedestrian movements than existing human motion generation methods and achieve better zero-shot generalization by generating high-quality context-aware movements. Additional experiments and ablation studies demonstrate the effectiveness of PedGen in addressing noisy labels and incorporating the key context factors. It enables the application of forecasting pedestrian movements in the real world and populating simulated environments with realistic pedestrians. We summarize our contributions as follows: 1) A new task of context-aware pedestrian movement generation from web videos with unique challenges in dealing with label noises and modeling various motion contexts. 2) A new large-scale real-world pedestrian movement dataset CityWalkers with pseudo-labels of diverse pedestrian movements and motion contexts. 3) The context-aware generative model PedGen that can learn from noisy pseudo-labels to generate diverse pedestrian movements.
+
+# 2 RELATED WORK
+
+Pedestrian Movement Analysis. Pedestrian behaviors have been extensively studied in transportation and social science. A hierarchical structure is defined for pedestrian behavior analysis (Hoogendoorn & Bovy, 2004; Feng et al., 2021) from the high-level strategic behavior, the middle-level tactical behavior, to the low-level operational behavior. Pedestrian movements belong to operational behaviors, where pedestrians continuously make short-term movement decisions on their route to respond
+
+to their immediate environment (Daamen, 2002; Duives, 2016). Many works analyze different factors influencing pedestrian movements (Dridi, 2015). Some critical environmental factors include types of space (Sime, 1995), objects in the environment (Daamen & Hoogendoorn, 2003), and movement of other pedestrians (van den Berg, 2016). Pedestrian movements also depend on their biometric data, like age (Ostrosky et al., 1994), gender (Yamasaki et al., 1991), and body size (Heglund & Taylor, 1988). Different from most existing works in pedestrian movement analysis that collect data from field observations (Shields & Boyce, 2000) or controlled experiments (Haghani & Sarvi, 2018) and analyze them using statistical approaches (Tong & Bode, 2023), we extract pedestrian movements from web videos and learn a generative model to facilitate pedestrian movement modeling.
+
+Human Motion Datasets. AMASS (Mahmood et al., 2019) is one of the most popular human motion datasets with diverse motions annotated with SMPL (Loper et al., 2023) parameters. The main issue of AMASS is the lack of context information related to the motion. Later datasets have focused on adding more human subjects (Cheng et al., 2023b), text descriptions (Guo et al., 2022), human-object interactions (Bhatnagar et al., 2022) and human-scene interactions (Hassan et al., 2019; Huang et al., 2022; Jiang et al., 2024) to the labels. Nevertheless, most of these datasets are captured in a controlled environment with the subjects asked to follow specific action instructions, and many motions, such as jump jacks and martial arts, are rarely seen in urban scenes. In-the-wild videos serve as a more suitable source for studying human movement with their richness in individuals and environments. Another line of work (Von Marcard et al., 2018; Guzov et al., 2021; Dai et al., 2023; Kaufmann et al., 2023) focuses on collecting human motion in outdoor places from in-the-wild videos, but their additional sensor requirements like IMUs limit their scalability to collect large-scale datasets. Zhu et al. (2021); Zheng et al. (2022) propose benchmarks for gait recognition in the wild, but these datasets lack diverse scene contexts. Some other datasets (Vendrow et al., 2023; Robicquet et al., 2016) label human social behaviors and trajectories from street videos as key points or bounding boxes. Still, these labels have worse motion granularity than the SMPL parameters for pedestrian movement analysis. Our proposed dataset, CityWalkers, consists of large-scale web videos of pedestrians in diverse urban environments. CityWalkers provides both SMPL movement labels and context pseudo-labels, including the body shape of the pedestrians, their route destinations, and the semantics and geometry of the scene.
+
+Human Motion Generation. Human motion generation has been accelerated by large-scale motion datasets and rapid advancements in generative models, with diffusion models (Ho et al., 2020) being the most successful architecture for its high generation quality and multi-modal modeling capacity. To generate motion with more fine-grained control, various input conditions have been used, including action labels (Tevet et al., 2022), texts (Zhang et al., 2024), audios (Dabral et al., 2023), and history motions (Chen et al., 2023a). The most relevant motion generation models to ours are the ones that condition on the indoor scenes (Huang et al., 2023; Yi et al., 2024; Jiang et al., 2024). However, there is a huge gap between indoor and outdoor environments, and generating pedestrian movement requires considering more context factors other than the surrounding environment, like route destinations and pedestrian characteristics. Some works (Tripathi et al., 2024; Xue & Seo, 2024) propose to condition the motion on the body shape but do not consider the scene context. Rempe et al. (2023) animate pedestrian movements by generating high-level trajectories and then training a policy to control the pedestrian movements in simulation. Nevertheless, their movement data for training the RL policy comes from a subset of AMASS (Mahmood et al., 2019) with limited diversity. Wang et al. (2024) can generate diverse pedestrian animations while following the given trajectory, but the motion is not learned from real-world pedestrians. Shan et al. (2023) tackle pedestrian movement generation with rule-based approaches by combining path planning algorithms (Treuille et al., 2006) and manually designed animations, lacking the diversity and realism of real-world pedestrian movements. On the contrary, our proposed PedGen model learns to generate diverse and realistic pedestrian movements conditioned on the context factors from large-scale real-world data with noisy labels.
+
+# 3 CAPTURING DIVERSE REAL-WORLD PEDESTRIAN MOVEMENTS
+
+This section introduces our effort to capture real-world pedestrian movements from web videos. Existing human motion datasets rarely capture natural pedestrian movements, lack diversity in scenes and human subjects, and do not provide the critical context factors of pedestrian movements, such as surrounding environments, individual characteristics, and route destinations. To support the task
+
+![](images/38cb0da8ac84cc1e555e9b41a6287783c331450da03f054dccbc8558af65acde.jpg)  
+Figure 2: Samples in the CityWalker dataset. Top Left: The diverse pedestrian movements. Top Right: The diverse body shapes of the pedestrians. Bottom Left: The diverse surrounding environments from depth-unprojected images and the 4D pedestrian movement labels. Bottom Right: The diverse route destinations shown on the depth labels and semantic maps of the scene. The background showcases pedestrians in bustling cities from where we construct the CityWalker dataset.
+
+of context-aware pedestrian movement generation, we construct CityWalkers, a large-scale dataset with real-world pedestrian movements in diverse urban environments annotated by pseudo-labeling techniques. Sec. 3.1 shows an overview of CityWalkers, and Sec. 3.2 describes our data collection and annotation procedure. Please refer to the appendix for more details about CityWalkers.
+
+# 3.1 CITYWALKERS DATASET
+
+The CityWalkers dataset is collected from YouTube's city tour videos. Our data source consists of high-quality web videos of walking in cities worldwide posted by content creators on YouTube, such as the YouTube channel POPtravel (Sczepansky, 2024). We manually picked videos so they incorporate a wide spectrum of urban public places in different regions and cultures with various scene contexts and real-world pedestrian movements. We label each video's scene attributes, including weather, location, crowd density, and time of the day, using an off-the-shelf VLM (Chen et al., 2023b). We then detect (Jocher et al., 2023) and track (Cheng et al., 2023a) each pedestrian in the video, and label the pedestrian movements as SMPL parameters and scene context as depth and semantic segmentation maps. These labels contain the crucial context factors for learning pedestrian movements. In total, CityWalker contains 30.8 hours of high-quality videos, including 120,914 pedestrians and 16,215 scenes across 227 cities and 49 countries, making it the most diverse human motion dataset regarding scene context and human subjects. As shown in Fig. 2, CityWalkers contains a variety of pedestrian movements, scene contexts, pedestrian body shapes, and route destinations.
+
+# 3.2 DATA PROCESSING AND AUTOLABELING
+
+We adopt WHAM (Shin et al., 2023) to recover 4D pedestrian movement pseudo-labels. Given tracked pedestrians and their body key points, WHAM outputs their global human motion with SMPL (Loper et al., 2023) parameters  $\{t_t, \phi_t, \theta_t, \beta\}$  at every timestep  $t$ .  $t_t \in \mathbb{R}^3$ ,  $\phi_t \in SO(3)$  are the root translation and orientation of the SMPL model, and  $\theta_t \in SO(3)^{23}$ ,  $\beta \in \mathbb{R}^{10}$  are the body pose and shape parameters. We further use the state-of-the-art monocular depth estimation model ZoeDepth (Bhat et al., 2023) and semantic segmentation model SegFormer (Xie et al., 2021) to automatically label scene depth and semantics. To filter low-quality pedestrian movement labels brought by model noise, we threshold the detection confidence score, human bounding box size, and keypoint confidence scores. We also discard the occluded motion by thresholding the number of human key points that are within the 2D human segmentation mask. To further clean the labels and curate a more accurate evaluation dataset, we manually check the label quality and adjust the labels, such as 2D key points, if necessary. We want to stress that though we have tried our best to improve
+
+the quality of pseudo labels by using state-of-the-art models and filtering wrong predictions, label noise from web videos is still inevitable. We show in the experiments that learning from the noisy labels still benefits pedestrian movement generation. We also apply techniques in our generation model to further mitigate the effect of label noise. A benchmark result of the accuracy of our data autolabeling pipeline is provided in the appendix.
+
+The final label includes future pedestrian movements and their corresponding context factors.  $\{\pmb {x} =$ $\{(t_t,\phi_t,\pmb {\theta}_t)\} ,\pmb {y} = [\mathcal{I},\mathcal{I}^d,\mathcal{I}^s,\beta ,t_1,t_T]\}$  , where  $\pmb{x}$  is the pedestrian movement label and  $\pmb{y}$  is the context label.  $\mathcal{I},\mathcal{I}^d,\mathcal{I}^s$  are images, depth labels, and segmentation maps of the scene,  $\beta$  represents the pedestrian's personal characteristics, and  $t_1,t_T$  are the starting and goal positions.
+
+# 4 GENERATING CONTEXT-AWARE PEDESTRIAN MOVEMENTS FROM NOISY LABELS
+
+This section introduces our method, PedGen, to address the label noise and incorporate the context factors from the CityWalkers dataset. PedGen is a diffusion-based generative model and the first method for the new task of context-aware pedestrian movement generation. An overview of PedGen is shown in Fig. 3. Sec. 4.1 defines the task of context-aware pedestrian movement generation. We introduce the overall architecture of PedGen in Sec. 4.2. Sec. 4.3 introduces two simple and effective strategies to deal with the low-quality and incomplete labels from web videos, respectively. As it is crucial to model the context factors, including the surrounding environments, the pedestrian's characteristics, and the goal points during generation, we design a novel Context Encoder in Sec. 4.4.
+
+# 4.1 TASK DEFINITION
+
+We define the task of context-aware pedestrian movement generation as follows: Given the initial 3D position  $\pmb{t}_1 = [x_1, y_1, z_1]$  of a pedestrian and the 2D image of an urban scene, our goal is to generate the pedestrian's future movements  $\pmb{x} = [t_t, \phi_t, \theta_t]_{t=1}^T$ . The movement at each timestep is represented as the SMPL root translation  $\pmb{t}_t$ , root orientation  $\phi_t$ , and body pose  $\pmb{\theta}_t$ . The following context factors are also provided to help generation: 1). The urban scene context represented as the 2D image  $\mathcal{I}$ , semantic mask  $\mathcal{I}^s$ , and depth label  $\mathcal{I}^d$ . 2). The SMPL human shape parameter  $\beta$ , which is a latent representation that can indicate a pedestrian's characteristics, such as height, weight, and body shape. 3). The 3D goal position  $\pmb{t}_T = [x_T, y_T, z_T]$  as the pedestrian's route destination in the scene.
+
+# 4.2 PEDGEN MODEL
+
+PedGen follows the conditional diffusion framework (Ho et al., 2020; Ho & Salimans, 2022). Given a sampled movement  $\pmb{x}$  from the dataset, the forward diffusion is a Markov noising process  $\{\pmb{x}^k\}_{k=0}^K$  defined from the noise scheduling parameter  $\{\alpha^k\}_{k=1}^K$ . The goal is to train a reverse denoising model  $\hat{\pmb{x}} = F(\hat{\pmb{x}}^k, k, c)$  that predicts the sampled movement  $\pmb{x}$  given the noisy movement  $\hat{\pmb{x}}^k$ , the diffusion timestep  $k$ , and the condition factor  $c$ . Our denoising model architecture follows state-of-the-art transformer-based human motion generation models (Tevet et al., 2022; Jiang et al., 2024; Chen et al., 2023a), where movements at different timesteps are encoded as separate tokens in the transformer. We represent  $\pmb{x}$  as velocity  $\pmb{v}_t = \pmb{t}_t - \pmb{t}_{t-1}$  and rotation  $\phi_t, \theta_t$  with 6D rotation (Zhou et al., 2019) representation. Different from existing approaches that encode all parts of the motion into a single token, we find it helpful to treat the velocity and the rotation of the motion as different tokens in the transformer, as they have different representations and scales in the data. Our loss function for training the denoising model  $F$  is as follows:
+
+$$
+\mathcal {L} (\boldsymbol {x}, \hat {\boldsymbol {x}}) = \mathbb {E} _ {k \in [ 1, K ], (\boldsymbol {x}, \boldsymbol {c}) \in \mathcal {D}} \left[ w _ {\text {r e c}} \mathcal {L} _ {\text {r e c}} + w _ {\text {t r a j}} \mathcal {L} _ {\text {t r a j}} + w _ {\text {g e o}} \mathcal {L} _ {\text {g e o}} \right], \tag {1}
+$$
+
+where  $\mathcal{D}$  is the training dataset.  $\mathcal{L}_{\mathrm{rec}} = \| \pmb{x} - \hat{\pmb{x}}\|_2^2$  is the standard diffusion reconstruction loss.  $\mathcal{L}_{\mathrm{traj}} = \sum_{t=1}^{T}\|\pmb{t}_t - \hat{\pmb{t}}_t - \pmb{t}_0\|_1$  is the trajectory loss on the reconstructed global translation  $\hat{\pmb{t}}_t = \sum_{t=1}^{T}\hat{\pmb{v}}_t$ , and  $\mathcal{L}_{\mathrm{geo}} = \sum_{t=1}^{T}\|\mathrm{FK}(\pmb{\theta}_t) - \mathrm{FK}(\hat{\pmb{\theta}}_t)\|_2$  is the geometric loss (Tevet et al., 2022) that computes the joint position error with forward kinematics on the SMPL body pose. In ablation experiments, we will show the effectiveness of separating velocity and rotation tokens, as well as adding the trajectory and geometry losses.
+
+During inference, we start from the  $k = K$  step by sampling  $\hat{\pmb{x}}^K$  from the standard Gaussian distribution. We then pass it to the denoising model to predict the clean motion  $\hat{\pmb{x}}$  and add noise again to obtain the noised sample  $\hat{\pmb{x}}^{K - 1}$ . This process is repeated  $K$  times to get the final generated pedestrian movement  $\hat{\pmb{x}}$ . Our model can also facilitate the efficient generation of infinitely long movements by concatenating the short motion intervals. Please refer to the appendix for more details.
+
+# 4.3 ADDRESSING NOISY LABELS
+
+Pseudo-labels from pre-trained predictors have inevitable noise. Unlike existing human motion generation approaches that focus on datasets with clean MoCap data, our PedGen model aims to address noisy labels from web videos. There are two sources of label noise in CityWalkers. The first is the inherent compound noise from the data and models used for pseudo-labeling, leading to low-quality labels. As it is difficult to examine the label quality of 4D human motion from only 2D videos, many anomaly labels remain in the dataset even after rule-based filtering and manual checks. We propose to automatically identify these low-quality labels using reconstruction-based unsupervised anomaly detection techniques (Livernoche et al., 2023; Wolleb et al., 2022). Specifically, we first train a PedGen model without context on the training data of CityWalkers. We then partially add noise with half of the diffusion steps  $K / 2$  for each sample and denoise it using the trained model. We use the reconstruction error between the original and the denoised sample as a metric for anomaly labels and filter labels with errors greater than a certain threshold. We then iterate the above process by re-training the model with the remaining labels and filtering based on the new reconstruction error.
+
+Another source of label noise is the discontinuous or lost tracks of pedestrians due to occlusions and missed detections. As a result, more than half of the labels are incomplete and only annotated on a subset of frames. These labels provide partial supervision that can benefit learning more diverse context-aware pedestrian movements. To train with these partial labels, we replace the missing timesteps for these partial labels with a learnable motion mask embedding  $m$  to the denoising transformer. We first define a label mask  $M$ , where  $M_{t} == 1$  indicates the label at timestep  $t$  is missing. Then we add the mask embedding to the original noisy sample as  $x^{k} = x^{k}(1 - M) + m \cdot M$  to replace the missing timesteps with the mask embedding  $m$  and feed to the network to output the denoised prediction  $\hat{x}$  similar to Sec. 4.2. We then update the loss with the masked predictions and masked ground truth as  $L = L(\boldsymbol{x}(M), \hat{\boldsymbol{x}}(M))$  so it only operates on the labels at the available timesteps. All losses will guide the training with the masked outputs.
+
+# 4.4 CONTEXT ENCODER
+
+Our context encoder module outputs a condition embedding  $c$  from the provided context factors. To encode the scene context, we first unproject the 2D depth label  $\mathcal{I}^d$  and semantic map  $\mathcal{I}^s$  into a 3D point cloud  $\mathcal{P} = \{\mathbf{p} = [p_x, p_y, p_z, p_c]\}$ , where  $p_c$  is the semantic label. Next, we extract points within a local neighborhood of the starting location  $t_1$ , resulting in the local point cloud  $\mathcal{P}_{\mathrm{local}} = \{\mathbf{p} \in \mathcal{P} \mid \| p_x - x_1 \| < \Delta_x, \| p_y - y_1 \| < \Delta_y, \| p_z - z_1 \| < \Delta_z\}$  and voxelize it into a 3D grid. The class label of each voxel is either empty or determined by majority voting of the points within the voxel. The voxel is then processed with a single cross-attention layer to get the scene context embedding. The scene encoding is then added with embeddings from other conditions, including the human shape  $\beta$  and the goal position  $t_T$ , to get the final context embedding  $c$ .
+
+As web videos only contain 2D images, a natural idea is to directly encode the scene context using the 2D image feature extractors (e.g., Dino-V2 (Oquab et al., 2023)). We find that our proposed context encoder can encode the scene context more effectively than encoding the scene context in 2D. Since pedestrian movement is represented in the 3D space and generating it requires 3D information about the scene, it is easier to reason about the surrounding environment using a 3D representation than 2D image features. Furthermore, it is hard to disentangle the pedestrian and its surrounding context from the 2D image features. As a result, the model may exploit the ego pedestrian instead of the scene context to generate future movements. Our context encoder could address this issue by unprojecting the pixels into 3D point clouds and discarding point clouds belonging to the ego pedestrian. In our ablation experiments, we show the importance of using the 3D scene representation and appending the semantic labels in encoding the scene context.
+
+![](images/c499ab206310a7032b61c08e46557dee6331a66a88f016dae90ec83221861b0c.jpg)  
+Figure 3: Our method. We discard the anomaly labels with an iterative automatic label filtering procedure and add the partial labels to training data. We then train PedGen with a Context Encoder to represent crucial context factors. The scene context is obtained by lifting the 2D depth and semantic labels to the 3D space and converting them into a local voxel representation. The encoded scene context is combined with other context factors, including the body shape and the goal to get the context embedding  $\pmb{c}$ . The context embedding  $\pmb{c}$  and the timestep embedding  $\pmb{k}$  are then used to guide the Denoising Transformer to predict the clean motion from the noised one. We use a learnable motion mask embedding  $\pmb{m}$  to address the partial labels during training.
+
+# 5 EXPERIMENTS
+
+We compare the performance of PedGen to the other baselines on the real-world Citywalkers and Waymo datasets and in simulated CARLA environments in Sec. 5.1. Sec. 5.2 shows experiments on the effect of training with noisy labels and different context factors. Sec. 5.3 demonstrates the ablation study of our data and model. The diverse pedestrian movements generated by PedGen are shown in Fig. 4. We provide additional experimental details in the appendix.
+
+Datasets. We train PedGen on the proposed CityWalkers dataset. For each pedestrian movement trajectory, we sample the initial timestep at an interval of 30 frames and keep at most the future 60 frames (2 seconds) as the ground truth movement. The training set has 104,192 samples, including 53,405 partial labels that have at least 30 frames of annotation. The validation set has 13,039 samples and only contains complete labels. We split the validation set to contain novel scenes with completely different locations and human subjects never seen in the training set.
+
+To verify the performance of PedGen on real-world pedestrian movement prediction with ground truth labels, we use the Waymo open dataset (Sun et al., 2020) with human-annotated 3D human keypoint labels, which is only used for testing. Waymo is critical as it is the largest urban dataset that captures diverse pedestrian motions with sparse 3D keypoint labels at  $10\mathrm{Hz}$ . In total, we selected 80 test samples that (1) spans 2 seconds and (2) includes at least 6 sparse human keypoint labels.
+
+We also collect an additional test set in the CARLA Simulator (Dosovitskiy et al., 2017) to demonstrate the application of PedGen in simulation. We sample initial locations, goal locations, and camera view angles in different maps and render the ground truth images, depth labels, and segmentation maps in simulation. We manually check that all the sampled locations are valid and that all the camera views are not occluded by obstacles, resulting in a simulated test set with 262 diverse samples.
+
+Evaluation metrics. For experiments on the CityWalkers and Waymo dataset, we compare the generated pedestrian movement with the ground truth, and follow the metrics used in human motion prediction (Chen et al., 2023a) to compute Average Displacement Error (ADE) and Final Displacement Error (FDE) on the realism of the predicted motion. We generate 50 movements for each data sample and report both minimum and average ADEs (mADE, aADE) and FDEs (mFDE, aFDE) among all samples. For experiments on the simulated test set, we evaluate the context awareness and the physical plausibility of the generated motion. Our metrics include the Collision Rate (CR), which measures the ratio of the generated movement that collides with other objects in the environment, and the Foot Floating Rate (FFR), which measures the ratio of the generated movement whose feet are either floating or penetrating with the ground greater than a given threshold  $(20\mathrm{cm})$
+
+Baselines. We compare our method with several recent human motion diffusion models, which have different input conditions and motion representations. Note that since context-aware pedestrian
+
+![](images/2f8c1b114f75e4173ef8bf2b78b65a4dd6ef572a0a884693bbb70113d54682e0.jpg)  
+Figure 4: Visualizations of the generated pedestrian movements. The top row shows results in real scenes from the CityWalkers dataset, the middle row shows results in the real-world Waymo test set, and the bottom row shows results in simulated scenes from the CARLA test set.
+
+Table 1: Comparison to baselines. We consider two cases where the model is given the goal condition or not. We evaluate on the validation set of CityWalkers, the real-world Waymo test set, and the simulated CARLA test set. We evaluate our model (PedGen) compared with the other baselines.  
+
+<table><tr><td rowspan="2">Goal Condition</td><td rowspan="2">Method</td><td colspan="4">CityWalkers</td><td colspan="4">Waymo</td><td colspan="2">CARLA</td></tr><tr><td>mADE ↓</td><td>aADE ↓</td><td>mFDE ↓</td><td>aFDE ↓</td><td>mADE ↓</td><td>aADE ↓</td><td>mFDE ↓</td><td>aFDE ↓</td><td>CR ↓</td><td>FFR ↓</td></tr><tr><td>X</td><td>HumanMAC</td><td>1.31</td><td>4.67</td><td>1.86</td><td>8.65</td><td>3.19</td><td>5.29</td><td>5.61</td><td>10.36</td><td>2.5%</td><td>10.2%</td></tr><tr><td>X</td><td>MDM</td><td>1.33</td><td>4.55</td><td>1.93</td><td>8.41</td><td>3.03</td><td>5.35</td><td>5.66</td><td>10.60</td><td>2.1%</td><td>3.2%</td></tr><tr><td>X</td><td>PedGen</td><td>1.13</td><td>4.08</td><td>1.61</td><td>7.56</td><td>2.90</td><td>5.15</td><td>5.52</td><td>10.11</td><td>1.6%</td><td>2.6%</td></tr><tr><td>✓</td><td>TRUMANS</td><td>0.73</td><td>1.26</td><td>0.56</td><td>1.13</td><td>2.01</td><td>2.37</td><td>1.41</td><td>1.94</td><td>0.6%</td><td>0.6%</td></tr><tr><td>✓</td><td>PedGen</td><td>0.59</td><td>1.08</td><td>0.46</td><td>0.99</td><td>1.91</td><td>2.18</td><td>0.78</td><td>1.04</td><td>0.0%</td><td>0.0%</td></tr></table>
+
+movement generation is a newly defined task, no models could directly support it; we thus made minimum adjustments to make them compatible with our problem setting. MDM (Tevet et al., 2022) uses texts or action labels as the condition and proposes several geometric losses to improve motion quality. We did not use the foot contact loss proposed in their method as CityWalkers does not provide ground-truth foot contact labels. HumanMac (Chen et al., 2023a) conditions the future motion generation on the history motion sequences, which uses the DCT transform to ensure smooth motion generation. TRUMANS (Jiang et al., 2024) uses the initial human pose, the 2D BEV goal position, the indoor scene context, and the frame-wise action labels as the input condition. We modify TRUMANS so it is compatible with our scene context and goal point conditions.
+
+# 5.1 COMPARISON TO BASELINES
+
+We compare PedGen with the other baselines on the validation set of CityWalkers, the real-world Waymo test set, and the simulated CARLA test set. We separately evaluate whether the goal condition is given, as it is a deterministic factor for pedestrian movement. The benchmarking results are shown in Tab. 1. It can be observed that PedGen outperforms other baseline models by a clear margin on CityWalkers. PedGen also achieves the best zero-shot generalization ability on Waymo and CARLA test sets, further showing the superiority of PedGen to the other baseline.
+
+# 5.2 EFFECT OF NOISY LABELS AND CONTEXT FACTORS
+
+Table 2a demonstrates the effectiveness of PedGen in addressing noisy labels. We can see that removing the anomaly labels with the proposed automatic label filtering can help generate more realistic pedestrian movements and improve aADE by  $2.9\%$ . Moreover, adding the partial labels as additional training data can improve the aADE by  $5.8\%$ , highlighting the value of partial labels in web videos. Combining both strategies for noisy labels leads to the best performance of a 4.08 aADE.
+
+![](images/48dcc36a14eca05353829126b355d002a43fa52a3dd520ef7ad8b3d46da6c5a0.jpg)  
+a)  
+No Context
+
+![](images/495a1155cbc4818b210e67938032145004dc5cab02c3d9b5f77d9004fc510d57.jpg)  
+With Context
+
+![](images/d1258fa776be134ff6c05cc1b5675ef51232fc819bba950eb8c8ec9f4c941202.jpg)
+
+![](images/9b338d826e13cb517c4e05071597beaa59f2f60c6da1435d023e46bc3b7cc2a7.jpg)  
+No Context  
+Figure 5: Qualitative comparison of training with context factors. We compare the generated movements of PedGen trained with or without context factors in real-world environments (a) and in simulation (b).
+
+![](images/1601fd8022e65243800fcf27e448ac43c385b63c5e452c806cff7f48fc16a160.jpg)  
+With Context
+
+![](images/212d17aa010344c8c2c7777e6d574b3217f3fe9fcb37f689dbbeed96491512b5.jpg)  
+With Context (Long-Term)  
+No Context
+
+![](images/587e0ac0e59155867ad6816d3697efff53fe9af722601354d3b09bae0f36d439.jpg)  
+With Context
+
+(a) Evaluation of training with noisy labels. We evaluate PedGen with no context trained with or without anomaly and partial labels.
+
+(b) Evaluation of the context factors. We evaluate PedGen conditioned on each context factor, including the surrounding environment (scene), the pedestrian's own characteristics (human), and the goal points (goal).
+
+Table 2: Results with noisy labels and context factors. We experiment on the CityWalkers validation set and the CARLA test set and study the effect of training with noisy labels (a) and different context factors (b).  
+
+<table><tr><td rowspan="2">Anomaly Labels</td><td rowspan="2">Partial Labels</td><td colspan="4">CityWalkers</td></tr><tr><td>mADE ↓</td><td>aADE ↓</td><td>mFDE ↓</td><td>aFDE ↓</td></tr><tr><td>✓</td><td>✘</td><td>1.17</td><td>4.45</td><td>1.64</td><td>8.31</td></tr><tr><td>✘</td><td>✘</td><td>1.13</td><td>4.32</td><td>1.60</td><td>8.09</td></tr><tr><td>✓</td><td>✓</td><td>1.15</td><td>4.19</td><td>1.62</td><td>7.79</td></tr><tr><td>✘</td><td>✓</td><td>1.13</td><td>4.08</td><td>1.61</td><td>7.56</td></tr></table>
+
+<table><tr><td rowspan="2">Idx</td><td rowspan="2">Context Factor</td><td colspan="4">CityWalkers</td><td colspan="2">CARLA</td></tr><tr><td>mADE ↓</td><td>aADE ↓</td><td>mFDE ↓</td><td>aFDE ↓</td><td>CR ↓</td><td>FFR ↓</td></tr><tr><td>1</td><td>No</td><td>1.13</td><td>4.08</td><td>1.61</td><td>7.56</td><td>2.1%</td><td>5.2%</td></tr><tr><td>2</td><td>Scene</td><td>1.11</td><td>3.75</td><td>1.55</td><td>6.92</td><td>1.6%</td><td>2.6%</td></tr><tr><td>3</td><td>Human</td><td>1.09</td><td>3.24</td><td>1.61</td><td>5.95</td><td>1.9%</td><td>0.7%</td></tr><tr><td>4</td><td>Scene + Human</td><td>1.00</td><td>3.12</td><td>1.43</td><td>5.65</td><td>1.5%</td><td>0.3%</td></tr><tr><td>5</td><td>Goal</td><td>0.60</td><td>1.09</td><td>0.47</td><td>1.00</td><td>0.0%</td><td>0.0%</td></tr><tr><td>6</td><td>Scene + Goal</td><td>0.59</td><td>1.08</td><td>0.46</td><td>0.99</td><td>0.0%</td><td>0.0%</td></tr><tr><td>7</td><td>Human + Goal</td><td>0.55</td><td>1.01</td><td>0.43</td><td>0.93</td><td>0.0%</td><td>0.0%</td></tr><tr><td>8</td><td>Scene + Human + Goal</td><td>0.54</td><td>0.96</td><td>0.43</td><td>0.91</td><td>0.0%</td><td>0.0%</td></tr></table>
+
+We further examine the effect of each context factor on the pedestrian movement generation performance in Tab. 2b. The comparisons between "setting 1 vs. 2" and between "setting 1 vs. 3" show that adding the surrounding environment and the pedestrian's own characteristics as conditions are helpful to pedestrian movement generation and can reduce the aADE by  $6.7\%$  and  $19.4\%$ , respectively. Setting 4 demonstrates that incorporating both context factors achieves even better results. From settings 1 and 5, we find that the goal points are the most important context factor for the final performance and can significantly reduce aADE by  $72.9\%$ . The comparison between "setting 5 vs. 6" and between "setting 5 vs. 7" further proves that the other two context factors (scene and human) can also help movement generation after the goal is provided. Finally, setting 8 demonstrates that using all three context factors leads to the smallest generation errors. From the results of the CARLA test set, we can see that the scene context can contribute to both a lower collision rate of  $1.6\%$  and a lower foot floating rate of  $2.6\%$ , while the human context is more useful in reducing the foot floating rate to only  $0.7\%$ . Adding both the scene and the human context can further improve the physical plausibility of the generated movements. Using the goal context is the most crucial factor and can reduce the failure rate to 0. This further proves the effectiveness of each context factor.
+
+Some qualitative comparison results in real-world environments are shown in Fig. 5 (a). We can observe that PedGen trained without context factors generate arbitrary movements that walk off the sidewalk. By conditioning the generation on the context factors, the movement becomes context-aware, and the model can further generate long-term pedestrian walking behaviors on the sidewalk. Visualization results in simulation are shown in Fig. 5 (b). From the left two figures, we can see that without the context factors, the generated movements have the feet floating from the ground, while adding context factors fix this issue. From the right two figures, we see that PedGen trained without context generated a walking pose that is in collision with the bench, while it successfully generated a sitting pose on the bench after considering the context factors.
+
+Table 3: Ablation experiment results. We ablate on the training data of PedGen (a) and the PedGen model's key components (b) on the CityWalkers validation set.  
+(a) Ablation on training data of PedGen.  
+
+<table><tr><td rowspan="2">Context Factor</td><td rowspan="2">Training Data</td><td colspan="4">Metric</td></tr><tr><td>mADE ↓</td><td>aADE ↓</td><td>mFDE ↓</td><td>aFDE ↓</td></tr><tr><td rowspan="3">No</td><td>SLOPER4D</td><td>1.61</td><td>6.04</td><td>2.42</td><td>11.65</td></tr><tr><td>CityWalkers(50%)</td><td>1.16</td><td>4.19</td><td>1.66</td><td>7.76</td></tr><tr><td>CityWalkers(100%)</td><td>1.13</td><td>4.08</td><td>1.61</td><td>7.56</td></tr><tr><td rowspan="3">Scene</td><td>SLOPER4D</td><td>1.47</td><td>6.45</td><td>2.19</td><td>12.50</td></tr><tr><td>CityWalkers(50%)</td><td>1.14</td><td>4.02</td><td>1.57</td><td>7.47</td></tr><tr><td>CityWalkers(100%)</td><td>1.11</td><td>3.75</td><td>1.55</td><td>6.92</td></tr><tr><td rowspan="3">Human</td><td>SLOPER4D</td><td>3.82</td><td>10.39</td><td>7.11</td><td>20.62</td></tr><tr><td>CityWalkers(50%)</td><td>1.13</td><td>3.34</td><td>1.65</td><td>6.18</td></tr><tr><td>CityWalkers(100%)</td><td>1.09</td><td>3.24</td><td>1.61</td><td>5.95</td></tr></table>
+
+(b) Ablation on model components of PedGen.  
+
+<table><tr><td rowspan="2">Context Factor</td><td rowspan="2">Method</td><td colspan="4">Metric</td></tr><tr><td>mADE ↓</td><td>aADE ↓</td><td>mFDE ↓</td><td>aFDE ↓</td></tr><tr><td rowspan="3">No</td><td>PedGen (-traj/geo)</td><td>1.33</td><td>4.47</td><td>1.97</td><td>8.29</td></tr><tr><td>PedGen (-sep. token)</td><td>1.32</td><td>4.26</td><td>2.01</td><td>7.95</td></tr><tr><td>PedGen</td><td>1.13</td><td>4.08</td><td>1.61</td><td>7.56</td></tr><tr><td rowspan="3">Scene</td><td>PedGen (-3D rep.)</td><td>1.13</td><td>4.26</td><td>1.60</td><td>7.95</td></tr><tr><td>PedGen (-semantic)</td><td>1.12</td><td>3.86</td><td>1.56</td><td>7.09</td></tr><tr><td>PedGen</td><td>1.11</td><td>3.75</td><td>1.55</td><td>6.92</td></tr></table>
+
+# 5.3 ABLATION STUDY
+
+Ablation on training data of PedGen. Table 3a shows an ablation study on the training data of PedGen. To demonstrate the effectiveness and necessity of using web videos for pedestrian movement generation, we train PedGen on SLOPER4D (Dai et al., 2023). SLOPER4D is one of the largest outdoor 4D human datasets annotated from LiDAR point clouds. Still, its scale and diversity in human subjects and scene contexts are much less than CityWalkers, and its motion is captured from human actors instead of real-world pedestrians. We can observe that training on CityWalkers leads to significant performance improvement compared to training on SLOPER4D, even though CityWalkers is annotated with pseudo-labels. Notably, training with the human context on SLOPER4D only achieves  $3.82\mathrm{mADE}$ , as the model can easily overfit the 12 human subjects in the dataset and results in degraded generalization performance. On the contrary, with more training data of CityWalkers, models conditioned on different context factors all perform better than those trained with less data, highlighting the value of capturing large-scale pedestrian movement data using web videos.
+
+Ablation on model components of PedGen. Table 3b shows ablations on key model components of PedGen. -traj/geo means the model is trained only with the diffusion reconstruction loss without the trajectory and geometry losses. -sep. token means the motion at each timestep is represented as a single token in the transformer instead of separate tokens for the rotation and velocity. -3D rep. means the scene context is encoded with 2D feature using a depth anything (Yang et al., 2024) pre-trained image backbone DINOv2 (Oquab et al., 2023). -semantic means the 3D voxel only encodes the occupancy without semantic labels. The results show the importance of using 3D scene context with semantic labels for pedestrian movement generation, the effectiveness of separating velocity and rotation tokens, and adding the geometry and trajectory losses in the motion diffusion model.
+
+# 6 CONCLUSION
+
+We study a new task of generating context-aware pedestrian movements by learning from web videos with noisy labels. To facilitate this study, we collect a large-scale dataset CityWalkers with diverse real-world pedestrian movements in urban scenes. We further propose PedGen, a generative model that addresses the noisy labels in CityWalkers and models the three important context factors: the scene context, the personal characteristics, and the goal position. Experiments show that PedGen can generate realistic context-aware pedestrian movements in both real-world and simulation environments. We hope this study will present new opportunities and facilitate future research on modeling pedestrian movements in real-world settings.
+
+Limitations. PedGen only considers static scene context at the starting frame, while pedestrian movement also depends on dynamic scene contexts such as the history trajectories of other pedestrians. Modeling these dynamic objects would be an interesting future direction. In addition, PedGen only generates the movements of a single pedestrian at one time, while modeling group activities in urban scenes can help generate more realistic real-world behaviors.
+
+# ETHICS STATEMENT
+
+We are fully aware of the potential privacy and ethical issues when using street-view videos, and we take these concerns as our highest priority. We acknowledge that we will be solely responsible for any legal violations with respect to our collected dataset and that we accept all the associated risks. We have taken several measures to mitigate these issues:
+
+Legal Compliance with YouTube's Terms of Service and Creative Commons Licensing: Our dataset comprises YouTube videos that are explicitly labeled under the Creative Commons Attribution (CC-BY-  $\mathbf{SA}^{1}$  ) license. This license allows for redistribution and adaptation of the material for any purpose, including research, as long as proper attribution is provided to the original creators. We have carefully reviewed YouTube's Terms of Service and Creative Commons guidelines  $^2$  to ensure that our use of these videos is legally permissible. YouTube's Terms of Service explicitly permit redistribution of Creative Commons-licensed content, provided proper attribution is given. Our dataset strictly excludes videos marked under the "Standard YouTube License" or any content that carries additional restrictions. We do not download videos through unauthorized means but instead rely on publicly available Creative Commons content accessible via YouTube's official API, ensuring compliance with both YouTube's platform policies and broader copyright laws.
+
+Privacy Protection and Anonymization Measures: We follow standard protocols used by street datasets such as Waymo and Ego4D (Sun et al., 2020; Grauman et al., 2022) to ensure complete anonymization of identifiable features. As the first step of our data preprocessing, we apply mosaics to faces and license plates to prevent re-identification of individuals in the videos. Additionally, we omit any copyright-related information, such as logos, channel owner details, or other copyrighted materials, during data processing to protect intellectual property rights. Our model is trained exclusively on preprocessed images, ensuring that identifiable features and copyrighted materials are neither stored nor accessible within our dataset. Furthermore, our IRB-approved protocol ensures compliance with human subjects protection guidelines, giving individuals in the videos the right to request data removal if privacy concerns arise.
+
+Ethical Use and Responsible Data Sharing: Upon data release, we will provide clear and transparent terms of use for the dataset, outlining ethical guidelines, usage restrictions, and legal obligations that users must comply with. We will implement safeguards on our dataset webpage, including detailed user agreements, encryption measures to protect personal information, and access controls to monitor and prevent unauthorized data use. To align with privacy protection best practices from existing YouTube datasets such as YouTube-8M and YouTube-VOS (Abu-El-Haija et al., 2016; Xu et al., 2018), we will not provide processed video clips directly. Instead, users will be redirected to the original YouTube videos via a link and will be required to follow our preprocessing scripts to prepare the dataset themselves. Additionally, as our labels are generated using off-the-shelf models, we will not release direct annotations but will provide scripts that allow users to generate their own pseudo-labels. To further prevent misuse, we will only release the trained model weights, allowing researchers to utilize the dataset for downstream tasks such as pedestrian movement simulation without exposing raw video data.
+
+Institutional Review Board (IRB) Oversight and Legal Compliance: Our dataset will be overseen by the UCLA Institutional Review Board (IRB) with security protocols in place to protect against unauthorized access and misuse. The UCLA IRB will review our data collection, ensuring compliance with ethical and legal standards. Specifically, the IRB will assess the legal permissibility of using publicly available, CC-licensed videos and will confirm that our approach aligns with applicable laws and institutional policies. The human subjects in the videos have the right to view, correct, and request deletion of personal information in the dataset. The review board will continue to evaluate the collection, use, and sharing of the data to ensure alignment with best practices in data privacy and ethics. Any legal concerns raised post-publication will be addressed under UCLA's data governance framework, ensuring continued compliance and ethical responsibility.
+
+By implementing these measures, we ensure that our dataset adheres to both legal requirements and ethical best practices. We remain committed to continuously reviewing and improving our compliance protocols in response to emerging ethical and legal considerations.
+
+# ACKNOWLEDGMENTS
+
+The project was supported by the NSF grants CNS-2235012 and IIS-2339769.
+
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+# LEARNING TO HELP IN MULTI-CLASS SETTINGS
+
+Yu Wu $^{\S}$ , Yansong Li $^{\ddagger=}$ , Zeyu Dong $^{\dagger=}$ , Nitya Sathyavageeswaran $^{\S}$ , Anand D. Sarwate $^{\S*}$
+
+$\S$  Rutgers University,  $\ddagger$ University of Illinois Chicago,  $\dagger$ Stony Brook University
+
+# ABSTRACT
+
+Deploying complex machine learning models on resource-constrained devices is challenging due to limited computational power, memory, and model retrainability. To address these limitations, a hybrid system can be established by augmenting the local model with a server-side model, where samples are selectively deferred by a rejector and then sent to the server for processing. The hybrid system enables efficient use of computational resources while minimizing the overhead associated with server usage. The recently proposed Learning to Help (L2H) model proposed training a server model given a fixed local (client) model. This differs from the Learning to Defer (L2D) framework which trains the client for a fixed (expert) server. In both L2D and L2H, the training includes learning a rejector at the client to determine when to query the server. In this work, we extend the L2H model from binary to multi-class classification problems and demonstrate its applicability in a number of different scenarios of practical interest in which access to the server may be limited by cost, availability, or policy. We derive a stage-switching surrogate loss function that is differentiable, convex, and consistent with the Bayes rule corresponding to the 0-1 loss for the L2H model. Experiments show that our proposed methods offer an efficient and practical solution for multi-class classification in resource-constrained environments.
+
+# 1 INTRODUCTION
+
+Machine Learning (ML) models deployed on local devices often face significant limitations in terms of computational resources and retrainability. Local devices are typically constrained by limited processing power, memory, and battery life (Ajani et al., 2021; Biglari & Tang, 2023), which can impede the model's ability to handle large-scale data or perform complex computations in real-time. Furthermore, once a local model is deployed, it may be difficult to retrain or update (Hanzlik et al., 2021), leading to a potential degradation in performance over time when data distribution drifts (Lu et al., 2019).
+
+To address these issues, one strategy is to augment the local machine learning system (a "client") with an external model hosted on a remote server. This approach, seen in recent applications like Apple Intelligence (Gunter et al., 2024), enhances the overall system performance by leveraging the server's superior computational power and capacity for model updates. Recent studies have shown that efficient fine-tuning methods, like few-shot learning (Brown et al., 2020) and parameter-efficient fine-tuning (Fu et al., 2023), can achieve competitive performance, making it a feasible choice for server-side model deployment. Those methods utilize the pre-trained model's capabilities while requiring minimal computational resources, allowing for easy and cost-effective updates to adapt to specific tasks or data distributions. Consequently, it enables the deployment of powerful models on servers without the high costs of comprehensive model training.
+
+While training a server model can be relatively inexpensive, extensive use of server-side models can be costly due to data transfer, latency, instability connection, and service fees. Moreover, server operators may wish to limit the frequency of offloaded inferences from clients by imposing costs
+
+or access constraints. In the presence of such restrictions, the practical solution for clients to use a rejector that selectively defers/offloads only the most challenging and uncertain samples to the server for processing, thereby optimizing the balance between usage cost and performance. Informally, the rejector is defined as a decision function to either select the local model or server model for a sample (a rigorous definition is given in Sec. 2).
+
+A framework for training a rejector and server model given a fixed "legacy" device-bound model was recently proposed for binary classification models under the name "Learning to Help" (L2H) (Wu & Sarwate, 2024). In this work, we extend and generalize this framework to encompass a variety of problems of more practical interest. We can identify several scenarios that can arise in the context of building ML systems in those settings with a fixed local model. In particular, we demonstrate how to handle various restrictions on the use of the server:
+
+- PAY-PER-REQUEST (PPR): the device must pay a cost each time the rejector defers to the server; as an example, the server can be thought of as a consultant who offers their services at a cost.  
+- INTERMITTENT AVAILABILITY (IA): in the PPR model, the rejection rule must also account for the server being potentially unavailable; for example, this can occur when the internet connection is unstable, or the server is busy servicing requests of other devices.  
+- BOUNDED REJECT RATE (BRR): instead of PPR, the rate of rejections/deferrals per unit time may not exceed a predefined upper limit; for example, there is a limit on the usage of some LLMs even with a paid subscription.
+
+The L2H model is a natural complement to prior works on two-party decision systems, including Learning with Abstention (LWA) and Learning to Defer (L2D). These models either assume that rejected samples can be discarded at a fixed cost or that the server is a pre-trained "expert" and optimize decision rules for the device/rejector. As ML.AI decision systems become integrated into physical devices and infrastructure, issues of sustainability will require newer server systems to support pre-existing legacy models which can allow only partial retraining. Relatively little attention has been given to this scenario: we address the extreme case where only the rejection rule can be updated with the server.
+
+Existing work (Wu & Sarwate, 2024) on the L2H model only studies binary classification in the PPR model. In this work, we further extend the L2H framework to multi-class problems and show how to train systems under the three scenarios mentioned above. More specifically, we propose a generalized (non-differentiable) 0-1 loss to measure the prediction system's performance and find the Bayes rule for this loss. We then provide a surrogate loss function which is convex and differentiable and show that its minimizer is consistent with the Bayes rule for the generalized 0-1 loss.
+
+We then design algorithms which can optimize rules for training multi-class predictors in the three scenarios mentioned earlier: PPR, IA, and BRR. To guarantee BRR we use a "post-hoc" method to control the rejection rate. We show experimentally that incorporating a server model with a rejector enhances the overall performance of the ML system in all three scenarios. Training with our surrogate loss function, the rejector helps in identifying the challenging and uncertain samples, while balancing the accuracy and usage of the server model.
+
+# RELATED WORKS
+
+Hybrid ML systems. The hybrid ML systems that consist of client side and server side, have been of substantial research in many fields, including federated learning (Zhang et al., 2021; McManan et al., 2017), distributed learning (Cao et al., 2023a; Horvath et al., 2023), and decentralized learning (Sun et al., 2022; Fang et al., 2022; Liu et al., 2024; Li & Han, 2023). However, those frameworks only focus on interaction between different sides during the training process. Once the training is done, no more communication is needed among them. In this work, we are interested in the collaboration between client and server both in the training and inference phases.
+
+Learning with Abstention (LWA). The foundational work that first considered extra options for recognition tasks was proposed by Chow (1957) and Chow (1970). Herbei & Wegkamp (2006) revisit the framework in classification tasks and propose a score-based reject method. Subsequent works extend this framework to different types of classifiers (Rigollet, 2007; Wegkamp, 2007;
+
+Bartlett & Wegkamp, 2008; Wegkamp & Yuan, 2011). Cortes et al. (2016) consider a separate function to make reject decisions in binary classification. Zhu & Nowak (2022b;a) introduce the reject option in active learning. Cortes et al. (2018) add abstention to online learning. Zhang et al. (2024) consider the rejection rule when the data distribution contains noisy labels. Yin et al. (2024) use rejection for ensuring fairness. Mao et al. (2024d) and Mao et al. (2024c) analyze the theoretical bounds and algorithms for score-based and predictor-rejector-based methods, respectively.
+
+Unlike the setting where rejection incurs an extra cost, few works focus on scenarios with no extra cost for rejection, but the reject rate is bounded. Pietraszek (2005) construct the rejection rule using ROC analysis. Denis & Hebiri (2020) derive the Bayes optimal classifier for the bounded reject rate setting. Shekhar et al. (2020) consider binary classification with bounded reject rates in the active learning setting. In LWA, although there are works that can identify data samples that are uncertain or challenging for the local model to predict, the subsequent action is merely to discard those samples without providing an answer. While this framework can indeed improve prediction accuracy, the system still cannot handle the samples that are challenging for the local model.
+
+Learning to Defer (L2D). Building upon LWA, Madras et al. (2018) were the first to consider a subsequent expert can process the rejected samples. Raghu et al. (2019) consider binary classification with expert deferral using uncertainty estimators. Mozannar & Sontag (2020a) propose the first method that jointly trains the local model and rejector in multi-class classification. Verma & Nalisnick (2022); Mozannar et al. (2023); Hemmer et al. (2023); Cao et al. (2023b) propose different surrogate loss functions for L2D. Okati et al. (2021) and Narasimhan et al. (2022) add post-hoc algorithms to the cases where the reject decision incurs extra cost.
+
+Seeking help from multiple experts has been explored in different machine learning models (Kerrigan et al., 2021; Keswani et al., 2021; Corvelo Benz & Gomez Rodriguez, 2022; Hemmer et al., 2022). Verma et al. (2022) extend the one-vs-all-parameterized L2D to the multiple experts case. Mao et al. (2023) study a two-stage scenario for L2D with multiple experts. Mozannar et al. (2023) provide a linear-programming formulation that optimally solve L2D in the linear setting. Verma et al. (2023) incorporate a conformal inference technique for multiple experts. Tailor et al. (2024) formulate a L2D system that can cope with never-before-seen experts. Mao et al. (2024b) introduce regression with deferral to multiple experts. Mao et al. (2024a) present a theoretical study of surrogate losses and algorithms for L2D with multiple experts. L2D complements the missing part of LWA; that is, after a reject decision, instead of being discarded, samples are sent to remote experts for predictions. However, in L2D, the experts on the server side are assumed to be either human or well-trained ML models, which are fixed during the training process. L2D focuses on training the local model and the rejector under the existence of expert models. This framework cannot handle the cases described in Sec. 1, where local models are legacy systems that have been deployed to the client side, and retraining the local models is not available.
+
+Learning to Help (L2H). Learning to Help (L2H) (Wu & Sarwate, 2024) is a complementary model to L2D whose aim is to train a server model and rejector to enhance the usability of systems containing a "legacy" local ML model that is unavailable for updating or retraining. The prior work focused on binary classification in the PPR setting and proposes a surrogate loss function that requires model-based calibration. This cannot be directly extended to multi-class problems due to issues with differentiability. In this work we generalize this model to multi-class problems and more scenarios (see Appendix A for a comparison).
+
+# 2 PROBLEM FORMULATION
+
+We consider a decision system with two parties: a client and the server. We consider the client device with relatively limited computational power while the server has access to more computing resources. We study a multi-class problem in which the goal is to learn a prediction function  $f\colon \mathcal{X}\to \mathcal{Y}$ , where  $\mathcal{X}\subset \mathbb{R}^l$  is a space of instances/feature vectors with  $l$  dimensions and  $\mathcal{Y} = [K]\triangleq \{1,2,\dots ,K\}$  is a set of labels.
+
+The L2H decision system (See Fig. 1) is constrained to have a certain architecture: in operation, the client receives an input  $x$  and has two options: it can either make a prediction  $\hat{y}_{\mathrm{local}}$  locally on the client or forward the input ("defer")  $x$  to the remote server which produces a response  $\hat{y}_{\mathrm{remote}}$ .
+
+![](images/c5ae4ca455d0abbd8c405576292b59542e64b7881ea9f64e176e8e9ba7d32f5e.jpg)  
+Figure 1: Diagram of learning to help framework.
+
+We can represent this system by a tuple of functions  $(r(x), m(x), e(x))$ , where the client side has a reject function  $r(x)$  and prediction function  $m(x)$  and the server has a prediction function  $e(x)$ . The function  $r(x)$  makes a binary choice representing the client's choice to use from those two options and then lets either  $m(x)$  or  $e(x)$  make a prediction for  $x$ .
+
+We assume that the training algorithm has access to training set  $\{(x_i, y_i) : i \in [n]\}$  of  $n$  feature-label pairs in  $\mathcal{X} \times \mathcal{Y}$ . For the purposes of analysis we make the standard assumption (Vapnik, 2010; Shalev-Shwartz & Ben-David, 2014) that random variables of feature vector  $X$  and label  $Y$  are sampled independently and identically distributed (i.i.d.) according to an unknown distribution  $\mathcal{D}$ .
+
+The function  $r \colon \mathcal{X} \to \{\text{LOCAL}, \text{REMOTE}\}$ , known as the rejection rule or rejector, determines whether the client should label the input locally or send it to the remote server for labeling. Formally, if  $r(x) = \text{LOCAL}$ , the client uses the local classifier  $m(x)$  to provide a label. Conversely, if  $r(x) = \text{REMOTE}$ , the client forwards sample  $x$  to the server, which then labels the input using remote classifier  $e(x)$ .
+
+For the functions  $r(x)$ ,  $m(x)$ , and  $e(x)$ , there are four possible outcomes for a given sample  $(x, y)$ . The first event occurs when  $r(x) = \text{LOCAL}$  and  $m(x) = y$ , representing a correct decision by the client. The second event is when  $r(x) = \text{LOCAL}$  and  $m(x) \neq y$  indicate an error by the client. The third event occurs when  $r(x) = \text{REMOTE}$  and  $e(x) = y$ , representing a correct decision by the server. The fourth event is when  $r(x) = \text{REMOTE}$  and  $e(x) \neq y$ , indicate an error by the server. We define the costs associated with each of these outcomes as  $c_{\text{cc}}$ ,  $c_{\text{ce}}$ ,  $c_{\text{sc}}$ , and  $c_{\text{se}}$ , respectively. For the scenarios discussed in Sec. 1, our goal is to provide accurate predictions, satisfying  $c_{\text{cc}} \leq c_{\text{ce}}$  and  $c_{\text{sc}} \leq c_{\text{se}}$ , as well as instant predictions, satisfying  $c_{\text{cc}} \leq c_{\text{sc}}$  and  $c_{\text{ce}} \leq c_{\text{se}}$ . Based on four outcomes and corresponding costs, the general loss function of this two-side framework given sample  $(x, y)$  is defined as:
+
+$$
+\begin{array}{l} L (r, e, x, y, m) = c _ {\mathrm {c c}} \mathbf {1} _ {m (x) = y} \mathbf {1} _ {r (x) = \text {L O C A L}} + c _ {\mathrm {c e}} \mathbf {1} _ {m (x) \neq y} \mathbf {1} _ {r (x) = \text {L O C A L}} \tag {1} \\ + c _ {\mathrm {s c}} \mathbf {1} _ {e (x) = y} \mathbf {1} _ {r (x) = \text {R E M O T E}} + c _ {\mathrm {s e}} \mathbf {1} _ {e (x) \neq y} \mathbf {1} _ {r (x) = \text {R E M O T E}}, \\ \end{array}
+$$
+
+where  $\mathbf{1}_{[\cdot ]}$  is the indicator function. The expected loss, also known as risk, is defined as  $R(r,m,e)\triangleq \mathbf{E}_{(X,Y)\sim \mathcal{D}}[L(r,e,x,y,m)]$  . The Bayes Classifiers then defined as:  $r^B,m^B,e^B\in$  arg  $\min_{r,m,e}R(r,m,e)$  . Jointly training all three functions is trivial because the Bayes classifiers satisfy  $m^{B} = e^{B} = \arg \max_{i}\eta_{i}(x)$  , where
+
+$$
+\eta_ {i} (x) = P (Y = i \mid X = x) \tag {2}
+$$
+
+represents the regression function for  $i$ -th class. The proof is similar with Proposition 2 derived by Mozannar & Sontag (2020b). Since the Bayes client classifier equals the Bayes server classifier, together with the cost constraints stated above, we have  $r^B(x) = \mathrm{LOCAL}$ , for all  $x$ . No input will be sent to the server.
+
+Since jointly training three functions leads to a trivial solution, we focus on a constrained but still interesting case, learning to help (L2H), where the client classifier cannot be updated or retrained (Fig. 1). While the prior work on L2H only considers binary classification, in this paper we extend the approach to multi-class problems.
+
+# 2.1 GENERALIZED 0-1 LOSS FOR MULTI-CLASS CLASSIFICATION
+
+To extend L2H to multi-class classification, we consider a special case of the general loss function (equation 1), often referred to as the generalized 0-1 loss in binary classification for L2H. In the
+
+generalized 0-1 loss, the parameters are defined as follows:  $c_{\mathrm{cc}} = 0$ ,  $c_{\mathrm{ce}} = 1$ ,  $c_{\mathrm{sc}} = c_{\mathrm{e}}$ , and  $c_{\mathrm{se}} = c_{\mathrm{e}} + c_{1}$ . The idea behind this setup is that there is no cost when the local prediction is correct, but a cost of 1 is incurred when the local prediction is incorrect. Therefore, we set  $c_{\mathrm{cc}} = 0$  and  $c_{\mathrm{ce}} = 1$ . Additionally, recall the interesting settings we mentioned in Sec. 1; requesting assistance from a remote server is not free. Once a sample is sent to the server, a constant reject cost  $c_{\mathrm{e}}$  is incurred, regardless of the server's prediction. Therefore,  $c_{\mathrm{sc}} = c_{\mathrm{e}}$ . Furthermore, if the server prediction is incorrect, an additional penalty inaccuracy cost  $c_{1}$  is imposed to account for the mistake. It is important to note that  $c_{1}$  may be greater than 1, especially in critical scenarios such as medical diagnoses, where expert misdiagnoses can lead to more severe consequences. As a result, we set  $c_{\mathrm{se}} = c_{\mathrm{e}} + c_{1}$ . In summary, the generalized 0-1 loss for multi-class classification for L2H is defined as:
+
+$$
+\begin{array}{l} L _ {\text {g e n e r a l}} (r, e, x, y; m) = \mathbf {1} _ {m (x) \neq y} \mathbf {1} _ {r (x) = \text {L O C A L}} \\ + c _ {\mathrm {e}} \mathbf {1} _ {e (x) = y} \mathbf {1} _ {r (x) = \text {R E M O T E}} + \left(c _ {\mathrm {e}} + c _ {1}\right) \mathbf {1} _ {e (x) \neq y} \mathbf {1} _ {r (x) = \text {R E M O T E}}. \tag {3} \\ \end{array}
+$$
+
+As discussed previously, we assume that  $m(x)$  is fixed while jointly training  $r(x)$  and  $e(x)$ . The risk of equation 3 is defined as  $R_{\mathrm{general}} \triangleq \mathbf{E}_{(X,Y) \sim \mathcal{D}}[L_{\mathrm{general}}(r,e,x,y;m)]$  and the Bayes Classifiers for multi-class L2H is defined as:
+
+$$
+r ^ {B}, e ^ {B} \in \underset {r, e} {\arg \min } R _ {\text {g e n e r a l}} (r, e; m). \tag {4}
+$$
+
+The existence of these two Bayes classifiers will be proved in the next subsection.
+
+# 2.2 BAYES OPTIMAL REJECTOR AND SERVER CLASSIFIER
+
+In this subsection, we derive the Bayes classifiers as defined in equation 4 for both rejector and server classifier with a fixed client classifier  $m(x)$  under generalized 0-1 loss function in equation 3. As mentioned at the beginning of Sec. 2, the task we considered is multi-class classification. The client classifier  $m(x)$  and server classifier  $e(x)$  are multi-class classifiers with output to be one label among  $K$  classes. The rejector  $r(x)$  is a binary classifier with two possible labels: LOCAL and REMOTE, which means either making a prediction on the client or on the server. The rejector  $r(x)$  will give a label LOCAL to  $x$  if  $r(x) > 0$  and give a label REMOTE to  $x$  when  $r(x) \leq 0$ . Without loss of generality, we assume that for the Bayes classifier of rejector,  $r^B(x) \in \{+1, -1\}$ .
+
+As discussed after equation 3, we consider the case where the client classifier  $m$  is given and fixed during training. The output of the client classifier depends on input sample  $x$ , which can either be deterministic or stochastic, depending on the machine learning model of  $m$ . We formally consider the case where the client classifiers consist of  $K$  sub-functions, say  $[m_1(x), m_2(x), \dots, m_K(x)]$ . The  $i$ -th sub-function  $m_i(x)$  represents the score for  $i$ -th class on sample  $x$ . Then the prediction of client classifier is  $\arg \max_i m_i(x)$ . For simplicity, we assume that  $\arg \max_i m_i(x)$  is unique. Choosing the class that has the highest output score as prediction is a standard operation for multiclass classification. In the following analysis, we assume that the output of client is a random variable  $M$  conditioned on the input  $x$  (Neal, 2012).
+
+Theorem 2.1. Given a client classifier  $m(x)$ , the solutions of Bayes classifiers (defined in equation 4) for generalized 0-1 loss under the space of all measurable functions are:
+
+$$
+e ^ {B} = \arg \max  _ {i} \eta_ {i} (x), \tag {5}
+$$
+
+where  $\eta_i(x)$  is defined in equation 2 and
+
+$$
+r ^ {B} = \mathbf {1} \left[ \eta_ {j ^ {*} (x)} (x) > \left(1 - c _ {e} - c _ {1}\right) + c _ {1} \max  _ {i} \eta_ {i} (x) \right] \cdot 2 - 1, \tag {6}
+$$
+
+where  $j^{*}(x)\triangleq \arg \max_{j}m_{j}(x)$
+
+The proof is given in Appendix C. By Theorem 2.1, we find that the Bayes classifier for the servers  $e^{B}$  is the same as the Bayes classifier for single classifiers (right side of equation 5). Also, the Bayes classifier for rejector  $r^{B}$  compares the posterior risk (expected loss) for two different decisions: predicting on the client classifier or predicting on the server classifier. The function  $\eta_{j^{*}(x)}(x)$  is the regression function for class predicted by client classifier  $m(x)$ , while  $\max_{i}\eta_{i}(x)$  is the regression
+
+function of the class predicted by server classifier. A larger value of the regression function tends to pull a sample to the classifier corresponding to the  $\eta_{i}(x)$  because the regression function here means how likely that a sample belongs to the class predicted either by  $m(x)$  or  $e^{B}(x)$ . If we write the right-hand side of the inequality inside the indicator (from equation 6) in another form:  $(1 - c_{e} - c_{1}) + c_{1}\max_{i}\eta_{i}(x) = 1 - c_{e} - c_{1}(1 - \max_{i}\eta_{i}(x))$ , it's clear that larger  $c_{e}$  and  $c_{1}$  would impede reject decision. This result coincides with the intuition in scenarios we are interested in: if inquiring servers are likely to be more costly, we tend to finish tasks without asking for help too often.
+
+However, we cannot directly get the Bayes classifiers  $(e^{B}, r^{B})$  in real-world tasks. To see this, in reality, we don't have the knowledge of the distribution  $\mathcal{D}$  of the data set. Furthermore, we cannot approach the Bayes optimal classifiers through gradient-based methods because the generalized 0-1 loss (equation 3) is not differentiable since it contains variables in the indicator function. To solve this problem, we propose a surrogate loss function in Sec. 3. We show that the surrogate loss function is differentiable and convex, and optimal solutions of this surrogate loss function are consistent with the Bayes optimal classifiers.
+
+# 3 STAGE-SWITCHING SURROGATE LOSS FUNCTION
+
+In Sec. 2.2, the Bayes classifiers we derived minimize the risk of generalized 0-1 loss (equation 3). Since the distribution of the data set is unknown and the loss function is not differentiable, it's computationally intractable to get  $(e^{B}, r^{B})$  by solving equation 4. Another potential concern comes from the framework of the client-server system. In this system, the rejector is placed on the client side while the server classifier is placed on the server side, which requires synchronous updates between two sides while searching for the solution of the problem equation 4. In scenarios where the connection between the client and server is unstable, or bandwidth is limited as discussed in Sec. 1, continuously synchronizing the client and server during training becomes costly. These conditions necessitate that both the rejector and server classifier be capable of being trained in an asynchronous setting.
+
+Similar with the definition of client classifier  $m(x)$  in Sec. 2.2, we consider the case where server classifier  $e(x)$  also consists of  $K$  sub-functions, denoted as  $[e_1(x), e_2(x), \dots, e_K(x)]$ . The prediction of the server classifier is  $\arg \max_i e_i(x)$  and assumed to be unique. We further consider the case that the rejector consists of 2 sub-functions, say  $[r_1(x), r_2(x)]$ , where  $r_1$  stands for the score of LOCAL while  $r_2$  stands for the score of REMOTE. In accordance with the space of  $r^B$  as stated in Sec. 2.2, the output of the rejector is defined as  $r(x) \triangleq \mathbf{1}[r_1(x) > r_2(x)] \cdot 2 - 1$ . Based on the definitions stated above, we propose a stage-switching surrogate loss function, which is differentiable and can be used in both synchronous and asynchronous settings. The surrogate loss function is defined as:
+
+$$
+L _ {\mathrm {S}} (r, e, x, y; m) = L _ {1} (e, x, y) + L _ {2} (r, e, x, y; m) \tag {7}
+$$
+
+where
+
+$$
+L _ {1} (e, x, y) = - \ln \frac {\exp \left(e _ {y} (x)\right)}{\sum_ {j = 1} ^ {K} \exp \left(e _ {j} (x)\right)}, \tag {8}
+$$
+
+and
+
+$$
+\begin{array}{l} L _ {2} (r, e, x, y; m) = - \left(1 - c _ {e} - c _ {1} + c _ {1} \mathbf {1} _ {e (x) = y}\right) \ln \frac {\exp \left(r _ {2} (x)\right)}{\exp \left(r _ {2} (x)\right) + \exp \left(r _ {1} (x)\right)} \tag {9} \\ - \mathbf {1} _ {m (x) = y} \ln \frac {\exp (r _ {1} (x))}{\exp (r _ {2} (x)) + \exp (r _ {1} (x))}. \\ \end{array}
+$$
+
+Similarly, the surrogate risk is defined as
+
+$$
+R _ {\mathrm {S}} \triangleq \mathbf {E} _ {(X, Y) \sim \mathcal {D}} [ L _ {\mathrm {S}} (r, e, x, y; m) ]. \tag {10}
+$$
+
+The surrogate loss function is designed as the summation of two sub-loss functions,  $L_{1}$  and  $L_{2}$ . In the training process, we iteratively update the rejector or the server classifier while keeping the other
+
+fixed under each sub-loss function at one stage. Specifically, in server stage, we update the parameters of the server classifier under the sub-loss function  $L_{1}$  while keeping the rejector unchanged, and in rejector stage, we update the parameters of rejector  $r$  under sub-loss function  $L_{2}$  while keeping server classifier unchanged. The way of switching stages depends on specific synchronous or asynchronous settings. Notice that the client classifier  $m(x)$  is always fixed in all stages in the L2H framework. Detailed comparison with previous surrogate loss function for binary L2H is in Appendix A and comparison with direct extensions from other proposed surrogate loss functions in LWA and L2D is attached in Appendix B. The results show that previous works can not fully cover the settings that we are interested in.
+
+Based on the stage-switching training method, the surrogate loss function is differentiable in all stages. In the server stage, the sub-surrogate loss function  $L_{1}$  is the same as the cross-entropy loss function. In the rejector stage, for each given input sample  $x$  with true label  $y$ , the indicator  $\mathbf{1}_{e(x) = y}$  and  $\mathbf{1}_{e(x) = y}$  are constants, that is, either one or zero. In either case,  $L_{2}$  is differentiable since only the  $r_{1}$  and  $r_{2}$  are variables in this stage, and we can calculate the gradient with respect to  $r_{1}$  and  $r_{2}$ . What's more, both  $L_{1}$  and  $L_{2}$  are convex w.r.t.  $e$  as shown in the following subsection.
+
+# 3.1 CONVEXITY AND MONOTONICITY OF THE SURROGATE LOSS FUNCTION
+
+In the following proposition, we show that in both stages, the sub-surrogate loss function  $L_{1}$  and  $L_{2}$  are both convex or monotone.
+
+Proposition 3.1. For each given  $(x,y)$ , the loss function  $L_{1}$  is convex over  $e_i(x)$ , for any  $i\in [K]$ ; and the loss function  $L_{2}$  is:
+
+- convex over  $r_1(x)$  and  $r_2(x)$ , when  $1 - c_e - c_1 + c_1\mathbf{1}_{e = y} > 0$ ;  
+- monotonically decreasing over  $r_1$  and monotonically increasing over  $r_2$  when  $1 - c_e - c_1 + c_1\mathbf{1}_{e = y} \leq 0$ .
+
+The proof is given in Appendix D. The convexity of a function ensures that we can find the global minimizer through gradient-related optimization methods (Boyd & Vandenberghe, 2004; Nesterov, 2014). As for the monotonicity of  $L_{2}$  when  $1 - c_{e} - c_{1} + c_{1}\mathbf{1}_{e = y} \geq 0$ , we will show that in Theorem 3.2, this property can help for solving the corner case in the proof. Based on differentiability, convexity, or monotonicity, we prove in Sec. 3.2 that our proposed surrogate loss function is consistent, that is, any minimizer of the surrogate risk (equation 10) also minimizes the generalized 0-1 risk of equation 3, referring to the consistency defined in Section 2.2 by Cao et al. (2023b).
+
+# 3.2 CONSISTENCY OF SURROGATE LOSS FUNCTION
+
+In this subsection, we verify the consistency between the surrogate loss function (equation 7) and the generalized 0-1 loss function (equation 3). Formally, we prove the following theorem:
+
+Theorem 3.2. Under the space of all measurable functions, the surrogate loss function (equation 7) is consistent with the generalized 0-1 loss function (equation 3), that is, the minimizer of the risk of surrogate loss function also minimizes the risk of original loss function:
+
+$$
+r^{*},e^{*}\in \operatorname *{arg  min}_{r,e}R_{\text{general}}(r,e;m),
+$$
+
+for all  $r^*, e^* \in \arg \min_{r, e} R_{\mathrm{S}}(r, e; m)$ .
+
+The proof is given in Appendix E. As shown in Theorem 3.2, the function spaces for server classifier  $e$  and rejector  $r$  are both spaces of all measurable functions; training our model with this stage-switching surrogate loss function would eventually lead to Bayes classifiers, which is the global optimal solution.
+
+In summary, we propose a stage-switching surrogate loss function, which is differentiable, convex, or monotone, and theoretically prove that this surrogate loss function is consistent. Based on those properties of our surrogate loss function, practical optimization methods can be conducted to derive the empirical minimizer of surrogate risk  $R_{\mathrm{S}}$ , as defined in equation 10, with a finite number of data samples in real-world scenarios. In the next section, we propose computationally feasible algorithms for different settings PPR, IA, and BRR discussed in Sec. 1.
+
+# 4 COMPUTATIONALLY FEASIBLE ALGORITHMS FOR LEARNING TO HELP
+
+The stage-switching surrogate loss function that we propose in equation 7 ensures the usability of gradient-based optimization methods to train the rejector and server classifiers with flexibility in three settings we discuss in Section 1: PPR, IA, and BRR settings.
+
+The training in the server stage is the same for all three settings since  $L_{1}$  is only a function of the server classifier; no knowledge of the rejector or client classifier is needed. In the rejector stage, we set up different algorithms based on the constraints in different settings. Besides, for BRR settings, we propose post-hoc Algo. 4 and 5 after standard training. Since the stage-switching surrogate loss function is differentiable, any gradient-based optimization method can be used for training our model. To reduce the computation, we use Stochastic Gradient Descent (SGD) for presentation.
+
+Pay-Per-Request (PPR) Recall the scenarios we depict in Section 1, in the PPR setting, we consider that the client is always connected with the server; that is, the rejector has instant access to the latest version of server classifier  $e(x)$  during the whole training phase. This setting works for scenarios where the connection is in real-time, and the bandwidth is ample and free during training, but each request conducts payment.
+
+We design a synchronized algorithm, Algo. 1, that works for the PPR setting. The algorithm iteratively runs in the server stage, and in the rejector stage for each pair of samples and labels  $(x,y)$ . In the server stage, the server classifier  $e(x)$  is updated by the sub-loss  $L_{1}$ . In the rejector stage, the current  $e(x)$  is instantly shared with the rejector. Together with the outputs of the fixed client classifier and current server classifier, the sub-loss  $L_{2}$  is calculated to update the rejector.
+
+Algorithm 1 Optimization With Our Surrogate Loss Function  
+Input: Training set  $\{(x_i,y_i):i\in [n]\}$  , Fixed client classifier  $m$  , Rejector  $r^0$  , Sever classifier  $e^0$  , Constant cost  $c_{1}$  and  $c_{e}$  .   
+1: for  $t = 1$  to n do   
+2:  $L_1^t (x_t,y_t) = -\ln \frac{\exp(e_y_t(x_t))}{\sum_j\exp(e_j(x_t))}$    
+3:  $e^{t}\gets \mathrm{SGD}(L_{1}^{t}(x_{t},y_{t}),e^{t - 1})$    
+4:  $L_2^t (x_t,y_t)\quad = \quad -\mathbf{1}_{m(x_t) = y_t}\ln \frac{\exp(r_1(x_t))}{\exp(r_2(x_t)) + \exp(r_1(x_t))}\quad -\quad (1\quad -\quad c_e\quad -\quad c_1\quad +$ $c_{1}\mathbf{1}_{e^{t}(x_{t}) = y_{t}})\ln \frac{\exp(r_{2}(x_{t}))}{\exp(r_{2}(x_{t})) + \exp(r_{1}(x_{t}))}$    
+5:  $r^t\gets \mathrm{SGD}(L_2(x_t,y_t),r_{t - 1})$    
+6: end for   
+7: return  $r^n$ $e^n$
+
+Intermittent Availability (IA) In the IA setting, the connection between the client and the server is not always available. This scenario may happen in any distributed system that is connected to the public network because of traffic control. Also in private networks, connection may be lost due to instability of the network. If we still train our model with synchronized algorithm 1, the training process will be postponed once the connection is delayed or lost. Holding the status of both the rejector and server classifier will cause a waste of resources and time.
+
+Our stage-switching surrogate loss function in equation 7 consists of two sub-loss functions. An asynchronous algorithm is best to tackle those issues. The idea is to separately train the server classifier with  $L_{1}$  on the server side and train the rejector with  $L_{2}$  while keeping a temporary version of the server classifier  $e^{-}$  on the client side. The  $e^{-}$  will be updated only when the intermittent connection between client and server is built. Considering that the client and server only connect every  $S$  time slot we propose the asynchronous training algorithm in Algo. 2. In line 4 of Algo. 2, the server connects with the client and sends the current server classifier  $e$  as the latest  $e^{-}$ . In line 5, we calculate the  $L_{2}$  loss with stored server classifier  $e^{-}$ .
+
+Bounded-Reject-Rate (BRR) While a few works focus on the BRR setting in L2D and L2H frameworks, the setting is indeed realistic. For example, subscribers of ChatGPT or other AI assistants don't pay extra money for each inquiry to advanced models, but they are allowed to only
+
+Algorithm 2 Asynchronous Optimization With Our Surrogate Loss Function
+
+Input: Training set  $\{(x_i, y_i) \mid i \in [n]\}$ , Fixed client classifier  $m$ , Rejector  $r^0$ , Sever classifier  $e^0$ , Constant cost  $c_1$  and  $c_e$ , Synchronization interval  $S$ .
+
+1: for  $t = 1$  to  $n$  do  
+2:  $L_1^t (x_t,y_t) = -\ln \frac{\exp(e_{y_t}(x_t))}{\sum_j\exp(e_j(x_t))}$  
+3:  $e^t \gets \operatorname{SGD}(L_1^t(x_t, y_t), e^{t-1})$  
+4: if  $t \bmod S = 1$  then  $e^{-} \gets e^{t}$  end if  
+5:  $L_2^t (x_t,y_t) = -(1 - c_e - c_1 + c_1\mathbf{1}_{e^-(x_t) = y_t})\ln \frac{\exp(r_2(x_t))}{\exp(r_2(x_t)) + \exp(r_1(x_t))} -$ $\mathbf{1}_{m(x_t) = y_t}\ln \frac{\exp(r_1(x_t))}{\exp(r_2(x_t)) + \exp(r_1(x_t))}$  
+6:  $r^t\gets \mathsf{SGD}(L_2(x_t,y_t),r_{t - 1})$  
+7: end for  
+8: return  $r^n$ ,  $e^n$
+
+![](images/f39a717bc2c2794d43365c93d3c1154943673796b4f00a7b7805b99ce8625cf2.jpg)
+
+![](images/69631c5ab839dbfe92dae7813cb957b5cc338206c87f5284b5573eae72060949.jpg)
+
+![](images/435659e5c392a272d1c13f392b164b80e0111b9c410ead32329fc61d8deb31f8.jpg)
+
+![](images/dee0d6eb3a61ca1741dd48ef758e6c4dfa6cdab0fa08f99a369950dcdc8a61b9.jpg)
+
+![](images/54cd29bde794bbd982107f7dbf763f7643cf1ddd373c39e8a2d9c86c096ca342.jpg)  
+Figure 2: Impact of  $c_{e}$  nad  $c_{1}$  on accuracy and reject rate. First row: change of accuracy as reject rate changes; Second row: change of reject rate as reject cost  $c_{e}$  changes.
+
+![](images/f4df6bc4c256d1d5cc56b48cd6e171a11bf7f3873bf9ae848134d3ec19df4de6.jpg)
+
+![](images/499ce87dd320a261e016ab106ba21e401845dd5624a807399d7ae531231e093a.jpg)
+
+![](images/781d6ae6cd8e2f114d9f38448ddb0ce9d8134e91434d3106952ef3ba8a77df59.jpg)
+
+ask an assigned number of questions each hour during peak hours. In this setting, we consider that making a reject decision and sending samples to the server is free, but the number of samples that can be sent to the server in units of time, i.e., reject rate, has a fixed upper bound  $q$ . There are two concerns in this setting: a) the rejector should actively pick up the samples that have higher accuracy on the server classifier compared to the client classifier; b) the actual reject rate in the inference phase should be bounded.
+
+To address the concerns stated above, we propose stochastic post-hoc algorithms in Algo. 3, 4, and 5. These algorithms sample random variables to decide either sending extra samples to server when empirical reject rate is lower than upper bound  $q$  or keep a portion of rejected samples locally when it's higher than upper bound  $q$ . By conducting this stochastic post-hoc algorithm, we can ensure the bounded reject rate while making the best use of server classifier. The detailed discussions and algorithms are in Appendix F.
+
+# 5 EXPERIMENT
+
+In this section, we test the proposed surrogate loss function in equation 7 and algorithms for different settings on CIFAR-10 (Krizhevsky & Hinton, 2009), SVHN (Netzer et al., 2011), and CIFAR-100 (Krizhevsky & Hinton, 2009) datasets.
+
+In our experiments, the base network structure for the client classifier and the rejector is LeNet-5, and the server classifier is either AlexNet or ViT. The client classifier is trained solely and works as a fixed model in the following process. Since Algo. 1 is the basic algorithm for our multi-class L2H framework, we start from the evaluation for the PPR setting.
+
+Trade-off accuracy and reject rate on reject cost and inaccuracy cost In this experiment, we evaluate the impact of reject cost  $c_{e}$  and inaccuracy cost  $c_{1}$  on the overall accuracy and reject rate,
+
+Table 1: Contrastive Evaluation Results with  $c_{1} = 1.25$  and  $c_{e} = 0.25$  
+
+<table><tr><td rowspan="2"></td><td colspan="4">cifar10 (%)</td><td colspan="4">SVHN (%)</td></tr><tr><td>ratio</td><td>m</td><td>e</td><td>differ.</td><td>ratio</td><td>m</td><td>e</td><td>differ.</td></tr><tr><td>data with r(x) = LOCAL</td><td>44.11</td><td>73.9</td><td>81.9</td><td>8.0</td><td>91.71</td><td>90.6</td><td>93.3</td><td>2.7</td></tr><tr><td>data with r(x) = REMOTE</td><td>55.9</td><td>54.5</td><td>67.7</td><td>13.2</td><td>8.29</td><td>61.2</td><td>72.8</td><td>11.6</td></tr></table>
+
+![](images/106515177d14704ac323d2dd70ca75d19537b4fcfdedd7983c3866bf3ab4c783.jpg)  
+Figure 3: Comparison of the training loss for different synchronized settings.
+
+![](images/69e8e60a5142a1484f899d4fce2926991dbdb5e36e7eaa52705f567e36f7639e.jpg)
+
+respectively. Specifically, we choose  $c_{e}$  from an interval between [0, 0.5] with fixed inaccuracy costs  $c_{1} = 1$  and  $c_{1} = 1.25$ . From Fig. 2 we see that, incorporating a server classifier indeed helps increase the overall accuracy, while both the cost of rejection and inaccuracy on the server will balance the usage of the client classifier and server classifier.
+
+Contrastive evaluation over client and server classifiers To see what the rejector learns in training and how it works in the inference phase, we conduct a contrastive evaluation, where in the inference phase, we split the testing set into a rejected subset and non-rejected subset, according to the output of  $r(x)$ . We test the accuracy of the rejected and non-rejected subset on both the client classifier  $m(x)$  and the server classifier  $e(x)$ . The result is shown in Table 1, which implies that the rejector mostly only sends the samples that are predicted inaccurately on  $m(x)$  while predicted more accurately on  $e(x)$  to the server end. Column "ratio" indicates the percentage of samples over dataset have the same  $r(x)$ . For different  $c_{e}$  and  $c_{1}$ , the results are similar as shown in Appendix G.
+
+Convergence rate comparison over synchronization and asynchronization We compare the convergence rate and accuracy with models trained by Algo. 1 for PPR and that trained by Algo. 2 for IA. The accuracy performances are similar in both algorithms, as compared Table 1 with Table 7 in Appendix G. This result coincides with the comparison in Fig. 3, where a larger synchronization interval  $S$  causes a slower convergence rate but converges to the same level of loss. The experiment results are in line with our expectation since the minimizers of our surrogate loss function derived in Theorem 3.2, are independent to specific optimization algorithms.
+
+Experiments for stochastic post-hoc Algorithm 3, 4, and 5, its comparison with randomly reject under BRR settings, experiments on CIFAR-100 with server classifier to be ViT, together with other additional experiment materials, are in Appendix G.
+
+# 6 CONCLUSION
+
+In this paper, we extend the Learning to Help framework to address multi-class classification with different resource constraints: PAY-PER-REQUEST, INTERMITTENT AVAILABILITY, and BOUNDED REJECT RATE. By introducing a stage-switching surrogate loss function, we enabled effective training of both the server-side classifier and the rejector with computationally feasible algorithms for three settings. Our experimental results demonstrate that the proposed methods offer a practical and efficient solution for multi-class classification, aligning with the theoretical guarantees we established. This work opens new opportunities for further exploration of multi-party collaboration in hybrid machine learning systems, especially in the era of LLM.
+
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+
+# A COMPARISON WITH BINARY SURROGATE LOSS
+
+In this section, we compare the surrogate loss function proposed in binary case by Wu & Sarwate (2024) with our proposed method in Section 3.
+
+In this work, the formula of the surrogate loss function is:
+
+$$
+L _ {\mathrm {W u}} (r, e, x, y) = c _ {1} \exp \left(\frac {\beta}{2} (- e y - r)\right) + c _ {e} \exp (- r) + 1 _ {m (x) y \leq 0} \exp (\alpha r)
+$$
+
+where  $\alpha$  and  $\beta$  are parameters for calibration. If we directly make the extension to multi-class in our setting, we can keep the same definition of rejector while the output of client classifier and server classifier becomes  $\arg \max_{i}m_{i}(x)$  and  $\arg \max_{i}e_{i}(x)$ , which results in the surrogate loss:
+
+$$
+L _ {\mathrm {W u}} ^ {'} (r, e, x, y) = c _ {1} \exp \left(\frac {\beta}{2} (- \underset {i} {\arg \max} e _ {i} (x) y - r)\right) + c _ {e} \exp (- r) + 1 _ {\arg \max _ {i} m _ {i} (x) y \leq 0} \exp (\alpha r)
+$$
+
+The client classifier part works well since the client classifier is fixed during training. However, this part  $\arg \max_{i}m_{i}(x)y$  will lead to trouble since both  $\arg \max_{i}m_{i}(x)$  and  $y$  are index numbers of class in multi-class and therefore, the function is not differentiable with respect to  $e_i(x)$ .
+
+Furthermore, this surrogate loss function requires to calculate calibration parameters  $\alpha$  and  $\beta$ , which depends on the estimation of the accuracy of the client classifier, for each input sample  $x$  as shown in Theorem 2 in this work (Wu & Sarwate, 2024), which adds more calculations and estimation errors to practical experiments. This surrogate loss function is not available in the IA setting since it's jointly training the rejector and server classifier at the same time.
+
+Compared to the extension of the surrogate loss function proposed by Wu & Sarwate (2024) with our stage-switching surrogate loss function, our method is differentiable in multi-class and more flexible for fitting different settings.
+
+# B COMPARISON WITH DIFFERENT SURROGATE LOSS FUNCTIONS
+
+In this section, we justify that previous surrogate loss functions, proposed from LWA and L2D, can not be directly extended to L2H by comparing those surrogate loss functions with our method, shown in Table 2, on the properties that are needed in the settings that we are interested in.
+
+Confidence-LWA (or score-based-LWA) represents the methods in LWA that use the output  $m(x)$  or function of output  $f(m(x))$  of the client classifier as a metric of confidence on current prediction. If the value of the metric is smaller than a threshold, then the input sample  $x$  will be discarded as a result of rejection. The theoretical framework of Confidence-LWA was first proposed by Herbei & Wegkamp (2006). There is no separate rejector in those methods. The rejected samples will be abandoned since there will be no more actions after rejection. Separate rejector-LWA represents those surrogate loss functions in LWA that assume a separate rejector to independently decide whether to reject or not. This kind of method was first proposed by Cortes et al. (2016) and proves the inability of confidence-based methods in special cases, referring to Sec. 2.2 of this work. However, in the LWA framework, the rejection decision only depends on the local classifier  $m$  and does not consider the issues after rejection.
+
+In the L2D framework, samples are sent to an expert after rejection. Softmax-L2D, firstly proposed by Mozannar & Sontag (2020b), and OvA-L2D, firstly proposed by Verma & Nalisnick (2022) are two different classes of surrogate loss functions. Softmax-L2D adds a reject option as another class of task to the cross-entropy loss function, while OvA-L2D uses surrogate loss functions that transform multi-class classification into a combination of multi-binary classifiers. However, the output of the rejector on both methods also depends on the output of each class  $m_{i}(x)$  on mobile classifier  $m(x)$ . Moreover, during training, the client and the server must keep connection with each other.
+
+Two-stage (TS) confidence-L2D and two-stage (TS) separate rejector-L2D are those kinds of surrogate loss functions that consist of different sub-loss functions and can be trained in corresponding stages for both confidence and separate rejector cases, firstly proposed by Mao et al. (2023). Those two methods can be used for IA settings where the training is asynchronized since their surrogate loss
+
+function can be easily split and run in different locations. However, in two-stage confidence-L2D methods, the rejector takes the output of  $m(x)$  as input, while in two-stage separate rejector-L2D methods, the rejector only considers the performance of experts on different servers.
+
+For those methods that do not have a separate rejector, the limitation has been proved by Cortes et al. (2016). What's more, usually ML classifiers are more complex than rejectors in scale. In the scenarios of our PPR and IA settings, when the latency is a concern, a system with a separate reject is faster since in a non-separate rejector system, all samples are firstly sent to the client classifier and then decided by the rejector, which may take longer time.
+
+For those methods that the rejector does not learn from client classifier  $m(x)$  during training, the reject decision only depends on the performance of server classifier  $e(x)$ . For those methods that the rejector does not learn from server classifier  $e(x)$ , the reject decision only depends on the performance of client classifier  $m(x)$ . In our PPR and IA settings, asking the server for help is not free, so we have to balance the use of the server. A desired rejector should learn the ability to compare the performance of  $m(x)$  and  $e(x)$  for each input  $x$  as shown in the Bayes classifier by 2.1.
+
+Those methods that can not train in asynchronous fields may fail in the IA setting since the connection between client and server is unstable and intermittent.
+
+Based on the settings that are realistic and interesting in real-world scenarios, we require a surrogate loss function for a multi-class L2H system to have a separate rejector, and the rejector learns from the knowledge of both the client classifier and server classifier during training and can operate in asynchronous fashion. Previous works that stem from LWA and L2H can not be directly extended to L2H because they are either technically non-differentiable for training server classifiers or conceptually unmatched with the required properties stated above.
+
+Table 2: Difference among surrogate loss function proposed in LWA and L2D  
+
+<table><tr><td></td><td>Separate rejector</td><td>Rejector learns m(x)</td><td>Rejector learns e(x)</td><td>Async</td></tr><tr><td>Confidence-LWA</td><td>×</td><td>✓</td><td>×</td><td>-</td></tr><tr><td>Separate rejector-LWA</td><td>✓</td><td>✓</td><td>×</td><td>-</td></tr><tr><td>Softmax-L2D</td><td>×</td><td>✓</td><td>✓</td><td>×</td></tr><tr><td>OvA-L2D</td><td>×</td><td>✓</td><td>✓</td><td>×</td></tr><tr><td>TS confidence-L2D</td><td>×</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>TS separate rejector-L2D</td><td>✓</td><td>×</td><td>✓</td><td>✓</td></tr><tr><td>Ours</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr></table>
+
+# C PROOF OF THEOREM 2.1
+
+The standard approach to deriving the Bayes classifier for any given loss function is to compare the posterior risk associated with each possible decision of the classifier. The decision that minimizes the posterior risk is then chosen as the output. Recall in Section 2.1, the regression function is  $\eta_{i}(x) = P(Y = i|X = x)$  and the generalized 0-1 loss function defined in equation 3 is:
+
+$$
+\begin{array}{l} L _ {\text {g e n e r a l}} (r, e, x, y; m) = \mathbf {1} _ {m (x) \neq y} \mathbf {1} _ {r (x) = \text {L O C A L}} \\ + c _ {e} \mathbf {1} _ {e (x) = y} \mathbf {1} _ {r (x) = \text {R E M O T E}} + \left(c _ {e} + c _ {1}\right) \mathbf {1} _ {e (x) \neq y} \mathbf {1} _ {r (x) = \text {R E M O T E}}. \\ \end{array}
+$$
+
+Note that  $m(x)$  is also stochastic, as discussed in Section 2.2. The random variable corresponding to the client classifier is denoted by  $M$ . Then, for any given sample  $x$ , the posterior risk over the conditional distribution of  $Y$  and  $M$  of any decision  $r' \in \{\text{REMOTE}, \text{ LOCAL}\}$  and  $e' \in [K]$  is:
+
+$$
+\begin{array}{l} \mathbb {E} _ {y, m \mid x} L _ {\text {g e n e r a l}} \left(r ^ {\prime}, e ^ {\prime}, x, y; m\right) \\ = \mathbb {E} _ {y, m \mid x} \left(\mathbf {1} _ {m (x) \neq y} \mathbf {1} _ {r ^ {\prime} = \text {L O C A L}} + c _ {\mathrm {e}} \mathbf {1} _ {e ^ {\prime} = y} \mathbf {1} _ {r ^ {\prime} = \text {R E M O T E}} + \left(c _ {e} + c _ {1}\right) \mathbf {1} _ {e ^ {\prime} \neq y} \mathbf {1} _ {r ^ {\prime} = \text {R E M O T E}}\right) \\ = \mathbb {E} _ {m \mid x} (P (Y = e ^ {\prime} | X = x) (\mathbf {1} _ {m (x) \neq e ^ {\prime}} \mathbf {1} _ {r ^ {\prime} = \text {L O C A L}} + c _ {\mathrm {e}} \mathbf {1} _ {e ^ {\prime} = e ^ {\prime}} \mathbf {1} _ {r ^ {\prime} = \text {R E M O T E}}) \\ + P \left(Y \in \mathcal {Y} _ {e ^ {\prime}} \mid X = x\right) \left(\mathbf {1} _ {m (x) \notin \mathcal {Y} _ {e ^ {\prime}}} \mathbf {1} _ {r ^ {\prime} = \text {L O C A L}} + \left(c _ {\mathrm {e}} + c _ {1}\right) \mathbf {1} _ {e ^ {\prime} \notin \mathcal {Y} _ {e ^ {\prime}}} \mathbf {1} _ {r ^ {\prime} = \text {R E M O T E}}\right)), \\ \end{array}
+$$
+
+where  $\mathcal{Y}_{e'} = \mathcal{Y} \setminus \{e'\}$  is the set of all labels except for  $e'$ . Then,
+
+$$
+\begin{array}{l} \mathbb {E} _ {y, m \mid x} L _ {\text {g e n e r a l}} \left(r ^ {\prime}, e ^ {\prime}, x, y; m\right) \\ = \mathbb {E} _ {m \mid x} \left(\eta_ {e ^ {\prime}} (x) \left(\mathbf {1} _ {m (x) \neq e ^ {\prime}} \mathbf {1} _ {r ^ {\prime} = \text {L O C A L}} + c _ {\mathrm {e}} \mathbf {1} _ {r ^ {\prime} = \text {R E M O T E}}\right) \right. \\ + \left(1 - \eta_ {e ^ {\prime}} (x)\right) \left(\mathbf {1} _ {m (x) \notin \mathcal {Y} _ {e ^ {\prime}}} \mathbf {1} _ {r ^ {\prime} = \text {L O C A L}} + \left(c _ {\mathrm {e}} + c _ {1}\right) \mathbf {1} _ {r ^ {\prime} = \text {R E M O T E}}\right)) \\ = \mathbb {E} _ {m \mid x} \left(\mathbf {1} _ {r ^ {\prime} = \text {L O C A L}} \left(\eta_ {e ^ {\prime}} (x) \mathbf {1} _ {m (x) \neq e ^ {\prime}} + (1 - \eta_ {e ^ {\prime}} (x)) \mathbf {1} _ {m (x) \notin \mathcal {Y} _ {e ^ {\prime}}}\right) \right. \\ + \mathbf {1} _ {r ^ {\prime} = \text {R E M O T E}} \left(c _ {e} \eta_ {e ^ {\prime}} (x) + \left(c _ {e} + c _ {1}\right) \left(1 - \eta_ {e ^ {\prime}} (x)\right)\right)) \\ = \mathbb {E} _ {m \mid x} \left(\mathbf {1} _ {r ^ {\prime} = \text {L O C A L}} \left(\eta_ {e ^ {\prime}} (x) \mathbf {1} _ {m (x) \neq e ^ {\prime}} + \left(1 - \eta_ {e ^ {\prime}} (x)\right) \mathbf {1} _ {m (x) \notin \mathcal {Y} _ {e ^ {\prime}}}\right) \right. \\ + \mathbf {1} _ {r ^ {\prime} = \text {R E M O T E}} \left(c _ {e} + c _ {1} \left(1 - \eta_ {e ^ {\prime}} (x)\right)\right)) \\ = \mathbf {1} _ {r ^ {\prime} = \text {L O C A L}} \left(\eta_ {e ^ {\prime}} (x) P \left(M \neq e ^ {\prime} \mid X = x\right) + \left(1 - \eta_ {e ^ {\prime}} (x)\right) P \left(M \notin \mathcal {Y} _ {e ^ {\prime}} \mid X = x\right)\right) \\ + \mathbf {1} _ {r ^ {\prime} = \text {R E M O T E}} \left(c _ {e} + c _ {1} \left(1 - \eta_ {e ^ {\prime}} (x)\right)\right) \\ = \mathbf {1} _ {r ^ {\prime} = \text {L O C A L}} P (M \neq Y \mid X = x) + \mathbf {1} _ {r ^ {\prime} = \text {R E M O T E}} \left(c _ {e} + c _ {1} \left(1 - \eta_ {e ^ {\prime}} (x)\right)\right). \tag {11} \\ \end{array}
+$$
+
+Also, for any  $e'$ , the following inequality holds:
+
+$$
+\begin{array}{l} \mathbf {1} _ {r ^ {\prime} = \text {L O C A L}} P (M \neq Y \mid X = x) + \mathbf {1} _ {r ^ {\prime} = \text {R E M O T E}} \left(c _ {e} + c _ {1} \left(1 - \eta_ {e ^ {\prime}} (x)\right)\right) \\ \geq \mathbf {1} _ {r ^ {\prime} = \text {L O C A L}} P (M \neq Y \mid X = x) + \mathbf {1} _ {r ^ {\prime} = \text {R E M O T E}} \left(c _ {e} + c _ {1} \left(1 - \max  _ {i} \eta_ {i} (x)\right)\right) \tag {12} \\ \end{array}
+$$
+
+The equality in Inequality 12 holds if and only if the decision  $e'$  satisfies  $\eta_{e'}(x) = \max_i \eta_i(x)$ . By combining this with equation 11, the Bayes classifier  $e^B$  defined in equation 4, which minimizes  $\mathbb{E}_{y,m|x} L_{\text{general}}(r', e', x, y; m)$ , satisfies  $\eta_{e^B} = \max_i \eta_i(x)$ . Therefore, the Bayes classifier for the local client is:
+
+$$
+e ^ {B} = \underset {i} {\arg \max} \eta_ {i} (x).
+$$
+
+Recalling the lower bound given by equation 12, the following lower bound can be derived:
+
+$$
+\begin{array}{l} \mathbf {1} _ {r ^ {\prime} = \mathrm {L O C A L}} P (M \neq Y \mid X = x) + \mathbf {1} _ {r ^ {\prime} = \mathrm {R E M O T E}} \left(c _ {e} + c _ {1} \left(1 - \max  _ {i} \eta_ {i} (x)\right) \right. \\ \geq \min  \left\{P (M \neq Y \mid X = x), c _ {e} + c _ {1} \left(1 - \max  _ {i} \eta_ {i} (x) \right. \right\} \tag {13} \\ \end{array}
+$$
+
+Inequality 13 implies that for a given  $x$ , if  $P(M \neq Y \mid X = x) \geq c_{e} + c_{1}(1 - \max_{i} \eta_{i}(x))$ , the posterior risk reaches the lower bound when  $r' = \text{REMOTE}$ . Conversely, if  $P(M \neq Y \mid X = x) < c_{e} + c_{1}(1 - \max_{i} \eta_{i}(x))$ , the posterior risk reaches the lower bound when  $r' = \text{ LOCAL}$ . By the definition of the rejector, one way to construct the Bayes classifier of the rejector is:
+
+$$
+\begin{array}{l} r ^ {B} = \mathbf {1} [ P (M \neq Y | X = x) <   c _ {e} + c _ {1} (1 - \max _ {i} \eta_ {i} (x)) ] \cdot 2 - 1 \\ = \mathbf {1} [ 1 - P (M = Y | X = x) <   c _ {e} + c _ {1} (1 - \max  _ {i} \eta_ {i} (x)) ] \cdot 2 - 1 \\ = \mathbf {1} \left[ 1 - \eta_ {\operatorname {a r g} \max  _ {j} m _ {j} (x)} (x) <   c _ {e} + c _ {1} \left(1 - \max  _ {i} \eta_ {i} (x)\right) \right] \cdot 2 - 1 \\ = \mathbf {1} \left[ \eta_ {\arg \max _ {j} m _ {j} (x)} (x) > (1 - c _ {e} - c _ {1}) + c _ {1} \max _ {i} \eta_ {i} (x) \right] \cdot 2 - 1. \\ \end{array}
+$$
+
+In summary, the Bayes classifiers for 0-1 generalized loss are
+
+$$
+e ^ {B} = \arg \max _ {i} \eta_ {i} (x)
+$$
+
+and
+
+$$
+r ^ {B} = \mathbf {1} [ \eta_ {j ^ {*} (x)} (x) > (1 - c _ {e} - c _ {1}) + c _ {1} \max _ {i} \eta_ {i} (x) ] \cdot 2 - 1,
+$$
+
+where  $j^{*}(x)\triangleq \arg \max_{j}m_{j}(x)$
+
+# D PROOF OF PROPOSITION 3.1
+
+We first show the convexity of  $L_{1}$  and  $L_{2}$  w.r.t  $e$ . For  $L_{1}$  in equation 8, we notice that it equals to cross-entropy loss function for multi-class tasks. Referring to the proof by Cover & Thomas (2005), in which the cross-entropy is transformed into the form of KL divergence, which is convex, we can
+
+directly draw the conclusion that the  $L_{1}$  is a convex function over  $e_i(x)$  for any  $i \in [K]$  given  $(x, y)$ .
+
+Next, we prove the convexity of  $L_{2}$  w.r.t  $e$ . Recall the definition of  $L_{2}$  in equation 9:
+
+$$
+L _ {2} = - (1 - c _ {e} - c _ {1} + c _ {1} \mathbf {1} _ {e = y}) \ln \frac {\exp (r _ {2})}{\exp (r _ {2}) + \exp (r _ {1})} - \mathbf {1} _ {m = y} \ln \frac {\exp (r _ {1})}{\exp (r _ {2}) + \exp (r _ {1})}.
+$$
+
+As described in Section 3, once in the rejector stage, only the rejector  $r$  is variable. For convenience, we rewrite  $L_{2}$  in this form:
+
+$$
+f (r _ {1}, r _ {2}) = - a \ln \frac {\exp (r _ {2})}{\exp (r _ {2}) + \exp (r _ {1})} - b \ln \frac {\exp (r _ {1})}{\exp (r _ {2}) + \exp (r _ {1})},
+$$
+
+where  $a = 1 - c_{e} - c_{1} + c_{1}\mathbf{1}_{e = y}$  and  $b = \mathbf{1}_{m = y}$ , which are depends on  $(x,y)$ . The analysis can be divided into two cases:
+
+- case (1):  $a \leq 0$  
+- case (2):  $a > 0$ .
+
+In case (1), the partial derivative of  $f(r_{1}, r_{2})$  are:
+
+$$
+\frac {\partial f \left(r _ {1} , r _ {2}\right)}{\partial r _ {1}} = a \frac {\exp \left(r _ {1}\right)}{\left(\exp \left(r _ {2}\right) + \exp \left(r _ {1}\right)\right)} - b \frac {\exp \left(r _ {2}\right)}{\left(\exp \left(r _ {2}\right) + \exp \left(r _ {1}\right)\right)},
+$$
+
+and
+
+$$
+\frac {\partial f \left(r _ {1} , r _ {2}\right)}{\partial r _ {2}} = - a \frac {\exp \left(r _ {1}\right)}{\left(\exp \left(r _ {2}\right) + \exp \left(r _ {1}\right)\right)} + b \frac {\exp \left(r _ {2}\right)}{\left(\exp \left(r _ {2}\right) + \exp \left(r _ {1}\right)\right)}.
+$$
+
+Since  $b = \{+1, -1\}$ , we have that:
+
+$$
+\frac {\partial f \left(r _ {1} , r _ {2}\right)}{\partial r _ {1}} \leq a \frac {\exp \left(r _ {1}\right)}{\left(\exp \left(r _ {2}\right) + \exp \left(r _ {1}\right)\right)} \leq 0,
+$$
+
+and
+
+$$
+\frac {\partial f \left(r _ {1} , r _ {2}\right)}{\partial r _ {2}} \geq - a \frac {\exp \left(r _ {1}\right)}{\left(\exp \left(r _ {2}\right) + \exp \left(r _ {1}\right)\right)} \geq 0
+$$
+
+always holds, which means  $f(r_{1},r_{2})$  (or  $L_{2}$ ), are non-increasing function of  $r_1$  and nondecreasing function of  $r_2$ . For the case  $c_{\mathrm{e}} > c_{1} = 1$ , it is impossible for  $a = 0$ . Thus,  $f(r_{1},r_{2})$  (or  $L_{2}$ ), are monotonically decreasing function of  $r_1$  and monotonically increasing function of  $r_2$ .
+
+In case (2), where  $a > 0$ , we have that  $a + b > 0$  always holds. Then we calculate  $\frac{f(r_1, z_1)}{2}$  and  $\frac{f(r_1, z_2)}{2}$  for any  $z_1$  and  $z_2$ :
+
+$$
+\begin{array}{l} (f (r _ {1}, z _ {1}) + f (r _ {1}, z _ {2})) / 2 \\ = \frac {1}{2} \left(- a \left(z _ {1} - \ln \left(e ^ {z _ {1}} + e ^ {r _ {1}}\right)\right) \right. \\ - b \left(r _ {1} - \ln \left(e ^ {r _ {1}} + e ^ {z _ {1}}\right)\right) - a \left(z _ {2} - \ln \left(e ^ {z _ {2}} + e ^ {r _ {1}}\right)\right) - b \left(r _ {1} - \ln \left(e ^ {r _ {1}} + e ^ {z _ {2}}\right)\right)) \\ = \frac {1}{2} \left(- a z _ {1} + a \ln \left(e ^ {z _ {1}} + e ^ {r _ {1}}\right) - 2 b r _ {1} \right. \\ + b \ln \left(e ^ {r _ {1}} + e ^ {z _ {1}}\right) - a z _ {2} + a \ln \left(e ^ {z _ {2}} + e ^ {r _ {1}}\right) + b \ln \left(e ^ {z _ {2}} + e ^ {r _ {1}}\right)) \\ = \frac {1}{2} (- a (z _ {1} + z _ {2}) - 2 b r _ {1} + (a + b) \ln (e ^ {z _ {1} + z _ {2}} + e ^ {r _ {1} + z _ {2}} + e ^ {r _ {1} + z _ {1}} + e ^ {2 r _ {1}})), \\ \end{array}
+$$
+
+and
+
+$$
+\begin{array}{l} f \left(r _ {1}, \left(z _ {1} + z _ {2}\right) / 2\right) = - a \left(\left(z _ {1} + z _ {2}\right) / 2 - \ln \left(e ^ {\left(z _ {1} + z _ {2}\right) / 2} + e ^ {r _ {1}}\right)\right) - b \left(r _ {1} - \ln \left(e ^ {r _ {1}} + e ^ {\left(z _ {1} + z _ {2}\right) / 2}\right)\right) \\ = \frac {- a \left(z _ {1} + z _ {2}\right)}{2} + a \ln \left(e ^ {\left(z _ {1} + z _ {2}\right) / 2} + e ^ {r _ {1}}\right) - b r _ {1} + b \ln \left(e ^ {r _ {1}} + e ^ {\left(z _ {1} + z _ {2}\right) / 2}\right). \\ \end{array}
+$$
+
+The subtraction between them is:
+
+$$
+\begin{array}{l} \left(f \left(r _ {1}, z _ {1}\right) + f \left(r _ {1}, z _ {2}\right)\right) / 2 - f \left(r _ {1}, \left(z _ {1} + z _ {2}\right) / 2\right) \tag {14} \\ = \frac {1}{2} (a + b) \ln \left(e ^ {z _ {1} + z _ {2}} + e ^ {r _ {1} + z _ {2}} + e ^ {r _ {1} + z _ {1}} + e ^ {2 r _ {1}}\right) \\ - (a + b) \ln \left(e ^ {\left(z _ {1} + z _ {2}\right) / 2} + e ^ {r _ {1}}\right) \\ = (a + b) \ln \frac {\sqrt {\left(e ^ {z _ {1}} + e ^ {r _ {1}}\right) \left(e ^ {z _ {2}} + e ^ {r _ {1}}\right)}}{e ^ {(z _ {1} + z _ {2}) / 2} + e ^ {r _ {1}}} \\ \end{array}
+$$
+
+We notice that exponential function  $\exp (z)$  is convex function such that
+
+$$
+\frac {e ^ {z _ {1}} + e ^ {z _ {2}}}{2} \geq e ^ {(z _ {1} + z _ {2}) / 2}
+$$
+
+holds for any  $z_{1},z_{2}$  . Since
+
+$$
+\begin{array}{l} \frac {e ^ {x} + e ^ {y}}{2} \geq e ^ {(x + y) / 2} \\ \Leftrightarrow e ^ {r _ {1} + y} + e ^ {r _ {1} + x} \geq 2 e ^ {(x + y) / 2 + r _ {1}} \\ \Leftrightarrow \left(e ^ {x} + e ^ {r _ {1}}\right) \left(e ^ {y} + e ^ {r _ {1}}\right) \geq \left(e ^ {(x + y) / 2} + e ^ {r _ {1}}\right) ^ {2} \\ \Leftrightarrow \frac {\sqrt {(e ^ {x} + e ^ {r _ {1}}) (e ^ {y} + e ^ {r _ {1}})}}{e ^ {(x + y) / 2} + e ^ {r _ {1}}} \geq 1, \\ \end{array}
+$$
+
+and  $a + b > 0$ , we prove that the for equation 14:
+
+$$
+(f (r _ {1}, z _ {1}) + f (r _ {1}, z _ {2})) / 2 - f (r _ {1}, (z _ {1} + z _ {2}) / 2) \geq 0
+$$
+
+always holds, that is  $f(r_1, r_2)$  is a convex function over  $r_2$ . Following similar steps, we also prove that  $f(r_1, r_2)$  is a convex function over  $r_1$ .
+
+# E PROOF OF THEOREM 3.2
+
+The surrogate loss function defined in equation 7 consists of  $L_{1}$  and  $L_{2}$ , which are targeted to the server classifier and rejector, respectively. Since the rejector and server classifier are trained in a stage-switching manner as described in Section 3, parameters of  $e$  are only updated by the gradient derived from  $L_{1}$ , and that of  $r$  is only updated by the gradient derived from  $L_{2}$ . Therefore, the minimizer of  $e$  is derived from the risk of  $L_{1}$  in the server stage, and the minimizer of  $r$  is derived from the risk of  $L_{2}$  (with the current server classifier) in the rejector stage. The risks are:
+
+$$
+\mathbb {E} _ {y, m \mid x} L _ {1} = - \sum_ {y \in \mathcal {Y}} \eta_ {y} (x) \log \left(\frac {\exp \left(e _ {y} (x)\right)}{\sum_ {y ^ {\prime} \in \mathcal {Y}} \exp \left(e _ {y ^ {\prime}} (x)\right)}\right)
+$$
+
+and
+
+$$
+\mathbb {E} _ {y, m \mid x} L _ {2} = - \mathbb {E} _ {y, m \mid x} (1 - c _ {e} - c _ {1} + c _ {1} \mathbf {1} _ {e = y}) \ln \frac {\exp {(r _ {2})}}{\exp {(r _ {2})} + \exp {(r _ {1})}} - \mathbb {E} _ {y, m \mid x} \mathbf {1} _ {m = y} \ln \frac {\exp {(r _ {1})}}{\exp {(r _ {2})} + \exp {(r _ {1})}}
+$$
+
+Since the Bayes classifiers are optimal for any  $x$  according to the definition in equation 4, it's equivalent to deriving the minimizer of risk of each sub-surrogate loss function for any given  $x$ .
+
+Note that  $L_{1}$  is a cross-entropy loss function for server classifier  $e(x)$ ; we can directly get the minimizers for each  $e_{i}$  by the arguments in Section 10.6 proposed by Hastie et al. (2009). Let's denote the minimizers of each sub-function  $e_{i}$ , which is defined in Section 3, of server classifier by  $e_{i}^{*}$ . The minimizes  $e_{i}^{*}$  satisfy the following condition:
+
+$$
+\frac {\exp \left(e _ {i} ^ {*} (x)\right)}{\sum_ {j} \exp \left(e _ {j} ^ {*} (x)\right)} = \eta_ {i} (x), \quad \forall i,
+$$
+
+where  $\eta_{i}(x)$  is the regression function for  $i$ -th class. Therefore, the  $e^{*}(x) = \arg \max_{i} e_{i}^{*}(x) = \arg \max_{i} \eta_{i}(x) = e^{B}(x)$ . The minimizer of server classifier  $e^{*}$  is the same as the Bayes classifier for server  $e^{B}$ .
+
+Next, we consider the rejector  $r(x)$ . Since  $e(x) = \arg \max_{j} e_{j}(x)$ , when we plug in the minimizers  $e_{i}^{*}(x)$  to  $L_{2}$ , the risk of  $L_{2}$  conditioned on  $x$  becomes:
+
+$$
+\begin{array}{l} \mathbb {E} _ {y, m \mid x} L _ {2} = \mathbb {E} _ {y, m \mid x} \left(- \mathbf {1} _ {e = y} \ln \frac {\exp (r _ {2})}{\exp (r _ {2}) + \exp (r _ {1})} - \mathbf {1} _ {m = y} \ln \frac {\exp (r _ {1})}{\exp (r _ {2}) + \exp (r _ {1})}\right) \\ = - \left(1 - c _ {e} - c _ {1} + c _ {1} \max  _ {i} \eta_ {i} (x)\right) \ln \frac {\exp \left(r _ {2}\right)}{\exp \left(r _ {2}\right) + \exp \left(r _ {1}\right)} \\ - \eta_ {j ^ {*} (x)} \ln \frac {\exp (r _ {1})}{\exp (r _ {2}) + \exp (r _ {1})}, \\ \end{array}
+$$
+
+where  $j^{*}(x) = \arg \max_{i}m_{i}(x)$  as defined in Theorem 2.1. The following analysis will be divided into two cases:
+
+- case (a):  $\left( {1 - {c}_{e} - {c}_{1}}\right)  + {c}_{1}\mathop{\max }\limits_{{i}}{\eta }_{i}\left( x\right)  \leq  0$  ,  
+- case (b):  $\left( {1 - {c}_{e} - {c}_{1}}\right)  + {c}_{1}\mathop{\max }\limits_{i}{\eta }_{i}\left( x\right)  > 0$  .
+
+In case (a), when  $(1 - c_{e} - c_{1}) + c_{1}\max_{i}\eta_{i}(x)\leq 0$ ,  $\partial \mathbb{E}_{y,m|x}L_2 / \partial r_1\leq 0$  always holds, such that  $\mathbb{E}_{y,m|x}L_2$  is monotonically decreasing function of  $r_1$ . And  $\partial \mathbb{E}_{y,m|x}L_2 / \partial r_2 > 0$  always holds such that  $\mathbb{E}_{y,m|x}L_2$  monotonically increasing function of  $r_2$ . In fact, we can also draw the same conclusions by using the monotonicity from Proposition 3.1 since the expectation of monotone function is still monotone. Therefore, in this case,
+
+$$
+\frac {\exp \left(r _ {1} ^ {*}\right)}{\left(\exp \left(r _ {2} ^ {*}\right) + \exp \left(r _ {1} ^ {*}\right) \right.} \longrightarrow 1 \quad \text {a n d} \quad \frac {\exp \left(r _ {2} ^ {*}\right)}{\left(\exp \left(r _ {2} ^ {*}\right) + \exp \left(r _ {1} ^ {*}\right) \right.} \longrightarrow 0
+$$
+
+That means  $r_1^* > r_2^*$  always holds. Therefore, the rejector always satisfies:
+
+$$
+r ^ {*} (x) = \mathbf {1} \left[ r _ {1} ^ {*} (x) > r _ {2} ^ {*} (x) \right] \cdot 2 - 1 = 1
+$$
+
+Also, in this case, the Bayes classifier for rejector is  $r^B = \mathbf{1}[\eta_{\arg \max_i m_i(x)}(x) > (1 - c_e - c_1) + c_1\max_i\eta_i(x)]\cdot 2 - 1 = 1\cdot 2 - 1 = 1$ . Therefore,  $r^{*}(x) = r^{B}(x)$ .
+
+Then we consider the case (b) when  $(1 - c_{e} - c_{1}) + c_{1}\max_{i}\eta_{i}(x) > 0$ . Since  $L_{1}$  is convex according to Proposition 3.1, and expectation of convex function is still convex proved by Boyd & Vandenberghe (2004),  $\mathbb{E}_{y|x}L_{2}$  is convex, and its minimizers are achieved when partial derivative equal to 0. We take the partial derivative over  $r_1$  and  $r_2$ :
+
+$$
+\frac {\partial \mathbb {E} _ {y | x} L _ {2}}{\partial r _ {1}} = (1 - c _ {e} - c _ {1} + c _ {1} \max  _ {i} \eta_ {i} (x)) \frac {\exp (r _ {1})}{(\exp (r _ {2}) + \exp (r _ {1}))} - \eta_ {j ^ {*} (x)} \frac {\exp (r _ {2})}{(\exp (r _ {2}) + \exp (r _ {1}))},
+$$
+
+and
+
+$$
+\frac {\partial \mathbb {E} _ {y | x} L _ {2}}{\partial r _ {2}} = - (1 - c _ {e} - c _ {1} + c _ {1} \max  _ {i} \eta_ {i} (x)) \frac {\exp (r _ {1})}{(\exp (r _ {2}) + \exp (r _ {1}))} + \eta_ {j ^ {*} (x)} \frac {\exp (r _ {2})}{(\exp (r _ {2}) + \exp (r _ {1}))}.
+$$
+
+Setting the partial derivatives to 0, we have that
+
+$$
+\frac {\exp \left(r _ {1} ^ {*}\right)}{\exp \left(r _ {2} ^ {*}\right)} = \frac {\eta_ {j ^ {*} (x)} (x)}{1 - c _ {e} - c _ {1} + c _ {1} \operatorname* {m a x} _ {i} \eta_ {i} (x)}.
+$$
+
+Then, the rejector becomes:
+
+$$
+\begin{array}{l} r ^ {*} (x) = \mathbf {1} \left[ r _ {1} ^ {*} (x) > r _ {2} ^ {*} (x) \right] \cdot 2 - 1 \\ = \mathbf {1} \left[ \exp \left(r _ {1} ^ {*} (x)\right) > \exp \left(r _ {2} ^ {*} (x)\right) \right]) \cdot 2 - 1 \\ = \mathbf {1} \left[ \frac {\exp \left(r _ {1} ^ {*} (x)\right)}{\exp \left(r _ {2} ^ {*} (x)\right)} > 1 \right]) \cdot 2 - 1 \\ = \mathbf {1} \left[ \eta_ {j ^ {*} (x)} (x) > \left(1 - c _ {e} - c _ {1}\right) + c _ {1} \max  _ {i} \eta_ {i} (x) \right] \cdot 2 - 1 \\ = r ^ {B} (x). \\ \end{array}
+$$
+
+We prove that in both cases (a) and (b), the minimizer of  $L_{2}$ ,  $r^{*}(x)$ , equals the Bayes Classifier of rejector  $r^{B}(x)$ .
+
+In summary, we prove that  $(r^{B}(x),e^{B}(x)) = (r^{*}(x),e^{*}(x))$  for any  $x$ , so the stage-switching surrogate loss function is consistent with the generalized 0-1 loss function.
+
+# F STOCHASTIC POST-HOC ALGORITHM FOR BOUNDED REJECT RATE SETTING
+
+We propose a stochastic post-hoc method, which is shown in Algo. 3, Algo. 4 and Algo 5. In this algorithm for BRR settings, we firstly train our rejector and server classifier following the process as shown in Algo. 3, which is similar to algorithms for PPR or IA settings (can also be trained with asynchronous algorithm 2). After training, we use a calibration set  $D_{\mathrm{cal}}$  to calculate the empirical reject rate  $q_{1}$ , based on the output of rejector  $r(x)$ . Once we get the empirical reject rate, we compare it with the bounded reject rate  $q$ .
+
+If  $q$  is greater than  $q_{1}$ , that means we have to sacrifice several rejected samples to force them to make prediction locally at client classifier  $m(x)$ ; the input samples are following the Algo. 4, where each rejected samples are made a final decision by the value of a uniform random variable. Once the value is below the ratio  $p$ , the sample will still be sent to the server. Otherwise, it may be predicted by the client classifier. When the empirical reject rate is higher than the bounded reject rate, after adding this post-hoc mechanism, the bounded reject rate  $q$  can just ensured.
+
+If  $q$  is smaller than or equal to  $q_{1}$ , that means we can reject more times than the rejector  $r(x)$  request. Then, we follow a similar mechanism in Algo. 5, which uses a random variable to decide which samples the rejector sends to the server classifier. After using the post-hoc mechanism, we can ensure the bounded reject rate while making the best use of the server classifier.
+
+Algorithm 3 Training Process of Stochastic Post-hoc Method  
+Input: Training set  $D = \{(x_{i},y_{i}):i\in [n]\}$  , Calibration set  $D_{\mathrm{cal}} = \{(x_i,y_i):i\in [k]\}$  , Fixed client classifier m, Rejector  $r^0$  , Sever classifier  $e^0$  , Bounded reject rate q.   
+1: for t = 1 to n do   
+2:  $L_1^t (x_t,y_t) = -\ln \frac{\exp(e_{yt}(x_t))}{\sum_j\exp(e_j(x_t))}$    
+3:  $e^{t}\gets \operatorname {SGD}(L_{1}^{t}(x_{t},y_{t}),e^{t - 1})$    
+4:  $L_2^t (x_t,y_t)\quad = \quad -(1 - c_e\quad -\quad c_1\quad +\quad c_1\mathbf{1}_{e^t (x_t) = y_t})\ln \frac{\exp(r_2(x_t))}{\exp(r_2(x_t))+\exp(r_1(x_t))}\quad -$    
+1m(xt)=yt ln exp(r1(xt)) exp(r2(xt))+exp(r1(xt))   
+5:  $r^t\gets \operatorname {SGD}(L_2(x_t,y_t),r_{t - 1})$    
+6: end for   
+7: q1<-EmpiricalRejectRate(Dcali)   
+8: return rn, en, q, q1
+
+Algorithm 4 Inference Phase of Stochastic Post-hoc Method When  $q < q_{1}$  
+Input: Client Classifier  $m$ , Trained Rejector  $r^n$ , Trained Sever Classifier  $e^n$ , Input Sample  $x$ , Bounded reject rate  $q$ , empirical reject rate  $q_1$ .  
+1:  $p = q / q_1$   
+2: if  $r(x) \leq 0$  then  
+3: Sample  $i$  from (0,1) uniform distribution.  
+4: if  $i \leq p$  then  
+5:  $\hat{y} \gets e^n(x)$   
+6: else  
+7:  $\hat{y} \gets m(x)$   
+8: end if  
+9: else  
+10:  $\hat{y} \gets m(x)$   
+11: end if  
+12: return  $\hat{y}$
+
+Algorithm 5 Inference Phase of Stochastic Post-hoc Method when  $q \geq q_1$  
+Input: Client Classifier  $m$ , Trained Rejector  $r^n$ , Trained Sever Classifier  $e^n$ , Input Sample  $x$ , Bounded reject rate  $q$ , empirical reject rate  $q_1$ .  
+1:  $p = (q - q_1) / (1 - q_1)$   
+2: if  $r(x) > 0$  then  
+3: Sample  $i$  from  $(0,1)$  uniform distribution.  
+4: if  $i \leq p$  then  
+5:  $\hat{y} \gets e^n(x)$   
+6: else  
+7:  $\hat{y} \gets m(x)$   
+8: end if  
+9: else  
+10:  $\hat{y} \gets e^n(x)$   
+11: end if  
+12: return  $\hat{y}$
+
+# G EXPERIMENT DESCRIPTION AND ADDITIONAL RESULTS
+
+Datasets We test our proposed surrogate loss function 7 and algorithms for different settings on CIFAR-10 (Krizhevsky & Hinton, 2009), SVHN (Netzer et al., 2011) and CIFAR-100 (Krizhevsky & Hinton, 2009) data sets. CIFAR-10 consists of  $32 \times 32$  color images drawn from 10 classes and is split into 50000 training and 10000 testing images. CIFAR-100 has 100 classes containing 600 images each. SVHN is obtained from house numbers with over 600000 photos in Google Street View images and has 10 classes with  $32 \times 32$  images centered around a single character. The experiments are conducted in RTX 3090. It takes approximately 5 minutes to train the local model and 30 minutes to train each remote server classifier with the rejector when server classifier is set to be AlexNet or 5 hours to train each remote server classifier with the rejector when server classifier is set to be ViT.
+
+Comparison over synchronization and asynchronizaton We conduct additional experiments training the server model and rejector under both synchronized, as shown in Algo 1 and asynchronous settings as shown in Algo 2. The model is evaluated in the same way as in Section 5. The contrastive evaluation and loss curve are shown in Table 3 and Figure 4.
+
+Result for different  $c_{1}$  and  $c_{e}$  on CIFAR-10 and SVHN Besides the result we show in Section 5, we add extensive result for contrastive evaluation with different inaccuracy cost  $c_{1}$ , reject cost  $c_{e}$  and synchronization interval  $S$  (for IA setting) for CIFAR-10 and SVHN with local classifier and rejector being LeNets and remote classifier being AlexNet. The results are shown in Table 3.
+
+Result for different  $c_{1}$  and  $c_{e}$  on CIFAR-100 Since our stage-switching surrogate loss function doesn't set up constraints on the size of dataset and structures of machine learning methods, we test the Contrastive Evaluation on dataset CIFAR100 with local classifier and rejector being LeNet and remote classifier being ViT, a transformer-based vision neural network. The results are shown in Table 4. Both higher  $c_{1}$  and  $c_{e}$  can reduce the ratio of numbers that are sent to remote classifier, but in our experimental results, the reject rate is more sensitive to  $c_{e}$ . When  $c_{e}$  is close to 1, almost no sample is sent to remote classifier, which makes sense because  $c_{e} = 1$  means that the cost of asking for help is always greater than locally prediction.
+
+Samples partitioned by rejector We choose one case where the rejector and remote classifier AlexNet are trained on SVHN when  $c_{1} = 1.25$  and  $c_{e} = 0.25$ . Then we let rejector partition the test set and randomly pick images from the subset with  $r(x) = \text{REMOTE}$  and  $r(x) = \text{LOCAL}$ , respectively. The results are shown in Fig. 5 and Fig. 6. It's obvious that samples kept locally are clearer, contrast, focused, with high-resolution, while the samples that are supposed to send to the server are blurry, unfocused, with low-resolution. Without any manual adjustment, our rejector learns to assess "difficulty" using the same metrics as humans, by interacting with both local model and server model during the training process.
+
+Table 3: Contrastive Evaluation Results for different cost and synchronization interval  
+
+<table><tr><td rowspan="2">S</td><td rowspan="2">c1</td><td rowspan="2">ce</td><td rowspan="2"></td><td colspan="3">cifar10 (%)</td><td colspan="3">SVHN (%)</td></tr><tr><td>m</td><td>e</td><td>differ.</td><td>m</td><td>e</td><td>differ.</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">1.0</td><td rowspan="2">0.25</td><td>r(x) = LOCAL</td><td>74.8</td><td>82.8</td><td>8.0</td><td>90.1</td><td>93.2</td><td>3.1</td></tr><tr><td>r(x) = REMOTE</td><td>54.5</td><td>68.6</td><td>14.1</td><td>62.0</td><td>72.5</td><td>10.5</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">1.25</td><td rowspan="2">0.25</td><td>r(x) = LOCAL</td><td>73.9</td><td>81.9</td><td>8.0</td><td>90.6</td><td>93.3</td><td>2.7</td></tr><tr><td>r(x) = REMOTE</td><td>54.5</td><td>67.7</td><td>13.2</td><td>61.2</td><td>72.8</td><td>11.6</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">1.25</td><td rowspan="2">0.0</td><td>r(x) = LOCAL</td><td>82.3</td><td>86.8</td><td>4.6</td><td>97.2</td><td>97.9</td><td>0.7</td></tr><tr><td>r(x) = REMOTE</td><td>62.6</td><td>73.8</td><td>11.2</td><td>87.6</td><td>90.8</td><td>3.2</td></tr><tr><td rowspan="2">100</td><td rowspan="2">1.0</td><td rowspan="2">0.25</td><td>r(x) = LOCAL</td><td>75.9</td><td>84.3</td><td>8.4</td><td>90.6</td><td>92.9</td><td>2.3</td></tr><tr><td>r(x) = REMOTE</td><td>55.3</td><td>69.5</td><td>14.1</td><td>63.6</td><td>72.7</td><td>9.1</td></tr><tr><td rowspan="2">100</td><td rowspan="2">1.25</td><td rowspan="2">0.25</td><td>r(x) = LOCAL</td><td>74.5</td><td>83.3</td><td>8.8</td><td>91.4</td><td>93.5</td><td>2.2</td></tr><tr><td>r(x) = REMOTE</td><td>55.5</td><td>69.9</td><td>14.4</td><td>65.4</td><td>73.9</td><td>8.5</td></tr><tr><td rowspan="2">100</td><td rowspan="2">1.25</td><td rowspan="2">0.0</td><td>r(x) = LOCAL</td><td>81.4</td><td>87.4</td><td>6.0</td><td>97.0</td><td>97.8</td><td>0.8</td></tr><tr><td>r(x) = REMOTE</td><td>62.6</td><td>75.3</td><td>12.7</td><td>87.3</td><td>90.6</td><td>3.3</td></tr><tr><td rowspan="2">1000</td><td rowspan="2">1.0</td><td rowspan="2">0.25</td><td>r(x) = LOCAL</td><td>74.8</td><td>83.1</td><td>8.2</td><td>90.5</td><td>93.2</td><td>2.7</td></tr><tr><td>r(x) = REMOTE</td><td>54.7</td><td>69.2</td><td>14.6</td><td>64.3</td><td>72.9</td><td>8.6</td></tr><tr><td rowspan="2">1000</td><td rowspan="2">1.25</td><td rowspan="2">0.25</td><td>r(x) = LOCAL</td><td>75.2</td><td>83.3</td><td>8.1</td><td>90.8</td><td>93.9</td><td>3.1</td></tr><tr><td>r(x) = REMOTE</td><td>56.0</td><td>69.7</td><td>13.7</td><td>65.1</td><td>75.6</td><td>10.5</td></tr><tr><td rowspan="2">1000</td><td rowspan="2">1.25</td><td rowspan="2">0.0</td><td>r(x) = LOCAL</td><td>80.3</td><td>87.7</td><td>7.4</td><td>97.3</td><td>98.0</td><td>0.8</td></tr><tr><td>r(x) = REMOTE</td><td>62.8</td><td>74.5</td><td>11.7</td><td>86.9</td><td>90.7</td><td>3.7</td></tr><tr><td rowspan="2">|D|</td><td rowspan="2">1.0</td><td rowspan="2">0.25</td><td>r(x) = LOCAL</td><td>74.1</td><td>82.9</td><td>8.9</td><td>90.1</td><td>93.0</td><td>2.9</td></tr><tr><td>r(x) = REMOTE</td><td>56.8</td><td>71.1</td><td>14.3</td><td>62.9</td><td>73.3</td><td>10.4</td></tr><tr><td rowspan="2">|D|</td><td rowspan="2">1.25</td><td rowspan="2">0.25</td><td>r(x) = LOCAL</td><td>72.8</td><td>82.9</td><td>10.1</td><td>90.8</td><td>93.7</td><td>2.9</td></tr><tr><td>r(x) = REMOTE</td><td>57.1</td><td>70.4</td><td>13.3</td><td>63.8</td><td>74.1</td><td>10.3</td></tr><tr><td rowspan="2">|D|</td><td rowspan="2">1.25</td><td rowspan="2">0.0</td><td>r(x) = LOCAL</td><td>78.9</td><td>86.8</td><td>7.9</td><td>96.2</td><td>97.5</td><td>1.3</td></tr><tr><td>r(x) = REMOTE</td><td>62.8</td><td>74.9</td><td>12.0</td><td>87.5</td><td>90.9</td><td>3.3</td></tr></table>
+
+Table 4: Contrastive Evaluation Results for different costs on CIFAR-100 with remote classifier being ViT  
+
+<table><tr><td rowspan="2">S</td><td rowspan="2">c1</td><td rowspan="2">ce</td><td rowspan="2"></td><td colspan="5">cifar100 (%)</td></tr><tr><td>ratio</td><td>m</td><td>e</td><td>differ.</td><td>joint accuracy</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">1.0</td><td rowspan="2">0.2</td><td>r(x) = LOCAL</td><td>0.1</td><td>78.6</td><td>100.0</td><td>21.4</td><td rowspan="2">88.1</td></tr><tr><td>r(x) = REMOTE</td><td>99.9</td><td>27.5</td><td>88.1</td><td>60.6</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">1.0</td><td rowspan="2">0.4</td><td>r(x) = LOCAL</td><td>1.2</td><td>69.8</td><td>94.8</td><td>25.0</td><td rowspan="2">88.1</td></tr><tr><td>r(x) = REMOTE</td><td>98.8</td><td>27.1</td><td>88.3</td><td>61.2</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">1.0</td><td rowspan="2">0.6</td><td>r(x) = LOCAL</td><td>13.3</td><td>46.9</td><td>90.9</td><td>43.9</td><td rowspan="2">82.4</td></tr><tr><td>r(x) = REMOTE</td><td>86.7</td><td>24.6</td><td>87.9</td><td>63.3</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">1.0</td><td rowspan="2">0.8</td><td>r(x) = LOCAL</td><td>76.4</td><td>30.7</td><td>88.8</td><td>58.0</td><td rowspan="2">43.4</td></tr><tr><td>r(x) = REMOTE</td><td>23.6</td><td>17.2</td><td>84.7</td><td>67.5</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">1.0</td><td rowspan="2">0.95</td><td>r(x) = LOCAL</td><td>100.0</td><td>27.6</td><td>88.2</td><td>60.7</td><td rowspan="2">27.6</td></tr><tr><td>r(x) = REMOTE</td><td>0.0</td><td>N/A</td><td>N/A</td><td>N/A</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">1.9</td><td rowspan="2">0.5</td><td>r(x) = LOCAL</td><td>4.2</td><td>58.6</td><td>92.7</td><td>34.1</td><td rowspan="2">86.8</td></tr><tr><td>r(x) = REMOTE</td><td>95.8</td><td>26.2</td><td>88.0</td><td>61.9</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">1.9</td><td rowspan="2">0.75</td><td>r(x) = LOCAL</td><td>50.8</td><td>35.7</td><td>89.5</td><td>53.8</td><td rowspan="2">60.7</td></tr><tr><td>r(x) = REMOTE</td><td>49.2</td><td>19.1</td><td>86.6</td><td>67.5</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">2.0</td><td rowspan="2">0.2</td><td>r(x) = LOCAL</td><td>0.1</td><td>100.0</td><td>100.0</td><td>0.0</td><td rowspan="2">88.1</td></tr><tr><td>r(x) = REMOTE</td><td>99.9</td><td>27.5</td><td>88.1</td><td>60.6</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">2.0</td><td rowspan="2">0.4</td><td>r(x) = LOCAL</td><td>1.1</td><td>67.3</td><td>94.4</td><td>27.1</td><td rowspan="2">87.6</td></tr><tr><td>r(x) = REMOTE</td><td>98.9</td><td>27.1</td><td>87.8</td><td>60.7</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">2.0</td><td rowspan="2">0.6</td><td>r(x) = LOCAL</td><td>13.5</td><td>47.2</td><td>90.9</td><td>43.7</td><td rowspan="2">82.1</td></tr><tr><td>r(x) = REMOTE</td><td>86.5</td><td>24.5</td><td>87.6</td><td>63.1</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">2.0</td><td rowspan="2">0.8</td><td>r(x) = LOCAL</td><td>76.4</td><td>30.8</td><td>89.1</td><td>58.3</td><td rowspan="2">43.6</td></tr><tr><td>r(x) = REMOTE</td><td>23.6</td><td>17.0</td><td>85.2</td><td>68.2</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">2.0</td><td rowspan="2">0.95</td><td>r(x) = LOCAL</td><td>100.0</td><td>27.6</td><td>88.0</td><td>60.4</td><td rowspan="2">27.6</td></tr><tr><td>r(x) = REMOTE</td><td>0.0</td><td>N/A</td><td>N/A</td><td>N/A</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">3.0</td><td rowspan="2">0.2</td><td>r(x) = LOCAL</td><td>0.1</td><td>100.0</td><td>100.0</td><td>0.0</td><td rowspan="2">87.8</td></tr><tr><td>r(x) = REMOTE</td><td>99.9</td><td>27.5</td><td>87.8</td><td>60.3</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">3.0</td><td rowspan="2">0.4</td><td>r(x) = LOCAL</td><td>1.1</td><td>66.1</td><td>93.9</td><td>27.8</td><td rowspan="2">87.8</td></tr><tr><td>r(x) = REMOTE</td><td>98.9</td><td>27.1</td><td>88.1</td><td>61.0</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">3.0</td><td rowspan="2">0.6</td><td>r(x) = LOCAL</td><td>13.9</td><td>46.9</td><td>90.1</td><td>43.3</td><td rowspan="2">82.2</td></tr><tr><td>r(x) = REMOTE</td><td>86.1</td><td>24.4</td><td>87.9</td><td>63.5</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">3.0</td><td rowspan="2">0.8</td><td>r(x) = LOCAL</td><td>78.5</td><td>30.5</td><td>88.8</td><td>58.3</td><td rowspan="2">42.3</td></tr><tr><td>r(x) = REMOTE</td><td>21.5</td><td>16.7</td><td>85.4</td><td>68.7</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">3.0</td><td rowspan="2">0.95</td><td>r(x) = LOCAL</td><td>100.0</td><td>27.6</td><td>88.0</td><td>60.4</td><td rowspan="2">27.6</td></tr><tr><td>r(x) = REMOTE</td><td>0.0</td><td>N/A</td><td>N/A</td><td>N/A</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">4.0</td><td rowspan="2">0.2</td><td>r(x) = LOCAL</td><td>0.1</td><td>100.0</td><td>100.0</td><td>0.0</td><td rowspan="2">60.4</td></tr><tr><td>r(x) = REMOTE</td><td>99.9</td><td>27.5</td><td>87.9</td><td>60.4</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">4.0</td><td rowspan="2">0.4</td><td>r(x) = LOCAL</td><td>1.1</td><td>65.7</td><td>95.2</td><td>29.5</td><td rowspan="2">87.8</td></tr><tr><td>r(x) = REMOTE</td><td>99.0</td><td>27.1</td><td>88.0</td><td>60.8</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">4.0</td><td rowspan="2">0.6</td><td>r(x) = LOCAL</td><td>13.6</td><td>46.4</td><td>90.5</td><td>44.1</td><td rowspan="2">81.9</td></tr><tr><td>r(x) = REMOTE</td><td>86.4</td><td>24.6</td><td>87.5</td><td>62.9</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">4.0</td><td rowspan="2">0.8</td><td>r(x) = LOCAL</td><td>78.5</td><td>30.6</td><td>88.7</td><td>58.1</td><td rowspan="2">42.4</td></tr><tr><td>r(x) = REMOTE</td><td>21.5</td><td>16.5</td><td>85.3</td><td>68.8</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">4.0</td><td rowspan="2">0.95</td><td>r(x) = LOCAL</td><td>100.0</td><td>27.6</td><td>87.8</td><td>60.2</td><td rowspan="2">27.6</td></tr><tr><td>r(x) = REMOTE</td><td>0.0</td><td>N/A</td><td>N/A</td><td>N/A</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">5.0</td><td rowspan="2">0.2</td><td>r(x) = LOCAL</td><td>0.1</td><td>100.0</td><td>100.0</td><td>0.0</td><td rowspan="2">87.9</td></tr><tr><td>r(x) = REMOTE</td><td>99.9</td><td>27.5</td><td>87.9</td><td>60.4</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">5.0</td><td rowspan="2">0.4</td><td>r(x) = LOCAL</td><td>1.1</td><td>68.7</td><td>95.7</td><td>27.0</td><td rowspan="2">87.6</td></tr><tr><td>r(x) = REMOTE</td><td>98.9</td><td>27.1</td><td>87.8</td><td>60.7</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">5.0</td><td rowspan="2">0.5</td><td>r(x) = LOCAL</td><td>4.1</td><td>57.8</td><td>93.4</td><td>35.7</td><td rowspan="2">86.6</td></tr><tr><td>r(x) = REMOTE</td><td>95.9</td><td>26.3</td><td>87.8</td><td>61.6</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">5.0</td><td rowspan="2">0.6</td><td>r(x) = LOCAL</td><td>13.6</td><td>49.4</td><td>90.6</td><td>41.1</td><td rowspan="2">82.5</td></tr><tr><td>r(x) = REMOTE</td><td>86.4</td><td>24.1</td><td>87.7</td><td>63.6</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">5.0</td><td rowspan="2">0.8</td><td>r(x) = LOCAL</td><td>79.3</td><td>30.5</td><td>88.8</td><td>58.3</td><td rowspan="2">41.9</td></tr><tr><td>r(x) = REMOTE</td><td>20.7</td><td>16.4</td><td>85.4</td><td>69.1</td></tr><tr><td rowspan="2">sync.</td><td rowspan="2">5.0</td><td rowspan="2">0.95</td><td>r(x) = LOCAL</td><td>100.0</td><td>27.6</td><td>87.9</td><td>60.4</td><td rowspan="2">27.6</td></tr><tr><td>r(x) = REMOTE</td><td>0.0</td><td>N/A</td><td>N/A</td><td>N/A</td></tr></table>
+
+![](images/5b36aa5adde61e9838a7218d5f684ce471bb35fb90d0be329fb974903b596d0f.jpg)
+
+![](images/b8e5911c634091b3e48b8a04a06d295e73df8e24c9d71f15c1a293dbb83ff9d8.jpg)
+
+![](images/a50995276205aea058960d84095f096c0048367f735ba14a0c728df8ea557748.jpg)  
+Figure 4: Comparison of synchronization and synchronization with different parameters
+
+![](images/d4e8ca498b6796a7579aad971f03611e68443dd641ae613b0fd3a5c03bd6b60c.jpg)
+
+Table 5: Accuracy of classifiers on CIFAR-10 when client classifier is pre-trained without "truck" class  
+
+<table><tr><td></td><td>“truck” (%)</td><td>other classes (%)</td><td>all classes (%)</td></tr><tr><td>only local classifier</td><td>0</td><td>64.4</td><td>58.0</td></tr><tr><td>only remote classifier</td><td>87.5</td><td>72.5</td><td>74.0</td></tr><tr><td>jointly work</td><td>85.2</td><td>70.5</td><td>72.0</td></tr><tr><td>rejected rate under jointly work</td><td>96.1</td><td>63.1</td><td>72.7</td></tr></table>
+
+Experiments on imbalance dataset We set up an additional experiment to evaluate the impact of imbalance dataset issues for our multi-class L2H. We choose one case where the rejector and remote classifier AlexNet are trained on CIFAR-10 or SVHN when  $c_{1} = 1.25$  and  $c_{e} = 0.25$ . In this experiment, we pre-train the local model with the reduced dataset that doesn't contain any sample from one certain class (no "truck" in CIFAR-10 and no "9" in SVHN). Based on that, we train rejector and edge model with a full dataset. The results in Table 5 and Table 6 show that almost all the samples from missing class are identified by rejector and sent to server model. The overall accuracy doesn't get undermined compared to standard setting. The training of the rejector makes the multi-class L2H robust to imbalance dataset or distribution drifting issues. And after training, the remote classifier becomes expert that is specialized on the missing class.
+
+Table 6: Accuracy of classifiers on SVHN when client classifier is pre-trained without "9" class  
+
+<table><tr><td></td><td>“9” (%)</td><td>other classes (%)</td><td>all classes(%)</td></tr><tr><td>only local classifier</td><td>0</td><td>92.2</td><td>83.0</td></tr><tr><td>only remote classifier</td><td>94.5</td><td>89.5</td><td>90.0</td></tr><tr><td>jointly work</td><td>90.0</td><td>88.9</td><td>89.0</td></tr><tr><td>rejected rate under jointly work</td><td>93.5</td><td>12.0</td><td>17.8</td></tr></table>
+
+![](images/bade66620496a82d99517f1271f707db6248d0de64230178db299a49e5eced62.jpg)  
+Figure 5: Samples from SVHN with  $r(x) = \text{LOCAL}$
+
+![](images/1a4e31471989a9d8f97f5e0b4c35122adeda16b3b12695edc7cadcf895cd07d5.jpg)  
+Figure 6: Samples from SVHN with  $r(x) = \mathrm{REMOTE}$
+
+Table 7: Contrastive Evaluation on Training Asynchronously with  $S = {100},{c}_{1} = {1.25}$  and  ${c}_{e} = {0.25}$  
+
+<table><tr><td rowspan="2"></td><td colspan="3">cifar10 (%)</td><td colspan="3">SVHN (%)</td></tr><tr><td>m</td><td>e</td><td>differ.</td><td>m</td><td>e</td><td>differ.</td></tr><tr><td>data with rejector r(x) = LOCAL</td><td>74.5</td><td>83.3</td><td>8.8</td><td>91.4</td><td>93.5</td><td>2.2</td></tr><tr><td>data with rejector r(x) = REMOTE</td><td>55.5</td><td>69.9</td><td>14.4</td><td>65.4</td><td>73.9</td><td>8.5</td></tr></table>
+
+![](images/698ae8ec3262cd202da3242800fc17ac54e8085995da524bca056e5a1915881e.jpg)  
+Figure 7: Comparison with randomly reject after Stochastic Post-hoc Algorithm when  $c_{e} = 0.25$  and  $c_{1} = 1.12$
+
+![](images/2b36647f252ad41f5e5815a1d8c6976d93ad3f1d384ae511fb83c083c1652518.jpg)
+
+Accuracy comparison over stochastic post-hoc and randomly reject on bounded reject rate setting To show that the client classifier under Bounded Reject Rate settings would benefit from training with Algo. 3 and testing with Algo. 4 and Algo. 5, we set up a randomly reject method as a baseline. In this randomly reject method, we assume that there is no rejector, the sample is randomly sent to server classifier according the value of bounded reject rate, e.g., with bounded reject rate 0.5, half of the samples will be randomly chosen and sent to server for inference. We train our rejector and server classifier by following Algo. 3. After that, given different bounded reject rate, we use either Algo. 4, or Algo. 5 to strictly control the actual reject rate below the bound. The results of the accuracy-vs-bounded reject rate for the Stochastic Post-hoc method and the random reject method are shown in Fig. 7. The result demonstrates that by using the Stochastic Post-hoc Algorithm, the system can still actively pick up the samples that fit the server classifier even under the constraints of a bounded reject rate.
\ No newline at end of file
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+# LEARNING TO PLAN BEFORE ANSWERING: SELF-TEACHING LLMS TO LEARN ABSTRACT PLANS FOR PROBLEM SOLVING
+
+Jin Zhang $^{1,2}$ , Flood Sung $^{2}$ , Zhilin Yang $^{2}$ , Yang Gao $^{1}$ , Chongjie Zhang $^{3}$
+
+$^{1}$ Institute for Interdisciplinary Information Sciences, Tsinghua University, China
+
+$^{2}$ Moonshot AI
+
+3Washington University in St. Louis
+
+jin-zhan20@mails.tsinghua.edu.cn
+
+# ABSTRACT
+
+In the field of large language model (LLM) post-training, the effectiveness of utilizing synthetic data generated by the LLM itself has been well-presented. However, a key question remains unaddressed: what essential information should such self-generated data encapsulate? Existing approaches only produce step-by-step problem solutions, and fail to capture the abstract meta-knowledge necessary for generalization across similar problems. Drawing insights from cognitive science, where humans employ high-level abstraction to simplify complex problems before delving into specifics, we introduce a novel self-training algorithm: LEarning to Plan before Answering (LEPA). LEPA trains the LLM to formulate anticipatory plans, which serve as abstract meta-knowledge for problem-solving, before engaging with the intricacies of problems. This approach not only outlines the solution generation path but also shields the LLM from the distraction of irrelevant details. During data generation, LEPA first crafts an anticipatory plan based on the problem, and then generates a solution that aligns with both the plan and the problem. LEPA refines the plan through self-reflection, aiming to acquire plans that are instrumental in yielding correct solutions. During model optimization, the LLM is trained to predict both the refined plans and the corresponding solutions. By efficiently extracting and utilizing the anticipatory plans, LEPA demonstrates remarkable superiority over conventional algorithms on various challenging natural language reasoning benchmarks.
+
+# 1 INTRODUCTION
+
+Large Language Models (LLMs) have revolutionized the field of natural language processing, demonstrating remarkable capabilities in handling complex language tasks (Achiam et al., 2023; Zhao et al., 2023; Yang et al., 2024; Shahriar et al., 2024). While post-training optimization of LLMs demands a substantial volume of data (Xiao et al., 2023; Wang et al., 2024b), recent works reveal that LLMs obtain the potential of generating high-quality synthetic data themselves (Zelikman et al., 2022; Gulcehre et al., 2023; Singh et al., 2023; Bansal et al., 2024). These works, known as self-training methods, improve the LLM by iterating between generating data with LLMs and optimizing LLMs with the generated data. Self-training methods alleviate the requirement of expensive human annotations and make post-training much more scalable.
+
+A central challenge in self-training is, what essential information should such self-generated synthetic data encapsulate? Despite remarkable progress, this problem has not been well studied. Previous works only generate step-by-step problem solutions, and train the LLM to maximize the log-likelihood of generating these solutions (Zelikman et al., 2022; Singh et al., 2023). This approach only trains the LLM to memorize knowledge about task-specific solutions, and fails to capture the high-level abstract meta-knowledge necessary for generalization across similar problems. As a con
+
+sequence, previous self-training methods obtain only limited generalization abilities, and struggle on difficult natural language tasks such as Hendrycks MATH (Hendrycks et al., 2021).
+
+![](images/df6fbaf58b25c67c654ce97f32d12317b139ff479899f412494d3fd5c13d1775.jpg)  
+Figure 1: A didactic example demonstrating how LEPA outperforms baseline methods by learning to generate anticipatory plans before answering. (a) An example problem in the Hendrycks MATH test set. (b) An incorrect solution given by the LLM trained with a baseline method, ReST. The model fails to generate correct reasoning steps. (c) A correct solution given by the LLM trained with our proposed method, LEPA. The model generates high-quality plans, and then follows the plan to solve the problem correctly.
+
+To tackle this challenge, we draw insights from cognitive science (Wang & Chiew, 2010; Raduntz, 2020): humans simplify complex problems through high-level abstraction before engaging with details (Ross, 2009). Such abstraction not only lightens the cognitive load but also distills high-level meta-knowledge that is transferable to analogous problems. This idea is also evidenced by recent advances in meta-learning (Finn et al., 2017; Rakelly et al., 2019), which learn generalizable meta-knowledge that enables fast adaptation to similar problems. We propose a novel self-training algorithm, LEarning to Plan before Answering (LEPA), that learns to generate anticipatory plans before generating detailed step-by-step problem solutions. The anticipatory plans serve as high-level abstract meta-knowledge that outlines the solution generation path and shields the LLM from the distraction of irrelevant details. During data generation, LEPA prompts the LLM to first devise an anticipatory plan that encapsulates the high-level problem-solving steps, and then generate a solution that aligns with both the problem and the plan. If the solution is correct, the plan-solution pair is stored into the training dataset. Otherwise, the LLM is asked to reflect on the plan and the incorrect solution, and refine the plan until it successfully prompts the LLM to generate correct solutions. With this self-reflection mechanism, LEPA acquires plans that are instrumental in yielding correct solutions. During model optimization, we utilize supervised fine-tuning (SFT) to train the LLM to predict both the plans after self-reflection and the corresponding solutions. As shown in Figure 1, after self-training with LEPA, the LLM generates helpful abstract anticipatory plans that outline the solution steps and are generalizable to similar problems, thus achieving better performance than baseline algorithms. LEPA is extensively evaluated on various challenging language reasoning benchmarks including Hendrycks MATH, and significantly outperforms baseline methods.
+
+To summarize, our main contributions are listed as follows:
+
+1. We present the fundamental problem of what information should self-generated data encapsulate in the field of LLM self-training.  
+2. We propose a novel self-training algorithm, LEPA, that learns to generate anticipatory plans, which serves as high-level abstract meta-knowledge guiding solution generation, before generating detailed problem solutions.
+
+![](images/48f3ae0300deb3b6891c3ce31e22c848d303a4c754b7bdfa00ce40524f64038c.jpg)  
+(a) Baseline algorithms' data generation procedure.
+
+![](images/3d8ceece7190cbf0044db5d32161ddcf5352c752d508e9db6f0b6a8cca5a168c.jpg)  
+(b) LEPA's data generation procedure.  
+Figure 2: Comparison between baseline algorithms' and LEPA's data generation procedure. (a) Baseline algorithms only generate step-by-step solutions to each problem, lacking high-level abstract meta-knowledge that guides solution generation. (b) LEPA generates anticipatory plans before generating detailed problem solutions. These plans are optimized with self-reflection, and encapsulate the high-level abstract problem-solving steps. The plans efficiently guide the LLM to generate correct solutions.
+
+3. We evaluate LEPA on several challenging language reasoning benchmarks and demonstrate LEPA's superior performance compared to based algorithms.
+
+# 2 LEARNING TO PLAN BEFORE ANSWERING (LEPA)
+
+This section introduces LEPA, a novel self-training algorithm that self-trains the LLM to devise high-level anticipatory plans, which serve as abstract solution-generation blueprints, before generating detailed problem solutions. LEPA iterates between a data generation phase and a model optimization phase. In the data generation phase, LEPA generates high-quality plan-solution pairs with self-reflection. In the model optimization phase, LEPA fine-tunes the LLM with the generated data using SFT. Finally, we discuss multiple advantages that the anticipatory plans offer for enhancing the self-training process.
+
+# 2.1 DATA GENERATION PHASE
+
+LEPA operates within the common self-training framework, which involves an initial LLM denoted as  $\theta_0$ , a set of prompts containing  $N$  problems  $\mathcal{D}_{prompt} = \{x_i\}_{i=0}^{N-1}$ , and a binary scoring function  $f_{cor}(x_i, y_i)$  that evaluates the correctness of a solution  $y_i$  with a score of either 0 or 1.
+
+In each iteration  $t$ , as depicted in Figure 2, LEPA differs from previous methods in that it does not directly prompt the LLM to generate step-by-step solutions to problems. Instead, LEPA instructs the LLM to first generate an anticipatory plan  $p_i^t$  that serves as an abstract blueprint for solution generation, and then generate the actual solutions  $y_i^t$  based on the plan and the problem. To avoid the degenerate case of generating plans containing detailed step-by-step problem solutions, LEPA
+
+stresses in the prompt that the plan should be general high-level meta-knowledge that is applicable to similar problems, and should not contain any problem-specific information such as detailed calculations. If the solution is correct, i.e.,  $r_{cor}(x_i,y_i) = 1$ , then the problem-plan-solution tuple  $(x_{i},p_{i}^{t},y_{i}^{t})$  is added to the training dataset  $\mathcal{D}_{train}^{t}$ . Otherwise, LEPA refines the plan with self-reflection. The LLM is prompted with the problem, the previous plan, the corresponding incorrect solution, and the correct answer (if accessible). Then LEPA instructs the LLM to reflect on why the previous plan fails to guide itself to generate correct solutions, and then generate a new plan based on its reflection results. To avoid information bypassing, LEPA also stresses in the reflection prompt that the reflected plan should not contain problem-specific information, including detailed calculation and the correct answer. LEPA evaluates the refined plan by instructing the LLM to solve the problem with the refined plan. If the generated solution is correct, the problem-plan-solution tuple  $(x_{i},p_{i}^{t},y_{i}^{t})$  is added to the training dataset. Otherwise, LEPA repeats the self-reflection process, unless either a correct solution is generated or the number of trials reaches a certain limit  $l$ . The self-reflection process empowers LLMs to enhance anticipatory plans based on correctness feedback and analysis of unsuccessful attempts, thus efficiently seeking out superior plans.
+
+# 2.2 MODEL OPTIMIZATION PHASE
+
+In each iteration, after acquiring the training dataset  $\mathcal{D}_{train}^{t}$ , LPEA optimizes the model with SFT. LEPA formats data into a two-round conversation. In the first round, The user inputs the problem  $x_{i}$  and requires the LLM to generate an anticipatory plan, and the assistant output is the plan  $p_i^t$ . In the second round, the user instructs the LLM to solve the problem based on the plan it proposed, and the assistant output is the solution  $y_{i}^{t}$ . The training objective is to minimize the following negative log-likelihood loss:
+
+$$
+\mathcal {L} _ {S F T} \left(\theta_ {t}, \mathcal {D} _ {t r a i n} ^ {t}\right) = - \mathbb {E} _ {\left(x _ {i}, p _ {i} ^ {t}, y _ {i} ^ {t}\right) \sim \mathcal {D} _ {t r a i n} ^ {t}} \left[ \log p _ {\theta_ {t}} \left(p _ {i} ^ {t}, y _ {i} ^ {t} \mid x _ {i}\right) \right]. \tag {1}
+$$
+
+While we employ SFT for algorithm simplicity, LEPA is also compatible with more sophisticated reinforcement learning (RL) algorithms such as Direct Policy Optimization (DPO) (Rafailov et al., 2024) and Proximal Policy Optimization (PPO) (Schulman et al., 2017). We believe RL algorithms can further boost LEPA's performance, and are important future directions. The pseudo-code for LEPA is presented in Algorithm 1. Detailed prompts and hyper-parameters used by LEPA is deferred to Appendix A.
+
+# 2.3 WHY IS THE ANTICIPATORY PLAN BENEFICIAL?
+
+Central to LEPA's efficacy is the anticipatory plan, offering multiple advantages for self-training. This subsection discusses these benefits in detail.
+
+Reducing cognitive workload. As demonstrated in Figure 1, without the anticipatory plans, the LLM may get lost in the problem-solving process, leading to erroneous solution steps. In contrast, the anticipatory plans serve as blueprints that outline the necessary problem-solving steps, and shield the LLM from the distraction of irrelevant details. Consequently, when generating detailed problem solutions, the LLM is conscious of what to do at the current step, and successfully solves the problem. Research in cognitive science (Wang & Chiew, 2010; Raduntz, 2020) supports the notion that such a structured approach significantly eases cognitive load and improves learning efficiency.
+
+Learning generalizable high-level meta-knowledge. The anticipatory plans are abstract high-level meta-knowledge that does not involve problem specifics, and is thus generalizable across similar problems. For example, the plan demonstrated in Figure 1 can be readily adapted to a variety of combinatorial mathematical problems with similar underlying structures but different parameters. From the meta-learning perspective, LEPA can be interpreted as a meta-learning algorithm that extracts the meta-knowledge in the form of anticipatory plans. The learned meta-knowledge empowers the LLM to solve similar problems more effectively.
+
+Learning generalizable high-level meta-knowledge. When the correct answer is accessible, the anticipatory plans enable self-reflection that avoids the pitfall of information bypassing. Previous methods like STaR (Zelikman et al., 2022) directly modify incorrect solutions by referring to the
+
+Algorithm 1 LEPA: LEarning to Plan before Answering  
+1: Require: An initial LLM  $\theta_0$ , a set of problems  $\mathcal{D}_{prompt} = \{x_i\}_{i=0}^{N-1}$ , a binary scoring function  $f_{cor}(x_i, y_i)$ , number of iterations  $T$ , maximum self-reflection trials  $l$ , learning rate  $\alpha$   
+2: for  $t \gets 0$  to  $T - 1$  do // In each iteration do  
+3: Initialize an empty training set  $\mathcal{D}_{train}^t$   
+4: for  $i \gets 0$  to  $N - 1$  do // For each problem do  
+5: Ask  $\theta_t$  to generate anticipatory plan  $p_i^{t,0}$  to problem  $x_i$   
+6: Ask  $\theta_t$  to generate solution  $y_i^{t,0}$  based on  $x_i$  and  $p_i^{t,0}$   
+7: if  $f_{cor}(x_i, y_i^{t,0}) == 1$  then // Solution is correct, add to training set  
+8: Add  $\{x_i, p_i^{t,0}, y_i^{t,0}\}$  to  $\mathcal{D}_{train}^t$   
+9: else  
+10: for  $j \gets 1$  to  $l$  do // Self-reflection iterations  
+11: Ask  $\theta_t$  to self-reflect on  $p_i^{t,j-1}$  and  $y_i^{t,j-1}$ , and generate  $p_i^{t,j}$   
+12: Ask  $\theta_t$  to generate solution  $y_i^{t,j}$  based on  $x_i$  and  $p_i^{t,j}$   
+13: if  $f_{cor}(x_i, y_i^{t,j}) == 1$  then  
+14: Add  $\{x_i, p_i^{t,j}, y_i^{t,j}\}$  to  $\mathcal{D}_{train}^t$   
+15: Break // Solution is correct, stop self-reflection  
+16: end if  
+17: end for  
+18: end if  
+19: end for  
+20:  $\theta_{t+1} \gets \theta_t - \alpha \nabla_{\theta_t} \mathcal{L}_{SFT}(\theta_t, \mathcal{D}_{train}^t)$  // Model Optimization with SFT
+
+correct answer, and are very likely to cheat by only modifying the final answer and ignoring the consistency between intermediate steps and the final answer (Singh et al., 2023). In contrast, as LEPA requires the anticipatory plans to not include any problem-specific information including the final correct answer, it isolates the correct answer from solution generation. The model must generate correct solutions without seeing the correct answer, preventing the model from cheating during solution generation.
+
+# 3 EXPERIMENTS
+
+To demonstrate the effectiveness of LEPA, we evaluate on several challenging reasoning benchmarks, including Hendrycks MATH (challenging math problems) (Hendrycks et al., 2021), Hellaswag (sentence completion reasoning) (Zellers et al., 2019), BoolQ (paragraph understanding and reasoning) (Clark et al., 2019), and PIQA (physics reasoning) (Bisk et al., 2020). For Hendrycks MATH, we evaluate solution correctness with the function provided by the dataset creators (https://github.com/hendrycks/math). We utilize Llama 3.8B Instruct (Dubey et al., 2024) as the initial LLM. LEPA is compared against several representative self-training algorithms:  $ReST$  (Gulcehre et al., 2023),  $ReST^{EM}$  (Singh et al., 2023), and STaR (Zelikman et al., 2022). All these baseline methods only generate step-by-step solutions to problems. Both  $ReST$  and  $ReST^{EM}$  generate solutions with rejection sampling. In each iteration,  $ReST$  fine-tunes the model trained after the previous iteration, while  $ReST^{EM}$  instead fine-tunes from the initial LLM. STaR generates solutions by prompting the LLM to modify incorrect solutions with the aid of correct answers, and also fine-tunes from the initial LLM in each iteration. We demonstrate algorithms' test accuracy at convergence<sup>1</sup>. For a fair comparison, all methods do not utilize few-shot examples in their prompts. We also demonstrate the initial LLM's efficacy, with either a zero-shot CoT prompt (Kojima et al., 2022) or a LEPA prompt that instructs it to first generate an anticipatory plan before answering.
+
+Table 1: Test accuracy of LEPA and various baselines on four challenging reasoning benchmarks. "CoT" and "Plan+CoT" refer to the initial LLM's performance with a zero-shot CoT prompt and the LEPA prompt, respectively. LEPA demonstrates superior accuracy in comparison to all other algorithms on each of the benchmarks. Numbers in the parentheses are LEPA's performance improvement over the best-performing baseline algorithm on each benchmark.  
+
+<table><tr><td></td><td>CoT</td><td>Plan+CoT</td><td>ReST</td><td>ReSTEM</td><td>STaR</td><td>LEPA</td></tr><tr><td>Hellaswag</td><td>60.8%</td><td>56.1%</td><td>86.3%</td><td>86.4%</td><td>85.7%</td><td>91.2% (+4.8%)</td></tr><tr><td>Hendrycks MATH</td><td>19.5%</td><td>22.1%</td><td>28.2%</td><td>27.2%</td><td>25.9%</td><td>30.2% (+2.0%)</td></tr><tr><td>BoolQ</td><td>77.3%</td><td>80.8%</td><td>84.5%</td><td>86.3%</td><td>85.8%</td><td>88.4% (+2.1%)</td></tr><tr><td>PIQA</td><td>67.0%</td><td>75.7%</td><td>81.4%</td><td>83.5%</td><td>84.2%</td><td>85.9% (+1.7%)</td></tr><tr><td>Average</td><td>56.1%</td><td>58.7%</td><td>70.1%</td><td>70.8%</td><td>70.4%</td><td>73.9% (+3.1%)</td></tr></table>
+
+# 3.1 MAIN RESULTS
+
+Table 1 presents a comparative analysis of algorithm performance across the four reasoning benchmarks. Notably, in the absence of self-training, the LEPA prompt (Plan+CoT) enhances the initial LLM's performance on three benchmarks when compared to the traditional zero-shot CoT prompt (CoT). This suggests that the practice of formulating anticipatory plans before generating detailed solutions can significantly improve model efficacy. However, on the Hellaswag benchmark, Plan+CoT falls short of CoT, implying that such enhancement is not uniformly achievable across different tasks, potentially due to the initial LLM's lack of calibration for producing high-quality anticipatory plans. As for self-training performance, baseline self-training algorithms only train the LLM to predict step-by-step solutions, lacking abstract high-level meta-knowledge about problem-solving. As a consequence, these algorithms perform poorly on these benchmarks. In contrast, LEPA efficiently extracts high-level abstract meta-knowledge with the anticipatory plans, thereby surpassing all baseline algorithms consistently across all benchmarks.
+
+Figure 3 illustrates algorithms' learning curve across learning iterations. LEPA's superior performance is evident across all benchmarks. Specifically, on Hellaswag, LEPA lags initially during the early iterations (0-10), where the LEPA prompt is slightly less effective than the zero-shot CoT prompt. However, as training progresses, LEPA's performance incrementally surpasses that of the baseline algorithms, suggesting that self-training is instrumental in awakening the LLM's capacity to conceive and leverage anticipatory plans effectively. On the remaining three benchmarks, LEPA acquires better initial performance and converges at higher test accuracies, demonstrating the effectiveness of introducing the anticipatory plans. We also observe a great performance drop of STaR on Hendrycks MATH. This is because STaR is very likely to generate false-positive solutions, i.e., solutions with wrong rationales but correct final answers (Singh et al., 2023), and greatly hinders learning on complex reasoning benchmarks like Hendrycks MATH.
+
+# 3.2 ABLATION STUDIES
+
+LEPA consists of three key components: the anticipatory plan, plan optimization with self-reflection, and utilizing more inference compute to achieve better performance. This subsection discusses the necessity of each component with ablation studies.
+
+Anticipatory plans. We test a variant of LEPA that does not introduce anticipatory plans in the data generation phase, and only trains the LLM to predict the step-by-step solutions optimized with self-reflection. As shown in Table 2, this variant ("Without Plan") under-performs LEPA. There are two reasons for this degrade in performance. Firstly, without the anticipatory plans, the LLM does not learn abstract high-level meta-knowledge about problem-solving. Secondly, as discussed in Section 2.3, directly performing self-reflection on the solutions is very likely to generate false-positive solutions, which greatly hinders learning.
+
+Self-reflection. To demonstrate the necessity of self-reflection in LEPA's plan optimization, we test a variant that instead utilizes rejection sampling (Singh et al., 2023) to sample plan-answer pairs. As shown in Table 2, this variant ("Without Self-Reflection") also performs worse than LEPA.
+
+![](images/7195991663ce787a9b67e874731d123566dee0953905b2597c772e199b9767f9.jpg)
+
+![](images/b8793b4453ccffb0a1f213eb3bd5eaf9e617be6c17ea8a947d6da226c996d470.jpg)
+
+![](images/d7fe035370d95343944be10b23a67addbbec69fd2fb4a5025fc3a23099c4c9c5.jpg)  
+Figure 3: Algorithms' learning curves on the four benchmarks. LEPA achieves better performance than baseline algorithms.
+
+![](images/30e9fa7e9388584135275f54d977650692d42a96d4aeae669f82c9db928fb1f7.jpg)
+
+Table 2: Ablation study on the anticipatory plan and self-reflection. We also demonstrate the performance of  $ReST^{EM}$ , the baseline with the highest average test accuracy. "Without Plan" is LEPA without anticipatory plans, and "Without Self-Reflection" is LEPA without self-reflection.  
+
+<table><tr><td></td><td>RESTEM</td><td>LEPA</td><td>Without Plan</td><td>Without Self-Reflection</td></tr><tr><td>Hendrycks MATH</td><td>27.2%</td><td>30.2%</td><td>24.3%</td><td>28.8%</td></tr><tr><td>BoolQ</td><td>86.3%</td><td>88.4%</td><td>84.8%</td><td>86.9%</td></tr><tr><td>PIQA</td><td>84.2%</td><td>85.9%</td><td>84.5%</td><td>84.8%</td></tr></table>
+
+This result implies that self-reflection is more effective than rejection sampling in optimizing the anticipatory plans, as it gives linguistic feedback for LLMs to improve the previous plans.
+
+Different ways of utilizing inference compute. LEPA generates both anticipatory plans and problem solutions, utilizing more compute at inference time. It is worth discussing how much contribution the extra compute makes, and whether the anticipatory plan is an effective way to utilize inference compute. For the first question, as discussed in Section 3.2, without self-training, utilizing inference compute with anticipatory plans can improve performance on three of the four benchmarks, and degrade performance on one benchmark. In contrast, after self-training, the anticipatory plans can consistently help LEPA outperform baseline methods. This result demonstrates that extra inference compute contributes a part to LEPA's performance, and self-training is also vital for unlocking the LLM's ability to efficiently utilize these extra compute. For the second question, we test three other variants that train the LLM to utilize inference compute in different ways. The first variant adds silence tokens in the solution to give the LLM more compute to generate answers (Goyal et al., 2023). The second variant trains the LLM to first output a solution, and then outputs a new solution if it finds the original solution incorrect. For data generation of this variant, solutions are generated with rejection sampling, analogous to  $ReST$ . We synthesize training data by appending correct solutions to the end of incorrect solutions. The third variant simply asks the LLM to generate long solutions. All variants fine-tune the LLM with  $ReST$ . As shown in Table 3, LEPA is the only method that successfully utilizes additional inference compute to outperform baseline methods. In
+
+Table 3: Ablation study on ways of utilizing inference compute. We test on the Hendrycks MATH dataset."Silence token" is the variant that adds silence tokens in the solution. "Correction" is the variant that trains the LLM to output new solutions if it finds its initial solution incorrect. "Long Solution" is the variant that instructs the LLM to generate long solutions. "#of Tokens" is the average token length of the LLM's responses to test problems, and "Accuracy" is the LLM's test accuracy. LEPA is the only method that efficiently utilizes additional inference compute to outperform baseline methods. We put the results in two rows due to the page width limit.  
+
+<table><tr><td colspan="2">STaR</td><td colspan="2">ReST</td><td colspan="2">LEPA</td></tr><tr><td># of Tokens</td><td>Accuracy</td><td># of Tokens</td><td>Accuracy</td><td># of Tokens</td><td>Accuracy</td></tr><tr><td>175.1</td><td>25.9%</td><td>477.8</td><td>28.2%</td><td>826.4</td><td>30.2%</td></tr><tr><td colspan="2">Silence Tokens</td><td colspan="2">Correction</td><td colspan="2">Long Solution</td></tr><tr><td># of Tokens</td><td>Accuracy</td><td># of Tokens</td><td>Accuracy</td><td># of Tokens</td><td>Accuracy</td></tr><tr><td>869.3</td><td>28.3%</td><td>979.4</td><td>27.8%</td><td>1409.7</td><td>25.4%</td></tr></table>
+
+contrast, the first variant performs similarly to the  $ReST$  baseline, suggesting that silence tokens offer limited benefits for the challenging Hendrycks MATH benchmark. Both the second and the third variant underperform  $ReST$ , as the LLM is trained to predict tokens with scant correlation to correct solution formulation. The results above implies that introducing the anticipatory plans is a more efficient way to generate long self-training data compared to the aforementioned alternatives. Detailed implementation of these variants are deferred to Appendix B.
+
+Incorporation with RL algorithms. To demonstrate that LEPA is also applicable to more advanced RL optimization algorithms, we test a variant of LEPA that utilizes REINFORCE (Zhang et al., 2021b) as the underlying optimization algorithm, which is called LEPA+REINFORCE. The only difference between LEPA and LEPA+REINFORCE is that LEPA+REINFORCE labels data with rewards of either 1 or -1 (based on the final answer correctness), and optimizes the LLM with the labelled data using the REINFORCE algorithm. On Hendrycks MATH, LEPA+REINFORCE achieves a test accuracy of  $30.6\%$ , while the original LEPA achieves  $30.2\%$ . This performance improvement demonstrates the potential of incorporating LEPA with more advanced optimization algorithms like RL, and is a promising future direction.
+
+LPEA with RL. To investigate the potential of incorporating LPEA with RL algorithms, we demonstrate a variant of LEPA that utilizes REINFORCE as the optimization algorithm in the model optimization phase. This variant is named LEPA+REINFORCE. The only difference from LEPA is that LEPA+REINFORCE does not discard failure data. Instead, it utilizes the solution correctness as the reward (1 for correct solutions, -1 for incorrect solutions). This implementation makes no modification to the data generation process. On Hendrycks MATH, LEPA+REINFORCE achieves a test accuracy of  $30.6\%$ , while LEPA achieves  $30.2\%$ . This performance improvement demonstrates the feasibility and effectiveness of incorporating LEPA with RL algorithms. Additional ablation studies including algorithm performance on OOD benchmarks, other LLMs, additional benchmarks, and evaluation with Simple-Eval are deferred to Appendix C.
+
+# 3.3 CASE STUDY
+
+We present a case to demonstrate how LEPA's self-reflection mechanism optimizes the anticipatory plans and the solutions. As shown in Figure 4, the initial plan generated by the model is too vague, and cannot provide enough guidance for the model to solve the problem correctly. Consequently, during solution generation, the model generates irrelevant steps, makes a mistake in the symbol of the expression, and fails to answer correctly. In the self-reflection process, the model finds out that the previous answer failed to calculate the correct symbols. So it modifies the plan to contain more detailed instructions on how to solve this problem. Note that the plan after self-reflection is still general meta-knowledge that is applicable to a wide range of similar problems. With this modified plan, the model pays more attention to signs, generates only necessary steps, and successfully generates a correct solution.
+
+Question:
+
+What is the value of  $\sqrt{15 - 6\sqrt{6}} + \sqrt{15 + 6\sqrt{6}}$ ?
+
+Initial Plan:
+
+Analyze the problem carefully. Identify the mathematical object or expression involved, and determine the relevant mathematical field or discipline it belongs to. In this case, identify the type of expression, its structure, and the mathematical operations involved. Consider the properties and relations between the given numbers, expressions, and variables. Look for any opportunities to apply mathematical concepts, theorems, or formulas to simplify or solve the problem.
+
+Answer guided by the initial plan:
+
+Analyzing the problem carefully, I identify the mathematical object as a radical expression involving square roots. The relevant mathematical field is algebra, specifically rationalizing the denominators and simplifying expressions.
+
+Step 1: I consider the structure of the expression and identify the perfect squares within the radicals. Recognizing that  $6\sqrt{6}$  can be rewritten as  $\sqrt{36}\sqrt{6} = 6\sqrt{6}$ , I can rewrite the expression as  $\sqrt{15 - 6\sqrt{6}} + \sqrt{15 + 6\sqrt{6}} = \sqrt{9 + 6 - 6\sqrt{6}} + \sqrt{9 + 6 + 6\sqrt{6}}$ .
+
+I apply the difference of squares formula to the first radical, obtaining  $\sqrt{15 - 6\sqrt{6}} = \sqrt{\left(3 + \sqrt{6}\right)^2} = 3 + \sqrt{6}$ . Similarly, I apply the difference of squares formula to the second radical, obtaining  $\sqrt{15 + 6\sqrt{6}} = \sqrt{\left(3 + \sqrt{6}\right)^2} = 3 + \sqrt{6}$ .
+
+Step 3:
+
+I combine the expressions, adding the like terms, to obtain  $3 + \sqrt{6} + 3 + \sqrt{6} = 6 + 2\sqrt{6}$ . Answer:  $6 + 2\sqrt{6}$
+
+![](images/6f8d9dd4ea1f1bcc4001e0d1684a6611e2f459ce89272ce3bb61bde96d2907db.jpg)
+
+Self-reflection:
+
+The model fails to solve the problem because it incorrectly applies the difference of squares formula, resulting in the same expression for both radicals. This is likely due to the model's lack of attention to the signs and the correct application of the formula.
+
+The new instruction should focus on the correct application of the difference of squares formula and the importance of considering the signs.
+
+Plan after self- reflection:
+
+When simplifying radical expressions, consider the structure of the radicand and the signs.  
+1. Identify the radical expression and its components.  
+2. Simplify each radical separately, considering the signs  
+3. Combine like terms to obtain the simplified expression.  
+4. Verify the result by checking the original expression and the simplified expression for equivalence.
+
+Answer guided by the plan after self-reflection:
+
+Step 1: Identify the radical expression and its components.
+
+The radical expression is  $\sqrt{15 - 6\sqrt{6}} + \sqrt{15 + 6\sqrt{6}}$ , which consists of two radical terms.
+
+Step 2: Simplify each radical separately, considering the signs
+
+We can start by simplifying the first radical:  $\sqrt{15 - 6\sqrt{6}} = \sqrt{\left(3 - \sqrt{6}\right)^2} = 3 - \sqrt{6}$ .
+
+Similarly, we can simplify the second radical:  $\sqrt{15 + 6\sqrt{6}} = \sqrt{\left(3 + \sqrt{6}\right)^2} = 3 + \sqrt{6}$ .
+
+Step 3: Combine like terms to obtain the simplified expression.
+
+The simplified expression is  $3 - \sqrt{6} + 3 + \sqrt{6} = 6$ .
+
+Step 4: Verify the result by checking the original expression and the simplified expression for equivalence.
+
+The original expression and the simplified expression are equivalent, so our answer is correct.
+
+Answer: 6
+
+![](images/5be68a46decce46ed611036192860c4c99a215595364b07ac49def788bc6ad10.jpg)  
+Figure 4: A case study demonstrating how LEPA optimizes the anticipatory plans and the solutions with self-reflection. The initial plan is too broad and lacks detail, and fails to provide enough guidance to generate correct answers. The self-reflection process successfully analyses what is wrong, and generates a new, high-quality plan that provides more guidance while maintaining generalizability. With the new plan after self-reflection, the model successfully generates correct solutions.
+
+# 4 RELATED WORKS
+
+Self-training. With the fast development of LLMs, the thirst for data continues to grow. A promising way is to generate high-quality data with the LLM itself. A branch of works mainly focus on designing the data generation progress. STaR (Zelikman et al., 2022) operates by initially prompting the LLM to produce step-by-step solutions, followed by an adjustment phase where the LLM corrects its errors with the aid of the correct answers. One severe limitation of STaR is that the modification process makes it very possible to generate false-positive solutions, i.e., solutions with wrong rationales but correct final answers. RFT (Yuan et al., 2023),  $ReST$  (Gulcehre et al., 2023), and  $ReST^{EM}$  (Singh et al., 2023) instead adopt rejection sampling for data generation, and suffer less from the false-positive issue. TRICE (Hoffman et al., 2024) improves over STaR by utilizing a Markov-chain Monte Carlo expectation-maximization algorithm to sample solutions, and introducing a control-variate method to control gradient variance. Re-ReST (Dou et al., 2024) utilizes self-reflection to correct the generated wrong answers. LMSI (Huang et al., 2022) considers the scenario where the correctness of model-generated data cannot be verified during training, and filers data with majority voting. Apart from these methods, SPAG (Cheng et al., 2024) generates data by asking LLMs to self-play in adversarial games. These previous methods above only generate step-by-step solutions to problems, and lack high-level meta-knowledge that are generalizable across
+
+similar problems. In contrast, LEPA learns abstract meta-knowledge in the form of anticipatory plans, and achieves better performance on complex benchmarks.
+
+Scaling inference compute. As proposed by Snell et al. (2024) and confirmed by the recent inspiring GPT O1 model (Hu et al., 2024), scaling inference compute can further boost LLM performance. Similar to LEPA, PS Prompting (Wang et al., 2023b) also scales inference compute by asking the LLM to first generate a plan before answering, but does not consider how to generate data and fine-tune the LLM. Moreover, it does not consider how to automatically optimize the anticipatory plans. HSP (Fu et al., 2024) is the most relevant work to ours, which trains the LLM to output hints before solving the problem. However, HSP's hints are pre-collected rather than self-generated, and induce additional data collection costs. PHP (Zheng et al., 2023) utilizes previously generated answers as hints, and encourages the LLM to answer with reference to its previous answers. LEPA efficiently utilizes inference compute by training the LLM to generate helpful anticipatory plans, which contain high-level meta-knowledge on problem-solving, before generating actual problem solutions. These plans are automatically optimized by the LLM itself, and do not require additional human design.
+
+Meta-learning. Meta-learning aims at "learning to learn", i.e., designing meta-algorithms that optimize learning algorithms automatically (Finn et al., 2017; Sung et al., 2017; Rakelly et al., 2019; Zhang et al., 2021a; Wang et al., 2023a). LEPA can be interpreted as a meta-learning algorithm that learns the meta-knowledge of designing the anticipatory plans for each problem, rather than designing plans with human effort. The most relevant work is Quiet-STaR (Zelikman et al., 2024), which meta-learns meta-tokens that help the LLM to predict the next token. LEPA considers the setting of problem-solving rather than general next-token prediction, and meta-learns the generation of anticipatory problem-solving plans.
+
+Planning in LLMs. Recently, several works have demonstrated the effectiveness of integrating planning in LLMs. ReAct (Yao et al., 2022) and DEPS (Wang et al., 2024c) generate plans before dealing with decision-making problems, and LUMOS (Yin et al., 2023) fine-tunes the LLM on pre-collected datasets containing planning data. To our best knowledge, LEPA is the first work to integrate planning in the process of self-training, and improves the LLM's planning ability by training on self-generated data.
+
+Self-reflection. Self-reflection enables LLMs to reflect on their mistakes and generate better responses. It can be viewed as a process of in-context optimization to produce better responses. Previous works demonstrate that self-reflection can significantly improve LLM response quality (Renze & Guven, 2024; Shinn et al., 2024; Madaan et al., 2024). LEPA utilizes self-reflection to optimize plans and solutions in the data generation phase, and acquires data of higher quality.
+
+# 5 CONCLUSION
+
+This paper presents the fundamental problem of what data should be generated in self-training algorithms. Inspired by cognitive science research and recent meta-learning advances, we propose a novel idea of learning abstract meta-knowledge in the form of anticipatory problem-solving plans. Based on this idea, we propose a novel self-training algorithm, LEPA, which automatically generates and learns the anticipatory plans. Experiment results on several challenging reasoning benchmarks demonstrate the effectiveness of LEPA. An interesting future direction is to incorporate LEPA with more advanced model optimization methods such as RL. It is also worth exploring how well can LEPA perform on larger and more advanced LLMs, and how to scale LEPA to utilize more inference compute. Furthermore, as LLMs may solve simple problems without planning, an important future direction is to automatically identify complex problems that require planning from simple problems that can be easily solved without planning. This identification can avoid wasting compute resources and help the LLM solve problems more efficiently.
+
+# ACKNOWLEDGEMENT
+
+This work is supported by National Natural Science Foundation of China (62176135), the National Key R&D Program of China (2022ZD0161700), Shanghai Qi Zhi Institute Innovation Program SQZ202306 and the Tsinghua University Dushi Program.
+
+# ETHICS STATEMENT
+
+Concerns about safety and reliability are key points of discussion in the LLM community. The use of anticipatory plans in LLMs is a step towards making the models' actions more understandable and transparent to people. Yet, LEPA cannot guarantee that every solution will strictly match the plans it creates, which means further work is needed to solidify the trustworthiness of LLMs.
+
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+Wayne Xin Zhao, Kun Zhou, Junyi Li, Tianyi Tang, Xiaolei Wang, Yupeng Hou, Yingqian Min, Beichen Zhang, Junjie Zhang, Zican Dong, et al. A survey of large language models. arXiv preprint arXiv:2303.18223, 2023.  
+Chuanyang Zheng, Zhengying Liu, Enze Xie, Zhenguo Li, and Yu Li. Progressive-hint prompting improves reasoning in large language models. arXiv preprint arXiv:2304.09797, 2023.
+
+# A DETAILED PROMPTS AND HYPER-PARAMETERS
+
+This section demonstrates the detailed prompts and the hyper-parameters used by LEPA and baseline algorithms. Figure 5 presents the prompts used by LEPA and baseline algorithms.
+
+As for hyper-parameters, for a fair comparison, we ensure that all algorithms have the same number of trials (5) in the data generation phase. LEPA is allowed to have maximally 4 self-reflection processes for each problem. For  $ReST$  and  $ReST^{EM}$ , 5 solutions are sampled for each question. For STaR, it has maximally 4 opportunities to modify the previous incorrect answer. All algorithms fine-tunes the LLM for one epoch in each model optimization phase. For the data generation phase of all algorithms, we use a temperature of 0.5 for sampling. We use a temperature of 0.0005 for all test results. We use 3e-7 as the learning rate for all learning algorithms.
+
+# Prompt for anticipatory plan generation:
+
+You are an expert at designing plans for large language models to solve problems. The problem to be solved is:
+
+[Question]
+
+Output the plan you design. Note that the plan should be general knowledge that help solve similar problems, so do not contain any question-specific
+
+information. Also, the content will be directly added to the prompt, so pay attention to its format. The plan should be concise, no longer than 1024 tokens.
+
+Output only the plan. Do not output any other words.
+
+# Prompt for solution generation:
+
+Based on the plan you propose, solve the problem step by step. In each step of your solution, explain how the plan affects you to form your answers. The last
+
+line of your response should be of the form Answer: $ANSWER (without quotes) where $ANSWER is the answer to the problem. Remember to put your answer
+
+on its own line after "Answer:" , and you do not need to use a \boxed command. Your response should be concise, no longer than 1024 tokens. The problem is:
+
+[Question]
+
+# Prompt for self-reflection:
+
+You are an expert in designing plans for large language models to solve problems. You have found that the original plan fails to solve a problem. You need to
+
+Analyze the failure case, and design a new plan. The new plan should help the large language model to solve the failure case. You are encouraged to design
+
+plans distinct from the original plan to better explore high-quality plans.
+
+The problem is:
+
+[Question]
+
+The original plan is:
+
+[Original Plan]
+
+The incorrect solution given by the large language model under the original plan is:
+
+[Original solution]
+
+The desired correct final answer is:
+
+[Correct Answer]
+
+Analyze the information above. Why does the model fail to solve the problem? What is wrong in the answer? How to design a new plan so that the model can
+
+correctly solve the problem? How distinct should the new plan be from the original plan? What contents should the new plan obtain? Pay special attention to
+
+the formatting requirements. Does the model's output strictly follow the required output format? Answer concisely, no longer than 2560 tokens.
+
+# Prompt for new plan generation after self-reflection:
+
+Based on the analysis above, output the new plan. Note that the new plan should be general knowledge that help solve similar problems, so do not contain
+
+any task-specific information. You must not contain the correct final answer in the plan. You are encouraged to design plans distinct from the original plan to
+
+better explore high-quality plans. Also, the content will be directly added to prompt, so pay attention to its format. The content should be short and concise,
+
+no longer than 1024 tokens. Output only the plan. Do not output any other words.
+
+# (a) LEPA prompt.
+
+# Prompt for solution generation:
+
+Solve the following problem step by step. The last line of your response should be of the form Answer: $ANSWER (without quotes) where $ANSWER is the
+
+answer to the problem. Remember to put your answer on its own line after "Answer:" , and you do not need to use a \\boxed command. Your response should
+
+be concise, no longer than 1024 tokens. The problem is:
+
+[Question]
+
+Prompt for solution modification (only used in STaR):
+
+Your solution is wrong. The correct answer is:
+
+[Correct Answer]
+
+Modify your previous solution to get the correct answer. Output the modified solution only. Do not output any other words.
+
+# (b) Prompt used by baseline methods.
+
+Figure 5: Detailed prompts used by (a) LEPA and (b) baseline algorithms.
+
+# B ABLATION DETAILS
+
+This section presents the details of the variants discussed in the "Different ways of utilizing inference compute" part of Section 3.2.
+
+For the second variant, we first sample correct and incorrect solutions for each problem with rejection sampling. Then we synthesize training data by first adding a sentence of "Oops, I made a mistake. The correct solution is: " to the end of incorrect solutions. Then we append a correct solution to the end of this sentence.
+
+For the third variant, we explicitly instruct the LLM to output solutions that are approximately 2,000 words long. We observe that the LLM generates verbose responses that obscure the important steps in solving the problem.
+
+# C ADDITIONAL ABLATION STUDIES
+
+OOD performance. We evaluate OOD performance by training on Hendrycks MATH and testing on the Math category of MMLU-Pro (Wang et al., 2024a). As shown in Table 4, LEPA consistently outperforms baseline algorithms in this OOD setting.
+
+<table><tr><td></td><td>CoT</td><td>Plan+CoT</td><td>ReST</td><td>ReSTEM</td><td>STaR</td><td>LEPA</td></tr><tr><td>Performance</td><td>30.4%</td><td>33.9%</td><td>35.1%</td><td>35.3%</td><td>35.8%</td><td>38.9%</td></tr></table>
+
+Other LLMs. We additionally evaluate algorithm performance on Llama 3.1 8B Instruct. As shown in Table 5, on the Hendrycks MATH dataset, the LEPA prompt can slightly improve over the zero-shot CoT prompt on the initial LLM. As for self-training, LEPA significantly outperforms the baseline algorithm. These empirical results are consistent with our main results presented in Section 3.1.
+
+Additional Benchmarks. We additionally evaluate on CSQA (Saha et al., 2018) and MMLU (Hendrycks et al., 2020), and results are shown in Table 6. LEPA consistently outperforms baseline algorithms on these benchmarks.
+
+Table 4: Performance of different algorithms training on Hendrycks MATH and testing on the Math category of MMLU-Pro. "CoT" and "Plan+CoT" refer to the initial LLM's performance with a zero-shot CoT prompt and the LEPA prompt, respectively. LEPA achieves better generalization than baseline algorithms.  
+
+<table><tr><td>Algorithm</td><td>CoT</td><td>Plan+CoT</td><td>ReST</td><td>ReSTEM</td><td>STaR</td><td>LEPA</td></tr><tr><td>Performance</td><td>37.2%</td><td>38.4%</td><td>45.3%</td><td>46.9%</td><td>45.0%</td><td>49.6%</td></tr></table>
+
+Table 5: Algorithm performance on Hendrycks MATH, with Llama 3.1 8B Instruct as the initial LLM. "CoT" and "Plan+CoT" refer to the initial LLM's performance with a zero-shot CoT prompt and the LEPA prompt, respectively  
+
+<table><tr><td></td><td>CoT</td><td>Plan+CoT</td><td>ReST</td><td>ReSTEM</td><td>STaR</td><td>LEPA</td></tr><tr><td>CSQA</td><td>67.1%</td><td>69.3%</td><td>73.2%</td><td>74.0%</td><td>74.1%</td><td>75.2%</td></tr><tr><td>MMLU</td><td>61.9%</td><td>60.1%</td><td>64.3%</td><td>65.6%</td><td>65.8%</td><td>66.1%</td></tr></table>
+
+Table 6: Performance comparison of different methods on CSQA and MMLU benchmarks. LEPA achieves higher performance than baseline algorithms.
+
+Evaluation with Simple-Eval. We re-evaluate Hendrycks MATH performance with Simple-Eval, and the results are demonstrated in Table 7. With the new evaluation, LEPA still outperforms baseline algorithms.
+
+<table><tr><td></td><td>CoT</td><td>Plan+CoT</td><td>ReST</td><td>ReSTEM</td><td>STaR</td><td>LEPA</td></tr><tr><td></td><td>26.1%</td><td>28.5%</td><td>31.2%</td><td>31.4%</td><td>29.2%</td><td>33.7%</td></tr></table>
+
+Table 7: Hendrycks MATH performance evaluated with Simple-Eval. With the new evaluation, LEPA still outperforms baseline algorithms.
\ No newline at end of file
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+# LEARNING TO SELECT NODES IN BRANCH AND BOUND WITH SUFFICIENT TREE REPRESENTATION
+
+Sijia Zhang, Shuli Zeng, Shaoang Li, Feng Wu *, Xiang-Yang Li *
+
+School of Computer Science and Technology, University of Science and Technology of China {sxzsj, zengshuli0130,lishaoa}  $@$  mail.ustc.edu.cn {wufeng02, xiangyangli}@ustc.edu.cn
+
+# ABSTRACT
+
+Branch-and-bound methods are pivotal in solving Mixed Integer Linear Programming (MILP), where the challenge of node selection arises, necessitating the prioritization of different regions of the space for subsequent exploration. While machine learning techniques have been proposed to address this, two crucial problems concerning (P1) how to sufficiently extract features from the branch-and-bound tree, and (P2) how to assess the node quality comprehensively based on the features remain open. To tackle these challenges, we propose to tackle the node selection problem employing a novel Tripartite graph representation and Reinforcement learning with a Graph Neural Network model (TRGNN). The tripartite graph is theoretically proved to encompass sufficient information for tree representation in information theory. We learn node selection via reinforcement learning for learning delay rewards and give more comprehensive node metrics. Experiments show that TRGNN significantly improves the efficiency of solving MILPs compared to human-designed and learning-based node selection methods on both synthetic and large-scale real-world MILPs. Moreover, experiments demonstrate that TRGNN well generalizes to MILPs that are significantly larger than those seen during training.
+
+# 1 INTRODUCTION
+
+Mixed-integer linear programming (MILP) is a general optimization formulation for a wide range of important real-world applications (Zhang et al., 2023), such as industrial process scheduling (Floudas & Lin, 2005), resources allocation (Ren & Gao, 2010), and logistic operations (Paschos, 2014), etc. Modern MILP solvers (Gurobi Optimization, LLC, 2023; Bestuzheva et al., 2021) rely on the branch-and-bound (B&B) algorithm (Land & Doig, 1960), which recursively divides the search space into a tree, solving relaxations of the problem until an integral solution is found and proven optimal. Throughout this procedure, numerous decisions must be repeatedly made (Linderoth & Savelsbergh, 1999), including node selection, and branching variable selection, etc. These decisions often dramatically impact the overall performance (Kianfar, 2010). Traditionally, these would be made according to hard-coded expert heuristics implemented in solvers. To further improve the efficiency of MILP solvers, more and more attention is given to statistical learning approaches for replacing and improving upon those heuristics (Bengio et al., 2021).
+
+Here, we focus on the node selection problem, which has a significant impact on the overall solver performance (Achterberg, 2007). Current machine learning-based node selection methods (He et al., 2014; Song et al., 2018; Yilmaz & Yorke-Smith, 2020; Labassi et al., 2022) predominantly extract features from a subset of candidate nodes, supplemented by manually extracted global tree features such as global upper and lower bounds. These methods primarily employ imitation learning to mimic an oracle which prioritizes a node if it contains the optimal solution.
+
+However, the existing learning-based node selection methods (Labassi et al., 2022; Yilmaz & Yorke-Smith, 2020; He et al., 2014; Song et al., 2018) suffer from two limitations. Firstly, while individual MILP problems can be effectively represented using bipartite graphs (Gasse et al., 2019;
+
+Ding et al., 2020; Nair et al., 2020; Gupta et al., 2020; Paulus et al., 2022; Khalil et al., 2022; Shen et al., 2021; Labassi et al., 2022), node selection problem requires more than just representing a single MILP problem. It necessitates the representation of the entire branch-and-bound tree, which includes numerous nodes, each of which can be considered an individual MILP problem. However, representing all nodes in the tree as a bi-
+
+partite graph is not feasible for large-scale problems due to the exponential increase in node count. For instance, the "Anonymous" dataset, rooted in a largescale industrial context (Gasse et al., 2022), necessitates that SCIP handle upwards of 50,000 nodes. The previous work compares the nodes among the candidate nodes two by two to select the optimal node.
+
+Table 1: Comparison between the expert strategies and SCIP in  $\mathrm{GISP}(n\in [70,80])$  
+
+<table><tr><td>Method</td><td>Time(s)</td><td>Wins</td><td>Nodes</td></tr><tr><td>SCIP</td><td>7.54 ± 1.39</td><td>26/50</td><td>397.46 ± 2.59</td></tr><tr><td>Expert</td><td>7.82 ± 1.44</td><td>24/50</td><td>276.71 ± 2.82</td></tr></table>
+
+They only represent the two currently selected nodes to be compared as a bipartite graph, supplemented by manually extracted global features, such as global upper and lower bounds and the depth of nodes. However, these manually extracted features do not fully capture the solving process of the branch-and-bound tree. Secondly, the recent work (Labassi et al., 2022; Yilmaz & Yorke-Smith, 2020; He et al., 2014; Song et al., 2018), which is based on the imitation learning approach, is limited by the performance of the expert itself. However, the oracle sometimes fails to outperform SCIP, as observed in datasets like GISP (Colombi et al., 2017), as shown in Table 1. If the expert does not beat a baseline, imitating it is unlikely to bring gains.
+
+To overcome the aforementioned challenges, we propose to tackle the node selection problem employing a Tripartite graph representation and Reinforcement learning with a Graph Neural Network model (TRGNN). Firstly, we propose a tripartite graph to represent the branch-and-bound tree, which is theoretically proved to be sufficient in information theory. Secondly, we introduce a reinforcement learning framework for learning delay rewards in the branch-and-bound process and give more comprehensive node metrics. These metrics include the global gap, the node potential and the overhead of path switching, which is the transition between the current node and the subsequent selected one.
+
+We evaluate our approach on six NP-hard MILP problem benchmarks, which consist of three classical synthetic MILP problems and three real-world MILP problems from diverse application areas. Experiments show that TRGNN significantly outperforms human-designed and learning-based node selection methods in terms of solving time. We achieve up to  $31.44\%$  reduction on synthetic instances and up to  $59.42\%$  reduction on real-world instances. Notably, TRGNN is the only known learning-based node selection method to surpass the default SCIP across all of the six datasets in terms of solving time. Moreover, experiments show that TRGNN can well generalize to MILPs that are significantly larger than those seen during training.
+
+# 2 RELATED WORK
+
+He et al. (2014) were the first to explore node selection heuristics in branch-and-bound by applying imitation learning. They trained a support vector machine (SVM) to replicate the decisions of an oracle that selects nodes along the optimal solution path. Their model leveraged hand-crafted features, including node-specific attributes and global metrics like the number of solutions found and the current global bounds. However, their approach primarily focused on combining node selection with a learned pruning model, which aimed to cut off unpromising branches in the branch-and-bound tree. Their strategy was more akin to a primal heuristic, emphasizing the quick discovery of high-quality feasible solutions rather than guaranteeing optimality. They reported improvements in the optimality gap against SCIP under a time limit and Gurobi under a node limit on four benchmarks.
+
+Song et al. (2018) proposed a RankNet model combined with a novel retrospective imitation learning method. In contrast to He et al. (2014), they focused solely on node selection without an additional pruning operator. They retrospectively correct the node selection trajectory into a shortest path to the best solution found. When the solver runs until optimality, the trajectory mirrors those generated by a diving oracle. They generated these trajectories using Gurobi and trained the model with the DAGGER (Ross et al., 2011) and SMILE (Ross & Bagnell, 2010) imitation learning algorithms. Their method demonstrated significant improvements in the optimality gap under a node limit on path planning integer programs. However, they reported less favorable results on
+
+more challenging benchmarks, such as combinatorial auctions, indicating limitations in the model's generalizability to more complex problems.
+
+Yilmaz & Yorke-Smith (2020) extended the work of He et al. (2014) by proposing a neural network-based node selection operator for branch-and-bound. Their method can be viewed as a refinement of Song et al. (2018)'s neural network node selector, with the key difference being that it only evaluates children of the current node and defaults to depth-first search otherwise. Using the state encoding from Gasse et al. (2019), they trained their model to mimic an oracle that prioritizes nodes leading to one of the top k solutions, essentially generalizing He et al. (2014)'s oracle. On three benchmarks, they reported improvements in both time and number of nodes, and in some cases, reduced node counts compared to SCIP. However, their approach still lagged behind SCIP in terms of solving time, showing limitations in overall efficiency despite the node count improvements.
+
+Inspired by the work of Gasse et al. (2019) for variable selection in branch-and-bound, Labassi et al. (2022) considered a node selection strategy that compared nodes in the candidate queue pair by pair to identify the optimal node. They represented only the two currently selected nodes as bipartite graphs, supplemented by manually extracted global features such as the global upper and lower bounds and the nodes' depth. On three benchmarks, they reported reductions in the node counts of the branch-and-bound tree and demonstrated time improvements over SCIP on one dataset. However, like earlier efforts, their method underperformed on other datasets in terms of solving time. Notably, even the oracle failed to outperform SCIP's default rule on certain datasets. Consequently, it is unsurprising that no other machine learning approach was able to surpass SCIP in these cases.
+
+Some works have sought to provide a full tree representation. Scavuzzo et al. (2022) address the variable selection problem by proposing tree Markov Decision Processes that offers a more suitable framework for learning to branch. On the other hand, Mattick & Mutschler (2023) focus on node selection, addressing the complexity of batching variable and constraint structures. Instead of creating  $k$  vertices for  $k$  variables, they represent variable values as a distribution with 10 buckets. This method allows for scaling to much larger instances than typically possible, although it sacrifices some detail in representation, thus leaving room for optimization and fine-tuning.
+
+# 3 PRELIMINARIES
+
+# 3.1 MIXED INTEGER LINEAR PROGRAMMING
+
+A general MILP problem is defined by a set of decision variables, where a subset or all variables are required to be integers. The objective is to maximize a linear function under a series of linear constraints, as formulated below:
+
+$$
+\max  \boldsymbol {c} ^ {\top} \boldsymbol {x}, \quad \text {s . t .} \quad \boldsymbol {A} \boldsymbol {x} \geq \boldsymbol {b}, \boldsymbol {x} \in \mathbb {R} ^ {l}, \quad x _ {j} \in \mathbb {Z}, \forall j \in \mathcal {I}, \tag {1}
+$$
+
+For simplicity, we assume that the objective of our MILP problems is to seek the maximum value.
+
+An MILP problem can be effectively represented as a weighted bipartite graph  $G = (V \cup C, E)$  (Nair et al., 2020; Gasse et al., 2019). Each vertex in  $V$  corresponds to a variable of the MILP, and each vertex in  $C$  represents a constraint. An edge  $(v_i, c_j)$  connects a variable vertex  $v_i$  with a constraint vertex  $c_j$  if the variable is involved in the constraint. The edge set  $E \in \mathbb{R}^{l \times m \times e}$  represents the edge features, where  $l$  and  $m$  denote the number of variables and constraints, respectively, and  $e$  indicates the dimension of the edge attributes.
+
+# 3.2 BRANCH AND BOUND
+
+The branch and bound (Land & Doig, 1960) algorithm can be described as follows. At every node, the linear program (LP) relaxation is solved, where all variables are treated as continuous. If the LP relaxation solution  $x_{N_s}$  of the selected node  $N_{s}$  violates the original integrality constraints, the problem "branches" into two sub-MILPs (child nodes) by adding constraints that force a fractional variable to round up or down. Specifically, the leaf node is added with constraints  $x_{i}\leq \lfloor (x_{N_{s}})_{i}\rfloor$  and  $x_{i}\geq \lceil (x_{N_{s}})_{i}\rceil$ , respectively, where  $x_{i}$  denotes the  $i$ -th variable,  $(x_{N_s})_i$  denotes the  $i$ -th variable of the LP relaxation solution  $x_{N_s}$ . If the solution  $x_{N_s}$  is integer (and feasible for the original MILP as per Equation 1), and its objective value surpasses the current best integer feasible solution, it is designated as the new global lower bound. If the objective value  $f(x_{N_s})$  (i.e., the node upper bound) is lower than the global lower bound, or if the LP problem is infeasible, the node is pruned.
+
+![](images/49468772f8415e0b6087960b4cfcb0556099f20608e07bb3d60f4852056b4fa3.jpg)  
+Figure 1: Example of tripartite graph representation. The root node (red) is conceptualized as a bipartite graph, consisting of variable and constraint vertices, while the leaf nodes (grey) embody sets of added branching constraints. The features of the edges, which connect the variable vertices to the node constraint vertices, delineate the constraint space of the leaf nodes.
+
+# 3.3 INFORMATION THEORY
+
+In information theory (Reza, 1994), mutual information  $I(X;Y)$  is the amount of uncertainty in  $X$  due to the knowledge of  $Y$ . Mathematically, mutual information is defined as
+
+$$
+I (X; Y) = \sum_ {x, y} p (x, y) \log \left(\frac {p (x , y)}{p (x) p (y)}\right) \tag {2}
+$$
+
+where  $p(x,y)$  is the joint probability distribution function of  $X$  and  $Y$ , and  $p(x)$  and  $p(y)$  are the marginal probability distribution functions for  $X$  and  $Y$ . We can also say
+
+$$
+I (X; Y) = H (X) - H (X | Y) \tag {3}
+$$
+
+where  $H(X)$  is the marginal entropy,  $H(X|Y)$  is the conditional entropy. If  $H(X)$  represents the measure of uncertainty about a random variable, then  $H(X|Y)$  measures what  $Y$  does not say about  $X$ . This is the amount of uncertainty in  $X$  after knowing  $Y$  and this substantiates the intuitive meaning of mutual information as the amount of information that knowing either variable provides about the other.
+
+# 4 TRIPARTITE GRAPH REPRESENTATION OF BRANCH AND BOUND TREE
+
+While a sufficient representation of every node within a branch-and-bound tree  $T$  would theoretically capture the entire scope of tree information, such an approach is not feasible for large-scale problems due to the exponential increase in node count. For instance, the Anonymous dataset (Gasse et al., 2022), rooted in a large-scale industrial context, necessitates that SCIP handle upwards of 50,000 nodes. We define the complete tree feature vector,  $F = \bigoplus_{N \in T} J_N$ , by aggregating features  $J_N \in \mathcal{I}$  for each node  $N$ . This raises a critical question: Which node features within  $F$  are essential and should be selectively represented to effectively model the tree? Addressing this, we introduce a tripartite graph structure for tree representation in Section 4.1.
+
+In order to judge whether a tree representation effectively can capture all relevant node features necessary for node quality assessment, we first need to define what a sufficient tree representation is. Let  $\phi(\cdot)$  denote a feature extraction function, which depends on the specific methods employed for tree representation. The feature vector,  $\phi(F)$ , is derived from the tree complete feature  $F$ . Shannon's information theory provides a suitable formalism for quantifying the above concepts. The information about the quality vector  $Q$  contained in the feature vector  $\phi(F)$  is then  $I(Q; \phi(F))$ . As a consequence of the data processing inequality,  $I(Q; \phi(F)) \leq I(Q; F)$ . A feature vector  $\phi(F)$  is defined to be sufficient if the inequality above is an equality.
+
+Definition 4.1 (Definition of Sufficient Tree Representation). A tree feature representation  $\phi(F)$  is considered sufficient if:
+
+$$
+I (Q; \phi (F)) = I (Q; F)
+$$
+
+In Section 4.2, we theoretically demonstrate that the tree information captured within this graph is sufficient for node selection.
+
+# 4.1 TREE REPRESENTATION
+
+In this section, we introduce a tripartite graph for tree representation. An advantage of our representation is that it is theoretically proved to be sufficient in Section 4.2. This theoretical suffi
+
+ciency ensures that our representation can comprehensively capture the necessary features across the branch-and-bound tree for effective node quality assessment. Additionally, to enhance practical applicability in machine learning, we tailor the representation to facilitate feature learning. Instead of directly representing each node with its variables and constraints, which would redundantly present at the root node, we focus on depicting the unique branch constraints that distinguish each node. This approach emphasizes the subtle, often overlooked differences among nodes, thereby facilitating machine learning algorithms to more effectively learn and discriminate node features.
+
+We represent the branch-and-bound tree as a weighted tripartite graph  $G = (V \cup C \cup LN, E^C \cup E^{LN})$ . The vertex set  $V \cup C \cup LN$  is divided into three subsets: the variable vertex set  $V$  with  $|V| = l$ , the constraint vertex set  $C$  with  $|C| = m$  and the leaf node vertex set  $LN$  with  $|LN| = n$ . The vertex sets  $V, C$  and  $LN$  are mutually exclusive:  $V \cap C = V \cap LN = C \cap LN = \emptyset$ . We denote the variable vertices as  $V = \{v_1, v_2, \ldots, v_l\}$ , constraint vertices as  $C = \{c_1, c_2, \ldots, c_m\}$  and leaf node vertices  $LN = \{ln_1, ln_2, \ldots, ln_n\}$ . The leaf node vertices represent candidate leaf nodes for selection strategies within the branch-and-bound tree. After a node is selected, the branch-and-bound algorithm divides it into two new leaf nodes, adding constraints of the form  $\{x_i \leq z\}$  or  $\{x_i \geq z\}$ , where  $x_i$  is the variable chosen for branching and  $z \in \mathbb{Z}$ . The constraint edges  $E^C$  connect variable vertices  $V$  to constraint vertices  $C$  and the branching edges  $E^B$  connect variable vertices  $V$  to leaf node vertices  $LN$ , indicating branching decisions. The edges are denoted as  $E_{i,j}^C = (v_i, c_j)$  and  $E_{j,k}^B = (c_j, ln_k)$ , with  $|E^C| = e_1, |E^B| = e_2$ , for  $i \in \{1, 2, \ldots, l\}, j \in \{1, 2, \ldots, m\}, k \in \{1, 2, \ldots, n\}$ . We denote the collection of all such weighted tripartite graphs  $G = (V \cup C \cup LN, E^C \cup E^{LN})$  with  $|V| = l, |C| = m$  and  $|LN| = n$  as  $G_{l,m,n}$ .
+
+Each vertex in the tripartite graph is associated with a feature vector: vertices in  $V$  (variables),  $C$  (constraints), and  $LN$  (leaf nodes) hold features  $h_i^V \in \mathcal{H}^V$ ,  $h_j^C \in \mathcal{H}^C$ , and  $h_k^{LN} \in \mathcal{H}^{LN}$ , respectively. We define the feature spaces for the sets of all variable vertices, constraint vertices, and leaf node vertices as  $\mathcal{H}_l^V \coloneqq (\mathcal{H}^V)^l$ ,  $\mathcal{H}_m^C \coloneqq (\mathcal{H}^C)^m$ , and  $\mathcal{H}_n^{LN} \coloneqq (\mathcal{H}^{LN})^n$ . Edges in the graph are imbued with specific attributes that detail the interactions between vertices. For constraint edges that connect constraint vertices  $c_i$  to variable vertices  $v_j$ , each edge feature  $f_{i,j}^C \in \mathcal{F}^C$  is one-dimensional and directly represents the coefficient  $A_{ij}$  of variable  $x_j$  in constraint  $c_i$ . Branching edges connect variable vertices  $v_j$  to leaf node vertices  $ln_k$ , and these edges carry two-dimensional features that specify the branching conditions for the variable  $x_j$  at the leaf node. Each branching edge feature encapsulates  $x_j \geq a$  and  $x_j \leq b$ , represented as  $f_{j,k}^B = (a,b) \in \mathcal{F}^B$ . We define  $\mathcal{F}_{e_1}^C \coloneqq (\mathcal{F}^C)^{e_1}$  and  $\mathcal{F}_{e_2}^B \coloneqq (\mathcal{F}^B)^{e_2}$ . The exact features of the graph are described in Appendix C.
+
+We extract an MILP solving search tree's features via a tripartite graph  $\phi_t(F) = (V \cup C \cup LN, E^C \cup E^B) \in \mathcal{G}_{l,m,n} \times \mathcal{H}_l^V \times \mathcal{H}_m^C \times \mathcal{H}_n^{LN} \times \mathcal{F}_{e_1}^C \times \mathcal{F}_{e_2}^B$ , where  $F$  denotes the aggregate of node features. In Figure 1, we illustrate the process of converting a branch-and-bound tree from an MILP instance into a tripartite graph. The conversion begins with the root node, where variables and constraints are represented as "variable vertices" and "constraint vertices", respectively. As the branching process progresses, each step adds new constraints to the leaf nodes, encapsulated as "leaf node vertices" in the graph. These additional constraints differentiate each leaf node's MILP problem from the root node's problem. Edges connect these leaf node vertices to variable vertices, forming a path that represents the sequence of branching decisions made from the root node to each leaf node. For example, in Figure 1, the candidate leaf node in the lower left undergoes two branching steps, adding constraints  $y \leq 1$  and  $x \leq 1$ , represented by edges  $e_1$  and  $e_3$ , respectively.
+
+# 4.2 THEORETICAL PROOF OF SUFFICIENCY
+
+In this section, we prove our main theorems and present a sketch of our proof lines. The full proof lines are presented in the Appendix A. We first present with the following theorem that the tripartite graph is sufficient to represent an MILP search tree.
+
+Theorem 4.1. Given the node  $N_{i}$  with features  $J_{N_{i}}$ , the tripartite graph representation  $\phi_t(F)$  satisfies:  $I(Q; \phi_t(F)) = I(Q; F)$ , where  $F = \bigoplus_{i=0}^{m+n} J_{N_i}$  and  $Q$  denotes the quality vector.
+
+This theorem asserts that the information uncertainty in  $Q$ , when knowing the tripartite graph representation  $\phi_t(F)$ , is equivalent to that when knowing the features of all nodes in the tree  $F$ .
+
+Then, we analyze the relationship between the features of child nodes and their parent node:
+
+![](images/777ee852c8e67595e8261fdfea555eeb8340e559cfb5759c3a6db48349e30066.jpg)  
+Figure 2: Illustration of our proposed RL framework for learning node selection policies. In this framework, the search tree is represented as a tripartite graph (see Section 4), serving as the environment, and the TRGNN model acts as the agent(see Section 5). We first apply the heuristic algorithm "BestEstimate" to pre-select  $n$  candidate nodes and incorporate their estimate values into the node features. Subsequently, the TRGNN model processes these features and outputs a Q-value vector, from which the nodes with the highest Q-values are selected as the final choices.
+
+Theorem 4.2. Given a node  $N_{p}$  and its two child nodes  $N_{l}$  and  $N_{r}$ , it holds that  $I(Q; J_{N_{p}} \oplus J_{N_{l}} \oplus J_{N_{r}}) = I(Q; J_{N_{l}} \oplus J_{N_{r}})$ , where  $J_{N}$  denotes the extracted feature of the node  $N$ .
+
+This theorem illustrates that the information provided by the features of the parent node can be completely represented by the features of its child nodes. The proof proceeds as follows: We start with the obvious inequality:  $I(Q; J_{N_p} \oplus J_{N_l} \oplus J_{N_r}) \geq I(Q; J_{N_l} \oplus J_{N_r})$  according to the data processing inequality. To establish the reverse inequality, we analyze the node features, encompassing both MILP characteristics, denoted by  $J_m$ , including the variables, constraints and objective function, and the global attributes within the tree, denoted by  $J_g$ . Thus, we define:  $J_N = (J_m)_N \oplus (J_g)_N$ . We demonstrate that functions  $f_m$  and  $f_g$  exist such that  $(J_m)_{N_p} = f_m((J_m)_{N_l} \oplus (J_m)_{N_r})$  and  $(J_g)_{N_p} = f_g((J_g)_{N_l} \oplus (J_g)_{N_r})$ . This relationship establishes a Markov chain:  $Q \to J_{N_l} \oplus J_{N_r} \to J_{N_p} \oplus J_{N_l} \oplus J_{N_r}$ , allowing the use of the data processing inequality to confirm  $I(Q; J_{N_p} \oplus J_{N_l} \oplus J_{N_r}) \leq I(Q; J_{N_l} \oplus J_{N_r})$ .
+
+By invoking Theorem 4.2 recursively, we illustrate that the features of all leaf nodes, denoted by  $L$ , encapsulate the necessary information to represent the whole tree, thereby establishing that  $I(Q;L) = I(Q;F)$ . This finding is pivotal for proving Theorem 4.1.
+
+In proving Theorem 4.1, it is essential to validate that the information flow  $Q \to \phi_t(F) \to L$  holds. The tripartite graph representation  $\phi_t(F)$  includes the root node's MILP features  $(J_m)_{N_0}$  and each leaf node's branching constraints  $B_N$  and global features  $(J_g)_{N}$ , ensures that from  $(J_m)_{N}$  and  $B_N$ , the MILP features  $(J_m)_{N}$  of the leaf node  $N$  can be deduced. Employing the data processing inequality again, we establish the sufficiency of  $\phi_t(F)$  as a representation, proving Theorem 4.1.
+
+# 5 LEARNING NODE SELECTION VIA TRIPARTITE GRAPH
+
+Building upon the tripartite graph framework detailed in Section 4 for extracting observable features from the branch-and-bound tree (P1), this section delves into (P2): assessing node quality and selecting a node based on the tripartite graph representation.
+
+The challenges in node selection are twofold. Firstly, the branch-and-bound process involves sequential decision-making, where each choice impacts subsequent decisions. This creates a scenario of delayed rewards, where the consequences of early decisions may not become apparent until later stages. Secondly, there is a distinct lack of comprehensive metrics for evaluating node quality.
+
+To address these issues, we frame the node selection problem within a Markov Decision Process (MDP) framework and leverage Reinforcement Learning (RL) to learn node selection policies, as detailed in Section 5.1. Following this, we introduce three node quality metrics that assess the impact of nodes in the solving process, further discussed in Section 5.2.
+
+# 5.1 REINFORCEMENT LEARNING FORMULATION
+
+We formulate an MILP solver as the environment and the RL model as the agent. We consider an MDP defined by the tuple  $(S, \mathcal{A}, r, \pi)$ . Specifically, we specify the state space  $S$ , the action space  $\mathcal{A}$ , the reward function  $r: S \times \mathcal{A} \to \mathbb{R}$ , the transition function  $\pi$ , and the terminal state in the following. (1) The state space  $S$ . As delineated in Section 4, the core information for node selection is represented by a tripartite graph. (2) The action space  $\mathcal{A}$ . The action space is intended to include all nodes that are potentially selectable. However, the dynamic nature of the selection process means the number of selectable nodes is subject to change due to the addition of newly expanded nodes and the removal of pruned nodes. To address this variability, we employ the heuristic node selection algorithm called Estimate in modern solvers to pre-select nodes, choosing the top  $n$  nodes, where  $n$  is a predetermined value, to form a set of node candidates. If the initial set of candidates is less than  $n$ , placeholders are used to fill the remaining slots, ensuring a consistent set size. We define the action space as this set of node candidates with a size of  $n$ . (3) The reward function  $r$ . The reward function, as discussed in Section 5.2, encompasses the gap update reward, the optimal solution path reward, and the path switching penalty. (4) The transition function  $\pi$ . The transition function maps the current state  $s$  and the action  $a$  to the next state  $s'$ , representing the ensuing search tree post the expansion of node  $a$ . (5) The terminal state. The process reaches a terminal state when the gap attains zero or no remaining candidate nodes are in the set. Within this framework, the trajectory probability  $\tau = (s_0, \dots, s_T)$  depends on the node selection policy  $\pi$  and the other solver components, formulated as  $p_{\pi}(\tau) = p(s_0) \prod_{t=0}^{T-1} \sum_{a \in \mathcal{A}} \pi(a|s_t)p(s_{t+1}|s_t,a)$ .
+
+TRGNN Model. Figure 2 delineates the architecture of our proposed model. The features of the variable, constraint, and leaf node vertices on the tripartite graph undergo an initial transformation via a 32-dimensional embedding layer. This layer is pivotal for normalizing and refining the input features before they traverse through three subsequent graph convolutional layers, each with dimensions 8, 4, and 4, and each utilizing a ReLU activation function to capture complex, non-linear relationships. Post convolution, the refined representations of the variable, constraint, and node constraint vectors are separately averaged, reducing dimensionality and mitigating overfitting risks. These averaged representations are then amalgamated with the global node features, creating a comprehensive feature vector encapsulating both localized and global information. This amalgamated feature vector is then processed through a linear layer and a sigmoid activation layer, culminating in a Q-value vector of predetermined dimension  $n$ . This vector quantitatively represents the value of each node in the pre-selected candidate node set, serving as a decisive metric for action selection. The action is determined by selecting the node corresponding to the maximum Q-value, directing the model's focus towards the most promising regions of the search space.
+
+A significant advantage of our model is higher learning efficiency for different node features. Experiments demonstrate that TRGNN requires far fewer training examples compared to conventional imitation learning models (Labassi et al., 2022), thus conserving computational resources. For instance, with the GISP datasets, our model needs only 250 instances for training, whereas traditional imitation learning methods may require up to 1000 instances.
+
+# 5.2 REWARD FUNCTION
+
+The reward function  $R$  for the selected node  $N$  is defined as:
+
+$$
+R (N) = w _ {1} R _ {g} (N) + w _ {2} R _ {o} (N) - w _ {3} R _ {s} (N) \tag {4}
+$$
+
+where  $R_{g}$  represents the updates to the global gap,  $R_{o}$  represents the potential of a node to lead to an optimal solution, and  $R_{s}$  signifies the normalized reward for path-switching steps. Due to space constraints, the normalizing methods and parameters are detailed in the Appendix.
+
+In branch-and-bound algorithms, node selection is crucial for optimizing the solving process. The overarching goal is to efficiently converge to the optimal solution by narrowing the primal and dual bounds. Each node is critically evaluated for its potential to: (1) update the global lower bound, finding an improved feasible solution, (2) influence the global upper bound, reducing the gap between the current best known solution and the optimal solution, or (3) generate promising descendant nodes that might yield feasible solutions.
+
+The updates to the global lower and upper bounds are quantitatively assessed by changes in the global gap. We denote the optimal solution reward  $R_{o}$ , a methodology endorsed by prior re
+
+search (He et al., 2014; Yilmaz & Yorke-Smith, 2020; Labassi et al., 2022).
+
+$$
+R _ {o} (N) = \left\{ \begin{array}{l l} 1 & \text {i f} L B _ {N} \leq x ^ {*} \leq U B _ {N}, \\ 0 & \text {o t h e r w i s e .} \end{array} \right. \tag {5}
+$$
+
+where  $LB_{N}$  and  $UB_{N}$  denotes the local lower and upper bound of node  $N$ ,  $x^{*}$  denotes the optimal solution. Each node defines a unique segment of the search space; If the optimal solution resides within the current node's domain, the reward  $R_{o}$  is 1. If the optimal solution is not within the domain of the current node, the reward for this component is zero. This method enables the model to learn from node features to discern which nodes are likely to lead to potentially promising solutions.
+
+However, one often-overlooked aspect of node selection is the time taken to transition from the last selected node to the newly chosen one. This transition time can significantly impact overall performance. For instance, in the MaxSAT dataset, where  $n$  ranges from 70 to 80, the path switching phase consumes, on average,  $5.2\%$  of the total solving time. In the branch-and-bound process, solvers like SCIP (Bestuzheva et al., 2021) navigate the path from the last selected node to the newly chosen one, known as path switching.
+
+Figure 3 illustrates how each node traces back through the tree to find a common ancestor. Although this process might seem straightforward, it introduces significant computational overhead. We employ the "Eventhdlr" from PySCIPOpt (Maher et al., 2016). We utilize the "PY_SCIP_EVENTTYPE" states "NODESOLVED" and "NODEFOCUSED" to track the time from when one node is completely solved ("NODESOLVED") to when the solver shifts its focus to another node ("NODEFOCUSED"). Since the path switch time is positively correlated with the number of path switch steps, we incorporate the steps required for a path switch as a component of the reward func
+
+![](images/2f4931a11647c314ab0fbd2f02ed790fe744fa5f3498a7cc68054218fe67421e.jpg)  
+(a)  
+Figure 3: Path switching process. Figure (a) depicts the current state  $s_t$  with node  $N_t$  selected and figure (b) shows the subsequent state  $s_{t+1}$  with node  $N_{t+1}$  chosen.
+
+![](images/e8acf84369f4fb2d331244ecf00c8f3bb10d6d26ab7c6c07c8ca124ca92b8ebf.jpg)  
+(b)
+
+tion in our reinforcement learning model. We calculate the number of steps between nodes by adding the number of steps each node takes to reach their nearest common ancestor.
+
+# 6 EXPERIMENTS
+
+Table 2: Policy evaluation on the synthetic and real-world datasets. The best performance is marked in bold. We report the 1-shifted geometric mean (standard deviation) of the Time.  
+
+<table><tr><td rowspan="2">Methods</td><td colspan="3">FCMCNF</td><td colspan="3">MaxSAT</td><td colspan="3">GISP</td></tr><tr><td>Time(s)</td><td>wins</td><td>Im(%)</td><td>Time(s)</td><td>wins</td><td>Im(%)</td><td>Time(s)</td><td>wins</td><td>Im(%)</td></tr><tr><td>SCIP</td><td>21.15 ± 2.42</td><td>20/100</td><td>0.00</td><td>8.22 ± 1.90</td><td>23/100</td><td>0.00</td><td>3.88 ± 1.32</td><td>27/100</td><td>0.00</td></tr><tr><td>SVM</td><td>22.17 ± 2.44</td><td>14/100</td><td>-4.82</td><td>8.15 ± 1.76</td><td>8/100</td><td>+0.85</td><td>4.98 ± 1.78</td><td>3/100</td><td>-28.35</td></tr><tr><td>RankNet</td><td>22.39 ± 2.24</td><td>14/100</td><td>-5.86</td><td>8.29 ± 1.83</td><td>9/100</td><td>-0.85</td><td>5.03 ± 1.81</td><td>2/100</td><td>-29.64</td></tr><tr><td>GNN</td><td>19.63 ± 2.13</td><td>18/100</td><td>+7.19</td><td>8.05 ± 1.98</td><td>17/100</td><td>+2.07</td><td>4.19 ±1.79</td><td>13/100</td><td>-7.99</td></tr><tr><td>TRGNN (Ours)</td><td>14.50 ± 2.21</td><td>34/100</td><td>+31.44</td><td>7.60 ± 1.86</td><td>43/100</td><td>+7.54</td><td>3.34 ± 1.37</td><td>55/100</td><td>+13.92</td></tr></table>
+
+<table><tr><td rowspan="2">Methods</td><td colspan="3">MIK</td><td colspan="3">CORLAT</td><td colspan="3">Anonymous</td></tr><tr><td>Time(s)</td><td>wins</td><td>Im(%)</td><td>Time(s)</td><td>wins</td><td>Im(%)</td><td>Time(s)</td><td>wins</td><td>Im(%)</td></tr><tr><td>SCIP</td><td>29.45 ± 1.76</td><td>0/18</td><td>0.00</td><td>6.81 ± 5.00</td><td>6/50</td><td>0.00</td><td>215.28 ± 37.90</td><td>0/20</td><td>0.00</td></tr><tr><td>SVM</td><td>39.60 ± 1.79</td><td>0/18</td><td>-34.46</td><td>4.78 ± 4.42</td><td>4/50</td><td>+29.81</td><td>1966.14 ± 25.06</td><td>0/20</td><td>-8.13</td></tr><tr><td>RankNet</td><td>48.00 ± 1.95</td><td>0/18</td><td>-62.99</td><td>4.52 ± 3.81</td><td>5/50</td><td>+33.63</td><td>1840.65 ± 48.86</td><td>0/20</td><td>-7.55</td></tr><tr><td>GNN</td><td>40.08 ± 1.69</td><td>2/18</td><td>-36.99</td><td>4.71 ± 3.90</td><td>8/50</td><td>+30.84</td><td>271.40 ± 15.38</td><td>0/20</td><td>-26.07</td></tr><tr><td>TRGNN (Ours)</td><td>11.95 ± 1.45</td><td>16/18</td><td>+59.42</td><td>4.23 ± 4.10</td><td>19/50</td><td>+37.89</td><td>105.68 ± 20.87</td><td>20/20</td><td>+50.91</td></tr></table>
+
+Our experiments have three main parts: Experiment (1) Evaluate our approach on three classical MILP problems and three challenging MILP problem benchmarks from diverse application areas. Experiment (2) Perform carefully designed ablation studies to provide further insight into TRGNN.
+
+Experiment (3) Test whether TRGNN can generalize to instances significantly larger than those seen during training. The codes are modified from Labassi et al. (2022).
+
+# 6.1 SETUP
+
+**Benchmarks.** We evaluate our approach on six NP-hard MILP problem benchmarks, which consist of three classical synthetic MILP problems and three real-world MILP problems from diverse application areas. (1) Synthetic datasets comprise three widely used synthetic MILP problem benchmarks: Fixed Charge Multicommodity Network Flow (FCMCNF) (Hewitt et al., 2010), Maximum Satisfiability (MAXSAT) (Ansótegui & Gabás, 2017) and Generalized Independent Set (GISP) (Colombi et al., 2017). We artificially generate instances following Béjar et al. (2009); Chmiela et al. (2021); Labassi et al. (2022). (2) Real-world datasets comprise MIK (Atamürk, 2003), CORLAT (Gomes et al., 2008), and the Anonymous problem, inspired by a large-scale industrial application (Gasse et al., 2022). Details of datasets are provided in the Appendix D.1. Throughout all experiments, we use SCIP 8.0.4 (Bestuzheva et al., 2021) as the backend solver, which is the state-of-the-art open source solver, and is widely used in research of machine learning for combinatorial optimization (Chmiela et al., 2021; Gasse et al., 2019; Turner et al., 2022; Wang et al., 2023). We keep all the other SCIP parameters to default and emphasize that all of the SCIP solver's advanced features, such as presolve and heuristics, are open.
+
+Baselines. We compare against the state-of-the-art best estimate node selection rule (Bénichou et al., 1971; Forrest et al., 1974). This is the default method in SCIP (Bestuzheva et al., 2021). In addition, we compare against three machine learning approaches: the Support Vector Machine (SVM) approach (He et al., 2014), the RankNet feedforward neural network approach (Song et al., 2018), and the approach based on Graph Neural Networks (GNN) (Labassi et al., 2022).
+
+Training. We generate extensive synthetic datasets, consisting of 10,000 training samples and 1,000 test samples. From these, we randomly select 1,000 samples for the SVM, RankNet, and GNN models training and 100 samples for testing within each problem. Additionally, we generate 1,000 larger-scale datasets, randomly selecting 50 samples from each to evaluate the transfer capability of our method. For the real-world dataset, each problem set is split into training and test sets with  $80\%$  and  $20\%$  of the instances. Specifically, we train model with 1,000 samples for Corlat, 72 samples for MIK and 65 samples for Anonymous dataset. The model implemented in PyTorch (Paszke et al., 2019) and optimized using Adam (Kingma & Ba, 2014) with training batch size of 16. The training process is conducted on a single machine that contains eight GPU devices(NVIDIA GeForce RTX 4090) and two AMD EPYC 7763 CPUs.
+
+We observe that our model performs remarkably well even when trained with fewer samples, achieving results comparable to those obtained with larger datasets. This demonstrates its efficiency and effectiveness in utilizing limited data. The ability to maintain such performance with a reduced dataset size is particularly valuable, given the significant time and computational resources required to collect training samples by solving multiple MILP instances. For instance, collecting 90 samples for the Anonymous problem alone took over 13.89 hours. In the Appendix D.5, we provide a comparison between our TRGNN model and a GNN baseline trained with the same limited dataset, referred to as GNN_light.
+
+Evaluation Metrics. We employ two widely recognized evaluation metrics: (1) Time: running times in seconds (lower is better) and (2) Wins: number of times each node selection strategy resulted in the fastest solving time, over total number of solved instances. We limit the solution time for all problems to a maximum of 3600 seconds. Any time exceeding this limit is recorded as 3600 seconds. We assess node selection methods in terms of the 1-shifted geometric mean over the instances, accompanied by the geometric standard deviation. Furthermore, to evaluate different node selection methods compared to the default SCIP, we propose an (3) Improvement metric. Specifically, we define the metric by  $\text{Im} = \frac{T(SCIP) - T(M)}{T(SCIP)}$ , where  $T(SCIP)$  represents the solving time of SCIP, and  $T(M)$  represents the solving time of the compared method. Due to the space limits, we report the size of the branch-and-bound tree (Nodes) in the Appendix D.4.
+
+# 6.2 COMPARATIVE EXPERIMENT
+
+For each problem, machine learning models are trained on instances of the same size as the test instances. The results in Table 2 suggest TRGNN significantly outperforms all baseline methods.
+
+Importantly, it is the only known learning-based node selection method to consistently outperform the default SCIP in terms of solving time across all six datasets. Compared to SCIP, TRGNN demonstrates notable efficiency improvements across all tested problems, being  $31.44\%$  faster in FCMCNF,  $7.54\%$  in MaxSAT,  $13.92\%$  in GISP,  $59.42\%$  in MIK,  $37.89\%$  in CORLAT,  $50.91\%$  in Anonymous.
+
+# 6.3 ABLATION STUDY
+
+We present an ablation study conducted to evaluate the contribution of different metrics in Sec. 5. We report the performance of TRGNN, TRGNN-1 and TRGNN-2 with different reward strategies in Table 3 on FCMCNF, GISP, MIK, and Anonymous. TRGNN-1 utilizes our RL framework with a reward function based on the optimal solution, as established in prior research (Labassi et al., 2022).
+
+TRGNN-2 incorporates gap change into the node metrics, while TRGNN, our full model, additionally accounts for path switching overhead. The results in Table 3 reveal that both TRGNN-1 and TRGNN-2 outperform SCIP on GISP, MIK, and Anonymous datasets, highlighting the benefits of integrating the reinforcement learning model with our tripartite graph representation. Furthermore, TRGNN significantly surpasses TRGNN-1, TRGNN-2, and other baselines across all four datasets. The
+
+Table 3: Comparison between TRGNN, TRGNN with two metrics (TRGNN-2) and TRGNN with one metric (TRGNN-1).  
+
+<table><tr><td>Methods</td><td>FCMCNF</td><td>GISP</td><td>MIK</td><td>Anonymous</td></tr><tr><td>SCIP</td><td>21.15 s</td><td>3.88 s</td><td>29.45 s</td><td>215.28 s</td></tr><tr><td>TRGNN-1</td><td>23.28 s</td><td>3.48 s</td><td>13.27 s</td><td>183.62 s</td></tr><tr><td>TRGNN-2</td><td>22.97 s</td><td>3.36 s</td><td>12.08 s</td><td>124.05 s</td></tr><tr><td>TRGNN(Ours)</td><td>14.50 s</td><td>3.34 s</td><td>11.95 s</td><td>105.68 s</td></tr></table>
+
+results demonstrate that both of the gap update and path switching overhead is important for efficient exploration in complex tasks.
+
+# 6.4 GENERALIZATION
+
+We evaluate the ability of TRGNN to generalize across larger sizes of MILPs. We evaluate the generalization ability on FCMCNF, MaxSAT and GISP datasets, as we can artificially generate the larger transfer instances.
+
+Table 4: The generalization ability of TRGNN. The best performance is marked in bold. We report the 1-shifted geometric mean (standard deviation) of the Time.  
+
+<table><tr><td rowspan="2">Methods</td><td colspan="3">FCMCNF</td><td colspan="3">MaxSAT</td><td colspan="3">GISP</td></tr><tr><td>Time(s)</td><td>wins</td><td>Im(%)</td><td>Time(s)</td><td>wins</td><td>Im(%)</td><td>Time(s)</td><td>wins</td><td>Im(%)</td></tr><tr><td>SCIP</td><td>1474.60 ± 4.19</td><td>3/50</td><td>0.00</td><td>76.12 ± 1.83</td><td>4/50</td><td>0.00</td><td>427.29 ± 1.65</td><td>0/50</td><td>0.00</td></tr><tr><td>SVM</td><td>1857.29 ± 3.99</td><td>0/50</td><td>-25.95</td><td>84.78 ± 2.66</td><td>1/50</td><td>-11.37</td><td>557.52 ± 2.39</td><td>0/50</td><td>-30.48</td></tr><tr><td>RankNet</td><td>1560.22 ± 4.18</td><td>7/50</td><td>-5.80</td><td>101.45 ± 2.15</td><td>0/50</td><td>-33.27</td><td>701.62 ± 2.02</td><td>0/50</td><td>-64.20</td></tr><tr><td>GNN</td><td>1503.83 ± 3.90</td><td>11/50</td><td>-1.98</td><td>69.44 ± 1.57</td><td>19/50</td><td>+8.78</td><td>534.72 ± 1.90</td><td>0/50</td><td>-25.14</td></tr><tr><td>TRGNN (Ours)</td><td>1243.39 ± 3.64</td><td>29/50</td><td>+15.68</td><td>48.20 ± 1.55</td><td>26/50</td><td>+36.67</td><td>339.66 ± 1.85</td><td>50/50</td><td>+20.51</td></tr></table>
+
+The results in Table 4 indicate that TRGNN significantly surpasses the baselines in terms of both solving time and number of wins, demonstrating its superior generalization ability. Notably, TRGNN demonstrates an even more pronounced improvement in solving time on larger datasets compared to smaller ones, underscoring the importance of effective node selection strategies, particularly for large-scale MILP instances.
+
+# 7 CONCLUSIONS
+
+We addressed two pivotal problems in node selection including the sufficient tree representation and assessment of the node quality with delayed rewards. We introduce an innovative tripartite graph representation for the branch-and-bound search tree and provide theoretical evidence. Subsequently, we introduce more comprehensive metrics for node selection and develop a novel TRGNN model, leveraging reinforcement learning to acquire node selection policies. Experiments show that TRGNN significantly outperforms human-designed and learning-based baselines in terms of solving efficiency on three synthetic MILP problems and three real-world MILP problems.
+
+# ACKNOWLEDGMENTS
+
+The research is partially supported by National Key R&D Program of China under Grant No. 2021ZD0110400, Innovation Program for Quantum Science and Technology 2021ZD0302900 and China National Natural Science Foundation with No. 62132018, 62231015, “Pioneer” and “Leading Goose” R&D Program of Zhejiang, 2023C01029, and 2023C01143.
+
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+
+# A THEORETICAL ANALYSIS
+
+We enumerate our theoretical findings and provide detailed proofs to substantiate them.
+
+First, we list the definition of a Markov chain and the data processing inequality (Reza, 1994) to aid in the proof of the subsequent theorem.
+
+Definition A.1. Consider three random variables  $X$ ,  $Y$ , and  $Z$ . They form a Markov chain  $X \to Y \to Z$  if it holds that:
+
+$$
+p (x, y, z) = p (x) p (y | x) p (z | y),
+$$
+
+where  $p(x,y,z)$  is the joint probability density function of  $X$ ,  $Y$ , and  $Z$ .
+
+Theorem A.1 (Data processing inequality). If  $X \to Y \to Z$  forms a Markov chain, then:
+
+$$
+I (X; Y) \geq I (X; Z).
+$$
+
+Theorem A.2 (Restatement of Theorem 4.2). Given a node  $N_{p}$  and its two child nodes  $N_{l}$  and  $N_{r}$ , it holds that  $I(Q; J_{N_{p}} \oplus J_{N_{l}} \oplus J_{N_{r}}) = I(Q; J_{N_{l}} \oplus J_{N_{r}})$ , where  $J_{N}$  denotes the extracted feature of the node  $N$ .
+
+Proof of Theorem 4.2. First, note that  $J_{N_l} \oplus J_{N_r} \subseteq J_{N_p} \oplus J_{N_l} \oplus J_{N_r}$ , hence  $Q \to J_{N_l} \oplus J_{N_r} \to J_{N_p} \oplus J_{N_l} \oplus J_{N_r}$  is a Markov chain. According to the data processing inequality (Theorem A.1), we have:
+
+$$
+I (Q; J _ {N _ {p}} \oplus J _ {N _ {l}} \oplus J _ {N _ {r}}) \geq I (Q; J _ {N _ {l}} \oplus J _ {N _ {r}}).
+$$
+
+To complete the proof, it is necessary to show that:
+
+$$
+I (Q; J _ {N _ {p}} \oplus J _ {N _ {l}} \oplus J _ {N _ {r}}) \leq I (Q; J _ {N _ {l}} \oplus J _ {N _ {r}}).
+$$
+
+Assuming  $J_{N_l} \oplus J_{N_r} \to J_{N_p}$ , then  $Q \to J_{N_p} \oplus J_{N_l} \oplus J_{N_r} \to J_{N_l} \oplus J_{N_r}$  also forms a Markov chain. Thus, by applying the data processing inequality once again, we establish the desired equality.
+
+To prove the Markov chain, we analyze the relationship between the parent node  $N_{p}$  and the child nodes  $N_{l}$  and  $N_{r}$ . The node features encompass both the MILP characteristics, denoted by  $J_{m}$ , including the variables, constraints and its objective function, and the global attributes within the tree, denoted by  $J_{g}$ . Thus, we have:  $J_{N} = (J_{m})_{N} \oplus (J_{g})_{N}$ .
+
+In order to prove  $J_{N_l} \oplus J_{N_r} \to J_{N_p}$ , we only need to prove  $(J_m)_{N_l} \oplus (J_m)_{N_r} \to (J_m)_{N_p}$  and  $(J_g)_{N_l} \oplus (J_g)_{N_r} \to (J_g)_{N_p}$ .
+
+We consider the MILP characteristics, which is only related to their problem structures. The variables and objective function in the branch-and-bound tree is the same as the root node  $(J_{m})_{N_{0}}$ . We denote the common variables and objective function feature as  $CF$ , the variable set as  $V = \{x_{i} \mid i \in \{0,1,\ldots,l\}\}$ , with  $|V| = l$ , and the constraint set of the node  $N$  as  $S_{N}$ . Thus, we have  $(J_{m})_{N} = S_{N} \oplus CF$ .
+
+Because the constraints of both the child nodes  $N_{l}$  and child nodes  $N_{r}$  is the constraint of the parent node  $N_{p}$  added by the branching constraints. Denote  $N_{p}$ 's LP relax solution as  $x_{N_p}$ , and let  $z = \left\lfloor (x_{N_p})_i\right\rfloor$ , where  $(x_{N_p})_i$  denotes the value of variable  $x_{i}$  in the LP relax solution  $x_{N_p}$ .
+
+The branching constraints are denoted as  $x_{i} \leq z$  for  $N_{l}$  and  $x_{i} \geq z + 1$  for  $N_{r}$ , where  $x_{i} \in V$  is the selected branch variable on the parent node  $N_{p}$ .
+
+Consequently, the sets of constraints for  $N_{l}$  and  $N_{r}$  can be represented as:
+
+$$
+S _ {N _ {l}} = S _ {N _ {p}} \cup \{x _ {i} \leq z \},
+$$
+
+and
+
+$$
+S _ {N _ {r}} = S _ {N _ {p}} \cup \{x _ {i} \geq z + 1 \}.
+$$
+
+which leads to a significant overlap:  $S_{N_l} \cap S_{N_r} = S_{N_P}$ . This implies that the combined MILP characteristics  $J_{m}$  for  $N_{l}$  and  $N_{r}$  encapsulate the MILP characteristics of  $N_{p}$ :
+
+$$
+\begin{array}{l} \left(J _ {m}\right) _ {N _ {l}} \cap \left(J _ {m}\right) _ {N _ {r}} = \left(S _ {N _ {l}} \oplus C F\right) \cap \left(S _ {N _ {r}} \oplus C F\right) \\ = \left(S _ {N _ {l}} \cap S _ {N _ {r}}\right) \oplus C F \\ = \left(S _ {N _ {p}} \oplus C F\right) \\ = \left(J _ {m}\right) _ {N _ {p}}. \\ \end{array}
+$$
+
+Thus, we establish that a function relationship  $f_{m}$  exists such that:  $(J_{m})_{N_{p}} = f_{m}((J_{m})_{N_{l}} \oplus (J_{m})_{N_{r}})$ .
+
+Considering the mutual information equations:
+
+$$
+\begin{array}{l} I \left(Q; J _ {N _ {p}} \oplus \left(J _ {N _ {l}} \oplus J _ {N _ {r}}\right)\right) = I \left(Q; \left(\left(J _ {m}\right) _ {N _ {p}} \oplus \left(J _ {g}\right) _ {N _ {p}}\right) \oplus \left(J _ {N _ {l}} \oplus J _ {N _ {r}}\right)\right) \\ \leq I (Q; f _ {m} \left(\left(J _ {m}\right) _ {N _ {l}} \oplus \left(J _ {m}\right) _ {N _ {r}}\right) \oplus \left(J _ {g}\right) _ {N _ {p}} \oplus \left(J _ {N _ {l}} \oplus J _ {N _ {r}}\right)) \\ \leq I (Q; \left(\left(J _ {m}\right) _ {N _ {l}} \oplus \left(J _ {m}\right) _ {N _ {r}}\right) \oplus \left(J _ {g}\right) _ {N _ {p}} \oplus \left(J _ {N _ {l}} \oplus J _ {N _ {r}}\right)) \\ \leq I (Q; \left(\left(J _ {m}\right) _ {N _ {l}} \oplus \left(J _ {m}\right) _ {N _ {r}}\right) \oplus \left(J _ {g}\right) _ {N _ {p}} \oplus \left(\left(J _ {m}\right) _ {N _ {l}} \oplus \left(J _ {g}\right) _ {N _ {l}}\right) \\ \oplus \left(\left(J _ {m}\right) _ {N _ {r}} \oplus \left(J _ {g}\right) _ {N _ {r}}\right)) \\ = I (Q; \left(J _ {g}\right) _ {N _ {p}} \oplus \left(J _ {N _ {l}} \oplus J _ {N _ {r}}\right)). \\ \end{array}
+$$
+
+Next, we turn our attention to the global attributes  $J_{g}$ , which contain the solving process characteristics of each node and the positional characteristics of each node to represent the whole tree.
+
+The solving process characteristics originates from the an improved feasible integer solution or the LP relaxation solution, denoted as the update of the local upper and lower bounds. We denote the local upper and lower bounds on the node  $N$  as  $UB_{N}$  and  $LB_{N}$ . We denote the node  $N$ 's positional characteristics as  $p_{N}$ , which can represent the position of the node  $N$  in the tree. Thus, we have  $(J_{g})_{N} = (LB_{N},UB_{N})\oplus p_{N}$ .
+
+Given that the LP relaxation solutions of the child nodes,  $N_{l}$  and  $N_{r}$ , are also solutions of the node  $N_{p}$ . The local upper bound only updates when finding a tighter LP relaxation solution. Thus, it is established that  $UB_{N_l} \leq UB_{N_p}$  and  $UB_{N_r} \leq UB_{N_p}$ . According to Lemma A.1, it is inferred that the infeasible integer solution of the node  $N_{p}$  is also a solution of either  $N_{l}$  or  $N_{r}$ . Hence, we deduce that  $LB_{N_l} \leq LB_{N_p}$  and  $LB_{N_r} \leq LB_{N_p}$ . This establishes that:
+
+$$
+\left(L B _ {N _ {p}}, U B _ {N _ {p}}\right) \subseteq \left(L B _ {N _ {l}}, U B _ {N _ {l}}\right) \cup \left(L B _ {N _ {r}}, U B _ {N _ {r}}\right).
+$$
+
+Since child nodes are derived from the parent node, positional attributes of  $p_{N_p}$  of the parent node can be deduced by the two child nodes' positional features  $p_{N_l}$  and  $p_{N_r}$ . We denote a function  $f_p$  such that
+
+$$
+p _ {N _ {p}} = f _ {p} (p _ {N _ {l}}, p _ {N _ {r}}),
+$$
+
+demonstrating that:
+
+$$
+(J _ {g}) _ {N _ {p}} \subseteq (L B _ {N _ {l}}, U B _ {N _ {l}}) \cup (L B _ {N _ {r}}, U B _ {N _ {r}}) \oplus p _ {N _ {p}} = (L B _ {N _ {l}}, U B _ {N _ {l}}) \cup (L B _ {N _ {r}}, U B _ {N _ {r}}) \oplus f _ {p} (p _ {N _ {l}}, p _ {N _ {r}}).
+$$
+
+Considering the mutual information equations:
+
+$$
+\begin{array}{l} I \left(Q; J _ {N _ {p}} \oplus J _ {N _ {l}} \oplus J _ {N _ {r}}\right) \leq I \left(Q; \left(J _ {g}\right) _ {N _ {p}} \oplus \left(J _ {N _ {l}} \oplus J _ {N _ {r}}\right)\right) \\ = I (Q; \left(\left(L B _ {N _ {p}}, U B _ {N _ {p}}\right) \oplus p _ {N _ {p}}\right) \oplus \left(J _ {N _ {l}} \oplus J _ {N _ {r}}\right)) \\ \leq I (Q; \left(\left(L B _ {N _ {l}}, U B _ {N _ {l}}\right) \cup \left(L B _ {N _ {r}}, U B _ {N _ {r}}\right) \oplus f _ {p} \left(p _ {N _ {l}}, p _ {N _ {r}}\right)\right) \\ \oplus \left(J _ {N _ {l}} \oplus J _ {N _ {r}}\right)) \\ \leq I (Q; \left(\left(L B _ {N _ {l}}, U B _ {N _ {l}}\right) \oplus \left(L B _ {N _ {r}}, U B _ {N _ {r}}\right) \oplus p _ {N _ {l}} \oplus p _ {N _ {r}}\right) \\ \oplus \left(J _ {N _ {l}} \oplus J _ {N _ {r}}\right)) \\ = I (Q; \left(\left(J _ {g}\right) _ {N _ {l}} \oplus \left(J _ {g}\right) _ {N _ {r}}\right) \oplus \left(J _ {N _ {l}} \oplus J _ {N _ {r}}\right)) \\ = I (Q; J _ {N _ {l}} \oplus J _ {N _ {r}}). \\ \end{array}
+$$
+
+![](images/52b3cf7e5d76c7be1ffe0157e620b5a5353baba54d2ecf0988f8967ff254c2f6.jpg)
+
+Lemma A.1. Given a node  $N_{p}$  and its two child nodes  $N_{l}$  and  $N_{r}$ , any infeasible integer solution of node  $N_{p}$  is also an infeasible solution of either  $N_{l}$  or  $N_{r}$ .
+
+Proof of Lemma A.1. Suppose there exists an infeasible integer solution  $x$  of node  $N_0$  that is not an infeasible solution of either child nodes,  $N_1$  or  $N_2$ .
+
+Let  $S_{N_0}, S_{N_1}$ , and  $S_{N_2}$  represent the constraint spaces of nodes  $N_0, N_1$ , and  $N_2$ , respectively. By assumption,  $x \in S_{N_0}$  but  $x \notin S_{N_1} \cup S_{N_2}$ .
+
+Given that  $N_{1}$  and  $N_{2}$  are child nodes of  $N_{0}$ , we can express the constraint spaces of  $N_{1}$  and  $N_{2}$  as
+
+$$
+\mathcal {S} _ {N _ {1}} = \mathcal {S} _ {N _ {0}} \cap \{x \in \mathcal {C} | x _ {i} \geq z + 1, z \in \mathbb {Z} \}
+$$
+
+and
+
+$$
+\mathcal {S} _ {N _ {2}} = \mathcal {S} _ {N _ {0}} \cap \{x \in \mathcal {C} | x _ {i} \leq z, z \in \mathbb {Z} \}.
+$$
+
+Thus,  $x$  must belong to the set difference  $\mathcal{S}_{N_0} \setminus (\mathcal{S}_{N_1} \cup \mathcal{S}_{N_2})$ , which is equivalent to
+
+$$
+\mathcal {S} _ {N _ {0}} \cap \{x \in \mathcal {C} | z <   x _ {i} <   z + 1, z \in \mathbb {Z} \}.
+$$
+
+However, since  $x$  is an integer solution, it cannot belong to the set  $\{x \in \mathcal{C} | z < x_i < z + 1, z \in \mathbb{Z}\}$ , leading to a contradiction. Therefore, any infeasible integer solution of node  $N_0$  must also be an infeasible solution of either  $N_1$  or  $N_2$ .
+
+Theorem A.3 (Restatement of Theorem 4.1). Given the node  $N_i$  with features  $J_{N_i}$ , the tripartite graph representation  $\phi_t(F)$  satisfies:  $I(Q; \phi_t(F)) = I(Q; F)$ , where  $F = \bigoplus_{i=0}^{m+n} J_{N_i}$ .
+
+Proof of Theorem 4.1. The leaf nodes are denoted as  $N_{j_1}, N_{j_2}, \ldots, N_{j_n}$ . The set of the leaf nodes' features is denoted as  $L = \bigoplus_{i=1}^{n} J_{N_{j_i}}$ .
+
+First, let us prove  $I(Q; \phi_t(F)) = I(Q; L)$ , where  $\phi_t(F) = (J_m)_{N_0} \oplus (\bigoplus_{i=1}^n B_{N_{j_i}}) \oplus (\bigoplus_{i=1}^n (J_g)_{N_{j_i}})$ , where  $J_N$  denotes the extracted feature of the node  $N$ ,  $B_N$  denotes the set of branching constraints added from the root node to the node  $N$ , and  $(J_g)_N$  denotes the global features in the tree of the node  $N$ , include both the local upper and lower bounds and its positional characteristics.
+
+Thus, we only need to prove  $Q \to \phi_t(F) \to L$  is a Markov chain.
+
+We have  $J_{N_{j_i}} = (J_m)_{N_{j_i}} \oplus (J_g)_{N_{j_i}}$  and  $(J_m)_{N_{j_i}} = S_{N_{j_i}} \oplus CF = (S_{N_0} \cup (\bigoplus_{i=1}^n B_{N_{j_i}})) \oplus CF = (J_m)_{N_0} \cup ((\bigoplus_{i=1}^n B_{N_{j_i}}) \oplus CF)$ . Thus,  $L = \bigoplus_{i=1}^n J_{N_{j_i}} = \bigoplus_{i=1}^n (J_m)_{N_0} \cup ((\bigoplus_{i=1}^n B_{N_{j_i}}) \oplus CF)) \oplus (J_g)_{N_{j_i}}$ .
+
+From  $\phi_t(F)$ , we can deduce  $(J_m)_{N_0}, B_{N_{j_i}}$ , and  $(J_g)_{N_{j_i}}$ , for all  $i = 1,2,\ldots,n$ . Due to  $(J_m)_{N_0} = S_{N_0} \oplus CF$ , where  $S_{N_0}$  denotes the set of the root node's constraints, we can know  $CF$  from  $\phi_t(F)$ . Then, we can denote there exists a function  $f_1$  that  $f_1(\phi_t(F)) = L$ , which shows  $Q \to \phi_t(F) \to L$  is a Markov chain.
+
+At the same, we can also known  $(J_{g})_{N_{j_{i}}}$ ,  $(J_{m})_{N_{0}}$ , and  $B_{N_{j_i}}$ , for all  $i = 1,2,\ldots ,n$ , from the representation  $L$ . Thus, there exists a function  $f_{2}$  that  $f_{2}(L) = \phi_{t}(F)$ , which shows  $Q\to L\to \phi_t(F)$  is a Markov chain.
+
+Next, we prove  $I(Q; \bigoplus_{i=1}^{n} J_{N_{j_i}}) = I(Q; F)$ . Because  $I(Q; L) \leq I(Q; F)$  obviously satisfies, we only need to prove  $I(Q; L) \geq I(Q; F)$ . We prove this step by mathematical induction. When expanding the root node for the first time, the conclusion holds according to Theorem 4.2.
+
+Suppose that the conclusion  $I(Q;L) \geq I(Q;F)$  holds after the  $k$ -th selection of nodes. That is, we have  $Q \to L \to F$  is a Markov chain. We denote the set of all of the nodes in the tree as  $\mathcal{T}_k$ . When expanding the leaf nodes for the  $k + 1$ -th time, we denote the selected leaf node as  $N_s$ . We denote the set of the other candidate nodes as  $\mathcal{N}'$ . Thus, we have  $Q \to (\bigoplus_{N \in \mathcal{N}'} J_N) \oplus J_{N_s} \to \bigoplus_{N \in \mathcal{T}_k} J_N$  is a Markov chain. We denote the selected node is expanded into two child nodes. To simplify, we denote them as  $N_{s,l}$  and  $N_{s,r}$ . The tree node set turns to be  $T_{k + 1} = T_k \cup \{N_{s,l}, N_{s,r}\}$ . According to Theorem 4.2, we have  $Q \to J_{N_{s,l}} \oplus J_{N_{s,r}} \to J_{N_s} \oplus J_{N_{s,l}} \oplus J_{N_{s,r}}$  is a Markov chain. Thus,  $Q \to (\bigoplus_{N \in \mathcal{N}'} J_N) \oplus J_{N_{s,l}} \oplus J_{N_{s,r}} \to \bigoplus_{N \in \mathcal{T}_k} J_N \oplus J_{N_{s,l}} \oplus J_{N_{s,r}}$  is a Markov chain.
+
+According to the Data processing inequality(Theorem A.1), we can prove  $I(Q;L) \geq I(Q;F)$ .
+
+□
+
+# B MORE DETAILS OF NODE SELECTION PROCESS
+
+Node selection process. Algorithm 1 details a general procedure for node selection within this MDP formulation. It describes the primary operations carried out by the node selector in the B&B process. The specific implementation details may vary depending on the chosen node selection strategy, but the fundamental concept remains consistent. Initially, a node is chosen based on the branching strategy. After selection, the node is evaluated, which involves updating its bounds and assessing whether it should be pruned. If the node is not pruned, the tree is branched to enable further exploration.
+
+Algorithm 1 General Node Selection Procedure  
+Require: Node list  $L$ , parent node bounds, probability Prob of heuristic integer solution calculation  
+1: for each node  $P$  in  $L$  do  
+2: Select node  $P$  according to the node selection strategy.  
+3: Update the lower bound of  $P$ ,  $LB(P) = lb_{parent}$ , where  $lb_{parent}$  is the lower bound of parent node of  $P$ .  
+4: Solve the LP relaxation of node  $P$ , getting the solution Sol1 and objective function value Obj1.  
+5: Update the upper bound of  $P$ ,  $UB(P) = Obj1$ . Propagate the updated upper bound upward.  
+6: if Obj1  $\leq$  LB(P) then  
+7: Prune node  $P$ .  
+8: else  
+9: if Sol1 is an integer solution then  
+10: Update the lower bound of  $P$ ,  $LB(P) = Obj1$ . Propagate the updated lower bounds upward.  
+11: end if  
+12: Call the heuristic to calculate an integer solution with probability Prob, resulting in Sol2 and Obj2.  
+13: if Obj2 > LB(P) is an integer solution then  
+14: Update the lower bound of  $P$ ,  $LB(P) = Obj2$ . Propagate the updated lower bounds upward.  
+15: end if  
+16: Branch at node  $P$  and add the children nodes to list  $L$ .  
+17: end if  
+18: end for
+
+# C MORE DETAILS OF TRIPARTITE GRAPH.
+
+Graph representation. Having sufficient information is crucial to infer the optimal node for selection. But what qualifies as "sufficient" in this context? An ideal problem representation should be capable of incorporating information that affects node selection, which includes the inherent attributes of the original problem  $(A, b, c)$  and the attributes of the explored space. Actually, such problem has a strong mathematical structure (Chen et al., 2022). For instance, if we swap the positions of the  $i, j$ -th variable in 1, elements in vectors  $b, c$  and columns of matrix  $A$  will be reordered. The reordered features  $(\hat{A}, \hat{b}, \hat{c})$  actually represent an exactly equivalent MILP problem with the original one  $(A, b, c)$ . Such property is named as permutation invariance. If we do not explicitly restrict ML models with a permutation invariant structure, the models may overfit to the variable/constraint orders of instances in the training set. Motivated by this point, we adopt the graph representation that are permutation invariant naturally in Section 4. A list of the features included in our tripartite graph representation is given as Table 5.
+
+Table 5: Description of the constraint, variable and leaf node vertex features, and edge features in our tripartite state representation.  
+
+<table><tr><td>Category</td><td>Feature</td><td>Description</td></tr><tr><td rowspan="4">variable vertex</td><td>lb</td><td>Lower bound.</td></tr><tr><td>ub</td><td>Upper bound.</td></tr><tr><td>objective_coeff</td><td>Objective coefficient.</td></tr><tr><td>var_type</td><td>Type (binary, integer and continuous) as a one-hot encoding.</td></tr><tr><td>constraint edge</td><td>coef</td><td>Constraint coefficient.</td></tr><tr><td rowspan="2">constraint vertex</td><td>rhs</td><td>Right-hand side of the constraint.</td></tr><tr><td>cons_type</td><td>Constraint type feature (eq, geq) as a one-hot encoding.</td></tr><tr><td rowspan="4">leaf node vertex</td><td>leaf.lb</td><td>Lower bound of the leaf node.</td></tr><tr><td>leaf ub</td><td>Upper bound of the leaf node.</td></tr><tr><td>depth</td><td>Depth of the leaf node.</td></tr><tr><td>estimate</td><td>Estimate value of the leaf node.</td></tr><tr><td rowspan="2">branching constraint edge</td><td>bc.lb</td><td>Lower bound of the variable.</td></tr><tr><td>bc ub</td><td>Upper bound of the variable.</td></tr></table>
+
+# D MORE DETAILS OF EXPERIMENTS
+
+The details of experiments are provided in this section.
+
+# D.1 BENCHMARK
+
+Synthetic datasets. We similarly employ three synthetic instance families, just like Labassi et al. (2022) in the latest node selection work. The first benchmark is composed of Fixed Charge Multicommodity Network Flow (FCMCNF) instances (Hewitt et al., 2010), generated from the code of Chmiela et al. (2021). We train and test on instances with  $n = 20$  and  $m = 1.5 \times n$  commodities, and also evaluate on larger transfer instances with  $n = 30$  nodes. The second benchmark is composed of MaxSAT (Ansótegui & Gabàs, 2017) instances, generated following the scheme of Béjar et al. (2009). We train and test on instances with a uniformly sampled number of nodes  $n \in [80, 100]$  and transfer on instances with  $n \in [120, 150]$ . Our third benchmark is composed of Generalized Independent Set (GISP) instances (Colombi et al., 2017), generated from the code of Chmiela et al. (2021). We train and test on instances with a uniformly sampled number of nodes  $n \in [60, 70]$  and transfer on instances with  $n \in [120, 150]$ . All these families require an underlying graph: we use in each case Erdős-Rényi random graphs with the prescribed number of nodes, with edge probability  $p = 0.33$  for FCMCNF and  $p = 0.66$  for MaxSAT and GISP.
+
+Table 6: Average value of the size of variables and constraints for MILP problems (Train and Test).  
+
+<table><tr><td>Number</td><td>GISP</td><td>MaxSAT</td><td>FCMCNF</td><td>MIK</td><td>Corlat</td><td>Anonymous</td></tr><tr><td>Avg. Binary</td><td>65.0</td><td>1611.70</td><td>125.61</td><td>79.17</td><td>100.0</td><td>14747.35</td></tr><tr><td>Avg. Int</td><td>0.0</td><td>0.0</td><td>0.0</td><td>420.83</td><td>0.0</td><td>1268.35</td></tr><tr><td>Avg. Continuous</td><td>625.48</td><td>0.0</td><td>3185.67</td><td>18.33</td><td>366.0</td><td>21865.6</td></tr><tr><td>Avg. Cons</td><td>1249.78</td><td>1542.16</td><td>633.36</td><td>439.17</td><td>486.46</td><td>49603.6</td></tr></table>
+
+Table 7: Average value of the size of variables and constraints for MILP problems (Transfer).  
+
+<table><tr><td>Number</td><td>GISP</td><td>MaxSAT</td><td>FCMCNF</td></tr><tr><td>Avg. Binary</td><td>134.14</td><td>3353.61</td><td>287.23</td></tr><tr><td>Avg. Int</td><td>0.0</td><td>0.0</td><td>0.0</td></tr><tr><td>Avg. Continuous</td><td>2690.61</td><td>0.0</td><td>10754.84</td></tr><tr><td>Avg. Cons</td><td>5378.52</td><td>3218.40</td><td>1410.72</td></tr></table>
+
+Real-world datasets. Following He et al. (2014); Nair et al. (2020), our dataset consists of the following components: MIK (Atamtürk, 2003), a set of MILP problems with knapsack constraints, and CORLAT (Gomes et al., 2008), a real dataset used for the construction of a wildlife corridor for grizzly bears in the Northern Rockies region. In addition to these datasets, we have introduced a more challenging dataset into our experiments: the Anonymous problem, sourced from the NeurIPS 2021 ML4CO competition (Gasse et al., 2022). Each problem set is split into training and test sets with  $80\%$  and  $20\%$  of the instances.
+
+# D.2 BASELINES
+
+The SVM and RankNet methods utilize a multilayer perceptron; the latter varies for one benchmark where they use three hidden layers, while for simplicity, we use a multilayer perceptron with a hidden layer of 32 neurons for all benchmarks (MLP). The GNN method uniquely leverages the structure of the graph to guide node selection. The features used in these papers are roughly similar; again, for simplicity, we adopt the fixed-dimensional features of He et al. (2014) for both the SVM and RankNet method.
+
+# D.3 HYPER-PARAMETER SETTINGS
+
+The hyperparameters primarily consist of those associated with the reward function and the preselection size  $n$  determined by the "BestEstimate" heuristic algorithm. In our evaluation, we set  $n = 5$
+
+and more detailed analysis of how variations in  $n$  affect the final node selection results is provided in Section D.7.
+
+Reward function hyper-parameters. We denote a parameter  $m$  to normalize the path switching steps. The path switching steps are denoted as  $s$ , the normalized reward for path switching steps is  $R_{s} = \frac{s}{m} - 1$ . In this function, if the path switching steps are less than  $m$ ,  $R_{s} > 0$ ; otherwise, the model receives a penalty. The value of  $m$  depends on the size of the problem. We use  $m = 15$  for the FCMCNF, MaxSAT, GISP, MIK and CORLAT datasets and  $m = 50$  for the Anonymous dataset. Moreover, we take the weight values  $w_{1} = w_{2} = w_{3} = \frac{1}{3}$ .
+
+Additionally, we follow Labassi et al. (2022) to limit the number of times the learning-based node selector is called. Consistent with the setup in Labassi et al. (2022), the learning-based node selection method is only called before the primal bound is updated four times. After four updates, we switch to the Breadth-First Search method.
+
+# D.4 MORE RESULTS OF NODES
+
+Table 8: Policy evaluation on the synthetic and real-world datasets. The best performance is marked in bold. We report the 1-shifted geometric mean (standard deviation) of the Nodes and Im_N (in percentage).  
+
+<table><tr><td rowspan="2">Methods</td><td colspan="2">FCMCNF</td><td colspan="2">MaxSAT</td><td colspan="2">GISP</td></tr><tr><td>Nodes</td><td>Im_N</td><td>Nodes</td><td>Im_N</td><td>Nodes</td><td>Im_N</td></tr><tr><td>SCIP</td><td>105.58 ± 5.50</td><td>0.00%</td><td>128.43 ± 2.33</td><td>0.00%</td><td>100.77 ± 3.32</td><td>0.00%</td></tr><tr><td>SVM</td><td>101.28 ± 4.62</td><td>+3.01%</td><td>224.28 ± 2.01</td><td>-74.63%</td><td>127.15 ± 3.09</td><td>-26.17%</td></tr><tr><td>RankNet</td><td>92.48 ± 3.70</td><td>+12.40%</td><td>204.54 ± 2.18</td><td>-59.26%</td><td>119.78 ± 3.25</td><td>-18.86%</td></tr><tr><td>GNN</td><td>78.65 ± 4.21</td><td>+25.50%</td><td>194.22 ± 2.31</td><td>-51.22%</td><td>117.56 ± 3.05</td><td>-16.66%</td></tr><tr><td>TRGNN (Ours)</td><td>230.74 ± 4.50</td><td>-118.54%</td><td>78.31 ± 3.48</td><td>+39.02%</td><td>101.07 ± 3.19</td><td>-0.30%</td></tr><tr><td rowspan="2">Methods</td><td colspan="2">MIK</td><td colspan="2">CORLAT</td><td colspan="2">Anonymous</td></tr><tr><td>Nodes</td><td>Im_N</td><td>Nodes</td><td>Im_N</td><td>Nodes</td><td>Im_N</td></tr><tr><td>SCIP</td><td>896.19 ± 4.56</td><td>0.00%</td><td>273.81 ± 35.66</td><td>0.00%</td><td>51037.30 ± 31.49</td><td>0.00%</td></tr><tr><td>SVM</td><td>870.12 ± 4.64</td><td>+2.90%</td><td>160.23 ± 26.51</td><td>+41.48%</td><td>52369.21 ± 297.75</td><td>-2.60%</td></tr><tr><td>RankNet</td><td>937.15 ± 5.53</td><td>-4.57%</td><td>156.60 ± 21.65</td><td>+42.80%</td><td>53881.66 ± 32.55</td><td>-5.57%</td></tr><tr><td>GNN</td><td>757.99 ± 4.32</td><td>+15.42%</td><td>173.82 ± 22.73</td><td>+36.51%</td><td>16039.64 ± 39.03</td><td>+68.57%</td></tr><tr><td>TRGNN (Ours)</td><td>953.56 ± 4.58</td><td>-6.38%</td><td>276.36 ± 28.88</td><td>-0.93%</td><td>19445.26 ± 36.45</td><td>+61.89%</td></tr></table>
+
+Table 9: The generalization ability of TRGNN. The best performance is marked in bold. We report the 1-shifted geometric mean (standard deviation) of the Time.  
+
+<table><tr><td rowspan="2">Methods</td><td colspan="2">FCMCNF</td><td colspan="2">MaxSAT</td><td colspan="2">GISP</td></tr><tr><td>Nodes</td><td>Im_N</td><td>Nodes</td><td>Im_N</td><td>Nodes</td><td>Im_N</td></tr><tr><td>SCIP</td><td>2427.33 ± 6.17</td><td>0.00%</td><td>390.92 ± 2.66</td><td>0.00%</td><td>31552.14 ± 1.89</td><td>0.00%</td></tr><tr><td>SVM</td><td>3036.70 ± 5.66</td><td>-25.10%</td><td>582.88 ± 2.88</td><td>-49.10%</td><td>23675.73 ± 2.26</td><td>+24.96%</td></tr><tr><td>RankNet</td><td>1766.20 ± 5.66</td><td>+27.23%</td><td>591.18 ± 2.50</td><td>-51.23%</td><td>31214.54 ± 1.99</td><td>+1.07%</td></tr><tr><td>GNN</td><td>1710.77 ± 5.21</td><td>+29.52%</td><td>503.43 ± 2.07</td><td>-28.78%</td><td>27909.15 ± 2.01</td><td>+11.55%</td></tr><tr><td>TRGNN (Ours)</td><td>1812.40 ± 8.33</td><td>+25.33%</td><td>531.73 ± 2.72</td><td>-36.02%</td><td>26386.49 ± 2.21</td><td>+16.37%</td></tr></table>
+
+Table 8 and Table 9 illustrate the results of various methods solving MILPs, with the number of nodes (Nodes) used as the evaluation metric. To evaluate different node selection methods compared to the default SCIP, we propose an Improvement metric. Specifically, we define the metric by  $\text{Im}_N = \frac{\text{Nodes}(\text{SCIP}) - \text{Nodes}(M)}{\text{Nodes}(\text{SCIP})}$ , where  $\text{Nodes}(\text{SCIP})$  represents the nodes of SCIP, and  $\text{Nodes}(M)$  represents the nodes of the compared method.
+
+Due to the imposition of a 3600-second time limit, instances exceeding this duration were capped at 3600 seconds for statistical purposes. We provide the results which averaging the node counts at the point of interruption for instances. Some instances in Anonymous dataset could not be solved within
+
+this time frame, rendering comparisons of node counts less meaningful, as RankNet only completed 9/18 and SVM completed 8/18 instances within the time constraint.
+
+Although TRGNN does not consistently yield the smallest tree size, it illustrates a pivotal observation: a reduction in solving time does not invariably equate to a decrease in the number of nodes in the search tree. This reinforces our proposition that the influence of node selection on solving efficiency is multifaceted, intertwining with elements like the cost of path switching, discussed in detail in Section 5.2.
+
+# D.5 RESULTS WITH DIFFERENT SAMPLE SIZES
+
+We conducted tests to assess the impact of training with an increased number of samples. The results demonstrate a consistent upward trend in performance.
+
+Table 10: Comparison of Solving Time (s) for GNN and TRGNN with sample sizes.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">GISP</td><td colspan="2">MaxSAT</td><td colspan="2">FCMCNF</td></tr><tr><td>Test</td><td>Transfer</td><td>Test</td><td>Transfer</td><td>Test</td><td>Transfer</td></tr><tr><td>SCIP</td><td>4.27 ± 1.72</td><td>77.82 ± 1.31</td><td>11.11 ± 2.22</td><td>322.73 ± 1.76</td><td>15.89 ± 1.68</td><td>2102.49 ± 1.99</td></tr><tr><td>GNN_light (250 instances)</td><td>4.57 ± 1.76</td><td>82.34 ± 1.25</td><td>11.28 ± 1.36</td><td>420.44 ± 1.76</td><td>14.59 ± 1.59</td><td>2311.61 ± 1.83</td></tr><tr><td>GNN (1000 instances)</td><td>4.18 ± 1.44</td><td>71.53 ± 1.58</td><td>10.08 ± 1.41</td><td>400.96 ± 1.86</td><td>13.57 ± 1.94</td><td>1968.54 ± 2.19</td></tr><tr><td>TRGNN (250 instances)</td><td>2.89 ± 1.40</td><td>62.02 ± 1.46</td><td>8.63 ± 1.47</td><td>300.00 ± 1.72</td><td>11.62 ± 1.63</td><td>1410.13 ± 1.74</td></tr><tr><td>TRGNN (500 instances)</td><td>2.93 ± 1.40</td><td>61.98 ± 1.46</td><td>8.54 ± 1.44</td><td>299.19 ± 1.71</td><td>11.65 ± 1.76</td><td>1417.93 ± 1.74</td></tr><tr><td>TRGNN (1000 instances)</td><td>2.81 ± 1.42</td><td>62.01 ± 1.46</td><td>8.51 ± 1.40</td><td>306.80 ± 1.71</td><td>11.63 ± 1.62</td><td>1406.12 ± 1.74</td></tr></table>
+
+# D.6 ABLATION STUDY ON TRIPARTITE AND BIPARTITE GRAPH REPRESENTATIONS
+
+With the inclusion of leaf nodes, the tripartite graph has a complexity of  $O(|V| + |C| + |LN| + |E| + |V||LN|)$ , compared to the bipartite graph's  $O(|V| + |C| + |E|)$ . While this increases the space overhead, the tripartite representation is more space-efficient during node selection. For every candidate leaf node, the bipartite approach requires a complete graph representation, leading to a space complexity of  $O(|LN|(|V| + |C| + |E|))$ . In problems where the number of constraints is similar to or exceeds the number of variables, such as GISP and MaxSAT, the tripartite graph saves space by avoiding redundant representations of constraints and variables.
+
+We conducted an ablation experiment on the bipartite graph approach. In this study, we used the BestEstimate heuristic to pre-filter candidate nodes and then represented each candidate node with a bipartite graph within our reinforcement learning framework (BRGNN). The training times for the MaxSAT and GISP datasets are listed in Table 11. The results indicate that the computational overhead primarily arises from reinforcement learning rather than the tripartite graph itself. Moreover, the tripartite graph representation can save  $29.37\%$  and  $20.16\%$  of training time on MaxSAT and GISP, respectively.
+
+Table 11: Comparison of training time (s) for GNN, BRGNN and TRGNN.  
+
+<table><tr><td>Method</td><td>MaxSAT</td><td>GISP</td></tr><tr><td>GNN (IL)</td><td>62.64</td><td>54.07</td></tr><tr><td>BRGNN</td><td>14391.13</td><td>3658.54</td></tr><tr><td>TRGNN (Ours)</td><td>10164.75</td><td>2921.07</td></tr></table>
+
+We added tests for the average graph construction time, and inference time per instance on the MaxSAT and GISP datasets. Although we acknowledged that reinforcement learning required more time during the training phase compared to imitation learning (IL), it did not add to inference time. Additionally, the tripartite graph approach did not increase additional time overhead compared to the bipartite method.
+
+To demonstrate the impact of the tripartite graph representation on solving efficiency compared to bipartite graphs, we conducted tests on the GISP and MaxSAT datasets. Table 16 shows that
+
+Table 12: Comparison of graph construction time (s) for GNN, BRGNN and TRGNN.  
+
+<table><tr><td>Method</td><td>MaxSAT</td><td>GISP</td></tr><tr><td>GNN (IL)</td><td>0.02</td><td>0.01</td></tr><tr><td>BRGNN</td><td>0.01</td><td>0.00</td></tr><tr><td>TRGNN (Ours)</td><td>0.01</td><td>0.00</td></tr></table>
+
+Table 13: Comparison of inference time(s) for GNN, BRGNN and TRGNN.  
+
+<table><tr><td>Method</td><td>MaxSAT</td><td>GISP</td></tr><tr><td>GNN (IL)</td><td>0.16</td><td>0.05</td></tr><tr><td>BRGNN</td><td>0.13</td><td>0.01</td></tr><tr><td>TRGNN (Ours)</td><td>0.20</td><td>0.01</td></tr></table>
+
+the "BestEstimate" method performs similarly to the default SCIP settings. Our tripartite graph approach (TRGNN) improves over SCIP and BestEstimate. Notably, the tripartite graph method outperforms the bipartite method, achieving improvements of  $19.4\%$  on the MaxSAT dataset and  $2.5\%$  on the GISP dataset.
+
+Table 14: Comparison of TRGNN and BRGNN on GISP and MaxSAT datasets.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">MaxSAT (n ∈ [80, 100])</td><td colspan="3">GISP (n ∈ [60, 70])</td></tr><tr><td>Time (s)</td><td>Wins</td><td>Nodes</td><td>Time (s)</td><td>Wins</td><td>Nodes</td></tr><tr><td>SCIP</td><td>8.15 ± 1.64</td><td>3/50</td><td>147.51 ± 2.02</td><td>3.35 ± 1.30</td><td>6/50</td><td>94.81 ± 3.67</td></tr><tr><td>BestEstimate</td><td>7.98 ± 1.68</td><td>11/50</td><td>177.64 ± 2.13</td><td>3.53 ± 1.30</td><td>8/50</td><td>102.99 ± 3.18</td></tr><tr><td>BRGNN</td><td>7.92 ± 1.67</td><td>5/50</td><td>200.68 ± 2.05</td><td>3.20 ± 1.32</td><td>17/50</td><td>97.27 ± 3.59</td></tr><tr><td>TRGNN (Ours)</td><td>6.38 ± 1.85</td><td>31/50</td><td>134.93 ± 2.83</td><td>3.03 ± 1.27</td><td>19/50</td><td>83.72 ± 3.51</td></tr></table>
+
+# D.7 ANALYSIS OF PRE-SELECTION SIZE USING THE HEURISTIC ALGORITHM
+
+In our work, we initially use the heuristic algorithm "BestEstimate" to pre-select a fixed candidate set of size  $n$ . To determine the potential impact of the heuristic algorithm on solution quality, we increased  $n$  from 5 to 10. We recorded the position of the nodes selected by our TRGNN within the rankings provided by the heuristic.
+
+![](images/0f6e4c4222493798522ff5ef60fae354509fc678b6edb23d104587906eaee40b.jpg)  
+Figure 4: Histograms showing the distribution of positions picked by TRGNN in "BestEstimate," with GISP on the left and MaxSAT on the right.
+
+![](images/7dd076589992e471debcc9f9c7c54a4c48150320bc6ac1f5d6b2168f4989e3f0.jpg)
+
+The results show that in MaxSAT and GISP, nodes ranked higher than 5 by "BestEstimate" are never chosen by TRGNN. This indicates that nodes not ranked among the top by "BestEstimate" have a
+
+very low probability of being selected. This is because "BestEstimate" primarily relies on the node estimate value which represents an optimistic prediction of the best feasible solution that can be found within the subtree of that node. Nodes ranked toward the end generally have poorer estimates, indicating a smaller probability of leading to optimal solutions.
+
+# D.8 ANALYSIS OF THE IMPACT OF THE HEURISTIC ALGORITHM ON THE SOLVING RESULTS
+
+To ensure fair comparisons, we tested a GNN algorithm based on imitation learning with heuristic pre-selection (GNN+BestEstimate) to eliminate the impact of "BestEstimate" on the solving results. We found that the filtered results were consistent with the original ones because nodes ranked lower by the heuristic "BestEstimate" were seldom chosen by the GNN (see Appendix D.7).
+
+Table 15: Comparison of TRGNN and GNN+BestEstimate on GISP and MaxSAT datasets.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">MaxSAT (n ∈ [80, 100])</td><td colspan="2">GISP (n ∈ [60, 70])</td></tr><tr><td>Time (s)</td><td>Nodes</td><td>Time (s)</td><td>Nodes</td></tr><tr><td>SCIP</td><td>8.15 ± 1.64</td><td>147.51 ± 2.02</td><td>3.12 ± 1.30</td><td>77.10 ± 3.79</td></tr><tr><td>GNN+BestEstimate</td><td>7.71 ± 1.93</td><td>230.72 ± 3.30</td><td>3.26 ± 1.35</td><td>96.99 ± 3.66</td></tr><tr><td>TRGNN (Ours)</td><td>6.38 ± 1.85</td><td>134.93 ± 2.83</td><td>2.88 ± 1.28</td><td>80.50 ± 3.39</td></tr></table>
+
+# D.9 RESULTS WITH THE PURE BRANCH-AND-BOUND FRAMEWORK
+
+While our TRGNN demonstrates significant improvements over the modern solver SCIP, our tripartite graph representation is specifically tailored for the branch-and-bound framework. Our graph does not account for external solver features, such as restarts or cut pools. An intriguing area for future research lies in designing enhanced representations for cutting planes within branch-and-cut algorithms. We believe this is a promising direction.
+
+To better evaluate our node selection method in a branch-and-bound context, we tested SCIP with cutting planes and restarts disabled to better evaluate our node selection method within branch-and-bound. The results showed further improvements for both the GNN and our TRGNN methods compared to SCIP. For example, on MaxSAT, the improvement in solving time for TRGNN over SCIP increased from  $7.54\%$  to  $12.71\%$ , and on GISP from  $13.92\%$  to  $18.67\%$ .
+
+Table 16: Comparison of TRGNN and GNN in a branch-and-bound context.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">MaxSAT (n ∈ [80, 100])</td><td colspan="2">GISP (n ∈ [60, 70])</td></tr><tr><td>Time (s)</td><td>Nodes</td><td>Time (s)</td><td>Nodes</td></tr><tr><td>SCIP</td><td>4.56 ± 1.65</td><td>78.48 ± 2.66</td><td>3.16 ± 1.30</td><td>1119.03 ± 3.90</td></tr><tr><td>GNN</td><td>4.42 ± 1.70</td><td>205.15 ± 3.01</td><td>2.72 ± 1.18</td><td>1753.04 ± 1.52</td></tr><tr><td>TRGNN (Ours)</td><td>3.98 ± 1.70</td><td>156.42 ± 3.24</td><td>2.57 ± 1.17</td><td>1334.94 ± 1.62</td></tr></table>
\ No newline at end of file
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+# LEARNING TO STEER MARKOVIAN AGENTS UNDER MODEL UNCERTAINTY
+
+Jiawei Huang†, Vinzenz Thoma‡, Zebang Shen†, Heinrich H. Nax§, Niao He†
+
+$\dagger$  Department of Computer Science, ETH Zurich {jiawei.huang, zebang.shen, niao.he}@inf.ethz.ch  $\ddagger$  ETH AI Center vinzens.z.thoma@ai.ethz.ch  $\S$  University of Zurich heinrich.nax@uzh.ch
+
+# ABSTRACT
+
+Designing incentives for an adapting population is a ubiquitous problem in a wide array of economic applications and beyond. In this work, we study how to design additional rewards to steer multi-agent systems towards desired policies without prior knowledge of the agents' underlying learning dynamics. Motivated by the limitation of existing works, we consider a new and general category of learning dynamics called Markovian agents. We introduce a model-based non-episodic Reinforcement Learning (RL) formulation for our steering problem. Importantly, we focus on learning a history-dependent steering strategy to handle the inherent model uncertainty about the agents' learning dynamics. We introduce a novel objective function to encode the desiderata of achieving a good steering outcome with reasonable cost. Theoretically, we identify conditions for the existence of steering strategies to guide agents to the desired policies. Complementing our theoretical contributions, we provide empirical algorithms to approximately solve our objective, which effectively tackles the challenge in learning history-dependent strategies. We demonstrate the efficacy of our algorithms through empirical evaluations.
+
+# 1 INTRODUCTION
+
+Many real-world applications can be formulated as Markov Games (Littman, 1994) where the agents repeatedly interact and update their policies based on the received feedback. In this context, different learning dynamics and their convergence properties have been studied extensively (see, for example, Fudenberg and Levine (1998)). Because of the mismatch between the individual short-run and collective long-run incentives, or the lack of coordination in decentralized systems, agents following standard learning dynamics may not converge to outcomes that are desirable from a system designer perspective, such as the Nash Equilibria (NE) with the largest social welfare. An interesting class of games that exemplify these issues are so-called "Stag Hunt" games (see Fig. 1-(a)), which are used to study a broad array of real-world applications including collective action, public good provision, social dilemma, team work and innovation adoption (Skyrms, 2004)<sup>1</sup>. Stag Hunt games have two pure-strategy NE, one of which is 'payoff-dominant', that is, both players obtain higher payoffs in that equilibrium than in the other. Typical algorithms may fail to reach the payoff-dominant equilibrium  $(\mathbb{H}, \mathbb{H})$  (LHS Fig. 1-(b)). Indeed, the other equilibrium  $(\mathbb{G}, \mathbb{G})$  is typically selected when it is risk-dominant (Harsanyi and Selten, 1988; Newton, 2021).
+
+This paper focuses on situations when an external "mediator" exists, who can influence and steer the agents' learning dynamics by modifying the original rewards via additional incentives. This kind of mediator can be conceptualized in various ways. In particular, we can think of a social planner who provides monetary incentives for joint ventures or for adoption of an innovative technology via individual financial subsidies. As illustrated on the RHS of Fig. 1-(b), with suitable steering, agents' dynamics can be directed to the best outcome. Our primary objective is to steer the agents to some desired policies, that is, to minimize the steering gap vis-a-vis the target outcome. As a secondary
+
+objective, the payments to agents regarding the steering rewards should be reasonable, that is, the steering cost should be low.
+
+To our knowledge, Zhang et al. (2023) is the first work studying a similar steering problem as ours. They assume the agents are no-regret learners and may act in arbitrarily adversarial ways. In some natural settings, such assumption may not be practical, because no-regret criterion typically requires careful processing of the entire history of learning. In settings with limited cognitive resources and bounded rationality, it is natural to favor models where the agents only process a subset of the available information (Camerer, 2011). In particular, humans have been widely shown to rely overproportionally on recent experiences in decision making, known as 'recency bias' (Costabile and Klein, 2005; Page and Page, 2010; Durand et al., 2021). Besides, there is evidence that behavioral dynamics that only rely on the most recent experience are able to fit behavioral data well in certain situations (Más and Nax, 2016). Motivated by these insights, we therefore study steering a different category of learning dynamics called Markovian agents, where the agents' policy updates only depend on their current policy and the (modified) reward function. Our model complements
+
+![](images/ba8edb7191f6d56053a98d44ac2f28c4040cd4f6611f44aa42decbe64f0ace6d.jpg)  
+Figure 1: Example: The "Stag Hunt" Game
+
+(a) Payoff Matrix of the Two-Player "Stag Hunt" Game. H and G stand for two actions Hunt and Gather. Both (H, H) and (G, G) are NE and (H, H) is payoff-dominant.
+
+![](images/27070d0b3fd8bb34dc59333c30fe9b3d786b931dedfe9e8104de18c0efe108db.jpg)
+
+![](images/e777e2bafbe19dc2c35c33c9824b23818cec96c1d0a0ec123e19e6b4464e75e6.jpg)
+
+(b) Dynamics of Agents Policies without/with Steering. Agents follow natural policy gradient (replicator dynamics) for policy update.  $x$  and  $y$  axes correspond to the probability to take action  $\mathbb{H}$  by the row and column players. Red curves represent the dynamics of agents' policies starting from different initializations (black dots).
+
+the prior work on no-regret agents, and serves as the first abstraction of behavior based on limited cognitive abilities with recency bias in steering setting. Theoretically, Markovian agents subsumes a broader class of popular policy-based methods as concrete examples (Giannou et al., 2022; Ding et al., 2022), which are not covered by no-regret assumptions. We note that a concurrent work (Canyakmaz et al., 2024) considers a similar setting as ours, and we defer to Sec. 1.1 for further discussion.
+
+In practice, learning the right steering strategies encounters two main challenges. First, the agents may not disclose their learning dynamics model to the mediator. As a result, this creates fundamental model uncertainty, which we will tackle with appropriate Reinforcement Learning (RL) techniques to trading-off exploration and exploitation. Second, it may be unrealistic to assume that the mediator is able to force the agents to "reset" their policies in order to generate multiple steering episodes with the same initial state. This precludes the possibility of learning steering strategies through episodic trial-and-error. Therefore, the most commonly-considered, fixed-horizon episodic RL (Dann and Brunskill, 2015) framework is not applicable here. Instead, we will consider a finite-horizon non-episodic setup, where the mediator can only generate one finite-horizon episode, in which we have to conduct both the model learning and steering of the agents simultaneously. Motivated by these considerations, we would like to address the following question in this paper:
+
+# How to learn desired steering strategies for Markovian agents in the non-episodic setup under model uncertainty?
+
+We consider a model-based setting where the mediator can get access to a model class  $\mathcal{F}$  containing the agents' true learning dynamics  $f^{*}$ . We summarize and highlight our key contributions as follow:
+
+- Conceptual Contributions: In Sec. 3, we formulate steering as a non-episodic RL problem, and propose a novel optimization objective in Obj. (1), where we explicitly tackle the inherent model uncertainty by learning history-dependent steering strategies. As we show in Prop. 3.3, under certain conditions, even without prior knowledge of  $f^{*}$ , the optimal solution to Obj. (1) achieves not only low steering gap, but also "Pareto Optimality" in terms of both steering costs and gaps.  
+- Theoretical Contributions: In Sec. 4, we provide sufficient conditions under which there exists steering strategies achieving low steering gap. These results in turn justify our chosen objective and problem formulation.
+
+- Algorithmic Contributions: Learning a history-dependent strategy presents challenges due to the exponential growth in the history space. We propose algorithms to overcome these issues.
+
+- When the model class  $|\mathcal{F}|$  is small, in Sec. 5.1, we approach our objective from the perspective of learning in a Partially Observable MDP, and propose to to learn a policy over the model belief state space instead of over the history space.  
+- For the case when  $|\mathcal{F}|$  is large, exactly solving Obj. (1) can be challenging. Instead, we focus on approximate solutions to trade-off optimality and tractability. In Sec. 5.2, we propose a First-Explore-Then-Exploit (FETE) framework. Under some conditions, we can still ensure the directed agents converge to the desired outcome.
+
+- Empirical Validation: In Sec. 6, we evaluate our algorithms in various representative environments, and demonstrate their effectiveness under model uncertainty.
+
+# 1.1 CLOSELY RELATED WORKS
+
+We discuss the works most closely related to ours in this section, and defer the others to Appx. B.2.
+
+Steering Learning Dynamics As mentioned in the introduction, Zhang et al. (2023) are the first to introduce the "steering problem", but their setting differs quite fundamentally from ours in several key aspects. Firstly, they assume that agents behave as no-regret and arbitrarily adversarial learners, which may be unrealistic in settings with limited information and feedback, and owed to agents' limited cognitive resources (Camerer, 2011) including recency bias (Costabile and Klein, 2005). Motivated by this, we instead focus on a broad class of Markovian dynamics. Secondly, the mediator's objective in Zhang et al. (2023) is to steer agents such that the average policy converges to the target NE while maintaining a sublinear accumulative budget, motivated by their infinite-horizon setup. In contrast, we consider the finite-horizon setting, and therefore, we are concerned with minimizing the steering gap of the terminal policy and the cumulative steering cost. Thirdly, when the desired NE is not pure, Zhang et al. (2023) require the mediator to be able to "give advice" to the players to facilitate coordination, while we do not allow the mediator to do this. Because of these differences, the methods and results obtained by Zhang et al. (2023) and us are not directly comparable, yet complement one another depending on application considered.
+
+Perhaps the closest to ours is a concurrent work by Canyakmaz et al. (2024). They consider a similar finite-horizon non-episodic setup as ours, and experimentally investigate the use of control methods to direct game dynamics towards desired outcomes, in particular allowing for model uncertainty. Compared to their work, we handle the model uncertainty in a more principled way by employing history-dependent steering strategies, since history can serve as sufficient information set for decision making under uncertainty. This leads to several differences in the design principles between our algorithms and Canyakmaz et al. (2024). Concretely, we propose a learning objective for history-dependent strategies in Obj. (1), and two algorithms for low uncertainty (small  $\mathcal{F}$ ) and high uncertainty (large  $\mathcal{F}$ ) settings, respectively. In the former case, we contribute a belief-state based algorithm that can exactly solve Obj. (1), offering a stronger solution due to the theoretical guarantee in Prop. 3.3. For the latter, although both our FETE and SIAR-MPC (Canyakmaz et al., 2024) share a two-phase (exploration + exploitation) structure, ours represents a more general framework with a more advanced exploration strategy (see more explanation in Sec. 5.2). Besides, we develop additional novel theory regarding the existence of strategies with low steering gap.
+
+# 2 PRELIMINARY
+
+In the following, we formally define the finite-horizon Markov Game that we will focus on. We summarize all the frequently used notations in this paper in Appx. A.
+
+Finite Horizon Markov Game A finite-horizon  $N$ -player Markov Game is defined by a tuple  $G \coloneqq \{\mathcal{N}, s_1, H, S, \mathcal{A} \coloneqq \{\mathcal{A}^n\}_{n=1}^N, \mathbb{P}, r \coloneqq \{r^n\}_{n=1}^N\}$ , where  $\mathcal{N} \coloneqq \{1, 2, \dots, N\}$  is the indices of agents,  $s_1$  is the fixed initial state,  $H$  is the horizon length,  $\mathcal{S}$  is the finite shared state space,  $\mathcal{A}^n$  is the finite action space for agent  $n$ , and  $\mathcal{A}$  denotes the joint action space. Besides,  $\mathbb{P} \coloneqq \{\mathbb{P}_h\}_{h \in [H]}$  with  $\mathbb{P}_h: \mathcal{S} \times \mathcal{A} \to \Delta(\mathcal{S})$  denotes the transition function of the shared state, and  $r^n \coloneqq \{r_h^n\}_{h \in [H]}$  with  $r_h^n: \mathcal{S} \times \mathcal{A} \to [0,1]$  denotes the reward function for agent  $n$ . For each agent  $n$ , we consider the nonstationary Markovian policies  $\Pi^n \coloneqq \{\pi^n = \{\pi_1^n, \dots, \pi_H^n\} | \forall h \in [H], \pi_h^n: \mathcal{S} \to \Delta(\mathcal{A}^n)\}$ . We denote
+
+$\Pi \coloneqq \Pi^1 \times \ldots \times \Pi^N$  to be the joint policy space of all agents. Given a policy  $\pi \coloneqq \{\pi^1, \dots, \pi^N\} \in \Pi$ , a trajectory is generated by:  $\forall h \in [H]$ ,  $\forall n \in [N]$ ,  $a_h^n \sim \pi^n(\cdot|s_h)$ ,  $r_h^n \gets r_h^n(s_h, \boldsymbol{a}_h)$ ,  $s_{h+1} \sim \mathbb{P}_h(\cdot|s_h, \boldsymbol{a}_h)$ , where  $\boldsymbol{a}_h \coloneqq \{a_h^n\}_{n \in [N]}$  denotes the collection of all actions. Given a policy  $\pi$ , we define the value functions by:  $Q_{h|\boldsymbol{r}}^{n,\pi}(\cdot, \cdot) \coloneqq \mathbb{E}_{\pi}[\sum_{h'=h}^{H} r_{h'}^n(s_{h'}, \boldsymbol{a}_{h'})|s_h = \cdot, \boldsymbol{a}_h = \cdot]$ ,  $V_{h|\boldsymbol{r}}^{n,\pi}(\cdot) \coloneqq \mathbb{E}_{\pi}[\sum_{h'=h}^{H} r_{h'}^n(s_{h'}, \boldsymbol{a}_{h'})|s_h = \cdot]$ , where we use  $|r|$  to specify the reward function associated with the value functions. In the rest of the paper, we denote  $A_{|\boldsymbol{r}}^{n,\pi} = Q_{|\boldsymbol{r}}^{n,\pi} - V_{|\boldsymbol{r}}^{n,\pi}$  to be the advantage value function, and denote  $J_{|\boldsymbol{r}}^{n}(\pi) \coloneqq V_{1|\boldsymbol{r}}^{n,\pi}(s_1)$  to be the total return of agent  $n$  w.r.t. policy  $\pi$ .
+
+# 3 THE PROBLEM FORMULATION OF THE STEERING MARKOVIAN AGENTS
+
+We first introduce our definition of Markovian agents. Informally, the policy updates of Markovian agents are independent of the interaction history conditioning on their current policy and observed rewards. This encompasses a broader class of popular policy-based methods as concrete examples (Giannou et al., 2022; Ding et al., 2022; Xiao, 2022; Daskalakis et al., 2020).
+
+Definition 3.1 (Markovian Agents). Given a game  $G$ , a finite and fixed  $T$ , the agents are Markovian if their policy update rule  $f$  only depends on the current policy  $\pi_t$  and the reward function  $r$ :
+
+$$
+\forall t \in [ T ], \quad \boldsymbol {\pi} _ {t + 1} \sim f (\cdot | \boldsymbol {\pi} _ {t}, \boldsymbol {r}).
+$$
+
+Here we only highlight the dependence on  $\pi_t$  and  $r$ , and omit other dependence (e.g. the transition function of  $G$ ). It is worth to note that we do not restrict whether the updates of agents' policies are independent or correlated with each other, deterministic or stochastic. We assume  $T$  is known to us.
+
+In the steering problem, the mediator has the ability to change the reward function  $\pmb{r}$  via the steering reward  $\pmb{u}$ , so that the agents' dynamics are modified to:
+
+$$
+\forall t \in [ T ], \quad \boldsymbol {u} _ {t} \sim \psi_ {t} (\cdot | \boldsymbol {\pi} _ {1}, \boldsymbol {u} _ {1}, \dots , \boldsymbol {\pi} _ {t - 1}, \boldsymbol {u} _ {t - 1}, \boldsymbol {\pi} _ {t}), \quad \boldsymbol {\pi} _ {t + 1} \sim f (\cdot | \boldsymbol {\pi} _ {t}, \boldsymbol {r} + \boldsymbol {u} _ {t}),
+$$
+
+Here  $\psi \coloneqq \{\psi_t\}_{t\in [T]}$  denotes the mediator's "steering strategy" to generate  $u_{t}$ . We consider history-dependent strategies to handle the model uncertainty, which we will explain later. Besides,  $\pmb{u}_t\coloneqq \{u_{t,h}^n\}_{h\in [H],n\in [N]}$ , where  $u_{t,h}^n:\mathcal{S}\times \mathcal{A}\to [0,U_{\mathrm{max}}]$  is the steering reward for agent  $n$  at game horizon  $h$  and steering step  $t$ .  $U_{\mathrm{max}} < + \infty$  denotes the upper bound for the steering reward. With a bit of abuse of notation, here we treat both  $\pmb{r}$  and  $\pmb{u}_t$  as vectors with length  $HN|S||\mathcal{A}|$  and use  $\pmb{r} + \pmb{u}_t$  to denote elementwise addition. For practical concerns, we follow the standard constraints that the steering rewards are non-negative.
+
+The mediator has a terminal reward function  $\eta^{\mathrm{goal}}$  and a cost function  $\eta^{\mathrm{cost}}$ . First,  $\eta^{\mathrm{goal}}: \Pi \to [0, \eta_{\max}]$  assesses whether the final policy  $\pi_{T+1}$  aligns with desired behaviors—this encapsulates our primary goal of a low steering gap. Note that we consider the general setting and do not restrict the maximizer of  $\eta^{\mathrm{goal}}$  to be a Nash Equilibrium. For instance, to steer the agents to a desired policy  $\pi^{*}$ , we could choose  $\eta^{\mathrm{goal}}(\pi) := -\|\pi - \pi^{*}\|_{2}$ . Alternatively, in scenarios focusing on maximizing utility,  $\eta^{\mathrm{goal}}(\pi)$  could be defined as the total utility  $\sum_{n \in [N]} J_{|r}^{n}(\pi)$ . For  $\eta^{\mathrm{cost}}: \Pi \to \mathbb{R}_{\geqslant 0}$ , it is used to quantify the steering cost incurred while steering. In this paper, we fix  $\eta^{\mathrm{cost}}(\pi, u) := \sum_{n \in [N]} J_{|u}^{n}(\pi^{n})$  to be the total return related to  $\pi$  and the steering reward  $u$ . Note that we always have  $0 \leqslant \eta^{\mathrm{cost}}(\pi, u) \leqslant U_{\max} NH$ .
+
+Steering Dynamics as a Markov Decision Process (MDP) Given a game  $G$ , the agents' dynamics  $f$  and  $(\eta^{\mathrm{cost}},\eta^{\mathrm{goal}})$ , the steering dynamics can be modeled by a finite-horizon MDP.  $M\coloneqq \{\pi_1,T,\Pi ,\mathcal{U},f,(\eta^{\mathrm{cost}},\eta^{\mathrm{goal}})\}$  with initial state  $\pi_{1}$ , horizon length  $T$ , state space  $\Pi$ , action space  $\mathcal{U}\coloneqq [0,U_{\max}]^{HN|S||A|}$ , stationary transition  $f$ , running reward  $\eta^{\mathrm{cost}}$  and terminal reward  $\eta^{\mathrm{goal}}$ . For completeness, we defer to Appx. B.3 for an introduction of finite-horizon MDP
+
+Steering under Model Uncertainty In practice, the mediator may not have precise knowledge of agents learning dynamics model, and the uncertainty should be taken into account. We will only focus on handling the uncertainty in agents' dynamics  $f$ , and assume the mediator has the full knowledge of  $G$  and the reward functions  $\eta^{\mathrm{goal}}$  and  $\eta^{\mathrm{cost}}$ . We consider the model-based setting where the mediator only has access to a finite model class  $\mathcal{F}(|\mathcal{F}| < +\infty)$  satisfying the following assumption:
+
+Assumption A (Realizability). The true learning dynamics  $f^{*}$  is realizable, i.e.  $f^{*} \in \mathcal{F}$ .
+
+A Finite-Hoziron Non-Episodic Setup and Motivation As motivated previously, we formulate steering as a finite-horizon non-episodic RL problem. To our knowledge, in contrast to our finite-horizon setting, most of the non-episodic RL settings consider the infinite-horizon setup with stationary or non-stationary transitions, and therefore, they are also not suitable here. We provide more discussion in Appx. B.2.
+
+Definition 3.2 (Finite Horizon Non-Episodic Steering Setting). The mediator can only interact with the real agents for one episode  $\{\pi_1, u_1, \dots, \pi_T, u_T, \pi_{T+1}\}$ , where  $\pi_{t+1} \sim f^*(\cdot|\pi_t, r + u_t)$  for all  $t \in [T]$ . Nonetheless, the mediator can get access to the simulators for all models in  $\mathcal{F}$ , and it can sample arbitrary trajectories and do episodic learning with those simulators to decide the best steering actions  $u_1, u_2, \dots, u_T$  to deploy.
+
+The Learning Objective Motivated by the model-based non-episodic setup, we propose the following objective function, where we search over the set of all history-dependent strategies, denoted by  $\Psi$ , to optimize the average performance over all  $f \in \mathcal{F}$ .
+
+$$
+\psi^ {*} \leftarrow \arg \max  _ {\psi \in \Psi} \frac {1}{| \mathcal {F} |} \sum_ {f \in \mathcal {F}} \mathbb {E} _ {\psi , f} \left[ \beta \cdot \eta^ {\text {g o a l}} \left(\boldsymbol {\pi} _ {T + 1}\right) - \sum_ {t = 1} ^ {T} \eta^ {\text {c o s t}} \left(\boldsymbol {\pi} _ {t}, \boldsymbol {u} _ {t}\right) \right], \tag {1}
+$$
+
+Here we use  $\mathbb{E}_{\psi, f}[\cdot] \coloneqq \mathbb{E}[\cdot|\forall t \in [T], \boldsymbol{u}_t \sim \psi_t(\cdot|\{\pi_{t'}, \boldsymbol{u}_{t'}\}_{t'=1}^{t-1}, \boldsymbol{\pi}_t), \boldsymbol{\pi}_{t+1} \sim f(\cdot|\boldsymbol{\pi}_t, \boldsymbol{r} + \boldsymbol{u}_t)]$  to denote the expectation over trajectories generated by  $\psi$  and  $f \in \mathcal{F}$ ;  $\beta > 0$  is a regularization factor. Next, we explain the rationale to consider history-dependent strategies. As introduced in Def. 3.2, we only interact with the real agents once. Therefore, the mediator needs to use the interaction history with  $f^*$  to decide the appropriate steering rewards to deploy, since the history is the sufficient information set including all the information regarding  $f^*$  availale to the mediator.
+
+We want to clarify that in our steering framework, we will first solve Obj. (1), and then deploy  $\psi^{*}$  to steer real agents. The learning and optimization of  $\psi^{*}$  in Obj. (1) only utilizes simulators of  $\mathcal{F}$ . Besides, after deploying  $\psi^{*}$  to real agents, we will not update  $\psi^{*}$  with the data generated during the interaction with real agents. This is seemingly different from common online learning algorithms which conduct the learning and interaction repeatedly(Dann and Brunskill, 2015). But we want to highlight that, given the fact that  $\psi^{*}$  is history-dependent, it is already encoded in  $\psi^{*}$  how to make decisions (or say, learning) in the face of uncertainty after gathering data from real agents. In other words, one can interpret that, in Obj. (1), the  $\psi^{*}$  to be optimized is an "online algorithm" which can "smartly" decide the next steering reward to deploy given the past interaction history. As we will justify in the following, our Obj. (1) can indeed successfully handle the model uncertainty.
+
+Justification for Objective (1) We use  $C_{\psi, T}(f) \coloneqq \mathbb{E}_{\psi, f}[\sum_{t=1}^{T} \eta^{\mathrm{cost}}(\pmb{\pi}_t, \pmb{u}_t)]$  and  $\Delta_{\psi, T}(f) \coloneqq \mathbb{E}_{\psi, f}[\max_{\pi} \eta^{\mathrm{goal}}(\pmb{\pi}) - \eta^{\mathrm{goal}}(\pmb{\pi}_{T+1})]$  as short notes of the steering cost and the steering gap (of the terminal policy  $\pmb{\pi}_{T+1}$ ), respectively. Besides, we denote  $\Psi^{\varepsilon} \coloneqq \{\psi \in \Psi | \max_{f \in \mathcal{F}} \Delta_{\psi, T}(\mathcal{F}) \leqslant \varepsilon\}^2$  to be the collection of all steering strategies with  $\varepsilon$ -steering gap. Based on these notations, we introduce two desiderata, and show how an optimal solution  $\psi^*$  of Obj. (1) can achieve them.
+
+Desideratum 1 (ε-Steering Gap). We say  $\psi$  has  $\varepsilon$ -steering gap, if  $\max_{f \in \mathcal{F}} \Delta_{\psi, T}(f) \leqslant \varepsilon$ .
+
+Desideratum 2 (Pareto Optimality). We say  $\psi$  is Pareto Optimal if there does not exist another  $\psi' \in \Psi$ , such that (1)  $\forall f \in \mathcal{F}$ ,  $C_{\psi', T}(f) \leqslant C_{\psi, T}(f)$  and  $\Delta_{\psi', T}(f) \leqslant \Delta_{\psi, T}(f)$ ; (2)  $\exists f' \in \mathcal{F}$ , s.t. either  $C_{\psi', T}(f') < C_{\psi, T}(f')$  or  $\Delta_{\psi', T}(f') < \Delta_{\psi, T}(f')$ .
+
+Proposition 3.3. [Justification for Obj. (1)] By solving Obj. (1): (1)  $\psi^{*}$  is Pareto Optimal; (2) Given any  $\varepsilon, \varepsilon' > 0$ , if  $\Psi^{\varepsilon / |\mathcal{F}|} \neq \emptyset$  and  $\beta \geqslant \frac{U_{\max NHT}|\mathcal{F}|}{\varepsilon'}$ , we have  $\psi^{*} \in \Psi^{\varepsilon + \varepsilon'}$ ;
+
+Next, we give some interpretation. As our primary desideratum, we expect the agents converge to some desired policy that maximizes the goal function  $\eta^{\mathrm{goal}}$  after being steered for  $T$  steps, regardless of the true model  $f^{*}$ . Therefore, we restrict the worst case steering gap to be small. As stated in Prop. 3.3, for any accuracy level  $\varepsilon > 0$ , as long as  $\varepsilon / |\mathcal{F}|$ -steering gap is achievable, by choosing  $\beta$  large enough, we can approximately guarantee  $\psi^{*}$  has  $\varepsilon$ -steering gap. For the steering cost, although it is not our primary objective, Prop. 3.3 states that at least we can guarantee the Pareto Optimality: competing with  $\psi^{*}$ , there does not exist another  $\psi'$ , which can improve either the steering cost or gap for some  $f' \in \mathcal{F}$  without deteriorating any others.
+
+Given the above discussion, one natural question is that: when is  $\Psi^{\varepsilon}$  non-empty, or equivalently, when does a strategy  $\psi$  with  $\varepsilon$ -steering gap exist? In Sec. 4, we provide sufficient conditions and concrete examples to address this question in theory. Notably, we suggest conditions where  $\Psi^{\varepsilon}$  is non-empty for any  $\varepsilon > 0$ , so that the condition  $\Psi^{\varepsilon / |\mathcal{F}|} \neq \emptyset$  in Prop. 3.3 is realizable, even for large  $|\mathcal{F}|$ . After that, in Sec. 5, we introduce algorithms to solve our Obj. (1).
+
+# 4 EXISTENCE OF STEERING STRATEGY WITH  $\varepsilon$ -STEERING GAP
+
+In this section, we identify sufficient conditions such that  $\Psi^{\varepsilon}$  is non-empty. In Sec. 4.1, we start with the special case when  $f^{*}$  is known, i.e.  $\mathcal{F} = \{f^{*}\}$ . The results will serve as basis when we study the general unknown model setting with  $|\mathcal{F}| > 1$  in Sec. 4.2.
+
+# 4.1 EXISTENCE WHEN  $f^{*}$  IS KNOWN: NATURAL POLICY GRADIENT AS AN EXAMPLE
+
+In this section, we focus on a popular choice of learning dynamics called Natural Policy Gradient (NPG) dynamics (Kakade, 2001; Agarwal et al., 2021) (a.k.a. the replicator dynamics (Schuster and Sigmund, 1983)) with direct policy parameterization. NPG is a special case of the Policy Mirror Descent (PMD) (Xiao, 2022). For the readability, we stick to NPG in the main text, and in Appx. D.1, we formalize PMD and extend the results to the general PMD, which subsumes other learning dynamics, like the online gradient ascent (Zinkevich, 2003).
+
+Definition 4.1 (Natural Policy Gradient). For any  $n \in [N], t \in [T], h \in [H], s_h \in S$ , the policy is updated by:  $\pi_{t+1,h}^n(\cdot|s_h) \propto \pi_{t,h}^n(\cdot|s_h) \exp(\alpha \hat{A}_{h|r^n + u_t^n}^n(s_h, \cdot))$ . Here  $\hat{A}_{h|r^n + u_t^n}^n$  is some random estimation for the advantage value  $A_{h|r^n + u_t^n}^{n,\pi_t}$  with  $\mathbb{E}_{\pi^n}[\hat{A}_{h|r^n + u_t^n}^n(s_h, \cdot)] = 0$ .
+
+We use  $\widehat{A}_{|r + u}^{\pi}$  (and  $A_{|r + u}^{\pi}$ ) to denote the concatenation of the values of all agents, horizon, states and actions. We only assume  $\widehat{A}_{|r + u}^{\pi_t}$  is controllable and has positive correlation with  $A_{|r + u}^{\pi_t}$  but could be biased, which we call the "general incentive driven" agents.
+
+Assumption B (General Incentive Driven Agents).
+
+$$
+\forall t \in [ T ], \quad \langle \mathbb {E} [ \widehat {A} _ {| \pmb {r} + \pmb {u} _ {t}} ^ {\pmb {\pi} _ {t}} ], A _ {| \pmb {r} + \pmb {u} _ {t}} ^ {\pmb {\pi} _ {t}} \rangle \geqslant \lambda_ {\min } \| A _ {| \pmb {r} + \pmb {u} _ {t}} ^ {\pmb {\pi} _ {t}} \| _ {2} ^ {2}, \| \widehat {A} _ {| \pmb {r} + \pmb {u} _ {t}} ^ {\pmb {\pi} _ {t}} \| _ {2} ^ {2} \leqslant \lambda_ {\max } ^ {2} \| A _ {| \pmb {r} + \pmb {u} _ {t}} ^ {\pmb {\pi} _ {t}} \| _ {2} ^ {2},
+$$
+
+Note that the exact NPG is captured by the special case with  $\lambda_{\mathrm{min}} = \lambda_{\mathrm{max}} = 1$ . For NPG, since the policy is always bounded away from 0. We will use  $\Pi^{+} := \{\pi|\forall n,h,a_{h},s_{h}:\pi_{h}^{n}(a_{h}|s_{h}) > 0\}$  to denote such feasible policy set. We state our main result below.
+
+Theorem 4.2 (Informal). Suppose  $\eta^{\mathrm{goal}}$  is Lipschitz in  $\pi$ , given any initial  $\pi_1 \in \Pi^+$ , for any  $\varepsilon > 0$ , if the agents follow Def. 4.1 under Assump.  $B$ , and  $T$  and  $U_{\mathrm{max}}$  are large enough, we have  $\Psi^{\varepsilon} \neq \emptyset$ .
+
+Our result is strong in indicating the existence of a steering path for any feasible initialization. The proof is based on construction. The basic idea is to design the  $\boldsymbol{u}_t$  so that  $A_{|\boldsymbol{r} + \boldsymbol{u}_t}^{\pi_t} \propto \log \frac{\pi^*}{\pi_t}$ , for some target policy  $\pi^* \in \Pi^+$  (approximately) maximizing  $\eta^{\mathrm{goal}}$ , then we can guarantee the convergence of  $\pi_t$  towards  $\pi^*$  under Assump. B. The main challenge here would be the design of  $u_t$ . We defer the details and the formal statements to Appx. D.
+
+# 4.2 EXISTENCE WHEN  $f^*$  IS UNKNOWN: THE IDENTIFIABLE MODEL CLASS
+
+Intuitively, when  $f^*$  is unknown, if we can first use a few steering steps  $\widetilde{T} < T$  to explore and identify  $f^*$ , and then steer the agents from  $\pi_{\widetilde{T}}$  to the desired policy within  $T - \widetilde{T}$  steps given the identified  $f^*$ , we can expect  $\Psi^\varepsilon \neq \emptyset$ . Motivated by this insight, we introduce the following notion.
+
+Definition 4.3  $((\delta, T_{\mathcal{F}}^{\delta})$ -Identifiable). Given  $\delta \in (0,1)$ , we say  $\mathcal{F}$  is  $(\delta, T_{\mathcal{F}}^{\delta})$ -identifiable, if  $\max_{\psi} \min_{f \in \mathcal{F}} \mathbb{E}_{\psi, f}[\mathbb{I}[f = f_{\mathrm{MLE}}]] \geqslant 1 - \delta$ , where  $\mathbb{I}[\mathcal{E}] = 1$  if  $\mathcal{E}$  is true and otherwise  $0$ ;  $f_{\mathrm{MLE}} := \arg \max_{f \in \mathcal{F}} \sum_{t=1}^{T_{\mathcal{F}}^{\delta}} \log f(\pmb{\pi}_{t+1} | \pmb{\pi}_t, \pmb{u}_t)$ .
+
+Intuitively,  $\mathcal{F}$  is  $(\delta, T_{\mathcal{F}}^{\delta})$ -identifiable, if  $\exists \psi$ , s.t. after  $T_{\mathcal{F}}^{\delta}$  steering steps, the hidden model  $f$  can be identified by the Maximal Likelihood Estimation (MLE) with high probability. Next, we provide a general set of  $(\delta, T_{\mathcal{F}}^{\delta})$ -identifiable function classes with  $T_{\mathcal{F}}^{\delta}$  upper bounded for any  $\delta \in (0,1)$ .
+
+Example 4.4. [One-Step Difference] If  $\forall \pi \in \Pi$ , there exists a steering reward  $u_{\pi} \in \mathcal{U}$ , s.t.  $\min_{f, f' \in \mathcal{F}} \mathbb{H}^2(f(\cdot | \pi, r + u_{\pi}), f'(\cdot | \pi, r + u_{\pi})) \geqslant \zeta$ , for some universal  $\zeta > 0$ , where  $\mathbb{H}$  is the Hellinger distance, then for any  $\delta \in (0,1)$ ,  $\mathcal{F}$  is  $(\delta, T_{\mathcal{F}}^{\delta})$ -identifiable with  $T_{\mathcal{F}}^{\delta} = O(\zeta^{-1} \log(|\mathcal{F}| / \delta))$ .
+
+Based on Def. 4.3, we provide a sufficient condition when  $\Psi^{\varepsilon}$  is non-empty.
+
+Theorem 4.5. [A Sufficient Condition for Existence] Given any  $\varepsilon >0$ ,  $\Psi_T^\varepsilon (\mathcal{F};\pi_1)^3\neq \emptyset$ , if  $\exists \widetilde{T} < T$  s.t., (1)  $\mathcal{F}$  is  $(\frac{\varepsilon}{2\eta_{\mathrm{max}}},\widetilde{T})$ -identifiable, (2)  $\Psi_{T - \widetilde{T}}^{\varepsilon /2}(\mathcal{F};\pi_{\widetilde{T}})\neq \emptyset$  for any possible  $\pi_{\widetilde{T}}$  generated at step  $\widetilde{T}$  during the steering.
+
+We conclude this section by noting that, by Thm. 4.2, the above condition (2) is realistic for NPG (or more general PMD) dynamics. The proofs for all results in this section are deferred to Appx. E.
+
+# 5 LEARNING (APPROXIMATELY) OPTIMAL STEERING STRATEGY
+
+In this section, we investigate how to solve Obj. (1). Comparing with the episodic RL setting, the main challenge is to learn a history-dependent policy. Since the history space grows exponentially in  $T$ , directly solving Obj. (1) can be computationally intractable for large  $T$ . Therefore, the main focus of this section is to design tractable algorithms to overcome this challenge.
+
+As a special case, when the model is known, i.e.  $\mathcal{F} = \{f^{*}\}$ , by the Markovian property, Obj. (1) reduces to a normal RL objective, and a state-dependent steering strategy  $\psi : \Pi \to \mathcal{U}$  is already enough. For completeness, we include the algorithm but defer to Alg. 3 in Appx. B.4. In the rest of this section, we focus on the general case  $|\mathcal{F}| > 1$ . In Sec. 5.1, we investigate the solutions when  $|\mathcal{F}|$  is small, and in Sec. 5.2, we study the more challenging case when  $|\mathcal{F}|$  is large.
+
+# 5.1 SMALL MODEL CLASS: DYNAMIC PROGRAMMING WITH MODEL BELIEF STATE
+
+A Partially Observable MDP Perspective In fact, we can interpret Obj. (1) as learning the optimal policy in a POMDP, in which the hidden state is  $(\pi_t,f)$ , i.e. a tuple containing the policy and the hidden model  $f$  uniformly sampled from  $\mathcal{F}$ , and the mediator can only partially observe the policy  $\pi_t$ . It is well-known that any POMDP can be lifted to the belief MDP, where the state is the belief state of the original POMDP. Then, the optimal policy in the belief MDP is exactly the optimal history-dependent policy in the original POMDP (Ibe, 2013). In our case, for each step  $t\in [T]$ , the belief state is  $(\pi_t,b_t)$ , where  $b_{t}:= [\operatorname*{Pr}(f|\{\pmb{\pi}_{t^{\prime}},\pmb{u}_{t^{\prime}}\}_{t^{\prime} = 1}^{t},\pmb{\pi}_{t})]_{f\in \mathcal{F}}$  is the "model belief state" defined to be the posterior distribution of models given the history of observations and actions. When  $|\mathcal{F}|$  is small, the model belief state  $b_{t}\in \mathcal{R}^{|\mathcal{F}|}$  is low dimensional and computable. Learning  $\psi^{*}$  is tractable by running any RL algorithm on the lifted MDP. In Proc. 1, we show how to steer in this setting. We defer the detailed algorithm of learning such belief-state dependent strategy to Alg. 4 in Appx. B.5.
+
+# Procedure 1: The Steering Procedure when  $|\mathcal{F}|$  is Small
+
+1 Input: Model Set  $\mathcal{F}$ ; Total step  $T$ ;  
+2 Solving Obj. (1) by learning a belief state-dependent strategy  $\psi_{\mathrm{Belief}}^*$  by Alg. 4 with  $\mathcal{F}$  and  $T$
+
+3 Deploy  $\psi_{\mathrm{Belief}}^{*}$  to steer the real agents for  $T$  steps.
+
+# 5.2 LARGE MODEL CLASS: A FIRST-EXPLORE-THEN-EXPloit FRAMEWORK
+
+When  $|\mathcal{F}|$  is large, the method in Sec. 5.1 is inefficient since the belief state  $b_{t}$  is high-dimensional. In fact, the above POMDP interpretation implies the intractability of Obj. (1) for large  $|\mathcal{F}|$ : the number of hidden states of the POMDP scales with  $|\mathcal{F}|$ . Therefore, instead of exactly solving Obj. (1), we turn to the First-Explore-Then-Exploit (FETE) framework as stated in Procedure 2.
+
+The first  $\tilde{T} < T$  steps are the exploration phase, where we learn and deploy an exploration policy  $\psi^{\mathrm{Explore}}$  maximizing the probability of identifying the hidden model with the MLE estimator. The
+
+remaining  $T - \widetilde{T}$  steps belong to the exploitation stage. We first estimate the true model by the MLE with the interaction history with real agents. Next, we learn an exploitation strategy to steer real agents for the rest  $T - \widetilde{T}$  steps by solving Obj. (1) with  $\mathcal{F} = \{f_{\mathrm{MLE}}\}$ , time  $T - \widetilde{T}$  and the initial policy  $\pi_{\widetilde{T} + 1}$ , as if  $f_{\mathrm{MLE}}$  is the true model.
+
+Justification for FETE We cannot guarantee that Desiderata 1&2 are achievable, because we do not exactly solve Obj. 1. However, if  $\mathcal{F}$  is  $(\delta /|\mathcal{F}|,T_{\mathcal{F}}^{\delta /|\mathcal{F}|})$  -identifiable (Def. 4.3) and we choose  $\widetilde{T}\geqslant T_{\mathcal{F}}^{\delta /|\mathcal{F}|}$ , we can verify  $\operatorname *{Pr}(f_{\mathrm{MLE}} = f^{*})\geqslant 1 - \delta$  in FETE. Therefore, we can still expect the exploitation policy  $\psi^{\mathrm{Exploit}}$  steer the agents to approximately maximize  $\eta^{\mathrm{goal}}(\pi_{T + 1})$  with reasonable steering cost for the rest  $T - \widetilde{T}$  steps.
+
+Procedure 2: The Steering Procedure when  $|\mathcal{F}|$  is Large (The FETE Framework)  
+1 Input: Model Set  $\mathcal{F}$ ; Total step  $T$ ; Exploration horizon  $\widetilde{T}$ ;  
+2 /* Exploration Phase */  
+3 Learn an exploration strategy  $\psi^{\mathrm{Explore}} \gets \arg \max_{\psi} \frac{1}{|\mathcal{F}|} \sum_{f \in \mathcal{F}} \mathbb{E}_{\boldsymbol{\pi}_1', \boldsymbol{u}_1', \dots, \boldsymbol{\pi}_{\widetilde{T} + 1}'} \sim_{\psi, f} [\mathbb{I}[f = \arg \max_{f' \in \mathcal{F}} \sum_{t=1}^{\widetilde{T}} \log f'(\boldsymbol{\pi}_{t+1}' | \boldsymbol{\pi}_t', \boldsymbol{u}_t')]]$ .
+
+4 Deploy  $\psi^{\mathrm{Explore}}$  to steer the real agents and collect  $\{\pi_1, u_1, \dots, \pi_{\tilde{T}}, u_{\tilde{T}}, \pi_{\tilde{T} + 1}\}$   
+5 /* Exploitation Phase */
+
+Estimate  $f_{\mathrm{MLE}} \gets \arg \max_{f \in \mathcal{F}} \sum_{t=1}^{\tilde{T}} \log f(\pmb{\pi}_{t+1} | \pmb{\pi}_t, \pmb{u}_t)$
+
+7 Deploy  $\psi^{\mathrm{Exploit}}\gets \arg \max_{\psi}\mathbb{E}_{\psi ,f_{\mathrm{MLE}}}[\beta \cdot \eta^{\mathrm{goal}}(\pmb{\pi}_{T - \tilde{T} +1}) - \sum_{t = 1}^{T - \tilde{T}}\eta^{\mathrm{cost}}(\pmb {\pi}_t,\pmb {u}_t)|\pmb {\pi}_1 = \pmb{\pi}_{\tilde{T} +1}]$
+
+We conclude this section by highlighting the computational tractability of FETE. Note that when computing  $\psi^{\text{Exploit}}$ , we treat  $f_{\text{MLE}}$  as the true model, so an history-independent  $\psi^{\text{Exploit}}$  is enough. Therefore, the only part where we need to learn a history-dependent strategy is in the exploration stage, and the maximal history length is at most  $\widetilde{T}$ , which can be much smaller than  $T$ . Moreover, in some cases, it is already enough to just learn a history-independent  $\psi^{\text{Explore}}$  to do the exploration (for example, the model class in Example 4.4).
+
+Comparison with Canyakmaz et al. (2024) Both SIAR-MPC in Canyakmaz et al. (2024) and our FETE (Procedure 2) adopt a first-explore-then-exploit structure. We examine the differences between two algorithms from two aspects: exploration strategy and model estimation strategy. Regarding exploration strategy, SIAR-MPC uses noise-based random exploration, whereas we adopt a more strategic approach, which uses the identification success rate as a signal to learn the exploration policy. Empirical results in Sec. 6.2 demonstrate the higher efficiency of our methods. In terms of model estimation strategy, SIAR-MPC estimates the hidden model by solving a constrained regression problem (Eq. (8) in Canyakmaz et al. (2024)), while we solve a MLE objective. In principle, our MLE objective is more general and can recover the regression problem in SIAR-MPC if choosing  $\mathcal{F}$  to be the function class including Gaussian noise perturbed dynamics with side-information constraints introduced in Canyakmaz et al. (2024).
+
+# 6 EXPERIMENTS
+
+In this section, we discuss our experimental results. For more details of all experiments in this section (e.g. experiment setup and training details), we defer to Appx. G. The steering horizon is set to be  $T = 500$ , and all the error bar shows  $95\%$  confidence level. We denote  $[x]^+ := \max \{0, x\}$ .
+
+# 6.1 LEARNING STEERING STRATEGIES WITH KNOWLEDGE OF  $f^{*}$
+
+Normal-Form Stag Hunt Game In Fig. 1-(b), we compare the agents' dynamics with/without steering, where the agents learn to play the Stag Hunt Game in Fig. 1-(a). We report the experiment setup here. Both agents follow the exact NPG (Def. 4.1 with  $\hat{A}^{\pi} = A^{\pi}$ ) with fixed learning rate  $\alpha = 0.01$ . For the steering setup, we choose the total utility as  $\eta^{\mathrm{goal}}$ , and use PPO to train the steering strategy (one can choose other RL or control algorithms besides PPO). We also conduct experiments in a representative zero-sum game 'Matching Pennies', which we defer the details to Appx. G.2.
+
+Grid World Stag Hunt Game: Learning Steering Strategy with Observations on Agents' Behaviors In the previous experiments, we consider the direct parameterization and the state space  $\mathcal{X} = \Pi \subset \mathbb{R}^4$  has low dimension. In real-world scenarios, the policy space  $\Pi$  can be extremely rich and high-dimensional if the agents consider neural networks as policies. In addition, the mediator may not get access to the agents' exact policy  $\pi$  because of privacy issues.
+
+This motivates us to investigate the possibility of steering agents with observations on agents' behavior only (e.g. trajectories of agents in a game  $G$ ), instead of the full observation of  $\pi$ . In Appx. F, we justify this setup and formalize it as a partially observable extension of our current framework. We consider the evaluation in a grid-world version of the Stag Hunt Game as shown in Fig. 2-(a). In this setting, the state space in game  $G$  becomes pixel-based images, and both agents (blue and red) will adopt Convolutional Neural Networks (CNN) based policies with thousands of parameters and update with PPO. We train a steering strategy, which only takes the agents' recent trajectories as input to infer the steering reward. As shown in Fig. 2-(b), without direct usage of the agents' policy, we can still train a steering strategy towards desired solution
+
+![](images/9718942b5ffc28efb74625d9e2feb882eced8ca83f751ac098200e228433e305.jpg)  
+Figure 2: Grid-World Version of Stag Hunt Game. Left: Illustration of game. Right: The performance of agents with/without steering. Without steering, the agents converge to go for hares, which has sub-optimal utility. Under our learned steering strategy, the agents converge to a better equilibrium and chase the stag.
+
+![](images/299a6a9caa9aca0deda2854a99d876ea516e8aa999be3ce38c23e5a3bff7a105.jpg)
+
+# 6.2 LEARNING STEERING STRATEGIES WITHOUT KNOWLEDGE OF  $f^{*}$
+
+Small Model Set  $|\mathcal{F}|$ : Belief State Based Steering Strategy In this part, we evaluate Proc. 1 designed for small  $\mathcal{F}$ . We consider the same normal-form Stag Hunt game and setup as Sec. 6.1, while the agents update by the NPG with a random learning rate  $\alpha = [\xi]^+$ , where  $\xi \sim \mathcal{N}(\mu, 0.3^2)$ . Here the mean value  $\mu$  is unknown to the mediator, and we consider a model class  $\mathcal{F} := \{f_{0.7}, f_{1.0}\}$  including two possible values of  $\mu \in \{0.7, 1.0\}$ . We report our experimental results in Table 1.
+
+Table 1: Evaluation for Proc. 1 (Averaged over 25 different initial  $\pi_1$ , see Appx. G.1).  
+(a) Performance in  $f_{0.7}$  
+
+<table><tr><td></td><td>p(Δψ,T ≤ ε)</td><td>Cψ,T</td></tr><tr><td>ψ0.7*</td><td>0.99 ± 0.01</td><td>10.6 ± 0.3</td></tr><tr><td>ψ1.0*</td><td>0.13 ± 0.02</td><td>7.6 ± 0.2</td></tr><tr><td>ψBelief*</td><td>0.87 ± 0.05</td><td>10.5 ± 0.4</td></tr></table>
+
+(b) Performance in  $f_{1,0}$  
+
+<table><tr><td></td><td>p(Δψ,T ≤ ε)</td><td>Cψ,T</td></tr><tr><td>ψ*0.7</td><td>1.00 ± 0.00</td><td>8.2 ± 0.2</td></tr><tr><td>ψ*1.0</td><td>1.00 ± 0.00</td><td>5.6 ± 0.2</td></tr><tr><td>ψ*Belief</td><td>0.99 ± 0.01</td><td>6.1 ± 0.3</td></tr></table>
+
+Firstly, we demonstrate the suboptimal behavior if the mediator ignores the model uncertainty and just randomly deploys the optimal strategy of  $f_{0.7}$  or  $f_{1.0}$ . To do this, we train the (history-independent) optimal steering strategy by Alg. 3, as if we know  $f^{*} = f_{0.7}$  (or  $f^{*} = f_{1.0}$ ), which we denote as  $\psi_{0.7}^{*}$  (or  $\psi_{1.0}^{*}$ ). To meet with our Desideratum 1, we first set the accuracy level  $\varepsilon = 0.01$ , and search the minimal  $\beta$  so that the learned steering strategy can achieve  $\varepsilon$ -steering gap (see Appx. G.3.1). Because of the difference in  $\mu$ , we have  $\beta = 70$  and  $\beta = 20$  in training  $\psi_{0.7}^{*}$  and  $\psi_{1.0}^{*}$ , respectively, and empirically, we observe that  $\psi_{0.7}^{*}$  requires much larger steering reward than  $\psi_{1.0}^{*}$ . As we marked in red in Table 1-(a) and (b), because of the difference in the steering signal,  $\psi_{0.7}^{*}$  consumes much higher steering cost to achieve the same accuracy level in  $f_{1.0}$ , and  $\psi_{1.0}^{*}$  may fail to steer agents with  $f_{0.7}$  to the desired accuracy. Next, we train another strategy  $\psi_{\mathrm{Belief}}^{*}$  via Alg. 4, which predicts the steering reward based on both the agents' policy  $\pi$  and the belief state of the model. As we can see,  $\psi_{\mathrm{Belief}}^{*}$  can almost always achieve the desired  $\varepsilon$ -steering gap with reasonable steering cost.
+
+Large Model Set  $|\mathcal{F}|$ : The FETE Framework In this part, we evaluate the FETE framework (Proc. 2 in Sec. 5.2). We consider an cooperative setting with  $N = 10$  players. Each agent has two actions  $\mathbb{A}$  and  $\mathbb{B}$ , and the mediator only receives non-zero utility when all the agents cooperate together to take action  $\mathbb{A}$ , i.e.  $\eta^{\mathrm{goal}}(\pi) \coloneqq \prod_{n=1}^{N} \pi^n(\mathbb{A})$ . The agents do not have intrinsic rewards ( $r = 0$ ), but the mediator's can steer them to maximize its own utility by providing additional steering rewards.
+
+We consider "avaricious agents" with varying degrees of greediness, who tend to decrease the learning rates if the payments by mediator are high. Consequently, they require more steering steps to converge to the desired policies, potentially earning more incentive payments from the mediators. More concretely, the learning rate of agent  $n$  is  $\alpha_{n} := [\xi_{n}]^{+}$  with  $\xi_{n} \sim \mathcal{N}(1.5 - \beta^{n} \cdot [V_{|u}^{n,\pi} - \lambda^{n}]^{+}, 0.5^{2})$ , where  $\beta^{n} > 0$  is a scaling factor and  $\lambda^{n} > 0$  is the threshold to exhibit
+
+![](images/092cb77755dac2d4001c76f286b8e7938dd63d736b37d005a7acf9c3f6c028c5.jpg)  
+Figure 3: Evaluation for Proc. 2. Left: Accuracy of MLE estimator  $(\lambda_{\mathrm{MLE}}^n)$  after doing exploration for  $t$  steps. Ours can achieve near  $100\%$  accuracy after 30 steering steps, while the random exploration takes more than 300 steps. Middle and Right: Average steering gap and steering cost of Oracle, FETE and FETE-RE. Our FETE achieves competitive performance comparing with Oracle, and significantly outperforms FETE-RE (adaption of SIAR-MPC (Canyakmaz et al., 2024) to our setting) in terms of steering gap.
+
+![](images/f563ff249051aea489108323972061b94071eb978c3e7570cd9f964c2e534611.jpg)
+
+![](images/0f38ed21653b8bbe78e5f8fdd389f2f6e86231bdee06454c64bf8e17a14353db.jpg)
+
+avaricious behavior. In our experiments, the model uncertainty comes from multiple possible realization of  $\lambda^n\in \{0.25,0.75, + \infty \}$ , which results in an extremely large model class  $\mathcal{F}$  with  $|\mathcal{F}| = 3^{10}$ . Here  $\lambda^n = +\infty$  corresponds to normal agents whose learning rates are stable. The mediator does not know the agents' types  $\{\lambda^n\}_{n\in [N]}$  in advance, and it can only observe one learning rate samples  $\{\alpha_{n}\}_{n\in [N]}$  of agents per iteration  $t\in [T]$  and estimate the true types from those samples. We consider the fixed initial policy with  $\forall n\in [N]$ ,  $\pi_1^n (\mathbb{A}) = 1 - \pi_1^n (\mathbb{B}) = 1 / 3$ , and set the maximal steering reward  $U_{\mathrm{max}} = 1.0$ .
+
+To understand the exploration challenge, note that, during the exploration phase, if the steering signal  $\pmb{u}$  is not strong enough, i.e.  $V_{|u}^{n,\pi} < \lambda^n$ , the mediator may fail to distinguish those avaricious agents from the normal ones, because they behave exactly the same. Such failure can lead to undesired outcomes in the exploitation phase: higher steering rewards can accelerate the convergence of normal agents, but can lead to larger steering gaps for avaricious agents.
+
+We provide the evaluation results in Fig. 3. First, we compare the exploration efficiency. We can see the clear advantage of our strategic exploration in FETE (Procedure 2) compared with noise-based random exploration (Canyakmaz et al., 2024). Next, we compare the steering gaps and costs of three methods: (i) FETE; (ii) FETE-RE; (iii) Oracle - if the mediator knows  $f^{*}$  in advance and solving Obj. (1) with  $\mathcal{F} = \{f^{*}\}$ . Here FETE-RE can be regarded as adaption of SIAR-MPC (Canyakmaz et al., 2024) to our case by replacing strategic exploration in FETE with random exploration (see Appx. G.4 for more explanation). We choose exploration horizon  $\tilde{T} = 30$  suggested by the previous exploration experiment, and report results for three realizations of  $f^{*} \in \{f_{1}, f_{2}, f_{3}\}$ . For  $f_{1}$  and  $f_{2}$ , all the agents share  $\lambda^{n} = 0.75$  and  $+\infty$ , respectively.  $f_{3}$  is a mixed setup where  $\lambda^{n} = 0.75$  for  $1 \leqslant n \leqslant 5$  and  $\lambda^{n} = +\infty$  for  $5 < n \leqslant 10$ . As we can see, comparing with Oracle, both the steering gap and cost of our FETE are competitive. Moreover, thanks to our strategic exploration method, FETE exhibits significant advantage over Canyakmaz et al. (2024) in terms of steering gaps.
+
+# 7 CONCLUSION
+
+In this paper, we introduce the problem of steering Markovian agents under model uncertainty. We provide theoretical foundations for this problem by formulating a novel optimization objective and providing existence results. Moreover, we design several algorithmic approaches suitable for varying degrees of model uncertainty in this problem class. We test their performances in different experimental settings and show their effectiveness. Our work opens up avenues for compelling open problems that merit future investigation. Firstly, future work could aim to identify superior optimization objectives that guarantee strictly better performances in terms of steering gap and cost than ours. Secondly, when applying our strategies in real-world applications, constraints on the steering reward budget could be added. Finally, the framework could be generalized to permit non-Markovian agents.
+
+# ACKNOWLEDGEMENT
+
+The work is supported by ETH research grant and Swiss National Science Foundation (SNSF) Project Funding No. 200021-207343 and SNSF Starting Grant. VT is supported by an ETH AI Center Doctoral Fellowship. HHN is supported by the Swiss National Science Foundation under Eccellenza Grant 'Markets and Norms,' as well as under NCCR Automation Grant 51NF40-180545.
+
+# REPRODUCIBILITY STATEMENT
+
+The code of all the experiments in this paper and the instructions for running can be found in https://github.com/jiaweiihuang/SteeringMARKOVIAgents.
+
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+
+# CONTENTS
+
+1 Introduction 1  
+1.1 Closely Related Works 3  
+2 Preliminary 3  
+3 The Problem Formulation of the Steering Markovian Agents 4  
+4 Existence of Steering Strategy with  $\varepsilon$ -Steering Gap 6
+
+4.1 Existence when  $f^{*}$  is Known: Natural Policy Gradient as an Example 6  
+4.2 Existence when  $f^{*}$  is Unknown: the Identifiable Model Class 6
+
+5 Learning (Approximately) Optimal Steering Strategy 7
+
+5.1 Small Model Class: Dynamic Programming with Model Belief State 7  
+5.2 Large Model Class: A First-Explore-Then-Exploit Framework 7
+
+6 Experiments 8
+
+6.1 Learning Steering Strategies with Knowledge of  $f^{*}$  8  
+6.2 Learning Steering Strategies without Knowledge of  $f^{*}$  9
+
+7 Conclusion 10
+
+A Frequently Used Notations 18  
+B Missing Details in the Main Text 18
+
+B.1 A Real-World Scenario that Can be Modeled as a Stag Hunt Game 18  
+B.2 Additional Related Works 18  
+B.3 A Brief Introduction to Markov Decision Process 20  
+B.4 Algorithm for Learning Optimal (History-Independent) Strategy when  $f^{*}$  is Known 20  
+B.5 Algorithm for Learning Belief-State Dependent Steering Strategy 20
+
+C Missing Proofs in Section 3 20  
+D Missing Proofs for Existence when the True Model  $f^{*}$  is Known 22
+
+D.1 More Details about Policy Mirror Descent 22  
+D.2 Proofs for the Existence of Desired Steering Strategy 23
+
+D.2.1 Special Case: PMD with Exact Advantage-Value 23  
+D.2.2 The General Incentive Driven Agents under Assump. B 25
+
+E Missing Proofs for Existence when the True Model  $f^{*}$  is Unknown 26  
+F Generalization to Partial Observation MDP Setup 28  
+F.1 POMDP Basics 28
+
+F.2 Steering Process as a POMDP 29
+
+# G Missing Experiment Details 29
+
+G.1 About Initialization in Evaluation 29  
+G.2 Experiments for Known Model Setting 30
+
+G.2.1 Experiment Details in Normal-Form Stag Hunt Game 30  
+G.2.2 Experiment Details in Grid-World Version of Stag Hunt Game 30  
+G.2.3 Experiments in Matching Pennies 31
+
+G.3 Experiments for Unknown Model Setting 32
+
+G.3.1 Details for Experiments with Small Model Set  $\mathcal{F}$  32  
+G.3.2 Details for Experiments with Large Model Set  $\mathcal{F}$  33
+
+G.4 Explanation of the Consistency of the Adaption 33  
+G.5 A Summary of the Compute Resources by Experiments in this Paper 33
+
+# H Additional Discussion about Generalizing our Results 34
+
+A FREQUENTLY USED NOTATIONS  
+
+<table><tr><td>Notation</td><td>Description</td></tr><tr><td>G</td><td>A finite-horizon general-sum Markov Game</td></tr><tr><td>N</td><td>The number of agents</td></tr><tr><td>S, A</td><td>State space and action space of the game G</td></tr><tr><td>H</td><td>The horizon of the game G</td></tr><tr><td>P</td><td>Transition function of the game G</td></tr><tr><td>r</td><td>Reward function of the game G</td></tr><tr><td>π</td><td>The agents&#x27; policy (collection of policies of all agents)</td></tr><tr><td>π1</td><td>The initial policy</td></tr><tr><td>M</td><td>A finite-horizon Markov Decision Process (the steering MDP)</td></tr><tr><td>X, U</td><td>State space and action space of M</td></tr><tr><td>T</td><td>The horizon of M (i.e. the horizon of the steering dynamics)</td></tr><tr><td>T</td><td>(Stationary) Transition function of M</td></tr><tr><td>ηcost</td><td>The steering cost function of M</td></tr><tr><td>ψ</td><td>The history-dependent steering strategy by mediator</td></tr><tr><td>u (or ut for a specific horizon t)</td><td>The steering reward function</td></tr><tr><td>Umax</td><td>The upper bound for steering reward</td></tr><tr><td>f</td><td>Agents learning dynamics (T = f in the steering MDP)</td></tr><tr><td>ηgoal</td><td>The goal function of M</td></tr><tr><td>F</td><td>The model class of agents dynamics (with finite candidates)</td></tr><tr><td>β</td><td>Regularization coefficient in Obj. (1)</td></tr><tr><td>Cψ,T(f)</td><td>The total expected steering cost Eψ,f[∑t=1Tηcost(πt, ut)]</td></tr><tr><td>Δψ,T(f)</td><td>The steering gap: Eψ,f[ maxηηgoal(π) - ηηgoal(πT+1)]</td></tr><tr><td>Ψ</td><td>The collection of all history dependent policies</td></tr><tr><td>Ψε as a short note of ΨT,Umax(F; π1)</td><td>{ψ ∈ Ψ|Eψ,f[ maxηηgoal(π) - ηηgoal(πT+1)|π1] ≤ ε}</td></tr><tr><td>Qn,r+u, Vn,r+u, Ah,r+u</td><td>The Q-value, V-value and advantage value functions for agent n</td></tr><tr><td>fMLE</td><td>The Maximal Likelihood Estimator (introduced in Def. 4.3)</td></tr><tr><td>bt</td><td>Model belief state [Pr(f|{πt&#x27;, ut&#x27;}t&#x27;=1, πt)]f∈F∈R|F|</td></tr><tr><td>ψExplore/ψExploit</td><td>The exploration/exploitation policy in FETE framework.</td></tr><tr><td>O(·), Ω(·), Θ(·), O(·), Ω(·), Θ(·)</td><td>Standard Big-O notations, (·) omits the log terms.</td></tr></table>
+
+# B MISSING DETAILS IN THE MAIN TEXT
+
+# B.1 A REAL-WORLD SCENARIO THAT CAN BE MODELED AS A STAG HUNT GAME
+
+As a real-world example, the innovation adaption can be modeled as a (multi-player) Stag Hunt game. Consider a situation involving a coordination problem where people can choose between an inferior/unsustainable communication or transportation technology that is cheap (the Gather action) and a superior technology that is sustainable but more expensive (the Hunt action). If more and more people buy products by the superior technology, the increasing profits can lead to the development of that technology and the decrease of price. Eventually, everyone can afford the price and benefit from the sustainable technology. In contrast, if people are trapped by the products of the inferior technology due to its low price, the long-run social welfare can be sub-optimal. The mediator's goal is to steer the population to adopt the superior technology.
+
+# B.2 ADDITIONAL RELATED WORKS
+
+We first complements the comparison with Zhang et al. (2023) in Sec. 1.1 by noting a minor but worth to be mentioned difference between our setting and (Zhang et al., 2023) in terms of incentive
+
+schemes. While they consider that the mediator influences the agents' learning dynamics through a scalar payment function, we operate with additional steering rewards in a multi-dimensional reward vector space. As a result, there may not exist direct ways to translate the steering strategies between both settings, especially in the bandit feedback setting where only the sampled actions of agents can be observed (Zhang et al., 2023).
+
+**Opponent Shaping** In the RL literature a line of work focuses on the problem of opponent shaping, where agents can influence each other's learning by handing out rewards (Foerster et al., 2018; Yang et al., 2020; Willi et al., 2022; Lu et al., 2022; Willis et al., 2023; Zhao et al., 2022). Although the ways of influencing agents are similar to our setting, we study the problem of a mediator that acts outside the Markov Game and steers all the agents towards desired policies, while in opponent shaping the agents themselves learn to influence each other for their own interests.
+
+Learning Dynamics in Multi-Agent Systems In multi-agent setting, it is an important question to design learning dynamics and understand their convergence properties (Hernandez-Leal et al., 2017). Previous works have established near-optimal convergence guarantees to equilibra (Daskalakis et al., 2021; Cai et al., 2024). When the transition model of the multi-agent system is unknown, many previous works have studied how to conduct efficient exploration and learn equilibria under uncertainty (Jin et al., 2021; Bai et al., 2020; Zhang et al., 2021; Leonardos et al., 2021; Yardim et al., 2023; Huang et al., 2024b;a). However, most of these results only have guarantees on solving an arbitrary equilibrium when multiple equilibria exist, and it is unclear how to build algorithms based on them to reach some desired policies to maximize some goal functions.
+
+Mathematical Programming with Equilibrium Constraints (MPEC) MPEC generalises bilevel optimization to problems where the lower level consists of solving an equilibrium problem (Luo et al., 1996). (Li et al., 2020; Liu et al., 2022; Wang et al., 2022; 2023; Yang et al., 2022). These works consider variants of an MPEC and present gradient based approaches, most of which rely on computing hypergradients via the implicit function theorem and thus strong assumptions on the lower level problem, such as uniqueness of the equilibrium. Most games fail to satisfy such constraints. In contrast, our work makes no assumptions on the equilibrium structure and instead mild assumptions on the learning dynamics.
+
+Game Theory and Mechanism Design In Game Theory, a setup such as ours can be modelled as a Stackelberg game. Several works have considered finding Stackelberg equilibria using RL (Gerstgrasser and Parkes, 2023; Zhong et al., 2024) or gradient-based approaches (Fiez et al., 2020). Deng et al. (2019) showed how agents can manipulate learning algorithms to achieve more reward, as if they were playing a Stackelberg game. Related problems are implementation theory (Monderer and Tennenholtz, 2004) and equilibrium selection (Harsanyi and Selten, 1992). Moreover, the field of mechanism design has been concerned with creating economic games that implement certain outcomes as their equilibria. Several recent works have considered mechanism design on Markov Games (Curry et al., 2024; Baumann et al., 2020; Guo et al., 2023). In the case of congestion games, mechanisms have been proposed to circumvent the price of anarchy (Balcan et al., 2013; Paccagnan and Gairing, 2021; Roughgarden and Tardos, 2004), i.e. equilibria with low social welfare.
+
+There is also a line of work has focused on control strategies for evolutionary games (Gong et al., 2022; Paarporn et al., 2018). However, the game and learning dynamics differ significantly from our setting. For a full survey of control-theoretic approaches, we refer the reader to Ratliff et al. (2019); Riehl et al. (2018).
+
+Bilevel Reinforcement Learning Bilevel RL considers the problem of designing an MDP—by for example changing the rewards—with a desirable optimal policy. Recently, several works have studied gradient-based approaches to find such good MDP configurations (Chen et al., 2022; Chakraborty et al., 2023; Shen et al., 2024; Thoma et al., 2024). While similar in some regards, in this setting we assume the lower level is a Markov Game instead of just an MDP. Moreover, our aim is not to design a game with a desirable equilibrium from scratch, but to take a given game and agent dynamics and steer them with minimal additional rewards to a desired outcome within a certain amount of time. Therefore our upper-level problem is a strategic decision-making problem, solved by RL instead of running gradient descent on some parameter space.
+
+Episodic RL and Non-Episodic RL Most of the existing RL literature focus on the episodic learning setup, where the entire interaction history can be divided into multiple episodes starting from the same initial state distribution(Dann and Brunskill, 2015; Dann et al., 2017). Comparing with this setting, our finite-horizon non-episodic setting is more challenging because the mediator cannot simply learn from repeated trial-and-error. Therefore, the learning criterions (e.g. no-regret (Azar et al., 2017; Jin et al., 2018) or sample complexity (Dann and Brunskill, 2015)) in episodic RL setting is not suitable in our case, which targets at finding a near-optimal policy in maximizing return. This motivates us to consider the new objective (Obj. (1)).
+
+To our knowledge, most of the previous works use "non-episodic RL" to refer to the learning in infinite-horizon MDP. One popular setting is the infinite-horizon MDPs with stationary transitions, where people consider the discounted (Schulman et al., 2017; Dong et al., 2019) or average return (Auer et al., 2008; Wei et al., 2020). The infinit-horizon setting with non-stationary dynamics is known as the continual RL (Khetarpal et al., 2022; Abel et al., 2024), where the learners "never stops learning" and continue to adapt to the dynamics. Since we focus on the steering problem with fixed and finite horizon, the methodology in those works cannot be directly applied here.
+
+Most importantly, we are also the first work to model the steering problem as a RL problem.
+
+# B.3 A BRIEF INTRODUCTION TO MARKOV DECISION PROCESS
+
+A finite-horizon Markov Decision Process is specified by a tuple  $M \coloneqq \{x_1, T, \mathcal{X}, \mathcal{U}, \mathbb{T}, (\eta, \eta^{\mathrm{term}})\}$ , where  $x_1$  is the fixed initial state,  $T$  is the horizon length,  $\mathcal{X}$  is the state space,  $\mathcal{U}$  is the action space. Besides,  $\mathbb{T} \coloneqq \{\mathbb{T}_t\}_{t \in [T]}$  with  $\mathbb{T}_t : \mathcal{X} \times \mathcal{U} \to \Delta(\mathcal{X})$  denoting the transition function<sup>4</sup>,  $\eta \coloneqq \{\eta_t\}_{t \in [T]}$  with  $\eta_t : \mathcal{X} \times \mathcal{U} \to [0,1]$  is the normal reward function and  $\eta^{\mathrm{term}} : \mathcal{X} \times \mathcal{U} \to [0,1]$  denotes the additional terminal reward function. In this paper, without further specification, we will consider history-dependent non-stationary policies  $\Psi \coloneqq \{\psi \coloneqq \{\psi_1, ..., \psi_T\} | \forall t \in [T], \psi_t : (\mathcal{X} \times \mathcal{U})^{t-1} \times \mathcal{X} \to \Delta(\mathcal{U})\}$ . Given a  $\psi \in \Psi$ , an episode of  $M$  is generated by:  $\forall t \in [T]$ ,  $\boldsymbol{u}_t \sim \psi_t(\cdot | \{x_{t'}, \boldsymbol{u}_{t'}\}_{t'=1}^{t-1}, x_t)$ ,  $\eta_t \gets \eta_t(x_t, \boldsymbol{u}_t)$ ,  $x_{t+1} \sim \mathbb{T}_t(\cdot | x_t, \boldsymbol{u}_t)$ ;  $\eta^{\mathrm{term}} \gets \eta^{\mathrm{term}}(x_{T+1})$ ;
+
+# B.4 ALGORITHM FOR LEARNING OPTIMAL (HISTORY-INDEPENDENT) STRATEGY WHEN  $f^{*}$  IS KNOWN
+
+Algorithm 3: Learning with Known Steering Dynamics  
+1 Input: Model Set  $\mathcal{F} \coloneqq \{f^{*}\}$ ; Initial steering strategy  $\psi^1 \coloneqq \{\psi_t^1\}_{t \in [T]}$ ; Regularization coefficient  $\beta$ ; Iteration number  $K$ ;  
+2 for  $k = 1, 2, \ldots, K$  do  
+3 Agents initialize with policy  $\pi_1^k$ .  
+4 Sample trajectories with  $\psi_{\zeta_k}, \forall t \in [T]$ :  
+    -  $\pmb{u}_t^k \sim \psi_t^k (\cdot |\pi_t^k)$ ,  $\pi_{t+1}^k \sim f^*(\cdot|\pi_t^k, r + \pmb{u}_t^k)$ ,  $\eta_t^k = -\eta^{\mathrm{cost}}(\pi_t^k, \pmb{u}_t^k)$ .  
+5 Update  $\psi^{k+1} \gets \text{RLAlgorithm}(\psi^k, \{\pi_t^k, \pmb{u}_t^k, \eta_t^k\}_{t=1}^T) \cup \{\beta \cdot \eta^{\mathrm{goal}}(\pi_{T+1}^k)\})$ .  
+6 end  
+7 Output  $\widehat{\psi}^* \gets \psi_{\zeta_K}$ .
+
+# B.5 ALGORITHM FOR LEARNING BELIEF-STATE DEPENDENT STEERING STRATEGY
+
+# C MISSING PROOFS IN SECTION 3
+
+Proposition 3.3. [Justification for Obj. (1)] By solving Obj. (1): (1)  $\psi^{*}$  is Pareto Optimal; (2) Given any  $\varepsilon, \varepsilon' > 0$ , if  $\Psi^{\varepsilon / |\mathcal{F}|} \neq \emptyset$  and  $\beta \geqslant \frac{U_{\max NHT}|\mathcal{F}|}{\varepsilon'}$ , we have  $\psi^{*} \in \Psi^{\varepsilon + \varepsilon'}$ ;
+
+Algorithm 4: Solving Obj. (1) by Learning Belief State-Dependent Strategy  
+1 Input: Model Set  $\mathcal{F}$ ; Regularization coefficient  $\beta$ ; Initial steering strategy  $\psi^1 \coloneqq \{\psi_t^1\}_{t=1}^T$ ; Iteration number  $K$ ;  
+2 for  $k = 1,2,\ldots,K$  do  
+3 Sample  $f \sim \mathrm{Uniform}(\mathcal{F})$ ; Initialize  $\pi_1^k = \pi_1$ .  
+4 Sample trajectories with  $\psi^k$  from simulator of  $f$ :  
+5  $\forall t \in [T] \quad b_t^k \coloneqq \operatorname*{Pr}(\cdot|\pi_1^k, \boldsymbol{u}_1^k, \dots, \pi_{t-1}^k, \boldsymbol{u}_{t-1}^k, \pi_t^k), \quad \boldsymbol{u}_t^k \sim \psi_t^k(\cdot|b_t^k, \pi_t^k)$ ,  
+6  $\pi_{t+1}^k \sim f(\cdot|\pi_t^k, \boldsymbol{r} + \boldsymbol{u}_t^k), \quad \eta_t^k \gets -\eta^{\mathrm{cost}}(\pi_t^k, \boldsymbol{u}_t^k)$   
+7 Update  $\psi^{k+1} \gets \text{RLAlgorithm}(\psi^k, (\pi_t^k, b_t^k), \boldsymbol{u}_t^k, \eta_t^k)_{t=1}^T \cup \{\beta \cdot \eta^{\mathrm{goal}}(\pi_{T+1}^k)\}$ .  
+8 end  
+9 return  $\widehat{\psi}^* \coloneqq \psi^K = \{\psi_t^K\}_{t=1}^T$
+
+Proof. Suppose  $\Psi^{\varepsilon /|\mathcal{F}|}$  is non-empty, we denote  $\psi^{\varepsilon /|\mathcal{F}|}$  as one of the elements in  $\Psi^{\varepsilon /|\mathcal{F}|}$ . By definition, since  $\max_{\pi} \eta^{\mathrm{goal}}(\pi)$  is fixed, we have:
+
+$$
+\psi^ {*} \leftarrow \arg \max  _ {\psi} \frac {1}{| \mathcal {F} |} \sum_ {f \in \mathcal {F}} - \beta \Delta (\psi , f, T) - C (\psi , f, T).
+$$
+
+If  $\beta \geqslant \frac{U_{\mathrm{max}}NHT|\mathcal{F}|}{\varepsilon'}$ , by definition,
+
+$$
+\begin{array}{l} 0 \leqslant \left(\frac {1}{| \mathcal {F} |} \sum_ {f \in \mathcal {F}} - \beta \Delta (\psi^ {*}, f, T) - C (\psi^ {*}, f, T)\right) - \left(\frac {1}{| \mathcal {F} |} \sum_ {f \in \mathcal {F}} - \beta \Delta (\psi^ {\varepsilon / | \mathcal {F} |}, f, T) - C (\psi^ {\varepsilon / | \mathcal {F} |}, f, T)\right) \\ \leqslant \frac {1}{| \mathcal {F} |} \sum_ {f \in \mathcal {F}} \beta \left(\Delta \left(\psi^ {\varepsilon / | \mathcal {F} |}, f, T\right) - \Delta \left(\psi^ {*}, f, T\right)\right) + U _ {\max } N H T \\ \end{array}
+$$
+
+(the steering reward  $u\in [0,U_{\mathrm{max}}]$ )
+
+$$
+\begin{array}{l} \leqslant \frac {1}{| \mathcal {F} |} \sum_ {f \in \mathcal {F}} \beta \left(\frac {\varepsilon}{| \mathcal {F} |} - \Delta (\psi^ {*}, f, T)\right) + U _ {\max } N H T \quad \left(\psi^ {\varepsilon / | \mathcal {F} |} \in \Psi_ {T, U _ {\max }} ^ {\varepsilon} (\mathcal {F})\right) \\ \leqslant \frac {U _ {\max } N H T}{\varepsilon^ {\prime}} \left(\frac {\varepsilon}{| \mathcal {F} |} - \frac {1}{| \mathcal {F} |} \sum_ {f \in \mathcal {F}} \Delta (\psi^ {*}, f, T)\right) + U _ {\max } N H T \\ \end{array}
+$$
+
+As a direct observation, if  $\mathbb{E}_{f\sim \mathrm{Unif}(\mathcal{F})}[\Delta (\psi^{*},f,T)] = \frac{1}{|\mathcal{F}|}\sum_{f\in \mathcal{F}}\Delta (\psi^{*},f,T) > \frac{\varepsilon + \varepsilon^{\prime}}{|\mathcal{F}|}$ , the RHS will be strictly less than 0, which results in contradiction. Therefore, we must have
+
+$$
+\forall f \in \mathcal {F}, \quad \Delta (\psi^ {*}, f, T) \leqslant | \mathcal {F} | \cdot \mathbb {E} _ {f \sim \operatorname {U n i f} (\mathcal {F})} [ \Delta (\psi^ {*}, f, T) ] \leqslant \varepsilon + \varepsilon^ {\prime}.
+$$
+
+which implies  $\psi^{*}\in \Psi^{\varepsilon +\varepsilon^{\prime}}$
+
+Next, we show the Pareto Optimality. If there exists  $\psi$  and  $f$  such that
+
+- For all  $f' \in \mathcal{F}$  with  $f \neq f'$ ,  $C(\psi^*, f, T) \geqslant C(\psi, f, T)$  and  $\Delta(\psi^*, f, T) \geqslant \Delta(\psi, f, T)$ ;  
+- For  $f$ , either  $C(\psi^{*}, f, T) > C(\psi, f, T)$  and  $\Delta(\psi^{*}, f, T) \geqslant \Delta(\psi, f, T)$  or  $C(\psi^{*}, f, T) \geqslant C(\psi, f, T)$  and  $\Delta(\psi^{*}, f, T) > \Delta(\psi, f, T)$ .
+
+Therefore, we must have:
+
+$$
+\frac {1}{| \mathcal {F} |} \sum_ {f \in \mathcal {F}} \beta \Delta (\psi , f, T) - C (\psi , f, T) <   \frac {1}{| \mathcal {F} |} \sum_ {f \in \mathcal {F}} \beta \Delta (\psi^ {*}, f, T) - C (\psi^ {*}, f, T),
+$$
+
+which conflicts with the optimality condition of Obj. (1).
+
+Remark C.1. Intuitively, Pareto optimality ensures the steering rewards predicted by  $\psi^{*}$  are not necessarily large. For example, consider agents following NPG dynamics (Def. 4.1). Given any steering strategy  $\psi$  and a constant  $c > 0$ , consider another steering strategy  $\psi'(\cdot) \coloneqq \psi(\cdot) + c$ , i.e., by increasing the steering reward predicted by  $\psi$  with  $c$ . Note that the learning dynamics under both  $\psi$  and  $\psi'$  are the same, and therefore the final steering gaps are the same. Pareto optimality implies that  $\psi'$  will never be more preferable than  $\psi$ .
+
+# D MISSING PROOFS FOR EXISTENCE WHEN THE TRUE MODEL  $f^{*}$  IS KNOWN
+
+In this section, we study the Policy Mirror Descent as a concrete example. In Appx. D.1, we provide more details about PMD. Then, we study the PMD with exact updates and stochastic updates in Appx. D.2.1 and D.2.2, respectively. The theorems in Sec. 4.1 will be subsumed as special cases.
+
+# D.1 MORE DETAILS ABOUT POLICY MIRROR DESCENT
+
+Definition D.1 (Policy Mirror Descent). For each agent  $n \in [N]$ , the updates at step  $t \in [T]$  follows:
+
+$$
+\begin{array}{l} \forall h \in [ H ], s _ {h} \in \mathcal {S}, \quad \theta_ {t + 1, h} ^ {n} (\cdot | s _ {h}) \leftarrow \theta_ {t, h} ^ {n} (\cdot | s _ {h}) + \alpha \widehat {A} _ {h} ^ {n, \pi_ {t}} (s, \cdot), \quad (U p d a t e \text {i n t h e m i r r o r s p a c e}) \\ z _ {t + 1, h} ^ {n} (\cdot | s _ {h}) \leftarrow (\nabla \phi^ {n}) ^ {- 1} \left(\theta_ {t + 1, h} ^ {n} (\cdot | s _ {h})\right) \quad (\text {M a p} \theta \text {b a c k t o t h e p r i m a l s p a c e}) \\ \pi_ {t + 1, h} ^ {n} (\cdot | s _ {h}) \leftarrow \underset {z \in \Delta (\mathcal {A} ^ {n})} {\arg \min } D _ {\phi^ {n}} (z, z _ {t + 1, h} ^ {n} (\cdot | s _ {h})), \quad \text {(P r o j e c t i o n)} \\ \end{array}
+$$
+
+Similar to Def. 4.1, here  $\widehat{A}_{h|r^n +u_t^n}^n$  is some random estimation for the advantage value  $A_{h|r^n +u_t^n}^n$  with  $\mathbb{E}_{\pi^n}[\widehat{A}_{h|r^n +u_t^n}^n (s_h,\cdot)] = 0$ . Besides,  $\theta_{t,h}^{n}\in \mathbb{R}^{|S||A|}$  denotes the variable in the dual space.  $\phi^n:\mathrm{dom}(\phi^n)\to \mathbb{R}$  is a function satisfying Assump. C below, which gives the mirror map  $\nabla \phi^n$ ;  $(\nabla \phi^n)^{-1}$  is the inverse mirror map;  $D_{\phi^n}(z,\widetilde{z}):= \phi^n (z) - \phi^n (\widetilde{z}) - \langle \nabla \phi^n (\widetilde{z}),z - \widetilde{z}\rangle$  is the Bregman divergence regarding  $\phi^n$ .
+
+Assumption C. We assume for all  $n \in [N]$ ,  $\phi^n$  is  $\mu$ -strongly convex and essentially smooth, i.e. differentiable and  $\| \nabla \phi^n(z^k) \| \to +\infty$  for any sequence  $z^k \in \mathrm{dom}(\phi^n)$  converging to a point on the boundary of  $\mathrm{dom}(\phi^n)$ .
+
+By Pythagorean Theorem and the strictly convexity of  $D_{\phi^n}$ , the projection  $\pi$  in Def. D.1 is unique.
+
+Lemma D.2. Given a convex set  $\mathcal{C}$  and a function  $\phi$  which is  $\mu$ -strongly convex on  $\mathcal{C}$ , we have
+
+$$
+\| \arg \min  _ {z \in \mathcal {C}} D _ {\phi} (z, (\nabla \phi) ^ {- 1} (\theta_ {1})) - \arg \min  _ {z \in \mathcal {C}} D _ {\phi} (z, (\nabla \phi) ^ {- 1} (\theta_ {2})) \| \leqslant \frac {1}{\mu} \| \theta_ {1} - \theta_ {2} \| _ {2}.
+$$
+
+Proof. Given any dual variables  $\theta_{1}$  and  $\theta_{2}$ , and their projection  $z_{1} := \arg \min_{z \in \mathcal{C}} D_{\phi}(z, (\nabla \phi)^{-1}(\theta_{1}))$  and  $z_{2} := \arg \min_{z \in \mathcal{C}} D_{\phi}(z, (\nabla \phi)^{-1}(\theta_{2}))$ , by the first order optimality condition, we have:
+
+$$
+\forall z \in \mathcal {C}, \quad \left\langle \nabla \phi \left(z _ {1}\right) - \theta_ {1}, z - z _ {1} \right\rangle \geqslant 0,
+$$
+
+$$
+\left\langle \nabla \phi \left(z _ {2}\right) - \theta_ {2}, z - z _ {2} \right\rangle \geqslant 0
+$$
+
+If we choose  $z = z_{2}$  in the first equation and  $z = z_{1}$  in the second equation, and sum together, we have:
+
+$$
+\left\langle \theta_ {1} - \nabla \phi (z _ {1}) + \nabla \phi (z _ {2}) - \theta_ {2}, z _ {1} - z _ {2} \right\rangle \geqslant 0,
+$$
+
+By strongly convexity of  $\phi$ , the above implies:
+
+$$
+\left\langle \theta_ {1} - \theta_ {2}, z _ {1} - z _ {2} \right\rangle \geqslant \left\langle \nabla \phi (z _ {1}) - \nabla \phi (z _ {2}), z _ {1} - z _ {2} \right\rangle \geqslant \mu \cdot \| z _ {1} - z _ {2} \| ^ {2}
+$$
+
+Therefore,
+
+$$
+\mu \| z _ {1} - z _ {2} \| \leqslant \| \theta_ {1} - \theta_ {2} \|,
+$$
+
+and we finish the proof.
+
+Next, we discuss some concrete examples.
+
+Example D.3 (Natural Policy Gradient). If we consider the mirror map and Bregman Divergence generated by  $\phi^n(z) \coloneqq \sum_{a^n \in \mathcal{A}^n} z(a^n) \log z(a^n)$ , we have  $D_\phi^n(z_1, z_2) = \mathrm{KL}(z_1 \| z_2)$ , and recover the NPG in Def. 4.1. Note that  $\phi^n$  is 1-strongly convex on the convex set  $\Delta(\mathcal{A}^n)$ , Assump. C is satisfied with  $\mu = 1$ .
+
+Example D.4 (Online Gradient Ascent (Zinkevich, 2003)). If we consider the Euclidean distance generated by  $l_{2}$ -norm  $\phi^n(z) = \frac{1}{2} \|z\|_2^2$ , we recover the projected gradient ascent
+
+Definition D.5. For each agent  $n \in [N]$ , the updates at step  $t \in [T]$  follows:
+
+$$
+\forall h \in [ H ], s _ {h} \in \mathcal {S}, \pi_ {t + 1, h} ^ {n} (\cdot | s _ {h}) \gets \mathrm {P r o j} _ {\Delta (\mathcal {A} ^ {n})} (\pi_ {t, h} ^ {n} (\cdot | s _ {h}) + \alpha \widehat {A} _ {h | r ^ {n} + u _ {t} ^ {n}} ^ {n, \pi_ {t}} (s, \cdot)),
+$$
+
+Note that the projection with Euclidean distance is 1-Lipschitz, Assump.  $C$  is satisfied with  $\mu = 1$ .
+
+Other Notations and Remarks In the following, we use  $\Pi^{+}$  to be the "feasible policy set" (for NPG in Def. 4.1,  $\Pi^{+}$  refers to be set of policies bounded away from 0), such that for any  $\pi \in \Pi^{+}$ , there exists a dual variable  $\theta$  corresponding to  $\pi$ , i.e.,
+
+$$
+\forall n \in [ N ], h \in [ H ], s _ {h} \in \mathcal {S}, \quad \pi_ {h} ^ {n} (\cdot | s _ {h}) \leftarrow \operatorname * {a r g   m i n} _ {z \in \Delta (\mathcal {A} ^ {n})} D _ {\phi^ {n}} (z, (\nabla \phi^ {n}) ^ {- 1} (\theta_ {h} ^ {n} (\cdot | s _ {h}))).
+$$
+
+In the following Lem. D.6, we show that constant shift in  $\theta_{t,h}^n (\cdot |s_h)$  does not change the projection result. Therefore, when we say the dual variable  $\theta$  associated with a given policy  $\pi$ , we only consider those  $\theta$  satisfying  $\mathbb{E}_{a_h^n\sim \pi_h^n}[\theta_h^n (a_h^n |s_h)] = 0$ .
+
+Lemma D.6 (Constant Shift does not Change the Projection). For any  $n \in [N]$ , regularizer  $\phi^n$  satisfying conditions in Assump.  $C$ , and any  $\theta \in \mathbb{R}^{|\mathcal{A}^n|}$ , consider the constant vector  $c\mathbf{I}$ , where  $c \in \mathbb{R}$  is a constant and  $\mathbf{I} = \{1, 1, \dots, 1\} \in \mathbb{R}^{|\mathcal{A}^n|}$ , we have:
+
+$$
+\operatorname *{arg  min}_{z\in \Delta (\mathcal{A}^{n})}D_{\phi^{n}}(z,(\nabla \phi^{n})^{-1}(\theta)) = \operatorname *{arg  min}_{z\in \Delta (\mathcal{A}^{n})}D_{\phi^{n}}(z,(\nabla \phi^{n})^{-1}(\theta +c\boldsymbol {I}))
+$$
+
+Proof.
+
+$$
+\begin{array}{l} \arg \min  _ {z \in \Delta (\mathcal {A} ^ {n})} D _ {\phi^ {n}} (z, (\nabla \phi^ {n}) ^ {- 1} (\theta + c \mathbf {1})) \\ = \arg \min  _ {z \in \Delta (\mathcal {A} ^ {n})} \phi^ {n} (z) - \langle \theta + c \mathbf {1}, z \rangle \\ = \arg \min  _ {z \in \Delta (\mathcal {A} ^ {n})} \phi^ {n} (z) - \langle \theta , z \rangle + c \quad \text {(w e h a v e c o n s t r a i n t s t h a t} z \in \Delta (\mathcal {A} ^ {n})) \\ = \arg \min  _ {z \in \Delta (\mathcal {A} ^ {n})} \phi^ {n} (z) - \langle \theta , z \rangle \\ = \operatorname *{arg  min}_{z\in \Delta (\mathcal{A}^{n})}D_{\phi^{n}}(z,\theta). \\ \end{array}
+$$
+
+![](images/41836143c0a6e68adf457e545d222002182a03acf24c27f5dbe23f049d3fe728.jpg)
+
+# D.2 PROOFS FOR THE EXISTENCE OF DESIRED STEERING STRATEGY
+
+We first formally introduce the Lipschitz condition that Thm. 4.2 requires.
+
+Assumption D ( $\eta^{\mathrm{goal}}$  is  $L$ -Lipschitz). For any  $\pi, \pi' \in \Pi$ ,  $|\eta^{\mathrm{goal}}(\pi) - \eta^{\mathrm{goal}}(\pi')| \leqslant L\|\pi - \pi'\|_2$ .
+
+In the following, in Appx. D.2.1, as a warm-up, we start with the exact case when the estimation  $\hat{A}^{\pi}$  is exactly the true advantage value  $A^{\pi}$  (which can be regarded as a special case of Assump. B). Then, in Appx. D.2.2, we study the general setting and prove Thm. 4.2 as a special case of PMD.
+
+# D.2.1 SPECIAL CASE: PMD WITH EXACT ADVANTAGE-VALUE
+
+Lemma D.7 (Existence of Steering Path between Feasible Policies). Consider two feasible policies  $\pi, \widetilde{\pi}$  which are induced by dual variables  $\{\theta_{1,h}^n\}_{h \in [H], n \in [N]}$  and  $\{\widetilde{\theta}_h^n\}_{h \in [H], n \in [N]}$ , respectively. If the agents follow Def. D.1 with exact  $Q$  value and start with  $\pi_1 = \pi$ , as long as  $U_{\max} \geqslant 2H + \frac{2}{\alpha T} (\max_{n,h,s_h,a_h^n} |\widetilde{\theta}_h^n(a_h^n|s_h) - \theta_h^n(a_h^n|s_h) - \mathbb{E}_{\bar{a}_h^n \sim \pi_{t,h}^n(|s_h^n|)} [\widetilde{\theta}_h^n(\bar{a}_h^n|s_h) - \theta_h^n(\tilde{a}_h^n|s_h)]|)$ , there exists a (history-independent) steering strategy  $\psi := \{\psi_t\}_{t \in [T]}$  with  $\psi_t: \Pi^+ \to \mathcal{U}$ , s.t.,  $\pi_{T+1} = \widetilde{\pi}$ .
+
+Proof. For agent  $n \in [N]$ , given a  $\pi_t$ , we consider the following steering reward functions
+
+$$
+u _ {t, h} ^ {n} (s _ {h}, a _ {h} ^ {n}) = \nu_ {t, h} ^ {n} (s _ {h}, a _ {h} ^ {n}) - A _ {h | r ^ {n}} ^ {n, \pmb {\pi} _ {t}} (s _ {h}, a _ {h} ^ {n}) - \mathbb {E} _ {\tilde {a} ^ {n} \sim \pi_ {t, h} ^ {n} (\cdot | s _ {h})} [ \nu_ {t, h} ^ {n} (s _ {h}, \tilde {a} _ {h} ^ {n}) - A _ {h | r ^ {n}} ^ {n, \pmb {\pi} _ {t}} (s _ {h}, \tilde {a} _ {h} ^ {n}) ]
+$$
+
+$$
+- \min _ {\bar {s} _ {h}, \bar {a} _ {h} ^ {n}} \{\nu_ {t, h} ^ {n} (\bar {s} _ {h}, \bar {a} _ {h} ^ {n}) - A _ {h | r ^ {n}} ^ {n, \pi_ {t}} (\bar {s} _ {h}, \bar {a} _ {h} ^ {n}) - \mathbb {E} _ {\tilde {a} ^ {n} \sim \pi_ {t, h} ^ {n} (\cdot | s _ {h})} [ \nu_ {t, h} ^ {n} (\bar {s} _ {h}, \tilde {a} _ {h} ^ {n}) - A _ {h | r ^ {n}} ^ {n, \pi_ {t}} (\bar {s} _ {h}, \tilde {a} _ {h} ^ {n}) ] \},
+$$
+
+where  $\nu_{t,h}^n:\mathcal{S}\times \mathcal{A}^n\to \mathbb{R}$  will be defined later. By construction, we have:
+
+$$
+\mathbb {E} _ {a _ {h} ^ {n} \sim \pi_ {t, h} ^ {n} (\cdot | s _ {h})} \left[ u _ {t, h} ^ {n} \left(s _ {h}, a _ {h} ^ {n}\right) \right] \tag {2}
+$$
+
+$$
+= - \min  _ {\bar {s} _ {h}, \bar {a} _ {h} ^ {n}} \left\{\nu_ {t, h} ^ {n} \left(\bar {s} _ {h}, \bar {a} _ {h} ^ {n}\right) - A _ {h | r ^ {n}} ^ {n, \pi_ {t}} \left(\bar {s} _ {h}, \bar {a} _ {h} ^ {n}\right) - \mathbb {E} _ {\tilde {a} ^ {n} \sim \pi_ {t, h} ^ {n} (\cdot | s _ {h})} \left[ \nu_ {t, h} ^ {n} \left(\bar {s} _ {h}, \tilde {a} _ {h} ^ {n}\right) - A _ {h | r ^ {n}} ^ {n, \pi_ {t}} \left(\bar {s} _ {h}, \tilde {a} _ {h} ^ {n}\right) \right] \right\}, \tag {3}
+$$
+
+which is a constant and independent w.r.t.  $s_h, a_h^n$ . Besides, by definition, we can ensure the nonnegativity of  $u_{t,h}^n$ . As a result,
+
+$$
+\forall t \in [ T ], Q _ {h | r ^ {n} + u _ {t} ^ {n}} ^ {t, \pmb {\pi} _ {t}} (s _ {h}, a _ {h} ^ {n})
+$$
+
+$$
+= A _ {h \mid r ^ {n}} ^ {t, \pi_ {t}} \left(s _ {h}, a _ {h} ^ {n}\right) + u _ {t, h} ^ {n} \left(s _ {h}, a _ {h} ^ {n}\right) + C _ {h} \left(s _ {h}\right) \tag {Eq. (3)}
+$$
+
+$$
+= \nu_ {t, h} ^ {n} \left(s _ {h}, a _ {h} ^ {n}\right) + C _ {h} ^ {\prime} \left(s _ {h}\right). \tag {4}
+$$
+
+where we use  $C_h(s_h)$  and  $C_h'(s_h)$  to denote some state-dependent but action-independent value. According to Lem. D.6, under the above steering reward design, the dynamics of  $\pi_1, \dots, \pi_t, \dots, \pi_T$  can be described by the following dynamics:
+
+$$
+\forall t \in [ T ], \forall n \in [ N ], h \in [ H ], s _ {h} \in \mathcal {S}: \theta_ {t + 1, h} ^ {n} (\cdot | s _ {h}) \leftarrow \theta_ {t, h} ^ {n} (\cdot | s _ {h}) + \alpha \nu_ {t, h} ^ {n} (s _ {h}, a _ {h} ^ {n}) \tag {5}
+$$
+
+$$
+\pi_ {t + 1, h} ^ {n} (\cdot | s _ {h}) \leftarrow \underset {z \in \Delta (\mathcal {A} ^ {n})} {\arg \min } D _ {\phi^ {n}} (z, \theta_ {t + 1, h} ^ {n} (\cdot | s _ {h})), \quad (6)
+$$
+
+Now we consider the following choice of  $\nu_{t,h}^n$ :
+
+$$
+\nu_ {t, h} ^ {n} (s _ {h}, a _ {h} ^ {n}) = \frac {\widetilde {\theta} _ {h} ^ {n} (a _ {h} ^ {n} | s _ {h}) - \theta_ {h} ^ {n} (a _ {h} ^ {n} | s _ {h})}{\alpha T},
+$$
+
+which implies  $\theta_{T + 1} = \widetilde{\theta}$ , and therefore,  $\pi_{T + 1} = \widetilde{\pi}$ . Besides, the steering reward function can be upper bounded by:
+
+$$
+\begin{array}{l} u _ {t, h} ^ {n} \left(s _ {h}, a _ {h} ^ {n}\right) \leqslant 2 \max  _ {\bar {s} _ {h}, \bar {a} _ {h} ^ {n}} \left| \nu_ {t, h} ^ {n} \left(\bar {s} _ {h}, \bar {a} _ {h} ^ {n}\right) - A _ {h | r ^ {n}} ^ {n, \pi_ {t}} \left(\bar {s} _ {h}, \bar {a} _ {h} ^ {n}\right) - \mathbb {E} _ {\tilde {a} ^ {n} \sim \pi_ {t, h} ^ {n} \left(\cdot | s _ {h}\right)} \left[ \nu_ {t, h} ^ {n} \left(\bar {s} _ {h}, \tilde {a} _ {h} ^ {n}\right) - A _ {h | r ^ {n}} ^ {n, \pi_ {t}} \left(\bar {s} _ {h}, \tilde {a} _ {h} ^ {n}\right) \right] \right| \\ \leqslant 2 H + \frac {2}{\alpha T} \big (\max _ {n, h, s _ {h}, a _ {h} ^ {n}} | \widetilde {\theta} _ {h} ^ {n} (a _ {h} ^ {n} | s _ {h}) - \theta_ {h} ^ {n} (a _ {h} ^ {n} | s _ {h}) | \big), \\ \end{array}
+$$
+
+which implies the appropriate choice of  $U_{\mathrm{max}}$ .
+
+Theorem D.8. Under Assump.  $D$ , given the initial  $\pi_1 \coloneqq \pi \in \Pi^+$ , for any  $T \geqslant 1$  and  $\varepsilon > 0$ , if the agents follow Def. 4.1 with exact  $Q$  value, then  $\Psi_{T,U_{\max}}^{\varepsilon} \neq \emptyset$  if the following conditions are satisfied:
+
+- There exists feasible  $\widetilde{\pi} \in \Pi^{+}$  such that  $\eta^{goal}(\widetilde{\pi}) \geqslant \max_{\pi} \eta^{goal}(\pi) - \varepsilon$  
+- Denote  $\theta$  and  $\widetilde{\theta}$  as the dual variables associated with  $\pi$  and  $\widetilde{\pi}$ , respectively. We require  $U_{\max} \geqslant 2H + \frac{2}{\alpha T} (\max_{n,h,s_h,a_h^n} |\widetilde{\theta}_h^n(a_h^n | s_h) - \theta_h^n(a_h^n | s_h)|)$
+
+Proof. The proof is a directly application of Lem. D.7.
+
+NPG as a Special Case For NPG, we have the following results.
+
+Lemma D.9. Given  $\forall \pi, \widetilde{\pi} \in \Pi^{+}$ ,  $T \geqslant 1$ , if the agents follow Def. 4.1 with exact adv-value and start from  $\pi_{1} = \pi$ , by choosing  $U_{\mathrm{max}}$  appropriately, there exists a (history-independent) steering strategy  $\psi := \{\psi_t\}_{t \in [T]}$  with  $\psi_t: \Pi^+ \to \mathcal{U}$ , s.t.,  $\pi_{T + 1} = \widetilde{\pi}$ .
+
+Theorem D.10. Under Assump.  $D$ , given any initial  $\pi_1 \in \Pi^+$ , for any  $T \geqslant 1$  and  $\varepsilon > 0$ , if the agents follow Def. 4.1 with exact  $Q$  value, by choosing  $U_{\mathrm{max}}$  appropriately, we have  $\Psi^{\varepsilon} \neq \emptyset$ .
+
+Proof for Lem. D.9 and Thm. D.10 The proof is by directly applying Lem. D.7 and Thm. D.8 since NPG is a special case of PMD with KL-Divergence as Bregman Divergence. For any  $\pi, \widetilde{\pi} \in \Pi^{+}$ , we consider the dual variables  $\theta, \widetilde{\theta}$  such that:
+
+$$
+\theta_ {h} ^ {n} (\cdot | s _ {h}) = \log \pi_ {h} ^ {n} (\cdot | s _ {h}) - \mathbb {E} _ {a _ {h} ^ {n} \sim \pi_ {h} ^ {n}} [ \log \pi_ {h} ^ {n} (a _ {h} ^ {n} | s _ {h}) ], \quad \widetilde {\theta} _ {h} ^ {n} (\cdot | s _ {h}) = \log \widetilde {\pi} _ {h} ^ {n} (\cdot | s _ {h}) - \mathbb {E} _ {a _ {h} ^ {n} \sim \pi_ {h} ^ {n}} [ \log \widetilde {\pi} _ {h} ^ {n} (a _ {h} ^ {n} | s _ {h}) ]. \tag {7}
+$$
+
+Choice of  $U_{\mathrm{max}}$  in Lem. D.9 By applying Lem. D.7 and Thm. D.8, we consider the following choice of  $U_{\mathrm{max}}$
+
+$$
+U _ {\max } \geqslant 2 H + \frac {2}{\alpha T} \left(\max  _ {n, h, s _ {h}, a _ {h}} \left| \log \frac {\widetilde {\pi} _ {h} ^ {n} \left(s _ {h} , a _ {h} ^ {n}\right)}{\pi_ {h} ^ {n} \left(s _ {h} , a _ {h} ^ {n}\right)} - \mathbb {E} _ {\widetilde {a} _ {h} ^ {n} \sim \pi_ {h} ^ {n}} \left[ \log \frac {\widetilde {\pi} _ {h} ^ {n} \left(s _ {h} , \widetilde {a} _ {h} ^ {n}\right)}{\pi_ {h} ^ {n} \left(s _ {h} , \widetilde {a} _ {h} ^ {n}\right)} \right] \right|\right). \tag {8}
+$$
+
+Choice of  $U_{\mathrm{max}}$  in Thm. D.10 We denote  $\pi^{*}\in \arg \max_{\pi \in \Pi}\eta^{\mathrm{goal}}(\pi)\notin \Pi^{+}$
+
+When  $\pi^{*} \in \Pi^{+}$ , we can directly apply Thm. D.8 with  $\widetilde{\pi} \gets \pi^{*}$ , and choosing  $U_{\mathrm{max}}$  correspondingly following Eq. (8).
+
+However, in some cases,  $\pi^{*} \notin \Pi^{+}$  because it takes deterministic action in some states. In that case, since  $\eta^{\mathrm{goal}}$  is  $L$ -Lipschitz in  $\pi$ , we can consider the mixture policy  $\widetilde{\pi} := (1 - O(\frac{\varepsilon}{L}))\pi^{*} + O(\frac{\varepsilon}{L})\pi_{\mathrm{Uniform}}$ , where  $\pi_{\mathrm{Uniform}}$  is the uniform policy. As a result, we have  $\widetilde{\pi} \in \Pi^{+}$  as well as  $\eta^{\mathrm{goal}}(\widetilde{\pi}) \geqslant \max_{\pi \in \Pi} \eta^{\mathrm{goal}}(\pi) - \varepsilon$ . Then the  $U_{\mathrm{max}}$  can be chosen following Eq. (8).
+
+# D.2.2 THE GENERAL INCENTIVE DRIVEN AGENTS UNDER ASSUMP. B
+
+Theorem D.11 (Formal Version of Thm. 4.2 for the general PMD). Under Assump.  $D$  and Assump.  $C$ , given the initial  $\pi_1 \coloneqq \pi \in \Pi^+$ , for any  $\varepsilon > 0$ , if the agents follow Def. 4.1 under the Assump.  $B$ , then  $\Psi_{T,U_{\max}}^{\varepsilon} \neq \emptyset$  if the following conditions are satisfied:
+
+- There exists feasible  $\widetilde{\pi} \in \Pi^{+}$  such that  $\eta^{\text{goal}}(\widetilde{\pi}) \geqslant \max_{\pi} \eta^{\text{goal}}(\pi) - \frac{\varepsilon}{2}$ .  
+- Denote  $\theta$  and  $\widetilde{\theta}$  as the dual variables associated with  $\pi$  and  $\widetilde{\pi}$ , respectively. We require  $U_{\max} \geqslant 2(H + \frac{\lambda_{\min}}{\alpha \lambda_{\max}^2} (1 + \frac{\lambda_{\min}}{\lambda_{\max}})^T \| \widetilde{\theta} - \theta \|_2)$  and  $T = \Theta (\frac{\lambda_{\max}^2}{\lambda_{\min}^2} \log \frac{L \| \widetilde{\theta} - \theta \|_2}{\mu \varepsilon})$ .
+
+Remark D.12. In Thm. D.2.2, our bound for  $U_{\mathrm{max}}$  here is just a worst-case bound to handle the noisy updates in the worst case. With high probability, the dual variable  $\theta_t$  will converge to  $\widetilde{\theta}$  and the steering reward does not have to be as large as  $U_{\mathrm{max}}$ .
+
+Proof. Given a  $\pi_t$ , we consider the following steering reward  $u_t$ :
+
+$$
+\begin{array}{l} u _ {t, h} ^ {n} (s _ {h}, a _ {h} ^ {n}) = \nu_ {t, h} ^ {n} (s _ {h}, a _ {h} ^ {n}, \pmb {\pi} _ {t}) - A _ {h | r ^ {n}} ^ {n, \pmb {\pi} _ {t}} (s _ {h}, a _ {h} ^ {n}) - \mathbb {E} _ {\tilde {a} ^ {n} \sim \pi_ {t, h} ^ {n} (\cdot | s _ {h})} [ \nu_ {t, h} ^ {n} (s _ {h}, \tilde {a} _ {h} ^ {n}, \pmb {\pi} _ {t}) - A _ {h | r ^ {n}} ^ {n, \pmb {\pi} _ {t}} (s _ {h}, \tilde {a} _ {h} ^ {n}) ] \\ - \min _ {\bar {s} _ {h}, \bar {a} _ {h} ^ {n}} \{\nu_ {t, h} ^ {n} (\bar {s} _ {h}, \bar {a} _ {h} ^ {n}, \pmb {\pi} _ {t}) - A _ {h | r ^ {n}} ^ {n, \pmb {\pi} _ {t}} (\bar {s} _ {h}, \bar {a} _ {h} ^ {n}) - \mathbb {E} _ {\tilde {a} ^ {n} \sim \pi_ {t, h} ^ {n} (\cdot | s _ {h})} [ \nu_ {t, h} ^ {n} (\bar {s} _ {h}, \tilde {a} _ {h} ^ {n}, \pmb {\pi} _ {t}) - A _ {h | r ^ {n}} ^ {n, \pmb {\pi} _ {t}} (\bar {s} _ {h}, \tilde {a} _ {h} ^ {n}) ] \}, \\ \end{array}
+$$
+
+Here we choose  $\nu_{t,h}^n (s_h,a_h^n,\pi_t)\coloneqq \frac{1}{\gamma}\cdot (\widetilde{\theta}_h^n (a_h^n |s_h) - \theta_{t + 1,h}^n (a_h^n |s_h))$ , where  $\widetilde{\theta}$  denotes the dual variable of policy  $\widetilde{\pi}$  and  $\gamma$  will be determined later. Comparing with the design in the proof of Thm. D.8, here the "driven term"  $\nu_{h}^{n}$  need to depend on  $\pi_t$  because of the randomness in updates.
+
+As we can see,  $u_{t,h}^n (s_h,a_h^n)\geqslant 0$ , and for each  $t$ , we have:
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \| \widetilde {\theta} - \theta_ {t + 1} \| _ {2} ^ {2} \right] = \mathbb {E} \left[ \| \widetilde {\theta} - \theta_ {t} \| _ {2} ^ {2} \right] - 2 \mathbb {E} \left[ \langle \widetilde {\theta} - \theta_ {t}, \theta_ {t + 1} - \theta_ {t} \rangle \right] + \mathbb {E} \left[ \| \theta_ {t + 1} - \theta_ {t} \| _ {2} ^ {2} \right] \\ = \mathbb {E} \left[ \| \widetilde {\theta} - \theta_ {t} \| _ {2} ^ {2} \right] - 2 \alpha \mathbb {E} \left[ \langle \widetilde {\theta} - \theta_ {t}, \widehat {A} _ {| \boldsymbol {r} + \boldsymbol {u} _ {t}} ^ {\pi_ {t}} \rangle \right] + \mathbb {E} \left[ \| \widehat {A} _ {| \boldsymbol {r} + \boldsymbol {u} _ {t}} ^ {\pi_ {t}} \| _ {2} ^ {2} \right] \\ \leqslant \left(1 - 2 \lambda_ {\min } \frac {\alpha}{\gamma} + \lambda_ {\max } ^ {2} \frac {\alpha^ {2}}{\gamma^ {2}}\right) \mathbb {E} \left[ \| \widetilde {\theta} - \theta_ {t} \| _ {2} ^ {2} \right], \\ \end{array}
+$$
+
+which implies
+
+$$
+\mathbb {E} \left[ \| \widetilde {\theta} - \theta_ {T + 1} \| _ {2} ^ {2} \right] \leqslant \left(1 - 2 \lambda_ {\min } \frac {\alpha}{\gamma} + \lambda_ {\max } ^ {2} \frac {\alpha^ {2}}{\gamma^ {2}}\right) ^ {T} \| \widetilde {\theta} - \theta \| _ {2} ^ {2}.
+$$
+
+We consider the choice  $\gamma = \frac{\lambda_{\mathrm{max}}^2\alpha}{\lambda_{\mathrm{min}}}$ , which implies,
+
+$$
+\mathbb {E} \big [ \| \widetilde {\theta} - \theta_ {T + 1} \| _ {2} ^ {2} \big ] \leqslant \big (1 - \frac {\lambda_ {\mathrm {m i n}} ^ {2}}{\lambda_ {\mathrm {m a x}} ^ {2}} \big) ^ {T} \| \widetilde {\theta} - \theta \| _ {2} ^ {2}.
+$$
+
+When  $T = 2c_{0}\frac{\lambda_{\mathrm{max}}^{2}}{\lambda_{\mathrm{min}}^{2}}\log \frac{2L\|\widetilde{\theta} - \theta\|_{2}}{\mu\varepsilon} \geqslant c_{0}\log_{1 - \frac{\lambda_{\mathrm{min}}^{2}}{\lambda_{\mathrm{max}}^{2}}}(\frac{\nu^{2}\varepsilon^{2}}{2L^{2}\|\widetilde{\theta} - \theta\|_{2}^{2}})$  for some constant  $c_{0}$ , we have:
+
+$$
+\mathbb {E} \left[ \| \widetilde {\theta} - \theta_ {T + 1} \| _ {2} \right] \leqslant \frac {\mu \varepsilon}{2 L},
+$$
+
+which implies,
+
+$$
+\mathbb {E} \left[ \eta \left(\boldsymbol {\pi} ^ {*}\right) - \eta^ {\text {g o a l}} \left(\boldsymbol {\pi} _ {T + 1}\right) \right] \leqslant \frac {\varepsilon}{2} + L \mathbb {E} \left[ \| \widetilde {\boldsymbol {\pi}} - \boldsymbol {\pi} _ {T + 1} \| _ {2} \right] \leqslant \frac {\varepsilon}{2} + \frac {L}{\mu} \mathbb {E} \left[ \| \widetilde {\boldsymbol {\theta}} - \theta_ {T + 1} \| _ {2} \right] = \varepsilon .
+$$
+
+Next, we discuss the choice of  $U_{\mathrm{max}}$ , by Assump. B, we know,
+
+$$
+\begin{array}{l} \left\| \widetilde {\theta} - \theta_ {t + 1} \right\| _ {2} = \left\| \widetilde {\theta} - \theta_ {t} - \alpha \widehat {A} _ {| \boldsymbol {r} + \boldsymbol {u} _ {t}} ^ {\pi_ {t}} \right\| _ {2} \leqslant \left\| \widetilde {\theta} - \theta_ {t} \right\| _ {2} + \alpha \left\| \widehat {A} _ {| \boldsymbol {r} + \boldsymbol {u} _ {t}} ^ {\pi_ {t}} \right\| _ {2} \\ \leqslant \| \widetilde {\theta} - \theta_ {t} \| _ {2} + \alpha \lambda_ {\max } \| A _ {| \boldsymbol {r} + \boldsymbol {u} _ {t}} ^ {\pi_ {t}} \| _ {2} \\ \leqslant \left(1 + \frac {\lambda_ {\operatorname* {m i n}}}{\lambda_ {\operatorname* {m a x}}}\right) \| \widetilde {\theta} - \theta_ {t} \| _ {2} \\ \end{array}
+$$
+
+where we use the fact that  $\| A_{|\pmb{r} + \pmb{u}_{\tau}}^{\pmb{\pi}_{\tau}}\|_{2} = \frac{1}{\gamma}\|\widetilde{\theta} - \theta_{\tau}\|_{2}$  and our choice of  $\gamma$ . Therefore, for all  $t \in [T]$ ,  $\|\widetilde{\theta} - \theta_t\|_2 \leqslant (1 + \frac{\lambda_{\min}}{\lambda_{\max}})^T\|\widetilde{\theta} - \theta\|_2$ . To ensure our design of  $u_{t,h}^{n}$  is feasible, we need to set:
+
+$$
+\begin{array}{l} U _ {\max } = 2 \left(H + \frac {1}{\gamma} \left(1 + \frac {\lambda_ {\min }}{\lambda_ {\max }}\right) ^ {T} \| \widetilde {\theta} - \theta \| _ {2}\right) \\ = 2 \left(H + \frac {\lambda_ {\operatorname* {m i n}}}{\alpha \lambda_ {\operatorname* {m a x}} ^ {2}} \left(1 + \frac {\lambda_ {\operatorname* {m i n}}}{\lambda_ {\operatorname* {m a x}}}\right) ^ {T} \| \widetilde {\theta} - \theta \| _ {2}\right). \\ \end{array}
+$$
+
+![](images/2216ef8efd720b4bc0c27446444b4d498aa5ccb94b835f1ea84dd3fb1e35d355.jpg)
+
+Proof for Thm. 4.2 As we discuss in Example. D.3, Assump. C is satisfied with  $\mu = 1$ . The proof is a direct application of Thm. D.8 with the same choice of dual variables as Eq. (7).
+
+# E MISSING PROOFS FOR EXISTENCE WHEN THE TRUE MODEL  $f^{*}$  IS UNKNOWN
+
+In the following, we establish some technical lemmas for the maximal likelihood estimator. Given a steering dynamics model class  $\mathcal{F}$  and the true dynamics  $f^{*} \sim p_{0}$  and a steering strategy  $\psi: \Pi \to \mathcal{U}$ , we consider a steering trajectory  $\tau_{T_0} := \{\pmb{\pi}_1, \pmb{u}_1, \dots, \pmb{\pi}_{T_0}, \pmb{u}_{T_0}, \pmb{\pi}_{T_0 + 1}\}$  generated by:
+
+$$
+\forall t \in [ T _ {0} ], \quad \boldsymbol {u} _ {t} \leftarrow \psi (\boldsymbol {\pi} _ {t}), \boldsymbol {\pi} _ {t + 1} \sim f ^ {*} (\cdot | \boldsymbol {\pi} _ {t}, \boldsymbol {u} _ {t}), \tag {9}
+$$
+
+where  $\pi_{t + 1}$  is independent w.r.t.  $\pi_{t'}$  for  $t' < t$  conditioning on  $\pi_t$ . In the following, we will denote  $\tau_t := \{\pmb{\pi}_1, \pmb{u}_1, \dots, \pmb{\pi}_t, \pmb{u}_t, \pmb{\pi}_{t + 1}\}$  to be the trajectory up to step  $t$ .
+
+For any  $f\in \mathcal{F}$  , we define:
+
+$$
+p _ {f} \left(\tau_ {T _ {0}}\right) := \prod_ {t = 1} ^ {T _ {0}} f \left(\boldsymbol {\pi} _ {t + 1} \mid \boldsymbol {\pi} _ {t}, \boldsymbol {u} _ {t}\right). \tag {10}
+$$
+
+Given  $\tau_{T_0}$ , we use  $\bar{\tau}_{T_0}$  to denote the "tangent" trajectory  $\{(\pmb{\pi}_t, \pmb{u}_t, \bar{\pmb{\pi}}_{t+1})\}_{t=1}^{T_0}$  where  $\bar{\pi}_{t+1} \sim f^*(\cdot|\pmb{\pi}_t, \pmb{u}_t)$  is independently sampled from the same distribution as  $\pmb{\pi}_{t+1}$  conditioning on the same  $\pmb{\pi}_t$  and  $u_t$ .
+
+Lemma E.1. Let  $l: \Pi \times \mathcal{U} \times \Pi \to \mathbb{R}$  be a real-valued loss function. Define  $L(\tau_{T_0}) := \sum_{t=1}^{T_0} l(\pmb{\pi}_t, \pmb{u}_t, \pmb{\pi}_{t+1})$  and  $L(\bar{\tau}_{T_0}) := \sum_{t=1}^{T_0} l(\pmb{\pi}_t, \pmb{u}_t, \bar{\pmb{\pi}}_{t+1})$ . Then, for arbitrary  $t \in [T_0]$ ,
+
+$$
+\mathbb {E} \bigl [ \exp (L (\tau_ {t}) - \log \mathbb {E} _ {\bar {\tau} _ {T _ {0}}} \bigl [ \exp (L (\bar {\tau} _ {t})) | \tau_ {t} \bigr ]) \bigr ] = 1.
+$$
+
+Proof. We denote  $E^{i} \coloneqq \mathbb{E}_{\pi_{i + 1}}[\exp (l(\pi_{i},u_{i},\bar{\pi}_{i + 1}))|\pi_{i},u_{i},f^{*}]$ . By definition, we have:
+
+$$
+\mathbb {E} _ {\bar {\tau} _ {t}} \left[ \exp \left(\sum_ {i = 1} ^ {t} l \left(\boldsymbol {\pi} _ {i}, u _ {i}, \bar {\boldsymbol {\pi}} _ {i + 1}\right)\right) | \tau_ {t} \right] = \prod_ {i = 1} ^ {k} E ^ {i}.
+$$
+
+Therefore,
+
+$$
+\begin{array}{l} \mathbb {E} _ {\tau_ {T _ {0}}} \left[ \exp \left(L \left(\tau_ {T _ {0}}\right) - \log \mathbb {E} _ {\bar {\tau} _ {T _ {0}}} \left[ \exp \left(L \left(\bar {\tau} _ {T _ {0}}\right)\right) \mid \tau_ {T _ {0}} \right]\right) \right] \\ = \mathbb {E} _ {\tau_ {T _ {0} - 1} \cup \left\{\boldsymbol {\pi} _ {T _ {0}}, \boldsymbol {u} _ {T _ {0}} \right\}} \left[ \mathbb {E} _ {\boldsymbol {\pi} _ {T _ {0} + 1}} \left[ \frac {\exp \left(\sum_ {t = 1} ^ {T _ {0}} l \left(\boldsymbol {\pi} _ {t} , \boldsymbol {u} _ {t} , \boldsymbol {\pi} _ {t + 1}\right)\right)}{\mathbb {E} _ {\bar {\tau} _ {T _ {0}}} \left[ \exp \left(\sum_ {t = 1} ^ {T _ {0}} l \left(\boldsymbol {\pi} _ {t} , \boldsymbol {u} _ {t} , \boldsymbol {\pi} _ {t + 1}\right)\right) \mid \tau_ {T _ {0}} \right]} \mid \tau_ {T _ {0} - 1} \cup \left\{\boldsymbol {\pi} _ {T _ {0}}, \boldsymbol {u} _ {T _ {0}} \right\} \right] \right] \\ = \mathbb {E} _ {\tau_ {T _ {0} - 1} \cup \left\{\boldsymbol {\pi} _ {T _ {0}}, \boldsymbol {u} _ {T _ {0}} \right\}} \left[ \mathbb {E} _ {\boldsymbol {\pi} _ {T _ {0} + 1}} \left[ \frac {\exp \left(\sum_ {t = 1} ^ {T _ {0}} l \left(\boldsymbol {\pi} _ {t} , \boldsymbol {u} _ {t} , \boldsymbol {\pi} _ {t + 1}\right)\right)}{\prod_ {t = 1} ^ {T _ {0}} E ^ {t}} \mid \tau_ {T _ {0} - 1} \cup \left\{\boldsymbol {\pi} _ {T _ {0}}, \boldsymbol {u} _ {T _ {0}} \right\} \right] \right] \\ = \mathbb {E} _ {\tau_ {T _ {0} - 1} \cup \left\{\boldsymbol {\pi} _ {T _ {0}}, \boldsymbol {u} _ {T _ {0}} \right\}} \left[ \frac {\exp \left(\sum_ {t = 1} ^ {T _ {0} - 1} l \left(\boldsymbol {\pi} _ {t} , \boldsymbol {u} _ {t} , \boldsymbol {\pi} _ {t + 1}\right)\right)}{\prod_ {t = 1} ^ {T _ {0} - 1} E ^ {t}} \cdot \mathbb {E} _ {\boldsymbol {\pi} _ {T _ {0} + 1}} \left[ \frac {l \left(\boldsymbol {\pi} _ {T _ {0}} , \boldsymbol {u} _ {T _ {0}} , \boldsymbol {\pi} _ {T _ {0} + 1}\right)}{E ^ {T _ {0}}} \mid \tau_ {T _ {0} - 1} \cup \left\{\boldsymbol {\pi} _ {T _ {0}}, \boldsymbol {u} _ {T _ {0}} \right\} \right] \right] \\ = \mathbb {E} _ {\tau T _ {0} - 1} \left[ \frac {\exp \left(\sum_ {t = 1} ^ {T _ {0} - 1} l \left(\boldsymbol {\pi} _ {t} , \boldsymbol {u} _ {t} , \boldsymbol {\pi} _ {t + 1}\right)\right)}{\prod_ {t = 1} ^ {T _ {0} - 1} E ^ {t}} \right] = \dots = 1. \\ \end{array}
+$$
+
+![](images/4937aef0e7e080da3bf31f2715f8a368ca9d51929622d90455ad7d9940572bb0.jpg)
+
+Lemma E.2. [Property of the MLE Estimator] Under the condition in Prop. 4.4, given the true model  $f^{*}$  and any deterministic steering strategy  $\psi : \Pi \to \mathcal{U}$ , define  $f_{MLE} \gets \arg \max_{f \in \mathcal{F}} \sum_{t=1}^{T_0} \log f(\pi_{t+1} | \pi_t, \boldsymbol{u}_t)$ , where the trajectory is generated by:
+
+$$
+\forall t \in [ T _ {0} ], \quad \boldsymbol {u} _ {t} \leftarrow \psi (\boldsymbol {\pi} _ {t}), \boldsymbol {\pi} _ {t + 1} \sim f ^ {*} (\cdot | \boldsymbol {\pi} _ {t}, \boldsymbol {u} _ {t}),
+$$
+
+then, for any  $\delta \in (0,1)$ , w.p. at least  $1 - \delta$ , we have:
+
+$$
+\sum_ {t = 1} ^ {T _ {0}} \mathbb {H} ^ {2} \left(f _ {M L E} (\cdot | \boldsymbol {\pi} _ {t}, \boldsymbol {u} _ {t}), f ^ {*} (\cdot | \boldsymbol {\pi} _ {t}, \boldsymbol {u} _ {t})\right) \leqslant \log \left(\frac {| \mathcal {F} |}{\delta}\right).
+$$
+
+Proof. Given a model  $f \in \mathcal{F}$ , we consider the loss function:
+
+$$
+l _ {M} (\boldsymbol {\pi}, u, \boldsymbol {\pi} ^ {\prime}) := \left\{ \begin{array}{l l} \frac {1}{2} \log \frac {f (\boldsymbol {\pi} ^ {\prime} | \boldsymbol {\pi} , u)}{f ^ {*} (\boldsymbol {\pi} ^ {\prime} | \boldsymbol {\pi} , u)}, & \text {i f} f ^ {*} (\boldsymbol {\pi} ^ {\prime} | \boldsymbol {\pi}, u) \neq 0 \\ 0, & \text {o t h e r w i s e} \end{array} \right.
+$$
+
+Considering the event  $\mathcal{E}$ :
+
+$$
+\mathcal {E} := \left\{- \log \mathbb {E} _ {\bar {\tau} _ {T _ {0}}} \left[ \exp L _ {M} \left(\bar {\tau} _ {T _ {0}}\right) \mid \tau_ {T _ {0}} \right] \leqslant - L _ {M} \left(\tau_ {T _ {0}}\right) + \log \left(\frac {| \mathcal {F} |}{\delta}\right), \quad \forall f \in \mathcal {F} \right\}.
+$$
+
+where we define  $L_{M}(\tau_{T_{0}}) \coloneqq \sum_{t=1}^{T_{0}} l_{M}(\pmb{\pi}_{t}, \pmb{u}_{t}, \pmb{\pi}_{t+1})$  and  $L_{M}(\bar{\tau}_{T_{0}}) \coloneqq \sum_{t=1}^{T_{0}} l_{M}(\pmb{\pi}_{t}, \pmb{u}_{t}, \bar{\pmb{\pi}}_{t+1})$ . Besides, by applying Lem. E.1 on  $l_{M}$  defined above and applying Markov inequality and the union bound over all  $f \in \mathcal{F}$ , we have  $\operatorname{Pr}(\mathcal{E}) \geqslant 1 - \delta$ . On the event  $\mathcal{E}$ , we have:
+
+$$
+\begin{array}{l} - \log \mathbb {E} _ {\bar {\tau} _ {T _ {0}}} \left[ \exp L _ {f _ {\mathrm {M L E}}} \left(\bar {\tau} _ {T _ {0}}\right) | \tau_ {T _ {0}} \right] \\ \leqslant - L _ {f _ {\mathrm {M L E}}} \left(\tau_ {T _ {0}}\right) + \log \left(\frac {| \mathcal {F} |}{\delta}\right) \\ \leqslant l _ {\mathrm {M L E}} \left(f ^ {*}\right) - l _ {\mathrm {M L E}} \left(f _ {\mathrm {M L E}}\right) + \log \left(\frac {| \mathcal {F} |}{\delta}\right) \\ \leqslant \log (\frac {| \mathcal {F} |}{\delta}). \quad (f _ {\mathrm {M L E}} \text {m a x i m i z e s t h e l o g - l i k e l i h o o d}) \\ \end{array}
+$$
+
+Therefore,
+
+$$
+\log \left(\frac {| \mathcal {F} |}{\delta}\right) \geqslant - \sum_ {t = 1} ^ {T _ {0}} \log \mathbb {E} _ {\bar {\tau} _ {T _ {0}}} \left[ \sqrt {\frac {f (\bar {\pi} _ {t + 1} | \boldsymbol {\pi} _ {t} , \boldsymbol {u} _ {t})}{f ^ {*} (\bar {\pi} _ {t + 1} | \boldsymbol {\pi} _ {t} , \boldsymbol {u} _ {t})}} | \boldsymbol {\pi} _ {t}, \boldsymbol {u} _ {t}, f ^ {*} \right]
+$$
+
+$$
+\begin{array}{l} \geqslant \sum_ {t = 1} ^ {T _ {0}} 1 - \mathbb {E} _ {\bar {\pi} _ {t + 1}} [ \sqrt {\frac {f (\bar {\pi} _ {t + 1} | \boldsymbol {\pi} _ {t} , \boldsymbol {u} _ {t})}{f ^ {*} (\bar {\pi} _ {t + 1} | \boldsymbol {\pi} _ {t} , \boldsymbol {u} _ {t})}} | \boldsymbol {\pi} _ {t}, \boldsymbol {u} _ {t}, f ^ {*} ] \quad (- \log x \geqslant 1 - x) \\ = \sum_ {t = 1} ^ {T _ {0}} \mathbb {H} ^ {2} \left(f (\cdot | \boldsymbol {\pi} _ {t}, \boldsymbol {u} _ {t}), f ^ {*} (\cdot | \boldsymbol {\pi} _ {t}, \boldsymbol {u} _ {t})\right). \\ \end{array}
+$$
+
+![](images/eb9ad8e5e2da1c854cec6eac39b0247980aeb3c3cf6e9ecbd61ae79c3a22cbbe.jpg)
+
+Example 4.4. [One-Step Difference] If  $\forall \pi \in \Pi$ , there exists a steering reward  $u_{\pi} \in \mathcal{U}$ , s.t.  $\min_{f, f' \in \mathcal{F}} \mathbb{H}^2(f(\cdot | \pi, r + u_{\pi}), f'(\cdot | \pi, r + u_{\pi})) \geqslant \zeta$ , for some universal  $\zeta > 0$ , where  $\mathbb{H}$  is the Hellinger distance, then for any  $\delta \in (0,1)$ ,  $\mathcal{F}$  is  $(\delta, T_{\mathcal{F}}^{\delta})$ -identifiable with  $T_{\mathcal{F}}^{\delta} = O(\zeta^{-1} \log(|\mathcal{F}| / \delta))$ .
+
+Proof. Consider the steering strategy  $\psi(\pi) = u_{\pi}$ . Given any  $f \in \mathcal{F}$ , and the trajectory sampled by  $\psi$  and  $f$ , by Lem. E.2, w.p.  $1 - \frac{\delta}{|\mathcal{F}|}$ , if  $f_{\mathrm{MLE}} \neq f$ , we have:
+
+$$
+2 \log \left(\frac {| \mathcal {F} |}{\delta}\right) \geqslant \sum_ {t = 1} ^ {T _ {0}} \mathbb {H} ^ {2} \left(f (\cdot | \boldsymbol {\pi} _ {t}, \boldsymbol {u} _ {t}), f _ {\mathrm {M L E}} (\cdot | \boldsymbol {\pi} _ {t}, \boldsymbol {u} _ {t})\right) \geqslant T _ {0} \zeta .
+$$
+
+By union bound over all  $f \in \mathcal{F}$ , if  $T_0 = \left\lceil \frac{4}{\zeta} \log \frac{|\mathcal{F}|}{\delta} \right\rceil + 1$ , the following holds:
+
+$$
+\max _ {f \in \mathcal {F}} \mathbb {E} _ {f, \psi} \left[ \mathbb {I} [ f = f _ {\mathrm {M L E}} ] \right] = \max _ {f \in \mathcal {F}} \mathbb {E} _ {f, \psi} \left[ \mathbb {I} [ f = \underset {f ^ {\prime} \in \mathcal {F}} {\arg \max} \sum_ {t = 1} ^ {T _ {\delta}} \log f ^ {\prime} (\pmb {\pi} _ {t + 1} | \pmb {\pi} _ {t}, \pmb {u} _ {t}) ] \right] \geqslant 1 - \delta .
+$$
+
+![](images/8ecb0eccc05864c8ace7102c7a8ee0e418a5f8c9ccc973746491f0c36e53ea3f.jpg)
+
+Theorem 4.5. [A Sufficient Condition for Existence] Given any  $\varepsilon >0$ ,  $\Psi_T^\varepsilon (\mathcal{F};\pi_1)^5\neq \emptyset$ , if  $\exists \widetilde{T} < T$ , s.t., (1)  $\mathcal{F}$  is  $(\frac{\varepsilon}{2\eta_{\max}},\widetilde{T})$ -identifiable, (2)  $\Psi_{T - \widetilde{T}}^{\varepsilon /2}(\mathcal{F};\pi_{\widetilde{T}})\neq \emptyset$  for any possible  $\pi_{\widetilde{T}}$  generated at step  $\widetilde{T}$  during the steering.
+
+Proof. We denote  $\psi_{\mathrm{Explore}} \coloneqq \{\psi_{\mathrm{Explore},t}\}_{t \in [T]}$  to be the exploration strategy to identify  $f^*$ . Given a  $\pi_{\widetilde{T}}$ , we denote  $\psi_{\pi_{\widetilde{T}}}^{\varepsilon/2} \coloneqq \{\psi_{\pi_{\widetilde{T}},t}\}_{t \in [T]} \in \Psi_{T - \widetilde{T}}^{\varepsilon/2}(\pi_{\widetilde{T}})$  to be one of the steering strategy with  $\varepsilon$ -optimal gap starting from  $\pi_{\widetilde{T}}$ .
+
+We consider the history-dependent steering strategy  $\psi \coloneqq \{\psi_t\}_{t\in [T]}$ , such that for  $t\leqslant \widetilde{T}$ ,  $\psi_{t} = \psi_{\mathrm{Explore},t}$ , and for all  $t > \widetilde{T}$ , we have  $\psi_{t} = \psi_{\pi_{\widetilde{T}},t}^{\varepsilon /2}$ .
+
+As a result, for any  $f \in \mathcal{F}$ , the final gap would be:
+
+$$
+\Delta_ {\psi , T} (f) = \Pr (f _ {\mathrm {M L E}} = f) \cdot \frac {\varepsilon}{2} + \Pr (f _ {\mathrm {M L E}} \neq f) \cdot \eta_ {\max } \leqslant \varepsilon ,
+$$
+
+which implies  $\psi \in \Psi_T^\varepsilon (\mathcal{F};\pi_1)$
+
+![](images/78c84e8c567ad850905452adcd3250d008f893e02aa52b87a416fde1ba6e6b1f.jpg)
+
+# F GENERALIZATION TO PARTIAL OBSERVATION MDP SETUP
+
+# F.1 POMDP BASICS
+
+Partial Observation Markov Decision Process A (finite-horizon) Partial-Observation Markov Decision Process (with hidden states) can be specified by a tuple  $M \coloneqq \{\nu_1, T, \mathcal{X}, \mathcal{U}, \mathcal{O}, \mathbb{T}, \eta, \mathbb{O}\}$ . Here  $\nu_1$  is the initial state distribution,  $L$  is the maximal horizon length,  $\mathcal{X}$  is the hidden state space,  $\mathcal{U}$  is the action space,  $\mathcal{O}$  is the observation space. Besides,  $\mathbb{T}: \mathcal{X} \times \mathcal{U} \to \mathcal{X}$  denotes the stationary transition function,  $\mathbb{O}: \mathcal{X} \to \Delta(\mathcal{O})$  denotes the stationary emission model, i.e. the probability of some observation conditioning on some state. We will denote  $\mathcal{H}_h \coloneqq \mathcal{O}_1 \times \mathcal{U}_1 \dots \times \mathcal{O}_h$  to be the history space, and use  $\tau_h \coloneqq \{o_1, u_1, \ldots, o_h\}$  to history observation up to step  $h$ . We consider the history dependent policy  $\psi \coloneqq \{\psi_1, \ldots, \psi_H\}$  with  $\psi_h: \mathcal{H}_h \to \Delta(\mathcal{U})$ . Starting from the initial state  $x_1$ , the trajectory induced by a policy  $\psi$  is generated by:
+
+$$
+\forall h \in [ H ], \quad o _ {h} \sim \mathbb {O} (\cdot | x _ {h}), \quad u _ {h} \sim \psi_ {h} (\cdot | \tau_ {h}), \quad \eta_ {h} \sim \eta_ {h} (o _ {h}, u _ {h}), \quad x _ {h + 1} \sim \mathbb {T} (\cdot | x _ {h}, u _ {h}).
+$$
+
+![](images/f098716abd17f93ff311ba4f963b0c36f152827167ab16c72177ba08d397f8e3.jpg)  
+Figure 4: Probabilistic Graphic Model (PGM) of the POMDP formulation of the steering process. Starting with the initial state  $x_{1} \coloneqq (\pi_{1},\tau_{1})$ , for all  $t \geqslant 1$ , the mediator receives observation  $o_{t} \sim \mathbb{O}(\cdot |x_{t})$  and output the steering reward given the history  $u_{t} \sim \psi (\cdot |o_{1},u_{1},\dots,o_{t})$ . The agents then update their policies following the dynamics  $f$  and the modified reward function  $r + u_{t}$ .
+
+# F.2 STEERING PROCESS AS A POMDP
+
+Given a game  $G$ , we consider the following Markovian agent dynamics:
+
+$$
+\forall t \in [ T ], \quad \tau_ {t} \sim \pi_ {t}, \quad \pi_ {t + 1} \sim f (\cdot | \pi_ {t}, \tau_ {t}, r),
+$$
+
+where  $\tau_{t} := \{s_{1}^{t,k}, \boldsymbol{a}_{1}^{t,k}, \dots, s_{H}^{t,k}, \boldsymbol{a}_{H}^{t,k}\}_{k=1}^{K}$  is several trajectories generated by the policy  $\pi_{t}$ .
+
+In each step  $t$ , we assume the agents first collect trajectories  $\tau_{t}$  with policy  $\pi_{t}$ , and then optimize their policies following some update rule  $f(\cdot|\pi_t, \tau_t, r)$ . Comparing with the Markovian setup in Sec. 3, here  $f$  has additional dependence on the trajectories  $\tau_{t}$ .
+
+Based on this new formulation, the dynamics given the steering strategy is defined by:
+
+$$
+\forall t \in [ T ], \quad \tau_ {t} \sim \pi_ {t}, \quad u _ {t} \sim \psi_ {t} (\cdot | \tau_ {1}, u _ {1},..., \tau_ {t - 1}, u _ {t - 1}, \pi_ {t}), \quad \pi_ {t + 1} \sim f (\cdot | \pi_ {t}, r + u _ {t}),
+$$
+
+In Fig. 4, we illustrate the steering dynamics by Probabilistic Graphical Model (PGM). Here we treat the joint of  $\pi_t$  and  $\tau_t$  as the hidden state at step  $t$ , and the trajectory  $\tau_t$  is the partial observation  $o_t$  received by the mediator. Next, we introduce the notion of decodable POMDP, where the hidden state is determined by a short history.
+
+Definition F.1 (m-Decodable POMDP). Given a POMDP  $M$ , we say it is  $m$ -decodale, if there exists a decoder  $\phi$ , such that,  $x_{h} = \phi(o_{h - m}, u_{h - m}, \ldots o_{h - 1}, u_{h - 1}, o_{h})$ ,
+
+In our steering setting, if for any  $f \in \mathcal{F}$ ,  $f$  is  $m$ -decodable, we just need to learn a steering strategy  $\psi := (\mathcal{O} \times \mathcal{U})^m \times \mathcal{O} \to \mathcal{U}$ , which predicts the steering reward given the past  $m$ -step history. This is the motivation for our experiment setup in the Grid World Stag Hunt game in Sec. 6.1. More concretely, we assume the agents trajectories in the past few steps can be used as sufficient statistics for the current policy, and use them as input of the steering strategy (see Appx. G.2.2 for more details).
+
+# G MISSING EXPERIMENT DETAILS
+
+# G.1 ABOUT INITIALIZATION IN EVALUATION
+
+In some experiments, we will evaluate our steering strategies with multiple different initial policy  $\pi_1$ , in order to make sure our evaluation results are representative.
+
+Here we explain how we choose the initial policies  $\pi_1$ . We will focus on games with two actions which is the only case we use this kind of initialization. For each player, given an integer  $i$ , we construct an increasing sequence with common difference  $\mathrm{Seq}_i := \left(\frac{1}{2i}, \frac{3}{2i}, \dots, \frac{2i - 1}{2i}\right)$ . Then, we consider the initial policies  $\pi_1$  such that  $\pi^1(a^1) = 1 - \pi^1(a^2) \in \mathrm{Seq}_i$ ,  $\pi^2(a^1) = 1 - \pi^2(a^2) \in \mathrm{Seq}_i$ . In this way, we obtain a set of initial policies uniformly distributed in grids with common difference
+
+$\frac{1}{i}$ . As a concrete example, the initial points in Fig. 1-(b) marked in color black is generated by the above procedure with  $i = 10$ .
+
+# G.2 EXPERIMENTS FOR KNOWN MODEL SETTING
+
+# G.2.1 EXPERIMENT DETAILS IN NORMAL-FORM STAG HUNT GAME
+
+We provide the missing experiment details for the steering experiments in Fig. 1-(b).
+
+Choice of  $\eta^{\mathrm{goal}}$  We consider the total utility as the goal function. But for the numerical stability, we choose  $\eta^{\mathrm{goal}}(\pi) = \sum_{n\in [N]}J_{|r}^{n}(\pi) - 10$  where we shift the reward via the maximal utility value 10.
+
+The Steering Strategy The steering strategy is a 2-layer MLP with 256 hidden layers and tanh as the activation function. Given a time step  $t$  and the policy  $\pi_t \coloneqq \{\pi_t^1, \pi_t^2\}$  with  $\pi_t^n(\mathsf{H}) + \pi_t^n(\mathsf{G}) = 1$  for  $n \in \{1, 2\}$ , the input of the steering strategy is
+
+$$
+\left(\log \sqrt {\frac {\pi_ {t} ^ {1} (\mathrm {H})}{\pi_ {t} ^ {1} (\mathrm {G})}}, - \log \sqrt {\frac {\pi_ {t} ^ {1} (\mathrm {H})}{\pi_ {t} ^ {1} (\mathrm {G})}}, \log \sqrt {\frac {\pi_ {t} ^ {2} (\mathrm {H})}{\pi_ {t} ^ {2} (\mathrm {G})}}, - \log \sqrt {\frac {\pi_ {t} ^ {2} (\mathrm {H})}{\pi_ {t} ^ {2} (\mathrm {G})}}, \frac {T - t}{1 0 0}\right). \tag {11}
+$$
+
+Here the first (second) two components correspond to the "dual variable" of the policy  $\pi_t^1 (\mathbb{H})$  and  $\pi_t^1 (\mathsf{G})(\pi_t^2 (\mathsf{H})$  and  $\pi_t^2 (\mathsf{G}))$ , respectively; the last component is the time embedding because our steering strategy is time-dependent.
+
+The steering strategy will output a vector with dimension 4, which corresponds to the steering rewards for two actions of two players. Note that here the steering reward function  $u^1: \mathcal{S} \times \mathcal{A}^1[0, U_{\max}]$  (for agent 1) and  $u^2 := \mathcal{S} \times \mathcal{A}^2 \to [0, U_{\max}]$  (agent 2) is defined on the joint of state space and individual action space. This can be regarded as a specialization of the setup in our main text, where we consider  $u^n: \mathcal{S} \times \mathcal{A} \to [0, U_{\max}] \forall n \in [N]$ , which is defined on the joint of state space and the entire action space.
+
+Training Details The maximal steering reward  $U_{\mathrm{max}}$  is set to be 10, and we choose  $\beta = 25$ . We use the PPO implementation of StableBaseline3 (Raffin et al., 2021). The training hyper-parameters can be found in our codes in our supplemental materials.
+
+During the training, the initial policy is randomly selected from the feasible policy set, in order to ensure the good performance in generalizing to unseen initialization points. Another empirical trick we adopt in our experiments is that, we strengthen the learning signal of the goal function by including  $\eta_{t}^{\mathrm{goal}}(\pi_t)$  for each step  $t\in [T]$ . In another word, we actually optimize the following objective function:
+
+$$
+\psi^ {*} \leftarrow \arg \max  _ {\psi \in \Psi} \frac {1}{| \mathcal {F} |} \sum_ {f \in \mathcal {F}} \mathbb {E} _ {\psi , f} \left[ \beta \cdot \eta^ {\text {g o a l}} \left(\boldsymbol {\pi} _ {T + 1}\right) + \sum_ {t = 1} ^ {T} \beta \cdot \eta^ {\text {g o a l}} \left(\boldsymbol {\pi} _ {t}\right) - \eta^ {\text {c o s t}} \left(\boldsymbol {\pi} _ {t}, \boldsymbol {u} _ {t}\right) \right]. \tag {12}
+$$
+
+The main reason is that here  $T = 500$  is very large, and if we only have the goal reward at the terminal step, the learning signal is extremely sparse and the learning could fail.
+
+Other Experiment Results In Fig. 5, we investigate the trade-off between steering gap and the steering cost when choosing different coefficients  $\beta$ . In general, the larger  $\beta$  can result in lower steering gap and higher steering cost.
+
+# G.2.2 EXPERIMENT DETAILS IN GRID-WORLD VERSION OF STAG HUNT GAME
+
+We recall the illustration in LHS of Fig. 2. We consider a  $3 \times 3$  grid world environment with two agents (blue and red). At the bottom-left and up-right blocks, we have 'stag' and 'hares', respectively, whose positions are fixed during the game. At the beginning of each episode, agents start from the up-left and bottom-right blocks, respectively.
+
+For each time step  $h \in [H]$ , every agent can take four actions {up, down, left, right} to move to the blocks next to their current blocks. But if the agent hits the wall after taking the action (e.g. the agent locates at the most right column and takes the action right), it will not move. As long as one
+
+![](images/73454bf4763cdcab467721fdcc25228390c732d7695608bf7082a176fc85f3d1.jpg)  
+Figure 5: Trade-off between Steering Gap (Left) and Steering Cost (Right). (averaged over 5x5 uniformly distributed grids as initializations of  $\pi_1$ , see Appx. G.1).
+
+![](images/1ce2b5c969047e05a91bfd3b8a99c5acd22f1affc3a0c1918be12c344756a08e.jpg)
+
+agent reaches the block with either stag or hare, the agents will receive rewards and be reset to the initial position (up-left and bottom-right blocks). The reward is defined by the following.
+
+- If both agents reach the block with stag at the same time, each of them receive reward 0.25.  
+- If both agents reach the block with hares at the same time, each of them receive reward 0.1.  
+- If one agent reaches the block with hares, it will get reward 0.2 and the other gets reward 0.  
+- In other cases, the agents receive reward 0.
+
+We choose  $H = 16$ . The best strategy is that all the agents move together towards the block with Stag, so within one episode, the agents can reach the Stag  $16 / 2 = 8$  times, and the maximal total return would be  $8 * 0.25 = 4.0$ .
+
+In the following, we introduce the training details. Our grid-world environment and the PPO training algorithm is built based on the open source code from (Lu et al., 2022).
+
+Agents Learning Dynamics The agents will receive a 3x3x4 image encoding the position of all objects to make the decision. The agents adopt a CNN, and utilize PPO to optimize the CNN parameters with learning rate 0.005.
+
+Steering Setup and Details in Training Steering Strategy Our steering strategy is another CNN, which takes the agents recent trajectories as input. More concretely, for each steering iteration  $t$ , we ask the agents to interact and generated 256 episodes with length  $H$ , and concatenate them together to a tensor with shape  $[256 * H, 3, 3, 4]$ . The mediator takes that tensor as input and output an 8-dimension steering reward vector. Here the steering rewards correspond to the additional rewards given to the agents when one of them reach the blocks with stag or hares (we do not provide individual incentives for states and actions before reaching those blocks). To be more concrete, the 8 rewards correspond to the additional reward for blue and red agents for the following 4 scenarios: (1) both agents reach stag together (2) both agents reach hares together (3) this agent reach stag while the other does not reach stag (4) this agent reach hares while the other does not reach hares.
+
+The steering strategy is also trained by PPO. We choose  $\beta = 25$  and learning rate 0.001. We consider the total utility as the goal function, and we adopt the similar empirical trick as the normal-form version, where we include  $\eta^{\mathrm{goal}}$  into the reward function for every  $t\in [T]$  (Eq. (12)). The results in Fig. 2 is the average of 5 steering strategies trained by different seeds for 80 iterations. The two-sigma error bar is shown.
+
+# G.2.3 EXPERIMENTS IN MATCHING PENNIES
+
+Matching Pennies is a two-player zero-sum game with two actions  $\mathrm{H} =$  Head and  $\mathrm{T} =$  Tail and its payoff matrix is presented in Table 2.
+
+Choice of  $\eta^{\mathrm{goal}}$  In this game, the unique Nash Equilibrium is the uniform policy  $\pi^{\mathrm{NE}}$  with  $\pi^{n,\mathrm{NE}}(\mathbb{H}) = \pi^{n,\mathrm{NE}}(\mathbb{T}) = \frac{1}{2}$  for all  $n\in \{1,2\}$ . We consider the distance with  $\pi^{\mathrm{NE}}$  as the goal function, i.e.  $\eta^{\mathrm{goal}} = -\| \pi -\pi^{\mathrm{NE}}\| _2$ .
+
+Experiment Setups We follow the same steering strategy and training setups for Stag Hunt Game in Appx. G.2.1. The agents follow NPG to update the policies with learning rate  $\alpha = 10$ .
+
+Table 2: Payoff Matrix of Two-Player Game Matching Pennies. Two actions H and T stand for Head and Tail, respectively.  
+
+<table><tr><td colspan="2">H T</td></tr><tr><td>(1,-1)</td><td>(-1,1)</td></tr><tr><td>(-1,1)</td><td>(1,-1)</td></tr></table>
+
+![](images/8f77c2419092cbfc007b7de07eca0b0731df5e75df7dedf66048da1655e19787.jpg)  
+(a) Dynamics of Agents Policies.
+
+![](images/98b5e236cc89d770c5f773d5eecbaf34da3b7018330dc913a66864b50119a182.jpg)  
+(b) Trade-off between Accuracy and Cost by  $\beta$
+
+![](images/e89e9c21d03e1a4ad1cd1b8be163e6b19320f865f8fdef116475b080af3ab7b6.jpg)  
+Figure 6: Experiments in MatchingPennies. (a)  $x$  and  $y$  axes correspond to the probability that agents take Head. Black dots mark the initial policies, and red curves represent the trajectories of agents policies. The steering strategy to plot the figure is trained with  $\beta = 25$ . (b) We compare  $\beta = 10, 25, 100$ . Error bar shows  $95\%$  confidence intervals. (averaged over 5x5 uniformly distributed grids as initializations of  $\pi_1$ , see Appx. G.1)
+
+![](images/cbd08de40994f6126258a4fa42141212d964a342e66f5fc5957e76fead6c87a4.jpg)
+
+Experiment Results As shown in Fig. 6-(a), we can observe the cycling behavior without steering guidance (Akin and Losert, 1984; Mertikopoulos et al., 2018). In contrast, our learned steering strategy can successfully guide the agents towards the desired Nash. In Fig. 6-(b), we also report the trade-off between steering gap and steering cost with different choice of  $\beta$ .
+
+# G.3 EXPERIMENTS FOR UNKNOWN MODEL SETTING
+
+# G.3.1 DETAILS FOR EXPERIMENTS WITH SMALL MODEL SET  $\mathcal{F}$
+
+The results in Table 1 is averaged over 5 seeds and the error bars show  $95\%$  confidence intervals.
+
+Training Details for  $\psi_{0.7}^{*}$  and  $\psi_{1.0}^{*}$  The training of  $\psi_{0.7}^{*}$  and  $\psi_{1.0}^{*}$  follow the similar experiment setup as Appx. G.2.2, except here the agents adopt random learning rates. For the choice of  $\beta$ , we train the optimal steering strategy with  $\beta \in \{10, 20, 30, 40, 50, 60, 70, 80, 90, 100\}$  for both  $f_{0.7}$  and  $f_{1.0}$ , and choose the minimal  $\beta$  such that the resulting steering strategy can achieve almost  $100\%$  accuracy (i.e.  $\Delta_{\psi, f} \leqslant \varepsilon$  for almost all  $5 \times 5$  uniformly distributed initial policies generated by process in Appx. G.1). As we reported in the main text, we obtain  $\beta = 70$  for  $f_{0.7}$  and  $\beta = 20$  for  $f_{1.0}$ .
+
+Training Details for  $\psi_{\text{Belief}}^{*}$  For the training of  $\psi_{\text{Belief}}^{*}$ , the input of the steering strategy is the original state (Eq. (11)) appended by the belief state of the model. In each steering step  $t \in [T]$ , we assume the mediator can observe a learning rate sample  $\alpha$ , and use it to update the model belief state correspondingly. The regularization coefficient  $\beta$  for the training of  $\psi_{\text{Belief}}$  is set to be the expected regularization coefficient over the belief state  $\beta = b(f_{0.7}) \cdot 70 + b(f_{1.0}) \cdot 20$ . In another word, we use the sum of the coefficient of two models weighted by the belief state. This is reasonable by the definition of the reward function in the belief state MDP lifted from the original POMDP.  $\psi_{\text{Belief}}^{*}$  is trained the PPO algorithm.
+
+During the training of  $\psi_{\mathrm{Belief}}^*$ , we find that the train is not very stable, possibly because the chosen  $\beta$  for two models are quite different. Therefore, we keep tracking the steering gap of the steering strategy during the training and save the model as long as it outperforms the previous ones in steering gap. Our final evaluation is based on that model.
+
+# G.3.2 DETAILS FOR EXPERIMENTS WITH LARGE MODEL SET  $\mathcal{F}$
+
+We set  $U_{\mathrm{max}} = 1.0$ , and the random exploration strategy (red curve in the left sub-plot in Fig. 3) will sample the steering reward uniformly from the interval  $[0, U_{\mathrm{max}}]$ . We use the PPO (Raffin et al., 2021) to train of exploration policy and also the steering strategy given hidden model. To amplify the exploration challenge, we set  $\beta^n = 1$  when  $\pi^n(\mathbb{A}) \leqslant 0.5$  and increase to  $\beta^n = 10$  when  $\pi^n(\mathbb{A}) \leqslant 0.5$ . As a result, if the mediator follows first-explore-then-exploit strategy and fail to distinguish avaricious agents from the normal ones, adopting large steering reward can lead to much worse performance.
+
+For the training of exploration policy, although the learning signal  $\mathbb{I}[f = f_{\mathrm{MLE}}]$  in Proc. 2 is supported by theory, it contains much less information than the posterior probability  $[\operatorname*{Pr}(f|\pmb{\pi}_1,u_1,\dots,\pmb{\pi}_T,u_T,\pmb{\pi}_{T + 1})]_{f\in \mathcal{F}}$ . Therefore, empirically, we instead train a history-independent steering strategy to maximize the posterior probability of  $f$ :
+
+$$
+\psi^ {\text {E x p l o r e}} \leftarrow \arg \max  _ {\psi} \frac {1}{| \mathcal {F} |} \sum_ {f \in \mathcal {F}} \mathbb {E} _ {\psi , f} [ \sum_ {n \in [ N ]} \Pr (\lambda^ {n} | \pi_ {1}, u _ {1}, \dots , \pi_ {\tilde {T}}, u _ {\tilde {T}} \pi_ {\tilde {T} + 1}) ]. \tag {13}
+$$
+
+Here we use the sum of posteriors of  $\lambda^n$ s since the  $\lambda^n$ s are independent for all  $n\in [N]$ . We observe it results in better performance, and it is doable by keep tracking the model belief state of each agent. Besides, similar to Stag Hunt games, we observe that using the posterior  $\operatorname*{Pr}(\lambda^n |\pmb {\pi}_1,u_1,\dots,\pmb {\pi}_t,u_t,\pmb{\pi}_{t + 1})$  as rewards in the non-terminal steps  $t < T$  increase the performance, and we use the same trick (Eq. (12)).
+
+To plot the results in the middle and right sub-plots in Fig. 3, for each model  $f^{*} \in \{f_{1}, f_{2}, f_{3}\}$ , we train three steering strategies (with the same state design in Eq. (11)). The first one is the oracle strategy, which starts with  $\pi_{1}$  and steering for  $T = 500$  steps. The second one is the FETE strategy, including an exploration policy and another exploitation policy. The exploration policy is trained following the above Eq. (13) with exploration horizon  $\tilde{T} = 30$ . Then, we estimate the model from samples generated by the exploration policy, and train another exploitation policy following the remaining step of FETE with exploitation horizon  $T = 470$  (Procedure 2). The third strategy FETE-RE is the same as FETE except we just use random policy as the exploration policy, and estimate the model by interaction samples generated by that. During the evaluation, for FETE and FETE-RE, we steer with the exploration policy for the first 30 steps, and execute the exploitation policy for the rest. The results are averaged over 5 seeds and two-sigma error bar is shown.
+
+# G.4 EXPLANATION OF THE CONSISTENCY OF THE ADPTION
+
+We first want to highlight it is not easy to have an apples-to-apples comparison with (Canyakmaz et al., 2024). First, because the experiment setting in (Canyakmaz et al., 2024) does not present significant exploration challenges, we design and decide to evaluate both methods in the "avaricious agents" setting (Fig. 3). Second, SIAR-MPC is specialized for polynomial function classes and the dynamics tractable by MPC, making it difficult to generalize beyond that setting. Therefore, we have to do some necessary adaption. To ensure the fair comparison, we consider to use FETE-RE as the adaption of SIAR-MPC in our setting. For the exploration stage, FETE-RE aligns with SIAR-MPC in using random exploration. For the model identification and exploitation phase, FETE-RE adopts MLE estimation and RL methods to train the exploitation policy, which inherit the same inspirits as SIAR-MPC and also aligned with our original FETE.
+
+From another perspective, the main focus of our empirical comparison between FETE and (Canyakmaz et al., 2024) is the impact of different exploration strategies on the final steering gap and steering costs. This is reasonable. Because from the discussion in Sec. 5.2, we can conclude that our FETE is more general compared with SIAR-MPC in (Canyakmaz et al., 2024) in terms of both the model estimation strategy and exploitation strategy. The main distinction and improvement of our FETE compared with (Canyakmaz et al., 2024) is our strategic exploration strategy.
+
+# G.5 A SUMMARY OF THE COMPUTE RESOURCES BY EXPERIMENTS IN THIS PAPER
+
+Experiments on Two-Player Normal-Form Games For the experiments in 'Stag Hunt' and 'Matching Pennies' (illustrated in Fig. 1, 5, 6), we only use CPUs (AMD EPYC 7742 64-Core Processor). It takes less than 5 hours to finish the training.
+
+Experiments on Grid-World Version of 'Stag Hunt' For the experiments in grid-world 'Stag Hunt' (illustrated in Fig. 2), we use one RTX 3090 and less than 5 CPUs (AMD EPYC 7742 64-Core Processor). The training (per seed) takes around 48 hours.
+
+Experiments on  $N$ -Player Normal-Form Cooperative Games For the experiments in cooperative games (illustrated in Fig. 3), we only use CPUs (AMD EPYC 7742 64-Core Processor). It takes less than 10 hours to finish the training.
+
+# H ADDITIONAL DISCUSSION ABOUT GENERALIZING OUR RESULTS
+
+In this section, we discuss some extensions of the principle and algorithms in this paper to more general settings.
+
+Non-Tabular Setting When the game is non-tabular and its state and action spaces are infinite, the steering problem itself is fundamentally challenging without additional assumptions, since the policies are continuous distributions with infinite dimension and the learning dynamics can be arbitrarily complex. Nonetheless, when the agents' policies and steering rewards are parameterized by finite variables, our methods and algorithms can still be generalized by treating the parameters as representatives.
+
+As a concrete example, the Linear-Quadratic (LQ) game is a popular model involving countinous state and action spaces (Jacobson, 1973; Bāsār and Bernhard, 2008; Zhang et al., 2019). In zero-sum LQ game, the game dynamics are characterized by a linear system:
+
+$$
+x _ {t + 1} = A x _ {t} + B y _ {t} + C z _ {t},
+$$
+
+with one-step reward function
+
+$$
+r ^ {1} \left(x _ {t}, y _ {t}, z _ {t}\right) = - r ^ {2} \left(x _ {t}, y _ {t}, z _ {t}\right) = x _ {t} ^ {\top} Q x _ {t} + y _ {t} ^ {\top} R ^ {u} y _ {t} - z _ {t} ^ {\top} R ^ {v} z _ {t}.
+$$
+
+Here  $x_{t}, x_{t+1} \in \mathbb{R}^{d}$  are the system states,  $y_{t} \in \mathbb{R}^{m_{1}}$  and  $z_{t} \in \mathbb{R}^{m_{2}}$  denote the actions of two agents. Besides, the agents' policies are parameterized by matrices  $K_{t} \in \mathbb{R}^{m_{1} \times d}, L_{t} \in \mathbb{R}^{m_{2} \times d}$ , i.e.
+
+$$
+y _ {t} = - K _ {t} x _ {t}, \quad z _ {t} = - L _ {t} x _ {t}.
+$$
+
+Following the quadratic form of the original reward, one may consider quadratic steering reward functions with parameters  $\Theta_t \coloneqq (\Theta_t^Q, \Theta_t^u, \Theta_t^v)$  and  $\Xi_t \coloneqq (\Xi_t^Q, \Xi_t^u, \Xi_t^v)$ , such that, the steering reward for two agents at step  $t$  is specified by:
+
+$$
+u _ {t} ^ {1} (x _ {t}, y _ {t}, z _ {t}) = x _ {t} ^ {\top} \Theta_ {t} ^ {Q} x _ {t} + y _ {t} ^ {\top} \Theta_ {t} ^ {u} y _ {t} - z _ {t} ^ {\top} \Theta_ {t} ^ {v} z _ {t},
+$$
+
+$$
+u _ {t} ^ {2} \left(x _ {t}, y _ {t}, z _ {t}\right) = x _ {t} ^ {\top} \Xi_ {t} ^ {Q} x _ {t} + y _ {t} ^ {\top} \Xi_ {t} ^ {u} y _ {t} - z _ {t} ^ {\top} \Xi_ {t} ^ {v} z _ {t},
+$$
+
+and the reward after modification would be:
+
+$$
+r ^ {1} (x _ {t}, y _ {t}, z _ {t}) + u _ {t} ^ {1} (x _ {t}, y _ {t}, z _ {t}) = x _ {t} ^ {\top} \left(\Theta_ {t} ^ {Q} + Q\right) x _ {t} + y _ {t} ^ {\top} \left(\Theta_ {t} ^ {u} + R ^ {u}\right) y _ {t} - z _ {t} ^ {\top} \left(\Theta_ {t} ^ {v} + R ^ {v}\right) z _ {t},
+$$
+
+$$
+r ^ {2} (x _ {t}, y _ {t}, z _ {t}) + u _ {t} ^ {2} (x _ {t}, y _ {t}, z _ {t}) = x _ {t} ^ {\top} (\Theta_ {t} ^ {Q} - Q) x _ {t} + y _ {t} ^ {\top} (\Theta_ {t} ^ {u} - R ^ {u}) y _ {t} - z _ {t} ^ {\top} (\Theta_ {t} ^ {v} - R ^ {v}) z _ {t}.
+$$
+
+Although the state, action and steering reward spaces are continuous, both the policies and steering reward are determined by their parameters. Therefore, the agents' learning dynamics can be modeled by a function  $f^{*}$  mapping between those parameters instead:
+
+$$
+(K_{t + 1},L_{t + 1})\sim f^{*}(\cdot | \underbrace{(K_{t},L_{t})}_{\text{agents' policies}},\underbrace{(\Theta_{t}^{Q} + Q,\Theta_{t}^{u} + R^{u},\Theta_{t}^{u} + R^{v},\Xi_{t}^{Q} - Q,\Xi_{t}^{u} - R^{u},\Xi_{t}^{u} - R^{v})}_{\text{modified rewards}}).
+$$
+
+Besides, the learning of steering strategy is equivalent to learning a function mapping  $\psi$  from parameters in history  $\{(K_{\tau},L_{\tau},\Theta_{\tau},\Xi_{\tau})\}_{\tau = 1}^{t - 1}\cup \{(K_{t},L_{t})\}$  to the next steering reward parameter  $(\Theta_t,\Xi_t)$ . Since both the policy parameters and steering reward parameters have finite dimension, this problem is tractable under our frameworks.
+
+Uncountable function class  $\mathcal{F}$  Our results can be extended to cases where the model class  $\mathcal{F}$  is infinite but has a finite covering number. We denote  $\mathcal{F}_{\varepsilon_0}$  as the  $\varepsilon_0$ -cover for  $\mathcal{F}$ , s.t.
+
+$$
+\forall f \in \mathcal {F}, \quad \exists f ^ {\prime} \in \mathcal {F} _ {\varepsilon_ {0}}, \text {s . t .} \max  _ {\boldsymbol {\pi} \in \Pi , \boldsymbol {u}} \mathbb {T} \mathbb {V} \Big (f (\cdot | \boldsymbol {\pi}, \boldsymbol {u}) - f ^ {\prime} (\cdot | \boldsymbol {\pi}, \boldsymbol {u}) \Big) \leqslant \varepsilon_ {0}.
+$$
+
+where  $\mathbb{T}\mathbb{V}$  denotes the total variation distance. If  $\mathcal{F}$  is uncountable but  $\mathcal{F}_{\varepsilon_0}$  is finite, we run our algorithms with  $\mathcal{F}_{\varepsilon_0}$  instead of  $\mathcal{F}$ . Under Assump. A, we denote  $f_{\varepsilon_0}^* \in \mathcal{F}_{\varepsilon_0}$  is the function  $\varepsilon_0$  close to  $f^*$ . By simulation lemma, then we have:
+
+$$
+\left| \mathbb {E} _ {\psi , f ^ {*}} \left[ \eta^ {\text {g o a l}} \left(\boldsymbol {\pi} _ {T + 1}\right) \right] - \mathbb {E} _ {\psi , f _ {\varepsilon_ {0}} ^ {*}} \left[ \eta^ {\text {g o a l}} \left(\boldsymbol {\pi} _ {T + 1}\right) \right] \right| \leqslant T \cdot \varepsilon_ {0} \cdot \eta_ {\max }
+$$
+
+$$
+| \mathbb {E} _ {\psi , f ^ {*}} [ \sum_ {t = 1} ^ {T} \eta^ {\mathrm {c o s t}} (\boldsymbol {\pi} _ {t}, \boldsymbol {u} _ {t}) ] - \mathbb {E} _ {\psi , f _ {\varepsilon_ {0}} ^ {*}} [ \sum_ {t = 1} ^ {T} \eta^ {\mathrm {c o s t}} (\boldsymbol {\pi} _ {t}, \boldsymbol {u} _ {t}) ] | \leqslant T ^ {2} \cdot \varepsilon_ {0} \cdot \max _ {\boldsymbol {\pi}, \boldsymbol {u}} \eta^ {\mathrm {c o s t}} (\boldsymbol {\pi}, \boldsymbol {u}).
+$$
+
+As we can see, we can still optimize the objective in Eq. (1) with  $\mathcal{F}_{\varepsilon_0}$ , and then transfer guarantees on steering gap and cost for  $f_{\varepsilon_0}^*$  (e.g. the worst case guarantees in Prop. 3.3) to  $f^*$  with additional  $O(T^{2} \cdot \varepsilon_{0})$  errors, which is ignorable when  $\varepsilon_0$  is small enough.
+
+Non-Markovian Learning Dynamics In general, non-Markovian learning dynamics is intractable, as implied by the fundamental difficulty in learning optimal policies in POMDPs. However, when some special structures exhibit, our methods for Markovian agents can be generalized. One example is the non-Markovian agents with finite-memory, i.e.,
+
+$$
+\boldsymbol {\pi} _ {t + 1} \sim f ^ {*} (\cdot | \boldsymbol {\pi} _ {t - m + 1}, \boldsymbol {r} + \boldsymbol {u} _ {t - m + 1}, \dots , \boldsymbol {\pi} _ {t}, \boldsymbol {r} + \boldsymbol {u} _ {t}).
+$$
+
+This can be reformulated by a Markovian dynamics
+
+$$
+\boldsymbol {x} _ {t + 1} \sim F ^ {*} (\cdot | \boldsymbol {x} _ {t}, \boldsymbol {r} + \boldsymbol {u} _ {t}),
+$$
+
+with the same steering rewards as actions but a new definition of "state":  $\pmb{x}_t \coloneqq \{\pmb{\pi}_{t - m + 1}, \pmb{r} + \pmb{u}_{t - m + 1}, \dots, \pmb{\pi}_{t - 1}, \pmb{r} + \pmb{u}_{t - 1}, \pmb{\pi}_t\}$ . Comparing with Def. 3.1, the dimension of the state space is expanded by  $m$  times, which is still tractable for small  $m$ .
+
+Neural Networks as Model Class to Approximate Complex  $f^{*}$  The main principle for choosing  $\mathcal{F}$  is to ensure our "realizability" assumption holds with high probability, i.e. the true model  $f^{*} \in \mathcal{F}$ . The concrete choice of  $\mathcal{F}$  depends on the prior knowledge we have about the agents' learning dynamics. In general, the less prior knowledge we have, the larger  $\mathcal{F}$  should be to ensure realizability, and vice versa.
+
+In practice, one "safe-choice" can be considered a class of parameterized neural networks as  $\mathcal{F}$ , since it has been proven in deep RL and supervised learning literature that neural networks have powerful approximation ability when  $f^{*}$  is potentially very complex. Because in our formulation, we allow the randomness of next policy  $\pi_{t + 1}$  (instead of a deterministic output) given  $\pi_t$  and  $\boldsymbol {r} + \boldsymbol{u}_t$ , we may consider a neural network taking the concatenation of  $\pi_t$ ,  $\boldsymbol {r} + \boldsymbol{u}_t$  and another random Gaussian vector  $\xi$  as inputs. Here the noise vector is introduced to model the stochasticity of  $\pi_t$ .
+
+The parameters of neural networks are in general continuous variables, which implies the model class is uncountable. However, if the parameters has bounded value range, we can show the finite covering number on the parameter space. If we consider Lipschitz continuous activation functions (which is most of the cases), it implies the bounded covering number.
+
+Besides, when considering neural networks, the resulting model class  $\mathcal{F}$  can be extremely large and the MLE-based strategic exploration in Procedure 2 will be inefficient. We highlight that we design such exploration step in order to align with the main principle: the algorithm design should be supported by theoretical guarantees on the performance of the learned steering strategy. This focus on theoretical rigor is the main factor limiting the scalability of our algorithms in more complex settings. Conversely, if we relax the requirements on theoretical guarantees, it is not very challenging to adapt our algorithms to complex scenarios. For example, we can instead consider more scalable exploration methods, such as Random Network Distillation (RND) (Burda et al., 2018) or Bootstrapped DQN (Osband et al., 2016), although without theoretical guarantees.
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+# LEARNING TO ENGINEER PROTEIN FLEXIBILITY
+
+Petr Kouba $^{1,2}$  Joan Planas-Iglesias $^{2,3}$  Jiri Damborsky $^{2,3}$
+
+Jiri Sedlar $^{1}$  Stanislav Mazurenko $^{2,3}$  Josef Sivic $^{1}$
+
+$^{1}$ Czech Institute of Informatics, Robotics and Cybernetics, Czech Technical University
+
+$^{2}$ Loschmidt Laboratories, Department of Experimental Biology and RECETOX, Masaryk University
+
+$^{3}$ International Clinical Research Centre, St. Anne's University Hospital Brno
+
+# ABSTRACT
+
+Generative machine learning models are increasingly being used to design novel proteins for therapeutic and biotechnological applications. However, the current methods mostly focus on the design of proteins with a fixed backbone structure, which leads to their limited ability to account for protein flexibility, one of the crucial properties for protein function. Learning to engineer protein flexibility is problematic because the available data are scarce, heterogeneous, and costly to obtain using computational as well as experimental methods. Our contributions to address this problem are three-fold. First, we comprehensively compare methods for quantifying protein flexibility and identify data relevant to learning. Second, we design and train flexibility predictors utilizing sequential or both sequential and structural information on the input. We overcome the data scarcity issue by leveraging a pre-trained protein language model. Third, we introduce a method for fine-tuning a protein inverse folding model to steer it toward desired flexibility in specified regions. We demonstrate that our method Flexpert-Design enables guidance of inverse folding models toward increased flexibility. This opens up new possibilities for protein flexibility engineering and the development of proteins with enhanced biological activities.
+
+# 1 INTRODUCTION
+
+The goal of this study is to develop a tool that integrates protein flexibility into computational protein design. Computational protein engineering and design are crucial technologies that have shown immense potential in creating next-generation enzymes for a wide range of applications, from the production of fine chemicals, pharmaceuticals, and food ingredients to sustainable, environmentally friendly solutions in green energy and biodegradation (Planas-Iglesias et al., 2021). These approaches aim to propose new amino acid sequences, either from scratch (de novo) or by modifying sequences of existing natural proteins to alter their properties in a desired way. Despite the long history of these efforts, protein engineering and design remain labor-intensive and time-consuming, with uncertain outcomes in each case (Listov et al., 2024). Recent advances in generative machine learning have led to several powerful tools for designing proteins with specific topologies (the inverse folding problem), offering the potential for universal protein design platforms (Notin et al., 2024; Kortemme, 2024). However, the number of successful case studies using these tools remains relatively low. One of the major obstacles hindering their broader application is their inability to account for protein flexibility (Chu et al., 2024; Kouba et al., 2023).
+
+Proteins are highly dynamic biomolecules, and the presence of flexible regions is critical for their biological function (Corbella et al., 2023; Lemay-St-Denis et al., 2022). In particular, fine-tuning conformational dynamics of loops close to the active site has been shown to be an important method for modulating substrate specificities (Romero-Rivera et al., 2022), turnover rates (Crean et al., 2021), and pH dependency (Shen et al., 2021) of enzymes. In addition, the biological function of many proteins often requires that a small molecule is transported through their structures, for example via tunnels leading to the active site, whose dynamical properties are thus critical for protein function (Jurcik et al., 2018). Modulating the flexibility of protein parts even far away from the active sites has been shown to be an efficient strategy for improving proteins (Karamitros et al., 2022). However, quantification of protein flexibility remains challenging despite recent progress in both experimental and computational approaches. Experimental methods, such as X-ray crystallography, nuclear magnetic resonance, and hydrogen-deuterium exchange coupled to mass spectroscopy,
+
+require expensive equipment, specialized biophysical expertise, and time-consuming sample preparation and analysis (Peacock & Komives, 2021). Consequently, they lack the high throughput needed for systematic protein flexibility evaluation. Computational methods provide a broad spectrum of approaches for assessing protein flexibility, from rapid coarse-grained modeling to more accurate but highly computationally intensive molecular dynamics (MD) simulations (Kmiecik et al., 2016). However, the available datasets are limited, and systematic comparisons of different methods across large protein datasets are thus lacking. More importantly, it remains unclear how to effectively integrate these computational methods into the state-of-the-art generative tools, increasingly used in protein engineering and design (Kortemme, 2024).
+
+In this work, we aim to fill these gaps by proposing a new method for designing protein flexibility. First, we perform a comprehensive comparison of methods for quantification of protein flexibility and identify relevant data for learning (Section 3). Second, we address the data scarcity issue by proposing new flexibility predictors utilizing just sequential (Flexpert-Seq) or both sequential and structural (Flexpert-3D) information on the input (Section 4.1). Importantly, our goal is to create a fast predictor, which is crucial for its integration into state-of-the-art ML-guided protein engineering pipelines. Third, we introduce a new method (Flexpert-Design) for fine-tuning a protein inverse folding model to make it steerable toward desired flexibility in specified positions (Section 4.2). Our results show that it is possible to predict flexibility from sequence, and that adding structural information further improves the prediction. Furthermore, using our protein flexibility predictor, we demonstrate that inverse folding models can be steered toward generating protein sequences with increased flexibility, suggesting a potential strategy for engineering of protein flexibility.
+
+# 2 RELATED WORK
+
+Experimental methods for measuring protein flexibility. Common experimental methods for determining protein flexibility include X-ray crystallography, Nuclear Magnetic Resonance (NMR), and Hydrogen-Deuterium Exchange coupled to Mass Spectroscopy (HDX-MS) (Peacock & Komives, 2021). X-ray crystallography provides flexibility estimates based on the consistency of protein conformations in a crystalline state. The regularity of atomic positions across crystal lattice cells determines the precision of atom location, quantified by the B-factor (or temperature factor), which indicates mobility (Wlodawer et al., 2008). Details on NMR and HDX-MS are given in Appendix A.
+
+Computational methods for measuring protein flexibility. Protein flexibility can be predicted using faster, more economical in silico methods (Rueda et al., 2007). A common approach involves deriving flexibility from Molecular Dynamics (MD) simulations, which apply Newton's laws to atoms to compute their motion over time. From these simulations, Root Mean Square Fluctuations (RMSF) can be computed per residue to quantify their flexibility as observed in the simulations. While effective, MD is time-consuming, as it must explore a wide range of protein conformations. Elastic Network Models (ENMs) offer a faster alternative by modeling proteins as a system of beads and springs, using Normal Mode Analysis to predict motions (López-Blanco & Chacón, 2016; Tirion, 1996). ENMs were refined by Bahar et al. to include Gaussian and anisotropic movements (Bahar et al., 1997; Atilgan et al., 2001) and are implemented in tools like ProDy (Bakan et al., 2011; Zhang et al., 2021). More details on ENMs are given in Appendix C. Recent machine learning (ML)-based methods, such as AlphaFold (Jumper et al., 2021) and ESMFold (Lin et al., 2022), have been adapted for flexibility prediction. Combining these models with flow matching techniques in AlphaFlow has enabled sampling of broader conformational spaces at about 10 times the speed of MD, though with reduced structural precision (Jing et al., 2024).
+
+Protein flexibility engineering. Engineering flexibility in proteins remains a significant challenge. Traditional approaches often involve quantum mechanics (QM), QM/MM, molecular dynamics (MD), and Monte-Carlo simulations, along with extensive experimental testing (Planas-Iglesias et al., 2021). For example, Yu & Dalby (2018) employed MD simulations to analyze correlated motions in enzyme dynamics. Similarly, Osuna's group developed the Shortest Path Map (SPM) algorithm to identify key conformational hotspots in enzymes like retro-aldolase and monoamine oxidase (Casadevall et al., 2024; Romero-Rivera et al., 2017). Kamerlin's lab also utilized MD to study enzyme dynamics, particularly in  $(\beta \alpha)_{8}$  barrel enzymes (Romero-Rivera et al., 2022). Another approach to flexibility design involves creating chimeras, where regions of one protein are transplanted into another. Notable examples include chimeric TEM1/PSE4  $\beta$ -lactamases studied
+
+with NMR (Morin et al., 2010) and Kamerlin's transplantation of the WPD loop in phosphatases (Moise et al., 2018; Shen et al., 2022). Damborsky's lab has focused on transplanting dynamic loops to explore flexibility and catalytic mechanisms in luciferases (Schenkmayerova et al., 2023; Chaloupkova et al., 2019), leading to the development of LoopGrafter, a web tool for in silico protein flexibility design (Planas-Iglesias et al., 2022).
+
+Protein inverse folding. The task of predicting a sequence that folds into a given backbone structure (protein inverse folding) has become a key area where machine learning significantly contributes to protein design, along with structure prediction. ProteinMPNN (Dauparas et al., 2022), based on the graph neural network architecture introduced by Ingraham et al. (2019), is one of the most widely used tools for this task. Recent benchmarks, such as ProteinInvBench (Gao et al., 2023c), feature newer methods, e.g., KWDesign and PiFold (Gao et al., 2023b;a), which outperform ProteinMPNN in standard metrics such as sequence recovery on datasets like CATH4.3 (Pearl et al., 2003). Despite these advancements, ProteinMPNN remains the community standard, largely due to its extensive experimental validation. This highlights the limitations of such metrics as sequence recovery, which do not fully capture a model's ability to design functional or stable proteins (Wang et al., 2023). As a result, there is growing interest in developing more comprehensive evaluation metrics and tasks that better reflect real-world protein design challenges.
+
+# 3 ANALYSIS OF DATA SOURCES FOR PROTEIN FLEXIBILITY
+
+One of the key challenges in the engineering of protein flexibility is the lack of consistent and affordable methods for flexibility measurement or estimation. This hinders the development of flexibility datasets useful for learning. The most reliable type of data come from experiments and from physics-based simulations, in particular molecular dynamics simulations. However, experimental data are costly, time-consuming, and highly heterogeneous due to biases and different calibrations of the experiments. Similarly, molecular dynamics data are time-consuming and subject to many human-made choices, such as the choice of force field. Our goal in this section is to analyze the available data and data generation methods for protein flexibility that could be utilized for learning a fast flexibility predictor. We first specify how we quantify protein flexibility using relevant experimental and computational methods (Section 3.1). Then we critically compare these methods to identify a relevant data source for learning (Section 3.2).
+
+# 3.1 METHODS FOR QUANTIFICATION OF PROTEIN FLEXIBILITY
+
+In this work we focus on flexibility as a quantity describing the ability of individual residues (mainly their backbone) to move within the protein structure. To this end, we introduce in this section the relevant methods for quantification of protein flexibility, and in particular, the metrics used with each method for reporting per-residue protein flexibility values.
+
+MD. Molecular Dynamics (MD) simulations can be used to quantify protein flexibility by simulating the atomic movements in a given modeled solvent environment. MD simulations are considered an accurate physics-based method and we thus regard them as the gold standard for flexibility estimation. However, their limitations should still be noted: (i) the simulation of the solvent is only approximate, affecting mainly the simulation accuracy of side-chain atoms, and (ii) potential interactions driven by molecules commonly present in the natural environment but not explicitly present in the simulation might affect the flexibility. For working with data on MD simulation, we select the recently published ATLAS dataset (Vander Meersche et al., 2023), particularly due to its excellent level of data curation. ATLAS consists of MD trajectories of 1390 proteins, each simulated in three replicas for 100ns per replica. For each protein and its residues, we quantify their flexibility by computing their root mean square fluctuations (RMSF) averaged over the three simulation replicas (see Appendix B for details on RMSF).
+
+B-factors. The crystallographic B-factor carries information on protein flexibility, as it describes the inherent thermal vibrations of atoms in a crystal lattice. In such an organized environment, however, the natural movement of the protein is diminished, and crystal packing artifacts may lead to an underestimation of the real protein motions (Eyal et al., 2005). In this paper, we report B-factors obtained from the Protein Data Bank (PDB) (Berman et al., 2000). Note that for a few cases from PDB, we also kept the B-factors originating from NMR experiments, whose nature is different from those obtained in crystallographic experiments (for details, see Appendix A).
+
+Table 1: Pearson correlation coefficients of flexibility predictions obtained by PDB B-factors and by computational methods (see Section 3.1 for details of the studied flexibility quantification methods). The reported coefficients are averaged over the 1383/1390 proteins from the ATLAS dataset (some were skipped due to missing pieces of structure resulting in NaNs from ENMs). As the table is symmetric, we show only values in the upper triangle. The best result in terms of correlation to MD simulations is highlighted in bold, and the second best is underlined.
+
+<table><tr><td></td><td>MD</td><td>B-factors</td><td>AlphaFold2</td><td>ESMFold</td><td>GNM</td><td>ANM</td></tr><tr><td>MD</td><td>-</td><td>0.51</td><td>0.71</td><td>0.57</td><td>0.76</td><td>0.77</td></tr><tr><td>B-factors</td><td></td><td>-</td><td>0.52</td><td>0.38</td><td>0.55</td><td>0.51</td></tr><tr><td>AlphaFold2</td><td></td><td></td><td>-</td><td>0.66</td><td>0.55</td><td>0.56</td></tr><tr><td>ESMFold</td><td></td><td></td><td></td><td>-</td><td>0.45</td><td>0.43</td></tr><tr><td>GNM</td><td></td><td></td><td></td><td></td><td>-</td><td>0.94</td></tr><tr><td>ANM</td><td></td><td></td><td></td><td></td><td></td><td>-</td></tr></table>
+
+AlphaFold2. The protein structure prediction model AlphaFold2 (AF2) (Jumper et al., 2021) contains a confidence model predicting pLDDT scores. While the pLDDT score is primarily a measure of estimated confidence by AlphaFold2 (that may be induced by sequence variability in the region, for instance), it has been shown to be negatively correlated with flexibility (Ruff & Pappu, 2021; Guo et al., 2022; Saldano et al., 2022) and exploited to approximate flexibility as measured by Nuclear Magnetic Resonance (NMR) (Ma et al., 2023). Therefore, to quantify flexibility with AF2, we use (1- pLDDT). Since the pLDDT prediction cannot be easily decoupled from the structure prediction, its runtime is the same as for the structure prediction, resulting in a relatively slow speed.
+
+ESMFold. The protein structure prediction model ESMFold (Lin et al., 2022) also provides a pLDDT confidence model but with a significantly faster runtime than AlphaFold2. As in the case of AlphaFold2, we use (1-pLDDT) to quantify flexibility with ESMFold.
+
+GNM. Gaussian Network Models (GNM) are a type of Elastic Network Models that model the atomic motions as isotropic, i.e., they model only the magnitudes of the displacements. To quantify flexibility, we use GNM to estimate the RMSF of each residue using the input structure from the ATLAS dataset that underwent a short MD relaxation. For details on the computation of RMSF for GNMs, see Appendix B.
+
+ANM. In contrast, Anisotropic Network Models (ANM) model the atomic motions as anisotropic, i.e. they model also the direction of the displacement. As in the case of GNM, we quantify flexibility using RMSF estimated from an MD relaxed structure. For details on the computation of RMSF for ANMs, see Appendix B.
+
+# 3.2 COMPARISON OF PROTEIN FLEXIBILITY QUANTIFICATION METHODS
+
+Next, we compare the selected methods for quantification of protein flexibility on the ATLAS dataset (see Table 1). This is done via computing the correlation between the per residue flexibility profiles obtained through the different methods. The results of this analysis are discussed next.
+
+Table 1 shows that the B-factors obtained for the ATLAS proteins from the PDB database do not correlate well with the other methods. We hypothesize that this is due to the crystal packing effect. The root mean square fluctuations (RMSF) from MD simulations thus appear to be a more reliable proxy for the true flexibility of single proteins (i.e., not in a crystal). We further support this conclusion by training a predictor of the B-factor data, which reached only modest performance, hinting at B-factor data being of insufficient quality for learning (see Appendix D). Hence, in the rest of the paper, we regard the flexibility computed using RMSF from the MD simulations (denoted further as MD) as the gold standard, which we will try to approach using learning.
+
+Measured by the correlation to the RMSFs from the MD simulations, the best-performing computational methods are the Anisotropic Network Models (ANM), closely followed by the Gaussian Network Models (GNM) (see highlighted row in Table 1). Here, both ANM and GNM are evaluated over the MD relaxed structures from the ATLAS dataset and both result in mutually highly correlated RMSFs (Pearson correlation coefficient 0.94).
+
+Having each simulation in three replicas in the ATLAS dataset allows us to evaluate the level of reliability of the RMSF computed using the short ATLAS MD trajectories. To this end, we compute
+
+the correlation of RMSF between different simulation replicas of the same protein. We find that the different replicas mutually correlate on average with the Pearson correlation coefficient (PCC) of 0.88. We consider this number to be the indicative upper bound for the average correlation of any flexibility predictor trained and evaluated on the ATLAS dataset.
+
+ANMs require a protein (backbone) structure on the input, which can be a limiting factor if such a structure is not available. While the ATLAS dataset provides MD relaxed structures, we also want to know how the ANMs perform outside of the ATLAS setting, for example, on structures coming from (i) the PDB, (ii) protein folding of the original sequence by ESMFold, or (iii) protein folding of an alternative sequence obtained with ProteinMPNN (Dauparas et al., 2022). Therefore, we investigate the effect of the type of input structure on the performance of ANMs in Table 6 in Appendix E, where we show that redesigning the sequence of a protein with ProteinMPNN does not have a significant effect on the performance of ANM.
+
+# 4 METHODS
+
+In this section, we introduce our new methods for predicting protein flexibility (Section 4.1) and the new Flexpert-Design framework demonstrating the applicability for the problem of engineering protein flexibility (Section 4.2).
+
+# 4.1 LEARNING TO PREDICT PROTEIN FLEXIBILITY
+
+Our above analysis of methods for protein flexibility quantification (see Section 3.2) motivates the effort to develop a new protein flexibility predictor as there is still enough room for improvement in terms of correlation to MD (compare the PCC of 0.77 for ANM with the indicative upper bound on PCC of 0.88). Here we address the challenge by introducing a novel and fast predictor of protein flexibility in two versions: Flexpert-Seq, taking a protein sequence on the input, and Flexpert-3D, taking both sequence and structure on the input. In both versions, the method learns to predict the RMSF of MD simulations in a supervised manner using the ATLAS dataset. The two predictors are described next.
+
+Flexpert-Seq: Predicting flexibility from sequence. The goal is to develop a predictor of protein flexibility that uses only a protein sequence on the input and learns to predict the protein sequence flexibility as close as possible to the predictions from MD trajectories. In detail, given a dataset  $\mathcal{D}_{Seq} = \{(s_i, y_i)\}_{i \in D}$  of protein sequences  $s_i \in \mathcal{A}^{N_i}$  and per residue flexibility labels  $y_i \in \mathbb{R}^{N_i}$ , where  $D$  is the set indexing the protein in the dataset,  $\mathcal{A}$  is the alphabet of 20 natural amino acids and  $N_i$  is the number of amino acids of protein  $i$ , the goal is to train a regression model  $\mathcal{F}_{Seq}: \mathcal{A}^{N_i} \to \mathbb{R}^{N_i}$  to predict the per residue flexibility from a protein sequence. To overcome the limitations coming from the small amount of annotated data (1390 proteins in the ATLAS dataset), we leverage the pretrained protein language model ProtTrans (Elnaggar et al., 2022) to obtain
+
+![](images/acb1c75c3d16e51b1b70d7d40be700ba53e43dbaa3554bbfa4c0656f83ddb0f3.jpg)  
+Figure 1: Overview of the architecture of our protein flexibility prediction methods Flexpert-Seq (the red box) and Flexpert-3D (the whole architecture). Icons of snowflake and flame denote frozen and unfrozen layers, respectively, in modules containing trainable parameters. Trainable (orange) and fixed (blue) modules denote whether any trainable unfrozen parameters are present in the module.
+
+per-residue embeddings, on top of which we learn a linear layer to regress a single scalar value for each residue. An alternative choice for the backbone protein language model could be the ESM-2 model, which was shown to perform similarly to ProtTrans in fine-tuning for diverse downstream
+
+tasks (Schmirler et al., 2024). Following the procedure proposed by Schmirler et al. (2024), we employ LoRA (Hu et al., 2021a) to fine-tune ProtTrans together with training the regression layer for our downstream task. See Figure 1 for an overview of the method.
+
+Flexpert-3D: Predicting flexibility from sequence and structure. Since inverse folding pipelines usually have access to protein backbones on the input, we also explore the strategy of using both protein sequence and protein backbone structure on the input to predict the protein sequence flexibility as close as possible to the predictions from MD trajectories. More formally, given a dataset  $\mathcal{D}_{3D} = \{(s_i,\chi_i,y_i)\}_{i\in D}$  of protein sequences  $s_i\in \mathcal{A}^{N_i}$ , protein backbone structures  $\chi_{i}\in \mathbb{R}^{4\times 3N_{i}}$  (4 atoms per residue backbone) and per residue flexibility labels  $y_{i}\in \mathbb{R}^{N_{i}}$ , where  $D$  is the set indexing the protein in the dataset,  $\mathcal{A}$  is the alphabet of 20 natural amino acids and  $N_{i}$  is the number of amino acids of protein  $i$ , the goal is to train a regression model  $\mathcal{F}_{3D}:A^{N_i}\times \mathbb{R}^{4\times 3N_i}\to \mathbb{R}^{N_i}$  to predict the per residue flexibility from protein sequence and backbone structure. As ANMs can be run relatively fast if the protein backbone is available and they were the best performing method as evaluated in Section 3.2, we tackle the task by combining our sequence-based method Flexpert-Seq with ANM-based flexibility values. This strategy is thus akin to learning how to correct the crude flexibility annotations returned by ANMs to match more accurate values derived from MD trajectories. To this end, we enhance the architecture of Flexpert-Seq by a CNN head acting as an adaptor of the ANM-predicted flexibility into the embedding space of Flexpert-Seq (see Figure 1). We train the CNN Adaptor together with the LoRA layers by fine-tuning the ProtTrans Encoder of Flexpert-Seq and, together with the regression layer, mapping the combined embeddings to per-residue flexibility.
+
+# 4.2 FLEXPERT-DESIGN: FLEXIBILITY ENGINEERING VIA FLEXIBILITY-AWARE INVERSE FOLDING
+
+In this section, we apply our flexibility predictor Flexpert-3D for the task of engineering protein flexibility. The goal is to generate a protein sequence that respects a given protein backbone structure and a set of flexibility instructions.
+
+More formally, we are given a dataset  $\mathcal{D}_{bb} = \{(s_i,\chi_i)\}_{i\in D}$  of protein sequences  $s_i\in \mathcal{A}^{N_i}$  and protein backbone atom coordinates  $\chi_{i}\in \mathbb{R}^{4\times 3N_{i}}$  (4 atoms per residue backbone), where  $D$  is the set indexing the protein in the dataset,  $\mathcal{A}$  is the alphabet of 20 natural amino acids and  $N_{i}$  is the number of amino acids of protein  $i$ . We are also given a set of flexibility instructions  $F = \{\pmb {f}_i\}_{i\in D}$  where  $\pmb {f}_i\in \mathbb{R}^{N_i}$ . The goal is to train model  $\mathcal{P}_F:\mathbb{R}^N\times \mathbb{R}^{4\times 3N}\to \mathring{\mathcal{A}}^N$ , which takes  $N$  flexibility instructions and a backbone structure of a protein of length  $N$  and maps it to a protein sequence of the length  $N$ , respecting the input backbone and the flexibility instructions.
+
+We tackle the problem by leveraging the established protein inverse folding model ProteinMPNN and modifying it to take the flexibility instruction on the input. We then devise a learning strategy where we utilize our flexibility predictor Flexpert-3D to fine-tune the inverse folding model toward respecting the flexibility instructions. Details are given next.
+
+Teaching inverse folding model to engineer flexibility. The critical aspects of our flexibility engineering task are the choice of the flexibility engineering instructions  $F$  and the design of the learning strategy steering the inverse folding model to follow the instructions.
+
+Our novel training strategy is to inform the inverse folding model about the flexibility of the native sequence and to optimize the model toward generating sequences with flexibility preserved with respect to the input. We hypothesize that such training makes the model pay attention to the flexibility instructions  $\pmb{f} \in F$  provided on the input. During inference, this awareness enables us to exploit the flexibility input of the model to pass our actual instructions for modifying the flexibility with respect to the native one.
+
+For learning the flexibility of native sequences, we propose the following sequence of steps. First, we construct the set  $F^{native}$  of flexibility pseudolabels obtained with Flexpert-3D as
+
+$$
+F ^ {\text {n a t i v e}} = \left\{\boldsymbol {f} _ {i} ^ {\text {n a t i v e}} \mid \boldsymbol {f} _ {i} ^ {\text {n a t i v e}} = \mathcal {F} _ {3 D} \left(\boldsymbol {s} _ {i}, \boldsymbol {\chi} _ {i}\right), \left(\boldsymbol {s} _ {i}, \boldsymbol {\chi} _ {i}\right) \in D _ {b b}, i \in D \right\}, \tag {1}
+$$
+
+where the Flexpert-3D predictor is used to predict flexibility for all tuples of sequences  $s_i$  and backbone coordinates  $\chi_i$  from dataset  $\mathcal{D}_{bb}$ . These predicted flexibilities  $f_i^{native}$ , which we call "native", are used in training to inform the inverse folding model about flexibility.
+
+![](images/d858946caf0813bfcd826f9d34579b77875684d9a277b4d3c878f81fa9617329.jpg)  
+Figure 2: Overview of the Flexpert-Design method for fine-tuning of inverse folding models toward steerability by requirements on protein flexibility. The inverse folding model ProteinMPNN, taking protein backbone structure on the input, is modified to additionally take on the input flexibility instructions. This is implemented by adding a flexibility instruction to the zero-initialized node feature for each residue. The predicted sequence is sampled in a differentiable way using Gumbel-Softmax and passed to Flexpert-3D together with the input structure. Note that we are making use of the assumption that ProteinMPNN does not modify the backbone structure as we are matching together the newly predicted sequence with the ground truth backbone structure. The flexibility of the sequence predicted by Flexpert-3D is passed to the loss function  $L_{Flex}$  which computes the distance between the input flexibility instructions and the flexibility of the predicted sequence. Icons of snowflake and flame denote frozen and unfrozen layers, respectively, in modules containing trainable parameters. Trainable (orange) and fixed (blue) modules denote whether any trainable unfrozen parameters are present in the module.
+
+Second, we take a vanilla ProteinMPNN inverse folding model  $\mathcal{P}:\mathbb{R}^{4\times 3N}\to \mathcal{A}^N$ , which maps protein backbones of length  $N$  to protein sequences of the same length, and modify it to a flexibility-aware ProteinMPNN model  $\mathcal{P}_F:\mathbb{R}^N\times \mathbb{R}^{4\times 3N}\to \mathcal{A}^N$ , which accepts flexibility instructions  $\pmb {f}\in F\subset \mathbb{R}^N$  on the input as well (see Figure 2).
+
+Third, we introduce a new loss function  $\mathcal{L}_{Flex}$  for guidance of the inverse folding model  $\mathcal{P}_F$  toward preservation of the input flexibility by matching the input flexibility instructions  $\pmb{f}$  to the output flexibility  $\mathcal{F}_{3D}(\hat{s},\chi)$ :
+
+$$
+\mathcal {L} _ {F l e x} = \mathcal {L} \left(\mathcal {F} _ {3 D} (\hat {\boldsymbol {s}}, \boldsymbol {\chi}), \boldsymbol {f}\right), \tag {2}
+$$
+
+where  $\mathcal{L}$  is a regression loss (MSE or L1),  $\chi$  is the input backbone structure, and  $\hat{s} = \mathcal{P}_F(f,\chi)$  is the sequence predicted by the flexibility-aware inverse folding model  $\mathcal{P}_F$ . We illustrate our method, including the loss function  $\mathcal{L}_{Flex}$ , in Figure 2. Using the pseudolabels of flexibility  $F^{native}$  obtained by Flexpert-3D as the flexibility instructions in training, the  $L_{Flex}$  loss ensures in the training process that the model learns to preserve flexibility it has seen on the input. During inference, the learned flexibility preservation is exploited to pass non-native flexibility instructions to the model to steer the inverse folding toward generating sequences with the required flexibility.
+
+Training to recover flexibility and sequence. To train ProteinMPNN toward flexibility awareness, we still need to ensure it also works for its original objective of generating a protein sequence that fits the input backbone. Therefore, we first train ProteinMPNN using its standard Cross-Entropy loss  $\mathcal{L}_{Seq}$  and optimize it for maximal sequence recovery. Taking the pre-trained ProteinMPNN model weights, we further fine-tune them using the combined  $\mathcal{L}_{Flexpert}$  loss:
+
+$$
+\mathcal {L} _ {\text {F l e x p e r t}} = \theta \cdot \mathcal {L} _ {\text {F l e x}} + (1 - \theta) \cdot \mathcal {L} _ {\text {S e q}}, \tag {3}
+$$
+
+where  $\theta \in [0,1]$  is the parameter mixing the two losses  $\mathcal{L}_{Flex}$  and  $\mathcal{L}_{Seq}$  into a single loss. The choice of  $\theta$  and the sequence recovery of the trained models is discussed in Appendix H.
+
+Table 2: Pearson correlation coefficients between fast sequence-based baselines (AlphaFold2 and ESMFold) and our method Flexpert-Seq (columns) and the RMSF values from MD simulations and from Anisotropic Network Models (ANM) evaluated over MD relaxed structures (rows). Note that an upper bound on correlation to MD is estimated to be 0.88. Evaluation performed for 139 proteins from ATLAS testset. Our approach outperforms the AlphaFold2 and ESMFold baselines and obtains the best correlation with the flexibility estimates obtained from Molecular Dynamics (see the value in bold). Interestingly, the correlation coefficient of our approach and Anisotropic Network Models (ANM) is just 0.66, suggesting that both our approach and ANM might be complementary, motivating the development of a predictor combining the sequence-based predictor with ANM.
+
+<table><tr><td></td><td>AF2</td><td>ESMFold</td><td>Flexpert-Seq (ours)</td></tr><tr><td>MD</td><td>0.72</td><td>0.58</td><td>0.78</td></tr><tr><td>ANM</td><td>0.56</td><td>0.43</td><td>0.66</td></tr></table>
+
+Preserving differentiability through discrete language model input. An interesting challenge in utilizing the protein language model-based predictor Flexpert-3D inside of an end-to-end training pipeline is the question of preserving differentiability at the interface of the inverse folding model output (continuous probability distribution) and the language model input (discrete tokens) (see Figure 2). Our solution to this problem is to sample the learned distribution of amino acids using the Gumbel-Softmax trick (Jang et al., 2017). In particular, we use its straight-through variant, which takes the argmax of the input distribution in the forward pass but preserves the gradients by approximating the argmax with the Gumbel-Softmax distribution in the backward pass. This enables us to sample a discrete sequence for the language model while preserving the gradients and enabling end-to-end training of the entire pipeline.
+
+# 5 RESULTS
+
+In this section, we report the performance of our methods Flexpert-Seq and Flexpert-3D in terms of protein flexibility prediction (Section 5.1) and the performance of Flexpert-Design in terms of increasing flexibility in engineered protein regions (Section 5.2).
+
+# 5.1 FLEXIBILITY PREDICTION
+
+Experimental setup. We use the sequences and MD relaxed structures from the ATLAS dataset as our inputs. The flexibility labels are the per residue RMSFs averaged over the 3 MD replicas for each protein. We split the dataset using topology splitting, with the additional requirement that no topologies present in the CATH4.3 test set are present in our ATLAS training set. The reason is to prevent potential leakage when combining the flexibility predictors into the Flexpert-Design pipeline which is trained using CATH4.3. We evaluate the Pearson correlation coefficient (PCC) between the predicted flexibility and the ground truth flexibility (RMSF) for each protein in the ATLAS test set. We report the average PCC across the test set.
+
+Results. Our predictors Flexpert-Seq and Flexpert-3D outperform the baselines when using only the sequence on the input and when also using the structure, respectively. See Table 2 for the sequence case and Table 3 for the case when using sequence and structure. Note that our predictor Flexpert-3D, which partly relies on ANM, improves over them significantly and does not rely on them exclusively (see PCC of 0.78 to ANM).
+
+We further tested our predictors Flexpert-Seq and Flexpert-3D on a separate dataset called mdCATH (Mirarchi et al., 2024), which features MD trajectories obtained at various simulation temperatures ranging from  $320\mathrm{K}$  to  $450\mathrm{K}$ . Using this data, we test the effect of the simulation temperature on the performance of our models trained on the ATLAS dataset simulated at a physiological temperature of  $300\mathrm{K}$ . The experiment confirmed that the further the temperature is from the simulation temperature in the training set, the lower the Pearson correlation coefficients are between the model predictions and the RMSFs, see Appendix F.
+
+In Appendix G, we provide qualitative examples of Flexpert-Seq and Flexpert-3D predictions and how these predictions compare to the ground truth RMSF and to the predictions of ANM.
+
+Table 3: Pearson correlation coefficients between fast structure-based baselines (Anisotropic Network Models and Gaussian Network Models) and our method Flexpert-3D (columns) and the RMSF values from MD simulations and from Anisotropic Network Models (ANM) evaluated over MD relaxed structures (rows). Evaluation is performed for 139 proteins from ATLAS test set. Note that an upper bound on correlation to MD is estimated to be 0.88. Also, note that our approach outperforms the ANM and GNM baselines and obtains the best correlation with the flexibility estimates obtained from Molecular Dynamics (row MD). The correlation coefficient of our model with ANM is 0.78 (see the bottom row), indicating that our model relies on ANM input only partially and actually learns novel information from the sequence.
+
+<table><tr><td></td><td>ANM</td><td>GNM</td><td>Flexpert-3D (ours)</td></tr><tr><td>MD</td><td>0.76</td><td>0.76</td><td>0.83</td></tr><tr><td>ANM</td><td>1.00</td><td>0.93</td><td>0.78</td></tr></table>
+
+# 5.2 PROTEINMPNN CAN BE STEERED BY FLEXIBILITY INSTRUCTIONS
+
+Experimental setup. The evaluation is done using the CATH4.3 dataset. The set of flexibility-increasing instructions  $F_{\uparrow}^{\tau, S}$  is constructed in the following way: (i) The flexibility instructions are initialized by the native flexibilities  $F^{native}$  (as defined in Equation (1)); (ii) For each protein independently, a random contiguous segment  $S$  of length  $|S|$  is subselected from the sequence. In case  $|S| \geq N$ , where  $N$  is the length of the protein, the whole protein is selected as the segment; (iii) For each residue  $i$  in each selected segment  $S$ , its flexibility instruction  $f_{i}$  is incremented by  $\tau > 0$ , aiming at increasing the residues flexibility:
+
+$$
+f _ {i} = f _ {i} ^ {\text {n a t i v e}} + \tau \cdot \mathbb {1} _ {i \in S}, \tag {4}
+$$
+
+where  $\mathbb{1}_{i\in S}$  is the indicator function for the set of amino acids corresponding to the segment  $S$
+
+We then take the set of flexibility increasing instructions  $F_{\uparrow}^{\tau, S}$  together with the corresponding backbones  $\mathcal{D}_{bb}$  from the CATH4.3 test set and pass it to the flexibility-aware ProteinMPNN  $\mathcal{P}_F$  trained using our Flexpert-Design method to obtain predicted protein sequences  $\hat{s}$ :
+
+$$
+\hat {\boldsymbol {s}} _ {j} = \mathcal {P} _ {F} \left(\boldsymbol {f} _ {j}, \boldsymbol {\chi} _ {j}\right), \forall \left(\boldsymbol {f} _ {j}, \boldsymbol {\chi} _ {j}\right) \in F _ {\uparrow} ^ {\tau , S} \times \mathcal {D} _ {b b}. \tag {5}
+$$
+
+We evaluate the predicted sequences using the metrics described next. For each predicted sequence  $\hat{s}_j$ , we identify the mutated residues  $\mathcal{M}_j \subset S_j$  inside the segment  $S_j$  engineered for increased flexibility. For such residues, we define the flexibility enrichment ratio  $r_{ij}$ :
+
+$$
+r _ {i j} = \frac {\mathcal {F} _ {3 D} \left(\hat {\boldsymbol {s}} _ {j} , \boldsymbol {\chi} _ {j}\right) _ {i}}{f _ {i j} ^ {\text {n a t i v e}}}, \tag {6}
+$$
+
+where  $\hat{s}_j$  is the predicted sequence for protein  $j$  with backbone  $\chi_j$ ,  $i \in \mathcal{M}_j$  indexes the mutated residues in the engineered region  $S_j$ , and  $f_{ij}^{native}$  is the native flexibility of the residue (see Equation (1)). The flexibility enrichment ratios  $r_{ij}$  are then used to report the median enrichment ratio and the proportion of flexibility-increasing mutations, i.e., those for which  $r_{ij} > 1$ .
+
+Results. Table 4 shows that our method Flexpert-Design successfully managed to engineer increased flexibility in the engineered region. This result is presented in more detail in Figure 3, which shows that Flexpert-Design can shift the flexibility distribution in the engineered segment toward higher flexibility. Furthermore, Figure 3 shows the change in the distribution of amino acid types in the engineered region. The observation that the proportion of Alanine (A) and Glycine (G) increases goes in line with the biochemical understanding that these amino acids are smaller and tend to be more flexible. The sensitivity of Flexpert-Design to the size of the engineered region  $|S|$  and the value of the flexibility increasing instruction  $\tau$  is studied in Appendix J.
+
+Interestingly, the most contributing component of Flexpert-Design is the inclusion of flexibility input and training with the Flexpert-3D generated flexibility pseudolabels. The additional optimization using  $\mathcal{L}_{Flex}$  brings only relatively modest improvements in steerability (see Table 4 and Figure 8). Please compare "Flexpert-Design (without loss)", which does not use  $\mathcal{L}_{Flex}$ , and the full model denoted "Flexpert-Design". The results also clearly demonstrate significant improvements of our
+
+![](images/561d63124cc11491f3da4e723e2a22c2c1f81d3d5b50d88b65b839b7b4b1446a.jpg)  
+Figure 3: Comparison of the enrichment ratio distributions (left) and the amino acid type distributions (right) for the flexibility-increasing experiment with segment lengths  $|S| = 50$  and  $\tau = 5$ . Our Flexpert-Design model is compared to its variant that was trained without the  $\mathcal{L}_{Flex}$  loss but with the flexibility on the input. Vanilla ProteinMPNN is presented as a baseline. On the left, the median values for each distribution are denoted with a vertical line. On the right, the native amino acid distributions in the examined segments are presented for reference.
+
+![](images/dd97af3f276637d40917b525017e25609cb5e4c2dcd11232f70515b3e812799c.jpg)
+
+Table 4: Comparison of the ability of flexibility engineering models to increase flexibility of engineered segments. The presented median enrichment ratios  $\mathrm{Med}(r_{ij})$  and the proportion of flexibility-increasing mutations were obtained using the CATH4.3 test set with engineered segments of length  $|S| = 50$ , and flexibility engineering instructions corresponding to native flexibility incremented by  $\tau = 5$  in the engineered segment. The best results in each category are highlighted in bold, second best are underlined.  
+
+<table><tr><td></td><td>Median enrich. ratio Med(rij) ↑</td><td>Prop. of flexibility incr. mutations ↑</td></tr><tr><td>Flexpert-Design (ours)</td><td>1.52</td><td>0.83</td></tr><tr><td>Flexpert-Design (w/o loss)</td><td>1.43</td><td>0.80</td></tr><tr><td>ProteinMPNN (baseline)</td><td>1.07</td><td>0.61</td></tr></table>
+
+model in the ability to engineer flexibility compared to the ProteinMPNN baseline. Additional results in Section H of the Appendix also demonstrate virtually no loss in sequence recovery metrics of our approach compared to the ProteinMPNN baseline. The analysis of the structure preservation of the sequences with engineered increased flexibility is given in Appendix I.
+
+We also investigated the use of our model for engineering of decrease in flexibility, results of which we discuss in Appendix K.
+
+# 6 CONCLUSIONS
+
+In this work, we address the problem of engineering protein flexibility—a highly relevant task due to its potential impact on the field of protein design. Firstly, we analyze the available data obtained in wet-lab experiments, physics-based simulations, or as a byproduct of machine learning-based structure prediction models. We identify the flexibility estimate by molecular dynamics simulation as the best learning target, set an upper bound on the potential performance of the learned predictor, and analyze the effect of the type of input protein structure on the performance of Anisotropic Network Models. Secondly, we develop a protein language model-based predictor of protein flexibility working either with protein sequence (Flexpert-Seq) or with both sequence and protein backbone structure on the input (Flexpert-3D) and show that our predictors outperform the relevant baselines. Thirdly, we propose a learning strategy Flexpert-Design, which uses the Flexpert-3D flexibility predictor to steer an inverse folding model toward a generation of sequences with increased flexibility. We demonstrate that inverse folding models can be steered toward generating protein sequences with increased flexibility as measured by our protein flexibility predictor. This opens up new possibilities for development of proteins with enhanced biological activities. In some practical protein engineering tasks, it would also be of interest to engineer protein sequences with decreased flexibility, where the ability of our current Flexpert-Design model seems limited (see Appendix K). This motivates future development of new models, which would provide a tighter control of the designed sequence's flexibility and which would be successful regardless of whether the flexibility was instructed to increase or decrease. An interesting future direction could be to use Flexpert-3D or Flexpert-Seq for guidance of discrete flow matching models.
+
+# REPRODUCIBILITY STATEMENT
+
+The code, together with the instructions on how to download the data and trained weights for the reproduction of this work, can be found at https://github.com/KoubaPetr/Flexpert.
+
+# ACKNOWLEDGEMENTS
+
+This work was supported by the Ministry of Education, Youth and Sports of the Czech Republic through projects ELIXIR (No. LM2023055), ESFRI RECETOX RI (No. LM2023069), and OP JAK CLARA (No. 02_23_029). We acknowledge VSB - Technical University of Ostrava, IT4Innovations National Supercomputing Center, Czech Republic, for awarding this project access to the LUMI supercomputer, owned by the EuroHPC Joint Undertaking, hosted by CSC (Finland) and the LUMI consortium through the Ministry of Education, Youth and Sports of the Czech Republic through the e-INFRA CZ (No. 90254). This work was also supported by the European Union (ERC project FRONTIER (No. 101097822), ELIAS (No. 101120237), CLARA (No. 101136607)), the Technology Agency of the Czech Republic under the NCC Programme from the state budget (No. RETEMED TN02000122) and under the NRP from the EU RRF (No. TEREP TN02000122/001N), and the CETOCOEN EXCELLENCE (No. 857560) projects supported from the European Union's Horizon 2020 research and innovation programme. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. Petr Kouba is a holder of the Brno Ph.D. Talent scholarship funded by the Brno City Municipality and the JCMM.
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+Ruidan Shen, Rory M Crean, Keith J Olsen, Marina Corbella, Ana R Calixto, Teisha Richan, Tiago A S Brandão, Ryan D Berry, Alex Tolman, J Patrick Loria, Sean J Johnson, Shina C L Kamerlin, and Alvan C Hengge. Insights into the importance of WPD-loop sequence for activity and structure in protein tyrosine phosphatases. Chem. Sci., 13(45):13524-13540, November 2022.
+
+M M Tirion. Large amplitude elastic motions in proteins from a single-parameter, atomic analysis. Phys. Rev. Lett., 77(9):1905-1908, August 1996.  
+Lukas Uhrik, Tomas Henek, Joan Planas-Iglesias, Josef Kucera, Jiri Damborsky, Martin Marek, and Lenka Hernychova. Study of protein conformational dynamics using hydrogen/deuterium exchange mass spectrometry. Methods Mol. Biol., 2652:293-318, 2023.  
+Yann Vander Meersche, Gabriel Cretin, Aria Gheeraert, Jean-Christophe Gelly, and Tatiana Galochkina. ATLAS: protein flexibility description from atomistic molecular dynamics simulations. Nucleic Acids Research, 52(D1):D384-D392, 11 2023. ISSN 0305-1048. doi: 10.1093/nar/gkad1084. URL https://doi.org/10.1093/nar/gkad1084.  
+Chuanrui Wang, Bozitao Zhong, Zuobai Zhang, Narendra Chaudhary, Sanchit Misra, and Jian Tang. Pdb-struct: A comprehensive benchmark for structure-based protein design, 2023. URL https://arxiv.org/abs/2312.00080.  
+Alexander Wlodawer, Wladek Minor, Zbigniew Dauter, and Mariusz Jaskolski. Protein crystallography for non-crystallographers, or how to get the best (but not more) from published macromolecular structures. FEBS J., 275(1):1-21, January 2008.  
+Lei Yang, Guang Song, and Robert L Jernigan. Protein elastic network models and the ranges of cooperativity. Proceedings of the National Academy of Sciences, 106(30):12347-12352, 2009.  
+Haoran Yu and Paul A Dalby. Coupled molecular dynamics mediate long- and short-range epistasis between mutations that affect stability and aggregation kinetics. Proc. Natl. Acad. Sci. U. S. A., 115(47):E11043-E11052, November 2018.  
+She Zhang, James M Krieger, Yan Zhang, Cihan Kaya, Burak Kaynak, Karolina Mikulska-Ruminska, Pemra Doruker, Hongchun Li, and Ivet Bahar. ProDy 2.0: increased scale and scope after 10 years of protein dynamics modelling with python. Bioinformatics, 37(20):3657-3659, October 2021.
+
+# APPENDIX
+
+# A EXPERIMENTAL MEASUREMENT OF PROTEIN FLEXIBILITY BY NMR AND HDX-MS
+
+In contrast to crystallographic methods directly obtaining the B-factors (temperature factors), NMR reads chemical shift perturbations, in other words, how the proximity of different atoms affects the chemical environment of each other. As a product of these measurements, atom-to-atom distance restraints can be inferred. Protein flexibility can be derived as the change in the intensity and amplitude of the read signals over time; thus, different sets of restraints can be inferred for different time points (Hu et al., 2021b). Hence, NMR usually provides an ensemble of structural models, from which atomistic Root Mean Square Fluctuations (RMSFs) can be derived, which can be converted to B-factors by the following equation:
+
+$$
+B _ {i} = \frac {8 \pi^ {2}}{3} \left(R M S F _ {i}\right) ^ {2}, \tag {7}
+$$
+
+where  $B$  is the B-factors and  $i$  represents an individual atom (Emperador et al., 2010).
+
+The HDX-MS relies on the exchange of isotopic hydrogen in a deuterium-rich medium over a given span of time to detect the protein segments that are more exposed to solvent. Flexible regions (which often lay on the protein surface away from the hydrophobic tightly packed protein core) allow for higher solvent penetrance and, thus, attain higher deuteration values when compared to less flexible areas (Uhrik et al., 2023).
+
+# B ROOT MEAN SQUARE FLUCTUATIONS (RMSF) AS A MEASURE OF FLEXIBILITY
+
+The Root Mean Square Fluctuations (RMSF) are a way of quantifying protein flexibility, typically based on simulated data of dynamics or based on simplified models of dynamics such as the Elastic Network Models (ENM). In this work, we use RMSF computed on top of both MD simulations and ENM (Anisotropic Network Models and Gaussian Network Models). These RMSFs were obtained in the following way:
+
+RMSF from MD simulations. We used the values provided directly with the ATLAS dataset, which provides the per residue RMSF for each of its simulation replicas. To represent each protein's flexibility profile, we used the per residue RMSF averaged over all 3 simulation replicas of each protein. The values provided by the ATLAS dataset were computed as standard deviation of the  $\alpha$ -carbon displacements, using the GROMACS software. That is, the RMSF for MD simulations were computed using the following formula:
+
+$$
+\rho_ {i} = \sqrt {\left\langle \left| \left| r _ {i} - \left\langle r _ {i} \right\rangle \right| \right| _ {2} ^ {2} \right\rangle}, \tag {8}
+$$
+
+where  $\rho_{i}$  denotes the RMSF of residue  $i$ ,  $r_i$  stands for the coordinates of the  $\alpha$ -carbon of residue  $i$  in the global reference frame, the angle brackets  $\langle \rangle$  stand for the average over all the frames of the simulation and  $||\cdot ||_2$  denotes the Euclidean norm.
+
+RMSF for Elastic Network Models (Anisotropic Network Models). We used the ProDy (Bakan et al., 2011) package to compute the ENMs and the corresponding RMSF values. Firstly, we used ProDy to build the Hessian matrix, based on the  $\alpha$ -carbon coordinates of our input structure and the cutoff of 16 Angstrom, then the non-trivial modes of the Hessian were calculated, and consequently these modes were used to solve the problem of finding the inverse of the Hessian to compute the fluctuations according to the formula for the ENMs:
+
+$$
+\rho_ {i} = \sqrt {\frac {1}{C} \operatorname {T r} \left([ H ^ {- 1} ] _ {i i}\right)}, \tag {9}
+$$
+
+Table 5: Pearson correlation coefficients of MD flexibility and Flexpert-Seq predictions, when trained on B-factors (B) and per protein normalized B-factors ( $B^{norm}$ ).  
+
+<table><tr><td></td><td>Flexpert-Seq (trained on B)</td><td>Flexpert-Seq (trained on Bnorm)</td></tr><tr><td>MD</td><td>0.47</td><td>0.56</td></tr></table>
+
+where  $\rho_{i}$  denotes the RMSF of residue  $i$ ,  $C$  denotes a constant that is typically equal to the product of the Boltzmann constant and the temperature ( $C = k_{B}T$ ), but for us this is arbitrary as we are interested in the correlations of the fluctuations and the correlations are not affected by the scaling by a constant. The  $\mathrm{Tr}$  in the formula stands for the trace of the matrix  $[H^{-1}]_{ii}$  which is the 3x3 submatrix of the inverse Hessian corresponding to the residue  $i$  (the 3x3 submatrix comes from the fact that the  $\alpha$ -carbon positions are represented in 3D).
+
+RMSF for Elastic Network Models (Gaussian Network Models). For Gaussian Network models, the procedure is analogous to the case of Anisotropic Network Models. The only difference is that instead of Hessian matrix, the Kirchoff matrix is used:
+
+$$
+\rho_ {i} = \sqrt {\frac {1}{C} \operatorname {T r} \left([ \Gamma^ {- 1} ] _ {i i}\right)}, \tag {10}
+$$
+
+where  $\Gamma$  is the Kirchhoff matrix constructed for the  $\alpha$ -carbon coordinates of the input structure considering the cutoff of 16 Angstrom.
+
+More details on calculation of fluctuations from ENMs (both Anisotropic and Gaussian Network Models) can be found in (Yang et al., 2009).
+
+# C FINER-GRAINED ELASTIC NETWORK MODELS
+
+The complexity of the ENM model can increase by considering different types of beads and springs (i.e., each amino acid and all the possible combinations of pairs) or making them finer-grained (i.e., considering more than one bead per amino acid), approaching the model at the atomistic resolution. The order of the algorithm increases at least quadratically with the complexity of the model, making these solutions more accurate but less efficient. Some implementations of modified ENMs are available in ENCoM (Frappier & Najmanovich, 2014) and DynaMut (Rodrigues et al., 2021).
+
+# D THE DIFFICULTY OF LEARNING FROM B-FACTORS COMPARED TO MD TRAJECTORIES
+
+We also investigated the potential of B-factors to be used as data for learning to predict protein flexibility. To this end, we retrained Flexpert-Seq using B-factors instead of the RMSFs from MD trajectories. Because the absolute values of B-factors strongly depend on the calibration of the experiment, we trained both on the raw B-factors as well as on the B-factors standardized over each protein chain separately, using the following equation:
+
+$$
+B _ {i} ^ {\text {n o r m}} = \frac {B _ {i} - \bar {B}}{\sigma (B)} \tag {11}
+$$
+
+where  $B_{i}$  is the raw B-factor of residue  $i$ ,  $\bar{B} = \frac{1}{N}\sum_{j=1}^{N}B_{j}$  is the average raw B-factor of a protein of size  $N$  and  $\sigma B = \sqrt{\frac{1}{N-1}\sum_{j=1}^{N}(B_{j} - \bar{B})^{2}}$  is the standard deviation of the raw B-factors across protein  $i$ .
+
+Table 5 reports the performance of Flexpert-Seq trained to predict B-factors and normalized B-factors. We see the performance is significantly compromised compared to the results of Flexpert-Seq trained on MD data (Pearson correlation coefficient to MD of 0.76). We suspect that the experimental data suffer from the crystal packing effect, and they are too noisy to reliably learn from.
+
+Table 6: Pearson correlation coefficients between root mean square fluctuations (RMSF) from molecular dynamics (MD) simulations (row) and flexibility predictions obtained by Anisotropic Network Models (ANM) over differently processed input structures (columns).  
+
+<table><tr><td></td><td>ANM
+(MD str.)</td><td>ANM
+(PDB str.)</td><td>ANM
+(ESMFold str.)</td><td>ANM
+(pMPNN + ESMFold str.)</td></tr><tr><td>MD</td><td>0.77</td><td>0.73</td><td>0.73</td><td>0.71</td></tr></table>
+
+# E THE EFFECT OF TYPE OF INPUT STRUCTURE ON THE PERFORMANCE OF ANISOTROPIC NETWORK MODELS
+
+Table 6 shows that the PDB structure obscures some of the flexibility, probably due to the crystal packing effect present in most cases. Similarly, the ANM performance is diminished when a structure obtained by ESMFold is given on the input. Interestingly, preceding the ESMFold by applying ProteinMPNN to generate an alternative sequence leads to only a small additional drop in performance. This is explained by the fact that the standard ANMs implemented in ProDy (Bakan et al., 2011), which we used, do not consider the amino-acid identities and only work with the coordinates of the backbone. And since ProteinMPNN is optimized to preserve the backbone geometry, it has only a marginal effect on the performance of ANMs.
+
+# F EVALUATION OF FLEXPERT-3D AND FLEXPERT-SEQ ON MDCATH DATASET
+
+To evaluate the generalization capability of Flexpert-3D and Flexpert-Seq predictors outside of the ATLAS dataset, we performed an additional experiment evaluating the predictors on the mdCATH dataset (Mirarchi et al., 2024).
+
+The recently introduced mdCATH dataset presents a challenging test of generalizability for our predictors trained on the ATLAS dataset. The mdCATH dataset is larger in terms of the number of data points (5398 compared to 1390 in ATLAS) and the simulated data correspond to individual protein domains (i.e., they are shorter proteins, on average with 137 residues compared to 236 residues in ATLAS). Furthermore, the simulation times in mdCATH are longer than in ATLAS (on average 464 ns for mdCATH compared to 100 ns for ATLAS) and the simulations were performed at different temperatures (320 K, 348 K, 379 K, 413 K, and 450 K) than the simulations in the ATLAS dataset (300 K). The results of the evaluation are presented in Table 7.
+
+Table 7: Evaluation of Flexpert-3D and Flexpert-Seq predictors on MD simulations from the mdCATH dataset. Both predictors were evaluated on the full dataset (5398 proteins, columns "Full data") as well as on its subset excluding topologies present in the ATLAS training set (4013 proteins, columns "Topo. filtered"). The reported numbers are Pearson correlation coefficients between the model predictions and the RMSF from the dataset averaged over all the proteins in the dataset.  
+
+<table><tr><td>Simulation temperature</td><td>Flexpert-3D
+Full data / Topo. filtered</td><td>Flexpert-Seq
+Full data / Topo. filtered</td></tr><tr><td>T = 320 K</td><td>0.69 / 0.66</td><td>0.64 / 0.64</td></tr><tr><td>T = 348 K</td><td>0.64 / 0.63</td><td>0.63 / 0.63</td></tr><tr><td>T = 379 K</td><td>0.59 / 0.57</td><td>0.60 / 0.59</td></tr><tr><td>T = 413 K</td><td>0.49 / 0.48</td><td>0.52 / 0.52</td></tr><tr><td>T = 450 K</td><td>0.34 / 0.33</td><td>0.38 / 0.38</td></tr></table>
+
+Despite some performance drop with respect to the performance achieved on the ATLAS dataset (0.82 for Flexpert-3D and 0.76 for Flexpert-Seq), the performance of our predictors remains reasonable for the simulations performed at the temperature  $T = 320 \, \mathrm{K}$  (0.66 for Flexpert-3D and 0.64 for Flexpert-Seq, for the topology filtered dataset). Considering that the temperature in this part of the mdCATH dataset is  $20 \, \mathrm{K}$  away from the temperature of the simulations in the ATLAS training
+
+![](images/f2669dd5e6f5443d9f46225d0c090b150b8f5c5f0bd43df5e6ec002ddc54e773.jpg)  
+Flexpert-Seq  $(\mathrm{R} = 0.91)$
+
+![](images/4065cc76fd13eb0861c4edac18a3adbc8f7cf32cb9ca920e682d644b92910556.jpg)  
+Flexpert-3D  $(\mathrm{R} = 0.95)$
+
+![](images/98b2e0d18cdc716a30c158a23ce6203112027fdc2705399be3e7170761210d6b.jpg)  
+Figure 4: ATLAS datapoint 5jca_S. Flexpert-Seq and Flexpert-3D seem to improve over the prediction of ANMs mainly by correctly capturing the higher flexibility of the helix at the terminus.
+
+![](images/9be631f44dfe041a04ca8ec6ecad1683dc72c32ab2db61ad3ffdc8547564f013.jpg)
+
+![](images/ab81afea038f026ad39a426dea7ab7f4f74a5fa80f85070fdc012dcff3d7d9d9.jpg)  
+Rigid
+
+set, this appears as a reasonable generalization. We also notice that Flexpert-3D experiences a more dramatic decrease in performance than Flexpert-Seq for higher temperatures in mdCATH simulations, suggesting that the sole reliance on sequence information makes the method more robust to the change in the simulation temperature. In addition, the proteins are possibly getting unfolded in the high-temperature simulations of mdCATH, leading to non-physiological degrees of flexibility.
+
+# G QUALITATIVE ANALYSIS OF FLEXPERT-SEQ AND FLEXPERT-3D FLEXIBILITY PREDICTIONS
+
+In this section, we present some qualitative examples of Flexpert-Seq and Flexpert-3D predictions in the form of visualization of protein structures colored by the flexibility. The visualized flexibility is predicted by Flexpert-Seq and Flexpert-3D or estimated by MD (considered as the ground truth) or by Anisotropic Network Models (ANM). See Figures 4 to 7.
+
+# H BALANCING SEQUENCE RECOVERY AND FLEXIBILITY AWARENESS IN FLEXPERT-DESIGN TRAINING
+
+When training Flexpert-Design using the combined loss  $\mathcal{L}_{\text{Flexpert}} = \theta \cdot \mathcal{L}_{\text{Flex}} + (1 - \theta) \cdot \mathcal{L}_{\text{Seq}}$ , where  $\theta \in [0, 1]$  is the parameter balancing the two losses. We studied the effect of  $\theta$  on sequence recovery of the fine-tuned ProteinMPNN model. On one hand, we want  $\theta$  to be as high as possible to focus on learning flexibility awareness; on the other hand, we still need the fine-tuned inverse folding model to retain the sequence recovery so that it still has its inverse folding ability. We observed a drop of only one percentage point in terms of sequence recovery, when we trained with  $\theta = 0.8$ , with the recovery further dropping for higher  $\theta$ , see Table 8. Therefore, we selected  $\theta = 0.8$  as the parameter that still sufficiently maintains the original performance of the vanilla ProteinMPNN.
+
+When balancing the  $\mathcal{L}_{\text{Flex}}$  loss, we also tried to "effectively normalize" the  $\mathcal{L}_{\text{Flex}}$  and  $\mathcal{L}_{\text{Seq}}$  losses to have better control with the parameter  $\theta$ . This we did by normalization of the gradients in the backward pass and then using the parameter  $\theta$  to mix at the level of gradients instead of the level of losses. Such training, however, exhibited high instability, so we did not use it for the final version.
+
+![](images/d7a346f6d3e5afd71f0fdf15ca65a53eb3941c3edcab5c22566ae282fee5b16d.jpg)  
+RMSF (Ground Truth)
+
+![](images/3b8cfc2a457aab0ba1b3e1c92fb8c887564e812970df16a0e81eeb4797204b85.jpg)  
+Flexpert-Seq  $(\mathrm{R} = 0.80)$
+
+![](images/297206477ace9a308bf1b0cb77325fa5367e885a348322e1d9d05897b84acce0.jpg)  
+ANM  $(\mathrm{R} = 0.54)$
+
+![](images/451d1e5ad4199f22d5fe5898f468d0c70b3e2dc2912df043acbfd2b45d6c20d8.jpg)  
+Flexpert-3D  $(\mathrm{R} = 0.77)$
+
+![](images/16ed637af65cfee09168f2aa75e8bb5ae0df397301939d218701722fc5043525.jpg)  
+Flexible  
+Figure 5: ATLAS datapoint 1c52_A. Anisotropic Network Models perform relatively poorly, which likely propagates to Flexpert-3D causing it to underperform (Pearson  $R = 0.77$ ) with respect to Flexpert-Seq (Pearson  $R = 0.8$ ).
+
+![](images/f279a1b11c1ac0548845cdc9df40bcab6df3280e6d94fee319c3144ee43d70f6.jpg)  
+RMSF (Ground Truth)
+
+![](images/d78a9d5d4954bdc9aba5e6c1f1c0542c0422fc75e5ac6bfab1c8ad160ede9532.jpg)  
+Flexpert-Seq (R=0.28)
+
+![](images/3ceca1ee6f43ba07cf13cfdab1c8ecb6c6ed156cbe6b700de9c7d5c91a0a99c0.jpg)  
+ANM  $(\mathrm{R} = 0.88)$
+
+![](images/90cabe7477632a4a4f8e12ea3766b6a6447fd4134ebba8e47e7bdf4e47b68ece.jpg)  
+Flexpert-3D  $(R = 0.59)$
+
+![](images/fb0360d77417e6ef5d3b6d75e63dfd024217887f0b0b341b54c764c6c7adcd52.jpg)  
+Flexible  
+Figure 6: ATLAS datapoint 6o2v_A. This is a failure case for Flexpert-Seq (Pearson  $R = 0.28$ ), which predicts one beta strand together with the terminal coil to be more flexible than what was observed in MD simulation (see RMSF subfigure). Anisotropic Network Models perform better in this case, possibly helping Flexpert-3D improve over Flexpert-Seq.
+
+![](images/e7f03e29bc7ed5ebf83f29c01e38f2cb935c65482b15467cf43398d6e0d5e2e9.jpg)  
+RMSF (Ground Truth)
+
+![](images/f034ad6463606c12aebac50a29f5744ca59963faa52dcdabbfe1ca88b01c38ee.jpg)  
+Flexpert-Seq (R=0.75)
+
+![](images/8d1ddaf9965e6f7c606aa2010b9cacd1f790e0a99f21396d81a7a1f77de608fc.jpg)  
+ANM  $(\mathrm{R} = 0.87)$
+
+![](images/57db989f3a4e3fdc59e46b90973f9db1735badd0ee3f70faf3af9692a7bb028d.jpg)  
+Flexpert-3D  $(R = 0.92)$
+
+![](images/bf6e835ba3f7891ac1eb41c7d387bf72602fbac80a8e52fc76c21d49aad64cf0.jpg)  
+Figure 7: ATLAS datapoint legw_B. This datapoint features relatively high variance in the flexibility, posing a challenge for Flexpert-Seq (Pearson R = 0.75). The Anisotropic Network Models work better (Pearson R=0.87) and presumably help Flexpert-3D to excel (Pearson R=0.92).
+
+Table 8: Effect of the parameter  $\theta$  for mixing of sequence recovery and flexibility awareness on the sequence recovery of the fine-tuned ProteinMPNN model. The vanilla ProteinMPNN is denoted as  $\mathcal{P}.\mathcal{P}_F^{\theta = X}$  denotes Flexpert-Design trained with  $\theta$  value of  $X$ .  
+
+<table><tr><td></td><td>P</td><td>PFθ=0</td><td>PFθ=0.8</td><td>PFθ=0.9</td><td>PFθ=0.95</td></tr><tr><td>Sequence recovery</td><td>0.49</td><td>0.48</td><td>0.48</td><td>0.47</td><td>0.47</td></tr></table>
+
+Training and fine-tuning. We tried to train Flexpert-Design from scratch using the  $\mathcal{L}_{\text{Flexpert}}$  loss, but it seemed that the  $\mathcal{L}_{\text{Flex}}$  loss was not informative in the early stages. Therefore, we decided to first train ProteinMPNN in a standard way for 100 epochs on topology split CATH4.3 dataset. Second, we finetuned the model for 12 hours on 1 GPU using the  $\mathcal{L}_{\text{Flexpert}}$ , which resulted in additional 8-11 epochs.
+
+# I EVALUATION OF STRUCTURE PRESERVATION IN FLEXPERT-DESIGN GENERATED SEQUENCES WITH INCREASED FLEXIBILITY
+
+To evaluate how Flexpert-Design alters the structure of the proteins by the engineering of the increased flexibility, we predicted the structures of the sequences with engineered flexibility using AlphaFold2 inside Colabfold (Jumper et al., 2021; Mirdita et al., 2022). We compared the predicted backbone structures with the input backbone structure and evaluated the confidence of the structure prediction. The experiment was run for 700 randomly selected proteins from the CATH test set, for which we ran the Flexpert-Design predictions with the flexibility increasing instructions and the segment length  $S = 50$  (the same choice as in the experiment reported in Figure 3 and Table 4).
+
+The predicted sequences were compared to the ground truth structures at the level of  $C_{\alpha}$  using the root mean square deviation between the structures (RMSD) (Table 9). Furthermore, the pLDDT confidence overall across the whole protein and separately inside and outside the engineered region (where the flexibility increasing instructions were provided) was analyzed, see Table 9.
+
+We see that the Flexpert-Design generated sequences show lower recovery of the ground truth structure compared to ProteinMPNN, which could suggest that Flexpert-Design produces less stable structures or that it introduces notable conformational changes. A particularly significant drop in
+
+Table 9: Evaluation of the Flexpert-Design generated protein sequences using the alignment of the folded backbone structures to the ground truth backbone structures (RMSD column) and the AF2 predicted pLDDT of the whole protein, the engineered region (eng. region) and of the outside of the engineered region (non-eng. region).  
+
+<table><tr><td>Model</td><td>RMSD (Å) ↓</td><td>ø pLDDT ↑ (whole protein)</td><td>ø pLDDT ↑ (eng. region)</td><td>ø pLDDT ↑ (non-eng. region)</td></tr><tr><td>ProteinMPNN</td><td>2.00</td><td>0.77</td><td>0.81</td><td>0.73</td></tr><tr><td>Flexpert-Design (no loss)</td><td>3.31</td><td>0.61</td><td>0.60</td><td>0.62</td></tr><tr><td>Flexpert-Design (ours)</td><td>3.36</td><td>0.60</td><td>0.58</td><td>0.63</td></tr></table>
+
+![](images/e517a37f22dac6ec6656cd853af1417515c00f52b7aa0ed610a72507b7057279.jpg)  
+Figure 8: The effect of the flexibility increasing parameter  $\tau$  (left) on the median enrichment ratio of our Flexpert-Design model and its variant trained without  $\mathcal{L}_{Flex}$  compared to the vanilla Protein-MPNN as a baseline. The effect of the length of the engineered segment on the median enrichment ratio (right).
+
+![](images/ea87f960139e2d72a09c883a8d29b5ec5996798f94169ee2110c92156792eccc.jpg)
+
+the pLDDT is in the engineered regions (which tended to be well-structured in the ground truth, as suggested by the high pLDDT for ProteinMPNN sequences), which might hint at the increased flexibility (which was shown by our predictor in Figure 3 and Table 4) causing the drop in pLDDT and increase in RMSD.
+
+# J THE EFFECT OF FLEXIBILITY INSTRUCTION VALUE AND THE SIZE OF THE ENGINEERED SEGMENT ON FLEXIBILITY ENRICHMENT
+
+The effect of the size of the engineered segment  $|S|$  and of the parameter  $\tau$  used for increasing the flexibility is presented in Figure 8. The graphs suggest that our model achieves the best median enrichment ratio in the setting of  $|S| = 50$  and  $\tau = 5$ . At the same time, for parameters within the range  $\tau \in [2.5, 10]$  and  $|S| \in [10, 100]$ , the results of our model are comparable to the optimal setting demonstrating reasonable robustness of the model to setting these parameters.
+
+# K LIMITATION: ENGINEERING DECREASE IN FLEXIBILITY
+
+Analogically to the flexibility-increasing experiment presented in Section 5.2, we also investigated the possibility of decreasing flexibility using our approach with parameter  $\tau < 0$ . We present the results for  $|S| = 50$  and  $\tau = -10$  in Table 10.
+
+Table 10: Comparing the ability of flexibility engineering models to decrease the flexibility of engineered segments. The presented median enrichment ratios  $Med(r_{ij})$  and the proportion of flexibility decreasing mutations were obtained using the CATH4.3 test set with engineered segments of length  $|S| = 50$ , and flexibility engineering instructions corresponding to native flexibility incremented by  $\tau = -10$  in the engineered segment. The best results in each category are highlighted in bold, second best are underlined.
+
+<table><tr><td></td><td>Median enrich. ratio r ↓</td><td>Prop. of flexibility decreasing mutations ↑</td></tr><tr><td>Flexpert-Design (MSE loss)</td><td>1.06</td><td>0.40</td></tr><tr><td>Flexpert-Design (L1 loss)</td><td>0.99</td><td>0.51</td></tr><tr><td>Flexpert-Design (w/o loss)</td><td>1.04</td><td>0.46</td></tr><tr><td>ProteinMPNN (vanilla)</td><td>1.07</td><td>0.39</td></tr></table>
+
+Table 10 also presents the variant of Flexpert-Design trained using L1 loss instead of MSE, which seems to work better for the case of decreasing flexibility. However, despite achieving a modest decrease in flexibility, we consider this task as a limitation of Flexpert-Design. Our explanation is that Flexpert-Design is biased toward positive flexibility engineering instructions because it is trained on the native distribution of flexibilities, which are mostly positive as they approximate the RMSF quantity, which is by definition non-negative. Similarly, we suspect that the Flexpert-3D predictor might be slightly biased toward overpredicting flexibility since it predicts a flexibility increase even for the vanilla ProteinMPNN model. The possible reason is the effect of outliers in the training set. Outliers can exist in the dataset for positive values, but cannot exist for negative due to the non-negative nature of RMSF. Such disbalance can potentially skew the learning distribution toward higher values.
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+# LEARNING UNDER TEMPORAL LABEL NOISE
+
+Sujay Nagaraj<sup>1</sup> Walter Gerych<sup>2</sup> Sana Tonekaboni<sup>3</sup> Anna Goldenberg<sup>1</sup> Berk Ustun<sup>4</sup> Thomas Hartvigsen<sup>5</sup>
+
+<sup>1</sup>University of Toronto <sup>2</sup>MIT <sup>3</sup>Broad Institute <sup>4</sup>UCSD <sup>5</sup>University of Virginia
+
+Corresponding authors: s.nagaraj@mail.utoronto.ca, hartvigsen@virginia.edu
+
+# ABSTRACT
+
+Many time series classification tasks, where labels vary over time, are affected by label noise that also varies over time. Such noise can cause label quality to improve, worsen, or periodically change over time. We first propose and formalize temporal label noise, an unstudied problem for sequential classification of time series. In this setting, multiple labels are recorded over time while being corrupted by a time-dependent noise function. We first demonstrate the importance of modeling the temporal nature of the label noise function and how existing methods will consistently underperform. We then propose methods to train noise-tolerant classifiers by estimating the temporal label noise function directly from data. We show that our methods lead to state-of-the-art performance under diverse types of temporal label noise on real-world datasets. $^{1}$
+
+# 1 INTRODUCTION
+
+Supervised learning datasets often contain incorrect, or noisy, observations of ground truth labels. Such label noise commonly arises during human annotation [1, 28] when labelers lack expertise [31, 77], labeling is hard [13, 30, 77], labels are subjective [49, 57, 62], or during automatic annotation where there are systematic issues like measurement error [32, 52]. These factors, among others, make label noise prevalent in supervised learning. Label noise is a key vulnerability in supervised learning [17, 25, 75]. Intuitively, models trained with noisy labels may learn to predict noise - such models then underperform at test time when attempting to predict the ground truth [26, 42].
+
+Label noise has been studied extensively, producing a stream of works on static prediction tasks where we predict single outcomes [3, 43, 45, 67]. Many non-static prediction tasks (i.e., time series) have features and labels that evolve over time. In these cases, it is only natural for label noise to also evolve over time – label noise can be temporal. This notion of time-varying noise has yet to be formalized, yet it is found in numerous real-world tasks. For example:
+
+- Human Activity Recognition. Wearable device studies often ask participants to annotate their activities (e.g., exercise) over time. Participants often mislabel their activities due to recall bias, time of day, or labeling-at-random for monetized studies [27, 61].  
+- Self-Reported Outcomes for Mental Health: Mental health studies often collect self-reported survey data (e.g., depression) over long periods of time. Such self-reporting is known to be biased [5, 53, 55, 72, 46], as participants are more or less likely to report certain outcomes. For example, the accuracy of self-reported alcohol consumption is often seasonal [8].  
+- Clinical Measurement Error: Clinical prediction models often predict outcomes (e.g., mortality) derived from clinician notes in electronic health records. These labels may capture noisier annotations during busier times – e.g., when a patient is deteriorating [76].
+
+Addressing the problem of temporal label noise is challenging from a technical perspective. Existing methods for learning under static label noise [48, 50] are not flexible enough to account for noise rates varying over time – these approaches have no mechanism to capture temporal label noise and therefore implicitly misspecify the noise model. Additionally, the label noise function, temporal or otherwise, is often unknown; most datasets lack an indication for which instances or time steps are
+
+![](images/1893d5865da37d8f474e172df6640c91d9821406af43263590e301ddc3021fa8.jpg)  
+Figure 1: In time series tasks, label quality can vary over time due to temporal label noise. Existing methods assume noise can perform poorly as they assume a static noise model over time. We propose that accurate modeling of temporal label noise can improve performance. We demonstrate this by showing performance improvements on an activity recognition task where we compare reconstruction error and accuracy between static and temporal methods subject to  $30\%$  temporal label noise across 10 draws (see results for moving dataset in Appendix D.4 for details).
+
+![](images/ab78b61c983eec43cc81226709fa20eeb5f0365ed032a4e36282ae5c8f8a02e0.jpg)
+
+more likely to be accurate. Another practical limitation is the lack of ground truth data in real-world noisy datasets.
+
+To address these challenges, we propose objective functions for sequence-to-sequence classification tasks in the context of time series data. These objectives learn the temporal noise function directly from time series data with temporally noisy labels. Our methods leverage a temporal loss function that is robust to temporal label noise. Our algorithms can be used out of the box and allow practitioners to learn noise-tolerant time series classifiers, even with unknown, temporal label noise. Our main contributions are:
+
+1. We formalize the problem of learning from noisy labels in temporal settings.  
+2. We propose a novel loss function for training models that are robust to temporal label noise. On its own, we show this can already be used to improve prior methods.  
+3. We develop a suite of methods to learn time series classifiers under temporal label noise. Our methods can overcome the practical challenges of this regime - i.e., by estimating an unknown noise function from time series data.  
+4. We show the consequences of not accounting for temporal label noise by simply using existing methods for noisy labels - these methods perform poorly under temporal noise. Our results highlight the necessity of accounting for temporal noise, as our methods lead to more noise-robust classifiers - temporal or otherwise.
+
+# 2 RELATED WORK
+
+Our work is motivated by a series of problems that arise in machine learning in healthcare [20, 40, 46]. Existing approaches primarily rely on supervised learning to model changes in latent states over time [14, 35, 64, 68, 73, 85]. In practice, these underlying latent states can reflect clinical states that patients can transition in and out of (e.g., sick vs healthy). Existing approaches for this problem typically rely on modeling sequences with deep neural networks [11, 39, 65] and using attention-based mechanisms to prioritize relevant segments of data [71]. An understudied problem in this domain is label noise – what happens when inaccurate labels arise in real-world time series datasets?
+
+The vast majority of work on noisy labels studies label noise in static prediction tasks where we predict a single outcome and where label noise cannot change over time [3, 43, 45, 67, 47] We consider how label noise can arise in time series - a prediction task where the noise rates can change over time. Recent work in this setting focuses on identifying noisy instances in time series, specifically by exploiting the notion that labels at neighboring time steps are unlikely to be corrupted together [see e.g., 2, 12]. In contrast, we focus on developing algorithms to train models that are robust to temporal label noise. With respect to existing work, our approach can account for label noise in tasks where labels at neighboring time steps are more likely to be corrupted together (e.g., due to
+
+seasonal fluctuations in annotator error). We develop methods for empirical risk minimization with noise-robust loss functions [c.f., 21, 38, 43, 48, 74]. Our work establishes the potential to learn from noisy labels in this setting when we have knowledge of the underlying noise process [c.f., 48, 50], and when we are able to learn the noise model from noisy data [see e.g., 37, 38, 50, 79, 82, 86].
+
+# 3 FRAMEWORK
+
+Preliminaries We consider a temporal classification task over  $C$  classes and  $T$  time steps. Each instance is characterized by a triplet of sequences over  $T$  time steps  $(\mathbf{x}_{1:T},\mathbf{y}_{1:T},\tilde{\mathbf{y}}_{1:T})\in \mathcal{X}\times \mathcal{Y}\times \mathcal{Y}$  where  $\mathcal{X}\subseteq \mathbb{R}^{d\times T}$  represents a multivariate time-series and  $\mathcal{Y} = \{1,\dots ,C\} ^T$  represents a sequence of labels. Here,  $\mathbf{x}_{1:T},\mathbf{y}_{1:T}$ , and  $\tilde{\mathbf{y}}_{1:T}$  are sequences of instances, clean labels and noisy labels, respectively. For example, we can capture settings where  $\mathbf{x}_{1:T},\mathbf{y}_{1:T}$  are recordings from an accelerometer with true activity labels, and  $\tilde{\mathbf{y}}_{1:T}$  represent noisy annotations of activity.
+
+Under temporal label noise, the true label sequence  $\mathbf{y}_{1:T}$  is unobserved, and we only have access to a set of  $n$  noisy instances  $D = \{(\pmb{x}_{1:T},\tilde{\pmb{y}}_{1:T})_i\}_{i=1}^n$ . We assume that each sequence in  $D$  is generated i.i.d. from a joint distribution  $P_{\mathbf{x}_{1:T},\mathbf{y}_{1:T},\tilde{\mathbf{y}}_{1:T}}$ , where the label noise process can vary over time. This distribution obeys two standard assumptions in temporal modeling and label noise [4, 10, 69]:
+
+Assumption 1 (Future Independence). A label at time  $t$  depends only on the past sequence of feature vectors up to  $t$ :  $p(\mathbf{y}_{1:T} \mid \mathbf{x}_{1:T}) = \prod_{t=1}^{T} p(\mathbf{y}_t | \mathbf{x}_{1:t})$
+
+Assumption 2 (Feature Independence). The sequence of noisy labels is conditionally independent of the features given the true labels:  $\tilde{\mathbf{y}}_{1:t} \perp \mathbf{x}_{1:t} \mid \mathbf{y}_{1:t}$  for  $t = 1, \ldots, T$
+
+These assumptions are relatively straightforward, as they require that the current observation is independent of future observations and assume a feature-independent noise regime. Assumption 1 allows the joint sequence distribution to factorize as:  $p(\tilde{\mathbf{y}}_{1:T}|\mathbf{x}_{1:T}) = \prod_{t=1}^{T} q_t(\tilde{\mathbf{y}}_t|\mathbf{x}_{1:t})$ . Here, we introduce  $q_t$ , a slight abuse of notation, to denote a probability that is time-dependent. Assumption 2 allows for the noisy label distribution at  $t$  to be further decomposed as:
+
+$$
+q _ {t} \left(\tilde {\mathbf {y}} _ {t} \mid \mathbf {x} _ {1: t}\right) = \sum_ {y \in \mathcal {Y}} q _ {t} \left(\tilde {\mathbf {y}} _ {t} \mid \mathbf {y} _ {t} = y\right) p \left(\mathbf {y} _ {t} = y \mid \mathbf {x} _ {1: t}\right) \tag {1}
+$$
+
+# 3.1 LEARNING FROM TEMPORAL LABEL NOISE
+
+Our goal is to learn a temporal classification model  $\pmb{h}_{\theta}:\mathbb{R}^{d\times t}\to \mathbb{R}^{C}$  with model parameters  $\theta \in \Theta$  and  $t\leq T$ . Here,  $h_\theta (\mathbf{x}_{1:t})$  returns an estimate of  $p(\mathrm{y}_t|\mathbf{x}_{1:t})$ . To infer the label at time step  $t$ ,  $h_{\theta}$  takes as input a sequence of feature vectors up to  $t$ , and outputs a sequence of labels by taking the arg max of the predicted distribution for each time step (see e.g., [4, 69]). We estimate parameters  $\hat{\theta}$  for a model robust to noise, by maximizing the expected accuracy as measured in terms of the clean labels:
+
+$$
+\hat {\theta} = \underset {\theta \in \Theta} {\operatorname {a r g m a x}} \mathbb {E} _ {\mathbf {y} _ {1: T} \mid \mathbf {x} _ {1: T}} \prod_ {t = 1} ^ {T} p \left(\mathbf {y} _ {t} = h _ {\theta} \left(\mathbf {x} _ {1: t}\right) \mid \mathbf {x} _ {1: t}\right)
+$$
+
+However, during training time we only have access to sequences of noisy labels. To demonstrate how we can sidestep this limitation, we first need to introduce a flexible way to model noise rates varying over time. Existing methods assume that noise is time-invariant (Fig. 1). To relax this assumption, we capture the temporal nature of noisy labels using a temporal label noise function, in Def. 1.
+
+Definition 1. Given a temporal classification task with  $C$  classes and noisy labels, the temporal label noise function is a matrix-valued function  $Q: \mathbb{R}_+ \to [0,1]^{C \times C}$  that specifies the label noise distribution at any time  $t > 0$ .
+
+We denote the output of the temporal noise function at time  $t$  as  $Q_{t} \coloneqq Q(t)$ . This is a  $C \times C$  matrix whose  $i, j^{\mathrm{th}}$  entry encodes the flipping probability of observing a noisy label  $j$  given clean label  $i$  at time  $t$ :  $q_{t}(\tilde{y}_{t} = j \mid y_{t} = i)$ . We observe that  $Q_{t}$  is positive, row-stochastic, and diagonally dominant - ensuring that  $Q_{t}$  encodes a valid probability distribution [38, 50].
+
+$Q_{\omega}$  denotes a temporal noise function parameterized by a function with parameters  $\omega$ . This parameterization can be constructed to encode essentially any temporal noise function. As shown in Table 1, we can capture a wide variety of temporal noise using this representation.
+
+<table><tr><td>Pattern</td><td>Depiction</td><td>Noise Model Qij</td><td>Parameters</td><td>Applications</td></tr><tr><td rowspan="3">Periodic</td><td rowspan="3">Noise Rate (%)</td><td rowspan="3">1/2 + 1/2 sin(αijt + φij)</td><td>ωij = (αij, φij)</td><td>Annotation reliability varies over time of day – e.g., due to changes in annotator attentiveness over the day [16].</td></tr><tr><td>α controls frequency</td><td rowspan="2"></td></tr><tr><td>φ controls shift</td></tr><tr><td rowspan="3">Decay</td><td rowspan="3">Noise Rate (%)</td><td rowspan="3">αij exp(-βijt)</td><td>ωij = (αij, λij)</td><td>Annotation reliability improves rapidly (e.g., diagnostic accuracy of COVID-19 rapidly improved at the onset of the pandemic [51]) or improves and plateaus (e.g., irreducible uncertainty in labels [54])</td></tr><tr><td>α controls initial noise</td><td rowspan="2"></td></tr><tr><td>β controls decay rate</td></tr><tr><td rowspan="4">Growth</td><td rowspan="4">Noise Rate (%)</td><td rowspan="4">αij 1+exp(-βij(t-γij))</td><td>ωij = (αij, βij, γij)</td><td rowspan="4">Annotation reliability decreases abruptly – e.g., due to rapid adoption of improved clinical guidelines [66]. Or, it decreases gradually – e.g., due to underdiagnosis during the pandemic due to health-care disruptions [18]</td></tr><tr><td>α controls limit</td></tr><tr><td>β controls growth rate</td></tr><tr><td>γ controls inflection point</td></tr></table>
+
+Table 1: Overview of temporal noise functions for time series classification tasks (see Appendix D for other examples). We show the noise model  $\mathbf{Q}(t)$  and parameters  $\omega$  to model  $q(\tilde{\mathbf{y}}_t = j\mid \mathbf{y}_t = i)$ . Practitioners can account for temporal variation by choosing a parametric class to capture effects and fit parameters from data.
+
+# 3.2 LOSS CORRECTION
+
+Modeling temporal label noise is the first piece of the puzzle in training time series classifiers robust to temporal label noise. However, we still need to consider how to leverage these noise models during empirical risk minimization. It remains unclear if and how existing loss correction techniques work for time series. Here, we present theoretical results showing that learning is possible in our setting when we know the true temporal noise function  $Q(t)$ . We include proofs in Appendix A.
+
+We begin by treating the noisy posterior as the matrix-vector product of a noise transition matrix and a clean class posterior Eq. (1). To this effect, we define the forward temporal loss:
+
+Definition 2. Given a temporal classification task over  $T$  time steps, a noise function  $Q(t)$ , and a proper composite loss function  $\ell_t$  [58], the forward temporal loss of a model  $h_\theta$  on an instance  $(\tilde{\pmb{y}}_{1:T},\pmb{x}_{1:T})$  is:
+
+$$
+\overrightarrow {\ell} _ {s e q} \left(\tilde {\boldsymbol {y}} _ {1: T}, \boldsymbol {x} _ {1: T}, \boldsymbol {h} _ {\theta}\right) := \sum_ {t = 1} ^ {T} \ell_ {t} \left(\tilde {y} _ {t}, \boldsymbol {Q} _ {t} ^ {\top} \boldsymbol {h} _ {\theta} \left(\boldsymbol {x} _ {1: t}\right)\right)
+$$
+
+An intriguing property of the forward temporal loss is that its minimizer over the noisy labels maximizes the likelihood of the data over the clean labels. This suggests that the forward temporal loss is robust to label noise:
+
+Proposition 1. A classifier that minimizes the empirical forward temporal loss over the noisy labels maximizes the empirical likelihood of the data over the clean labels.
+
+$$
+\underset {\theta \in \Theta} {\operatorname {a r g m i n}} \mathbb {E} _ {\tilde {\mathbf {y}} _ {1: T}, \mathbf {x} _ {1: T}} \overrightarrow {\ell} _ {s e q} (\tilde {\mathbf {y}} _ {1: T}, \mathbf {x} _ {1: T}, \boldsymbol {h} _ {\theta}) = \underset {\theta \in \Theta} {\operatorname {a r g m i n}} \sum_ {t = 1} ^ {T} \mathbb {E} _ {\mathbf {y} _ {1: t}, \mathbf {x} _ {1: t}} \ell_ {t} (\mathbf {y} _ {t}, \boldsymbol {h} _ {\theta} (\mathbf {x} _ {1: t}))
+$$
+
+Proposition 1 implies that we can train on the noisy distribution and learn a noise-tolerant classifier in expectation.
+
+# 4 METHODOLOGY
+
+Our results in the previous section show that we can train models that are robust to label noise. However, the temporal noise function  $Q$  is not available to the practitioner and must be learned from data. Our algorithm seeks to first learn this function from noisy data then account for noise using Def. 2. This strategy can be applied in multiple ways. In what follows, we propose a method where we jointly learn the temporal noise function and the model. We also present extensions for different use-cases in Section 4.3 and discuss how to choose between methods in Section 4.4
+
+# 4.1 FORMULATION
+
+We start with a method that simultaneously learns a time series classifier and temporal noise function. Given a noisy dataset, we learn these elements by solving the following optimization problem  $\forall t\in [1,T]$ :
+
+$$
+\min  _ {\omega , \theta} \operatorname {V o l} \left(\boldsymbol {Q} _ {\omega} (t)\right) \tag {2}
+$$
+
+$$
+\begin{array}{l} \text {s . t .} \quad Q _ {\omega} (t) ^ {\top} h _ {\theta} (\boldsymbol {x} _ {1: t}) = p (\tilde {y} _ {t} \mid \boldsymbol {x} _ {1: t}) \end{array}
+$$
+
+Eq. (2) is designed to return a faithful representation of the noise function  $\hat{\omega}$  by imposing the minimum-volume simplex assumption [38], and a noise-tolerant temporal classifier  $\hat{\theta}$  by minimizing the forward temporal loss in Def. 2.
+
+Here, the objective minimizes the volume of the noise matrix, denoted as  $\mathrm{Vol}(Q_t)$ . This returns a matrix  $Q_{t}$ , at each time step, that obeys the minimum-volume simplex assumption, which is a standard condition used to ensure identifiability of label noise in static tasks [see e.g., 38, 83]. In practice, this assumption ensures that  $Q_{t}$  encloses the noisy conditional data distribution at time  $t$ :  $p(\tilde{\mathbf{y}}_t\mid \mathbf{x}_{1:t})$ . Containment ensures that the estimated noise matrix could have generated each point in the noisy dataset - i.e., so that the corresponding noisy probabilities  $p(\tilde{\mathbf{y}}_t\mid \mathbf{x}_{1:t})$  obey Eq. (1). Our use of this assumption guarantees the identifiability of  $Q_{t}$  when the posterior distribution is sufficiently-scattered over the unit simplex [see also 38, for details].
+
+# 4.2 ESTIMATION PROCEDURE
+
+The formulation above applies to any generic matrix-valued function according to Def. 1 with parameters  $\omega$ . Because time series classification tasks can admit many types of temporal label noise functions (Table 1), we must ensure  $\omega$  has sufficient representational capacity to handle many noise functions. Therefore in practice, we instantiate our solution as a fully connected neural network with parameters  $\omega$ ,  $Q_{\omega}(\cdot): \mathbb{R} \to [0,1]^{C \times C}$ , adjusted to meet Def. 1 (see Appendix D.3 for implementation details). We can now model any temporal label noise function owing to the universal approximation properties of neural networks.
+
+Provided the function space  $\Theta$  defines autoregressive models of the form  $p(y_{t}|\boldsymbol{x}_{1:t})$  (e.g., RNNs, Transformers, etc.), we can solve Eq. (2) using an augmented Lagrangian method for equality-constrained optimization problems [6]:
+
+$$
+\mathcal {L} (\theta , \omega) = \frac {1}{T} \sum_ {t = 1} ^ {T} \left[ \| Q _ {\omega} (t) \| _ {F} + \lambda R _ {t} (\theta , \omega) + \frac {c}{2} | R _ {t} (\theta , \omega) | ^ {2} \right] \tag {3}
+$$
+
+Here:  $\| Q_{\omega}(t)\| _F$  denotes the Frobenius norm of  $Q_{\omega}(t)$ , which acts as a convex surrogate for  $\mathrm{Vol}(Q_{\omega}(t))$  [7]. Likewise,  $R_{t}(\theta ,\omega) = \frac{1}{n}\sum_{i = 1}^{n}\ell_{t}(\tilde{y}_{t,i},Q_{\omega}(t)^{\top}h_{\theta}(\boldsymbol{x}_{1:t,i}))$  denotes the violation of the equality constraint for each  $t = 1\dots T$ .  $\lambda \in \mathbb{R}_+$  is the Lagrange multiplier and  $c > 0$  is a penalty parameter. Both these terms are initially set to a default value of 1, we gradually increase the penalty parameter until the constraint holds and  $\lambda$  converges to the Lagrangian multiplier for the optimization problem Eq. (2) [6]. This approach recovers the best-fit parameters to the optimization problem in Eq. (2). We call this approach Continuous Estimation and provide additional details on our implementation in Appendix B.
+
+# 4.3 ALTERNATIVE APPROACHES
+
+We describe two alternative approaches that extend existing label noise methods to temporal settings and may be valuable to use in tasks where we cannot assume continuity or have access to additional information.
+
+Discontinuous Estimation Another approach is to assume that there is no temporal relationship between each  $Q_{t}$  across time and treat each time step independently. This approach is well suited for tasks where time steps are unevenly spaced (e.g., some clinical data involves labels collected over years or at different frequencies [56]). We can address such tasks through an estimation procedure where we learn a model that achieves the same objective without assuming continuity. In contrast to
+
+Eq. (3), this approach  $\hat{Q}_t$  is parameterized with a separate set of trainable real-valued weights, fitting the parameters at each time step using the data from that time step. This approach can be generalized to extend any state-of-the-art technique for noise transition matrix estimation in the static setting (e.g., Li et al. [38], Yong et al. [83], etc.).
+
+Plug-In Estimation It is also advantageous to have a simple, plug-in estimator of the temporal noise function. Plug-in estimators are model-agnostic, can flexibly be deployed to other models, and can be efficient to estimate. We can construct a plug-in estimator of temporal label noise using anchor points, instances whose labels are known to be correct. Empirical estimates of the class probabilities of anchor points can be used to estimate the noise function. This estimate can then be plugged into Def. 2 to train a noise-tolerant time series classifier. Formally, in a temporal setting, anchor points [41, 50, 79] are instances that maximize the probability of belonging to class  $i$  at time step  $t$ :
+
+$$
+\bar {\boldsymbol {x}} _ {t} ^ {i} = \underset {\boldsymbol {x} _ {t}} {\arg \max } p \left(\tilde {\mathbf {y}} _ {t} = i \mid \boldsymbol {x} _ {1: t}\right) \tag {4}
+$$
+
+Since  $p(\mathbf{y}_t = i \mid \bar{\boldsymbol{x}}_{1:t}^i) \approx 1$  for the clean label, we can express each entry of the label noise matrix as:
+
+$$
+\hat {\boldsymbol {Q}} (t) _ {i, j} = p \left(\tilde {\mathbf {y}} _ {t} = j \mid \tilde {\boldsymbol {x}} _ {1: t} ^ {i}\right) \tag {5}
+$$
+
+We construct a plug-in estimate of  $\hat{Q}(t)$  using a two-step approach: we identify anchor points for each class and time step  $t = 1, \dots, T$  and replace each entry of  $\hat{Q}(t)$  with the empirical estimate defined in Eq. (5). This is a generalization of the approach of Patrini et al. [50] developed for static prediction tasks.
+
+# 4.4 DISCUSSION
+
+All three methods in Section 4.2 and 4.3 improve performance in temporal classification tasks by accounting for temporal label noise (see e.g., Fig. 1 and Section 5). However, they each have their strengths and limitations. Here we provide guidance for users to choose between methods:
+
+- Continuous estimation imposes continuity across time steps – assuming that nearby points likely have similar noise levels i.e.,  $\hat{Q}(t) \approx \hat{Q}(t + 1)$ . This assumption can improve reconstruction in tasks with multiple time steps by reducing the effective number of model parameters. Conversely, it may also lead to misspecification in settings that exhibit discontinuity.  
+- Discontinuous estimation can handle discontinuous temporal noise processes – assuming the noise levels of nearby points are independent of one another. This approach requires fitting more parameters, which scales according to  $T$ . This can lead to computational challenges and overfitting, especially for long sequences.  
+- Plug-In estimation has a simpler optimization problem. This is useful in that we can use separate datasets to estimate the noise model and to train the classifier. In practice, the main challenges in this approach stem from verifying the existence of anchor points [79].
+
+# 5 EXPERIMENTS
+
+We benchmark our methods on a collection of temporal classification tasks from real-world applications. Our goal is to evaluate methods in terms of robustness to temporal label noise, and characterize when it is important to consider temporal variation in label noise. We include additional details on setup and results in Appendix D and code to reproduce our results on GitHub.
+
+# 5.1 SETUP
+
+We work with four real-world datasets from healthcare. Each dataset reflects binary classification tasks over a complex feature space, with labeled examples across multiple time steps, where the labels are likely to exhibit label noise. The tasks include:
+
+1. moving: human activity recognition task where we detect movement states (e.g., walking vs. sitting) using temporal accelerometer data in adults [59].  
+2. senior: similar human activity recognition task as above but in senior citizens [44]
+
+3. sleeping: sleep state detection (e.g., light sleep vs. REM) task using continuous EEG data [22]  
+4. blinking: eye movement (open vs. closed) detection task using continuous EEG data [60] from continuous EEG data.
+
+We consider a semi-synthetic setup where we treat labels in the training sample as ground truth labels and create noisy labels by corrupting them with a noise function for different noise rates. This setup reflects a standard approach used to benchmark algorithms for learning under label noise [see e.g., 48, 50]. We consider six different label noise functions:
+
+- Linear: noise rates linearly fall over time.  
+- Decay: noise rates decay exponentially over time.  
+- Growth: noise rates rapidly rise over time.  
+Periodic: noise rates vary in a seasonal fashion.  
+- Mixed: noise rates for  $y_{t} = 0$  grow and noise rates for  $y_{t} = 1$  fall over time.  
+- Static: noise rates are constant over time.
+
+We present two sample noise functions in Fig. 2, and a complete list in Fig. 4 in Appendix D.
+
+We split each dataset into a noisy training sample (80%, used to train the models and correct for label noise) and a clean test sample (20%, used to compute unbiased estimates of out-of-sample performance).
+
+We train time series classifiers on the aforementioned datasets using the three techniques described in Section 4: Continuous, Discontinuous, and Plug-In. We compare these models to baseline models that ignore label noise (Ignore) to loss-tolerant methods that assume a static noise model across time steps: Anchor [41, 50, 79] and VolMinNet [38].
+
+We evaluate each model in terms of the following metrics:
+
+- Test Accuracy measures how well each method performs on held-out test data with clean labels.  
+- Approximation Error characterizes how well each method learns the temporal noise function, using the  $\operatorname{Error}(\mathbf{Q}_t, \hat{\mathbf{Q}}_t) \coloneqq \frac{1}{T} \sum_{t=1}^{T} \| \mathbf{Q}_t - \hat{\mathbf{Q}}_t \|$  between the true  $\mathbf{Q}_t$  and estimated  $\hat{\mathbf{Q}}_t$  for all  $t$ .
+
+# 5.2 RESULTS
+
+We summarize our results for two temporal noise functions in Table 2 and Table 3. We evaluate the performance of methods across different noise models and attribute the gains to our ability to capture the underlying temporal noise function in Fig. 2. We include additional results for other noise models and for multi-class classification tasks in Appendix D.4. In what follows we discuss key results.
+
+On the Performance Gains of Modeling Temporal Noise First, we show clear value in accounting for temporal label noise. Table 2 shows the performance of each method on all four datasets. We
+
+![](images/26c2f61c62411e62866e8cdb56a4d9fd0e5a9e3ce6e7c1b3caa3fa549d4315f7.jpg)  
+Figure 2: Comparison of the ground truth unseen temporal noise function  $\mathbf{Q}(t)$  and its estimate  $\hat{\mathbf{Q}}(t)$  from each Temporal method on the moving data. We show the noise rate for the negative class only for clarity. We show results for Periodic and Mixed noise regimes. As shown, our Temporal methods can learn the true label noise function across different noise patterns. The resulting noise models have lower reconstruction error and are superior to static approaches.
+
+<table><tr><td rowspan="2">Dataset</td><td rowspan="2">Metric</td><td rowspan="2">Ignore</td><td colspan="2">Static</td><td colspan="3">Temporal</td></tr><tr><td>Anchor</td><td>VolMinNet</td><td>Plug-In</td><td>Discontinuous</td><td>Continuous</td></tr><tr><td>moving [59]</td><td>Test Error</td><td>29.4 ± 1.7%</td><td>20.9 ± 2.6%</td><td>14.9 ± 2.7%</td><td>20.0 ± 1.8%</td><td>15.9 ± 2.7%</td><td>4.2 ± 2.2%</td></tr><tr><td>n = 192, d = 14, T = 50</td><td>Approx. Error</td><td>-</td><td>42.4 ± 3.4%</td><td>35.3 ± 0.8%</td><td>36.8 ± 1.7%</td><td>32.6 ± 0.5%</td><td>10.3 ± 3.9%</td></tr><tr><td>senior [44]</td><td>Test Error</td><td>22.7 ± 1.7%</td><td>20.7 ± .01%</td><td>19.0 ± 0.7%</td><td>18.8 ± 1.1%</td><td>13.6 ± 1.2%</td><td>11.0 ± 0.3%</td></tr><tr><td>n = 444, d = 6, T = 100</td><td>Approx. Error</td><td>-</td><td>35.9 ± 2.5%</td><td>36.3 ± 0.4%</td><td>26.9 ± 1.2%</td><td>21.7 ± 0.2%</td><td>6.4 ± 0.8%</td></tr><tr><td>blinking [60]</td><td>Test Error</td><td>34.1 ± 2.0%</td><td>34.1 ± 2.3%</td><td>29.6 ± 2.2%</td><td>29.6 ± 2.8%</td><td>29.9 ± 3.0%</td><td>29.6 ± 2.3%</td></tr><tr><td>n = 299, d = 14, T = 50</td><td>Approx. Error</td><td>-</td><td>35.3 ± 0.8%</td><td>35.2 ± 0.7%</td><td>19.6 ± 1.1%</td><td>26.6 ± 0.9%</td><td>14.9 ± 2.3%</td></tr><tr><td>sleeping [22]</td><td>Test Error</td><td>28.7 ± 0.8%</td><td>24.9 ± 1.1%</td><td>26.8 ± 1.4%</td><td>20.4 ± 1.8%</td><td>19.6 ± 0.8%</td><td>16.3 ± 0.4%</td></tr><tr><td>n = 964, d = 7, T = 100</td><td>Approx. Error</td><td>-</td><td>34.3 ± 1.8%</td><td>41.8 ± 0.1%</td><td>19.1 ± 3.5%</td><td>22.4 ± 0.2%</td><td>4.9 ± 0.5%</td></tr></table>
+
+Table 2: Model performance and approximation error for all methods and datasets when labels are corrupted using a Mixed noise function (average  $30\%$  label noise across all time steps, one class with decreasing noise rate, one class with increasing noise rate). We report the clean test error  $(\%)$  and mean approximation error of  $\hat{Q}(t) \pm \text{st.dev}$  over 10 runs. The best-performing methods are highlighted in Green. Continuous outperforms all baselines.  
+
+<table><tr><td rowspan="2">Dataset</td><td rowspan="2">Metric</td><td rowspan="2">Ignore</td><td colspan="2">Static</td><td colspan="3">Temporal</td></tr><tr><td>Anchor</td><td>VolMinNet</td><td>Plug-In</td><td>Discontinuous</td><td>Continuous</td></tr><tr><td>moving [59]</td><td>Test Error</td><td>24.0 ± 5.1%</td><td>18.0 ± 3.6%</td><td>13.5 ± 6.0%</td><td>18.5 ± 4.3%</td><td>14.0 ± 5.7%</td><td>1.7 ± 0.6%</td></tr><tr><td>n = 192, d = 14, T = 50</td><td>Approx. Error</td><td>-</td><td>49.9 ± 4.7%</td><td>43.5 ± 3.0%</td><td>48.1 ± 4.4</td><td>33.5 ± 5.0%</td><td>9.1 ± 1.6%</td></tr><tr><td>senior [44]</td><td>Test Error</td><td>22.5 ± 2.2%</td><td>20.1 ± 1.5%</td><td>16.4 ± 2.9%</td><td>19.9 ± 1.3%</td><td>14.1 ± 2.6%</td><td>10.6 ± 0.7%</td></tr><tr><td>n = 444, d = 6, T = 100</td><td>Approx. Error</td><td>-</td><td>47.3 ± 4.0%</td><td>42.4 ± 3.9%</td><td>46.9 ± 3.9%</td><td>20.5 ± 2.1%</td><td>9.7 ± 2.7%</td></tr><tr><td>blinking [60]</td><td>Test Error</td><td>37.4 ± 3.1%</td><td>35.8 ± 2.7%</td><td>33.2 ± 2.8%</td><td>35.8 ± 2.3%</td><td>32.0 ± 2.9%</td><td>30.4 ± 3.1%</td></tr><tr><td>n = 299, d = 14, T = 50</td><td>Approx. Error</td><td>-</td><td>49.3 ± 4.2%</td><td>42.4 ± 3.9%</td><td>46.4 ± 4.4%</td><td>24.7 ± 3.3%</td><td>10.5 ± 3.3%</td></tr><tr><td>sleeping [22]</td><td>Test Error</td><td>23.3 ± 3.1%</td><td>22.8 ± 2.7%</td><td>22.4 ± 3.0%</td><td>22.1 ± 2.0%</td><td>20.1 ± 3.5%</td><td>15.2 ± 0.8%</td></tr><tr><td>n = 964, d = 7, T = 100</td><td>Approx. Error</td><td>-</td><td>44.9 ± 4.0%</td><td>42.6 ± 4.1%</td><td>43.9 ± 4.3%</td><td>19.1 ± 1.8%</td><td>8.8 ± 2.6%</td></tr></table>
+
+Table 3: Model performance and approximation error for all methods and datasets when labels are corrupted using the Periodic noise function (average  $30\%$  label noise across all time steps). We report the clean test error  $(\%)$  and mean approximation error of  $\hat{Q}(t) \pm \text{st.dev}$  over 10 runs. The best-performing methods are highlighted in green. Continuous outperforms all baselines.
+
+find that the temporal methods are consistently more accurate than their non-temporal counterparts, highlighting the importance of modeling temporal noise. For example, Plug-In is a temporal extension of Anchor, and we see that Plug-In outperforms Anchor in most settings. For example, on moving, the temporal method reduces the test error by over  $10\%$  compared to the nearest-performing static counterpart. This shows that our methods are robust to label noise despite being trained on data that is  $30\%$  corrupted with noisy labels, with no prior knowledge of the noise. This can have important consequences – e.g., in healthcare, where the noise is unknown and accurate models support life-or-death decisions.
+
+Among the temporal methods, Continuous achieves the best performance in comparison to Plug-In and Discontinuous. Continuous' superiority is even clearer when compared to the static methods. Comparing the results for Table 2 (which assumes a Mixed noise model) and Table 3 (which assumes a Periodic noise model), we can see that these results hold across multiple types of temporal label noise. Fig. 2 demonstrates the ability of our methods to accurately reconstruct the underlying label noise. More importantly, the benefit of Continuous becomes more evident as the amount of noise increases in the data. In all these cases, we observe that temporal methods are consistently more robust to both temporal and static label noise. Overall, these findings suggest that we can improve performance by explicitly modeling how noise varies across time instead of assuming it is distributed uniformly in time.
+
+On Learning the Temporal Noise Function Our results show that the performance increases hand in hand with our ability to estimate the noise model. In particular, we observe that low values of reconstruction consistently lead to low values of error rates.
+
+We note that Continuous has the best reconstruction error (Approx. Error) on all datasets (Table 2). We qualitatively compare estimated noise functions and the ground truth for Continuous, Discontinuous,
+
+and other baselines in Fig. 2. We use a DNN to model  $Q$ , which consistently estimates the noise function with lower mean absolute error across different families of noise functions. Appendix D.4 has extended results on all other datasets (including multi-class).
+
+On the Risk of Misspecification As shown in Section 3.2, our ability to learn under temporal label noise depends on our ability to correctly specify the temporal noise process. However, existing  $Q$ -estimators assume static, time-invariant label noise – which means that they will always fail to capture the correct noise function when it is temporal. In Fig. 5 (Appendix D), we validate this experimentally by showing how a model that assumes static noise compares to the forward temporal loss model learned using the true temporal noise function (see Appendix D.4 for other results). Our results highlight that there is no downside to modeling temporal effects. As shown, performance does not change when the temporal noise process is truly static. In contrast, performance drops when temporal effects exist. Using our temporal approaches they can relax this assumption to admit different types of temporal label noise. There is no cost to doing so, as our methods are competitive even if the noise model is truly static.
+
+# 6 DEMONSTRATION ON STRESS DETECTION
+
+We now demonstrate the viability of our approach on a real-world time series application where we have both clean and noisy labels. We focus on the task of stress detection, which is a common feature in modern smartwatches [9, 19] and is increasingly used to support interventions to mitigate clinical burnout [23, 29, 15]. Although many models use self-reported stress labels, these labels are subject to temporal label noise which arises from subjectivity, forgetfulness, and seasonal patterns in self-reported outcomes [53]. Given these effects, objective physiological measures (e.g., heart rate variability, temperature, etc.) are the standard for measuring stress and guiding interventions.
+
+Setup Our goal is to learn a time series classifier using the noisy labels (i.e., self-reported stress labels) that will generalize the clean labels (i.e., a physiological indicator of stress). We use a dataset from Goodday [24] monitoring stress in healthcare workers using physiological measures and self-reported measures. The dataset contains  $n = 289$  unique individuals over  $T = 50$  days. Each sequence has  $d = 9$  features. Here, the noisy label is:  $\tilde{y}_{i,t} = 1$  if person  $i$  self-reports stress on day  $t$ . The clean label is  $y_{i,t} = 1$  if the objective physiological measure of stress for person  $i$  indicates stress on day  $t$ . We train sequential classifiers to predict stress over time. Our setup is identical to the one in Section 5, except that our training labels reflect real-world noisy labels and we have clean labels representing the ground truth stress.
+
+![](images/9f64d779887b1a5076043a16c8bc0f1a485fbcda9b94185dd7d8589873858786.jpg)  
+Figure 3: Temporal effects in label noise in a real-world stress detection task. We show noise rates when individuals are stressed (left) and not stressed (right). True noise rate is the average disagreement rates between clean and noisy labels over time. We can see clear, temporal label noise patterns – our temporal label noise methods do a superior job of approximating it.
+
+Results We report our results in Fig. 3 and Table 4. In Fig. 3, the black line indicates the average disagreement rates between the noisy self-reported label of stress and the clean physiological label over time. We can see clear temporal patterns in label noise, where participants under-report stress in a seasonal manner.
+
+Our results are consistent with those in Section 5.2. In particular, we find that accounting for temporal label noise improves both training and test accuracy (Table 4). These results also highlight
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">What It Represents</td><td>Noise Estimation</td><td colspan="2">Model Performance</td></tr><tr><td>Approx. Error</td><td>Train Error</td><td>Test Error</td></tr><tr><td>Ignore</td><td>ignoring label noise completely</td><td>17.6 ± 0.0%</td><td>31.0 ± 0.3%</td><td>31.1 ± 4.3%</td></tr><tr><td>Static</td><td>static label noise correction per time step</td><td>16.4 ± 1.1%</td><td>31.0 ± 0.3%</td><td>29.5 ± 4.0%</td></tr><tr><td>Discontinuous</td><td>temporal label noise correction with discontinuity</td><td>11.4 ± 0.8%</td><td>27.1 ± 0.7%</td><td>26.0 ± 2.5%</td></tr><tr><td>Continuous</td><td>temporal label noise correction imposing continuity</td><td>9.7 ± 1.3%</td><td>21.1 ± 0.5%</td><td>25.5 ± 1.7%</td></tr></table>
+
+Table 4: Noise rate estimation error and model performance for stress detection. We train sequential classifiers to predict stress using the training setup in Section 5. We report results for four methods to highlight the mechanism driving performance improvements through an ablation study. We show the approximation error noise rates (left); and the clear label error rate on the training sample (middle) and test sample (right) All values correspond to the mean 10-CV estimates  $\pm$  st.dev.
+
+the mechanism for these effects. Specifically, we see that lower approximation error leads to lower training error (via noise correction) and lower training error leads to lower test error (via generalization). In this case, all model classes are the same and the gains in Continuous are driven by its ability to learn the underlying temporal noise function in a joint optimization procedure.
+
+Discussion This demonstration shows the effectiveness of our methods in regimes without synthetic noise injection. We identified clean and noisy labels in this dataset using physiological measures and self-reports, respectively. In this scenario, temporal label noise represents the time-dependent process that models the discrepancy between these labels.
+
+# 7 CONCLUDING REMARKS
+
+Temporal label noise is an under-explored problem facing many real-world time series tasks. In such tasks, we must predict a sequence of labels, each of which can be subject to label noise whose distribution can change over time. Our work shows that existing methods for label noise substantially underperform when the distribution of label noise changes over time. As we show, we can achieve improved performance by accounting for temporal label noise. Given that this noise function is unknown, we also highlight the viability of learning the noise model from data. Finally, we perform a demonstration on a real-world source of temporal label noise – self-reported labels in digital health studies. In this setting, our methods continue to outperform methods that ignore such noise. Overall, temporal label noise presents a challenge in modeling time series – our paper underscores the necessity of explicitly modeling and accounting for temporal noise to achieve reliable and robust time series classification.
+
+**Limitations** We assume that all time series in a dataset have the same underlying label noise function. This may not be true as different annotators can introduce differing noise patterns. We can potentially relax this assumption by considering models that represent differences across annotators [see e.g., 63, 36, 70]. Additionally, we work with stationary time series, where the predictive distribution is stable over time. Though this is a standard assumption in time series work [see e.g., 80, 81, 78, 84], it is a limitation that can be addressed. For example, our loss-correction technique can be extended with learning theory from the study of non-stationary time series [33, 34]. Finally, the lack of time series datasets with both noisy and clean labels are few and far between. Without clean labels, it is difficult to verify that the temporal noise function learned is accurate or verify that a model trained on noisy labels generalizes to clean labels. We hope this paper will inspire the curation of more time series datasets containing both sets of labels.
+
+Acknowledgements This work was supported by funding from the National Science Foundation IIS 2040880, IIS 2313105, and the NIH Bridge2AI Center Grant U54HG012510. SN was supported by a CIHR Vanier Scholarship. AG is a CIFAR AI Chair. Resources used in preparing this research were provided, in part, by the Province of Ontario, the Government of Canada through CIFAR, and companies sponsoring the Vector Institute. We also thank Abhishek Moturu and Vijay Giri for valuable discussions about the project.
+
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+
+# Supplementary Materials
+
+# A PROOFS
+
+In what follows, we use vector notation for completeness and clarity of exposition. We describe key quantities and notation that we use in Table 5. We will also use Assumption 1 and Assumption 2 to factor the noisy label distribution as follows:
+
+$$
+\boldsymbol {p} \left(\tilde {\mathbf {y}} _ {1: T} \mid \mathbf {x} _ {1: T}\right) = \prod_ {t = 1} ^ {\top} \boldsymbol {q} _ {t} \left(\tilde {\mathbf {y}} _ {t} \mid \mathbf {x} _ {1: t}\right). \tag {6}
+$$
+
+<table><tr><td>Notation</td><td>Description</td></tr><tr><td>p(yt | x1:t) := [yt = c | x1:t]Tc=1:C</td><td>Vector of probabilities for each label value, for the clean label distribution p(yt | x1:t) ∈ RC×1</td></tr><tr><td>p(ŷt | x1:t) := [yt = c | x1:t]Tc=1:C</td><td>Vector of probabilities for each possible label value, for the noisy label distribution p(ŷt | x1:t) ∈ RC×1</td></tr><tr><td>hθ(x1:t) = pθ(yt | x1:t = x1:t)</td><td>Classifier that predicts label distribution at t given preceding observations Rd×t → RC</td></tr><tr><td>hθ(x1:t) = ψ-1(gθ(x1:t))</td><td>When hθ is a deep network, gθ is the final logits, and ψ: ΔC-1 → RC represents an invertible link function (e.g., softmax)</td></tr><tr><td>Qt := [qt(ŷt = k | yt = j)]j,k</td><td>The temporal noise matrix at time t: Qt ∈ RC×C</td></tr><tr><td>lt(yt, hθ(x1:t)) = - log pθ(yt = yt | x1:t = x1:t)</td><td>Loss at t: Y × RC → R</td></tr><tr><td>lψ,t(yt, hθ(x1:t)) = l(t(yt, ψ-1hθ(x1:t))</td><td>A composite loss function using a link function ψ</td></tr><tr><td>lt(hθ(x1:t)) = [lt(c, hθ(x1:t))]Tc=1:C</td><td>Vector of NLL losses for each possible value of the ground truth RC → RC</td></tr><tr><td>t,ψ(c, hθ(x1:t)) = lt(c, QtT·ψ-1(gθ))</td><td>Forward loss for class c</td></tr><tr><td>seq,ψ(y1:T, hθ(x1:t)) = ∑t=1Tt,ψ(c, hθ(x1:t))</td><td>Sequence forward loss</td></tr></table>
+
+Table 5: Quantities and definitions used for the following proof
+
+# Proof of Proposition 1
+
+Proof. Our goal is to show:
+
+$$
+\underset {\theta} {\operatorname {a r g m i n}} \mathbb {E} _ {\bar {\mathbf {y}} _ {1: T}, \mathbf {x} _ {1: T}} \overrightarrow {\ell} _ {s e q, \psi} (\mathbf {y} _ {1: T}, \boldsymbol {g} _ {\theta} (\mathbf {x} _ {1: T})) = \underset {\theta} {\operatorname {a r g m i n}} \sum_ {t = 1} ^ {\top} \mathbb {E} _ {\mathbf {y} _ {1: t}, \mathbf {x} _ {1: t}} \ell_ {t, \phi} (\mathbf {y} _ {1: T}, \boldsymbol {g} _ {\theta} (\mathbf {x} _ {1: T})).
+$$
+
+First, note that:
+
+$$
+\begin{array}{l} \vec {\ell} _ {t, \psi} \left(\mathbf {y} _ {t}, \boldsymbol {h} _ {\theta} \left(\mathbf {x} _ {1: t}\right)\right) = \ell_ {t} \left(\mathbf {y} _ {t}, \boldsymbol {Q} _ {t} ^ {\top} \boldsymbol {\psi} ^ {- 1} \left(\boldsymbol {g} _ {\theta} \left(\mathbf {x} _ {1: t}\right)\right)\right) (7) \\ = \ell_ {\phi_ {t}, t} \left(\mathbf {y} _ {t}, \boldsymbol {g} _ {\theta} \left(\mathbf {x} _ {1: t}\right)\right), (8) \\ \end{array}
+$$
+
+where  $\phi_t^{-1} = \psi^{-1} \circ Q_t^\top$ . Thus,  $\phi_t: \Delta^{C-1} \to \mathbb{R}^C$  is invertible, and is thus a proper composite loss [58].
+
+Thus, as shown in Patrini et al. [50]:
+
+$$
+\begin{array}{l} \underset {\theta} {\operatorname {a r g m i n}} \mathbb {E} _ {\tilde {\mathbf {y}} _ {t}, \mathbf {x} _ {1: t}} \ell_ {\phi , t} \left(\mathbf {y} _ {t}, \boldsymbol {g} _ {\theta} \left(\mathbf {x} _ {1: t}\right)\right) = \underset {\theta} {\operatorname {a r g m i n}} \mathbb {E} _ {\tilde {\mathbf {y}} _ {t} | \mathbf {x} _ {1: t}} \ell_ {\phi_ {t}, t} \left(\mathbf {y} _ {t}, \boldsymbol {g} _ {\theta} \left(\mathbf {x} _ {1: t}\right)\right) (9) \\ = \phi_ {t} \left(\boldsymbol {p} \left(\tilde {\mathbf {y}} _ {t} \mid \mathbf {x} _ {1: t}\right)\right) \quad \text {(p r o p e r t y) o f p r o p e r c o m p o s i t e l o s s e s)} \\ = \psi \left(\left(\boldsymbol {Q} _ {t} ^ {- 1}\right) ^ {\top} \boldsymbol {p} \left(\tilde {\mathbf {y}} _ {t} \mid \mathbf {x} _ {1: t}\right)\right) (10) \\ = \psi \left(\boldsymbol {p} \left(\mathbf {y} _ {t} \mid \mathbf {x} _ {1: t}\right)\right) (11) \\ \end{array}
+$$
+
+The above holds for the minimizer at a single time step, not the sequence as a whole. To find the minimizer of the loss over the entire sequence:
+
+$$
+\begin{array}{l} \underset {\theta} {\operatorname {a r g m i n}} \mathbb {E} _ {\mathbf {x} _ {1: T}, \tilde {\mathbf {y}} _ {1: T}} \overrightarrow {\ell} _ {s e q, \psi} \left(\tilde {\mathbf {y}} _ {1: T}, \boldsymbol {g} _ {\theta} \left(\mathbf {x} _ {1: T}\right)\right) = \underset {\theta} {\operatorname {a r g m i n}} \mathbb {E} _ {\tilde {\mathbf {y}} _ {1: T} \mid \mathbf {x} _ {1: T}} \overrightarrow {\ell} _ {s e q, \psi} \left(\tilde {\mathbf {y}} _ {1: T}, \boldsymbol {g} _ {\theta} \left(\mathbf {x} _ {1: T}\right)\right) (12) \\ = \underset {\theta} {\operatorname {a r g m i n}} \mathbb {E} _ {\tilde {\mathbf {y}} _ {1: T} \mid \mathbf {x} _ {1: T}} \sum_ {t = 1} ^ {\top} \vec {\ell} _ {t, \psi} \left(\tilde {\mathbf {y}} _ {t}, \boldsymbol {g} _ {\theta} \left(\mathbf {x} _ {1: t}\right)\right) (13) \\ = \underset {\theta} {\operatorname {a r g m i n}} \sum_ {t = 1} ^ {\top} \mathbb {E} _ {\tilde {\mathbf {y}} _ {1: T} \mid \mathbf {x} _ {1: T}} \overrightarrow {\ell} _ {t, \psi} \left(\tilde {\mathbf {y}} _ {t}, \boldsymbol {g} _ {\theta} \left(\mathbf {x} _ {1: t}\right)\right) (14) \\ = \underset {\theta} {\operatorname {a r g m i n}} \sum_ {t = 1} ^ {\top} \mathbb {E} _ {\tilde {\mathbf {y}} _ {t} | \mathbf {x} _ {1: t}} \overrightarrow {\ell} _ {t, \psi} \left(\tilde {\mathbf {y}} _ {t}, \boldsymbol {g} _ {\theta} \left(\mathbf {x} _ {1: t}\right)\right) (15) \\ = \underset {\theta} {\operatorname {a r g m i n}} \sum_ {t = 1} ^ {\top} \mathbb {E} _ {\tilde {\mathbf {y}} _ {t} | \mathbf {x} _ {1: t}} \ell_ {t, \phi} \left(\tilde {\mathbf {y}} _ {t}, \boldsymbol {g} _ {\theta} \left(\mathbf {x} _ {1: t}\right)\right) (16) \\ \end{array}
+$$
+
+As the minimizer of the sum will be the function that minimizes each element of the sum, then  $\mathrm{argmin}_{\theta} \mathbb{E}_{\tilde{\mathbf{y}}_{1:T}, \mathbf{x}_{1:T}} \overline{\ell}_{seq,\psi}(\mathbf{y}_{1:T}, \pmb{g}_{\theta}(\mathbf{x}_{1:T})) = \psi(\pmb{p}(\mathbf{y}_{1:T} \mid \mathbf{x}_{1:T}))$ . Note that the  $\mathrm{argmin}_{\theta} \sum_{t=1}^{\top} \mathbb{E}_{\mathbf{y}_{1:t}, \mathbf{x}_{1:t}} \ell_{t,\phi}(\mathbf{y}_{1:T}, \pmb{g}_{\theta}(\mathbf{x}_{1:T})) = \psi(\pmb{p}(\mathbf{y}_{1:T} \mid \mathbf{x}_{1:T}))$ , because the minimizer of the NLL is the data distribution. Thus,  $\mathrm{argmin}_{\theta} \mathbb{E}_{\tilde{\mathbf{y}}_{1:T}, \mathbf{x}_{1:T}} \overline{\ell}_{seq,\psi}(\mathbf{y}_{1:T}, \pmb{g}_{\theta}(\mathbf{x}_{1:T})) = \mathrm{argmin}_{\theta} \sum_{t=1}^{\top} \mathbb{E}_{\mathbf{y}_{1:t}, \mathbf{x}_{1:t}} \ell_{t,\phi}(\mathbf{y}_{1:T}, \pmb{g}_{\theta}(\mathbf{x}_{1:T}))$ .
+
+![](images/4c5e8072763e283e2cded985ff31666167e685b141bad70fffcaefee80a864a9.jpg)
+
+# B CONTINUOUS LEARNING ALGORITHM
+
+We summarize the augmented lagrangian approach to solving the continuous objective in Algorithm 1
+
+Algorithm 1 Continuous Learning Algorithm  
+Input: Noisy Training Dataset  $D$  , hyperparameters  $\gamma$  and  $\eta$    
+Output: Model  $\theta$  , Temporal Noise Function  $\omega$ $c\gets 1$  and  $\lambda \gets 1$    
+for  $k = 1,2,3,\ldots$  do  $\theta^k,\omega^k = \arg \min_{\theta ,\omega}\mathcal{L}(\theta ,\omega)$  Computed with SGD using the Adam optimizer  $\lambda \leftarrow \lambda +c*R_t(\theta^k,\omega^k)$  Update Lagrange multiplier if  $k > 0$  and  $R_{t}(\theta^{k},\omega^{k}) > \gamma R_{t}(\theta^{k - 1},\omega^{k - 1})$  then  $c\gets \eta c$  else  $c\gets c$  end if if  $R_{t}(\theta^{k},\omega^{k}) = = 0$  then break end if end for
+
+For all experiments we set  $\lambda = 1, c = 1$ ,  $\gamma = 2$ , and  $\eta = 2$ .  $k$  and the maximum number of SGD iterations are set to 15 and 10, respectively. This is to ensure that the total number of epochs is 150, which is the max number of epochs used for all experiments.
+
+# C PLUG-IN PROCEDURE
+
+1. Fit a probabilistic classifier to predict noisy labels from the observed data.  
+2. For each class  $y\in \mathcal{V}$  and time  $t\in [1\dots T]$
+
+i Identify anchor points for class  $y$ :  $\bar{\pmb{x}}_t^j = \arg \max_{\pmb{x}_t} p(\bar{y}_t = y \mid \pmb{x}_{1:t})$ .
+
+ii Set  $\hat{\pmb{Q}}(t)_{y,y'}$  as the probability of classifier predicting class  $y'$  at time  $t$  given  $\bar{x}_t^j$ .
+
+# D SUPPORTING MATERIAL FOR EXPERIMENTAL RESULTS
+
+In what follows, we describe details for reproducing the experiments shown in the paper. Our code is available on GitHub.
+
+# D.1 DATASET DETAILS
+
+<table><tr><td>Dataset</td><td>Classification Task</td><td>n</td><td>d</td><td>T</td></tr><tr><td>blinking [60]</td><td>Eye Open vs Eye Closed</td><td>299</td><td>14</td><td>50</td></tr><tr><td>sleeping [22]</td><td>Sleep vs Awake</td><td>964</td><td>7</td><td>100</td></tr><tr><td>moving [59]</td><td>Walking vs Not Walking</td><td>192</td><td>9</td><td>50</td></tr><tr><td>senior [44]</td><td>Walking vs Not Walking</td><td>444</td><td>6</td><td>100</td></tr></table>
+
+Table 6: Datasets used in the experiments. Classification tasks, number of samples  $(n)$ , dimensionality at each time step  $(d)$ , and sequence length  $(T)$  are shown.
+
+moving from UC Irvine [59] consists of inertial sensor readings of 30 adult subjects performing activities of daily living. The sensor signals are already preprocessed and a vector of features at each time step are provided. We apply z-score normalization at the participant-level, then split the dataset into subsequences of a fixed size 50. We use a batchsize of 64.
+
+senior from UC Irvine [44] consists of inertial sensor readings of 18 elderly subjects performing activities of daily living. The sensor signals are already preprocessed and a vector of features at each time step are provided. We apply z-score normalization at the participant-level, then split the dataset into subsequences of a fixed size 100. We use a batchsize of 256.
+
+sleeping from Physionet [22] consists of EEG data measured from 197 different whole nights of sleep observation, including awake periods at the start, end, and intermittently. We apply z-score normalization at the whole night-level. Then downsample the data to have features and labels each minute, as EEG data is sampled at  $100\mathrm{Hz}$  and labels are sampled at  $1\mathrm{Hz}$ . We then split the data into subsequences of a fixed size 100. We use a batchsize of 512.
+
+blinking from UC Irvine [60] consists of data measured from one continuous participant tasked with opening and closing their eyes while wearing a headset to measure their EEG data. We apply z-score normalization for the entire sequence, remove outliers ( $>5$  SD away from mean), and split into subsequences of a fixed size 50. We use a batchsize of 128.
+
+synthetic We generate data for binary and multiclass classification with  $n = 1000$  samples and  $d = 50$  features over  $T = 100$  time steps. We generate the class labels and observations for each time step using a Hidden Markov Model (HMM). The transition matrix generating the markov chain is uniform ensuring an equal likelihood of any state at any given time. We corrupted them using multidimensional (50) Gaussian emissions. The mean of the gaussian for state/class  $c$  is set to  $c$
+
+with variance 1.5 (i.e. class 1 has mean 1 and variance 1.5). The high-dimensionality and overlap in feature-space between classes makes this a sufficiently difficult task, especially under label noise. We use a batchsize of 256
+
+# D.2 NOISE INJECTION AND NOISE REGIMES
+
+We create noise labels for each dataset by applying one of the six temporal noise functions shown in Fig. 4. We choose functions to capture different functional forms, assumptions on time-variance, and class-dependence.
+
+![](images/adbb77c5b5cb32669cc453e67b55287a38401a844ddf613ccdb2257a32607305.jpg)
+
+![](images/cb63a0f1eac88ee7f3d3cec707845fbd96ce2ddd06aa01d46f5e8353aac5027c.jpg)
+
+![](images/2e01bd459b4f8b4e450342132d25049006f6a18b9fc39e1ff701716e42b51340.jpg)
+
+![](images/8f090fc46905289cbf513e9e0e32fe60d153da9426d1f2653ba41bf1b38c99df.jpg)
+
+Figure 4: Temporal label noise function  $Q(t)$  used in the experiments. We present six examples for binary classification task (from top-left clockwise): static, decay periodic, mixed, linear, growth. Each plot shows the off-diagonal entries of various parameterized forms of  $Q(t)$ .  
+![](images/6367ea53e7957abdede4e48dec75c0841132962f183c7feae7e0d42655c4ca8b.jpg)  
+P(yt=0|yt=1) P(yt=1|yt=0)
+
+![](images/33dfd33c21bb19f1d5971107f660b8ddeb04968f659e22d588032577bce5adc1.jpg)
+
+# D.3 DETAILS ON TRAINING ALGORITHMS
+
+GRU the GRU  $r: \mathbb{R}^d \times \mathbb{Z} \to \mathbb{R}^C \times \mathbb{Z}$  produces an output vector such that the output of  $r(\boldsymbol{x}_t, \boldsymbol{z}_{t-1})$  is our model for  $h_\theta(\boldsymbol{x}_{1:t})$ , and a hidden state  $z_t \in \mathbb{Z}$  that summarizes  $x_{1:t}$ . We use a softmax activation on the output vector of the GRU to make it a valid parameterization of  $p_\theta(y_t \mid x_{1:t})$ . The GRU has a single hidden layer with a 32 dimension hidden state.
+
+Continuous Continuous uses an additional fully-connected neural network with 10 hidden layers that outputs a  $C * C$ -dimensional vector to represent each entry of a flattened  $\hat{Q}_t$ . To ensure the output of this network is valid for Def. 1, we reshape the prediction to be  $C \times C$ , apply a row-wise softmax function, add this to the identity matrix to ensure diagonal dominance, then rescale the rows to be row-stochastic. These operations are all differentiable, ensuring we can optimize this network with standard backpropagation.
+
+VolMinNet and Discontinuous We do a similar parameterization for VolMinNet and Discontinuous, using a set of differentiable weights to represent the entries of  $Q_{t}$  rather than a neural network.
+
+Anchor and Plug-In Patrini et al. [50] show that in practice taking the 97th percentile anchor points rather than the maximum yield better results, so we use that same approach in our experiments. They also describe a two-stage approach: 1) estimate the anchor points after a warmup period 2) use the anchor points to train the classifier with forward corrected loss. We set the warmup period to 25 epochs.
+
+Hyperparameters We train each model for 150 epochs using the adam optimizer with default parameters and a learning rate of 0.01. We train all models using the same set of hyperparameters for experiment, set the batch size manually for each dataset, and avoid hyperparameter tuning to avoid label leakages. For VolMinNet, Discontinuous, and Continuous we use adam optimizer with default parameters and a learning rate of 0.01 to optimize each respective  $\hat{Q}_t$ -estimation technique.  $\lambda$  was set to  $1e - 4$  for VolMinNet and Discontinuous for all experiments following the setup of Li et al. [38].
+
+# D.4 ADDITIONAL RESULTS
+
+# D.4.1 STATIC APPROXIMATION
+
+Here we study the performance of forward temporal loss where we know the noise function  $Q(t)$  – that is, even if we could perfectly estimate the noise process – and where we have a static estimate of  $Q(t)$  (the average over time). We find that even if the noise process is perfectly estimated, accounting for temporal noise outperforms a static estimate.
+
+![](images/3b3fe9eb3b4e34cfdb16280de26301deba9917497e31792240e542ed50b12baa.jpg)
+
+![](images/4dcffe685e5bf35af53a38ad98a8ea5620762f88ba8c7e7bd9d85fb3641ea11a.jpg)
+
+![](images/1143a64a87ea480dee7c4e92d3483ea687ca8078f9a97319a49d29978e614ad8.jpg)
+
+![](images/86a426c530b055e9ffa18fc6678bcce582f9a7c99edc5c8912bbf26f8ecfaf82.jpg)
+
+![](images/3eccd2ffe2d9b34422a473f6dcaf77280e0a3d493c7e23750bb7a02a18eff659.jpg)  
+Figure 5: Comparing performance of models trained with forward temporal loss vs no noise correction on synth with varying degrees of temporal label noise using either the true temporal noise function (Temporal) or the average temporal noise function (Static). Error bars are st. dev. over 10 runs.
+
+![](images/75ae7a5a11ff5c5dc64c604312405bc41c19d3ea9c3a6af2b56811bd9e75a338.jpg)
+
+![](images/b509ff4d5ef8a8523fcaca7b0fe4f6b081365a0853c1706ab9e04b6c390cfbb1.jpg)
+
+# D.4.2 MULTIClass Classification
+
+![](images/31dc0b86109d6a0ca352efbec2dbad467643dd044799c037f6d98dc2c2b14f3f.jpg)
+
+![](images/403c3a8215e5af5673066d8026595bc6ff26c6a55772417770e06058a3cf2215.jpg)
+
+![](images/a0273d6252390500ee33538eca25498fd5c4bf0086abc9441aef1f49eb4a87f7.jpg)
+
+![](images/84fbb46cf72f1de08400039943685b66ad14a93d672e0e1183451c71f930c815.jpg)
+
+Figure 6: Comparison of clean test set Accuracy  $(\%)$  for synth across varying degrees of temporal label noise comparing all methods for 3-class classification. Error bars are st. dev. over 10 runs.  
+![](images/b5fa355722b84e82670341b47612e251f6cdfe818fe250dba8390732a4ddfac5.jpg)  
+- Ignore - Anchor - Plug-In VolMinNet Discontinuous Continuous
+
+![](images/7cccb0a67109d7ec29a4a020f2403f5ba00b2f288fac164c4b54369a32c9c53b.jpg)
+
+![](images/ab4ee6308d20acb93bd559cc63a462133e91b9fc725dbd48ace0fdf30cb58879.jpg)
+
+![](images/a04330de121125adb9230c71490364e6e7c4dc5d1571955f6d0d53d5897fbc01.jpg)
+
+![](images/bf3ca4f1e68eb41d29695f0a86493db0883da1933a944fdee206725b4aebe42b.jpg)
+
+![](images/2a671472aff230b03154086d21836c03df35829e9e1e773b32b47d91637d7268.jpg)
+
+![](images/954f0bd93bc553bfa6f4ec592890cdcdcbb2a69717c97a7d0191bf49f31c64fa.jpg)  
+Figure 7: Comparison of noisy function reconstruction Mean Absolute Error (MAE) for synth across varying degrees of temporal label noise comparing all methods for 3-class classification. Error bars are st. dev. over 10 runs.
+
+![](images/816c821469c4eb9ae331aeb88322ab79b925b79c2128c62404aef2c7d4a04b67.jpg)
+
+![](images/aa31d282c02b790b33a2f37f8757316ad23dec8c3676b781a712603b37fc7ee7.jpg)
+
+![](images/95866017d85966ccb47fa25e843774568e2a60b547bf407fb614bfc86f58686c.jpg)
+
+![](images/3e6f122427b8c02d108867910cf349230a9bd3a55c6e9c4d938cc5e4aeb5628c.jpg)
+
+![](images/1bc73baa89ef05c0e89414ad2d67dac965e98a68c7d0ff25b995dbc3707b4c44.jpg)
+
+![](images/f5125e579b1c9f8f1f8eff909e503b8dc14ae13d787bb37f2ec4fbef1f8c4155.jpg)
+
+![](images/788f29d9d6a88b4a3ef6e8a6ea796f0629f85e46fd4ee1dd57c6876badc517ad.jpg)  
+Figure 8: Comparison of clean test set Accuracy  $(\%)$  for sleeping across varying degrees of temporal label noise comparing all methods for 3-class classification. Error bars are st. dev. over 10 runs.
+
+![](images/07b8525e029bea8b4c5e2489e223edaa391ae28a12e85f84882a19b2eea0a2e9.jpg)  
+- Ignore - Anchor - Plug-In - VolMinNet - Discontinuous - Continuous
+
+![](images/01be33ae7d7cbef8fab0b75b861f3f660a6d9f3e8b81addf396d8aa37b223e64.jpg)
+
+![](images/bb92f534fb8a7c58e856a63a9b22d83c94d6ef1fcae78d05fddb872a33bd49fa.jpg)
+
+![](images/a717463b44bc38055fa5cb761142ccdf771338e827381003b3a419a90452d9db.jpg)
+
+![](images/88acd3de2a7f3890fc38fe0b1e1037d6b0e99388c093648130691676babfb774.jpg)
+
+![](images/e8fe7387091ba322bcdb30e92dc50ee14b517c93f323f3046f3bf0f0af76c5f2.jpg)  
+Figure 9: Comparison of noisy function reconstruction Mean Absolute Error (MAE) for sleeping across varying degrees of temporal label noise comparing all methods for 3-class classification. Error bars are st. dev. over 10 runs.
+
+![](images/550257857fc0207ed54e1b7cf247e785156818cd7682bad4f5ab33d3fc40ce83.jpg)
+
+![](images/682cad8df4659ad8171ea5f4155fb34344afad8e70d77567894acc7a73598426.jpg)
+
+![](images/b66b7c02f6af696b0ffb60fb7568430165faccc488d5a4fc467f6a39db138fad.jpg)
+
+![](images/5d30801f9d578135241fd2d64082238bbb72d51747fa72d2e15e2bc40aacdff2.jpg)
+
+![](images/92b5b8946bcfd6639e4d1dca17e1cf89c2328671638a036c9f63322eb6ce4660.jpg)
+
+![](images/dd63f4817aaedea1a624bd51a0c09659318e4cac0103a494ba0915784db0df18.jpg)
+
+![](images/fa557b4c3eed332bd928387ad41df72057da9cde0c7e35337afa738194045b08.jpg)  
+Figure 10: Comparison of clean test set Accuracy  $(\%)$  for moving across varying degrees of temporal label noise comparing all methods for 4-class classification. Error bars are st. dev. over 10 runs.
+
+![](images/6ff3b326fd5c045eb54e42d7808d3d53ae4edeb348453ca9085a9f2306dce704.jpg)
+
+![](images/3a923108c9c6391b5e3e60fc568e02882d88bca0e32d7a8e023244edb6ebbb4f.jpg)
+
+![](images/1b3473a1967c7757c3742f39f6707769f9c3787d25b05f8fd6f4a742175fc042.jpg)
+
+![](images/9fd887881355c6a1bac3aa0852d2127b45f949437b20deaf779f644eed0e487f.jpg)
+
+![](images/3d3bbc1bf08d50ac26014462d9003bfa752edda4946a20297874319d5b1032bd.jpg)
+
+![](images/39d7f2e17992aa9968f1dffd296500d705b5f028691df3f8126503e72e8e6964.jpg)  
+Figure 11: Comparison of noisy function reconstruction Mean Absolute Error (MAE) for har across varying degrees of temporal label noise comparing all methods for 4-class classification. Error bars are st. dev. over 10 runs.
+
+![](images/9d77e66bd84dabd4d4e04b809d86fbd15bd78c74103dd56c2a36a2626d826481.jpg)
+
+![](images/3ec541728a7d3a58e1292028ed5bd2743f257e6652b8cf71d7f4dec409c6ba94.jpg)
\ No newline at end of file
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+# LEARNING-AUGMENTED SEARCH DATA STRUCTURES
+
+Chunkai Fu* Brandon G. Nguyen* Jung Hoon Seo* Ryan Zesch* Samson Zhou†
+
+# ABSTRACT
+
+We study the integration of machine learning advice to improve upon traditional data structure designed for efficient search queries. Although there has been recent effort in improving the performance of binary search trees using machine learning advice, e.g., Lin et. al. (ICML 2022), the resulting constructions nevertheless suffer from inherent weaknesses of binary search trees, such as complexity of maintaining balance across multiple updates and the inability to handle partially-ordered or high-dimensional datasets. For these reasons, we focus on skip lists and KD trees in this work. Given access to a possibly erroneous oracle that outputs estimated fractional frequencies for search queries on a set of items, we construct skip lists and KD trees that provably provides the optimal expected search time, within nearly a factor of two. In fact, our learning-augmented skip lists and KD trees are still optimal up to a constant factor, even if the oracle is only accurate within a constant factor. We also demonstrate robustness by showing that our data structures achieves an expected search time that is within a constant factor of an oblivious skip list/KD tree construction even when the predictions are arbitrarily incorrect. Finally, we empirically show that our learning-augmented search data structures outperforms their corresponding traditional analogs on both synthetic and real-world datasets.
+
+# 1 INTRODUCTION
+
+As efficient data management has become increasingly crucial, the integration of machine learning (ML) has significantly improved the design and performance of traditional algorithms for many big data applications. Kraska et al. (2018) first showed that ML could be incorporated to create data structures that support faster look-up operations while also saving an order-of-magnitude of memory compared to optimized data structures oblivious to such ML heuristics. Subsequently, learning-augmented algorithms (Mitzenmacher & Vassilvitskii, 2020) have been shown to achieve provable worst-case guarantees beyond the limitations of oblivious algorithms for a wide range of settings. For example, ML predictions have been utilized to achieve more efficient data structures (Mitzenmacher, 2018; Lin et al., 2022), algorithms with faster runtimes (Dinitz et al., 2021; Chen et al., 2022c; Davies et al., 2023), mechanisms with better accuracy-privacy tradeoffs (Khodak et al., 2023), online algorithms with better performance than information-theoretic limits (Purohit et al., 2018; Gollapudi & Panigrahi, 2019; Lattanzi et al., 2020; Wang et al., 2020; Wei & Zhang, 2020; Bamas et al., 2020; Anand et al., 2020; Almanza et al., 2021; Anand et al., 2021; Im et al., 2021; Lykouris & Vassilvitskii, 2021; Aamand et al., 2022; Anand et al., 2022; Azar et al., 2022; Grigorescu et al., 2022; Khodak et al., 2022; Jiang et al., 2022; Scully et al., 2022; Antoniadis et al., 2023b;a; Shin et al., 2023), streaming algorithms with better accuracy-space tradeoffs (Hsu et al., 2019; Indyk et al., 2019; Jiang et al., 2020; Chen et al., 2022b;a; Li et al., 2023), and polynomial-time algorithms beyond hardness-of-approximation limits, e.g., NP-hardness (Ergun et al., 2022; Nguyen et al., 2023; C. S. et al., 2024).
+
+In this paper, we focus on the consolidation of ML advice to improve data structures for the fundamental problem of searching for elements among a large dataset. For this purpose, tree-based structures stand out as a popular choice among other structures, particularly for their logarithmic average performance. However, these structures often have weaknesses for specific use cases that make them sub-optimal for various applications, which we now discuss.
+
+Skip lists. One weakness of tree-based structures is that they need to be balanced for optimal performance, and thus their effectiveness is often closely tied to the order of element insertions. For example, the motivation of Lin et al. (2022) to study learning-augmented binary search trees noted that although previous results already characterized the statically optimal tree if the underlying distribution is known (Knuth, 1971; Mehlhorn, 1977), these methods do not handle dynamic insertion operations. In contrast, skip lists, introduced by Pugh (1990a), maintain balance probabilistically, offering a simpler implementation while delivering substantial speed enhancements (Pugh, 1990b). Skip lists are generally built iteratively in levels. The bottom levels of the skip list is an ordinary-linked list, in which the items of the dataset are organized in order. Each higher level serves to accelerate the search for the lower levels, by storing only a subset of the items in the lower levels, also as an ordered link list. Traditional skip lists are built by promoting each item in a level to a higher level randomly with a fixed probability  $p \in (0,1)$ .
+
+Querying for a target element begins at the first element in the highest level and continues by searching along the linked list in the highest level until finding an item whose value is at least that of the target element. If the found item is greater than the target element, the process is repeated after returning to the previous element and dropping to a lower list. It can be shown that the expected number of steps in the search is  $\mathcal{O}\left(\frac{1}{p}\log_{1/p}n\right)$  so that  $p$  serves as a trade-off parameter between the search time and the storage costs.
+
+In many modern applications, skip lists are used because of their excellent search runtime and their space efficiency. Skip lists are often preferred over binary search trees due to their simplicity of implementation, their support for efficient range query, and their amenability to concurrent processes (Shavit & Lotan, 2000; Lindén & Jonsson, 2013), high efficiency for dynamic datasets (Ge & Zdonik, 2008; Pittard & Tharp, 2010), network routing (Hu et al., 2003; Avin et al., 2020), and real-time analytics (Basin et al., 2020; Zhou et al., 2023). Thus while binary search trees have been a long-standing choice for querying ordered elements, skip lists offer a simpler, more efficient, and in some cases, necessary alternative.
+
+KD trees. Another weakness of tree-based data structures is that they generally require the data to obey an absolute ordering. However, in many cases, e.g., geometric applications or multidimensional data, the input points can only be partially ordered. Thus in 1975, KD trees, which stand for  $k$ -dimensional trees, were proposed as a more efficient alternative to binary search trees for searching in higher-dimensional spaces in procedures such as nearest neighbor search or ray tracing for applications in computational geometry or computer vision. A KD tree works by picking a data point and splitting along some spatial dimension to partition the space. This process is repeated until every data point is included in the tree, creating a hierarchical tree structure that enables quick access to specific data points or ranges within the dataset.
+
+Skewed distributions. Traditional search data structures treat each element equally when promoting the elements to higher levels. This balancing behavior facilitates good performance in expectation when a query to the skip list is equally likely to be any dataset element. On the other hand, this behavior may limit the performance of the data structure when the incoming queries are from an unbalanced probability distribution.
+
+Real-world applications can feature a diverse range of distribution patterns. One particularly common distribution is the Zipfian distribution, which is a probability distribution that is a discrete counterpart of the continuous Pareto distribution, and is characterized by the principle that a small number of events occur very frequently, while a large number of events occur rarely.
+
+In a Zipfian distribution, the frequency of an event  $N(k; \alpha, N)$  is inversely proportional to its rank  $k$ , raised to the power of  $\alpha$  (where  $\alpha$  is a positive parameter), in a dataset of  $N$  elements. In particular, we have  $N(k; \alpha, N) = \frac{1 / k^{\alpha}}{\sum_{n=1}^{N}(1 / n^{\alpha})}$ . The value of  $\alpha$  determines the steepness of the distribution so that a smaller  $\alpha$  value, i.e., closer to 0, makes it more uniform, while a larger  $\alpha$  increases skewness.
+
+Zipfian distributions provide a simple means for understanding phenomena in various fields involving rank and frequency, ranging from linguistics to economics, and from urban studies to information technology. Indeed, they appear in many applications such as word frequencies in natural language (Wang & Wang, 2016; Blocki et al., 2018), city populations (Gabaix, 1999; Vitanov & Ausloos, 2015), biological cellular distributions (Lazzardi et al., 2023), income distribution (Sandmo, 2015), etc.
+
+Unfortunately, although Zipfian distributions are common in practice, their properties are generally not leveraged by traditional search data structures, which are oblivious to any information about the query distributions. To improve this performance bottleneck, we propose the augmentation of traditional skip lists and KD trees with "learned" advice, which (possibly erroneously) informs the data structure in advance about some useful statistics on the incoming queries. Although we model the data structure as having oracle access to the advice, in practice, such advice can often easily be acquired from machine learning heuristics trained for these statistics.
+
+# 1.1 OUR CONTRIBUTIONS
+
+We propose the incorporation of ML advice into the design of skip lists and KD trees to improve upon traditional data structure design. For ease of discussion in this section, we assume the items that may appear either in the data set or the query set can be associated with an integer in  $[N] := \{1, \dots, N\}$ , which also in the case of high-dimensional data, may be associated with a  $k$ -dimensional point. We allow the algorithm access to a possibly erroneous oracle that, for each  $i \in [N]$ , outputs a quantity  $p_i$ , which should be interpreted as an estimation for the proportion of search queries that will be made to the data structure for the item  $i$ . Hence, for each  $i \in [N]$ , we assume that  $p_i \in [0, 1]$  and  $p_1 + \ldots + p_n = 1$ . Note that these constraints can be easily enforced upon the oracle as a pre-processing step prior to designing the skip list or KD tree data structure. We also assume that the oracle is readily accessible so that there is no cost for each interaction with the oracle. Consequently, we assume the algorithm has access to the predicted frequency  $p_i$  by the oracle for all  $i \in [N]$ . On the other hand, we view a sequence of queries as defining a probability distribution over the set of queries, so that  $f_i$  is the true proportion of queries to item  $i$ , for each  $i \in [N]$ . Although  $f_i$  is the ground truth, our algorithms only have access to  $p_i$ , which may or may not accurately capture  $f_i$ .
+
+Consistency for accurate oracles. We introduce construction for a learning-augmented skip list and KD trees, which gives expected search time at most  $2C + 2\sum_{i=1}^{n} f_i \cdot \min \left(\log \frac{1}{p_i}, \log n\right)$ , for some constant  $C > 0$ . On the other hand, we show that any skip list or KD tree construction requires an expected search time of at least the entropy  $H(f)$  of the probability vector  $f$ . We recall that the entropy  $H(f)$  is defined as  $H(f) = \sum_{i=1}^{n} f_i \cdot \log \frac{1}{f_i}$ .
+
+Thus, our results indicate that within nearly a factor of two, our learning-augmented search data structures are optimal for any distribution of queries, provided that the oracle is perfectly accurate. Moreover, even if the oracle on each estimated probability  $p_i$  is only accurate up to a constant factor, then our learning-augmented search data structures are still optimal, up to a constant factor.
+
+Implications to Zipfian distributions. We describe the implications of our results to queries that follow a Zipfian distribution; analogous results hold for other skewed distributions, e.g., the geometric distribution. It is known that if the  $r$ -th most common query/item has proportion  $\frac{z}{r^s}$  for some  $s > 1$ , then the entropy of the corresponding probability vector is a constant. Consequently, if the set of queries follows a Zipfian distribution and the oracle is approximately accurate within a constant factor, then the expected search time for an item by our search data structures is only a constant, independent of the total number of items, i.e.,  $\mathcal{O}(1)$ . By comparison, a traditional skip list or KD tree will have expected search time  $\mathcal{O}(\log n)$ .
+
+Robustness to erroneous oracles. So far, our discussions have centered around an oracle that either produces estimated probabilities  $p_i$  such that  $p_i = f_i$  or  $p_i$  is within a constant factor of  $f_i$ . However, in some cases, the machine learning algorithm serving as the oracle can be completely wrong. In particular, a model that is trained on a dataset before a distribution change, e.g., seasonal trends or other temporal shifts, can produce wildly inaccurate predictions. We show that our search data structures are robust to erroneous oracles. Specifically, we show that our algorithms achieve an expected search time that is within a constant factor of an oblivious skip list or KD tree construction when the predictions are incorrect. Therefore, our data structure achieves both consistency, i.e., good algorithmic performance when the oracle is accurate, and robustness, i.e., standard algorithmic performance when the oracle is inaccurate.
+
+Empirical evaluations. Finally, we analyze our learning-augmented search data structures list on both synthetic and real-world datasets. Firstly, we compare the performance of traditional skip lists with our learning-augmented skip lists on synthetically generated data following Zipfian distributions with various tail parameters. The dataset is created using four distinct  $\alpha$  values ranging from 1.01
+
+to 2, along with a uniform dataset. During the assessment, we query a specified number of  $n$  items selectively chosen based on their frequency weights. Our results match our theory, showing that learning-augmented skip lists have faster query times, with an average speed-up factor ranging from 1.33 up to 7.76, depending on the different skewness parameters.
+
+We then consider various datasets for internet traffic data, collected by AOL and by CAIDA, observed over various durations. For each dataset, we split the overall observation time into an early period, which serves as the training set for the oracle, and a later period, which serves as the query set for the skip list. The oracle trained using the IP addresses in the early periods outputs the probability of the appearance of a given node, and then the position of each node is determined.
+
+Our learning-augmented skip list outperforms traditional skip lists with an average speed-up factor of 1.45 for the AOL dataset and 1.63 for the CAIDA dataset. Moreover, the insertion time of our learning-augmented skip list is comparable with that of traditional skip lists on both synthetic and real-world datasets. We also observe that our history-based oracle demonstrates good robustness against temporal change, with little shift in the dominant element set. The adopted datasets show that the set of the top frequent elements does not change much across the time intervals in the datasets used herein.
+
+We similarly perform evaluations on our learning-augmented KD tree data structure. We evaluate our KD tree data structure on Zipfian distributions with various tail parameters, and provide a heatmap of average lookup times for elements. We find that for a large variety of Zipfian parameters, our method is able to provide large improvements over traditional KD trees. We perform a similar experiment under Zipfian distributions with added noise, and find our data structure still provides considerable improvements in query time.
+
+We additionally evaluate our method on point cloud samples taken from a 3D model. We bin these samples in space, and create our learning-augmented KD tree on these binned samples. When querying this tree with new binned point samples, we find a decrease in average query time as compared to a traditional KD tree under the same conditions. In addition to this experiment, we provide results on real world datasets of n-grams and neuron activation, and similarly find improvements over traditional KD trees.
+
+Concurrent and independent work. We mention that concurrent and independent of our work, Zeynali et al. (2024) used similar techniques to achieve the same guarantees on the performance of learning-augmented skip lists that are robust to erroneous predictions. However, they do not show optimality for their learning-augmented skip lists and arguably perform less exhaustive empirical evaluations. They also do not consider KD trees at all, which forms a significant portion of our contribution, both theoretically and empirically.
+
+Comparison to Lin et al. (2022). Our work was largely inspired by Lin et al. (2022), who observed that classical literature characterizing statically optimal binary search trees (Knuth, 1971; Mehlhorn, 1977) no longer apply in the dynamic setting, as elements arrive iteratively over time. Thus, they designed the construction of dynamic learning-augmented binary search trees (BSTs). Their analysis for the expected search time utilized the notion of pivots within their trees and thus were somewhat specialized to BSTs. Therefore, Lin et al. (2022) explicitly listed skip trees and advanced tree data structures as interesting open directions. Qualitatively, our results are similar to Lin et al. (2022), as are those of Zeynali et al. (2024). This is not quite altogether surprising because the main difference between these data structures is not necessarily the search time, but the either the ease of construction in the setting of skip lists, or the ability to handle multi-dimensional data in the setting of KD trees.
+
+# 2 LEARNING-AUGMENTED SKIP Lists
+
+In this section, we describe our construction for a learning-augmented skip list and show various consistency properties of the data structure. In particular, we show that up to a factor of two, our algorithm is optimal, given a perfect oracle. More realistically, if the oracle provides a constant-factor approximation to the probabilities of each element, our algorithm is still optimal up to a constant factor.
+
+We first describe our learning-augmented skip list, which utilizes predictions  $p_i$  for each item  $i \in [n]$ , from an oracle. Similar to a traditional skip list, the bottom level of our skip list is an ordinary-linked
+
+list that contains the sorted items of the dataset. As before, the purpose of each higher level is to accelerate the search for an item, but the process for promoting an item from a lower level to a higher level now utilizes the predictions. Whereas traditional skip lists promote each item in a level to a higher level randomly with a fixed probability  $p \in (0,1)$ , we automatically promote the item  $i$  to a level  $\ell$  if its predicted frequency  $p_i$  satisfies  $p_i \geq \frac{2^\ell - 1}{n}$ . Otherwise, we promote the item with probability  $\frac{1}{2}$ . This adaptation ensures that items with high predicted query frequencies will be promoted to higher levels of the skip list and thus be more likely to be found quickly.
+
+It is worth addressing a number of other natural approaches and their shortcomings. For example, one natural approach would be to use the "median" frequency across the items as a threshold to promote elements to higher levels. However, this promotion scheme is not ideal because computing the median frequency at each time would either require an additional data structure for fast update time or increase the insertion time. A potential approach to resolve this issue would be to use a separate threshold probability is set for each level so that only nodes with a probability higher than the corresponding threshold are promoted to the next level. However, this approach seems to result in an unnecessarily large number of created levels if some item appears with small probability, e.g.,  $\frac{1}{2^n}$ . We can thus first filters out the low-frequency elements and place them remain on the bottom level of the skip list and then proceed with using a separate threshold probability for each level. Unfortunately, this approach utterly fails to even match the search time performance of oblivious skip lists when the distribution is uniform, because all items will be in the same level, resulting in an expected search time of  $\Omega(n)$ . Hence, we ensure that each element still has a chance of being promoted to higher levels even when their probability is less than the corresponding threshold.
+
+We again emphasize that due to the dynamic nature of the updates, existing results on statically optimal binary search trees (Knuth, 1971; Mehlhorn, 1977) do not apply, as observed by Lin et al. (2022). We give the full details in Algorithm 1. For the sake of presentation, we focus on the setting where the queries are made to items in the dataset. However, we remark that our results generalize to the setting where queries can be made on the search space rather than the items in the dataset, provided the oracle is also appropriately adjusted to estimate the query distribution, using the approach we describe in Section 3.
+
+Algorithm 1 Learning-augmented skip list  
+Require: Predicted frequencies  $p_1,\dots ,p_n$  for each item in [n]   
+Ensure: Learning-augmented skip list   
+1: Insert all items at level 0   
+2: for each  $\ell$  do   
+3: if there are no items at level  $\ell -1$  then   
+4: return the skip list   
+5: else   
+6: for each  $i\in [n]$  do   
+7: if predicted frequency  $p_i\geq \frac{2^{\ell - 1}}{n}$  then   
+8: Insert  $i$  into level  $\ell$    
+9: else if  $i$  is in level  $\ell -1$  then   
+10: Insert  $i$  into level  $\ell$  with probability  $\frac{1}{2}$
+
+We first show an upper bound on the expected search time of our learning-augmented skip-list.
+
+Theorem 2.1. For each  $i \in [n]$ , let  $f_i$  and  $p_i$  be the proportion of true and predicted queries to item  $i$ . Then with probability at least 0.99 over the randomness of the construction of the skip list, the expected search time over the choice of queries at most  $20 + 2\sum_{i=1}^{n} f_i \cdot \min \left(\log \frac{1}{p_i}, \log n\right)$ .
+
+To achieve Theorem 2.1, we first show that each item  $i \in [n]$  must be contained at some level  $\max(0, 1 + \lfloor \log(np_i) \rfloor)$ , depending on the predicted frequency  $p_i$  of the item. We also show that with high probability, the total number of levels in the skip list is at most  $\mathcal{O}(\log n)$ . This allows us to upper bound the expected search time for item  $i$  by at most  $2C + 2\min\left(\log \frac{1}{p_i}, \log n\right)$ . We can then analyze the expected search time across the true probability distribution  $f_i$ . Putting these steps together, we obtain Theorem 2.1.
+
+We also prove a lower bound on the expected search time of an item drawn from a probability distribution  $f$  for any skip list.
+
+Theorem 2.2. Given a random variable  $X \in [n]$  so that  $X = i$  with probability  $f_{i}$ , let  $T(X)$  denote the search time for  $X$  in a skip list. Then  $\mathbb{E}[T(x)] \geq H(f)$ , where  $H(f)$  is the entropy of  $f$ .
+
+Theorem 2.2 uses standard entropy arguments that have been previously used to lower bound the optimal constructions of data structures such as Huffman codes. We next upper bound the entropy of a probability vector that satisfies a Zipfian distribution with parameter  $s$ .
+
+Lemma 2.3. Let  $s, z > 0$  be fixed constants and let  $f$  be a frequency vector such that  $f_{i} = \frac{z}{i^{s}}$  for all  $i \in [n]$ . If  $s > 1$ , then  $H(f) = \mathcal{O}(1)$  and otherwise if  $s \leq 1$ , then  $H(f) \leq \log n$ .
+
+By Theorem 2.1 and Lemma 2.3, we thus have the following corollary for the expected search time of our learning-augmented skip list on a set of search queries that follows a Zipfian distribution.
+
+Corollary 2.4. With high probability, the expected search time on a set of queries that follows a Zipfian distribution with exponent  $s$  is at most  $\mathcal{O}(1)$  for  $s > 1$  and  $\mathcal{O}(\log n)$  for  $s \leq 1$ .
+
+Next, we show that our learning-augmented skip list construction is robust to somewhat inaccurate oracles. Let  $f$  be the true-scaled frequency vector so that for each  $i \in [n]$ ,  $f_i$  is the probability that a random query corresponds to  $i$ . Let  $p$  be the predicted frequency vector, so that for each  $i \in [n]$ ,  $p_i$  is the predicted probability that a random query corresponds to  $i$ . For  $\alpha, \beta \in (0,1)$ , we call an oracle  $(\alpha, \beta)$ -noisy if for all  $i \in [n]$ , we have  $p_i \geq \alpha \cdot f_i - \beta$ . Then we have the following guarantees for an  $(\alpha, \beta)$ -noisy oracle:
+
+Lemma 2.5. Let  $\alpha$  be a constant and  $\beta < \frac{\alpha}{4n}$ . A learning-augmented skip list with a set of  $(\alpha, \beta)$ -noisy predictions has performance that matches that of a learning-augmented learned with a perfect oracle, up to an additive constant.
+
+To achieve Lemma 2.5, we parameterize our analysis in Theorem 2.1. Due to the guarantees of the  $(\alpha, \beta)$ -noisy oracles, we can write  $p_i \geq \frac{\alpha}{2} \cdot f_i$ , which allows us to express the search time  $\log \frac{1}{p_i}$  in terms of the true entropy of the distribution and a small additive constant that stems from  $\log \frac{1}{\alpha}$ . In fact, we remark that even when the predictions are arbitrarily inaccurate, our learning-augmented skip list still has expected query time  $\mathcal{O}(\log n)$ , since the total number of levels is at most  $\mathcal{O}(\log n)$  with high probability. Since the expected query list of an oblivious skip list is also  $\mathcal{O}(\log n)$ , then the expected query time of our learning-augmented skip list is within a constant multiplicative factor, even with arbitrarily poor predictions.
+
+# 3 LEARNING-AUGMENTED KD TREES
+
+In this section we present details on our novel approach to KD tree construction. First, we present the algorithm that constructs a learning-augmented KD tree. We focus on the setting where queries can be made on the search space rather than the items in the dataset, which is much more interesting for high-dimensional datasets, since even building a balanced tree on the search space could result in prohibitively high query time, as the height of the tree would already be at least the dimension  $d$ . Nevertheless, assuming that we have a query probability prediction  $p_i$  for element  $i$  of our dataset, the intuition of our method is straightforward. Whereas a learning-augmented binary search tree would attempt to find a value such that the probability of a query being on either branch of the tree is balanced, high-dimensional datasets do not have an absolute ordering.
+
+Thus, instead of relying on standard techniques to determine the splitting point of our dataset, we find a specific dimension in which there exists a balanced split such that the probability of a query being on either branch of the tree is balanced. However, there can still be high frequency queries that are not in the dataset, which can cause significantly high query time if not optimized. Hence, we also add to the tree construction high frequency queries that are not data points, in order to reject these negative queries more quickly.
+
+We prove the following guarantees on the performance of our learning-augmented KD tree, first assuming that our oracle is perfect.
+
+Theorem 3.1. Suppose  $[\Delta]^d$  is the space of possible input points and queries. Let  $N = \Delta^d$  and  $p_i$  be the probability that a random query is made to  $i \in [N]$ , given the natural mapping between
+
+Algorithm 2 Learning-augmented KD tree construction  
+1: function BUILNODE(x)  
+2:  $T \gets \emptyset$   
+3: if  $|x| = 1$  then  
+4:  $T = x$   
+5: return T  
+6: best  $\gets \emptyset$   
+7: for each dimension  $i$  of  $x$  do  
+8: for each element  $x$  do  
+9: Compute the probability of points to the left of  $x$  on axis  $i$   
+10: If the probability is closer to 0.5 than best, update best  
+11:  $T$  .axis = best-axis  
+12:  $T$  .left = BUILNODE  $(x : x[\text{axis}] \leq \text{best.value})$   
+13:  $T$  .right = BUILNODE  $(x : x[\text{axis}] > \text{best.value})$   
+14: return T
+
+Algorithm 3 Learning-augmented KD tree  
+1: function BUILD(dataset, queries)  
+2: dataset  $f \gets \{x \in dataset \mid x.\text{prob} > \frac{1}{n^2}\}$   
+3: queries  $f \gets \{x \in queries \mid x.\text{prob} > \frac{1}{n^2}\}$   
+4:  $T \gets \text{BUILDNODE}(dataset_f \cup queries_f)$   
+5: Insert  $\{x \in dataset \mid x.\text{prob} \leq \frac{1}{n^2}\}$  into  $T$  using standard balanced KD tree construction  
+6: return  $T$
+
+$[N]$  and  $[\Delta]^d$ . Let  $p = (p_1, \ldots, p_N) \in \mathbb{R}^N$  be the probability vector and  $H(p)$  be its entropy. Then given a set of  $n$  input points, the expected query time for the tree  $T$  created by Algorithm 3 is  $\mathcal{O}(\min(H(p), \log n))$ .
+
+The analysis of Theorem 3.1 corresponding to our learning-augmented KD tree follows from a similar structure as the proof of Theorem 2.1. However, the crucial difference is that the universe size is now  $N$ , which is exponential in  $d$ . Thus constructions that consider distributions over all of  $[N]$  may suffer  $\mathcal{O}(\log N) = \mathcal{O}(d\log \Delta)$  query time, which can be prohibitively expensive for large  $d$ , e.g., high-dimensional data. Hence, our algorithm requires a bit more care in the truncation of queries with low probability and instead, we build a balanced KD tree for any item with less than  $\frac{1}{n^2}$  probability of being queried, so that each of their query times is at most  $\mathcal{O}(\log n)$ . We further remark this implies robustness of our data structure to arbitrarily poor predictions, by a similar argument as in Section 2.
+
+We next prove a lower bound on the expected search time of an item drawn from a probability distribution  $f$  for any KD tree.
+
+Theorem 3.2. Given a random variable  $X \in [n]$  so that  $X = i$  with probability  $f_{i}$ , let  $D(X)$  denote the depth for  $X$  in a learning-augmented KD tree. Then  $\mathbb{E}[D(X)] \geq H(f)$ , where  $H(f)$  is the entropy of  $f$ .
+
+By Theorem 3.1 and Lemma 2.3, we thus have the following corollary for the expected query time on our learning-augmented KD tree on a set of search queries that follows a Zipfian distribution.
+
+Corollary 3.3. With high probability, the expected query time on a set of queries that follows a Zipfian distribution with exponent  $s$  is at most  $\mathcal{O}(1)$  for  $s > 1$  and  $\mathcal{O}(\log n)$  for  $s \leq 1$ .
+
+Finally, we show near-optimality when given imperfect predictions from a  $(\alpha, \beta)$ -noisy oracle:
+
+Lemma 3.4. Let  $\alpha$  be a constant and let  $\beta \leq \frac{\alpha}{n^2}$ . Then the query time for our learning-augmented KD tree with  $(\alpha, \beta)$ -noisy prediction matches the performance of a learning-augmented KD tree constructed using a perfect oracle up to an additive constant.
+
+# 4 EMPIRICAL EVALUATIONS
+
+In this section, we describe a number of empirical evaluations demonstrating the efficiency of our learning-augmented search data structures on both synthetic and real-world datasets. We provide additional experiments in Appendix D.
+
+Skip lists on CAIDA dataset. In the CAIDA datasets (CAIDA, 2016), the receiver IP addresses from one minute of the internet flow data are extracted for testing, which contains over  $650\mathrm{k}$  unique IP addresses of the 30 million queries. Given that the log-log plot of the frequency of all nodes in the CAIDA datasets follows approximately a straight line in Figure 1, the CAIDA datasets can be approximately characterized by an  $\alpha$  factor of 1.37. The insertion time is similar between classic and augmented skip lists, while Figure 2 shows that query time is almost halved when using the learning augmented skip lists at different query sizes. These results assume that the predicted frequency of all items in the query stream is accurate, i.e., the probability vector that is used to build the skip list matches exactly the query stream. The speed-up between the query times of the largest learning-augmented and the oblivious skip lists in Figure 2 is roughly  $1.86\times$ , which is surprisingly and perhaps coincidentally close to our theoretical speed-up of roughly  $1.81\times$  on a Zipfian dataset with exponent 1.37.
+
+![](images/ee20c94e2560061099873694ff6d3b21e881d5d8445396560a451813f8526a3b.jpg)  
+(a) CAIDA data distribution
+
+![](images/2277a9d1482e1825a966f9b7dc1a53fd292e8ec2a73b12862744c8c7c52b161b.jpg)  
+(b)Zipfian fit  $(\alpha = 1.37)$
+
+![](images/21b9d971cf5bea99dd369442362261e23ad868a8462681eba04cfdbd7eff4085.jpg)  
+Figure 1: CAIDA datasets distribution characterization in Figure 1a. The nearly straight-fitted curve in Figure 1b implies that a Zipfian distribution with  $\alpha = 1.37$  is a good fit to the CAIDA dataset distribution.  
+Insertion Time of Vanilla and Learned Skiplist  
+(a) Insert time on CAIDA  
+Figure 2: Comparison of insertion and query time on CAIDA for classic and learning-augmented skip lists. This figure compares the insertion and query times under varying numbers of top frequently accessed unique IPs between classic and augmented implementations. The horizontal axis in the two subfigures depicts the same scheme of IP selection, represented in two different ways, e.g., the top 29.9 million queries contain 665210 unique IPs, the next 29.5 million queries comprise 296384 unique IPs, etc.
+
+![](images/343a427117a7b700ca778c0f42ba99b9bba3217b2e9a973813d53183d224327e.jpg)  
+Query Time of Vanilla and Learned Skiplist  
+(b) Query time on CAIDA
+
+Next, we demonstrate that our proposed algorithm still manages to outperform the classic skip list even when temporal change exists in the probability vector by comparing the query time for the same set of query elements with different probability vectors being used to guide the building of the structure. For the skip list augmented by a noisy probability vector, the probability vector of elements
+
+during a period of T1 is used as the predicted frequencies. The skip list being augmented by this probability vector has its own set of elements to be organized into the target skip list. Suppose the historic data from T1 contains a set of elements S1, and some future query stream contains a set of elements S2. For each element in our target set S2, if the element is present in S1, then the occurrence probability of this element from S1 will be used to build S2; otherwise, if the element has not shown up during T1 (i.e., in S1), then we assume its probability to be 0. After this, the probability vector is normalized to sum to 1, resulting in a predicted probability vector to be used to build a skip list based on the historic element frequency. Since there is temporal changes in the frequency of elements being queried, the predicted probability vector will show a discrepancy with the true probability vector. The results are presented in Figure 3a and we show that even when the prediction is not perfect, the augmented skip list still performs better than a conventional skip list.
+
+![](images/41c33f717cf0296352e1271782f1db208283e0f006dfc3fc05ee5fcd0e15fbb1.jpg)  
+(a) Robustness test on CAIDA datasets
+
+![](images/794e84ef3a5912d3b6bdc537aa2fcc26fad381bb145cc51b6b0a20d9a2c5fd38.jpg)  
+(b) Oracle credibility on CAIDA datasets  
+Figure 3: Robustness of our learning-augmented skip list to erroneous oracles. In Figure 3b, the labels on the axis indicate the time stamp that the internet trace data is collected, e.g., 130100 means the collection starts at 13:01:00 and lasts for 1 minute.
+
+Figure 3a shows that the skip list with perfect learning shows the best performance, while the skip list augmented with noisy learning performs very close to the scenario with perfect predictions. Moreover, the closer the test data is to the reference data chronologically, the closer the noisy-augmented skip list will perform to the perfect learning skip list. The CAIDA datasets used in this study contain 12 minutes of internet flow data, which totals around 444 million queries. The indices on the x-axis in Figure 3a means:
+
+- 10_2: the first 10 minutes of data are used to create the reference (i.e., oracle) and the last 2 minutes are used to build and test the total query time using the former as reference.  
+- 2_2: the 9th and 10th minutes data is used as reference and the last 2 minutes are used for testing.  
+- 3_3: the 1st, 2nd and 3rd minutes of data are used to create reference and the 4th, 5th and 6th minutes of data are used for testing.  
+- 6_6: the first 6 minutes are used to create the reference and the last 6 minutes are used for testing.
+
+Further analysis of the temporal change of item frequency shows the reason behind the good performance of the history-based oracle. Figure 3b shows the change of intersection index between any 2 given minutes among the 12 minutes of CAIDA data. The intersection index is defined as the ratio of the number of shared queries to the total number of queries of any given 2 minutes of queries. Figure 3b shows that the number of intersects queries has decreased by about  $6\%$  after 12 minutes, which indicates that the probability of the majority of the elements will be predicted with good accuracy, resulting in good oracle performance.
+
+KD trees on synthetic datasets. KD Trees are commonly used in the field of computer graphics, with applications in collision detection, ray-tracing, and reconstruction. We first generate datasets of  $2^{12}$  points in 3-dimensional space, with frequencies given by a fixed Zipfian distribution with parameters  $a = 5, b = 2$  - parameters at which our method greatly outperforms a standard KD tree. In order to simulate constructing the tree on noisy data, we multiply the ground truth query probabilities by
+
+![](images/c3c9aeb7abb7b8e040be6731782e265e5356cb0d5df037616914d84334533f61.jpg)
+
+![](images/49d065a10c8150f185e188a855692b4f4c4e08d59262397c2e8cbb5999fd3b2a.jpg)
+
+![](images/4c520de91b6bb063bd597f94fd8a70b0063a21a7a113a99064a115a56d66cfff.jpg)  
+Figure 4: Query time comparison for standard and learning-augmented KD trees with various noise.
+
+numbers sampled uniformly from 1 to  $M$ , and then add numbers uniformly sampled from 0 to  $A$ , before renormalizing to form a valid probability distribution. We query the tree  $2^{14}$  times, with point queries selected by the ground truth Zipfian distribution. We repeat this process 32 times, and report the median of the average query depth across all runs in Figure 4. We find that our method continues to outperform traditional KD trees under moderate amounts of noise, and at worst, performs on-par with a traditional KD tree.
+
+Next, we generate datasets of  $2^{12}$  points in 3-dimensional space, with frequencies given by a Zipfian distribution with parameters  $a, b$ . In the left plot, we assign these Zipfian weights randomly. In the right plot, however, we assign Zipfian weights with ranks decreasing with the distance to some random data point. We then query the tree  $2^{14}$  times, with point queries selected by the same Zipfian distribution. We repeat this process 32 times, and report the median of the average query depth across all runs. We find that, when points frequencies are distributed smoothly over space, our method's performance increases on less skew distributions, as seen in this Figure 5.
+
+![](images/43bfb194d39d31cab471318ba5c694eddbf200a3cd2da9bf119d78658d619785.jpg)
+
+![](images/2685c1cb25842d44ff2d486268b3661d95fab84aaa3c379cb3f872763200aad1.jpg)  
+Figure 5: Comparison of query time on learning-augmented KD trees with and without smooth spatial distribution across various Zipfian parameters
+
+KD trees on 3D point-cloud datasets. Finally, we evaluate our method on point cloud data generated from the Stanford Lucy mesh (Stan), with dimensions  $\sim 1000 \times 500 \times 1500$ . We first uniformly sample  $2^{22}$  points along the mesh surface, and bin points with resolution 10, and assign lookup frequencies by the number of bin occupants. This results in 32k bins. Note, the resulting frequency distribution for binned cells is not highly skewed.
+
+We then generate a new set of  $2^{16}$  surface samples on the mesh, binning them and assigning frequencies in the same way. When looking up with the new samples, our method yields an average query depth of 15.1, while a traditional KD tree yields an average lookup depth of 17.6.
+
+# ACKNOWLEDGEMENTS
+
+Samson Zhou is supported in part by NSF CCF-2335411. The work was conducted in part while Samson Zhou was visiting the Simons Institute for the Theory of Computing as part of the Sublinear Algorithms program.
+
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+
+# A ADDITIONAL RELATED WORKS
+
+In this section, we discuss a number of related works in addition to those mentioned in Section 1. This paper builds upon the increasing body of research in learning-augmented algorithms, data-driven algorithms, and algorithms with predictions. For example, learning-augmented algorithms have been applied to a number of problems in the online setting, where the input arrives sequentially and the goal is to achieve algorithmic performance competitive with the best solution in hindsight, i.e., an algorithm that has the complete input on hand. Among the applications in the online model, learning-augmented algorithms have been developed for ski rental problem and job scheduling (Purohit et al., 2018), caching (Lykouris & Vassilitskii, 2021), and matching (Antoniadis et al., 2023b). Learning-augmented algorithms have also been used to improve the performance of specific data structures such as Bloom filters (Mitzenmacher, 2018), index structures (Kraska et al., 2018), CountMin and CountSketch (Hsu et al., 2019). Specifically, (Kraska et al., 2018) proposes substituting B-Trees (or other index structures) with trained models for querying databases. In their approach, rather than traversing the B-Tree to locate a record, they use a neural network to directly identify its position. Our work differs in that we retain the desired data structures, i.e., skip lists and kd trees, and focus on optimizing their structures to enable faster queries, which allows us to continue supporting standard operations specific to the data structures such as traversal, order statistics, merging, and joining, among others. Our work uses the frequency estimation oracle trained in (Hsu et al., 2019) on the AOL search query dataset and the CAIDA IP traffic monitoring dataset.
+
+Perhaps the works most closely related to ours in the area of learning-augmented algorithms are those of (Lin et al., 2022; Cao et al., 2023; Zeynali et al., 2024). Lin et al. (2022) noted that traditional theory on statically optimal binary search trees (Knuth, 1971; Mehlhorn, 1977) is no longer applicable in dynamic settings, where elements are added incrementally over time. Hence, they developed learning-augmented binary search trees (BSTs) and showed that their expected search time is nearoptimal. Cao et al. (2023) then extended these techniques to general search trees, allowing for nodes with more than two children. Cao et al. (2023) also studied the setting where the predictions may be updated, while ultimately still utilizing a data structure that requires rebalancing as data is dynamically changing. Zeynali et al. (2024) also consider the performance of learning-augmented skip lists that are robust to erroneous predictions; we elaborate more on the differences from Zeynali et al. (2024) in Section 1.1. We also note that none of these works consider KD trees at all, which is an important data structure with applications in computer vision and computational geometry, thus forming a basis of our work. For a more comprehensive source of related works in learning-augmented algorithms, see https://algorithms-with-predictions.github.io/.
+
+Beyond the context of learning-augmented algorithms, there is a large body of works that study design of data structures that are optimal for their inputs. For example, while standard binary search trees use  $\mathcal{O}(\log n)$  query time, optimal static trees can be constructed using dynamic programming or efficient greedy algorithms (Mehlhorn, 1977; Yao, 1982; Karpinski et al., 1996), given access frequencies. However, the computational cost of these methods often exceeds the cost of directly querying the tree. As a result, a key objective is to construct a tree whose cost is within a constant factor of the entropy of the data. Several approaches have achieved this either for worst-case data (Mehlhorn, 1977) or when the input follows particular distributions (Allen & Munro, 1978).
+
+More recent works have considered using results from learning theory to estimate the query frequencies, rather than assuming explicit access to their values. For example, Cayton & Dasgupta (2007)
+
+studied how to obtain such an oracle for learning-augmented data structures. In particular, they study generalization bounds in the context of learning theory, analyzing the number of samples from an underlying distribution necessary to produce an oracle with a small error rate. On the other hand, Ailon et al. (2011) studied algorithms for sorting and clustering that can improve their expected performance given access to multiple instances sampled from a fixed distribution. Although the high-level goal of improving algorithmic performance using auxiliary information is the same as ours, the specifics of the paper seem quite different than ours, as the paper focuses on techniques for sorting and clustering. Similarly, Ciriani et al. (2002) considers self-adjusting data structures, including skip lists, which can dynamically change as the sequence of queries arrive. However, their methods are catered specifically to the setting where there is access to the queries, whereas our data structures must be constructed without such access and must therefore be able to handle erroneous predictions. Finally, we remark that utilizing techniques for learning theory, there are standard results about the learnability of oracles for the purposes of learning-augmented algorithms (Ergun et al., 2022; Izzo et al., 2021).
+
+# B MISSING PROOFS FROM SECTION 2
+
+In this section, we give the missing proofs from Section 2.
+
+# B.1 EXPECTED SEARCH TIME
+
+We first show that each item is promoted to a higher level with probability at least  $\frac{1}{2}$ .
+
+Lemma B.1. For each item  $i \in [n]$  at level  $\ell$ , the probability that  $i$  is in level  $\ell + 1$  is at least  $\frac{1}{2}$ .
+
+Proof. Note that if  $p_i \geq \frac{2^\ell}{n}$ , then  $i$  will be placed in level  $\ell + 1$ . Otherwise, conditioned on the item  $i \in [n]$  being at level  $\ell$ , then Algorithm 1 places  $i$  at level  $\ell + 1$  with probability  $\frac{1}{2}$ . Thus, the probability that  $i$  is in level  $\ell + 1$  is at least  $\frac{1}{2}$ .
+
+We next upper bound the expected search time for any item at any fixed level, where the randomness is over the construction of the skip list.
+
+Lemma B.2. In expectation, the search time for item  $i \in [n]$  at level  $\ell$  is at most 2.
+
+Proof. Suppose item  $i \in [n]$  is in level  $\ell$ . Let  $S_{<i}^{\ell} \subseteq [n]$  be the subset of items in level  $\ell$  that are less than  $i$ . Note that by Lemma B.1, each item of  $S_{\leq i}^{\ell}$  is promoted to level  $\ell + 1$  with probability at least  $\frac{1}{2}$ . Thus, the search time for item  $i$  at level  $\ell$  is  $t$  if and only if the previous  $t$  items in  $S_{\leq i}^{\ell}$  were all not promoted, which can only happen with probability at most  $\frac{1}{2^t}$ . Hence, the expected search time  $T$  for item  $i \in [n]$  at level  $\ell$  is at most
+
+$$
+\mathbb {E} [ T ] \leq 1 \cdot \frac {1}{2} + 2 \cdot \frac {1}{2 ^ {2}} + \dots + n \cdot \frac {1}{2 ^ {n}} \leq \sum_ {t = 1} ^ {\infty} \frac {t}{2 ^ {t}} \leq 2.
+$$
+
+We now show that each item  $i$  must be contained at some level depending on the predicted frequency  $p_i$  of the item.
+
+Lemma B.3. Each item  $i$  is included in level max  $(0,1 + \lfloor \log (np_{i})\rfloor)$
+
+Proof. First, observe that all items are inserted at level 0. Next, note that Algorithm 1 inserts item  $i$  into level  $\ell$  if  $p_i \geq \frac{2^{\ell - 1}}{n}$  or equivalently  $\log (np_i) \geq \ell - 1$ . Thus, each item  $i$  is included in level  $\max \left(0, 1 + \lfloor \log (np_i) \rfloor\right)$ .
+
+We next analyze the expected search time for each item  $i$ .
+
+Lemma B.4. Suppose the total number of levels is at most  $C + \log n$  for some constant  $C > 0$ . Then the expected search time for item  $i$  is at most  $2C + 2\min \left(\log \frac{1}{p_i}, \log n\right)$ .
+
+Proof. By Lemma B.3, item  $i$  is included in level  $\max(0, 1 + \lfloor \log(np_i) \rfloor)$ . By Lemma B.2 the expected search time at each level is at most 2. Thus, in expectation, the total search time is at most  $2(C + \log n - \max(0, 1 + \lfloor \log(np_i) \rfloor)) \leq 2C + 2\min\left(\log \frac{1}{p_i}, \log n\right)$ .
+
+Finally, we analyze the expected search time across the true probability distribution  $f_{i}$ .
+
+Lemma B.5. Suppose the total number of levels is at most  $C + \log n$  for some constant  $C > 0$ . For each  $i \in [n]$ , let  $f_{i}$  be the proportion of queries to item  $i$ . Then the expected search time at most  $2C + 2\sum_{i=1}^{n} f_{i} \cdot \min \left( \log \frac{1}{p_{i}}, \log n \right)$ .
+
+Proof. For each query, the probability that the query is item  $i$  is  $f_{i}$ . Conditioned on the total number of levels being at most  $C + \log n$ , then by Lemma B.4, the expected search time for item  $i$  is at most  $2C + 2\max \left(\log \frac{1}{p_i},\log n\right)$ . Thus, the expected search time at most
+
+$$
+\begin{array}{l} 2 C \left(f _ {1} + \dots + f _ {n}\right) + 2 f _ {1} \min  \left(\log \frac {1}{p _ {1}}, \log n\right) + \dots + 2 f _ {n} \min  \left(\log \frac {1}{p _ {n}}, \log n\right) \\ = 2 C + 2 \sum_ {i = 1} ^ {n} f _ {i} \min  \left(\log \frac {1}{p _ {i}}, \log n\right). \\ \end{array}
+$$
+
+![](images/567b92dcfc98b68ced8dbc84a3e3f656569e8742a3d50547f6a34e95f1cfe8dd.jpg)
+
+We now show that with high probability, the total number of levels in the skip list is at most  $\mathcal{O}(\log n)$ .
+
+Lemma B.6. With probability at least 0.99, the total number of levels in the skip list is at most  $10 + \log n$ .
+
+Proof. For each level  $\ell$ , let  $n_{\ell}$  be the number of items  $i \in [n]$  that are deterministically promoted to exactly level  $\ell$ , i.e.,  $p_i \in \left[\frac{2^{\ell} - 1}{n}, \frac{2^{\ell}}{n}\right)$ . Note that for each fixed  $i \in [n]$ , the highest level it remains is a geometric random variable with parameter  $\frac{1}{2}$ , beyond the highest level at which it is deterministically placed. This is because the item is promoted to each higher level with probability  $\frac{1}{2}$ . Hence with probability  $1 - \frac{1}{2^k}$ ,  $i$  is not placed at least  $k$  levels above its highest deterministic placement. Therefore, the probability that an item at level  $\ell$  is placed at level  $10 + \log n$  is at most  $\frac{2^{\ell}}{1024n}$ . Since no fixed  $i$  will have predicted frequency more than 1, then no item will be deterministically placed at level  $2 + \log n$ . Hence by a union bound over all  $\ell \in [2 + \log n]$ , the probability that an item is placed at level  $10 + \log n$  is at most
+
+$$
+\sum_ {\ell = 0} ^ {2 + \log n} \frac {n _ {\ell} \cdot 2 ^ {\ell}}{1 0 2 4 n}.
+$$
+
+On the other hand, we have  $\sum_{i=1}^{n} p_i = 1$ , so that
+
+$$
+\sum_ {\ell = 0} ^ {2 + \log n} n _ {\ell} \cdot 2 ^ {\ell} \leq 2 n.
+$$
+
+Therefore, with probability at least 0.99, the total number of levels in the skip list is at most  $10 + \log n$ .
+
+Thus, putting together Lemma B.5 and Lemma B.6, we get:
+
+Theorem 2.1. For each  $i \in [n]$ , let  $f_{i}$  and  $p_i$  be the proportion of true and predicted queries to item  $i$ . Then with probability at least 0.99 over the randomness of the construction of the skip list, the expected search time over the choice of queries at most  $20 + 2\sum_{i = 1}^{n}f_{i}\cdot \min \left(\log \frac{1}{p_{i}},\log n\right)$ .
+
+# B.2 NEAR-OPTIMALITY
+
+We first recall the construction of a Huffman code, a type of variable-length code that is often used for data compression. The encoding for a Huffman is known to be an optimal prefix code and can be represented by a binary tree, which we call the Huffman tree (Huffman, 1952).
+
+To construct a Huffman code, we first create a min-heap priority queue that initially contains all the leaf nodes sorted by their frequencies, so that the least frequent items have the highest priority. The algorithm then iteratively removes the two nodes with the lowest frequencies from the priority queue, which become the left and right children of a new internal node that is created to represent the sum of the frequencies of the two nodes. This internal node is then added back to the priority queue. This process is continued until there only remains a single node left in the priority queue, which is then the root of the Huffman tree.
+
+A binary code is then assigned to the paths from the root to each leaf node in the Huffman tree, so that each movement along a left edge in the tree corresponds to appending a 0 to the codeword, and each movement along a right edge in the tree corresponds to appending a 1 to the codeword. Thus, the resulting binary code for each item is the path from the root to the leaf node corresponding to the item.
+
+Huffman coding is a type of symbol-by-symbol coding, where each individual item is separately encoded, as opposed to alternatives such as run-length encoding. It is known that Huffman coding is optimal among symbol-by-symbol coding with a known input probability distribution (Huffman, 1952) and moreover, by Shannon's source coding theorem, that the entropy of the probability distribution is an upper bound on the expected length of a codeword of a symbol-by-symbol coding:
+
+Theorem B.7 (Shannon's source coding theorem). (Shannon, 2001) Given a random variable  $X \in [n]$  so that  $X = i$  with probability  $f_{i}$ , let  $L(X)$  denote the length of the codeword assigned to  $X$  by a Huffman code. Then  $\mathbb{E}[L(x)] \geq H(f)$ , where  $H(f)$  is the entropy of  $f$ .
+
+We now prove our lower bound on the expected search time of an item drawn from a probability distribution  $f$ .
+
+Theorem 2.2. Given a random variable  $X \in [n]$  so that  $X = i$  with probability  $f_{i}$ , let  $T(X)$  denote the search time for  $X$  in a skip list. Then  $\mathbb{E}[T(x)] \geq H(f)$ , where  $H(f)$  is the entropy of  $f$ .
+
+Proof. Let  $\mathcal{L}$  be a skip list. We build a symbol-by-symbol encoding using the search process in  $\mathcal{L}$ . We begin at the top level. At each step, we either terminate, move to the next item at the current level, or move down to a lower level. Similar to the Huffman coding, we append a 0 to the codeword when we move down to a lower level, and we append a 1 to the codeword when we move to the next item at the current level. Now, the search time for an item  $x$  in  $\mathcal{L}$  corresponds to the length of the codeword of  $x$  in the symbol-by-symbol encoding. By Theorem B.7 and the optimality of Huffman codes among symbol-by-symbol encodings, we have that  $\mathbb{E}[T(x)] \geq H(f)$ , where  $f$  is the probability distribution vector of  $x$ .
+
+# B.3 ZIPFIAN DISTRIBUTION
+
+In this section, we briefly describe the implications of our data structure to Zipfian distributions.
+
+We first recall the following entropy upper bound for a probability distribution with support at most  $n$ .
+
+Theorem B.8. (Cover, 1999) Let  $f$  be a probability distribution on a support of size  $[n]$ . Then  $H(f) \leq \log n$ .
+
+We can then upper bound the entropy of a probability vector that satisfies a Zipfian distribution with parameter  $s$ .
+
+Lemma 2.3. Let  $s, z > 0$  be fixed constants and let  $f$  be a frequency vector such that  $f_{i} = \frac{z}{i^{s}}$  for all  $i \in [n]$ . If  $s > 1$ , then  $H(f) = \mathcal{O}(1)$  and otherwise if  $s \leq 1$ , then  $H(f) \leq \log n$ .
+
+Proof. Since  $f$  is a probability distribution on the support of size  $[n]$ , then by Theorem B.8, we have that  $H(f) \leq \log n$ . Thus, it remains to consider the case where  $s > 1$ . Since  $z \leq 1$ , we have
+
+$$
+\begin{array}{l} h (f) = \sum_ {i = 1} ^ {n} \frac {z}{i ^ {s}} \log \frac {i ^ {s}}{z} \\ \leq s \sum_ {i = 1} ^ {n} \frac {\log i}{i ^ {s}}. \\ \end{array}
+$$
+
+Note that there exists an integer  $\gamma > 0$  such that for  $i > \gamma$ , we have  $\frac{\log i}{i^s} < \frac{1}{i^{(s+1)/2}}$ . Since  $s > 1$ , then  $\frac{s+1}{2} > 1$  and thus
+
+$$
+\sum_ {i = \gamma} ^ {n} \frac {1}{i ^ {(s + 1) / 2}} \leq \sum_ {i = 1} ^ {\infty} \frac {1}{i ^ {(s + 1) / 2}} = \mathcal {O} (1).
+$$
+
+Hence,
+
+$$
+h (f) \leq s \sum_ {i = 1} ^ {\gamma - 1} \frac {\log i}{i ^ {s}} + s \sum_ {\gamma} ^ {\infty} \frac {\log i}{i ^ {s}} = \mathcal {O} (1).
+$$
+
+![](images/3970177aae445ac786df73dbbb5b827d01bafe43ccfa7e54bce1a684e36c137e.jpg)
+
+By Theorem 2.1 and Lemma 2.3, we have the following statement about the performance of our learning-augmented skip list on a set of search queries that follows a Zipfian distribution.
+
+Corollary 2.4. With high probability, the expected search time on a set of queries that follows a Zipfian distribution with exponent  $s$  is at most  $\mathcal{O}(1)$  for  $s > 1$  and  $\mathcal{O}(\log n)$  for  $s \leq 1$ .
+
+# B.4 NOISY ROBUSTNESS
+
+In this section, we show that our learning-augmented skip list construction is robust to somewhat inaccurate oracles. Let  $f$  be the true-scaled frequency vector so that for each  $i \in [n]$ ,  $f_i$  is the probability that a random query corresponds to  $i$ . Let  $p$  be the predicted frequency vector, so that for each  $i \in [n]$ ,  $p_i$  is the predicted probability that a random query corresponds to  $i$ . For  $\alpha, \beta \in (0,1)$ , we call an oracle  $(\alpha, \beta)$ -noisy if for all  $i \in [n]$ , we have  $p_i \geq \alpha \cdot f_i - \beta$ .
+
+Lemma 2.5. Let  $\alpha$  be a constant and  $\beta < \frac{\alpha}{4n}$ . A learning-augmented skip list with a set of  $(\alpha, \beta)$ -noisy predictions has performance that matches that of a learning-augmented learned with a perfect oracle, up to an additive constant.
+
+Proof. Suppose the total number of levels is at most  $C + \log n$  for some constant  $C > 0$ . Note that this occurs with a high probability for a learning-augmented skip list with a set of  $(\alpha, \beta)$ -noisy predictions. For each  $i \in [n]$ , let  $f_{i}$  be the proportion of queries to item  $i$  and let  $p_{i}$  be the predicted proportion of queries to item  $i$ . By Lemma B.5, the expected search time at most
+
+$$
+2 C + 2 \sum_ {i = 1} ^ {n} f _ {i} \cdot \min  \left(\log \frac {1}{p _ {i}}, \log n\right).
+$$
+
+Since the oracle is  $(\alpha, \beta)$ -noisy then we have  $p_i \geq \alpha \cdot f_i - \beta$  for all  $i \in [n]$ .
+
+We first note that in the expected search time for  $i$  is proportional to  $\min \left( \log \frac{1}{f_i}, \log n \right)$ . Thus, for expected search time for item  $i$ , it suffices to assume  $f_i > \frac{1}{2n}$  for all  $i$ .
+
+Observe that for  $f_{i} > \frac{1}{2n}$  and  $\beta < \frac{\alpha}{4n}$ , then  $p_{i} \geq \alpha \cdot f_{i} - \beta$  implies
+
+$$
+p _ {i} \geq \alpha \cdot f _ {i} - \beta \geq \alpha \cdot f _ {i} - \frac {\alpha}{4 n} \geq \frac {\alpha}{2} \cdot f _ {i}.
+$$
+
+Hence, we have  $\frac{1}{p_i} \leq \frac{2}{\alpha} \cdot \frac{1}{f_i}$  so that the expected search time for item  $i$  is at most
+
+$$
+2 C + 2 \cdot \min  \left(\log \frac {1}{f _ {i}} + \log \frac {2}{\alpha}, \log n\right).
+$$
+
+Therefore, the expected search time is at most
+
+$$
+\begin{array}{l} 2 C + 2 \sum_ {i = 1} ^ {n} f _ {i} \cdot \min  \left(\log \frac {1}{p _ {i}}, \log n\right) \leq 2 C + 2 \sum_ {i = 1} ^ {n} \left(f _ {i} \cdot \min  \left(\log \frac {1}{f _ {i}}, \log n\right) + f _ {i} \cdot \log \frac {2}{\alpha}\right) \\ \leq 2 C + 2 \log \frac {2}{\alpha} + 2 \sum_ {i = 1} ^ {n} f _ {i} \cdot \min  \left(\log \frac {1}{f _ {i}}, \log n\right). \\ \end{array}
+$$
+
+Since the perfect oracle would achieve runtime  $2C + 2\sum_{i=1}^{n} f_i \cdot \min \left( \log \frac{1}{f_i}, \log n \right)$ , then it follows that a learning-augmented skip list with a set of  $(\alpha, \beta)$ -noisy predictions has performance that matches that of a learning-augmented learned with a perfect oracle, up to an additive constant.
+
+# C MISSING PROOFS FROM SECTION 3
+
+First, we will show that the expected depth of a given query depends on the probability of that query, and that high frequency queries must be found close to the root of our tree.
+
+Lemma C.1. Suppose  $[\Delta]^d$  is the space of possible input points and queries. Let  $N = \Delta^d$  and  $p_i$  be the probability that a random query is made to  $i \in [N]$ , given the natural mapping between  $[N]$  and  $[\Delta]^d$ . Then the level at which  $i$  resides in the tree is at most  $\mathcal{O}\left(\log \frac{1}{p_i}\right)$ .
+
+Proof. First, consider only the high-frequency query points and data points for which we base our construction off.
+
+In constructing the learning-augmented KD tree, we balance the contents of the children nodes such that a query to that node has a probability of  $\frac{1}{2}$  of belonging to each of the children. Therefore, at a depth of  $d$ , the probability of belonging to either child is  $\frac{1}{2^d}$ . In particular, a query  $i$  with probability  $p_i$  at a depth  $d$  must satisfy  $p_i > \frac{1}{2^d}$ . Thus, we have that the depth of  $i$  is  $\mathcal{O}\left(\log \frac{1}{p_i}\right)$ , as desired.
+
+Since the lowest probability of a high-frequency data point is  $\frac{1}{n^2}$ , this tree must have a depth of at most  $2\log n$ .
+
+Now, consider a low-frequency data point, which we add to the bottom of the tree. By construction, the learned point of our tree has depth at most  $2\log n$ . Then, when inserting the additional data points as a balanced KD tree, we can accumulate at most an additional depth of  $\log n$ . Note,  $p < \frac{1}{n^2}$  implies  $\log n < \log \frac{1}{p}$ . Thus, this low-frequency data point will have a depth of at most  $3\log n = \mathcal{O}\left(\log \frac{1}{p}\right)$ , as desired.
+
+Similarly, if  $i$  is not a data point and is low frequency, we achieve the same bound of  $\mathcal{O}\left(\log \frac{1}{p}\right)$ . In this case, we simply terminate at a leaf node and determine that the desired query is not in the dataset. In summary, any query which has high frequency can be found in  $\mathcal{O}\left(\log \frac{1}{p}\right)$  time. Low-frequency data points can similarly be found in  $\mathcal{O}\left(\log \frac{1}{p}\right)$  time, and low frequency queries can be determined to not exist in  $\mathcal{O}\left(\log \frac{1}{p}\right)$  time.
+
+Lemma C.2. Suppose  $[\Delta]^d$  is the space of possible input points and queries. Let  $N = \Delta^d$  and  $p_i$  be the probability that a random query is made to  $i \in [N]$ , given the natural mapping between  $[N]$  and  $[\Delta]^d$ . Then the level at which  $i$  resides in the tree is at most  $\mathcal{O}(\log n)$ .
+
+Proof. This follows directly from the analysis in Llemma C.1.  $\square$
+
+Now, we have demonstrated the depth of a given query point  $i$  is bounded by both  $\mathcal{O}(\log n)$  and  $\mathcal{O}\left(\log \frac{1}{p_i}\right)$ . Using this fact, we will now show that the expected query time of our algorithm is bounded by both the entropy of the dataset  $H(p)$  in addition to  $\log n$ .
+
+We now analyze the performance of our learning-augmented KD tree.
+
+Theorem 3.1. Suppose  $[\Delta]^d$  is the space of possible input points and queries. Let  $N = \Delta^d$  and  $p_i$  be the probability that a random query is made to  $i \in [N]$ , given the natural mapping between  $[N]$  and  $[\Delta]^d$ . Let  $p = (p_1, \ldots, p_N) \in \mathbb{R}^N$  be the probability vector and  $H(p)$  be its entropy. Then given a set of  $n$  input points, the expected query time for the tree  $T$  created by Algorithm 3 is  $\mathcal{O}(\min(H(p), \log n))$ .
+
+Proof. Following C.1, the points  $i$  in  $[\Delta]^d$  with non-negligible probability  $p_i \geq \frac{1}{n^2}$  are guaranteed to exist in the learning-augmented KD tree  $\mathcal{T}$  with depth at most  $\log \frac{1}{p_i}$ . For points in the dataset  $[n]$  with negligible probability, they exist in the tree and have depth in  $\mathcal{O}(\log n)$ . For all other points not contained in the KD tree, the query will terminate at a depth of  $\mathcal{O}(\log n)$ .
+
+For any point  $i$  in  $[\Delta]^d$ , the depth that a query to  $i$  will terminate in the tree is
+
+$$
+\operatorname {D e p t h} (i) = \mathcal {O} \left(\min  \left(\log \frac {1}{p _ {i}}, \log n\right)\right). \tag {1}
+$$
+
+Then, the expected search time  $T$  is the expected depth of a given point,
+
+$$
+\mathbb {E} [ T ] = \sum_ {i \in [ \Delta ] ^ {d}} p _ {i} \cdot \operatorname {D e p t h} (i) = \sum_ {i \in [ \Delta ] ^ {d}} p _ {i} \cdot \mathcal {O} \left(\min  \left(\log \frac {1}{p _ {i}}, \log n\right)\right) = \mathcal {O} (\min  (H (p), \log n)). \tag {2}
+$$
+
+![](images/095bd8891a6126223d61ff31bc9195ca9cd26807e3ef5aa1c8eb134dadaef428.jpg)
+
+Thus, we have shown that the expected query time of our algorithm is bounded by the entropy of the dataset. In particular, when the dataset has a highly skew distribution,  $H(p)$  can be far less than  $\log n$ .
+
+# C.1 NEAR-OPTIMALITY
+
+Near-optimality of our learning-augmented KD trees uses a similar argument to the proof of the near-optimality of our learning-augmneted skip lists. In particular, we again utilize Shannon's source coding theorem from Theorem B.7. We then have the following:
+
+Theorem 3.2. Given a random variable  $X \in [n]$  so that  $X = i$  with probability  $f_{i}$ , let  $D(X)$  denote the depth for  $X$  in a learning-augmented KD tree. Then  $\mathbb{E}[D(X)] \geq H(f)$ , where  $H(f)$  is the entropy of  $f$ .
+
+Proof. In a learning-augmented KD tree, the search path to an element  $i$  can be encoded as a  $0 - 1$  codeword, with entries indicating whether the lower or upper branch is taken at each node traversal. Moreover, the length of this codeword in the symbol-by-symbol encoding corresponds to the depth of element  $i$ . Then, by Theorem B.7 and the optimality of Huffman codes in symbol-by-symbol encodings, we have that  $\mathbb{E}[D(X)] \geq H(f)$ , as desired.
+
+# C.2 NOISY ROBUSTNESS
+
+In this section, we analyze the performance of the learning-augmented KD tree under noisy data conditions.
+
+Previously, we analyzed the performance of the learning-augmented KD-tree assuming access to a perfect prediction oracle.
+
+Now we analyze the performance with a noisy oracle. That is, for each  $i \in [\Delta]^d$  there is a true-scaled frequency  $f_{i}$  that the point will be queried and that  $p_i$  is a prediction made by the noisy oracle.
+
+First, we analyze the multiplicative robustness of the algorithm. In this case, the oracle predicts  $f_{i}$  up to some multiplicative constant  $\alpha \in \mathbb{R}^{+}$  such that  $f_{i} = \alpha p_{i}$ .
+
+Lemma C.3. Suppose  $[\Delta]^d$  is the space of possible input points and queries. Let  $N = \Delta^d$  and  $p_i$  be the probability that a random query is made to  $i \in [N]$  during tree construction. Suppose during runtime that the true probability of querying  $i$  is  $f_i = \alpha p_i$  for some  $\alpha \in \mathbb{R}^+$ . Then the level at which  $i$  resides in the tree is at most  $\mathcal{O}\left(\log \frac{1}{f_i} + \log \frac{1}{\alpha}\right)$ .
+
+Proof. If  $\alpha \leq 1$ , this is immediate. If this is the case, in construction we expected  $i$  to be queried more often than it actually is, so our construction placed the point  $i$  higher in the tree than is necessary. Thus, the depth is at most the previously shown  $\frac{1}{p_i} \leq \frac{1}{f_i}$ .
+
+Now, suppose  $\alpha > 1$ . In this case, we must have placed  $i$  deeper in the tree than we should have, as our construction frequency is less than the true query frequency. Then, as in C.1, the depth of  $i$  is  $\mathcal{O}\left(\log \frac{1}{p_i}\right)$ . Now, we have that
+
+$$
+\mathcal {O} \left(\log \frac {1}{p _ {i}}\right) = \mathcal {O} \left(\log \frac {\alpha}{f _ {i}}\right) = \mathcal {O} \left(\log \frac {1}{f _ {i}} + \log \alpha\right). \tag {3}
+$$
+
+![](images/5124775f96d1fbe6b8c6a760f6bb6c6f4c83c9a55f1d4dbceb29a75a77c353d1.jpg)
+
+Having shown multiplicative robustness of the learning-augmented KD tree we next analyze the additive-multiplicative robustness of the method. An oracle is  $(\alpha, \beta)$ -noisy if the prediction satisfies  $p_i \geq \alpha f_i - \beta$  for constants  $\alpha, \beta \in (0,1)$ .
+
+Lemma 3.4. Let  $\alpha$  be a constant and let  $\beta \leq \frac{\alpha}{n^2}$ . Then the query time for our learning-augmented KD tree with  $(\alpha, \beta)$ -noisy prediction matches the performance of a learning-augmented KD tree constructed using a perfect oracle up to an additive constant.
+
+Proof. From Lemma C.1 we have that the depth of a point  $i$  in the learning-augmented KD tree is  $\mathcal{O}\left(\frac{1}{p_i}\right)$  where  $p_i$  is the predicted frequency of  $i$ .
+
+Then, with an  $(\alpha, \beta)$ -noisy oracle we have that the prediction is bounded from below by  $p_i \geq \alpha f_i - \beta$ . As  $i$  is a point in the tree, we can assume that the true-scaled frequency  $f_i$  has a lower-bound of  $1/n^2$ . By choosing  $\beta \leq \alpha/n^2$  we ensure that the predicted  $p_i$  is always nonnegative. Then, let  $\beta = \alpha/2n^2$
+
+$$
+p _ {i} \geq \alpha f _ {i} - \beta \geq \alpha f _ {i} - \frac {\alpha}{2 n ^ {2}} \geq \frac {\alpha}{2} f _ {i} \tag {4}
+$$
+
+Then, applying Lemma C.1 again
+
+$$
+\mathcal {O} \left(\log \frac {1}{p _ {i}}\right) = \mathcal {O} \left(\log \frac {2}{\alpha f _ {i}}\right) = \mathcal {O} \left(\log \frac {1}{f _ {i}} + \log \frac {1}{\alpha}\right). \tag {5}
+$$
+
+![](images/9819d6846014b06731890c2679bb300affb9bfaea25cd5d0e0d20ddffd7bd544.jpg)
+
+# D ADDITIONAL EMPIRICAL EVALUATIONS
+
+# D.1 SKIP LISTS
+
+In this section, we perform empirical evaluations comparing the performance of our learning-augmented skip list to that of traditional skip lists, on both synthetic and real-world datasets. Firstly, we compare the performance of traditional skip lists with our learning-augmented skip lists on synthetically generated data following Zipfian distributions. The proposed learning-augmented skip lists are evaluated empirically with both synthetic datasets and real-world internet flow datasets from the Center for Applied Internet Data Analysis (CAIDA) and AOL. In the synthetic datasets, a diverse range of element distributions, which are characterized by the skewness of the datasets, are evaluated to assess the effectiveness of the learning augmentation. In the CAIDA datasets, the  $\alpha$  factor is calculated to reflect the skewness of the data distribution.
+
+The metrics of performance evaluations include insertion time and query time, representing the total time it takes to insert all elements in the query stream and the time it takes to find all elements in the query stream using the data structure, respectively.
+
+The computer used for benchmarking is a Lenovo Thinkpad P15 with an intel core i7-11800H@2.3GHz, 64GB RAM, and 1TB of Solid State Drive. The tests were conducted in a Ubuntu 22.04.3 LTS OS. GNOME version 42.9.
+
+# D.1.1 SYNTHETIC DATASETS
+
+In the synthetic datasets, both the classic and augmented skip lists are tested against different element counts and  $\alpha$  values. In terms of the distribution of the synthetic datasets, the uniform distribution and a Zipfian distribution of  $\alpha$  between 1.01 and 2 with query counts up to 4 million are evaluated. It is worth noting that the number of unique element queries could vary for the same query count at different  $\alpha$  values in the Zipfian distribution, which may affect the insertion time.
+
+Table 1: Speed up factor of augmented skip list over classic skip list under different synthetic distributions  
+
+<table><tr><td rowspan="2">Distribution</td><td colspan="12">Query size of synthetic data (unit: thousand)</td></tr><tr><td>0.5</td><td>10</td><td>100</td><td>500</td><td>1000</td><td>1500</td><td>2000</td><td>2500</td><td>3000</td><td>3500</td><td>4000</td><td>Average</td></tr><tr><td>uniform</td><td>3.02</td><td>0.84</td><td>1.01</td><td>1.05</td><td>1.11</td><td>1.14</td><td>1.17</td><td>1.21</td><td>1.22</td><td>1.42</td><td>1.4</td><td>1.33</td></tr><tr><td>α=1.01</td><td>3.63</td><td>2.6</td><td>1.04</td><td>1.24</td><td>1.03</td><td>1.21</td><td>1.2</td><td>1.14</td><td>1.3</td><td>1.18</td><td>1.3</td><td>1.53</td></tr><tr><td>α=1.25</td><td>3.28</td><td>3.74</td><td>5.87</td><td>2.89</td><td>2.47</td><td>3.21</td><td>2.95</td><td>3.34</td><td>3.55</td><td>3.16</td><td>3.12</td><td>3.42</td></tr><tr><td>α=1.5</td><td>2.42</td><td>8.97</td><td>6.93</td><td>6.54</td><td>7.99</td><td>5.83</td><td>4.65</td><td>3.8</td><td>4.92</td><td>5.34</td><td>5.93</td><td>5.76</td></tr><tr><td>α=1.75</td><td>12.43</td><td>10.4</td><td>5.76</td><td>9.78</td><td>6.76</td><td>7.13</td><td>7.31</td><td>7.09</td><td>6.63</td><td>5.07</td><td>6.98</td><td>7.76</td></tr><tr><td>α=2</td><td>8.19</td><td>2.5</td><td>5.56</td><td>10.1</td><td>4.47</td><td>3.91</td><td>7.26</td><td>5.33</td><td>9.29</td><td>7.65</td><td>5.55</td><td>6.35</td></tr></table>
+
+Table 2: Node count for each distribution configuration in the 4 million dataset  
+
+<table><tr><td>α</td><td>Unique node count</td></tr><tr><td>1.01</td><td>2886467</td></tr><tr><td>1.25</td><td>259892</td></tr><tr><td>1.75</td><td>8386</td></tr><tr><td>2</td><td>2796</td></tr></table>
+
+Table 1 shows the speed-up factor, defined as the time taken by the augmented skip list over the classic skip list for the same query stream. We can observe a progressive improvement in the performance of our augmented skip lists as the dataset skewness increases. It also suggests that our augmented skip list will perform at least as good as the traditional skip list and will outperform a traditional skip list by a factor of up to 7 times depending on the skewness of the datasets.
+
+Figure 6 shows that the insertion time decreases with more skewed datasets for the same size of the query stream. This is attributed to the reduced number of nodes in the datasets, as shown in Table 2.
+
+The query time of augmented skip lists is also reduced greatly compared to the classic skip lists as shown in Figure 7.
+
+In addition, we conduct experiments to compare the performance of standard binary search trees and standard skip lists. In particular, we generate datasets of size  $n \in \{5000, 10000, 15000, 20000, 25000, 30000, 35000, 40000, 45000, 50000\}$ . For each fixed value of  $n$ , each element of the dataset is generated uniformly at random in  $[2n]$ , i.e., uniformly at random from  $\{1, 2, \dots, 2n\}$ . Because the dataset is generated uniformly at random, then learning-augmented data structures will perform similar to oblivious data structures. We measure the construction time of the data structures, based on the input dataset. Our results demonstrate that as expected, skip lists perform significantly better than balanced binary search trees across all values of  $n$ , due to the latter's necessity of constantly rebalancing the data structure. In fact, skip lists performed almost  $4 \times$  better than BSTs in some cases, e.g.,  $n = 20000$ . We illustrate our results in Figure 8.
+
+# D.1.2 AOL DATASET
+
+The AOL dataset (G. Pass, 2006) features around 20M web queries collected from 650k users over three months. The distribution of the queries is shown in Figure 9. The AOL dataset is a less skewed dataset than CAIDA with an alpha value of 0.75.
+
+Figure 9 shows the distribution of the AOL queries with an estimated alpha value of 0.75. The AOL dataset resembles more to a slightly skewed uniform distribution with very few highly frequent items, which accounts for a lower improvement as in the case of AOL shown in Figure 10. The total number of queries for items with higher than 1000 frequency accounts for only  $5\%$  of the total number of queries for the AOL datasets. The learning-augmented skip list still outperforms the traditional skip
+
+![](images/3c719344082ee9a7d7f3f8537842d3db46bb50d6ba8afee5d201fc8c78bce54d.jpg)
+
+![](images/9c378211fc4bbccd5a44201064d24e0196664579055f024a2bc6fc3a6aacf02a.jpg)
+
+![](images/cd2a879cf873fca0c53763f5fefb644146d2d02dc66bc0f425923a10296ba73b.jpg)  
+(a) Uniform Distribution  
+(c)  $\alpha = 1.5$
+
+![](images/19e9e7606677502d3809f0457b168db7821d988b7f4b842cc11fa02fabb9c230.jpg)  
+(b)  $\alpha = 1.25$  
+(d)  $\alpha = 2$  
+Figure 6: Insertion time for synthetic datasets with a uniform distribution and under different  $\alpha$  values of the Zipfian distribution for both classic and augmented skip lists. This figure illustrates the insertion time on the synthetic data for both the uniform distribution and the Zipfian distribution at different  $\alpha$  values. Generally, higher skewness of the datasets results in less insertion time when using the augmented structure. The decrease in insertion time is proportional to the increase in the  $\alpha$  value, as a higher  $\alpha$  value leads to a reduction in the number of unique nodes, as illustrated in Table 2.
+
+list on this slightly skewed dataset. This result is also in line with the results from the synthetic data shown in Table 1 where lower alpha values have resulted in a lower speedup factor.
+
+# D.2 KD TREES
+
+For KD Trees, first describe our methodology for evaluating our data structure on synthetic data. We then describe our empirical evaluations on real-world datasets.
+
+The computer used for KD tree benchmarking is a desktop machine with an Intel Core i9-14900KF@ 3200MHz, with 64GB RAM, and 2TB of Solid State Drive. The tests were conducted in Windows 10 Enterprise, version 10.0.19045 Build 19045.
+
+# D.2.1 SYNTHETIC DATASETS WITH PERFECT KNOWLEDGE
+
+First, consider a dataset with a Zipfian distribution. In order to construct this dataset, we first select  $n$  unique data points in  $[\Delta]^d$  uniformly. We then generate Zipfian frequencies,  $f_i \approx \frac{1}{(i + b)^a}$ , and randomly pair the frequencies to the data points to serve as both the construction and query frequencies. We construct either a traditional KD tree, or our learned KD tree on this dataset. Then, we evaluate the performance of querying by sampling the known datapoints with probabilities given by their Zipfian probabilities.
+
+In Table 3, we use a Zipfian distribution with parameters  $a = 1$  and  $b = 2.7$ . This data demonstrates that our method outperforms a traditional KD tree, given that the Zipfian distribution has the given parameters. In Figure 11, we vary the Zipfian parameters of our dataset. As expected, our learned KD tree performance increases with how skew the distribution is. Moreover, we find that our method outperforms the traditional KD tree on all tested Zipfian distributions.
+
+![](images/2f02c6c2bf3624d7f8dd9bd3655a4fcc7adfb729b380f2d9b1e38fbd0cc50b74.jpg)
+
+![](images/d8a1e9cdc965a5637b462aa07823247bbd1ce483fa3127ae525837b204dfcc75.jpg)
+
+![](images/8a8dc07261b832affa1237d6d52a2d9ea8b66995c0fa2e6947590311e569a6b7.jpg)  
+(a) Uniform Distribution  
+(c)  $\alpha = 1.5$
+
+![](images/7c280f4cc55e6b552673d27438349794157a5b10de39524143db5c513e802fc7.jpg)  
+(b)  $\alpha = 1.25$  
+(d)  $\alpha = 2$  
+Figure 7: Query time for synthetic datasets with a uniform distribution and under different  $\alpha$  values of the Zipfian distribution for both classic and augmented skip lists. This figure compares the query time of the classic and augmented skip lists for both uniform distribution and Zipfian distribution at various  $\alpha$  values. Similar to insertion, the query time is significantly reduced under different query sizes with the implemented augmentation. The performance enhancement is especially pronounced for the high skewness of the dataset.
+
+<table><tr><td>type</td><td>dim</td><td>const_time</td><td>avg query(s)</td><td>avg query depth</td></tr><tr><td>traditional</td><td>1</td><td>1.267E-01</td><td>2.645E-05</td><td>1.330E+01</td></tr><tr><td>learned</td><td>1</td><td>2.387E-01</td><td>2.181E-05</td><td>1.086E+01</td></tr><tr><td>traditional</td><td>2</td><td>1.266E-01</td><td>2.709E-05</td><td>1.341E+01</td></tr><tr><td>learned</td><td>2</td><td>3.232E-01</td><td>2.221E-05</td><td>1.086E+01</td></tr><tr><td>traditional</td><td>3</td><td>1.253E-01</td><td>2.700E-05</td><td>1.334E+01</td></tr><tr><td>learned</td><td>3</td><td>3.981E-01</td><td>2.264E-05</td><td>1.097E+01</td></tr><tr><td>traditional</td><td>4</td><td>1.185E-01</td><td>2.724E-05</td><td>1.336E+01</td></tr><tr><td>learned</td><td>4</td><td>4.549E-01</td><td>2.256E-05</td><td>1.089E+01</td></tr><tr><td>traditional</td><td>5</td><td>1.300E-01</td><td>2.743E-05</td><td>1.336E+01</td></tr><tr><td>learned</td><td>5</td><td>5.286E-01</td><td>2.296E-05</td><td>1.096E+01</td></tr><tr><td>traditional</td><td>10</td><td>1.277E-01</td><td>2.896E-05</td><td>1.340E+01</td></tr><tr><td>learned</td><td>10</td><td>8.642E-01</td><td>2.403E-05</td><td>1.091E+01</td></tr><tr><td>traditional</td><td>20</td><td>1.295E-01</td><td>3.121E-05</td><td>1.344E+01</td></tr><tr><td>learned</td><td>20</td><td>1.543E+00</td><td>2.564E-05</td><td>1.085E+01</td></tr><tr><td>traditional</td><td>40</td><td>1.361E-01</td><td>3.558E-05</td><td>1.340E+01</td></tr><tr><td>learned</td><td>40</td><td>2.911E+00</td><td>2.902E-05</td><td>1.078E+01</td></tr></table>
+
+Table 3: We construct and query KD trees with our method and with a traditional KD tree on synthetic datasets of various dimensionality. We construct our tree on 10k points, and query 1M times. We find that, independent of the data dimensionality, our method produces lower average query depths.
+
+# D.2.2 SYNTHETIC DATASETS WITH NOISY KNOWLEDGE
+
+In reality, it is rarely the case that we have perfect knowledge in constructing a model. In order to evaluate the performance of our method on noisy data, we create a synthetic dataset with noisy training information.
+
+As before, we generate world points and assign them Zipfian weights. When constructing the tree, each weight us updated to be  $Mp_{i} + A$ , where  $M$  and  $A$  are drawn from uniform distributions. This
+
+![](images/098ba0b4c6f2ec9938763af1e75dc5e081595fc6ee02a68aea6e6969ae783973.jpg)  
+Comparison of BST and Skiplist
+
+![](images/51a0e9b09a4c41a3baa36cbf8a847b5a6eab8956de10d94e05e2998fe58a033f.jpg)  
+(a) AOL data distribution  
+Figure 9: AOL datasets distribution characterization. This figure illustrates how the  $\alpha$  value of 0.75 is obtained for the AOL dataset. The AOL dataset shows a much smaller  $\alpha$  value compared to the CAIDA dataset so AOL almost resembles a uniform distribution despite very few high-frequency nodes. This also explains why the performance of the augmented skip list is close to the classic implementation.
+
+![](images/8b5f2cf2e042349418cd1a9fb03ca7331b88be88053d55aab05a301b427e5d7c.jpg)  
+Figure 8: BST vs skip list construction times  
+(b)Zipfian fit  $(\alpha = 0.75)$
+
+new noisy distribution is normalized in order to form a valid probability distribution. Then, points are queried many times with their ground truth Zipfian probabilities. For the fixed Zipfian distribution with  $a = 5$ ,  $b = 2$ , we plot the effects of different ranges of  $M$  and  $A$  to demonstrate the effect noise has in Fig. 4. This figure demonstrates that, even with moderate amounts of noise, our method still outperforms a traditional KD tree. Moreover, our method still remains on par with the traditional KD tree when significant noise is present, due to our robustness guarantees.
+
+![](images/591ae83aef551ab1d08cfe3e4872f6f529996f5dfed43238e9f9c6a95e4a95d8.jpg)  
+(a) Insert time on AOL
+
+![](images/f1e2c7ffc25a9736cffcd3d4eed7787e4628e194536e06c77f0b3ceb10ca7b18.jpg)  
+(b) Query time on AOL  
+Figure 10: Insertion and query time on AOL of classic and augmented skip lists
+
+![](images/d09f7d0988e54c6425ac3265591ea027154b4f02a244fb2ffc75d8190705c771.jpg)  
+Figure 11: We generate datasets of  $2^{12}$  points in 3-dimensional space, with frequencies given by a Zipfian distribution with parameters  $a, b$ . We then query the tree  $2^{14}$  times, with point queries selected by the same Zipfian distribution. We repeat this process 32 times, and report the median of the average query depth across all runs. We find that our method is able to outperform the traditional KD tree method across all tested Zipfian distributions. Moreover, our performances increase as  $a$  increases and  $b$  decreases, making the Zipfian distribution most skew. Notably, when  $a = 0$ , all points have uniform weights, at which point our method performs equivalently to the traditional KD tree.
+
+![](images/b4e025f8f8479d6d0742f30bf45a3ae7d4f95549d59932590346c8f2d5e65efc.jpg)
+
+# D.2.3 REAL-WORLD DATASETS
+
+In addition to evaluating our method on synthetic data, we also evaluate results on real-world data.
+
+First, we consider  $n$ -grams in various languages. We test on a pre-processed subset of the Google N-Gram dataset (Goog; Google, 2012).
+
+In order to evaluate our method, we convert an  $n$ -gram to a vector in  $\mathbb{Z}^n$  with each entry indexing the words in the  $n$ -gram. We construct the learning-augmented and traditional KD trees, and show the performance of lookup with queries weighted by their ground truth frequency in Table 4. In all cases, we find that the learning-augmented KD tree outperforms the traditional KD tree at average lookup depth.
+
+Additionally, we test our method on a dataset of neuron activity, as provided by Aitchison et al. (2014), which has shown to be Zipfian. This dataset consists of vectors in  $\{0, 1\}^{30}$ , indicating which of 30 cells fire at agiven time. As in their work, we bin in 20ms increments when constructing these
+
+vectors. We similarly ignore the time of observations, and build our learning-augmented KD tree with frequencies given by the rate of appearance of vectors. We find that, when querying with probabilities equivalent to the training distribution, a traditional KD tree has an average query depth of 23.7. Using our learning-augmented KD tree, however, we are able to achieve an average query depth of 14.9.
+
+<table><tr><td>Dataset</td><td>Traditional Avg Query Depth</td><td>Learned Avg Query Depth</td></tr><tr><td>2grams_chinese_simplified</td><td>14.8388</td><td>12.1144</td></tr><tr><td>2grams_english-fiction</td><td>14.216</td><td>11.6232</td></tr><tr><td>2grams_english</td><td>14.7253</td><td>11.6647</td></tr><tr><td>2grams_french</td><td>14.5123</td><td>11.5999</td></tr><tr><td>2grams_german</td><td>13.9473</td><td>11.7066</td></tr><tr><td>2grams_hbrew</td><td>13.5322</td><td>12.0007</td></tr><tr><td>2grams_italian</td><td>14.5231</td><td>11.8027</td></tr><tr><td>2grams_russian</td><td>14.5769</td><td>11.8082</td></tr><tr><td>2grams-spanish</td><td>15.1763</td><td>11.6582</td></tr><tr><td>3grams_chinese_simplified</td><td>12.343</td><td>11.4211</td></tr><tr><td>3grams_english-fiction</td><td>13.5264</td><td>11.3183</td></tr><tr><td>3grams_english</td><td>15.002</td><td>11.3632</td></tr><tr><td>3grams_french</td><td>14.9235</td><td>11.3529</td></tr><tr><td>3grams_german</td><td>14.129</td><td>11.4285</td></tr><tr><td>3grams_hbrew</td><td>13.0887</td><td>9.6663</td></tr><tr><td>3grams_italian</td><td>13.3839</td><td>11.3616</td></tr><tr><td>3grams_russian</td><td>15.2475</td><td>11.1415</td></tr><tr><td>3grams-spanish</td><td>15.3855</td><td>11.3635</td></tr><tr><td>4grams_chinese_simplified</td><td>11.5823</td><td>9.8661</td></tr><tr><td>4grams_english-fiction</td><td>12.5141</td><td>9.7589</td></tr><tr><td>4grams_english</td><td>13.4464</td><td>9.7793</td></tr><tr><td>4grams_french</td><td>12.6383</td><td>9.835</td></tr><tr><td>4grams_german</td><td>12.3911</td><td>9.8765</td></tr><tr><td>4grams_hbrew</td><td>9.3198</td><td>7.4493</td></tr><tr><td>4grams_italian</td><td>11.3456</td><td>9.6834</td></tr><tr><td>4grams_russian</td><td>13.2406</td><td>9.6274</td></tr><tr><td>4grams-spanish</td><td>12.9712</td><td>9.8191</td></tr><tr><td>5grams_chinese_simplified</td><td>11.5373</td><td>9.8982</td></tr><tr><td>5grams_english-fiction</td><td>12.8027</td><td>9.8069</td></tr><tr><td>5grams_english</td><td>12.5607</td><td>9.5967</td></tr><tr><td>5grams_french</td><td>12.402</td><td>9.835</td></tr><tr><td>5grams_german</td><td>11.6912</td><td>9.8432</td></tr><tr><td>5grams_hbrew</td><td>7.097</td><td>6.2143</td></tr><tr><td>5grams_italian</td><td>11.3061</td><td>9.7771</td></tr><tr><td>5grams_russian</td><td>12.885</td><td>9.6805</td></tr><tr><td>5grams-spanish</td><td>12.111</td><td>9.8079</td></tr></table>
+
+Table 4: We construct traditional and learning-augmented KD trees for  $n$  -grams for various languages, and of various lengths  $n$  . Our method outperforms a traditional KD tree in all cases.
\ No newline at end of file
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+# LEARNING-GUIDED ROLLING HORIZON OPTIMIZATION FOR LONG-HORIZON FLEXIBLE JOB-SHOP SCHEDULING
+
+Sirui Li, Wenbin Ouyang, Yining Ma & Cathy Wu  
+MIT  
+{siruil, oywenbin, yiningma, cathywu}@mit.edu
+
+# ABSTRACT
+
+Long-horizon combinatorial optimization problems (COPs), such as the Flexible Job-Shop Scheduling Problem (FJSP), often involve complex, interdependent decisions over extended time frames, posing significant challenges for existing solvers. While Rolling Horizon Optimization (RHO) addresses this by decomposing problems into overlapping shorter-horizon subproblems, such overlap often involves redundant computations. In this paper, we present L-RHO, the first learning-guided RHO framework for COPs. L-RHO employs a neural network to intelligently fix variables that in hindsight did not need to be re-optimized, resulting in smaller and thus easier-to-solve subproblems. For FJSP, this means identifying operations with unchanged machine assignments between consecutive subproblems. Applied to FJSP, L-RHO accelerates RHO by up to  $54\%$  while significantly improving solution quality, outperforming other heuristic and learning-based baselines. We also provide in-depth discussions and verify the desirable adaptability and generalization of L-RHO across numerous FJSP variates, distributions, online scenarios and benchmark instances. Moreover, we provide a theoretical analysis to elucidate the conditions under which learning is beneficial.
+
+# 1 INTRODUCTION
+
+Enhancing the efficiency and scalability of solving Combinatorial Optimization Problems (COPs) has been a central focus of both the Operations Research (OR) (Schrijver et al., 2003) and the emerging Neural Combinatorial Optimization (NCO) communities (Bengio et al., 2021). Numerous methods have been proposed to decompose large-scale problems into smaller, more tractable subproblems, with the majority focusing on spatial or structural decomposition of decision variables. Despite their success, these methods often face limitations in addressing the unique temporal challenges of long-horizon COPs - frequently encountered by industrial practitioners - which involve optimizing complex decisions over extended time horizons. Such unique challenges, coupled with the inherent NP-hardness and large-scale nature of the problems, call for advanced temporal decomposition strategies (Du and Pardalos, 1998; Hentenryck and Bent, 2006; Yang et al., 2013).
+
+Building on successes in control for complex dynamical systems (Welikala and Cassandras, 2021; Franze and Lucia, 2015; Castaman et al., 2021), Rolling Horizon Optimization (RHO) has emerged as a natural temporal decomposition technique for long-horizon COPs (Glomb et al., 2022). RHO breaks the problem into overlapping subproblems with shorter planning horizons rolling forward over time, allowing for much better scalability. The key to RHO is its temporal overlap, which enhances decision-making at the boundaries of consecutive subproblems, mitigating myopic decisions and facilitating modelling interdependencies across the temporal dimension (Sethi and Sorger, 1991; Mattingley et al., 2011). However, such overlaps often lead to redundant computations that reduce the efficiency especially when only a small subset of variables needs re-optimization. This presents an opportunity to accelerate RHO by identifying such redundancies, an approach that, to our knowledge, has not yet been explored in the combinatorial optimization context.
+
+To this end, this work introduces a novel learning-based RHO framework, termed L-RHO, designed to accelerate RHO for long-horizon COPs by identifying overlapping decision variables that do
+
+not need to be re-optimized between consecutive iterations. We demonstrate the effectiveness of our L-RHO on the long-horizon Flexible Job-Shop Scheduling Problem (FJSP), a proof-of-concept example with numerous applications in manufacturing (Liu et al., 2017), healthcare (Addis et al., 2016), transportation (Zhang et al., 2012), and logistics (Xie et al., 2019). Long-horizon FJSPs represent a complex class of COPs involving interdependent assignment and scheduling decisions over extended time horizons, significantly increasing the difficulty compared to the basic JSP variant. The core of L-RHO is to intelligently fix operations assigned to the same machine across successive RHO iterations. By employing a customized network with tailored contextual embeddings, L-RHO learns to imitate a look-ahead oracle, effectively reducing the size of subproblems, significantly improving solve times, and delivering notable improvements in solution quality.
+
+Moreover, to gain deeper insights into the research question - "To what extent can machine learning accelerate RHO in the context of long-horizon COPs such as FJSP?" - we validate and discuss our framework, L-RHO, through both computational experiments and theoretical analysis. Empirically, we show that L-RHO reduces RHO solve times by up to  $54\%$  and improves solution quality by up to  $21\%$  compared to various heuristic and learning-based baselines (with and without decomposition), across a range of FJSP settings and distributions. Moreover, we further discuss the unique potential of our L-RHO in online FJSP settings, where FJSPs operate with limited initial information and require ongoing but more efficient re-optimization as new orders arrive with dynamic environments. Theoretically, we formalize analytical criteria under which L-RHO outperforms the heuristic baselines; the analysis incorporates factors including the data distribution and the False Positive (FP) and False Negative (FN) Rates of the learning method, closely aligning with the empirical results.
+
+Our contributions are fourfold: (1) We introduce L-RHO, the first learning-guided rolling-horizon method for COPs; by learning which overlapping solutions in consecutive RHO iterations do not require re-optimization, it significantly reduces RHO subproblem sizes and solve times. (2) Validated on long-horizon FJSP, L-RHO effectively accelerates RHO by up to  $54\%$  while largely improving the solution quality. (3) Extensive experiments show that L-RHO consistently outperforms various state-of-the-art heuristic and learning-based baselines (w/ or w/o decomposition), and demonstrates robust effectiveness across diverse FJSP settings, distributions, and online scenarios. (4) We further provide a principled probabilistic analysis to identify conditions where L-RHO surpasses heuristic RHO baselines, offering theoretical justifications that are aligned with the empirical observations.
+
+# 2 RELATED WORKS
+
+Decomposition for COPs. Despite advances in traditional solvers (Pisinger and Ropke, 2019; Vidal, 2022; Xiong et al., 2022), learning-guided solvers (Labassi et al., 2022; Scavuzzo et al., 2022; Wang et al., 2023b; Li et al., 2024) and neural solvers (Kool et al., 2018; Ma et al., 2024; Zhang et al., 2024c; Ye et al., 2024a) for COPs, scalability and adaptability to real-world complexities remain challenging. Various decomposition strategies have been explored by OR community, such as variable partitioning (Helsgaun, 2017), adaptive randomized decomposition (ARD) (Pacino and Van Hentenryck, 2011), and sub-problem constraint relaxation (Pisinger and Sigurd, 2007). More recently, machine learning has been exploited to guide the decomposition by selecting subproblems (Li et al., 2021; Zong et al., 2022; Huang et al., 2022) or to auto-regressively solve decomposed subproblems (Wang et al., 2021; Ye et al., 2024b; Luo et al., 2024), leading to notable improvements. However, they primarily emphasize spatial or problem-structural decomposition (e.g., for routing problems), which is not suitable for long-horizon time-structured COPs involving complex, interdependent decision variables and constraints spanning extended time horizons. This highlights the need for effective temporal decomposition. We note such temporal decomposition can be orthogonal to other existing ones, and future work could combine them to improve scalability and flexibility.
+
+RHO for Long-horizon COPs. RHO is a temporal decomposition method originating from Model Predictive Control (MPC) (Garcia et al., 1989). It improves the scalability by dividing the time-structured problem into overlapping subproblems. While such overlap improves boundary decision-making, it can introduce redundant computations. Thus, many control and robotics studies leverage previous decisions to reduce the computations of the current subproblem. This is done, e.g., through hand-crafted methods, such as recording repeat computations (Hespanhol et al., 2019) or tightening primal and dual bounds (Marcucci and Tedrake, 2020), and learning-based models that predict active constraints (Bertsimas and Stellato, 2022) or solutions for discrete variables (Cauligi et al., 2021).
+
+However, these methods are not tailored for long-horizon COPs. Recently, RHO has been initially adapted for long-horizon COPs, with wide applications such as scheduling (Bischi et al., 2019), lot-sizing problems (Glomb et al., 2022), railway platform scheduling (Lu et al., 2022), stochastic supply chain management (Fattahi and Govindan, 2022), and pickup and delivery with time windows (Kim et al., 2023). However, they often overlook the redundancies, and none of them have integrated machine learning to address this issue, leaving a gap in accelerating RHO for COPs.
+
+FJSPs. FJSP is a complex class of COPs that involve interdependent assignment and scheduling decisions over extended time horizons, making it more challenging than the basic JSP, which only addresses scheduling (Dauzère-Pérès et al., 2023). Recently, learning-based solvers have shown power to outperform traditional methods such as Constraint Programming (Perron and Didier) and genetic algorithms (Li and Gao, 2016); for example, see Zhang et al. (2020; 2024b;a) for JSP and Wang et al. (2023c); Song et al. (2022) for FJSP. However, they are typically limited to small-scale instances (fewer than 200 operations) and struggle to scale to real-world, long-horizon scenarios. Although decomposition (e.g., the above ARD) has been explored for FJSP, their efficiency remains constrained. Moreover, these methods are offline, requiring full problem information, limiting their use in online setting. While some works have studied dynamic, online FJSP Luo et al. (2021); Lei et al. (2023), they are still restricted to small-scale experiments. Consequently, large research gaps remain in developing more scalable and flexible decomposition methods for long-horizon (and online) FJSP.
+
+# 3 PRELIMINARIES
+
+FJSP Definition. A FJSP instance consists of a set  $\mathcal{T}$  of jobs, a set  $\mathcal{M}$  of machines, and a set  $\mathcal{O}$  of operations. Each job  $j\in \mathcal{T}$  consists of a set of  $n_j$  operations  $\{O_{j,k}\}_{k = 1}^{n_j}\subseteq \mathcal{O}$  required to be processed in a precedence order  $O_{j,1}\to O_{j,2}\to \ldots \to O_{j,n_j}$ . Each operation  $O_{j,k}$  can be processed by any of the compatible machines  $\mathcal{M}_{j,k}\subseteq \mathcal{M}$ ; the process duration of operation  $O_{j,k}$  by machine  $\bar{m}\in \mathcal{M}_{j,k}$  is denoted by  $p_{j,k}^{\bar{m}}$ . A solution to the FJSP, denoted as  $\Pi = (m,\pi)$ , consists of (i) an assignment  $m:\mathcal{O}\rightarrow \mathcal{M}$  that represents the machine assignment of each operation, with  $m(O_{j,k})\in \mathcal{M}_{j,k}$  for all  $j,k$ , and (ii) a schedule  $\pi :\{O_{j,k}\mid \forall j,k\} \to \mathbb{N}$  that represents the process start time of each operation  $O_{j,k}$  as  $\pi (O_{j,k})$ ; with the process duration as  $p_{j,k}^{m(O_{j,k})}$ , the process end time of the corresponding operation is  $\pi_t(O_{j,k})\coloneqq \pi (O_{j,k}) + p_{j,k}^{m(O_{j,k})}$ . Many FJSP objectives exist in the literature, and we consider testing our method on a variety of objectives including makespan (Xie et al., 2019), total start delays (Pham and Klinkert, 2008; Lee and Yih, 2014), and end delays (Zhou et al., 2009; Andrade-Pineda et al., 2020). Formally, the makespan objective can be expressed as  $\max_{O_{j,k}}\pi_t(O_{j,k})$ . For the delay-based objectives, let each operation  $O_{j,k}$  be further associated with (i) a release time  $s_{j,k}$  with a constraint  $\pi (O_{j,k})\geq s_{j,k}$  such that the operation can only be processed after the release time, and (ii) a target end time  $t_{j,k}$ ; both respect the operations' precedence orders within the same job, that is,  $\forall$  job  $j$  and operations  $k_1 < k_2$ , we assume  $s_{j,k_1}\leq s_{j,k_2}$  and  $t_{j,k_1}\leq t_{j,k_2}$ . The total start delay objective is expressed as  $\sum_{O_{j,k}}\pi (O_{j,k}) - s_{j,k}$ , and the total end delay objective as  $\sum_{O_{j,k}}\max (\pi_t(O_{j,k}) - t_{j,k},0)$ . Appendix A.1 provides a detailed list of notations.
+
+RHO for Long-Horizon FJSP. The temporal structure of FJSP enables the use of RHO to decompose a long-horizon FJSP into a sequence of shorter-horizon FJSP subproblems.
+
+Our RHO utilizes a planning window size  $H$  but executes only a step size  $S$  with  $S \leq H$ . Each subproblem is limited to solving for  $T$  seconds. Given the long-horizon FJSP instance, we first sort the operations  $\mathcal{O}$  into  $\{O^{(1)}, \dots, O^{(|\mathcal{O}|)}\}$  based on the precedence order within each job or the associated release time. These operations are then divided into overlapping subproblems. Specifically, the  $r^{th}$  RHO iteration considers a FJSP subproblem  $P_r$  given by a subset of operations  $\mathcal{O}_{plan,r}$ , consisting of the next  $H$  non-executed operations according to the sorted RHO sequence order; we introduce additional constraints in  $P_r$  to handle boundary conditions – for example, each operation in  $\mathcal{O}_{plan,r}$  should start after all previously executed operations from the same job. After solving  $P_r$  under the time limit  $T$ , we obtain the subproblem solution  $\Pi_r = (m_r, \pi_r)$ . We then execute the first  $S$  operations with the earliest process start time  $\pi_r(O)$  following the solution  $\Pi_r$ , while deferring the remaining  $H - S$  operations for replanning in future iterations. The procedure is then repeated until we solve the full FJSP  $P$ , resulting in  $|O| / S$  subproblems. A detailed algorithm can be found in Appendix A.2. RHO is also well-suited for online settings with limited visibility, solving early operations while deferring the rest until future batches in the next iteration (see Sec. 5.2).
+
+![](images/9abfd801e32301e46c11214171723d2caeb54b70439b56bae1660b126d07e780.jpg)  
+Figure 1: One iteration of our L-RHO pipeline. (a) Construct a subproblem  $P_r$  with  $H$  non-executed operations in  $\mathcal{O}_{plan,r}$ . This includes a set of overlapping operations  $\mathcal{O}_{overlap,r} \subseteq \mathcal{O}_{plan,r}$ , each associated with a solution from the previous iteration given by the assignment  $m_{r-1}$  and the schedule  $\pi_{r-1}$ . (b) Identify operations  $\mathcal{O}_{fix,r} \subseteq \mathcal{O}_{overlap,r}$  that in hindsight, their assignments did not need to be re-optimized; during training, a Look-Ahead Oracle determines  $\mathcal{O}_{fix,r}$  by solving  $P_r$  for  $Q$  times; During inference, our learned neural network selects it. (c) Create a restricted subproblem  $\hat{P}_r$  by fixing the machine assignments for  $\mathcal{O}_{fix,r}$ . (d) Feed  $\hat{P}_r$  to a subsolver, solve for up to  $T$  seconds, and (e) Obtain an updated solution  $\Pi_r = (m_r, \pi_r)$ . (f) Execute a subset of  $S$  operations in  $\mathcal{O}_{exec,r} \subseteq \mathcal{O}_{plan,r}$  based on the solution  $\Pi_r$ . We then repeat steps (a)-(f).
+
+# 4 LEARNING-GUIDED ROLLING HORIZON OPTIMIZATION (L-RHO)
+
+We now introduce our learning framework, L-RHO, that accelerates RHO for long-horizon FJSP. We refer to the original RHO, RHO for training data collection, and RHO for inference as  $RHO_0$ ,  $RHO_{data}$ , and  $RHO_{test}$ . In the standard  $RHO_0$ , each consecutive iterations  $r-1$  and  $r$  share overlapping operations  $\mathcal{O}_{\text{overlap}, r} = \mathcal{O}_{\text{plan}, r} \cap \mathcal{O}_{\text{plan}, r-1}$ . While new operations  $\mathcal{O}_{\text{new}, r} = \mathcal{O}_{\text{plan}, r} \setminus \mathcal{O}_{\text{plan}, r-1}$  in subproblem  $P_r$  may change the solution of the overlap parts  $\mathcal{O}_{\text{overlap}, r}$ , we find two key observations that, for various FJSP distributions evaluated in Sec. 5:
+
+(1) A significant subset of the overlapping operations retain the same machine assignment between the consecutive optimizations. Formally, the shared operations are given by  $\mathcal{O}_{fix,r}^{*} = \{O\in$ $O_{overlap,r}\mid m_r(O) = m_{r - 1}(O)\}$ , where  $\Pi_r = (m_r,\pi_r)$  and  $\Pi_{r - 1} = (m_{r - 1},\pi_{r - 1})$  are the solutions of the unrestricted subproblem  $P_{r}$  and  $P_{r - 1}$  in  $RHO_0$ .  
+(2) Let  $\hat{P}_r^*$  be the restricted subproblem where we leverage the solution of the  $r - 1^{th}$  RHO iteration and fix the machine assignments of the shared operations  $\mathcal{O}_{fix,r}^*$ . The solve time of  $\hat{P}_r^*$  is significantly reduced from that of the unrestricted subproblem  $P_r$ .
+
+Assignment-Based Subproblem Restriction. Based on these observations, we propose restricting the subproblem by fixing machine assignments for a set of operations  $\mathcal{O}_{fix,r} \subseteq \mathcal{O}_{overlap,r}$ , selected through oracle, learning, or heuristic methods. This results in the restricted subproblem  $\hat{P}_r$ , with a formal formulation provided in Appendix A.5.2. Notably, this approach can be easily extended to other subproblem restriction methods, which we leave for future work.
+
+Our L-RHO Pipeline. We design the L-RHO pipeline (which is shown in Fig. 1) as follows: First, we collect training data via a Look-Ahead Oracle by solving the unrestricted subproblem  $P_r$  and identifying  $\mathcal{O}_{fix,r}^*$  as the set of overlapping operations with the same assignments in  $m_r$  and  $m_{r-1}$ . A neural network  $f_\theta$  is then trained with such collected labels. During inference,  $f_\theta$  predicts a subset  $\mathcal{O}_{fix,r}$ , from which we form the assignment-based restricted subproblem  $\hat{P}_r$ . This largely speeds up the inference time optimization by replacing the expensive  $P_r$  with the easier-to-solve  $\hat{P}_r$ .
+
+Training Data Collection (RHOdata). At each iteration  $r$ , we solve the unrestricted subproblem  $P_r$  for  $Q \geq 1$  times under a time limit, resulting in  $Q$  possibly different solutions  $\{\Pi_r^q = (m_r^q, \pi_r^q)\}_{q=1}^Q$ . We then use the highest overlapping machine assignment as the training labels. That is, let  $q^* = \arg \max_{q \in \{1, \dots, Q\}} \sum_{O' \in \mathcal{O}_{\text{overlap},r}} y_{O'}^q$ , with  $y_{O'}^q = \mathbb{1}_{m_r^q(O') = m_{r-1}(O')}$ . We set the training labels as  $y_O^* = y_O^{q^*}$  for each operation  $O \in \mathcal{O}_{\text{overlap},r}$ . Intuitively, the solve time of FJSP subproblem reduces when more assignments are fixed. Empirically, we find the labels  $\{y_O^*\}_{O \in \mathcal{O}_{\text{overlap},r}}$  not only lead to short solve times but also high solution quality. We then obtain the set  $\mathcal{O}_{fix,r}^* = \{O \in \mathcal{O}_{\text{overlap},r} | y_O^* = 1\}$  and construct the assignment-based restricted subproblem  $\hat{P}_r^*$  accordingly. Fig. 1(b) top illustrates the data collection procedure RHOdata. Comparing with RHO0 that solves one unrestricted  $P_r$  at each iteration, RHOdata solves  $P_r$  for  $Q$  times and the restricted  $\hat{P}_r$  one time, where  $Q$  trades-off data collection efficiency and the training label quality.
+
+Input Features. The state  $\mathbf{s}_r$  at each RHO iteration  $r$  consists of both the current FJSP subproblem  $P_r$  and the previous solution  $\Pi_{r-1} = (m_{r-1}, \pi_{r-1})$ . We design a rich set of input features, capturing FJSP state details for overlapping operations  $\mathcal{O}_{\text{overlap}, r}$ , new operations  $\mathcal{O}_{\text{new}, r}$ , and machines  $\mathcal{T}$ . For  $\mathcal{O}_{\text{overlap}, r}$  and  $\mathcal{T}$ , we further design features to capture the previous solution's information, including previous assignment, duration and end time (see Appendix A.3 for details).
+
+Neural Architecture. Our neural network  $f_{\theta}$  predicts the probability of whether each overlapping operation  $O \in \mathcal{O}_{\text{overlap},r}$  should be fixed at each state  $\mathbf{s}_r$ . It consists of three parts: (1) Input Embedding: generate embeddings for each operation  $\mathcal{O}_{\text{overlap},r} \cup \mathcal{O}_{\text{new},r}$  and each machine  $\mathcal{M}$ , with separate MLPs. (2) Machine/Operation Concatenation: concatenate the hidden feature of each overlapping operation  $O \in \mathcal{O}_{\text{overlap},r}$  with both the hidden feature of the previously assigned machine  $m_{r-1}(O) \in \mathcal{M}$ , and a global feature obtained by a mean pooling. The concatenated features then pass through another MLP. (3) Output: each overlap
+
+![](images/092ae853ea1ee4a7d4db929549919314142d15e32e9812319464bbf6be364dc8.jpg)  
+Figure 2: Our neural architecture  $f_{\theta}$  to predict the probability of whether each overlapping operation should fix the assignment.
+
+ping operation  $\mathcal{O}_{\text{overlap},r}$  is passed through a final MLP layer to output the probability prediction. Notably, the purpose of (2) is to augment each operation with both local information from the previously assigned machine, as well as global information from all entities, which include overlapping operations, new operations, and the machines. The illustration of  $f_{\theta}$  is depicted in Fig. 2. The current simple architecture shows strong performance, and our preliminary comparisons with advanced architectures, such as Multi-head Attention, show no significant improvement (see Appendix A.6.3); we leave the exploration of more advanced designs in future work.
+
+Loss Function. We use weighted binary cross-entropy loss where  $w_{pos} > 0$  scales positive labels.
+
+$$
+l \left(f _ {\theta} \left(\mathbf {s} _ {r}; O\right), y _ {O}\right) = - \left[ w _ {p o s} y _ {O} \log f _ {\theta} \left(\mathbf {s} _ {r}; O\right) + \left(1 - y _ {O}\right) \log \left(1 - f _ {\theta} \left(\mathbf {s} _ {r}; O\right)\right) \right] \tag {1}
+$$
+
+The weight  $w_{pos}$  balances False Positive (FPR) and False Negative Rate (FNR). A  $w_{pos} < 1$  emphasizes avoiding predicting False Positives, i.e., cases where the network incorrectly fixes changing operations, which can improve the objective but may increase solve time (see our analysis in Sec. 6).
+
+Inference (RHO test). The inference is illustrated in Fig. 1 (b) bottom, where we use the network  $f_{\theta}$  to predict  $\mathcal{O}_{fix,r}$ : for each operation  $O \in \mathcal{O}_{overlap,r}$ , we predict  $\tilde{y}_O = \mathbb{1}_{f_\theta(\mathbf{s}_r;O) \geq 0.5}$ , which results in the fixed operation set  $\mathcal{O}_{fix,r} = \{O \in \mathcal{O}_{overlap,r} \mid \tilde{y}_O = 1\}$  and the associated assignment-based restricted subproblem  $\hat{P}_r$  solved at RHO iteration  $r$ . By replacing the unrestricted  $P_r$  with the restricted  $\hat{P}_r$ , the computation time can be greatly reduced in comparison with  $RHO_0$ .
+
+# 5 EXPERIMENT
+
+In Sec. 5.1, we evaluate L-RHO in a standard offline setting, comparing it to various baselines for long-horizon FJSP. Next, in Sec. 5.2, we dive into detailed analysis of L-RHO under different FJSP variants including online settings. We provide additional experimental results in Appendix A.6, including a deep dive on the architecture design and evaluation on a real-world dataset. We aim to
+
+answer: (1) To what extent does L-RHO improve upon competitive offline baseline methods and RHO variants? (2) Can L-RHO adapt to different FJSP distributions and objectives? Moreover, a unique benefit of RHO is its adaptability to online settings with limited visibility. As a step toward realistic online settings, we then study (3) Is L-RHO robust under noisy observation of process durations? (4) is L-RHO reliable under machine breakdowns?
+
+FJSP Data Distribution and Solver. We generate synthetic large-scale FJSP instances following common procedure in (Behnke and Geiger, 2012), characterizes by  $(|\mathcal{M}|, |\mathcal{T}|, n)$ , where each job  $j$  has  $|\{\mathcal{O}_{j,k}\}_{k=1}^{n_j}| = n$  operations. The total number of operations  $|\mathcal{O}|$  ranges from 600 to 2000. Notably, the problem horizon in this work (e.g., 600-2000) is considerably larger than those in previous studies (Wang et al., 2023a; Behnke and Geiger, 2012; Dauzère-Pérès and Paulli, 1997; Hurink et al., 1994; Brandimarte, 1993), which typically consider fewer than 500 operations. Following the conventions (Zhang et al., 2024a;b), we employ the competitive constraint programming (CP) solver OR-Tools CP-SAT (Perron and Didier) to solve FJSP, with its detailed CP formulation provided in Appendix A.5.2. We consider makespan objective in Sec. 5.1, and delay-based objectives in Sec. 5.2 (start and end delay) for comprehensive evaluation. More details are in Appendix A.5.1.
+
+Proposed Method (L-RHO) Setup. We provide descriptions of the training set up in Appendix A.4, and the details on the choices of RHO parameters in Appendix A.5.4 and A.5.6.
+
+Baselines. We include a range of traditional and learning-based baselines, with and without decomposition for comprehensive evaluation (see Appendix A.5.3 for more details about baselines):
+
+(1) Traditional solver w/o decomposition - We include the widely used benchmark solver CP-SAT (Google, 2023) and Genetic Algorithm (GA) (Li and Gao, 2016).  
+(2) Learning-based solver w/o decomposition - We compare with the state-of-the-art DRL constructive method for FJSP (Wang et al., 2023a), which we denote as DRL-20K (using 20,000 training instances as in the original paper). We further compare with two recent learning methods: DRL-Echeverria (Echeverria et al., 2023) and DRL-Ho (Ho et al., 2023).  
+(3) Traditional solver w/ decomposition - We include the time decomposition method ARD-LNS (Time-based) and machine decomposition method ARD-LNS (Machine-based) in (Pacino and Van Hentenryck, 2011), which decompose large neighborhood search (LNS) by randomly fixing temporal intervals or machine subsets as subproblems, respectively.  
+(4) Learning-guided solver w/ decomposition - While learning-guided decomposition exist, e.g., for routing problems (Li et al., 2021), they are not adaptable to long-horizon COPs like FJSP. We thus opt to include an Oralcle-LNS (Time-based) baseline to estimate the upper bound of enhancing ARD-LNS. It samples  $K$  subproblems and selects the best one by looking ahead.  
+(5) RHO decomposition - We include Default RHO, where each iteration solves an unrestricted FJSP subproblem  $P_r$  without fixing any variables, and Warm-Start RHO, where previous machine assignments of overlapping operations  $\{m_{r-1}(O) \forall O \in \mathcal{O}_{\text{overlap}, r}\}$  are provided as hints (not fixed) to CP-SAT as a warm start, akin to (Hespanhol et al., 2019) in control systems.
+
+Evaluation Metric. We compare the objective and solve time in Table 1. Objective improvement  $(\mathrm{OI}\%)$  and time improvement  $(\mathrm{TI}\%)$  over Default RHO are reported in Table 2. For a solver with an objective obj and a solve time  $t$ , its OI and TI are calculated as  $(obj_0 - obj) / obj_0 \times 100\%$  and  $(t_0 - t) / t_0 \times 100\%$ , respectively, where negative values indicate degradation.
+
+# 5.1 CANONICAL OFFLINE FJSP UNDER MAKESPAN OBJECTIVE
+
+We consider the canonical offline FJSP with makespan as the objective, with horizons up to 2000 operations (significantly longer than literature), following a similar synthetic data distribution as in Wang et al. (2023a). The number of jobs and machines are set to 20 and 10, respectively. For both L-RHO and DRL, we train separate models for 30, 40, and 60 operations per job, testing each in its respective setting and transferring the 60 operations-per-job model to a large-scale test with 100 operations per job. Results are gathered in Table 1. We highlight the following findings:
+
+Comparison with baseline solvers w/o decomposition. L-RHO outperforms both traditional solvers (CP-SAT, GA) and the learning-based DRL solver in the standard (offline) FJSP setting. Specifically, (1) L-RHO achieves the best objective, outperforming DRL and all heuristic baselines in both in-domain and zero-shot generalization to larger scales; (2) We find the performance of DRL-Echeverria and DRL-Ho poor; (3) Comparing with DRL-20K (Wang et al., 2023a), we see that DRL Greedy is the fastest method but has a worse performance than Default RHO. (4) DRL
+
+Table 1: Offline FJSP under Makespan Objective. Each column corresponds to a different FJSP size given by the format: total number of operations, followed by a tuple of (number of machines  $|\mathcal{M}|$ , jobs  $|\mathcal{T}|$ , and operations per job  $n$ ). L-RHO outperforms all types of baselines.  
+
+<table><tr><td rowspan="2"></td><td colspan="2">600 (10, 20, 30)</td><td colspan="2">800 (10, 20, 40)</td><td colspan="2">1200 (10, 20, 60)</td><td colspan="2">2000 (10, 20, 100), Transfer</td></tr><tr><td>Time (s) ↓</td><td>Makespan ↓</td><td>Time (s) ↓</td><td>Makespan ↓</td><td>Time (s) ↓</td><td>Makespan ↓</td><td>Time (s) ↓</td><td>Makespan ↓</td></tr><tr><td>CP-SAT (10 hours)</td><td>36000</td><td>1583 ± 65</td><td>36000</td><td>2128 ± 75</td><td>36000</td><td>3206 ± 87</td><td>36000</td><td>18821 ± 1986</td></tr><tr><td>CP-SAT</td><td>1800</td><td>2274 ± 147</td><td>1800</td><td>4017 ± 413</td><td>1800</td><td>10925 ± 1013</td><td>1800</td><td>39585 ± 2707</td></tr><tr><td>GA</td><td>1800</td><td>3659 ± 87</td><td>1800</td><td>5150 ± 92</td><td>1800</td><td>8086 ± 111</td><td>1800</td><td>10406 ± 640</td></tr><tr><td>ARD-LNS (Time-based)</td><td>300</td><td>2100 ± 269</td><td>400</td><td>3298 ± 365</td><td>600</td><td>5115 ± 558</td><td>1000</td><td>14258 ± 1522</td></tr><tr><td>ARD-LNS (Machine-based)</td><td>300</td><td>3974 ± 633</td><td>400</td><td>6594 ± 1227</td><td>600</td><td>12764 ± 4206</td><td>1000</td><td>52225 ± 1793</td></tr><tr><td>Oracle-LNS (Time-based)</td><td>300</td><td>1789 ± 104</td><td>400</td><td>2663 ± 120</td><td>600</td><td>4470 ± 151</td><td>1000</td><td>7275 ± 785</td></tr><tr><td>DRL-Echeverria, Greedy</td><td>84 ± 27</td><td>2347 ± 189</td><td>115 ± 32</td><td>3105 ± 161</td><td>105 ± 7</td><td>4570 ± 218</td><td>213 ± 10</td><td>7673 ± 356</td></tr><tr><td>DRL-Echeverria, Sampling</td><td>157 ± 8</td><td>2351 ± 139</td><td>217 ± 11</td><td>3131 ± 165</td><td>427 ± 20</td><td>4657 ± 192</td><td>603 ± 41</td><td>7774 ± 271</td></tr><tr><td>DRL-Ho, Greedy</td><td>4 ± 2</td><td>3030 ± 69</td><td>6 ± 4</td><td>4038 ± 87</td><td>11 ± 6</td><td>6045 ± 99</td><td>22 ± 3</td><td>10044 ± 115</td></tr><tr><td>DRL-Ho, Sampling</td><td>174 ± 1</td><td>2907 ± 46</td><td>168 ± 14</td><td>3907 ± 56</td><td>385 ± 12</td><td>5870 ± 71</td><td>517 ± 10</td><td>9848 ± 92</td></tr><tr><td>DRL-20K, Greedy</td><td>4 ± 0.02</td><td>1628 ± 72</td><td>6 ± 0.04</td><td>2128 ± 80</td><td>10 ± 0.1</td><td>3141 ± 97</td><td>16 ± 0.2</td><td>5184 ± 114</td></tr><tr><td>DRL-20K, Sample 100</td><td>29 ± 0.3</td><td>1551 ± 60</td><td>47 ± 1</td><td>2048 ± 69</td><td>110 ± 3</td><td>3063 ± 81</td><td>302 ± 10</td><td>5082 ± 98</td></tr><tr><td>DRL-20K, Sample 500</td><td>146 ± 5</td><td>1537 ± 61</td><td>261 ± 8</td><td>2031 ± 68</td><td>597 ± 16</td><td>3045 ± 79</td><td>1738 ± 16</td><td>5062 ± 98</td></tr><tr><td>Default RHO (Long)</td><td>599 ± 55</td><td>1529 ± 58</td><td>728 ± 98</td><td>2044 ± 75</td><td>1099 ± 108</td><td>3002 ± 87</td><td>2871 ± 244</td><td>4994 ± 114</td></tr><tr><td>Default RHO</td><td>244 ± 21</td><td>1558 ± 73</td><td>348 ± 26</td><td>2103 ± 78</td><td>545 ± 36</td><td>3136 ± 91</td><td>862 ± 42</td><td>5207 ± 114</td></tr><tr><td>Warm Start RHO</td><td>203 ± 23</td><td>1521 ± 67</td><td>278 ± 22</td><td>2055 ± 75</td><td>420 ± 33</td><td>3081 ± 96</td><td>716 ± 41</td><td>5057 ± 106</td></tr><tr><td>L-RHO (450)</td><td>126 ± 19</td><td>1513 ± 70</td><td>160 ± 23</td><td>2015 ± 86</td><td>259 ± 37</td><td>3011 ± 106</td><td>473 ± 52</td><td>4982 ± 132</td></tr></table>
+
+Sampling improves its performance but significantly increases computational time, with DRL-20K (500 Samples) taking longer and performing worse than L-RHO. Additionally, the solve time of DRL heavily relies on batch-decoding and GPU availability, whereas our L-RHO has a competitive inference time even solely running on CPUs. (5) L-RHO requires significantly fewer training instances than DRL-20K, making it advantageous when obtaining FJSP instances is costly.
+
+Comparison with baseline solvers w/ decomposition. We first observe that, (1) for ARDLNS, time-based decomposition outperforms the machine-based variant but still results in a worse makespan and longer solve time compared to Default RHO. This suggests that effective temporal decomposition is crucial for time-structured, long-horizon COPs. In contrast, (2) L-RHO significantly outperforms both LNS methods in terms of solving time and objective. Furthermore, (3) while Oracle-LNS improves performance by simulating learning-based decomposition, it still underperforms compared to Default RHO. This suggests that RHO is better suited for long-horizon FJSP, partly due to complex temporal dependencies as follows: LNS begins with a complete solution and refines it iteratively. However, the makespan objective only improves when the last operation on the critical path is included. This temporal delay in propagating local improvements contrasts with prior learning-based LNS methods for Mixed Integer Programming (Huang et al., 2023) or Vehicle Routing (Li et al., 2021), where subproblem improvements directly impact the objective. In contrast, RHO builds the solution progressively, maintaining a tighter makespan by following the natural temporal order of operations. Lastly, we note that L-RHO's temporal decomposition is orthogonal to other strategies and could be combined in future work to enhance scalability and flexibility.
+
+Discussion - RHO for online settings. Unlike offline baselines (CP-SAT, GA, ARD, DRL) that require complete information, RHO constructs solutions progressively using only near-future information, making it suitable for online settings. Next, we explore FJSP variants, including observation noise and machine breakdowns, to evaluate the applicability of L-RHO to batch-online FJSP.
+
+# 5.2 DETAILED COMPARISON WITH RHO BASELINESE UNDER DIFFERENT FJSP VARIANTS
+
+We now benchmark L-RHO against diverse RHO baselines across FJSP variants. We evaluate objectives such as makespan, start delay, and end delay, while also testing its adaptability under higher system congestion, observation noise, and machine breakdowns to simulate real-world dynamics.
+
+We include two more RHO baselines, following the  $RHO_{test}$  procedure but obtaining the fixed operations  $\mathcal{O}_{fix,r}$  with the following heuristics: (a) Random  $\sigma_{\mathbf{R}} \in [0,1]$ :  $\mathcal{O}_{fix,r}$  is obtained by selecting each operation in  $\mathcal{O}_{overlap,r}$  with a probability  $\sigma_{R}$  uniformly at random, (b) First  $\sigma_{\mathbf{F}} \in [0,1]$ :  $\mathcal{O}_{fix,r}$  is obtained as the first  $\sigma_{F}$  fraction of operations in  $\mathcal{O}_{overlap,r}$  with the earliest RHO sequence order based on the precedence order (for makespan) or release time (for delay-based objectives).
+
+New Objective: Start Delay. In Table 2, under the new start delay objective, our L-RHO consistently outperforms all RHO variants in both solve time and objective across time horizons. Notably,
+
+Table 2: Detailed Comparison of L-RHO with RHO variants under Start Time Delay Objective. We present the time and objective improvements (TI%, OI%) of RHO variants relative to Default RHO. The best TI% and OI% are bolded. L-RHO significantly outperforms all baseline RHO variants.  
+
+<table><tr><td rowspan="2"></td><td colspan="2">600 (25, 25, 24)</td><td colspan="2">1050 (35, 35, 30)</td><td colspan="2">1600 (40, 40, 40)</td></tr><tr><td>TI %↑</td><td>OI %↑</td><td>TI %↑</td><td>OI %↑</td><td>TI %↑</td><td>OI %↑</td></tr><tr><td rowspan="2">Default RHO</td><td>946.9s ± 31.6s</td><td>1790.5 ± 145.3</td><td>1119.86s ± 35.5s</td><td>2960.4 ± 215.7</td><td>2349.3s ± 111.7s</td><td>4148.6 ± 356.8</td></tr><tr><td>(0.0% ± 0.0%)</td><td>(0.0% ± 0.0%)</td><td>(0.0% ± 0.0%)</td><td>(0.0% ± 0.0%)</td><td>(0.0% ± 0.0%)</td><td>(0.0% ± 0.0%)</td></tr><tr><td>Warm Start RHO</td><td>38.6% ± 1.0%</td><td>-7.5% ± 2.8%</td><td>-12.9% ± 1.6%</td><td>11.2% ± 2.7%</td><td>5.2% ± 2.4%</td><td>6.8% ± 4.8%</td></tr><tr><td>First 20%</td><td>26.1% ± 1.3%</td><td>2.9% ± 2.5%</td><td>10.2% ± 1.7%</td><td>-30.5% ± 10.7%</td><td>31.0% ± 2.4%</td><td>-94.1% ± 24.3%</td></tr><tr><td>First 30%</td><td>33.6% ± 1.3%</td><td>-7.0% ± 2.5%</td><td>13.2% ± 1.7%</td><td>-48.9% ± 7.2%</td><td>31.7% ± 2.1%</td><td>-129.4% ± 19.1%</td></tr><tr><td>First 40%</td><td>34.8% ± 1.3%</td><td>-32.3% ± 3.7%</td><td>14.1% ± 2.1%</td><td>-100.4% ± 8.7%</td><td>34.7% ± 3.0%</td><td>-274.7% ± 28.1%</td></tr><tr><td>Random 10%</td><td>23.0% ± 1.3%</td><td>-11.9% ± 2.5%</td><td>-15.5% ± 1.7%</td><td>-12.9% ± 3.4%</td><td>10.0% ± 3.0%</td><td>-55.2% ± 9.6%</td></tr><tr><td>Random 20%</td><td>23.8% ± 1.7%</td><td>-27.6% ± 3.4%</td><td>-2.0% ± 2.6%</td><td>-86.1% ± 7.9%</td><td>10.2% ± 3.5%</td><td>-115.6% ± 14.5%</td></tr><tr><td>L-RHO (Ours)</td><td>53.0% ± 1.0%</td><td>16.0% ± 1.9%</td><td>35.1% ± 1.3%</td><td>21.0% ± 2.4%</td><td>47.3% ± 2.2%</td><td>13.1% ± 4.4%</td></tr></table>
+
+![](images/cb6ae72e697e023c666083bb2b833d7e73f3b7972ba91e16b4d3d7519e0c4936.jpg)  
+Figure 3: L-RHO under different FJSP variants. Left: Increased Congestion level with more jobs  $|\mathcal{T}| \in \{30,32,34\}$ . We circle the baselines for these three settings with green, yellow and red ellipsoids, using different markers to represent each setting. L-RHO is plotted in pink. Middle: Multi-objective and Noisy Observation (online). We analyze FJSP (30, 30, 24) with the (i) start delay objective (ii) start and end delay objective, and (iii) start and end delay objective plus observation noise. The arrows illustrate the performance changes of each method across (i)-(ii)-(iii). Right: Machine Breakdown (online). We simulate machine breakdowns during RHO's process, varying intensity (low, mid, high) by adjusting event frequency and machine availability.
+
+it achieves a  $35\% - 53\%$  speed up from Default RHO with a further  $13\% - 21\%$  improvement in the objective (total start delay), demonstrating the effectiveness of our learning method in accelerating RHO. Comparing with the baseline methods, we observe that: (1) Warm Start RHO does not significantly outperform Default RHO, indicating the inadequacy of simple warm start techniques for the complex long-horizon FJSP. (2) The assignment-based subproblem restriction heuristics, namely First and Random, improve the solve time from Default RHO, albeit at the expense of poorer objective. Moreover, First  $\sigma_{F}$  emerges as a more effective heuristic than Random  $\sigma_{R}$ , with a better solve time and objective improvement when  $\mathcal{O}_{fix,r}$  is of a comparable size ( $\sigma_{F} = \sigma_{R}$ ).
+
+Data Distribution: Higher Congestion. We raise the congestion level by increasing the number of jobs  $|\mathcal{T}| \in \{30, 32, 34\}$  while keeping the number of machines  $|\mathcal{M}| = 30$  and operations per machine to be 25 under the start delay objective. This results in a higher job-to-machine ratio, with congestion build-up over time. In Fig. 3 (left), we observe that L-RHO achieves solve time and objective improvements  $\mathrm{TI}\% \in \{41\%, 39\%, 28\% \}$  and  $\mathrm{OI}\% \in \{14\%, 20\%, 27\% \}$  across the different settings, demonstrating the robustness of the learning method to system congestion. In contrast, the baselines (First and Random) have limited solve time improvement and substantial objective degradation for all settings. This highlights L-RHO's ability to enhance system throughput by enabling the system to process more jobs than Default and baseline RHO methods.
+
+Multi-Objective: Start and End Delay. While the total start delay objective emphasizes adhering to the release time, reducing the total end delay is crucial in many real world scenarios where following each operation's scheduled due time is important. Here, we use the sum of the start and end delay as the objective and generate random target end time for each operation (see Appendix A.5.1). In Fig. 3, middle (ii), we observe that L-RHO maintains solve time and objective improvements
+
+with a  $\mathrm{TI\%}$  of  $39\%$  and  $\mathrm{OI\%}$  of  $4\%$  under this combined objective. Interestingly, Random and First also improve relative to Default RHO (better  $\mathrm{TI\%}$  and  $\mathrm{OI\%}$ , see Random  $30\%$ , for example). Empirically, we find more overlapping operations maintain the same assignment under this distribution, our analysis in Sec. 6 indicates the advantageous for Random and First under such a scenario.
+
+Online FJSP: Observation Noise. In real-world FJSP deployments, practitioners often face incomplete and noisy operation data. To simulate this, we introduce observation noise into the RHO process, moving towards a more realistic online FJSP. Specifically, we assume that in each RHO subproblem, only the durations of operations with earlier release times are accurately observed. For other operations, we only have access to noisy estimates, and RHO optimizes subproblems based on this noisy information. Thus, the observed durations in  $\hat{P}_r$  may differ from the true execution duration during execution (see Appendix A.5.1 for details). In Fig. 3, middle (iii), the Random and First strategies degrade significantly under observation noise, with a more pronounced effect on First. Our analysis in Sec. 6 captures such a behavior for First and Random, as the observation noise leads to (1) more overlapping operations changing assignments (2) a less informative distribution to determine which operation in the subproblem will change assignments. In contrast, L-RHO remains effective under observation noise, with a  $\mathrm{TI}\%$  of  $25\%$  and  $\mathrm{OI}\%$  of  $4\%$  over Default RHO.
+
+Online FJSP: Machine Breakdown. In real-world FJSP scenarios, unexpected disruptions such as machine breakdowns can occur (Zhang and Wong, 2017). We simulate breakdowns, and combine the periodic RHO optimization with event-based rescheduling: when a breakdown or recovery occurs, we re-optimize the RHO subproblem with updated machine availability. This leads to two major changes in the RHO pipeline (1) RHO subproblems are solved more frequently due to the breakdown (or recovery) events, and (2) operations with all compatible machines down are deferred until the breakdown ends, thereby resulting in longer scheduling horizon (see Appendix A.5.1 for details). In Fig. 3 (Right), we test L-RHO under varying machine breakdown intensities by adjusting breakdown frequency and the proportion of downed machines, using makespan as the objective with 10 machines, 20 jobs, and 800 operations. Our L-RHO demonstrates significant time improvements  $\mathrm{TI\%} \in \{47\%, 52\%, 37\% \}$  over Default RHO across all intensity levels, while also achieving a better makespan. This highlights L-RHO's reliability under unexpected disruptions. We also observe that, at high breakdown intensity, First loses its advantageous to Random. This can be similarly explained in our analysis next, as unexpected disruptions reduce the amount of information First can leverage.
+
+# 6 THEORETICAL PROBABILISTIC ANALYSIS
+
+In this section, we present a theoretical probabilistic analysis to identify when RHO can benefit from machine learning. Intuitively speaking, the more irregular the operations to be fixed  $(\mathcal{O}_{fix}^{*})$ , the more advantageous L-RHO can be, because of the greater potential gain from prediction. Also, L-RHO must balance both False Positive (FP) and False Negative (FN) errors: fixing something that should not have been (FP) harms the objective but helps the solve time, while failing to fix something that should have been (FN) harms the solve time and also indirectly harms the objective (under a fixed time limit). Thus, an ideal L-RHO method will balance the two errors (see Fig. 4).
+
+Notation. We analyze a generic subproblem  $\hat{P}_r\coloneqq \hat{P}$  at each iteration  $r$  , and drop the  $r$  subscript for ease of notation. We consider a fixed set of RHO parameters and denote  $W = H - S = |\mathcal{O}_{\text{overlap}}|$  Given  $\mathcal{O}_{fix}^{*}$  and  $\mathcal{O}_{fix}$  from the Oracle and a method in  $\{\mathrm{Random~}\sigma_R,\mathrm{First~}\sigma_F,\mathrm{L - RHO}\}$  , we denote  $\mathbb{E}[n_{fix}^{*}] = \mathbb{E}[|\mathcal{O}_{fix}^{*}|],\mathcal{O}_{fn} = \mathcal{O}_{fix}^{*}\backslash \mathcal{O}_{fix},$  and  $\mathcal{O}_{fp} = \mathcal{O}_{fix}\backslash \mathcal{O}_{fix,r}^{*}$  . We define the FN and FP errors  $\mathbb{E}[n_{fn}] = \mathbb{E}[|\mathcal{O}_{fn}|]$  and  $\mathbb{E}[n_{fp}] = \mathbb{E}[|\mathcal{O}_{fp}|]$  . The (expected) FP and FN rates of each method are  $\mathbb{E}[n_{fp}] / \mathbb{E}[n_{fix}^{*}],\mathbb{E}[n_{fn}] / (W - \mathbb{E}[n_{fix}^{*}])$  , with L-RHO's FPR and FNR denoted as  $(\alpha ,\beta)$
+
+We leverage an intuitive linear decay assumption, validated in Appendix A.7.2: in a given FJSP subproblem, the later an operation appears in the RHO sequence, the less likely its assignment should be fixed. This assumption allows us to formally relate the FP and FN errors of different methods (L-RHO versus Random versus First, see Prop. 1 and Fig. 4 (middle)).
+
+Assumption 1 (Linear Decay). Let the probability that each operation  $O^{(i)} \in \mathcal{O}_{\text{overlap}}$  is included in  $\mathcal{O}_{fix}^*$  be denoted as  $p_{fix}^*(i) \in [0,1]$ , where the index  $i \in \{1,\dots,W\}$  follows the RHO sequence order. We consider linearly decreasing  $p_{fix}^*(i) = b - m \cdot \frac{i}{W}$ , where constants  $0 \leq b, m \leq 1$  reflect average behavior across RHO iterations. We have  $\mathbb{E}[n_{fix}^*] = \sum_i p_{fix}^*(i) = (b - \frac{m}{2})W - \frac{m}{2}$ .
+
+![](images/81bdfcce0a434321face7528041847359af69773daa823f5cfcc2ebb75c66e0a.jpg)  
+Figure 4: Left: We analyze FN and FP errors of each RHO method relative to Oracle and interpret their impact on objective and solve time. Middle: We show the FN and FP errors for Random and First with  $\sigma_R, \sigma_F \in [0,1]$  under Assump. 1, fixing  $b - \frac{m}{2} = 0.5$  while increasing the slope  $m$ . Higher  $m$  enhances the performance of First relative to Random (with equal  $\sigma$ ) by reducing both errors. We plot  $\sigma_R, \sigma_F \in \{0,0.1,\dots,1\}$  using circles and squares, with darker colors for smaller values. Right: For FJSP (25, 25, 24) with the total start delay objective, we depict the FN and FP errors of {Random, First, and L-RHO} in the low FP region, using an empirically validated  $p_{fix}^{*}(i)$  distribution. By reducing both  $\mathbb{E}[n_{fp}]$  and  $\mathbb{E}[n_{fn}]$ , L-RHO empirically outperforms First  $\sigma_F \leq 60\%$  in solve time and all baselines in objective (pink region). Transforming the coordinates (right and top axes) can further reveal the FP and FN rates L-RHO should achieve for learning to be effective.
+
+Proposition 1. Under Assump. 1, the FN and FP errors for each method is given in closed-form as
+
+- Random:  $\mathbb{E}[n_{fn}^{Random}] = (1 - \sigma_R)\mathbb{E}[n_{fix}^*],\mathbb{E}[n_{fp}^{Random}] = \sigma_R(W - \mathbb{E}[n_{fix}^*])$  
+- First:  $\mathbb{E}[n_{fn}^{First}] = (1 - \sigma_F)(\mathbb{E}[n_{fix}^*] - \frac{m}{2}\sigma_FW),\mathbb{E}[n_{fp}^{First}] = \sigma_F\big(W - \mathbb{E}[n_{fix}^*] - \frac{m}{2} (1 - \sigma_F)W\big);$  
+-  $L$ -RHO:  $\mathbb{E}[n_{fn}^{L-RHO}] = \beta \mathbb{E}[n_{fix}^*], \mathbb{E}[n_{fp}^{L-RHO}] = \alpha (W - \mathbb{E}[n_{fix}^*])$ .
+
+Furthermore, ignoring the  $\frac{m}{2}$  term in  $\mathbb{E}[n_{fix}^*]$  for ease of exposition, the FP and FN Rates of First  $\sigma_F$  and Random  $\sigma_R$  are  $(\alpha_F, \beta_F) = (\frac{\sigma_F(1 - b + \frac{m}{2}\sigma_F)}{1 - b + \frac{m}{2}}, 1 - \frac{\sigma_F(b - \frac{m}{2}\sigma_F)}{b - \frac{m}{2}})$  and  $(\alpha_R, \beta_R) = (\sigma_R, 1 - \sigma_R)$ .
+
+Interpretation. The  $p_{fix}^{*}(i)$  distribution determines the strength of the Random and First baselines and thus the potential for improvement by learning methods. (1) When  $\mathbb{E}[n_{fix}^{*}]$  is close to 0 or 1, Random achieves low FP and FN errors by setting  $\sigma_R$  close to  $\mathbb{E}[n_{fix}^{*}]$ . (2) Our analysis captures the empirical observation that First often outperforms Random: with a strong linear decay (a large slope  $m$ ), First further reduces the FP and FN errors from Random for each  $\sigma_F = \sigma_R$ . (3) L-RHO is advantageous when its FP and FN rates  $(\alpha, \beta)$  is lower than those for First and Random, as given in the Prop. This can provide practitioners a valuable way to assess the effectiveness of the learned model prior to deployment. We refer to Appendix A.7 for detailed proofs and interpretation.
+
+# 7 CONCLUSION
+
+We contribute the first learning-guided rolling-horizon method for COPs, which we call L-RHO. By learning which overlapping solutions in consecutive RHO iterations do not need to be re-optimized, our method substantially reduces RHO subproblem sizes and thus accelerates their solve time. L-RHO scales and accelerates RHO by up to  $54\%$ , improves solution quality, and outperforms a wide range of baselines, both with and without decomposition. We also analyze L-RHO's performance across various FJSP settings, distributions, and online scenarios, highlighting its flexibility and adaptability. We further provide a probability analysis to identify conditions where learning-guided RHO is most beneficial. One limitation is that L-RHO alone may not achieve state-of-the-art performance when extended to other long-horizon COPs. Future work could extend L-RHO to -1) other long-horizon COPs, e.g., multi-item inventory management and vehicle routing with time windows, 2) training on diverse distributions for better generalization, 3) integration with other decomposition or subproblem restriction methods, and 4) addressing more complex real-world online scenarios. Our theoretical analysis may also guide the development of more effective RHO warm-start techniques with applications to control and robotics, e.g., accelerating Hybrid MPCs. We believe L-RHO offers a valuable tool for long-horizon COPs and will inspire future work in various domains. Our code is publicly available at https://github.com/mit-wu-lab/l-rho.
+
+# 8 ACKNOWLEDGEMENT
+
+The authors would like to thank Sean Lo for insightful discussions. This work was supported by the National Science Foundation (NSF) CAREER award (#2239566), MIT's Research Support Committee, an Amazon Robotics Ph.D. Fellowship, and a gift from Mathworks. The authors acknowledge the MIT SuperCloud and Lincoln Laboratory Supercomputing Center for providing HPC resources that have contributed to the research results reported within this paper.
+
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+
+# A SUPPLEMENTARY MATERIAL: Learning-Guided Rolling Horizon Optimization for Long-Horizon Flexible Job-Shop Scheduling
+
+# CONTENTS
+
+A.1 Notations 16  
+A.2 Algorithm: Learning-Guided Rolling Horizon Optimization 18  
+A.3 Input Features. 19  
+A.4 Architecture, train and evaluation setup. 23  
+A.5 Experimental Details 24
+
+A.5.1 FJSP Instance Distribution, Machine Breakdown and Observation Noise. 24  
+A.5.2 FJSP Formulation 27  
+A.5.3 Baseline Details 30  
+A.5.4 Rolling Horizon Parameter Search 32  
+A.5.5 Performance of RHO methods for FJSP (30, 30, 24). 33  
+A.5.6 Performance of RHO methods under the same RHO parameter. 33
+
+A.6 Additional Experimental Results 35
+
+A.6.1 Performance on Real World Dataset (Dauzère-Pérès and Paulli (1997) 35  
+A.6.2 Accuracy, TPR, TNR, Precision, and Recall of the Learning Method. 35  
+A.6.3 Ablation Study: A Detailed Comparison with Attention-Based Architecture. 35  
+A.6.4 Performance Comparison with DRL under the same number of FJSP training instance. 38  
+A.6.5 Performance of Default RHO with Longer Time Limits 38
+
+A.7 Detailed Theoretical Probabilistic Analysis 39
+
+A.7.1 Details of all considered RHO methods 39  
+A.7.2 Empirical validation of the linear decay assumption for  $p_{fix}^{*}(i)$  40  
+A.7.3 Derivations of  $\mathbb{E}[n_{fp}]$  and  $\mathbb{E}[n_{fn}]$  for each method 40  
+A.7.4 Qualitative Relationship between  $(\mathbb{E}[n_{fp}]$  and  $\mathbb{E}[n_{fn}])$  and (objective, solve time) 41  
+A.7.5 Detailed Derivation of Table 20. 44  
+A.7.6 Interpretation of Prop. 1 45
+
+# A.1 NOTATIONS
+
+We provide detailed lists of notations for the Flexible Job-shop Scheduling Problem (FJSP) and Rolling Horizon Optimization (RHO) in Table 3 and Table 4.
+
+Table 3: Notations for the Flexible Job-shop Scheduling Problem (FJSP).  
+
+<table><tr><td colspan="3">FJSP Related</td></tr><tr><td rowspan="8">Problem P</td><td>M</td><td>A set of machines. The machines are indexed by m ∈ M.</td></tr><tr><td>T</td><td>A set of job. The jobs are indexed by j ∈ T.</td></tr><tr><td>O</td><td>A set of operations. The operations are indexed by Oj,k ∈ O. Each job j ∈ T consists of a set of nj operations {Oj,k}nk=1nj required to be processed in a predefined precedence order Oj,1 → Oj,2 → ... → Oj,nj. After sorting the operations by their respective release time (see below), each operation O(i) ∈ O = {O(1), ..., O(|O|)} is indexed by i.</td></tr><tr><td>Compatible Machines Mj,k</td><td>Each operation Oj,k can be processed on any machine among a subset Mj,k ⊆ M of compatible machines.</td></tr><tr><td>Process Duration pmj,k</td><td>pmj,k denotes the process duration of operation Oj,k on machine m ∈ Mj,k.</td></tr><tr><td>Release Time sj,k (Target Start Time)</td><td>Under delay-based objectives, each operation Oj,k has a release time sj,k, which respects the precedence orders of operations within the same job: for operations Oj,k1 and Oj,k2 with ∀k1 &lt; k2, we assume sj,k1 ≤ sj,k2. Each operation can only be processed after its release time.</td></tr><tr><td>Target End Time tj,k</td><td>Under the total end delay objective, each operation Oj,k is associated with a target end time tj,k, which represents a target time to finish the operation and similarly respects the operation precedence orders.</td></tr><tr><td>Operations&#x27; Ordering {O(1), ..., O(|O|)}</td><td>Ordering of the operations O = {Oj,k}j,k used to divide the full operation set O into overlapping RHO subproblems. • For the makespan objective, the operations are ordered by their relative position in the jobs based on the precedence order: For job j with nj operations in precedence order Oj,1 → ... → Oj,nj, we give operation Oj,k a score of k/nj, and choose each RHO subproblem as the first H non-executed operations with the lowest score. • For delay-based objectives, the operations are ordered by their release time s(1) ≤ ... ≤ s(|O|).</td></tr><tr><td rowspan="3">Solution Π = (m,π)</td><td>Assignment m</td><td>m : O → M represents the machine assignment of each operation, with m(Oj,k) ∈ Mj,k for all j,k.</td></tr><tr><td>Schedule π (Process Start Time)</td><td>π : {Oj,k | ∀j,k} → N, where the process start time of each operation Oj,k satisfies π(Oj,k) ≥ sj,k.</td></tr><tr><td>Process End Time πt</td><td>The process end time of operation Oj,k is πt(Oj,k) := π(Oj,k) + pm(Oj,k), where pm(Oj,k) is the process duration.</td></tr><tr><td rowspan="3">Objective</td><td>Makespan</td><td>max πt(Oj,k), see Alg. 7.</td></tr><tr><td>Total Start Delay</td><td>∑ Oj,k∈O π(Oj,k) - sj,k, see Alg. 7.</td></tr><tr><td>Total Start and End Delay</td><td>∑ Oj,k∈O π(Oj,k) - sj,k + max(πt(Oj,k) - tj,k, 0), see Alg. 7.</td></tr></table>
+
+Table 4: Notations for the Rolling Horizon Optimization (RHO).  
+
+<table><tr><td></td><td></td><td>RHO Related: RHO Iteration r</td></tr><tr><td rowspan="4">Parameters</td><td>H</td><td>A planning window size (H operations in the current planning window).</td></tr><tr><td>S</td><td>An execution step size (S ≤ H operations executed at each iteration).</td></tr><tr><td>T</td><td>A solve time limit of T seconds per RHO subproblem.</td></tr><tr><td>Ts</td><td>We early terminate the solver if the value of the best feasible solution does not improve within Ts seconds. (see Appendix. A.5.4)</td></tr><tr><td rowspan="2">FJSP Optim. Subproblem</td><td>Unrestricted Pr</td><td>FJSP subproblem Pr given by a subset of operations Oplan,r, with additional constraints to ensure Oplan,r are processed after the previously executed operations within the same job or the same machine (see Appendix A.5.2 Alg. 8).</td></tr><tr><td>Restricted P_r</td><td>Given a subset Ofix,r, we can construct a simpler, restricted FJSP subproblem P_r from Pr by setting the compatible machines Mj,k = {m_{r-1}(Oj,k)} for Oj,k ∈ Ofix,r.</td></tr><tr><td rowspan="6">Relevant Operations Subset</td><td>Oplan,r</td><td>Operations included in the current RHO subproblem Pr or P_r: The first H non-executed operations with earliest RHO sequence order. See Appendix Sec. A.2 Alg. 2.</td></tr><tr><td>Ostep,r</td><td>Executed operations based on the solution of Pr or P_r: The first S operations with the earliest process start time Earliest-S_Operations{ {πr(O)|O ∈ Oplan,r}}. See Alg. 3.</td></tr><tr><td>Ooverlap,r</td><td>Overlapping operations in consecutive RHO iterations (r-1 and r): Ooverlap,r = Oplan,r ∩ Oplan,r-1</td></tr><tr><td>Onew,r</td><td>New operations in iteration r but not in r-1: Onew,r = Oplan,r \ Oplan,r-1</td></tr><tr><td>Ofix,r</td><td>The (look-ahead Oracle) subset of operations in Ooverlap,r with the same machine assignment in the consecutive unrestricted RHO subproblems Pr and Pr-1: Ofix,r = {O ∈ Ooverlap,r | mr(O) = mr-1(O)} That is, given the solution to the previous subproblem Pr-1, we lookahead by solving the current subproblem Pr and identify operations with the same machine assignments.</td></tr><tr><td>Ofix,r</td><td>The fixed operations subset given by a specific method. Given Ofix,r ⊆ Ooverlap,r, we can construct P_r from Pr by setting the compatible machines Mj,k = {m_{r-1}(Oj,k)} for operations Oj,k ∈ Ofix,r.</td></tr><tr><td rowspan="2">Subproblem Solution Πr = (mr, πr)</td><td>Subproblem m_r Assignment Schedule π_r</td><td>m_r: Oplan,r → M represents the machine assignment of each operation in the subproblem, with m(Oj,k) ∈ Mj,k for all Oj,k ∈ Oplan,r.</td></tr><tr><td>Subproblem π_r</td><td>π_r: Oplan,r → N, where the (solution) start time of each operation in the subproblem Oj,k ∈ Oplan,r is π(Oj,k) ≥ sj,k.</td></tr><tr><td rowspan="6">Methods</td><td>Default (Cold Start)</td><td>Each RHO iteration solves an unrestricted FJSP subproblems Pr from scratch (cold start, Ofix,r ∈ ∅).</td></tr><tr><td>Random σ_R</td><td>Each RHO iteration solves a restricted FJSP subproblem P_r associated with the fixed operation set Ofix,r, constructed by sampling each operation in Ooverlap,r with a probability σ_R uniformly at random.</td></tr><tr><td>First σ_F</td><td>Each RHO iteration solves a restricted FJSP subproblem P_r associated with the fixed operation set Ofix,r obtained as the first σ_F fraction of operations in Ooverlap,r with the earliest RHO sequence order (determined by operation precedence for makespan or release time for delay-based objectives).</td></tr><tr><td>Warm Start</td><td>At each RHO iteration, we apply CP-SAT&#x27;s internal warm-start techniques to solve the unrestricted FJSP subproblems Pr. Specifically, we provide the machine assignments of the overlapping operations in the previous iteration {mr-1(O) ∀ O ∈ Ooverlap,r} as hints to the solver. See: “https://developers.google.com/optimization/reference/ python/sat/python/cp_model#addhint”.</td></tr><tr><td>Oracle</td><td>Follow our data collection pipeline (Fig. 1 (b) top, omit the solve time for the unrestricted Pr), where each RHO iteration solves a restricted FJSP subproblem P_r associated with the oracle fixed operation set Ofix,r.</td></tr><tr><td>L-RHO (Ours)</td><td>Follow the inference procedure RHO test in Fig. 1 (b) bottom, where each RHO iteration solves a restricted FJSP subproblem P_r associated with the fixed operation set Ofix,r predicted by the learning model fθ.</td></tr></table>
+
+# A.2 ALGORITHM: LEARNING-GUIDED ROLLING HORIZON OPTIMIZATION
+
+The inference procedure for the learning-guided rolling horizon algorithm (Fig. 1) is provided as follows, where at each RHO iteration, we use the neural network  $f_{\theta}$  to identify a set of fixed operation  $\mathcal{O}_{fix,r}$  (Line 10), resulting in the machine assignment-based restricted subproblem  $\hat{P}_r$  (Line 15). We note that the heuristic RHO baselines considered in Sec. 5 and 6 (e.g. First  $\sigma_F$  and Random  $\sigma_R$ ), we follow the same algorithm but use the associated heuristics to find the set  $\mathcal{O}_{fix,r}$  instead (Line 10).
+
+Algorithm 1: Learning-Guided Rolling Horizon Optimization for FJSP  
+Input: FJSP problem instance  $P = (\mathcal{M},\mathcal{T},\mathcal{O},\{p_{j,k}^{m}\} ,\{s_{j,k}\} ,\{t_{j,k}\})$  , RHO planning window size  $H$  operations, execution step size  $S$  operations, and solve time limit  $T$  per subproblem;. Output: Solution  $\Pi = (m,\pi)$  of the FJSP problem instance  $P$    
+1 Initialize  $m\gets \{\}$ $\pi \leftarrow \{\}$  .   
+for Iteration  $r = 1:\mathcal{O}|$  /S do   
+//Obtain first  $H$  non-executed operations   
+ $\mathcal{O}_{plan,r}\gets$  GetPlanOperations(O,H,r,m,pi);   
+// Generate FJSP subproblem (see Appendix A.5.2)   
+ $P_{r}\gets$  FJSP_Subproblem(P,Oplan,r,m,pi);   
+if Iteration  $r\geq 2$  then   
+// Learn to identify fixed operations in  $\mathcal{O}_{\text{overlap},r}$ $\mathcal{O}_{fix,r}\gets f_{\theta}(P_r,\mathcal{O}_{\text{overlap},r},P_{r - 1},m_{r - 1},\pi_{r - 1})$  .   
+else   
+ $\mathcal{O}_{fix,r}\gets \emptyset$    
+end if   
+//Obtain the assignment-based restricted FJSP subproblem   
+ $\hat{P}_r\gets$  Restricted_FJSP_Subproblem  $(P_r,\mathcal{O}_{fix,r})$  .   
+// Solve the restricted FJSP subproblem   
+ $(m_r,\pi_r)\gets \mathrm{FJSP\_Solver}(\hat{P}_r;T)$    
+// Execute first  $S$  operations in  $\mathcal{O}_{\text{plan},r}$  with earliest start times   
+ $\mathcal{O}_{step,r}\gets$  GetStepOperations(Oplan,r,S,r,mr,  $\pi_r$  .   
+// Fix final solution for executed operations   
+for  $O^{(i)}\in \mathcal{O}_{step,r}$  do   
+ $m(O^{(i)})\gets m_r(O^{(i)})$  .   
+ $\pi (O^{(i)})\gets \pi_r(O^{(i)})$  .   
+end for
+
+Algorithm 2: GetPlanOperations  
+Input: Operations  $\mathcal{O} = \{O^{(1)},\dots,O^{(|\mathcal{O}|)}\}$  ordered based on the precedence or release time (Table 3), RHO planning window size (H operations), RHO current iteration  $r$  , executed solutions  $(m,\pi)$    
+Output: Operations in the current RHO planning window  $\mathcal{O}_{plan,r}$ $\mathcal{O}_{plan,r}\gets [];i\gets 0$    
+// select the first H non-executed operations based on the sorted order   
+while  $|\mathcal{O}_{plan,r}| <   H$  and  $i <   |\mathcal{O}|$  do if  $O^{(i)}\notin (m,\pi)$  then  $\begin{array}{rl}{\mid}&{\mathcal{O}_{plan,r}.append(O^{(i)})}\\&{endif}\\{i}\end{array}$    
+end while
+
+# Algorithm 3: GetStepOperations
+
+Input: Operations in the current window  $\mathcal{O}_{plan,r}$ , RHO execution step size ( $S$  operations), RHO current iteration  $r$ , solution  $\Pi_r = (m_r, \pi_{s,r})$  of the  $r^{th}$  FJSP subproblem
+
+Output: Operations executed after the  $r^{th}$  RHO iteration  $\mathcal{O}_{step,r}$
+
+1 // select the first  $S$  non-executed operations with the earliest process start time  
+$\tilde{\mathcal{O}}_{plan,r}\gets$  Sort  $\mathcal{O}_{plan,r}$  by the process start time  $\{\pi_{s,r}(O^{(i)})\}_{O^{(i)}\in \mathcal{O}_{plan,r}}$  
+3  $\mathcal{O}_{step,r}\gets \{\tilde{\mathcal{O}}_{plan,r}^{(1)},\dots,\tilde{\mathcal{O}}_{plan,r}^{(S)}\}$
+
+# A.3 INPUT FEATURES.
+
+This section provides a detailed description of all input features, which can be categorized as the following three main parts:
+
+(1) For each operation in  $\mathcal{O}_{plan,r} = \mathcal{O}_{overlap,r} \cup \mathcal{O}_{new,r}$ , we design input features to capture the FJSP instance information such as the earliest process start time of the operation (considering previously executed operations within the same job), the duration (mean / min / max / std) to complete the operation by the compatible machines, and the operation's release time (for delay-based objectives);  
+(2) For each operation in  $\mathcal{O}_{\text{overlap},r}$ , we design additional input features to encode the solution information from the previous RHO iteration, including the duration, machine assignment of the operation and end time / delay.  
+(3) For each machine in  $\mathcal{M}$ , we further design input features that include the earliest process start time of the machine (considering previously executed operations by the machine), and the duration (mean / min / max / std) of the overlapping operations  $\mathcal{O}_{\text{overlap},r}$  previously assigned to the machine.
+
+As we consider the following different FJSP scenarios in our experiments (Sec. 5), we adjust the input features accordingly as follows
+
+- Makespan objective without observation noise: we use all input features as listed in Table 5.  
+- Makespan objective under machine breakdowns: we use all input features as listed in Table 5, and the additional input features as listed in Table 6 to incorporate the machine breakdown information.  
+- Total Start Delay objective without observation noise: we use all input features as listed in Table 7, where the delay input feature is the start time delay for each operation based on the previous RHO iteration's solution, i.e.  $\pi_{r - 1}(O_{j,k}) - s_{j,k}$ .  
+- Total Start and End Delay objective without observation noise: we use all input features as listed in Table 7, where the delay input feature is the start and end time delay for each operation based on the previous RHO iteration's solution, i.e.  $\pi_{r - 1}(O_{j,k}) - s_{j,k} + \max (\pi_{t,r - 1}(O_{j,k}) - t_{j,k},0)$  where  $\pi_{r - 1}(O_{j,k}) = \pi_{r - 1}(O_{j,k}) + p_{j,k}^{m_{r - 1}(O_{j,k})}$ . We further consider the end time related input features as listed in Table 8.  
+- Total Start and End Delay objective with observation noise: we use all input features in Table 7 and 8. Additionally, as described in Appendix A.5, we obtain noisy observations of each operation's duration at each RHO iteration (the observation of the same operation may vary across different RHO iterations). That is, given the same machine assignment and start time  $(\tilde{m}_{r - 1},\tilde{\pi}_{r - 1})$  from the previous  $(r - 1^{th})$  iteration, the observed duration and end time for each overlapping operation can differ between the previous  $(r - 1^{th})$  and current  $(r^{th})$  iteration. As a result, we compute the delay and end time delay in Table 7 and 8 using the noisy observation from the previous RHO iteration, and we further re-evaluate the delay, duration, and end time for the overlapping operations based on the current RHO iteration's noisy observation, as detailed in Table 9.
+
+We normalize all input features with mean and standard deviation computed on the training set  $\mathcal{K}_{data}$ .
+
+Table 5: Input Features for the Operations and Machines for the makespan objective.  
+
+<table><tr><td>Type</td><td>Feature</td><td>Description</td></tr><tr><td rowspan="22">Operations</td><td>job_start_time</td><td>The earliest start time of the job j associated with the operation Oj,k, given by ...</td></tr><tr><td>avg_dur</td><td>... the latest process end time of all executed operations for the same job</td></tr><tr><td>std_dur</td><td>Average duration of all compatible machine assignments for the operation</td></tr><tr><td>min_dur</td><td>Duration standard deviation of all compatible machine assignments for the operation</td></tr><tr><td>max_dur</td><td>Minimum duration of all compatible machine assignments for the operation</td></tr><tr><td>job_id</td><td>Maximum duration of all compatible machine assignments for the operation</td></tr><tr><td>ops_id</td><td>Job id for the operation (j ∈ {1,...|T|} before normalization)</td></tr><tr><td>in_overlap</td><td>Operation id for the operation (k ∈ {1,...n_j} before normalization)</td></tr><tr><td></td><td>The operation Oj,k ∈ OOverlap,r at the current RHO iteration r</td></tr><tr><td>prev-machine</td><td>Machine assignment of the operation in the solution of...</td></tr><tr><td>(-1 if not in_overlap)</td><td>... the previous RHO iteration (m(Oj,k) ∈ {0,...|M-1|} before normalization)</td></tr><tr><td>prev_duration</td><td>Operation&#x27;s process duration given by the previous machine assignment prj,k</td></tr><tr><td>(-1 if not in_overlap)</td><td>End time of the operation in the solution of ...</td></tr><tr><td>prev_end_time</td><td>... the previous RHO iteration (πt,r-1(Oj,k))</td></tr><tr><td>alt_avg_dur</td><td>Average process duration of all other machines ...</td></tr><tr><td>(-1 if not in_overlap)</td><td>... (not prevmachine) to process the operation</td></tr><tr><td>alt_std_dur</td><td>Standard deviation of duration of all other machines ...</td></tr><tr><td>(-1 if not in_overlap)</td><td>... (not prevmachine) to process the operation</td></tr><tr><td>alt_min_dur</td><td>Minimum duration of all other machines ...</td></tr><tr><td>(-1 if not in_overlap)</td><td>... (not prevmachine) to process the operation</td></tr><tr><td>alt_max_dur</td><td>Maximum duration of all other machines ...</td></tr><tr><td>(-1 if not in_overlap)</td><td>... (not prevmachine) to process the operation</td></tr><tr><td rowspan="20">Machines</td><td>machine_start_time</td><td>The earliest start time of the machine m, given by the latest process ...</td></tr><tr><td>num_overlap</td><td>... end time of all executed operations assigned to machine m</td></tr><tr><td></td><td>Number of overlapping operations assigned to the machine m ...</td></tr><tr><td></td><td>... in the solution of the previous RHO iteration ∑O∈OOverlap,r 1{mr-1(O)=m}</td></tr><tr><td>avg_end_time</td><td>Average end time of overlapping operations assigned to the machine ...</td></tr><tr><td>(-1 if num_overlap is 0)</td><td>... in the solution of the previous RHO iteration</td></tr><tr><td>std_end_time</td><td>Standard deviation of the end time of overlapping operations assigned to ...</td></tr><tr><td>(-1 if num_overlap is 0)</td><td>... the machine in the solution of the previous RHO iteration</td></tr><tr><td>max_end_time</td><td>Maximum end time of overlapping operations assigned to the machine ...</td></tr><tr><td>(-1 if num_overlap is 0)</td><td>... in the solution of the previous RHO iteration</td></tr><tr><td>min_end_time</td><td>Minimum end time of overlapping operations assigned to the machine ...</td></tr><tr><td>(-1 if num_overlap is 0)</td><td>... in the solution of the previous RHO iteration</td></tr><tr><td>avg_duration</td><td>Average duration of overlapping operations assigned to the machine ...</td></tr><tr><td>(-1 if num_overlap is 0)</td><td>... in the solution of the previous RHO iteration</td></tr><tr><td>std_duration</td><td>Standard deviation of the duration of overlapping operations assigned to ...</td></tr><tr><td>(-1 if num_overlap is 0)</td><td>... the machine in the solution of the previous RHO iteration</td></tr><tr><td>max_duration</td><td>Maximum duration of overlapping operations assigned to the machine ...</td></tr><tr><td>(-1 if num_overlap is 0)</td><td>... in the solution of the previous RHO iteration</td></tr><tr><td>min_duration</td><td>Minimum duration of overlapping operations assigned to the machine ...</td></tr><tr><td>(-1 if num_overlap is 0)</td><td>... in the solution of the previous RHO iteration</td></tr></table>
+
+Table 6: Additional Input Features for the Machine Breakdown.  
+
+<table><tr><td>Type</td><td>Feature</td><td>Description</td></tr><tr><td rowspan="3">Operations</td><td>is_break_down</td><td>The current RHO subproblem has some machine breakdown</td></tr><tr><td>isOps_break_down</td><td>The machine assigned to the operation in the previous RHO subproblem ... 
+... is broken down in the current RHO subproblem</td></tr><tr><td>isOps_recovered</td><td>The operation&#x27;s machine assignment choices was restricted ... 
+... in the previous RHO subproblem due to machine breakdown, ... 
+... but the breakdown is recovered in the current subproblme</td></tr><tr><td rowspan="6">Machines</td><td>is_break_down</td><td>The current RHO subproblem has some machine breakdown</td></tr><tr><td>ismachine_break_down</td><td>The machine itself is broken down in the current RHO subproblem</td></tr><tr><td>prev_start</td><td>Process start time of the operation in the solution of ...</td></tr><tr><td>(-1 if not in_overlap)</td><td>... the previous RHO iteration πr-1(Oj,k)</td></tr><tr><td>prev_end</td><td>Process end time of the operation in the solution of ...</td></tr><tr><td>(-1 if not in_overlap)</td><td>... the previous RHO iteration πt,r-1(Oj,k) = πr-1(Oj,k) + pmr-1(Oj,k)</td></tr></table>
+
+Table 7: Input Features for the Operations and Machines for delay-based objectives.  
+
+<table><tr><td>Type</td><td>Feature</td><td>Description</td></tr><tr><td rowspan="22">Operations</td><td>job_start_time</td><td>The earliest start time of the job j associated with the operation Oj,k, given by ...</td></tr><tr><td>ops_release_time</td><td>The release time sj,k) of the operation Oj,k</td></tr><tr><td>avg_dur</td><td>Average duration of all compatible machine assignments for the operation</td></tr><tr><td>std_dur</td><td>Duration standard deviation of all compatible machine assignments for the operation</td></tr><tr><td>min_dur</td><td>Minimum duration of all compatible machine assignments for the operation</td></tr><tr><td>max_dur</td><td>Maximum duration of all compatible machine assignments for the operation</td></tr><tr><td>job_id</td><td>Job id for the operation (j ∈ {1,...|T}) before normalization)</td></tr><tr><td>ops_id</td><td>Operation id for the operation (k ∈ {1,...nj} before normalization)</td></tr><tr><td>in_overlap</td><td>The operation Oj,k ∈ Ooverlap,r at the current RHO iteration r</td></tr><tr><td>prev_delay</td><td>Delay of the operation in the solution of ...</td></tr><tr><td>(-1 if not in_overlap)</td><td>... the previous RHO iteration (πr-1(Oj,k) - sj,k)</td></tr><tr><td>prevmachine</td><td>Machine assignment of the operation in the solution of ...</td></tr><tr><td>(-1 if not in_overlap)</td><td>... the previous RHO iteration (m(Oj,k) ∈ {0,...|M-1}) before normalization)</td></tr><tr><td>prev_duration</td><td>Operation&#x27;s process duration given by the previous machine assignment pmr-1(Oj,k)</td></tr><tr><td>(-1 if not in_overlap)</td><td>Average duration of all other machines ...</td></tr><tr><td>alt_avg_dur</td><td>... (not prevmachine) to process the operation</td></tr><tr><td>(-1 if not in_overlap)</td><td>Standard deviation of duration of all other machines ...</td></tr><tr><td>alt_std_dur</td><td>... (not prevmachine) to process the operation</td></tr><tr><td>alt_min_dur</td><td>Minimum duration of all other machines ...</td></tr><tr><td>(-1 if not in_overlap)</td><td>... (not prevmachine) to process the operation</td></tr><tr><td>alt_max_dur</td><td>Maximum duration of all other machines ...</td></tr><tr><td>(-1 if not in_overlap)</td><td>... (not prevmachine) to process the operation</td></tr><tr><td rowspan="10">Machines</td><td>machine_start_time</td><td>The earliest start time of the machine m, given by the latest process ...</td></tr><tr><td>num_overlap</td><td>... end time of all executed operations assigned to machine m</td></tr><tr><td>avg_delay</td><td>Number of overlapping operations assigned to the machine m ...</td></tr><tr><td>(-1 if num_overlap is 0)</td><td>... in the solution of the previous RHO iteration ∑O∈Ooverlap,r 1{mr-1(O) = m}</td></tr><tr><td>std_delay</td><td>Average delay of overlapping operations assigned to the machine ...</td></tr><tr><td>(-1 if num_overlap is 0)</td><td>... in the solution of the previous RHO iteration</td></tr><tr><td>max_delay</td><td>Standard deviation of the delay of overlapping operations assigned to ...</td></tr><tr><td>(-1 if num_overlap is 0)</td><td>... the machine in the solution of the previous RHO iteration</td></tr><tr><td>min_delay</td><td>Maximum delay of overlapping operations assigned to the machine ...</td></tr><tr><td>(-1 if num_overlap is 0)</td><td>... in the solution of the previous RHO iteration</td></tr></table>
+
+Table 8: Additional Input Features for the Total Start and End Delay Objective.  
+
+<table><tr><td>Type</td><td>Feature</td><td>Description</td></tr><tr><td rowspan="3">Operations</td><td>ops_target_due_time</td><td>The target due time \(t_{j,k}\) of the operation \(O_{j,k}\)</td></tr><tr><td>prev_start(-1 if not in_overlap)</td><td>Process start time of the operation in the solution of ... ... the previous RHO iteration \(\pi_{r-1}(O_{j,k})\)</td></tr><tr><td>prev_end(-1 if not in_overlap)</td><td>Process end time of the operation in the solution of ... ... the previous RHO iteration \(\pi_{t,r-1}(O_{j,k})=\pi_{r-1}(O_{j,k})+p_{j,k}^{m_{r-1}(O_{j,k})}\)</td></tr></table>
+
+Table 9: Additional Input Features for Observation Noise: given the solution from the previous RHO iteration, we re-evaluate the delay, duration, end time of the overlapping operation based on the noisy observation at the current RHO iteration.  
+
+<table><tr><td>Type</td><td>Feature</td><td>Description</td></tr><tr><td rowspan="3">Operations</td><td>prev_duration_reeval(-1 if not in_overlap)</td><td>Currently observed process duration of the operation O ... ... (given the machine assignment from the previous RHO iteration)</td></tr><tr><td>prev_end_reeval(-1 if not in_overlap)</td><td>Process end time of the operation (given the machine ... ... assignment from the previous RHO iteration), ... reevaluated based on the current observed process duration</td></tr><tr><td>prev_delay_reeval(-1 if not in_overlap)</td><td>Delay of the operation (given the machine ... ... assignment from the previous RHO iteration) ... ... , reevaluated based on the current observed process duration</td></tr><tr><td rowspan="6">Machines</td><td>avg_delay_reeval(-1 if num_overlap is 0)</td><td>Average delay of overlapping operations assigned to the machine ... ... in the solution of the previous RHO iteration, ... ... , reevaluated based on the current observed process duration</td></tr><tr><td>std_delay_reeval(-1 if num_overlap is 0)</td><td>Standard deviation of the delay of overlapping operations assigned to ... ... the machine in the solution of the previous RHO iteration, ... ... , reevaluated based on the current observed process duration</td></tr><tr><td>max_delay_reeval(-1 if num_overlap is 0)</td><td>Maximum delay of overlapping operations assigned to the machine ... ... in the solution of the previous RHO iteration, ... ... , reevaluated based on the current observed process duration</td></tr><tr><td>min_delay_reeval(-1 if num_overlap is 0)</td><td>Minimum delay of overlapping operations assigned to the machine ... ... in the solution of the previous RHO iteration ... ... , reevaluated based on the current observed process duration</td></tr><tr><td>machine_end_time(-1 if num_overlap is 0)</td><td>Maximum process end time of overlapping operations ... ... assigned to the machine in the solution of the previous RHO iteration, ... ... , based on the previous iteration&#x27;s observed process duration</td></tr><tr><td>machine_end_time_reeval(-1 if num_overlap is 0)</td><td>Maximum process end time of overlapping operations ... ... assigned to the machine in the solution of the previous RHO iteration, ... ... , reevaluated based on the current observed process duration</td></tr></table>
+
+Table 10: Architecture hyperparameters. The input dimensions  $(d_o,d_m) = (15,11),(18,13),(16,6),(19,6),(22,12)$  for the five settings detailed in Table 5, 6, 8 and 9.  
+
+<table><tr><td colspan="2">Input Dimension</td><td>MLP Layers</td><td>2</td></tr><tr><td>Operation |Oplan,r| × do</td><td>R(H-S)×do</td><td rowspan="2">Activation</td><td rowspan="2">ReLU</td></tr><tr><td>Machine |T| × dm</td><td>R|T|×dm</td></tr><tr><td>Embedding dimension dhidden</td><td>64</td><td>Output Dimension</td><td>R(H-S)×1</td></tr></table>
+
+Table 11: Training hyperparameters.  
+
+<table><tr><td>Optimizer</td><td>Adam</td></tr><tr><td>Learning rate</td><td>10-3</td></tr><tr><td>Batch size</td><td>64</td></tr><tr><td>Positive Label Weight wpos</td><td>0.5</td></tr><tr><td>Num. of 
+Gradient Steps</td><td>5 × 105</td></tr></table>
+
+# A.4 ARCHITECTURE, TRAIN AND EVALUATION SETUP.
+
+We use the neural architecture  $f_{\theta}$  in Fig. 2 to identify the fixed operation subset  $\mathcal{O}_{fix,r}$ . The network takes as input a set of input features associated with each operation and machine, with the aggregated size  $\mathbb{R}^{| \mathcal{O}_{plan,r} | \times d_o}$  and  $\mathbb{R}^{| \mathcal{T} | \times d_m}$ , and outputs the predicted probability  $f_{\theta}(\mathbf{s}_r) \in [0,1]^{|\mathcal{O}_{overlap,r}|}$  of whether each overlapping operation  $O \in \mathcal{O}_{overlap,r}$  should be included in  $\mathcal{O}_{fix,r}$ . The architecture consists of the following four main components:
+
+(1) Input Embedding: We first embed the input vector of the operations  $\mathcal{O}_{plan,r} = \mathcal{O}_{overlap,r} \cup \mathcal{O}_{new,r}$  and machines  $\mathcal{T}$ , each with a input dimension of  $\mathbb{R}^{d_o}$  and  $\mathbb{R}^{d_m}$ , into hidden representations, each with a hidden dimension of  $\mathbb{R}^{d_{hidden}}$ . We use two separate MLPs for the operations and machines.  
+(2) Machine/Operation Concatenation: We then concatenate the hidden feature of each operation  $O \in \mathcal{O}_{\text{plan},r}$  with the hidden feature of the previously assigned machine  $m_{r-1}(O)$  (the local feature), and a global feature obtained by a mean pooling on the hidden features from all operations and machines. The concatenation results in a hidden dimension of  $\mathbb{R}^{3 \cdot d_{\text{hidden}}}$  for each operation, and we apply another MLP to project the dimension down to  $\mathbb{R}^{d_{\text{hidden}}}$ .  
+(3) Output: Lastly, we pass the hidden feature of each overlapping operation  $\mathcal{O}_{\text{overlap},r}$  through another MLP, which outputs the predicted probability that each operation  $O \in \mathcal{O}_{\text{overlap},r}$  should be included in  $\mathcal{O}_{\text{fix},r}$ .
+
+We train the proposed architecture with Adam optimizer with a learning rate of  $1e^{-3}$  and a batch size of 64 for 500 epochs with around  $5 \times 10^5$  gradient steps. All hyperparameters are selected on the validation set and frozen before evaluating on the test set. Table 10 and 11 provide a list of hyperparameters.
+
+Training and Evaluation Setup. For each FJSP setting, we collect a small set  $\mathcal{K}_{ps}$  of 10 instances for parameter search on all RHO methods (default, baseline and learning), and a large training set  $\mathcal{K}_{train}$  of 450 instances for neural network training. By default, we hold out a validation set  $\mathcal{K}_{val}$  of 20 instances and a test set  $\mathcal{K}_{test}$  of 100 instances each. We collect data, train, validate and test all methods on a distributed compute cluster using nodes equipped with 48 Intel AVX512 CPUs. A Nvidia Volta V100 GPU is used to train all neural networks. The model training time is within 12 hours for all FJSP settings. The parameter search for all RHO methods (Appendix A.5.4) requires approximately 48-96 hours for each FJSP setting in the experiment Sec. 5, based on the problem size. Furthermore, as the FJSP constraint programming solver CP-SAT allows multi-processing, we use 12 CPUs when solving each RHO subproblem  $(P_r / \hat{P}_r)$  for all RHO methods (Default, Baselines and Learning) for both training and testing; we also use 12 CPUs to evaluate the Full FJSP baseline. When collecting training labels, we solve each unrestricted subproblem  $P_r$  five times ( $Q = 5$ ); the training data collection takes around 24-72 hours for each FJSP setting based on the problem size. In the experiment section, we report the solve time for different methods on the test set  $\mathcal{K}_{test}$  (Table 1, Table 2, Fig. 3, and Appendix A.6).
+
+# A.5 EXPERIMENTAL DETAILS
+
+# A.5.1 FJSPINSTANCE DISTRIBUTION, MACHINE BREAKDOWN AND OBSERVATION NOISE.
+
+# Makespan-based Data Distribution.
+
+We follow the a similar data distribution as in the previous learning literature (Wang et al., 2023a) but significantly increase the FJSP problem sizes from 200 operations (previously) to up to 2000 operations (this work). Each FJSP instance is generated by sampling the following attributes:
+
+-  $(n_{machines}, n_{jobs}, n_{operations\_per\_job}) = (|\mathcal{M}|, |\mathcal{T}|, |\mathcal{O} / \mathcal{T}|)$ : we fix  $|\mathcal{M}| = 10, |\mathcal{T}| = 20$  and consider a range of  $n_{operations\_per\_job} \in \{30, 40, 60, 100\}$ . That is, we have  $(|\mathcal{M}|, |\mathcal{T}|, |\mathcal{O} / \mathcal{T}|) \in \{(10, 20, 30), (10, 20, 40), (10, 20, 60), (10, 20, 100)\}$ , with the total number of operations ranging from 600 to 2000.  
+- Compatible Machines  $\mathcal{M}_{j,k}$ : For each operation  $O_{j,k} \in \mathcal{O}$ , we randomly select a subset of machines  $\mathcal{M}_{j,k} \subseteq \mathcal{M}$  with size  $|\mathcal{M}_{j,k}| \sim \mathrm{Uniform}\{1, \dots, |\mathcal{M}|\}$  as the compatible machines for the operation.  
+- Process Duration  $p_{j,k}^{m}$ : we sample the duration of operation  $O_{j,k} \in \mathcal{O}$  processed by machine  $m \in \mathcal{M}$  from a uniform distribution  $U[1,99]$ .
+
+Machine Breakdown. As a step towards online / dynamics FJSP, ours experiments in Fig. 3 (right) accounts for machine breakdowns during the RHO solution process. The intensity of machine breakdowns is determined by two factors: 1) the frequency of breakdowns 2) the proportion of unavailable machines during each breakdown. For each FJSP instance, we simulate a series of machine breakdown events represented by a set of time intervals  $\{t_i^b\}_{i = 1,2,\ldots}$ . For each breakdown event, we randomly sample a subset of machines  $\mathcal{M}_i^b \subseteq \mathcal{M}$ , which become unavailable during the event. Specifically, the first breakdown event starts at time  $t_0^b \sim U[50,150]$  and lasts for a duration of dur. Each subsequent breakdown occurs at time  $t_i^b \sim t_{i - 1}^b + dur + U[w_{lb},w_{ub}]$  and similarly lasts for a duration of dur. We sample the subset of breakdown machines  $\mathcal{M}_i^b$  by selecting each machine in  $\mathcal{M}$  i.i.d. with a probability  $p^b$ . In our experiments, we explore three levels of breakdown intensities: 1) Low: we set dur = 100,  $w_{lb} = 400$ ,  $w_{ub} = 600$ ,  $p^b = 0.2$ , 2) Mid: we set dur = 100,  $w_{lb} = 175$ ,  $w_{ub} = 300$ ,  $p^b = 0.35$ , and 3) High: we set dur = 50,  $w_{lb} = 100$ ,  $w_{ub} = 200$ ,  $p^b = 0.5$ .
+
+Following dynamic FJSP literature (Zhang and Wong, 2017), we combine periodic RHO subproblem optimization with event-based rescheduling: when a machine breakdown event starts or ends, we re-optimize the updated RHO subproblem using the updated machine availability. This event-based rescheduling affects (1) the RHO subproblem step size, which may be reduced due to breakdown-triggered re-optimizations. Specifically, for each RHO iteration, we execute the set of operations  $\mathcal{O}_{\text{exec},r}$  in order of their scheduled end times. If a breakdown occurs during an operation's execution window, we halt that operation and any subsequent ones, and reformulate the RHO subproblem with the updated set of the first  $W$  unexecuted operations, incorporating the updated machine availability for re-optimization. (2) If all compatible machines for an operation are down, we defer that operation, along with any subsequent operations in the same job, until the breakdown ends.
+
+Implementation-wise, (1) is formally described in Alg. 5, which replaces Alg. 3 for the RHO operation execution step; (2) is formally described in Alg. 4, which replaces Alg. 2 to construct each RHO subproblem.
+
+Delay-based Data Distribution. Following common FJSP instance generation procedure in the literature (Behnke and Geiger, 2012), we generate each FJSP instance by sampling the following attributes:
+
+-  $(n_{machines},n_{jobs},n_{operations\_per\_job}) = (|\mathcal{M}|,|\mathcal{T}|,|\mathcal{O} / \mathcal{T}|)$ : we consider a series of large scale settings  $\{(25,25,24), (30,30,24), (35,35,30), (40,40,40)\}$ , with the total number of operations ranging from 600 to 1600. We further set the compatible machines  $\mathcal{M}_{j,k} = \mathcal{M}$  for all operations  $O \in \mathcal{O}$ .  
+- Process Duration  $p_{j,k}^{m}$ : we sample the duration of operation  $O_{j,k} \in \mathcal{O}$  processed by machine  $m \in \mathcal{M}$  from a uniform distribution  $U[l_{low}, l_{high}]$ , with  $l_{low} \sim \mathrm{Uniform}(\{3,5,7,9\})$  and  $l_{high} \sim l_{low} + \mathrm{Uniform}(\{9,12,15,18,21\})$ , resulting in the duration range  $p_{j,k}^{m} \in [3,30]$ ;
+
+- Release time: we sample the release time for all operations  $\{O_{j,k}\}_{k=1}^{n_j}$  within each job  $j$  as  $s_{j,1} \sim U([0,15])$  and  $s_{j,k} = s_{j,k-1} + U([0,15])$  for all  $k > 1$ ;  
+- Target end time: we sample the target end time for each operation  $O_{j,k}$  as  $t_{j,k} \sim s_{j,k} + U([0,30])$ ; this attribute is used when the objective contains total end delay (Sec. 5.2).
+
+Observation noise. In the observation noise setting (Sec. 5.2), we have noisy observation of the process duration  $p_{j,k}^{m}$ . Specifically, we generate the clean (true) process duration  $p_{j,k}^{m}$  for all operations and machines (which is only partially observed) following the procedure described above. Then, at each RHO iteration, we perturb process duration as follows: consider the operations  $\mathcal{O}_{plan,r} = \{O^{(1)},\dots,O^{(H)}\}$  (ordered by their release times) in the current iteration's planning horizon. We assume perfect observation of the first  $S \leq H$  operations with earlier release times  $\mathcal{O}_{clean,r} = \{O^{(1)},\dots,O^{(S)}\}$ , and we have noisy observation for a randomly selected subset  $\mathcal{O}_{perturb,r}$  of the remaining operations with later release times, where we select each operation in  $\mathcal{O}_{plan,r} \backslash \mathcal{O}_{clean,r}$  to be included in  $\mathcal{O}_{perturb,r}$  uniformly at random with a probability of  $\epsilon = 20\%$ . We then set the noisy observation of the process duration for each operation  $O_{j,k} \in \mathcal{O}_{plan,r}$  by machine  $m \in \mathcal{M}_{j,k}$  at the current RHO iteration  $r$  as:
+
+$$
+\tilde {p} _ {j, k} ^ {m, r} = \left\{ \begin{array}{l l} p _ {j, k} ^ {m} & \text {i f o p e r a t i o n O _ {j , k} \in \mathcal {O} _ {c l e a n , r}} \\ \operatorname {c l i p} (p _ {j, k} ^ {m} + U [ - 5, 5 ], 3, 3 0) & \text {i f o p e r a t i o n O _ {j , k} \in \mathcal {O} _ {p e r t u r b , r}}. \end{array} \right.
+$$
+
+Notably, we may have different noisy observations for the same overlapping operation in different RHO iterations. That is, we may have  $\tilde{p}_{j,k}^{m,r-1} \neq \tilde{p}_{j,k}^{m,r}$  for the same operation  $O_{j,k}$  and machine  $m$ .
+
+Operation execution under observation noise. After obtaining the solutions  $(\tilde{m}_r,\tilde{\pi}_r)$  to the noisy FJSP subproblem, we execute a subset of  $S$  operation and update the final full FJSP solution  $(m,\pi)$  as follows (replacing Line 17-21 in Alg. 1 with the procedure in Alg. 6): We first determine  $\mathcal{O}_{step,r}$  as the  $S$  operations with the earliest process start time according to the noisy solution  $\tilde{\pi}_r$ . Then, for each operation  $O_{j,k} \in \mathcal{O}_{step,r}$  (ordered by the solution start time  $\tilde{\pi}_r(O_{j,k})$ ), we assign the operation to the machine  $m(O_{j,k}) := \tilde{m}_r(O_{j,k})$ , which takes a true duration of  $p_{j,k}^{\tilde{m}_r(O_{j,k})}$  process the operation. Due to discrepancy between observed duration at planning and the actual duration at execution, the operation may not be able to start at its solution start time  $\tilde{\pi}_r(O_{j,k})$  given by the solution. We hence further adjust the actual start time  $\pi(O_{j,k})$  at execution to maintain solution feasibility. The detailed execution procedure is provided in Alg. 6.
+
+Algorithm 4: GetPlanOperations Under Machine Breakdown  
+Input: Operations  $\mathcal{O} =$  {O(1),...,O(|O|)} ordered based on the precedence order or release time (Table 3), RHO planning window size (H operations), RHO current iteration  $r$ , executed solutions  $(m,\pi)$  Output: Operations in the current RHO planning window  $\mathcal{O}_{plan,r}$    
+1  $\mathcal{O}_{plan,r}\gets []$  .  $i\gets 0$    
+2 Ignored_Jobs  $=$  Set()   
+3 // select the first H non-executed operations based on the sorted order   
+4 while  $|\mathcal{O}_{plan,r}| <   H$  and  $i <   |\mathcal{O}|$  do   
+5 if  $O^{(i)}\notin (m,\pi)$  then   
+6 if All compatible machines of  $O^{(i)}$  are down then. Ignored_Jobs.add(Job_of_O(i)) continue   
+8 end if   
+9 if Job_of_O(i)  $\in$  Ignored_Jobs then continue   
+11 end if   
+13  $\mathcal{O}_{plan,r}$  append(O(i))   
+end if   
+15  $i\gets i + 1$    
+end while
+
+Algorithm 5: Execute Operations Under Machine Breakdown  
+Input: Operations in the current window  $\mathcal{O}_{\text{plan},r}$ , RHO execution step size ( $S$  operations), RHO current iteration  $r$ , solution  $\Pi_r = (m_r, \pi_{s,r})$  of the  $r^{\text{th}}$  FJSP subproblem  
+Output: Operations executed after the  $r^{\text{th}}$  RHO iteration  $\mathcal{O}_{\text{step},r}$ $\tilde{\mathcal{O}}_{\text{plan},r} \gets \text{Sort } \mathcal{O}_{\text{plan},r}$  by the process start time  $\{\pi_{s,r}(O^{(i)})\}_{O^{(i)} \in \mathcal{O}_{\text{plan},r}}$ $\mathcal{O}_{\text{step},r} \gets []$   
+for  $O_{j,k} \in \mathcal{O}_{\text{step},r}$  (sorted by the solution process start time) do  
+if machine breakdown starts or ends during the operation's execution window  
+ $[\pi(O_{j,k}), \pi_t(O_{j,k})]$  then  
+// End the current RHO subproblem. Do not execute any more operations.  
+break  
+end if  
+else  
+ $\mathcal{O}_{\text{step},r}$ .append $(O_{j,k})$   
+end if  
+end for
+
+Algorithm 6: Execute Operations Under Observation Noise  
+Input: Operations in the current window  $\mathcal{O}_{plan,r}$ ; RHO execution step size ( $S$  operations); RHO current iteration  $r$ ; (partial) solution  $\Pi = (m, \pi)$  of the full FJSP problem instance  $P$  before the current iteration  $r$ ; solution  $\tilde{\Pi}_r = (\tilde{m}_r, \tilde{\pi}_r)$  of the  $r^{th}$  FJSP subproblem under the noisy observation; clean process duration  $\{p_{j,k}^m\}_{m \in \mathcal{M}}$  Output: Updated (partial) solution  $\Pi = (m, \pi)$  of the FJSP problem instance  $P$  after the current RHO iteration  $r$  based on the clean process duration  
+1 // Find executed operations for the current RHO iteration, using  $(\tilde{m}_r, \tilde{\pi}_r)$   
+2  $\tilde{\mathcal{O}}_{plan,r} \gets \text{Sort } \mathcal{O}_{plan,r}$  by the process start time  $\{\tilde{\pi}_r(O^{(i)})\}_{O^{(i)} \in \mathcal{O}_{plan,r}}$   
+3  $\mathcal{O}_{step,r} \gets \{\tilde{\mathcal{O}}_{plan,r}^{(1)}, \dots, \tilde{\mathcal{O}}_{plan,r}^{(S)}\}$   
+4 // Auxiliary variables to track solution feasibility  
+5 machine_end_time  $\leftarrow \{alt: \max_{\substack{O_{j,k} \in (m, \pi), \\ m(O_{j,k}) = alt}} \pi_t(O_{j,k}) \text{ for each machine } alt \in \mathcal{M}\}$   
+6 job_end_time  $\leftarrow \{j: \max_{O_{j,k} \in (m, \pi)} \pi_t(O_{j,k}) \text{ for each job } j \in \mathcal{T}\}$   
+7 // Fix the final solution for the executed operations based on the clean process duration  
+8 for  $O_{j,k} \in \mathcal{O}_{step,r}$  (sorted by the solution process start time) do  
+9  $m(O_{j,k}) \gets \tilde{m}_r(O_{j,k})$   
+10 // Update the process start time to the earliest feasible start time  
+11  $\pi(O_{j,k}) \gets \max(\tilde{\pi}_r(O_{j,k}), \text{ machine_end_time}[m(O_{j,k})], \text{ job_end_time}[j])$   
+12 // Re-evaluate the process end time based on the clean process duration  
+13  $\pi_t \gets \pi(O_{j,k}) + p_{j,k}^{m(O_{j,k})}$   
+14 // Update the auxiliary variables to track feasibility  
+15 machine_end_time[m(O_{j,k})]  $\leftarrow$  max (machine_end_time[m(O_{j,k})],  $\pi_t$ )  
+16 job_end_time[j]  $\leftarrow$  max (job_end_time[j],  $\pi_t$ )  
+17 end for
+
+# A.5.2 FJSP FORMULATION
+
+Decision Variables. Given a FJSP problem instance
+
+$$
+P = (\mathcal{M},\mathcal{T},\mathcal{O},\text{process duration}\{p_{j,k}^{m}\}_{\substack{\mathcal{O}_{j,k}\in \mathcal{O}\\ m\in \mathcal{M}_{j,k}}},
+$$
+
+$$
+\text {(o p t i o n a l)} \quad \text {r e l e a s e} \left\{s _ {j, k} \right\} _ {\mathcal {O} _ {j, k} \in \mathcal {O}}, \tag {2}
+$$
+
+$$
+\left(\text {o p t i o n a l}\right) \text {t a r g e t e n e t} \left\{t _ {j, k} \right\} _ {\mathcal {O} _ {j, k} \in \mathcal {O}}
+$$
+
+The constraint programming formulation for the FJSP is obtained by introducing additional boolean variables denoting whether an operation is assigned to a machine<sup>1</sup>. That is, we define the following set of decision variables
+
+$$
+X = \left( \right.\left\{\pi \left(O _ {j, k}\right)\right\} _ {O _ {j, k} \in \mathcal {O}}, \left\{d u r \left(O _ {j, k}\right)\right\} _ {O _ {j, k} \in \mathcal {O}}, \left\{\pi_ {t} \left(O _ {j, k}\right)\right\} _ {O _ {j, k} \in \mathcal {O}}, / / \text {p r o c e s s s t a r t t i m e}, d u r a t i o n, a n d e n d t i m e
+$$
+
+$$
+\{l _ {a l t} (O _ {j, k}) \} _ {\substack {O _ {j, k} \in \mathcal {O} \\ a l t \in \mathcal {M} _ {j, k}}} / / \text {boolean variable: whether} O _ {j, k} \text {is assigned to machine alt}
+$$
+
+$$
+\{\pi_{alt}(O_{j,k})\}_{\substack{O_{j,k}\in \mathcal{O}\\ alt\in \mathcal{M}_{j,k}}},\quad / / \text{process start time if} O_{j,k}\text{is assigned to machine alt}
+$$
+
+$$
+\{d u r _ {a l t} (O _ {j, k}) \} _ {\substack {O _ {j, k} \in \mathcal {O} \\ a l t \in \mathcal {M} _ {j, k}}} \quad / / \text {process duration if} O _ {j, k} \text {is assigned to machine alt}
+$$
+
+$$
+\left. \left\{\pi_ {t, a l t} \left(O _ {j, k}\right) \right\} _ {\substack {O _ {j, k} \in \mathcal {O} \\ a l t \in \mathcal {M} _ {j, k}}}\right) / / \text {p r o c e s s e n d t i m e i f} O _ {j, k} \text {i s a s s i g n e d t o m a c h i n e} a l t \tag{3}
+$$
+
+where the target end time and the decision variables related to the process end time are only used when the objective contains total end delay. From the above decision variables, the machine assignment of operation  $O_{j,k}$  can be found as  $m(O_{j,k}) = alt \in \mathcal{M}_{j,k}$  when  $l_{alt}(O_{j,k}) = 1$ , and the process start time of the operation can be found as  $\pi (O_{j,k})$ .
+
+FJSP Full and Subproblem Formulation. Given the above decision variables, the Constraint Programming (CP) formulation for the full FJSP problem  $P$  is given in Alg. 7.
+
+- Unrestricted Subproblem  $P_r$ : Given the set of operations in the current planning horizon  $\mathcal{O}_{plan,r}$ , we can obtain the subproblem instance  $P_r$  and the set of decision variables  $X_r$  associated with the restricted operation set  $\mathcal{O}_{plan,r}$ . Then we use the same CP formulation as in Alg. 7 on the decision variables  $X_r$ ; we additionally introduce new constraints in Alg. 8, which requires operations in  $\mathcal{O}_{plan,r}$  to be processed after all previously executed operations within the same job or the same machine.  
+- Assignment-Based Restricted Subproblem  $\hat{P}_r$ : Given a set of operations in the current planning horizon  $\mathcal{O}_{plan,r}$ , we follow the same CP formulation in Alg. 7 and 8 as for the unrestricted subproblem  $P_r$ . Given the fixed operations  $\mathcal{O}_{fix,r} \subseteq \mathcal{O}_{overlap,r}$ , we further restrict the compatible machines for each fixed operation as the single machine from the previous RHO iteration's solution, i.e. we restrict  $\mathcal{M}_{j,k} = \{m_{r-1}(O_{j,k})\}$  for all operations  $O_{j,k} \in \mathcal{O}_{fix,r}$ .
+
+Algorithm 7: FJSP Constraint Programming Formulation  
+Decision Variables:  $X$  (see Eq. equation 3)   
+Objective:   
+Makespan: min max  $\pi_t(O_{j,k})$    
+Total start delay: min  $\sum_{O_{j,k}\in \mathcal{O}}\pi (O_{j,k}) - s_{j,k}$  (4)   
+Total start and end delay: min  $\sum_{O_{j,k}\in \mathcal{O}}\pi (O_{j,k}) - s_{j,k} + \max (\pi_t(O_{j,k}) - t_{j,k},0)$    
+Constraints:   
+// Machine assignment definition for each operation   
+for each operation  $O_{j,k}\in \mathcal{O}$  do   
+// Operation end time definition   
+ $\pi_t(O_{j,k}) = \pi (O_{j,k}) + dur(O_{j,k})$  (5)   
+for each possible machine assignment alt  $\in \mathcal{M}_{j,k}$  do   
+// Operation duration definition for machine assignment   
+ $dur_{alt}(O_{j,k}) = p_{j,k}^{alt}$  (6)   
+// Operation end time definition for alternative assignment   
+ $\pi_{t,alt}(O_{j,k}) = \pi_{alt}(O_{j,k}) + dur_{alt}(O_{j,k})$  (7)   
+// Enforce solution linking if alt is selected   
+ $(\pi (O_{j,k}) = \pi_{alt}(O_{j,k}))$ .EnforceIf(lalt(Oj,k)=1) (8)   
+ $(dur(O_{j,k}) = dur_{alt}(O_{j,k}))$ .EnforceIf(lalt(Oj,k)=1) (9)   
+ $(\pi_t(O_{j,k}) = \pi_{t,alt}(O_{j,k}))$ .EnforceIf(lalt(Oj,k)=1) (10)   
+end for   
+// Only one machine assignment alternative can be selected   
+ExactlyOne(\{lalt(Oj,k)=1\}alt\in M_{j,k}) (11)   
+end for   
+// Operation precedence constraints for each job   
+for each job  $j\in \mathcal{T}$  do   
+for each operation  $O_{j,k}$  with  $1\leq k\leq n_j$  do   
+ $\pi_t(O_{j,k})\leq \pi (O_{j,k + 1})$  (12)   
+end for   
+end for   
+// Non-overlapping Operation constraints on each machine   
+for each machine alt  $\in \mathcal{M}$  do   
+NoOverlap({Interval[πalt(Oj,k),πt,alt(Oj,k)]}Oj,k∈O) (13)
+
+Algorithm 8: Additional Constraints for FJSP RHO Subproblems  $P_r$  
+1 Decision Variables:  $X_{r}$  for the operation set  $\mathcal{O}_{\text{overlap}, r}$  in the planning horizon (see Eq. equation 3)  
+2 Solutions of Previously Executed Operations:  $(m, \pi)$   
+3 Additional Constraints:  
+4 // Compliance with previously executed operations (if exists)  
+5 if  $(m, \pi)$  is not empty then  
+6 prev-machine_end_time  $\leftarrow$  {alt: max  $\pi_{t}(O_{j,k})\}$  for each machine alt  $\in \mathcal{M}$   
+7 prev_job_end_time  $\leftarrow$  {j: max  $\pi_{t}(O_{j,k})$ } for each job  $j \in \mathcal{T}$   
+8 for each operation  $O_{j,k} \in \mathcal{O}$  do  
+9 // Ensure processing after all executed operations within the same job  
+10  $\pi(O_{j,k}) \geq$  prev_job_end_time[j]  
+11 // Ensure processing after all executed operations on the same machine  
+12 for each possible machine assignment alt  $\in \mathcal{M}_{j,k}$  do  
+13 |  $\pi_{\text{alt}}(O_{j,k}) \geq$  prevmachine_end_time[alt]  
+14 end for  
+15 end for  
+16 end if
+
+![](images/4255dc66de96c041fc7f685d7d433bd9489d49f5c40a2989754e8aa423ab39de.jpg)
+
+![](images/0eb36a642d89cb2f919ed7a8a925b54fed24c4b6996a8c73b3cacb681a4398fb.jpg)
+
+![](images/65a1059400540a8bc3001a4f542e5c42114f551b255a4c1d988ea31ebaf52765.jpg)  
+Figure 5: The Large Neighborhood Search (LNS) Pipeline from (Pacino and Van Hentenryck, 2011). (a) At the  $r^{th}$  LNS iteration, we start with a complete solution  $\Pi$  for the full FJSP. The solution at the first iteration is given by solving the full FSJP for a short duration (30s). (b) We construct a FJSP subproblem by selecting a subset of FJSP operations to update their solution. Two subproblem selection methods are considered based on time and machine based decomposition methods. When constructing the FJSP subproblem, we include additional constraints on the operation's start time and each machine's available time so that the subproblem's solution is compatible with the solution of the non-selected FJSP operations. (c) We feed the FJSP subproblem into a subsolver to get a new solution  $\Pi_r$ . The old solution of the subproblem is given to warm start the solve, as it empirically improves the performance. (d) We update the complete solution  $\Pi$  with the new subproblem's solution  $\Pi_r$ , and repeat (a)-(d).
+
+![](images/88ce42bafcdf59bd4e14b0ff987f7415cff2b53d4f8c5afa8eaebf092b4cc534.jpg)
+
+# A.5.3 BASELINE DETAILS
+
+Traditional solver w/o decomposition The CP-SAT and Genetic Algorithm (GA) results are, respectively, obtained by applying the competitive constraint programming solver OR-Tools CP-SAT (Perron and Didier), and the widely used hybrid genetic algorithm with tabu search metaheuristics (Li and Gao, 2016), to solve the full FJSPs without decomposition. These comparisons emphasize the significance of decomposition in solving long-horizon FJSP.
+
+Learning-based solver w/o decomposition. For each setting, we train a DRL model (Wang et al., 2023c) on the same set of 450 training instances as L-RHO and also on a larger set of 20,000 instances, following the original paper<sup>2</sup>. At inference time, we decode the DRL model using Greedy and Sampling strategies (100 and 500 samples).
+
+Traditional solver w/ decomposition. Fig. 5 provides an illustration of the Large Neighborhood Search (LNS) algorithm that we compare as a baseline in the offline setting. In particular, (Pacino and Van Hentenryck, 2011) introduces Time Decomposition and Machine Decomposition to select different LNS neighborhoods for large-scale FJSP. The implementation details for each LNS iteration are as follows:
+
+- The ARD-LNS (Time-based) leverages temporal locality to select a random start and end time of a contiguous time interval to define the LNS subproblem. It imposes additional machine availability constraints and operation precedence constraints in defining the subproblem. The former restricts the start and end time of each machine to be compatible of fixed operations, and the later restricts the start and end time of each selected operations to be compatible with the fixed operations within the same job. It further includes (1) adaptive LNS subproblem size: The contiguous time interval starts at a length of  $20\%$  of the horizon. If no improvement is found within 5 iterations, the size of time window increases by  $5\%$ , with the longest time window length capped at  $50\%$ . If improvement is found, the size returns to  $20\%$ . (2) New objective for each LNS subproblem: Instead of makespan, it instead maximizes the distance between
+
+each operation's end time and its latest feasible completion time as the surrogate objective when solving each LNS subproblem.
+
+- The ARD-LNS (Machine-based) in each LNS iteration selects a subset of machines and define the subproblem as the set of operations on those machines. As we select all operations on each machine, the subproblem only requires the operation precedence constraint. The subproblem is solved with Makespan as the objective.
+
+In the original paper by (Pacino and Van Hentenryck, 2011), the LNS procedures are benchmarked on the dataset from (Hurink et al., 1994), where the largest instance consists of 225 operations (15 machines, 15 jobs, and 15 operations per job). We successfully reproduced their reported results on these instances; however, when applying their algorithm to the significantly larger-scale setting considered in Table 1, we observed a dramatic degradation in LNS performance. In contrast, the RHO-based algorithm maintains competitive performance on these long-horizon FJSP instances.
+
+Learning-guided solver w/ decomposition: Oracle-LNS (Time-based). We use a LookAhead Oracle similar to the one in previous learning-guided LNS studies (Huang et al., 2022; Li et al., 2021). In each LNS iteration, we first select  $K = 10$  Time Decomposition LNS subproblems as candidates. We then look ahead by solving these subproblems and choose the one that gives the biggest improvement in makespan. We randomly decide whether to evaluate the improvement based on the local makespan (for the current subproblem) or the global makespan (for the full FJSP problem), as this approach works better than focusing on just one objective. The local objective helps LNS prioritize subproblems needing the most fine-tuning, even though updating them doesn't immediately impact the final makespan. Meanwhile, the global objective encourages selecting subproblems toward the end of the time horizon to optimize the overall makespan. This differs from prior studies on Vehicle Routing (Li et al., 2021), Multi-Agent Path Finding (Huang et al., 2022) or Graph-Based Mixed Integer Linear Programming (Huang et al., 2023), where improving the local subproblem directly enhances the global objective.
+
+RHO decomposition. Detailed descriptions of different RHO baselines (Default, Warm Start, Random, First) can be found in Table 4. In Table 1, for Default RHO, Warm Start RHO, and L-RHO, we set the RHO parameters  $(H,S,T,T_{es}) = (80,30,60,3)$ . That is, each RHO subproblem involves  $H = 80$  operations, and each execution steps move forward  $S = 30$  operations. The time limit of solving each RHO subproblem is  $T = 60s$ , and we early terminate the CP-SAT (with a callback) if the best objective does not change after  $T_{es} = 3s$ . For Default RHO (Long), we use a more generous early stop time  $T_{es} = 10s$ , where we see the Default RHO's performance improves at the expense of substantially longer solve time. In Table 2, the RHO parameters are determined based on an extensive parameter grid search procedure, as described in Appendix A.5.4.
+
+![](images/0a1a1c8581836f007b6d691c49e7b679cb8e953a338c667f1c9e725490cfd0b6.jpg)  
+Figure 6: The objective and solve time for a variety of methods and RHO parameter settings, average across a set  $\mathcal{K}_{ps}$  of FJSP instances with sizes (25, 25, 24) on the left and (30, 30, 24) on the right. Each dot represents a method and a RHO parameter setting  $(H, S, T, T_{es})$ . We label the selected best parameter setting for each method with text.
+
+# A.5.4 ROLLING HORIZON PARAMETER SEARCH
+
+We perform a grid search to find the best parameter setting for each RHO variant (default, baselines and our learning method). We consider the following four parameters as a joint parameter setting
+
+{RHO Planning Window Size  $H$  , RHO Execution Step Size  $S$
+
+Time Limit  $T$  of each RHO FJSP Subproblem,
+
+Early Stopping Time  $T_{es}$  for each RHO Subproblem}
+
+In particular, the Early Stopping Time  $T_{es}$  represents the time to early terminate the CP-SAT (with a callback) if the best objective does not change after  $T_{es}$  seconds. We introduce this parameter due to the empirical observation that under a (relaxed) time limit, the objective of the best feasible solution obtained by CP-SAT may stuck at a given value for a long period of time.
+
+We perform a grid search to evaluate the performances on a set  $\mathcal{K}_{ps}$  of FJSP instances for each FJSP data distribution in the experiment Sec. 5, on the following parameter settings
+
+$\mathcal{H} =$  (Window Size, Step Size)  $\times$  (Time Limit, Early Stop Time)  
+$= \{(50,15),(50,20),(50,25),(50,30),$  
+(80, 20), (80, 25), (80, 30), (80, 35), (80, 40), (14)  
+(100,20),(100,30),(100,40),(100,50)}  
+$\times \{(15s,2s),(30s,3s),(60s,3s)\}$
+
+We visualize the objective and solve time for all methods and parameters in Fig. 6, where each dot represents the objective and solve time (averaged over  $\kappa_{ps}$ ) of each method evaluated under a parameter setting. For each method, the dots of all parameters define a Pareto frontier for the objective and solve time trade-off. As we aim to reduce solve time with minimal impact on the objective, we use the following procedure to identify the individual best parameter setting of each method:
+
+- Let  $obj^*$  be the best objective across all methods and all parameter settings. We empirically find that  $obj^*$  is typically achieved by our Oracle data collection method, which offers an objective improvement from default RHO.  
+- For each method  $M$ , we perform a line search with a step size  $r_s = 0.1$  to find the best parameter setting: starting with iteration  $i = 0$ , let  $\mathcal{H}_i^M \subseteq \mathcal{H}$  be the parameters whose objectives are within the  $i^{th}$  bucket  $[(1 + r_s \times i) \times obj^*, (1 + r_s \times (i + 1)) \times obj^*]$ . If  $\mathcal{H}_i^M \neq \emptyset$ , we then select the parameter setting  $H^{M*} \in \mathcal{H}_i$  with the shortest solve time as the individual best parameter setting for the method  $M$ ; if  $\mathcal{H}_i^M = \emptyset$ , we then move on to the next iteration  $i + 1$  and repeat the procedure, until we find the individual best parameter setting.
+
+The line search procedure ensures that if there are two parameter settings with similar objectives, but one results in a much shorter solve time than the other, we would select the parameter setting with a shorter solve time as the best parameter setting for the method. In Table 12, We provide a list of the best parameter setting for each method, identified through the line search procedure.
+
+In Table 2, we apply the best parameter setting selected for each RHO variant to evaluate the method's performance. L-RHO is evaluated using the same parameters as Oracle. Since the best parameters for Default, Warm Start, and Oracle are consistent across different settings, we use a fixed set of RHO parameters  $(H,S,T,T_{es}) = (80,30,60,3)$  in Table 1 to avoid additional computation for parameter search.
+
+Table 12: Best RHO parameter settings  $(H,S,T,T_{es})$  of various RHO methods for the different FJSP settings presented in Table 2. Parameter settings highlighted in bold are the same as the Oracle's best parameter setting for each FJSP setting.  
+
+<table><tr><td></td><td>25, 25, 24</td><td>30, 30, 24</td><td>35, 35, 30</td><td>40, 40, 40</td></tr><tr><td>Default</td><td>(80, 20, 60, 3)</td><td>(80, 25, 60, 3)</td><td>(80, 30, 60, 3)</td><td>(80, 20, 60, 3)</td></tr><tr><td>Warm Start</td><td>(80, 35, 60, 3)</td><td>(80, 25, 60, 3)</td><td>(80, 25, 60, 3)</td><td>(80, 20, 60, 3)</td></tr><tr><td>Fix first 20%</td><td>(80, 20, 60, 3)</td><td>(100, 30, 60, 3)</td><td>(100, 40, 60, 3)</td><td>(100, 40, 60, 3)</td></tr><tr><td>Fix first 30%</td><td>(80, 20, 60, 3)</td><td>(100, 30, 60, 3)</td><td>(100, 40, 60, 3)</td><td>(100, 40, 60, 3)</td></tr><tr><td>Fix first 40%</td><td>(80, 20, 60, 3)</td><td>(100, 30, 60, 3)</td><td>(100, 30, 30, 3)</td><td>(100, 30, 30, 3)</td></tr><tr><td>Fix first 50%</td><td>(80, 30, 30, 3)</td><td>(100, 20, 30, 3)</td><td>(100, 40, 30, 3)</td><td>(100, 40, 30, 3)</td></tr><tr><td>Random 10%</td><td>(80, 25, 60, 3)</td><td>(80, 25, 60, 3)</td><td>(80, 25, 60, 3)</td><td>(80, 25, 60, 3)</td></tr><tr><td>Random 20%</td><td>(80, 20, 30, 3)</td><td>(80, 25, 60, 3)</td><td>(80, 25, 30, 3)</td><td>(80, 25, 60, 3)</td></tr><tr><td>Random 30%</td><td>(80, 20, 30, 3)</td><td>(80, 30, 60, 3)</td><td>(100, 30, 30, 3)</td><td>(80, 25, 30, 3)</td></tr><tr><td>Oracle</td><td>(80, 25, 60, 3)</td><td>(80, 25, 60, 3)</td><td>(80, 25, 60, 3)</td><td>(80, 20, 60, 3)</td></tr></table>
+
+# A.5.5 PERFORMANCE OF RHO METHODS FOR FJSP (30, 30, 24).
+
+In Table 13, we provide the detailed results of all RHO baseline methods' performance under both the individual best RHO parameter setting (Individual Best Param.) and a fixed RHO parameter setting (the Oracle's best parameter, Same Param.) for FJSP (30, 30, 24) with the total start delay objective and no observation noise, which is used as the base setting in Fig. 3 middle, (ii), of Sec. 5.2.
+
+Table 13: Time and Objective improvements (TI%, OI%) of different RHO variants for FJSP (30, 30, 24) under the start delay objective. We report the mean values (higher the better, negative indicates degradation) and two standard errors. The best TI% and OI% among the RHO methods (excluding Oracle) are bold-faced. L-RHO significantly outperforms all baseline methods in the objective with substantially reduced solve time. Additionally, L-RHO closely aligns with the performance of the Look-Ahead Oracle, demonstrating its learning effectiveness.
+
+<table><tr><td rowspan="2">30, 30, 24</td><td colspan="2">Individual Best Param.</td><td colspan="2">Same Param.</td></tr><tr><td>TI %</td><td>OI %</td><td>TI %</td><td>OI %</td></tr><tr><td>Default</td><td>853.9s ± 26.9s(0.0% ± 0.0%)</td><td>1736.9 ± 116.8(0.0% ± 0.0%)</td><td>0.0% ± 0.0%</td><td>0.0% ± 0.0%</td></tr><tr><td>Warm Start All</td><td>-0.1% ± 1.3%</td><td>0.8% ± 2.2%</td><td>-0.1% ± 1.3%</td><td>0.8% ± 2.2%</td></tr><tr><td>Fix first 20%</td><td>3.6% ± 1.5%</td><td>-14.3% ± 2.7%</td><td>19.4% ± 1.5%</td><td>-23.1% ± 3.8%</td></tr><tr><td>Fix first 30%</td><td>12.4% ± 1.6%</td><td>-21.5% ± 3.8%</td><td>20.9% ± 1.7%</td><td>-57.9% ± 5.1%</td></tr><tr><td>Fix first 40%</td><td>19.5% ± 1.7%</td><td>-41.2% ± 4.4%</td><td>17.8% ± 2.0%</td><td>-131.4% ± 7.7%</td></tr><tr><td>Fix first 50%</td><td>7.9% ± 2.0%</td><td>-78.1% ± 5.5%</td><td>19.0% ± 2.1%</td><td>-238.5% ± 13.0%</td></tr><tr><td>Fix first 60%</td><td>33.6% ± 1.7%</td><td>-225.8% ± 11.8%</td><td>26.9% ± 2.3%</td><td>-416.0% ± 19.9%</td></tr><tr><td>Fix first 80%</td><td>51.0% ± 1.7%</td><td>-727.1% ± 35.4%</td><td>50.2% ± 1.6%</td><td>-874.8% ± 46.3%</td></tr><tr><td>Random 10%</td><td>1.8% ± 1.7%</td><td>-20.2% ± 3.1%</td><td>1.80% ± 1.69%</td><td>-20.2% ± 3.1%</td></tr><tr><td>Random 20%</td><td>0.3% ± 2.0%</td><td>-54.7% ± 4.6%</td><td>1.8% ± 1.7%</td><td>-20.2% ± 3.1%</td></tr><tr><td>Random 30%</td><td>9.9% ± 2.3%</td><td>-153.6% ± 8.1%</td><td>-0.3% ± 2.4%</td><td>-103.8% ± 7.2%</td></tr><tr><td>Random 40%</td><td>-10.9% ± 3.3%</td><td>-105.6% ± 7.2%</td><td>2.0% ± 2.5%</td><td>-164.68% ± 9.43%</td></tr><tr><td>(Oracle)</td><td>(49.9% ± 1.1%)</td><td>(15.3% ± 2.22%)</td><td>(49.9% ± 1.1%)</td><td>(15.3% ± 2.2%)</td></tr><tr><td>L-RHO (Ours)</td><td>41.5% ± 6.4%</td><td>14.0% ± 10.2%</td><td>41.5% ± 6.4%</td><td>14.0% ± 10.2%</td></tr></table>
+
+# A.5.6 PERFORMANCE OF RHO METHODS UNDER THE SAME RHO PARAMETER.
+
+In Table 14, we present the performance of all RHO variants under the same RHO parameter setting (the Oracle's best parameter), for all three FJSP settings in Table 2. The  $\mathrm{TI\%}$  and  $\mathrm{OI\%}$  is relative to Default RHO under Default's best parameter (same as in Table 2). From Table 13 and 14, we
+
+observe that although the First  $\sigma_f$  baseline can substantially reduce the solve time when  $\sigma_f$  is high, the associated objective drastically deteriorates at the same time. In contrast, our learning method L-RHO not only significantly significantly accelerates Default RHO, but it is also the only method that simultaneously achieves a notable improvement in the objective.
+
+Table 14: Time and Objective improvements (TI%, OI%) of different RHO methods in comparison with Default RHO (with Default's best parameter) for three FJSP settings in Table 2; all the methods are evaluated on the same RHO parameter setting (Oracle's best RHO parameter). We report the mean values (higher the better, negative indicates degradation) and two standard errors. The best TI% and OI% among the RHO methods (excluding Oracle) are bold-faced. L-RHO significantly outperforms all baseline methods in the objective with substantially reduced solve time.
+
+<table><tr><td rowspan="2"></td><td colspan="2">25, 25, 24</td><td colspan="2">35, 35, 30</td><td colspan="2">40, 40, 40</td></tr><tr><td>TI %</td><td>OI %</td><td>TI %</td><td>OI %</td><td>TI %</td><td>OI %</td></tr><tr><td>Default</td><td>17.6% ± 1.3%</td><td>-6.9% ± 4.3%</td><td>-16.0% ± 2.1%</td><td>5.1% ± 3.8%</td><td>0.0% ± 0.0%</td><td>0.0% ± 0.0%</td></tr><tr><td>Warm Start</td><td>20.0% ± 1.3%</td><td>-1.7% ± 2.5%</td><td>-12.9% ± 1.6%</td><td>11.2% ± 2.7%</td><td>5.2% ± 2.4%</td><td>6.8% ± 4.8%</td></tr><tr><td>Fix first 20%</td><td>36.2% ± 1.2%</td><td>-3.78% ± 2.7%</td><td>0.3% ± 1.6%</td><td>-33.4% ± 4.0%</td><td>4.6% ± 3.5%</td><td>-78.3% ± 9.0%</td></tr><tr><td>Fix first 30%</td><td>41.6% ± 1.2%</td><td>-14.6% ± 3.0%</td><td>-0.9% ± 1.7%</td><td>-91.5% ± 5.4%</td><td>1.5% ± 4.2%</td><td>-207.7% ± 17.3%</td></tr><tr><td>Fix first 40%</td><td>42.1% ± 1.3%</td><td>-46.8% ± 4.2%</td><td>-5.1% ± 2.6%</td><td>-222.2% ± 11.1%</td><td>7.7% ± 4.7%</td><td>-423.2% ± 30.6%</td></tr><tr><td>Fix first 50%</td><td>43.8% ± 1.2%</td><td>-100.9% ± 6.8%</td><td>3.2% ± 3.0%</td><td>-359.2% ± 16.6%</td><td>24.5% ± 3.7%</td><td>-748.0% ± 46.2%</td></tr><tr><td>Fix first 60%</td><td>46.9% ± 1.4%</td><td>-218.4% ± 12.1%</td><td>17.2% ± 2.7%</td><td>-600.4% ± 29.1%</td><td>40.7% ± 2.5%</td><td>-1118.0% ± 80.6%</td></tr><tr><td>Fix first 80%</td><td>59.5% ± 1.4%</td><td>-594.5% ± 32.5%</td><td>49.6% ± 1.8%</td><td>-1105.2% ± 55.3%</td><td>73.2% ± 1.1%</td><td>-1892.8% ± 138.9%</td></tr><tr><td>Random 10%</td><td>23.0% ± 1.3%</td><td>-11.9% ± 2.5%</td><td>-15.5% ± 1.7%</td><td>-12.9% ± 3.4%</td><td>-0.3% ± 2.7%</td><td>-16.8% ± 5.8%</td></tr><tr><td>Random 20%</td><td>24.2% ± 1.5%</td><td>-32.8% ± 3.6%</td><td>-18.8% ± 2.0%</td><td>-58.0% ± 5.7%</td><td>-1.6% ± 3.1%</td><td>-61.8% ± 10.0%</td></tr><tr><td>Random 30%</td><td>23.6% ± 1.7%</td><td>-75.1% ± 5.9%</td><td>-17.4% ± 2.3%</td><td>-104.1% ± 7.3%</td><td>-0.8% ± 3.3%</td><td>-131.8% ± 12.8%</td></tr><tr><td>Random 40%</td><td>27.4% ± 2.0%</td><td>-153.3% ± 9.1%</td><td>-14.4% ± 3.1%</td><td>-204.9% ± 11.4%</td><td>1.4% ± 3.6%</td><td>-202.6% ± 20.9%</td></tr><tr><td>(Oracle)</td><td>(60.3% ± 1.0%)</td><td>(18.5% ± 1.7%)</td><td>(40.5% ± 1.0%)</td><td>(22.4% ± 2.3%)</td><td>(53.2% ± 1.3%)</td><td>(12.8% ± 4.4%)</td></tr><tr><td>L-RHO (Ours)</td><td>53.0% ± 1.0%</td><td>16.0% ± 1.9%</td><td>35.1% ± 1.3%</td><td>21.0% ± 2.4%</td><td>47.3% ± 2.2%</td><td>13.13% ± 4.4%</td></tr></table>
+
+# A.6 ADDITIONAL EXPERIMENTAL RESULTS
+
+# A.6.1 PERFORMANCE ON REAL WORLD DATASET (DAUZÈRE-PÉRÉS AND PAULLI (1997)
+
+We find most of the traditional benchmarks on small scale ( $< 500$  operations). We benchmark the performance on Dauzère-Pérès and Paulli (1997), 1997, which is the largest traditional benchmarks available, containing 16 instances evenly divided to three different sizes (respectively, 196, 293, and 387 operations). We exclude 4 instances from the dataset, which is simple and take  $< 2s$  for RHO methods to solve. To mimic large scale settings with more operations, We further replicate all operations for each job  $k$  times, with  $k = 3, 6, 9$  to mimic long horizon scale settings with more operations. We provide the performance comparison in Table 15. For DRL and L-RHO, we both directly transfer the model learned on 600 operations (10, 20, 30) setting to the real world instances. The best known solution for the Dauzère original setting has an average makespan of 2120 with standard deviation 78.
+
+We find that the direct transfer setting, our L-RHO outperforms DRL in the real world dataset, achieving a close gap with respect to best known solution in the literature for the smallest nonaugmented setting, and the performance advantage further holds for longer-horizon augmented settings. Furthermore, L-RHO significantly speeds up Default RHO in all settings, demonstrating the benefit of introducing learning to accelerate RHO. We further note there exists slight mismatch in the training distribution and the real world dataset, and it is an interesting future work is to train the model on a diverse set of distribution to further improve L-RHO's transfer learning performance.
+
+Table 15: Performance Comparison on the Dauzère-Pérès and Paulli (1997) dataset. The best known solution of the original dataset (the first result column) has a makespan of  $2120 \pm 78$ . In the last three result columns, we randomly sample (with replacement)  $\times k$  operations for each job to mimic the long horizon settings.  
+
+<table><tr><td rowspan="2"></td><td colspan="2">Original</td><td colspan="2">Augment 3</td><td colspan="2">Augment 6</td><td colspan="2">Augment 9</td></tr><tr><td>Time</td><td>Makespan</td><td>Time</td><td>Makespan</td><td>Time</td><td>Makespan</td><td>Time</td><td>Makespan</td></tr><tr><td>DRL-20K (Wang) - Greedy</td><td>2 ± 0.1</td><td>2493 ± 44</td><td>6 ± 0.5</td><td>7279 ± 69</td><td>12 ± 1.0</td><td>14337 ± 218</td><td>22 ± 2.6</td><td>21218 ± 250</td></tr><tr><td>DRL-20K (Wang) - Sample 100</td><td>8 ± 1</td><td>2279 ± 19</td><td>57 ± 9.6</td><td>6946 ± 83</td><td>235 ± 42</td><td>13935 ± 164</td><td>534 ± 95.2</td><td>20784 ± 226</td></tr><tr><td>DRL-20K (Wang) - Sample 500</td><td>35 ± 6</td><td>2253 ± 21</td><td>321 ± 58</td><td>6887 ± 79</td><td>1341 ± 231</td><td>13884 ± 162</td><td>3014 ± 516</td><td>20722 ± 216</td></tr><tr><td>DRL-20K (Wang) - Sample 1000</td><td>72 ± 13</td><td>2247 ± 20</td><td>670 ± 116</td><td>6858 ± 83</td><td>2679 ± 458</td><td>13859 ± 156</td><td>6068 ± 1038</td><td>20706 ± 217</td></tr><tr><td>Default RHO</td><td>83 ± 33</td><td>2174 ± 68</td><td>290 ± 84</td><td>6505 ± 232</td><td>509 ± 150</td><td>13085 ± 428</td><td>794 ± 241</td><td>19689 ± 592</td></tr><tr><td>Warm Start RHO</td><td>75 ± 25</td><td>2167 ± 68</td><td>241 ± 67</td><td>6490 ± 237</td><td>450 ± 141</td><td>13034 ± 432</td><td>684 ± 210</td><td>19589 ± 609</td></tr><tr><td>L-RHO Transfer</td><td>65 ± 23</td><td>2185 ± 66</td><td>199 ± 70</td><td>6499 ± 254</td><td>345 ± 134</td><td>13015 ± 504</td><td>492 ± 200</td><td>19593 ± 725</td></tr></table>
+
+# A.6.2 ACCURACY, TPR, TNR, PRECISION, AND RECALL OF THE LEARNING METHOD.
+
+We provide the accuracy, true positive rate (TPR), true negative rate (TNR) precision, and recall of the learning architecture (Fig. 2) in Table 16.
+
+# A.6.3 ABLATION STUDY: A DETAILED COMPARISON WITH ATTENTION-BASED ARCHITECTURE.
+
+We compare the architecture in Fig. 2 with a more advanced attention based architecture shown in Fig. 7. We make the following two modifications to obtain the new architecture:
+
+- First, at the machine/operation concatenation block, for each new operation  $O \in \mathcal{O}_{new,r}$  that does not have a machine assignment in  $m_{r - 1}$ , we concatenate it a dummy machine embedding (all-zero feature) and the global embedding from mean pooling on all entities (operations and machines).  
+- Then, after the machine/operation concatenation block of the original architecture, we perform a Multi-Head Attention between the overlapping operations  $\mathcal{O}_{\text{overlap},r}$  and the new operations  $\mathcal{O}_{\text{new},r}$  (with 4 attention heads), and we also include a residual connection for the overlapping operations.
+
+For the Multi-Head Attention, we take the overlapping operations' embeddings as queries  $(Q)$ , and consider the following three setup for the attention keys  $(K)$  and values  $(V)$ : (1) New operations only, (2) Overlapping operations only, and (3) Both overlapping and new operations.
+
+Table 16: Accuracy, True Positive Rate (TPR), True Negative Rate (TNR), Precision, and Recall of the learning architecture in Fig. 2 on the hold out validation set for each setting in Table 1 and 2.  
+
+<table><tr><td></td><td>Accuracy</td><td>True Positive Rate</td><td>True Negative Rate</td><td>Precision</td><td>Recall</td></tr><tr><td colspan="6">Table 1</td></tr><tr><td>600 (10, 20, 30)</td><td>0.77</td><td>0.81</td><td>0.67</td><td>0.86</td><td>0.81</td></tr><tr><td>800 (10, 20, 40)</td><td>0.76</td><td>0.79</td><td>0.67</td><td>0.86</td><td>0.79</td></tr><tr><td>1200 (10, 20, 60)</td><td>0.76</td><td>0.77</td><td>0.73</td><td>0.87</td><td>0.77</td></tr><tr><td colspan="6">Table 2</td></tr><tr><td>625 (25, 25, 25)</td><td>0.78</td><td>0.71</td><td>0.85</td><td>0.83</td><td>0.71</td></tr><tr><td>1225 (35, 35, 35)</td><td>0.81</td><td>0.72</td><td>0.87</td><td>0.79</td><td>0.72</td></tr><tr><td>1600 (40, 40, 40)</td><td>0.81</td><td>0.69</td><td>0.90</td><td>0.83</td><td>0.69</td></tr></table>
+
+In Table 17, we find that for option (1) and (2) the introduction of the attention architecture leads to a slightly higher True Negative Rate (TNR) but lower True Positive Rate (TPR) and a slight reduction accuracy, when compared with the architecture without attention. Furthermore, when evaluating the performance on 600 operations FJSP (10, 20, 30) in Table 1, we see that option (1) and (2), results in a longer solve time but an improved makespan from the architecture without attention. We also note that option (3) is strictly dominated by the performance of the architecture without attention.
+
+We note that the TNR-TPR tradeoff on the performance and solve time aligns with our theoretical analysis, as fixing something that should not have been (low TNR) harms the objective but helps the solve time, while failing to fix something that should have been (low TPR) harms the solve time and also indirectly harms the objective (under a fixed time limit).
+
+Due to the time benefit of the architecture without attention and the relatively competitive objective, we believe it makes sense to keep the simpler architecture without attention in the main paper.
+
+![](images/ed0dcd8b5935c24e903e250e7c714600877ce4bef4348ae098e8833ecbce77e9.jpg)  
+Figure 7: Ablation neural architecture: Attention among the overlapping and new operations. The architecture follows Fig. 2, but introduces an additional cross attention among the overlapping and new operations before output the predicted probability for each overlapping operation.
+
+Table 17: Comparison of the learning architecture in Fig. 2 and the attention-based learning architecture in Fig. 7 on the hold out validation set for the 600 operations setting (10, 20, 30) in Table 1.  
+
+<table><tr><td></td><td>Accuracy</td><td>TPR</td><td>TNR</td><td>Precision</td><td>Recall</td><td>Time (s)</td><td>Makespan</td></tr><tr><td>No Attention</td><td>0.77</td><td>0.81</td><td>0.67</td><td>0.86</td><td>0.81</td><td>126 ± 19</td><td>1513 ± 70</td></tr><tr><td>Attention (K&amp;V: New Ops.)</td><td>0.76</td><td>0.77</td><td>0.74</td><td>0.88</td><td>0.77</td><td>170 ± 18</td><td>1504 ± 68</td></tr><tr><td>Attention (K&amp;V: Overlapping Ops.)</td><td>0.75</td><td>0.75</td><td>0.75</td><td>0.88</td><td>0.75</td><td>179 ± 21</td><td>1507 ± 57</td></tr><tr><td>Attention (K&amp;V: Overlapping and New Ops.)</td><td>0.76</td><td>0.76</td><td>0.74</td><td>0.88</td><td>0.76</td><td>133 ± 14</td><td>1527 ± 68</td></tr></table>
+
+![](images/c52cabb775f65a4918c18ad5f7a28bc51e62cc5a8e88be3f17824db007bb244e.jpg)
+
+![](images/f45030ed8654103300bada5fd0ddaf6c5b025da8cb2467e60f72fbc794fae119.jpg)
+
+![](images/0aea524221d2c29d294131861d33493c602a7034b02ae99a1605d65f1d8b6402.jpg)
+
+![](images/3882499bb063214f21dc1a0078686c82bba7b387507b3a6f5d71e8144acb39c4.jpg)
+
+![](images/c2197a75f24117d0779fbc74767b0a10864b0fd1e9bc31e88b2bfdcde77a045f.jpg)
+
+![](images/356455d5f1a8110b704dac0e64d80a5edda4332444d503743159187a167b82c9.jpg)  
+(a) Only the new operations as keys  $K$  and values  $V$ .
+
+![](images/5d82e30c7b0b27843e5ec80b5cf0563667d4c5aeb9f7c813ac2b45ddba99ff76.jpg)
+
+![](images/d36248320c037b650ffad26e6225856db8296a0541eb215b1221306d79ce40ed.jpg)
+
+![](images/7b25d4d3f0b0cd779b7eda9d22a6a3d823f168302aeeee84356f9f06ab81f7fa.jpg)
+
+![](images/7efe02e3d459b569fb9dbcd3bf72d6f03152ba06edcc4dc80d6300d7ffe3b332.jpg)  
+(b) Only the overlapping operations as keys  $K$  and values  $V$ .
+
+![](images/a8cd7520dc0826c6e017af1462538359a7532850199e5a5f56bbee6b51589810.jpg)
+
+![](images/a9bc50ec78a911022108fcbba9a04044a67ba44a8572fd2ec636daa5edf91a6b.jpg)
+
+![](images/2aaa888316821b48757282aac250a97c0722a9d58dd1f672e31b2e540111dc92.jpg)
+
+![](images/064dd4da24744a6857c3540bd9e7adda24bae1e2900b04328fe6c4da68e6e99d.jpg)  
+Figure 8: Comparison of Learned Attention Maps. We set the overlapping operations as the query  $Q$  (the rows) and consider three different types of keys  $K$  and values  $V$  (the columns).
+
+![](images/6473c760b672cf9c61f16003b8a06b87f172696c82a0f01de1ee572c9943a2ab.jpg)  
+(c) Both overlapping and new operations as keys  $K$  and values  $V$ .
+
+# A.6.4 PERFORMANCE COMPARISON WITH DRL UNDER THE SAME NUMBER OF FJSP TRAINING Instance.
+
+In Table 18, we compare L-RHO with DRL (Wang et al., 2023a) when training on the same set of 450 FJSP instances. We observe that DRL's performance degrades with 450 training instances, especially at larger scales, highlighting L-RHO's unique advantage with a lighter training set. This can be beneficial under situations where FJSP training instances are harder to acquire.
+
+Table 18: Compare L-RHO with DLR Wang et al. (2023a) under the same number of FJSP training instances (DRL-450).  
+
+<table><tr><td rowspan="2"></td><td colspan="2">600 (10, 20, 30)</td><td colspan="2">800 (10, 20, 40)</td><td colspan="2">1200 (10, 20, 60)</td><td colspan="2">2000 (10, 20, 100), Transfer</td></tr><tr><td>Time (s) ↓</td><td>Makespan ↓</td><td>Time (s) ↓</td><td>Makespan ↓</td><td>Time (s) ↓</td><td>Makespan ↓</td><td>Time (s) ↓</td><td>Makespan ↓</td></tr><tr><td>CP-SAT</td><td>1800</td><td>2274 ± 147</td><td>1800</td><td>4017 ± 413</td><td>1800</td><td>10925 ± 1013</td><td>1800</td><td>39585 ± 2707</td></tr><tr><td>CP-SAT (10 hours)</td><td>36000</td><td>1583 ± 65</td><td>1800</td><td>2128 ± 75</td><td>1800</td><td>3206 ± 87</td><td>1800</td><td>18821 ± 1986</td></tr><tr><td>DRL-450, Greedy</td><td>4 ± 0.04</td><td>1635 ± 75</td><td>6 ± 0.04</td><td>2178 ± 85</td><td>11 ± 0.2</td><td>3307 ± 115</td><td>17 ± 0.5</td><td>5415 ± 134</td></tr><tr><td>DRL-450, Sample 100</td><td>30 ± 1</td><td>1563 ± 62</td><td>48 ± 1</td><td>2093 ± 70</td><td>111 ± 3</td><td>3237 ± 91</td><td>301 ± 8</td><td>5341 ± 108</td></tr><tr><td>DRL-450, Sample 500</td><td>143 ± 4</td><td>1546 ± 50</td><td>261 ± 8</td><td>2070 ± 68</td><td>610 ± 25</td><td>3210 ± 94</td><td>1751 ± 30</td><td>5311 ± 111</td></tr><tr><td>DRL-20K, Greedy</td><td>4 ± 0.02</td><td>1628 ± 72</td><td>6 ± 0.04</td><td>2128 ± 80</td><td>10 ± 0.1</td><td>3141 ± 97</td><td>16 ± 0.2</td><td>5184 ± 114</td></tr><tr><td>DRL-20K, Sample 100</td><td>29 ± 0.3</td><td>1551 ± 60</td><td>47 ± 1</td><td>2048 ± 69</td><td>110 ± 3</td><td>3063 ± 81</td><td>302 ± 10</td><td>5082 ± 98</td></tr><tr><td>DRL-20K, Sample 500</td><td>146 ± 5</td><td>1537 ± 61</td><td>261 ± 8</td><td>2031 ± 68</td><td>597 ± 16</td><td>3045 ± 79</td><td>1738 ± 16</td><td>5062 ± 98</td></tr><tr><td>Default RHO</td><td>244 ± 21</td><td>1558 ± 73</td><td>348 ± 26</td><td>2103 ± 78</td><td>545 ± 36</td><td>3136 ± 91</td><td>862 ± 42</td><td>5207 ± 114</td></tr><tr><td>Warm Start RHO</td><td>203 ± 23</td><td>1521 ± 67</td><td>278 ± 22</td><td>2055 ± 75</td><td>420 ± 33</td><td>3081 ± 96</td><td>716 ± 41</td><td>5057 ± 106</td></tr><tr><td>L-RHO (450)</td><td>126 ± 19</td><td>1513 ± 70</td><td>160 ± 23</td><td>2015 ± 86</td><td>259 ± 37</td><td>3011 ± 106</td><td>473 ± 52</td><td>4982 ± 132</td></tr></table>
+
+# A.6.5 PERFORMANCE OF DEFAULT RHO WITH LONGER TIME LIMITS
+
+As described in Appendix A.5.4, two parameters control the time limit per RHO subproblem: 1) total time limit, where solving terminates if the time exceeds  $T$  seconds; 2) early stop time, where solving terminates early if the objective does not improve within  $T_{es}$  seconds. In Table ??, we evaluate the default RHO with increased  $T$  and  $T_{es}$ . The results show that, as the time limit increases by over 10x, the default RHO achieves better objectives than our L-RHO, albeit with prohibitively long runtime. Our L-RHO delivers superior performance under practical time constraints, offering benefits especially in online cases.
+
+Table 19: Comparison of Default RHO's performance across different time limit parameters  $T$  and  ${T}_{es}$  for RHO.  
+
+<table><tr><td rowspan="2">T, Tes</td><td colspan="2">600 (10, 20, 30)</td><td colspan="2">800 (10, 20, 40)</td><td colspan="2">1200 (10, 20, 60)</td></tr><tr><td>Time (s)</td><td>Makespan</td><td>Time (s)</td><td>Makespan</td><td>Time (s)</td><td>Makespan</td></tr><tr><td>60, 3 (reported)</td><td>244 ± 21</td><td>1558 ± 73</td><td>348 ± 26</td><td>2103 ± 78</td><td>545 ± 36</td><td>3136 ± 91</td></tr><tr><td>60, 10</td><td>599 ± 55</td><td>1529 ± 58</td><td>728 ± 98</td><td>2044 ± 75</td><td>1109 ± 108</td><td>3002 ± 87</td></tr><tr><td>60, 30</td><td>923 ± 71</td><td>1502 ± 66</td><td>1099 ± 130</td><td>2028 ± 72</td><td>1697 ± 175</td><td>2963 ± 78</td></tr><tr><td>60, 60</td><td>819 ± 117</td><td>1494 ± 67</td><td>1096 ± 156</td><td>2018 ± 71</td><td>1714 ± 154</td><td>2968 ± 89</td></tr><tr><td>120, 60</td><td>1476 ± 312</td><td>1497 ± 70</td><td>1811 ± 241</td><td>1997 ± 70</td><td>2809 ± 308</td><td>2932 ± 75</td></tr><tr><td>180, 120</td><td>2142 ± 461</td><td>1490 ± 70</td><td>2509 ± 346</td><td>1994 ± 76</td><td>3932 ± 590</td><td>2924 ± 83</td></tr><tr><td>360, 180</td><td>3631 ± 816</td><td>1476 ± 64</td><td>3673 ± 797</td><td>1987 ± 68</td><td>5663 ± 900</td><td>2924 ± 86</td></tr><tr><td>L-RHO</td><td>126 ± 19</td><td>1513 ± 70</td><td>160 ± 23</td><td>2015 ± 86</td><td>259 ± 37</td><td>3011 ± 106</td></tr></table>
+
+![](images/bd486e08840bcb6759d80500f51a033a8b10b179299e68a997c8772cccaece4c.jpg)  
+(a) (25, 25, 24) Start Delay
+
+![](images/329b4ebbd6162460e587accf97657e9ea24b9120a7168c70d4e77b65b7081491.jpg)  
+(b) (30, 30, 24) Start Delay
+
+![](images/4b6b391b3becb17c22c3c5b84efe27d0057dcab2480871ff41a3c1d11ed40dc9.jpg)
+
+![](images/bb9e2df2a78a480b282b6cf06bb59be68bd2c14f325e4d5a2161709bd0bfff81.jpg)  
+(d) (30, 30, 24) Start and End Delay & Observation Noise
+
+![](images/029beefe9077f9e1f3c671e4516818b696953d095dece6036e6edc8e97dddc07.jpg)  
+(c) (30, 30, 24) Start + End Delay  
+(e) (35, 35, 30) Start Delay
+
+![](images/04091ffe64ea400b7526580aec65574d12bc8286115a318c69614a1705f5249d.jpg)  
+(f) (40, 40, 40) Start Delay  
+Figure 9: The empirical  $p_{fix}^{*}(i)$  distribution across a variety of FJSP settings. We plot the empirical distribution calculated by Eq. equation 15 with the solid green curve, and the fitted linear approximation  $p_{fix}^{*}(i) = b - m\frac{i}{H - S}$  with the dashed red line. We also plot the empirical standard error (the shaded green area) computed across all RHO iterations, with  $N_{iter}$  ranging from 20 to 75 (based on the FJSP problem size). We observe a consistent linear decreasing trend across all FJSP distributions, and the low standard errors further indicate consistency across RHO iterations.
+
+# A.7 DETAILED THEORETICAL PROBABILISTIC ANALYSIS
+
+Following the main paper, we analyze a generic subproblem  $\hat{P}_r\coloneqq \hat{P}$  at each RHO iteration  $r$ . For ease of notation, we omit the  $r$  subscript for the rest of this section when it is clear from the context.
+
+# A.7.1 DETAILS OF ALL CONSIDERED RHO METHODS
+
+We consider a set of assignment-based RHO warm start procedure, where we follow the  $RHO_{test}$  procedure in Fig. 1 (b), bottom but obtain  $\mathcal{O}_{fix}$  at each RHO iteration with different methods described as follows. For tractability analysis, we consider the same RHO window size  $H$  and step size  $S$ , which results in  $H - S$  overlapping operations  $|\mathcal{O}_{overlap}| = H - S$  in consecutive RHO iterations.
+
+- Random  $\sigma_{\mathbf{R}}$ :  $\mathcal{O}_{fix}$  is obtained by selecting each operation in  $\mathcal{O}_{overlap}$  with a probability  $\sigma_R$  uniformly at random.  
+- First  $\sigma_{\mathbf{F}}$ :  $\mathcal{O}_{fix}$  is obtained as the first  $\sigma_F$  fraction of operations in  $\mathcal{O}_{overlap}$  with the earliest RHO sequence order (determined by release time for delay-based objectives or operation precedence for makespan). Notably, Random/First  $\sigma = 0$  exactly matches the default RHO procedure.  
+- L-RHO (Ours):  $\mathcal{O}_{fix}$  is predicted by the learning model  $f_{\theta}$  as described in Sec. 4 (Fig. 1, (b) bottom).  
+- Oracle: the look-ahead Oracle with  $\mathcal{O}_{fix}^{*}$ , which is empirically obtained by applying the same procedure as in Fig. 1 (b) top but only includes the solve time of the restricted subproblem  $\hat{P}^{*}$ . As illustrated in Fig. 4 (left), we analyze the performance of each method (Random, First, L-RHO) based on the closeness of  $\mathcal{O}_{fix}$  to the Oracle  $\mathcal{O}_{fix}^{*}$ .
+
+# A.7.2 EMPIRICAL VALIDATION OF THE LINEAR DECAY ASSUMPTION FOR  $p_{fix}^{*}(i)$
+
+We compute the distribution of  $p_{fix}^{*}(i)$  using the training set  $\mathcal{K}_{train}$ . Specifically, for each FJSP training instance  $P$  and RHO iteration  $r$ , we have a training label with the form  $y^{P,r*} = \{y_{O(i)}^{P,r*}\}_{O(i)\in \mathcal{O}_{\text{overlap},r}}$ , where the index  $i\in \{1,\dots,H - S\}$  of each operations  $O^{(i)}$  follows the RHO sequence order, and the associated  $\mathcal{O}_{fix,r}^{*} = \{O^{(i)}\in \mathcal{O}_{\text{overlap},r}:y_{O(i)} = 1\}$ . For each RHO iteration  $r$ , we calculate the empirical
+
+$$
+p_{fix,r}^{*emp}(i) = \frac{1}{N_{P}}\sum_{\substack{\text{FJSP instance $P$}}}y_{O^{(i)}}^{P,r*},
+$$
+
+where  $N_P$  is the number of training instances, from which we obtain the empirical distribution averaged across all RHO iterations and the associated standard error
+
+$$
+p _ {f i x} ^ {* e m p} (i) = \frac {1}{N _ {i t e r}} \sum_ {\text {R H O i t e r a t i o n} r} p _ {f i x, r} ^ {* e m p} (i), \tag {15}
+$$
+
+$$
+s. e. \left(p _ {f i x} ^ {* e m p} (i)\right) = s. t. d \left(\left\{p _ {f i x, r} ^ {* e m p} (i) \right\} _ {\text {R H O i t e r a t i o n} r}\right) / \sqrt {N _ {i t e r}},
+$$
+
+where  $N_{iter}$  is the number of RHO iterations. In Fig 9, we visualize the empirical  $p_{fix}^{*emp}(i)$  distributions (and the associated two standard errors) across a variety of FJSP settings, under the Oracle's best RHO parameter setting from grid search ( $H = 80$ ,  $S = 25$  for most settings). We also plot the linear approximation  $p_{fix}^{*}(i) = b - m\frac{i}{H - S}$  through fitting the empirical distribution, resulting in  $\mathbb{E}[n_{fix}^{*}] = \sum_{i=1}^{H-S} p_{fix}^{*}(i) = b - m\sum_{i=1}^{H-S}\frac{i}{H - S} = (b - \frac{m}{2})(H - S) - \frac{m}{2}$ .
+
+Observations from the  $p_{fix}^{*}$  distribution in Fig. 9. We observe a consistent linear decreasing trend across all distributions, with varying slopes and intercepts. The empirical distribution closely aligns with our fitted linear approximation, hence validating our Assump. 1; the low standard error further reflects the consistency across RHO iterations. Intuitively, this is because the new operations  $\mathcal{O}_{new}$  in the current RHO subproblem  $\hat{P}$  are closer in the precedence order or release time with the later overlapping operations in  $\mathcal{O}_{overlap}$ , making them more likely to be scheduled around similar time and thus have more effect on each other's solution.
+
+Moreover,  $\mathbb{E}[n_{fix}^*]$  is around  $0.5(H - S)$  for all settings, with a higher  $\mathbb{E}[n_{fix}^*]$  in (c) for (30, 30, 24) with Start and End Delay, but lower  $\mathbb{E}[n_{fix}^*]$  for larger FJSP sizes (35, 35, 30) and (40, 40, 40) in (e) and (f). We have the following noteworthy comparisons:
+
+1. Comparing FJSP (30, 30, 24) (c) Start and End Delay and (b) Start Delay, our analysis indicates that the larger number of Oracle fixed operations (high  $\mathbb{E}[n_{fix}^*]$ ) in (c) enhances the performance of the Random and First heuristics (reducing performance gap relative to Default RHO) compared with (b). This aligns with our empirically observation when comparing (i) and (ii) of Fig 3 (Right).  
+2. Comparing FJSP (30, 30, 24) (d) Start and End Delay & Observation Noise with (c), our analysis indicates a performance decline of Random and First under observation noise, as (d) has a significantly lower  $\mathbb{E}[n_{fix,r}^*]$  than (c). This is consistent with our empirical observation when comparing (iii) and (ii) in Fig 3 (Right). Moreover, comparing (d) with (b), we see that (d) has a similar  $\mathbb{E}[n_{fix,r}^*]$  but a notably lower slope  $m$  than (b). Our analysis indicates that this reduces the performance advantage of First over Random, which is reflected by our empirical comparison of (iii) and (i) in Fig. 3 (Right).
+
+# A.7.3 DERIVATIONS OF  $\mathbb{E}[n_{fp}]$  AND  $\mathbb{E}[n_{fn}]$  FOR EACH METHOD
+
+Proposition 1. The closed-form of  $\mathbb{E}[n_{fp}]$  and  $\mathbb{E}[n_{fn}]$  for Random  $\sigma_R$ , First  $\sigma_F$  and L-RHO can be found in Table 20 for a general  $p_{fix}^{*}(i)$  and a linearly decreasing  $p_{fix}^{*}(i)$  distribution (Assump. 1). Furthermore, ignoring an  $\frac{m}{2}$  term in  $\mathbb{E}[n_{fix}^{*}]$  for ease of exposition, The FPR and FNR of First  $\sigma_F$  and Random  $\sigma_R$  are  $(\alpha_F, \beta_F) = (\frac{\sigma_F(1 - b + \frac{m}{2}\sigma_F)}{1 - b + \frac{m}{2}}, 1 - \frac{\sigma_F(b - \frac{m}{2}\sigma_F)}{b - \frac{m}{2}})$  and  $(\alpha_R, \beta_R) = (\sigma_R, 1 - \sigma_R)$ .
+
+Table 20: Closed-form expressions of  $\mathbb{E}[n_{fp}]$  and  $\mathbb{E}[n_{fn}]$  for Default, Random, First, and the learning method L-RHO with False Positive and Negative Rates  $(\alpha, \beta)$ .  
+
+<table><tr><td colspan="2">Random σR</td><td colspan="2">First σF</td><td>L-RHO (α,β)</td></tr><tr><td colspan="5">General p*fix(i)</td></tr><tr><td>E[nfp]</td><td>σR∑1≤i≤H-S(1-p*fix(i))</td><td colspan="2">∑1≤i≤σF(H-S)(1-p*fix(i))</td><td>α ∑1≤i≤H-S(1-p*fix(i))</td></tr><tr><td>E[nfn]</td><td>(1-σR) ∑1≤i≤H-Sp*fix(i)</td><td colspan="2">∑σF(H-S)+1≤i≤(H-S)p*fix(i)</td><td>β ∑1≤i≤H-Sp*fix(i)</td></tr><tr><td colspan="5">Linear Decreasing p*fix(i) = b - m i/H-S</td></tr><tr><td>E[nfp]</td><td>σR[(1-b+ m/2)(H-S)+ m/2]</td><td colspan="2">σF[(1-b+ m/2σF)(H-S)+ m/2]</td><td>α[(1-b+ m/2)(H-S)+ m/2]</td></tr><tr><td>E[nfn]</td><td>(1-σR)[(b-m/2)(H-S)-m/2]</td><td colspan="2">(1-σF)[(b-m/2-m/2σF)(H-S)-m/2]</td><td>β[(b-m/2)(H-S)-m/2]</td></tr><tr><td colspan="5">Equivalently, Given E[n*fix] = ∑i p*fix(i) = (b-m/2)(H-S) - m/2</td></tr><tr><td>E[nfp]</td><td>σR(H-S-E[n*fix])</td><td colspan="2">σF(H-S-E[n*fix]-m/2(1-σF)(H-S))</td><td>α(H-S-E[n*fix])</td></tr><tr><td>E[nfn]</td><td>(1-σR)E[n*fix]</td><td colspan="2">(1-σF)(E[n*fix]-m/2σR(H-S))</td><td>βE[n*fix]</td></tr></table>
+
+Proof. See Appendix A.7.5 for the detailed derivation of Table 20. Since we have  $\mathbb{E}[n_{fix}^*] = \sum_i p_{fix}^*(i) = (b - \frac{m}{2})(H - S) - \frac{m}{2}$ , we can substitute the corresponding terms in the table with  $\mathbb{E}[n_{fix}^*]$  and arrive at the last equivalent closed-forms as in the main paper Prop. 1.
+
+Ignoring the  $\frac{m}{2} \in [0,1]$  term in  $\mathbb{E}[n_{fix}^*]$ , the coordinate transformation from the FP and FN errors  $(\mathbb{E}[n_{fp}], \mathbb{E}[n_{fn}])$  to the FP and FN rates  $(\alpha, \beta)$  involves scaling the axes by  $H - S - \mathbb{E}[n_{fix}^*] = (1 - b + \frac{m}{2})(H - S)$  and  $\mathbb{E}[n_{fix}^*] = (b - \frac{m}{2})(H - S)$ , respectively, from which we obtain the FPR and FNR for First  $\sigma_F$  and Random  $\sigma_R$ . For example,
+
+$$
+\begin{array}{l} \beta_ {F} = \frac {\mathbb {E} [ n _ {f n} ]}{\mathbb {E} [ n _ {f i x} ^ {*} ]} = \frac {(1 - \sigma_ {F}) (b - \frac {m}{2} - \frac {m}{2} \sigma_ {F}) (H - S)}{(b - \frac {m}{2}) (H - S)} = \frac {b - \frac {m}{2} - \frac {m}{2} \sigma_ {F} - b \sigma_ {F} + \frac {m}{s} \sigma_ {F} + \frac {m}{2} \sigma_ {F} ^ {2}}{b - \frac {m}{2}} \\ = \frac {b - \frac {m}{2} - b \sigma_ {F} + \frac {m}{2} \sigma_ {F} ^ {2}}{b - \frac {m}{2}} = 1 - \frac {\sigma_ {F} (b - \frac {m}{2} \sigma_ {F})}{b - \frac {m}{2}}. \\ \end{array}
+$$
+
+![](images/b356f5c0e59176b6440c826cf2c9f944af0a412690904ae6ce65f9c05295a3b2.jpg)
+
+# A.7.4 QUALITATIVE RELATIONSHIP BETWEEN  $(\mathbb{E}[n_{fp}]$  AND  $\mathbb{E}[n_{fn}])$  AND (OBJECTIVE, SOLVE TIME)
+
+Based on empirical observation and analytical insights, we interpret the effects of  $\mathbb{E}[n_{fp}]$  and  $\mathbb{E}[n_{fn}]$  on the objective and solve time in the table below (and also in Fig. 4 left), where  $+$  and  $-$  reflects positive and negative correlations, respectively.
+
+<table><tr><td></td><td>Objective</td><td>Solve Time</td></tr><tr><td>E[nfp]</td><td>+</td><td>+(small E[nfp]) or - (large E[nfp])</td></tr><tr><td>E[nfn]</td><td>+</td><td>+</td></tr></table>
+
+# Specifically,
+
+- The objective is significantly affected by  $\mathbb{E}[n_{fp}]$ , the number of operations with incorrectly fixed machine assignments. A higher  $\mathbb{E}[n_{fp}]$  indicates fixing more sub-optimal decision variables in the restricted subproblem  $\hat{P}_r$ , and hence worsens the objective value (often exponentially, as observed in Appendix A.6 when  $\sigma$  increases for Random / First). Therefore, reducing  $\mathbb{E}[n_{fp}]$  improves the objective.
+
+- The solve time is heavily influenced by  $\mathbb{E}[n_{fn}]$ , the number of operations that have the same machine assignments but are failed to be identified. A higher  $\mathbb{E}[n_{fn}]$  indicates a larger search space of the optimization subproblem  $\hat{P}_r$ , hence increasing the solve time (often exponentially, as observed when  $\sigma$  decreases for Random / First). Hence, a lower  $\mathbb{E}[n_{fn}]$  reduces the solve time.  
+-  $\mathbb{E}[n_{fp}]$  has a mixed contribution to the solve time. On one hand, an increasing  $\mathbb{E}[n_{fp}]$  reduces the size of the subproblem  $\hat{P}_r$ , which could reduce the solve time; on the other hand, an increasing  $\mathbb{E}[n_{fp}]$  leads to more sub-optimal assignment fixing in  $\hat{P}_r$ , which potentially complicates the optimization landscapes and hence can increase the solve time, especially when  $\mathbb{E}[n_{fp}]$  is small. Generally speaking, we observe that a higher  $\mathbb{E}[n_{fp}]$  slightly increases the solve time in the low  $\mathbb{E}[n_{fp}]$  regime (positive correlation) but reduces the solve time when  $\mathbb{E}[n_{fp}]$  is close to 1 (negative correlation). Specifically, in Appendix A.6, we see that when  $\mathbb{E}[n_{fp}]$  is low, L-RHO typically solves faster than First  $\sigma_f$  baselines with a similar  $\mathbb{E}[n_{fn}]$  but a higher  $\mathbb{E}[n_{fp}]$ ; likewise, Oracle generally has a similar solve time as First  $80\%$ , whose  $\mathbb{E}[n_{fp}]$  is much higher than Oracle.  
+- Reducing  $\mathbb{E}[n_{fn}]$  can improve the objective under a given time limit, as the correctly restricted subproblem  $\hat{P}_r$  from a low  $\mathbb{E}[n_{fn}]$  can simplify the optimization landscape. This effect is especially evident in the low  $\mathbb{E}[n_{fp}]$  regime, where the baseline RHO methods often reach the time limit with a suboptimal solution, whereas our L-RHO with a lower  $\mathbb{E}[n_{fn}]$  can often solve the subproblem  $\hat{P}_r$  optimally with a shorter solve time.
+
+Therefore, to achieve objectives comparable to Default RHO while reducing the solve time, an effective RHO method should first reduce  $\mathbb{E}[\mathbf{n}_{\mathrm{fp}}]$  to mitigate objective degradation, and subsequently reduce  $\mathbb{E}[\mathbf{n}_{\mathrm{fn}}]$ , which can further reduces both solve time and objective in the small  $\mathbb{E}[n_{fp}]$  regime.
+
+More Visualizations. We provide additional visualizations of different FJSP variants in Fig. 10, using the same plotting procedure as in Fig. 4 (Right). The  $(\alpha, \beta)$  of L-RHO is selected as the False Positive and Negative Rates on the validation set  $\mathcal{K}_{val}$ . Empirically, our L-RHO outperforms all baseline methods (Random, First, Default RHO) with a lower objective and has a similar solve time as First  $60\%$ ; the pink regions in Fig. 10 illustrates the empirical dominance relationship of L-RHO over the respective First and Random baselines. Based on the above analysis, the effectiveness of our learning method L-RHO is due to its ability to simultaneously achieve a small  $\mathbb{E}[n_{fp}]$  (which ensures a low objective) and a small  $\mathbb{E}[n_{fn}]$  (which improves both the objective and solve time). In contrast, the heuristic baselines Random or First trades-off  $\mathbb{E}[n_{fn}]$  and  $\mathbb{E}[n_{fp}]$  by adjusting  $\sigma$ , but cannot obtain a low  $\mathbb{E}[n_{fn}]$  and a  $\mathbb{E}[n_{fp}]$  at the same time.
+
+![](images/83f6ecd6e79be9867f82000e735c3cd1e01e26b00b7c07a98d2a8a8658fe44a9.jpg)
+
+![](images/42ae40ad1877621540a38836a9fa4fac7ba798aa3ff838c1d788bf02a7874466.jpg)
+
+![](images/c2e1fa8d8a8c1c3d6ee4a877cab33e9696708825f1156168e9c463e0baae8dbb.jpg)  
+(a) (25, 25, 24) Start Delay
+
+![](images/6c7dcb0d843bc682d645361edd105330247bc9a4224e3aca18f1035f5458ac89.jpg)  
+(b) (30, 30, 24) Start Delay
+
+![](images/afddff8f8f81e44d2f8b91d26bdd3fc95538c14a34d9cd0e20bf4232d6911747.jpg)  
+(c) (30, 30, 24) Start and End Delay  
+(e) (35, 35, 30) Start Delay  
+Figure 10: For a variety of FJSP distributions, we depict  $\mathbb{E}[n_{fn}]$  and  $\mathbb{E}[n_{fp}]$  of Random, First, and L-RHO in the low  $\mathbb{E}[n_{fp}]$  region, using empirically validated  $p_{fix}^{*}(i)$  distribution from Fig. 9 and L-RHO FPR-FNR  $(\alpha, \beta)$  on the validation set. The pink region illustrates the empirical dominating region where our L-RHO has a better solve time and objective than all the baseline methods in the region; in particular, L-RHO has a better objective than all baseline methods, and a better solve time for a large range of  $\sigma$  values (up to First  $50\% - 60\%$ ), as it can achieve low  $\mathbb{E}[n_{fp}]$  and  $\mathbb{E}[n_{fn}]$  simultaneously. Additionally, we can transform the coordinates to the FP and FN rates  $(\alpha, \beta)$ , as depicted by the right and top axes, to provide insights on what FP and FN rates should L-RHO achieve for learning to be effective.
+
+![](images/1086dbc4192788e3d3f31d792522f69d0caf0f0ad5bf48b4cac3edd69161668a.jpg)  
+(d) (30, 30, 24) Start and End Delay & Obs. Noise  
+(f) (40, 40, 40) Start Delay
+
+# A.7.5 DETAILED DERIVATION OF TABLE 20.
+
+Proof. (1) General  $p_{fix}^{*}(i)$  distribution.
+
+- Random  $\sigma_R$ : by independence from the i.i.d. random sampling procedure, we have
+
+$$
+\begin{array}{l} \mathbb {E} \left[ n _ {f p, r} ^ {\text {R a n d o m} \sigma_ {R}} \right] = \sum_ {1 \leq i \leq H - S} P r (\text {o p e r a t i o n} O ^ {i} \in \mathcal {O} _ {f i x, r} \text {a n d} i \notin \mathcal {O} _ {f i x, r} ^ {*}), \\ = \sum_ {1 \leq i \leq H - S} P r (\text {o p e r a t i o n} O ^ {(i)} \in \mathcal {O} _ {\text {f i x}, r}) (1 - P r (O ^ {(i)} \in \mathcal {O} _ {\text {f i x}, r} ^ {*})) \\ = \sum_ {1 \leq i \leq H - S} \sigma_ {R} (1 - p _ {f i x} ^ {*} (i)), \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \mathbb {E} \left[ n _ {f n, r} ^ {\text {R a n d o m} \sigma_ {R}} \right] = \sum_ {1 \leq i \leq H - S} P r (\text {o p e r a t i o n} O ^ {(i)} \notin \mathcal {O} _ {f i x, r} \text {a n d} O ^ {(i)} \in \mathcal {O} _ {f i x, r} ^ {*}) \\ = \sum_ {1 \leq i \leq H - S} P r (\text {o p e r a t i o n} O ^ {(i)} \notin \mathcal {O} _ {f i x, r}) P r (O ^ {(i)} \in \mathcal {O} _ {f i x, r} ^ {*}) \\ = \sum_ {1 \leq i \leq H - S} (1 - \sigma_ {R}) p _ {f i x} ^ {*} (i), \\ \end{array}
+$$
+
+where the sum is over all overlapping operations in  $\mathcal{O}_{\text{overlap},r}$ .
+
+- First  $\sigma_F$ : as it always selects the first  $\sigma_F(H - S)$  operations, we have
+
+$$
+\mathbb {E} \left[ n _ {f p, r} ^ {\text {F i r s t} \sigma_ {F}} \right] = \sum_ {1 \leq i \leq \sigma_ {F} (H - S)} P r (\text {o p e r a t i o n} O ^ {(i)} \notin \mathcal {O} _ {f i x, r} ^ {*}) = \sum_ {1 \leq i \leq \sigma_ {F} (H - S)} \left(1 - p _ {f i x} ^ {*} (i)\right),
+$$
+
+$$
+\mathbb {E} [ n _ {f n, r} ^ {\text {F i r s t} \sigma_ {F}} ] = \sum_ {\sigma_ {F} (H - S) \leq i \leq (H - S)} P r (\text {o p e r a t i o n} O ^ {(i)} \in \mathcal {O} _ {f i x, r} ^ {*}) = \sum_ {\sigma_ {F} (H - S) \leq i \leq (H - S)} p _ {f i x} ^ {*} (i),
+$$
+
+where the sum is over the first  $k(H - S)$  overlapping operations in  $\mathcal{O}_{\text{overlap}, r}$ .
+
+- L-RHO  $(\alpha, \beta)$ : Given the relationship between FP, FN rates and FP, FN errors, we have
+
+$$
+\mathbb {E} [ n _ {f p, r} ^ {\mathrm {L - R H O}} ] = \alpha (H - S - \mathbb {E} [ n _ {f i x, r} ^ {*} ]) = \sum_ {1 \leq i \leq H - S} \alpha (1 - p _ {f i x} ^ {*} (i)),
+$$
+
+$$
+\mathbb {E} \left[ n _ {f n, r} ^ {\mathrm {L} \cdot \mathrm {R H O}} \right] = \beta \mathbb {E} \left[ n _ {f i x, r} ^ {*} \right] = \sum_ {1 \leq i \leq H - S} \beta p _ {f i x} ^ {*} (i).
+$$
+
+(2) Linearly decreasing  $p_{fix}^{*}(i) = b - m \cdot \frac{i}{H - S}$ .
+
+We have  $1 - p_{fix}^{*}(i) = (1 - b) + m \cdot \frac{i}{H - S}$ , and we first compute the following quantities
+
+$$
+\sum_ {1 \leq i \leq H - S} \frac {i}{H - S} = \frac {1 + H - S}{2}; \quad \sum_ {1 \leq i \leq \sigma_ {F} (H - S)} \frac {i}{H - S} = \frac {\sigma_ {F} + \sigma_ {F} ^ {2} (H - S)}{2}
+$$
+
+$$
+\sum_ {\sigma_ {F} (H - S) \leq i \leq (H - S)} \frac {i}{H - S} = \sum_ {1 \leq i \leq H - S} \frac {i}{H - S} - \sum_ {1 \leq i \leq \sigma_ {F} (H - S)} \frac {i}{H - S} = \frac {(1 - \sigma_ {F}) + (1 - \sigma_ {F} ^ {2}) (H - S)}{2}
+$$
+
+$$
+\sum_ {1 \leq i \leq H - S} p _ {f i x} ^ {*} (i) = \sum_ {1 \leq i \leq H - S} b - m \cdot \frac {i}{H - S} = (b - \frac {m}{2}) (H - S) - \frac {m}{2}
+$$
+
+$$
+\sum_ {1 \leq i \leq \sigma_ {F} (H - S)} p _ {f i x} ^ {*} (i) = \sum_ {1 \leq i \leq \sigma_ {F} (H - S)} b - m \cdot \frac {i}{H - S} = \sigma_ {F} (b - \frac {m}{2} \sigma_ {F}) (H - S) - \sigma_ {F} \frac {m}{2}
+$$
+
+$$
+\sum_ {\sigma_ {F} (H - S) + 1 \leq i \leq (H - S)} p _ {f i x} ^ {*} (i) = (1 - \sigma_ {F}) (b - \frac {m}{2} - \frac {m}{2} \sigma_ {F}) (H - S) - (1 - \sigma_ {F}) \frac {m}{2}
+$$
+
+$$
+\sum_ {1 \leq i \leq H - S} \left(1 - p _ {f i x} ^ {*} (i)\right) = \sum_ {1 \leq i \leq (H - S)} (1 - b) + m \cdot \frac {i}{H - S} = (1 - b + \frac {m}{2}) (H - S) + \frac {m}{2}
+$$
+
+$$
+\sum_ {1 \leq i \leq \sigma_ {F} (H - S)} \left(1 - p _ {f i x} ^ {*} (i)\right) = \sum_ {1 \leq i \leq \sigma_ {F} (H - S)} (1 - b) + m \cdot \frac {i}{H - S} = \sigma_ {F} \left(1 - b + \frac {m}{2} \sigma_ {F}\right) (H - S) + \sigma_ {F} \frac {m}{2}
+$$
+
+We hence have
+
+- Random  $\sigma_R$ :
+
+$$
+\mathbb {E} [ n _ {f p} ] = \sum_ {1 \leq i \leq H - S} \sigma_ {R} (1 - p _ {f i x} ^ {*} (i)) = \sigma_ {R} (1 - b + \frac {m}{2}) (H - S) + \sigma_ {R} \frac {m}{2},
+$$
+
+$$
+\mathbb {E} [ n _ {f n} ] = \sum_ {1 \leq i \leq H - S} (1 - \sigma_ {R}) p _ {f i x} ^ {*} (i) = (1 - \sigma_ {R}) (b - \frac {m}{2}) (H - S) - (1 - \sigma_ {R}) \frac {m}{2}.
+$$
+
+- First  $\sigma_F$ :
+
+$$
+\mathbb {E} \left[ n _ {f p, r} ^ {\text {F i r s t} \sigma_ {F}} \right] = \sum_ {1 \leq i \leq \sigma_ {F} (H - S)} \left(1 - p _ {f i x} ^ {*} (i)\right) = \sigma_ {F} \left(1 - b + \frac {m}{2} \sigma_ {F}\right) (H - S) + \sigma_ {F} \frac {m}{2},
+$$
+
+$$
+\mathbb {E} [ n _ {f n, r} ^ {\mathrm {F i r s t} \sigma_ {F}} ] = \sum_ {\sigma_ {F} (H - S) + 1 \leq i \leq (H - S)} p _ {f i x} ^ {*} (i) = (1 - \sigma_ {F}) (b - \frac {m}{2} - \frac {m}{2} \sigma_ {F}) (H - S) - (1 - \sigma_ {F}) \frac {m}{2}.
+$$
+
+L-RHO  $(\alpha ,\beta)$
+
+$$
+\mathbb {E} \left[ n _ {f p, r} ^ {\mathrm {L - R H O}} \right] = \sum_ {1 \leq i \leq H - S} \alpha \left(1 - p _ {f i x} ^ {*} (i)\right) = \alpha \left(1 - b + \frac {m}{2}\right) (H - S) + \alpha \frac {m}{2},
+$$
+
+$$
+\mathbb {E} [ n _ {f n, r} ^ {\mathrm {L - R H O}} ] = \sum_ {1 \leq i \leq H - S} \beta p _ {f i x} ^ {*} (i) = \beta (b - \frac {m}{2}) (H - S) - \beta \frac {m}{2}.
+$$
+
+![](images/f485a7bc618a1b8b34d09874c0b9fe519598663854944af75fc024f2d8100472.jpg)
+
+# A.7.6 INTERPRETATION OF PROP. 1
+
+We obtain the following insights by interpreting Prop. 1.
+
+1. The  $p_{fix}^{*}(i)$  distribution determines the strength of the Random and First baselines and thus the potential for improvement by learning methods. Specifically,  $\mathbb{E}[n_{fix}^{*}]$  governs the performance of Random: when  $\mathbb{E}[n_{fix}^{*}] \approx 0.5$ , Random suffers from either high FP error or high FN error across all  $\sigma_R$ , but when  $\mathbb{E}[n_{fix}^{*}] = 0$  (or 1), then Random  $\sigma_R = 0$  (or  $100\%$ ) can simultaneously achieve zero FP and FN errors. Moreover, the magnitude of the slope  $m$  indicates how much First can improve over Random, with a larger  $m$  leading to greater improvement (Fig. 4 (middle)).  
+2. The qualitative relationship between the FP & FN errors and objective & solve time is illustrated in Fig. 4 (left) and further elaborated in Appendix A.7.4. Specifically, the FP error represents sub-optimal assignments incorrectly fixed in  $\hat{P}$ , which harms the objective, whereas the FN error corresponds to (near)-optimal assignments failed to be fixed in  $\hat{P}$ , which negatively impacts both metrics (solve time and objective) when a time limit is imposed. Thus, effective RHO methods should maintain low a FP error for a competitive objective and a low FN error to further improve both metrics; this is indeed achieved by our L-RHO, as evident in Fig. 4 (right).  
+3. We can further convert the FP and FN errors  $(\mathbb{E}[n_{fp}],\mathbb{E}[n_{fn}])$  into the FP and FN rates  $(\alpha ,\beta)$  as shown by the right and top axes in Fig. 4 (right). In this case,  $\alpha_{F}$  and  $\beta_{F}$  in Prop. 1 can be interpreted as the FPR and FNR that L-RHO's learned model should achieve to match the performance of First  $\sigma_F$ . This provides a valuable estimate to identify what  $(\alpha ,\beta)$  should we aim for when training L-RHO and to assess the effectiveness of the learned model before deployment.
\ No newline at end of file
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+# LEAVE-ONE-OUT STABLE CONFORMAL PREDICTION
+
+Kiljae Lee, Yuan Zhang
+
+The Ohio State University
+
+lee.10428@osu.edu, yzhanghf@stat.osu.edu
+
+# ABSTRACT
+
+Conformal prediction (CP) is an important tool for distribution-free predictive uncertainty quantification. Yet, a major challenge is to balance computational efficiency and prediction accuracy, particularly for multiple predictions. We propose Leave-One-Out Stable Conformal Prediction (LOO-StabCP), a novel method to speed up full conformal using algorithmic stability without sample splitting. By leveraging leave-one-out stability, our method is much faster in handling a large number of prediction requests compared to existing method RO-StabCP based on replace-one stability. We derived stability bounds for several popular machine learning tools: regularized loss minimization (RLM) and stochastic gradient descent (SGD), as well as kernel method, neural networks and bagging. Our method is theoretically justified and demonstrates superior numerical performance on synthetic and real-world data. We applied our method to a screening problem, where its effective exploitation of training data led to improved test power compared to state-of-the-art method based on split conformal.
+
+# 1 INTRODUCTION
+
+Conformal prediction (CP) offers a powerful framework for distribution-free prediction. It is useful for a variety of machine learning tasks and applications, including computer vision (Angelopoulos et al., 2020), medicine (Vazquez & Facelli, 2022; Lu et al., 2022), and finance (Wisniewski et al., 2020), where reliable uncertainty quantification for complex and potentially mis-specified models is in much need. Initially introduced by Vovk et al. (2005), conformal prediction has gained renewed interest. Numerous studies significantly enriched the conformal prediction toolbox and deepened theoretical understandings (Lei et al., 2018; Gibbs & Candes, 2021; Barber et al., 2023).
+
+Given data  $\mathcal{D} = \{(X_i,Y_i)\}_{i = 1}^n$ , where  $(X_{i},Y_{i})\in (\mathcal{X},\mathcal{Y})\stackrel {\mathrm{i.i.d.}}{\sim}P_{X,Y}$ , the goal is to predict the unobserved responses  $\{Y_{n + j}\}_{j = 1}^{m}$  on the test data  $\mathcal{D}_{\mathrm{test}} = \{(X_{n + j},Y_{n + j} = ?)\}_{j = 1}^{m}\stackrel {\mathrm{i.i.d.}}{\sim}P_{X,Y}$ . Conformal prediction constructs prediction intervals  $\mathcal{C}_{\alpha}(X_{n + j})$  at any given level  $\alpha \in (0,1)$ , such that
+
+$$
+\mathbb {P} \left(Y _ {n + j} \in \mathcal {C} _ {\alpha} \left(X _ {n + j}\right)\right) \geq 1 - \alpha , \quad \text {f o r a l l} j = 1, \dots , m. \tag {1}
+$$
+
+A highlighted feature of conformal prediction is distribution-free: even when a wrong model is used for prediction, the coverage validity (1) still holds (but the prediction interval will become wider).
+
+A primary challenge in conformal prediction lies in balancing computation cost with prediction accuracy. Among the variants of conformal prediction, full conformal is the most accurate (i.e., shortest predictive intervals) but also the slowest; split conformal greatly speeds up by a data-splitting scheme, but decreases accuracy and introduces variability that heavily depends on the particular split (Angelopoulos & Bates, 2021; Vovk, 2015; Barber et al., 2021). Derandomization (Solari & Djordjilovic, 2022; Gasparin & Ramdas, 2024) addresses the latter issue but increases computational cost and may make the method more conservative (Ren & Barber, 2024).
+
+Algorithmic stability is an important concept in machine learning theory (Bousquet & Elisseeff, 2002). It measures the sensitivity of a learning algorithm to small changes in the input data. Numerous studies have focused on techniques that induce algorithmic stability, such as regularized loss minimization (Shalev-Shwartz et al., 2010; Shalev-Shwartz & Ben-David, 2014) and stochastic gradient descent (Hardt et al., 2016; Bassily et al., 2020). Recent research has applied the concept of algorithmic stability to the field of conformal prediction (Ndiaye, 2022; Liang & Barber, 2023).
+
+Ndiaye (2022) proposed replace-one stable conformal prediction (RO-StabCP) that effectively accelerates full conformal by leveraging algorithmic stability. While it accelerates full conformal without introducing data splitting, thus preserving prediction accuracy, the prediction model needs to be refit for each  $X_{n+j}$ , lowering its computational efficiency for multiple predictions.
+
+In this paper, we introduce Leave-One-Out Stable Conformal Prediction (LOO-StabCP), which represents a distinct form of algorithmic stability type than that in RO-StabCP. The key innovation is that our stability correction no longer depends on the predictor at the test point  $X_{n+j}$ . As a result, our method only needs one model fitting using the training data  $\mathcal{D}$ . This enables our method to effectively handle a large number of prediction requests. Theoretical and numerical studies demonstrate that our method achieves competitive prediction accuracy compared to existing method, while preserving the finite-sample coverage validity guarantee.
+
+# 2 PRIOR WORKS ON CONFORMAL PREDICTION (CP)
+
+To set up notation and introduce our method, we begin with a brief review of classical CP methods.
+
+Full conformal. We begin by considering the prediction of a single  $Y_{n+j}$ . Fix  $j \in [m] = \{1, \ldots, m\}$  and let  $y$  denote a guessed value of the unobserved  $Y_{n+j}$ . We call  $\mathcal{D}_j^y = \mathcal{D} \cup \{(X_{n+j}, y)\}$  the augmented data and train a learning algorithm  $f$  (such as linear regression) on  $\mathcal{D}_j^y$ . To emphasize that the outcome of the training depends on both  $X_{n+j}$  and  $y$ , we denote the fitted model by  $\widehat{f}_j^y$ . Here we require that the training algorithm is permutation-invariant, meaning that  $\widehat{f}_j^y$  remains unchanged if any two data points  $(X_i, Y_i)$  and  $(X_{i'}, Y_{i'})$  are swapped for  $i, i' \in [n] \cup \{n+j\}$ . Next, for each  $i = 1, \ldots, n, n+j$ , we define a non-conformity score  $S_{i,j}^y = S(Y_i, \widehat{f}_j^y(X_i)) = |Y_i - \widehat{f}_j^y(X_i)|$  to measure  $\widehat{f}_i^y$ 's goodness of prediction on the  $i$ th data point. Notice that  $S_{i,j}^y$  also depends on  $X_{n+j}$  through  $\widehat{f}_j^y(X_i)$ , thus its subscript  $j$ . For simplicity, we set  $j = 1$  and write  $S_{i,j}^y$  as  $S_i^y$  only for this part. Now, plugging in  $y = Y_{n+1}$ , by symmetry, all non-conformity scores  $\{S_i^{Y_{n+1}}\}_{i=1}^n \cup \{S_{n+1}^{Y_{n+1}}\}$  are exchangeable, and then the rank of  $S_{n+1}^{Y_{n+1}}$  (in ascending order) is uniformly distributed over  $\{1, \ldots, n+1\}$ , implying that
+
+$$
+\mathbb {P} \left(S _ {n + 1} ^ {Y _ {n + 1}} \leq \mathcal {Q} _ {1 - \alpha} \left(\left\{S _ {i} ^ {Y _ {n + 1}} \right\} _ {i = 1} ^ {n} \cup \{\infty \}\right)\right) \geq 1 - \alpha ,
+$$
+
+where  $\mathcal{Q}_p(S) \coloneqq \inf \{x | F_S(x) \geq p\}$  denotes the lower- $p$  sample quantile of data  $S$ . This implies coverage validity  $\mathbb{P}(Y_{n+1} \in \mathcal{C}_{\alpha}^{\mathrm{full}}(X_{n+1})) \geq 1 - \alpha$  of the prediction set  $\mathcal{C}_{\alpha}^{\mathrm{full}}(X_{n+1})$  defined as
+
+$$
+\mathcal {C} _ {\alpha} ^ {\text {f u l l}} \left(X _ {n + 1}\right) = \left\{y \in \mathcal {Y}: S _ {n + 1} ^ {y} \leq \mathcal {Q} _ {1 - \alpha} \left(\left\{S _ {i} ^ {y} \right\} _ {i = 1} ^ {n} \cup \{\infty \}\right) \right\}. \tag {2}
+$$
+
+This leads to the full conformal (FullCP) method: compute  $\mathcal{C}_{\alpha}^{\mathrm{full}}(X_{n + 1})$  by a grid search over  $\mathcal{V}$ .
+
+Split conformal. The grid search required by full conformal is expensive. The key to acceleration is to decouple the prediction function  $\widehat{f}$ , thus most non-conformity scores  $\{S_i^y\}_{i=1}^n$ , from both  $j$  and  $y$ : if  $S_{n+1}^y$  is the only term that depends on  $y$ , then the prediction set can be analytically solved from (2). Split conformal (SplitCP) (Papadopoulos et al., 2002; Vovk, 2015) randomly splits  $\mathcal{D}$  into the training data  $\mathcal{D}_{\mathrm{train}}$  and the calibration data  $\mathcal{D}_{\mathrm{calib}}$ , train  $\widehat{f}$  only on  $\mathcal{D}_{\mathrm{train}}$ , and compute and rank non-conformity scores only on  $\mathcal{D}_{\mathrm{calib}} \cup \{(X_{n+j}, y)\}$ . While split conformal effectively speeds up computation, the flip side is the reduced amount of data used for both training and calibration, leading to wider predictive intervals and less stable output.
+
+Replace-one stable conformal. Ndiaye (2022) accelerated FullCP by leveraging algorithmic stability. From now on, we will switch back to the full notation for  $S$  and no longer abbreviate  $S_{i,j}^{y}$  as  $S_{i}^{y}$ . To decouple the non-conformity scores from  $y$ , Ndiaye (2022) evaluate these scores using  $\widetilde{y}$ , an arbitrary guess of  $y$ . Therefore, we call his method replace-one stable conformal prediction (RO-StabCP). To bound the impact of guessing  $y$ , he introduced the replace-one (RO) stability.
+
+Definition 1 (Replace-One Algorithmic Stability). A prediction method  $\widehat{f}$  is replace-one stable, if for all  $j\in [m]$  and  $i\in [n]\cup \{n + j\}$ , there exists  $\tau_{i,j}^{\mathrm{RO}} < \infty$ , such that
+
+$$
+\sup  _ {y, \widetilde {y}, z \in \mathcal {Y}} | S (z, \widehat {f} _ {j} ^ {y} (X _ {i})) - S (z, \widehat {f} _ {j} ^ {\widetilde {y}} (X _ {i})) | \leq \tau_ {i, j} ^ {\mathrm {R O}},
+$$
+
+where  $\widehat{f}_j^\mathfrak{H}$  is trained on  $\mathcal{D} \cup \{(X_{n + j},\mathfrak{n})\}$ , for  $\mathfrak{n} = y$  or  $\widetilde{y}$ .
+
+Recall that  $S_{i,j}^{\widetilde{y}} = |Y_i - \widehat{f_j^{\widetilde{y}}} (X_i)|$  denote the non-conformity score computed using  $\widetilde{y}$ . Then it suffices to build a predictive interval that contains  $\mathcal{C}_{j,\alpha}^{\mathrm{full}}(X_{n + j})$  in (2). By Definition 1, the following inequality  $|y - \widehat{f_j^{\widetilde{y}}} (X_{n + j})| - \tau_{n + j,j}^{\mathrm{RO}}\leq S_{n + j,j}^{y}\leq \mathcal{Q}_{1 - \alpha}(\{S_{i,j}^{y}\}_{i = 1}^{n}\cup \{\infty \})\leq \mathcal{Q}_{1 - \alpha}(\{S_{i,j}^{\widetilde{y}} + \tau_{i,j}^{\mathrm{RO}}\}_{i = 1}^{n}\cup \{\infty \})$  holds true for any  $y\in \mathcal{C}_{j,\alpha}^{\mathrm{full}}(X_{n + j})$ . Consequently, the RO stable prediction set
+
+$$
+\mathcal {C} _ {j, \alpha} ^ {\mathrm {R O}} \left(X _ {n + j}\right) = \left\{y \in \mathcal {Y}: | y - \widehat {f _ {j} ^ {\widetilde {y}}} \left(X _ {n + j}\right) | - \tau_ {n + j, j} ^ {\mathrm {R O}} \leq \mathcal {Q} _ {1 - \alpha} \left(\left\{S _ {i, j} ^ {\widetilde {y}} + \tau_ {i, j} ^ {\mathrm {R O}} \right\} _ {i = 1} ^ {n} \cup \{\infty \}\right) \right\} \tag {3}
+$$
+
+contains  $\mathcal{C}_{j,\alpha}^{\mathrm{full}}(X_{n + j})$  as a subset, thus also has valid coverage. The numerical studies in Ndiaye (2022) demonstrated that RO-StabCP computes as fast as SplitCP while stably producing more narrower predictive intervals (i.e., higher prediction accuracy).
+
+# 3 LOO-STABCP: LEAVE-ONE-OUT STABLE CONFORMAL PREDICTION
+
+# 3.1 LEAVE-ONE-OUT (LOO) STABILITY AND GENERAL FRAMEWORK
+
+When predicting one  $Y_{n+j}$ , RO-StabCP has accelerated full conformal to the speed comparable to split conformal. However, its non-conformity scores  $S_{i,j}^{\widetilde{y}}$ 's still depend on  $X_{n+j}$ . Consequently, in order to predict  $\{Y_{n+j}\}_{j=1}^{m}$ , RO-StabCP would refit the model  $m$  times, once for each  $X_{n+j}$ .
+
+This naturally motivates our approach: can we let all predictions be based off a common model  $\widehat{f}$ , which only depends on  $\mathcal{D} = \{(X_i,Y_i)\}_{i=1}^n$ , but not any of  $\{X_{n+j}\}_{j=1}^m$ ? Interestingly, the idea might appear similar to a beginner's mistake when learning FullCP, overlooking that the model fitting in FullCP should also include  $(X_{n+j},y)$ , not just  $\mathcal{D}$ . Clearly, to ensure a valid method, we must correct for errors inflicted by using  $\widehat{f}$  in lieu of  $\widehat{f}_j^y$ . Since these two model fits (ours vs FullCP) are computed on similar sets of data, with the only difference of whether to consider  $(X_{n+j},y)$ , our strategy is to study the leave-one-out (LOO) stability of the prediction method.
+
+Definition 2 (Leave-One-Out Algorithmic Stability). A prediction method is leave-one-out stable, if for all  $j \in [m]$  and  $i \in [n] \cup \{n + j\}$ , there exists  $\tau_{i,j}^{\mathrm{LOO}} < \infty$ , such that
+
+$$
+\sup _ {y, z \in \mathcal {Y}} | S (z, \widehat {f} _ {j} ^ {y} (X _ {i})) - S (z, \widehat {f} (X _ {i})) | \leq \tau_ {i, j} ^ {\mathrm {L O O}}.
+$$
+
+The  $\tau_{i,j}^{\mathrm{LOO}}$  's appearing in Definition 2 are called LOO stability bounds. Their values can often be determined by analysis of the specific learning algorithm  $f$ . For each  $j$ , we used a different set of LOO stability bounds  $\{\tau_{i,j}^{\mathrm{LOO}}\}_{i\in [n]\cup \{n + j\}}$ . This approach is adopted to achieve sharper bounds compared to using a uniformly bound for all  $j$ . We clarify that the concept of algorithmic stability is well-defined for parametric or non-parametric  $f$  's alike. For an  $f$  parameterized by some  $\theta$ , the stability bound does not focus on the whereabout of the optimal  $\theta$ , but on how much impact leaving out  $(n + j)$ th data point will have on the trained  $f$ , possibly via quantifying its impact on the estimated  $\theta$ . We will elaborate using concrete examples in Section 3.2. For now, we assume that  $\tau_{i,j}^{\mathrm{LOO}}$  's are known and present the general framework of our method, called leave-one-out stable conformal prediction (LOO-StabCP), as Algorithm 1.
+
+The implementation requires computation of  $\mathcal{O}(mn)$  many  $\tau_{i,j}^{\mathrm{LOO}}$  values. However, these computations are relatively inexpensive and do not significantly impact the overall time cost. In many examples (such as SGD, see Section 3.2.2), the main computational cost comes from model fitting, especially for complex models. We empirically confirm this in Section 4 through various numerical experiments.
+
+Table 1 presents a comparison of the computational complexity for conformal prediction methods. The concept of leave-one-out perturbations in conformal prediction has been studied in Liang & Barber (2023), but their angle is very different from ours. They focused on studying the LOO as a part of Jackknife+ (Barber et al., 2021), which fits  $n$  models, one for each  $\mathcal{D} \setminus \{(X_i,Y_i)\}$ . Then all these  $n$  models are used simultaneously for each prediction. In contrast, we use LOO technique in a very different way, developing a fast algorithm that fit only one model to  $\mathcal{D}$  (without deletion). The "one" in our leave-one-out refers to  $((X_{n + j},y))$  in  $\mathcal{D} \cup \{(X_{n + j},y)\}$ , for each  $j$ .
+
+# Algorithm 1: (LOO-StabCP) Leave-One-Out Stable Conformal Prediction Set
+
+Input: Training set  $\mathcal{D} = \{(X_i,Y_i)\}_{i = 1}^n$ , test points  $\{X_{n + j}\}_{j = 1}^{m}$ , desired coverage  $1 - \alpha$
+
+Output: Prediction interval  $\mathcal{C}_{j,\alpha}^{\mathrm{LOO}}(X_{n + j})$  for each  $j\in [m]$
+
+1. Fit the prediction function  $\widehat{f}$  on  $\mathcal{D}$ ;  
+2. Compute (uncorrected) non-conformity scores on  $\mathcal{D}$ :  $S_{i} = |Y_{i} - \widehat{f}(X_{i})|$  for  $i \in [n]$ ;
+
+for  $j\in [m]$  do
+
+3. Compute stability bounds  $\tau_{i,j}^{\mathrm{LOO}}$  for  $i\in [n]\cup \{n + j\}$  
+4. Compute prediction interval:
+
+$$
+\mathcal {C} _ {j, \alpha} ^ {\mathrm {L O O}} (X _ {n + j}) = \left[ \widehat {f} (X _ {n + j}) \pm \left\{\mathcal {Q} _ {1 - \alpha} \left(\left\{S _ {i} + \tau_ {i, j} ^ {\mathrm {L O O}} \right\} _ {i = 1} ^ {n} \cup \{\infty \}\right) + \tau_ {n + j, j} ^ {\mathrm {L O O}} \right\} \right];
+$$
+
+end
+
+<table><tr><td></td><td>FullCP</td><td>SplitCP</td><td>RO-StabCP</td><td>LOO-StabCP</td></tr><tr><td># of model fits</td><td>|Y|·m</td><td>1</td><td>m</td><td>1</td></tr><tr><td># of prediction evaluations</td><td>(n+1)·|Y|·m</td><td>n+m</td><td>(n+1)·m</td><td>n+m</td></tr><tr><td># of stability bounds</td><td>Not applicable</td><td>Not applicable</td><td>(n+1)·m</td><td>(n+1)·m</td></tr></table>
+
+Table 1: Computational complexities of our method and benchmarks, where  $|\mathcal{V}|$  is the size of the search grid used in FullCP. We emphasize that in many examples, such as SGD (Section 3.2.2), one model fitting is much more costly than one prediction or computation of one stability bound.
+
+Next, we provide the theoretical guarantee of our algorithm's coverage validity.
+
+Theorem 1. If the prediction method is leave-one-out stable as in Definition 2. Then for each  $j \in [m]$ , the prediction set  $\mathcal{C}_{j,\alpha}^{\mathrm{LOO}}(X_{n + j})$  constructed by Algorithm 1 satisfies
+
+$$
+\mathbb {P} \left(Y _ {n + j} \in \mathcal {C} _ {j, \alpha} ^ {\mathrm {L O O}} \left(X _ {n + j}\right)\right) \geq 1 - \alpha .
+$$
+
+# 3.2 LOO STABLE ALGORITHMS
+
+So far, we have been treating the stability bounds  $\tau_{i,j}^{\mathrm{LOO}}$  as given without showing how to obtain them. In this section, we derive these bounds for two important machine learning tools: Regularized Loss Minimization (RLM) and Stochastic Gradient Descent (SGD). Many machine learning tasks aim to minimize a loss function  $\ell(y, f_{\theta}(x))$  over training data. Empirical Risk Minimization (ERM) is a common approach, which seeks to minimize  $\frac{1}{n}\sum_{i=1}^{n}\ell(Y_i, f_{\theta}(X_i))$  with respect to  $\theta$ . However, the objective function is often highly nonconvex, making the optimization challenging. RLM alleviates nonconvexity by adding an explicit penalty (e.g., ridge and LASSO) to the objective function (Hoerl & Kennard, 1970; Tibshirani, 1996). Alternatively, SGD implicitly regularizes the optimization procedure (Robbins & Monro, 1951) by iteratively updating model parameters using one data point at a time. Its computational efficiency makes it a preferred method in deep learning (LeCun et al., 2015; He et al., 2016).
+
+# 3.2.1 EXAMPLE 1: REGULARIZED LOSS MINIMIZATION (RLM)
+
+To derive the LOO stability bound, we compare two versions of RLM, only differing by their training data. The first is trained on  $\mathcal{D}$ , producing  $\widehat{\theta} = \arg \min_{\theta \in \Theta}[\frac{1}{n}\sum_{i=1}^{n}\ell(Y_i,f_\theta(X_i)) + \Omega(\theta)]$ , where  $\Theta$  is the parameter space and  $\Omega(\theta)$  is the explicit penalty term; while the second is trained on the augmented data  $\mathcal{D}_j^y$  (recall  $\mathcal{D}_j^y = \mathcal{D} \cup \{(X_{n+j},y)\}$ ), producing  $\widehat{\theta}_j^y = \arg \min_{\theta \in \Theta}[\frac{1}{n+1}\{\sum_{i=1}^{n}\ell(Y_i,f_\theta(X_i)) + \ell(y,f_\theta(X_{n+j})\} + \Omega(\theta)]$ . The LOO stability for RLM is described by Definition 2, with  $\widehat{f}(\cdot) = f_{\widehat{\theta}}(\cdot)$  and  $\widehat{f}_j^y(\cdot) = f_{\widehat{\theta}_j^y}(\cdot)$ . To state our main result, we need some concepts from optimization.
+
+Definition 3 ( $\rho$ -Lipschitz). A continuous function  $g: \mathbb{R}^p \to \mathbb{R}^q$  is  $\rho$ -Lipschitz, if
+
+$$
+\| g (x) - g (y) \| \leq \rho \| x - y \|, \quad f o r a n y x, y \in \mathbb {R} ^ {p}.
+$$
+
+Definition 4 (Strong Convexity). A function  $g: \mathbb{R}^p \to \mathbb{R}$  is  $\lambda$ -strongly convex, if
+
+$$
+g (t x + (1 - t) y) \leq t g (x) + (1 - t) g (y) - \frac {\lambda}{2} \| x - y \| ^ {2}, \quad f o r a n y x, y \in \mathbb {R} ^ {p} a n d t \in (0, 1).
+$$
+
+In addition, a function  $g$  is convex if it is 0-strongly convex.
+
+Now we are ready to formulate the LOO stability bounds for RLM.
+
+Theorem 2. Suppose: 1) for each  $i \in [n + m]$  and given any  $y$ , the loss function  $\ell(y, f_{\theta}(X_i))$  is convex and  $\rho_i$ -Lipschitz in  $\theta$ ; 2) the penalty term  $\Omega$  is  $\lambda$ -strongly convex; 3) for each  $i \in [n + m]$ , the prediction function  $f_{\theta}(X_i)$  is  $\nu_i$ -Lipschitz in  $\theta$ ; and 4) given any  $y$ , the non-conformity score  $S(y, z)$  is  $\gamma$ -Lipschitz in  $z$ . Then, RLM has the following LOO and RO stability bounds.
+
+$$
+\tau_ {i, j} ^ {\mathrm {L O O}} = \frac {2 \gamma \nu_ {i} (\rho_ {n + j} + \bar {\rho})}{\lambda (n + 1)}, \quad a n d \quad \tau_ {i, j} ^ {\mathrm {R O}} = \frac {4 \gamma \nu_ {i} \rho_ {n + j}}{\lambda (n + 1)}, \tag {4}
+$$
+
+where  $i$  ranges in  $[n] \cup \{n + j\}$  for each  $1 \leq j \leq m$ , and  $\bar{\rho} = n^{-1} \sum_{i=1}^{n} \rho_i$ .
+
+As a side remark, a uniform RO stability bound has been established in Ndiaye (2022) (Corollary 3.10). Our RO bound in (4) is non-uniform (not maximizing over  $X_{i}$ ) and potentially sharper.
+
+# 3.2.2 EXAMPLE 2: STOCHASTIC GRADIENT DESCENT (SGD)
+
+For simplicity, we recap how SGD operates when trained on  $\mathcal{D}$ . It starts with an initial parameter value  $\theta_0$  and runs for  $R$  epochs. In each epoch, generate a random permutation  $\pi = (\pi_1, \dots, \pi_n)$  of  $[n]$ . Then for each  $i \in [n]$ , update the model parameter by  $\theta = \theta - \eta \nabla_{\theta} \ell(Y_{\pi_i}, f_{\theta}(X_{\pi_i}))$ , where  $\eta > 0$  is a user-selected learning rate. After a total of  $Rn$  updates, the output  $\hat{\theta}$  is used for prediction. Like in RLM, our LOO stability bound compares two versions of SGD, trained on  $\mathcal{D}$  and  $\mathcal{D}_j^y$ , respectively.
+
+Theorem 3. Suppose: 1) for each  $i \in [n + m]$ , the loss function  $\ell(y, f_{\theta}(X_i))$  is convex,  $\rho_i$ -Lipschitz in  $\theta$ , and its gradient  $\nabla_{\theta} \ell(y, f_{\theta}(X_i))$  is  $\varphi_i$ -Lipschitz in  $\theta$ , for any  $y$ ; 2) for each  $i \in [n + m]$ , the prediction function  $f_{\theta}(X_i)$  is  $\nu_i$ -Lipschitz in  $\theta$ ; and 3) the non-conformity score  $S(y, z)$  is  $\gamma$ -Lipschitz in  $z$ , for any  $y$ . Then, with learning rate  $\eta \leq \frac{2}{\max\{\varphi_i\}}$ , SGD has the following LOO and RO stability bounds.
+
+$$
+\tau_ {i, j} ^ {\mathrm {L O O}} = R \eta \cdot \gamma \nu_ {i} \rho_ {n + j}, \quad a n d \quad \tau_ {i, j} ^ {\mathrm {R O}} = 2 R \eta \cdot \gamma \nu_ {i} \rho_ {n + j}, \tag {5}
+$$
+
+where  $i$  ranges in  $[n] \cup \{n + j\}$  for each  $1 \leq j \leq m$ .
+
+Readers may have noticed that for SGD,  $\tau_{i,j}^{\mathrm{LOO}}$  is only half of  $\tau_{i,j}^{\mathrm{RO}}$ , which is very different from the case for RLM (c.f. Theorem 2). The gap here stems from the iterative nature of SGD. Recall that in each epoch, SGD performs  $n$  (or  $n + 1$ ) gradient descent (GD) updates, with each update depending on a single data point. Consequently, leaving out one data point results in one fewer GD update. In contrast, replacing one data point means performing one GD update differently – in the worst-case scenario, this update may move in opposite directions before and after the replacement, doubling the stability bound.
+
+SGD's iterative nature makes it an excellent example where the number of model fits is the main bottleneck in scaling a crucial learning technique. For SGD, each model fit requires  $O(Rn)$  gradient updates, while each prediction costs  $O(1)$  time, and evaluating stability bounds for each prediction costs  $O(n)$  time. Combining this understanding with Table 1, we see that our method provides significantly faster stable conformal prediction than RO-StabCP for performing a large number of predictions.
+
+# 3.2.3 TOWARDS BROADER APPLICABILITY OF LOO-STABCP
+
+Kernel method: The kernel method (or "kernel trick") (Scholkopf, 2002) is a commonly used technique in statistical learning. It implicitly transforms data into complex spaces through a kernel function  $k(x,x')$ . This leads to the reformulated optimization problem:  $\widehat{\theta} = \arg \min_{\theta \in \mathbb{R}^n} \frac{1}{n} \sum_{i=1}^{n} \ell(Y_i, k_i^T \theta) + \lambda \theta^T \mathbf{K} \theta$ , where  $\mathbf{K}$  is a positive-definite kernel matrix  $K_{i,j} = k(X_i, X_j)$ , and  $k_i$  denotes its  $i$ -th row. It is not difficult to verify that the kernel method is a special case of RLM, thus Theorem 2 applies to the kernel method.
+
+Neural networks: The stability bounds for RLM and SGD rely on convexity assumptions that might not always hold in practice, such as in (deep) neural networks. Here, we analyze the LOO stability of SGD as a popular optimizer for neural networks, without assuming convexity.
+
+Theorem 4. Assume the conditions of Theorem 3, except that the loss function  $\ell(y, f_{\theta}(X_i))$  is not required to be convex in  $\theta$ . Then, for the same range of  $(i,j)$ , SGD has the following LOO and RO stability bounds:
+
+$$
+\tau_ {i, j} ^ {\mathrm {L O O}} = R ^ {+} \eta \cdot \gamma \nu_ {i} \rho_ {n + j}, \quad a n d \quad \tau_ {i, j} ^ {\mathrm {R O}} = 2 R ^ {+} \eta \cdot \gamma \nu_ {i} \rho_ {n + j},
+$$
+
+where  $R^{+} = \sum_{r = 1}^{R}\kappa^{r}$  and  $\kappa = \prod_{i = 1}^{n}(1 + \eta \varphi_{i})$
+
+While Theorem 4 does provide a rigorous theoretical justification for neural networks, in practice, the term  $\kappa$  may be large if the learning rate  $\eta$  is not sufficiently small or the activation function is not very smooth, leading to large Lipschitz constants  $\varphi_{i}$ 's. Therefore, this stability bound may turn out to be conservative. Similar to Hardt et al. (2016), we also observed that the empirical stability of SGD for training neural networks is often far better than the worst-case bound described by Theorem 4, see our numerical results in Section 5. This suggests practitioners to still apply the stability bound in Theorem 3, dismissing non-convexity. It is an intriguing but challenging future work to narrow the gap between theory and practice here.
+
+Bagging: Bagging (bootstrap aggregating) (Soloff et al., 2024) is a general framework that averages over  $B$  models trained on resamples of size  $m$  from  $\mathcal{D}$ . Random forest (Wang et al., 2023) is a popular special case of bagging, in which each  $f^{(b)}(x)$  is a regression tree. Therefore, we focus on studying bagging. It predicts by  $f^{B}(x) = \frac{1}{B}\sum_{b=1}^{B} f^{(b)}(x)$ , where  $f^{(b)}(x)$  indicates the individual model trained on the  $b$ th resample. For simplicity, here we analyze a "derandomized bagging" (Soloff et al., 2024), i.e., setting  $B \to \infty$ . The prediction function becomes  $f^{\infty}(x) = \mathbb{E}[f^{(b)}(x)]$ . Below is the LOO stability of derandomized bagging. Here, we denote  $f_{j}^{y,(b)}(x)$  and  $f^{(b)}(x)$  as the individual models obtained from  $\mathcal{D}_{j}^{y}$  and  $\mathcal{D}$ , respectively.
+
+Theorem 5. Assume that 1) for any  $j\in [m]$  and  $(x,y)\in \mathcal{X}\times \mathcal{Y}$ , all individual prediction functions  $f_{j}^{y,(b)}(x)$  and  $f^{(b)}(x)$  are bounded within a range of width  $w_{j}$ ; 2) the nonconformity score  $S(y,z)$  is  $\gamma$ -Lipschitz in  $z$  for any  $y$ . Then, derandomized bagging achieves the following LOO stability bound:
+
+$$
+\tau_ {i, j} ^ {\mathrm {L O O}} = \frac {\gamma w _ {j}}{2} \sqrt {\frac {p}{1 - p}},
+$$
+
+where  $p = 1 - \left(1 - \frac{1}{n}\right)^m$  and  $i\in [n]\cup \{n + j\}$  for  $j = 1,\ldots ,m$
+
+From above, note that the only assumption about the prediction model is bounded output. For example, regression trees satisfy this assumption.
+
+Due to page limit, we relegate more results and discussion to Appendix A.3.
+
+# 4 SIMULATION
+
+In this simulation, we compare several CP methods serving RLM and SGD. We set  $n = m = 100$ ,  $\alpha = 0.1$  and generated synthetic data using  $X_{i} \stackrel{\mathrm{i.i.d.}}{\sim} \mathcal{N}(0, \frac{1}{d}\pmb{\Sigma})$  with  $d = 100$ , where  $\pmb{\Sigma}_{i,j} = \rho^{|i - j|}$  (i.e., AR(1)). In particular, we chose  $\rho = 0.5$  in this experiment. For the response variable we set  $Y_{i} = \mu(X_{i}; \beta) + \epsilon_{i}$ , where  $\epsilon_{i} \stackrel{\mathrm{i.i.d.}}{\sim} \mathcal{N}(0, 1)$ .
+
+We considered two models for  $\mu (\cdot ;\beta)$  : linear  $\mu (x;\beta) = \sum_{j = 1}^{d}\beta_{j}x_{j}$  and nonlinear  $\mu (x;\beta) = \sum_{j = 1}^{d}\beta_{j}e^{x_{j} / 10}$ . In both models, set  $\beta_{j}\propto (1 - j / d)^{5}$  for  $j\in [d]$ , and normalize:  $\| \beta \| _2^2 = d$ . To fit the model, we used robust linear regression, equipped with Huber loss:
+
+$$
+\ell (y, f _ {\theta} (x)) = \left\{ \begin{array}{l l} \frac {1}{2} (y - f _ {\theta} (x)) ^ {2}, & \text {i f} | y - f _ {\theta} (x) | \leq \epsilon , \\ \epsilon | y - f _ {\theta} (x) | - \frac {1}{2} \epsilon^ {2}, & \text {i f} | y - f _ {\theta} (x) | > \epsilon , \end{array} \right.
+$$
+
+where  $f_{\theta}(x) = x^{T}\theta$  and we set  $\epsilon = 1$  throughout. We used absolute residual as non-conformity scores. In RLM, we set  $\Omega(\theta) = \| \theta \|^2$  and solved it using gradient descent (Diamond & Boyd,
+
+2016). Throughout, we ran SGD for  $R = 15$  epochs for all methods, except  $R = 5$  for the very slow FullCP. For both RLM and SGD, we set the learning rate to be  $\eta = 0.001$ . For more implementation details, see Appendix G.1. Each experiment was repeated 100 times.
+
+We compare our method to the following benchmarks: 1) OracleCP (Appendix C.1, Algorithm 4): an impractical method that uses the true  $\{Y_{n + j}\}_{j = 1}^{m}$  to predict; 2) FullCP; 3) SplitCP:  $70\%$  training  $+30\%$  calibration; and 4) RO-StabCP. The performance of each method is evaluated by three measures: 1) coverage probability (method validity); 2) length of predictive interval (prediction accuracy); and 3) computation time (speed).
+
+![](images/08d76949e654ca59dc9e5f980e8ba892b6e0b8489d14f0ce8e7083bba98f98fb.jpg)  
+Figure 1: Comparison of CP methods. Our method (LOO-StabCP) achieves competitive prediction accuracy and computes at the speed comparable to SplitCP, while maintaining coverage validity.
+
+Figure 1 presents the results of our simulation. In the plots for coverage and length, the horizontal dashed lines represent the desired coverage level  $(1 - \alpha)$  and the length of the tightest possible prediction band obtained from the true distribution of the data $^3$ , respectively. As expected, all methods maintain valid coverage. Our method shows competitive prediction accuracy, comparable to those of OracleCP, FullCP, and RO-StabCP. These four methods exhibit more consistent and overall superior accuracy compared to SplitCP. In terms of computational efficiency, our method performs on par with SplitCP and is clearly faster than other methods. Notably, our method significantly outperforms RO-StabCP in handling a large number of prediction requests.
+
+# 5 DATA EXAMPLES
+
+We showcase the use of our method on the two real-world data examples analyzed in Ndiaye (2022). The Boston Housing data (Harrison Jr & Rubinfeld, 1978) contain 506 different areas in Boston, each area has 13 features as predictors, such as the local crime rate and the average number of rooms. The goal is to predict the median house value in that area. The Diabetes data (Efron et al., 2004) measured 442 individuals at their "baseline" time points for 10 variables, including age, BMI, and blood pressure, aiming to predict diabetes progression one year after baseline. Both datasets are complete, with no missing entries. All continuous variables have been normalized, and no outliers were identified.
+
+For each data set, we randomly held out  $m$  data points (as the test data) for performance evaluation and released the rest to all methods for training/calibration. We tested two settings:  $m = 1$  and  $m = 100$ , two model fitting algorithms: RLM and SGD; and repeated each experiment 100 times.
+
+The other configurations, including the model fitted to data  $(\ell, \epsilon, \Omega, \text{etc.})$ , the list of compared benchmarks and the performance measures, all remained unchanged from Section 4.
+
+![](images/b63b0947af1a5037cdf5f70d69a98d10b40cd094b0aa2e64102295fa97c97e71.jpg)  
+Figure 2: Comparison of prediction interval lengths, under choices of  $m = 1$  and  $m = 100$ .
+
+Figure 2 shows the result. While most interpretations are consistent with that of the simulation, we observe two significant differences between the settings  $m = 1$  and  $m = 100$ . First, under  $m = 1$ , RO-StabCP and our method take comparable time, but when  $m$  increases to 100, our method exhibits remarkable speed advantage, as expected. Second, with an increased  $m$ , the amount of available data for prediction/calibration also decreases. This leads to wider prediction intervals for all methods. Also, SplitCP continues to produce more variable and lengthier prediction intervals compared to most other methods for  $m \in \{1, 100\}$ . In summary, our method LOO-StabCP exhibits advantageous performances in all aspects across different settings. The empirical coverage rates are consistent with those in the previous experiments and are provided in Appendix G.3.
+
+![](images/4634226b3f49348fdbdba9f22e609ff7212c29e905ee2c4bf4b3451c19cf2f08.jpg)  
+Figure 3: Comparison of CP methods with neural networks with single hidden layer under choice of  $m = 100$ . LOO-StabCP continues to closely achieve the target coverage while exhibiting lower variability in prediction intervals.
+
+To further evaluate the performance of LOO-StabCP with non-convex learning methods, we conducted experiments with a neural network of a single hidden layer of 20 nodes and a sigmoid activation function. We set  $\eta = 0.001$  and  $R = 30$ . For stability bounds, we borrowed from the practical guidance in Hardt et al. (2016) and Ndiaye (2022) and set  $\tau_{i,j}^{\mathrm{LOO}} \approx R\eta \cdot \gamma \|X_i\| \|X_{n+j}\|$  for LOO-StabCP and  $\tau_{i,j}^{\mathrm{RO}} \approx 2R\eta \cdot \gamma \|X_i\| \|X_{n+j}\|$  for RO-StabCP, respectively. This choice is elaborated in Appendix A.2, see (8). Figure 3 presents the results. LOO-StabCP maintained valid
+
+coverage across all scenarios. These findings highlight the robustness of LOO-StabCP in handling complex models like neural networks.
+
+Finally, since one could consider derandomization by aggregating results across multiple different splits to reduce variation of SplitCP (Solari & Djordjilovic, 2022; Gasparin & Ramdas, 2024), we also numerically compared our method to this approach. The result demonstrated that our LOO-StabCP is computationally faster and less conservative than two popular derandomized SplitCP methods Solari & Djordjilovic (2022); Gasparin & Ramdas (2024). Due to page limit, we relegate all details of this study to Appendix B.
+
+# 6 APPLICATION: CONFORMALIZED SCREENING
+
+Many decision-making processes, such as drug discovery and hiring, often involve a screening stage to filter among a large number of candidates, prior to more resource-intensive stages like clinical trials and on-site interviews. The data structure is what we have been studying in this paper: training data  $\mathcal{D} = \{(X_i,Y_i)\}_{i=1}^n$  and a large number of test points  $\{X_{n+j}\}_{j=1}^m$  without observing  $Y_{n+j}$ 's. Suppose higher values of  $Y$  are of interests. Jin & Candès (2023) formulated this as the following randomized hypothesis testing problem:
+
+$$
+H _ {0 j}: Y _ {n + j} \leq c _ {j} \quad \text {v s} \quad H _ {1 j}: Y _ {n + j} > c _ {j}, \quad \text {f o r} j \in [ m ],
+$$
+
+where  $c_{j}$ 's are user-selected thresholds (e.g., qualifying score for phone interviews). Then screening candidates means simultaneously testing these  $m$  randomized hypotheses. To control for error in multiple testing, one popular criterion is the false discovery rate (FDR), defined as the expected false discovery proportion (FDP) among all rejections.
+
+$$
+\mathrm {F D R} = \mathbb {E} [ \mathrm {F D P} ], \quad \text {w h e r e F D P} = \frac {\sum_ {j = 1} ^ {m} \mathbf {1} \left\{H _ {0 j} \text {i s i n c o r r e c t l y r e j e c t e d} \right\}}{1 \vee \sum_ {j = 1} ^ {m} \mathbf {1} \left\{H _ {0 j} \text {i s r e j e c t e d} \right\}}.
+$$
+
+Jin & Candès (2023) proposed a method called cfBH based on SplitCP. Our narration will build upon non-conformity scores without repeating details about model fitting. In this context, the nonconformity score should be defined differently, for instance:  $S(y,z) = y - z$  without the absolute value, where  $y$  is the observed response and  $z$  is the fitted value. On the calibration data,  $S_{i} = S(Y_{i},\widehat{f}(X_{i}))$ , whereas on the test data, we would consider  $S_{n+j}^{c_{j}} = S(c_{j},\widehat{f}(X_{n+j}))$ . Jin & Candès (2023)'s cfBH method computes the following conformal p-value:
+
+$$
+p _ {j} ^ {\text {s p l i t}} = \frac {\sum_ {i \in \mathcal {I} _ {\mathrm {c a l i b}}} \mathbf {1} \left\{S _ {i} <   S _ {n + j} ^ {c _ {j}} \right\} + 1}{\left| \mathcal {I} _ {\mathrm {c a l i b}} \right| + 1}, \tag {6}
+$$
+
+where  $\mathcal{I}_{\mathrm{calib}}$  denotes the index set corresponding to the calibration data. To intuitively understand (6), notice that  $p_j^{\mathrm{split}} < \alpha$  if and only if  $c_{j}$  falls outside the level-  $(1 - \alpha)$  (one-sided) split conformal prediction interval for  $Y_{n + j}$ . Finally, plugging  $\{p_j^{\mathrm{split}}\}_{j = 1}^m$  into a Benjamini-Hochberg (BH) procedure (Benjamini & Hochberg, 1995) controls the FDR at a desired level  $q$ : compute  $k^{\star} = \max \{k:\sum_{j = 1}^{m}\mathbf{1}\{p_j^{\mathrm{split}}\leq qk / m\} \geq k\}$ , and reject all  $H_{0j}$ 's with  $p_j^{\mathrm{split}} < qk^{\star} / m$ .
+
+While Jin & Candès (2023)'s method effectively controls FDR and computes fast, the data splitting mechanism leaves space for more thoroughly exploiting available information for model fitting. To this end, we propose a new approach, called LOO-cfBH built upon our main method LOO-StabCP. We compute stability-adjusted p-values as follows:
+
+$$
+p _ {j} ^ {\mathrm {L O O}} = \frac {\sum_ {i = 1} ^ {n} \mathbf {1} \left\{S _ {i} - \tau_ {i , j} ^ {\mathrm {L O O}} <   S _ {n + j} ^ {c _ {j}} + \tau_ {n + j , j} ^ {\mathrm {L O O}} \right\} + 1}{n + 1}. \tag {7}
+$$
+
+Algorithm 2 describes the full details of our method.
+
+To numerically compare our method to existing approaches, we used the recruitment data set Ganatara (2020) that was also analyzed in Jin & Candès (2023). It contains 215 individuals, each measured on 12 features such as education, work experience, and specialization. The binary response indicates whether the candidate receives a job offer. We import the robust regression from Section 5 as the prediction method, optimized by SGD. Since the task is classification, we use the
+
+Algorithm 2: (LOO-cfBH) Conformal Selection by Prediction with Leave-One-Out p-values
+
+Input: Training set  $\mathcal{D}$ , test points  $\{X_{n + j}\}_{j = 1}^{m}$ , thresholds  $\{c_j\}_{j = 1}^m$ , FDR level  $q$ . Output: Set of rejected null hypotheses
+
+1. Fit the prediction function  $\widehat{f}$  on  $\mathcal{D}$ ;  
+2. Compute (uncorrected) non-conformity scores on  $\mathcal{D}$ :  $S_{i} = S(Y_{i},\widehat{f} (X_{i}))$  for  $i\in [n]$ ; for  $j\in [m]$  do
+
+3. Compute stability bounds  $\tau_{i,j}^{\mathrm{LOO}}$  for  $i\in [n]\cup \{n + j\}$  
+4. Compute LOO conformal p-values  $p_j^{\mathrm{LOO}}$  as in (7);
+
+# end
+
+5. Implement BH Procedure:  $k^{\star} = \max \{k:\sum_{j = 1}^{m}\mathbf{1}\{p_{j}^{\mathrm{LOO}}\leq qk / m\} \geq k\}$  and reject all  $H_{0j}$ 's satisfying  $p_j^{\mathrm{LOO}} < qk^\star /m$ .
+
+clip function in Jin & Candès (2023) as the non-conformity score:  $S(y, \widehat{f}) = 100y - \widehat{f}$ . For illustration, we also formulated a benchmark RO-cfBH, following the spirit of Ndiaye (2022) and using replace-one stability. RO-cfBH replaces all  $\tau^{\mathrm{LOO}}$  terms in (7) by the corresponding  $\tau^{\mathrm{RO}}$  terms; it is otherwise identical to LOO-cfBH. We repeated the experiment 1000 times, each time leaving out 20% data points as the test data. In cfBH, the data was split into 70% for training and 30% for calibration. We tested three target FDR levels  $q \in \{0.1, 0.2, 0.3\}$  and consider three performance measures: 1) FDP; 2) test power, defined as  $\left( \sum_{j=1}^{m} \mathbf{1}\{H_{0j} \text{ is rejected}\} \right) / \left( \sum_{j=1}^{m} \mathbf{1}\{H_{1j} \text{ is true}\} \right)$ ; and 3) time cost.
+
+![](images/a362cb07e856a7267153397f0740d5bcc754c6f92967746bc854a6ccf8c35422.jpg)  
+Figure 4: Comparison of screening methods on recruitment data. Time cost does not vary with  $q$ .
+
+![](images/5d0b1c0d62fd844268dded7e343a3e267980ec79ff638355e197ad9586030d9c.jpg)
+
+![](images/8120149e43c88f61bc030ad17d9c575eb50d21b67d667b11a48ae69734fbcc79.jpg)
+
+Figure 4 shows the result. Our method achieves valid FDP control for all tested  $q$ . Compared to cfBH, our method is more powerful, due to improved exploitation of available data for prediction. The performance measures also reflect that our method LOO-cfBH produces more stable prediction intervals, while sample splitting introduces additional artificial random variations to the result of cfBH. Compared to RO-cfBH, we highlight our method's significant speed advantage. Moreover, as we showed in Theorem 3, for SGD, our LOO approach achieves a tighter stability bound than RO. As a result, our method is less conservative and more powerful compared to RO-cfBH.
+
+# 7 CONCLUSION AND FUTURE WORK
+
+In this paper, we propose a novel approach to stable conformal prediction. Our method significantly improves computational efficiency for multiple prediction requests, compared to the classical stable conformal prediction (Ndiaye, 2022). Here, we mention three directions for future work. First, while we have derived stability bounds for RLM, SGD, neural networks and bagging, improving the tightness of bounds for complex methods remains an important avenue for future research. Second, we have been focusing on continuous responses. It would be an intriguing future work to expand our theory to classification. A third direction is to investigate algorithmic stability for adaptive conformal prediction.
+
+# SUPPLEMENTAL MATERIALS
+
+The Supplemental Materials contains all proofs and additional numerical results. The code for reproducing numerical results is available at: https://github.com/Kiljael/LOO-StabCP.
+
+# ACKNOWLEDGMENT
+
+The authors thank Eugene Ndiaye for helpful discussion and the anonymous reviewers for constructive criticism. This research was supported by NSF grant DMS-2311109.
+
+# REFERENCES
+
+Anastasios Angelopoulos, Stephen Bates, Jitendra Malik, and Michael I Jordan. Uncertainty sets for image classifiers using conformal prediction. arXiv preprint arXiv:2009.14193, 2020.  
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+Rina Foygel Barber, Emmanuel J. Candès, Aaditya Ramdas, and Ryan J. Tibshirani. Predictive inference with the jackknife+. The Annals of Statistics, 49(1):486 - 507, 2021. doi: 10.1214/20-AOS1965. URL https://doi.org/10.1214/20-AOS1965.  
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+
+# Supplemental Materials for "Leave-one-out Stable Conformal Prediction"
+
+Kiljae Lee and Yuan Zhang
+
+# A DETAILED INSIGHTS ON PRACTICAL EXTENSIONS OF LOO-STABCP
+
+In this appendix, we provide further numerical and theoretical analysis to support the approaches discussed in Section 3.2.3.
+
+# A.1 NUMERICAL EXPERIMENTS USING KERNEL TRICK
+
+The key insight of the kernel trick is that by transforming the data into a higher-dimensional feature space using a kernel function, the original optimization problem
+
+$$
+\widehat {\beta} = \arg \min _ {\beta \in \mathbb {R} ^ {d}} \frac {1}{n} \sum_ {i = 1} ^ {n} \ell (Y _ {i}, X _ {i} ^ {T} \beta) + \lambda \| \beta \| ^ {2},
+$$
+
+can be reformulated as
+
+$$
+\widehat {\theta} = \arg \min _ {\theta \in \mathbb {R} ^ {n}} \frac {1}{n} \sum_ {i = 1} ^ {n} \ell (Y _ {i}, k _ {i} ^ {T} \theta) + \lambda \theta^ {T} \mathbf {K} \theta .
+$$
+
+This reformulation using the kernel trick does not violate the assumptions required for RLM and SGD, as the transformation maintains the core structure of the optimization problem. Specifically, the kernel matrix  $\mathbf{K}$  implicitly defines the high-dimensional feature space through the kernel function  $k(X_{i},X_{j})$ , without requiring explicit computation of the transformed features. This ensures that the problem remains computationally tractable.
+
+For RLM, the regularization term  $\| \beta \| ^2$  in the original formulation translates directly to  $\theta^T\mathbf{K}\theta$  in the kernelized version, preserving the strong convexity of the optimization problem as long as we use a positive definite kernel (e.g. radial basis kernel, polynomial kernel, etc.). Similarly, for SGD, the smoothness and Lipschitz continuity of the loss function are preserved, as the transformation affects only the inner product computations, which is linear, and does not alter the fundamental properties of the objective function. Thus, if our original problem satisfies the conditions of LOO stability of RLM and SGD, the kernel trick enables the model to capture nonlinear patterns in the data while ensuring that the theoretical guarantees remain intact.
+
+By integrating the kernel trick, we revisit the scenarios in Section 4, where we initially considered synthetic data examples using standard robust linear regression methods. For our experiments, we employed the radial basis function (RBF) kernel  $k_{\mathrm{RBF}}(x,x^{\prime}) = \exp \left(-\frac{\|x - x^{\prime}\|^{2}}{2\sigma^{2}}\right)$  and the polynomial kernel  $k_{\mathrm{Poly}}(x,x^{\prime}) = (x^{T}x^{\prime} + c)^{d}$ , both chosen for their ability to model complex nonlinear relationships effectively. For hyperparameters, we chose  $\sigma = 0.1$ ,  $c = 1$ , and  $d = 2$ . We compared these results to the outcomes in Section 4 and this can be theoretically viewed as a special case of kernel robust regression using a linear kernel.
+
+As shown in Figure 5, LOO-StabCP continues to perform reliably under both settings settings without any loss of coverage, validating its adaptability to more sophisticated model structures. Moreover, compared to the linear setting, the use of the kernel trick in nonlinear settings leads to a notable reduction in prediction interval length. This reduction highlights the ability of LOO-StabCP with kernel trick to provide more precise predictions while capturing the complex patterns inherent in data, thereby enhancing its practical utility.
+
+# A.2 DETAILED INSIGHTS INTO NONCONVEX OPTIMIZATION
+
+In Section 3.2.3, we derived stability bounds for SGD under nonconvex settings. Here, we provide additional details on the derivation and implications of these bounds.
+
+![](images/80c2db1795c5d733c4fb3f10ec0042af491c58d0d76b6f89f669616920cbebe2.jpg)
+
+![](images/b3d4619c18feb7ae0898fdb0b8ecc32a15ca219f88ccea8778daf1ee745bdd41.jpg)
+
+![](images/79f8fbc98b3be7bb606e66a45591da0bc6b96a7a14ca01e3eae91287fa2b047f.jpg)
+
+![](images/2cd3063c619d30ec768e01b9ef873f1d3ec21650e5ece9621ed723d54c613bdf.jpg)
+
+![](images/36f55bbc0e11997b54e1b6bccfc27ea2708a1ea71f04d7f702090bddec5e0aa8.jpg)  
+Figure 5: Comparison of CP methods using kernelized technique.
+
+![](images/2ce9ea6edf04045f1c0744da691d5b641eeb0f343553b9eb6f0ec2d3b9abaf48.jpg)
+
+In the convex case (Theorem 3), the stability bounds for SGD are given by:
+
+$$
+\tau_ {i, j} ^ {\mathrm {L O O}} = R \eta \cdot \gamma \nu_ {i} \rho_ {n + j}, \quad \tau_ {i, j} ^ {\mathrm {R O}} = 2 R \eta \cdot \gamma \nu_ {i} \rho_ {n + j}.
+$$
+
+On the other hand, for the nonconvex case (Theorem 4), these bounds are modified to include the term  $R^{+}$ :
+
+$$
+\tau_ {i, j} ^ {\mathrm {L O O}} = R ^ {+} \eta \cdot \gamma \nu_ {i} \rho_ {n + j}, \quad \tau_ {i, j} ^ {\mathrm {R O}} = 2 R ^ {+} \eta \cdot \gamma \nu_ {i} \rho_ {n + j},
+$$
+
+where  $R^{+} = \sum_{r=1}^{R} \kappa^{r}$  and  $\kappa = \prod_{i=1}^{n} (1 + \eta \varphi_{i})$ . Note that the only distinction in the nonconvex case is that  $R^{+}$  replaces  $R$ . Hence,  $R^{+}$  can be interpreted as representing the cumulative effect of nonconvexity. This term is influenced by the learning rate  $\eta$  and the Lipschitz constants of the gradients  $\varphi_{i}$ . Specifically, if  $\kappa \approx 1$ ,  $R^{+}$  approximates  $R$ , aligning with the convex optimization scenario. However, when  $\kappa$  is significantly greater than 1,  $R^{+}$  can grow exponentially with  $R$ , resulting in overly loose bounds. The practical implication of this result is that  $\eta$  and  $\varphi_{i}$  significantly influence the tightness of stability bounds. While smaller values of  $\eta$  can mitigate this issue, they may also slow down convergence, creating a trade-off between theoretical stability and computational efficiency.
+
+As described in Section 5, we conducted experiments with a neural network featuring a single hidden layer and employed approximated stability bounds:
+
+$$
+\tau_ {i, j} ^ {\mathrm {L O O}} \approx R \eta \cdot \gamma \| X _ {i} \| \| X _ {n + j} \| \quad \text {a n d} \quad \tau_ {i, j} ^ {\mathrm {R O}} \approx 2 R \eta \cdot \gamma \| X _ {i} \| \| X _ {n + j} \|. \tag {8}
+$$
+
+Although these terms approximate our problem as if it were convex, they still capture the interaction between the training and test points in our dataset, providing a practical measure of stability. By using these approximations, we adapted our conformal prediction framework without relying on overly conservative worst-case bounds. Alongside the results in Section 5, Figure 6 shows outcomes for a two-hidden-layer network with 10 and 5 nodes, respectively, under the same settings.
+
+Our empirical results, shown in Figure 3 and Figure 6, demonstrate that these approximations, despite their theoretical looseness, do not compromise the validity of Loo-StabCP. These findings are consistent with prior observations (Hardt et al., 2016; Ndiaye, 2022), where theoretical stability bounds in nonconvex settings are often pessimistic, yet empirical results tend to outperform these expectations.
+
+![](images/9e2e04c70b52cefe5f468e6eaf1e5c47e805e49221b38aab38324a8367ae6c9d.jpg)  
+Figure 6: Comparison of CP methods with neural networks with two hidden layers under choice of  $m = 100$ .
+
+# A.3 ADDITIONAL RESULTS AND DISCUSSION ON BAGGING
+
+While derandomized bagging discussed in 3.2.3 provides conceptual insights into stability, practical bagging methods have internal randomness induced by the resampling scheme. To account for this randomness, here, we provide theoretical results on the LOO stability of bagging in practice.
+
+The randomness in bagging is two-fold. One source is the resampling process, where datasets are created by sampling with replacement from the original dataset. Another arises from the base learning algorithm itself, as seen in random forest, where random feature subsets are selected in each bag. The latter can be characterized by a random variable  $\xi \sim \mathcal{U}(0,1)$ . Algorithm 3 illustrates the implementation of bagging.
+
+# Algorithm 3: Bagging
+
+Input : Training set  $\mathcal{D} = \{(X_i,Y_i)\}_{i = 1}^n$  , Number of bags  $B$  , Number of samples in each bag m
+
+Output: Prediction function  $\widehat{f} (\cdot) = f^{B}(\cdot)\coloneqq \frac{1}{B}\sum_{b = 1}^{B}f^{(b)}(\cdot).$
+
+for  $b\in [B]$  do
+
+1. Sample bag  $r^{(b)} = (i_1^{(b)},\dots ,i_m^{(b)})$  where  $i_j^{(b)}\stackrel {\mathrm{i.i.d.}}{\sim}\mathcal{U}([n])$  for  $j\in [m]$  
+2. Sample seed  $\xi^{(b)}\sim \mathcal{U}([0,1])$  
+3. Fit model  $f^{(b)}$  with  $r^{(b)}$  and  $\xi^{(b)}$ ;
+
+# end
+
+The prediction function of bagging is inherently random, making it challenging to derive a deterministic stability bound. Nonetheless, based on Theorem 5, we can deduce with high probability that bagging is LOO stable. In the context of bagging,  $\widehat{f}_j^y (x) = \frac{1}{B}\sum_{b = 1}^{B}f_j^{y,(b)}(x)$  and  $\widehat{f} (x) = \frac{1}{B}\sum_{b = 1}^{B}f^{(b)}(x)$ , where the sample average replaces the expectation compared to derandomized bagging. The following theorem provides a probabilistic guarantee on the LOO stability bounds for bagging.
+
+Theorem 6. Suppose the conditions of Theorem 5 hold. Then, for any  $\delta \in (0,1)$ , bagging has the following LOO stability bounds with probability at least  $1 - \delta$ .
+
+$$
+\tau_ {i, j} ^ {\mathrm {L O O}} (\delta) = \gamma w _ {j} \left\{\frac {1}{2} \sqrt {\frac {p}{1 - p}} + \sqrt {\frac {2}{B} \log \left(\frac {4}{\delta}\right)} \right\}
+$$
+
+with  $p = 1 - (1 - \frac{1}{n})^m$  where  $i$  ranges in  $[n] \cup \{n + j\}$  for each  $1 \leq j \leq m$ .
+
+The implications of Theorem 5 and Theorem 6 are as follows. Note that the above theorem requires only the minimal assumption that the base model fitting algorithm used in bagging has bounded output. This suggests that Loo-StabCP can be applied to a wide range of algorithms. For example, building on the stability of bagging, Wang et al. (2023) extended the results to the stability of random forest. Their key insight was that random forest utilize weak decision trees as their base model fitting algorithm, and the final output of a decision tree is always determined as the average of the responses
+
+in the training data it uses. As a result, it is straightforward to see that the output of a decision tree cannot exceed the range of the responses in the training set. Moving forward, these insights can serve as a foundation for exploring stability guarantees in other complex learning algorithms.
+
+# B COMPARISON WITH DERANDOMIZATION APPROACHES
+
+In this appendix, we extend our numerical experiments to include comparisons with derandomization approaches (Solari & Djordjilovic, 2022; Gasparin & Ramdas, 2024), which are potential alternatives to LOO-StabCP in terms of reducing the variability of SplitCP. Specifically, these methods differ from SplitCP, which relies on a single data split, by merging multiple prediction intervals constructed from various splits into one final prediction interval.
+
+![](images/412056fab7e7b72cb316cd12e8a8bb2fd8a239e9a7a4fd32de7b88bada119203.jpg)  
+Figure 7: Comparison of CP methods including derandomization approaches on synthetic datasets.
+
+Among these, two notable methods have been proposed in the literature. Solari & Djordjilovic (2022) were the first to propose this approach. They generated split conformal prediction intervals with  $1 - \frac{\alpha}{2}$  validity from multiple data splits and then derived a final prediction interval through majority voting (i.e., the range covered by more than half of these intervals). They showed that this final interval maintains  $1 - \alpha$  validity. We refer to their method as MM-SplitCP (Majority Multi-Split Conformal Prediction). Meanwhile, Gasparin & Ramdas (2024) focused on the exchangeability of each prediction interval derived through MM-SplitCP. Building on this property, they proposed an alternative aggregation technique that produces tighter yet still valid prediction intervals by applying a majority vote correction. We denote this method as EM-SplitCP (Exchangeable Multi-Split Conformal Prediction). For further details on these methods, we refer readers to Solari & Djordjilovic (2022); Gasparin & Ramdas (2024).
+
+We compare the performance of these two derandomization techniques with our proposed method, LOO-StabCP. To this end, we applied the methods to the settings described in Section 4 and Section 5. For MM-SplitCP and EM-SplitCP, we merged 30 splits. Figures 7 and 8 present the results on synthetic and real datasets, respectively. From the results, we observe that the variability in coverage and interval length produced by MM-SplitCP and EM-SplitCP is noticeably lower than that of SplitCP, indicating that these derandomization techniques effectively reduce the internal variability of data-splitting approaches.
+
+However, we also find that the average coverage of MM-SplitCP and EM-SplitCP is generally higher than the predetermined level, suggesting that these derandomization techniques tend to produce conservative intervals. Furthermore, both methods require significantly more computational time, which can be attributed to their reliance on multiple model fits, unlike SplitCP and
+
+![](images/2eecb405e5f06d8bb921b1a6aa14ffcdbae1227f5702d1719235f8c81341cc00.jpg)  
+Figure 8: Comparison of CP methods including derandomization approaches on real datasets.
+
+LOO-StabCP, which rely on a single model fit. In contrast, LOO-StabCP produces tight and stable intervals while maintaining reasonable coverage. These results underscore the computational efficiency and precision of LOO-StabCP. These findings are consistent across both synthetic and real data scenarios, showcasing the adaptability and efficiency of LOO-StabCP compared to other derandomization methods.
+
+# C IMPLEMENTATIONS OF CONFORMAL PREDICTION METHODS
+
+# C.1 ORACLE CONFORMAL PREDICTION
+
+Algorithm 4: (OracleCP) Oracle Conformal Prediction Set  
+```txt
+Input : Training set  $\mathcal{D} = \{(X_i,Y_i)\}_{i = 1}^n$  test sets  $\mathcal{D}_{\mathrm{test}} = \{(X_{n + j},Y_{n + j})\}_{j = 1}^{m}$  desired coverage  $1 - \alpha$
+```
+
+for  $j\in [m]$  do  
+```txt
+Output: Prediction interval  $\mathcal{C}_{j,\alpha}^{\mathrm{oracle}}(X_{n + j})$  for each  $j\in [m]$
+```
+
+1. Fit the prediction function  $\widehat{f}_j$  on  $\mathcal{D}_j^{Y_{n + j}}$ ;  
+2. Compute non-conformity scores on  $\mathcal{D}_j^{Y_{n + j}}\colon S_{i,j} = |Y_i - \widehat{f}_j(X_i)|$  for  $i\in [n]\cup \{n + j\}$  
+3. Compute prediction interval:
+
+$$
+\mathcal {C} _ {j, \alpha} ^ {\text {o r a c l e}} \left(X _ {n + j}\right) = \left[ \widehat {f} _ {j} \left(X _ {n + j}\right) \pm \mathcal {Q} _ {1 - \alpha} \left(\left\{S _ {i, j} \right\} _ {i = 1} ^ {n} \cup \left\{S _ {n + j, j} \right\}\right) \right];
+$$
+
+end
+
+# C.2 FULL CONFORMAL PREDICTION
+
+Algorithm 5: (FullCP) Full Conformal Prediction Set  
+Input: Training set  $\mathcal{D} = \{(X_i,Y_i)\}_{i = 1}^n$  , test points  $\{X_{n + j}\}_{j = 1}^{m}$  , search grid  $\mathcal{G}$  , desired coverage  $1 - \alpha$    
+Output: Prediction interval  $\mathcal{C}_{j,\alpha}^{\mathrm{full}}(X_{n + j})$  for each  $j\in [m]$    
+for  $j\in [m]$  do for  $y\in \mathcal{G}$  do 1. Fit the prediction function  $\widehat{f}_j^y$  on  $\mathcal{D}_j^y$  . 2. Compute non-conformity scores on  $\mathcal{D}$  ..  $S_{i,j}^{y} = |Y_{i} - \widehat{f}_{j}^{y}(X_{i})|$  for  $i\in [n]$  . 3. Compute quantile value  $\mathcal{Q}_{1 - \alpha}(\{S_{i,j}^{y}\}_{i = 1}^{n}\cup \{\infty \})$  end 4. Compute prediction interval  $\mathcal{C}_{j,\alpha}^{\mathrm{full}}(X_{n + j}) = \left\{y\in \mathcal{G}:S_{n + j,j}^{y}\leq \mathcal{Q}_{1 - \alpha}(\{S_{i,j}^{y}\}_{i = 1}^{n}\cup \{\infty \})\right\}$  end
+
+# C.3 SPLIT CONFORMAL PREDICTION
+
+Algorithm 6: (SplitCP) Split Conformal Prediction Set  
+Input: Training set  $\mathcal{D}_{\mathrm{train}} = \{(X_i,Y_i)\}_{i\in \mathcal{I}_{\mathrm{train}}}$ , calibration set  $\mathcal{D}_{\mathrm{calib}} = \{(X_i,Y_i)\}_{i\in \mathcal{I}_{\mathrm{calib}}}$ , test points  $\{X_{n + j}\}_{j = 1}^{m}$ , desired coverage  $1 - \alpha$ .  
+Output: Prediction interval  $\mathcal{C}_{j,\alpha}^{\mathrm{split}}(X_{n + j})$  for each  $j\in [m]$ .  
+1. Fit the prediction function  $\widehat{f}_{\mathrm{train}}$  on  $\mathcal{D}_{\mathrm{train}}$ ;  
+2. Compute non-conformity scores on  $\mathcal{D}_{\mathrm{calib}}$ :  $S_{i} = |Y_{i} - \widehat{f}_{\mathrm{train}}(X_{i})|$  for  $i\in \mathcal{I}_{\mathrm{calib}}$ ;  
+for  $j\in [m]$  do  
+3. Compute prediction interval  $\mathcal{C}_{j,\alpha}^{\mathrm{split}}(X_{n + j}) = [\widehat{f}_{\mathrm{train}}(X_{n + j})\pm \mathcal{Q}_{1 - \alpha}(\{S_{i}\}_{i\in \mathcal{I}_{\mathrm{calib}}}\cup \{\infty \})]$ ;  
+end
+
+# C.4 REPLACE-ONE STABLE CONFORMAL PREDICTION
+
+Algorithm 7: (RO-StabCP) Replace-One Stable Conformal Prediction Set  
+Input: Training set  $\mathcal{D} = \{(X_i,Y_i)\}_{i = 1}^n$  test points  $\{X_{n + j}\}_{j = 1}^{m}$  , initial guesses  $\{\widetilde{y}_{n + j}\}_{j = 1}^{m}$  desired coverage  $1 - \alpha$    
+Output: Prediction interval  $\mathcal{C}_{j,\alpha}^{\mathrm{RO}}(X_{n + j})$  for each  $j\in [m]$    
+for  $j\in [m]$  do 1. Fit the prediction function  $\widehat{f}_j$  on  $\mathcal{D}_j^{\widetilde{y}_j}$  . 2. Compute (guessed) non-conformity scores on  $\mathcal{D}$  ..  $S_{i} = |Y_{i} - \widehat{f}_{j}(X_{i})|$  for  $i\in [n]$  . 3. Compute stability bounds  $\tau_{i,j}^{\mathrm{RO}}$  for  $i\in [n]\cup \{n + j\}$  . 4. Compute prediction interval:  $\mathcal{C}_{j,\alpha}^{\mathrm{RO}}(X_{n + j}) = [\widehat{f}_j(X_{n + j})\pm (\mathcal{Q}_{1 - \alpha}(\{S_i + \tau_{i,j}^{\mathrm{RO}}\}_{i = 1}^n\cup \{\infty \}) + \tau_{n + j,j}^{\mathrm{RO}})];$    
+end
+
+# D IMPLEMENTATIONS OF LOO STABLE ALGORITHMS
+
+# D.1 REGULARIZED LOSS MINIMIZATION
+
+Algorithm 8: (RLM) Regularized Loss Minimization  
+```txt
+Input: Training set  $\mathcal{D} = \{(X_i,Y_i)\}_{i = 1}^n$  Output: Prediction function  $\widehat{f} (\cdot) = f_{\widehat{\theta}}(\cdot)$
+```
+
+```txt
+1. Compute optimal parameter  $\widehat{\theta} := \arg \min_{\theta \in \Theta} \left\{ \frac{1}{n} \sum_{i=1}^{n} \ell(Y_i, f_\theta(X_i)) + \Omega(\theta) \right\}$ ;
+```
+
+# D.2 STOCHASTIC GRADIENT DESCENT
+
+Algorithm 9: (SGD) Stochastic Gradient Descent (with Random Reshuffling)  
+```txt
+Input: Training set  $\mathcal{D} = \{(X_i,Y_i)\}_{i = 1}^n$ , number of epochs  $R$ , step size  $\eta$ , initial value  $\theta_0$ . Output: Prediction function  $\widehat{f} (\cdot) = f_{\widehat{\theta}}(\cdot)$ .
+```
+
+1. Initialize parameter  $\theta \coloneqq \theta_0$  
+for  $r\in [R]$  do
+
+2. Sample a permutation  $\pi$  of  $[n]$  uniformly at random; for  $i\in [n]$  do
+
+3. Update parameter  $\theta \coloneqq \theta -\eta \nabla_{\theta}\ell (Y_{\pi_i},f_\theta (X_{\pi_i}))$  end
+
+end
+
+4. Set the final parameter  $\widehat{\theta} \coloneqq \theta$ ;
+
+# E USEFUL LEMMAS
+
+Lemma 1 (Lemma 13.5 in Shalev-Shwartz & Ben-David (2014)). 1. Let  $g, h: \mathbb{R}^p \to \mathbb{R}$  be convex function and  $\lambda$ -strongly convex function, respectively. Then,  $g + h$  is  $\lambda$ -strongly convex. 2. Let  $g: \mathbb{R}^p \to \mathbb{R}$  be  $\lambda$ -strongly convex and  $y$  minimize  $g$ , then, for any  $x$ ,
+
+$$
+g (x) - g (y) \geq \frac {\lambda}{2} \| x - y \| ^ {2}.
+$$
+
+Lemma 2 (Lemma 3.6 in Hardt et al. (2016)). Let  $g: \mathbb{R}^p \to \mathbb{R}$  be a function such that  $\nabla g: \mathbb{R}^p \to \mathbb{R}^p$  is a  $\varphi$ -Lipschitz. Define  $h: \mathbb{R}^p \to \mathbb{R}^p$  such that  $h(\theta) = \theta - \alpha \nabla g(\theta)$  with  $\alpha \leq 2/\varphi$ . Then,  $h$  is  $(1 + \eta \varphi)$ -Lipschitz. If  $g$  is in addition convex,  $h$  is 1-Lipschitz.
+
+# F PROOFS OF THEOREMS
+
+# F.1 PROOF OF THEOREM 1
+
+Proof. By Definition 2, we have  $S_{i,j}^{y} \leq S_{i} + \tau_{i,j}^{\mathrm{LOO}}$ , for  $i \in [n]$  and  $j = [m]$ . Similarly, for any  $j$ , we have  $|y - \widehat{f}(X_{n+j})| - \tau_{n+j,j}^{\mathrm{LOO}} \leq S_{n+j,j}^{y}$ . Therefore, for any  $j$ , the following holds for all  $y$  contained in  $\mathcal{C}_{j,\alpha}^{\mathrm{full}}(X_{n+j})$ :
+
+$$
+\left| y - \widehat {f} (X _ {n + j}) \right| - \tau_ {n + j, j} ^ {\mathrm {L O O}} \leq \mathcal {Q} _ {1 - \alpha} (\{S _ {i} + \tau_ {i, j} ^ {\mathrm {L O O}} \} _ {i = 1} ^ {n} \cup \{\infty \}),
+$$
+
+which is equivalent to
+
+$$
+y \in [ \widehat {f} (X _ {n + j}) \pm (\mathcal {Q} _ {1 - \alpha} (\{S _ {i} + \tau_ {i, j} ^ {\mathrm {L O O}} \} _ {i = 1} ^ {n} \cup \{\infty \}) + \tau_ {n + j, j} ^ {\mathrm {L O O}}) ].
+$$
+
+This directly implies  $\mathcal{C}_{j,\alpha}^{\mathrm{LOO}}(X_{n + j})\supseteq \mathcal{C}_{j,\alpha}^{\mathrm{full}}(X_{n + j})$  and hence
+
+$$
+\mathbb {P} \left(Y _ {n + j} \in \mathcal {C} _ {j, \alpha} ^ {\mathrm {L O O}} \left(X _ {n + j}\right)\right) \geq \mathbb {P} \left(Y _ {n + j} \in \mathcal {C} _ {j, \alpha} ^ {\mathrm {f u l l}} \left(X _ {n + j}\right)\right) \geq 1 - \alpha ,
+$$
+
+for any choice of  $\alpha$ .
+
+# F.2 PROOF OF THEOREM 2
+
+Proof. For  $y \in \mathcal{V}$  and  $j \in [m]$  define  $F_{j}^{y}(\theta) = \frac{1}{n + 1}\{\sum_{i = 1}^{n}\ell(Y_{i},f_{\theta}(X_{i})) + \ell(y,f_{\theta}(X_{n + j}))\} + \Omega(\theta)$  and  $F(\theta) = \frac{1}{n}\{\sum_{i = 1}^{n}\ell(Y_{i},f_{\theta}(X_{i}))\} + \Omega(\theta)$ . Then,  $\widehat{\theta}_{j}^{y} = \arg \min_{\theta \in \Theta}F_{j}^{y}(\theta)$  and  $\widehat{\theta} = \arg \min_{\theta \in \Theta}F(\theta)$ .
+
+We begin by proving the LOO algorithmic stability. Fix  $y$  and  $j$  and suppose that  $\| \widehat{\theta}_j^y -\widehat{\theta}\| \leq \frac{2(\rho_{n + j} + \bar{\rho})}{\lambda(n + 1)}$ . Then, for all  $i\in [n]\cup \{n + j\}$  and  $z\in \mathcal{V}$ ,
+
+$$
+\begin{array}{l} | S (z, \widehat {f} _ {j} ^ {y} (X _ {i})) - S (z, \widehat {f} (X _ {i})) | \leq \gamma | \widehat {f} _ {j} ^ {y} (X _ {i}) - \widehat {f} (X _ {i}) | \\ = \gamma | f _ {\widehat {\theta} _ {j} ^ {y}} (X _ {i}) - f _ {\widehat {\theta}} (X _ {i}) | \\ \leq \gamma \nu_ {i} \| \widehat {\theta} _ {j} ^ {y} - \widehat {\theta} \| \\ \leq \frac {2 \gamma v _ {i} (\rho_ {n + j} + \bar {\rho})}{\lambda (n + 1)}. \\ \end{array}
+$$
+
+The first and the second inequalities follow from the Lipschitz property of non-conformity score function and the prediction function, respectively. Therefore, it suffices to obtain the bound of  $\|\widehat{\theta}^y - \widehat{\theta}\|$  as assumed above. By the first part of Lemma 1,  $F_j^y$  and  $F$  are  $\lambda$ -strongly convex functions of  $\theta$ . Using the second part of the Lemma 1, we have:
+
+$$
+\begin{array}{l} \frac {\lambda}{2} \left\| \widehat {\theta} - \widehat {\theta} _ {j} ^ {y} \right\| ^ {2} \leq F _ {j} ^ {y} (\widehat {\theta}) - F _ {j} ^ {y} \left(\widehat {\theta} _ {j} ^ {y}\right) \\ = \frac {n}{n + 1} \left\{F (\widehat {\theta}) - F \left(\widehat {\theta} _ {j} ^ {y}\right) \right\} \\ + \frac {1}{n + 1} \left\{\ell \left(y, f _ {\widehat {\theta}} \left(X _ {n + j}\right)\right) - \ell \left(y, f _ {\widehat {\theta} _ {j} ^ {y}} \left(X _ {n + j}\right)\right) + \Omega (\widehat {\theta}) - \Omega \left(\widehat {\theta} _ {j} ^ {y}\right) \right\} \tag {9} \\ \leq \frac {1}{n + 1} \{\ell (y, f _ {\widehat {\theta}} (X _ {n + j})) - \ell (y, f _ {\widehat {\theta} _ {j} ^ {y}} (X _ {n + j})) + \Omega (\widehat {\theta}) - \Omega (\widehat {\theta} _ {j} ^ {y}) \}. \\ \end{array}
+$$
+
+The last inequality follows from the optimality of  $\widehat{\theta}$ . Now, by the Lipschitz property of the loss function, we have:
+
+$$
+\ell \left(y, f _ {\widehat {\theta}} \left(X _ {n + j}\right)\right) - \ell \left(y, f _ {\widehat {\theta} _ {j} ^ {y}} \left(X _ {n + j}\right)\right) \leq \rho_ {n + j} \| \widehat {\theta} - \widehat {\theta} _ {j} ^ {y} \|. \tag {10}
+$$
+
+On the other hand, again by the optimality of  $\widehat{\theta}$ , it holds that
+
+$$
+0 \leq F (\widehat {\theta} _ {j} ^ {y}) - F (\widehat {\theta}) = \frac {1}{n} \sum_ {i = 1} ^ {n} \{\ell (Y _ {i}, f _ {\widehat {\theta} _ {j} ^ {y}} (X _ {i})) - \ell (Y _ {i}, f _ {\widehat {\theta}} (X _ {i})) + \Omega (\widehat {\theta} ^ {y}) - \Omega (\widehat {\theta}),
+$$
+
+which implies
+
+$$
+\Omega (\widehat {\theta}) - \Omega \left(\widehat {\theta} _ {j} ^ {y}\right) \leq \bar {\rho} \| \widehat {\theta} _ {j} ^ {y} - \widehat {\theta} \|, \tag {11}
+$$
+
+by the Lipschitz property of the loss function. Finally, combining (10) and (11) to (9), we get  $\| \widehat{\theta}_j^y -\widehat{\theta}\| \leq \frac{2(\rho_{n + j} + \bar{\rho})}{\lambda(n + 1)}.$
+
+For the RO algorithmic stability, fix  $y$ ,  $\widetilde{y}$ , and  $j$ . By the similar arguments as for (9), we have
+
+$$
+\begin{array}{l} \frac {\lambda}{2} \| \widehat {\theta_ {j} ^ {\widetilde {y}}} - \widehat {\theta} _ {j} ^ {y} \| ^ {2} \leq \frac {1}{n + 1} \left\{\ell \left(y, f _ {\widehat {\theta} _ {j} ^ {\widetilde {y}}} \left(X _ {n + j}\right)\right) - \ell \left(y, f _ {\widehat {\theta} _ {j} ^ {y}} \left(X _ {n + j}\right)\right) \right\} \\ + \frac {1}{n + 1} \left\{\ell \left(\widetilde {y}, f _ {\widehat {\theta} _ {j} ^ {y}} \left(X _ {n + j}\right)\right) - \ell \left(\widetilde {y}, f _ {\widehat {\theta} _ {j} ^ {\widetilde {y}}} \left(X _ {n + j}\right)\right) \right\} \\ \leq \frac {2 \rho_ {n + j}}{n + 1} \| \widehat {\theta} _ {j} ^ {y} - \widehat {\theta} _ {j} ^ {\widetilde {y}} \|, \\ \end{array}
+$$
+
+and this implies  $\| \widehat{\theta}_j^y -\widehat{\theta}_j^y\| \leq \frac{4\rho_{n + j}}{\lambda(n + 1)}$ . The rest of the proof is similar to the previous case.
+
+# F.3 PROOF OF THEOREM 3
+
+Proof. We start with proving the case of RO algorithmic stability first, for clearer presentation. Also, we only prove the case of  $j = 1$  and  $R = 1$  since extending to the case of  $j > 1$  or  $R > 1$  is straightforward. Let  $\pi$  be an arbitrary permutation of  $[n + 1]$  and  $k$  be such that  $\pi_k = n + 1$ . Fix  $y$  and  $\widetilde{y}$ . Let  $(\theta_0^y, \theta_1^y, \ldots, \theta_{n + 1}^y)$  and  $(\theta_0^{\widetilde{y}}, \theta_1^{\widetilde{y}}, \ldots, \theta_{n + 1}^{\widetilde{y}})$  be the updating sequences of SGD sharing  $\pi$  for  $\mathcal{D}_j^y$  and  $\mathcal{D}_j^{\widetilde{y}}$ , respectively. Note that  $\widehat{f}_j^y = \widehat{f}_1^y = f_{\theta_{n + 1}^y}$  and  $\widehat{f}_j^{\widetilde{y}} = \widehat{f}_1^{\widetilde{y}} = f_{\theta_{n + 1}^{\widetilde{y}}}$ . As in the proof of Theorem 2, we first bound the distance between the two terminal parameters,  $\| \theta_{n + 1}^y - \theta_{n + 1}^{\widetilde{y}} \|$ .
+
+Let us first consider the case of  $k = n + 1$ . Then, by the SGD update rule, we can see that  $\theta_i^y = \theta_i^{\overline{y}}$  for all  $i = [n]$  since for SGD update, the two sequences share the first  $n$  data points as well as the initial parameter. Therefore, we have
+
+$$
+\begin{array}{l} \| \theta_ {n + 1} ^ {y} - \theta_ {n + 1} ^ {\widetilde {y}} \| = \| \{\theta_ {n} ^ {y} - \eta \nabla_ {\theta} \ell (y, f _ {\theta_ {n} ^ {y}} (X _ {n + 1})) \} - \{\theta_ {n} ^ {\widetilde {y}} - \eta \nabla_ {\theta} \ell (\widetilde {y}, f _ {\theta_ {n} ^ {\widetilde {y}}} (X _ {n + 1})) \} \| \\ \leq \eta \| \nabla_ {\theta} \ell \left(y, f _ {\theta_ {n} ^ {y}} \left(X _ {n + 1}\right)\right) \| + \eta \| \nabla_ {\theta} \ell \left(\widetilde {y}, f _ {\theta_ {n} ^ {\widetilde {y}}} \left(X _ {n + 1}\right)\right) \| \\ \leq 2 \eta \rho_ {n + 1}. \\ \end{array}
+$$
+
+Here, we used triangle inequality, and then the Lipschitz property of loss function.
+
+Now, consider the case of  $k < n + 1$ . If  $i = k$ , by the similar argument, we can show that  $\| \theta_i^y - \theta_i^{\widetilde{y}} \| \leq \| \theta_{i-1}^y - \theta_{i-1}^{\widetilde{y}} \| + 2\eta \rho_{n+1}$ . Otherwise, if  $i \neq k$ , by Lemma 2 with the choice of  $\alpha \coloneqq \eta$ ,  $\varphi \coloneqq \varphi_{\pi_i} \leq \frac{2}{\eta}$ , and  $g(\theta) \coloneqq \ell(Y_{\pi_i}, f_\theta(X_{\pi_i}))$ , we have:
+
+$$
+\begin{array}{l} \left\| \theta_ {i} ^ {y} - \theta_ {i} ^ {\widetilde {y}} \right\| = \left\| \left\{\theta_ {i - 1} ^ {y} - \eta \nabla_ {\theta} \ell \left(Y _ {\pi_ {i}}, f _ {\theta_ {i} ^ {y}} \left(X _ {\pi_ {i}}\right)\right) \right\} - \left\{\theta_ {i - 1} ^ {\widetilde {y}} - \eta \nabla_ {\theta} \ell \left(Y _ {\pi_ {i}}, f _ {\theta_ {i} ^ {\widetilde {y}}} \left(X _ {\pi_ {i}}\right)\right) \right\} \right\| \tag {12} \\ \leq \left\| \theta_ {i} ^ {y} - \theta_ {i} ^ {\bar {y}} \right\|, \\ \end{array}
+$$
+
+since  $\ell(Y_{\pi_i}, f_\theta(X_{\pi_i}))$  is convex. Unraveling the recursion from the above two inequalities, we get  $\| \theta_{n+1}^y - \theta_{n+1}^{\widetilde{y}} \| \leq \| \theta_0^y - \theta_0^{\widetilde{y}} \| + 2\eta \rho_{n+1} = 2\eta \rho_{n+1}$ . The last equality holds since the two updating sequences share the common initial value. The remaining parts follow similarly to the proof of Theorem 2.
+
+Next, to prove the LOO algorithmic stability, fix  $y$  and let  $\pi' = (\pi_1', \ldots, \pi_n')$  be the sequence obtained from  $\pi$  by excluding the  $k$ th entry. For example, if we choose  $n = 4$  and  $\pi = (3, 2, 5, 1, 4)$ , then  $k = 3$  and  $\pi' = (3, 2, 1, 4)$ . Then, it can be shown that  $\pi'$  is an arbitrary permutation of  $[n]$ . Define an updating sequence of SGD,  $(\theta_0, \theta_1, \dots, \theta_n)$  for  $\mathcal{D}$  induced by  $\pi'$ , i.e.,  $\widehat{f} = f_{\theta_n}$ . Note that  $\theta_0 = \theta_0^y$ . As the case of the RO algorithmic stability, it suffices to show that  $\| \theta_{n+1}^y - \theta_n \| \leq \eta \rho_{n+1}$ .
+
+If  $k = n + 1$ , then we have  $\theta_i^y = \theta_i$  for  $i = [n]$ . Therefore, it follows that
+
+$$
+\begin{array}{l} \left\| \theta_ {n + 1} ^ {y} - \theta_ {n} \right\| = \left\| \left\{\theta_ {n} ^ {y} - \eta \nabla_ {\theta} \ell \left(y, f _ {\theta_ {n} ^ {y}} \left(X _ {n + 1}\right)\right) \right\} - \theta_ {n} \right\| \\ \leq \eta \| \nabla_ {\theta} \ell (y, f _ {\theta_ {n} ^ {y}} (X _ {n + 1})) \| \\ \leq \eta \rho_ {n + j}. \\ \end{array}
+$$
+
+For the case of  $k < n + 1$  and further remaining parts, we can follow the same procedure used in the RO algorithmic stability.
+
+# F.4 PROOF OF THEOREM 4
+
+Proof. The overall structure of the proof is almost identical to that of Theorem 3. Again, let us focus on the proof of RO stability with  $R = 1$  first. Recall that in that proof, the convexity assumption was used only in (12). Since we have discarded the convexity assumption of  $\ell(Y_{\pi_i}, f_\theta(X_{\pi_i}))$  by Lemma 2 again, the Lipschitz constant of  $h(\theta) = \theta - \eta \nabla_\theta \ell(Y_{\pi_i}, f_\theta(X_{\pi_i}))$  is replaced from 1 to  $1 + \eta \varphi_{\pi_i}$ . That is, we obtain the following recursive inequalities:
+
+$$
+\left\| \theta_ {i} ^ {y} - \theta_ {i} ^ {\widetilde {y}} \right\| \leq \left\{ \begin{array}{l l} \| \theta_ {i - 1} ^ {y} - \theta_ {i - 1} ^ {\widetilde {y}} \| + 2 \eta \rho_ {n + 1} & \text {i f} \quad i = k, \\ (1 + \eta \varphi_ {\pi_ {i}}) \| \theta_ {i - 1} ^ {y} - \theta_ {i - 1} ^ {\widetilde {y}} \| & \text {i f} \quad i <   k. \end{array} \right. \tag {13}
+$$
+
+Considering  $\| \theta_0^y - \theta_0^{\widetilde{y}} \| = 0$ , unraveling these inequalities yields
+
+$$
+\begin{array}{l} \left\| \theta_ {n + 1} ^ {y} - \theta_ {n + 1} ^ {\widetilde {y}} \right\| \leq \left(\prod_ {i = k + 1} ^ {n + 1} \left(1 + \eta \varphi_ {\pi_ {i}}\right)\right) \cdot 2 \eta \rho_ {n + j} \\ \leq \left(\prod_ {i = 1} ^ {n + 1} \left(1 + \eta \varphi_ {\pi_ {i}}\right)\right) \cdot 2 \eta \rho_ {n + j} \\ = \left(\prod_ {i = 1} ^ {n + 1} (1 + \eta \varphi_ {i})\right) \cdot 2 \eta \rho_ {n + j} \\ = 2 \kappa \eta \rho_ {n + j}, \\ \end{array}
+$$
+
+since  $\eta \varphi_{i} > 0$  by definitions. Extending this to the case of  $R > 1$  is not as straightforward as the proof of Theorem 3, hence we also present the corresponding proof. In this case, we can use induction. Set  $R > 1$  and let  $r\in [R]$ . Suppose that up to  $(r - 1)$ th epoch, the difference of parameter is bounded by  $2\left(\sum_{s = 1}^{r - 1}\kappa^s\right)\eta \rho_{n + j}$ . Then,  $r$ th iteration can be treated as the case of  $R = 1$  with  $\| \theta_0^y -\theta_0^{\widetilde{y}}\| \leq 2\left(\sum_{s = 1}^{r - 1}\kappa^s\right)\eta \rho_{n + j}$ . In this case, unraveling (13) yileds
+
+$$
+\begin{array}{l} \left\| \theta_ {n + 1} ^ {y} - \theta_ {n + 1} ^ {\widetilde {y}} \right\| \leq \left(\prod_ {i = k + 1} ^ {n + 1} \left(1 + \eta \varphi_ {\pi_ {i}}\right)\right) \left[ \left(\prod_ {i = 1} ^ {k - 1} \left(1 + \eta \varphi_ {\pi_ {i}}\right)\right) \| \theta_ {0} ^ {y} - \theta_ {0} ^ {\widetilde {y}} \| + 2 \eta \rho_ {n + j} \right] \\ \leq \left(\prod_ {i = k + 1} ^ {n + 1} \left(1 + \eta \varphi_ {\pi_ {i}}\right)\right) \left[ \left(\prod_ {i = 1} ^ {k - 1} \left(1 + \eta \varphi_ {\pi_ {i}}\right)\right) 2 \left(\sum_ {s = 1} ^ {r - 1} \kappa^ {s}\right) \eta \rho_ {n + j} + 2 \eta \rho_ {n + j} \right] \\ \leq 2 \eta \rho_ {n + j} \left[ \left(\prod_ {i \neq k} (1 + \eta \varphi_ {\pi_ {i}})\right) \left(\sum_ {s = 1} ^ {r - 1} \kappa^ {s}\right) + \left(\prod_ {i = k + 1} ^ {n + 1} (1 + \eta \varphi_ {\pi_ {i}})\right) \right] \\ \leq 2 \kappa \eta \rho_ {n + j} \left[ \left(\sum_ {s = 1} ^ {r - 1} \kappa^ {s}\right) + 1 \right] \\ = 2 \left(\sum_ {s = 1} ^ {r} \kappa^ {s}\right) \eta \rho_ {n + j}. \\ \end{array}
+$$
+
+Since we already proved the case of  $r = 1$ , this completes proof for RO stability. For the LOO stability, we can use the same reasoning.
+
+# F.5 PROOF OF THEOREM 5
+
+Proof. Fix  $(x,y) \in \mathcal{X} \times \mathcal{Y}$  and  $j \in [m]$ . Due to the symmetry of the resampling scheme, i.e., sampling uniformly with replacement, we have
+
+$$
+\widehat {f} (x) = \mathbb {E} \left[ f _ {j} ^ {y, (b)} (x) \Big | n + 1 \notin r \right].
+$$
+
+Therefore, using the above facts along with the Lipschitz property of the non-conformity score function, we get
+
+$$
+\begin{array}{l} \left| S (z, \widehat {f} _ {j} ^ {y} (x)) - S (z, \widehat {f} (x)) \right| \leq \gamma \left| \widehat {f} _ {j} ^ {y} (x) - \widehat {f} (x) \right| \\ = \gamma \left| \mathbb {E} \left[ f _ {j} ^ {y, (b)} (x) \right] - \mathbb {E} \left[ f _ {j} ^ {y, (b)} (x) \mid n + 1 \notin r \right] \right| \\ = \gamma \left| \mathbb {E} \left[ f _ {j} ^ {y, (b)} (x) - \mathbb {E} \left[ f _ {j} ^ {y, (b)} (x) \right] \mid n + 1 \notin r \right] \right|. \\ \end{array}
+$$
+
+Next, by the definitions of conditional expectation and covariance,
+
+$$
+\begin{array}{l} \mathbb {E} \left[ f _ {j} ^ {y, (b)} (x) - \mathbb {E} \left[ f _ {j} ^ {y, (b)} (x) \right] | n + 1 \notin r \right] = \frac {1}{\mathbb {P} (n + 1 \notin r)} \mathbb {E} \left[ \left\{f _ {j} ^ {y, (b)} (x) - \mathbb {E} \left[ f _ {j} ^ {y, (b)} (x) \right] \right\} \mathbf {1} \{n + 1 \notin r \} \right] \\ = \frac {1}{\mathbb {P} (n + 1 \notin r)} \operatorname {C o v} \left(f _ {j} ^ {y, (b)} (x), \mathbf {1} \{n + 1 \notin r \}\right). \\ \end{array}
+$$
+
+Combining the above results, we have
+
+$$
+\left| S \left(z, \widehat {f} _ {j} ^ {y} (x)\right) - S \left(z, \widehat {f} (x)\right) \right| \leq \frac {1}{1 - p} \left| \operatorname {C o v} \left(f _ {j} ^ {y, (b)} (x), \mathbf {1} \{n + 1 \notin r \}\right) \right|, \tag {14}
+$$
+
+where  $p = \mathbb{P}(n + 1\in r) = 1 - (1 - \frac{1}{n})^m$  . Furthermore, it holds that
+
+$$
+\begin{array}{l} \left| \operatorname {C o v} \left(f _ {j} ^ {y, (b)} (x), \mathbf {1} \{n + 1 \notin r \}\right) \right| \leq \left[ \operatorname {V a r} \left(f _ {j} ^ {y, (b)} (x)\right) \operatorname {V a r} \left(\mathbf {1} \{n + 1 \notin r \}\right) \right] ^ {\frac {1}{2}} \\ \leq \sqrt {\frac {w _ {j} ^ {2}}{4} p (1 - p)} \\ = \frac {w _ {j}}{2} \sqrt {p (1 - p)}. \\ \end{array}
+$$
+
+Here, the first inequality follows from the Cauchy-Schwarz inequality. For the second inequality, we apply Popoviciu's inequality for variance and the properties of the Bernoulli distribution. Substituting this bound into (14) completes the proof.
+
+# F.6 PROOF OF THEOREM 6
+
+Proof. Fix  $(x,y) \in \mathcal{X} \times \mathcal{Y}$  and  $j \in [m]$ . Let  $\widehat{f}_j^y(x)$  and  $\widehat{f}(x)$  denote the predictions corresponding to bagging, and let  $\widehat{f}_j^{y,\infty}(x)$  and  $\widehat{f}^\infty(x)$  denote the predictions corresponding to derandomized bagging. Then,
+
+$$
+\begin{array}{l} | S (z, \widehat {f} _ {j} ^ {y} (x)) - S (z, \widehat {f} (x)) | \leq \gamma \left| \widehat {f} _ {j} ^ {y} (x) - \widehat {f} (x) \right| \\ = \gamma \left| \left\{\widehat {f} _ {j} ^ {y} (x) - \widehat {f} _ {j} ^ {y, \infty} (x) \right\} + \left\{\widehat {f} _ {j} ^ {y, \infty} (x) - \widehat {f} ^ {\infty} (x) \right\} + \left\{\widehat {f} ^ {\infty} (x) - \widehat {f} (x) \right\} \right| \\ \leq \gamma \left[ \left| \widehat {f} _ {j} ^ {y} (x) - \widehat {f} _ {j} ^ {y, \infty} (x) \right| + \left| \widehat {f} _ {j} ^ {y, \infty} (x) - \widehat {f} ^ {\infty} (x) \right| + \left| \widehat {f} ^ {\infty} (x) - \widehat {f} (x) \right| \right]. \tag {15} \\ \end{array}
+$$
+
+Consider each term on the last line of (15). For the first term, note that
+
+$$
+\left| \widehat {f} _ {j} ^ {y} (x) - \widehat {f} _ {j} ^ {y, \infty} (x) \right| = \left| \frac {1}{B} \sum_ {b = 1} ^ {B} f _ {j} ^ {y, (b)} (x) - \mathbb {E} \left[ f _ {j} ^ {y, (b)} (x) \right] \right|.
+$$
+
+Since each single prediction  $f_{j}^{y,(b)}(x)$  is almost surely bounded within an interval of range  $w_{j}$ , by Hoeffding's inequality, we have
+
+$$
+\mathbb {P} \left(\left| \widehat {f} _ {j} ^ {y} (x) - \widehat {f} _ {j} ^ {y, \infty} (x) \right| \leq t\right) \geq 1 - 2 \exp \left(- 2 B t ^ {2} / w _ {j} ^ {2}\right),
+$$
+
+for any  $t > 0$ . Setting  $\frac{\delta}{2} = 2\exp(-2Bt^2 / w_j^2)$  yields
+
+$$
+\mathbb {P} \left(\left| \widehat {f} _ {j} ^ {y} (x) - \widehat {f} _ {j} ^ {y, \infty} (x) \right| \leq \sqrt {\frac {w _ {j} ^ {2}}{2 B} \log \left(\frac {4}{\delta}\right)}\right) \geq 1 - \frac {\delta}{2}.
+$$
+
+Similarly, for the third term, we obtain an identical bound:
+
+$$
+\mathbb {P} \left(\left| \widehat {f} (x) - \widehat {f} ^ {\infty} (x) \right| \leq \sqrt {\frac {w _ {j} ^ {2}}{2 B} \log \left(\frac {4}{\delta}\right)}\right) \geq 1 - \frac {\delta}{2}.
+$$
+
+For the second term, a direct application of Theorem 5 gives the following deterministic bound:
+
+$$
+\left| \widehat {f} _ {j} ^ {y, \infty} (x) - \widehat {f} ^ {\infty} (x) \right| \leq \frac {\gamma w _ {j}}{2} \sqrt {\frac {p}{1 - p}}.
+$$
+
+Combining all the bounds for the three terms in (15) using the union bound, we have that, with probability at least  $1 - \delta$ ,
+
+$$
+\begin{array}{l} | S (z, \widehat {f} _ {j} ^ {y} (x)) - S (z, \widehat {f} (x)) | \leq \gamma \left[ \sqrt {\frac {w _ {j} ^ {2}}{2 B} \log \left(\frac {4}{\delta}\right)} + \frac {w _ {j}}{2} \sqrt {\frac {p}{1 - p}} + \sqrt {\frac {w _ {j} ^ {2}}{2 B} \log \left(\frac {4}{\delta}\right)} \right] \\ = \gamma w _ {j} \left\{\frac {1}{2} \sqrt {\frac {p}{1 - p}} + \sqrt {\frac {2}{B} \log \left(\frac {4}{\delta}\right)} \right\}. \\ \end{array}
+$$
+
+![](images/b7a1401927c32f77a71588054c55679ae50a4d41b4d284209d3eaf10c7b5afd9.jpg)
+
+# G DETAILS OF NUMERICAL EXPERIMENTS
+
+# G.1 DETAILS OF ALGORITHMS
+
+The configurations satisfy the assumptions of Theorem 2 and Theorem 3, allowing us to compute the stability bounds concretely. First, for RLM, the following stability bounds were used. For  $i \in [n] \cup \{n + j\}$  for each  $j \in [m]$ ,
+
+$$
+\tau_ {i, j} ^ {\mathrm {L O O}} = \frac {2 \epsilon \| X _ {i} \|}{\lambda (n + 1)} \left(\| X _ {n + j} \| + \frac {1}{n} \sum_ {i = 1} ^ {n} \| X _ {i} \|\right), \quad \tau_ {i, j} ^ {\mathrm {R O}} = \frac {4 \epsilon \| X _ {i} \| \| X _ {n + j} \|}{\lambda (n + 1)}.
+$$
+
+Next, the stability bounds for SGD are as follows:
+
+$$
+\tau_ {i, j} ^ {\mathrm {L O O}} = R \eta \epsilon \| X _ {i} \| \| X _ {n + j} \|, \quad \tau_ {i, j} ^ {\mathrm {R O}} = 2 R \eta \epsilon \| X _ {i} \| \| X _ {n + j} \|.
+$$
+
+# G.2 ADDITIONAL RESULTS FROM SECTION 4
+
+<table><tr><td colspan="3"></td><td>OracleCP</td><td>FullCP</td><td>SplitCP</td><td>RO-StabCP</td><td>LOO-StabCP</td></tr><tr><td rowspan="6">Linear</td><td rowspan="3">RLM</td><td>Coverage</td><td>0.903 (0.040)</td><td>0.896 (0.043)</td><td>0.903 (0.060)</td><td>0.910 (0.039)</td><td>0.910 (0.039)</td></tr><tr><td>Length</td><td>3.272 (0.250)</td><td>3.300 (0.257)</td><td>3.455 (0.514)</td><td>3.442 (0.257)</td><td>3.442 (0.257)</td></tr><tr><td>Time</td><td>3.201 (0.172)</td><td>176.783 (15.704)</td><td>0.017 (0.006)</td><td>3.190 (0.176)</td><td>0.035 (0.008)</td></tr><tr><td rowspan="3">SGD</td><td>Coverage</td><td>0.903 (0.041)</td><td>0.896 (0.043)</td><td>0.900 (0.059)</td><td>0.911 (0.040)</td><td>0.906 (0.040)</td></tr><tr><td>Length</td><td>3.252 (0.250)</td><td>3.300 (0.257)</td><td>3.420 (0.557)</td><td>3.464 (0.259)</td><td>3.405 (0.259)</td></tr><tr><td>Time</td><td>0.720 (0.087)</td><td>19.320 (2.018)</td><td>0.005 (0.003)</td><td>0.720 (0.075)</td><td>0.009 (0.005)</td></tr><tr><td rowspan="6">Nonlinear</td><td rowspan="3">RLM</td><td>Coverage</td><td>0.892 (0.045)</td><td>0.886 (0.047)</td><td>0.893 (0.059)</td><td>0.897 (0.044)</td><td>0.897 (0.044)</td></tr><tr><td>Length</td><td>3.659 (0.317)</td><td>3.690 (0.340)</td><td>3.812 (0.554)</td><td>3.828 (0.344)</td><td>3.827 (0.344)</td></tr><tr><td>Time</td><td>3.289 (0.219)</td><td>163.101 (22.120)</td><td>0.017 (0.005)</td><td>3.275 (0.221)</td><td>0.038 (0.010)</td></tr><tr><td rowspan="3">SGD</td><td>Coverage</td><td>0.892 (0.045)</td><td>0.886 (0.047)</td><td>0.895 (0.062)</td><td>0.900 (0.043)</td><td>0.894 (0.044)</td></tr><tr><td>Length</td><td>3.641 (0.318)</td><td>3.690 (0.340)</td><td>3.855 (0.612)</td><td>3.849 (0.345)</td><td>3.789 (0.345)</td></tr><tr><td>Time</td><td>0.732 (0.074)</td><td>17.425 (2.206)</td><td>0.005 (0.003)</td><td>0.746 (0.093)</td><td>0.009 (0.005)</td></tr></table>
+
+Table 2: Mean (and standard deviation) of empirical coverage, average prediction interval length, and execution time across 100 iterations for each scenario in simulation.
+
+G.3 ADDITIONAL RESULTS FROM SECTION 5  
+
+<table><tr><td colspan="2"></td><td colspan="2">OracleCP</td><td>FullCP</td><td>SplitCP</td><td>RO-StabCP</td><td>LOO-StabCP</td></tr><tr><td rowspan="4">m=1</td><td rowspan="2">Boston</td><td>RLM</td><td>0.920 (0.273)</td><td>0.910 (0.288)</td><td>0.910 (0.288)</td><td>0.920 (0.273)</td><td>0.920 (0.273)</td></tr><tr><td>SGD</td><td>0.900 (0.302)</td><td>0.920 (0.273)</td><td>0.910 (0.288)</td><td>0.900 (0.302)</td><td>0.900 (0.302)</td></tr><tr><td rowspan="2">Diabetes</td><td>RLM</td><td>0.910 (0.288)</td><td>0.900 (0.302)</td><td>0.890 (0.314)</td><td>0.910 (0.288)</td><td>0.910 (0.288)</td></tr><tr><td>SGD</td><td>0.920 (0.273)</td><td>0.910 (0.288)</td><td>0.920 (0.273)</td><td>0.930 (0.256)</td><td>0.920 (0.273)</td></tr><tr><td rowspan="4">m=100</td><td rowspan="2">Boston</td><td>RLM</td><td>0.906 (0.031)</td><td>0.897 (0.031)</td><td>0.898 (0.040)</td><td>0.905 (0.030)</td><td>0.905 (0.030)</td></tr><tr><td>SGD</td><td>0.905 (0.029)</td><td>0.901 (0.032)</td><td>0.901 (0.036)</td><td>0.910 (0.028)</td><td>0.906 (0.029)</td></tr><tr><td rowspan="2">Diabetes</td><td>RLM</td><td>0.900 (0.035)</td><td>0.889 (0.037)</td><td>0.894 (0.043)</td><td>0.900 (0.035)</td><td>0.900 (0.035)</td></tr><tr><td>SGD</td><td>0.902 (0.031)</td><td>0.890 (0.036)</td><td>0.902 (0.037)</td><td>0.914 (0.030)</td><td>0.906 (0.031)</td></tr></table>
+
+Table 3: The mean (and the standard deviation) of empirical coverage over 100 iterations for each scenario on Boston Housing and Diabetes datasets.  
+G.4 ADDITIONAL RESULTS FROM SECTION 6  
+
+<table><tr><td></td><td></td><td>cfBH</td><td>RO-cfBH</td><td>LOO-cfBH</td></tr><tr><td rowspan="3">q=0.1</td><td>FDP</td><td>0.0928 (0.0713)</td><td>0.0038 (0.0115)</td><td>0.0657 (0.0617)</td></tr><tr><td>Power</td><td>0.6319 (0.2053)</td><td>0.3041 (0.1452)</td><td>0.6744 (0.1295)</td></tr><tr><td>Time</td><td>0.0037 (0.0006)</td><td>0.2976 (0.0114)</td><td>0.0060 (0.0011)</td></tr><tr><td rowspan="3">q=0.2</td><td>FDP</td><td>0.2000 (0.0807)</td><td>0.0602 (0.0813)</td><td>0.1836 (0.0560)</td></tr><tr><td>Power</td><td>0.9277 (0.0838)</td><td>0.6522 (0.1450)</td><td>0.9430 (0.0486)</td></tr><tr><td>Time</td><td>0.0037 (0.0005)</td><td>0.2971 (0.0100)</td><td>0.0060 (0.0002)</td></tr><tr><td rowspan="3">q=0.3</td><td>FDP</td><td>0.2882 (0.0806)</td><td>0.2483 (0.1212)</td><td>0.2837 (0.0934)</td></tr><tr><td>Power</td><td>0.9923 (0.0198)</td><td>0.9627 (0.0347)</td><td>0.9917 (0.0136)</td></tr><tr><td>Time</td><td>0.0037 (0.0002)</td><td>0.2970 (0.0101)</td><td>0.0060 (0.0003)</td></tr></table>
+
+Table 4: Mean (and standard deviation) of FDP, power, and execution time for three conformal selection methods.
\ No newline at end of file
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+# LESS IS MORE: MASKING ELEMENTS IN IMAGE CONDITION FEATURES AVOIDS CONTENT LEAKAGES IN STYLE TRANSFER DIFFUSION MODELS
+
+Lin Zhu $^{1}$ , Xinbing Wang $^{1}$ , Chenghu Zhou $^{2}$ , Qinying Gu $^{3}$ , Nanyang Ye $^{1}$  *
+
+$^{1}$  Shanghai Jiao Tong University,  $^{2}$  Chinese Academy of Sciences,
+
+$^{3}$  Shanghai Artificial Intelligence Laboratory  
+{zhulin_sjtu, xwang8, ynylincoln}@sjtu.edu.cn, zhouchsjtu@gmail.com, guqinying@pjlab.org.cn
+
+# ABSTRACT
+
+Given a style-reference image as the additional image condition, text-to-image diffusion models have demonstrated impressive capabilities in generating images that possess the content of text prompts while adopting the visual style of the reference image. However, current state-of-the-art methods often struggle to disentangle content and style from style-reference images, leading to issues such as content leakages. To address this issue, we propose a masking-based method that efficiently decouples content from style without the need of tuning any model parameters. By simply masking specific elements in the style reference's image features, we uncover a critical yet under-explored principle: guiding with appropriately-selected fewer conditions (e.g., dropping several image feature elements) can efficiently avoid unwanted content flowing into the diffusion models, enhancing the style transfer performances of text-to-image diffusion models. In this paper, we validate this finding both theoretically and experimentally. Extensive experiments across various styles demonstrate the effectiveness of our masking-based method and support our theoretical results.
+
+# 1 INTRODUCTION
+
+![](images/5fa2e07524780992bcf71c077e5be82919270ea5ff5b168d01d01270a960b120.jpg)  
+Figure 1: Given a style-reference image, our method is capable of synthesizing new images that resemble the style and are faithful to text prompts simultaneously. Previous methods often face issues of either content leakages or style degradation. We mark the results with significant content leakages, style degradation, and loss of text fidelity with red, green, and blue boxes, respectively.
+
+Recently, text-to-image diffusion models (Zhao et al., 2024b; Sahara et al., 2022; Zhang et al., 2023a; Zhu et al., 2023; Zhang et al., 2023b) have achieved notable success in generating high-quality images, especially for tasks requiring personalized image creation that preserves specific stylistic elements. By incorporating a style-reference image as an additional input, recent approaches (Park et al., 2024; Hamazaspyan & Navasardyan, 2023; Chung et al., 2024; Wang et al., 2023b; Qi et al., 2024; Zhang et al., 2023c) have effectively synthesized images that not only align with the content described in text prompts but also adopt the visual style of the reference image. However, despite these advancements, content leakages from style-reference images remain a persistent issue (Wang et al., 2024a; Ruiz et al., 2023; Jeong et al., 2024), as illustrated in Figure 1. Content leakages occurs when intensifying the style transfer causes unintended non-stylistic elements from the reference image to be incorporated into the generated output (Wang et al., 2024a). Conversely, reducing style intensity to prevent content leakage can result in style degradation, hindering effective style transfer (Jeong et al., 2024). These challenges highlight the difficulty of disentangling styles from contents in style-reference images.
+
+Some approaches (Zhang et al., 2018a;b; Qi et al., 2024) try to achieve the disentanglement by constructing paired datasets in which images share the same subject but exhibit distinct styles, facilitating the extraction of disentangled style and content representations. Other works (Sohn et al., 2024; Liu et al., 2023; Zhang et al., 2024) optimize some or all of the model parameters using large sets of diverse style images, allowing them to isolate and integrate these stylistic elements into diffusion models. However, due to the inherently ambiguous nature of style, building comprehensive style datasets is resource-intensive and limits the model's capacity to generalize to styles not present in the dataset. To address this issue, InstantStyle (Wang et al., 2024a) proposed a training-free strategy to separate style from content by subtracting content-related features from image features. Although this approach is simple and training-free, feature subtraction across different modalities inevitably introduces the image-text misalignment issue (Kim et al., 2023; Gordon et al., 2023), which hinders accurate disentanglement of content and style. As illustrated in Figure 1, although InstantStyle mitigates content leakage, it comes at the cost of significant style degradation.
+
+To overcome all these limitations, we propose a simple and effective training-free method that efficiently decouples content from style, without requiring tuning any model parameters. Unlike InstantStyle, which subtracts features across different modalities, our approach removes content from the style-reference image by masking the image feature elements associated with the content. Specifically, we identify these content-related elements through clustering the element-wise product of the style-reference image features and the content text features, and then set their values to zero. The theoretical evidence for the effectiveness of this identification approach is presented in Proposition 1.
+
+By simply masking specific elements in the style reference's image features, we uncover a critical yet under-explored principle: guiding with appropriate masked conditions (e.g., masking several image feature elements) can prevent undesired content information from leaking into diffusion models, thereby improving style transfer performance. We further present theoretical evidence for this principle. As demonstrated in Theorem 1 and Theorem 2, diffusion models guided by fewer appropriately selected conditions (e.g., the masked image feature and the text feature) achieve a lower divergence between the generated and real image distributions compared to models relying on more conditions that are less coherent (e.g., unfiltered image features combined with text and additional content features). This result aligns with the concept that "Less is more". Extensive experiments across various styles, along with comparisons to state-of-the-art methods, validate the effectiveness of our approach and support our theoretical findings.
+
+# 2 PRELIMINARIES
+
+Because of the portability and efficiency, we illustrate the proposed masking-based method based on the baseline module IP-Adapter (Ye et al., 2023). In this section, we present the background knowledge and key observations from our initial experiments as follows:
+
+Conditional Diffusion Models Diffusion models consist of two processes: a diffusion process (forward process), which incrementally adds Gaussian noise  $\epsilon$  to the data  $x_0$  through a Markov chain. Additionally, a denoising process generates samples from Gaussian noise  $x_{T}\sim N(0,1)$  with a learnable denoising model  $\hat{\epsilon}_{\theta}(x_t,t,c)$  parameterized by  $\theta$ . This denoising model  $\epsilon_{\theta}(\cdot)$  is imple
+
+![](images/41e422c42ef03e52c719b4e26a9d2329228470c19fbdd1ba0ac290a9dc34c4d1.jpg)  
+Figure 2: Top: The differences in the conditions between IP-Adapter (Ye et al., 2023), InstantStyle (Wang et al., 2024a), and Ours. We elaborate on how to select masked elements in Section 3.1. Bottom: Illustration of the content leakages issue.
+
+mented with U-Net and trained with a mean-squared loss derived by a simplified variant of the variational bound:  $\mathcal{L} = \mathbb{E}_{t,\pmb{x}_0,\epsilon}\left[\|\pmb{\epsilon} - \hat{\pmb{\epsilon}}_\theta(\pmb{x}_t, t, \pmb{c})\|^2\right]$ , where  $\pmb{x}_0$  represents the real data with an additional condition  $\pmb{c}$ ,  $t \in [0, T]$  denotes the time step of diffusion process,  $\pmb{x}_t = \alpha_t \pmb{x}_{t-1} + \sigma_t \pmb{\epsilon}$  is the noisy data at  $t$  step, and  $\alpha_t, \sigma_t$  are predefined functions of  $t$  that determine the diffusion process. For conditional learning, classifier-free guidance (Ho & Salimans, 2022) is often employed, which uses a single neural network to parameterize both the conditional model and unconditional model, where for the unconditional model one can simply input a null token  $\varnothing$  for the text features  $c$  when predicting the noise, i.e.  $\pmb{\epsilon}_\theta(\pmb{x}_t, t) = \pmb{\epsilon}_\theta(\pmb{x}_t, t, c = \varnothing)$ . Then one can perform sampling using the following linear combination of the conditional and unconditional noise estimates:  $\tilde{\epsilon}_\theta(\pmb{x}_t, t, c) = \omega \pmb{\epsilon}_\theta(\pmb{x}_t, t, c) + (1 - \omega) \pmb{\epsilon}_\theta(\pmb{x}_t, t)$ . Once the model  $\epsilon_\theta(\cdot)$  is trained, images can be generated from random noises in an iterative manner.
+
+IP-Adapter As "an image is worth a thousand words", IP-Adapter (Ye et al., 2023) proposed an effective and lightweight adapter to achieve image prompt capability for the pre-trained text-to-image diffusion models. It uses two decoupled cross-attention modules to process text and image conditions and finally performs linear weighting. Given the query features  $\mathbf{Z}$ , the text features  $c_{t}$  and the image features  $c_{i}$ , the final formulation of the two cross-attention modules is defined as:
+
+$$
+\mathbf {Z} ^ {\text {n e w}} = \operatorname {S o f t m a x} \left(\frac {\mathbf {Q} \mathbf {K} ^ {\top}}{\sqrt {d}}\right) \mathbf {V} + \operatorname {S o f t m a x} \left(\frac {\mathbf {Q} \left(\mathbf {K} ^ {\prime}\right) ^ {\top}}{\sqrt {d}}\right) \mathbf {V} ^ {\prime} \tag {1}
+$$
+
+$$
+w h e r e \mathbf {Q} = \mathbf {Z} \mathbf {W} _ {q}, \mathbf {K} = c _ {t} \mathbf {W} _ {k}, \mathbf {V} = c _ {t} \mathbf {W} _ {v}, \mathbf {K} ^ {\prime} = c _ {i} \mathbf {W} _ {k} ^ {\prime}, \mathbf {V} ^ {\prime} = c _ {i} \mathbf {W} _ {v} ^ {\prime}
+$$
+
+where  $\mathbf{Q}$ ,  $\mathbf{K}$  and  $\mathbf{V}$  are the query, key, and values matrices from the text features;  $\mathbf{K}'$  and  $\mathbf{V}'$  are the key, and values matrices from the image features. IP-Adapter used the same query for image cross-attention as for text cross-attention. The weight matrices  $\mathbf{W}_q$ ,  $\mathbf{W}_k$ , and  $\mathbf{W}_v$  correspond to text cross-attention and remain frozen, consistent with the original pre-trained model. Only the weight matrices in image cross-attention,  $\mathbf{W}_k'$  and  $\mathbf{W}_v'$ , are trainable. In the inference stage, one can also adjust the weight of the image condition:
+
+$$
+\mathbf {Z} ^ {\text {n e w}} = \operatorname {A t t e n t i o n} (\mathbf {Q}, \mathbf {K}, \mathbf {V}) + \lambda_ {i} \cdot \operatorname {A t t e n t i o n} (\mathbf {Q}, \mathbf {K} ^ {\prime}, \mathbf {V} ^ {\prime}) \tag {2}
+$$
+
+where  $\lambda_{i}$  is the coefficient of image conditions, and the model becomes the original text-to-image diffusion model if  $\lambda_{i} = 0$ .
+
+InstantStyle and Limitations Fully compatible with IP-Adapter, InstantStyle (Wang et al., 2024a) employs block-specific injection techniques to achieve style transfer. Additionally, it proposes an efficient method to decouple content and style from style references, highlighting that straightforward subtraction of content text features from image features can effectively reduce content leakages. However, this approach has some limitations: 1) On the one hand, the feature subtraction is based on CLIP's embeddings, and it relies on the assumption that CLIP global features provide a robust characterization for explicit decoupling. It means that the process necessitates good alignment between CLIP's image-text features for all style references, which may be unrealistic for complex style
+
+references. 2) On the other hand, tuning the coefficient of image condition (i.e.,  $\lambda_{i}$  in Equation 2) is important for its effect in addressing content leakages, but it is labour-intensive and very tricky<sup>1</sup>.
+
+To comprehensively understand the content leakages issue, we present visualization examples for IP-Adapter and InstantStyle in the context of style transfer. Given a style-reference image as the additional image condition, the models are tasked with generating images that reflect the text prompt while incorporating the style of the image reference. Based on Stable Diffusion V1.5 (SD1.5, (Rombach et al., 2022)), we adopt DDIM sampler (Song et al., 2022) with 30 steps, set the guidance scale to 7.5, and the coefficient of image condition to 1.0. As illustrated in Figure 2, IP-Adapter struggles with maintaining the presence of objects in text prompts. Due to image-text misalignment, InstantStyle also fails to achieve seamless synthesis of text prompts and reference styles.
+
+Effectiveness of Less Condition To mitigate image-text misalignment, we focus on feature manipulation within the image feature space instead of introducing feature subtraction between different modalities. Specifically, we propose to eliminate content from the style-reference image by discarding the image feature elements associated with that content. To achieve this, we mask the content-related elements by setting their values to zero. The content-related elements are identified through clustering the element-wise product of the style reference image features and the content text features. As shown in Figure 2, our method successfully achieves more accurate style transfer by masking certain elements in the style reference. Moreover, as illustrated in Figure 10, compared to InstantStyle, our method can achieve more stable results across various coefficient values of  $\lambda_{i}$ , particularly in high-coefficient scenarios. Thus, we arrive at the key motivation of this paper:
+
+The experiment results in Figure 1 and Figure 2 suggest that leveraging appropriately-selected fewer conditions, such as the masked image features, surprisingly effectively avoids content leakages, thereby enhancing text-to-image models in style transfer.
+
+Motivated by this observation, we theoretically and experimentally explore the role of the masking strategy in eliminating content leakages in text-to-image diffusion models. In this paper, we demonstrate that our masking-based method can outperform recent state-of-the-art methods without the need for tuning model parameters or the coefficient value of  $\lambda_{i}$ , which we fix at 1.0.
+
+# 3 METHODOLOGY
+
+In Section 3.1, we elaborate on our novel masking-based method for efficiently decoupling content from style. This approach utilizes a masking strategy for image features, where the masked image feature elements are identified through clustering on the element-wise product of image features and content text features. We also provide supporting evidence for the effectiveness of this masked element selection method in Proposition 1. Furthermore, in section 3.2, we theoretically demonstrate that our method surpasses InstantStyle's feature subtraction by achieving a smaller divergence between the generated image distribution and the real image distribution (See Theorem 1). To delve deeper, we also investigate whether the effectiveness of appropriately fewer conditions holds in tuning-based models. We present the theoretical results in Theorem 2.
+
+# 3.1 THE PROPOSED MASKING-BASED METHOD FOR DECOUPLING CONTENT FORM STYLE
+
+Before delving into the details of our method, we first present the important notations as follows:
+
+Notations 1 Let  $q(\pmb{x} | c)$  be the joint distribution for the data  $\pmb{x}$  and the condition  $c$  and  $q(\pmb{x}) = \sum_{c} q(\pmb{x} | c)$ . Let  $p_{\theta, e}(\pmb{x})$  be a model parameterized by  $\pmb{\theta} \in \Theta$  and  $e \in E$ , where  $\pmb{\theta}$  denotes the model parameters and  $e$  is the embedding for a condition. In the context of style transfer, let  $c_1, c_2,$  and  $c_3$  represent the style reference, the content text in style reference, and the target text prompt, respectively. Models are tasked with generating a plausible target image by combining the content of  $c_3$  with the style of  $c_1$ , while ensuring that the unwanted content  $c_2$  does not transfer into the generated result. The corresponding embeddings of  $c_1, c_2,$  and  $c_3$  are denoted as  $e_1, e_2,$  and  $e_3$ . Each of these embeddings is a d-dimensional feature. Here,  $e_1$  is an image feature that is embedded
+
+![](images/4237ec4e09896c37a79004c34ff2f929eaa9436788f4c76e3588a06a50dfaf98.jpg)  
+Figure 3: (a) The proposed content-related elements identification method: we cluster the element-wise product between image and text features and directly discard elements in the high-means cluster; (b) Illustration of tuning-based models, which we detail in Section 3.2. Text-Adapter and Image-Adapter learn the content feature from the text content feature and image feature, respectively. Only the newly added feature adapter modules (denoted as "Linear+LN") are trained while the pre-trained diffusion model is frozen.
+
+![](images/f885be2fa16137bb28ac02f00ba14eebc517491670e4a32c8aee5b9e6dc6d735.jpg)
+
+by IP-Adapter's image encoder;  $e_2$  and  $e_3$  are text features embedded by the CLIP model. We denote the element-wise product feature between  $e_1$  and  $e_2$  as  $e_p$ , i.e.,  $e_p^i = e_1^i \cdot e_2^i$ .
+
+To mitigate the image-text misalignment issue, we conduct feature manipulation within the latent space of image features. We propose to eliminate content from the style reference by discarding the specific elements corresponding to that content. To achieve this, we drop out these elements by setting their values to zero. As illustrated in Figure 3 (a), the process unfolds as follows:
+
+1) We first compute the element-wise products between the image feature  $e_1$  and the corresponding content text feature  $e_2$ , denoting the results as  $e_p^i (i = 1, \dots, d)$  where  $e_p^i = e_1^i \cdot e_2^i$ . 2) Next, we cluster these elements  $e_p^i (i = 1, \dots, d)$  into  $K$  classes. 3) We then generate a masking vector  $m$  based on the clustering results. For the element  $e_p^i$  in the highest-means cluster, we set the corresponding element  $m^i$  to 0. 4) Finally, we apply the mask vector  $m$  to the image feature  $e_1$  by computing  $e_1' = e_1 \odot m$ . This masked image feature  $e_1'$ , along with the text feature  $e_3$ , is then incorporated into the cross-attention module of IP-Adapter.
+
+The complete algorithm can be found in Algorithm 1 in Appendix A.3. With Algorithm 1, we explicitly remove the image feature elements that contribute to the similarity between the image feature and content text feature. This means we identify and remove those elements that are most correlated with the content text feature in the feature space, thereby effectively reducing content leakages. Additionally, we provide a detailed explanation of why we utilize clustering on  $e_1^i \cdot e_2^i$  to capture content-related elements, rather than relying on the absolute difference between  $e_1^i$  and  $e_2^i$  or other metrics. Through theoretical exploration, we discovered that masking strategy with on  $e_1^i \cdot e_2^i$  leads to the highest energy score for content text feature  $e_2$ , thus effectively decoupling content from style. We prove this advantage in Proposition 1). Before the proposition, we first give a brief introduction to Energy Diffusion Guidance as follows:
+
+Energy Diffusion Guidance Given the noisy image  $\mathbf{x}_t$  and the domain of the given conditions  $c$ , Yu et al. (Yu et al., 2023) proposed a energy diffusion guidance to model the gradient  $\nabla_{\mathbf{x}_t}\log p(c|\mathbf{x}_t)$  by resorting to the energy function:  $p(c|\mathbf{x}_t) = \frac{\exp(-\lambda\mathcal{E}(c,\mathbf{x}_t))}{Z}$ , where  $\lambda$  denotes the positive temperature coefficient and  $Z > 0$  denotes the normalizing constant, computed as  $Z = \int_{c\in \mathcal{C}}\exp \{-\lambda \mathcal{E}(c,\mathbf{x}_t)\}$ ,  $\mathcal{E}(c,\mathbf{x}_t)$  is an energy function that measures the compatibility between the condition  $c$  and the noisy image  $\mathbf{x}_t$ . The value of the energy function will be smaller when  $c$  is more compatible with  $\mathbf{x}_t$ . Therefore, the gradient  $\nabla_{\mathbf{x}_t}\log p(c|\mathbf{x}_t)$  can be implemented with the following:  $\nabla_{\mathbf{x}_t}\log p(c|\mathbf{x}_t)\propto -\nabla_{\mathbf{x}_t}\mathcal{E}(c,\mathbf{x}_t)$ , which is referenced to the energy guidance. Following (Yu et al., 2023), we use a time-independent distance measuring functions  $\mathcal{D}_{\theta}(c,\mathbf{x}_0)$  to approximate the energy function  $\mathcal{E}(c,\mathbf{x}_0)$ :
+
+$$
+\mathcal {E} (c, \mathbf {x} _ {t}) \approx \mathbb {E} _ {p (\mathbf {x} _ {0} | \mathbf {x} _ {t})} \mathcal {D} _ {\boldsymbol {\theta}} (c, \mathbf {x} _ {0}) \tag {3}
+$$
+
+where  $\theta$  defines the model parameters of the diffusion model.  $\mathcal{D}_{\theta}(c,\mathbf{x}_0)$  computes the cosine similarity between the CLIP embeddings of the given condition  $c$  and image  $\mathbf{x}_0$ .
+
+Proposition 1 [The superiority of the proposed masked element selection method] We denote the masked elements in the image feature as  $e_1^{s+1}, \dots, e_1^d$  and denote the feature composed by these elements as  $e_1^m$ , i.e.,  $e_1^m \coloneqq [e_1^{s+1}, \dots, e_1^d]$ . Incorporating the masking strategy, i.e.,  $e_1^m = \emptyset$  leads to  $\nabla_{\mathbf{x}_t} \log p(e_1^m | \mathbf{x}_t, e_3) = 0$ . According to  $\nabla_{\mathbf{x}_t} \log p(c | \mathbf{x}_t) \propto -\nabla_{\mathbf{x}_t} \mathcal{E}(c, \mathbf{x}_t)$ , we have approximated
+
+the local maximization of  $\mathcal{D}_{\theta}(e_1^m, \mathbf{x}_{0|t})$  since  $\mathcal{E}(e_1^m, \mathbf{x}_t) \approx \mathcal{D}_{\theta}(e_1^m, \mathbf{x}_{0|t})$ . The proposed masking strategy enforces the selected component  $e_1^m$  to have the closest distance with the content text feature  $e_2$ . Therefore, according to the relation between energy and distance as defined in Equation 3, the masking strategy with clustering on  $e_1^i \cdot e_2^i$  can lead to the highest energy score for the content text feature  $e_2$  when compared to other masking methods.
+
+This proposition indicates that our proposed masking-based method not only mitigates the image-text misalignment issue by manipulating embeddings within the image feature space, but also achieves the highest energy score for content text feature compared to other masked element selection methods. This effectively reduces the likelihood of content text features, leading to superior performance in content removal.
+
+# 3.2 THEORETICAL EVIDENCE OF FEWER CONDITIONS IN ENHANCING STYLE TRANSFER
+
+In this section, we present the theoretical evidence supporting our method's superiority over InstantStyle and IP-Adapter. Theorem 1 illustrates the advantages of our masking-based approach, suggesting that fewer conditions can achieve a smaller divergence between the generated image distribution and the real image distribution. To delve deeper, we also investigate whether a tuning-based model guided by fewer conditions can yield improved results. Our findings indicate that training models with fewer conditions can also enhance style transfer, as illustrated in Theorem 2.
+
+Notations 2 We use  $p_{\theta, e}(\pmb{x})$  to approximate the conditional data distribution  $q(\pmb{x}|c)$ . Let  $p_{\theta, \phi}(\pmb{x}|c) = p_{\theta, e}(\pmb{x})|_{e = \phi(c)}$ , where  $e$  denotes the embedding of the given condition  $c$ . We denote a certain statistics divergence as  $\mathcal{D}$  (or more loosely a divergence upper bound).
+
+Theorem 1 [Why the masking strategy is better] Suppose the divergence  $\mathcal{D}$  is convex, and the elements in the image feature are independent of each other. We denote  $e_1 \coloneqq e_1^{i,\dots,d}$ ,  $e_2 \coloneqq e_2^{i,\dots,d}$ , and  $e_3 \coloneqq e_3^{i,\dots,d}$ . Thus, the divergence between the generated and ground-truth image distribution of the InstantStyle model is:
+
+$$
+D _ {1} = \mathbb {E} _ {q (\pmb {e} _ {1} ^ {i, \dots , d}, \pmb {e} _ {2} ^ {i, \dots , d}, \pmb {e} _ {3} ^ {i, \dots , d})} \mathcal {D} (q (\pmb {x} | \pmb {e} _ {1}, \pmb {e} _ {2}, \pmb {e} _ {3}) \| p _ {\pmb {\theta}} (\pmb {x} | \pmb {e} _ {1} ^ {i, \dots , d}, \pmb {e} _ {2} ^ {i, \dots , d}, \pmb {e} _ {3} ^ {i, \dots , d}))
+$$
+
+We denote the masked element in the image feature as  $e_1^{s+1}, \dots, e_1^d$ . Thus, the divergence result of the proposed masking strategy is:
+
+$$
+D _ {2} = \mathbb {E} _ {q (\pmb {e} _ {1} ^ {i, \dots , d}, \pmb {e} _ {2} ^ {i, \dots , d}, \pmb {e} _ {3} ^ {i, \dots , d})} \mathcal {D} (q (\pmb {x} | \pmb {e} _ {1}, \pmb {e} _ {2}, \pmb {e} _ {3}) \| p _ {\pmb {\theta}} (\pmb {x} | \pmb {e} _ {1} ^ {i, \dots , s}, \pmb {e} _ {2} ^ {i, \dots , d}, \pmb {e} _ {3} ^ {i, \dots , d}))
+$$
+
+With the assumption:
+
+$$
+\mathbb {E} _ {q (\boldsymbol {e} _ {1} ^ {i, \dots , d})} \mathcal {D} (q (\boldsymbol {x} | \boldsymbol {e} _ {1} ^ {i, \dots , d}) \| p _ {\theta} (\boldsymbol {x} | \boldsymbol {e} _ {1} ^ {i, \dots , s})) \leq \mathbb {E} _ {q (\boldsymbol {e} _ {1} ^ {i, \dots , s})} \mathcal {D} (q (\boldsymbol {x} | \boldsymbol {e} _ {1} ^ {i, \dots , s}) \| p _ {\theta} (\boldsymbol {x} | \boldsymbol {e} _ {1} ^ {i, \dots , s}))
+$$
+
+and by Jensen's inequality, we have  $D_2 \leq D_1$ .
+
+This theorem indicates that: compared to InstantStyle, masking certain elements (i.e.,  $e_1^{s+1}, \dots, e_1^d$ ) of the image feature achieves a smaller divergence between the generated and ground-truth image distribution. Further, we also investigate whether a tuning-based model conditioned on appropriately fewer conditions can yield improved results. We begin by formalizing the learning paradigms of tuning-based models, as illustrated in Figure 3 (b), as follows:
+
+Learning Paradigms Text-Adapter: Given the style-reference image  $c_{1}$ , the content in style reference  $c_{2}$ , and the target text prompt  $c_{3}$ , models are tasked with generating images that reflect the text prompt  $c_{3}$  while incorporating the style of  $c_{1}$  and avoiding the presence of content  $c_{2}$ . Without loss of generality, we keep the parameters in condition encoders frozen and perform adapter tuning for content text feature  $e_{2}$  with adapter  $\phi$ . The style reference's image feature  $e_{1}$  is subtracted by the content feature  $\phi(e_{2})$  to avoid the presence of content  $c_{2}$ . The text adapter  $\phi$  can be optimized by:
+
+$$
+\min  _ {\phi} \mathbb {E} _ {q \left(c _ {1}, c _ {2}, c _ {3}\right)} \mathcal {D} \left(q \left(\boldsymbol {x} \mid c _ {1}, c _ {2}, c _ {3}\right) \| p _ {\boldsymbol {\theta}, \phi} \left(\boldsymbol {x} \mid c _ {1}, c _ {2}, c _ {3}\right)\right)
+$$
+
+Image-Adapter: Instead of tuning the content text feature, we directly extract the content feature from the style-reference image. We denote the extraction function (image adapter) as  $\psi$ . Then we
+
+have the final image feature incorporated into the cross-attention module represented as  $e_1 - \psi(e_1)$ . Thus, the optimization objective is:
+
+$$
+\min  _ {\boldsymbol {\psi}} \mathbb {E} _ {q (c _ {1}, c _ {2}, c _ {3})} \mathcal {D} (q (\boldsymbol {x} | c _ {1}, c _ {2}, c _ {3}) \| p _ {\boldsymbol {\theta}, \boldsymbol {\psi}} (\boldsymbol {x} | c _ {1}, c _ {3}))
+$$
+
+In practice, the feature adapters (denoted as "Liner+LN") in Figure 3 (b) are trained through image reconstruction using mean squared error (MSE) loss to predict reconstruction errors. The specific algorithm is provided in Algorithm 2 in Appendix A.3. We instantiate the two optimization objectives, as shown in the 9th line of Algorithm 2, for the Text-Adapter and Image-Adapter, respectively.
+
+Assumption 1 Suppose the condition  $c_{1} \in \mathcal{C}_{1}$  and  $c_{3} \in \mathcal{C}_{3}$  are independent of each other, and the condition  $c_{2}$  is dependent on the style reference  $c_{1}$ . Given the condition  $c_{1} = x$ , the content information  $c_{2} = y$  is uniquely determined.
+
+Under Assumption 1, we reveal that a tuning-based model conditioned on appropriately fewer conditions can yield improved style transfer results:
+
+Theorem 2 [The superiority of tuning-based model conditioned on fewer conditions] Suppose the divergence  $\mathcal{D}$  is convex, and the function space  $\Phi$  and  $\Psi$  ( $\phi \in \Phi$  and  $\psi \in \Psi$ ) includes all measurable functions. Under Assumption 1 and by Jensen's inequality, we have:
+
+$$
+\begin{array}{l} \min _ {\boldsymbol {\psi}} \mathbb {E} _ {q (c _ {1}, c _ {2}, c _ {3})} \mathcal {D} (q (\boldsymbol {x} | c _ {1}, c _ {2}, c _ {3}) \| p _ {\boldsymbol {\theta}, \boldsymbol {\psi}} (\boldsymbol {x} | c _ {1}, c _ {3})) \\ \leq \min  _ {\phi} \mathbb {E} _ {q \left(c _ {1}, c _ {2}, c _ {3}\right)} \mathcal {D} \left(q \left(\boldsymbol {x} \mid c _ {1}, c _ {2}, c _ {3}\right) \| p _ {\boldsymbol {\theta}, \phi} \left(\boldsymbol {x} \mid c _ {1}, c _ {2}, c _ {3}\right)\right) \tag {4} \\ \end{array}
+$$
+
+This theorem indicates that learning content features based on the text feature results in a larger divergence between the generated and real image distribution, compared to learning content features directly within the feature space of image features. Overall, these theoretical results demonstrate that appropriately fewer conditions boost better text-to-image diffusion models in style transfer.
+
+# 4 EXPERIMENTS
+
+In this section, we first demonstrate the proposed theoretical results. Previous evaluation datasets do not contain explicitly defined references' contents, thus making it inaccurate in evaluating content leakages. Instead, we consider an evaluation dataset comprising various defined reference contents and styles for comprehensively assessing models' capability in addressing content leakages. In this paper, we construct the evaluation dataset consisting of 10 content objects and 21 image styles. Extensive experimental results demonstrate that both tuning-free and tuning-based models, conditioned on appropriately selected fewer conditions, achieve higher text fidelity and style similarity, which aligns well with Theorem 1-2. Next, we report our method's performance across various styles and compare it with existing approaches using the StyleBench (Gao et al., 2024) benchmark. Experimental results demonstrate the proposed method's effectiveness in avoiding content leakages.
+
+# 4.1 QUANTITATIVE ANALYSIS OF OUR METHOD IN ADDRESSED CONTENT LEAKAGES
+
+Evaluation Dataset We construct the evaluation dataset using the 10 classes of CIFAR-10 (Krizhevsky et al., 2009). Leveraging the code of MACE (Lu et al., 2024), we generate 21 distinct styles for each class, each containing 8 variations. The dataset is divided into two subsets based on image style for training and testing. Using these generated images as references, we train tuning-based models (i.e., Image-Adapter and Text-Adapter) through image reconstruction. During inference, we utilize the test dataset as image references to conduct text-driven style transfer for 5 text prompts. Additional details about the datasets are provided in Table 4 of Appendix A.4.
+
+Model Configuration For the tuning-based models, we update adapter weights for 2500 steps using Adam optimizer (Kingma, 2014) with a learning rate of 0.00001. We adopt the same adapter layer structure for Image-Adapter and Text-Adapter, which consists of a linear layer and a batch normalization. Subsequently, leveraging the test data as image references, the trained model is utilized to generate stylized images for 5 text prompts. For all experiments, we adopt Stable Diffusion V1.5 as our base text-to-image model, and we set the clustering classes to 2 for our masking-based method.
+
+Table 1: Quantitative comparison with advanced text-driven style transfer methods. We mark the Best results and underscore the second best results.  
+
+<table><tr><td>Method</td><td>StyleCrafter</td><td>StyleAligned</td><td>StyleDrop</td><td>DEADiff</td><td>InstantStyle</td><td>StyleShot</td><td>Ours</td></tr><tr><td>style score ↑</td><td>0.245</td><td>0.244</td><td>0.240</td><td>0.230</td><td>0.290</td><td>0.267</td><td>0.273</td></tr><tr><td>fidelity score ↑</td><td>0.858</td><td>0.662</td><td>0.889</td><td>0.916</td><td>0.820</td><td>0.956</td><td>0.972</td></tr><tr><td>leakage score ↓</td><td>0.589</td><td>0.720</td><td>0.600</td><td>0.523</td><td>0.596</td><td>0.543</td><td>0.478</td></tr><tr><td>Human Preference ↑</td><td>1.7%</td><td>4.7%</td><td>19.6%</td><td>4.7%</td><td>8.3 %</td><td>24.3%</td><td>36.7%</td></tr></table>
+
+Evaluation Metrics Human Preference: Following (Liu et al., 2023; Qi et al., 2024), we conduct user preference studies to evaluate models' style transfer ability. Compared to other methods, our method achieves the highest human preferences by a large margin, demonstrating robust stylization across various styles and responsiveness to text prompts. CLIP-Based Scores: We also assess the quality of generated images using CLIP (Radford et al., 2021) with ViT-H/14 as the image encoder. We perform binary classification using the CLIP model Radford et al. (2021) on the generated images to distinguish between the reference's content text and the text prompt. The computed classification accuracy is referred to as the fidelity score  $\uparrow$ . We also calculate the similarity between the generated images and the reference's content text, calibrated by the similarity between the reference images and the content text, termed the leakage score  $\downarrow$ . Finally, we assess the similarity between the generated images and the style reference, adjusted by subtracting the leakage score, which we refer to as the style score  $\uparrow$ . The fidelity score measures the fidelity to the text instructions, the style score assesses style similarity with the style reference, and the leakage score indicates content leakages from the style reference, where a lower score is preferable. More details of these scores are provided in Appendix A.4.
+
+Experiment Results We provide visual comparisons between the proposed masking-based method and state-of-the-art methods in Figure 4, demonstrating that our approach can mitigate content leakages without introducing style degradations. These results align well with the theoretical findings of Theorem 1, showcasing that fewer conditions more effectively address both content leakages and style degradations.
+
+![](images/f32f0776d4d2605232ef9e0faf8cc99dcfd9c06420ae653f849f1ff6529e1bdf.jpg)
+
+![](images/1535e87079aa38cddbce747f2c6c89fe839692ae6a58ff9b0fcbe9acd589ab95.jpg)
+
+![](images/c79524c38c68bc6a0ba92188321ee8e9e7a26076480d1904082fe550364f0de2.jpg)  
+Style Reference
+
+![](images/21dfebfb99c23d17dfe39ed36d1f0a9e86f89354a93ef4c2eed3170d695753aa.jpg)
+
+![](images/5da127f9b16c7ffe8e6e57722928368390db497d45177465284482c92299eb81.jpg)
+
+![](images/2a5d83d904dcf80221ca0d3fe58e964cfdfd0f08a18a2250e53366d9465b5a2f.jpg)  
+Ours(ZS)
+
+![](images/9ff5e92257a1d4eea12765bc063ae5de8b1cc25e6a81b86da2143ca461d0022f.jpg)
+
+![](images/da4f22679e76a8223bd97db98d7fba0029a2e1ccc394d5efd85dae8871c8460a.jpg)
+
+![](images/b8de1c569a2c9a3e73ce6008d368df9c05df969021741acbf20969d29e2b4824.jpg)  
+Ours(FT)
+
+![](images/dcc3a34d1e4e9196975b9fa9825ba659e82250ee0a4760721995d7f8439d2fcb.jpg)
+
+![](images/00df9483c3778ba5fc0731421459f6b64ad189a39ade8e9fcbdf33096792e990.jpg)
+
+![](images/46dd838f17c3ceae6ffc85a78f2e703b95303335c190b951f2865a99cb688727.jpg)  
+InstantStyle
+
+![](images/138885523f81f18cf0f337f7c1592a1a460b1d0b5ac557f3fe273e0daea7789a.jpg)
+
+![](images/7af4b37c3fbfff42dd3b89e509511c54178f545169e9dd64b155258161ae479b.jpg)
+
+![](images/2a05f8a159ef1e75b12d1736076016526426e37ff638b94e725f85af0f206cdd.jpg)  
+StyleShot
+
+![](images/955fabe5cdeae301e9a421b227d540a24ddc0811ada09c2e749ccd66c8c2be08.jpg)
+
+![](images/8fda00f321cbdac3c7389ba8a07dc2e1daba3a84012ef68e0093b6dcf797d4d7.jpg)
+
+![](images/ce198ef237b6cc6aa0fd37f8da3eb366bd58b817d41f4be26b75cbb120aed7fb.jpg)  
+DEADiff
+
+![](images/cddfa9b17dcf0e7681c59a132d42e992f992135d3afb968d424136610fde1858.jpg)
+
+![](images/848beeabed4d69dd5696a766790ad66098bded5ac4d2cfeef9bb6100352b1513.jpg)
+
+![](images/4f72db9f4986cd146f892cb9b06c4c518d91eff45376c00fd858cd9ce2153022.jpg)  
+StyleDrop
+
+![](images/fae54da7881874c9217ad91173e63a1d148a28b8e8120245a7a4c8ab75460c67.jpg)
+
+![](images/60990200c90113490c1f830f926950828940dedf705f931a675716a526b79623.jpg)
+
+![](images/e74fc8302d597e348e7b54676a5d28493d253d3373353940fc6df4a54a8fdde1.jpg)  
+StyleAligned
+
+![](images/de963a613e2b8d261f23007c4a779c0f20a83fcefaa6e3249179840b11f9ec09.jpg)
+
+![](images/0611d35a32971f670416f7563f3ba8a4163ca96fba53594b6856aadc53fec1b4.jpg)
+
+![](images/85e1f1faa028f53f23e30207104e98fae16b18cecb56b9208c5d05770fb09c0f.jpg)  
+StyleCrafter
+
+![](images/6dfd0ee2966ffb8867c6205c034f61a55ff71acccca6e16b4cea15aefe14f721.jpg)  
+Figure 4: In the figure, the text prompt is "A human". Leveraging appropriately fewer conditions, Ours(ZS) and Ours(FT) denote the proposed masking-based method and the tuning-based Image-Adapter method, respectively. Our methods successfully transfer the references' styles without content leakages. More results can be found in Figure 9 and Figure 12-16 in Appendix A.6.  
+(a) Image and text alignment results during training  
+Figure 5: Comparison between the Image-Adapter and Text-Adapter model. (a) Following (Gao et al., 2024), we report the image and text alignment scores alongside training steps. We also present the tuning-free models' (i.e., IP-Adapter, InstantStyle, and our masking-based method) fidelity scores in the figure. (b) Visual comparisons between Image-Adapter and Text-Adapter.
+
+![](images/d43708a9abba28f18abecce7dc9d7ec178f3647ec8c63c8c08736a0f72c24956.jpg)  
+(b) Visualization results of Image-Adapter and Text-Adapter
+
+Table 2: Quantitative comparison with state-of-the-art text-driven style transfer methods. The highest human preference scores demonstrate the effectiveness of our method.  
+
+<table><tr><td>Method</td><td>StyleCrafter</td><td>DEADiff</td><td>StyleDrop</td><td>InST</td><td>StyleAligned</td><td>StyleShot</td><td>InstantStyle</td><td>Ours</td></tr><tr><td>text alignment ↑</td><td>0.202</td><td>0.232</td><td>0.220</td><td>0.204</td><td>0.213</td><td>0.219</td><td>0.275</td><td>0.265</td></tr><tr><td>image alignment ↑</td><td>0.706</td><td>0.597</td><td>0.621</td><td>0.623</td><td>0.680</td><td>0.640</td><td>0.575</td><td>0.657</td></tr><tr><td>Human Preference ↑</td><td>4.2%</td><td>10.1%</td><td>2.6%</td><td>5.7%</td><td>6.3%</td><td>21.1%</td><td>7.9%</td><td>42.1%</td></tr></table>
+
+For the tuning-based models, we present their image and text alignment scores and generation results along with the training steps in Figure 5. The observations are as follows: 1) Guided by appropriately-selected fewer conditions, the Image-Adapter model outperforms the Text-Adapter in both style scores and fidelity scores, indicating a smaller distribution divergence between the generated images and real images, consistent with the theoretical results of Theorem 2. 2) As shown in Figure 5 (b), while the Text-Adapter reduces content leakages, it leads to significant style degradation as the training steps increase. In contrast, by leveraging fewer conditions, the Image-Adapter successfully avoids the image and text modal misalignment, with no content leakages while achieving style enhancement.
+
+Overall, we provide quantitative comparisons with recent methods, such as StyleCrafter (Liu et al., 2023), StyleAligned (Hertz et al., 2024), StyleDrop (Sohn et al., 2024), DEADiff (Qi et al., 2024), StyleShot (Gao et al., 2024) and so on. We present the CLIP-based score results in Table 1. Additional visual comparisons can be found in Figure 9 and Figure 12-16 of Appendix A.6. The overall best performances in CLIP-based scores and human preferences further demonstrate the effectiveness of our method in balancing content leakage mitigation with style enhancement.
+
+# 4.2 COMPARISON WITH STATE-OF-THE-ART METHODS ON STYLEBENCH
+
+![](images/f2182c6c0b8052eb3891d10db8766cdab118df3ee4c6e80f88ae7266c64951c4.jpg)  
+Figure 6: Visual comparison between recent state-of-the-art methods, including RB-Modulation (Rout et al., 2024), CSGO (Xing et al., 2024), for addressing content leakage, and ours in text-driven style transfer. We mark the results with significant content leakages, style degradation, and loss of text fidelity with red, green, and blue boxes, respectively. The proposed masking-based method does not require content knowledge of the image reference; instead, we leverage the CLIP text feature of "person, animal, plant, or object in the foreground" to identify the elements that need to be masked. More comparison examples are in Figure 11 of Appendix A.6.
+
+Experiment Details To comprehensively evaluate the effectiveness of the proposed masking-based method, we conduct evaluations on the recent style transfer benchmark StyleBench (Gao et al., 2024), which covers 73 distinct styles, ranging from paintings, flat illustrations to sculptures with varying materials. For InstantStyle and our method, we employ the feature subtraction and masking strategy, respectively, on the extracted image features by StyleShot.
+
+Experiment Results Following StyleShot (Gao et al., 2024), we report the quantitative comparison on text and image alignment with state-of-the-art text-driven style transfer methods in Table 2. Figure 6 displays our results and baselines of four distinct style images, each corresponding to the same
+
+pair of text prompts. As shown in Figure 6, we observe that InstantStyle (Wang et al., 2024a) and the most recent method StyleShot (Gao et al., 2024) retain the image style but may fail to generate the target semantic information. In contrast, our method can improve text fidelity for text prompts without sacrificing style enhancement, avoiding content leakages and achieving style enhancement. As shown in the last column of Figure 6, we also present ablation study results in the last column, where we retain the identified elements to be discarded while masking the other features. Consequently, there is almost no style information identical to the reference image, further confirming that our method can efficiently and accurately decouple content from style.
+
+# 4.3 ABLATION STUDIES
+
+# Effectiveness in Image-Driven Style Transfer
+
+The proposed method also excels at transferring style onto target content images. We compare our method with the recent SOTA method StyleShot (Gao et al., 2024) to showcase the superiority of our method in efficiently mitigating content leakages. As shown in Figure 7, StyleShot usually generates unsatisfied results when the style reference image consists of a human face. In contrast, according to the clustering results on the element-wise product feature  $e_{p}$ , by only masking 1-5 elements, our method can successfully mitigate content leakages and style degradation of StyleShot.
+
+![](images/808b2139312cd8cad5e52b0bcaba4efba31d41f9d64fc86225e2b89417be14a9.jpg)  
+Figure 7: Visual comparison between StyleShot and ours in image-driven style transfer. Results with content leakages and style degradation are highlighted with red and green boxes, respectively. More results are in Figure 19.
+
+Moreover, we also compare our method with
+
+InstantStyle's block-specific injection technique based on different diffusion models and style encoders. Due to space limitations, we provide additional results in Appendix A.7.
+
+Ablation Studies on Clustering Number In Table 3, we conduct ablation studies clustering number  $K$  in text-driven style transfer using the StyleBench dataset. Due to space limitations, we provide additional visualization results in Appendix A.8. The results show that a smaller  $K$ , such as  $K = 2$ , can lead to higher text alignment scores, as more content-related elements in the style reference are masked. This is particularly evident in styles such as 3D models, Anime, and Baroque art, which contain more human-related images, resulting in more effective content leakage avoidance.
+
+# 5 CONCLUSION
+
+Table 3: Ablation study results on clustering number  $K$  .  
+
+<table><tr><td></td><td>K</td><td>3D Model</td><td>Anime</td><td>Baroque</td></tr><tr><td rowspan="4">image alignment</td><td>2</td><td>0.474</td><td>0.372</td><td>0.384</td></tr><tr><td>3</td><td>0.478</td><td>0.381</td><td>0.393</td></tr><tr><td>4</td><td>0.485</td><td>0.390</td><td>0.404</td></tr><tr><td>5</td><td>0.487</td><td>0.380</td><td>0.411</td></tr><tr><td rowspan="4">text alignment</td><td>2</td><td>0.213</td><td>0.234</td><td>0.257</td></tr><tr><td>3</td><td>0.206</td><td>0.232</td><td>0.253</td></tr><tr><td>4</td><td>0.189</td><td>0.231</td><td>0.253</td></tr><tr><td>5</td><td>0.188</td><td>0.229</td><td>0.252</td></tr></table>
+
+In this paper, we propose a masking-based method, that efficiently decouples content from style without requiring tuning any model parameters. By masking (zeroing out) certain elements in the image feature corresponding to that content, we effectively eliminate content leakages from style references across various evaluation datasets. More importantly, we have theoretically proved that our model, under the guidance of appropriately selected fewer conditions, achieves a smaller divergence between the generated image distribution and the real image distribution, outperforming those conditioned on larger, yet less
+
+coherent conditions. Extensive experiments across various styles and targets have demonstrated the effectiveness of our proposed method.
+
+# 6 ETHICS STATEMENT
+
+This work aims to make a positive impact on the field of AI-driven image generation. Our method builds upon existing open-source diffusion models, which already include some built-in safeguards, such as the NSFW ("not safe for work") Filter. We believe that our contributions in generating images with diverse styles, while preventing content leakages, can further positively help avoid stereotypes and discriminations. Further details regarding our ethics statement are provided in Appendix A.11
+
+# ACKNOWLEDGEMENTS
+
+This work is supported by National Natural Science Foundation of China under Grant (No.62106139).
+
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+
+# A APPENDIX
+
+# A.1 RELATED WORKS
+
+Stylized Image Generation Stylized image generation, commonly referred to as image style transfer, involves transferring the stylistic or aesthetic attributes from a reference image to a target image. Thanks to the significant advancements in diffusion models (Ho et al., 2020; Podell et al., 2023; Song et al., 2022; Li et al., 2024; Rombach et al., 2022; Ho & Salimans, 2022; Ramesh et al., 2022; Sahara et al., 2022; Nichol et al., 2021), numerous methods (Sun et al., 2023; Xu et al., 2024; Lu et al., 2023; Lin et al., 2024) have been developed to ensure style consistency across images generated. Among inversion-based approaches (Zhang et al., 2023c; Gal et al., 2022; Hertz et al., 2024) project style images into a learnable embedding in the text token space to guide style-specific generation. Unfortunately, these methods can lead to information loss due to the mapping from visual to text modalities. Cross-attention manipulation (Le & Carlsson, 2022; Hertz et al., 2024; Chung et al., 2024; Hertz et al., 2024; Wang et al., 2023b) is another method for achieving style transfer, involving the manipulation of features within self-attention layers. In contrast, IP-Adapter (Ye et al., 2023) and Style-Adapter (Wang et al., 2023b) introduce a distinct cross-attention mechanism that de-couples the attention layers for text and image features, allowing for a coarse control over the style transfer process. Although these methods have achieved significant advancements, they often struggle with content leakages from style-reference images.
+
+Methods Addressing Content Leakages Some approaches (Zhang et al., 2018a;b; Qi et al., 2024) aim to tackle the content leakages issue by constructing paired datasets where images share the same subject matter but exhibit distinct styles, facilitating the extraction of disentangled style and content representations. DEADiff (Qi et al., 2024) stands out by extracting disentangled representations of content and style using a paired dataset, facilitated by the Q-Former (Li et al., 2023) technique. Other works (Sohn et al., 2024; Liu et al., 2023) optimize some or all model parameters using extensive style images, embedding their visual style into the model's output domain. However, the inherently underdetermined nature of style makes the creation of large-scale paired datasets or style datasets both resource-intensive and limited in the diversity of styles it can capture. To address this issue, InstantStyle (Wang et al., 2024a), a recent innovation, employs block-specific injection and feature subtraction techniques to implicitly achieve decoupling of content and style, offering a nuanced approach to style transfer. In the context of image-driven style transfer, InstantStyle-Plus (Wang et al., 2024b) further proposed several techniques to prioritize the integrity of the content image while seamlessly integrating the target style. Although the InstantStyle approach achieved significant advancements, feature manipulation across different modalities inevitably introduces the image-text misalignment issue (Kim et al., 2023; Gordon et al., 2023), which hinders accurate disentanglement of content and style. StyleDiffusion (Wang et al., 2023a) introduced a CLIP-based style disentanglement loss coordinated with a style reconstruction to decouple content from style in the CLIP image space. However, this framework required a training process to disentangle style from each style image, achieving this by providing approximately 50 content images for training. DiffuseIT (Kwon & Ye, 2022) introduced a novel diffusion-based unsupervised image translation method for decoupling content from style, but it also requires complex loss regularization. More recent and stronger models, such as RB-Modulation (Rout et al., 2024), have been proposed to alleviate the content leakage problem. RB-Modulation uses attention-based feature aggregation and different descriptors to decouple content and style. It is training-free and is reported to outperform InstantStyle. CSGO (Xing et al., 2024) is another recent approach that employs a separately trained style projection layer to mitigate content leakage. Additionally, Zhao et al. (Zhao et al., 2024a) proposed a method to identify and address the issue of conditional content leakage in image-to-video (I2V) generation. Several studies (Motamed et al., 2023; Huang et al., 2024; Le & Carlsson, 2022) focus on concept disentanglement, but they are not specifically aimed at style transfer.
+
+Masking Mechanism in Synthesizing High-Quality Images Although several studies (Couairon et al., 2022; Gao et al., 2023; Hansen-Estruch et al., 2024; Pan et al., 2023; Lei et al., 2023) have explored the effectiveness of masking mechanisms, our method differs from these approaches in several key aspects: 1) No coupled denoising processes: Our method avoids the need for two denoising processes, thus saving computational resources. For instance, the DIFFEDIT method (Couairon et al., 2022) requires two denoising processes—one conditioned on the query text and the other conditioned on a reference text. By contrasting the predictions of the two diffusion models, DIFFEDIT generates a mask that locates the regions needing editing to match the query text. 2) Masking in the
+
+latent space: Unlike DIFFEDIT (Couairon et al., 2022), which operates on the pixel level to generate a mask highlighting the regions of the input image that need editing, our method performs masking in the latent space, bypassing pixel-level operations and patch-level manipulations. 3) Focus on content leakage in style transfer: While the MDT method (Gao et al., 2023) introduces a latent masking scheme to enhance the DPMs' ability to learn contextual relations among object semantics in an image, it focuses on predicting randomly masked tokens from unmasked ones. In contrast, our method targets content leakage in style transfer. We mask feature elements that are related to unwanted content from the style reference, guided by clustering results on the element-wise product.
+
+# A.2 PROOF
+
+Proof of Proposition 1 We denote the masked elements in the image feature as  $e_1^{s+1}, \dots, e_1^d$  and denote the feature composed by these elements as  $e_1^m$ , i.e.,  $e_1^m \coloneqq [e_1^{s+1}, \dots, e_1^d]$ . Following (Yu et al., 2023), we use a time-independent distance measuring functions  $\mathcal{D}_{\theta}(c, \mathbf{x}_0)$  to approximate the energy function  $\mathcal{E}(c, \mathbf{x}_0)$ :
+
+$$
+\mathcal {E} (c, \mathbf {x} _ {t}) \approx \mathbb {E} _ {p (\mathbf {x} _ {0} | \mathbf {x} _ {t})} \mathcal {D} _ {\boldsymbol {\theta}} (c, \mathbf {x} _ {0}) \tag {5}
+$$
+
+where  $\theta$  defines the model parameters of the diffusion model.  $\mathcal{D}_{\theta}(c,\mathbf{x}_0)$  computes the cosine similarity between the CLIP embeddings of the given condition  $c$  and image  $\mathbf{x}_0$ .
+
+Based on the classifier-free guidance (Ho & Salimans, 2022), incorporating the masking strategy, i.e.,  $e_1^m = \emptyset$  leads to  $\nabla_{\mathbf{x}_t} \log p(e_1^m | \mathbf{x}_t, e_3) = 0$ .
+
+Building upon the energy-based assumption in (Yu et al., 2023), i.e.,  $\nabla_{\mathbf{x}_t}\log p(c|\mathbf{x}_t)\propto -\nabla_{\mathbf{x}_t}\mathcal{E}(c,\mathbf{x}_t)$ , we have approximated the local maximization of  $\mathcal{E}(e_1^m,\mathbf{x}_t)$ . Again since  $\mathcal{E}(e_1^m,\mathbf{x}_t)\approx \mathcal{D}_{\pmb{\theta}}(e_1^m,\mathbf{x}_{0|t})$ , we have approximated the local maximization of  $\mathcal{D}_{\pmb{\theta}}(e_1^m,\mathbf{x}_{0|t})$ .
+
+Based on the clustering result on element-wise product feature  $e_p(e_p^i = e_1^i \cdot e_2^i)$ , we mask (drop out) the high-value elements, denoted as  $e_1^m$ , in feature  $e_p$ . With a fixed masking proportion, our proposed strategy differs from other methods—such as those relying on the absolute difference between  $e_1^i$  and  $e_2^i$ —by ensuring that the selected component  $[0, \dots, 0, e_1^{s+1}, \dots, e_1^d]$  have the highest cosine similarity with the content text feature  $e_2$ . This approach can lead to  $\max \mathcal{D}_{\theta}(e_2, \mathbf{x}_{0|t})$  when compared to other masking methods.
+
+Note that the energy function for  $e_2$  satisfies:  $\mathcal{E}(e_2, \mathbf{x}_t) = \mathcal{D}_{\theta}(\mathbf{e}_2, \mathbf{x}_{0|t})$ . Therefore, according to the relation between energy and distance as defined in Equation 3, the masking strategy with clustering on  $e_1^i \cdot e_2^i$  can lead to the highest energy score for the content text feature  $e_2$ .
+
+Proof of Theorem 1 We denote the element in the image feature and text feature as  $e_1^i (i \in \{1, \dots, d\})$  and  $e_2^i (i \in \{1, \dots, d\})$ , respectively. Thus, the divergence between the generated and ground-truth image distribution of the InstantStyle model is:
+
+$$
+D _ {1} = \mathbb {E} _ {q (\pmb {e} _ {1} ^ {i, \dots , d}, \pmb {e} _ {2} ^ {i, \dots , d}, \pmb {e} _ {3} ^ {i, \dots , d})} \mathcal {D} (q (\pmb {x} | \pmb {e} _ {1}, \pmb {e} _ {2}, \pmb {e} _ {3}) \| p _ {\pmb {\theta}} (\pmb {x} | \pmb {e} _ {1} ^ {i, \dots , d}, \pmb {e} _ {2} ^ {i, \dots , d}, \pmb {e} _ {3} ^ {i, \dots , d}))
+$$
+
+We denote the masked element in the image feature as  $e_1^{s + 1}, \dots, e_1^d$ , and the divergence result of the proposed masking strategy is:
+
+$$
+D _ {2} = \mathbb {E} _ {q (\pmb {e} _ {1} ^ {i, \dots , d}, \pmb {e} _ {2} ^ {i, \dots , d}, \pmb {e} _ {3} ^ {i, \dots , d})} \mathcal {D} (q (\pmb {x} | \pmb {e} _ {1}, \pmb {e} _ {2}, \pmb {e} _ {3}) \| p _ {\pmb {\theta}} (\pmb {x} | \pmb {e} _ {1} ^ {i, \dots , s}, \pmb {e} _ {2} ^ {i, \dots , d}, \pmb {e} _ {3} ^ {i, \dots , d}))
+$$
+
+With the assumption:
+
+$$
+\mathbb {E} _ {q (\pmb {e} _ {1} ^ {i, \dots , d})} \mathcal {D} (q (\pmb {x} | \pmb {e} _ {1} ^ {i, \dots , d}) \| p _ {\theta} (\pmb {x} | \pmb {e} _ {1} ^ {i, \dots , s})) \leq \mathbb {E} _ {q (\pmb {e} _ {1} ^ {i, \dots , s})} \mathcal {D} (q (\pmb {x} | \pmb {e} _ {1} ^ {i, \dots , s}) \| p _ {\theta} (\pmb {x} | \pmb {e} _ {1} ^ {i, \dots , s}))
+$$
+
+and by Jensen's inequality (Ash & Doleans-Dade, 2000), we have
+
+$$
+\begin{array}{l} \mathbb {E} _ {q (\pmb {e} _ {1} ^ {i, \dots , d}, \pmb {e} _ {2} ^ {i, \dots , d}, \pmb {e} _ {3} ^ {i, \dots , d})} \mathcal {D} (q (\pmb {x} | \pmb {e} _ {1}, \pmb {e} _ {2}, \pmb {e} _ {3}) \| p _ {\pmb {\theta}} (\pmb {x} | \pmb {e} _ {1} ^ {i, \dots , s}, \pmb {e} _ {2} ^ {i, \dots , d}, \pmb {e} _ {3} ^ {i, \dots , d})) \\ \leq \mathbb {E} _ {q \left(\boldsymbol {e} _ {1} ^ {i, \dots , s}, \boldsymbol {e} _ {2} ^ {i, \dots , d}, \boldsymbol {e} _ {3} ^ {i, \dots , d}\right)} \mathcal {D} \left(\mathbb {E} _ {q \left(\boldsymbol {e} _ {1} ^ {s + 1, \dots , d}\right)} q (\boldsymbol {x} | \boldsymbol {e} _ {1}, \boldsymbol {e} _ {2}, \boldsymbol {e} _ {3}) \| \mathbb {E} _ {q \left(\boldsymbol {e} _ {1} ^ {s + 1, \dots , d}\right)} p _ {\boldsymbol {\theta}} (\boldsymbol {x} | \boldsymbol {e} _ {1} ^ {i, \dots , d}, \boldsymbol {e} _ {2} ^ {i, \dots , d}, \boldsymbol {e} _ {3} ^ {i, \dots , d})\right) \\ \leq \mathbb {E} _ {q \left(\boldsymbol {e} _ {1} ^ {s + 1, \dots , d}\right)} \mathbb {E} _ {q \left(\boldsymbol {e} _ {1} ^ {i, \dots , s}, \boldsymbol {e} _ {2} ^ {i, \dots , d}, \boldsymbol {e} _ {3} ^ {i, \dots , d}\right)} \mathcal {D} (q (\boldsymbol {x} | \boldsymbol {e} _ {1}, \boldsymbol {e} _ {2}, \boldsymbol {e} _ {3}) \| p _ {\boldsymbol {\theta}} (\boldsymbol {x} | \boldsymbol {e} _ {1} ^ {i, \dots , d}, \boldsymbol {e} _ {2} ^ {i, \dots , d}, \boldsymbol {e} _ {3} ^ {i, \dots , d})) \\ = \mathbb {E} _ {q \left(e _ {1} ^ {i, \dots , d}, e _ {2} ^ {i, \dots , d}, e _ {3} ^ {i, \dots , d}\right)} \mathcal {D} \left(q \left(\boldsymbol {x} \mid e _ {1}, e _ {2}, e _ {3}\right) \| p _ {\boldsymbol {\theta}} \left(\boldsymbol {x} \mid e _ {1} ^ {i, \dots , d}, e _ {2} ^ {i, \dots , d}, e _ {3} ^ {i, \dots , d}\right)\right) \tag {6} \\ \end{array}
+$$
+
+The last line is because of the assumption that the elements of the image feature are independent of each other. Thus we complete the proof of  $D_{2} \leq D_{1}$ .
+
+Proof of Theorem 2 Suppose the divergence  $\mathcal{D}$  is convex, and the function space  $\Phi$  and  $\Psi$  ( $\phi \in \Phi$  and  $\psi \in \Psi$ ) includes all measurable functions. Under Assumption 1 and by Jensen's inequality, we have:
+
+$$
+\begin{array}{l} \mathcal {D} (q (\boldsymbol {x} | c _ {1}, c _ {3}) \| p _ {\boldsymbol {\theta}, \psi} (\boldsymbol {x} | c _ {1}, c _ {3})) = \mathcal {D} (\mathbb {E} _ {q (c _ {2})} q (\boldsymbol {x} | c _ {1}, c _ {2}, c _ {3}) \| \mathbb {E} _ {q (c _ {2})} p _ {\boldsymbol {\theta}, \psi} (\boldsymbol {x} | c _ {1}, c _ {2}, c _ {3})) \\ \begin{array}{l} = \mathcal {D} \left(\mathbb {E} _ {q (c _ {2})} q (\boldsymbol {x} | c _ {1}, c _ {2}, c _ {3}) \| \mathbb {E} _ {q (c _ {2})} p _ {\boldsymbol {\theta}, \boldsymbol {\psi}} (\boldsymbol {x} | c _ {1}, c _ {2}, c _ {3})\right) \\ \leq \mathbb {E} _ {q (c _ {2})} \mathcal {D} \left(q (\boldsymbol {x} | c _ {1}, c _ {2}, c _ {3}) \| p _ {\boldsymbol {\theta}, \boldsymbol {\psi}} (\boldsymbol {x} | c _ {1}, c _ {2}, c _ {3})\right) \end{array} \tag {7} \\ \end{array}
+$$
+
+Under Assumption 1, the condition  $c_{1} \in \mathcal{C}_{1}$  and  $c_{3} \in \mathcal{C}_{3}$  are independent of each other, and the condition  $c_{2}$  is dependent on the style reference  $c_{1}$ . Given the condition  $c_{1} = x$ , the content information  $c_{2} = y$  is uniquely determined. Thus, we have  $q(c_{2})q(c_{1},c_{3}) \geq q(c_{1},c_{2},c_{3})$ , since  $q(c_{1} = x,c_{2} = y,c_{3} = z) = q(c_{1} = x,c_{3} = z)$  and  $q(c_{1} = x,c_{2} \neq y,c_{3} = z) = 0$ .
+
+By the Tower Law (Ash & Doleans-Dade, 2000) and non-negativity of  $\mathcal{D}$ , we have
+
+$$
+\begin{array}{l} \mathbb {E} _ {q \left(c _ {1}, c _ {2}, c _ {3}\right)} \mathcal {D} \left(q \left(\boldsymbol {x} \mid c _ {1}, c _ {2}, c _ {3}\right) \| p _ {\theta , \boldsymbol {\psi}} (\boldsymbol {x} \mid c _ {1}, c _ {3})\right) \\ \leq \mathbb {E} _ {q (c _ {2} | c _ {1}, c _ {3})} \mathbb {E} _ {q (c _ {1}, c _ {3})} \mathcal {D} \left(q (\boldsymbol {x} | c _ {1}, c _ {2}, c _ {3}) \| p _ {\theta , \boldsymbol {\psi}} (\boldsymbol {x} | c _ {1}, c _ {3})\right) \tag {8} \\ = \mathbb {E} _ {q \left(c _ {1}, c _ {3}\right)} \mathcal {D} \left(q \left(\boldsymbol {x} \mid c _ {1}, c _ {3}\right) \| p _ {\theta , \psi} (\boldsymbol {x} \mid c _ {1}, c _ {3})\right) \\ \end{array}
+$$
+
+It is straightforward to extend this to the minimization case, leading to the following inequality:
+
+$$
+\min  _ {\psi} \mathbb {E} _ {q \left(c _ {1}, c _ {2}, c _ {3}\right)} \mathcal {D} \left(q \left(\boldsymbol {x} \mid c _ {1}, c _ {2}, c _ {3}\right) \| p _ {\boldsymbol {\theta}, \psi} (\boldsymbol {x} \mid c _ {1}, c _ {3})\right) \leq \min  _ {\psi} \mathbb {E} _ {q \left(c _ {1}, c _ {3}\right)} \mathcal {D} \left(q \left(\boldsymbol {x} \mid c _ {1}, c _ {3}\right) \| p _ {\boldsymbol {\theta}, \psi} (\boldsymbol {x} \mid c _ {1}, c _ {3})\right) \tag {9}
+$$
+
+Combining Equation 7 and Equation 9, we have:
+
+$$
+\begin{array}{l} \min _ {\boldsymbol {\psi}} \mathbb {E} _ {q (c _ {1}, c _ {2}, c _ {3})} \mathcal {D} (q (\boldsymbol {x} | c _ {1}, c _ {2}, c _ {3}) \| p _ {\boldsymbol {\theta}, \boldsymbol {\psi}} (\boldsymbol {x} | c _ {1}, c _ {3})) \\ \leq \min  _ {\phi} \mathbb {E} _ {q \left(c _ {1}, c _ {2}, c _ {3}\right)} \mathcal {D} \left(q \left(\boldsymbol {x} \mid c _ {1}, c _ {2}, c _ {3}\right) \| p _ {\boldsymbol {\theta}, \phi} (\boldsymbol {x} \mid c _ {1}, c _ {2}, c _ {3})\right) \tag {10} \\ \end{array}
+$$
+
+Thus complete the proof.
+
+![](images/489ec220e6f98ebbfb37f52960878cdb4ac471fcac052578092c3496b4749d8b.jpg)  
+Figure 8: Visual examples of the style-reference images in our constructed dataset. The training sets are only used for learning Image-Adapter and Text-Adapter.
+
+# A.3 ALGORITHM
+
+As shown in the 9th line of Algorithm 2, we highlight the optimization objective for the Text-Adapter and Image-Adapter, respectively. These optimization objectives are used to minimize the prediction error of noise in reconstructing style reference while maximizing the difference between the model conditioned on the style reference's content and that conditioned on the target prompt.
+
+Algorithm 1 Algorithm of the proposed Tuning-Free masking-based method  
+1: Input: the style reference  $c_{1}$ , default style reference's content text prompt  $c_{2}$ , target text prompt  $c_{3}$ , VAE-encoder  $\mathcal{E}$ , pre-defined parameters  $\alpha_{t}$ , the repeat times of time travel  $T$ .  
+2: for  $t = 1$  to  $T$  do  
+3:  $\epsilon \sim \mathcal{N}(\mathbf{0},\mathbf{I}), \mathbf{z}_0 = \mathcal{E}(c_1)$   
+4:  $\mathbf{z}_{t} = \alpha_{t}\mathbf{z}_{t - 1} + (1 - \alpha_{t})\epsilon$  Add noise to the latent feature  
+5: end for  
+6: for  $t = T$  to 1 do  
+7: Calculate noise prediction (InstantStyle performs (a) and ours performs (b)):  
+(a)  $\epsilon_{\theta}(\mathbf{z}_{t},t,\pmb {e}_{1} - \pmb{e}_{2},\pmb{e}_{3})$  (InstantStyle)  
+(b) Perform clustering (such as K-means clustering (Jin & Han, 2010)) on element product  $e_1^i\cdot e_2^i$ ; Generate masking vector  $\mathbf{m}$ , where we set the element value to 1 for those elements in the highest-means cluster; Calculate  $\epsilon_{\theta}(\mathbf{z}_{t},t,\pmb {e}_{1}\odot \mathbf{m},\pmb{e}_{3})$  (Ours)  
+8: Denoise diffusion model using predicted noise;  
+9: end for  
+10: Output: decode the reversed latent code  $\mathbf{z}_0$  to image space and output the generated image
+
+Algorithm 2 Algorithm of the Tuning-Based model: Image-Adapter and Text-Adapter  
+1: Input: the data distribution of style reference  $c_{1}$ , style reference's content text prompt  $c_{2}$  and target text prompt  $c_{3}$ , text adapter  $\phi_{\theta_1}(\cdot)$ , image adapter  $\psi_{\theta_2}(\cdot)$ , VAE-encoder  $\mathcal{E}$ , pre-defined parameters  $\alpha_{t}$ , hyper-parameter  $\lambda$ , maximum training step  $M$  and the repeat times of time travel  $T$ .  
+2: for  $m = 1$  to  $M$  do  
+3:  $(c_{1}, c_{2}, c_{3}) \sim p(c_{1}, c_{2}, c_{3}) \triangleright$  Sample conditions from data distribution  
+4: for  $t = 1$  to  $T$  do  
+5:  $\epsilon \sim \mathcal{N}(\mathbf{0}, \mathbf{I}), \mathbf{z}_0 = \mathcal{E}(c_1)$   
+6:  $\mathbf{z}_t = \alpha_t \mathbf{z}_{t-1} + (1 - \alpha_t) \epsilon \triangleright$  Add noise to the latent feature  
+7: end for  
+8: for  $t = T$  to 1 do  
+9: Take the gradient step (Text-Adapter performs (a) and Image-Adapter performs (b)):  
+(a)  $\nabla_{\theta_1}[\|\epsilon_\theta(\mathbf{z}_t, t, e_1 - \phi_{\theta_1}(e_2), e_3) - \epsilon\|^2 - \lambda\|\epsilon_\theta(\mathbf{z}_t, t, e_1 - \phi_{\theta_1}(e_2), e_3) - \epsilon_\theta(\mathbf{z}_t, t, e_1 - \phi_{\theta_1}(e_2), e_3)\|^{2}$  (Text-Adapter)  
+(b)  $\nabla_{\theta_2}[\|\epsilon_\theta(\mathbf{z}_t, t, e_1 - \psi_{\theta_2}(e_1), e_3) - \epsilon\|^2 - \lambda\|\epsilon_\theta(\mathbf{z}_t, t, e_1 - \psi_{\theta_2}(e_1), e_3) - \epsilon_\theta(\mathbf{z}_t, t, e_1 - \psi_{\theta_2}(e_1), e_3)\|^{2}$  (Image-Adapter)  
+10: Update the model parameter  $\theta_1$  or  $\theta_2$   
+11: end for  
+12: end for  
+13: Output: The Image-Adapter and Text-Adapter model
+
+# A.4 EVALUATION DATASETS AND METRICS
+
+# A.4.1 EVALUATION DATASETS
+
+Previous evaluation datasets do not contain explicitly defined references' contents, thus making it inaccurate in evaluating content leakages. Instead, we consider an evaluation dataset comprising various defined reference contents and styles for comprehensively assessing models' capability in addressing content leakages. In this paper, we construct the evaluation dataset consisting of 10 content objects and 21 image styles. We present detailed information on the constructed dataset in Table 4. Visual examples of the style-reference images are provided in Figure 8.
+
+# A.4.2 EVALUATION METRICS
+
+Following previous studies (Sohn et al., 2024; Gao et al., 2024), we report the image alignment and text alignment scores in Table 2 and Figure 5.
+
+Image alignment and text alignment  $\uparrow$ : Image alignment refers to the cosine similarity between the CLIP embeddings of the generated images and the style reference images, while text alignment
+
+measures the cosine similarity between the CLIP embeddings of the generated images and the target text prompts.
+
+Our method is specifically designed to address content leakage in style transfer diffusion models. In addition to the image alignment and text alignment scores used in previous studies (Sohn et al., 2024; Gao et al., 2024), we also introduce three quantitative metrics to comprehensively assess the quality of the generated images from the perspectives of style similarity, text fidelity, and content leakages from style references. We denote the CLIP image feature of the generated image and the style-reference image as  $e_g$  and  $e_1$ , respectively. Given the class name of style reference's content object and the target text prompt, we denote the CLIP text feature of reference's content object and text prompt as  $e_2$  and  $e_3$ , respectively.
+
+Fidelity score  $\uparrow$ : We perform binary classification on the generated images to differentiate between the reference's content object and the text prompt, computing the classification accuracy, referred to as the fidelity score  $\uparrow$ . Specifically, we denote the cosine similarity between the CLIP image feature of the generated image and the CLIP text feature of reference's content object as  $\frac{\langle e_2,e_g\rangle}{|e_2|\cdot|e_g|}$ . Similarly, we denote the cosine similarity between the CLIP image feature of the generated image and the CLIP text feature of text prompt as  $\frac{\langle e_3,e_g\rangle}{|e_3|\cdot|e_g|}$ . If  $\frac{\langle e_2,e_g\rangle}{|e_2|\cdot|e_g|} < \frac{\langle e_3,e_g\rangle}{|e_3|\cdot|e_g|}$ , the generated image is considered correctly classified, meaning it contains the target content rather than the content of the style reference. Therefore, the fidelity score mainly reflects the models' control ability of text prompts.
+
+Leakage score  $\downarrow$ : We calculate the similarity between the generated images and the reference's content text, calibrated by the similarity between the reference images and the content text, termed the leakage score  $\downarrow$ . Thus, the leakage score is calculated by:
+
+$$
+\left\{ \begin{array}{l l} <   e _ {g}, e _ {2} > / <   e _ {1}, e _ {2} > & \text {i f} e _ {g} \text {i s a c c u r a t e l y c l a s s i f i e d} \\ 1 & \text {e l s e} \end{array} \right. \tag {11}
+$$
+
+The leakage score indicates content leakages from the style reference, where a lower score is preferable.
+
+Style score  $\uparrow$ : Finally, we assess the similarity between the generated images and the style reference, adjusting by subtracting the similarity between the generated images and the reference's content text, which we refer to as the style score  $\uparrow$ . The style score is calculated as follows:
+
+$$
+\left\{ \begin{array}{l l} <   e _ {g}, e _ {1} > - <   e _ {g}, e _ {2} > & \text {i f} e _ {g} \text {i s a c c u r a t e l y c l a s s i f i e d} \\ 0 & \text {e l s e} \end{array} \right. \tag {12}
+$$
+
+Compared to the image alignment score, the style score more accurately reflects the style similarity between the generated images and the style references.
+
+Human preference score  $\uparrow$ : In addition to objective evaluations, we have also designed a user study to subjectively assess the practical performance of various methods. In Section 4.1, the constructed dataset consists of 10 content objects (from CIFAR-10) and 21 image styles (11 for training and 10 for testing) for each content object, with 8 variations per style. This results in a total of  $10 \times 11 \times 8 = 880$  style references. For each style reference, we perform style transfer for 5 target text prompts, with 4 generations per target text prompt, leading to  $880 \times 4 = 3520$  generations per text prompt. We randomly sample 50 images from the 3520 generated images for each target text prompt. In total, this gives us  $50 \times 5 = 250$  images from each method to evaluate. The same procedure is applied in the evaluation presented in Section 4.2. We asked 10 users from diverse backgrounds to evaluate the generated results in terms of text fidelity, content leakages and style similarity, and to provide their overall preference considering these three aspects. Finally, the final average results are displayed in Table 1 and Table 2.
+
+# A.5 COEFFICIENT-TUNING RESULTS
+
+We present the coefficient-tuning results of the IP-Adapter and InstantStyle in Figure 10. For IP-Adapter and InstantStyle, lowering the coefficient for image condition helps to enhance the control ability of text prompts, but it also comes with style degradation. Moreover, the coefficient-tuning process is quite labour-intensive. In contrast, our method can achieve much more stable results across various coefficient values. Figure 10 highlights the limitations of the InstantStyle approach in decoupling content and style, particularly regarding labour-intensive coefficient tuning.
+
+Table 4: Evaluation Datasets in Section 4.1.  
+
+<table><tr><td rowspan="7">Train</td><td rowspan="6">Style References</td><td colspan="6">Content</td></tr><tr><td>An automobile</td><td>An airplane</td><td>A bird</td><td>A cat</td><td>A deer</td><td>A dog</td></tr><tr><td>A horse</td><td>A frog</td><td>A ship</td><td>A truck</td><td></td><td></td></tr><tr><td colspan="6">Styles</td></tr><tr><td>natural environment</td><td>rainy day</td><td>snowy day</td><td>3D model</td><td>abstract</td><td>cyberpunk</td></tr><tr><td>oil painting</td><td>realism</td><td>watercolor</td><td>beautiful landscape</td><td>watercolor</td><td></td></tr><tr><td>Text Prompts</td><td colspan="6">Image reconstruction: use the content of style reference</td></tr><tr><td rowspan="7">Test</td><td rowspan="6">Style References</td><td colspan="6">Content</td></tr><tr><td>An automobile</td><td>An airplane</td><td>A bird</td><td>A cat</td><td>A deer</td><td>A dog</td></tr><tr><td>A horse</td><td>A frog</td><td>A ship</td><td>A truck</td><td></td><td></td></tr><tr><td colspan="6">Styles</td></tr><tr><td>lego toy</td><td>statue</td><td>steampunk</td><td>stick figure</td><td>tattoo</td><td>line art</td></tr><tr><td>wool toy</td><td>surround with colored smoke</td><td>pop art</td><td>surround with white smoke</td><td></td><td></td></tr><tr><td>Text Prompts</td><td>A bench</td><td>A human</td><td>A laptop</td><td>A rocket</td><td>A flower</td><td></td></tr></table>
+
+# A.6 MORE VISUALIZATION RESULTS FOR SECTION 4
+
+We present more visualization results for Section 4. In Figure 12-16, we mark the results with significant content leakages, style degradation, and loss of text fidelity with red, green, and blue boxes, respectively.
+
+# A.7 COMPARISON BETWEEN OUR MASKING STRATEGY AND INSTANTSTYLE
+
+To comprehensively demonstrate the effectiveness of our method in image-driven style transfer, we compare InstantStyle's feature subtraction (i.e., directly subtracting the content text feature from the image feature) and our masking-based method based on StyleShot's style encoder, providing visualization results in Figure 17 (a). Due to image-text misalignment, InstantStyle may disrupt the style information extracted by StyleShot's style encoder. In contrast, guided by appropriately-selected fewer conditions and without introducing additional content text features during the denoising process, our method successfully preserves the style features. Furthermore, we conduct ablation studies on the backbone of the diffusion model, as shown in Figure 17 (b). Using the SDXL diffusion model (Podell et al., 2023), we compare our method with InstantStyle's block-specific injection technique, which injects image features into style-specific blocks in the text-driven style transfer task. While InstantStyle encounters style disruption, our method significantly alleviates this problem by leveraging fewer appropriately-selected conditions.
+
+# A.8 ABLATION STUDIES ON CLUSTERING NUMBER
+
+We ablate on cluster number in the text-driven style transfer based on the StyleBench dataset. We report the image alignment and text alignment results based on three different CLIP backbones in Table 5 and provide the visual comparisons of our masking-based method with varying cluster numbers in Figure 18. It is shown that a smaller  $K$ , such as  $K = 2$ , can lead to better performance in avoiding content leakage, as more content-related elements in the style reference are masked. This is particularly evident in styles such as 3D models, Anime, and Baroque art, which contain more human-related images. In these cases, a smaller  $K$  results in higher text alignment scores and more effective avoidance of content leakage.
+
+# A.9 SIMULATION RESULTS OF PROPOSITION 1
+
+We conduct simulation experiments based on our constructed dataset to demonstrate Proposition 1. Using the energy score proposed by (Liu et al., 2021), we calculate the energy scores of the masked image features for two different masking approaches: one based on clustering the product of  $e_1^i$  and  $e_2^i$  ( $e_1^i \cdot e_2^i, i \in \{1, \dots, d\}$ ) and the other based on clustering the absolute difference of  $e_1^i$  and  $e_2^i$  ( $|e_1^i - e_2^i|, i \in \{1, \dots, d\}$ ). For both methods, we report the 0th, 25th, 50th, 75th, and 100th percentiles of the energy scores. As shown in Table 6, our method consistently generates higher energy scores when discriminating content  $c_2$ , confirming the results outlined in Proposition 1.
+
+# A.10 VISUAL COMPARISON IN TRADITIONAL STYLE TRANSFER
+
+In Figure 21, we provide additional results comparing the performance of our method with previous approaches in traditional style transfer, where non-object style images are used as style references.
+
+Table 5: Ablation study results on clustering number  $K$  .  
+
+<table><tr><td></td><td colspan="4">CLIP Model</td></tr><tr><td></td><td>K</td><td>ViT-B/32</td><td>ViT-L/14</td><td>ViT-H/14</td></tr><tr><td rowspan="4">image alignment</td><td>2</td><td>0.657</td><td>0.608</td><td>0.403</td></tr><tr><td>3</td><td>0.656</td><td>0.611</td><td>0.410</td></tr><tr><td>4</td><td>0.657</td><td>0.615</td><td>0.415</td></tr><tr><td>5</td><td>0.657</td><td>0.614</td><td>0.415</td></tr><tr><td rowspan="4">text alignment</td><td>2</td><td>0.265</td><td>0.212</td><td>0.258</td></tr><tr><td>3</td><td>0.264</td><td>0.211</td><td>0.253</td></tr><tr><td>4</td><td>0.265</td><td>0.210</td><td>0.252</td></tr><tr><td>5</td><td>0.264</td><td>0.210</td><td>0.252</td></tr><tr><td></td><td>Style</td><td>3D Model</td><td>Anime</td><td>Baroque</td></tr><tr><td></td><td>K</td><td></td><td></td><td></td></tr><tr><td rowspan="2">image alignment</td><td>2</td><td>0.474</td><td>0.372</td><td>0.384</td></tr><tr><td>3</td><td>0.478</td><td>0.381</td><td>0.393</td></tr><tr><td rowspan="2">based on ViT-H/14</td><td>4</td><td>0.485</td><td>0.390</td><td>0.404</td></tr><tr><td>5</td><td>0.487</td><td>0.380</td><td>0.411</td></tr><tr><td rowspan="2">text alignment</td><td>2</td><td>0.213</td><td>0.234</td><td>0.257</td></tr><tr><td>3</td><td>0.206</td><td>0.232</td><td>0.253</td></tr><tr><td rowspan="2">based on ViT-H/14</td><td>4</td><td>0.189</td><td>0.231</td><td>0.253</td></tr><tr><td>5</td><td>0.188</td><td>0.229</td><td>0.252</td></tr></table>
+
+Table 6: The energy scores of the masked image features for two different masking approaches. We report the 0th, 25th, 50th, 75th, and 100th percentiles of the energy scores for each approach.  
+
+<table><tr><td>Masking Proportion</td><td>Method</td><td>0th</td><td>25th</td><td>50th</td><td>75th</td><td>100th</td></tr><tr><td>5%</td><td>clustering on ei·i·2(Ours)</td><td>-10.78</td><td>-6.59</td><td>-5.36</td><td>-4.08</td><td>1.64</td></tr><tr><td>5%</td><td>clustering on |ei1 - ei2|</td><td>-13.87</td><td>-9.63</td><td>-8.56</td><td>-7.54</td><td>-1.60</td></tr><tr><td>10%</td><td>clustering on ei·i·2(Ours)</td><td>-9.15</td><td>-5.15</td><td>-4.00</td><td>-2.77</td><td>2.02</td></tr><tr><td>10%</td><td>clustering on |ei1 - ei2|</td><td>-11.57</td><td>-8.80</td><td>-7.88</td><td>-7.05</td><td>-2.18</td></tr><tr><td>20%</td><td>clustering on ei·i·2(Ours)</td><td>-7.46</td><td>-3.46</td><td>-2.36</td><td>-1.35</td><td>2.66</td></tr><tr><td>20%</td><td>clustering on |ei1 - ei2|</td><td>-10.73</td><td>-7.57</td><td>-6.91</td><td>-6.20</td><td>-3.16</td></tr><tr><td>30%</td><td>clustering on ei·i·2(Ours)</td><td>-6.05</td><td>-2.62</td><td>-1.58</td><td>-0.63</td><td>2.87</td></tr><tr><td>30%</td><td>clustering on |ei1 - ei2|</td><td>-9.19</td><td>-6.73</td><td>-6.13</td><td>-5.59</td><td>-3.31</td></tr><tr><td>40%</td><td>clustering on ei·i·2(Ours)</td><td>-5.71</td><td>-2.23</td><td>-1.29</td><td>-0.43</td><td>2.70</td></tr><tr><td>40%</td><td>clustering on |ei1 - ei2|</td><td>-8.04</td><td>-5.99</td><td>-5.51</td><td>-5.07</td><td>-3.61</td></tr><tr><td>50%</td><td>clustering on ei·i·2(Ours)</td><td>-5.24</td><td>-1.94</td><td>-1.13</td><td>-0.37</td><td>2.69</td></tr><tr><td>50%</td><td>clustering on |ei1 - ei2|</td><td>-7.26</td><td>-5.37</td><td>-4.99</td><td>-4.60</td><td>-3.63</td></tr><tr><td>60%</td><td>clustering on ei·i·2(Ours)</td><td>-4.93</td><td>-1.72</td><td>-0.92</td><td>-0.22</td><td>2.72</td></tr><tr><td>60%</td><td>clustering on |ei1 - ei2|</td><td>-6.23</td><td>-4.85</td><td>-4.53</td><td>-4.19</td><td>-3.29</td></tr><tr><td>70%</td><td>clustering on ei·i·2(Ours)</td><td>-3.91</td><td>-1.25</td><td>-0.53</td><td>0.15</td><td>2.86</td></tr><tr><td>70%</td><td>clustering on |ei1 - ei2|</td><td>-5.62</td><td>-4.32</td><td>-4.06</td><td>-3.77</td><td>-2.98</td></tr><tr><td>80%</td><td>clustering on ei·i·2(Ours)</td><td>-3.11</td><td>-0.77</td><td>-0.14</td><td>0.53</td><td>2.20</td></tr><tr><td>80%</td><td>clustering on |ei1 - ei2|</td><td>-4.72</td><td>-3.77</td><td>-3.57</td><td>-3.36</td><td>-2.57</td></tr><tr><td>90%</td><td>clustering on ei·i·2(Ours)</td><td>-2.40</td><td>-0.67</td><td>-0.18</td><td>0.37</td><td>2.12</td></tr><tr><td>90%</td><td>clustering on |ei1 - ei2|</td><td>-4.00</td><td>-3.32</td><td>-3.15</td><td>-3.01</td><td>-2.55</td></tr></table>
+
+As shown in Figure 21, when using non-object style references, previous methods such as StyleDrop (Sohn et al., 2024) and StyleShot (Gao et al., 2024) may suffer from style degradation or a loss of text fidelity. As pointed out in the original paper (Rout et al., 2024), "The inherent limitations of the style descriptor or diffusion model might propagate into our framework", RB-Modulation Rout et al. (2024) may fail to preserve the style of the reference when the style description does not align well with the image reference, as illustrated in the 3rd and 4th lines in Figure 21. As shown in the 9th and 10th lines in Figure 21, CSGO (Xing et al., 2024) may also suffer from style degradation or loss of text fidelity, showing inferior performance compared to our method. In contrast, from Figure 6 and Figure 21, the proposed method demonstrates superior stylization, performing better in both object-centered style references and non-object style references.
+
+# A.11 ETHICS STATEMENT
+
+This work aims to make a positive impact on the field of AI-driven image generation. We aim to facilitate the creation of images with diverse styles, and we expect all related processes to comply with laws and be used responsibly.
+
+The use of AI to generate human-related images, particularly those involving characteristics such as skin color, gender, age, and other demographic factors, raises complex ethical questions. We are aware that the generation of images involving these attributes must be handled with care to avoid reinforcing stereotypes, perpetuating discrimination, or contributing to the misrepresentations of certain groups. We take these concerns very seriously and believe that AI should be used in a way that promotes fairness, inclusion, and respect for all individuals. Here, we give several examples of text prompts containing different genders, skin colors, and ages, as shown in Figure 22. We observe that in most cases, our method is able to generate images with diversity. However, there are certain cases that general image-generation methods can be misused, and caution must be exercised to mitigate any potential harm.
+
+Our methodology builds upon existing open-source diffusion models (e.g., Stable Diffusion V1.5 (Rombach et al., 2022) and SDXL (Podell et al., 2023)), which already include some built-in safeguards, such as the NSFW ("not safe for work") Filter. We believe that our contributions in generating images with diverse styles, while preventing content leakages, can further positively help avoid stereotypes and discriminations. Moreover, successfully preventing the leakage of style reference' content into the generated image can avoid the unintended blending of the target subject with the style reference' content, thereby helping to mitigate malicious or offensive generations. Furthermore, we are committed to ongoing research in this area, aiming to enhance safety mechanisms and prevent the creation of malicious or offensive imagery. Ultimately, we believe that the responsible use of AI-driven image-generation tools is essential to fostering ethical and equitable outcomes in the field.
+
+![](images/a3e6520be1eb3469ac8a73151aec130e2ae971028ab1ca07e1912d184d1b74d8.jpg)  
+Figure 9: Effectiveness of the proposed masking strategy over feature subtraction in avoiding content leakages. The superior performance of our method showcases that the appropriately-selected fewer conditions can more efficiently avoid content leakages.
+
+![](images/c63f59b131facfec78add24cebea8947e5e5e6b3c6b38839389b21a39eb0e7d7.jpg)  
+Figure 10: The coefficient-tuning results of IP-Adapter (Ye et al., 2023), InstantStyle (Wang et al., 2024a) and our proposed method. We use different coefficients for the image condition (i.e.,  $\lambda_{i}$  in Equation 2). We highlight the satisfactory results with green boxes.
+
+![](images/d47a0d7aaeedb44643a6ef91adec4ccc68c52dbe38e3ac71be21658b630396e6.jpg)  
+Figure 11: Visual comparison between recent state-of-the-art methods and ours in text-driven style transfer on StyleBench. The proposed masking-based method does not require content knowledge of the image reference; instead, we leverage the CLIP text feature of "person, animal, plant, or object in the foreground" to identify the elements that need to be masked.
+
+![](images/d2c347ae5c11e2a82cbe99228b3a90c908d089960766bbf0f5d71e0a8611d4bb.jpg)  
+Text prompt: A human  
+Figure 12: Visual comparison between recent state-of-the-art methods and ours in text-driven style transfer, where the text prompt is "A human".
+
+![](images/8e48781232158759a9ff4fdaf123db5f8a24c4c393c1609789b38be3f74c64fd.jpg)  
+Text prompt: A bench  
+Figure 13: Visual comparison between recent state-of-the-art methods and ours in text-driven style transfer, where the text prompt is "A bench".
+
+![](images/b2cdadab224ce1f8df558dcf433551508788db6e481dceb9502ba867c1740428.jpg)  
+Text prompt: A laptop  
+Figure 14: Visual comparison between recent state-of-the-art methods and ours in text-driven style transfer, where the text prompt is "A laptop".
+
+![](images/0317f543191fad9ed433f8e23925c29a088f00b64d2be7eabc0374851d0c3925.jpg)  
+Text prompt: A flower  
+Figure 15: Visual comparison between recent state-of-the-art methods and ours in text-driven style transfer, where the text prompt is "A flower".
+
+![](images/f526b6f8b7f72ef8a5787ae913525d7f23cb49ae1e19114985a351fb33f3e0fa.jpg)  
+Text prompt: A rocket
+
+Target Content
+
+Style reference
+
+constantStyle
+
+Ours
+
+Style Reference
+
+nstantStyle
+
+Ours
+
+S Refs
+
+lence
+
+A bench
+
+A car
+
+A laptoc
+
+A house
+
+with a preparing meals tree beside in kitchen
+
+![](images/25a5612bbb134c75ddf722a45dd7ef877fdb67dd6fc6c145e42df56ddb77fe17.jpg)
+
+![](images/6930feb86fbac5a61e92128d90372bd9bb1ef56d868531f146c93d0fa0052e72.jpg)
+
+![](images/eb376b002eb5ae2b20184fec1d5a21c50a7e84a951e29220d2a491a65a03524a.jpg)  
+(a) Visual comparison in image-driven style transfer
+
+![](images/cee4073e3d6409b1699bb2115384396764174b995acbbbb019156faf56bbdefb.jpg)  
+Figure 16: Visual comparison between recent state-of-the-art methods and ours in text-driven style transfer, where the text prompt is "A rocket".
+
+![](images/582c49ec1456f07753fcb65715a759bdad4985201d8d914bdf4537be859ddfba.jpg)  
+Figure 17: (a) Visual comparison between InstantStyle's feature subtraction and ours in image-driven style transfer. (b) Visual comparison between InstantStyle's block-specific injection techniques and ours in text-driven style transfer based on SDXL.
+
+![](images/f31e0b2cb098d0b7f83716b57d3574e130d9c5a685e699e9f52625a6dbaeace3.jpg)
+
+![](images/185f012e95c226afac50488593135ac56582c9bea30a96763d0cfa5b1be3ac1d.jpg)
+
+![](images/9c46e3f901385de69ca1e67c7409dfefc753b380727ceeadbe8a1d6d7d797824.jpg)
+
+![](images/f8daced707f46baca92fad4183c73ecf7fd069eb9c72f58777f685e6822c83cb.jpg)
+
+![](images/fdb39604154cfdae55c9f6267510940464bc0d6d1df2f0505cdd08e9a428eaca.jpg)
+
+![](images/b723543edffbf99eb75712672a31a15454940054dc4b70214352071faebc33a4.jpg)
+
+![](images/f67a921b4e011e1db900666f564c6122cbf9670b4801328876895f87687407a6.jpg)
+
+![](images/9ae3f4f7ddb823c6b571cd02ef4cf99cf7b979969635d7da0c26cad44ca57620.jpg)  
+(b) Visual comparison in text-driven style transfer based on SDXL
+
+![](images/23a93fc9077ca1d5932fb90541877c1d7a9c2f2ae9df1e98e07ac8d6ef94b921.jpg)
+
+![](images/106c194f57189c1ab4c53414cbf6cc89b28a7ddfe802333197bdb0b8e78a22bb.jpg)
+
+![](images/f657d83f90ffb12fc7a4b0c7900484bcfbf056c36ae90121ec3fc26b810be022.jpg)
+
+![](images/17edb434d190a0973070ae32eecba51a367b671742da9880e8d479796fa0d2b4.jpg)  
+向西区 2023 高联竞赛班笔记-1 的第 8 页
+
+![](images/556a19cbf0dca2c940ec9dc67059e3df23a3c8533bead9ad45c38127a1c0ebca.jpg)
+
+![](images/b74b86d4fa93838606afbbb8fbe51d148afe92cc7a6c5dac3f7b67e70009a6e4.jpg)
+
+![](images/5f3ae4b154c0f92871666bb5b19e76e513def6c0a9a62c7782e87d4446d9bfc4.jpg)
+
+![](images/391148e1f90cae5000e9d67839fee563b60bab6a72754cd252b31ab19e4b9fdd.jpg)
+
+![](images/67e8f56823b838ece8f59e76184d44c6d6e4e1745e033d2a0d1ca9af19b8d3d2.jpg)
+
+![](images/871a974fbdabe844129946c972e76e79ccb6106909fefc5f5336095f136fc341.jpg)
+
+![](images/687433a13a2f2605d85e229d549c9ab05521205bcc970f76babd09e7ee7bb558.jpg)
+
+![](images/43879e4122b453c8a826cad3e05dab430282d7347d5d0a4e147aa32abcae314b.jpg)
+
+![](images/ac409da1c042ad328058a0b6661696448735aa5408558b3edbffce5a08706c56.jpg)  
+Figure 18: Visual comparison of the proposed masking-based method with varying cluster numbers. It is shown that a smaller  $K$ , such as  $K = 2$ , can lead to better performance in avoiding content leakage, as more content-related elements in the style reference are masked. This is particularly evident in styles such as 3D models, Anime, and Baroque art, which contain more human-related images. In these cases, a smaller  $K$  results in higher text alignment scores and more effective avoidance of content leakage.
+
+![](images/7f531fc9290705c5a103e9c5b4e3f6b4ab72d7b215706c5f3e556e6da4ecca84.jpg)  
+Figure 19: Visual comparison between StyleShot and ours in image-driven style transfer. Results with content leakages and style degradation are highlighted with red and green boxes, respectively.
+
+![](images/009b2bdcdd2da149c53c06a24aa986138ee1855e7a2c899d565fe3003d42bf5f.jpg)  
+Figure 20: The one-to-one image generation results between the proposed masking-based method and StyleShot. We provide multi-sample generalization results for each combination of style reference and target text prompt. To mitigate the influence of randomness, we ensure that all model configurations remain consistent, including the random seed, guidance seed, denoising steps, and other parameters. From the one-to-one comparison, we observe that our method significantly reduces content leakage and alleviates loss of text fidelity, consistently refining StyleShot's results across all combinations. We mark the results with significant content leakages and loss of text fidelity with red and blue boxes, respectively.
+
+![](images/08ee0851d15b17ed5417e89c99cc2a51c6f9779384814d7e703e4ff1ed34928c.jpg)  
+Figure 21: Visual comparison between recent state-of-the-art methods in traditional style transfer.
+
+![](images/c0be0ffc87b121a8688b82e8f54e34d9955c8eac6c9ee8f3dadb2c4e438f3cc2.jpg)  
+Figure 22: Examples of text prompts contain different genders, skin colors, and ages.
\ No newline at end of file
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+# LET ME GROK FOR YOU: ACCELERATING GROKKING VIA EMBEDDING TRANSFER FROM A WEAKER MODEL
+
+Zhiwei  $\mathbf{X}\mathbf{u}^{\dagger*}$ , Zhiyu  $\mathbf{N}\mathbf{i}^{\ddagger*}$ , Yixin Wang $^{\dagger\diamond}$ , Wei  $\mathbf{H}\mathbf{u}^{\dagger\diamond}$
+
+†University of Michigan, ‡University of California, Berkeley
+
+{zhiweixu,yixinw,vvh}@umich.edu,zhiyuni@berkeley.edu
+
+# ABSTRACT
+
+"Grokking" (Power et al., 2022) is a phenomenon where a neural network first memorizes training data and generalizes poorly, but then suddenly transitions to near-perfect generalization after prolonged training. While intriguing, this delayed generalization phenomenon compromises predictability and efficiency. Ideally, models should generalize directly without delay. To this end, this paper proposes GrokTransfer, a simple and principled method for accelerating grokking in training neural networks, based on the key observation that data embedding plays a crucial role in determining whether generalization is delayed. GrokTransfer first trains a smaller, weaker model to reach a nontrivial (but far from optimal) test performance. Then, the learned input embedding from this weaker model is extracted and used to initialize the embedding in the target, stronger model. We rigorously prove that, on a synthetic XOR task where delayed generalization always occurs in normal training, GrokTransfer enables the target model to generalize directly without delay. Moreover, we demonstrate that, across empirical studies of different tasks, GrokTransfer effectively reshapes the training dynamics and eliminates delayed generalization, for both fully-connected neural networks and Transformers.
+
+# 1 INTRODUCTION
+
+"Grokking" is an intriguing phenomenon recently discovered by Power et al. (2022), where a neural network first memorizes the training dataset but has poor test performance, and after much longer training, it suddenly transitions to near-perfect generalization. Initially reported for Transformer models trained on modular arithmetic tasks, the grokking phenomenon has since been observed in other settings such as learning group operations (Chughtai et al., 2023), sparse parity (Barak et al., 2022), and image classification (Liu et al., 2023).
+
+While grokking is an interesting phenomenon, it introduces unpredictability into the training process and compromises its practical efficiency. When the model has interpolated the training data with small training loss but still performed poorly on the validation set, it becomes difficult to predict whether or when the model will eventually generalize. Ideally, we would like the model to make continuous progress during training, keeping the gap between training and validation errors minimal. This raises the question:
+
+How can we effectively modify the training dynamics so that the model generalizes without delay?
+
+In this work, we show that data embedding plays a crucial role in determining the training dynamics; an informative embedding enables continuous progress during training. To obtain such an informative embedding without excessive computational cost, we propose a novel method called GrokTransfer, which leverages the embedding learned by a weaker, smaller model to accelerate the generalization of a larger target model. See Figure 1a for an overview of GrokTransfer.
+
+Specifically, GrokTransfer involves two main steps: (1) Train a weaker model until it groks to non-trivial test performance; (2) Extract the weak model's learned embedding and use a linear
+
+![](images/59d35d718bdae4f191bec318563c72a2e94055e1570c2125a52b57a15f0c321e.jpg)  
+(a)
+
+![](images/ef80979de16851e445bf25572c6bea44c6a04b1a3b661d0590fe694a2b0bd28d.jpg)  
+(b)
+
+![](images/eddbd1fc9960a740176c09cf50a2d38b505218fa86e18874bd8904c3da6db596.jpg)  
+Figure 1: (a) Overview of the GrokTransfer framework. (b) Comparison of the training dynamics of a model trained using GrokTransfer versus one trained from scratch. There is a clear phase transition between memorization and generalization if we train the model from scratch (blue lines). GrokTransfer (red lines) enables the model to make continuous progress, significantly reducing the gap between memorization and generalization. See Appendix A.3 for the detailed experimental setup.
+
+mapping of this embedding to initialize the embedding of the target model. Then, proceed to train the target model. We theoretically study GrokTransfer in the setting of a two-layer neural network trained on a high-dimensional XOR classification task, where normal training exhibits grokking. We prove that GrokTransfer enables the target model to directly generalize without delay. We further empirically verify the effectiveness of GrokTransfer on typical algorithmic tasks that show grokking. This is done for both fully-connected neural networks with trainable embeddings and Transformers. Figure 1b shows typical training curves of GrokTransfer vs. training a target model from scratch, on a modular addition task. It shows that GrokTransfer effectively eliminates grokking and significantly improves efficiency.
+
+In summary, our contributions are as follows:
+
+- We propose a novel method, GrokTransfer, which leverages the embedding learned from a smaller, weaker model to accelerate grokking in the target model.  
+- We theoretically justify GrokTransfer in an XOR classification task. We further empirically validate our method on several algorithmic tasks that exhibit grokking in normal training, demonstrating that GrokTransfer can effectively eliminate delayed generalization.
+
+# 1.1 RELATED WORK
+
+Our work draws on two themes around grokking and weak-to-strong knowledge transfer.
+
+Grokking. Liu et al. (2022) reported that the model starts grokking when it learns the hidden structure of the data. Gromov (2023) showed that grokking is robust to different optimizers such as vanilla gradient descent and Adam; and regularization methods including no regularization, weight decay, and dropout. Davies et al. (2023) hypothesized that grokking and double descent, another surprising phenomenon, are caused by the same hidden mechanism. Nanda et al. (2023) reverse-engineered a grokked transformer model for modular addition and reported that the learned algorithm is a composition of trigonometric and inverse trigonometric functions. Merrill et al. (2023) and Varma et al. (2023) contributed to the occurrence of grokking to the competition of sparse (generalizing) and dense (complementary) subnetworks during training. Zhu et al. (2024) showed that models only grok when the training data exceeds some critical size. Liu et al. (2023) attributed grokking to large initialization scale and induced grokking on real-world datasets such as MNIST and IMDb by initializing models with large weight norm. Further work (Miller et al., 2023; Humayun et al., 2024) showed that grokking can also be observed in other scenarios such as Gaussian Process regression and multi-class classification with adversarial samples. A series of theoretical papers have established rigorous results for grokking/delayed generalization in several settings outside of algorithmic tasks: linear regression with linear models (Zunković & Ilievski, 2022), and binary classification with neural networks (Lyu et al., 2024; Xu et al., 2024). Lyu et al. (2024) proved that grokking can be induced by a sharp phase transition from kernel regime to rich regime. Mallinar et al. (2024) trained Recursive
+
+Feature Machines on algorithmic tasks and found its training dynamics similar to neural networks, showing that grokking is not restricted to neural networks. He et al. (2024); Wang et al. (2024) found transformers achieve out-of-distribution generalization on some tasks through grokking. Doshi et al. (2024) provided analytical solutions for complex modular arithmetic tasks and hypothesized that some complex modular polynomial tasks cannot be learned by shallow neural networks. Mohamadi et al. (2024) showed that learning modular addition is fundamentally hard for neural networks in the kernel regime. A related phenomenon, termed "sudden drop in the loss" (Chen et al., 2024; Gopalani et al., 2025; Yang et al., 2025), describes an abrupt drop in loss after an extended plateau during online training.
+
+Recent work has proposed several methods to accelerate grokking. Liu et al. (2023) explained grokking through the concept of a "Goldilocks zone", a spherical shell of weights, and found that restricting the weight norm to a sphere of the appropriate radius during training can accelerate generalization. However, this method introduces instability in the training process and still involves a phase transition. Furuta et al. (2024) suggested initializing the model with weights or embeddings from another model that has already generalized on a different task may accelerate grokking, which needs to train the same model on additional data, while our method do not need additional data. Lee et al. (2024) decomposed the gradient at each step and accelerated grokking by amplifying part of the gradient. Interestingly, Minegishi et al. (2024) demonstrated that the gap between memorization and generalization can be nearly eliminated if a lottery ticket, a set of sparse mask matrices, is applied to the model during training. However, this lottery ticket can only be obtained by first training the same model under the same initialization till generalization. In contrast, our approach can nearly eliminate the phase transition without requiring additional data or pretraining on the same model.
+
+Weak to strong knowledge transfer. Burns et al. (2023) proposed a method where a small model is first fine-tuned as a teacher model. This teacher model is then used to generate pseudo-labels to fine-tune a larger student model. Surprisingly, the student model can outperform the teacher. Wang et al. (2023) designed a learned linear growth operator, which uses a learnable linear map of a pretrained small model's weights as the initialization for the large model's weights, to accelerate the training of large models. In contrast to these works, our method focuses on transferring the embedding layer from a weaker model to the target model and reshaping the training dynamics to accelerate grokking.
+
+# 1.2 NOTATION
+
+For a set  $S$  with finite elements, we denote its cardinality by  $|S|$  and use  $\mathrm{Uniform}(S)$  to represent the uniform distribution over  $S$ . We denote the set  $\{1,2,\dots ,n\}$  by  $[n]$ . We use  $\operatorname{sgn}(x)$  to represent the sign of a scalar  $x$ . For a matrix  $A\in \mathbb{R}^{n\times m}$ , we denote by  $A_{i,} = [A_{i,1},\dots ,A_{i,m}]$  the  $i$ -th row,  $A_{i:j,\cdot} = [A_{i,\cdot}^{\top},\dots ,A_{j,\cdot}^{\top}]^{\top}\in \mathbb{R}^{(j - i + 1)\times m}$  the  $i$ -th to  $j$ -th rows, and  $\| A\|_{\mathrm{F}}$  the Frobenius norm. We use  $\phi (x) = \max \{0,x\}$  to represent the ReLU activation function. We denote the inner product between two vectors  $a,b$  by  $\langle a,b\rangle$ . For two sequences  $\{x_n\}$  and  $\{y_n\}$ , we say  $x_{n} = O(y_{n})$  if there exists some constant  $C > 0$  such that  $x_{n}\leq Cy_{n}$  for all  $n$  and  $x_{n} = \Omega (y_{n})$  if  $y_{n} = O(x_{n})$ .
+
+# 2 ACCELERATING GROKKING VIA EMBEDDING TRANSFER FROM A WEAKER MODEL
+
+# 2.1 MOTIVATION: THE ROLE OF DATA EMBEDDING
+
+To demonstrate the pivotal role of data embedding in shaping training dynamics, we examine the modular addition task  $a + b \mod p$ . Following settings in Nanda et al. (2023) and Liu et al. (2023), we take  $p = 113$ . The dataset consists of  $\{(a,b),y)\}_{0\leq a,b\leq p - 1}$  with label  $y = (a + b) \mod p$ .  $25\%$  of the dataset is randomly sampled as the training set. We evaluate four types of embeddings:
+
+- One-hot embedding: Each integer  $a \in  \lbrack 0,p - 1\rbrack$  is represented by its one-hot encoding.  
+- Binary embedding: Each  $a$  is encoded in binary, padded with zeros to the maximum length  $\lfloor \log_2(p - 1) \rfloor + 1$ .  
+- Fourier embedding: Each  $a$  is encoded as a vector of trigonometric functions:  $\left[\cos \left(\frac{2\pi i_1a}{p}\right), \sin \left(\frac{2\pi i_1a}{p}\right), \dots, \cos \left(\frac{2\pi i_ka}{p}\right), \sin \left(\frac{2\pi i_ka}{p}\right)\right]$ , where  $i_1, \dots, i_k \in \mathbb{N}$  are predetermined frequencies.
+
+![](images/80f1e60157a2722529fc5edf37a45953fc070258df4b8b69de0a8a7bc9579fa9.jpg)  
+Figure 2: FNN training dynamics using different embeddings for the modular addition task ( $p = 113$ ). The training dynamics vary significantly across different embeddings. The one-hot embedding and GPT embedding exhibit sharp phase transition. See Appendix A.3 for details of the experimental setup.
+
+![](images/a38b88a8be2c450c62d5a0847b6b923aa91fdbe49d66f1d96cfe5630abf3af8c.jpg)
+
+- GPT embedding: Each  $a$  is embedded using OpenAI's text-embedding-3-small model (OpenAI, 2024)
+
+One-hot embeddings contain no prior information about the data, while binary embeddings capture the ordinal information of integers. Fourier embeddings, inspired by the analytical solutions learned by neural networks (Nanda et al., 2023; Morwani et al., 2024), encode task-specific information. GPT embeddings encode general information about integers. Figure 2 shows the training dynamics of a feed-forward neural network using these embeddings. The training dynamics with one-hot and GPT embeddings exhibit clear grokking behavior, whereas those with binary and Fourier embeddings show continuous generalization progress. Notably, Fourier embeddings enable the model to simultaneously achieve memorization and perfect generalization. We observe that general embeddings like one-hot and GPT embeddings suffer from generalization delay, while embeddings encoded with task-related information allow the model to generalize continuously.
+
+A series of works (Liu et al., 2023; Kumar et al., 2024; Lyu et al., 2024; Mohamadi et al., 2024) found that the default initialization scale is relatively large and causes generalization delay. They observed that reducing the initialization scale can accelerate grokking and hypothesized that grokking arises from a time gap between the Neural Tangent Kernel (NTK) regime and the feature-learning regime. However, our empirical findings indicate that grokking persists even after carefully tuning the initialization scale (see Appendix A.3.1). This suggests that grokking occurs even when the model is not initialized in the kernel regime, implying that the kernel regime may not be the sole cause of grokking. In Figure 3, we compare the changes in the empirical NTK (Mohamadi et al., 2023)
+
+![](images/03ff20ace143ad7e72b416b71251415e6f4e9fdba7613d8ad302a8f0f168d55b.jpg)  
+Figure 3: Change of empirical NTK.
+
+corresponding to the dynamics in Figure 2. The change of empirical NTK evolves similarly across all four types of embeddings (see Appendix A.3 for details).
+
+In conclusion, the choice of embedding significantly impacts training dynamics, and an informative embedding can close the gap between memorization and generalization. However, finding such an informative embedding for specific tasks is not always straightforward. Binary embedding, for example, reduces the sharp phase transition for modular addition but fails to do so for modular multiplication. In the next section, we will show that constructing a task-specific embedding from training data can be a promising approach to obtaining an informative embedding that can accelerate grokking. The embedding construction can be achieved by training a much smaller, weaker model. Here "small" refers to smaller model expressivity. This weaker model can learn an informative embedding without achieving optimal generalization. This embedding can then be used to positively influence the training dynamics of the larger target model.
+
+# 2.2 OUR METHOD: GROKTRANSFER
+
+We propose GrokTransfer, a simple and principled method for accelerating grokking in training neural networks. In more detail, given a specific task and a training set  $\mathcal{G}$ , we consider a target model  $f_{T}$  that has a trainable embedding layer  $\mathrm{E}_T$  with vocabulary size  $d_v$  and embedding dimension  $d_T$ . Our proposed method GrokTransfer works as follows:
+
+1. Train a Weaker Model: Train a weaker model  $f_{W}$  with a trainable embedding table  $\mathrm{E}_W \in \mathbb{R}^{d_v \times d_W}$  on  $\mathcal{G}$ , where  $d_{W}$  is the embedding dimension in the weak model. Train  $f_{W}$  until it grows to a non-trivial performance on the validation set.  
+2. Train the Target Model: Initialize  $A = \mathrm{E}_W$  and randomly initialize a matrix  $B \in \mathbb{R}^{d_W \times d_T}$ . Train the target model with an embedding layer set to  $\mathrm{E}_T = A \cdot B$ , where both  $A$  and  $B$  are trainable.
+
+By training a weaker model, the first step aims to obtain an informative embedding that aids the training of the target model. In practice, the weak model can be much smaller than the target model or can even have a different architecture (e.g., the weak model can be a fully-connected network when the target model is a Transformer; see Section 4). As a result, training a weak model greatly reduces the computational cost of acquiring an informative embedding. This contrasts with the method proposed in Minegishi et al. (2024), which requires the target model to be trained till perfect generalization first. In the next sections, we will demonstrate, both theoretically and empirically, that even if the weak model only partially generalizes (i.e., has a non-trivial but non-optimal test error), its embedding still allows the large model to generalize optimally without delay.
+
+In the second step, we impose a low-rank structure  $A \cdot B$  on the embedding  $\mathrm{E}_T$  while training the target model. This constraint alters the empirical risk landscape and provides a favorable initialization for the embedding table. The intuition behind our method is as follows: by initializing with an informative embedding from the weak model, the target model can bypass the initial phase of pure memorization. Instead, it can start generalizing almost immediately as it begins to optimize the training loss.
+
+# 3 CASE STUDY: GROKTRANSFER ON XOR CLUSTER DATA
+
+In this section, we theoretically study an XOR classification task and prove that GrokTransfer can eliminate grokking for this task.
+
+# 3.1 THE SETUP OF XOR CLUSTER DATA
+
+We study the setting where the data  $x = [x_{1}, x_{2}, \dots, x_{p}]^{\top} = [x_{\mathrm{signal}}^{\top}, x_{\mathrm{noise}}^{\top}]^{\top} \in \mathbb{R}^{p}$ ,  $x_{\mathrm{signal}} \sim \mathrm{Uniform}(\{\pm 1\}^{2})$ ,  $x_{\mathrm{noise}} \sim \mathrm{Uniform}(\{\pm \varepsilon\}^{p - 2})$ , and the label  $y = x_{1}x_{2}$ . Here  $\varepsilon$  is the parameter that controls the scale of the noise. We denote this data distribution by  $P$  and consider  $n$  training datapoints  $\{(x_{i}, y_{i})\}_{i=1}^{n}$  drawn i.i.d. from the distribution  $P$ . We assume the sample size  $n$  to be sufficiently large, specifically larger than any universal constant mentioned in this paper. The data distribution comprises four feature vectors (see Figure 5a for a projected visualization), and the model need learn all four features to achieve perfect generalization.
+
+We denote a width- $m$  two-layer neural network by  $f(x) = \sum_{j=1}^{m} a_j \phi(\langle w_j, x \rangle)$ , where  $w_j \in \mathbb{R}^p$ ,  $j \in [m]$  are neurons in the hidden layer and  $a_j \in \mathbb{R}$ ,  $j \in [m]$  are second-layer weights. The model is randomly initialized by
+
+$$
+w _ {j} \stackrel {{i. i. d}} {{\sim}} N (0, w _ {\mathrm {i n i t}} ^ {2} I _ {p}), \quad a _ {j} \stackrel {{i. i. d}} {{\sim}} N (0, a _ {\mathrm {i n i t}} ^ {2}), \quad j \in [ m ].
+$$
+
+Define the empirical risk with the exponential loss as:  $\widehat{L}(f) = \sum_{i=1}^{n} l(y_i, f(x_i)) / n$ , where  $l(y, \widehat{y}) = \exp(-y\widehat{y})$ . We use gradient descent (GD) with weight decay  $\theta_j^{(t+1)} = (1 - \lambda)\theta_j^{(t)} - \alpha \nabla_{\theta_j}\widehat{L}(f^{(t)})$  to update both layers  $\{w_j, a_j\}_{j=1}^m$ , where  $\lambda$  is the coefficient of  $L_2$  regularization.
+
+Setting  $p = 80000$ ,  $n = 400$ ,  $\varepsilon = 0.05$ , this configuration approximates one of the distributions explored in Xu et al. (2024), where grokking was observed. Under this setup, we train a two-layer neural network on  $\{(x_i, y_i)\}_{i=1}^n$  with default PyTorch initialization. We observe grokking, as shown in Figure 4(a), where overfitting is achieved by the fifth epoch and generalization begins around the
+
+80-th epoch. Below we will show how our method GrokTransfer constructs a new embedding and eliminates the observed delay in generalization in subsequent sections.
+
+(a)  
+![](images/95c01dc8d3709ce6268d3f15092fa11a1aea5ae4663cbd09081ceaaba014cb66.jpg)  
+Train Accuracy Test Accuracy Train Loss Test Loss
+
+(b)  
+![](images/ee6aabd625b913563d592f5ac12d241e5a9797245951093f63a5ab76e6b94243.jpg)  
+Neuron 0 Neuron 1 Neuron 2
+
+![](images/a3f39b9b4d3e301d82c541e532524705591678849603f3e043fcde166854f90a.jpg)  
+(c)  
+Figure 4: (a) Training dynamics of a two-layer neural network with a hidden width of 2048, where grokking is observed. (b) Training dynamics of a two-layer neural network with a hidden width of 3. The model can only achieve around  $75\%$  validation accuracy and a phase transition near 100th epoch is observed. (c) Visualization of individual neuron weights from the model trained in (b). It shows three distinct patterns and each corresponds to a feature direction of the XOR data distribution. See Appendix A.3 for details of the experimental setup.
+
+# 3.2 EMPIRICAL ANALYSIS OF THE WEAKER MODEL
+
+Applying GrokTransfer, we first train a small two-layer neural network with only 3 neurons  $f_{S}(x) = \sum_{j=1}^{3} a_{j} \phi(\langle w_{j}, x \rangle)$  till convergence (Figure 4(b)). Denote the first-layer weight matrix by  $W = [w_{1}, w_{2}, w_{3}] \in \mathbb{R}^{p \times 3}$ , the number of training steps by  $T$ , and the model after training by  $f_{S}^{(T)}$ . Due to the complexity of the training dynamics, it is hard to derive the closed form of  $f_{S}^{(T)}$  and  $W^{(T)}$ . Below we empirically investigate what information the model has gained and how well it learns.
+
+Figure 4(b) shows that, after training, this weak model has non-trivial performance with test accuracy around  $75\%$ . The neurons  $\{w_{j}^{(T)}\}_{j = 1}^{3}$  are visualized in Figure 4(c), displaying patterns  $[-1,1,0,\dots ,0],[1, - 1,0,\dots ,0]$ , and  $[-1, - 1,0,\dots ,0]$ . Note that the specific features learned by the model are sensitive to its initialization. Nevertheless, we find that empirically, the learned features are always three among the four features  $[\pm 1,\pm 1,0,\ldots ,0]$ , provided the test accuracy is around  $75\%$ .
+
+Notice that an optimal function for this classification task is
+
+$$
+f (x) = \operatorname {s g n} \left(\phi \left(x _ {1} + x _ {2}\right) + \phi \left(- x _ {1} - x _ {2}\right) - \phi \left(- x _ {1} + x _ {2}\right) - \phi \left(x _ {1} - x _ {2}\right)\right),
+$$
+
+which needs four neurons to represent all features  $[\pm 1, \pm 1]$ . It thus follows intuitively that the weak model  $f_{S}$  cannot achieve better generalization with only three neurons. Formally, we establish the following lemma regarding the expressive power of  $f_{S}$ .
+
+Lemma 3.1. For any  $f(x) = \sum_{j=1}^{3} a_j \phi(w_j^\top x)$ , where  $\phi$  is the ReLU activation function, we have
+
+$$
+\mathbb {P} _ {(x, y) \sim P} (y = \operatorname {sgn} (f (x))) \leq 75 \%
+$$
+
+Although the model  $f_{S}^{(T)}$  fails to generalize perfectly due to the inherent limitation of capacity, it has correctly selected the subset that contains features after training as shown in Figure 4(c). Consequently, for any input  $x \sim P$ ,  $W^{(T)\top}x$  becomes a high-quality embedding for  $x$  in a much lower dimensional space. Figure 5a shows that, with this new embedding, data points are well-separated in a three-dimensional space with a relatively high signal-to-noise ratio (SNR) compared to the original embedding.
+
+Next, we empirically examine the order of the ratio between the norm of the complementary subnetwork and the norm of the generalizing subnetwork. This will be used to estimate the SNR of  $P$  with the new embedding. Given the structure of the XOR cluster data, the first two rows of  $W^{(T)}$  correspond to the generalizing subnetwork. We define the norm ratio between the complementary
+
+and generalizing subnetwork as follows:
+
+$$
+r _ {W} = \frac {\| W _ {3 : p , .} ^ {(T)} \| _ {\mathrm {F}} / \sqrt {p - 2}}{\| W _ {1 : 2 , .} ^ {(T)} \| _ {\mathrm {F}} / \sqrt {2}}.
+$$
+
+Figure 5b and 5c show that the norm ratio is proportional to  $\varepsilon$  and  $1 / \sqrt{n}$ , i.e.  $r_W \propto \varepsilon / \sqrt{n}$ . We will use this property to show that, under mild assumptions, the target model can learn this low-dimensional XOR task with just one step of gradient descent.
+
+![](images/814b011fb6e790c86c0ae80567887ef688b9e1c7f4d2ab9ba2dfc9d169d8d35e.jpg)  
+(a)
+
+![](images/ca06dfbe3d612778a192939090326907af235775367b326c56318ee0997d91ac.jpg)  
+(b)  
+Figure 5: (a) 3-D Visualization of the distribution  $P$  with the embedding from the weak model. The clusters are well-separated under the new embedding. (b) Norm ratio  $r_W$  for different values of  $p$  and  $\varepsilon$  with fixed sample size  $n$ , indicating that  $r_W$  does not depend on  $p$ . (c) Norm ratio  $r_W$  for different values of  $n$  and  $\varepsilon$  with fixed feature dimension  $p$ . For each  $\epsilon$ , the slope is around  $-1/2$ , indicating that  $r_W$  is proportional to  $1/\sqrt{n}$ . See Appendix A.3 for details of the experimental setup.
+
+![](images/ca87990b986f92b553fea9284a32427c7f4d2b84dad908ac2b62bd3b33672e88.jpg)  
+(c)
+
+# 3.3 THEORETICAL ANALYSIS OF THE TARGET MODEL
+
+In this section, we theoretically analyze the behavior of GrokTransfer on the XOR cluster data. We consider the target model as a large model with width  $m$  of the form  $f_{L}(x) = \sum_{j=1}^{m} a_{j} \phi(\langle v_{j}, U^{\top} x \rangle)$ , where  $U = [u_{1}, u_{2}, u_{3}] \in \mathbb{R}^{p \times 3}$  comes from the first-layer weight matrix  $W^{(T)}$  learned by the weak model (visualized in Figure 4(c)). Here,  $U$  is the embedding matrix being transferred from the weak model  $f_{S}$ , which will then go through another linear transformation (given by  $v_{j}$ 's) to form the embedding in the target model. Following our observation in Section 3.2, we can write
+
+$$
+u _ {1} = \left[ \mu_ {2} ^ {\top}, \delta_ {1} ^ {\top} \right] ^ {\top}, \quad u _ {2} = \left[ - \mu_ {2} ^ {\top}, \delta_ {2} ^ {\top} \right] ^ {\top}, \quad u _ {3} = \left[ - \mu_ {1} ^ {\top}, \delta_ {3} ^ {\top} \right] ^ {\top},
+$$
+
+where  $\mu_1 = [1,1]^\top$ ,  $\mu_2 = [-1,1]^\top$  are two orthogonal features of  $P$ , and  $\delta_j = [\delta_{j,1},\dots,\delta_{j,p-2}]^\top \in \mathbb{R}^{p-2}(j \in [3])$ . Here we let  $\delta = [\delta_1,\delta_2,\delta_3] = W_{3:p,'}^{(T)}$ . Given a universal constant  $C > 1$ , we assume
+
+(A1) The noise scale  $\varepsilon \leq (n / (p\log^3 n))^{1 / 4}$ .  
+(A2) The norm of the complementary subnetwork satisfies  $\| \delta \|_{\mathrm{F}}\leq C\varepsilon \sqrt{p / n}$ .  
+(A3) The initialization scale  $v_{\mathrm{init}} \leq C \log^{-3/2}(n)$ .  
+(A4) The step size  $\sqrt{m} v_{\mathrm{init}} / C \leq \alpha \leq \sqrt{m} v_{\mathrm{init}}$  
+(A5) The number of neurons satisfies  $m \geq 2\log^3 n$ .
+
+Here Assumption (A2) corresponds to the finding that  $r_W \propto \varepsilon / \sqrt{n}$  in Section 3.2. Assumptions (A1) and (A2) together ensure that the SNR of the distribution  $P$  in the new embedding space is large enough. Assumption (A3) controls the initial weight norm of the target model such that the empirical risk starts within a reasonable range. Assumption (A4) guarantees that the step size is appropriately balanced; it is neither too small to prevent meaningful updates after a single-step gradient descent nor
+
+too large to cause overly drastic movements. Assumption (A5) ensures that the model's width is large enough to ensure certain concentration results about the random initialization. All assumptions are satisfied in the empirical setup discussed in Section 3.2.
+
+We denote  $a = [a_1, \dots, a_m]^\top \in \mathbb{R}^m$  and  $V = [v_1, \dots, v_m] \in \mathbb{R}^{3 \times p}$ . We initialize  $a$  and  $V$  as follows:
+
+$$
+a _ {j} \stackrel {{i. i. d}} {{\sim}} \operatorname {U n i f o r m} (\{\pm 1 / \sqrt {m} \}), \quad v _ {j} \stackrel {{i. i. d}} {{\sim}} \operatorname {U n i f o r m} (\{\pm v _ {\text {i n i t}} \} ^ {3}), \quad j \in [ m ],
+$$
+
+and keep  $a$  and  $U$  fixed during the training process. Following the training method outlined in Section 3.1, we use gradient descent  $V^{(t + 1)} = V^{(t)} - \alpha \nabla_V\widehat{L} (f_L^{(t)})$  at step  $t$  to update the linear layer  $V$ , where  $\alpha$  is the step size and the empirical risk  $\widehat{L} (\cdot)$  is defined in Section 3.1. With the assumptions and initializations, we state the theorem that characterizes the train and test error of the target model after one step.
+
+Theorem 3.2. Suppose that Assumptions (A1)-(A5) hold. With probability at least  $1 - O(1 / n^2)$  over the generation of the training data and initial weights of  $f_{L}$ , after one step of training, the classifier  $\operatorname{sgn}(f_L^{(1)}(x))$  can correctly classify all training datapoints and generalize with test error no greater than  $\exp(-\Omega (\log^2 n))$ .
+
+Theorem 3.2 shows that with GrokTransfer, after just one step of gradient descent, the target model overfits all training data and achieves near perfect test accuracy. Notably, this is not in a kernel regime but a feature learning regime. Since models with normal training cannot achieve generalization in one step (Figure 4(a)), this result indicates that our method GrokTransfer effectively boosts the generalization speed of the target model and eliminates the time gap between overfitting and generalization. Empirically, the model continues to generalize with further training (see Figure 9 in Appendix A.3). Given that the weaker model  $f_{S}$  has only three neurons, the computational cost of training  $f_{S}$  is negligible compared to the cost of training the target model  $f_{L}$  with sufficiently large width. This implies that GrokTransfer may reduce the overall computational cost. In the next section, we will compare the computational cost of our method to that of standard training procedures.
+
+# 4 EXPERIMENTS
+
+This section empirically studies GrokTransfer in modular addition and multiplication, as well as the sparse parity task. Our experiments verify that GrokTransfer effectively reshapes the training dynamics and eliminate delayed generalization for both fully-connected neural networks (FNN) and Transformers (TF). The AdamW optimizer (Loshchilov & Hutter, 2019) is used in all experiments in this section.
+
+# 4.1 FNN  $\rightarrow$  FNN
+
+We first consider a three-layer FNN as the target model and conduct GrokTransfer on tasks including modular addition, modular multiplication, and  $(q,k)$ -parity (Barak et al., 2022). These results are compared to training a target model from scratch. The modular addition task is introduced in Section 2.1, and modular multiplication is defined similarly with the label  $y = ab \bmod p$ . The  $(q,k)$ -parity task consists of a dataset  $\{(x,y): x \in \{\pm 1\}^q, y = \prod_{i \in S} x_i, |S| = k\}$ . Following the setting in Merrill et al. (2023), we choose  $q = 40, k = 3$ , and  $S = \{1,2,3\}$ .
+
+For the modular addition and multiplication tasks, we employ a two-layer neural network with a trainable embedding as the weak model, which we train for  $10^{4}$  epochs. We then initialize the target model by setting its layer  $A$  to the embedding learned by the weak model. Figure 6a and 6b show the training dynamics of the weak model, the target model trained via GrokTransfer, and the target model trained from scratch. Notably, GrokTransfer nearly eliminates the sharp phase transition observed in normal training. Here all training hyperparameters (initialization scale, learning rate, weight decay) are selected by grid search, and the best configuration is defined as the one that reaches  $99\%$  test accuracy the quickest. The oscillations of accuracies in the second row of Figure 6 are related to the "slingshot mechanism" (Thilak et al., 2022) and training instabilities associated with large learning rates (Wortsman et al., 2024). Since large learning rate and this kind of oscillation
+
+![](images/1b641a9e4c9ca8f04939c611b22e45c6595d2eb82af5732d54c77ece5ee23ca8.jpg)
+
+![](images/18b0f9ed32df2b45c861c2339e93894e4edcd46ba23b8a3e76649813f362efbf.jpg)  
+(a) Modular addition
+
+![](images/fbd1db095dd487c658ac1197b1021b4749ffe352665f5ace9b3440da65ee0cad.jpg)
+
+![](images/13c3811cc32b7567a92b6794d28088329d7e842c98e2bd36b31db6363223e4ad.jpg)  
+(b) Modular multiplication
+
+![](images/ac51471b255bfc05ae20db8d2404b53c7d21948a620307c1fbf180a922ab703a.jpg)
+
+![](images/1f2192a2cb273c70577e94c18db45af6176fb50f783f82aaac05cc33b8958876.jpg)  
+(c) (40,3)-parity  
+Figure 6: Training dynamics of FNNs on various tasks. The rows represent different models/training methods: The first row shows the dynamics of the weak model used in GrokTransfer, the second row shows the dynamics of the target model trained using GrokTransfer, and the target model trained from scratch. The columns represent different tasks: the first column is for the modular addition task, the second column is for the modular multiplication task, and the third column is for the (40, 3)-parity task. The comparison between the first and second rows shows that the target model trained via GrokTransfer can surpass the weak model's performance. The comparison within the second row shows that GrokTransfer eliminates the sharp phase transition and enables the model to make continuous progress. See Appendix A.3 for details of the experimental setup.
+
+are believed to help generalization (Damian et al., 2023; Lu et al., 2024), we do not change our configuration selection criteria.
+
+For the parity task, we use a three-layer FNN as the weak model, as empirical evidence suggests that a two-layer FNN without bias terms cannot generalize on this task. The weak model is trained until it achieves  $70\%$  test accuracy, after which the first layer's weight matrix is transferred to the target model. As shown in Figure 6c, the weak model undergoes a generalization delay, but the large model inheriting its embedding generalizes continuously.
+
+Ablation study: To further understand the empirical effectiveness of GrokTransfer, we perform an ablation study by varying the training epochs of the weak model in the modular addition task.
+
+We extract the embeddings of the weak model at epochs 100, 500, 800, 900, 1000, 1100, 1500, and 2000. For each embedding, we apply GrokTransfer to the target model and train it for  $10^{4}$  epochs. To measure the generalization delay of the target model, we define Time Gap as the difference between the first epoch that achieves  $95\%$  training accuracy and the first epoch that achieves  $95\%$  test accuracy. If the target model fails to reach  $95\%$  accuracy, we set  $1 / \text{Time Gap} = 0$ . Figure 7 shows that the test performance of the target model, initialized with the weak model's embedding, is positively correlated with the test performance of the weak model. A grokked weak model is essential for the target model to achieve near-perfect generalization with minimal generalization delay. We hypothesize that the target model can only generalize well after the weak model has grokked.
+
+![](images/b54141255f88ee60d7f19cb48b7751f35ee66ba869034bd8443b9af2c97612bd.jpg)  
+Figure 7: Ablation study showing the effect of the weak model's performance on the test accuracy of the target model (initialized via GrokTransfer and trained for  $10^{4}$  epochs).
+
+# 4.2 FNN  $\rightarrow$  TRANSFORMERS
+
+Interestingly, we find that the embeddings extracted from the weak FNN model can be transferred to the target model even when the target model is a Transformer comparable to the scale of GPT2-small (Radford et al., 2019). Under this FNN  $\rightarrow$  TF setting, GrokTransfer still mitigates the generalization delay of the target model. Specifically, we choose the target model to be a Transformer
+
+![](images/c75517c842415e80eee5909575003ccf510558689ef6accb381912db9cc2cfc8.jpg)  
+Figure 8: Training dynamics of Transformers on Modular Addition Task. The weak model is a three-layer FNN. (a) Dynamics of the target model (an 8-layer transformer) trained via GrokTransfer, and the target model trained from scratch. (b) Dynamics of the target model (a two-layer Transformer) trained via GrokTransfer, and the target model trained via GrokFast (Lee et al., 2024).
+
+![](images/c5ee69737187b2420cd05cd8fba1463b116a1bf4ae4d2ae01cc596f8adb089d5.jpg)
+
+with 8 attention layers,  $(d_{\mathrm{embed}}, d_{\mathrm{mlp}}, n_{\mathrm{head}}) = (512, 512, 4)$ . For each sample, the input is a sequence with two tokens  $(a, b)$ . We extract the embeddings of the weak model in Figure 6a at the point that it first reaches  $30\%$  test accuracy. Figure 8(a) shows that GrokTransfer enables the target model to generalize much faster than training from scratch and exhibits little generalization delay. Here both method suffer from training instability of large learning rates.
+
+In terms of the computation cost, we use wall-clock time as the measure. The computation cost of GrokTransfer comprises the training of weak model and the training of target model. Table 1 shows the total wall-clock time for weak model, target model with GrokTransfer, and target model trained from scratch. The time spent training the weak FNN model is negligible compared to training the target transformer model. The total wall-clock time of GrokTransfer is approximately five times faster than training from scratch.
+
+<table><tr><td>Model</td><td>Weak</td><td>Target (GrokTransfer)</td><td>Target (scratch)</td></tr><tr><td>Total Wall-clock time (ms)</td><td>2828</td><td>71079</td><td>392667</td></tr></table>
+
+Table 1: Comparison of total wall clock times (forward and backward passes) for different models. The weak model is a three-layer FNN. The target/large model is an 8-layer transformer.
+
+Lee et al. (2024) proposed a gradient amplification algorithm GrokFast to accelerate grokking. We compare GrokTransfer with GrokFast in Figure 8(b). The weak model embedding we transfer is the same as the one used in Figure 8(a). For the target model, we follow the model used in Lee et al. (2024), which is a two-layer decoder-only transformer with  $(d_{\mathrm{embed}}, d_{\mathrm{mlp}}, n_{\mathrm{head}}) = (128, 512, 4)$ . The Time Gap of GrokTransfer is 46 while the Time Gap of GrokFast is 1119.
+
+# 5 CONCLUSION
+
+To eliminate the unpredictability associated with grokking, we proposed GrokTransfer, a novel method that effectively accelerates grokking by transferring the embedding from a weaker model. Our method was inspired by the key observation that data embedding critically shapes training dynamics. We theoretically justified GrokTransfer on an XOR classification task. We also empirically evaluated it on various algorithmic tasks known to exhibit grokking under standard training. Our results showed that GrokTransfer can effectively modify training dynamics, enabling continuous progression in model performance.
+
+One limitation of our work is that the theoretical result only considers a relatively simple XOR task. For this task, after transferring the embedding from the smaller model, one step of gradient descent suffices for both memorization and generalization. Theoretical justification for more complex problems is an important future direction. Furthermore, our method focuses solely on accelerating grokking and was only investigated on problems where grokking occurs. It would be interesting to study whether similar ideas can be applied to improve training dynamics or enable weak-to-strong generalization in a broader context.
+
+# ACKNOWLEDGMENTS
+
+This work was supported in part by the Office of Naval Research under grant number N00014-23-1-2590, the National Science Foundation under Grant No. 2231174, No. 2310831, No. 2428059, No. 2435696, No. 2440954, and a Michigan Institute for Data Science Propelling Original Data Science (PODS) grant.
+
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+
+# A APPENDIX
+
+A.1 Proofs 15
+
+A.1.1 Proof of Lemma 3.1 15  
+A.1.2 Proof of Theorem 3.2 15
+
+A.2 Additional Experiments 25  
+A.3 Experimental details 25
+
+A.3.1 Experiments in Section 1 and 2.1 25  
+A.3.2 Experiments in Section 3 26  
+A.3.3 Experiments in Section 4 26
+
+A.4 Additional Discussion 27
+
+A.4.1 Forward Pass FLOPs estimation for Models in Figure 8 Left 27  
+A.4.2 Additional Experiments 27
+
+# A.1 PROOFS
+
+# A.1.1 PROOF OF LEMMA 3.1
+
+Lemma 3.1. For any  $f(x) = \sum_{j=1}^{3} a_j \phi(w_j^\top x)$ , where  $\phi$  is the ReLU activation function, we have
+
+$$
+\mathbb{P}_{(x,y)\sim P}(y = \operatorname {sgn}(f(x)))\leq 75\% .
+$$
+
+Proof. For any  $(x,y) \sim P$ , define  $x' = (x_1, -x_2, x_3, \dots, x_p)$  and  $y' = \operatorname{sgn}(x_1'x_2') = -y$ . It is sufficient to show that if  $y = \operatorname{sgn}(f(x))$ ,  $y = \operatorname{sgn}(f(-x))$ ,  $y' = \operatorname{sgn}(f(x'))$ , then  $y' \neq \operatorname{sgn}(f(-x'))$  with probability 1.
+
+Assume  $y = \operatorname{sgn}(f(x))$  and  $y = \operatorname{sgn}(f(-x))$ . Given  $\phi(z) \geq 0, \forall z$ ,  $y = \operatorname{sgn}(f(x))$  implies that there exists at least one  $i \in [3]$  such that  $a_i$  has the same sign as  $y$  and  $w_i^\top x > 0$ . Without loss of generality, assume  $\operatorname{sgn}(a_1) = y$ ,  $w_1^\top x > 0$ . Then for  $(-x, -y)$ , it follows that
+
+$$
+f (- x) = \sum_ {j = 1} ^ {3} a _ {j} \phi (- w _ {j} ^ {\top} x) = a _ {2} \phi (- w _ {2} ^ {\top} x) + a _ {3} \phi (- w _ {3} ^ {\top} x)
+$$
+
+has the same sign as  $y$ . Again without loss of generality, we assume  $\operatorname{sgn}(a_2) = y$ .
+
+If  $y' = \operatorname{sgn}(f(x'))$  and  $y' \neq \operatorname{sgn}(f(-x'))$  hold, following the same discussion, we have that at least two  $a_i$ 's have the same sign as  $y' = -y$ , which contradicts the previous assumption that  $\operatorname{sgn}(a_1) = \operatorname{sgn}(a_2) = y$ .
+
+# A.1.2 PROOF OF THEOREM 3.2
+
+Additional notations: For training dataset  $\{(x_i, y_i)\}_{i=1}^n$ , we denote the signal of  $x_i$  by  $\bar{x}_i = [x_{i,1}, x_{i,2}]^\top \in \{\pm \mu_1, \pm \mu_2\}$ . For each  $\mu \in \{\pm \mu_1, \pm \mu_2\}$ , define
+
+$$
+\mathcal {I} _ {\mu} = \{i \in [ n ]: \bar {x} _ {i} = \mu \}
+$$
+
+and  $n_{\mu} = |\mathcal{I}_{\mu}|$ . Denote the new embedding of the  $i$ -th datapoint by  $z_{i} = U^{\top}x_{i}, i \in [n]$ . Define
+
+$$
+\nu_ {i} = [ \mu_ {2}, - \mu_ {2}, - \mu_ {1} ] ^ {\top} \mu_ {i}, \quad i = 1, 2.
+$$
+
+Then  $\nu_{1} = [0,0, - 2],\nu_{2} = [2, - 2,0]$ , and  $\{\pm \nu_1,\pm \nu_2\}$  becomes the features for  $P$  with the new embedding. Denote the signal of  $z_{i}$  by  $\bar{z}_i = [\mu_2, - \mu_2, - \mu_1]^\top \bar{x}_i$ . Define the set of training data
+
+$$
+\mathcal {G} _ {\text {d a t a}} = \left\{\left\{\left(x _ {i}, y _ {i}\right) \right\} _ {i = 1} ^ {n}: \| z _ {i} - \bar {z} _ {i} \| \leq \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n, \text {f o r a l l} i \in [ n ] \right\}.
+$$
+
+By Lemma A.2,  $\mathbb{P}(\{(x_i,y_i)\}_{i = 1}^n\in \mathcal{G}_{\mathrm{data}})\geq 1 - \exp (-\Omega (\log^2 n))$  . We further define sets to separate the second-layer coefficients for the ease of discussion:
+
+$$
+\mathcal {J} _ {\mathrm {P o s}} = \{j \in [ m ]: a _ {j} > 0 \}; \quad \mathcal {J} _ {\mathrm {N e g}} = \{j \in [ m ]: a _ {j} <   0 \}.
+$$
+
+We divide the index of neurons by its initialization and define  $\mathcal{J}_e = \{j\in [m]:v_j^{(0)} = v_{\mathrm{init}}e\}$  for  $e\in \mathrm{Uniform}(\{\pm 1\})^3$ . We further define
+
+$$
+\mathcal {J} _ {\mathrm {P o s}, e} = \mathcal {J} _ {\mathrm {P o s}} \cap \mathcal {J} _ {e}; \quad \mathcal {J} _ {\mathrm {N e g}, e} = \mathcal {J} _ {\mathrm {N e g}} \cap \mathcal {J} _ {e}.
+$$
+
+For each initialization of  $v_{j}^{(0)}$ , we denote the set of datapoints which have positive inner product with it by
+
+$$
+\mathcal {I} _ {e, \mu} = \left\{i \in \mathcal {I} _ {\mu}: \langle e, z _ {i} \rangle > 0 \right\}, \quad e \in \operatorname {U n i f o r m} \left(\left\{\pm 1 \right\} ^ {3}\right), \mu \in \left\{\pm \mu_ {1}, \pm \mu_ {2} \right\}.
+$$
+
+Theorem 3.2. Suppose that Assumptions (A1)-(A5) hold. With probability at least  $1 - O(1/n^{2})$  over the generation of the training data and initial weights of  $f_{L}$ , after one step of training, the classifier  $\operatorname{sgn}(f_{L}^{(1)}(x))$  can correctly classify all training datapoints and generalize with test error no greater than  $\exp(-\Omega(\log^2 n))$ .
+
+Proof. For brevity, we omit the subscript  $L$  in  $f_{L}$  in the proof below.
+
+At step  $t = 0$ : for each  $(x_i, y_i)$ , we have
+
+$$
+f ^ {(0)} (x _ {i}) = \sum_ {j = 1} ^ {m} a _ {j} \phi (\langle v _ {j} ^ {(0)}, z _ {i} \rangle),
+$$
+
+where  $a_{j}\phi (\langle v_{j}^{(0)},z_{i}\rangle),j\in [m]$  are bounded random variables with zero mean. The absolute bound is
+
+$$
+| a _ {j} \phi (\langle v _ {j} ^ {(0)}, z _ {i} \rangle) | \leq \frac {\sqrt {3} v _ {\mathrm {i n i t}}}{\sqrt {m}} \left(\max  _ {i} \| \bar {z} _ {i} \| + \varepsilon^ {2} \sqrt {p / n} \log n\right) \leq 5 v _ {\mathrm {i n i t}} / \sqrt {m},
+$$
+
+where the first inequality uses Lemma A.2 and the second inequality uses  $\max_i\| \bar{z}_i\| = 2\sqrt{2}$  and Assumption (A1). Then by Hoeffding's inequality and law of total probability,
+
+$$
+\mathbb {P} (| f ^ {(0)} (x _ {i}) | > t) \leq \mathbb {P} (| f ^ {(0)} (x _ {i}) | > t | \mathcal {G} _ {\mathrm {d a t a}}) + \mathbb {P} (\mathcal {G} _ {\mathrm {d a t a}}) \leq 2 \exp \left(- \frac {2 t ^ {2}}{2 5 v _ {\mathrm {i n i t}} ^ {2}}\right) + \exp (- \Omega (\log^ {2} n)).
+$$
+
+Let  $t = v_{\mathrm{init}} \log n$ . It follows that
+
+$$
+\begin{array}{l} \mathbb {P} \left(\max  _ {i \in [ n ]} | f ^ {(0)} (x _ {i}) | \leq t\right) \geq 1 - \sum_ {i = 1} ^ {n} \mathbb {P} \left(| f ^ {(0)} (x _ {i}) | > t\right) \\ \geq 1 - 2 n \exp \left(- \frac {2 \log^ {2} n}{2 5}\right) - n \exp \left(- \Omega \left(\log^ {2} n\right)\right) = 1 - \exp \left(- \Omega \left(\log^ {2} n\right)\right). \tag {1} \\ \end{array}
+$$
+
+We define a set of training data and initial weights:
+
+$$
+\begin{array}{l} \mathcal {G} = \left\{\left(\left\{\left(x _ {i}, y _ {i}\right) \right\} _ {i = 1} ^ {n}, a, V ^ {(0)}\right): \left\{\left(x _ {i}, y _ {i}\right) \right\} _ {i = 1} ^ {n} \in \mathcal {G} _ {\text {d a t a}}, \text {c o n d i t i o n (1) a n d} \right. \\ \left. \text {a l l} A. 1 \text {a n d} A. 4 \text {h o l d} \right\}. \\ \end{array}
+$$
+
+Combining (1), Lemma A.1, A.2, and A.4 then applying the union bound, we have
+
+$$
+\mathbb {P} \big (\big (\{x _ {i}, y _ {i}) \} _ {i = 1} ^ {n}, a, V ^ {(0)} \big) \in \mathcal {G} \big) \geq 1 - \exp (- \Omega (\log^ {2} n)) - O (\frac {1}{n ^ {2}}) - O (\frac {1}{n ^ {4}}) = 1 - O (\frac {1}{n ^ {2}}).
+$$
+
+Denote  $l_i^{(t)} = l(y_i, f^{(t)}(x_i)) = \exp(-y_i f^{(t)}(x_i))$ . Conditioning on  $\mathcal{G}$ , the ratio between the maximum and minimum loss is bounded by:
+
+$$
+R ^ {(0)} := \frac {\max  _ {i \in [ n ]} l _ {i} ^ {(0)}}{\min  _ {i \in [ n ]} l _ {i} ^ {(0)}} \leq \exp (2 v _ {\text {i n i t}} \log n). \tag {2}
+$$
+
+For each  $j$ , below we will analyze the gradient descent update for all possible combinations of  $a_{j}^{(0)}, v_{j}^{(0)}$  conditioning on the event  $\mathcal{G}$ .
+
+(1) When  $a_j > 0$ : If  $v_j^{(0)} = v_{\mathrm{init}}[1,1,1]$ , then according to Lemma A.1, we have
+
+$$
+\mathcal {I} _ {[ 1, 1, 1 ], + \mu_ {1}} = \varnothing ; \quad \mathcal {I} _ {[ 1, 1, 1 ], - \mu_ {1}} = \mathcal {I} _ {- \mu_ {1}}; \quad \left| | \mathcal {I} _ {[ 1, 1, 1 ], \mu} | - \frac {n _ {\mu}}{2} \right| \leq \sqrt {n \log n}, \mu = \pm \mu_ {2}. \tag {3}
+$$
+
+Recall that the gradient descent update of  $v_{j}^{(t)}$  is
+
+$$
+v _ {j} ^ {(t + 1)} = v _ {j} ^ {(t)} + \frac {\alpha}{n} a _ {j} \sum_ {i = 1} ^ {n} y _ {i} \exp \left(- y _ {i} f ^ {(t)} \left(x _ {i}\right)\right) \phi^ {\prime} \left(\left\langle v _ {j} ^ {(t)}, z _ {i} \right\rangle\right) z _ {i}. \tag {4}
+$$
+
+It follows that
+
+$$
+\begin{array}{l} v _ {j, 3} ^ {(1)} = v _ {j, 3} ^ {(0)} + \frac {\alpha}{n} a _ {j} \sum_ {i = 1} ^ {n} y _ {i} l _ {i} ^ {(0)} \phi^ {\prime} (\langle v _ {j} ^ {(0)}, z _ {i} \rangle) z _ {i, 3} \\ = v _ {j, 3} ^ {(0)} + \frac {\alpha}{n} a _ {j} \sum_ {i \in \mathcal {I} _ {- \mu_ {1}}} y _ {i} l _ {i} ^ {(0)} z _ {i, 3} + \frac {\alpha}{n} a _ {j} \sum_ {i \in \mathcal {I} _ {[ 1, 1, 1 ], \mu_ {2}} \cup \mathcal {I} _ {[ 1, 1, 1 ], - \mu_ {2}}} y _ {i} l _ {i} ^ {(0)} z _ {i, 3} \\ \geq v _ {\text {i n i t}} + \frac {2 \alpha}{n \sqrt {m}} \sum_ {i \in \mathcal {I} _ {- \mu_ {1}}} l _ {i} ^ {(0)} - O \left(\frac {\alpha}{\sqrt {m}} \max  _ {i} l _ {i} ^ {(0)} \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n\right) \tag {5} \\ \geq v _ {\mathrm {i n i t}} + \frac {1 . 9 \alpha | \mathcal {I} _ {- \mu_ {1}} |}{n \sqrt {m}} \exp (- v _ {\mathrm {i n i t}} \log n) \geq v _ {\mathrm {i n i t}} + \frac {1 . 9 \alpha}{4 \sqrt {m}} (1 - \frac {4}{\log n}) (1 - v _ {\mathrm {i n i t}} \log n) \\ \geq v _ {\mathrm {i n i t}} + \frac {2 \alpha}{5 \sqrt {m}}, \\ \end{array}
+$$
+
+where the first inequality uses Lemma A.2 and  $\bar{z}_{i,3} = 0, i \in \mathcal{I}_{\pm \mu_2}$ ; the second inequality uses Assumption (A1), (A3) and (A4); the third inequality uses Lemma A.4 and  $\exp(x) \geq 1 + x$ . Further for  $l = 1, 2$ , we have
+
+$$
+\begin{array}{l} \left|v_{j,l}^{(1)} - v_{j,l}^{(0)}\right| = \left|\frac{\alpha}{n} a_{j}\sum_{i\in \mathcal{I}_{- \mu_{1}}}y_{i}l_{i}^{(0)}z_{i,l} + \frac{\alpha}{n} a_{j}\sum_{i\in \mathcal{I}_{[1,1,1],\mu_{2}}\cup \mathcal{I}_{[1,1,1], - \mu_{2}}}y_{i}l_{i}^{(0)}z_{i,l}\right| \\ = \frac{\alpha}{n} a_{j}\bigg|\sum_{\substack{i\in \mathcal{I}_{-\mu_{1}}\cup \mathcal{I}_{[1,1,1],\mu_{2}}\cup \mathcal{I}_{[1,1,1], - \mu_{2}}}y_{i}l_{i}^{(0)}(z_{i,l} - \bar{z}_{i,l}) \\ - \left[ \sum_ {i \in \mathcal {I} _ {[ 1, 1, 1 ], \mu_ {2}}} l _ {i} ^ {(0)} \bar {z} _ {i, l} + \sum_ {i \in \mathcal {I} _ {[ 1, 1, 1 ], - \mu_ {2}}} l _ {i} ^ {(0)} \bar {z} _ {i, l} \right] \\ \leq \frac {\alpha}{\sqrt {m}} \exp (v _ {\text {i n i t}} \log n) \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n + \frac {2 \alpha}{n \sqrt {m}} \left| \sum_ {i \in \mathcal {I} _ {[ 1, 1, 1 ], \mu_ {2}}} l _ {i} ^ {(0)} - \sum_ {i \in \mathcal {I} _ {[ 1, 1, 1 ], - \mu_ {2}}} l _ {i} ^ {(0)} \right| \tag {6} \\ \leq \frac {\alpha}{\sqrt {m}} \exp (v _ {\text {i n i t}} \log n) \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n + \frac {2 \alpha}{n \sqrt {m}} \exp (v _ {\text {i n i t}} \log n) \left(\frac {n}{8} + \frac {n}{2 \log n} \right. \\ + \sqrt {n \log n} - \exp (- 2 v _ {\text {i n i t}} \log n) \left(\frac {n}{8} - \frac {n}{2 \log n} - \sqrt {n \log n}\right) \\ \leq C \frac {\alpha \varepsilon^ {2} \sqrt {p}}{\sqrt {m n}} \log n + C \frac {\alpha}{n \sqrt {m}} \left(\frac {n}{\log n} + v _ {\mathrm {i n i t}} n \log n\right) \leq C \frac {\alpha}{\sqrt {m \log n}}, \\ \end{array}
+$$
+
+where the first inequality uses (2) and  $\bar{z}_{i,l} = -\bar{z}_{j,l}$  for  $i\in \mathcal{I}_{[1,1,1],\mu_2},j\in \mathcal{I}_{[1,1,1],-\mu_2}$ ; the second inequality uses (2), (3), and (B4) in Lemma A.4; the third inequality uses Assumption (A1)-(A5); and the last inequality uses Assumption (A1), (A3) and (A4).
+
+For a datapoint  $(x,y)\sim P$ , define  $z = [z_{1},z_{2},z_{3}]^{\top} = U^{\top}x$ . Applying Lemma A.3 we obtain
+
+$$
+\mathbb {P} (\| z - \bar {z} \| \leq \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n) \geq 1 - \exp (- \Omega (\log^ {2} n)).
+$$
+
+Conditioning on
+
+$$
+\left\| z - \bar {z} \right\| \leq \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n, \tag {7}
+$$
+
+if  $x_{\mathrm{signal}} = -\mu_1$  , we combine (5) and (6) and have
+
+$$
+\langle v _ {j} ^ {(1)}, z \rangle = \langle v _ {j} ^ {(1)}, \bar {z} \rangle + \langle v _ {j} ^ {(1)}, z - \bar {z} \rangle \geq 2 \left(v _ {\text {i n i t}} + \frac {2 \alpha}{5 \sqrt {m}}\right) - C v _ {\text {i n i t}} \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n \geq \frac {3}{2} \left(v _ {\text {i n i t}} + \frac {2 \alpha}{5 \sqrt {m}}\right). \tag {8}
+$$
+
+Further for any pair  $j_1, j_2$  with  $v_{j_1}^{(0)} = v_{j_2}^{(0)} = v_{\mathrm{init}}[1, 1, 1]$  and  $a_{j_1} > 0, a_{j_2} < 0$ :
+
+If  $\langle v_{j_2}^{(1)},z\rangle < 0$ , it follows that
+
+$$
+\begin{array}{l} z _ {3} \frac {\alpha}{n \sqrt {m}} \sum_ {i = 1} ^ {n} y _ {i} l _ {i} ^ {(0)} \phi^ {\prime} (\langle v _ {j _ {2}} ^ {(0)}, z _ {i} \rangle) z _ {i, 3} = - \langle v _ {j _ {2}} ^ {(1)}, z \rangle + z _ {3} v _ {j _ {2}, 3} ^ {(0)} + \sum_ {l = 1} ^ {2} z _ {l} v _ {j _ {1}, l} ^ {(1)} \\ \geq z _ {3} v _ {\text {i n i t}} - \sum_ {l = 1} ^ {2} \left| z _ {l} - \bar {z} _ {l} \right| \left| v _ {j _ {2}, l} ^ {(1)} \right| \tag {9} \\ \geq z _ {3} v _ {\text {i n i t}} - 2 \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n \left(v _ {\text {i n i t}} + C \frac {\alpha}{\sqrt {m \log n}}\right) \geq \frac {z _ {3}}{2} v _ {\text {i n i t}}, \\ \end{array}
+$$
+
+where the first inequality uses  $v_{j_2,3}^{(0)} = v_{\mathrm{init}}$  and  $\bar{z}_l = 0, l = 1, 2$ ; the second inequality uses (7); and the last inequality uses condition (7),  $\bar{z}_3 = 2$ , and Assumption (A1), (A3) and (A4). Combining (5) and (9), we have
+
+$$
+\frac {\alpha}{n \sqrt {m}} \sum_ {i = 1} ^ {n} y _ {i} l _ {i} ^ {(0)} \phi^ {\prime} (\langle v _ {j _ {2}} ^ {(0)}, z _ {i} \rangle) z _ {i, 3} \geq \max  \left\{\frac {v _ {\text {i n i t}}}{2}, \frac {2 \alpha}{5 \sqrt {m}} \right\},
+$$
+
+which together with (6) yield that
+
+$$
+\begin{array}{l} a _ {j _ {1}} \phi (\langle v _ {j _ {1}} ^ {(1)}, z \rangle) + a _ {j _ {2}} \phi (\langle v _ {j _ {2}} ^ {(1)}, z \rangle) = a _ {j _ {1}} \langle v _ {j _ {1}} ^ {(1)}, z \rangle \\ = \frac {1}{\sqrt {m}} \left[ \langle v _ {j _ {1}} ^ {(0)}, z \rangle + \frac {\alpha}{n \sqrt {m}} z _ {3} \sum_ {i = 1} ^ {n} y _ {i} l _ {i} ^ {(0)} \phi^ {\prime} (\langle v _ {j _ {2}} ^ {(0)}, z _ {i} \rangle) z _ {i, 3} + \sum_ {l = 1} ^ {2} (v _ {j _ {1}, l} ^ {(1)} - v _ {j _ {1}, l} ^ {(0)}) z _ {l} \right] \\ \geq \frac {1}{\sqrt {m}} \left[ v _ {\text {i n i t}} + \max  \left\{\frac {v _ {\text {i n i t}}}{2}, \frac {2 \alpha}{5 \sqrt {m}} \right\} - C v _ {\text {i n i t}} \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n - C \frac {\alpha}{\sqrt {m \log n}} \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n \right] \tag {10} \\ \geq \frac {1}{\sqrt {m}} \left[ v _ {\text {i n i t}} + \frac {v _ {\text {i n i t}}}{4} + \frac {\alpha}{5 \sqrt {m}} - C v _ {\text {i n i t}} \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n - C \frac {\alpha}{\sqrt {m \log n}} \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n \right] \\ \geq \frac {v _ {\mathrm {i n i t}}}{\sqrt {m}} + \frac {\alpha}{1 0 m}, \\ \end{array}
+$$
+
+where the second inequality uses  $\max (x,y)\geq (x + y) / 2$  and the last inequality uses the fact that  $n$  is sufficiently large.
+
+If  $\langle v_{j_2}^{(1)},z\rangle >0$  , we have
+
+$$
+\begin{array}{l} a _ {j _ {1}} \phi (\langle v _ {j _ {1}} ^ {(1)}, z \rangle) + a _ {j _ {2}} \phi (\langle v _ {j _ {2}} ^ {(1)}, z \rangle) = \frac {1}{\sqrt {m}} \langle v _ {j _ {1}} ^ {(1)} - v _ {j _ {2}} ^ {(1)}, z \rangle \\ = \frac {1}{\sqrt {m}} \langle v _ {j _ {1}} ^ {(1)} - v _ {j _ {1}} ^ {(0)}, z \rangle - \frac {1}{\sqrt {m}} \langle v _ {j _ {2}} ^ {(1)} - v _ {j _ {2}} ^ {(0)}, z \rangle \\ = \frac {1}{\sqrt {m}} \left[ 2 \frac {\alpha}{n \sqrt {m}} z _ {3} \sum_ {i = 1} ^ {n} y _ {i} l _ {i} ^ {(0)} \phi^ {\prime} \left(\left\langle v _ {j _ {1}} ^ {(0)}, z _ {i} \right\rangle\right) z _ {i, 3} + \sum_ {l = 1} ^ {2} \left(v _ {j _ {1}, l} ^ {(1)} - v _ {j _ {2}, l} ^ {(1)}\right) \left(z _ {l} - \bar {z} _ {l}\right) \right] \tag {11} \\ \geq \frac {1}{\sqrt {m}} \Big [ \frac {4 \alpha}{5 \sqrt {m}} - C \frac {\alpha}{\sqrt {m \log n}} \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n \Big ] \geq \frac {2 \alpha}{5 m}, \\ \end{array}
+$$
+
+where the second equation uses  $v_{j_1}^{(0)} = v_{j_2}^{(0)}$ ; the third equation uses (4); the first inequality uses (5); and the second inequality uses Assumption (A1). Combining (10) and (11), it follows that
+
+$$
+a _ {j _ {1}} \phi \left(\left\langle v _ {j _ {1}} ^ {(1)}, z \right\rangle\right) + a _ {j _ {2}} \phi \left(\left\langle v _ {j _ {2}} ^ {(1)}, z \right\rangle\right) \geq \frac {2 \alpha}{5 m} \tag {12}
+$$
+
+when  $v_{j_1}^{(0)} = v_{j_2}^{(0)} = v_{\mathrm{init}}[1,1,1]$  and  $x_{\mathrm{signal}} = -\mu_1$ .
+
+If  $x_{\mathrm{signal}} = +\mu_1$ , following the same procedure, we obtain that  $\langle v_j^{(1)}, z \rangle < 0$  for  $a_j > 0$ . For  $a_j < 0$  similar to (5), we have
+
+$$
+\begin{array}{l} v _ {j, 3} ^ {(1)} = v _ {j, 3} ^ {(0)} + \frac {\alpha}{n} a _ {j} \sum_ {i \in \mathcal {I} _ {- \mu_ {1}}} y _ {i} l _ {i} ^ {(0)} z _ {i, 3} + \frac {\alpha}{n} a _ {j} \sum_ {i \in \mathcal {I} _ {[ 1, 1, 1 ], \mu_ {2}} \cup \mathcal {I} _ {[ 1, 1, 1 ], - \mu_ {2}}} y _ {i} l _ {i} ^ {(0)} z _ {i, 3} \\ \geq v _ {\text {i n i t}} - \frac {2 \alpha}{n \sqrt {m}} \sum_ {i \in \mathcal {I} - \mu_ {1}} l _ {i} ^ {(0)} - O \left(\frac {\alpha}{\sqrt {m}} \max  _ {i} l _ {i} ^ {(0)} \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n\right) \tag {13} \\ \geq v _ {\mathrm {i n i t}} - \frac {2 . 1 \alpha | \mathcal {I} _ {- \mu_ {1}} |}{n \sqrt {m}} \exp (v _ {\mathrm {i n i t}} \log n) \geq v _ {\mathrm {i n i t}} - \frac {2 . 1 \alpha}{4 \sqrt {m}} (1 + \frac {4}{\log n}) (1 + 2 v _ {\mathrm {i n i t}} \log n) \\ \geq v _ {\mathrm {i n i t}} - \frac {3 \alpha}{4 \sqrt {m}} \geq \frac {v _ {\mathrm {i n i t}}}{4}, \\ \end{array}
+$$
+
+where the last inequality comes from Assumption (A4). Then  $\langle v_j^{(1)},z\rangle < 0$  also hold for  $a_{j} < 0$  following the same analysis. Thus we have
+
+$$
+a _ {j _ {1}} \phi \left(\left\langle v _ {j _ {1}} ^ {(1)}, z \right\rangle\right) + a _ {j _ {2}} \phi \left(\left\langle v _ {j _ {2}} ^ {(1)}, z \right\rangle\right) = 0 \tag {14}
+$$
+
+when  $v_{j_1}^{(0)} = v_{j_2}^{(0)} = v_{\mathrm{init}}[1,1,1]$  and  $x_{\mathrm{signal}} = +\mu_1$ .
+
+If  $x_{\mathrm{signal}} \in \{\pm \mu_2\}$ , combining (5) and (6), we have
+
+$$
+\begin{array}{l} | \langle v _ {j} ^ {(1)}, z \rangle | \leq | \langle v _ {j} ^ {(0)}, \bar {z} \rangle | + | \langle v _ {j} ^ {(0)}, z - \bar {z} \rangle | + | \langle v _ {j} ^ {(1)} - v _ {j} ^ {(0)}, \bar {z} \rangle | + | \langle v _ {j} ^ {(1)} - v _ {j} ^ {(0)}, z - \bar {z} \rangle | \\ \leq 0 + v _ {\mathrm {i n i t}} \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n + 0 + C \frac {\alpha}{\sqrt {m}} \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n \leq 2 v _ {\mathrm {i n i t}} \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n, \\ \end{array}
+$$
+
+where the last inequality uses Assumption (A3) and (A4). Thus
+
+$$
+\left| a _ {j} \left\langle v _ {j} ^ {(1)}, z \right\rangle \right| \leq 2 v _ {\text {i n i t}} \varepsilon^ {2} \sqrt {\frac {p}{n m}} \log n \leq \frac {2 v _ {\text {i n i t}}}{\sqrt {m \log n}}. \tag {15}
+$$
+
+Note that neurons initialized with  $v_{\mathrm{init}}[i,i,k]$ ,  $i, k \in \{\pm 1\}$  share very similar dynamics and following the same procedure, specifically, if  $k = +1$  (resp.  $-1$ ), the neurons align well with  $-\mu_1$  (resp.  $+\mu_1$ ). Additionally, the neurons do not align well with  $\pm \mu_2$  for both  $i = +1$  and  $i = -1$ . For brevity, we omit the analysis for  $v_j^{(0)} = v_{\mathrm{init}}[i,i,k]$ ,  $i, k \in \{\pm 1\} \backslash \{v_{\mathrm{init}}[1,1,1]\}$ .
+
+Next we analyze the one-step update of neuron  $v_{j}$  with initialization  $v_{\mathrm{init}}[1, - 1,1]$ .
+
+(2) If  $v_{j}^{(0)} = v_{\mathrm{init}}[1, -1, 1]$ , then according to Lemma A.1, we have
+
+$$
+\mathcal {I} _ {[ 1, - 1, 1 ], + \mu_ {1}} = \varnothing ; \quad \mathcal {I} _ {[ 1, - 1, 1 ], - \mu_ {1}} = \mathcal {I} _ {- \mu_ {1}}; \quad \mathcal {I} _ {[ 1, - 1, 1 ], \mu_ {2}} = \mathcal {I} _ {+ \mu_ {2}}; \quad \mathcal {I} _ {[ 1, - 1, 1 ], - \mu_ {2}} = \varnothing . \tag {16}
+$$
+
+Similar to (5), we have
+
+$$
+\begin{array}{l} \left| v _ {j, 3} ^ {(1)} - \left(v _ {j, 3} ^ {(0)} + \frac {\alpha}{2} a _ {j}\right) \right| = \left| \frac {\alpha}{n} a _ {j} \sum_ {i = 1} ^ {n} y _ {i} l _ {i} ^ {(0)} \phi^ {\prime} (\langle v _ {j} ^ {(0)}, z _ {i} \rangle) z _ {i, 3} - \frac {\alpha}{2} a _ {j} \right| \\ = \left| \frac {\alpha}{n} a _ {j} \left[ \sum_ {i \in \mathcal {I} _ {- \mu_ {1}}} y _ {i} l _ {i} ^ {(0)} z _ {i, 3} + \sum_ {i \in \mathcal {I} _ {+ \mu_ {2}}} y _ {i} l _ {i} ^ {(0)} z _ {i, 3} \right] - \frac {\alpha}{2} a _ {j} \right| \\ = \left| \frac {\alpha}{n \sqrt {m}} \left[ \sum_ {i \in \mathcal {I} _ {- \mu_ {1}}} l _ {i} ^ {(0)} z _ {i, 3} - \sum_ {i \in \mathcal {I} _ {+ \mu_ {2}}} l _ {i} ^ {(0)} z _ {i, 3} \right] - \frac {\alpha}{2 \sqrt {m}} \right| \tag {17} \\ = \left| \frac {\alpha}{n \sqrt {m}} \Big [ \sum_ {i \in \mathcal {I} _ {- \mu_ {1}}} l _ {i} ^ {(0)} \bar {z} _ {i, 3} + \sum_ {i \in \mathcal {I} _ {- \mu_ {1}}} l _ {i} ^ {(0)} (z _ {i, 3} - \bar {z} _ {i, 3}) - \sum_ {i \in \mathcal {I} _ {+ \mu_ {2}}} l _ {i} ^ {(0)} (z _ {i, 3} - \bar {z} _ {i, 3}) \Big ] - \frac {\alpha}{2 \sqrt {m}} \right| \\ \leq \frac {2 \alpha}{n \sqrt {m}} \left| \frac {n}{4} - n _ {- \mu_ {1}} \exp (- v _ {\mathrm {i n i t}} \log n) \right| + \frac {\alpha}{n \sqrt {m}} \left(n _ {- \mu_ {1}} + n _ {+ \mu_ {2}}\right) \exp (v _ {\mathrm {i n i t}} \log n) \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n \\ = O \left(\frac {\alpha}{\sqrt {m}} \left(\varepsilon^ {2} \sqrt {\frac {p}{n}} + v _ {\text {i n i t}}\right) \log n\right) = O \left(\frac {\alpha}{\sqrt {m} \log n}\right), \\ \end{array}
+$$
+
+where the first equation comes from the GD update; the second equation uses (16); the third equation uses  $|a_{j}| = 1 / \sqrt{m}$ ; the fourth equation uses  $\bar{z}_{i,3} = 0$  for  $i \in \mathcal{I}_{+\mu_2}$ ; the first inequality uses  $\bar{z}_{i,3} = 2$ ,  $i \in \mathcal{I}_{-\mu_1}$ , (2) and the definition of  $\mathcal{G}$ ; the fifth equation uses  $|n_{\mu} - n / 4| \leq n / \log n$  and  $|\exp(-v_{\mathrm{init}}\log n) - 1| \leq 2v_{\mathrm{init}}\log n \leq 2 / \sqrt{\log n}$  by Assumption (A3); and the last equation uses Assumption (A1) and (A3). Further for the first entry of  $v_{j}$ , we have
+
+$$
+\begin{array}{l} \left| v _ {j, 1} ^ {(1)} - \left(v _ {j, 1} ^ {(0)} - \frac {\alpha}{2} a _ {j}\right) \right| = \left| \frac {\alpha}{n} a _ {j} \sum_ {i = 1} ^ {n} y _ {i} l _ {i} ^ {(0)} \phi^ {\prime} (\langle v _ {j} ^ {(0)}, z _ {i} \rangle) z _ {i, 1} + \frac {\alpha}{2} a _ {j} \right| \\ = \left| \frac {\alpha}{n} a _ {j} \left[ \sum_ {i \in \mathcal {I} _ {- \mu_ {1}}} y _ {i} l _ {i} ^ {(0)} z _ {i, 1} + \sum_ {i \in \mathcal {I} _ {+ \mu_ {2}}} y _ {i} l _ {i} ^ {(0)} z _ {i, 1} \right] + \frac {\alpha}{2} a _ {j} \right| \\ = \left| \frac {\alpha}{n \sqrt {m}} \left[ \sum_ {i \in \mathcal {I} _ {- \mu_ {1}}} l _ {i} ^ {(0)} z _ {i, 1} - \sum_ {i \in \mathcal {I} _ {+ \mu_ {2}}} l _ {i} ^ {(0)} z _ {i, 1} \right] + \frac {\alpha}{2 \sqrt {m}} \right| \tag {18} \\ = \left| \frac {\alpha}{n \sqrt {m}} \left[ - \sum_ {i \in \mathcal {I} _ {+ \mu_ {2}}} l _ {i} ^ {(0)} \bar {z} _ {i, 1} + \sum_ {i \in \mathcal {I} _ {- \mu_ {1}}} l _ {i} ^ {(0)} (z _ {i, 1} - \bar {z} _ {i, 1}) - \sum_ {i \in \mathcal {I} _ {+ \mu_ {2}}} l _ {i} ^ {(0)} (z _ {i, 1} - \bar {z} _ {i, 1}) \right] + \frac {\alpha}{2 \sqrt {m}} \right| \\ \leq \frac {2 \alpha}{n \sqrt {m}} \left| 1 - n _ {+ \mu_ {2}} \exp (- v _ {\mathrm {i n i t}} \log n) \right| + \frac {\alpha}{n \sqrt {m}} \left(n _ {- \mu_ {1}} + n _ {+ \mu_ {2}}\right) \exp (v _ {\mathrm {i n i t}} \log n) \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n \\ = O (\frac {\alpha}{\sqrt {m}} (\varepsilon^ {2} \sqrt {\frac {p}{n}} + v _ {\mathrm {i n i t}}) \log n) = O (\frac {\alpha}{\sqrt {m \log n}}), \\ \end{array}
+$$
+
+where the inequality uses  $\bar{z}_{i,1} = 2$  for  $i\in \mathcal{I}_{+\mu_2}$ . And for the second entry of  $v_{j}$ , it follows similarly that
+
+$$
+\left| v _ {j, 2} ^ {(1)} - \left(v _ {j, 2} ^ {(0)} + \frac {\alpha}{2} a _ {j}\right) \right| = \left| \frac {\alpha}{n \sqrt {m}} \left[ \sum_ {i \in \mathcal {I} _ {- \mu_ {1}}} l _ {i} ^ {(0)} z _ {i, 2} - \sum_ {i \in \mathcal {I} _ {+ \mu_ {2}}} l _ {i} ^ {(0)} z _ {i, 2} \right] - \frac {\alpha}{2 \sqrt {m}} \right| = O (\frac {\alpha}{\sqrt {m \log n}}). \tag {19}
+$$
+
+Unifying (17), (17) and (18), we obtain
+
+$$
+\left| v _ {j, l} ^ {(1)} - \left(v _ {j, l} ^ {(0)} + \frac {\alpha}{2} a _ {j} \operatorname {s g n} \left(v _ {j, l} ^ {(0)}\right) \xi_ {l}\right) \right| = O \left(\frac {\alpha}{\sqrt {m \log n}}\right) \tag {20}
+$$
+
+for  $l = 1,2,3$ . Here  $\{\xi_l\}$  are defined as  $\xi_{l} = -1,l = 1,2$  and  $\xi_3 = 1$ .
+
+For a datapoint  $(x,y)\sim P$  with  $z = [z_{1},z_{2},z_{3}]^{\top} = U^{\top}x$ . We condition on the event
+
+$$
+\left\| z - \bar {z} \right\| \leq \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n.
+$$
+
+If  $x_{\mathrm{signal}} = -\mu_1$ : for each pair  $j_1, j_2$  with  $v_{j_1}^{(0)} = v_{j_2}^{(0)} = v_{\mathrm{init}}[1, -1, 1]$  and  $a_{j_1} > 0, a_{j_2} < 0$ , we have  $\langle v_{j_l}^{(1)}, z \rangle > 0, l = 1, 2$ , and
+
+$$
+\| v _ {j _ {1}} ^ {(1)} - v _ {j _ {2}} ^ {(1)} - \frac {\alpha}{\sqrt {m}} [ - 1, 1, 1 ] ^ {\top} \| = \| (v _ {j _ {1}} ^ {(1)} - v _ {j _ {1}} ^ {(0)}) - (v _ {j _ {2}} ^ {(1)} - v _ {j _ {2}} ^ {(0)}) - \frac {\alpha}{\sqrt {m}} [ - 1, 1, 1 ] ^ {\top} \| = O (\frac {\alpha}{\sqrt {m \log n}})
+$$
+
+by (20). It follows that
+
+$$
+\begin{array}{l} a _ {j _ {1}} \phi (\langle v _ {j _ {1}} ^ {(1)}, z \rangle) + a _ {j _ {2}} \phi (\langle v _ {j _ {2}} ^ {(1)}, z \rangle) = \frac {1}{\sqrt {m}} \langle v _ {j _ {1}} ^ {(1)} - v _ {j _ {2}} ^ {(1)}, z \rangle \\ = \frac {1}{\sqrt {m}} \left(\left\langle \frac {\alpha}{\sqrt {m}} [ - 1, 1, 1 ], \bar {z} \right\rangle + \left\langle v _ {j _ {1}} ^ {(1)} - v _ {j _ {2}} ^ {(1)} - \frac {\alpha}{\sqrt {m}} [ - 1, 1, 1 ], \bar {z} \right\rangle + \left\langle v _ {j _ {1}} ^ {(1)} - v _ {j _ {2}} ^ {(1)}, z - \bar {z} \right\rangle\right) \tag {21} \\ \geq \frac {1}{\sqrt {m}} \Big (\frac {2 \alpha}{\sqrt {m}} - O (\frac {\alpha}{\sqrt {m} \log n}) - O (\frac {\alpha}{\sqrt {m} \log n} \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n) \Big) \geq \frac {\alpha}{m}. \\ \end{array}
+$$
+
+If  $x_{\mathrm{signal}} = +\mu_1$  we have  $\langle v_{j_l}^{(1)},z\rangle < 0,l = 1,2,$  thus
+
+$$
+a _ {j _ {1}} \phi (\langle v _ {j _ {1}} ^ {(1)}, z \rangle) = a _ {j _ {2}} \phi (\langle v _ {j _ {2}} ^ {(1)}, z \rangle) = 0
+$$
+
+If  $x_{\mathrm{signal}} = +\mu_2$ : we have  $\langle v_{j_l}^{(0)}, z \rangle > 0, l = 1, 2$ . Applying (20) and Assumption (A4), we have  $\langle v_{j_l}^{(1)}, z \rangle > 0, l = 1, 2$ . It follows that
+
+$$
+\begin{array}{l} a _ {j _ {1}} \phi (\langle v _ {j _ {1}} ^ {(1)}, z \rangle) + a _ {j _ {2}} \phi (\langle v _ {j _ {2}} ^ {(1)}, z \rangle) = \frac {1}{\sqrt {m}} \langle v _ {j _ {1}} ^ {(1)} - v _ {j _ {2}} ^ {(1)}, z \rangle \\ = \frac {1}{\sqrt {m}} \left(\left\langle \frac {\alpha}{\sqrt {m}} [ - 1, 1, 1 ], \bar {z} \right\rangle + \left\langle v _ {j _ {1}} ^ {(1)} - v _ {j _ {2}} ^ {(1)} - \frac {\alpha}{\sqrt {m}} [ - 1, 1, 1 ], \bar {z} \right\rangle + \left\langle v _ {j _ {1}} ^ {(1)} - v _ {j _ {2}} ^ {(1)}, z - \bar {z} \right\rangle\right) \tag {22} \\ \leq \Big (- \frac {2 \alpha}{\sqrt {m}} + O (\frac {\alpha}{\sqrt {m} \log n}) + O (\frac {\alpha}{\sqrt {m} \log n} \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n) \Big) \leq - \frac {\alpha}{m} \\ \end{array}
+$$
+
+for sufficiently large  $n$ . Here the last inequality uses Assumption (A1).
+
+If  $x_{\mathrm{signal}} = -\mu_2$ : we have  $\langle v_{j_l}^{(0)}, z \rangle < 0, l = 1, 2$ . Applying (20) and Assumption (A4), we have  $\langle v_{j_l}^{(1)}, z \rangle < 0, l = 1, 2$ . It follows that
+
+$$
+a _ {j _ {1}} \phi \left(\left\langle v _ {j _ {1}} ^ {(1)}, z \right\rangle\right) + a _ {j _ {2}} \phi \left(\left\langle v _ {j _ {2}} ^ {(1)}, z \right\rangle\right) = 0. \tag {23}
+$$
+
+In conclusion, for datapoint  $(x,y)$  with  $x_{\mathrm{signal}} = -\mu_1$  conditioning on (7), the output of  $f^{(1)}$  is
+
+$$
+\begin{array}{l} f ^ {(1)} (x) = \sum_ {j = 1} ^ {m} a _ {j} \phi (\langle v _ {j} ^ {(1)}, z \rangle) = \sum_ {e \in \operatorname {U n i f o r m} (\{\pm 1 \} ^ {3})} \sum_ {j \in \mathcal {J} _ {e}} a _ {j} \phi (\langle v _ {j} ^ {(1)}, z \rangle) \\ = \sum_ {e: e _ {3} = 1} \left[ \sum_ {j \in \mathcal {J} _ {\mathrm {P o s}, e}} a _ {j} \phi \left(\left\langle v _ {j} ^ {(1)}, z \right\rangle\right) - \sum_ {j \in \mathcal {J} _ {\mathrm {N e g}, e}} a _ {j} \phi \left(\left\langle v _ {j} ^ {(1)}, z \right\rangle\right) \right] \tag {24} \\ \geq \sum_ {e: e _ {3} = 1} \left[ \min  \left\{\left| \mathcal {J} _ {\text {P o s}, e} \right|, \left| \mathcal {J} _ {\text {N e g}, e} \right| \right\} \frac {2 \alpha}{5 m} - \frac {4 \sqrt {m}}{\log n} \left(v _ {\text {i n i t}} + \frac {\alpha}{\sqrt {m}}\right) \right] \\ \geq \sum_ {e: e _ {3} = 1} \left[ \frac {\alpha}{4 0} - \frac {4 \sqrt {m}}{\log n} (C \frac {\alpha}{\sqrt {m}} + \frac {\alpha}{\sqrt {m}}) \right] > 0 \\ \end{array}
+$$
+
+for sufficiently large  $n$ . Here the first inequality uses (12) and (21), the property that  $||\mathcal{J}_{\mathrm{Pos},e}| - |\mathcal{J}_{\mathrm{Neg},e}|| \leq 2m / \log n$  from (B3) in Lemma A.4 and the property that  $\phi(\langle v_j^{(1)}, z \rangle) \leq 2(v_{\mathrm{init}} + \alpha / \sqrt{m})$ ; the second inequality uses (B3) and Assumption (A4). Similarly, we have that for datapoint  $(x, y)$  with  $x_{\mathrm{signal}} = +\mu_1$ , conditioning on (7), the output of  $f^{(1)}$  is
+
+$$
+f ^ {(1)} (x) = \sum_ {e: e _ {3} = - 1} \left[ \sum_ {j \in \mathcal {J} _ {\mathrm {P o s}, e}} a _ {j} \phi \left(\left\langle v _ {j} ^ {(1)}, z \right\rangle\right) - \sum_ {j \in \mathcal {J} _ {\mathrm {N e g}, e}} a _ {j} \phi \left(\left\langle v _ {j} ^ {(1)}, z \right\rangle\right) \right] > 0. \tag {25}
+$$
+
+For datapoint  $(x,y)$  with  $x_{\mathrm{signal}} = +\mu_2$  conditioning on (7), the output of  $f^{(1)}$  is
+
+$$
+\begin{array}{l} f ^ {(1)} (x) = \sum_ {j = 1} ^ {m} a _ {j} \phi (\langle v _ {j} ^ {(1)}, z \rangle) = \sum_ {e \in \mathrm {U n i f o r m} (\{\pm 1 \} ^ {3})} \sum_ {j \in \mathcal {J} _ {e}} a _ {j} \phi (\langle v _ {j} ^ {(1)}, z \rangle) \\ = \left(\sum_ {e: [ e _ {1}, e _ {2} ] = [ 1, - 1 ]} + \sum_ {e: e _ {1} = e _ {2}}\right) \left[ \sum_ {j \in \mathcal {J} _ {\mathrm {P o s}, e}} a _ {j} \phi \left(\langle v _ {j} ^ {(1)}, z \rangle\right) - \sum_ {j \in \mathcal {J} _ {\mathrm {N e g}, e}} a _ {j} \phi \left(\langle v _ {j} ^ {(1)}, z \rangle\right) \right] \\ \leq \sum_ {e: \left[ e _ {1}, e _ {2} \right] = [ 1, - 1 ]} \left[ - \min  \left\{\left| \mathcal {J} _ {\text {P o s}, e} \right|, \left| \mathcal {J} _ {\text {N e g}, e} \right| \right\} \frac {\alpha}{m} + \frac {4 \sqrt {m}}{\log n} \left(v _ {\text {i n i t}} + \frac {\alpha}{\sqrt {m}}\right) \right] + \sum_ {e: e _ {1} = e _ {2}} \frac {2 v _ {\text {i n i t}} \left| \mathcal {J} _ {\text {N e g} , e} \right|}{\sqrt {m \log n}} \\ \leq 2 \left(- \frac {\alpha}{1 6} + \frac {5 \sqrt {m}}{\log n} \left(v _ {\text {i n i t}} + \frac {\alpha}{\sqrt {m}}\right)\right) + \frac {v _ {\text {i n i t}} \sqrt {m}}{\sqrt {\log n}} \leq - \frac {\alpha}{8} + \frac {1 0 (C + 1) \alpha}{\log n} + \frac {C \alpha}{\sqrt {\log n}} <   0, \tag {26} \\ \end{array}
+$$
+
+where the first inequality uses (15), (22), (B3) and the property that  $\phi (\langle v_j^{(1)},z\rangle)\leq 2(v_{\mathrm{init}} + \alpha /\sqrt{m})$  the second inequality uses (B3); and the third inequality uses Assumption (A4). Similarly, we have that for datapoint  $(x,y)$  with  $x_{\mathrm{signal}} = -\mu_2$  conditioning on (7),  $f^{(1)}(x) < 0$  , which combined with (24), (25) and (26), yields that
+
+$$
+\operatorname {s g n} (f ^ {(1)} (x)) = y
+$$
+
+for any  $(x,y)\sim P,z = U^{\top}x$  satisfying
+
+$$
+\| z - \bar {z} \| \leq \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n.
+$$
+
+According to the definition of  $\mathcal{G}_{\mathrm{data}}$ , all  $(x_i, y_i)$  satisfy this condition. Thus conditioning on the event  $\mathcal{G}$ , the model  $f^{(1)}$  can correctly classify all training data points. And applying the law of total probability, we obtain that the test error is bounded by:
+
+$$
+\begin{array}{l} \mathbb {P} _ {(x, y) \sim P} (y \neq \operatorname {s g n} (f ^ {(1)} (x))) \leq \mathbb {P} _ {(x, y) \sim P} \Big (y \neq \operatorname {s g n} (f ^ {(1)} (x)) \mid \| z - \bar {z} \| \leq \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n \Big) \\ + \mathbb {P} _ {(x, y) \sim P} \Big (\| z - \bar {z} \| \leq \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n \Big) \\ = \mathbb {P} _ {(x, y) \sim P} \left(\| z - \bar {z} \| \leq \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n\right) \leq \exp (- \Omega (\log^ {2} n)), \\ \end{array}
+$$
+
+where the last inequality uses Lemma A.3.
+
+![](images/51ab1d29773b91dd964fbb63925ea73c4e7978c29a806b5c1533b58efe329f2f.jpg)
+
+Lemma A.1. Suppose that Assumption (A2) holds. With probability at least  $1 - O\left(\frac{1}{n^2}\right)$ , the following conditions hold:
+
+$$
+\begin{array}{l} \mathcal {I} _ {[ i, j, - 1 ], + \mu_ {1}} = \mathcal {I} _ {+ \mu_ {1}}; \quad \mathcal {I} _ {[ i, j, - 1 ], - \mu_ {1}} = \varnothing , \quad i, j \in \{\pm 1 \}; \\ \mathcal {I} _ {[ i, j, + 1 ], + \mu_ {1}} = \varnothing ; \quad \mathcal {I} _ {[ i, j, + 1 ], - \mu_ {1}} = \mathcal {I} _ {- \mu_ {1}}, \quad i, j \in \{\pm 1 \}; (27) \\ \mathcal {I} _ {[ + 1, - 1, k ], + \mu_ {2}} = \mathcal {I} _ {+ \mu_ {2}}; \quad \mathcal {I} _ {[ + 1, - 1, k ], - \mu_ {2}} = \varnothing , \quad k \in \{\pm 1 \}; \\ \mathcal {I} _ {[ - 1, + 1, k ], + \mu_ {2}} = \varnothing ; \quad \mathcal {I} _ {[ - 1, + 1, k ], - \mu_ {2}} = \mathcal {I} _ {- \mu_ {2}}, \quad k \in \{\pm 1 \}; \\ \left| \left| \mathcal {I} _ {[ i, i, k ], \mu} \right| - \frac {n _ {\mu}}{2} \right| \leq \sqrt {n \log n}, \quad i, k \in \{\pm 1 \}, \mu \in \{\pm \mu_ {2} \}. (28) \\ \end{array}
+$$
+
+Proof. For simplicity, we denote  $\mathbb{P}(\cdot \mid \{(x_i,y_i)\}_{i = 1}^n\in \mathcal{G}_{\mathrm{data}})$  as  $\mathbb{P}(\cdot)$  in the proof below.
+
+For  $v_{j}^{(0)} = [v_{\mathrm{init}}, v_{\mathrm{init}}, v_{\mathrm{init}}]$ , we first show that for  $\{(x_{i}, y_{i})\}_{i=1}^{n} \in \mathcal{G}_{\mathrm{data}}$ ,
+
+$$
+\left\langle v _ {j} ^ {(0)}, z _ {i} \right\rangle > 0, \quad \forall i \in \mathcal {I} _ {- \mu_ {1}}.
+$$
+
+According to the definition of  $\mathcal{G}_{\mathrm{data}}$ ,  $\| z_{i} - \bar{z}_{i}\| \leq \varepsilon^{2}\sqrt{p / n}\log n$  for all  $i\in \mathcal{I}_{-\mu_1}$ . Thus
+
+$$
+\langle v _ {j} ^ {(0)}, z _ {i} \rangle = \langle v _ {j} ^ {(0)}, \bar {z} _ {i} \rangle + \langle v _ {j} ^ {(0)}, z _ {i} - \bar {z} _ {i} \rangle \geq v _ {\mathrm {i n i t}} (2 - \| z _ {i} - \bar {z} _ {i} \|) > 0,
+$$
+
+where the first inequality uses  $\bar{z}_i = [0,0,2]$  when  $i\in \mathcal{I}_{-\mu_1}$  and the second inequality uses Assumption (A1). Similarly, we have
+
+$$
+\langle v _ {j} ^ {(0)}, z _ {i} \rangle <   0, \quad \forall i \in \mathcal {I} _ {+ \mu_ {1}}.
+$$
+
+Thus conditioning on  $\{(x_i,y_i)\}_{i = 1}^n\in \mathcal{G}_{\mathrm{data}}$  , we have
+
+$$
+\mathcal {I} _ {[ 1, 1, 1 ], - \mu_ {1}} = \mathcal {I} _ {- \mu_ {1}}; \mathcal {I} _ {[ 1, 1, 1 ], + \mu_ {1}} = \varnothing .
+$$
+
+For  $i \in \mathcal{I}_{\pm \mu_2}$ , recall that  $x_i = [x_{i,\mathrm{signal}}, x_{i,\mathrm{noise}}]^\top$  with  $x_{i,\mathrm{signal}} = [x_{i,1}, x_{i,2}]^\top$  and  $x_{i,\mathrm{noise}} = [x_{i,3}, \dots, x_{i,p}]^\top$ . We have
+
+$$
+\langle v _ {j} ^ {(0)}, z _ {i} \rangle = v _ {\mathrm {i n i t}} \sum_ {l = 1} ^ {3} z _ {i, l} = v _ {\mathrm {i n i t}} \left(\sum_ {l = 1} ^ {3} u _ {l}\right) ^ {\top} x _ {i} = v _ {\mathrm {i n i t}} \left[ - \mu_ {1} ^ {\top}, \left(\sum_ {l = 1} ^ {3} \delta_ {l}\right) ^ {\top} \right] x _ {i} = v _ {\mathrm {i n i t}} \left(\sum_ {l = 1} ^ {3} \delta_ {l}\right) ^ {\top} x _ {i, \text {n o i s e}}.
+$$
+
+It follows that
+
+$$
+\mathbb {P} \left(\langle v _ {j} ^ {(0)}, z _ {i} \rangle > 0\right) = \frac {1}{2}.
+$$
+
+Applying Hoeffding's inequality, we obtain
+
+$$
+\mathbb {P} \left(\left| \left| \mathcal {I} _ {[ 1, 1, 1 ], + \mu_ {2}} \right| - \frac {\left| \mathcal {I} _ {+ \mu_ {2}} \right|}{2} \right| > t\right) \leq 2 \exp \left(- \frac {2 t ^ {2}}{n}\right).
+$$
+
+Similarly we have
+
+$$
+\mathbb {P} \big (\big | \mathcal {I} _ {[ 1, 1, 1 ], - \mu_ {2}} \big | - \frac {\big | \mathcal {I} _ {- \mu_ {2}} \big |}{2} \bigg | > t \big) \leq 2 \exp (- \frac {2 t ^ {2}}{n}).
+$$
+
+Let  $t = \sqrt{n\log n}$ . We have
+
+$$
+\left| \left| \mathcal {I} _ {[ 1, 1, 1 ], \mu} \right| - \frac {n _ {\mu}}{2} \right| \leq \sqrt {n \log n}, \quad \mu \in \{\pm \mu_ {2} \}
+$$
+
+with probability at least  $1 - 4 / n^2$ . Following similar discussion, we have that
+
+$$
+\mathcal {I} _ {[ i, j, - 1 ], + \mu_ {1}} = \mathcal {I} _ {+ \mu_ {1}}; \quad \mathcal {I} _ {[ i, j, - 1 ], - \mu_ {1}} = \varnothing , \quad i, j \in \{\pm 1 \};
+$$
+
+$$
+\mathcal {I} _ {[ i, j, + 1 ], + \mu_ {1}} = \varnothing ; \quad \mathcal {I} _ {[ i, j, + 1 ], - \mu_ {1}} = \mathcal {I} _ {- \mu_ {1}}, \quad i, j \in \{\pm 1 \};
+$$
+
+$$
+\mathcal {I} _ {[ + 1, - 1, k ], + \mu_ {2}} = \mathcal {I} _ {+ \mu_ {2}}; \quad \mathcal {I} _ {[ + 1, - 1, k ], - \mu_ {2}} = \varnothing , \quad k \in \{\pm 1 \};
+$$
+
+$$
+\mathcal {I} _ {[ - 1, + 1, k ], + \mu_ {2}} = \varnothing ; \quad \mathcal {I} _ {[ - 1, + 1, k ], - \mu_ {2}} = \mathcal {I} _ {- \mu_ {2}}, \quad k \in \{\pm 1 \}
+$$
+
+hold with probability 1 given  $\{(x_i,y_i)\}_{i = 1}^n\in \mathcal{G}_{\mathrm{data}}$  . And
+
+$$
+\left| \left| \mathcal {I} _ {[ i, i, k ], \mu} \right| - \frac {n _ {\mu}}{2} \right| \leq \sqrt {n \log n}, \quad i, k \in \{\pm 1 \}, \mu \in \{\pm \mu_ {2} \}
+$$
+
+hold with probability at least  $1 - 16 / n^2$ . In total, the conditions above hold with probability at least  $1 - \exp(-\Omega(\log^2 n)) - O\left(\frac{1}{n^2}\right) = 1 - O\left(\frac{1}{n^2}\right)$ .
+
+![](images/44efdcc970f81faa68af881602ae3c94351bc2904c9b98c2d0ef5cdc0c4e0691.jpg)
+
+Lemma A.2. Suppose that Assumption (A2) holds. Let the training data  $\{x_{i},y_{i}\}_{i = 1}^{n}$  for model  $f_{L}$  be sampled i.i.d from  $P$ . With probability at least  $1 - \exp (-\Omega (\log^2 n))$ , we have
+
+$$
+\left\| z _ {i} - \bar {z} _ {i} \right\| \leq \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n, \quad \text {f o r a l l} i \in [ n ]. \tag {29}
+$$
+
+Proof. Applying Lemma A.3, we obtain
+
+$$
+\begin{array}{l} \mathbb {P} \left(\left\| z _ {i} - \bar {z} _ {i} \right\| \leq \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n, \forall i \in [ n ]\right) \geq 1 - \sum_ {i = 1} ^ {n} \mathbb {P} \left(\left\| z _ {i} - \bar {z} _ {i} \right\| > \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n\right) \\ \geq 1 - n \exp (- \Omega (\log^ {2} n)) = 1 - \exp (- \Omega (\log^ {2} n)). \\ \end{array}
+$$
+
+![](images/ff82df0ab1c5b3a688f731efc69d669541b0d4430ad888ff842a41dd8584a723.jpg)
+
+Lemma A.3. Suppose that Assumption (A2) holds. For  $x = [x_{1}, x_{2}, \dots, x_{p}] \sim P$  with  $[x_{1}, x_{2}]^{\top} = \mu, \mu \in \{\pm \mu_{1}, \pm \mu_{2}\}$ , we have
+
+$$
+\mathbb {P} \left(\left\| U ^ {\top} x - \nu \right\| _ {\max } > \varepsilon^ {2} \sqrt {\frac {p}{n}} \log n\right) \leq \exp \left(- \Omega \left(\log^ {2} n\right)\right), \tag {30}
+$$
+
+where  $\nu = [\mu_2, -\mu_2, -\mu_1]^\top \mu \in \mathbb{R}^3$
+
+Proof. We start our analysis with  $[x_1, x_2]^\top = \mu_1$ . Note that
+
+$$
+u _ {1} ^ {\top} x = \sum_ {i = 1} ^ {p - 2} \delta_ {1, i} x _ {i + 2}
+$$
+
+is a summation of independent bounded random variables with zero mean. By Hoeffding's inequality, we have
+
+$$
+\mathbb {P} (| u _ {1} ^ {\top} x | \geq t) \leq 2 \exp \left(- \frac {t ^ {2}}{2 \sum_ {i = 1} ^ {p - 2} \delta_ {1 , i} ^ {2} \varepsilon^ {2}}\right) \leq 2 \exp \left(- \frac {t ^ {2}}{2 \| \delta \| _ {\mathrm {F}} ^ {2} \varepsilon^ {2}}\right).
+$$
+
+Similarly, the concentration for  $u_{2}^{\top}x$  and  $u_{3}^{\top}x + 2$  are as follows:
+
+$$
+\mathbb {P} \left(\left| u _ {2} ^ {\top} x \right| \geq t\right) \leq 2 \exp \left(- \frac {t ^ {2}}{2 \| \delta \| _ {\mathrm {F}} ^ {2} \varepsilon^ {2}}\right);
+$$
+
+$$
+\mathbb {P} \left(\left| u _ {3} ^ {\top} x + 2 \right| \geq t\right) \leq 2 \exp \left(- \frac {t ^ {2}}{2 \| \delta \| _ {\mathrm {F}} ^ {2} \varepsilon^ {2}}\right).
+$$
+
+Combining these inequalities yields
+
+$$
+\mathbb {P} \left(\| U ^ {\top} x - \nu_ {1} \| _ {\max } > t\right) \leq 6 \exp \left(- \frac {t ^ {2}}{2 \| \delta \| _ {\mathrm {F}} ^ {2} \varepsilon^ {2}}\right) \leq 6 \exp \left(- \frac {n t ^ {2}}{2 C ^ {2} \varepsilon^ {4} p}\right), \tag {31}
+$$
+
+where the last inequality uses Assumption (A2). The proof concludes by letting  $t = \varepsilon^2\sqrt{p / n}\log n$ . The analysis for other values of  $[x_1,x_2]^\top$  follows similarly.
+
+Lemma A.4. Suppose that Assumption (A5) holds. Then the following conditions hold with probability at least  $1 - O(1/n^4)$ :
+
+(B1)  $\max_{k\in \{\mathrm{Pos,Neg}\}}\left|\left|\mathcal{J}_k\right| - \frac{m}{2}\right|\leq \frac{m}{\log n}.$  
+(B2)  $\max_{e\in \mathit{Uniform}(\{\pm 1\} ^3)}|\mathcal{I}_e - \frac{m}{8} |\leq \frac{m}{\log n}$  
+(B3)  $\max_{k\in \{\mathrm{Pos,Neg}\} ,e\in \mathrm{Uniform}(\{\pm 1\} ^3)}|\mathcal{J}_{k,e} - \frac{m}{16} |\leq \frac{m}{\log n}.$  
+(B4)  $\max_{\mu \in \{\pm \mu_1, \pm \mu_2\}} |n_\mu - \frac{n}{4}| \leq \frac{n}{\log n}$ .
+
+Proof. Note that  $|\mathcal{J}_{\mathrm{Pos}}| \sim \mathrm{Bin}(m, 1/2)$ . Applying Hoeffding's inequality, we have
+
+$$
+\mathbb {P} \left(\left| \left| \mathcal {J} _ {\mathrm {P o s}} \right| - \frac {m}{2} \right| \leq \frac {m}{\log n}\right) \leq 2 \exp \left(- \frac {2 m}{\log^ {2} n}\right) \leq \frac {2}{n ^ {4}},
+$$
+
+where the last inequality comes from Assumption (A5). And similarly
+
+$$
+\mathbb {P} \left(\left| \left| \mathcal {J} _ {\mathrm {N e g}} \right| - \frac {m}{2} \right| \leq 2 \exp \left(- \frac {2 m}{\log^ {2} n}\right) \leq \frac {2}{n ^ {4}}, \right.
+$$
+
+which completes the proof of (B1). Note that  $|n_{\mu}| \sim \mathrm{Bin}(n, 1/4)$ . Applying Hoeffding's inequality, we have
+
+$$
+\mathbb {P} \left(\left| | n _ {\mu} | - \frac {n}{4} \right| \leq \frac {n}{\log n}\right) \leq 2 \exp \left(- \frac {2 n}{\log^ {2} n}\right) = O \left(\frac {1}{n ^ {4}}\right), \quad \forall \mu \in \{\pm \mu_ {1}, \pm \mu_ {2} \}.
+$$
+
+(B2)-(B3) can be proved following the same procedure. We omit the proof here.
+
+![](images/8fc5057e5925b709ff3cff647b9c069d7afe57588fd0fa92f6fd83351ec76312.jpg)  
+Figure 9: Training dynamics of the model  $f_{L}$  discussed in Section 3.3.
+
+# A.2 ADDITIONAL EXPERIMENTS
+
+# A.3 EXPERIMENTAL DETAILS
+
+All experiments in the paper can be run on a single NVIDIA A100 GPU. The loss function for modular arithmetic tasks is cross-entropy loss and for (40, 3)-parity task is logistic loss. All models used in the paper, unless stated otherwise, set  $d_{mlp} = 4d_{embed}$ , where  $d_{mlp}$  is the MLP dimension and  $d_{embed}$  is the embedding dimension. All FNN models used are in the paper are homogeneous and do not have bias terms. Code is available at https://github.com/zhiweixx/groktransfer.
+
+# A.3.1 EXPERIMENTS IN SECTION 1 AND 2.1
+
+In Figure 1b, we use a two-layer FNN with trainable embedding layer as the weak model. We choose  $(d_{embed},\mathrm{width}) = (4,16)$  for the weak model. The target model is a three-layer FNN with trainable embedding layer. We choose  $(d_{embed},\mathrm{width}) = (128,512)$  for the target model. The hyperparameters (init scale, learning rate, weight decay) are selected by the following grid search:
+
+$$
+i n i t \text {s c a l e :} [ 0. 1, 0. 2, \dots , 1. 5 ]
+$$
+
+$$
+\text {l e a n n i n g r a t e :} [ 1 0 ^ {- 4}, 5 \times 1 0 ^ {- 4}, 1 0 ^ {- 3}, 5 \times 1 0 ^ {- 3}, 1 0 ^ {- 2}, 1 0 ^ {- 1} ]
+$$
+
+$$
+w e i g h t \quad d e c a y: [ 1 0 ^ {- 4}, 1 0 ^ {- 3}, 1 0 ^ {- 2}, 1 0 ^ {- 1}, 1, 2, 3, 4, 5 ].
+$$
+
+We select the configuration that first achieves  $90\%$  accuracy on the validation set. The best configuration for GrokTransfer is (0.3, 0.005, 3). For standard training, only learning rate and weight decay are tuned. They are selected by the following grid search:
+
+$$
+\text {l e a n n i n g r a t e :} [ 1 0 ^ {- 3}, 5 \times 1 0 ^ {- 3}, 1 0 ^ {- 2}, 5 \times 1 0 ^ {- 2}, 1 0 ^ {- 1} ]
+$$
+
+$$
+w e i g h t \quad d e c a y: [ 1 0 ^ {- 2}, 1 0 ^ {- 1}, 1, 2, 3, 4, 5 ],
+$$
+
+and the optimal configuration is (0.05, 3).
+
+In Figure 2, we set the dimension of the GPT embedding to be 128. For the Fourier embedding, we choose  $k = 7$  frequencies, and let  $i_j$  to be the  $j$ -th smallest prime number. For each type of embedding, we normalize the embedding of each integer to be 1. The FNN used in Figure 2 is a three layer dense neural network
+
+$$
+f (x) = W _ {3} \phi \left(W _ {2} \phi \left(W _ {1} x\right)\right),
+$$
+
+where  $W_{1} \in \mathbb{R}^{\mathrm{width} \times \mathrm{embed} \, \dim}$ ,  $W_{2} \in \mathbb{R}^{\mathrm{width} \times \mathrm{width}}$ ,  $W_{3} \in \mathbb{R}^{p \times \mathrm{width}}$ , width = 512. The hyperparameters (init scale, learning rate, weight decay) are selected by the following grid search:
+
+$$
+i n i t \text {s c a l e :} [ 0. 1, 0. 2, \dots , 1. 5 ]
+$$
+
+$$
+\text {l e a n n i n g r a t e :} [ 1 0 ^ {- 4}, 1 0 ^ {- 3}, 1 0 ^ {- 2}, 1 0 ^ {- 1}, 1 ]
+$$
+
+$$
+w e i g h t \quad d e c a y: [ 1 0 ^ {- 4}, 1 0 ^ {- 3}, 1 0 ^ {- 2}, 1 0 ^ {- 1}, 1, 5, 1 0 ].
+$$
+
+We select the configuration that first achieves  $90\%$  accuracy on the validation set. The best configuration (init, lr, wd) for the four embeddings are:
+
+One-hot: (0.2, 0.01, 5); Binary: (0.3, 0.01, 1); Fourier: (0.5, 0.1, 0.1) GPT: (1.3, 0.01, 1).
+
+In Figure 3, the distance between two empirical NTK is estimated following the method in Mohamadi et al. (2023). We denote  $\widehat{\Theta}_t$  as the pseudo-NTK of the model at epoch  $t$ , i.e.
+
+$$
+\widehat {\Theta} _ {t} \left(x _ {1}, x _ {2}\right) = \left[ \nabla_ {\theta} \sum_ {i = 1} ^ {p} f _ {\theta} ^ {(i)} \left(x _ {1}\right) \right] ^ {\top} \left[ \nabla_ {\theta} \sum_ {i = 1} ^ {p} f _ {\theta} ^ {(i)} \left(x _ {2}\right) \right] / p \in \mathbb {R}.
+$$
+
+We estimate the distance between the empirical NTK at step  $t$  and  $t - 1$  by  $\| \widehat{\Theta_t} - \widehat{\Theta_{t-1}} \|_{\mathbf{F}}$ .
+
+# A.3.2 EXPERIMENTS IN SECTION 3
+
+For experiments in Section 3, we let the sample size  $n = 400$ , feature dimension  $p = 80000$ , and noise level  $\varepsilon = 0.05$ . For Figure 4(a), the model is a two-layer neural network with width 2048. The optimizer is full-batch gradient descent with learning rate 0.1 and weight decay 0.1. In Figure 4(b), we train a small model with only three neurons. The initialization of the hidden layer follows i.i.d  $N(0,0.01)$ , and the initialization of the second layer follows i.i.d  $N(0,10^{-4})$ . The learning rate is 0.1 and weight decay is 0, 1. Figure 4(c) visualizes the hidden layer of that small model after training.
+
+Figure 5a generates 4000 i.i.d datapoints from the distribution  $P$ , and visualizes  $Ux$  for each  $x$ . Figure 5b fixes  $n = 1000$  and train the weak model for  $p = [4 \times 10^4, 8 \times 10^4, 16 \times 10^4, ], \epsilon = [1/40, 1/80, 1/160]$ . Figure 5c fixes  $p = 8 \times 10^4$  and train the weak model for  $n = [400, 800, 1600, 3200], \epsilon = [1/40, 1/80, 1/160]$ . Figure 9 takes  $v_{\mathrm{init}} = 0.4$ , learning rate 2.0 and zero weight decay.
+
+# A.3.3 EXPERIMENTS IN SECTION 4
+
+The attention layer used in this paper follows the same structure as that in Nanda et al. (2023). While Nanda et al. (2023) also suggested to set the precision to be float64 to mitigate the Slingshot phenomenon (Thilak et al., 2022), a fluctuation of accuracy and loss during training process, we still use float32 to control the computation cost.
+
+All modular tasks set  $p = 113$  and the fraction of training data being  $25\%$ .
+
+For the (40, 3)-parity task, we set the sample size  $n = 1000$ .
+
+Unless otherwise specified, we use the AdamW optimizer (Loshchilov & Hutter, 2019) for all experiments; we initialize the weights using the default PyTorch initialization scaled by a factor init scale  $>0$  to control the initial weight norm, as proposed by Liu et al. (2023).
+
+The hyperparameters (init scale, learning rate, weight decay) are selected by the following grid search:
+
+$$
+\text {i n i t s c a l e :} [ 0. 0 5, 0. 1, 0. 2, 0. 3, \dots , 1. 5 ]
+$$
+
+$$
+\text {l e a n n i n g r a t e :} [ 1 0 ^ {- 4}, 5 \times 1 0 ^ {- 4}, 1 0 ^ {- 3}, 5 \times 1 0 ^ {- 3}, 1 0 ^ {- 2}, 1 0 ^ {- 1}, 0. 5, 1. 0 ]
+$$
+
+$$
+w e i g h t \quad d e c a y: [ 1 0 ^ {- 4}, 1 0 ^ {- 3}, 1 0 ^ {- 2}, 1 0 ^ {- 1}, 1, 2, 3, 4, 5 ].
+$$
+
+In Figure 6(a),(b), the structure of weak and target model are the same as those in Figure 1b. In Figure 6(c), the weak model is a three-layer width  $= 16$  FNN, the target model is a three-layer width  $= 512$  FNN with  $d_{embed} = 128$ . In Figure 6(a), the optimal configuration for GrokTransfer is (0.3, 0.001, 1) and the optimal one for training from scratch is (0.1, 0.1, 2). In Figure 6(b), the optimal configuration for GrokTransfer is (0.3, 0.005, 3) and the optimal one for training from scratch is (0.1, 0.1, 2). In Figure 6(b), we have the number of training samples  $n = 1000$ . The model trained via GrokTransfer uses learning rate  $10^{-3}$  and weight decay  $10^{-3}$ ; the model trained from scratch uses learning rate  $10^{-2}$  and weight decay 1.
+
+In Figure 8(a), the weak model is a two-layer width-4 FNN, and the target model is an 8-layer transformer with  $d_{embed} = 512$ ,  $d_{mlp} = 512$ ,  $n_{head} = 4$ ,  $d_{head} = 128$ . The optimal configuration for target model trained via GrokTransfer is (0.7, 0.001, 1). The optimal configuration for target model trained from scratch is (0.4, 0.0005, 1). In Figure 8(b), the weak model remains the same, and the target model becomes an 2-layer transformer with  $d_{embed} = 128$ ,  $d_{mlp} = 128$ ,  $n_{head} = 4$ ,  $d_{head} = 32$ . The optimal configuration for target model trained via GrokTransfer is (0.6, 0.005, 0.1). The optimal configuration for GrokFast is (lr, wd) = (0.01, 1.0).
+
+# A.4 ADDITIONAL DISCUSSION
+
+# A.4.1 FORWARD PASS FLOPS ESTIMATION FOR MODELS IN FIGURE 8 LEFT
+
+For the weak model, a two-layer width-4 MLP:
+
+$$
+C _ {f o r w a r d} \approx 2 * (8 * 4 + 8 + 4 * 1 1 3 + 2 2 6) = 1 4 3 6 \sim 1 0 ^ {3}.
+$$
+
+For the target model, an 8-layer transformer, following Table 1 in Kaplan et al. (2020), we have
+
+$$
+N = 2 d _ {e m b e d} n _ {l a y e r} (2 d _ {a t t n} + d _ {m l p}) = 2 * 5 1 2 * 8 * (2 5 6 + 5 1 2) = 6 2 9 1 4 5 6, C _ {f o r w a r d} = 2 (N + 8 * 2 * 1 2 8) \sim 1 0 ^ {7}.
+$$
+
+Two-layer FNN takes around 2000 epochs to generalize. Target model trained by GrokTransfer takes around 1000 epochs and target model trained from scratch takes around 10000 epochs. Thus, the total flops of GrokTransfer is around  $10^{10} + 2 * 10^6$  and the total flops of training from scratch is around  $10^{11}$ .
+
+# A.4.2 ADDITIONAL EXPERIMENTS
+
+![](images/cc574bce4dcf02f3c41720b25f01a26c4bb2b87c897cc4c3233922906376c60e.jpg)  
+Figure 10: Left: Training dynamics of the one neuron weak model. Middle: Visualization of the neuron in the weak model. We can see it has learned the feature  $[1, -1]$ . Right: training dynamics of the target model with embedding transferred from the one-neuron weak model.
+
+![](images/b7ea74381b48b1ca307bd0c3bd9c41587b92b02f358e951b2ab8d8585c3c91fb.jpg)
+
+![](images/116fb1c56f560b404f8180dd9f88021a3449c076fdbcb4e8d391fd45d5062576.jpg)
+
+![](images/b157843bfd90edaa2468be915acb7864d5481fb87b86ec6fabab1ce5e802e73e.jpg)  
+Figure 11: Left: Visualization of the distribution  $P$  with the embedding from the one-neuron weak model. Middle: Visualization of the distribution  $P$  with the embedding from the two-neuron weak model. Right: Training dynamics of the target model with embedding transferred from a two-neuron weak model.
+
+![](images/7812c2d15418ebb506ac1486298bd429b1b5f8f92ad7339ce5921f5bf286e651.jpg)
+
+![](images/5859fe50061320eedc17a810d20f0d2175c5cbedbfa886a2c4097d3befb6c797.jpg)
+
+When the weak model only has one neuron, the target model has the form:
+
+$$
+f _ {L} (x) = \sum_ {j = 1} ^ {m} a _ {j} \phi \left(v _ {j} \cdot u ^ {\top} x\right).
+$$
+
+Denote  $z = u^{\top}x$ . Equivalently, we have
+
+$$
+f _ {L} (z) = \sum_ {j = 1} ^ {m} a _ {j} \phi \left(v _ {j} \cdot z\right).
+$$
+
+Note that  $\mathrm{sign}(f_L(z_1)) = \mathrm{sign}(f_L(z_2))$  if  $\mathrm{sign}(z_1) = \mathrm{sign}(z_2)$ . Since both positive and negative classes locate on both sides of the original point (Figure 11 Left), it is impossible for  $f_L$  to correctly classify these points (Figure 10 Right).
+
+![](images/0040c6d29312c0d0a29b4c8dede27ba1485e7695c46c916c5f2cf49edcf0b741.jpg)  
+Figure 12: 2D Visualization of the 4D embedding from the weak model trained on modular addition task.
+
+![](images/69d7b6c6850530635b91ba91cac1a4d5cc109709f39273e752f40ef268b9d6ef.jpg)
+
+![](images/2521c1d8c37075c0f055adc73613e4f8f27cef291a647c9767dd7704f2ba652e.jpg)  
+Figure 13: Left: Training dynamics of an 8-layer transformer on modular multiplication task. Right: Training dynamics of a 2-layer transformer and a 4-layer transformer trained on the  $(40,3)$ -parity task. Both do not have delayed generalization.
+
+![](images/a9e940121a9c8edac8e0d88516ab7eb3bcceac8f746d91535ebbc671f9ef3c9e.jpg)
+
+![](images/c3f046349bf6c09495753971760a74a42be01d726bbe6c8f628085c429706c36.jpg)  
+Figure 14: Training dynamics of a four-layer MLP on MNIST dataset. 200 samples are randomly selected as the training data, and 400 samples are randomly selected as the test data. The weak model is a two-layer MLP with  $40\%$  test accuracy.
+
+![](images/19c86b287fd8ae67ecd72d218a55df2c71af617949a69c0065ab14e94684173e.jpg)  
+Figure 15: (a) Training dynamics of the target models used in Figure 6a and Figure 8 Right, but with low-rank embedding  $A \cdot B$ , both  $A$  and  $B$  are randomly initialized. MLP represents a three-layer MLP and TF represents a two-layer transformer. (b) Merge  $A$  and  $B$  into  $E_{T}$  at 100-th epoch and keep training. (c) Set the embedding dimension of the weak model to be the same as that of the target model.
+
+![](images/f75a3f4a9dddb26912fb62d139d9687b043f83bde842ceceb6e389fceb76410c.jpg)
+
+![](images/520f1dd16f1d5abe0d5fa543e4159fdcab9b03a542a529d1f5f89899e93aba22.jpg)
+
+![](images/934a1c217ead8ff63509cd4fb4235e5e77342f30879230aaf7ec5ac035e57f06.jpg)  
+Figure 16: Dynamics of the target model (a two-layer Transformer) trained via GrokTransfer, and the target model trained via GrokFast on modular multiplication task.
\ No newline at end of file
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+# LET YOUR FEATURES TELL THE DIFFERENCES: UNDERSTANDING GRAPH CONVOLUTION BY FEATURE SPLITTING
+
+Yilun Zheng $^{1}$ , Xiang Li $^{2}$ , Sitao Luan $^{3}$ , Xiaojiang Peng $^{2}$ , Lihui Chen $^{1}$
+
+<sup>1</sup>Nanyang Technological University, Centre for Info. Sciences and Systems,
+
+$^{2}$ College of Big Data and Internet, Shenzhen University,
+
+$^{3}$ Mila - Quebec Artificial Intelligence Institute.
+
+# ABSTRACT
+
+Graph Neural Networks (GNNs) have demonstrated strong capabilities in processing structured data. While traditional GNNs typically treat each feature dimension equally important during graph convolution, we raise an important question: Is the graph convolution operation equally beneficial for each feature? If not, the convolution operation on certain feature dimensions can possibly lead to harmful effects, even worse than convolution-free models. Therefore, it is required to distinguish convolution-favored and convolution-disfavored features. Traditional feature selection methods mainly focus on identifying informative features or reducing redundancy, but they are not suitable for structured data as they overlook graph structures. In graph community, some studies have investigated the performance of GNN with respect to node features using feature homophily metrics, which assess feature consistency across graph topology. Unfortunately, these metrics do not effectively align with GNN performance and cannot be reliably used for feature selection in GNNs. To address these limitations, we introduce a novel metric, Topological Feature Informativeness (TFI), to distinguish GNN-favored and GNN-disfavored features, where its effectiveness is validated through both theoretical analysis and empirical observations. Based on TFI, we propose a simple yet effective Graph Feature Selection (GFS) method, which processes GNN-favored and GNN-disfavored features with GNNs and non-GNN models separately. Compared to original GNNs, GFS significantly improves the extraction of useful topological information from each feature with comparable computational costs. Extensive experiments show that after applying GFS to 8 baseline and state-of-the-art (SOTA) GNN architectures across 10 datasets,  $90\%$  of the GFS-augmented cases show significant performance boosts. Furthermore, our proposed TFI metric outperforms other feature selection methods for GFS. These results verify the effectiveness of both GFS and TFI. Additionally, we demonstrate that GFS's improvements are robust to hyperparameter tuning, highlighting its potential as a universally valid method for enhancing various GNN architectures. To facilitate reproducibility and further research, we have made our code publicly available at https://github.com/KTTRCDL/graph-feature-selection.
+
+# 1 INTRODUCTION
+
+Graph Neural Networks (GNNs) are widely used for processing graph-structured data, such as recommendation systems (Ong et al., 2023; Ong & Khong, 2024), social networks (Li et al., 2023a; Awasthi et al., 2023; Luan et al., 2019), telecommunication (Lu et al., 2024a) and bio-informatics (Zhang et al., 2021; Kang et al., 2022; Hua et al., 2024). Although graph convolution has been shown effective to enrich node features with topological information through message propagation, the performance gain is found to be restricted by the assumption of homophily, i.e., similar nodes are more likely to be connected in a graph (McPherson et al., 2001). On the other hand, when a graph exhibits low homophily, i.e., heterophily, the graph convolution operation can lead to performance
+
+![](images/7455baa3a7cd053f09e3ed7224df038a72d7da6252930f13ba7eea5263dff9b3.jpg)  
+Figure 1: Improvements in GNN performance at node level and feature level. Different colors denote node labels, while the direction and magnitude of arrows denote node features.
+
+![](images/2351187943dad96eefe156a6368ce653bf6abfc84ed2c413a3741c3d6c4ecdb8.jpg)
+
+![](images/786108d01937cab44d8fe143db9fa7df3a94c0358bc60c6085f7a20b0253d580.jpg)
+
+degradation and sometimes even underperform convolution-free models, such as Multi-Layer Perceptrons (MLPs) (Zhu et al., 2020a; Luan et al., 2022b; 2024c). Therefore, to measure the impact of graph convolution operation, label-based homophily metrics (Pei et al., 2020a; Zhu et al., 2020a) are proposed to measure the label consistency along graph topology. However, they neglect the effects of node features, which is crucial for graph learning. Feature homophily metrics (Yang et al., 2021; Jin et al., 2022) are then proposed to measure the feature consistency along the graph.
+
+Although these existing metrics can capture the feature similarity between connected nodes, they overlook that different feature dimensions may exhibit different levels of compatibility with graph structures, and thus may gain different amounts of benefits or negative impacts from graph convolution. For example, as illustrated in Figure 1 (right), the GNN-favored feature exhibits uniform values among intra-class nodes while have different values across inter-class nodes. This characteristic enables graph convolution operation to effectively increase the distinguishability among nodes from different classes, as demonstrated in (Luan et al., 2024b). Conversely, graph convolution on GNN-disfavored features may hinder the learning process. This example raises a crucial question: How can we determine whether graph convolution is beneficial or not for a specific feature dimension?
+
+To address this issue, in this paper, we propose Topological Feature Informativeness (TFI), which measures the mutual information between each dimension of aggregated node features and labels. Compared with previous feature selection metrics that mainly focus on selecting the most informative features or reducing redundancy for i.i.d. data (Li et al., 2017), our proposed TFI emphasizes differentiating between GNN-favored and GNN-disfavored features, which is essential for effective graph representation learning. To achieve this, we show that TFI can provably provide an upper bound on the performance gap between graph convolution and convolution-free models, which is supported by both theoretical analysis and empirical observations. Additionally, TFI overcomes the "good heterophily" issue (Ma et al., 2021; Luan et al., 2024c), which is a serious misalignment of existing feature homophily metrics and GNN performance.
+
+Motivated by the principle of "feed the right features to the right model" Luan et al. (2022a), we propose a simple yet effective method called Graph Feature Selection (GFS). GFS first uses TFI to identify GNN-favored and GNN-disfavored features. Then, to enhance the extraction of useful information, the GNN-favored features are processed by GNNs, while MLPs handle the GNN-disfavored features. Last, a final linear layer fuses the embeddings from both models to obtain the final node representation. GFS can be seamlessly integrated into almost any GNN architecture, improving overall model performance with comparable computational costs. Our experiments on real-world benchmark datasets demonstrate that GFS significantly boosts the performance of 8 GNNs across 10 datasets in node classification tasks and the improvement is robust to hyperparameter tuning. Moreover, we demonstrate that TFI outperforms other statistical and optimization-based metrics for feature selection in GNNs, validating its superiority. Besides, we surprisingly find that GFS is much more effective on node embeddings encoded by Pretrained Language Models (PLMs) than other methods. This implies that the advantages of PLMs in understanding graph-structured data might be rooted in their ability to split the topology-aware and topology-agnostic information into separate feature dimensions. In summary, our main contributions are as follows.
+
+- We introduce a novel metric, Topological Feature Informativeness (TFI), to distinguish GNN-favored and GNN-disfavored features. We validate its effectiveness through both empirical observations and theoretical analysis.
+
+- We propose Graph Feature Selection (GFS) based on TFI, a simple yet powerful method that significantly boosts GNN performance. To the best of our knowledge, we are the first to address feature selection problem based on the splitting of GNN-favored and GNN-disfavored features.  
+- We conduct extensive experiments by applying GFS to 8 baseline and state-of-the-art (SOTA) GNN architectures across 10 benchmark datasets. The results show a significant performance boost in  $90\%$  (72 out of 80) of the cases.
+
+# 2 PRELIMINARY
+
+We define a graph as  $\mathcal{G} = \{\mathcal{V},\mathcal{E}\}$ , where  $\mathcal{V}$  is the node set and  $\mathcal{E}$  is the edge set. For an undirected graph, its structure can be represented by an adjacency matrix  $A\in \mathbb{R}^{N\times N}$ , where  $N$  is the number of nodes,  $A_{u,v} = A_{v,u} = 1$  indicates the presence of an edge between nodes  $u$  and  $v$ , i.e.,  $e_{uv},e_{vu}\in \mathcal{E}$ , otherwise  $A_{u,v} = A_{v,u} = 0$ . The node classification task on graph aims to predict node labels  $\mathbf{Y}\in \mathbb{R}^N$  by utilizing node features  $\mathbf{X}\in \mathbb{R}^{N\times M}$  and topological information, where  $X_{u,m}$  denotes the  $m$ -th feature of node  $u$ . The node degree matrix is denoted as  $\pmb {D}\in \mathbb{R}^{N\times N}$ , where  $D_{u} = \sum_{v\in \mathcal{V}}A_{u,v}$  is the degree of node  $u$ . The adjacency matrix can be normalized as  $\hat{\pmb{A}} = D^{-\frac{1}{2}}\pmb {A}\pmb {D}^{-\frac{1}{2}}$ . The neighborhood set of node  $u$  are denoted by  $\mathcal{N}_u = \{v|e_{uv}\in \mathcal{E}\}$ .
+
+Graph Neural Networks. Graph Neural Networks (GNNs), such as GCN (Kipf & Welling, 2016), GAT (Velicković et al., 2017), and GraphSage (Hamilton et al., 2017), can effectively capture useful topological information from neighbors by message propagation mechanism. Specifically, for a node  $u$ , its embedding  $z_u^l$  in the  $l$ -th layer of a GNN can be expressed as:
+
+$$
+z _ {u} ^ {l} = \operatorname {U P D A T E} \left(z _ {u} ^ {l - 1}, \text {A G G R E G A T E} \left(z _ {u} ^ {l - 1}, \left\{z _ {v} ^ {l - 1} \mid v \in \mathcal {N} _ {u} \right\}\right)\right) \tag {1}
+$$
+
+where AGGREGATE(\cdot) represents the aggregation function, and UPDATE(\cdot) is the update function based on the ego node and the aggregated neighbor representations from the previous layer. For the first layer, the initial embedding is set as input node features.
+
+The representative graph convolution can be formulated as  $H^{l} = \sigma (\hat{A} H^{l - 1}W^{l - 1})$ , where  $H^l$  and  $W^l$  denote the node embeddings and learnable parameter matrix at the  $l$ -th layer, and  $\sigma (\cdot)$  is an activation function. To boost GNN performance, some architectures concatenate ego and neighborhood node embeddings as the input of the next layer (Abu-El-Haija et al., 2019; Zhu et al., 2020b), i.e.,  $[\hat{A} H^{l - 1},H^{l - 1}]$  or adding the aggregated node embeddings with the ego node embeddings (Luan et al., 2022a), i.e.,  $(\hat{A} +I)H^{l - 1}$ , which has been shown to improve GNN performance (Platonov et al., 2023), especially in the scenario of heterophily (Luan et al., 2024c; Lu et al., 2024b).
+
+Label Homophily. Homophily, a concept originating from social networks, is defined as the tendency of similar nodes are more likely to connect with each other (McPherson et al., 2001). In graph learning, people attempt to use homophily as a metric to measure whether graph convolution on a certain graph is beneficial for GNN or not. Higher homophily often implies that the topological structure can provide useful information, which can lead to better GNN performance. The commonly used label-based homophily metrics Luan et al. (2024a) are defined as follows:
+
+$$
+h _ {\text {e d g e}} (\mathcal {G}, \mathbf {Y}) = \frac {\left| \left\{e _ {u v} \mid e _ {u v} \in \mathcal {E} , Y _ {u} = Y _ {v} \right\} \right|}{| \mathcal {E} |}, h _ {\text {n o d e}} (\mathcal {G}, \mathbf {Y}) = \frac {1}{| \mathcal {V} |} \sum_ {v \in \mathcal {V}} \frac {\left| \left\{u \mid u \in \mathcal {N} _ {v} , Y _ {u} = Y _ {v} \right\} \right|}{\left| \mathcal {N} _ {v} \right|} \tag {2}
+$$
+
+Generally, these metrics only measure label consistency across graph topology. However, they overlook the feature aspect, which also plays a critical role in GNN performance (Zheng et al., 2024a). Therefore, some studies focus on the feature consistency across graph topology.
+
+Feature Homophily. To extend the conventional definition of label homophily to node features, feature homophily metrics are introduced and the general form is,
+
+$$
+h \left(\mathcal {G}, X _ {:, m}\right) = \frac {1}{\eta \left(X _ {: , m}\right)} \sum_ {e _ {u v} \in \mathcal {E}} \sin \left(X _ {u, m}, X _ {v, m}\right) \tag {3}
+$$
+
+where  $\eta(\cdot)$  is a normalization function and  $\mathrm{sim}(\cdot)$  is a similarity metric. The key difference between different feature homophily metrics lies in the choice of the similarity function  $\mathrm{sim}(\cdot)$ , which could be cosine similarity (Jin et al., 2022), dot-product (Yang et al., 2021), or Euclidean distance (Chen et al., 2023). Please refer to Appendix A for more detailed introduction to homophily metrics.
+
+Mutual Information. Mutual Information measures the amount of information obtained about one random variable given another variable (Reza, 1994). More specifically, given variable  $X$  and  $Y$ , the mutual information can be expressed as<sup>1</sup>:
+
+$$
+I (X; Y) = \sum_ {y \in \mathcal {Y}} \sum_ {x \in \mathcal {X}} p (x, y) \log \frac {p (x , y)}{p (x) p (y)} \tag {4}
+$$
+
+where  $p(x,y)$  is joint probability, and  $p(x)$  and  $p(y)$  are marginal probability distributions.
+
+# 3 TOPOLOGICAL FEATURE INFORMATIVENESS
+
+As illustrated in Figure 1, not all features in graphs benefit from graph convolution. This raises the question: how can we determine which feature can get benefit from graph convolution? Previous studies (Zheng et al., 2024a; Luan et al., 2024b; Wang et al., 2024a) have demonstrated that it is the stable topological patterns of intra-class nodes that enable effective message aggregation from neighbors, rather than relying solely on consistency-based homophily metrics. To address the limitations of feature homophily metrics, we propose Topological Feature Informativeness (TFI) to measure the informativeness of neighbors in graphs given the feature  $\mathbf{X}_{:,m}$ , which is:
+
+$$
+\operatorname {T F I} _ {m} = I (\boldsymbol {Y}; \tilde {\boldsymbol {X}} _ {:, m}) \tag {5}
+$$
+
+where  $\tilde{X}_{:,m} = (\hat{A})^k X_{:,m}$  is the aggregated node features from  $k$ -hop neighbors,  $\hat{A}$  is the adjacency matrix without self-loop, and we set  $k = 1$  by default.  $\mathrm{TFI}_m$  leverages mutual information to quantify the dependence between variables, including both linear and non-linear relationships.
+
+Then, we conduct a theoretical analysis to show how  $\mathrm{TFI}_m$  can effectively measure the benefit of a certain feature gained from graph convolution.
+
+Theorem 1. Given a graph  $\mathcal{G} = \{\mathcal{V},\mathcal{E}\}$  with  $C$  classes and  $\mathrm{TFI}_m$  measured on the  $m$ -th feature, the prediction accuracy  $P_A$  of a classifier on node labels  $\mathbf{Y}$  with aggregated features  $\tilde{\mathbf{X}}_{:,m}$  is upper bounded by:
+
+$$
+P _ {A} \leq \frac {\mathrm {T F I} _ {m} + \log 2}{\log C} \tag {6}
+$$
+
+The proof is given in Appendix B.
+
+From Theorem 1, we can see that  $\mathrm{TFI}_m$  can be used to estimate the effect of graph convolution on feature dimension  $m$  by relating it to model performance. More specifically, a higher  $\mathrm{TFI}_m$  means the prediction accuracy of a classifier with aggregated node feature  $m$  has a higher upper bound, which indicates that the graph convolution operation is more likely to be beneficial for feature  $m$ . To guide feature selection with  $\mathrm{TFI}_m$ , in the following theorem, we explain how it relates to the performance gap between graph-convoluted and convolution-free features.
+
+Theorem 2. The gap between  $I(\mathbf{Y}; \tilde{\mathbf{X}}_{:,m}, \mathbf{X}_{:,m})$  and  $I(\mathbf{Y}; \mathbf{X}_{:,m})$  is upper bounded by  $\mathrm{TFI}_m$ :
+
+$$
+I \left(\mathbf {Y}; \tilde {\mathbf {X}} _ {:, m}, \mathbf {X} _ {:, m}\right) - I (\mathbf {Y}; \mathbf {X} _ {:, m}) = I (\mathbf {Y}; \tilde {\mathbf {X}} _ {:, m} | \mathbf {X} _ {:, m}) \leq I (\mathbf {Y}; \tilde {\mathbf {X}} _ {:, m}) = \operatorname {T F I} _ {m} \tag {7}
+$$
+
+In Theorem 2, the mutual information  $I(\mathbf{Y};\mathbf{X}_{:,m})$  measures how well the ego node features capture the relevant information of labels, without making assumptions about the underlying data distribution. This makes  $I(\mathbf{Y};\mathbf{X}_{:,m})$  a non-parametric estimation of MLP performance. On the other hand,  $I(\mathbf{Y};\tilde{\mathbf{X}}_{:,m},\mathbf{X}_{:,m})$  serves as a non-parametric estimation of GNN performance, which utilizes both the ego node features  $\mathbf{X}_{:,m}$  and the aggregated features  $\tilde{\mathbf{X}}_{:,m}$  for prediction<sup>2</sup>. The subtraction of  $I(\mathbf{Y};\tilde{\mathbf{X}}_{:,m},\mathbf{X}_{:,m})$  and  $I(\mathbf{Y};\mathbf{X}_{:,m})$  indicates the information gap between graph-convoluted and convolution-free features w.r.t.  $\mathbf{Y}$ , which is upper bounded by  $\mathrm{TFI}_m$ . In other words, a feature  $m$  with larger  $\mathrm{TFI}_m$  means that feature  $m$  is more likely to be GNN-favored; otherwise, GNN-disfavored.
+
+To verify the effectiveness of the above claims about TFI, we conduct experiments on real-world benchmark datasets. Specifically, we first compute  $\mathrm{TFI}_m$  for all features in the graph, sort them,
+
+![](images/3e1c1afe0918c097e43719ee450d9d9b0e04b8b221a3c61cf6d5c3b8fc1cda86.jpg)  
+Figure 2: The performance gap between GCN and MLP with the increase of TFI.
+
+![](images/b183ba4d7c28bb63750224a6f63c74d28020ae4205d3cb1a5bfb0621fc358ba6.jpg)
+
+and divide them evenly into 10 bins. Next, we train GCN and MLP only with the features in a bin respectively, and report the performance difference of GCN and MLP.
+
+As shown in Figure 2, GCN underperforms MLP in bins with low TFI and outperforms MLP in bins with high TFI. This observation is consistent with Theorem 1 and 2, and indicates that TFI can effectively identify GNN-favored and GNN-disfavored features. Note that the estimation of  $\mathrm{TFI}_m$  does not require all of the labels  $Y$ . In a semi-supervised setting,  $\mathrm{TFI}_m$  can be estimated using only the training labels and features, i.e.,  $I(Y_{\mathrm{train}}; \tilde{X}_{\mathrm{train}, m})$ , without the need for pseudo labels.
+
+In the next section, we will introduce how to use  $\mathrm{TFI}_m$  for feature selection in GNNs.
+
+# 4 GRAPH FEATURE SELECTION
+
+In this section, we propose Graph Feature Selection (GFS), a TFI-based feature selection method. It is composed of three main components: (1) GNN-favored feature selection, (2) feature embedding, and (3) feature fusion.
+
+![](images/a6361ecfc7834fa934284464ddc817124f855f5d3231da20072c59d50dd23f1b.jpg)  
+Figure 3: Framework of GFS with TFI.
+
+As illustrated in Figure 3, for each feature dimension  $m$ , we first measure its corresponding  $\mathrm{TFI}_m$  using Eq. (5). Since a higher  $\mathrm{TFI}_m$  indicates stronger GNN performance compared to MLP, we apply a threshold to select the set of GNN-favored features as  $\{X_{\mathcal{G}} | X_{:,m} \in X_{\mathcal{G}}, \mathrm{TFI}_m \geq \delta(r)\}$ , where  $\delta(r)$  is the threshold for the top  $r$  percentile of all TFI values in features, and  $r \in (0,1)$ . The GNN-disfavored features are the remaining  $1 - r$  of features, defined as  $X_{\neg \mathcal{G}} = X \backslash \{X_{\mathcal{G}}\}$ .
+
+The GNN-favored features  $X_{\mathcal{G}}$  are expected to be more suitable for GNNs, whereas the GNN-disfavored features  $X_{\neg \mathcal{G}}$  are better suited for MLPs. Thus, we feed  $X_{\mathcal{G}}$  into a GCN (or any other GNN architecture) and  $X_{\neg \mathcal{G}}$  into an MLP to better leverage information from both neighbors and ego nodes. Specifically, in the  $l$ -th layer of GCN or MLP, we have:
+
+$$
+\operatorname {G C N}: \boldsymbol {H} _ {\mathcal {G}} ^ {l} = \sigma \left(\hat {\boldsymbol {A}} \boldsymbol {H} _ {\mathcal {G}} ^ {l - 1} \boldsymbol {W} _ {1} ^ {l - 1}\right), \text {M L P}: \boldsymbol {H} _ {\neg \mathcal {G}} ^ {l} = \sigma \left(\boldsymbol {H} _ {\neg \mathcal {G}} ^ {l - 1} \boldsymbol {W} _ {2} ^ {l - 1}\right) \tag {8}
+$$
+
+where  $H_{\mathcal{G}}^{0} = X_{\mathcal{G}}^{0}$  and  $H_{\neg \mathcal{G}}^{0} = X_{\neg \mathcal{G}}^{0}$ . During message passing, the representations of  $H_{\mathcal{G}}^{l}$  and  $H_{\neg \mathcal{G}}^{l}$  are derived independently, ensuring that  $X_{\mathcal{G}}^{l}$  and  $X_{\neg \mathcal{G}}^{l}$  can be encoded by the most appropriate model without interference.
+
+Lastly, to combine the node information from  $X_{\mathcal{G}}$  and  $X_{\neg \mathcal{G}}$  together, we concatenate the node representation from the last layer  $L$  of GNN and MLP, and feed them into a linear layer:
+
+$$
+\boldsymbol {H} = \left[ \boldsymbol {H} _ {\mathcal {G}} ^ {L}, \boldsymbol {H} _ {\neg \mathcal {G}} ^ {L} \right] \boldsymbol {W} + \boldsymbol {b} \tag {9}
+$$
+
+where  $\pmb{H}$  is the final node representation,  $\pmb{W}$  is the weight matrix, and  $\pmb{b}$  is the bias, respectively. This final node representation is then used for downstream graph tasks.
+
+Complexity Analysis. To compare the time complexity of GNN+GFS against standard GNNs, we use GCN as the backbone GNN and assume the embedding size  $F$  is consistent across all layers. First, estimating and sorting TFI introduces a complexity of  $O(KC + N(1 + \log N))$ , where  $K$  is a parameter related to mutual information estimation,  $C$  is the number of classes, and  $N$  is the number of nodes. Both  $K$  and  $C$  are much smaller than  $N$  or the number of edges  $|\mathcal{E}|$ , making this overhead negligible compared to GNN training complexity as below.
+
+After feature selection, we obtain GNN-favored features  $X_{\mathcal{G}} \in \mathbb{R}^{N \times \lceil rM \rceil}$  and GNN-disfavored features  $X_{\neg \mathcal{G}} \in \mathbb{R}^{N \times \lceil (1 - r)M \rceil}$ , where  $\lceil \cdot \rceil$  is the floor function. For the first layer of GCN+GFS, the time complexity is  $O(|\mathcal{E}| \lceil rM \rceil + N \lceil rM \rceil F)$  for the GNN channel and  $O(N \lceil (1 - r)M \rceil F)$  for the MLP channel. Their summation,  $O(|\mathcal{E}| \lceil rM \rceil + NM^2)$ , is smaller than the complexity of a standard GCN,  $O(|\mathcal{E}| M + NM^2)$ , for  $r < 1$ . This is because GFS reduces the input feature dimension for GCN.
+
+In subsequent layers, GCN+GFS has a complexity of  $O(|\mathcal{E}|F + 2NF^2)$ , slightly higher than the complexity of GCN  $O(|\mathcal{E}|F + NF^2)$  by an additional  $NF^2$  term due to the MLP layers. However, this additional cost  $O(NF^2)$  is smaller than the  $O(|\mathcal{E}|F)$  introduced by graph convolutions. Thus, the overall complexity of GCN+GFS is comparable to that of the original GCN.
+
+# 5 EXPERIMENTS
+
+To verify the effectiveness of our proposed Graph Feature Selection (GFS) and Topological Feature Informativeness (TFI), we try to answer the following questions with experiments in this section.
+
+RQ1: Can GFS really enhance the performance of GNNs? Does the improvement always hold?
+
+RQ2: How much does hyperparameter tuning influence the performance gains from GFS?
+
+RQ3: How does TFI compare with other statistic-based and optimization-based metrics for GFS?
+
+RQ4: Do the GNN-favored and GNN-disfavored features identified by TFI really fit better with GNNs or MLPs, respectively?
+
+# 5.1 EXPERIMENTAL SETUPS
+
+Data Preparation. The datasets used in our experiments include Children, Computers, Fitness, History, and Photo from Yan et al. (2023), and Amazon-Ratings, Minesweeper, Questions, Roman-Empire, and Tolokers from Platonov et al. (2023). These datasets exhibit varying levels of label homophily across different domains. The dataset statistics and descriptions are shown in Appendix C.1. For all datasets, we randomly split the data into training, validation, and test sets in a ratio of  $50\% : 25\% : 25\%$  for 10 runs. Note that the node features in datasets from Yan et al. (2023) are encoded by Pretrained Language Models (PLMs).
+
+Baselines. We implement GFS and compare its performance with 5 baseline GNNs, GCN (Kipf & Welling, 2016), GAT (Velicković et al., 2017), GraphSAGE (SAGE) (Hamilton et al., 2017), SGC (Wu et al., 2019) and APPNP (Gasteiger et al., 2018), and 3 SOTA GNNs, Graph Transformer (GT) (Shi et al., 2020), ACM-GCN (Luan et al., 2022a) and FAGCN (Bo et al., 2021). The GFS-augmented GNN is denoted as  $\mathrm{GNN + GFS}$  throughout this paper. To enhance performance, we incorporate skip connections (He et al., 2016) and layer normalization (Ba et al., 2016) for all the methods, as recommended in Platonov et al. (2023); Luo et al. (2024). The detailed descriptions of these GNNs are introduced in Appendix C.2. Additionally, we compare our proposed TFI with other statistic-based metrics, including  $h_{GE}$  (Jin et al., 2022),  $h_{attr}$  (Yang et al., 2021),  $h_{sim-cos}$  (Chen et al., 2023),  $h_{sim-euc}$  (Chen et al., 2023), and  $h_{CTF}$  (Lee et al., 2024). We also evaluate optimization-based metrics such as  $\theta_{Soft}$  and  $\theta_{Hard}$  within GFS. Detailed definitions of these statistic- and optimization-based metrics can be found in Appendix A and C.4, respectively.
+
+Training. During training, we utilize the Adam optimizer (Kingma & Ba, 2014) to update model parameters. Each model is trained for 1000 epochs. Model performance is evaluated on node
+
+classification tasks, measuring accuracy for multi-class datasets and AUC-ROC for binary-class datasets. The searching space for the ratio  $r$  is  $\{0.1, 0.2, \dots, 0.9\}$  in  $\mathrm{GFS}^3$  and  $k = 1$  for the number of the hop of neighbors in  $\mathrm{TFI}^4$ . See Appendix C.3 for more training details.
+
+# 5.2 PERFORMANCE (RQ1)
+
+The results and comparisons of the model performance are shown in Table 1, where the performance gap between GNN+GFS and GNN is denoted as  $\Delta$ . We observe that (1) GFS significantly boosts GNN performance on  $90\% (72/80)$  cases of the GFS-augmented GNNs, with an average increase of  $3.2\%$  in accuracy or AUC-ROC on node classification tasks with various homophily levels. This demonstrate the effectiveness of our proposed method. (2) We notice that the improvement of GFS varies by dataset, e.g., in Children and Computers, GCN+GFS achieves a  $5\%$  average performance improvement, while in Minesweeper and Questions, the improvement is less significant. This may be because most features in Minesweeper and Questions are already GNN-favored. Nevertheless, GFS remains effective on most datasets. (3) We find that GFS demonstrates greater improvements on datasets encoded by Pretrained Language Models (PLMs) (40/40 cases) than those encoded by some traditional methods (32/40 cases), e.g., one-hot, fasttext or statistics-based methods. This suggests that, compared to traditional feature encoding methods, which is widely used in graph learning, PLMs potentially enable the splitting of topology-aware and topology-agnostic information into separate feature dimensions, which helps GFS work easier. This is a surprising discovery and to our knowledge, we are the first to reveal this phenomenon. It can help explain why PLMs can assist the learning of graph models (Ye et al., 2024). In section 5.6, we will discuss more about the TFI distribution under PLM vs. traditional feature encoding methods.
+
+Table 1: The performance of GFS on GNN baselines. The improvement is highlighted as **bold** if there is an increase, i.e.,  $\Delta > 0$ , after applying GFS.  
+
+<table><tr><td>Model</td><td>Children</td><td>Comp.</td><td>Fitness</td><td>History</td><td>Photo</td><td>Amazon.</td><td>Mines.</td><td>Questions</td><td>Roman.</td><td>Tolokers</td></tr><tr><td>GCN</td><td>53.88±0.37</td><td>83.38±1.78</td><td>87.56±0.79</td><td>84.26±0.38</td><td>84.48±0.65</td><td>51.40±0.49</td><td>90.01±0.53</td><td>76.28±1.18</td><td>75.51±0.65</td><td>84.19±0.75</td></tr><tr><td>GCN+GFS</td><td>59.29±0.31</td><td>90.23±0.18</td><td>93.09±0.13</td><td>84.99±0.38</td><td>87.31±0.36</td><td>53.32±0.81</td><td>89.99±0.60</td><td>76.50±1.38</td><td>83.30±0.57</td><td>85.01±0.91</td></tr><tr><td>Δ</td><td>+5.41</td><td>+6.85</td><td>+5.53</td><td>+0.73</td><td>+2.83</td><td>+1.92</td><td>-0.02</td><td>+0.22</td><td>+7.79</td><td>+0.82</td></tr><tr><td>GAT</td><td>53.54±0.49</td><td>86.52±0.92</td><td>88.23±0.91</td><td>83.83±0.27</td><td>85.67±0.45</td><td>51.08±0.60</td><td>90.26±0.53</td><td>77.64±1.15</td><td>84.57±0.80</td><td>83.41±0.47</td></tr><tr><td>GAT+GFS</td><td>57.74±0.35</td><td>90.50±0.20</td><td>93.20±0.11</td><td>84.54±0.38</td><td>87.58±0.28</td><td>53.75±0.57</td><td>90.22±0.64</td><td>77.03±1.11</td><td>86.17±0.56</td><td>84.41±0.77</td></tr><tr><td>Δ</td><td>+4.20</td><td>+3.98</td><td>+4.97</td><td>+0.71</td><td>+1.91</td><td>+2.67</td><td>-0.04</td><td>-0.61</td><td>+1.60</td><td>+1.00</td></tr><tr><td>SAGE</td><td>54.68±0.84</td><td>86.08±0.50</td><td>88.65±1.22</td><td>84.06±0.42</td><td>85.08±0.64</td><td>53.80±0.56</td><td>90.74±0.59</td><td>74.91±1.06</td><td>82.81±0.61</td><td>82.77±0.38</td></tr><tr><td>SAGE+GFS</td><td>59.14±0.33</td><td>90.47±0.24</td><td>93.63±0.10</td><td>84.68±0.33</td><td>87.23±0.51</td><td>54.17±0.61</td><td>90.71±0.62</td><td>75.37±1.33</td><td>85.47±0.53</td><td>83.16±0.73</td></tr><tr><td>Δ</td><td>+4.46</td><td>+4.39</td><td>+4.98</td><td>+0.62</td><td>+2.15</td><td>+0.37</td><td>-0.03</td><td>+0.46</td><td>+2.66</td><td>+0.39</td></tr><tr><td>GT</td><td>51.20±0.38</td><td>85.63±1.16</td><td>87.37±1.38</td><td>83.61±0.43</td><td>83.65±0.59</td><td>51.30±0.73</td><td>90.11±0.57</td><td>77.57±1.09</td><td>84.95±0.54</td><td>83.20±0.60</td></tr><tr><td>GT+GFS</td><td>56.01±0.35</td><td>89.97±0.27</td><td>92.44±0.11</td><td>84.09±0.32</td><td>87.41±0.41</td><td>52.47±0.54</td><td>89.93±0.64</td><td>77.70±0.73</td><td>86.99±0.45</td><td>83.46±0.67</td></tr><tr><td>Δ</td><td>+4.81</td><td>+4.34</td><td>+5.07</td><td>+0.48</td><td>+3.76</td><td>+1.17</td><td>-0.18</td><td>+0.13</td><td>+2.04</td><td>+0.26</td></tr><tr><td>SGC</td><td>52.69±0.52</td><td>82.50±0.34</td><td>84.23±0.31</td><td>83.98±0.37</td><td>82.79±0.45</td><td>49.00±0.42</td><td>76.43±1.01</td><td>71.78±0.81</td><td>69.89±0.51</td><td>78.65±1.01</td></tr><tr><td>SGC+GFS</td><td>55.93±0.37</td><td>87.78±0.40</td><td>90.29±0.25</td><td>84.15±0.23</td><td>85.46±0.33</td><td>51.43±0.64</td><td>88.79±0.66</td><td>73.53±1.10</td><td>74.00±0.67</td><td>82.76±0.84</td></tr><tr><td>Δ</td><td>+3.24</td><td>+5.28</td><td>+6.06</td><td>+0.17</td><td>+2.67</td><td>+2.43</td><td>+12.36</td><td>+1.75</td><td>+4.11</td><td>+4.11</td></tr><tr><td>APPNP</td><td>50.63±0.89</td><td>83.67±0.90</td><td>86.76±1.18</td><td>83.37±0.26</td><td>82.20±1.41</td><td>48.73±0.61</td><td>81.61±0.79</td><td>75.29±1.06</td><td>71.48±0.65</td><td>79.82±1.17</td></tr><tr><td>APPNP+GFS</td><td>56.71±0.36</td><td>88.51±0.34</td><td>91.22±0.25</td><td>84.75±0.33</td><td>86.65±0.32</td><td>50.76±0.62</td><td>83.19±0.97</td><td>75.66±1.03</td><td>72.35±0.69</td><td>83.66±0.65</td></tr><tr><td>Δ</td><td>+6.08</td><td>+4.84</td><td>+4.46</td><td>+1.38</td><td>+4.45</td><td>+2.03</td><td>+1.58</td><td>+0.37</td><td>+0.87</td><td>+3.84</td></tr><tr><td>ACMGCN</td><td>54.60±0.50</td><td>85.94±0.72</td><td>89.10±0.98</td><td>84.22±0.34</td><td>84.99±0.34</td><td>51.91±0.39</td><td>90.59±0.58</td><td>76.66±1.27</td><td>85.27±0.57</td><td>83.61±0.83</td></tr><tr><td>ACMGCN+GFS</td><td>59.04±0.37</td><td>89.59±0.19</td><td>93.44±0.07</td><td>84.70±0.30</td><td>86.61±0.42</td><td>52.81±0.75</td><td>90.45±0.59</td><td>75.99±1.27</td><td>87.03±0.57</td><td>83.45±1.03</td></tr><tr><td>Δ</td><td>+4.44</td><td>+3.65</td><td>+4.34</td><td>+0.48</td><td>+1.62</td><td>+0.90</td><td>-0.14</td><td>-0.67</td><td>+1.76</td><td>-0.16</td></tr><tr><td>FAGCN</td><td>50.43±0.86</td><td>79.92±0.99</td><td>83.10±0.45</td><td>82.04±0.62</td><td>80.67±0.77</td><td>46.08±0.52</td><td>78.22±2.86</td><td>58.60±2.08</td><td>62.02±2.98</td><td>73.07±0.70</td></tr><tr><td>FAGCN+GFS</td><td>56.17±0.38</td><td>87.72±0.49</td><td>89.66±0.35</td><td>84.32±0.36</td><td>85.54±0.37</td><td>50.72±0.82</td><td>88.12±1.32</td><td>71.94±2.03</td><td>72.05±1.45</td><td>82.82±1.28</td></tr><tr><td>Δ</td><td>+5.74</td><td>+7.80</td><td>+6.56</td><td>+2.28</td><td>+4.87</td><td>+4.64</td><td>+9.90</td><td>+13.34</td><td>+10.03</td><td>+9.75</td></tr></table>
+
+# 5.3 SENSITIVITY TO HYPERPARAMETERS (RQ2)
+
+Some recent GNNs (Li et al., 2022; Liu et al., 2022) introduce more and more hyperparameters, which could be fragile to hyperparameter tuning (Luan et al., 2024c). To investigate the sensitivity of the superiority of GFS over GNNs to ratio  $r$  and other model hyperparameters, we compare the performance between GCN and GCN+GFS under varying settings. First, we test how model performance responds to the changes in ratio  $r$  while fixing all other hyperparameters, and Figure 4 shows the results on 4 datasets. Note that GFS collapses to normal GCN when  $r = 1.0$  and MLP when  $r = 0.0$ , as all features are sent to GNN or MLP, respectively. On Children, Computer, and Roman-Empire, GCN+GFS with  $r$  from 0.1 to 0.9 consistently outperforms GCN ( $r = 1.0$ ) and MLP ( $r = 0.0$ ), indicating that the performance gain obtained from GFS is robust to hyperparameter
+
+$r$ . We also observe that, although  $\mathrm{GCN + GFS}$  does not significantly outperform GCN on Tolokers, sending  $40\%$  ( $r = 0.4$ ) to  $90\%$  ( $r = 0.9$ ) features to GCN is at least not worse than sending all the features into GCN ( $r = 1.0$ ). This again demonstrates the robustness of  $\mathrm{GCN + GFS}$  on  $r$ . The above results highlight that we do not always need to convolute all the features, as neither  $r = 1.0$  nor  $r = 0.0$  yields optimal results on most datasets.
+
+![](images/9616cab3806aee5ba9477d47a6718497e3784bc9532b434fbabf9ed4439c4c49.jpg)  
+Figure 4: The impact of ratio  $r$  on the performance of GCN+GFS, where  $r$  features are sent to MLP and  $1 - r$  features are sent to GCN. The star  $\star$  represents the best performance.
+
+![](images/6101ed03d47f49636f0b94fd28e24656708c06c98410a4e1e18e703755ae74e0.jpg)
+
+![](images/bc780c4cd30e1c7b6969d5666102bd20ab310239572d87798de04b5988b1279e.jpg)
+
+![](images/48d573fa68b7e4d8ae212a59de316b98da93946e4830560057c7cd9b0e06b4de.jpg)
+
+Secondly, we analyze how other hyperparameters affect GFS performance. Figure 5 shows how GCN, MLP, and  $\mathrm{GCN + GFS}$  respond to the changes in the number of layers, dimension of hidden embeddings, learning rate, weight decay, and dropout rate on Computers (homophilic graph) and Amazon-Ratings (heterophilic graph). Results indicate that GCN+GFS consistently outperforms GCN and MLP across various hyperparameter values. This property allows GFS to be easily integrated into most baseline GNNs without much more hyperparameter tuning. Please refer to Appendix D.2 for more results on the sensitivity of GFS to hyperparameters on the other datasets.
+
+![](images/25851249cec8c42e2ea8494e29279b06c6e3da270674aef855f68f46db7ab1a6.jpg)
+
+![](images/caf4e60b3eb3bf24431c3da15c1a8ab336ff778ef34fdf5839c869277e08249b.jpg)
+
+![](images/814111090f234dcb2652effa0bbab68afb2682501048b650cd62f4849354d84e.jpg)
+
+![](images/9d30e3e49f0405ace2b7463cf64a64db5d3a3c64bfc14755e83276523df435ae.jpg)
+
+![](images/201931eb42ddb4826c2c2176bd83779fa578b79ef9c07871d2676d2486d74d74.jpg)
+
+![](images/4b2c54ad842469dbce8c65ab4602d5f2fac2e5b5ac1ca0c27d6c9689a91bf0b3.jpg)  
+Figure 5: Comparison of the response of GCN+GFS, GCN, and MLP to 5 different hyperparameters on Computers and Amazon-Ratings.
+
+![](images/eb4d7100061fd5810a73245a822cf2e96274cc085f897146c832921367d6b461.jpg)
+
+![](images/91a8c5f627f50f2169dbce4ba976ee5e36de852875b039f89db34369ff22ef98.jpg)
+
+![](images/1d2bdaccda78caa7dde733c237426df1c7a92cb29fc16a58de0e7e9d7ff39a98.jpg)
+
+![](images/83be407b51e7ff39538831cb0f145a864e792f97d2042abc379c4ef33895a8e0.jpg)
+
+# 5.4 EFFECTIVENESS OF TFI (RQ3)
+
+We show the superiority of TFI over other metric-based feature selection methods on GCN+GFS, including statistic-based metrics  $h_{GE}$  Jin et al. (2022),  $h_{attr}$  Yang et al. (2021),  $h_{sim-cos}$ ,  $h_{sim-euc}$  Chen et al. (2023), and  $h_{CTF}$  Lee et al. (2024) and optimization-based methods e.g.,  $\theta_{Soft}$  and  $\theta_{Hard}$ , where feature selection process is learned in an end-to-end manner. All methods follow the same hyperparameter tuning process as in GFS. Table 2 shows the performance of different metrics in feature selection for GCN, where TFI achieves the best performance according to average rank across 10 datasets. This is reasonable because other statistic-based metrics suffer from explaining "good heterophily" issue and TFI overcomes the issue by the measurement of mutual information Platonov et al. (2024). Besides, the optimization-based metrics underperform TFI, indicating the feature selection cannot be effectively handled automatically during the optimization process of GNNs. In addition, we find that even for the metrics that are inferior to TFI, e.g.,  $h_{attr}$ ,  $h_{sim-euc}$ , and  $h_{CTF}$ , applying them within our GFS framework can still improve the performance of baseline GCN, indicating the necessity of feature selection in GNNs.
+
+Table 2: The comparison of feature selection with feature homophily metrics and optimization-based methods.  
+
+<table><tr><td>Metrics</td><td>Children</td><td>Comp.</td><td>Fitness</td><td>History</td><td>Photo</td><td>Amazon.</td><td>Mines.</td><td>Questions</td><td>Roman.</td><td>Tolokers</td><td>Avg. Rank</td></tr><tr><td>TFI</td><td>59.29±0.31</td><td>90.23±0.18</td><td>93.09±0.13</td><td>84.99±0.38</td><td>87.11±0.42</td><td>53.32±0.81</td><td>89.99±0.60</td><td>76.50±1.38</td><td>83.30±0.57</td><td>85.01±0.91</td><td>1.40</td></tr><tr><td>hGE</td><td>55.00±0.25</td><td>83.72±1.32</td><td>87.06±0.58</td><td>83.88±0.26</td><td>83.36±0.94</td><td>51.71±0.53</td><td>89.75±0.68</td><td>75.73±1.53</td><td>79.11±0.48</td><td>83.31±0.77</td><td>6.70</td></tr><tr><td>hattr</td><td>55.15±0.32</td><td>84.62±0.61</td><td>87.88±0.53</td><td>84.22±0.25</td><td>83.22±0.52</td><td>51.46±0.70</td><td>90.18±0.70</td><td>76.40±1.39</td><td>78.90±0.58</td><td>84.04±0.82</td><td>4.70</td></tr><tr><td>hsim-cos</td><td>55.15±0.51</td><td>84.11±0.84</td><td>87.09±0.98</td><td>84.07±0.31</td><td>83.38±1.01</td><td>51.68±0.39</td><td>89.87±0.53</td><td>75.88±1.07</td><td>78.98±0.66</td><td>83.69±0.82</td><td>5.95</td></tr><tr><td>hsim-euc</td><td>56.32±0.26</td><td>89.08±0.20</td><td>87.52±1.04</td><td>85.00±0.38</td><td>86.71±0.30</td><td>51.35±0.49</td><td>90.19±0.69</td><td>76.51±1.17</td><td>78.96±0.73</td><td>84.03±0.83</td><td>3.35</td></tr><tr><td>hCTF</td><td>55.04±0.51</td><td>84.80±0.87</td><td>88.69±2.25</td><td>84.32±0.30</td><td>83.46±0.85</td><td>51.76±0.57</td><td>89.97±0.68</td><td>76.40±1.37</td><td>78.70±0.38</td><td>83.38±0.89</td><td>4.55</td></tr><tr><td>Soft</td><td>57.04±1.37</td><td>85.35±3.53</td><td>89.54±1.32</td><td>84.47±0.92</td><td>84.83±3.28</td><td>51.70±0.62</td><td>89.93±0.60</td><td>75.94±1.28</td><td>79.20±0.46</td><td>84.03±0.84</td><td>3.50</td></tr><tr><td>Hard</td><td>45.28±0.79</td><td>66.92±4.70</td><td>69.48±4.14</td><td>80.98±0.70</td><td>72.09±3.72</td><td>42.77±0.57</td><td>86.62±2.25</td><td>75.56±1.29</td><td>76.96±0.61</td><td>79.13±0.94</td><td>8.90</td></tr><tr><td>None</td><td>53.36±1.08</td><td>83.38±1.78</td><td>87.56±0.79</td><td>84.13±0.29</td><td>84.00±0.72</td><td>51.40±0.49</td><td>89.93±0.52</td><td>76.24±1.48</td><td>75.51±0.65</td><td>84.19±0.75</td><td>5.95</td></tr></table>
+
+Since TFI is not an unsupervised measurement, we investigate how the percentage of supervision in TFI (percentage of node labels used to calculate TFI in Eq. (5)) influences the model performance of GCN+GFS. As shown in Figure 6, as the supervision percentage increases, the model performance first increases and then stabilizes after  $30\%$ . This indicates that  $30\%$  of the label supervision achieves similar results as full labels in most datasets and it highlights the effectiveness of TFI in sparse label scenarios. Compared to some other methods (Zheng et al., 2024b; Li et al., 2023b) that require pseudo labels during training, TFI requires less preprocessing in semi-supervised node classification settings. Even with only  $10\%$  supervision, TFI can still enhance GCN+GFS performance on most datasets compared to the original GCN. Please refer to Appendix D.3 for more results on the influence of the supervision percentage in TFI on GFS performance.
+
+![](images/cf199ebb65f107911c9920457af77f7caf14bdb2c7793d9ddcf1b096e30d1b2d.jpg)  
+Figure 6: Influence of the supervision percentage in TFI on the model performance of GCN+GFS.
+
+![](images/b236345c189d60c96242c8b5b2c5c7bdf2ca1664dd54819aa5d858ea3fe7cb9a.jpg)
+
+![](images/ee9787136529093ab95f5188225a62f7dba170e78b3cca3756ea99b7c33bdddd.jpg)
+
+![](images/9795c6921b2a86fd97b91e78cd4eaac796e3740c29e820ffdedd213cf5091862.jpg)
+
+Based on TFI, we extend the feature selection from the raw node features to latent node embeddings, which may also be splitted into GNN-favored and GNN-disfavored. Specifically, we train a GCN or MLP on the training set and then get the node embeddings from the last layer. After that, we train  $\mathrm{GCN + GFS}$  on these node embeddings following the same process as on the original node features. As shown in Figure 7, applying  $\mathrm{GCN + GFS}$  on either the original node features,  $X$ , or pretrained node embeddings,  $\mathrm{MLP}(X)$  and  $\mathrm{GCN}(X,A)$ , improves GCN performance on most datasets. Furthermore, for some datasets, such as Computers, Questions, and Romain-Empire, the performance of  $\mathrm{GCN + GFS}$  could be further improved using pretrained node embeddings compared with original node features. Additionally, it would be interesting to explore the impact of graph feature selection on dynamically updated node embeddings in future research.
+
+![](images/363205f2f01b8546a28351d4373d5f40de981f8671c6aa9bb53bf7ae7dd8f22a.jpg)  
+Figure 7: GCN+GFS on node features  $\mathbf{X}$  and pretrained node embeddings  $\operatorname{GCN}(\mathbf{X},\mathbf{A})$  and  $\operatorname{MLP}(\mathbf{X})$ .
+
+![](images/b80eea4829e6a84e8f25e045815eafcb23402048b2b00b375eda383676813717.jpg)  
+Figure 8: Performance drop of GCN+GFS by swapping GNN-favored and GNN-disfavored features.
+
+# 5.5 GNN-FAVORED AND GNN-DISFAVORED FEATURES (RQ4)
+
+To validate whether features selected by a high TFI are truly GNN-favored, we swapped the feature selection process by feeding GNN-favored features into MLP and GNN-disfavored features into
+
+GNN. As shown in Figure 8, the performance of GCN+GFS drops significantly across all datasets after swapping. On some datasets, such as Minesweeper, there is a  $40\%$  drop in performance. These results show that TFI can reliably identify GNN-favored and GNN-disfavored features.
+
+![](images/99068effa73084854b34761b63f8f5b88c4e96653e5582bd87403346bdcba70b.jpg)
+
+![](images/9d8d7192de9324f9260c1f7964c349f4b0c79ee7ae03bafd354728cfda2d1b2a.jpg)
+
+![](images/12c8863c015fa5543be047e45094b5bd0657573892dabf56fe669d0c2d4419cb.jpg)
+
+![](images/fcec46d324a961bf0788af9d65c7e6c6216f7801cea0db0102fb8c73af2f3e2a.jpg)
+
+![](images/a38cecb1320b1bfaa1abb420952f7cda1bc83fa239aefc4de40868f42a139b31.jpg)
+
+![](images/3906fef594437ed22f2568c50bee1ac7e43b5710ea259989f6b353377a319483.jpg)  
+Figure 9: Comparison of histogram of TFI on 10 datasets.
+
+![](images/5a47087218f4b722374de986d66b75282b97310d9e03a1a867075eec36afe671.jpg)
+
+![](images/9f58bdb398561cdcdc9256d3218e7ea48e0e3c8b1d3faefa4bf7704f1bd912b9.jpg)
+
+![](images/49b88cec5c02ca498ab4203ae87cdc39ac24eb00b7dfb88fe05e82973b4c0a1b.jpg)
+
+![](images/1a215d8173230a79cf34b649f258e1d32e0d9041eed6cf857bee6a0bb903d4e9.jpg)
+
+# 5.6 TFI DISTRIBUTIONS: PLM vs. TRADITIONAL ENCODING METHODS
+
+To understand why GFS is much more effective on datasets with PLM encoded features, we conduct detailed analysis on the histogram of TFI distributions for PLM (first row of Figure 9) and traditional encoding methods (second row of Figure 9). We observe that,
+
+- PLM: The distributions share similar uni-modal and right-skewed shapes across different datasets. This means that most graph-disfavored feature dimensions are concentrated at the small TFI area. And the graph-favored features are long-tailed with a few feature dimensions containing the most graph-relevant information. Such distribution can help GFS efficiently split graph-aware and graph-agnostic feature dimensions and PLM can consistently produce encodings with this pattern for graph-related tasks.  
+- Traditional Methods: As shown in Appendix C.1, Amazon-Ratings, Questions, Roman-Empire are encoded by fasttext, Minesweeper is encoded by one-hot, and Tolokers is encoded by statistics-based method. Compared to PLM, these methods have lower average TFI values and show irregular TFI distributions. Although fasttext shows good performance and has an unimodal right-skewed TFI distribution on Roman-Empire, it fails to consistently generate high-quality features across different datasets.
+
+# 6 CONCLUSION AND FUTURE WORKS
+
+In this paper, we attempt to enhance GNN performance by identifying GNN-favored and GNN-disfavored features. To this end, we introduce Topological Feature Informativeness (TFI), which measures the mutual information between aggregated node features and labels. This metric effectively quantifies the performance gap between GNNs and Multi-Layer Perceptrons (MLPs), as supported by our empirical observations and theoretical analyses. We then propose a simple yet effective method, Graph Feature Selection (GFS), to incorporate TFI to improve the GNN performance. Our extensive experiments demonstrate that applying GFS to 8 GNN architectures across 10 datasets yields a significant performance boost in  $90\%$  (72 out of 80) of the cases, with an average increase of  $3.2\%$  in accuracy or AUC-ROC on node classification tasks. We also show that the performance gains from GFS are robust to hyperparameter tuning, indicating its potential as a universal method for enhancing various types of GNNs.
+
+However, the features selected by TFI introduce a new hyperparameter, ratio  $r$ , in GFS. Although GFS improves GNNs performance across a wide range of  $r$ , the best performance remains unknown due to the high complexity of the optimization process in GNNs and MLP. Therefore, it is interesting to explore the auto-selection of ratio  $r$  in the future. Furthermore, since TFI is a supervised, statistic-based metric and other types of metrics may also enhance GNN performance under GFS, it would be interesting to investigate more types of metrics, both supervised and unsupervised, for feature selection of GNNs in the future.
+
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+
+# A RELATED WORK
+
+The proposed Graph Feature Selection (GFS) in this paper mainly relates with feature selection and GNNs efficacy. Thus, in this section, we first introduce feature selection methods on non-GNNs models and GNNs. Then, we introduce the studies on the effectiveness of GNNs from a graph level, node level, or feature level.
+
+Feature Selection. Feature selection is crucial for improving model performance, preventing overfitting, and reducing computational complexity (Li et al., 2017). To preserve data similarity, Laplacian Score (He et al., 2005) and SPEC (Zhao & Liu, 2007) are proposed to select features that best preserve the data manifold structure. To maximize the correlation between feature and class labels, MIM (Lewis, 1992) is proposed based on information theory. To minimize the fitting errors in classification tasks, Hara & Maehara (2017); Zhu et al. (2003) propose to select features with larger weights in models with  $l_{1}$ -norm regularization. To reduce feature redundancy, T-Score (Davis et al., 1986) and Chi-Square Score (Liu & Setiono, 1995) are proposed to access whether features could distinguish different classes. All of the aforementioned methods only consider data in Euclidean space, which cannot be applied for non-Euclidean data, such as graph-structured data. To address this, Ye & Liu (2012); Kim & Xing (2009) use a Graph Lasso regularizer to select features consistent across connected nodes.
+
+While effective in traditional machine learning, these methods do not identify GNN-favored or GNN-disfavored features, as discussed in this paper. Our approach does not simply discard GNN-disfavored features; instead, it strategically utilizes them with a non-GNN model, driven by the principle of aligning the right features with the right model.
+
+Graph Homophily. Even if traditional GNNs (Kipf & Welling, 2016; Velicković et al., 2017) are believed to perform well on graph-related tasks, the performance of GNNs could be inferior than non-GNNs models in some graphs. Therefore, metrics of graph homophily, such as edge homophily (Zhu et al., 2020a), node homophily (Pei et al., 2020b), and class homophily (Lim et al., 2021), are proposed to determine when GNNs perform well. The definitions of these homophily metrics are given as:
+
+$$
+h _ {\text {e d g e}} (\mathcal {G}, \boldsymbol {Y}) = \frac {\left| \left\{e _ {u v} \mid e _ {u v} \in \mathcal {E} , Y _ {u} = Y _ {v} \right\} \right|}{| \mathcal {E} |}, h _ {\text {n o d e}} (\mathcal {G}, \boldsymbol {Y}) = \frac {1}{| \mathcal {V} |} \sum_ {v \in \mathcal {V}} \frac {\left| \left\{u \mid u \in \mathcal {N} _ {v} , Y _ {u} = Y _ {v} \right\} \right|}{\left| \mathcal {N} _ {v} \right|} \tag {10}
+$$
+
+$$
+h _ {\text {c l a s s}} (\mathcal {G}, \mathbf {Y}) = \frac {1}{C - 1} \sum_ {c = 1} ^ {C} \left[ \frac {\sum_ {u \in \mathcal {V} , Y _ {u} = c} | \{v \mid v \in \mathcal {N} _ {u} , Y _ {u} = Y _ {v} \} |}{\sum_ {u \in \{u | Y _ {u} = c \}} d _ {u}} - \frac {N _ {c}}{N} \right] _ {+} \tag {11}
+$$
+
+$$
+h _ {a d j} (\mathcal {G}, \mathbf {Y}) = \frac {h _ {e d g e} (\mathcal {G} , \mathbf {Y}) - \sum_ {c = 1} ^ {C} \frac {D _ {c} ^ {2}}{(2 | \mathcal {E} |) ^ {2}}}{1 - \sum_ {c = 1} ^ {C} \frac {D _ {c} ^ {2}}{(2 | \mathcal {E} |) ^ {2}}} \tag {12}
+$$
+
+Generally, these metrics measure the label consistency along the graph topology using an indicator function to determine if the connected nodes share the same labels. However, these metrics suffering from the "good heterophily" (Ma et al., 2021), leading to a misalignment with GNNs performance. Therefore, new metrics, such as label informativeness (Platonov et al., 2024), aggregation homophily (Luan et al., 2022a), and classifier-based homophily (Luan et al., 2024b) are proposed to mitigate this deficiency. Nevertheless, all these metrics neglect the feature aspect in graphs, which also constitute an important role in GNNs performance (Zheng et al., 2024a). Therefore, some feature homophily metrics, such as generalized edge homophily (Jin et al., 2022) is proposed to measure the feature consistency across the graph topology:
+
+$$
+h _ {G E} (\mathcal {G}, \boldsymbol {X}) = \frac {1}{| \mathcal {E} |} \sum_ {e _ {u v} \in \mathcal {E}} \frac {\boldsymbol {X} _ {u} \boldsymbol {X} _ {v}}{\| \boldsymbol {X} _ {u} \| \| \boldsymbol {X} _ {v} \|} \tag {13}
+$$
+
+Similarly, local similarity (Chen et al., 2023) is proposed to measure feature homophily at the node level by cosine similarity or Euclidean similarity:
+
+$$
+h _ {L S - c o s} (\mathcal {G}, \boldsymbol {X}) = \frac {1}{| \mathcal {V} |} \sum_ {u \in \mathcal {V}} \frac {1}{d _ {u}} \sum_ {v \in \mathcal {N} _ {u}} \frac {\boldsymbol {X} _ {u} \boldsymbol {X} _ {v}}{\left\| \boldsymbol {X} _ {u} \right\| \left\| \boldsymbol {X} _ {v} \right\|} \tag {14}
+$$
+
+$$
+h _ {L S - e u c} (\mathcal {G}, \boldsymbol {X}) = \frac {1}{| \mathcal {V} |} \sum_ {u \in \mathcal {V}} \frac {1}{d _ {u}} \sum_ {v \in \mathcal {N} _ {u}} (- \| \boldsymbol {X} _ {u} - \boldsymbol {X} _ {v} \| _ {2})
+$$
+
+Attribute homophily (Yang et al., 2021) also consider the feature homophily but with a different normalization on each feature:
+
+$$
+h _ {a t t r, m} (\mathcal {G}, \boldsymbol {X} _ {:, m}) = \frac {1}{\sum_ {u \in \mathcal {V}} X _ {u , m}} \sum_ {u \in \mathcal {V}} \left(X _ {u, m} \frac {\sum_ {v \in \mathcal {N} _ {u}} X _ {v , m}}{d _ {u}}\right) \tag {15}
+$$
+
+$$
+h _ {a t t r} (\mathcal {G}, \boldsymbol {X}) = \sum_ {m = 1} ^ {M} h _ {a t t r, m} (\mathcal {G}, \boldsymbol {X} _ {:, m})
+$$
+
+Class-controlled feature homophily (Lee et al., 2024) examines the relationship between graph topology and feature dependence by analyzing the difference in expected distances between a node and its neighbors compared to random nodes, defined as follows:
+
+$$
+h _ {C F} (\mathcal {G}, \boldsymbol {X}, \boldsymbol {Y}) = \frac {1}{| \mathcal {V} |} \sum_ {u \in \mathcal {V}} \frac {1}{d _ {u}} \sum_ {v \in \mathcal {N} _ {u}} \left(\boldsymbol {d} (v, \mathcal {V} \backslash \{u \}) - \boldsymbol {d} (v, \{u \})\right)
+$$
+
+$$
+\boldsymbol {d} (u, \mathcal {V} ^ {\prime}) = \frac {1}{| \mathcal {V} ^ {\prime} |} \sum_ {v \in \mathcal {V} ^ {\prime}} \| (\boldsymbol {X} _ {u} | \boldsymbol {Y}) - (\boldsymbol {X} _ {v} | \boldsymbol {Y}) \| \tag {16}
+$$
+
+$$
+\boldsymbol {X} _ {u} | \boldsymbol {Y} = \boldsymbol {X} _ {u} - \left(\frac {\sum_ {Y _ {u} = Y _ {v}} \boldsymbol {X} _ {v}}{| \{v | Y _ {u} = Y _ {v} , v \in \mathcal {V} \} |}\right)
+$$
+
+where  $X_{u}|Y$  represents class-controlled features and  $\pmb{d}(\cdot)$  denotes a distance function.
+
+All of these feature homophily metrics measure the feature consistency, which still suffer from the "good heterophily" issue as similar as node homophily or edge homophily. Our proposed Topological Feature Informativeness (TFI) addresses this limitation by the mutual information, which better aligns with GNNs performance on each feature in graphs.
+
+Heterophily-oriented GNNs For graphs with low homophily (heterophily), GNNs are likely to fail and even worse than non-GNNs models (Luan et al., 2024a). Therefore, many approaches (Wang et al., 2024b; Yang et al., 2021; Zheng et al., 2024b; Lu et al., 2024b; Luan et al., 2024c) are proposed to improve the performance of GNNs on heterophilous graphs, which can be categorized into graph level, node level, or feature level-based GNNs.
+
+For the graph level, FAGCN (Bo et al., 2021) introduces high frequency signals in graph convolution to capture local information to address heterophily. Similarly, ACMGCN (Luan et al., 2022a) and FB-GNNs (Luan et al., 2022b) proposes filterbanks to fuse identify, low-pass, and high-pass filter signals. To mitigate the limitation of local heterophilous neighbors, Mixhop (Abu-El-Haija et al., 2019) and H2GCN Zhu et al. (2020b) introduces multi-hop neighbors during the graph convolution. To extend local neighbors in heterophilous graphs to global neighbors, GloGNN (Li et al., 2022) employs a signed and learnable coefficient matrix, while HiGNN (Zheng et al., 2024b) incorporates new neighbors by utilizing neighbor distribution as heterophilous information.
+
+For the node level, SnoH (Wang et al., 2024b) modifies the message propagation for each node with varying receptive field in graph convolution. Similarly, CO-GNN (Finkelshtein et al., 2023) assigns each node with different types of directed message propagation in a cooperative manner, NoSAF (Wang et al., 2024c) provides each node with a node-specific layer aggregation with varying filter weights, and Node-MOE (Han et al., 2024) identifies heterophilous nodes with a feature inconsistency measurement, then treats these nodes differently with different filters. Different from these approaches, DisamGCL (Zhao et al., 2024) adopts a contrastive learning objective, by identifying ambiguous nodes with historical label predictions and treating these nodes as negative samples.
+
+To our best knowledge, only DMP (Yang et al., 2021) proposes to treat each feature differently in heterophily-oriented GNNs. DMP first defines the attribute homophily as feature consistency
+
+across the graph topology, then use a learnable weight to automatic learn the layer-wised weights for each features. However, this kind of attribute homophily suffers from "good heterophily" issue (Ma et al., 2021) and these homophily values haven't been used to guide the feature selection in graphs. Furthermore, these learnable weights in GNNs for feature selection still inferior than our statistic-based metrics, TFI.
+
+# B PROOF OF THEOREM 1
+
+Theorem 1. Given a graph  $\mathcal{G} = \{\mathcal{V},\mathcal{E}\}$  with node features  $\mathbf{X}_{\mathbf{m}}\in \mathbb{R}^{N}$  in the  $m$  -th dimension and uniform node labels  $\mathbf{Y}\in \mathbb{R}^N$  over  $\nu$  , we can obtain aggregated features by applying graph convolution  $k$  times, i.e.,  $\tilde{\mathbf{X}}_{\mathbf{m}} = (\hat{\mathbf{A}})^{k}\mathbf{X}_{\mathbf{m}}$  . For a classifier that predicts label  $Y$  using the aggregated features  $\tilde{X}_m$  , its accuracy rate  $P_A$  is upper bounded by:
+
+$$
+P _ {A} \leq \frac {I (Y ; \tilde {X} _ {m}) + \log 2}{\log (C)} \tag {17}
+$$
+
+Proof. For a node with its aggregated feature  $\tilde{X}_m$ , denote its true label as  $Y$  and the predicted label as  $\hat{Y} = f(\tilde{X}_m)$ , where  $f(\cdot)$  represents the classifier. The occurrence of an error  $E$  in the classifier can be expressed as:
+
+$$
+E := \left\{ \begin{array}{l l} 1 & \text {i f} \hat {Y} \neq Y, \\ 0 & \text {i f} \hat {Y} = Y. \end{array} \right. \tag {18}
+$$
+
+Considering the Markov chain  $Y \to \tilde{X} \to \hat{Y}$ , we apply Fano's inequality (Gerchinovitz et al., 2020) to obtain:
+
+$$
+H (Y | \tilde {X}) \leq H _ {b} \left(P _ {E}\right) + P _ {E} \log (C - 1) \tag {19}
+$$
+
+where  $P_{E}$  is the error rate and  $H_{b}(\cdot)$  is the binary entropy function.
+
+To express  $P_{E}$ , we rearrange the inequality:
+
+$$
+P _ {E} \geq \frac {H (Y | \tilde {X}) - H _ {b} \left(P _ {E}\right)}{\log (C - 1)} \tag {20}
+$$
+
+Noting that  $H(Y|\tilde{X}) = H(Y) - I(Y; \tilde{X}) = \log(C) - I(Y; \tilde{X})$  and that  $H_b(P_E) \leq \log 2$ , we can substitute these terms into the equation:
+
+$$
+P _ {E} \geq 1 - \frac {I (Y ; \tilde {X} _ {m}) + \log 2}{\log (C)} \tag {21}
+$$
+
+Finally, converting the expression to the accuracy rate, we find:
+
+$$
+P _ {A} = 1 - P _ {E} \leq \frac {I (Y ; \tilde {X} _ {m}) + \log 2}{\log (C)} \tag {22}
+$$
+
+This concludes the proof.
+
+# C IMPLEMENTATION DETAILS
+
+In this section, we provide all implementation details in Section 3 and Section 5. We have also made the code publicly available, which can be accessed here: https://github.com/KTTRCDL/graph-feature-selection
+
+# C.1 DATASETS
+
+We use Children, Computers, Fitness, History, Photo, Amazon-Ratings, Minesweeper, Questions, Roman-Empire, Tolokers as mentioned in Table 3 and Squirrel, Chameleon, Actor, Texas, Cornell, Wisconsin, Cora, CiteSeer, PubMed (Yang et al., 2016) as mentioned in Table 14 for all experiments. All datasets used in this work are in compliance with the MIT license. Tables 3 and 14 show the dataset statistics and the values of homophily metrics measured on these datasets. The detailed definition of these homophily measurements are shown in Appendix A. The descriptions of main datasets are given below:
+
+Table 3: Dataset Statistics  
+
+<table><tr><td>Dataset</td><td>#Nodes</td><td>#Edges</td><td>#Features</td><td>#Classes</td><td>Ave. Degrees</td><td>Domain</td><td>Feat. Modeling</td><td>hnode</td><td>hedge</td><td>hclass</td><td>hadj</td><td>Avg. TFI</td></tr><tr><td>Children</td><td>76,875</td><td>1,554,578</td><td>768</td><td>24</td><td>20.22</td><td>E-commerce</td><td>PLMs</td><td>0.4579</td><td>0.4220</td><td>0.2372</td><td>0.2913</td><td>0.0225</td></tr><tr><td>Comp.</td><td>87,229</td><td>721,081</td><td>768</td><td>10</td><td>8.27</td><td>E-commerce</td><td>PLMs</td><td>0.8469</td><td>0.8322</td><td>0.7601</td><td>0.7988</td><td>0.0208</td></tr><tr><td>Fitness</td><td>173,055</td><td>1,773,500</td><td>768</td><td>13</td><td>10.25</td><td>E-commerce</td><td>PLMs</td><td>0.8991</td><td>0.9004</td><td>0.7940</td><td>0.8528</td><td>0.0366</td></tr><tr><td>History</td><td>41,551</td><td>358,574</td><td>768</td><td>12</td><td>8.63</td><td>E-commerce</td><td>PLMs</td><td>0.7812</td><td>0.6626</td><td>0.2654</td><td>0.5463</td><td>0.0296</td></tr><tr><td>Photo</td><td>48,362</td><td>500,939</td><td>768</td><td>12</td><td>10.36</td><td>E-commerce</td><td>PLMs</td><td>0.7792</td><td>0.7491</td><td>0.7229</td><td>0.6892</td><td>0.0234</td></tr><tr><td>Amazon.</td><td>24,492</td><td>93,050</td><td>300</td><td>5</td><td>3.80</td><td>E-commerce</td><td>FastText</td><td>0.3793</td><td>0.3804</td><td>0.1270</td><td>0.1357</td><td>0.0177</td></tr><tr><td>Mines.</td><td>10,000</td><td>39,402</td><td>7</td><td>2</td><td>3.94</td><td>Games</td><td>One-hot</td><td>0.6832</td><td>0.6828</td><td>0.0094</td><td>0.0108</td><td>0.0202</td></tr><tr><td>Questions</td><td>48,921</td><td>153,540</td><td>301</td><td>2</td><td>3.14</td><td>Website</td><td>FastText</td><td>0.8963</td><td>0.8396</td><td>0.0722</td><td>0.2759</td><td>0.0049</td></tr><tr><td>Roman.</td><td>22,662</td><td>32,927</td><td>300</td><td>18</td><td>1.45</td><td>Website</td><td>FastText</td><td>0.0415</td><td>0.0469</td><td>0.0230</td><td>-0.0778</td><td>0.4870</td></tr><tr><td>tolokers</td><td>11,758</td><td>519,000</td><td>10</td><td>2</td><td>44.14</td><td>Social</td><td>Statistics</td><td>0.6331</td><td>0.5945</td><td>0.1867</td><td>0.0887</td><td>0.0044</td></tr></table>
+
+- Children and History (Yan et al., 2023) datasets are derived from the Amazon-Books dataset, consisting of items with the second-level label "Children" and "History", respectively. In both datasets, nodes represent books, and edges indicate frequent co-purchases or co-views between books. Each book's label corresponds to its third-level category. The node attributes are derived from the book's title and description using Pre-trained Language Models (PLMs). The task is to classify the books into 24 categories for Children and 12 categories for History. More details can be found on: https://github.com/sktsherlock/TAG-Benchmark.
+
+- Computers and Photo (Yan et al., 2023) datasets are extracted from the Amazon-Electronics dataset, including products with the second-level label "Computers" and "Photo", respectively. Nodes in these datasets represent electronics products, and edges signify frequent co-purchases or co-views. The labels correspond to the third-level category of the products. User reviews embedded by PLMs were used as the text attributes of the nodes, with the review having the highest number of votes being selected, or, if such a review was unavailable, a random review was chosen instead. The task is to classify the products into 10 categories for Computers and 12 categories for Photo. More details can be found on: https://huggingface.co/datasets/Sherirto/CSTAG/tree/main.
+
+- Fitness dataset (Yan et al., 2023) is derived from the Amazon-Sports dataset, consisting of fitness-related items with the second-level label "Fitness". Nodes represent fitness products, and edges indicate frequent co-purchases or co-views between products, encoded by PLM. The labels are based on the third-level category, and the task is to classify items into 13 categories. More details can be found on: https://huggingface.co/datasets/Sherirto/CSTAG/tree/main/Fitness.
+
+- Amazon-Ratings (Platonov et al., 2023) is derived from the Amazon product co-purchasing network metadata, provided by the SNAP (Leskovec & Krevl, 2014) Datasets, which nodes represent products and edges connect products that are frequently bought together. Node features are created using the mean of fastText (Grave et al., 2018) embeddings from product descriptions. The task is to predict the average rating of a product, grouped into five classes. More details can be found on: https://github.com/yandex-research/heterophilous-graphs/tree/main/data.
+
+- Minesweeper (Platonov et al., 2023) is inspired by the classic Minesweeper game and is synthetic in nature. It consists of a regular  $100 \times 100$  grid, where each node represents a cell connected to its eight neighboring cells (except for cells on the edges, which have fewer neighbors). In this setup,  $20\%$  of the nodes are randomly designated as mines. Each node has one-hot-encoded features representing the number of neighboring mines, with  $50\%$  of the nodes having missing features, indicated by a separate binary attribute. The task to recognize if a node is mine or not. More details can be found https://github.com/yandexresearch/heterophilous-graphs/tree/main/data.
+
+- Questions (Platonov et al., 2023) is based on user interactions on the Yandex Q question-answering platform. Nodes' labels represent users, and edges connect users if one answered the other's question during a one-year period. Node features are the mean of fast-Text (Grave et al., 2018) embeddings from user descriptions, with an additional binary feature indicating users without descriptions. The task is to predict which users remained active (i.e., not deleted or blocked) at the end of the period. More details can be found https://github.com/yandex-research/heterophilous-graphs/tree/main/data.
+
+- Roman-Empire (Platonov et al., 2023) is constructed from the "Roman Empire" article in the English Wikipedia. Nodes in the graph represent individual words in the article, and edges connect words that either appear consecutively in the text or are related via syntactic dependencies. The node's class is syntactic role obtained using spaCy (Honnibal
+
+et al., 2020), and fastText (Grave et al., 2018) are used for word embeddings of node features. More details can be found https://github.com/yandex-research/heterophilous-graphs/tree/main/data.
+
+- Tolokers (Platonov et al., 2023) is based on data from the Toloka crowdsourcing platform (Likhobaba et al., 2023). The nodes represent tolokers (workers) who have participated in at least one of 13 projects and edges connect workers who have collaborated on the same task. Node features are based on profile information and task performance statistics. The task is to predict which tolokers have been banned from a project. More details can be found https://github.com/yandex-research/heterophilous-graphs/tree/main/data.
+
+For all datasets, we randomly split the train, validation, and test set as  $50\% : 25\% : 25\%$  for 10 runs. Specifically, to investigate the impact of supervision percentage on the performance of GFS, in Figure 6, we combine the train, validation and test sets for TFI calculation with a given ratio  $r$ .
+
+# C.2 MODEL
+
+The details of all the GNN methods used in our experiments are introduced as follows:
+
+- GCN (Kipf & Welling, 2016) performs a layer-wise propagation of node features and aggregates features from neighboring nodes to capture local graph structures. Each layer of the network updates the node embeddings by applying a convolution operation over the graph, which combines the node's own features with the features of its neighbors. The authors' implementation is available at https://github.com/tkipf/gcn.
+
+- GAT (Velicković et al., 2017) leverages self-attention mechanisms to perform node classification on graph-structured data. The innovation of GAT lies in its ability to learn different attention coefficients for each neighboring node dynamically. Specifically, the attention coefficient between node  $i$  and its neighbor  $j$  is computed as:  $e_{ij} = \text{LeakyReLU}(\mathbf{a}^T[W\mathbf{h}_i\|W\mathbf{h}_j])$ , where  $\mathbf{h}_i$  and  $\mathbf{h}_j$  are the feature vectors of nodes  $i$  and  $j$ ,  $W$  is a shared weight matrix, and  $\mathbf{a}$  is a learnable attention vector. The coefficients are normalized using the softmax function:  $\alpha_{ij} = \frac{\exp(e_{ij})}{\sum_{k\in\mathcal{N}_i}\exp(e_{ik})}$ , where  $\mathcal{N}_i$  denotes the set of neighbors of node  $i$ . The final node representation is computed as a weighted sum of its neighbors' features:  $\mathbf{h}_i' = \sigma \left( \sum_{j\in \mathcal{N}_i} \alpha_{ij} W\mathbf{h}_j \right)$ . The multi-head attention mechanism improves stability and expressiveness by concatenating or averaging the outputs of multiple independent attention heads, which is set with num_heads = 8 in our experiments. This allows GAT to assign different importances to each neighboring node while maintaining computational efficiency. The authors' implementation is available at https://github.com/PetarV/GAT.
+
+- SAGE (Hamilton et al., 2017) introduces an inductive framework for node embeddings by aggregating features from a node's local neighborhood rather than requiring all nodes to be available during training. Its key innovation lies in the aggregation functions that can efficiently generate embeddings for unseen nodes. The aggregation process involves sampling and aggregating feature information from a node's neighborhood at each layer of the network. The general form of feature aggregation is:  $h_v^{(k)} = \sigma \left(W^{(k)} \cdot \text{AGGREGATE}\left(h_u^{(k-1)}, \forall u \in \mathcal{N}(v)\right)\right)$ , where  $\mathcal{N}(v)$  represents the set of neighbors of node  $v$ , and AGGREGATE is set to mean operation in our experiment. This allows the model to generalize across evolving graphs and unseen nodes. The authors' implementation is available at https://github.com/williamleif/GraphSAGE.
+
+- GT (Shi et al., 2020) is a transformer-based architecture designed for graph learning. It adapts the traditional Transformer model to graph data by incorporating node and edge features into the attention mechanism. The key innovation of GT is the use of multi-head attention to propagate node features across graph edges while also considering edge information. In our experiments, we employ the Graph Transformer with 8 attention heads. The attention coefficients are calculated as:  $\alpha_{ij} = \frac{\exp\left(\text{LeakyReLU}\left(\mathbf{q}_i^\top\mathbf{k}_j + \mathbf{e}_{ij}\right)\right)}{\sum_{k\in\mathcal{N}(i)}\exp\left(\text{LeakyReLU}\left(\mathbf{q}_i^\top\mathbf{k}_k + \mathbf{e}_{ik}\right)\right)}$  where  $\mathbf{q}_i$  and  $\mathbf{k}_j$  are the query and key vectors for nodes  $i$  and  $j$ , and  $\mathbf{e}_{ij}$  represents the edge feature between nodes  $i$  and  $j$ . This allows the model to aggregate both node and edge information. The author's implementation is available at https://github.com/PaddlePaddle/PGL/tree/main/ogb/examples/nodeproppred/unimp
+
+- SGC (Simplifying Graph Convolution) (Wu et al., 2019) is a simplified variant of GCN designed to reduce computational complexity while maintaining similar performance. SGC removes the non-linear activation functions between layers of a traditional GCN, collapsing multiple layers into a single linear transformation. This reduces the overall complexity of the model and results in faster training and inference times. The propagation of node features in SGC can be expressed as:  $\mathbf{H} = \mathbf{S}^{K}\mathbf{X}\mathbf{W}$ , where  $\mathbf{S}$  is the normalized adjacency matrix,  $\mathbf{X}$  is the input feature matrix,  $\mathbf{W}$  is the weight matrix, and  $K$  is the number of propagation steps (or layers). By precomputing  $\mathbf{S}^{K}\mathbf{X}$ , the model reduces to simple logistic regression on the preprocessed features, significantly speeding up the training process. In our experiments, we use  $K = 1$  for optimal performance. The authors' implementation is available at https://github.com/Tiiger/SGC.
+
+- APPNP (Gasteiger et al., 2018) builds upon Graph Convolutional Networks (GCNs) by utilizing personalized PageRank for improved propagation of node features while avoiding oversmoothing. The model separates the neural network prediction from the propagation process, allowing it to handle larger neighborhood sizes efficiently. The propagation is controlled by the following iterative equation:  $Z^{(k + 1)} = (1 - \alpha)\hat{A} Z^{(k)} + \alpha H$ , where  $\hat{A}$  is the normalized adjacency matrix,  $\alpha$  is the teleport probability (set as  $\alpha = 0.1$  in our experiments), and  $\hat{H}$  is the initial node feature set. After  $k$  (set default  $k = 2$ ) propagation steps, the output node features are computed as:  $Z = \mathrm{softmax}(Z^{(k)})$ . This structure permits the model to aggregate information from both local and distant nodes without increasing the depth of the neural network. The authors' implementation is available at https://github.com/klicperajo/ppnp.
+
+- ACMGCN (Luan et al., 2022a) addresses the heterophily problem in graph neural networks (GNNs) by introducing an Adaptive Channel Mixing (ACM) framework. This model adapts to both homophilic and heterophilic graphs by dynamically learning to balance information from three channels: aggregation, diversification, and identity. The key innovation lies in its ability to adaptively learn different weights for each node, allowing the model to exploit local graph structure and node feature similarities. The three channels are combined using learned weights, allowing the model to emphasize different types of information for different nodes:  $H^{(l)} = \alpha_{L}H_{L}^{(l)} + \alpha_{H}H_{H}^{(l)} + \alpha_{I}H_{I}^{(l)}$ , where  $H_{L}^{(l)}, H_{H}^{(l)}$ , and  $H_{I}^{(l)}$  represent the low-pass (aggregation), high-pass (diversification), and identity channels, respectively, and  $\alpha_{L}, \alpha_{H}, \alpha_{I}$  are learned weights that balance these channels. This flexible channel mixing enables ACMGCN to significantly outperform standard GNN models on heterophilic graphs while maintaining strong performance on homophilic graphs. The authors' implementation is available at https://github.com/SitaoLuan/ACM-GNN.
+
+- FAGCN (Bo et al., 2021) tackles the limitation of traditional GNNs that primarily focus on low-frequency signals. FAGCN introduces a self-gating mechanism to adaptively combine both low-frequency and high-frequency signals, allowing it to handle both assortative and disassortative networks effectively. The key innovation is its ability to dynamically adjust the contribution of each frequency type in the message-passing process. The aggregation of node features is expressed as:  $h_i' = \epsilon h_i + \sum_{j \in \mathcal{N}(i)} \alpha_{Lij} (F_L h_j) + \alpha_{Hij} (F_H h_j)$ , where  $F_L$  and  $F_H$  are low-pass and high-pass filters, respectively, and  $\alpha_{Lij}$  and  $\alpha_{Hij}$  are the learned attention coefficients for each type of signal. This enables FAGCN to adapt to different graph structures and alleviates the over-smoothing problem common in deep GNNs. The authors' implementation is available at https://github.com/bdy9527/FAGCN.
+
+# C.3 TRAINING DETAILS
+
+All models are implemented using the PyTorch (Ansel et al., 2024) framework and the DGL (Wang et al., 2019) library. We run experiments on a machine with 4 NVIDIA RTX A6000 GPUs, each with 48GB of memory. For most of the tables and figures illustrated in Section 5, we perform a grid search on all dataset and the hyperparameters used for search are list as follow:
+
+Number of layers:  $\{2,3\}$  
+- Hidden dimension:  $\{128, 256, 512\}$  
+- Learning rate:  $\{3 \times 10^{-5}, 10^{-4}, 3 \times 10^{-4}, 10^{-3}, 3 \times 10^{-3}, 10^{-2}\}$  
+Weight decay:  $\{0,10^{-5},10^{-3}\}$  
+- Dropout rate:  $\{0.1, 0.2, 0.4, 0.6, 0.8\}$
+
+Specially, for response of GCN+GFS, GCN, and MLP to number of layers and hidden dimension, as shown in Figure 5 and Figure 12, the hyperparameters range of set are list as follow:
+
+Number of layers:  $\{1,2,3,4,5,6,7,8,9,10\}$  
+- Hidden dimension:  $\{16,32,64,128,256,512,1024\}$  
+- Learning rate:  $\{10^{-5}, 3 \times 10^{-5}, 10^{-4}, 3 \times 10^{-4}, 10^{-3}, 3 \times 10^{-3}, 10^{-2}, 3 \times 10^{-2}\}$  
+Weight decay:  $\{0,10^{-5},3\times 10^{-5},10^{-4},3\times 10^{-4},10^{-3}\}$  
+- Dropout rate:  $\{0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9\}$
+
+To enhance the performance of all GNN models, we use skip connections (He et al., 2016) and layer normalization (Ba et al., 2016) across all methods.
+
+Table 4: Hyperparameters for GNN baselines and GNN+GFS on Children.  
+
+<table><tr><td>Dataset</td><td>Model</td><td>Num of Layers</td><td>Hidden Dim</td><td>Learning Rate</td><td>Weight Decay</td><td>Dropout</td></tr><tr><td rowspan="16">Children</td><td>GCN</td><td>2</td><td>256</td><td>3e-3</td><td>1e-3</td><td>0.2</td></tr><tr><td>GCN+GFS</td><td>2</td><td>256</td><td>3e-4</td><td>1e-4</td><td>0.2</td></tr><tr><td>GAT</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>GAT+GFS</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.2</td></tr><tr><td>SAGE</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>SAGE+GFS</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>GT</td><td>2</td><td>512</td><td>3e-4</td><td>1e-3</td><td>0.2</td></tr><tr><td>GT+GFS</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.4</td></tr><tr><td>SGC</td><td>2</td><td>512</td><td>3e-3</td><td>0</td><td>0.2</td></tr><tr><td>SGC+GFS</td><td>2</td><td>512</td><td>3e-3</td><td>0</td><td>0.2</td></tr><tr><td>APPNP</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>APPNP+GFS</td><td>2</td><td>512</td><td>3e-3</td><td>0</td><td>0.2</td></tr><tr><td>ACMGCN</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>ACMGCN+GFS</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.6</td></tr><tr><td>FAGCN</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>FAGCN+GFS</td><td>2</td><td>512</td><td>3e-3</td><td>0</td><td>0.2</td></tr></table>
+
+Table 5: Hyperparameters for GNN baselines and GNN+GFS on Computers.  
+
+<table><tr><td>Dataset</td><td>Model</td><td>Num of Layers</td><td>Hidden Dim</td><td>Learning Rate</td><td>Weight Decay</td><td>Dropout</td></tr><tr><td rowspan="16">Computers</td><td>GCN</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>GCN+GFS</td><td>3</td><td>512</td><td>1e-4</td><td>3e-5</td><td>0.2</td></tr><tr><td>GAT</td><td>2</td><td>128</td><td>3e-3</td><td>0</td><td>0.2</td></tr><tr><td>GAT+GFS</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.4</td></tr><tr><td>SAGE</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>SAGE+GFS</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.4</td></tr><tr><td>GT</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>GT+GFS</td><td>3</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>SGC</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.2</td></tr><tr><td>SGC+GFS</td><td>3</td><td>512</td><td>3e-3</td><td>0</td><td>0.2</td></tr><tr><td>APPNP</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>APPNP+GFS</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.2</td></tr><tr><td>ACMGCN</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>ACMGCN+GFS</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.4</td></tr><tr><td>FAGCN</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>FAGCN+GFS</td><td>3</td><td>512</td><td>3e-3</td><td>0</td><td>0.2</td></tr></table>
+
+# C.4 OPTIMIZATION-BASED GRAPH FEATURE SELECTION
+
+We implement optimization-based metrics  $\theta_{\mathrm{Soft}}$  and  $\theta_{\mathrm{Hard}}$  mentioned in Section 5.
+
+-  $\theta_{\mathrm{Soft}}$ : We use two learnable weight matrices,  $\mathbf{W}_{\mathrm{GNN}} \in \mathbb{R}^{M}$  and  $\mathbf{W}_{\mathrm{MLP}} \in \mathbb{R}^{M}$ , to guide feature selection for GNN and MLP. For the input node features  $\mathbf{X} \in \mathbb{R}^{N \times M}$ , we divide them into a GNN-favored part  $\mathbf{X}_{\mathcal{G}}$  and a GNN-disfavored part  $\mathbf{X}_{\neg \mathcal{G}}$ , computed as follows:
+
+$$
+\begin{array}{l} \mathbf {X} _ {\mathcal {G}} = \frac {\mathbf {W} _ {\mathrm {G N N}} ^ {2}}{\mathbf {W} _ {\mathrm {G N N}} ^ {2} + \mathbf {W} _ {\mathrm {M L P}} ^ {2} + \epsilon} \cdot \mathbf {X} (23) \\ \mathbf {X} _ {\neg \mathcal {G}} = \frac {\mathbf {W} _ {\mathrm {M L P}} ^ {2}}{\mathbf {W} _ {\mathrm {G N N}} ^ {2} + \mathbf {W} _ {\mathrm {M L P}} ^ {2} + \epsilon} \cdot \mathbf {X} (24) \\ \end{array}
+$$
+
+Table 6: Hyperparameters for GNN baselines and GNN+GFS on Fitness.  
+
+<table><tr><td>Dataset</td><td>Model</td><td>Num of Layers</td><td>Hidden Dim</td><td>Learning Rate</td><td>Weight Decay</td><td>Dropout</td></tr><tr><td rowspan="16">Fitness</td><td>GCN</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>GCN+GFS</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.2</td></tr><tr><td>GAT</td><td>2</td><td>128</td><td>3e-3</td><td>0</td><td>0.2</td></tr><tr><td>GAT+GFS</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.2</td></tr><tr><td>SAGE</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>SAGE+GFS</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.2</td></tr><tr><td>GT</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>GT+GFS</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.2</td></tr><tr><td>SGC</td><td>2</td><td>512</td><td>3e-3</td><td>0</td><td>0.2</td></tr><tr><td>SGC+GFS</td><td>3</td><td>512</td><td>3e-3</td><td>0</td><td>0.2</td></tr><tr><td>APPNP</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>APPNP+GFS</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.6</td></tr><tr><td>ACMGCN</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>ACMGCN+GFS</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.6</td></tr><tr><td>FAGCN</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>FAGCN+GFS</td><td>2</td><td>512</td><td>3e-3</td><td>0</td><td>0.2</td></tr></table>
+
+Table 7: Hyperparameters for GNN baselines and GNN+GFS on History.  
+
+<table><tr><td>Dataset</td><td>Model</td><td>Num of Layers</td><td>Hidden Dim</td><td>Learning Rate</td><td>Weight Decay</td><td>Dropout</td></tr><tr><td rowspan="16">History</td><td>GCN</td><td>2</td><td>256</td><td>3e-3</td><td>0</td><td>0.4</td></tr><tr><td>GCN+GFS</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>GAT</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>GAT+GFS</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>SAGE</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>SAGE+GFS</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>GT</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>GT+GFS</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.2</td></tr><tr><td>SGC</td><td>2</td><td>256</td><td>3e-3</td><td>0</td><td>0.2</td></tr><tr><td>SGC+GFS</td><td>3</td><td>512</td><td>3e-3</td><td>0</td><td>0.2</td></tr><tr><td>APPNP</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>APPNP+GFS</td><td>2</td><td>512</td><td>3e-3</td><td>0</td><td>0.2</td></tr><tr><td>ACMGCN</td><td>2</td><td>512</td><td>3e-4</td><td>0</td><td>0.2</td></tr><tr><td>ACMGCN+GFS</td><td>2</td><td>512</td><td>3e-4</td><td>0</td><td>0.6</td></tr><tr><td>FAGCN</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>FAGCN+GFS</td><td>3</td><td>512</td><td>3e-3</td><td>0</td><td>0.2</td></tr></table>
+
+Table 8: Hyperparameters for GNN baselines and GNN+GFS on Photo.  
+
+<table><tr><td>Dataset</td><td>Model</td><td>Num of Layers</td><td>Hidden Dim</td><td>Learning Rate</td><td>Weight Decay</td><td>Dropout</td></tr><tr><td rowspan="16">Photo</td><td>GCN</td><td>3</td><td>256</td><td>3e-3</td><td>0</td><td>0.4</td></tr><tr><td>GCN+GFS</td><td>3</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>GAT</td><td>2</td><td>128</td><td>3e-3</td><td>0</td><td>0.2</td></tr><tr><td>GAT+GFS</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.2</td></tr><tr><td>SAGE</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>SAGE+GFS</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.2</td></tr><tr><td>GT</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.2</td></tr><tr><td>GT+GFS</td><td>3</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>SGC</td><td>2</td><td>512</td><td>3e-3</td><td>1e-3</td><td>0.2</td></tr><tr><td>SGC+GFS</td><td>3</td><td>512</td><td>3e-3</td><td>0</td><td>0.2</td></tr><tr><td>APPNP</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>APPNP+GFS</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.4</td></tr><tr><td>ACMGCN</td><td>2</td><td>512</td><td>3e-3</td><td>0</td><td>0.4</td></tr><tr><td>ACMGCN+GFS</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.6</td></tr><tr><td>FAGCN</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>FAGCN+GFS</td><td>3</td><td>512</td><td>3e-3</td><td>0</td><td>0.2</td></tr></table>
+
+Table 9: Hyperparameters for GNN baselines and GNN+GFS on Amazon-Ratings.  
+
+<table><tr><td>Dataset</td><td>Model</td><td>Num of Layers</td><td>Hidden Dim</td><td>Learning Rate</td><td>Weight Decay</td><td>Dropout</td></tr><tr><td rowspan="16">Amazon-Ratings</td><td>GCN</td><td>2</td><td>512</td><td>3e-4</td><td>1e-3</td><td>0.4</td></tr><tr><td>GCN+GFS</td><td>2</td><td>512</td><td>1e-3</td><td>1e-3</td><td>0.4</td></tr><tr><td>GAT</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>GAT+GFS</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.4</td></tr><tr><td>SAGE</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>SAGE+GFS</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.4</td></tr><tr><td>GT</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>GT+GFS</td><td>3</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>SGC</td><td>3</td><td>512</td><td>3e-3</td><td>0</td><td>0.2</td></tr><tr><td>SGC+GFS</td><td>2</td><td>512</td><td>3e-3</td><td>0</td><td>0.2</td></tr><tr><td>APPNP</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>APPNP+GFS</td><td>2</td><td>512</td><td>3e-3</td><td>0</td><td>0.4</td></tr><tr><td>ACMGCN</td><td>2</td><td>512</td><td>3e-4</td><td>0</td><td>0.4</td></tr><tr><td>ACMGCN+GFS</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.6</td></tr><tr><td>FAGCN</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>FAGCN+GFS</td><td>2</td><td>512</td><td>3e-3</td><td>0</td><td>0.2</td></tr></table>
+
+Table 10: Hyperparameters for GNN baselines and GNN+GFS on Minesweeper.  
+
+<table><tr><td>Dataset</td><td>Model</td><td>Num of Layers</td><td>Hidden Dim</td><td>Learning Rate</td><td>Weight Decay</td><td>Dropout</td></tr><tr><td rowspan="16">Minesweeper</td><td>GCN</td><td>2</td><td>512</td><td>3e-4</td><td>0</td><td>0.2</td></tr><tr><td>GCN+GFS</td><td>2</td><td>128</td><td>1e-3</td><td>3e-4</td><td>0.2</td></tr><tr><td>GAT</td><td>2</td><td>512</td><td>3e-4</td><td>0</td><td>0.2</td></tr><tr><td>GAT+GFS</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>SAGE</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>SAGE+GFS</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>GT</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>GT+GFS</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>SGC</td><td>2</td><td>128</td><td>3e-3</td><td>0</td><td>0.2</td></tr><tr><td>SGC+GFS</td><td>3</td><td>512</td><td>3e-3</td><td>0</td><td>0.2</td></tr><tr><td>APPNP</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>APPNP+GFS</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.2</td></tr><tr><td>ACMGCN</td><td>3</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>ACMGCN+GFS</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>FAGCN</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>FAGCN+GFS</td><td>3</td><td>512</td><td>3e-3</td><td>0</td><td>0.2</td></tr></table>
+
+Table 11: Hyperparameters for GNN baselines and GNN+GFS on Questions.  
+
+<table><tr><td>Dataset</td><td>Model</td><td>Num of Layers</td><td>Hidden Dim</td><td>Learning Rate</td><td>Weight Decay</td><td>Dropout</td></tr><tr><td rowspan="16">Questions</td><td>GCN</td><td>2</td><td>512</td><td>3e-4</td><td>0</td><td>0.2</td></tr><tr><td>GCN+GFS</td><td>3</td><td>128</td><td>1e-4</td><td>1e-3</td><td>0.2</td></tr><tr><td>GAT</td><td>2</td><td>512</td><td>3e-4</td><td>0</td><td>0.2</td></tr><tr><td>GAT+GFS</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>SAGE</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>SAGE+GFS</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>GT</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>GT+GFS</td><td>3</td><td>512</td><td>3e-5</td><td>0</td><td>0.4</td></tr><tr><td>SGC</td><td>3</td><td>512</td><td>3e-3</td><td>0</td><td>0.8</td></tr><tr><td>SGC+GFS</td><td>3</td><td>512</td><td>3e-5</td><td>0</td><td>0.6</td></tr><tr><td>APPNP</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>APPNP+GFS</td><td>3</td><td>512</td><td>3e-5</td><td>0</td><td>0.6</td></tr><tr><td>ACMGCN</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>ACMGCN+GFS</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>FAGCN</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>FAGCN+GFS</td><td>3</td><td>512</td><td>3e-3</td><td>0</td><td>0.2</td></tr></table>
+
+Table 12: Hyperparameters for GNN baselines and GNN+GFS on Roman-Empire.  
+
+<table><tr><td>Dataset</td><td>Model</td><td>Num of Layers</td><td>Hidden Dim</td><td>Learning Rate</td><td>Weight Decay</td><td>Dropout</td></tr><tr><td rowspan="16">Roman-Empire</td><td>GCN</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>GCN+GFS</td><td>3</td><td>128</td><td>1e-2</td><td>1e-4</td><td>0.3</td></tr><tr><td>GAT</td><td>2</td><td>128</td><td>3e-3</td><td>0</td><td>0.4</td></tr><tr><td>GAT+GFS</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.6</td></tr><tr><td>SAGE</td><td>2</td><td>512</td><td>3e-3</td><td>0</td><td>0.6</td></tr><tr><td>SAGE+GFS</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.4</td></tr><tr><td>GT</td><td>2</td><td>512</td><td>3e-3</td><td>0</td><td>0.8</td></tr><tr><td>GT+GFS</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.6</td></tr><tr><td>SGC</td><td>2</td><td>128</td><td>3e-3</td><td>0</td><td>0.4</td></tr><tr><td>SGC+GFS</td><td>2</td><td>512</td><td>3e-3</td><td>0</td><td>0.2</td></tr><tr><td>APPNP</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>APPNP+GFS</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>ACMGCN</td><td>2</td><td>512</td><td>3e-3</td><td>0</td><td>0.8</td></tr><tr><td>ACMGCN+GFS</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.8</td></tr><tr><td>FAGCN</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>FAGCN+GFS</td><td>2</td><td>512</td><td>3e-3</td><td>0</td><td>0.2</td></tr></table>
+
+Table 13: Hyperparameters for GNN baselines and GNN+GFS on Tolokers.  
+
+<table><tr><td>Dataset</td><td>Model</td><td>Num of Layers</td><td>Hidden Dim</td><td>Learning Rate</td><td>Weight Decay</td><td>Dropout</td></tr><tr><td rowspan="16">Tolokers</td><td>GCN</td><td>2</td><td>512</td><td>3e-4</td><td>1e-5</td><td>0.2</td></tr><tr><td>GCN+GFS</td><td>3</td><td>128</td><td>1e-3</td><td>3e-5</td><td>0.2</td></tr><tr><td>GAT</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>GAT+GFS</td><td>3</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>SAGE</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>SAGE+GFS</td><td>3</td><td>512</td><td>3e-3</td><td>0</td><td>0.4</td></tr><tr><td>GT</td><td>3</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>GT+GFS</td><td>3</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>SGC</td><td>2</td><td>512</td><td>3e-3</td><td>0</td><td>0.2</td></tr><tr><td>SGC+GFS</td><td>3</td><td>512</td><td>3e-3</td><td>0</td><td>0.2</td></tr><tr><td>APPNP</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>APPNP+GFS</td><td>3</td><td>512</td><td>3e-3</td><td>0</td><td>0.2</td></tr><tr><td>ACMGCN</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.2</td></tr><tr><td>ACMGCN+GFS</td><td>3</td><td>512</td><td>3e-4</td><td>0</td><td>0.2</td></tr><tr><td>FAGCN</td><td>2</td><td>512</td><td>3e-5</td><td>0</td><td>0.2</td></tr><tr><td>FAGCN+GFS</td><td>3</td><td>512</td><td>3e-3</td><td>0</td><td>0.2</td></tr></table>
+
+where  $\epsilon$  is set to  $10^{-7}$  to prevent division by zero. In this formulation,  $\mathbf{W}_{\mathrm{GNN}}$  represents the proportion of each feature that is GNN-favored, while  $\mathbf{W}_{\mathrm{MLP}}$  represents the proportion that is MLP-favored. The sum of  $\mathbf{X}_{\mathcal{G}}$  and  $\mathbf{X}_{-\mathcal{G}}$  equals the original input  $\mathbf{X}$ , indicating that the features are divided based on the relative influence of GNN and MLP preferences. These weight parameters are updated during the training process.
+
+-  $\theta_{\mathrm{Hard}}$ : We use a learnable weight matrix,  $\mathbf{W}_{\mathrm{Hard}} \in \mathbb{R}^{M \times 2}$ , to guide feature selection for GNN and MLP. For the input node features  $\mathbf{X} \in \mathbb{R}^{N \times M}$ , we divide them into a GNN-favored part  $\mathbf{X}_{\mathcal{G}}$  and a GNN-disfavored part  $\mathbf{X}_{\neg \mathcal{G}}$ , computed as follows:
+
+$$
+\mathbf {M} _ {\text {H a r d}} = \text {G u m b e l - S o f t m a x} (\mathbf {W} _ {\text {H a r d}}) \tag {25}
+$$
+
+$$
+\mathbf {X} _ {\mathcal {G}} = \mathbf {X} \cdot \mathbf {M} _ {\text {H a r d}} [ :, 0 ] \tag {26}
+$$
+
+$$
+\mathbf {X} _ {\neg \mathcal {G}} = \mathbf {X} \cdot \mathbf {M} _ {\text {H a r d}} [ :, 1 ] \tag {27}
+$$
+
+In this case,  $\mathbf{M}_{\mathrm{Hard}} \in \mathbb{R}^{M \times 2}$  is the output of the Gumbel-Softmax applied to  $\mathbf{W}_{\mathrm{Hard}}$ . For each row in  $\mathbf{M}_{\mathrm{Hard}}$ , one value will be 1 and the other will be 0. A value of 1 indicates the selection of that feature, meaning that either the GNN-favored part or the MLP-favored part is chosen for each feature. This binary selection mechanism ensures that each feature is exclusively assigned to either  $\mathbf{X}_{\mathcal{G}}$  or  $\mathbf{X}_{\neg \mathcal{G}}$ , depending on which component is favored. The use of Gumbel-Softmax(\cdot) (Jang et al., 2017) allows for differentiable sampling, enabling the weight parameters to be updated during the training process.
+
+# C.5 DETAILS OF PRETRAINED NODE EMBEDDINGS
+
+We implement experiments using both GCN and MLP pretrained features to evaluate their effect when applying  $\mathrm{GCN + GFS}$  on either the original node features,  $X$ , or on the pretrained node embeddings,  $\mathrm{MLP}(X)$  and  $\mathrm{GCN}(X,A)$ . The results of this evaluation are presented in Figure 7 in the main text.
+
+-  $\mathrm{MLP}(X)$ : We pretrain a Multi-Layer Perceptron (MLP) model with the following setup: 2 layers, a hidden dimension of 128, trained for 1000 steps using the ELU activation function. For each of the 10 data splits, the MLP is trained separately with the Adam optimizer, using a learning rate of  $1e - 2$  and a weight decay of  $5e - 4$ . The pretrained MLP is then used to extract  $\mathrm{MLP}(X)$  corresponding to the training splits, which serves as the node embeddings for the experiment.  
+- GCN $(X, A)$ : We pretrain a Graph Convolutional Network (GCN) with 2 layers, a hidden dimension of 128, and trained for 1000 steps. The activation function used is ReLU, with a dropout rate of 0.5. Similarly, GCN is trained separately on each of the 10 data splits using the Adam optimizer, with a learning rate of  $1e - 2$  and a weight decay of  $5e - 4$ . The pretrained GCN is then used to extract  $\mathrm{GCN}(X, A)$ , corresponding to the training splits, as the node embeddings.
+
+# D MORE EXPERIMENTAL RESULTS
+
+# D.1 GCN+GFS ON ADDITIONAL DATASETS
+
+We report the performance of GCN+GFS on additional datasets including Squirrel, Chameleon, Actor, Texas, Cornell, Wisconsin, Cora, CiteSeer, and PubMed. The dataset statistics and descriptions are shown in Figure 14.
+
+Table 14: Additional Dataset Statistics  
+
+<table><tr><td>Dataset</td><td>#Nodes</td><td>#Edges</td><td>#Features</td><td>#Classes</td><td>Ave. Degrees</td><td>hnode</td><td>hedge</td><td>hclass</td><td>hadj</td><td>Avg. TFI</td></tr><tr><td>Squirrel</td><td>2223</td><td>46998</td><td>2089</td><td>5</td><td>21.14</td><td>0.1759</td><td>0.2072</td><td>0.0725</td><td>-0.0076</td><td>0.0173</td></tr><tr><td>Chameleon</td><td>890</td><td>8854</td><td>2325</td><td>5</td><td>9.95</td><td>0.2273</td><td>0.2361</td><td>0.0601</td><td>0.0360</td><td>0.0177</td></tr><tr><td>Actor</td><td>7600</td><td>26659</td><td>932</td><td>5</td><td>3.51</td><td>0.2197</td><td>0.2167</td><td>0.0074</td><td>0.0015</td><td>0.0029</td></tr><tr><td>Texas</td><td>183</td><td>279</td><td>1703</td><td>5</td><td>1.52</td><td>0.0748</td><td>0.0609</td><td>0.0017</td><td>-1.4628</td><td>0.0348</td></tr><tr><td>Cornell</td><td>183</td><td>277</td><td>1703</td><td>5</td><td>1.51</td><td>0.1246</td><td>0.1227</td><td>0.0482</td><td>-1.1984</td><td>0.0202</td></tr><tr><td>Wisconsin</td><td>251</td><td>450</td><td>1703</td><td>5</td><td>1.79</td><td>0.1934</td><td>0.1778</td><td>0.0447</td><td>-0.3947</td><td>0.0182</td></tr><tr><td>Cora</td><td>2708</td><td>10556</td><td>1433</td><td>7</td><td>3.90</td><td>0.8252</td><td>0.8100</td><td>0.7657</td><td>0.7717</td><td>0.0188</td></tr><tr><td>CiteSeer</td><td>3327</td><td>9228</td><td>3703</td><td>6</td><td>2.77</td><td>0.7166</td><td>0.7391</td><td>0.6267</td><td>0.6673</td><td>0.0103</td></tr><tr><td>PubMed</td><td>19717</td><td>88651</td><td>500</td><td>3</td><td>4.50</td><td>0.7924</td><td>0.8024</td><td>0.6641</td><td>0.6836</td><td>0.0440</td></tr></table>
+
+As shown in Figure 10, the performance of the GCN+GFS model surpasses the standard GCN across additional datasets. The selective feature processing approach provided by GFS allows GNNs to focus on beneficial features, leading to improved performance across diverse datasets.
+
+![](images/27fd7579cfe8b3f9171045421fa75050840d0e05a81647a39af55a593ecc119f.jpg)  
+Figure 10: Performance on additional datasets.
+
+# D.2 SENSITIVITY OF GFS
+
+Figure 11 shows the response of GCN+GFS to  $r$  on 6 other datasets, where GFS collapses to a GNN when  $r = 1.0$  or an MLP when  $r = 0.0$ , as all features are sent to GNN or MLP, respectively. Although in some datasets like Minesweeper and Questions, where all the features may favor GNNs, GFS doesn't significantly improve GCN performance, it doesn't diminish GFS's overall effectiveness on most datasets.
+
+Figure 12 shows how GCN, MLP, and  $\mathrm{GCN + GFS}$  respond to changes in number of layers, dimension of hidden embeddings, learning rate, weight decay, and dropout rate on a homophilous graph (Computers) and a heterophilous graph (Amazon-Ratings) on other 6 datasets. In most cases, GFS outperforms GCN, but for certain hyperparameter settings on the Questions dataset, GFS does not consistently outperform GCN. This is likely because the node features in these datasets are already highly graph-favored, making the additional feature selection offered by GFS less impactful. These results highlight the robustness of GFS under different hyperparameter settings.
+
+![](images/9f72982b2a3bd723e2dd5a73ec265e260bd05d13a09f4f2a0491eb309d6dea37.jpg)
+
+![](images/06952bf5b632f92f1caa9e2dadd5ab934990d95ed7433935682c86c068bf0ff5.jpg)
+
+![](images/1e670db5306e6da9cf1a055d69893239d1c56faf94b6f4a6a4fabec8d52778b7.jpg)
+
+![](images/750dfd25aef25152b6a99cdb34f0195f6cc8b284239881750ddc3f58b27fcd04.jpg)  
+Figure 11: The performance of GCN+GFS is shown as the ratio  $r$  of GNN-favored features in TFI increases. The point representing the best performance is highlighted as  $\star$ .
+
+![](images/d4511f551d83ffaf08135b981f2945e41db9b949b46a11973ac2a093c06b99eb.jpg)
+
+![](images/d752e3f36d167ed2a1868c668fe6b75d3063a27b495ad1fff9b3475198b2fd29.jpg)
+
+# D.3 PERCENTAGE OF SUPERVISION IN TFI
+
+In Figure 13, the influence of the supervision percentage in TFI on the performance of GCN+GFS across other 6 datasets is shown. The results highlight that as the supervision percentage increases, the performance of GCN+GFS generally improves or stabilizes. Even with only  $10\%$  supervision, GCN+GFS outperforms the standard GCN, demonstrating the robustness of the TFI approach. For datasets like Fitness and Amazon, there is a noticeable improvement in performance as the supervision percentage increases, but it stabilizes at around  $30\%$ . In datasets such as History, Photos, and Mines, the model performance is relatively stable, indicating that a small percentage of supervision is sufficient to achieve optimal performance. The performance on Questions shows more variability with increasing supervision, possibly due to the graph-favored dataset's features. This figure supports the claim that TFI requires minimal supervision to improve GCN+GFS performance, which is particularly useful in semi-supervised learning settings.
+
+# D.4 THE NUMBER OF K-HOP NEIGHBORS IN TFI
+
+We further investigate the influence of the number of neighbor hops in TFI on the performance of GCN+GFS. We set  $k$  from  $\{1,2,\dots,8\}$  as shown in Eq. (5). As illustrated in Figure 14, increasing the number of neighbor hops does not significantly affect model performance and results in only minor deviations in most datasets. This indicates that 1-hop neighbors are sufficient for TFI to select GNN-favored or GNN-disfavored features.
+
+# D.5 PERFORMANCE ON DATASETS WITH PUBLIC Splits
+
+We further conduct the experiments on these datasets with public splits as shown in Table 15. Accuracy results on node classification are reported for Cora, PubMed, and CiteSeer, each having one split. For the other datasets, which have ten splits, we provide both accuracy and standard deviation. The results demonstrate that GNN+GFS outperforms baseline GNN in most datasets for 4 types of GNN backbones, which include GCN, GAT, SGC, and GraphSAGE.
+
+Table 15: The performance of graph feature selection on GNN baselines in public split datasets.  
+
+<table><tr><td>Model</td><td>Cora</td><td>Pubmed.</td><td>Citeseer</td><td>Actor</td><td>Chameleon.</td><td>Cornell</td><td>Squirrel</td><td>Texas.</td><td>Wisconsin</td></tr><tr><td>GCN</td><td>81.28</td><td>78.06</td><td>70.60</td><td>33.68±0.64</td><td>62.41±1.97</td><td>60.81±6.53</td><td>48.41±1.52</td><td>66.49±6.14</td><td>77.65±4.91</td></tr><tr><td>GCN+GFS</td><td>81.94</td><td>78.46</td><td>71.74</td><td>35.30±0.85</td><td>62.68±2.02</td><td>68.11±6.47</td><td>49.13±1.78</td><td>77.57±6.50</td><td>80.39±4.89</td></tr><tr><td>Δ</td><td>+0.66</td><td>+0.40</td><td>+1.14</td><td>+1.62</td><td>+0.27</td><td>+7.30</td><td>+0.72</td><td>+11.08</td><td>+2.74</td></tr><tr><td>GAT</td><td>80.16</td><td>76.92</td><td>69.88</td><td>33.07±0.79</td><td>65.79±2.55</td><td>63.24±4.97</td><td>51.63±1.47</td><td>74.05±4.97</td><td>76.47±3.70</td></tr><tr><td>GAT+GFS</td><td>81.06</td><td>77.94</td><td>72.24</td><td>35.23±0.83</td><td>66.49±2.11</td><td>69.19±5.87</td><td>52.15±1.83</td><td>77.84±4.38</td><td>81.57±6.21</td></tr><tr><td>Δ</td><td>+0.90</td><td>+1.02</td><td>+2.36</td><td>+2.16</td><td>+0.70</td><td>+5.95</td><td>+0.52</td><td>+3.79</td><td>+5.10</td></tr><tr><td>SGC</td><td>80.74</td><td>76.96</td><td>69.84</td><td>30.64±0.75</td><td>54.39±1.79</td><td>43.24±7.64</td><td>39.91±1.56</td><td>54.86±4.42</td><td>53.73±5.64</td></tr><tr><td>SGC+GFS</td><td>81.48</td><td>76.38</td><td>71.18</td><td>35.62±1.20</td><td>54.63±2.14</td><td>67.30±4.84</td><td>42.19±1.12</td><td>71.62±4.27</td><td>78.63±6.43</td></tr><tr><td>Δ</td><td>+0.74</td><td>-0.58</td><td>+1.34</td><td>+4.98</td><td>+0.24</td><td>+24.06</td><td>+2.28</td><td>+16.76</td><td>+24.90</td></tr><tr><td>SAGE</td><td>81.06</td><td>76.80</td><td>68.52</td><td>36.20±0.81</td><td>58.42±2.53</td><td>70.00±7.03</td><td>39.32±2.06</td><td>74.59±5.73</td><td>81.18±2.11</td></tr><tr><td>SAGE+GFS</td><td>81.88</td><td>78.40</td><td>71.46</td><td>36.47±0.72</td><td>58.71±1.93</td><td>71.08±5.70</td><td>39.28±1.49</td><td>81.08±5.55</td><td>83.14±5.08</td></tr><tr><td>Δ</td><td>+0.82</td><td>+1.60</td><td>+2.94</td><td>+0.27</td><td>+0.29</td><td>+1.08</td><td>-0.04</td><td>+6.49</td><td>+1.96</td></tr></table>
+
+# D.6 IMPACT OF LABEL HOMOPHILY
+
+To investigate how GSF performs under varying label homophily, we conduct experiments on synthetic datasets using CSBM-H (Luan et al., 2024b; Zheng et al., 2024a) to control homophily levels. Specifically, in the CSBM-H model, for a node  $u$  labeled with  $y$ , its features  $\mathbf{X}_u \in \mathbb{R}^M$  are sampled from a class-wise Gaussian distribution, specifically  $\mathbf{X}_u \sim \mathcal{N}_{Y_u}(\boldsymbol{\mu}_{Y_u}, \boldsymbol{\Sigma}_{Y_u})$ . Each dimension of  $\mathbf{X}_u$  is independent of each other. To construct the graph structure  $\mathcal{G}$  with a specified homophily degree  $h$ , the node  $u$  has a probability  $h$  of connecting to nodes with the same label and a probability of  $\frac{1 - h}{C - 1}$  of connecting to nodes with different label. We randomly generate 10 graphs with different seeds in our experiments to reduce uncertainty. Each graph has 1000 nodes with 10 features and 5 classes. The node degrees are uniformly sampled from [2, 8].
+
+We demonstrate how the performance of GCN+GFS varies with label homophily across the range  $[0.1, 0.2, \dots, 0.9]$  in Table 16 and Figure 15 for node classification tasks. Generally, GCN+GFS outperforms both GCN and MLP across different levels of label homophily, indicating that GFS addresses the limitations of GCN under low homophily and of MLP under high homophily through effective feature selection. Furthermore, the relative increase  $\Delta$  (defined as the accuracy gap between GCN+GFS and GCN, divided by the accuracy of GCN) decreases as label homophily increases. This finding is expected, as higher homophily levels typically enhance the performance of all graph-aware models, thereby limiting the potential for improvement through GFS.
+
+Table 16: Impact of label homophily on GFS.  
+
+<table><tr><td>Label Homophily</td><td>0.1</td><td>0.2</td><td>0.3</td><td>0.4</td><td>0.5</td><td>0.6</td><td>0.7</td><td>0.8</td><td>0.9</td></tr><tr><td>MLP</td><td>73.12±4.06</td><td>71.24±5.81</td><td>68.44±6.93</td><td>70.36±5.30</td><td>66.92±6.36</td><td>67.52±4.64</td><td>70.12±5.49</td><td>69.88±4.77</td><td>69.32±4.68</td></tr><tr><td>GCN</td><td>50.12±3.09</td><td>50.56±3.28</td><td>56.00±5.49</td><td>61.24±5.30</td><td>65.84±5.11</td><td>72.32±2.35</td><td>80.32±3.44</td><td>87.64±2.23</td><td>90.64±2.32</td></tr><tr><td>GCN-GFS</td><td>73.24±3.85</td><td>69.96±7.24</td><td>69.04±6.76</td><td>71.36±6.17</td><td>69.68±7.69</td><td>73.84±4.65</td><td>82.76±6.65</td><td>89.36±4.20</td><td>91.20±6.04</td></tr><tr><td>Δ (relative)</td><td>23.12%</td><td>19.40%</td><td>13.04%</td><td>10.12%</td><td>3.84%</td><td>1.52%</td><td>2.44%</td><td>1.72%</td><td>0.56%</td></tr></table>
+
+# D.7 IMPACT OF FEATURE DIMENSION
+
+We investigate the impact of feature dimension size based on the CSBM-H model (Luan et al., 2024b; Zheng et al., 2024a), following the setup described in Section D.6. As shown in Table 17 and Figure 16, GFS demonstrates a pronounced advantage over GCN as the feature dimension increases. This outcome is expected, as more features allow GFS to exhibit higher fault tolerance during feature selection. In real-world scenarios, where the feature dimensions in datasets can reach hundreds or thousands, as indicated in Table 3, GFS is likely to perform well.
+
+Table 17: Impact of size of feature dim on GFS.  
+
+<table><tr><td>#Feature Dim</td><td>10</td><td>20</td><td>30</td><td>40</td><td>50</td></tr><tr><td>MLP</td><td>67.52±4.64</td><td>77.76±3.53</td><td>83.40±2.58</td><td>87.60±1.89</td><td>91.12±1.88</td></tr><tr><td>GCN</td><td>72.32±2.35</td><td>79.20±2.15</td><td>81.40±2.37</td><td>84.64±3.07</td><td>87.80±2.73</td></tr><tr><td>GCN-GFS</td><td>73.84±4.65</td><td>84.12±3.70</td><td>87.40±3.68</td><td>90.84±3.51</td><td>94.12±3.37</td></tr><tr><td>Δ (relative)</td><td>2.06%</td><td>5.85%</td><td>6.86%</td><td>6.83%</td><td>6.71%</td></tr></table>
+
+# D.8 IMPACT OF FEATURE SPARSENESS
+
+We investigate the impact of feature sparseness on the CSBM-H model (Luan et al., 2024b; Zheng et al., 2024a), following the setup outlined in Section D.6. To control sparseness in synthetic datasets, we randomly mask a certain percentage of node features. As shown in Table 18 and Figure 17, the results indicate that the advantage of GFS diminishes as feature sparseness increases. Although the relative increase rate of GFS decreases in this context, it still outperforms GCN. This is attributed to the presence of both GNN-favored and GNN-disfavored features, which allows GFS to remain effective in enhancing GNN performance.
+
+Table 18: Impact of sparseness on GFS.  
+
+<table><tr><td>Sparseness</td><td>0%</td><td>10%</td><td>20%</td><td>30%</td><td>40%</td><td>50%</td><td>60%</td><td>70%</td><td>80%</td><td>90%</td></tr><tr><td>GCN</td><td>54.16±5.58</td><td>46.48±3.79</td><td>48.88±3.43</td><td>45.68±1.93</td><td>42.12±2.88</td><td>39.52±2.86</td><td>34.88±5.59</td><td>32.08±2.05</td><td>29.92±2.75</td><td>25.12±3.07</td></tr><tr><td>GCN-GFS</td><td>67.96±7.72</td><td>59.84±6.83</td><td>60.04±5.49</td><td>54.92±5.47</td><td>50.64±4.15</td><td>46.20±5.26</td><td>39.52±6.72</td><td>35.68±3.05</td><td>31.36±5.09</td><td>27.24±3.65</td></tr><tr><td>Δ (relative)</td><td>25.48%</td><td>28.74%</td><td>22.83%</td><td>20.23%</td><td>20.23%</td><td>16.90%</td><td>13.30%</td><td>11.22%</td><td>4.81%</td><td>8.44%</td></tr></table>
+
+# D.9 IMPACT OF NOISES IN FEATURES
+
+To investigate the impact of feature noise on GFS, we conduct experiments on four real-world datasets: Children, Computer, Roman-Empire, and Tolokers, by introducing noise to the original node features. Specifically, after normalizing the input features, we add Gaussian noise multiple times. As shown in Table 19 and Figure 18, the relative increase  $\Delta$  (defined as the accuracy gap between  $\mathrm{GCN + GFS}$  and GCN, divided by the accuracy of GCN) remains relatively stable, indicating that the performance of GFS is robust against feature noise.
+
+Table 19: Impact of noise on GFS.  
+
+<table><tr><td>Datasets</td><td>Noise</td><td>1</td><td>5</td><td>10</td><td>15</td><td>20</td></tr><tr><td rowspan="3">Children</td><td>GCN</td><td>52.92±1.17</td><td>52.93±0.62</td><td>52.57±0.75</td><td>52.32±0.89</td><td>52.20±0.75</td></tr><tr><td>GCN-GFS</td><td>58.16±1.82</td><td>57.56±2.64</td><td>57.01±3.24</td><td>56.31±3.78</td><td>55.87±3.94</td></tr><tr><td>Δ (relative)</td><td>9.89%</td><td>8.75%</td><td>8.44%</td><td>7.65%</td><td>7.04%</td></tr><tr><td rowspan="3">Comp.</td><td>GCN</td><td>84.39±1.25</td><td>84.42±0.85</td><td>83.07±1.09</td><td>82.46±0.77</td><td>81.68±0.94</td></tr><tr><td>GCN-GFS</td><td>88.87±2.82</td><td>88.50±2.25</td><td>87.80±3.17</td><td>87.10±3.40</td><td>85.01±5.18</td></tr><tr><td>Δ (relative)</td><td>5.31%</td><td>4.84%</td><td>5.69%</td><td>5.63%</td><td>4.08%</td></tr><tr><td rowspan="3">Roman.</td><td>GCN</td><td>84.11±0.30</td><td>83.96±0.33</td><td>83.65±0.40</td><td>83.23±0.33</td><td>83.00±0.25</td></tr><tr><td>GCN-GFS</td><td>84.85±0.80</td><td>84.57±0.77</td><td>84.19±0.92</td><td>83.73±0.91</td><td>83.59±0.83</td></tr><tr><td>Δ (relative)</td><td>0.87%</td><td>0.73%</td><td>0.64%</td><td>0.60%</td><td>0.71%</td></tr><tr><td rowspan="3">Tolokers</td><td>GCN</td><td>82.83±1.41</td><td>83.28±0.73</td><td>82.81±0.79</td><td>82.50±0.77</td><td>82.12±0.62</td></tr><tr><td>GCN-GFS</td><td>86.52±1.85</td><td>85.99±1.79</td><td>85.48±1.54</td><td>84.99±0.87</td><td>84.44±1.75</td></tr><tr><td>Δ (relative)</td><td>4.46%</td><td>3.25%</td><td>3.23%</td><td>3.01%</td><td>2.81%</td></tr></table>
+
+# D.10 GOOD HETEROPHILY ON FEATURE HOMOPHILY
+
+In this section, we show the phenomenon of "good heterophily" occurs in feature homophily. First, we split node features into 10 bins according to the values of feature homophily. Then, we run GCN on these bins separately to see how the model performance changes with feature homophily. As shown in Table 20 and Figure 19, the GCN performance remains good under a low value of feature homophily, which includes attribute homophily (Yang et al., 2021) ( $h_{attr}$ ), local similarity (Chen et al., 2023) ( $h_{sim-euc}$ ), and class-controlled feature homophily (Lee et al., 2024) ( $h_{CTF}$ ). This result indicates these consistency-based feature homophily cannot align well with GNN performance, which is similar to the phenomenon in label homophily (Ma et al., 2021; Luan et al., 2024b; Zheng et al., 2024a). Conversely, GNN performance consistently increases with the increase of our proposed TFI, implying its effectiveness in selecting GNN-favored or GNN-disfavored features.
+
+# D.11 STATISTICAL OF SELECTED FEATURES
+
+As shown in Table 21, we present the statistics of values, sparsity, and TFI for all features, GNN-favored features, and GNN-disfavored features. The results indicate that: (1) the TFI of GNN-
+
+Table 20: GCN performance on different bins of feature homophily metrics.  
+
+<table><tr><td>Metrics</td><td>0.0-0.1</td><td>0.1-0.2</td><td>0.2-0.3</td><td>0.3-0.4</td><td>0.4-0.5</td><td>0.5-0.6</td><td>0.6-0.7</td><td>0.7-0.8</td><td>0.8-0.9</td><td>0.9-1.0</td></tr><tr><td>hattr</td><td>31.94±0.16</td><td>31.51±0.95</td><td>30.26±4.82</td><td>30.21±4.76</td><td>30.02±4.99</td><td>30.07±3.95</td><td>29.34±4.30</td><td>30.85±2.59</td><td>29.96±3.90</td><td>31.40±2.26</td></tr><tr><td>hsim-euc</td><td>31.92±0.18</td><td>29.52±5.15</td><td>31.13±2.02</td><td>31.21±1.61</td><td>30.75±3.65</td><td>30.16±4.75</td><td>31.38±1.40</td><td>31.19±1.86</td><td>31.22±1.61</td><td>30.89±3.10</td></tr><tr><td>hCTF</td><td>31.94±0.16</td><td>31.43±0.94</td><td>30.31±4.46</td><td>30.55±3.37</td><td>30.79±2.97</td><td>31.55±0.92</td><td>30.08±4.67</td><td>31.31±1.78</td><td>30.75±2.75</td><td>31.92±1.74</td></tr><tr><td>TFI</td><td>31.94±0.16</td><td>35.24±1.02</td><td>37.43±1.63</td><td>37.44±1.36</td><td>41.04±0.70</td><td>43.22±0.55</td><td>44.28±0.60</td><td>45.13±0.30</td><td>45.12±0.33</td><td>45.67±0.34</td></tr></table>
+
+favored features is significantly higher than that of GNN-disfavored features; (2) the datasets encoded by Pretrained Language Models (PLMs), including Children, Computers, Fitness, History, and Photo, exhibit similar values in features, yet their TFI varies considerably. Notably, a higher average TFI across all features correlates with better GFS performance; (3) GFS is less effective on datasets with higher sparsity, such as Minesweeper and Tolokers. (4) Datasets with lower homophily tend to identify more features as GNN-disfavored, whereas those with higher homophily identify more features as GNN-favored.
+
+Table 21: Summary Statistics of Values, TFI, and Sparseness for Selected Features.  
+
+<table><tr><td rowspan="2">Dataset</td><td colspan="3">X</td><td colspan="3">XG</td><td colspan="3">X-G</td><td rowspan="2">r</td></tr><tr><td>Value</td><td>Sparseness</td><td>TFI</td><td>Value</td><td>Sparseness</td><td>TFI</td><td>Value</td><td>Sparseness</td><td>TFI</td></tr><tr><td>Children</td><td>0.0202±0.4081</td><td>0.00%</td><td>0.0278±0.0158</td><td>0.0202±0.4081</td><td>0.00%</td><td>0.0396±0.0143</td><td>0.0202±0.4082</td><td>0.00%</td><td>0.0159±0.0039</td><td>50%</td></tr><tr><td>Comp.</td><td>0.0201±0.4068</td><td>0.00%</td><td>0.0965±0.0232</td><td>0.0201±0.4063</td><td>0.00%</td><td>0.1010±0.0239</td><td>0.0202±0.4090</td><td>0.00%</td><td>0.0785±0.0021</td><td>80%</td></tr><tr><td>Fitness</td><td>0.0195±0.3928</td><td>0.00%</td><td>0.1841±0.0220</td><td>0.0195±0.3929</td><td>0.00%</td><td>0.1906±0.0233</td><td>0.0195±0.3927</td><td>0.00%</td><td>0.1688±0.0019</td><td>70%</td></tr><tr><td>History</td><td>0.0201±0.4039</td><td>0.00%</td><td>0.0412±0.0261</td><td>0.0201±0.4034</td><td>0.00%</td><td>0.0594±0.0258</td><td>0.0201±0.4044</td><td>0.00%</td><td>0.0230±0.0057</td><td>50%</td></tr><tr><td>Photo</td><td>0.0202±0.4066</td><td>0.00%</td><td>0.0680±0.0278</td><td>0.0201±0.4052</td><td>0.00%</td><td>0.0817±0.0285</td><td>0.0202±0.4088</td><td>0.00%</td><td>0.0475±0.0037</td><td>60%</td></tr><tr><td>Amazon.</td><td>0.0003±0.0573</td><td>0.00%</td><td>0.0914±0.0065</td><td>0.0003±0.0572</td><td>0.00%</td><td>0.0957±0.0040</td><td>0.0003±0.0575</td><td>0.00%</td><td>0.0850±0.0036</td><td>60%</td></tr><tr><td>Mines.</td><td>0.1429±0.3499</td><td>85.71%</td><td>0.0369±0.0384</td><td>0.1429±0.3499</td><td>85.71%</td><td>0.0510±0.0368</td><td>0.1429±0.3499</td><td>85.71%</td><td>0.0014±0.0014</td><td>80%</td></tr><tr><td>Questions</td><td>-0.0007±0.0510</td><td>15.18%</td><td>0.0074±0.0011</td><td>-0.0007±0.0510</td><td>15.19%</td><td>0.0077±0.0010</td><td>-0.0007±0.0504</td><td>15.09%</td><td>0.0055±0.0004</td><td>90%</td></tr><tr><td>Roman.</td><td>0.0006±0.0986</td><td>0.27%</td><td>0.0874±0.0396</td><td>0.0007±0.1015</td><td>0.15%</td><td>0.1520±0.0388</td><td>0.0006±0.0979</td><td>0.30%</td><td>0.0712±0.0167</td><td>20%</td></tr><tr><td>tolokers.</td><td>0.3825±0.4567</td><td>48.03%</td><td>0.0295±0.0139</td><td>0.3826±0.4565</td><td>48.06%</td><td>0.0349±0.0135</td><td>0.3823±0.4570</td><td>47.98%</td><td>0.0169±0.0005</td><td>70%</td></tr></table>
+
+![](images/52e3f9e49e7231158981767d2946621f4595ca7ae700a2b5701f9bb508856a7d.jpg)
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+![](images/da24358cc622a8f253cbb9ac9f15f7680b97150e09648cd233e2dcb8108ab185.jpg)
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+![](images/802b1c9926caa93384c637e3ff05e5b4a93dc81ad64f8921debafe8e099a3696.jpg)
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+![](images/4840bc3a82c599f291f478f6c7542b45ca0e9c785874639e031862b8dc313b1a.jpg)
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+![](images/d5a42871572652b10755b26898e3fe5b3ab96bec0b15a1a0e0891c55db0c786d.jpg)
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+![](images/41b7bb52fad3ce652e70b2f3e9db1e592394d2cd0abab65de305b304eaa3a79a.jpg)
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+![](images/4fd558462c65c3221c13a7de38348f7a081e83b6a018504e878c507a2b598c23.jpg)
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+![](images/39c84af5071983ae4e2fbb45a966ec1645a0cd73efbb2166a4295d99d72850f1.jpg)
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+![](images/d092aa86d883fd1dab4b6730ace0c17abc5736bcedb9bf695f8962338fe10d03.jpg)
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+![](images/81ea4152e62d3d8215e0af73aa1b2e84ca64fcb3edd5e284b435bb3b171d20b7.jpg)
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+![](images/abebc0c763bf635a14178d443a051171c728e97867b745e70c6b73c057981c86.jpg)
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+![](images/1fbb09f4485f41e5f0bbe54eddc886e04023d0080a41b2aa184d8aa3e9e928e2.jpg)
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+![](images/1e6c4304cc3a4745a44a2644576b15dae48f172ec9ac8bda82a8f29d53e42db7.jpg)
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+![](images/ee3523a7d1821620efb0cffab4740cf06329ca1f983da428c57c70681bbe7897.jpg)
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+![](images/8a92d631cc2d712af0ecb594f0c1e061a366a5e5bd50a62b83477a74cee78795.jpg)
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+![](images/0deba404a2039ad27de5178f84a55cbf2926d52bf1c1f61e0f254ff558988d1c.jpg)
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+![](images/ec33c95d45119d63b30b38f59cb3aa1a0d6fadff8e2d8539311d2c19ff613ba7.jpg)
+
+![](images/2f045710821f7a85273e8e11f3731432ac86fcb7fb544133d98811810a554371.jpg)
+
+![](images/e2aa652808bceb59fa06107261043762fd28fe39c53d8ed878cab22b639e0627.jpg)
+
+![](images/bada3827065e0aa88db2b37c2956a7f0ea4e5494ca13663215ddc9e4d96a1cfe.jpg)
+
+![](images/2b66bfa22497486f167691196a180b4645062e4ebe1a4257bd9d5ecdc2e88515.jpg)
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+![](images/ce4ae1acf150ac421e90d63add482acc473487f16dee895f3433cff4c3c0e3e5.jpg)
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+![](images/57c2c18da2da878007eb5ae0463837c0da686a2f6fb7d4980ec0b532a2731962.jpg)
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+![](images/96bcf240f0810c0be419192d1a1957bc35db1075d25ce80be25b6e71afcd16df.jpg)
+
+![](images/942ac34c73396fc3dad2a4c1e3fc5bb2f21d7ccc719d55e7362554901896f864.jpg)
+
+![](images/7289455dfb17489c2afa16929f5e053e2d3046d10aff8f0d182f522c08244e5e.jpg)  
+Figure 12: Response of GCN+GFS, GCN, and MLP to 5 hyperparameters on more datasets.
+
+![](images/7584a13a5001e7e336366d4cb4ab197b3af5025a2d61d414d53770fd9d8e60b3.jpg)
+
+![](images/53db644051d8b6257f3e5c5fe274550ba8348320c08185e720bcf5d0b6f0b418.jpg)
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+![](images/3df18cb2f91f94f1819a4f8af7abfbb47cf003132bb768507ec12e22658d795d.jpg)
+
+![](images/24d4b4ecdc1f5de2c7df8a0f75be17b5bc756e1195f4c2d0d0786d2b4d264106.jpg)
+
+![](images/2e98ac5cff79f46d17fe82b1633fd7d7ca5e4e8a1665228720efd065ea6481d0.jpg)
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+![](images/5f6c511471f67546487589ddf716d43fb237f49402a53942d4d6ea888e744223.jpg)
+
+![](images/c0f784d517c139983206114e850e01c01934adb98a7457cc9f89b8c674874fe4.jpg)
+
+![](images/3fbc218ff657182d2d43bfb3cfc38d56a1c7db266bf9b5df67b13cf9a3a85063.jpg)  
+Figure 13: Influence of the percentage of the supervision in TFI on the model performance of GCN+GFS.
+
+![](images/8e5331e7833c1460eaa0d240e1df5e8a606641d0381b648d13fbb17ee837ad91.jpg)
+
+![](images/d989963245992ceeaf96f5af62d97823c9367ced8ae53b157cbbe98c1e3e37e3.jpg)
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+![](images/0b2a1ab08875ce8a35e65a0959d57f89d38ee6e6fec8212b5ac7f4e390fe4991.jpg)
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+![](images/bc0300b4e32e502684a15a400cb56b103bfe9169f542834577b37d3c6a66b2b4.jpg)
+
+![](images/9b96785c04d797633e70bf46c50ea2a7c298d72dd5b105853cc1608b31a1e397.jpg)
+
+![](images/6f917fb05524a7eddf70186f6448bd9e2bc75a5cfdf2213ca111513b737c0ee3.jpg)
+
+![](images/edf51747d397357f724386c58e8787c46b0b3fee115fe3b0416ea15d67677d5c.jpg)
+
+![](images/947edbafb341aa2e900cf9a6a33ccc556ee6c9f1c51911dfbbf926bcfbd8b6e0.jpg)  
+Figure 14: Influence of the number of neighbor hops in TFI on the performance of GCN+GFS.
+
+![](images/e18c39ff019b048137acacd31713f209c902ff51de0205b8ab69a6221a11ba5e.jpg)
+
+![](images/bde290afd26f2998f752b067de2c717bc5cbd3d9629a722d2c4893a6ba097d92.jpg)
+
+![](images/81ff5b5d2f6d4e6086a4a17db550d8d4e3716dab03cb4f806467efe5afc0b222.jpg)
+
+![](images/67d52945ddabbf5d5da8ff9c2c6e04beece744c5e3e6e650bb63fd6614f031ab.jpg)
+
+![](images/3ddeeeb48b8f4e03653e31f52b925dc3e2da22d920a3a103dc17039472663b4b.jpg)  
+Figure 15: Comparison of the performance of MLP, GCN, and GFS across varying levels of label homophily in synthetic datasets.
+
+![](images/e214c8777a6de0a164e1ecbd737df3a8ef85bb772773fcb33c2b8ec260f88aef.jpg)  
+Figure 16: Comparison of the performance of MLP, GCN, and GFS with the increase of the number of feature dimensions.
+
+![](images/f1deb5687cc8d0a3cc0034a42d1c20435bd0b50275c60cd8c30f66b35072e49d.jpg)  
+Figure 17: Comparison of the performance of MLP, GCN, and GFS with the sparseness in synthetic datasets
+
+![](images/d9b6444dbf6abc6ca158f6679bbdb7ba5556189a4604f30980c7672598511e62.jpg)  
+Figure 18: Comparison of the performance of MLP, GCN, and GFS with the increase of feature noises
+
+![](images/f8e3743db9261ab00e12cd54ff481389fbd372d2990971c2ba2be44bdaa7af2c.jpg)
+
+![](images/224b4f163106101d0fde0853bfd6017be739afb96e3888e7f9dc37d89169ce0a.jpg)
+
+![](images/275559ba64737f7255b237077b3ee1a3ecc04e0f62c6ef5a47c6677cab3ebb9a.jpg)
+
+![](images/99c042f7db75c90081a0e4c6feb2db55d5b02a74ced6e6001830240a9d763308.jpg)
+
+![](images/28b06cba79af34b3ca6e88957fb996fbf0bd8095b2755a3185f4c08f4d2f6d2e.jpg)
+
+![](images/3ff2146bdc87c6a6fb792c90237e5b0c695b96e734e3375f02b4a6a953f2d2aa.jpg)  
+Figure 19: Performance of GCN under varying levels of feature homophily or TFI.
+
+![](images/348797d203c36b6815bc864c27830f93b2489c676392785b83169143e9cdeb93.jpg)
\ No newline at end of file
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+# LET THE CODE LLM EDIT ITSELF WHEN YOU EDIT THE CODE
+
+Zhenyu He \*Jun Zhang \* Shengjie Luo Jingjing Xu Zhi Zhang \* Di He
+
+National Key Laboratory of General Artificial Intelligence, School of Intelligence Science and Technology, Peking University  $\spadesuit$  ByteDance Inc.
+
+# ABSTRACT
+
+In this work, we investigate a typical scenario in code generation where a developer edits existing code in real time and requests a code assistant, e.g., a large language model, to re-predict the next token or next line on the fly. Naively, the LLM needs to re-encode the entire KV cache to provide an accurate prediction. However, this process is computationally expensive, especially when the sequence length is long. Simply encoding the edited subsequence and integrating it to the original KV cache meets the temporal confusion problem, leading to significantly worse performance. We address this efficiency and accuracy trade-off by introducing Positional Integrity Encoding (PIE). Building upon the rotary positional encoding, PIE first removes the rotary matrices in the Key cache that introduce temporal confusion and then reapplies the correct rotary matrices. This process ensures that positional relationships between tokens are correct and requires only a single round of matrix multiplication. We validate the effectiveness of PIE through extensive experiments on the RepoBench-C-8k dataset, utilizing DeepSeek-Coder models with 1.3B, 6.7B, and 33B parameters. Our evaluation includes three real-world coding tasks: code insertion, code deletion, and multi-place code editing. Results demonstrate that PIE reduces computational overhead by over  $85\%$  compared to the standard full recomputation approach across all model sizes and tasks while well approximating the model performance. Code is available at https://github.com/zhenyuhe00/PIE.
+
+![](images/0a6dae0459ee5486e72e64c6e99996d5b1193b810f302b95f524653a51b13c7e.jpg)  
+Figure 1: Latency and accuracy comparison of the full recomputation approach and our PIE using DeepSeek-Coder 6.7B on the RepoBench-C-8k(XF-F) Python dataset on a single A100 GPU. The latency only records the time cost for the KV cache update.
+
+# 1 INTRODUCTION
+
+Large language models (LLMs) (Dettmers et al., 2022; Anil et al., 2023; Touvron et al., 2023; Zeng et al., 2023) have seen widespread adoption and achieved impressive results across various natural language processing (NLP) tasks. Despite these successes, LLMs face significant computational
+
+challenges, particularly in handling long sequences. To address this, numerous approaches have been proposed to accelerate the inference process, including lossless (e.g., memory and IO optimization (Dao et al., 2022; Kwon et al., 2023; Sheng et al., 2023), speculative decoding (Stern et al., 2018; Leviathan et al., 2023)) and lossy techniques (e.g., quantization (Frantar et al., 2022; Xiao et al., 2023) and KV cache eviction (Xiao et al., 2024; Zhang et al., 2024)). We refer to the above setting as the static setting, where the content is fixed, and the goal is to generate responses efficiently without compromising too much on performance.
+
+Besides the static setting, we observe there is a strong demand for an alternative, which we call the real-time editing setting, where users frequently edit the content and expect the LLM to generate correct responses based on the updated information. A typical scenario is the interactive coding assistant, where developers often make incremental changes to their existing code and require the AI copilot to correctly predict the next line or complete a partial code snippet on the fly. The standard approach is re-encoding the KV cache of the content after each edit and then making the prediction. However, as illustrated in Figure 1, this approach leads to considerable substantial computational overhead and latency when the content is long (Fu, 2024; Agrawal et al., 2024), making it impractical for real-time applications where quick and accurate responses are essential.
+
+In this paper, we aim to improve the efficiency of AI copilots in real-time editing scenarios, as illustrated in Figure 2. Naively, an efficient strategy is encoding only the edited subsequence and then directly integrating those keys and values into the original KV cache. However, this strategy results in temporal confusion between the pre-edit and post-edit sequences. Keys of certain positions either disappear or multiple keys at different positions share the same index, causing the model to attend to incorrect information, leading to poor next-token prediction performance in practice. To address this problem, we introduce Positional Integrity Encoding (PIE). PIE is built upon rotary positional encoding (RoPE) (Su et al., 2021), the de-facto standard component in modern LLMs. PIE first removes the rotary matrices in the Key cache that introduce temporal confusion and then reapplies the correct rotary matrix for each position through simple matrix multiplications. By ensuring that the positional relationships between tokens in the new sequence are unique and consecutive, PIE can help the model make accurate predictions. It is worth noting that the calculation of PIE requires only a single round of matrix operations to modify the KV cache, resulting in negligible computational overhead.
+
+We demonstrate the effectiveness of Positional Integrity Encoding (PIE) through extensive experiments conducted on the RepoBench-C-8k dataset, utilizing the DeepSeek-Coder (Guo et al., 2024) models with 1.3B, 6.7B, and 33B parameters. To rigorously evaluate PIE's performance, we curated three tasks designed to simulate real-world coding scenarios: code insertion, code deletion, and multi-place code edition. These tasks were chosen to reflect common operations that developers perform during interactive coding sessions, thereby providing a comprehensive assessment of PIE's practical utility. Our experimental results indicate that PIE achieves a reduction in computational overhead of over  $85\%$  for editing the KV cache across all model sizes and tasks without compromising performance compared to the naive full-recomputation approach. By leveraging PIE, developers can experience efficient interactions with AI coding assistants.
+
+# 2 RELATED WORK
+
+Positional Encodings Positional information is essential for modeling languages. The original Transformer model (Vaswani et al., 2017) encodes positional information using Absolute Positional Encoding (APE). In particular, a (learnable) real-valued embedding is assigned to each position  $i$ . Differently, Relative Positional Encodings (RPE) (Shaw et al., 2018; Dai et al., 2019; Raffel et al., 2020; Press et al., 2022; Su et al., 2021; Luo et al., 2021; 2022; Chi et al., 2022; Sun et al., 2023; Chi et al., 2023; Li et al., 2023; He et al., 2024) instead encode the relative distance  $i - j$  for each position pair  $(i,j)$ . One of the most widely used RPE in state-of-the-art LLMs is Rotary Position Encoding (RoPE) (Su et al., 2021). RoPE rotates the query and key vectors by an angle proportional to their absolute positions before the attention mechanism, resulting in the attention being a function of the relative distance between tokens.
+
+In the literature, relative positional encodings play essential roles across various tasks and data modalities, such as improving the length extrapolation capability of language models (Press et al., 2022; Sun et al., 2023; Chi et al., 2022; 2023; He et al., 2024) and enabling flexible modeling of
+
+![](images/154e4515997563968a2a30756a2e8a63e4707d35dd8876f610714c37d38d51ee.jpg)  
+Static setting
+
+![](images/ac2b659dd722b5f40c44d9a0b51b8c4398ba35df5ea60429bbce4b8d5e3d9ae5.jpg)  
+Real-time editing setting  
+Figure 2: Illustration of the KV cache mechanism in both static and real-time editing settings for large language models (LLMs). Top: In the static setting, the model processes a fixed input to generate predictions, leveraging precomputed Key/Value (KV) pairs stored in the cache. Bottom: In the real-time editing setting, the input code is frequently edited, necessitating updates to the KV cache to maintain accurate information to generate the correct next tokens. Our objective is to optimize the efficiency of the green arrow pathway, which represents the process of updating the KV cache in response to code edits.
+
+structural information beyond sequence data like images (Liu et al., 2021) and graphs (Ying et al., 2021; Zhang et al., 2023a; Luo et al., 2023). In this work, we develop the Positional Integrity Encoding based on RoPE to improve the efficiency of LLMs in the real-time editing setting.
+
+Transformer Efficiency Improving the efficiency of Transformer models has great significance in real-world applications. In the literature, existing approaches can be briefly categorized into (1) efficient attention, (2) model compression, and (3) system-architecture co-design.
+
+The attention module of Transformer needs to calculate pairwise correlations between all positions, resulting in quadratic time and memory cost with respect to the sequence length. To reduce the cost, many efficient attention variants have been proposed, such as (1) sparse attentions (Child et al., 2019; Beltagy et al., 2020; Qiu et al., 2020), which design either pre-defined or learnable patterns to reduce the amount of the key-value pairs that each query needs to attend to; (2) approximation-based attention (Katharopoulos et al., 2020; Wang et al., 2020; Choromanski et al., 2021; Kitaev et al., 2020; Tay et al., 2020; Roy et al., 2021), which use tailored approaches like low-rank projection or random features to approximate standard attention for efficient computation.
+
+Another perspective for Transformer efficiency is model compression, mainly including (1) pruning (Wang et al., 2021; Hubara et al., 2021; Ma et al., 2023; Frantar & Alistarh, 2023), which aims at removing redundant model parameters or layers for efficient deployment without scarifying performance; (2) quantization (Yao et al., 2022; Park et al., 2022; Dettmers et al., 2022; Frantar et al., 2022; Xiao et al., 2023; Liu et al., 2023), which uses post-processing to represent weights and activations via low-precision format for reducing time and memory costs; (3) knowledge distillation (Sanh et al., 2019; Gu et al., 2024), which uses a smaller model to learn knowledge from a large model for balancing efficiency-accuracy trade-offs.
+
+In the era of LLMs, the importance of system-architecture co-design has been highlighted to improve Transformer efficiency further. Many works begin to design more efficient approaches for serving Transformers regarding the characteristics of computer systems for real-world applications, such as FlashAttention (Dao et al., 2022; Dao, 2023), PagedAttention (Kwon et al., 2023), and FlexGen (Sheng et al., 2023) that are proposed for memory and I/O optimization. Additionally, to reduce functional calls during generation, speculative decoding (Stern et al., 2018; Leviathan et al., 2023; Chen et al., 2023; Miao et al., 2023; Spector & Re, 2023; Cai et al., 2024; Zhang et al., 2023b;
+
+He et al., 2023; Li et al., 2024) has been proposed. Our Positional Integrity Encoding is specially designed to improve the efficiency of LLMs in real-time editing settings, which can be seamlessly combined with all the above-introduced approaches to achieve further speed-ups.
+
+# 3 METHODS
+
+# 3.1 BACKGROUND
+
+Let  $s = (w_{1}, w_{2}, \ldots, w_{n})$  represent the input token sequence, where each  $w_{i}$  belongs to a fixed vocabulary. Let  $\theta_{\mathrm{LLM}}$  represent a Transformer-based large language model, which can calculate the conditional probability distribution of the next token  $p(w_{n+1}|s; \theta_{\mathrm{LLM}})$  and generate tokens iteratively. Typically, the input  $s$  is fixed. Therefore, the generation process of LLMs usually employs the KV Cache mechanism (Pope et al., 2023) to store previously computed Key/Value vectors during each layer's attention calculation. We denote the KV cache as  $K = (K_{1}, K_{2}, \ldots, K_{n})$  and  $V = (V_{1}, V_{2}, \ldots, V_{n})$ , where  $K_{i}$  and  $V_{i}$  are the keys and values associated with token  $w_{i}$ . When predicting the token at position  $n+1$ , we can use  $w_{n}$  as input and, in each layer, compute the attention between the current hidden representation and the stored KV cache, avoiding recomputing the hidden representation of previous tokens. Without any confusion, we also denote the next token probability distribution as  $p(w_{n+1}|K, V, w_{n}; \theta_{\mathrm{LLM}})$ .
+
+In this study, we aim to investigate a new scenario where the context  $s$  is real-time edited by users, which makes it impossible for the KV cache to predict the correct next token without any modification. We can model such real-time context as a sequence of steps. Each step can be formulated as an action where tokens from position  $i$  to  $j$  in  $s$  are edited, resulting in a modified sequence *todo* represent the new inputs that replace  $[w_i, \dots, w_j]$ . Our goal is to accurately and efficiently predict  $w_{n+1}$  given by  $\theta_{\text{LLM}}$  on  $s^{\text{edit}}$ . This problem is crucial in various scenarios. For instance, users can frequently edit their previous codes for different purposes and expect the code language model to swiftly adapt to these changes and predict the correct next line based on the updated information.
+
+As the context between position  $i$  and position  $j$  is edited, the KV cache corresponding to these tokens must be updated. Furthermore, these changes will impact the representations of subsequent tokens after position  $j$ , thereby necessitating updates to all subsequent KV cache. The naive approach involves a full-recomputation strategy: re-encoding all the KV cache for  $[a_1, a_2, \ldots, a_m, w_{j+1}, \ldots, w_n]$  layer by layer, followed by making predictions using the updated cache  $K^*$  and  $V^*$ . This approach ensures the KV cache is exact when predicting the next tokens. However, it is easy to see that it is computationally expensive, especially when the edits are light but the texts to be re-encoded are long. It's worth noting that the original  $K$  and  $V$  already encode rich information on  $[w_{j+1}, \ldots, w_n]$ , and a full recomputation may not be essential for practical problems. With this in mind, we seek to find ways to efficiently edit  $K$  and  $V$ , yielding  $K^{\mathrm{edit}}$  and  $V^{\mathrm{edit}}$ , which approximates  $p(w_{n+1}|K^*, V^*, w_n; \theta_{\mathrm{LLM}}) \approx p(w_{n+1}|K^{\mathrm{edit}}, V^{\mathrm{edit}}, w_n; \theta_{\mathrm{LLM}})$  in an effective way.
+
+# 3.2 POSITIONAL INTEGRITY ENCODING (PIE)
+
+When a user modifies  $s$  into  $s^{\mathrm{edit}}$ , KV cache associated with the first  $i$  tokens, i.e.,  $\pmb{K}_{[1:i]}$  and  $\pmb{V}_{[1:i]}$ , remains unchanged. As  $[a_1, a_2, \ldots, a_m]$  is the user's new input, we feed this subsequence to the LLM to obtain the keys and values from position  $i + 1$  to  $i + m$ . We denote this piece of new KV cache as  $\pmb{K}_{[i + 1:i + m]}^{\mathrm{edit}}$  and  $\pmb{V}_{[i + 1:i + m]}^{\mathrm{edit}}$ , and now have the edited KV cache as:
+
+$$
+\boldsymbol {K} ^ {\text {e d i t}} = \operatorname {C o n c a t} \left(\boldsymbol {K} _ {[ 1: i ]}, \boldsymbol {K} _ {[ i + 1: i + m ]} ^ {\text {e d i t}}, \boldsymbol {K} _ {[ j + 1: n ]}\right), \tag {1}
+$$
+
+$$
+\boldsymbol {V} ^ {\text {e d i t}} = \operatorname {C o n c a t} \left(\boldsymbol {V} _ {[ 1: i ]}, \boldsymbol {V} _ {[ i + 1: i + m ]} ^ {\text {e d i t}}, \boldsymbol {V} _ {[ j + 1: n ]}\right), \tag {2}
+$$
+
+where the red symbols indicate real-time calculations.
+
+Challenges. The key challenge lies in how to edit the succeeding KV cache  $\pmb{K}_{[j + 1:n]}$  and  $\pmb{V}_{[j + 1:n]}$ . Clearly, the modification of  $[a_1, a_2, \ldots, a_m]$  impacts the subsequent content in two ways: semantically and structurally. The semantic impact refers to the changes in the understanding of the subsequent text caused by the edited content. This can be a problem in natural language applications, such as dialog systems, where modifications to earlier conversations can significantly influence the
+
+generation of current responses. The other impact is structural, primarily concerning the temporal confusion between the pre-edit and post-edit sequences when  $j - i \neq m$ . This issue arises with common editing actions in code, such as additions and deletions (corresponding to  $j - i = 0$  or  $m = 0$ ). To be more concrete, imagine the original sequence has 5 tokens. If we add three tokens between the second and third position, it will occupy the positional index [3,4,5]. We calculate  $K_{[3:5]}^{\mathrm{edit}}$  and  $V_{[3:5]}^{\mathrm{edit}}$  in equation (1) and (2). However, in the original  $K_{[3:5]}$  and  $V_{[3:5]}$ , the positional index [3,4,5] is also occupied. If we integrate them together and take no actions during the next token prediction, the model will calculate similarity with multiple keys in the KV cache with the same index [3,4,5], causing confusion and potential prediction errors. Empirically, in code tasks, we find that the semantic impact is relatively small. Addressing temporal confusion for light edits alone can already lead to good performance (see the experiments in Section 4 for more details).
+
+Our approach. To mitigate the temporal confusion during real-time editing, we propose a simple yet effective solution: Positional Integrity Encoding (PIE), which ensures that positional information remains correctly ordered after editing without the need to re-encode the KV cache for subsequent tokens. PIE builds upon the rotary positional encoding (RoPE) (Su et al., 2021), which is the most widely used positional encoding in LLMs. Without loss of generality, given a query vector  $\boldsymbol{x}_i$  at position  $i$  and a key vector  $\boldsymbol{x}_j$  at position  $j$ , RoPE calculates the dot-product similarity using
+
+$$
+z _ {i j} = \boldsymbol {x} _ {i} ^ {T} W _ {q} ^ {T} \boldsymbol {R} _ {j - i} W _ {k} \boldsymbol {x} _ {j} \tag {3}
+$$
+
+where  $R_{j - i}$  is the rotary matrix parameterized by the relative distance  $j - i$ , and  $W_{q}$  and  $W_{k}$  are learnable projection matrices. By definition,  $R_{j - i}$  can be expressed by the multiplication of two rotary matrices:
+
+$$
+\boldsymbol {R} _ {j - i} = \boldsymbol {R} _ {i} ^ {T} \boldsymbol {R} _ {j} \tag {4}
+$$
+
+For practical implementation, during inference, we compute  $R_{\Theta,i}W_kx_i$  as the key on the fly and store it in the cache, and when a query arrives at a new position, we rotate the query using its corresponding rotary matrix and calculate its similarity with all the keys in the cache to obtain the attention scores.
+
+It can be easily seen that the positional information in the KV cache is encoded within the rotary matrix. When an edit occurs, the rotary matrix associated with the keys must be adjusted to reflect their post-edit locations. Leveraging the formulation of RoPE-based attention calculation, this challenge can be addressed by first removing the rotary matrices in  $\pmb{K}$  that introduce temporal confusion and then reapplying the correct rotary matrix. In detail, assume we would like to update the key vector  $\pmb{k}_{j'}^{l}$  for the original position  $j' \in [j + 1, n]$ , where  $l \in [1, L]$  is the layer index. We can simply edit the key vector by using
+
+$$
+\boldsymbol {k} _ {j ^ {\prime}} ^ {\text {e d i t}, l} = \boldsymbol {R} _ {i + m + j ^ {\prime} - j} \boldsymbol {R} _ {j ^ {\prime}} ^ {- 1} \boldsymbol {k} _ {j ^ {\prime}} ^ {l} \tag {5}
+$$
+
+where  $R_{j'}^{-1}$ , the inverse rotary matrix at position  $j'$ , is used to remove the incorrect positional information, and  $R_{i + m + j' - j}$  is used to encode the correct position  $i + m + j' - j$  in  $s^{\mathrm{edit}}$ . It can easily seen that the computation can be further simplified as
+
+$$
+\boldsymbol {k} _ {j ^ {\prime}} ^ {\text {e d i t}, l} = \boldsymbol {R} _ {i + m + j ^ {\prime} - j} \boldsymbol {R} _ {j ^ {\prime}} ^ {- 1} \boldsymbol {k} _ {j ^ {\prime}} ^ {l} = \boldsymbol {R} _ {i + m + j ^ {\prime} - j} \boldsymbol {R} _ {- j ^ {\prime}} \boldsymbol {k} _ {j ^ {\prime}} ^ {l} = \boldsymbol {R} _ {i + m - j} \boldsymbol {k} _ {j ^ {\prime}} ^ {l} \tag {6}
+$$
+
+Hence, the full editing process for  $K_{[j + 1:n]}$  is as follows:
+
+$$
+\boldsymbol {K} _ {[ j + 1: n ]} ^ {\mathrm {e d i t}} = \left[ K _ {j + 1} ^ {\mathrm {e d i t}}, \dots , K _ {n} ^ {\mathrm {e d i t}} \right] \tag {7}
+$$
+
+where each  $K_{j'}^{\mathrm{edit}} = \{\pmb{k}_{j'}^{\mathrm{edit},1},\dots ,\pmb{k}_{j'}^{\mathrm{edit},l},\dots ,\pmb{k}_{j'}^{\mathrm{edit},L}\} ,j'\in [j + 1,n],l\in [1,L]$
+
+each  $\pmb{k}_{j'}^{\mathrm{edit},l} = \pmb{R}_{i + m - j}\pmb{k}_{j'}^{l}$
+
+Unlike the full recomputation approach, the above calculation only requires a single round of matrix multiplication to directly modify the pre-computed KV cache, where the computational overhead can be considered negligible. By utilizing these transformations, we finally construct the edited KV cache as:
+
+$$
+\boldsymbol {K} ^ {\text {e d i t}} = \operatorname {C o n c a t} \left(\boldsymbol {K} _ {[ 1: i ]}, \boldsymbol {K} _ {[ i + 1: i + m ]} ^ {\text {e d i t}}, \boldsymbol {K} _ {[ j + 1: n ]} ^ {\text {e d i t}}\right), \tag {8}
+$$
+
+$$
+\boldsymbol {V} ^ {\text {e d i t}} = \operatorname {C o n c a t} \left(\boldsymbol {V} _ {[ 1: i ]}, \boldsymbol {V} _ {[ i + 1: i + m ]} ^ {\text {e d i t}}, \boldsymbol {V} _ {[ j + 1: m ]}\right), \tag {9}
+$$
+
+Table 1: Statistics of RepoBench-C-8k (Liu et al., 2024a) test set.  
+
+<table><tr><td>Language</td><td>XF-F</td><td>XF-R</td><td>IF</td><td>Average Number of Tokens</td></tr><tr><td>Python</td><td>18,000</td><td>7,500</td><td>10,500</td><td>3,967</td></tr><tr><td>Java</td><td>18,000</td><td>7,500</td><td>10,500</td><td>4,179</td></tr></table>
+
+where the red symbols indicate real-time calculations. The LLM then makes predictions based on  $p(x_{n+1} | \mathbf{K}^{\mathrm{edit}}, \mathbf{V}^{\mathrm{edit}}, x_n; \theta_{\mathrm{LLM}})$ . It is worth noting that PIE is compatible with KV cache eviction methods (Xiao et al., 2024; Zhang et al., 2024; Liu et al., 2024b). These KV cache eviction methods focus on reducing the memory usage of the KV cache during inference. PIE is designed to obtain the KV cache of the edited context with minimal overhead. By integrating PIE with KV cache eviction methods, it is possible to maintain efficient memory management while ensuring the integrity of the positional information in the real-time edit setting.
+
+# 4 EXPERIMENTS
+
+In this section, we empirically study the effectiveness of our proposed method. In particular, we aim at answering the following questions through experiments:
+
+- Question 1: Can our Positional Integrity Encoding maintain the prediction accuracy of full re-computation in code editing scenarios?  
+- Question 2: How much efficiency improvement can be achieved by using our Positional Integrity Encoding compared to existing approaches?  
+- Question 3: How large is the gap between our Positional Integrity Encoding and full re-computation in terms of LLM's predictions & representations?
+
+We will answer each question with carefully designed experiments in the following sub-sections. We also conduct additional experiments in Appendix B.
+
+# 4.1 EXPERIMENTAL SETUP
+
+Tasks. Our experiments are conducted on RepoBench-C-8k (Liu et al., 2024a). This benchmark focuses on the prediction of the next line of code, given a set of in-file context (including import statements and preceding lines before the target line), and cross-file context (comprising snippets from other files parsed by import statements). The detailed statistics of RepoBench-C-8k is shown in Table 1. To effectively evaluate next-line prediction performance of code LLMs, we follow Liu et al. (2024a) to use three task settings: (1) Cross-File-First (XF-F): mask the first appearance of a cross-file line within a file; (2) Cross-File-Random (XF-R): mask a random and non-first occurrence of a cross-file line; (3) In-File (IF): mask an in-file line that does not involve any cross-file modules. Moreover, we carefully design three real-world scenarios covering code insertion, code deletion, and code edition to comprehensively examine our approach. See Appendix A.1 for more detailed descriptions of tasks construction.
+
+Settings. In our experiments, we employ DeepSeek-Coder (Guo et al., 2024), a code LLM that achieves strong performance in handling repository-level code completion tasks (We also conduct experiments on CodeLlama (Roziere et al., 2023) in Appendix A.2). We use Transformers (Wolf et al., 2020) as our codebase. We benchmark our method on models of different sizes covering 1.3B, 6.7B, and 33B. During inference, the greedy decoding strategy is used to deterministically generate 64 tokens. For 1.3B and 6.7B models, all the experiments are conducted on a single NVIDIA A100 GPU. For 33B models, the time for encoding the context is conducted on two NVIDIA A100 GPUs and the full generation process is conducted on eight NVIDIA A100 GPUs. The first non-comment line in the output is truncated and used as the prediction. The batch size is set to 1. All experiments are repeated three times with different seeds and the averaged scores are reported.
+
+Evaluation. For comparison with our Positional Integrity Encoding, we choose two standard approaches as baselines: (1) Full-recomputation: re-compute the KV cache for all edited tokens
+
+Table 2: Performance comparisons of insertion experiments. In this task, for each next-line prediction target, we insert several lines of code into its context randomly to simulate real-world scenarios. EM and ES denote the Exact Match and Edit Similarity score respectively. All results demonstrate that our Positional Integrity Encoding approach brings substantial speed-ups without performance drops.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Model</td><td rowspan="2">Method</td><td colspan="3">XF-F</td><td colspan="3">XF-R</td><td colspan="3">IF</td></tr><tr><td>EM</td><td>ES</td><td>Time</td><td>EM</td><td>ES</td><td>Time</td><td>EM</td><td>ES</td><td>Time</td></tr><tr><td rowspan="3">Python</td><td>1.3B</td><td>Full-recomputation</td><td>22.42</td><td>65.26</td><td>192ms</td><td>35.41</td><td>72.96</td><td>193ms</td><td>28.78</td><td>69.22</td><td>193ms</td></tr><tr><td>1.3B</td><td>Conflict Fast Encoding</td><td>7.32</td><td>43.73</td><td>23ms</td><td>10.61</td><td>47.18</td><td>22ms</td><td>8.91</td><td>45.25</td><td>22ms</td></tr><tr><td>1.3B</td><td>PIE</td><td>22.3</td><td>65.2</td><td>29ms</td><td>35.33</td><td>72.88</td><td>28ms</td><td>28.69</td><td>69.13</td><td>29ms</td></tr><tr><td rowspan="3">Python</td><td>6.7B</td><td>Full-recomputation</td><td>28.95</td><td>70.11</td><td>561ms</td><td>40.89</td><td>76.19</td><td>564ms</td><td>35.26</td><td>72.73</td><td>562ms</td></tr><tr><td>6.7B</td><td>Conflict Fast Encoding</td><td>5.35</td><td>33.32</td><td>34ms</td><td>6.52</td><td>35.25</td><td>34ms</td><td>6.09</td><td>38.76</td><td>34ms</td></tr><tr><td>6.7B</td><td>PIE</td><td>28.83</td><td>70.01</td><td>50ms</td><td>40.77</td><td>76.14</td><td>50ms</td><td>35.2</td><td>72.72</td><td>50ms</td></tr><tr><td rowspan="3">Python</td><td>33B</td><td>Full-recomputation</td><td>35.75</td><td>73.46</td><td>2194ms</td><td>46.0</td><td>78.9</td><td>2199ms</td><td>39.75</td><td>75.12</td><td>2194ms</td></tr><tr><td>33B</td><td>Conflict Fast Encoding</td><td>3.96</td><td>30.13</td><td>126ms</td><td>5.41</td><td>32.32</td><td>121ms</td><td>3.92</td><td>35.56</td><td>127ms</td></tr><tr><td>33B</td><td>PIE</td><td>35.77</td><td>73.45</td><td>134ms</td><td>45.74</td><td>78.85</td><td>140ms</td><td>39.74</td><td>75.1</td><td>141ms</td></tr><tr><td rowspan="3">Java</td><td>1.3B</td><td>Full-recomputation</td><td>26.21</td><td>70.89</td><td>200ms</td><td>36.77</td><td>76.31</td><td>200ms</td><td>45.89</td><td>78.04</td><td>198ms</td></tr><tr><td>1.3B</td><td>Conflict Fast Encoding</td><td>0.29</td><td>3.12</td><td>22ms</td><td>0.57</td><td>3.19</td><td>23ms</td><td>0.7</td><td>2.63</td><td>23ms</td></tr><tr><td>1.3B</td><td>PIE</td><td>26.13</td><td>70.82</td><td>30ms</td><td>36.67</td><td>76.25</td><td>30ms</td><td>45.92</td><td>77.99</td><td>29ms</td></tr><tr><td rowspan="3">Java</td><td>6.7B</td><td>Full-recomputation</td><td>32.51</td><td>75.56</td><td>578ms</td><td>41.97</td><td>79.41</td><td>578ms</td><td>50.86</td><td>80.53</td><td>578ms</td></tr><tr><td>6.7B</td><td>Conflict Fast Encoding</td><td>0.47</td><td>2.77</td><td>34ms</td><td>0.76</td><td>2.78</td><td>35ms</td><td>0.67</td><td>2.48</td><td>33ms</td></tr><tr><td>6.7B</td><td>PIE</td><td>32.21</td><td>75.47</td><td>50ms</td><td>41.96</td><td>79.32</td><td>49ms</td><td>50.85</td><td>80.43</td><td>48ms</td></tr><tr><td rowspan="3">Java</td><td>33B</td><td>Full-recomputation</td><td>35.05</td><td>76.93</td><td>2269ms</td><td>44.95</td><td>80.87</td><td>2281ms</td><td>53.23</td><td>81.76</td><td>2270ms</td></tr><tr><td>33B</td><td>Conflict Fast Encoding</td><td>0.38</td><td>2.51</td><td>120ms</td><td>0.59</td><td>2.40</td><td>122ms</td><td>0.68</td><td>2.09</td><td>122ms</td></tr><tr><td>33B</td><td>PIE</td><td>34.78</td><td>76.78</td><td>138ms</td><td>45.01</td><td>80.95</td><td>133ms</td><td>53.16</td><td>81.66</td><td>139ms</td></tr></table>
+
+and subsequent tokens; (2) Conflict Fast Encoding: re-compute the KV cache for the edited tokens while keeping the rest of the cache intact (i.e., using equation (1,2). Following Lu et al. (2021), we use Exact Match (EM) and Edit Similarity (ES) (Svyatkovskiy et al., 2020) to evaluate the accuracy of the predicted code lines on code completion tasks. We also report the time required to encode the edited context for efficiency evaluation.
+
+# 4.2 MAIN RESULTS
+
+Positional Integrity Encoding perfectly preserves the full re-computation performance. Results of different code editing settings are presented in Table 2, 3 and 4 respectively. It can be easily seen that all metrics indicate that our Positional Integrity Encoding can perfectly maintain the prediction accuracy of full re-computation in different scenarios. This indicates that PIE effectively addresses temporal confusion, and the semantic impact of minor edits is relatively negligible. For example, in the most challenging XF-F setting requiring to handle long-range cross-file context, the maximum relative difference between our PIE and Full-recomputation across different model sizes and code languages is  $0.3\% / 0.15\%$ ,  $0.66\% / 0.79\%$ ,  $1.33\% / 2.24\%$  for code insertion, deletion, and edition in terms of EM/ES respectively, which is rather negligible. This thorough examination serves as a strong support of the reliability of our PIE approach in real-world code editing scenarios.
+
+Positional Integrity Encoding significantly reduces computational overhead. Moreover, we further benchmark the computational costs brought by different approaches. From all results in the above tables, it can be easily seen that our Positional Integrity Encoding can achieve substantial speed-up compared to full re-computation while preserving its performance simultaneously. In particular, for the code edition experiment that requires both insertion and deletion, the averaged reductions of computational overhead induced by our PIE are  $87.9\% / 88.2\%$ ,  $92.4\% / 92.8\%$ ,  $94.8\% / 95.2\%$  for 1.3B, 6.7B, and 33B models on Python/Java languages respectively. Furthermore, compared to Conflict Fast Encoding which induces minimal costs but largely hurts performance, our PIE only brings negligible overhead, showing its good accuracy-efficiency balance.
+
+In summary, our main results comprehensively demonstrate the superiority of our Positional Integration Encoding for code LLMs towards real-world code editing scenarios, which perfectly preserves the prediction accuracy and significantly addresses the crucial gap in the efficient deployment of LLMs in real-time dynamic scenarios.
+
+Table 3: Performance comparisons of deletion experiments. In this task, for each next-line prediction target, we delete several lines of code of its context randomly to simulate real-world scenarios. EM and ES denotes the Exact Match and Edit Similarity score respectively. All results demonstrate that our Positional Integrity Encoding approach brings substantial speed-ups without performance drops.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Model</td><td rowspan="2">Method</td><td colspan="3">XF-F</td><td colspan="3">XF-R</td><td colspan="3">IF</td></tr><tr><td>EM</td><td>ES</td><td>Time</td><td>EM</td><td>ES</td><td>Time</td><td>EM</td><td>ES</td><td>Time</td></tr><tr><td rowspan="3">Python</td><td>1.3B</td><td>Full-recomputation</td><td>22.42</td><td>65.26</td><td>192ms</td><td>35.41</td><td>72.96</td><td>193ms</td><td>28.78</td><td>69.22</td><td>193ms</td></tr><tr><td>1.3B</td><td>Conflict Fast Encoding</td><td>0.41</td><td>39.31</td><td>22ms</td><td>0.59</td><td>42.02</td><td>24ms</td><td>1.05</td><td>42.85</td><td>22ms</td></tr><tr><td>1.3B</td><td>PIE</td><td>22.31</td><td>65.19</td><td>29ms</td><td>35.25</td><td>72.81</td><td>27ms</td><td>28.8</td><td>69.16</td><td>27ms</td></tr><tr><td rowspan="3">Python</td><td>6.7B</td><td>Full-recomputation</td><td>28.95</td><td>70.11</td><td>561ms</td><td>40.89</td><td>76.19</td><td>564ms</td><td>35.26</td><td>72.73</td><td>562ms</td></tr><tr><td>6.7B</td><td>Conflict Fast Encoding</td><td>0.51</td><td>40.24</td><td>30ms</td><td>0.77</td><td>42.77</td><td>31ms</td><td>1.42</td><td>43.93</td><td>30ms</td></tr><tr><td>6.7B</td><td>PIE</td><td>28.86</td><td>70.01</td><td>43ms</td><td>40.77</td><td>76.15</td><td>42ms</td><td>35.09</td><td>72.67</td><td>42ms</td></tr><tr><td rowspan="3">Python</td><td>33B</td><td>Full-recomputation</td><td>35.75</td><td>73.46</td><td>2194ms</td><td>46.0</td><td>78.9</td><td>2199ms</td><td>39.75</td><td>75.12</td><td>2194ms</td></tr><tr><td>33B</td><td>Conflict Fast Encoding</td><td>1.42</td><td>43.93</td><td>105ms</td><td>1.55</td><td>44.22</td><td>108ms</td><td>2.4</td><td>45.04</td><td>108ms</td></tr><tr><td>33B</td><td>PIE</td><td>35.09</td><td>72.67</td><td>128ms</td><td>45.21</td><td>78.18</td><td>118ms</td><td>39.69</td><td>75.05</td><td>119ms</td></tr><tr><td rowspan="3">Java</td><td>1.3B</td><td>Full-recomputation</td><td>26.21</td><td>70.89</td><td>200ms</td><td>36.77</td><td>76.31</td><td>200ms</td><td>45.89</td><td>78.04</td><td>198ms</td></tr><tr><td>1.3B</td><td>Conflict Fast Encoding</td><td>0.33</td><td>33.48</td><td>22ms</td><td>0.55</td><td>36.24</td><td>22ms</td><td>1.35</td><td>38.65</td><td>22ms</td></tr><tr><td>1.3B</td><td>PIE</td><td>26.05</td><td>70.77</td><td>28ms</td><td>36.83</td><td>76.34</td><td>27ms</td><td>45.74</td><td>77.91</td><td>27ms</td></tr><tr><td rowspan="3">Java</td><td>6.7B</td><td>Full-recomputation</td><td>32.51</td><td>75.56</td><td>578ms</td><td>41.97</td><td>79.41</td><td>578ms</td><td>50.86</td><td>80.53</td><td>578ms</td></tr><tr><td>6.7B</td><td>Conflict Fast Encoding</td><td>0.49</td><td>33.6</td><td>29ms</td><td>0.8</td><td>36.17</td><td>29ms</td><td>1.84</td><td>38.82</td><td>29ms</td></tr><tr><td>6.7B</td><td>PIE</td><td>32.57</td><td>75.56</td><td>42ms</td><td>41.83</td><td>79.39</td><td>43ms</td><td>50.88</td><td>80.52</td><td>42ms</td></tr><tr><td rowspan="3">Java</td><td>33B</td><td>Full-recomputation</td><td>35.05</td><td>76.93</td><td>2269ms</td><td>44.95</td><td>80.87</td><td>2281ms</td><td>53.23</td><td>81.76</td><td>2270ms</td></tr><tr><td>33B</td><td>Conflict Fast Encoding</td><td>0.91</td><td>35.06</td><td>109ms</td><td>1.01</td><td>37.97</td><td>106ms</td><td>2.31</td><td>40.15</td><td>105ms</td></tr><tr><td>33B</td><td>PIE</td><td>34.82</td><td>76.76</td><td>117ms</td><td>44.93</td><td>80.86</td><td>120ms</td><td>53.19</td><td>81.71</td><td>119ms</td></tr></table>
+
+Table 4: Performance comparisons of edition experiments. In this task, for each next-line prediction target, we delete several lines of code of its context and simultaneously insert other lines of code randomly to simulate real-world scenarios. EM and ES denote the Exact Match and Edit Similarity score respectively. All results demonstrate that our Positional Integrity Encoding approach brings substantial speed-ups without performance drops.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Model</td><td rowspan="2">Method</td><td colspan="3">XF-F</td><td colspan="3">XF-R</td><td colspan="3">IF</td></tr><tr><td>EM</td><td>ES</td><td>Time</td><td>EM</td><td>ES</td><td>Time</td><td>EM</td><td>ES</td><td>Time</td></tr><tr><td rowspan="3">Python</td><td>1.3B</td><td>Full-recomputation</td><td>22.42</td><td>65.26</td><td>242ms</td><td>35.41</td><td>72.96</td><td>242ms</td><td>28.78</td><td>69.22</td><td>244ms</td></tr><tr><td>1.3B</td><td>Conflict Fast Encoding</td><td>8.80</td><td>50.08</td><td>23ms</td><td>13.83</td><td>54.01</td><td>22ms</td><td>11.29</td><td>51.99</td><td>22ms</td></tr><tr><td>1.3B</td><td>PIE</td><td>22.04</td><td>64.59</td><td>30ms</td><td>34.49</td><td>72.02</td><td>29ms</td><td>28.15</td><td>68.32</td><td>29ms</td></tr><tr><td rowspan="3">Python</td><td>6.7B</td><td>Full-recomputation</td><td>28.95</td><td>70.11</td><td>705ms</td><td>40.89</td><td>76.19</td><td>706ms</td><td>35.26</td><td>72.73</td><td>713ms</td></tr><tr><td>6.7B</td><td>Conflict Fast Encoding</td><td>11.26</td><td>51.63</td><td>34ms</td><td>12.57</td><td>53.27</td><td>34ms</td><td>12.75</td><td>51.45</td><td>34ms</td></tr><tr><td>6.7B</td><td>PIE</td><td>28.07</td><td>69.00</td><td>54ms</td><td>39.89</td><td>75.10</td><td>54ms</td><td>34.05</td><td>71.64</td><td>54ms</td></tr><tr><td rowspan="3">Python</td><td>33B</td><td>Full-recomputation</td><td>35.75</td><td>73.46</td><td>2766ms</td><td>46.00</td><td>78.90</td><td>2759ms</td><td>39.75</td><td>75.12</td><td>2787ms</td></tr><tr><td>33B</td><td>Conflict Fast Encoding</td><td>14.33</td><td>53.73</td><td>126ms</td><td>15.12</td><td>54.59</td><td>121ms</td><td>13.88</td><td>51.94</td><td>127ms</td></tr><tr><td>33B</td><td>PIE</td><td>34.62</td><td>72.47</td><td>146ms</td><td>44.59</td><td>77.69</td><td>142ms</td><td>38.83</td><td>74.08</td><td>141ms</td></tr><tr><td rowspan="3">Java</td><td>1.3B</td><td>Full-recomputation</td><td>26.21</td><td>70.89</td><td>251ms</td><td>36.77</td><td>76.31</td><td>253ms</td><td>45.89</td><td>78.04</td><td>249ms</td></tr><tr><td>1.3B</td><td>Conflict Fast Encoding</td><td>5.87</td><td>31.46</td><td>22ms</td><td>8.01</td><td>32.74</td><td>23ms</td><td>10.20</td><td>34.34</td><td>23ms</td></tr><tr><td>1.3B</td><td>PIE</td><td>25.29</td><td>68.93</td><td>30ms</td><td>35.59</td><td>74.17</td><td>30ms</td><td>44.51</td><td>76.06</td><td>29ms</td></tr><tr><td rowspan="3">Java</td><td>6.7B</td><td>Full-recomputation</td><td>32.51</td><td>75.56</td><td>733ms</td><td>41.97</td><td>79.41</td><td>736ms</td><td>50.86</td><td>80.53</td><td>728ms</td></tr><tr><td>6.7B</td><td>Conflict Fast Encoding</td><td>9.28</td><td>39.32</td><td>34ms</td><td>11.47</td><td>39.11</td><td>35ms</td><td>15.51</td><td>41.92</td><td>33ms</td></tr><tr><td>6.7B</td><td>PIE</td><td>31.32</td><td>73.83</td><td>52ms</td><td>40.89</td><td>77.65</td><td>53ms</td><td>49.44</td><td>78.80</td><td>53ms</td></tr><tr><td rowspan="3">Java</td><td>33B</td><td>Full-recomputation</td><td>35.05</td><td>76.93</td><td>2892ms</td><td>44.95</td><td>80.87</td><td>2894ms</td><td>53.23</td><td>81.76</td><td>2833ms</td></tr><tr><td>33B</td><td>Conflict Fast Encoding</td><td>8.02</td><td>32.93</td><td>120ms</td><td>9.67</td><td>33.29</td><td>122ms</td><td>12.10</td><td>34.70</td><td>122ms</td></tr><tr><td>33B</td><td>PIE</td><td>33.72</td><td>74.69</td><td>134ms</td><td>43.71</td><td>78.66</td><td>138ms</td><td>51.71</td><td>79.49</td><td>143ms</td></tr></table>
+
+# 4.3 MORE ANALYSIS
+
+In this subsection, we further present detailed analysis to investigate how large is the gap between our Positional Integrity Encoding and full re-computation in terms of context representations and predictions from LLMs, which provide additional insight of our approach.
+
+![](images/910c4c2bbb018b690309714358e7b8838aebfa424551e6794b7d4e5fefb7e69f.jpg)  
+Figure 3: Cosine similarity of key representations across model layers. The plots compare the cosine similarity between  $\mathbf{K}_{[j + 1:n]}$  and  $\mathbf{K}_{[j + 1:n]}^*$  (indicating temporal confusion of Conflict Fast Encoding) with the cosine similarity between  $\mathbf{K}_{[j + 1:n]}^{\mathrm{edit}}$  and  $\mathbf{K}_{[j + 1:n]}^*$  (showing the effectiveness of PIE).
+
+How large is the gap on context representations? In practical scenarios, real-time editing by users results in the modified sequence  $\mathbf{x}^{\mathrm{edit}}$ , requiring the KV cache to must be updated. In our analysis, we use the cosine similarity between context representations of full re-computation  $K_{[j + 1:n]}^{*}$  and (1) our Positional Integrity Encoding  $K_{[j + 1:n]}^{\mathrm{edit}}$ ; (2) Conflict Fast Encoding  $K_{[j + 1:n]}$ . We employ the DeepSeek-Coder 6.7B model on the Python subset of RepoBench. Averaged results are reported.
+
+In Figure 3, the cosine similarity between representations of full re-computation and our Positional Integraty Encoding is consistently around 1.0 across all layers. This high similarity demonstrates the effectiveness of PIE in preserving the contextual integrity of the key representations after editing, suggesting that PIE successfully mitigates the temporal confusion that typically arises when manipulating the KV cache. In contrast, the cosine similarity between representations of full re-computation and Conflict Fast Encoding is significantly lower, indicating the temporal confusion issue that hurts model performance a lot.
+
+![](images/294fe67817ecfa18f54b8a579f7ccd399ce036edda5119fea4b278ed3804a801.jpg)  
+Figure 4: KL divergence of the generated token distributions. The plots compare the KL divergence between the generated token distributions of PIE and Full-recomputation, and the KL divergence between the generated token distributions of Conflict Fast Encoding and Full-recomputation.
+
+How large is the gap on model predictions? Moreover, we further use Kullback-Leibler (KL) divergence as the metric to investigate the gap between model predictions of different approaches. Similarly, we employ the DeepSeek-Coder 6.7B model on the Python subset of RepoBench and report the averaged results. In Figure 4, the KL divergence between model predictions of full re-computation and our Positional Integrity Encoding remains consistently low, i.e., below 0.0002 across 64 tokens. However, the KL divergence between model predictions of full re-computation and Conflict Fast Encoding is substantially higher (2x larger). These findings again underscore the importance of maintaining positional integrity within the KV cache to ensure accurate generation results.
+
+# 5 CONCLUSION
+
+In this paper, we introduce Positional Integrity Encoding (PIE), a novel method designed to enhance the efficiency of large language models (LLMs) in the real-time editing setting. Our approach addresses the significant computational overhead associated with re-encoding contexts after small edits, a common scenario in interactive coding environments. Through extensive experiments,
+
+we demonstrated that PIE not only significantly reduces latency but also maintains high accuracy compared to the naive full re-computation method.  
+PIE represents a substantial step forward in the development of efficient LLMs, particularly in dynamic contexts where frequent edits are made. Future work could explore the integration of PIE with other optimization techniques and its application to a broader range of tasks beyond code generation. Our method paves the way for more responsive and resource-efficient AI assistants, enhancing their practicality and usability in various real-world scenarios.
+
+# ACKNOWLEDGEMENTS
+
+We thank all the anonymous reviewers for the very careful and detailed reviews as well as the valuable suggestions. Their help has further enhanced our work. Di He is supported by National Science Foundation of China (NSFC62376007).
+
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+
+# A EXPERIMENTAL DETAILS
+
+# A.1 EXPERIMENTAL SETUP
+
+Tasks Construction for Code Insertion. To simulate code insertion tasks, we start by randomly deleting five consecutive lines from each context. The resulting context, which lacks these five lines, is considered the original context. The complete context, which includes the previously deleted lines, is treated as the edited context. The tokens within the deleted lines are identified as the inserted tokens (around 64 tokens for Python and 51 tokens for Java). This setup allows us to evaluate the model's capability to accurately restore missing code segments, mimicking real-world scenarios where developers frequently insert blocks of code.
+
+Tasks Construction for Code Deletion. For code deletion tasks, we begin by randomly selecting a line within the context and then inserting five randomly sampled lines at this position. The context containing these additional lines is designated as the original context. The complete context, which excludes the inserted lines, is regarded as the edited context. The tokens in the inserted lines are treated as the deleted tokens (around 64 tokens for Python and 51 tokens for Java). This construction enables us to assess the model's performance in identifying and removing extraneous code, reflecting situations where developers need to clean up or refactor their codebase.
+
+Table 5: Performance comparisons of insertion experiments for CodeLlama. In this task, for each next-line prediction target, we insert several lines of code into its context randomly to simulate real-world scenarios. EM and ES denotes the Exact Match and Edit Similarity score respectively. All results demonstrate that our Positional Integrity Encoding approach brings substantial speed-ups without performance drops.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Model</td><td rowspan="2">Method</td><td colspan="3">XF-F</td><td colspan="3">XF-R</td><td colspan="3">IF</td></tr><tr><td>EM</td><td>ES</td><td>Time</td><td>EM</td><td>ES</td><td>Time</td><td>EM</td><td>ES</td><td>Time</td></tr><tr><td rowspan="3">Python</td><td>7B</td><td>Full-recomputation</td><td>25.72</td><td>66.49</td><td>561ms</td><td>38.87</td><td>73.81</td><td>564ms</td><td>33.55</td><td>70.26</td><td>562ms</td></tr><tr><td>7B</td><td>Conflict Fast Encoding</td><td>14.49</td><td>55.10</td><td>34ms</td><td>19.32</td><td>57.34</td><td>34ms</td><td>15.49</td><td>53.05</td><td>34ms</td></tr><tr><td>7B</td><td>PIE</td><td>25.42</td><td>66.40</td><td>50ms</td><td>38.76</td><td>73.83</td><td>50ms</td><td>33.30</td><td>70.11</td><td>50ms</td></tr><tr><td rowspan="3">Python</td><td>34B</td><td>Full-recomputation</td><td>31.04</td><td>69.48</td><td>2013ms</td><td>42.80</td><td>76.27</td><td>2029ms</td><td>37.69</td><td>72.36</td><td>1999ms</td></tr><tr><td>34B</td><td>Conflict Fast Encoding</td><td>11.56</td><td>50.23</td><td>113ms</td><td>16.06</td><td>52.59</td><td>110ms</td><td>11.58</td><td>48.36</td><td>112ms</td></tr><tr><td>34B</td><td>PIE</td><td>30.53</td><td>68.79</td><td>123ms</td><td>42.73</td><td>76.40</td><td>119ms</td><td>37.51</td><td>72.25</td><td>123ms</td></tr><tr><td rowspan="3">Java</td><td>7B</td><td>Full-recomputation</td><td>28.02</td><td>72.49</td><td>578ms</td><td>39.61</td><td>77.81</td><td>578ms</td><td>49.20</td><td>79.76</td><td>578ms</td></tr><tr><td>7B</td><td>Conflict Fast Encoding</td><td>10.45</td><td>37.69</td><td>34ms</td><td>14.28</td><td>38.45</td><td>35ms</td><td>16.95</td><td>38.15</td><td>33ms</td></tr><tr><td>7B</td><td>PIE</td><td>28.02</td><td>72.39</td><td>50ms</td><td>39.48</td><td>77.74</td><td>49ms</td><td>49.31</td><td>79.79</td><td>48ms</td></tr><tr><td rowspan="3">Java</td><td>34B</td><td>Full-recomputation</td><td>32.16</td><td>75.09</td><td>2179ms</td><td>43.45</td><td>79.89</td><td>2184ms</td><td>52.43</td><td>80.97</td><td>2197ms</td></tr><tr><td>34B</td><td>Conflict Fast Encoding</td><td>12.04</td><td>41.76</td><td>104ms</td><td>14.25</td><td>39.19</td><td>107ms</td><td>17.29</td><td>38.50</td><td>109ms</td></tr><tr><td>34B</td><td>PIE</td><td>31.45</td><td>74.64</td><td>119ms</td><td>43.54</td><td>79.80</td><td>118ms</td><td>52.42</td><td>81.00</td><td>115ms</td></tr></table>
+
+Tasks Construction for Multi-place Code Edition To comprehensively evaluate the model's performance in handling simultaneous code insertion and deletion, we construct a task scenario that integrates both operations. Initially, we randomly delete five consecutive lines from each context to simulate code insertion. The context without these lines is treated as the original context. The tokens in the deleted lines are identified as the inserted tokens.
+
+Simultaneously, we randomly select another line within the context and insert five randomly sampled lines at this position. The complete context, which includes all lines as they appear after both deletions and insertions, is regarded as the edited context. The tokens in the newly inserted lines are considered the deleted tokens. This dual operation setup allows us to evaluate the model's ability to handle complex, simultaneous edits, adding missing code segments while removing extraneous ones, reflecting the multifaceted nature of real-world coding environments where developers often perform multiple types of edits concurrently.
+
+# A.2 RESULTS ON CODELLAMA
+
+Results of different code editing settings on CodeLlama (Roziere et al., 2023) are presented in Table 5, 6 and 7 respectively. Similar to DeepSeek-Coder, CodeLlama with Positional Integrity Encoding demonstrates strong performance and fast speed across various editing scenarios. The metrics indicate that PIE effectively maintains prediction accuracy and addresses temporal confusion, ensuring the impact of minor edits is minimal.
+
+# B ADDITIONAL EXPERIMENTS
+
+# B.1 CODE GENERATION TASKS
+
+To further validate the effectiveness of PIE, we conduct experiments on code generation tasks, using HumanEval Chen et al. (2021) and its C++ version from HumanEval-X Zheng et al. (2023). We randomly choose three consecutive lines in the prompt as the user inserted code and report Pass@1 for different methods. It's important to note that the context (prompt) in HumanEval consists of only a few lines (with an average of 15 lines), making each edit semantically significant. The results in Table 8 demonstrate that PIE significantly outperforms the baseline methods, approaching the performance of full recomputation. Note that we also include a reuse baseline where the pre-edit context is reused without modification. Given the importance of these edits in such short contexts, updating the KV cache with PIE becomes essential for maintaining high prediction accuracy, whereas the performance of the reuse baseline drops substantially.
+
+Table 6: Performance comparisons of deletion experiments for CodeLlama. In this task, for each next-line prediction target, we delete several lines of code of its context randomly to simulate real-world scenarios. EM and ES denote the Exact Match and Edit Similarity score respectively. All results demonstrate that our Positional Integrity Encoding approach brings substantial speed-ups without performance drops.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Model</td><td rowspan="2">Method</td><td colspan="3">XF-F</td><td colspan="3">XF-R</td><td colspan="3">IF</td></tr><tr><td>EM</td><td>ES</td><td>Time</td><td>EM</td><td>ES</td><td>Time</td><td>EM</td><td>ES</td><td>Time</td></tr><tr><td rowspan="3">Python</td><td>7B</td><td>Full-recomputation</td><td>25.72</td><td>66.49</td><td>561ms</td><td>38.87</td><td>73.81</td><td>564ms</td><td>33.55</td><td>70.26</td><td>562ms</td></tr><tr><td>7B</td><td>Conflict Fast Encoding</td><td>10.14</td><td>52.80</td><td>30ms</td><td>15.24</td><td>57.88</td><td>31ms</td><td>13.04</td><td>54.53</td><td>30ms</td></tr><tr><td>7B</td><td>PIE</td><td>25.52</td><td>66.47</td><td>43ms</td><td>38.88</td><td>73.87</td><td>42ms</td><td>33.51</td><td>70.18</td><td>42ms</td></tr><tr><td rowspan="3">Python</td><td>34B</td><td>Full-recomputation</td><td>31.04</td><td>69.48</td><td>2013ms</td><td>42.80</td><td>76.27</td><td>2029ms</td><td>37.69</td><td>72.36</td><td>1999ms</td></tr><tr><td>34B</td><td>Conflict Fast Encoding</td><td>12.15</td><td>54.76</td><td>88ms</td><td>16.40</td><td>59.29</td><td>94ms</td><td>13.78</td><td>55.90</td><td>93ms</td></tr><tr><td>34B</td><td>PIE</td><td>31.01</td><td>69.44</td><td>100ms</td><td>42.76</td><td>76.13</td><td>100ms</td><td>37.60</td><td>72.38</td><td>102ms</td></tr><tr><td rowspan="3">Java</td><td>7B</td><td>Full-recomputation</td><td>28.02</td><td>72.49</td><td>578ms</td><td>39.61</td><td>77.81</td><td>578ms</td><td>49.20</td><td>79.76</td><td>578ms</td></tr><tr><td>7B</td><td>Conflict Fast Encoding</td><td>7.21</td><td>42.51</td><td>29ms</td><td>7.77</td><td>43.86</td><td>29ms</td><td>8.78</td><td>44.46</td><td>29ms</td></tr><tr><td>7B</td><td>PIE</td><td>28.01</td><td>72.45</td><td>42ms</td><td>39.51</td><td>77.78</td><td>42ms</td><td>49.30</td><td>79.80</td><td>42ms</td></tr><tr><td rowspan="3">Java</td><td>34B</td><td>Full-recomputation</td><td>32.16</td><td>75.09</td><td>2179ms</td><td>43.45</td><td>79.89</td><td>2184ms</td><td>52.43</td><td>80.97</td><td>2197ms</td></tr><tr><td>34B</td><td>Conflict Fast Encoding</td><td>8.76</td><td>45.44</td><td>92ms</td><td>8.67</td><td>45.57</td><td>92ms</td><td>10.11</td><td>46.64</td><td>95ms</td></tr><tr><td>34B</td><td>PIE</td><td>32.12</td><td>75.02</td><td>99ms</td><td>43.33</td><td>79.86</td><td>98ms</td><td>52.36</td><td>80.96</td><td>107ms</td></tr></table>
+
+Table 7: Performance comparisons of edition experiments for CodeLlama. In this task, for each next-line prediction target, we delete several lines of code of its context and simultaneously insert other lines of code randomly to simulate real-world scenarios. EM and ES denote the Exact Match and Edit Similarity score respectively. All results demonstrate that our Positional Integrity Encoding approach brings substantial speed-ups without performance drops.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Model</td><td rowspan="2">Method</td><td colspan="3">XF-F</td><td colspan="3">XF-R</td><td colspan="3">IF</td></tr><tr><td>EM</td><td>ES</td><td>Time</td><td>EM</td><td>ES</td><td>Time</td><td>EM</td><td>ES</td><td>Time</td></tr><tr><td rowspan="3">Python</td><td>7B</td><td>Full-recomputation</td><td>25.72</td><td>66.49</td><td>705ms</td><td>38.87</td><td>73.81</td><td>706ms</td><td>33.55</td><td>70.26</td><td>713ms</td></tr><tr><td>7B</td><td>Conflict Fast Encoding</td><td>19.97</td><td>61.46</td><td>34ms</td><td>29.79</td><td>67.31</td><td>34ms</td><td>24.70</td><td>63.28</td><td>34ms</td></tr><tr><td>7B</td><td>PIE</td><td>24.97</td><td>66.04</td><td>54ms</td><td>38.21</td><td>73.40</td><td>54ms</td><td>32.84</td><td>69.66</td><td>54ms</td></tr><tr><td rowspan="3">Python</td><td>34B</td><td>Full-recomputation</td><td>31.04</td><td>69.48</td><td>2574ms</td><td>42.80</td><td>76.27</td><td>2569ms</td><td>37.69</td><td>72.36</td><td>2581ms</td></tr><tr><td>34B</td><td>Conflict Fast Encoding</td><td>21.91</td><td>61.78</td><td>91ms</td><td>30.87</td><td>67.16</td><td>91ms</td><td>25.76</td><td>63.15</td><td>90ms</td></tr><tr><td>34B</td><td>PIE</td><td>29.92</td><td>68.7</td><td>126ms</td><td>41.74</td><td>75.37</td><td>126ms</td><td>37.01</td><td>71.76</td><td>127ms</td></tr><tr><td rowspan="3">Java</td><td>7B</td><td>Full-recomputation</td><td>28.02</td><td>72.49</td><td>733ms</td><td>39.61</td><td>77.81</td><td>736ms</td><td>49.20</td><td>79.76</td><td>728ms</td></tr><tr><td>7B</td><td>Conflict Fast Encoding</td><td>18.78</td><td>57.38</td><td>34ms</td><td>25.79</td><td>60.97</td><td>35ms</td><td>32.60</td><td>62.66</td><td>33ms</td></tr><tr><td>7B</td><td>PIE</td><td>27.43</td><td>71.60</td><td>52ms</td><td>38.89</td><td>76.96</td><td>53ms</td><td>48.60</td><td>78.94</td><td>53ms</td></tr><tr><td rowspan="3">Java</td><td>34B</td><td>Full-recomputation</td><td>32.16</td><td>75.09</td><td>2726ms</td><td>43.45</td><td>79.89</td><td>2807ms</td><td>52.43</td><td>80.97</td><td>2727ms</td></tr><tr><td>34B</td><td>Conflict Fast Encoding</td><td>21.25</td><td>59.70</td><td>104ms</td><td>27.83</td><td>61.91</td><td>107ms</td><td>32.98</td><td>62.03</td><td>109ms</td></tr><tr><td>34B</td><td>PIE</td><td>30.82</td><td>73.84</td><td>126ms</td><td>42.71</td><td>78.94</td><td>119ms</td><td>51.64</td><td>80.15</td><td>125ms</td></tr></table>
+
+# B.2 CONTEXTUALLY RELATED EDITS
+
+We design a contextually related insertion setting, where we use Edit Distance to identify lines in the context that are similar to the target line, simulating user insertions. We also increase the number of edited lines. We conduct experiments on the Cross-File-First setting of RepoBench-C-8k, keeping all other settings the same as Section 4. The results in Table 9 demonstrate that PIE is robust in handling contextually related edits and outperforms the baseline methods, approaching the performance of full recomputation. Note that we also include a reuse baseline where the pre-edit context is reused without modification.
+
+# B.3 MULTI-PLACE EDITS
+
+Users may edit a function definition followed by updates to all associated call sites. To simulate this scenario, we treat each line in the context as a potential site and use Edit Distance to identify multiple sites in the context that are similar to the target line, thereby simulating multi-place insertions. We conduct experiments in this contextually related multi-place editing setting on the Cross-File-First task of RepoBench-C-8k. The results in Table 10 demonstrate that PIE is robust in handling multi-place edits and outperforms the baseline methods.
+
+Table 8: Performance of DeepSeek-Coder 6.7B on HumanEval and HumanEval-X  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Pass@1</td></tr><tr><td>Python</td><td>C++</td></tr><tr><td>Full-recomputation</td><td>49.4</td><td>50.3</td></tr><tr><td>Conflict Fast Encoding</td><td>11.0</td><td>15.5</td></tr><tr><td>Reuse baseline</td><td>17.7</td><td>29.2</td></tr><tr><td>PIE</td><td>43.9</td><td>49.7</td></tr></table>
+
+Table 9: Performance of DeepSeek-Coder 6.7B on contextually-related insertion experiments  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Model</td><td rowspan="2">Method</td><td colspan="2">5 lines</td><td colspan="2">7 lines</td><td colspan="2">11 lines</td><td colspan="2">13 lines</td><td colspan="2">15 lines</td></tr><tr><td>EM</td><td>ES</td><td>EM</td><td>ES</td><td>EM</td><td>ES</td><td>EM</td><td>ES</td><td>EM</td><td>ES</td></tr><tr><td rowspan="4">Python</td><td>6.7B</td><td>Full-recomputation</td><td>28.95</td><td>70.11</td><td>28.95</td><td>70.11</td><td>28.95</td><td>70.11</td><td>28.95</td><td>70.11</td><td>28.95</td><td>70.11</td></tr><tr><td>6.7B</td><td>Conflict Fast Encoding</td><td>7.92</td><td>41.56</td><td>6.39</td><td>38.56</td><td>4.78</td><td>34.91</td><td>4.11</td><td>33.34</td><td>3.43</td><td>32.15</td></tr><tr><td>6.7B</td><td>Reuse baseline</td><td>25.58</td><td>68.23</td><td>25.29</td><td>68.03</td><td>25.23</td><td>67.95</td><td>25.13</td><td>67.83</td><td>24.71</td><td>67.74</td></tr><tr><td>6.7B</td><td>PIE</td><td>27.40</td><td>69.10</td><td>27.23</td><td>68.94</td><td>27.04</td><td>68.80</td><td>26.92</td><td>68.8</td><td>26.86</td><td>68.63</td></tr><tr><td rowspan="4">Java</td><td>6.7B</td><td>Full-recomputation</td><td>32.51</td><td>75.56</td><td>32.51</td><td>75.56</td><td>32.51</td><td>75.56</td><td>32.51</td><td>75.56</td><td>32.51</td><td>75.56</td></tr><tr><td>6.7B</td><td>Conflict Fast Encoding</td><td>2.81</td><td>15.46</td><td>1.89</td><td>12.16</td><td>1.10</td><td>7.89</td><td>0.87</td><td>6.52</td><td>0.74</td><td>5.57</td></tr><tr><td>6.7B</td><td>Reuse baseline</td><td>25.99</td><td>73.29</td><td>25.72</td><td>73.17</td><td>25.18</td><td>73.03</td><td>25.15</td><td>72.99</td><td>24.99</td><td>72.84</td></tr><tr><td>6.7B</td><td>PIE</td><td>30.04</td><td>74.46</td><td>30.00</td><td>74.44</td><td>30.13</td><td>74.47</td><td>29.87</td><td>74.38</td><td>29.76</td><td>74.29</td></tr></table>
+
+Table 10: Performance of DeepSeek-Coder 6.7B on contextually-related multi-place insertion experiments  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Model</td><td rowspan="2">Method</td><td colspan="2">5 lines</td><td colspan="2">7 lines</td><td colspan="2">11 lines</td><td colspan="2">13 lines</td><td colspan="2">15 lines</td></tr><tr><td>EM</td><td>ES</td><td>EM</td><td>ES</td><td>EM</td><td>ES</td><td>EM</td><td>ES</td><td>EM</td><td>ES</td></tr><tr><td rowspan="4">Python</td><td>6.7B</td><td>Full-recomputation</td><td>28.95</td><td>70.11</td><td>28.95</td><td>70.11</td><td>28.95</td><td>70.11</td><td>28.95</td><td>70.11</td><td>28.95</td><td>70.11</td></tr><tr><td>6.7B</td><td>Conflict Fast Encoding</td><td>7.43</td><td>39.95</td><td>5.83</td><td>37.00</td><td>4.06</td><td>33.02</td><td>3.48</td><td>31.71</td><td>2.97</td><td>30.59</td></tr><tr><td>6.7B</td><td>Reuse baseline</td><td>24.61</td><td>67.43</td><td>24.44</td><td>67.17</td><td>23.84</td><td>66.77</td><td>23.62</td><td>66.63</td><td>23.49</td><td>66.55</td></tr><tr><td>6.7B</td><td>PIE</td><td>26.97</td><td>68.85</td><td>26.77</td><td>68.67</td><td>26.33</td><td>68.34</td><td>26.15</td><td>68.222</td><td>25.9</td><td>68.23</td></tr><tr><td rowspan="4">Java</td><td>6.7B</td><td>Full-recomputation</td><td>32.51</td><td>75.56</td><td>32.51</td><td>75.56</td><td>32.51</td><td>75.56</td><td>32.51</td><td>75.56</td><td>32.51</td><td>75.56</td></tr><tr><td>6.7B</td><td>Conflict Fast Encoding</td><td>1.69</td><td>10.91</td><td>1.06</td><td>7.66</td><td>0.43</td><td>4.26</td><td>0.28</td><td>3.36</td><td>0.22</td><td>2.81</td></tr><tr><td>6.7B</td><td>Reuse baseline</td><td>24.46</td><td>72.35</td><td>23.99</td><td>72.05</td><td>23.11</td><td>71.69</td><td>22.97</td><td>71.56</td><td>22.75</td><td>71.38</td></tr><tr><td>6.7B</td><td>PIE</td><td>29.56</td><td>74.14</td><td>29.26</td><td>73.91</td><td>29.03</td><td>73.81</td><td>28.83</td><td>73.68</td><td>28.56</td><td>73.59</td></tr></table>
\ No newline at end of file
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+# LEVATTENTION: TIME, SPACE AND STREAMING EFFICIENT ALGORITHM FOR HEAVY ATTENTIONS
+
+Ravindran Kannan*
+
+Simons Institute, UC Berkeley
+
+kannan100@gmail.com
+
+Chiranjib Bhattacharya
+
+Indian Institute of Science chiru@iisc.ac.in
+
+Praneeth Kacham
+
+Google Research
+
+pkacham@google.com
+
+David P. Woodruff†
+
+Carnegie Mellon University
+
+dwoodruf@cs.cmu.edu
+
+# ABSTRACT
+
+A central problem related to transformers can be stated as follows: given two  $n \times d$  matrices  $Q$  and  $K$ , and a non-negative function  $f$ , define the matrix  $A$  as follows: (1) apply the function  $f$  to each entry of the  $n \times n$  matrix  $QK^T$ , and then (2) normalize each of the row sums of  $A$  to be equal to 1. The matrix  $A$  can be computed in  $O(n^2 d)$  time assuming  $f$  can be applied to a number in constant time, but the quadratic dependence on  $n$  is prohibitive in applications where it corresponds to long context lengths. For a large class of functions  $f$ , we show how to find all the "large attention scores", i.e., entries of  $A$  which are at least a positive value  $\varepsilon$ , in time with linear dependence on  $n$  (i.e.,  $n \cdot \mathrm{poly}(d / \varepsilon)$ ) for a positive parameter  $\varepsilon > 0$ . Our class of functions include all functions  $f$  of the form  $f(x) = |x|^p$ , as explored recently in transformer models. Using recently developed tools from randomized numerical linear algebra, we prove that for any  $K$ , there is a "universal set"  $U \subset [n]$  of size independent of  $n$ , such that for any  $Q$  and any row  $i$ , the large attention scores  $A_{i,j}$  in row  $i$  of  $A$  all have  $j \in U$ . We also find  $U$  in  $n \cdot \mathrm{poly}(d / \varepsilon)$  time. Notably, we (1) make no assumptions on the data, (2) our workspace does not grow with  $n$ , and (3) our algorithms can be computed in streaming and parallel settings. We call the attention mechanism that uses only the subset of keys in the universal set as LevAttention since our algorithm to identify the universal set  $U$  is based on leverage scores. We empirically show the benefits of our scheme for vision transformers, showing how to train new models that use our universal set while training as well, showing that our model is able to consistently select "important keys" during training. We also provide theoretical motivation by formulating a planted model in which our efficient algorithms provably identify relevant keys for each query.
+
+# 1 INTRODUCTION
+
+A transformer architecture is one of the most popular architectures for building foundation models, with applications to natural language processing, computer vision, and many other modalities and their combinations. It is well-known that exact computation of their attention layers naively requires quadratic (in the context length) time, which poses a huge problem for scalability. A large body of work has tried to improve the efficiency of computing attention layers under a variety of assumptions, including imposing sparsity constraints (Parmar et al., 2018; Child et al., 2019; Beltagy et al., 2020; Kitaev et al., 2020; Tay et al., 2020), kernel methods (Bello et al., 2021; Choromanski et al., 2021; Peng et al., 2021; Zheng et al., 2022), low rank assumptions (Wang et al., 2020; Xiong et al., 2021; Ma et al., 2021), and assumptions of bounded entries or conditions on column or row norms (Alman & Song, 2023; Han et al., 2024).
+
+In each attention layer, one receives as input an  $n \times d'$  input matrix  $X$ , which may be the embedding of the tokenization of the input, or an input from previous layers in the transformer. Here  $n$  is the context length and  $d'$  is the embedding dimension of each token, which is typically much smaller than  $n$ . From  $X$  we multiply by three  $d' \times d$  learned matrices  $W^Q$ ,  $W^K$ , and  $W^V$ , and define the query matrix  $Q = X \cdot W^K$ , the key matrix  $K = X \cdot W^K$ , and the value matrix  $V = X \cdot W^V$ . One then outputs the attention matrix, which is defined to be  $D^{-1} \cdot f(QK^T) \cdot V$ , where  $D$  and  $f(QK^T)$  are each  $n \times n$  matrices defined as follows. For a non-negative function  $f$ , we apply  $f$  entry-wise to the  $n \times n$  matrix  $QK^T$  to form  $f(QK^T)$ . Then we let  $D$  be a diagonal matrix with  $D_{i,i} = \sum_{j=1}^{n} f(\langle Q_i, K_j \rangle)$ , where  $Q_i$  is the  $i$ -th row of  $Q$  and  $K_j$  is the  $j$ -th row of  $K$ , respectively. Let  $A = D^{-1}f(QK^T)$ . Note that the entries of each row of  $A$  are non-negative and sum to 1, and each row of  $A \cdot V$  can be viewed as a non-negative combination of the rows of  $V$ , where the coefficients of the combination sum to 1.
+
+Instantiating the above framework with  $f(x) = e^{x / \sqrt{d}}$  corresponds to taking a softmax of each row of  $QK^T / \sqrt{d}$  and then multiplying by  $V$ . In this case, by appropriately scaling the hard instance in Alman & Song (2023), one can show that computing attention with high precision requires  $n^{2 - o(1)}$  time under a standard complexity-theoretic assumption. In an attempt to bypass this hardness, recent work has considered replacing  $f$  with other functions, such as  $f(x) = x^p$  for a positive even integer  $p$ . Indeed, both the PolySketchFormer (Kacham et al., 2024) as well as the work of Sarlós et al. (2023) consider using TensorSketch to speed up the computation of an approximate attention matrix for such functions  $f$ . Motivated by these works, we define the  $f$ -sensitivities  $\sigma_i^f$  of an  $n \times d$  matrix  $K$  for  $i = 1, 2, \ldots, n$  as follows:
+
+$$
+\sigma_ {i} ^ {f} (K) = \sup  _ {y \neq 0} \frac {f (\langle K _ {i} , y \rangle)}{\sum_ {j = 1} ^ {n} f (\langle K _ {j} , y \rangle)}.
+$$
+
+When  $f(x) = x^{2}$ , these are just the so-called leverage scores of the rows of matrix  $K$ , which are well-studied in randomized numerical linear algebra (see, e.g., Mahoney et al. (2011); Woodruff et al. (2014)). For general  $p$ , these are known as the  $\ell_p$ -sensitivities, which are also well-studied (see, e.g., Woodruff & Yasuda (2023); Padmanabhan et al. (2023)), and they can be bounded by the so-called  $\ell_p$ -Lewis weights of the matrix  $K$  (Cohen & Peng, 2015) (for background, see, e.g., Section 3.3 of Clarkson et al. (2019)). Many interesting properties of such scores are known. One such property is the following. Let  $\Psi^f = \sup_K \sum_{i=1}^n \sigma_i^f(K)$ . When  $f(x) = |x|^p$  for  $1 \leq p \leq 2$ , one has  $\Psi^f \leq d$ , and when  $f(x) = |x|^p$  for  $p \geq 2$ , one has  $\Psi^f \leq d^{p/2}$ . These bounds do not depend on the context length  $n$ .
+
+Critical to our work will be the observation in practice that the matrix  $A$  is often well-approximated by retaining only its large entries, i.e., preserving all entries above a certain threshold  $\varepsilon > 0$  and replacing the remaining entries with 0, or perhaps fitting a low rank approximation to the remaining entries (Gupta et al., 2021; Wang et al., 2022). The entries of  $A$  are called the attention scores or attention weights, and we will say a score is large if its value is at least  $\varepsilon$ .
+
+# 1.1 OUR CONTRIBUTIONS
+
+We outline our contributions below.
+
+Existence of a Universal Set. We prove that for a large class of functions  $f$ , for any key matrix  $K$ , there is a small "universal set"  $U \subset [n] = \{1, 2, \dots, n\}$  of size independent of  $n$ , such that for any query matrix  $Q$  and any  $i \in \{1, 2, \dots, n\}$ , the large attention scores  $A_{i,j}$  in row  $i$  of  $A$  all have  $j \in U$ . One of our results, which combines some ideas from Sections 2 and 3, is the following:
+
+Theorem 1.1. Let  $f$  be a non-negative function and let  $\Psi^f = \sup_K \sum_{i=1}^n \sigma_i^f(K)$ . There is a subset  $U \subset [n]$  of size  $\Psi^f / \varepsilon$  so that for any query matrix  $Q$  and  $i \in \{1,2,\dots,n\}$ , if  $A_{i,j} \geq \varepsilon$ , then  $j \in U$ .
+
+We note that for  $f(x) = x^2$ , the  $f$ -sensitivities are just the leverage scores of the matrix  $K$ , and it is known that  $\Psi^f = d$  (see, e.g., Mahoney et al. (2011); Woodruff et al. (2014)). In this case, by Theorem 1.1, we have  $|U| = d / \varepsilon$ . More generally, if  $f(x) = |x|^p$ , we have  $\Psi^f \leq d$  for  $1 \leq p \leq 2$  and  $\Psi^f \leq d^{p/2}$  for  $p > 2$  (see, e.g., Section 3.3 of Clarkson et al. (2019)), and so  $|U| \leq \max(d, d^{p/2}) / \varepsilon$ . Additional bounds are known for other functions  $f$ , e.g., from the work of
+
+Musco et al. (2022). There, for example, if  $f(x)$  is the Huber function  $f(x) = x^{2}$  for  $|x| \leq \tau$  and  $f(x) = |x|$  for larger  $|x|$ , then  $\Psi = O(d\log n)$ , or if  $f(x)$  is the Tukey function  $f(x) = x^{2}$  for  $|x| \leq \tau$ , and  $f(x) = \tau$  for larger  $|x|$ , then  $\Psi = O(d\log n)$ . Here  $\tau > 0$  is any specified threshold.
+
+We give an outline of the proof of Theorem 1.1 here. If  $A_{i,j}$  is a large attention score, then  $A_{i,j} = \frac{f(\langle Q_i, K_j \rangle)}{\sum_\ell f(\langle Q_i, K_\ell \rangle)} \geq \varepsilon$ . It follows that  $\sup_y \frac{f(\langle y, K_j \rangle)}{\sum_\ell f(\langle y, K_\ell \rangle)} > \varepsilon$  and so  $\sigma_i^f(K) \geq \varepsilon$ . If we define  $U$  to be the set of  $i$  for which  $\sigma_i^f(K) \geq \varepsilon$ , then  $i \in U$  and we have  $|U| \cdot \varepsilon \leq \Psi^f(K)$ , and so  $|U| \leq \Psi^f(K) / \varepsilon$ .
+
+We stress that our set  $U$  does not depend on any particular query or query matrix  $Q$ , i.e., for any possible future query  $q$ , any large attention scores it participates in necessarily involve keys in  $U$ .
+
+Fast Algorithm for Finding  $U$ . There are very efficient algorithms for computing the set  $U$  for many interesting functions  $f$ . For example, when  $f(x) = x^2$ , these are the large leverage scores and a simple way of computing them is by computing a QR-decomposition of the matrix  $K$  and finding all rows with squared norm at least  $\varepsilon$ . This takes  $O(nd^2)$  time and has the advantage of computing the leverage scores exactly, resulting in the smallest possible set  $U$ . There are also sketching techniques for more quickly finding the large leverage scores (Drineas et al., 2012; Clarkson & Woodruff, 2013) in time  $\mathrm{nnz}(K) + \mathrm{poly}(d / \varepsilon)$ , up to logarithmic factors, where  $\mathrm{nnz}$  denotes the number of non-zero entries of  $K$ . Similarly, for  $f(x) = |x|^p$  one can use the  $\mathrm{nnz}(K) + \mathrm{poly}(d / \varepsilon)$  time algorithm for finding  $\ell_p$ -Lewis weights in Cohen & Peng (2015), and the fact that a scaling of these weights bounds the  $f$ -sensitivities (see, e.g., Section 3.3 of Clarkson et al. (2019)).
+
+Time and Memory-Efficient Algorithm for Finding Large Attentions Given  $U$ . We do not make any assumptions on the data and our theorem holds for any query matrix  $Q$ , and thus given any potentially new query  $q$ , one only needs to search in the set  $U$  for the set of large attention scores involving  $q$ . Thus, instead of naively spending  $O(nd)$  time to walk through each possible key to find the large attention scores, one only needs to spend  $O(\Phi^fd / \varepsilon)$  time, assuming that  $f$  can be evaluated in constant time. Notice that our time per query does not grow with the context length  $n$ . Further, as we only store the keys in the set  $U$ , our workspace also does not grow with the context length.
+
+We also show how, for  $f(x) = x^{p}$  for even integers  $p$  it is possible to spend at most  $n \cdot \mathrm{poly}(d)$  time preprocessing  $K$  so that from  $U$  and  $K$ , given a query  $q$ , one can output the exact value of all large attention scores involving  $q$  in  $\mathrm{poly}(d / \varepsilon)$  time. Note that this is not as trivial as simply computing  $\langle q, K_i \rangle^p$  for each  $K_i \in U$ , since we also would like to compute the exact normalization factor in the attention matrix, and do not want to spend  $n$  time to do so. For  $p = 2$  this amounts to computing  $\| Kq \|_2^2$ , but by computing the SVD of  $K$  in a preprocessing step, this quantity can be computed in only  $O(d^2)$  time. For even integers  $p > 2$ , we can reduce to the case of  $p = 2$  by using Khatri-Rao products on the rows of  $K$ , as well as Khatri-Rao products of the query with itself. One can further speed up these computations by approximating the heavy attention values using sketching.
+
+Extension to Streaming Environments. It is also possible to compute the set  $U$  in a streaming environment. We illustrate the idea for  $f(x) = x^2$ , though using the tensor trick discussed above, it also applies to  $f(x) = x^p$  for any even integer  $p$ . The simplest algorithm is a two-pass algorithm over  $K$ , where we maintain the  $d \times d$  matrix  $K^T K$  by summing  $n$  outer products as we read each of the  $n$  keys. Then, in a second pass, the  $i$ -th leverage score is  $K_i^T (K^T K)^{-1} K_i$ , and we simply let  $U$  be the set of keys for which the leverage score is at least  $\varepsilon$ . The total memory is only  $O(d^2 / \varepsilon)$ .
+
+More interestingly, one can compute  $U$  in a single pass over  $K$  by again maintaining  $K^T K$  but also storing all keys whose online leverage score Cohen et al. (2016a) is at least  $\varepsilon$ . It is known that the online leverage scores are at least the leverage scores and sum to  $O(d\log \kappa^{OL})$ , where  $\kappa^{OL}$  is the online condition number, which can be bounded by  $n^{O(d)}$  assuming the entries of  $K$  have  $O(\log n)$  bits of precision. In this case, at the end of the stream, one simply evaluates  $K_j^T (K^T K)^{-1}K_j$  for each key  $K_j$  that was stored because it had a large online leverage score. The total memory is  $\mathrm{poly}(d / \varepsilon)$  and the algorithm is a single pass algorithm.
+
+Extension to Distributed Environments. It is possible to find  $U$  in a distributed environment, where the keys are distributed across multiple servers. In this case if server  $i$  holds an  $n_i \times d$  matrix  $K^i$ , so that  $K = [K^1; K^2; \ldots, K^s]$  if there are  $s$  servers, then the  $i$ -th server can communicate  $(K^i)^T K^i$  to the first server for each  $i$ , and the first server can add these up to compute  $K^T K$  and send  $K^T K$  to each server. Then the  $i$ -th server can send a set  $U^i$  of keys  $K_j$  that it holds for which
+
+$K_{j}^{T}(K^{T}K)^{-1}K_{j}\geq \varepsilon$ . Then  $U = \cup_{j}U_{j}$ . Although our discussion has focused on  $f(x) = x^{2}$ , similar ideas exist for  $f(x) = x^{p}$  for any  $p$ , as well as the functions studied in Musco et al. (2022).
+
+Planted Model. Finally, we formulate a planted model of keys and queries, where each query is a noisy linear combination of a small subset of keys. We show that in this model, we can find all keys relevant to a given query in sublinear time. We give deterministic conditions under which this is possible, and describe a natural stochastic setting where these conditions are realized.
+
+Experiments. We perform experiments on pretrained ViT models to empirically understand the structure of the attention matrices that arise for typical inputs to a model. Our experiments suggest that for many attention heads, a small subset of universal keys along with a set of local tokens capture a large fraction of attention weight for a large number of tokens. This motivates the identification problem that we study in this paper.
+
+We then evaluate the effectiveness of leverage score selection for the downstream task of image classification using the pretrained softmax model. In this experiment, at each attention head, we compute a subset of keys with the largest leverage scores and make the queries attend only to that subset of keys. We observe that while the accuracy of the model drops compared to the full attention model, the model still maintains non-trivial accuracy and that leverage score selection obtains better accuracy compared to e.g., row norm based selection, showing that leverage score based selection does compute a subset of important keys.
+
+We also trained multiple ViT models from scratch using the leverage score based attention mechanism and observe that the model quality improves significantly compared to doing inference on the softmax pretrained models using the leverage score mechanism. Across all the models, we observe that the model quality achieves  $>90\%$  accuracy of the full softmax attention while selecting the top 32 keys (out of 197 keys for L/16 and S/16 models and out of 785 keys for the L/8 model) using the leverage score mechanism at each attention head. On the negative side, we observe that models trained from scratch using squared row norm selection or even "random key selection", where we select a set of random key positions for each attention head and each query attends to only keys in those positions throughout the training/inference process, attains similar accuracies. However, unlike leverage score sampling, these latter methods do not have the "universality" property as discussed in the introduction for a relaxed definition of the attention mechanism. We leave it as an open question how to make effective use of leverage score information to train better models.
+
+Notation: For a matrix  $B$ , we use  $\mathrm{nnz}(B)$  to denote the number of non-zero entries of  $B$ . We let  $\omega$  be the exponent of matrix multiplication, so that multiplying two  $d \times d$  matrices takes  $d^{\omega}$  time.
+
+# 2 CONNECTION TO LEVERAGE SCORES
+
+We define the Generalized Attention Problem (GAP): let  $n$ , and  $D \geq d$  be positive integers. Consider two known maps:  $\psi, \phi \colon \mathbb{R}^d \to \mathbb{R}^D$ . We assume that  $\phi$  and  $\psi$  are computable in time  $O(D)$ . We are given matrices  $Q$  and  $K$ , where  $Q$  is  $n \times d$  and  $K$  is  $n \times d$ . GAP is the problem of computing the  $n \times n$  matrix  $A$  defined by  $A_{ij} = \frac{\langle\psi(Q_i), \phi(K_j)\rangle^2}{\sum_{\ell=1}^{n}\langle\psi(Q_i), \phi(K_\ell)\rangle^2}$ , where  $K_j$  denotes the  $j$ -th row of  $K$ .
+
+Remark 2.1. One special case is the SoftMax operation. In this case the value  $D$  is infinite, but Softmax can be approximated with finite  $D$ , see, e.g., Choromanski et al. (2021). Note that  $\phi = \psi$  in approximations to Softmax. Another set of interesting special cases occurs when  $\phi$  and  $\psi$  are polynomial maps, as in Sarlós et al. (2023); Kacham et al. (2024).
+
+GAP requires  $\Omega(n^2)$  time to write down the entries of  $A$ . We show how to improve this by finding only the large entries of  $A$ , i.e., those that are at least  $\varepsilon$ .
+
+Theorem 2.1. There is a deterministic algorithm  $\mathcal{A}$  that finds a subset  $U$  of rows of  $K$  (i.e.,  $U \subseteq [n]$ ) satisfying:  $|U| \leq \frac{D}{\varepsilon}$  and for all  $Q_{n \times d}$  and for all  $i \in \{1, 2, \ldots, n\} \colon \{j \colon A_{ij} \geq \varepsilon\} \subseteq U$ .
+
+We show that it suffices to prove this for the special case  $D = d$  and  $\phi, \psi$  are the identity maps.
+
+Lemma 2.2. Without loss of generality, we can assume in Theorem 1 that  $D = d$  and  $\phi, \psi$  are both the identity map (i.e.,  $\phi(x) = \psi(x) = x$ ).
+
+Proof. The proof follows from a kernel-type construction. From  $K$ , define an  $n \times D$  matrix  $\widehat{K}$  given by  $\widehat{K}_j = \phi(K_j)$ . For any  $n \times d$  matrix  $Q$ , define an  $n \times D$  matrix  $\widetilde{Q}$  by  $\widetilde{Q}_j = \psi(Q_j)$ . Further, define an  $n \times n$  matrix  $\widehat{A}$  by:  $\widehat{A}_{ij} = \frac{\langle\widehat{Q}_i,\widehat{K}_j\rangle^2}{\sum_{\ell=1}^{n}\langle\widehat{Q}_i,\widehat{K}_{\ell}\rangle^2}$ . It is easy to see that  $\widehat{A} = A$ . Assuming that Theorem 2.1 holds when  $D = d$  and  $\phi, \psi$  are both the identity map, we can find a  $\widehat{U}$  with  $\{j\colon \widehat{A}_{ij} \geq \varepsilon\} \subseteq \widehat{U}$ . Since  $\widehat{A} = A$ , we get that  $\{j\colon A_{ij} \geq \varepsilon\} \subseteq \widehat{U}$ , proving the lemma.
+
+Restricting to the case when  $D = d$  and  $\phi(x) = \psi(x) = x$ , we formally define the set  $LA$  for large attentions scores:  $LA(K, \varepsilon, v) = \{j \colon \langle K_j, v \rangle^2 \geq \varepsilon \sum_{\ell=1}^{n} \langle K_\ell, v \rangle^2\}$  for  $v \in \mathbb{R}^d$ . Let  $U(K, \varepsilon) = \bigcup_{v \in \mathbb{R}^d \setminus \{0\}} LA(K, \varepsilon, v)$ . The following is a simple characterization of  $U(K, \varepsilon)$ .
+
+Theorem 2.3. Let  $U(K, \varepsilon)$  be the set of rows of  $K$  with leverage score  $\geq \varepsilon$ , then  $|U(K, \varepsilon)| \leq d / \varepsilon$ .
+
+Proof. Observe that  $j \in U(k, \varepsilon)$  if and only if there is a non-zero vector  $v$  for which  $\frac{\langle K_j, v \rangle^2}{\sum_{\ell=1}^{m} \langle K_\ell, v \rangle^2} \geq \varepsilon$ , which means that the  $j$ -th  $f$ -sensitivity  $\sigma_j^f(K) \geq \varepsilon$ . Conversely, if  $\sigma_j^f(K) \geq \varepsilon$ , then there exists a non-zero vector  $v$  for which  $\frac{\langle K_j, v \rangle^2}{\sum_{\ell=1}^{m} \langle K_\ell, v \rangle^2} \geq \varepsilon$ , and so  $j \in U(K, \varepsilon)$ . It is a well-known fact that for  $f(x) = x^2$  that  $\sigma_j^f(K)$  is precisely the  $j$ -th leverage score of the matrix  $K$ , see, e.g., the solution to problem 2.2 in Woodruff (2021). Note also that  $j \in U(K, \varepsilon) \Rightarrow \tau(K, j) \geq \varepsilon \Rightarrow LS(K, j) \geq \varepsilon$ . As the sum of leverage scores is equal to  $d$  (see, e.g., Foster (1953)), we have  $|U(K, \varepsilon)| \leq \frac{d}{\varepsilon}$ .
+
+Thus, to compute  $LA$ , we simply need to compute the rows  $j$  of  $K$  with leverage score at least  $\varepsilon$ . Theorem 2.1 now follows from Lemma 2.2 and Theorem 2.3.
+
+Streaming Algorithms. The  $j$ -th leverage score of  $K$  is equal to  $K_{j}^{T}(K^{T}K)^{-1}K_{j}$  (see, e.g., Mahoney et al. (2011); Woodruff et al. (2014)), and so a simple 2-pass streaming algorithm using  $O(d^{2})$  memory would be to maintain  $K^T K$  as a sum of  $n$  outer products in a first pass over the keys of  $K$ , and then to compute  $K_{j}^{T}(K^{T}K)^{-1}K_{j}$  exactly in the second pass over keys  $K_{j}$ , and store those  $K_{j}$  for which this quantity is at least  $\varepsilon$ . This simple 2-pass streaming algorithm uses  $O(d^{2})$  words of memory and can be implemented in  $O(nd^{2} + d^{\omega})$  time.
+
+One can obtain a 1-pass algorithm to compute the rows  $j$  of  $K$  with leverage score at least  $\varepsilon$  with only slightly more memory. The idea is to store the set  $S$  of rows  $K_{j}$  with online leverage score Cohen et al. (2016a) at least  $\varepsilon$  in the first pass, and to also store  $K^T K$ . The  $j$ -th online leverage score is equal to  $K_j^T ((K^{j-1})^T (K^{j-1}))^{-1} K_j$ , where  $K^{j-1}$  denotes the prefix of the first  $j-1$  rows of  $K$ . The  $j$ -th online leverage score is at least the  $j$ -th leverage score since sensitivities cannot increase as more rows are added to  $K$ , but more interestingly, Theorem 2.2 of Cohen et al. (2016a) shows that the sum of online leverage scores is bounded by  $O(d \log \kappa^{OL})$ , where  $\kappa^{OL}$  is the online condition number, see Lemma 3.3 of Woodruff & Yasuda (2022). It is well-known for matrices with integer entries bounded by  $\mathrm{poly}(n)$ , which is an assumption often used to obtain meaningful memory bounds in a data stream, that the online condition number is at most  $n^{O(d)}$  (see, e.g., Lemma 4.1 of Clarkson & Woodruff (2009) which lower bounds the minimum non-zero singular value of an  $n \times d$  such matrix by  $n^{-O(d)}$ ), and thus the number of rows stored will be at most  $O(d^2 \log n)$ , and so the memory is bounded by  $O(d^3 \log n)$  words. One can also maintain  $K^T K$  in the stream as before. At the end of the stream, for each row  $K_j$  stored which had online leverage score at least  $\varepsilon$ , one can compute its exact leverage score  $K_j^T (K^T K)^{-1} K_j$  at the end of the stream in order to exactly find LA. Overall this is a 1-pass algorithm with memory poly  $(d \log n)$  words.
+
+Finding all Heavy Attentions for a Query. Given a query  $q$ , for  $f(x) = x^2$  we can compute all large attention score values it participates in exactly because we can first preprocess  $K$  in  $O(nd^2)$  time to compute its singular value decomposition (SVD)  $K = U'\Sigma V^T$ . Then, given a query  $q$ , we can compute  $\langle q, K_i \rangle^2$  for each  $K_i \in U$  in  $|U| \cdot d = O(d^2 / \varepsilon)$  time, and we can also compute the normalization  $\| Kq\|_2^2 = \| \Sigma V^T q\|_2^2$  in  $O(d^2)$  time since  $\Sigma V^T$  is a  $d \times d$  matrix. Thus, after an initial preprocessing of  $n \cdot \mathrm{poly}(d / \varepsilon)$  time, we can compute all large attention score values involving a query exactly and in only  $\mathrm{poly}(d / \varepsilon)$  time. Note that if one would like to instead approximate the large attention values up to a  $1 + \varepsilon$  factor, instead of computing the SVD of  $K$ , one can use a random sketching matrix  $S$  so that  $\| SKq\|_2^2 = (1 \pm \varepsilon)\| Kq\|_2^2$ . If one uses the Subsampled Randomized
+
+Hadamard Transform for example, then this improves the  $O(nd^2)$  time required to compute the SVD of  $K$  to only  $nd \cdot \mathrm{poly}(\log n / \varepsilon)$  time and incurring  $1 / \mathrm{poly}(n)$  failure probability, e.g., using the analysis of Cohen et al. (2016b). It is also possible to use CountSketch in  $O(nd + \mathrm{poly}(d))$  time with a  $1 / \mathrm{poly}(d)$  failure probability Clarkson & Woodruff (2013).
+
+This section was for  $f(x) = x^2$ , while the next section handles  $f(x) = |x|^p$  for any  $p \geq 1$ .
+
+# 3 ATTENTION SCORES USING  $f(x) = |x|^p$
+
+We next consider  $f$ -sensitivities for  $f(x) = |x|^p$ .
+
+Theorem 3.1. Let  $K$  be an  $n \times d$  matrix. Let  $p \in [1, \infty)$  and let  $f(x) = |x|^p$ . There exists a set  $S$  of rows of  $K$  containing all  $f$ -sensitivities at least  $\varepsilon$  with the following properties:
+
+1. For  $1 \leq p \leq 2$ ,  $|S| = O(d / \varepsilon)$ .  
+2. For  $p > 2$ ,  $|S| = O(d^{p / 2} / \varepsilon)$ .
+
+Moreover,  $S$  can be computed in  $(nnz(K) + poly(d / \varepsilon)poly(\log n)$  time.
+
+Proof. Fix  $p \geq 1$  and let  $f(x) = |x|^p$ . Let  $\tau_i$  denote the  $\ell_p$ -Lewis weights of  $K$  (Cohen & Peng, 2015). These can all be computed up to a constant factor in total time  $(\mathrm{nnz}(K) + d^{\omega})\mathrm{poly}(\log n)$  time (Cohen & Peng, 2015).
+
+For  $1 \leq p \leq 2$ , it is known that  $\tau_{i}$  upper bounds the  $i$ -th  $f$ -sensitivity  $\sigma_{i}^{f}$ . For  $p > 2$ , it is known that  $d^{p/2 - 1} \tau_{i}$  upper bounds the  $i$ -th  $f$ -sensitivity. The  $\ell_{p}$ -Lewis weights sum to at most  $d$ . For an exposition of these statements, see e.g., Section 3.3 of Clarkson et al. (2019) and references therein.
+
+Therefore, for  $1 \leq p \leq 2$ , the set  $S$  of all rows  $i$  with  $\tau_i \geq \varepsilon$  satisfies the desired properties. For  $p > 2$ , the set  $S$  of all rows  $i$  with  $d^{p/2-1} \tau_i \geq \varepsilon$  satisfies the desired properties. By approximating the  $\tau_i$  up to a factor of 2 and including all  $i$  whose approximate  $\tau_i$  is at least  $\varepsilon/2$  for  $1 \leq p \leq 2$ , while including all  $i$  whose approximate  $\tau_i$  is at least  $\varepsilon/(2d^{p/2-1})$  for  $p \geq 2$ , we will include all rows  $i$  whose  $f$ -sensitivity is at least  $\varepsilon$ . Further, if the approximate  $\tau_i$  is at least  $\varepsilon/2$ , then the actual  $\tau_i$  is at least  $\varepsilon/4$ , and so we have  $|S| = O(d/\varepsilon)$  for  $1 \leq p \leq 2$  while  $|S| = O(d^{p/2}/\varepsilon)$  for  $p > 2$ .
+
+We next show how to efficiently approximate the normalization term for all  $1 \leq p < \infty$ , in order to approximate all heavy attentions for a given query  $q$  by using the set  $U$ . We will later show how to compute the normalization term exactly for  $p$  an even integer.
+
+Theorem 3.2. Let  $Q$  be an  $n \times d$  matrix and  $K$  be an  $n \times d$  matrix. For  $1 \leq p < \infty$  and any  $j \in \{1, \dots, n\}$ , the normalization term  $\sum_{i=1}^{n} |\langle Q_j, K_i \rangle|^p$  can be estimated efficiently. Specifically, there exists a sampling and rescaling matrix  $S$  with the following properties:
+
+1. For  $1 \leq p \leq 2$ ,  $S$  has  $d \cdot \text{polylog}(d / \varepsilon) / \varepsilon^2$  columns,  
+2. For  $p > 2$ ,  $S$  has  $d^{p/2}$  polylog/ $\varepsilon^2$  columns,  
+3. With failure probability  $\frac{1}{poly(d)}$ , simultaneously for all  $x \in \mathbb{R}^d$ ,  $\| xK^T S\|_p^p = (1 \pm \varepsilon) \| xK^T\|_p^p$ .
+
+$S$  can be computed in  $(\text{nnz}(K) + \text{poly}(d / \varepsilon))\text{poly}(\log n)$  time, and after this one-time computation,  $\| xK^T S\| _p^p$  can be computed in  $\text{poly}(d / \varepsilon)$  time.
+
+Proof. The normalization term can be written as  $\| Q_j K^T \|_p^p$ . We can obtain the matrix  $S$  by Lewis weight sampling on  $K^T$ . This is a well-known technique for constructing subspace embeddings that preserve  $\ell_p$  norms simultaneously for all  $x$  Cohen & Peng (2015). Specifically, by Woodruff & Yasuda (2022), there exists an algorithm that computes and rescaling matrix  $S$  with the stated number of columns such that for all  $x \in \mathbb{R}^d$ ,  $\| xK^T S \|_p^p = (1 \pm \varepsilon) \| xK^T \|_p^p$ . The time to compute  $S$  is  $O(\mathrm{nnz}(K) + \mathrm{poly}(d / \varepsilon))$  poly(log n). Once  $S$  is computed, we can efficiently estimate
+
+the normalization term for any  $j$  as  $\| Q_j K^T S \|_p^p$ . Since  $S$  has  $\mathrm{poly}(d / \varepsilon)$  columns, computing this term takes  $\mathrm{poly}(d / \varepsilon)$  time.
+
+Finding all Heavy Attentions Exactly for a Query  $q$ . Although the above procedure allows for quickly estimating the normalization term up to a  $(1 + \varepsilon)$ -factor, one may be interested in the exact value of the attention score if the model is sensitive to slight perturbations. One can also do this with a slightly larger time complexity for even integers  $p > 2$ , as the idea for  $p = 2$  for computing heavy attention scores exactly extends to  $f(x) = x^p$  for even integers  $p$  using a tensor trick. Indeed, we can form the  $n \times d^{p/2}$  matrix  $K'$  by taking the Khatri-Rao product of each row of  $K$  with itself  $p/2$  times. Then we compute the SVD  $K' = U' \Sigma V^T$ . Given a query  $q$ , we can again compute  $\langle q, K_i \rangle^p$  for each  $K_i \in U$  in  $|U| \cdot d = O(d^{p/2+1}/\varepsilon)$  time, and now we can also compute the normalization  $\sum_{j} \langle K_j, q \rangle^p = \| K' q' \|_2^2 = \| \Sigma V^T q' \|_2^2$ , where  $q'$  is the Khatri-Rao product of  $q$  with itself  $p/2$  times. Thus, for constant even integers  $p$ , after an initial preprocessing of  $n \cdot \mathrm{poly}(d/\varepsilon)$  time, we can compute all large attentions for any particular  $q$  exactly in only  $\mathrm{poly}(d/\varepsilon)$  time.
+
+Remark 3.1. The choice of  $p$  in the definition of  $f$ -sensitivity can significantly impact the identification of large attention scores. Consider the following examples:
+
+Case  $p > 2$ : Suppose one entry in a row of the attention matrix is  $n^{1/p}$ , while all other entries are  $\Theta(1)$ . This entry is a large attention with constant  $\varepsilon$  for  $f(x) = |x|^p$ , but it is not a large attention score for  $f(x) = x^2$  unless  $\varepsilon$  is inverse polynomial in  $n$ .
+
+Case  $p < 2$ : Suppose one entry in a row is  $\varepsilon$ , another entry is 1, and the remaining entries are close to 0. The entry with value  $\varepsilon$  is a large attention score for  $f(x) = x^2$  with value  $\varepsilon^2$ , but it is a large attention score for  $f(x) = |x|$  with value  $\varepsilon$ . Thus, using  $\ell_1$ -sensitivities allows for faster identification and allows for storing a smaller set of large sensitivity rows of  $K$ .
+
+Remark 3.2. In Musco et al. (2022), a method for efficiently computing  $f$ -sensitivities with respect to a generic function  $f$  is presented. This function  $f$  is assumed to satisfy basic properties such as subadditivity and symmetry. Examples include the Huber and Tukey loss functions. Musco et al. (2022) show that the sum of these sensitivities is  $O(d\log n)$ , implying the existence of a subspace embedding  $S$  with  $\text{poly}(d)$  columns to facilitate fast computation of the normalization factor. Furthermore, the universal set  $U$  of rows of  $K$  now has size  $O(d\log n / \epsilon)$  and both  $U$  and  $S$  can be found in  $O(\text{nnz}(K) + \text{poly}(d))\text{poly}(\log n)$  time. These generalized loss functions may offer more flexibility and different efficiency versus accuracy tradeoffs.
+
+# 4 A PLANTED MODEL
+
+We formulate a planted model of keys and queries, where, each query is a noisy liner combination of a small set of keys; we call this set of keys, the "relevant keys for the query". We show that our algorithm, based on finding large attention scores, given  $K$  and a query satisfying model assumptions, finds the relevant set of keys. For a plausible set of model parameters, the running time of our algorithm per query, amortized over many queries, is sublinear in  $n$ . Throughout,  $K$  will be an  $n \times d$  matrix.  $K_{j}$  denotes the  $1 \times d$  vector (the  $j$ -th row of  $K$ ). Each row is a key.  $q$  is a  $1 \times d$  vector, and will denote a generic query. We assume there is a subset  $S$  of keys such that for  $i \in S$ , the correlation of  $K_{i}$  to other keys is upper bounded. We will assume later that query vectors are convex combinations of keys in  $S$ . We let  $\delta_{1}, \delta_{2}$  be parameters satisfying  $0 < \delta_{2} \leq \delta_{1} \leq 1/4$ . We assume there is a subset  $S$  of keys satisfying:
+
+$$
+\forall j, \ell \in S, j \neq \ell , | K _ {j} K _ {\ell} ^ {T} | \leq \delta_ {1} \cdot \min  \left(\| K _ {j} \| _ {2} ^ {2}, \| K _ {\ell} \| _ {2} ^ {2}\right) \tag {1}
+$$
+
+$$
+\forall j \in S, \ell \notin S, \left| K _ {\ell} K _ {j} ^ {T} \right| \leq \delta_ {2} \cdot \min  \left(\left\| K _ {j} \right\| _ {2} ^ {2}, \left\| K _ {\ell} \right\| _ {2} ^ {2}\right) \tag {2}
+$$
+
+Remark 4.1.  $S$  may be thought of as a "stand out" subset of keys, since the correlation of  $j \in S$  to other keys is upper bounded.
+
+We first argue that elements of  $S$  have high self-attention scores.
+
+Lemma 4.1. Let  $A = K K^T$ . For all  $i \in S$ ,  $(A_{ii})^2 \geq \left(\sum_{j=1}^{n}(A_{ij})^2\right) / (1 + \delta_1^2|S| + \delta_2^2n)$ .
+
+Proof.
+
+$$
+\left(A _ {i i}\right) ^ {2} = \left(K _ {i} K _ {i} ^ {T}\right) ^ {2} = \| K _ {i} \| _ {2} ^ {4}.
+$$
+
+$$
+\sum_ {j} (A _ {i j}) ^ {2} = | K _ {i} K _ {i} ^ {T} | ^ {2} + \sum_ {\ell \in S \backslash i} (K _ {\ell} K _ {i} ^ {T}) ^ {2} + \sum_ {\ell \not \in S} (K _ {\ell} K _ {i} ^ {T}) ^ {2} \leq \| K _ {i} \| _ {2} ^ {4} \left(1 + \delta_ {1} ^ {2} (| S | - 1) + \delta_ {2} ^ {2} n\right),
+$$
+
+using (1) and (2). Now, both of them together imply the lemma.
+
+![](images/afb16d595b29744f1dcc0e43c7efd3895b3c7657f673c91e053b321d6e1d0bb1.jpg)
+
+Note that  $\| K_i\| _2$  canceled out because of the normalization. This demonstrates the effectiveness of row normalization, which is a non-linear operation.
+
+# 4.1 ASSUMPTION ON QUERIES
+
+There is an unknown subset  $S(q)^1$  of  $S$  and unknown weights  $w_{j}(q)$  for  $j\in S(q)$  satisfying
+
+$$
+\sum_ {j \in S (q)} w _ {j} (q) \leq 1 \text {a n d} w _ {j} (q) \geq 4 \delta_ {1} \text {f o r a l l} j \in S (q) \tag {3}
+$$
+
+and an unknown  $d$ -dimensional vector  $z(q)^2$  with
+
+$$
+\left| z (q) K _ {\ell} ^ {T} \right| \leq \frac {\delta_ {1}}{4} \| K _ {\ell} \| _ {2} ^ {2} \text {f o r a l l} \ell \in [ n ], \text {w i t h} q = \sum_ {j \in S (q)} w _ {j} K _ {j} + z (q). \tag {4}
+$$
+
+We will just abbreviate  $w_{j}(q)$  by  $w_{j}$ .
+
+# 4.2 PROPERTY OF RELEVANT KEYS
+
+Theorem 4.2. We have  $\forall j\in S(q):qK_j^T\geq \frac{11}{4}\delta_1\| K_j\| _2^2$  and  $\forall j\notin S(q),qK_j^T\leq \frac{5}{4}\delta_1\| K_j\| _2^2.$
+
+Proof. For  $j \in S(q)$ :  $qK_{j}^{T} = w_{j}\| K_{j}\|_{2}^{2} + \sum_{\ell \in S(q)\backslash j}w_{\ell}(K_{\ell}K_{j}^{T}) + z(q)K_{j}^{T} \geq \| K_{j}\|_{2}^{2}(4\delta_{1} - \delta_{1} - \delta_{1}/4) = (11/4)\delta_{1}\| K_{j}\|^{2}$ , using (3), (4) and (1). For  $j \in S \setminus S(q)$ :  $|qK_{j}^{T}| = \left|\sum_{\ell \in S(q)}w_{\ell}(K_{\ell}K_{j}^{T}) + z(q)K_{j}^{T}\right| \leq \| K_{j}\|_{2}^{2}(\delta_{1} + (\delta_{1}/4)) = 5\delta_{1}\| K_{j}\|_{2}^{2}/4$ , using (3), (4) and (1). For  $j \notin S$ :  $|qK_{j}^{T}| = \left|\sum_{\ell \in S(q)}w_{\ell}(K_{\ell}K_{j}^{T}) + z(q)K_{j}^{T}\right| \leq \| K_{j}\|_{2}^{2}(\delta_{2} + (\delta_{1}/4)) \leq (5/4)\delta_{1}\| K_{j}\|_{2}^{2}$  since  $\delta_{2} \leq \delta_{1}$ .
+
+# 4.3 ALGORITHM TO LEARN RELEVANT KEYS
+
+Let  $\frac{1}{1 + \delta_1^2|S| + \delta_2^2n} = \rho$ . It is easy to see that  $\frac{A_{ii}^2}{\sum_{j=1}^{n}A_{ij}^2}$  is at most the  $i$ -th leverage score of  $K_i$  since the  $i$ -th leverage score is  $\sup_{y \neq 0} \frac{\langle y, K_i \rangle^2}{\sum_{j=1}^{n}\langle y, K_j \rangle^2}$ , which is at least as large as the value in this expression with the particular value  $y = K_i$ . Let  $U = \{i : LS(K_i) \geq \rho\}$ . Since the sum of all leverage scores is at most  $d$ , we have  $|U| \leq d(1 + \delta_1^2|S| + \delta_2^2n)$ . For a suitable setting of parameters, we have  $d(1 + \delta_1^2|S| + \delta_2^2n) \in o(n)$  and  $|U| \in o(n)$ . Let  $U' = \{i : \frac{(A_{ii})^2}{\left(\sum_{j=1}^{n}(A_{ij})^2\right)} \geq \rho\}$ .
+
+We can find  $U'$  using our algorithm for finding heavy leverage scores. We then compute  $\| K_i\|_2$  for all  $i$ . This takes  $O(nd)$  time. To estimate the row lengths of  $A = KK^T$ , we use a Johnson-Lindenstrauss sketch to find the row lengths of  $KK^T B$  for a random  $n \times O(\ln n)$  Gaussian matrix  $B$  (see, e.g., Woodruff et al. (2014) for background on Johnson Lindenstrauss sketches). When a query  $q$  arrives, we check for each  $i \in U'$  if  $qK_i^T \geq (11/4)\| K_i\|_2^2$ . This suffices since  $S \subseteq U'$ .
+
+# 4.4 A STOCHASTIC EXAMPLE
+
+We now present an example  $K$  in which the assumptions (1) and (2) hold. The example has  $K$  as a random matrix described below. There is a latent subspace  $V$  of dimension  $k \leq d / 4$ . For the conceptual description, we assume the first  $k$  of the  $d$  coordinates of a vector are in  $V$  and the remaining  $d - k$  coordinates are in  $V^{\perp}$ . Note that this is only for ease of description; we do not
+
+actually know  $V$ . Intuitively, for  $i \in S$ ,  $K_i$  is mostly in  $V$  with a small "noise" component in  $V^\perp$ , whereas, for  $i \notin S$ ,  $K_i$  is mostly in  $V^\perp$  with a small component in  $V$ . The matrix drawn below describes the distributions from which each part is generated. The coordinates are i.i.d. inside each of the four parts with variance differing between parts. We assume  $\delta_1 \geq (4/k)$  and  $\delta_2 \geq (4\varepsilon_1/k)$ . No condition on  $\epsilon_0$  is required. By standard concentration bounds, (1) and (2) hold with high probability, so that Lemma (4.1) and Theorem (4.2) hold and our algorithm applies.
+
+$$
+K _ {n \times d} = \left[ \begin{array}{c c} K ^ {1} & K ^ {2} \\ \mathcal {N} (0, 1 / k) & \mathcal {N} (0, \varepsilon_ {1} / (d - k)) \\ \hline K ^ {3} & K ^ {4} \\ \mathcal {N} (0, \varepsilon_ {1} / k) & \mathcal {N} (0, 1 / (d - k)) \end{array} \right]
+$$
+
+Here  $K^1$  is  $\varepsilon_0n\times k$ , and  $K^2$  is  $\varepsilon_0n\times (d - k)$ , and  $K^3$  is  $(1 - \varepsilon_0)n\times k$ , and  $K^4$  is  $(1 - \varepsilon_0)n\times (d - k)$ .
+
+# 5 EXPERIMENTS
+
+# 5.1 STRUCTURAL PROPERTIES OF TYPICAL ATTENTION MATRICES
+
+We consider a pretrained ViT model Dosovitskiy et al. (2021) for image classification and consider the properties of the attention matrices across all of the attention heads in the model. The model we consider has six layers and each of the layers has six attention heads. The inputs to the model are tokenized into 197 tokens - 196 tokens representing the input image and an additional token whose final representation is used for classifying the image (see Figure 1). Therefore each of the attention matrices that we consider is of size  $197 \times 197$ .
+
+For  $i,j$ , given the set of queries  $\{Q_1,\ldots ,Q_n\} \in \mathbb{R}^d$  and keys  $\{K_{1},\dots,K_{n}\} \in \mathbb{R}^{d}$ , then for  $i\in [n]$  and  $j\in [n]$ ,  $A_{i,j} = \frac{\exp(\langle Q_i,K_j\rangle / \sqrt{d})}{\sum_j\exp(\langle Q_i,K_j\rangle / \sqrt{d})}$ . For each  $i$ , we define  $\ell_i(1),\ldots ,\ell_i(n)$  such that  $A_{i,\ell_i(1)}\geq \dots \geq A_{i,\ell_i(n)}$ .
+
+Top Heaviness. We find that at many of the attention heads, the attention weight of a token is distributed in a "top heavy" manner, meaning that for many tokens  $i$ , the sum of the top-32 attention weights, defined as  $\sum_{j=1}^{32} A_{i,\ell_i(j)}$  is quite large. We show the distribution of top-32 attention weights for an image input in Figure 2 in the Appendix. For a given attention head, we observe that the distribution of top-32 attention weights remains similar across multiple inputs.
+
+Locality. We measure the amount of attention mass that is solely captured by the "neighboring" tokens (see Figure 1). We observe that only at a small fraction of heads, the attention mass of tokens is fully captured by the local tokens and that in general, a significant portion of the attention mass is distributed among non-local tokens. In Figure 3 in the Appendix, we show the histograms of the attention weights captured by local tokens at each of the attention heads in all the layers of the model for a typical input.
+
+Important Keys. For  $j \in [n]$ , we define  $W_{j}$  to be the non-local attention weight captured by  $j$  as  $\sum_{i:(i,j)}$  are not neighbors  $A_{i,j}$ . We define important keys to be the set of 32 keys with the largest  $W_{j}$  values. We find that in the initial layers of the models, at many attention heads, most of the attention weight is captured by the local tokens and a small set of "important keys". In Figure 4 in the Appendix, we show the distribution of the attention weight captured by the set of "important keys" along with the local tokens.
+
+![](images/e346d7e5bc01bebb0ed944bff5adc61579ccc509b77fbc1dad60c2a5928d1815.jpg)  
+Figure 1: Example of an image partitioned into 196 patches using a  $14 \times 14$  grid. The green patches represent the neighbors of the red patch within a distance of 3.
+
+# 5.2 INFERENCE FROM SOFTMAX VIT MODELS VIA LEVERAGE SCORE SELECTION
+
+We consider three ViT models from the work of Dosovitskiy et al. (2021): (i) S/16, a small 22M parameter model that splits an image into 196 patches each of size  $16 \times 16$  pixels, (ii) L/16, a large
+
+<table><tr><td>Model</td><td>Accuracy on validation set</td></tr><tr><td>S/16 (softmax)</td><td>76.47%</td></tr><tr><td>S/16 (LevAttention, top-32, pretrained w/ softmax)</td><td>13.3%</td></tr><tr><td>S/16 (l2norm selection, top-32, pretrained w/ softmax)</td><td>3.3%</td></tr><tr><td>S/16 (LevAttention, top-32)</td><td>68.30%</td></tr><tr><td>S/16 (LevAttention, top 64)</td><td>72.48%</td></tr><tr><td>L/16 (softmax)</td><td>78.83%</td></tr><tr><td>L/16 (LevAttention, top-32, pretrained w/ softmax)</td><td>48.58%</td></tr><tr><td>L/16 (l2norm selection, top-32, pretrained w/ softmax)</td><td>8.9%</td></tr><tr><td>L/16 (LevAttention, top-32)</td><td>75.12%</td></tr><tr><td>L/16 (LevAttention, top-64)</td><td>77.27%</td></tr><tr><td>L/16 (LevAttention, top-128)</td><td>77.17%</td></tr><tr><td>L/8 (softmax)</td><td>79.47%</td></tr><tr><td>L/8 (LevAttention, top-32, pretrained w/ softmax)</td><td>0.8%</td></tr><tr><td>L/8 (LevAttention, top-32)</td><td>71.96%</td></tr><tr><td>L/8 (LevAttention, top-64)</td><td>74.64%</td></tr><tr><td>L/8 (LevAttention, top-128)</td><td>76.69%</td></tr></table>
+
+Table 1: Accuracies of models with various attention mechanisms.
+
+305M parameter model that splits an image into 196 patches each of size  $16 \times 16$  pixels, and (iii) L/8, a large 305M parameter model that splits an image into 784 patches each of size  $8 \times 8$  pixels. Note that all the models have one token appended whose representation after processing through the transformer is used for classifying images. We train the models on the Imagenet-1k (Russakovsky et al., 2015) dataset using the same hyperparameters as in the original work (Dosovitskiy et al., 2021). We find that the S/16, L/16 and L/8 models achieve accuracies of  $76.47\%$ ,  $78.83\%$  and  $79.47\%$  respectively on the validation split of the Imagenet-1k dataset.
+
+We then estimate the performance of leverage score selection on these datasets. At each attention head, we compute the keys with the 32 largest  $\ell_2$  leverage scores and then make each query attend only to this set of keys in the attention mechanism. While we find that the accuracies drop significantly compared to the full softmax attention, our results show that the models still achieve nontrivial accuracies. In particular, the L/16 model has an accuracy of  $48.58\%$  using this mechanism. If we instead select top-32 keys based on the squared row norms, we observe that the accuracy of the L/16 model drops to  $8.9\%$ . This supports the idea that leverage scores are much better predictors of the importance of the keys compared to row norms.
+
+# 5.3 TRAINING VIT MODELS
+
+In the previous experiments, we used models trained using softmax attention and used LevAttention only at inference time. To see if the performance of the models improve if they are aware of LevAttention, we use the leverage score selection based attention mechanism to train the models as well. Using the same training setup as the softmax attention models, i.e., the same learning rate schedule, batch sizes, and optimizer, we see significant improvements in the validation accuracies. For the L/8 and L/16, we train for the initial  $15\%$  of the steps with full attention to obtain the warm start parameters and then train the remaining  $85\%$  of the steps using the leverage score selection based attention. We report the results in Table 1. To test if training the models with the awareness of "leverage score selection" truly is useful, we train models using (i) "row norm selection" where at each attention head, we pick 32 keys with the largest  $\ell_2$  norms, and (ii) "random selection", where at each attention head, we pick 32 random key positions and only make the queries attend to those positions throughout the training process. We observe that these models achieve similar performance to the models trained via "leverage score selection". This shows that the "selection aware" training procedure is currently unable to translate the usefulness of leverage scores, as identified in the previous section, into obtaining better models than those achieved by "row norm selection" and "random selection" attention mechanisms. We note that only leverage scores have some provable guarantees, while the other methods do not. We leave the important question of obtaining better models trained using LevAttention as a future research direction.
+
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+
+# 6 APPENDIX: EXPERIMENTAL RESULTS
+
+In this section, we present empirical observations on the behavior of attention matrices in a ViT model trained using softmax attention. We use the same setup as in (Dosovitskiy et al., 2021) to train a 6-layer transformer model with 6 attention heads in each of the layers on theImagenet-1k dataset. We consider a typical input to the model and in Figures 2, 3 and 4, we present the properties of the attention weights matrix across different layers and attention heads in the model.
+
+![](images/6f37a49101dfc8eb7803283ca7f93f961326b705d0d74dce139772f37492155a.jpg)  
+Figure 2: Histograms of top-32 attention weights. Each of the histograms plots the distribution of the sum of top-32 attention weights for query tokens. We note that at many attention heads, for many query tokens, the largest 32 attention weights constitute a significant fraction of the total attention weight.
+
+![](images/b78bbd0093a48f7198f534fb08fc448fc3f2df9793d9b49576eb160b0769a748.jpg)  
+Figure 3: Histograms of attention weights captured by local tokens. Each of the histograms plots the distribution of attention weight captured by keys within a Manhattan distance of 3.
+
+![](images/c278bc9e4829fce2771d9f4e8954164eb9866d1f69e1cd29d7a7dd26f69b0251.jpg)
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+![](images/54c4e32e582ab43a31275d11e6adec59a3d025ccec576975a6fb1fe7388445f5.jpg)
+
+![](images/96ddc5b126148a74710d689fec5d214de45ea0d552c03d07fd1c9f64a4ee47ce.jpg)
+
+![](images/f82bd37d88c7c28a388f3d55c6294d01ca84b3959ab6a1c50ff07feabb04dc89.jpg)
+
+![](images/7b53e6216ba9417bf62e5fc8b72bb5ba70688fa2042ddfdc93df835ac8ab1ee4.jpg)  
+Figure 4: Histograms of Attention weights captured by the important keys along with local tokens. As discussed in Section 5.1, important keys are defined as the 32 keys capturing the largest attention weight at a given attention head. The histogram shows that at a large number of attention heads, the important keys together with local keys are able to capture a significant fraction of the attention weight for a large number of queries.
+
+![](images/eea5cbe5d1a4b65faad00b1fc0517c56fd9a671bdf8563f70ad958cf05ee9f13.jpg)
+
+![](images/bdc35ef36ade6a4b4c86d540e3d25872abd35bb0f2e40d40ca28bd953f8ba846.jpg)
+
+![](images/db2c51746541ca7e338311e5f4e1a3ce8a5ce7c3ca84d567ecd924d76e3d7bc0.jpg)
+
+![](images/cb376ab63ac185e939758c4d3287e4d0b0f4306e95bb70b3ad1e4f626d5e49be.jpg)
+
+![](images/2efa78b8e0fe9a5d6233b98a0830248e0bcefa47a4943241bf7dba920f047d68.jpg)
+
+# 7 APPENDIX: OVERVIEW OF RECENT RELATED WORK FOR REDUCING COMPUTATIONAL COMPLEXITY
+
+Zaheer et al. (2020) propose Big Bird, a sparse attention mechanism that combines global and local attention to handle longer sequences in transformers. While Big Bird employs a fixed sparsity pattern, our work introduces a data-dependent sparsity approach by identifying a universal set of keys based on leverage scores. These approaches could be seen as complementary: Big Bird provides a general framework for handling long sequences, while our method offers a more fine-grained approach for identifying the most relevant information in the attention matrix. Combining these techniques could be an interesting direction for future work.
+
+Song et al. (2024) address the computational challenge of solving attention kernel regression problems, where the matrix exponential of the Gram matrix is the kernel. They propose using a preconditioner to accelerate the solution of these regression problems. While their work focuses on a specific formulation of the attention mechanism as a regression task, our work provides a more general method of approximating the attention matrix by identifying a universal set of important keys. Our approach could potentially complement their work by providing a reduced set of keys, which could further speed up their pre-conditioned solver.
+
+Gao et al. (2023) propose a fast optimization approach for training single-layer attention networks in LLMs. They reformulate the attention mechanism using tensor and SVM tricks to achieve a training time comparable to matrix multiplication. While they modify the training process itself to improve efficiency, our work focuses on identifying a core set of important keys, which can be used to speed up both inference and training. Both their approach and ours could potentially be used in conjunction with learned leverage scores during training. They could incorporate learned leverage scores into their tensor-based optimization framework, while we could use learned leverage scores to construct our universal sets. Exploring the interplay between these techniques could be an interesting avenue for future research.
+
+KDEFormer (Zandieh et al. (2023)) is a precursor to HyperAttention that uses fast kernel density estimation algorithms to approximate the softmax attention. These algorithms efficiently estimate the normalization factor for each row of  $\exp(QK^T)$ , as well as the sampling probabilities used in its approximate matrix multiplication. However, KDEFormer requires at least  $n^{1.173}$  time and relies on assumptions such as bounded diameter datasets and small stable rank to achieve its strongest results. In contrast, our method directly identifies the most important columns with theoretical guarantees and runs in linear time with respect to  $n$ , without requiring these assumptions. Furthermore, as discussed in our work, the universal set we identify can be used as input to HyperAttention, effectively reducing its runtime by focusing its computation on the most relevant tokens. This could potentially lead to both faster and more accurate attention mechanisms.
\ No newline at end of file
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+# LEVERAGING DRIVER FIELD-OF-VIEW FOR MULTIMODAL EGO-TRAJECTORY PREDICTION
+
+M. Eren Akbiyik<sup>1</sup>, Nedko Savov<sup>2</sup>, Danda Pani Paudel<sup>2</sup>, Nikola Popovic<sup>1,2</sup>, Christian Vater<sup>3</sup>, Otmar Hilliges<sup>1</sup>, Luc Van Gool<sup>2</sup> & Xi Wang<sup>1,4</sup>
+
+$^{1}$  ETH Zürich  $^{2}$  INSAIT, Sofia University "St. Kliment Ohridski"
+
+$^{3}$  University of Bern TU Munich
+
+# ABSTRACT
+
+Understanding drivers' decision-making is crucial for road safety. Although predicting the ego-vehicle's path is valuable for driver-assistance systems, existing methods mainly focus on external factors like other vehicles' motions, often neglecting the driver's attention and intent. To address this gap, we infer the ego-trajectory by integrating the driver's gaze and the surrounding scene. We introduce RouteFormer, a novel multimodal ego-trajectory prediction network combining GPS data, environmental context, and driver field-of-view—comprising first-person video and gaze fixations. We also present the Path Complexity Index (PCI), a new metric for trajectory complexity that enables a more nuanced evaluation of challenging scenarios. To tackle data scarcity and enhance diversity, we introduce GEM, a comprehensive dataset of urban driving scenarios enriched with synchronized driver field-of-view and gaze data. Extensive evaluations on GEM and DR(eye)VE demonstrate that RouteFormer significantly outperforms state-of-the-art methods, achieving notable improvements in prediction accuracy across diverse conditions. Ablation studies reveal that incorporating driver field-of-view data yields significantly better average displacement error, especially in challenging scenarios with high PCI scores, underscoring the importance of modeling driver attention. All data and code is available at meakbiyik.github.io/routeformer.
+
+# 1 INTRODUCTION
+
+Understanding the perception and decision-making process of drivers is crucial for road safety in autonomous and assisted driving. Statistics reveal that  $42\%$  of car-bicycle collisions result from driver inattention (Allenbach et al., 2021). In response, a driver-assistance system capable of integrating the driver's perspective into its decision-making would enable vehicles to make informed decisions (Schwarting et al., 2018). Although predicting the driver's path is critical for such systems, traditional methods focus mainly on inferring the paths of other vehicles (Salzmann et al., 2020; Gu et al., 2021; Ngiam et al., 2021; Varadarajan et al., 2022; Nayakanti et al., 2023). In contrast, we seek to predict the drivers' ego-trajectory by observing both the surrounding scene, and the perception of the ego-vehicle's drivers via their frontal view and gaze fixations.
+
+The central idea of this work builds on the insight that the perception of a person is tightly intertwined with their imminent and long-term goals, influenced by the surrounding traffic (Triesch et al., 2003; Hayhoe et al., 2003; Argyle et al., 1994). In driving, the attention of a driver can reveal otherwise inaccessible information about their decisions on the road. Complementing scene information with these cues enables the vehicles to resolve ambiguous situations. For instance, the head movements of the driver might be highly informative in determining the turn direction in a junction. The value of predicting ego-trajectories emerge in ambiguous, complex road paths, e.g., cases where the road is not trivially straight, or when predicting where the driver would turn at an intersection.
+
+Our focus is thus twofold: firstly, to develop a framework capable of predicting the drivers' future ego-trajectories in various driving scenarios, and secondly, to create a method for identifying complex yet rare trajectories and quantifying their complexity. Despite the presence of other driving datasets with gaze data (Gopinath et al., 2021; Xia et al., 2017; Fang et al., 2023), to the best of
+
+![](images/c29c6bf0e7395d1916c9bff1484b76de8f788c38903bb735f6e68882d5acce8c.jpg)  
+Figure 1: RouteFormer framework. Using the past GPS and the scene together with driver field-of-view, we predict the future ego-trajectory and visual features in driving with a novel loss scheme.
+
+our knowledge, only one public benchmark DR(eye)VE contains both ego-drivers' locations and their field-of-view (Palazzi et al., 2018). The scarcity of available data poses another challenge for our task. To this end, we present an end-to-end framework that predicts future ego-trajectories, a novel metric to measure the complexity of road scenarios, and a new dataset of real-world driving recordings that include drivers' field-of-view changes.
+
+First, we build a multi-modal framework, RouteFormer, for egocentric trajectory estimation that integrates scene data, driver FOV, and past trajectory, as shown in Figure 1. RouteFormer is crafted based on insights from time-series forecasting literature, offering the flexibility to incorporate additional data from other modalities, such as navigation paths and LIDAR. We train RouteFormer with auxiliary tasks of predicting future gaze and scene features for regularization, using a novel future-discounted loss formulation. To better quantify the complexity of various driving scenarios, we propose a new metric named Path Complexity Index (PCI), which measures the divergence of a driving trajectory from an extrapolation of the current path. It also indicates the difficulty of the corresponding prediction tasks and allows us to analyze model performance under different scenarios. Finally, we introduce a Gaze-assisted Ego Motion (GEM) dataset that captures diverse urban driving scenarios across 10 subjects, integrating GPS, high-resolution scene data from two front-view cameras, and driver field-of-view (FOV)—comprising first-person video and gaze fixations.
+
+Experimental results show that RouteFormer outperforms state-of-the-art (SOTA) methods on the GEM dataset. To test its generalization capabilities, we also evaluate RouteFormer on the DR(eye)VE dataset (Palazzi et al., 2018), and our method surpasses the SOTA approaches. Ablation studies show that incorporating drivers' field-of-view information improves the prediction quality, particularly in high-PCI situations.
+
+In summary, our contributions are:
+
+- RouteFormer, an end-to-end multimodal ego-motion prediction network that effectively utilizes FOV data with a novel loss design.  
+- PCI, a metric that measures the trajectory complexity.  
+- GEM, an ego-motion dataset with driver positions and perspective.
+
+# 2 RELATED WORK
+
+Ego-trajectory prediction. Predicting the future sequence of locations of a moving entity has been explored for humans (Lyu et al., 2022; Rodin et al., 2021; Singh et al., 2016; Park et al., 2016), road agents (Huang et al., 2022; Gulzar et al., 2021) and aircraft (Zeng et al., 2022). Autonomous driving has largely focused on estimating the trajectories of the many agents surrounding the ego-vehicle (Salzmann et al., 2020; Gu et al., 2021; Nayakanti et al., 2023; Varadarajan et al., 2022; Ngiam et al., 2021). Recent advances in multi-agent prediction leverage specialized modalities like HD maps (Shi et al., 2022a) and interaction graphs (Girgis et al., 2021; Vishnu et al., 2023), while emerging approaches exploit large language models (Mao et al., 2023; Zheng et al., 2024) and human-inspired perception (Liao et al., 2024a;b). The fusion of these diverse modalities has shown
+
+promise in improving prediction accuracy (Choi et al., 2021; Li et al., 2024a). However, predicting the ego-vehicles's trajectory, which is crucial for driving assistance systems (Schwarting et al., 2018; Jain et al., 2015; Vellenga et al., 2024; Kung et al., 2024), remains less explored. Kinetic models assuming constant velocity and turn rate have been utilized (Ammoun & Nashashibi, 2009). Kim et al. (2021) predict the ego-vehicle's trajectory with a VAE by conditioning on predicted driving style, while Baumann et al. (2018) use observations of the static environment, albeit neither with visual information, similarly with Kim et al. (2017) and Feng et al. (2019). Only Malla et al. (2020) proposes a vision module but requires hand-annotated actions. Taking a different direction, we focus on leveraging drivers' gaze and the scene to predict future ego-vehicle locations.
+
+Driving and attention. Individuals tend to focus on relevant objects in situations with a particular goal in view, acquiring necessary information even before they are needed (Land & Lee, 1994; Underwood et al., 1999; Zhang et al., 2022; Fathi et al., 2012; Admoni & Srinivasa, 2016; Argyle et al., 1994). In navigation tasks, individuals often fixate on an intended path or destination well before initiating the movement (Hayhoe et al., 2003). For example, Zheng et al. (2022) successfully shows that the gaze is a high-quality indicator for human motion with GIMO. Similarly, numerous studies have also employed eye tracking to study the driving behavior of individuals, focusing on aspects such as attention (Ahlström et al., 2021), cognitive load (Engström et al., 2005; Kountouriotis et al., 2015), and levels of fatigue (Heitmann et al., 2001; Gao et al., 2015), while many of them are conducted in a driving simulator. For driver maneuver classification, recent work has shown relative success with an in-cabin camera observing the driver (Jain et al., 2015; Ma et al., 2023; Vellenga et al., 2024; Li et al., 2024b), with some works exploring using gaze and scene information for ego-action prediction, e.g. turns or lane changes (Lee et al., 2021; Amadori et al., 2020; Wu et al., 2019; Yi et al., 2023). Han et al. (2023) predict maneuvers based on rider sensor head motion data. Yan et al. (2023) use head motion and headset gaze to forecast the driver's future path by fitting a polynomial, but their non-robust gaze data limits its effectiveness. Finally, Ma et al. (2023) propose a transformer model, which integrates driver behavior information for maneuver prediction. In contrast, we use driver's field of view together with the surrounding scenes for the trajectory prediction task, inspired by similar works in human motion such as GIMO (Zheng et al., 2022).
+
+Driving datasets with driver FOV. In the context of driving, there are datasets that provide in-vehicle footage of the driver's behavior as an indication of driver's attention (Ortega et al., 2020), zone-binned gaze (Ghosh et al., 2021; Ribeiro & Costa, 2019) (stationary), head pose (Schwarz et al., 2017). Jain et al. (2015) provide synchronized footage of the driver and the surrounding scene, yet without gaze data. Palazzi et al. (2018) provide a dataset of driving under various conditions, with gaze data from a tracking headset and synchronized GPS locations, however, with limited urban scenes. Given the limited amount of driving datasets with gaze, we introduce a new high-resolution dataset, where accurate eye tracker gaze location and vehicle GPS information are available. In contrast to others, we focus on city scenes with many agents where complex situations can arise.
+
+# 3 EGO-MOTION PREDICTION WITH DRIVER FIELD-OF-VIEW
+
+Driving is an interplay of short and long-term goals, influenced by exogenous factors. Although such external effects can be understood to some level through computer vision, drivers' behavior is mostly driven by self-established targets, which complicates the task of ego-motion prediction. To address this challenge, we propose a new framework named RouteFormer. At its core, our method is designed to exploit visual information from multiple cameras, the past trajectory of the vehicle, and, most notably, the driver's field-of-view—comprising first-person video and gaze fixations. Each modality contributes to understanding a different aspect: the surrounding scene, the motion of the vehicle, and the targets of the driver. See Figure 2 for an overview.
+
+RouteFormer offers two key contributions: (1) a novel architecture that fuses multimodal inputs via self- and cross-attention mechanisms for time-series prediction, and (2) an auxiliary loss and future-doubling to regularize long-term forecasting that is otherwise prone to over-fitting.
+
+# 3.1 TASK DEFINITION
+
+In the ego-trajectory prediction task, given measurements of the vehicle and the environment for  $T$  time steps, we aim to anticipate the future locations of the vehicle for each of the next  $T_{\mathrm{pred}}$  time
+
+![](images/04adf47501a4d5ac0dd3c1be3849c09363616d5d08def7e21c6a8b0a81f89802.jpg)  
+(a) Multimodal encoder for FOV and scenes
+
+![](images/d1cfe434b6be8c6ca68d3f7adcf4f0b47c77187fba29bde3cae22277aeaef4e7.jpg)  
+(b) Time-series module and loss configuration  
+Figure 2: Framework details. The multimodal architecture fuses FOV, scene, and motion data for ego-trajectory forecasting. (a) The videos are encoded with the scene encoder  $E_{S}$  frame-wise using a pre-trained vision backbone. FOV data is then encoded via a cross-modal transformer. The resulting tensors, all in the image feature domain, are stacked across time, self-attended, and concatenated with motion features for forecasting. (b) RouteFormer predicts the trajectory, as well as features from visual modalities concurrently to use them as auxiliary losses for regularization.
+
+steps (our prediction horizon). For the first  $T$  steps, we use as input the vehicle's motion  $M_{1:T}$  and scene representations in the form of video from K cameras of the surroundings  $S_{1:T}^{1:K}$ . Notably, in addition, we incorporate FOV data  $F_{1:T}$ . We build a model  $\Psi(M_{1:T}, S_{1:T}^{1:K}, F_{1:T}|\lambda) = M_{T+1:T_{\mathrm{pred}}}^{\prime}$  to predict the future motion, where  $\lambda$  are the model's parameters.
+
+We represent vehicle location with two GPS coordinates using the EPSG:3857 coordinate system (in meters) (Pridal & Pohanka, 2024). The  $t$ 'th element of the input motion  $M_{1:T} \in \mathbb{R}^{T \times 2}$  corresponds to the relative change of the coordinates compared to  $t - 1$ , as in Salzmann et al. (2021). For each time step  $t \in [0,T]$ , the scene  $S_{1:T}^{1:K} \in \mathbb{R}^{K \times T \times 3 \times H_S^K \times W_S^K}$  contains images from K external vehicle cameras, with width  $W_S^K$  and height  $H_S^K$ . FOV data has two components  $F_{1:T} = \{F_{1:T}^V; F_{1:T}^L\}$  representing a frontal video feed from a camera fitted to the driver's head  $F_{1:T}^V \in \mathbb{R}^{T \times 3 \times H_F \times W_F}$ , and the high-frequency gaze positions, respectively. The second component,  $F_{1:T}^L \in \mathbb{R}^{T \times 2}$ , consists of  $(x,y)$  coordinates of points gazed in the image  $F^V$ , relative to the bottom-left of the image.
+
+# 3.2 ROUTEFORMER ARCHITECTURE
+
+In our architecture, scene and FOV modalities are processed by separate encoders  $E_{S}(S_{1:T}^{1:K}) = s_{1:T}^{1:K}$  and  $E_{F}(F_{1:T}) = f_{1:T}$ , producing sequences of compact feature vectors of the same size. Next, all K scene features are fused with FOV features using a self-attention module  $E_{V}(s_{1:T}^{1:K}, f_{1:T}) = v \in \mathbb{R}^{T \times D}$ . Finally, a time-series forecasting module  $\mathcal{T}$  predicts the future  $T_{pred}$  data points - the future motion, in addition to feature vectors of other modalities used in training. We design two versions of this architecture: RouteFormer-Base,  $\Psi(M_{1:T}, S_{1:T}^{1:K} | \lambda)$ , which does not use the FOV features, and RouteFormer,  $\Psi(M_{1:T}, S_{1:T}^{1:K}, F_{1:T} | \lambda)$ , which incorporates all the available modalities, including the driver's perspective.
+
+Scene Module. Our shared scene module  $E_{S}$  encodes each frame in  $S_{1:T}^{1:K}$  using a pre-trained vision backbone, in our case SwinV2 (Liu et al., 2021). The features are then reduced through a self-attention layer to produce  $s_{1:T} \in \mathbb{R}^{T \times D}$  (see Figure 2 a). These features are used in subsequent layers for temporal reasoning with inter-frame attention.
+
+FOV Module. The frames  $F_{1:T}^{V}$  of the FOV data  $F_{1:T}$  are encoded using the same scene encoder  $E_{S}$ , in order to share the representation space and allow for implicit registration between the images. To account for high frame rate of gaze positions and refine the potentially noisy gaze positions, we perform a multi-head self-attention of  $F_{1:T}^{G}$ , following the decoder design in Zhou et al. (2021). After achieving two encoded feature vectors of the same shape for gaze and frontal video, we perform gaze-frame cross-attention between visual and gaze location features. The attention mechanism allows the model to "pick" the relevant scene features using the gaze positions across frames, thus accounting for the highly dynamic scene in a moving vehicle. The vector produced is  $f_{1:T} \in \mathbb{R}^{T \times D}$ .
+
+Modality Fusion. The next step is using an encoder  $E_V$  to fuse scene  $s_{1:T}^{1:K}$  and optionally FOV  $f_{1:T}$  features. To this end, we concatenate  $s_{1:T}^{1:K}$  and  $f_{1:T}$  across the time dimension, sum with the modality embeddings  $E_M$ , and apply a self-attention mechanism.  $E_M$  encodes the feature source (e.g. one of the scenes ( $k \in [0,K]$ ) or FOV). This self-attention layer performs implicit image registration and feature selection within and between the scene and FOV modalities. More precisely, we take the concatenation  $[s_{1:T}^{1:K}, f_{1:T}] \in \mathbb{R}^{(K+1)\cdot T \times D}$  and produce its fused encoding  $v \in \mathbb{R}^{T \times D}$ . In the case of RouteFormer-Base, where gaze data is not available, we only fuse the scene vectors.
+
+Time-Series Forecasting. Next, we merge (along the feature dimension) the motion features  $m_{1:T}$  and the aforementioned fused features  $v_{1:T}$  to obtain  $[m_{1:T}, v_{1:T}] \in \mathbb{R}^{T \times 2 \cdot D}$ . This forms the input for our time-series forecasting component  $\mathcal{T}$ , which takes the merged features  $[m_{1:T}, v_{1:T}]$  and produces a prediction of the future motion  $M_{T:T_{\mathrm{pred}}}^{\prime}$ . For this task, we experiment with the state-of-the-art long-term time-series prediction methods Informer (Zhou et al., 2021), Transformer (Vaswani et al., 2017), PatchTST (Nie et al., 2023) and NLinear/DLinear (Zeng et al., 2023). While the focus is on future trajectory prediction, we also predict future visual features  $v_{T:T_{\mathrm{pred}}}^{\prime}$ . The auxiliary outputs are essential for regularization during training by enforcing intermediate features to remain rich in scene- and FOV-related information, whenever available.
+
+# 3.3 TRAINING OBJECTIVES
+
+Trajectory prediction models are prone to over-fitting due to the inherent difficulty of the training regime: as the subsequent predictions depend on the correctness of the previous forecasts, models might learn from spurious correlations of the later sections of the trajectory. To tackle this problem, RouteFormer employs a novel loss formulation which we term as "future-discounted loss". The principle behind this loss is to consider predictions for future time steps with diminishing weight, particularly at the early stages of the training, making predictions for the immediate future more influential on the overall loss than those for the distant future. This also follows real-world use cases where immediate predictions often have a higher impact than long-term predictions.
+
+Future-discounted loss. Given a predicted sequence  $\mathbf{y}_{pred}$  and a ground truth  $\mathbf{y}_{gt}$ , the future-discounted loss  $\mathcal{L}_{\mathrm{fd}}$  over  $N$  time steps is then defined as:
+
+$$
+\mathcal {L} _ {\mathrm {f d}} (\mathbf {y} _ {p r e d}, \mathbf {y} _ {g t}) = \sum_ {i = 1} ^ {N} \gamma^ {i} \left(\mathbf {y} _ {p r e d, i} - \mathbf {y} _ {g t, i}\right) ^ {2}
+$$
+
+where  $\gamma$  is the discount factor with  $0 < \gamma < 1$ . This is inspired by reward discounting in reinforcement learning literature, where the discount rate is often in [0.9, 0.99]. On 6s at 5fps, the choice of 0.97, as used in our experiments, gives  $0.97^{30} = 0.4$  weight in the final time step.
+
+Loss composition. Both the primary trajectory loss and the auxiliary losses are subject to the future-discounting. Let  $\mathcal{L}_T$  denote the discounted trajectory loss and  $\mathcal{L}_V$  the discounted video visual. See Appendix A.1 for detailed definitions. The combined loss  $\mathcal{L}_{\mathrm{combined}}$  can be expressed as:
+
+$$
+\mathcal {L} _ {\text {c o m b i n e d}} = \mathcal {L} _ {T} + \alpha_ {V} \mathcal {L} _ {V}.
+$$
+
+The coefficient  $\alpha_{V}$  dynamically balances the primary and auxiliary losses. It is determined by:
+
+$$
+\alpha_ {V} = \rho_ {V} \times \frac {\left| \left| \mathcal {L} _ {T} \right| \right|}{\left| \left| \mathcal {L} _ {V} \right| \right|}.
+$$
+
+Here,  $\rho_{V}$  is a scalar hyperparameter and  $||\cdot ||$  represents the parameter's scalar value detached from gradients. This ensures auxiliary losses maintain a consistent proportion to  $\mathcal{L}_T$ , stabilizing training while preserving trajectory prediction as the primary objective.
+
+# 3.4 IMPLEMENTATION DETAILS
+
+RouteFormer is trained using the AdamW optimizer with a linear warm-up of 2 epochs and cosine annealing, over a total of 200 epochs. Maximum learning rate of  $1 \times 10^{-5}$  and weight decay of  $1 \times 10^{-4}$  are used with batch size 16. A full set of hyperparameters can be found in Appendix
+
+Regularization. We employ no dropout inside the transformers, but apply aggressive within-modality dropouts (e.g., dropping one of the two scene videos) to improve validation scores.
+
+Caching. To benefit from pre-trained vision backbones while keeping the training duration under control, we have implemented module-level caching that utilizes intermediate features extracted from these backbones. Details of the caching mechanism are explained in Appendix
+
+# 4 PATH COMPLEXITY INDEX
+
+A critical challenge in autonomous driving is tackling the long tail of driving scenarios - situations that are less frequent but potentially high-impact. These scenarios often encompass a diverse range of unpredictable driving conditions, such as navigating complex intersections or road obstructions. Traditional approaches often view the task as a classification problem, constrained by the limitations of hand labeling such events (Teeti et al., 2022; Girase et al., 2021; Palazzi et al., 2018). To automatically identify and quantify the complexity of driving situations, we introduce a novel metric termed Path Complexity Index (PCI). This metric evaluates the deviation from simple driving patterns, thereby providing a robust tool for assessing and improving trajectory prediction models.
+
+Given an input trajectory, the PCI metric computes the Fréchet distance between the target trajectory and a hypothetical simple trajectory in which the driver maintains the speed and direction exhibited in the last segments of the input trajectory. Formally, the PCI of an observed target trajectory  $\mathcal{T}_{target} \in \mathbb{R}^{T \times 2}$  given an input trajectory  $\mathcal{T}_{input} \in \mathbb{R}^{T' \times 2}$ , is given by:
+
+$$
+\begin{array}{l} P C I \left(\mathcal {T} _ {\text {t a r g e t}} \mid \mathcal {T} _ {\text {i n p u t}}\right) = \\ \inf  _ {\alpha , \beta \in [ 0, 1 ]} \max  \left(\sup  _ {t \in [ 0, 1 ]} \| \mathcal {T} _ {\text {t a r g e t}} (\alpha (t)) - \mathcal {T} _ {\text {s i m p l e}} (\beta (t)) \|\right) \\ \end{array}
+$$
+
+where  $\alpha : [0,1] \to [0,1]$  and  $\beta : [0,1] \to [0,1]$  are continuous non-decreasing functions that represent all path reparameterizations (Eiter & Mannila, 1994), and  $\mathcal{T}_{simple} \in \mathbb{R}^{T \times 2}$  is the simple trajectory derived by following the final velocity vector of  $\mathcal{T}_{input}$ :
+
+$$
+\mathcal {T} _ {\text {s i m p l e}} (t) = \mathcal {T} _ {\text {i n p u t}} (T) + v _ {\text {f i n a l}} \cdot t, \quad t \geq T
+$$
+
+Here,  $v_{final} = \mathcal{T}_{input}(T') - \mathcal{T}_{input}(T' - 1)$  represents the velocity vector estimated from the final two points of  $\mathcal{T}_{input}$ , and  $t$  is the time parameter extending beyond the duration of the input trajectory  $T'$ .  $\| \cdot \|$  denotes the Euclidean distance in the 2D plane.
+
+A high value of PCI indicates a significant deviation from the simple trajectory. Some generated examples of deviations from a straight input trajectory and their respective PCI range can be seen in Figure 3. A sample of real trajectories for these ranges are provided in Figure 4. We propose that a high PCI indicates intriguing events, which are often characterized as sudden changes in direction or speed in the literature (Girase et al., 2021). We provide more insights into this behavior in Appendix with a comprehensive breakdown of the PCI statistics in GEM dataset.
+
+Are all curved paths interesting? An essential task for trajectory prediction models is learning the road curvature from scene information. PCI is simplistically designed to not bias towards a particular type of curvature to encourage that while allowing us to eliminate instances of straight cruising that may dominate the training and evaluation phases.
+
+![](images/e08b1730c99364a210c51f48d909d58bac697e338949873cc0241eed7248e797.jpg)  
+Figure 3: Generated trajectories and their values. The black paths to the left are inputs, and the colored paths are targets generated exhaustively by varying the speed, turning angle, and turn curvature.
+
+![](images/96b069ff9ec7c5d21aee8be9dbd52fe0a4e864ae216fd125b66ad84712b19982.jpg)  
+Figure 4: Example trajectories with varying PCI. White is the input and red is the target trajectory.
+
+(a) Comparison of existing trajectory prediction datasets with ours. Only public datasets are considered. \*: hand-annotated.
+
+Table 1: Contribution of GEM. Our dataset is designed for ego-trajectory prediction with noise-free GPS positions and gaze data, in an entity-rich urban setting.  
+
+<table><tr><td>Dataset</td><td>Gaze</td><td>Noise-free pose</td><td>Scene cameras</td><td>Duration</td></tr><tr><td>Waymo (Ettinger et al., 2021)</td><td>X</td><td>✓</td><td>5</td><td>574h</td></tr><tr><td>nuScenes (Caesar et al., 2020)</td><td>X</td><td>✓</td><td>6</td><td>5.5h</td></tr><tr><td>Argoverse (Wilson et al., 2021)</td><td>X</td><td>✓*</td><td>7</td><td>305h</td></tr><tr><td>LOKI (Girase et al., 2021)</td><td>X</td><td>X</td><td>1</td><td>2.3h</td></tr><tr><td>DR(eye)VE (Palazzi et al., 2018)</td><td>✓</td><td>X</td><td>1 (+1 headcam)</td><td>6h</td></tr><tr><td>GEM (Ours)</td><td>✓</td><td>✓*</td><td>2 (+1 headcam)</td><td>5h</td></tr></table>
+
+(b) GEM vs. DR(eye)VE: percentage of frames with other agents, and total entity counts, both detected by YOLOv8 at 1 fps.  
+
+<table><tr><td></td><td>GEM</td><td>DR(eye)VE</td></tr><tr><td>% of frames with other drivers</td><td>89.7%</td><td>38.1%</td></tr><tr><td>% of frames with pedestrians</td><td>31.2%</td><td>22.5%</td></tr><tr><td># of cars</td><td>23331</td><td>3979</td></tr><tr><td># of pedestrians</td><td>3640</td><td>297</td></tr><tr><td># of buses</td><td>1215</td><td>155</td></tr><tr><td># of bicycles</td><td>357</td><td>18</td></tr></table>
+
+# 5 GAZE-ASSISTED EGO MOTION (GEM) DATASET
+
+The driver's gaze plays a significant role during driving, providing insights about their decisions on the road. Understanding this role may allow us to build systems to improve vehicular safety. However, training models in the gaze domain is constrained by data scarcity. The sole public dataset featuring ego-driver positions and gaze, DR(eye)VE, contains only 6 hours of data, only one-third of which is urban driving, hindering the capture of rare gaze-based cues. (Palazzi et al., 2018).
+
+We, therefore, propose GEM, a multimodal ego-motion with gaze dataset with synchronized multivacap video footage of the road scene and FOV data, with precise gaze locations from an eye-tracking headset, and GPS ego-vehicle locations, manually corrected for further accuracy. By coupling gaze data with the car's motion and the driving scene, our dataset provides a rich source of information for ego-trajectory prediction and modeling of drivers' behavior. The primary objective of our dataset is to enable the development and validation of models that can predict future ego-vehicle locations based on the driver head/eye movements and surrounding scenes.
+
+# 5.1 HARDWARE SETUP
+
+We mount high-resolution cameras on the vehicle to ensure high-quality recordings of the surrounding environment and record drivers' eye gaze using a head-mounted eye tracker.
+
+Frontal cameras and GPS tracking. To record the driving scene and car motion, we use two GoPro cameras with a 4K resolution and a  $30\mathrm{Hz}$  sampling rate. These cameras record the car's GPS positions in real-time and embed this information into the video streams. However, GPS data can have discrepancies due to various factors such as satellite interference or obstructions, which we further correct.
+
+Gaze tracking. We employ the Pupil Invisible glasses  ${}^{1}$  from Pupil Labs to track the driver's gaze. This device captures eye movements at  ${200}\mathrm{\;{Hz}}$  using two near-eye infrared cameras and includes a frontal video camera of the scene. The gaze tracker and GoPro cameras are fully synchronized in post-processing using the timestamp metadata.
+
+# 5.2 DATA COLLECTION
+
+10 participants were instructed to drive naturally on a variety of road types with a focus on urban areas, during different times of the day to capture a comprehensive range of driving conditions. Each participant drove for approximately 30 minutes, resulting in a total of 5 hours of collected data. See Appendix D.1 for a breakdown of the dataset.
+
+Procedure. Upon arrival, participants were briefed about the study and given a tutorial on the equipment. Before commencing the driving sessions, participants went through a gaze calibration process. They were asked to look at specific points on a board, ensuring that the eye-tracking device was accurately capturing their fixations. After fitting the Pupil Invisible glasses and ensuring calibration, participants were asked to start their driving session. A researcher was present in the car to oversee the equipment and address any unexpected situations.
+
+![](images/d4c51a9b80983a80a64c64fc8c05ddf5dd717a3107a35376ad07cc6631a99680.jpg)  
+Figure 5: Qualitative examples. RouteFormer shows higher confidence in sharp turns than other SOTA models using gaze, which tend to prefer the mean of the past trajectory (left). The turn confidence is lower when no driver FOV information is used (right).
+
+Post-processing. GPS technology inherently exhibits limitations, including potential positional inaccuracies up to  $1.82\mathrm{m}$  at the 95th percentile, as reported by the U.S. FAA (U.S. FAA NSTB/WAAS T&E Team, 2021). These inaccuracies are notably magnified in occluded environments, making the resulting data unsuitable for ground truth applications (see Appendix for examples). To mitigate this, we developed a user-operated application that allows for the manual correction of GPS markers to more accurate positions. This correction tool has been instrumental in enhancing the positional accuracy of all samples within the GEM dataset. The tool's source code will be released alongside the dataset. We provide the data split and detailed description of the dataset in the Appendix.
+
+# 5.3 COMPARISON WITH EXISTING DATASETS
+
+Most driving datasets to create road assistance systems lack either gaze data, noise-free vehicle trajectory, or sufficient quality scene information, as seen in Table 1 (a). To the best of the authors' knowledge, the only other public multimodal driving dataset with gaze data is DR(eye)VE, as presented by Palazzi et al. (2018). However, this dataset includes mostly rural/highway driving (63.5% of the data), while GEM is designed for complex urban trajectory prediction with many interacting agents. Our comparison in Table 1 (b), using YOLOv8 (Jocher et al., 2023), shows GEM's significant lead in terms of traffic-related content: it has other vehicles in 89.7% of frames (compared to DR(eye)VE's 38.1%), six times more cars, over ten times more pedestrians, and twenty times more bicycles than DR(eye)VE. For GEM, we further ensured the high quality of the GPS information, fixing the inherent GPS limitations through hand annotation, especially in occluded areas like tunnels (see in Appendix C for examples, and the dataset noise comparison).
+
+# 6 EXPERIMENTS
+
+We evaluate our method RouteFormer by measuring the predicted motion trajectories using a set of standard metrics. We compare against several state-of-the-art methods and baselines. See Appendix B for more results and examples.
+
+# 6.1 SETUP
+
+Datasets. We evaluate our method on our proposed GEM dataset. To demonstrate that the benefit from gaze extends beyond our dataset, we additionally train and evaluate our framework on a second dataset. DR(eye)VE (Palazzi et al., 2018) has 6 hours of driving along with gaze input, however, with a smaller number of traffic entities than GEM, and lower data quality.
+
+Task. Following the task setting in nuScenes (Caesar et al., 2020), our task uses 8-second input trajectories to predict the next 6-second motion. We sample synchronized sequences of location, videos, and gaze data with a sliding window of 2-second strides, resulting in approximately 16 hours of input for the training split. We choose different drivers for training and evaluation according to the data splits of either dataset (see Appendix D) to better demonstrate the benefit of FOV.
+
+Table 2: Comparison to baselines and SOTA. We use the test sets of both the proposed dataset and the DR(eye)VE dataset. Our model RouteFormer achieves the best performance on both datasets.  
+
+<table><tr><td>Method</td><td>ADE (m)</td><td>GEM ADE+20PCI (m)</td><td>DR(eye)VE ADE (m)</td><td>ADE+20PCI (m)</td></tr><tr><td>Stationary Baseline</td><td>32.04</td><td>30.22</td><td>64.11</td><td>61.41</td></tr><tr><td>Linear Baseline</td><td>7.37</td><td>13.37</td><td>10.16</td><td>15.54</td></tr><tr><td>GIMO (Zheng et al., 2022)</td><td>7.61</td><td>10.03</td><td>13.97</td><td>18.68</td></tr><tr><td>Multimodal Transformer (Li et al., 2021)</td><td>6.07</td><td>8.41</td><td>10.67</td><td>14.65</td></tr><tr><td>Ours - RouteFormer-Base (video-only)</td><td>6.13</td><td>8.14</td><td>9.95</td><td>13.15</td></tr><tr><td>Ours - RouteFormer (video+gaze)</td><td>5.99</td><td>7.70</td><td>8.75</td><td>12.26</td></tr></table>
+
+Table 3: Varying the prediction horizon. Average final displacement errors on GEM for predicting 1-6s in the future for test set samples with  $20+$  PCI, demonstrating RouteFormer's increasingly higher accuracy in long-term forecasting for complex samples, compared to RouteFormer-Base.  
+
+<table><tr><td>Method</td><td>FDE@1s</td><td>FDE@2s</td><td>FDE@3s</td><td>FDE@4s</td><td>FDE@5s</td><td>FDE@6s</td></tr><tr><td>RouteFormer-Base (video only)</td><td>1.74</td><td>4.19</td><td>7.08</td><td>10.56</td><td>14.45</td><td>18.45</td></tr><tr><td>RouteFormer (video+gaze)</td><td>1.68</td><td>4.04</td><td>6.72</td><td>9.93</td><td>13.61</td><td>17.37</td></tr><tr><td>Improvement</td><td>3.57%</td><td>3.71%</td><td>5.36%</td><td>6.34%</td><td>6.17%</td><td>6.21%</td></tr></table>
+
+Baselines. We use two simple baselines. Stationary baseline assumes that the vehicle remains stationary throughout the prediction window. Linear baseline assumes that the vehicle maintains a linear trajectory, following the vehicle's final direction and speed from the input.
+
+Motion prediction models. We compare our model with two gaze-assisted human motion prediction models adapted from the literature. Multimodal Transformer (Li et al., 2021) encodes the input modalities with linear layers and predicts the future trajectory with an encoder/decoder architecture. In addition, we implement GIMO (Zheng et al., 2022) to utilize scene videos using RouteFormer's own video encoder instead of 3D point clouds and predict the future trajectory with a cross-modal encoder architecture. The adaptations follow Zheng et al. (2022)'s adjustments in GIMO to compare with their baseline Multimodal Transformer (referred to in Tab. 2).
+
+Metrics. We use the standard evaluation metrics Average Displacement Error (ADE) and Final Displacement Error (FDE). ADE measures the average error between the ground truth and predicted trajectory across all time steps. FDE measures the displacement error in the final prediction time step, emphasizing the model's accuracy in predicting the endpoint of a trajectory.
+
+# 6.2 EVALUATION OF ROUTEFORMER
+
+Comparison to baselines and SOTA. We report results on the test sets of both our proposed dataset GEM and the DR(eye)VE dataset in Table 2. We train on 8 participants and test on 2 others to explore generalizability across people. Our model RouteFormer outperforms all baselines across both evaluation metrics and, therefore achieving state-of-the-art performance on both datasets.
+
+Qualitative evaluation. In Figure 5, we display the behavior of the proposed model compared to SOTA and baselines. RouteFormer shows better confidence in sharper turns compared to other models, predicts right exits in roundabouts, and overall has better performance in high-PCI situations.
+
+Evaluating the benefit of driver FOV. We compare RouteFormer with RouteFormer-Base, its version without driver FOV (Table 2). The additional modality in RouteFormer significantly improves performance, especially for later points in the prediction horizon (Table 3). This confirms the importance of driver field-of-view information in ego-trajectory prediction, particularly for long-term forecasts. Notably, RouteFormer achieves lane-level accuracy for shorter horizons even in challenging GEM samples, demonstrating potential for industrial short-term forecasting applications.
+
+Path complexity evaluation. We evaluated RouteFormer and RouteFormer-Base across different levels of path complexity using the PCI metric. Results in Figure 6 show that gaze data becomes increasingly valuable for more irregular trajectories. This demonstrates RouteFormer's ability to generalize driver FOV learning, improving predictions for complex paths across different drivers.
+
+Gaze fixation importance. Throughout this study, we consider driver's FOV as a composition of a first-person video feed and gaze fixations, as they are highly interconnected and novel remote tracking systems (such as Smart Eye Pro) can capture them together (Smart Eye AB, 2024). To verify the importance of gaze fixations beyond first-person video, we analyzed their attribution using Inte
+
+![](images/4f05b745c4ea445a3ffde4b8850728d4681c3e75d09b9f534177863525d65b68.jpg)  
+Figure 6: The effect of gaze over different PCI. Lower is better for ADE. We bin GEM's test set by PCI, displaying average ADE per group. The line represents the ADE difference between RouteFormer-Base and RouteFormer, with higher values favoring RouteFormer.
+
+![](images/930534aa6a09369fa7b6f07ca16f28dc96aac785fca10b50cdeb06f50dc7f951.jpg)  
+Figure 7: Attribution to gaze fixations across different PCI bins and prediction horizons. Gaze fixations increasingly affect the prediction for further away horizons and medium/high PCI values. Estimated using Integrated Gradients (Sundararajan et al., 2017).
+
+grated Gradients (Figure 7). Fixations significantly impact the final result, with increased reliance for later prediction points and higher PCI samples. This confirms that RouteFormer effectively utilizes both video and gaze data, despite having the same parameter count as RouteFormer-Base.
+
+# 6.3 ABLATION STUDIES
+
+We performed a set of ablation studies to choose the right vision backbones and time series modules for RouteFormer, and justify the discounted/auxiliary losses.
+
+Auxiliary and discounted losses. As one of the major contributions of our framework, we assess the effect of the future-discounted loss  $\mathcal{L}_{fd}$  and the auxiliary losses. Both additions show significant improvement over the standard architecture on the validation set (Table 4).
+
+Time-series module. Informer (Zhou et al., 2021) is a Transformer-based model and leverages the ProbSparse attention mechanism. Table 5 shows that Informer consistently outperforms later models such as Autoformer (Wu et al., 2021), LTSF-Linear (Zeng et al., 2023), and PatchTST (Nie et al., 2023). We used Informer in RouteFormer.
+
+Table 4: Loss ablations. Results over GEM validation set for  $20+$  PCI samples.  
+
+<table><tr><td>Method</td><td>Val. ADE (m)</td><td>Val. FDE (m)</td></tr><tr><td>Vanilla RouteFormer</td><td>8.85</td><td>58.54</td></tr><tr><td>+ auxiliary losses</td><td>8.68 (-0.17)</td><td>57.09 (-1.45)</td></tr><tr><td>+ future-discounted loss</td><td>8.27 (-0.41)</td><td>54.38 (-2.71)</td></tr></table>
+
+Table 5: Visual encoder and time-series module ablations. Results over GEM val. set for  $20+$  PCI samples. Time-series modules (top) perform prediction without scene or gaze data. Pretrained vision modules (bottom) use Informer.  
+
+<table><tr><td>Method</td><td>Val. ADE (m)</td><td>Val. FDE (m)</td></tr><tr><td>DLinear (Zeng et al., 2023)</td><td>10.67</td><td>81.40</td></tr><tr><td>NLinear (Zeng et al., 2023)</td><td>10.49</td><td>83.38</td></tr><tr><td>PatchTST (Nie et al., 2023)</td><td>10.23</td><td>79.80</td></tr><tr><td>Transformer (Vaswani et al., 2017)</td><td>9.97</td><td>67.88</td></tr><tr><td>Informer (Zhou et al., 2021)</td><td>9.62</td><td>66.14</td></tr><tr><td>Informer+SAM (Kirillov et al., 2023)</td><td>9.08</td><td>57.89</td></tr><tr><td>Informer+DinoV2 (Oquab et al., 2023)</td><td>8.69</td><td>60.83</td></tr><tr><td>Informer+SwinV2 (Liu et al., 2021)</td><td>8.51</td><td>55.56</td></tr></table>
+
+Pretrained vision backbone. In Table 5, we evaluate the impact of various visual encoders, including SAM (Segment Anything) (Kirillov et al., 2023), DinoV2 (Oquab et al., 2023) and SwinV2 (Liu et al., 2021). SwinV2 consistently demonstrated superior performance on the validation set with embedding sizes smaller than its counterparts.
+
+# 7 CONCLUSION
+
+Our results highlight the significance of incorporating driver FOV in predicting ego-vehicle trajectories. Through our innovative multimodal method, RouteFormer, which integrates FOV, scene and motion data, we achieve superior performance over existing baselines, pushing the boundaries of state-of-the-art achievements in this domain. Specifically, our approach demonstrates that incorporating first-person video and gaze fixations data enhances the prediction of complex non-linear trajectories. By establishing a new benchmark with our GEM dataset, we pave the way for further intuitive and human-centric developments in assisted driving technologies.
+
+# ACKNOWLEDGEMENTS
+
+INSAIT, Sofia University "St. Kliment Ohridski". Partially funded by the Ministry of Education and Science of Bulgaria's support for INSAIT as part of the Bulgarian National Roadmap for Research Infrastructure. This project was supported with computational resources provided by Google Cloud Platform (GCP). This project is also partially funded by the German Federal Ministry of Education and Research through the ExperTeam4KI funding program for UDance (Grant No. 01IS24064).
+
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+
+# APPENDIX
+
+# A RouteFormer Implementation Details 17
+
+A.1 RouteFormer Notation 17  
+A.2 Module-Level Caching with TORCHCACHED 18
+
+# B Experiments 19
+
+B.1 Evaluation Metrics 19  
+B.2 Hyperparameters 19  
+B.3 Analysis of Failure Cases 19  
+B.4 Ablation Studies on Modality Contributions 20  
+B.5 Extended Baseline Selection Discussion 21
+
+# C GEM Dataset 22
+
+C.1 Ground Truth Correction with TRAJAPP 22  
+C.2 GEM vs. DR(eye)VE Noise Comparison 22  
+C.3 Gaze Fixation Detection 22  
+C.4 Gaze Alignment and Robustness 24  
+C.5 Exploring Image Homography and Stitching 24
+
+# D PCI Metric 27
+
+D.1 Breakdown of GEM Dataset and PCI Insights 27  
+D.2 Visualizing Path Complexity Index 27  
+D.3 PCI Analysis on Urban Driving 28
+
+# A ROUTEFORMER IMPLEMENTATION DETAILS
+
+All experiments are carried out on a machine with an NVIDIA GPU with  $>10$  GB memory, 64 GB of RAM, and an Intel CPU. The framework is implemented using PyTorch 2.0.
+
+# A.1 ROUTEFORMER NOTATION
+
+Using the notation we have established, our full model can be described as:
+
+$$
+\begin{array}{l} \Psi \left(M _ {1: T}, S _ {1: T} ^ {1: K}, F _ {1: T} | \lambda\right) \equiv \\ \mathcal {T} \left(M _ {1: T}, E _ {P} \left(E _ {S} \left(S _ {1: T} ^ {1: K}\right), E _ {F} \left(F _ {1: T}\right)\right)\right) = \\ \left\{M _ {T: T _ {\text {p r e d}}} ^ {\prime}, v _ {T: T _ {\text {p r e d}}} ^ {\prime} \right\} \\ \end{array}
+$$
+
+Loss definitions. For the model outputs  $\Psi\left(M_{1:T}, S_{1:T}^{1:K}, F_{1:T}|\lambda\right) = \{M_{T:T_{\mathrm{pred}}}^{\prime}, v_{T:T_{\mathrm{pred}}}^{\prime}\}$ , let  $\mathcal{L}_T$  denote the discounted trajectory loss, and  $\mathcal{L}_V$  the discounted visual loss. The auxiliary losses are defined as:
+
+$$
+\begin{array}{l} \mathcal {L} _ {T} = \mathcal {L} _ {\mathrm {f d}} \left(M _ {T: T _ {\mathrm {p r e d}}} ^ {\prime}, M _ {T: T _ {\mathrm {p r e d}}}\right) \\ \mathcal {L} _ {V} = \mathcal {L} _ {\mathrm {f d}} \left(v _ {T: T _ {\mathrm {p r e d}}} ^ {\prime}, E _ {P} (E _ {S} (S _ {T: T _ {\mathrm {p r e d}}} ^ {1: K}), E _ {F} (F _ {T: T _ {\mathrm {p r e d}}}))\right). \\ \end{array}
+$$
+
+# A.2 MODULE-LEVEL CACHING WITH TORCHCACHE
+
+Caching is a crucial component of our training regime. During the initial epochs, the intermediate representations from the pre-trained vision backbones are computed for each input frame and stored in an efficient cache memory structure. When subsequent epochs require the same input data, the pre-computed representations are fetched from the cache, thereby eliminating the need for repetitive and costly forward passes through the vision backbones.
+
+To make this process repeatable, we implemented the TORCHCACHE utility to provide a generic caching mechanism. The TORCHCACHE library offers an easy way to cache the outputs of PyTorch modules, making it particularly suited for our need to efficiently cache the outputs of large, pre-trained vision transformers. The package of this mechanism is open-sourced at github.com/meakbiyik/torchcache.
+
+# A.2.1 KEY FEATURES
+
+The TORCHCACHE library offers several notable features:
+
+- Allows caching of PyTorch module outputs in two formats: in-memory for fast retrieval, and persistently on the disk for larger datasets.  
+- Adopt a decorator-based interface that simplifies its integration with PyTorch modules.  
+- Implemented an MRU ( most-recently-used) cache policy to manage and limit memory or disk consumption.
+
+# A.2.2 USAGE
+
+The library's decorator-based design ensures minimal modifications to the existing PyTorch codebase. By adding the @torchcache() decorator, the output of the module's forward method gets cached, as illustrated in the provided basic usage example:
+
+```python
+from torchcache import torchcache   
+@torchcache()   
+class MyModule(nn.Module): def init(self): super().init() self.Linear  $\equiv$  nn.Linear(10, 10) def forward(self, x): # This output will be cached return self.Linear(x)
+```
+
+More detailed examples, advanced usage patterns, and further documentation will be accessible at the official TORCHCACHE documentation.
+
+# A.2.3 OPERATIONAL ASSUMPTIONS
+
+For the TORCHCACHED library to operate seamlessly, there are certain assumptions that the PyTorch modules should adhere to:
+
+1. The module should be a subclass of nn-module.  
+2. The forward method of the module should be capable of accepting any number of positional arguments with shapes denoted by  $(B,*)$ , where  $B$  stands for the batch size, and * denotes any other dimensions.  
+3. All input tensors should be located on the same computational device and should share the same data type (dtype).  
+4. The method should return a single tensor with the shape  $(B,*)$ .
+
+The TORCHCACHED library has proven invaluable in our experiments by allowing us to efficiently utilize the outputs of our pre-trained vision transformers without incurring significant computational overhead.
+
+![](images/c4a117b61ac072bd741a3906c088c0ab1583a2c0b3968a72d39c2ee58f1b23be.jpg)  
+(a)  $t = 0\mathrm{s}$
+
+![](images/099b103e3882bf0e5645e59d88b77f4090eb284356ef7eca2f403c0f84f14de3.jpg)
+
+![](images/d1109a29185a5502807be1850aad82dff12926849b30068357f82e5e695701c7.jpg)  
+(c)  $t = 2\mathrm{s}$
+
+![](images/e11e93baba12114e92228d868b2235aabc1b292081c0d45d30a60c718ec0732b.jpg)  
+(d)  $t = 3s$
+
+![](images/bbbd67cdf362d14a1051001b0ad075e30291235eb0efa55df69f1fd82d0208e8.jpg)  
+(e)  $t = 4\mathrm{s}$
+
+![](images/ca814e9d9d7deff43096831c6ddb0db00381ccaea52e44f407665c9c3414f390.jpg)  
+(b)  $t = 1\mathrm{s}$
+
+![](images/7fbc1be6c3ba38b16a500ebe8125da7a2f0d97d1e2dc64f7fe213250d9eafd1c.jpg)  
+Figure 8: A sample from the GEM dataset. The top row shows frames with gaze heatmap overlay on top, at  $1\mathrm{Hz}$ . The bottom left image displays the trajectory of the vehicle, while the bottom right plot shows the speed changes. This is an example of gaze-indicated short-term intent: the driver decelerates as soon as their gaze meets with the brake lights of the vehicle in front.
+
+# B EXPERIMENTS
+
+# B.1 EVALUATION METRICS
+
+The Average Displacement Error (ADE) is formally defined as:
+
+$$
+\mathrm {A D E} = \frac {1}{T} \sum_ {t = 1} ^ {T} \| \mathbf {p} _ {t} ^ {\text {p r e d}} - \mathbf {p} _ {t} ^ {\mathrm {g t}} \| _ {2}, \tag {1}
+$$
+
+where  $T$  is the total number of prediction time steps,  $\mathbf{p}_t^{\mathrm{pred}}$  is the predicted position at time  $t$ , and  $\mathbf{p}_t^{\mathrm{gt}}$  is the ground truth position at the same time.
+
+The Final Displacement Error (FDE) is the displacement error at the final prediction time step:
+
+$$
+\mathrm {F D E} = \left\| \mathbf {p} _ {T} ^ {\text {p r e d}} - \mathbf {p} _ {T} ^ {\mathrm {g t}} \right\| _ {2}. \tag {2}
+$$
+
+# B.2 HYPERPARAMETERS
+
+The hyperparameters for the experiments can be found in Table 6, Table 7, Table 8, and Table 9.
+
+# B.3 ANALYSIS OF FAILURE CASES
+
+We observe two primary categories of prediction failures:
+
+1. Occlusions in right turns: As shown in Figure 9a and 9b, trees and parked vehicles occlude crucial parts of the upcoming path during right turns. While human drivers naturally compensate for such occlusions through experience, our model shows increased uncertainty in these scenarios. Reliance on additional information from the car, such as turn signals or LIDAR, can improve driver behavior prediction for such instances.  
+2. Complex intersections: Figure 9c and 9d demonstrate varying accuracy in predicting motion at complex intersections and turns. The model particularly struggles determining when the driver is going to stop or resume in complex intersections (Figure 9c) and when multiple path options are available (Figure 9d). This limitation suggests an opportunity to incorporate past driver behavior to model driver initiative/aggression.
+
+Table 6: Training configurations  
+
+<table><tr><td>Parameter</td><td>Value</td></tr><tr><td>Optimizer</td><td>AdamW</td></tr><tr><td>Learning rate</td><td>1e-5</td></tr><tr><td>Weight decay</td><td>1e-4</td></tr><tr><td>Epochs</td><td>200</td></tr><tr><td>Epsilon</td><td>1</td></tr><tr><td>Visual epsilon</td><td>0.3</td></tr><tr><td>Scene video FPS</td><td>1</td></tr><tr><td>Gaze video FPS</td><td>1</td></tr><tr><td>Batch size</td><td>16</td></tr><tr><td>Output FPS</td><td>5</td></tr><tr><td>Loss discount factor</td><td>0.97</td></tr></table>
+
+Table 7: Framework configurations  
+
+<table><tr><td>Parameter</td><td>Value</td></tr><tr><td>Encoder heads</td><td>8</td></tr><tr><td>Encoder layers</td><td>8</td></tr><tr><td>Feature dropout</td><td>0.05</td></tr><tr><td>Encoder hidden size</td><td>64</td></tr><tr><td>Gaze decoder layers</td><td>2</td></tr><tr><td>Gaze dropout</td><td>0.2</td></tr><tr><td>View dropout</td><td>0.6</td></tr><tr><td>Front video scaling factor</td><td>0.3</td></tr><tr><td>Scene video scaling factor</td><td>0.1</td></tr><tr><td>Gaze decoder dropout</td><td>0.05</td></tr><tr><td>Image embedding size</td><td>64</td></tr><tr><td>Dense visual loss ratio</td><td>0.5</td></tr></table>
+
+Table 8: GPS backbone (Informer) configurations  
+
+<table><tr><td>Parameter</td><td>Value</td></tr><tr><td>Distil</td><td>true</td></tr><tr><td>Factor</td><td>4</td></tr><tr><td>Model dimension</td><td>832</td></tr><tr><td>Dropout</td><td>0.0</td></tr><tr><td>Number of Heads</td><td>8</td></tr><tr><td>Sequence Length</td><td>40</td></tr><tr><td>Decoder layers</td><td>1</td></tr><tr><td>Encoder layers</td><td>6</td></tr><tr><td>Prediction Length</td><td>30</td></tr><tr><td>Label Length</td><td>40</td></tr><tr><td>Activation</td><td>relu</td></tr><tr><td>Individual</td><td>false</td></tr><tr><td>Moving average</td><td>25</td></tr></table>
+
+Table 9: Video backbone (SwinV2) configuration  
+
+<table><tr><td>Parameter</td><td>Value</td></tr><tr><td>Model Type</td><td>swinv2 window12to16</td></tr><tr><td>Pad to square</td><td>true</td></tr><tr><td>Train backbone</td><td>false</td></tr></table>
+
+# B.4 ABLATION STUDIES ON MODALITY CONTRIBUTIONS
+
+Following our architecture in Figure 2(a), we conducted ablation experiments by selectively removing modality branches from RouteFormer during inference. For FOV removal, we bypass the cross-modal transformer and use only scene features for self-attention. For scene removal, we retain only the FOV branch. When testing motion-only performance, we replace the entire visual feature tensor  $v_{1:T}$  with zeros.
+
+<table><tr><td>Method</td><td>ADE (m)</td><td>Δ ADE (%)</td><td>ADE+20PCI (m)</td><td>Δ ADE+20PCI (%)</td></tr><tr><td>RouteFormer (full)</td><td>5.99</td><td>Baseline</td><td>7.70</td><td>Baseline</td></tr><tr><td>motion+scene (w/o FOV)</td><td>6.60</td><td>+10.2</td><td>8.58</td><td>+11.4</td></tr><tr><td>motion+FOV (w/o scene)</td><td>7.83</td><td>+30.7</td><td>11.17</td><td>+45.1</td></tr><tr><td>motion only</td><td>8.19</td><td>+36.7</td><td>11.22</td><td>+45.7</td></tr></table>
+
+Table 10: Ablation study results showing degraded performance in removing different modalities.
+
+The results, show in Tab. 10, demonstrate the clear benefits of each modality. Notably, our scene-only variant maintains performance consistent with RouteFormer-Base (6.13m), and our FOV-only variant remains competitive with GIMO (7.61m). This stability across modality configurations is particularly valuable for real-world deployment, where sensors or gaze tracking might be temporarily unavailable.
+
+![](images/f150dd73198c1008fb905551ed92eb2512f4234bd868637cc55aa14044d0422a.jpg)
+
+![](images/d9b2be624cd138745944fa4eff1c1a43e3e5d6e8e675fb6b93eb3a93dcd9a201.jpg)  
+(b) A right turn occluded by trees
+
+![](images/5ef2775dd20d7a4ec15ec635cb79a3988b18b8678c815c0c4f8ef60cd1891def.jpg)  
+(a) An unanticipated right turn  
+(c) Driver stops while model predict right turn  
+Figure 9: Failure cases analysis showing different scenarios where the model prediction deviates from ground truth (green). Blue is RouteFormer, red is Routeformer-Base (without FOV modality), and white is the input trajectory.
+
+![](images/d5ad1eaa940c37254b5ddb7e07496ae08f884468db76a9208068d6383b4890e8.jpg)  
+(d) Driver does not stop while models do
+
+# B.5 EXTENDED BASELINE SELECTION DISCUSSION
+
+We detail here our approach to baseline selection and the challenges in adapting vehicle-specific models to our task. As the recent state-of-the-art methods work with multi-agent related traffic information, we consider such models (Autobots Girgis et al. (2021), MTR (Shi et al., 2022b)) as potential baselines.
+
+We adapted Autobots Girgis et al. (2021) to our datasets by providing past ego-trajectory (ego_in) and setting other inputs (agents_in, road graph) to zeros/placeholders. The model's performance was significantly worse than our baselines (ADE of 10.87m on GEM). This underperformance stems from the fundamental mismatch between the input modalities and architectural design of these models, which are optimized for rich environmental context from multi-agent interactions rather than driver-centric prediction. The key architectural differences between ego-trajectory and multi-agent prediction approaches are summarized in Tab. 11.
+
+Given the constraints, we focused on baselines that enable fair evaluation of our core contribution - integrating driver FOV data for trajectory prediction. GIMO and Multimodal Transformer
+
+<table><tr><td>Feature</td><td>MTR/Autobots</td><td>RouteFormer (Ours)</td></tr><tr><td>Primary Input</td><td>BEV, HD maps, LIDAR</td><td>Egocentric video, FOV data</td></tr><tr><td>Target Output</td><td>Multi-agent trajectories</td><td>Ego-vehicle trajectory</td></tr></table>
+
+Table 11: Architectural differences between ego-motion and multi-agent prediction approaches.
+
+were selected because they support similar input modalities while representing the state-of-the-art in attention-based trajectory prediction.
+
+# C GEM DATASET
+
+In this work, we proposed GEM, a multimodal ego-motion with gaze dataset with synchronized multi-cam video footage of the road scene, precise gaze locations from an eye-tracking headset and GPS ego-vehicle locations, manually corrected for further accuracy. We provide an additional sample scene from the dataset in Figure 8, demonstrating the connection between driver FOV and the short-term intentions of the driver.
+
+# C.1 GROUND TRUTH CORRECTION WITH TRAJAPP
+
+GPS data, particularly those collected using devices like GoPro cameras, can often be noisy and potentially inaccurate due to various reasons, ranging from satellite interference to obstructions like tall buildings or tunnels. Such inaccuracies in geospatial data can severely hamper the reliability and utility of a dataset, especially when the data is intended for detailed analysis and modeling. Examples of such cases, as well as milder issues, can be seen in Figure 10.
+
+In response to this challenge, we have developed the desktop tool TRAJAPP, designed to facilitate the correction of noisy GPS data. The primary strength of TRAJAPP lies in its unique user interface which juxtaposes the video feed and the associated GPS location points on a map. This side-by-side presentation enables users to cross-verify the video content with the GPS coordinates visually.
+
+Using the application is intuitive and straightforward, an example of which can be seen in Figure 11. As users watch the video playback, they can simultaneously observe the GPS path on the map. If discrepancies are noted, users can manually re-label the GPS data by simply clicking on the map and dragging the points to the desired accurate location. One key feature of TRAJAPP is its interpolation mechanism. Once users adjust certain key points on the map, the application automatically interpolates the GPS data between these points, ensuring smooth and consistent transitions.
+
+Designed with user-friendliness in mind, TRAJAPP is cross-platform, with versions available for Windows, MacOS, and Linux. This ensures that researchers and dataset curators across diverse technological environments can access and utilize the tool to enhance the accuracy of their geospatial data. The application can be downloaded for free at github.com/meakbiyik/traj-app.
+
+# C.2 GEM VS. DR(EYE)VE NOISE COMPARISON
+
+To demonstrate the difference in degree of noise between the proposed GEM dataset and the driving gaze saliency dataset from the literature, DR(eye)VE, we plotted the distribution of the GPS distances to the nearest road center for each dataset in Figure 12. The road centers are gathered from the public data in OpenStreetMap. For GEM, the distance distribution is centered around 1.25 meters, which is consistent with the average lane width of 3 meters in urban settings, while the distribution is considerably more heavy-tailed in the case of DR(eye)VE, indicating significant GPS noise.
+
+# C.3 GAZE FIXATION DETECTION
+
+For our fixation detection, we adopted the methodology from Pupil Labs, the creators of the glasses embedded with gaze detectors (Tonsen et al., 2020). Our criteria define gaze fixations as instances
+
+![](images/40459b44eb7ca1ecc3e1b259ff32f1a6a8e845a9d31b491ebbaadb67efd83a6f.jpg)
+
+![](images/62ff32424fe9423a165c6aba77fb3e288c073fcb3f6175d54471f331d9c27067.jpg)
+
+![](images/d424784976588ca431519a546936878d29b1d32b3a273fbb1fcd974d885aa056.jpg)  
+(a) A bad GPS sample from a scene where the car drives through a tunnel.  
+(c) Left turn with slow-down.
+
+![](images/b4eee6db762dfc12c4194323dfca7adeaed539cfdb11004af895ac526375090f.jpg)  
+(b) Consistent offset while driving straight.  
+(d) Lane change and a stop.
+
+![](images/1bbf56798c3934cbd27cc50ba0e0b83f3b4e523298b732f30ee877fad0f2c55b.jpg)  
+Figure 10: Example trajectory fixes with TRAJAPP. The original GPS points are in gray, and the hand-placed markers are blue. The new markers are interpolated with a cubic spline with a color that represents the instantaneous speed: red for slow, and green for fast movement.  
+Figure 11: User interface for GPS refinement. We show a video and its corresponding GPS locations on a map concurrently. An annotator places correct markers on the map such that the GPS locations align with the video observations.
+
+![](images/d769796365795288a3e60c860cbb70fe7bac86e1955f9d58d1795563c0947429.jpg)  
+Figure 12: GPS distances to the nearest road center. Estimated via OpenStreetMap. DR(eye)VE's GPS points are further away from road centers, indicating significant noise.
+
+where the gaze remains relatively stationary for a duration ranging from 80ms to 1 second. The upper limit of this range, 1 second, could also characterize a smooth pursuit, where the eyes smoothly track a moving object. Furthermore, we account for dispersion, which represents the spatial movement permitted within a single fixation. A gaze group is considered a fixation if the dispersion is within a limit of 1.5 degrees, ensuring that minor involuntary or random gaze movements do not undermine the detection of genuine fixations.
+
+A set of example fixations can be seen in Figure 14. Note that there are frames without any detected fixations, which is explained by saccades and blinks. For others, the accuracy is high: note how the gaze meets exactly with the brake light of the bus in front, in Scene 1, Frame 9. Similarly, the gaze over the motorcycle in front is well-centered in Scene 5 for most frames.
+
+# C.4 GAZE ALIGNMENT AND ROBUSTNESS
+
+Our framework processes driver gaze data following established practices in eye-tracking research (Hayhoe et al., 2003; Land, 2006). To align the high-frequency gaze measurements (200 Hz) with video frames (5 Hz), we use median downsampling. This approach ensures robust position estimates while preserving the temporal characteristics of driver attention patterns.
+
+To evaluate our model's robustness to potential measurement errors, we conducted additional experiments on the GEM dataset. As shown in Table 12, artificially introducing gaze position noise of  $\pm 50$  pixels (approximately  $5\%$  error) results in minimal degradation of prediction accuracy (0.01m increase in ADE). This resilience can be attributed to our "field-of-view" approach, which treats gaze as supplementary to the driver's head-mounted video feed rather than as a primary input.
+
+<table><tr><td>Configuration</td><td>ADE (m)</td><td>ADE+20PCI (m)</td></tr><tr><td>RouteFormer - without noise</td><td>5.99</td><td>7.70</td></tr><tr><td>RouteFormer - with ±50px noise</td><td>6.00</td><td>7.72</td></tr><tr><td>RouteFormer-Base (without FOV)</td><td>6.13</td><td>8.14</td></tr></table>
+
+Table 12: Impact of gaze measurement noise on prediction accuracy. The minimal change in ADE demonstrates our model's robustness to gaze uncertainty. Top and bottom rows are from Table 2
+
+These results demonstrate that our processing pipeline, built on standard gaze analysis techniques, effectively handles both natural gaze patterns and potential measurement uncertainties.
+
+# C.5 EXPLORING IMAGE HOMOGRAPHY AND STITCHING
+
+In our dataset, we provide synchronized videos from different viewpoints. In an attempt to provide a fully registered view from multiple cameras to our model, we have explored approaches for stitching them. However, we have discovered that using the separate videos brings benefits:
+
+- Using the current state of the art, it is not feasible to register the driver headcam and the scene videos within an acceptable error margin. We experimented with classical feature extractors such as DISK (Tyszkiewicz et al., 2020) and transformer-based detector-free local feature matching (Sun et al., 2021), using Kornia library (Riba et al., 2019).
+
+- We have observed that the redundancy from having multiple scene camera feeds can be exploited for training regularization. By dropping out one of the views at each frame during training, we improved the validation performance.
+
+![](images/865234babcc4275a197d22219c956610d99bd66616eef19ed6b431f0aa0b65e8.jpg)  
+Figure 13: Successful and failed stitching examples. We used LoFTR for feature-matching across scene videos and the head-cam.
+
+![](images/535f709e0e5654ff001ef18a0299fbf97aef71c9a3d8ce30ef477c8e5038d4ad.jpg)
+
+Example frames from the stitching experiments can be seen in Figure 13.
+
+(a) Frame 1 (b) Frame 2 (c) Frame 3 (d) Frame 4 (e) Frame 5 (f) Frame 6 (g) Frame 7 (h) Frame 8 (i) Frame 9 (j) Frame 10  
+![](images/84b7e8b677b66fa5f60b0e3550d76e8739223b284606c87db56db1149af02218.jpg)  
+Figure 14: Visual representation of gaze heatmaps across different scenes in the dataset. Each scene row is visualized with a sample of 10 frames with a one-second time difference, illustrating gaze patterns over time. Note that fixations are not always available due to blinks and saccades.
+
+![](images/07a486e703a6b1d9b663c1b9c8c9300529a2fc7d45a0ea7262e892422dea8df0.jpg)  
+Figure 15: Function of PCI with respect to the distance to an intersection. Speed range-the difference between the highest and lowest instantaneous speed throughout the trajectory-is chosen instead of speed as high yet constant speed without a change in direction (e.g. cruising on a highway) is a low-PCI event. Bins with multiple samples are displayed, with a median speed range per bin. PCI is highly correlated with distance to an intersection, but only if a vehicle changes its speed throughout the trajectory, which discards stoppings and straight driving cases.
+
+Table 13: Breakdown of our GEM dataset. We report duration, weather condition, and PCI for each participant.  
+
+<table><tr><td>Subject</td><td>Total Distance (m)</td><td>Rainy</td><td>PCI (mean)</td><td>PCI (max)</td><td>Split</td></tr><tr><td>001</td><td>13,112</td><td>✓</td><td>12.37</td><td>92.99</td><td>train</td></tr><tr><td>002</td><td>12,824</td><td>✓</td><td>10.76</td><td>81.22</td><td>val.</td></tr><tr><td>003</td><td>12,997</td><td>X</td><td>20.09</td><td>94.87</td><td>train</td></tr><tr><td>004</td><td>13,064</td><td>X</td><td>14.59</td><td>79.30</td><td>val.</td></tr><tr><td>005</td><td>12,872</td><td>X</td><td>17.29</td><td>86.39</td><td>train</td></tr><tr><td>006</td><td>13,127</td><td>X</td><td>13.82</td><td>88.00</td><td>train</td></tr><tr><td>007</td><td>12,973</td><td>X</td><td>16.42</td><td>77.47</td><td>train</td></tr><tr><td>008</td><td>13,700</td><td>✓</td><td>17.02</td><td>89.48</td><td>test</td></tr><tr><td>009</td><td>12,958</td><td>X</td><td>15.30</td><td>103.96</td><td>test</td></tr><tr><td>010</td><td>13,048</td><td>X</td><td>14.63</td><td>81.42</td><td>train</td></tr></table>
+
+# D PCIMETRIC
+
+# D.1 BREAKDOWN OF GEM DATASET AND PCI INSIGHTS
+
+For the information on the dataset split and the behavior of each participant, see Table 13.
+
+A deeper dive into our dataset, combined with synchronization with OpenStreetMap (OpenStreetMap contributors, 2017), reveals that the high PCI events indeed happen either close to the intersections, or on trajectories with a high-speed change, as observed in Figure 15.
+
+# D.2 VISUALIZING PATH COMPLEXITY INDEX
+
+Figure 16 illustrates the key components of PCI computation. Given an input trajectory (gray), we extrapolate a simple trajectory (dashed) by maintaining the final velocity vector. PCI measures the Fréchet distance (red) in meters between this simple trajectory and the actual target path (blue), effectively quantifying how much the actual path deviates from the simple projection. This deviation captures the complexity of the maneuver - higher deviations indicate more complex trajectories, requiring additional context to predict accurately. Unlike an alternative metric like mean-squared
+
+![](images/d3e603f5513214c9d2ab21201d4c260f7f7c7735e3e0203aa2cfd68c297ef051.jpg)  
+Figure 16: A simple representation of PCI as a concept.
+
+error, Fréchet distance is more stable against speed changes, better capturing differences in trajectory rather than point-by-point error.
+
+# D.3 PCI ANALYSIS ON URBAN DRIVING
+
+Figure 17 visualizes PCI values across a representative urban drive in GEM. High PCI regions (bright yellow) consistently align with challenging scenarios: roundabout exits (top-right,  $\mathrm{PCI} > 80$ ), lane merges after intersections (bottom-left,  $\mathrm{PCI} > 60$ ), and sharp turns exceeding 45 degrees (top-left and bottom-right). Notably, slower 90-degree turns exhibit lower PCI values due to reduced divergence from simple trajectories, demonstrating PCI's ability to distinguish the degree of challenge to predict each target curve.
+
+![](images/6de78a8e9aff5e661cb47f6c21b9d54163582764fb71e4ef105419a39f33800f.jpg)  
+Figure 17: Average PCI values of a full trajectory of a video in GEM. Estimated by running a sliding window of 1 second across the whole trajectory with 6s-8s input-target length, and averaging the PCI values assigned to each point. Brightest yellow parts map to  $\sim 80$  PCI while the dark blue is at 0 PCI.
\ No newline at end of file
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+# LEVERAGING FLATNESS TO IMPROVE INFORMATION-THEORETIC GENERALIZATION BOUNDS FOR SGD
+
+Ze Peng†, Jian Zhang†, Yisen Wang‡, Lei Qi◇, Yinghuan Shi†*, Yang Gao†
+
+† State Key Laboratory for Novel Software Technology, Nanjing University
+
+${}^{ \ddagger  }$  State Key Lab of General Artificial Intelligence,
+
+School of Intelligence Science and Technology, Peking University
+
+$\diamond$ School of Computer Science and Engineering, Southeast University
+
+{pengze, zhangjian7369}@smail.nju.edu.cn,yisen.wang@pku.edu.cn,
+
+qilei@seu.edu.cn, {syh, gaoy}@nju.edu.cn
+
+# ABSTRACT
+
+Information-theoretic (IT) generalization bounds have been used to study the generalization of learning algorithms. These bounds are intrinsically data- and algorithm-dependent so that one can exploit the properties of data and algorithm to derive tighter bounds. However, we observe that although the flatness bias is crucial for SGD's generalization, these bounds fail to capture the improved generalization under better flatness and are also numerically loose. This is caused by the inadequate leverage of SGD's flatness bias in existing IT bounds. This paper derives a more flatness-leveraging IT bound for the flatness-favoring SGD. The bound indicates the learned models generalize better if the large-variance directions of the final weight covariance have small local curvatures in the loss landscape. Experiments on deep neural networks show our bound not only correctly reflects the better generalization when flatness is improved, but is also numerically much tighter. This is achieved by a flexible technique called "omniscient trajectory". When applied to Gradient Descent's minimax excess risk on convex-Lipschitz-Bounded problems, it improves representative IT bounds'  $\Omega(1)$  rates to  $O(1/\sqrt{n})$ . It also implies a by-pass of memorization-generalization trade-offs.
+
+# 1 INTRODUCTION
+
+Over-parameterized deep models trained by Stochastic Gradient Descent (SGD) are observed to generalize well, contradicting classic statistical learning theories that over-parameterization leads to overfitting (Vapnik et al., 1998; Bartlett & Mendelson, 2002; Zhang, 2002). The drawback of these theories is that they are too general and unable to leverage the specific properties of learning algorithms and data (Nagarajan & Kolter, 2019). Therefore, modern learning theories have turned to data- and algorithm-dependent bounds that leverage the properties of data and popular algorithms (e.g., the limited hypothesis subset reached by SGD, various norms of matrix weights, low-rankness and sparsity of parameters or hidden representations, etc.) to derive tighter bounds specific to them (Brutzkus et al., 2018; Allen-Zhu et al., 2019; Arora et al., 2019; Cao & Gu, 2019; Neyshabur et al., 2019; Pesme et al., 2021; Muthukumar & Sulam, 2023; Alquier, 2023).
+
+Recently, generalization bounds have been developed using information-theoretic measures (Hellström et al., 2024), because these measures are defined with the data distribution and the conditional distributions of the output hypothesis given the training data and are naturally data- and algorithm-dependent. Representative examples include the PAC-Bayesian bounds (McAllester, 1999) and the bounds using mutual information (MI) between the training data and the output of the algorithm (Russo & Zou, 2020; Xu & Raginsky, 2017). Thanks to the dependence, PAC-Bayesian approaches have led to the first non-vacuous numerical generalization bound for deep networks (Dziugaite & Roy, 2017), later tightened (Pérez-Ortiz et al., 2021) and scaled up (Zhou et al., 2019; Lotfi et al., 2022; 2024). MI bounds, the focus of this paper, can be seen as PAC-Bayesian bounds with optimal priors (Alquier, 2023). Tighter variants of the MI bound have been developed, e.g., its chaining
+
+![](images/5edca2cffe415a7ed5e6fa67fcdcff51b173cdc303cd87406894815a5a1f0f11.jpg)  
+Figure 1: Wang & Mao (2022)'s bound for ResNet-18 on CIFAR-10 under varied flatness.
+
+variants (Asadi et al., 2018), its individual-sample variants (Bu et al., 2020), the conditional MI framework (CMI) (Steinke & Zakynthinou, 2020), the evaluated variants (Harutyunyan et al., 2021; Hellström & Durisi, 2022; Wang & Mao, 2023), and bounds using other measures like rate-distortion (Sefidgaran et al., 2022) and Wasserstein distance (Wang et al., 2019). By bounding the measures for specific algorithms, they have been applied to SGLD (Pensia et al., 2018; Negrea et al., 2019; Wang et al., 2021b; Futami & Fujisawa, 2023), discretized SDE (Wang & Mao, 2024b), and SGD (Neu et al., 2021; Wang & Mao, 2022).
+
+Another line of research on algorithmic properties finds SGD favors flat minima (Hochreiter & Schmidhuber, 1997; Keskar et al., 2017). Flat minima are minima at wide and flat basins of the loss landscape, and they are robust to loss landscape changes between the training and testing set. Consequently, flatness has been used to understand and improve the generalization of SGD-trained deep models (Achille & Soatto, 2018; Jiang et al., 2019; Foret et al., 2020; Cha et al., 2021; Zhao et al., 2022), where it is formulated with the Hessians  $\hat{H}_S$ ,  $H_\mu$  of empirical and population losses (Achille & Soatto, 2018; Orvieto et al., 2022). Therefore, as algorithm-dependent bounds, information-theoretic bounds for SGD should leverage the flatness as an important algorithmic property.
+
+However, existing information-theoretic bounds for SGD do not adequately leverage the flatness bias. By controlling flatness through varying learning rate and batch size (Jastrzewski et al., 2018; He et al., 2019; Wu et al., 2022), we empirically observe how the generalization under varying flatness is captured by Wang & Mao (2022)'s bound (which is tighter than Neu et al. (2021)'s
+
+$$
+[\text{Generalization Error} ]\leq \inf_{\sigma >0}\sqrt{\frac{2R^{2}}{n\sigma^{2}}\sum_{t\in[T]}\mathbb{V}\left[g_{t}\right]} + \frac{\sigma^{2}}{2}\cdot \underbrace{T\cdot\operatorname{tr}\mathbb{E}\left[\hat{H}_{S} - H_{\mu}\right]}_{\text{Flatness Term}},
+$$
+
+with  $g_{t}$  being the update at step  $t$ . As shown in Figure 1a, as batch size decreases, the actual generalization error decreases while Wang & Mao (2022)'s bound increases. That is, their result misaligns with the true generalization under varied flatness. The bound consists of two terms: the trajectory term and the flatness term. Figures 1b and 1c show the flatness term can capture the generalization to some extent while the trajectory term cannot, causing the misalignment. Unlike the flatness term that depends on Hessians, which are the explicit formulation of flatness, the trajectory term depends on gradient variance that is an implicit measure of flatness and requires extra conditions (e.g., near-zero losses) to approximate the Hessian traces (Zhu et al., 2019; Martens, 2020; Feng et al., 2023). Therefore, we postulate it is the trajectory term's implicit dependence on flatness that causes the misalignment, and we intend to make the whole bound fully explicitly depend on and leverage flatness. To this end, Proposition 8 of Neu et al. (2021) (see Proposition 2) is a good start, which involves finer-grained optimization and depends on more algorithmic properties, potentially including flatness. It is proved by a technique of auxiliary trajectory that is constructed by randomly perturbing the SGD trajectory with independent Gaussian noises of covariance  $\Sigma$ . The bound optimizes  $\Sigma$  as a parameter and takes the form
+
+$$
+[ \text {G e n e r a l i z a t i o n E r r o r} ] \leq \inf  _ {\Sigma} \mathbb {E} _ {W} [ f (W; \Sigma) ],
+$$
+
+i.e., a mixture of an optimization over  $\Sigma$  and an expectation over output weight  $W$ . By constructing  $\Sigma$  from the Hessians of empirical and population losses, the bound will fully explicitly depend on the flatness. However, we find the bound still has two drawbacks: Firstly, it is suboptimal because the optimization is outside the expectation and cannot depend on or leverage the specific properties of  $W$  instances. Secondly, accurate estimation is crucial for evaluating, comparing bounds and helping
+
+algorithm design. Yet, the bound is not easy to estimate: It is similar to optimizing the "population risk" of "sample"  $W$  over "hypothesis"  $\Sigma$ , which requires sampling some  $W$  and minimizing the "empirical risk" (ERM). Since only a few  $W$  can be sampled due to the computation and data cost of training deep models, the "ERM" is prone to "overfitting", i.e., negative bias in estimation.
+
+Both issues can be addressed by moving the optimization inside the expectation (i.e., bounds like  $\mathbb{E}_W[\inf_{\Sigma}f(W;\Sigma)])$  so that the optimization depends on  $W$ : The inside optimization can leverage the specific properties (e.g., population Hessian instance) of  $W$ , leading to a tighter bound; moreover, the empirical mean  $\frac{1}{k}\sum_{i = 1}^{k}\inf_{\Sigma^i}f(W^i;\Sigma^i)$  of the inside (instance-overfitting) optimization is an unbiased estimator and overfitting is no longer a source of negative bias in estimations. To implement this interchange, we find it is the independence of the Gaussian auxiliary perturbation that makes  $\Sigma$  the same when conditioned on any  $W$  instance, i.e.,  $\Sigma$  cannot adapt to  $W$  instances and unable to leverage their specific properties. Therefore, we extend the auxiliary trajectory technique by building it from an auxiliary perturbation that is no longer independent but at least depends on  $W$ . This is equivalent to moving the optimization inside (check  $\inf_{\Sigma (w)}\mathbb{E}\left[f(W;\Sigma (W))\right] = \mathbb{E}\left[\inf_{\Sigma}f(W;\Sigma)\right]$ ). Pushing this insight further, we make the perturbation depend on all random variables (e.g., the training data) in the SGD training process (e.g., to leverage the instance empirical Hessian that depends on training data), leading to the "omniscient trajectory"², which leads to tighter bounds if well optimized.
+
+The technique yields a tighter generalization error bound for SGD that explicitly and fully depends on the flatness of the SGD output instance. Intuitively, it indicates the algorithm generalizes well when the output weights are flat and the flatness aligns with the covariance of output weights. Here, the covariance of output weights is computed after independently training multiple models, and the alignment means that the variance is low along sharp directions and high along flat directions, as illustrated in Figures 2b and 2c. As discussed in Section 3.4, our alignment better leverages the flatness compared to similar notions of alignments (Wang & Mao, 2024b; Wang et al., 2021a). The bound is evaluated on ResNet-18 trained by CIFAR-10. When varying batch size, it aligns well with the actual generalization error, indicating better exploitation of flatness. Moreover, our bound is numerically tight by being only a few percentages looser than the truth across hyperparameters.
+
+Our omniscient trajectory technique has a simple nature, namely, interchanging the order of the expectation and optimization. Thanks to the simplicity, our technique can be flexibly combined with many existing techniques. Furthermore, recent works (Livni, 2024; Haghifam et al., 2023; Attias et al., 2024) have highlighted the limitations of information-theoretic bounds for having an  $\Omega(1)$  lower-bound for GD or any accurate learners on some convex-Lipschitz-Bounded (CLB) problems. It can also be seen as an information-accuracy trade-off highlighting the complex relationship between memorization and generalization. We find our simple and flexible technique yields an  $O(1/\sqrt{n})$  minimax rate for GD on CLB problems. Therefore, despite being simple, it provides asymptotic improvements and addresses a significant limitation of existing information-theoretic generalization theory. It also implies a by-pass of the trade-off: although accurate learners themselves memorize a lot, they are quite close to some oracle learners that memorize little.
+
+Our contributions are summarized as follows: 1) We derive an information-theoretic generalization bound for SGD that better leverages its flatness bias and is numerically tighter; 2) our bound shows how the direction of flatness affects generalization; 3) we introduce a flexible omniscient trajectory technique that also 4) yields an  $O(1 / \sqrt{n})$  information-theoretic bound for GD on CLB problems.
+
+# 2 PRELIMINARY
+
+We first introduce basic notations. To present existing information-theoretic bounds for SGD and discuss important insights behind them for our (re)use in Section 2.3, we first introduce the algorithm of interest in Section 2.1 and the formulation and properties of flatness in Section 2.2. For  $k \in \mathbb{N}^+$ , let  $[k] := \{1, 2, \ldots, k\}$ . For sequence  $a_0, a_1, \ldots$ , let  $a_{l:r} := (a_i)_{i=l}^r$  and let  $a_{-i}$  denote the rest after excluding  $a_i$ . For vector  $x \in \mathbb{R}^k$  and matrix  $A \in \mathbb{R}^{k \times k}$ , let  $\|x\| := \sqrt{x^\top x}$ , and, with an abuse,
+
+$\| x\| _A^2 \coloneqq x^\top Ax$ . Random variables, realizations, and domains are denoted by capital, lowercase, and calligraphic letters, respectively. For example,  $Z$  is a random sample,  $\mathcal{Z}$  is the sample space, and  $z$  is a realization. Let  $\mu$  over  $\mathcal{Z}$  be the sample distribution. Let  $X'$  denote an I.I.D. copy of random variable  $X$ . Let  $d$  be the number of model parameters and  $\mathcal{W} \subseteq \mathbb{R}^d$  be the hypothesis space. If not specified otherwise, assume  $\mathcal{W} = \mathbb{R}^d$ . Let  $S \coloneqq (Z_1,\dots,Z_n) \sim \mu^n$  be the training set of  $n$  I.I.D. samples. A stochastic learning algorithm is formulated as conditional distribution  $P_{W|S}$ .
+
+Let  $\ell : \mathcal{W} \times \mathcal{Z} \to \mathbb{R}$  be a loss function. The ultimate goal of the learning algorithm is to optimize the population risk  $\mathcal{L}_{\mu}(w, \ell) \coloneqq \mathbb{E}_{Z \sim \mu}[\ell(w, Z)]$  over  $w \in \mathcal{W}$ . Since  $\mu$  is not fully accessible, one uses empirical risk minimization (ERM) by sampling a training set  $S \in \mathcal{Z}^n$  from  $\mu$  and optimizing the empirical risk defined by  $\hat{\mathcal{L}}_s(w, \ell) \coloneqq \sum_{i=1}^{|s|} \ell(w, z_i) / |s|$ . The difference in expectation over training sets and algorithmic randomness is the (expected) generalization error (gap)  $\mathrm{gen}(\mu^n, P_{W|S}, \ell) \coloneqq \mathbb{E}_{(S,W) \sim \mu^n \circ P_{W|S}}\left[\mathcal{L}_\mu(W, \ell) - \hat{\mathcal{L}}_S(W, \ell)\right]$ , where  $\mu^n \circ P_{W|S}$  denotes the joint distribution determined by  $\mu^n$  and  $P_{W|S}$ . We may omit the loss function  $\ell$  when it is clear from the context. Let the loss difference under transnational perturbation  $\gamma \in \mathbb{R}^d$  and Gaussian perturbation  $\xi$  with covariance  $\Sigma \in \mathbb{R}^{d \times d}$  be  $\Delta_\gamma^\Sigma(w, s) \coloneqq \mathbb{E}_{\xi \sim \mathcal{N}(0,\Sigma)}\left[\hat{\mathcal{L}}_s(w + \gamma + \xi) - \hat{\mathcal{L}}_s(w)\right]$ . We write  $\sigma^2$  instead of  $\Sigma$  in the superscript if  $\Sigma = \sigma^2 I$ , and omit  $\gamma$  or  $\Sigma$  if they are zero.
+
+# 2.1 ITERATIVE STOCHASTIC ALGORITHMS
+
+To facilitate analysis, we assume an abstract form of iterative stochastic algorithms to hide unnecessary details and improve generality. We assume the algorithm first prepares an independent random variable  $V$  for internal randomness, then starts from an independent initial hypothesis  $W_0 \in \mathcal{W}$  and updates the hypothesis iteratively for  $T \in \mathbb{N}^+$  steps by  $W_t := W_{t-1} - g_t(W_{t-1}, S, V, W_{0:t-2}) \in \mathcal{W}$  for  $t \in [T]$ , and finally outputs  $W := W_T$ . Here,  $g_t$  is a deterministic function. The algorithm specifies a random process  $W_{0:T}$ , referred to as the original trajectory. We may omit  $g_t$ 's dependence on  $(W_{t-1}, S, V, W_{0:t-2})$  to save space. SGD with batch size  $b$  can be obtained by first generating the indices  $B_{1:T} \in ([n]^T)$  for each mini-batch, then saving them in  $V$  and finally letting  $g_t$  compute the gradients in the mini-batch defined by  $B_t$ . Lastly, with access to past weights  $W_{0:t-2}$  and all randomnesses  $V$ ,  $g_t$  can recover past gradients and covers momentum or Adam, etc. As a result, our theoretical results can be directly applied to these algorithms.
+
+# 2.2 FLATNESS
+
+The flatness at  $w \in \mathcal{W}$  is formulated by the Hessians  $\hat{H}_S(w)$  and  $H_{\mu}(w)$  of the empirical and population losses, respectively. Empirically, the flatness of the empirical loss is highly anisotropic for deep models: after SGD training, most empirical Hessian eigenvectors have small eigenvalues while only the rest few have large eigenvalues (Sagun et al., 2018; Papyan, 2019). We distinguish these eigenvectors by "flat" versus "sharp" directions (Jastrzebski et al., 2019; Wu et al., 2022). Perturbations in the weight space along the sharp directions cause large loss changes, while those along the flat directions cause only slight loss changes.
+
+# 2.3 INFORMATION-THEORETIC BOUNDS AND APPLICATION ON SGD
+
+A random variable  $X$  is  $R$ -sub-Gaussian if  $\mathbb{E}\left[e^{\lambda (X - \mathbb{E}[X])}\right] \leq e^{\lambda^2 R^2 /2}$  for any  $\lambda \in \mathbb{R}$ . Let  $I(A;B)$  be the mutual information (MI) between a pair of random variables  $(A,B)$  (Cover & Thomas, 2006). The MI between the training set and the output weight can bound the generalization error:
+
+Lemma 1 (Xu & Raginsky (2017)) Assume  $\ell(w, \cdot)$  is  $R$ -sub-Gaussian on  $\mu$  for any  $w \in \mathcal{W}$ . The generalization error of  $P_{W|S}$  on  $\mu$  is bounded by  $\mathrm{gen}(\mu^n, P_{W|S}) \leq \sqrt{2R^2 I(W; S)/n}$ .
+
+MI is hard to compute for SGD and one must bound it to apply Lemma 1. However, directly bounding MI on SGD is not enough because the MI of SGD can be infinite (Hellström et al., 2024), leading to a vacuous generalization bound. Neu et al. (2021) propose auxiliary trajectories to address this problem. Auxiliary trajectories are perturbed versions of the original trajectory, aiming to decrease the MI between the training set and the output weight. If well designed, the larger the perturbation is, the more the MI decreases. It is often easier to bound the MI of the auxiliary trajectory, which finally
+
+becomes the "trajectory term" in the bound. Given the auxiliary trajectory, to transform the bound of it into a bound for the original trajectory, one must pay a penalty term of the loss difference between their output weights. The penalty comprises of several instances of  $\Delta_{\gamma}^{\Sigma}$  terms, which are approximated by Taylor expansion to  $\Delta_{\gamma}^{\Sigma}(w,s)\approx \mathbb{E}\left[\gamma^{\top}\nabla \hat{\mathcal{L}}_s(w) + (\gamma +\xi)^{\top}\times \hat{H}_s(w)\times (\gamma +\xi) / 2\right]$ . As a result, the penalty is related to the flatness: It will be very large if the perturbation  $(\gamma +\xi)$  has large projections onto the top eigenvectors of  $\hat{H}_s$ , i.e., the sharp directions, while it is safe to have large projections onto the flat directions. Trading off between the trajectory and penalty terms, we have Insight 1 for designing auxiliary perturbation and trajectories.
+
+Insight 1 The ideal perturbation for the auxiliary trajectory should have large projections onto flat directions to reduce MI and the trajectory term while maintaining small projections onto sharp directions to keep the penalty term small.
+
+We now present the existing information-theoretic bounds for SGD with the help of auxiliary trajectories. Neu et al. (2021) propose isotropic Gaussian as the perturbation for the auxiliary trajectory:
+
+Definition 1 (SGLD-Like Trajectory) The SGLD-like trajectory  $\bar{W}_{0:T}$  of any trajectory  $\bar{W}_{0:T}$  is
+
+$$
+\tilde {W} _ {0} := \bar {W} _ {0}, \quad \tilde {W} _ {t} := \tilde {W} _ {t - 1} + (\bar {W} _ {t} - \bar {W} _ {t - 1}) + N _ {t},
+$$
+
+where  $N_{t}\sim \mathcal{N}(0,\sigma_{t}^{2}I)$  is an independent isotropic Gaussian noise with  $\sigma_t > 0$
+
+After building the SGLD-like auxiliary trajectory of the original trajectory, Neu et al. (2021) exploit the properties of Gaussian noises to bound the MI of the SGLD-like output weight. Based on the same SGLD-like trajectory, Wang & Mao (2022) improve the technique for bounding MI, providing a numerically tighter result in Proposition 1.
+
+Proposition 1 (Theorem 2 of Wang & Mao (2022)) SGD's generalization error is bounded by
+
+$$
+\mathrm {g e n} (\mu^ {n}, P _ {W | S}) \leq \sqrt {\frac {R ^ {2}}{n} \underbrace {\sum_ {t = 1} ^ {T} \frac {1}{\sigma_ {t} ^ {2}} \mathbb {E} \left[ \| g _ {t} - \mathbb {E} \left[ g _ {t} \right] \| ^ {2} \right]} _ {M I b o u n d (T r a j e c t o r y T e r m)}} + \underbrace {\mathbb {E} \left[ \Delta^ {\sum_ {t} \sigma_ {t} ^ {2}} (W _ {T} , S) - \Delta^ {\sum_ {t} \sigma_ {t} ^ {2}} (W _ {T} , S ^ {\prime}) \right]} _ {P e n a l t y f o r t h e S G L D - l i k e t r a j e c t o r y (F l a t n e s s T e r m)},
+$$
+
+where the expectations are over the randomness of training set sampling, initialization, and  $V$ .
+
+Since the trajectory term and the flatness term correspond to the MI bound and the penalty of the SGLD-like trajectory, respectively, these two terminologies will be used interchangeably. The SGLD-like trajectory decreases MI at the cost of the penalty term. The flatness is exploited to ensure the penalty is not very large. However, as shown in Figure 1, Proposition 1's trajectory term does not adequately exploit the flatness. This drawback is partially due to the isotropic Gaussian that adds perturbations of the same strength along all directions, violating Insight 1. To exploit the anisotropy of flatness, one can use non-isotropic Gaussian noises as in Proposition 2.
+
+Proposition 2 (Proposition 8 of Neu et al. (2021)) SGD's generalization error is bounded by
+
+$$
+\operatorname {g e n} (\mu^ {n}, P _ {W | S}) \leq \inf  _ {\Sigma \in \mathcal {S} _ {+}} \sqrt {\frac {R ^ {2}}{n} \mathbb {E} \left[ \| W _ {T} - \mathbb {E} \left[ W _ {T} \right] \| _ {\Sigma^ {- 1}} ^ {2} \right]} + \mathbb {E} \left[ \Delta^ {\Sigma} (W, S) - \Delta^ {\Sigma} (W, S ^ {\prime}) \right],
+$$
+
+where  $S_{+} \subseteq \mathbb{R}^{d \times d}$  is the set of symmetric positive definite matrices.
+
+Here,  $\Sigma$  is the covariance of the Gaussian. By setting, for example,  $\Sigma = \mathbb{E}\left[\hat{H}_S\right]^{-1}$ , the noise has large variances along the directions with small curvatures, better exploiting the flatness. Nevertheless, we find Proposition 2 still has room for improvement, as elaborated in Section 3.1.
+
+![](images/9a17564588fabe28d90ae39d1d7f9a07dc3d88773c54054b3ab434c20a90fb6b.jpg)  
+(a) The relationships between trajectories.
+
+![](images/7bec8f82acd3a5a7138e569ee80a8a42a61201c5e41f59d4504b3eb4d0c3ac0b.jpg)  
+(b) Aligned.  
+Figure 2: The relationship between the original, omniscient, and SGLD-like trajectories, and illustrative examples of of alignment and misalignment between flatness and output weight covariance. The two trajectories have decoupled roles: the omniscient trajectory optimizes the whole bound while the SGLD-like trajectory bounds the MI for the omniscient trajectory.
+
+![](images/6037ec0ba0ef6579fd0bb36adb0a72abe4a67561ed3ca373e4bcc216237b02a7.jpg)  
+(c) Not aligned.
+
+# 3 PROPOSED OMNISCIENT TRAJECTORY
+
+# 3.1 ANALYSIS ON EXISTING RESULTS' DRAWBACKS
+
+To see the inefficiencies of Proposition 2, we recall a generic principle:
+
+Principle 1 Being specific enhances optimization:  $\min_A\mathbb{E}_X[f(X;A)]\geq \mathbb{E}_X[\min_Af(X;A)]$
+
+Through the lens of this principle, Proposition 2 is closer to LHS of Principle 1 since  $\Sigma$  to be optimized is shared among all instances in the expectation, leading to trade-offs between these instances. As a result, it is suboptimal compared to a fully specific and dependent one.
+
+Moreover, Proposition 2 is prone to overfitting when estimated in numerical studies. Since the distribution of output weights is not fully accessible, one must sample some of them before optimizing over  $\Sigma$ . This procedure is identical to ERM, leading to potential overfitting to the sampled weights. Sampling output weights involves training multiple models on multiple training sets. The computational costs and data requirements of training deep models limit the number of samples. In contrast,  $S_{+}$  is a  $\Theta(d^{2})$ -dimensional manifold. Therefore, the overfitting is severe at least in the classic view and the estimated bound has a severe negative bias. Although the overfitting may be benign as it is in deep learning, a meta-generalization theory is needed to ensure it, which is too complex and may even induce a meta-meta-overfitting problem with more parameters. To our knowledge, Proposition 2 lacks numerical results, possibly due to the large parameter count and the overfitting.
+
+Some examples of the two drawbacks can be found in Appendix B.1. As indicated by Principle 1, both issues can be addressed by moving the optimization inside the expectation. Regarding the suboptimality, Principle 1 explicitly shows better tightness. Regarding overfitting, the RHS of Principle 1 is already the result of overfitting to instance trajectories, eliminating overfitting as a bias source. Furthermore, depending on more random variables, such as the training set, makes it easier to access and exploit the empirical flatness. Motivated by these benefits, we propose fully dependent auxiliary trajectories in the form of functions of all random variables in the training process. With knowledge of all random variables, the trajectories are termed "omniscient (auxiliary) trajectories".
+
+# 3.2 OMNISCIENT TRAJECTORY
+
+The omniscient trajectory that depends on all random variables in the training is defined as follows:
+
+Definition 2 (Omniscient Trajectory) The omniscient trajectory  $\tilde{W}_{0:T}$  of  $(S,V,g_{1:T},W_{0:T})$  specified by omniscient perturbation  $\Delta g_{1:T}$  is given by
+
+$$
+\check {W} _ {0} := W _ {0}, \check {W} _ {t} := \check {W} _ {t - 1} - (g _ {t} (W _ {t - 1}, S, V, W _ {0: t - 2}) - \Delta g _ {t} (S, V, g _ {1: T}, W _ {0: T})),
+$$
+
+where  $\Delta g_{t}$  is a deterministic function. Let  $\Gamma_t\coloneqq \sum_{\tau = 1}^t\Delta g_\tau (S,V,g_{1:T},W_{0:T})$  , then  $\tilde{W}_t = W_t + \Gamma_t$
+
+To bound the generalization error of the original SGD trajectory, we transform the generalization error of the original output weight into that of the omniscient output weight at the cost of a "penalty term" for the translational perturbation  $\Gamma_T$ . To bound generalization error and MI of the omniscient
+
+output weight, we construct an SGLD-like auxiliary trajectory  $\tilde{W}_{0:T}$  based on the omniscient trajectory and transform the generalization of the omniscient trajectory into that of the SGLD-like trajectory, at the cost of another penalty, i.e., the flatness term. We then bound the generalization error of the SGLD-like output weight using the MI  $I(\tilde{W};S)$ . The relationships between the trajectories and their generalization errors are summarized in Figure 2a. To bound the MI, we extend the key Lemma 4 of Wang & Mao (2022) that bounds the mutual information using variances in Lemma B.1 so that it can be used for the double-layered auxiliary trajectories despite its heavy dependence to  $S$ . Putting these problem transformations and bounds together, we obtain a generalization bound Theorem 1 using trajectory statistics similar to Proposition 1 but modified by the omniscient perturbation.
+
+Theorem 1 Assume  $R$ -sub-Gaussianity for  $\ell$ . For any  $\Delta g_{1:T}$  and any  $\sigma_{1:T} \in (\mathbb{R}^{>0})^T$ , we have
+
+$$
+\begin{array}{l} \operatorname {g e n} (\mu^ {n}, P _ {W | S}) \leq \sqrt {\frac {R ^ {2}}{n} \underbrace {\sum_ {t = 1} ^ {T} \frac {1}{\sigma_ {t} ^ {2}} \mathbb {E} \left[ \| g _ {t} - \mathbb {E} \left[ g _ {t} \right] - \Delta g _ {t} \| ^ {2} \right]} _ {M I b o u n d (T r a j e c t o r y t e r m)}} + \underbrace {\mathbb {E} \left[ \Delta_ {\Gamma_ {T}} (W _ {T} , S) - \Delta_ {\Gamma_ {T}} (W _ {T} , S ^ {\prime}) \right]} _ {P e n a l t y f o r t h e o m n i. t r a j e c t o r y (P e n a l t y T e r m)} \\ + \underbrace {\mathbb {E} \left[ \Delta^ {\sum_ {t} \sigma_ {t} ^ {2}} (W _ {T} + \Gamma_ {T} , S) - \Delta^ {\sum_ {t} \sigma_ {t} ^ {2}} (W _ {T} + \Gamma_ {T} , S ^ {\prime}) \right]} _ {\text {P e n a l t y f o r t h e S G L D - l i k e t r a j e c t o r y (F l a t n e s s T e r m)}}. \\ \end{array}
+$$
+
+Setting  $\Delta g_{t} \equiv 0$  recovers Proposition 1. We further optimize it to tighten the bound.
+
+# 3.3 OPTIMIZING THE OMNISCIENT TRAJECTORY AND EXPLOITING FLATNESS
+
+To avoid optimizing over  $T$  elements  $\Delta g_{1:T}$ , we first simplify Theorem 1 with  $\Delta g_t = (g_t - \mathbb{E}[g_t]) - \frac{1}{T} \sum_{\tau=1}^{T} (g_\tau - \mathbb{E}[g_\tau]) + \frac{1}{T} \Delta G$ . As a result, for any deterministic function  $\Delta G$  of  $(S, V, g_{1:T}, W_{0:T})$  and  $\sigma > 0$ , we have Corollary B.1 that can be informally stated as (with  $\Delta W_t := W_t - W_0$ )
+
+$$
+\mathrm {g e n} \leq \sqrt {\frac {R ^ {2}}{n \sigma^ {2} T} \mathbb {E} \left[ \left\| \Delta W _ {T} - \mathbb {E} \left[ \Delta W _ {T} \right] + \Delta G \right\| ^ {2} \right]} + \mathbb {E} \left[ \Delta_ {\Delta G} ^ {\sigma^ {2} T} (W _ {T}, S) - \Delta_ {\Delta G} ^ {\sigma^ {2} T} (W _ {T}, S ^ {\prime}) \right].
+$$
+
+Appendix B.4 verifies such  $\Delta g_{t}$  is optimal under the constraint  $\Gamma_T = \Delta G$ , and the generality is not harmed by this simplification. Setting  $\Delta G = 0$  recovers the isotropic version of Proposition  $2^{4}$ .
+
+We then optimize over  $\Delta G$ . It can decrease the trajectory term by canceling  $\Delta W_{T} - \mathbb{E}\left[\Delta W_{T}\right]$ . However, large  $\Delta G$  would increase the penalty term, leading to a trade-off. Fortunately, near flat minima, most directions are flat, and the penalty term is insensitive to perturbations along them. Following Insight 1, we confine  $\Delta G$  to align with the flat directions. This is done by approximating the penalty terms to the second order, where Hessians emerge, and solving an optimization problem formed by the output weights and the Hessians (see Appendix B.6). Theorem 2 presents the result, where the three expectations correspond to the penalty, flatness, and trajectory terms, respectively.
+
+Theorem 2 Assume  $\ell(w, \cdot)$  is  $R$ -sub-Gaussian on  $\mu$  for any  $w \in \mathcal{W}$ ,  $\ell(\cdot, z)$  and  $\mathcal{L}_{\mu}(\cdot)$  are thirdly continuously differentiable w.r.t.  $w$  for any  $z \in \mathcal{Z}$ , and there is a constant  $D > 0$  such that  $D \|\Delta w\|^4$  bounds the residuals in the third-order Taylor expansions of  $\ell(\cdot, z)$  and  $\mathcal{L}_{\mu}(\cdot)$  for any  $z \in \mathcal{Z}$ . Then for any  $\lambda > 0$  trading-off between the trajectory and penalty terms, we have
+
+$$
+\begin{array}{l} \operatorname {g e n} \left(\mu^ {n}, P _ {W | S}\right) \leq \mathbb {E} \left[ \Delta_ {\Delta G} \left(W _ {T}, S\right) - \Delta_ {\Delta G} \left(W _ {T}, S ^ {\prime}\right) \right] + \frac {3}{2} \sqrt [ 3 ]{\frac {R ^ {2}}{n} \mathbb {E} \left[ \operatorname {t r} \left(\Delta H \left(W _ {T} + \Delta G\right)\right) \right]} \\ \times \sqrt [ 3 ]{\mathbb {E} \left[ \left\| (I - (I + \tilde {H} _ {\mathrm {p e n}} / 2 \lambda C) ^ {- 1}) (\Delta W _ {T} - \mathbb {E} [ \Delta W _ {T} ]) - (2 \lambda C I + \tilde {H} _ {\mathrm {p e n}}) ^ {- 1} J \right\| ^ {2} \right]} + r, \tag {1} \\ \end{array}
+$$
+
+where  $\Delta H(w) \coloneqq \hat{H}_S(w) - H_\mu(w)$ ,  $\tilde{H}_{\mathrm{flat}}$  is chosen from  $\Delta H(W_T)$  and  $\hat{H}_S(W_T)$ ,  $\tilde{H}_{\mathrm{pen}}$  is chosen from  $\Delta H(W_T)$  and  $\hat{H}_S(W_T)$ , and  $J$  is chosen from  $\nabla (\hat{\mathcal{L}}_S(W_T) - \mathcal{L}_\mu(W_T))$  and  $\hat{\mathcal{L}}_S(W_T)$ ,  $C \coloneqq \frac{3}{2}\left(\frac{R^2}{n} |\mathrm{tr}(\tilde{H}_{\mathrm{flat}})|\right)^{1/3}$ ,  $\Delta G = -(2\lambda C I + \tilde{H}_{\mathrm{pen}})^{-1}(2\lambda C (\Delta W_T - \mathbb{E}[\Delta W_T]) + J)$ , and  $r = O(d^2 \sigma_*^4)$  is the residual in a second-order approximation. See Eq. (B.6) in Appendix B.7 for the form of  $\sigma_*$ .
+
+# 3.4 DISCUSSION AND COMPARISON
+
+To see the dependence on the flatness more clearly, we approximate the result of Theorem 2. To simplify the approximation, only for this subsection, we assume  $J \coloneqq \nabla \hat{\mathcal{L}}_S(W_T) - \nabla \mathcal{L}_\mu (W_T)$ , and the model is sufficiently trained so that  $\nabla \hat{\mathcal{L}}_S(W_T) \approx 0$ . We also assume  $\lambda$  is large enough so that  $2\lambda C$  surpasses  $\tilde{H}_{\mathrm{pen}}$ 's top singular value to approximate  $I - (I + \tilde{H}_{\mathrm{pen}} / 2\lambda C)^{-1} \approx \tilde{H}_{\mathrm{pen}} / 2\lambda C$ . Be noted that the above assumptions are used only in this subsection, i.e., Theorem 2 and experiments in Section 4 do not require them. Results without some of these assumption can be found in Appendix B.8. Approximating the penalty term to the second order leads to the following result:
+
+$$
+\begin{array}{l} \operatorname {g e n} \lessapprox \underbrace {\mathbb {E} \left[ \| \Delta W _ {T} - \Delta \mathbb {E} \left[ W _ {T} \right] \| _ {| \Delta H (W _ {T}) | - \Delta H (\mathbb {E} [ W _ {T} ])} ^ {2} + \| \nabla \mathcal {L} _ {\mu} (W _ {T}) \| _ {\frac {\left| \Delta H (W _ {T}) \right|}{4 \lambda^ {2} C ^ {2}} - \frac {I}{2 \lambda C}} ^ {2} \right]} _ {\text {C o r r e s p o n d i n g t o P e n a l t y T e r m}} \\ + \frac {3}{2} \sqrt [ 3 ]{\frac {2 R ^ {2}}{n} \underbrace {\mathbb {E} \left[ | \operatorname {t r} \left(\Delta H \left(W _ {T} + \Delta G\right)\right) | \right]} _ {\text {C o r r e s p o n d i n g t o F l a t n e s s T e r m}} \cdot \mathbb {E} \left[ \underbrace {\| \Delta W _ {T} - \mathbb {E} \left[ \Delta W _ {T} \right] \| _ {\frac {H _ {\text {p e n}} ^ {2}}{4 \lambda^ {2} C ^ {2}}} ^ {2} + \| \nabla \mathcal {L} _ {\mu} \left(W _ {T}\right) \| _ {\frac {J}{4 \lambda^ {2} C ^ {2}}} ^ {2}} _ {\text {C o r r e s p o n d i n g t o T r a j e c t o r y T e r m}} \right]}, \tag {2} \\ \end{array}
+$$
+
+where  $|\cdot|$  replaces the eigenvalues of a matrix with their absolute values. The details can be found in Appendix B.8. It can be seen that all terms except the two population gradient norms depend on Hessians. Particularly, the "norms" of deviation  $\Delta W_T - \mathbb{E}[\Delta W_T]$  are defined by the Hessians. As a result, the components of the deviation along flat directions contribute little to the bound. The flatter the minima are, the more and flatter the flat directions there are, and more components of the deviations contribute little to the bound. Moreover, if the flatness is aligned with the covariance, most large deviations can be found near the "flat subspace", leading to a smaller bound. As a result, generalization is better if minima are flat and the flatness aligns with the covariance.
+
+This focus on alignment is similar to the work by Wang & Mao (2024b) and Wang et al. (2021a). These alignments are compared in Appendix B.8.1 in detail. Briefly, the main difference is that our alignment directly relies on flatness. Our technique is also similar to the rate-distortion bound (Sefidgaran et al., 2022) as both involve weight-dependent perturbations. However, our omniscient trajectory depends on more random variables, such as the training data that is crucial for leveraging the empirical Hessian. Besides, the existing chaining (Asadi et al., 2018) and evaluated (Harutyunyan et al., 2021; Hellström & Durisi, 2022; Wang & Mao, 2023) bounds, which can leverage the similarity between adjacent hypotheses, are potentially useful because the flatness of a minimum reflects a similarity with neighboring weights. However, chaining requires partitioning the hypothesis space, which is difficult for deep neural networks; the evaluated bounds directly rely on model losses, obscuring insights in the language of weights and flatness.
+
+# 4 EXPERIMENTAL STUDY
+
+In this section, we experimentally show how our bound captures the true generalization error/gap measured by Cross Entropy (0-1 loss does not have Hessians and is motivationally incompatible) compared to the existing bounds. We vary the hyperparameter and train 6 independent ResNet-18 models on CIFAR-10 at each hyperparameter. To ensure sub-Gaussianity, capped cross-entropy (CE) is used in testing and bound estimation, while vanilla CE is used for training for efficient training. We estimate Theorem 2 with  $\lambda \in \{1,10^3,10^9\}$ ,  $\tilde{H}_{\mathrm{flat}} = \tilde{H}_{\mathrm{pen}} = \hat{H}_S(W_T)$  to make estimation easier and  $J = \nabla \hat{\mathcal{L}}_S(W_T) - \nabla \mathcal{L}_\mu (W_T)$  for numerical tightness as detailed in Appendix C.2. Estimating the bounds requires splitting datasets: The 6 models are trained by 6 random splits of the training set. Terms involving population statistics (e.g., the population Hessians in the flatness terms and the population gradient in the penalty term) are approximated to the second order and estimated on validation sets. But for existing bounds, we assume  $\mathcal{L}_{\mu}(W)\leq \mathbb{E}_{\xi \sim \mathcal{N}(0,\sigma^2 I)}[\mathcal{L}_{\mu}(W + \xi)]$  (Wang & Mao, 2022) and avoid the population Hessian. The true generalization error is estimated on a separate test set. See Appendix C for more details and the results with population Hessians.
+
+We first evaluate whether our bound can better capture the generalization under varying flatness. As shown in Figure 3, the bound correctly captures the tendency of generalization error under varied batch size and learning rate. Specifically, the tendency of the trajectory term w.r.t. the batch size
+
+![](images/08509ffc75d9bae2665db779350c65b794b9419921cf130bb94483a46653b2be.jpg)  
+Figure 3: Thm. 2 ( $\lambda = 10^{9}$ ) on CIFAR-10 and ResNet-18 with varied learning rate and batch size.
+
+![](images/1b75951550875f0285d70c6a43f45fadc09850d9bfc65b577c9885268c6f0dac.jpg)  
+Figure 4: Numerical results for ResNet-18 on CIFAR-10 under varied training data usage, label noise level, width, depth, and weight decay. The isotropic version of Proposition 2 is used.
+
+is corrected. The penalty term for the omniscient trajectory correlates well with the generalization error  $w.r.t$ . both learning rate and batch size, thanks to the Hessians in Eq. (2). The flatness term also generally has the correct tendency  $w.r.t$ . both hyperparameters. Unfortunately, the tendency of the trajectory term  $w.r.t$  learning rate is still incorrect. As shown in Section 3.4, the trajectory term is essentially the product between Hessians and the variance of the last-step weight. We conjecture it is the increased variance when learning rate increases that overpowers the improved flatness and makes the tendency uncorrected. In contrast, the variance is less sensitive to batch size. Detailed discussion can be found in Appendix C.4.1. Nevertheless, after multiplied together, they contribute little to the bound and the uncorrected tendency does not affect the bound very much.
+
+Figure 4 shows how our bound captures the generalization under varied generalization-critical hyperparameters. It can be seen that our bound with large  $\lambda$  (i.e., putting more weights on reducing the trajectory term) can well capture the generalization under these variations, while the some existing bounds fail to capture the improved generalization under stronger weight decay.
+
+Results for MLP on MNIST can be found in Appendix C.4. According to Figures 3 and 4 and results in the appendix, our bounds are numerically much tighter. The main improvement comes from the trajectory term, as a result of both Proposition 2 and the optimized omniscient trajectory leveraging flatness. See Appendix C.4.1 for detailed discussions. In Appendix C.6, we evaluate our bounds under weight scaling, where the generalization is still well captured.
+
+The trajectory term and the flatness almost diminish after multiplied together, and the bound mainly comprises of the penalty term. In Theorem 2 and Eq. (2), the term heavily relies on the population gradients. We consider this reliance to be the major limitation of our bound.
+
+# 5 EXTENSIONS
+
+Our technique is so flexible that it can be combined with existing techniques, such as the individual-sample technique and CMI framework. We can also combine it with the data-dependent prior technique of Negrea et al. (2019) for SGLD to yield an SGD bound. The combined results are listed in Appendix D. Notably, the omniscient trajectories in these results depend on all new random variables introduced by the variants. As a result, the omniscient trajectory can leverage the features of the variants and optimize more specifically.
+
+Our technique can also address the known limitations of representative existing information-theoretic bounds on convex-Lipschitz-Bounded (CLB) problems in stochastic convex optimization (SCO). An SCO problem is a triplet  $(\mathcal{W},\mathcal{Z},\ell)$ , where  $\mathcal{W}$  is convex and  $\ell (\cdot ,z)$  is convex given any  $z\in \mathcal{Z}$ . The CLB subclass  $\mathcal{C}_{L,D}$  of SCO further requires  $\mathcal{W}$  is closed and bounded with a diameter
+
+$D$ , while  $\ell(\cdot, z)$  is  $L$ -Lipschitz given any  $z \in \mathcal{Z}$ . The goal is to minimize the excess (population) risk  $\mathbb{E}_W[\mathcal{L}_\mu(W)] - \inf_{w \in \mathcal{W}} \mathcal{L}_\mu(w)$  under an unknown sample distribution, which can be bounded by the excess optimization error and the generalization error. The minimax excess risk reflects the worst-case generalization:  $\inf_{P_{W|S} \in \mathcal{A}} \sup_{\mu \in \mathcal{M}_1(\mathcal{Z})} \mathbb{E}_{W \sim \mu^n \circ P_{W|S}}[\mathcal{L}_\mu(W)] - \inf_{w \in \mathcal{W}} \mathcal{L}_\mu(w)$ , where  $\mathcal{A}$  is a family of algorithms and  $\mathcal{M}_1(\mathcal{Z})$  is the set of all distributions over  $\mathcal{Z}$ .
+
+An  $O(LD / \sqrt{n})$  minimax rate has been obtained for Gradient Descent (GD) using uniform stability. However, Haghifam et al. (2023) have shown there exist CLB instances where PAC-Bayesian bounds, the vanilla (C)MI bounds, their Gaussian-perturbed variants (Propositions 1 and 2 fall in this category if adapted to CLB settings), and representative variants all have an  $\Omega(1)$  lower-bound, meaning these representative bounds cannot explain the generalization of GD on CLB problems. Attias et al. (2024) have shown any accurate CLB learner with low excess generalization risk must have high CMI, i.e., the CMI-accuracy trade-off. Recently, Wang & Mao (2024a) have addressed the counterexample by Haghifam et al. (2023) and all CLB problems by augmenting CMI bounds with stability. However, a lot of complicated CMIs are involved and due to this complexity it is hard to intuitively interpret their relationship with the CMI-accuracy trade-off. Therefore, we try our technique to see whether it leads to a similar but simpler result. Combining individual-sample bounds with omniscient trajectory, we can easily resemble the stability argument:
+
+Theorem 3 (Informal) For CLB problem  $(\mathcal{W},\mathcal{Z},\ell)$  and distribution  $\mu$  over  $\mathcal{Z}$ , we have
+
+$$
+\begin{array}{l} \operatorname {g e n} \leq \inf  _ {\Delta G ^ {1: n}} \frac {1}{n} \sum_ {i = 1} ^ {n} \left(L D \sqrt {2 I (W _ {T} + \Delta G ^ {i} ; Z _ {i} \mid Z _ {- i})} + \mathbb {E} [ \Delta_ {\Delta G ^ {i}} (W _ {T}, Z _ {i}) - \Delta_ {\Delta G ^ {i}} (W _ {T}, S ^ {\prime}) ]\right) \\ \stackrel {\text {(f o r s t a b l e a l g o r i t h m s)}} {\leq} \frac {2 L}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \| W _ {T} - \mathbb {E} \left[ W _ {T} \mid Z _ {- i} \right] \| \right] \underset {\text {(w i t h T s t e p s o f s i z e \eta)}} {\overset {\text {(f o r G D)}} {\leq}} 8 L ^ {2} \sqrt {T} \eta + 8 L ^ {2} T \eta / n \\ \end{array}
+$$
+
+This bound recovers that of Bassily et al. (2020) up to a constant. Following similar steps of Haghi-fam et al. (2023), one can recover the best-achievable  $O(LD / \sqrt{n})$  excess risk bound. Theorem 3 and its proof essentially state the original trajectory has a vanishing distance (measured by loss differences) to the omniscient trajectory of zero individual MI. Therefore, Theorem 3 provides an alternative to the trade-off on GD or stable algorithms: although the accurate learners have high CMI, they are quite close to auxiliary (oracle) learners with low  $\mathrm{MI}^5$ . Extending this intuition for more general accurate learners in Theorem 4 of Appendix D.5, we explore to replace CMI with a new information measure induced by our technique and break through the information-accuracy trade-off: this generalization-bounding new information measure vanishes together with risks. However, Theorem 4 requires vanishing excess optimization error and is still partial and preliminary. Lastly, when losses are smooth and convex, we also recover a stability-based  $O(1 / n)$  rate for SGD, as detailed in Appendix D.6.
+
+# 6 CONCLUSION AND LIMITATION
+
+In this paper, we address the inadequate leverage of flatness in existing information-theoretic generalization bounds for SGD. By continuing the "being more specific" trend in generalization theory, we propose the omniscient trajectory that can be optimized depending on all random variables in the training process to better leverage the flatness. Our bound shows that an algorithm generalizes better if its output flatness aligns with its output covariance. When applied to deep neural networks, our bound aligns well with the generalization under varied flatness and is numerically tighter. When applied to CLB problems for GD, the technique yields an  $O(1\sqrt{n})$  minimax rate, addressing a recently highlighted limitation of information-theoretic generalization theory.
+
+However, our bound relies on population gradients and Hessians. Although tolerable for theories (e.g., Neu et al. (2021) and Wang & Mao (2022; 2024b) also rely on population statistics), this problem prevents the bound from being a part of self-certified algorithms (Pérez-Ortiz et al., 2021). Moreover, the most information-theoretic components vanish in both the CLB bound and the experiments (see Eq. (D.5) and Figures 3c and C.4c). We conjecture these two issues may be addressed in future works by information-theoretically bounding the generalization of higher-order statistics. More detailed discussions on the limitations can be found in Appendix E.
+
+# REPRODUCIBILITY STATEMENT
+
+All proofs and full versions of informally stated results can be found in the appendices for the corresponding sections. Details of experiments can be found in Appendix C. The codes for the experiments can be found in the supplementary material or at https://github.com/peng-ze/ omniscient-bounds.
+
+# ACKNOWLEDGMENTS
+
+This work was supported by the NSFC Project (62192783, 92370129, 62222604, 62206052, 62376010), China Postdoctoral Science Foundation (2024M750424), Fundamental Research Funds for the Central Universities (020214380120, 020214380128), State Key Laboratory Fund (ZZKT2024A14), the Postdoctoral Fellowship Program of CPSF (GZC20240252), Jiangsu Funding Program for Excellent Postdoctoral Talent (2024ZB242), Jiangsu Science and Technology Major Project (BG2024031), and Beijing Nova Program (20230484344, 20240484642).
+
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+Lei Wu, Mingze Wang, and Weijie Su. The alignment property of SGD noise and how it helps select flat minima: A stability analysis. In Advances in Neural Information Processing Systems, volume 35, pp. 4680-4693, 2022.  
+Aolin Xu and Maxim Raginsky. Information-theoretic analysis of generalization capability of learning algorithms. In Advances in Neural Information Processing Systems, volume 30, 2017.  
+Zhewei Yao, Amir Gholami, Kurt Keutzer, and Michael W. Mahoney. PyHessian: Neural Networks Through the Lens of the Hessian. In 2020 IEEE International Conference on Big Data, pp. 581-590, 2020.  
+Tong Zhang. Covering number bounds of certain regularized linear function classes. Journal of Machine Learning Research, 2(Mar):527-550, 2002.  
+Yang Zhao, Hao Zhang, and Xiuyuan Hu. Penalizing gradient norm for efficiently improving generalization in deep learning. In Proceedings of the 39th International Conference on Machine Learning, volume 162, pp. 26982-26992. PMLR, 2022.
+
+Wenda Zhou, Victor Veitch, Morgane Austern, Ryan P. Adams, and Peter Orbanz. Non-vacuous generalization bounds at the imagenet scale: a PAC-bayesian compression approach. In International Conference on Learning Representations. ICLR, 2019.  
+Zhanxing Zhu, Jingfeng Wu, Bing Yu, Lei Wu, and Jinwen Ma. The anisotropic noise in Stochastic Gradient Descent: Its behavior of escaping from sharp minima and regularization effects. In International Conference on Machine Learning, pp. 7654-7663. PMLR, 2019.
+
+# A TECHNICAL LEMMAS
+
+Lemma A.1 If  $X, Y \in \mathbb{R}^d$  are independent random variables, then we have  $\mathbb{E}\left[\| X - \mathbb{E}[Y]\|\right] \leq \mathbb{E}\left[\| X - Y\|\right]$ .
+
+Proof By the convexity of the  $L_{2}$  norm and the independence between  $X$  and  $Y$ , we have
+
+$$
+\mathbb {E} \left[ \| X - Y \| \right] = \mathbb {E} _ {X} \left[ \mathbb {E} _ {Y} \left[ \| X - Y \| \right] \right] \geq \mathbb {E} _ {X} \left[ \| \mathbb {E} _ {Y} \left[ X - Y \right] \| \right] = \mathbb {E} _ {X} \left[ \| X - \mathbb {E} \left[ Y \right] \| \right].
+$$
+
+Corollary A.1 If  $X_{1}, X_{2} \in \mathbb{R}^{d}$  are I.I.D. copies of  $X$ , then we have  $\mathbb{E}\left[\|X - \mathbb{E}[X]\|\right] \leq \mathbb{E}\left[\|X_{1} - X_{2}\|\right]$ .
+
+Lemma A.2 If  $X_{1}, X_{2} \in \mathbb{R}^{d}$  are I.I.D. copies of  $X$ , then we have  $\mathbb{E}\left[\|X - \mathbb{E}[X]\|^{2}\right] \leq \frac{1}{2}\mathbb{E}\left[\|X_{1} - X_{2}\|^{2}\right]$ .
+
+Proof
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \| X _ {1} - X _ {2} \| ^ {2} \right] = \mathbb {E} \left[ \| X _ {1} \| ^ {2} \right] - 2 \mathbb {E} \left[ X _ {1} ^ {\top} X _ {2} \right] + \mathbb {E} \left[ \| X _ {2} \| ^ {2} \right] \\ = \mathbb {E} \left[ \left\| X _ {1} \right\| ^ {2} \right] - 2 \mathbb {E} \left[ X _ {1} \right] ^ {\top} \mathbb {E} \left[ X _ {2} \right] + \mathbb {E} \left[ \left\| X _ {2} \right\| ^ {2} \right] \\ = \mathbb {E} \left[ \| X \| ^ {2} \right] - 2 \mathbb {E} [ X ] ^ {\top} \mathbb {E} [ X ] + \mathbb {E} \left[ \| X \| ^ {2} \right] \\ = 2 \mathbb {E} \left[ \| X - \mathbb {E} [ X ] \| ^ {2} \right]. \\ \end{array}
+$$
+
+Lemma A.3 Let  $f: \mathcal{W} \to \mathbb{R}_{\geq 0}$  be a non-negative convex function. Let  $W \in \mathcal{W}$  be a random variable, then we have
+
+$$
+\mathbb {E} \left[ | f (\mathbb {E} [ W ]) - f (W) | \right] \leq 2 \mathbb {E} \left[ f (W) \right].
+$$
+
+Proof
+
+$$
+\begin{array}{l} \mathbb {E} \left[ | f (\mathbb {E} [ W ]) - f (W) | \right] \leq \mathbb {E} \left[ | f (\mathbb {E} [ W ]) | \right] + \mathbb {E} \left[ | f (W) | \right] \\ = \mathbb {E} [ f (\mathbb {E} [ W ]) ] + \mathbb {E} [ f (W) ] \quad (f \geq 0) \\ \leq 2 \mathbb {E} [ f (W) ]. \quad (\text {c o n v e x i t y} f) \\ \end{array}
+$$
+
+# B PROOFS AND DETAILS FOR SEC. 3 PROPOSED OMNISCIENT TRAJECTORY
+
+# B.1 DETAILS FOR SECTION 3.1
+
+In this section, we present illustrative examples of suboptimality and overfitting of Proposition 2 mentioned in Section 3.1.
+
+# B.1.1 SUBOPTIMALITY
+
+Different  $(S,W)$  has different empirical and population Hessians with different sharp directions. For instance, assume  $v_{1}$  is the only sharp direction for  $(s_1,w_1)$  and  $v_{2}$  for  $(s_2,w_2)$ . Then the nonspecific  $\Sigma$  would apply small noises in both directions. If  $v_{1}$  and  $v_{2}$  are nearly orthogonal, the noises along  $v_{2}$  and  $v_{1}$  are insufficient for  $(s_1,w_1)$  and  $(s_2,w_2)$ , respectively.
+
+# B.1.2 OVERFITTING
+
+When  $k$  output weights are sampled, they lie in a  $k$ -dimensional subspace. As a result, noises along the remaining  $d - k$  directions will not decrease the trajectory term. The optimal  $\Sigma$  induces infinitely small noises along the remaining  $d - k$  directions to decrease the penalty terms. Therefore, the optimal  $\Sigma$  will have rank at most  $k$  when  $k$  samples are used. If one additional weight is sampled and it does not reside in the  $k$ -dimensional subspace,  $\Sigma$  cannot reduce the variance along the  $(k + 1)$ -th direction. As a result, the bound optimized for  $k$  samples is suboptimal for the  $k + 1$  samples, indicating overfitting.
+
+# B.2 LEMMA B.1
+
+Lemma B.1 extends Lemma 4 of Wang & Mao (2022) that bounds the mutual information using gradient variances.
+
+Lemma B.1 Let  $X, Y \in \mathbb{R}^{d_1}$ ,  $\Delta \in \mathbb{R}^{d_2}$  and  $O \in \mathbb{R}^{d_3}$  be arbitrary random variables. Let  $N \sim \mathcal{N}(0, I)$  be a  $d_1$ -dimensional Gaussian noise that is independent of  $(X, Y, \Delta, O)$ . Then for any  $\sigma > 0$ , for any deterministic  $\mathbb{R}^{d_1}$  function  $f$  of  $(X, Y, \Delta)$ , we have for any function  $\Omega$  solely of  $(Y, O)$ ,
+
+$$
+I (f (X, Y, \Delta) + \sigma N; X \mid Y, O) \leq \frac {1}{2 \sigma^ {2}} \mathbb {E} \left[ \| f (X, Y, \Delta) - \Omega (Y, O) \| ^ {2} \right], \tag {B.1}
+$$
+
+and for any function  $\Omega$  solely of  $Y$ ,
+
+$$
+I (f (X, Y, \Delta) + \sigma N; X \mid Y) \leq \frac {1}{2 \sigma^ {2}} \mathbb {E} \left[ \| f (X, Y, \Delta) - \Omega (Y) \| ^ {2} \right]. \tag {B.2}
+$$
+
+Proof We prove Eq. (B.1) first. Let  $0_{k} \in \mathbb{R}^{k}$  denote  $k$ -dimensional zero vector. Define  $X' := \left[ \begin{array}{ccc} X^{\top} & 0_{d_2}^{\top} & 0_{d_3}^{\top} \end{array} \right]^{\top}, Y' := \left[ \begin{array}{ccc} Y^{\top} & 0_{d_2} & O^{\top} \end{array} \right]^{\top}, \Delta' := \left[ \begin{array}{ccc} 0_{d_1}^{\top} & \Delta^{\top} & 0_{d_3}^{\top} \end{array} \right]^{\top}$  and  $f': \mathbb{R}^{d_1 + d_2 + d_3} \times \mathbb{R}^{d_1} \to \mathbb{R}^d$  and  $\Omega': \mathbb{R}^{d_1 + d_2 + d_3} \to \mathbb{R}^{d_1}$  by
+
+$$
+f ^ {\prime} (v, u) := f \left(u _ {1: d _ {1}}, v _ {1: d _ {1}}, v _ {d _ {1} + 1: d _ {1} + d _ {2}}\right), \Omega^ {\prime} (v) := \Omega \left(v _ {1: d _ {1}}, v _ {d _ {1} + d _ {2} + 1: d _ {1} + d _ {2} + d _ {3}}\right),
+$$
+
+where  $0_{d} \in \mathbb{R}^{d}$  is the zero vector. By assumption,  $(X', Y', \Delta')$  is independent of  $N$ . Consequently, by Lemma 4 of Wang & Mao (2022), we have
+
+$$
+\begin{array}{l} I (f ^ {\prime} (Y ^ {\prime} + \Delta^ {\prime}, X ^ {\prime}) + \sigma N; X ^ {\prime} \mid Y ^ {\prime}) \leq \frac {d}{2} \mathbb {E} \left[ \log \left(\frac {\mathbb {E} [ \| f ^ {\prime} (Y ^ {\prime} + \Delta^ {\prime} , X ^ {\prime}) - \Omega^ {\prime} (Y ^ {\prime}) \| \mid Y ^ {\prime} ]}{d \sigma^ {2}} + 1\right) \right] \\ \leq \frac {1}{2 \sigma^ {2}} \mathbb {E} \left[ \| f ^ {\prime} (Y ^ {\prime} + \Delta^ {\prime}, X ^ {\prime}) - \Omega^ {\prime} (Y ^ {\prime}) \| \right]. \\ \end{array}
+$$
+
+By construction, we have  $f'(Y' + \Delta', X') = f(X, Y, \Delta)$  and  $\Omega'(Y') = \Omega(Y, O)$ . Combined with the definition of  $Y'$ , they lead to Eq. (B.2) of Lemma B.1.
+
+To obtain Eq. (B.2), let  $O''$  be the constant  $0_{d_2}$  and  $\Omega''(y, o'') \coloneqq \Omega(y)$ . Then by Eq. (B.1), we have
+
+$$
+\begin{array}{l} I (f (X, Y, \Delta) + \sigma N; X \mid Y) = I (f (X, Y, \Delta) + \sigma N; X \mid Y, O ^ {\prime \prime}) \\ \leq \frac {1}{2 \sigma^ {2}} \mathbb {E} \left[ \| f (X, Y, \Delta) - \Omega^ {\prime \prime} (Y, O ^ {\prime \prime}) \| ^ {2} \right] \\ = \frac {1}{2 \sigma^ {2}} \mathbb {E} \left[ \| f (X, Y, \Delta) - \Omega (Y) \| ^ {2} \right], \\ \end{array}
+$$
+
+which is exactly Eq. (B.2).
+
+# B.3 PROOF OF THEOREM 1
+
+By the technique of a change of trajectory, we first transform the generalization error of the original trajectory into that of the omniscient and SGLD-like trajectories at the cost of penalty terms:
+
+$$
+\begin{array}{l} \mathrm {g e n} (\mu^ {n}, P _ {W | S}) \\ := \mathbb {E} \left[ \mathcal {L} _ {\mu} (W) - \hat {\mathcal {L}} _ {S} (W) \right] \\ = \mathbb {E} \left[ \mathcal {L} _ {\mu} \left(\check {W} _ {T}\right) - \hat {\mathcal {L}} _ {S} \left(\check {W} _ {T}\right) \right] + \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} \left(\check {W} _ {T}\right) - \hat {\mathcal {L}} _ {S} \left(W _ {T}\right) \right] - \mathbb {E} \left[ \mathcal {L} _ {\mu} \left(\check {W} _ {T}\right) - \mathcal {L} _ {\mu} \left(W _ {T}\right) \right] \\ = \mathbb {E} \left[ \mathcal {L} _ {\mu} (\tilde {W} _ {T}) - \hat {\mathcal {L}} _ {S} (\tilde {W} _ {T}) \right] + \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (\tilde {W} _ {T}) - \hat {\mathcal {L}} _ {S} (\breve {W} _ {T}) \right] - \mathbb {E} \left[ \mathcal {L} _ {\mu} (\tilde {W} _ {T}) - \mathcal {L} _ {\mu} (\breve {W} _ {T}) \right] \\ + \mathbb {E} \left[ \Delta_ {\Gamma_ {T}} \left(W _ {T}, S\right) - \Delta_ {\Gamma_ {T}} \left(W _ {T}, S ^ {\prime}\right) \right] \\ = \mathbb {E} \left[ \mathcal {L} _ {\mu} \left(\tilde {W} _ {T}\right) - \hat {\mathcal {L}} _ {S} \left(\tilde {W} _ {T}\right) \right] + \mathbb {E} \left[ \Delta^ {\sum_ {t} \sigma_ {t} ^ {2}} \left(W _ {T} + \Gamma_ {T}, S\right) - \Delta^ {\sum_ {t} \sigma_ {t} ^ {2}} \left(W _ {T} + \Gamma_ {T}, S ^ {\prime}\right) \right] \\ + \mathbb {E} \left[ \Delta_ {\Gamma_ {T}} \left(W _ {T}, S\right) - \Delta_ {\Gamma_ {T}} \left(W _ {T}, S ^ {\prime}\right) \right]. \tag {B.3} \\ \end{array}
+$$
+
+With the penalty terms matching those in Theorem 1, the goal reduces to bounding the generalization error of  $\tilde{W}_T$ . To this end, we apply Lemma 1 that bounds the error by mutual information and then decompose the mutual information using the chain rule. Since  $\ell(w, \cdot)$  is  $R$ -sub-Gaussian by assumption, we have
+
+$$
+\mathbb {E} \left[ \mathcal {L} _ {\mu} (\tilde {W} _ {T}) - \hat {\mathcal {L}} _ {S} (\tilde {W} _ {T}) \right] = \mathrm {g e n} (\mu^ {n}, P _ {\tilde {W} _ {T} | S}, \ell) \leq \sqrt {\frac {2 R ^ {2}}{n} I (\tilde {W} _ {T} ; S)},
+$$
+
+where
+
+$$
+I (\tilde {W} _ {T}; S) \leq I (\tilde {W} _ {0: T}; S)
+$$
+
+(Data-processing inequality)
+
+$$
+\begin{array}{l} = I \left(\tilde {W} _ {0}; S\right) + \sum_ {t = 1} ^ {T} I \left(\tilde {W} _ {t}; S \mid \tilde {W} _ {0: t - 1}\right) \quad (\text {C h a i n r u l e o f M I}) \\ = \sum_ {t = 1} ^ {T} I (\tilde {W} _ {t}; S \mid \tilde {W} _ {0: t - 1}) \quad (\tilde {W} _ {0} = W _ {0} \text {i s i n d e p e n d e n t o f} S) \\ = \sum_ {t = 1} ^ {T} I \left(\tilde {W} _ {t - 1} - \left(g _ {t} - \Delta g _ {t}\right) + N _ {t}; S \mid \tilde {W} _ {0: t - 1}\right). \tag {Definition1} \\ \end{array}
+$$
+
+To bound the stepwise conditional mutual information, we apply Lemma B.1 with  $X = S$ ,  $Y = \tilde{W}_{0:t-1}$ ,  $\Delta = (W_{0:T}, V)$  and
+
+$$
+f (X, Y, \Delta) = \tilde {W} _ {t - 1} - \left(g _ {t} \left(W _ {t - 1}, S, V, W _ {0: t - 2}\right) - \Delta g _ {t} \left(S, V, g _ {1: T}, W _ {0: T}\right)\right).
+$$
+
+Since  $(X,Y,\Delta)$  is a function of  $(S,V,g_{1:T},W_{0:T})$ , which is independent with the SGLD noise  $N_{t}\sim \mathcal{N}(0,\sigma^{2}I)$ , by Lemma B.1, we have
+
+$$
+\begin{array}{l} I \left(\tilde {W} _ {t}; S \mid \tilde {W} _ {0: t - 1}\right) = I \left(\tilde {W} _ {t - 1} - \left(g _ {t} - \Delta g _ {t}\right) + N _ {t}; S \mid \tilde {W} _ {0: t - 1}\right) \\ \leq \frac {1}{2 \sigma^ {2}} \mathbb {E} \left[ \left\| \tilde {W} _ {t - 1} - (g _ {t} - \Delta g _ {t}) - \Omega (\tilde {W} _ {0: t - 1}) \right\| ^ {2} \right] \\ \end{array}
+$$
+
+for any function  $\Omega$  of  $Y = \tilde{W}_{0:t - 1}$ . To make things simpler by removing  $\tilde{W}_{t - 1}$  and minimize the expected squared norm, we set  $\Omega (\tilde{w}_{0:t - 1}) = \tilde{w}_{t - 1} - \mathbb{E}[g_t]$  and obtain
+
+$$
+I \left(\tilde {W} _ {t}; S \mid \tilde {W} _ {0: t - 1}\right) \leq \frac {1}{2 \sigma^ {2}} \mathbb {E} \left[ \| g _ {t} - \mathbb {E} [ g _ {t} ] - \Delta g _ {t} \| ^ {2} \right]. \tag {B.4}
+$$
+
+Plugging Eq. (B.4) back to Eq. (B.3) leads to Theorem 1.
+
+Remark B.1 The bound can be further tightened by setting  $\Omega(\tilde{w}_{0:t-1}) = \tilde{w}_{t-1} - \mathbb{E}[g_t] + \mathbb{E}[\Delta g_t]$ . However, this eventually leads to an extra  $\mathbb{E}[\Delta G]$  in the trajectory terms of Corollary B.1. This extra term creates a "feedback" between instances that hinders optimization: when optimizing  $\Delta G$ , we must consider all instances of  $\Delta G(S,V,g_{1:T},W_{0:T})$  even if we are optimizing against a single instance of trajectory to form Theorem 2. For simplicity, we only set  $\Omega(\tilde{w}_{0:t-1}) = \tilde{w}_{t-1} - \mathbb{E}[g_t]$ .
+
+# B.4 SELECTION OF  $\Delta g_{t}$  GIVEN  $\Gamma_{T}$
+
+In the result of Theorem 1, the penalty and flatness terms only rely on  $\Gamma_T$  among omniscient-related variables. Therefore, we can first fix  $\Gamma_T$  and optimize  $\Delta g_{1:T}$ .
+
+Assume for this subsection that  $\Gamma_T$  is fixed as  $\Delta G$ , a function of  $(S, V, g_{1:T}, W_{0:T})$ , and that  $\Delta g_{1:T}$  is constrained to satisfy  $\Delta G = \sum_{i=1}^{T} \Delta g_t$ . Penalty and flatness terms are now fixed since they only rely on  $\Gamma_T$  and we only need to optimize the trajectory term
+
+$$
+\underset {\Delta g _ {1: T}: \sum_ {t} \Delta g _ {t} = \Delta G} {\arg \min } \sqrt {\frac {R ^ {2}}{n} \sum_ {t = 1} ^ {T} \frac {1}{\sigma_ {t} ^ {2}} \mathbb {E} \left[ \| g _ {t} - \mathbb {E} [ g _ {t} ] - \Delta g _ {t} \| ^ {2} \right]}
+$$
+
+over  $\Delta g_{1:T}$ . We then set  $\sigma_t = \sigma$ , throw the square root and coefficients away, and move the summation inside the expectation, arriving at
+
+$$
+\underset {\Delta g _ {1: T}: \sum_ {t} \Delta g _ {t} = \Delta G} {\arg \min } \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \| g _ {t} - \mathbb {E} [ g _ {t} ] - \Delta g _ {t} \| ^ {2} \right]
+$$
+
+Set  $\nu_{t} \coloneqq g_{t} - \mathbb{E}[g_{t}]$ . Since the constraints are instance-wise, we can also optimize  $\Delta g_{t}$  instances-wisely:
+
+$$
+\begin{array}{l} \underset {\Delta g _ {1: T}: \sum_ {t} \Delta g _ {t} = \Delta G} {\arg \min} \sum_ {t = 1} ^ {T} \| \nu_ {t} - \Delta g _ {t} \| ^ {2} = T \cdot \mathbb {E} _ {t \sim U [ T ]} \left[ \| \nu_ {t} - \Delta g _ {t} \| ^ {2} \right] \geq T \cdot \left\| \mathbb {E} _ {t \sim U [ T ]} \left[ \nu_ {t} - \Delta g _ {t} \right] \right\| ^ {2} \\ = T \cdot \left\| \mathbb {E} _ {t \sim U [ T ]} \left[ \nu_ {t} \right] - \frac {1}{T} \Delta G \right\| ^ {2}, \\ \end{array}
+$$
+
+where inequality is due to the convexity of the squared norm and takes equality if and only if  $\nu_{t} - \Delta g_{t}$  is constant (w.r.t.  $t$ ), say  $\Delta$ . Then we have  $\sum_{\tau} \Delta g_{\tau} = \left(\sum_{\tau} \nu_{\tau}\right) - T \cdot \Delta = \Delta G$ , which implies  $\Delta g_{t} = g_{t} - \mathbb{E}[g_{t}] - \frac{1}{T} \sum_{\tau} (g_{\tau} - \mathbb{E}[g_{\tau}]) + \frac{1}{T} \Delta G$ .
+
+# B.5 COROLLARY B.1
+
+Corollary B.1 Assume  $\ell(w, \cdot)$  is  $R$ -sub-Gaussian on  $\mu$  for any  $w \in \mathcal{W}$ . For any deterministic function  $\Delta G$  of  $(S, V, g_{1:T}, W_{0:T})$  and any  $\sigma > 0$ , we have
+
+$$
+\operatorname {g e n} \left(\mu^ {n}, P _ {W | S}\right) \leq \sqrt {\frac {R ^ {2}}{n \sigma^ {2}} \frac {\mathbb {E} \left[ \| \Delta W _ {T} - \mathbb {E} \left[ \Delta W _ {T} \right] + \Delta G \| ^ {2} \right]}{T}} + \mathbb {E} \left[ \Delta_ {\Delta G} ^ {\sigma^ {2} T} \left(W _ {T}, S\right) - \Delta_ {\Delta G} ^ {\sigma^ {2} T} \left(W _ {T}, S ^ {\prime}\right) \right],
+$$
+
+where  $\Delta W_{T} := W_{T} - W_{0}$  is the change before and after the training. If we give up further optimization, i.e.,  $\Delta G = 0$ , then we have
+
+$$
+\operatorname {g e n} \left(\mu^ {n}, P _ {W | S}\right) \leq \sqrt {\frac {R ^ {2}}{n T \sigma^ {2}} \mathbb {V} \left[ \Delta W _ {T} \right]} + \mathbb {E} \left[ \Delta^ {T \sigma^ {2}} \left(W _ {T}, S\right) - \Delta^ {T \sigma^ {2}} \left(W _ {T}, S ^ {\prime}\right) \right],
+$$
+
+where  $\mathbb{V}[X] := \mathbb{E}\left[\|X - \mathbb{E}[X]\|^2\right]$  denotes variance.
+
+Proof We prove Corollary B.1 by instantiating  $\Delta g_{t}$  of Theorem 1.
+
+Since  $\Delta g_{t}$  depend on  $g_{1:T}$ , we can use  $\nu_{t} := g_{t} - \mathbb{E}[g_{t}]$  and  $\bar{\nu} = \frac{1}{T}\sum_{t}\nu_{t}$  when constructing  $\Delta g_{t}$ . Let  $\Delta G$  be a deterministic function of  $(S,V,g_{1:T},W_{0:T})$  and  $\sigma > 0$  be a constant. Then  $\Delta g_{t} := (\nu_{t} - \bar{\nu}) + \frac{1}{T}\Delta G$  is indeed a deterministic function of  $(S,V,g_{1:T},W_{0:T})$ .
+
+With  $\Gamma_T \coloneqq \sum_t \Delta g_t = \sum_t \nu_t - T \cdot \bar{\nu} + \Delta G = \Delta G$  and  $\sigma_t$  set to constant  $\sigma$ , by Theorem 1 we obtain
+
+$$
+\begin{array}{l} \mathrm {g e n} (\mu^ {n}, P _ {W | S}) \leq \sqrt {\frac {R ^ {2}}{n \sigma^ {2}} \sum_ {t} \mathbb {E} \left[ \left\| \nu_ {t} - (\nu_ {t} - \bar {\nu}) - \frac {1}{T} \Delta G \right\| ^ {2} \right]} + \mathbb {E} \left[ \Delta_ {\Delta G} (W _ {T}, S) - \Delta_ {\Delta G} (W _ {T}, S ^ {\prime}) \right] \\ + \mathbb {E} \left[ \Delta^ {T \sigma^ {2}} (W _ {T} + \Delta G, S) - \Delta^ {T \sigma^ {2}} (W _ {T} + \Delta G, S ^ {\prime}) \right], \\ \end{array}
+$$
+
+where
+
+$$
+\begin{array}{l} \sum_ {t} \mathbb {E} \left[ \left\| \nu_ {t} - (\nu_ {t} - \bar {\nu}) - \frac {1}{T} \Delta G \right\| ^ {2} \right] = \sum_ {t} \mathbb {E} \left[ \left\| \frac {1}{T} \sum_ {\tau} (g _ {\tau} - \mathbb {E} [ g _ {\tau} ]) - \frac {1}{T} \Delta G \right\| ^ {2} \right] \\ = T \cdot \frac {1}{T ^ {2}} \mathbb {E} \left[ \left\| \sum_ {t} g _ {t} - \mathbb {E} \left[ \sum_ {t} g _ {t} \right] - \Delta G \right\| ^ {2} \right]. \\ \end{array}
+$$
+
+Since  $W_{T} = W_{0} - \sum_{t}g_{t}$ , we have  $\sum_{t}g_{t} = -\Delta W_{T}$  and
+
+$$
+\sum_ {t} \mathbb {E} \left[ \left\| \nu_ {t} - \left(\nu_ {t} - \bar {\nu}\right) - \frac {1}{T} \Delta G \right\| ^ {2} \right] = \frac {1}{T} \mathbb {E} \left[ \| \Delta W _ {T} - \mathbb {E} [ \Delta W _ {T} ] + \Delta G \| ^ {2} \right].
+$$
+
+Putting everything together leads to Corollary B.1.
+
+# B.6 DETAILS FOR SECTION 3.3
+
+Firstly, we need a explicit form of the penalty terms to balance the tradeoff. Therefore, the penalty terms are approximated to the second order in the surrogate optimization, where the Hessians at the output weight show up. The approximation is the same as Appendix B.7. Secondly, we assume the empirical and population Hessians in the penalty term for the SGLD-like trajectory does not change too much, namely  $\hat{H}_S(W_T + \Delta G) - H_\mu (W_T + \Delta G)\approx \hat{H}_S(W_T) - H_\mu (W_T)$ , to simplify the surrogate optimization. These simplifications lead to the following surrogate optimization target (be noted that we have not replacing terms with  $\tilde{H}_{\mathrm{flat}}$ ,  $\tilde{H}_{\mathrm{pen}}$  or  $J$ ):
+
+$$
+\begin{array}{l} \min _ {\Delta G} \frac {3}{2} \sqrt [ 3 ]{\frac {R ^ {2}}{n} \left| \mathbb {E} \left[ \mathrm {t r} \left(\hat {H} _ {S} (W _ {T}) - H _ {\mu} (W _ {T})\right) \right] \right|} \mathbb {E} \left[ \| \Delta W _ {T} - \mathbb {E} \left[ \Delta W _ {T} \right] + \Delta G \| ^ {2} \right] \\ + \mathbb {E} \left[ \Delta G ^ {\top} \nabla (\hat {\mathcal {L}} _ {S} (W _ {T}) - \mathcal {L} _ {\mu} (W _ {T})) + \frac {1}{2} \Delta G ^ {\top} (\hat {H} _ {S} (W _ {T}) - H _ {\mu} (W _ {T})) \Delta G \right], \\ \end{array}
+$$
+
+where  $\hat{H}_S(w)$  is the Hessian of the empirical loss at the trajectory terminal and  $H_{\mu}(w)$  is that of the population loss. If we optimize the above surrogate target, we will rely on testing/Validation sets to estimate the population gradient and Hessian. Therefore, one may want to partially avoid this reliance as much as possible by simply ignoring them. On the other hand, optimizing against validation sets leads to better numerical tightness. To accommodate both needs, we leave it as an option by letting  $\tilde{H}_{\mathrm{flat}} \in \left\{\Delta H(W_T), \hat{H}_S(W_T)\right\}$ ,  $\tilde{H}_{\mathrm{pen}} \in \left\{\Delta H(W_T), \hat{H}_S(W_T)\right\}$  and  $J \in \nabla (\hat{\mathcal{L}}_S(W_T) - \mathcal{L}_\mu(W_T))$ ,  $\nabla \hat{\mathcal{L}}_S(W_T)$  and replacing the empirical-population differences in the above surrogate losses. Another difficulty is that, one has to consider other independent runs when optimizing  $\Delta G$  for one trajectory. To further simplify optimization, we modify the surrogate optimization target so that only one trajectory is considered:
+
+$$
+\min _ {\Delta G} \frac {3}{2} \sqrt [ 3 ]{\frac {R ^ {2}}{n} \left| \mathrm {t r} \left(\tilde {H} _ {\mathrm {f l a t}}\right) \right| \left\| \Delta W _ {T} - \mathbb {E} \left[ \Delta W _ {T} \right] + \Delta G \right\| ^ {2}} + \Delta G ^ {\top} J + \frac {1}{2} \Delta G ^ {\top} \tilde {H} _ {\mathrm {p e n}} \Delta G,
+$$
+
+where  $\Delta G$  is a function of the trajectory. To obtain a convex optimization problem, we partially remove the cubic root, giving
+
+$$
+\min _ {\Delta G} \lambda C \left\| \Delta W _ {T} - \mathbb {E} \left[ \Delta W _ {T} \right] + \Delta G \right\| ^ {2} + \Delta G ^ {\top} J + \frac {1}{2} \Delta G ^ {\top} \tilde {H} _ {\mathrm {p e n}} \Delta G,
+$$
+
+where  $C \coloneqq \frac{3}{2}\sqrt[3]{\frac{R^2}{n}\left|\operatorname{tr}\left(\tilde{H}_{\mathrm{flat}}\right)\right|}$ . Bound parameter  $\lambda$  here is used to compensate the "weight" change after twisting the target: When the terms under the cubic root become smaller, the derivative of the cubic root increases fast; but after the cubic root is removed, the derivative of the norm becomes smaller when the term becomes smaller. Therefore, we should use large  $\lambda$  to emphasize the reduction of trajectory term. The convex optimization problem has a closed form solution
+
+$$
+\Delta G = - \left(2 \lambda C I + \tilde {H} _ {\text {p e n}}\right) ^ {- 1} \left(2 \lambda C \left(\Delta W _ {T} - \mathbb {E} \left[ \Delta W _ {T} \right]\right) + J\right). \tag {B.5}
+$$
+
+Since all variables in Eq. (B.5) is a function of weights, empirical and population gradients and Hessians, which are functions of weights, the training set and the population distribution, Eq. (B.5) indeed specifies a function of  $(S,V,g_{1:T},W_{0:T})$ . This omniscient trajectory is used in Theorem 2.
+
+# B.7 PROOF OF THEOREM 2
+
+Based on Corollary B.1, Theorem 2 considers the specific omniscient trajectory defined by  $\Delta G = -(2\lambda CI + \tilde{H}_{\mathrm{pen}})^{-1}(2\lambda C(\Delta W_T - \mathbb{E}[\Delta W_T]) + J)$  as Eq. (B.5). We first optimize  $\sigma$  given the specific  $\Delta G$ , for which we approximate the penalty for the SGLD-like trajectory to the third order. Specifically, for the empirical part we have
+
+$$
+\begin{array}{l} \Delta^ {\sigma^ {2} T} (W _ {T} + \Delta G, S) \\ := \mathbb {E} _ {\xi \sim \mathcal {N} (0, \sigma^ {2} T I)} \left[ \hat {\mathcal {L}} _ {S} \left(W _ {T} + \Delta G + \xi\right) - \hat {\mathcal {L}} _ {S} \left(W _ {T} + \Delta G\right) \right] \\ = \mathbb {E} _ {\xi} \left[ \xi^ {\top} \nabla \hat {\mathcal {L}} _ {S} \left(W _ {T} + \Delta G\right) + \frac {1}{2} \xi^ {\top} \hat {H} _ {S} \left(W _ {T} + \Delta G\right) \xi + \sum_ {i \leq j \leq k} a _ {i, j, k} \xi_ {i} \xi_ {j} \xi_ {j} + r _ {0} \right] \\ = \frac {\sigma^ {2} T}{2} \operatorname {t r} \left(\hat {H} _ {S} \left(W _ {T} + \Delta G\right)\right) + \mathbb {E} _ {\xi} \left[ r _ {0} \right]. \\ \end{array}
+$$
+
+where  $r_0$  is the residual at the third order and the coefficients  $a_{i,j,k}$  are independent  $\xi$ . By the assumption of Theorem 2, we have  $|r_0| \leq D\|\xi\|^4$  and  $|\mathbb{E}_{\xi}[r_0]| \leq D\mathbb{E}\left[\|\xi\|^4\right] = Dd(d + 2)\sigma^4$ . One can obtain similar results for the population loss difference, and combining the both we have
+
+$$
+\mathbb {E} \left[ \Delta^ {\sigma^ {2} T} (W _ {T} + \Delta G, S) - \Delta^ {\sigma^ {2} T} (W _ {T} + \Delta G, S ^ {\prime}) \right] = \frac {\sigma^ {2} T}{2} \operatorname {t r} \left(\hat {H} _ {S} (W _ {T} + \Delta G) - H _ {\mu} (W _ {T} + \Delta G)\right) + r,
+$$
+
+where  $|r| \leq Dd(d + 2)\sigma^4 = O(d^2\sigma^4)$ .
+
+Similarly to Theorem 2 of Wang & Mao (2022), we optimize  $\sigma > 0$  to balance
+
+$$
+\sqrt {\frac {R ^ {2}}{n \sigma^ {2}} \frac {\mathbb {E} \left[ \left\| \Delta W _ {T} - \mathbb {E} \left[ \Delta W _ {T} \right] + \Delta G \right\| ^ {2} \right]}{T}} =: \frac {A}{\sigma}.
+$$
+
+and
+
+$$
+\frac {\sigma^ {2} T}{2} \operatorname {t r} \left(\hat {H} _ {S} \left(W _ {T} + \Delta G\right) - H _ {\mu} \left(W _ {T} + \Delta G\right)\right) =: \sigma^ {2} B
+$$
+
+For  $A, B > 0$ ,  $A / \sigma + \sigma^2 B$  takes the minimum  $3(A / 2)^{2 / 3}B^{1 / 3}$  at
+
+$$
+\begin{array}{l} \sigma_ {*} = (A / 2 B) ^ {1 / 3} \\ = \left(\frac {\sqrt {\frac {R ^ {2}}{n} \mathbb {E} \left[ \left\| (I - (I + \tilde {H} _ {\mathrm {p e n}} / 2 \lambda C) ^ {- 1}) (W _ {T} - \Delta W _ {T}) - (2 \lambda C I + \tilde {H} _ {\mathrm {p e n}}) ^ {- 1} J \right\| ^ {2} \right]}}{T ^ {3 / 2} \operatorname {t r} (\Delta H (W _ {T} + \Delta G))}\right) ^ {1 / 3}. \tag {B.6} \\ \end{array}
+$$
+
+The proof is finished by setting  $\sigma$  to this value and putting everything together.
+
+# B.8 DETAILS FOR SECTION 3.4
+
+We take  $J \coloneqq \nabla (\hat{\mathcal{L}}_S(W_T) - \mathcal{L}_\mu (W_T))$  and sufficiently large  $\lambda > 0$  so that  $\left|\lambda_1(\tilde{H}_{\mathrm{pen}} / 2\lambda C)\right| \ll 1$ . We assume the training is sufficient so that  $\nabla \hat{\mathcal{L}}_S(W_T) \approx 0$ . We also assume that the initial weight  $W_0$  is fixed.
+
+By applying  $\| x + y\|^2 \leq 2\| x\|^2 + 2\| y\|^2$  and approximating the penalty for the omniscient trajectory to the second order, we obtain
+
+$$
+\mathrm {g e n} (\mu^ {n}, P _ {W | S})
+$$
+
+$$
+\begin{array}{l} \lesssim \frac {3}{2} \sqrt [ 3 ]{\frac {2 R ^ {2}}{n} \mathbb {E} \left[ | \operatorname {t r} \left(\Delta H (W _ {T} + \Delta G)\right) | \right]} \mathbb {E} \left[ \| \Delta W _ {T} - \mathbb {E} \left[ \Delta W _ {T} \right] \| _ {(I - (I + \tilde {H} _ {\mathrm {p e n}} / 2 \lambda C) ^ {- 1}) ^ {2}} ^ {2} + \| J \| _ {(2 \lambda C I + \tilde {H} _ {\mathrm {p e n}}) ^ {- 2}} ^ {2} \right] \\ + \mathbb {E} \left[ - 2 \lambda C (\nabla \hat {\mathcal {L}} _ {S} (W _ {T}) - \nabla \mathcal {L} _ {\mu} (W _ {T})) ^ {\top} (2 \lambda C I + \tilde {H} _ {\mathrm {p e n}}) ^ {- 1} (\Delta W _ {T} - \mathbb {E} [ \Delta W _ {T} ]) \right] \\ + \mathbb {E} \left[ - \big (\nabla \hat {\mathcal {L}} _ {S} (W _ {T}) - \mathcal {L} _ {\mu} (W _ {T}) \big) ^ {\top} (2 \lambda C I + \tilde {H} _ {\mathrm {p e n}}) ^ {- 1} J \right] \\ + \mathbb {E} \left[ \| \Delta W _ {T} - \mathbb {E} \left[ \Delta W _ {T} \right] \| _ {4 \lambda^ {2} C ^ {2} (2 \lambda C I + \tilde {H} _ {\mathrm {p e n}}) ^ {- 1} | \Delta H | (2 \lambda C I + \tilde {H} _ {\mathrm {p e n}}) ^ {- 1}} ^ {2} + \| J \| _ {(2 \lambda C I + \tilde {H} _ {\mathrm {p e n}}) ^ {- 1} | \Delta H | (2 \lambda C I + \tilde {H} _ {\mathrm {p e n}}) ^ {- 1}} ^ {2} \right]. \\ \end{array}
+$$
+
+By selection of  $\lambda$ , we have  $(2\lambda CI + \tilde{H}_{\mathrm{pen}})^{-1} \approx (2\lambda CI)^{-1} = I / 2\lambda C$  and  $I - (I + \tilde{H}_{\mathrm{pen}} / 2\lambda C)^{-1} = -\sum_{k=1}^{\infty} (-1)^{k} (\tilde{H}_{\mathrm{pen}} / 2\lambda C)^{k} \approx \tilde{H}_{\mathrm{pen}} / 2\lambda C$ . These approximations lead to
+
+$$
+\begin{array}{l} \mathrm {g e n} (\mu^ {n}, P _ {W | S}) \\ \lesssim \frac {3}{2} \sqrt [ 3 ]{\frac {2 R ^ {2}}{n} \mathbb {E} \left[ | \mathrm {t r} \left(\Delta H (W _ {T} + \Delta G)\right) | \right]} \mathbb {E} \left[ \left\| \Delta W _ {T} - \mathbb {E} \left[ \Delta W _ {T} \right] \right\| _ {\tilde {H} _ {\mathrm {p e n}} ^ {2} / 4 \lambda^ {2} C ^ {2}} ^ {2} + \left\| J \right\| ^ {2} / 4 \lambda^ {2} C ^ {2} \right] \\ + \mathbb {E} \left[ - (\nabla \hat {\mathcal {L}} _ {S} (W _ {T}) - \nabla \mathcal {L} _ {\mu} (W _ {T})) ^ {\top} (\Delta W _ {T} - \mathbb {E} [ \Delta W _ {T} ]) \right] \\ + \mathbb {E} \left[ - (\nabla \hat {\mathcal {L}} _ {S} (W _ {T}) - \mathcal {L} _ {\mu} (W _ {T})) ^ {\top} J / 2 \lambda C \right] \\ + \mathbb {E} \left[ \| \Delta W _ {T} - \mathbb {E} [ \Delta W _ {T} ] \| _ {| \Delta H |} ^ {2} + \| J \| _ {| \Delta H | / 4 \lambda^ {2} C ^ {2}} ^ {2} \right] \\ = \frac {3}{2} \sqrt [ 3 ]{\frac {2 R ^ {2}}{n} \mathbb {E} \left[ | \mathrm {t r} \left(\Delta H (W _ {T} + \Delta G)\right) | \right]} \mathbb {E} \left[ \left\| \Delta W _ {T} - \mathbb {E} \left[ \Delta W _ {T} \right] \right\| _ {\tilde {H} _ {\mathrm {p e n}} ^ {2} / 4 \lambda^ {2} C ^ {2}} ^ {2} + \left\| J \right\| ^ {2} / 4 \lambda^ {2} C ^ {2} \right] \\ - \mathbb {E} \left[ J ^ {\top} \left(\Delta W _ {T} - \mathbb {E} \left[ \Delta W _ {T} \right]\right) \right] \\ - \mathbb {E} \left[ \left\| J \right\| ^ {2} / 2 \lambda C \right] \\ + \mathbb {E} \left[ \| \Delta W _ {T} - \mathbb {E} [ \Delta W _ {T} ] \| _ {| \Delta H |} ^ {2} + \| J \| _ {| \Delta H | / 4 \lambda^ {2} C ^ {2}} ^ {2} \right]. \\ \end{array}
+$$
+
+The second term  $\mathbb{E}\left[\nabla \mathcal{L}_{\mu}(W_T)^\top (\Delta W_T - \mathbb{E}[\Delta W_T])\right]$  can also be transformed to Hessian-induced norms by exploiting the symmetry of the terminal deviation  $\Delta W_{T} - \mathbb{E}[\Delta W_{T}]$ . To this end, approximate the population gradient  $\nabla \mathcal{L}_{\mu}(W_T)$  at the mean terminal weight  $\mathbb{E}[W_T]$  by  $\nabla \mathcal{L}_{\mu}(W_T) \approx \nabla \mathcal{L}_{\mu}(\mathbb{E}[W_T]) + H_{\mu}(\mathbb{E}[W_T])(W_T - \mathbb{E}[W_T])$ . Then we have
+
+$$
+\begin{array}{l} \mathbb {E} \left[ J ^ {\top} \left(\Delta W _ {T} - \mathbb {E} \left[ \Delta W _ {T} \right]\right) \right] \\ \approx \mathbb {E} \left[ \left(\nabla \hat {\mathcal {L}} _ {S} (\mathbb {E} [ W _ {T} ]) - \nabla \mathcal {L} _ {\mu} (\mathbb {E} [ W _ {T} ]) + \Delta H (\mathbb {E} [ W _ {T} ]) (W _ {T} - \mathbb {E} [ W _ {T} ])\right) ^ {\top} (\Delta W _ {T} - \mathbb {E} [ \Delta W _ {T} ]) \right] \\ = \mathbb {E} \left[ \| \Delta W _ {T} - \mathbb {E} [ \Delta W _ {T} ] \| _ {\Delta H (\mathbb {E} [ W _ {T} ])} ^ {2} \right]. \\ \end{array}
+$$
+
+Plugging this back leads to
+
+$$
+\begin{array}{l} \mathrm {g e n} (\mu^ {n}, P _ {W | S}) \\ \lesssim \frac {3}{2} \sqrt [ 3 ]{\frac {2 R ^ {2}}{n} \mathbb {E} \left[ | \mathrm {t r} \left(\Delta H (W _ {T} + \Delta G)\right) | \right]} \mathbb {E} \left[ \| \Delta W _ {T} - \mathbb {E} \left[ \Delta W _ {T} \right] \| _ {\tilde {H} _ {\mathrm {p e n}} ^ {2} / 4 \lambda^ {2} C ^ {2}} ^ {2} + \| J \| ^ {2} / 4 \lambda^ {2} C ^ {2} \right] \\ - \mathbb {E} \left[ \| \Delta W _ {T} - \mathbb {E} [ \Delta W _ {T} ] \| _ {\Delta H (\mathbb {E} [ W _ {T} ])} ^ {2} \right] \\ - \mathbb {E} \left[ \| J \| ^ {2} / 2 \lambda C \right] + \mathbb {E} \left[ \| \Delta W _ {T} - \mathbb {E} [ \Delta W _ {T} ] \| _ {| \Delta H |} ^ {2} + \| J \| _ {| \Delta H | / 4 \lambda^ {2} C ^ {2}} ^ {2} \right]. \\ \end{array}
+$$
+
+The above results apply without the assumption that  $\nabla \hat{\mathcal{L}}_S(W_T) \approx 0$ .
+
+Using the assumption  $\nabla \hat{\mathcal{L}}_S(W_T) \approx 0$  leads to Eq. (2).
+
+# B.8.1 COMPARISONS WITH SIMILAR ALIGNMENTS WITH LOCAL GEOMETRY
+
+In Section 3.4, Theorem 2 is shown to connect with an alignment between terminal weight deviations/covariance and Hessians. This is similar to the work by Wang & Mao (2024b) and Wang et al. (2021a) on the alignment and the more fine-grained structure of local geometry.
+
+Wang & Mao (2024b) provides two types of generalization bounds for discretized SDE: the trajectory-state bound based on the statistics along the full trajectories and terminal-state bound based on the statistics after the last update, which are separately compared below.
+
+The trajectory-state bound focuses on the alignment between the population gradient noise covariance (GNC)  $\Sigma_t^\mu(W_t) \coloneqq \mathbb{V}_{Z' \sim \mu}[\nabla \ell(W_t, Z')]$  that depends on  $W_t$  and the empirical GNC  $C_t(W_t, S) \coloneqq \frac{n - b}{b(n - 1)} \mathbb{V}_{i \sim U[n]}[\nabla \ell(W_t, Z_i)]$  that depends on  $(W_t, S)$ . Therefore, this alignment and ours are both alignment between weight/gradient distribution and the local geometry. They also have the same dependence: the weight/gradient statistics are both  $W_{t-1}$ -dependent and the local geometric properties are both  $(W_{t-1}, S)$ -dependent. Differences mainly lie in the type of geometric properties used and how directly they reflect flatness. Wang & Mao (2024b)'s trajectory-state bound uses the inverted GNC  $C_t^{-1}$  to capture local geometry, which is an indirect measure of flatness. By comparing Wang & Mao (2024b)'s Figure 2 and 3, one can see that their bound does not correlate correctly with Hessian traces even along training steps. In contrast, we directly use Hessian, allowing us to better leverage the flatness as required by our motivation. Their choice of  $C_t^{-1}$  is a result of the discretized SDE algorithm instead of bound optimization ( $\Sigma_t^\mu$  is the ingredient corresponding to bound optimization). Therefore, it is determined by the algorithm and it is difficult to switch to Hessian or other statistics. Another result of the above determining is that  $C_t^{-1}$  can only reflect the flatness at step  $t$  instead of the flatness of terminal weights. In contrast, our alignment only involves the flatness at terminal weights. According to the empirical results from Wang & Mao (2024b)'s Figure 3 and Jastrzewski et al. (2019)'s Figure 2, the eigenvalues of Hessian during the middle of training are much larger than those at terminal. Therefore, it is better to only use terminal flatness, as it is in our alignment. Regarding Wang & Mao (2024b)'s terminal-state bound, both this bound and our bound involve terminal states. Still, their bound uses the inverted weight covariance instead of Hessian to reflect local geometries.
+
+Wang et al. (2021a) optimize the noise in SGLD, with a new information-theoretic generalization bound as a surrogate for real risks. They show the square root of expected GNC is greedily optimal under their bound and use this noise to improve the optimization and generalization of SGLD and closes the gap with SGD. Their results indicate the importance of direction and alignment of noises. If we put their algorithms and bounds together, then the result will be very similar with Wang & Mao (2024b)'s. Therefore, our alignment has similar differences with Wang et al. (2021a) as with Wang & Mao (2024b).
+
+Moreover, our omniscient trajectory can be seen as a surrogate algorithm, and optimizing it is very similar to designing new algorithms with better generalization. As a result, additional similarities and differences can be found on the goal of introducing local geometry. In Wang et al. (2021a)'s work, before Theorem 1, the only use of local geometry is in Constraint 1 on not rising empirical risks. Therefore, it is used to guide how to increasing noises without sacrificing the empirical risks. In our bound, as discussed in Insight 1, we use local geometry to guide how to pull terminal weights together/closer without changing the losses too much. Therefore, local geometry in alignments has a similar role between Wang et al. (2021a)'s results and ours of guiding information reduction without sacrificing losses. Regarding differences, Wang et al. (2021a) decrease information measures by adding noises, while we do so by mapping things together or closer.
+
+# C DETAILS FOR SEC. 4 EXPERIMENTAL STUDY
+
+Codes are available in the supplementary material or at https://github.com/peng-ze/ omniscient-bounds.
+
+# C.1 HESSIAN-RELATED DETAILS
+
+In experiments, all Hessian traces are computed using PyHessian (Yao et al., 2020).
+
+The optimized omniscient trajectory in Theorem 2 requires products between an Hessian-related inverse matrix  $(I + \tilde{H}_{\mathrm{pen}} / 2\lambda C)^{-1}$  and vectors, say,  $u$ . Since Hessians of deep models are too large to store and compute with, it is ubiquitous to directly compute the inverse-vector product (iVP). For iVP, following Dagréou et al. (2024), we use conjugate gradient method that only requires the product  $(I + \tilde{H}_{\mathrm{pen}} / 2\lambda C)v$  of the original matrix with an arbitrary vector  $v$ , which is supplied with PyHessian's hv-product. More specifically, we approximate  $v$  such that  $(I + \tilde{H}_{\mathrm{pen}} / 2\lambda C)v = u$  to obtain a  $\hat{v}$ . We run conjugate gradient method until the error  $\left\| (I + \tilde{H}_{\mathrm{pen}} / 2\lambda C)\hat{v} - u\right\|^2$  is less than  $1\%$  of  $\| u\|^2$ , or the maximum iteration 20 is reached.
+
+# C.2 BOUND ESTIMATION DETAILS
+
+This section presents the details in estimating Eq. (1) of Theorem 2. For the randomness in weights, we train  $k > 1$  models independently to obtain weights  $W_{T}^{1:k}$ . To ensure the training data for each model is I.I.D. sampled, we assume the whole training set is sampled in an I.I.D. manner, randomly partition the whole training set into  $k$  subsets  $\{S^i\}_{i=1}^k$  and use  $S^i$  to train  $W_{0:T}^i$ . For fair comparison, we then make unbiased or positively biased estimates for expectations in Eq. (1) separately and then compute the bound using these estimates. Being concave, the cubic root will bring negative bias. Nevertheless, this negative bias can also be found in baseline existing bounds. Moreover, in experiments, the expectation product under the cubic root is extremely small. Therefore, this negative bias will not influence the overall result too much and we leave it as it is for simplicity.
+
+When optimizing the bound to obtain  $\Delta G$ , we use a validation set  $S''$  to estimate  $J$ .
+
+For the penalty terms, i.e.,  $\mathbb{E}\left[\Delta_{\Delta G}(W_T,S) - \Delta_{\Delta G}(W_T,S')\right]$  and  $\mathbb{E}\left[\mathrm{tr}(\Delta H(W_T + \Delta G))\right]$ , we unbiasedly estimate them with the test set. The same test set is used for all weights to fully use the data. The loss differences are directly computed by modifying the parameters and forwarding the testing data  $S'$  instead of approximating them.
+
+For the flatness-optimized trajectory term, the estimate is more complicated, since we must also estimate  $\mathbb{E}\left[\Delta W_T\right]$  within the estimate and handle the matrix inverses and the population gradients in  $J$ .
+
+To estimate  $\mathbb{E}\left[\Delta W_T\right]$  in expectations, one should use samples of  $W_{T}$  that are independent of  $W_{T}^{i}$ . However, another draw of  $k$  weights results in further partitioning on the training set, and the data used for each weight is much less. To fully use the data, similarly to cross validation, we estimate the inner expectation by  $\Delta \bar{W}_T^{-i} \coloneqq \frac{1}{k - 1} \sum_{i' \in [k]}^{i' \neq i} \Delta W_T^{i'}$ , which is independent of  $W_{T}^{i}$ .
+
+To avoid the population Hessian in the inverses and make estimation easier, we select  $\tilde{H}_{\mathrm{pen}} \coloneqq \hat{H}_S(W_T)$ . In this way, the inverses solely depend on the training data and the terminal weight, and their interaction with the testing set is only linear, which is much more friendly to expectations. Define  $E \coloneqq I - (I + \tilde{H}_{\mathrm{pen}} / 2\lambda C)^{-1}$  and  $F \coloneqq (2\lambda CI + \tilde{H}_{\mathrm{pen}})^{-1}$ , which only depend on  $(S, W_T)$ . Initial experiments on CIFAR-10 indicate that it is necessary to include the population gradients for numerical tightness, and we set  $J \coloneqq \nabla \hat{\mathcal{L}}_S(W_T) - \nabla \mathcal{L}_\mu(W_T)$ . Under these specific settings, we will estimate
+
+$$
+\mathbb {E} \left[ \left\| E \left(\Delta W _ {T} - \mathbb {E} \left[ \Delta W _ {T} \right]\right) - F \left(\nabla \hat {\mathcal {L}} _ {S} \left(W _ {T}\right) - \nabla \mathcal {L} _ {\mu} \left(W _ {T}\right)\right) \right\| ^ {2} \right] \tag {C.1}
+$$
+
+Replacing the expectation with the empirical mean and the random variables with I.I.D. copies, we obtain an estimator of the trajectory term
+
+$$
+\frac {1}{k} \sum_ {i = 1} ^ {k} \left\| E ^ {i} (\Delta W _ {T} ^ {i} - \Delta \bar {W} _ {T} ^ {- i}) - F ^ {i} (\nabla \hat {\mathcal {L}} _ {S ^ {i}} (W _ {T} ^ {i}) - \nabla \mathcal {L} _ {S ^ {\prime \prime}} (W _ {T} ^ {i})) \right\| ^ {2},
+$$
+
+which is positively biased because
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \frac {1}{k} \sum_ {i = 1} ^ {k} \left\| E ^ {i} (\Delta W _ {T} ^ {i} - \Delta \bar {W} _ {T} ^ {- i}) - F ^ {i} (\nabla \hat {\mathcal {L}} _ {S ^ {i}} (W _ {T} ^ {i}) - \nabla \mathcal {L} _ {S ^ {\prime \prime}} (W _ {T} ^ {i})) \right\| ^ {2} \right] \\ = \frac {1}{k} \sum_ {i = 1} ^ {k} \mathbb {E} _ {S ^ {i}, W _ {T} ^ {i}} \left[ \mathbb {E} \left[ \left\| E ^ {i} \left(\Delta W _ {T} ^ {i} - \Delta \bar {W} _ {T} ^ {- i}\right) - F ^ {i} \left(\nabla \hat {\mathcal {L}} _ {S ^ {i}} \left(W _ {T} ^ {i}\right) - \nabla \mathcal {L} _ {S ^ {\prime \prime}} \left(W _ {T} ^ {i}\right)\right) \right\| ^ {2} \mid S ^ {i}, W _ {T} ^ {i} \right] \right] \\ \geq \frac {1}{k} \sum_ {i = 1} ^ {k} \mathbb {E} _ {S ^ {i}, W _ {T} ^ {i}} \left[ \left\| \mathbb {E} \left[ E ^ {i} (\Delta W _ {T} ^ {i} - \Delta \bar {W} _ {T} ^ {- i}) - F ^ {i} (\nabla \hat {\mathcal {L}} _ {S ^ {i}} (W _ {T} ^ {i}) - \nabla \mathcal {L} _ {S ^ {\prime \prime}} (W _ {T} ^ {i})) \mid S ^ {i}, W _ {T} ^ {i} \right] \right\| ^ {2} \right] \\ = \frac {1}{k} \sum_ {i = 1} ^ {k} \mathbb {E} _ {S ^ {i}, W _ {T} ^ {i}} \left[ \left\| E ^ {i} \big (\Delta W _ {T} ^ {i} - \mathbb {E} \left[ \Delta \bar {W} _ {T} ^ {- i} \mid S ^ {i}, W _ {T} ^ {i} \right] \big) - F ^ {i} (\nabla \hat {\mathcal {L}} _ {S ^ {i}} (W _ {T} ^ {i}) - \mathbb {E} \left[ \nabla \mathcal {L} _ {S ^ {\prime \prime}} (W _ {T} ^ {i}) \mid S ^ {i}, W _ {T} ^ {i} \right]) \right\| ^ {2} \right] \\ = \frac {1}{k} \sum_ {i = 1} ^ {k} \mathbb {E} _ {S ^ {i}, W _ {T} ^ {i}} \left[ \left\| E ^ {i} \left(\Delta W _ {T} ^ {i} - \mathbb {E} \left[ \Delta W _ {T} \right]\right) - F ^ {i} \left(\nabla \hat {\mathcal {L}} _ {S ^ {i}} \left(W _ {T} ^ {i}\right) - \nabla \mathcal {L} _ {\mu} \left(W _ {T} ^ {i}\right)\right) \right\| ^ {2} \right] \\ = \mathbb {E} \left[ \left\| E \left(\Delta W _ {T} - \mathbb {E} \left[ \Delta W _ {T} \right]\right) - F \left(\nabla \hat {\mathcal {L}} _ {S} \left(W _ {T}\right) - \nabla \mathcal {L} _ {\mu} \left(W _ {T}\right)\right) \right\| ^ {2} \right], \\ \end{array}
+$$
+
+where the second step is because  $\mathbb{E}\left[X^{\top}X\right]\geq \mathbb{E}\left[X\right]^{\top}\mathbb{E}\left[X\right]$
+
+Among the above estimations, the most loose step comes from the last estimation, i.e., the estimation on the trajectory term, because other expectations are estimated unbiasedly and this expectation is positively biased. In principle, we can estimate it without bias. To see this, set  $A \coloneqq E\Delta W_T - F(\nabla \hat{\mathcal{L}}_S(W_T) - \nabla \mathcal{L}_\mu (W_T))$  expand Eq. (C.1) as
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \left\| E (\Delta W _ {T} - \mathbb {E} [ \Delta W _ {T} ]) - F (\nabla \hat {\mathcal {L}} _ {S} (W _ {T}) - \nabla \mathcal {L} _ {\mu} (W _ {T})) \right\| ^ {2} \right] \\ = \mathbb {E} \left[ \| A - E \times \mathbb {E} \left[ \Delta W _ {T} ^ {\prime} \right] \| ^ {2} \right] \\ = \mathbb {E} \left[ \| A \| ^ {2} \right] - 2 \mathbb {E} \left[ \mathbb {E} \left[ \Delta W _ {T} ^ {\prime} \right] ^ {\top} E ^ {\top} A \right] + \mathbb {E} \left[ \| E \times \mathbb {E} \left[ \Delta W _ {T} ^ {\prime} \right] \| ^ {2} \right] \\ = \mathbb {E} \left[ \| A \| ^ {2} \right] - 2 \mathbb {E} \left[ \Delta W _ {T} ^ {\prime} \right] ^ {\top} \mathbb {E} \left[ E ^ {\top} A \right] + \mathbb {E} \left[ \| E \times \mathbb {E} \left[ \Delta W _ {T} ^ {\prime} \right] \| ^ {2} \right], \\ \end{array}
+$$
+
+which can be term by term estimated without bias. However, since  $E$  involves Hessian inversion, this unbiased estimator requires more Hessian iVPs (iHVP): Computing  $A$  requires two iHVPs, while  $E^{\top}A$  and  $E \times \mathbb{E}\left[\Delta W_{T}\right]$  require another two iHVPs. As a result, the unbiased estimator requires  $4k$  iHVPs. In contrast, the positively biased estimate only requires  $2k$  iHVPs. Since iVPs, especially iHVPs, are extremely time-consuming, we still use the positively biased estimator in the experiments.
+
+# C.3 TRAINING DETAILS
+
+For both MNIST and CIFAR-10, we train  $k = 6$  independent models from the same randomly chosen initial weight, which means each model receives 10,000 training samples. We use 2-layer MLPs for MNIST as Wang & Mao (2022) while ResNet-18 for CIFAR-10. Since a bounded loss is naturally sub-Gaussian (Xu & Raginsky, 2017), we employ cross-entropy (CE) losses capped at  $12\log c$  for both testing and bound estimation, where  $c$  represents the class number. Here,  $\log c$  corresponds to the expected cross-entropy at random initialization, which are heuristically the maximum of the average loss during the training. However, we still use vanilla CE losses for training. As Wang & Mao (2022), we use SGD with momentum of 0.9. The use of momentum does not violate the assumptions in Section 2.1 because each update can access history weights to recover gradients and compute momentum. ResNet-18 models are trained for 200 epochs while 2-layer MLP models are trained for 500 epochs as Wang & Mao (2022). We start from a base hyperparameter, where the learning rate is 0.01, the batch size is 60, and no dropout or weight decay is used. For the 2-layer MLP, the base hidden width is 512. To enhance the generalization of the networks on CIFAR-10 for evaluating the bounds, we use random horizontal flip and random resized crop. This use of data
+
+augmentations does not violate the assumptions in Section 2.1, as they can be achieved by passing in  $V$  the random seeds for the augmentations and letting  $g_{t}$  augment the samples using these seeds before computing the gradients.
+
+For single-hyperparameter variations in Figures 4 and C.5, we vary that single hyperparameter while keeping others the same as the base. For Figures 3 and C.4, we vary the learning rate and the batch size in a grid-search manner while keeping others the same as the base. When the learning rate is too small or too large, or the batch size is too large, the model even cannot fit the training data well. Therefore, we exclude the training records whose final weight has a training accuracy less than  $95\%$ .
+
+Models are trained on 12 NVidia RTX4090D GPUs for 2 day with auto mixed precision (BF16) and torch.compile() to save memory and increase parallelization. To ensure consistency of model and training details with previous results, our codes are modified based on Wang & Mao (2022) with bound estimation modules totally re-implemented.
+
+We must remark that the test accuracy of our models on CIFAR-10 is relatively low, because each model is trained by a one-sixth subset of CIFAR-10.
+
+# C.4 MORE EXPERIMENTAL RESULTS
+
+# C.4.1 MORE RESULTS FOR RESNET-18 ON CIFAR-10 AND MORE DISCUSSIONS
+
+Here, we display more results of existing bounds for ResNet-18 on CIFAR-10. Figure C.1 displays the result of the isotropic Proposition 2 of Neu et al. (2021).
+
+From Figure C.1, we find that the trajectory term of isotropic Proposition 2 is insensitive to batch size, which is an improvement over the incorrect tendency of Proposition 1. However, this improvement is only partial compared to our trajectory term that decreases as the batch size decreases. Thanks to the flatness term that has the correct tendency w.r.t. batch size under large learning rates, the bound also has the correct tendency w.r.t. batch size if learning rate is large enough. Compared to this result, our bound can even better capture the generalization under all learning rates by having a more aligned trajectory term.
+
+Now we focus on the tendency w.r.t. learning rate. Our bound fails to fully correct the wrong tendency of the trajectory term of Propositions 1 and 2 w.r.t. learning rate. Nevertheless, the wrong tendency seems partially corrected because the lines corresponding to the trajectory terms under different learning rates are well separated in Figures 1b and C.1b of the existing bounds, while some of those from our bounds twist together in Figure 3b. However, this partial correction is not enough to invert the wrong tendency, unlike the tendency w.r.t. batch size. We conjecture it is because the trajectory term is not solely determined by flatness, but is a "product" between the Hessians and the output weight variance (see Eq. (2)). Increasing the learning rate increases the step size, which then increases the variance of the output weight. This variance increase is empirically confirmed by Figure C.1b: we display the trajectory term with  $\sigma^{-1}$  excluded in this figure. As a result, what is displayed in this figure is proportional to the output weight variance. It clearly shows that the variance increases as the learning rate increases. Therefore, there is a competition between the increased variance and the improved flatness. Unfortunately, it seems the increased variance overpowers the improved flatness in this process and our bound fails to invert the wrong tendency w.r.t. learning rate. On the other hand, decreasing the batch size seems to also increase the variance and we must explain why the tendency w.r.t. batch size is fixed. The reasons is that, the variance is, in fact, very insensitive to batch size: From Figure C.1b, we can see that the variance is almost fixed when batch size changes. As a result, the trajectory term is dominated by the improved flatness when the batch size decreases.
+
+Regarding the tightness improvement (mainly due to the trajectory term), Proposition 2 has a much smaller trajectory term than Proposition 1 ( $\sim 10^{-2}$  v.s.  $\sim 10^{0}$ ). It mainly happens because gradients of SGD are noisy and gradients from different steps often partially cancel. In Proposition 1, the canceled parts are still counted in the trajectory term while in Proposition 2 where gradients are summed together before taking norms. The components canceled in the trajectory indeed cancel in the trajectory term and do not contribute to the bound. This improvement is pushed further by the omniscient trajectory and the optimization under flatness ( $\sim 10^{-2}$  v.s.  $\sim 10^{-8}$ ). Since Theorem 2 can be seen as the combination of Proposition 2 and the omniscient trajectory (see Corollary B.1 and
+
+![](images/5224102e8ce3d71ac59e55c3af3011504ad9ba63e106087fef8f4a5192bd0b1a.jpg)  
+Generalization error bound Estimated generalization error  
+(a) Bound
+
+![](images/b1ea825f8ed1809af157e7bb3ea55b084af395251928aca1a77a55b38a28cc96.jpg)  
+(b) Trajectory Term  
+Figure C.1: the isotropic version of Proposition 2 for ResNet-18 on CIFAR-10 under varied flatness.
+
+![](images/8b992aa57df557f8ab2508d11634b7dd9a576af07234d1a34c3e369bd51ef5db.jpg)  
+(c) Flatness Term
+
+![](images/c953fe8fa1772c11a996fe7b8fa1c0cc5f33b174a183d765bd9aa0e9c601fac0.jpg)  
+Generalization error bound Estimated generalization error  
+(a) Bound  
+(b) Trajectory Term  
+(c) Flatness Term  
+Figure C.2: Wang & Mao (2022)'s bound for 2-layer MLP on MNIST under varied flatness.
+
+how it connects to Proposition 2 by setting  $\Delta G \equiv 0$ ), the improved tightness of our bound compared to Proposition 1 is two-folded, done by both Proposition 2 and the omniscient trajectory.
+
+# C.4.2 RESULTS FOR 2-LAYER MLP ON MNIST
+
+The numerical results on MNIST (LeCun et al., 1998) with the 2-layer MLP are displayed in Figures C.2 to C.5.
+
+# C.5 EXPERIMENTS OF EXISTING BOUNDS WITH POPULATION HESSIAN
+
+In Figures 1, 4 and C.3, we have displayed results of existing bounds without using population Hessian. That is, we are computing the bounds whose (approximated) flatness term is
+
+$$
+\frac {\sigma^ {2} T}{2} \mathbb {E} \left[ \mathrm {t r} \left(\hat {H} _ {S} (W _ {T})\right) \right]
+$$
+
+![](images/7be83b26043c170157677cd53e7c6817c65fa306524b745384cfd481017541c6.jpg)  
+Generalization error bound Estimated generalization error  
+(a) Bound  
+(b) Trajectory Term  
+(c) Flatness Term  
+Figure C.3: the isotropic version of Proposition 2 for 2-layer MLP on MNIST under varied flatness.
+
+![](images/b08bef456b9722f4aabc1b2d1a6a53be8aed0836f797e179a15dff6373f94540.jpg)  
+Figure C.4: Numerical results of Theorem 2 with  $\lambda = 10^{9}$  on MNIST and a 2-layer MLP with varied learning rate and batch size.
+
+![](images/b9db0ab16b48a90efd83c004a49ecaef85d524688cdef53fdf556daa5a089776.jpg)  
+Figure C.5: Numerical results for a 2-layer MLP on MNIST under varied label noise level, width, and weight decay. The isotropic version of Neu et al. (2021)'s Proposition 8 is used.
+
+instead of  $\frac{\sigma^2T}{2}\mathbb{E}\left[\mathrm{tr}\left(\hat{H}_S(W_T)\right) - \mathrm{tr}\left(H_\mu (W_T)\right)\right]$ . In contrast, our bound depends heavily on population statistics like the mean gradient and Hessians. One may doubts whether this comparison is unfair due to our extra dependence on population Hessians. This concern is amplified by the fact that the population Hessian potentially improves the numerical tightness of existing bounds: If one assumes as Wang & Mao (2022) that  $\mathcal{L}_{\mu}(W_T)\leq \mathbb{E}_{\xi \sim \mathcal{N}(0,\sigma^2 I)}[\mathcal{L}_{\mu}(W_T + \xi)]$ , one is essentially assuming  $H_{\mu}(W_T)$  is positive semi-definite, whose trace is non-negative. In Propositions 1 and 2, the non-negative traces are subtracted from the bound, which is tighter than the case where it is not subtracted.
+
+To address this concern, we also allow the existing bounds to depend on the population statistics, especially the population Hessian, and evaluate them in experiments. The results are surprising and confusing: the population Hessians indeed have positive traces but their traces are orders of magnitude larger than the traces of the empirical Hessians. As a result, the flatness term becomes negative. If one optimize the bound parameter  $\sigma$ , one would arrive at a  $-\infty$  bound of the generalization error, which is obviously incorrect. We conjecture this is due to the second order approximation used in the expectation, which becomes inaccurate for population loss changes under the SGLD-like noise of variance  $\sigma^2 TI$ . This is also a drawback of existing bounds to require the large noises to decrease the impact of the trajectory term in the  $\sigma$ -optimized bound. In contrast, our bound has an orders-of-magnitude smaller trajectory term and requires orders of magnitude smaller noises. This is supported by empirical results in Table 1, where the value of (average)  $\sigma_*^2 (\sigma_*)$  is the optimal values of  $\sigma_t$  in  $\sum_{t}\sigma_{t}^{2}$  of Proposition 1, the  $\sigma$  in  $\Sigma = \sigma^2 TI$  of isotropic Proposition 2, or the  $\sigma$  in  $\sigma^2 T$  in Corollary B.1 / Theorem 2) in different experiments are presented. Be noted that although  $\sigma_*^2$  values themselves seem already small for all experiments, the actual noises in the flatness term has covariance  $\sigma_*^2 TI$ , where  $T$  is the number of training steps ( $\sim 10^4$  for CIFAR10) and identity matrix  $I$  has a size equal to the number of parameters ( $\sim 10^6$  for ResNets). As a result, the magnitude of  $\sigma_*^2$  have a significant impact on the second-order approximation in the flatness terms and the second-order approximation is much more accurate for evaluating our bound.
+
+To alleviate this problem, we take absolute values of the flatness terms to make them positive as in the main results of Wang & Mao (2022) and Neu et al. (2021). That is, we now estimate the bound with flatness term replaced by
+
+$$
+\frac {\sigma^ {2} T}{2} \left| \mathbb {E} \left[ \operatorname {t r} \left(\hat {H} _ {S} (W _ {T})\right) - \operatorname {t r} \left(H _ {\mu} (W _ {T})\right) \right] \right|.
+$$
+
+<table><tr><td></td><td>l.r. and b.s.</td><td>label noise</td><td>width</td><td>weight decay</td></tr><tr><td>Prop. 1 w/o pop. Hess.</td><td>15381.627 × 10-9</td><td>15372.131 × 10-9</td><td>19904.060 × 10-9</td><td>17674.105 × 10-9</td></tr><tr><td>Prop. 2 w/o pop. Hess.</td><td>1434.559 × 10-9</td><td>1278.293 × 10-9</td><td>1660.805 × 10-9</td><td>1341.889 × 10-9</td></tr><tr><td>Prop. 1 w/ pop. Hess.</td><td>6659.479 × 10-9</td><td>6572.998 × 10-9</td><td>9175.748 × 10-9</td><td>9149.648 × 10-9</td></tr><tr><td>Prop. 2 w/ pop. Hess.</td><td>676.226 × 10-9</td><td>545.839 × 10-9</td><td>767.169 × 10-9</td><td>643.772 × 10-9</td></tr><tr><td>Thm. 2 (λ = 1)</td><td>291.674 × 10-9</td><td>595.630 × 10-9</td><td>303.843 × 10-9</td><td>307.434 × 10-9</td></tr><tr><td>Thm. 2 (λ = 103)</td><td>4.354 × 10-9</td><td>6.269 × 10-9</td><td>3.866 × 10-9</td><td>3.167 × 10-9</td></tr><tr><td>Thm. 2 (λ = 109)</td><td>0.193 × 10-9</td><td>0.204 × 10-9</td><td>0.244 × 10-9</td><td>0.167 × 10-9</td></tr></table>
+
+Table 1: The average value of  $\sigma_{*}^{2}$  across bounds and CIFAR-10 experiments varying different hyperparameters. It can be seen that our bound requires orders of magnitude smaller noises for the SGLD-like trajectory's Gaussian noises when  $\lambda$  is suitably selected. As a result, the distortion due to the second-order approximation has less impact to our bound.
+
+![](images/63f0e67f7f19fc906368754be5f13a2901446aa24553cbd0146b3afbfed825c3.jpg)  
+Figure C.6: Wang & Mao (2022)'s bound with population Hessian for ResNet-18 on CIFAR-10 under varied flatness.
+
+The population Hessian is estimated on a new validation set. After this modification, we display the numerical results for Propositions 1 and 2 in Figures C.6 and C.7.
+
+Since the modification only involves the flatness term, the trajectory term is almost the same as those in Figures 1b and C.1b of the existing bounds without the population Hessian. Therefore, we mainly focus on the flatness term.
+
+Since the subtracted population Hessian traces are orders of magnitude larger than the empirical Hessian, the flatness terms with absolute value are dominated by the population Hessian traces and are orders of magnitude larger than those in Figures 1c and C.1c. As a result, the existing bounds even become looser if fed with validation sets.
+
+When it comes to the tendency, the flatness terms' tendencies w.r.t. batch size also become wrong when batch size is large or learning rate is small. As a result, Proposition 2's (Neu et al., 2021) partial improvement of trajectory term w.r.t. batch size can not compensate the wrong tendency of the flatness term, and the whole bound scales incorrectly w.r.t. batch size.
+
+![](images/de8ca71c5b75289e34122c262774450b9a4e0410980b9ddaf40c111cb16cda4f.jpg)  
+Figure C.7: The isotropic version of Proposition 2 with population Hessians for ResNet-18 on CIFAR-10 under varied flatness.
+
+![](images/f7495be9a0e1ab1981374b31dc1d93ccbedc80595e926271407b7c998cbdf2db.jpg)  
+Figure C.8: Numerical results of existing bounds with population Hessians for ResNet-18 on CIFAR-10 under varied training data usage, label noise level, width, depth, and weight decay. The tendency w.r.t. weight scaling is also added. However, the results are taken from (unbiased) MLP on MNIST because ResNets are more complicated models and it takes time to develop a weight scaling scheme that does not essentially change the prediction. The isotropic version of Proposition 2 (Neu et al. (2021)'s Prop. 8) is used. If a model has a channel scaling factor of  $c$ , it means the model has  $c$  times the number of channels at each layer compared to a standard ResNet-18.
+
+![](images/fcbd33ed89c3f7c6017613fc523add53702d1062aa643a502af09092286fcfbd.jpg)  
+(a) Generalization gap on all samples. (b) Gen. gap on correct classification.  
+Figure C.9: Cross Entropy generalization gap and bounds under weight scaling of MLPs on MNIST. Figure C.9a displays the generalization gap on all testing samples. Figure C.9b displays the generalization gap on only the correctly classified testing samples. Since Figure C.9b involves negative values, we use log-scale above  $10^{-4}$  and linear scale below  $10^{-4}$ .
+
+The correct tendency of the flatness term without population Hessians and the wrong tendency of that with population Hessian indicate population Hessians have wrong tendencies when batch size is large or the learning rate is small. This somehow points to the notion the generalization and overfitting of higher order statistics, as the empirical Hessian has the correct tendency as shown in Figures 1c and C.1c yet the population Hessian has the wrong tendency. Small batch size and large learning rate seem also helpful for this higher-order generalization.
+
+As shown in Figure C.8, when weight decay is used, the existing bounds with population Hessian can better capture the tendency of generalization under regularization. However, when the hyperparameters in Figure C.8 are varied, the existing bounds with population Hessians are times looser than those without the population Hessians.
+
+To sum up, the existing bounds cannot fully exploit the population statistics even if we allow them to access the validation set. Therefore, the comparison between our bound and the existing bounds is still fair, at least for existing bounds in its current form estimated after the conventional (Wang & Mao, 2022) second-order approximation.
+
+# C.6 EXPERIMENTS ON WEIGHT SCALING
+
+To test whether our flatness-related generalization bound captures the generalization under weight scaling, we train homogenous two-layer MLPs, whose output layer has no bias, on MNIST and scale the weight at the end as the last step of the training. The results are displayed in Figure C.9a, where generalization gap is computed for clipped Cross Entropy losses. In this case, our bounds with large  $\lambda$  can well capture the generalization.
+
+However, the results in Figure C.9a out of expectation. One would expect the generalization improves as the weight is scaled up because most MNIST samples would be correctly classified and the
+
+generalization risk measured by Cross Entropy would be decreased by increasing the "confidence". However, in Figure C.9a, Cross Entropy increases when scaling up the weights. We conjecture it is because we split the training set into  $k = 6$  subsets, the generalization of these MLPs are in fact harmed and about  $4\%$  of testing samples are misclassified. As a result, scaling the weight also increases the generalization risk on the misclassified testing samples. The increase on misclassified testing samples are arguably much faster than those on correctly classified samples, because weight scaling on misclassification pushes the loss to infinity while weight scaling on correct classification pushes the loss to zero. As a result, the generalization gap increases in Figure C.9a.
+
+To see how our bound captures the improved generalization on fully correctly classified scenarios, we filter the testing sets and only keep the correctly classified samples, simulating the cases where all samples are correctly classified. We implement this simulation using the following loss:
+
+$$
+\underbrace{\mathbb{I}[\arg\max\logit_{i}=y]}_{\substack{i\\ \text{Filtering the correct samples}}}\cdot\underbrace{\min(\text{CrossEntropy}(\logit, y), 12\log c)}_{\substack{\text{Clipped Cross Entropy}}},
+$$
+
+where  $c$  is the number of classes. It is bounded, thus sub-Gaussian and can be rigorously used in the existing and our bounds, whose results can be found in Figure C.9b. In Figure C.9b, when weights are scaled down, the training loss increases and increases faster (maybe the faster increase is related to overfitting), decreasing the gap. However, when weights are sufficiently scaled up, the generalization gap indeed decreases with the decreases of scaling (maybe because now the training loss is almost zero and its decrease slows down). In the latter phase, the bound from Wang & Mao (2022) and Neu et al. (2021) all have weak or wrong dependency w.r.t. weight scaling. In contrast, our bound still well captures the improved generalization in both cases, if  $\lambda$  is suitably selected. Our bounds even captures the strange negative generalization error. By inspecting the raw data, we find the negative value mainly comes from the empirical part of the penalty term, indicating that the negativeness of the bound value is not mainly due to the dependence on the validation set.
+
+To sum up, our bounds can well capture the generalization under weight scaling in our experiments.
+
+# D PROOFS AND DETAILS FOR SEC. 5 EXTENSIONS
+
+In this section, we present and prove the combined results of the omniscient trajectory with existing results to demonstrate the flexibility of the technique. The proofs generally follow those of the existing results. But note that the omniscient and the SGLD-like trajectories may depend on random variables more than the training set  $S$ . Since the existing results often assume the trajectory solely depends on  $S$ , we must re-prove some of them instead of directly applying them to handle the extra dependence. Sometimes we must rearrange the order of the steps of the original proofs to insert the omniscient trajectory. Setting the omniscient perturbation to zero will recover the existing results, which we will not bother to restate.
+
+# D.1 COMBINATION WITH THE INDIVIDUAL-SAMPLE TECHNIQUE
+
+Corollary D.1 Assume  $\ell(w, \cdot)$  is  $R$ -sub-Gaussian on  $\mu$  for any  $w \in \mathcal{W}$ . Recall  $n$  is the number of training samples. Then for any family  $\{\Delta g_{1:T}^i\}_{i=1}^n$  of omniscient perturbations and any  $\sigma_{1:T} \in (\mathbb{R}^{>0})^T$ , we have
+
+$$
+\begin{array}{l} \mathrm {g e n} (\mu^ {n}, P _ {W | S}) \leq \frac {R}{n} \sum_ {i = 1} ^ {n} \sqrt {2 I (\tilde {W} _ {T} ^ {i} ; Z _ {i})} + \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \Delta_ {\Gamma_ {T} ^ {i}} ^ {\sum_ {t} ^ {t} \sigma_ {t} ^ {2}} (W _ {T}, Z _ {i}) - \Delta_ {\Gamma_ {T} ^ {i}} ^ {\sum_ {t} ^ {t} \sigma_ {t} ^ {2}} (W _ {T}, S ^ {\prime}) \right] \\ \leq \frac {R}{n} \sum_ {i = 1} ^ {n} \sqrt {2 I (\tilde {W} _ {T} ^ {i} ; Z _ {i} \mid Z _ {- i})} + \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \Delta_ {\Gamma_ {T} ^ {i}} ^ {\sum_ {t} ^ {t} \sigma_ {t} ^ {2}} (W _ {T}, Z _ {i}) - \Delta_ {\Gamma_ {T} ^ {i}} ^ {\sum_ {t} ^ {t} \sigma_ {t} ^ {2}} (W _ {T}, S ^ {\prime}) \right] \\ \leq \frac {R}{n} \sum_ {i = 1} ^ {n} \sqrt {\sum_ {t = 1} ^ {T} \frac {1}{\sigma_ {t} ^ {2}} \mathbb {E} \left[ \left\| g _ {t} - \mathbb {E} [ g _ {t} \mid Z _ {- i} ] - \Delta g _ {t} ^ {i} \right\| ^ {2} \right]} \\ + \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \Delta_ {\Gamma_ {T} ^ {i}} ^ {\sum_ {t} \sigma_ {t} ^ {2}} \left(W _ {T}, Z _ {i}\right) - \Delta_ {\Gamma_ {T} ^ {i}} ^ {\sum_ {t} \sigma_ {t} ^ {2}} \left(W _ {T}, S ^ {\prime}\right) \right], \tag {D.1} \\ \end{array}
+$$
+
+where  $\Delta g_t^i$  is a deterministic function of  $(S, V, g_{1:T}, W_{0:T}, Z_{-i})$ , and  $\Gamma_t^i \coloneqq \sum_{\tau=1}^t \Delta g_\tau^i$ .
+
+Remark D.1 The omniscient trajectory additionally depends on  $i$  and  $Z_{-i}$ , the two featuring random variables of the individual-sample bounds.
+
+Proof The crux of the individual-sample technique is to first extract the summation in the empirical risk and change its order with the expectation, before other things like using the MI bound or building the auxiliary trajectories:
+
+$$
+\begin{array}{l} \operatorname {g e n} (\mu^ {n}, P _ {W | S}) \\ := \mathbb {E} \left[ \mathcal {L} _ {\mu} \left(W _ {T}\right) - \frac {1}{n} \sum_ {i = 1} ^ {n} \ell \left(W _ {T}, Z _ {i}\right) \right] = \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \mathcal {L} _ {\mu} \left(W _ {T}\right) - \ell \left(W _ {T}, Z _ {i}\right) \right] \\ = \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \mathcal {L} _ {\mu} (\breve {W} _ {T} ^ {i}) - \ell (\breve {W} _ {T} ^ {i}, Z _ {i}) \right] + \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \Delta_ {\Gamma_ {T} ^ {i}} (W _ {T}, Z _ {i}) - \Delta_ {\Gamma_ {T} ^ {i}} (W _ {T}, S ^ {\prime}) \right], \\ \end{array}
+$$
+
+where  $\check{W}_t^i\coloneqq W_t + \Gamma_T^i$  is the  $i$ -th omniscient trajectory. It can be seen that the omniscient penalty term has matched that in Eq. (D.1). Thus, we will then bound the individual-sample generalization error of the omniscient trajectories. The process is similar to the proof of Theorem 1, i.e., adding Gaussian noises to obtain the SGLD trajectories:
+
+$$
+\begin{array}{l} \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \mathcal {L} _ {\mu} \left(\check {W} _ {T} ^ {i}\right) - \ell \left(\check {W} _ {T} ^ {i}, Z _ {i}\right) \right] = \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \mathcal {L} _ {\mu} \left(\tilde {W} _ {T} ^ {i}\right) - \ell \left(\tilde {W} _ {T} ^ {i}, Z _ {i}\right) \right] \\ + \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \Delta^ {\sum_ {t} \sigma_ {t} ^ {2}} (W _ {T} + \Gamma_ {T} ^ {i}, Z _ {i}) - \Delta^ {\sum_ {t} \sigma_ {t} ^ {2}} (W _ {T} + \Gamma_ {T} ^ {i}, S ^ {\prime}) \right], \\ \end{array}
+$$
+
+where  $\tilde{W}_t^i \coloneqq \breve{W}_t^i + \sum_{\tau=1}^t N_t^i$ , and  $N_t^i \sim \mathcal{N}(0, \sigma_t^2 I)$  is an independent Gaussian noise. Again we throw away the already matched penalty terms and focus on bounding the individual-sample generalization error of the SGLD-like trajectory. To this end, we see each term in the error as the generalization error of an algorithm that takes one sample (but stealthily samples  $n-1$  samples as algorithm's internal randomness) and apply Lemma 1:
+
+$$
+\begin{array}{l} \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \mathcal {L} _ {\mu} (\tilde {W} _ {T} ^ {i}) - \ell (\tilde {W} _ {T} ^ {i}, Z _ {i}) \right] = \frac {1}{n} \sum_ {i = 1} ^ {n} \operatorname * {g e n} \left(\mu^ {n}, P _ {\tilde {W} _ {T} ^ {i} | Z _ {i}}, \ell\right) \leq \frac {1}{n} \sum_ {i = 1} ^ {n} \sqrt {\frac {2 R ^ {2}}{1} I (\tilde {W} _ {T} ^ {i} ; Z _ {i})} \\ \leq \frac {R}{n} \sum_ {i = 1} ^ {n} \sqrt {2 I (\tilde {W} _ {T} ^ {i} ; Z _ {i} \mid Z _ {- i})}. \\ \end{array}
+$$
+
+We proceed by a similar argument as in Theorem 1 to bound the mutual information: for any  $i \in [n]$ , we have
+
+$$
+\begin{array}{l} I (\tilde {W}; Z _ {i} \mid Z _ {- i}) \leq I (\tilde {W} _ {0} ^ {i}; Z _ {i} \mid Z _ {- i}) + \sum_ {t = 1} ^ {T} I (\tilde {W} _ {t} ^ {i}; Z _ {i} \mid \tilde {W} _ {0: t - 1} ^ {i}, Z _ {- i}) \\ = \sum_ {t = 1} ^ {T} I \left(\tilde {W} _ {t - 1} ^ {i} - \left(g _ {t} - \Delta g _ {t} ^ {i}\right) + N _ {t} ^ {i}; Z _ {i} \mid \tilde {W} _ {0: t - 1} ^ {i}, Z _ {- i}\right). \\ \end{array}
+$$
+
+By instantiating  $X^{i} = Z_{i},Y^{i} = \tilde{W}_{0:t - 1}^{i},\Delta^{i} = (W_{0:T},V,Z_{-i}),O^{i} = Z_{-i}$  and
+
+$$
+\begin{array}{l} f ^ {i} \left(X ^ {i}, Y ^ {i}, \Delta^ {i}\right) = \tilde {W} _ {t - 1} ^ {i} - \left(g _ {t} \left(W _ {t - 1}, \left(Z _ {1}, \dots , Z _ {i - 1}, Z _ {i}, Z _ {i + 1}, \dots , Z _ {n}\right), V, W _ {0: t - 2}\right) \right. \\ \left. - \Delta g _ {t} ^ {i} (S, V, g _ {1: T}, W _ {0: T}, Z _ {- i})\right) \\ \end{array}
+$$
+
+and applying Eq. (B.1) in Lemma B.1, for any deterministic function  $\Omega^i$  of solely  $(\tilde{W}_{0:t-1}^i, Z_{-i})$ , respectively, we have
+
+$$
+I (\tilde {W} _ {t - 1} ^ {i} - (g _ {t} - \Delta g _ {t} ^ {i}) + N _ {t} ^ {i}; Z _ {i} | \tilde {W} _ {0: t - 1} ^ {i}, Z _ {- i}) \leq \frac {1}{2 \sigma_ {t} ^ {2}} \mathbb {E} \left[ \left\| \tilde {W} _ {t - 1} ^ {i} - (g _ {t} - \Delta g _ {t} ^ {i}) - \Omega^ {i} (\tilde {W} _ {0: t - 1} ^ {i}, Z _ {- i}) \right\| ^ {2} \right].
+$$
+
+Setting  $\Omega^i (\tilde{w}_{0:t - 1}^i,z_{-i}) = \tilde{w}_{t - 1}^i -\mathbb{E}\left[g_t\mid Z_{-i} = z_{-i}\right]$  leads to
+
+$$
+I (\tilde {W} _ {t - 1} ^ {i} - (g _ {t} - \Delta g _ {t} ^ {i}) + N _ {t} ^ {i}; Z _ {i} | \tilde {W} _ {0: t - 1} ^ {i}, Z _ {- i}) \leq \frac {1}{2 \sigma_ {t} ^ {2}} \mathbb {E} \left[ \left\| g _ {t} - \mathbb {E} [ g _ {t} | Z _ {- i} ] - \Delta g _ {t} ^ {i} \right\| ^ {2} \right].
+$$
+
+After putting everything together, the corollary is proved.
+
+# D.2 COMBINATION WITH THE CMI FRAMEWORK
+
+Conditional MI (CMI) framework (Steinke & Zakynthinou, 2020) is developed to fundamentally solve the potential unboundedness of MI on some algorithms. CMI framework considers the following random process: Firstly, a supersample  $\tilde{S} = \tilde{Z}_{1:n,0:1} \in \mathcal{Z}^{n \times 2}$  of  $2n$  samples are sampled as an effective discrete "sample space". Then,  $n$  independent indices  $U_{1:n} \in \{0,1\}^n$  are sampled as the discrete "sample". The training set is constructed by  $S := (\tilde{Z}_{i,U_i})_{i=1}^n$  using the supersample and indices. CMI then bounds the generalization by how much the output reveals the indices given the supersample. We can combine the omniscient trajectory with CMI framework and give the following result:
+
+Corollary D.2 Assume  $\ell(w, \cdot)$  is  $R$ -sub-Gaussian on  $\mu$  for any  $w \in \mathcal{W}$ . Then for any omniscient trajectory  $\Delta g_{1:T}$  that additionally depends on  $(\tilde{S}, U)$ , and any  $\sigma_{1:T} \in (\mathbb{R}^{>0})^T$ , we have
+
+$$
+\begin{array}{l} \operatorname {g e n} \left(\mu^ {n}, P _ {W | S}\right) \leq 2 \sqrt {\frac {2 R ^ {2}}{n} I \left(\tilde {W} _ {T} ; U \mid \tilde {S}\right)} + \mathbb {E} \left[ \Delta_ {\Gamma_ {T}} ^ {\sum_ {t} \sigma_ {t} ^ {2}} \left(W _ {T}, Z _ {i}\right) - \Delta_ {\Gamma_ {T}} ^ {\sum_ {t} \sigma_ {t} ^ {2}} \left(W _ {T}, S ^ {\prime}\right) \right] \\ \leq 2 \sqrt {\frac {R ^ {2}}{n} \sum_ {t = 1} ^ {T} \frac {1}{\sigma_ {t} ^ {2}} \mathbb {E} \left[ \left\| g _ {t} - \mathbb {E} \left[ g _ {t} \mid \tilde {S} \right] - \Delta g _ {t} \right\| ^ {2} \right]} \\ + \mathbb {E} \left[ \Delta_ {\Gamma_ {T}} ^ {\sum_ {t} \sigma_ {t} ^ {2}} (W _ {T}, Z _ {i}) - \Delta_ {\Gamma_ {T}} ^ {\sum_ {t} \sigma_ {t} ^ {2}} (W _ {T}, S ^ {\prime}) \right], \\ \end{array}
+$$
+
+where  $\Delta g_{t}$  is a deterministic function of  $(S,V,g_{1:T},W_{0:T},\tilde{S},U)$ .
+
+Remark D.2 The omniscient perturbation additionally depends on  $(\tilde{S}, U)$ , which are the featuring random variables of CMI bounds.
+
+Proof After paying the penalties for changes of trajectories, we can focus on bounding the generalization error on the SGLD-like trajectory. With the omniscient trajectory depending on  $(\tilde{S}, U)$ , the distribution of the SGLD-like trajectory is determined by  $P_{\tilde{W}_{0:T}|\tilde{S},U}$ . We will apply the following re-proved CMI bound that handles the extra dependence on  $(\tilde{S}, U)$  of  $\tilde{W}_T$ :
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \mathcal {L} _ {\mu} (\tilde {W} _ {T}) - \hat {\mathcal {L}} _ {S} (\tilde {W} _ {T}) \right] = 2 \mathbb {E} \left[ \hat {\mathcal {L}} _ {\tilde {S}} (\tilde {W} _ {T}) - \hat {\mathcal {L}} _ {S} (\tilde {W} _ {T}) \right] \\ = 2 \mathbb {E} _ {\tilde {S}} \left[ \mathbb {E} \left[ \hat {\mathcal {L}} _ {\tilde {S}} (\tilde {W} _ {T}) - \hat {\mathcal {L}} _ {S} (\tilde {W} _ {T}) \mid \tilde {S} \right] \right] = 2 \mathbb {E} _ {\tilde {S}} \left[ \operatorname {g e n} (P _ {U}, P _ {\tilde {W} _ {T} | U, \tilde {S}}) \mid \tilde {S} \right] \\ \leq 2 \mathbb {E} _ {\tilde {S}} \left[ \sqrt {\frac {2 R ^ {2}}{n} I (\tilde {W} _ {T} ; U \mid \tilde {S} = \tilde {S})} \mid \tilde {S} \right] \tag {Lemma1} \\ \leq 2 \sqrt {\frac {2 R ^ {2}}{n} \mathbb {E} _ {\tilde {S}} \left[ I (\tilde {W} _ {T} ; U \mid \tilde {S} = \tilde {S}) \right]} = 2 \sqrt {\frac {2 R ^ {2}}{n} I (\tilde {W} _ {T} ; U \mid \tilde {S})}, \\ \end{array}
+$$
+
+where the first step is because
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \hat {\mathcal {L}} _ {\tilde {S}} (\tilde {W} _ {T}) - \hat {\mathcal {L}} _ {S} (\tilde {W} _ {T}) \right] \\ = \mathbb {E} \left[ \frac {1}{n} \sum_ {i = 1} ^ {n} \frac {1}{2} \left(\hat {\mathcal {L}} _ {Z _ {2 (i - 1) + U _ {i}}} (\tilde {W} _ {T}) + \hat {\mathcal {L}} _ {Z _ {2 (i - 1) + 1 - U _ {i}}} (\tilde {W} _ {T})\right) - \frac {1}{n} \sum_ {i = 1} ^ {n} \hat {\mathcal {L}} _ {Z _ {2 (i - 1) + U _ {i}}} (\tilde {W} _ {T}) \right] \\ = \frac {1}{2} \mathbb {E} \left[ \frac {1}{n} \sum_ {i = 1} ^ {n} \hat {\mathcal {L}} _ {Z _ {2 (i - 1) + 1 - U _ {i}}} (\tilde {W} _ {T}) - \hat {\mathcal {L}} _ {S} (\tilde {W} _ {T}) \right] = \frac {1}{2} \mathbb {E} \left[ \mathcal {L} _ {\mu} (\tilde {W} _ {T}) - \hat {\mathcal {L}} _ {S} (\tilde {W} _ {T}) \right]. \\ \end{array}
+$$
+
+The following steps are similar to the proof of Theorem 1 and Corollary D.1, including an application of the chain rule to the conditional MI and an application of Lemma B.1, where  $X = S, Y = \tilde{W}_{0:t-1} O = (W_{0:T}, V, \tilde{S}, U)$  and  $\Omega = \tilde{W}_t - \mathbb{E}\left[g_t \mid \tilde{S}\right]$ .
+
+# D.3 COMBINATION WITH NEGREA ET AL.(2019)'S DATA-DEPENDENT PRIOR
+
+To state Corollary D.3, we need extra notations. For an array  $a_{1:l}$  of length  $l$  and a list  $U \in [l]^k$  of indices, define  $a_U := (a_{U_i})_{i=1}^k$ . For two lists of non-repeated indices  $U^1 \in \mathbb{N}^{k_1}$  and  $U^2 \in \mathbb{N}^{k_2}$ , let  $U^1 \setminus U^2$  be the list of indices that can be found in  $U^1$  but not in  $U^2$ , ordered as in  $U^1$ . Let  $U^1 \cap U^2$  be their intersection ordered as in  $U^1$ .
+
+Corollary D.3 Assume  $\ell(w, \cdot)$  is  $R$ -sub-Gaussian on  $\mu$  for any  $w \in \mathcal{W}$ . Assume the following specific form for the algorithm:
+
+- The randomness  $V = B_{1:T}$ , where  $B_t \in [n]^b$  specifies the samples in the  $b$ -sized batch at step  $t$  with no repeated entries;  
+- The update function can be written as
+
+$$
+g _ {t} (W _ {t - 1}, S, V, W _ {0: t - 2}) = \bar {\nabla} \hat {\mathcal {L}} _ {S _ {B _ {t}}} (W _ {t - 1}),
+$$
+
+where  $\bar{\nabla}$  stands for samplewisely clipped gradient operator (i.e., compute gradients for each sample, clip and then average).
+
+Let  $U \in [n]^m$  be the result of  $m$  uniform (over the remaining indices) sample indices without replacement in  $[n]$ , independent of previous random variables. Define
+
+$$
+\xi_ {t} := \frac {b - | U \cap B _ {t} |}{b} \left(\bar {\nabla} \hat {\mathcal {L}} _ {S _ {B _ {t} \setminus U}} (W _ {t - 1}) - \bar {\nabla} \hat {\mathcal {L}} _ {S _ {U}} (W _ {t - 1})\right)
+$$
+
+to be the gradient incoherence (see it as the difference between the true update at step  $t$  and a prediction of the update using only samples in  $U$ ). For omniscient trajectories defined by bounded  $\Delta g_{1:T}$  that additionally depends on  $U$ , then we have
+
+$$
+\mathrm {g e n} (\mu^ {n}, P _ {W | S}) \leq \sqrt {\frac {R ^ {2}}{n - m} \sum_ {t = 1} ^ {T} \frac {1}{\sigma_ {t} ^ {2}} \mathbb {E} \left[ \| \xi_ {t} - \Delta g _ {t} \| ^ {2} \right]} + \mathbb {E} \left[ \Delta_ {\Gamma_ {T}} ^ {\sum_ {t} \sigma_ {t} ^ {2}} (W _ {T}, S _ {U} ^ {c}) - \Delta_ {\Gamma_ {T}} ^ {\sum_ {t} \sigma_ {t} ^ {2}} (W _ {T}, S ^ {\prime}) \right].
+$$
+
+Corollary D.3 is an application of the following more general result:
+
+Corollary D.4 Assume  $\ell(w, \cdot)$  is  $R$ -sub-Gaussian on  $\mu$  for any  $w \in \mathcal{W}$ . Assume samples have no interactions in updates in the following sense:
+
+- The randomness  $V$  in the update is  $V = B_{1:T}$ , where  $B_t \in [n]^b$  specifies the samples in the  $b$ -sized batch at step  $t$  with no repeated entries;  
+- The update function can be written as
+
+$$
+g _ {t} \left(W _ {t - 1}, S, V, W _ {0: t - 2}\right) = p _ {t} \left(W _ {t - 1}, S _ {B _ {t}}, W _ {0: t - 2}\right),
+$$
+
+where  $p_t:\mathcal{W}\times \bigcup_{i = 0}^{\infty}\mathcal{Z}^i\times \mathcal{W}^{t - 1}\to \mathcal{W}$  is defined by
+
+$$
+p _ {t} (w _ {t - 1}, s, w _ {0: t - 2}) := \left\{ \begin{array}{l l} 0 & | s | = 0, \\ \frac {1}{| s |} \sum_ {i = 1} ^ {| s |} f _ {t} (w _ {t - 1}, s _ {i}, w _ {0: t - 2}) & | s | > 0, \end{array} \right.
+$$
+
+and  $f_{t}:\mathcal{W}\times \mathcal{Z}\times \mathcal{W}^{t - 1}\to \mathcal{W}$  is a deterministic function.
+
+Also, let  $U \in [n]^m$  be the result of  $m$  uniform (over the remaining indices) samples without replacement in  $[n]$ , independent of previous random variables. Lastly, assume that given  $(U, S_U, S_U^c, V, \tilde{W}_{0:t-1})$ , then  $p_t(W_{t-1}, S_{B_t}, W_{0:t-2}), p_t(W_{t-1}, S_U, W_{0:t-2})$  and
+
+$p_t(W_{t-1}, S_{B_t \cap U}, W_{0:t-2})$  all have bounded second moments (e.g., by samplewise gradient clipping).
+
+Define
+
+$$
+\xi_ {t} := \frac {b - | U \cap B _ {t} |}{b} \left(p _ {t} \left(W _ {t - 1}, S _ {B _ {t} \backslash U}, W _ {0: t - 2}\right) - p _ {t} \left(W _ {t - 1}, S _ {U}, W _ {0: t - 2}\right)\right)
+$$
+
+to be the gradient incoherence. For omniscient trajectories defined by  $\Delta g_{1:T}$  that additionally depend on  $U$ , if  $\Delta g_t$  also satisfies the above conditional bounded second moments (e.g., bounded in value), then we have
+
+$$
+\mathrm {g e n} (\mu^ {n}, P _ {W | S}) \leq \sqrt {\frac {R ^ {2}}{n - m} \sum_ {t = 1} ^ {T} \frac {1}{\sigma_ {t} ^ {2}} \mathbb {E} \left[ \| \xi_ {t} - \Delta g _ {t} \| ^ {2} \right]} + \mathbb {E} \left[ \Delta_ {\Gamma_ {T}} ^ {\sum_ {t} \sigma_ {t} ^ {2}} (W _ {T}, S _ {U} ^ {c}) - \Delta_ {\Gamma_ {T}} ^ {\sum_ {t} \sigma_ {t} ^ {2}} (W _ {T}, S ^ {\prime}) \right].
+$$
+
+Remark D.3 The omniscient trajectory additionally depends on  $U$  and then  $\xi_{t}$ , the two featuring random variables of Negrea et al. (2019)'s SGLD bound.
+
+Proof In this proof, we combine Theorem 2.4 and Theorem 3.1 of Negrea et al. (2019) with our technique. As Negrea et al. (2019), let  $S_U^c \coloneqq S_{(1,2,\ldots,n)\setminus U}$ . Due to extra dependence on  $(U,V)$  of the omniscient trajectory, the SGLD-like trajectory is determined by  $P_{\tilde{W}_{0:T}|(U,S,V)}$ , violating the assumptions of Negrea et al. (2019)'s Theorem 2.4. Therefore we must re-prove this data-dependent prior bound, during which we insert the auxiliary trajectories:
+
+$$
+\begin{array}{l} \operatorname {g e n} (\mu^ {n}, P _ {W | S}) = \mathbb {E} \left[ \mathcal {L} _ {\mu} (W _ {T}) - \hat {\mathcal {L}} _ {S _ {U} ^ {c}} (W _ {T}) \right] \\ = \mathbb {E} \left[ \mathcal {L} _ {\mu} (\tilde {W} _ {T}) - \hat {\mathcal {L}} _ {S _ {U} ^ {c}} (\tilde {W} _ {T}) \right] + \mathbb {E} \left[ \Delta_ {\Gamma_ {T}} ^ {\sum_ {t} \sigma_ {t} ^ {2}} (W _ {T}, S _ {U} ^ {c}) - \Delta_ {\Gamma_ {T}} ^ {\sum_ {t} \sigma_ {t} ^ {2}} (W _ {T}, S ^ {\prime}) \right], \\ \end{array}
+$$
+
+where the first equality is because given  $(S,W_T)$ ,  $\hat{\mathcal{L}}_{S_U^c}(W_T)$  is an unbiased estimator on the empirical loss. With the penalty term matching with the statement of the corollary, we focus on the generalization error of the SGLD-like trajectory, which can be bounded by
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \mathcal {L} _ {\mu} (\tilde {W} _ {T}) - \hat {\mathcal {L}} _ {S _ {U} ^ {c}} (\tilde {W} _ {T}) \right] = \mathbb {E} _ {U, S _ {U}, V} \left[ \mathbb {E} \left[ \mathcal {L} _ {\mu} (\tilde {W} _ {T}) - \hat {\mathcal {L}} _ {S _ {U} ^ {c}} (\tilde {W} _ {T}) \mid U, S _ {U}, V \right] \right] \\ = \mathbb {E} _ {U, S _ {U}, V} \left[ \operatorname {g e n} \left(\mu^ {n - m}, P _ {\tilde {W} _ {T} | U, S _ {U} ^ {c}, S _ {U}, V}\right) \right] \\ \leq \mathbb {E} _ {U, S _ {U}, V} \left[ \sqrt {\frac {2 R ^ {2}}{n - m} I \left(\tilde {W} _ {T} ; S _ {U} ^ {c} \mid U = U , S _ {U} = S _ {U} , V = V\right)} \right]. \quad (L e m m a 1) \\ \end{array}
+$$
+
+By the "golden formula" of MI (Eq.(8.7) of Csiszar & Korner (2011)) stating  $I(X;Y) = \mathbb{E}_Y\left[\mathcal{D}_{\mathrm{KL}}\left(P_{X|Y}\parallel P_X\right)\right] \leq \mathbb{E}_Y\left[\mathcal{D}_{\mathrm{KL}}\left(P_{X|Y}\parallel Q_X\right)\right]$ , for any data-dependent prior  $Q_{\tilde{W}_{0:T}^{\prime}}(u,s_u,v)$  over  $\mathcal{W}^{T + 1}$  under the Definition 2.1 of Negrea et al. (2019), we have
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \mathcal {L} _ {\mu} (\tilde {W} _ {T}) - \hat {\mathcal {L}} _ {S _ {U} ^ {c}} (\tilde {W} _ {T}) \right] \\ \leq \mathbb {E} _ {U, S _ {U}, V} \left[ \sqrt {\frac {2 R ^ {2}}{n - m} \mathbb {E} _ {S _ {U} ^ {c} | U , S _ {U} , V} \left[ \mathcal {D} _ {\mathrm {K L}} \left(P _ {\tilde {W} _ {0 : T} | U , S _ {U} , S _ {U} ^ {c} , V} \parallel Q _ {\tilde {W} _ {0 : T} ^ {\prime}} (U , S _ {U} , V)\right) \right]} \right] \\ \end{array}
+$$
+
+Now we turn to bound the KL divergence given any realization  $(u, s_u, s_u^c, v)$  of the conditioned variables. We restrict the data-dependent prior to have the same distribution as the real distribution for  $\tilde{W}_0$ , i.e.,  $P_{\tilde{W}_0|u, s_u, s_u^c, v} = P_{\tilde{W}_0} = Q_{\tilde{W}_0'}(u, s_u, v)$ . By Proposition 2.6 of Negrea et al. (2019), we have
+
+$$
+\begin{array}{l} \mathcal {D} _ {\mathrm {K L}} \left(P _ {\tilde {W} _ {0: T} | u, s _ {u}, s _ {u} ^ {c}, v} \| Q _ {\tilde {W} _ {T}} (u, s _ {u}, v)\right) \leq \mathcal {D} _ {\mathrm {K L}} \left(P _ {\tilde {W} _ {0: T} | u, s _ {u}, s _ {u} ^ {c}, v} \| Q _ {\tilde {W} _ {0: T} ^ {\prime}} (u, s _ {u}, v)\right) \\ \leq \sum_ {t = 1} ^ {T} \mathbb {E} _ {\tilde {W} _ {0: t - 1} \sim P _ {\tilde {W} _ {0: t - 1} | u, s _ {u}, s _ {u} ^ {c}, v}} \left[ \mathcal {D} _ {\mathrm {K L}} \left(P _ {\tilde {W} _ {t} | \tilde {W} _ {0: t - 1}, u, s _ {u}, s _ {u} ^ {c}, v} \| Q _ {\tilde {W} _ {t} ^ {\prime} | \tilde {W} _ {0: t - 1}} (u, s _ {u}, v)\right) \right] \tag {D.2} \\ \end{array}
+$$
+
+We fix a condition  $c = (u, s_u, s_u^c, v, \tilde{w}_{0:t-1})$  and turn to bound the KL divergences in the above expectation. We also instantiate the data-dependent prior to the prior designed by Negrea et al. (2019) for SGLD, which uses samples in  $S_U$  to predict the update using  $B_t$  so that the error between the predicted and the true update eventually becomes the trajectory term. The gradients and updates depend on the model weight and so do the predictions. Therefore, the data-dependent prior must somehow access the  $(t-1)$ -th weight of the original trajectory to make predictions. In Negrea et al. (2019)'s proof for SGLD, the data-dependent prior technique is directly applied to the original trajectory and thus the  $(t-1)$ -th original weight can be directly found in the condition  $c$ . However, in our case where the technique is applied to the auxiliary SGLD-like trajectory, we only have the latest step of the SGLD-like trajectory in the condition, which is not the original SGD weight but corresponds to many  $W_{t-1}$  or  $\tilde{W}_{t-1}$  on the SGD or the omniscient trajectory, given that the SGLD-like trajectory is constructed by adding Gaussian noises to the omniscient trajectory. Directly predicting and comparing gradients at the SGLD-like trajectory will make the final result related to the SGLD-like trajectory. Switching back using local smoothness along the trajectory will result in a "local gradient sensitivity" term as Neu et al. (2021) that accumulates very fast along the training process (Wang & Mao, 2022). The ideal adapted proof would be somehow "tracing back" to the SGD weight  $W_{t-1}$  that leads to the SGLD weight in the condition, making predictions based on the SGD weight  $W_{t-1}$ , and comparing it with the update  $g_t(W_{t-1})$  on the SGD trajectory, followed by averaging over the distribution of  $W_{t-1}$  given the SGLD-like trajectory weight. This intuition suggests a coupling between the true and the predicted update through  $W_{t-1}$ , which reminds us of a lemma by Neu et al. (2021):
+
+Lemma D.1 (Lemma 4 of Neu et al. (2021)) Let  $X$  and  $Y$  be random variables taking values in  $\mathbb{R}^d$  with bounded second moments and let  $\sigma > 0$ . Letting  $\epsilon \sim \mathcal{N}(0, \sigma^2 I)$  be independent of  $(X, Y)$ , the KL divergence between the distributions of  $X + \epsilon$  and  $Y + \epsilon$  is bounded as
+
+$$
+\mathcal {D} _ {\mathrm {K L}} \left(P _ {X + \epsilon} \parallel P _ {Y + \epsilon}\right) \leq \frac {1}{2 \sigma^ {2}} \mathbb {E} \left[ \left\| X - Y \right\| ^ {2} \right],
+$$
+
+where the expectation is taken over any joint distribution with any coupling between  $X$  and  $Y$  as long as the marginals are still  $P_{X}, P_{Y}$ , respectively.
+
+The arbitrariness of the coupling allows a specific one through  $W_{t - 1}$ . Therefore, we turn to construct the coupling of the data-dependent prior.
+
+Since the coupling involves multiple random variables, it must start from a joint distribution. For any condition  $(u,s_u,s_u^c,v,\tilde{w}_{0:t - 1})$ , let  $Q_{M_{0:T},\check{M}_t,\check{M}_t',\tilde{M}_t,\tilde{M}_t'|u,s_u,s_u^c,v,\tilde{w}_{0:t - 1}}$  be the joint distribution over  $\mathcal{W}^5$  given by the Markov chain  $M_{0:T}\to (\check{M}_t,\check{M}_t')\to (\check{M}_t,\check{M}_t')$ , where  $M_{0:T}$  is sampled from  $P_{W_{0:T}|u,s_u,s_u^c,v,\tilde{w}_{0:t - 1}}$  and
+
+$$
+\begin{array}{l} \tilde {M} _ {t} := \tilde {w} _ {t - 1} - \left(g _ {t} \left(M _ {t - 1}, s, v, M _ {0: t - 2}\right) - \Delta g _ {t} (s, v, g _ {1: T}, M _ {0: T}, u)\right), \\ \check {M} _ {t} ^ {\prime} := \tilde {w} _ {t - 1} - \left(\frac {| u _ {t} |}{b} p _ {t} \left(M _ {t - 1}, s _ {u _ {t}}, M _ {0: t - 2}\right) + \frac {b - | u _ {t} |}{b} p _ {t} \left(M _ {t - 1}, s _ {u}, M _ {0: t - 2}\right)\right), \tag {D.3} \\ \tilde {M} _ {t} = \breve {M} _ {t} + N _ {t} ^ {\prime}, \tilde {M} _ {t} ^ {\prime} = \breve {M} _ {t} ^ {\prime} + N _ {t} ^ {\prime}, \\ \end{array}
+$$
+
+where  $j_{t} \coloneqq j \cap b_{t}$  is the samples in  $j$  contained in the current batch,  $N_{t}^{\prime} \sim \mathcal{N}(0, \sigma_{t}^{2}I)$  is independent of other random variables in the random process defining  $Q$ . By assumption, terms defining  $\check{M}_{t}$  and  $\check{M}_{t}^{\prime}$  have bounded second moments, satisfying the assumption of Lemma D.1. It can be easily verified that the marginal distribution  $Q_{\tilde{M}_t|c} = P_{\tilde{W}_t|c}$ . Moreover, all used samples in Eq. (D.3) are contained in  $S_U$ . As a result, the marginal distribution of  $M_t^\prime$  is independent of  $S_U^c$ . Therefore, by setting the data-dependent prior  $Q_{\tilde{W}_t^{\prime}|\tilde{W}_{0:t - 1}}(u,s_u,v) = Q_{\tilde{M}_t^{\prime}|u,s_u,(s_u^c)'(u,\tilde{w}_{0:t - 1},s_u,v),v,\tilde{w}_{0:t - 1}}$ , where  $(s_u^c)^\prime (\cdot)$  is a function that outputs some (conditionally) supported realization of  $S_U^c$ , we have  $Q_{\tilde{W}_t^{\prime}|\tilde{W}_{0:t - 1}}(u,s_u,v) = Q_{\tilde{M}_t^{\prime}|c}$ . With the help of this coupling and Lemma D.1, we can bound the
+
+KL divergence in Eq. (D.2) by
+
+$$
+\begin{array}{l} \mathcal {D} _ {\mathrm {K L}} \left(P _ {\tilde {W} _ {t} | \boldsymbol {c}} \| Q _ {\tilde {W} _ {t} ^ {\prime} | \tilde {W} _ {0: t - 1}} (u, s _ {u}, v)\right) = \mathcal {D} _ {\mathrm {K L}} \left(Q _ {\tilde {M} _ {t} | \boldsymbol {c}} \| Q _ {\tilde {M} _ {t} ^ {\prime} | \boldsymbol {c}}\right) \\ = \mathcal {D} _ {\mathrm {K L}} \left(Q _ {\check {M} _ {t} + N _ {t} ^ {\prime} | c} \| Q _ {\check {M} _ {t} ^ {\prime} + N _ {t} ^ {\prime} | c}\right) \leq \frac {1}{2 \sigma_ {t} ^ {2}} \mathbb {E} _ {\left(\check {M} _ {t}, \check {M} _ {t} ^ {\prime}\right) \sim Q _ {\check {M} _ {t}, \check {M} _ {t} ^ {\prime} | c}} \left[ \left\| \check {M} _ {t} - \check {M} _ {t} ^ {\prime} \right\| ^ {2} \right] \quad (\text {L e m m a D . 1}) \\ = \frac {1}{2 \sigma_ {t} ^ {2}} \mathbb {E} _ {Q} \left[ \left\| g _ {t} \left(M _ {t - 1}, s, v, M _ {0: t - 2}\right) - \Delta g _ {t} (s, v, g _ {1: T}, M _ {0: T}, u) \right. \right. \\ \left. - \left(\frac {\left| u _ {t} \right|}{b} p _ {t} \left(M _ {t - 1}, s _ {u _ {t}}, M _ {0: t - 2}\right) + \frac {b - \left| u _ {t} \right|}{b} p _ {t} \left(M _ {t - 1}, s _ {u}, M _ {0: t - 2}\right)\right) \| ^ {2} | c \right] \quad \text {E q .} \tag {D.3} \\ = \frac {1}{2 \sigma_ {t} ^ {2}} \mathbb {E} _ {P} \left[ \left\| g _ {t} \left(W _ {t - 1}, s, v, W _ {0: t - 2}\right) - \Delta g _ {t} (s, v, g _ {1: T}, W _ {0: T}, u) \right. \right. \\ \left. - \left(\frac {\left| u _ {t} \right|}{b} p _ {t} \left(W _ {t - 1}, s _ {u _ {t}}, W _ {0: t - 2}\right) + \frac {b - \left| u _ {t} \right|}{b} p _ {t} \left(W _ {t - 1}, s _ {u}, W _ {0: t - 2}\right)\right) \| ^ {2} \mid c \right] \\ = \frac {1}{2 \sigma_ {t} ^ {2}} \mathbb {E} \left[ \| \xi_ {t} - \Delta g _ {t} \| ^ {2} \mid \boldsymbol {c} \right], \\ \end{array}
+$$
+
+where the penultimate step is because, by construction, we have  $Q_{M_0:T|\mathbf{e}} = P_{W_0:T|\mathbf{e}}$ .
+
+After plugging everything together, interchanging the order of the expectation and the square root by the latter's concavity, and putting expectations together, the corollary is proved.
+
+# D.4 GENERALIZATION OF STABLE ALGORITHMS AND GD ON CLB PROBLEMS
+
+Theorem 3 Assume  $n, d \in \mathbb{N}^+$ ,  $L, D > 0$  and  $\eta > 0$ ,  $T \in \mathbb{N}^+$ . Assume the algorithm starts from a fixed initialization  $W_0 := w_0 \in \mathcal{W}$  and the whole trajectory remains within  $\mathcal{W}$ , i.e.,  $W_{0:T} \in \mathcal{W}^{T + 1}$ . For any CLB problem  $(\mathcal{W},\mathcal{Z},\ell) \in \mathcal{C}_{L,D}$  and any data distribution  $\mu \in \mathcal{M}_1(\mathcal{Z})$ , we have the following data-dependent and -agnostic bounds that recover stability bounds:
+
+$$
+\begin{array}{l} \mathrm {g e n} (\mu^ {n}, P _ {W | S}) \\ \leq \inf  _ {\Delta G ^ {1: n}} \frac {L D}{n} \sum_ {i = 1} ^ {n} \sqrt {2 I (W _ {T} + \Delta G ^ {i} ; Z _ {i} \mid Z _ {- i})} + \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} [ \Delta_ {\Delta G ^ {i}} (W _ {T}, Z _ {i}) - \Delta_ {\Delta G ^ {i}} (W _ {T}, S ^ {\prime}) ] \\ \leq \frac {2 L}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \left\| W _ {T} - \mathbb {E} \left[ W _ {T} \mid Z _ {- i} \right] \right\| \right]. \\ \end{array}
+$$
+
+If the algorithm is a GD algorithm using projected subgradients of step size  $\eta$  and step count  $T$ , we have
+
+$$
+\mathrm {g e n} (\mu^ {n}, P _ {W | S}) \leq 8 L ^ {2} \sqrt {T} \eta + \frac {8 L ^ {2} T \eta}{n}.
+$$
+
+Therefore, we have the following worst-case generalization error bound for CLB and GD:
+
+$$
+\sup_{(\mathcal{W},\mathcal{Z},\ell)\in \mathcal{C}_{L,D}}\sup_{\mu \in \mathcal{M}_{1}(\mathcal{Z})}\operatorname {gen}(\mu^{n},P_{W|S}^{GD_{n,\eta ,T}})\leq 8L^{2}\sqrt{T}\eta +\frac{8L^{2}T\eta}{n}.
+$$
+
+Proof Let  $(\mathcal{W},\mathcal{Z},\ell)\in \mathcal{C}_{L,D}$  be an SCO problem and  $\mu \in \mathcal{M}_1(\mathcal{Z})$  be a distribution of samples. Therefore, the sample loss  $\ell$ , the empirical loss  $\hat{\mathcal{L}}_S(\cdot)$  and the population loss  $\hat{\mathcal{L}}_S(\cdot)$  are  $L$ -Lipschitz w.r.t. the weight and  $\mathcal{W}$  is bounded with a diameter  $D$  and convex. Let  $W_{0:T}$  be the trajectory given by GD, where  $W_{t}\in \mathcal{W}$ .
+
+The proof essentially resembles the uniform stability argument of Bassily et al. (2020) through the superior expressivity added to the MI bounds by the omniscient trajectory. Since uniform stability
+
+considers replacing one sample in the training set, we need to focus on one sample instead of the entire training set. To keep other samples "unchanged" in the replacement, we also need to condition on other samples. Therefore, we will start from the "individual sample" technique (Bu et al., 2020). This observation motivates us to use Corollary D.1. However, losses in CLB problems are not sub-Gaussian in general. Therefore, we need variants similar to Theorem 11 of Haghifam et al. (2023) that replaces sub-Gaussianity with Lipschitzness and boundedness. By repeating the proof of Corollary D.1 but with Theorem 11 of Haghifam et al. (2023) instead of Lemma 1, we obtain
+
+$$
+\begin{array}{l} \operatorname {g e n} \left(\mu^ {n}, P _ {W | S}\right) \leq \frac {L D}{n} \sum_ {i = 1} ^ {n} \sqrt {2 I \left(W _ {T} + \sum_ {t = 1} ^ {T} \Delta g _ {t} ^ {i} ; Z _ {i} \mid Z _ {- i}\right)} \\ + \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \Delta_ {\Gamma_ {T} ^ {i}} \left(W _ {T}, Z _ {i}\right) - \Delta_ {\Gamma_ {T} ^ {i}} \left(W _ {T}, S ^ {\prime}\right) \right], \tag {D.4} \\ \end{array}
+$$
+
+where  $\{\Delta g_{1:T}^i\}_{i=1}^n$  is a family of omniscient perturbations, which moves the terminal within  $\mathcal{W}$ , i.e.,  $W_T + \Gamma_T^i \in \mathcal{W}$ .
+
+Remark D.4 The SGLD-like trajectory is not used in this proof. This is because the SGLD-like trajectory is used only to bound the MI of the omniscient trajectory. As one will see later, the omniscient trajectory in this proof has a very special form (constant) with a trivial MI bound (0). As a result, an MI bound through the SGLD-like trajectory is no longer needed.
+
+This proof proceeds by bounding the penalty terms in Eq. (D.4) with the help of the inherited Lipschitzness of  $\ell$  in the empirical and the population loss:
+
+$$
+\begin{array}{l} \left| \Delta_ {\Gamma_ {T} ^ {i}} (W _ {T}, Z _ {i}) \right| := \left| \hat {\mathcal {L}} _ {Z _ {i}} (W _ {T} + \Gamma_ {T} ^ {i}) - \hat {\mathcal {L}} _ {Z _ {i}} (W _ {T}) \right| \leq L \left\| \Gamma_ {T} ^ {i} \right\|, \\ \left| \mathbb {E} \left[ \Delta_ {\Gamma_ {T} ^ {i}} (W _ {T}, S ^ {\prime}) \right] \right| := \left| \mathbb {E} \left[ \mathcal {L} _ {\mu} (W _ {T} + \Gamma_ {T} ^ {i}) - \mathcal {L} _ {\mu} (W _ {T}) \right] \right| \leq L \mathbb {E} \left[ \| \Gamma_ {T} ^ {i} \| \right]. \\ \end{array}
+$$
+
+Therefore, we have the following bound for the penalty terms:
+
+$$
+\left| \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \Delta_ {\Gamma_ {T} ^ {i}} \left(W _ {T}, Z _ {i}\right) - \Delta_ {\Gamma_ {T} ^ {i}} \left(W _ {T}, S ^ {\prime}\right) \right] \right| \leq \frac {2 L}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \| \Gamma_ {T} ^ {i} \| \right].
+$$
+
+Plugging this bound to Eq. (D.4) leads to
+
+$$
+\operatorname {g e n} \left(\mu^ {n}, P _ {W | S}\right) \leq \frac {L D}{n} \sum_ {i = 1} ^ {n} \sqrt {I \left(W _ {T} + \Gamma_ {T} ^ {i} ; Z _ {i} \mid Z _ {- i}\right)} + \frac {2 L}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \| \Gamma_ {T} ^ {i} \| \right].
+$$
+
+By setting  $\Delta g_t^i = g_t - \mathbb{E}[g_t\mid Z_{-i}]$ , we have  $W_{T} + \Gamma_{T}^{i} = \mathbb{E}\left[W_{T}\mid Z_{-i}\right]$ . By convexity of  $\mathcal{W}$  and that  $W_{T}\in \mathcal{W}$ , we have  $W_{T} + \Gamma_{T}^{i}\in \mathcal{W}$ . Notably, given  $Z_{-i}$ ,  $W_{T} + \Gamma_{T}^{i}$  is constant regardless of  $Z_{i}$ . Therefore, we have  $I(W_{T} + \Gamma_{T}^{i};Z_{i}\mid Z_{-i}) = 0$  and
+
+$$
+\operatorname {g e n} \left(\mu^ {n}, P _ {W | S}\right) \leq \frac {2 L}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \left\| \Gamma_ {T} ^ {i} \right\| \right]. \tag {D.5}
+$$
+
+Note that  $\mathbb{E}\left[\| W_T - \mathbb{E}\left[W_T\mid Z_{-i}\right]\| \mid Z_{-i}\right]$  is the expected distance to the center given  $Z_{-i}$ . By Corollary A.1, this value is smaller than  $\mathbb{E}\left[\| W_T - W_T'\| \mid Z_{-i}\right]$ , where  $W_T'$  is the terminal weight trained by the same  $Z_{-i}$  and an independently sampled  $i$ -th sample.
+
+Now we turn to GD. By Theorem 3.2 of Bassily et al. (2020), given  $Z_{-i}$ ,  $\| W_T - W_T' \|$  of GD has a data-agnostic upperbound  $4L\sqrt{T}\eta + \frac{4LT\eta}{n}$ . Plugging this bound, we obtain
+
+$$
+\begin{array}{l} \operatorname {g e n} \left(\mu^ {n}, P _ {W | S}\right) \leq \frac {2 L}{n} \sum_ {i = 1} ^ {n} \mathbb {E} _ {Z _ {- i}} \left[ \mathbb {E} \left[ \| W _ {T} - \mathbb {E} \left[ W _ {T} \mid Z _ {- i} \right] \| \mid Z _ {- i} \right] \right] \\ \leq \frac {2 L}{n} \sum_ {i = 1} ^ {n} \mathbb {E} _ {Z _ {- i}} \left[ \mathbb {E} \left[ \| W _ {T} - W _ {T} ^ {\prime} \| \mid Z _ {- i} \right] \right] \\ \leq 8 L ^ {2} \sqrt {T} \eta + \frac {8 L ^ {2} T \eta}{n}. \\ \end{array}
+$$
+
+Remark D.5 Although adding the omniscient trajectory can address the limitation of representative information-theoretic bounds on CLB problems, we currently do not know whether it is exactly our technique that makes it happen. This is because, between the Gaussian-perturbed individual-sample (C)MI bound considered by Haghifam et al. (2023) and the omniscient bound, there exist bounds derived by non-isotropic Gaussian perturbations, bounds by non-Gaussian but general independent perturbations, and bounds by general weight-dependent perturbations (Rate-Distortion bounds (Se-fidgaran et al., 2022)). It is possible some of these bounds can already address the limitation but we have not obtained any positive or negative results for them before the submission deadline. Nevertheless, our technique makes the proof extremely simple and without it the proof will be at least much longer. For example, it will be harder to make the (conditional) MI diminish without knowing  $Z_{-i}$ .
+
+# D.5 EXTENSION TO  $\epsilon$ -LEARNERS ON CLB PROBLEMS
+
+Attias et al. (2024) have found that any  $\epsilon$ -learners have CMI of at least  $\Omega(1/\epsilon)$  on CLB problems, indicating a CMI-accuracy trade-off. Here, a learning algorithm is an  $\epsilon$ -learner if for every data distribution  $\mu$  over  $\mathcal{Z}$ , the excess generalization risk of the algorithm is at most  $\epsilon$ .
+
+Our Theorem 3 and its proof have given an intuitive alternative to the trade-off: although GD or stable learners have large CMI themselves, they are quite close to learners with low CMI. However, Theorem 3 assumes GD or stable algorithms, yet Attias et al. (2024)'s trade-off covers more general algorithms. Therefore, we explore whether our technique and alternative can extend to more (expectation-)  $\epsilon$ -learners in Theorem 4. It states our technique can indeed extends to more  $\epsilon$ -learners under some assumptions. It also states if one sees the omniscient trajectory augmented MI as a new information measure, then the information-accuracy trade-off is penetrated because both can vanish as  $n \to \infty$ . However, the result is still partial and preliminary, because we only covers  $\epsilon$ -learners that are also  $O(\epsilon)$ -optimizers, i.e., they are "well-behaved" in the sense that its excess optimization error is not too large compared to the excess generalization error  $\epsilon$ . Nevertheless, we believe this assumption is rather gentle for well-behaved learners used practically, e.g., deep models.
+
+Theorem 4 Assume  $d \in \mathbb{N}^+$ ,  $L, D > 0$  and let  $(\mathcal{W}, \mathcal{Z}, \ell) \in \mathcal{C}_{L,D}$  be a CLB problem. Assume for any  $w \in \mathcal{W}$  and  $z \in \mathcal{Z}$ , we have  $\ell(w, z) \in [-LD, +LD]$ . If not, shift the loss functions. This shifting does not affect the excess generalization risks or the excess optimization errors, which this theorem mainly assume on.
+
+Let  $\epsilon > 0$  be a function of sample number  $n$ . Let  $\{\mathcal{A}_n : \mathcal{Z}^n \to \mathcal{W}\}$  be a family of expectation- $\epsilon$ -learners for  $(\mathcal{W}, \mathcal{Z}, \ell)$ . That is, for every sufficiently large  $n$ , for every distribution  $\mu$  over  $\mathcal{Z}$ , one has  $\mathbb{E}_{S \sim \mu^n}[\mathcal{L}_\mu(\mathcal{A}_n(S))] - \inf_{w \in \mathcal{W}}\mathcal{L}_\mu(w) \leq \epsilon$ .
+
+Assume  $\{\mathcal{A}_n\}$  is also an  $O(\epsilon)$ -optimizer. That is,  $\mathcal{A}_n$  has an excess optimization error satisfying  $\mathbb{E}_{S \sim \mu^n} \left[ \hat{\mathcal{L}}_S(\mathcal{A}_n(S)) - \inf_w \hat{\mathcal{L}}_S(w) \right] \leq O(\epsilon)$  for any sufficiently large  $n$ .
+
+Assume  $n \in \mathbb{N}^{+}$  is large enough so that  $\epsilon$  bounds the excess generalization risk and  $O(\epsilon)$  bounds the excess optimization error. Then, we have
+
+$$
+\begin{array}{l} \operatorname {g e n} \left(\mu^ {n}, P _ {W | S}\right) \leq \inf  _ {\left\{\Delta G ^ {i} \right\} _ {i = 1} ^ {n}} \frac {L D}{n} \sum_ {i = 1} ^ {n} \sqrt {2 I \left(W + \Delta G ^ {i} ; Z _ {i} \mid Z _ {- i}\right)} + \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \Delta_ {\Delta G ^ {i}} \left(W, Z _ {i}\right) - \Delta_ {\Delta G ^ {i}} \left(W, S ^ {\prime}\right) \right] \\ \leq O (\epsilon) + O \left(\frac {L D}{\sqrt {n}}\right), \\ \end{array}
+$$
+
+where  $W \coloneqq \mathcal{A}_n(S)$  is the output of the algorithm, and  $\{\Delta G^i\}_{i=1}^n$  is a family of (one-step) omniscient perturbations that additionally depend on  $i$  and  $Z_{-i}$ .
+
+Remark D.6 We assume the algorithm  $\{\mathcal{A}_n\}$  is deterministic but the extension to random ones is straightforward.
+
+Proof After repeating the initial steps of the proof and obtaining Eq. (D.4), we have
+
+$$
+\begin{array}{l} \operatorname {g e n} \left(\mu^ {n}, P _ {W | S}\right) \leq \frac {L D}{n} \sum_ {i = 1} ^ {n} \sqrt {2 I \left(W + \Delta G ^ {i} ; Z _ {i} \mid Z _ {- i}\right)} \\ + \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \Delta_ {\Delta G ^ {i}} (W, Z _ {i}) - \Delta_ {\Delta G ^ {i}} (W, S ^ {\prime}) \right] \\ \leq \frac {L D}{n} \sum_ {i = 1} ^ {n} \sqrt {2 I (W + \Delta G ^ {i} ; Z _ {i} \mid Z _ {- i})} + \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \left| \hat {\mathcal {L}} _ {S} (W + \Delta G ^ {i}) - \hat {\mathcal {L}} _ {S} (W) \right| \right] \\ + \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \left| \mathcal {L} _ {\mu} (W + \Delta G ^ {i}) - \mathcal {L} _ {\mu} (W) \right| \right], \\ \end{array}
+$$
+
+After applying a similar omniscient perturbation  $\Delta G^i \coloneqq \mathbb{E}[W \mid Z_{-i}] - W$  as in the proof of Theorem 3, we have
+
+$$
+\begin{array}{l} \operatorname {g e n} \left(\mu^ {n}, P _ {W | S}\right) \leq \frac {L D}{n} \sum_ {i = 1} ^ {n} \sqrt {2 I \left(\mathbb {E} \left[ W \mid Z _ {- i} \right] ; Z _ {i} \mid Z _ {- i}\right)} + \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \left| \hat {\mathcal {L}} _ {S} \left(\mathbb {E} \left[ W \mid Z _ {- i} \right]\right) - \hat {\mathcal {L}} _ {S} (W) \right| \right] \\ + \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \left| \mathcal {L} _ {\mu} \left(\mathbb {E} \left[ W \mid Z _ {- i} \right]\right) - \mathcal {L} _ {\mu} (W) \right| \right] \\ = \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \left| \hat {\mathcal {L}} _ {S} (\mathbb {E} [ W \mid Z _ {- i} ]) - \hat {\mathcal {L}} _ {S} (W) \right| \right] + \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ | \mathcal {L} _ {\mu} (\mathbb {E} [ W \mid Z _ {- i} ]) - \mathcal {L} _ {\mu} (W) | \right]. \tag {D.6} \\ \end{array}
+$$
+
+We first bound the population loss difference by applying Lemma A.3 to the difference with convex  $f(w) = \mathcal{L}_{\mu}(w) - \mathcal{L}_{\mu}(w^{*}) \geq 0$ , where  $w^{*}$  is the weight with optimal generalization risk:
+
+$$
+\begin{array}{l} \mathbb {E} \left[ | \mathcal {L} _ {\mu} (\mathbb {E} [ W \mid Z _ {- i} ]) - \mathcal {L} _ {\mu} (W) | \right] = \mathbb {E} _ {Z _ {- i}} \left[ \mathbb {E} \left[ | (\mathcal {L} _ {\mu} (\mathbb {E} [ W \mid Z _ {- i} ]) - \mathcal {L} _ {\mu} (w ^ {*})) - (\mathcal {L} _ {\mu} (W) - \mathcal {L} _ {\mu} (w ^ {*})) | \mid Z _ {- i} ] \right] \right. \\ \leq 2 \mathbb {E} _ {Z _ {- i}} [ \mathbb {E} [ f (W) \mid Z _ {- i} ] ] = 2 (\mathbb {E} [ \mathcal {L} _ {\mu} (W) ] - \mathcal {L} _ {\mu} (w ^ {*})) \leq 2 \epsilon . \\ \end{array}
+$$
+
+Now we bound the empirical loss difference. Let  $w_{s}^{*}$  be the empirical risk minimizer of training set  $s$ . The main difficulty is that different  $W$  corresponds to different  $S$ , and one cannot find an  $f$  to be  $\hat{\mathcal{L}}_S(\cdot) - \hat{\mathcal{L}}_S(w_S^*)$  and  $\hat{\mathcal{L}}_{S'}(\cdot) - \hat{\mathcal{L}}_{S'}(w_{S'}^*)$  at the same time. Fortunately, with the individual technique, we can put weights corresponding to  $Z_{-i}$  together. Most of their training set is the same, while the only different sample only contributes  $1 / n$  of the loss, which vanishes as  $n\to \infty$ . Therefore, we have
+
+$$
+\mathbb {E} \left[ \left| \hat {\mathcal {L}} _ {S} (\mathbb {E} [ W | Z _ {- i} ]) - \hat {\mathcal {L}} _ {S} (W) \right| \right] = \mathbb {E} _ {Z _ {- i}} \left[ \mathbb {E} \left[ \left| \hat {\mathcal {L}} _ {S} (\mathbb {E} [ W | Z _ {- i} ]) - \hat {\mathcal {L}} _ {S} (W) \right| | Z _ {- i} \right] \right],
+$$
+
+where
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \left| \hat {\mathcal {L}} _ {S} (\mathbb {E} [ W | Z _ {- i} ]) - \hat {\mathcal {L}} _ {S} (W) \right| | Z _ {- i} \right] \leq \frac {n - 1}{n} \mathbb {E} \left[ \left| \hat {\mathcal {L}} _ {Z _ {- i}} (\mathbb {E} [ W | Z _ {- i} ]) - \hat {\mathcal {L}} _ {Z _ {- i}} (W) \right| | Z _ {- i} \right] \\ + \frac {1}{n} \mathbb {E} \left[ \left| \hat {\mathcal {L}} _ {Z _ {i}} (\mathbb {E} [ W | Z _ {- i} ]) - \hat {\mathcal {L}} _ {Z _ {i}} (W) \right| | Z _ {- i} \right] \\ \end{array}
+$$
+
+(Lipschitzness and boundedness of CLB problems)
+
+$$
+\leq \frac {n - 1}{n} \mathbb {E} \left[ \left| \hat {\mathcal {L}} _ {Z _ {- i}} (\mathbb {E} [ W \mid Z _ {- i} ]) - \hat {\mathcal {L}} _ {Z _ {- i}} (W) \right| \mid Z _ {- i} \right] + \frac {L D}{n}.
+$$
+
+Applying Lemma A.3 with convex  $f(w) \coloneqq \hat{\mathcal{L}}_{Z_{-i}}(w) - \hat{\mathcal{L}}_{Z_{-i}}(w_{Z_{-i}}^*) \geq 0$  leads to
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \left| \hat {\mathcal {L}} _ {S} (\mathbb {E} [ W \mid Z _ {- i} ]) - \hat {\mathcal {L}} _ {S} (W) \right| \mid Z _ {- i} \right] \\ \leq \frac {n - 1}{n} \cdot 2 \mathbb {E} [ f (W) \mid Z _ {- i} ] + \frac {L D}{n} \\ \leq \frac {n - 1}{n} \cdot 2 \mathbb {E} \left[ \hat {\mathcal {L}} _ {Z _ {- i}} (W) - \hat {\mathcal {L}} _ {Z _ {- i}} \left(w _ {Z _ {- i}} ^ {*}\right) \mid Z _ {- i} \right] + \frac {L D}{n} \\ \left(\frac {1}{n} \mathbb {E} \left[ \hat {\mathcal {L}} _ {Z _ {i}} (W) - \hat {\mathcal {L}} _ {Z _ {i}} \left(w _ {Z _ {i}} ^ {*}\right) \mid Z _ {- i} \right] \geq 0\right) \\ \leq 2 \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (W) - \left(\frac {n - 1}{n} \hat {\mathcal {L}} _ {Z _ {- i}} (w _ {Z _ {- i}} ^ {*}) + \frac {1}{n} \hat {\mathcal {L}} _ {Z _ {i}} (w _ {Z _ {i}} ^ {*})\right) | Z _ {- i} \right] + \frac {L D}{n} \\ \end{array}
+$$
+
+Taking expectation over  $Z_{-i}$  leads to
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \left| \hat {\mathcal {L}} _ {S} (\mathbb {E} [ W | Z _ {- i} ]) - \hat {\mathcal {L}} _ {S} (W) \right| \right] \\ \leq 2 \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (W) \right] - 2 \left(\frac {n - 1}{n} \mathbb {E} \left[ \hat {\mathcal {L}} _ {S _ {n - 1}} \left(w _ {S _ {n - 1}} ^ {*}\right) \right] + \frac {1}{n} \mathbb {E} \left[ \hat {\mathcal {L}} _ {Z} \left(w _ {Z} ^ {*}\right) \right]\right) + \frac {L D}{n}, \tag {D.7} \\ \end{array}
+$$
+
+where  $S_{n - 1}\sim \mu^{n - 1}$ . We need to relate  $\frac{n - 1}{n}\mathbb{E}\left[\hat{\mathcal{L}}_{S_{n - 1}}(w_{S_{n - 1}}^*)\right] + \frac{1}{n}\mathbb{E}\left[\hat{\mathcal{L}}_Z(w_Z^*)\right]$  to  $\mathbb{E}\left[\hat{\mathcal{L}}_S(w_S^*)\right]$ . We can bound  $\frac{1}{n}\mathbb{E}\left[\hat{\mathcal{L}}_Z(w_Z^*)\right]$  by  $\frac{LD}{n}$  again by the assumption that losses are bounded in  $[-LD, + LD]$ , leaving  $\frac{n - 1}{n}\mathbb{E}\left[\hat{\mathcal{L}}_{S_{n - 1}}(w_{S_{n - 1}}^*)\right]$  and  $\mathbb{E}\left[\hat{\mathcal{L}}_S(w_S^*)\right]$ , which somehow forms the stability of the empirical risk minimizer. Since (projected) GD with fixed initialization has been proved to be stable and has been proved to approximate the empirical minimizer, we use (projected) GD to bridge the empirical minimizers.
+
+According to Orabona (2020) and Eq.(1) of Haghifam et al. (2023), on CLB problems, (projected) GD algorithm  $\mathrm{GD}_{\eta ,T}:\mathcal{Z}^*\to \mathcal{W}$  with step size  $\eta$  and step count  $T$  and a fixed initialization has an excess optimization error  $\frac{D^2}{2\eta T} +\frac{(\log T + 2)\eta L^2}{2}$ . As a result, we can approximate the empirical minimizers using GDs with errors
+
+$$
+\begin{array}{l} \left| \mathbb {E} \left[ \hat {\mathcal {L}} _ {S _ {n - 1}} (w _ {S _ {n - 1}} ^ {*}) \right] - \mathbb {E} \left[ \hat {\mathcal {L}} _ {S _ {n - 1}} (\mathbf {G D} _ {\eta_ {n - 1}, T _ {n - 1}} (S _ {n - 1})) \right] \right| \leq \frac {D ^ {2}}{2 \eta_ {n - 1} T _ {n - 1}} + \frac {(\log T _ {n - 1} + 2) \eta_ {n - 1} L ^ {2}}{2}, \\ \left| \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (w _ {S} ^ {*}) \right] - \mathbb {E} \left[ \hat {\mathcal {L}} _ {S _ {n}} (\mathbf {G D} _ {\eta_ {n}, T _ {n}} (S)) \right] \right| \leq \frac {D ^ {2}}{2 \eta_ {n} T _ {n}} + \frac {(\log T _ {n} + 2) \eta_ {n} L ^ {2}}{2}, \\ \end{array}
+$$
+
+for any  $(\eta_{n - 1},T_{n - 1})$  and any  $(\eta_n,T_n)$ . We assign  $\eta_{n - 1} = \eta_n, T_{n - 1} = T_n$  and select them suitably as in Haghifam et al. (2023)'s Eq.(3), which bounds the approximation errors by  $O(LD / \sqrt{n})$  at the same time:
+
+$$
+\begin{array}{l} \left| \mathbb {E} \left[ \hat {\mathcal {L}} _ {S _ {n - 1}} \left(w _ {S _ {n - 1}} ^ {*}\right) \right] - \mathbb {E} \left[ \hat {\mathcal {L}} _ {S _ {n - 1}} \left(\mathrm {G D} _ {\eta_ {n}, T _ {n}} \left(S _ {n - 1}\right)\right) \right] \right| \leq O \left(\frac {L D}{\sqrt {n}}\right), \\ \left| \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (w _ {S} ^ {*}) \right] - \mathbb {E} \left[ \hat {\mathcal {L}} _ {S _ {n}} (\mathrm {G D} _ {\eta_ {n}, T _ {n}} (S)) \right] \right| \leq O \left(\frac {L D}{\sqrt {n}}\right). \\ \end{array}
+$$
+
+We then need to relate  $\mathbb{E}\left[\hat{\mathcal{L}}_{S_{n - 1}}(\mathrm{GD}_{\eta_n,T_n}(S_{n - 1}))\right]$  and  $\mathbb{E}\left[\hat{\mathcal{L}}_{S_n}(\mathrm{GD}_{\eta_n,T_n}(S))\right]$ , which is the removal-based stability of GD. To this end, for each  $s_{n - 1}$ , we construct an artificial sample  $z_{S_{n - 1}}$  such that
+
+$$
+\ell (w, z _ {S _ {n - 1}}) := \hat {\mathcal {L}} _ {S _ {n - 1}} (w),
+$$
+
+and denote  $S_{n - 1}^{+} \coloneqq S_{n - 1} \cup \{z_{S_{n - 1}}\}$ . By construction, we have  $\hat{\mathcal{L}}_{S_{n - 1}^{+}}(w) = \hat{\mathcal{L}}_{S_{n - 1}}(w)$  for any  $w \in \mathcal{W}$ , i.e., the optimizations using  $S_{n - 1}$  and  $S_{n - 1}^{+}$  happen on the same loss landscape. Since GD relies on (sub-)gradients and thus only relies on loss landscape, we have  $\mathrm{GD}_{\eta_n,T_n}(S_{n - 1}) =$
+
+$\mathrm{GD}_{\eta_n,T_n}(S_{n - 1}^+)$ . If we pair  $S_{n - 1}$  and  $S_{n}$ , then  $S_{n - 1}^{+}$  and  $S_{n}$  only differs by one sample, allowing us to apply the replacement-based uniform stability for GD on CLB from Bassily et al. (2020):
+
+$$
+\begin{array}{l} \left| \mathbb {E} \left[ \hat {\mathcal {L}} _ {S _ {n - 1}} \left(\mathrm {G D} _ {\eta_ {n}, T _ {n}} \left(S _ {n - 1}\right)\right) \right] - \mathbb {E} \left[ \hat {\mathcal {L}} _ {S _ {n}} \left(\mathrm {G D} _ {\eta_ {n}, T _ {n}} (S)\right) \right] \right| \\ \leq \mathbb {E} _ {S _ {n - 1}} \left[ \mathbb {E} \left[ \left| \hat {\mathcal {L}} _ {S _ {n - 1}} \left(\mathrm {G D} _ {\eta_ {n}, T _ {n}} \left(S _ {n - 1}\right)\right) - \hat {\mathcal {L}} _ {S _ {n}} \left(\mathrm {G D} _ {\eta_ {n}, T _ {n}} (S)\right) \right| \mid S _ {n - 1} \right] \right] \\ \leq L \cdot \mathbb {E} _ {S _ {n - 1}} [ \mathbb {E} [ \| \mathbf {G D} _ {\eta_ {n}, T _ {n}} (S _ {n - 1}) - \mathbf {G D} _ {\eta_ {n}, T _ {n}} (S) \| | S _ {n - 1} ] ] \\ = L \cdot \mathbb {E} _ {S _ {n - 1}} \left[ \mathbb {E} \left[ \left\| \mathbf {G D} _ {\eta_ {n}, T _ {n}} \left(S _ {n - 1} ^ {+}\right) - \mathbf {G D} _ {\eta_ {n}, T _ {n}} (S) \right\| \mid S _ {n - 1} \right] \right] \\ \leq O \left(L ^ {2} \sqrt {T _ {n}} \eta_ {n} + \frac {L ^ {2} T _ {n} \eta_ {n}}{n}\right). \\ \end{array}
+$$
+
+With the same selection of  $(T_{n},\eta_{n})$ , the above difference can be bounded by  $O\left(\frac{LD}{\sqrt{n}}\right)$ .
+
+As a result, Eq. (D.7) can be bounded by
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \left| \hat {\mathcal {L}} _ {S} (\mathbb {E} [ W | Z _ {- i} ]) - \hat {\mathcal {L}} _ {S} (W) \right| \right] \\ \leq 2 \left(\mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (W) \right] - \frac {n - 1}{n} \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} \left(w _ {S} ^ {*}\right) \right]\right) + O \left(\frac {L D}{n}\right) + O \left(\frac {L D}{\sqrt {n}}\right) \\ \leq 2 \left(\mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (W) \right] - \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} \left(w _ {S} ^ {*}\right) \right]\right) + O \left(\frac {L D}{n}\right) + O \left(\frac {L D}{\sqrt {n}}\right) \\ \leq 2 \cdot O (\epsilon) + O \left(\frac {L D}{n}\right) + O \left(\frac {L D}{\sqrt {n}}\right). \\ \end{array}
+$$
+
+Plugging everything back to Eq. (D.6) finishes the proof.
+
+# D.6 EXTENSION TO SGD AND SMOOTH LOSSES
+
+In this subsection, we apply our technique to SGD under smooth losses. We will prove an omniscient information-theoretic bound and then show it recovers some existing stability-based bounds.
+
+To prove the omniscient bound, we need a basic form of information-theoretic bounds like Lemma 1 and Theorem 11 of Haghifam et al. (2023). This is done by results from Lemma D.2 to Corollary D.5. After that, we make the basic bound omniscient in Theorem 5. Finally, we bound the omniscient bound by the stability-based bound in Proposition 3.
+
+Lemma D.2 (2-Wasserstain Distance Generalization Bound under Smoothness) Assume for any sample  $z \in \mathcal{Z}$ ,  $\ell(\cdot, z)$  is non-negative, differentiable in  $\mathbb{R}^d$  and  $\beta$ -smooth, i.e., for any  $w, w' \in \mathbb{R}^d$
+
+$$
+\left\| \nabla \ell (w, z) - \nabla \ell \left(w ^ {\prime}, z\right) \right\| \leq \beta \| w - w ^ {\prime} \|.
+$$
+
+Then we have the following 2-Wasserstain-based information-theoretic (individual-sample) bound:
+
+$$
+\operatorname {g e n} \left(\mu^ {n}, P _ {W | S}\right) \leq \frac {\beta}{\gamma} \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (W) \right] + \frac {\beta + \gamma}{2 n} \sum_ {i = 1} ^ {n} \mathbb {E} _ {Z _ {i}} \left[ \mathbb {W} _ {2} \left(P _ {W | Z _ {i}}, P _ {W}\right) \right],
+$$
+
+where  $\gamma > 0$  is a constant, and  $\mathbb{W}_2(\cdot, \cdot)$  denotes 2-Wasserstein distance, the optimal transport under squared  $L_2$  norm.
+
+Proof Let constant  $\gamma > 0$ . Given any index  $i$  and any instance of the  $i$ -th training sample  $z_{i}$ , let  $\pi_{z_i}^{\epsilon}$  be the coupling that approximates the 2-Wasserstein distance  $\mathbb{W}_2(P_{W|Z_i = z_i}, P_W)$  between
+
+$P_{W|Z_i = z_i}$  and  $P_W$  by an error at most  $\epsilon > 0$ . Then for any  $\epsilon > 0$ , we have
+
+$$
+\begin{array}{l} \operatorname {g e n} \left(\mu^ {n}, P _ {W | S}\right) = \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} _ {\left(W, Z _ {i}\right), W ^ {\prime}} \left[ \ell \left(W ^ {\prime}, Z _ {i}\right) - \ell \left(W, Z _ {i}\right) \right] \\ = \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} _ {Z _ {i}} \left[ \mathbb {E} _ {(W, W ^ {\prime}) \sim \pi_ {Z _ {i}} ^ {\epsilon}} \left[ \ell (W ^ {\prime}, Z _ {i}) - \ell (W, Z _ {i}) \right] \right] \\ \leq \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} _ {Z _ {i}} \left[ \mathbb {E} _ {(W, W ^ {\prime}) \sim \pi_ {Z _ {i}} ^ {\epsilon}} \left[ (W ^ {\prime} - W) ^ {\top} \nabla \ell (W, Z _ {i}) + \frac {\beta}{2} \| W ^ {\prime} - W \| ^ {2} \right] \right]. \\ \end{array}
+$$
+
+We then follow Lei & Ying (2020) to handle the inner product as in their Appendix B:
+
+$$
+\begin{array}{l} \left(W ^ {\prime} - W\right) ^ {\top} \nabla \ell \left(W, Z _ {i}\right) \leq \left\| W ^ {\prime} - W \right\| \cdot \left\| \nabla \ell \left(W, Z _ {i}\right) \right\| \\ \leq \frac {\gamma}{2} \left\| W ^ {\prime} - W \right\| ^ {2} + \frac {1}{2 \gamma} \left\| \nabla \ell \left(W, Z _ {i}\right) \right\| ^ {2}. \\ \end{array}
+$$
+
+Thanks to the self-bounding property of positive smooth functions (Lemma A.1 of Lei & Ying (2020)), we have  $\| \nabla \ell(W, Z_i) \|^2 \leq 2\beta \cdot \ell(W, Z_i)$  and
+
+$$
+\left(W ^ {\prime} - W\right) ^ {\top} \nabla \ell \left(W, Z _ {i}\right) \leq \frac {\gamma}{2} \left\| W ^ {\prime} - W \right\| ^ {2} + \frac {\beta}{\gamma} \ell \left(W, Z _ {i}\right).
+$$
+
+Plugging this back leads to
+
+$$
+\begin{array}{l} \operatorname {g e n} (\mu^ {n}, P _ {W | S}) \leq \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} _ {Z _ {i}} \left[ \mathbb {E} _ {(W, W ^ {\prime}) \sim \pi_ {Z _ {i}} ^ {\epsilon}} \left[ \frac {\beta}{\gamma} \ell (W, Z _ {i}) + \frac {\beta + \gamma}{2} \| W ^ {\prime} - W \| ^ {2} \right] \right] \\ \leq \frac {\beta}{\gamma} \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (W) \right] + \frac {\beta + \gamma}{2 n} \sum_ {i = 1} ^ {n} \mathbb {E} _ {Z _ {i}} \left[ \mathbb {W} _ {2} \left(P _ {W | Z _ {i}}, P _ {W}\right) + \epsilon \right]. \\ \end{array}
+$$
+
+By arbitrariness of  $\epsilon > 0$ , we have
+
+$$
+\operatorname {g e n} \left(\mu^ {n}, P _ {W | S}\right) \leq \frac {\beta}{\gamma} \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (W) \right] + \frac {\beta + \gamma}{2 n} \sum_ {i = 1} ^ {n} \mathbb {E} _ {Z _ {i}} \left[ \mathbb {W} _ {2} \left(P _ {W | Z _ {i}}, P _ {W}\right) \right].
+$$
+
+Lemma D.3 For any random variables  $(X,Y,Z)$  such that  $Y$  is independent of  $Z$ , we have
+
+$$
+\mathbb {E} _ {Y} \left[ \mathbb {W} _ {2} \left(P _ {X | Y}, P _ {X}\right) \right] \leq \mathbb {E} _ {Z} \left[ \mathbb {E} _ {Y \sim P _ {Y | Z}} \left[ \mathbb {W} _ {2} \left(P _ {X | Y, Z}, P _ {X | Z}\right) \right] \right].
+$$
+
+Proof Let  $\pi_{y,z}^{\epsilon}$  be the coupling that approximates the 2-Wasserstein distance  $\mathbb{W}_2\left(P_{X|y,z},P_{X|y}\right)$  between  $P_{X|y,z}$  and  $P_{X|y}$  by an error at most  $\epsilon >0$ . Then for any  $\epsilon >0$ , we have
+
+$$
+\begin{array}{l} \mathbb {E} _ {Z} \left[ \mathbb {E} _ {Y \sim P _ {Y \mid Z}} \left[ \mathbb {W} _ {2} \left(P _ {X \mid Y, Z}, P _ {X \mid Z}\right) \right] \right] = \mathbb {E} _ {Y} \left[ \mathbb {E} _ {Z} \left[ \mathbb {W} _ {2} \left(P _ {X \mid Y, Z}, P _ {X \mid Z}\right) \right] \right] \quad (\text {I n d e p e n d e n c e b e t w e e n} Y, Z) \\ \geq \mathbb {E} _ {Y} \left[ \mathbb {E} _ {Z} \left[ \mathbb {E} _ {(X, X ^ {\prime}) \sim \pi_ {Y, Z} ^ {\epsilon}} \left[ \| X - X ^ {\prime} \| ^ {2} \right] - \epsilon \right] \right] \\ = \mathbb {E} _ {Y} \left[ \mathbb {E} _ {(Z, X, X ^ {\prime}) \sim P _ {Z} \circ \pi_ {Y, Z} ^ {\epsilon}} \left[ \| X - X ^ {\prime} \| ^ {2} \right] \right] - \epsilon \\ \geq \mathbb {E} _ {Y} \left[ \mathbb {W} _ {2} \left(P _ {X | Y}, P _ {X}\right) \right] - \epsilon . \\ \end{array}
+$$
+
+The lemma follows the arbitrariness of  $\epsilon > 0$ .
+
+Lemma D.2 is very similar to the fact that  $I(X;Y) \leq I(X;Y \mid Z)$  if  $Y$  is independent of  $Z$ . Following this similarity, we write (conditional) expected 2-Wasserstain distances similar to (conditional) MI, or equivalently, replace the KL-divergence in MI with the Wasserstain distance to compare the prior and posterior:
+
+Definition 3 For any random variables  $(X,Y,Z)$ , let
+
+$$
+I _ {\mathbb {W} _ {2}} (X; Y) := \mathbb {E} _ {Y} \left[ \mathbb {W} _ {2} \left(P _ {X | Y}, P _ {X}\right) \right],
+$$
+
+$$
+I _ {\mathbb {W} _ {2}} (X; Y \mid Z) := \mathbb {E} _ {Z} \left[ \mathbb {E} _ {Y \sim P _ {Y \mid Z}} \left[ \mathbb {W} _ {2} \left(P _ {X \mid Y, Z}, P _ {X \mid Z}\right) \right] \right].
+$$
+
+Lemma D.4 If  $X$  and  $Y$  are independent given  $Z$ , then  $I_{\mathbb{W}_2}(X;Y|Z) = 0$ .
+
+It leads to the following corollary:
+
+Corollary D.5 (Stability-Style 2-Wasserstain Generalization Bound) Under the same assumptions as Lemma D.2, we have
+
+$$
+\begin{array}{l} \operatorname {g e n} \left(\mu^ {n}, P _ {W | S}\right) \leq \frac {\beta}{\gamma} \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (W) \right] + \frac {\beta + \gamma}{2 n} \sum_ {i = 1} ^ {n} I _ {\mathbb {W} _ {2}} (W; Z _ {i}) \\ \leq \frac {\beta}{\gamma} \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (W) \right] + \frac {\beta + \gamma}{2 n} \sum_ {i = 1} ^ {n} I _ {\mathbb {W} _ {2}} (W; Z _ {i} \mid Z _ {- i}) \\ \end{array}
+$$
+
+Proof  $Z_{i}$  is independent of  $Z_{-i}$ .
+
+Theorem 5 (Omniscient 2-Wasserstain Bound under Smoothness) Assume  $\ell$  is non-negative, differentiable and  $\beta$ -smooth. Let  $\gamma_1 > 0, \gamma_2 > \beta$ . Let  $\{\Delta G^i\}_{i=1}^n$  be a family of omniscient (output-weight) perturbations, each of which additionally depends on  $Z_{-i}$ . Then we have
+
+$$
+\begin{array}{l} \operatorname {g e n} \left(\mu^ {n}, P _ {W | S}\right) \leq \frac {\beta}{\gamma_ {1}} \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \ell \left(W + \Delta G ^ {i}, Z _ {i}\right) \right] + \frac {\beta + \gamma_ {1}}{2 n} \sum_ {i = 1} ^ {n} I _ {\mathbb {W} _ {2}} \left(W + \Delta G ^ {i}; Z _ {i} \mid Z _ {- i}\right) \\ + \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \Delta_ {\Delta G ^ {i}} \left(W, Z _ {i}\right) - \Delta_ {\Delta G ^ {i}} \left(W, Z ^ {\prime}\right) \right], \\ \end{array}
+$$
+
+and
+
+$$
+\begin{array}{l} \operatorname {g e n} \left(\mu^ {n}, P _ {W | S}\right) \leq \frac {1}{1 - \frac {\beta}{\gamma_ {2}}} \left(\frac {\beta}{\gamma_ {1} n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \ell \left(W + \Delta G ^ {i}, Z _ {i}\right) \right] + \frac {\beta + \gamma_ {1}}{2 n} \sum_ {i = 1} ^ {n} I _ {\mathbb {W} _ {2}} \left(W + \Delta G ^ {i}; Z _ {i} \mid Z _ {- i}\right) \right. \\ \left. + \frac {2 \beta}{\gamma_ {2}} \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (W) \right] + \frac {\beta + \gamma_ {2}}{n} \sum_ {i = 1} ^ {n} \| \Delta G ^ {i} \| ^ {2}\right). \tag {D.8} \\ \end{array}
+$$
+
+Proof This proof is very similar to the proof of Theorem 3. By repeating the proof of Corollary D.1 but with Corollary D.5 instead of Lemma 1, we obtain the first inequality in the theorem statement:
+
+$$
+\begin{array}{l} \operatorname {g e n} \left(\mu^ {n}, P _ {W | S}\right) \leq \frac {\beta}{\gamma_ {1}} \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \ell \left(W + \Delta G ^ {i}, Z _ {i}\right) \right] + \frac {\beta + \gamma_ {1}}{2 n} \sum_ {i = 1} ^ {n} I _ {\mathbb {W} _ {2}} \left(W + \Delta G ^ {i}; Z _ {i} \mid Z _ {- i}\right) \\ + \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \Delta_ {\Delta G ^ {i}} \left(W, Z _ {i}\right) - \Delta_ {\Delta G ^ {i}} \left(W, Z ^ {\prime}\right) \right]. \\ \end{array}
+$$
+
+The penalty terms can be bounded the same way as in Lemma D.2:
+
+$$
+\begin{array}{l} | \Delta_ {\Delta G ^ {i}} (W, z) | \leq \left| (\Delta G ^ {i}) ^ {\top} \nabla \ell (W, z) \right| + \frac {\beta}{2} \left\| \Delta G ^ {i} \right\| ^ {2} \\ \leq \left\| \Delta G ^ {i} \right\| \left\| \nabla \ell (W, z) \right\| + \frac {\beta}{2} \left\| \Delta G ^ {i} \right\| ^ {2} \\ \leq \frac {\gamma_ {2}}{2} \left\| \Delta G ^ {i} \right\| ^ {2} + \frac {\beta}{\gamma_ {2}} \ell (W, z) + \frac {\beta}{2} \left\| \Delta G ^ {i} \right\| ^ {2} \\ = \frac {\beta}{\gamma_ {2}} \ell (W, z) + \frac {\beta + \gamma_ {2}}{2} \| \Delta G ^ {i} \| ^ {2}. \\ \end{array}
+$$
+
+Therefore, we have
+
+$$
+\begin{array}{l} \operatorname {g e n} \left(\mu^ {n}, P _ {W | S}\right) \leq \frac {\beta}{\gamma_ {1}} \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \ell \left(W + \Delta G ^ {i}, Z _ {i}\right) \right] + \frac {\beta + \gamma_ {1}}{2 n} \sum_ {i = 1} ^ {n} I _ {\mathbb {W} _ {2}} \left(W + \Delta G ^ {i}; Z _ {i} \mid Z _ {- i}\right) \\ + \frac {\beta}{\gamma_ {2}} \left(\mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (W) \right] + \mathbb {E} \left[ \mathcal {L} _ {\mu} (W) \right]\right) + (\beta + \gamma_ {2}) \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \left\| \Delta G ^ {i} \right\| ^ {2} \right]. \\ \end{array}
+$$
+
+Population loss  $\mathbb{E}\left[\mathcal{L}_{\mu}(W)\right]$  appears at the right of the inequality. To move it to the left, we pair it with a virtual empirical loss term and moving the consequent  $\mathrm{gen}(\mu^n,P_{W|S})$  to the left:
+
+$$
+\begin{array}{l} \operatorname {g e n} \left(\mu^ {n}, P _ {W | S}\right) \leq \frac {\beta}{\gamma_ {1}} \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \ell \left(W + \Delta G ^ {i}, Z _ {i}\right) \right] + \frac {\beta + \gamma_ {1}}{2 n} \sum_ {i = 1} ^ {n} I _ {\mathbb {W} _ {2}} \left(W + \Delta G ^ {i}; Z _ {i} \mid Z _ {- i}\right) \\ + \frac {\beta}{\gamma_ {2}} \left(2 \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (W) \right] + \mathrm {g e n} (\mu^ {n}, P _ {W | S})\right) + \frac {\beta + \gamma_ {2}}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \left\| \Delta G ^ {i} \right\| ^ {2} \right], \\ \end{array}
+$$
+
+$$
+\begin{array}{l} (1 - \frac {\beta}{\gamma_ {2}}) \operatorname {g e n} (\mu^ {n}, P _ {W | S}) \leq \frac {\beta}{\gamma_ {1}} \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \ell (W + \Delta G ^ {i}, Z _ {i}) \right] + \frac {\beta + \gamma_ {1}}{2 n} \sum_ {i = 1} ^ {n} I _ {\mathbb {W} _ {2}} (W + \Delta G ^ {i}; Z _ {i} \mid Z _ {- i}) \\ + \frac {2 \beta}{\gamma_ {2}} \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (W) \right] + \frac {\beta + \gamma_ {2}}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \left\| \Delta G ^ {i} \right\| ^ {2} \right]. \\ \end{array}
+$$
+
+Restricting  $\gamma_{2} > \beta$  allows us to divide the inequality by  $1 - \frac{\beta}{\gamma_2}$  without changing the direction of the inequality:
+
+$$
+\begin{array}{l} \operatorname {g e n} \left(\mu^ {n}, P _ {W | S}\right) \leq \frac {1}{1 - \frac {\beta}{\gamma_ {2}}} \left(\frac {\beta}{\gamma_ {1} n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \ell \left(W + \Delta G ^ {i}, Z _ {i}\right) \right] + \frac {\beta + \gamma_ {1}}{2 n} \sum_ {i = 1} ^ {n} I _ {\mathbb {W} _ {2}} \left(W + \Delta G ^ {i}; Z _ {i} \mid Z _ {- i}\right) \right. \\ \left. + \frac {2 \beta}{\gamma_ {2}} \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (W) \right] + \frac {\beta + \gamma_ {2}}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \left\| \Delta G ^ {i} \right\| ^ {2} \right]\right). \\ \end{array}
+$$
+
+Now that we have proved the omniscient bound for smooth losses, we turn to recovering some existing stability-based bounds.
+
+Proposition 3 Under the same setting as Theorem 5, we have
+
+$$
+\begin{array}{l} \operatorname {g e n} \left(\mu^ {n}, P _ {W | S}\right) \leq \inf  _ {\left\{\Delta G ^ {i} \right\}, \gamma_ {1} > 0, \gamma_ {2} > \beta} [ R H S o f E q. (D. 8) ] \\ \leq \inf _ {\gamma_ {2} > \beta} \frac {1}{1 - \frac {\beta}{\gamma_ {2}}} \left(\frac {2 \beta}{\gamma_ {2}} \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (W) \right] + \frac {\beta + \gamma_ {2}}{2} \cdot \epsilon^ {\mathrm {s t a b i l i t y}}\right), \\ \end{array}
+$$
+
+where
+
+$$
+\epsilon^ {\mathrm {s t a b i l i t y}} := \mathbb {E} _ {Z _ {- i}, V} \left[ \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \| W ^ {\prime} - W \| ^ {2} \mid Z _ {- i}, V \right] \right]
+$$
+
+(rephrased in our notation) is exactly the  $\ell_2$  on-average model stability in Definition 4 of Lei & Ying (2020).
+
+Remark D.7 Proposition 3 recovers the relationship between stability and generalization in Lei & Ying (2020)'s Theorem 2(b) up to constants.
+
+Proof After setting  $\Delta G^{i} = -W + \mathbb{E}\left[W\mid Z_{-i},V\right]$  as a function of  $Z_{-i}$  and  $V$ , we have  $W + \Delta G^{i} = \mathbb{E}\left[W\mid Z_{-i},V\right]$ , which is a function of  $Z_{i}$ -independent  $Z_{-i}$  and  $V$ . As a result,  $W + \Delta G^{i}$  is
+
+independent of  $Z_{i}$  and  $I_{\mathbb{W}_2}(W + \Delta G^i;Z_i\mid Z_{-i}) = 0$  according to Lemma D.4. Therefore, we have
+
+$$
+\begin{array}{l} \operatorname {g e n} \left(\mu^ {n}, P _ {W | S}\right) \leq \inf  _ {\left\{\Delta G ^ {i} \right\}, \gamma_ {1} > 0, \gamma_ {2} > 0} [ \mathrm {R H S} \text {o f E q . (D . 8)} ] \\ \leq \inf  _ {\gamma_ {1} > 0, \gamma_ {2} > \beta} \frac {1}{1 - \frac {\beta}{\gamma_ {2}}} \left(\frac {\beta}{\gamma_ {1} n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \ell (W + \Delta G ^ {i}, Z _ {i}) \right] + \frac {2 \beta}{\gamma_ {2}} \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (W) \right] + \frac {\beta + \gamma_ {2}}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \| \Delta G ^ {i} \| ^ {2} \right]\right) \\ \leq \inf  _ {\gamma_ {2} > \beta} \frac {1}{1 - \frac {\beta}{\gamma_ {2}}} \left(\frac {2 \beta}{\gamma_ {2}} \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (W) \right] + \frac {\beta + \gamma_ {2}}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \| \mathbb {E} [ W | Z _ {- i}, V ] - W \| ^ {2} \right]\right) (\gamma_ {1} \rightarrow + \infty) \\ \end{array}
+$$
+
+With a closer look, one can find on-average model stability (Lei & Ying, 2020) term at RHS:
+
+$$
+\begin{array}{l} \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \| \mathbb {E} \left[ W \mid Z _ {- i}, V \right] - W \| ^ {2} \right] = \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} _ {Z _ {- i}, V} \left[ \mathbb {E} \left[ \| \mathbb {E} \left[ W ^ {\prime} \mid Z _ {- i}, V \right] - W \| ^ {2} \mid Z _ {- i}, V \right] \right] \\ = \frac {1}{2 n} \sum_ {i = 1} ^ {n} \mathbb {E} _ {Z _ {- i}, V} \left[ \mathbb {E} \left[ \| W ^ {\prime} - W \| ^ {2} \mid Z _ {- i}, V \right] \right] = \frac {1}{2} \underbrace {\mathbb {E} _ {S , S ^ {\prime} , V} \left[ \frac {1}{n} \sum_ {i = 1} ^ {n} \left\| \mathcal {A} (S , V) - \mathcal {A} (S ^ {(i)} , V) \right\| ^ {2} \right]} _ {\ell_ {2} \text {o n - a v e r a g e m o d e l s t a b i l i t y i n D e f i n i t i o n 4 o f L e i & Y i n g (2 0 2 0)}} \\ \end{array}
+$$
+
+where the second step follows Lemma A.2,  $\mathcal{A}(\cdot ,v)$  denotes the SGD when the random seed is  $v$  and  $S^{(i)}$  means replacing the  $i$ -th sample of  $S$  with the  $i$ -th sample from  $S^{\prime}$ .
+
+![](images/ed2257f68d22ea03da9eeec525317acd7c3d3efe9d991b75573b592c71f91302.jpg)
+
+Now that we have recovered stability arguments, we can directly borrow stability of SGD to derive excess risk bounds. The following results are based on the on-average model stability bound derived by Lei & Ying (2020).
+
+Proposition 4 Assume the loss is non-negative, convex and  $\beta$ -smooth. Assume the training algorithm is projected SGD that starts from a fixed  $W_0 \coloneqq w_0 \in \mathcal{W}$  and runs  $T$  steps with non-increasing step sizes  $\{\eta_t\}_{t=1}^{T+1}$  such that  $\eta_t \leq 1/2\beta$ . Then for any  $\gamma > \beta$ , we have the following excess risk bound:
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \mathcal {L} _ {\mu} \left(W ^ {\mathrm {a c c}}\right) - \mathcal {L} _ {\mu} \left(w ^ {*}\right) \right] \\ \leq \frac {2 \beta}{\gamma_ {2} - \beta} \mathcal {L} _ {\mu} (w ^ {*}) + \frac {1 + T / n}{n} \frac {4 e \gamma_ {2} \beta (\beta + \gamma_ {2})}{\gamma_ {2} - \beta} \left(\eta_ {1} \| w ^ {*} \| ^ {2} + 2 \sum_ {t = 0} ^ {T} \eta_ {t + 1} \left(\sum_ {\tau = 0} ^ {t - 1} \eta_ {\tau + 1} ^ {2} \mathcal {L} _ {\mu} (w ^ {*})\right) / \sum_ {\tau = 0} ^ {T} \eta_ {\tau + 1}\right) \\ + \frac {\gamma_ {2} + \beta}{\gamma_ {2} - \beta} \left(\left(1 / 2 + \beta \eta_ {1}\right) \| w ^ {*} \| ^ {2} + 2 \beta \sum_ {t = 0} ^ {T} \eta_ {t + 1} ^ {2} \mathcal {L} _ {\mu} (w ^ {*})\right) / \sum_ {\tau = 0} ^ {T} \eta_ {\tau + 1}, \\ \end{array}
+$$
+
+where  $W^{\mathrm{acc}}$  is the accumulated weight
+
+$$
+W ^ {\mathrm {a c c}} := \frac {\sum_ {t = 0} ^ {T} \eta_ {t + 1} W _ {t}}{\sum_ {t = 0} ^ {T} \eta_ {t + 1}}.
+$$
+
+Remark D.8 In separable settings, i.e., when  $\mathcal{L}_{\mu}(w^{*}) = 0$ , the excess risk bound simplifies to
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \mathcal {L} _ {\mu} \left(W ^ {\mathrm {a c c}}\right) - \mathcal {L} _ {\mu} \left(w ^ {*}\right) \right] \\ \leq \frac {1 + T / n}{n} \frac {4 e \gamma_ {2} \beta (\beta + \gamma_ {2})}{\gamma_ {2} - \beta} \left(\eta_ {1} \| w ^ {*} \| ^ {2}\right) + \frac {\gamma_ {2} + \beta}{(\gamma_ {2} - \beta) \sum_ {\tau = 0} ^ {T} \eta_ {\tau + 1}} \left((1 / 2 + \beta \eta_ {1}) \| w ^ {*} \| ^ {2}\right). \\ \end{array}
+$$
+
+After setting  $\eta_t = \eta \leq 1 / 2\beta$  and reparameterizing  $\gamma_2 = k\beta$  for  $k > 1$ , we have
+
+$$
+\mathbb {E} \left[ \mathcal {L} _ {\mu} (W ^ {\mathrm {a c c}}) - \mathcal {L} _ {\mu} (w ^ {*}) \right] \leq \left(\frac {4 e k \beta^ {2} \eta}{n} + \frac {T}{n ^ {2}} \cdot 4 e k \beta^ {2} \eta + \frac {1 / 2 + \beta \eta}{T \eta}\right) \left(\frac {k + 1}{k - 1} \| w ^ {*} \| ^ {2}\right).
+$$
+
+By minimizing over  $T$ , we can obtain the following:
+
+$$
+\mathbb {E} \left[ \mathcal {L} _ {\mu} (W ^ {\mathrm {a c c}}) - \mathcal {L} _ {\mu} (w ^ {*}) \right] \leq \left(\frac {4 e k \beta^ {2} \eta}{n} + \frac {2}{n} \cdot \sqrt {4 e k \beta^ {2} \cdot (1 / 2 + \beta \eta)}\right) \left(\frac {k + 1}{k - 1} \| w ^ {*} \| ^ {2}\right).
+$$
+
+Now set  $\eta = 1 / 2\beta$  to obtain the following:
+
+$$
+\mathbb {E} \left[ \mathcal {L} _ {\mu} \left(W ^ {\mathrm {a c c}}\right) - \mathcal {L} _ {\mu} \left(w ^ {*}\right) \right] \leq \inf  _ {k > 1} \frac {2 \beta}{n} \left(e k + 2 \sqrt {e k}\right) \left(\frac {k + 1}{k - 1} \| w ^ {*} \| ^ {2}\right) = O (\beta \| w ^ {*} \| ^ {2} / n),
+$$
+
+which indicates an  $O(1/n)$  sample complexity for smooth, convex and separable settings. This result recovers the Theorem 5 and the  $O(1/n)$  rate in Lei & Ying (2020) up to constants.
+
+Proof This proof is adapted from Appendix C.2 of Lei & Ying (2020). The excess risk can be decomposed into (excess) optimization error and generalization error. The optimization error bound is directly borrowed from Lei & Ying (2020). The generalization error is bounded by combining the recovered stability bound Proposition 3 and the stability of SGD from Lei & Ying (2020).
+
+Let  $w^{*}$  be the weight that achieves the optimal population loss.
+
+# D.6.1 EXCESS OPTIMIZATION ERROR
+
+According to Lemma A.2(c) of Lei & Ying (2020), if the loss is non-negative, convex and  $\beta$ -smooth, and  $\eta_t \leq 1/2L$  and non-increasing, then for any constant  $\bar{w}$  and constant  $s$ , one has
+
+$$
+\sum_ {\tau = 0} ^ {t} \eta_ {\tau + 1} \mathbb {E} \left[ \hat {\mathcal {L}} _ {s} (W _ {\tau}) - \hat {\mathcal {L}} _ {s} (\bar {w}) \mid S = s \right] \leq \left(1 / 2 + \beta \eta_ {1}\right) \| \bar {w} \| ^ {2} + 2 \beta \sum_ {\tau = 0} ^ {t} \eta_ {\tau + 1} ^ {2} \hat {\mathcal {L}} _ {s} (\bar {w}).
+$$
+
+After setting  $\bar{w}$  to  $w^{*}$  and taking expectation over training sets, we have
+
+$$
+\sum_ {\tau = 0} ^ {t} \eta_ {\tau + 1} \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} \left(W _ {\tau}\right) - \hat {\mathcal {L}} _ {S} \left(w ^ {*}\right) \right] \leq (1 / 2 + \beta \eta_ {1}) \| w ^ {*} \| ^ {2} + 2 \beta \sum_ {\tau = 0} ^ {t} \eta_ {\tau + 1} ^ {2} \mathcal {L} _ {\mu} \left(w ^ {*}\right). \tag {D.9}
+$$
+
+Since the excess training error bound is only given after summing over steps, one has to sum the generalization error bound over steps as well.
+
+# D.6.2 STABILITY AND GENERALIZATION ERROR
+
+Theorem 3 of Lei & Ying (2020) states that if the loss is non-negative, convex and  $\beta$ -smooth, and SGD has step size  $\eta_t \leq 2 / L$ , the for any  $p > 0$  one has
+
+$$
+\epsilon_ {t} ^ {\mathrm {s t a b i l i t y}} \leq \frac {8 (1 + 1 / p) \beta}{n} \sum_ {\tau = 0} ^ {t - 1} (1 + p / n) ^ {t - 1 - \tau} \eta_ {\tau + 1} ^ {2} \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (W _ {\tau}) \right],
+$$
+
+where  $\epsilon_t^{\mathrm{stability}}$  is the  $\ell_2$  on-average model stability at step  $t$ :
+
+$$
+\epsilon_ {t} ^ {\mathrm {s t a b i l i t y}} := \mathbb {E} _ {Z _ {- i}, V} \left[ \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} \left[ \left\| W _ {t} ^ {\prime} - W _ {t} \right\| ^ {2} \mid Z _ {- i}, V \right] \right]
+$$
+
+Let  $\gamma_{2} > 0$  be a constant. Plugging this stability bound into Proposition 3 leads to
+
+$$
+\begin{array}{l} \operatorname {g e n} \left(\mu^ {n}, P _ {W _ {t} | S}\right) \\ \leq \frac {1}{1 - \beta / \gamma_ {2}} \left(\frac {2 \beta}{\gamma_ {2}} \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (W _ {t}) \right] + \frac {\beta + \gamma_ {2}}{2} \epsilon_ {t} ^ {\mathrm {s t a b i l i t y}}\right) \\ \leq \frac {1}{1 - \beta / \gamma_ {2}} \left(\frac {2 \beta}{\gamma_ {2}} \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (W _ {t}) \right] + \frac {\beta + \gamma_ {2}}{2} \frac {8 (1 + 1 / p) \beta}{n} \sum_ {\tau = 0} ^ {t - 1} (1 + p / n) ^ {t - 1 - \tau} \eta_ {\tau + 1} ^ {2} \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (W _ {\tau}) \right]\right) \\ \leq \frac {1}{1 - \beta / \gamma_ {2}} \left(\frac {2 \beta}{\gamma_ {2}} \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (W _ {t}) \right] + \frac {4 (1 + 1 / p) (\beta + \gamma_ {2}) \beta (1 + p / n) ^ {t - 1}}{n} \sum_ {\tau = 0} ^ {t - 1} \eta_ {\tau + 1} ^ {2} \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (W _ {\tau}) \right]\right) \\ \end{array}
+$$
+
+There are empirical losses on the trajectory at RHS of the above inequality. They can be bounded by Eq. (A.5) of Lei & Ying (2020), which states given any training set  $s$  and any constant  $\bar{w} \in \mathcal{W}$ , one has
+
+$$
+\sum_ {\tau = 0} ^ {t - 1} \eta_ {\tau + 1} ^ {2} \mathbb {E} \left[ \hat {\mathcal {L}} _ {s} (W _ {\tau}) \mid S = s \right] \leq \eta_ {1} \| \bar {w} \| ^ {2} + 2 \sum_ {\tau = 0} ^ {t - 1} \eta_ {\tau + 1} ^ {2} \hat {\mathcal {L}} _ {s} (\bar {w}).
+$$
+
+Setting  $\bar{w}$  to  $w^{*}$  and taking expectation over training sets, we have
+
+$$
+\sum_ {\tau = 0} ^ {t - 1} \eta_ {\tau + 1} ^ {2} \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (W _ {\tau}) \right] \leq \eta_ {1} \| w ^ {*} \| ^ {2} + 2 \sum_ {\tau = 0} ^ {t - 1} \eta_ {\tau + 1} ^ {2} \mathcal {L} _ {\mu} (w ^ {*}).
+$$
+
+Plugging it back leads to
+
+$$
+\mathrm {g e n} (\mu^ {n}, P _ {W _ {t} | S}) \leq \frac {1}{1 - \beta / \gamma_ {2}} \left(\frac {2 \beta}{\gamma_ {2}} \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (W _ {t}) \right] + \frac {4 (1 + 1 / p) (\beta + \gamma_ {2}) \beta (1 + p / n) ^ {t - 1}}{n} \left(\eta_ {1} \| w ^ {*} \| ^ {2} + 2 \sum_ {\tau = 0} ^ {t - 1} \eta_ {\tau + 1} ^ {2} \mathcal {L} _ {\mu} (w ^ {*})\right)\right)
+$$
+
+As in Lei & Ying (2020), one can choose  $p = n / T$  to have  $(1 + p / n)^{t - 1} \leq (1 + p / n)^{T - 1} = (1 + 1 / T)^{T - 1} < e$ . As a result, we have
+
+$$
+\operatorname {g e n} (\mu^ {n}, P _ {W _ {t} | S}) \leq \frac {1}{1 - \beta / \gamma_ {2}} \left(\frac {2 \beta}{\gamma_ {2}} \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (W _ {t}) \right] + \frac {4 (1 + T / n) (\beta + \gamma_ {2}) \beta e}{n} \left(\eta_ {1} \| w ^ {*} \| ^ {2} + 2 \sum_ {\tau = 0} ^ {t - 1} \eta_ {\tau + 1} ^ {2} \mathcal {L} _ {\mu} (w ^ {*})\right)\right)
+$$
+
+To align the generalization error bounds with the weighted summation form of the optimization error, we weight them by  $\eta_{t + 1}$  and sum over steps the above inequality:
+
+$$
+\begin{array}{l} \sum_ {t = 0} ^ {T} \eta_ {t + 1} \operatorname {g e n} \left(\mu^ {n}, P _ {W _ {t} | S}\right) \\ \leq \sum_ {t = 0} ^ {T} \frac {\eta_ {t + 1}}{1 - \beta / \gamma_ {2}} \left(\frac {2 \beta}{\gamma_ {2}} \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (W _ {t}) \right] + \frac {4 (1 + T / n) (\beta + \gamma_ {2}) \beta e}{n} \left(\eta_ {1} \| w ^ {*} \| ^ {2} + 2 \sum_ {\tau = 0} ^ {t - 1} \eta_ {\tau + 1} ^ {2} \mathcal {L} _ {\mu} (w ^ {*})\right)\right) \\ = \frac {1}{1 - \beta / \gamma_ {2}} \left(\frac {2 \beta}{\gamma_ {2}} \sum_ {t = 0} ^ {T} \eta_ {t + 1} \mathbb {E} \left[ \hat {\mathcal {L}} _ {S} (W _ {t}) \right] + \frac {4 (1 + T / n) (\beta + \gamma_ {2}) \beta e}{n} \sum_ {t = 0} ^ {T} \eta_ {t + 1} \left(\eta_ {1} \| w ^ {*} \| ^ {2} + 2 \sum_ {\tau = 0} ^ {t - 1} \eta_ {\tau + 1} ^ {2} \mathcal {L} _ {\mu} (w ^ {*})\right)\right) \\ \end{array}
+$$
+
+To get rid of the empirical losses  $\sum_{t=0}^{T} \eta_{t+1} \mathbb{E}\left[\hat{\mathcal{L}}_S(W_t)\right]$ , we apply Eq. (D.9) again and obtain
+
+$$
+\begin{array}{l} \sum_ {t = 0} ^ {T} \eta_ {t + 1} \operatorname {g e n} \left(\mu^ {n}, P _ {W _ {t} | S}\right) \\ \leq \frac {1}{1 - \beta / \gamma_ {2}} \left(\frac {2 \beta}{\gamma_ {2}} \left((1 / 2 + \beta \eta_ {1}) \| w ^ {*} \| ^ {2} + 2 \beta \sum_ {t = 0} ^ {T} \eta_ {t + 1} ^ {2} \mathcal {L} _ {\mu} (w ^ {*}) + \sum_ {t = 0} ^ {T} \eta_ {t + 1} \mathcal {L} _ {\mu} (w ^ {*})\right) \right. \\ \left. + \frac {4 (1 + T / n) (\beta + \gamma_ {2}) \beta e}{n} \sum_ {t = 0} ^ {T} \eta_ {t + 1} \left(\eta_ {1} \| w ^ {*} \| ^ {2} + 2 \sum_ {\tau = 0} ^ {t - 1} \eta_ {\tau + 1} ^ {2} \mathcal {L} _ {\mu} (w ^ {*})\right)\right). \\ \end{array}
+$$
+
+After obtaining the above generalization error bound, the bound for excess risks can be obtained by summing up it and the optimization error bound Eq. (D.9):
+
+$$
+\begin{array}{l} \sum_ {t = 0} ^ {T} \eta_ {t + 1} \mathbb {E} \left[ \mathcal {L} _ {\mu} \left(W _ {t}\right) - \mathcal {L} _ {\mu} \left(w ^ {*}\right) \right] \\ \leq \frac {1}{1 - \beta / \gamma_ {2}} \frac {2 \beta}{\gamma_ {2}} \left(\left(1 / 2 + \beta \eta_ {1}\right) \| w ^ {*} \| ^ {2} + 2 \beta \sum_ {t = 0} ^ {T} \eta_ {t + 1} ^ {2} \mathcal {L} _ {\mu} (w ^ {*}) + \sum_ {t = 0} ^ {T} \eta_ {t + 1} \mathcal {L} _ {\mu} (w ^ {*})\right) \\ + \frac {1}{1 - \beta / \gamma_ {2}} \frac {4 (1 + T / n) (\beta + \gamma_ {2}) \beta e}{n} \sum_ {t = 0} ^ {T} \eta_ {t + 1} \left(\eta_ {1} \| w ^ {*} \| ^ {2} + 2 \sum_ {\tau = 0} ^ {t - 1} \eta_ {\tau + 1} ^ {2} \mathcal {L} _ {\mu} (w ^ {*})\right) \\ + \left(1 / 2 + \beta \eta_ {1}\right) \| w ^ {*} \| ^ {2} + 2 \beta \sum_ {t = 0} ^ {T} \eta_ {t + 1} ^ {2} \mathcal {L} _ {\mu} (w ^ {*}) \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \leq \frac {2 \beta}{\gamma_ {2} - \beta} \sum_ {t = 0} ^ {T} \eta_ {t + 1} \mathcal {L} _ {\mu} (w ^ {*}) + \frac {1 + T / n}{n} \frac {4 e \gamma_ {2} \beta (\beta + \gamma_ {2})}{\gamma_ {2} - \beta} \sum_ {t = 0} ^ {T} \eta_ {t + 1} \left(\eta_ {1} \| w ^ {*} \| ^ {2} + 2 \sum_ {\tau = 0} ^ {t - 1} \eta_ {\tau + 1} ^ {2} \mathcal {L} _ {\mu} (w ^ {*})\right) \\ + \frac {\gamma_ {2} + \beta}{\gamma_ {2} - \beta} \left(\left(1 / 2 + \beta \eta_ {1}\right) \| w ^ {*} \| ^ {2} + 2 \beta \sum_ {t = 0} ^ {T} \eta_ {t + 1} ^ {2} \mathcal {L} _ {\mu} (w ^ {*})\right) \\ \end{array}
+$$
+
+The stated inequality can be obtained by dividing  $\sum_{t=0}^{T} \eta_{t+1}$  and applying the Jensen's inequality to the convex  $\hat{\mathcal{L}}_S(\cdot)$ .
+
+# E DISCUSSION ON THE LIMITATION
+
+The major limitation of our bound is that our bound still relies on population gradients and Hessians. This limitation harms the applicability of our bound to self-certified algorithms (Pérez-Ortiz et al., 2021).
+
+However, the limitation is not unique to our bound, but is an inherent limitation of the auxiliary trajectory technique. The most essential step of this technique is to switch from the original trajectory to the auxiliary trajectory with better properties. However, one must relate the auxiliary trajectory back to the original trajectory by adding their differences into the bound. In this process, the loss differences are used to measure such differences, resulting in the population loss difference and population statistics. As a result, previous representative works all have explicit reliance on population statistics. See  $S'$  in Propositions 1 and 2. This reliance can be alleviated through some assumptions like  $\mathcal{L}_{\mu}(W_T) \leq \mathbb{E}_{\xi \sim \mathcal{N}(0,\sigma I)}[\mathcal{L}_{\mu}(W_T + \xi)]$  of Wang & Mao (2022). Nevertheless, one still must verify this assumptions on the population set to rigorously apply them, especially when the model is under-fitted or the generalization is bad so that the output weight is far from local minima of the population loss. To sum up, existing representative results based on auxiliary trajectory must rely on population statistics at least implicitly.
+
+In terms of dependence to population/validation statistics, we also optimize the omniscient trajectory using validation statistics, which is a heavier dependence. This may forms unfair comparison with existing bounds. However, we have tried to make the comparison fair by allowing the existing bounds to rely on the validation statistics (see Appendix C.5). Even with full access to validation sets, the existing bounds cannot exploit them and are still much numerically looser than ours. Lastly, the results of the existing and our bounds can be seen as not only competitors, but also different trade-offs between the dependence on validation set and bound tightness. Our bound demonstrates how tight a bound can be if one allows heavy dependence on validation sets, while previous works show the looseness when one controls the access to validation sets. Future works can start from these two extremes to achieve a better trade-off or even break it.
\ No newline at end of file
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+# LEVERAGING SUB-OPTIMAL DATA FOR HUMAN-IN-THE-LOOP REINFORCEMENT LEARNING
+
+Calarina Muslimani
+
+University of Alberta
+
+musliman@ualberta.ca
+
+Matthew E. Taylor
+
+University of Alberta
+
+Alberta Machine Intelligence Institute
+
+matthew.e.taylor@ualberta.ca
+
+# ABSTRACT
+
+To create useful reinforcement learning (RL) agents, step zero is to design a suitable reward function that captures the nuances of the task. However, reward engineering can be a difficult and time-consuming process. Instead, human-in-the-loop RL methods hold the promise of learning reward functions from human feedback. Despite recent successes, many of the human-in-the-loop RL methods still require numerous human interactions to learn successful reward functions. To improve the feedback efficiency of human-in-the-loop RL methods (i.e., require less human interaction), this paper introduces Sub-optimal Data Pre-training, SDP, an approach that leverages reward-free, sub-optimal data to improve scalar- and preference-based RL algorithms. In SDP, we start by pseudo-labeling all low-quality data with the minimum environment reward. Through this process, we obtain reward labels to pre-train our reward model without requiring human labeling or preferences. This pre-training phase provides the reward model a head start in learning, enabling it to recognize that low-quality transitions should be assigned low rewards. Through extensive experiments with both simulated and human teachers, we find that SDP can at least meet, but often significantly improve, state of the art human-in-the-loop RL performance across a variety of simulated robotic tasks.
+
+# 1 INTRODUCTION
+
+In reinforcement learning (RL), an agent's objective is to interact with an environment and maximize its total (discounted) expected reward. The reward hypothesis further maintains that a well-specified reward function is sufficient for an agent to learn to solve a task (Sutton & Barto, 2018). However, defining a reward function that precisely captures all task complexities is often tedious and nontrivial (Booth et al., 2023). There have been notable examples of reward misspecification, in which RL agents discovered and exploited unintended shortcuts in the reward function (Skalse et al., 2022). One notorious example is the CoastRunners game, in which the goal should be to finish a boat race as fast as possible — an RL agent instead gained the most reward by spinning its boat in a circle despite concurrently catching on fire and crashing into other boats (Clark & Amodei, 2016).
+
+A promising alternative is to learn reward functions directly from human feedback. In this paradigm, humans can provide feedback in the form of preferences or scalar signals, which can then be used to learn a reward function that is consistent with human desires (Daniel et al., 2014; Christiano et al., 2017). Despite recent progress, existing preference- and scalar-based RL methods still suffer from high human labeling costs that can require thousands of human queries to learn an adequate reward function (Christiano et al., 2017). Prior work attempts to mitigate this issue through several mechanisms, including active learning (Lee et al., 2021a), data augmentation (Park et al., 2022), semi-supervised learning (Park et al., 2022), and meta-learning (Hejna III & Sadigh, 2023).
+
+Alternatively, our work draws on recent advances in offline RL that have demonstrated the value of low-quality data (Yu et al., 2021). However, its potential in human-in-the-loop RL remains unexplored. As low-quality data is often readily accessible or easy to obtain, this work addresses this gap by asking the question:
+
+# Can we leverage sub-optimal, unlabeled data to improve learning in human-in-the-loop RL methods?
+
+To that end, we present Sub-optimal Data Pre-training, SDP, a tool for human-in-the-loop RL algorithms to increase human feedback efficiency. SDP leverages sub-optimal trajectories by pseudolabeling all transitions with the minimum environment reward. The now pseudo-labeled sub-optimal data serves two purposes. First, we pre-train a regression-based reward model by applying standard supervised learning to minimize the mean squared loss. Intuitively, this pre-training provides the reward model a head start, biasing it towards assigning lower reward values to these low-quality transitions. Second, we initialize the RL agent's replay buffer with the sub-optimal data and make learning updates to the RL agent. This process changes the RL agent's policy and provides different behaviors for the human to provide feedback on (relative to learning with no initial sub-optimal data). This ensures that when the human teacher provides feedback, their time is used efficiently, avoiding redundant feedback on the existing sub-optimal data. Afterward, we follow the standard preference- or scalar-based RL protocol.
+
+This paper's core contribution is showing that we can harness the availability of low-quality, reward-free data for human-in-the-loop RL approaches by pseudo-labeling it with minimum rewards and treating it as a prior for learning reward models. We first validate the utility of SDP in extensive simulated teacher experiments, combining it with four scalar- and preference-based RL algorithms. These experiments show that SDP significantly improves the feedback efficiency in complex tasks from both the DeepMind Control (DMControl) (Tassa et al., 2018) and Meta-World (Yu et al., 2020) suites. Crucially, we further highlight the real-world applicability of SDP by demonstrating its success with human teachers in a 16-person user study. Overall, this work takes an important step toward considering how human-in-the-loop RL approaches can take advantage of readily available sub-optimal data.
+
+# 2 RELATED WORK
+
+Human-in-the-Loop RL Several approaches in human-in-the-loop RL allow agents to leverage human feedback to adapt or learn new behavior. Learning from demonstration is one such methodology that allows a human to provide examples of desired agent behavior (Argall et al., 2009). Human demonstration data has been used to shape the environment's reward function (Brys et al., 2015), develop a reward function from scratch (Abbeel & Ng, 2004), or bias the agent's policy towards certain actions (Taylor et al., 2011). Although demonstrations can be a rich source of feedback, they are often expensive to obtain and may require domain experts (Dragan & Srinivasa, 2012).
+
+Another approach is learning from preference-based feedback where a teacher provides preferences between two or more sets of agent behavior (Christiano et al., 2017). Preference learning has been popularized in recent years as it can require less effort and expertise compared to providing demonstrations. To further reduce the amount of human interaction required, several strategies have been introduced. This has included combining preferences with demonstrations (Ibarz et al., 2018; Biryik et al., 2022), unsupervised pre-training (Lee et al., 2021a), bi-level optimization (Liu et al., 2022), semi-supervised learning (Park et al., 2022), data augmentation (Park et al., 2022), uncertainty-based exploration (Liang et al., 2022), meta-learning (Hejna III & Sadigh, 2023), and active learning approaches (Hu et al., 2024). Despite its popularity, some argue that comparison feedback might not capture the full intricacies of human preferences, as oftentimes the human is limited to choosing between two options (Daniel et al., 2014; White et al., 2024).
+
+As a result, another body of work focuses on learning from scalar feedback where human teachers can provide scalar signals to evaluate an agent's behavior (Knox & Stone, 2009; Griffith et al., 2013; Loftin et al., 2016; White et al., 2024). Several works use scalar feedback to either learn a reward model (Daniel et al., 2014; Cabi et al., 2020) or an action-value function (Knox & Stone, 2009; 2013; Warnell et al., 2018) via regression.
+
+Learning from Sub-Optimal Data SDP aims to leverage sub-optimal data for scalar- and preference-based RL algorithms. However, learning from low-quality data or negative examples has been applied in other areas of RL and imitation learning (Chen et al., 2021; Tangkaratt et al., 2021). In standard RL, several works use sub-optimal demonstrations to initialize a policy (Taylor et al., 2011; Hester et al., 2018; Gao et al., 2019). In goal-conditioned RL, Hindsight-Experience-Replay
+
+uses failed episodes by treating them as a success with respect to a different goal (Andrychowicz et al., 2017). In inverse reinforcement learning (IRL), Shiarlis et al. (2016) proposed a constrained optimization formulation that can accommodate both successful and failed demonstrations. Brown et al. (2019) makes use of ranked demonstrations to learn a reward function in IRL. Later work in IRL automatically generates ranked trajectories by adding increasing amounts of noise to a learned policy (Brown et al., 2020). Lastly, in offline RL, Singh et al. (2020) leverages sub-optimal transitions from multiple prior tasks and assigns reward labels according to the current task reward function. Our work is most closely related to Yu et al. (2022), which leverages reward-free, suboptimal data in the offline RL setting by pseudo-labeling all transitions with zero and adding them to the RL agent's replay buffer. However, we found that directly applying this approach to the human-in-the-loop RL setting was ineffective (see Appendix D.5).
+
+# 3 BACKGROUND
+
+In the RL paradigm, agents interact with an environment to maximize the total (discounted) expected reward it can achieve. This interaction is modeled as a Markov Decision Process (MDP) which consists of  $\langle S, \mathcal{A}, T, r, \gamma \rangle$ . At every time-step  $t$ , the agent receives a state  $s_t \in S$  from the environment and chooses an action  $a_t \in \mathcal{A}$ . The transition function,  $T$ , determines the probability of transitioning to state  $s_{t+1}$  and receiving reward  $r_{t+1}$ , given the agent was in state  $s_t$  and executed action  $a_t$ . The environment then provides the agent  $r_{t+1}$ . The agent attempts to learn a policy,  $\pi: S \to \mathcal{A}$ , that maximizes the expected return  $\mathbb{E}[G] = \sum_{k=0}^{\infty} \gamma^k r_{t+k+1}$ , which is defined as the expected sum of discounted future rewards with discount factor  $\gamma \in [0,1)$ .
+
+# 3.1 REWARD LEARNING FROM HUMAN FEEDBACK
+
+This paper assumes that we are in a reward-free paradigm, an MDP/R setting, where our goal is to (1) learn a reward function,  $\hat{r}$ , from human feedback and (2) learn a policy that maximizes the total (discounted) expected  $\hat{r}$ . We follow the standard reward learning framework that uses supervised learning to learn a parameterized reward function,  $\hat{r}_{\theta}$ , with parameters  $\theta$  (Christiano et al., 2017). In both scalar- and preference-based settings, we consider trajectory segments  $\sigma$ , where  $\sigma$  consists of a sequence of states and actions:  $\{s_t, a_t, s_{t+1}, a_{t+1}, \dots, s_{t+k}, a_{t+k}\}$ , with  $k$  as the segment size.
+
+Preference-based Reward Learning In preference-based learning, two segments,  $\sigma^0$  and  $\sigma^1$ , are compared by a teacher, yielding  $y \in \{0, 0.5, 1\}$ . Specifically, if the teacher preferred segment  $\sigma^1$  over segment  $\sigma^0$ , then  $y$  is set to 1, and if the converse is true  $y$  is set to 0. If both segments are equally preferred, then  $y$  is set to 0.5. As feedback is collected, it is stored as tuples  $(\sigma^0, \sigma^1, y)$  in the reward model data set  $D_{RM}$ . In general, if  $\sigma^i > \sigma^j$ , then the segment  $\sigma^i$  is preferred by the teacher over segment  $\sigma^j$ . We follow the Bradley-Terry model (Bradley & Terry, 1952) to define a preference predictor using the reward function  $\hat{r}_{\theta}$ :
+
+$$
+P _ {\theta} \left(\sigma^ {1} > \sigma^ {0}\right) = \frac {\exp \left(\sum_ {t} \hat {r} _ {\theta} \left(s _ {t} ^ {1} , a _ {t} ^ {1}\right)\right)}{\sum_ {i \in \{0 , 1 \}} \exp \left(\sum_ {t} \hat {r} _ {\theta} \left(s _ {t} ^ {i} , a _ {t} ^ {i}\right)\right)} \tag {1}
+$$
+
+Intuitively, this model assumes that the probability of the teacher preferring a segment depends exponentially on the total sum of predicted rewards along the segment. To train the reward function, we can use supervised learning where the teacher provides the labels  $y$ . More specifically, we update  $\hat{r}_{\theta}$  by minimizing the standard binary cross-entropy objective:
+
+$$
+L ^ {C E} (\theta , D) = - E _ {(\sigma^ {0}, \sigma^ {1}, y) \sim D} \left[ (1 - y) \log P _ {\theta} \left(\sigma^ {0} > \sigma^ {1}\right) + y \log P _ {\theta} \left(\sigma^ {1} > \sigma^ {0}\right) \right] \tag {2}
+$$
+
+Scalar-based Reward Learning The primary difference between scalar and preference-based reward learning is that in scalar-based learning, the human teacher assigns numerical ratings to trajectory segments. In this setting, the comparisons between segments are implicit. More concretely, a teacher assigns a scalar value  $y$  to a segment  $\sigma^i$ , and as feedback is collected, it is stored as tuples  $(\sigma^i, y)$  in the reward model data set  $D_{RM}$ . We then apply standard regression and update  $\hat{r}_{\theta}$  by minimizing the mean squared error:
+
+$$
+L ^ {M S E} (\theta , D) = E _ {(\sigma^ {i}, y) \sim D} \left[ \left(y - \sum_ {t} \hat {r} _ {\theta} \left(s _ {t} ^ {i}, a _ {t} ^ {i}\right)\right) ^ {2} \right] \tag {3}
+$$
+
+Human Studies in Human in the Loop RL To evaluate human-in-the-loop RL algorithms, a common protocol is the use of simulated teachers, where feedback is provided according to a ground truth reward function. This can be useful, as it offers an efficient and controlled evaluation setting. However, we argue that it is essential to evaluate human-in-the-loop RL algorithms with real human feedback. This is especially important in light of recent work finding discrepancies between preference learning algorithms when using simulated teacher feedback versus human feedback (Metcalf et al., 2024).
+
+# 4 SUB-OPTIMAL DATA PRE-TRAINING
+
+In this section, we present SDP, a tool that leverages sub-optimal trajectories to improve the feedback efficiency for human-in-the-loop RL. We refer to sub-optimal trajectories as sequences of  $(s,a)$  pairs such that:
+
+$$
+r \left(s _ {t}, a _ {t}\right) - r _ {\min } <   \epsilon \quad \forall t \in [ t, t + k ] \quad \text {f o r s o m e s m a l l} \quad \epsilon > 0, \tag {4}
+$$
+
+where  $k$  is the segment size and  $r_{\mathrm{min}}$  is the minimum possible environment reward. Equation 4 essentially expresses that the rewards achieved along a sub-optimal trajectory should be close to  $r_{\mathrm{min}}$ . However, it is important to note that in practice we do not have access to the ground truth reward. This prevents us from directly identifying sub-optimal trajectories using Equation 4. Instead, we rely on selecting trajectories that we estimate will align with this criterion, such as gathering trajectories via a random policy.
+
+Once sub-optimal trajectories are collected, we take the approach of pseudo-labeling all transitions with  $r_{\mathrm{min}}$ . The goal of SDP is then to use this pseudo-labeled data to create a prior for reward models in human-in-the-loop RL methods (see Figure 1). Its simplicity enables SDP to be used in conjunction with any off-the-shelf human-in-the-loop RL algorithm that learns a reward function from human feedback.
+
+SDP comprises two phases: (1) the reward model pre-training phase and (2) the agent update phase. In the reward model pre-training phase, we first gather a data set,  $D_{\mathrm{sub}}$ , of  $N$  sub-optimal state, action transitions. We then pseudo-label all transitions in  $D_{\mathrm{sub}}$  with rewards of  $r_{\mathrm{min}}$ , resulting in  $D_{\mathrm{sub}} = \{s_i, a_i, r_{\mathrm{min}}\}_{i=1}^N$ .  $D_{\mathrm{sub}}$  is then used to optimize the reward model  $\hat{r}_\theta$  with the mean squared loss in Equation 3. As a result, the reward model  $\hat{r}_\theta$  learns to associate all sub-optimal transitions with a low reward. Without such a prior, the reward model would initially have random estimates for these transitions; while the only way to improve such estimates is to obtain feedback from a teacher. Therefore, the reward model pre-training phase provides a valuable reward initialization without requiring any human feedback.
+
+Next, in the agent update phase, we initialize the RL agent's replay buffer  $D_{\mathrm{agent}}$  with  $D_{\mathrm{sub}}$ . The RL agent then briefly interacts with its environment and performs gradient updates according to its loss functions. The agent update process changes the RL agent's policy and generates new transitions, which are then stored in both the agent's replay buffer  $D_{\mathrm{agent}}$  and the reward model's data set  $D_{\mathrm{RM}}$ . It is important to note that in standard scalar- and preference-based reward learning, we query the teacher for feedback on trajectory segments sampled from  $D_{\mathrm{RM}}$ . Therefore, adding new transitions into  $D_{\mathrm{RM}}$  during the agent update phase is necessary to ensure that the teacher does not provide redundant feedback to the original sub-optimal transitions (as  $D_{\mathrm{RM}}$  was empty prior to the agent update phase). When it is time for the teacher to provide their first set of feedback, the feedback can cover a different region of the state and action space, relative to the original sub-optimal data. In Appendix D.1, Figure 8 we empirically show that the agent update phase changes the RL agent's policy by performing policy rollouts and analyzing the differences in state distributions. See Algorithm 1 for the complete pseudocode.
+
+At first glance, labeling sub-optimal transitions with an incorrect reward may seem problematic, as incorrect labels do introduce statistical bias into the reward model and the RL agent's value network. However, as the transitions are sub-optimal, we observe that the bias for using an incorrect reward is low (see Figure 9 in Appendix D.2). Moreover, by using the sub-optimal transitions, we increase the overall amount of data used by both models, which can decrease the models' variance, as shown in the offline RL setting (Yu et al., 2022).
+
+![](images/b934a16fca79a08e016e33b4b61487f0460f20328fbeca04015693b00540874f.jpg)  
+Figure 1: Overview of SDP: After obtaining a data set of sub-optimal trajectories, we pseudo-label the transitions with rewards of  $r_{\mathrm{min}}$  (e.g.,  $r_{\mathrm{min}} = 0$ ). We then pre-train the reward model  $\hat{r}_{\theta}$  using this data set. During the agent update phase, we initialize the RL agent's replay buffer with the same pseudo-labeled data set. The agent then interacts in the environment and makes learning updates to obtain new behaviors for a teacher to give feedback.
+
+# Algorithm 1 SDP
+
+Require: Reward model  $\hat{r}_{\theta} \gets \theta$  randomly initialized, Reward model data set  $D_{\mathrm{RM}} \gets \emptyset$ , RL agent with replay buffer  $D_{\mathrm{agent}} \gets \emptyset$ , Sub-optimal data set  $D_{\mathrm{sub}}$  with reward labels  $r_{\mathrm{min}}$
+
+1: // REWARD MODEL PRE-TRAIN PHASE  
+2: for each gradient step do  
+3: Optimize  $\hat{r}_{\theta}$  on  $D_{\mathrm{sub}}$  with  $L^{MSE}$  (Equation 3)  
+4: end for  
+5: // AGENT UPDATE PHASE  
+6:  $D_{\mathrm{agent}} \gets D_{\mathrm{sub}}$  
+7: for each time-step  $t$  do  
+8: Collect  $s_{t+1}$  by taking action  $a_t \sim \pi(s_t)$ .  
+9: Store  $(s_t, a_t, \hat{r}_\theta, s_{t+1})$  in  $D_{\text{agent}}$  
+10: Store  $(s_t, a_t)$  in  $D_{\mathrm{RM}}$  
+11: Update RL agent with  $D_{\mathrm{agent}}$  
+12: end for  
+13: Begin scalar- or preference-based RL using pre-trained  $\hat{r}_{\theta}$ , RL agent, and  $D_{\mathrm{agent}}$ ,  $D_{\mathrm{RM}}$
+
+# 5 EXPERIMENTS
+
+This section considers the following four research questions (RQ's):
+
+RQ 1: Can SDP improve upon existing scalar- and preference-based RL methods?  
+RQ 2: Can SDP effectively leverage sub-optimal trajectories from different tasks to improve performance on a target task?  
+RQ 3: Can SDP be used with real human feedback?  
+RQ 4: How sensitive is SDP to various hyperparameters?
+
+# 5.1 EXPERIMENTAL DESIGN
+
+To demonstrate the versatility of SDP, we apply SDP to both preference and scalar-based RL approaches. However, as preference feedback can be less time-consuming than scalar feedback, we primarily concentrate on preference-based RL in our experiments, exploring scalar feedback in a smaller capacity. For the preference-based experiments, we combine SDP with four contemporary preference-based algorithms: PEBBLE (Lee et al., 2021a), RUNE (Liang et al., 2022), SURF (Park et al., 2022), and MRN (Liu et al., 2022). We benchmark the performance of the four algorithms augmented with SDP against their original versions without SDP, as well as against SAC. We treat SAC (Haarnoja et al., 2018) as an oracle (i.e., upper bound) because it learns while accessing the ground truth reward function, which is unavailable to the other algorithms. For the scalar-based experiments, we combine SDP with R-PEBBLE (a regression variant of PEBBLE). We compare SDP + R-PEBBLE against R-PEBBLE, Deep TAMER (Warnell et al., 2018) (a scalar feedback RL algorithm), and SAC. We note that SAC is the core RL algorithm used across all baselines.
+
+Implementation Details For SDP, we collected sub-optimal trajectories via a random policy. In particular, we used 50000 state, action transitions for all experiments in Section 5.2. Note that we do not require explicit access to a sub-optimal policy; we only require state, action transitions from said policy. Moreover, to ensure a fair comparison across algorithms, we maintained equal feedback budgets for all algorithms within each environment, while adjusting the budget across environments to reflect their degree of difficulty. See Appendix A for a complete overview of the implementation process and specific hyperparameters for all algorithms.
+
+Evaluation We show average offline performance (i.e., freeze the policy and evaluate it with no exploration) over ten episodes using either the ground truth reward function (DMControl experiments) or the success rate (Meta-World experiments). It is important to note that only SAC has access to the ground truth reward function. We perform this evaluation every 10000 training steps. To systemically evaluate performance, we use a simulated teacher that provides either a scalar rating of a single trajectory segment or preferences between two trajectory segments according to the ground truth reward function. To thoroughly test the effectiveness of SDP, we perform evaluations on four robotic locomotion tasks from the DMControl Suite: Walker-walk, Cheetah-run, Quadruped-walk, and Cartpole-swingup, and five robotic manipulation tasks from Meta-World: Hammer, Door-unlock, Door-lock,Drawer-open, and Window-open. In our experiments, the results are averaged over five seeds with shaded regions or error bars indicating  $95\%$  confidence intervals. To test for significant differences in final performance (i.e., the undiscounted return) and learning efficiency (i.e., the total area under the return curve, AUC), we perform Welch  $t$ -tests (equal variances not assumed) with a p-value of 0.05. See Appendices D.8 and D.9, Tables 9-14 for a summary of final performance and AUC across all experiments.
+
+# 5.2 LOCOMOTION AND MANIPULATION RESULTS
+
+Preference Feedback Experiments We first address RQ 1 by evaluating the utility of SDP in the preference-based RL setting. Considering all four preference-based algorithms in the nine environments, SDP significantly  $(p < 0.05)$  improved learning (i.e., either final performance or AUC) in 23 out of the 36 experiments (see Figure 2). In the remaining experiments, there were no significant differences in the performance between SDP and the baseline algorithms. The addition of SDP (i.e., SDP + base algorithm) never statistically hurt performance.
+
+Scalar Feedback Experiments Continuing our investigation into RQ 1, we now evaluate the performance of SDP in the scalar-based RL setting. We performed evaluations in Walker-walk, Cheetah-run, and Quadruped-walk. In Figure 3, we found that SDP (purple curve) significantly improves either the final performance or AUC compared to R-PEBBLE (navy curve) and Deep TAMER (yellow curve). More impressively, we found that SDP achieves comparable final performance to SAC (red curve), which uses the ground truth reward function, using as little as 60 feedback queries.
+
+Leveraging Different Task Data in SDP Our previous experiments showed that SDP can leverage sub-optimal data from the target task to improve human-in-the-loop RL methods. Now, addressing RQ 2, we investigate whether SDP can similarly use sub-optimal data from related tasks to
+
+![](images/2403ea9bf6074b33b55af5a16c0370fb03ef98b846957e88ca4e44daa7e240b3.jpg)  
+Figure 2: Results from the preference feedback experiments in the DMControl and Meta-World suites show mean AUC  $\pm 95\%$  confidence intervals. * indicates that SDP + the base preference learning algorithm achieves a statistically greater score than the base preference learning algorithm.
+
+![](images/36290cc3af9116cb28719795f4c7ff1d3f794b2422a3ac4c9d0bcd9b9ad72d0c.jpg)  
+Figure 3: In the scalar feedback experiments on the DMControl environments, SDP significantly outperforms R-PEBBLE and Deep TAMER and achieves comparable performance to SAC.
+
+improve performance on the target task. We perform three preference learning experiments, comparing PEBBLE with SDP + PEBBLE, using sub-optimal data from a different prior task that has the same virtual robot: (1) Walker-stand for Walker-walk, (2) Quadruped-walk for Quadruped-run, and (3) Drawer-open for Door-open. To obtain the sub-optimal data for the prior tasks, we gathered transitions from partially trained policies as opposed to using random policies. Each partially trained policy achieved a final score of approximately  $15 - 20\%$  of that achieved by a fully trained policy. This ensured that the distribution of sub-optimal data differed between the prior and target tasks. See Appendix A.2 for further details on the experiment setup. Figure 4 demonstrates that in all three tested environments, SDP can successfully leverage sub-optimal data from related tasks (green curve) as it achieved similar performance to SDP when leveraging target task data (purple curve).
+
+# 5.3 PREFERENCE LEARNING WITH HUMAN FEEDBACK
+
+To evaluate human-in-the-loop RL algorithms, we argue that it is critical to understand their efficacy with human teachers. However, human user studies do not appear to be widely adopted in the current literature. To empirically investigate this, we conducted a survey of 45 preference learning studies
+
+![](images/532908636b5468d550b40a67870e5f4982535887b2f2431ea738ffd031c41040.jpg)  
+Figure 4: This highlights that SDP can leverage sub-optimal data from different prior tasks as it performed comparable to SDP when using target task sub-optimal data.
+
+![](images/ead2e27104d39d983d549fe456690decc7d409e0248318b2674fa4449a5adaa4.jpg)
+
+![](images/b1bc6ea5f72b746175d23b82c9f5311144df3b789674830e53cb00bdb459ab51.jpg)
+
+from 2012 to  $2024^{1}$  to understand the prevalence of human user studies in the preference learning literature. We found that fewer than  $50\%$  tested their proposed algorithms with human participants who were not also the authors. Moreover, of those studies that did involve human participants, only  $41\%$  included non-expert individuals, while none provided sufficient demographic information about their participants (e.g., gender, race/ethnicity, level of education). These limitations raise significant concerns about whether and how current preference learning algorithms generalize to different populations.
+
+To that end, we now address RQ 3 and evaluate the efficacy of SDP with real human feedback. We conduct an ethics-approved human-subject study of 16 participants (9 male, 7 female). Age ranges were collected: 18-24 (6), 25-30 (7), and 50-70 (3). Participants self-identified their membership in racial groups: South Asian (6), East Asian (3), White (4), Middle Eastern or North African (2), and multi-racial (1). The participants' highest educational attainments were: high school diploma (3), Bachelor's degrees (8), and Master's degrees (5), and their expertise varied across AI/ML (12), other computer science topics (1), and non-computer science fields (3). This highlights the diversity of our participants in terms of demographics and expertise.
+
+This user study compares the performance of SDP and PEBBLE in two DMControl environments: Pendulum-swingup (with 7 participants) and Cartpole-swingup (with 12 participants). We focused our comparison to PEBBLE as it performed comparably to the other preference learning baselines in Section 5.2. We use a between-subjects experimental design — each participant provides preferences for a single seed of both SDP and PEBBLE algorithms. The preference budgets for Pendulum-swingup and Cartpole-swingup were 40 and 48, respectively. We selected these environments specifically because they could be solved with fewer preferences, aiming to reduce the overall time commitment required from participants. Each trial for a single environment lasted approximately 1–1.5 hours. See Appendix B for more details including user instructions and interface.
+
+We visualize both algorithms' final performance and AUC in Figure 5. We found that in both environments SDP (purple plots) maintains significant  $(p < 0.05)$  performance gains over PEBBLE (blue plots) in terms of either final performance or AUC. In Cartpole-swingup, we also observed consistent performance from SDP regardless of whether the teacher was human or simulated. This suggests that our prior results with simulated teachers can generalize to settings where human feedback is provided. Moreover, we observed no significant differences in SDP's effectiveness across demographic factors, including gender, age, educational, and computer science background (see Appendix C, Tables 7 and 8). This finding is particularly encouraging for the potential real-world deployment of SDP, highlighting its usability for non-expert users across diverse demographics.
+
+# 5.4 ABLATION AND SENSITIVITY STUDIES
+
+To further understand the effectiveness of SDP, we perform further analysis of SDP across three dimensions: (1) the phases of SDP, (2) the number of feedback queries, and (3) the amount of sub-optimal data. This analysis aims to address RQ 4 and provide a deeper understanding of the factors influencing SDP's performance. For these experiments, we focus on SDP + R-PEBBLE in the Walker-walk environment. Additional results for Cheetah-run can be found in Appendix D.5.
+
+![](images/14fc53d27b0ebc860eb96b79bd41ed8ddd02252563a610d48061b5db4ec9080d.jpg)  
+Figure 5: This demonstrates that SDP can significantly outperform PEBBLE in terms of both AUC (left) and final performance (right) even when human teachers are providing preferences. * denotes a statistically significant difference between SDP and PEBBLE.
+
+![](images/9f9e943ed8ed1fb0d5eb36f712a004f3f6d8b2b320f16545a12918da2d413de9.jpg)  
+Figure 6: These figures show ablation and sensitivity studies of SDP in Walker-walk.
+
+SDP Component Analysis First, we evaluate the effect of each phase of SDP individually, the reward model pre-train phase, and the agent update phase. Figure 6 (leftmost) demonstrates the importance of using both phases in SDP for scalar-based RL approaches. We found that the SDP variants that only use one of the phases (green and gray curves) result in worse performance than the full SDP (purple curve).
+
+Effect of Feedback Amount We evaluate SDP and R-PEBBLE with feedback budgets  $\in$  [60, 100, 200] to analyze the impact of feedback quantity on performance. As shown in Figure 6 (middle), SDP (purple curves) consistently outperforms R-PEBBLE (navy curves), further demonstrating its effectiveness across varying feedback levels.
+
+Effect of Sub-Optimal Data Amount We evaluate the performance of SDP using varying amounts of sub-optimal transitions  $\in$  [5000, 15000, 50000]. Figure 6 (rightmost) indicates that while 5000 transitions (gray curve) led to the poorest performance, increasing this amount to 15000 or 50000 (green and purple curves) yielded comparable or improved results, suggesting that more sub-optimal data can benefit SDP.
+
+# 6 CONCLUSION
+
+In this work, we present SDP, an approach that improves the feedback efficiency for human-in-the-loop RL algorithms. SDP is specifically designed to leverage reward-free, sub-optimal data for scalar- and preference-based RL approaches. By pseudo-labeling low-quality data with the minimum environment rewards, we can pre-train the reward model without the need for human labeling. This provides the reward model with a head start in learning. This head start allows the reward model to learn to associate low-quality transitions with low reward values, even before receiving any actual
+
+human feedback. Our simulated teacher experiments in DMControl and Meta-World suites demonstrate that SDP can significantly improve a variety of preference- and scalar-based reward learning algorithms. Importantly, we further validate the real-world applicability of SDP by demonstrating its success in a 16-person user study. This work takes an important step towards considering how sub-optimal data can be leveraged for human-in-the-loop RL.
+
+# ACKNOWLEDGMENTS
+
+Part of this work has taken place in the Intelligent Robot Learning Lab at the University of Alberta, which is supported in part by research grants from Alberta Innovates; Alberta Machine Intelligence Institute (Amii); a Canada CIFAR AI Chair, Amii; Digital Research Alliance of Canada; Mitacs; and the National Science and Engineering Research Council.
+
+# ETHICS STATEMENT
+
+The goal of preference and scalar-based RL is to learn reward functions that encode human preferences. However, this necessitates interaction with human users, raising concerns about potential biases in the collected preferences. If these preferences are primarily drawn from a non-representative group, the resulting RL system may unfairly prioritize the desires or needs of that group over others. To that end, we performed a human subject study that was reviewed and approved by an external ethics committee. This ensured the responsible and ethical collection of human preferences. In addition, we took care to gather participants from a wide range of both demographic and educational backgrounds, which we detail in Section 5.3.
+
+# REPRODUCIBILITY STATEMENT
+
+To ensure the reproducibility of our work, we provide a link to our code repository: https://github.com/cmuslima/SDP_ICLR. We also provide pseudocode in Section 4. Moreover, we outline extensive training and evaluation details in Section 5.1 and Appendix A.2. We also outline all hyperparameters used for all algorithms and environments in Appendix A.2, Tables 1-5. Lastly, for the human subject study, we provide details on the instructions provided to the participants in Appendix B as well as a description of the preference interface used. In our released codebase, we also provide the specific code files used to run the experiments in the user study. In addition, an anonymized version of the human subject data will be released, as stated in our ethics approval.
+
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+Gokul Swamy, Christoph Dann, Rahul Kidambi, Zhiwei Steven Wu, and Alekh Agarwal. A Minimaximalist Approach to Reinforcement Learning from Human Feedback. In International Conference on Machine Learning, 2024.
+
+Voot Tangkaratt, Nontawat Charoenphakdee, and Masashi Sugiyama. Robust Imitation Learning from Noisy Demonstrations. In International Conference on Artificial Intelligence and Statistics, 2021.  
+Yuval Tassa, Yotam Doron, Alistair Muldal, Tom Erez, Yazhe Li, Diego de Las Casas, David Budden, Abbas Abdelmaleki, Josh Merel, Andrew Lefrancq, et al. Deepmind Control Suite. CoRR, 2018.  
+Matthew E Taylor, Halit Bener Suay, and Sonia Chernova. Integrating Reinforcement Learning with Human Demonstrations of Varying Ability. In International Conference on Autonomous Agents and Multiagent Systems, 2011.  
+Mudit Verma and Katherine Metcalf. Hindsight PRIORs for Reward Learning from Human Preferences. In International Conference on Learning Representations, 2024.  
+Mudit Verma, Siddhant Bhambri, and Subbarao Kambhampati. Exploiting Unlabeled Data for Feedback Efficient Human Preference based Reinforcement Learning. In Workshop on Representation Learning for Responsible Human-Centric AI at AAAI Conference on Artificial Intelligenc, 2023.  
+Ngo Anh Vien, Wolfgang Ertl, and Tae Choong Chung. Learning via Human Feedback in Continuous State and Action Spaces. Applied Intelligence, 2013.  
+Ren-Jian Wang, Ke Xue, Yutong Wang, Peng Yang, Haobo Fu, Qiang Fu, and Chao Qian. Diversity from Human Feedback. In Workshop on Agent Learning in Open-Endness at Neural Information Processing Systems, 2023.  
+Ruiqi Wang, Weizheng Wang, and Byung-Cheol Min. Feedback-efficient Active Preference Learning for Socially Aware Robot Navigation. In International Conference on Intelligent Robots and Systems, 2022.  
+Xiaofei Wang, Kimin Lee, Kourosh Hakhamaneshi, Pieter Abbeel, and Michael Laskin. Skill Preferences: Learning to Extract and Execute Robotic Skills from Human Feedback. In Conference on Robot Learning, 2021.  
+Garrett Warnell, Nicholas Waytowich, Vernon Lawhern, and Peter Stone. Deep TAMER: Interactive Agent Shaping in High-Dimensional State Spaces. In AAAI Conference on Artificial Intelligence, 2018.  
+Devin White, Mingkang Wu, Ellen Novoseller, Vernon J Lawhern, Nicholas Waytowich, and Yongcan Cao. Rating-based Reinforcement Learning. In AAAI Conference on Artificial Intelligence, 2024.  
+Nils Wilde, Erdem Bıyık, Dorsa Sadigh, and Stephen L Smith. Learning Reward Functions from Scale Feedback. In Conference on Robot Learning, 2021.  
+Aaron Wilson, Alan Fern, and Prasad Tadepalli. A Bayesian Approach for Policy Learning from Trajectory Preference Queries. In Neural Information Processing Systems, 2012.  
+Wanqi Xue, Bo An, Shuicheng Yan, and Zhongwen Xu. Reinforcement Learning from Diverse Human Preferences. In International Joint Conference on Artificial Intelligence, 2024.  
+Tianhe Yu, Deirdre Quillen, Zhanpeng He, Ryan Julian, Karol Hausman, Chelsea Finn, and Sergey Levine. Meta-world: A Benchmark and Evaluation for Multi-task and Meta Reinforcement Learning. In Conference on Robot Learning, 2020.  
+Tianhe Yu, Aviral Kumar, Yevgen Chebotar, Karol Hausman, Sergey Levine, and Chelsea Finn. Conservative Data Sharing for Multi-task Offline Reinforcement Learning. Neural Information Processing Systems, 2021.  
+Tianhe Yu, Aviral Kumar, Yevgen Chebotar, Karol Hausman, Chelsea Finn, and Sergey Levine. How to Leverage Unlabeled Data in Offline Reinforcement Learning. In International Conference on Machine Learning, 2022.
+
+Guoxi Zhang and Hisashi Kashima. Learning State Importance for Preference-based Reinforcement Learning. Machine Learning, 2024.  
+Jiawei Zhao, Florian Tobias Schaefer, and Anima Anandkumar. ZerO Initialization: Initializing Neural Networks with only Zeros and Ones. Transactions on Machine Learning Research, 2022.  
+Tianchen Zhu, Yue Qiu, Haoyi Zhou, and Jianxin Li. Decoding Global Preferences: Temporal and Cooperative Dependency Modeling in Multi-Agent Preference-Based Reinforcement Learning. In AAAI Conference on Artificial Intelligence, 2024.
+
+# APPENDIX
+
+# A SIMULATED EXPERIMENT DETAILS
+
+# A.1 BENCHMARKS
+
+PEBBLE PEBBLE has two primary components: unsupervised exploration and off-policy learning with relabeling. The purpose of the unsupervised exploration phase is to collect diverse experiences for a human teacher to provide feedback on. More specifically, PEBBLE optimizes state entropy to explore the environment. Furthermore, PEBBLE uses off-policy reinforcement learning to learn a policy. PEBBLE specifically uses an off-policy RL algorithm as they are more sample efficient compared to their on-policy counterparts. Then as the reward model changes, PEBBLE relabels all transitions in the RL agent's replay buffer with the latest reward model. This is integral as the reward model is non-stationary, and relabeling the transitions stabilizes the learning process.
+
+To adapt PEBBLE to the scalar feedback setting, we make one minor change to the reward model. In the scalar feedback setting, we use a scripted teacher that provides a scalar rating of a single trajectory segment. Therefore, the only update to PEBBLE is with respect to the loss function. Instead of using the cross-entropy loss in Equation 2, we use the mean-squared error loss in Equation 3.
+
+RUNE RUNE is a preference learning algorithm (built on top of PEBBLE) that uses an uncertainty-based exploration strategy to improve feedback efficiency. To encourage exploration for the SAC agent, RUNE adds an intrinsic reward component based on the standard deviation in the reward model.
+
+SURF SURF is another preference learning algorithm (built on top of PEBBLE) that improves feedback efficiency by using semi-supervised and data augmentation approaches. To incorporate semi-supervised learning, SURF generates pseudo-labels for unlabeled trajectories by querying the learned reward model. If the reward model confidently (e.g., low output standard deviation) predicts the pseudo-label, then the trajectory, label pair is added to the reward model training data set. Further, SURF proposes a new data augmentation technique that crops sub-sequences of trajectories.
+
+MRN MRN is a preference learning algorithm also integrated with PEBBLE. Unlike other PbRL algorithms, MRN is a bi-level optimization algorithm in which the actor and critic are updated in the inner loop, and the reward model is updated in the outer loop. Importantly, the reward model takes into account the performance of the critic on the preference data.
+
+Deep TAMER In our scalar feedback experiments, we also consider the Deep TAMER benchmark. In Deep TAMER, scalar feedback is used to learn a human reward function via regression. Then the agent acts greedily with respect to this reward function. Furthermore, the original implementation of Deep TAMER was built on top of DQN. Therefore, there was no separate actor-network. In addition, Deep TAMER only used discrete feedback  $\in [-1,0,1]$ .
+
+To make Deep TAMER a fair benchmark, we made a few adjustments. To start, we allow Deep TAMER to learn from real-valued feedback as done in the other scalar-based experiments. However, instead of using the ground truth reward function as feedback, we use the state-action values from a fully-trained SAC agent. We do this because, in TAMER (and Deep TAMER), the teacher is intended to provide feedback representative of the return. Secondly, in Deep TAMER, feedback is provided per (state, action) pair. Therefore, to make sure Deep TAMER received the same amount of feedback as the other baselines, we used trajectory segment size  $\times$  feedback amount for Deep TAMER only. The other benchmarks receive a scalar feedback value that is the sum of rewards along a trajectory segment. Third, to learn the reward model we use standard regression as described in Section 3.1. Lastly, as our testing environments are continuous state and action, we learn a separate actor policy, similarly done in (Vien et al., 2013).
+
+# A.2 TRAINING DETAILS
+
+In all of our experiments, we use the hyperparameters in Table 1 for the reward models used in all benchmarks. For the agent update phase of SDP, an additional hyperparameter is associated with the number of environment interactions made before the standard preference/scalar feedback learning loop begins. However, for simplicity, we kept the same value as the existing feedback frequency hyperparameter, as feedback frequency also dictates the number of environment interactions made between feedback sessions
+
+Furthermore, we use most of the existing reward model hyperparameters used in PEBBLE, however, we adjusted the following four hyperparameters: feedback frequency, amount of feedback per session, trajectory segment size (only for Meta-world), and activation function for the final NN layer. We adjusted the first two hyperparameters because PEBBLE originally used a significantly larger feedback budget, therefore we wanted the feedback schedule to better reflect a smaller feedback budget. We used a different trajectory segment size for Meta-world because we wanted to keep the segment sizes the same across both the DMControl and Meta-world environments. Moreover, we found that the output activation function could significantly affect learning, therefore we tested all benchmarks using both Tanh (original activation used) and Leaky-ReLU and chose the reward model that achieved better final performance. For the RUNE and SURF baselines, we use any hyperparameters associated with their specific algorithm according to the original paper (see Table 2). For a fair comparison with SDP, we provide all human-in-the-loop baselines (e.g., PEBBLE, R-PEBBLE, Deep TAMER, RUNE, and SURF) with the sub-optimal data set to be used in both the reward model and by the RL agent.
+
+Furthermore, to select trajectory segments for the teacher to provide feedback on, we use uniform sampling in the DMControl tasks and disagreement sampling in the Meta-world tasks. Disagreement sampling is a popular active learning approach in which trajectories with higher uncertainty (based on an ensemble of neural networks) are more likely to be sampled (Christiano et al., 2017). As for the SAC hyperparameters, we use the values found in Tables 3-4.
+
+For the experiments in which we leveraged sub-optimal data from a different task (i.e., Walker-stand, Quadruped-walk, Drawer-open), we gathered 50,000 transitions from partially trained policies. We note that for these experiments, we purposely did not use transitions gathered from a random policy. In these experiments, the prior and target tasks were environments in which the simulated robot was identical. The only difference is the environmental reward. Therefore, the random policy for both environments would be the same. To truly demonstrate transfer, we wanted to ensure we obtained low-quality transitions of the prior task that were different from the target task.
+
+Each partially trained policy achieved a final score of approximately  $15 - 20\%$  of that achieved by a fully trained policy. More specifically, we used the following procedure to train the SAC policies. First, for Walker-stand, we trained a SAC policy for 5,000 time steps, and the average final performance was approximately 194. Second, for Quadruped-walk, we trained a SAC policy for 100,000 timesteps, and the average final performance was approximately 184. Lastly, for Drawer-open, we trained a SAC policy for 50,000 time steps, and the average final success rate was approximately  $14\%$ .
+
+<table><tr><td>HYPERPARAMETER</td><td>VALUE</td></tr><tr><td>SEGMENT SIZE</td><td>50</td></tr><tr><td>RANDOM STEPS(I.E., SUB-OPTIMAL DATA TRANSITIONS)</td><td>50000</td></tr><tr><td>UNSUPERVISED EXPLORATION STEPS</td><td>9000</td></tr><tr><td>CONSIDER UNSUPERVISED EXPLORATIONSAS SUB-OPTIMAL</td><td>FALSE (WALKER-WALK, CARTPOLE-SWINGUP, QUADRUPED-WALK - PBRL)TRUE (OTHERS)</td></tr><tr><td>FREQUENCY OF FEEDBACK</td><td>20000 (DMCONTROL)10000 (WINDOW-OPEN, DOOR-UNLOCK, DOOR-LOCK)5000 (HAMMER, DRAWER-OPEN)</td></tr><tr><td>FEEDBACK BUDGETS IN ABLATIONS</td><td>2000 (DOOR-OPEN)1000 (CHEETAH-RUN, WALKER-WALK)</td></tr><tr><td>FEEDBACK QUERIES PER SESSION</td><td>50 (HAMMER, DRAWER-OPEN)8 (CARTPOLE-SWINGUP)20 (OTHERS)</td></tr><tr><td>SAMPLING SCHEME</td><td>DISAGREEMENT SAMPLING (METAWORLD)UNIFORM SAMPLING (DMCONTROL)</td></tr><tr><td>TRAINING EPOCHS</td><td>200 (WINDOW-OPEN, SURF AND SDP + SURF)50 (OTHERS)</td></tr><tr><td>LEARNING RATE</td><td>3 ×10-4</td></tr><tr><td>INTERMEDIATE NEURAL NETWORK ACTIVATION</td><td>LEAKY RELU</td></tr><tr><td>BATCH SIZE</td><td>128</td></tr><tr><td>HIDDEN LAYERS</td><td>4</td></tr><tr><td>NEURONS PER HIDDEN LAYER</td><td>128</td></tr><tr><td>LOSS FUNCTION</td><td>MEAN SQUARED ERROR (SCALAR FEEDBACK)CROSS ENTROPY LOSS (PREFERENCE FEEDBACK)</td></tr><tr><td>OPTIMIZER</td><td>ADAM</td></tr><tr><td>For Reward Model Pre-training Phase in SDPLAST LAYER NEURAL NETWORK ACTIVATION</td><td>TANH</td></tr><tr><td>For Human-in-the-Loop AlgorithmLAST LAYER NEURAL NETWORK ACTIVATION</td><td>LEAKY RELU (PEBBLE, RUNE, SURFTANH (SDP - HAMMER, DRAWER-OPEN, WALKER-WALK)LEAKY RELU (SDP - OTHERS)</td></tr></table>
+
+Table 1: Hyperparameters for the reward model used in all experiments (both preference and scalar feedback variants).
+
+<table><tr><td>HYPERPARAMETER</td><td>VALUE</td></tr><tr><td colspan="2">Specific RUNE Hyperparameters</td></tr><tr><td>BETA SCHEDULE</td><td>LINEAR DECAY</td></tr><tr><td>BETA INIT</td><td>0.05</td></tr><tr><td>BETA DECAY</td><td>10-5</td></tr><tr><td colspan="2">Specific SURF Hyperparameters</td></tr><tr><td>THRESHOLD λ</td><td>0.99</td></tr><tr><td>UNLABELED BATCH RATIO</td><td>4</td></tr><tr><td>LOSS WEIGHT</td><td>1</td></tr><tr><td>MIN/MAX LENGTH OF CROPPED SEGMENT</td><td>[45, 55]</td></tr><tr><td>SEGMENT LENGTH BEFORE CROPPING</td><td>60</td></tr><tr><td colspan="2">Specific MRN Hyperparameters</td></tr><tr><td>META STEPS</td><td>1000 (WALKER-WALK)</td></tr><tr><td></td><td>3000 (QUADRUPED-WALK)</td></tr><tr><td></td><td>10000 (DOOR-OPEN)</td></tr><tr><td></td><td>5000 (OTHERS)</td></tr></table>
+
+Table 2: Baseline Specific Hyperparameters.  
+
+<table><tr><td>HYPERPARAMETER</td><td>VALUE</td></tr><tr><td>OPTIMIZER</td><td>ADAM (Kingma &amp; Ba, 2015)</td></tr><tr><td>DISCOUNT</td><td>0.99</td></tr><tr><td>ALPHA LEARNING RATE</td><td>10-4</td></tr><tr><td>ACTOR BETAS</td><td>0.9, 0.999</td></tr><tr><td>CRITIC BETAS</td><td>0.9, 0.999</td></tr><tr><td>ALPHA BETAS</td><td>0.9, 0.999</td></tr><tr><td>TARGET SMOOTHING COEFFICIENT</td><td>0.005</td></tr><tr><td>ACTOR UPDATE FREQUENCY</td><td>1</td></tr><tr><td>CRITIC TARGET UPDATE FREQUENCY</td><td>2</td></tr><tr><td>INIT TEMPERATURE</td><td>0.1</td></tr><tr><td>NETWORK TYPE</td><td>MLP</td></tr><tr><td>NONLINEARITY</td><td>RELU</td></tr><tr><td>GRADIENT UPDATES PER STEP</td><td>1</td></tr></table>
+
+Table 3: Hyperparameters for SAC that were shared by all algorithms. SAC is the standard RL algorithm that learns from the environment's true reward signal, not via preference learning.
+
+<table><tr><td>HYPERPARAMETER</td><td>VALUE</td></tr><tr><td>DMControl</td><td></td></tr><tr><td>BATCH SIZE</td><td>512 (CARTPOLE-SWINGUP)1024 (OTHERS)</td></tr><tr><td>HIDDEN LAYERS</td><td>2</td></tr><tr><td>NEURONS PER HIDDEN LAYER</td><td>256 (CARTPOLE-SWINGUP)1024 (OTHERS)</td></tr><tr><td>ACTOR/CRITIC LEARNING RATE</td><td>5 × 10-5(CHEETAH-RUN PREFERENCE FEEDBACK)10-4(CHEETAH-RUN SCALAR FEEDBACK)5 × 10-4(WALKER-WALK)10-4(OTHERS)</td></tr><tr><td>TRAINING STEPS</td><td>0.5 × 106(WALKER-WALK, CARTPOLE-SWINGUP)106(OTHERS)</td></tr><tr><td>Metaworld</td><td></td></tr><tr><td>BATCH SIZE</td><td>512</td></tr><tr><td>HIDDEN LAYERS</td><td>3</td></tr><tr><td>NEURONS PER HIDDEN LAYER</td><td>256</td></tr><tr><td>ACTOR/CRITIC LEARNING RATE</td><td>3 × 10-4</td></tr><tr><td>TRAINING STEPS</td><td>0.5 × 106(DOOR-LOCK AND WINDOW-OPEN)106(OTHERS)</td></tr></table>
+
+Table 4: Specific SAC hyperparameters that were tuned for the DMControl and Metaworld experiments. The majority of these hyperparameters were selected from the PEBBLE repo (Lee et al., 2021a).  
+
+<table><tr><td>HYPERPARAMETER</td><td>VALUE</td></tr><tr><td>SAC</td><td></td></tr><tr><td>TRAINING STEPS</td><td>0.3 × 106</td></tr><tr><td>BATCH SIZE</td><td>1024</td></tr><tr><td>HIDDEN LAYERS</td><td>2</td></tr><tr><td>NEURONS PER HIDDEN LAYER</td><td>1024</td></tr><tr><td>ACTOR/CRITIC LEARNING RATE</td><td>10-4</td></tr><tr><td>Reward Model</td><td></td></tr><tr><td rowspan="2">CONSIDER UNSUPERVISED EXPLORATIONS AS SUB-OPTIMAL FREQUENCY OF FEEDBACK</td><td>FALSE</td></tr><tr><td>20000</td></tr><tr><td>FEEDBACK QUERIES PER SESSION</td><td>10 (PENDULUM-SWINGUP)8 (CARTPOLE-SWINGUP)</td></tr><tr><td>SAMPLING SCHEME</td><td>UNIFORM SAMPLING</td></tr><tr><td>BATCH SIZE</td><td>128</td></tr><tr><td>LAST LAYER NEURAL NETWORK ACTIVATION</td><td>LEAKY RELU</td></tr></table>
+
+Table 5: Tuned hyperparameters for human feedback experiments.
+
+# B HUMAN SUBJECT STUDY DETAILS
+
+Our study was approved by an external ethics committee. In total, 16 users participated in our study, 3 of which participated in both Pendulum-swingup and Cartpole-swingup tasks. As for the amount of participation per task, 7 users provided preferences in Pendulum-swingup and 12 users provided preferences in Cartpole-swingup. During the timeline of our user study, we initially had participants interacting in Pendulum-swingup, however we found that it was very simple, therefore we decided to include Cartpole-swingup, as a more difficult task. Each participant provided preferences for a single run of the environment, as each session took approximately 1 to 1.5 hours. Table 5 shows the specific hyperparameters used.
+
+# B.1 PROTOCOL STEPS
+
+The study took place in person, where the participant was first provided the consent from to review and sign. Then, participants and the researcher, together, reviewed the instructions outlining the objective of the study. The instructions had two components. First, they outlined participant's goal in the study. Second, as done in Christiano et al. (2017), we provided a guide on what constitutes good and bad behavior in each domain.
+
+Pendulum-swingup The first set of instructions for the study are as follows:
+
+1. Objective: for the pole to swing up and balance.  
+2. Agent controls how much torque (or force or twist) to apply  
+3. It does not matter which way the pole swings, left or right, as long as the pole balances.  
+4. Your task: You will see two clips of behaviors, and your job is to select the clip you think is better given the above objective. If you think both clips are identical in behavior, you can select equally preferable.
+
+The second set of instructions (i.e., advice) for the study are as follows:
+
+1. The first priority is for the pole to begin swinging back and forth (i.e., picking up momentum). Therefore, the video clip where the pole has swung higher is better. Even if the agent is not behaving well in either clips, if you can tell that the pole is higher in one clip than the other, it is better to prefer that clip.  
+2. In general, if the pole is swinging rapidly in a circle (e.g., complete 360), then this is usually worse behavior than if the pole is barely moving.
+
+Cartpole-swingup The first set of instructions for the study are as follows:
+
+1. Pole is attached to a cart.  
+2. Agent can move the cart left and right.  
+3. Goal is for the agent to move the cart such that the pole swings up and balances  
+4. Your task: You will see two clips of behaviors, and your job is to select the clip you think is better given the above objective. If you think both clips are identical in behavior, you can select equally preferable.
+
+The second set of instructions (i.e., advice) for the study are as follows:
+
+1. The first priority is for the pole to begin swinging back and forth (i.e., picking up momentum). Therefore, the video clip where the pole has swung higher is better. Even if the agent is not behaving well in either clips, if you can tell that the pole is higher in one clip than the other, it is better to prefer that clip.  
+2. In general, if the pole is swinging rapidly in a circle (e.g., complete 360), then this is usually worse behavior than if the pole is barely moving.  
+3. If the pole is balancing then falls over, then is is better behavior than if the pole is barely moving.
+
+During the instruction period, participants also watched video clips of what successful and unsuccessful behavior looks like. In addition, participants were able to ask questions about the environment's objective.
+
+Once the instruction period was complete, participants completed a practice round in providing preferences. This was included so users could gain familiarity with the interface. Figure 7 shows a screenshot of the interface used. Participants would use keyboard input to move from one video clip to the next. Then, participants had the opportunity to rewatch either clips as many times as they liked. Afterwards, participants chose which clip they preferred (or equally preferred) via keyboard input. Each video clip consisted of a 50 step segment, which was approximately 2 seconds long. The total study time was  $\sim$  (1 - 1.5) hours. After the study was complete, participants filled out a demographic survey which included questions pertaining to their age, race/ethnicity, education level completed, and area of expertise.
+
+![](images/f13c175289ad7893626948a8e0cb3ef8cbe95ed8ab18f56e8b4b6da9e101ba0a.jpg)  
+Figure 7: This shows the user interface used for the human subject study. Participants would view each video clip, sequentially, then decide which clip they preferred.
+
+# B.2 PREFERENCE LEARNING SURVEY DETAILS
+
+Table 6 provides a list of all papers included in our preference learning survey. We now describe the criteria for inclusion and exclusion in our survey. To find papers, we used Google Search with key words: preference learning, reinforcement learning from human feedback, and preference based reinforcement learning. We also found papers by reviewing those that cited popular preference learning works. This included Christiano et al. (2017) and Lee et al. (2021a), as these were two of the first papers that popularized preference learning. We included conference, journal and workshop papers in our survey. We did not include papers that were strictly on Arxiv. To keep the survey within scope, we did not include any preference learning works related to large language or foundation models. We also did not include any works that were entirely theoretical (e.g., no empirical results were provided).
+
+Table 6: Articles used in Preference Learning Survey from Section 5.3. NA indicates no information was provided.  
+
+<table><tr><td>PAPER REFERENCE</td><td>HUMAN USERS (NON-AUTHOR)</td><td>NUMBER OF USERS</td></tr><tr><td>(Myers et al., 2023)</td><td>Yes</td><td>22</td></tr><tr><td>(Metcalf et al., 2024)</td><td>Yes</td><td>50</td></tr><tr><td>(Myers et al., 2022)</td><td>Yes</td><td>50</td></tr><tr><td>(Christiano et al., 2017)</td><td>Yes</td><td>NA</td></tr><tr><td>(Hejna III &amp; Sadigh, 2023)</td><td>Yes</td><td>4</td></tr><tr><td>(Hwang et al., 2023b)</td><td>Yes</td><td>5</td></tr><tr><td>(Biyik &amp; Sadigh, 2018)</td><td>Yes</td><td>10</td></tr><tr><td>(Hwang et al., 2023a)</td><td>Yes</td><td>5</td></tr><tr><td>(Biyik et al., 2020)</td><td>Yes</td><td>10</td></tr><tr><td>(Wilde et al., 2021)</td><td>Yes</td><td>18</td></tr><tr><td>(Biyik et al., 2022)</td><td>Yes</td><td>15</td></tr><tr><td>(Ibarz et al., 2018)</td><td>Yes</td><td>NA</td></tr><tr><td>(Knox et al., 2023)</td><td>Yes</td><td>143</td></tr><tr><td>(Holk et al., 2024)</td><td>Yes</td><td>32</td></tr><tr><td>(Metcalf et al., 2023)</td><td>Yes</td><td>40</td></tr><tr><td>(Marta et al., 2024)</td><td>Yes</td><td>NA</td></tr><tr><td>(Marta et al., 2023a)</td><td>Yes</td><td>70</td></tr><tr><td>(Marta et al., 2023b)</td><td>Yes</td><td>20</td></tr><tr><td>(Ren et al., 2022)</td><td>Yes</td><td>NA</td></tr><tr><td>(Mehta &amp; Losey, 2024)</td><td>Yes</td><td>15</td></tr><tr><td>(White et al., 2024)</td><td>Yes</td><td>20</td></tr><tr><td>(Wang et al., 2022)</td><td>Yes</td><td>10</td></tr><tr><td>(Wang et al., 2023)</td><td>No</td><td></td></tr><tr><td>(Zhang &amp; Kashima, 2024)</td><td>No</td><td></td></tr><tr><td>(Lee et al., 2021a)</td><td>No</td><td></td></tr><tr><td>(Liang et al., 2022)</td><td>No</td><td></td></tr><tr><td>(Park et al., 2022)</td><td>No</td><td></td></tr><tr><td>(Hu et al., 2024)</td><td>No</td><td></td></tr><tr><td>(Liu et al., 2022)</td><td>No</td><td></td></tr><tr><td>(Liu et al., 2023)</td><td>No</td><td></td></tr><tr><td>(Xue et al., 2024)</td><td>No</td><td></td></tr><tr><td>(Daniels-Koch &amp; Freedman, 2022)</td><td>No</td><td></td></tr><tr><td>(Verma et al., 2023)</td><td>No</td><td></td></tr><tr><td>(Barnett et al., 2023)</td><td>No</td><td></td></tr><tr><td>(Verma &amp; Metcalf, 2024)</td><td>No</td><td></td></tr><tr><td>(Metcalf et al., 2023)</td><td>No</td><td></td></tr><tr><td>(Swamy et al., 2024)</td><td>No</td><td></td></tr><tr><td>(Lee et al., 2021b)</td><td>No</td><td></td></tr><tr><td>(Wang et al., 2021)</td><td>No</td><td></td></tr><tr><td>(Maxence Hussonnois &amp; Rana, 2023)</td><td>No</td><td></td></tr><tr><td>(Wilson et al., 2012)</td><td>No</td><td></td></tr><tr><td>(Liu et al., 2024)</td><td>No</td><td></td></tr><tr><td>(Giovannielli et al., 2024)</td><td>No</td><td></td></tr><tr><td>(Zhu et al., 2024)</td><td>No</td><td></td></tr><tr><td>(Cheng et al., 2024)</td><td>No</td><td></td></tr></table>
+
+C ADDITIONAL HUMAN TEACHER RESULTS  
+
+<table><tr><td>TASK</td><td>FEEDBACK</td><td>BACKGROUND</td><td>AUC</td><td>P VALUE</td><td>FINAL PERFORMANCE</td><td>P VALUE</td></tr><tr><td rowspan="2">CARTPOLE-SWINGUP</td><td rowspan="2">48</td><td>Non-CS</td><td>8258.44 ± 1898.47</td><td rowspan="2">0.183</td><td>648.88 ± 35.17</td><td rowspan="2">0.692</td></tr><tr><td>CS</td><td>10208.11 ± 2311.16</td><td>614.14 ± 114.81</td></tr><tr><td rowspan="2">PENDULUM-SWINGUP</td><td rowspan="2">40</td><td>Non-CS</td><td>7677.84 ± 3540.70</td><td rowspan="2">0.258</td><td>327.65 ± 407.80</td><td rowspan="2">0.23</td></tr><tr><td>CS</td><td>10188.70 ± 3858.94</td><td>641.68 ± 237.76</td></tr></table>
+
+Table 7: This table shows the AUC and final performance of SDP (mean ± 95% confidence intervals) for CS and non-CS participants in the human subject study.  
+
+<table><tr><td>Task</td><td>Feedback</td><td>Demographic</td><td>AUC</td><td>P Value</td></tr><tr><td rowspan="2">CARTPOLE-SWINGUP</td><td rowspan="2">48</td><td>FEMALE</td><td>8720.03 ± 3185.4</td><td rowspan="2">0.085</td></tr><tr><td>MALE</td><td>11770.22 ± 1669.96</td></tr><tr><td rowspan="2">PENDULUM-SWINGUP</td><td rowspan="2">40</td><td>FEMALE</td><td>8587.13 ± 2772.96</td><td rowspan="2">0.331</td></tr><tr><td>MALE</td><td>10134.67 ± 4822.22</td></tr><tr><td rowspan="2">CARTPOLE-SWINGUP</td><td rowspan="2">48</td><td>AGE 18-30</td><td>10109.54 ± 2370.61</td><td rowspan="2">0.313</td></tr><tr><td>AGE 50-70</td><td>10923.05 ± 1292.83</td></tr><tr><td rowspan="2">PENDULUM-SWINGUP</td><td rowspan="2">40</td><td>AGE 18-30</td><td>8619.07 ± 3381.48</td><td rowspan="2">0.318</td></tr><tr><td>AGE 50-70</td><td>10607.93 ± 5249.72</td></tr><tr><td rowspan="2">CARTPOLE-SWINGUP</td><td rowspan="2">48</td><td>EDUCATION: BACHELORS OR HIGHER</td><td>11044.55 ± 1783.25</td><td rowspan="2">0.794</td></tr><tr><td>EDUCATION: HIGH SCHOOL DIPLOMA</td><td>6247.99 ± 5186.47</td></tr><tr><td rowspan="2">PENDULUM-SWINGUP</td><td rowspan="2">40</td><td>EDUCATION: BACHELORS OR HIGHER</td><td>10188.88 ± 3858.95</td><td rowspan="2">0.742</td></tr><tr><td>EDUCATION: HIGH SCHOOL DIPLOMA</td><td>7677.85 ± 3540.7</td></tr></table>
+
+Table 8: This table shows the AUC of SDP (mean ± 95% confidence intervals) for different demographic conditions. This table highlights that SDP can be effective irrespective of demographic background (gender, age, and educational background).
+
+Tables 7-8 highlight that there is no significant difference in the performance of SDP when human teachers have differing backgrounds. This includes a background in CS, educational level, age, and gender. We note that we did not perform statistical analysis comparing racial groups as some experiments had only one participant of a particular racial background, making comparisons less meaningful.
+
+# D ADDITIONAL SIMULATED TEACHER RESULTS
+
+For simplicity, in all additional experiments in this section, we only compare SDP + PEBBLE with PEBBLE (or R-PEBBLE).
+
+# D.1 STUDY OF AGENT UPDATE PHASE
+
+Figure 8 emphasizes how the agent update phase does result in new transitions, therefore the teacher provides feedback to transitions that are different from the original sub-optimal transitions used for pre-training.
+
+![](images/c2c4b53232cd793cd54a7e3fdb110b956b1d439f2f76408104c4252514aa4e14.jpg)
+
+![](images/01efd51ec664c86eebc614f60ca4a905db2e956a0f824604cc25c0823a00251e.jpg)
+
+![](images/28f4c5a6e94cf55849a6456fcfe24c71a2737b6fe4eceb11a679168423fd62ca.jpg)
+
+![](images/74825d5075ddc9b7b74e61b871463fdab668d9f3f41876b21049ca35ec83cd89.jpg)  
+Figure 8: Door open, preference learning exp. These plots show how the agent's policy has changed from the reward model pre-training phase (purple histograms) to the agent update phase (blue histograms), thereby resulting in a different distribution for the state features.
+
+![](images/172a5a571e5dbea90c5d0679a151b53c60f00b993de605ea6e30eda0848b1811.jpg)
+
+![](images/746b083d0fb5018b37f4471b076a9bdaeddd60d14ec47447d8b1b8838c881210.jpg)
+
+# D.2 TRUE REWARD VALUES OF RANDOM POLICY
+
+Figure 9 shows the true reward values for sub-optimal data gathered through a random policy in the DMControl suite. This emphasizes that the true reward value is close to the value we pseudo-label the sub-optimal transitions with (i.e., zero), therefore SDP should not yield a large incorrect reward bias.
+
+![](images/5aa8c4f7eaffbab47db2113a346b919b0d02f3550a76537390f04fd4c9344623.jpg)  
+(a) Walker-walk.
+
+![](images/30a0aab10b4381aa3bd5c1f3cd2c12c734d98c54adbd59afba430aca67f94927.jpg)  
+(b) Cheetah-run.
+
+![](images/2ce1d6bcab3868ea6067dbc37993101787a39c2be599a48530000502afcb9cef.jpg)  
+(c) Quadruped-walk.  
+Figure 9: Distribution of true reward values for transitions obtained with a random policy
+
+# D.3 ZERO WEIGHT STUDY
+
+To understand the effect of each component of SDP, we performed two additional ablations. First, in the reward model pre-training phase, the goal is for the reward model to learn to output zero. However, a trivial means to achieve an output of zero is to set all weights and biases in the neural network to zero. Therefore, we compare the full SDP to SDP using a zero-weight initialization as a replacement for the reward model pre-training. We found that using a zero-weight initialization for the reward model in place of the pre-training phase results in significantly degraded performance (see the green curve in Figure 10-right). This is not surprising, as previous works have found that a zero-weight initialization can negatively affect the training of neural networks (Blumenfeld et al., 2020; Zhao et al., 2022). In addition, Figure 11 demonstrates that the reward model pre-training phase does not produce reward model weights of zero, emphasizing why we do not experience the same performance degradation that occurs if we use a zero-weight initialization.
+
+# D.4 FAKE INPUT STUDY
+
+Second, SDP makes use of state, action transitions that are gathered through a sub-optimal policy. Therefore, these are transitions that an agent experiences while interacting with its environment. However, as previously noted, the goal of the reward model pre-training phase is for the reward model to learn to output zero. Therefore, is it necessary for the reward model to pre-train on inputs that are real environment transitions? Instead, can we pre-train the reward model on transitions that did not result from an agent-environment interaction? To test this, we created "fake" inputs of size  $\dim(state) + \dim(action)$ , and for each input dimension, we randomly sampled a value from  $\mathcal{N}(0,1)$ . We obtained 50,000 "fake" transitions and used this data for the reward model pre-training phase. In this experiment, our goal is to understand the effect of the type of inputs on the reward model pre-training phase, therefore we kept the agent update phase as is (i.e., provided the true sub-optimal data for this phase). We then compared SDP using true transitions to SDP using "fake" transitions in Figures 10-left and 12. We found that the full SDP (purple curve) in which we pre-train the reward model on true sub-optimal transitions yields significantly greater final performance than SDP using "fake" transitions (green curve). This highlights the importance of using true sub-optimal transitions in SDP.
+
+Figure 10: Zero weights and fake input studies in Walker-walk  
+![](images/91e96c7656b1c359a4d1e4d0de8528d38ad4f2fb665b0cd924357a3fa47adfd5.jpg)  
+R-PEBBLE SDP, 'Fake transitions SDP
+
+![](images/4f8e9fcaf073156b32639bf21bab0c1c2ecfdda582662f5dfca258da8a68d8ca.jpg)  
+R-PEBBLE SDP,Zero weight init. SDP
+
+![](images/256e146bd09c0d4d23813f38b242d505aabd7846ba1515f4e8671a7d3a2e8256.jpg)  
+(a) Scalar feedback experiment in Walker-walk.
+
+![](images/e14105fd35862bbb52383007097cbdf6fa51c46cbbaf3d2503d842e76d5e18ed.jpg)  
+Figure 11: This figure shows the reward model weights of SDP after the reward model pre-training phase. This demonstrates that the reward model pre-train phase of SDP does not result in zero neural network weights.  
+Cheetah-run, Fake Input Ablation  
+Figure 12: Fake Input Study in Cheetah Run Preference Learning
+
+# D.5 SDP COMPONENT STUDIES
+
+Figure 13 shows additional phase ablations in the preference learning experiments done on the Cheetah-run environment. This highlights the importance of using both phases in SDP. In addition, we ran an experiment in Walker-walk where we directly apply the procedure Yu et al. (2022) uses for leveraging sub-optimal data in the offline RL setting. They simply store the sub-optimal transitions in the RL agent's replay buffer with the pseudo reward label of 0 and follow standard offline RL. We repeated this with the other difference of following standard preference learning (PEBBLE) afterwards. We found that the average final performance was 104.32 with a  $95\%$  confidence interval of 46.12, which is close to 4 times less than the final performance of SDP.
+
+![](images/b8b173328faa413a6084d36ff97ccee199b1c314447e756da30a174c12578be9.jpg)  
+Figure 13: Phase Ablation Study in Cheetah Run Preference Learning
+
+![](images/6b5b095757bf6bc215b94ab148f70bd4a973db29cf728bc2133ba42f14c6a6b8.jpg)  
+Figure 14 shows an additional ablation over the amount of prior data used in SDP. We observed similar performance gains as described in section 5.4.  
+Figure 14: Prior Data Amount Study in Cheetah Run Preference Learning
+
+# D.6 EFFECT OF RELABELLING REPLAY BUFFER
+
+SDP is combined with four preference learning algorithms, PEBBLE, RUNE, SURF, and MRN. A core feature of these algorithms is that every time the reward model is updated, all transitions inside the RL agent's replay buffer are updated using the latest reward model. Figure 15 shows the effects of not relabeling the sub-optimal data (i.e., the reward labels) with the latest learned reward model. This means the reward labels remain frozen at zero throughout the entire training process. This ablation was to demonstrate that if the sub-optimal data in the agent's replay buffer is not updated with the latest reward model, then performance will suffer. This is likely the case because the incorrect reward bias from the pseudo-labeling process persists, whereas when we relabel the transitions with the updated reward model, the incorrect reward bias may reduce over time.
+
+![](images/26a4b33660a95d7996fe6dcf1781f7a573eb1006292ff69a98c0f8e99fec6e5c.jpg)  
+Walker-walk, Relabeling Sub-opt Data Abla  
+Figure 15: Relabeling sub-opt data study: Walker-walk, scalar feedback
+
+# D.7 EFFECT OF TRANSITION QUALITY IN SDP
+
+Moreover, we show that the effectiveness of SDP relies on the use of sub-optimal data transitions. If we use high-quality data transitions (i.e., transitions that came from a fully trained RL agent policy), SDP will fail (see Figure 16). This unsurprising result confirms that pseudo-labeling high-reward transitions with zero can significantly hurt the reward model and the agent's performance
+
+![](images/bf600bcc1e2dca0ff90c27b491755c7d3c5db36d6c71ad2d5db430e309093547.jpg)  
+Figure 16: Using high-quality data in SDP study: Cheetah-run, preference feedback
+
+# D.8 SCALAR-BASED EXPERIMENT STATISTICS
+
+Tables 9 and 10 provide a summary of the mean final performance and the mean area under the curve for all environments and benchmarks in the scalar feedback setting  
+
+<table><tr><td colspan="4">LEARNING FROM SCALAR FEEDBACK</td></tr><tr><td>Task</td><td>SDP</td><td>R-PEBBLE</td><td>DEEP TAMER</td></tr><tr><td>WALKER-WALK</td><td>931.31 ± 36.59</td><td>362.99 ± 181.23*</td><td>33.10 ± 11.10*</td></tr><tr><td>CHEETAH-RUN</td><td>862.72 ± 49.13</td><td>522.10 ± 238.87*</td><td>30.71 ± 16.26*</td></tr><tr><td>QUADRUPED-WALK</td><td>777.38 ± 156.74</td><td>543.92 ± 193.21*</td><td>73.74 ± 48.38*</td></tr></table>
+
+Table 9: This table shows the mean final performance plus and minus the  $95\%$  confidence interval. \* indicates SDP achieves a significantly greater mean final performance.  
+
+<table><tr><td colspan="4">LEARNING FROM SCALAR FEEDBACK</td></tr><tr><td>Task</td><td>SDP</td><td>R-PEBBLE</td><td>DEEP TAMER</td></tr><tr><td>WALKER-WALK</td><td>34981.65 ± 1559.24</td><td>13147.1 ± 6102.31*</td><td>1823.42 ± 578.48*</td></tr><tr><td>CHEETAH-RUN</td><td>55144.89 ± 4266.94</td><td>33557.58 ± 15185.47*</td><td>2459.23 ± 1226.34*</td></tr><tr><td>QUADRUPED-WALK</td><td>58370.85 ± 10759.00</td><td>37076.52 ± 11968.48*</td><td>7365.68 ± 3556.27*</td></tr></table>
+
+Table 10: This table shows the mean area under the learning curve (AUC) plus and minus the  $95\%$  confidence interval. * indicates SDP achieves a significantly greater AUC.
+
+# D.9 PREFERENCE LEARNING EXPERIMENT STATISTICS
+
+Tables 11-14 provide a summary of the mean final performance and the mean area under the curve for all environments and benchmarks in the preference feedback setting.  
+
+<table><tr><td>TASK</td><td>FEEDBACK</td><td>METHOD</td><td>FINAL RETURN</td><td>P VALUE</td></tr><tr><td rowspan="8">CARTPOLE-SWINGUP</td><td rowspan="8">48</td><td>PEBBLE</td><td>226.4 ± 136.35</td><td>0.007*</td></tr><tr><td>SDP + PEBBLE</td><td>533.94 ± 57.71</td><td></td></tr><tr><td>RUNE</td><td>263.48 ± 176.2</td><td></td></tr><tr><td>SDP + RUNE</td><td>561.77 ± 108.17</td><td>0.021*</td></tr><tr><td>SURF</td><td>83.12 ± 44.53</td><td></td></tr><tr><td>SDP + SURF</td><td>324.19 ± 179.8</td><td>0.039*</td></tr><tr><td>MRN</td><td>151.05 ± 76.31</td><td></td></tr><tr><td>SDP + MRN</td><td>644.03 ± 57.99</td><td>0.0*</td></tr><tr><td rowspan="8">WALKER-WALK</td><td rowspan="8">100</td><td>PEBBLE</td><td>173.9 ± 100.26</td><td>0.001*</td></tr><tr><td>SDP + PEBBLE</td><td>490.07 ± 50.07</td><td></td></tr><tr><td>RUNE</td><td>160.6 ± 61.85</td><td>0.0*</td></tr><tr><td>SDP + RUNE</td><td>575.71 ± 104.52</td><td></td></tr><tr><td>SURF</td><td>202.1 ± 95.17</td><td>0.005*</td></tr><tr><td>SDP + SURF</td><td>435.99 ± 39.45</td><td></td></tr><tr><td>MRN</td><td>215.14 ± 58.09</td><td>0.0*</td></tr><tr><td>SDP + MRN</td><td>672.3 ± 85.53</td><td></td></tr><tr><td rowspan="8">CHEETAH-RUN</td><td rowspan="8">200</td><td>PEBBLE</td><td>582.51 ± 67.62</td><td>0.062</td></tr><tr><td>SDP + PEBBLE</td><td>704.63 ± 102.27</td><td></td></tr><tr><td>RUNE</td><td>681.59 ± 68.05</td><td>0.645</td></tr><tr><td>SDP + RUNE</td><td>616.78 ± 279.34</td><td></td></tr><tr><td>SURF</td><td>642.52 ± 84.3</td><td>0.266</td></tr><tr><td>SDP + SURF</td><td>682.34 ± 65.08</td><td></td></tr><tr><td>MRN</td><td>728.36 ± 38.69</td><td>0.305</td></tr><tr><td>SDP + MRN</td><td>748.99 ± 55.59</td><td></td></tr><tr><td rowspan="8">QUADRUPED-WALK</td><td rowspan="8">500</td><td>PEBBLE</td><td>350.24 ± 228.91</td><td>0.017*</td></tr><tr><td>SDP + PEBBLE</td><td>753.18 ± 104.79</td><td></td></tr><tr><td>RUNE</td><td>363.65 ± 163.93</td><td>0.009*</td></tr><tr><td>SDP + RUNE</td><td>704.1 ± 102.16</td><td></td></tr><tr><td>SURF</td><td>550.51 ± 211.19</td><td>0.089</td></tr><tr><td>SDP + SURF</td><td>767.78 ± 140.67</td><td></td></tr><tr><td>MRN</td><td>200.12 ± 63.04</td><td>0.0*</td></tr><tr><td>SDP + MRN</td><td>733.83 ± 62.87</td><td></td></tr></table>
+
+Table 11: This table shows the final performance (mean ± 95% confidence intervals) for all DM-Control preference learning experiments. * indicates significant differences.
+
+<table><tr><td>TASK</td><td>FEEDBACK</td><td>METHOD</td><td>AUC</td><td>P VALUE</td></tr><tr><td rowspan="8">CARTPOLE-SWINGUP</td><td rowspan="8">48</td><td>PEBBLE</td><td>8248.14 ± 2148.59</td><td>0.0*</td></tr><tr><td>SDP + PEBBLE</td><td>21519.9 ± 2435.02</td><td></td></tr><tr><td>RUNE</td><td>8817.21 ± 3534.57</td><td>0.004*</td></tr><tr><td>SDP + RUNE</td><td>20970.54 ± 4643.96</td><td></td></tr><tr><td>SURF</td><td>5593.44 ± 1099.56</td><td>0.074</td></tr><tr><td>SDP + SURF</td><td>10288.22 ± 4584.16</td><td></td></tr><tr><td>MRN</td><td>6774.95 ± 1783.95</td><td>0.0*</td></tr><tr><td>SDP + MRN</td><td>22564.32 ± 3846.6</td><td></td></tr><tr><td rowspan="8">WALKER-WALK</td><td rowspan="8">100</td><td>PEBBLE</td><td>7571.79 ± 2903.09</td><td>0.001*</td></tr><tr><td>SDP + PEBBLE</td><td>18458.81 ± 1224.69</td><td></td></tr><tr><td>RUNE</td><td>6525.52 ± 2566.88</td><td>0.0*</td></tr><tr><td>SDP + RUNE</td><td>21059.76 ± 3056.28</td><td></td></tr><tr><td>SURF</td><td>7675.24 ± 2857.99</td><td>0.001*</td></tr><tr><td>SDP + SURF</td><td>16700.51 ± 1449.62</td><td></td></tr><tr><td>MRN</td><td>7711.05 ± 1295.46</td><td>0.0*</td></tr><tr><td>SDP + MRN</td><td>25184.26 ± 3071.65</td><td></td></tr><tr><td rowspan="8">CHEETAH-RUN</td><td rowspan="8">200</td><td>PEBBLE</td><td>39245.94 ± 5189.16</td><td>0.006*</td></tr><tr><td>SDP + PEBBLE</td><td>51121.68 ± 3488.87</td><td></td></tr><tr><td>RUNE</td><td>45824.63 ± 4459.04</td><td>0.792</td></tr><tr><td>SDP + RUNE</td><td>37202.31 ± 16330.83</td><td></td></tr><tr><td>SURF</td><td>40522.18 ± 2989.22</td><td>0.218</td></tr><tr><td>SDP + SURF</td><td>42777.1 ± 3775.07</td><td></td></tr><tr><td>MRN</td><td>47569.45 ± 2191.57</td><td>0.106</td></tr><tr><td>SDP + MRN</td><td>49968.24 ± 2187.97</td><td></td></tr><tr><td rowspan="8">QUADRUPED-WALK</td><td rowspan="8">500</td><td>PEBBLE</td><td>20584.95 ± 9814.49</td><td>0.018*</td></tr><tr><td>SDP + PEBBLE</td><td>42044.34 ± 11183.84</td><td></td></tr><tr><td>RUNE</td><td>22861.89 ± 9861.31</td><td>0.082</td></tr><tr><td>SDP + RUNE</td><td>32664.79 ± 3929.32</td><td></td></tr><tr><td>SURF</td><td>30884.8 ± 11660.52</td><td>0.074</td></tr><tr><td>SDP + SURF</td><td>44176.44 ± 8404.76</td><td></td></tr><tr><td>MRN</td><td>14805.07 ± 1860.39</td><td>0.002*</td></tr><tr><td>SDP + MRN</td><td>35209.32 ± 6225.09</td><td></td></tr></table>
+
+Table 12: This table shows the AUC (mean ± 95% confidence intervals) for all DMControl preference learning experiments. * indicates significant differences.
+
+<table><tr><td>TASK</td><td>FEEDBACK</td><td>METHOD</td><td>FINAL RETURN</td><td>P VALUE</td></tr><tr><td rowspan="8">HAMMER</td><td rowspan="8">7500</td><td>PEBBLE</td><td>10.0 ± 13.58</td><td>0.003*</td></tr><tr><td>SDP + PEBBLE</td><td>66.0 ± 21.18</td><td></td></tr><tr><td>RUNE</td><td>4.0 ± 7.01</td><td>0.001*</td></tr><tr><td>SDP + RUNE</td><td>68.0 ± 17.0</td><td></td></tr><tr><td>SURF</td><td>0.0 ± 0.0</td><td>0.033*</td></tr><tr><td>SDP + SURF</td><td>36.0 ± 25.16</td><td></td></tr><tr><td>MRN</td><td>4.0 ± 7.01</td><td>0.027*</td></tr><tr><td>SDP + MRN</td><td>46.0 ± 27.5</td><td></td></tr><tr><td rowspan="8">DOOR-UNLOCK</td><td rowspan="8">500</td><td>PEBBLE</td><td>42.0 ± 34.35</td><td>0.066</td></tr><tr><td>SDP + PEBBLE</td><td>80.0 ± 15.68</td><td></td></tr><tr><td>RUNE</td><td>26.0 ± 22.59</td><td>0.02*</td></tr><tr><td>SDP + RUNE</td><td>36.0 ± 38.65</td><td></td></tr><tr><td>SURF</td><td>20.0 ± 35.06</td><td>0.354</td></tr><tr><td>SDP + SURF</td><td>80.0 ± 22.17</td><td></td></tr><tr><td>MRN</td><td>48.0 ± 38.17</td><td>0.399</td></tr><tr><td>SDP + MRN</td><td>56.0 ± 37.02</td><td></td></tr><tr><td rowspan="8">DOOR-LOCK</td><td rowspan="8">500</td><td>PEBBLE</td><td>38.0 ± 30.06</td><td>0.035*</td></tr><tr><td>SDP + PEBBLE</td><td>80.0 ± 7.84</td><td></td></tr><tr><td>RUNE</td><td>62.0 ± 32.51</td><td>0.793</td></tr><tr><td>SDP + RUNE</td><td>40.0 ± 30.87</td><td></td></tr><tr><td>SURF</td><td>70.0 ± 31.85</td><td>0.684</td></tr><tr><td>SDP + SURF</td><td>60.0 ± 13.58</td><td></td></tr><tr><td>MRN</td><td>66.0 ± 26.93</td><td>0.268</td></tr><tr><td>SDP + MRN</td><td>78.0 ± 17.88</td><td></td></tr><tr><td rowspan="8">DRAWER-OPEN</td><td rowspan="8">1000</td><td>PEBBLE</td><td>4.0 ± 7.01</td><td>0.045*</td></tr><tr><td>SDP + PEBBLE</td><td>36.0 ± 25.16</td><td></td></tr><tr><td>RUNE</td><td>0.0 ± 0.0</td><td>0.001*</td></tr><tr><td>SDP + RUNE</td><td>66.0 ± 14.24</td><td></td></tr><tr><td>SURF</td><td>20.0 ± 35.06</td><td>0.018*</td></tr><tr><td>SDP + SURF</td><td>80.0 ± 14.67</td><td></td></tr><tr><td>MRN</td><td>20.0 ± 35.06</td><td>0.136</td></tr><tr><td>SDP + MRN</td><td>56.0 ± 40.21</td><td></td></tr><tr><td rowspan="8">WINDOW-OPEN</td><td rowspan="8">200</td><td>PEBBLE</td><td>34.0 ± 37.44</td><td>0.234</td></tr><tr><td>SDP + PEBBLE</td><td>54.0 ± 26.35</td><td></td></tr><tr><td>RUNE</td><td>12.0 ± 21.04</td><td>0.014*</td></tr><tr><td>SDP + RUNE</td><td>38.0 ± 24.42</td><td></td></tr><tr><td>SURF</td><td>14.0 ± 20.44</td><td>0.098</td></tr><tr><td>SDP + SURF</td><td>66.0 ± 26.35</td><td></td></tr><tr><td>MRN</td><td>46.0 ± 32.61</td><td>0.208</td></tr><tr><td>SDP + MRN</td><td>68.0 ± 31.06</td><td></td></tr></table>
+
+Table 13: This table shows the final performance (mean ± 95% confidence intervals) for all Meta-world preference learning experiments. * indicates significant differences.
+
+<table><tr><td>TASK</td><td>FEEDBACK</td><td>METHOD</td><td>AUC</td><td>P VALUE</td></tr><tr><td rowspan="8">HAMMER</td><td rowspan="8">7500</td><td>PEBBLE</td><td>424.0 ± 175.18</td><td>0.017*</td></tr><tr><td>SDP + PEBBLE</td><td>2222.0 ± 1009.47</td><td></td></tr><tr><td>RUNE</td><td>490.0 ± 300.85</td><td>0.001*</td></tr><tr><td>SDP + RUNE</td><td>2520.0 ± 627.47</td><td></td></tr><tr><td>SURF</td><td>482.0 ± 215.59</td><td>0.006*</td></tr><tr><td>SDP + SURF</td><td>3042.0 ± 1073.54</td><td></td></tr><tr><td>MRN</td><td>494.0 ± 311.76</td><td>0.016*</td></tr><tr><td>SDP + MRN</td><td>2494.0 ± 1108.52</td><td></td></tr><tr><td rowspan="8">DOOR-UNLOCK</td><td rowspan="8">500</td><td>PEBBLE</td><td>2142.0 ± 1738.06</td><td>0.031*</td></tr><tr><td>SDP + PEBBLE</td><td>4838.0 ± 1251.13</td><td></td></tr><tr><td>RUNE</td><td>1272.0 ± 1275.71</td><td>0.047*</td></tr><tr><td>SDP + RUNE</td><td>1836.0 ± 2063.68</td><td></td></tr><tr><td>SURF</td><td>1580.0 ± 2462.83</td><td>0.348</td></tr><tr><td>SDP + SURF</td><td>4802.0 ± 1537.81</td><td></td></tr><tr><td>MRN</td><td>3082.0 ± 2232.43</td><td>0.342</td></tr><tr><td>SDP + MRN</td><td>3842.0 ± 2230.38</td><td></td></tr><tr><td rowspan="8">DOOR-LOCK</td><td rowspan="8">500</td><td>PEBBLE</td><td>876.0 ± 647.87</td><td>0.008*</td></tr><tr><td>SDP + PEBBLE</td><td>2300.0 ± 135.0</td><td></td></tr><tr><td>RUNE</td><td>1342.0 ± 590.05</td><td>0.809</td></tr><tr><td>SDP + RUNE</td><td>884.0 ± 633.87</td><td></td></tr><tr><td>SURF</td><td>1892.0 ± 842.85</td><td>0.429</td></tr><tr><td>SDP + SURF</td><td>1986.0 ± 184.83</td><td></td></tr><tr><td>MRN</td><td>1966.0 ± 751.51</td><td>0.306</td></tr><tr><td>SDP + MRN</td><td>2236.0 ± 479.9</td><td></td></tr><tr><td rowspan="8">DRAWER-OPEN</td><td rowspan="8">1000</td><td>PEBBLE</td><td>314.0 ± 533.02</td><td>0.117</td></tr><tr><td>SDP + PEBBLE</td><td>1702.0 ± 1706.12</td><td></td></tr><tr><td>RUNE</td><td>38.0 ± 49.71</td><td>0.004*</td></tr><tr><td>SDP + RUNE</td><td>3794.0 ± 1306.6</td><td></td></tr><tr><td>SURF</td><td>696.0 ± 1167.88</td><td>0.015*</td></tr><tr><td>SDP + SURF</td><td>4078.0 ± 1839.07</td><td></td></tr><tr><td>MRN</td><td>1550.0 ± 2690.98</td><td>0.211</td></tr><tr><td>SDP + MRN</td><td>3206.0 ± 2114.16</td><td></td></tr><tr><td rowspan="8">WINDOW-OPEN</td><td rowspan="8">200</td><td>PEBBLE</td><td>794.0 ± 1015.04</td><td>0.131</td></tr><tr><td>SDP + PEBBLE</td><td>1698.0 ± 822.8</td><td></td></tr><tr><td>RUNE</td><td>230.0 ± 334.14</td><td>0.01*</td></tr><tr><td>SDP + RUNE</td><td>948.0 ± 631.92</td><td></td></tr><tr><td>SURF</td><td>242.0 ± 248.44</td><td>0.064</td></tr><tr><td>SDP + SURF</td><td>2048.0 ± 867.93</td><td></td></tr><tr><td>MRN</td><td>1252.0 ± 796.8</td><td>0.223</td></tr><tr><td>SDP + MRN</td><td>1728.0 ± 666.48</td><td></td></tr></table>
+
+Table 14: This table shows the area under the curve (mean ± 95% confidence intervals) for all Metaworld preference learning experiments. * indicates significant differences.
\ No newline at end of file
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+# LEVERAGING SUBMODULE LINEARITY ENHANCES TASK ARITHMETIC PERFORMANCE IN LLMS
+
+Rui Dai  $^{1}$  Sile Hu  $^{2}$  Xu Shen  $^{*2}$  Yonggang Zhang  $^{3}$  Xinmei Tian  $^{*1}$  Jieping Ye  $^{2}$
+
+$^{1}$ National Engineering Laboratory for Brain-Inspired Intelligence Technology and Application, University of Science and Technology of China  
+$^{2}$ Independent Researcher  
+$^{3}$ Hong Kong Baptist University  
+*shenxuustc@gmail.com, xinmei@ustc.edu.cn
+
+# ABSTRACT
+
+Task arithmetic is a straightforward yet highly effective strategy for model merging, enabling the resultant model to exhibit multi-task capabilities. Recent research indicates that models demonstrating linearity enhance the performance of task arithmetic. In contrast to existing methods that rely on the global linearization of the model, we argue that this linearity already exists within the model's submodules. In particular, we present a statistical analysis and show that submodules (e.g., layers, self-attentions, and MLPs) exhibit significantly higher linearity than the overall model. Based on these findings, we propose an innovative model merging strategy that independently merges these submodules. Especially, we derive a closed-form solution for optimal merging weights grounded in the linear properties of these submodules. Experimental results demonstrate that our method consistently outperforms the standard task arithmetic approach and other established baselines across different model scales and various tasks. This result highlights the benefits of leveraging the linearity of submodules and provides a new perspective for exploring solutions for effective and practical multi-task model merging.
+
+# 1 INTRODUCTION
+
+In recent years, the growing scale of large language models has significantly increased the demand for data and training costs in fine-tuning multi-task models. To integrate the capabilities of various existing single-task models into a unified model, various model merging techniques have been developed (Yu et al., 2024; Jin et al., 2023; Matena & Raffel, 2022; Yadav et al., 2023). Task arithmetic (Ilharco et al., 2023) is one of the simplest and most efficient model merging strategies. It involves a straightforward weighted combination of the weights from single-task models, allowing the final merged model to exhibit multi-task capabilities without the need for additional training or data.
+
+Recent studies (Ortiz-Jiménez et al., 2023; Zhou et al., 2024) have revealed a connection between the linearity of fine-tuned models and the effectiveness of the weighting operations in task arithmetic. Here, linearity specifically refers to the linear relationship between the differences in model weights and the differences in output features caused by fine-tuning $^{1}$  (Ortiz-Jiménez et al., 2023; Zhou et al., 2024), which is very different from the Traditional Linear Properties $^{2}$ . This linearity intuitively aligns with the linear weighting operations in task arithmetic (Ilharco et al., 2023). Research (Ortiz-Jiménez et al., 2023; Tang et al., 2024; Jin et al., 2024; Liu et al., 2024) has shown that models exhibiting linearity retain their individual task performance better when merged using task arithmetic, leading to superior multi-task models. However, models produced through conventional fine-tuning often lack this ideal linear property (Ortiz-Jiménez et al., 2023). To address this issue, existing methods (Ortiz-Jiménez et al., 2023; Tang et al., 2024; Jin et al., 2024; Liu et al., 2024) have
+
+![](images/0413e94615b17d468672b1cf8fa5144b1e54660ef595f8e8071bd3d17dd7888b.jpg)  
+(a) Non-linearity on fine-tuned models
+
+![](images/0f708ba09613e6569a531f5514302b1482a8c361e9aaf5af7a3891e5062096dd.jpg)  
+(b) Non-linearity in the context of model merging  
+Figure 1: (a) Comparison of non-linearity in full models and submodules for fine-tuned models, measured using Non-linearity Score defined in Definition 2. A lower value indicates better linearity, based on data from three fine-tuned models detailed in Section 4.1. (b) Comparison of non-linearity in full models and submodules within the context of model merging, assessed via Eq.8 (Zhou et al., 2024). A lower value indicates better linearity, based on data from three merging combinations outlined in the caption of Figure 3. Error bars represent standard deviation.
+
+explored adjustments to the fine-tuning paradigm, retraining the model using original training data to enhance the linearity and the final merging outcomes.
+
+In contrast to existing methods that rely on global linearization of the model, which impose requirements for retraining with the original data, we argue that this linearity has already existed within the model's submodules. Intuitively, the most straightforward approach to seek linearity from a complex nonlinear function is to decompose it into simpler ones. Following this idea, we divide the model into multiple shallower submodules (e.g., layers, self-attentions, MLPs) and discover that these submodules exhibit a level of linearity that significantly surpasses that of the overall model. Figure 1 compares the linearity characteristics of full models and their constituent modules. In panel (a), the non-linearity of full models is significantly higher than that of the decomposed submodules, which exhibit scores close to zero, indicating their strong linearity. In panel (b), we further evaluate the linear properties in the context of model merging $^3$  using the metric introduced by Zhou et al. (2024). The results reveal a distinct gap in non-linearity between full models and submodules, suggesting that the latter can demonstrate good linearity even when the full models do not.
+
+Inspired by these observations, we propose an innovative training-free model merging strategy: 1) Decompose the models into submodules that exhibit linear characteristics. 2) Compute the optimal merging weights for each submodule by leveraging their linear properties. 3) Apply these derived weights to facilitate the merging of the models. To obtain the optimal merging weights under the linearity assumption of the modules, we derived a closed-form solution by minimizing the distance between the output features of the merged module and those of the corresponding module across fine-tuned tasks (Jin et al., 2023). Notably, this computational process necessitates only a minimal amount of data (30 samples per task) and a limited number of model inference iterations.
+
+To validate the effectiveness of our proposed method, we conducted merging experiments across models fine-tuned from Llama-2-7B and Llama-2-13B (Touvron et al., 2023) for three distinct tasks: Math, Coding, and Translate. The experimental results demonstrate that our method significantly outperforms various baseline techniques, including standard arithmetic approaches, across different model scales and diverse tasks, particularly with respect to the decomposition level of layers and attn/MLPs. These findings provide strong evidence that leveraging the linearity inherent in submodules holds substantial potential for enhancing model merging performance with task arithmetic.
+
+# 2 ANALYZING THE LINEARITY OF MODEL AND MODULES
+
+In this section, we first introduce some definitions regarding linearity (Ortiz-Jiménez et al., 2023) and task arithmetic (Ilharco et al., 2023). We then present a score that can quantitatively compare the
+
+linearity of models and modules. Subsequently, following the insight outlined in the introduction, we divide the model into submodules to examine linearity, discovering that the linearity of the module surpasses that of the model. Finally, we further validate the linearity when merging multiple modules, laying a foundation for the methodologies proposed in the next section.
+
+# 2.1 PRELIMINARY
+
+Notation. Let  $f: X \times \Theta \to Y$  be a neural network taking inputs  $x \in X$  and parameterized by a set of weights  $\theta \in \Theta$ . Consider  $T$  tasks, where  $D_{t} \subseteq X$  is a dataset used to fine-tune the models starting from the pre-trained weights  $\theta_0$  and obtain the fine-tuned weights  $\theta_{t}$ .
+
+Task Arithmetic. (Ilharco et al., 2023) Task arithmetic is a simple, efficient, and widely used model merging strategy. It assigns multi-task performance to the merged model through a simple weighted combination of the weights of models, without the need for additional training or data support.
+
+Task vector for task  $t$  is defined as the difference between the fine-tuned and the pre-trained weights, namely,  $\tau_{t} = \theta_{t} - \theta_{0}$ . The approach of task arithmetic involves a weighted combination of the task vectors, which is then added to the pre-trained model  $\theta_{0}$ . This can be expressed as
+
+$$
+\theta_ {\text {m e r g e}} = \theta_ {0} + \sum_ {t = 1} ^ {T} \alpha_ {t} \tau_ {t}, \tag {1}
+$$
+
+where  $\alpha_{t}$  represents the weight corresponding to  $\tau_{t}$ .
+
+Here, linearity specifically refers to the linear relationship between the differences in model weights and the differences in output features caused by fine-tuning
+
+Definition 1 (Linearity). Let  $\theta$  be a fine-tuned model and  $\theta_0$  be its corresponding pre-trained model. If we assert that  $\theta$  exhibits linearity, the differences in model weights caused by fine-tuning are linearly related to the differences in output features caused by fine-tuning for any  $x \in X$ , i.e.
+
+$$
+f (x; \theta_ {0} + \alpha \tau) \approx f (x; \theta_ {0}) + \alpha \Delta f (x; \theta_ {0} + \tau) \tag {2}
+$$
+
+where  $\tau = \theta -\theta_0$  is the differences in model weights before and after fine-tuning, and  $\Delta f(x;\theta_0+$ $\tau) = f(x;\theta_0 + \tau) - f(x;\theta_0)$  is the differences in model output features before and after fine-tuning. It is very different from the Traditional Linear Properties.4
+
+Research (Ortiz-Jiménez et al., 2023; Jin et al., 2024) has shown that models demonstrating linearity retain higher single-task performance when merged via task arithmetic, leading to enhanced multi-task models. However, models developed through conventional fine-tuning methods typically lack this desired linearity (Ortiz-Jiménez et al., 2023).
+
+Proposed Linearity Metric. The interpolation model  $\theta_{\mathrm{inter}}$  between the model  $\theta$  and the pre-trained model  $\theta_0$  lies on the line connecting  $\theta$  and  $\theta_0$  in the parameter space. If the model  $\theta$  satisfies linearity in Definition 1, then its output features  $f(\theta_{\mathrm{inter}},x)$  should also lie on the line formed by the output features  $f(\theta ,x)$  and  $f(\theta_0,x)$ . Moreover, the distance between  $f(\theta_{\mathrm{inter}},x)$  and  $f(\theta_0,x)$  should be proportional to the interpolation weight. Based on this idea, to measure and compare the linearity of models quantitively, we propose the Non-linearity Score.
+
+Definition 2 (Non-linearity Score). For a fine-tuned model  $\theta$  and its corresponding pre-trained model  $\theta_0$ , if  $\theta$  satisfies linearity in Definition 1 under any distance metric  $\mathcal{D}(\boldsymbol{a},\boldsymbol{b}) = \| \boldsymbol{a} - \boldsymbol{b}\|$  in the feature space, where  $\| \cdot \|$  denotes the norm operation, then for any  $\alpha, \beta \in [0,1]$ , the following approximation holds:
+
+$$
+\mathcal {D} \left(f \left(x; \theta_ {0} + \alpha \tau\right), f \left(x; \theta_ {0} + \beta \tau\right)\right) \approx | \alpha - \beta | \mathcal {D} \left(f (x; \theta), f (x; \theta_ {0})\right) \tag {3}
+$$
+
+where  $\tau = \theta -\theta_0$
+
+When we replace the approximation in Eq.3 with equality, it indicates a state of perfect linearity. Therefore, we define the Non-linearity Score of model  $\theta$  as the measure of the deviation of model  $\theta$  from this ideal linear condition:
+
+$$
+\text {N o n - l i n e a r i t y S c o r e} = \int_ {0} ^ {1} \int_ {0} ^ {1} \left(\frac {\mathcal {D} (f (x ; \theta_ {0} + \alpha \tau) , f (x ; \theta_ {0} + \beta \tau))}{\mathcal {D} (f (x ; \theta) , f (x ; \theta_ {0}))} - | \alpha - \beta |\right) ^ {2} d \alpha d \beta \tag {4}
+$$
+
+We calculated the Non-linearity Score  ${}^{5}$  for models fine-tuned on various tasks and presented the results in Figure 2. Our observations indicate that nearly all models exhibit a lack of satisfactory linearity at the model level. This finding aligns with conclusions drawn in other research OrtizJiménez et al. (2023) employing different methodologies.
+
+# 2.2 ANALYZING THE LINEARITY OF SINGLE FINE-TUNED MODELS
+
+Following the idea of seeking linearity by decomposition, we divide the entire model according to its architecture into multiple levels, including layers, self-attention, MLP, and head levels. Analogous to Definition 1, the linearity at the module level can be expressed as follows:
+
+Property 1 (Linearity of Module). Let  $\theta^i$  be a module weight from the fine-tuned model  $\theta$  and  $\theta_0^i$  be its corresponding module's weight from the pre-trained model. If we assert that  $\theta^i$  exhibits linearity, the differences in module weights caused by fine-tuning are linearly related to the differences in this module's output features caused by fine-tuning for any  $x \in H^i$ , i.e.
+
+$$
+f \left(x; \theta_ {0} ^ {i} + \alpha \tau^ {i}\right) \approx f \left(x; \theta_ {0} ^ {i}\right) + \alpha \Delta f \left(x; \theta_ {0} ^ {i} + \tau^ {i}\right) \tag {5}
+$$
+
+where  $\tau^i = \theta^i -\theta_0^i$  is the differences in module weight before and after fine-tuning, and  $\Delta f(x;\theta_0^i +$ $\tau^i) = f(x;\theta_0^i +\tau^i) - f(x;\theta_0^i)$  is the differences in module output features before and after fine-tuning with same input  $x\in H^{i}$ $H^{i}$  corresponds to the input feature space of module i.6
+
+![](images/9eeb8905bbb0ec15e6abc08188d6c086a654015fcde4760c79fedc0229f3fd48.jpg)  
+(a) Llama-2-7B
+
+![](images/74652e38f9e3b7b29a7b79adaec967e01c4987d942c07ae049ba817a0c5c08e2.jpg)  
+(b) Llama-2-13B  
+Figure 2: Comparison of Non-linearity Score in full models and submodules for models fine-tuned from tasks of Math, Coding, and Translate in Llama-2-7B and Llama-2-13B. A lower value indicates better linearity. Error bars represent standard deviation across different modules.
+
+Linearity in Models and Modules. We can employ the same Non-Linearity Score defined in 2 to assess the degree of satisfaction concerning this property for multiple fine-tuned models and different sets of corresponding submodules. In Figures 2(a) and (b), we compare the average linearity at both the module level and model level across models fine-tuned on three distinct tasks, on two different scales of Llama-2 backbones. Several observations can be derived from these figures: 1) All submodules exhibit a degree of linearity that significantly exceeds that of the overall model. 2) The Non-Linearity scores for all submodules are near zero, suggesting an adequate approximation as indicated in Eq.5. 3) The pronounced linearity disparity between the model level and the module level remains consistent across all three fine-tuned tasks and both scales of the model.
+
+In conclusion, these observations provide compelling evidence that the modules can exhibit linearity even when the models do not. This suggests a considerable opportunity to leverage the linear properties at the submodule level when integrating fine-tuned models with task arithmetic, even if the overall model does not exhibit the same degree of linearity.
+
+# 2.3 ANALYZING THE LINEARITY OF MERGING MULTIPLE FINE-TUNED MODELS
+
+Property 1 pertains to the linearity of modules within a single fine-tuned model. However, in the context of model merging, multiple fine-tuned models are involved. Consequently, we extend Property 1 to encompass multiple models:
+
+Property 2 (Linearity of Merged Modules). Let  $\theta^i$  be a module weight from the fine-tuned model  $\theta$  and  $\theta_0^i$  be its corresponding module's weight from the pre-trained model. If we assert that  $\theta^i$  exhibits linearity of merging, we imply that the differences in output features originating from different models can be linearly separated, i.e., for any  $x \in H^i$ ,
+
+$$
+f \left(x; \theta_ {0} ^ {i} + \sum_ {t = 1} ^ {T} \alpha_ {t} ^ {i} \tau_ {t} ^ {i}\right) \approx f \left(x; \theta_ {0} ^ {i}\right) + \sum_ {t = 1} ^ {T} \alpha_ {t} ^ {i} \Delta f \left(x; \theta_ {0} ^ {i} + \tau_ {t} ^ {i}\right) \tag {6}
+$$
+
+where  $\tau_{t}^{i} = \theta_{t}^{i} - \theta_{0}^{i}$  is the differences in module  $i$  weight from model  $t$  and  $\Delta f(x;\theta_0^i +\tau_t^i) = f(x;\theta_0^i +\tau_t^i) - f(x;\theta_0^i)$  is the differences in module output features before and after fine-tuning.
+
+Assesing Linearity in the Context of Model Merging. Property 2 is only feasible in the presence of Property 1 for all corresponding models; however, its occurrence is not guaranteed. The combination of multiple models fine-tuned for distinct tasks may introduce additional non-linearity due to the approximations inherent in these equations. To ascertain whether the previously mentioned observations regarding linearity remain applicable within the context of model merging, we adopted the methodology delineated by Zhou et al. (2024), which evaluates linearity using a feature space metric. For clarity and convenience, we refer to this metric as Projection Distance in this paper.
+
+Specifically, we firstly calculate the cosine similarity between two sets of feature differences: the differences between the output features of the merged module (derived from multiple fine-tuned models) and those of the pre-trained module, denoted as  $\Delta f(x; \theta_0^i + \sum_{t=1}^{T} \alpha_t^i \tau_t^i) = f(x; \theta_0^i + \sum_{t=1}^{T} \alpha_t^i \tau_t^i) - f(x; \theta_0^i)$ , as well as the corresponding weighted combination of the differences in output features from each fine-tuned model, represented as  $\sum_{t=1}^{T} \alpha_t^i \Delta f(x; \theta_0^i + \tau_t^i) = \sum_{t=1}^{T} \alpha_t(f(x; \theta_0^i + \tau_t^i) - f(x; \theta_0^i))$ . This similarity is denoted by the following equation
+
+$$
+\operatorname {c o s i n e} _ {\text {m e r g e}} (x; \alpha_ {1 \dots T} ^ {i}) = \cos \left(\Delta f \left(x; \theta_ {0} ^ {i} + \sum_ {t = 1} ^ {T} \alpha_ {t} ^ {i} \tau_ {t} ^ {i}\right), \sum_ {t = 1} ^ {T} \alpha_ {t} ^ {i} \Delta f \left(x; \theta_ {0} ^ {i} + \tau_ {t} ^ {i}\right)\right) \tag {7}
+$$
+
+where  $\cos (\pmb {a},\pmb {b}) = \pmb {a}\cdot \pmb {b} / (\| \pmb {a}\| \| \pmb {b}\|)$  is the Cosine Similarity. As follow, the Projection Distance  $\mathrm{proj}_{merge}^{\mathbb{E}}(\alpha_{1\dots T}^{i})$  between these two sets of feature deltas is represented as
+
+$$
+\operatorname {p r o j} _ {\text {m e r g e}} ^ {\mathbb {E}} \left(\alpha_ {1 \dots T} ^ {i}\right) = \left| 1 - \mathbb {E} _ {x \in H ^ {i}} \frac {\left\| \Delta f \left(x ; \theta_ {0} ^ {i} + \sum_ {t = 1} ^ {T} \alpha_ {t} ^ {i} \tau_ {t} ^ {i}\right) \right\| \operatorname {c o s i n e} _ {\text {m e r g e}} \left(x ; \alpha_ {1 \dots T} ^ {i}\right)}{\left\| \sum_ {t = 1} ^ {T} \alpha_ {t} ^ {i} \Delta f \left(x ; \theta_ {0} ^ {i} + \tau_ {t} ^ {i}\right)\right) \|} \right| \tag {8}
+$$
+
+where  $\mathbb{E}$  represents expectation and  $H^{i}$  represents the input feature set corresponding to module  $i$ , with further details provided in Appendix A.3.3.
+
+Results of Projection Distance<sup>7</sup>. In Figure 3 (a) and (b), we present the projection distances resulting from the merging of two and three models. Both configurations reveal a trend that submodules exhibit superior linearity compared to the model level, consistent with the observations made for single models presented in Figure 2. It is noteworthy that the projection distance for merging three models is significantly lower than that observed for the merging of two models across layers, attention, and MLPs. This suggests that there exists an even more favorable condition for linear merging
+
+![](images/c4add3ff2e7fae4d1f90507f070e6f7bb598686a7a458d121ad9db28cacc939c.jpg)  
+Figure 3: Comparison of Projection Distances in full models and submodules within the context of model merging. A lower value indicates better linearity. For each backbone, we computed the results across all possible combinations of three fine-tuned models, alongside the corresponding merging weights  $\alpha_{1}$  and  $\alpha_{2}$  (and  $\alpha_{3}$ ). In the case of merging two models, there are three combinations of fine-tuned models, yielding 25 possible configurations for merging weights  $\alpha_{t} \in [0.2, 0.4, 0.6, 0.8, 1]$  where  $t \in [1, 2]$ . For merging three models, there's only one combination of fine-tuned models, resulting in 27 configurations for merging weights  $\alpha_{t} \in [0.3, 0.5, 0.7]$  with  $t \in [1, 2, 3]$ . Error bars represent standard deviations.
+
+![](images/b6abd66f0197fa88201a6ab1fd1837d38246408bc32d050d6b9d67469cd7062b.jpg)
+
+when three models are considered compared to when only two models are merged, particularly when analyzed at these granular levels.
+
+In summary, these findings validate the applicability of Property 2 within submodules, demonstrating its relevance to the merging of both two and three models. As illustrated in Section 3, this property facilitates a closed-form solution for determining the optimal merging weights, thereby enhancing the task arithmetic performance while minimizing associated costs.
+
+# 3 MERGING MODULES WITH LINEARITY
+
+The selection of merging weights is a pivotal factor influencing model merging performance in task arithmetic. In (Ilharco et al., 2023), the authors propose determining this parameter through ablation while maintaining equal weights across different models to constrain the search space within a manageable range. Although this method demonstrates a commendable balance between cost and efficiency at the model level, it becomes impractical when merging models at the module level. In such cases, the search space expands exponentially relative to the number of modules; for instance, abating  $n$  candidate weights results in a search space of  $n^m$  for  $m$  modules. Fortunately, by leveraging Property 2, we can derive a closed-form solution for the optimal merging weights, thereby circumventing the cumbersome ablation process and expanding the solution space by relaxing the constraint of equal weights. Here we first introduce a commonly used objective in model merging (Jin et al., 2023; Ortiz-Jimenez et al., 2023), which is also regarded as the objective of task arithmetic (Ortiz-Jimenez et al., 2023).
+
+The Objective of Model Merging. Let  $D = \{D_t \subset X\}_{t \in [T]}$  be a collection of  $T$  datasets, with  $\theta_1, \ldots, \theta_T$  representing the parameters of models fine-tuned from  $\theta_0$  using the corresponding datasets. The parameters resulting from the merging process are denoted by  $\theta_{\mathrm{merge}}$ . The objective of model merging is defined as:
+
+$$
+f (x; \theta_ {\text {m e r g e}}) = \left\{ \begin{array}{l l} f (x; \theta_ {t}) & \text {i f} x \in D _ {t} \\ f (x; \theta_ {0}) & \text {i f} x \notin \bigcup_ {t = 1} ^ {T} D _ {t} \end{array} \right. \tag {9}
+$$
+
+Next, we will independently apply the objective to each module that satisfies linearity, using standard weighted operations of task arithmetic, i.e.,  $\theta_{\mathrm{merge}}^i = \theta_0^i +\sum_{t = 1}^T\alpha_t^i\tau_t^i$
+
+Objectives Function for Each Module. For a module located at position  $i$  (which could be a layer/attention/MLP/head), let  $\theta_t^i$  represent the weights of this module from model  $t$ , and  $f(x; \theta_t^i)$  denote the output of this module given input  $x$ . The merging parameters  $\alpha_1^i, \alpha_2^i, \ldots, \alpha_T^i$  for  $\theta_1^i, \theta_2^i, \ldots, \theta_T^i$  can be obtained by
+
+$$
+\alpha_ {1} ^ {i}, \alpha_ {2} ^ {i}, \dots , \alpha_ {T} ^ {i} = \arg \min  \sum_ {t = 1} ^ {T} \mathbb {E} _ {x \in H _ {t} ^ {i}} \left\| f \left(x; \theta_ {0} ^ {i} + \sum_ {t ^ {\prime} = 1} ^ {T} \alpha_ {t ^ {\prime}} ^ {i} \tau_ {t ^ {\prime}} ^ {i}\right) - f (x; \theta_ {t} ^ {i}) \right\| ^ {2} \tag {10}
+$$
+
+where  $\tau_t^i = \theta_t^i -\theta_0^i$ ,  $H_{t}^{i}$  represent the set of input features of module  $i$  corresponding to  $D_{t}$ . In this paper,  $H_{t}^{i}$  is obtained by inputting all  $x\in D_{t}$  into the model  $\theta_0$  and collecting the input features corresponding to module  $i$ , more details please refer to Appendix A.3.3.
+
+If module  $i$  satisfies linearity, by applying Property 1 and Property 2, we have
+
+$$
+\alpha_ {1} ^ {i}, \alpha_ {2} ^ {i}, \dots , \alpha_ {T} ^ {i} = \arg \min  \sum_ {t = 1} ^ {T} \mathbb {E} _ {x \in H _ {t} ^ {i}} \left\| \sum_ {t ^ {\prime} = 1, t ^ {\prime} \neq t} ^ {T} \alpha_ {t ^ {\prime}} ^ {i} \Delta f (x; \theta_ {t ^ {\prime}} ^ {i}) + (\alpha_ {t} ^ {i} - 1) \Delta f (x; \theta_ {t} ^ {i}) \right\| ^ {2} \tag {11}
+$$
+
+where  $\Delta f(x;\theta_t^i) = f(x;\theta_t^i) - f(x;\theta_0^i)$
+
+Solving this equation allows us to directly obtain the closed-form solutions for  $\alpha_{1}^{i},\alpha_{2}^{i},\ldots ,\alpha_{T}^{i}$
+
+$$
+\left[ \alpha_ {1} ^ {i}, \alpha_ {2} ^ {i}, \dots , \alpha_ {T} ^ {i} \right] ^ {\top} = \boldsymbol {A} ^ {- 1} \boldsymbol {b} \tag {12}
+$$
+
+where  $A\subseteq \mathbb{R}^{T\times T}$  and  $\pmb {b}\subseteq \mathbb{R}^T$
+
+$$
+\boldsymbol {A} _ {j, k} = \sum_ {t = 1} ^ {T} \boldsymbol {B} _ {t j k} ^ {i}, \quad \boldsymbol {b} _ {j} = \sum_ {t = 1} ^ {T} \boldsymbol {B} _ {t j t} ^ {i}, \tag {13}
+$$
+
+and
+
+$$
+\boldsymbol {B} _ {a b c} ^ {i} = \mathbb {E} _ {x \in H _ {a} ^ {i}} \sum_ {k = 1} ^ {d} \Delta f (x; \theta_ {b} ^ {i}) _ {k} \Delta f (x; \theta_ {c} ^ {i}) _ {k}, \tag {14}
+$$
+
+$$
+\Delta f (x; \theta_ {t} ^ {i}) = f (x; \theta_ {t} ^ {i}) - f (x; \theta_ {0} ^ {i}).
+$$
+
+where  $d$  denote the dimension of vector  $f(x;\theta^i)$ . The detailed derivation process can be found in the Appendix A.3.1.
+
+We present our merging process at the layer-level decomposition detailed in Algorithm 1. Initially, the input feature sets  $H_{t}^{i}$  are compiled for all layers by feeding a randomly sampled subset from the task-related datasets into the pre-trained model  $\theta_0$ . Subsequently, for each layer, we apply Eq.12 to derive the optimal merging weights, which are then utilized to perform linear merging. The adaptation of this algorithm to other granularities is straightforward, as described in Appendix A.3.3.
+
+# Algorithm 1 Merging Modules with Linearity at Layer Level
+
+Input:  $T$  Fine-tuned Models with  $L$  layers  $\theta_{1} = \{\theta_{1}^{1},\theta_{1}^{2},\dots,\theta_{1}^{L}\} ,\dots,\theta_{T} = \{\theta_{T}^{1},\theta_{T}^{2},\dots,\theta_{T}^{L}\}$
+
+Pre-trained Model  $\theta_0 = \{\theta_0^1, \theta_0^2, \dots, \theta_0^L\}$ , Task-Related Datasets  $D_1, D_2, \dots, D_T$
+
+Output: Merged Model  $\theta_{merge} = \{\theta_{merge}^{1},\theta_{merge}^{2},\dots,\theta_{merge}^{L}\}$
+
+for  $t = 1$  to  $T$  do
+
+$d_{t}\gets \{x\in D_{t}  |   x$  is sampled randomly,  $|d_t| = N\}$  ▷ Sample a small set of task-related data
+
+$$
+H _ {t} ^ {1} \leftarrow d _ {t}
+$$
+
+for  $i = 1$  to  $L - 1$  do
+
+$$
+H _ {t} ^ {i + 1} \leftarrow f (x; \theta_ {0} ^ {i}), \forall x \in H _ {t} ^ {i}
+$$
+
+for  $i = 1$  to  $L$  do
+
+$$
+\alpha_ {1} ^ {i}, \alpha_ {2} ^ {i}, \dots , \alpha_ {T} ^ {i} \leftarrow \text {E q . 1 2}
+$$
+
+$$
+\theta_ {m e r g e} ^ {i} \leftarrow \theta_ {0} ^ {i} + \sum_ {t = 1} ^ {T} \alpha_ {t} ^ {i} \left(\theta_ {t} ^ {i} - \theta_ {0} ^ {i}\right)
+$$
+
+Prepare input feature sets for Eq.12
+
+$\triangleright$  Calculate optimal merging weights with Eq.12
+
+Merge modules from different models linearly
+
+# 4 EXPERIMENTS
+
+# 4.1 EXPERIMENTS SETUP
+
+Models, Datasets and Evaluation Metrics. We utilize Llama-2-7B and Llama-2-13B as our backbone models. Fine-tuning is conducted across three distinct tasks: mathematics, coding, and translation. For the mathematics task, we employ the GSM8K (Cobbe et al., 2021) training set. For the coding task, we utilize the Code Alpaca (Chaudhary, 2023) dataset. Lastly, for the translation task, we apply the zh  $\leftrightarrow$  en dataset from Xu et al. (2024a). During training, we adopt the prompt of the
+
+Table 1: Results of merging fine-tuned models by different methods with Llama-2-13b as the backbone. For each setting, we fully replicate our method three times with different data sample seeds and compute the mean and standard deviation of the three results. The best and second-best results are highlighted in bold and underlined, respectively. For the hyperparameters of other baselines, we conducted a grid search in each merging setting based on the recommendations in the original paper and reported the best results. We only report the average results of different task evaluations here; for more detailed results, please refer to the Appendix A.1.2.  
+
+<table><tr><td>Methods</td><td>Math &amp; Coding</td><td>Math &amp; Translate</td><td>Coding &amp; Translate</td><td>Math &amp; Coding &amp; Translate</td></tr><tr><td>Fine-tuned Model</td><td>34.02</td><td>65.16</td><td>51.26</td><td>50.14</td></tr><tr><td>Weight Avg</td><td>33.57</td><td>64.57</td><td>52.28</td><td>46.95</td></tr><tr><td>DARE</td><td>32.97</td><td>62.47</td><td>54.33</td><td>47.51</td></tr><tr><td>Task Arithmetic</td><td>34.60</td><td>64.72</td><td>53.22</td><td>48.24</td></tr><tr><td>Ours Layer Level</td><td>34.47±0.17</td><td>66.08±0.14</td><td>52.42±0.15</td><td>50.80±0.17</td></tr><tr><td>Ours Attn/MLP Level</td><td>35.11±0.29</td><td>66.11±0.11</td><td>52.51±0.26</td><td>51.05±0.16</td></tr></table>
+
+FastChat (Zheng et al., 2023) template and fine-tune the models for 3 epochs with a batch size of 128 and a learning rate of  $2 \times 10^{-5}$ .
+
+For evaluation, we test mathematical capability using the GSM8K (Cobbe et al., 2021) test set, coding capability with the HumanEval (Chen et al., 2021), and translation capability with the tools and datasets from (Xu et al., 2024a).
+
+Algorithm Implementation Details. In practice, we observed varying levels of bias in the feature magnitudes across different datasets. To mitigate potential adverse impact, we implemented a mechanism to balance the influence of each sample, thereby resulting in a minor modification of the original formula. For further details, please refer to the Appendix A.3.2. Due to the constraints of our compact computational and storage budget, we use a limited dataset of only 30 samples per task for calculating the merging weights. Surprisingly, this minimal data size demonstrates a commendable performance, underscoring the efficiency of our method in terms of data requirements and computational costs.
+
+# 4.2 MAIN RESULTS
+
+We compare our proposed method with several baseline approaches: Weight Average (Wortsman et al., 2022a;b): Weight Average is the simplest and most direct model merging method, which involves taking the average of the parameters of fine-tuned models. Task Arithmetic(Ilharco et al., 2023): Task arithmetic entails a straightforward weighted combination of the task vectors and weight of the pre-trained model. DARE(Yu et al., 2024): DARE builds upon task arithmetic by employing random dropout of parameters within the task vectors to reduce conflicts between different task vectors during the merging process.
+
+We present the results of merging fine-tuned models employing various methodologies, utilizing Llama-2-13b and Llama-2-7b (Touvron et al., 2023) as the fine-tuning backbones. The findings are detailed in Table 1 and Table 2, respectively. Each value indicates the average evaluation results across multiple related tasks. It is evident that our approach surpasses the baseline in most merging settings, particularly in the case of merging three models, where a performance improvement of  $3\%$  is observed, albeit slightly lower results were achieved in the merging of code and translation models. This validates the effectiveness of our method. Furthermore, the similar outcomes on both Llama-2-7b and Llama-2-13b demonstrate that our method has consistent performance.
+
+# 4.3 ABLATION AND DISCUSSION
+
+Since the optimal merging weights are derived through a closed-form solution, our proposed merging method lacks adjustable hyperparameters. Consequently, our discussion centers on the influence
+
+Table 2: Results of merging fine-tuned models by different methods with Llama-2-7b as the backbone. The details of this table are the same as Table 1. For more results, please refer to the Appendix A.1.2.  
+
+<table><tr><td>Methods</td><td>Math &amp; Coding</td><td>Math &amp; Translate</td><td>Coding &amp; Translate</td><td>Math &amp; Coding &amp; Translate</td></tr><tr><td>Fine-tuned Model</td><td>32.35</td><td>62.62</td><td>50.99</td><td>48.65</td></tr><tr><td>Weight Avg</td><td>25.14</td><td>53.70</td><td>49.25</td><td>38.30</td></tr><tr><td>DARE</td><td>19.11</td><td>50.77</td><td>50.15</td><td>39.00</td></tr><tr><td>Task Arithmetic</td><td>27.45</td><td>54.65</td><td>51.11</td><td>39.59</td></tr><tr><td>Ours Layer Level</td><td>27.18±0.04</td><td>49.79±0.43</td><td>50.64±0.64</td><td>42.68±0.26</td></tr><tr><td>Ours Attn/MLP Level</td><td>27.33±0.10</td><td>55.86±0.48</td><td>49.80±0.23</td><td>42.88±0.27</td></tr></table>
+
+Table 3: Ablation results of merging fine-tuned models by different levels in our method. For each setting, we fully replicate our method three times with different seeds and compute the mean and standard deviation of the three results.  
+
+<table><tr><td>Level</td><td>Math &amp; Coding</td><td>Math &amp; Translate</td><td>Coding &amp; Translate</td><td>Math &amp; Coding &amp; Translate</td></tr><tr><td colspan="5">Llama-2-7B</td></tr><tr><td>Ours Model Level</td><td>28.34±0.22</td><td>13.38±0.57</td><td>46.51±0.15</td><td>35.95±0.51</td></tr><tr><td>Ours Layer Level</td><td>27.18±0.04</td><td>49.79±0.43</td><td>50.64±0.64</td><td>42.68±0.26</td></tr><tr><td>Ours Attn/MLP Level</td><td>27.33±0.10</td><td>55.86±0.48</td><td>49.80±0.23</td><td>42.88±0.27</td></tr><tr><td>Ours Head/MLP Level</td><td>13.08±9.33</td><td>14.17±1.14</td><td>41.14±12.39</td><td>8.16±1.30</td></tr><tr><td colspan="5">Llama-2-13B</td></tr><tr><td>Ours Model Level</td><td>31.27±0.43</td><td>18.90±2.45</td><td>15.75±7.60</td><td>50.68±0.10</td></tr><tr><td>Ours Layer Level</td><td>34.47±0.17</td><td>66.08±0.14</td><td>52.42±0.15</td><td>50.80±0.17</td></tr><tr><td>Ours Attn/MLP Level</td><td>35.11±0.29</td><td>66.11±0.11</td><td>52.51±0.26</td><td>51.05±0.16</td></tr><tr><td>Ours Head/MLP Level</td><td>12.91±15.60</td><td>30.17±19.36</td><td>13.71±3.50</td><td>22.59±19.65</td></tr></table>
+
+of decomposition granularity on our approach. In addition to the Layer-Level and Attention/MLP-Level configurations previously illustrated, we also apply our method at the Model-Level, which serves as a baseline setting without any decomposition, as well as at the Head/MLP-Level, where attentions are further vertically divided into individual heads. $^{8}$
+
+Results at the Model Level. We also attempted to directly apply our method for model merging at the model level. The results in Table 3 indicate that, in most cases, it failed to yield reasonable outcomes. A visualization of the parameters calculated at the model level can be found in Table 13. The failure at the model level can be attributed to the lack of sufficient linear correlation between the weight differences caused by fine-tuning and the differences in output features resulting from fine-tuning. The absence of linearity prevents us from leveraging feature-level computations to reasonably merge parameters, ultimately leading to unpredictable results. This implies that our method is only effective under linear conditions.
+
+Results at the Head/MLP Level. Despite exhibiting good linearity at the Head level, the merging with our method at this level reveals unstable outcomes, shown in Table 3. We also provide a visual example of the parameters during actual Head level merging in Figure 5.
+
+We offer a conjectural explanation for the occurrence of this situation here. Many studies (Wang et al., 2023; Chen et al., 2024; Zhang et al., 2024) indicate that each head in the model serves different functions during inference. After fine-tuning various tasks, certain heads may take on distinct roles, while our objective function may ask the head to simultaneously average close to multiple heads with different functionalities. This requirement can be quite challenging for an individual head, ultimately leading to a collapse of functionality of the merged head.
+
+However, this may not indicate that the linearity of the head is unhelpful for the merging process. Considering the specific functionality of the head (or the particular task associated with that head) during the merging to redefine a more reasonable merging objective could be a valuable direction for future research.
+
+# 5 RELATED WORK
+
+Weight Interpolation and Task Arithmetic. In recent years, the increasing scale of large language models has significantly heightened the demand for data and training costs associated with fine-tuning multi-task models. To merge the capabilities of various existing single-task models into a unified framework, several model merging techniques (Daheim et al., 2024; Jin et al., 2023; Wan et al., 2024; Yang et al., 2024b) have been developed. The simplest and most intuitive approach is weight interpolation(Frankle et al., 2020; Izmailov et al., 2018; Ramé et al., 2023; 2022), which has been applied to enhance model generalization(Wortsman et al., 2022b), improve specific single-task performance(Wortsman et al., 2022a), and boost multi-task effectiveness(ILharco et al., 2022; Li et al., 2022). Subsequently, Task Arithmetic(ILharco et al., 2023) was introduced, enabling the merging of multiple models through a weighted operation on the parameter difference  $\tau$ . Many subsequent methods(Yang et al., 2024c; Yu et al., 2024; Yadav et al., 2023; Zhang et al., 2023) have been proposed based on the principles of Task Arithmetic.
+
+Linearity and Task Arithmetic. The research conducted by Ortiz-Jiménez et al. (2023) begins with the relevant theories of the Neural Tangent Kernel (NTK)(Jacot et al., 2018; Chizat et al., 2019) to explore the connection between linearity and task arithmetic. Although experimental validations indicate that fine-tuned models do not comply with NTK's conclusion regarding the approximate one-order Taylor expansion of models, Ortiz-Jiménez et al. (2023) observes that linearity still can aid task arithmetic. To enhance model linearity, Ortiz-Jiménez et al. (2023) proposes a constraint on the parameter update space during fine-tuning, thereby fine-tuning a model that achieves linearity, ultimately improving the efficiency of model merging. To expedite this training process, various studies have attempted to linearize only a subset of parameters during training. For instance, Jin et al. (2024) suggest linearly fine-tuning solely the linear layers in the attention modules. Tang et al. (2024) combine linearly fine-tuning with Parameter-Efficient Fine-Tuning (PEFT) techniques by linearizing only the Adapter modules, thereby reducing computational costs. Liu et al. (2024) derives a closed-form linearized solution for efficiently fine-tuning Transformer networks.
+
+In conclusion, modifying fine-tuning methods to achieve a more linear model can reduce interference during model merging. However, these methods still require training data and retraining, which is often challenging in practical scenarios.
+
+# 6 CONCLUSION
+
+In this study, we performed a statistical analysis to investigate the linearity of model-level and decomposed submodule-level components. Our findings indicate that the missing linearity in full models can be found in the submodules. This observation motivated us to develop an innovative, training-free approach to enhance task arithmetic performance. Specifically, we first decompose the model into its constituent submodules and then compute the optimal merging weights for each module based on a closed-form solution derived from linear properties. Subsequently, we perform a linear merging of all submodules using the corresponding merging weights. Our experimental results demonstrate that this approach significantly outperforms various baseline techniques, including standard task arithmetic methods, across different model scales and diverse task scenarios. These findings underscore the advantages of leveraging the linearity of submodules and present a novel perspective for investigating effective and practical solutions for multi-task model merging.
+
+# ACKNOWLEDGEMENTS
+
+This work was supported in part by NSFC No. 62222117.
+
+# ETHICS STATEMENT
+
+We proposed a novel approach to enhance model merging performance in large language models (LLMs). Our methodology utilizes publicly available language datasets and leverages pre-trained language models for experimental validation. We do not believe that our code or method is inherently subject to concerns such as discrimination, bias, fairness, inappropriate potential applications, impact, privacy and security issues, legal compliance, or research integrity concerns. However, language datasets and models may possess intrinsic biases that could be inherited by models merged using our approach.
+
+# REPRODUCIBILITY STATEMENT
+
+To ensure the reproducibility of our approach, we provide key information from the main text and Appendix as follows.
+
+Algorithm. We provide our algorithm in Algorithm 1.
+
+Experimental Details. We provide our experimental details in Section 4. Moreover, we provide detailed experiment settings in Appendices A.2 A.3.1, A.3.2, A.3.3 and A.3.5.
+
+Moreover, we are committed to providing the source code of our approach, if accepted.
+
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+
+# A APPENDIX
+
+# A.1 MORE RESULTS
+
+# A.1.1 MORE RESULTS IN ANALYZING THE LINEARITY OF SINGLE FINE-TUNED MODELS
+
+To ascertain whether the previously mentioned observations regarding linearity remain applicable within the context of model merging, we followed the methodology outlined in Zhou et al. (2024). We analyzed two types of linearity metrics introduced in their work, cosine similarity and projection distance in feature space.
+
+Specifically, we analyze the cosine similarity between two sets of feature deltas: the differences between the output features of the merged module (derived from multiple fine-tuned models) and those of the pre-trained module, denoted as  $\Delta f(x; \theta_0^i + \sum_{t=1}^{T} \alpha_t^i \tau_t^i) = f(x; \theta_0^i + \sum_{t=1}^{T} \alpha_t^i \tau_t^i) - f(x; \theta_0^i)$ , as well as the corresponding weighted combination of the differences in output features from the fine-tuned models, represented as  $\sum_{t=1}^{T} \alpha_t^i \Delta f(x; \theta_0^i + \tau_t^i) = \sum_{t=1}^{T} \alpha_t(f(x; \theta_0^i + \tau_t^i) - f(x; \theta_0^i))$ . This similarity is denoted by the following equation:
+
+$$
+\operatorname {c o s i n e} _ {\text {m e r g e}} \left(x; \alpha_ {1 \dots T} ^ {i}\right) = \cos \left(\Delta f \left(x; \theta_ {0} ^ {i} + \sum_ {t = 1} ^ {T} \alpha_ {t} ^ {i} \tau_ {t} ^ {i}\right), \sum_ {t = 1} ^ {T} \alpha_ {t} ^ {i} \Delta f \left(x; \theta_ {0} ^ {i} + \tau_ {t} ^ {i}\right)\right) \tag {15}
+$$
+
+and
+
+$$
+\operatorname {c o s i n e} _ {\text {m e r g e}} ^ {\mathbb {E}} \left(\alpha_ {1 \dots T} ^ {i}\right) = \mathbb {E} _ {x \in H ^ {i}} \operatorname {c o s i n e} _ {\text {m e r g e}} \left(x; \alpha_ {1 \dots T} ^ {i}\right). \tag {16}
+$$
+
+Following Zhou et al. (2024), we compare these cosine similarities against a natural baseline: the average cosine similarity of the feature deltas induced by fine-tuning across all possible pairs of the tuned models, denoted as:
+
+$$
+\cos \mathrm {i n e} _ {\text {b a s e}} ^ {\mathbb {E}} = \mathbb {E} _ {x \in H ^ {i}} \frac {1}{T (T - 1)} \sum_ {1 \leq t <   t ^ {\prime} \leq T} \cos \left(\Delta f \left(x; \theta_ {0} ^ {i} + \tau_ {t} ^ {i}\right), \Delta f \left(x; \theta_ {0} ^ {i} + \tau_ {t ^ {\prime}} ^ {i}\right)\right) \tag {17}
+$$
+
+![](images/0f913ef1dc78ffdf9b9426eee1b8e91a9c5636bb4ba3b1c5c843a3807d7b9c6c.jpg)  
+(a) Cosine Similarity in Llama-2-7b
+
+![](images/3c4f218a605b93c1deef2924188df2db237927222b3aeddca6e0fba7b086dbc5.jpg)  
+(b) Cosine Similarity in Llama-2-13b  
+Figure 4: Comparison of Cosine Similarity in full models and submodules within the context of model merging. A lower value indicates better linearity. For each backbone, we computed the results across all possible combinations of three fine-tuned models, alongside the corresponding merging weights  $\alpha_{1}$  and  $\alpha_{2}$  (and  $\alpha_{3}$ ). In the case of merging two models, there are three combinations of fine-tuned models, yielding 25 possible configurations for merging weights  $\alpha_{t} \in [0.2, 0.4, 0.6, 0.8, 1]$  where  $t \in [1, 2]$ . For merging three models, there's only one combination of fine-tuned models, resulting in 27 configurations for merging weights  $\alpha_{t} \in [0.3, 0.5, 0.7]$  with  $t \in [1, 2, 3]$ . Error bars represent standard deviations.
+
+Results of Cosine Similarity. As illustrated in Figure 4 (a) and (b), we observe cosine similarity results consistent with Zhou et al. (2024), revealing a significant disparity between  $1 - \mathbb{E}_{\alpha}\cos \mathrm{e}^{\mathbb{E}}_{merge}$  and the baseline  $1 - \cos \mathrm{e}_{base}^{\mathbb{E}}$ . Furthermore, a greater gap and reduced values are noted at the module level compared to the model level, suggesting a more favorable condition for linear merging.
+
+# A.1.2 MORE RESULTS IN MERGING MODULES WITH LINEARITY
+
+In this section, we report more detailed results of Tables 1 and 2 in Tables 4 and 5, as well as the outcomes of our approach on additional models and datasets. The results of model merging on the
+
+Qwen-2-0.5B(Yang et al., 2024a) model, along with datasets Gsm8k(Cobbe et al., 2021), Cosmos QA (Huang et al., 2019), Ropes (Lin et al., 2019), and Winogrande (Sakaguchi et al., 2020), are presented in Table 6. We also report result on Qwen-2-7B(Qwen Team, 2024) model. During training, we adopt the prompt template from FastChat (Zheng et al., 2023), fine-tuning the models for 3 epochs with a batch size of 128 and a learning rate of  $2 \times 10^{-5}$ .
+
+Table 4: Detailed experimental results on LLama-2-13B.  
+
+<table><tr><td rowspan="2">Method
+Llama-2-13B</td><td colspan="6">Merging Two Models</td></tr><tr><td>Math</td><td>Coding</td><td>Avg</td><td>Math</td><td>Translate</td><td>Avg</td></tr><tr><td>Fine-tuned Model</td><td>47.91</td><td>20.12</td><td>34.02</td><td>47.91</td><td>82.40</td><td>65.16</td></tr><tr><td>Weight Avg</td><td>42.76</td><td>24.39</td><td>33.57</td><td>45.11</td><td>84.04</td><td>64.57</td></tr><tr><td>DARE</td><td>41.55</td><td>24.39</td><td>32.97</td><td>40.94</td><td>83.99</td><td>62.47</td></tr><tr><td>Task Arithmetic</td><td>45.41</td><td>23.78</td><td>34.60</td><td>45.87</td><td>83.58</td><td>64.72</td></tr><tr><td>Ours Model Level</td><td>48.12±0.29</td><td>14.43±0.57</td><td>31.27±0.43</td><td>0.20±0.09</td><td>37.60±4.91</td><td>18.90±2.45</td></tr><tr><td>Ours Layer Level</td><td>46.37±0.34</td><td>22.56±0.00</td><td>34.47±0.17</td><td>48.42±0.25</td><td>83.75±0.04</td><td>66.08±0.14</td></tr><tr><td>Ours Attn/MLP Level</td><td>46.85±0.00</td><td>23.37±0.57</td><td>35.11±0.29</td><td>48.55±0.20</td><td>83.68±0.03</td><td>66.11±0.11</td></tr><tr><td>Ours Head/MLP Level</td><td>15.87±20.86</td><td>9.96±10.45</td><td>12.91±15.60</td><td>16.27±17.88</td><td>44.07±20.93</td><td>30.17±19.36</td></tr></table>
+
+<table><tr><td rowspan="2">Method
+Llama-2-13B</td><td colspan="3">Merging Two Models</td><td rowspan="2">Math</td><td colspan="3">Merging Three Models</td></tr><tr><td>Coding</td><td>Translate</td><td>Avg</td><td>Coding</td><td>Translate</td><td>Avg</td></tr><tr><td>Fine-tuned Model</td><td>20.12</td><td>82.40</td><td>51.26</td><td>47.91</td><td>20.12</td><td>82.40</td><td>50.14</td></tr><tr><td>Weight Avg</td><td>20.12</td><td>84.43</td><td>52.28</td><td>33.13</td><td>23.78</td><td>83.95</td><td>46.95</td></tr><tr><td>DARE</td><td>24.39</td><td>84.27</td><td>54.33</td><td>34.87</td><td>23.78</td><td>83.88</td><td>47.51</td></tr><tr><td>Task Arithmetic</td><td>21.95</td><td>84.48</td><td>53.22</td><td>38.36</td><td>22.56</td><td>83.80</td><td>48.24</td></tr><tr><td>Ours Model Level</td><td>0.00±0.00</td><td>31.50±15.19</td><td>15.75±7.60</td><td>47.08±0.68</td><td>23.58±0.29</td><td>81.38±0.43</td><td>50.68±0.10</td></tr><tr><td>Ours Layer Level</td><td>20.53±0.29</td><td>84.31±0.02</td><td>52.42±0.15</td><td>45.21±0.26</td><td>23.37±0.29</td><td>83.81±0.04</td><td>50.80±0.17</td></tr><tr><td>Ours Attn/MLP Level</td><td>20.73±0.50</td><td>84.29±0.04</td><td>52.51±0.26</td><td>45.59±0.09</td><td>23.78±0.50</td><td>83.78±0.01</td><td>51.05±0.16</td></tr><tr><td>Ours Head/MLP Level</td><td>0.00±0.00</td><td>27.43±7.00</td><td>13.71±3.50</td><td>14.99±20.24</td><td>8.13±11.07</td><td>44.66±27.65</td><td>22.59±19.65</td></tr></table>
+
+Table 5: Detailed experimental results on LLama-2-7B.  
+
+<table><tr><td rowspan="2">Method
+Llama-2-7B</td><td colspan="6">Merging Two Models</td></tr><tr><td>Math</td><td>Coding</td><td>Avg</td><td>Math</td><td>Translate</td><td>Avg</td></tr><tr><td>Fine-tuned Model</td><td>43.97</td><td>20.73</td><td>32.35</td><td>43.97</td><td>81.26</td><td>62.62</td></tr><tr><td>Weight Avg</td><td>31.99</td><td>18.29</td><td>25.14</td><td>25.02</td><td>82.38</td><td>53.70</td></tr><tr><td>DARE</td><td>21.15</td><td>17.07</td><td>19.11</td><td>19.41</td><td>82.14</td><td>50.77</td></tr><tr><td>Task Arithmetic</td><td>37.83</td><td>17.07</td><td>27.45</td><td>27.07</td><td>82.24</td><td>54.65</td></tr><tr><td>Ours Model Level</td><td>43.47±0.50</td><td>13.21±0.76</td><td>28.34±0.22</td><td>2.27±0.34</td><td>24.48±0.80</td><td>13.38±0.57</td></tr><tr><td>Ours Layer Level</td><td>35.25±0.21</td><td>19.11±0.29</td><td>27.18±0.04</td><td>17.77±0.84</td><td>81.82±0.03</td><td>49.79±0.43</td></tr><tr><td>Ours Attn/MLP Level</td><td>35.96±0.09</td><td>18.70±0.29</td><td>27.33±0.10</td><td>30.00±0.98</td><td>81.73±0.03</td><td>55.86±0.48</td></tr><tr><td>Ours Head/MLP Level</td><td>17.41±13.21</td><td>8.74±6.57</td><td>13.08±9.33</td><td>0.58±0.72</td><td>27.77±1.57</td><td>14.17±1.14</td></tr></table>
+
+<table><tr><td rowspan="2">Method
+Llama-2-7B</td><td colspan="3">Merging Two Models</td><td rowspan="2">Math</td><td colspan="3">Merging Three Models</td></tr><tr><td>Coding</td><td>Translate</td><td>Avg</td><td>Coding</td><td>Translate</td><td>Avg</td></tr><tr><td>Fine-tuned Model</td><td>20.73</td><td>81.26</td><td>50.99</td><td>43.97</td><td>20.73</td><td>81.26</td><td>48.65</td></tr><tr><td>Weight Avg</td><td>15.85</td><td>82.64</td><td>49.25</td><td>17.66</td><td>15.24</td><td>81.99</td><td>38.30</td></tr><tr><td>DARE</td><td>17.68</td><td>82.62</td><td>50.15</td><td>19.64</td><td>15.24</td><td>82.13</td><td>39.00</td></tr><tr><td>Task Arithmetic</td><td>19.51</td><td>82.70</td><td>51.11</td><td>22.29</td><td>14.63</td><td>81.86</td><td>39.59</td></tr><tr><td>Ours Model Level</td><td>11.79±0.29</td><td>81.23±0.18</td><td>46.51±0.15</td><td>41.57±0.62</td><td>14.43±0.57</td><td>51.84±1.54</td><td>35.95±0.51</td></tr><tr><td>Ours Layer Level</td><td>18.90±1.32</td><td>82.38±0.03</td><td>50.64±0.64</td><td>30.50±0.54</td><td>15.85±0.50</td><td>81.68±0.05</td><td>42.68±0.26</td></tr><tr><td>Ours Attn/MLP Level</td><td>17.28±0.57</td><td>82.33±0.12</td><td>49.80±0.23</td><td>32.02±0.31</td><td>14.84±0.57</td><td>81.79±0.03</td><td>42.88±0.27</td></tr><tr><td>Ours Head/MLP Level</td><td>16.06±2.74</td><td>66.22±22.41</td><td>41.14±12.39</td><td>0.00±0.00</td><td>0.00±0.00</td><td>24.48±3.90</td><td>8.16±1.30</td></tr></table>
+
+# A.1.3 ABLATION STUDY ON NUMBER OF MODEL TO BE MERGED
+
+To demonstrate the advantages of our method in scenarios where more models are merged simultaneously, we increase the number of task-specific models merged simultaneously gradually and calculate the average performance of the merged models.
+
+Table 6: Detailed experimental results on Qwen-2-0.5B.  
+
+<table><tr><td rowspan="2">Method Qwen-2-0.5B</td><td rowspan="2">gsm8k</td><td rowspan="2">cosmos_qa</td><td rowspan="2">ropes</td><td colspan="5">Merging Three Models</td></tr><tr><td>Avg</td><td>gsm8k</td><td>cosmos_qa</td><td>winogrande</td><td>Avg</td></tr><tr><td>Fine-tuned Model</td><td>31.61</td><td>65.01</td><td>45.65</td><td>47.42</td><td>31.61</td><td>65.01</td><td>56.59</td><td>51.07</td></tr><tr><td>Weight Avg</td><td>25.09</td><td>51.91</td><td>37.50</td><td>38.17</td><td>17.74</td><td>49.31</td><td>51.82</td><td>39.62</td></tr><tr><td>DARE</td><td>34.57</td><td>40.09</td><td>41.61</td><td>38.76</td><td>31.24</td><td>45.14</td><td>51.34</td><td>42.57</td></tr><tr><td>Task Arithmetic</td><td>26.54</td><td>50.83</td><td>37.56</td><td>38.31</td><td>22.52</td><td>48.63</td><td>51.82</td><td>40.99</td></tr><tr><td>Ours Model Level</td><td>31.35±0.11</td><td>22.71±2.15</td><td>46.95±0.09</td><td>33.67±0.65</td><td>14.56±0.00</td><td>47.95±0.00</td><td>52.96±0.00</td><td>38.49±0.00</td></tr><tr><td>Ours Layer Level</td><td>30.52±0.11</td><td>51.76±0.05</td><td>38.07±0.03</td><td>40.12±0.01</td><td>27.71±0.04</td><td>51.00±0.03</td><td>52.31±0.00</td><td>43.67±0.02</td></tr><tr><td>Ours Attn/MLP Level</td><td>30.55±0.08</td><td>52.07±0.08</td><td>37.62±0.06</td><td>40.08±0.03</td><td>27.48±0.27</td><td>51.13±0.07</td><td>52.55±0.00</td><td>43.72±0.11</td></tr><tr><td>Ours Head/MLP Level</td><td>29.19±0.83</td><td>50.07±1.98</td><td>37.53±0.33</td><td>38.93±1.05</td><td>15.05±12.09</td><td>39.77±12.24</td><td>50.57±0.61</td><td>35.13±8.31</td></tr></table>
+
+<table><tr><td rowspan="2">Method Qwen-2-0.5B</td><td rowspan="2">gsm8k</td><td rowspan="2">winogrande</td><td rowspan="2">ropes</td><td colspan="5">Merging Three Models</td></tr><tr><td>Avg</td><td>winogrande</td><td>cosmos_qa</td><td>ropes</td><td>Avg</td></tr><tr><td>Fine-tuned Model</td><td>31.61</td><td>56.59</td><td>45.65</td><td>44.62</td><td>56.59</td><td>65.01</td><td>45.65</td><td>55.75</td></tr><tr><td>Weight Avg</td><td>10.84</td><td>50.93</td><td>49.09</td><td>36.96</td><td>51.42</td><td>44.36</td><td>45.23</td><td>47.00</td></tr><tr><td>DARE</td><td>33.97</td><td>52.39</td><td>48.85</td><td>45.07</td><td>51.82</td><td>42.91</td><td>46.20</td><td>46.97</td></tr><tr><td>Task Arithmetic</td><td>15.54</td><td>52.06</td><td>48.67</td><td>38.76</td><td>51.90</td><td>44.84</td><td>45.35</td><td>47.36</td></tr><tr><td>Ours Model Level</td><td>7.77±0.11</td><td>53.68±0.00</td><td>31.34±0.06</td><td>30.93±0.02</td><td>52.39±0.00</td><td>48.00±0.05</td><td>30.95±0.45</td><td>43.78±0.13</td></tr><tr><td>Ours Layer Level</td><td>24.30±0.19</td><td>51.46±0.04</td><td>48.79±0.06</td><td>41.52±0.03</td><td>51.78±0.04</td><td>50.10±0.02</td><td>43.21±0.27</td><td>48.36±0.08</td></tr><tr><td>Ours Attn/MLP Level</td><td>24.18±0.45</td><td>51.94±0.04</td><td>48.25±0.00</td><td>41.46±0.17</td><td>51.62±0.04</td><td>50.64±0.08</td><td>42.48±0.03</td><td>48.25±0.05</td></tr><tr><td>Ours Head/MLP Level</td><td>24.00±0.27</td><td>51.58±0.08</td><td>47.98±0.21</td><td>41.18±0.04</td><td>50.93±0.16</td><td>50.22±0.20</td><td>43.21±0.03</td><td>48.12±0.02</td></tr></table>
+
+Table 7: Experimental results on LLaMA-2-7B with More Dataset: Winogrande, Cosmos qa and Ropes  
+
+<table><tr><td>Method</td><td colspan="9">Merging Two Models</td></tr><tr><td>Llama-2-7b</td><td>Winogrande</td><td>Cosmos qa</td><td>Avg</td><td>Winogrande</td><td>Ropes</td><td>Avg</td><td>Cosmos qa</td><td>Ropes</td><td>Avg</td></tr><tr><td>Fine-tuned Model</td><td>82.26</td><td>84.72</td><td>83.49</td><td>82.26</td><td>75.42</td><td>78.84</td><td>84.72</td><td>75.42</td><td>80.07</td></tr><tr><td>Task Arithmetic</td><td>72.22</td><td>71.85</td><td>72.03</td><td>72.14</td><td>54.28</td><td>63.21</td><td>73.48</td><td>58.75</td><td>66.11</td></tr><tr><td>Ours Attn/MLP Level</td><td>68.42</td><td>84.18</td><td>76.3</td><td>69.47</td><td>69.98</td><td>69.72</td><td>73.07</td><td>68.59</td><td>70.83</td></tr></table>
+
+<table><tr><td>Method</td><td colspan="4">Merging Three Models</td></tr><tr><td>Llama-2-7b</td><td>Winogrande</td><td>Cosmos qa</td><td>Ropes</td><td>Avg</td></tr><tr><td>Fine-tuned Model</td><td>82.26</td><td>84.72</td><td>75.42</td><td>80.8</td></tr><tr><td>Task Arithmetic</td><td>71.25</td><td>64.13</td><td>55.85</td><td>63.7433</td></tr><tr><td>Ours Attn/MLP Level</td><td>71.49</td><td>68.70</td><td>56.70</td><td>65.63</td></tr></table>
+
+In the table 10, we can observe that as the number of models increases, our method gradually surpasses Task Arithmetic and the performance gap becomes larger, which demonstrates the efficiency of our method particularly when more models are merged simultaneously.
+
+# A.1.4 ABLATION ON THE NUMBER OF DATA
+
+From the table 11, we can see that our method shows consistent performance across different data quantities, sometimes achieving good results with as few as 3 data per task. As a result, if one wishes to use our method in a situation of absolutely no data, we believe that simply synthesizing a few data for the corresponding task through manual methods or with the help of LLMs is feasible.
+
+# A.2 MORE DETAILS ABOUT NON-LINEARITY SCORE
+
+Overall, we achieve the practical implementation of the Non-linearity Score through discretization and using the Euclidean distance as distance metric  $\mathcal{D}$  in the feature space.
+
+Given a fine-tuned model  $\theta$ , we interpolate between this model and its corresponding pre-trained model  $\theta_0$  using  $k \in \{0, 1, 2, \ldots, N-1, N\}$ . This results in  $N+1$  models  $\theta_k = \frac{k}{N} \theta + \frac{1-k}{N} \theta_0 = \theta_0 + \frac{k}{N} (\theta - \theta_0)$ . If  $\theta$  satisfies linearity in Definition 1 under a distance metric  $\mathcal{D}$  in the feature space, then for any  $i, j \in \{0, 1, 2, \ldots, N-1, N\}$ , the following approximation holds:
+
+$$
+\mathcal {D} \left(f \left(x; \theta_ {i}\right), f \left(x; \theta_ {j}\right)\right) \approx \frac {(i - j)}{N} \mathcal {D} \left(f (x; \theta), f (x; \theta_ {0})\right) \tag {18}
+$$
+
+Table 8: Experimental results on Qwen-2.5-7B  
+
+<table><tr><td rowspan="2">Method Qwen-2.5-7B</td><td colspan="9">Merging Two Models</td><td colspan="4">Merging Three Models</td></tr><tr><td>Math</td><td>Coding</td><td>Avg</td><td>Math</td><td>Translate</td><td>Avg</td><td>Coding</td><td>Translate</td><td>Avg</td><td>Math</td><td>Coding</td><td>Translate</td><td>Avg</td></tr><tr><td>Fine-tuned Model</td><td>71.49</td><td>60.36</td><td>65.92</td><td>71.49</td><td>85.83</td><td>78.66</td><td>60.36</td><td>85.83</td><td>73.09</td><td>71.49</td><td>60.36</td><td>85.83</td><td>72.56</td></tr><tr><td>Task Arithmetic</td><td>75.13</td><td>67.68</td><td>71.40</td><td>78.69</td><td>86.43</td><td>82.56</td><td>64.02</td><td>86.30</td><td>75.16</td><td>78.92</td><td>65.24</td><td>86.56</td><td>76.90</td></tr><tr><td>Ours Layer Level</td><td>78.01</td><td>64.02</td><td>71.01</td><td>77.93</td><td>86.55</td><td>82.24</td><td>64.63</td><td>86.59</td><td>75.61</td><td>81.04</td><td>64.02</td><td>86.65</td><td>77.23</td></tr><tr><td>Ours Attn/MLP Level</td><td>77.86</td><td>64.63</td><td>71.24</td><td>78.01</td><td>86.58</td><td>82.29</td><td>64.02</td><td>86.62</td><td>75.32</td><td>80.66</td><td>65.24</td><td>86.64</td><td>77.51</td></tr></table>
+
+Table 9: Experimental results with more baseline  
+
+<table><tr><td>Method</td><td colspan="9">Merging Two Models</td><td colspan="4">Merging Three Models</td></tr><tr><td>Llama-2-7B</td><td>Math</td><td>Coding</td><td>Avg</td><td>Math</td><td>Translate</td><td>Avg</td><td>Coding</td><td>Translate</td><td>Avg</td><td>Math</td><td>Coding</td><td>Translate</td><td>Avg</td></tr><tr><td>Fine-tuned Model</td><td>43.97</td><td>20.73</td><td>32.35</td><td>43.97</td><td>81.26</td><td>62.62</td><td>20.73</td><td>81.26</td><td>50.99</td><td>43.97</td><td>20.73</td><td>81.26</td><td>48.65</td></tr><tr><td>Weight Avg</td><td>31.99</td><td>18.29</td><td>25.14</td><td>25.02</td><td>82.38</td><td>53.7</td><td>15.85</td><td>82.64</td><td>49.25</td><td>17.66</td><td>15.24</td><td>81.99</td><td>38.3</td></tr><tr><td>DARE</td><td>21.15</td><td>17.07</td><td>19.11</td><td>19.41</td><td>82.14</td><td>50.77</td><td>17.68</td><td>82.62</td><td>50.15</td><td>19.64</td><td>15.24</td><td>82.13</td><td>39</td></tr><tr><td>Task Arithmetic</td><td>37.83</td><td>17.07</td><td>27.45</td><td>27.07</td><td>82.24</td><td>54.65</td><td>19.51</td><td>82.7</td><td>51.11</td><td>22.29</td><td>14.63</td><td>81.86</td><td>39.59</td></tr><tr><td>Breadcrumbs</td><td>18.11</td><td>18.90</td><td>18.51</td><td>16.22</td><td>82.24</td><td>49.23</td><td>19.51</td><td>82.26</td><td>50.89</td><td>7.96</td><td>18.90</td><td>82.78</td><td>36.55</td></tr><tr><td>TIES</td><td>34.49</td><td>18.90</td><td>26.69</td><td>32.97</td><td>82.11</td><td>57.54</td><td>19.51</td><td>82.75</td><td>51.13</td><td>29.03</td><td>15.85</td><td>81.49</td><td>42.12</td></tr><tr><td>Consensus TA</td><td>27.21</td><td>17.68</td><td>22.44</td><td>25.39</td><td>82.47</td><td>53.93</td><td>18.29</td><td>82.51</td><td>50.40</td><td>26.38</td><td>17.68</td><td>82.36</td><td>42.14</td></tr><tr><td>Ours Layer Level</td><td>35.25</td><td>19.11</td><td>27.18</td><td>17.77</td><td>81.82</td><td>49.79</td><td>18.90</td><td>82.38</td><td>50.64</td><td>30.5</td><td>15.85</td><td>81.68</td><td>42.68</td></tr><tr><td>Ours Attn/MLP Level</td><td>35.96</td><td>18.70</td><td>27.33</td><td>30.00</td><td>81.73</td><td>55.86</td><td>17.28</td><td>82.33</td><td>49.8</td><td>32.02</td><td>14.84</td><td>81.79</td><td>42.88</td></tr></table>
+
+Since  $f(x; \theta)$  is a vector, we utilize the Euclidean distance as distance metric  $\mathcal{D}$ , to measure the distance between features. To assess the discrepancy between the actual situation regarding  $\theta$  and the Eq.18, the Non-linearity Score is defined as follows:
+
+$$
+\text {N o n - l i n e a r i t y} = \sum_ {i \in \{0,.., N \}} \sum_ {j \in \{0,.., N \}} \left(\frac {\mathcal {D} (f (x ; \theta_ {i}) , f (x ; \theta_ {j}))}{\mathcal {D} (f (x ; \theta) , f (x ; \theta_ {0}))} - \frac {| i - j |}{N}\right) ^ {2} \tag {19}
+$$
+
+We set  $N = 10$  in this paper. Some visualization of  $\frac{\mathcal{D}(f(x;\theta_i),f(x;\theta_j))}{\mathcal{D}(f(x;\theta),f(x;\theta_0))}$  is shown in Figure 6.
+
+We can also extend the Non-linearity Score for analyzing the linearity of merging multiple fine-tuned models, maintaining the same functionality as the Projection Distance metric discussed in Section 2.3. Due to the limitations of our compact computational and storage budget, we employ Projection Distance in Section 2.3 for its higher efficiency.
+
+# A.3 MORE DETAILS ABOUT OUR PROPOSED METHOD
+
+# A.3.1 THE DERIVATION OF THE CLOSED-FORM SOLUTION
+
+Objectives Function. For a module located at position  $i$  (which could be a layer/attention/MLP), let  $\theta_t^i$  represent the weights of this module from model  $t$ , and  $f(x; \theta_t^i)$  denote the output of this module given input  $x$ . The merging parameters  $\alpha_1^i, \alpha_2^i, \ldots, \alpha_T^i$  for  $\theta_1^i, \theta_2^i, \ldots, \theta_T^i$  can be obtained by
+
+$$
+\alpha_ {1} ^ {i}, \alpha_ {2} ^ {i}, \dots , \alpha_ {T} ^ {i} = \arg \min  \sum_ {t = 1} ^ {T} \mathbb {E} _ {x \in H _ {t} ^ {i}} \left\| f \left(x; \theta_ {0} ^ {i} + \sum_ {t ^ {\prime} = 1} ^ {T} \alpha_ {t ^ {\prime}} ^ {i} \tau_ {t ^ {\prime}} ^ {i}\right) - f (x; \theta_ {t} ^ {i}) \right\| ^ {2} \tag {20}
+$$
+
+where  $\tau_{t}^{i} = \theta_{t}^{i} - \theta_{0}^{i}$ ,  $H_{t}^{i}$  represent the set of input features of module  $i$  corresponding to  $D_{t}$ . In this paper,  $H_{t}^{i}$  is obtained by inputting all  $x\in D_t$  into the model  $\theta_0$  and collecting the input features corresponding to module  $i$ .
+
+If module  $i$  satisfies linearity, by applying Property 2, we have
+
+Table 10: Ablation study on Number of Model to be Merged  
+
+<table><tr><td>Avg. Acc</td><td>Coding &amp; Translate</td><td>Coding &amp; Translate &amp; Cosmos_qa</td><td>Coding &amp; Translate &amp; Cosmos_qa &amp; Ropes</td><td>Coding &amp; Translate &amp; Cosmos_qa &amp; Ropes &amp; Winogrande</td></tr><tr><td>Task Arithmetic</td><td>0.5111</td><td>0.5643</td><td>0.5257</td><td>0.4666</td></tr><tr><td>Ours</td><td>0.4980</td><td>0.5581</td><td>0.5382</td><td>0.5489</td></tr><tr><td>Gap Δ</td><td>-2.6%</td><td>-1.1%</td><td>+2.3%</td><td>+14.9%</td></tr></table>
+
+Table 11: Ablation on the Number of Data  
+
+<table><tr><td></td><td>Data Num per Task</td><td>1</td><td>3</td><td>10</td><td>30</td><td>50</td></tr><tr><td rowspan="3">Layer Level Merge Three Models</td><td>Math</td><td>29.92 ± 1.53</td><td>31.19 ± 0.36</td><td>30.83 ± 0.28</td><td>30.50 ± 0.54</td><td>30.81 ± 0.48</td></tr><tr><td>Coding</td><td>16.87 ± 1.75</td><td>15.24 ± 0.50</td><td>15.65 ± 0.57</td><td>15.85 ± 0.50</td><td>15.65 ± 0.29</td></tr><tr><td>Translate</td><td>81.78 ± 0.09</td><td>81.67 ± 0.02</td><td>81.72 ± 0.02</td><td>81.68 ± 0.05</td><td>81.80 ± 0.03</td></tr><tr><td rowspan="3">Attn/MLP Level Merge Three Models</td><td>Math</td><td>32.17 ± 0.73</td><td>32.78 ± 0.68</td><td>32.85 ± 0.60</td><td>32.02 ± 0.31</td><td>32.68 ± 0.06</td></tr><tr><td>Coding</td><td>15.24 ± 0.86</td><td>14.63 ± 0.01</td><td>14.63 ± 0.50</td><td>14.84 ± 0.57</td><td>14.43 ± 0.29</td></tr><tr><td>Translate</td><td>81.84 ± 0.05</td><td>81.74 ± 0.04</td><td>81.72 ± 0.06</td><td>81.79 ± 0.03</td><td>81.79 ± 0.05</td></tr></table>
+
+$$
+\begin{array}{l} f \left(x; \theta_ {0} ^ {i} + \sum_ {t ^ {\prime} = 1} ^ {T} \alpha_ {t ^ {\prime}} ^ {i} \tau_ {t ^ {\prime}} ^ {i}\right) - f (x; \theta_ {t} ^ {i}) \\ = \sum_ {t ^ {\prime} = 1, t ^ {\prime} \neq t} ^ {T} \alpha_ {t ^ {\prime}} ^ {i} \left(f \left(x; \theta_ {0} ^ {i} + \tau_ {t ^ {\prime}} ^ {i}\right) - f \left(x; \theta_ {0} ^ {i}\right)\right) + \left(\alpha_ {t} ^ {i} - 1\right) \left(f \left(x; \theta_ {0} ^ {i} + \tau_ {t} ^ {i}\right) - f \left(x; \theta_ {0} ^ {i}\right)\right) \tag {21} \\ = \sum_ {t ^ {\prime} = 1, t ^ {\prime} \neq t} ^ {T} \alpha_ {t ^ {\prime}} ^ {i} \Delta f (x; \theta_ {t ^ {\prime}} ^ {i}) + (\alpha_ {t} ^ {i} - 1) \Delta f (x; \theta_ {t} ^ {i}) \\ \end{array}
+$$
+
+where  $\Delta f(x;\theta_t^i) = f(x;\theta_t^i) - f(x;\theta_0^i)$
+
+Consider  $f(x; \theta)$  as a  $d$ -dimensional vector. We have
+
+$$
+\alpha_ {1} ^ {i}, \alpha_ {2} ^ {i}, \dots , \alpha_ {T} ^ {i} = \arg \min  \sum_ {t = 1} ^ {T} \mathbb {E} _ {x \in H _ {t} ^ {i}} \sum_ {k = 1} ^ {d} \left(\sum_ {t ^ {\prime} = 1, t ^ {\prime} \neq t} ^ {T} \alpha_ {t ^ {\prime}} ^ {i} \Delta f (x; \theta_ {t ^ {\prime}} ^ {i}) _ {k} + (\alpha_ {t} ^ {i} - 1) \Delta f (x; \theta_ {t} ^ {i}) _ {k}\right) ^ {2} \tag {22}
+$$
+
+Taking the derivative of  $\alpha_{j}^{i}$  and setting the derivative equal to zero, we have
+
+$$
+\begin{array}{l} 2 \sum_ {t = 1, t \neq j} ^ {T} \mathbb {E} _ {x \in D _ {t}} \sum_ {k = 1} ^ {d} \Delta f (x; \theta_ {j} ^ {i}) _ {k} \left(\sum_ {t ^ {\prime} = 1, t ^ {\prime} \neq t} ^ {T} \alpha_ {t ^ {\prime}} ^ {i} \Delta f (x; \theta_ {t ^ {\prime}} ^ {i}) _ {k} + (\alpha_ {t} ^ {i} - 1) \Delta f (x; \theta_ {t} ^ {i}) _ {k}\right) \tag {23} \\ + 2 \mathbb {E} _ {x \in D _ {j}} \sum_ {k = 1} ^ {d} \Delta f (x; \theta_ {j} ^ {i}) _ {k} \left(\sum_ {t ^ {\prime} = 1, t ^ {\prime} \neq j} ^ {T} \alpha_ {t ^ {\prime}} ^ {i} \Delta f (x; \theta_ {t ^ {\prime}} ^ {i}) _ {k} + (\alpha_ {j} ^ {i} - 1) \Delta f (x; \theta_ {j} ^ {i}) _ {k}\right) = 0 \\ \end{array}
+$$
+
+Let
+
+$$
+\boldsymbol {B} _ {a b c} = \mathbb {E} _ {x \in D _ {a}} \sum_ {k = 1} ^ {d} \Delta f (x; \theta_ {b} ^ {i}) _ {k} \Delta f (x; \theta_ {c} ^ {i}) _ {k} \tag {24}
+$$
+
+We have
+
+$$
+\begin{array}{l} \sum_ {t = 1, t \neq j} ^ {T} \left(\sum_ {t ^ {\prime} = 1, t ^ {\prime} \neq t} ^ {T} \alpha_ {t ^ {\prime}} ^ {i} \boldsymbol {B} _ {t j t ^ {\prime}} + \left(\alpha_ {t} ^ {i} - 1\right) \boldsymbol {B} _ {t j t}\right) \tag {25} \\ + \left(\sum_ {t ^ {\prime} = 1, t ^ {\prime} \neq j} ^ {T} \alpha_ {t ^ {\prime}} ^ {i} \boldsymbol {B} _ {j j t ^ {\prime}} + (\alpha_ {j} ^ {i} - 1) \boldsymbol {B} _ {j j j}\right) = 0 \\ \end{array}
+$$
+
+Then we have
+
+$$
+\sum_ {t = 1, t \neq j} ^ {T} \sum_ {t ^ {\prime} = 1, t ^ {\prime} \neq t} ^ {T} \alpha_ {t ^ {\prime}} ^ {i} \boldsymbol {B} _ {t j t ^ {\prime}} + \sum_ {t = 1, t \neq j} ^ {T} \alpha_ {t} ^ {i} \boldsymbol {B} _ {t j t} + \sum_ {t ^ {\prime} = 1, t ^ {\prime} \neq j} ^ {T} \alpha_ {t ^ {\prime}} ^ {i} \boldsymbol {B} _ {j j t ^ {\prime}} + \alpha_ {j} ^ {i} \boldsymbol {B} _ {j j j} = \sum_ {t = 1, t \neq j} ^ {T} \boldsymbol {B} _ {t j t} + \boldsymbol {B} _ {j j j} \tag {26}
+$$
+
+We have
+
+$$
+\sum_ {m = 1, m \neq j} ^ {T} \left(\sum_ {t = 1, t \neq j} ^ {T} \boldsymbol {B} _ {t j m} - \boldsymbol {B} _ {m j m} + \boldsymbol {B} _ {m j m} + \boldsymbol {B} _ {j j m}\right) \alpha_ {m} ^ {i} + \left(\sum_ {t = 1, t \neq j} ^ {T} \boldsymbol {B} _ {t j j} + \boldsymbol {B} _ {j j j}\right) \alpha_ {j} ^ {i} = \sum_ {t = 1} ^ {T} \boldsymbol {B} _ {t j t} \tag {27}
+$$
+
+Then
+
+$$
+\sum_ {m = 1} ^ {T} \left(\sum_ {t = 1} ^ {T} \boldsymbol {B} _ {t j m}\right) \alpha_ {m} ^ {i} = \sum_ {t = 1} ^ {T} \boldsymbol {B} _ {t j t} \tag {28}
+$$
+
+Therefore, by differentiating with respect to each  $\alpha_{j}^{i}$ , where  $j\in [1,\dots ,T]$ , and setting the derivative equal to zero, we obtain:
+
+$$
+\sum_ {m = 1} ^ {T} (\sum_ {t = 1} ^ {T} \boldsymbol {B} _ {t 0 m}) \alpha_ {m} ^ {i} = \sum_ {t = 1} ^ {T} \boldsymbol {B} _ {t 0 t}
+$$
+
+$$
+\sum_ {m = 1} ^ {T} \left(\sum_ {t = 1} ^ {T} \boldsymbol {B} _ {t 1 m}\right) \alpha_ {m} ^ {i} = \sum_ {t = 1} ^ {T} \boldsymbol {B} _ {t 1 t} \tag {29}
+$$
+
+.
+
+$$
+\sum_ {m = 1} ^ {T} (\sum_ {t = 1} ^ {T} \boldsymbol {B} _ {t T m}) \alpha_ {m} ^ {i} = \sum_ {t = 1} ^ {T} \boldsymbol {B} _ {t T t}
+$$
+
+We have
+
+$$
+\boldsymbol {A} \left[ \alpha_ {1} ^ {i}, \alpha_ {2} ^ {i}, \dots , \alpha_ {T} ^ {i} \right] ^ {\top} = \boldsymbol {b} \tag {30}
+$$
+
+where
+
+$$
+\boldsymbol {A} _ {j, k} = \sum_ {\substack {t = 1 \\ T}} ^ {T} \boldsymbol {B} _ {t j k}, \boldsymbol {A} \subseteq \mathbb {R} ^ {T \times T} \tag{31}
+$$
+
+$$
+\boldsymbol {b} _ {j} = \sum_ {t = 1} ^ {T} \boldsymbol {B} _ {t j t}, \boldsymbol {b} \subseteq \mathbb {R} ^ {T}
+$$
+
+Finally we have
+
+$$
+\left[ \alpha_ {1} ^ {i}, \alpha_ {2} ^ {i}, \dots , \alpha_ {T} ^ {i} \right] ^ {\top} = \boldsymbol {A} ^ {- 1} \boldsymbol {b} \tag {32}
+$$
+
+In the practical implementation, we utilize np.linalg.solve to solve Eq.32.
+
+# A.3.2 BALANCE THE INFLUENCE OF EACH SAMPLE
+
+In practical applications of the formula for calculating merging parameters, we observed discrepancies in biases present in data from different tasks. For instance, the average token length varies across tasks (e.g., code data exhibits a greater average length than translation data), and there are differing token distributions (e.g., common token distributions differ between mathematical and code datasets). Consequently, this leads to varying norms of the output feature differences  $(\| \Delta f(x; \theta) \|^2)$  of the models and modules.
+
+To balance the influence of each sample within the formula, we introduce a denominator for each data-related term:
+
+$$
+\alpha_ {1} ^ {i}, \alpha_ {2} ^ {i}, \dots , \alpha_ {T} ^ {i} = \arg \min  \sum_ {t = 1} ^ {T} \mathbb {E} _ {x \in D _ {t}} \frac {\left\| f \left(x ; \theta_ {0} ^ {i} + \sum_ {t ^ {\prime} = 1} ^ {T} \alpha_ {t ^ {\prime}} ^ {i} \tau_ {t ^ {\prime}} ^ {i}\right) - f (x ; \theta_ {t} ^ {i}) \right\| ^ {2}}{\frac {1}{T} \sum_ {t ^ {\prime \prime} = 1} ^ {T} \| \Delta f (x ; \theta_ {0} ^ {i} + \tau_ {t ^ {\prime \prime}} ^ {i}) \| ^ {2}} \tag {33}
+$$
+
+Finally, we obtain that
+
+$$
+B _ {a b c} = \mathbb {E} _ {x \in D _ {a}} \frac {\sum_ {k = 1} ^ {d} \Delta f (x ; \theta_ {b} ^ {i}) _ {k} \Delta f (x ; \theta_ {c} ^ {i}) _ {k}}{\frac {1}{T} \sum_ {t ^ {\prime \prime} = 1} ^ {T} \| \Delta f (x ; \theta_ {0} ^ {i} + \tau_ {t ^ {\prime \prime}} ^ {i}) \| ^ {2}} \tag {34}
+$$
+
+The remaining content is identical to that of the previous section.
+
+# A.3.3 GATHER INPUT FEATURE SET FOR SUBMODULE LEVEL ANALYZING AND MERGING
+
+When analyzing the linearity of modules and calculating the merging parameters for module level merging (including layer level, attn/mlp level and head/mlp leve), We should collect the feature set  $H_{t}^{i}$  corresponding to the input position of each module  $i$  first.
+
+In practice, we input the data from dataset  $D_{t}$  into the pre-trained model  $\theta_0$  all at once, performing a single forward inference to collect all intermediate features, which serve as the input for each module at the corresponding position.
+
+Especially, in layer level's calculation, the input of  $i^{th}$  layer should be the output of the  $(i - 1)^{th}$  layer in  $\theta_0$ . In attn/mlp level's calculation, the input of  $i^{th}$  atm should be the output of the  $(i - 1)^{th}$  layer in  $\theta_0$ , and the input of  $i^{th}$  mlp should be the output of the  $(i)^{th}$  atm together with output of  $(i - 1)^{th}$  layer in  $\theta_0$ . In head level's calculation, the input of head in layer  $i^{th}$  should be the output of the  $(i - 1)^{th}$  layer in  $\theta_0$ .
+
+# A.3.4 MORE DETAILS ABOUT THE BASELINE
+
+For Task arithmetic, We explored the hyperparameter merging weights  $\alpha \in [0.1, 0.2, \dots, 0.9, 1.0]$  and selected the best results for reporting in the table.
+
+For DARE, following the recommendations in the paper, we examined various combinations of the hyperparameter dropout probability  $drop\_ratio \in [0.6,\dots,0.9]$  and merging weights  $\alpha \in [0.6,0.8,1.0]$ , reporting the best results in the table.
+
+The optimal hyperparameters corresponding to the best results in Table 1 and 2 obtained in practice are reported in Table 12.
+
+Table 12: The optimal hyperparameters corresponding to the best results obtained in practice for baslines.  
+
+<table><tr><td>Hyper-parameters</td><td>Math &amp; Code</td><td>Math &amp; Translate</td><td>Coding &amp; Translate</td><td>Math &amp; Coding &amp; Translate</td></tr><tr><td colspan="5">Llama-2-7B</td></tr><tr><td>DARE</td><td>0.8 1.0</td><td>0.8 1.0</td><td>0.8 1.0</td><td>0.8 1.0</td></tr><tr><td>Task Arithmetic</td><td>0.6</td><td>0.5</td><td>0.4</td><td>0.5</td></tr><tr><td colspan="5">Llama-2-13B</td></tr><tr><td>DARE</td><td>0.7 1.0</td><td>0.7 1.0</td><td>0.8 1.0</td><td>0.7 1.0</td></tr><tr><td>Task Arithmetic</td><td>0.6</td><td>0.6</td><td>0.4</td><td>0.5</td></tr></table>
+
+# A.3.5 MORE DETAILS AT THE HEAD/MLP LEVEL
+
+For the Head/MLP level, due to the architecture of the transformer, the output features corresponding to a single head cannot be directly obtained: The o-proj module will gather the intermediate features of each head and map them into a comprehensive attention module output, which can not be directly divided by the arrangement of the heads.
+
+Therefore, we employed certain techniques to achieve this in practice: For each layer, we first collect the input of oProj. Subsequently, we divid oProj into n_head linear layers according to the parameter relationships associated with each head. The corresponding segments of the collected input are then fed into these corresponding linear layers, producing outputs that serve as the final output features for each head, which will be used for subsequent calculations of the merging parameters.
+
+# A.4 VISUALIZATION RESULTS
+
+Table 13: The actual parameters obtained at the model level. This is one of the results from three random trials.  
+
+<table><tr><td>Method</td><td>Math &amp; Code</td><td>Math &amp; Translate</td><td>Coding &amp; Translate</td><td>Math &amp; Coding &amp; Translate</td></tr><tr><td colspan="5">Llama-2-7B</td></tr><tr><td>Ours Model Level</td><td>[0.851 0.259]</td><td>[3.431 -1.537]</td><td>[0.237 0.882]</td><td>[0.758 0.480 -0.180]</td></tr><tr><td colspan="5">Llama-2-13B</td></tr><tr><td>Ours Model Level</td><td>[1.039 0.132]</td><td>[-1.083 1.652]</td><td>[-0.463 1.923]</td><td>[0.562 0.422 0.062 ]</td></tr></table>
+
+# A.5 MORE DISCUSSIONS
+
+# A.5.1 DIFFERENCES BETWEEN THE LINEARLY PROPERTIES STUDIED IN THIS PAPER AND TRADITIONAL LINEAR PROPERTIES
+
+The linearity properties we studied in this paper differ from the Traditional Linear Properties, as illustrated in the following table:
+
+Table 14: Differences between the Linearly Properties Studied in this Paper and Traditional Linear Properties  
+
+<table><tr><td></td><td>Traditional Linear Properties</td><td>Linearity Properties Studied in This Paper</td></tr><tr><td>Definition</td><td>Linear relationship between the difference in output features of different input fθ(x) - fθ(x0) and the difference in input x - x0. fθ() denotes model/module) with parameter θ</td><td>Linear relationship between the difference in output features of the model (module) before and after fine-tuning fθ(x) - fθ0(x) and the difference in parameters before and after fine-tuning θ - θ0.</td></tr><tr><td>Formula Form</td><td>fθ(x) - fθ(x0) ≈ C(x - x0) where C is a constant determined by θ.</td><td>fθ(x) - fθ0(x) ≈ C(θ - θ0) where C is a constant determined by θ0 and x.</td></tr><tr><td>Determinants</td><td>Determined by the model (module) architecture of θ.</td><td>Determined by both the alteration introduced by fine-tuning process and the model (module) architecture of θ0.</td></tr></table>
+
+We also present an example to further illustrate that there is no particularly intuitive connection between the linearity of the module and the linearity of its submodules:
+
+Consider two contiguous submodules,  $\theta_{1}$  and  $\theta_{2}$ . Let  $\{\theta_1;\theta_2\}$  denote the composed module.
+
+If  $\theta_{1}$  and  $\theta_{2}$  both satisfy the Traditional Linear Properties, it is obvious that the module  $\{\theta_1;\theta_2\}$  which formed by these submodules will also adhere to the Traditional Linear Properties.
+
+However, if  $\theta_{1}$  and  $\theta_{2}$  both satisfy the linearity property studied in this paper, for an arbitrary  $x$ , the final output results after applying perturbations  $\Delta \theta_{1}$  and  $\Delta \theta_{2}$  to the two modules is:
+
+<table><tr><td>Input</td><td>x</td></tr><tr><td>Output of the first submodule θ1 + Δθ1</td><td>fθ1(x) + Cθ1,xΔθ1</td></tr><tr><td>Output of the second submodule θ2 + Δθ2</td><td>fθ2(fθ1(x) + Cθ1,xΔθ1) + Cθ2,fθ1(x)+Cθ1,xΔθ1Δθ2... (1)</td></tr></table>
+
+![](images/7218fa964833bc68c3477dd3692514709d01d6118defb23dc009ec1611fed338.jpg)  
+Figure 5: A visual representation of the merging parameters obtained at different levels during the actual merging process is presented. Here, we showcase a result from the merging of three finetuned models. Additionally, only a portion of the head's merging parameters is displayed, while the distribution of the merging parameters for the other heads is similar to those presented.
+
+$C_{\theta ,x}$  denote a constant determined by  $\theta$  and  $x$ . Now, if  $\{\theta_1;\theta_2\}$  as a whole satisfies linearity property studied in this paper, then its output should be  $f_{\theta_2}(f_{\theta_1}(x)) + C_{\{\theta_1;\theta_2\},x}\{\Delta \theta_1;\Delta \theta_2\} \ldots (2)$  when input  $x$ .
+
+We can observe a significant difference between formulas (1) and (2), indicating that there is no particularly intuitive connection between the linearity of the module and the linearity of its submodules, which also necessitates further experimental and observational analysis, as what we do in this paper.
+
+![](images/b3b4071ca81a4964065833f81b101a13278352885b3841c97dea5811bb72df7f.jpg)  
+Figure 6: A sample of Visualization of the  $\frac{\mathcal{D}(f(x;\theta_i),f(x;\theta_j))}{\mathcal{D}(f(x;\theta),f(x;\theta_0))}$ . The image is sourced from the linearity validation of the Llama-2-13B model after fine-tuning for mathematical tasks. From left to right, it displays the model-level linearity validation, the layer-level validation at the 9th layer, and the MLP-level validation at the 9th layer. The latter two validations are already very close to ideal linearity.
+
+# A.5.2 COMPARISON WITH WEMOE AND CAT
+
+Table 15: Comparison with WEMoE and CAT.  
+
+<table><tr><td>Categories</td><td>Ours</td><td>BTX(Sukhbaatar et al., 2024)(std MoE)</td><td>WEMoE(Tang et al.)</td><td>CAT (Prabhakar et al., 2024)</td></tr><tr><td>Requirements for the Model to be Merged</td><td>Same pretrain model, conventional fine-tuning with different task</td><td>Same pretrain model, conventional fine-tuning with different task</td><td>Same pretrain model, conventional fine-tuning with different task</td><td>Same pretrain model, LoRA fine-tuning with different task.</td></tr><tr><td>Which Part to be Merged</td><td>All modules</td><td>All modules except MLP</td><td>All modules except MLP, The MLP will be merged during inference.</td><td>All LoRA&#x27;s Parameter</td></tr><tr><td>Hyperparameters or Additional Parameters Required for Merging</td><td>Weights for each submodules (such as Layer/Attn/MLP)</td><td>Router Network for each MLP</td><td>Router Network for each MLP</td><td>Weights for each Lora Martix</td></tr><tr><td>Optimization Methods</td><td>Closed-form Solution (Training-free)</td><td>Training</td><td>Training</td><td>Training</td></tr><tr><td>The Architecture of the Merged Model</td><td>Same as Model to be Merged</td><td>MoE Architecture (Router for mlp&#x27;s output Merging Weights)</td><td>MoE Architecture (Router for Parameter Merging Weights)</td><td>Same as Model to be Merged</td></tr></table>
+
+In our approach, merging is performed directly, which may result in different merging outcomes due to the selection of varying data. Our method requires only minimal data and does not necessitate training, yielding a model with a standard architecture. In comparison to our method, WEMoE constructs a MoE architecture model and employs a Router during inference to determine the merging weights for each MLP based on the current input, which also utilizes a linear merging form. As a result, WEMoE may get better merging weights and achieve better results but incurs additional training and inference costs.
+
+# A.5.3 DISCUSSION ON SECOND-ORDER INFORMATION
+
+In our initial setup, considering that the original form of Task Arithmetic is first-order linear, we did not give much consideration to the involvement of second-order information for consistency. As mentioned in the comments, we recognize that further considering the incorporation of second-order information is a very meaningful direction. If we can find methods to utilize second-order information, we may be able to further refine our definitions of linearity and formula expressions, and estimate the linearity of each module with lower error.
+
+Additionally, we might consider iteratively calculating the weight of each module; for example, first calculate the weight of the first layer, then use the fused output features of the first layer to guide the weight calculation of the second layer. This might further exploit the potential connections between modules in different positions. By utilizing second-order information, we might be able to design a new loss function that merges all the weights to be optimized into a simultaneous optimization
+
+process. In future work, we will explore more applications of second-order information in our method.
+
+# A.5.4 COMPARISON WITH GIT RE BASIN, ZIPIT AND MUDSC
+
+Table 16: Comparison with Git Re Basin,Zipit and MuDSC  
+
+<table><tr><td>Comparison Table</td><td>Applicable Module</td><td>The Type of Linearity Relied upon during the Merging</td><td>Key method for merging</td></tr><tr><td>Git Re Basin (Ainsworth et al.)</td><td>Linear Layer (full connected layer) in Network</td><td>Traditional Linearity Properties</td><td>Combinatorial optimization</td></tr><tr><td>Zipit! (Stoica et al., 2023)</td><td>Linear Layer (full connected layer) in Network</td><td>Traditional Linearity Properties</td><td>Align features by similarity in the activation space</td></tr><tr><td>MuDSC (Xu et al., 2024b)</td><td>Linear Layer (full connected layer) in Network</td><td>Traditional Linearity Properties</td><td>Align units by similarity in both the parameter space and the activation space</td></tr><tr><td>Ours</td><td>Each submodule level (including layer, attn and MLP) that exhibit Linearity introduced in this paper</td><td>Linearity Properties Studied in This Paper</td><td>Closed-form solution of optimal merging weights</td></tr></table>
+
+As shown in the table, all these methods relies on the Traditional Linearity Property, and can only apply to linear layers. In summary, we believe that our method has several novel contributions that make it distinct from Git Re Basin (Ainsworth et al.), Zipit!(Stoica et al., 2023) and MuDSC(Xu et al., 2024b).
+
+# A.5.5 DISCUSSION ON FINELY-GRAINED COMPONENTS
+
+Further decomposing the model into more finely-grained components is indeed an intuitive and promising direction for exploration. In the paper, we conducted some related experiments. Based on the structure of the LLM model, we further divided the attention layer into individual heads, independently estimating their linearity and ultimately testing their performance, but revealing unstable outcomes at this level (as shown in Table 3).
+
+Here we propose a potential explanation for the occurrence of this situation. Numerous studies (Wang et al., 2023; Chen et al., 2024; Zhang et al., 2024) suggest that each head in the model performs different roles during inference. Following fine-tuning across various tasks, certain heads may adopt distinct functions. However, our objective function may necessitate a single head to simultaneously average among multiple heads with diverse functionalities. This requirement can be quite challenging for an individual head and may ultimately result in a collapse of functionality in the merged head.
+
+However, this may not imply that the linearity of the head or more finely-grained components is unhelpful for the merging process. As we mentioned above, the relationship between the linearity of the module and the linearity of its submodules is not straightforward; therefore, further experiments and analyses are required. We will further explore whether there are better ways to address the aforementioned issues in future work. For instance, based on the architecture of the LLM, we could further decompose each module into individual linear layers (decompose attention into qkvo projection linear layers, and MLP into two linear layers). Alternatively, we could incorporate interpretability methods(Wang et al., 2023) to improve understanding and identify which head is most relevant to a specific task.
+
+# A.5.6 DISCUSSION ABOUT ACTIVATION FUNCTIONS
+
+In this paper, we did not make any changes to the architecture of any part of the model, and all nonlinear activation functions remained unaltered. Regarding the achievement of such linearity without modifying the activation functions, we believe this is due to the many differences between the Traditional Linear Properties and the Linearity Properties Studied in This Paper discussed above.
+
+The connection between nonlinear activation functions and Traditional Linear Properties has been widely studied, but the connection between nonlinear activation functions and the Linearity Properties Studied in This Paper still awaits exploration. We believe that activation functions are likely to play a crucial role in the emergence of linearity.
+
+Currently, the vast majority of LLMs (including Qwen and Llama-2) use the SwiGLU activation function. Compared to other activation functions such as ReLU and GELU, SwiGLU is more complex and nonlinear, yet it also enhances performance. We hypothesize that replacing SwiGLU with more linear activation functions could improve overall linearity, which would be beneficial for our method. However, we must carefully consider the balance between the immediate performance drop caused by the replacement and the performance gains achieved through the merging, which requires further experiments and analysis.
+
+However, it is difficult to find other identical LLMs, differing only in their activation function, for fair comparison due to the enormous cost of training a LLM. We plan to conduct experiments related to activation functions in a controlled environment as described in Allen-Zhu & Li (2023) in future work.
+
+# A.5.7 DISCUSSION ABOUT HOW TO LEVERAGE LINEARITY AT THE HEAD LEVEL
+
+1. Our initial proposal involves collaborating with interpretability methods Wang et al. (2023). Initially, we can use interpretability methods to determine which head is actually more relevant to a particular task. By obtaining weights through relevance, we can further adjust the originally explored weights to make full use of the capability of each head.  
+2. The second proposal involves modifying our loss function: our original loss is quite simple and only explores the output feature level. We can refer to other approaches (Stoica et al., 2023; Xu et al., 2024b) to also explore parameter space, making the explored weights more reasonable and enhancing the use of linearity.  
+3. The third proposal is to use our linearity to analyze which task a given head is actually more related to (since our linearity is introduced by fine-tuning). This might potentially become a new method of interpretability.
+
+# A.5.8 DISCUSSION ABOUT THE EFFECTS OF MODEL DEPTH
+
+For a fair and convincing comparison, we utilized existing backbones like Qwen and Llama to examine how different depths impact linearity, using the Non-linearity Score as an indicator, which is introduced in our paper and with smaller score indicating closer adherence to ideal linearity.
+
+Table 17: Non-linearity Score for Different Models  
+
+<table><tr><td>Level</td><td>Qwen-2-0.5B</td><td>Qwen-2.5-7B</td><td>Llama-2-7B</td><td>Llama-2-13B</td></tr><tr><td>Model</td><td>2.1130 ± 0.5250</td><td>4.2559 ± 0.7037</td><td>2.7637 ± 0.3442</td><td>2.1929 ± 0.2271</td></tr><tr><td>Layer</td><td>0.4430 ± 0.7331</td><td>0.1412 ± 0.3019</td><td>0.1645 ± 0.4329</td><td>0.1675 ± 0.3760</td></tr><tr><td>Attn</td><td>0.2439 ± 0.1631</td><td>0.0649 ± 0.0592</td><td>0.0229 ± 0.0107</td><td>0.0250 ± 0.0115</td></tr><tr><td>MLP</td><td>0.3112 ± 0.8353</td><td>0.0810 ± 0.2068</td><td>0.2030 ± 0.5655</td><td>0.1622 ± 0.4295</td></tr></table>
+
+From the table, we can observe:
+
+1. At the model level, linearity does not seem to have a clear relationship with model depth. For the Qwen model, the linearity of the 0.5B model is better than the deeper 7B model, while for Llama, linearity is greater in the 7B model compared to the deeper 13B.  
+2. At the Layer/Attn/MLP level, within the transformer architecture of decoder language models, the depth of each Layer/Attn/MLP across different model sizes can be regarded as the same, with variations occurring in the width. Overall, wider modules tend to exhibit slightly better linearity, which aligns with conclusions in NTK-related works such as Jacot et al. (2018) and Chizat et al. (2019).
+
+# A.5.9 DISCUSSION ABOUT THE PHENOMENON THAT QWEN-2.5-7B HAS MUCH HIGHER NON-LINEARITY SCORE THAN OTHERS
+
+In order to provide a possible explanation for the phenomenon that the non-linearity score for Qwen-2.5-7B is much higher than others, we conducted further experiments based on Property 1 from the paper.
+
+Specifically, we first computed the parameter difference before and after fine-tuning for each model, denoted as  $\tau = \theta -\theta_0$ . We then used a parameter  $\alpha$  to weight this parameter difference and added it back to the pre-trained model  $\theta_0$  to obtain the interpolated model  $\theta_0 + \alpha \tau$ . The L2 distance between the features output by this model  $\theta_0 + \alpha \tau$  and the features output by the fine-tuned model  $\theta$  was computed and is presented in the table below. (When  $\alpha = 0$ , the interpolated model becomes the pre-trained model  $\theta_0$ . When  $\alpha = 1$ , the interpolated model becomes the fine-tuned model  $\theta$ , the distance will be 0. All distances are normalized. The Non-linearity Score is directly proportional to the difference between the value for the specific model and the value for ideal linearity in the table.)
+
+Table 18: Model Linearity and Corresponding Non-linearity Score  
+
+<table><tr><td>Model</td><td>α = 0.0</td><td>α = 0.1</td><td>α = 0.2</td><td>α = 0.3</td><td>α = 0.4</td><td>α = 0.5</td><td>α = 0.6</td><td>α = 0.7</td><td>α = 0.8</td><td>α = 0.9</td><td>α = 1.0</td><td>Non-linearity Score</td></tr><tr><td>Ideal Linearity</td><td>1.000</td><td>0.900</td><td>0.800</td><td>0.700</td><td>0.600</td><td>0.500</td><td>0.400</td><td>0.300</td><td>0.200</td><td>0.100</td><td>0.000</td><td>0</td></tr><tr><td>Qwen-2.5-7B</td><td>1.000</td><td>0.911</td><td>0.879</td><td>0.841</td><td>0.805</td><td>0.731</td><td>0.429</td><td>0.277</td><td>0.169</td><td>0.085</td><td>0.000</td><td>4.2559</td></tr><tr><td>Qwen-2-0.5B</td><td>1.000</td><td>0.923</td><td>0.828</td><td>0.763</td><td>0.719</td><td>0.655</td><td>0.570</td><td>0.427</td><td>0.262</td><td>0.125</td><td>0.000</td><td>2.1130</td></tr><tr><td>Llama-2-7B</td><td>1.000</td><td>0.871</td><td>0.726</td><td>0.637</td><td>0.545</td><td>0.483</td><td>0.370</td><td>0.263</td><td>0.169</td><td>0.080</td><td>0.000</td><td>2.7637</td></tr><tr><td>Llama-2-13B</td><td>1.000</td><td>0.805</td><td>0.760</td><td>0.635</td><td>0.531</td><td>0.435</td><td>0.339</td><td>0.247</td><td>0.150</td><td>0.073</td><td>0.000</td><td>2.1929</td></tr></table>
+
+For a model that satisfies linearity, the distance between the interpolated model's output features and the fine-tuned model's output features should decrease linearly as  $\alpha$  increases (as shown in the first row of the table). However, we observe from the table that the Qwen-2.5-7B model experiences a very steep decline between  $\alpha = 0.5$  and  $\alpha = 0.6$ , which does not occur with other backbones. Such a steep decline does not satisfy linearity well, resulting in the non-linearity score for Qwen-2.5-7B in model level being much higher than others.
+
+This is a very interesting phenomenon. We find that the performance of Qwen-2.5-7B is very similar to a sigmoid function: when the parameter difference (or  $\alpha$ ) is far from the threshold, the change in its output features is relatively slow, whereas the change is rapid when the parameter difference (or  $\alpha$ ) approaches the threshold.
+
+We suspect that this phenomenon is related to the model's foundational capabilities, as Qwen-2.5-7B is a model with significantly stronger performance in various aspects compared to other models (according to official website data). To explore the specific reasons for this intriguing phenomenon, we will conduct more experiments and analyses in our future work.
\ No newline at end of file
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+# LEVERAGING VARIABLE SPARSITY TO REFINE PARETO STATIONARITY IN MULTI-OBJECTIVE OPTIMIZATION
+
+Zeou Hu
+
+University of Waterloo, Vector Institute
+
+zeou.hu@uwaterloo.ca
+
+Yaoliang Yu
+
+University of Waterloo, Vector Institute
+
+yaoliang.yu@uwaterloo.ca
+
+# ABSTRACT
+
+Gradient-based multi-objective optimization (MOO) is essential in modern machine learning, with applications in e.g., multi-task learning, federated learning, algorithmic fairness and reinforcement learning. In this work, we first reveal some limitations of Pareto stationarity, a widely accepted first-order condition for Pareto optimality, in the presence of sparse function-variable structures. Next, to account for such sparsity, we propose a novel solution concept termed Refined Pareto Stationarity (RPS), which we prove is always sandwiched between Pareto optimality and Pareto stationarity. We give an efficient partitioning algorithm to automatically mine the function-variable dependency and substantially trim non-optimal Pareto stationary solutions. Then, we show that gradient-based descent algorithms in MOO can be enhanced with our refined partitioning. In particular, we propose Multiple Gradient Descent Algorithm with Refined Partition (RP-MGDA) as an example method that converges to RPS, while still enjoying a similar per-step complexity and convergence rate. Lastly, we validate our approach through experiments on both synthetic examples and realistic application scenarios where distinct function-variable dependency structures appear. Our results highlight the importance of exploiting function-variable structure in gradient-based MOO, and provide a seamless enhancement to existing approaches.
+
+# 1 INTRODUCTION
+
+Multi-objective optimization (MOO) aims to optimize several objective functions simultaneously over shared parameters, seeking solutions where no objective can be improved without worsening another. Many modern machine learning applications naturally involve multiple objectives. For example, in fair machine learning, one needs to balance model utility with various fairness notions, while in multi-task learning, models are trained across different tasks with shared representations. Similarly, federated learning and reinforcement learning often involve optimizing the performance of multiple agents or entities distributed across environments. The solutions to such problems are known as Pareto optimal (PO) solutions, representing a balance across the competing objectives. However, finding PO solutions becomes challenging in complex (e.g. high-dimensional) problems.
+
+Among various approaches for solving MOO problems, one that has gained significant attention recently is the Multiple Gradient Descent Algorithm (MGDA) (Mukai, 1980; Fliege and Svaiter, 2000; Désideri, 2012). MGDA extends classical gradient descent to optimize multiple objectives simultaneously. Unlike traditional linear scalarization, which requires predetermined weights for each objective, MGDA directly optimizes the vector-valued problem without such weighting. At each iteration, MGDA identifies a common descent direction that improves all objectives, a feature not guaranteed by scalarization methods. MGDA's ability to converge to a Pareto stationary solution, which serves as a first-order necessary condition for Pareto optimality, has made it widely applicable in areas like multi-task learning (Sener and Koltun, 2018), federated learning (Hu et al., 2022), and generative modeling (Albuquerque et al., 2019). While Pareto stationarity provides a solid theoretical foundation (as shown in Lemma 1 below), it can fall short in complex, especially sparse settings, creating room for a more refined approach, as we illustrate in Section 4.
+
+![](images/aa4ce9972a77662da347912a6c80120d74391ec73286131bcc805340a669c5e1.jpg)  
+Figure 1: A simple illustration of our main idea. From left to right: adjacency matrix showing the function-variable dependency structure; the underlying bipartite graph with a cycle highlighted in red; merging the variables in the cycle to a subset and contracting the graph; adjacency matrix showing the new dependency structure. The process is iterated until the graph becomes acyclic.
+
+![](images/923d53ef73fad05478b4c163b502c0fe56fff61b10f6f87a26aedce9756e19f8.jpg)
+
+![](images/46cc8ff94368c87fce1ec716c4ca5ce71952af40f8369d63d72dd47aa3490242.jpg)
+
+![](images/a547dfd3693d755fa7391d0a2132095b33e7174a5de39255b7eac95b5021d92d.jpg)
+
+In this work, we reveal that Pareto stationarity (PS), while necessary for Pareto optimality, is often insufficient. Even in convex settings, PS can frequently lead to sub-optimal PO solutions, especially when sparse function-variable dependencies cause strict convexity to fail. To address this, we propose Refined Pareto Stationarity (RPS), which leverages sparsity to identify a valid partition of variables and refines PS with respect to this partition. We prove that RPS is (strictly) sandwiched between PS and Pareto optimality (PO), making it a more useful and practical goal. Additionally, under partial strict convexity, we prove RPS reduces exactly to Pareto optimality, whereas the widely-used PS does not. These advantages are illustrated through intuitive examples and validated experimentally.
+
+Since RPS is defined with respect to a refined partition, we propose an efficient partitioning procedure (REFINED_PARTITION) to automatically discover, if needed, the correct refined partition of variables, based on an input adjacency matrix encoding function-variable dependencies. Figure 1 illustrates our main idea through a simple example. More importantly, our results enable practitioners to enhance existing gradient-based descent algorithms for MOO by incorporating this partitioning routine and the refined stationary condition RPS. Specifically, we introduce Multiple Gradient Descent Algorithm with Refined Partition (RP-MGDA) to exploit these advances and we formally prove its convergence.
+
+We summarize our contributions below, with further context provided in the next section:
+
+- We present illustrative examples to reveal some limitations of Pareto stationarity, and motivate the need for a valid partitioning of variables when function-variable dependency is sparse.  
+- We propose RPS, a novel solution concept that refines Pareto stationarity, together with an efficient partitioning procedure REFINED_PARTITION to automatically discover the refined partition.  
+- We introduce RP-MGDA, an example of enhancing gradient-based descent algorithms in MOO by incorporating refined partitioning. We establish the convergence and complexity of RP-MGDA.  
+- We validate the effectiveness of our approach through experiments on synthetic and benchmark datasets, showcasing its application to ML scenarios with diverse dependency structures.
+
+Our code is available at https://github.com/watml/RP-MGDA.
+
+# 2 RELATED WORKS
+
+Multi-objective optimization (MOO) and Pareto solution have been extensively studied, with traditional approaches such as evolutionary algorithms (Deb et al., 2002). However, these techniques face challenges when applied to modern machine learning problems, which are often better addressed by gradient-based methods. Consequently, in this work we focus on gradient-based MOO.
+
+Gradient-based MOO. Gradient-based MOO involves optimizing multiple objectives using gradient information. A foundational algorithm in this regard is MGDA (Mukai, 1980; Fliege and Svaiter, 2000; Désideri, 2012), which computes a common descent direction for all objectives by solving for the minimum-norm vector in the convex hull of gradients. Fliege et al. (2019) offered a detailed convergence analysis of MGDA, while subsequent works have also extended classical single-objective
+
+methods to the multi-objective setting, including Newton's method (Fliege et al., 2009), (accelerated) proximal gradient (Tanabe et al., 2019; Tanabe et al., 2023), conditional gradient (Assunção et al., 2021), and subgradient method (O. Montonen and Mäkelä, 2018), among others. Another important line of research focuses on stochastic variants of MGDA, e.g. Mercier et al. (2018), due to their practical relevance in machine learning, particularly for mini-batch training of deep neural networks. Liu and Vicente (2021) introduced SMG as a direct stochastic counterpart of MGDA, albeit with a biased estimate of the common descent direction. Subsequent works of Zhou et al. (2022) and Fernando et al. (2023) track historical information to reduce bias, while Chen et al. (2023) and Xiao et al. (2023) propose unbiased estimate of the update direction through double sampling.
+
+MOO in Multi-task Learning (MTL). MTL aims to train a common model that can solve multiple, potentially related tasks. Sener and Koltun (2018) first framed MTL as a multi-objective optimization problem and addressed it using MGDA. Since then, numerous works in MTL have approached this problem within the general MOO framework, treating task losses as objectives and proposing novel gradient aggregation schemes to mitigate task conflicts while addressing various aspects of MTL (e.g., Lin et al., 2019). PCGrad (Yu et al., 2020) projects each gradient onto the normal plane of other conflicting gradients and aggregates them. CAGrad (Liu et al., 2021a) balances minimizing the average task loss and improving the worst-performing objective by constraining the search region for the common direction in MGDA. Nash-MTL (Navon et al., 2022) formulates MTL as a bargaining game to weigh the gradients. I-MTL (Liu et al., 2021b) separately searches for the magnitude and direction of the common gradient. FairGrad (Ban and Ji, 2024) proposes a new gradient aggregation scheme to ensure  $\alpha$ -fairness in MTL.
+
+MOO in Fair machine learning (FML). FML is inherently a multi-objective problem, due to the trade-off between utility and various notions of fairness that need not be compatible with each other (Kleinberg et al., 2017). A common strategy in FML is to linearly combine the multiple objectives through scalar weights (Kamishima et al., 2011), corresponding to the well-known scalarization technique in MOO, while later work, such as Padh et al. (2021), applies MGDA to handle these competing objectives more directly.
+
+The works mentioned above either 1) adopt the general MOO formulation that treats all learnable parameters as a single variable  $\mathbf{w} \in \mathbb{R}^d$ , thereby disregarding any underlying function-variable structure, or 2) are confined to the MTL setting presented in Sener and Koltun (2018), whose variable partition scheme can be viewed as a special case of our more general framework. To our best knowledge, our work is the first to systematically investigate and refine Pareto stationary solutions in gradient-based MOO, by taking into account the function-variable structure. Our results are complementary to the above approaches and potentially integrable with many of them.
+
+# 3 PRELIMINARIES
+
+In this section, we briefly review the concepts of Pareto optimality and Pareto stationarity in multi-objective optimization. We then introduce the widely used MGDA method.
+
+# 3.1 MULTI-OBJECTIVE MINIMIZATION (MOO)
+
+In mathematical terms, a Multi-Objective Minimization (MOO) problem can be written as<sup>1</sup>
+
+$$
+\min  _ {\mathbf {w} \in \mathbb {R} ^ {d}} \mathbf {f} (\mathbf {w}), \text {w h e r e} \mathbf {f} (\mathbf {w}) := \left(f _ {1} (\mathbf {w}), f _ {2} (\mathbf {w}), \dots , f _ {m} (\mathbf {w})\right) \tag {1}
+$$
+
+and the minimum is defined w.r.t. the partial ordering $^2$  :
+
+$$
+\mathbf {f} (\mathbf {w}) \leq \mathbf {f} (\mathbf {z}) \iff \forall i = 1, \dots , m, f _ {i} (\mathbf {w}) \leq f _ {i} (\mathbf {z}). \tag {2}
+$$
+
+Unlike single objective optimization, with multiple objectives it is possible that
+
+$$
+\mathbf {f} (\mathbf {w}) \not \leq \mathbf {f} (\mathbf {z}) \text {a n d} \mathbf {f} (\mathbf {z}) \not \leq \mathbf {f} (\mathbf {w}), \tag {3}
+$$
+
+in which case we say  $\mathbf{w}$  and  $\mathbf{z}$  are not comparable. As a result, there is usually a set of solutions that are all optimal (a.k.a. Pareto Optimal) for a given MOO problem, whose objective values form the Pareto front.
+
+# 3.2 PARETO OPTIMALITY AND PARETO STATIONARITY
+
+Definition 1 (Pareto Optimality). We call  $\mathbf{w}_{*}$  a Pareto optimal solution of (1) if its objective value  $\mathbf{f}(\mathbf{w}_{*})$  is a minimum element w.r.t. the partial ordering in (2); equivalently,
+
+$$
+\forall \mathbf {w}, \mathbf {f} (\mathbf {w}) \leq \mathbf {f} \left(\mathbf {w} _ {*}\right) \Longrightarrow \mathbf {f} (\mathbf {w}) = \mathbf {f} \left(\mathbf {w} _ {*}\right). \tag {4}
+$$
+
+In other words, it is not possible to improve any component objective in  $\mathbf{f}(\mathbf{w}_*)$  without compromising some other objective.
+
+Definition 2 (Weakly Pareto Optimality). We call  $\mathbf{w}_{*}$  a weakly Pareto optimal solution if there does not exist any  $\mathbf{w}$  such that  $\mathbf{f}(\mathbf{w}) < \mathbf{f}(\mathbf{w}_{*})$ , i.e., it is not possible to improve all component objectives in  $\mathbf{f}(\mathbf{w}_{*})$  simultaneously.
+
+Clearly, any Pareto optimal solution is also weakly Pareto optimal but the converse may not hold.
+
+Next, we recall the concept of Pareto Sationarity (also referred to as Pareto Criticality), which is the first order necessary condition for Pareto optimality.
+
+Definition 3 (Pareto Stationarity). We call  $\mathbf{w}_{*}$  Pareto stationary iff
+
+$$
+\mathbf {0} \in \operatorname {c o n v} \left\{\nabla f _ {1} \left(\mathbf {w} _ {*}\right), \dots , \nabla f _ {m} \left(\mathbf {w} _ {*}\right) \right\}, \tag {5}
+$$
+
+i.e., there exists some  $\lambda \geq 0$  such that  $\sum_{i}\lambda_{i} = 1$  and  $\sum_{i}\lambda_{i}\nabla f_{i}(\mathbf{w}^{*}) = \mathbf{0}$
+
+The relevance of Pareto stationarity is captured in the following lemma:
+
+Lemma 1 (e.g., Mukai, 1980, Theorem 1). Any Pareto optimal solution is Pareto stationary. Conversely, if all functions are convex (resp., strictly convex), then any Pareto stationary solution is weakly Pareto optimal (resp., Pareto optimal).
+
+# 3.3 MULTIPLE GRADIENT DESCENT ALGORITHM (MGDA)
+
+MGDA has been independently proposed by Mukai (1980) and Fliege and Svaiter (2000) and Désidéri (2012) as a gradient-based method to find Pareto stationary solutions of an MOO problem. It has gained notable attention in machine learning in recent years, largely because of its gradient-based nature, in contrast to traditional MOO techniques. Compared with the scalarization technique (i.e., linearly combining different objectives), MGDA removes the need to pre-determine any weight vector. More importantly, MGDA is guaranteed to improve all objectives of any starting solution while scalarization cannot (Mukai, 1980).
+
+In each iteration, MGDA seeks a new solution that minimizes the maximum change among all objectives, i.e.,
+
+$$
+\tilde {\mathbf {w}} ^ {t + 1} = \underset {\mathbf {w}} {\operatorname {a r g m i n}} \max  _ {\boldsymbol {\lambda} \in \Delta} \boldsymbol {\lambda} ^ {\top} \left(\mathbf {f} (\mathbf {w}) - \mathbf {f} \left(\mathbf {w} ^ {t}\right)\right), \tag {6}
+$$
+
+where  $\Delta$  is the standard simplex. Next, we upper bound  $\mathbf{f}(\mathbf{w})$  by the usual quadratic approximation:
+
+$$
+\min  _ {\mathbf {w}} \max  _ {\lambda \in \Delta} \boldsymbol {\lambda} ^ {\top} J _ {\mathbf {f}} ^ {\top} \left(\mathbf {w} ^ {t}\right) \left(\mathbf {w} - \mathbf {w} ^ {t}\right) + \frac {1}{2 \eta} \| \mathbf {w} - \mathbf {w} ^ {t} \| ^ {2}. \tag {7}
+$$
+
+where  $J_{\mathbf{f}} = [\nabla f_1, \dots, \nabla f_m] \in \mathbb{R}^{d \times m}$  is the Jacobian and  $\eta > 0$  is the step size. This approximation (7) is (strongly) convex in  $\mathbf{w}$  (even for nonconvex  $\mathbf{f}$ ) and concave in  $\lambda$ , and hence strong duality holds. Swapping min with max in (7) and setting the derivative w.r.t.  $\mathbf{w}$  to  $\mathbf{0}$ , we deduce the following "minimum-norm" direction:
+
+$$
+\mathbf {d} ^ {t} = J _ {\mathbf {f}} \left(\mathbf {w} ^ {t}\right) \boldsymbol {\lambda} _ {*} ^ {t}, \text {w h e r e} \quad \boldsymbol {\lambda} _ {*} ^ {t} = \underset {\boldsymbol {\lambda} \in \Delta} {\operatorname {a r g m i n}} \| J _ {\mathbf {f}} \left(\mathbf {w} ^ {t}\right) \boldsymbol {\lambda} \| ^ {2}. \tag {8}
+$$
+
+MGDA then performs the update along the direction  $\mathbf{d}^t$ :
+
+$$
+\mathbf {w} ^ {t + 1} = \mathbf {w} ^ {t} - \eta \mathbf {d} ^ {t}. \tag {9}
+$$
+
+Common stopping criteria include  $\| \mathbf{d}^t\| \leq \mathrm{tol}$  for some tolerance tol or by capping at a maximum number of iterations.
+
+When the step size  $\eta$  is sufficiently small (so that (7) is indeed an upper bound on (6)), the MGDA update (9) simultaneously decreases all objectives (seen easily by the feasible choice  $\mathbf{w} = \mathbf{w}_t$  in (7)). It is clear that the fixed points of MGDA are exactly Pareto stationary solutions. In fact, Fliege et al. (2019) proved that the minimum norm  $\| \mathbf{d}^t\|$  converges to 0 at rate  $O(1 / \sqrt{t})$ , matching the well-known result in single objective optimization.
+
+![](images/40e16ee3307aad6b3302570738eb6bd6983c0d870d53627d9834b5c16948b790.jpg)
+
+![](images/87e5e172ce379432bb703bdee44f48781baa31c819d147b60d8c4c6dede01018.jpg)  
+Figure 2: Illustration of Example 1 and Example 2. Left: function-variable dependency structure of Example 1. There is no cycle in the underlying bipartite graph so our method partitions each variable into its own block. Right: function-variable dependency structure of Example 2. There is a cycle (highlighted in red) in the underlying bipartite graph so our method merges both variables into a single block. In both examples, MGDA treats all variables as a single block.
+
+![](images/3c84fc9c755a7d17d452dba868181105a10472bef90a02f8a3a3d55c36d98c24.jpg)
+
+![](images/9ded91922d61e4c2f2effaf9b080007ee715798365860d593b808c41c73fbbe7.jpg)
+
+# 4 ILLUSTRATIVE EXAMPLES
+
+We construct two examples that illustrate the various definitions above and reveal some subtle issues about Pareto stationarity and the (coordinate-wise) MGDA update.
+
+Example 1. Let  $f_{1}(\mathbf{w}) = w_{0}^{2} + w_{1}^{2}$ ,  $f_{2}(\mathbf{w}) = w_{0}^{2} + w_{2}^{2}$  and  $f_{3}(\mathbf{w}) = w_{0}^{2} + w_{3}^{2}$ . Clearly, all component functions are convex. There is a unique Pareto optimal solution  $\mathbf{w}_{\star} = (0,0,0,0)$ , but there are infinitely many Pareto stationary (in fact, weakly Pareto optimal) solutions:
+
+$$
+\{(0, 0, w, z), (0, w, 0, z), (0, w, z, 0): w, z \in \mathbb {R} \}. \tag {10}
+$$
+
+It is clear the MGDA update (8)-(9) could get trapped at any of the Pareto stationary solutions. However, if we update the variables separately, the coordinate-wise MGDA converges to the unique Pareto optimal solution. See Figure 2 for a pictorial illustration.
+
+We note that Example 1 does not contradict Lemma 1, since none of the component functions is strictly convex. In general, strict convexity, or even convexity, is rarely satisfied in deep learning.
+
+It is then tempting to always partition the variables and update them separately using MGDA (i.e., a coordinate-wise MGDA). However, the following example shows this idea may also lead us astray.
+
+Example 2. Let  $f_{1}(\mathbf{w}) = (w_{1} - w_{2})^{2} + w_{1}^{2}$  and  $f_{2}(\mathbf{w}) = (w_{1} - w_{2})^{2} + w_{2}^{2}$ . Clearly, both functions are strictly convex. There is a unique Pareto stationary (and hence Pareto optimal) solution  $\mathbf{w}_{\star} = (0,0)$ . However, if we apply MGDA to each variable separately, it may converge to any point on the diagonal  $\{(w,w):w\in \mathbb{R}\}$ . On the other hand, the vanilla MGDA (treating both variables as a whole) converges to the unique Pareto stationary solution. See Figure 2 for a pictorial illustration.
+
+In the next section, by leveraging on the (sparse) function-variable dependency structure, we prove that it is possible to identify a valid partition of variables and define refined Pareto stationary solutions that are sandwiched by Pareto optimal solutions and Pareto stationary solutions.
+
+# 5 REFINED PARETO STATIONARY SOLUTIONS
+
+Motivated by the aforementioned examples, we now consider partitioning the variables in the MOO problem (1). Let  $\mathcal{P} = \{P_1,\dots ,P_k\}$  be a partition of  $[d]\coloneqq \{1,\ldots ,d\}$ . For a subset  $P$  of variables, we use  $\mathbf{f}^P$  to denote the set of functions  $f_{i}$  that depend on some variable in  $P$ .
+
+Definition 4 (Generalized Pareto Stationarity). We call  $\mathbf{w}_{*}$  Pareto stationary w.r.t. the partition  $\mathcal{P}$  iff
+
+$$
+\mathbf {0} \in \prod_ {j = 1} ^ {k} \operatorname {c o n v} \left\{\nabla_ {\mathbf {w} _ {P _ {j}}} \mathbf {f} ^ {P _ {j}} \left(\mathbf {w} _ {*}\right) \right\} (\text {t h e C a r t e s i a n p r o d u c t}), \tag {11}
+$$
+
+i.e., for all  $j \in [k]$ , there exists  $\lambda^j \in \Delta$  such that  $\sum_{i} \lambda_i^j \nabla_{\mathbf{w}_{P_j}} f_i(\mathbf{w}^*) = \mathbf{0}$  and  $\lambda_i^j = 0$  if  $f_i$  does not depend on the variables in  $P_j$ .
+
+When  $\mathcal{P} = [d]$  is the trivial partition, our definition reduces to Pareto stationarity in Definition 3. The following lemma slightly generalizes one direction of Lemma 1.
+
+Lemma 2. Any Pareto optimal solution is Pareto stationary w.r.t. any partition.
+
+(All proofs can be found in Appendix A.) However, the converse part of Lemma 2 under (strict) convexity is harder to extend: we need to find the right partition first. Indeed, Example 2 already showed Pareto stationarity w.r.t. the finest partition is not equal to Pareto optimality, or put it slightly differently, Pareto stationary w.r.t. an arbitrary finer partition does not necessarily imply Pareto stationary w.r.t. a coarser partition<sup>3</sup>.
+
+Consider the adjacency matrix  $A \in \{0,1\}^{m \times d}$  of a bipartite graph, with
+
+$$
+A _ {i j} = \left\{ \begin{array}{l l} 1, & \text {i f} f _ {i} \text {d e p e n d s o n} w _ {j} \\ 0, & \text {o t h e r w i s e} \end{array} . \right. \tag {12}
+$$
+
+Our next goal is to find the (finest) partition  $\mathcal{P}$  of variables (i.e., columns of  $A$ ) so that Pareto stationary w.r.t.  $\mathcal{P}$  is also Pareto stationary (w.r.t. the trivial partition). This is particularly useful when the dependency matrix  $A$  is sparse, since it enables us to largely reduce the set of Pareto stationary solutions (without missing any Pareto optimal ones); see Example 1.
+
+The main idea is to find cycles in the underlying bipartite graph, merge and contract the resulting variables, and repeat until the graph becomes acyclic; see Figure 1 for an illustration and Appendix A.1 for pseudo-code. We recall from Cormen et al. (2009, Exercise 22.4-3, p. 615) that detecting a cycle in an undirected graph using depth-first search costs  $O(m + d)$  (the number of nodes) for our bipartite graph. Since merging the variables in a cycle reduces the graph size by at least 1, the process can repeat at most  $O(d)$  times. Therefore, the entire process, which we denote as REFINED_PARTITION ( $A$ ), costs  $O(d(m + d))$  to output a partition  $\mathcal{P} \coloneqq \{P_1, \ldots, P_d\}$  of the variables.
+
+We formally justify the above partition procedure in:
+
+Theorem 1. Let  $\mathcal{P} = \mathrm{REFINED\_PARTITION}(A)$  where  $A$  is defined in (12). Then, any  $\mathbf{w}_*$  that is Pareto stationary w.r.t.  $\mathcal{P}$ , which we call refined Pareto stationary from now, is Pareto stationary.
+
+We are finally ready to provide a converse of Lemma 2, under (partial) strict convexity.
+
+Theorem 2. Let  $\mathcal{P} = \{P_1, \ldots, P_k\}$  be returned by REFINED_PARTITION. Suppose each  $f_i$  is strictly convex in  $\mathbf{w}_{I_i}$  where  $I_i \subseteq \mathcal{P}$  is the set of blocks of variables that  $f_i$  depends on. Then, any refined Pareto stationary solution (w.r.t.  $\mathcal{P}$ ) is Pareto optimal.
+
+To summarize, refined Pareto stationarity (based on our partition procedure) is sandwiched between Pareto optimality and Pareto stationarity; Figure 3 illustrates the relationships between these concepts. In particular, we have identified a valid partition to sharpen Lemma 1 in its full extent. Importantly, applying our refined Pareto stationarity to Example 1 and Example 2 successfully identifies the unique Pareto optimal solution while eliminating infinitely many spurious Pareto stationary solutions.
+
+![](images/32057a5acb76e8c678dfdede88e7500b1ae62ddf81a352cfaaa06a3c6925ee46.jpg)  
+Figure 3: Euler diagram for Pareto Optimal (PO), Refined Pareto Stationary (RPS) and Pareto Stationary (PS).
+
+# 6 IMPROVING EXISTING ALGORITHMS WITH THE REFINED PARTITION
+
+The refined Pareto stationarity introduced in the previous section is a solution concept. Broadly, it can be used to enhance existing gradient-based descent algorithms for MOO. Indeed, we can simply cycle through or randomly pick the blocks of variables identified by REFINED_PARTITION, and update each block with a chosen gradient-based descent algorithm in turn. The resulting algorithm is guaranteed to be descending, i.e., it improves all objectives in each step.
+
+Interestingly, for certain MOO algorithms we can even afford to update all blocks simultaneously. We demonstrate this possibility and its practical value by modifying and analyzing the MGDA algorithm from Section 3.3, enabling it to converge to a refined Pareto stationary solution. The full algorithm, which we call RP-MGDA, is summarized in Algorithm 1. The algorithm again terminates when  $\|\mathbf{d}\|_2 < \mathfrak{t}01$ , or when a maximum number of iterations is achieved.
+
+Algorithm 1: Multiple Gradient Descent Algorithm with Refined Partition (RP-MGDA)  
+Input: function-variable dependency structure  $A$ , functions  $\mathbf{f}$ , initializer  $\mathbf{w} \in \mathbb{R}^d$ , learning rate  $\pmb{\eta}$   
+1  $(P_1, \ldots, P_k) = \text{REFINED_PARTITION}(A)$  // partition variables based on  $A$   
+2 for  $t = 1, 2, \ldots$  do  
+3 parfor  $j = 1, 2, \ldots, k$  do // in parallel  
+4  $\begin{array}{l} J^j(\mathbf{w}) \leftarrow \nabla_{\mathbf{w}_{P_j}} \mathbf{f}^{P_j}(\mathbf{w}) \\ \lambda^j \leftarrow \operatorname*{argmin}_{\lambda \in \Delta} \|J^j(\mathbf{w})\lambda\|^2 \\ \mathbf{d}_{P_j}^t \leftarrow J^j(\mathbf{w})\lambda^j \\ \mathbf{w}_{P_j} \leftarrow \mathbf{w}_{P_j} - \eta_j \cdot \mathbf{d}_{P_j}^t \end{array}$  // compute gradients  
+5 // solve the dual subproblem  
+6 // minimum-norm direction in  $P_j$   
+7
+
+Output: final weights w
+
+Following Fliege et al. (2019) we prove that the direction  $\mathbf{d}$  in RP-MGDA (Algorithm 1) converges to  $\mathbf{0}$  at the rate  $O(1 / \sqrt{t})$ , and hence matching the usual result for single objective optimization.
+
+Theorem 3. Suppose the functions  $f_{i}$  are bounded from below and Lipschitz smooth with constant  $L_{i}$ . Set  $\eta_{j}\equiv \eta \leq \min_{i}\frac{1}{L_{i}}$ . Then,  $\min_{1\leq t\leq T}\| \mathbf{d}^t\| \leq O(\frac{1}{\sqrt{T}})$ .
+
+Combining Theorem 1 and Theorem 3 we immediately have the following result:
+
+Corollary 1. Assuming RP-MGDA converges, then it converges to a Pareto stationary solution at the rate  $O(1 / \sqrt{t})$ .
+
+A few remarks regarding RP-MGDA are in order: 1) When the sublevel set  $\{\mathbf{w} : \mathbf{f}(\mathbf{w}) \leq \mathbf{f}(\mathbf{w}^1)\}$  is compact, RP-MGDA admits a limit point that is refined Pareto stationary and hence Pareto stationary, thanks to its descending property. 2) In practice we do not need to know the Lipschitz constants  $L_i$ : a standard line search procedure on  $\eta$  to test the quadratic upper bound (7) suffices to achieve the same convergence rate. 3) Theorem 1 and hence Corollary 1 continue to hold if we replace the partition  $\mathcal{P}$  with any coarser partition (such as the trivial partition  $[d]$ ). Thus, our results strictly generalize those about MGDA (e.g., Fliege et al., 2019).
+
+Complexity. We emphasize the inner loop of Algorithm 1 is executed in parallel, as they correspond to updating different blocks of  $\mathbf{w}$  simultaneously. We have analyzed the complexity of the subroutine Refined_PARTITION in the previous section, and we note that this one-time cost can often be avoided if one knows the domain application well, see our experiments in Section 7 for examples. Barring the one-time overhead in line 1, Algorithm 1 is theoretically cheaper than the vanilla MGDA, since the most costly step in line 5 (solving the dual subproblem) often has superlinear dependence on  $m$  (the number of objectives) and hence dividing it into blocks of smaller sizes reduces the runtime.
+
+# 7 EXPERIMENTS
+
+In this section, we present experiments comparing RP-MGDA and MGDA across both synthetic examples and realistic benchmark datasets. These experiments explore varying variable dependency structures and convexity assumptions, demonstrating RP-MGDA's effectiveness and versatility. More experimental details and discussions can be found in Appendix B.
+
+# 7.1 SYNTHETIC EXAMPLES
+
+We examine two settings. First, quadratic MOO problems with randomly generated adjacency matrices, focusing on the behaviour of REFINED_PARTITION and the effectiveness of RP-MGDA. Second, in Appendix B.1, we consider a two-objective convex problem, where we visualize and compare solution fronts, observing how often MGDA converges to sub-optimal PS solutions.
+
+Randomly generated sparse adjacency matrix. We randomly generate  $m \times d$  adjacency matrices with varying degrees of sparsity to set up the function-variable dependency structures of quadratic MOO problems with  $m$  objectives and  $d$  input variables. The sparsity of function-variable dependencies is controlled by a 'density' parameter, which determines the percentage of non-zero entries in the adjacency matrix. We use our proposed REFINED_PARTITION procedure to automatically
+
+![](images/8756c3057615974b68a0d6d12c0315a79a4f2374fc31add3f9bd3291fb450020.jpg)  
+Figure 4: Experiments on randomly generated sparse adjacency matrices. Left:  $[20 \times 100$  matrix]. The bar chart compares the solution from MGDA with that post-processed by RP-MGDA. Red bars indicate the magnitude by which RP-MGDA outperforms MGDA in objective value, while green bars (if any) indicate the opposite. Right:  $[100 \times 100$  matrices]. The impact of adjacency matrix density on variable merging in REFINED_PARTITION, where  $y$ -axis shows the final number of variable blocks returned. A value of 1 indicates all variables are merged into a single block, and a value of 100 (represented as 10 on the square root scale) indicates that all variables are fully separated.
+
+![](images/ba26a07e5967814394c4b01343f586c9c9482ecd708ad67aa6c1ff26dc81e115.jpg)
+
+![](images/7a075908032558190dcf141e9ea9b68ed7df251b43ade9c3eeedcf014ae1ec3f.jpg)  
+Figure 5: Different function-variable dependency structures for experiments on benchmark datasets. Left: Personalized federated learning; Middle: Hierarchical classification; Right: Multi-objective learning with partial information.
+
+![](images/651dd14ce6f19ade9cd4fc0d062753c9b1d02958f3d487ccff4d320d06ad5bce.jpg)
+
+![](images/be46c6e15df19108645a4ee5ecc54ad1e6cb5f9aa36ba8b4fe9ededd36b1c567.jpg)
+
+generate refined partitions, especially for larger adjacency matrices (e.g.  $100 \times 100$ ) where manual partitioning becomes cumbersome. Details of the setup can be found in Appendix B.2.
+
+Results. From Figure 4 (Left) we observe that by post-processing the outputs of MGDA, i.e., using them to initialize RP-MGDA, we can improve the solutions to Pareto dominate the original ones. Conversely, we point out that MGDA cannot improve the stationary solutions of RP-MGDA, due to Theorem 1. In the demonstration with  $100 \times 100$  matrix (Figure 4, Right), we note that the number of final blocks returned by REFINED_PARTITION decreases as the density increases, starting at  $1.5\%$ , where all variables are separated, and eventually merging all variables into a single block when the density reaches  $7\%$ . We also empirically verified Theorem 3 in Appendix B.2.
+
+# 7.2 ML BENCHMARK DATASETS
+
+We examine three problems with different function-variable dependency structures derived from real-world application scenarios. These examples also serve as a 'proof of concept' for applying refined partitioning to realistic ML problems. The structures and their corresponding refined partitions are illustrated in Figure 5.
+
+# 7.2.1 PERSONALIZED FEDERATED LEARNING ('MTL' STRUCTURE)
+
+Setup. We consider a personalized federated learning setting with  $m = 4$  clients, each holding distinct non-i.i.d. data  $D_{i}$  sampled from the training dataset (MNIST/CIFAR-10). The objectives  $f_{i}$  are defined as empirical cross-entropy losses over the clients' data distributions. As in standard federated learning, we have a shared global model with learnable parameters  $\mathbf{w}_0$ . To handle data heterogeneity and enable personalization, instead of directly feeding inputs (e.g., images) into the global model  $\mathbf{w}_0$ , each client  $i$  has a local personalized network, parameterized by  $\mathbf{w}_{i}$ , which extracts
+
+![](images/c172c10f084ce50eaed27a4429a27fc91a3bdae184ede79cdbdbc8844e49a37d.jpg)  
+Figure 6: Radar graph of final cross-entropy losses  $f_{i}$  in the personalized federated learning setting, using MGDA and RP-MGDA respectively. Lower is better.
+
+![](images/ee60f4aa03e61ce16fb988bd09c1eb7c9a53c5fad8132564d0bdcd3287469e55.jpg)
+
+client-specific representations. These representations are then forwarded to the global model for prediction (e.g., Liang et al., 2020). Formally,  $f_{i}(\mathbf{w}_{0},\mathbf{w}_{i}) = \mathbb{E}_{\mathbf{x}\sim D_{i}}[\ell (\mathrm{NN}(\mathrm{NN}(\mathbf{x};\mathbf{w}_{i});\mathbf{w}_{0}))]$ , thus having the dependency structure illustrated in Figure 5 (Left), which is also commonly seen in multi-task learning (MTL), see e.g., Sener and Koltun (2018).
+
+Refined Partition. For this 'MTL' dependency structure, the optimal refined partition is  $\{\{\mathbf{w}_0\}, \{\mathbf{w}_1\}, \ldots, \{\mathbf{w}_m\}\}$ . Consequently, at each iteration, RP-MGDA updates  $\mathbf{w}_0$  by descending along the minimum-norm direction, while  $\mathbf{w}_i$  (for  $i \in [m]$ ) is updated with gradient descent simultaneously. This partitioning (a) simplifies the dual sub-problem by reducing its dimensionality and (b) allows for more flexible updates to the personalized networks.
+
+Results. As shown in Figure 6 and Appendix B.3.2, we observe that RP-MGDA converges faster and reaches better solutions than MGDA, particularly for  $f_{2}, f_{3}$  and  $f_{4}$ , while for  $f_{1}$ , the performances are similar. Further details are provided in Appendix B.3.
+
+# 7.2.2 HIERARCHICAL CLASSIFICATION ('LADDER' STRUCTURE)
+
+Setup. We consider a multi-objective hierarchical classification problem with  $m = 3$  levels of objectives that predict image categories, from coarse to fine labels, on the CIFAR-10 dataset, using a network structure similar to Branch-CNN (Zhu and Bain, 2017). Here,  $\mathbf{w}_1, \mathbf{w}_2$  and  $\mathbf{w}_3$  represent the network weights of feature extraction layers 1, 2 and 3, respectively. The three objectives,  $f_1, f_2$  and  $f_3$ , are all cross-entropy losses but defined for different prediction tasks. The first task is binary classification, categorizing images as 'living' or 'non-living,' using only the representations from feature extraction layer 1, and thus depending solely on  $\mathbf{w}_1$ . The second task classifies images into four categories: 'ground vehicle,' 'non-ground vehicle,' 'land animal,' or 'non-land animal,' using representations from layers 1 and 2, and thus depending on  $\mathbf{w}_1$  and  $\mathbf{w}_2$ . The third task classifies images into the original CIFAR-10 categories, using representations from all three layers, and thus depending on  $\mathbf{w}_1, \mathbf{w}_2$ , and  $\mathbf{w}_3$ . This hierarchical structure allows for progressively more complex predictions as we move from coarse to fine labels.
+
+Refined Partition. The corresponding dependency structure in Figure 5 (Middle) features an optimal refined partition of  $\{\{\mathbf{w}_1\mathbf{w}_2\} ,\{\mathbf{w}_3\} \}$ , where RP-MGDA optimizes  $\mathbf{W}_3$  using gradient descent and the block  $\{\mathbf{w}_1\mathbf{w}_2\}$  using MGDA. This partitioning should be advantageous for Objective 3, as it allows for more flexible updates to  $\mathbf{W}_3$  compared to MGDA.
+
+Results. Indeed, we observe that the training loss curves for Objectives 1 and 2 are nearly indistinguishable; however, RP-MGDA demonstrates faster and better convergence for Objective 3, as illustrated in Figure 7.
+
+![](images/1f1b91305fa7a35cf291f0bd8177dd8f77a853e8b740fa925f973939a553c647.jpg)  
+Figure 7: Loss curves of MGDA and RP-MGDA for the three objectives in hierarchical classification.
+
+![](images/8f529cba3516560807b1078f03796795b7ea237dc2067508efe60c6e44c124be.jpg)
+
+![](images/434ee870962b53557c95c952036f482dfb8a3988744b10bb958d9ffc3ba9d2c1.jpg)
+
+# 7.2.3 MULTI-OBJECTIVE LEARNING WITH PARTIAL INFORMATION (‘CHAIN’ STRUCTURE)
+
+Setup. We simulate a scenario inspired by vertical federated learning (Liu et al., 2024) where each party holds different feature sets for the same instances, akin to multiple institutions possessing various attributes for the same users. In particular, we explore the setting where each party only has access to consecutive variables, e.g., a company in a supply chain having only adjacent upstream and downstream data. In this context, clients with heterogeneous data collaborate to learn a stronger model without directly sharing sensitive user data. Specifically, we consider an  $m = 4$ -objective learning problem on the California housing dataset (Pace and Barry, 1997), where we employ a linear model to predict the median house value. The original 8 features are grouped into 3 sets<sup>5</sup>, each associated with linear regression weights  $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3$ . The objectives  $f_i$  represent the mean-squared error (MSE) for different parties, all trained on the same dataset but with each objective depending on different subsets of features, as illustrated in Figure 5 (Right).
+
+Refined Partition. For this 'chain' dependency structure, the optimal refined partition is  $\{\{\mathbf{w}_1\}, \{\mathbf{w}_2\}, \{\mathbf{w}_3\}\}$ , allowing RP-MGDA to apply block-wise MGDA to  $\mathbf{w}_1, \mathbf{w}_2$  and  $\mathbf{w}_3$  separately. This results in simpler dual sub-problems and improved convergence.
+
+Results. As shown in Figure 8, RP-MGDA demonstrates greater stability and lower average losses across the 20 trials. We also observe that the stationary solutions of MGDA can be improved by post-processing with RP-MGDA, in 18 out of 20 trials. Additionally, the conditions of Theorem 2 are met in this setup, ensuring that RP-MGDA achieves Pareto optimality.
+
+![](images/9d097596665de6564126c3c8075288e42d75be080f5eadc2551350de1189f880.jpg)  
+Figure 8: We calculate the mean and standard deviation of the 4 losses across 20 random trials (different seeds) for both MGDA and RP-MGDA, and plot the means, with shaded regions representing the standard deviation.
+
+![](images/dcbe6e9cc5b1d1eeaf97ff25b54f8aa67cee2a4200d013a518ded6227868b25b.jpg)
+
+![](images/6aa2056e4133c8f76e89e17f89bb70e3e23f333d4cbb2254272b6ccaf5da0b23.jpg)
+
+![](images/3ed70d1b433fe2aa98be15a3f8e00ea39cad0817d6fdbf10379c918aece5b1e8.jpg)
+
+# 8 CONCLUSION AND DISCUSSION
+
+In this work, we introduced Refined Pareto Stationarity (RPS) with an efficient partitioning procedure (REFINED_PARTITION) to address limitations of Pareto stationarity in multi-objective optimization (MOO). We proposed RP-MGDA, a novel MOO algorithm based on refined partitioning that converges to RPS, supported by both theoretical and empirical justification. Future work may explore applying our idea to other MOO algorithms and demonstrate the superior performance on other real problems with (sparse) dependency structures.
+
+# ACKNOWLEDGEMENTS
+
+We thank the reviewers and the area chair for their constructive comments that have improved the presentation of this work. Zeou Hu would like to thank Dihong Jiang and Haoye Lu for helpful discussions. YY gratefully acknowledges CIFAR and NSERC for funding support.
+
+# REFERENCES
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+
+# A PROOFS AND PSEUDO-CODE OMITTED FROM THE MAIN TEXT
+
+Lemma 2. Any Pareto optimal solution is Pareto stationary w.r.t. any partition.
+
+Proof. Let  $\mathbf{w}^*$  be Pareto optimal w.r.t.  $\mathbf{f}$ . Consider the function  $\mathbf{g}(\mathbf{z}) \coloneqq \mathbf{f}^{P_1}(\mathbf{z}, \mathbf{w}_{P_2}^*, \ldots, \mathbf{w}_{P_k}^*)$ . Clearly,  $\mathbf{w}_{P_1}^*$  is Pareto optimal w.r.t.  $\mathbf{g}$ . Applying Lemma 1 we know  $\mathbf{0} \in \mathrm{conv}\{\nabla \mathbf{g}(\mathbf{w}_{P_1}^*)\} = \mathrm{conv}\{\nabla_{\mathbf{w}_{P_1}} \mathbf{f}^{P_1}(\mathbf{w}^*)\}$ . Iterating the argument proves (11).
+
+Theorem 1. Let  $\mathcal{P} = \mathrm{REFINED\_PARTITION}(A)$  where  $A$  is defined in (12). Then, any  $\mathbf{w}_*$  that is Pareto stationary w.r.t.  $\mathcal{P}$ , which we call refined Pareto stationary from now, is Pareto stationary.
+
+Proof. For each  $j \in [k]$ , let us define
+
+$$
+F _ {j} := \{i \in [ m ]: f _ {i} \text {i s c o n n e c t e d t o} P _ {j} \text {i n t h e f i n a l g r a p h} \}, \tag {13}
+$$
+
+namely, the set of functions that the variable set  $P_{j}$  "sees." We claim that for all  $j \neq l$ ,
+
+$$
+\left| F _ {j} \cap F _ {l} \right| \leq 1. \tag {14}
+$$
+
+Suppose not, let  $a \neq b \in F_j \cap F_l$ . Then, in our final graph,  $f_a$  and  $f_b$  are connected to both  $P_j$  and  $P_l$ , creating a cycle of 4 and hence contradicting to the fact that  $P_j$  and  $P_l$  were not merged.
+
+Since  $\mathbf{w}_*$  is Pareto stationary w.r.t.  $\mathcal{P}$ , for each  $j\in [k]$  there exists  $\lambda^j\in \Delta$  such that
+
+$$
+\nabla_ {\mathbf {w} _ {P _ {j}}} \mathbf {f} \left(\mathbf {w} _ {*}\right) \cdot \boldsymbol {\lambda} ^ {j} = \mathbf {0}, \quad \text {w h e r e} \quad \lambda_ {i} ^ {j} = 0 \text {i f} i \notin F _ {j}. \tag {15}
+$$
+
+Our goal is to construct  $\lambda \in \Delta$  (independent of the index  $j$ ) such that  $\nabla \mathbf{f}(\mathbf{w}_*) \cdot \boldsymbol{\lambda} = \mathbf{0}$ , i.e.,  $\mathbf{w}_*$  is Pareto stationary.
+
+W.l.o.g. we make the following assumptions:
+
+1. The graph is connected, as we can consider each connected component separately.  
+2. The variable sets are rearranged according to their distance to  $P_{1}$  (ties broken arbitrarily).  
+3. (positivity)  $\lambda_i^j = 0$  if and only if  $i \notin F_j$ ; we will see how to reduce to this case later.
+
+For each  $j \in [k]$ , consider any  $n \geq j + 1$  such that there exists a (unique)  $i_n \in F_j \cap F_n$ , and perform the rescaling
+
+$$
+\boldsymbol {\lambda} ^ {n} \leftarrow \boldsymbol {\lambda} ^ {n} \cdot \frac {\lambda_ {i _ {n}} ^ {j}}{\lambda_ {i _ {n}} ^ {n}}, \tag {16}
+$$
+
+where we note that  $\lambda_{i_n}^n\neq 0$  due to the positivity assumption. Importantly, rescaling (by a positive number) does not affect any of the stationary conditions in (15), other than the normalization constraint  $\mathbf{1}^{\top}\pmb{\lambda}^{j} = 1$ , which we will take care of at the end.
+
+We claim that each  $\lambda^n$  is rescaled at most once. Suppose not so there exist  $j_1 < j_2 < n$  and  $i_{j_1} \in F_{j_1} \cap F_n$  and  $i_{j_2} \in F_{j_2} \cap F_n$ , i.e., we have the edges shown on Figure 9 (left). Due to their appearance order, we must have
+
+$$
+\operatorname {d i s t} \left(P _ {j _ {1}}, P _ {1}\right) \leq \operatorname {d i s t} \left(P _ {j _ {2}}, P _ {1}\right) \leq \operatorname {d i s t} \left(P _ {n}, P _ {1}\right). \tag {17}
+$$
+
+Noting the edges from  $i_{j_1}$  to  $P_{j_1}$  and  $P_{n}$  and from  $i_{j_2}$  to  $P_{j_2}$  and  $P_{n}$ , we must have
+
+$$
+\operatorname {d i s t} \left(P _ {j _ {1}}, P _ {1}\right) = \operatorname {d i s t} \left(P _ {j _ {2}}, P _ {1}\right) <   \operatorname {d i s t} \left(P _ {n}, P _ {1}\right), \tag {18}
+$$
+
+for otherwise there will be a cycle (due to a back edge, see Cormen et al., 2009). We can now trace back the ancestors of  $P_{j_1}$  and  $P_{j_2}$  until we finally arrive at the same variable set  $P$  (since only  $P_1$  has 0 distance to  $P_1$ ) that intersects both  $P_{j_1}'$  and  $P_{j_2}'$  (here prime denotes the final ancestor pair). But this creates a cycle (see Figure 9, right), contradiction.
+
+Lastly, we take component-wise maximum to construct
+
+$$
+\boldsymbol {\lambda} = \vee_ {j = 1} ^ {k} \boldsymbol {\lambda} ^ {j}. \tag {19}
+$$
+
+![](images/7c852f89ba96a2d35d3e6ac8c6e0725b61c92937d6e930e5d15297d90b5736c2.jpg)  
+Figure 9: Illustration of the claim that each  $\lambda^n$  is rescaled at most once.
+
+Clearly,  $\lambda \neq 0$  so we may normalize it so that  $\lambda \in \Delta$ . Since each  $\lambda^j$  is re-scaled at most once (and nonzero entries of different  $\lambda^j$  and  $\lambda^n$  intersect at most once, see (14)), for any  $j$  and  $\lambda_i^j \neq 0$ , we have
+
+$$
+\lambda_ {i} = \lambda_ {i} ^ {j}. \tag {20}
+$$
+
+Since (15) holds for any  $j$  (other than the normalization constraint), it follows that  $\nabla \mathbf{f}(\mathbf{w}_*)\cdot \boldsymbol {\lambda} = \mathbf{0}$
+
+We are left to justify the positivity assumption. Consider the "adjacency" matrix
+
+$$
+\Lambda = \left[ \begin{array}{l l l l} \boldsymbol {\lambda} ^ {1} & \boldsymbol {\lambda} ^ {2} & \dots & \boldsymbol {\lambda} ^ {k} \end{array} \right], \tag {21}
+$$
+
+where we put an edge between  $f_{i}$  and  $P_{j}$  if the corresponding  $\lambda_{i}^{j}$  value is nonzero. Suppose  $\lambda_{i}^{j} = 0$  for some  $i \in F_{j}$ . Then, for all variable subsets  $P_{l}$ , other than  $P_{j}$ , that can be reached by  $f_{i}$ , we set their  $\lambda^{l} = 0$  (namely entire columns in  $\Lambda$ ) and  $F_{l} = \emptyset$ . Note that we do not change  $\lambda^{j}$  (for if we can reach  $P_{j}$  from  $i$  but not through the edge  $\{i, P_{j}\}$ , we will have a cycle) so  $\Lambda$  remains nonzero after our operation. Moreover, the stationary conditions (15), other than the normalization constraint, are maintained. Thus, we may repeat this process until the positivity assumption holds.
+
+The proof is now complete.
+
+![](images/92e7503323063f6f8e2802915bd97ed832c5fbe7575b1547b079d15f237a27ef.jpg)
+
+Remark 1. We point out that REFINED_PARTITION(A) returns a unique solution (irrespective of the order of the cycles we contract). Indeed, when we merge the variables and contract the graph, any existing cycle (other than the one we are contracting) will remain as a cycle. Another way to see this is to think of contracting a cycle as replacing it with a complete bipartite subgraph.
+
+Remark 2. We note that Theorem 1 continues to hold if we replace  $\mathcal{P}$  by any of its coarser partition  $\mathcal{Q}$  (i.e., for any  $P\in \mathcal{P}$  there exists  $Q\in \mathcal{Q}$  such that  $P\subseteq Q$ ). In particular, setting  $\mathcal{Q} = [d]$  we recover Pareto stationarity in Definition 3.
+
+We will need the following elementary result:
+
+Lemma 3. If  $g(\mathbf{w}, \mathbf{u})$  and  $h(\mathbf{w}, \mathbf{v})$  are both strictly convex in  $(\mathbf{w}, \mathbf{u})$  and  $(\mathbf{w}, \mathbf{v})$ , respectively, then  $f(\mathbf{w}, \mathbf{u}, \mathbf{v}) \coloneqq g(\mathbf{w}, \mathbf{u}) + h(\mathbf{w}, \mathbf{v})$  is strictly convex in  $(\mathbf{w}, \mathbf{u}, \mathbf{v})$ .
+
+Proof. Indeed, consider  $(\mathbf{w}_0,\mathbf{u}_0,\mathbf{v}_0)\neq (\mathbf{w}_1,\mathbf{u}_1,\mathbf{v}_1)$ . For any  $\lambda \in (0,1)$ , denote  $\mathbf{w}_{\lambda}\coloneqq (1 - \lambda)\mathbf{w}_0 + \lambda \mathbf{w}_1$  and similarly for  $\mathbf{u}_{\lambda}$  and  $\mathbf{v}_{\lambda}$ , we have
+
+$$
+g \left(\mathbf {w} _ {\lambda}, \mathbf {u} _ {\lambda}\right) \leq (1 - \lambda) g \left(\mathbf {w} _ {0}, \mathbf {u} _ {0}\right) + \lambda g \left(\mathbf {w} _ {1}, \mathbf {u} _ {1}\right) \tag {22}
+$$
+
+$$
+h \left(\mathbf {w} _ {\lambda}, \mathbf {v} _ {\lambda}\right) \leq (1 - \lambda) h \left(\mathbf {w} _ {0}, \mathbf {v} _ {0}\right) + \lambda h \left(\mathbf {w} _ {1}, \mathbf {v} _ {1}\right), \tag {23}
+$$
+
+where at least one of the inequalities is strict. Thus,  $f(\mathbf{w}_{\lambda}, \mathbf{u}_{\lambda}, \mathbf{v}_{\lambda}) < (1 - \lambda)f(\mathbf{w}_{0}, \mathbf{u}_{0}, \mathbf{v}_{0}) + \lambda f(\mathbf{w}_{1}, \mathbf{u}_{1}, \mathbf{v}_{1})$ .
+
+Theorem 2. Let  $\mathcal{P} = \{P_1, \ldots, P_k\}$  be returned by REFINED_PARTITION. Suppose each  $f_i$  is strictly convex in  $\mathbf{w}_{I_i}$  where  $I_i \subseteq \mathcal{P}$  is the set of blocks of variables that  $f_i$  depends on. Then, any refined Pareto stationary solution (w.r.t.  $\mathcal{P}$ ) is Pareto optimal.
+
+Proof. Let  $\mathbf{w}^*$  be an arbitrary refined Pareto stationary solution w.r.t.  $\mathcal{P}$ . For the sake of contradiction, consider any Pareto optimal solution  $\mathbf{z}$  that dominates  $\mathbf{w}^*$ , i.e.,  $\mathbf{f}(\mathbf{z}) \preceq \mathbf{f}(\mathbf{w}^*)$ .
+
+Applying Theorem 1 we know  $\mathbf{w}^*$  is Pareto stationary, i.e., there exists some  $\lambda \in \Delta$  such that
+
+$$
+\sum_ {i} \lambda_ {i} \nabla f _ {i} \left(\mathbf {w} ^ {*}\right) = 0. \tag {24}
+$$
+
+Restricting to the subset  $F^{+} \coloneqq \{i : \lambda_{i} \neq 0\} \neq \emptyset$ , we know from Lemma 3 that the function  $h(\mathbf{w}) \coloneqq \sum_{i \in F^{+}} \lambda_{i} f_{i}(\mathbf{w})$  is strictly convex in  $\mathbf{w}_{I_h}$  that it depends on (where  $I_h \subseteq \mathcal{P}$ ). Since  $h(\mathbf{z}) \leq h(\mathbf{w}^*)$ , we necessarily have
+
+$$
+\mathbf {z} _ {\cup I _ {h}} = \mathbf {w} _ {\cup I _ {h}} ^ {*}. \tag {25}
+$$
+
+We are now ready to reduce. We fix  $\mathbf{w}_{\cup I_h}^*$  and remove all blocks of variables in  $I_{h}$  (and the corresponding edges in our bipartite graph). Clearly, the bipartite graph remains acyclic and  $\mathbf{w}_{\mathcal{P}\backslash I_h}^*$  remains to be refined Pareto stationary w.r.t. the reduced problem. Iterating the argument in the previous paragraph, we deduce that  $\mathbf{z}$  agrees with  $\mathbf{w}^{*}$  in at least one more block of variables. Since there are only  $k$  blocks in total, after at most  $k$  iterations we must have  $\mathbf{z} = \mathbf{w}^{*}$ , which contradicts to the assumption  $\mathbf{f}(\mathbf{z})\preceq \mathbf{f}(\mathbf{w}^{*})$ . Thus,  $\mathbf{w}^{*}$  is Pareto optimal and the proof is complete.
+
+Remark 3. Since Theorem 1 also holds for any partition  $\mathcal{Q}$  that is coarser than  $\mathcal{P}$ , the same is true for Theorem 2. In particular, choosing  $\mathcal{Q} = [m]$ , refined Pareto stationarity reduces to Pareto stationarity and Theorem 2 reduces to the well-known result in MOO (see Lemma 1).
+
+Example 3 (Personalized federated learning). Let us give an example where Theorem 2 strictly improves Lemma 1. Consider the personalized federated learning setting with  $f_{i}(\mathbf{w}_{0},\mathbf{w}_{i}), i \in [m]$ . It is clear that none of the functions  $f_{i}$  is strictly convex w.r.t.  $(\mathbf{w}_0,\mathbf{w}_1,\dots ,\mathbf{w}_m)$ . Thus, Lemma 1 is not applicable and not every Pareto stationary solution is Pareto optimal.
+
+On the other hand, REFINED_PARTITION returns  $\mathcal{P} = \{\{\mathbf{w}_0\}, \{\mathbf{w}_1\}, \ldots, \{\mathbf{w}_m\}\}$  and it is possible for each function  $f_i$  to be strictly convex w.r.t. the (blocks of) variables  $(\mathbf{w}_0, \mathbf{w}_i)$  that it depends on. (This can also be achieved by regularizing each function  $f_i$ .) Therefore, Theorem 2 is applicable. In fact, we can apply Theorem 2 to both our Example 1 and Example 2 in the main text and conclude that the refined Pareto stationary solution indeed coincides with the unique Pareto optimal solution.
+
+To be more concrete, consider
+
+$$
+f _ {1} \left(w _ {0}, w _ {1}, w _ {2}, w _ {3}\right) = w _ {0} ^ {2} + w _ {1} ^ {2} \tag {26}
+$$
+
+$$
+f _ {2} \left(w _ {0}, w _ {1}, w _ {2}, w _ {3}\right) = \left(w _ {0} - 1\right) ^ {2} + w _ {2} ^ {2} \tag {27}
+$$
+
+$$
+f _ {3} \left(w _ {0}, w _ {1}, w _ {2}, w _ {3}\right) = \left(w _ {0} - 2\right) ^ {2} + w _ {3} ^ {2}. \tag {28}
+$$
+
+We run simulations with step size  $\eta = 0.01$  and 1000 iterations. With initialization  $\mathbf{w}^1 = (1.5,1,0.1,0.1)$ , MGDA converges to  $\mathbf{w}^{1000} = (1.5,1,0,0)$  with function values  $\mathbf{f}(\mathbf{w}^{1000}) = (\frac{13}{4},\frac{1}{4},\frac{1}{4})$ , which is not Pareto optimal since it is dominated by  $\mathbf{f}^* = (\frac{9}{4},\frac{1}{4},\frac{1}{4})$ , achieved at  $\mathbf{w}^* = (1.5,0,0,0)$ . In contrast, under the same setting, RP-MGDA converges to the Pareto optimal solution  $\mathbf{w}^* = (1.5,0,0,0)$ .
+
+From Theorem 2, we also see how MGDA fails to reach the Pareto optimal solution in this example. When MGDA converges to  $\mathbf{w}^{1000} = (1.5, 1, 0, 0)$ , the corresponding  $\lambda_{1} = 0$  while  $\nabla_{w_{1}}f_{1}(w_{0}^{*}, w_{1}^{*}) \neq 0$ , which means MGDA stops even though there is a descent direction along  $w_{1}$ .
+
+Theorem 3. Suppose the functions  $f_{i}$  are bounded from below and Lipschitz smooth with constant  $L_{i}$ . Set  $\eta_{j}\equiv \eta \leq \min_{i}\frac{1}{L_{i}}$ . Then,  $\min_{1\leq t\leq T}\| \mathbf{d}^t\| \leq O(\frac{1}{\sqrt{T}})$ .
+
+Proof. Denote
+
+$$
+I _ {i} := \{j: f _ {i} \text {d e p e n d s o n} P _ {j} \} \tag {29}
+$$
+
+and apply smoothness to upper bound the  $i$ -th objective as (for step size  $\eta \leq \frac{1}{L_i}$ ,  $L_i$  being the Lipschitz constant of  $\nabla f_i$ ):
+
+$$
+\begin{array}{l} f _ {i} \left(\mathbf {w} ^ {t + 1}\right) \leq f _ {i} \left(\mathbf {w} ^ {t}\right) + \nabla f _ {i} \left(\mathbf {w} ^ {t}\right) \cdot \left(\mathbf {w} ^ {t + 1} - \mathbf {w} ^ {t}\right) + \frac {1}{2 \eta} \| \mathbf {w} ^ {t + 1} - \mathbf {w} ^ {t} \| _ {I _ {i}} ^ {2} (30) \\ = f _ {i} \left(\mathbf {w} ^ {t}\right) + \sum_ {j \in I _ {i}} \eta \cdot \left[ \nabla_ {P _ {j}} f _ {i} \left(\mathbf {w} ^ {t}\right) \cdot \mathbf {d} _ {P _ {j}} ^ {t} + \frac {1}{2} \| \mathbf {d} _ {P _ {j}} ^ {t} \| _ {2} ^ {2} \right] (31) \\ \leq f _ {i} \left(\mathbf {w} ^ {t}\right) - \sum_ {j \in I _ {i}} \frac {\eta}{2} \cdot \left\| \mathbf {d} _ {P _ {j}} ^ {t} \right\| _ {2} ^ {2}, (32) \\ \end{array}
+$$
+
+where the last inequality follows from the definition of  $\mathbf{d}_{P_j}^t$ :
+
+$$
+\mathbf {d} _ {P _ {j}} ^ {t} = \underset {\mathbf {d} _ {P _ {j}}} {\operatorname {a r g m i n}} \max  _ {i \in F _ {j}} \left[ \nabla_ {P _ {j}} f _ {i} \left(\mathbf {w} ^ {t}\right) \cdot \mathbf {d} _ {P _ {j}} + \frac {1}{2} \| \mathbf {d} _ {P _ {j}} \| _ {2} ^ {2} \right]. \tag {33}
+$$
+
+Indeed, we note that the objective above is 1-strongly convex in  $\mathbf{d}_{P_j}$ , so setting  $\mathbf{d}_{P_j} = \mathbf{0}$  and  $\mathbf{d}_{P_j} = \mathbf{d}_{P_j}^t$  we obtain from e.g. Nesterov (2018, Corollary 3.2.3, p. 210):
+
+$$
+0 \geq \max  _ {i \in F _ {j}} \left[ \nabla_ {P _ {j}} f _ {i} (\mathbf {w} ^ {t}) \cdot \mathbf {d} _ {P _ {j}} ^ {t} + \frac {1}{2} \| \mathbf {d} _ {P _ {j}} ^ {t} \| _ {2} ^ {2} \right] + \frac {1}{2} \| \mathbf {0} - \mathbf {d} _ {P _ {j}} ^ {t} \| _ {2} ^ {2}. \tag {34}
+$$
+
+Rearranging and summing over  $t$  we obtain:
+
+$$
+\sum_ {t = 1} ^ {T} \sum_ {j \in I _ {i}} \| \mathbf {d} _ {P _ {j}} ^ {t} \| _ {2} ^ {2} \leq \frac {2}{\eta} \left[ f _ {i} \left(\mathbf {w} ^ {1}\right) - f _ {i} \left(\mathbf {w} ^ {T + 1}\right) \right]. \tag {35}
+$$
+
+Dividing both sides by  $T$  we obtain:
+
+$$
+\min  _ {1 \leq t \leq T} \sum_ {j \in I _ {i}} \| \mathbf {d} _ {P _ {j}} ^ {t} \| _ {2} ^ {2} \leq \frac {1}{T} \sum_ {t = 1} ^ {T} \sum_ {j \in I _ {i}} \| \mathbf {d} _ {P _ {j}} ^ {t} \| _ {2} ^ {2} \leq \frac {2}{\eta T} [ f _ {i} (\mathbf {w} ^ {1}) - f _ {i} (\mathbf {w} ^ {T + 1}) ]. \tag {36}
+$$
+
+Finally, we sum over any subset  $F$  ( $e.g., F = [m]$ ) of functions that together depend on all variables:
+
+$$
+\frac {2}{\eta} \sum_ {i \in F} \left[ f _ {i} \left(\mathbf {w} ^ {1}\right) - f _ {i} \left(\mathbf {w} ^ {T + 1}\right) \right] \geq \sum_ {t = 1} ^ {T} \sum_ {i \in F} \sum_ {j \in I _ {i}} \| \mathbf {d} _ {P _ {j}} ^ {t} \| _ {2} ^ {2} \geq \sum_ {t = 1} ^ {T} \sum_ {j = 1} ^ {k} \| \mathbf {d} _ {P _ {j}} ^ {t} \| _ {2} ^ {2} = \sum_ {t = 1} ^ {T} \| \mathbf {d} ^ {t} \| _ {2} ^ {2}. \tag {37}
+$$
+
+Dividing both sides by  $T$  we obtain:
+
+$$
+\frac {2}{\eta T} \sum_ {i \in F} \left[ f _ {i} \left(\mathbf {w} ^ {1}\right) - f _ {i} \left(\mathbf {w} ^ {T + 1}\right) \right] \geq \frac {1}{T} \sum_ {t = 1} ^ {T} \| \mathbf {d} ^ {t} \| _ {2} ^ {2} \geq \min  _ {t} \| \mathbf {d} ^ {t} \| _ {2} ^ {2}. \tag {38}
+$$
+
+Since all functions are assumed to be bounded from below, the claim is proved.
+
+# A.1 PSEUDO-CODE FOR REFINED_PARTITION
+
+In the pseudocode below, a cycle  $C$  is a collection of column indices (i.e. variable indices) of the adjacency matrix  $A \in \mathbb{R}^{m \times d}$ . We can use depth-first search (DFS) or networkx for detect_cycle. The loop can continue for at most  $d$  iterations, with each iteration costing  $O(m + d)$ .
+
+# Algorithm 2:REFINED_PARTITION
+
+Input: Adjacency matrix  $A$  representing the function-variable dependencies (bipartite graph)
+
+Output: Refined partition  $\mathcal{P}$  of variables
+
+Initialize  $\mathcal{P}\gets \{\{w_1\} ,\{w_2\} ,\ldots ,\{w_d\} \}$  // each variable in its own group while there exists a cycle do  $C\gets$  detect_cycle(A) // e.g., use networkx package or DFS  $P\gets \cup_{j\in C}\mathcal{P}_j$  // merge variables in the cycle 5  $\mathbf{a}\gets \bigvee_{j\in C}A_{:j}$  // update the edges 6  $A_{:C}\gets []$ $A\gets [A,\mathbf{a}]$  // contract the graph 7  $\mathcal{P}_C\gets []$ $\mathcal{P}\gets \mathcal{P}\cup \{P\}$  // update the partition   
+return P
+
+# B EXPERIMENT DETAILS
+
+# B.1 SYNTHETIC EXAMPLES
+
+![](images/c2e2ad276ba831d2fc13962618b2b9122648c0c5cf5fd306cc9cfc43f8e2357a.jpg)  
+Figure 10: Comparisons of stationary solution fronts generated by MGDA and RPMGDA. (1) and (2) for the quadratic example, (3) and (4) for the non-strictly convex example.
+
+![](images/ab346add6d903c0e98918eb9f8b999c310c49811242c1b3029e138130f2243ba.jpg)
+
+![](images/1c6cc4e7364d3d95492145c696920bb29c8966218d3f8c98a7d0998358664a46.jpg)
+
+![](images/d8cd99e6670419246df9247e3340039eb9255ad583f7943f2fd140f54d98f739.jpg)
+
+Two-objective toy examples. We examine two examples with the same underlying function-variable dependency structure but different convexity conditions. The quadratic one is given by:  $f_{1}(\mathbf{w}) = w_{1}^{2} + w_{2}^{2}$ ,  $f_{2}(\mathbf{w}) = (w_{2} - 1)^{2} + (w_{3} - 1)^{2}$ . The non-strictly convex one is:  $f_{1}(\mathbf{w}) = w_{1}^{2} + |w_{2}|$ ,  $f_{2}(\mathbf{w}) = |w_{2} - 1| + (w_{3} - 1)^{2}$ . Initialized with a grid search over  $[-1,1]^{3}$ , we plot the solution fronts in terms of function values  $[f_{1}(\mathbf{w}^{*}), f_{2}(\mathbf{w}^{*})]$  for MGDA and RP-MGDA in Figure 10.
+
+We observe that RP-MGDA always reaches Pareto optimal solutions regardless of initialization, while MGDA may converge to sub-optimal PS solutions (i.e., those with  $f_{1} = 0$ ,  $f_{2} > 1$  and  $f_{1} > 1$ ,  $f_{2} = 0$ ). Moreover, by post-processing the sub-optimal solutions of MGDA with RP-MGDA, we can refine these solutions and bring them to the Pareto front.
+
+Adding regularization to induce strict convexity. We have discussed in Section 4 the importance of joint strict convexity for the performance of Pareto stationary solutions, particularly as highlighted in the second part of Lemma 1. We note that the functions in the two-objective toy example setting in Appendix B.1 are not strictly convex w.r.t.  $\mathbf{w}$ , though they are strictly convex with respect to their dependent variables (in the quadratic case). One natural attempt is to add regularization to the objectives to induce strict (indeed strong) convexity. For the two-objective toy example setting in Appendix B.1, we regularize the objectives with L2-norm on  $\mathbf{w} = (w_{1},w_{2},w_{3})$  with different magnitudes of regularization  $\gamma$ , and examine the new stationary solution fronts generated by MGDA. Specifically, we consider  $f_{1}^{\gamma}(\mathbf{w})\coloneqq w_{1}^{2} + w_{2}^{2} + \gamma \| \mathbf{w}\|_{2}^{2}$ ,  $f_{2}^{\gamma}(\mathbf{w}) = (w_{2} - 1)^{2} + (w_{3} - 1)^{2} + \gamma \| \mathbf{w}\|_{2}^{2}$ .
+
+We observe in Figure 11 that bigger regularization yields solutions that are further away from Pareto front; while smaller regularization gradually recover the previous (undesired) stationary solution set of MGDA (as in Figure 10), thus still having those inferior weakly Pareto solutions (i.e., those on the tails). To summarize, adding a regularization term, whether large or small, to induce strong convexity, does not alleviate the issues with MGDA, further highlighting the effectiveness of our refined partitioning for MGDA.
+
+![](images/4ad02d755db4f1c03ccb4fcf8f751346404d5a736028e4f3422246275fb8fbf9.jpg)  
+Figure 11: Stationary solution fronts of MGDA with regularized objectives, compared to the true Pareto front (red). From left to right: regularization magnitude  $\gamma = 1$ , 0.1 and 0.01.
+
+![](images/66d9e153d956f29ee76057b0be1d6e7b7e658bb7c2e589380b41fbfede11585c.jpg)
+
+![](images/d6d947440cfa4bca3fcb5943813339619e63ef1f2b2a871a8f82e03869527212.jpg)
+
+# B.2 RANDOMLY GENERATED SPARSE ADJACENCY MATRIX
+
+Setup. The problem is formulated by first generating a random sparse adjacency matrix of  $m \times d$ , with density  $\rho$ . In detail, we do this by randomly assigning  $md\rho$  (rounded to integer) non-zero entries to a zero  $m \times d$  matrix. Afterwards, we define our objectives  $f_{i} = \sum_{w_{j} \in v(f_{i})} (w_{j} - i)^{2}, i \in [m]$ , where  $v(f_{i})$  is the set of variables that  $f_{i}$  depends on.
+
+More results. We compare the solution of MGDA and RP-MGDA with the same random initialization  $\mathbf{w}_0$  (see Figure 12 Left), and notice that RP-MGDA outperforms MGDA in most objectives. Although RP-MGDA is theoretically superior, their different update rules can lead to incomparable solutions, even when the solution of RP-MGDA is Pareto optimal and MGDA is not.
+
+We track the norm  $\| \mathbf{d}^t\|$  of RP-MGDA and MGDA during optimization, and observe that both converge to zero. Note that the gradient norm  $\| \mathbf{d}^t\|$  of RP-MGDA is much larger than MGDA in the beginning, which is not surprising since MGDA finds the overall min-norm element. Moreover, a descent direction  $\mathbf{d}^t$  with a larger norm generally indicates a greater improvement for each update.
+
+![](images/06c9e5f829d77fe0712dfb5c03f3c9a2cf1b8e247c94f57d32e68b38bd40a77e.jpg)  
+Figure 12: Left: compare MGDA and RP-MGDA solution, with same random initialization  $\mathbf{w}_0$ . Right: The norm of  $\mathbf{d}^t$  over time converges to zero, empirically verifying Theorem 3.
+
+![](images/c18c3203f1a6cbcefbdf3d9720551684a1c4738c41316a212713117621ef9ab5.jpg)
+
+We also present a  $5 \times 5$  toy example below, which maybe useful for readers.
+
+Toy  $5 \times 5$  dependency matrix. Here we consider the quadratic objectives:  $f_{i} = \sum_{w_{j} \in v(f_{i})} (w_{j} - i)^{2}, i \in [5]$ , where  $v(f_{i})$  is the set of variables that  $f_{i}$  depends on. The variable dependency structure is given in Figure 32, Appendix B.7. The refined partition for this problem is  $\{w_{1}, w_{2}, [w_{3} w_{4} w_{5}]\}$ , where RP-MGDA is performing (8) w.r.t.  $\mathbf{w}' = (w_{3}, w_{4}, w_{5})$  and gradient descent w.r.t  $w_{1}$  and  $w_{2}$ . We observe from Figure 13 that the solutions of RP-MGDA dominate those of MGDA, although all being Pareto stationary (with corresponding convex combination coefficients plotted).
+
+![](images/7959c8f6d5dd67fb78fb4b54b8a22320784794e57d079b5e650132d47ff8fb1e.jpg)
+
+![](images/d534af47e17ebe44a119be3982579333479ba75264a91abd0a3a61bae7ae76a1.jpg)
+
+![](images/b14cca079060c9a61dffbdc9db181c6d36f1dec79438603ec5630ac9f48239e7.jpg)  
+Figure 13: Comparison of MGDA and RP-MGDA solutions, with different initializations. (i) Colored lines connect the final objective values of each  $f_{i}$ , where blue represents RP-MGDA, and orange represents MGDA. (ii) Red markers (with values on the right  $y$ -axis) are the corresponding dual coefficients  $\lambda_{i}$  of the solutions, showing the differences of solutions from a dual perspective.
+
+![](images/31bb8f9298e141e970c192d742af5157c4cb9e152f1891b7b817abef339613a0.jpg)
+
+# B.3 PERSONALIZED FEDERATED LEARNING (THE 'MTL' STRUCTURE)
+
+In this section, we present detailed experiment setup and more results regarding the personalized federated learning (PFL) experiments.
+
+# B.3.1 DETAILED EXPERIMENT SETUP
+
+Data. For the PFL setting, we conducted experiments on both MNIST and CIFAR-10 dataset. In order to create a non-i.i.d. dataset, we follow a similar sampling procedure as in McMahan et al. (2017): First we sort all data points according to their classes. Then, they are split consecutively into shards (300 shards for MNIST, 250 shards for CIFAR-10), with 200 images per shard, each shard contains images from only one class. Each client is randomly assigned 10 different shards, totaling 2000 instances per client. The data distribution for each client varies, with each having access to different subsets of class labels. For example, client 1's data includes class labels  $[0', 2', 4', 6', 7', 8', 9]$ , while lacking data from class labels  $1'$ ,  $3'$  and  $5'$ .
+
+Model. For MNIST, we use Multi-Layer Perceptrons (MLP) for both the lower personalized networks and the top global network. For CIFAR-10, the lower personalized networks are Convolutional Neural Networks (CNNs), while the top global network remains an MLP. Input data first passes through the lower personalized networks, and the resulting representations are then forwarded to the top global model for predictions. See Table 1 and 2 for configurations.
+
+Misc. We adopt a warmstart procedure by first running RP-MGDA for 50 epochs on a given random initialization, using the resulting weights as the starting point for both MGDA and RP-MGDA training. This approach helps improve stability in some cases while still ensuring a fair comparison. We also employ a periodic exponential decay learning rate scheme for the CIFAR-10 experiments.
+
+Table 1: Network architecture for PFL MNIST experiments.  
+
+<table><tr><td>Network</td><td>Input Dim</td><td>Hidden Dim</td><td>Output Dim</td></tr><tr><td>Lower Personalized MLP</td><td>img_size</td><td>256</td><td>128</td></tr><tr><td>Top Global MLP</td><td>128</td><td>128</td><td>10</td></tr></table>
+
+Table 2: Network architecture for PFL CIFAR-10 experiments.  
+
+<table><tr><td>Layer Type</td><td>Configuration</td><td>Activation</td></tr><tr><td colspan="3">Lower Personalized CNN Network</td></tr><tr><td>Conv2D</td><td>3 × 32, kernel size 5</td><td>ReLU</td></tr><tr><td>BatchNorm2D</td><td>32 channels</td><td>-</td></tr><tr><td>MaxPool2D</td><td>kernel size 2, stride 2</td><td>-</td></tr><tr><td>Conv2D</td><td>32 × 32, kernel size 5</td><td>ReLU</td></tr><tr><td>BatchNorm2D</td><td>32 channels</td><td>-</td></tr><tr><td>MaxPool2D</td><td>kernel size 2, stride 2</td><td>-</td></tr><tr><td>Fully Connected</td><td>32 × 5 × 5 → 384</td><td>ReLU</td></tr><tr><td colspan="3">Top Global MLP Network</td></tr><tr><td>Fully Connected</td><td>384 → 192</td><td>ReLU</td></tr><tr><td>Fully Connected</td><td>192 → 10</td><td>-</td></tr></table>
+
+# B.3.2 MORE RESULTS
+
+![](images/02897eeda69b7bab2cf6450fe685077004b9c015d2044ba42b3af10a2a6a24dd.jpg)
+
+![](images/f980757a37e4e9840b9283541177ee675baed9d74b78cae1cf8aec18fb1599b7.jpg)
+
+![](images/a6afe36d866cc7f289aeffd927609a655b14941135757712fcea7f1d2036c2a5.jpg)  
+Figure 14: Personalized federated learning problem on MNIST. Comparing 4 objectives using RP-MGDA and MGDA. Random seed 1.
+
+![](images/efaae678c228943a6c6da7d40d8e71c3ef6665883aff8db16b48d4dd854b3c5f.jpg)
+
+![](images/df346e4c3691f2bef4ad0cc017ec5e6336c9810abdaaa33098e1a32f4ea50fa9.jpg)
+
+![](images/652c80968045a1c2ccbbbff6e4fe8c49e6f9dc7520b36c4e5ad10b2aac976862.jpg)
+
+![](images/bc19acb22ec49a99ccbf67eb3d4eff22022b643482186d497a3b3cbcf30cbde7.jpg)  
+Objective 3  
+Figure 15: Personalized federated learning problem on MNIST. Comparing 4 objectives using RP-MGDA and MGDA. Random seed 2.
+
+![](images/e9032e4d7861ceeaddbce20bde9b1122f816150d454acbe4a74b792ded9c857d.jpg)  
+Objective 4
+
+![](images/7f08eb7acf64b963d7125846f42bd147101d934b2084b7709f7663eac5c7ab22.jpg)
+
+![](images/7fc7b5fcee3e6de1ec81545cd95e582f0f8b90f789270f6ebadd59c54476968d.jpg)
+
+![](images/0c1b2c281a1e4eebe641c86096ad332ee8e9e63ac5dc611153310935325bea89.jpg)  
+Objective 3  
+Figure 16: Personalized federated learning problem on CIFAR10. Comparing 4 objectives using RP-MGDA and MGDA. First random seed.
+
+![](images/00d7eb1c963b9466cf392acbe233e21b6bd77dc8ff89428cdd1764d2faf988e8.jpg)  
+Objective 4
+
+![](images/b1044e76e0abbfc99849fce9d6340752841e5102dc3630837597807e06115903.jpg)
+
+![](images/7cdf5a6e5bf13b6c0e1f9ef9299a14681393d6d67269bf1521715b879facf87e.jpg)
+
+![](images/6c86dcd390d4016cd19b26c322b5053b7ed413e2bdd61689580436cdf98e4e20.jpg)  
+Objective 3  
+Figure 17: Personalized federated learning problem on CIFAR10. Comparing 4 objectives using RP-MGDA and MGDA. Random seed 2.
+
+![](images/20f4b9c09774ccafb9edc4ce868a6f2aa6f91de73a36e79548b640f02010defa.jpg)  
+Objective 4
+
+# B.3.3 ELU ACTIVATION
+
+The purpose of these experiments is to verify whether replacing ReLU with differentiable ELU activations yields similar results and conclusions. We observed that the performance of both RP-MGDA and MGDA remains consistent with their performance when using ReLU, with RP-MGDA still outperforming MGDA.
+
+![](images/eb5f21c11c85a558d87bdbd39d5b89adedd4bfa65966d498b994a6b966ae6e56.jpg)  
+Figure 18: With ELU activation. Radar graph of final cross-entropy losses  $f_{i}$  in the personalized federated learning setting, using MGDA and RP-MGDA respectively. Lower is better.
+
+![](images/3ea3133f2c352c6446c82f0c11297b758ee14691dcdc1a331b29300a001edde8.jpg)
+
+![](images/30b760a52ef81353ea35de9711a7b079b347ad94b701d53c54cfb7ad52ea0cd4.jpg)
+
+![](images/a3e21b932bb2bddde2527954f9ab8526e3b9b9a2c65ce8cc12e5fbeeb3796417.jpg)
+
+![](images/5d61a26448bf67972f832fe037922a871ef2cd43f92f47e3315cf80b0ac9139a.jpg)  
+Objective 3  
+Figure 19: Personalized federated learning problem on MNIST. Comparing 4 objectives using RP-MGDA and MGDA. ELU activation.
+
+![](images/e311b50adb3ef9a62f9e22a2b27b7371ee7e016ee017b35a66d737c7305f41af.jpg)  
+Objective 4
+
+![](images/8fb1f5936719f532dabc0aaecf45d294de6357bdb52743d378365a1bed51f696.jpg)
+
+![](images/fe911bb3f0390413a5a8faaeb849d01d9549dc6de301462bae064952407e1748.jpg)
+
+![](images/baa86f7af3cce2ddb7ff6976b4981a0f9fd092d6b2a4e05b0bf1b08ebfc8ff6a.jpg)  
+Objective 3  
+Figure 20: Personalized federated learning problem on CIFAR-10. Comparing 4 objectives using RP-MGDA and MGDA. ELU activation.
+
+![](images/80da9bb4c1672937cd3296a272dbb308f57e4262f0aff718d7440dc258cca2d0.jpg)  
+Objective 4
+
+# B.3.4 STOCHASTIC GRADIENTS
+
+We evaluate both RP-MGDA and MGDA in the stochastic setting, where only stochastic gradients are available due to mini-batch training. The experiments are run for 500 epochs, with each epoch corresponding to a full pass through the dataset. Mini-batch size=200. For each epoch, we report the loss at the start of the epoch. For radar graph, we report the loss averaged across mini-batches of the last epoch.
+
+We observe that RP-MGDA continues to outperform MGDA in performance. However, neither algorithm guarantees descent in every iteration, and the performance of both methods drops slightly, potentially due to bias in the descent direction.
+
+![](images/59c2990b79d6e2e8d1466ad1ac0f0b3572eaba61e70d8bffee83b7b6ba73d1f2.jpg)  
+Figure 21: Stochastic gradients. Radar graph of final cross-entropy losses  $f_{i}$  in the personalized federated learning setting, using MGDA and RP-MGDA respectively. Lower is better.
+
+![](images/3f2f4ef0e8ca7a44abf8ff120c388cdd3ab9a97e5a94544868f92ff03b0880d7.jpg)
+
+![](images/9e795aafa10fae8ea8b6891d9f169e53c14a10427c647b6a1a210678f17d2111.jpg)
+
+![](images/e6c328aaa8f0636fe85de5153830982c9bc4e5fa7bf2a737457b99a19c4d5b22.jpg)
+
+![](images/47f7e3edbb6f0ce8b5a021f3132a4e63404916ba39dd8b9ebf18a0d1e163e677.jpg)  
+Figure 22: Personalized federated learning problem on MNIST. Comparing 4 objectives using stochastic RP-MGDA and stochastic MGDA. Report loss at the start of each epoch.
+
+![](images/7632d1823044260d60d1a822a2f3d891a81d7b758706fef061cc3042ad6217eb.jpg)
+
+![](images/666fb8b68fae0138dc31bee5db760da707e9930d9576dd76d86292e5793a886d.jpg)
+
+![](images/3afda2f005b814d606bd0c745ae8382e7dd91e7e3e66302bcff1a69c3e38ebf4.jpg)
+
+![](images/5353e26b3e627c4f40df5311ee3304f856b6148dafcc9f489a104c81dc9b85e3.jpg)  
+Objective 3
+
+![](images/b064bb325a095bd150ffc60da63817e2389108fc5d501ef21fe23bc88750dc4f.jpg)  
+Objective 4  
+Figure 23: Personalized federated learning problem on MNIST. Comparing 4 objectives using stochastic RP-MGDA and stochastic MGDA. Report loss at the start of each epoch.
+
+# B.3.5 PCGRAD+ AND RP-PCGRAD+
+
+In this section, we further validate the effectiveness of Refined-Partitioning (RP) by conducting experiments on the PCGrad+ algorithm, both with and without RP, in the PFL setting. PCGrad+ is a looped version of the PCGrad algorithm (Yu et al., 2020), where the iterative projection and correction step (Lines 4-7 in Algorithm 1 of (Yu et al., 2020)) is repeated until the 'surgery' gradient  $\mathbf{g}_i^{PC}$  achieves a non-negative inner product with all other gradients. This modification ensures that PCGrad+ is a common descent algorithm for all objectives even when  $m \geq 3$ .
+
+For MNIST, we run 1000 epochs with lr=0.1 for both algorithms; for CIFAR-10, we run 5000 epochs with lr=0.05 and a step decay of 0.9 every 200 epochs for both algorithms. Other settings follow those of MGDA and RP-MGDA. We observe that RP-PCGrad+ outperforms PCGrad+.
+
+![](images/3e55efc9c4dfa86deb9227a47814635eae9d00c0c94cef4a8850d2eeafe88880.jpg)  
+Figure 24: Radar graph of final cross-entropy losses  $f_{i}$  in the personalized federated learning setting, using PCGrad+ and RP-PCGrad+ respectively. Lower is better.
+
+![](images/5f1b13eb85951062d8ec283a8f4f0307bbfd3735460233e564876319651f9e7a.jpg)
+
+![](images/81aad6db70698f57328015be9a8f68a254264bb3340706203f1bf31e5d1a65f1.jpg)
+
+![](images/1ba2d4e93cc8173f4f8a94d8d0deec7df752f683ab4787550627432ca4bfd413.jpg)
+
+![](images/03ff5c4b7fccd8a86e5eb35f67ebdfe73aaab21fe3654e8779f58c7a48d143d3.jpg)  
+Objective 3  
+Figure 25: Personalized federated learning problem on MNIST. Comparing 4 objectives using RP-PCGrad+ and PCGrad+.
+
+![](images/81ff2be8280e108ff24d0dfd19c8741838f1000d471010f70faf3740d0ad7061.jpg)  
+Objective 4
+
+![](images/b6dfe1cced8e9e5c71d3d51ba6418914a0205e6ea1347eec99cbc4cb54f2dcc9.jpg)
+
+![](images/db727996e1e66cae3f9ff9e10a543c4ddf7f0e3542e7a068694d23edafe4870b.jpg)
+
+![](images/4f934c4b8d471becd0789ad4f896810d3d8b388cc2fe7660e69b336b491b68d1.jpg)  
+Objective 3
+
+![](images/43d4f392d3dba09fdad0299856c3e98e9fc582a045caabfe2834373652b0d540.jpg)  
+Objective 4  
+Figure 26: Personalized federated learning problem on CIFAR-10. Comparing 4 objectives using RP-PCGrad+ and PCGrad+.
+
+# B.4 HIERARCHICAL CLASSIFICATION (THE 'LADDER' STRUCTURE)
+
+In this section, we present detailed experiment setup.
+
+Data. In addition to the original 10 classes in the CIFAR-10 dataset, we define broader categories with binary labels: ['non-living', 'living'], as well as mid-level categories: ['ground-vehicle', 'non-ground-vehicle', 'land-animal', 'non-land-animal']. These serve as coarser labels for the original classes, which include ['plane', 'car', 'bird', 'cat', 'deer', 'dog', 'frog', 'horse', 'ship', 'truck']. Consequently, each image instance in the dataset with an original label is also assigned a binary label and a mid-level label for the tasks of binary classification  $(f_{1})$  and four-class classification  $(f_{2})$ , respectively. For instance, a 'plane' is labeled as 'non-living' 'non-ground-vehicle', while a 'deer' is labeled as 'living' 'land-animal'. We take a smaller subset of the original CIFAR-10 dataset as our training set.
+
+Model. We employ a CNN architecture similar in spirit to Branch-CNN (Zhu and Bain, 2017), featuring three layers for feature extraction and three 'off-ramp' linear classifiers that branch out after the first, second, and third feature extraction layers, respectively. The first task,  $f_{1}$ , processes input through the shared feature extraction layer 1 followed by classifier 1. The second task,  $f_{2}$ , processes input through the shared feature extraction layers 1 and 2, followed by classifier 2. The third task,  $f_{3}$ , processes input through the shared feature extraction layers 1 and 2, then through the non-shared feature extraction layer 3, and finally classifier 3. See Table 3.
+
+Misc. We adopt a warmstart procedure similar to the previous section by first running RP-MGDA for 5 epochs. We also employ a periodic exponential decay learning rate scheme for the experiments.
+
+Table 3: Network architecture for hierarchical classification experiments.  
+
+<table><tr><td>Layer Type</td><td>Configuration</td><td>Activation</td></tr><tr><td colspan="3">First Feature Extraction Layer</td></tr><tr><td>Conv2D</td><td>3 × 32, kernel size 5</td><td>ReLU</td></tr><tr><td>BatchNorm2D</td><td>32 channels</td><td>-</td></tr><tr><td>MaxPool2D</td><td>Kernel size 2, stride 2</td><td>-</td></tr><tr><td>Conv2D</td><td>32 × 32, kernel size 5</td><td>ReLU</td></tr><tr><td>BatchNorm2D</td><td>32 channels</td><td>-</td></tr><tr><td>MaxPool2D</td><td>Kernel size 2, stride 2</td><td>-</td></tr><tr><td colspan="3">Second Feature Extraction Layer</td></tr><tr><td>Fully Connected (FC)</td><td>32 × 5 × 5 → 384</td><td>ReLU</td></tr><tr><td colspan="3">Third Feature Extraction Layer</td></tr><tr><td>Fully Connected (FC)</td><td>384 → 192</td><td>ReLU</td></tr><tr><td colspan="3">First Classifier</td></tr><tr><td>Fully Connected (FC)</td><td>32 × 5 × 5 → 2</td><td>-</td></tr><tr><td colspan="3">Second Classifier</td></tr><tr><td>Fully Connected (FC)</td><td>384 → 4</td><td>-</td></tr><tr><td colspan="3">Third Classifier</td></tr><tr><td>Fully Connected (FC)</td><td>192 → 10</td><td>-</td></tr></table>
+
+# B.5 MULTI-OBJECTIVE LEARNING WITH PARTIAL INFORMATION (THE 'CHAIN' STRUCTURE)
+
+In this section, we present detailed experiment setup.
+
+Data. We conduct experiments on the California housing dataset, which consists of 20,640 samples from the 1990 U.S. Census, aiming to predict median house values. The dataset includes 8 features, ['Longitude', 'Latitude', 'AveOccup', 'Population', 'MedInc', 'HouseAge', 'AveRooms', 'AveBedrms']. We group 'Longitude' and 'Latitude' as  $\mathbf{w}_1$ ; 'AveOccup', 'Population' and 'MedInc' as  $\mathbf{w}_2$ ; 'HouseAge', 'AveRooms' and 'AveBedrms' as  $\mathbf{w}_3$ . To implement the function-variable dependency structure, we use feature masks on X during training.
+
+Model. We consider a linear model with 8 weights, each corresponding to one of the 8 features in the dataset. We set the bias  $b$  to be the mean of Y train.
+
+Misc. We standardize each feature to have zero mean and unit variance to improve training.
+
+![](images/1580df50f7b4f27115bb8d261028381af426993fbf3ee39c5283f39a0f73b1f4.jpg)
+
+![](images/02fecd6309e2b4f987d6f5d14eff5f44ef35b53d63edd9ae5a9dc2ea12e4cc0e.jpg)
+
+![](images/1c733e9a4b3d359dbdad4c6be3ad54cff6122d8ef9c5df77ba655a6c8d7c10e0.jpg)  
+Figure 27: We calculate the mean and standard deviation of the 4 losses across the 20 trials (different seeds) for both MGDA and RP-MGDA, and plot the means, with shaded regions representing the standard deviations. Although the solution of RP-MGDA for a single trial does not dominate MGDA's. RP-MGDA achieves lower average losses for all 4 objectives and has much smaller deviations.
+
+![](images/c7c80df1ed49379b732e4dedd0102a8c8208ce0cd9f02d62f480cfe43d5b880f.jpg)
+
+# B.6 JACOBI VS. GAUSS-SEIDEL UPDATE
+
+In Section 6, we discussed the approach of cycling through the blocks of variable identified by REFINED_PARTITION for updates, also known as the 'Gauss-Seidel' update. The main RP-MGDA algorithm presented in Algorithm 1, which updates all blocks simultaneously, is referred to as the 'Jacobi' update.
+
+In this section we empirically compare the Jacobi (default) and Gauss-Seidel versions of RP-MGDA updates, using a refined partition of 5 blocks for the PFL setting on both the MNIST and CIFAR-10 dataset. We find no significant difference overall when the number of variable updates is kept the same.
+
+![](images/894abf7c93cfa6fad9ac26f059e693db7fc0110ade82276d472c8fdb6dc99260.jpg)
+
+![](images/a9e51c360be105e01515614624877ea7f8d7fe6e674b9b2e0217af8b178cb5fd.jpg)
+
+![](images/afe3b686ac80a323de7e06c5dfce0d53378ebfd3d237232606e3fc459e2d6cf1.jpg)  
+Objective 3  
+Figure 28: A look at the first 100 epochs. Gauss-Seidel update exhibits some 'zig-zags' and converges slower in terms of pure number of epochs. CIFAR-10 dataset.
+
+![](images/d9f7ab1d253c9bb94f878915ee4eae302c02ba1e8593a478b82d6debf5ce6a0d.jpg)  
+Objective 4
+
+![](images/9c3ca0ff09d8e441bbb4aa0b53ab3dc024f72c923337a0a70dac6f125d307262.jpg)
+
+![](images/28fa336ff6f1717749434ac86bb312b9b250e0eed1bfc80f4047bafb8eb710ab.jpg)
+
+![](images/9fdabca5c248ce4948a5da11fc305b895cbcfbf230848e623dd83eb59866249c.jpg)  
+Objective 3  
+Figure 29: Overall convergence trends are similar if we plot the loss of Gauss-Seidel update every 5 epochs instead, to ensure the total number of variable updates are the same for Jacobi and Gauss-Seidel at each time stamp in the plot. CIFAR-10 dataset.
+
+![](images/179dcef1f5aa8c4dec0e4f83d2bc545cf8d80c86317473e34647c9b9c70c1edc.jpg)  
+Objective 4
+
+![](images/8be93870ee912ab0036f334187979459c4bd5e20c756925fa02e8ac62c68724d.jpg)
+
+![](images/1be80110f1592dfdece1d48cd9b5541bc0635175010e6ec4cb3c6f56b8228268.jpg)
+
+![](images/e8efe0beeb732249fb065bdce0a29902337ed2f711f9c9c87d797928f3d65cd7.jpg)  
+Objective 3  
+Figure 30: A look at the first 100 epochs. MNIST dataset, PFL setting.
+
+![](images/e3a1640edd93628cb10467d5cbd28c785794afd7a41debcd72657def9e2040a6.jpg)  
+Objective 4
+
+![](images/af6ed2abf1a0c58ad6b75f1eccf7a776b8bfe80ab5e5707bad8dbf866324bc61.jpg)
+
+![](images/6d30bf246bdd690580e8fa499d330f8fd7c994b6216cfef69700b1d5ef7cd90b.jpg)
+
+![](images/a4a6d94a4933640c71687f3a58e40f20657ddced0db5710cb700be21f066d043.jpg)  
+Objective 3  
+Figure 31: Overall convergence in 1000 epochs, with Gauss-Seidel update plotted every 5 epochs to ensure the total number of variable updates are the same. MNIST dataset, PFL setting.
+
+![](images/c1b146278015641780f89eeea76e09da83157dd63ba03506bd97d49ead2349b6.jpg)  
+Objective 4
+
+# B.7 MISC
+
+![](images/a3ef97acb523387b85ad3a664a304ca9ca34958a738ad60051f37325ec1d8b7f.jpg)  
+Figure 32: Variable dependency structures of synthetic toy examples in §B.1 and §B.2.
+
+![](images/9d0ba1b9811f510e9911851f6eb97ba81dc88bd91e1f6a9b4117f83610a1288a.jpg)
\ No newline at end of file
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+# LINES: POST-TRAINING LAYER SCALING PREVENTS FORGETTING AND ENHANCES MODEL MERGING
+
+Ke Wang*
+
+EPFL
+
+k.wang@epfl.ch
+
+Nikolaos Dimitriadis*
+
+EPFL
+
+nikolaos.dimitriadis@epfl.ch
+
+Alessandro Favero
+
+EPFL
+
+alessandro.favero@epfl.ch
+
+Guillermo Ortiz-Jimenez
+
+Google DeepMind
+
+gortizj@google.com
+
+François Fleuret
+
+University of Geneva, Meta FAIR
+
+francois.fleuret@unige.ch
+
+Pascal Frossard
+
+EPFL
+
+pascal.frossard@epfl.ch
+
+# ABSTRACT
+
+Fine-tuning pre-trained models has become the standard approach to endow them with specialized knowledge, but it poses fundamental challenges. In particular, (i) fine-tuning often leads to catastrophic forgetting, where improvements on a target domain degrade generalization on other tasks, and (ii) merging fine-tuned checkpoints from disparate tasks can lead to significant performance loss. To address these challenges, we introduce LiNeS, Layer-increasing Network Scaling, a post-training editing technique designed to preserve pre-trained generalization while enhancing fine-tuned task performance. LiNeS scales parameter updates linearly based on their layer depth within the network, maintaining shallow layers close to their pre-trained values to preserve general features while allowing deeper layers to retain task-specific representations. In multi-task model merging scenarios, layer-wise scaling of merged parameters reduces negative task interference. LiNeS demonstrates significant improvements in both single-task and multi-task settings across various benchmarks in vision and natural language processing. It mitigates forgetting, enhances out-of-distribution generalization, integrates seamlessly with existing multi-task model merging baselines improving their performance across benchmarks and model sizes, and can boost generalization when merging LLM policies aligned with different rewards via RLHF. Our method is simple to implement, computationally efficient and complementary to many existing techniques. Our source code is available at github.com/wang-kee/LiNeS.
+
+# 1 INTRODUCTION
+
+Pre-trained models have become the backbone of modern machine learning pipelines (Bommasani et al., 2021; Touvron et al., 2023). Their introduction has shifted the paradigm from end-to-end training to fine-tuning (Zhuang et al., 2020), leading to the proliferation of thousands of fine-tuned checkpoints derived from a few foundation models (Rombach et al., 2022; Team et al., 2023). To improve downstream performance across multiple tasks or align with multiple preferences (Singh & Jaggi, 2020; Matena & Raffel, 2022; Ilharco et al., 2023; Yadav et al., 2023; Ramé et al., 2024a), model merging techniques combine available checkpoints, avoiding the costly process of joint fine-tuning (Ilharco et al., 2023; Yadav et al., 2023). However, specializing models introduces trade-offs, such as the forgetting of previously acquired knowledge (Aghajanyan et al., 2021) - a phenomenon known as catastrophic forgetting (McCloskey & Cohen, 1989). Furthermore, merging checkpoints fine-tuned on different tasks can lead to significant performance degradation due to task interference (Yadav et al., 2023; Wang et al., 2024).
+
+To mitigate catastrophic forgetting, many works propose regularizing the fine-tuning process (Aghajanyan et al., 2021; Kumar et al., 2022; Gouk et al., 2021; Razdaibiedina et al., 2023). Leveraging the insight that shallow layers capture generalizable representations (Yosinski et al., 2014; Neyshabur et al., 2020), Howard & Ruder (2018); Dong et al. (2022) apply lower learning rates to the shallow layers to retain general features. However, modifying the fine-tuning process can be complex and computationally expensive. This motivates the development of post-training model editing and model merging methods that directly edit the checkpoints in the weight space. For instance, Wortzman et al. (2022b); Ramé et al. (2022) mitigate catastrophic forgetting by interpolating weights between pre-trained and fine-tuned models. In multi-task settings, Yadav et al. (2023); Wang et al. (2024) propose methods to reduce interference among tasks when merging multiple checkpoints. Yet, significant performance degradation persists when merging multiple models, leaving this as an open challenge.
+
+Most model merging methods, however, treat all layers equally, overlooking the earlier insight that shallow layers should remain close to their pre-trained weights to avoid losing the general representations they encode. In this paper, we explore whether this insight can be leveraged post-training. We find that reducing the magnitude of shallow-layer updates after fine-tuning can retain single-task performance gains while significantly mitigating forgetting.
+
+We propose LiNeS, Layer-increasing Network Scaling, a post-training, plug-and-play method that directly edits the residual, i.e., the difference between the fine-tuned and pre-trained checkpoint, by applying a scaling coefficient that linearly increases with layer depth. This scaling effectively preserves the general features captured in the shallow layers of the pre-trained model while retaining task-specific features in the deep layers of the fine-tuned model. Moreover, we extend LiNeS to the multi-task model merging setting, where contributions from one task distort the general features also required by other tasks. By preserving the general features in the shallow layers, LiNeS mitigates task interference and improves multi-task performance.
+
+LiNeS demonstrates remarkable performance on diverse test scenarios and is orthogonal to existing post-training merging algorithms. It modifies the fine-tuned checkpoint to consistently retrieve nearly full performance on the fine-tuned task while significantly restoring generalization on other tasks. Furthermore, it can be seamlessly integrated with existing weight interpolation methods for improving out-of-distribution generalization (Wortsman et al., 2022b). When merging multiple models, LiNeS improves baseline methods for merging checkpoints fine-tuned on multiple tasks in both computer vision and NLP benchmarks (Ilharco et al., 2023; Yadav et al., 2023; Wang et al., 2024) and also enhances performance when merging checkpoints fine-tuned on the same task (Wortsman et al., 2022a) and merging LLM policies aligned with different rewards (Ramé et al., 2024a) via Reinforcement Learning with Human Feedback (RLHF) (Christiano et al., 2017).
+
+# Our contributions are as follows:
+
+- We propose LiNeS, a post-training editing technique that preserves the zero-shot generalization of pre-trained models while retaining fine-tuned knowledge by applying layer-wise scaling on parameter updates. For example, in image-classification tasks with CLIP ViT-B/32 checkpoints, LiNeS maintains on average  $99.8\%$  of performance on the fine-tuned task while preserving  $97.9\%$  performance of the pre-trained model on other control tasks, effectively mitigating catastrophic forgetting.  
+- We demonstrate that LiNeS significantly enhances multi-task model merging baselines, consistently improving performance across benchmarks and architectures in both vision and NLP domains. For instance, we observe a  $3.1\%$  and  $4.0\%$  improvement over Task Arithmetic (Ilharco et al., 2023) and Ties-merging (Yadav et al., 2023) respectively, for a 20-task computer vision benchmark with ViT-L/14.  
+- We show that LiNeS can be applied to enhance existing weight interpolation methods across various scenarios, improving out-of-distribution generalization, merging multiple checkpoints fine-tuned on the same task with different hyper-parameter configurations, and merging LLM policies aligned with different rewards.
+
+Our proposed method is simple to implement<sup>1</sup>, orthogonal to many existing approaches, and improves performance in a wide variety of settings.
+
+# 2 RELATED WORK
+
+Representation collapse and regularized fine-tuning Pre-trained models such as CLIP exhibit strong zero-shot performance across diverse data distributions due to the robust and transferable feature representations learned during pre-training (Radford et al., 2021; Jia et al., 2021). However, fine-tuning on specific tasks often harms the zero-shot generalization performance on distributions different from the fine-tuning domain (Wortsman et al., 2022b; Goyal et al., 2023; Aghajanyan et al., 2021). This degradation arises from the distortion of pre-trained features during fine-tuning (Kumar et al., 2022), a phenomenon referred to as representation collapse by Aghajanyan et al. (2021). To mitigate representation collapse, many works have proposed to regularize the fine-tuning process to preserve the general pre-trained features (Kumar et al., 2022; Goyal et al., 2023; Gouk et al., 2021; Zhang et al., 2022; Razdaibiedina et al., 2023; Shen et al., 2021; Lee et al., 2022). Some of these approaches take into account that different layers of a model learn distinct features, with the shallower layers capturing more general features and deeper layers specializing in task-specific representations (Neyshabur et al., 2020; Yosinski et al., 2014; Adilova et al., 2024). Specifically, they apply layer-wise learning rate decay, preserving more of the pre-trained features in the shallow layers while allowing deeper layers to specialize for the target domain (Clark et al., 2020; Bao et al., 2022; Dong et al., 2022; Howard & Ruder, 2018; Zhang et al., 2021). However, modifying the fine-tuning process is orders of magnitude more computationally expensive compared to post-training merging methods.
+
+Weight interpolation and model merging Garipov et al. (2018); Draxler et al. (2018) showed that two solutions derived from separate training runs can be connected by nonlinear paths of low loss, while linear mode connectivity (Frankle et al., 2020) extended the paths to the linear case. These insights enabled the transfer of the benefits regarding robustness of (traditional) output ensembles (Hansen & Salamon, 1990; Lakshminarayanan et al., 2017) to weight ensembles, reconciling the bias-variance trade-off (Belkin et al., 2019) while eliminating the computational cost of multiple inferences (Fort et al., 2020). These findings can be leveraged to improve performance on single-task (Izmailov et al., 2018; Wortsman et al., 2021; Ramé et al., 2022; Wortsman et al., 2022a; Jang et al., 2024), out-of-distribution (Wortsman et al., 2022b; Ramé et al., 2023), multi-task (Ilharco et al., 2022; Dimitriadis et al., 2023; 2025) and multi-objective alignment (Zhong et al., 2024; Ramé et al., 2024b) settings. Furthermore, model merging can also applied as a scalable approach to unify multiple task-specific models into a single model with multi-task capabilities (Ilharco et al., 2023; Yadav et al., 2023), despite performance loss compared to individual models. Several methods have tried to improve multi-task model merging by preserving the important parameters defined via the Fisher Information Matrix (Matena & Raffel, 2022; Tam et al., 2024), using heuristics (Davari & Belilovsky, 2023; Luo et al., 2023; Jin et al., 2023), randomly dropping and rescaling the task vector parameters (Yu et al., 2024) or by focusing on resolving weight interference caused by sign disagreements and redundant parameters (Yadav et al., 2023; Wang et al., 2024). Recent works use gradient descent to learn the layer-specific merging coefficients per task, e.g., Ada-merging (Yang et al., 2024) minimizes entropy in unlabeled test data while aTLAS (Zhang et al., 2024) optimizes using cross-entropy loss on validation data. Compared to LiNeS, these methods do not incorporate any prior knowledge on early vs. deep layers and require training, resulting in significant computational overheads.
+
+# 3 POST-TRAINING LAYER-WISE SCALING MITIGATES FORGETTING
+
+In this section, we present the key insight of our work: Scaling down the updates of shallow layers after fine-tuning can mitigate catastrophic forgetting and restore zero-shot generalization while preserving performance on the target task.
+
+Notation We consider a pre-trained model  $\theta_0\in \mathbb{R}^N$  with  $N$  parameters. Fine-tuning on a specific task  $t$  results in the fine-tuned weights  $\pmb{\theta}_t$ . The difference between these two sets of weights,  $\pmb{\tau}_t = \pmb{\theta}_t - \pmb{\theta}_0$ , is referred to as the task vector or residual for task  $t$  (Ilharco et al., 2023) and represents the updates made during fine-tuning.
+
+Fine-tuning leads to catastrophic forgetting We quantitatively demonstrate the phenomenon of catastrophic forgetting with the following experiments. Consider the 8-task image classification benchmark studied in Ilharco et al. (2023). We fine-tune a CLIP ViT-B/32 model on each task,
+
+![](images/65100315a88bd7777a18bce9aff995009015821d3add1e8c838ba028f178f064.jpg)  
+Figure 1: Downscaling the shallow layers maintains the fine-tuned performance on target tasks (orange line, left), while restoring zero-shot performance from pre-trained model on control tasks (orange line, right). The performance for downscaling deep layers instead is presented in blue lines, which underperforms downscaling shallow layers in both cases.  $\gamma$  represents the minimum scaling factor applied to the layers, where a smaller  $\gamma$  leads to stronger downscaling strength, with  $\gamma = 1$  restoring the original fine-tuned model.
+
+![](images/eef6438665993507b7b340eedfaf8f400b9ceeb5b426dbdf3601a03bf07b1b2a.jpg)
+
+measuring performance on the fine-tuned task – referred to as the target task – and the remaining 7 tasks – the control tasks. The averaged results over all target and control task combinations, shown in Table 1, demonstrate that while fine-tuning significantly improves accuracy on the target task, it drastically reduces accuracy on the control tasks, underscoring the loss of the model's zero-shot generalization abilities.
+
+Shallow-layer updates impact minimally on target task accuracy Most parameter updates during the fine-tuning process are redundant, as similar performance is achievable without updating most pre-trained weights (Yadav et al., 2023; Wang et al., 2024; He et al., 2025). Moreover, prior work shows that task-specific features are often concentrated in deeper layers of the network (Neyshabur et al., 2020; Yosinski et al., 2014; Raghu et al., 2019). Based on these observations, we hypothesize that updates to the
+
+shallow layers contribute minimally to target tasks. To test this, we progressively downscale the updates to shallow layers after fine-tuning. Specifically, we apply a scaling factor to the updates to the  $\ell$ -th layer  $\tau^{(\ell)}$ , defined as:  $\lambda^{(\ell)} = \gamma + (1 - \gamma) \frac{\ell - 1}{L - 1}$ ,  $\forall \ell \in [L]$ ,  $\gamma \in [0,1]$ . This linearly scales the updates from a factor of  $\gamma$  for the first layer to 1 for the last one. As a result, fine-tuning updates to the shallow layers are scaled down more aggressively, with later layers experiencing progressively smaller reductions. We then reintroduce the scaled task vector into the pre-trained model and measure its performance on the fine-tuned task. Figure 1 (left) shows the results of this experiment for the CLIP ViT-B/32 checkpoint fine-tuned across the 8 tasks, where  $\gamma$  is progressively decreased to strengthen the downscaling effect. We observe that, even with strong downscaling of shallow layers, the target task accuracy remains nearly unaffected. In contrast, when we downscale the deeper layers, target task accuracy drops significantly. These results support our hypothesis that shallow-layer updates are largely unnecessary for maintaining accuracy on the target task.
+
+Table 1: Fine-tuning harms generalization on control tasks. Our proposed post-training edition leads to a superior trade-off between performance on target and control tasks.  
+
+<table><tr><td>Model / Accuracy</td><td>Target</td><td>Control</td></tr><tr><td>Pre-trained</td><td>48.3</td><td>48.3</td></tr><tr><td>Fine-tuned</td><td>90.5</td><td>38.0</td></tr><tr><td>Fine-tuned+LiNeS(ours)</td><td>90.3</td><td>48.0</td></tr></table>
+
+Shallow-layer updates undermine zero-shot generalization While shallow-layer updates have minimal impact on target-task accuracy, they distort the general features learned during pre-training, which reside primarily in the shallow layers (Neyshabur et al., 2020; Yosinski et al., 2014; Raghu et al., 2019). We hypothesize that the degradation of performance on control tasks is largely due to these distortions in the shallow layers. Using the same experimental setup, we now evaluate the zero-shot performance on the control tasks, i.e., the other 7 unseen tasks. As shown in Figure 1 (right), as the strength of the shallow-layer downscaling increases, the accuracy on control tasks approaches the original pre-trained model's performance. This shows that by reducing the shallow-layer updates, we can restore most of the zero-shot performance that is lost during fine-tuning.
+
+Improved trade-off between target and control performance To optimize the trade-off between target and control task performance, we select a scaling coefficient  $\gamma$  for each model that maximizes a weighted balance between these two objectives, as detailed in Appendix C.1. After selecting the optimal scaling coefficient, the test results are shown in the final row of Table 1. Our post-training method preserves target task accuracy with a minimal  $0.2\%$  difference while improving control task performance by  $10\%$ , compared to the fine-tuned model.
+
+We further apply the same method to a 20-task computer vision benchmark (Wang et al., 2024). For evaluation, we report both the target task normalized accuracy and the control task normalized accuracy on the 19 tasks, where accuracy is normalized by the performance of the fine-tuned model for the target task and the
+
+![](images/f5dc044aa9615a1293d151369e1cb1d99a68fa091cdd2d1dad9b65a9a079a7e0.jpg)  
+Figure 2: Our linear scaling (LiNeS) retains performance on both control and fine-tuned target tasks. Each dot represents a different model.
+
+zero-shot accuracy of the pre-trained model for the control tasks. We compare to fine-tuned models on each task and the pre-trained model as baselines. Figure 2 shows that fine-tuning degrades zero-shot generalization, as indicated by the performance drop on control tasks. In contrast, our post-training scaling method significantly improves generalization while maintaining near-full target task accuracy. On average, our method achieves a target task normalized accuracy of  $99.8\%$  and a control task normalized accuracy  $97.9\%$ . This demonstrates its effectiveness in preserving both task-specific knowledge from fine-tuned checkpoints and the generalization capabilities of the pre-trained model. The full breakdown of results by task is available in Figure 6 in Appendix.
+
+In Appendix, we show that catastrophic forgetting happens with models fine-tuned with LoRA (Hu et al., 2022) as well. As shown in Table 11, higher expressivity in the form of higher ranks increases target accuracy for LoRA but at the cost of lower performance on control tasks. Still, LiNeS significantly improves control performance while minimally affecting target accuracy. Furthermore, in Figure 10 of the appendix, we show that similar benefits can be observed for convolutional architectures such as ConvNeXt (Liu et al., 2022). Finally, we provide a performance comparison between editing models with LiNeS and regularized-fine-tuning-based methods in Appendix C.8, including applying different learning rates per layer. Also in these cases, LiNeS demonstrates superior performance on control tasks, while being much more computationally efficient.
+
+# 4 METHOD
+
+Motivated by the results of the previous section for mitigating forgetting, we propose LiNeS for Layer-increasing Network Scaling, a simple post-training technique that linearly rescales the updates of different layers in the task vector based on their depth in the network. LiNeS is designed to retain general features in the shallow layers while preserving the task-specific adaptations in the deeper layers.
+
+Given a task vector  $\pmb{\tau}$  with  $L$  layer blocks we apply the layer-wise linear scaling to adjust the contributions of shallow and deep layers using the following formulation:
+
+$$
+\boldsymbol {\tau} _ {\text {L i N e S}} = \operatorname {c o n c a t} \left(\lambda^ {(1)} \boldsymbol {\tau} ^ {(1)}, \dots , \lambda^ {(L)} \boldsymbol {\tau} ^ {(L)}\right), \quad \text {w h e r e} \lambda^ {(\ell)} = \alpha + \beta \frac {\ell - 1}{L - 1}, \quad \forall \ell \in [ L ]. \tag {1}
+$$
+
+As a result, the layers in  $\pmb{\tau}$  are progressively scaled with a factor between  $\alpha$  for the first layer and  $\alpha +\beta$  for the last layer, with intermediate layers scaled with a linearly increasing schedule depending on their depth. The final model  $\pmb{\theta}$  is then obtained by summing the pre-trained model weights and the edited task vector, i.e.,  $\pmb {\theta} = \pmb {\theta}_0 + \pmb{\tau}_{\mathrm{LiNeS}}$ . Notice that, in Equation 1,  $\pmb{\tau}$  can correspond to either a single-task residual or, in the context of model merging, a multi-task vector obtained by merging the residuals of multiple checkpoints fine-tuned starting from a common initialization. Additional
+
+![](images/a12afdd6450679d860a8fa735a58a4a3616575bdbc71e753cbe1adfe08a370a4.jpg)
+
+![](images/087e06a2bcbf08ebc122e81546b8df0a11bd5894694c633f81f2222f5a7ea9d1.jpg)
+
+![](images/a404acae79bdc82dcb33edd2669fbb565c094437a76ae69d6d56c9127dd2bbde.jpg)
+
+Figure 3: Application of LiNeS to WiSE-FT (Wortsman et al., 2022b) improves performance on ImageNet and five different distribution shifts, resulting in a dominating Pareto Front over WiSE-FT. LP initialization refers to the model initialized from a linear probe (Kumar et al., 2022).  
+![](images/b4dbd505ca23efd1c72c496776e9cb5e6572819208852d49bd8ad4d02a6f47e2.jpg)  
+WiSE-FT WiSE-FT + LiNeS ★ Fine-tuned Fine-tuned + LiNeS LP initialization
+
+![](images/38c6bd5951a4aacbd9116a4e4dd664f4daf27e5c9a0912efba2edf98698414c5.jpg)
+
+details on this process are provided in the next section. Setting  $\alpha = \beta = 0$  corresponds to the pretrained model, while  $\alpha = 1, \beta = 0$  is the fine-tuned model in the case that  $\pmb{\tau}$  is a single-task vector.
+
+In practice, we find that tuning just one hyper-parameter (either  $\alpha$  or  $\beta$ ) is often sufficient to achieve a good balance between target task performance and generalization. Specific details on hyper-parameter tuning for different applications are provided in the experimental sections. The linear scaling method introduced in Section 3 corresponds to LiNeS by setting  $\alpha = \gamma$  and  $\beta = 1 - \gamma$ . This formulation generalizes our previous approach, offering a flexible way to adjust the contributions of different layers based on the task requirements.
+
+# 5 MODEL MERGING EXPERIMENTS
+
+We empirically verify the effectiveness of applying LiNeS across diverse application domains. Section 5.1 presents results for improving robust fine-tuning (Wortsman et al., 2022b) for OOD generalization; Section 5.2 focuses on improving existing multi-task merging methods (Ilharco et al., 2023; Yadav et al., 2023; Wang et al., 2024) in both vision and NLP benchmarks. In Section 5.3, we apply LiNeS and improve the merging of single-task fine-tuned models within the setting of Model Soups (Wortsman et al., 2022a), and finally, we enhance merging foundation models fine-tuned on different rewards (Ramé et al., 2024a) in Section 5.4.
+
+# 5.1 IMPROVING ROBUST FINE-TUNING FOR OOD GENERALIZATION
+
+We first consider the setting of robust fine-tuning or WiSE-FT (Wortsman et al., 2022b), where linearly interpolating between the pre-trained and the fine-tuned weights improves model performance on OOD datasets. The interpolation is equivalent to scaling the residual  $\pmb{\tau}$ :  $(1 - \gamma)\pmb{\theta}_0 + \gamma\pmb{\theta} = \pmb{\theta}_0 + \gamma\pmb{\tau}$  for  $\gamma \in [0,1]$ . We apply LiNeS to the residual  $\pmb{\tau}$ . Following Wortsman et al. (2022b), we evaluate CLIP models fine-tuned on ImageNet (Deng et al., 2009), considering 5 OOD datasets, namely ImageNetSketch (Wang et al., 2019), ImageNet-A (Hendrycks et al., 2021), ImageNet-R (Hendrycks et al., 2020), ObjectNet (Barbu et al., 2019), ImageNet-V2 (Recht et al., 2019).
+
+We apply this LiNeS to each of the 70 fine-tuned checkpoints<sup>2</sup> provided by Wortsman et al. (2022a) setting  $\alpha = \beta = 0.5$ . We present the average results in Figure 3, comparing the performance of WiSE-FT with and without applying LiNeS on the 5 OOD datasets. Without applying WiSE-FT,
+
+Table 2: Results for multi-task model merging in vision classification benchmarks of 8 tasks (Ilharco et al., 2023), 14 tasks, and 20 tasks (Wang et al., 2024) for different vision transformer architectures. Applying LiNeS improves baseline performance for all benchmark/architecture combinations.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">with LiNeS</td><td colspan="3">ViT-B/32</td><td colspan="3">ViT-L/14</td></tr><tr><td>8 tasks</td><td>14 tasks</td><td>20 tasks</td><td>8 tasks</td><td>14 tasks</td><td>20 tasks</td></tr><tr><td>Zero-shot</td><td></td><td>48.3</td><td>57.3</td><td>56.1</td><td>64.8</td><td>68.3</td><td>65.3</td></tr><tr><td>Fine-tuned</td><td></td><td>90.5</td><td>89.5</td><td>90.4</td><td>94.0</td><td>93.3</td><td>94.0</td></tr><tr><td rowspan="2">Task Arithmetic</td><td>X</td><td>69.7</td><td>65.0</td><td>60.3</td><td>84.0</td><td>79.2</td><td>74.0</td></tr><tr><td>✓</td><td>74.2 (+4.5)</td><td>69.1 (+4.1)</td><td>63.4 (+3.1)</td><td>86.5 (+2.5)</td><td>82.2 (+3.0)</td><td>77.1 (+3.1)</td></tr><tr><td rowspan="2">Ties-Merging</td><td>X</td><td>73.6</td><td>67.6</td><td>63.1</td><td>85.6</td><td>79.3</td><td>75.6</td></tr><tr><td>✓</td><td>77.2 (+3.6)</td><td>72.1 (+4.5)</td><td>67.2 (+4.1)</td><td>88.0 (+2.4)</td><td>82.5 (+3.2)</td><td>79.6 (+4.0)</td></tr><tr><td rowspan="2">Consensus Merging</td><td>X</td><td>74.5</td><td>70.1</td><td>65.3</td><td>85.2</td><td>81.9</td><td>78.7</td></tr><tr><td>✓</td><td>77.6 (+3.1)</td><td>73.6 (+3.5)</td><td>68.6 (+3.3)</td><td>87.3 (+2.1)</td><td>84.0 (+2.1)</td><td>81.0 (+2.3)</td></tr></table>
+
+LiNeS already enhances both the ID and OOD performance of the fine-tuned models by a notable margin. Starting from this edited model and applying the WiSE-FT interpolation with the pre-trained weights leads to a Pareto Front (Caruana, 1997) that consistently dominates the one by WiSE-FT across all distribution shifts, illustrating the applicability of the proposed method across various distribution shifts. A granular result for applying LiNeS to each of the 70 checkpoints is provided in Appendix C.10.2, further highlighting its universal effectiveness across models. We also report similar findings in Figure 11 in Appendix for a CLIP ViT-B/16 checkpoint fine-tuned on ImageNet, using the same hyper-parameters as Wortsman et al. (2022b).
+
+# 5.2 IMPROVING MULTI-TASK MODEL MERGING
+
+In this section, we extend LiNeS to improve multi-task merging algorithms, aiming to combine multiple models fine-tuned independently on different tasks into a single model (Matena & Raffel, 2022; Ilharco et al., 2023; Ortiz-Jimenez et al., 2023; Yadav et al., 2023; Hazimeh et al., 2024). Task arithmetic (Ilharco et al., 2023) proposed to decouple the contributions of the pre-trained model and individual task vectors, first generating a multi-task vector  $\pmb{\tau}_{\mathrm{MTL}} = g(\pmb{\tau}_1,\dots,\pmb{\tau}_T)$  with a merging function  $g:\mathbb{R}^N\times \dots \times \mathbb{R}^N\mapsto \mathbb{R}^N$ , and then adding back to the pre-trained checkpoint with a scaling factor to construct a multi-task model  $\pmb{\theta} = \pmb{\theta}_0 + \lambda \cdot \pmb{\tau}_{\mathrm{MTL}}$ . The scalar coefficient  $\lambda$  is tuned using a held-out validation set. Recent works (Yadav et al., 2023; Wang et al., 2024) follow the same protocol while improving the merging function  $g$  for retaining more task information. We refer to Appendix B.1 for a more detailed explanation of these methods.
+
+However, significant performance loss occurs between the merged multi-task model and the original fine-tuned checkpoints. This performance decrease partially stems from interference (Yadav et al., 2023; Wang et al., 2024) among task vectors, where the contribution of one task negatively impacts performance on others, leading to overall degradation. Task interference is linked to catastrophic forgetting, as the individual task vectors lose a significant amount of generalization ability to other tasks after fine-tuning and merging them leads to interference among each other. Therefore, we can edit each task vector with LiNeS before merging to restore the generalization to other tasks, or for simplicity, edit directly the merged multi-task vector to preserve the shallow and general features that are beneficial across tasks.
+
+We enhance the merging methods by applying LiNeS on the merged multi-task vector  $\tau_{\mathrm{MTL}}$ . For the linear scaling schedule, we tune only  $\beta$  and set  $\alpha$  using a heuristic that adjusts based on both the number of merged models and the merging method. Specifically, for task arithmetic which aggregates the individual task vectors through a simple summation operation:  $\tau_{\mathrm{sum}} = \sum_{i=1}^{N_{\mathrm{models}}} \tau_i$ , we set  $\alpha = 1 / N_{\mathrm{models}}$ . For other merging strategies which result in  $\tau_{\mathrm{MTL}}$  with different magnitudes of norms, e.g., aggregation with summation leads to a norm  $\times N_{\mathrm{models}}$  larger compared to averaging, we further multiply by a scaling term  $\|\tau_{\mathrm{sum}}\| / \|\tau_{\mathrm{MTL}}\|$  to normalize their norm to simple summation. Overall, we set the intercept  $\alpha$  to:
+
+$$
+\alpha = \frac {1}{N _ {\text {m o d e l s}}} \frac {\left\| \boldsymbol {\tau} _ {\text {s u m}} \right\|}{\left\| \boldsymbol {\tau} _ {\text {M T L}} \right\|}, \text {w h e r e} \boldsymbol {\tau} _ {\text {s u m}} = \sum_ {i = 1} ^ {N _ {\text {m o d e l s}}} \boldsymbol {\tau} _ {i} \tag {2}
+$$
+
+Table 3: Results for multi-task model merging methods in three NLP benchmarks with T5-large model. LiNeS improves baseline performance across merging methods and benchmarks.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">with LiNeS</td><td colspan="3">T5-large (Lester et al., 2021)</td></tr><tr><td>7 NLP tasks (Yadav et al., 2023)</td><td>8 QA tasks (Zhou et al., 2022)</td><td>11 NLP tasks (Wang et al., 2024)</td></tr><tr><td>Zero-shot</td><td></td><td>44.9</td><td>33.1</td><td>36.9</td></tr><tr><td>Fine-tuned</td><td></td><td>85.9</td><td>80.7</td><td>78.7</td></tr><tr><td rowspan="2">Task Arithmetic</td><td>X</td><td>71.9</td><td>63.8</td><td>63.6</td></tr><tr><td>✓</td><td>76.4 (+4.5)</td><td>67.6 (+3.8)</td><td>66.2 (+2.6)</td></tr><tr><td rowspan="2">Ties-Merging</td><td>X</td><td>71.6</td><td>63.0</td><td>64.0</td></tr><tr><td>✓</td><td>72.0 (+0.4)</td><td>66.0 (+3.0)</td><td>66.4 (+2.4)</td></tr><tr><td rowspan="2">Consensus Merging</td><td>X</td><td>73.5</td><td>68.6</td><td>67.5</td></tr><tr><td>✓</td><td>75.4 (+1.9)</td><td>69.3 (+0.7)</td><td>67.5 (+0.0)</td></tr></table>
+
+Therefore, we only tune  $\beta$  and search over the same range as the constant scaling  $\lambda$  used by the aforementioned merging techniques. As a result, LiNeS shares the same computational requirements as the baseline merging methods; we provide more details for the hyper-parameters in Appendix B.2.1, as well as a sensitivity analysis to the hyper-parameters in Appendix C.6. Specifically, we consider various multi-task model merging baselines, namely Task Arithmetic (Ilharco et al., 2023), Ties-merging (Yadav et al., 2023), Consensus Merging (Wang et al., 2024), enhancing them with LiNeS and evaluate on both computer vision and NLP benchmarks.
+
+# 5.2.1 COMPUTER VISION
+
+We experiment with the 8-task image classification benchmark proposed by Ilharco et al. (2023), as well as the more challenging 14-task and 20-task benchmarks from Wang et al. (2024). Detailed descriptions of task composition appear in Appendix B.2.2. We also examine the efficacy of LiNeS across the model scale axis, studying three vision transformer (Dosovitskiy et al., 2021), namely ViT-B/32, ViT-B/16 and ViT-L/14, as CLIP visual encoders (Radford et al., 2021).
+
+Table 2 presents the results for ViT-B/32 and ViT-L/14, while Appendix C.4 contains the ViT-B/16 experiments. We observe that LiNeS provides a significant improvement to all baseline merging methods across all tested scenarios, regardless of model sizes and total number of tasks. For example, for the 8-task benchmark with ViT-B/32, LiNeS improves task arithmetic by  $4.5\%$ , Ties-merging by  $3.6\%$  and consensus merging by  $3.1\%$ . For the challenging 20-task benchmark with ViT-L/14, LiNeS leads to consistent and significant improvements, improving task arithmetic by  $3.1\%$ , Ties-merging by  $4.0\%$  and consensus merging by  $2.3\%$ . The detailed performance on individual tasks for each tested scenario is presented in Appendix C.11.
+
+# 5.2.2 NATURAL LANGUAGE PROCESSING
+
+We also evaluate the effectiveness of LiNeS in NLP domain, including a 7-task NLP benchmark (Yadav et al., 2023), an 8-task Question-Answering benchmark (Zhou et al., 2022), and their combined 11-task benchmark (Wang et al., 2024). Appendix B details the experimental settings. Following Tam et al. (2024), we adopt a variant of T5-large model (Raffel et al., 2020), namely T5-large-LM-Adapt (Lester et al., 2021), and use their provided checkpoints. While T5-large contains both encoder and decoder networks, we apply LiNeS only to the decoder, as our findings in Appendix C.5 indicate that applying the edition to the decoder leads to similar observations to vision.
+
+The performance of applying LiNeS to baseline methods with T5-large across various NLP tasks is summarized in Table 3. LiNeS consistently improves multi-task performance across baseline merging methods and benchmarks with a notable margin. For example, on the 7 NLP tasks benchmark, LiNeS improves task arithmetic by 4.5 points, and consensus merging by 1.9 points. Meanwhile, LiNeS outperforms Ties-merging by  $3.0\%$  and  $2.4\%$  for the 8-QA benchmark and 11-NLP benchmark, respectively.
+
+# 5.3 IMPROVING MODEL SOUPS FOR MERGING SINGLE-TASK MODELS
+
+Averaging in weight space multiple models fine-tuned on the same task derived from the same pre-trained model has been shown to increase target performance (Wortsman et al., 2022a; Ramé
+
+Table 4: LiNeS improves performance over Model Soups (Wortsman et al., 2022a), for both uniform and greedy soup in merging multiple checkpoints fine-tuned on ImageNet with different hyper-parameter configurations.  
+
+<table><tr><td>Method</td><td>Enhancements</td><td>ImageNet Acc.</td></tr><tr><td>Averaged accuracy</td><td>/</td><td>77.98</td></tr><tr><td>Best individual model</td><td>/</td><td>80.36</td></tr><tr><td rowspan="3">Uniform soup</td><td>/</td><td>79.99</td></tr><tr><td>Task Arithmetic</td><td>80.17</td></tr><tr><td>LiNeS</td><td>80.47 (+0.48)</td></tr><tr><td rowspan="3">Greedy soup</td><td>/</td><td>81.01</td></tr><tr><td>Task Arithmetic</td><td>81.01</td></tr><tr><td>LiNeS</td><td>81.16 (+0.15)</td></tr></table>
+
+![](images/b488b85c94213f8a1aec47e1a352c18aa0374b47d8b843e6b6d7a773a11b6322.jpg)  
+Figure 4: Applying LiNeS to Rewarded Soups (Ramé et al., 2023) improves merging of LLM policies RL fine-tuned on different rewards with a dominating Pareto Front.
+
+et al., 2022). In this section, we investigate whether LiNeS can enhance the test performance when merging single-task models.
+
+We follow the setting in Model Soups (Wortsman et al., 2022a) and merge 70 CLIP ViT-B/32 checkpoints fine-tuned on ImageNet (Deng et al., 2009) using different hyper-parameters, plus the pre-trained checkpoint. We consider both variants introduced in Wortsman et al. (2022a), namely uniform and greedy soup. We refer to Appendix B.3 for details regarding these methods and experimental settings. For both cases, the weight-averaging process can be decomposed as follows:
+
+$$
+\boldsymbol {\theta} _ {\text {s o u p}} = \boldsymbol {\theta} _ {0} + \boldsymbol {\tau} _ {\text {s o u p}}, \text {w h e r e} \boldsymbol {\tau} _ {\text {s o u p}} = \frac {1}{N _ {\text {m o d e l s}}} \sum_ {i = 1} ^ {N _ {\text {m o d e l s}}} \left(\boldsymbol {\theta} _ {i} - \boldsymbol {\theta} _ {0}\right) \tag {3}
+$$
+
+We apply LiNeS to  $\tau_{\mathrm{soup}}$  fixing  $\alpha = 1$  and searching over  $\beta$ . As a baseline, we also consider task arithmetic, where we search for a constant scaling factor on  $\tau_{\mathrm{soup}}$ . Note that both settings introduce one hyper-parameter to vanilla model soups, and refer to Appendix B.3 for a detailed description of the modifications. Table 4 summarizes the results and shows that LiNeS improves over vanilla soups and task arithmetic for both uniform and greedy soup by  $0.48\%$  and  $0.15\%$  on ImageNet, respectively. We report the best-performing model and the average performance as baselines. Finally, our proposed method compounds the gains from the greedy soup and leads to the best-performing model.
+
+# 5.4 IMPROVING REWARDED SOUPS
+
+In this section, we explore the effectiveness of LiNeS for merging foundation models fine-tuned on different rewards. We consider the Rewarded Soups setting (Ramé et al., 2023), which interpolates the weights  $\theta_{1}$  and  $\theta_{2}$  of two LLM policies, each optimized for a distinct reward  $R_{1}$  and  $R_{2}$ , respectively.
+
+Starting with an LLM parameterized by weights  $\theta_0$ , we first fine-tune it using supervised fine-tuning (SFT) on labeled demonstrations. From the resulting weights  $\theta_{\mathrm{SFT}}$ , we then apply Reinforcement Learning from Human Feedback (RLHF) (Christiano et al., 2017; Ouyang et al., 2022), training two independent policies via Proximal Policy Optimization (PPO) (Schulman et al., 2017) to maximize the rewards  $R_{1}$  and  $R_{2}$  respectively. To merge these policies, we linearly interpolate the residuals  $\tau_{1} = \theta_{1} - \theta_{\mathrm{SFT}}$  and  $\tau_{2} = \theta_{2} - \theta_{\mathrm{SFT}}$ , defining a continuous set of rewarded policies:
+
+$$
+\boldsymbol {\theta} _ {R S} = \boldsymbol {\theta} _ {\mathrm {S F T}} + \lambda \boldsymbol {\tau} _ {1} + (1 - \lambda) \boldsymbol {\tau} _ {2}, \quad \lambda \in [ 0, 1 ], \tag {4}
+$$
+
+where the coefficient  $\lambda$  models the user's preferences. We apply LiNeS to the weighted-sum residual:  $\lambda \pmb{\tau}_{1} + (1 - \lambda)\pmb{\tau}_{2}$ , fixing  $\alpha = \beta = 1$  for computational reasons.
+
+In our experiment, we use LLaMA-2 7B (Touvron et al., 2023) and the Reddit Summary task (Stiennon et al., 2020), which consists of  $14.9\mathrm{k}$  post-summary pairs. We fine-tune the model using LoRA (Hu et al., 2022) with  $r_{\mathrm{LoRA}} = 64$ ,  $\alpha_{\mathrm{LoRA}} = 128$ , and 0.05 dropout. We employ two reward models:
+
+GPT2-reward-summarization – which scores summaries based on human preferences – and BART-faithful-summary-detector (Chen et al., 2021) – which evaluates the faithfulness of the generated summary to the source post. To evaluate the models, we use a subset of 1k samples from the test set, generate the responses, and compute the average score for each reward dimension.
+
+In Table 4, we present the empirical Pareto Fronts for both Rewarded Soups and Rewarded Soups+LiNeS. LiNeS consistently outperforms the vanilla Rewarded Soups across the full preference space, Pareto dominating the baseline. This result highlights the generality of LiNeS.
+
+# 6 DISCUSSION
+
+We compare LiNeS with prior work that optimizes the scaling coefficients via backpropagation. Specifically, Ada-merging (Yang et al., 2024) minimizes the entropy loss of the predictions on the test set, while aTLAS (Zhang et al., 2024) minimizes a cross entropy loss on validation samples. Both methods operate on a more fine-grained level and introduce coefficients per layer and per task, requiring all  $T + 1$  checkpoints, for task vectors and pre-trained model respectively, to be stored in memory during their finetuning process.
+
+We consider the 8-task computer vision benchmark with the ViT-B/32 visual encoder, and present the per-layer scalings in Figure 5. For aTLAS and Ada-merging, we report the average optimized scaling coefficients for attention and linear layers in each block
+
+across tasks. Without requiring training, LiNeS leverages the inductive bias of neural networks to achieve scaling very close to Ada-merging or aTLAS, but with much less computational cost. Apart from the excessive memory overhead, both aTLAS and Ada-merging require multiple training epochs, making it challenging to scale for large models. As we demonstrate in Section 5.4, LiNeS efficiently scales to large models like LLaMA (Touvron et al., 2023).
+
+![](images/4bd7eb8ee983b30f4eff63ef0ef21709d644eae65710de501b22d2e16a3d1747.jpg)  
+Figure 5: Comparison of the scalings obtained by different methods on 8-task merging benchmark with CLIP ViT-B/32.
+
+# 7 CONCLUSION
+
+In this work, we presented LiNeS, a novel method designed to mitigate catastrophic forgetting after fine-tuning process. By reducing the magnitude for parameter updates in the shallower layers, LiNeS improves the generalization performance of the edited model on control tasks while almost fully preserving performance on the fine-tuned tasks. Furthermore, we demonstrated the versatility of LiNeS in addressing task interference in multi-task model merging, where it consistently improves the baseline model merging methods across vision and NLP benchmarks. Our experiments confirm the broad applicability of LiNeS across various scenarios, from improving OOD generalization to enhancing multi-task and single-task model merging strategies, as well as improving merging LLM policies aligned with different rewards. Given its simplicity and ease of integration with existing methods, LiNeS offers a practical and inexpensive solution for boosting the generalization and robustness of fine-tuned models in diverse application domains.
+
+# ACKNOWLEDGMENTS
+
+The authors thank Adam Hazimeh and the anonymous reviewers for their constructive discussions and comments.
+
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+
+# Appendix
+
+# Table of Contents
+
+A LiNeS Pseudocode 20
+
+B Experimental Details 21
+
+B.1 Descriptions of baseline model merging methods 21  
+B.2 Experimental details for multi-task model merging 21  
+B.3 Experimental details for single-task model merging 22
+
+C Additional Results 23
+
+C.1 Different trade-offs for retention of task and control task performance 23  
+C.2 Detailed labels for Figure 2 23  
+C.3 Ablations of different choices of scaling function 23  
+C.4 Results for ViT-B/16 for multi-task merging 24  
+C.5 Results for editing T5 with LiNeS 25  
+C.6 Sensitivity analysis for hyper-parameters 25  
+C.7 Experiments with CNN architectures 26  
+C.8 Experiments with Regularized Fine-Tuning 26  
+C.9 Experiments with LoRA Fine-tuning 27  
+C.10 Additional results for improving WiSE-FT with LiNeS 27  
+C.11 Detailed performance on individual tasks for multi-task model merging 28
+
+# A LINESPSEUDOCODE
+
+We provide here a python pseudocode for the scaling the task vectors.
+
+```python
+def line_scaling(task_vector, alpha=0.0, beta=1.0, num_blocks=12):
+    ""
+    Progressively scales the task vector based on layer depth.
+Parameters:
+    task_vector: dict
+        A dictionaryves control performantween the fine-tuned checkpoint and the pre-trained checkpoint.
+alpha: float
+        The minimum scaling factor for the blocks.
+beta: float
+        The maximum scaling coefficient difference between the last and first block.
+num_blocks: int
+        The total number of layer blocks in the model.
+Returns:
+        scale_task_vector: dict
+        A copy of 'task_vector' where each key is scaled based on the layer depth.
+    ""
+import copy
+# Deep copy the task vector to avoid modifying the original scaled_task_vector = copy.deepcopy(task_vector)
+# Generate the key blocks corresponding to the layers of the model key_blocks = [f".layer{i}." for i in range(num_blocks)]
+# Create a scaling dictionary to store the scaling factor for each key scalingDIC = {}
+for k in task_vector.keys():
+    # Find the layer block in the key and assign scaling factor based on layer depth
+    for layer, block in enumerate(key_blocks):
+        if block in k:
+            scalingDIC[k] = alpha + beta * (layer / (num_blocks - 1))
+        break
+# Scale the task vector based on the scaling dictionary scaled_task_vector/vector = {
+    # Use alpha if layer is outside residual blocks
+    k: task_vector.element[k] * scalingDIC.get(k, alpha)
+    for k in task_vector_keys()
+} return scaled_task_vector
+# example: scale single-task fine-tuned residual task_vector = {k: theta_t[k] - theta_0[k] for k in theta_0_keys())
+scaled_task_vector = line_scaling(task_vector, alpha=gamma, beta=1.0 - gamma, num_blocks=12)
+# example: Scale the multi-task vectors mtv = {
+    k: sum(theta_ft[k] - theta_0[k] for theta_ft in ft_models)
+    for k in theta_0_keys()
+} scaled_mtv = line_scaling(mtv, alpha=1 / len(ft_models), beta=beta, num_blocks=12)
+```
+
+# B EXPERIMENTAL DETAILS
+
+# B.1 DESCRIPTIONS OF BASELINE MODEL MERGING METHODS
+
+- Task Arithmetic (Ilharco et al., 2023) generates a multi-task vector by summing the individual task vectors for each task. This multi-task vector is then added to the pre-trained checkpoint, with a scaling factor chosen based on validation set performance.  
+- Ties-Merging (Yadav et al., 2023) resolves parameter conflicts during model merging by first pruning parameters with lower magnitudes from the individual task vectors, followed by addressing sign mismatches, and finally merging parameters with consistent signs with averaging operation. The resulting multi-task vector is then added to the pre-trained checkpoint using a scaling factor determined from the validation set.  
+- Consensus Merging (Wang et al., 2024) enhances existing model merging techniques by eliminating redundant weights in the multi-task vector. It first identifies the relevant subset of parameters for each task, then filters out weights that are relevant to either none or only one task. After removing these redundant weights, the refined multi-task vector is added to the pre-trained checkpoint with a scaling factor selected from the validation set.
+
+While consensus merging can be applied to various merging methods, in all our experiments, we evaluate only its application to task arithmetic.
+
+# B.2 EXPERIMENTAL DETAILS FOR MULTI-TASK MODEL MERGING
+
+# B.2.1 HYPER-PARAMETERS TUNING
+
+We list here the hyper-parameter search space for each model merging method in Table B.2.1, while we suggest the authors to the original papers for a detailed description of these hyper-parameters. We highlight that applying LiNeS does not introduce extra computational cost in hyper-parameter search for the baseline merging methods.
+
+<table><tr><td>Method</td><td>With LiNeS</td><td>Hyper-parameter search space</td></tr><tr><td rowspan="2">Task Arithmetic</td><td>X</td><td>constant scaling term for multi-task vector: [0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0]</td></tr><tr><td>✓</td><td>scaling term β in Eq. 1 for the multi-task vector: [0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0]</td></tr><tr><td rowspan="2">Ties-Merging</td><td>X</td><td>constant scaling term for multi-task vector: [0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0,1.1,1.2,1.3,1.4,1.5]</td></tr><tr><td>✓</td><td>scaling term β in Eq. 1 for the multi-task vector: [0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0,1.1,1.2,1.3,1.4,1.5]</td></tr><tr><td rowspan="2">Consensus Merging</td><td>X</td><td>constant scaling term for multi-task vector: [0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0]; weight-pruning threshold: [1,2]</td></tr><tr><td>✓</td><td>scaling term β in Eq. 1 for the multi-task vector: [0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0]; weight-pruning threshold: [1,2]</td></tr></table>
+
+# B.2.2 BENCHMARKS
+
+Image classification For the benchmarks used in image classification, we utilized the 8-task benchmark initially proposed by Ilharco et al. (2023), as well as the 14-task and 20-task benchmarks expanded by Wang et al. (2024).
+
+- The 8-task benchmark comprises the following tasks: Cars (Krause et al., 2013), DTD (Cimpoi et al., 2014), EuroSAT (Helber et al., 2019), GTSRB (Stallkamp et al., 2011), MNIST (LeCun, 1998), RESISC45 (Cheng et al., 2017), SUN397 (Xiao et al., 2016), and SVHN (Netzer et al., 2011).  
+- The 14-task benchmark includes the original eight tasks plus additional ones: CIFAR100 (Krizhevsky & Hinton, 2009), STL10 (Coates et al., 2011), Flowers102 (Nilsback & Zisserman, 2008), OxfordIIITPet (Parkhi et al., 2012), PCAM (Veeling et al., 2018), and FER2013 (Goodfellow et al., 2013).  
+- The 20-task benchmark builds on the 14-task benchmark with the addition of: EMNIST (Cohen et al., 2017), CIFAR10 (Krizhevsky & Hinton, 2009), Food101 (Bossard et al., 2014), FashionMNIST (Xiao et al., 2017), RenderedSST2 (Socher et al., 2013; Radford et al., 2019), and KMNIST (Clanuwat et al., 2018).
+
+Natural Language Processing For our NLP experiments, we utilized benchmarks established by Yadav et al. (2023), Tam et al. (2024), and Wang et al. (2024).
+
+- The 7 NLP Tasks benchmark, as explored in Yadav et al. (2023), includes the following datasets: QASC (Khot et al., 2020), QuaRTz (Tafjord et al., 2019), PAWS (Zhang et al., 2019), Story Cloze (Sharma et al., 2018), WikiQA (Yang et al., 2015), Winogrande (Sakaguchi et al., 2021), and WSC (Levesque et al., 2012).  
+- The 8 QA Tasks (Tam et al., 2024) comprises the following datasets: CosmosQA Huang et al. (2019), QASC (Khot et al., 2020), QuAIL Rogers et al. (2020), QuaRTz (Tafjord et al., 2019), PAWS (Zhang et al., 2019), ROPES Lin et al. (2019), SocialIQA Sap et al. (2019), and WikiQA (Yang et al., 2015).  
+- The 11 NLP Tasks benchmark is a union of these two benchmarks, as studied in Wang et al. (2024). It contains the following tasks: QASC (Khot et al., 2020), QuaRTz (Tafjord et al., 2019), PAWS (Zhang et al., 2019), Story Cloze (Sharma et al., 2018), WikiQA (Yang et al., 2015), Winogrande (Sakaguchi et al., 2021), WSC (Levesque et al., 2012), CosmosQA Huang et al. (2019), QuAIL Rogers et al. (2020), ROPES Lin et al. (2019), and SocialIQA Sap et al. (2019).
+
+# B.3 EXPERIMENTAL DETAILS FOR SINGLE-TASK MODEL MERGING
+
+# B.3.1 DESCRIPTION OF MODEL SOUPS
+
+Model soups (Wortsman et al., 2022a) is a model merging method which averages the weights of multiple fine-tuned models with different hyper-parameter configurations, improving accuracy of the merged model without increasing inference or memory costs. The authors of model soups proposed two methods:
+
+- Uniform soup: Averages the weights of all fine-tuned checkpoints, providing a simple and efficient way to improve performance.  
+- Greedy soup: Starting with the best-performing checkpoint, greedily and iteratively adds the next best-performing checkpoint to the soup, keeping those that improve accuracy of current collection of model checkpoints.
+
+# B.3.2 EXPERIMENTAL DETAILS FOR MODIFICATIONS TO MODEL SOUPS
+
+We describe in detail the modifications to model soups, namely task arithmetic and our proposed LiNeS. For reference, model soups merges the checkpoints by averaging the weights of the individual checkpoints:
+
+$$
+\boldsymbol {\theta} _ {\text {s o u p}} ^ {\text {v a n i l l a}} = \boldsymbol {\theta} _ {0} + \boldsymbol {\tau} _ {\text {s o u p}} \tag {5}
+$$
+
+Enhancing Model Soups with Task Arithmetic We enhance model soups with task arithmetic, by introducing a scaling factor  $\lambda_{\mathrm{ta}}$  to  $\tau_{\mathrm{soup}}$  in Equation 5. We search for this hyper-parameter within the range of [1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0]. Note that  $\lambda_{\mathrm{ta}} = 1.0$  yields the vanilla model soups.
+
+$$
+\boldsymbol {\theta} _ {\text {s o u p}} ^ {\mathrm {t a}} = \boldsymbol {\theta} _ {0} + \lambda_ {\mathrm {t a}} \cdot \boldsymbol {\tau} _ {\text {s o u p}} \tag {6}
+$$
+
+Enhancing Model Soups with LiNeS We enhance model soups with LiNeS, by applying LiNeS to  $\pmb{\tau}_{\mathrm{soup}}$  in Equation 5. For the scaling, we apply directly the scaling introduced in Equation 1 to create a scaled task vector  $\pmb{\tau}_{\mathrm{soup}}^{\mathrm{LiNeS}}$ , fixing  $\alpha$  to 1 while searching the value for  $\beta$  within the range of [0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0]. Note that  $\beta = 0.0$  yields the vanilla model soups.
+
+$$
+\boldsymbol {\theta} _ {\text {s o u p}} ^ {\text {L i N e S}} = \boldsymbol {\theta} _ {0} + \boldsymbol {\tau} _ {\text {s o u p}} ^ {\text {L i N e S}} \tag {7}
+$$
+
+We further note that, both Task Arithmetic and our proposed method introduce only one hyperparameter to model soups, while the computational cost for hyper-parameter search is the same.
+
+When applying the modifications to greedy soup, we only apply them directly to the selected subset of checkpoints after the greedy selection process.
+
+We search for the hyper-parameter within the validation set and report the performance on test set with the best hyper-parameter based on validation performance.
+
+# C ADDITIONAL RESULTS
+
+# C.1 DIFFERENT TRADE-OFFS FOR RETENTION OF TASK AND CONTROL TASK PERFORMANCE
+
+In Section 3, we need to balance two competing objectives: maximizing accuracy on the target task while preserving performance on the control task. To account for different user preferences, we scalarize these objectives by assigning varying weights to the target task accuracy. This weighting scheme can be adjusted depending on the scenario to reflect different priorities. Let  $w_{\mathrm{target}}$  represent the weight assigned to the target task accuracy, and  $\mathbf{M}_{\mathrm{target}}$  and  $\mathbf{M}_{\mathrm{control}}$  denote the normalized accuracies for the target and control tasks, respectively. The optimal value of  $\gamma$  is selected to maximize the following weighted trade-off on the validation set:
+
+$$
+w _ {\text {t a r g e t}} \mathbf {M} _ {\text {t a r g e t}} + \mathbf {M} _ {\text {c o n t r o l}}
+$$
+
+To account for the high variance in control task performance and to emphasize the target task, we assign it a weight of 2, signifying that its accuracy is prioritized twice as much as the control task's accuracy.
+
+Table 5: Validation results on the target vs control performance benchmark, presented in Section 3, averaged over the 8 tasks. We balance two competing objectives with various scalarization weights  $w_{\text{target}}$ . In the main text, we use  $w_{\text{target}} = 2$ .  
+
+<table><tr><td rowspan="2">w_target</td><td colspan="2">Averaged normalized accuracy (%)</td></tr><tr><td>Target task</td><td>Control tasks</td></tr><tr><td>1</td><td>99.8</td><td>101.9</td></tr><tr><td>2</td><td>100.0</td><td>101.5</td></tr><tr><td>5</td><td>100.2</td><td>100.8</td></tr></table>
+
+# C.2 DETAILED LABELS FOR FIGURE 2
+
+We provide in Figure 6 the detailed labels corresponding to each scatter dot. Each scatter dot corresponds to applying a specific model (FT for fine-tuned model; PT for pre-trained model; LS for fine-tuned model edited with LiNeS) on different tasks.
+
+# C.3 ABLATIONS OF DIFFERENT CHOICES OF SCALING FUNCTION
+
+We provide in this section an ablation study on applying different scaling functions for LiNeS. In LiNeS we used directly  $\lambda^{(\ell)} = \alpha +\beta \cdot \frac{\ell - 1}{L - 1}$  to scale different layers. Here we test the performance on multi-task model merging in vision benchmarks with the following choices for scaling functions  $f(\cdot)$ : linear scaling, quadratic scaling and square root scaling:
+
+- linear scaling:  $\lambda^{(\ell)} = \alpha +\beta \cdot \frac{\ell - 1}{L - 1}$  
+square root scaling:  $\lambda^{(\ell)} = \alpha +\beta \cdot \left(\frac{\ell - 1}{L - 1}\right)^{\frac{1}{2}}$  
+quadratic scaling:  $\lambda^{(\ell)} = \alpha +\beta \cdot \left(\frac{\ell - 1}{L - 1}\right)^{2}$
+
+We provide in Table C.3 the performance of different choices of scaling on vision benchmarks with ViT-B/32. While using quadratic scaling sometimes outperforms using identify function, especially with a larger number of tasks during merging, the improvement is not substantial. Therefore, we choose the linear scaling to keep the method simple and general.
+
+![](images/348a5fd6343b21661d53b3d803c94facf31cb6f9efbadf37987fbce4645cdbbc.jpg)  
+Figure 6: Figure 2 with detailed label information. Each scatter dot corresponds to applying a specific model (FT for fine-tuned model; PT for pre-trained model; LS for fine-tuned model edited with LiNeS) on different task.
+
+<table><tr><td>FT-Cars</td><td>PT-Cars</td><td>LS-Cars</td></tr><tr><td>FT-DTD</td><td>PT-DTD</td><td>LS-DTD</td></tr><tr><td>FT-EuroSAT</td><td>PT-EuroSAT</td><td>LS-EuroSAT</td></tr><tr><td>FT-GTSRB</td><td>PT-GTSRB</td><td>LS-GTSRB</td></tr><tr><td>FT-MNIST</td><td>PT-MNIST</td><td>LS-MNIST</td></tr><tr><td>FT-RESISC45</td><td>PT-RESISC45</td><td>LS-RESISC45</td></tr><tr><td>FT-SVHN</td><td>PT-SVHN</td><td>LS-SVHN</td></tr><tr><td>FT-SUN397</td><td>PT-SUN397</td><td>LS-SUN397</td></tr><tr><td>FT-STL10</td><td>PT-STL10</td><td>LS-OxfordIIITPet</td></tr><tr><td>FT-OxfordIIITPet</td><td>PT-OxfordIIITPet</td><td>LS-Flowers102</td></tr><tr><td>FT-Flowers102</td><td>PT-Flowers102</td><td>LS-CIFAR100</td></tr><tr><td>FT-CIFAR100</td><td>PT-CIFAR100</td><td>LS-PCAM</td></tr><tr><td>FT-PCAM</td><td>PT-PCAM</td><td>LS-FER2013</td></tr><tr><td>FT-FER2013</td><td>PT-FER2013</td><td>LS-Food101</td></tr><tr><td>FT-CIFAR10</td><td>PT-CIFAR10</td><td>LS-FashionMNIST</td></tr><tr><td>FT-Food101</td><td>PT-Food101</td><td>LS-RenderedSST2</td></tr><tr><td>FT-FashionMNIST</td><td>PT-FashionMNIST</td><td>LS-EMNIST</td></tr><tr><td>FT-RenderedSST2</td><td>PT-RenderedSST2</td><td>LS-KMNIST</td></tr><tr><td>FT-EMNIST</td><td>PT-EMNIST</td><td>LS-STL10</td></tr><tr><td>FT-KMNIST</td><td>PT-KMNIST</td><td>LS-CIFAR10</td></tr></table>
+
+Table 6: Ablation study for applying different scalings for LiNeS on vision benchmarks with ViT-B/32.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Scaling function</td><td colspan="3">ViT-B/32</td></tr><tr><td>8 tasks</td><td>14 tasks</td><td>20 tasks</td></tr><tr><td rowspan="3">Task Arithmetic</td><td>linear</td><td>74.2</td><td>69.1</td><td>63.4</td></tr><tr><td>square root</td><td>73.9</td><td>67.5</td><td>62.4</td></tr><tr><td>quadratic</td><td>73.8</td><td>69.2</td><td>64.6</td></tr><tr><td rowspan="3">Ties-Merging</td><td>linear</td><td>77.2</td><td>72.1</td><td>67.2</td></tr><tr><td>square root</td><td>76.9</td><td>70.4</td><td>65.6</td></tr><tr><td>quadratic</td><td>76.1</td><td>71.6</td><td>67.4</td></tr><tr><td rowspan="3">Consensus Merging</td><td>linear</td><td>77.6</td><td>73.6</td><td>68.6</td></tr><tr><td>square root</td><td>77.1</td><td>72.5</td><td>67.3</td></tr><tr><td>quadratic</td><td>77.1</td><td>73.9</td><td>69.0</td></tr></table>
+
+# C.4 RESULTS FOR VIT-B/16 FOR MULTI-TASK MERGING
+
+We provide in Table C.4 the results complementary to Table 2 for using ViT-B/16 as the image encoder, where we observe similar performance gains and observations by using LiNeS as in Table 2.
+
+Table 7: Complementary to Table 2, for results obtained with ViT-B/16 as image encoder.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">with LiNeS</td><td colspan="3">ViT-B/16</td></tr><tr><td>8 tasks</td><td>14 tasks</td><td>20 tasks</td></tr><tr><td>Zero-shot</td><td></td><td>55.5</td><td>61.4</td><td>59.8</td></tr><tr><td>Fine-tuned</td><td></td><td>92.6</td><td>91.6</td><td>92.3</td></tr><tr><td rowspan="2">Task Arithmetic</td><td>X</td><td>74.6</td><td>70.4</td><td>65.7</td></tr><tr><td>✓</td><td>77.6 (+3.0)</td><td>72.7 (+2.3)</td><td>67.7 (+2.0)</td></tr><tr><td rowspan="2">Ties-Merging</td><td>X</td><td>79.1</td><td>73</td><td>68.1</td></tr><tr><td>✓</td><td>79.9 (+0.8)</td><td>75.2 (+2.2)</td><td>71.2 (+3.1)</td></tr><tr><td rowspan="2">Consensus Merging</td><td>X</td><td>78.9</td><td>73.9</td><td>70.2</td></tr><tr><td>✓</td><td>79.5 (+0.6)</td><td>75.8 (+1.9)</td><td>72.0 (+1.8)</td></tr></table>
+
+# C.5 RESULTS FOR EDITING T5 WITH LINEs
+
+We repeat here similar experiments we performed on ViT-B/32 model in Section 3 with T5-large (Raffel et al., 2020). T5-large contains both encoder and decoder structure, with sequential residual blocks in both structures. We investigate separately how the shallow-layer updates in the encoder and decoder infect the target and control task accuracy.
+
+We consider the 8-question-answering benchmark (Zhou et al., 2022), and plot in Figure 7 the averaged target and control task accuracy after applying LiNeS to
+
+1. only the decoder part (left),  
+2. only the encoder part (middle),  
+3. both the encoder and decoder part (right).
+
+![](images/97b16c05e6f84293b53339d770535e5d0463b27427d40cae8cb4da4177e1a250.jpg)  
+Figure 7: The impact of downscaling the shallower-layer parameter updates on T5-large model within the 8-question-answering benchmark (Raffel et al., 2020). Downscaling only on the decoder (left) architecture preserves full target performance, while slightly improving the control tasks performance. Downscaling on the encoder leads to performance degradation on target tasks.
+
+![](images/9b3e731ab4a0973a6db7105d32315cf3592bb9da1da57601051748025552cde9.jpg)
+
+![](images/c92e325cf80e0e5b5c8c0f2a0d6775c96c18060c963fdad2e8562167efd445d7.jpg)
+
+We observe that, only downscaling on the decoder architecture fully preserves full target performance, while downscaling on the encoder, or on both encoder and decoder, leads to performance drop on the target tasks. On the other hand, downscaling on the decoder part slight improves control generalization of the fine-tuned model, which we do not observe from downscaling on the encoder, or simultaneously on the encoder and decoder. We also note that, unlike the case in vision, the fine-tuned checkpoints on this NLP benchmark actually improve over the zero-shot performance of the pre-trained model on control tasks.
+
+These results motivate us to apply LiNeS to only the decoder part of T5-large when merging multiple checkpoints, which preserves full target task accuracy while slightly improving control task performance, leading to similar observation in applying LiNeS to the ViT-B/32 architecture in vision,
+
+# C.6 SENSITIVITY ANALYSIS FOR HYPER-PARAMETERS
+
+We provide in this section the sensitivity analysis for the hyper-parameters of LiNeS. Specifically, we consider the setting in multi-task merging in the 8-task vision classification benchmark with ViT-B/32 CLIP model.
+
+As explained in Section 5.2, LiNeS fixes  $\alpha$  with a heuristic value by Equation 2 and only tunes  $\beta$  for multi-task merging. The slope hyper-parameter  $\beta$  is tuned within the same range as the uniform scaling coefficient  $\lambda$  for the baseline merging methods. We compare in Figure 8 the sensitivity of averaged multi-task validation accuracy to the respective hyper-parameters, i.e., to  $\lambda$  for baseline merging methods and  $\beta$  for the LiNeS-enhanced merging methods. The results show that, for all three merging methods, including Task arithmetic, Ties-merging and Consensus, enhancing with LiNeS is less sensitive to hyper-parameter choices compared to the corresponding baseline method.
+
+Furthermore, we perform an ablation treating  $\alpha$  as a hyper-parameter and analyze the sensitivity to both  $\alpha$  and  $\beta$  for LiNeS in a two-dimensional grid for the same benchmark. The results are presented in Figure 9. The results clearly demonstrate the necessity for applying layer-increasing scaling, as the optimal performance is obtained with both  $\alpha > 0$  and  $\beta > 0$  for all three merging methods. Note that the optimal configurations found by the ablation study are very close to the configurations found in our method, as shown in Table 8, by setting  $\alpha$  via the heuristic and searching only for  $\beta$ .
+
+![](images/a88a95c058eb30bc659e7e1d894b4fa6142e4ba962d0f4a8cfa6b2d4bc194c69.jpg)  
+Figure 8: Sensitivity to hyper-parameters in multi-task merging for the 8-task benchmark with CLIP ViT-B/32 model. The y-axis represents the averaged multi-task validation accuracy and x-axis represents the hyper-parameter value, i.e.,  $\lambda$  for the baseline method and  $\beta$  for the method enhanced with LiNeS.
+
+![](images/d9a7c17913356261f50f5ade3276ff66e628856c35b1acdda518cbb63f7833b0.jpg)
+
+![](images/8bd217f4b47993ca6a0d10204dc038a47f3bbd23edb1e4f8123fd854f6b76896.jpg)
+
+![](images/dd978b180f0aa7c8ac35e82a46c58fccaf89e5b3ddb4ffd2d82ab5155f881e40.jpg)  
+Figure 9: Sensitivity to both  $\alpha$  and  $\beta$  in multi-task merging for the 8-task benchmark with CLIP ViT-B/32 model. The heatmap represents the averaged multi-task validation accuracy, while x and y axis represent the  $\beta$  and  $\alpha$  respectively. The optimal configuration is annotated with a red box.
+
+![](images/742ac6ff13aabdf32af05bd556e6f2cf95f9d2fecc4764dccee85aa518041ad2.jpg)
+
+![](images/e5b4556a95d79345c4a242021c416fb670f29949b9bb16dcc7495dbbb4d8abb9.jpg)
+
+# C.7 EXPERIMENTS WITH CNN ARCHITECTURES
+
+In this section, we apply LiNeS to CNN architectures. Specifically, we consider the ConvNeXt (Liu et al., 2022) architecture. First, we repeat the experiments presented in Section 3 regarding mitigating catastrophic forgetting. The final results are presented in Figure 10, where we observe similar findings with CLIP ViTs, i.e., LiNeS greatly improves the performance on control tasks when applied to the fine-tuned checkpoints while preserving most of the accuracy on target tasks. Furthermore, we present in Table 9 the results on multi-task model merging, following the experimental protocol established in Section 5.2. Again, we see that LiNeS improves the performance of baseline merging methods.
+
+# C.8 EXPERIMENTS WITH REGULARIZED FINE-TUNING
+
+In this section, we evaluate LiNeS against several regularized fine-tuning methods, focusing on their ability to preserve general features and mitigate catastrophic forgetting. The regularization strategies applied during fine-tuning are described below:
+
+1. Fine-tuning with Linear Layer-Wise Learning Rate Decay (LinLR): Applies a linear learning rate schedule where the learning rate linearly increases from 0.0 to the maximum value for all the layers.  
+2. Fine-tuning with Exponential Layer-Wise Learning Rate Decay (ExpLR): Applies an exponential learning rate schedule where the learning rate is set to maximum for the deepest layers and decays by a factor of 0.5 by each layer for the shallower layers.  
+3. Fine-tuning with First Half of Blocks Frozen (HalfFT): Freezes the parameters of the first half of the model's blocks during training.  
+4. Fine-tuning only the Final Block (LastFT): Freezes all blocks except the final block of the feature encoder.
+
+Table 8:  $\alpha$  and  $\beta$  values for  $\alpha$  set by our proposed heuristic in Equation 2.  
+
+<table><tr><td>Method</td><td>α</td><td>β</td></tr><tr><td>Task Arithmetic</td><td>0.125</td><td>0.3</td></tr><tr><td>Ties-merging</td><td>0.21</td><td>1.4</td></tr><tr><td>Consensus merging</td><td>0.21</td><td>0.9</td></tr></table>
+
+![](images/eee60c2c305c060be86dd7d8dab6354b6c419e79434463ef48a1d89b79535d84.jpg)  
+Figure 10: Our linear scaling (LiNeS) retains performance on both control and fine-tuned target tasks for ConvNext architecture.
+
+We present the results in Table 10. We observe that LiNeS, as a post-training editing method, outperforms the regularized fine-tuning methods in terms of restoring the zero-shot performance on the control tasks. We further emphasize that compared with the regularized fine-tuning methods, LiNeS benefits from many advantages such as efficiency, flexibility and computational cost.
+
+# C.9 EXPERIMENTS WITH LORA FINE-TUNING
+
+We explore the applicability of the method on models fine-tuned with LoRA (Hu et al., 2022). For a layer with pre-trained weights  $\mathbf{W}_0 \in \mathbb{R}^{m \times n}$ , LoRA adds trainable matrices  $\mathbf{A} \in \mathbb{R}^{m \times r}$  and  $\mathbf{B} \in \mathbb{R}^n$ , for rank  $r \ll \min(n, m)$ . The weights of the layer become:
+
+$$
+\boldsymbol {W} = \boldsymbol {W} _ {0} + \frac {\alpha}{r} \boldsymbol {B} \boldsymbol {A}
+$$
+
+for  $\alpha \in \mathbb{R}$ . Following common practice, we set  $\alpha = r$  and fine-tune with the same protocol used for full fine-tuning. We consider only the case of ViT-B/32 fine-tuned on 8 tasks and replicate the experiment presented in Table 1. Specifically, for each of the 8 LoRA-fine-tuned models, we compute the accuracy on the same (target) task as well as the average performance for each of the 7 remaining control tasks. Table 11 reports the average over the 8 cases for ranks  $r \in \{16, 32, 64, 128\}$ . We observe that LoRA fine-tuning has lower target performance compared to full fine-tuning and that increasing target performance comes at the cost of more forgetting. In all the cases, LiNeS restores control performance while minimally affecting target performance.
+
+# C.10 ADDITIONAL RESULTS FOR IMPROVING WISE-FT WITH LINE S
+
+# C.10.1 RESULTS FOR USING VIT-B/16 AS VISUAL ENCODER
+
+We provide in Figure 11 the results for applying LiNeS for improving WiSE-FT, using ViT-B/16 as visual encoder. The ViT-B/16 checkpoint obtained through fine-tuning the CLIP checkpoint on ImageNet with the same hyper-parameter configurations in Wortsman et al. (2022b). From Figure 11
+
+Table 9: Multi-task model merging results using a ConvNeXt architecture. LiNeS improves the results compared to uniform scaling for both Task Arithmetic and Ties-merging.  
+
+<table><tr><td>Method</td><td>LiNeS</td><td>Acc (%)</td><td>Norm. Acc (%)</td></tr><tr><td rowspan="2">Task Arithmetic</td><td>✘</td><td>77.9</td><td>83.8</td></tr><tr><td>✓</td><td>79.0 [+1.1]</td><td>84.8 [+1.0]</td></tr><tr><td rowspan="2">Ties-merging</td><td>✘</td><td>79.7</td><td>85.8</td></tr><tr><td>✓</td><td>80.3 [+0.6]</td><td>86.3 [+0.5]</td></tr></table>
+
+Table 10: Performance of different methods on target and control tasks, averaged over all target and control task combinations in the 8-task vision benchmark (Ilharco et al., 2023).  
+
+<table><tr><td></td><td>pre-trained</td><td>fine-tuned</td><td>FT+LiNeS</td><td>FT+LinLR</td><td>FT+ExpLR</td><td>FT+HalfFT</td><td>FT+LastFT</td></tr><tr><td>Target (%)</td><td>48.3</td><td>90.5</td><td>90.3</td><td>90.7</td><td>89.6</td><td>90.4</td><td>85.6</td></tr><tr><td>Control (%)</td><td>48.3</td><td>38.0</td><td>48.0</td><td>46.9</td><td>46.0</td><td>46.8</td><td>46.6</td></tr></table>
+
+we observe that LiNeS improves over WiSE-FT for both ID and OOD accuracies, leading to similar observations as the results obtained with ViT-B/32.
+
+![](images/e703e3d10f66403cc735fc0c6df52539d7b3d6263e6a5ab722349b6f4d758e0a.jpg)
+
+![](images/290636899efee34760ff67a835c13befcc649cbe3c64137c7bc6d9979a908713.jpg)
+
+![](images/346fce64aad100c023e5dc7021d34418041b78feaba9ac951ae38c39da7a2a02.jpg)
+
+Figure 11: Results for improving WiSE-FT with LiNeS on with ViT-B/16 model fine-tuned on ImageNet.  
+![](images/89e8d42537270f61aa1e8a37a03d048bb728776827a278e5ffc02256462e8cc6.jpg)  
+WiSE-FT WiSE-FT + LiNeS ★ Fine-tuned Fine-tuned + LiNeS LP initialization
+
+![](images/c9f8b7665a26565808b436f40f04ec68015221507f76b05c1d569a533f21f0bc.jpg)
+
+# C.10.2 INDIVIDUAL RESULTS FOR 70 CHECKPOINTS
+
+We provide in Figure 12 individual results separately for the experiments on the 70 individual model checkpoints. Note that here y-axis represents the averaged accuracy over 5 OOD datasets. From the figure, we observe that LiNeS consistently improves WiSE-FT in terms of both ID and OOD accuracies for most of the individual checkpoints.
+
+# C.11 DETAILED PERFORMANCE ON INDIVIDUAL TASKS FOR MULTI-TASK MODEL MERGING
+
+Image Classification We provide the detailed performance on each individual task for multi-task model merging in image classification benchmarks, complementary to the results in Table 2 and Table C.4 where the accuracies are averaged on the individual tasks.
+
+Table 11: Similar to Table 1, LoRA Fine-tuning harms generalization on control tasks. Increased target performance results in higher levels of forgetting. Still, our proposed method LiNeS restores control performance for all ranks considered while minimally affecting target performance.  
+
+<table><tr><td></td><td>Target</td><td>Control</td></tr><tr><td>Pre-trained</td><td>48.3</td><td>48.3</td></tr><tr><td>Fine-tuned</td><td>90.5</td><td>38.0</td></tr><tr><td>Fine-tuned +LiNeS</td><td>90.3</td><td>48.0</td></tr><tr><td>r = 16</td><td>84.4</td><td>44.2</td></tr><tr><td>r = 16 + LiNeS</td><td>84.3</td><td>46.7</td></tr><tr><td>r = 32</td><td>85.8</td><td>42.8</td></tr><tr><td>r = 32 + LiNeS</td><td>85.5</td><td>46.7</td></tr><tr><td>r = 64</td><td>86.6</td><td>41.6</td></tr><tr><td>r = 64 + LiNeS</td><td>86.4</td><td>46.2</td></tr><tr><td>r = 128</td><td>87.5</td><td>41.6</td></tr><tr><td>r = 128 + LiNeS</td><td>87.2</td><td>46.3</td></tr></table>
+
+The single-task performance is presented in Figure 13 for ViT-B/32, Figure 14 for ViT-B/16, and Figure 15 for ViT-L/14. From the results we observe that our method demonstrates a noticeable improvement over baseline merging techniques across individual tasks in all test scenarios.
+
+Natural Language Processing We provide in Figure 16 the detailed single-task performance for the three NLP benchmarks using T5-large, complementary to the results in Table 3. Similar to the observation in vision, our method provides a consistent improvement over baselines across individual tasks.
+
+![](images/81160279a160c89dc81977dd07d585c81fc15d39853b6332457151834c837fa6.jpg)
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+![](images/957d00c10bc7648af75b9612aaaecc7a0c5d3569f435f1fb2b0c7dcb04c3f6ae.jpg)
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+![](images/7e6f185709a35ee15f8b440e2786532cba922e59cf28f699f8c7b71f01cd6b48.jpg)
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+
+![](images/a06224922278cb2120a3a17593b4101f08127d02d5f9d31c3ac1ca7416b37f53.jpg)
+
+![](images/b9c8444e4d6e38cb2c2c6a10da311d426f9fea1be8de01f67fca935a1c472542.jpg)
+
+![](images/f14c659fafc6787ea862995c86b60d1657a1b06e6494a7f6469bdf6f80c33083.jpg)
+
+![](images/6f66e94643bc60b5729f6d840fe76e7705a9020e9fa3bbba3cd9759b219a3e5a.jpg)
+
+![](images/f1ba58acdc0100609a9a78313e7e6218460d1b0db94a07c9582868302f69e22c.jpg)
+
+![](images/cbebdc6f72ceb4b669b5324d38569f1a7eb0b51b3b3989f4bf95592ea125c7d6.jpg)
+
+![](images/7a45f9383bb12f6d6fa593765557a9757bcae429f371e5b71875a0aaa223e94e.jpg)
+
+![](images/d82479ded5d596f59e88f86edeba67319afd6c7d9b240cae70950d636566a021.jpg)
+
+![](images/ac8887f2fffdde1e1f7a9ee5331032e03ac85d412d7e9c74998e9aff6798e039.jpg)
+
+![](images/a8860c00602b9a8dfc2d5ae281241a18d74cc0b00a4c5d0cafde702e8c7b2279.jpg)
+
+![](images/87b388d309d53913164030f86c0a964d30506cc178141cd73fca422f42034e1b.jpg)  
+Figure 12: Performance of applying LiNeS to WiSE-FT to each ViT-B/32 checkpoint fine-tuned on ImageNet.
+
+![](images/61ae0cdb719947cc7fed06258e4a88b6d50e36097f86ac98839a88a66e9f205d.jpg)  
+WiSE-FT WiSE-FT + LiNeS ★ Fine-tuned Fine-tuned + LiNeS LP initialization
+
+![](images/57a962337203aa6139d926975ab49779de7aefe8ce74e8c6e6a57b38bde8cc90.jpg)
+
+![](images/87b9bfed48fe798ea6711217bdb935ba79cb22cb1ad370085ebb1b9b4725760c.jpg)
+
+![](images/a73d983df570e60fc3106a07c12b41e86dfd120b7472910b0526343473390813.jpg)
+
+![](images/d1267ec8b57644bc1b81750125900ab4b66e1cbb397697724a3af0158e4e99b4.jpg)  
+Fine-tuned  
+Zero-shot
+
+![](images/980fadd31962a265b2b280e07e9180acfa4920ab121b96c62fa092b8e2f19a06.jpg)  
+TA  
+TA+LiNeS
+
+![](images/029d64919315bc8db5543dc5e8ca83bd337b0dde3e6bc0ca88b33479d283c1b3.jpg)  
+TIES  
+TIES+LiNeS
+
+![](images/287fa7a737d68641bec6a95ef0bd00cebd65110127b8ce5a271dbd9913352ca6.jpg)  
+Consensus
+
+![](images/28a0c2e339bf75b9a2fa764bbe46fd3ce6c0bf9b45169807ed92ac577a4ac6b3.jpg)  
+Consensus+LiNeS
+
+![](images/021bc8dd674a3dc28a0419e93a73dd74cbd45c63754934c3cf5005213ecb1980.jpg)  
+(a) Task Arithmetic - 8 Tasks
+
+![](images/6bd6601a5be54034e31ab7716afea610f5ccfac738f91e0756e75290c36b47a3.jpg)  
+(b) Task Arithmetic - 14 Tasks
+
+![](images/0a1b407af36152e4028b7ab4f8ca6176c08366f1d955ee069423ae4bb429e012.jpg)  
+(c) Task Arithmetic - 20 Tasks
+
+![](images/e09e39b47aa457098bee4c33ae1517aad600e79f920c0fb81c25615b5477b56f.jpg)  
+SUN397
+
+![](images/9ebae72906434402b73041237c472d3843267b18ec5a98036992d62c589fd9eb.jpg)  
+PCAM STL10
+
+![](images/9fdd108928da3e846e8bf398668fee4e5b97840feb23717888032c6bd0776981.jpg)  
+(c) TIES - 20 Tasks
+
+![](images/93936ccaf758ffa5c7404c9df4325e3e0f7179b70a4f717eec46b11d0c49dbd3.jpg)  
+(a) TIES - 8 Tasks  
+(a) Consensus - 8 Tasks  
+Figure 13: Single-task accuracies for multi-task merging on image classification benchmarks for ViT-B/32.
+
+![](images/4624a52f8076806d4f2a7169fef0b51e2def4283b4f23a2e01019e10a2db4861.jpg)  
+(b) TIES - 14 Tasks  
+(b) Consensus - 14 Tasks
+
+![](images/d5c5f3e7ac00e78de8729e9ba3b065612bc7c3d7a868d59c165b6956886558b6.jpg)  
+(c) Consensus - 20 Tasks
+
+![](images/6d3b5f239c3a8f9c54e2df2ca8516c134c9625c9fc375ce855e532bcf76e442d.jpg)  
+Fine-tuned  
+Zero-shot
+
+![](images/814dc4d5370217716f75ebe7d04c26034d24e444cbfc7481228d51e42a815643.jpg)  
+TA  
+TA+LiNeS
+
+![](images/ad30464ee68f18127b99e1c85ad9762bd4f95bdb3a71bae53f8125a1221100e6.jpg)  
+TIES  
+TIES+LiNeS
+
+![](images/17960b0e0ee8ca8fef17e94b81fdc9e32672dd22b1f7247fe2a1a229a04e299c.jpg)  
+Consensus
+
+![](images/fe126222ab1de7526bb91a88b84814fddd1122b1f94d34256bc44e3426f7b3c5.jpg)  
+Consensus+LiNeS
+
+![](images/15e229e605a8ccc5f6b5f471f036d3a51e07a7d0c10d38f7d6f3a78cf8a71f9e.jpg)  
+(a) Task Arithmetic - 8 Tasks
+
+![](images/afc6f3ca8fe7a27beb9f0ced769b9c8bfad30f827ccff292e11450b28bd8d458.jpg)  
+(b) Task Arithmetic - 14 Tasks
+
+![](images/7af6cf6f62d2a49b143ddad2b42bd1a5af391e0d8e3702ef78cf9c96bd1f7836.jpg)  
+(c) Task Arithmetic - 20 Tasks
+
+![](images/1dabf90effafb7b91b26f240d9ed972b24e3ed3361cc2cd921bf2191beeeb68b.jpg)  
+SUN397
+
+![](images/e286fe87ac3d8cd9b63422c09da52dc67edc12988e489aaa2f42b4525a326cdc.jpg)  
+PCAM STL10
+
+![](images/3a7a3c07d6fa65eef9c1c584c1eb4b5995cb1a70cae4269e9dffaf25300f66b1.jpg)  
+(c) TIES - 20 Tasks
+
+![](images/dc06ab5ad7069a2cc5ac249fc635ebb3c679d21693abf8a74b522ca004f4947c.jpg)  
+(a) TIES - 8 Tasks  
+(a) Consensus - 8 Tasks
+
+![](images/26eee87586be7aa2f691258753474df7db4336ee780f2c3748610d82e4d607d6.jpg)  
+(b) TIES - 14 Tasks  
+PCAM STL10  
+(b) Consensus - 14 Tasks
+
+![](images/5c9644682ca646ca569846093a05dd2e268cf749592a1a14a4cc4e0d9a447766.jpg)  
+(c) Consensus - 20 Tasks  
+Figure 14: Single-task accuracies for multi-task merging on image classification benchmarks for ViT-B/16.
+
+![](images/7205b66ab33d36b9706369817a4e3544b7e8de7257e20e083bfdc137a38d5315.jpg)  
+Fine-tuned  
+Zero-shot
+
+![](images/f037cc9a02367290bcae73d2130d22d027582bca58cabdfa82fe341723fbfc9d.jpg)  
+TA  
+TA+LiNeS
+
+![](images/97a07a028afb173e74e2eb6389f88ea8f331ee4f207b384f046a33582e6a9a4c.jpg)  
+TIES  
+TIES+LiNeS
+
+![](images/7c3bd624ab18386abe11315eb94ec7ff72d3fbd39192aa622770cb2745ccc2bb.jpg)  
+Consensus
+
+![](images/731fd78af6787a06e47bc8300487767e83a84f00b3bd85ae50abcf07e3eb0798.jpg)  
+Consensus+LiNeS
+
+![](images/078ba0f3ea0ecc443cb72704225f1c667a7527c318466018db7a11025e39562e.jpg)  
+(a) Task Arithmetic - 8 Tasks
+
+![](images/c27cee4211cebd92c194c2a83bd59e6a6fa72eddc2081f24bce3d9ff97038151.jpg)  
+(b) Task Arithmetic - 14 Tasks
+
+![](images/ad5f4f90a3c78bd71f85b639330616bbaad50f57931e47930c9754728c0d3615.jpg)  
+(c) Task Arithmetic - 20 Tasks
+
+![](images/06c5252acac1212dcf085e563638209f06306a8ab55cc073498838a5d46c2983.jpg)  
+(a) TIES - 8 Tasks
+
+![](images/492e48e981c462bf38ba6d741d93d91f8dd0712dda3ae633204fa036b4a761a5.jpg)  
+(b) TIES - 14 Tasks
+
+![](images/cd876d794c3c55fa604189d0fdd4b9d05cccf77089abe89303a456a87af7315d.jpg)  
+(c) TIES - 20 Tasks
+
+![](images/47c9c1668c270a1e068c91d854ab190dbd9f614f759d4ee23afd964b80d4fac6.jpg)  
+(a) Consensus - 8 Tasks  
+Figure 15: Single-task accuracies for multi-task merging on image classification benchmarks for ViT-L/14.
+
+![](images/bf6c958a735b417fcba328372c730a5759d46055f12ef1104542dfdeb9ee74d8.jpg)  
+(b) Consensus - 14 Tasks
+
+![](images/bbaf92c79a203994438a2e669592377eb87532d671f470535a6699fa7d13072e.jpg)  
+(c) Consensus - 20 Tasks
+
+![](images/44d107472e0c8d3c58ca801213f5dc937c626af11ffec2a8e4619370e3d80c02.jpg)  
+Fine-tuned  
+Zero-shot
+
+![](images/7f1b98c6b3813b5e3d193df1ee25fce66cfdcd1b6babd3d3df93f0133b4f4eed.jpg)  
+TA  
+TA+LiNeS
+
+![](images/e98553b2cc2bbd2f309e46f9841c50f7d23c0d03b03a2c054ad5e8b427623d0b.jpg)  
+TIES+LiNeS
+
+![](images/d715591e9c13cec9eae23c50763d29421f9c1a3f540a62eca1a5c43915d2f414.jpg)  
+Consensus+LiNeS
+
+![](images/b75b64d084018af7ceccd693dd03f4a4b80c0657fa72d3eb6c6d8b82d684538c.jpg)  
+(a) Task Arithmetic - 7 NLP tasks
+
+![](images/3b38ecce6f8fcaefa26631a3e855d076754cbf2b2d93a55659606bb2a1c3b19e.jpg)  
+(b) Task Arithmetic - 8 QA tasks
+
+![](images/ad4a20e32fee9a32972dd59af88957c06ed326d7fd177f4ea37d89cf95d54724.jpg)  
+(c) Task Arithmetic - 11 NLP tasks
+
+![](images/30b7143c520c3bb98c688a588e70d9b39e9f3c44e45e7079ff8a2f491ddfa42a.jpg)  
+(a) TIES - 7 NLP tasks
+
+![](images/17e431b9bd04f72c4bb5b7df29fc4f462bb3bb3f1590cc598ee09af9fd1875e4.jpg)  
+(b) TIES - 8 QA tasks
+
+![](images/08a2ab307a22b6b837d86fbe2e6974b2cadfdff4a933a0932755d05902d93412.jpg)  
+(c) TIES - 11 NLP tasks
+
+![](images/5da6e16205555d8ef4c86c2d5450cf08e79cd3f2e871b931267f4753764deb00.jpg)  
+Winogrande  
+(a) Consensus - 7 NLP tasks
+
+![](images/a507f5b4b49714a6eef26b74f539e91274e35e0cd5023015fdc95c3496ed3180.jpg)  
+(b) Consensus - 8 QA tasks  
+Figure 16: Single-task accuracies for multi-task merging on NLP benchmarks for T5-large.
+
+![](images/3f0c943d38d9c3a56938c8e56596eff1e9750940ec90d3c9def439227d783cc2.jpg)  
+(c) Consensus - 11 NLP tasks
\ No newline at end of file
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@@ -0,0 +1,1558 @@
+# LIE ALGEBRA CANONICALIZATION: EQUIVARIANT NEURAL OPERATORS UNDER ARBITRARY LIE GROUPS
+
+Zakhar Shumaylov*,1, Peter Zaika*,1, James Rowbottom1, Ferdia Sherry1, Melanie Weber2, Carola-Bibiane Schonlieb1
+
+<sup>1</sup>University of Cambridge, <sup>2</sup>Harvard University, *Equal contributions
+
+# ABSTRACT
+
+The quest for robust and generalizable machine learning models has driven recent interest in exploiting symmetries through equivariant neural networks. In the context of PDE solvers, recent works have shown that Lie point symmetries can be a useful inductive bias for Physics-Informed Neural Networks (PINNs) through data and loss augmentation. Despite this, directly enforcing equivariance within the model architecture for these problems remains elusive. This is because many PDEs admit non-compact symmetry groups, oftentimes not studied beyond their infinitesimal generators, making them incompatible with most existing equivariant architectures. In this work, we propose Lie aLgebrA Canonicalization (LieLAC), a novel approach that exploits only the action of infinitesimal generators of the symmetry group, circumventing the need for knowledge of the full group structure. To achieve this, we address existing theoretical issues in the canonicalization literature, establishing connections with frame averaging in the case of continuous non-compact groups. Operating within the framework of canonicalization, LieLAC can easily be integrated with unconstrained pre-trained models, transforming inputs to a canonical form before feeding them into the existing model, effectively aligning the input for model inference according to allowed symmetries. LieLAC utilizes standard Lie group descent schemes, achieving equivariance in pre-trained models. Finally, we showcase LieLAC's efficacy on tasks of invariant image classification and Lie point symmetry equivariant neural PDE solvers using pre-trained models.
+
+# 1 INTRODUCTION
+
+The deep learning revolution of the 2010s has demonstrated the remarkable effectiveness of deep learning (DL) models in various domains, from text generation to computer vision. This widespread success has led to the adoption of DL in numerous scientific fields, including biology (Abramson et al., 2024), physics (Merchant et al., 2023) and engineering (Degrave et al., 2022). The training of such models often requires access to large amounts of high-quality data and computing resources, which can be challenging to obtain in practice. However, many tasks in these domains involve structured data, including data with symmetries induced by fundamental physical principles. Physics-informed neural networks (PINNs) seek to leverage such structure to mitigate the aforementioned challenges.
+
+DL models have been observed to significantly benefit from the inductive bias of symmetries. Notable examples include translation invariance in imaging data and time-series, translation and rotation symmetries of molecules, resulting in position and momentum conservation. Incorporating these symmetries into the design of neural networks can enhance their performance and generalization capabilities. This has been analyzed empirically (Esteves et al., 2023; Wang et al., 2020) and through the lens of learning-theoretic trade-offs (Mei et al., 2021; Bietti et al., 2021; Kiani et al., 2024b).
+
+Prior work on equivariant neural networks focuses on "simple" groups, resulting in frameworks that are often not rich enough to encode the complex geometric structure found in scientific applications. For example, the heat equation in Appendix D.3 was recently shown by Koval & Popovych (2023) to admit a group of point symmetries  $\mathrm{SL}(2,\mathbb{R})\ltimes_{\phi}\mathrm{H}(1,\mathbb{R})$ , which is not only non-compact, it also does not act in a representation on the underlying jet space. This renders most of the currently available equivariant architectures unusable. To overcome this, we introduce a novel framework that is able to encode a wider range of symmetries, not requiring a priori knowledge of the full group structure. This flexibility allows for integration with existing models, notably pre-trained models, with the potential of leveraging the benefits of geometric inductive biases beyond classical equivariant architectures.
+
+# 1.1 EQUIVARIANCE IN NEURAL NETWORKS
+
+Incorporating group symmetries inside neural networks, in the form of equivariance, is one of the fundamental tasks in geometric deep learning (Bronstein et al., 2021). The translational equivariance of convolutional neural networks (CNNs) (Fukushima & Miyake, 1982; LeCun et al., 1989) has been suggested as an important reason for their successes in tasks such as image classification (Krizhevsky et al., 2012; He et al., 2016). Because datasets may have symmetries more intricate than just translations, the past decade has seen a push towards generalizing this approach to allow for other (Lie group) symmetries, starting from works on roto-translationally equivariant CNNs (Dieleman et al., 2015; 2016; Cohen & Welling, 2016), to more recent methods, efficiently utilizing knowledge of the Lie algebra for more general groups (Kumar et al., 2024; MacDonald et al., 2022; Finzi et al., 2020a; McNeela, 2023) and computationally efficient methods (Mironenko & Forre, 2024). Notably, convolution-based approaches to equivariance satisfy a type of local equivariance too (Weiler & Cesa, 2019): with compactly supported filters, if features in the data are sufficiently separated, these networks are equivariant to transformations localized to the features. Despite all of the work in this direction, construction of efficient and accurate equivariant architectures is not simple, see Appendix A.
+
+More recently, there has been a drive towards publicly available foundational models (Bommasani et al., 2021) for various tasks, pretrained on massive datasets. These models, however, are often not equivariant. For example, Alphafold 3 (Abramson et al., 2024) no longer utilises equivariant architectures, unlike the initial version (Jumper et al., 2021). For this reason a direction of research has been investigating the possibility of utilizing the generalization offered by equivariance, while being able to use pre-trained models. The main two approaches to achieve this are canonicalization $^{1}$  Kaba et al. (2023); Mondal et al. (2023) and frame averaging Puny et al. (2022); Basu et al. (2023) (also referred to as symmetrization). However, there still remain a number of open theoretical questions regarding the possibility of continuous canonicalization (Dym et al., 2024), connections between frame averaging and canonicalizations Ma et al. (2024) and constructibility of frames Dym et al. (2024); Puny et al. (2022).
+
+![](images/19dfc4de246f5834e9753dcdec46c882fc88d32bcf278895bf5d93f5997c2562.jpg)
+
+![](images/bdb6705ea3dd006a416d669e1c14341a605ad2a087538eeb1b53198c550a2a02.jpg)
+
+![](images/09f1a2bf6c87f4e3089a8f93188e2829db0d625359860bea0519d1949ec033a1.jpg)  
+Figure 1: Effect of canonicalization on decision boundaries in  $k$ -NN classification for separating the inner and the outer rings Section 4.1.
+
+![](images/670474e0a7033287ee8af6538de9eca495d300f8b75e06aa85869b9e090df6da.jpg)
+
+# 1.2 PHYSICS-INFORMED MACHINE LEARNING
+
+In a separate direction, deep neural networks are being increasingly used in scientific computing, particularly in simulating physical systems modeled by partial differential equations (PDEs). This surge in interest is largely driven by limitations of traditional numerical solvers, as they often struggle with demanding scenarios, such as those involving non-linear PDEs on complex domains necessitating fine discretization, or high-dimensional problems where the curse of dimensionality kicks in (Han et al., 2018). This is further exacerbated if there is a need to solve PDEs across varying domain geometries, parameters, and initial/boundary conditions, requiring independent simulations for each unique configuration.
+
+The main prominent approach that has emerged is based on integrating physical laws and constraints into the learning process to solve PDEs (Raissi et al., 2019), called physics-informed neural networks (PINNs), bypasses the need for direct discretization. However, classical PINNs are not without
+
+![](images/05ec134e5af389bdc49e8233e16cfa512d18bf74431dee0e18e76c2987cd89ad.jpg)  
+(a) DeepONet operator for the Heat equation Appendix D.3.
+
+![](images/562dbc2080210963976b2b649b17c49af7e12aca95487ab5325c5882c75de3ce.jpg)  
+(b) DeepONet operator for the Burgers' equation Appendix D.4.
+
+![](images/3ac28575db004e3251c451243e2fa16f6847be6010c2677ff09a7266b59123e7.jpg)  
+(c) POSEIDON operator for the Allen-Cahn equation Section 4.3.1.  
+Figure 2: Canonicalization pipeline for numerical PDE evolution Section 4.3.
+
+limitations (Grossmann et al., 2023; Krishnapriyan et al., 2021). See Appendix A for a further discussion. To address the challenge of solving PDEs across diverse scenarios, Neural Operator models have emerged as a promising solution (Li et al., 2021a; Raonic et al., 2024). Unlike standard DNNs, which approximate functions over finite-dimensional vector spaces, neural operators learn mappings between infinite-dimensional function spaces. These learned operators map parameters or IBCs to corresponding PDE solutions. Combining such Neural Operator models with a physics-informed loss has resulted in physics-informed neural operators (Li et al., 2021b; Wang et al., 2021b), oftentimes improving generalization and physical validity, compared to their physics un-informed variants. In what follows, we concern ourselves precisely with this class of approaches.
+
+# 1.3 EQUIVARIANCE IN PINNS
+
+Based on the discussions above, a natural question arises - can generalization of physics-informed neural networks be improved by utilizing symmetries arising in the study of PDEs? A few strides have been made towards understanding ways in which to take advantage of the Lie point symmetries (LPS) of differential equations in neural solvers. The simplest ways to add symmetry information into neural PDE solvers is through data augmentation (Brandstetter et al., 2022; Li et al., 2022), resulting in more expensive training, or loss augmentation (Akhound-Sadegh et al., 2023; Li et al., 2023; Zhang et al., 2023), resulting in more difficult training. Both of these, however, oftentimes result in improved generalization. Similar benefits have also been observed in Gaussian process based solvers Dalton et al. (2024). Alternatively, exact equivariance can be seeked. As an example, in Wang et al. (2021a), neural PDE solvers were combined with standard approaches to designing equivariant architectures for simple symmetries such as translations, roto-translations and scalings. Still, differential equations may have significantly more complicated Lie point symmetry groups than the groups for which there are off-the-shelf equivariant architectures. The concept of moving frames has been applied to the problem of using PINNs to solve ODEs with LPS in Arora et al. (2024). Here, the approach taken was to invariate the ODE in question, and apply a PINN to this equation, before mapping back to an (approximate) solution of the original ODE. Similarly, Lagrave & Tron (2022) uses equivariant neural networks to first approximate differential invariants, before passing to a classical numerical scheme for evolving the solution. Overall, the literature so far has illustrated that Lie point symmetries can provide useful inductive biases for Neural PDE solvers, and from a practical point of view can even be approximated from observed data (Gabel et al., 2024). However it still remains an open question, whether it is possible to make existing physics-informed approaches equivariant. In this paper we directly tackle this question. Preliminaries are provided in Appendix A.
+
+# 1.4 OVERVIEW OF CONTRIBUTIONS
+
+Motivated by limitations of existing equivariant architectures, we turn to energy-based canonicalization as a means for inducing equivariance in existing models. This approach was first introduced in Kaba et al. (2023), but remains largely unexplored. In this paper we revisit and extend this approach, and explore its utility in scientific machine learning applications. Noting that many existing
+
+definitions and results in the prior literature do not extend readily to the general setting, we extend and refine existing definitions to handle non-discrete and non-compact groups. Our key contribution is an energy-based canonicalization approach that provides a unifying theoretical framework for constructing frames and canonicalizations (Section 2.2). To define energy-based canonicalization we introduce a generalization of standard canonicalization, allowing use with non-compact Lie groups, which we call weighted canonicalization. This new framework encapsulates prior work of Dym et al. (2024) on weighted frames, and our energy-based canonicalizations live naturally in this framework.
+
+We emphasize that our proposed approach is generic, in that it has the potential to apply to a wide range of models and learning tasks. The energy canonicalization framework developed here is readily applicable for an arbitrary group with an arbitrary action, including those not acting linearly, therefore being extendable to gauge and local symmetries. For Lie groups with smooth actions, one only requires knowledge of the action of some basis of the Lie algebra, without requiring a global parametrization of the group.
+
+This flexibility stems from the choice of energy functional, which directly influences empirical performance and the size of resulting canonicalizations. We offer guidelines for constructing energy functionals and for performing Lie group optimization in Section 3. We showcase the effectiveness of our method on image classification tasks under homography and affine group invariance in Section 4.2. We further discuss turning neural operators Lie point symmetry equivariant on three examples, including the pre-trained POSEIDON foundation model (Herde et al., 2024) in Section 4.3.
+
+# 2 GENERAL FRAMEWORK FOR CANONICALIZATIONS AND FRAMES
+
+For this section let  $G$  be a Lie group acting smoothly on a manifold  $X$ . Let  $C_b^0 (X)$  be the bounded continuous  $\mathbb{R}$ -valued functions on  $X$ . Let  $\mathrm{PMeas}(X)$  be the set of Radon probability measures on  $X$ , with respect to the Borel  $\sigma$ -algebra on  $X$ . We will set  $\phi \in C_b^0 (X)$  some function to be acted on.
+
+The two main ways of imposing equivariance on arbitrary models: frame averaging and canonicalization, both arise from the study of the Reynolds operator. For finite groups  $G$ , defined as
+
+$$
+\mathcal {R} _ {G} (\phi) (x) = \frac {1}{| G |} \sum_ {g \in G} g \cdot \phi (g ^ {- 1} \cdot x).
+$$
+
+Evaluating this equivariant function  $\mathcal{R}_G(\phi)$  can become prohibitively expensive, requiring summation over the entire group for every input  $x$ . Therefore one seeks ways to restrict the range of this summand while preserving the equivariance of the resulting function. Frames are defined, for each element  $x \in X$ , as a subset of the group  $G$  for which to average over, with an equivariance condition detailed below to ensure equivariance of the resulting function. Canonicalizations<sup>2</sup> are the opposite perspective, for each element  $x \in X$ , directly providing a subset of  $X$  over which to directly average the function over with an invariance condition to ensure the resulting function is equivariant. The presentation here was reduced to fit the page limit; for a full description see Appendix H.
+
+# 2.1 FRAMES
+
+Definition 2.1. Frames are functions  $\mathcal{F}: X \to 2^G \setminus \{\varnothing\}$  such that for any  $x \in X$  and  $g \in G$ :  $\mathcal{F}(gx) = g\mathcal{F}(x)$ . We call the space of frames  $\mathrm{Fra}_G(X)$ .
+
+A frame  $\mathcal{F}$  acts on  $\phi$  by frame averaging  $\mathcal{F}(\phi)(x) = \frac{1}{|\mathcal{F}(x)|}\sum_{g\in \mathcal{F}(x)}\phi (g^{-1}x)$ . As  $\mathcal{F}(x) = G_x\mathcal{F}(x)$  for any frame  $\mathcal{F}$ , frame averaging is inherently limited to the case that the frames are finite at each point, i.e. that the set  $\mathcal{F}(x)$  is finite  $\forall x\in X$ . For frames this can only be the case when the group acts with finite stabilizers at all points. This leads to the following definition of Dym et al. (2024). A weighted frame  $\mu_{x}$  acts on  $\phi$  similarly by averaging  $\mu (\phi)(x) = \int_{G}\phi (g^{-1}\cdot x)d\mu_{x}(g)$ .
+
+Definition 2.2. (Weakly)-Equivariant Weighted Frames are functions  $\mu_{[\cdot]}: X \to \mathrm{PMeas}(G)$  s.t.  $\forall x \in X, g \in G: \mu_{gx} = g_*\mu_x$  (respectively:  $(\pi_{gx})_*\mu_{gx} = (\pi_{gx})_*g_*\mu_x$  where  $\pi_{gx}: G \to G / G_{gx}$  is the quotient map. See Lemma F.2). We call the space of weakly-equivariant weighted frames  $\mathrm{WFr}_{G}(X)$ .
+
+$^{2}$ Original proposal of Kaba et al. (2023) relied on these being singletons, requiring authors to introduce relaxed equivariance. Treating these as sets turns out to be natural for both energy canonicalizations (Section 2.3), and for providing (equivalence) isomorphism of (weighted) canonicalizations with (weighted) frames (Theorem G.3 and Ma et al. (2024)).
+
+# 2.2 CANONICALIZATIONS
+
+We briefly recall the definitions of (finite) canonicalizations from Ma et al. (2024).
+
+Definition 2.3. A canonicalization is a function  $\mathcal{C}: X \to 2^X \setminus \{\varnothing\}$  s.t.  $\forall x \in X, g \in G: \mathcal{C}(gx) = \mathcal{C}(x)$ . We call the space of canonicalizations  $\operatorname{Can}_G(X)$ .
+
+Definition 2.4. An orbit canonicalizations is a canonicalization s.t.  $\forall x\in X,\mathcal{C}(x)\subseteq Gx$  . We call the space of orbit canonicalizations  $\mathrm{OCan}_G(X)$
+
+A canonicalization  $\mathcal{C}$  acts on  $\phi$  similarly to frames by  $\mathcal{C}(\phi)(x) = \frac{1}{|\mathcal{C}(x)|}\sum_{y\in \mathcal{C}(x)}\phi (y)$ . By the orbit-stabilizer theorem, orbit canonicalizations are equivalent to frames. In this isomorphism, frames are always larger by a factor equal to the size of the stabilizer group. Thus from a computational perspective orbit canonicalizations are preferred, and we will consider the canonicalization perspective from now on. We shall also see that this is indeed the correct perspective to work with when  $G$  is a non-compact Lie group.
+
+# 2.3 ENERGY-BASED CANONICALIZATION
+
+We are now in a position to introduce the definition of our energy-based canonicalization method. We start with an energy function  $E: X \to [0, +\infty]$  which we then minimize over the orbits of the group action  $Gx$ . First, in the case where  $G$  is finite, every orbit in  $X$  necessarily has a minimum of  $E$ , and we note that energy minimization naturally results in a frame (Proposition G.1):
+
+$$
+\mathcal {F} _ {E} (x) = \underset {g \in G} {\arg \min } E \left(g ^ {- 1} x\right) \tag {1}
+$$
+
+Moving onto canonicalizations, we want to note that intuitively the best energy would be some approximation of the negative log likelihood for the sampled distribution. Therefore finding the minimizers of the energy within an orbit has the effect of finding those examples in the orbit which are most likely to have been trained on. In keeping with this intuition, we would like to define the energy minimizing canonicalization to be:
+
+$$
+\mathcal {C} _ {E} (x) = \underset {y \in G x} {\arg \min } E (y) \tag {2}
+$$
+
+In the non-compact cases we wish to consider, this definition isn't suitable for two main reasons: the set of minima may not be finite or there may be no minima. Dealing with the first naturally leads us to defining weighted canonicalizations, in analogy with weighted frames:
+
+Definition 2.5. A weighted canonicalization is a  $G$ -invariant function  $\kappa_{[\cdot]}: X \to \mathrm{PMeas}(X)$ . We call the space of weighted canonicalizations  $\mathrm{WCan}_G(X)$ .
+
+Definition 2.6. A weighted orbit canonicalization is a weighted canonicalization  $\kappa$  such that for every  $x\in X$ ,  $\operatorname {supp}\kappa_{x}\subseteq Gx$ . We denote the set of weighted orbit canonicalizations as  $\mathrm{WOCan}_G(X)$
+
+A weighted canonicalization  $\kappa_{x}$  acts on  $\phi$  similarly as  $\kappa (\phi)(x) = \int_{X}\phi d\kappa_{x}$ . As in the non-weighted case, we have a similar equivalence between weighted orbit canonicalizations and weakly-equivariant weighted frames given by Theorem G.3. Weighted canonicalizations do not address the second issue however, as orbits of a non-compact Lie group are not necessarily compact or even closed. The non-compactness of these orbits is easy to solve by making the standard assumption that  $E$  is topologically coercive: for any  $N\in \mathbb{R}^{+}$  there is a compact  $K\subseteq X$  such that  $E|_{X\backslash K} > N$ . The non-closedness of orbits is more subtle, as shown by the following example.
+
+Example. Let  $X = \mathbb{R}^2$  and  $G = \mathbb{R}_{>0}$  with its standard multiplicative Lie group structure. Note that  $G$  is a non-compact Lie group. Let  $t \in G$  act by  $t \cdot x = tx$ . Then one sees that the orbits of this action are exactly the non-closed rays  $\mathbb{R}_{>0}x$  emanating from origin as well as simply the origin. In practice, this might represent a scale-invariant learning problem.
+
+Now consider the energy minimization problem defined by taking the energy to be  $E(x) = |x|^2$ . The global minima of this function is clearly the origin, which isn't in the domain. On the other hand, any of the non-closed rays have the origin as their limit points, therefore  $E$  does not have a minima on any of these orbits.
+
+To get around the non-closedness issue we introduce the following notion. This may seem like an artificial solution to the non-closedness problem, but in fact weighted closed canonicalizations is the most natural extension, as they are simply limits of weighted orbit canonicalizations by Theorem G.5.
+
+Definition 2.7. A weighted closed canonicalization is a weighted canonicalization  $\kappa : X \to \mathrm{PMeas}(X)$  such that for every  $x \in X$ ,  $\operatorname{supp} \kappa_x \subseteq \overline{Gx}$ . Call the set of weighted closed canonicalizations  $\mathrm{WCCan}_G(X)$ .
+
+By taking the closures of orbits, we solve the problem of  $E$  not having minima, so we are naturally led to constructing weighted closed canonicalizations. First, we define the energy-minimizing set,  $\mathcal{M}_E(x) = \arg \min_{y \in G_x} E(y)$ , set of minima of  $E$  on the closure of the orbit of  $x$ .
+
+Proposition 2.1. (Proposition G.2)  $\mathcal{M}_E(x)$  is  $G$ -invariant and non-empty.
+
+If  $\mathcal{M}_E(x)$  is finite for each  $x\in X$ , then we define the (non-weighted) closed canonicalization  $\kappa_{E}\in \mathrm{WCCan}_{G}(X)$  by taking  $\kappa_{E}(x)$  to be the normalized counting measure on  $\mathcal{M}_E(x)$  as usual.
+
+For a general  $E$ , we may not be able to put any reasonable probability measure on the set  $\mathcal{M}_E(x)$ , for example if  $\mathcal{M}_E(x)$  has highly fractal geometry. We therefore must assume that  $E$  is such that for all  $x$ , the Hausdorff measure of  $\mathcal{M}_E(x)$  is non-zero. As each  $\mathcal{M}_E(x)$  is compact, the Hausdorff measure is finite, hence combined with this assumption we may normalize it to obtain a probability measure on  $\mathcal{M}_E(x)$  which is a weighted canonicalization  $\kappa_E(x) \in \mathrm{WCCan}_G(X)$ . Refer to Appendix E for more discussion on the Hausdorff measure and on why this assumption is reasonable. Note that if  $\mathcal{M}_E(x)$  is finite, then the normalized Hausdorff measure constructed agrees with the normalized counting measure. Hence this construction is a generalization of the purely finite case.
+
+# 2.4 CONTINUITY PRESERVING CANONICALIZATIONS
+
+Using the setup above we may now write down our main diagram connecting all the different notions of canonicalization and framing together, shown in Figure 3.
+
+In practice, it is often desired for continuity to be preserved under canonicalization/frame averaging. In Dym et al. (2024) it is shown that there are many topological obstructions to constructing continuous canonicalization functions. In analogy with Dym et al. (2024), we may now ask whether energy canonicalizations presented here take continuous functions to continuous functions. In our framework this is a simple condition to add.
+
+![](images/85c0aae1a1866b3492814e16dad200bd6b20bfeb7691cc1d235a6a33c62a66a6.jpg)  
+Figure 3: Connections between various notions introduced previously and in this work. Top row: finite frames and canonicalizations, mapping vertically into weighted versions via normalized counting measures. Inside each, contained a sequentially closed subspace of those that preserve continuity.
+
+Definition 2.8. A weighted canonicalization  $\kappa$  is continuous if for every  $\phi \in C_b^0 (X)$ ,  $\kappa (\phi)$  is continuous. Call the space of continuous weighted canonicalizations  $\mathrm{WCan}_G^{\mathrm{cts}}(X)$ . We similarly define continuous weighted (orbit) canonicalizations and weighted frames.
+
+Proposition 2.2. A weighted canonicalization  $\kappa$  is continuous if and only if for every convergent sequence  $x_{n}\to x$  in  $X$ , the sequence of measures  $\kappa_{x_n}$  converges weakly to  $\kappa_{x}$ .
+
+It turns out, that for energy canonicalizations this completely provides a characterization of which energies give rise to continuous weighted canonicalizations.
+
+Lemma 2.3. (Lemma G.7) Let  $E$  be a lower semi-continuous energy function. Then  $\kappa_{E}$  is continuous if and only if for every convergent sequence  $x_{n} \to x$  we have  $\operatorname{Li} \mathcal{M}_{E}(x_{n}) = \mathcal{M}_{E}(x)$ .
+
+# 3 PRACTICAL CONSIDERATIONS
+
+In Section 2.3 we proposed energy canonicalization as a constructive way of achieving equivariance, but a question remains - how to choose the energy? Unfortunately there is no unique way to choose the energy or the optimization routine to find the minima. In this section we instead outline key considerations to guide these choices, with Section 4.1 serving as a guiding example. In complex settings of Sections 4.2 and 4.3.1 we choose it to be a neural network, as explained in Appendix B, while for Section 4.1 and appendices D.3 and D.4, the energy is hand-crafted.
+
+As we may not directly have access to parameterizations of the orbits for Equation (2), we instead implement group descent algorithms for Equation (1), e.g. illustrated in Algorithm 1, and then consider the resulting canonicalizations, as this are much smaller than frames and often.
+
+In what follows, we clarify the interplay between the four primary problem aspects. We assume to be given a specific choice of group and action on a domain, as well as information about the domain itself (either through samples or in the case of synthetic data, full knowledge is available). Based on these, we need to construct two things: (1) the energy function, based on the domain information; and (2) the group-based optimization routine for the problem in Equation (2), based on the domain information and the energy choice (potentially leading to problem relaxation as e.g. in for iterative optimization as e.g. in Appendix D.3).
+
+<table><tr><td>Algorithm 1: Canonicalization with a global retraction</td></tr><tr><td>Data: Non-canonical input x</td></tr><tr><td>parameters: N, steps ηi, ξ0 ∈ g, Energy E: X → R, Retraction τ: g → G</td></tr><tr><td>Result: Canonicalized input x̂ = g-1·x; inverse g-1; canonicalizing group element g.</td></tr><tr><td># Do gradient descent on ξ ∈ g</td></tr><tr><td>ξ ← ξ0;</td></tr><tr><td>for i = 0...N do</td></tr><tr><td>|ξ ← ξ - ηi∇ξE(τ(ξ) · x)</td></tr><tr><td>end</td></tr><tr><td>return τ(ξ) · x; τ(ξ); [τ(ξ)]-1</td></tr></table>
+
+Section 4.3.1 or improved initialization
+
+Domain information Models trained on a specific domain oftentimes do not generalize beyond their training data. To ensure the canonicalized input remains within the domain of the trained model, domain-specific information should be directly incorporated into the energy function. To give a concrete example, consider a setting in which some neural network is trained on samples living on a compact set  $\mathcal{M}$ . In such a setting we would only be interested in canonicalizations, which are either close to the set, or live on the set itself. Thus, choosing the energy function to include the characteristic function of the set  $\chi_{\mathcal{M}}$  is particularly suitable when the action is regular (or transitive). Otherwise, there may not be any elements in the set  $\mathcal{M}$  that can be mapped to, in which case a relaxation using the distance function  $d_{\mathcal{M}}(x) = \min_{y \in \mathcal{M}} \| x - y \|$  can be used. Such choices turn out to be beneficial, especially in the setting of Section 4.3.
+
+In the case of data sampled from an underlying distribution  $x \sim p_{\mathcal{X}}$ , one natural choice would be to consider as energy function the negative log likelihood  $E(x) = -\log p_{\mathcal{X}}(x)$ , which may be available in closed form if trained on synthetic data. Otherwise, an approximation can be used. For this, there exist quite a few choices, including normalizing flows (Dinh et al., 2016) or score-based diffusion models (Song et al., 2020) to approximate the true likelihood or its gradient. In this paper, for examples in Sections 4.2 and 4.3.1 we opt for a simpler variational autoencoder (Kingma, 2013), using the ELBO, being a lower bound, instead of the likelihood for the energy.
+
+Non-convexity and Flatness Even with a convex energy, minimization with respect to  $g \in G$  introduces non-convexity. The orbit  $Gx$  is often non-convex, e.g. see Figure 5, where this is a circle. Invoking a parameterization of a group element through its Lie algebra and considering its action on the space leads to further non-convexity. In the worst cases, such as those considered in PDE examples in Section 4.3, we may not have access to a global group parameterization, significantly complicating the process. What is worse, the energy may exhibit many flat regions, with the gradients being noisy, as in Figure 5.
+
+Thus the problem requires the use of non-convex optimization methods. We find, that in all examples in Section 3, utilizing a multi-initialization strategy with a fixed number of gradient steps is sufficient for convergence. Alternatively, relaxation to the problem in Equation (2) to be performed over  $x$ , subject to an orbit distance constraint can be used.
+
+![](images/2b0ed8b5c2cd8692344531920cd610157dc9b7e0e52859171c5b681818b7de64.jpg)  
+Figure 4:  $t$ -SNE plots: For MNIST, homNIST and canonicalized homNIST (Section 4.2).
+
+![](images/a58fa500bcd54e2f0943b6adb5f074c96b6b8068e5717d5f253481c266254ee1.jpg)
+
+![](images/eabd089e92350c51635da3a79b5de113bb27885d2fdb9de4cc713140ed93115a.jpg)
+
+To address the problem of flat zero-likelihood regions, alternative energy functions can be considered, e.g. empirical distance  $E(x) = \mathbb{E}_{X\sim p_X}d(X,x)$ . However, in Sections 4.2 and 4.3.1 we make a simpler choice to utilize extra regularization. Due to the difficulty in handcrafting regularization for various specific cases, we utilize learned convex regularization, trained adversariably on real and partially transformed data, based on the recent proposals in adversarial regularization Mukherjee et al. (2024); Shumaylov et al. (2024; 2023). Ideally, the regularization would also ensure uniqueness of minimizers to the problem in Equation (2), however it is not clear how to achieve this. Generically, we would want to construct an energy such that level sets are transverse to group orbits, similar to Panigrahi & Mondal (2024), which is indeed achieved by utilizing adversarial regularization.
+
+Non-parameterizability Moving from compact groups to non-compact presents further challenges. Even with access to the exponential mapping, it may not be surjective, making global parameterization of the group using the Lie algebra potentially impossible, if the group is not Noetherian. However, Lie algebras of many PDE symmetry groups are solvable, which can be used for group parameterizations that simplify the optimization problem: if the Lie algebra is solvable, any group element can be decomposed using the derived subalgebras (Varadarajan, 2013, Theorem 3.18.11) as a train of exponentials. For example, for the 1D heat equation (Appendix D.3), we can represent any element of the group as  $\exp(a_6v_6)\ldots\exp(a_1v_1)$ , reducing optimization over the group to optimization over  $(a_{1},\ldots,a_{6})\in\mathbb{R}^{\dim(\mathfrak{g})}$ . In this case one can utilize Algorithm 1.
+
+When the algebra is not solvable, one has to revert to Lie algebra based descent schemes. However, for Lie point symmetry groups, oftentimes only actions of specific basis vectors of the Lie algebra are available, making it necessary to use a coordinate descent (CD) approach. CD has received considerable attention in classical optimization (see Wright (2015)) and has recently been studied in geometric settings such as Riemannian manifolds and Lie groups (Han et al., 2024). The algorithms for both of these cases can be found in Algorithms 2 and 3. If a global parameterization is available, then classical schemes may be utilized in the same way (Lezcano Casado, 2019; Absil et al., 2008).
+
+# 4 EXPERIMENTS
+
+# 4.1 2D TOY EXAMPLE
+
+To illustrate the universality of the approach, we consider one simple two dimensional example without any machine-learned components. We consider the problem of classification under inherent rotation invariance illustrated on Figure 5. To mimic a realistic setting, we assume to have samples only. In this case, we can approximate the density using kernel density estimation with a Gaussian kernel, illustrated in the second column on Figure 5. As one would expect, without knowledge of the symmetries of the distribution, the approximation is far from the ground truth. In what follows, the energy is chosen to be the negative log-likelihood of the approximated distribution.
+
+This example illustrates the main issue with energy based canonicalization, which also extensively arises in more complex examples: even when the energy is convex, it is not convex when viewed as a function of  $g$ . This results in a non-convex optimization problem with (potentially many) local
+
+minima, which one can get stuck in when running standard descent schemes, and as discussed in Section 3, one has to revert to more complex optimization routines. In this case, as shown in Figure 5, it is possible to find global minimizers, despite the non-convexity, by running gradient descent for a fixed number of iterations, but for a number of different initializations. The power of canonicalization can then be seen in Figure 1, turning a non-equivariant kNN classifier rotation equivariant.
+
+# 4.2 INVARIANT CLASSIFICATION
+
+To illustrate the effectiveness of LieLAC in achieving invariance of pre-trained models, we first test it on invariant image classification with respect to the groups of affine and homography transformations. We take MacDonald et al. (2022) as the baseline for comparison due to the similarity in the required knowledge of the group required for implementation.
+
+Here, we take a pre-trained convolutional classifier, trained on the original MNIST dataset. The combination with the canonicalizer, as described in Section 3, is then tested on the affine-perturbed MNIST
+
+Table 1: MNIST test accuracy for Affine and Homography groups, compared with affConv and homConv of MacDonald et al. (2022).  
+
+<table><tr><td>Name</td><td>MNIST</td><td>affNIST</td></tr><tr><td>CNN</td><td>0.985</td><td>0.629</td></tr><tr><td>LieLAC [CNN]</td><td>0.979</td><td>0.972</td></tr><tr><td>affConv</td><td>0.982</td><td>0.943</td></tr><tr><td>Name</td><td>MNIST</td><td>homNIST</td></tr><tr><td>CNN</td><td>0.985</td><td>0.644</td></tr><tr><td>LieLAC [CNN]</td><td>0.982</td><td>0.960</td></tr><tr><td>homConv</td><td>0.980</td><td>0.927</td></tr></table>
+
+(affNIST) (Gu & Tresp, 2020) and the homography-perturbed MNIST (homNIST) (MacDonald et al., 2022). The overall performance results for this example are reported in Table 1. We can see that the canonicalization based approach achieves improved equivariance compared to the baseline, as illustrated by improved test accuracy on the datasets. We can also directly analyse achieved equivariance, as it appears only through canonicalization. We provide examples, visualising achieved equivariance in Figure 6. We can also analyze the effect of canonicalization globally by considering a  $t$ -SNE plot (Van der Maaten & Hinton, 2008) in Figure 4, showing that energy canonicalization turns the dataset into the usual MNIST form, clearly separating the classes, unlike homNIST.
+
+# 4.3 EQUIVARIANT PDE EVOLUTION
+
+In analogy with Akhound-Sadegh et al. (2023), we first consider two simple examples with non-trivial symmetry groups: the 1D Heat and Burgers' equations, which also admit global parameterisations of the group elements (Koval & Popovych, 2023). Due to space restrictions, evaluation can be found in Appendix D.3 for the Heat and Appendix D.4 for the Burgers' equation. Full description of the experimental setting, further visualisations and theoretical background can also be found in the appendix. Examples of canonicalized solutions are shown in Figure 2.
+
+We explicitly emphasize that in this paper we only consider symmetries for initial value problems, where there is no dependence of  $g \cdot x$  and  $g \cdot t$  on  $u$ . I.e. for  $(x', t', u') = g \cdot (x, t, u)$ , we have  $x'(x, t, u) = x'(x, t)$ . In this case the action on  $x, t$  can be separated from the action on  $u$ , making the desired equivariance for the physics-informed neural operator to be:
+
+$$
+\mathcal {O} _ {\theta} \left[ u _ {0}, x _ {0}, t _ {0} \right] \left(x _ {f}, t _ {f}\right) = g \cdot \mathcal {O} _ {\theta} \left[ g ^ {- 1} \cdot u _ {0}, g ^ {- 1} \cdot x _ {0}, g ^ {- 1} \cdot t _ {0} \right] \left(g ^ {- 1} \cdot x _ {f}, g ^ {- 1} \cdot t _ {f}\right) \tag {3}
+$$
+
+# 4.3.1 ALLEN-CAHN EQUATION
+
+We test the POSEIDON (Herde et al., 2024) foundation model, fine-tuned on the Allen-Cahn equation (ACE) on data it was trained on, and on transformed data (using Lie-point symmetry group of ACE).
+
+$$
+u _ {t} = u _ {x x} + u _ {y y} - \epsilon^ {2} u (u ^ {2} - 1) \tag {4}
+$$
+
+We consider solving Equation (4) on a 2D domain with periodic boundary conditions for  $t \in [0,1]$ ,  $x \in \Omega = [0,1]^2$ . This equation admits a 4-dimensional Lie algebra, see Appendix D.5, with SE(2) as the point symmetry group.
+
+Choice of Energy In order to justify the choice of energy, we need to understand the POSEIDON (Herde et al., 2024) model itself. Firstly,  $x_0$  and  $t_0$  in Equation (3) are implicitly defined in the model to be fixed to the standard domain  $x_0 = \Omega$ , and  $t_0 = 0$ . As such, the only input is the initial condition  $u_0|_{\Omega}$ . Therefore, we are going to have to only consider transformations, that preserve  $x_0$  and  $t_0$ . This
+
+can be imposed directly within the energy canonicalization framework as described in Section 3 by considering the characteristic functions as follows:
+
+$$
+E _ {\mathrm {A C}} \left(x _ {0}, t _ {0}, u _ {0}, x _ {f}, t _ {f}\right) = \chi_ {\{\Omega \}} \left(x _ {0}\right) + \chi_ {\{0 \}} \left(t _ {0}\right) + \chi_ {[ 0, 1 ]} \left(t _ {f}\right) + \chi_ {\Omega} \left(x _ {f}\right) + E \left(u _ {0}\right) \tag {5}
+$$
+
+Note, that this is able to significantly reduce the computational cost of optimization, as the group generated by  $v_{4}$  in 28, resulting in rotations of the underlying domain, can be reduced to a discrete subgroup  $\mathrm{C}_4 \leq \mathrm{SO}(2)$ , as the only transformations such that  $g^{-1}\Omega = \Omega$ . Further information on the training and testing data is available in Appendix D.5. The overall performance is reported in Table 2.
+
+Finetuning As seen in prior works (Mondal et al., 2023), the canonicalized data may end up misaligned with the original data. For this reason finetuning of the model/canonicalization may be desired.
+
+Table 2: ACE Test error evaluated over 90 trajectories for in-distribution (ID) and out-of-distribution (OOD). All errors are  $(\times 10^{-4})$ .  
+
+<table><tr><td>Name \( \left( {\times {10}^{-4}}\right) \)</td><td>ID Error eq. (29)</td><td>OOD Error eq. (30)</td><td>Avg</td></tr><tr><td>POSEIDON</td><td>\( {6.93} \pm  {2.83} \)</td><td>\( {75.76} \pm  {6.80} \)</td><td>41.35</td></tr><tr><td>LieLAC [POSEIDON]</td><td>\( {16.69} \pm  {5.42} \)</td><td>\( {29.19} \pm  {5.64} \)</td><td>22.94</td></tr><tr><td>POSEIDON + ft (can).</td><td>\( {8.45} \pm  {2.87} \)</td><td>\( {20.09} \pm  {3.32} \)</td><td>14.27</td></tr><tr><td>LieLAC [POSEIDON+ ft.]</td><td>\( {10.23} \pm  {3.36} \)</td><td>\( {11.34} \pm  {2.92} \)</td><td>10.79</td></tr><tr><td>POSEIDON + ft. (data aug.)</td><td>\( {8.41} \pm  {3.48} \)</td><td>\( {8.50} \pm  {3.11} \)</td><td>8.46</td></tr></table>
+
+It is important to emphasize that finetuning
+
+is not necessary, and even without finetuning the out of distribution errors go down significantly, as seen in Table 2. All of the examples above, including kNN (Section 4.1), MNIST (Section 4.2), Heat and Burgers (Appendices D.3 and D.4) did not require finetuning. That is due to tasks not being as complex, and not requiring the same level of precision. POSEIDON, being pre-trained in a regressive manner, turns out to be very good to begin with, and finetuning is performed to illustrate that baseline accuracy of data-augmentation (Brandstetter et al., 2022) can be matched.
+
+While it may seem that finetuning the resulting operator on canonicalized data is as hard as augmenting the original data, it turns out to not be, as fine-tuning can be done on a very small data-pool (in this case only 100 trajectories). This stems from the energy minima in Equation (2) residing near, but not necessarily on, training data. If the orbits are well-sampled in training, finetuning is not necessary, e.g. seen in Section 4.2. However, based on the training data in eq. (29), the orbits under  $\mathrm{C_4}$  are well-sampled, while the orbits under  $\mathrm{SE}(2)$  are not, illustrated in Table 2, wherein the error falls 2/3-fold after finetuning. An example is shown on Figure 2c, with further illustrations in Figure 13.
+
+# 5 CONCLUSION
+
+In this paper we provided a unified theoretical framework for defining frames and canonicalizations over non-compact and infinite groups. We showed that energy-based canonicalization can be used constructively to obtain both, and provide a Lie algebra based approach for turning any pre-trained network equivariant. We showcased that the proposed method is effective in turning image classifiers invariant, as well as neural operators Lie point symmetry equivariant.
+
+Future work Our example in Section 4.3.1 illustrates the promise of the proposed framework for fine-tuning foundation models. Due to the typically high cost of (re-)training foundation models, further exploration of such applications is an important direction for future work.
+
+Of particular interest are "realistic" scientific machine learning applications: In such applications collecting sufficiently large training data sets may be prohibitively expensive or technically impossible, i.e., available data not sufficient for retraining existing models/ using standard fine-tuning approaches. Encoding symmetries explicitly allows for more efficient fine-tuning, which may be more accessible in settings with little data.
+
+Limitations Energy-based canonicalization suffers from slow inference (resulting in  $5 - 30 \times$  slowdown). Improvements to the optimization are possible via non-convex optimization techniques (e.g., simulated annealing (Baudoin et al., 2008; Rutenbar, 1989), particle swarm (Huang et al., 2023), consensus-based (Totzeck, 2021; Fornasier et al., 2024)), particularly applicable as Lie groups are oftentimes low-dimensional, while the ambient space is not. The majority of these, however, are not easily adaptable to the setting of Lie groups.
+
+Implementing Lie point symmetries in existing neural operators is difficult due to hard-coded boundary conditions. Future work should focus on foundation models handling initial/boundary conditions and parameters, or explore alternative PDE symmetries (Brandstetter et al., 2022). This could include studying infinite-dimensional symmetry groups, which would also naturally arise when studying equivariance to local transformations.
+
+# ACKNOWLEDGMENTS
+
+The authors would like to thank Olga Obolenets, Lucy Richman, Hong Ye Tan and Yury Korolev for useful discussions and feedback. ZS acknowledges support from the Cantab Capital Institute for the Mathematics of Information, Christs College and the Trinity Henry Barlow Scholarship scheme. FS acknowledges support from the EPSRC advanced career fellowship EP/V029428/1. MW was supported by NSF Awards DMS-2406905 and CBET-2112085, and a Sloan Research Fellowship. CBS acknowledges support from the Philip Leverhulme Prize, the Royal Society Wolfson Fellowship, the EPSRC advanced career fellowship EP/V029428/1, EPSRC Grants EP/S026045/1 and EP/T003553/1, EP/N014588/1, EP/T017961/1, the Wellcome Innovator Awards 215733/Z/19/Z and 221633/Z/20/Z, the European Union Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant agreement No. 777826 NoMADS, the Cantab Capital Institute for the Mathematics of Information and the Alan Turing Institute.
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+
+# Appendices
+
+# A Extended Background 19
+
+A.1 Equivariant convolutions 19  
+A.2 Physics-Informed Machine Learning 19  
+A.3 Why Lie point symmetries? 20  
+A.4 Symmetries of Partial Differential Equations 21  
+A.5 Deep Learning For PDEs 22  
+A.6 Lie Groups 22
+
+# B Choosing the energy 23
+
+# C Further MNIST Visualisation 24
+
+# D Further PDE discussion 26
+
+D.1 Canonicalization as a unifying view 26  
+D.2 On Deep Learning for PDEs 26  
+D.3 Heat equation 27  
+D.4 Burgers' equation 30  
+D.5 Allen-Cahn equation 32
+
+# E Measure Theory Background 35
+
+# F More on Frames and Canonicalization 36
+
+F.1 Weakly-Equivariant Frames 36  
+F.2 On projections 38  
+F.3 Energies and Closed Canonicalizations 38
+
+# G Proofs 38
+
+# H Overview of previous concepts 44
+
+H.1 Weighted Canonicalization 45  
+H.2 Energy Minimizing Canonicalization 46  
+H.3 Almost everywhere continuity is not as impossible 46
+
+# I Algorithms for solvable and non-solvable Lie algebras 48
+
+![](images/e4f3e01deafbe838777fb77eb02f125741554bed210e6229d88e159e1c211d9f.jpg)  
+Figure 5: LieLAC optimised sample from a mix of Gaussian distributions.
+
+# A EXTENDEDBACKGROUND
+
+# A.1 EQUIVARIANT CONVOLUTIONS
+
+An important fact that comes up repeatedly in the design of equivariant neural networks is that ordinary convolutions on Euclidean spaces can be generalized to any group on which we can integrate against an invariant measure (as is the case for any Lie group, for example), giving rise to equivariant linear maps in the form of group convolutions. Furthermore, converse results can be established, showing that a linear equivariant map (under general conditions) must take the form of a group convolution. This is the approach taken in Cohen & Welling (2016) for discrete groups of roto-translations acting on images. Since group convolutions are applied to signals defined on the group, it is necessary to apply an initial equivariant linear lifting operation to convert a signal on the original domain to a signal on the group. Incompatibility of continuous groups and grid-based discretizations of signals is a major issue in methods that take this approach, and for general groups, it is not necessarily clear how one should discretize the integral defining a group convolution. For Lie groups, there have been various proposals for approximation methods for these integrals based on finite samples, including the works in Bekkers (2020); Finzi et al. (2020b) and works following them, which focus more heavily on using the Lie algebra for approximating group convolutions (MacDonald et al., 2022; Kumar et al., 2024). The corresponding methods have been shown to result in more efficient use of training data and more robustness to transformations not seen during training, and have been applied successfully in areas such as medical image analysis (Bekkers et al., 2018; Winkels & Cohen, 2019), computational molecular sciences (Batzner et al., 2022; Batatia et al., 2022) and inverse problems in imaging (Celledoni et al., 2021; Chen et al., 2023). Beyond construction of equivariant neural networks, information about the symmetries can be extracted directly from data (Moskalev et al., 2022; Gruver et al.).
+
+It is worth noting that some specific groups allow for construction of more efficient equivariant architectures. For example, recent work on geometric and clifford algebra networks Zhdanov et al. (2024); Ruhe et al. (2024) provides an efficient parameterization for groups of symmetries of vector spaces, like Euclidean (Brehmer et al., 2024) and Minkowski groups (Spinner et al., 2024).
+
+# A.2 PHYSICS-INFORMED MACHINE LEARNING
+
+In a separate direction, deep neural networks are being increasingly used in scientific computing, particularly in simulating physical systems modeled by partial differential equations (PDEs). This surge in interest is largely driven by limitations of traditional numerical solvers, as they often struggle with demanding scenarios, such as those involving non-linear and non-smooth PDEs necessitating
+
+fine discretization, or high-dimensional problems where the curse of dimensionality kicks in (Han et al., 2018). This is further exacerbated if there is a need to solve PDEs across varying domain geometries, parameters, and initial/boundary conditions, requiring independent simulations for each unique configuration.
+
+The main prominent approach that has emerged is based on integrating physical laws and constraints into the learning process to solve PDEs (Raissi et al., 2019). These neural networks have been dubbed PINNs, and have been shown to be useful for solving forward and inverse problems for PDEs (Raissi et al., 2019; Sun et al., 2020; Zhu et al., 2019; Karumuri et al., 2020; Sirignano & Spiliopoulos, 2018), and regularizing learning (Zhang et al., 2024).
+
+PINNs are able to handle irregular domains and complex geometries, overcoming some of the challenges of classical methods. However, classical PINNs are not without limitations (Grossmann et al., 2023; Krishnapriyan et al., 2021). The original method is unable to handle varying parameters or input and boundary conditions (IBCs), and often faces challenges with training due to unbalanced gradients and non-convex loss landscapes, requiring careful hyperparameter tuning or preconditioning Müller & Zeinhofer (2024).
+
+To address the challenge of solving PDEs across diverse scenarios, neural operator models have emerged as a promising solution. Unlike standard DNNs, which approximate functions over finite-dimensional vector spaces, neural operators learn mappings between infinite-dimensional function spaces. These learned operators map parameters or IBCs to corresponding PDE solutions, overcoming the computational burden of performing independent simulations for each scenario. Combining such neural operator models with a physics informed loss has resulted in physics-informed neural operators (Li et al., 2021b; Wang et al., 2021b), oftentimes improving generalization and physical validity, compared to their physics un-informed variants. In what follows, we concern ourselves precisely with this class of approaches.
+
+# A.3 WHY LIE POINT SYMMETRIES?
+
+What is appealing about LPS, is that for a given PDE, under certain regularity conditions, all such (simply connected) symmetries can be derived in the form of one-parameter transformation groups (Olver, 1993). For many classic PDEs these have been derived and can e.g. be found in (Ibragimov, 1995). There further exists symbolic software packages, e.g. Baumann (1998), to derive symmetries for general PDEs.
+
+# A.4 SYMMETRIES OF PARTIAL DIFFERENTIAL EQUATIONS
+
+Partial differential equations (PDEs) are the most generic mathematical model used to describe dynamics of various systems in sciences. In what follows, we will be working primarily with open subsets of euclidean space, but these ideas generically extend to more complicated spaces. The formal presentation here closely follows Olver (1993), and an interested reader is referred there for more in-depth discussions of the concepts.
+
+This subsection introduces the concept of PDE point symmetries – transformations that map solutions of a given PDE to other solutions of the same PDE. To analyze these symmetries, traditionally focus is shifted to Jet spaces, where the problem of studying PDE symmetries is transformed into the simpler task of studying symmetries of algebraic equations. By employing one-parameter groups of transformations, it becomes possible to systematically derive all such symmetries by examining their infinitesimal actions.
+
+Suppose we are considering a system  $\Delta$  of  $n$ -th order differential equations involving  $p$  independent variables  $x = (x^{1},\ldots ,x^{p})\in X = \mathbb{R}^{p}$ , and  $q$  dependent variables  $u = (u^{1},\dots,u^{q})\in U = \mathbb{R}^{q}$ . PDEs define relations between various derivatives of the function, for denoting which we use the following multi-index notation for  $J = (j_{1},\ldots ,j_{k})$ , with  $\partial_Jf(x) = \frac{\partial^kf(x)}{\partial x^{j_1}\partial x^{j_2}\cdots\partial x^{j_k}}$ .
+
+Jet spaces and Prolongations: Consider  $f: X \to U$  smooth and let  $U_k$  be the Euclidean space, endowed with coordinates  $u_J^\alpha = \partial_J f^\alpha(x)$  so as to represent the above derivatives. Furthermore, set  $U^{(n)} = U \times U_1 \times \dots \times U_n$  to be the Cartesian product space, whose coordinates represent all the derivatives of functions  $u = f(x)$  of all orders from 0 to  $n$ . The total space  $X \times U^{(n)}$ , whose coordinates represent the independent variables, the dependent variables and the derivatives of the dependent variables up to order  $n$  is called the  $n$ -th order jet space of the underlying space  $X \times U$ . Given a smooth function  $u = f(x)$ , there is an induced function  $u^{(n)} = \operatorname{pr}^{(n)} f(x)$ , called the  $n$ -th prolongation of  $f$ , defined by the equations  $u_J^\alpha = \partial_J f^\alpha(x)$ . Thus  $\operatorname{pr}^{(n)} f$  is a function from  $X$  to the space  $U^{(n)}$ , and for each  $x$  in  $X$ ,  $\operatorname{pr}^{(n)} f(x)$  is a vector representing values of  $f$  and all its derivatives up to order  $n$  at the point  $x$ .
+
+PDE Symmetry groups: With the notation defined, we can now write down the PDE, given as a system of equations  $\Delta(x, u^{(n)}) = 0$ , with  $\Delta : X \times U^{(n)} \to \mathbb{R}^l$  smooth. The differential equations themselves tell where the given map  $\Delta$  vanishes on  $X \times U^{(n)}$ , and thus determine a subvariety
+
+$$
+S _ {\Delta} = \left\{\left(x, u ^ {(n)}\right): \Delta \left(x, u ^ {(n)}\right) = 0 \right\} \subset X \times U ^ {(n)}
+$$
+
+Solutions of the system will be of the form  $u = f(x)$ . A symmetry group of the system  $\Delta$  will be a local group of transformations,  $G^{\Delta}$ , acting on some open subset  $M \subset X \times U$  in such that "G transforms solutions of  $\Delta$  to other solutions of  $\Delta$ ".
+
+Prolongations of group actions and vector fields Now suppose  $G$  is a local group of transformations acting on an open subset  $M \subset X \times U$  of the space of independent and dependent variables. There is an induced local action of  $G$  on the  $n$ -jet space  $M^{(n)}$ , called the  $n$ -th prolongation of  $G$  denoted  $\operatorname{pr}^{(n)}G$ . This prolongation is defined so that it transforms the derivatives of functions  $u = f(x)$  into the corresponding derivatives of the transformed function  $\tilde{u} = \tilde{f}(\tilde{x})$ . Consider now  $\mathbf{v}$  a vector field on  $M$ , with corresponding (local) one-parameter group  $\exp(\varepsilon \mathbf{v})$ . The  $n$ -th prolongation of  $\mathbf{v}$ , denoted  $\operatorname{pr}^{(n)}\mathbf{v}$ , will be a vector field on the  $n$ -jet space  $M^{(n)}$ , and is defined to be the infinitesimal generator of the corresponding prolonged one-parameter group  $\operatorname{pr}^{(n)}[\exp(\varepsilon \mathbf{v})]$ . In other words,
+
+$$
+\left. \operatorname {p r} ^ {(n)} \right| _ {\left(x, u ^ {(n)}\right)} = \left. \frac {d}{d \varepsilon} \right| _ {\varepsilon = 0} \operatorname {p r} ^ {(n)} [ \exp (\varepsilon \mathbf {v}) ] \left(x, u ^ {(n)}\right)
+$$
+
+for any  $(x,u^{(n)})\in M^{(n)}$ . These can be derived using Olver (1993, Theorem 2.36).
+
+Theorem A.1 (2.31 in Olver (1993)). Suppose  $\Delta(x, u^{(n)}) = 0$ , is a system of differential equations of maximal rank defined over  $M \subset X \times U$ . If  $G$  is a local group of transformations acting on  $M$ , and
+
+$$
+\Pr^ {(n)} \mathrm {v} \left[ \Delta (x, u ^ {(n)}) \right] = 0, \quad \text {w h e n e v e r} \quad \Delta (x, u ^ {(n)}) = 0
+$$
+
+for every infinitesimal generator  $\mathbf{v}$  of  $G$ , then  $G$  is a symmetry group of the system.
+
+# A.5 DEEP LEARNING FOR PDES
+
+The abstract discussion of symmetries of PDEs above only concerned itself with transformations leaving the set of solutions invariant. In practice, we are often interested in problems, for which the solution is unique due to either initial or boundary conditions (IBCs). In this paper, we primarily concern ourselves with temporal PDEs in the form of
+
+$$
+\begin{array}{l} \Delta = u _ {t} + \mathcal {D} _ {x} [ u ] = 0, \text {a n d} u (0, x) = f (x), \quad x \in \Omega , \quad t \in [ 0, T ] \\ u (t, x) = g (t, x), \quad x \in \partial \Omega , \quad t \in [ 0, T ] \\ \end{array}
+$$
+
+where  $u(t,x) \in \mathbb{R}^{\dim u}$  is the solution to the PDE,  $t$  denotes the time, and  $x$  is a vector of spatial coordinates.  $\Omega \subset \mathbb{R}^{\dim x}$  is the domain and  $\mathcal{D}_x$  is a (potentially non-linear) differential operator.  $f(x)$  is known as the initial condition function and  $g(t,x)$  describes the boundary conditions. As discussed in Section 1.2, we are interested in physics-informed approaches, wherein  $u(t,x)$  is parameterised as a neural network  $u_{\theta}(t,x)$  with parameters  $\theta$ , and the network is trained by minimizing a loss comprised of two parts:  $\mathcal{L}(\theta) = \mathcal{L}_{\mathrm{PDE}} + \mathcal{L}_{\mathrm{Data}}$ . The first term encourages the network to satisfy the PDE, while data-fit is a supervised loss, enforcing IBC and ensuring a good fit with the training data. Specific forms of the losses, and further discussion can be found in Appendix D.2.
+
+# A.6 LIE GROUPS
+
+In this section we briefly recall some necessary notions from Lie group theory. For a more comprehensive introduction, see Kirillov (2008). A Lie group  $G$  is a group with a smooth manifold structure such that both the inverse and multiplication maps are smooth. A Lie algebra is a finite dimensional vector space  $V$  with an alternating bilinear map  $[\cdot, \cdot] : V \times V \to V$  called the Lie bracket, satisfying the Jacobi identity. Given a Lie group  $G$ , the tangent space at the identity  $\mathfrak{g} = T_eG$  naturally has the structure of a Lie algebra. A 1-parameter subgroup of  $G$  is simply a group homomorphism  $\mathbb{R} \to G$ , where  $\mathbb{R}$  is a group under addition. For every Lie algebra vector  $v$  there is a unique 1-parameter subgroup  $\phi_v$  such that  $\phi_v(0)$  is the identity and  $\phi_v'(0)(\partial_t) = v$ . There is a smooth map  $\exp : \mathfrak{g} \to G$  which sends  $v$  to  $\phi_v(1)$ . This has image contained inside  $G_0$ , the connected component of the identity  $e \in G$ . If  $G_0$  is compact then  $\exp$  surjects onto  $G_0$ , but in the non-compact case it doesn't necessarily surject.
+
+For a Lie group  $G$  acting smoothly on a manifold  $X$  we have a version of the orbit-stabilizer theorem.
+
+Theorem A.2. The stabilizer  $G_{x} \subseteq G$  is a closed Lie subgroup, and the natural map  $G / G_{x} \to X$  is an injective immersion with image  $Gx$ .
+
+# B CHOOSING THE ENERGY
+
+Based on discussion in Section 3, we wish to construct an energy, which contains information about the domain (or dataset) and the energy levels are transverse to orbits. The first we achieve by using a variational autoencoder, and choosing the ELBO as the energy itself; and second by training an appropriate (convex) adversarial regularizer. Thus the overall energy  $E = \mathcal{L} + \alpha R$ , for some constant  $\alpha > 0$  and terms below. We emphasize once again that this is simply a construction, which have observed to work well in practice. Effectively all that is required is to align minimas of such an energy with the training set - doing this directly is not that simple and should be a fruitful direction for future work.
+
+Variational Autoencoders Variational Autoencoders (VAEs) (Kingma, 2013) are a class of generative models that learn the underlying probability distribution of a dataset. VAEs consist of two main components:
+
+1. Encoder: This component maps an input data point  $x$  to a lower-dimensional latent representation  $z$ . Instead of directly mapping to a single point in latent space, the encoder outputs parameters for a probability distribution, typically a Gaussian distribution,  $q(z|x) = \mathcal{N}(z; \mu(x), \sigma^2(x))$ . Here,  $\mu(x)$  and  $\sigma^2(x)$  are the mean and variance of the latent distribution, respectively, and are outputs of the encoder network.  
+2. Decoder: This component takes a point in the latent space  $z$  and maps it back to the original data space, reconstructing the input data  $\hat{x}$ . This is achieved through a probability distribution  $p(x|z)$ , also often modeled as a Gaussian.
+
+VAEs are trained by maximizing a lower bound on the data log-likelihood, known as the Evidence Lower Bound (ELBO):
+
+$$
+\mathcal {L} (x) = \mathbb {E} _ {q _ {\theta} (z | x)} [ \log p _ {\theta} (x | z) ] - D _ {\mathrm {K L}} \left(q _ {\theta} (z | x) | | p (z)\right) \tag {6}
+$$
+
+The first term in the ELBO encourages the decoder to reconstruct the input data accurately. The second term is the Kullback-Leibler (KL) divergence between the learned latent distribution  $q(z|x)$  and a prior distribution  $p(z)$ , typically a standard normal distribution. This term acts as a regularizer, encouraging the latent space to be well-structured and preventing overfitting.
+
+Adversarial Regularization Adversarial regularization Lunz et al. (2018); Mukherjee et al. (2024) is a framework for solving inverse problems that incorporates a discriminator network to learn and impose data-driven regularization. The discriminator network is trained to distinguish between unregularized reconstructions and real solutions obtained from ground truth data. The regularizer is trained with the objective of favoring solutions that are similar to the ground-truth images in the training dataset and penalizing reconstructions with artifacts.
+
+The adversarial loss function for the regularizer can be formulated as:
+
+$$
+\mathcal {L} _ {\mathrm {a d v}} (R) = \mathbb {E} _ {p _ {X} (x)} [ R (x) ] - \mathbb {E} _ {p _ {n} (\hat {x})} [ R (\hat {x}) ] \tag {7}
+$$
+
+where  $R$  is the regularizer,  $x$  is a ground truth image,  $\hat{x}$  is a noisy measurement,  $p_{X}$  and  $p_{n}$  are the data distributions. This loss encourages the regularizer to produce a small output when a true image is given as input and a large output when it is presented with an unregularized reconstruction.
+
+In our setting, the noisy measurement distribution is chosen to be  $\hat{x} = g\cdot x$ , for  $x\sim p_{X}$  and  $g\in G$  sampled randomly. Thus the regularizer learns to discern between true looking images and group transformed ones - leading to level sets lying transverse to the orbits.
+
+Adversarial regularization has been shown to provide a flexible data-driven approach to impose priors in inverse problems, which lends it readily to be used in our problem.
+
+![](images/6ed18a4c3c0198d671fc5991909b39b8ba02a1f261a369189b82ef0bf89d5e4c.jpg)  
+(a) Affine MNIST examples  
+Figure 6: MNIST canonicalization images for both Affine and Homography groups, as described in Section 4.2. From left to right: Original MNIST image; Canonicalization of the original image; Original image distorted by a random homography transformation; Canonicalization of the distorted image. This figure illustrates the equivariance properties of energy canonicalization.
+
+![](images/4d017b3c0d887cd4bb0e70ce80ff0bcb60af9a9e07e1f7605652bf48f647d0a8.jpg)  
+(b) Homography MNIST examples
+
+# C FURTHER MNIST VISUALISATION
+
+Of particular interest are the failure modes of canonicalization, warranting further theoretical investigation. We present the failure modes in Figure 7c, and as we can see from the examples provided - in majority of the cases the reason why failure modes exists is due to existence of group transformations resulting in completely different digits! As such, the overall distribution is not exactly equivariant.
+
+![](images/0d99969067c964889e7960931d2ce994b29a136a2b3970ae44344211c998f5b3.jpg)  
+(c) Misclassification examples  
+Figure 7: Expanded Visualisations
+
+![](images/bb31417edfafaf7df17affcfd51a7c552d12d41d00c2430a6e5d793a0be8f4ba.jpg)  
+(a) TSNE post-affine canonicalization
+
+![](images/d273cdd90a462d521ba626e6074c6b91652b9e3056bf03760979b1a94ccb5f01.jpg)  
+Figure 8: TSNE plots
+
+![](images/6c59338272676f59c622c218f65489247a716ba8217a08f3ab8ff79778ecb234.jpg)  
+(b) TSNE post-homography canonicalization
+
+![](images/3995b975e215b5be89925753221048eb1537cbd18f98c65a7478ae1a54ed3b7c.jpg)
+
+# D FURTHER PDE DISCUSSION
+
+# D.1 CANONICALIZATION AS A UNIFYING VIEW
+
+The idea of putting the evolutionary equation and initial into a canonical form prior to evolving is not novel, and in fact quite standard in all numerical analysis textbooks. The first step in solving an ODE numerically is to normalize the equations and remove units and normalize the domain. For example, when solving a time evolution equation on the domain  $x \in [0,15]$ , one would first normalize the  $x$  variable by rescaling it to be in  $[0,1]$ , and solve on this domain instead. Floating point representations are much coarser for smaller values and as such, this can lead to lower numerical errors.
+
+Another example is the recent work on achieving exact unit equivariance - which is precisely scaling group canonicalization (Villar et al., 2023). A similar problem also often arises in the problems of image registration (Kanter & Lellmann, 2022) with an infinite-dimensional Lie algebra of diffeomorphisms. Another particular example being Procrustes analysis (Gower, 1975), when considering only the homography group.
+
+In this work, we present that canonicalization can be seen as the unifying approach, especially through the lens of energy canonicalization. Insights from the problems above can then be used to improve the optimization, and used for PDE symmetry groups.
+
+# D.2 ON DEEP LEARNING FOR PDES
+
+In what follows we demonstrate the effectiveness of canonicalization across three example PDEs Heat, Burgers, Allen-Cahn with baseline models taken from three recent state of the art physics-informed machine learning papers Akhound-Sadegh et al. (2023); Wang et al. (2021b); Herde et al. (2024) respectively. In each case we take the pre-trained neural PDE solver and demonstrate a failure mode when evaluating on out-of-distribution initial conditions. We then optimize for a group action derived from the Lie algebra based energy minimization and report results after the canonicalization technique is applied. Results and additional experimental details for each PDE are provided in the following sections. In PINNs, the PDE solution  $u(t,x)$  is parameterised as a neural network  $u_{\theta}(t,x)$  with parameters  $\theta$ . The loss function is then compromised of two parts, with different weightings:
+
+$$
+\mathcal {L} (\theta) = \alpha_ {\text {P I N N}} \mathcal {L} _ {\text {P D E}} + \gamma_ {\text {d a t a}} \mathcal {L} _ {\text {d a t a - f i t}}
+$$
+
+The first term is the physics-informed objective of Raissi et al. (2019), ensuring the neural network satisfies the PDE. The PDE loss is the calculated as the residual, with derivatives are calculated using automatic differentiation, as calculated on a finite set of points  $(t,x)_{1:N_{\mathrm{dom}}}$  which are sampled from inside the domain  $[0,T]\times \Omega$  to obtain the PDE loss:
+
+$$
+\mathcal {L} _ {\mathrm {P D E}} = \frac {1}{N _ {\mathrm {d o m}}} \sum_ {i = 1} ^ {N _ {\mathrm {d o m}}} \| \partial_ {t} u _ {\theta} \left(t _ {i}, x _ {i}\right) + \mathcal {D} _ {x} \left[ u _ {\theta} \left(t _ {i}, x _ {i}\right) \right] \| _ {2} ^ {2} \tag {8}
+$$
+
+The second term, is a supervised loss enforcing initial and boundary conditions:
+
+$$
+\mathcal {L} _ {\text {d a t a - f i t}} = \frac {1}{N _ {\mathrm {i c}}} \sum_ {i = 1} ^ {N _ {\mathrm {i c}}} \left\| u _ {\theta} \left(0, x _ {i} ^ {0}\right) - f \left(x _ {i} ^ {0}\right) \right\| _ {2} ^ {2} + \frac {1}{N _ {\mathrm {b c}}} \sum_ {i = 1} ^ {N _ {\mathrm {b c}}} \left\| u _ {\theta} \left(t _ {i} ^ {b}, x _ {i} ^ {b}\right) - g \left(t _ {i} ^ {b}, x _ {i} ^ {b}\right) \right\| _ {2} ^ {2}, \tag {9}
+$$
+
+with  $\left(x^{0}\right)_{1:N_{\mathrm{ic}}} \in \Omega$  sample at which the initial condition function  $f$  is evaluated.  $\left(t^{b}, x^{b}\right)_{1:N_{\mathrm{bc}}}$  are  $N_{\mathrm{bc}}$  points sampled on the boundary  $[0, T] \times \partial \Omega$ . The two loss terms are then weighted by  $\alpha_{\mathrm{PINN}}$  for PDE loss equation 8 and  $\gamma_{\mathrm{data}}$  for data-fit equation 9 to be consistent with Akhound-Sadegh et al. (2023). Within the field of neural operators, it is desirable that the trained network generalizes across various initial conditions, and the above mentioned loss is evaluated for a number of initial conditions  $N_{f,\mathrm{train}}$ .
+
+# D.3 HEAT EQUATION
+
+This and following sections primarily recap the relevant facts regarding the symmetry groups of the PDEs considered, which act to motivate the need for new equivariant architectures due to the non-compactness of the resulting symmetry groups, and the difficulty of constructing global parametrisations of either the action or the group considered.
+
+# D.3.1 POINT SYMMETRIES
+
+Simplest PDE example is the heat equation, admitting a non-trivial group of Lie point symmetries, beyond space isometries.
+
+$$
+u _ {t} - \nu u _ {x x} = 0 \tag {10}
+$$
+
+This equation admits the following 6-dimensional Lie algebra (note the typo in Akhound-Sadegh et al. (2023)), see Olver (1993); Koval & Popovych (2023):
+
+$$
+v _ {1} = \partial_ {x}, \quad v _ {2} = \partial_ {t}, \quad v _ {3} = \frac {1}{\nu} u \partial_ {u}, \quad v _ {4} = 2 t \partial_ {t} + x \partial_ {x} - \frac {1}{2} u \partial_ {u}, \tag {11}
+$$
+
+$$
+v _ {5} = t \partial_ {x} - \frac {1}{2 \nu} x u \partial_ {u}, \quad v _ {6} = t ^ {2} \partial_ {t} + t x \partial_ {x} - \frac {1}{4 \nu} (x ^ {2} + 2 \nu t) u \partial_ {u}. \tag {12}
+$$
+
+The resulting one parameter flows are:
+
+$$
+\exp \left[ v _ {1} \right] (\epsilon): \quad \tilde {t} = t, \qquad \tilde {x} = x + \epsilon , \quad \tilde {u} = u,
+$$
+
+$$
+\exp \left[ v _ {2} \right] (\epsilon): \quad \tilde {t} = t + \epsilon , \quad \tilde {x} = x, \qquad \tilde {u} = u,
+$$
+
+$$
+\exp \left[ v _ {3} \right] (\epsilon): \quad \tilde {t} = t, \qquad \tilde {x} = x, \qquad \tilde {u} = \mathrm {e} ^ {\epsilon / \nu} u,
+$$
+
+$$
+\exp \left[ v _ {4} \right] (\epsilon): \quad \tilde {t} = \mathrm {e} ^ {2 \epsilon} t, \quad \tilde {x} = \mathrm {e} ^ {\epsilon} x, \quad \tilde {u} = \mathrm {e} ^ {- \frac {1}{2} \epsilon} u,
+$$
+
+$$
+\exp \left[ v _ {5} \right] (\epsilon): \quad \tilde {t} = t, \quad \quad \tilde {x} = x + \epsilon t, \quad \tilde {u} = \mathrm {e} ^ {- \frac {1}{4 \nu} \left(\epsilon^ {2} t + 2 \epsilon x\right)} u,
+$$
+
+$$
+\exp [ v _ {6} ] (\epsilon): \quad \tilde {t} = \frac {t}{1 - \epsilon t}, \quad \tilde {x} = \frac {x}{1 - \epsilon t}, \quad \tilde {u} = \sqrt {| 1 - \epsilon t |} \mathrm {e} ^ {\frac {\epsilon x ^ {2}}{4 \nu (\epsilon t - 1)}} u,
+$$
+
+The only non-zero Lie brackets are:
+
+$$
+\left[ v _ {4}, v _ {2} \right] = - 2 v _ {2}, \quad \left[ v _ {4}, v _ {6} \right] = 2 v _ {6}, \quad \left[ v _ {2}, v _ {6} \right] = v _ {4}, \tag {13}
+$$
+
+$$
+\left[ v _ {2}, v _ {5} \right] = v _ {1}, \quad \left[ v _ {4}, v _ {5} \right] = v _ {5}, \quad \left[ v _ {4}, v _ {1} \right] = - v _ {1}, \quad \left[ v _ {6}, v _ {1} \right] = - v _ {5}, \tag {14}
+$$
+
+$$
+\left[ v _ {5}, v _ {1} \right] = \frac {1}{2} v _ {3}. \tag {15}
+$$
+
+In particular, these would be enough for LieLAC, however in this particular example it is possible to show more. To be precise, as shown in Koval & Popovych (2023), the global structure, as explained below, of the symmetry group (the finite simply connected component) is  $G^{H} = \mathrm{SL}(2,\mathbb{R})\ltimes_{\phi}\mathrm{H}(1,\mathbb{R})$ , where  $\mathrm{H}(1,\mathbb{R})$  is the rank one polarized Heisenberg group.
+
+Theorem D.1 (Koval & Popovych (2023) for  $\nu \neq 1$ ). The point symmetry pseudogroup  $G$  of the  $(1 + 1)$ -dimensional linear heat equation is constituted by the point transformations of the form
+
+$$
+\tilde {t} = \frac {\alpha t + \beta}{\gamma t + \delta}, \quad \tilde {x} = \frac {x + \lambda_ {1} t + \lambda_ {0}}{\gamma t + \delta},
+$$
+
+$$
+\tilde {u} = \sigma \sqrt {| \gamma t + \delta |} \exp \left(\frac {\gamma (x + \lambda_ {1} t + \lambda_ {0}) ^ {2}}{4 \nu (\gamma t + \delta)} - \frac {\lambda_ {1}}{2 \nu} x - \frac {\lambda_ {1} ^ {2}}{4 \nu} t\right) (u + h (t, x)),
+$$
+
+where  $\alpha, \beta, \gamma, \delta, \lambda_1, \lambda_0$  and  $\sigma$  are arbitrary constants with  $\alpha \delta - \beta \gamma = 1$  and  $\sigma \neq 0$ , and  $h$  is an arbitrary solution of the heat equation.
+
+We now describe the product on  $G$ . First we note that there is a subgroup  $R$  consisting of those elements for which  $\lambda_1 = \lambda_0 = 0$  and  $\sigma = 1$ . This is evidently isomorphic to  $\operatorname{SL}(2, \mathbb{R})$ . Therefore we shall group the  $\alpha, \beta, \gamma, \delta$  terms into a matrix  $A = \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix}$ . Further we note that there are two connected components of this group, corresponding to sign of  $\sigma$ , and the group can be written as a direct product corresponding to this sign
+
+$$
+G = G ^ {\prime} \times \mathbb {Z} _ {2} \tag {16}
+$$
+
+Thus we may simply describe the product on  $G'$ . For this we may assume that the two elements we multiply have  $\sigma > 0$ . For clarity, we shall write out the product rule in terms of the logarithm of  $\sigma$ . The product can now be written as
+
+$$
+\left(A ^ {\prime}, \lambda_ {1} ^ {\prime}, \lambda_ {0} ^ {\prime}, \ln \sigma^ {\prime}\right) \cdot \left(A, \lambda_ {1}, \lambda_ {0}, \ln \sigma\right) = \left(A ^ {\prime} A, \lambda_ {1} + \alpha \lambda_ {1} ^ {\prime} + \gamma \lambda_ {0} ^ {\prime}, \lambda_ {0} + \beta \lambda_ {1} ^ {\prime} + \delta \lambda_ {0} ^ {\prime}, \right.
+$$
+
+$$
+\ln \sigma + \ln \sigma^ {\prime} + \frac {1}{4} (\lambda_ {0} ^ {\prime} \lambda_ {1} ^ {\prime} - (\alpha \lambda_ {1} ^ {\prime} + \gamma \lambda_ {0} ^ {\prime}) (\beta \lambda_ {1} ^ {\prime} + \delta \lambda_ {0} ^ {\prime})) - \frac {\lambda_ {0}}{2} (\alpha \lambda_ {1} ^ {\prime} + \gamma \lambda_ {0} ^ {\prime}))
+$$
+
+We see that the subgroup  $F$  consisting of those elements for which  $\alpha = \delta = 1, \beta = \gamma = 0$  and  $\sigma > 0$  is isomorphic to the rank one polarized Heisenberg group  $\mathrm{H}(1, \mathbb{R})$  via
+
+$$
+\left(\lambda_ {1}, \lambda_ {0}, \ln \sigma\right) \mapsto \left( \begin{array}{c c c} 1 & - \frac {1}{2} \lambda_ {1} & \ln \sigma \\ 0 & 1 & \lambda_ {0} \\ 0 & 0 & 1 \end{array} \right) \tag {17}
+$$
+
+where we use the standard embedding of  $\mathrm{H}(1,\mathbb{R})$  into the group of  $3\times 3$  real valued matrices. Note, that while the parameterization here looks the same as the one from Koval & Popovych (2023), it is different (resulting in different  $\phi$ ), as in that paper, the Heisenberg group and the polarized Heisenberg group are conflated, making the product rule unclear. Here we fix a specific form of the Heisenberg group as a subgroup of the group of  $3\times 3$  real matrices, with standard matrix multiplication as the product rule. The product formula can then be described in terms of  $F$  and  $R$  as well. Let  $\phi :\mathrm{SL}(2,\mathbb{R})\to \mathrm{Aut}(\mathrm{H}(1,\mathbb{R}))$  be the antihomomorphism defined by
+
+$$
+\phi (A) = \left( \begin{array}{c} \lambda_ {1} \\ \lambda_ {0} \\ \ln \sigma \end{array} \right) \mapsto \left( \begin{array}{c} \alpha \lambda_ {1} + \gamma \lambda_ {0} \\ \beta \lambda_ {1} + \delta \lambda_ {0} \\ \ln \sigma + \frac {1}{4} \lambda_ {0} \lambda_ {1} - \frac {1}{4} (\alpha \lambda_ {1} + \gamma \lambda_ {0}) (\beta \lambda_ {1} + \delta \lambda_ {0}) \end{array} \right) \tag {18}
+$$
+
+where write  $\mathrm{H}(1,\mathbb{R})$  using the identification  $F\cong \mathrm{H}(1,\mathbb{R})$ . Then we can write
+
+$$
+G ^ {\prime} = \operatorname {S L} (2, \mathbb {R}) \ltimes_ {\phi} \mathrm {H} (1, \mathbb {R}) \tag {19}
+$$
+
+As the semidirect product with an antihomomorphism is rarely written out in literature, we write out explicitly what this entails. We have that the underlying set of  $G'$  is  $\mathrm{SL}(2,\mathbb{R}) \times \mathrm{H}(1,\mathbb{R})$  and the product is
+
+$$
+\left(A ^ {\prime}, H ^ {\prime}\right) \cdot (A, H) = \left(A ^ {\prime} A, \phi (A) \left(H ^ {\prime}\right) H\right) \tag {20}
+$$
+
+In order to decanonicalize, we also require access to the inverse group element. Particularly, for our implementation we can either find the action of the inverse of  $\exp(e_6v_6)\ldots\exp(e_1v_1)$  as  $\exp(-e_1v_1)\ldots\exp(-e_6v_6)$ , but given access to a global parametrization of the group, we can write down an inverse of a given element in the form of  $A,H$ . This is  $\left(A^{-1},\phi(A^{-1})\left(H^{-1}\right)\right)$ . I.e. for  $\alpha,\beta,\gamma,\delta,\lambda_0,\lambda_1,\sigma$ , the inverse is
+
+$$
+A ^ {- 1} = \left( \begin{array}{c c} \delta & - \beta \\ - \gamma & \alpha \end{array} \right) \quad H ^ {- 1} = \left( \begin{array}{c c c} 1 & \frac {1}{2} \lambda_ {1} & - \ln \sigma - \frac {1}{2} (\lambda_ {1} \lambda_ {0}) \\ 0 & 1 & - \lambda_ {0} \\ 0 & 0 & 1 \end{array} \right)
+$$
+
+Writing this out in coordinates this becomes
+
+$$
+\delta , - \beta , - \gamma , \alpha , \beta \lambda_ {1} - \alpha \lambda_ {0}, - \delta \lambda_ {1} + \gamma \lambda_ {0}, - \ln \sigma - \frac {1}{4} \lambda_ {0} \lambda_ {1} - \frac {1}{4} (- \delta \lambda_ {1} + \gamma \lambda_ {0}) (\beta \lambda_ {1} - \alpha \lambda_ {0})
+$$
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">Heat (Original)</td><td colspan="2">Heat (+ data aug.)</td><td colspan="2">Burgers (Original)</td></tr><tr><td>ID (21)</td><td>OOD (21)</td><td>ID (21)</td><td>OOD (21)</td><td>ID (D.4.2)</td><td>OOD (D.4.2)</td></tr><tr><td>DeepONet</td><td>0.0498 ± 0.0072</td><td>0.6572 ± 0.1235</td><td>0.0504 ± 0.0014</td><td>0.0687 ± 0.0044</td><td>0.0832 ± 0.0547</td><td>0.8369 ± 0.0987</td></tr><tr><td>LieLAC [DeepONet]</td><td>0.0443 ± 0.0027</td><td>0.0435 ± 0.0017</td><td>0.0500 ± 0.0003</td><td>0.0500 ± 0.0003</td><td>0.0916 ± 0.0632</td><td>0.1006 ± 0.0637</td></tr></table>
+
+Table 3:  $L_{2}$  relative error for Heat equation and Burgers' equation averaged over  $[0,T]$ . Original refers to model trained on in-distribution (ID) data. For heat equation this corresponds to with fixed  $A_{k}^{\mathrm{Train}} = 1$  in Equation (21). Data aug refers to model trained on out of distribution data achieved via symmetry transformations, resulting in  $A_{k}^{\mathrm{Train}} \in \{0.5,5.0\}$ . For testing ID refers to  $A_{k}^{\mathrm{Test}} \in \{0.95,1.05\}$ , while OOD refers to  $A_{k}^{\mathrm{Test}} \in \{0.5,5.0\}$ . For Burgers' equation, ID corresponds to gaussian random fields as in Appendix D.4.2, while OOD refers to ID plus a constant shift.
+
+# D.3.2 NUMERICAL RESULTS
+
+For the numerical experiments, we consider solving Equation (10) with diffusivity  $\nu = 0.1$  on the spatial domain  $[0,2\pi]$ , with periodic boundary conditions for  $t \in [0,16]$ . We train a physics informed DeepONet (Lu et al., 2019; Wang et al., 2021b) with initial conditions of the form following Akhound-Sadegh et al. (2023); Brandstetter et al. (2022)
+
+$$
+u _ {0} (x) = \sum_ {i = 1} ^ {K} A _ {k} \sin \left(2 \pi l _ {k} x / L + \phi_ {k}\right). \tag {21}
+$$
+
+To focus on the effects of data augmentation and canonicalization, we train the model under two regimes. In the first regime (which we refer to as in-distribution), the amplitude parameter is fixed to  $A_{k} = 1$ , with the other parameters as  $K = 1$ ,  $l_{k} = 2$ ,  $\phi_{k} = 0$ , analogous to PINN training. In the second regime, the amplitude parameter is sampled from  $A \sim \mathcal{U}[0.5, 5.0]$ , representing a broader operator training distribution.
+
+The model is then trained using the physics loss with sampling weights of  $\alpha_{\mathrm{PINN}} = 150$ ,  $\gamma_{\mathrm{data}} = 20$ , we use a learning rate of 0.001 and batch size of 8 and train for 100k epochs following Akhound-Sadegh et al. (2023). Training data consists of  $N_{f,\mathrm{train}}$  initial condition functions, evaluated at  $N_{\mathrm{ic}} = 200$  points at  $t = 0$  to evaluate  $u_0$ ,  $N_{\mathrm{bc}} = 100$  points on the boundaries  $x = 0$  or  $x = 2\pi$  to enforce the periodic boundary conditions and finally  $N_{\mathrm{dom}} = 500$  points in the interior of the domain where the residual physics loss is enforced. The official implementation of DeepONet from the code and examples provided by the library DeepXDE (Lu et al., 2021) was used.
+
+We report results for both training regimes, averaged over 10 random seeds, in Table 3. The first rows in both tables show the accuracy of the standard DeepONet. In both cases, the first column shows a slight improvement with LieLAC canonicalizing to the average of the tight distribution. In Table 3 the second column shows DeepONet's failure to generalize when tested on  $Nf_{test} = 10$  test functions sampled  $A \sim \mathcal{U}[0.5, 5.0]$ , which is outside of the training range. Applying LieLAC, following the canonicalization pipeline shown in Figure 2a, restores test accuracy to in-distribution levels. In Table 3, extending the training range to  $A \sim \mathcal{U}[0.5, 5.0]$  improves generalization by exposing the model to a broader range of amplitudes. However, even in this setting, LieLAC outperforms data augmentation by leveraging the canonicalizing group action.
+
+**Choice of Energy** Given the initial conditions in Equation (21), we see that the distribution of initial conditions can be described by the bounding box of the jet. The energy is therefore chosen to be the distance to the trained domain. Letting  $\mathrm{jet}_0 = (u_0, x_0, t_0)$  and  $\mathrm{jet}_{\Omega} = (1, \Omega, 0)$ , the energy is:
+
+$$
+\begin{array}{l} E _ {\text {H e a t}} \left(x _ {0}, t _ {0}, u _ {0}, x _ {f}, t _ {f}\right) = \operatorname {d i s t} \left[ \max  \left(\mathrm {j e t} _ {0}\right), \max  \left(\mathrm {j e t} _ {\Omega}\right) \right] + \operatorname {d i s t} \left[ \min  \left(\mathrm {j e t} _ {0}\right), \min  \left(\mathrm {j e t} _ {\Omega}\right) \right] \tag {22} \\ + \operatorname {d i s t} [ x _ {f}, \Omega ] + \operatorname {d i s t} [ t _ {f}, 1 6 ] \\ \end{array}
+$$
+
+# D.3.3 FURTHER HEAT ILLUSTRATIONS
+
+Figure 9 shows canonicalicalisation of the initial conditions, out-of-distribution and canonicalized DeepONet predictions.
+
+![](images/1b788b720b5f02682cdb563255e038503cf16f6236a3c378d16098e399f3f487.jpg)  
+Figure 9: Canonicalization for the Heat equation given out of distribution  $u_{0}(x)$  and improved equivariant prediction
+
+![](images/42119d6d6494e9ffaad422ccb278accce17837389c10c8d1b048e09990e075cf.jpg)
+
+![](images/109c34b40e1140b1b866f7eecf7c3cf11782c2f2444771b98812d37d0e14e018.jpg)
+
+![](images/e143d91917b26436e81516b8d77659df98b055392f155c5602d4de5c1afaa9f7.jpg)
+
+![](images/bebcf4c1f2195b1e7e4112ffb590e4b673ead35b5ab593618f1aaf2eca4051dd.jpg)
+
+# D.4 BURGERS' EQUATION
+
+# D.4.1 POINT SYMMETRIES
+
+Burgers' Equation combines diffusion and non-linear advection, which can be related to the linear heat equation above via the Cole-Hopf transformation. The nonlinearity of the equation results in more complex dynamics, e.g. via shock formation, while still admitting non-trivial Lie point symmetries.
+
+$$
+u _ {t} + u u _ {x} - \nu u _ {x x} = 0 \tag {23}
+$$
+
+This equation admits the following 5-dimensional Lie algebra (note the typo in Akhound-Sadegh et al. (2023)), see Freire (2010); Koval & Popovych (2023):
+
+$$
+v _ {1} = \partial_ {x}, \quad v _ {3} = 2 t \partial_ {t} + x \partial_ {x} - u \partial_ {u}, \quad v _ {5} = t ^ {2} \partial_ {t} + t x \partial_ {x} + (x - t u) \partial_ {u}, \tag {24}
+$$
+
+$$
+v _ {2} = \partial_ {t}, \quad v _ {4} = t \partial_ {x} + \partial_ {u}.
+$$
+
+The resulting one parameter flows, for  $\epsilon$  arbitrary constant, are:
+
+$$
+\exp \left[ v _ {1} \right] (\epsilon): \quad \tilde {t} = t, \qquad \tilde {x} = x + \epsilon , \quad \tilde {u} = u,
+$$
+
+$$
+\exp \left[ v _ {2} \right] (\epsilon): \quad \tilde {t} = t + \epsilon , \quad \tilde {x} = x, \qquad \tilde {u} = u,
+$$
+
+$$
+\exp \left[ v _ {3} \right] (\epsilon): \quad \tilde {t} = \mathrm {e} ^ {2 \epsilon} t, \quad \tilde {x} = \mathrm {e} ^ {\epsilon} x, \quad \tilde {u} = \mathrm {e} ^ {- \epsilon} u,
+$$
+
+$$
+\exp \left[ v _ {4} \right] (\epsilon): \quad \tilde {t} = t, \qquad \tilde {x} = x + \epsilon t, \quad \tilde {u} = u + \epsilon ,
+$$
+
+$$
+\exp \left[ v _ {5} \right] (\epsilon): \quad \tilde {t} = \frac {t}{1 - \epsilon t}, \quad \tilde {x} = \frac {x}{1 - \epsilon t}, \quad \tilde {u} = u (1 - \epsilon t) + \epsilon x,
+$$
+
+The non-zero brackets are:
+
+$$
+\left[ v _ {3}, v _ {2} \right] = - 2 v _ {2}, \quad \left[ v _ {3}, v _ {5} \right] = 2 v _ {5}, \quad \left[ v _ {2}, v _ {5} \right] = v _ {3}, \tag {25}
+$$
+
+$$
+\left[ v _ {2}, v _ {4} \right] = v _ {1}, \quad \left[ v _ {3}, v _ {4} \right] = v _ {4}, \quad \left[ v _ {3}, v _ {1} \right] = - v _ {1}, \quad \left[ v _ {5}, v _ {1} \right] = - v _ {4} \tag {26}
+$$
+
+Once again, this would be enough for the LieLAC formulation, however in this particular example, we can say more:
+
+Theorem D.2 (Koval & Popovych (2023) for  $\nu \neq 1$ ). The point symmetry group  $G^{\mathrm{B}}$  of the Burgers' equation consists of the point transformations of the form
+
+$$
+\tilde {t} = \frac {\alpha t + \beta}{\gamma t + \delta}, \quad \tilde {x} = \frac {x + \lambda_ {1} t + \lambda_ {0}}{\gamma t + \delta}, \quad \tilde {u} = (\gamma t + \delta) u - \gamma x + \lambda_ {1} \delta - \lambda_ {0} \gamma ,
+$$
+
+where  $\alpha, \beta, \gamma, \delta, \lambda_1$  and  $\lambda_0$  are arbitrary constants with  $\alpha \delta - \beta \gamma = 1$ .
+
+And similar to before, the global structure of the group can be inferred, with the group  $G^{\mathrm{B}}$  isomorphic to the group  $\operatorname{SL}(2,\mathbb{R}) \ltimes_{\check{\varphi}} (\mathbb{R}^2, +)$ , where  $\check{\varphi} : \operatorname{SL}(2,\mathbb{R}) \to \operatorname{Aut}\left(\left(\mathbb{R}^2, +\right)\right)$  is the group antihomomorphism defined by  $\check{\varphi}(\alpha, \beta, \gamma, \delta) = (\lambda_1, \lambda_0) \mapsto (\lambda_1, \lambda_0) \varrho_1(\alpha, \beta, \gamma, \delta)$ . Thus, the group  $G^{\mathrm{B}}$  is connected.
+
+$$
+\varrho_ {1} = (\alpha , \beta , \gamma , \delta) _ {\alpha \delta - \beta \gamma = 1} \mapsto \left( \begin{array}{c c} \alpha & \beta \\ \gamma & \delta \end{array} \right), \quad (\lambda_ {1}, \lambda_ {0}) \mapsto (\lambda_ {1}, \lambda_ {0}).
+$$
+
+The standard conjugacy action of the subgroup  $\check{F} < G^{\mathrm{B}}$  on the normal subgroup  $\check{R} \triangleleft G^{\mathrm{B}}$  is given by  $(\tilde{\lambda}_1, \tilde{\lambda}_0) = (\lambda_1, \lambda_0) \varrho_1(\alpha, \beta, \gamma, \delta)$ . The inverse can be found similar to Appendix D.3.
+
+# D.4.2 NUMERICAL RESULTS
+
+For the numerical experiments, we consider solving Equation (23) on the domain  $x \in \Omega = [0,1]$ , with periodic boundary conditions on  $t \in [0,1]$ . An example of a canonicalized solution is shown in Figure 2, with further illustrations in Figures 10 and 11.
+
+For Burgers' equation we take the code and pretrained Physics Informed DeepONet (PI-DeepONet) model weights provided by Wang et al. (2021b). This includes a model trained on  $N_{f,\mathrm{train}} = 1000$  initial conditions sampled from a Gaussian random field (GRF) parameterised as  $\mathcal{N}\left(0,25^2 (-\Delta +5^2 I)^{-4}\right)$  on the periodic domain  $\Omega = [0,L]\times [0,T] = [0,1]\times [0,1]$  discretized by a meshgrid with  $101\times 101$  points.
+
+In order to sample out-of-distribution initial conditions from the GRF we apply a uniform 0.2 shift to  $u_{0}$  away from the mean of 0. Figure 11 shows surprisingly even this small perturbation is enough to break the pre-trained PI DeepONet. Interestingly in both in and out of distribution cases LieLAC learns a group action that is able to redistribute around zero mean, leading to a more accurate prediction.
+
+Choice of Energy Given the slightly different form of initial conditions to the heat equation, now sampled as GRFs we adjust the energy to focus on the bounding box of the domain and mean of  $u_{0}$ . The energy is therefore chosen to be the distance to the training domain. Letting  $\mathrm{trunk}_0 = (x_0,t_0)$  and  $\mathrm{trunk}_{\Omega} = (\Omega ,0)$ , the energy is:
+
+$$
+\begin{array}{l} E _ {\text {B u r g e r s}} \left(x _ {0}, t _ {0}, u _ {0}, x _ {f}, t _ {f}\right) = \operatorname {d i s t} \left[ \max  \left(x _ {0}\right) - \min  \left(x _ {0}\right), 1\right) \tag {27} \\ + \operatorname {d i s t} [ t _ {0}, \Omega ] + \operatorname {d i s t} [ \operatorname {m e a n} [ u _ {0} ], 0 ] \\ + \operatorname {d i s t} [ x _ {f}, \Omega ] + \operatorname {d i s t} [ t _ {f}, \Omega ] \\ \end{array}
+$$
+
+# D.4.3 FURTHER BURGERS' ILLUSTRATIONS
+
+Figure 10 shows canonicalicalisation of the initial conditions, out-of-distribution and canonicalized DeepONet predictions.
+
+![](images/070bc281bd7f9116e808be52d9487290f0661f08da3dbcfeb5d73241bbbdf229.jpg)  
+Figure 10: Illustration of Burgers' equation for in distribution GRF  $u_{0}(x)$
+
+![](images/6bffeca293db9f446e03ac49c9a125797c301a496c26d7a0d570247a01151efe.jpg)
+
+![](images/61f394f4fdb10ad6a11fb882c399c057a58b2d3b797c6eca571045a3c9e5fa8e.jpg)
+
+![](images/bfcd2b2c173336d2c9ff4488e8cf9df9d9728e7142e8fcc0ab30647e9d70f203.jpg)  
+Figure 11 shows canonicalicalisation of the initial conditions, out-of-distribution and canonicalized DeepONet predictions.  
+Figure 11: Canonicalization for Burgers' equation given out of distribution  $u_0(x)$  and improved equivariant prediction
+
+# D.5 ALLEN-CAHN EQUATION
+
+The Lie point symmetries of Equation (4) can be found either by hand or using the computational software of Baumann (1998). Both yield a 4-dimensional Lie algebra with basis vectors:
+
+$$
+v _ {1} = \partial_ {t}, \quad v _ {2} = \partial_ {x}, \quad v _ {3} = \partial_ {y}, \quad v _ {4} = - y \partial_ {x} + x \partial_ {y} \tag {28}
+$$
+
+Noting that there is no coupling between the space, time and the dependent components, we can identify this group as the connected component of the group of isometries of the underlying space. It is worth noting here that the full symmetry group can be observed to be  $\mathrm{E}(2)$ , while the group generated by vectors in 28 is  $\mathrm{SE}(2)$ , i.e. the identity connected component of  $\mathrm{E}(2)$ .
+
+Training data The POSEIDON (Herde et al., 2024) model is trained on random initial conditions in the form of:
+
+$$
+u _ {0} (x, y) = \sum_ {i, j = 1} ^ {K} a _ {i j} \cdot \left(i ^ {2} + j ^ {2}\right) ^ {- r} \sin (\pi i x) \sin (\pi j y), \quad \forall x, y \in (0, 1), \tag {29}
+$$
+
+where  $K$  is an integer drawn uniformly at random from [16, 32], and  $r \sim U_{[0.7,1.0]}$  and  $a_{ij} \sim U_{[-1,1]}$ . Based on this, one obvious limitation of the training data is that all sampled initial conditions are zero on the boundary. However, due to the semi-group property based training of Herde et al. (2024), as trajectories are sampled at different points, the values on the boundary are not guaranteed to remain zero over time. As a result, we expect the model to not be equivariant under the group of symmetries of ACE, as illustrated on Figure 12, but we do not have an obvious way to construct the energy. Based on the positive results in Section 4.2, we employ the same approach, by training a variational autoencoder and an adversarial regulariser. For the experiments, we considered the TINY POSEIDON model from Herde et al. (2024).
+
+Testing data To test the equivariance, we use a dataset of modified initial conditions, achieved by transformations of Equation (29).
+
+$$
+u _ {0} (x, y) = \sum_ {i, j = 1} ^ {K} a _ {i j} \cdot \left(i ^ {2} + j ^ {2}\right) ^ {- r} \sin \left(\pi i \{x - x _ {0} \}\right) \sin \left(\pi j \{y - y _ {0} \}\right), \quad \forall x, y \in (0, 1), \tag {30}
+$$
+
+where  $\{\cdot\}$  denotes the sawtooth function, with parameters as in Equation (29), with  $x_0, y_0 \sim U_{[0,1]}$ .
+
+![](images/1e300c5c7a0d352feae2ef965e0b5ccc43808b766e49a8e105068d3c0915452b.jpg)  
+Figure 12: Example of failure of equivariance of the original POSEIDON model.
+
+# D.5.1 FURTHER ACE ILLUSTRATIONS
+
+![](images/e551bdc20a4e6fc155d46748cca4bca7a298d28eec3fd34653b61aedf0299864.jpg)  
+Figure 13: Further canonicalization illustration for the ACE.
+
+# E MEASURE THEORY BACKGROUND
+
+Let  $X$  be a set. Then a  $\sigma$ -algebra  $\Sigma$  is a subset of  $2^{X}$  such that
+
+$X\in \Sigma$  
+- If  $A \in \Sigma$  then  $X \backslash A \in \Sigma$  
+- For any countable collection  $(A_{n})_{n = 1}^{\infty}$  of sets in  $\Sigma$ , the union  $\bigcup_{n = 1}^{\infty}A_{n}\in \Sigma$
+
+A set  $A \in \Sigma$  is called measurable. A pair  $(X, \Sigma)$  of a set and a  $\sigma$ -algebra is called a measure space. Suppose  $(X, \Sigma), (Y, \Sigma')$  are measure spaces. Then a function  $f: X \to Y$  is measurable if for every measurable set  $A \in \Sigma'$  the set  $f^{-1}(A)$  is measurable.
+
+Given a measure space, a measure on it is a set function  $\mu : \Sigma \to [0, \infty]$  such that
+
+$\mu (\varnothing) = 0$  
+- For all countable collections  $(A_{n})_{n = 1}^{\infty}$  of pairwise disjoint sets in  $\Sigma$
+
+$$
+\mu \left(\bigcup_ {n = 1} ^ {\infty} A _ {n}\right) = \sum_ {i = 1} ^ {\infty} \mu \left(A _ {n}\right) \tag {31}
+$$
+
+We may also define signed and complex measures, which simply replace the codomain of a measure with either  $\mathbb{R} \cup \{-\infty, +\infty\}$  or  $\mathbb{C}$  respectively.
+
+If  $f:(X,\Sigma)\to (Y,\Sigma^{\prime})$  is measurable, and  $\mu$  is a measure on  $(X,\Sigma)$ , we define the pushforward measure  $f_{*}\mu$  as follows
+
+$$
+f _ {*} \mu (A) = \mu \left(f ^ {- 1} (A)\right) \tag {32}
+$$
+
+There is always a measure on  $(X, \Sigma)$  which simply sends every measurable set to 0, we call this the zero measure. A finite measure is a measure  $\mu$  such that  $\mu(X) < \infty$ . A probability measure is a measure  $\mu$  such that  $\mu(X) = 1$ . Given a finite non-zero measure  $\mu$  on  $(X, \Sigma)$  we may always define a probability measure  $\mu'$  on  $(X, \Sigma)$  as follows
+
+$$
+\mu^ {\prime} (A) = \frac {\mu (A)}{\mu (X)} \tag {33}
+$$
+
+We will often call this the normalized measure associated to  $\mu$ .
+
+If  $X$  is a topological space there is a canonical  $\sigma$ -algebra associated to it, known as the Borel  $\sigma$ -algebra  $\mathcal{B}(X)$ . It is the smallest  $\sigma$ -algebra such that all open sets are measurable. Then any continuous function between  $X$  and  $Y$  is measurable when  $X$  and  $Y$  are equipped with their respective Borel  $\sigma$ -algebras. We often call the pair  $(X, \mathcal{B}(X))$  a Borel space and when clear we will simply refer to this pair by  $X$ .
+
+For a topological space  $S$  and a real number  $s \geq 0$  there is an associated Hausdorff measure of dimension  $s$  which we shall denote  $\mathcal{H}_S^s$ . To  $S$  we can also define what is known as the Hausdorff dimension,  $\dim_{\mathcal{H}}(S)$ , which is always non-negative. There is then a Hausdorff measure of that dimension, which we write as  $\mathcal{H}_S$ . All Borel sets are measurable for this measure. If  $S$  is compact then  $\mathcal{H}_S(S) < \infty$ , but it may still be the zero measure. If it is not zero then we may as above construct the normalized Hausdorff measure. In this case this will be a Radon measure. For  $S \cong [0,1]^n$  the Hausdorff dimension is  $n$  and the normalized Hausdorff measure is exactly the Lebesgue measure. Similarly for  $S$  finite and discrete, the Hausdorff dimension is 0 and the normalized Hausdorff measure is the normalized counting measure.
+
+The Hausdorff measure is important to consider because of Frostman's Lemma Mattila (1995).
+
+Lemma E.1 (Frostman's Lemma). Let  $A$  be a Borel subset of  $\mathbb{R}^n$  and let  $s > 0$ . Then the following are equivalent.
+
+$\mathcal{H}^s (A) > 0$  
+- There is an unsigned Borel measure  $\mu$  on  $\mathbb{R}^n$  satisfying  $\mu(A) > 0$  and such that
+
+$$
+\mu (B (x, r)) \leq r ^ {s} \tag {34}
+$$
+
+for all  $x\in \mathbb{R}^n,r > 0$
+
+![](images/149a6d71d60c0f7fbf5a582b8baaa8ab8f39bc115481473a9cd4cc814349dc5a.jpg)  
+Figure 14: This diagram illustrates connections between the various notions of frames and canonicalizations introduced previously and in this work. The top row are the finite frames and canonicalizations, which all map vertically into their weighted versions by taking normalized counting measures. Inside of each we have the sequentially closed subspaces of those that persevere continuity.
+
+Let  $A$  be compact with Hausdorff dimension greater than zero. Then we know that the Hausdorff measure at the Hausdorff dimension is finite. By Frostman's Lemma, the Hausdorff measure of  $A$  being zero means that we cannot find any probability measure on  $A$  that interacts nicely with the metric structure. This shows that our assumption of not allowing sets whose Hausdorff measure is the zero measure is reasonable.
+
+# F MORE ON FRAMES AND CANONICALIZATION
+
+Recall that a frame is defined as a function  $\mathcal{F}:X\to 2^G\backslash \varnothing$  such that for  $x\in X$  and  $g\in G$
+
+$$
+\mathcal {F} (g x) = g \mathcal {F} (x) \tag {35}
+$$
+
+Lemma F.1. Let  $\mathcal{F}$  be frame for the action of a group  $G$  on a set  $X$ . Then  $\forall x \in X$ ,  $\mathcal{F}(x)$  is a union of left cosets of  $G_x \subseteq G$ .
+
+Proof. Indeed, let  $x \in X$  and let  $g \in G_x$ . Then
+
+$$
+\mathcal {F} (x) = \mathcal {F} (g x) = g \mathcal {F} (x) \tag {36}
+$$
+
+Hence we see that
+
+$$
+\mathcal {F} (x) = G _ {x} \mathcal {F} (x) \tag {37}
+$$
+
+Therefore we see that  $\mathcal{F}(x)$  is a union of left cosets of  $G_{x}$ .
+
+# F.1 WEAKLY-EQUIVARIANT FRAMES
+
+In Dym et al. (2024) weakly equivariant frames are introduced. These are maps  $\mu : X \to \mathrm{PMeas}(G)$  satisfying equivariance up to averaging over stabilizers.
+
+In the review process it was noted that this bears similarity to Nguyen et al., and indeed, in our framework their method is exactly the weighted frame defined by minimizing the energy given by minimizing orbit distance. Because the symmetry groups they consider are compact and act algebraically, they are able to obtain more concrete descriptions of these frames than we are able to in the general case. We do expect that in the compact case with algebraic actions, similar results should hold for energy minimizing canonicalizations.
+
+First, for any  $x \in X$  and Radon probability measure  $\tau$  Dym et al. (2024) define for every measurable  $A \subseteq G$ ,
+
+$$
+\langle \tau \rangle_ {x} (A) = \int_ {s \in G _ {x}} s _ {*} \tau (A) d s \tag {38}
+$$
+
+where  $ds$  is the normalized Haar measure on  $G_{x}$ . For a frame  $\mu$  we let  $\overline{\mu}_x = \langle \mu_x\rangle_x$ . Then  $\mu$  is weakly equivariant if for all  $x\in X$  and  $g\in G$ ,
+
+$$
+\bar {\mu} _ {g x} = g _ {*} \bar {\mu} _ {x} \tag {39}
+$$
+
+This definition does not extend to the case where the action has non-compact stabilizers.
+
+To deal with this we propose imposing what we call  $\pi_*$ -equivariance. Again we consider simply a map  $\mu : X \to \mathrm{PMeas}(G)$ . For a point  $x \in X$ , let  $\pi_x : G \to G / G_x$  be the quotient. We require that
+
+$$
+\left(\pi_ {g x}\right) _ {*} \mu_ {g x} = \left(\pi_ {g x}\right) _ {*} g _ {*} \mu_ {x} \tag {40}
+$$
+
+Lemma F.2. If the stabilizer  $G_{x}$  is compact then  $\langle \mu \rangle_{x} = \langle \mu^{\prime}\rangle_{x}$  if and only if  $(\pi_x)_*\mu = (\pi_x)_*\mu '$
+
+Proof. For ease of notation let  $\pi = \pi_{x}$ . First suppose that  $\langle \mu \rangle_{x} = \langle \mu \rangle_{x}$ . Let  $A \subseteq G / G_{x}$  be a measurable set. Note that  $\pi^{-1}(A)$  is a union of cosets of  $G_{x}$ . Hence it is invariant under multiplication by elements in  $G_{x}$ . Therefore for all  $s \in G_{x}$  we have  $s^{-1}\pi^{-1}(A) = \pi^{-1}(A)$ . Thus
+
+$$
+\langle \mu \rangle_ {x} (A) = \int_ {s \in G _ {x}} s _ {*} \mu \left(\pi^ {- 1} (A)\right) d s = \int_ {s \in G _ {x}} \mu \left(\pi^ {- 1} (A)\right) d s = \mu \left(\pi^ {- 1} (A)\right) \tag {41}
+$$
+
+and similarly,  $\langle \mu^{\prime}\rangle_{x}(\pi^{-1}(A)) = \mu^{\prime}(\pi^{-1}(A))$ . Hence,
+
+$$
+\mu \left(\pi^ {- 1} (A)\right) = \mu^ {\prime} \left(\pi^ {- 1} (A)\right) \tag {42}
+$$
+
+Therefore,  $\pi_{*}\mu = \pi_{*}\mu^{\prime}$
+
+Now suppose that  $\pi_{*}\mu = \pi_{*}\mu^{\prime}$ . Then for any measurable set  $A\subseteq G$ , we have that
+
+$$
+\mu \left(\pi^ {- 1} (\pi (A))\right) = \pi_ {*} \mu (\pi (A)) = \pi_ {*} \mu^ {\prime} (\pi (A)) = \mu^ {\prime} \left(\pi^ {- 1} (\pi (A)\right) \tag {43}
+$$
+
+We have
+
+$$
+\pi^ {- 1} (\pi (A)) = \bigcup_ {s \in G _ {x}} s A \tag {44}
+$$
+
+Using Fubini's theorem and the fact that both  $s$  and  $\mu$  are probability measure, we may write Equation (38) as follows
+
+$$
+\begin{array}{l} \langle \mu \rangle_ {x} = \int_ {s \in G _ {x}} s _ {*} \mu (A) d s (45) \\ = \int_ {s \in G _ {x}} \int_ {G} \chi_ {A} (s g) d \mu (g) d s (46) \\ = \int_ {G} \int_ {s \in G _ {x}} \chi_ {A} (s g) d s d \mu (g) (47) \\ \end{array}
+$$
+
+Consider the inner integral as a function  $\phi$  on  $G$ . We claim that  $\phi$  is  $G_{x}$ -invariant. Indeed, let  $g' = s'g$  for some  $s' \in G_{x}$ . Then
+
+$$
+\phi \left(g ^ {\prime}\right) = s \left(A g ^ {- 1} \left(s ^ {\prime}\right) ^ {- 1}\right) = s \left(A g ^ {- 1}\right) = \phi (g) \tag {49}
+$$
+
+as  $s$  is the normalized Haar measure, hence is right-invariant. Therefore  $\phi$  factors through the quotient map  $\pi$ . Let  $\phi'$  be the corresponding function on  $G / G_x$ , then  $\phi = \phi' \circ \pi$ . Hence
+
+$$
+\begin{array}{l} \langle \mu \rangle_ {x} = \int_ {G} \phi (g) d \mu (g) (50) \\ = \int_ {G / G _ {x}} \phi^ {\prime} ([ g ]) d \left(\pi_ {*} \mu\right) ([ g ]) (51) \\ = \int_ {G / G _ {x}} \phi^ {\prime} ([ g ]) d \left(\pi_ {*} \mu^ {\prime}\right) ([ g ]) (52) \\ = \int_ {G} \phi (g) d \mu^ {\prime} (g) = \left\langle \mu^ {\prime} \right\rangle_ {x} (53) \\ \end{array}
+$$
+
+□
+
+Corollary F.2.1. If the  $G$ -action on  $X$  has compact stabilizers at each point, then  $\pi_*$ -equivariance is equivalent to weak equivariance.
+
+# F.2 ON PROJECTIONS
+
+For a weighted orbit canonicalization  $\mu, \mathcal{P}_{\mu}: B(X, \mathbb{R}) \to B(X, \mathbb{R})^G$  turns out to be a projection, as
+
+$$
+\begin{array}{l} \mathcal {P} _ {\mu} \mathcal {P} _ {\mu} (\phi) (x) = \int_ {X} \mathcal {P} _ {\mu} (\phi) (s) d \mu_ {x} (s) \\ = \int_ {G x} \underbrace {\mathcal {P} _ {\mu} (\phi) (s)} _ {\text {G - i n v a r i a n t}} d \mu_ {x} (s) = \mathcal {P} _ {\mu} (\phi) (x) \tag {54} \\ \end{array}
+$$
+
+# F.3 ENERGIES AND CLOSED CANONICALIZATIONS
+
+By the construction of energy minimizing frames, we have a function from energy functions to weighted closed canonicalizations. We now describe the image of this.
+
+Theorem F.3. Let  $\kappa \in \mathrm{WCCan}_G(X)$  satisfying the following:
+
+-  $\kappa_{x}$  is the normalized Hausdorff measure on its support.  
+- For all  $y \in \overline{Gy}$ ,  $\operatorname{supp} \kappa_y = \operatorname{supp} \kappa_x \cap \overline{Gx}$ .
+
+Then there is an energy function  $E$  such that  $\kappa_{E} = \kappa$ .
+
+Proof. Suppose  $\kappa$  is a closed canonicalization satisfying the requirements of the theorem. Let  $E_{\kappa}(x) = 1 - \chi_{\mathrm{supp}\kappa_x}(x)$ .
+
+Now we consider the minimum sets
+
+$$
+\mathcal {M} _ {E _ {\kappa}} (x) = \min  _ {y \in \overline {{G x}}} E _ {\kappa} (y) \tag {55}
+$$
+
+We have that  $\operatorname{supp} \kappa_x$  is a non-empty closed subset of  $\overline{Gx}$ , hence the minimum of  $E_{\kappa}$  over  $\overline{Gx}$  is zero. Therefore  $\mathcal{M}_{E_{\kappa}}(x) = E_{\kappa}^{-1}(0) \cap \overline{Gx}$ . In other words it is those  $y \in \overline{Gx}$  such that  $y \in \operatorname{supp} \kappa_y$ . Further, by assumption, for any  $y \in \overline{Gx}$ ,
+
+$$
+\operatorname {s u p p} \mu_ {y} = \operatorname {s u p p} \kappa_ {x} \cap \overline {{G y}} \tag {56}
+$$
+
+Therefore  $y \in \operatorname{supp} \kappa_y$  if and only if  $y \in \operatorname{supp} \kappa_x$ . Hence we see that
+
+$$
+\mathcal {M} _ {E _ {\kappa}} (x) = \operatorname {s u p p} \kappa_ {x} \tag {57}
+$$
+
+But the by the assumption that  $\kappa_{x}$  is simply the normalized Hausdorff measure on  $\operatorname{supp} \kappa_{x}$ , we get that the  $\kappa_{E_{\kappa}} = \kappa$ .
+
+# G PROOFS
+
+Proposition G.1.  $\mathcal{F}_E$  defines an equivariant frame, i.e.  $\mathcal{F}_E(gx) = g\mathcal{F}_E(x)$  for any  $g\in G$ ,  $x\in X$ .
+
+Proof. Suppose that  $x \in \mathcal{X}$  and  $g \in G$ . Then
+
+$$
+\begin{array}{l} \mathcal{F}_{E}(gx) = \operatorname *{arg  min}_{h\in G}E(h^{-1}gx) \\ = \operatorname * {a r g   m i n} _ {g h \text {s . t .} h \in G} E ((g h) ^ {- 1} g x) \\ = \operatorname * {a r g   m i n} _ {g h \text {s . t .} h \in G} E (h ^ {- 1} x) \\ = g\bigl[\operatorname *{arg  min}_{h\in G}E(h^{-1}x)\bigr ] = g\mathcal{F}(x). \\ \end{array}
+$$
+
+Here, we have simply used that group multiplication is bijective when either of the arguments is fixed.
+
+Proposition G.2.  $\mathcal{M}_E(x)$  is  $G$ -invariant and non-empty.
+
+Proof.  $\mathcal{M}_E(x)$  is  $G$ -invariant, as the infimum is taken over the orbit  $\overline{Gx}$ , and this orbit is  $G$ -invariant. By the coercivity assumption on  $E$ , this is non-empty. Indeed, let  $x \in X$  and let  $N$  be large. Let  $K$  be a compact set of  $X$  such that  $E(y) > N$  for  $y \in X \setminus K$ . Taking  $N$  large enough, we can assume that  $K \cap \overline{Gx}$  is non-empty. Then we see that  $\mathcal{M}_E(x) \subseteq K \cap \overline{Gx}$ .  $K \cap \overline{Gx}$  is a closed subset of the compact set  $K$ , hence is compact. Therefore  $E$  attains its minimum on this set, hence  $\mathcal{M}_E(x)$  is non-empty.
+
+Theorem G.3. Let  $G$  be a Lie group acting smoothly on  $X$  a manifold. Let  $j_{x}:G\to X$  be defined as  $g\mapsto g^{-1}\cdot x$ . Then we have a map  $\alpha :\mathrm{WFra}_G(X)\to \mathrm{WOCan}_G(X)$  defined by
+
+$$
+\alpha (\mathcal {F}) (x) = \left(j _ {x}\right) _ {*} (\mathcal {F} (x)) \tag {58}
+$$
+
+Then for any  $\mathcal{F},\mathcal{F}^{\prime}\in \mathrm{WFr}_{G}(X)$  such that  $\alpha (\mathcal{F}) = \alpha (\mathcal{F}^{\prime})$  we have that  $(\pi_x)_*\mathcal{F} = (\pi_x)_*\mathcal{F}^\prime$  for every  $x\in X$  where  $\pi_x:G\to G / G_x$ . If each  $G_{x}$  has a probability measure  $P_{x}$ , then  $\alpha$  has image exactly the set of weighted orbit canonicalizations with compact support.
+
+Proof. First we check that  $\alpha$  as defined above lands in  $\mathrm{WOCan}_G(X)$ . Let  $\mathcal{F} \in \mathrm{WFra}_G(X)$ . Then  $\alpha(\mathcal{F})(x)$  is  $G$ -invariant as for any measurable  $A \subseteq X$ .
+
+$$
+\begin{array}{l} \alpha (\mathcal {F}) (g x) (A) = \left(j _ {g x}\right) _ {*} (\mathcal {F} (g x)) (A) \\ = \mathcal {F} (g x) (\{h \in G \mid h ^ {- 1} \cdot (g x) \in A \} \\ = \mathcal {F} (g x) (\{h \in G \mid h ^ {- 1} \cdot (g x) \in A \}) \\ = \pi_ {*} (\mathcal {F} (g x)) \left(\pi \left(\left\{h \in G \mid h ^ {- 1} \cdot (g x) \in A _ {J} ^ {\prime}\right)\right)\right) \\ = \pi_ {*} g _ {*} (\mathcal {F} (x)) (\pi (\{h \in G \mid h ^ {- 1} \cdot (g x) \in A \})) \\ = g _ {*} (\mathcal {F} (x)) \left(\left\{h \in G \mid h ^ {- 1} \cdot (g x) \in A \right\}\right) \\ = \mathcal {F} (x) \left(\left\{h \in G \mid (g h) ^ {- 1} \cdot (g x) \in A \right\}\right) \\ = \mathcal {F} (x) \left(\left\{h \in G \mid h ^ {- 1} \cdot x \in A \right\}\right) = \alpha (\mathcal {F}) (x) (A) \\ \end{array}
+$$
+
+Thus we see that  $\alpha(\mathcal{F}) \in \mathrm{WCan}_G(X)$ , as  $\alpha(\mathcal{F})(x)(X) = \mathcal{F}(x)(G) = 1$ , as  $\mathcal{F}(x)$  is a probability measure. To check that it is a weighted orbit canonicalization, first note that it is clear that  $\operatorname{supp} \alpha(\mathcal{F})(x) \subseteq \overline{Gx}$ , as for any  $y \notin \overline{Gx}$  we can find an open neighborhood  $U$  of  $y$  such that  $j_x^{-1}(U) = \emptyset$  which is always  $\mathcal{F}(x)$ -measure zero. Suppose that  $y \in \overline{Gx} \backslash Gx$ .  $\mathcal{F}(x)$  is a weighted frame, hence it is supported on some compact set  $K \subseteq G$ . We claim that there is an open set  $U$  containing  $y$  such that  $j_x^{-1}(U) \subseteq G \backslash K$ . Suppose not, then for every open neighborhood  $U$  of  $y$  there is an  $g_U \in K$  such that  $j_x(g) \in U$ . Then the collection  $(g_U)$  over the neighborhoods of  $y$  forms a net in  $K$ . As  $K$  is compact, this net must have a convergent subsequence. Call this subsequence  $(g_n)$  and let  $g$  be the limit. As the action of  $G$  on  $X$  is continuous we have
+
+$$
+g \cdot x = \lim  _ {n \rightarrow \infty} g _ {n} \cdot x \tag {59}
+$$
+
+For any  $U$  an open neighborhood of  $y$  we may, by definition of a net, find an  $N$  such that for any  $n \geq N$ ,  $g_n \cdot x \in U$ . As  $X$  is Hausdorff, we then must have that the limit  $\lim g_n \cdot x$  is simply  $y$ . But this contradicts  $y \notin Gx$ . Hence there is some open neighborhood  $U$  of  $y$  such that  $j_x^{-1}(U) \subseteq G \setminus K$ . Then  $j_x^{-1}(U)$  is measure zero with respect to  $\mathcal{F}(x)$ . Hence  $\alpha(\mathcal{F})(x)(U) = 0$ , so  $y \notin \operatorname{supp} \alpha(\mathcal{F})$ . Therefore we see that  $\alpha(\mathcal{F}) \in \mathrm{WOCan}_G(X)$ .
+
+Suppose that  $\mathcal{F},\mathcal{F}^{\prime}$  are weighted frames such that  $\alpha (\mathcal{F}) = \alpha (\mathcal{F}^{\prime})$  . Let  $x\in X$  . Then for any measurable  $A\subseteq X$
+
+$$
+\mathcal {F} (x) \left(j _ {x} ^ {- 1} (A)\right) = \alpha (\mathcal {F}) (x) (A) = \alpha \left(\mathcal {F} ^ {\prime}\right) (x) (A) = \mathcal {F} ^ {\prime} (x) \left(j _ {x} ^ {- 1} (A)\right) \tag {60}
+$$
+
+Both  $\mathcal{F}$  and  $\mathcal{F}'$  are compactly supported, say on compact subsets  $K$ ,  $K'$  respectively. Consider the natural map  $\overline{j_x}: G / G_x \to X$ . This is injective for group-theoretic reasons. Let  $\pi: G \to G / G_x$  be the quotient map. Then  $\pi(K)$  and  $\pi(K')$  are compact subsets of  $G / G_x$ , as they are continuous images of compacts. Now as  $\pi(K)$  and  $\pi(K')$  are compacts which inject into a Hausdorff space  $X$ ,  $\overline{j_x}$  restricts to homeomorphisms  $\pi(K) \cong \overline{j_x}(\pi(K))$  and similar for  $K'$ . Using Equation (60) we see that for any  $U \subseteq G / G_x$  open we have
+
+$$
+\mathcal {F} (x) \left(\pi^ {- 1} (U)\right) = \mathcal {F} (x) \left(\pi^ {- 1} (U)\right) \tag {61}
+$$
+
+Therefore  $\pi_{*}\mathcal{F}$  and  $\pi_{*}\mathcal{F}^{\prime}$  agree on the open sets of  $G / G_{x}$ , but then that implies that they agree on the entire Borel  $\sigma$ -algebra of  $G / G_{x}$ . Hence  $\pi_{*}\mathcal{F} = \pi_{*}\mathcal{F}^{\prime}$  as desired.
+
+Finally, let  $v \in \mathrm{WOCan}_G(X)$  have compact support. Let  $x \in X$ . Then  $v_x$  is some probability measure on  $X$  with supp  $v_x \subseteq K \subseteq G_x$  for some compact  $K \subseteq G_x$ . By Theorem A.2 we have that  $\overline{j_x} : G / G_x \to X$  is a immersion with image  $Gx$ .  $\overline{j_x}$  restricts to a homeomorphism on  $K$ , and using this we can define a measure  $v_x'$  on  $G / G_x$  which agrees with  $v_x$  under this homeomorphism. It is supported on the preimage  $K'$  of  $K$  which is also compact.  $\pi : G \to G / G_x$  is a principal bundle with fiber  $G_x$ . Therefore, locally on  $G / G_x$ ,  $\pi$  is the trivial fibration. We may cover  $K'$  by finitely many open sets over which  $\pi$  is trivial, call them  $U_1, \ldots, U_n$ . Let  $V_i = \pi^{-1}(U_i) \cong U_i \times G_x$ . Now define
+
+$$
+V _ {i} ^ {\prime} = V _ {i} \backslash \left(\bigcup_ {j <   i} V _ {j}\right) \tag {62}
+$$
+
+Then these are disjoint measurable sets which cover  $\pi^{-1}(K')$ . They are all trivially fibered over the base. Now we define a measure  $\nu$  using these,  $\nu$  will be supported on  $\pi^{-1}(K')$ . Let  $A$  be a measurable set in  $G$ . Firstly we may assume that  $A \subseteq \pi^{-1}(K')$ . Let  $A_i = A \cap V_i'$ . Then the  $A_i$  are disjoint measurable sets which cover  $A$ . So the value of  $\nu$  on  $A$  is determined by the value of  $\nu$  on each  $A_i$ . We define this value to be
+
+$$
+\nu \left(A _ {i}\right) = \left(v _ {x} ^ {\prime} \times P _ {x}\right) \left(A _ {i}\right) \tag {63}
+$$
+
+where here we use the homeomorphism  $V_{i}^{\prime} \cong \pi(V_{i}^{\prime}) \times G_{x}$  to put the product measure on this set.  $\pi_{*}\nu = v_{x}^{\prime}$ , and this is equivariant as  $G_{gx} = g^{-1}G_{x}$ . Hence  $\nu$  is  $\pi_{*}$ -equivariant.
+
+Note that for the above theorem to work, we need to assume that each  $G_{x}$  has a probability measure. This is automatic in the case that each  $G_{x}$  is compact, as in that case we may take that normalized Haar measure on it. It will also hold in the case that each  $G_{x}$  has non-zero Hausdorff measure.
+
+Theorem G.4. The sequential closure of  $\mathrm{WOCan}_G(X)$  is  $\mathrm{WCCan}_G(X)$
+
+Proof. First we show  $\mathrm{WCCan}_G(X)$  is sequentially closed. Suppose that  $(f^n)$  is a sequence in  $\mathrm{WCCan}_G(X)$  converging to  $f\in \mathrm{WCan}_G(X)$ . Fix  $x\in X$  and  $y\in X\backslash \overline{Gx}$ . Then it is sufficient to show that  $y\notin \operatorname {supp}f_x$ . As  $X$  is normal Hausdorff, by Urysohn's lemma there exists continuous  $\phi :X\to [0,1]$  such that  $\phi (\overline{Gx}) = 1$  and  $\phi (\{y\}) = 0$ .
+
+As  $f^n \to f$ , for any  $\varepsilon > 0$ , we can find  $N$  such that for any  $n \geq N$  we have the following
+
+$$
+\left\| f _ {n} (\phi) - f (\phi) \right\| <   \varepsilon \tag {64}
+$$
+
+We may easily compute  $f_{n}(\phi)(x)$  by the fact that  $\operatorname{supp} f_{n}(x) \subseteq \overline{Gx}$ .
+
+$$
+f _ {n} (\phi) (x) = \int_ {\overline {{G x}}} \phi d f _ {x} ^ {n} = 1 \tag {65}
+$$
+
+Similarly,
+
+$$
+f _ {n} (\phi) (y) = \int_ {\overline {{G y}}} \phi d f _ {y} ^ {n} = 0 \tag {66}
+$$
+
+Hence, by taking the limit we compute that  $f(\phi)(x) = 1$  and  $f(\phi)(y) = 0$ . Let  $U = \phi^{-1}([0,1/2])$ , this is an open neighborhood of  $y$ .
+
+$$
+\begin{array}{l} 1 = f (\phi) (x) = \int_ {X} \phi d f _ {x} (67) \\ = \int_ {X \backslash U} \phi d f _ {x} + \int_ {U} \phi d f _ {x} (68) \\ \leq f _ {x} (X \backslash U) + \frac {1}{2} f _ {x} (U) (69) \\ = 1 - f _ {x} (U) + \frac {1}{2} f _ {x} (U) (70) \\ = 1 - \frac {1}{2} f _ {x} (U) (71) \\ \end{array}
+$$
+
+Hence,  $f_{x}(U) = 0$ . Therefore  $y \notin \operatorname{supp} f_{x}$ .
+
+Clearly,  $\mathrm{WOCan}_G(X) \subseteq \mathrm{WCCan}_G(X)$ , hence the sequential closure of  $\mathrm{WOCan}_G(X)$  is contained inside  $\mathrm{WCCan}_G(X)$ . We now show the reverse inclusion. Fix an arbitrary metric on  $X$ , this is possible as  $X$  is assumed metrizable.
+
+Suppose we have a weighted closed canonicalization  $f$ . Let  $h$  be the left invariant Haar measure on  $G$ . Let  $\mu_x = (i_x)_*h$ , this is a measure supported on  $\overline{Gx}$ . Further it is a  $\sigma$ -finite measure as  $G$  is locally compact. Therefore we can use the Lebesgue decomposition theorem to decompose  $f(x)$  as
+
+$$
+f _ {x} = f _ {x} ^ {a c} + f _ {x} ^ {s c} + f _ {x} ^ {s d} \tag {72}
+$$
+
+where  $f_x^{ac}$  is absolutely continuous with respect to  $\mu_x$ ,  $f_x^{sc}$  is singular with respect to  $\mu_x$  and continuous, and  $f_X^{sd}$  is singular with respect to  $\mu_x$  and discrete.
+
+First we construct a sequence of weighted closed canonicalizations  $f^n$  converging to  $f$  such that upon decomposing each  $f^n$  as above, there is no singular continuous factor.
+
+$f^{sc}$  is singular continuous with respect to  $\mu_x$ , hence there is a measurable set  $S \subseteq \overline{Gx}$  such that  $\mu_x(S) = 0$  and  $f_x^{sc}(S^c) = 0$ . Fix a point  $y \in S$ . For  $l > 0$ , let  $K_l = \overline{B(y, l)} \cap \overline{S}$ . Then we have
+
+$$
+\bigcup_ {l} K _ {l} = \bar {S} \tag {73}
+$$
+
+By continuity of measure
+
+$$
+\lim  _ {l \rightarrow \infty} f _ {x} ^ {s c} (K _ {l}) = f _ {x} ^ {s c} (\bar {S}) = f _ {x} ^ {s c} (S) + f _ {x} ^ {s c} (\bar {S} \cap S ^ {c}) = f _ {x} ^ {s c} (S) \tag {74}
+$$
+
+Fix  $l$  large enough such that  $f_{x}^{sc}(K_{l}) > f_{x}^{sc} - \frac{1}{n}$ .  $K_{l}$  is compact so we may find finitely many points  $x_{1},\ldots ,x_{p(n)}$  such that the  $\frac{1}{n}$  balls around these points cover  $K_{l}$ . Call these open balls  $U_{i}$ . Let  $U_{i}' = U_{i}\backslash \bigcup_{j < i}U_{j}$ . Then each  $U_{i}'$  contains only  $x_{i}$  and none of the other  $x_{j}$ . Define a discrete measure  $f_{n}^{\prime}$  which is non-zero only on the  $x_{i}$  and taking the following values at those points
+
+$$
+\hat {f} ^ {n} (\{x _ {i} \}) = f _ {x} ^ {s c} (U _ {i} ^ {\prime}) \tag {75}
+$$
+
+Then one easily sees that
+
+$$
+\hat {f} ^ {n} (S) = f _ {x} ^ {s c} \left(U _ {1} ^ {\prime}\right) + f _ {x} ^ {s c} \left(U _ {2} ^ {\prime}\right) + \dots + f _ {x} ^ {s c} \left(U _ {p (n)} ^ {\prime}\right) = f _ {x} ^ {s c} \left(K _ {l}\right) \tag {76}
+$$
+
+Now let  $f^n = f_x^{ac} + \hat{f}^n + f_x^{sd}$ . This is then a measure with no singular discrete part with respect to  $\mu_x$ . We now check that  $f^n \to f$ . Let  $\phi \in C_b^0(X)$ . We may assume that  $\phi$  is non-negative on  $X$  by splitting it up as  $\phi = \phi^+ + \phi^-$  for  $\phi^+$  non-negative and  $\phi^-$  non-positive.
+
+$$
+\begin{array}{l} f _ {n} (\phi) (x) - f (\phi) (x) = \int \phi d \left(f _ {n} (x) - f (x)\right) \\ = \int \phi d (f _ {s c} (x) - f _ {n} ^ {\prime} (x)) \\ = \sum_ {i = 1} ^ {p} \left(\int_ {U _ {i} ^ {\prime}} \phi d f _ {s c} (x) - \phi \left(x _ {i}\right) f _ {n} ^ {\prime} \left(U _ {i} ^ {\prime}\right)\right) + \int_ {S \backslash K _ {n}} \phi d f _ {s c} (x) \\ \end{array}
+$$
+
+Therefore we see that, with  $C = \max \phi$
+
+$$
+\left| f _ {n} (\phi) (x) - f (\phi) (x) \right| \leq \max  \left(\sum_ {i = 1} ^ {p (n)} f _ {s c} \left(U _ {i} ^ {\prime}\right) \left(\max  _ {U _ {i} ^ {\prime}} \phi - \phi \left(x _ {i}\right)\right), \sum_ {i = 1} ^ {p} f _ {s c} \left(U _ {i} ^ {\prime}\right) \left(\phi \left(x _ {i}\right) - \min  _ {U _ {i} ^ {\prime}} \phi\right)\right) + \frac {C}{n} \tag {77}
+$$
+
+Let  $\epsilon > 0$ . Take  $N$  large enough such that for all  $n > N$ , and  $i = 1, \ldots, p$
+
+$$
+\left| \max  _ {U _ {i} ^ {\prime}} \phi - \min  _ {U _ {i} ^ {\prime}} \phi \right| <   \epsilon \tag {78}
+$$
+
+Recall that we can do this as there are finitely many  $U_{i}^{\prime}$ , each of which is contained in the  $\frac{1}{n}$  ball around  $x_{i}$ . Plugging this into Equation (77) we get
+
+$$
+\left| f _ {n} (\phi) (x) - f (\phi) (x) \right| \leq \frac {C}{n} + \sum_ {i = 1} ^ {p (n)} f _ {s c} \left(U _ {i} ^ {\prime}\right) \epsilon = \frac {C}{n} + \epsilon f _ {s c} \left(K _ {n}\right) \leq \frac {C}{n} + \epsilon f _ {s c} (S) \tag {79}
+$$
+
+Taking  $n$  large and  $\epsilon$  small gives the convergence result.
+
+Now we show that we can approximate a weighted closed canonicalization with no singular continuous part by weighted orbit canonicalizations. As before, decompose  $f_{x} = f_{x}^{ac} + f_{x}^{sd}$  with respect to  $\mu_{x}$ . For any  $n$ , choose  $V_{n}(x)$  to be the  $\frac{1}{n}$  open ball around  $\overline{Gx} \backslash Gx$ .  $f^{sd}$  is singular and discrete, hence it is supported on a (possibly infinite) set of points  $x_{1}, x_{2}, \ldots$  in  $\overline{Gx}$ . For each  $x_{i}$  choose a sequence of points  $(x_{in})$  in  $Gx$  converging to  $x_{i}$ . We may choose these such that for a fixed  $n$ , each of the  $x_{in}$  are different for every  $i$ . Further we assume that these are chosen such that for every  $i$
+
+$$
+d \left(x _ {i}, x _ {i n}\right) <   \frac {1}{n} \tag {80}
+$$
+
+Then define the singular measure  $\hat{f}^n$  which is non-zero only on the  $x_{in}$  for every  $i$ , taking value
+
+$$
+\hat {f} ^ {n} \left(\left\{x _ {i n} \right\}\right) = f _ {x} ^ {s d} \left(\left\{x _ {i} \right\}\right) \tag {81}
+$$
+
+Define the absolutely continuous measure  $\tilde{f}^n$  by
+
+$$
+\tilde {f} ^ {n} (U) = f ^ {a c} \left(U \backslash V _ {n} (x)\right). \tag {82}
+$$
+
+Then let
+
+$$
+f ^ {n} (x) = \hat {f} ^ {n} + \tilde {f} _ {n} \tag {83}
+$$
+
+This is then a measure supported on  $Gx$  and by construction it is  $G$ -invariant. Hence this is a weighted orbit canonicalization. Now we show that  $f^n \to f$ . Let  $\phi \in C_b^0 (X)$  and  $x \in X$ . Then
+
+$$
+\begin{array}{l} \left| f ^ {n} (\phi) (x) - f (\phi) (x) \right| = \left| \int_ {\overline {{G x}}} \phi d f _ {x} - \int_ {G x} \phi d f _ {x} ^ {n} \right| (84) \\ = \left| \int_ {\overline {{G x}}} \phi d f _ {x} ^ {a c} - \int_ {G x} \phi d \tilde {f} _ {x} ^ {n} \right| + \left| \int \phi d f _ {x} ^ {s d} - \int \phi d \hat {f} _ {x} ^ {n} \right| (85) \\ = \left| \int_ {V _ {n} (x)} \phi d f _ {x} ^ {a c} \right| + \left| \sum_ {i} f _ {x} ^ {s d} \left(\left\{x _ {i} \right\}\right) (\phi (x _ {i}) - \phi (x _ {i n}) \right| (86) \\ \leq C f _ {x} ^ {a c} \left(V _ {n} (x)\right) + \sum_ {i} f _ {x} ^ {s d} \left(\left\{x _ {i} \right\}\right) | \phi \left(x _ {i}\right) - \phi \left(x _ {i n}\right) | (87) \\ \end{array}
+$$
+
+Now fix  $\epsilon > 0$ . As  $f_x^{sd}$  is a finite measure, there is some  $I$  such that
+
+$$
+\sum_ {i = I} ^ {\infty} f _ {x} ^ {s d} \left(\left\{x _ {i} \right\}\right) <   \epsilon \tag {88}
+$$
+
+Thus we get
+
+$$
+\begin{array}{l} \left| f ^ {n} (\phi) (x) - f (\phi) (x) \right| \leq C f _ {x} ^ {a c} \left(V _ {n} (x)\right) + \left(\sum_ {i = 1} ^ {I - 1} + \sum_ {i = I} ^ {\infty}\right) f _ {x} ^ {s d} \left(\left\{x _ {i} \right\}\right) \left| \phi \left(x _ {i}\right) - \phi \left(x _ {i n}\right) \right| (89) \\ \leq C f _ {x} ^ {a c} \left(V _ {n} (x)\right) + \sum_ {i = 1} ^ {I - 1} f _ {x} ^ {s d} \left(\left\{x _ {i} \right\}\right) \left| \phi \left(x _ {i}\right) - \phi \left(x _ {i n}\right) \right| + 2 C \epsilon (90) \\ \end{array}
+$$
+
+Now choose  $n$  large enough such that  $f^{ac}(V_n(x)) < \epsilon$  and for each  $i = 1, \ldots, I - 1$ ,  $|\phi(x_i) - \phi(x_{in})| < \epsilon$ . Then we get a bound
+
+$$
+\left| f ^ {n} (\phi) (x) - f (\phi) (x) \right| \leq 3 C \epsilon + D \epsilon \tag {91}
+$$
+
+Hence,  $f^n \to f$ .
+
+Theorem G.5.  $\operatorname{WCan}_G(X)^{\mathrm{cont}}$  is sequentially closed.
+
+Proof. Suppose that  $(\mu^n)_n$  is a convergent sequence of continuous weighted canonicalizations, converging to a weighted canonicalization  $\mu$ . We will show that  $\mu$  is continuous. To do this, fix a
+
+convergent sequence  $(x_{m})_{m}$  in  $X$  converging to  $x$ . Let  $\phi \in C_b^0 (X)$ . Then we have the following bound
+
+$$
+| \mu (\phi) (x) - \mu (\phi) \left(x _ {m}\right) | \leq | \mu (\phi) (x) - \mu_ {n} (\phi) (x) | + | \mu_ {n} (\phi) (x) - \mu_ {n} (\phi) \left(x _ {m}\right) | + | \mu_ {n} (\phi) \left(x _ {m}\right) - \mu (\phi) \left(x _ {m}\right) | \tag {92}
+$$
+
+Fix  $\epsilon > 0$ . As  $\mu_{n}$  converges to  $\mu$  we may find a large enough  $n$  such that both the first and third terms are less than  $\epsilon$ . For this  $n$ , as  $\mu_{n}$  is continuous, we may find  $m$  large enough such that the middle term is less than  $\epsilon$  as well. Hence the right hand side is bounded by  $3\epsilon$ , so we see that  $\mu$  is continuous.
+
+Proposition G.6. Let  $Y$  be a topological space. Let  $f: Y \to [0, +\infty]$  be lower semi-continuous. Then the set
+
+$$
+M = \{y \in Y | f (y) = \inf  _ {Y} f \} \tag {93}
+$$
+
+is closed.
+
+Proof. Let  $\alpha$  be the infimum of  $f$ . We have that  $M = f^{-1}(\{\alpha\})$ . As  $\alpha$  is the infimum over  $Y$ , we see that in fact  $M = f^{-1}([0, \alpha])$ . Then as  $f$  is lower semi-continuous, this set is closed in  $Y$ .
+
+Lemma G.7. Let  $E$  be a lower semi-continuous energy function. Then  $\kappa_{E}$  is continuous if and only if for every convergent sequence  $x_{n} \to x$  we have  $\operatorname{Li} \mathcal{M}_{E}(x_{n}) = \mathcal{M}_{E}(x)$ .
+
+Proof. Let  $E: X \to [0, +\infty]$  be an energy function such that the corresponding energy minimizing canonicalization  $\kappa_E$  is continuous. Consider a convergent sequence  $(x_n)$  in  $X$ , converging say to  $x$ .  $\mathcal{M}_E(x)$  is closed inside  $\overline{Gx}$ , being the set of minimizers of a lower semi-continuous function. As  $\overline{Gx}$  is closed in  $X$ ,  $\mathcal{M}_E$  is closed in  $X$  as well. As  $X$  is metrizable, we may choose a countable neighborhood basis of  $\mathcal{M}_E(x)$ , which we shall write as a decreasing sequence  $(U_m)$  of open neighborhoods such that
+
+$$
+\bigcap U _ {m} = \mathcal {M} _ {E} (x) \tag {94}
+$$
+
+Let  $C_m = X \backslash U_m$ , this is closed. As  $X$  is normal Hausdorff, we may find  $\phi_m : X \to [0,1]$  continuous such that  $\phi_m^{-1}(\{0\}) = C_n$  and  $\phi_m^{-1}(\{1\}) = \mathcal{M}_E(x)$ .
+
+Now as  $\kappa_{E}$  is continuous we have that the measures  $\kappa_{E}(x_{n})$  weakly converge to  $\kappa_{E}(x)$ . Hence for every  $m$  we have that
+
+$$
+\lim  _ {n \rightarrow \infty} \int_ {\mathcal {M} _ {E} \left(x _ {n}\right)} \phi_ {m} d \mu_ {x _ {n}} \rightarrow \int_ {\mathcal {M} _ {E} (x)} \phi_ {m} d \mu_ {x} = 1 \tag {95}
+$$
+
+Hence
+
+$$
+\lim  _ {n \rightarrow \infty} \int_ {\mathcal {M} _ {E} \left(x _ {n}\right)} \phi_ {n} d \mu_ {x _ {n}} = 1 \tag {96}
+$$
+
+Let  $\epsilon > 0$ . We may find  $N$  such that for all  $n > N$ , the left hand side is greater than  $1 - \epsilon$ . As  $\mu_{x_n}(\mathcal{M}_E(x_n)) = 1$ , we must have that the minimum of  $\phi_n$  over  $\mathcal{M}_E(x_n)$  is greater than  $1 - \epsilon$ . As  $C_n = \phi_n^{-1}(\{0\})$ , we then must have that  $\mathcal{M}_E(x_n) \subseteq U_n$ . Hence by the definition of Kuratowski convergence we have  $\operatorname{Li} \mathcal{M}_E(x_n) \subseteq \mathcal{M}_E(x)$ .
+
+Now suppose that  $y \in \mathcal{M}_E(x)$ . Then we may as above choose a decreasing sequence  $(U_m)$  of open neighborhoods of  $y$  which form a neighborhood basis. Then fix for each  $m$ ,  $\phi_m : X \to [0,1]$  continuous such that  $\phi_m^{-1}(\{0\}) = X \setminus U_m$  and  $\phi_m^{-1}(\{1\}) = \{y\}$ . By the same estimates as before we must have that for large enough  $n$ , each  $\mathcal{M}_E(x_n)$  intersects  $U_m$  non-trivially. Hence  $y \in \operatorname{Li} \mathcal{M}_E(x_n)$ . So  $\operatorname{Li} \mathcal{M}_E(x_n) = \mathcal{M}_E(x)$ .
+
+Conversely suppose that we are given that for every convergent sequence  $x_{n} \to x$ ,  $\operatorname{Li} \mathcal{M}_{E}(x_{n}) = \mathcal{M}_{E}(x)$ . Let  $\phi \in C_b^0 (X)$ . We may assume that  $\phi$  is non-negative. Fix  $\epsilon > 0$ . As  $\mathcal{M}_E(x)$  is compact, we may find an open set  $U$  containing  $\mathcal{M}_E(x)$  such that
+
+$$
+\max  _ {y \in U} \phi (y) - \min  _ {y \in U} \phi (y) <   \epsilon \tag {97}
+$$
+
+As  $\operatorname{Li} \mathcal{M}_E(x_n) \subseteq \mathcal{M}_E(x)$ , we may find  $N$  large enough such that for all  $n \geq N$ ,  $\mathcal{M}_E(x_n) \subseteq U$ . Then
+
+$$
+\min  _ {y \in \mathcal {M} _ {E} (x)} \phi (y) \leq \int_ {\mathcal {M} _ {E} (x)} \phi d \mu_ {x} \leq \max  _ {y \in \mathcal {M} _ {E} (x)} \phi (y) \tag {98}
+$$
+
+and similarly
+
+$$
+\min  _ {y \in \mathcal {M} _ {E} \left(x _ {n}\right)} \phi (y) \leq \int_ {\mathcal {M} _ {E} \left(x _ {n}\right)} \phi d \mu_ {x _ {n}} \leq \max  _ {y \in \mathcal {M} _ {E} \left(x _ {n}\right)} \phi (y) \tag {99}
+$$
+
+Hence we see both of these integrals are bounded between the min and max of  $\phi$  on  $U$ . Hence
+
+$$
+\left| \int_ {\mathcal {M} _ {E} (x)} \phi d \mu_ {x} - \int_ {\mathcal {M} _ {E} \left(x _ {n}\right)} \phi d \mu_ {x _ {n}} \right| \leq 2 \epsilon \tag {100}
+$$
+
+Therefore this converges and we have weak convergence.
+
+![](images/9428502bad064036ed59d06a4efb82f8e39590f63273ed0fef1876415e78257c.jpg)
+
+# H OVERVIEW OF PREVIOUS CONCEPTS
+
+Frames and canonicalization both represent different ways to impose group equivariance or invariance on arbitrary functions. The most basic approach is simply to average over all group elements, sometimes referred to as the Reynolds operator. For a finite group  $G$  and a continuous bounded function  $\phi$  it is given by  $\mathcal{R}_G(\phi)(x) = \frac{1}{|G|} \sum_{g \in G} \phi(g^{-1} \cdot x)$ . This can also be modified to impose  $G$ -equivariance by changing the integrand to  $g \cdot \phi(g^{-1} \cdot x)$ . Frames allow one to not have to integrate over the full group  $G$  while still giving an invariant or equivariant function as output. Specifically, a frame is a function  $\mathcal{F}: X \to 2^G \backslash \emptyset$  such that for any  $x \in X$  and  $g \in G$ :  $\mathcal{F}(gx) = g\mathcal{F}(x)$ . In order to define an action on continuous groups, one must assume that the frame is finite, i.e. that for every  $x \in X$  the set  $\mathcal{F}(x)$  is finite. This is immediate if the group is finite, but if the group is non-finite this is a very strong restriction, as it implies that the stabilizer groups of the action are finite, as  $\mathcal{F}(x) = G_x\mathcal{F}(x)$ . This is detailed further in Appendix F.
+
+In order to deal with this restriction, Dym et al. (2024), introduced the notion of weighted weakly equivariant frames. These generalize frames to be  $G$ -equivariant maps from  $X$  to probability measures on  $G$ . Then for any continuous bounded function  $\phi$  we may apply a weighted frame  $\mu_x$  by  $\mu(\phi)(x) = \int_G \phi(g^{-1} \cdot x) d\mu_x(g)$ . This is still not enough in the setting of non-compact groups as discussed in the subsequent section<sup>5</sup>. Even worse, there is an inherent limitation to using frames: the input to  $\phi$  may end up outside the training distribution, resulting in worsened performance. The overall network would therefore need to be retrained, which may be exponentially hard (Kiani et al., 2024a).
+
+To deal with this, Kaba et al. (2023) proposed canonicalization as simply a  $G$ -invariant function  $f: X \to X$  such that for each  $x \in X$ ,  $f(x) \in Gx$ , which upon pre-composing with a function would make it invariant. In order to preserve continuity of functions, one needs to impose that  $f$  itself is continuous. In Dym et al. (2024) it is shown that there are many topological obstructions to constructing such continuous canonicalization functions.
+
+To get around these obstructions Ma et al. (2024) proposes canonicalization to now be set-valued  $G$ -invariant function  $\mathcal{C}: X \to 2^{X} \setminus \emptyset$  for  $\mathcal{C}(x)$  finite, acting on continuous functions  $\phi$  by  $\mathcal{C}(\phi)(x) = \frac{1}{|\mathcal{C}(x)|} \sum_{y \in \mathcal{C}(x)} \phi(y)$ . Contractive canonicalizations, which we refer to instead as orbit canonicalizations, are a specific kind of canonicalization where for every  $x \in X$ ,  $\mathcal{C}(x) \subseteq Gx$ . The orbit-stabilizer theorem applies to show these are equivalent to frames. Namely, there is a map  $\alpha$  which sends equivariant frames to orbit canonicalizations. For an equivariant frame  $\mathcal{F}$  the corresponding orbit canonicalization  $\alpha(\mathcal{F})$  is defined as  $\alpha(\mathcal{F})(x) = \{g^{-1} \cdot x | g \in \mathcal{F}(x)\}$ . Ma et al. (2024, Theorem 3.1) show that for  $G$  finite,  $\alpha$  is an isomorphism between the set of equivariant frames and orbit canonicalizations. Energy-based canonicalization was first introduced in Kaba et al. (2023). In energy-based canonicalization, given an energy function  $E$ , the canonicalizing element  $c: X \to G$  is defined as  $c(x) \in \arg \min_{g \in G} E(g^{-1}x)$ . As shown in the subsequent section, considering the full set of minimizers, instead of a singleton, naturally gives rise to a frame, and naturally results in set-valued canonicalizations. Extending these to non-compact groups naturally gives rise to weighted closed canonicalizations.
+
+# H.1 WEIGHTED CANONICALIZATION
+
+For this section let  $G$  be a Lie group acting smoothly on a manifold  $X$ . Let  $C_b^0 (X)$  be the bounded continuous  $\mathbb{R}$ -valued functions on  $X$ . Let  $\mathrm{PMeas}(X)$  be the set of Radon probability measures on  $X$ , with respect to the Borel  $\sigma$ -algebra on  $X$ . Let  $B(X,\mathbb{R})$  be the space of bounded functions on  $X$  and  $B(X,\mathbb{R})^G$  is the subspace of  $G$ -invariant such functions.
+
+Definition H.1. A weighted canonicalization is a  $G$ -invariant function  $\kappa_{[.]}: X \to \mathrm{PMeas}(X)$ . We call the space of weighted canonicalizations  $\mathrm{WCan}_G(X)$ .
+
+Given a weighted canonicalization  $\kappa$  we may define an operator  $\mathcal{P}_{\kappa}:B(X,\mathbb{R})\to B(X,\mathbb{R})^{G}$  via
+
+$$
+\left(\mathcal {P} _ {\kappa} (\phi)\right) (x) = \int_ {X} \phi d \kappa_ {x}. \tag {101}
+$$
+
+We may write the function  $\mathcal{P}_{\kappa}(\phi)$  simply as  $\kappa (\phi)$ . As  $\kappa_{x}\in \mathrm{PMeas}(X)$  for each  $x\in X$ , we have  $\| \mathcal{P}_{\kappa}(\phi)\|_{\infty}\leq \| \phi \|_{\infty}$ .  $\kappa (\phi)$  is  $G$ -invariant as  $\kappa$  is. We also define the function  $P_{\phi}(\kappa) = \mathcal{P}_{\kappa}(\phi)$ .
+
+The entire space  $\mathrm{WCan}_G(X)$  is much too general to be practical. We would like to consider a much more natural subset from the perspective of canonicalization, namely orbit canonicalizations. We now define a corresponding weighted notion.
+
+Definition H.2. A weighted orbit canonicalization is a weighted canonicalization  $\kappa : X \to \mathrm{PMeas}(X)$  such that for every  $x \in X$ ,  $\operatorname{supp} \kappa(x) \subseteq Gx$ . We denote the set of weighted orbit canonicalizations as  $\mathrm{WOCan}_G(X)$ .
+
+In particular, for weighted orbit canonicalizations both operators become projections, see Appendix F.2. As in the finite case, there is a function  $\alpha: \mathrm{WFr}_G(X) \to \mathrm{WOCan}_G(X)$ , which is surjective and injective up to weak-equivalence, see Appendix F.1.
+
+Given the action on continuous functions, we may equip  $\mathrm{WCan}_G(X)$  with a certain weak topology, namely the coarsest topology such that all the  $P_{\phi}$  for  $\phi$  continuous and bounded become continuous functions. Specifically this means that a sequence of weighted canonicalizations  $\kappa_{n}$  converges to  $\kappa$  if and only if for every  $\phi \in C_b^0 (X)$  the sequence  $\kappa_{n}(\phi)$  converges to  $\kappa (\phi)$  pointwise. For the canonicalization problem this is a very natural topology to consider, as we only care about canonicalization through their respective actions on continuous functions.
+
+So far, we have presented a generalization of canonicalization, which is readily applicable to finite or compact Lie groups. However, non-compact Lie groups may have non-closed orbits, and the space of weighted orbit canonicalizations is not closed under taking limits in the weak topology. To understand the closure under taking limits, we introduce the concept of a weighted closed canonicalization.
+
+Definition H.3. A weighted closed canonicalization is a weighted canonicalization  $\kappa : X \to \mathrm{PMeas}(X)$  such that for every  $x \in X$ ,  $\operatorname{supp} \kappa(x) \subseteq \overline{Gx}$ . Call the set of weighted closed canonicalizations  $\mathrm{WCCan}_G(X)$ .
+
+The following theorem shows that these allow us to fully characterize the behaviour of weighted orbit canonicalizations under taking limits.
+
+Theorem H.1. (Theorem G.4) The sequential closure of  $\mathrm{WOCan}_G(X)$  is  $\mathrm{WCCan}_G(X)$
+
+Using this theorem and the setup above we may now write down our main diagram connecting all the different notions of canonicalization and framing together, shown in Figure 3. In analogy with Dym et al. (2024), we may now ask whether such canonicalizations take continuous functions to continuous functions. In the presented framework this is a simple condition to add.
+
+Definition H.4. A weighted canonicalization  $\kappa : X \to \mathrm{PMeas}(X)$  is continuous if for every  $\phi \in C_b^0(X)$ ,  $\kappa(\phi)$  is continuous. Call the space of continuous weighted canonicalizations  $\mathrm{WCan}_G^{\mathrm{cts}}(X)$ . We similarly define continuous weighted orbit canonicalizations and continuous weighted canonicalizations.
+
+Proposition H.2. A weighted canonicalization  $\kappa$  is continuous if and only if for every convergent sequence  $x_{n}\to x$  in  $X$ , the sequence of measures  $\kappa_{x_n}$  converges weakly to  $\kappa_{x}$ .
+
+Theorem H.3. (Theorem G.5)  $\mathrm{WCan}_G^{cts}(X)$  is sequentially closed inside  $\mathrm{WCan}_G(X)$ .
+
+# H.2 ENERGY MINIMIZING CANONICALIZATION
+
+In Section 2 so far, we have provided a generalized theoretical understanding of frames and canonicalization, encompassing various scenarios with compact/non-compact and discrete/continuous groups. However, their relevance and how these can be used in practice remains unaddressed. We now move on to the concept of energy frames and canonicalizations, for general Lie groups. We define an energy function on  $X$  to be a non-constant function  $E: X \to [0, +\infty]$ . We further assume that  $E$  is topologically coercive: for any  $N \in \mathbb{R}^+$  there is a compact  $K \subseteq X$  such that  $E|_{X \setminus K} > N$ .
+
+In the case where  $G$  is finite, every orbit in  $X$  necessarily has a minimum of  $E$ , and we note that energy minimization naturally results in a frame
+
+$$
+\mathcal {F} _ {E} (x) = \underset {g \in G} {\arg \min } E \left(g ^ {- 1} x\right) \tag {102}
+$$
+
+Proposition H.4.  $\mathcal{F}_E$  defines an equivariant frame, i.e.  $\mathcal{F}_E(gx) = g\mathcal{F}_E(x)$  for any  $g\in G$ ,  $x\in X$ .
+
+However when  $G$  is not finite this set is not guaranteed to be finite. If  $G$  is additionally non-compact, then  $E$  does not necessarily achieve its minimum on every orbit in  $X$ , making this optimization problem not well-defined. For this reason, we instead turn to weighted canonicalizations. By taking the closures of orbits, we solve the problem of  $E$  not having minima, so we are naturally led to constructing weighted closed canonicalizations. First, we define the energy-minimizing set,  $\mathcal{M}_E(x) = \{x' \in \overline{Gx} \mid E(x') = \inf_{y \in \overline{Gx}} E(y)\}$ , set of minima of  $E$  on the closure of the orbit of  $x$ .
+
+Proposition H.5. (Proposition G.2)  $\mathcal{M}_E(x)$  is  $G$ -invariant and non-empty.
+
+If  $\mathcal{M}_E(x)$  is finite for each  $x\in X$ , then we define the (non-weighted) closed canonicalization  $\kappa_{E}\in \mathrm{WCCan}_{G}(X)$  by taking  $\kappa_{E}(x)$  to be the normalized counting measure on  $\mathcal{M}_E(x)$  as usual.
+
+For a general  $E$ , we may not be able to put any reasonable probability measure on the set  $\mathcal{M}_E(x)$ , for example if  $\mathcal{M}_E(x)$  has highly fractal geometry. We therefore must assume that  $E$  is such that for all  $x$ , the Hausdorff measure of  $\mathcal{M}_E(x)$  is non-zero. As each  $\mathcal{M}_E(x)$  is compact, the Hausdorff measure is finite, hence combined with this assumption we may normalize it to obtain a probability measure on  $\mathcal{M}_E(x)$  which is a weighted canonicalization  $\kappa_E(x) \in \mathrm{WCCan}_G(X)$ . Refer to Appendix E for more discussion on the Hausdorff measure and on why this assumption is reasonable. Note that if  $\mathcal{M}_E(x)$  is finite, then the normalized Hausdorff measure constructed agrees with the normalized counting measure. Hence this construction is a generalization of the purely finite case.
+
+For lower semi-continuous energy functions, we have the following characterization of which energies give rise to continuous weighted canonicalizations.
+
+Lemma H.6. (Lemma G.7) Let  $E$  be a lower semi-continuous energy function. Then  $\kappa_{E}$  is continuous if and only if for every convergent sequence  $x_{n} \to x$  we have  $\operatorname{Li} \mathcal{M}_{E}(x_{n}) = \mathcal{M}_{E}(x)$ .
+
+We have the following theorem telling us which weighted closed canonicalizations arise from energy minimizations.
+
+Theorem H.7. (Theorem F.3) Let  $\kappa \in \mathrm{WCCan}_G(X)$  be such that  $\kappa_{x}$  is the normalized Hausdorff measure on its support and for all  $y\in \overline{Gy}$ ,  $\operatorname {supp}\kappa_y = \operatorname {supp}\kappa_x\cap \overline{Gx}$ . Then there is an energy function  $E$  such that  $\kappa_{E} = \kappa$ .
+
+# H.3 ALMOST EVERYWHERE CONTINUITY IS NOT AS IMPOSSIBLE
+
+Canonicalization Let  $X$  be a space with a group  $G$  acting on it by homeomorphisms. Then a continuous canonicalization function is a continuous  $G$ -equivariant map  $v: X \to X$  which sends any element  $x$  to an element of the orbit  $Gx$ . In Dym et al. (2024) it is shown that in general these do not exist, as it is much too strong of a condition. In fact, because a continuous canonicalization function will be a right-inverse to the quotient map  $X \to X / G$ , a necessary condition to the existence of  $v$  is that the map  $\pi_1(X) \to \pi_1(X / G)$  induced by the quotient must be a surjection. If the quotient  $X \to X / G$  is a covering map then this only holds if  $G$  is trivial. For a concrete example of this, we can consider quotienting  $\mathbb{R}^2$  by the translation action of  $\mathbb{Z}^2$ . Then  $\pi_1(\mathbb{R}^2) = 0$  while  $\pi_1(\mathbb{R}^2 / \mathbb{Z}^2) = \mathbb{Z}^2$ , showing that a continuous canonicalization function can't exist.
+
+On the other hand, global existence of a continuous canonicalization function is unnecessary for machine learning applications. As long as we can construct a canonicalization function which is
+
+continuous away from a set of measure zero, this will suffice for our applications. Specifically we introduce the notion of an almost everywhere (a.e.) continuous canonicalization function.
+
+Definition H.5. An a.e. continuous canonicalization function is a function  $v: X \to X$  such that there exists a set  $S \subset X$  of measure-zero such that
+
+(i)  $G(S)\subseteq S$  
+(ii)  $v$  is continuous away from  $S$  
+(iii)  $\forall x\in S,\exists g\in G$  st.  $v(x) = gx$  
+(iv)  $v$  is  $G$ -invariant
+
+As with the regular canonicalization functions, we can use such a function  $v$  to construct equivariant and invariant functions. Given an arbitrary function  $f: X \to \mathbb{R}$  it is easy to see that
+
+$$
+\mathcal {I} _ {v} (f) (x) = f (v (x)) \tag {103}
+$$
+
+is  $G$ -invariant. Further it is easy to check that this sends a.e. continuous functions to a.e. continuous functions.
+
+The added freedom of the set  $S$  where  $v$  can take arbitrary values allows us to avoid the topological restrictions that prevent the existence of globally continuous canonicalization functions. For a simple example consider again the presentation of a torus as the quotient of  $\mathbb{R}^2$  by the translation action. Then an easy example of an a.e. continuous canonicalization function is the function
+
+$$
+v (x, y) = ([ x ], [ y ])
+$$
+
+where  $[a]$  is the decimal part of a real number  $a\in \mathbb{R}$ . This is discontinuous whenever either  $x$  or  $y$  are integers, but this set has measure zero.
+
+For another example, consider the action of the orthogonal group  $O(d)$  on the space of matrices  $\mathbb{R}^{d \times n}$ . The action preserves the rank of a matrix  $A \in \mathbb{R}^{d \times n}$ , so the subset of full rank matrices is preserved by  $O(d)$ . This is an open dense set, and if we take  $\mu$  to just be the Lebesgue measure, its complement has measure zero. So take  $S$  to be the set of matrices with non-full rank. Let  $A$  be a matrix of full rank. Then one of the  $d \times d$  minors of  $A$  is invertible, we may assume that it is the first one. Thus we can write  $A = (A_0A_1)$  with  $A_0$  an invertible  $d \times d$  matrix. Applying the Gram-Schmidt procedure to  $A_0$ , we obtain  $A_0 = QR$  with  $Q$  orthogonal and  $R$  upper-triangular. Now we take  $v(A) = Q^T A$ .  $Q$  is given by a polynomial in the elements of  $A$ , hence  $v$  is smooth. Further, if  $O \in O(d)$ , then we can decompose  $OA$  as  $OQR$  with  $OQ$  orthogonal. Hence,
+
+$$
+v (O A) = (O Q) ^ {T} O A = Q ^ {T} O ^ {T} O A = Q ^ {T} A = v (A)
+$$
+
+Therefore  $v$  is an a.e. continuous canonicalization function, with  $S$  the set of non-full rank matrices in  $\mathbb{R}^{d \times n}$ .
+
+Invariant Theory Finding and computing  $G$ -invariant functions is a rich topic which comes up in many different situations. In algebraic geometry, one wants to find generators for  $G$ -invariant polynomials, i.e. a set of  $G$ -invariant polynomials  $\sigma_1, \ldots, \sigma_n$  such that any other  $G$ -invariant polynomial  $f$  can be written as
+
+$$
+f = g \left(\sigma_ {1}, \dots , \sigma_ {n}\right)
+$$
+
+for some polynomial  $g$ . For linearly reductive groups acting on  $\mathbb{C}^n$ , there is an algorithm for computing these generators, which is described in Derksen & Kemper (2015).
+
+In the smooth case, there is a classic result of Schwarz (Schwarz, 1975), telling us that if  $G$  is a compact Lie group acting on a smooth manifold  $X$ , with generators for the space of  $G$ -invariant polynomials being  $\sigma_1, \ldots, \sigma_n$ , then any  $G$ -invariant smooth functions  $f$  can be written as
+
+$$
+f = g \left(\sigma_ {1}, \dots , \sigma_ {n}\right)
+$$
+
+where  $g$  is an some smooth function on  $\mathbb{R}^n$ .
+
+Algorithm 2: Canonicalization function via Lie algebra descent  
+Data: Non-canonical input  $x$    
+parameters:Nouter,Ninner,step-sizes  $\eta_{i}$  Energy  $E:\mathcal{X}\to \mathbb{R}$    
+Result: Canonicalized input  $\hat{x} = g^{-1}\cdot x$  ; inverse of canonicalizing group element  $g^{-1}$  . canonicalizing group element  $g$ $g^{-1}\gets \mathrm{id};$    
+steps  $= \{\}$    
+for  $k = 0\dots N_{\text{outer}}$  do # Do gradient descent on  $\xi \in \mathfrak{g}$ $\xi \leftarrow 0$  for  $i = 0\dots N_{\text{inner}}$  do  $\begin{array}{r}\mid \xi \leftarrow \xi -\eta_i\nabla_\xi E(g^{-1}\exp (-\xi)\cdot x) \end{array}$  end steps.append(  $\xi$  );  $g^{-1}\gets g^{-1}\exp (-\xi)$  #Recalibrate to be in the right tangent plane   
+end   
+ $g\gets \mathrm{id}\quad \# \mathrm{Reconstruct}$  the element  $g$    
+for  $k = N_{\text{outer}}\ldots 0$  do  $g\gets \exp (\mathrm{steps}[k])g$    
+end   
+return  $g^{-1}\cdot x;g^{-1};g$
+
+# I ALGORITHMS FOR SOLVABLE AND NON-SOLVABLE LIE ALGEBRAS
+
+As discussed in the main body, in certain cases it is possible to significantly simplify the algorithmic approach for minimizing Equation (2). We will distinguish to specific cases:
+
+If the group is compact or the algebra is solvable In either of these cases, we are able to parameterise the group globally using the Liealgebra. Denoting the basis as  $v_{i}\in \mathfrak{g}$ , and the coefficients as  $\alpha_{i}\in \mathbb{R}$ , we have that we can represent any  $g\in G$  as
+
+$$
+g = \exp \sum_ {i} \alpha_ {i} v _ {i},
+$$
+
+if  $G$  is compact, or as
+
+$$
+g = \prod_ {i} \exp \alpha_ {i} v _ {i},
+$$
+
+if the Lie algebra is solvable with  $v_{i}$  ordered as to represent the nested subalgebras. In both of these cases we can reduce the optimization to be done over the vector of coefficients  $\alpha \in \mathbb{R}^{\dim \mathfrak{g}}$ . Both of these can be viewed as specific types of global retraction mappings  $\tau : \mathfrak{g} \to G$ . For this case we use Algorithm 2
+
+If the group is non-compact or the algebra is not solvable Such a setting is particularly important for PDE-based groups, as these naturally result in groups that are non-compact, see section 4.3. In this case it is no longer possible to find a global retraction, and a natural question arises - can the infinitesimal actions ever be enough? It turns out that the answer is yes, quite generically, resulting in Algorithm 1:
+
+Proposition I.1 (Proposition 1.24 from Olver (1993)). Let  $G$  be a connected Lie group and  $U \subset G$  a neighbourhood of the identity. Also, let  $U^k \equiv \{g_1 \cdot g_2 \cdot \ldots \cdot g_k : g_i \in U\}$  be the set of  $k$ -fold products of elements of  $U$ . Then
+
+$$
+G = \bigcup_ {k = 1} ^ {\infty} U ^ {k}.
+$$
+
+In other words, every group element  $g \in G$  can be written as a finite product of elements of  $U$ .
+
+Algorithm 3: Canonicalization function via coordinate Lie algebra descent  
+```txt
+Data: Non-canonical input  $x$    
+parameters:N, step-sizes  $\eta_{i}$  , Energy  $E:\mathcal{X}\to \mathbb{R},\tau_k$  if using proximal   
+Result: Canonicalized input  $\hat{x} = g^{-1}\cdot x$  ; inverse of canonicalizing group element  $g^{-1}$  . canonicalizing group element  $g$ $g^{-1}\gets \mathrm{id};$    
+steps  $= \{\}$  .   
+for  $k = 0\dots N$  do # Do coordinate descent on  $\xi \in \mathfrak{g}$  , by picking index  $j$ $j\gets$  pick[0,...,dim  $\mathfrak{g}]$ $\lambda_{k}\leftarrow$  arg  $\min_{\lambda}E(g^{-1}\exp (-\lambda v_j)\cdot x)$  # May be desirable to use proximal descent #  $\lambda_{k}\gets$  arg  $\min_{\lambda}E(g^{-1}\exp (-\lambda v_{j})\cdot x) + \frac{1}{2\tau_{k}} ||\lambda ||^{2}$  .  $\xi \leftarrow \lambda_{k}v_{j}$  steps.append(ξ);  $g^{-1}\gets g^{-1}\exp (-\xi)$  #Recalibrate to be in the right tangent plane   
+end   
+ $g\gets \mathrm{id}$  #Reconstruct the element  $g$    
+for  $k = N\dots 0$  do  $g\gets$  exp(steps[k])g   
+end   
+return  $g^{-1}\cdot x;g^{-1};g$
+```
+
+If the group only provides information about basis vector action In most examples of PDE symmetries, integrating the action for a generic  $v\in \mathfrak{g}$  is impossible, and instead only the action with respect to certain basis  $\{v_i\}$  is known. In this case coordinate descent schemes must be used. For this we use Algorithm 3.
\ No newline at end of file
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+# LIFT YOUR MOLECULES: MOLECULAR GRAPH GENERATION IN LATENT EUCLIDEAN SPACE
+
+Mohamed Amine Ketata*, Nicholas Gao*, Johanna Sommer*, Tom Wollschlager, Stephan Gunnemann
+
+Department of Computer Science & Munich Data Science Institute
+
+Technical University of Munich
+
+{a.ketata,n.gao, jm.sommer,t.wollschaeger,s.guennemann}@tum.de
+
+# ABSTRACT
+
+We introduce a new framework for 2D molecular graph generation using 3D molecule generative models. Our Synthetic Coordinate Embedding (SYCo) framework maps 2D molecular graphs to 3D Euclidean point clouds via synthetic coordinates and learns the inverse map using an  $\mathrm{E}(n)$ -Equivariant Graph Neural Network (EGNN). The induced point cloud-structured latent space is well-suited to apply existing 3D molecule generative models. This approach simplifies the graph generation problem into a point cloud generation problem followed by node and edge classification tasks, without relying on molecular fragments nor autoregressive decoding. Further, we propose a novel similarity-constrained optimization scheme for 3D diffusion models based on inpainting and guidance. As a concrete implementation of our framework, we develop EDM-SYCo based on the E(3) Equivariant Diffusion Model (EDM). EDM-SYCo achieves state-of-the-art performance in distribution learning of molecular graphs, outperforming the best non-autoregressive methods by more than  $26\%$  on ZINC250K and  $16\%$  on the GuacaMol dataset while improving conditional generation by up to 3.9 times. $^1$
+
+# 1 INTRODUCTION
+
+Machine learning-based drug discovery has recently shown great potential to accelerate the drug development process while reducing costs associated with clinical trials (Jiménez-Luna et al., 2020; Kim et al., 2020). Among the steps of this process, generating novel molecules or optimizing properties such as drug-likeness or synthesizeability are of crucial importance. Molecular graph generation is an active area of research that aims to solve these tasks (Gómez-Bombarelli et al., 2018).
+
+Molecular graph generation is challenging due to the discrete and sparse nature of graphs as well as the presence of symmetric substructures such as rings. Many methods have been proposed to tackle this problem, offering different tradeoffs (Jin et al., 2018; Shi et al., 2020). A common approach is to train a Variational Autoencoder (VAE) to encode the graph into a continuous fixed-size latent space, then decode it back – oftentimes autoregressively – node-by-node (Liu et al., 2018) or fragment-by-fragment (Jin et al., 2018; 2020; Maziarz et al., 2021). While such methods achieve good generation and optimization performance, they have inherent limitations. First, using a fixed-sized latent space does not account for the variable size of molecular graphs, which can go from a few atoms for small molecules to thousands for macromolecules. Second, autoregressive decoding requires choosing a concrete generation order (Schneider et al., 2015), which prevents learning permutation-invariant graph distributions. Fueled by recent developments in diffusion models (Ho et al., 2020; Austin et al., 2021), new approaches to molecular graph generation have emerged that overcome these limitations (Jo et al., 2022; Vignac et al., 2022; Jo et al., 2023). These methods define a discrete diffusion process directly on the node and edge attributes and learn to reverse it to generate graphs all at once, i.e., not autoregressively. However, they still lack behind autoregressive methods, struggling to accurately estimate the joint distribution of nodes and edges.
+
+![](images/618db0aec39049efcaaf842ce502acf45d16ac0636355e4aa7e53164a49db7e6.jpg)  
+Figure 1: Overview of EDM-SYCo. (Training) First, the autoencoder is trained to map between molecular graphs and latent Euclidean point clouds. Then, the diffusion model is trained on the fixed latent space. (Sampling) Starting with a Gaussian sample, the diffusion model denoises it for  $T$  steps to predict the clean point cloud, which is mapped to a molecular graph using the decoder.
+
+In this work, we develop a method that combines the benefits of all-at-once generation with the benefits of a continuous latent space while avoiding the information bottleneck of fixed-sized latent spaces. Inspired by recent developments in the adjacent field of 3D molecule generation, we propose to embed molecular graphs as latent 3D point clouds that implicitly encode the discrete graph structure in synthetic Euclidean coordinates and use point cloud generative models to learn their distribution. We call this framework Synthetic Coordinate Embedding (SYCo).
+
+When combined with EDM (Hoogeboom et al., 2022), our framework gives rise to an atom-based, all-at-once, and permutation-invariant generative model for molecular graphs, which we call EDM-SYCo, depicted in Figure 1. On the ZINC250K dataset, EDM-SYCo outperforms existing all-at-once diffusion-based models by more than  $26\%$  on distribution learning benchmarks and by up to 15.6 times on conditional generation benchmarks, closing the gap to autoregressive models and highlighting the benefit of our continuous latent space on generation and optimization. In addition, we propose a new similarity-constrained optimization procedure for diffusion models on point clouds without retraining or specialized architectures, visualized in Figure 3. Notably, our proposed algorithm enables the addition of new atoms during the reverse diffusion process, addressing a well-known limitation of prior 3D diffusion models, namely the fixed number of nodes.
+
+# 2 RELATED WORK
+
+Molecular Graph Generation Early work on molecule generation (Gómez-Bombarelli et al., 2018; Segler et al., 2018) used language models to generate SMILES (Weininger, 1988), a text-based representation of molecules. However, this type of representation cannot capture the inherent structure of molecules well, which are more accurately depicted as graphs. For instance, small changes in the graph structure may correspond to large differences in the SMILES string (Jin et al., 2018). Therefore, recent methods have focused on graph-based approaches that can be broadly classified into two categories. All-at-once generation methods generate nodes and edges in parallel, often in a multi-step process such as diffusion models (Niu et al., 2020; Jo et al., 2022; Vignac et al., 2022). Autoregressive methods specify a generation order and generate molecules either atom-by-atom (Li et al., 2018; Shi et al., 2020) or fragment-by-fragment based on a pre-computed fragment vocabulary (Jin et al., 2018; 2020; Maziarz et al., 2021; Kong et al., 2022; Geng et al., 2023). EDM-SYCo is an all-at-once molecular graph generative model that operates on the atom level.
+
+Molecule Generation in 3D Another line of work aims to generate molecules in 3D Euclidean space and solve a point cloud generation problem (Gebauer et al., 2019; Garcia Satorras et al., 2021; Ayadi et al., 2025). For instance, EDM (Hoogeboom et al., 2022) and GeoLDM (Xu et al., 2023) are diffusion-based models for 3D molecule generation that jointly generate atomic features and coordinates. While tangentially related, these methods solve a different problem from molecular graph generation and require molecules with 3D information for training. Our proposed SyCo framework is an attempt to connect these two lines of work by enabling the training of 3D generative models on graph datasets through synthetic coordinates, formally introducing new graph generation methods. While we are the first to explore synthetic coordinates for generative models, Gasteiger et al. (2021) demonstrated that synthetic coordinates improve molecular property prediction tasks.
+
+Latent Generative Models The idea of defining the generative model on a space different from the original data space is reminiscent of latent generative models (Dai & Wipf, 2019; Vahdat et al.,
+
+2021). Recently, this approach has become increasingly popular with latent diffusion models. Stable Diffusion models (Rombach et al., 2022) achieve impressive results on text-guided image generation, and GeoLDM (Xu et al., 2023) extends this to molecular point cloud generative models. While EDM-SyCo operates on a similar latent space to GeoLDM, their fundamental difference lies in the original data space. Our model maps discrete molecular graphs to continuous point clouds and can be seen as a cross-modality latent generative model. In contrast, GeoLDM embeds point clouds as new point clouds with a reduced feature dimension.
+
+# 3 SYNTHETIC COORDINATE EMBEDDING (SYCO) FRAMEWORK
+
+Our goal is to model molecular graph distributions through distributions of Euclidean point clouds that implicitly encode the graph structure into their coordinates. For this, we propose a novel autoencoder architecture that maps between molecular graphs and 3D point cloud representations. By training it using a reconstruction objective, we can reformulate the problem of generating discrete molecular graphs to an equivalent problem of generating 3D point clouds defined on the induced latent space.
+
+Note on equivariance When dealing with physical objects like molecules, it is important to consider equivariance. Formally, a function  $f: A \to B$  is equivariant to the action of a group  $G$  iff  $f(T_g^A(z)) = T_g^B(f(z))$  for all  $g \in G$ , where  $T_g^A$  and  $T_g^B$  are the representations of the group element  $g$  in  $A$  and  $B$  respectively (Serre et al., 1977). As a special case, it is invariant iff  $f(T_g^A(z)) = f(z)$ . In the context of generative modeling, we aim to learn distributions invariant to the symmetries of the data, namely graph permutations in our case. In practice, this has the benefit of not relying on any generation order and enabling efficient likelihood estimation since an invariant likelihood is identical for all permutations of the same graph. We leverage equivariant architectures such as EGNNs to learn such distributions, removing the need for data augmentation during training, further increasing efficiency. EGNNs (Satorras et al., 2021) are a special type of graph neural networks that learn functions equivariant to rotations, translations, reflections, and permutations of their input point cloud. More details on this architecture are given in Appendix D. Also note that due to the reflection equivariance, our model is invariant to chirality.
+
+Notation We introduce helpful notation for the two molecule representations used in this work. Let  $N$  denote the number of atoms of a given molecule. On the one hand, the molecular graph  $\mathcal{G} = (h,A)\in \mathbb{G}$  consists of atoms as nodes and chemical bonds as edges. Each atom has one of  $a$  atom types and one of  $c$  formal charges, while each bond has one of  $b$  bond types. We represent atoms as stacked one-hot encodings  $h\in \{0,1\}^{N\times (a + c)}$  and chemical bonds as one-hot vectors in an adjacency matrix  $A\in \{0,1\}^{N\times N\times b}$ . Note that the molecular graph is sometimes called a "2D molecule" as it can be typically drawn as a planar graph. On the other hand, the molecular point cloud  $\mathcal{P} = (h,\pmb {x})\in \mathbb{P}$  is a 3D point cloud, where  $\pmb {x}\in \mathbb{R}^{N\times 3}$  represents the coordinates of atoms' nuclei in Euclidean space. The point cloud encodes the bond information implicitly in the atomic coordinates. We now present our autoencoder model, whose overview is shown in Figure 2.
+
+Encoding The encoder  $\mathcal{E}_{\phi}:\mathbb{G}\to \mathbb{R}^{N\times (d + 3)}$  with parameters  $\phi$  maps a molecule's graph representation  $\mathcal{G} = (\pmb {h},\pmb {A})$  to its point cloud representation  $\mathcal{P} = (\pmb {h},\pmb {x})$  then further compresses it into a node-structured latent representation  $\pmb {z} = (z^{(h)},z^{(x)})$  , where  $z^{(h)}\in \mathbb{R}^{N\times d}$  and  $\pmb {z}^{(x)}\in \mathbb{R}^{N\times 3}$  are continuous embeddings for  $\pmb{h}$  and  $\pmb{x}$  with  $d <   a + c$  and  $\pmb {z}\in \mathbb{R}^{N\times (d + 3)}$  is their concatenation.
+
+In the first encoding step, the coordinates  $\pmb{x}$  are computed using the ETKDG algorithm (Riniker & Landrum, 2015), which first computes a molecule's distance bounds matrix based on typical bond lengths between different atom types, then produces atomic coordinates that satisfy these bounds, which are further optimized using a molecular force field. Crucially, this procedure allows to leverage chemical domain knowledge as an inductive bias into the
+
+![](images/1e895aac781d3e0b2b5e042b389be05554f840b4d2df70dd128662b6cda4b6a5.jpg)  
+Figure 2: Autoencoder architecture. The encoder maps a molecular graph to a point cloud, and the decoder learns the inverse. Both are trained jointly to minimize the reconstruction loss.
+
+model and get chemically meaningful 3D conformers. These coordinates encode the bond information of the molecular graph and ideally define an injective mapping from  $\mathbb{G}$  to  $\mathbb{P}$ . Note that this step is not learned and, thus, not required to be differentiable. In practice, we generate conformers for the whole training dataset in a preprocessing step and reuse them throughout training.
+
+Then, we apply an EGNN that maps the intermediate representation  $\mathcal{P} = (h, x)$  to the latent representation  $z = (z^{(h)}, z^{(x)})$ , defining the space of the generative model. The main benefit of this is to incorporate the neighborhood information into each atom's embeddings and coordinates and mapping the discrete features  $h$  into a continuous and lower-dimensional embedding  $z^{(h)}$ .
+
+Decoding The decoder  $\mathcal{D}_{\xi}:\mathbb{R}^{N\times (d + 3)}\to \mathbb{G}$  with parameters  $\xi$  aims to approximate the inverse function  $\mathcal{E}_{\phi}^{-1}$  and maps the latent point cloud  $\pmb{z}$  of a molecule back to its graph representation  $\mathcal{G}$ . It consists of an atom and a bond classifier. For each node  $i$ , the decoder applies a 2-layer Multi-Layer Perceptron (MLP) on its latent embedding to predict the logits for the atom type and the formal charge as  $\hat{h}_i = \mathrm{MLP}_{node}(z_i^{(h)})$ . For each pair of nodes  $i$  and  $j$ , the bond type is inferred by running a different 2-layer MLP on their edge features to obtain the corresponding logits as follows:
+
+$$
+\hat {A} _ {i j} = \frac {1}{2} \left(\mathbf {M L P} _ {e d g e} \left(\left[ \begin{array}{l} z _ {i} ^ {(h)} \\ z _ {j} ^ {(h)} \\ d _ {i j} ^ {2} \end{array} \right]\right) + \mathbf {M L P} _ {e d g e} \left(\left[ \begin{array}{l} z _ {j} ^ {(h)} \\ z _ {i} ^ {(h)} \\ d _ {i j} ^ {2} \end{array} \right]\right)\right),
+$$
+
+where  $d_{ij} = \| z_i^{(x)} - z_j^{(x)}\| _2$  and we use the simple averaging trick to ensure that  $\hat{\pmb{A}}$  is symmetric.
+
+Autoencoder Training The encoder and decoder are jointly optimized in a variational autoencoder (VAE) framework. We define probabilistic encoding and decoding processes as  $q_{\phi}(z|\mathcal{G}) = \mathcal{N}(\mathcal{E}_{\phi}(\mathcal{G}),\sigma_0^2 I)$  and  $p_{\xi}(\mathcal{G}|z) = \prod_{i = 1}^{N}p_{\xi}(h_i|z)\prod_{j = 1}^{N}p_{\xi}(A_{ij}|z)$ , respectively, where  $\sigma_0\in \mathbb{R}$  controls the variance in the latent space. The VAE is trained by minimizing the loss function
+
+$$
+\mathcal {L} _ {V A E} = - \mathbb {E} _ {q (\mathcal {G}) q _ {\phi} (z | \mathcal {G})} [ p _ {\xi} (\mathcal {G} | z) ], \tag {1}
+$$
+
+which, in practice, is computed as the cross-entropy loss. The decoder performs three classification tasks to reconstruct a molecular graph: atom types, formal charges, and bond types. The total loss is the unweighted sum of the corresponding three cross-entropy terms.
+
+Before continuing with our approach, we present an ablation study on our autoencoder architecture. The task consists of predicting all the bond types of a molecule from its 3D coordinates. We compare our EGNN-based approach with (i) an MLP and (ii) a rule-based system akin to Hoogeboom et al. (2022). More details on the experimental setup are in Appendix G.1. Results on ZINC250K are shown in Table 1, and similar results are observed for GuacaMol. Our autoencoder model achieves an almost perfect reconstruction accuracy, highlighting the importance of the EGNN in our architecture. This experiment supports our assumptions (i) that the used mapping from  $\mathbb{G}$  to
+
+Table 1: Accuracy of predicting bond types from 3D coordinates on ZINC250K. Existing methods use rule-based algorithms, while we introduce an EGNN-based algorithm.
+
+<table><tr><td>METHOD</td><td>ACCURACY</td></tr><tr><td>EGNN + MLP</td><td>99.93%</td></tr><tr><td>MLP</td><td>11.77%</td></tr><tr><td>RULE-BASED</td><td>11.78%</td></tr></table>
+
+$\mathbb{P}$  is injective and (ii) that the decoder can accurately approximate its inverse.
+
+# 4 EDM-SYCo
+
+Our SYCo framework lifts the graph generation problem to a point cloud generation problem in 3D. Concretely, we are interested in learning the distribution of latent point clouds  $z \sim q_{\phi}(z|\mathcal{G})q(\mathcal{G})$  defined by the underlying molecular graph distribution and the trained encoder by approximating it with a parametric distribution  $p_{\theta}$ . While this generation problem can, in principle, be approached by any 3D molecular generative method, this work adapts EDM under the SYCo framework due to its simplicity and empirical performance in 3D molecule generation tasks (Hoogeboom et al., 2022).
+
+# 4.1 EQUIVARIANT DIFFUSION ON LATENT POINT CLOUDS
+
+A diffusion model generates samples by reversing a diffusion process, the process of adding noise to data (Sohl-Dickstein et al., 2015; Ho et al., 2020). For ease of notation, we flatten the latent repre
+
+sentation of the point clouds  $z$ , namely their latent coordinates and features, into a one-dimensional vector denoted as  $z_0 \in \mathbb{R}^{d'}$  with  $d' = N(d + 3)$ . The diffusion or forward process is a fixed Markov chain of Gaussian updates that transform  $z_0$  into latent variables  $z_1, \ldots, z_T$  of the same dimension as  $z_0$ , defined as
+
+$$
+q \left(\boldsymbol {z} _ {1: T} \mid \boldsymbol {z} _ {0}\right) := \prod_ {t = 1} ^ {T} q \left(\boldsymbol {z} _ {t} \mid \boldsymbol {z} _ {t - 1}\right), \text {w i t h} q \left(\boldsymbol {z} _ {t} \mid \boldsymbol {z} _ {t - 1}\right) := \mathcal {N} \left(\boldsymbol {z} _ {t}; \sqrt {1 - \beta_ {t}} \boldsymbol {z} _ {t - 1}, \beta_ {t} \boldsymbol {I}\right), \tag {2}
+$$
+
+where  $\beta_{t}$  is typically chosen such that  $q(z_{T})$  converges to  $\mathcal{N}(\pmb {z}_T;\pmb {0},\pmb {I})$ . By defining  $\alpha_{t}\coloneqq 1 - \beta_{t}$  and  $\bar{\alpha}_{t}\coloneqq \prod_{i = 1}^{t}\alpha_{i}$ , we can also map  $\pmb {z}_0$  directly to  $\pmb {z}_t$  through
+
+$$
+q \left(\boldsymbol {z} _ {t} \mid \boldsymbol {z} _ {0}\right) = \mathcal {N} \left(\boldsymbol {z} _ {t}; \sqrt {\bar {\alpha} _ {t}} \boldsymbol {z} _ {0}, (1 - \bar {\alpha} _ {t}) \boldsymbol {I}\right). \tag {3}
+$$
+
+To reverse the above process, we start from a Gaussian noise sample  $\mathbf{z}_T$  and iteratively sample from the posterior distributions  $q(\mathbf{z}_{t-1} | \mathbf{z}_t)$ , which is also Gaussian if  $\beta_t$  is small (Sohl-Dickstein et al., 2015). However, estimating them requires using the entire dataset. Therefore, diffusion models learn a reverse process  $p_\theta(\mathbf{z}_{0:T}) \coloneqq p(\mathbf{z}_T) \prod_{t=1}^T p_\theta(\mathbf{z}_{t-1} | \mathbf{z}_t)$ , with  $p(\mathbf{z}_T) = \mathcal{N}(\mathbf{z}_T; \mathbf{0}, \mathbf{I})$  and
+
+$$
+p _ {\theta} \left(\boldsymbol {z} _ {t - 1} \mid \boldsymbol {z} _ {t}\right) := \mathcal {N} \left(\boldsymbol {z} _ {t - 1}; \boldsymbol {\mu} _ {\theta} \left(\boldsymbol {z} _ {t}, t\right), \sigma_ {t} ^ {2} \boldsymbol {I}\right), \tag {4}
+$$
+
+where the mean  $\pmb{\mu}_{\theta}(\pmb{z}_t, t) \in \mathbb{R}^{d'}$  is parametrized by a neural network.
+
+We follow Ho et al. (2020) and, instead of directly predicting the mean, predict the Gaussian noise added to the latent point cloud representation during the diffusion process. The mean then becomes  $\mu_{\theta}(z_t,t) = \frac{1}{\sqrt{\alpha_t}}\Big(z_t - \frac{1 - \alpha_t}{\sqrt{1 - \bar{\alpha}_t}}\epsilon_\theta (z_t,t)\Big)$ , where  $\epsilon_{\theta}(\pmb {z}_t,t)\in \mathbb{R}^{d^{\prime}}$  is the output of the neural network. Following Hoogeboom et al. (2022), we parametrize  $\epsilon_{\theta}$  via an EGNN (Satorras et al., 2021). Additionally, the reverse process ensures equivariance to translations by defining the learned distribution  $p_{\theta}(z_0)$  on the zero center of gravity subspace.
+
+This network can then be trained by optimizing the  $L_{2}$  loss between the sampled Gaussian noise at each step and the network's output,
+
+$$
+\mathcal {L} _ {D M} = \mathbb {E} _ {\boldsymbol {z} _ {0}, \epsilon , t} \left[ \| \epsilon - \epsilon_ {\theta} (\boldsymbol {z} _ {t}, t) \| _ {2} ^ {2} \right]. \tag {5}
+$$
+
+During sampling, we start from a Gaussian sample  $\mathbf{z}_T \sim p(\mathbf{z}_T)$  and iteratively denoise it by sampling  $\epsilon_t \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$  and transforming  $z_t$  to  $z_{t-1}$ :
+
+$$
+\boldsymbol {z} _ {t - 1} = \underbrace {\frac {1}{\sqrt {\alpha_ {t}}} \left(\boldsymbol {z} _ {t} - \frac {1 - \alpha_ {t}}{\sqrt {1 - \bar {\alpha} _ {t}}} \boldsymbol {\epsilon} _ {\theta} \left(\boldsymbol {z} _ {t} , t\right)\right)} _ {\boldsymbol {\mu} _ {\theta} \left(\boldsymbol {z} _ {t}, t\right)} + \underbrace {\frac {\sqrt {1 - \bar {\alpha} _ {t - 1}}}{\sqrt {1 - \bar {\alpha} _ {t}}} \sqrt {\beta_ {t}}} _ {\sigma_ {t}} \boldsymbol {\epsilon} _ {t}. \tag {6}
+$$
+
+# 4.2 TRAINING AND SAMPLING FROM EDM-SYCO
+
+After training the autoencoder as described in Section 3, EDM is trained in a second stage by optimizing the loss function defined in Equation 5. We describe the full training procedure of both parts in Algorithm 1. Consistent with previous work on latent diffusion models (Rombach et al., 2022; Xu et al., 2023), we found this two-stage training procedure to perform better than jointly training the autoencoder and the diffusion model.
+
+For sampling, we need access to the trained EDM and the decoder but not the encoder. First, starting with a sample  $\boldsymbol{z}_T \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$  representing a random 3D point cloud, we iteratively denoise it with the diffusion model (Equation 6) to obtain the clean sample  $\boldsymbol{z}_0$ . Then, we apply the decoder to get the corresponding molecular graph  $\boldsymbol{h}$ ,  $\boldsymbol{A} = \mathcal{D}_{\xi}(\boldsymbol{z}_0)$ . Algorithm 2 formally describes this procedure, and Figure 1 further illustrates it. This sampling procedure defines a permutation-invariant molecular graph distribution. The following proposition, which we prove in Appendix C.2, formalizes this:
+
+Proposition 4.1. The marginal distribution of molecular graphs  $p_{\theta, \xi}(\mathcal{G}) = \mathbb{E}_{p_{\theta}(\mathbf{z}_0)}[p_{\xi}(\mathcal{G}|\mathbf{z}_0)]$  defined by the EDM and the decoder, is an  $S_N$ -invariant distribution, i.e. for any molecular graph  $\mathcal{G} = (\pmb{h}, \pmb{A})$ ,  $p_{\theta, \xi}(\pmb{h}, \pmb{A}) = p_{\theta, \xi}(Ph, PAP)$  for any permutation matrix  $P \in S_N$ .
+
+So far, we have assumed that the number of atoms  $N$  is fixed across all molecules. In practice, to generate molecules with different sizes, we follow previous work (Hoogeboom et al., 2022) and estimate the empirical distribution of the number of atoms in the training set  $p(N)$  and use it to sample different numbers of atoms before running the sampling procedure described above. In Appendix F.2, we describe the use of conditional distributions to sample  $N$  in a constrained generation or optimization.
+
+# 5 CONTROLLABLE GENERATION AND SIMILARITY-CONSTRAINED OPTIMIZATION
+
+So far, we have introduced our approach to distribution learning on molecular graphs. However, many tasks in drug discovery require generating molecules with specific conditions such as chemical properties, similarity constraints, or the presence of certain desirable substructures. In this section, we describe how we can adapt the sampling procedure of EDM-SyCo (or other models combined with SyCo) – trained in an unconditional setting – to different conditioning modalities. While previous work on 3D molecule generation tackled conditional generation (Hoogeboom et al., 2022; Bao et al., 2022) and scaffolding (Schneuing et al., 2022), the similarity-constrained optimization task (Jin et al., 2020) has not been addressed in the 3D molecule generation literature. We fill this gap by introducing a novel sampling algorithm for diffusion models to perform similarity-constrained optimization.
+
+Property-Conditional Generation via Diffusion Guidance In a conditional generation setting, we want to generate a molecule with a specific property  $c$  by sampling from the conditional distribution  $q(\boldsymbol{z}|\boldsymbol{c})$ . To achieve this, we adapt the classifier guidance algorithm (Dhariwal & Nichol, 2021), initially proposed to guide image diffusion models using a classifier. Instead, we use a property regressor to guide the generation towards molecules with specific continuous properties similar to Bao et al. (2022). One can show that the denoising network's output relates to the score function of the data distribution via  $\nabla_{\boldsymbol{z}_t}\log q(\boldsymbol{z}_t)\approx -\frac{1}{\sqrt{1 - \bar{\alpha}_t}}\epsilon_\theta (\boldsymbol{z}_t,t)$  (Dhariwal & Nichol, 2021) and the sampling procedure of diffusion models (Equation 6) is equivalent to running Langevin dynamics using the score function  $\nabla_{\boldsymbol{z}_t}\log q(\boldsymbol{z}_t)$  (Welling & Teh, 2011; Song & Ermon, 2019). This provides a way to sample from the distribution  $q$  only through its score function. For the case of  $q(\boldsymbol{z}_t|c)$ , its score function can be written using Bayes' rule and approximated as:
+
+$$
+\begin{array}{l} \nabla_ {\boldsymbol {z} _ {t}} \log q (\boldsymbol {z} _ {t} | c) = \nabla_ {\boldsymbol {z} _ {t}} \log q (\boldsymbol {z} _ {t}) + \nabla_ {\boldsymbol {z} _ {t}} \log q (c | \boldsymbol {z} _ {t}) \\ \approx - \frac {1}{\sqrt {1 - \bar {\alpha} _ {t}}} (\pmb {\epsilon} _ {\theta} (\pmb {z} _ {t}, t) - \sqrt {1 - \bar {\alpha} _ {t}} \nabla_ {\pmb {z} _ {t}} \log p _ {\eta} (c | \pmb {z} _ {t})), \\ \end{array}
+$$
+
+where  $p_{\eta}(c|\boldsymbol{z}_t) \sim \exp(-s(g_{\eta}(\boldsymbol{z}_t) - c)^2)$  is a Gaussian distribution approximating the probability of molecule  $\boldsymbol{z}_t$  having property  $c$ ,  $g_{\eta}$  is a property regressor trained to predict the property  $c$  given a noisy molecule  $\boldsymbol{z}_t$ , and  $s$  is a scale factor that controls the skewness of the distribution. With this, we can rewrite the conditional score function as  $\nabla_{\boldsymbol{z}_t}\log q(\boldsymbol{z}_t|c) \approx -\frac{1}{\sqrt{1 - \bar{\alpha}_t}} (\epsilon_\theta (\boldsymbol{z}_t,t) + \sqrt{1 - \bar{\alpha}_t} s\nabla_{\boldsymbol{z}_t}(g_{\eta}(\boldsymbol{z}_t) - c)^2)$ . To sample from this distribution, we use the same sampling algorithm defined by Equation 6 and replace  $\epsilon_{\theta}(\boldsymbol{z}_t,t)$  by
+
+$$
+\epsilon_ {\theta} \left(\boldsymbol {z} _ {t}, t\right) + \sqrt {1 - \bar {\alpha} _ {t}} s \nabla_ {\boldsymbol {z} _ {t}} \left(g _ {\eta} \left(\boldsymbol {z} _ {t}\right) - c\right) ^ {2}. \tag {7}
+$$
+
+Intuitively, this can be seen as minimizing the loss  $(g_{\eta}(\pmb{z}_t) - c)^2$ . Appendix E.2 provides additional details and discussions about the equivariance of the regressor.
+
+Unconstrained Property Optimization via Diffusion Guidance With the formulation introduced above, we can sample molecules with high property values by simply using the denoising step
+
+$$
+\epsilon_ {\theta} \left(\boldsymbol {z} _ {t}, t\right) - \sqrt {1 - \bar {\alpha} _ {t}} s \nabla_ {\boldsymbol {z} _ {t}} g _ {\eta} \left(\boldsymbol {z} _ {t}\right). \tag {8}
+$$
+
+Notice that we replaced the loss with the negative gradient of the regressor.
+
+Scaffold-Constrained Generation via Inpainting For tasks such as structure-based drug design, it is often desirable to generate molecules that contain a specific substructure, usually called a scaffold (Schuffenhauer et al., 2007; Maziarz et al., 2021). This problem can be viewed as a completion task and is reminiscent of inpainting, which aims to complete missing parts of an image (Song et al., 2020; Lugmayr et al., 2022). Lugmayr et al., 2022 propose RePaint, a simple modification to the sampling procedure of denoising diffusion models, enabling them to perform inpainting. This approach has already been successfully applied to molecule generation in 3D (Schneuing et al., 2022).
+
+The core idea of RePaint is to start the sampling procedure from a random sample  $\mathbf{z}_T$  and, at each step, enforce the part that we want to be present in the final sample. To illustrate this for the case of molecules, let  $\tilde{\mathbf{z}}_0$  denote the point cloud of a molecule containing the desired scaffold. Let  $m$  be a binary mask that indicates which nodes from  $\tilde{\mathbf{z}}_0$  belong to the scaffold such that  $m \odot \tilde{\mathbf{z}}_0$  denotes the
+
+![](images/eb88cceb913fcd368e48984bb9471ae6a3fd663d7d18ab31dc27ff615b7ae646.jpg)  
+Figure 3: Overview of our constrained optimization procedure. Based on a noising/denoising approach, we run the first steps of the reverse diffusion process using the inpainting algorithm to add new atoms and the remaining steps using the guidance algorithm to increase the target property. The depicted molecules have QED values of 0.79 (initial) and 0.91 (optimized), with a  $53\%$  similarity.
+
+scaffold point cloud and  $(1 - m)\odot \tilde{z}_0$  the unknown part. We aim to sample a new point cloud  $z$  containing this scaffold while extending it to get a harmonized point cloud. Formally, we require that  $m\odot z_0 = m\odot \tilde{z}_0$ . To achieve this, we iteratively apply the following modified sampling step starting from  $t = T$ :
+
+$$
+z _ {t - 1} ^ {\text {s c a f f o l d}} \sim \mathcal {N} \left(\sqrt {\bar {\alpha} _ {t - 1}} \tilde {z} _ {0}, (1 - \bar {\alpha} _ {t - 1}) I\right) \tag {9}
+$$
+
+$$
+\mathbf {z} _ {t - 1} ^ {\text {u n k n o w n}} \sim \mathcal {N} \left(\boldsymbol {\mu} _ {\theta} \left(\mathbf {z} _ {t}, t\right), \sigma_ {t} ^ {2} \boldsymbol {I}\right) \quad (\text {a s i n E q u a t i o n 6}) \tag {10}
+$$
+
+$$
+\boldsymbol {z} _ {t - 1} = m \odot \boldsymbol {z} _ {t - 1} ^ {\text {s c a f f o l d}} + (1 - m) \odot \boldsymbol {z} _ {t - 1} ^ {\text {u n k n o w n}} \tag {11}
+$$
+
+We refer to Appendix E.3 for more implementation details.
+
+Similarity-Constrained Optimization In this task, we aim to improve the target properties of a given molecule while satisfying a similarity constraint. We propose a novel approach to this task by combining the regressor guidance algorithm and the inpainting algorithm discussed above. The key idea is to add noise to the initial molecule and then denoise it with the regressor guidance algorithm to improve the target property. Formally, starting from the initial point cloud  $z_0$  with  $N$  atoms, we sample a noised  $z_t$  following Equation 3 for  $t \in (0, T)$ .  $t$  is a hyperparameter that controls the optimization quality. On the one hand, increasing  $t$  results in more reverse diffusion steps with the guidance algorithm, thus increasing the chance of improving the target property. On the other hand, it also results in losing information about the initial molecule due to noise. Intuitively, the best  $t$  yields the highest property improvement while satisfying the similarity constraints.
+
+However, since the diffusion model operates on  $N$  points, the denoised molecule also has  $N$  atoms. This limits the optimization of a molecule's property as it prevents adding new atoms. To overcome this limitation, we sample a Gaussian point cloud  $\mathbf{z}_T$  with  $N'$  atoms, where  $N' > N$ , and run the inpainting algorithm from  $T$  to  $t$  with  $\mathbf{z}_t$  as the scaffold, meaning that  $\mathbf{z}_t$  will replace  $m \odot \tilde{\mathbf{z}}_0$  in the inpainting algorithm described above. This results in a new intermediate representation  $\mathbf{z}_t'$  whose first  $N$  points correspond to  $\mathbf{z}_t$ , and the rest are the newly added atoms, as desired. Finally,  $\mathbf{z}_t'$  is denoised from  $t$  to 0 using the guidance algorithm with maximization objective (Equation 8) to get the point cloud  $\mathbf{z}_0'$  of the optimized molecule, which is then decoded to the graph using the decoder  $\mathcal{D}_{\xi}$ . An overview of this procedure and an example test molecule are shown in Figure 3.
+
+# LIMITATIONS
+
+EDM-SYCo can be seen as an extension to EDM, which carries the limitations of diffusion models like high inference cost. To mitigate this issue in future work, we can incorporate more efficient sampling strategies (Karras et al., 2022; 2023), or use more efficient models such as recent conditional flow matching approaches (Tong et al., 2023; Song et al., 2023). In addition, our sampling approach requires specifying the number of nodes a priori. While this works well empirically, we leave tackling the fundamental issue of generating variable-sized graphs or point clouds to future work. Lastly, our constrained optimization procedure cannot remove atoms during inference. Our experiments found that the decoder tends to predict disconnected subgraphs, which we treat as removing atoms, but a rigorous solution may improve performance.
+
+# 6 EXPERIMENTS
+
+# 6.1 COMPARISON TO 3D MOLECULE GENERATIVE MODEL GEOLDM
+
+EDM-SyCo leverages EDM to learn the distribution of 3D molecules in latent space, and combines it with an autoencoder that maps between 2D and 3D molecules. The overall architecture is therefore similar to GeoLDM in that both are EDM-based latent diffusion models for molecules. The main differences lie, however, in the design of the autoencoder and in the training and sampling procedure. In this section, we explain the need for such changes in order to adapt 3D molecule generative models like GeoLDM to the different modality of 2D molecules and validate the effect of these changes experimentally. In essence, there are two main challenges when it comes to applying GeoLDM or any other 3D molecule generative models to 2D molecule generation:
+
+(i) GeoLDM requires datasets with 3D atomic coordinates for training. To enable training on 2D datasets like ZINC250K or GuacaMol, we leverage the conformer generation component introduced in Section 3. Crucially, this allows to inject chemical domain knowledge as an inductive bias into the model by leveraging typical bond lengths and molecular force fields when computing 3D conformers.
+
+(ii) GeoLDM uses a simple rule-based algorithm to predict 2D molecules from 3D molecules, which fails even on medium-sized molecules. This algorithm predicts bond types from inter-atomic distances using a lookup table. In our experiments, this algorithm can correctly predict all bond types of a molecule only on  $11.78\%$  of the ZINC250K dataset (See Table 1). To overcome this limitation, we introduce our EGNN-based bond type predictor, which can leverage the richer neighborhood information of an atom and does not rely on hard-coded rules. To highlight the effect of this change on the quality of the generated 2D graphs, we compare GeoLDM and EDM-SyCo, both trained on ZINC250K using the same synthetic coordinates and report the results in Table 2.
+
+Table 2: Comparison between GeoLDM and EDM-SYCo on ZINC250K. To enable the training and evaluation of GeoLDM, we use the same synthetic coordinates as EDM-SYCo and use GeoLDM's bond type prediction algorithm.
+
+<table><tr><td></td><td>GEOLDM</td><td>EDM-SYCO</td></tr><tr><td>FCD (↑)</td><td>0.17</td><td>0.85</td></tr><tr><td>KL (↑)</td><td>0.79</td><td>0.96</td></tr><tr><td>NOVELTY (↑)</td><td>1.0</td><td>1.0</td></tr><tr><td>UNIQUENESS (↑)</td><td>1.0</td><td>1.0</td></tr><tr><td>VALIDITY (↑)</td><td>0.12</td><td>0.88</td></tr></table>
+
+From the above discussion, we can conclude that (i) applying 3D models off-the-shelf as 2D baselines is either not directly possible (for datasets without 3D information) or yields limited performance due to the inability of the used bond type prediction algorithm to produce large and valid molecules, and (ii) combining EDM or GeoLDM with our proposed components would yield exactly our model since EDM and GeoLDM both share the same diffusion framework, which we also use in EDM-SyCo.
+
+# 6.2 COMPARISON TO 2D MOLECULE GENERATIVE MODELS
+
+In the following set of experiments, we compare the performance of EDM-SYCo to several autoregressive and to all-at-onc diffusion-based baselines on de novo and conditional generation tasks as well as on a constrained optimization task. Specifically, comparing EDM-SYCo with the diffusion-based baselines highlights the benefit of defining the diffusion process on the latent 3D space compared to directly defining it on the 2D space.. More experiments are discussed in Appendix G. We use two datasets of different sizes and complexities: ZINC250K (Irwin et al., 2012) containing  $250\mathrm{K}$  molecules with up to 40 atoms, and GuacaMol (Brown et al., 2019) containing  $\approx 1.5\mathrm{M}$  drug-like molecules with up to 88 atoms. We train EDM-SYCo as described in Sections 3 and 4 on these two datasets. Training details and hyperparameters are given in Appendix F.
+
+De novo Generation We leverage the GuacaMol benchmark (Brown et al., 2019), an evaluation framework for de novo molecular graph generation. We consider two metrics that measure the similarity between the generated molecules and the training set. The Fréchet ChemNet Distance (FCD) compares the hidden representations of ChemNet, a neural network capturing chemical and biological features of molecules (Preuer et al., 2018), and the KL divergence (KL) compares the probability distributions of a variety of physiochemical descriptors. We also report novelty, uniqueness, and validity scores, which measure the fraction of generated molecules that are not in the training set, that are pairwise different, and that are chemically valid according to RDKit, respectively.
+
+Table 3: Evaluation metrics on the ZINC250K dataset. Results for methods with * are taken from their original papers. Methods that do no report the KL metric are denoted with N.A.. We split the methods into autoregressive and all-at-once methods. The best scores within each category are in bold. EDM-SyCO outperforms all all-at-once baselines by more than  $26\%$  on the FCD metric. Standard deviations are across 3 different training runs of the models.  
+
+<table><tr><td></td><td>METHOD</td><td>FCD (↑)</td><td>KL (↑)</td><td>NOVELTY (↑)</td><td>UNIQUENESS (↑)</td><td>VALIDITY (↑)</td></tr><tr><td rowspan="8">AUTOREGRESSIVE</td><td>JT-VAE (2018)</td><td>0.75 ± 0.01</td><td>0.93 ± 0.00</td><td>1.00 ± 0.00</td><td>1.00 ± 0.00</td><td>1.00 ± 0.00</td></tr><tr><td>GRAPHAF (2020)</td><td>0.05 ± 0.00</td><td>0.67 ± 0.01</td><td>0.91 ± 0.01</td><td>0.91 ± 0.01</td><td>1.00 ± 0.00</td></tr><tr><td>HIERVAE (2020)</td><td>0.50 ± 0.14</td><td>0.92 ± 0.00</td><td>0.96 ± 0.01</td><td>0.96 ± 0.01</td><td>1.00 ± 0.00</td></tr><tr><td>MOLER (2021)</td><td>0.83 ± 0.00</td><td>0.97 ± 0.00</td><td>0.99 ± 0.00</td><td>0.99 ± 0.00</td><td>1.00 ± 0.00</td></tr><tr><td>PS-VAE (2022)</td><td>0.28 ± 0.01</td><td>0.84 ± 0.00</td><td>1.00 ± 0.00</td><td>1.00 ± 0.00</td><td>1.00 ± 0.00</td></tr><tr><td>GRAPHARM* (2023)</td><td>0.04 ± 0.00</td><td>N.A.</td><td>1.00 ± 0.00</td><td>0.99 ± 0.00</td><td>0.88 ± 0.00</td></tr><tr><td>MICAM (2023)</td><td>0.63 ± 0.02</td><td>0.94 ± 0.00</td><td>0.98 ± 0.00</td><td>0.98 ± 0.00</td><td>1.00 ± 0.00</td></tr><tr><td>MAGNET (2023)</td><td>0.76 ± 0.00</td><td>0.95 ± 0.00</td><td>0.99 ± 0.00</td><td>0.99 ± 0.00</td><td>1.00 ± 0.00</td></tr><tr><td rowspan="6">ALL-AT-ONCE</td><td>GDSS* (2022)</td><td>0.10 ± 0.01</td><td>N.A.</td><td>1.00 ± 0.00</td><td>1.00 ± 0.00</td><td>0.97 ± 0.01</td></tr><tr><td>DIGRESS (2022)</td><td>0.65 ± 0.00</td><td>0.91 ± 0.00</td><td>0.99 ± 0.00</td><td>0.99 ± 0.00</td><td>0.85 ± 0.01</td></tr><tr><td>GEOLDM (2023)</td><td>0.17 ± 0.00</td><td>0.79 ± 0.00</td><td>1.00 ± 0.00</td><td>1.00 ± 0.00</td><td>0.12 ± 0.00</td></tr><tr><td>GRUM* (2023)</td><td>0.64 ± 0.01</td><td>N.A.</td><td>1.00 ± 0.00</td><td>1.00 ± 0.00</td><td>0.99 ± 0.00</td></tr><tr><td>SWINGNN* (2023)</td><td>0.67 ± 0.00</td><td>N.A.</td><td>0.96 ± 0.00</td><td>1.00 ± 0.00</td><td>0.91 ± 0.00</td></tr><tr><td>EDM-SYCO (OURS)</td><td>0.85 ± 0.01</td><td>0.96 ± 0.00</td><td>1.00 ± 0.00</td><td>1.00 ± 0.00</td><td>0.88 ± 0.01</td></tr></table>
+
+All scores, including FCD and KL, are normalized to lie between 0 and 1 and such that a higher score corresponds to a better performance. More details on these metrics are in Appendix F.8. De novo generation is a prerequisite for more complex tasks such as conditional generation and optimization.
+
+Tables 3 and 4 show the performance of EDM-SYCo and the baselines on the ZINC250K and GuacaMol datasets, respectively. Notably, EDM-SYCo outperforms all at-onc diffusion based models on the FCD metric by more than  $26\%$  on ZINC250K and  $16\%$  on GuacaMol. This shows the capacity of our model to more accurately capture the joint distribution of nodes and edges encoded in the latent features and coordinates. EDM-SYCo sets the new state-of-the-art FCD score on ZINC250K, outperforming all autoregressive and fragment-based baselines, which incorporate more domain knowl
+
+Table 4: FCD and KL scores on GuacaMol. * means corresponding numbers are from (Maziarz et al., 2021). DiGress results are from Vignac et al. (2022). For our model, we additionally report standard deviation across 3 validations of the same model.
+
+<table><tr><td></td><td></td><td>FCD (↑)</td><td>KL (↑)</td></tr><tr><td colspan="2">CGVAE* (2018)</td><td>0.26</td><td>N.A.</td></tr><tr><td colspan="2">HIERVAE* (2020)</td><td>0.62</td><td>N.A.</td></tr><tr><td colspan="2">JT-VAE* (2018)</td><td>0.73</td><td>N.A.</td></tr><tr><td colspan="2">MOLER* (2021)</td><td>0.81</td><td>N.A.</td></tr><tr><td rowspan="2">AAO</td><td>DIGRESS (2022)</td><td>0.68</td><td>0.93</td></tr><tr><td>EDM-SYCO (OURS)</td><td>0.79 ± 0.002</td><td>0.93 ± 0.006</td></tr></table>
+
+edge. Note that due to computational constraints, we use the same hyperparameters from ZINC250K to train our model on GuacaMol, while DiGress and MoLeR have been tuned for both datasets. Regarding validity, all autoregressive baselines achieve  $100\%$  validity scores due to valency checks after each intermediate step, which is usually not done for all-at-once methods. We show some sample molecules generated by our model in Figures 4 and 5 for the two datasets. We include an ablation study on the inductive bias of our architecture in Appendix G.10, compare to an EGNN-free encoder setting in Appendix G.11, and provide a runtime analysis in Appendix G.12.
+
+# Property-Conditioned Generation
+
+To evaluate the conditional generation performance of our approach, we follow Ninniri et al. (2023) and condition the generation of 1000 molecules on LogP and molecular weight (MW) values sampled from ZINC250K. We
+
+Table 5: Conditional generation results.  
+
+<table><tr><td rowspan="2">METHOD</td><td colspan="2">LOGP</td><td colspan="2">MW</td></tr><tr><td>MAE (↓)</td><td>VALID (↑)</td><td>MAE (↓)</td><td>VALID (↑)</td></tr><tr><td>DIGRESS</td><td>0.49 ± 0.05</td><td>65%</td><td>60.30 ± 6.10</td><td>73%</td></tr><tr><td>FREEGRESS</td><td>0.15 ± 0.02</td><td>85%</td><td>15.18 ± 2.71</td><td>75%</td></tr><tr><td>EDM-SYCO</td><td>0.14 ± 0.00</td><td>76%</td><td>3.86 ± 0.08</td><td>88%</td></tr></table>
+
+report the mean absolute error (MAE) between the target and estimated values for the generated molecules (using RDKit) and the validity rate. More details on the experimental setup are in Appendix F.6. We run the regressor guidance algorithm, discussed in Section 5. As baselines, we compare to DiGress, which uses a discrete guidance scheme, and FreeGress, which proposes a
+
+Table 6: Similarity-constrained optimization results from (Jin et al., 2020). We split all learning-based baslines into translation and optimization approaches.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">METHOD</td><td colspan="2">LOGP (SIM ≥ 0.4)</td><td colspan="2">LOGP (SIM ≥ 0.6)</td><td colspan="2">QED (SIM ≥ 0.4)</td><td colspan="2">DRD2 (SIM ≥ 0.4)</td></tr><tr><td>IMPROV.</td><td>DIV.</td><td>IMPROV.</td><td>DIV.</td><td>SUCC.</td><td>DIV.</td><td>SUCC.</td><td>DIV.</td></tr><tr><td rowspan="4">TRANSL.</td><td>SEQ2SEQ</td><td>3.37 ± 1.75</td><td>0.471</td><td>2.33 ± 1.17</td><td>0.331</td><td>58.5%</td><td>0.331</td><td>75.9%</td><td>0.176</td></tr><tr><td>JTNN</td><td>3.55 ± 1.67</td><td>0.480</td><td>2.33 ± 1.24</td><td>0.333</td><td>59.9%</td><td>0.373</td><td>77.8%</td><td>0.156</td></tr><tr><td>ATOMG2G</td><td>3.98 ± 1.54</td><td>0.563</td><td>2.41 ± 1.19</td><td>0.379</td><td>73.6%</td><td>0.421</td><td>75.8%</td><td>0.128</td></tr><tr><td>HIERG2G</td><td>3.98 ± 1.46</td><td>0.564</td><td>2.49 ± 1.09</td><td>0.381</td><td>76.9%</td><td>0.477</td><td>85.9%</td><td>0.192</td></tr><tr><td rowspan="4">OPTIM.</td><td>JT-VAE</td><td>1.03 ± 1.39</td><td>-</td><td>0.28 ± 0.79</td><td>-</td><td>8.8%</td><td>-</td><td>3.4%</td><td>-</td></tr><tr><td>CG-VAE</td><td>0.61 ± 1.09</td><td>-</td><td>0.25 ± 0.74</td><td>-</td><td>4.8%</td><td>-</td><td>2.3%</td><td>-</td></tr><tr><td>GCPN</td><td>2.49 ± 1.30</td><td>-</td><td>0.79 ± 0.63</td><td>-</td><td>9.4%</td><td>0.216</td><td>4.4%</td><td>0.152</td></tr><tr><td>EDM-SYCo</td><td>3.11 ± 1.27</td><td>0.559</td><td>1.51 ± 1.10</td><td>0.388</td><td>46.4%</td><td>0.352</td><td>27.3%</td><td>0.303</td></tr></table>
+
+regressor-free guidance mechanism Ninniri et al. (2023). Table 5 shows the results. While EDM-SYCo achieves a comparable score to FreeGress on LogP, it outperforms it by a factor of 3.9 on MW. Compared to DiGress, it improves on LogP and MW by factors 3.5 and 15.6, respectively. This experiment highlights the benefit of our continuous latent space on conditional generation compared to operating directly on the discrete graph space. It also shows that graph-level properties are represented sufficiently well in this space. We include additional related experiments in Appendix G.4 and Appendix G.5.
+
+Similarity-Constrained Optimization We follow Jin et al. (2020) and evaluate our optimization procedure from Section 5 with EDM-SYCO on their constrained optimization tasks: LogP, QED, and DRD2 on ZINC250K. These tasks consist of improving the properties of a set of 800 test molecules under the constraint that the Tanimoto similarity with Morgan fingerprints (Rogers & Hahn, 2010) between the optimized and initial molecule is above a given threshold. We report the average improvement for the LogP tasks and success rates for the binary tasks of QED and DRD2 consisting of translating molecules from a low range into a higher range. Additionally, we report the average diversity among the optimized molecules. More details are given in Appendix F.7.
+
+We split the learning-based baselines from (Jin et al., 2020) into optimization and translation approaches. Translation approaches are specifically designed for such tasks and are trained on pairs of molecules exhibiting the improvement and similarity constraints, giving them an advantage over optimization methods. Since we leverage the same generative model and regressor from previous experiments without further training on this task, optimization approaches are the main point of comparison. Table 6 shows that EDM-SYCo outperforms all optimization baselines by an average factor of 3.5 across all tasks while achieving comparable performance to the translation approaches. These results further support the effectiveness of our SYCo framework and our novel constrained optimization algorithm. Figures 6 and 7 illustrate some of EDM-SYCo's optimized molecules. Thanks to our atom-based approach, the optimized molecules frequently only change in a few atoms from the starting molecule, which would be harder to achieve for fragment-based models.
+
+# 7 CONCLUSION
+
+Despite the common practice of using fragment-based autoregressive models for molecular graph generation, we demonstrated with SYCo that a mapping between molecular graphs and latent Euclidean point clouds enables atom-based all-at-once approaches to be competitive with special-purpose molecular graph generators. Further, we introduced a similarity-constrained optimization procedure for 3D diffusion models based on guidance and inpainting. Based on this framework, we developed EDM-SYCo, which sets a new state-of-the-art FCD score on ZINC250K. In our conditional generation and optimization experiments, we found our guidance-based approach to accurately guide the generation process, outperforming all baselines. With these results, we conclude that latent Euclidean generative models hold significant promise for advancing molecular graph generation and accelerating drug discovery. We discuss our work's broader impact in Appendix B.
+
+# ACKNOWLEDGMENTS
+
+This project is supported by the DAAD programme Konrad Zuse Schools of Excellence in Artificial Intelligence, sponsored by the Federal Ministry of Education and Research. Further, it is funded by the Federal Ministry of Education and Research (BMBF) and the Free State of Bavaria under the Excellence Strategy of the Federal Government and the Länder. This project is also supported by the Bavarian Ministry of Economic Affairs, Regional Development and Energy with funds from the Hightech Agenda Bayern.
+
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+
+# A VISUALIZATIONS FOR GENERATED AND OPTIMIZED MOLECULES BY EDM-SYCo
+
+We visualize some of the molecules generated by EDM-SyCo. For de novo generation, we show molecules of the model trained on ZINC250K in Figure 4, and of the one trained on GuacaMol in Figure 5. For the similarity-constrained optimization task, we show some molecules found by our optimization approach in the LogP task in Figure 6, and in the QED task in Figure 7.
+
+![](images/34900e83ff521b46b072dfc88db2e1522d3f33f6fff6b3f1bec96cc697d13ae5.jpg)  
+Figure 4: Sample molecules generated by EDM-SYCo trained on ZINC250K.
+
+![](images/48e3f24a84cfbfc85aeec4a1b10155f4fc176d525086cdb56983d50c998d0702.jpg)  
+Figure 5: Sample molecules generated by EDM-SYCo trained on GuacaMol.
+
+![](images/ef4f401269ba3319987a1aff2f635e5d857be897c9c48f18df74d6a1c24857ae.jpg)  
+Figure 6: Molecules with the highest improvement in the LogP-constrained optimization task. Each row corresponds to one test molecule and 4 successful optimization results. We show the LogP value of the initial molecules, and for the optimized molecules, the achieved improvement (imp.) and the Tanimoto similarity to the initial molecule (sim.).
+
+![](images/e24f6bd76a0473ba8f552f981f27a0329f95bba26dfca1958ad5cba01efcb489.jpg)  
+Figure 7: Molecules with highest improvement in the QED constrained optimization task. Each row corresponds to one test molecule and 4 successful optimization results. We show the QED value of the initial molecules, and for the optimized molecules, the achieved improvement (imp.) and the Tanimoto similarity to the initial molecule (sim.).
+
+# B BROADER IMPACT
+
+This paper advances the field of computational drug discovery by proposing new efficient methods for generative models. While there may be unintended consequences, such as the misuse of chemical weapons, we firmly believe that their benefits significantly outweigh the minuscule chance of misuse.
+
+# C INVARIANCE AND EQUIVARIANCE
+
+In this work, we are dealing with two types of symmetries arising from how we represent the data and/or physical symmetries. Specifically, graphs are invariant to permuting their nodes, meaning that a single graph of  $N$  nodes can be represented by  $N!$  different representations corresponding to  $N!$  different orderings of its nodes. We refer to this transformation as the elements of the symmetric group  $S_N$ . Similarly, a point cloud representing a molecule in 3D can be arbitrarily rotated, reflected, and/or translated and still refer to the same molecule. This corresponds to an infinite number of different representations for the same object. We refer to this transformation as the action of the Euclidean group  $E(3)$ .
+
+# C.1  $S_{N}$  AND  $E(3)$ -INVARIANT TRAINING PROCESS
+
+To efficiently learn from such data, we need to develop training methods invariant to these symmetries without needing data augmentation techniques. Concretely, we must ensure that the gradient updates used to train our model do not change if the molecular graph is permuted and/or the latent point cloud representation is rotated/reflected/translated. We require two ingredients to achieve this: an equivariant architecture and an invariant loss (Vignac et al., 2022). Based on the model architecture and the loss function described in the main text, we can show that EDM-SyCo satisfies both requirements and that its training process is, thus, invariant to the action of the permutation group  $S_N$  and the Euclidean group  $E(3)$ .
+
+As our ultimate goal is to generate molecular graphs, we provide more details on the permutation invariance properties of the distribution of molecular graphs learned by EDM-SyCo in the next section.
+
+# C.2 PROOF OF PROPOSITION 4.1 ( $S_N$ -INVARIANT MARGINAL DISTRIBUTION)
+
+In this section, we provide a formal proof for the statement from the main text that the molecular graph distribution defined by EDM-SyCo is invariant to permutations. For convenience, we provide the proposition again:
+
+Proposition C.1. The marginal distribution of molecular graphs  $p_{\theta, \xi}(\mathcal{G}) = \mathbb{E}_{p_{\theta}(\mathbf{z}_0)}[p_{\xi}(\mathcal{G}|\mathbf{z}_0)]$  defined by the EDM and the decoder, is an  $S_N$ -invariant distribution, i.e. for any molecular graph  $\mathcal{G} = (\pmb{h}, \pmb{A})$ ,  $p_{\theta, \xi}(\pmb{h}, \pmb{A}) = p_{\theta, \xi}(Ph, PAP)$  for any permutation matrix  $P \in S_N$ .
+
+Note: This invariance property is required for efficient likelihood computation, as the likelihood of a graph is the sum of the likelihoods of its  $N!$  permutations, which is intractable to compute. However, when ensuring that all these permutations are assigned equal likelihood, it suffices to compute the likelihood of an arbitrary permutation.
+
+Proof. The idea of the proof is when the initial distribution of the diffusion model  $p_{\theta}(z_T)$  is invariant and the transition distributions  $p_{\theta}(z_{t - 1} | z_t)$  are equivariant, then all marginal distributions at all diffusion time steps  $p_{\theta}(z_t)$  are invariant, including  $p_{\theta}(z_0)$  (Xu et al., 2022; 2023; Hoogeboom et al., 2022). With the same logic and with the equivariant decoder taking the place of the transition distribution, the graph distribution  $p_{\theta, \xi}(\mathcal{G})$  will be invariant.
+
+We prove this result by induction:
+
+Base case:  $p_{\theta}(\boldsymbol{z}_T) = \mathcal{N}(\boldsymbol{z}_T; \mathbf{0}, \boldsymbol{I})$  is permutation-invariant, i.e.  $p_{\theta}(\boldsymbol{z}_T) = p_{\theta}(P\boldsymbol{z}_T)$  for all permutation matrices  $P \in \{0,1\}^{N \times N}$  acting on  $\boldsymbol{z}_T \in \mathbb{R}^{N \times (d + 3)}$  by permuting its rows. This holds because all rows are i.i.d., as  $p_{\theta}(\boldsymbol{z}_T)$  has a diagonal covariance matrix.
+
+Induction step: Assume  $p_{\theta}(\boldsymbol{z}_t)$  is permutation-invariant. We show that if  $p_{\theta}(\boldsymbol{z}_{t - 1}|\boldsymbol{z}_t)$  is permutation-equivariant, i.e.  $p_{\theta}(\boldsymbol{z}_{t - 1}|\boldsymbol{z}_t) = p_{\theta}(P\boldsymbol{z}_{t - 1}|P\boldsymbol{z}_t)$  (which holds when using a permutation-equivariant architecture  $\epsilon_{\theta}$ ), then  $p_{\theta}(\boldsymbol{z}_{t - 1})$  will be permutation-invariant.
+
+$$
+\begin{array}{l} p _ {\theta} \left(P \boldsymbol {z} _ {t - 1}\right) = \int_ {\boldsymbol {z} _ {t}} p _ {\theta} \left(P \boldsymbol {z} _ {t - 1} | \boldsymbol {z} _ {t}\right) p _ {\theta} \left(\boldsymbol {z} _ {t}\right) \quad (\text {C h a i n r u l e o f p r o b a b i l i t y}) \\ = \int_ {\boldsymbol {z} _ {t}} p _ {\theta} (P \boldsymbol {z} _ {t - 1} | P P ^ {- 1} \boldsymbol {z} _ {t}) p _ {\theta} (P P ^ {- 1} \boldsymbol {z} _ {t}) \quad (P P ^ {- 1} = \boldsymbol {I}) \\ = \int_ {\mathbf {z} _ {t}} p _ {\theta} \left(\mathbf {z} _ {t - 1} \mid P ^ {- 1} \mathbf {z} _ {t}\right) p _ {\theta} \left(P ^ {- 1} \mathbf {z} _ {t}\right) \quad \text {(E q u i v a r i a n c e a n d i n v a r i a n c e)} \\ = \int_ {\boldsymbol {u}} p _ {\theta} (\boldsymbol {z} _ {t - 1} | \boldsymbol {u}) p _ {\theta} (\boldsymbol {u}) \underbrace {| \det  P |} _ {= 1} \\ \end{array}
+$$
+
+(Change of variables  $\pmb{u} = P^{-1}\pmb{z}_t$  and  $P$  is orthogonal, so det  $P = \pm 1$ )
+
+$$
+= p _ {\theta} \left(z _ {t - 1}\right)
+$$
+
+By induction,  $p_{\theta}(\boldsymbol{z}_T), \ldots, p_{\theta}(\boldsymbol{z}_0)$  are all permutation-invariant.
+
+Finally, since the decoder  $p_{\xi}(\mathcal{G}|z_0)$  is permutation-equivariant (as it applies the same operation on each node and each pair of nodes, thus not depending on the nodes ordering), with the same derivation, we also conclude that the final induced distribution  $p_{\theta,\xi}(\mathcal{G})$  is permutation-invariant.
+
+Note: A similar derivation can also be applied to show the invariance of  $p_{\theta}(z_0)$  to rotations and reflections of the coordinates, as the only requirement on  $P$  for the derivation to hold is to be orthogonal, which is the case for rotation and reflection matrices  $R$ .
+
+# D  $E(n)$  EQUIVARIANT GRAPH NEURAL NETWORKS (EGNNs)
+
+To perform equivariant updates in our latent space, we use the  $E(n)$  Equivariant Graph Neural Network (EGNN) architecture (Satorras et al., 2021).
+
+An EGNN operates on a point cloud of size  $N$ , where each point has features  $h_i \in \mathbb{R}^d$  and coordinates  $x_i \in \mathbb{R}^3$  associated with it. Let  $\pmb{h} \in \mathbb{R}^{N \times d}$  and  $\pmb{x} \in \mathbb{R}^{N \times 3}$  be the stacked features and coordinates, respectively, of all points. An EGNN is composed of Equivariant Graph Convolutional Layers (EGCLs)  $\pmb{h}^{l+1}, \pmb{x}^{l+1} = \mathrm{EGCL}(\pmb{h}^l, \pmb{x}^l)$ . A single EGCL layer is defined as
+
+$$
+m _ {i j} = \phi_ {e} \left(h _ {i} ^ {l}, h _ {j} ^ {l}, d _ {i j} ^ {2}, a _ {i j}\right) \tag {12}
+$$
+
+$$
+h _ {i} ^ {l + 1} = \phi_ {h} \left(h _ {i} ^ {l}, \sum_ {j \neq i} \tilde {e} _ {i j} m _ {i j}\right) \tag {13}
+$$
+
+$$
+x _ {i} ^ {l + 1} = x _ {i} ^ {l} + \sum_ {j \neq i} \frac {x _ {i} ^ {l} - x _ {j} ^ {l}}{d _ {i j} + 1} \phi_ {x} \left(h _ {i} ^ {l}, h _ {j} ^ {l}, d _ {i j} ^ {2}, a _ {i j}\right), \tag {14}
+$$
+
+where  $l$  denotes the layer index,  $d_{ij} = \| x_i^l -x_j^l\| _2$  is the Euclidean distance between points  $i$  and  $j$ , and  $a_{ij}$  are optional edge attributes which we set to  $\| x_i^0 -x_j^0\| _2^2$ .  $\tilde{e}_{ij}$  serves as an attention mechanism that infers a soft estimation of the edges  $\tilde{e}_{ij} = \phi_{inf}(m_{ij})$ . Note that to update each point's features and coordinates, the EGCL layers consider all the other nodes, effectively treating the point cloud as a fully connected graph. All learnable components  $(\phi_e,\phi_h,\phi_x,$  and  $\phi_{inf})$  are fully connected neural networks. An EGNN architecture is then composed of  $L$  EGCL layers, denoted as  $\pmb {h}^L,\pmb {x}^L = \mathrm{EGNN}(\pmb {h}^0,\pmb {x}^0)$  with  $\pmb {h}^0 = \pmb{h}$  and  $\pmb{x}^0 = \pmb{x}$ . The main hyperparameters of the EGNN architectures are the number of layers  $L$  and a feature dimension  $n_f$  used to control the width of the fully connected neural networks.
+
+# E ADDITIONAL METHOD DETAILS
+
+In this section, we provide additional details for our method introduced in Sections 3, 4, and 5.
+
+# E.1 TRAINING AND SAMPLING
+
+We start by describing in more detail the training and sampling algorithms for EDM-SYCo in Algorithms 1 and 2.
+
+Algorithm 1 Training Algorithm  
+Input: a dataset of molecular graphs  $\mathcal{G} = (h,A)$    
+Initial: Encoder network  $\mathcal{E}_{\phi}$  , Decoder network  $\mathcal{D}_{\xi}$  , denoising network  $\epsilon_{\theta}$    
+First Stage: Autoencoder Training   
+repeat   
+ $\mu_z\gets \mathcal{E}_\phi (h,A)$  {Encoder}   
+ $\epsilon_z\sim \mathcal{N}(0,I)$  and subtract center of mass from  $\epsilon_z^{(x)}$ $z\leftarrow \mu_z + \sigma_0\epsilon_z$ $\hat{\boldsymbol{h}},\hat{\boldsymbol{A}}\gets \mathcal{D}_{\xi}(\boldsymbol {z})$ $\mathcal{L}_{VAE}(\phi ,\xi)\gets$  cross-entropy  $\begin{array}{r}\left(\left[\hat{\mathrm{h}}\right],\left[\mathrm{h}\right]\right) \end{array}$    
+Take an optimizer step on  $\mathcal{L}_{AE}(\phi ,\xi)$    
+until  $\phi$  and  $\xi$  have converged   
+Second Stage: Diffusion Model Training   
+Fix Autoencoder parameters  $\phi$  and  $\xi$    
+repeat   
+ $z_0\sim q_\phi (z|h,A)$ $\epsilon_z\sim \mathcal{N}(\mathbf{0},\mathbf{I})$  and subtract center of mass from  $\epsilon_z^{(x)}$ $t\sim \mathcal{U}(0,1,\dots,T)$ $z_{t}\gets \sqrt{\bar{\alpha}_{t}} z_{0} + \sqrt{1 - \bar{\alpha}_{t}}\epsilon_{z}$ $\mathcal{L}_{DM}(\theta)\gets \| \epsilon_z - \epsilon_\theta (z_t,t)\|^2$    
+Take an optimizer step on  $\mathcal{L}_{DM}(\theta)$    
+until  $\theta$  has converged   
+return  $\mathcal{E}_{\phi},\mathcal{D}_{\xi},\epsilon_{\theta}$
+
+Algorithm 2 Sampling Algorithm  
+Input: Decoder network  $\mathcal{D}_{\xi}$ , denoising network  $\epsilon_{\theta}$ $\boldsymbol{z}_T \sim \mathcal{N}(\boldsymbol{0}, \boldsymbol{I})$  and subtract center of mass from  $\boldsymbol{z}_T^{(x)}$   
+for  $t = T$  downto 1 do  
+ $\epsilon_{z} \sim \mathcal{N}(\boldsymbol{0}, \boldsymbol{I})$  and subtract center of mass from  $\epsilon_{z}^{(x)}$ $\boldsymbol{z}_{t-1} \gets \frac{1}{\sqrt{\alpha_t}} \left( \boldsymbol{z}_t - \frac{1 - \alpha_t}{\sqrt{1 - \alpha_t}} \epsilon_{\theta}(\boldsymbol{z}_t, t) \right) + \sigma_t \epsilon_z$   
+end for  
+ $h, A \gets \mathcal{D}_{\xi}(z_0)$   
+return 2D molecule  $\mathcal{G} = (h, A)$
+
+# E.2 CONDITIONAL GENERATION
+
+Here, we detail the regressor guidance algorithm we developed in Section 5. We showed in the main text that we can sample from the conditional distribution  $q(\boldsymbol{z}_t|c)$  by replacing the predicted noise of the denoising neural network by Equation 7. By putting this back into the sampling equation, Equation 6, we get the following modified denoising step
+
+$$
+\boldsymbol {z} _ {t - 1} = \boldsymbol {\mu} _ {\theta} (\boldsymbol {z} _ {t}, t) - \frac {1 - \alpha_ {t}}{\sqrt {\alpha_ {t}}} s \nabla_ {\boldsymbol {z} _ {t}} \left(g _ {\eta} (\boldsymbol {z} _ {t}) - c\right) ^ {2} + \sigma_ {t} \boldsymbol {\epsilon} _ {t}. \tag {15}
+$$
+
+A more general formulation for the new guidance term is given by  $-s(t)\nabla_{\boldsymbol{z}_t}l(\boldsymbol{z}_t,c)$ , where  $s(t)$  is a time-dependent scaling function and  $l$  can be any invariant loss function. In the specific case that we derived,  $s(t) = \frac{1 - \alpha_t}{\sqrt{\alpha_t}} s$  and  $l(\boldsymbol{z}_t,c) = (g_\eta (\boldsymbol{z}_t) - c)^2$ .  $l$  can be replaced by any other loss function that we want to optimize, or in the case of property maximization, it can be the prediction of the regressor
+
+directly (Equation 8). In our experiments, we found a simple linear schedule  $s(t) = at$  to perform well, where  $a \in \mathbb{R}$  is a hyperparameter that we tune for each task.
+
+To ensure the equivariance of the distribution even with guidance, the new term  $s(t)\nabla_{\boldsymbol{z}_t}l(\boldsymbol{z}_t,c)$  needs to be  $E(3)$ -equivariant. This is done by subtracting its center of mass at each sampling step and by choosing an invariant loss function  $l$ , as the differential operator of an invariant function yields an equivariant function (Chmiela et al., 2017; Schutt et al., 2017).
+
+# E.3 SCAFFOLD-CONSTRAINED GENERATION
+
+Here, we provide additional details for the inpainting algorithm described in Section 5, where we described an approach that allows to generate a point cloud that contains a chosen scaffold.
+
+Lugmayr et al. (2022) observed that the version of their algorithm described in the main text leads to inconsistent completions and proposes to run each sampling step multiple times to allow the model to harmonize its generations. Concretely, each denoising step is repeated  $r$  times to improve the generations' quality and allow the model to harmonize the unknown part to the scaffold. This is achieved by applying the forward diffusion model (Equation 2) after each denoising step. Lugmayr et al. (2022) also propose to go more than one forward step with the diffusion model to allow the model to better harmonize its generations. This choice is fixed by the jump length  $j$ . Note that this procedure increases the computation cost of running the model  $r$  times, which might be a limiting factor. For this reason, we combine this approach with the approach described in Appendix G.9 that allows to reduce the number of reverse diffusion steps to run.
+
+The procedure described in the main text gives a way to condition the diffusion model, which operates on the latent point cloud space. However, we are ultimately interested in generating a molecular graph containing a scaffold defined by a subset of atoms and the bonds between them. We map the molecular scaffold to its point cloud representation using our encoder to get  $m \odot z_0$ . This enables running the inpainting algorithm described above. Then, we decode the sampled 3D point cloud to a molecular graph. Since our decoder model operates on each node and edge separately, the final molecular graph is guaranteed to contain the original scaffold as long as the scaffold in isolation can be correctly reconstructed with our autoencoder.
+
+# F IMPLEMENTATION DETAILS
+
+# F.1 LICENSES FOR USED CODE, MODELS, AND DATASETS
+
+We start by listing the assets we used in our work and their respective licenses. We use the following tools, codebases, and datasets:
+
+- RDKit (Landrum et al., 2006) (BSD 3-Clause License)  
+- PyTorch (Paszke et al., 2019) (BSD 3-Clause License)  
+- EDM (Hoogeboom et al., 2022) (MIT license)  
+- GeoLDM (Xu et al., 2023) (MIT license)  
+ZINC250K dataset (Irwin et al., 2012) (Custom license)  
+- GuacaMol dataset (Brown et al., 2019) (MIT license)
+
+# F.2 ARCHITECTURE DETAILS
+
+We train two sets of models with the same architectures on the two used datasets, ZINC250K and GuacaMol. The EDM, together with the autoencoder, has 9.2M total parameters, while the regressor has 4.2M parameters.
+
+Encoder. The first part of the encoder  $\mathcal{E}$ , introduced in Section 3, is the conformer generation method, while the second part is an EGNN with 1 layer and 128 hidden features (See Appendix D). We found that having at least 1 EGNN layer is crucial to achieve high reconstruction accuracy for the autoencoder, and as this allows us to reach more than  $99\%$  accuracy, we did not use more layers. This EGNN takes as input a point cloud with the one-hot encodings of the atom types and formal charges
+
+as features  $h$  and the synthetic coordinates computed by the conformer generation part as coordinates  $x$ . The output is a processed point cloud with continuous node embeddings of dimension  $d = 2$  and updated coordinates.
+
+Decoder. The decoder  $\mathcal{D}$ , also introduced in Section 3, consists of 2 fully connected networks  $\mathrm{MLP}_{node}$  and  $\mathrm{MLP}_{edge}$ , both having 256 hidden dimensions.
+
+EDM. The denoising network of EDM is parametrized with an EGNN with 9 layers and 256 hidden features. The EDM has  $T = 1000$  diffusion steps.
+
+Regressor. The regressor used for the conditional generation and optimization experiments (see Section 6.2) is also an EGNN with 4 layers and 256 hidden features, followed by a sum pooling operation over the node features and then a fully connected network with 2 layers and a hidden dimension of 256 that maps the pooled graph embedding to the target values of the properties.
+
+# F.3 TRAINING DETAILS
+
+We use the original train/validation/test splits of the used datasets (Irwin et al., 2012; Brown et al., 2019). The autoencoder is trained in the first stage to minimize the cross-entropy loss between the ground truth and predicted graphs for a maximum of 100 epochs, with early stopping if the validation accuracy does not improve for 10 epochs. To deal with the class imbalance problem caused by the sparsity of the graph adjacency matrices and the dominance of some atom types and formal charge values, we scale each term of the cross-entropy loss with a class-specific value based on the statistics of the training set (King & Zeng, 2001). In the second training stage, the EDM model is trained for 1000 epochs on ZINC250 and approximately 300 epochs on GuacaMol.
+
+The regressor is trained with the  $L_{1}$  loss to predict the molecular properties on the noisy latent point cloud representations of the molecules by applying the same noise schedule of the EDM. The regressor is trained for 500 epochs, with early stopping if the MAE on the validation set does not improve after 50 epochs.
+
+All models are trained with a batch size of 64 and using the recent Prodigy optimizer (Mishchenko & Defazio, 2023) with  $d_{\text{coef}} = 0.1$ , which we found to be a very important hyperparameter for the stability of training.
+
+We train all models on ZINC250K on a single Nvidia A100 GPU, and on GuacaMol, we use multi-GPU training on 4 Nvidia A100 GPUs.
+
+# F.4 DATASET DETAILS
+
+We use two datasets in our experiments: ZINC250K (Irwin et al., 2012) and GuacaMol (Brown et al., 2019). For both datasets, we represent a molecule as a graph. The nodes are atoms with the one-hot encoding of their atom type and their formal charge as features. In addition to modeling all heavy atoms, we also model explicit H atoms, which we found to increase the validity of the generated molecules as it reduces some kekulization errors produced by rdkit  $^{2}$ . The edges are chemical bonds with the one-hot encoding of the bond type as features. We model the following bond types: no-bond, single, double, triple, and aromatic. We model the absence of a bond as a separate bond type to allow the autoencoder to reconstruct the full bond information. These graphs present very unbalanced scales for the atom types, formal charges, and bond types. We use class-specific weights to train the autoencoder, as outlined in Section 3.
+
+ZINC250K has a total of 250K molecules and 220,011 training molecules. It has 10 atom types (including H) and 3 formal charge values  $(-1, 0, 1)$ , and the largest molecule has 40 atoms.
+
+GuacaMol has a total of  $\approx 1.5\mathrm{M}$  molecules, of which we use 1,273,104 for training. It has 13 atom types (including H) and 5 formal charge values (-1, 0, 1, 2, 3), and the largest molecule has 88 atoms.
+
+We compute the synthetic coordinates needed for our model in a preprocessing step for both datasets and use the computed coordinates throughout the training.
+
+# F.5 SAMPLING THE NUMBER OF ATOMS
+
+In this work, the diffusion generative model operates on a fixed-size point cloud  $N$ . During sampling, the first step is to sample this number  $N$ . To sample molecules of similar sizes to those in the training set, we approximate the distribution of the number of nodes using a categorical distribution.
+
+To this end, we count the number of occurrences of each molecule size in the training set and use the categorical distribution parameterized by the normalized counts to sample the number of atoms of generated molecules. This simple procedure works well for the task of unconditional generation. However, in the property-conditioned generation, we must consider that some properties are highly correlated to molecule size, such as molecular weight (MW). Therefore, for this task, we start by dividing the range of property values from the training set into 100 equal-sized bins. We define a different categorical distribution for each bin that counts the sizes of molecules falling in that bin. Given the conditioning value, we first specify the bin it falls into and use the corresponding categorical distribution to sample the number of nodes. Finally, for molecule optimization, we curate a dataset of molecule pairs that exhibit the desired property improvement and similarity constraints and create different conditional categorical distributions, one for each size of the initial molecule.
+
+Note that these different procedures are needed due to the inherent limitation of diffusion models on graphs that require the number of nodes to be known and fixed in advance. Fundamentally, solving this issue would be a valuable contribution to the field, and we leave it for future work.
+
+# F.6 PROPERTY-CONDITIONAL GENERATION EXPERIMENT
+
+Following Ninniri et al. (2023), for each task, we sample 100 molecules from the test set and condition on their property values to generate 10 valid molecules each and report the mean absolute error between the target and the estimated values for the generated molecules measured by RDKit. We experimented with different values of the guidance scale  $s$  and with using the  $l_{1}$  and  $l_{2}$  losses to compute the guidance loss but found the  $l_{2}$  loss with scale values  $s = 2.5$  and  $s = 1.0$  for LogP and MW, respectively, to work best.
+
+# F.7 CONSTRAINED MOLECULE OPTIMIZATION
+
+Here, we provide more details on the constrained optimization tasks described in Section 6. Given a molecule  $\mathcal{G}$ , these tasks aim to find a different molecule  $\mathcal{G}'$  with higher property values satisfying the similarity constraint  $sim(\mathcal{G},\mathcal{G}') \geq 0.4$  (or 0.6), where  $sim$  computes the Tanimoto similarity with Morgan fingerprints (Rogers & Hahn, 2010). For the LogP task, the new molecule  $\mathcal{G}'$  needs to have a higher LogP value and we report the average improvement over 800 molecules from the test set with the lowest LogP values. For the QED task, the goal is to optimize another 800 molecules from the test with QED values in the low range [0.7, 0.8] into the high range [0.9, 1.0], and we report the success rate over these molecules. For DRD2, the goal is to translate inactive compounds ( $p \leq 0.05$ ) into active ones ( $p \geq 0.5$ ), and we similarly report the success rate over these molecules. In addition, for all tasks, we report the diversity as the average pairwise distance between the successfully optimized molecules for each initial test molecule. The distance is defined as  $dist(\mathcal{G},\mathcal{G}') = 1 - sim(\mathcal{G},\mathcal{G}')$ .
+
+Choosing  $t$ . As described in Section 5, the intermediate time step  $t$  controls the trade-off between the similarity to the initial molecule and the capacity to improve its target properties. In principle, for each test molecule, there is an optimal  $t$  yielding the highest improvement under the similarity constraints. However, finding the optimal  $t$  for each single molecule is computationally expensive. Therefore, we use a subset of 20 molecules and run our optimization algorithm for different values of  $t$  on these molecules to find the value of  $t$  that yields the highest average improvement under the considered similarity constraints. We use this value for all test molecules. Our experiments found that a value of  $t$  in the range [500, 700] usually works best with the total  $T = 1000$ . Figure 8 shows the effect of the time step  $t$  on the similarity and property value for one molecule from the test set. Table 7 lists the exact value of  $t$  used for each task.
+
+![](images/6857ef0ea76dc489b195927f42200df0a9b25addf865a768f914151e6f7404cb.jpg)
+
+![](images/3889fdfc917a6fd5d900f30906c2305738ca634be4fed5d3e0c1537a32d36e65.jpg)
+
+![](images/fd6d6f224df48c162b062807cc82118ae1e5f12ef6a451e5bed3da256e6caf92.jpg)  
+Figure 8: Effect of using different time steps  $t$  on the similarity and improvement values in the similarity-constrained optimization task of PLogP. We visualize 4 different molecules from the test set.
+
+![](images/b6bb63e0a0ccc622fba91ebcb6e92cafeb94381e22b5bde625c35c7d89e4e16a.jpg)
+
+Table 7: Values of the time step  $t$  used in all optimization tasks.  
+
+<table><tr><td></td><td>TIME STEP t</td></tr><tr><td>PLOGP (SIM ≥ 0.4)</td><td>600</td></tr><tr><td>PLOGP (SIM ≥ 0.6)</td><td>500</td></tr><tr><td>QED</td><td>650</td></tr><tr><td>DRD2</td><td>650</td></tr></table>
+
+# F.8 EVALUATION DETAILS
+
+We provide additional details about the computation of the FCD and KL metrics (Brown et al., 2019) and additional insights into the novelty and uniqueness metrics. Additionally, we provide a detailed analysis of the validity rate of our model.
+
+- KL: The following descriptors are computed for the generated and reference molecules: BertzCT, MolLogP, MolWt, TPSA, NumHAcceptors, NumHDonors, NumRotatableBonds, NumAliphaticRings, NumAromaticRings, and similarity to nearest neighbor with ECFP4 fingerprints. The KL divergence  $D_{KL,i}$  is computed for each descriptor between the two sets of molecules and aggregated to a final normalized score via  $\frac{1}{k} \sum_{i=1}^{k} \exp(-D_{KL,i})$ .  
+- FCD: The FCD is computed based on the hidden representations of molecules in ChemNet, trained for predicting biological activities, similarly to the FID usually applied to image generative models. Concretely, the means and covariances of the last hidden activations of ChemNet are computed for the reference and generated molecules and the Frechet distance between them is computed. This distance is then normalized via  $\exp(-0.2 \cdot FCD)$  to lie between 0 and 1.
+
+- Novelty & Uniqueness: Interestingly, most of the methods used in this work achieve a novelty and uniqueness scores close to  $100\%$  on the two datasets ZINC250K and GuacaMol. These high scores are mainly due to the larger molecular sizes in our datasets compared to smaller datasets such as QM9. The number of possible molecules grows exponentially with the molecule size, making it increasingly unlikely to sample exact replica of the training molecules or sample the exact molecule twice (note than even if a single atom or bond type is different, the molecules are considered different).  
+- Validity: We analyze the invalid molecules generated by our model and find the following common errors raised by RDKit: Kekulization errors  $(55\%)$ $^{3}$ , Valency errors  $(33\%)$ $^{4}$ , A non-ring atom is marked aromatic  $(12\%)$ .
+
+All these errors are due to inconsistent bond type predictions for a given molecule. Because the autoencoder achieves a near-perfect reconstruction accuracy on the training and test sets, this inconsistency is likely due to a mismatch between the training distribution and the generated 3D molecules distribution. This could be improved by using more powerful 3D generative models, or one could try to jointly train the autoencoder and the latent diffusion model to adapt the decoder to the distribution of generated molecules.
+
+# G ADDITIONAL EXPERIMENTS
+
+In this section, we provide additional experiments to showcase the performance of our model and analyze some of its properties.
+
+# G.1 BOND TYPE PREDICTION FROM SYNTHETIC COORDINATES
+
+A crucial part of successfully applying our method is to be able to accurately reconstruct the molecular graph from the Euclidean point cloud. This section provides an ablation study to compare our approach to other methods.
+
+Formally, we are given a point cloud with features  $\pmb{h} \in \{0,1\}^{N \times (a + c)}$  containing the atom type and formal charge of each atom and coordinates  $\pmb{x} \in \mathbb{R}^{N \times 3}$  computed using the conformer generation algorithm ETKDG (Riniker & Landrum, 2015). The goal is to reconstruct the bond types stored in the adjacency matrix  $A \in \{0,1\}^{N \times N \times b}$ . We test the following approaches:
+
+- **Rule-based:** This approach is widely used by generative models for molecules in 3D (Hoogeboom et al., 2022; Xu et al., 2023) and consists of using typical bond type lengths to infer the bond types between two atoms based on their features and the inter-atomic distances between them. It uses a set of if-else statements and predicts a specific bond type if the length falls within a specific range that takes the features of the two atoms into account.  
+- MLP: We develop this baseline to generalize the lookup table approach and replace the set of if-else statements by an MLP. Formally, we construct edge features between atoms  $i$  and  $j$  as  $[h_i, h_j, d_{ij}]$ , where  $d_{ij}$  is the Euclidean distance between the two atoms. This is also the same information that the lookup table method above uses to infer the bond types. Instead, we train an MLP to predict the bond type from this edge feature in a classification setting. Note that this MLP is defined on the edge level, so a molecule with  $N$  atoms provides  $N^2$  training examples for the MLP to learn from. Also note that we do not need to have an extra MLP for the atom type as they are directly available in this case since we do not run an EGNN in the first step.  
+- EGNN + MLP: The major limitation of the above approaches is that to predict the bond type between two atoms they only consider the distance between them and their respective features, without considering other atoms. However, not all molecules exhibit typical bond lengths between their atoms (Hoogeboom et al., 2022). To overcome this limitation, we combine the MLP approach with an EGNN that first computes updated coordinates and
+
+atom features considering the whole graph before running the local MLP to predict the bond type. Here, we add an extra MLP to predict the node labels (type and formal charge). This is the same architecture presented in the main text.
+
+We evaluate these three approaches on ZINC250K using the synthetic coordinates computed as outlined in Section 3. We report the molecule reconstruction accuracy on the validation set, where a molecule is considered correctly reconstructed if all of its atom features (atom types and formal charges) and bond types are correctly reconstructed. Table 1 shows that the EGNN approach achieves an almost perfect reconstruction accuracy. This shows that (i) the synthetic coordinates contain all the needed information to reconstruct the bond types, which might not be directly clear if we are limited to the other methods, and (ii) incorporating the global graph information into the prediction task is crucial to accurately reconstruct the bonds. We adopt the EGNN approach into our model.
+
+# G.2 CONFORMER GENERATION COMPARISON
+
+To analyze the effect of different conformer generation algorithms on the performance of our model, we perform an ablation study using 2 different algorithms/packages from RDKit to compute the synthetic coordinates: the ETKDG algorithm and the Pharm3D package. We perform two training runs using the same setup except for the conformer generation method. For these two runs, we use smaller models (we halve the number of layers) than those reported in the main text and train for 500 epochs, so they are not directly comparable. Results are shown in Table 8.
+
+Table 8: Model performance on ZINC using different conformer generation methods from RDKit.  
+
+<table><tr><td></td><td>ETKDG</td><td>PHARM3D</td></tr><tr><td>VAE RECONSTRUCTION ACCURACY</td><td>99.92%</td><td>99.90%</td></tr><tr><td>KL</td><td>0.94</td><td>0.93</td></tr><tr><td>FCD</td><td>0.75</td><td>0.70</td></tr></table>
+
+# G.3 COMPLETE BENCHMARK RESULTS
+
+We provide complete benchmark results for the GuacaMol benchmark (Brown et al., 2019) and the MOSES benchmark (Polykovskiy et al., 2020) in Tables 9 and 10, respectively. In addition to the baselines reported in the main text, we add two SMILES-based baselines and we add different flavours of MoLeR with different vocabulary sizes, denoted as MoLeR-  $V$ , where  $V$  denotes the vocabulary size.
+
+Table 9: GuacaMol benchmark metrics on the ZINC250K dataset. We report mean and standard deviations across 3 random training runs for all methods and metrics.  
+
+<table><tr><td>METRIC</td><td>FCD (↑)</td><td>KL (↑)</td><td>NOVELTY (↑)</td><td>UNIQUENESS (↑)</td><td>VALIDITY (↑)</td></tr><tr><td>CHARVAE</td><td>0.17 ± 0.08</td><td>0.78 ± 0.04</td><td>0.99 ± 0.00</td><td>0.99 ± 0.00</td><td>0.09 ± 0.01</td></tr><tr><td>SMILES-LSTM</td><td>0.93 ± 0.00</td><td>1.00 ± 0.00</td><td>0.98 ± 0.00</td><td>1.00 ± 0.00</td><td>0.96 ± 0.01</td></tr><tr><td>GRAPHAF</td><td>0.05 ± 0.00</td><td>0.67 ± 0.01</td><td>0.91 ± 0.01</td><td>0.91 ± 0.01</td><td>1.00 ± 0.00</td></tr><tr><td>HIERVVAE</td><td>0.50 ± 0.14</td><td>0.92 ± 0.00</td><td>0.96 ± 0.01</td><td>0.96 ± 0.01</td><td>1.00 ± 0.00</td></tr><tr><td>MICAM</td><td>0.63 ± 0.02</td><td>0.94 ± 0.00</td><td>0.98 ± 0.00</td><td>0.98 ± 0.00</td><td>1.00 ± 0.00</td></tr><tr><td>JTVAE</td><td>0.75 ± 0.01</td><td>0.93 ± 0.00</td><td>1.00 ± 0.00</td><td>1.00 ± 0.00</td><td>1.00 ± 0.00</td></tr><tr><td>PSVAE</td><td>0.28 ± 0.01</td><td>0.84 ± 0.00</td><td>1.00 ± 0.00</td><td>1.00 ± 0.00</td><td>1.00 ± 0.00</td></tr><tr><td>MAGNET</td><td>0.76 ± 0.00</td><td>0.95 ± 0.00</td><td>0.99 ± 0.00</td><td>0.99 ± 0.00</td><td>1.00 ± 0.00</td></tr><tr><td>MOLER-5</td><td>0.40 ± 0.01</td><td>0.93 ± 0.01</td><td>0.99 ± 0.00</td><td>0.97 ± 0.00</td><td>1.00 ± 0.00</td></tr><tr><td>MOLER-350</td><td>0.80 ± 0.01</td><td>0.98 ± 0.00</td><td>1.00 ± 0.00</td><td>1.00 ± 0.00</td><td>1.00 ± 0.00</td></tr><tr><td>MOLER-2000</td><td>0.83 ± 0.00</td><td>0.97 ± 0.00</td><td>0.99 ± 0.00</td><td>0.99 ± 0.00</td><td>1.00 ± 0.00</td></tr><tr><td>DIGRESS</td><td>0.65 ± 0.00</td><td>0.91 ± 0.00</td><td>0.99 ± 0.00</td><td>0.99 ± 0.00</td><td>0.85 ± 0.01</td></tr><tr><td>EDM-SYCo</td><td>0.85 ± 0.01</td><td>0.96 ± 0.00</td><td>1.00 ± 0.00</td><td>1.00 ± 0.00</td><td>0.88 ± 0.01</td></tr></table>
+
+Table 10: MOSES benchmark metrics on the ZINC250K dataset. We report mean and standard deviations across 3 random training runs for all methods and metrics.  
+
+<table><tr><td>METRIC</td><td>FILTERS (↓)</td><td>INTDIV (↑)</td><td>INTDIV2 (↑)</td><td>QED (↓)</td><td>SA (↓)</td><td>LOGP (↓)</td><td>WEIGHT (↓)</td></tr><tr><td>CHARVAE</td><td>0.43 ± 0.02</td><td>0.88 ± 0.01</td><td>0.87 ± 0.01</td><td>0.06 ± 0.03</td><td>0.48 ± 0.13</td><td>0.87 ± 0.14</td><td>34.23 ± 07.05</td></tr><tr><td>SMILES-LSTM</td><td>0.59 ± 0.01</td><td>0.87 ± 0.00</td><td>0.86 ± 0.00</td><td>0.00 ± 0.00</td><td>0.04 ± 0.02</td><td>0.12 ± 0.01</td><td>02.62 ± 00.16</td></tr><tr><td>GRAPHAF</td><td>0.47 ± 0.03</td><td>0.93 ± 0.00</td><td>0.91 ± 0.00</td><td>0.22 ± 0.01</td><td>0.88 ± 0.10</td><td>0.41 ± 0.02</td><td>96.93 ± 04.99</td></tr><tr><td>HIERVAE</td><td>0.58 ± 0.05</td><td>0.87 ± 0.01</td><td>0.86 ± 0.01</td><td>0.03 ± 0.00</td><td>0.19 ± 0.17</td><td>0.35 ± 0.21</td><td>18.97 ± 10.43</td></tr><tr><td>MICAM</td><td>0.54 ± 0.02</td><td>0.87 ± 0.00</td><td>0.87 ± 0.00</td><td>0.08 ± 0.00</td><td>0.51 ± 0.03</td><td>0.20 ± 0.05</td><td>51.19 ± 06.58</td></tr><tr><td>JTVAE</td><td>0.57 ± 0.00</td><td>0.86 ± 0.00</td><td>0.86 ± 0.00</td><td>0.01 ± 0.00</td><td>0.35 ± 0.01</td><td>0.29 ± 0.01</td><td>02.93 ± 00.07</td></tr><tr><td>PSVAE</td><td>0.85 ± 0.01</td><td>0.89 ± 0.00</td><td>0.88 ± 0.00</td><td>0.05 ± 0.00</td><td>1.16 ± 0.06</td><td>0.34 ± 0.01</td><td>38.26 ± 02.74</td></tr><tr><td>MAGNET</td><td>0.70 ± 0.00</td><td>0.87 ± 0.00</td><td>0.87 ± 0.00</td><td>0.01 ± 0.00</td><td>0.13 ± 0.00</td><td>0.23 ± 0.01</td><td>14.52 ± 00.40</td></tr><tr><td>MOLER-5</td><td>0.57 ± 0.01</td><td>0.86 ± 0.00</td><td>0.85 ± 0.00</td><td>0.01 ± 0.00</td><td>0.11 ± 0.04</td><td>0.17 ± 0.10</td><td>10.67 ± 04.13</td></tr><tr><td>MOLER-350</td><td>0.56 ± 0.00</td><td>0.87 ± 0.00</td><td>0.86 ± 0.00</td><td>0.01 ± 0.01</td><td>0.06 ± 0.01</td><td>0.13 ± 0.02</td><td>05.96 ± 01.16</td></tr><tr><td>MOLER-2000</td><td>0.55 ± 0.01</td><td>0.86 ± 0.00</td><td>0.86 ± 0.00</td><td>0.02 ± 0.00</td><td>0.08 ± 0.02</td><td>0.11 ± 0.00</td><td>07.62 ± 01.80</td></tr><tr><td>DIGRESS</td><td>0.77 ± 0.01</td><td>0.86 ± 0.00</td><td>0.86 ± 0.00</td><td>0.05 ± 0.00</td><td>0.09 ± 0.01</td><td>0.61 ± 0.02</td><td>32.51 ± 00.20</td></tr><tr><td>EDM-SYCo</td><td>0.59 ± 0.00</td><td>0.87 ± 0.00</td><td>0.87 ± 0.00</td><td>0.003 ± 0.00</td><td>0.19 ± 0.01</td><td>0.06 ± 0.02</td><td>04.49 ± 00.22</td></tr></table>
+
+# G.4 CONDITIONAL VS UNCONDITIONAL GENERATION
+
+In addition to the conditional generation experiment from the main text, we perform an extra experiment to evaluate our approach and compare it to the unconditional model, which does not consider the conditioning value. We consider four properties: (1) the penalized LogP score (PLogP), a compound's solubility and synthetic accessibility; (2) QED, a compound's drug-likeness; (3) DRD2, an estimate of the biological activity against dopamine type 2 receptors; (4) TPSA, a drug's ability to permeate cells.
+
+We generate 10,000 valid molecules for each property and report the mean absolute error between the targets and the estimated values for the generated molecules measured by RDKit. The results are shown in Table 11. Our guidance mechanism achieves an average improvement factor of more than 5.5 times compared to the unconditional model across all properties.
+
+Table 11: Mean absolute error between the targets and the property values of 10,000 generated molecules using regressor guidance, and the improvement factor compared to unconditional generation. We report standard deviation across 3 runs with different random seeds.  
+
+<table><tr><td></td><td>PLOGP</td><td>QED</td><td>DRD2</td><td>TPSA</td></tr><tr><td>UNCONDITIONAL</td><td>2.2440 ± 0.0018</td><td>0.1530 ± 0.0002</td><td>0.0148 ± 0.0000</td><td>25.9017 ± 0.0206</td></tr><tr><td>GUIDANCE</td><td>0.3557 ± 0.0042</td><td>0.0350 ± 0.0004</td><td>0.0051 ± 0.0001</td><td>03.0382 ± 0.0164</td></tr><tr><td>IMP. FACTOR (↑)</td><td>6.3087</td><td>4.3714</td><td>2.9020</td><td>08.5253</td></tr></table>
+
+# G.5 CONDITIONAL GENERATION WITH CONDITIONAL DIFFUSION MODELS AND REGRESSOR-FREE GUIDANCE
+
+In this section, we benchmark additional approaches for conditional generation presented in Section 6.2. In the main text, we reported results with the regressor guidance algorithm on top of an unconditional diffusion model inspired by the classifier guidance algorithm (Dhariwal & Nichol, 2021). Here, we additionally test a conditional diffusion model, which is a simple extension to the unconditional model, obtained by adding the target property to the node features and training a new model (Hoogeboom et al., 2022), referred to as Conditional Diffusion in Table 12. Inspired by the classifier-free guidance algorithm (Ho & Salimans, 2022), we train another variant of the conditional model by dropping the target property  $15\%$  of the time during training. This allows us to sample from the same model in a conditional or unconditional way, and by combining the two modes, we can guide the generation process towards the target properties, similar to the regressor-guidance case. We refer to this case as Regressor-Free Guidance in Table 12. We additionally test guiding the conditional model with an external regressor model (Regressor Guidance Conditional). All results for these models are shown in Table 12. Note that we experimented with smaller models than the one reported in the previous experiment due to the computational costs associated with training and running these models. Therefore, these numbers are not directly comparable to the numbers from the previous
+
+experiment. However, note that the best model presented here, the Regressor Guidance Conditional, has an MAE of 0.36, which is very close to the number from the previous experiment (0.3581), even though the model here is smaller. We expect that guiding the conditional model performs better if compared properly. However, it has the drawback of training a new diffusion model for each target property, compared to using the same diffusion model and one regressor that predicts all properties of interest.
+
+Table 12: MAE between condition value  $c$  and the property value computed on the generated molecules. The considered property in PLogP.  $\bar{s} = \sqrt{1 - {\bar{\alpha}}_{t}s}$  .  
+
+<table><tr><td>METHOD</td><td>SCORE FUNCTION</td><td>MAE</td></tr><tr><td>UNCONDITIONAL</td><td>-</td><td>2.24</td></tr><tr><td>CONDITIONAL DIFFUSION</td><td>εθ(zt,t,c)</td><td>0.60</td></tr><tr><td>REGRESSOR-FREE GUIDANCE</td><td>(s+1)εθ(zt,t,c) + sεθ(zt,t)</td><td>0.48</td></tr><tr><td>REGRESSOR GUIDANCE CONDITIONAL</td><td>εθ(zt,t,c) + s∇zt(gη(zt,t) - c)2</td><td>0.36</td></tr></table>
+
+# G.6 UNCONSTRAINED MOLECULE OPTIMIZATION
+
+This task aims to generate molecules from scratch with as high property values as possible without any other constraints. We use this task to demonstrate the performance of the proposed guidance mechanism for property maximization, described in Equation 8. Similarly to the conditional generation task, we start from Gaussian noise and run the modified sampling algorithm to generate molecules with high properties. Table 13 shows the top-3 scores of our model's generated molecules and for all baselines from (Kong et al., 2022). Note that we achieved the best scores compared to all these baselines, even though we only generated 1,000 molecules compared to 10,000 molecules generated by these baselines. This illustrates the high performance of our method and motivates its use for the harder task of constrained optimization.
+
+Table 13: Top-3 property values for molecules generated in the unconstrained optimization task. Results for all baselines are taken from (Kong et al., 2022). All baselines generate 10,000 molecules, while we only generate 1,000 for this task.  
+
+<table><tr><td rowspan="2">METHOD</td><td colspan="3">PLOGP</td><td colspan="3">QED</td></tr><tr><td>1ST</td><td>2ND</td><td>3RD</td><td>1ST</td><td>2ND</td><td>3RD</td></tr><tr><td>JT-VAE</td><td>5.30</td><td>4.93</td><td>4.49</td><td>0.925</td><td>0.911</td><td>0.910</td></tr><tr><td>GCPN</td><td>7.98</td><td>7.85</td><td>7.80</td><td>0.948</td><td>0.947</td><td>0.946</td></tr><tr><td>MRNN</td><td>8.63</td><td>6.08</td><td>4.73</td><td>0.844</td><td>0.796</td><td>0.736</td></tr><tr><td>GRAPHAF</td><td>12.23</td><td>11.29</td><td>11.05</td><td>0.948</td><td>0.948</td><td>0.947</td></tr><tr><td>GRAPHDF</td><td>13.70</td><td>13.18</td><td>13.17</td><td>0.948</td><td>0.948</td><td>0.948</td></tr><tr><td>GA</td><td>12.25</td><td>12.22</td><td>12.20</td><td>0.946</td><td>0.944</td><td>0.932</td></tr><tr><td>MARS</td><td>7.24</td><td>6.44</td><td>6.43</td><td>0.944</td><td>0.943</td><td>0.942</td></tr><tr><td>FREED</td><td>6.74</td><td>6.65</td><td>6.42</td><td>0.920</td><td>0.919</td><td>0.908</td></tr><tr><td>H-VAE</td><td>11.41</td><td>9.67</td><td>9.31</td><td>0.947</td><td>0.946</td><td>0.946</td></tr><tr><td>F-VAE</td><td>13.50</td><td>12.62</td><td>12.40</td><td>0.948</td><td>0.948</td><td>0.947</td></tr><tr><td>PS-VAE</td><td>13.95</td><td>13.83</td><td>13.65</td><td>0.948</td><td>0.948</td><td>0.948</td></tr><tr><td>EDM-SYCo</td><td>14.73</td><td>14.31</td><td>14.21</td><td>0.948</td><td>0.948</td><td>0.948</td></tr></table>
+
+# G.7 FRAGMENT OCCURRENCES
+
+In this section, we test our method's ability to generate molecules containing fragments of different occurrence frequencies in the training set and compare it with several baselines. Table 14 shows the results of this experiment. Even though our model is not motif-based, it can match the distribution of fragments very well.
+
+Table 14: Frequency of occurrence of different fragments computed on the generated molecules from different methods. Methods are split into SMILES-based, graph-based motif-level, and graph-based atom-level. The method that achieves the closest percentage to that in the training set is highlighted. SYCO can successfully reproduce the percentages of different fragments in its generated molecules without relying on motifs.  
+
+<table><tr><td>FRAGMENT</td><td>C1-CC-CC-C1</td><td>HN
+C1CCNCC1</td><td>HN
+C1-C(N)-HCl</td><td>C1ex2(2)C1CON2</td><td>C1-CCCC1</td><td>HN
+C1CCNCC1</td><td>C1CCCCC1</td><td>C1CCXXXCC1</td></tr><tr><td>TRAINING</td><td>76.47%</td><td>14.15%</td><td>9.37%</td><td>1.19%</td><td>1.09%</td><td>0.97%</td><td>0.39%</td><td>0.27%</td></tr><tr><td>CHARVAE</td><td>52.98%</td><td>4.06%</td><td>1.09%</td><td>0.13%</td><td>0.40%</td><td>2.31%</td><td>1.21%</td><td>0.00%</td></tr><tr><td>SM-LSTM</td><td>75.73%</td><td>14.02%</td><td>9.73%</td><td>1.01%</td><td>1.09%</td><td>1.10%</td><td>0.34%</td><td>0.24%</td></tr><tr><td>PS-VAE</td><td>71.31%</td><td>13.08%</td><td>5.58%</td><td>2.84%</td><td>1.07%</td><td>2.61%</td><td>0.27%</td><td>0.12%</td></tr><tr><td>HIERVAE</td><td>75.78%</td><td>12.80%</td><td>7.92%</td><td>0.71%</td><td>0.45%</td><td>0.56%</td><td>0.03%</td><td>0.00%</td></tr><tr><td>MICAM</td><td>79.47%</td><td>11.37%</td><td>5.74%</td><td>0.79%</td><td>0.94%</td><td>2.16%</td><td>0.37%</td><td>0.06%</td></tr><tr><td>JT-VAE</td><td>80.15%</td><td>13.84%</td><td>9.70%</td><td>1.99%</td><td>1.07%</td><td>1.48%</td><td>0.60%</td><td>0.41%</td></tr><tr><td>MAGNET</td><td>69.01%</td><td>11.54%</td><td>7.58%</td><td>0.99%</td><td>1.15%</td><td>0.87%</td><td>0.23%</td><td>0.19%</td></tr><tr><td>MOLER</td><td>78.47%</td><td>15.12%</td><td>7.85%</td><td>1.26%</td><td>0.64%</td><td>1.24%</td><td>0.25%</td><td>0.21%</td></tr><tr><td>GCPN</td><td>53.37%</td><td>2.03%</td><td>0.33%</td><td>0.44%</td><td>3.53%</td><td>0.38%</td><td>0.89%</td><td>0.13%</td></tr><tr><td>GRAPHAF</td><td>34.16%</td><td>1.27%</td><td>2.29%</td><td>0.11%</td><td>0.29%</td><td>0.25%</td><td>0.03%</td><td>0.00%</td></tr><tr><td>DIGRESS</td><td>83.96%</td><td>11.63%</td><td>8.03%</td><td>1.18%</td><td>0.52%</td><td>1.49%</td><td>0.38%</td><td>0.10%</td></tr><tr><td>EDM-SYCo</td><td>75.57%</td><td>12.84%</td><td>9.17%</td><td>1.02%</td><td>0.56%</td><td>1.89%</td><td>0.33%</td><td>0.16%</td></tr></table>
+
+# G.8 MODEL RUNTIME
+
+To understand the computational cost associated with our model, we report the average runtime for training and inference in Table 15. On the ZINC250K dataset, our model is trained for 1000 epochs, which takes approximately 10 days (considering intermediate evaluations every 10 epochs), with an average of 740 seconds for training per epoch. During sampling, our model greatly benefits from running batch-wise and takes, on average, 86 seconds to generate a batch of 100 molecules, which corresponds to an amortized runtime of 0.86 seconds per single molecule. Generating molecules conditioned on target values further increases the runtime because it requires running one forward and one backward pass of the regressor model at each diffusion step, increasing the average runtime to 1.8 seconds per molecule. All numbers were computed using a single Nvidia A100 GPU.
+
+Table 15: Average runtime of EDM-SYCo. Numbers are on the ZINC250K dataset, the training set contains 220K molecules, and the sampling times are obtained using a batch size of 100.  
+
+<table><tr><td></td><td>RUNTIME</td></tr><tr><td>TRAINING (EPOCH)</td><td>740 SECOND / EPOCH</td></tr><tr><td>TRAINING (TOTAL)</td><td>~ 10 DAYS</td></tr><tr><td>UNCONDITIONAL SAMPLING</td><td>0.86 SECOND / MOLECULE</td></tr><tr><td>CONDITIONAL SAMPLING (GUIDANCE)</td><td>1.8 SECOND / MOLECULE</td></tr></table>
+
+# G.9 SAMPLING SPEED VERSUS GENERATION QUALITY TRADE-OFF
+
+Our diffusion model is trained with  $T = 1000$  diffusion steps, which can be a limiting factor in some practical applications due to the resulting slow sampling process. One simple way to accelerate this process is to run a strided sampling schedule Nichol & Dhariwal (2021). The key idea is that to reduce the sampling steps from  $T$  to  $S$ , we run the denoising network (Equation 6) every  $\lceil T / S \rceil$  steps and update the noise schedule accordingly Nichol & Dhariwal (2021). In this experiment, we evaluate the quality of the molecules generated by running the model, initially trained with  $T = 1000$  steps, using only 500, 250, 200, and 100 steps evenly spaced between 1 and  $T$  (without any additional training). Figure 9 shows that the model performs well even with only 200 steps, which offers 5x speedup (if we neglect the cost of running the decoder, which is less than 1 step of reverse diffusion).
+
+![](images/22229854499c69732964d4157274c9fa27a23cff1ee5c215da8f556da6e73ae7.jpg)  
+Figure 9: Effect of varying the number of reverse diffusion steps on the performance, using the same model trained for 1000 steps.
+
+# G.10 ANALYSIS OF 3D INDUCTIVE BIAS
+
+We analyze the inductive bias of our architecture in two steps and perform an ablation study to highlight it:
+
+- Chemical bias of RDKit-generated coordinates: we train a new variant of EDM-SyCo on 3D coordinates produced by the Fruchterman-Reingold force-directed algorithm $^6$ , which generates 3D positions solely based on the graph structure without any chemical inductive bias. We refer to it as EDM-SyCo-graph-setup.  
+- Representation bias of the 3D latent embedding: we highlight baselines that operate directly on the graphs, through discrete diffusion (DiGress) or continuous diffusion by adding Gaussian noise to the adjacency matrix (GDSS).
+
+Results are presented in Table 16. We see that our base model performs better than EDM-SyCograph-Layout across all metrics, which in turn outperforms GDSS and Digress on FCD and KL.
+
+Table 16: Performance comparison of different model classes. Higher values indicate better performance for all metrics.  
+
+<table><tr><td>Model Class</td><td>FCD (↑)</td><td>KL (↑)</td><td>Novelty (↑)</td><td>Uniqueness (↑)</td><td>Validity (↑)</td></tr><tr><td>3D diffusion on RDKit-generated coordinates (EDM-SyCo-RDKit)</td><td>0.85</td><td>0.96</td><td>1.00</td><td>1.00</td><td>0.88</td></tr><tr><td>3D diffusion on graph-based coordinates (EDM-SyCo-graph layout)</td><td>0.73</td><td>0.94</td><td>1.00</td><td>1.00</td><td>0.54</td></tr><tr><td>Continuous diffusion on graph (GDSS)</td><td>0.10</td><td>N.A.</td><td>1.00</td><td>1.00</td><td>0.97</td></tr><tr><td>Discrete diffusion (DiGress)</td><td>0.65</td><td>0.91</td><td>0.99</td><td>0.99</td><td>0.85</td></tr></table>
+
+# G.11 MOVING THE EGNN FROM THE ENCODER TO THE DECODER
+
+To understand the effect of training the diffusion model on the latent space defined by the encoder's EGNN versus on the raw synthetic coordinates, we train a new diffusion model directly on the ETKDG-generated coordinates and move the EGNN to the decoder. We note that also in this setup, training the diffusion model is necessary because, for the used datasets, there is no public ground-truth coordinates and thus no pretrained 3D diffusion models. Although both model variants achieve similar unconditional generation performance, one benefit of having an MLP-only decoder is in a
+
+scaffold-constrained generation setting: Given a scaffold that can be reconstructed by the autoencoder, we perform inpainting in the latent 3D space and generate a completion for this scaffold. The issue with having the EGNN in the decoder is that the bond prediction between the atoms in the scaffold will depend on the newly added atoms, and the final decoded graph is no longer guaranteed to contain the original 2D scaffold, in contrast to having an MLP-only decoder. We validate this argument by generating 1000 valid completions for a 6-Carbon-ring (smiles string: C1CCCCC1, present in  $7\%$  of the training molecules) and computing how often the decoded molecules actually contain the scaffold (Scaffolding success rate). Results are shown in Table 17.
+
+Table 17: Performance comparison of different models on FCD, KL divergence, and Scaffolding Success Rate. Higher values indicate better performance.  
+
+<table><tr><td>Model</td><td>FCD</td><td>KL</td><td>Scaffolding</td><td>Success Rate (%)</td></tr><tr><td>Diffusion model on ETKDG output + EGNN-MLP-decoder</td><td>0.84</td><td>0.95</td><td>94.2</td><td></td></tr><tr><td>Diffusion model on EGNN output + MLP-only-decoder</td><td>0.85</td><td>0.96</td><td>100.0</td><td></td></tr></table>
+
+# G.12 MODEL SCALING WITH MOLECULE SIZE
+
+To check the scalability of our model with respect to the molecule size compared to previous diffusion models, we compare the average runtime required to generate a single molecule using DiGress and our model. DiGress is a discrete diffusion model that operates on 2D graphs. For both models, we generate a batch of 100 molecules of size N, for an increasing N, and report the average runtime to generate one molecule. Results are shown in Table 18.
+
+Table 18: Comparison of generation times (in seconds) for DiGress and EDM-SyCo across different molecule sizes.  
+
+<table><tr><td># Atoms</td><td>10</td><td>20</td><td>30</td><td>40</td><td>50</td><td>60</td></tr><tr><td>DiGress (time in seconds)</td><td>4.05</td><td>12.42</td><td>27.04</td><td>46.87</td><td>70.73</td><td>114.14</td></tr><tr><td>EDM-SyCo (time in seconds)</td><td>0.15</td><td>0.49</td><td>1.02</td><td>1.75</td><td>2.69</td><td>3.66</td></tr></table>
+
+# G.13 MOLECULE SIZE EXTRAPOLATION
+
+To evaluate generalization, we generate molecules larger than every other molecule from the training set and report the validity rate in Table 19. The better performance on GuacaMol compared to ZINC250K can be explained by the higher diversity of training molecule sizes in GuacaMol. We outline some ideas that can improve the generalization capability:
+
+- instead of modeling point clouds as fully-connected graphs for the diffusion model's EGNN, use distance cutoffs to make the number of neighbors of an atom roughly independent of the total number of atoms, as e.g. done in [3].  
+- modify the forward diffusion process to avoid that all atoms cluster around the CoM, e.g. by increasing the variance of the transition kernels.
+
+Table 19: Generalization performance of models on molecules larger than the training set, with results reported as validity rates (\%).  
+
+<table><tr><td>Molecule Size</td><td>nmax+5</td><td>nmax+10</td><td>nmax+20</td></tr><tr><td>ZINC250K (nmean=23.6, std=4.4, max=40)</td><td>45.3</td><td>24.5</td><td>21.1</td></tr><tr><td>GuacaMol (nmean=28.0, std=7.9, max=88)</td><td>70.1</td><td>55.4</td><td>15.9</td></tr></table>
\ No newline at end of file
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+# LIGHTNING-FAST IMAGE INVERSION AND EDITING FOR TEXT-TO-IMAGE DIFFUSION MODELS
+
+Dvir Samuel $^{1,3}$  * Barak Meiri $^{1,2}$  Haggai Maron $^{4,5}$  Yoad Tewel $^{2,5}$  Nir Darshan $^{1}$
+
+Shai Avidan
+
+Gal Chechik $^{3,5}$
+
+Rami Ben-Ari
+
+$^{1}$ OriginAI,  $^{2}$ Tel-Aviv University,  $^{3}$ Bar-Ilan University,  $^{4}$ Technion,  $^{5}$ NVIDIA Research
+
+Israel
+
+Inversion and Editing in 0.4 sec
+
+![](images/4a8bfc47eb83e43280143d18b28c753ba78ecc36236c9f1cceee454ebac8b0f1.jpg)  
+Figure 1: Consecutive real image inversions and editing using our GNRI with Flux.1-schnell (BlackForest, 2024) (0.4 sec on an A100 GPU).
+
+# ABSTRACT
+
+Diffusion inversion is the problem of taking an image and a text prompt that describes it and finding a noise latent that would generate the exact same image. Most current deterministic inversion techniques operate by approximately solving an implicit equation and may converge slowly or yield poor reconstructed images. We formulate the problem by finding the roots of an implicit equation and develop a method to solve it efficiently. Our solution is based on Newton-Raphson (NR), a well-known technique in numerical analysis. We show that a vanilla application of NR is computationally infeasible while naively transforming it to a computationally tractable alternative tends to converge to out-of-distribution solutions, resulting in poor reconstruction and editing. We therefore derive an efficient guided formulation that fastly converges and provides high-quality reconstructions and editing. We showcase our method on real image editing with three popular open-sourced diffusion models: Stable Diffusion, SDXL-Turbo, and Flux with different deterministic schedulers. Our solution, Guided Newton-Raphson Inversion, inverts an image within 0.4 sec (on an A100 GPU) for few-step models (SDXL-Turbo and Flux.1), opening the door for interactive image editing. We further show improved results in image interpolation and generation of rare objects.
+
+# 1 INTRODUCTION
+
+Text-to-image diffusion models (Rombach et al., 2022; Sahara et al., 2022; Ramesh et al., 2022; Balaji et al., 2022) can generate diverse and high-fidelity images based on user-provided text prompts. These models are further used in several important tasks that require inversion, namely, discovering an initial noise (seed) that, when subjected to a backward (denoising) diffusion process along
+
+with the prompt, generates the input image. Inversion is used in various tasks including image editing (Hertz et al., 2022), personalization (Gal et al., 2023b;a), seed noise interpolation for semantic augmentation (Samuel et al., 2023) and generating rare concepts (Samuel et al., 2024).
+
+As inversion became a critical building block in various tasks, several inversion methods have been suggested. Denoising Diffusion Implicit Models (DDIM) (Song et al., 2021) introduced a deterministic and fast sampling technique for image generation with diffusion models. DDIM inversion transforms an image back into a latent noise representation by approximating the inversion equation. Although this approximation makes it very fast, it also introduces an approximation error (as explained in section 3), causing noticeable distortion artifacts in the reconstructed images. This is particularly noticeable in few-step diffusion models (Luo et al., 2023; Sauer et al., 2023; Esser et al., 2024), with large gaps between the diffusion time-steps, and where inference is achieved with only 2 - 4 steps. Several efforts have been made to address inconsistencies in DDIM Inversion, which result in poor reconstruction quality (Mokady et al., 2023; Wallace et al., 2023; Pan et al., 2023). For example, (Pan et al., 2023; Garibi et al., 2024) used fixed-point iterations to solve the DDIM-inversion equation. Another approach in (Hong et al., 2024) tackled the minimization of residual error in DDIM-inversion via gradient descent. Although these methods demonstrate improvements over previous approaches, their editing quality and computational speed are still limited.
+
+In this paper, we frame the deterministic diffusion inversion problem as finding the roots of an implicit function. We propose a solution based on the Newton-Raphson (NR) numerical scheme (Burden et al., 2015) - a very fast and well-tested optimization method for root finding. We further define a scalar NR scheme for this problem that can be computed efficiently. However, NR has a main disadvantage in our context. When applied to our highly non-convex function, it tends to find roots that drift outside the distribution of latents that the diffusion model was trained on. To mitigate this, we introduce a guidance term that leverages prior knowledge of the likely solution locations, steering the NR iterations toward in-distribution roots. This is done by adding this prior knowledge on the noise distribution at each diffusion step. Ultimately, our approach enables rapid inversion while maintaining state-of-the-art reconstruction and editing quality.
+
+We name our approach GNRI, for Guided Newton Raphson Inversion. GNRI converges in just a few iteration steps at each diffusion time step and achieves high-quality image inversions (in terms of reconstruction accuracy). In practice, 1-2 iterations are sufficient for convergence that yields significantly more accurate results than other inversion methods. Importantly, GNRI requires no model training, fine-tuning, prompt optimization, or additional parameters, and is compatible with all pre-trained diffusion and flow-matching models with deterministic schedulers. We demonstrate its effectiveness for inverting different deterministic schedulers used with latent diffusion model (Stable Diffusion) (Rombach et al., 2022), few-step latent diffusion model (SDXL-turbo) (Sauer et al., 2023) and few-step flow matching model (Flux.1) (Black-Forest, 2024). Figure 1 demonstrates the quality and speed of GNRI for iterative editing using few-step diffusion models. Using such models models that require 4 denoising steps, our approach can edit real images within 0.4 seconds. This allows users to edit images on the fly, using text-to-image models.
+
+We conduct a comprehensive evaluation of GNRI. First, we directly assess the quality of inversions found with GNRI by measuring reconstruction errors compared to deterministic inversion approaches. Our method suppresses all methods with  $\times 2$  to  $\times 40$  speedup gain. We then demonstrate the benefit of GNRI in two downstream tasks (1) In Image editing, GNRI smoothly changes fine details in the image in a consistent and coherent way, whereas previous methods struggle to do so. (2) In Seed Interpolation and Rare concept generation (Samuel et al., 2023) that require diffusion inversion. In both of these tasks, GNRI yields more accurate seeds, resulting in superior generated images, both qualitatively and quantitatively.
+
+# 2 RELATED WORK
+
+Text-to-image diffusion models (Rombach et al., 2022; Sahara et al., 2022; Ramesh et al., 2022; Balaji et al., 2022; Esser et al., 2024) translate random samples (seeds) from a high-dimensional space, guided by a user-supplied text prompt, into corresponding images. DDIM (Song et al., 2021) is a widely used deterministic scheduler, that demonstrates the inversion of an image to its latent noise seed. When applied to the inversion of text-guided diffusion models, DDIM inversion suffers from low reconstruction accuracy that is reflected in further tasks, particularly when applied to few-
+
+step diffusion models that require only 3-4 denoising steps. This happens because DDIM inversion relies on a linear approximation, causing the propagation of errors that result in inaccurate image reconstruction. Recent studies (Mokady et al., 2023; Wallace et al., 2023; Pan et al., 2023; Hong et al., 2024) address this limitation. Null-text inversion (Mokady et al., 2023) optimizes the embedding vector of an empty string. This ensures that the diffusion process calculated using DDIM inversion, aligns with the reverse diffusion process. (Miyake et al., 2023) replace the null-text embedding with a prompt embedding instead. This enhances convergence time and reconstruction quality but results in inferior image editing performance. In both (Mokady et al., 2023) and (Miyake et al., 2023), the optimized embedding must be stored, resulting in nearly 3 million additional parameters for each image (using 50 denoising steps of StableDiffusion (Rombach et al., 2022)).
+
+EDICT (Wallace et al., 2023) introduced invertible neural network layers, specifically Affine Coupling Layers, to calculate both backward and forward diffusion paths. While effective, it comes at the cost of prolonging inversion time. BDIA (Zhang et al., 2023) introduced a novel approximation tailored for EDICT, enhancing its computational efficiency while maintaining the accuracy in diffusion inversion. However, it still demands considerably more time, approximately ten times longer than DDIM Inversion. TurboEdit (Wu et al., 2024) introduced an encoder-based iterative inversion technique for few-step diffusion models, enabling precise image inversion and disentangled image editing. DirectInv (Ju et al., 2023) separates the source and target diffusion branches for improved text-guided image editing, leading to better content preservation and edit fidelity. AIDI (Pan et al., 2023) and ReNoise (Garibi et al., 2024) used a fixed-point iteration technique at each inversion step to find a better solution for the implicit function posed by DDIM equations and thereby achieving improved accuracy. ExactDPM (Hong et al., 2024), similar to (Pan et al., 2023; Garibi et al., 2024) proposed gradient-based methods for finding effective solutions for the implicit function.
+
+Alternative methods proposed in (Huberman et al., 2023; Brack et al., 2024; Deutch et al., 2024) use a stochastic DDPM inversion instead of a deterministic one. Although it can recover the exact image during reconstruction, the stochastic nature of these approaches usually causes the method to struggle with reconstructing fine details during editing. These methods require many steps for high-quality editing, making the process time-consuming. Additionally, they demand a larger memory footprint, as they need to store  $T + 1$  latents for each inverted image, restricting their use only to reconstruction and editing tasks. We also compare our approach with these methods in terms of editing quality and demonstrate that our method outperforms them all.
+
+# 3 PRELIMINARIES
+
+We first establish the fundamentals of Denoising Diffusion Implicit Models (DDIMs). In this model, a backward pass (denoising) is the process that generates an image from a seed noise. A forward pass is the process of adding noise gradually to an image until it becomes pure Gaussian noise. Inversion (Song et al., 2021) is similar to the forward pass but the goal is to end with a specific Gaussian noise that would generate the image if denoised.
+
+Forward Pass in Diffusion Models. Diffusion models (Rombach et al., 2022) learn to generate images through a systematic process of iteratively adding Gaussian noise to a latent data sample until the data distribution is mostly noise. The data distribution is subsequently gradually restored through a reverse diffusion process initiated with a random sample (noise seed) from a Gaussian distribution. In more detail, the process of mapping a (latent) image to noise is a Markov chain that starts with  $z_0$ , and gradually adds noise to obtain latent variables  $z_1, z_2, \ldots, z_T$ , following a distribution  $q(z_1, z_2, \ldots, z_T | z_0) = \Pi_{t=1}^T q(z_t | z_{t-1})$ , where  $\forall t: z_t \in \mathbb{R}^d$  with  $d$  denoting the dimension of the space. Each step in this process is a Gaussian transition, that is,  $q$  follows a Gaussian distribution
+
+$$
+q \left(z _ {t} \mid z _ {t - 1}\right) := \mathcal {N} \left(z _ {t}; \mu_ {t} = \gamma_ {t} z _ {t - 1}, \Sigma_ {t} = \beta_ {t} I\right), \tag {1}
+$$
+
+parameterized by a schedule  $(\beta_0,\gamma_0),\ldots ,(\beta_T,\gamma_T)\in (0,1)^2$  . As discussed below, in DDIM,  $\gamma_{t} = \sqrt{1 - \beta_{t}}$  and in Euler scheduling  $\beta_{t} = \gamma_{t} = 1\forall t$
+
+Deterministic schedule diffusion models. It has been shown that one can "sample" in a deterministic way from the diffusion model, and this accelerates significantly the denoising process. Several deterministic schedulers have been proposed (Song et al., 2021; Lu et al., 2022; Karras et al., 2022), we describe here two popular ones, DDIM and Euler schedulers.
+
+Denoising Diffusion Implicit Models (DDIM). Sampling from diffusion models can be viewed as solving the corresponding diffusion Ordinary Differential Equations (ODEs) (Lu et al., 2022). DDIM (Song et al., 2021), a popular deterministic scheduler, proposed to denoise a latent by
+
+$$
+z _ {t - 1} = \sqrt {\frac {\alpha_ {t - 1}}{\alpha_ {t}}} z _ {t} - \sqrt {\alpha_ {t - 1}} \cdot \Delta \psi (\alpha_ {t}) \cdot \epsilon_ {\theta} (z _ {t}, t, p), \tag {2}
+$$
+
+where  $\alpha_{t} = 1 - \beta_{t}$ ,  $\psi (\alpha) = \sqrt{\frac{1}{\alpha} - 1}$ , and  $\Delta \psi (\alpha_{t}) = \psi (\alpha_{t}) - \psi (\alpha_{t - 1})$  and  $\epsilon_{\theta}(z_t,t,p)$  is the output of a network that was trained to predict the noise to be removed.
+
+Euler schedulers. Euler schedulers follow a similar deterministic update rule
+
+$$
+z _ {t - 1} = z _ {t} + \left(\sigma_ {t - 1} - \sigma_ {t}\right) v _ {\theta} \left(z _ {t}, t, p\right), \tag {3}
+$$
+
+here  $\sigma_t, \sigma_t - 1$  are scheduling parameters and  $v_\theta$  is the output of the network that was trained to predict the velocity.
+
+Diffusion inversion. We now focus on inversion in the latent representation. We describe our approach first for DDIM, and extension to other schedulers is discussed below. Given an image representation  $z_0$  and its corresponding text prompt  $p$ , we seek a noise seed  $z_T$  that, when denoised, reconstructs the latent  $z_0$ . Several approaches were proposed for this task. DDIM inversion rewrites Eq. (2) as:
+
+$$
+z _ {t} = f (z _ {t})
+$$
+
+$$
+f (z _ {t}) := \sqrt {\frac {\alpha_ {t}}{\alpha_ {t - 1}}} z _ {t - 1} + \sqrt {\alpha_ {t}} \cdot \Delta \psi (\alpha_ {t}) \cdot \epsilon_ {\theta} (z _ {t}, t, p). \tag {4}
+$$
+
+DDIM inversion approximates this implicit equation in  $z_{t}$  by replacing  $z_{t}$  with  $z_{t - 1}$
+
+$$
+f \left(z _ {t}\right) \approx \sqrt {\frac {\alpha_ {t}}{\alpha_ {t - 1}}} z _ {t - 1} + \sqrt {\alpha_ {t}} \cdot \Delta \psi \left(\alpha_ {t}\right) \cdot \epsilon_ {\theta} \left(z _ {t - 1}, t, p\right). \tag {5}
+$$
+
+The quality of the approximation depends on the difference  $z_{t} - z_{t-1}$  (a smaller difference would yield a small error) and on the sensitivity of  $\epsilon_{\theta}$  to that  $z_{t}$ . See (Dhariwal & Nichol, 2021; Song et al., 2021) for details. By applying Eq. (5) repeatedly for every denoising step  $t$ , one can invert an image latent  $z_{0}$  to a latent  $z_{T}$  in the seed space.
+
+DDIM inversion is fast, but the approximation of Eq. (5) inherently introduces errors at each time step. As these errors accumulate, they cause the whole diffusion process to become inconsistent in the forward and the backward processes, leading to poor image reconstruction and editing (Mokady et al., 2023; Wallace et al., 2023; Pan et al., 2023). This is particularly noticeable in few-step and consistency models with a small number of denoising steps (typically 2-4 steps), where there's a significant gap between  $z_{t}$  and  $z_{t - 1}$  Garibi et al. (2024).
+
+This inversion technique can also be applied to other deterministic schedulers. For instance, for Euler scheduler, one defines  $z_{t} = f(z_{t})$ ,  $f(z_{t}) := z_{t-1} + (\sigma_{t} - \sigma_{t-1}) v_{\theta}(z_{t}, t, p)$ , and this is approximated by  $z_{t-1} + (\sigma_{t} - \sigma_{t-1}) v_{\theta}(z_{t-1}, t, p)$ .
+
+Iterative inversion and optimization methods. Several papers proposed to improve the approximation using iterative methods. AIDI (Pan et al., 2023) and ReNoise (Garibi et al., 2024) proposed to directly solve Eq. (4) using fixed-point iterations (Burden, 1985), a widely-used method in numerical analysis for solving implicit functions. In a related way, (Hong et al., 2024) solves a more precise inversion equation, obtained by employing higher-order terms, using gradient descent.
+
+Newton-Raphson Root finding. The Newton-Raphson method is a widely used numerical technique for finding roots of a real-valued function  $F$  (Burden et al., 2015). It is particularly effective for solving equations of the form  $F(z) = 0$ , when  $F: \mathbb{R}^D \to \mathbb{R}^D$  for an arbitrary dimension  $D$ . It provides fast convergence, typically quadratic, by requiring the evaluation of the function and the inversion of its Jacobian matrix. It has also been shown that when initialized near a local extremum, NR may converge to that point (oscillating around it) (Kaw et al., 2003). The NR scheme in general form, is given by
+
+$$
+z _ {t} ^ {k + 1} = z _ {t} ^ {k} - J \left(z _ {t} ^ {k}\right) ^ {- 1} F \left(z _ {t} ^ {k}\right), \tag {6}
+$$
+
+where  $J(z_{t}^{k})^{-1}$  presents the inverse of a Jacobian matrix  $J \in \mathbb{R}^{D \times D}$  for  $F$  (all derivatives are w.r.t to  $z$ ), and  $k$  stands for the iteration number. The iteration starts with an initial guess,  $z = z^{0}$ .
+
+![](images/3f9edc25d7e4b19e755f7c12ea3e509acae2ec37e6c43ca03062c24c73f179fe.jpg)  
+Figure 2: Newton-Raphson Inversion iterates over an implicit function Eq. 4 using Eq. 10 scheme, at every time-step in the inversion path. Initialized with  $z_{t}^{0} = z_{t - 1}$  it converges within  $\approx 2$  iterations, to  $z_{t}$ . Each box denotes one inversion step; black circles correspond to intermediate latents in the denoising process; green circles correspond to intermediate Newton-Raphson iterations.
+
+# 4 OUR METHOD: GUIDED NEWTON RAPHSON INVERSION
+
+Existing inversion methods are either fast-but-inaccurate, like DDIM inversion, or more precise-but-slow, like (Pan et al., 2023) and (Hong et al., 2024). This paper presents a method that achieves a balance of speed and improved precision compared to these existing approaches.
+
+To achieve these features we introduce two key ideas. First, we frame inversion as a scalar root-finding problem that can be solved efficiently, and use the well-known Newton-Raphson root-finding method. Second, we make the observation that an inversion step is equivalent to finding a specific noise vector, and we use a strong prior obtained from the distribution of that noise. This prior during the iterative solution guides Newton-Raphson towards finding a root that adheres to the correct distribution of latents. See Figure 2 for illustration.
+
+# 4.1 FRAMING INVERSION AS EFFICIENT ROOT-FINDING
+
+Diffusion inversion can be done by finding the fixed point of the function  $f(z)$  in Eq. 4. This is equivalent to finding the zero-crossings or roots,  $z_{t}$  of the residual function  $r: \mathbb{R}^{D} \to \mathbb{R}^{D}$  defined as  $r(z_{t}) := z_{t} - f(z_{t})$ . One could now apply Newton-Raphson (NR) to find a root. However, in our case,  $z_{t}$  is a high dimensional latent  $D \approx 16K$  in StableDiffusion (Rombach et al., 2022), making it computationally infeasible to compute the Jacobian (Eq. 6), explicitly and invert it. To address this computational limitation, we apply a norm over  $r(z_{t})$ ,  $\hat{r}: \mathbb{R}^{d} \to \mathbb{R}_{0}^{+}$  yielding a multi-variable, scalar function, which has the same set of roots:
+
+$$
+\hat {r} \left(z _ {t}\right) := \left| \left| z _ {t} - f \left(z _ {t}\right) \right| \right| _ {1}, \tag {7}
+$$
+
+and seek the roots  $\hat{r}(z_t) = 0$  where  $||\cdot||$  denotes a  $L_1$  norm which simply sum over all absolute values of  $\hat{r}$ . Applying the norm on the residual has been done in previous work (Hong et al., 2024; Garibi et al., 2024), however here it is used to reduce the Jacobian to a vector and that can be computed quickly (see derivation in Appendix A). We call this method Newton-Raphson Inversion, NRI.
+
+This method is efficient and converges quickly in practice. However, NRI in our case has a key limitation: Eq. (4), and thus  $\hat{r}$ , can have multiple solutions or local minima. The accelerated convergence of NR due to linear "extrapolation", can cause it to "jump" to solutions outside the latent variable distribution for the diffusion model. Indeed, in practice, we find that solving Eq. (7) for  $\hat{r}(z_t) = 0$  often converges quickly to a solution that results with poor image reconstruction. We quantify and illustrate this phenomenon in section 5.1. Next, we describe a remedy to this problem.
+
+# 4.2 GUIDED NEWTON RAPHSON INVERSION
+
+The NR procedure discussed above may find roots that are far from the modes of the latent distribution. How can we guide NR to converge to useful roots among all possible ones?
+
+Note that determining a  $z_{t}$  given  $z_{t-1}$ , is equivalent to finding which noise term is added during the forward pass of the diffusion process. Luckily, the distribution of that noise is known by the design of the diffusion process and we can use it as prior information. More precisely, since each step in the diffusion process follows a Gaussian distribution  $q(z_{t}|z_{t-1})$  (Eq. 1), we define a positive function
+
+that obtains a zero value at the maximum likelihood of the  $q$  distribution by taking the negative log of the exponent part of the distribution (without the normalizing factor). This can be simplified to
+
+$$
+G \left(z _ {t}\right) = \left\{ \begin{array}{l l} \frac {1}{\beta_ {t}} \left| \left| z _ {t} - \mu_ {t} \right| \right| & \text {D D I M}, \\ \left| \left| z _ {t} - \mu_ {t} \right| \right| & \text {E u l e r}. \end{array} \right. \tag {8}
+$$
+
+where  $\mu_t, \beta_t$  were defined in Section 3, see Eq (1). To guide NR to select an "in-distribution" solution, we define a new objective  $\mathcal{F}: \mathbb{R}^D \to \mathbb{R}_0^+$ :
+
+$$
+\mathcal {F} \left(z _ {t}\right) := \left| \left| z _ {t} - f \left(z _ {t}\right) \right| \right| _ {1} + \lambda G \left(z _ {t}\right). \tag {9}
+$$
+
+Here,  $\lambda > 0$  is a hyperparameter weighting factor for the guidance term. See Appendix E for ablation of  $\lambda$  values. Note that this guidance term incorporates prior knowledge about the probable locations of the solutions in  $z$  space.
+
+We then seek to drive  $\mathcal{F}(z_t)$  near zero. Applying the Newton Raphson iteration scheme for Eq. (9) then emerges as (see derivation in Appendix A):
+
+$$
+z _ {t} ^ {0} = z _ {t - 1}
+$$
+
+$$
+z _ {t} ^ {k + 1} (i) = z _ {t} ^ {k} (i) - \frac {1}{D} \frac {\mathcal {F} \left(z _ {t} ^ {k}\right)}{g _ {i} \left(z _ {t} ^ {k}\right) + \eta}. \tag {10}
+$$
+
+Here  $g_{i} \coloneqq \frac{\partial\mathcal{F}(z_{t})}{\partial z_{t}(i)}$  is the partial derivative of  $\mathcal{F}$  with respect to the variable (descriptor)  $z_{t}(i)$ . The index  $i \in [0,1,\dots,D]$  indicates  $z$ 's components and  $k$  stands for the iteration number, while  $\eta$  is a small constant added for numerical stability. The function  $g$  can be computed efficiently using automatic differentiation engines. We initialize the process with  $z$  from the previous diffusion timestep  $z_{t-1}$ . We call our approach GNRI for Guided Newton Raphson Inversion. For discussion on applying NR on our objective function and the guidance term see Appendix B.
+
+# 5 EXPERIMENTS
+
+Here, we first provide motivation and analysis for the guidance term presented in Section 4.2. Then, we show evaluation results of GNRI on three main tasks: (1) Image inversion and reconstruction: Here, We assess the inversion fidelity by evaluating the quality of the reconstructed images. (2) Image Editing: We demonstrate the efficacy of our inversion scheme in image editing, highlighting its ability to facilitate real-time editing. (3) Rare Concept Generation: We illustrate how our method can be applied to improve the generation of rare concepts.
+
+Compared Methods: We compared our approach with the following methods. Standard DDIM Inversion (Song et al., 2021), SDEdit Meng et al. (2022), Null-text (Mokady et al., 2023) and EDICT (Wallace et al., 2023). Two fixed-point based methods: AIDI (Pan et al., 2023) and ReNoise (Garibi et al., 2024) with further comparison to Anderson Acceleration Pan et al. (2023) shown in Appendix I. Gradient-based method ExactDPM (Hong et al., 2024). Wu2024Turbo (Wu et al., 2024) and two DDPM inversion methods: EditFriendly Huberman et al. (2023) and Gal2024Turbo (Deutch et al., 2024). In all experiments, we used code and parameters provided by the respective authors.
+
+Implementation details: To demonstrate the versatility of our approach, we conducted experiments on latent diffusion model SD2.1 Rombach et al. (2022) and two few-step models: SDXL-turbo (Sauer et al., 2023) and the newly introduced flow matching Flux.1-schnell model (Black-Forest, 2024). Sampling steps were set to 50 for SD2.1 and 4 for SDXL-turbo and Flux.1. All methods ran until convergence. All methods were tested on a single A100 GPU for a fair comparison. PyTorch's built-in gradient calculation was used for computing derivatives of Eq. (9). Editing for all iterative methods was done using Prompt-to-Prompt (Hertz et al., 2022).
+
+# 5.1 GUIDANCE TERM MOTIVATION AND COMPARISON TO OTHER NUMERICAL SCHEMES
+
+To illustrate the motivation behind our guidance term, we compare the reconstruction quality of non-guided NRI a "vanilla" NR inversion method with two iterative numerical schemes: Fixed-point Iteration (FPI) ((Pan et al., 2023) and Gradient Descent (GD) (Hong et al. (2024). We test the
+
+![](images/f6b1555aa72bc2e8f008cda1bd99750d8bdf2ddb68b20ac2f618d5ab81dc0349.jpg)  
+Figure 3: The effect of GNRI guiding term on NR inversion, and comparison to other iterative inversion methods. All results are averages computed with SDXL-Turbo applied to 5,000 COCO images. (a) Average residuals throughout optimization. NR-based methods are the fastest to converge. Gradient-descent was run with the largest learning rate that was stable but still is slowest. (b) Reconstruction quality (PSNR). Adding guidance (blue) to NRI (green) significantly improves the quality of the converged solution. (c) Likelihood of inferred noise. Without the guiding term, NRI (green) finds solutions that are substantially different from those found by other methods, which explains the low reconstruction quality.
+
+![](images/9f97c8dca6ad58949b3b9df29773f64b565b3d33dd2041a2a3380db1b06e8038.jpg)  
+Figure 4: (Left) Reconstruction qualitative results: Comparing image inversion-reconstruction performance. While all baseline methods struggle to preserve the original image, GNRI successfully excels in accurately reconstructing it. (Right) Inversion Results: Mean reconstruction quality (y-axis, PSNR) and runtime (x-axis, seconds) on the COCO2017 validation set. Our method achieves high PSNR while reducing inversion-reconstruction time by a factor of  $\times 2$  (compared to DDIM) and up to  $\times 40$  (compared to ExactDPM) on SDXL-turbo and  $\times 10$  to  $\times 40$  on Flux.1, compared to other approaches.
+
+reconstruction of 5,000 image-caption pairs from COCO Lin et al. (2014), presenting three metrics in Fig. 3. Fig 3a shows that FPI, GD and non-guided NRI, all converge to low residual values but the reconstruction values for NRI (green line) in Fig. 3b are notably lower. This suggests that solutions reached by NRI are in areas of the latent space that yield poor results. Comparing the blue curve in Fig 3 to the green curve highlights the significant impact of adding a guidance term. It accelerates convergence (Fig. 3a), improves reconstruction (PSNR) (Fig 3b) and reaches a better likelihood (Fig. 3c).
+
+![](images/67566364184fa2814e766f5f2280f550a1bb6788a800b1e7307cbb242eef3caa.jpg)  
+Figure 5: (Left) Qualitative results of image editing. GNRI edits images more naturally while preserving the structure of the original image. All baselines were executed until they reached convergence. (Right) Evaluation of editing performance: GNRI achieves superior CLIP and LPIPS scores, indicating better compliance with text prompts and higher structure preservation.
+
+# 5.2 IMAGE RECONSTRUCTION
+
+To evaluate the fidelity of our approach, we measure the PSNR of images reconstructed from seed inversions. Specifically, we used the entire set of 5000 images from the MS-COCO-2017 validation dataset (Lin et al., 2014), along with their corresponding captions. For each image-caption pair, we first found the inverted latent and then used it to reconstruct the image using the same caption. Given that the COCO dataset provides multiple captions for each image, we used the first listed caption as a conditioning prompt. Figure 4 (left) demonstrates a qualitative comparison of reconstructed images. Figure 4 (right) illustrates the PSNR of reconstructed images versus inversion time, highlighting the performance of our method compared to SoTA inversion techniques. The dashed black line represents the upper bound set by the Stable Diffusion VAE, showing that our method achieves the highest PSNR among SoTA methods and approaches the VAE upper bound, while also being the fastest. Additional qualitative comparisons for SD2.1 and further analysis are in Appendix H.
+
+# 5.3 REAL-TIME IMAGE EDITING
+
+Current state-of-the-art methods for editing images start by inverting a real image into latent space and then operating on the latent. As a result, the quality of editing depends strongly on the quality of inversion (Mokady et al., 2023).
+
+We now evaluate the effect of inverting images with GNRI, compared to SoTA inversion baselines. We show that GNRI outperforms all baselines in editing performance while requiring the shortest time. This opens the door for real-time editing capabilities. We present below both qualitative and quantitative results. For Neural Function Evaluations (NFE) comparison see Appendix H.
+
+Table 1: Image editing comparison using descriptive text in PIE-Bench (Ju et al., 2023) dataset with SDXL-turbo. The efficiency is measured in a single A100 GPU. Our method achieves the best background preservation and clip similarity while being significantly faster than other methods.  
+
+<table><tr><td rowspan="2">PIE-Bench</td><td colspan="2">Background</td><td colspan="2">Preservation</td><td colspan="2">CLIP Similarity</td><td colspan="2">Efficiency</td></tr><tr><td>PSNR↑</td><td>LPIPS103↓</td><td>MSE104↓</td><td>SSIM102↑</td><td>Whole↑</td><td>Edited↑</td><td>Inverse(s)↓</td><td>Forward(s)↓</td></tr><tr><td>DDIM</td><td>18.59</td><td>177.96</td><td>184.69</td><td>66.86</td><td>23.62</td><td>21.20</td><td>0.454</td><td>0.445</td></tr><tr><td>ReNoise (Garibi et al., 2024)</td><td>27.11</td><td>49.25</td><td>31.23</td><td>72.3</td><td>23.98</td><td>21.26</td><td>5.41</td><td>0.445</td></tr><tr><td>ExactDPM (Hong et al., 2024)</td><td>24.54</td><td>59.88</td><td>36.49</td><td>69.18</td><td>23.77</td><td>21.23</td><td>15.11</td><td>0.445</td></tr><tr><td>EditFriendly (Huberman et al., 2023)</td><td>24.55</td><td>91.88</td><td>95.58</td><td>81.57</td><td>23.97</td><td>21.03</td><td>32.3</td><td>23.1</td></tr><tr><td>Direct Inv (Ju et al., 2023)</td><td>27.22</td><td>54.55</td><td>62.86</td><td>84.76</td><td>25.02</td><td>22.10</td><td>4.3</td><td>50</td></tr><tr><td>TurboEdit (Wu et al., 2024)</td><td>29.52</td><td>44.74</td><td>26.08</td><td>91.59</td><td>25.05</td><td>22.34</td><td>0.981</td><td>0.569</td></tr><tr><td>GNRI (Ours)</td><td>30.22</td><td>40.15</td><td>20.10</td><td>98.66</td><td>26.15</td><td>23.26</td><td>0.521</td><td>0.445</td></tr></table>
+
+<table><tr><td></td><td colspan="2">Interpolation</td><td colspan="2">Centroid</td></tr><tr><td></td><td>ACC↑</td><td>FID↓</td><td>ACC↑</td><td>FID↓</td></tr><tr><td>DDIM Inversion</td><td>51.59</td><td>6.78</td><td>67.24</td><td>5.48</td></tr><tr><td>ExactDPM (Hong et al., 2024)</td><td>52.01</td><td>6.13</td><td>68.14</td><td>5.32</td></tr><tr><td>ReNoise (Garibi et al., 2024)</td><td>52.51</td><td>6.0</td><td>69.14</td><td>5.22</td></tr><tr><td>GNRI (ours)</td><td>54.98</td><td>5.91</td><td>70.18</td><td>4.59</td></tr></table>
+
+Table 2: Image interpolation and centroid finding. In interpolation, two images  $x^{1}, x^{2}$  are inverted to generate images between their seeds  $z_{T}^{1}, z_{T}^{2}$ . In centroid-finding, a set of images is inverted to find their centroid. Best performance is achieved with our GNRI.
+
+Qualitative results. Figure 5(left) gives a qualitative comparison between SoTA approaches and GNRI in the context of real image editing. GNRI excels in accurately editing with target prompts producing images with both high fidelity to the input image and adherence to the target prompt. The examples illustrate how alternative approaches may struggle to retain the original structure or tend to overlook crucial components specified in the target prompt. As an example, in the first row of Figure 5(left), GNRI exclusively converts the cat into a dog without changing the text in the sign whereas other methods either fail to generate a proper dog or change the text in the sign. Results for SD2.1 in Appendix H. For additional qualitative results see also Appendix H,G and K. User Study. We further evaluated editing quality using human raters. We followed the evaluation process of Mokady et al. (2023) and asked human raters to rank edits made by the three leading methods: Ours, ReNoise (Garibi et al., 2024), and ExactDPM (Hong et al., 2024). Unfortunately, the code for TurboEdit (Wu et al., 2024) was not available at the time of our evaluation. 100 images were rated, 12 images were provided by the authors of (Mokady et al., 2023) and the rest were randomly selected from COCO (Lin et al., 2014) and PieBench (Ju et al., 2023) benchmarks. Three raters for each image were recruited through Amazon Mechanical Turk and were tasked to choose the image that better applies the requested text edit while preserving most of the original image. GNRI was preferred with  $71.6\%$  preference compared to ReNoise's (Garibi et al., 2024)  $15.9\%$  and ExactDPM (Hong et al., 2024)  $12.5\%$ .
+
+Quantitative results. Following (Wallace et al., 2023; Huberman et al., 2023; Garibi et al., 2024), we evaluate the results using two complementary metrics: LPIPS (Zhang et al., 2018) to quantify the extent of structure preservation (lower is better) and a CLIP-based score to quantify how well the generated images comply with the text prompt (higher is better). Metrics are averaged across 100 MS-COCO images. Figure 5(right) illustrates that editing with GNRI yields a superior CLIP and LPIPS score, demonstrating the ability to perform state-of-the-art real image editing with superior fidelity. Following (Wu et al., 2024) we also evaluated our approach on the newly introduced PieBench (Ju et al., 2023) dataset. Table 1 shows that our method more effectively follows the text guidance while preserving the background, outperforming current SoTA techniques. These evaluations further affirm the findings derived from the user study.
+
+# 5.4 SEED INTERPOLATION AND RARE CONCEPT GENERATION
+
+We evaluate the benefit of our inversion in a second task, of selecting latent seeds for image generation. It has been shown (Samuel et al., 2024; 2023) that text-to-image diffusion models poorly
+
+Table 3: Inversion Quality Impact on Rare Concept Generation: We assess image generation using a pre-trained classifier's accuracy, comparing NAO (Samuel et al., 2023) (with DDIM inversion), ReNoise (Garibi et al., 2024), ExactDPM Zhang et al. (2023) and GNRI. We report average per-class accuracy for Head (over 1M samples), Medium, and Tail (rare classes  $< 10\mathrm{K}$  samples). GNRI enhances rare and medium concept accuracy without sacrificing overall performance.  
+ImageNet1k in LAION2B  
+
+<table><tr><td rowspan="2">Methods</td><td>Head
+n=235</td><td>Medium
+n=509</td><td>Tail
+n=256</td><td rowspan="2">Total Acc</td><td rowspan="2">FID</td><td rowspan="2">TInit
+(sec)</td><td rowspan="2">TOpt
+(sec)</td></tr><tr><td>#&gt;1M</td><td>1M&gt;#&gt;10K</td><td>10K&gt;#</td></tr><tr><td>DDIM inversion</td><td>98.5</td><td>96.9</td><td>85.1</td><td>94.3</td><td>6.4</td><td>25</td><td>29</td></tr><tr><td>ExactDPM (Hong et al., 2024)</td><td>98.1</td><td>96.8</td><td>85.3</td><td>94.1</td><td>7.1</td><td>240</td><td>32</td></tr><tr><td>ReNoise (Garibi et al., 2024)</td><td>98.5</td><td>97.0</td><td>85.3</td><td>94.4</td><td>6.9</td><td>24</td><td>28</td></tr><tr><td>GNRI (ours)</td><td>98.6</td><td>97.9</td><td>89.1</td><td>95.8</td><td>6.3</td><td>17</td><td>25</td></tr></table>
+
+generated rare concepts, and this can be improved by selecting better noise latent  $z_{T}$ . To address this issue, SeedSelect (Samuel et al., 2024) takes a few images of a rare concept as input and uses a diffusion inversion module to iteratively refine the obtained seeds. These refined seeds are then used to generate new plausible images of the rare concept. NAO (Samuel et al., 2023) extends this by introducing new paths and centroids for seed initialization. Both methods rely on DDIM Inversions, crucial for initial seed evaluation. GNRI provides an alternative for precise inversion seeds, aiming for improved image quality and semantic accuracy. It's important to highlight that most inversion techniques (Mokady et al., 2023; Wallace et al., 2023; Huberman et al., 2023; Brack et al., 2024) are not applicable in this context as they necessitate extra parameters for image reconstruction, which impedes the straightforward implementation of interpolation.
+
+Interpolation and centroid finding: We evaluate GNRI by following the experimental protocol of (Samuel et al., 2023) and compare images generated by four methods: DDIM, ReNoise (Garibi et al., 2024), ExactDOM (Hong et al., 2024) and GNRI. Evaluation is conducted based on FID score and classification accuracy (by a pre-trained classifier). Results are presented in Table 2. Notably, initializing with GNRI seeds consistently results in higher-quality images both in interpolation paths and seed centroids.
+
+Rare concept generation: We further show the effect of our seed inversions on the performance of NAO centroids with SeedSelect for rare concept generation. Specifically, we compared images generated by SeedSelect, initialized with NAO using DDIM inversion, ReNoise, ExactDPM, and GNRI, as shown in Table 3). We followed the evaluation protocol of (Samuel et al., 2024; 2023) on ImageNet1k classes arranged by their prevalence in the LAION2B dataset (Schuhmann et al., 2022). The results, summarized in Table 3, demonstrate that our inversion method significantly boosts performance, both in classification accuracy and FID, compared to other methods. Furthermore, our inversions yield a more precise and effective initialization point for SeedSelect (Samuel et al., 2024), resulting in notably quicker convergence without compromising performance, in all categories (head, medium, and tail) along with a high gap at tail.
+
+# 6 SUMMARY
+
+Image inversion in diffusion models is vital for various applications like image editing, semantic augmentation, and generating rare concept images. Current methods often sacrifice inversion quality for computational efficiency, requiring significantly more compute resources for high-quality results. This paper presents Guided Newton-Raphson Inversion (GNRI), a novel iterative approach that balances rapid convergence with superior accuracy, execution time, and memory efficiency. Using GNRI opens the door for high-quality real-time image editing due to its efficiency and speed. GNRI requires no model training, fine-tuning, prompt optimization, or additional parameters, and is compatible with all pre-trained diffusion and flow-matching models with deterministic schedulers.
+
+# ACKNOWLEDGMENTS
+
+We would like to express our sincere gratitude to Prof. Nir Sochen for the intriguing discussion during the course of this study.
+
+# REFERENCES
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+Daiki Miyake, Akihiro Iohara, Yu Saito, and Toshiyuki Tanaka. Negative-prompt inversion: Fast image inversion for editing with text-guided diffusion models. arXiv preprint arXiv:2305.16807, 2023.  
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+Axel Sauer, Dominik Lorenz, Andreas Blattmann, and Robin Rombach. Adversarial diffusion distillation, 2023.  
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+
+# Appendix
+
+# A NEWTON METHOD FOR MULTIVARIABLE SCALAR FUNCTION
+
+In this section, we present the formulation of Newton's method for zero crossing of a multi-variable scalar function. For a vector  $\mathbf{x} \in \mathbb{R}^D$  and a function  $f(\mathbf{x}) : \mathbb{R}^D \to \mathbb{R}$  we are looking for the roots of the equation  $f(\mathbf{x}) = 0$ . Assuming that the function  $f$  is differentiable, we can use Taylor expansion:
+
+$$
+f (\mathbf {x} + \delta) = f (\mathbf {x}) + \nabla f \delta^ {T} + o (| | \delta | | ^ {2}) \tag {A1}
+$$
+
+For every  $\delta, \mathbf{x} \in \mathbb{R}^D$ . The idea of Newton's method in higher dimensions is very similar to the one-dimensional scalar function; Given an iterate  $\mathbf{x}^k$ , we define the next iterate  $\mathbf{x}^{k+1}$  by linearizing the equation  $f(\mathbf{x}^k) = 0$  around  $\mathbf{x}^k$  as above, and solving the linearized equation  $f(\mathbf{x}^k + \delta) = 0$ . Dropping the higher orders we then solve the remaining linear equation for  $\delta$ . In fact, in the scalar case, the solution for  $\delta$  is not unique, with each solution potentially determining a different direction of propagation of the iterate  $\mathbf{x}^k$  in the high-dimensional input space. The optimal direction cannot be known in advance, and some have proposed selecting the propagation direction by imposing constraints, such as minimizing  $\delta$  (Polyak & Tremba, 2020). We adopt a simple, non-sparse, yet effective solution, that allows the scheme flexibility in deriving the direction of propagation:
+
+$$
+\delta = - \frac {1}{D} \left[ \frac {1}{\partial f _ {1}}, \frac {1}{\partial f _ {2}}, \dots , \frac {1}{\partial f _ {D}} \right] f \left(\mathbf {x} ^ {k}\right) \tag {A2}
+$$
+
+where  $x_{i}$  indicates the  $i$ -th component of vector  $\mathbf{x}$ . The above can be easily proven as a solution by substituting  $\delta$  into Eq. A1, dropping the higher orders setting it equal to zero. Using the relation,  $\delta = \mathbf{x}^{k + 1} - \mathbf{x}^k$  we get the final iterative scheme:
+
+$$
+x _ {i} ^ {k + 1} = x _ {i} ^ {k} - \frac {1}{D} \frac {f \left(\mathbf {x} ^ {k}\right)}{\frac {\partial f}{\partial x _ {i}} \left(\mathbf {x} ^ {k}\right)} \tag {A3}
+$$
+
+Note that this equation is component-wise i.e. each component is updated separately, facilitating the solution.
+
+# B DISCUSSION ON THE NEWTON-RAPHSON METHOD FOR FUNCTIONS WITHOUT ZERO CROSSINGS
+
+Note that our objective function in Eq. 9 is a non-negative function. When the target function  $\mathcal{F} \geq 0$  and does not have zero crossings,
+
+the NR procedure may converge to a (non-zero) minima. A common example is the function  $f(x) = x^2 + 1$  that has no real roots, but NR converges to its minimum. Note that while the Newton-Raphson method is typically used to find zero crossings of a function, our approach applies it to drive convergence toward local minima of the objective function (being locally convex around this point). The analysis of this type of convergence within the Newton-Raphson framework is left for future numerical analysis research.
+
+# C GNRI SCHEME CONTRACTION MAPPING
+
+Fixed-point iterations are guaranteed to converge if the operator is contractive in the region of interest. Formally, a mapping  $T$  is contractive on its domain if there exists  $\rho < 1$  (also called Lipschitz constant) such that  $\| T(x) - T(y) \| \leq \rho \| x - y \|$ ,  $\forall (x, y) \in X$ , where  $\| \cdot \|$  denotes a norm. In the following we will demonstrate empirically that our GNRI scheme introduces contraction mapping, a property needed for convergence of the scheme. Contractive mappings ensure that each iteration moves closer to the limit, with the distance shrinking at a predictable rate. The rate of contraction (the Lipschitz constant) can be used to give bounds on how fast the error decreases, helping to assess the efficiency of the scheme. Next, we show empirically that, the mapping is contractive in our region of interest.
+
+![](images/732740b7938f947da1038b7d78801362fe4f4cdc4a73f50502c46f3d4e190fd6.jpg)  
+Figure C1: Contraction Mapping. Contraction mapping for two fast diffusion models are shown (in log-scale), at each time-step. The scheme converges in 2-3 iterations with a notable contraction occurring in the first iteration, assisting the rapid convergence.  $t = 0$  is close to the image side and  $t = 3$  close to seed.
+
+![](images/0264be2024771db2fcc03fcbdb9393a757ac9ee3faeebab6cb523ada26b732d3.jpg)
+
+According to Eq. 10 the GNRI scheme for the solution  $\mathcal{F}(z) = 0$  (defined in Eq. 9), at each diffusion time-step, is given by:
+
+$$
+z _ {i} ^ {k + 1} = z _ {i} ^ {k} - \frac {1}{D} \frac {\mathcal {F} \left(z ^ {k}\right)}{g _ {i} \left(z ^ {k}\right)} \tag {C1}
+$$
+
+where  $g_{i} \coloneqq \frac{\partial\mathcal{F}}{\partial z_{i}}$  is the partial-derivative of  $\mathcal{F}$  with respect to the variable  $z_{i}$ , while  $i \in [0,1,\dots,D]$  indicates  $z$ 's components. We can rewrite the NR iteration as the fixed-point iteration  $\tilde{f}(z) = z$  by setting  $\tilde{f}(z)$  as:
+
+$$
+\tilde {f} _ {i} \left(z ^ {k}\right) := z _ {i} ^ {k} - \frac {1}{D} \frac {\mathcal {F} \left(z ^ {k}\right)}{g _ {i} \left(z ^ {k}\right)} \tag {C2}
+$$
+
+Note that the fixed point of the newly introduced  $\tilde{f}$  is the solution of  $\mathcal{F}(z) = 0$ .
+
+Let us now define a new residual function:
+
+$$
+\tilde {r} \left(z ^ {k}\right) := \left| \left| \tilde {f} \left(z ^ {k}\right) - z ^ {k} \right| \right| \tag {C3}
+$$
+
+By substitution we get:
+
+$$
+\tilde {r} \left(z ^ {k + 1}\right) = \left| \left| \tilde {f} \left(z ^ {k + 1}\right) - z ^ {k + 1}\right) \right| | = \left| \left| f \left(z ^ {k + 1}\right) - \tilde {f} \left(z ^ {k}\right) \right| \right| \tag {C4}
+$$
+
+$$
+\tilde {r} (z ^ {k}) = | | \tilde {f} (z ^ {k}) - z ^ {k} | | = | | z ^ {k + 1} - z ^ {k} | |
+$$
+
+Therefore contraction mapping requirement for  $\tilde{f}$ :  $||\tilde{f}(z^{k+1}) - \tilde{f}(z^k)|| \leq \rho ||z^{k+1} - z^k||$  can be written as:
+
+$$
+\tilde {r} \left(z ^ {k + 1}\right) \leq \rho \tilde {r} \left(z ^ {k}\right) \tag {C5}
+$$
+
+and the contraction mapping is satisfied if  $\tilde{r}$  follows an exponential decay  $(r(z^k)\leq \rho^k r(z^0))$ , with some  $\rho < 1$ . Note that  $\tilde{r} (z^{k})$  given in Eq. C3 can be computed at each iteration step. In Fig. C1 we plot  $\tilde{r} (z^k)$ , tested on 5,000 image caption pairs from the COCO dataset (the same as used in Sec. 5.1) showing that  $\tilde{r} (z^k)$  indeed decays exponentially (linearly in log-scale), till convergence. This behaviour therefore implies that GNRI iteration is a contraction mapping. Note that our contraction mapping does not converge to zero, namely the fixed point, since our objective function  $\mathcal{F}$  may not have an exact solution. Computing contraction rate  $\rho$  for GNRI shows  $\rho \approx 0.50$  (at the first iteration) while  $\rho \approx 0.90$  for FPI indicating the faster convergence of GNRI.
+
+Note that we further show in the main paper the convergence of the equation residual  $\hat{r} = ||f(z_t) - z_t||$  (EQ. 7) as an indicator for convergence toward the inversion solution.
+
+![](images/d16b02081148c6a693b3c13c941202a35af612ed9121b2e61066836f542abd90.jpg)  
+"Your guess is as good as mine as to what these objects are."
+
+![](images/86840083998ee2e5984d29679c718e06d77d69b04ab4ab71f2f7e60aa0959c0c.jpg)  
+"A thing is in the outline and it shows up like something"
+
+![](images/2440dcfd91288d1b4ac09f950f20b57018bb568467a337fe28ead74746a67ad5.jpg)  
+"Food on a train with a pie and some vegetables."
+
+![](images/e3256935f6a0f611ab17543425a714e222b63b20b7a72754ce30e3215efb42ac.jpg)  
+Figure D1: Failure cases: GNRI fails to converge where the prompt and image do not align.  
+Figure D2: The influence of  $\lambda$  on reconstruction performance. We observe that employing no regularization  $(\lambda = 0)$  leads to poor reconstruction, while  $\lambda = 0.1$  typically yields the highest reconstruction accuracy. As  $\lambda$  increases, reconstruction accuracy declines, possibly because assigning excessive weight to the prior undermines the root-finding objective.
+
+# D FAILURE CASES
+
+Here we provide information about an analysis to find failure cases for convergence. We ran inversion and reconstruction, in scale, on COCO2017 (118k caption-image pairs) and found that in  $95.4\%$  of cases, GNRI successfully converged to a solution. Specifically, residuals went down from  $\sim 1$  (for DDIM) to  $< 10^{-4}$  indicating convergence, and the PSNR was  $>25.7$  indicating good solutions. The remaining  $4.6\%$  were samples with incorrect captions, see Fig. SD1. These results imply that lack of convergence may indicate text and image miss-alignment which we consider for future work.
+
+# E PRIOR TRADE-OFF PARAMETER
+
+Figure D2 shows the impact of the weight parameter  $\lambda$  on the reconstruction accuracy. We observe that using no regularization ( $\lambda = 0$ ) results in poor reconstruction, whereas setting  $\lambda = 0.1$  achieves the highest reconstruction accuracy, underscoring the importance of the prior in the objective function. There is a gradual decrease with larger  $\lambda$  values, where at  $\lambda = 1$  the PSNR starts to decline faster, likely because placing too much emphasis on the prior compromises the residual root-finding objective.
+
+![](images/a0ae10df48b86cfdfffc0bb5d3fffed082db1bcbb497e3b2689d1966dc669d26.jpg)  
+Figure E1: (Left) Inversion Results: Mean reconstruction quality (y-axis, PSNR) and runtime (x-axis, seconds) on the COCO2017 validation set. Our method achieves high PSNR while reducing inversion-reconstruction time by a factor of  $\times 4$  to  $\times 14$ . (Right) Evaluation of editing performance: GNRI achieves superior CLIP and LPIPS scores, indicating better compliance with text prompts and higher structure preservation.
+
+![](images/f2f591734e4e5cf5e1923f2a3abe8153ad4cbf794b51502834f3890103f93697.jpg)
+
+# F SEED EXPLORATION & RARE CONCEPT GENERATION
+
+![](images/a6b9f581ba2dca40982559790739af013e93491cca9f6008a5d6ad27750b05c9.jpg)  
+Figure F1: Generating rare concepts based on a few examples with the method of (Samuel et al., 2023) that heavily depends on the diffusion-inversion quality. In our comparison, we evaluate GNRI alongside ReNoise (refer to the discussion in the main paper). ReNoise frequently struggles to produce a realistic representation of certain objects (such as Oxygen-mask or Patas Monkey) or an accurate depiction of specific concepts (like Tiger-cat or Pay-phone). However, the use of GNRI rectifies these issues in the results. For a detailed quantitative comparison, please refer to the main paper. See the main paper for a quantitative comparison.
+
+Figure SF1 further displays results for the rare-concept generation task introduced in (Samuel et al., 2024; 2023). The objective is to enable the diffusion model to generate concepts rarely seen in its training set by using a few images of that concept. Both methods in (Samuel et al., 2024; 2023) utilize diffusion inversion for this purpose. In the main paper, we presented experiments showcasing the impact of our new inversion process on the outcomes of (Samuel et al., 2024; 2023).
+
+![](images/7c4b8e62841e7aa01a69cb48efc5eec1a20576837b30d8544f8896740415af3a.jpg)  
+Figure F2: Qualitative comparison for image editing on the PIE-benchmark. Illustrates how alternative approaches may struggle to preserve the original structure or fail to edit the image, while GNRI succeeds in editing the image properly.
+
+We provide qualitative results based on NAO (Samuel et al., 2023) centroid computation when utilizing GNRI, contrasting them with those obtained through DDIM inversions. The illustrations demonstrate that GNRI is capable of identifying high-quality centroids with correct semantic content compared to centroids found by DDIM.
+
+# G ADDITIONAL QUALITATIVE RESULTS ON PIE-BENCHMARK
+
+Additional quantitative results on the PIE-Benchmark. Figure F2 provides quantitative results on the PIE-Benchmark using Flux.1. It highlights how alternative methods often fail to maintain the original structure or miss key elements specified in the target prompt, whereas GNRI effectively achieves accurate and proper image editing.
+
+Quantitative results with Stable Diffusion 2.1. We present a quantitative comparison between GNRI and baseline methods, utilizing latent diffusion SD2.1 (Rombach et al., 2022). Figure E1(a) illustrates the image inversion-reconstruction performance, while Figure E1(b) highlights the editing performance, evaluated using LPIPS and CLIP-score metrics. The results demonstrate that GNRI surpasses all other inversion methods in terms of reconstruction quality, which subsequently improves editing performance.
+
+![](images/216851f6ae09644fc4f645c74c33d60eefdea1488a0c71f65ea6da23724db1b4.jpg)  
+Input Image
+
+![](images/0026688dce31ce2f773b5519f9021ddfb74e997489f9e0ecd9ec826e29befad6.jpg)  
+Null Text  
+EDICT
+
+![](images/55ac5c4c475f5ba0b60c39db5f0e3500ff33072ddc2852c7988b1365c570cc8b.jpg)
+
+![](images/b78d64d5d0a7e7a59b96d6ff53e4358dc2dc1ad060754fa00ec43a7ddf0ef936.jpg)
+
+![](images/0145565e3ec231e463ea2375af4e3cf6f8d2ce9e5d2e11488ece43dda557f234.jpg)  
+Ours
+
+![](images/1e1243335463d13155decfebdf48a2af38a1926e98fa1b6fb29682beacc98018.jpg)
+
+![](images/f85ef37b0d7ab23a34bcc9a8dcb3db1ca38e2eb88013886b5ef52c41ab4cfc25.jpg)  
+"Wooden cutting board with a slice of pizza bread"
+
+![](images/d5b60bbbb0ee786c103713c546ef805dce617520ee945c0999848c8531beeb99.jpg)  
+"An angry man with a puffy hair"
+
+![](images/ac534f90d8570d709c940cfd873614bd1da7894083fe4de0dde7ad9ceaef7d0f.jpg)
+
+![](images/62bf2223a01bd9fe268123ab5f45ea130be8285c4b5bd56c3c7163dde53f480a.jpg)
+
+![](images/08bbf9a6c1367430e51b34336281093bfaae3802c76f9d0130e691ffbcd9ff14.jpg)
+
+![](images/3cf73213c755d04657767ea5e492dc11b8641270cd6afa86605c36e43825b9fd.jpg)  
+"A lady with bananas oranges on her head"
+
+![](images/264e6b3961d5f301274d2ac09b433da9077cd93bd245d92760da09f0a92c57d1.jpg)
+
+![](images/54bd75a32f77b27f6eea9c536490aaaf5d0d036c05cfc860ae896facac97db7d.jpg)
+
+![](images/8da1ac54c603022e11207122c907ce649fdccffc022acaeca476a3ff12a415a6.jpg)
+
+![](images/b76713c73704ee8c592f7ce750a42b6f984cefe55d2dbd70ebdaa7b84c5cb84d.jpg)
+
+![](images/35e32bc38d0c6fc8e449e072ed912780ddb68bcffb0c18356146d706b21ad10d.jpg)
+
+![](images/e788a1a2595eb929ca329101e2da6c2af90c5e03cdf6e760170dd31220ed8e3b.jpg)
+
+![](images/2942f125ad2dd5a8cefc30aa1c6aedf132521f61098caa7ec55e4a1584a8d401.jpg)
+
+![](images/b03e03c15291fc8d4df91c0f773e64bfb1ef391beeb830437c09fc0fad13acc5.jpg)
+
+![](images/bf732ee1be4ae7a725837f1eccb4a7d9fa0db90df9d4c9747f9d0ded3f645133.jpg)
+
+![](images/e7cde0b737c1abcd401d6ae18867494477564e401dfe0ab5ead18a72763ec25c.jpg)  
+"People walking on sidewalk with yellow red fire hydrant"  
+"A dog pants next to a bowl of water food"
+
+![](images/094e9452eb5ca588ab952d76acd82940d19d06d5c2f1e0163057f93e87652689.jpg)
+
+![](images/5bbdb8c8b46ec7f5213fe46efdb2de69c7689ffc062fa7cf7067205e43061813.jpg)
+
+![](images/02d936c26e1823974111c388098e8e37dc314b22fe8c9b4612bcffdaf26d18c6.jpg)
+
+![](images/e20b991f0d6bd0897e326ac99ee15528d7569dcccd459fad2f74270a25b9ba9b.jpg)
+
+![](images/856496be2dd41ff7b1b195857def82f94b613c0008918a303da3b1021284cea7.jpg)
+
+![](images/f82b2160f4a53d53f91274a7d501593d7fe3371f615cc8471a24579f7c85a173.jpg)  
+"An oreo cake on a table"
+
+![](images/21536e0c427da55575da62981346eb8635c12db6131ceb240743505bbb859d3c.jpg)
+
+![](images/31b0a6f3028d97879cfa459d67f21e64668f25dc851121db3dc3b1ae10c6e1ef.jpg)
+
+![](images/a8ec90f778b610e659c061226ab5508a09f797348d07419b9f02051c6a883d99.jpg)  
+Figure F3: Qualitative comparison for image editing: Illustrates how alternative approaches may struggle to preserve the original structure or overlook crucial components specified in the target prompt, while GNRI succeeds in editing the image properly.
+
+![](images/96228eef66578e820a33193b2a679f53ef131dfd9b1fcd2e5f123e2784bdfbb7.jpg)
+
+![](images/55973fdafe4b4f4452ac083a8318a8de933bc1cf71bd66c086ecf7da7a9c1504.jpg)  
+"A plate with a cake hat"
+
+![](images/6e6e5bb85efd82ba5a4d68b00b877a9a596bfa992199d150109359549c404ae6.jpg)
+
+![](images/7eaab71c02e891a4f5c5b706d3d0d893d327c28db1f86ff2a7b77117d210e8f0.jpg)
+
+![](images/0117daa53bd1acf3a1ce6995132310e5fc71ddddccf9c47e94733cea42dfed13.jpg)  
+Input Image
+
+![](images/a38343ad47a32400d352a51d359790554af6627b6571f14322fb9b0f6842b538.jpg)  
+Input Caption: "A cat sitting on a chair"  
+cat  $\rightarrow$  giraffe
+
+![](images/5840040b25bbdbd9ef47bbe1c8d2dc60f1883e534f530d1dafeffb42ab0b7d36.jpg)  
+cat→boy  
+... with sunglasses
+
+![](images/1fa7f91a3a32d271ee403a252a6e6f4844bfa41fc0b8c6dcade3699d7507f2b9.jpg)  
+chair→sofa
+
+![](images/3aa2f4a0ffdcd15765da79358e896e93148fbf212a7b69b36c33af9ccd954b05.jpg)  
+Input Image  
+Figure F4: Various editing with same input: Note the GNRI capability in both subtle and extensive changes as one would expect from the particular prompt change.
+
+![](images/b6a08b72444d28f180bfd8b18086f468ca008d667e65a88ed1364fa093be28e6.jpg)  
+Input Caption: "An angry man"  
+angry→happy
+
+![](images/47479d6e134145802d3c6ceb59d26aa0156cd3c8dc19b1b42f287d986a82f600.jpg)  
+... with helmet  
+... with joker mask  
+... with silky hair
+
+Table H1: PSNR vs. NFE results using SDXL Turbo on the COCO validation set. Our method achieves the highest PSNR with the fewest NFEs, compared to other methods.  
+
+<table><tr><td></td><td>PSNR</td><td>NFE</td></tr><tr><td>ReNoise Garibi et al. (2024)</td><td>18.1</td><td>36</td></tr><tr><td>ExactDPM Hong et al. (2024)</td><td>20.0</td><td>40</td></tr><tr><td>TurboEdit Wu et al. (2024)</td><td>22</td><td>8</td></tr><tr><td>GNRI (Ours)</td><td>23.9</td><td>6</td></tr></table>
+
+Table H2: Inversion comparison with Edit Friendly on SDXL Turbo (COCO validation set). While DDPM achieves higher PSNR through stochastic inversion, our deterministic approach (GNRI) provides competitive results with significantly faster inversion time (0.521s vs 32.3s).  
+
+<table><tr><td></td><td>PSNR</td><td>Inv Time</td></tr><tr><td>EditFriendly Huberman et al. (2023)</td><td>26.1</td><td>32.3</td></tr><tr><td>GNRI (Ours)</td><td>23.9</td><td>0.521</td></tr></table>
+
+# H ADDITIONAL QUALITATIVE AND QUANTITATIVE RESULTS -COMPARISON TO BASELINES
+
+In this section, we present additional qualitative comparisons involving GNRI and baseline methods.
+
+Figures F3 provide further comparisons for real-image editing, using SD2.1. These figures illustrate how alternative approaches may struggle to preserve the original structure or overlook crucial components specified in the target prompt. For example, n the first row of Figure F3, GNRI correctly transforms the pizza into bread, while other methods generate a bad looking bread. Row three shows an example of GNRI success (bananas to oranges), where other alternatives totally fail.
+
+Figure F4 shows the outcomes of various edits on the same image, providing additional confirmation that inversion with GNRI yields modification of the pertinent parts in the image while maintaining the original structure.
+
+Table H1 provides PSNR vs NFE (Neural Function Evaluations) results using SDXL Turbo on the COCO validation set. Our method requires 6 NFEs, calculated as 4 steps with an average of 1.5 NR iterations per step (some steps needing 1 iteration, others 2). As shown in the table below, our approach achieves the highest PSNR while using the fewest NFEs compared to other methods.
+
+Table I1: Comparison of Anderson Acceleration and the Guidance term for out-of-distribution COCO samples. A higher likelihood is better.  
+
+<table><tr><td></td><td colspan="2">mean likelihood qzt|zt-1)</td><td>PSNR</td><td>Inv Time</td></tr><tr><td>Anderson Acceleration</td><td colspan="2">0.34</td><td>18.3</td><td>Inf</td></tr><tr><td>Guidance Regularizer (Ours)</td><td colspan="2">0.82</td><td>25.8</td><td>0.4 s</td></tr></table>
+
+Table J1: Memory usage and convergence time for different methods on A100 and RTX 3090Ti.  
+
+<table><tr><td></td><td colspan="2">A100 (VRAM 40GB)</td><td colspan="2">RTX 3090Ti (VRAM 24GB)</td></tr><tr><td></td><td>Memory</td><td>Inv Time</td><td>Memory</td><td>Inv Time</td></tr><tr><td>AIDI Pan et al. (2023)</td><td>15GB</td><td>5 sec</td><td>16 GB</td><td>13 sec</td></tr><tr><td>ExactDPM Hong et al. (2024)</td><td>30GB</td><td>17 sec</td><td>20 GB</td><td>25 sec</td></tr><tr><td>GNRI (Ours)</td><td>28GB</td><td>0.4 sec</td><td>19 GB</td><td>1 sec</td></tr></table>
+
+Table H2 compares inversion performance with Edit Friendly Huberman et al. (2023) using SDXL Turbo on the COCO validation set. Edit Friendly achieves the maximum PSNR possible given the VAE distortion, because it does a stochastic DDPM inversion, saving additional noise matrices at every step and "overfitting" the inverted image. As a result, it requires extensive inversion time (more than 30 seconds), whereas our GNRI reaches a relatively high PSNR in under a second. Our approach solves deterministic inversion (DDIM or Euler) making it suitable for applications like interpolation and rare concept generation, where DDPM is not feasible.
+
+# I COMPARISON TO ANDERSON ACCELERATION
+
+The guiding term introduced in Sec 4.2 might appear similar to Anderson acceleration used in Pan et al. (2023), which aims to allow more gradual changes from previous estimates. However, One key difference is that our guidance term introduced in Eq.(9) does not pull  $z_{t}$  toward  $z_{t-1}$  but towards the mean  $\mu_{t}$ . By that, it directs the Newton-Raphson method toward solutions having a high likelihood, using "side-information" about the noise. In contrast, Anderson acceleration relies on previous values of the implicit function  $f(z_{i})$  and residuals but lacks awareness of the latent distribution (see for example, line 13 in Algorithm 1 of (Pan et al., 2023)).
+
+In GNRI, keeping  $z_{t}$  close to  $\mu$ , encourages solutions that are more "in-distribution" and aligned with the diffusion model. In other words, if  $z_{t-1}$  starts far out-of-distribution, keeping  $z_{t}$  close to  $z_{t-1}$  is likely to yield worse solutions compared to guiding  $z_{t}$  toward inputs that the model expects to receive.
+
+We illustrate this effect in table I1. We analyzed the inversion of 300 COCO images that the model would consider to be non-typical by selecting the samples with the lowest likelihood under  $q(z_0)$  (Eq.(1)). The table shows that GNRI drives the latents toward significantly higher likelihood solutions (mean  $q(z_{t}|z_{t - 1})$ ), leading to substantially better reconstructions (higher PSNR) than Anderson acceleration.
+
+More broadly, our guidance term could potentially be applied to other methods, such as fixed-point iteration approaches like those proposed by Pan et al. It would also be interesting to find methods that can include more domain knowledge into the inversion process.
+
+# J MEMORY USAGE
+
+We compared the memory usage of our approach, with AIDI Pan et al. (2023) and ExactDPM (Zhang et al. (2023)) on a A100 GPU (40 GB VRAM) and a RTX 3090Ti GPU (24GB VRAM). Results are given in table J1. Our approach consumes about the same GPU memory as ExactDPM, however, it converges faster about x43 times than ExactDPM on an A100 GPU, and x25 faster on an RTX 3090 GPU.
+
+# K MULTI OBJECT EDITING
+
+We conducted several qualitative experiments to evaluate the capabilities of our method in editing multiple objects simultaneously. Our results demonstrate that the method can effectively handle complex scenes, allowing for the independent manipulation of several objects without compromising the quality of individual edits. As illustrated in Figure K1, our GNRI model successfully performs multi-object transformations while maintaining scene coherence and natural composition - from converting multiple dogs to cats while preserving the scene layout, to simultaneously transforming different food items (steak to apple, eggs to pancakes) on a plate. These results underscore GNRI's robustness in handling multiple concurrent edits while maintaining high visual quality and semantic coherence across the transformed scenes.
+
+![](images/63c8df585d4162e313e4f15bebf17e8e715dbd60a406e0e167c2923af5950801.jpg)  
+Figure K1: Multi-object editing capabilities of GNRI. Top: Simultaneous transformation of multiple dogs into cats. Bottom: Concurrent food item conversion on a plate (steak  $\rightarrow$  apple, eggs  $\rightarrow$  pancakes), demonstrating preservation of scene composition and context.
\ No newline at end of file
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+# LIGHTWEIGHT PREDICTIVE 3D GAUSSIAN SPLATS
+
+Junli Cao $^{1,2}$  Vidit Goel $^{2}$  Chaoyang Wang $^{2}$  Anil Kag $^{2}$  Ju Hu $^{2}$   
+Sergei Korolev $^{2}$  Chenfanfu Jiang $^{1}$  Sergey Tulyakov $^{2}$ $^{1}$ University of California, Los Angeles  $^{2}$ Snap, Inc.
+
+![](images/0f9027a78164871953af12ba5ed9f4869dad23fbeed8758b212a8d06796e783e.jpg)  
+Figure 1: Top: We show point clouds of the Garden scene (Barron et al., 2022a) obtained using different methods, where we feature the smallest number of points to store. Bottom: Images rendered using the compared methods. Ours shows the best PSNR. We magnify a region highlighted with blue, showing that despite significantly smaller storage requirements, we achieve the highest fidelity and can reconstruct the detailed structure of the image. Zoom-in for greater detail.
+
+# ABSTRACT
+
+Recent approaches representing 3D objects and scenes using Gaussian splats show increased rendering speed across a variety of platforms and devices. While rendering such representations is indeed extremely efficient, storing and transmitting them is often prohibitively expensive. To represent large-scale scenes, one often needs to store millions of 3D Gaussian, which can occupy up to gigabytes of storage. This creates a significant practical barrier, preventing widespread adoption on resource-constrained devices. In this work, we propose a new representation that dramatically reduces the hard drive footprint while featuring similar or improved quality when compared to the standard 3D Gaussian splats. This representation leverages the inherent feature sharing among splats in the close proximity using a hierarchical tree structure, with which only the parent splats need to be stored. We present a method for constructing tree structures from naturally unstructured point clouds. Additionally, we propose the adaptive tree manipulation to prune the redundant trees in the space, while spawn new ones from the significant children splats during the optimization process. On the benchmark datasets, we achieve  $20 \times$  storage reduction in hard-drive footprint with improved fidelity compared to the vanilla 3DGS and  $2 \times -5 \times$  reduction compared to the exiting compact solutions. More importantly, we demonstrate the practical application of our method in real-world rendering on mobile devices and AR glasses in our Webpage.
+
+# 1 INTRODUCTION
+
+Gaussian Splatting (3DGS)-based methods are taking the graphics and vision communities by a storm (Luiten et al., 2023; Wu et al., 2023; Yang et al., 2023). They strike the right balance between high-fidelity rendering, fast convergence, and efficient inference (Kerbl et al., 2023). The latter two benefits make 3DGS-based methods superior to Neural Radiance Fields (NeRFs)-based
+
+techniques (Mildenhall et al., 2020; Martin-Brualla et al., 2021; Barron et al., 2022b). Indeed, while NeRFs (Barron et al., 2022a) show high-fidelity renderings too, apart from several exceptions (Cao et al., 2023; Wang et al., 2022; Chen et al., 2023b; Müller et al., 2022a), their training and inference time is often prohibiting real-time and edge-based applications. 3DGS-based approaches represent a 3D scene using an explicit, point-based representation (Aliev et al., 2020). The 3D Gaussians are efficiently rasterized to 2D images, with much faster rendering than neural volumetric rendering approaches (Kerbl et al., 2023). However, to represent sophisticated geometry and texture, especially for large-scale scenes, a significant amount of splats along with their attributes need to be stored, which can amount to even gigabytes of storage.
+
+In a world of connected devices, real-time experiences and applications, this storage requirement imposes a heavy toll on the hard-drive and the transmission bandwidth. Hence, several initial solutions have been proposed to reduce the storage for 3DGS, such as incorporating a sparse voxel grid (Lu et al., 2023) or applying more aggressive pruning of the 3D points (Fan et al., 2024; Lee et al., 2024). Yet, existing studies still suffer either from large storage requirements (Lu et al., 2023) or inferior rendering quality compared to 3DGS (Fan et al., 2024; Lee et al., 2024).
+
+In this work, we present a lightweight hierarchical Gaussian splats representation that takes advantage of the spatial relationships among unstructured and isolated splats, offering improved rendering quality while significantly reducing storage requirements. Intuitively, splats in close proximity exhibit similar geometry and texture. Therefore, we leverage feature sharing among nearby splats and propose structuring them into a hierarchical tree, where the parent splats are employed to neural-predict splats that share similar features. We call these neural-predicted splats the children splats. Note that children splats do not have to be stored and can be neural-predicted on-the-fly instead. We use hash-grid (Müller et al., 2022b) to encode the offsets that are used to estimate the 3D locations of children splats. In addition, within the same hash grid, we first query the features of both the parent and children splats and apply an attention-based mechanism to attend to them. This attention is crucial for facilitating feature sharing within the tree. The attended features are then input into a shallow MLP to predict the Gaussian attributes. We opt for the hash-grid due to its ability to facilitate feature sharing in close proximity by interpolating spatially adjacent feature vectors. Our representation is independent of grid-based structures; any representation that encourages feature sharing can be utilized (e.g., K-plane (Fridovich-Keil et al., 2023)).
+
+To build such tree structures, we first allow every point obtained from SfM to be considered as a parent splat, and be used to predict its children splats. Since the splats in our representation are structured and treated as a cohesive unit, we further introduce the Adaptive Tree Manipulation(ATM) module to manage the tree during the optimization process. Specifically, we do not impose a limit on the depth of the tree. This means that a children splat can serve as a parent in the next optimization iteration and has its own children splats if it is deemed significant. Additionally, insignificant parent splats are pruned along with their insignificant children. Note that significant children are promoted to parent regardless of the significance of their parent. For instance, an insignificant parent may be removed in the next optimization iteration, but it can still have significant children that are promoted to parent. This flexible tree manipulation enables certain areas with complex geometry to include more splats for more accurate modeling.
+
+Fig. 1 shows the Garden scene (Barron et al., 2022a) reconstructed by the standard Gaussian Splats (Kerbl et al., 2023), Compact GS (Lee et al., 2024), Light GS (Fan et al., 2024) and the proposed approach. First, we observe a significantly reduced density of points in the point cloud reconstructed by our approach. This, and the predicting of the attributes instead of storing them, significantly reduces the storage requirement for our method. Second, we show improved PSNR scores and visual quality, when we zoom-in into the details of the rendered images. We summarize our contributions as follows:
+
+1. We propose a hierarchical tree structure to model the inherent spatial relationships among splats and an attention mechanism to enhance the relationship within the hierarchy.  
+2. We propose Adaptive Tree Manipulation in conjunction with the hierarchical representation to effectively refine the tree for improved modeling.  
+3. Our representation achieves  $20 \times$  reduction on average in hard-drive footprint, with improved PSNR and comparable SSIM and LPIPS comparing to 3DGS and  $2 \times -5 \times$  storage reduction comparing the exiting works. Additionally, we showcase the practical real-world rendering applications of our method on mobile devices and AR glasses.
+
+# 2 RELATED WORK
+
+Novel View Synthesis. Research on rendering scenes from unseen viewpoints with photorealism has evolved over several decades (Greene, 1986; Chen & Williams, 2023; Levoy & Hanrahan, 2023; Buehler et al., 2023; Srinivasan et al., 2019). Traditional approaches typically rely on explicit depth estimation to warp pixels for generating novel views (Kalantari et al., 2016; Penner & Zhang, 2017; Choi et al., 2019; Riegler & Koltun, 2021). However, the accuracy of depth estimation algorithms is critical, and handling disocclusions during rendering adds complexity. An alternative approach involves Multiplane Images (MPI) (Zhou et al., 2018; Srinivasan et al., 2019; Flynn et al., 2019), which learn a representation associating objects within the scene with fronto-parallel layers. This structured representation facilitates efficient rendering from different viewpoints while preserving depth relationships and occlusions. More recently, Neural Radiance Fields (NeRF) (Mildenhall et al., 2020) have gained popularity for their ability to achieve highly realistic rendering, even in scenarios involving complex view-dependent lighting effects such as transparency and reflectance. However, the weakness of NeRF lies in its volumetric rendering formulation, which necessitates sampling a large number of points per ray to render a single pixel. This high computational cost limits the usage of NeRF for real-time or on-device applications. While efforts to reduce computational requirements for volumetric rendering have been a focus of recent research (Liu et al., 2020; Neff et al., 2021; Garbin et al., 2021; Reiser et al., 2021; Lindell et al., 2021; Yu et al., 2021; Müller et al., 2022a; Fridovich-Keil et al., 2022; Lombardi et al., 2021; Cao et al., 2023; Gupta et al., 2024), point-based rendering, particularly 3D Gaussian Splatting (3DGS) (Kerbl et al., 2023), presents another promising direction for real-time view synthesis.
+
+Efficient Representation for 3D Gaussian Splatting. Despite the benefits of 3DGS (Kerbl et al., 2023), the disadvantages of bulky storage are noteworthy. As a result, several approaches (Fan et al., 2024; Lee et al., 2024; Lu et al., 2023; Girish et al., 2023; Niedermayr et al., 2024; Morgenstern et al., 2023; Navaneet et al.) have been proposed for compressing 3DGS. Several compression techniques have been explored such as pruning the redundant Gaussian (Fan et al., 2024; Lee et al., 2024) and utilizing codebooks (Fan et al., 2024; Lee et al., 2024; Niedermayr et al., 2024; Navaneet et al.). LightGS (Fan et al., 2024) introduces a point pruning and recovery process to minimize redundancy in Gaussian splats, utilizes distillation and pseudo-view augmentation to distill spherical harmonics to a lower degree, and employs quantization to further reduce storage. While LightGS achieves considerable storage reduction, it results in noticeable fidelity degradation compared to the original Gaussian splatting due to quantization. CompactGS (Lee et al., 2024) proposes using a grid-based neural field to implicitly represent view-dependent colors rather than explicitly storing spherical harmonics per point, offering promising storage efficiency without significant fidelity loss. Eagles (Girish et al., 2023) utilizes quantized embedding to quantize the per-point attributes and pruning strategy to remove redundant Gaussian, leading to lower storage memory. ScaffoldGS (Lu et al., 2023) exploits the spatial feature sharing by distributing local splats using anchor points, reparameterizing splats positions relative to these anchors to enable anchor-based point growing and pruning strategies for redundancy reduction in 3DGS. Our method shares similarities with CompactGS (Lee et al., 2024) and ScaffoldGS (Lu et al., 2023) while exhibits crucial differences. First, unlike CompactGS, we utilize a combination of neural fields and self-attention layers to predict not only view-dependent colors but also geometric properties. Second, in contrast to both approaches which explicitly store the position of every splat in the point cloud, our method only stores a small subset of splats, referred to as parent, while predicting the remaining points on-the-fly during rendering. This substantially reduces memory footprint. Third, anchor-based representation essentially creates a tree structure but restricts the depth of the tree to one. The growth strategy focuses solely on the anchor points, neglecting the growth directly from the splats. In contrast, our hierarchical tree representation combined with the proposed ATM take into account the significance of both parent and children splats, allowing for a strategy of sub-tree expansion.
+
+# 3 PRELIMINARIES
+
+3D Gaussian Splatting (3DGS) (Kerbl et al., 2023) represents a scene with 3D points  $\mathbf{x}$ . The points are initialized with a coarse point cloud obtained using Structure-from-Motion (SfM) (Schonberger & Frahm, 2016). These Gaussians,  $G(\mathbf{x})$ , serve as the anisotropic volumetric splats defined by their position (mean  $\mu$ ) and 3D covariance ( $\Sigma$ ) as  $G(\mathbf{x}) = e^{-\frac{1}{2} (\mathbf{x} - \mu)^T \Sigma^{-1} (\mathbf{x} - \mu)}$ . To ensure  $\Sigma$  remaining
+
+![](images/4fbee712ca83ec0419b1121a3543fdd9a1d0ac4e076d8b3580e062eec8600eae.jpg)  
+Figure 2: Top: Overview of our tree construction pipeline with the initial parent splats derived from SfM. The children spats are inferred on-the-fly from the parent splats via querying the displacement hash-grid. To estimate the Gaussian attributes like scale, rotation, color, and opacity, attribute features  $f_{\mathrm{a}}$  and  $f_{\mathrm{ak}}$  obtained on-the-fly from the attributes hash-grid are aggregated with self-attention. Bottom: Tree manipulation through ATM. The significant children splats are promoted to parent (regardless of the status of their parent, e.g., pruned) such that they have their own children in the next iteration. Bad trees (e.g., transparent parent) are removed together with the insignificant children.
+
+positive semi-definite during optimization, it is represented with an equivalent yet effective formulation with the scaling matrix  $S$  and the rotation matrix  $R$ , such that  $\Sigma = RSS^T R^T$ . The attributes of the 3D splats (e.g., location, covariance, and opacity) together with the directional appearance of the radiance filed, represented via the spherical harmonics (SH) (Sara Fridovich-Keil and Alex Yu et al., 2022), are end-to-end learned using optimization.
+
+To render an image, 3D  $G(\mathbf{x})$  are first transformed into 2D Gaussians (denoted as  $G^{\prime}(\mathbf{x})$ ) (Zwicker et al., 2001). 3DGS uses an efficient tiled-based rasterizer that presorts primitives for the entire image, allowing fast  $\alpha$ -blending of anisotropic splats. The color  $C$  of a pixel is computed by blending  $N$  2D Gaussians that overlap at the pixel as:  $C = \sum_{i\in N}c_i\alpha_iG_i^{\prime}(\mathbf{x})\prod_{j = 1}^{i - 1}(1 - \alpha_jG_j^{\prime}(\mathbf{x}))$ , where  $c_{i}$  represents view-dependent colors for each splat,  $\alpha_{i}$  is the opacity. With the highly optimized rasterizer for modern GPUs, 3DGS render high-fidelity scenes in real-time across many platforms. These benefits come with a cost. 3DGS require a significant number of 3D Gaussians, sometimes needing gigabytes for complex large-scale scenes. This requirement limits their application on edge devices, as downloading gigabytes over the network and storing them is hardly feasible or practical.
+
+# 4 METHOD
+
+We show a high-level overview of our approach in Fig. 2. Our primary motivation is to use a hierarchical representation(i.e., tree) to model the spatial relationships among the splats. We show that the locations of children splats and associated attributes —position, color, scale, etc.— can be derived from the parent using a small neural network. This allows us to store only the parent splats along with the weights of the neural network. To achieve this, we initially represent a 3D scene as a forest of depth-1 tree structures where the parent splats are initialized from SfM (Schönberger & Frahm, 2016) and the children are neural-predicted on-the-fly. The trees are then refined and expanded to sub-trees during the optimization process using Adaptive Tree Manipulation. Formally, we represent a scene using  $\mathcal{S} = \{\mathcal{X}_1, \mathcal{X}_2, \ldots, \mathcal{X}_n\}$ , where  $\mathcal{X}_i$  is tree and each node contains the attributes, such as position  $(x)$ , color  $(c)$ , opacity  $(\alpha)$ , scale  $(s)$ , and rotation  $(r)$ . This representation can be stored very efficiently, as for each tree we need to save only the positions and scales of parent splats and small neural network shared across the trees, to predict all the other attributes of the tree.
+
+# 4.1 NEURAL REPRESENTATION FOR LIGHTWEIGHT PREDICTIVE SPLATS
+
+We model close relationship between a parent and children nodes. Specifically, we assume that the children nodes are in the vicinity of parent node and have similar geometric and appearance attributes such as shape, color and opacity. We satisfy these requirements by using a hash-grid based approach (Müller et al., 2022b; Chen et al., 2023a) as our representation, which has an inherent
+
+property to return similar features when queried with the points located nearby via feature interpolation. Below we describe how a tree  $(\mathcal{X}_i)$  can be represented in storage efficient manner. In what follows we drop the index  $i$ . We use the notation node to refer the splat in the context of a tree.
+
+For a hash-grid  $\mathcal{H}(\cdot)$  shared across the trees and parent node positions  $x_{p}$ , we query the features as  $f = \mathcal{H}(x_p)$  and use them to predict the displacement of children and attributes of the tree. We divide  $f$  into two halves  $f\equiv \{f_{\Delta}\in \mathbb{R}^{D / 2},f_a\in \mathbb{R}^{D / 2}\}$ , where the first half  $(f_{\Delta})$  represents displacement and is used to predict the position of children. The second half  $(f_{a})$  is used to predict other attributes.
+
+Predicting Position. We want children and parent nodes to represent similar geometry and appearance. Hence, children should be located in the vicinity of the parents nodes. We model the position of children as their displacement from their parent nodes. For the parent we predict the position of  $k^{\text{th}}$  child using  $x_{k} = x_{p} + g_{\text{pos}}(f_{\Delta})[k]$  where  $g_{\text{pos}}$  is an MLP with output shape  $K \times 3$ .
+
+Having the positions of all nodes in the tree, we can predict the rest of the attributes, such as scale, rotation, color, and opacity. We reuse the hash-grid to get the attribute feature  $(f_{\mathrm{a}_k})$  for  $k^{\mathrm{th}}$  child node using  $\mathcal{H}(x_k)$ . A naive approach to extract the remaining attributes using  $f_{\mathrm{a}}$  and  $f_{\mathrm{a}_k}$  is to pass the latter to an MLP get scale, rotation, color and opacity. We found such approach to be sub optimal. A hash-grid representation implicitly makes the representation of spatially points similar. There is no mechanism to share information between the features after they are computed. Since there is relation between physical attributes of the parent and children nodes, having such information sharing mechanism is beneficial. To this end, we propose a modified self-attention mechanism to better capture the inter-dependencies between children and parent nodes. To do so, we first obtain the aggregated feature  $\mathcal{F}_{\mathrm{a}} \in \mathbb{R}^{K + 1 \times D / 2}$  by concatenating features of all the nodes in the tree, such as  $\mathcal{F}_{\mathrm{a}} = \text{Concat}(\{f_{\mathrm{a}}, (f_{\mathrm{a}_1}, \ldots, f_{\mathrm{a}_K}\})$ , where  $\text{Concat}$  is a concatenation operation. We then apply a modified self-attention operation on  $\mathcal{F}_{\mathrm{a}}$  to get the final feature  $\mathcal{F}_{\mathrm{a}}'$ :
+
+$$
+\mathcal {F} _ {\mathrm {a}} ^ {\prime} = \mathcal {F} _ {\mathrm {a}} + \lambda \sigma \left(\frac {\mathcal {P} _ {1} \left(\mathcal {F} _ {\mathrm {a}}\right) * \mathcal {P} _ {2} \left(\mathcal {F} _ {\mathrm {a}}\right) ^ {T}}{\sqrt {d}}\right) * \mathcal {F} _ {\mathrm {a}}, \tag {1}
+$$
+
+where  $\sigma(\cdot)$  is a Softmax function,  $\mathcal{P}_i(\cdot)$  is a projection matrix,  $d$  is a scaling factor set as  $D/2$ ,  $\lambda$  is a hyper-parameter for balancing the information trade-off from the attention mechanism and * denotes the matrix multiplication. Different from vanilla attention (Vaswani et al., 2017), we do not apply positional embedding, so that Eq. 1 is permutation invariant which is an important property to maintain while working with point clouds (Qi et al., 2016). Further, we use the unprojected  $\mathcal{F}_{\mathrm{a}}$  when multiplying with  $\sigma(\cdot)$ , since we empirically found no performance gain by projecting  $\mathcal{F}_{\mathrm{a}}$ . Next, we split  $\mathcal{F}_{\mathrm{a}}'$  in  $K + 1$  attribute feature vector to predict the remaining attributes for each node in the tree.
+
+Predicting Scale and Rotation. It is vital to properly initialize the scale of Gaussians for stable training. For instance, Gaussians with small scales make minimal contributions to the rendering quality, mainly because of their limited volume. In contrast, large Gaussians can potentially contribute to every pixel during rasterization, leading to a significant amount of GPU memory. Hence, to make training stable and minimize storage needs at the same time, we adopt a middle-ground strategy. More specifically, we represent the scales of children as a scaled version of their parents  $(s_{\mathrm{p}}): s_k = \hat{s}_k$ $s_p$  where  $\hat{s}_k$  is predicted by an MLP. In case of rotation, we directly regress it for both parents and children nodes using the corresponding attribute feature vector. We share the weights of the MLP to regress both scale and rotation. We experimentally found, that including position of node  $(x_{k})$ , the distance of the point to the center of the axis aligned bounding box  $(b_{k})$  along with attribute feature  $((f_{\mathrm{ak}}^{\prime})$  improves performance:  $\hat{s}_k,r_k = g_{rs}(f_{\mathrm{ak}}^{\prime},x_k,b_k)$
+
+Predicting Color and Opacity. 3DGS uses degree-3 spherical harmonics (SH) for view-dependent color representation (Kerbl et al., 2023). However, we find such design is unnecessary and the color can be directly predicted using from feature vectors and a viewing direction. We use an MLP that takes them as an input and directly predicts the color as output,  $c_{k} = g_{c}(f_{\mathrm{ak}}^{\prime},d_{k})$  where  $d_{k}$  is the viewing direction of the node in the tree. This reduces the storage by a significant amount as previously each splat storing the spherical harmonics individually. To predict opacity, we use another MLP with inputs as  $f_{\mathrm{ak}}^{\prime}$  and the position of the node to get corresponding opacity,  $o_{k} = g_{o}(f_{\mathrm{ak}}^{\prime},x_{k})$ .
+
+We described all the operations above for a single tree. The same operation is extended for all the trees. Further, the neural networks for all the operations share their weights across all the trees. To summarize, the proposed representation efficiently represents the tree structure with hash-grid based neural representations  $\mathcal{H}(\cdot)$  and a few MLPs. We only store position and scale of parent nodes and the weights of our neural networks, while the rest of the properties is regressed as described above.
+
+# 4.2 ADAPTIVE TREE MANIPULATION
+
+3DGS (Kerbl et al., 2023) starts by using an initial point cloud from SfM. To allow for some flexibility in the point cloud structure they propose several techniques to add and delete the Gaussians during optimization. These techniques work effectively for individual splats, but naively applying them to our tree (i.e., applying directly to the parent nodes of a tree) results in sub-optimal or incorrect outcomes(see Tab. 2a and Fig. 8). This is because such an approach implicitly limits the depth of all trees to one, overlooking the significance of the children nodes and preventing sub-tree expansion and consequently hindering the quality. This consideration necessitates a new strategy that incorporates the importance of the children into the process. On the other hand, a structural representation facilitates feature sharing, while simultaneously entangling the parent and children nodes into a cohesive unit due to the nature of the tree. Consequently, operations on the parent directly affect the children (e.g., pruning the parent results in the removal of the children). Considering these factors, we propose the following strategy (see Fig. 2):
+
+- Promotion A children can become a parent if it is deemed significant during training, and it can have its own children in the next iteration.  
+- Pruning Removing or pruning the parent operates only on itself and its insignificant children. Significant children are unaffected.  
+- Cloning Cloning the parent operates on the entire tree (i.e., all children are cloned as well).
+
+To determine the significance of a children, we track the position gradient of all the children splats. When the gradient of a child node is above a certain threshold  $\tau_{\mathrm{pos}}^c$  (i.e.,  $\nabla_{pos}^{c} > \tau_{pos}^{c}$ ) where  $\tau_{pos}^{c}$  is a hyper-parameter, we consider the children significant. Then we promote the children node to become a new parent in next training iteration (satisfying Promotion). This is crucial to represent complicated regions in the scene where there might not be many parent nodes. Once the children has been promoted to parent node we apply clone and split operations to all the parents following similar practices in 3DGS (satisfying Cloning). To address Pruning, we first check if any nodes meet the criteria outlined in Promotion and promote them if needed. Then to delete the tree we only rely on the statistics of parent node. This is because we assume if a children node was important then it would have been already promoted to become a new parent. Hence, we can safely delete the current parent that will in turn delete all the corresponding insignificant children nodes. Specifically, for deleting the trees we track the scale and opacity of the parent nodes and delete them if they are below a certain threshold similar to 3DGS.
+
+# 4.3 TRAINING
+
+Our model, including the hash-grid and MLPs, is end-to-end learnable guided by the  $\mathcal{L}_1$  loss between the rendered images and ground-truth images along with a D-SSIM loss, such that:
+
+$$
+\mathcal {L} = (1 - \beta) \mathcal {L} _ {1} + \beta \mathcal {L} _ {\mathrm {D} - \mathrm {S S I M}}, \tag {2}
+$$
+
+where  $\beta$  is set as 0.2 following the setting in (Kerbl et al., 2023).
+
+We use a warm-up training scheme that helps in convergence of the model (Kerbl et al., 2023). The warm-up consists of training the model in a low resolution setting, eventually moving to higher resolution after a certain number of steps have been completed. We found that the warm-up strategy is crucial to correctly position the splats and densify the regions. Without the warm-up, the model struggles to populate enough splats in the background area, despite the importance of the area resulting in substandard performance. Please refer to Appendix material for details.
+
+# 5 EXPERIMENTS
+
+Dataset and Metrics. We evaluate our method using seven scenes from the Mip-NeRF  $360^{\circ}$  dataset (Barron et al., 2022b), two scenes from Tank&Temples (Knapitsch et al., 2017), and two scenes from Deep Blending (Hedman et al., 2018). We use the widely adopted metrics like PSNR, SSIM (Wang et al., 2004), and LPIPS (Zhang et al., 2018) to assess the quality for image reconstruction. We also report the storage size (in MB) for various methods along with their on-device capabilities. We benchmark the Gaussian Splitting based methods on iPhone 14 with our implementation of the mobile application. We report three configurations of our method named C1, C2, and C3 by varying feature dimension  $D$  of the hash-grid  $\mathcal{H}$ . C1 is our smallest model with  $D = 32$
+
+followed by C2 with  $D = 48$  and C3 is largest with  $D = 64$ . Since our framework adds and removes points during optimization, the final storage for each model can vary. For each dataset we report the average size of all the scenes within one configuration. The metrics, too, are averaged over all scenes of each dataset. Per-scene quantitative results are in the Appendix.
+
+Quality vs. Storage. First, we show that our approach provides a practical means of satisfying diverse technical requirements. We can reduce or increase the feature dimension of the hash-grid and the number of points, while still maintaining similar or superior rendering quality. In Fig. 3, we plot PSNR, evaluated on the dataset introduced by (Barron et al., 2022a), for contemporary models as well as for three configurations of our approach. Our smallest configuration is almost  $50\%$  smaller than the smallest prior work (LightGS (Fan et al., 2024)), and shows the same rendering quality. Our largest configuration, which is still  $32\%$  smaller than the smallest existing work, shows significantly increased PSNR. To give the reader a better perspective, we also plot conventional works with large hard-drive footprint (Lu et al., 2023; Kerbl et al., 2023). Our largest configuration, which uses only  $20\%$  of ScaffoldGS (Lu et al., 2023) and only  $4.5\%$  of 3DGS (Kerbl et al., 2023) storage, shows higher quality than both of these much larger works. These advantages of our method are crucial for mobile deployment. Less disk storage also helps in speeding up transmission that significantly impacts user experience when sharing content.
+
+![](images/a5c76ce9dcf6c3596ff1fa764c42cd3a5833c62ee5a5e7192ef8f4ca3aa73858.jpg)  
+Figure 3: Comparisons of PSNR and storage computed on the dataset from Barron et al. (2022a).
+
+# 5.1 COMPARISON RESULTS
+
+Quantitative Results. Tab. 1 shows the quantitative results on real-world scenes, spanning from large-scale urban landscapes to intricate indoor and outdoor environments. We compare our approach against 3DGS (Kerbl et al., 2023) and concurrent works (i.e., LightGS (Fan et al., 2024), CompactGS (Lee et al., 2024), Eagles (Girish et al., 2023), CompGS (Navaneet et al.) and ScaffoldGS (Lu et al., 2023)). On the Mip-NeRF  $360^{\circ}$  dataset, we achieve the best PSNR among all the approaches. Compared with 3DGS (Kerbl et al., 2023), we obtain a significant storage reduction, i.e.,  $19.5\times$ , and require  $3.5\times$  fewer 3D points. On the Tank&Temples (Knapitsch et al., 2017) dataset, although ScaffoldGS (Lu et al., 2023) has better PSNR than our approach, our storage is almost  $2.4\times$  smaller than ScaffoldGS. Compared with 3DGS (Kerbl et al., 2023) on this dataset, we require  $1.9\times$  fewer 3D points and  $11.3\times$  less storage. On the Deep Blending (Hedman et al., 2018) dataset, our method has higher PSNR and  $19\times$  storage reduction than 3DGS (Kerbl et al., 2023).
+
+Fig. 4 demonstrates the high-quality rendering of our method produced using C3 configuration across 5 example scenes covering all the datasets. We see various examples where our method outperforms previous compression works. We can see our models can better capture background details (row 3, 5), better capture reflections (row 2) while being the smallest or of comparable size. It can also capture intricate details where other methods fail such as ceilings (row 1, 4).
+
+On-Device Capability. We explore the feasibility of running splatting based methods on mobile devices. We use iPhone14 and Snap AR glasses Spectacles to implement the applications. For fair comparison, we unpack splats from all methods to a standard 3DGS format (Kerbl et al., 2023) for
+
+Table 1: Quantitative comparisons on three widely used benchmark datasets, including Mip-NeRF  $360^{\circ}$  dataset (Barron et al., 2022a), Tanks&Temples (Knapitsch et al., 2017), and Deep Blending (Hedman et al., 2018). We report the image quality metrics, such as PSNR, SSIM, and LPIPS, and the required storage. We also report the on-device capability of each Gaussian Splitting based work (On-Device in the table), where OOM denotes Out-of-Memory error and  $\checkmark$  denotes the real-time capability ( $>30$  fps) on our tested device, i.e., iPhone14. and - denotes unknown of on-device capability. The evaluation results on other works are obtained from their papers. Compared with the methods that are capable to run on mobile devices, our models (Ours-C1, C2, C3) can obtain smaller model size with higher rendering quality (i.e., PSNR).  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">On-Device</td><td colspan="4">Mip-NeRF 360° Dataset</td><td colspan="4">Tank&amp;Temples</td><td colspan="4">Deep Blending</td></tr><tr><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>Storage ↓</td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>Storage ↓</td><td>PSNR ↑</td><td>SSIM ↑</td><td>LPIPS ↓</td><td>Storage ↓</td></tr><tr><td>ScaffoldGS (Lu et al., 2023)</td><td>OOM</td><td>29.02</td><td>0.848</td><td>0.220</td><td>156MB</td><td>23.96</td><td>0.853</td><td>0.177</td><td>87MB</td><td>30.21</td><td>0.906</td><td>0.254</td><td>66MB</td></tr><tr><td>3DGS (Kerbl et al., 2023)</td><td>OOM</td><td>28.69</td><td>0.870</td><td>0.182</td><td>693MB</td><td>23.14</td><td>0.841</td><td>0.183</td><td>411MB</td><td>29.41</td><td>0.903</td><td>0.243</td><td>676MB</td></tr><tr><td>CompGS (Navaneet et al.)</td><td>-</td><td>27.16</td><td>0.808</td><td>0.228</td><td>50.30MB</td><td>23.47</td><td>0.840</td><td>0.188</td><td>27.97MB</td><td>29.75</td><td>0.903</td><td>0.247</td><td>42.77MB</td></tr><tr><td>Eagles (Girish et al., 2023)</td><td>✓</td><td>27.15</td><td>0.808</td><td>0.238</td><td>68.89MB</td><td>23.41</td><td>0.840</td><td>0.200</td><td>34MB</td><td>29.91</td><td>0.910</td><td>0.250</td><td>62MB</td></tr><tr><td>LightGS (Fan et al., 2024)</td><td>✓</td><td>28.45</td><td>0.857</td><td>0.210</td><td>42.48MB</td><td>22.83</td><td>0.807</td><td>0.242</td><td>22.43MB</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>CompactGS (Lee et al., 2024)</td><td>✓</td><td>28.60</td><td>0.855</td><td>0.211</td><td>46.98MB</td><td>23.32</td><td>0.831</td><td>0.201</td><td>39.43MB</td><td>29.79</td><td>0.901</td><td>0.258</td><td>43.21MB</td></tr><tr><td>Ours-C1</td><td>✓</td><td>28.45</td><td>0.837</td><td>0.235</td><td>23.40 MB</td><td>23.19</td><td>0.810</td><td>0.239</td><td>22.00 MB</td><td>29.32</td><td>0.895</td><td>0.282</td><td>22.90MB</td></tr><tr><td>Ours-C2</td><td>✓</td><td>28.86</td><td>0.851</td><td>0.217</td><td>29.50 MB</td><td>23.47</td><td>0.820</td><td>0.228</td><td>29.05MB</td><td>29.61</td><td>0.896</td><td>0.277</td><td>29.15MB</td></tr><tr><td>Ours-C3</td><td>✓</td><td>29.11</td><td>0.857</td><td>0.210</td><td>35.60MB</td><td>23.82</td><td>0.829</td><td>0.210</td><td>35.32MB</td><td>29.89</td><td>0.902</td><td>0.267</td><td>35.40MB</td></tr></table>
+
+![](images/1338771550678f6095e1b2a0f3c3eb1df213daa78a506c3344f8c8d6c35c6c4a.jpg)  
+Figure 4: Visual comparisons with methods offering efficient GS representations (Lee et al. (2024); Fan et al. (2024)). We magnified regions to show qualitative differences. Our approach (C3) can render images with high-quality while greatly saving the storage. Zoom-in for greater detail.
+
+rendering. We observe Out-of-Memory error when running all scenes from the three benchmark datasets for 3DGS and ScaffoldGS, owing to their large number of splats. Our method can successfully run on device (as in Fig. 5), and achieves smaller and better rendering quality compared to LightGS (Fan et al., 2024), CompactGS (Lee et al., 2024) and Eagles (Girish et al., 2023). Check our Webpage for video demos.
+
+(a) Ablation Study on the propose contributions. FE stands for frequency encoding. Full refers to the full settings.
+
+Table 2: Ablation analysis on core design elements. We report the PSNR for each experiment.  
+
+<table><tr><td></td><td>FE</td><td>w/o Agg.</td><td>w/o ATM.</td><td>w/o Contract.</td><td>Full</td></tr><tr><td>Bicycle</td><td>19.72</td><td>22.80</td><td>22.66</td><td>22.88</td><td>23.68</td></tr><tr><td>Playroom</td><td>23.36</td><td>28.65</td><td>28.74</td><td>-</td><td>29.27</td></tr></table>
+
+(b) Analysis of self-attention. We use the attention with different number of heads (H) and attention head dimension (F), and different  $\lambda$  in Eqn. 1.
+
+<table><tr><td></td><td>H1-F32</td><td>H2-F16</td><td>H4-F8</td><td>λ=0.1</td><td>λ=0.5</td><td>λ=1.0</td></tr><tr><td>Bicycle</td><td>23.68</td><td>23.31</td><td>23.20</td><td>23.55</td><td>23.68</td><td>23.53</td></tr><tr><td>Playroom</td><td>29.27</td><td>29.15</td><td>29.10</td><td>29.14</td><td>29.27</td><td>29.22</td></tr></table>
+
+Storage Analysis. The storage of our method consists of four components: the hash-grid, parent splat locations and scales, and MLPs. The hash-grid is stored in 8 bits, while the other components are stored in 16 bits. Notably, the hash-grid occupies half of the total storage. However, our representation is independent of feature encoding module (i.e., hash-grid encoding) thus it allows for easy adjustments to more efficient representations if necessary. For instance, in our C1 configuration using the dataset Barron et al. (2022a), the storage allocation for each component is as follows:  $12.5\mathrm{MB}$  for the hash-grid,  $5.2\mathrm{MB}$  for the parent splat locations,  $5.2\mathrm{MB}$  for the scales, and  $0.5\mathrm{MB}$  for the MLPs.
+
+Inference Phase Optimization. Predicting the attributes brings overheads during the inference. However, we note that only the at-
+
+tribute color is view-dependent and the rest remain the same for all frames. Therefore, to minimize the computation cost, we opt to run the color prediction (small MLPs) only and the rest of the attributes are retrieved from the first frame. Thus, we get real-time rendering on the mobile phones and achieve comparable speed as 3DGS (Kerbl et al., 2023). We report two large-scale complex scenes Bicycle and Garden from Barron et al. (2022a) on Nvidia A100: the rendering FPS are 61, 55 for our method and 77, 63 for 3DGS, respectively.
+
+![](images/f0b1e26413b08a3be737fcad80c9c1515558f71d9a80a097134d724470f5a020.jpg)  
+Figure 5: Demo of running our method on iPhone 14 and Snap AR glasses Spectacles.
+
+# 5.2 ABLATION ANALYSIS
+
+We perform comprehensive analysis on various components of our methods using our C3 configuration. Here we choose two representative scenes to perform experiments: one unbounded outdoor scene Bicycle from Mip-NeRF  $360^{\circ}$  dataset (Barron et al., 2022a) and one indoor scene Playroom from the Deep Blending dataset (Hedman et al., 2018). We report the best PSNR that is achieved within 10K steps for all experiments.
+
+Importance of Hash Grid. We replace the hash grid with the frequency encoding of the 3D position followed by a 2-layer MLP to output a  $D = 64$ -dimensional feature vector, which has the same dimension as the one from hash grid. We denote the setting as  $FE$ . Without hash-grid we see a significant drop in the performance, highlighting the importance of the feature alignment encoded within the spatial hash grid.
+
+Importance of Attention Mechanism. When we remove self attention mechanism between the nodes of the tree, it is detrimental to the performance (Tab. 2a w/o Attn). This validates our motivation that there is relation between various physical attributes of the nodes of tree hence there needs to be a mechanism to facilitate the sharing of information. Additionally, adding attention mechanism reduces the number of parent splats making our method storage efficient: 884K v.s 1.06M averaged across all scenes in dataset (Barron et al., 2022a). We hypothesise that a configuration with attention can pull information from nearby splats, allowing the method to reduce the number of splats to store and represent the scene efficiently. Additionally, we ablated various configurations by varying  $\lambda$  in Eq. 1 and the number of heads in attention to find the best configuration (Tab. 2b). We see a right balance between the input features and attention features is important for best performance.
+
+Adaptive Tree Manipulation (ATM). We remove Adaptive Tree Manipulation (ATM) and add or delete the trees based only on parent nodes statistics and observe a drop in PSNR (Tab. 2a) also visible in rendered images Fig. 8. This is because there is no mechanism to promote important children to parent that might hinder in populating trees correctly and failing to represent complex
+
+![](images/8ec54c0753fe5e2cfc70425f26fa61ad52a687fa7c645ff1833bcd728275e8b7.jpg)
+
+![](images/2ab7e42c119b8dd9d9436211464d1530b95cbceda4379b3ee54851bf8575a243.jpg)  
+Figure 6: Visual comparison of model trained with and without ATM. We can see that model trained without ATM fails to model intricate details in the scene.  
+Figure 7: The effect of Adaptive Tree Manipulation (ATM). Yellow points indicate the splats who have not changed the parent status during entire optimization. Green points represent former children that have been promoted to parents. Around  $80\%$  of parents are from our ATM operation.
+
+scenes effectively. On the other hand, this might also lead to deletion of important children nodes when deleting a parent. Our proposed ATM method can effectively alleviate these issues.
+
+Additionally, in Fig. 7 we show the point clouds of three scenes. Green points represent parents promoted from children during the optimization. Yellow points show parents that stayed parents during entire optimization. It is clearly seen that the majority of the parent nodes are formed by promoting children nodes. Further note that parts with relatively flat geometry exhibit more yellow, while sophisticated geometry with high frequency details contain more green. Hence, ATM brings a further benefit of being able to fit sophisticated geometry better.
+
+Inputs of MLP for attribute prediction. Tab. 3 shows the analysis for the inputs used to predict the attributes. We conduct the experiments of without using the distance from points to the center of AABB (denoted as  $w/o \, Distance$ ) and without using the 3D position information (denoted as  $w/o \, Position$ ) to predict attributes. Position is very crucial for training while distance further improves the performance. Encoding on the view directions by performing degree from Degree of 3 gives the best performance as it has more capacity.
+
+Table 3: Analysis of the inputs used for attributes prediction.  
+
+<table><tr><td></td><td>w/o Distance</td><td>w/o Position</td><td>SH D1</td><td>SH D2</td><td>SH D3</td></tr><tr><td>Bicycle</td><td>23.32</td><td>9.72</td><td>23.05</td><td>23.60</td><td>23.68</td></tr><tr><td>Playroom</td><td>29.18</td><td>6.19</td><td>29.23</td><td>29.15</td><td>29.27</td></tr></table>
+
+We also analyze the degrees of the SH in 1 to 3 (denoted as  $SHD1$  to  $SHD3$ ). Identity to model complicated light effects.
+
+# 6 CONCLUSION
+
+This paper introduces predictive 3D Gaussian splats, a lightweight representation that dramatically reduces storage for large-scale scenes compared to 3DGS, while maintaining high-fidelity rendering results. We propose an efficient structural representation, i.e., parent-children structure to model the inherent spatial relationship among nearby splats. The children splats and most Gaussian attributes can be estimated during rendering by utilizing parent. Additionally, we leverage a hash grid and self-attention on aggregated features to enforce connectivity for parent and children nodes. We conduct extensive experiments on benchmark datasets to validate our design and demonstrate the our advantages of storage saving and high-quality novel view synthesis. For future work, since our representation is orthogonal to other compression techniques, we can combine it with methods like adaptive Gaussian pruning to achieve greater efficiency. This integration could enhance storage performance and reduce computational overhead, making our system more robust and scalable. Exploring these synergies will be a valuable direction for optimizing representation and improving overall efficiency in various applications.
+
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+
+# A APPENDIX
+
+# A.1 VISUAL EXPLANATION OF ADAPTIVE TREE MANIPULATION
+
+![](images/3bf2e333dfd8b811c1ac08a181db433d757e7ec942d7175fe7e8b803de31fe66.jpg)  
+Figure 8: Demonstration of Promotion, Pruning, and Cloning operations in ATM.
+
+As illustrated in Fig. 8, a significant child splot is promoted to the parent in the next iteration (i.e.,  $N + 1$ ) if satisfying the criterion (i.e.,  $\nabla_{pos}^{c} > \tau_{pos}^{c}$ ). In the iteration  $N + 1$ , the newly promoted parent has its own children splats (e.g., the green node in the tree). When pruning the parent splats, significant splats (e.g., the green node) and insignificant splats (e.g., the red node) are handled differently, where the insignificant splats are pruned together with the parent whereas the significant splat turns to a new tree in iteration  $N + 1$ . Lastly, the cloning operates on the tree level where children splats are also cloned.
+
+# A.2 IMPLEMENTATION DETAILS
+
+In this section, we provide more details for our training. We first provide the hyper-parameters used during the training in A.2.1. We discuss the architecture and training details in A.2.2. Next we show the visual illustration and the implementation details of the contraction in Sec. A.2.3. Then, we analyze the effectiveness of the warm-up training strategy employed in our method in Sec. A.2.4 and the convergence in Sec. A.2.5. Lastly, we discuss the pre-filtering of parent points and its implementation in Sec. A.2.6.
+
+# A.2.1 SETTINGS OF HYPER-PARAMETERS
+
+We employ different learning schedules for different modules. For the hash grid, we start with a learning rate of  $2e^{-3}$  and end with a rate of  $2e^{-5}$ . For opacity, we start with  $1e^{-3}$  and end with  $2e^{-5}$ . The scale and rotation parameters utilize a constant learning rate of  $1e^{-4}$ . Additionally, we maintain a constant learning rate of  $2e^{-4}$  for the attention module. We apply a standard exponential decay scheduling (Kerbl et al., 2023; Sara Fridovich-Keil and Alex Yu et al., 2022) to all modules.
+
+# A.2.2 ARCHITECTURE AND TRAINING DETAILS
+
+We use Instant-NGP (Müller et al., 2022b) as our hash-grid owing to its compact and efficient design and 2 layers MLP for all the MLPs. Following the practices in (Barron et al., 2022a; Müller et al., 2022b) we use scene contraction to map the position into [0, 1] before feeding it to the Instant-NGP. This helps bring the splats that are occasionally outside the Axis-Aligned-Bounding-Box (AABB) due to the densification of splatsalong with the position updates. We estimate the AABB with the initial COLMAP (Schönberger & Frahm, 2016) point cloud. We set  $\lambda = 0.5$  for all the experiments and train the model for  $30K$  steps with  $7.5K$  steps warm-up stage. In our experiments, we initially use lower resolution images to train the model  $7.5K$  steps for warm-up, after which we transit to high-resolution images. More precisely, following the setting from 3DGS (Kerbl et al., 2023), we employ  $32\times$  downsampling for the Mip-NeRF  $360^{\circ}$  dataset and  $4\times$  downsampling for the Tank&Temples (Knapitsch et al., 2017) and Deep Blending (Hedman et al., 2018) datasets in the warm-up stage. The number of children splats  $(K)$  used in our experiment varies across scenes. We empirically found that 2 children are enough for most scenes and we believe this is because our sub-tree expansion allows the tree to grow deeper and compensate the breadth requirement. Please refer to A.4 material for details.
+
+![](images/b6a0c3faebcf73e8a53754189880a2de0a92d849ca0199e0549b746ccc0a0690.jpg)  
+Figure 9: Illustration for our implemented contraction.
+
+# A.2.3 DETAILS FOR CONTRACTION
+
+We illustrate the process of contraction (described in Sec. 3.5 of the main paper) in Fig. 9 and Alg. 1. We calculate the inscribed and circumscribed spheres (i.e.,  $S_{inner}$  and  $S_{outer}$ ) with radius  $R_{inner}$  and  $R_{outer}$  of the initialized Axis-Aligned Bounding Box (AABB), which is estimated from the point cloud generated from COLMAP (Schönberger & Frahm, 2016). The estimated AABB is the circumscribed cube of the outer sphere  $S_{outer}$ . Points falling outside of the outer sphere are brought back to  $S_{outer}$ .
+
+<table><tr><td colspan="2">Algorithm 1 AABB Estimation and Contraction</td></tr><tr><td colspan="2">Require: Initialized AABB: AABBinit, point cloud: PC</td></tr><tr><td>Sinner← Inscribed sphere of AABBinit</td><td>▷ centered at O</td></tr><tr><td>Rinner← Radius of Sinner</td><td></td></tr><tr><td>Souter← Circumscribed sphere of AABBinit</td><td>▷ centered at O</td></tr><tr><td>Router← Radius of Souter</td><td></td></tr><tr><td>AABBest← Circumscribed cube of Souter</td><td></td></tr><tr><td>for p in PC do</td><td>▷ contract the points</td></tr><tr><td>if ||p-O|| ≤ Rinner then</td><td></td></tr><tr><td>p← p</td><td></td></tr><tr><td>else if ||p-O|| &gt; Rinner then</td><td></td></tr><tr><td>p← (Router - 1/||p-O|)(p-O/||p-O|) + O</td><td>▷ infinity is on Souter</td></tr><tr><td>end if</td><td></td></tr><tr><td>end for</td><td></td></tr></table>
+
+# A.2.4 ANALYSIS OF WARM-UP
+
+We run two experiments on the Garden (Barron et al., 2022a) scene in 10K steps to show the effectiveness of the warm-up in our method.
+
+We have found that using warm-up in training with low resolution images at early stages helps the points populate the empty areas, especially when the COLMAP (Schönberger & Frahm, 2016) initialization is poor. Fig. 10 shows the point cloud and corresponding rendered images from different training approaches. As can be seen, the warm-up training (second row) has a better reconstruction and rendering quality for the background scene, which is poorly initialized from COLMAP (Schönberger & Frahm, 2016).
+
+![](images/5303c75c24887730675a0c88e4107709fbdc02c46f0a44001001c86bf6257a43.jpg)  
+Figure 10: Analysis of warm-up. First row: training without warm-up. Second row: training with warm-up. Left column: points distribution. Right column: rendered images with the point cloud.
+
+![](images/21bce651d7879df629e67dab2b5f6627eab5fdd1363ea8c495b91cca39d04947.jpg)
+
+# A.2.5 ANALYSIS OF CONVERGENCE
+
+Leveraging the hash-grid structure increases the per-step training time compared to 3DGS (Kerbl et al., 2023). For example, 3DGS requires approximately 23 minutes to reach a PSNR of 30.63 at 30K steps on scene room from MipNeRF-360 dataset (Barron et al., 2022a), our method takes about 27.6 minutes, and HAC (Chen et al., 2024) requires a similar training time of 27.1 minutes. Nonetheless, our approach strikes a good balance between size, speed, and quality. Furthermore, thanks to our inference phase optimization strategy in Sec. 5.1, the hash-grid structure has a minimal impact on inference time, allowing our approach to run in real-time on mobile devices.
+
+# A.2.6 VIEW FRUSTUM CULLING
+
+We apply pre-filtering on the parent points before querying features for attributes prediction by culling the view frustum with depth, leading to the computation reduction and the training speedup.
+
+We empirically observe that  $15\% - 25\%$  points are removed across scenes. The implementation is shown in Alg. 2.
+
+# Algorithm 2 View Frustum Culling
+
+<table><tr><td>Require: points P : N × 3, view matrix M : 4 × 4</td></tr><tr><td>P_homo ← Concat(P, ones)</td></tr><tr><td>P_view ← M * (P_homo)^T</td></tr><tr><td>mask ← P_view[2,:] &gt; 0.201</td></tr><tr><td>P_filtered ← P[mask]</td></tr></table>
+
+<table><tr><td>▷ P_homo : N × 4</td></tr><tr><td>▷ * is matrix multiplication</td></tr><tr><td>▷ depth &gt; 0.201</td></tr></table>
+
+# A.3 MORE ABLATION STUDY
+
+Scene Contraction. We analyze the proposed contraction technique applied on the unbounded scene (Tab. 3 w/o Contract.). Compared with Full, we get inferior performance (0.8 PSNR drop), and tend to have training instability issues because the points occasionally move outside the Axis-Aligned Bounding box.
+
+# A.4 PER-SCENE QUANTITATIVE RESULTS
+
+We provide the per-scene results on the benchmark datasets. Tab. 4 shows the results on the Mip-NeRF  $360^{\circ}$  dataset (Barron et al., 2022a). Tab. 5 demonstrates the results on the Tank&Temples dataset (Knapitsch et al., 2017) and the Deep Blending dataset (Hedman et al., 2018). We report the per-scene storage (in MB), the number of parents and children, and the metrics for image quality evaluation.
+
+Table 4: Per-scene metrics for our approach on the Mip-NeRF  ${360}^{ \circ  }$  dataset Barron et al. (2022a).  
+
+<table><tr><td></td><td>Metric</td><td>Garden</td><td>Bicycle</td><td>Stump</td><td>Room</td><td>Counter</td><td>Kitchen</td><td>Bonsai</td></tr><tr><td rowspan="4">Compact GS</td><td>PSNR</td><td>26.81</td><td>24.77</td><td>26.46</td><td>30.88</td><td>28.71</td><td>30.480</td><td>32.08</td></tr><tr><td>SSIM</td><td>0.832</td><td>0.723</td><td>0.757</td><td>0.919</td><td>0.902</td><td>0.919</td><td>0.939</td></tr><tr><td>LPIPS</td><td>0.161</td><td>0.286</td><td>0.278</td><td>0.209</td><td>0.205</td><td>0.131</td><td>0.193</td></tr><tr><td>Storage (MB)</td><td>62.78</td><td>62.99</td><td>54.66</td><td>34.21</td><td>34.34</td><td>44.45</td><td>35.44</td></tr><tr><td rowspan="4">Light GS</td><td>PSNR</td><td>26.73</td><td>24.96</td><td>26.70</td><td>31.27</td><td>28.11</td><td>30.40</td><td>31.01</td></tr><tr><td>SSIM</td><td>0.836</td><td>0.738</td><td>0.768</td><td>0.926</td><td>0.893</td><td>0.914</td><td>0.944</td></tr><tr><td>LPIPS</td><td>0.155</td><td>0.265</td><td>0.261</td><td>0.220</td><td>0.218</td><td>0.147</td><td>0.204</td></tr><tr><td>Storage (MB)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td rowspan="4">Scaffold GS</td><td>PSNR</td><td>27.17</td><td>24.50</td><td>26.27</td><td>31.93</td><td>29.34</td><td>31.30</td><td>32.70</td></tr><tr><td>SSIM</td><td>0.842</td><td>0.705</td><td>0.784</td><td>0.925</td><td>0.914</td><td>0.928</td><td>0.946</td></tr><tr><td>LPIPS</td><td>0.146</td><td>0.306</td><td>0.284</td><td>0.202</td><td>0.191</td><td>0.126</td><td>0.185</td></tr><tr><td>Storage (MB)</td><td>271.00</td><td>248.00</td><td>493.00</td><td>133.00</td><td>194.00</td><td>173.00</td><td>258.00</td></tr><tr><td rowspan="4">3D GS</td><td>PSNR</td><td>27.25</td><td>25.10</td><td>26.66</td><td>31.50</td><td>29.11</td><td>31.53</td><td>32.16</td></tr><tr><td>SSIM</td><td>0.856</td><td>0.747</td><td>0.756</td><td>0.925</td><td>0.914</td><td>0.932</td><td>0.946</td></tr><tr><td>LPIPS</td><td>0.122</td><td>0.244</td><td>0.243</td><td>0.198</td><td>0.184</td><td>0.117</td><td>0.181</td></tr><tr><td>Storage (MB)</td><td>1331.33</td><td>1350.78</td><td>1073.60</td><td>350.14</td><td>276.52</td><td>411.76</td><td>295.08</td></tr><tr><td rowspan="4">Ours-C1</td><td>PSNR</td><td>27.17</td><td>24.32</td><td>25.75</td><td>31.62</td><td>28.54</td><td>30.47</td><td>31.32</td></tr><tr><td>SSIM</td><td>0.832</td><td>0.672</td><td>0.768</td><td>0.913</td><td>0.889</td><td>0.910</td><td>0.923</td></tr><tr><td>LPIPS</td><td>0.169</td><td>0.355</td><td>0.312</td><td>0.229</td><td>0.226</td><td>0.147</td><td>0.206</td></tr><tr><td>Storage (MB)</td><td>26.90</td><td>25.71</td><td>34.84</td><td>16.85</td><td>17.92</td><td>23.68</td><td>17.74</td></tr><tr><td rowspan="4">Ours-C2</td><td>PSNR</td><td>27.38</td><td>24.78</td><td>26.41</td><td>31.82</td><td>28.75</td><td>30.71</td><td>32.14</td></tr><tr><td>SSIM</td><td>0.842</td><td>0.701</td><td>0.751</td><td>0.916</td><td>0.894</td><td>0.913</td><td>0.935</td></tr><tr><td>LPIPS</td><td>0.156</td><td>0.325</td><td>0.260</td><td>0.224</td><td>0.218</td><td>0.146</td><td>0.192</td></tr><tr><td>Storage (MB)</td><td>33.01</td><td>31.72</td><td>41.09</td><td>23.02</td><td>24.04</td><td>30.05</td><td>23.61</td></tr><tr><td rowspan="4">Ours-C3</td><td>PSNR</td><td>27.63</td><td>24.90</td><td>26.43</td><td>31.84</td><td>29.10</td><td>31.27</td><td>32.67</td></tr><tr><td>SSIM</td><td>0.847</td><td>0.717</td><td>0.753</td><td>0.917</td><td>0.900</td><td>0.918</td><td>0.941</td></tr><tr><td>LPIPS</td><td>0.147</td><td>0.303</td><td>0.267</td><td>0.220</td><td>0.212</td><td>0.137</td><td>0.186</td></tr><tr><td>Storage (MB)</td><td>39.40</td><td>37.81</td><td>47.24</td><td>28.95</td><td>30.02</td><td>35.92</td><td>29.84</td></tr><tr><td></td><td># of Parents</td><td>1.20M</td><td>1.06M</td><td>1.86M</td><td>330K</td><td>419K</td><td>913K</td><td>403K</td></tr><tr><td></td><td># of Children (k)</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>1</td><td>2</td></tr></table>
+
+Table 5: Per-scene metrics for our approach on the Tank&Temples dataset Knapitsch et al. (2017) and the Deep Blending dataset Hedman et al. (2018).  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Metric</td><td colspan="2">Tank&amp;Temples</td><td colspan="2">Deep Blending</td></tr><tr><td>Truck</td><td>Train</td><td>Drjohnson</td><td>Playroom</td></tr><tr><td rowspan="4">Compact GS</td><td>PSNR</td><td>25.070</td><td>21.560</td><td>29.260</td><td>30.320</td></tr><tr><td>SSIM</td><td>0.871</td><td>0.792</td><td>0.9000</td><td>0.902</td></tr><tr><td>LPIPS</td><td>0.163</td><td>0.240</td><td>0.258</td><td>0.258</td></tr><tr><td>Storage (MB)</td><td>41.57</td><td>37.29</td><td>47.98</td><td>38.45</td></tr><tr><td rowspan="4">Light GS</td><td>PSNR</td><td>24.561</td><td>21.095</td><td>-</td><td>-</td></tr><tr><td>SSIM</td><td>0.855</td><td>0.760</td><td>-</td><td>-</td></tr><tr><td>LPIPS</td><td>0.188</td><td>0.296</td><td>-</td><td>-</td></tr><tr><td>Storage (MB)</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td rowspan="4">Scaffold GS</td><td>PSNR</td><td>25.77</td><td>22.15</td><td>29.80</td><td>30.62</td></tr><tr><td>SSIM</td><td>0.883</td><td>0.822</td><td>0.907</td><td>0.904</td></tr><tr><td>LPIPS</td><td>0.147</td><td>0.206</td><td>0.250</td><td>0.258</td></tr><tr><td>Storage (MB)</td><td>107.00</td><td>66.00</td><td>69.00</td><td>63.00</td></tr><tr><td rowspan="4">3D GS</td><td>PSNR</td><td>25.350</td><td>22.070</td><td>29.060</td><td>29.870</td></tr><tr><td>SSIM</td><td>0.878</td><td>0.812</td><td>0.899</td><td>0.901</td></tr><tr><td>LPIPS</td><td>0.148</td><td>0.208</td><td>0.247</td><td>0.247</td></tr><tr><td>Storage (MB)</td><td>608.70</td><td>255.82</td><td>773.61</td><td>553.03</td></tr><tr><td rowspan="4">Ours-C1</td><td>PSNR</td><td>24.93</td><td>21.44</td><td>28.89</td><td>29.75</td></tr><tr><td>SSIM</td><td>0.856</td><td>0.763</td><td>0.894</td><td>0.895</td></tr><tr><td>LPIPS</td><td>0.196</td><td>0.283</td><td>0.280</td><td>0.284</td></tr><tr><td>Storage (MB)</td><td>23.11</td><td>20.90</td><td>23.59</td><td>22.21</td></tr><tr><td rowspan="4">Ours-C2</td><td>PSNR</td><td>25.22</td><td>21.72</td><td>28.93</td><td>30.28</td></tr><tr><td>SSIM</td><td>0.862</td><td>0.777</td><td>0.902</td><td>0.902</td></tr><tr><td>LPIPS</td><td>0.184</td><td>0.272</td><td>0.287</td><td>0.268</td></tr><tr><td>Storage (MB)</td><td>30.73</td><td>27.36</td><td>29.84</td><td>28.46</td></tr><tr><td rowspan="4">Ours-C3</td><td>PSNR</td><td>25.45</td><td>22.18</td><td>29.34</td><td>30.44</td></tr><tr><td>SSIM</td><td>0.866</td><td>0.792</td><td>0.898</td><td>0.905</td></tr><tr><td>LPIPS</td><td>0.182</td><td>0.240</td><td>0.270</td><td>0.265</td></tr><tr><td>Storage (MB)</td><td>36.01</td><td>34.63</td><td>35.80</td><td>35.00</td></tr><tr><td></td><td># of Parents</td><td>1M</td><td>900K</td><td>900K</td><td>834K</td></tr><tr><td></td><td># of Children (k)</td><td>1</td><td>1</td><td>2</td><td>2</td></tr></table>
\ No newline at end of file
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+# LIMITS OF DEEP LEARNING: SEQUENCE MODELING THROUGH THE LENS OF COMPLEXITY THEORY
+
+Nikola Zubić<sup>1</sup>, Federico Soldá<sup>2,*</sup>, Aurelio Sulser<sup>2,*</sup>, Davide Scaramuzza<sup>1</sup>
+
+$^{1}$ Robotics and Perception Group, University of Zurich, Switzerland zubic@ifi.uzh.ch, sdavide@ifi.uzh.ch
+
+$^{2}$ Algorithms and Optimization Group, ETH Zurich, Switzerland federico.solda@inf.ethz.ch, asulser@student.ethz.ch
+
+*Equal contribution
+
+# ABSTRACT
+
+Despite their successes, deep learning models struggle with tasks requiring complex reasoning and function composition. We present a theoretical and empirical investigation into the limitations of Structured State Space Models (SSMs) and Transformers in such tasks. We prove that one-layer SSMs cannot efficiently perform function composition over large domains without impractically large state sizes, and even with Chain-of-Thought prompting, they require a number of steps that scale unfavorably with the complexity of the function composition. Also, the language of a finite-precision SSM is within the class of regular languages. Our experiments corroborate these theoretical findings. Evaluating models on tasks including various function composition settings, multi-digit multiplication, dynamic programming, and Einstein's puzzle, we find significant performance degradation even with advanced prompting techniques. Models often resort to shortcuts, leading to compounding errors. These findings highlight fundamental barriers within current deep learning architectures rooted in their computational capacities. We underscore the need for innovative solutions to transcend these constraints and achieve reliable multi-step reasoning and compositional task-solving, which is critical for advancing toward general artificial intelligence.
+
+# 1 INTRODUCTION
+
+Deep learning has revolutionized numerous fields, achieving remarkable success in natural language processing (OpenAI, 2023; Google, 2024; Touvron et al., 2023), computer vision (Nguyen et al., 2022; Zubić et al., 2024; Zhu et al., 2024), scientific computing (Merchant et al., 2023; Hansen et al., 2023), and autonomous systems (Kaufmann et al., 2023; Bousmalis et al., 2024). The pursuit of general artificial intelligence now stands as the new frontier, aiming to develop Large Language Models (LLMs) capable of solving novel and complex tasks across diverse domains such as mathematics, coding, vision, medicine, law, and psychology, approaching human-level performance (Bubeck et al., 2023). Mastery of function composition is essential for this objective, as tasks like mathematical problem-solving (Li et al., 2023), learning discrete algorithms (Thomm et al., 2024; Velicković & Blundell, 2021), logical reasoning (Liu et al., 2023b), and dynamic programming (Dziri et al., 2023) are deeply compositional. However, despite impressive capabilities on various language tasks, deep learning models continue to struggle with tasks requiring complex reasoning over sequences, particularly those involving function composition and compositional reasoning (Peng et al., 2024; Dziri et al., 2023).
+
+These tasks necessitate breaking down problems into simpler sub-problems and composing the solutions to these subtasks. Current Transformer models (Vaswani et al., 2017), including advanced ones like GPT-4, find it challenging to handle tasks demanding deep compositionality (Dziri et al., 2023). For instance, we demonstrate that GPT-4 achieves only about  $27\%$  accuracy on basic tasks like 4-by-3-digit multiplication. One explanation for this limitation is the Transformer's inability to express simple state-tracking problems (Merrill & Sabharwal, 2023a). Structured State Space Models
+
+(SSMs) (Gu et al., 2022; Gu & Dao, 2023) have been introduced as an alternative to Transformers, aiming to achieve similar expressive power to Recurrent Neural Networks (RNNs) for handling problems that are naturally sequential and require state tracking. While SSMs have demonstrated impressive capabilities on various sequential tasks (Goel et al., 2022; Schiff et al., 2024), they exhibit similar limitations to Transformer models in solving function composition problems. For the same 4-by-3-digit multiplication task, Jamba (Lieber et al., 2024), an SSM-Attention hybrid model, achieves only  $17\%$  accuracy.
+
+Existing research has experimentally confirmed the inability of Transformers to perform function composition and compositional tasks (Dziri et al., 2023; Zhao et al., 2024), leading to issues such as hallucinations—responses that are incompatible with training data and prompts. Complexity theory analysis further reveals that Transformers belong to a weak complexity class, logspace-uniform  $\mathbf{TC}^0$  (Merrill & Sabharwal, 2023a), as do SSMs (Merrill et al., 2024), emphasizing their inherent limitations. While the impossibility of function composition for Transformers has been theoretically studied (Peng et al., 2024), a similar theoretical understanding for SSMs remains lacking.
+
+In this paper, we address this gap with two main contributions:
+
+1. We provide a theoretical framework using complexity theory to explain the limitations of SSMs in sequence modeling, particularly in their inability to perform function composition efficiently. We prove that one-layer SSMs cannot solve function composition problems over large domains without an impractically large state size (Theorem 1). Additionally, we show that even with Chain-of-Thought prompting, SSMs require a polynomially growing number of steps to solve iterated function composition problems (Theorem 2).  
+2. We extend our theoretical analysis to multi-layer SSMs, demonstrating that the computation of an  $L$ -layer SSM on a prompt of length  $N$  can be carried out using  $O(L \log N)$  bits of memory, positioning SSMs within the complexity class  $\mathbf{L}$  (logarithmic space). This implies that SSMs cannot solve problems that are  $\mathbf{NL}$ -complete unless  $\mathbf{L} = \mathbf{NL}$ , which is widely believed to be false (Peng et al., 2024). We further discuss that SSMs share this limitation with Transformers, highlighting a fundamental barrier in current deep learning architectures (Theorem 3).
+
+Our critical insight is the formal proof that SSMs cannot solve iterated function composition problems without a polynomially growing number of Chain-of-Thought steps (Theorems 1 and 2) and that even multi-layer finite-precision SSMs are limited to recognizing regular languages due to their essential equivalence to finite-state machines (Theorem 4). While CoT prompting can, to some extent, enable complex problem-solving by breaking down tasks into intermediate steps, it introduces a trade-off between the model's state size and the number of input passes required, leading to increased resource demands, which is not optimal.
+
+These findings underscore the need for innovative solutions beyond current deep learning paradigms to achieve reliable multi-step reasoning and compositional task-solving in practical applications.
+
+# 2 EQUIVALENCE OF SSMs WITH OTHER DEEP LEARNING MODELS
+
+Recent advancements in deep learning architectures have unveiled significant connections between SSMs and other prevalent models such as Linear Transformers. Notably, Dao & Gu (2024) have demonstrated equivalence between Linear Transformers and SSMs, indicating that the computational processes of these models are fundamentally related. Moreover, SSMs can be trained like Convolutional Neural Networks (CNNs) and inferred as Recurrent Neural Networks (RNNs), leveraging the benefits of both convolutional and recurrent architectures. This duality allows SSMs to efficiently capture long-range dependencies like RNNs while benefiting from the parallelism during training characteristic of CNNs.
+
+Additionally, Merrill et al. (2024) have shown that SSMs and Transformers belong to the same computational complexity class, specifically logspace-uniform  $\mathbf{TC}^0$ . This alignment in computational capacity reinforces the notion that the limitations observed in SSMs indicate inherent challenges within the broader landscape of deep learning models. Therefore, by focusing our theoretical and empirical analysis on SSMs, we effectively cover the representational capabilities of current deep-learning models, including Transformers and CNNs. This comprehensive coverage justifies our
+
+exploration of the limits of deep learning in sequence modeling through the lens of complexity theory. Our findings highlight the specific shortcomings of SSMs and shed light on the fundamental constraints of deep learning architectures in handling tasks that require reliable multi-step reasoning and compositional task-solving.
+
+# 3 BACKGROUND
+
+For two natural numbers  $n \leq m$ , we denote  $[n] = 1, 2, \ldots, n$  and  $[n, m] = n, n + 1, \ldots, m$ , with  $[0] = [n, n - 1] = \emptyset$ . We refer to the number of bits used in each computation as computational precision  $p$ . Given two domains  $B, C$ , we denote by  $C^B$  the set of all functions from  $B$  to  $C$ .
+
+Definition 1 (SSM layer). Given an input sequence  $\mathbf{x}_1, \ldots, \mathbf{x}_n \in \mathbb{R}^m$ , an SSM layer  $\mathcal{L}$  is defined in terms of a series of matrices  $\mathbf{A}_t \in \mathbb{R}^{d \times d}$ ,  $\mathbf{B}_t \in \mathbb{R}^{d \times m}$ ,  $\mathbf{C}_t \in \mathbb{R}^{m \times d}$ , and  $\mathbf{D}_t \in \mathbb{R}^{m \times m}$  for  $t \in [n]$ .  $\mathcal{L}$  defines a sequence of states  $\mathbf{h}_1, \ldots, \mathbf{h}_n \in \mathbb{R}^d$  as
+
+$$
+\boldsymbol {h} _ {t} = \boldsymbol {A} _ {t} \boldsymbol {h} _ {t - 1} + \boldsymbol {B} _ {t} \boldsymbol {x} _ {t}; \tag {1}
+$$
+
+and outputs the sequence  $\mathbf{y}_1, \ldots, \mathbf{y}_n \in \mathbb{R}^m$  as
+
+$$
+\boldsymbol {y} _ {t} = \boldsymbol {C} _ {t} \boldsymbol {h} _ {t} + \boldsymbol {D} _ {t} \boldsymbol {x} _ {t}. \tag {2}
+$$
+
+Generally, the matrices  $\mathbf{A}_t = \mathbf{A}(\mathbf{x}_t)$ ,  $\mathbf{B}_t = \mathbf{B}(\mathbf{x}_t)$ ,  $\mathbf{C}_t = \mathbf{C}(\mathbf{x}_t)$ , and  $\mathbf{D}_t = \mathbf{D}(\mathbf{x}_t)$  are functions of the input vector  $\mathbf{x}_t$  for each  $t \in [n]$ . In the special case when  $\mathbf{A}_t, \mathbf{B}_t, \mathbf{C}_t$ , and  $\mathbf{D}_t$  are independent from the input sequence  $\mathbf{x}_1, \ldots, \mathbf{x}_n$ , we call  $\mathcal{L}$  a linear SSM layer. Moreover, we call  $d$  the embedding dimension.
+
+Remark: Although SSMs can be linked to streaming algorithms due to their limited hidden state, applying communication complexity to analyze their limitations in function composition involves intricate considerations unique to SSMs. No known streaming lower bound directly applies to our specific setting. Our analysis accounts for the particular architectural constraints of SSMs, providing a better understanding of their capabilities than general streaming algorithms.
+
+# 4 FUNCTION COMPOSITION REQUIRES WIDE ONE-LAYER MODELS
+
+Our analysis considers one-layer SSMs to establish fundamental limitations in function composition tasks. The insights gained at the single-layer level highlight critical challenges that persist even in deeper architectures. The function composition problem has been introduced in (Peng et al., 2024) to provide a theoretical understanding of the causes of the hallucination of Transformer models. The aim is to evaluate the model's capability to combine relational information in the data to understand language, which is the core competence of large language models. Indeed to correctly answer questions like 'what is the birthday of Frédéric Chopin's father?' given the information that 'the father of Frédéric Chopin was Nicolas Chopin' and that 'Nicolas Chopin was born on April 15, 1771', the model needs to be able to compose the functions 'birthday-of' and 'father-of' (Peng et al., 2024), (Guan et al., 2024). Our analysis focuses on function compositions where the functions map elements from one finite, discrete domain to another, such as mapping individuals to their parents or birthdates. These functions operate over discrete sets, like persons and dates, and not over real-valued or continuous domains. Although this function composition task resembles a database join operation, it is important to note that our analysis focuses on how SSMs handle such compositions given natural language prompts. These prompts specify functions in an informal and potentially incomplete manner, lacking the full intensional knowledge present in formal database schemas. We aim to assess the model's ability to perform reasoning over such natural language prompts despite their potential incompleteness.
+
+Next, we give a precise formulation of the function composition problem due to (Peng et al., 2024). Consider two functions,  $g$  mapping a domain  $A$  to a domain  $B$ , and  $f$  mapping  $B$  to another domain  $C$ . These functions will be described in a prompt  $X$ . The  $N$  tokens of  $X$  are divided into four parts:
+
+1. the zeroth part describes the argument  $x \in A$  
+2. the first part describes the function  $g$  through  $|A|$  sentences in simple, unambiguous language separated by punctuation, e.g., 'the father of Frédéric Chopin is Nicolas Chopin',
+
+3. the second part consists of  $|B|$  sentences describing the function  $f$ , e.g., 'the birthday of Nicolas Chopin is April 15, 1771',  
+4. the third part is the query question asking for the value of  $f(g(x))$ .
+
+In this section, we discuss the theoretical limitations of SSMs for solving the function composition problem. In our analysis, the concept of domain size is crucial. While we primarily consider discrete domains, such as finite sets like  $[n] = \{1, 2, \dots, n\}$ , it is important to discuss what domain size means in other contexts. For continuous domains like the interval  $[1, n]$ , representing general functions would require infinitely many bits, making function composition intractable for models like SSMs and Transformers. Therefore, in practical settings, the maximum meaningful domain size is constrained by the total number of tokens and the prompt length, as the model's input capacity is limited. In our composition tasks, the functions are described within the prompt, so the prompt length effectively serves as an upper bound on the domain size.
+
+Theorem 1. Consider a function composition problem with input domain size  $|A| = |B| = n$  and an SSM layer  $\mathcal{L}$  with embedding dimension  $d$  and computation precision  $p$ . Let  $R = n\log n - (d^2 + d)p \geq 0$ , then the probability that  $\mathcal{L}$  answers the query incorrectly is at least  $R / (3n\log n)$  if  $f$  is sampled uniformly at random from  $C^B$ .
+
+The proof is based on a reduction from a famous problem in communication complexity (Peng et al., 2024), (Yao, 1979). Additional background on Communication Complexity and relevant problem classes can be seen in the Appendix A. We have three agents dubbed Faye, Grace, and Xavier. We assume that the agents have unbounded computational capabilities but, the only communication allowed is from Faye and Grace to Xavier. Faye knows a function  $f:[n]\mapsto [n]$  and the argument  $x\in [n]$ , Grace knows a function  $g:[n]\mapsto [n]$  and the argument  $x$ , while Xavier only knows the argument  $x\in [n]$ . The goal is for Xavier to compute the value of  $f(g(x))$ , minimizing the total number of bits communicated from Faye to Xavier and from Grace to Xavier.
+
+We report a lemma from (Peng et al., 2024), which gives a hardness result for the abovementioned problem.
+
+Lemma 1 (Lemma 1 from (Peng et al., 2024)). Consider the problem described above: if fewer than  $n \log n$  bits are communicated by Faye to Xavier, then Xavier cannot know the value  $f(g(x))$ . In particular, if only  $n \log n - R$  bits are communicated for some  $R \geq 0$ , then the probability that the composition is computed incorrectly is at least  $R / (3n \log n)$  if  $f$  is sampled uniformly at random from  $C^B$ .
+
+Now, we prove the theorem based on the Lemma above.
+
+Proof of Theorem 1. To establish the bound on  $q$ , we give a reduction of the communication problem above to the function composition problem. Let  $\mathcal{L}$  be an SSM layer that can solve the function composition problem with probability  $q$ .
+
+Suppose we have Faye, Grace, and Xavier as in the settings above, and Xavier wants to find the value  $f(g(x))$ . We construct the following prompt: the zeroth token  $x_0$  is 'the argument of the function is  $x$ ', for  $i \in [1, n]$  let  $x_i$  be the token 'g applied to  $i$  is  $g(i)$ ', where the information is provided by Grace, and for  $i \in [n + 1, 2n]$  let  $x_i$  be the token string 'f applied to  $i$  is  $f(i)$ ', where the information is provided by Faye. Xavier provides the last token string  $x_{2n + 1} =$  'what is the value of  $f(g(x))$ '. Since the SSM layer  $\mathcal{L}$  can solve the composition task with probability  $q$ , we have that:
+
+$$
+\boldsymbol {y} _ {2 n + 1} = \boldsymbol {C} _ {2 n + 1} \boldsymbol {h} _ {2 n + 1} + \boldsymbol {D} _ {2 n + 1} \boldsymbol {x} _ {2 n + 1} = f (g (x)) \tag {3}
+$$
+
+with probability  $q$
+
+But this allows us to construct the following communication protocol. Since Grace knows  $g$  and the argument  $x$ , she knows the values of  $\boldsymbol{x}_i$  for  $i \in [0, n]$  and she iteratively computes:
+
+$$
+\boldsymbol {h} _ {i} = \boldsymbol {A} _ {i} \boldsymbol {h} _ {i - 1} + \boldsymbol {B} \boldsymbol {x} _ {i}, \tag {4}
+$$
+
+and then sends  $h_n$  to Xavier. On the other hand, Faye knows  $f$  and hence the values of  $x_i$  for  $i \in [n + 1, 2n]$ , she computes the matrix:
+
+$$
+\mathcal {A} = \prod_ {j = n + 1} ^ {2 n} \mathbf {A} _ {j}, \tag {5}
+$$
+
+then the vector:
+
+$$
+\boldsymbol {b} = \sum_ {i = n + 1} ^ {2 n} \left(\prod_ {j = n + 1} ^ {2 n - i} \boldsymbol {A} _ {j}\right) \boldsymbol {B} _ {i} \boldsymbol {x} _ {i}, \tag {6}
+$$
+
+and she sends them to Xavier. At this point, Xavier computes:
+
+$$
+\boldsymbol {h} _ {2 n + 1} = \boldsymbol {A} _ {2 n + 1} \cdot (\boldsymbol {\mathcal {A}} \cdot \boldsymbol {h} _ {n} + \boldsymbol {b}) + \boldsymbol {B} _ {2 n + 1} \boldsymbol {x} _ {2 n + 1}. \tag {7}
+$$
+
+and finds the value of  $f(g(x))$  with probability  $q$  by computing  $y_{2n+1} = C_{2n+1} \cdot h_{2n} + D_{2n+1} \cdot x_{2n+1}$ . The total number of bits of communication between Faye and Xavier is  $(d^2 + d) \cdot p$ . By Lemma 1, it follows that  $q \leq R / (3n \log n)$ .
+
+Our theoretical results in Theorem 1 highlight that SSMs, like other deep neural networks, approximate functions rather than perform symbolic reasoning. Specifically, the probability bound indicates that if we attempt to compose functions over domains of size  $n$  with an SSM of embedding dimension  $d$  and computational precision  $p$  such that  $(d^2 + d)p < n\log n / 2$ , the model will output the incorrect result with a probability of at least  $1 / 6$ . To achieve a high probability of correctness (e.g.,  $99\%$ ),  $(d^2 + d)p$  must be significantly larger than  $n\log n / 2$ . This establishes a strong lower bound on the model's width, demonstrating that to accurately perform function composition over large domains, the model's capacity must increase substantially.
+
+While Theorem 1 addresses the limitations of one-layer SSMs, a natural question arises: Can deeper SSMs overcome these limitations? We conjecture that any SSM with a constant number of layers would still be unable to resolve the iterated composition task (as formalized in our Chain-of-Thought section 5). This is because accurately communicating token embeddings between layers becomes increasingly challenging as the depth grows. The difficulty in preserving and transmitting the necessary information across layers suggests that simply increasing the number of layers without a corresponding increase in model capacity does not address the fundamental limitations identified.
+
+# 5 MANY THOUGHT STEPS ARE NEEDED
+
+A chain of thought (CoT) is a series of intermediate natural language reasoning steps that lead to the final output. In this section, we focus on language models that can generate a similar chain of thought—a coherent series of intermediate reasoning steps that lead to the final answer for a problem. In (Wei et al., 2022), it was observed that CoT can mitigate the issue of hallucinations by encouraging the LLM to generate prompts that break down the task into smaller steps, eventually leading to the correct answer. In this section, we prove that, in general, many CoT steps are needed to break down compositional tasks.
+
+We start the discussion with the formal definition of an SSM with  $k$  CoT steps. It adapts the definition for the Transformer model of (Merrill & Sabharwal, 2024) to the case of SSMs.
+
+Definition 2 (SSM with CoT). Let  $\phi : (\mathbb{R}^m)^* \to \mathbb{R}^m$  be a function mapping a prefix of tokens to a new token. The function  $\phi$  is parametrized by an SSM layer  $\mathcal{L}$ .
+
+Given an input sequence  $\pmb{x}_1, \pmb{x}_2, \dots, \pmb{x}_n \in \mathbb{R}^m$ , we call:
+
+$$
+\phi_ {k} \left(\boldsymbol {x} _ {1}, \boldsymbol {x} _ {2}, \dots , \boldsymbol {x} _ {n}\right) = \phi_ {k - 1} \left(\boldsymbol {x} _ {1}, \boldsymbol {x} _ {2}, \dots , \boldsymbol {x} _ {n}\right) \cdot \phi \left(\phi_ {k - 1} \left(\boldsymbol {x} _ {1}, \boldsymbol {x} _ {2}, \dots , \boldsymbol {x} _ {n}\right), \boldsymbol {x} _ {1}, \boldsymbol {x} _ {2}, \dots , \boldsymbol {x} _ {n}\right),
+$$
+
+where  $\phi_1(\pmb{x}_1, \pmb{x}_2, \dots, \pmb{x}_n) = \phi(\pmb{x}_1, \pmb{x}_2, \dots, \pmb{x}_n)$  and  $\cdot$  denotes concatenation, the output of the SSM layer  $\mathcal{L}$  with  $k$  CoT-steps.
+
+In this section, we want to prove that while this procedure could help SSM layers with compositional tasks, it might require many CoT steps to be effective. In particular, we focus on the iterated function composition problem and show a lower bound on the number of CoT steps needed by an SSM layer to solve this problem correctly.
+
+In the iterated function composition problem we are given  $k$  functions  $f_{1}, f_{2}, \ldots, f_{k} : [n] \mapsto [n]$ , and we need to calculate  $f_{k}(f_{k-1}(\dots f_{2}(f_{1}(x))\dots))$  for  $x \in [n]$ . Here we restrict to the case when  $f_{1} = f_{2} = \dots = f_{k}$ , we define  $f^{(k)}(x) \coloneqq f(f(\dots f(x)))$ , and we call this  $k$ -iterated function composition problem.
+
+Theorem 2. Consider an iterated composition problem with domain size  $n$ , computation precision  $p$  and embedding dimension  $d$ . An SSM layer requires  $\Omega\left(\frac{\sqrt{n\log n}}{dp}\right)$  CoT steps for answering correctly iterated function composition prompts.
+
+The proof relies on reducing the iterated function composition problem from the pointer chasing problem (Papadimitriou & Sipser, 1982), a classical problem in communication complexity. In the  $k$ -steps pointer chasing problem, we have two agents dubbed Alice and Bob; Alice knows a function  $f_{A}:[n]\mapsto [n]$  and Bob knows a function  $f_{B}:[n]\mapsto [n]$ . We then define the pointers:
+
+$$
+z _ {1} = 1, \quad z _ {2} = f _ {A} (z _ {1}), \quad z _ {3} = f _ {B} (z _ {2}), \quad z _ {4} = f _ {A} (z _ {3}), \quad z _ {5} = f _ {B} (z _ {4}), \quad \ldots .
+$$
+
+The communication proceeds for  $2k$  rounds, with Alice starting. The goal is for Bob to output the binary value of  $z_{2k + 2} \mod 2$ . Following, we prove that an SSM layer with  $R$  CoT steps solving the iterated function composition problem can be used to design a communication protocol for the pointer chasing problem where the number of transmitted bits scales with  $R$ . The next fundamental Lemma in communication complexity gives a lower bound on the number of bits that need to be communicated in any such communication protocol, thus allowing the lower bound to be derived on the CoT steps.
+
+Lemma 2 (Theorem 1.1 (Yehudayoff, 2020)). Any randomized protocol for the  $k$ -steps pointer chasing problem with error probability  $1/3$  under the uniform distribution must involve the transmission of at least  $n/(2000k) - 2k \log n$  bits.
+
+Before we begin with the actual proof, let us introduce some notation. We note that  $\phi_k$  is a string of  $k$  tokens of  $\mathbb{R}^m$ . Moreover, to compute the new token  $\phi(\phi_{k-1}(\pmb{x}_1, \pmb{x}_2, \dots, \pmb{x}_n), \pmb{x}_1, \pmb{x}_2, \dots, \pmb{x}_n)$  the SSM layer  $\mathcal{L}$  computes  $n + (k-1)$  hidden states. We denote the  $i$ -th hidden state by  $\phi_{k,i}(\pmb{x}_1, \dots, \pmb{x}_n)$ .
+
+Proof of Theorem 2. The proof is similar to the proof of Theorem 2 in (Peng et al., 2024). We reduce the pointer chasing problem to the iterated composition problem with CoT prompts. In particular, we show that if the SSM  $\mathcal{L}$  can solve the  $k$ -iterated function composition problem with  $R$  CoT steps, then we can construct a protocol solving the  $(k - 1)$ -steps pointer chasing problem using  $2Rdp$  bits of communication.
+
+Fix a  $(k - 1)$ -steps pointer chasing problem for the function  $f_{A}, f_{B} : [n] \mapsto [n]$ . Define the function  $f : [2n] \mapsto [2n]$  as:
+
+$$
+f (i) = \left\{ \begin{array}{l l} f _ {A} (i) + n, & i \in [ 1, n ]; \\ f _ {B} (i - n), & i \in [ n + 1, 2 n ]. \end{array} \right. \tag {8}
+$$
+
+We point out that  $f^{(k)}(i) = (f_B \circ f_A)^{(k)}(i)$ . Consider the  $k$ -iterated function composition problem for  $f$  and suppose that there exists an SSM  $\mathcal{L}$  that solves it using  $R$  CoT steps.
+
+We construct the following prompt: for  $i \in [1, n]$  let  $\pmb{x}_i$  be the token 'f applied to  $i$  is  $f(i)$ ', where the information  $f(i)$  is provided by Alice, and for  $i \in [n + 1, 2n]$  let  $\pmb{x}_i$  the token string 'f applied to  $i$  is  $f(i)$ ', where the information  $f(i)$  is provided by Bob. The last token string  $x_{2n + 1}$  is given by 'what is the value of  $f^{(k)}$  applied to  $x$ '. Since the SSM layer  $\mathcal{L}$  can solve the  $k$ -iterated function composition task with  $R$  CoT steps, we have that  $\phi_R(\pmb{x}_1, \dots, \pmb{x}_{2n})$  is the right answer for  $f^k(x)$ . We will use this fact to construct a communication protocol transmitting at most  $2 \cdot Rdp$  bits. The communication protocol lasts for  $R$  rounds.
+
+In the  $r$ -th round Alice computes  $\phi_{r,n+k}(\pmb{x}_1, \ldots, \pmb{x}_{2n})$  from  $\phi_{r-1}(\pmb{x}_1, \ldots, \pmb{x}_{2n})$  (where  $\phi_0(\pmb{x}_1, \ldots, \pmb{x}_{2n})$  is the empty string of tokens) and  $\pmb{x}_1, \ldots, \pmb{x}_n$  and communicates it with Bob. Bob on the other hand computes  $\phi_r(\pmb{x}_1, \ldots, \pmb{x}_{2n})$  from  $\phi_{r,n+k}(\pmb{x}_1, \ldots, \pmb{x}_{2n})$  and  $\pmb{x}_{n+1}, \ldots, \pmb{x}_{2n}$  and transmits it to Alice. In each iteration, at most  $dp$  bits are communicated from Alice to Bob and from Bob to Alice.
+
+After  $R$  rounds, Bob knows the value of  $\phi_R(\pmb{x}_1, \dots, \pmb{x}_{2n})$ . By hypothesis, this is the solution to the  $(k - 1)$ -steps pointer chasing problem. Notice that the total number of bits communicated by the protocol is  $2Rdp$ . In conclusion, we fix  $k = \frac{1}{100}\sqrt{\frac{n}{\log n}} + 1$  and by Lemma 2 we get that  $2Rdp \geq n / (2000k) - 2k\log n$  which gives  $R \geq \frac{3}{100}\frac{\sqrt{n\log n}}{dp}$
+
+# 6 SSMs ARE LIMITED TO REGULAR LANGUAGES
+
+In Peng et al. (2024), it is suggested to analyze the computational capability of LLMs on the computational problems below. The empirical compositional tasks studied in later sections—multiplication of multi-digit integers, dynamic programming, and logic puzzles such as "Einstein's Riddle"—can be expressed in terms of these computational problems (Peng et al., 2024).
+
+Circuit evaluation: Given the description of a circuit with gates, which can be either Boolean or arithmetic operations, as well as the values of all input gates of the circuit, evaluate the output(s) of the circuit. Multiplying decimal integers with multiple digits is an example of such a circuit.
+
+Derivability: Given a finite domain  $S$  and a relation  $D \subseteq S \times S$ . For a given initial set  $I \subseteq S$  and a final set  $F \subseteq S$ , answer the question whether there are elements  $a_1, a_2, \ldots, a_k \in S$  such that (a)  $a_0 \in I$ , (b)  $a_k \in F$ , and (c) for all  $j$  such that  $0 < j \leq k$ ,  $(a_{j-1}, a_j) \in D$ .
+
+Logical reasoning: Logic puzzles like 'Einstein's Riddle' can typically be formulated as satisfiability (or SAT) instances. This problem is NP-complete. However, most common-sense reasoning can be expressed by one of the three tractable exceptional cases of SAT: 2-SAT, Horn SAT, and Mod 2 SAT.
+
+In Peng et al. (2024), it was noted that Derivability and 2-SAT are NL-complete, while Horn SAT and Circuit Evaluation are P-complete problems. Since the log-precision Transformer model lies in the complexity class log-uniform  $\mathbf{TC}^0 \subseteq \mathbf{L}$  (Merrill & Sabharwal, 2023b), these problems cannot be solved by a log-precision Transformer model provided  $\mathbf{NL} \neq \mathbf{L}$  (which is a widely believed hypothesis in computational complexity). For Mod 2 SAT, the result is valid provided the weaker statement  $\mathbf{L} \neq \mathrm{Mod} 2\mathbf{L}$ . For Horn SAT and Circuit Evaluation, the result holds unless the stronger statement  $\mathbf{L} = \mathbf{P}$  holds. Very recently, in Merrill et al. (2024), it was established that log-precision linear and S6-SSMs (Gu & Dao, 2023) are also part of the complexity class log-uniform  $\mathbf{TC}^0$ , which yields the following theorem similar to the case of Transformers.
+
+Theorem 3. The problems of Derivability and 2-SAT cannot be solved by log-precision linear or S6-SSMs provided  $\mathbf{L} \neq \mathbf{NL}$ . For Mod 2 SAT, the result is valid provided the weaker statement  $\mathbf{L} \neq \operatorname{Mod} 2\mathbf{L}$  holds. For Horn SAT and Circuit Evaluation, the result holds unless the stronger statement  $\mathbf{L} = \mathbf{P}$  holds.
+
+So far, we have explored SSMs' computational capabilities and limitations in various settings. In particular, SSMs face fundamental challenges when handling tasks requiring complex reasoning or computations beyond their inherent architectural constraints. Building on these insights, we now examine the impact of finite precision arithmetic on the computational power of SSMs. In practical implementations, SSMs operate with finite precision due to hardware constraints, typically using fixed-point or floating-point representations with a limited number of bits. This finite precision restricts the range and granularity of values that the model's parameters and hidden states can represent. Let us denote by  $S$  the vectors that the hidden states can take. Consequently, it is reasonable to assume that an SSM does not embed or store all the information provided during training. Instead, it must extract and retain only a subset of relevant information within a memory of much smaller capacity than the input.
+
+Since SSMs have a fixed hidden dimension  $d$  and operate with finite precision, the set  $\mathcal{S}$  is finite. This finiteness imposes significant restrictions on the types of languages that SSMs can compute or recognize. To formalize this limitation, we introduce the following definition:
+
+Definition 3. Given a finite-precision SSM, we say the SSM accepts an input sequence  $\mathbf{x}_1, \ldots, \mathbf{x}_n$  if the hidden state  $\mathbf{h}_{n+1} \neq \mathbf{0}$ . We call the set of accepted input sequences the language of the SSM.
+
+The following theorem relates the computational power of a finite precision SSM to a Finite-State Machine (FSM).
+
+Theorem 4. The language of a finite-precision SSM is within the class of regular languages.
+
+Proof. We construct the Finite-State Machine  $M = (Q, \Sigma, \delta, q_0, F)$ , where:
+
+1. the set of states  $Q$  is the set of all possible vectors  $f$ ,  
+2. the finite input alphabet  $\Sigma$  is as well  $S$ ,
+
+3. the transition function  $\delta: Q \times \Sigma^2 \to Q$  is defined by the SSM's state update equations, i.e.,  $\delta(\pmb{h}, \pmb{x}) = \pmb{A}(\pmb{x})\pmb{h} + \pmb{B}(\pmb{x})\pmb{x}$  
+4.  $q_{0}$  is the initial state corresponding to the initial hidden state  $h_{0}$ ,  
+5.  $F = Q\setminus \{\mathbf{0}\}$  is the set of accepting states.
+
+Since the SSM has finite precision, the set  $\mathcal{V}$  is finite, and this defines a valid FSM. It is immediate that the FSM models the transitions of the SSM and accepts the same language.
+
+This result relates the computational power of SSMs to FSMs. Consequently, under finite precision constraints, SSMs cannot recognize languages beyond regular languages. Regarding computational limitations, tasks requiring computational models with greater expressive power, such as context-free grammars or context-sensitive grammars, cannot be efficiently solved by SSMs with finite precision. Examples of such tasks include recognizing balanced parentheses, detecting palindromic sequences, and performing more complex logical inference that necessitates memory beyond finite states.
+
+These limitations are significant because they highlight the boundaries of what SSMs can achieve in practical settings. Regarding practical considerations, since real-world implementations of SSMs operate on hardware with finite memory and finite precision arithmetic, these theoretical limitations directly apply to SSMs used in actual applications. Therefore, when designing systems for tasks that require processing beyond regular languages, it becomes clear that SSMs with finite precision do not suffice, and alternative architectures or computational mechanisms need to be considered to overcome these inherent constraints.
+
+# 7 EXPERIMENTS
+
+Our theoretical results suggest that SSMs inherently struggle with function composition and multi-step reasoning tasks due to their architectural limitations. To validate these findings, we empirically assess SSMs' performance on practical tasks requiring these capabilities.
+
+We evaluate the inability of various sequence models to address function composition tasks by examining three axes of composition: spatial, temporal, and relational (Appendix B.1). This evaluation uses four datasets designed to test function composition. Subsequently, we proceed to compositional tasks involving multi-digit multiplication, dynamic programming, and Einstein's puzzle. We investigate the effects of Chain-of-Thought (CoT) prompting (Appendix B.2) and conduct a thorough error analysis to understand the failure points and underlying reasons for the erroneous behavior (Appendix B.3).
+
+We conducted GPT experiments using the ChatGPT API (OpenAI, 2023) and performed all experiments with the GPT-4 model as of June 2024, while other models were evaluated on machines equipped with 2x NVIDIA A100 80 GB GPUs. We used Jamba version 1. Unless otherwise specified, each task is evaluated three times with 500 test samples per evaluation to ensure consistency and minimize variance. All other experimental details, including prompts and additional results, are provided in the Appendix.
+
+# 8 RELATED WORK
+
+Limitations in Function Composition and Reasoning Recent studies have underscored the limitations of deep learning models, particularly Transformers, in handling tasks requiring deep compositionality and multi-step reasoning (Peng et al., 2024; Dziri et al., 2023). These tasks are crucial in applications like mathematical problem-solving (Li et al., 2023), algorithm learning (Thomm et al., 2024; Velicković & Blundell, 2021), logical reasoning (Liu et al., 2023b), and dynamic programming (Dziri et al., 2023). Despite their capabilities, Transformers have been shown to struggle with function composition, which is essential for understanding relational information in data (Guan et al., 2024).
+
+Research has highlighted architectural and training limitations that prevent these models from maintaining accuracy over multiple reasoning steps, leading to issues like hallucinations and reasoning errors (Merrill & Sabharwal, 2023a; Zhao et al., 2023). Studies by Merrill et al. (2024) and Peng et al.
+
+(2024) have identified that both Transformers and SSMs belong to weak complexity classes, such as logspace-uniform  $\mathbf{TC}^0$ , which limits their computational abilities. However, prior work primarily focused on Transformers, with SSMs not thoroughly investigated theoretically and empirically concerning their ability to perform function composition and compositional tasks. Our contribution fills this gap by providing a comprehensive theoretical framework and empirical analysis specific to SSMs.
+
+Chain-of-Thought Prompting The Chain-of-Thought (CoT) prompting method has been proposed to improve reasoning capabilities in large language models by breaking down complex tasks into smaller, intermediate steps (Wei et al., 2022). CoT prompting aims to mitigate issues like hallucinations and enhance multi-step reasoning by encouraging models to generate intermediate reasoning steps. While CoT has shown promise in specific contexts, recent research indicates that even with CoT prompting, current models remain inadequate for solving deeply compositional tasks (Merrill & Sabharwal, 2023a; Liu et al., 2023a). Our work supports these findings, demonstrating that CoT prompting does not overcome the fundamental computational limitations of SSMs and Transformers in tasks requiring complex reasoning.
+
+While advanced methods like tree search algorithms (Trinh et al., 2024; Polu & Sutskever, 2020; Lample et al., 2022) and self-correction techniques (Wang et al., 2024; Kumar et al., 2024) have been proposed to improve reasoning by integrating external mechanisms, our work focuses on the inherent computational limitations of SSMs and Transformers when used without such augmentations. These external engines can mitigate some limitations by leveraging additional resources, but they do not address the core architectural constraints we have identified.
+
+Expressive Power and Complexity of Neural Networks A growing body of work is exploring the expressive power of neural network architectures and their limitations from a computational complexity perspective. Weiss et al. (2018) and Siegelmann & Sontag (1992) examined the capabilities of recurrent neural networks in relation to Turing machines. Pérez et al. (2019) investigated the Turing completeness of Transformers under certain conditions.
+
+More recently, Merrill et al. (2020) analyzed the relationship between network depth, parameter size, and computational expressivity. Bhattamishra et al. (2020) explored the computational limitations of Transformers concerning formal languages. Our work contributes to this line of research by analyzing SSMs within the framework of computational complexity, specifically their placement within the class  $\mathbf{L}$  and implications for their reasoning capabilities.
+
+Alternative Approaches to Complex Reasoning Given the limitations of current architectures, researchers have explored alternative approaches to enhance models' reasoning abilities. Methods include integrating external memory modules (Graves et al., 2016), incorporating symbolic reasoning components (Gaunt et al., 2017), and developing neuro-symbolic models (Dai et al., 2019). These approaches aim to combine the strengths of neural networks with symbolic computation to overcome the shortcomings in tasks requiring complex, multi-step reasoning. Our findings underscore the necessity for such innovative solutions, suggesting that overcoming the fundamental limitations identified requires moving beyond traditional deep learning paradigms.
+
+# 9 CONCLUSION
+
+In this work, we have demonstrated both theoretically and empirically that Structured State Space Models (SSMs) and Transformers face fundamental limitations in performing function composition and complex reasoning tasks. Our theoretical analysis shows that overcoming these limitations would require architectures beyond finite-state machines. SSMs with fixed hidden dimensions and layers are equivalent to finite-state machines and thus limited to regular languages (Theorem 4). This limitation explains their inability to handle tasks that require computational power beyond regular languages, such as context-free languages or problems that are NL-complete.
+
+Our empirical evaluations confirm these findings, revealing significant performance degradation as task complexity increases, even when employing advanced prompting techniques. Models often resort to shortcuts, leading to errors in multi-step reasoning processes. These results highlight that current deep learning architectures are fundamentally limited in their ability to perform reliable
+
+multi-step reasoning and compositional task-solving due to their architectural constraints. This underscores the necessity for innovative architectural solutions or computational frameworks that can handle such tasks more efficiently. Future research should explore new directions, such as integrating symbolic reasoning components, improving memory and state-tracking capabilities, or developing hybrid models that transcend the limitations of existing architectures. Addressing these challenges is crucial for advancing toward general artificial intelligence capable of sophisticated reasoning and problem-solving across diverse domains.
+
+# 10 ACKNOWLEDGMENT
+
+This work was supported by the European Research Council (ERC) under grant agreement No. 864042 (AGILEFLIGHT).
+
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+
+# Appendices
+
+# A Background on Communication Complexity and Computational Classes 16
+
+A.1 Communication Complexity 16  
+A.2 Computational Complexity Classes 16  
+A.3 Key Problems and Their Complexity 17
+
+# B Main Experiments 18
+
+B.1 Function composition and compositional tasks 18  
+B.2 CoT experiments 20  
+B.3 Error analysis 20  
+B.4 Learning Algorithmic Compositions 22
+
+# C Compositional Tasks Details 22
+
+C.1 Multiplication 22  
+C.2 Einstein's Puzzle 22  
+C.3 Dynamic Programming 24
+
+# D Details of CoT experiments 25
+
+D.1 Main CoT experiments 25  
+D.2 Performance of other models on multiplication and puzzle tasks 26  
+D.3 Few-shot prompting multiplication results 26
+
+# E Algorithmic Compositions 27
+
+E.1 Prompts for SSM and Attention-based models for PEN task 28  
+E.2 PEN generation code 30  
+E.3 PER Multi-generation code 31
+
+# A BACKGROUND ON COMMUNICATION COMPLEXITY AND COMPUTATIONAL CLASSES
+
+To provide a solid foundation for our theoretical results, we offer an overview of key concepts in communication complexity and computational complexity theory. This background is essential for understanding the limitations of SSMs and other deep learning architectures in sequence modeling tasks that require complex reasoning.
+
+# A.1 COMMUNICATION COMPLEXITY
+
+Communication complexity studies the amount of communication required between two or more parties to compute a function whose input is distributed among them. It provides lower bounds on the communication needed for distributed computation.
+
+Communication Protocols: A communication protocol specifies the rules by which parties exchange messages to compute a function collaboratively. The primary goal is to minimize the total number of bits exchanged.
+
+# Key Problems in Communication Complexity:
+
+- Function Composition Problem: Two parties, Faye and Grace, hold functions  $f: B \to C$  and  $g: A \to B$ , respectively, along with a common input  $x \in A$ . Their goal is to compute  $f(g(x))$  with minimal communication to a third party, Xavier. This problem models scenarios where composing functions over large domains requires significant communication, highlighting the challenges in function composition tasks for sequence models.  
+- Pointer Chasing Problem: This involves two parties who alternately apply functions to an initial input over several rounds. It is a fundamental problem used to establish lower bounds in communication complexity. It demonstrates that certain computations inherently require a substantial amount of communication, regardless of the protocol used.  
+- Set Disjointness Problem: Two parties each hold a subset of a universal set and wish to determine if their subsets intersect without revealing additional information. This problem is notable for having high communication complexity, serving as a basis for proving lower bounds in various computational models.
+
+Relevance to Sequence Modeling: Communication complexity provides tools to prove theoretical limits on the capabilities of computational models, including neural networks. By reducing problems in communication complexity to tasks performed by sequence models, we can establish lower bounds on the resources required (e.g., hidden state size, number of layers) for these models to perform certain computations. This approach helps in understanding why models like SSMs struggle with tasks requiring complex reasoning or function composition.
+
+# A.2 COMPUTATIONAL COMPLEXITY CLASSES
+
+Computational complexity theory classifies problems based on the resources required to solve them, such as time or memory space. Understanding these classes is crucial for characterizing the limitations of computational models.
+
+# Key Complexity Classes:
+
+- L (Logarithmic Space): The class of decision problems solvable by a deterministic Turing machine using logarithmic amount of memory space with respect to the input size. Problems in L are considered efficiently solvable with very limited memory.  
+- NL (Nondeterministic Logarithmic Space): Consists of decision problems solvable by a nondeterministic Turing machine using logarithmic space. NL is a broader class than L, as nondeterminism allows guessing and verifying solutions using limited memory.  
+- P (Polynomial Time): Contains decision problems solvable by a deterministic Turing machine in polynomial time. It represents problems that are efficiently solvable in terms of time, without specific memory constraints.
+
+- Regular Languages: The class of languages recognizable by finite automata or equivalently, by regular expressions. They are the simplest class in the Chomsky hierarchy and can be recognized using constant memory.  
+- Context-Free Languages: Recognizable by pushdown automata, these languages can handle nested structures and require memory that grows with the input size.  
+-  $\mathbf{TC}^0$  (Constant Depth Threshold Circuits): A class of problems solvable by constant-depth, polynomial-size circuits with threshold gates. These circuits can compute certain functions very efficiently in parallel.
+
+# Relationships Between Classes:
+
+$$
+\text {R e g u l a r L a n g u a g e s} \subseteq \mathbf {L} \subseteq \mathbf {N L} \subseteq \mathbf {P} \tag {9}
+$$
+
+It's widely believed that these inclusions are strict (e.g.,  $\mathbf{L} \neq \mathbf{NL}$ ), meaning each class strictly contains the previous one.
+
+Relevance to Sequence Modeling: By placing computational models within these complexity classes, we can formalize their computational power and limitations. For instance:
+
+- Finite-State Machines (FSMs): Equivalent to models that recognize regular languages. They have a finite number of states and cannot handle tasks requiring memory that scales with input size.  
+- Pushdown Automata: Recognize context-free languages and can handle nested or recursive structures due to their use of a stack.  
+- SSMs and Transformers: Our analysis shows that SSMs with fixed hidden dimensions and layers are equivalent to FSMs, limiting them to regular languages. Similarly, Transformers have been shown to have limitations corresponding to the class  $\mathrm{TC}^0$  or L under certain conditions.
+
+Implications for SSMs: Understanding that SSMs are limited to regular languages explains why they struggle with tasks requiring more computational power, such as:
+
+- Function Composition: Requires the ability to maintain and manipulate information over long sequences, which exceeds the capabilities of finite-state models.  
+- Complex Reasoning Tasks: Problems like multi-digit multiplication, logical puzzles, and dynamic programming necessitate memory and computational resources beyond what is available in models limited to regular languages.
+
+By grounding our analysis in communication complexity and computational complexity theory, we establish a theoretical foundation for the limitations of SSMs. This background enables us to formalize the challenges faced by sequence models in handling tasks that require computational resources beyond regular languages and logarithmic space.
+
+# A.3 KEY PROBLEMS AND THEIR COMPLEXITY
+
+To further contextualize the limitations of SSMs, we briefly describe some computational problems and their associated complexity classes:
+
+- Derivability (NL-Complete): Given a finite set and a relation, determine if there is a sequence of elements satisfying certain conditions. This problem requires nondeterministic logarithmic space and cannot be solved by models limited to  $\mathbf{L}$  unless  $\mathbf{L} = \mathbf{NL}$ .  
+- 2-SAT (NL-Complete): A satisfiability problem where each clause has at most two literals. It is solvable in polynomial time but is NL-complete, indicating it requires more than deterministic logarithmic space.  
+- Horn SAT and Circuit Evaluation (P-Complete): Problems that are as hard as any problem in P. Solving these efficiently would require polynomial time computation, beyond the capabilities of FSMs.
+
+- Mod 2 SAT (Beyond L): Involves solving satisfiability problems modulo 2. Requires computational resources beyond deterministic logarithmic space.
+
+Relevance to Our Work: The inability of SSMs to solve these problems stems from their equivalence to finite-state machines. Since FSMs cannot utilize memory that grows with input size, they are inherently incapable of solving problems that require maintaining and processing an unbounded amount of information. The limitations highlighted by these complexity classes and problems suggest that to handle complex reasoning tasks effectively, sequence models need architectures that go beyond finite-state computations. This could involve models that can simulate pushdown automata or Turing machines, allowing them to recognize context-free languages or perform computations requiring more substantial memory resources.
+
+# B MAIN EXPERIMENTS
+
+# B.1 FUNCTION COMPOSITION AND COMPOSITIONAL TASKS
+
+In the context of Large Language Models (LLMs), compositional tasks differ from function composition. Function composition  $f_{K}(f_{K - 1}(\ldots (f_{1}(x))))$  is a mathematical process where the output of one function serves as the input for another across multiple functions  $f_{1}, f_{2}, \ldots, f_{K}$ . Conversely, LLM compositional tasks involve breaking down complex inputs into simpler parts and integrating the results to generate an overall output. Examples include (i) combining linguistic elements to generate coherent text, (ii) solving multi-step reasoning problems, and (iii) decomposing complex tasks (e.g., multi-turn conversations, summarization) into manageable sub-tasks.
+
+Solving compositional tasks necessitates the capability to perform function composition (Peng et al., 2024; Dziri et al., 2023) and demands additional competencies such as contextual understanding, multi-step reasoning, and the integration of diverse information types. A model's proficiency in function composition is a critical prerequisite for tackling complex compositional tasks (Lu et al., 2023). For instance, if an SSM-powered LLM cannot evaluate  $f(g(x))$ , it will be inadequate for tasks involving multi-step arithmetic or logical operations that depend on nested functions.
+
+Composition tasks We begin with three fundamental composition tasks: spatial, temporal, and relationship compositions. These axes are crucial as they encapsulate core aspects of comprehending and interacting with the world. Spatial composition entails integrating information about the positions and orientations of objects. Temporal composition involves reasoning over sequences and durations of events. Relationship composition focuses on understanding the connections between entities, such as those in a genealogy tree.
+
+Number of Parameters We conducted experiments using Jamba (Lieber et al., 2024) (joint Mamba and Attention) with 7B parameters, Mamba (Gu & Dao, 2023) with 2.8B parameters, S4-H3 (Gu et al., 2022; Fu et al., 2023) with 2.7B parameters, GPT-4 (OpenAI, 2023), and GPT-4o models. Qualitative results are presented in Fig. 1. As illustrated in Fig. 1, all models failed to answer questions across the three composition axes correctly.
+
+![](images/5b0cd30bc9e739e21fc5dcd00d8f18e6c758be3e458d6c09ba7273bfd81101aa.jpg)  
+Figure 1: Qualitative example of zero-shot inference on prominent SSM and Attention-based models. None of the models successfully resolved the problems across any of the composition axes.
+
+# Problem 2 - Temporal axis
+
+Question: Anne is the younger sister of Erwin, Erwin is the elder brother of Daniel. Is Anne younger than Daniel?
+
+![](images/df8c49258e0403332b68392d88091f3d543189aed48b30d130ea70f86f12edf7.jpg)
+
+Correct: Not enough information.
+
+# Problem 3 - Relationship axis
+
+Question: Alan is the son of Marco, Joe is the son of Alan. Does Alan have any grandchildren?
+
+![](images/a8add640cd0d62664f6fb925efce4e5e23cb7808b5dd29a4a2b51b8491ce22dd.jpg)
+
+Correct: Not enough information.
+
+To quantitatively assess the limitations of models, including the latest GPT-4o (OpenAI, 2023), in solving function composition tasks, we evaluate their performance on four datasets specifically designed to test these capabilities. Unless otherwise specified, each model is tested on 500 samples.
+
+Composition datasets Math-QA dataset, derived from (Li et al., 2023), includes 25 math topics. We focus on the first 100 samples from Algebra, Calculus, Combinatorics, Game Theory, and Trigonometry. Problems involve solving function compositions and temporal reasoning. BIG-Bench Hard (Suzgun et al., 2022) dataset features 250 Boolean expressions that the model must evaluate. In Temporal-NLI (Thukral et al., 2021) dataset, each sample consists of a premise (e.g., "They got married on Saturday") and a hypothesis (e.g., "They got married before Friday"), requiring the model to determine if the relationship is entailment, contradiction, or neutral. SpaRTUN (Mirzaee & Kordjamshidi, 2022) dataset is designed for spatial reasoning, and it includes stories describing the spatial positions of objects, followed by questions about the orientation of one object relative to another (e.g., left, right, inside, above).
+
+<table><tr><td></td><td>GPT-4o</td><td>GPT-4</td><td>Jamba</td><td>Mamba</td><td>S4-H3</td></tr><tr><td>Math-QA</td><td>51.8%</td><td>51.0%</td><td>42.2%</td><td>35.0%</td><td>28.6%</td></tr><tr><td>BIG-Bench Hard</td><td>56.8%</td><td>58.4%</td><td>78.2%</td><td>67.0%</td><td>60.6%</td></tr><tr><td>Temporal-NLI</td><td>79.4%</td><td>77.2%</td><td>69.8%</td><td>59.2%</td><td>54.6%</td></tr><tr><td>SpaRTUN</td><td>80.8%</td><td>61.4%</td><td>50.8%</td><td>42.2%</td><td>35.2%</td></tr></table>
+
+Table 1: Performance of Attention, SSM and Attention-SSM based models on various function composition tasks involving logical expressions, temporal reasoning, spatial reasoning, and math tasks.  
+
+<table><tr><td></td><td>GPT-4o</td><td>GPT-4</td><td>Jamba</td><td>Mamba</td><td>S4-H3</td></tr><tr><td>Algebra</td><td>51%</td><td>47%</td><td>42%</td><td>36%</td><td>29%</td></tr><tr><td>Calculus</td><td>50%</td><td>48%</td><td>41%</td><td>34%</td><td>28%</td></tr><tr><td>Combinatorics</td><td>88%</td><td>70%</td><td>48%</td><td>38%</td><td>33%</td></tr><tr><td>Game theory</td><td>30%</td><td>40%</td><td>50%</td><td>41%</td><td>32%</td></tr><tr><td>Trigonometry</td><td>40%</td><td>50%</td><td>30%</td><td>26%</td><td>21%</td></tr></table>
+
+Table 2: Performance of models on various topics within the MathQA dataset. Input dependency consistently improves performance, with Mamba consistently outperforming S4-H3.
+
+The results presented in Tables 1 and 2 highlight several critical observations regarding the performance of various models across different composition tasks. Notably, Mamba (Gu & Dao, 2023) consistently outperforms the S4-H3 (Gu et al., 2022; Fu et al., 2023) model, despite both having almost the same number of parameters. This performance gap underscores the importance of input-dependence in model design, as Mamba's architecture better leverages input information to achieve superior results. Additionally, while GPT-4o is the most performant overall, it struggles with many tasks, including those that seem simple to humans, such as logical expression chaining, as evidenced by its performance on the BIG-Bench Hard (Suzgun et al., 2022) benchmark. This indicates that even state-of-the-art models like GPT-4o have limitations in solving complex composition tasks, which numerically justifies our theoretical findings. Accuracy for all models is calculated as the number of correct answers divided by the total number of samples.
+
+Compositional tasks Having demonstrated that models encounter difficulties even with more straightforward composition tasks, we now examine their performance on more complex compositional tasks. Given their proven inability to perform function composition, as established in Theorem 1, it is entirely anticipated that their performance on these tasks will be suboptimal. We explore three compositional tasks: (i) multi-digit multiplication, (ii) dynamic programming, and (iii) Einstein's puzzle.
+
+For the multi-digit multiplication task, we generate question-answer pairs such as "What is 5 times 90?" with the answer being "450". This task involves multiplying two numbers,  $x = (x_{1}, x_{2}, \ldots, x_{k})$  and  $y = (y_{1}, y_{2}, \ldots, y_{k})$ , where each number can have up to  $k$  digits. Consequently, there are  $9 \times 10^{(k-1)}$  possible combinations for each number. In our experiments, we set  $k$  to 5 and found that both Attention and SSM-based models are unable to solve the 5-by-5 digit multiplication task, even in the case of GPT-4o with CoT prompting (A-12).
+
+![](images/c4cc369b18915f32a9d1d7d4093bba0902978712d59e7ffecc179a1da8ffbd00.jpg)  
+Figure 2: Jamba Lieber et al. (2024) performance on multiplication, DP and puzzle tasks. For DP various models are shown. All struggle with compositional tasks, especially for larger input size.
+
+![](images/0027f6afbfceb11cc9ba6502d8f6c8f1a658bd40e768e94df76d67bdb0b25ad0.jpg)
+
+![](images/cc0ced035a10962c357365dfbfa6a43c96eeac213377ec9f1996253b84c64044.jpg)
+
+Dynamic programming (DP) recursively decomposes complex problems into simpler sub-problems, making solutions compositional by nature. We consider a classic relaxation of the NP-complete Maximum Weighted Independent Set problem (Kleinberg & Tardos, 2005): Given a sequence of integers, find a subsequence with the highest sum such that no two numbers in the subsequence are adjacent in the original sequence. DP can solve This relaxation in  $O(n)$  time. For our experiments, we restrict each integer to the range  $[-5, 5]$  and follow the generation steps as in (Dziri et al., 2023), with an input list containing from 2 to 6 elements. Prompting details are shown in the A-C.3.
+
+Einstein's puzzle is a well-known logic puzzle commonly used as a benchmark for solving constraint satisfaction problems (Prosser, 1993). It involves a series of houses with various attributes, and the objective is to determine which attributes correspond to each home by interpreting a set of predefined natural language clues or constraints. The solution to the puzzle is represented as a matrix of size  $H \times A$ , where  $H$  denotes the number of houses and  $A$  represents the number of attributes. As  $H$  and  $A$  increase, synthesizing partial solutions that satisfy individual constraints becomes increasingly compositionally complex. Qualitative examples and details about data generation for this task are provided in the A-C.2.
+
+# B.2 COTEXPERIMENTS
+
+Next, we evaluate how the popular chain-of-thought (CoT) prompting method (Wei et al., 2022) affects the performance of GPT-4o (OpenAI, 2023), Jamba (Lieber et al., 2024), Mamba (Gu & Dao, 2023) and S4-H3 (Gu et al., 2022) models on compositional tasks from Sec. B.1. CoT improves performance but does not solve the problem. Details of the experiments and examples of full prompts can be found in the A-D.
+
+# B.3 ERROR ANALYSIS
+
+We focus on graph analysis of errors, emphasizing multi-digit multiplication because this problem is easier to interpret and understand. From this analysis, we obtain a few interesting conclusions about how errors happen and then propagate inside SSM-based LLMs (Fu et al., 2023; Gu & Dao, 2023; Mirzaee & Kordjamshidi, 2022).
+
+Computation Graph To study the propagation of errors and its dependency on input size, we define  $A$  as a deterministic algorithm (function) and  $\mathcal{F}_A$  as the set of primitives (functions) the algorithm employs during execution. Given inputs  $\mathbf{x}$  to the algorithm  $A$ , we define the static computation graph of  $A(\mathbf{x})$ , denoted as  $G_{A(\mathbf{x})}$ , as  $G_{A(\mathbf{x})} = (V,E,s,op)$ , a directed acyclic graph.
+
+Nodes  $V$  represent all variable values during  $A$ 's execution, where each node  $v \in V$  has an associated value  $s(v) \in \mathbb{R}$ . Edges  $E$  represent function arguments involved in computations: for each non-source node  $v \in V$ , let  $U = \{u_1, \dots, u_j\} \subset V$  be its parent nodes. Then,  $s(v) = f(u_1, \dots, u_j)$  for some  $f \in \mathcal{F}_A$ . Since each node  $v$  is uniquely determined by the computation of a single primitive  $f$ , we define  $op : V \to \mathcal{F}_A$ ,  $op(v) = f$  as the operator function that yields  $s(v)$ . Let  $S \subset V$  be the source nodes of  $G_{A(\mathbf{x})}$ , and without loss of generality, let  $o \in V$  be its sole leaf node. By definition,  $S \equiv \mathbf{x}$  and  $A(\mathbf{x}) = s(o)$ , representing the input and output of  $A$ , respectively. To evaluate a language model's ability to follow algorithm  $A$ , we must linearize  $G_{A(\mathbf{x})}$  (arrange the nodes in a linear sequence that respects the dependencies). This means if a node  $u$  is a parent of node  $v$ , the  $u$  should appear before  $v$  in the sequence. Since we only consider autoregressive models, this linearization must also be a topological ordering. A topological order ensures that every node appears after its parent nodes, maintaining the correct order of computations. This is crucial for correctly following the sequence of operations defined by the algorithm  $A$ .
+
+To instantiate  $G_{A(\mathbf{x})}$ , let  $\mathcal{F}_A = \{\text{one-digit multiplication, sum, mod } 10, \text{ carry over, concatenation}\}$ . Source nodes  $S$  are digits of input numbers, leaf node  $o$  is the final output, and intermediate nodes  $v$  are partial results generated during the execution of the long-form multiplication algorithm (see Fig. 3). The corresponding algorithm is on the left of the Fig. 3 - Alg. 1.
+
+Error propagation We examine errors in SSMs, focusing on how they propagate through computation steps. Fig. 4 shows an example from the S4-H3 model performing multi-digit multiplication using CoT prompting. In this case, the model multiplies 9 by 63. It correctly computes  $9 \times 3 = 27$  but mistakenly carries over '3' instead of '2', leading to an incorrect middle digit in the final answer
+
+Algorithm 1 Multiply two numbers  
+1: function MULTIPLY(x[1..p], y[1..q])  
+    multiply x for each y[i]  
+2: for i = q to 1 do  
+3: carry = 0  
+4: for j = p to 1 do  
+5: t = x[j] × y[i]  
+6: t += carry  
+7: carry = t ÷ 10  
+8: digits [j] = t mod 10  
+9: end for  
+10: summands[i] = digits  
+11: end for  
+12: product = ∑i=1qsummands[q + 1 - i].  
+10i-1  
+13: return product  
+14: end function
+
+![](images/cebf5808634adde11863de8f7275d295334819d2ae8e9496cc54ed1e52dc50f6.jpg)  
+Figure 3: Example of 2-by-1 digit multiplication  $(87 \times 9)$ . Operations on graph include: inputs, multiply 1-digit, carry, sum, mod 10 and output.
+
+despite correct addition in later steps. This highlights propagation errors, where an initial mistake affects later steps. Our analysis shows these errors are 2-4 times more common than local errors, consistent with findings from Dziri et al. (2023). This suggests SSMs handle single-step tasks well but struggle with multi-step reasoning, leading to compounded errors.
+
+![](images/fc555f58cbffb857173b14bed7b64b210daba7ceeafe0e9240a0a66e242a8b5a.jpg)  
+Figure 4: Error Propagation. Carry operation outputs number 3 instead of 2 from node '27', and that error is further propagated, yielding incorrect solution in the middle digit, although all other steps were done right.
+
+![](images/e529c22b74082cf25d5f4f1636a094b5b6d588788ce8aa2241eb80a857d14cbd.jpg)  
+Figure 5: SSMs and Transformers learn shortcuts that seem to solve function composition but fail with larger inputs and out-of-distribution data.
+
+SSMs learn shortcuts The performance of SSMs provides valuable insights into their behavior. These models often predict partially correct answers even when the overall response is incorrect. For example, using Mamba (Gu & Dao, 2023) for 2-by-2 digit multiplication, the first and last digits are usually accurate. The first two and last two digits in larger multiplications tend to be correct. Using Relative Information Gain (RIG) analysis (Dziri et al., 2023), we find that SSMs learn shortcuts, performing fewer operations (illustrated by the red and green subgraphs in Fig. 5). This allows them to frequently predict peripheral digits correctly. For instance, the model multiplies 8 and 5 to compute the last digit, carrying 0 to the end, mimicking human multiplication, and accurately predicting the last digit. RIG analysis reveals a strong correlation between the first digit (or first two digits) of the output and the first digit (or first two digits) of the input numbers.
+
+These models leverage task distribution shortcuts to guess partial answers without whole multi-step reasoning. Increasing the number of Chain-of-Thought (CoT) steps doesn't constantly improve results, especially for larger input sizes (deeper computation graphs). If the model encounters relevant subgraphs during training, its inference seems highly compositional but is based on shortcuts (Geirhos et al., 2020; Liu et al., 2023a; Tang et al., 2023; Du et al., 2022). These experiments indicate that when an output element heavily relies on a few input features, SSMs recognize this correlation during training and map these features to predict the output during testing. This gives the false impression of performing compositional reasoning while bypassing rigorous multi-hop reasoning (Yang et al., 2024).
+
+```txt
+To multiply two numbers, start by multiplying the rightmost digit of the multiplicand by each digit of the multiplier, writing down the products and carrying over any remainders. Repeat this process for each digit of the multiplicand, and then add up all the partial products to obtain the final result. Here are examples:  
+Question: what's 32 times 8? Answer 256.  
+Question: what's 69 times 3? Answer 207.  
+Question: what's 93 times 6? Answer 558.  
+Question: what's 76 times 8? Answer:
+```
+
+Figure 6: Example prompt for the multiplication task used for the few-shot prompting.
+
+# B.4 LEARNING ALGORTHMIC COMPOSITIONS
+
+Finally, we conduct a comprehensive analysis of the capabilities of SSM-based models, along with GPT-4o (OpenAI, 2023), to "learn" discrete algorithms. This analysis is performed using two tasks that require the composition of multiple discrete sub-tasks. By empirically examining the models' algorithmic learning through compositionality testing, we observe their inability to effectively perform these tasks, even when provided with few-shot prompts and CoT examples (Wei et al., 2022). This suggests that within the framework of in-context learning, SSM and Transformer-based models fail to attain compositional learning when constrained to a fixed number of samples. Details in the A-E.
+
+# C COMPOSITIONAL TASKS DETAILS
+
+# C.1 MULTIPLICATION
+
+We show examples of few-shot and CoT prompting methods for multiplication task (Figs. 6 & 7).
+
+# C.2 EINSTEIN'S PUZZLE
+
+Data Construction Following Dziri et al. (2023), we first define a set of properties such as "Color", "PhoneModel", and "Pet", along with their corresponding values in natural language templates (e.g., "The house has a red color"). We then create a basic and clear set of clue types:
+
+1. found_at: For example, "Alice lives in House 2."  
+2. same_house: For example, "The person who is a cat lover lives in the house that has a red color."  
+3. direct_left: For example, "The person who has a dog as a pet lives to the left of the person who lives in a red house."  
+4. besides: For example, "The person who has a dog as a pet and the person who has a red house live next to each other."
+
+Additionally, we introduce more challenging clue types for auxiliary experiments, such as not_at, left_of (not necessarily directly left), and two_house_between. These harder clues are used to test the robustness and versatility of our models.
+
+Graph Construction To address the complex compositional reasoning required for a logical grid puzzle, we utilize existing puzzle solvers de Moura & Bjørner (2008) to generate the computation graph. Our algorithm follows a basic greedy principle: it applies the minimum number of rules necessary to solve any cell. Specifically, if a single rule can solve a cell, that rule is applied.
+
+The algorithm iterates through all clues in the clue set, seeking combinations that can solve any cell in the table. Although this approach may not be the most efficient, it enables models to have explicit scratchpad verbalization via an intuitive computation graph. Fig 8 shows an example of a scratchpad.
+
+Question: What is 904 times 74?
+
+Scratchpad: Let's perform the multiplication step by step:
+
+Let's multiply 904 by the digit in the ones place of 74, which is 4.
+
+1. Multiply 4 by the digit in the ones place of 904, which is 4. This gives  $4 \times 4 = 16$ . Write down the result 6 and carry over the 1 to the next step.  
+2. Multiply 4 by the digit in the tens place of 904, which is 0. Add the carryover from the previous step to account for this. This gives (0 x 4) + 1 = 1. Write down the result 1.  
+3. Multiply 4 by the digit in the hundreds place of 904, which is 9. This gives  $9 \times 4 = 36$ . Write down the result 36.  
+4. The partial product for this step is  $A = 3616$  which is the concatenation of the digits we found in each step.
+
+Now, let's multiply 904 by the digit in the tens place of 74, which is 7.
+
+5. Multiply 7 by the digit in the ones place of 904, which is 4. This gives  $4 \times 7 = 28$ . Write down the result 8 and carry over the 2 to the next step.  
+6. Multiply 7 by the digit in the tens place of 904, which is 0. Add the carryover from the previous step to account for this. This gives (0 x 7) + 2 = 2. Write down the result 2.  
+7. Multiply 7 by the digit in the hundreds place of 904, which is 9. This gives  $9 \times 7 = 63$ . Write down the result 63.  
+8. The partial product for this step is  $B = 6328$  which is the concatenation of the digits we found in each step.
+
+Now, let's sum the 2 partial products A and B, and take into account the position of each digit: A=3616 (from multiplication by 4) and B=6328 (from multiplication by 7 but shifted one place to the left, so it becomes 63280). The final answer is  $3616 \times 1 + 6328 \times 10 = 3616 + 63280 = 66896$ .
+
+Figure 7: A sample scratchpad for the multiplication task.
+
+<table><tr><td>This is a logic puzzle. There are 3 houses (numbered 1 on the left, 3 on the right). Each has a different person in them. They have different characteristics:
+- Each person has a unique name: peter, eric, arnold
+- People have different favorite sports: soccer, tennis, basketball
+- People own different car models: Tesla model 3, ford f150, Toyota camry</td></tr><tr><td>1. The person who owns a Ford F-150 is the person who loves tennis.
+2. Arnold is in the third house.
+3. The person who owns a Toyota Camry is directly left of the person who owns a Ford F-150.
+4. Eric is the person who owns a Toyota Camry.
+5. The person who loves basketball is Eric.
+6. The person who loves tennis and the person who loves soccer are next to each other.</td></tr><tr><td>Let&#x27;s think step by step. Please first briefly talk about your reasoning and show your final solution by filling the blanks in the below table.
+$ House: __ $ Name: __ $ Sports: __ $ Car: __
+$ House: __ $ Name: __ $ Sports: __ $ Car: __
+$ House: __ $ Name: __ $ Sports: __ $ Car: __</td></tr><tr><td>Reasoning:
+Step 1: First apply clue &lt;Arnold is in the third house.&gt; We know that The Name in house 3 is arnold.
+Step 2: Then combine clues: &lt;The person who loves tennis and the person who loves soccer are next to each other.&gt; &lt;The person who loves basketball is Eric.&gt; Unique Values Rules and the fixed table structure. We know that The Name in house 1 is eric. The FavoriteSport in house 1 is basketball. The Name in house 2 is peter.
+Step 3: Then apply clue &lt;Eric is the person who owns a Toyota Camry.&gt; We know that The CarModel in house 1 is Toyota camry.
+Step 4: Then apply clue &lt;The person who owns a Toyota Camry is directly left of the person who owns a Ford F-150.&gt; and Unique Values We know that The CarModel in house 2 is ford f150. The CarModel in house 3 is Tesla model 3.
+Step 5: Then apply clue &lt;The person who owns a Ford F-150 is the person who loves tennis.&gt; and Unique Values We know that The FavoriteSport in house 2 is tennis. The FavoriteSport in house 3 is soccer.
+The puzzle is solved.</td></tr><tr><td>Final solution:
+$ House: 1 $ Name: Eric $ Sports: Basketball $ Car: Camry
+$ House: 2 $ Name: Peter $ Sports: Tennis $ Car: Ford
+$ House: 3 $ Name: Arnold $ Sports: Soccer $ Car: Tesla</td></tr></table>
+
+Figure 8: A sample scratchpad for the Einstein's puzzle task.
+
+# C.3 DYNAMIC PROGRAMMING
+
+We show examples of zero/few-shot and CoT prompting methods for dynamic programming task (Figs. 9 & 10), following Dziri et al. (2023).
+
+Given a sequence of integers, find a subsequence with the highest sum, such that no two numbers in the subsequence are adjacent in the original sequence.
+
+Output a list with "1" for chosen numbers and "2" for unchosen ones. If multiple solutions exist, select the lexicographically smallest. input = [3, 2, 1, 5, 2].
+
+Figure 9: Example prompt for the DP task, used for zero-shot and few-shot settings.
+
+```txt
+Question: Let's solve input  $= [3,2,1,5,2]$    
+Scratchpad: dp[4]  $=$  max(input[4],0)  $=$  max(2,0)  $= 2$  dp[3]  $=$  max(input[3],input[4],0)  $=$  max(5,2,0)  $= 5$  dp[2]  $=$  max(dp[3],input[2]  $+$  dp[4],0)  $=$  max(5,1+2,0)  $= 5$  dp[1]  $=$  max(dp[2],input[1]  $+$  dp[3],0)  $=$  max(5,2+5,0)  $= 7$  dp[0]  $=$  max(dp[1],input[0]  $+$  dp[2],0)  $=$  max(7,3+5,0)  $= 8$  Finally, we reconstruct the lexicographically smallest subsequence that fulfills the task objective by selecting numbers as follows. We store the result on a list named "output".   
+Let can_use_next_item  $\equiv$  True.   
+Since dp[0]  $= =$  input[0]  $+$  dp[2] (8  $= = 3 + 5$  ) and can_use_next_item  $= =$  True, we store output[0]  $= 1$  . We update can_use_next_item  $\equiv$  False.   
+Since dp[1]  $! =$  input[1]  $+$  dp[3] (7  $! = 2 + 5$  ) or can_use_next_item  $= =$  False, we store output[1]  $= 2$  . We update can_use_next_item  $\equiv$  True.   
+Since dp[2]  $! =$  input[2]  $+$  dp[4] (5  $! = 1 + 2$  ) or can_use_next_item  $= =$  False, we store output[2]  $= 2$  . We update can_use_next_item  $\equiv$  True.   
+Since dp[3]  $= =$  input[3] (5  $= = 5$  ) and can_use_next_item  $\equiv =$  True, we store output[3]  $= 1$  . We update can_use_next_item  $\equiv$  False.   
+Since dp[4]  $! =$  input[4] (2  $! = 2$  ) or can_use_next_item  $\equiv =$  False, we store output[4]  $= 2$  Reconstructing all together, output=[1,2,2,1,2].
+```
+
+Figure 10: A sample scratchpad for the DP task.
+
+# D DETAILS OF COT EXPERIMENTS
+
+# D.1 MAIN COT EXPERIMENTS
+
+We plot the performance of Jamba Lieber et al. (2024) on multiplication and puzzle tasks and various models on DP tasks after using CoT.
+
+The leftmost heatmap on Fig. 11 represents the Jamba Lieber et al. (2024) model's multiplication performance, showing a consistently high performance for multipliers of 1 and 2, but a noticeable decline as the multipliers increase, particularly beyond 3. The middle heatmap compares the performance of four models—GPT-4o OpenAI (2023), Jamba Lieber et al. (2024), Mamba Gu & Dao (2023), and S4-H3 Gu et al. (2022); Fu et al. (2023)—on dynamic programming tasks with CoT prompting Wei et al. (2022). GPT-4o OpenAI (2023) consistently outperforms the other models, maintaining high performance even for larger problem list sizes, while the performance of the other models decreases more rapidly. The rightmost heatmap displays Jamba's puzzle-solving performance, indicating high accuracy for simpler puzzles with fewer attributes but a steep decline as the complexity increases. These visualizations highlight that while CoT prompting Wei et al. (2022)
+
+![](images/d2e81702649a4cf947d670091bef4b5833cab5f72a70e0c203c30f2036bc0ec8.jpg)  
+Figure 11: Jamba's Lieber et al. (2024) performance on multiplication and puzzle tasks improves with CoT, though not fully solved. Other models were tested on the DP task, where they failed at higher input sizes, despite CoT.
+
+generally enhances model performance; its effectiveness varies significantly across different models and task complexities.
+
+# D.2 PERFORMANCE OF OTHER MODELS ON MULTIPLICATION AND PUZZLE TASKS
+
+We observe the same pattern on both tasks, for all the models - Figs. 12 & 13. GPT-4o OpenAI (2023) is always the best model, followed by Jamba Lieber et al. (2024), then Mamba Gu & Dao (2023), then S4-H3 Fu et al. (2023); Gu et al. (2022). While CoT helps, it is not enough to solve the task.
+
+![](images/e4b2963d99bec6872a6d9310c6d8bf4fe1e57af710fb023ce468d6828774bf30.jpg)  
+Figure 12: Comparison of different models on multiplication task using CoT.
+
+![](images/fb3e9a53b39d168e8d00180f95e88e83680a2da0a19865f88f174a340a286e55.jpg)  
+Figure 13: Comparison of different models on puzzle task using CoT.
+
+# D.3 FEW-SHOT PROMPTING MULTIPLICATION RESULTS
+
+We investigate whether few-show prompting (giving a model few input/output pairs) and then asking for the answer to the new problem help. Fig. 14 shows the results, and consistently CoT outperforms Few-shot prompting, and Few-shot prompting outperforms Zero-shot prompting.
+
+![](images/4b06e4f5960add1d4bdcf495bc362f65707d2a05e598508d50ede7d51693aa51.jpg)  
+Figure 14: Comparison of different models on multiplication task using few-shot prompting.
+
+![](images/fb866af32defbab11c7ad5de503960d5145984c3c0f19023b91ff4e410a41aa7.jpg)
+
+![](images/1e4a5e02f865719ff590a32892350c06f9a45c19c32676b92237d10b7b5814f9.jpg)
+
+# E ALGORITHMIC COMPOSITIONS
+
+Following Thomm et al. (2024), we evaluate the models using the PE's Neighbour (PEN) task. This task involves navigating from one word to the next based on a specified matching criterion and outputting all the neighbors encountered along the way. The PEN task, inspired by Abnar et al. (2023) and rooted in the Pointer Value Retrieval framework Zhang et al. (2022), is particularly compelling due to its four sub-tasks, which test the necessary sub-operations for PEN. These sub-tasks are: (i) Copy words, (ii) Reverse Copy (copying words in reverse order, where words consist of multiple tokens), (iii) PE (outputting words in the matching chain instead of neighbors), and (iv) PE Verbose (PEV) - outputting both the words of the matching sequence and their neighbors. These sub-tasks are essential because, to predict the next word, the model must: take the current word in the answer, obtain the left neighbor (learned in Reverse Copy), match it (learned in PE), and then obtain the right neighbor (learned in Copy). PEV is considered a sub-task because it requires solving the same problem as PEN but with the added complexity of providing both matching words and their neighbors. PEN, on the other hand, only requires outputting the neighbors. For accurate next-token prediction the model cannot simply replicate the last matching sequence word from the previous answers, it must first infer it from the neighbour. To increase the task's complexity, "attention traps" or "doppelgangers" are introduced. These traps create additional matching possibilities by allowing each neighbor to match two other words, thus tempting the model to match from the neighbor of a matching sequence instead. This added layer of difficulty further challenges the models' ability to learn and compose discrete algorithms effectively. Pointer Execution Reverse Multicount (PER Multi) shares conceptual similarities with the PEN task; however, instead of matching forward and predicting the current word or its neighbor, the task involves first outputting the last word in the matching sequence and then proceeding backward. Consequently, to accurately predict the first word, the model must identify the end of the matching sequence and output that word. The model needs to count the total number of matchings and the number of matchings that align to the left in the given word order. The answer requires multiplying these two counts, introducing a non-linearity. For this task, we omit any attention traps, as there are no neighbors involved. In the A-E we show concrete prompt examples and share the code.
+
+We conducted extensive evaluations on 500 test samples using various models under different conditions: zero-shot, few-shot (providing a limited number of input-output pairs), and CoT prompting Wei et al. (2022). Remarkably, none of the models, including the state-of-the-art GPT-4o OpenAI (2023), succeeded in solving the PEN task Thomm et al. (2024). Typically, models correctly generated the initial strings but then halted prematurely or produced random strings. The same pattern of failure was observed with the PER Multi-task. Specifically, GPT-4o achieved only  $1\%$  and  $9\%$  accuracy using few-shot and CoT prompting, respectively, failing to solve the task. The marginal success of GPT-4o is attributed to its substantially larger parameter count compared to SSM-based models (B.1).
+
+Table 3: Model Accuracy for PEN task  
+
+<table><tr><td>Model</td><td>Prompt Setting</td><td>Accuracy [%]</td></tr><tr><td rowspan="3">GPT-4o</td><td>Zero-shot</td><td>0.00</td></tr><tr><td>Few-shot</td><td>0.00</td></tr><tr><td>CoT</td><td>0.00</td></tr><tr><td rowspan="3">Jamba</td><td>Zero-shot</td><td>0.00</td></tr><tr><td>Few-shot</td><td>0.00</td></tr><tr><td>CoT</td><td>0.00</td></tr><tr><td rowspan="3">Mamba</td><td>Zero-shot</td><td>0.00</td></tr><tr><td>Few-shot</td><td>0.00</td></tr><tr><td>CoT</td><td>0.00</td></tr><tr><td rowspan="3">S4-H3</td><td>Zero-shot</td><td>0.00</td></tr><tr><td>Few-shot</td><td>0.00</td></tr><tr><td>CoT</td><td>0.00</td></tr></table>
+
+Table 4: Model Accuracy for PER Multi task  
+
+<table><tr><td>Model</td><td>Prompt Setting</td><td>Accuracy [%]</td></tr><tr><td rowspan="3">GPT-4o</td><td>Zero-shot</td><td>0.00</td></tr><tr><td>Few-shot</td><td>0.01</td></tr><tr><td>CoT</td><td>0.09</td></tr><tr><td rowspan="3">Jamba</td><td>Zero-shot</td><td>0.00</td></tr><tr><td>Few-shot</td><td>0.00</td></tr><tr><td>CoT</td><td>0.00</td></tr><tr><td rowspan="3">Mamba</td><td>Zero-shot</td><td>0.00</td></tr><tr><td>Few-shot</td><td>0.00</td></tr><tr><td>CoT</td><td>0.00</td></tr><tr><td rowspan="3">S4-H3</td><td>Zero-shot</td><td>0.00</td></tr><tr><td>Few-shot</td><td>0.00</td></tr><tr><td>CoT</td><td>0.00</td></tr></table>
+
+In the following subsections, we focus on showing the prompts in few-shot and CoT settings for PEN and PER Multitasks. Moreover, we show the code we used to generate the samples.
+
+# E.1 PROMPTS FOR SSM AND ATTENTION-BASED MODELS FOR PEN TASK
+
+```txt
+Example: eg jy vm3zc si2zf nn4ll zf5ka ki7xd ew0si xp3og il5jsxn6yx my7ec xu2gb if2my fy3so ec2il ob5ch kt5if zc4xp ka3mj oglud zf2ka yh3ux hx2kt vc2pf jy4qd lj1xu wy5hx bd4xa my4ec at1kb jy3qd ux1fl ew3si ds2qz qd7ew xalay silzf ch4lj js3rf fl6xn mj7wy zy6rq zh2gu bj3rb if0my pg5ds yv3hs zu3ob ta7qi ji2bj mjlwy rq7ul mn3fw ay4qu kt2if kr3qb pr0ah tg0at uclvx xd1pd wy4hx dr6fy mk0vj sm0pg jl2mo rb1bd il2js vn6kr km4aq eg7nn ka6mj qu4vc hx7kt ll21b ec6il ud2vn di3xs pd6ji qd6ew yx7zu rh4qn lb1ki js5rf iv3yh jj0fa kb3sm lh6yk so0iv bx6rs qzlvm mw7bm qb2xo uy0ms qb2zyzm0pz xo4tg zx5jm Answer: jy ka6mj zf5ka ec6il js5rf ew0si wy4hx qd6ew mjlwy if0my il2js my4ec silzf kt2if hx7kt jy4qd <FEW MORE EXAMPLES> Your question: ey wt kj5yo jz0aa nu4yw gp2ro mv6kj nk2qz tr3mp ro7rk tu5xj rk0sj ad2lx up3vd ta7rv qz6ob rc7nt aa4nk mb6mm ob7us jew5wb wt4jz nn4sr wt0jz ev0fa gplro sr1nu sj0ku xs0ta us5up mp6jw vd1gp xj3cs sj7ku ol3vv vd3gp wd2mv wr4cz dg0py ro5rk jt6ev bv0cf yb2qv ch2ss xa3be nb5id lx4jt dz5ht wb5wd fb3ax fa0tu jn5ps rv7qj qa7el rn7ad lz3fk mm1tr yd3lv nt0xs lh4zk mr3ou ja5sn gi5ub rk4sj wn7zm jz3aa be4mb kw3bh qj4xa cg0mi jl2rn kv1wg qt5mr ye3kg yr5ol nk7qz ub1dg ob3us cs7so gw4vk ey4wm qz2ob qv4jl xz4hc li0yb oy4qu zm2yr up7vd ou7li rx4wc yw7gi aa2nk yo3qt yz5cx vv6nn us7up Clearly mark your answer by writing 'Answer: <your answer>' as last line.
+```
+
+Figure 15: Prompt for the PEN task, showcasing few-shot learning examples. Each word's start and end are encoded as distinct tokens, so a model can pattern-match the respective token to do the matching operation.
+
+# Prompt for the PEN task with few-shot CoT examples and a description.
+
+I give you a sequence of words. Each word has four characters plus a middle, words are separated by spaces. Start with the leftmost word. Output its neighbor. Then, match the last two characters of the current word (i.e., not the neighbor) to the word starting with those two characters. Again, output the neighbor. Do this until your current word (not the neighbor) has no match anymore.
+
+Example: xh jz qw4se zs1qh xv4vn me3af vs1nh ok3ks sn6iv qh1va da5gy ks1ew tw7ik em5zs xs5q ft3me gt3bc em3zs zn5qv ks5ew by7kn me7af je0wt cb0ft pw6hg rk7cb dv2sn ew3rk yg1by valcqu qut7pf qh4va vn5zn ok1ks cc7tw rk0cb bc7qi jz7em qz2cs ew6rk qv6gt ft7me fp1qw saok sd7pn jz3em wi3da cq7saiv olv zs7qh v12kc va5cq fe5wi x1zh hg0dv cq4sa ja2nb wh5vv ot4sh qe0jx yt6xs vc0qn xb1am rf2zl kn5hq xg5hk m7yq az3pw shw 90ak wt50t io6hd pn1je lo6vx hq5cc wp6fc cs7fe yw2ka gy3sd nr0ry am3yt pl0rl ik0tn ub5tq shoja ee2it nh6qz xz1ma se0rx is7rn kc1xv cb6fxt rx2mz wf7qf.
+
+The leftmost word is xh. Its right neighbor is jz, so the first output word is jz.
+
+Now, we need to find a word that starts with xh. The word is xh7pw. Its right neighbour is sa7ok, so the next output word is sa7ok.
+
+Now, we need to find a word that starts with pw. The word is pw6hg. Its right neighbour is rk7cb, so the next output word is rk7cb
+
+Now, we need to find a word that starts with  $\mathbf{hg}$ . The word is  $\mathbf{hg0dv}$ . Its right neighbour is  $\mathbf{cq4sa}$ , so the next output word is  $\mathbf{cq4sa}$ .
+
+Now, we need to find a word that starts with dv. The word is dv2sn. Its right neighbour is ew3rk, so the next output word is ew3rk.
+
+Now, we need to find a word that starts with sn. The word is sn6iv. Its right neighbour is qh1va, so the next output word is qh1va.
+
+Now, we need to find a word that starts with iv. The word is iv0vl. Its right neighbour is zs7qh, so the next output word is zs7qh.
+
+Now, we need to find a word that starts with v1. The word is v12kc. Its right neighbour is va5cq, so the next output word is va5cq.
+
+Now, we need to find a word that starts with kc. The word is kclxv. Its right neighbour is cb6ft, so the next output word is cb6ft.
+
+Now, we need to find a word that starts with xv. The word is xv4vn. Its right neighbour is me3af, so the next output word is me3af.
+
+Now, we need to find a word that starts with vn. The word is vn5zn. Its right neighbour is ok1ks, so the next output word is ok1ks.
+
+Now, we need to find a word that starts with  $\mathbf{z} \mathbf{n}$ . The word is  $\mathbf{z} \mathbf{n}5 \mathbf{q} \mathbf{v}$ . Its right neighbour is  $\mathbf{k} \mathbf{s}5 \mathbf{e} \mathbf{w}$ , so the next output word is  $\mathbf{k} \mathbf{s}5 \mathbf{e} \mathbf{w}$ .
+
+Now, we need to find a word that starts with qv. The word is qv6gt. Its right neighbour is ft7me, so the next output word is ft7me.
+
+Now, we need to find a word that starts with gt. The word is gt3bc. Its right neighbour is em3zs, so the next output word is em3zs.
+
+Now, we need to find a word that starts with bc. The word is bc7qi. Its right neighbour is jz7em, so the next output word is jz7em.
+
+There is no word that starts with qi, so we are done with the matching.
+
+Therefore the answer is: jz sa7ok rk7cb cq4sa ew3rk qh1va zs7qh va5cq cb6ft me3af ok1ks ks5ew ft7me em3zs jz7em.
+
+# <FEWMOREEXAMPLES>
+
+Your question: ap cb ch5ya GB6lt uu6le vn0pc og0ef md6ki jx0ph md4ki mq5ox vp1rx zp1xj is5am uq5fb te3rz eq3he cb0md he2zp fe2re ef6yp vn5pc ui3yt kb1ji qg2mq am4vp ez3eq lt5fi hw4eg Iz2te wn5kd kb2ji le6wk vp3x yt31q rx6gb ey4dx ji3fe lq1dq lx0te wk7sl am6vp zi0up ki5kb ek7uu re0v qcs3z vq 5zdx6se xt3f xp2km fe3re bz7hw rx2gb yp6qgb4lt at4cs fi7vn ox1nl fi5vn ph3zi rz4is kd2bz jilfe nl3kk k2ibk yoy6e ter1zd qb7ia bn2xp cd4m ya2wn gd7sq xj2jrg p6bl ap1bn is4am se5ui re5vq eg4uq csf6f fb6jx ll4ic sl4ch qs3nf sp5fd qj6bf dq1og rz1is km6yo vq3z up5sp wc5iv
+
+Reason step by step. Clearly mark your answer by writing 'Answer: <your answer> as last line.
+
+# E.2 PEN GENERATION CODE
+
+```python
+import itertools   
+import numpy as np   
+letter_char = list("abcdefghiJKLMnnpqrstuvwxyz")   
+big_letter_char = list("ABCDEFGHJKLMNOPQRSTUVWXYZ")   
+number_char = list("0123456789")   
+class DataConfig: def_init(self,min_len,max_len,min_hops,max_hops,learn_mode,ambiguous,no_green_confusion): self.min_len  $\equiv$  min_len self.max_len  $=$  max_len self.min_hops  $=$  min_hops self.max_hops  $=$  max_hops self. learn mode  $=$  learn mode self. ambiguous  $=$  ambiguous self.no_green_confusion  $=$  no_green_confusion def get(self,key,default): return getattr(self,key,default)   
+class PointerExecutionNeighbour: def_init(self,data_cfg): self.length_low  $\equiv$  data_cfg.min_len self.length_high  $\equiv$  data_cfg.max_len  $+1$  self.hops_low  $\equiv$  data_cfg.min_hops self.hops_higher  $\equiv$  data_cfg.max_hops  $+1$  self.all_2tuples  $\equiv$  ["join(t) for t in itertools/product(letter_chars,repeat=2)] self. learn mode  $\equiv$  data_cfg.get("learn_mode","next") self.data Choices  $\equiv$  list(numberchars[8]) self.ambiguous  $\equiv$  data_cfg.get("ambiguous",False) self.no_green_confusion  $\equiv$  data_cfg.get("no_green_confusion",False) def generate-double_pointer_execution(self,n_samples): lengths  $=$  np.arange(self.length_low,self.length_high) samples  $=$  [] answers  $=$  [] while len(samples)  $<$  n_samples: length  $=$  np.random.choice(lengths) n_MATCHing_hops  $=$  np.random.choice(np.arange(self.hops_low,min(self.hops_higher,length//2))) tuple Choices  $=$  np.random.choice(self.all_2tuples,length  $^+$  7,replace  $\equiv$  False) #select the positions where the green matching sequence will be positions  $=$  np.random.choice(np.arange(1,length),size=n_MATCHing_hops,replace  $\equiv$  False) cnt  $= 0$  question_words1  $=$  ["for_in range(length)] question_word2  $=$  ["for_in range(length)] remaining_posits  $=$  np.random.permutation([i for i in range(1,length) if i not in positions]) question_word1[0]  $=$  tuple Choices[cnt] answer_ learnseq  $\equiv$  [question_word1[0]] for pos in positions: question_word1[pos]  $=$  (tuple Choices[cnt] + np.random.choice(self.data Choices) + tuple Choices[cnt + 1]) answer_ learnseq.append(question_word1[pos]) cnt  $+= 1$  cnt  $+= 1$  cnt_confuse  $=$  cnt  $+$  length positions_next  $=$  np.random.permutation positions) question_word2[0]  $=$  tuple Choices[cnt] answer  $=$  [question_word2[0]] #select the positions where the doppelgangers of the neighbours will be positions_confuse  $=$  np.setdiff1d(np.arange(1,length),positions_next)[0:len positions_next)] np.random shuffle(placeums_confuse) for i, pos in enumerate(placeums_next): twoBIGletters  $\equiv$  np.random.choice(self.data Choices,size=2,replace  $\equiv$  self.ambiguous) question_word2[pos]  $=$  (tuple Choices[cnt] + twoBIGletters[0] + tuple Choices[cnt + 1]) question_word2[positions_confuse[i]]  $=$  (tuple Choices[cnt] + twoBIGletters[1] + tuple Choices[cnt + 1]) answer.append(question_word2[pos]) cnt  $+= 1$  cnt_confuse  $+= 1$  cnt  $=$  max(cnt,cnt_confuse)+1 remaining_nextPositions  $=$  np.random.permutation([i for i in range(1,length) if i not in positions_next and \\ i not in positions_confuse]) for pos in remainingPositions: question_word1[pos]  $=$  (tuple Choices[cnt] + np.random.choice(self.data Choices) + tuple Choices[cnt + 1]) cnt  $+= 1$  if self.no_green_confusion: cnt  $+= 1$  cnt  $+= 1$  for pos in remaining_nextPositions: question_word2[pos]  $=$  (tuple Choices[cnt] + np.random.choice(self.dataChoices) + tuple Choices[cnt + 1]) cnt  $+= 2$  answer_ learnnext  $=$  [question_word2[0]] for pos in positions: answer_ learnnext.append(question_word2[pos]) answer_seqnext  $=$  [] for i in range(len(answer_ learnseq)): answer_seqnext.append unanswered_ learnseq[i]) answer_seqnext.append unanswered_ learnnext[i]) answer_reverse() question_word  $\equiv$  [] for i in range(length): question_word.append(question_word1[i]) question_word.append(question_word2[i]) question_str  $=$  (f"pe{self.train_mode}:"+" .join(['".join(x) for x in question_words] " + " answer:" ) samples.append(question_str) if self.train_mode  $\equiv$  "seg": answers.append(" .join(assistant)}) elif self.train_mode  $\equiv$  "segnext": answers.append(" .join(assistant)) else self.train_mode  $\equiv$  "next": answers.append(" .join(assistant)) return samples, answers def generate(self,n_samples): samples, answers  $=$  selfgenerateDouble_pointer_execution(n_samples) return samples, answers
+```
+
+Figure 16: Code utilized for generating instances of the PEN task and its associated subtasks. The hyperparameters employed include a length ranging between [40, 50] and a number of hops ranging between [10, 20].
+
+# E.3 PER MULTI-GENERATION CODE
+
+```python
+import itertools   
+import numpy as np   
+letter_chars \(=\) list("abcdefghiJKLMnopqrstuwxyz")   
+class DataConfig: def__init_(self,min_len,max_len,logyname,learn_mode \(\equiv\) "seq"): self.min_len \(=\) min_len self.max_len \(=\) max_len self.logname \(=\) logname self. learn_mode \(=\) learn_mode get(self,key,default): return getattr(self,key,default)   
+class PointerExecutionReverseMulticount: def__init_(self,data_cfg): self.length_low \(=\) data_cfg.min_len self.length_higher \(=\) data_cfg.max_len \(+1\) self.logname \(=\) data_cfg.logname self.all_2tuples \(=\) ["."join(t) for t in itertools/product(letter_chars,repeat=2)] self. learn_mode \(=\) data_cfg.get("learn_mode","seq") assert self. learn_mode in["seq","multisegq","segrev","multisegrev"]   
+def generate_samples(self,n_samples): lengths \(=\) np.arange(self.length_low,self.length_higher) samples \(=\) [] answers \(=\) [] for_in range(n_samples): length \(=\) np.random.choice(lengths) tuple Choices \(=\) np.random.choice(self.all_2tuples,length \(+3\) ,replace \(\equiv\) False) last_word \(=\) tuple Choices[-3] \(^+\) tuple Choices[-2] shuffled_tuple Choices1 \(=\) np.random.permutation(tuple Choices[:=-3]) shuffled_tuple Choices2 \(=\) np.random.permutation(tuple Choices[:=-3]) words \(=\) [chl + ch2 for chl,ch2 in zip(shuffled_tuple Choices1, shuffled_tuple Choices2)] start \(=\) np.random.choice(words) words.append(last_word) if"rev" not in self. learn_mode: answer \(=\) self.solve_seqnext(words,start,self. learn_mode) else: # change the 2tuple of the start of the start word to a random one idx \(=\) words.index(start) words[idx] \(=\) tuple Choices[-1] \(^+\) words[idx][2:] start \(=\) words[idx] answer,answer_n_left \(=\) self.solve_seqnext(words,start,self. learn_mode) if self. learn_mode \(\equiv =\) "segrev": answer \(=\) reversed([Fw]for i,w in enumerate(answer)]) if self. learn_mode \(\equiv =\) "multisegrev": answer \(=\) reversed([Fw{w{i+n}]for i,(w,n)in enumerate(zip(answer,answer_n_left))) question \(=\) (f"prand {self. learn_mode}): "+".join(words)+""|+"start+"answer:" samples.append(question) answers.append("\).join(answers)) return samples, answers   
+def solve_seqnext(self,words,start,mode): answer_next \(=\) [] matching_seq \(=\) [] current_word \(=\) start idx \(=\) words.index(current_word) n_left \(= 0\) answer_n_left \(= [\cdot ]\) while True: matching_seq.append(current_word) answer_next.append( \(\mathrm{[width\text{i} + 1]}\) ) answer_n_left.append(n_left) next_word \(= [(\mathbf{W},\mathbf{i})\) for i,w in enumerate( \(\mathrm{word(s)}\) if w.startswith(current_word[-2:]]） if len(next_word \(\equiv = 0\) and"rev"in mode: break assert len(next_word)\(\equiv = 1\) current_word,new_idx \(\equiv\) next_word[0] if new_idx<idx: n_left +=1 idx = new_idx if current_word in matching_seq: break if"rev"in mode: return matching_seq,answer_n_left   
+if"multi"in mode: answer \(=\) [] for i,(w,n)in enumerate( \(\mathrm{zip(matching\_seq,}\) answer_n_left)): answer.append(f"\{\boldsymbol{w}.\{i+n\}") return answer   
+return matching_seq   
+def generate(self,n_samples): samples, answers \(=\) self.create_samples(n_samples) return samples, answers
+```
+
+Figure 17: Code employed for generating instances of the Pointer Execution Reverse Multicount task and its associated subtasks. The hyperparameters employed include a length ranging between [10, 20].
\ No newline at end of file
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+# LINEAR COMBINATION OF SAVED CHECKPOINTS MAKES CONSISTENCY AND DIFFUSION MODELS BETTER
+
+Enshu Liu\*1,2, Junyi Zhu\*3, Zinan Lin\*4, Xuefei Ning\*1, Shuaiqi Wang5, Matthew B. Blaschko3, Sergey Yekhanin4, Shengen Yan2, Guohao Dai2,6, Huazhong Yang1, Yu Wang\*1
+
+$^{1}$ Tsinghua University  $^{2}$ Infinigence-AI  $^{3}$ KU Leuven  $^{4}$ Microsoft Research
+
+5Carnegie Mellon University 6Shanghai Jiao Tong University
+
+# ABSTRACT
+
+Diffusion Models (DM) and Consistency Models (CM) are two types of popular generative models with good generation quality on various tasks. When training DM and CM, intermediate weight checkpoints are not fully utilized and only the last converged checkpoint is used. In this work, we find proper checkpoint merging can significantly improve the training convergence and final performance. Specifically, we propose LCSC, a simple, effective, and efficient method to enhance the performance of DM and CM, by combining checkpoints along the training trajectory with coefficients deduced from evolutionary search. We demonstrate the value of LCSC through two use cases: (a) Reducing training cost. With LCSC, we only need to train DM/CM with fewer number of iterations and/or lower batch sizes to obtain comparable sample quality with the fully trained model. For example, LCSC achieves considerable training speedups for CM  $(23\times$  on CIFAR-10 and  $15\times$  on ImageNet-64). (b) Enhancing pre-trained models. When full training is already done, LCSC can further improve the generation quality or efficiency of the final converged models. For example, LCSC achieves better FID using 1 number of function evaluation (NFE) than the base model with 2 NFE on consistency distillation, and decreases the NFE of DM from 15 to 9 while maintaining the generation quality. Applying LCSC to large text-to-image models, we also observe clearly enhanced generation quality.
+
+# 1 INTRODUCTION
+
+Diffusion Models (DMs) (Sohl-Dickstein et al., 2015; Ho et al., 2020; Song et al., 2020b) as a generative modeling paradigm have rapidly gained widespread attention in the last several years, showing excellent performance in various tasks like image generation (Ho et al., 2020; Dhariwal & Nichol, 2021; Rombach et al., 2022), video generation (Ho et al., 2022; Blattmann et al., 2023), and 3D generation (Poole et al., 2022; Lin et al., 2023). DM requires an iterative denoising process in the generation process, which could be slow. Consistency Models (CMs) (Song et al., 2023) are proposed to handle this dilemma, which provides a better generation quality under one or few-step generation scenarios and is also broadly applied (Luo et al., 2023a;b; Wang et al., 2023).
+
+In this paper, we investigate the training process of DM and CM. We find that checkpoints—the model weights periodically saved during the training process—have under-exploited potential in boosting the performance of DM and CM. In particular, within the metric landscape, we observe numerous high-quality basins located near any given point along the optimization trajectory. These high-quality basins cannot be reliably reached through Stochastic Gradient Descent (SGD), including its advanced variants like Adam (Kingma & Ba, 2014). However, we find that an appropriate linear combination of different checkpoints can locate these basins.
+
+![](images/d80143540e3b66292b05f36642b0f9f7589cf3ef92544ca120d67b5abb31d459.jpg)  
+Figure 1: Comparison of LCSC and vanilla training. LCSC achieves more than  $14\times$  training speed up on CIFAR-10 with Consistency Distillation (CD) and more than  $15\times$  training speed up compared to official released model on ImageNet-64 with CD. LCSC can also enhance the final converged model significantly and achieves  $1.7\times$  inference speedup for DM.
+
+![](images/4e05d4b0e80bf60b6e71148e1af22cc4bd77c88147b76d0058526e792967ffab.jpg)
+
+![](images/736649b97f7b02059eae0dd2f82f9d7ab68f4d9dc1677569ad890be484b2ffce.jpg)
+
+![](images/f7eed8de6b27e73ac6f29db50541646f0f8b7f3a9963c3bec0ca18b538a43208.jpg)  
+(a) Vanilla training  
+Figure 2: Comparison between the images generated by LCSC and vanilla training (on LCM-LoRA model (Luo et al., 2023a;b)). The prompts for the left and right images are "A car that seems to be parked illegally behind a legally parked car" and "Two women waiting at a bench next to a street" respectively. The vanilla method produces images with unrealistic front wheels (left) and an unnatural sitting posture (right), whereas LCSC produces more realistic images.
+
+![](images/62a36b1ac365ee06200a3c65151be8bbc15e6373c873eac522ed0b6f7f3c0129.jpg)  
+(b) LCSC
+
+![](images/27b15688395c1972754156249337b03d53aa3297d10162c9e921f3639bbb0e67.jpg)  
+(c) Vanilla training
+
+![](images/373d9aa16fc8c7e6eed65bf2cfc91dbc139644048d5cc991eb961cb6d0a47fbf.jpg)  
+(d) LCSC
+
+Inspired by the observation, we propose Linear Combination of Saved Checkpoints (LCSC). In particular, for a set of checkpoints saved until any point of the training process, LCSC searches for the optimal linear combination coefficients that optimize certain metrics (e.g., FID (Heusel et al., 2017)) using an evolutionary algorithm. LCSC optimizes only in a low-dimension space and needs no back-propagation, thus could be faster than training. Additionally, LCSC can optimize objectives whose gradients are hard to compute, under which gradient-based optimization is not applicable. We note that the widely used Exponential Moving Average (EMA) (Szegedy et al., 2016) in DM and CM can be viewed as a linear combination method whose combination coefficients are determined heuristically. We will theoretically prove and empirically demonstrate that EMA coefficients are sub-optimal and are worse than LCSC. While this paper focuses on DM and CM, from a broader perspective, LCSC is a general method and may be expanded to other tasks and neural networks.
+
+We argue that LCSC is beneficial in all steps in the DM/CM production stage and can be used to: (a) Reduce training cost. The training process of DM and CM is very costly. On ImageNet-64, SOTA DMs and CMs (Dhariwal & Nichol, 2021; Karras et al., 2022; Song et al., 2023) require more than 10K GPU hours on Nvidia A100. For higher resolution tasks, the resource demand soars dramatically again: Stable Diffusion (Rombach et al., 2022) costs more than 150K GPU hours (approx. 17 years) on Nvidia A100. By applying LCSC at the end, we can train CM/DM with significantly fewer iterations or smaller batch sizes and reach similar generation quality with the fully trained model, thereby reducing the computational cost of training. (b) Enhance model performance. If the full training is already done, LCSC can still be applied to get a model that is better than any model in the training process. For model developers with saved checkpoints, LCSC can be directly applied. For users who can only access the final released checkpoint, they can fine-tune the checkpoint for a few more iterations and apply LCSC on these checkpoints. As DM/CM provides a flexible trade-off between generation quality and the number of generation steps, the enhanced model from LCSC could lead to either better generation quality or faster generation.
+
+The reminder of this paper is organized as follows: In Sec. 2, we introduce the background and related works. In Sec. 3, we visualize the metric landscape of DM and CM, demonstrating the potential of checkpoint merging. Additionally, we provide theoretical analyses showing that EMA is suboptimal, and that a more flexible method for setting the merging coefficients is preferable. In Sec. 4, we present our efficient method LCSC, which flexibly adjusts the merging coefficients to find the best model through evolutionary search. A schematic diagram of LCSC is shown in Fig. 3. In Sec. 5, we conduct experiments to validate the effectiveness of LCSC on both DM and CM. The results show that LCSC can accelerate the training process of CM by up to  $23\times$ . Moreover, LCSC
+
+![](images/bd9c6d61d94c3ea16e7d99283bfd4e47bbb5713e6a5cbaeb30ec756d347e1f3f.jpg)  
+Figure 3: A schematic diagram of LCSC. Given a set of checkpoints from training (left), LCSC use evolutionary search to find the optimal linear combination coefficients (middle). LCSC can be applied on checkpoints from a training process with fewer training iterations or batch sizes and still gets similar performance (abbreviated as "Perf."), thus reducing training cost and enabling faster training. LCSC can also enhance the final model in terms of generation quality or speed (right).
+
+can decrease the NFE of DM from 15 to 9 while keeping the sample quality. We highlight some of the results in Fig. 1. Additionally, we demonstrate that LCSC can be applied to text-to-image models to further improve generation quality, with some images presented in Fig. 2. In Sec. 6, we discuss the searched patterns of coefficients to inspire further research on checkpoints merging.
+
+# 2 BACKGROUND AND RELATED WORK
+
+In this section, we briefly introduce the foundational concepts and related works necessary for understanding this paper. A more comprehensive discussion is provided in App. B
+
+Diffusion Probabilistic Model. Let us denote the data distribution by  $p_{data}$ , diffusion models (Sohl-Dickstein et al., 2015; Song et al., 2020a; Ho et al., 2020; Nichol & Dhariwal, 2021; Song et al., 2020b) learn a process that perturbs  $p_{data}$  with a stochastic differential equation:
+
+$$
+d \mathbf {x} _ {t} = \boldsymbol {\mu} (\mathbf {x} _ {t}, t) d t + \sigma (t) d \boldsymbol {w} _ {t}, \tag {1}
+$$
+
+where  $\pmb{\mu}(\cdot, \cdot)$  and  $\sigma(\cdot)$  represent the drift and diffusion coefficients, respectively,  $\pmb{w}_t$  denotes the standard Brownian motion, and  $t \in [0, T]$  indicates the time step.  $t = 0$  stands for the real data distribution.  $\pmb{\mu}(\cdot, \cdot)$  and  $\sigma(\cdot)$  are designed to make sure  $p_T(\pmb{x})$  becomes pure Gaussian noise.
+
+To train a diffusion model, we can construct a network  $s_{\theta}(\boldsymbol{x}_t,t)$  to approximate the score function of the perturbed data distribution  $\nabla \log p_t(\boldsymbol{x})$  by minimizing:
+
+$$
+\mathbb {E} _ {t \sim \mathcal {U} (0, T ]} \mathbb {E} _ {\mathbf {y} \sim p _ {\mathrm {d a t a}}} \mathbb {E} _ {\mathbf {x} _ {t} \sim \mathcal {N} (\mathbf {y}, \sigma (t) ^ {2} \mathbf {I})} \lambda (t) \| \mathbf {s} _ {\boldsymbol {\theta}} (\mathbf {x} _ {t}, t) - \nabla_ {\mathbf {x} _ {t}} \log p (\mathbf {x} _ {t} | \mathbf {y}) \|, \tag {2}
+$$
+
+where  $\lambda (t)$  represents the loss weighting and  $\pmb{y}$  denotes a training image.
+
+Consistency Models. During generation, diffusion models require multiple time steps, which results in low efficiency. Consistency models (CM) are proposed to enable single-step generation (Song et al., 2023), by mapping any point  $\boldsymbol{x}_t$  at any given time  $t$  along a probability flow trajectory directly to the trajectory's initial point  $\boldsymbol{x}_t$ . Training of CM can follow one of two methodologies: consistency distillation (CD) or direct consistency training (CT). In the case of CD, the model  $f_{\theta}$  leverages knowledge distilled from a pre-trained DM  $\phi$ . The distillation loss can be formulated as follows:
+
+$$
+\mathbb {E} _ {k \sim \mathcal {U} [ 1, K - 1 ]} \mathbb {E} _ {\boldsymbol {y} \sim p _ {d a t a}} \mathbb {E} _ {\boldsymbol {x} _ {t _ {k + 1}} \sim \mathcal {N} (\boldsymbol {y}, t _ {k + 1} ^ {2} \boldsymbol {I})} \lambda (t _ {k}) d \left(f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {t _ {k + 1}}, t _ {k + 1}\right), f _ {\boldsymbol {\theta} ^ {-}} \left(\hat {\boldsymbol {x}} _ {t _ {k}} ^ {\phi}, t _ {k}\right)\right), \tag {3}
+$$
+
+where  $f_{\theta^{-}}$  refers to the target model and  $\pmb{\theta}^{-}$  is computed through EMA of the historical weights of  $\pmb{\theta}$ ,  $\hat{\pmb{x}}_{t_k}^{\phi}$  is estimated by the pre-trained diffusion model  $\phi$  through one-step denoising based on  $\pmb{x}_{t_{k + 1}}$ , and  $\bar{d}$  is a metric such as  $\ell_2$  distance. Alternatively, in the CT case, the models  $f_{\theta}$  are developed independently, without relying on any pre-trained DM:
+
+$$
+\mathbb {E} _ {k \sim \mathcal {U} [ 1, K - 1 ]} \mathbb {E} _ {\boldsymbol {y} \sim p _ {d a t a}} \mathbb {E} _ {\boldsymbol {z} \sim \mathcal {N} (\boldsymbol {0}, \boldsymbol {I})} \lambda (t _ {k}) d \left(f _ {\boldsymbol {\theta}} \left(\boldsymbol {y} + t _ {k + 1} \boldsymbol {z}, t _ {k + 1}\right), f _ {\boldsymbol {\theta} ^ {-}} \left(\boldsymbol {y} + t _ {k} \boldsymbol {z}, t _ {k}\right)\right), \tag {4}
+$$
+
+where the target model  $f_{\theta^{-}}$  is set to be the same as the model  $f_{\theta}$  in the latest improved version of the consistency training (Song & Dhariwal, 2024), i.e.,  $\theta^{-} = \theta$ .
+
+Weight Averaging. A line of work (Ruppert, 1988; Polyak & Juditsky, 1992; Izmailov et al., 2018) explores the integration of a running average of the weights in the context of convex optimization and stochastic gradient descent (SGD). More recently, Wortsman et al. (2022) demonstrate the potential of averaging models that have been fine-tuned with various hyperparameter configurations. In early DM studies, the use of Exponential Moving Average (EMA) is found to significantly enhance the
+
+quality of generation. This empirical strategy has since been adopted in most, if not all, subsequent research endeavors. Consequently, CM have also incorporated this technique, discovering that EMA models perform substantially better. EMA employs a specific averaging form that uses the exponential rate  $\gamma$ :  $\tilde{\theta}_n = \gamma \tilde{\theta}_{n-1} + (1 - \gamma)\theta_n$ , where  $n = 1,\dots,N$  denotes the number of training iterations,  $\tilde{\theta}$  represents the EMA model, and is initialized with  $\tilde{\theta}_0 = \theta_0$ . For Large Language Models (LLMs), many works have proposed more advanced weight averaging strategies aiming at merging models fine-tuned for different downstream tasks to create a new model with multiple capabilities (Ilharco et al., 2022; Yadav et al., 2024; Jin et al., 2022; Yu et al., 2023).
+
+Unlike these approaches, our work focuses on accelerating model convergence and achieving better performance within a standalone training process. Additionally, our methodology for determining merging coefficients is novel, thereby distinguishing our method from these related works.
+
+Search-based Methods for Diffusion Models. Recently, many studies have proposed discrete optimization dimensions for DMs and employed search methods to discover optimal solutions. This includes using search methods to optimize model schedule for DMs (Liu et al., 2023a; Li et al., 2023; Yang et al., 2023), appropriate strategies for diffusion solvers (Liu et al., 2023b), or quantization settings for DMs (Zhao et al., 2024b;a). However, these works do not involve modification to model weights and are only applicable to DMs.
+
+# 3 THEORETICAL AND EMPIRICAL MOTIVATION OF LCSC
+
+In this section, we first conduct a theoretical analysis to understand the effectiveness of EMA in Sec. 3.1. A key insight from this analysis is that the optimal EMA rate is not fixed but depends on the number of training iterations, raising concerns about how to tune EMA effectively. We then present empirical evidence in Sec. 3.2, demonstrating that EMA is not special in its effectiveness. In fact, many linear combination results yield better performance on the trained models. Specifically, based on the empirical evidence, we provide further theoretical analysis to prove that EMA is a suboptimal method. These observations motivate the development of a more flexible merging method.
+
+# 3.1 THEORETICAL CONVERGENCE ANALYSIS
+
+Analysis Framework. Previous works have established various frameworks for analyzing DMs and CMs while considering their respective objectives (Sohl-Dickstein et al., 2015; Ho et al., 2020; Song et al., 2020a; Karras et al., 2022; Song et al., 2023). In this section, we follow the general optimization analysis, which represents the essential form of network optimization (Rakhlin et al., 2011; Shamir & Zhang, 2013; Harvey et al., 2019). Specifically, we denote  $f(\pmb{\theta}_n)$  as the computed loss of the network parameters at  $n$ -th iteration  $\pmb{\theta}_n$ . For each training iteration  $n = 1, \dots, N$ , we obtain an unbiased random estimate  $\hat{g}_n$  of the gradient  $\nabla f(\pmb{\theta}_n)$ , such that  $\mathbb{E}[\hat{g}_n] = \nabla f(\pmb{\theta}_n)$ . Additionally, we denote  $\mathbb{E}[\| \hat{g}_n\|^2] = \sigma_n^2$  and assume that the expected square sum is upper bounded, i.e.  $\forall n, \mathbb{E}[\| \hat{g}_n\|^2] \leq G^2$ . We emphasize that due to the high-variance characteristic of DM and CM (see App. F.4 for a more detailed discussion),  $\sigma_n^2$  does not diminish during training and may be significant, potentially distinguishing the training of DM and CM from other tasks. To conduct the analysis, we assume  $f$  is  $\beta$ -strongly convex (cf. Eq. (9)). Moreover, we consider  $f$  as a non-smooth function, since natural images often exhibit features like edges and textures that lead to abrupt changes and discontinuities in their distribution. Proofs of the following analyses are provided in App. A.
+
+Based on the above analytical framework, Shamir & Zhang (2013) prove the following theorem and show that the model of the last training iteration has a convergence rate of  $O(\log(N)/N)$ .
+
+Theorem 3.1 (Shamir & Zhang (2013)) Suppose  $f$  is  $\beta$ -strongly convex, and that  $\mathbb{E}[\| \hat{\pmb{g}}_n\|^2] \leq G^2$  for all  $n = 1, \dots, N$ . Consider SGD with step sizes  $\eta_n = 1 / \beta n$ , then for any  $N > 1$ , it holds that:  $\mathbb{E}[f(\pmb{\theta}_N) - f(\pmb{\theta}^*)] \leq \frac{17G^2(1 + \log(N))}{\beta N}$ , where  $\pmb{\theta}^*$  denotes the optimal model.
+
+The  $O(\log (N) / N)$  convergence rate of the last-iter model is proven to be tight, as a lower bound of  $\Omega (\log (N) / N)$  is found by Rakhlin et al. (2011). Next, we investigate the impact of merging the historical weights using the form:  $\bar{\theta}_N^{\alpha} = \frac{1}{\mathrm{A}}\sum_{n = 1}^{N}\alpha_n\theta_n$  , where  $\alpha = \{\alpha_{1},\dots ,\alpha_{N}\}$  are the merging coefficients, and  $\mathrm{A} = \sum_{n = 1}^{N}\alpha_{n}$  normalizes the sum of coefficients to 1. When EMA is applied, the coefficients can be represented as  $\alpha_{n} = \gamma^{N - n}$  , where  $\gamma$  is the exponential rate.* We prove that EMA reduces significant terms in the convergence bound to  $O(1 / N)$  for large  $N$
+
+*This formulation differs slightly from the existing EMA implementation regarding the treatment of the initial model. However, it provides mathematical convenience in the analysis. Our experimental results also show that they achieve the same performance, as the discrepancy in their coefficients is negligible.
+
+![](images/bc6010639f36f507a7d0ca74c318dd57dfb76d498fb80ba6bf0635d5eac9dffc.jpg)
+
+![](images/ad5fb3c9cf21da60fcba4c85c3c1b6ce5156e38e73a78fa5dd97e851d3b129b4.jpg)
+
+![](images/8c995399dc9d3a5f56f9b20ff465ab94a9aff1882c530ba6db8534a3ae774406.jpg)  
+(a) Metric Landscape of DM.
+
+![](images/9c28013dfc7779caecd1b46ce5b8d87cb836c583acef2b2235110fa11e4d48f0.jpg)  
+Figure 4: The metric landscape of DM and CM. Selected checkpoints  $\theta_{n_0},\theta_{n_1}$ , and  $\theta_{n_2}$  are aligned sequentially along the training trajectory, with  $n_0 < n_1 < n_2$  denoting the progression in the number of training iterations. The origin point  $(0,0)$  corresponds to the checkpoint  $\theta_{n_1}$ , while the X and Y axes quantify the differences between  $\theta_{n_1} - \theta_{n_0}$  and  $\theta_{n_2} - \theta_{n_0}$ , respectively. A weight located at coordinate  $(x,y)$  is formulated as  $\theta_{(x,y)} = \theta_{n_0} + x(\theta_{n_1} - \theta_{n_0}) + y(\theta_{n_2} - \theta_{n_0})$ . Additional visualizations are provided in App. D.
+
+![](images/8d24a898de005f81c7867477ac407b0a944da592e7dac5e7ba1360041d930581.jpg)
+
+![](images/f307ce765ce6b9b47db9465b43ea999c855b80053642534648e7c39b1f19bfe2.jpg)
+
+![](images/4be4a549c160bdf08f8e21e55932a81db12c9bf92a653f262da469da0939aac5.jpg)  
+(b) Metric Landscape of CM.
+
+![](images/32365a44fde8626422a23ff75cb5df46445037622b4b946882ea6fbfc6ada082.jpg)
+
+Theorem 3.2 Suppose  $f$  is  $\beta$ -strongly convex, and that  $\mathbb{E}[\| \hat{\pmb{g}}_n\|^2] \leq G^2$  for all  $n$ . Consider SGD with step sizes  $\eta_n = 1 / \beta n$  and EMA with factor  $\gamma \in (0,1)$ . Then for any  $N > \lceil -\frac{1}{\ln\sqrt{\gamma}}\rceil$ , we have:
+
+$$
+\mathbb {E} \left[ f \left(\bar {\boldsymbol {\theta}} _ {N} ^ {\boldsymbol {\alpha}}\right) - f \left(\boldsymbol {\theta} ^ {*}\right) \right] \leq \frac {G ^ {2}}{\beta} \left(\frac {1}{\gamma \left(1 - \gamma^ {N - 1}\right) (N + 1)} + \frac {v (\gamma)}{2 \left(1 - \gamma^ {N - 1}\right)} \gamma^ {N} + 1 - \gamma\right), \tag {5}
+$$
+
+where  $v(\gamma) = \sum_{j=1}^{\lceil -\frac{1}{\ln \sqrt{\gamma}} \rceil - 1} \frac{1 - \gamma}{\gamma^{j} j} - \frac{1 - \gamma}{\gamma^{\lceil - \frac{1}{\ln \sqrt{\gamma}} \rceil}}$  is a constant given  $\gamma$  and independent from  $N$ .
+
+We observe that the first term in Thm. 3.2 decays as  $O(1 / N)$ , while the second term decays exponentially as  $O(\gamma^{N})$ . Additionally, there is a residual of  $1 - \gamma$ . Considering  $\gamma$  is close to 1, we conclude that the significant terms (the first two terms) decay at least as fast as  $O(1 / N)$  and the residual term is small. We hypothesize that the improvement of these significant terms leads to an acceleration in convergence compared to the overall  $O(\log (N) / N)$  rate of the last-iter model. Furthermore, we note that for sufficiently large  $N$ , it is preferable for  $\gamma$  to be large so that the residual approaches zero, while the enlarged factors in the first two terms can be compensated by a large  $N$ . This insight aligns with practical experience. For example, Karras et al. (2022; 2024) implemented EMA rates that increase with the number of training iterations. Notably, the optimal EMA rate is difficult to predefine based on Thm. 3.2 without having the exact optimization landscape and the least upper bound. This motivates the search for EMA rates during or after the training. Moreover, in the next section, we will further show that EMA is suboptimal of merging coefficients per se.
+
+# 3.2 THE EFFECTIVENESS OF WEIGHT MERGING AND SUBOPTIMALITY OF EMA
+
+To visualize the impact of linear combination of weights on the generation quality, we select 3 checkpoints  $\theta_{n_0},\theta_{n_1},\theta_{n_2}$  from the same training trajectory. Then we sweep across the 2D space spanned by these three points and assess the FID scores. The results are shown in the first row of Fig. 4. As we can see, there exists substantial opportunities to enhance the model's performance at any training phase through the linear combination of existing weight checkpoints, which proves that the effectiveness of EMA is not special. Furthermore, we conduct additional experiments using EMA weights (see the second row of Fig. 4). Notably, we observe that linear combinations of the three EMA weights could also achieve superior performance. It is important to highlight that the linear combination of three EMA weights at various training iterations cannot be replicated by any single EMA weight according to Thm. 3.3. Consequently, this suggests that EMA is indeed a suboptimal solution, indicating possibilities for further improvement in this area.
+
+Theorem 3.3 Assume  $\theta_{1},\ldots ,\theta_{N}$  are linearly independent. Denote  $\theta_1^\gamma ,\dots ,\theta_N^\gamma$  as the results of EMA using the exponential rate  $\gamma \in (0,1)$ . and select three indices  $n_1,n_2,n_3$  s.t.  $1 < n_{1} < n_{2} < n_{3}$ . Consider three arbitrary merging coefficients  $c_{1},c_{2},c_{3}\in (0,1)$ , s.t.  $c_{1} + c_{2} + c_{3} = 1$ , we have:
+
+$$
+\nexists \gamma^ {\prime} \in (0, 1): \quad c _ {1} \boldsymbol {\theta} _ {n _ {1}} ^ {\gamma} + c _ {2} \boldsymbol {\theta} _ {n _ {2}} ^ {\gamma} + c _ {3} \boldsymbol {\theta} _ {n _ {3}} ^ {\gamma} \in \left\{\boldsymbol {\theta} _ {1} ^ {\gamma^ {\prime}}, \dots , \boldsymbol {\theta} _ {N} ^ {\gamma^ {\prime}} \right\}. \tag {6}
+$$
+
+Based on the above insights, we conclude that the optimal weight merging coefficients are difficult to predefined, and the performance of merged models can potentially be further improved with better merging coefficients. Therefore, we propose LCSC, which introduces an optimization method to search for the optimal merging coefficients.
+
+# 4 METHOD
+
+Next, we present the details of LCSC. The search problem is defined in Sec. 4.1. In Sec. 4.2, we elaborate on our algorithm. Finally, we list several typical use cases of our method in Sec. 4.3.
+
+# 4.1 DEFINITION OF THE SEARCH PROBLEM
+
+Let  $\{1,2,\ldots ,N\}$  denote the set of training iterations. Suppose  $\{n_1,n_2,\dots ,n_K\}$  is a subsequence selected from  $\{1,2,\ldots ,N\}$ . Define  $\Theta$  as the set of checkpoints saved at these specific training iterations:  $\Theta = \{\pmb {\theta}_{n_1},\pmb {\theta}_{n_2},\pmb {\theta}_{n_3},\dots ,\pmb {\theta}_{n_K}\}$ . Given  $\Theta$ , we aim to find a group of coefficients  $\alpha = \{\alpha_{1},\alpha_{2},\alpha_{3},\dots ,\alpha_{K}\}$ , that achieves the best utility when linearly combine all checkpoints:
+
+$$
+\underset {\alpha_ {1}, \alpha_ {2}, \alpha_ {3}, \dots , \alpha_ {K} \in \mathbb {R}} {\arg \min } \mathcal {F} \left(\alpha_ {1} \boldsymbol {\theta} _ {n _ {1}} + \alpha_ {2} \boldsymbol {\theta} _ {n _ {2}} + \alpha_ {3} \boldsymbol {\theta} _ {n _ {3}} + \dots + \alpha_ {K} \boldsymbol {\theta} _ {n _ {K}}\right); \quad \text {s . t .} \quad \sum_ {i = 1} ^ {K} \alpha_ {i} = 1, \tag {7}
+$$
+
+where  $\mathcal{F}(\theta)$  denotes an evaluation function that measures the generation quality of  $\theta$  regarding a specific metric, indicating higher quality with a smaller value. The constraint  $\sum_{i=1}^{K} \alpha_i = 1$  reduces the dimension of the search space by 1. However, we find it enables a more effective exploration for the search algorithm. Additionally, since the basin area in Fig. 4 extends beyond the convex hull of the model weights, we allow  $\alpha_i < 0$ , distinguishing LCSC from existing weight averaging methods.
+
+The trained model can also be a Low Rank Adapter (LoRA) (Hu et al., 2021). In this case, each checkpoint is a product of two low-rank matrices:  $\pmb{\theta}_{n_i} = \pmb{B}_{n_i}\pmb{A}_{n_i}$ . The weighted sum of all checkpoints by the coefficients is:  $\alpha_{1}\pmb{B}_{n_{1}}\pmb{A}_{n_{1}} + \alpha_{2}\pmb{B}_{n_{2}}\pmb{A}_{n_{2}} + \alpha_{3}\pmb{B}_{n_{3}}\pmb{A}_{n_{3}} + \dots +\alpha_{K}\pmb{B}_{n_{K}}\pmb{A}_{n_{K}}$
+
+# 4.2 EVOLUTIONARY SEARCH
+
+As discussed in Sec. 3, the optimal coefficients are difficult to predefine. To solve the problem in Sec. 4.1, we propose to use evolutionary search (Real et al., 2019; Liu et al., 2023a; Li et al., 2023).
+
+Our detailed algorithm is provided in Alg. 1 and an illustration is given in Fig. 3. Given a set of checkpoints  $\Theta$ , we can apply LCSC at any training iteration  $n$  along the training trajectory. The key points of our algorithm are summarized as follows: (a) For the stability during search, we subtract the first checkpoint from each subsequent checkpoint and search for the coefficients of their difference. (b) We initialize the population using EMA coefficients of several different rates. (c) We then conduct an evolutionary search which views a group of coefficients as an individual. In each search iteration, only the top-performing individuals are selected as parents to reproduce the next generation through crossover and mutation. Specifically, during crossover, we randomly mix the coefficients of the two parents or select one set entirely. For mutation, we introduce random Gaussian noise to each coefficient. This stochastic element ensures that beneficial adaptations are conserved and advanced to the subsequent generation, whereas detrimental modifications are discarded. (d) We repeat this reproduction process for a predetermined number of times within each search iteration and update the whole population with all newly generated individuals. (e) Upon the completion of the search process, we choose the best individual as the output of our search algorithm.
+
+Since evolutionary search only performs model inference and is thus gradient-free, LCSC achieves significant savings in GPU memory usage (applicable to both DM and CM) and computational time (particularly for CM, given that the inference process for DM is still resource-intensive). Furthermore, LCSC has the unique advantage of optimizing models directly for non-differentiable metrics. The applicability of LCSC could potentially extend to various other tasks and models as well.
+
+# 4.3 USE CASES
+
+Finally, we delve into the use cases of LCSC. (a) Decrease Training Cost. Through our investigation, we have discerned that the training process for DM and CM can be categorized into two distinct phases, with the second phase taking the majority of training iterations but converging slowly (further discussion is deferred to App. E.2). Our method can be employed at the onset of the second phase and yield performance on par with or even surpasses the final converged model. Additionally, SOTA DMs and CMs necessitate training with large batch sizes, which places a substantial demand on computational resources (Karras et al., 2022; Dhariwal & Nichol, 2021; Song et al., 2023). Decreasing the batch size often leads to worse convergence speed. LCSC can be used for low batch
+
+size training and achieve equivalent performance to models trained with full batch size. These applications aims to expedite the training process. (b) Enhance Converged Models. LCSC can be applied to the checkpoints saved during the final stage of training to refine the converged model. It can enhance the generation quality under the same NFE, or maintain the generation quality with fewer NFE, thereby reducing the inference cost during deployment. Additionally, for users who only have access to released pre-trained models, we suggest fine-tuning the model for a few iterations and saving the intermediate checkpoints. Then LCSC can be used to enhance the released model.
+
+# 5 EXPERIMENTS
+
+In Sec. 5.1, we introduce the experimental settings. In Sec. 5.2 and 5.3, we demonstrate results for the two use cases. In Sec. 5.4 we demonstrate the potential of merging LoRA checkpoints. We further apply our method on text-to-image task in Sec. 5.5. Finally, we discuss the generalization ability of LCSC in Sec. 5.6. We ablate several important hyper-parameters in our workflow in App. E.4.
+
+# 5.1 EXPERIMENTAL SETUP
+
+We follow previous work to configure the training process. Details of the training and search configurations are provided in App. E.1.
+
+Evaluation. We choose the most commonly used metric, FID (Heusel et al., 2017), as the search objective. To evaluate the combination coefficients during the evolutionary search, we sample 5K images for ImageNet-64 with DM and LSUN datasets, while 10K for other settings, using a fixed group of initial noise. For the final model evaluation, we generate 50K samples using a different group of initial noises. To demonstrate that LCSC searches based on FID but achieves overall improvement, we also report PickScore (Kirstain et al., 2023), ImageReward (Xu et al., 2023), and the winning rates based on these two metrics for the text-to-image task, and IS (Salimans et al., 2016), precision, and recall (Kynkänniemi et al., 2019) for other cases. In Sec. 5.6, we further report FCD (the variant of FID using CLIP features instead of Inception features) and KID (Binkowski et al., 2018) to show that LCSC also brings improvements in other feature spaces or metrics.
+
+Baselines. Since EMA is the default setting in almost all works of DM and CM, we report the performance of the EMA weights for full-model training using the rate reported by official papers (Song et al., 2023; 2020a; Nichol & Dhariwal, 2021) as our main baselines. For CM models on ImageNet-64 and LSUN datasets, we additionally download the official models trained with full batch size and test their performance for a complete comparison. We further conduct a grid search of EMA rate for the final model as a stronger baseline, denoted as EMA*, to prove the sub-optimality of EMA and that LCSC is a more effective method. More experimental details are provided in App. E.
+
+# 5.2 RESULTS ON REDUCING THE TRAINING COST
+
+We conduct a series of experiments on CD and CT to show that LCSC can significantly reduce their training costs. The efficiency of LCSC has considered both training and search cost on the CPU and GPU, with detailed information available in App. E.3. Generated images are visualized in App. H.
+
+LCSC can be applied at an early training stage to reduce the number of training iterations. As illustrated in Tab. 1, applying LCSC to the CD model with only  $50\mathrm{K}$  training iterations achieves a better FID than the final model trained with  $800\mathrm{K}$  iterations on CIFAR-10 (3.10 vs. 3.66), which accelerates the training by approximately  $14\times$ . For CT, we only achieve a converged FID of 9.87 with vanilla training and are unable to fully reproduce the result of  $\mathrm{FID} = 8.70$  reported by Song et al. (2023). However, by applying our method at  $400\mathrm{K}$  training iterations, we can achieve a similar FID to the reported one from  $800\mathrm{K}$  training iterations, achieving around a  $1.9\times$  speedup. On ImageNet-64 (Tab. 2), we decrease the batch size from 2048 to 256 due to limited resources. For CD, LCSC achieves better performance (5.51 vs. 7.30 in FID) than the final converged model with only half of the training iterations (300K vs. 600K). For CT, LCSC also significantly reduces the final converged FID (10.5 vs. 15.6) with fewer training iterations (600K vs. 1000K).
+
+LCSC can also handle smaller training batch sizes to achieve higher speedups. As illustrated in Tab. 1, LCSC with a batch size of 128 outperforms the final converged model with a batch size of 512 for both CD and CT (3.21 vs. 3.66 with CD and 8.54 vs. 9.87 with CT), achieving overall speedups of  $23 \times$  and  $7 \times$ , respectively. On ImageNet-64, we test the official models of Song et al. (2023), which were trained with a 2048 batch size for both CD and CT. With the assistance of LCSC,
+
+Table 1: Generation quality of CMs on CIFAR-10. The training speedup is compared against the standard training with 800K iterations and 512 batch size. Our results that beat or match the standard training (the "released" model for FID & IS, our reproduced results for Prec. & Rec.) are underlined.  
+
+<table><tr><td>Model</td><td>Method</td><td>Training Iter</td><td>Batch Size</td><td>NFE</td><td>FID(↓)</td><td>IS(↑)</td><td>Prec.(↑)</td><td>Rec.(↑)</td><td>Speed(↑)</td></tr><tr><td rowspan="11">CD</td><td rowspan="4">EMA(released)</td><td>200K</td><td>512</td><td>1</td><td>4.08</td><td>9.18</td><td>0.68</td><td>0.56</td><td></td></tr><tr><td>800K</td><td>512</td><td>1</td><td>3.66</td><td>9.35</td><td>0.68</td><td>0.57</td><td></td></tr><tr><td>850K</td><td>512</td><td>1</td><td>3.65</td><td>9.32</td><td>0.68</td><td>0.57</td><td></td></tr><tr><td>850K</td><td>512</td><td>2</td><td>2.89</td><td>9.55</td><td>0.69</td><td>0.58</td><td></td></tr><tr><td>EMA*(released)</td><td>800K</td><td>512</td><td>1</td><td>3.55</td><td>9.48</td><td>-</td><td>-</td><td></td></tr><tr><td>(released)</td><td>800K</td><td>512</td><td>2</td><td>2.93</td><td>9.75</td><td>-</td><td>-</td><td></td></tr><tr><td>EMA*</td><td>800K</td><td>512</td><td>1</td><td>3.51</td><td>9.37</td><td>0.68</td><td>0.57</td><td></td></tr><tr><td rowspan="4">LCSC</td><td>50K</td><td>512</td><td>1</td><td>3.10</td><td>9.50</td><td>0.66</td><td>0.58</td><td>~14×</td></tr><tr><td>800K</td><td>512</td><td>1</td><td>2.44</td><td>9.82</td><td>0.67</td><td>0.60</td><td>-</td></tr><tr><td>800+40K</td><td>512</td><td>1</td><td>2.50</td><td>9.70</td><td>0.68</td><td>0.59</td><td>-</td></tr><tr><td>100K</td><td>128</td><td>1</td><td>3.21</td><td>9.48</td><td>0.66</td><td>0.58</td><td>~23×</td></tr><tr><td rowspan="7">CT</td><td rowspan="3">EMA(released)</td><td>400K</td><td>512</td><td>1</td><td>12.1</td><td>8.52</td><td>0.67</td><td>0.43</td><td></td></tr><tr><td>800K</td><td>512</td><td>1</td><td>9.87</td><td>8.81</td><td>0.69</td><td>0.42</td><td></td></tr><tr><td>800K</td><td>512</td><td>1</td><td>8.70</td><td>8.49</td><td>-</td><td>-</td><td></td></tr><tr><td>EMA*</td><td>800K</td><td>512</td><td>1</td><td>9.70</td><td>8.81</td><td>0.69</td><td>0.42</td><td></td></tr><tr><td rowspan="3">LCSC</td><td>400K</td><td>512</td><td>1</td><td>8.89</td><td>8.79</td><td>0.67</td><td>0.47</td><td>~1.9×</td></tr><tr><td>800+40K</td><td>512</td><td>1</td><td>7.05</td><td>9.01</td><td>0.70</td><td>0.45</td><td>-</td></tr><tr><td>450K</td><td>128</td><td>1</td><td>8.54</td><td>8.66</td><td>0.69</td><td>0.44</td><td>~7×</td></tr></table>
+
+Table 2: Generation quality of consistency models on ImageNet-64. For CD, the speedup is compared against the standard training with 600K iterations and 2048 batch size. For CT, the speedup is compared against the standard training with 800K iterations and 2048 batch size. Our results that beat the standard training ("released") are underlined.  
+
+<table><tr><td>Model</td><td>Method</td><td>Training Iter</td><td>Batch Size</td><td>NFE</td><td>FID(↓)</td><td>IS(↑)</td><td>Prec.(↑)</td><td>Rec.(↑)</td><td>Speed(↑)</td></tr><tr><td rowspan="7">CD</td><td rowspan="4">EMA(released)</td><td>300K</td><td>256</td><td>1</td><td>7.70</td><td>37.0</td><td>0.67</td><td>0.62</td><td></td></tr><tr><td>600K</td><td>256</td><td>1</td><td>7.30</td><td>37.2</td><td>0.67</td><td>0.62</td><td></td></tr><tr><td>650K</td><td>256</td><td>1</td><td>7.17</td><td>37.7</td><td>0.67</td><td>0.62</td><td></td></tr><tr><td>600K</td><td>2048</td><td>1</td><td>6.31</td><td>39.5</td><td>0.68</td><td>0.63</td><td></td></tr><tr><td>EMA*</td><td>600K</td><td>256</td><td>1</td><td>7.17</td><td>37.7</td><td>0.67</td><td>0.62</td><td></td></tr><tr><td rowspan="2">LCSC</td><td>300K</td><td>256</td><td>1</td><td>5.51</td><td>39.8</td><td>0.68</td><td>0.62</td><td>~15×</td></tr><tr><td>600+20K</td><td>256</td><td>1</td><td>5.07</td><td>42.5</td><td>0.69</td><td>0.62</td><td>~7.6×</td></tr><tr><td rowspan="7">CT</td><td rowspan="4">EMA(released)</td><td>600K</td><td>256</td><td>1</td><td>16.6</td><td>30.6</td><td>0.62</td><td>0.54</td><td></td></tr><tr><td>800K</td><td>256</td><td>1</td><td>15.8</td><td>31.1</td><td>0.64</td><td>0.55</td><td></td></tr><tr><td>1000K</td><td>256</td><td>1</td><td>15.6</td><td>31.2</td><td>0.64</td><td>0.55</td><td></td></tr><tr><td>800K</td><td>2048</td><td>1</td><td>13.1</td><td>29.2</td><td>0.70</td><td>0.47</td><td></td></tr><tr><td>EMA*</td><td>1000K</td><td>256</td><td>1</td><td>15.6</td><td>31.2</td><td>0.64</td><td>0.55</td><td></td></tr><tr><td rowspan="2">LCSC</td><td>600K</td><td>256</td><td>1</td><td>10.5</td><td>36.8</td><td>0.66</td><td>0.56</td><td>~11.4×</td></tr><tr><td>800K</td><td>256</td><td>1</td><td>9.02</td><td>38.8</td><td>0.68</td><td>0.55</td><td>~7.3×</td></tr></table>
+
+models trained with a 256 batch size consistently outperform the official models (5.51, 5.07 vs. 6.31 with CD and 9.02, 10.5 vs. 13.1 with CT), achieving overall acceleration ratios up to  $15 \times$ .
+
+# 5.3 RESULTS ON ENHANCING PRE-TRAINED MODELS
+
+Our experiments on both DM and CM show that LCSC can significantly boost the performance of final converged models and reduce the required sampling steps.
+
+To simulate scenarios where users have access only to pre-trained models, we fine-tune the final converged model for a few iterations and then apply LCSC. As illustrated in the  $800 + 40\mathrm{K}$  LCSC row with CD and CT in Tab. 1, the  $600 + 20\mathrm{K}$  LCSC row with CD in Tab. 2 and the  $1000 + 20\mathrm{K}$  LCSC on both datasets in Tab. 3, this approach significantly enhances the performance of the already converged model. For model developers, they can directly utilize the checkpoints close to the final iteration, as demonstrated in the  $800\mathrm{K}$  LCSC row with CD in Tab. 1 and  $800\mathrm{K}$  LCSC row with CT in Tab. 2. LCSC leads to notable improvements in all these scenarios. Notably, on CIFAR-10, LCSC even outperforms the  $800\mathrm{K}$  and  $850\mathrm{K}$  models that use 2-step sampling, achieving better performance with 1-step sampling (2.44, 2.50 FID vs. 2.93, 2.89 FID), thereby doubling the inference speed. More experiments with Stable Diffusion checkpoints can be found at App. E.8
+
+Same phenomenon has also been observed for DM, as shown in Tab. 4: applying LCSC at the final stage of training can obtain a model that performs significantly better than the final model of EMA. Moreover, we observe that the model found by LCSC can match the performance of the best EMA
+
+Table 3: Generation quality of models on LSUN-bedroom and LSUN-cat with consistency training (CT). We get the checkpoints by fine-tuning the released official model.  
+
+<table><tr><td>Dataset</td><td>Method</td><td>Training Iter</td><td>Batch Size</td><td>NFE</td><td>FID(↓)</td><td>Prec.(↑)</td><td>Rec.(↑)</td></tr><tr><td rowspan="3">LSUN-Cat</td><td rowspan="2">EMA</td><td>1000K(released)</td><td>2048</td><td>1</td><td>20.8</td><td>0.53</td><td>0.46</td></tr><tr><td>1000+20K(fine-tuned)</td><td>256</td><td>1</td><td>21.0</td><td>0.58</td><td>0.44</td></tr><tr><td>LCSC</td><td>1000+20K</td><td>256</td><td>1</td><td>17.8</td><td>0.63</td><td>0.48</td></tr><tr><td rowspan="3">LSUN-Bedroom</td><td rowspan="2">EMA</td><td>1000K(released)</td><td>2048</td><td>1</td><td>16.1</td><td>0.60</td><td>0.17</td></tr><tr><td>1000+20K(fine-tuned)</td><td>256</td><td>1</td><td>16.0</td><td>0.60</td><td>0.17</td></tr><tr><td>LCSC</td><td>1000+20K</td><td>256</td><td>1</td><td>13.5</td><td>0.59</td><td>0.37</td></tr></table>
+
+<table><tr><td>Dataset</td><td>Method</td><td>Training Iter.</td><td>NFE</td><td>FID(↓)</td><td>IS(↑)</td><td>Prec.(↑)</td><td>Rec.(↑)</td></tr><tr><td rowspan="6">CIFAR10</td><td rowspan="2">EMA</td><td>150K</td><td>15</td><td>6.28</td><td>8.74</td><td>0.61</td><td>0.59</td></tr><tr><td>800K</td><td>15</td><td>4.16</td><td>9.37</td><td>0.64</td><td>0.60</td></tr><tr><td>EMA*</td><td>800K</td><td>15</td><td>3.96</td><td>9.50</td><td>0.64</td><td>0.60</td></tr><tr><td rowspan="3">LCSC</td><td>150K</td><td>15</td><td>4.76</td><td>8.97</td><td>0.61</td><td>0.59</td></tr><tr><td>800K</td><td>15</td><td>3.18</td><td>9.59</td><td>0.64</td><td>0.61</td></tr><tr><td>800K</td><td>9</td><td>3.97</td><td>9.50</td><td>0.63</td><td>0.60</td></tr><tr><td rowspan="6">ImageNet</td><td rowspan="2">EMA</td><td>150K</td><td>15</td><td>22.3</td><td>15.0</td><td>0.55</td><td>0.45</td></tr><tr><td>500K</td><td>15</td><td>19.8</td><td>16.9</td><td>0.58</td><td>0.59</td></tr><tr><td>EMA*</td><td>500K</td><td>15</td><td>18.1</td><td>17.3</td><td>0.59</td><td>0.59</td></tr><tr><td rowspan="3">LCSC</td><td>150K</td><td>15</td><td>19.1</td><td>15.3</td><td>0.56</td><td>0.56</td></tr><tr><td>500K</td><td>15</td><td>15.3</td><td>17.6</td><td>0.59</td><td>0.59</td></tr><tr><td>500K</td><td>12</td><td>17.2</td><td>17.2</td><td>0.57</td><td>0.59</td></tr></table>
+
+Table 4: Generation quality of diffusion models. Our results that beat the standard training are underlined.  
+Table 5: Single-step generation quality of LoRA models.  
+
+<table><tr><td>Dataset</td><td>Model</td><td>Method</td><td>Training Iter</td><td>Batch Size</td><td>FID(↓)</td><td>IS (↑)</td><td>Prec.(↑)</td><td>Rec.(↑)</td></tr><tr><td rowspan="6">ImageNet</td><td rowspan="3">CD</td><td>EMA</td><td>600K</td><td>256</td><td>7.30</td><td>37.21</td><td>0.67</td><td>0.62</td></tr><tr><td>LoRA</td><td>600+20K (fine-tuned with LoRA)</td><td>256</td><td>6.79</td><td>37.74</td><td>0.67</td><td>0.63</td></tr><tr><td>LCSC</td><td>600+20K</td><td>256</td><td>4.21</td><td>43.22</td><td>0.68</td><td>0.64</td></tr><tr><td rowspan="3">CT</td><td>EMA</td><td>800K</td><td>256</td><td>15.75</td><td>31.08</td><td>0.64</td><td>0.55</td></tr><tr><td>LoRA</td><td>800+20K (fine-tuned with LoRA)</td><td>256</td><td>15.14</td><td>31.59</td><td>0.64</td><td>0.56</td></tr><tr><td>LCSC</td><td>800+20K</td><td>256</td><td>4.90</td><td>45.33</td><td>0.67</td><td>0.65</td></tr><tr><td rowspan="2">LSUN-Bedroom</td><td rowspan="2">CT</td><td>EMA</td><td>1000K</td><td>2048</td><td>16.10</td><td>-</td><td>0.60</td><td>0.17</td></tr><tr><td>LCSC</td><td>1000+20K</td><td>256</td><td>14.49</td><td>-</td><td>0.60</td><td>0.20</td></tr></table>
+
+Table 6: Text-to-image generation quality of LCM LoRA. PKS and IR stand for PickScore and ImageReward. WR stands for winning rate among all generated images compared to baselines.  
+
+<table><tr><td>Method</td><td>Training Iter</td><td>Batch Size</td><td>Search NFE</td><td>Eval NFE</td><td>FID(↓)</td><td>PKS(↑)</td><td>WR@PKS(↑)</td><td>IR(↑)</td><td>WR@IR(↑)</td><td>CLIP-Score</td></tr><tr><td>Vanilla training</td><td>6K</td><td>12</td><td>-</td><td>4</td><td>32.52</td><td>0.46</td><td>34%</td><td>-2.20</td><td>35%</td><td>26.02</td></tr><tr><td>LCSC</td><td>6K</td><td>12</td><td>4</td><td>4</td><td>28.30</td><td>0.54</td><td>66%</td><td>-2.19</td><td>65%</td><td>26.39</td></tr><tr><td>Vanilla training</td><td>6K</td><td>12</td><td>-</td><td>2</td><td>43.32</td><td>0.46</td><td>33%</td><td>-2.22</td><td>23%</td><td>25.16</td></tr><tr><td>LCSC</td><td>6K</td><td>12</td><td>2</td><td>2</td><td>30.39</td><td>0.54</td><td>67%</td><td>-2.20</td><td>77%</td><td>26.01</td></tr><tr><td>Vanilla training</td><td>6K</td><td>12</td><td>-</td><td>2</td><td>43.32</td><td>0.47</td><td>34%</td><td>-2.22</td><td>25%</td><td>25.16</td></tr><tr><td>LCSC</td><td>6K</td><td>12</td><td>4</td><td>2</td><td>33.13</td><td>0.53</td><td>66%</td><td>-2.20</td><td>75%</td><td>25.89</td></tr></table>
+
+model with fewer NFE during inference. Specifically, our method requiring  $\mathrm{NFE} = 9$  is on par with  $\mathrm{EMA}^*$  using  $\mathrm{NFE} = 15$  on CIFAR-10. Similarly, our method with  $\mathrm{NFE} = 12$  is competitive with  $\mathrm{EMA}^*$  utilizing  $\mathrm{NFE} = 15$  on ImageNet-64. This also highlights the potential for inference speedup.
+
+# 5.4 RESULTS WITH LORA CHECKPOINTS
+
+As discussed in Sec. 4.1, we can fine-tune the model with LoRA and linearly combine the LoRA checkpoints, which significantly reduces the demand for memory and storage compared to full-model LCSC. Results are reported in Tab. 5. Fine-tuning with LoRA checkpoints achieves performance comparable to full-model fine-tuning with LCSC. On ImageNet-64 dataset with CT, we surprisingly find that applying LCSC to LoRA checkpoints outperforms the full model by a substantial margin, highlighting the effectiveness of combining LCSC with LoRA.
+
+# 5.5 RESULTS ON TEXT-TO-IMAGE TASK
+
+To further validate the practicality of LCSC, we apply it to the training process of LCM-LoRA (Luo et al., 2023a;b). We train the model on CC12M dataset and use 1k extra image-text pair in this dataset to conduct search. We apply 4-step sampling and 2-step sampling and observe that the search results from 4-step sampling can generalize to 2-step evaluation. We test the search result using 10k data in MS-COCO dataset with FID, PickScore, and ImageReward. Since PickScore and ImageReward are both image-wise metrics, we also report the winning rate of LCSC compared to baseline for these two metrics. More details of the setting can be found in App. E.1.3. Results are shown at Tab. 6, several examples are visualized in Fig. 2 and more are provided in Fig. 16. LCSC can significantly
+
+Table 7: Evaluation results using FID, FCD on the training dataset and KID on the test dataset. See Tab. 18 for test set FID and Tab. 17 for results on CT.  
+
+<table><tr><td colspan="6">(a) Results on CIFAR-10.</td><td colspan="6">(b) Results on ImageNet-64.</td></tr><tr><td>Method</td><td>Training Iter</td><td>Batch Size</td><td>FID(↓)</td><td>FCD(↓)</td><td>KID(↓)</td><td>Method</td><td>Training Iter</td><td>Batch Size</td><td>FID(↓)</td><td>FCD(↓)</td><td>KID(↓)</td></tr><tr><td>CD</td><td>800k</td><td>512</td><td>3.66</td><td>18.0</td><td>1.38e-3</td><td>CD</td><td>300k</td><td>256</td><td>7.70</td><td>30.0</td><td>3.98e-3</td></tr><tr><td>CD</td><td>800k</td><td>512</td><td>3.65</td><td>17.3</td><td>1.39e-3</td><td>CD</td><td>600k</td><td>256</td><td>7.30</td><td>29.1</td><td>3.65e-3</td></tr><tr><td>LCSC</td><td>100k</td><td>128</td><td>3.21</td><td>23.4</td><td>7.88e-4</td><td>CD</td><td>600k</td><td>2048</td><td>6.31</td><td>25.5</td><td>3.37e-3</td></tr><tr><td>LCSC</td><td>250k</td><td>512</td><td>2.66</td><td>14.3</td><td>4.43e-4</td><td>LCSC</td><td>300k</td><td>256</td><td>5.51</td><td>23.9</td><td>3.14e-3</td></tr><tr><td>LCSC</td><td>800k</td><td>512</td><td>2.44</td><td>14.7</td><td>4.32e-4</td><td>LCSC</td><td>800+40k</td><td>256</td><td>5.07</td><td>23.2</td><td>1.97e-3</td></tr><tr><td>LCSC</td><td>800+40k</td><td>512</td><td>2.50</td><td>14.0</td><td>4.11e-4</td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+![](images/b9adf163a85192af5ee6c0d6b779186228898980522be74e7fc1779ba227079e.jpg)  
+Figure 5: Visualization of weight combination coefficients obtained using LCSC and EMA.
+
+![](images/380da79730907b27f2dd575db30da5c191cab8035d1825f4c88b713baea137c3.jpg)
+
+![](images/fb2c2d686d91941b9b293928082780a1d212145bda1a230b0fc37b55dc42525e.jpg)
+
+boost the FID. For human evaluation based metrics, LCSC achieves both better average value and higher winning rate, indicating the potential of LCSC to be used in more complicated tasks.
+
+# 5.6 DISCUSSION ON THE GENERALIZATION OF LCSC
+
+Even if LCSC searches based on evaluation metrics (e.g., FID), we show here that LCSC generalizes across metrics and data.
+
+Metric Generalization. In Sec. 5.2 to 5.5, we have evaluated the search results across various metrics besides FID, including IS, Precision, Recall, PickScore, and ImageReward. Additionally, we further evaluate the search results on the test set using KID (Binkowski et al., 2018) and FCD (i.e., replacing Inception Net with CLIP as the feature extractor for FID); see Tab. 7 and Tab. 17. These metrics differ from FID in various aspects, including feature extractors (FCD, PickScore, ImageReward), calculation formulas (IS, Precision, Recall, KID, PickScore, and ImageReward), application scenarios (PickScore and ImageReward for human-based evaluation). We observe that most metrics are consistently improved, demonstrating the generalizability of LCSC across metrics.
+
+Data Generalization. (a) Generalization of initial noise. During the search process, a fixed group of initial noise is used. When performing evaluation, we use a different set of noise. Therefore, the search result of LCSC generalizes to different initial noise. (b) Generalization of data. Since FID is computed using the training set as the ground truth, we calculate additional metrics based on the test set to validate that the search results of LCSC generalize across data; see Tabs. 7 and 18. Additionally, for text-to-image task (see Sec. 5.5), LCSC was searched based on CC12M but evaluated on the test dataset of MS-COCO, demonstrating generalization to a closely related data distribution.
+
+# 6 DISCUSSION AND CONCLUSION
+
+Analysis of Search Patterns. Since our method searches for the optimal coefficients, it would be interesting to check if the search pattern aligns with previous methods using fixed forms, such as EMA, and to derive insights that could inspire further research on checkpoint merging. We visualize several searched combination coefficients in Fig. 5. (a) First, we observe that earlier checkpoints can also be important, as some of them have large coefficients. (b) Moreover, we find that LCSC tend to assign large coefficients to a small subset of weights, whereas the coefficients for the majority of weights are nearly zero. Further investigation suggests smaller and homogeneous solution also exists but may be difficult to be found by LCSC (seeApp. F). (c) Finally, the presence of multiple significant negative coefficients highlights that certain weights can act as critical negative examples. This finding implies traditional weight-averaging methods, such as EMA, are suboptimal. Since they commonly confine the resulting model within the convex hull of all weights, which excludes the discovered solutions. More search patterns and other insights are provided in App. F.
+
+In this work, we investigate linearly combining saved checkpoints during training to achieve better performance for DM and CM. We demonstrate the common benefits of checkpoint merging and provide a theoretical analysis to clarify that the current standard merging method is suboptimal, emphasizing the need for flexible merging coefficients. We then propose using an evolutionary algorithm to search for the optimal coefficients, which runs efficiently. Through extensive experiments, we demonstrate two uses of our method: reducing training costs and enhancing generation quality.
+
+# ACKNOWLEDGEMENT
+
+This work was supported by National Natural Science Foundation of China (No. 62325405, 62104128, U19B2019, U21B2031, 61832007, 62204164), Flemish Government (AI Research Program) and the Research Foundation - Flanders (FWO) through project number G0G2921N, Tsinghua EE Xilinx AI Research Fund, and Beijing National Research Center for Information Science and Technology (BNRist). We thank the anonymous reviewers for their valuable feedback and suggestions. We thank Yiran Shi for his help with experiments and all the support from Infinigence-AI.
+
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+
+# A PROOF OF THEORETICAL ANALYSIS
+
+Lemma A.1 Suppose  $f$  is  $\beta$ -strongly convex, and that  $\mathbb{E}[\| \hat{\pmb{g}}_n\|^2] \leq G^2$  for all  $n = 1,\dots ,N$ . Consider SGD with step sizes  $\eta_{n} = 1 / \beta n$ , then for any  $N > 1$ , it holds that:
+
+$$
+\mathbb {E} \left[ \| \boldsymbol {\theta} ^ {N} - \boldsymbol {\theta} ^ {*} \| ^ {2} \right] \leq \frac {2 G ^ {2}}{\beta^ {2} n} \tag {8}
+$$
+
+Proof of Lem. A.1 basically follows Rakhlin et al. (2011), while we improve the upper bound with a factor of 2.
+
+By the strong convexity of  $f$ , we have:
+
+$$
+\left\langle \boldsymbol {g} _ {n}, \boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*} \right\rangle \geq f \left(\boldsymbol {\theta} _ {n}\right) - f \left(\boldsymbol {\theta} ^ {*}\right) + \frac {\beta}{2} \| \boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*} \| ^ {2}, \tag {9}
+$$
+
+$$
+f \left(\boldsymbol {\theta} _ {n}\right) - f \left(\boldsymbol {\theta} ^ {*}\right) \geq \frac {\beta}{2} \left\| \boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*} \right\| ^ {2}. \tag {10}
+$$
+
+Based on the above inequalities, we can derive:
+
+$$
+\mathbb {E} \left[ \left\| \boldsymbol {\theta} _ {n + 1} - \boldsymbol {\theta} ^ {*} \right\| ^ {2} \right] = \mathbb {E} \left[ \left\| \boldsymbol {\theta} _ {n} + \eta_ {n} \hat {\boldsymbol {g}} _ {n} - \boldsymbol {\theta} ^ {*} \right\| \right] \tag {11}
+$$
+
+$$
+= \mathbb {E} [ \| \boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*} \| ^ {2} ] + \eta_ {n} ^ {2} \mathbb {E} [ \| \hat {\boldsymbol {g}} _ {n} \| ^ {2} ] - 2 \eta_ {n} \mathbb {E} [ \langle \boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*}, \hat {\boldsymbol {g}} _ {n} \rangle ] \tag {12}
+$$
+
+$$
+= \mathbb {E} \left[ \| \boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*} \| ^ {2} \right] + \eta_ {n} ^ {2} G ^ {2} - 2 \eta_ {n} \mathbb {E} \left[ \left\langle \boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*}, \boldsymbol {g} _ {n} \right\rangle \right] \tag {13}
+$$
+
+$$
+\leq \mathbb {E} [ \| \boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*} \| ^ {2} ] + \eta_ {n} ^ {2} G ^ {2} - 2 \eta_ {n} (f (\boldsymbol {\theta} _ {n}) - f (\boldsymbol {\theta} ^ {*}) + \frac {\beta}{2} \| \boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*} \| ^ {2}) \tag {14}
+$$
+
+$$
+\leq^ {1 0} \mathbb {E} \left[ \left\| \boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*} \right\| ^ {2} \right] + \eta_ {n} ^ {2} G ^ {2} - 2 \eta_ {n} \beta \left(\left\| \boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*} \right\| ^ {2}\right). \tag {15}
+$$
+
+Plugging in  $\eta_{n} = 1 / \beta n$ , we obtain:
+
+$$
+\mathbb {E} \left[ \| \boldsymbol {\theta} _ {n + 1} - \boldsymbol {\theta} ^ {*} \| ^ {2} \right] \leq \left(1 - \frac {2}{n}\right) \mathbb {E} \left[ \| \boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*} \| ^ {2} \right] + \frac {G ^ {2}}{\beta^ {2} n ^ {2}}. \tag {16}
+$$
+
+When  $n = 1$ , we have:
+
+$$
+\mathbb {E} \left[ \left\| \boldsymbol {\theta} _ {2} - \boldsymbol {\theta} ^ {*} \right\| ^ {2} \right] \leq - \mathbb {E} \left[ \left\| \boldsymbol {\theta} _ {1} - \boldsymbol {\theta} ^ {*} \right\| ^ {2} \right] + \frac {G ^ {2}}{\beta^ {2}} \leq \frac {G ^ {2}}{\beta^ {2}}. \tag {17}
+$$
+
+Since  $\mathbb{E}[\|\pmb{\theta}_2 - \pmb{\theta}^*\|^2] \geq 0$ , we can also obtain:
+
+$$
+\mathbb {E} \left[ \left\| \boldsymbol {\theta} _ {1} - \boldsymbol {\theta} ^ {*} \right\| ^ {2} \right] \leq \frac {G ^ {2}}{\beta^ {2}}. \tag {18}
+$$
+
+Therefore, when  $n = 1,2$ , Lem. A.1 is satisfied. For  $n \geq 2$ , we can further prove it using induction agreement:
+
+$$
+\mathbb {E} \left[ \| \boldsymbol {\theta} _ {n + 1} - \boldsymbol {\theta} ^ {*} \| ^ {2} \right] \leq \left(1 - \frac {2}{n}\right) \mathbb {E} \left[ \| \boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*} \| ^ {2} \right] + \frac {G ^ {2}}{\beta^ {2} n ^ {2}} \tag {19}
+$$
+
+$$
+\leq \left(1 - \frac {2}{n}\right) \frac {2 G ^ {2}}{\beta^ {2} n} + \frac {G ^ {2}}{\beta^ {2} n ^ {2}} \tag {20}
+$$
+
+$$
+\leq \frac {(2 n - 3) G ^ {2}}{\beta^ {2} n ^ {2}} \tag {21}
+$$
+
+$$
+\leq \frac {2 G ^ {2}}{\beta^ {2} (n + 1)} \tag {22}
+$$
+
+# A.1 PROOF OF THM. 3.2
+
+Rearrange Eq. (12):
+
+$$
+\mathbb {E} [ \langle \boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*}, \boldsymbol {g} _ {n} \rangle ] = \frac {1}{2 \eta_ {n}} (\mathbb {E} [ \| \boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*} \| ^ {2} ] - \mathbb {E} [ \| \boldsymbol {\theta} _ {n + 1} - \boldsymbol {\theta} ^ {*} \| ^ {2} ]) + \frac {\eta_ {n}}{2} \mathbb {E} [ \| \hat {\boldsymbol {g}} _ {n} \| ^ {2} ]. \tag {23}
+$$
+
+Plugging in Eq. (9) and  $\eta_{n} = 1 / \beta n$ , we obtain:
+
+$$
+\mathbb {E} \left[ f \left(\boldsymbol {\theta} _ {n}\right) - f \left(\boldsymbol {\theta} ^ {*}\right) \right] \leq \frac {\beta}{2} \left((n - 1) \mathbb {E} \left[ \| \boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*} \| ^ {2} \right] - n \mathbb {E} \left[ \| \boldsymbol {\theta} _ {n + 1} - \boldsymbol {\theta} ^ {*} \| ^ {2} \right]\right) + \frac {1}{2 \beta n} G ^ {2}. \tag {24}
+$$
+
+Summing over  $N$  iterations using coefficients  $\alpha$ :
+
+$$
+\begin{array}{l} \mathbb {E} \left[ f \left(\bar {\boldsymbol {\theta}} _ {n} ^ {\boldsymbol {\alpha}}\right) - f \left(\boldsymbol {\theta} ^ {*}\right) \right] (25) \\ \leq \frac {\beta}{2 \mathrm {A}} \sum_ {n = 1} ^ {N} \alpha_ {n} \left((n - 1) \mathbb {E} [ \| \boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*} \| ^ {2} ] - n \mathbb {E} [ \| \boldsymbol {\theta} _ {n + 1} - \boldsymbol {\theta} ^ {*} \| ^ {2} ]\right) + \frac {G ^ {2}}{2 \beta \mathrm {A}} \sum_ {n = 1} ^ {N} \frac {\alpha_ {n}}{n} (26) \\ = \frac {\beta}{2 \mathrm {A}} \left(\sum_ {n = 2} ^ {N} \left(\alpha_ {n} - \alpha_ {n - 1}\right) (n - 1) \mathbb {E} \left[ \| \boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*} \| ^ {2} \right] - N \mathbb {E} \left[ \| \boldsymbol {\theta} _ {N + 1} - \boldsymbol {\theta} ^ {*} \| ^ {2} \right]\right) + \frac {G ^ {2}}{2 \beta \mathrm {A}} \sum_ {n = 1} ^ {N} \frac {\alpha_ {n}}{n} (27) \\ \leq \frac {\beta}{2 \mathrm {A}} \sum_ {n = 2} ^ {N} \left(\alpha_ {n} - \alpha_ {n - 1}\right) (n - 1) \mathbb {E} \left[ \| \boldsymbol {\theta} _ {n} - \boldsymbol {\theta} ^ {*} \| ^ {2} \right] + \frac {G ^ {2}}{2 \beta \mathrm {A}} \sum_ {n = 1} ^ {N} \frac {\alpha_ {n}}{n}. (28) \\ \end{array}
+$$
+
+Plugging in Lem. A.1, we have:
+
+$$
+\begin{array}{l} \mathbb {E} \left[ f \left(\bar {\boldsymbol {\theta}} _ {n} ^ {\boldsymbol {\alpha}}\right) - f \left(\boldsymbol {\theta} ^ {*}\right) \right] \leq \frac {G ^ {2}}{\beta \mathrm {A}} \sum_ {n = 2} ^ {N} \frac {\left(\alpha_ {n} - \alpha_ {n - 1}\right) (n - 1)}{n} + \frac {G ^ {2}}{2 \beta \mathrm {A}} \sum_ {n = 1} ^ {N} \frac {\alpha_ {n}}{n} (29) \\ \leq \frac {G ^ {2}}{\beta \mathrm {A}} \sum_ {n = 2} ^ {N} \left(\alpha_ {n} - \alpha_ {n - 1}\right) + \frac {G ^ {2}}{2 \beta \mathrm {A}} \sum_ {n = 1} ^ {N} \frac {\alpha_ {n}}{n}. (30) \\ \end{array}
+$$
+
+Plugging in  $\alpha_{n} = \gamma^{N - n}$ ,  $\mathrm{A} = \frac{1 - \gamma^{N - 1}}{1 - \gamma}$ , we have
+
+$$
+\begin{array}{l} \mathbb {E} \left[ f \left(\bar {\boldsymbol {\theta}} _ {n} ^ {\boldsymbol {\alpha}}\right) - f \left(\boldsymbol {\theta} ^ {*}\right) \right] \leq \frac {G ^ {2}}{\beta \mathrm {A}} \sum_ {n = 2} ^ {N} \left(\gamma^ {N - n} - \gamma^ {N - n + 1}\right) + \frac {G ^ {2}}{2 \beta \mathrm {A}} \sum_ {n = 1} ^ {N} \frac {\gamma^ {N - n}}{n} (31) \\ = \frac {G ^ {2} (1 - \gamma)}{\beta \mathrm {A}} \sum_ {n = 2} ^ {N} \gamma^ {N - n} + \frac {G ^ {2}}{2 \beta \mathrm {A}} \gamma^ {N} \sum_ {n = 1} ^ {N} \frac {1}{\gamma^ {n} n} (32) \\ = \frac {G ^ {2} (1 - \gamma)}{\beta \mathrm {A}} \frac {1 - \gamma^ {N - 2}}{1 - \gamma} + \frac {G ^ {2}}{2 \beta \mathrm {A}} \gamma^ {N} \sum_ {n = 1} ^ {N} \frac {1}{\gamma^ {n} n} (33) \\ = \frac {G ^ {2} (1 - \gamma)}{\beta} \frac {1 - \gamma^ {N - 2}}{1 - \gamma^ {N - 1}} + \frac {G ^ {2} (1 - \gamma)}{2 \beta (1 - \gamma^ {N - 1})} \gamma^ {N} \sum_ {n = 1} ^ {N} \frac {1}{\gamma^ {n} n} (34) \\ \leq \frac {G ^ {2} (1 - \gamma)}{\beta} + \frac {G ^ {2} (1 - \gamma)}{2 \beta (1 - \gamma^ {N - 1})} \gamma^ {N} \sum_ {n = 1} ^ {N} \frac {1}{\gamma^ {n} n}. (35) \\ \end{array}
+$$
+
+Next, we focus on deriving an upper bound for the second term in Eq. (35), since there is no simple closed-form expression for it. We notice that  $\gamma^n$  decays faster than  $n$  grows. Therefore it is more important to evaluate  $\frac{1}{\gamma^n n}$  when  $n$  approaches  $N$ .
+
+We will slightly abuse the notation  $x$  to denote a positive real number. First, we note that:
+
+$$
+\begin{array}{l} \mathrm {d} \frac {1}{\gamma^ {x} x} = - (\gamma^ {x} x) ^ {- 2} \left(\gamma^ {x} + x \gamma^ {x} \ln \gamma\right) (36) \\ = - \frac {1 / x + \ln \gamma}{\gamma^ {x} x}. (37) \\ \end{array}
+$$
+
+If  $x \geq -\frac{1}{\ln \sqrt{\gamma}}$ , we have  $1 / x \leq \ln 1 / \sqrt{\gamma}$ . Plug this into Eq. (37):
+
+$$
+\begin{array}{l} i f x \geq - \frac {1}{\ln \sqrt {\gamma}}, \quad \mathrm {d} \frac {1}{\gamma^ {x} x} \geq - \frac {\ln 1 / \sqrt {\gamma} + \ln \gamma}{\gamma^ {x} x} (38) \\ \geq \frac {- \ln \sqrt {\gamma}}{\gamma^ {x} x}. (39) \\ \end{array}
+$$
+
+Integrating both sides from  $x = \left\lceil -\frac{1}{\ln\sqrt{\gamma}}\right\rceil$  to  $x = N + 1$ , we obtain:
+
+$$
+\begin{array}{l} \int_ {x = \lceil - \frac {1}{\ln \sqrt {\gamma}} \rceil} ^ {x = N + 1} \mathrm {d} \frac {1}{\gamma^ {x} x} = \frac {1}{\gamma^ {N + 1} (N + 1)} - \frac {1}{\gamma^ {\lceil - \frac {1}{\ln \sqrt {\gamma}} \rceil} \lceil - \frac {1}{\ln \sqrt {\gamma}} \rceil} (40) \\ \geq \int_ {x = \left\lceil - \frac {1}{\ln \sqrt {\gamma}} \right\rceil} ^ {x = N + 1} \frac {- \ln \sqrt {\gamma}}{\gamma^ {x} x} d x. (41) \\ \end{array}
+$$
+
+Since the derivative in Eq. (39) is always positive, the function  $\frac{1}{\gamma^x x}$  is monotonically increasing when  $x \geq -1 / \ln \sqrt{\gamma}$ . In such case, the lower Riemann sums underestimate the integral, we have:
+
+$$
+\begin{array}{l} \sum_ {n = \lceil - \frac {1}{\ln \sqrt {\gamma}} \rceil} ^ {N} \frac {1}{\gamma^ {n} n} \leq \int_ {x = \lceil - \frac {1}{\ln \sqrt {\gamma}} \rceil} ^ {x = N + 1} \frac {1}{\gamma^ {x} x} d x (42) \\ \stackrel {4 1} {\leq} \frac {1}{- \ln \sqrt {\gamma}} \left(\frac {1}{\gamma^ {N + 1} (N + 1)} - \frac {1}{\gamma^ {\lceil - \frac {1}{\ln \sqrt {\gamma}} \rceil} \lceil - \frac {1}{\ln \sqrt {\gamma}} \rceil}\right) (43) \\ \leq - \frac {1}{\gamma^ {N + 1} (N + 1) \ln \sqrt {\gamma}} - \frac {1}{\gamma^ {\lceil - \frac {1}{\ln \sqrt {\gamma}} \rceil}}. (44) \\ \end{array}
+$$
+
+Plugging Eq. (44) into Eq. (35), we have:
+
+$$
+\begin{array}{l} \mathbb {E} \left[ f \left(\boldsymbol {\theta} _ {n} ^ {\boldsymbol {\alpha}}\right) - f (\boldsymbol {\theta} *) \right] \leq \frac {G ^ {2} (1 - \gamma)}{\beta} + \frac {G ^ {2} (1 - \gamma)}{2 \beta (1 - \gamma^ {N - 1})} \gamma^ {N} \left(\sum_ {n = \lceil - \frac {1}{\ln \sqrt {\gamma}} \rceil} ^ {N} \frac {1}{\gamma^ {n} n} + \sum_ {j = 1} ^ {\lceil - \frac {1}{\ln \sqrt {\gamma}} \rceil - 1} \frac {1}{\gamma^ {j} j}\right) (45) \\ \leq \frac {G ^ {2} (1 - \gamma)}{\beta} + \frac {G ^ {2} (1 - \gamma)}{2 \beta (1 - \gamma^ {N - 1})} \gamma^ {N} ( \\ - \frac {1}{\gamma^ {N + 1} (N + 1) \ln \sqrt {\gamma}} - \frac {1}{\gamma^ {\lceil - \frac {1}{\ln \sqrt {\gamma}} \rceil}} + \sum_ {j = 1} ^ {\lceil - \frac {1}{\ln \sqrt {\gamma}} \rceil - 1} \frac {1}{\gamma^ {j} j}) (46) \\ = \frac {G ^ {2} (1 - \gamma)}{\beta} + \frac {G ^ {2}}{2 \beta (1 - \gamma^ {N - 1})} \left(\frac {1 - \gamma}{\gamma (N + 1) \ln 1 / \sqrt {\gamma}} + \gamma^ {N} v (\gamma)\right) (47) \\ \leq \frac {G ^ {2} (1 - \gamma)}{\beta} + \frac {G ^ {2}}{2 \beta (1 - \gamma^ {N - 1})} \left(\frac {2}{\gamma (N + 1)} + \gamma^ {N} v (\gamma)\right) (48) \\ = \frac {G ^ {2}}{\beta} \left(\frac {1}{\gamma (1 - \gamma^ {N - 1}) (N + 1)} + \frac {v (\gamma)}{2 (1 - \gamma^ {N - 1})} \gamma^ {N} + 1 - \gamma\right), (49) \\ \end{array}
+$$
+
+where  $v(\gamma) = \sum_{j=1}^{\lceil -\frac{1}{\ln\sqrt{\gamma}} \rceil - 1} \frac{1 - \gamma}{\gamma^j j} - \frac{1 - \gamma}{\gamma^{\lceil -\frac{1}{\ln\sqrt{\gamma}} \rceil}}$ . To derive Eq. (48), we first identify that  $(1 - \gamma) / \ln 1 / \sqrt{\gamma}$  is monotonically increasing for  $\gamma \in (0,1)$ . We then compute the limit as  $\gamma \to 1$  using L'Hôpital's rule.
+
+# A.2 PROOF OF THM. 3.3
+
+We prove Thm. 3.3 using a proof by contradiction.
+
+Assume  $\exists \gamma' \in (0,1)$ , s.t.  $c_{1}\pmb{\theta}_{n_{1}}^{\gamma} + c_{2}\pmb{\theta}_{n_{2}}^{\gamma} + c_{3}\pmb{\theta}_{n_{3}}^{\gamma} \in \{\pmb{\theta}_{1}^{\gamma'}, \dots, \pmb{\theta}_{N}^{\gamma'}\}$ . Denote  $n^{*} \in \{1 \dots N\}$  such that:
+
+$$
+c _ {1} \boldsymbol {\theta} _ {n _ {1}} ^ {\gamma} + c _ {2} \boldsymbol {\theta} _ {n _ {2}} ^ {\gamma} + c _ {3} \boldsymbol {\theta} _ {n _ {3}} ^ {\gamma} = \boldsymbol {\theta} _ {n ^ {*}} ^ {\gamma^ {\prime}}. \tag {50}
+$$
+
+Since  $\theta_1^\gamma, \ldots, \theta_N^\gamma$  are the EMA of  $\theta_1, \ldots, \theta_N$ , which are linearly independent, and  $\theta_{n^*}^{\gamma'}$  is an EMA of  $\theta_1, \ldots, \theta_{n^*}$ , we can derive:  $n^* = n_3$ .
+
+Substituting the EMA models with the original models, we have:
+
+$$
+\begin{array}{l} c _ {1} \pmb {\theta} _ {n - 1} ^ {\gamma} + c _ {2} \pmb {\theta} _ {n _ {2}} ^ {\gamma} + c _ {3} \pmb {\theta} _ {n _ {3}} ^ {\gamma} = (c _ {1} \gamma^ {n _ {1} - 1} + c _ {2} \gamma^ {n _ {2} - 1} + c _ {3} \gamma^ {n _ {3} - 1}) \pmb {\theta} _ {1} + \dots \\ + (1 - \gamma) \left(c _ {1} + c _ {2} \gamma^ {n _ {2} - n _ {1}} + c _ {3} \gamma^ {n _ {3} - n _ {1}}\right) \boldsymbol {\theta} _ {n _ {1}} + \dots \\ + (1 - \gamma) \left(c _ {2} + c _ {3} \gamma^ {n _ {3} - n _ {2}}\right) \boldsymbol {\theta} _ {n _ {2}} + \dots + (1 - \gamma) c _ {3} \boldsymbol {\theta} _ {n _ {3}}, \tag {51} \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \pmb {\theta} _ {n _ {3}} ^ {\gamma^ {\prime}} = \gamma^ {\prime n _ {3} - 1} \pmb {\theta} _ {1} + \ldots + (1 - \gamma^ {\prime}) \gamma^ {\prime n _ {3} - n _ {1}} \pmb {\theta} _ {n _ {1}} + (1 - \gamma^ {\prime}) \gamma^ {\prime n _ {3} - n _ {2}} \pmb {\theta} _ {n _ {2}} + \ldots \\ + \left(1 - \gamma^ {\prime}\right) \boldsymbol {\theta} _ {n _ {3}}. \tag {52} \\ \end{array}
+$$
+
+Note that we apply the practical implementation of EMA for Thm. 3.3, which is different from Thm. 3.2. However, their discrepancy is negligible. Since  $\theta_1^\gamma, \ldots, \theta_N^\gamma$  are linearly independent,
+
+coefficients of all models  $\theta_{1},\ldots ,\theta_{n_{3}}$  in Eqs. (51) and (52) should be aligned with each other. Based on the last term in Eqs. (51) and (52), we have:
+
+$$
+(1 - \gamma) c _ {3} = 1 - \gamma^ {\prime} \Rightarrow \gamma^ {\prime} = 1 - (1 - \gamma) c _ {3}. \tag {53}
+$$
+
+Case 1:  $n_2 < n_3 - 1$  First, we consider the case  $n_2 < n_3 - 1$ , we have the following equations for the index  $n_3 - 1$ :
+
+$$
+(1 - \gamma) \gamma c _ {3} \boldsymbol {\theta} _ {n _ {3} - 1} = (1 - \gamma^ {\prime}) \gamma^ {\prime} \boldsymbol {\theta} _ {n _ {3} - 1} \tag {54}
+$$
+
+$$
+(1 - \gamma) \gamma c _ {3} = (1 - \gamma^ {\prime}) \gamma^ {\prime} \tag {55}
+$$
+
+$$
+(1 - \gamma) \gamma c _ {3} \stackrel {5 3} {=} (1 - \gamma) c _ {3} \gamma^ {\prime} \tag {56}
+$$
+
+$$
+\gamma = \gamma^ {\prime} \tag {57}
+$$
+
+$$
+\gamma^ {\prime} \stackrel {5 3} {=} 1 - (1 - \gamma^ {\prime}) c _ {3} \tag {58}
+$$
+
+$$
+\gamma^ {\prime} \left(1 - c _ {3}\right) = 1 - c _ {3} \tag {59}
+$$
+
+$$
+\gamma^ {\prime} = 1. \tag {60}
+$$
+
+Eq. (60) contradicts with the assumption  $\gamma, \gamma' \in (0,1)$ .
+
+Case 2:  $n_2 = n_3 - 1$  and  $n_1 < n_2 - 1$ . Next, we consider the case  $n_2 = n_3 - 1$  and  $n_1 < n_2 - 1$ . We have the following equation for the index  $n_2$ :
+
+$$
+(1 - \gamma^ {\prime}) \gamma^ {\prime} \boldsymbol {\theta} _ {n _ {2}} = (1 - \gamma) \gamma c _ {3} \boldsymbol {\theta} _ {n _ {2}} + (1 - \gamma) c _ {2} \boldsymbol {\theta} _ {n _ {2}}. \tag {61}
+$$
+
+Additionally, we have the equations below for the index  $n_2 - 1$ :
+
+$$
+\left(1 - \gamma^ {\prime}\right) \gamma^ {\prime 2} = (1 - \gamma) \gamma^ {2} c _ {3} + (1 - \gamma) \gamma c _ {2} \tag {62}
+$$
+
+$$
+\left((1 - \gamma) \gamma c _ {3} + (1 - \gamma) c _ {2}\right) \gamma^ {\prime} \stackrel {6 1} {=} \left((1 - \gamma) \gamma c _ {3} + (1 - \gamma) c _ {2}\right) \gamma \tag {63}
+$$
+
+$$
+\gamma^ {\prime} = \gamma \tag {64}
+$$
+
+$$
+\gamma^ {\prime} ^ {5 7 - 6 0} 1. \tag {65}
+$$
+
+Eq. (65) contradicts with the assumption  $\gamma, \gamma' \in (0,1)$ .
+
+Case 3:  $n_2 = n_3 - 1$  and  $n_1 = n_2 - 1$ . Finally, we consider the case  $n_2 = n_3 - 1$  and  $n_1 = n_2 - 1$ . We have the following equation for the index  $n_1$ :
+
+$$
+\left(1 - \gamma^ {\prime}\right) \gamma^ {\prime 2} = (1 - \gamma) \gamma^ {2} c _ {3} + (1 - \gamma) \gamma c _ {2} + (1 - \gamma) c _ {1}. \tag {66}
+$$
+
+Additionally, we have the equations below for the index  $n_1 - 1$ :
+
+$$
+(1 - \gamma^ {\prime}) \gamma^ {\prime 3} = (1 - \gamma) \gamma^ {3} c _ {3} + (1 - \gamma) \gamma^ {2} c _ {2} + (1 - \gamma) \gamma c _ {1} \tag {67}
+$$
+
+$$
+\left((1 - \gamma) \gamma^ {2} c _ {3} + (1 - \gamma) \gamma c _ {2} + (1 - \gamma) c _ {1}\right) \gamma^ {\prime} \stackrel {6 6} {=} \left((1 - \gamma) \gamma^ {2} c _ {3} + (1 - \gamma) \gamma c _ {2} + (1 - \gamma) c _ {1}\right) \gamma \tag {68}
+$$
+
+$$
+\gamma^ {\prime} = \gamma \tag {69}
+$$
+
+$$
+\gamma^ {\prime} ^ {5 7 - 6 0} = 1. \tag {70}
+$$
+
+Eq. (70) contradicts with the condition  $\gamma, \gamma' \in (0,1)$ .
+
+Since the condition  $\gamma, \gamma' \in (0,1)$  always leads to a contradiction, we prove that  $\nexists \gamma' \in (0,1)$ , s.t.  $c_{1}\pmb{\theta}_{n_{1}}^{\gamma} + c_{2}\pmb{\theta}_{n_{2}}^{\gamma} + c_{3}\pmb{\theta}_{n_{3}}^{\gamma} \in \{\pmb{\theta}_{1}^{\gamma'}, \dots, \pmb{\theta}_{N}^{\gamma'}\}$ .
+
+# B EXTENDED BACKGROUND AND RELATED WORK
+
+# B.1 DIFFUSION PROBABILISTIC MODEL
+
+Let us denote the data distribution by  $p_{data}$  and consider a diffusion process that perturbs  $p_{data}$  with a stochastic differential equation (SDE) (Song et al., 2020b):
+
+$$
+d \mathbf {x} _ {t} = \boldsymbol {\mu} (\mathbf {x} _ {t}, t) d t + \sigma (t) d \boldsymbol {w} _ {t}, \tag {71}
+$$
+
+where  $\pmb{\mu}(\cdot, \cdot)$  and  $\sigma(\cdot)$  represent the drift and diffusion coefficients, respectively,  $\pmb{w}_t$  denotes the standard Brownian motion, and  $t \in [0, T]$  indicates the time step.  $t = 0$  stands for the real data distribution.  $\pmb{\mu}(\cdot, \cdot)$  and  $\sigma(\cdot)$  are designed to make sure  $p_T(\pmb{x})$  becomes pure Gaussian noise.
+
+Diffusion models (DM) (Sohl-Dickstein et al., 2015; Song et al., 2020a; Ho et al., 2020; Nichol & Dhariwal, 2021; Song et al., 2020b; Karras et al., 2022; Dhariwal & Nichol, 2021) undertake the reverse operation by initiating with  $\boldsymbol{x}_T$  sampled from pure Gaussian noise and progressively denoising it to reconstruct the image  $\boldsymbol{x}_0$ . Importantly, SDE has its corresponding "probability flow" Ordinary Differential Equation (PF ODE) (Song et al., 2020b;a), which delineates a deterministic pathway that yields the same distribution  $p_t(\boldsymbol{x})$  for  $\forall t$ , thereby offering a more efficient sampling mechanism:
+
+$$
+d \mathbf {x} _ {t} = \left[ \boldsymbol {\mu} (\mathbf {x} _ {t}, t) - \frac {1}{2} \sigma (t) ^ {2} \nabla \log p _ {t} (\mathbf {x}) \right] d t, \tag {72}
+$$
+
+where  $\nabla \log p_t(x)$  is referred to as the score function of  $p_t(x)$  (Hyvärinen, 2005). Various ODE solvers have been introduced to further expedite the sampling process utilizing Eq. (72) or minimizing the truncation error (Liu et al., 2022; Jolicoeur-Martineau et al., 2021; Karras et al., 2022; Lu et al., 2022).
+
+A pivotal insight within diffusion models is the realization that  $\nabla \log p_t(\pmb{x})$  can be approximated by a neural network  $s_\theta(\pmb{x}_t, t)$ , which can be trained using the following objective:
+
+$$
+\mathbb {E} _ {t \sim \mathcal {U} (0, T ]} \mathbb {E} _ {\mathbf {y} \sim p _ {\mathrm {d a t a}}} \mathbb {E} _ {\mathbf {x} _ {t} \sim \mathcal {N} (\mathbf {y}, \sigma (t) ^ {2} \mathbf {I})} \lambda (t) \| \mathbf {s} _ {\boldsymbol {\theta}} (\mathbf {x} _ {t}, t) - \nabla_ {\mathbf {x} _ {t}} \log p (\mathbf {x} _ {t} | \mathbf {y}) \|, \tag {73}
+$$
+
+where  $\lambda(t)$  represents the loss weighting and  $\pmb{y}$  denotes a training image.
+
+# B.2 CONSISTENCY MODELS
+
+Drawing inspiration from DM theory, consistency models (CM) have been proposed to enable single-step generation (Song et al., 2023; Song & Dhariwal, 2024). Whereas DM incrementally denoises an image, e.g., via the PF ODE, CM denoted by  $f_{\theta}$  is designed to map any point  $x_{t}$  at any given time  $t$  along a PF ODE trajectory directly to the trajectory's initial point  $x_{t}$  in a single step.
+
+CMs are usually trained through discretized time steps, so we consider segmenting the time span from  $[\epsilon, T]$  into  $K - 1$  sub-intervals, with  $\epsilon$  being a small value approximating zero. Training of CM can follow one of two primary methodologies: consistency distillation (CD) or direct consistency training (CT). In the case of CD, the model  $f_{\theta}$  leverages knowledge distilled from a pre-trained DM  $\phi$ . The distillation loss can be formulated as follows:
+
+$$
+\mathbb {E} _ {k \sim \mathcal {U} [ 1, K - 1 ]} \mathbb {E} _ {\boldsymbol {y} \sim p _ {d a t a}} \mathbb {E} _ {\boldsymbol {x} _ {t _ {k + 1}} \sim \mathcal {N} (\boldsymbol {y}, t _ {k + 1} ^ {2} \boldsymbol {I})} \lambda (t _ {k}) d \left(f _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {t _ {k + 1}}, t _ {k + 1}\right), f _ {\boldsymbol {\theta} ^ {-}} \left(\hat {\boldsymbol {x}} _ {t _ {k}} ^ {\phi}, t _ {k}\right)\right), \tag {74}
+$$
+
+where  $f_{\theta^{-}}$  refers to the target model and  $\pmb{\theta}^{-}$  is computed through EMA of the historical weights of  $\pmb{\theta}$ ,  $\hat{\pmb{x}}_{t_k}^{\phi}$  is estimated by the pre-trained diffusion model  $\phi$  through one-step denoising based on  $\pmb{x}_{t_{k + 1}}$ , and  $\bar{d}$  is a metric implemented by either the  $\ell_2$  distance or LPIPS (Zhang et al., 2018).
+
+Alternatively, in the CT case, the models  $f_{\theta}$  are developed independently, without relying on any pre-trained DM:
+
+$$
+\mathbb {E} _ {k \sim \mathcal {U} [ 1, K - 1 ]} \mathbb {E} _ {\boldsymbol {y} \sim p _ {\text {d a t a}}} \mathbb {E} _ {\boldsymbol {z} \sim \mathcal {N} (\boldsymbol {0}, \boldsymbol {I})} \lambda (t _ {k}) d \left(f _ {\boldsymbol {\theta}} \left(\boldsymbol {y} + t _ {k + 1} \boldsymbol {z}, t _ {k + 1}\right), f _ {\boldsymbol {\theta} ^ {-}} \left(\boldsymbol {y} + t _ {k} \boldsymbol {z}, t _ {k}\right)\right), \tag {75}
+$$
+
+where the target model  $f_{\theta^{-}}$  is set to be the same as the model  $f_{\theta}$  in the latest improved version of the consistency training (Song & Dhariwal, 2024), i.e.  $\pmb{\theta}^{-} = \pmb{\theta}$ .
+
+# B.3 WEIGHT AVERAGING
+
+The integration of a running average of the weights by stochastic gradient descent (SGD) was initially explored within the realm of convex optimization (Ruppert, 1988; Polyak & Juditsky, 1992). This concept was later applied to the training of neural networks (Szegedy et al., 2016). Izmailov et al.(Izmailov et al., 2018) suggest that averaging multiple weights over the course of training can yield better generalization than SGD. Wortsman et al.(Wortsman et al., 2022) demonstrate the potential of averaging weights that have been fine-tuned with various hyperparameter configurations. The Exponential Moving Average (EMA) employs a specific form that uses the exponential rate  $\gamma$  as a smoothing factor:
+
+$$
+\tilde {\boldsymbol {\theta}} _ {n} = \gamma \tilde {\boldsymbol {\theta}} _ {n - 1} + (1 - \gamma) \boldsymbol {\theta} _ {n}, \tag {76}
+$$
+
+where  $n = 1,\dots ,N$  denotes the number of training iterations,  $\hat{\theta}$  represents the EMA model, and is initialized with  $\tilde{\theta}_0 = \theta_0$ .
+
+Practitioners often opt for advanced optimizers such as Adam (Kingma & Ba, 2014) for different tasks and network architectures, which might reduce the need for employing a running average. However, the use of EMA has been noted to significantly enhance the quality of generation in early DM studies (Song et al., 2020b; Dhariwal & Nichol, 2021; Nichol & Dhariwal, 2021; Ho et al., 2020; Song et al., 2020a). This empirical strategy has been adopted in most, if not all, subsequent research endeavors. Consequently, CM have also incorporated this technique, discovering EMA models that perform substantially better.
+
+Recently, several works of more advanced weight averaging strategies have been proposed for Large Language Models (LLMs). Most of them focus on merging models fine-tuned for different downstream tasks to create a new model with multiple capabilities (Ilharco et al., 2022; Yadav et al., 2024; Jin et al., 2022; Yu et al., 2023). Different from these approaches, our work aims to accelerate the model convergence and achieve better performance in a standalone training process. Moreover, the methodology we employ for determining averaging coefficients is novel, thereby distinguishing our method from these related works. Further details can be found in Sec. 3 and Sec. 4.
+
+# B.4 SEARCH-BASED METHODS FOR DIFFUSION MODELS
+
+Search algorithms are widely used across various domains like Neural Architecture Search (NAS) (Real et al., 2019; Ning et al., 2020), where they are employed to identify specific targets. Recently, many works studies have set discrete optimization dimensions for DMs and utilized search methods to unearth optimal solutions. For example, Liu et al. (Liu et al., 2023a), Li et al. (Li et al., 2023) and Yang et al. (Yang et al., 2023) use search methods to find the best model schedule for DMs. Liu et al. (Liu et al., 2023b) apply search methods to find appropriate strategies for diffusion solvers. However, these works do not involve modification to model weights during search and are only applicable to DMs.
+
+# C LCSC ALGORITHM
+
+The detailed algorithm of LCSC is provided in Alg. 1.
+
+# D ADDITIONAL EXAMPLES OF THE METRIC LANDSCAPE
+
+We previously introduced the metric landscape of the DM and CD models in Sec. 3. In this section, we extend our analysis by evaluating additional metric landscapes using the FID. Specifically, we incorporate an additional intermediate training iteration and explore the metric landscape of the CT model. The comprehensive landscapes of the DM, CD, and CT models are depicted in Fig. 7.
+
+Since we use grid search to get the performance and using interpolation to get the smooth landscapes, we further provide some examples of the original performance heat-map in grids, as shown in Fig. 6
+
+# EXPERIMENTAL DETAILS AND ADDITIONAL RESULTS
+
+# E.1 EXPERIMENTAL DETAILS
+
+# E.1.1 TRAINING SETTING
+
+Since LCSC employs weight checkpoints, we endeavored to replicate the training processes of the baseline models.
+
+For DM, we follow DDIM (Song et al., 2020a) for the evaluation on CIFAR10 (Krizhevsky et al., 2009) and iDDPM (Nichol & Dhariwal, 2021) for the evaluation on ImageNet-64 (Deng et al., 2009). To improve the inference efficiency, we adopt DPM-Solver (Lu et al., 2022). For CM (Song et al., 2023), we evaluate LCSC with both CD and CT on CIFAR-10 and ImageNet-64, and CT on LSUN datasets. For CIFAR-10 and ImageNet-64, We train models with our own implementation. We follow all the settings reported by the official paper except ImageNet-64, on which we decrease the batch size to 256 on CM and 512 on DM due to the limited resources. We apply LCSC at
+
+Algorithm 1 Evolutionary Search for Combination Coefficients Optimization  
+Require:  
+```latex
+$\Theta = \{\pmb{\theta}_{n_1}, \pmb{\theta}_{n_2}, \pmb{\theta}_{n_3}, \dots, \pmb{\theta}_{n_K}\}$ : the set of saved checkpoints until training iter  $T$ $\mathcal{F}$ : the metric evaluator.
+```
+
+Symbols:  
+Hyperparameters:  
+```txt
+$P$  : The whole Population of model schedule.   
+ $CP$  : The Candidate Parents set of each loop, from which a parent coefficients is selected.   
+NG: The Next Generation newly mutated from the parent coefficients in each loop.   
+ $\alpha$  : A group of combination coefficients denoted as  $\alpha = \{\alpha_{1},\dots ,\alpha_{K}\}$ $\pmb {\alpha}\cdot \Theta$  : Equal to  $(1 - \sum_{i = 2}^{K}\alpha_{i})\pmb{\theta}_{n_{1}} + \sum_{i = 2}^{K}\alpha_{i}\pmb{\theta}_{n_{i}}$
+```
+
+Search Process:  
+```txt
+Epoch: Number of loops for the entire search process.  
+ $\mathbf{M}_{CP}$ : Maximum size of the candidate parents set  $CP$ .  
+Iter: Maximum number of mutations in each loop.
+```
+
+```latex
+1:  $P\gets \emptyset$    
+2: Initialize a group of coefficients  $\alpha_{init}$  with EMA weights   
+3:  $P\leftarrow P\cup \{(\pmb{\alpha}_{init},\mathcal{F}(\pmb{\alpha}_{init}\cdot \Theta))\}$    
+4: for  $t = 1,\dots ,$  Epochh do   
+5:  $NG\gets \emptyset$    
+6: for  $i = 1,\dots ,$  Iter do   
+7:  $CP\gets \{\pmb {\alpha}_i|\mathcal{F}(\pmb {\alpha}_i\cdot \Theta)$  ranks within the top min(MCP,  $|P|$  ) in  $P\}$    
+8:  $\alpha_{f},\alpha_{m}\xleftarrow{\mathrm{Random~Sample}}CP$    
+9:  $\alpha_{new}\gets$  Mutate(Crossover  $(\alpha_{f},\alpha_{m})$    
+10:  $NG\gets NG\cup \{(\alpha_{new},\mathcal{F}(\alpha_{new}\cdot \Theta))\}$    
+11: end for   
+12:  $P\gets P\cup NG$    
+13: end for   
+14:  $\alpha^{*}\gets \arg \min_{\alpha \in P}\mathcal{F}(\alpha \cdot \Theta)$    
+15: return  $\alpha^{*}\cdot \Theta$
+```
+
+different training stages. For LSUN datasets, we fine-tune the released official models and apply LCSC. We further train a LCM-LoRA model following the official setting and use LCSC with the final checkpoints. Specifically, for any selected iteration, we utilize checkpoints from every predetermined interval of iterations within a defined window size. Then we run a search process to find the optimal combination coefficients.
+
+For DMs, we were able to reproduce the results reported in the original papers successfully. However, for CMs, our training outcomes on CIFAR10 were slightly inferior to those documented in the original papers. Additionally, due to resource constraints, we opted for a smaller batch size than the original configuration when training on ImageNet. In the results tables, we present both our training outcomes and the results reported in the original papers, with the latter indicated as released. The specifics of the experimental setups are detailed below.
+
+For the experiments with DM, we utilize the DDIM (Song et al., 2020a) codebase (https: //github.com/ermongroup/ddim) on CIFAR-10, adhering to the default configuration settings. For ImageNet-64, we employ the iDDPM (Nichol & Dhariwal, 2021) codebase (https: //github.com/openai/improved-diffusion), setting the batch size to 512, the noise schedule to cosine, and maintaining other hyperparameters at their default values. During sampling, the DPM-Solver (Lu et al., 2022) (https://github.com/LuChengTHU/dpm-solver) is applied, conducting 15 timesteps of denoising using the default configuration for the respective setting. The reproduced results are on par with or slightly surpass those achieved using Euler integration, as reported in the original papers.
+
+For the experiments involving CM, we adhere closely to the configurations detailed in Table 3 of the original CM paper (Song et al., 2023). On CIFAR-10, instead of using the NCSNPP model (Song et al., 2020b) from CM's official implementation, we opt for the EDM (Karras et al.,
+
+![](images/3813d4e3d6167557fa53ef3d134b878823e26b4cf1e73b3def0bcca22889d04d.jpg)
+
+![](images/e8f4d9a9398a75eaef6ba88ff79941afeded4fde11954861c3d147099a93a941.jpg)
+
+![](images/ae4a682227a59ca5cb0ff86cef8b277643b772930b8664846ce3724cf3f9b075.jpg)  
+Figure 6: Metric landscapes in grid for CD.
+
+![](images/c592645e20e3e7707d3e5452a8ce586a61a38cc87da0c18d924b8e29ab0bc20d.jpg)
+
+2022) architecture. This decision stems from the official CM code for CIFAR-10 being implemented in JAX while our experiments are conducted using PyTorch. For further details, refer to https://github.com/openai/consistency_models_cifar10. This difference in implementation may partly explain the discrepancies between our replication results and those reported in Table 1 of CM's original paper (Song et al., 2023). On ImageNet-64 and LSUN, we follow the official implementation, employing the ADM (Dhariwal & Nichol, 2021) architecture. Due to resource constraints, we train the model with a smaller batch size of 256 instead of 2048 as used in the original study, resulting in worse outcomes. However, as indicated in Tab. 2, applying LCSC to models trained with a reduced batch size can still achieve performance comparable to models trained with a larger batch size.
+
+# E.1.2 SEARCH SETTING
+
+At each selected training iteration, historical weights are leveraged within designated window sizes and intervals. For CM, checkpoints have a window size of 40K with an interval of 100 on CIFAR-10, and a window size of 20K with an interval of 100 on ImageNet-64 and LSUN datasets. DM is assigned a window size of 50K and an interval of 200 for both datasets. FID calculation for DM on ImageNet-64 and CM on LSUN datasets utilizes a sample of 5K images, while 10K images are used for all other configurations. An evolutionary search spanning 2K iterations is applied consistently across all experimental setups.
+
+# E.1.3 TEXT-TO-IMAGE TASK
+
+For text-to-image task, we fine-tune a LoRA based on the Stable Diffusion v1-5 model (Rombach et al., 2022) on CC12M dataset. We use a batch size of 12 and train the LoRA for 6k steps. For LCSC, we use the checkpoints saved between 4k and 6k steps with an interval of 20. When sampling, we insert LoRA to Dreamshaper-7, which is a fine-tuned version of Stable Diffusion v1-5, and use 1k
+
+![](images/3f143cb0087915a28fe69aaf95a689f872aa9946fad15327d5ee145ddacd829e.jpg)
+
+![](images/6c1a7449c064b128f6e941620bf3b7b9d29a5b0efc3afc61da9be5a2bf92494e.jpg)
+
+![](images/e4df4651b79b4c992a3c2daeeffa5f2bc84f55938ff24473c99687493b90e320.jpg)
+
+![](images/3a0449a5faf6547952cf2a6c60d9323663b45a16ebcb13a3d5665c9388f9c049.jpg)
+
+![](images/e6bf2248270209dfd2728e606538f93966d48756ed3133b9f2ffd6935731f6eb.jpg)
+
+![](images/b2763e42334ad4949a1f57e4135e0b1e910b4cb2413e95f099d988c720754e5c.jpg)
+
+![](images/b8e1989dcaa3c337979616182a0c2b4ba2923b10a1eb5717be107a2a78ab13b8.jpg)
+
+![](images/b5e0634d8aad2aaa7937d55dd7a1417e2adc7c68d84394b7fa3441ffbd0631d3.jpg)  
+(a) Metric landscapes for DM.
+
+![](images/7b0dcbeccd5c8f25c2d4e9a5f5424bc68a5f1d2aa381a5f6bd1cdc08574f31e9.jpg)
+
+![](images/e0ae73a07b9b33a00e1b03312ea618a3a0bd1dd0f9ea55f3befbdebbc809bd92.jpg)  
+(b) Metric landscapes for CD.
+
+![](images/57aa7ae0a6ecfac0bb2f1b091432c951970987ad6f58a2068d7dc745fe0980ab.jpg)
+
+![](images/9fdf405e466ee22082c757c98bccdee99fcf3b4ab2c7a5d2a0857e3fcfe8d7e5.jpg)
+
+![](images/4d6919e74fa8dc0611bd30eb2a2dbacebfd5d84fed0dce5271bcce19048aec46.jpg)
+
+![](images/f37d6c49ff64f18ef53e89a106d1f852e64ffaa354dd2f07f30e2f0802d575f3.jpg)
+
+![](images/7b03f90d88c22b2e64f15224e85d4e3ca5871ba9e5a907d5de71da183c1c4a21.jpg)
+
+![](images/064e25e0848708c345421c51a5936c1db28c3267ab4d56349f5f525097c15af3.jpg)
+
+![](images/bd20ec646dd69bbd3d16205628a368d6e117030b166b59e67f03eda6b9a921e3.jpg)  
+(c) Metric landscapes for CT.
+
+![](images/9dc41d6e2613635a69468f3875b6657591edd733b9faa95c83ae78b59fe1277b.jpg)  
+Figure 7: Metric landscapes for DM, CD and CT on CIFAR-10.
+
+samples from CC12M to calculate FID. Finally, we randomly sample 10k samples from MS-COCO dataset to evaluate the search results.
+
+Table 8: Search time consumption per search iteration (in seconds). Results marked with * denote the CPU time cost, which is excludable from the overall time cost through parallel processing. "U-Net" refers to the denoising sampling process, "Inception" refers to the computation of inception features, "Merging" refers to the averaging of weights, and "FID" refers to the computation of FID statistics.  
+
+<table><tr><td>Model</td><td>Dataset</td><td>U-Net</td><td>Inception</td><td>Merging*</td><td>FID*</td></tr><tr><td rowspan="2">CM</td><td>CIFAR-10</td><td>7.34</td><td>4.28</td><td>5.57</td><td>2.49</td></tr><tr><td>ImageNet-64</td><td>34.7</td><td>4.32</td><td>12.0</td><td>2.37</td></tr><tr><td rowspan="2">DM</td><td>CIFAR-10</td><td>40.9</td><td>4.39</td><td>4.61</td><td>2.49</td></tr><tr><td>ImageNet-64</td><td>114.5</td><td>2.24</td><td>13.8</td><td>2.53</td></tr></table>
+
+For baselines, we find that the performance is sensitive to the scale of LoRA. Therefore, we first conduct a coarse scan to determine the approximate range of the optimal LoRA scale, followed by a fine sweep. We find 0.15 is the optimal scale for LoRA and use it as the baseline.
+
+# E.2 TWO-PHASE CONVERGENCE
+
+Fig. 8 shows that the generation quality convergence of DM and CM can be divided into two phases. The initial phase is relatively brief, during which DM and CM rapidly acquire the capability to generate visually satisfactory images. In contrast, the second phase is characterized by a slower optimization of models, focusing on the enhancement of sample quality. It is noteworthy that the majority of training iterations belongs to the second phase.
+
+![](images/e7b7463412fac80fcfc7f2c86510511f7b3beeb744c354b13589ea9eac9631bf.jpg)
+
+![](images/9849164617d00f9151edaf9e677185eeadad674ab25051c86a227bb2742d4f03.jpg)  
+Figure 8: Training curves on CIFAR-10 with CM and DM. P1, P2 represent the first phase and the second phase.
+
+# E.3 SEARCH COST ESTIMATION
+
+We detail the estimation of search costs, covering both CPU and GPU time consumption. Saved checkpoints are loaded into CPU memory and then transferred to the GPU after averaging. Consequently, CPU time comprises the duration for merging weight checkpoints and calculating the FID statistic. Meanwhile, GPU time is dedicated to the sampling and evaluation processes involving the diffusion U-Net and the Inception network. All experiments are performed on a single NVIDIA A100 GPU, paired with an Intel Xeon Platinum 8385P CPU.
+
+First, we profile the durations of each function across all experimental settings and document the time consumed in a single search iteration in Tab. 8. We note that the CPU time cost is significantly lower than that on the GPU. Therefore, while conducting sampling with the current merged weights on the GPU, we can simultaneously perform the FID statistic computation from the previous iteration and merge the checkpoints for the next iteration on the CPU. As a result, the CPU time cost can be effectively excluded from the overall time cost.
+
+Next, we examine the overall training and search time costs as outlined in Tab. 9, and we delve into the speedup ratio detailed in Tabs. 10 and 11. Given that DMs necessitate multiple timesteps for sampling, the overall search cost for DM proves to be non-negligible when compared to the training cost. This results in no observable speedup in convergence upon applying LCSC. However, this search cost is considered manageable and warranted for the anticipated improvements to the final model convergence.
+
+Table 9: Overall search time and training time consumption (in hours).  
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Dataset</td><td colspan="2">Search</td><td colspan="3">Training</td></tr><tr><td>Iteration</td><td>Time</td><td>Iteration</td><td>Batch Size</td><td>Time</td></tr><tr><td rowspan="2">CD</td><td>CIFAR-10</td><td>2K</td><td>6.45</td><td>800K</td><td>512</td><td>818</td></tr><tr><td>ImageNet-64</td><td>2K</td><td>21.7</td><td>600K</td><td>2048</td><td>7253</td></tr><tr><td rowspan="2">CT</td><td>CIFAR-10</td><td>2K</td><td>6.45</td><td>800K</td><td>512</td><td>640</td></tr><tr><td>ImageNet-64</td><td>2K</td><td>21.7</td><td>800K</td><td>2048</td><td>7040</td></tr><tr><td rowspan="2">DM</td><td>CIFAR-10</td><td>2K</td><td>25.2</td><td>800K</td><td>128</td><td>29.3</td></tr><tr><td>ImageNet-64</td><td>2K</td><td>64.8</td><td>500K</td><td>512</td><td>372</td></tr></table>
+
+Table 10: Accurate training speedup of consistency models on CIFAR-10. The speedup is compared against the standard training with 800K iterations and 512 batch size.  
+
+<table><tr><td>Model</td><td>Method</td><td>Training Iter</td><td>Batch Size</td><td>Speedup(↑)</td></tr><tr><td rowspan="4">CD</td><td>EMA</td><td>800K</td><td>512</td><td>-</td></tr><tr><td rowspan="3">LCSC</td><td>50K</td><td>512</td><td>14.27×</td></tr><tr><td>250K</td><td>512</td><td>3.12×</td></tr><tr><td>100K</td><td>128</td><td>25.55×</td></tr><tr><td rowspan="3">CT</td><td>EMA (released)</td><td>800K</td><td>512</td><td>-</td></tr><tr><td rowspan="2">LCSC</td><td>400K</td><td>512</td><td>1.96×</td></tr><tr><td>450K</td><td>128</td><td>6.64×</td></tr></table>
+
+Table 11: Accurate training speedup of consistency models on ImageNet-64. For CD, the speedup is compared against standard training with 600K iterations and 2048 batch size. For CT, the speedup is compared against standard training with 800K iterations and 2048 batch size.  
+
+<table><tr><td>Model</td><td>Method</td><td>Training Iter</td><td>Batch Size</td><td>Speedup(↑)</td></tr><tr><td rowspan="4">CD</td><td>EMA(released)</td><td>600K</td><td>2048</td><td>-</td></tr><tr><td rowspan="3">LCSC</td><td>150K</td><td>256</td><td>29.20×</td></tr><tr><td>300K</td><td>256</td><td>15.27×</td></tr><tr><td>620K</td><td>256</td><td>7.55×</td></tr><tr><td rowspan="3">CT</td><td>EMA(released)</td><td>800K</td><td>2048</td><td>-</td></tr><tr><td rowspan="2">LCSC</td><td>600K</td><td>256</td><td>10.33×</td></tr><tr><td>1000K</td><td>256</td><td>6.27×</td></tr></table>
+
+In contrast, for CMs, the search cost is substantially lower than the training cost, which often results in a markedly faster convergence when combining training with LCSC than with training alone. The precise speedup ratios for CM training are detailed in Tabs. 10 and 11. It should be noted that the figures in these tables may exhibit slight discrepancies from those reported in Sec. 5 due to the latter being profiled in an environment with more variables and potential fluctuations.
+
+# E.4 HYPERPARAMETER STUDY
+
+The hyperparameters of LCSC includes the window size  $(n_K - n_1)$  for retrieving the historical weights, the interval  $(n_k - n_{k-1})$  for two adjacent weight checkpoints, the number of samples for computing FID and the number of search iterations, and the number of timesteps for DM.
+
+Tab. 12 illustrates the impact of varying the number of samples, search iterations, and the size of the interval. We observe that increasing the number of samples and search iterations leads to higher performance, though this comes at the expense of increased search cost. In practice, we find that 10K
+
+Table 12: Hyperparameter study of LCSC with (a) DM and (b) CM. The models are trained on CIFAR10 with 250K iterations. We evaluate three values of each hyperparameter and compare them with our adopted setting highlighted in gray. The varied hyperparameter is in bold. For DM the window size is 50K, for CM it is 40K.  
+(a) Results on DM  
+
+<table><tr><td>Samples</td><td>Search Iters.</td><td>Interval</td><td>FID(↓)</td><td>IS(↑)</td></tr><tr><td>10K</td><td>2K</td><td>200</td><td>3.87</td><td>9.27</td></tr><tr><td>2K</td><td>2K</td><td>200</td><td>9.23</td><td>9.11</td></tr><tr><td>5K</td><td>2K</td><td>200</td><td>4.04</td><td>9.09</td></tr><tr><td>10K</td><td>1K</td><td>200</td><td>4.01</td><td>9.17</td></tr><tr><td>10K</td><td>4K</td><td>200</td><td>3.79</td><td>9.28</td></tr><tr><td>10K</td><td>2K</td><td>100</td><td>3.82</td><td>9.15</td></tr><tr><td>10K</td><td>2K</td><td>500</td><td>4.41</td><td>9.11</td></tr></table>
+
+(b) Results on CM  
+
+<table><tr><td>Samples</td><td>Search Iters.</td><td>Interval</td><td>FID(↓)</td><td>IS(↑)</td></tr><tr><td>10K</td><td>2K</td><td>100</td><td>2.76</td><td>9.71</td></tr><tr><td>2K</td><td>2K</td><td>100</td><td>3.40</td><td>9.39</td></tr><tr><td>5K</td><td>2K</td><td>100</td><td>2.79</td><td>9.66</td></tr><tr><td>10K</td><td>1K</td><td>100</td><td>3.03</td><td>9.57</td></tr><tr><td>10K</td><td>4K</td><td>100</td><td>2.69</td><td>9.70</td></tr><tr><td>10K</td><td>2K</td><td>200</td><td>2.84</td><td>9.71</td></tr><tr><td>10K</td><td>2K</td><td>50</td><td>2.71</td><td>9.79</td></tr></table>
+
+Table 13: Hyperparameter analysis of LCSC, supplementing the study in Tab. 12. Models are trained on CIFAR10 for 250K iterations. We assess three different values for each hyperparameter, contrasting these with our chosen setting, which is highlighted in gray. The hyperparameter under variation is indicated in bold. For DM, the interval between checkpoints is set to 200, whereas for CM, it is 100. We perform 2K search iterations and sample 10K images to compute the FID score at each iteration.  
+
+<table><tr><td>Method</td><td>Window Size</td><td>NFE</td><td>FID(↓)</td><td>IS(↑)</td></tr><tr><td rowspan="3">CM</td><td>40K</td><td>1</td><td>2.76</td><td>9.71</td></tr><tr><td>10K</td><td>1</td><td>2.89</td><td>9.60</td></tr><tr><td>50K</td><td>1</td><td>2.73</td><td>9.66</td></tr><tr><td rowspan="5">DM</td><td>50K</td><td>15</td><td>3.87</td><td>9.27</td></tr><tr><td>10K</td><td>15</td><td>4.41</td><td>9.11</td></tr><tr><td>30K</td><td>15</td><td>3.82</td><td>9.15</td></tr><tr><td>50K</td><td>7</td><td>3.91</td><td>9.10</td></tr><tr><td>50K</td><td>10</td><td>3.84</td><td>9.17</td></tr></table>
+
+samples and 2K search iterations can efficiently identify strong models. With a fixed window size, the interval between two checkpoints determines the dimension of the search space. Our findings suggest that search performance generally improves as the search space expands, but limiting the search dimension to fewer than 200 can detrimentally affect search performance.
+
+For the impact of window size  $(n_K - n_1)$  and sampling NFE of DM, the findings are detailed in Tab. 13. The results reveal that, despite earlier models being further from convergence, a sufficiently large window for accessing historical weights proves advantageous. Furthermore, conducting a lower NFE during the search in the DM context results in a similar reduction in FID but a smaller improvement in IS. This suggests that the search output's generality across different metrics diminishes when the generated samples during the search are less accurate, i.e., exhibit significant truncation error.
+
+# E.5 DETAILED RESULTS OF EMA RATE GRID SEARCH
+
+For the final training model of each experimental configuration, we conduct a comprehensive sweep across a broad range of EMA rates, presenting the optimal outcomes in Sec. 5. The exhaustive results are detailed in Tab. 14. In the majority of scenarios, the default EMA rate employed in the official implementations of our baseline models (Song et al., 2023; Ho et al., 2020; Nichol & Dhariwal, 2021) yields slightly inferior performance compared to the best EMA rate identified. Nevertheless, the findings demonstrate that LCSC consistently surpasses all explored EMA rates, underscoring the limitations of EMA as a strategy for weight averaging.
+
+Table 14: EMA rate grid search outcomes. Asterisks (*) indicate the results using the default rates from the official paper. Listed FID scores for each EMA rate correspond to fully trained models: CIFAR-10 models at 800K iterations; ImageNet-64 CD/CT/DM models at 600K/1000K/500K iterations, respectively. Iteration counts for models employed by LCSC are shown in parentheses.  
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Dataset</td><td colspan="7">EMA rate</td><td rowspan="2">LCSC</td></tr><tr><td>0.999</td><td>0.9995</td><td>0.9999</td><td>0.999943</td><td>0.99995</td><td>0.99997</td><td>0.99999</td></tr><tr><td rowspan="2">CD</td><td>CIFAR-10</td><td>4.35</td><td>4.13</td><td>3.66*</td><td>3.56</td><td>3.58</td><td>3.51</td><td>3.54</td><td>2.42 (800K)</td></tr><tr><td>ImageNet-64</td><td>7.45</td><td>7.38</td><td>7.19</td><td>7.17*</td><td>7.17</td><td>7.22</td><td>7.41</td><td>5.54 (620K)</td></tr><tr><td rowspan="2">CT</td><td>CIFAR-10</td><td>9.80</td><td>9.78</td><td>9.70*</td><td>9.70</td><td>9.69</td><td>9.71</td><td>9.77</td><td>8.60 (400K)</td></tr><tr><td>ImageNet-64</td><td>15.7</td><td>15.7</td><td>15.6</td><td>15.6*</td><td>15.6</td><td>15.6</td><td>15.7</td><td>12.1 (600K)</td></tr><tr><td rowspan="2">DM</td><td>CIFAR-10</td><td>5.73</td><td>5.18</td><td>4.16*</td><td>3.99</td><td>3.96</td><td>4.04</td><td>5.04</td><td>3.18 (800K)</td></tr><tr><td>ImageNet-64</td><td>23.1</td><td>20.3</td><td>19.8*</td><td>19.0</td><td>18.1</td><td>18.1</td><td>18.5</td><td>15.3 (500K)</td></tr></table>
+
+Table 15: The comparison of different formulations for base checkpoints on CIFAR-10 dataset. LCSC-Diff stands for the formulation we use in our main experiments. LCSC-Direct stands for using the checkpoints themselves as the weighted base.  
+
+<table><tr><td>Model</td><td>Method</td><td>Training Iter</td><td>Batch Size</td><td>NFE</td><td>FID(↓)</td><td>IS(↑)</td><td>Prec.(↑)</td><td>Rec.(↑)</td><td>Speed(↑)</td></tr><tr><td rowspan="10">CD</td><td rowspan="5">LCSC-Direct</td><td>50K</td><td>512</td><td>1</td><td>3.18</td><td>9.60</td><td>0.67</td><td>0.58</td><td>~14×</td></tr><tr><td>250K</td><td>512</td><td>1</td><td>2.76</td><td>9.71</td><td>0.67</td><td>0.59</td><td>~3.1×</td></tr><tr><td>800K</td><td>512</td><td>1</td><td>2.42</td><td>9.76</td><td>0.67</td><td>0.60</td><td>-</td></tr><tr><td>800+40K</td><td>512</td><td>1</td><td>2.38</td><td>9.70</td><td>0.67</td><td>0.60</td><td>-</td></tr><tr><td>100K</td><td>128</td><td>1</td><td>3.34</td><td>9.51</td><td>0.67</td><td>0.57</td><td>~23×</td></tr><tr><td rowspan="5">LCSC-Diff</td><td>50K</td><td>512</td><td>1</td><td>3.10</td><td>9.50</td><td>0.66</td><td>0.58</td><td>~14×</td></tr><tr><td>250K</td><td>512</td><td>1</td><td>2.66</td><td>9.64</td><td>0.67</td><td>0.59</td><td>~3.1×</td></tr><tr><td>800K</td><td>512</td><td>1</td><td>2.44</td><td>9.82</td><td>0.67</td><td>0.60</td><td>-</td></tr><tr><td>800+40K</td><td>512</td><td>1</td><td>2.50</td><td>9.70</td><td>0.68</td><td>0.59</td><td>-</td></tr><tr><td>100K</td><td>128</td><td>1</td><td>3.21</td><td>9.48</td><td>0.66</td><td>0.58</td><td>~23×</td></tr><tr><td rowspan="6">CT</td><td rowspan="3">LCSC-Direct</td><td>400K</td><td>512</td><td>1</td><td>8.60</td><td>8.89</td><td>0.67</td><td>0.47</td><td>~1.9×</td></tr><tr><td>800+40K</td><td>512</td><td>1</td><td>8.05</td><td>8.98</td><td>0.70</td><td>0.45</td><td>-</td></tr><tr><td>450K</td><td>128</td><td>1</td><td>8.33</td><td>8.67</td><td>0.69</td><td>0.44</td><td>~7×</td></tr><tr><td rowspan="3">LCSC-Diff</td><td>400K</td><td>512</td><td>1</td><td>8.89</td><td>8.79</td><td>0.67</td><td>0.47</td><td>~1.9×</td></tr><tr><td>800+40K</td><td>512</td><td>1</td><td>7.05</td><td>9.01</td><td>0.70</td><td>0.45</td><td>-</td></tr><tr><td>450K</td><td>128</td><td>1</td><td>8.54</td><td>8.66</td><td>0.69</td><td>0.44</td><td>~7×</td></tr></table>
+
+# E.6 DIFFERENT FORMULATION OF THE BASE CHECKPOINTS
+
+As discussed in Sec. 4.2, we subtract the first checkpoint from each checkpoint and use these differences as the weighted base. In this section, we explore an alternative approach for defining the base checkpoints: using all the checkpoints themselves as the base. In this case, the final combined weight is:
+
+$$
+\boldsymbol {\alpha} \cdot \boldsymbol {\Theta} = \sum_ {i = 1} ^ {K} \alpha_ {i} \boldsymbol {\theta} _ {n _ {i}}. \tag {77}
+$$
+
+We apply this formulation in our search, with the results presented in Tab. 15 and Tab. 16. We can see that the difference search method outperforms the checkpoint search method in all cases on the ImageNet-64. On CIFAR-10, the performance of the two methods are similar with each other.
+
+# E.7 EVALUATION WITH OTHER METRICS
+
+# E.7.1 EVALUATION WITH FCD AND KID
+
+In this section, we present the results of FCD and KID using CT method on CIFAR-10 and ImageNet-64 dataset. The results are shown in Tab. 17.
+
+Table 16: The comparison of different formulations for base checkpoints on ImageNet-64 dataset. LCSC-Diff stands for the formulation we use in our main experiments. LCSC-Direct stands for using the checkpoints themselves as the weighted base.  
+
+<table><tr><td>Model</td><td>Method</td><td>Training Iter</td><td>Batch Size</td><td>NFE</td><td>FID(↓)</td><td>IS(↑)</td><td>Prec.(↑)</td><td>Rec.(↑)</td><td>Speed(↑)</td></tr><tr><td rowspan="4">CD</td><td rowspan="2">LCSC-Direct</td><td>300K</td><td>256</td><td>1</td><td>5.71</td><td>41.8</td><td>0.68</td><td>0.62</td><td>~15×</td></tr><tr><td>600+20K</td><td>256</td><td>1</td><td>5.54</td><td>40.9</td><td>0.68</td><td>0.62</td><td>~7.6×</td></tr><tr><td rowspan="2">LCSC-Diff</td><td>300K</td><td>256</td><td>1</td><td>5.51</td><td>39.8</td><td>0.68</td><td>0.62</td><td>~15×</td></tr><tr><td>600+20K</td><td>256</td><td>1</td><td>5.07</td><td>42.5</td><td>0.69</td><td>0.62</td><td>~7.6×</td></tr><tr><td rowspan="4">CT</td><td rowspan="2">LCSC-Direct</td><td>600K</td><td>256</td><td>1</td><td>12.1</td><td>35.1</td><td>0.67</td><td>0.54</td><td>~10.4×</td></tr><tr><td>800K</td><td>256</td><td>1</td><td>11.1</td><td>35.7</td><td>0.65</td><td>0.57</td><td>~6.3×</td></tr><tr><td rowspan="2">LCSC-Diff</td><td>600K</td><td>256</td><td>1</td><td>10.5</td><td>36.8</td><td>0.66</td><td>0.56</td><td>~10.4×</td></tr><tr><td>800K</td><td>256</td><td>1</td><td>9.02</td><td>38.8</td><td>0.68</td><td>0.55</td><td>~7.3×</td></tr></table>
+
+Table 17: Evaluation results with FCD and KID metrics on CIFAR-10 and ImageNet-64 datasets.  
+
+<table><tr><td colspan="7">(a) Results on CIFAR-10.</td><td colspan="6">(b) Results on ImageNet-64.</td></tr><tr><td>Method</td><td>Training Iter</td><td>Batch Size</td><td>FID(↓)</td><td>FCD(↓)</td><td>KID(↓)</td><td></td><td>Method</td><td>Training Iter</td><td>Batch Size</td><td>FID(↓)</td><td>FCD(↓)</td><td>KID(↓)</td></tr><tr><td>CT</td><td>400k</td><td>512</td><td>12.1</td><td>43.0</td><td>7.47e-3</td><td>CT</td><td>600k</td><td>256</td><td>16.6</td><td>49.8</td><td>9.22e-3</td><td></td></tr><tr><td>CT</td><td>800k</td><td>512</td><td>9.87</td><td>35.8</td><td>5.14e-3</td><td>CT</td><td>800k</td><td>256</td><td>15.8</td><td>47.5</td><td>8.69e-3</td><td></td></tr><tr><td>LCSC</td><td>450k</td><td>128</td><td>8.54</td><td>27.2</td><td>4.65e-3</td><td>CT</td><td>800k</td><td>2048</td><td>13.1</td><td>47.5</td><td>8.55e-3</td><td></td></tr><tr><td>LCSC</td><td>400k</td><td>512</td><td>8.89</td><td>34.4</td><td>3.96e-3</td><td>LCSC</td><td>600k</td><td>256</td><td>10.5</td><td>38.8</td><td>4.41e-3</td><td></td></tr><tr><td>LCSC</td><td>800+40k</td><td>512</td><td>7.05</td><td>24.9</td><td>2.90e-3</td><td>LCSC</td><td>800k</td><td>256</td><td>9.02</td><td>30.0</td><td>3.87e-3</td><td></td></tr></table>
+
+# E.7.2 EVALUATION WITH FID ON TEST DATASET
+
+In this section, we report the FID calculated on the test dataset to validate that LCSC does not overfit on the training data. Results are shown in Tab. 18. We can see that LCSC also achieves significant improvement on test FID, indicating its generalization ability across different data.
+
+# E.8 SEARCH RESULTS WITH STABLE DIFFUSION MODELS
+
+To further demonstrate the effectiveness of LCSC in enhancing DMs, we conduct experiments on Stable Diffusion (Rombach et al., 2022) checkpoints. Specifically, we fine-tune the Stable Diffusion v1-5 model on CC12M using LoRA for 20k iterations. We then apply LCSC to the saved checkpoints at intervals of 100 iterations. The results, presented in Tab. 19, show that LCSC achieves a significant improvement compared to the released Stable Diffusion checkpoints with the same sampling NFE.
+
+To investigate whether LCSC can accelerate the Stable Diffusion model, we further test PickScore (Kirstain et al., 2023) between the LCSC model with 10 NFE and the Stable Diffusion model with 15 NFE. The results are 0.49 and 0.51 for Stable Diffusion and LCSC, respectively, with a  $57\%$  winning rate for LCSC. This further demonstrate the ability of LCSC to accelerate inference speed for diffusion models.
+
+# F MORE INSIGHTS AND ANALYSIS
+
+# F.1 ANALYSIS OF SEARCH PATTERNS
+
+# F.1.1 MORE EXAMPLES OF SEARCH PATTERNS
+
+We demonstrate search results on ImageNet-64 in Fig. 9, which share similar patterns with results on CIFAR-10 (refer to Fig. 5). Additionally, Fig. 10 shows that the checkpoints assigned with larger coefficients often has lower FID than the checkpoints with small coefficients.
+
+# F.1.2 SEARCH PATTERN OF DIFFERENT RANDOM SEED
+
+We compare search patterns across different random seeds. As shown in Fig. 11, although each seed's dominant coefficients map to distinct subsets of weight checkpoints, they produce similar outcomes in terms of FID as provided in the caption. These results suggest multiple high-quality
+
+Table 18: Evaluation results on CIFAR-10 test set.  
+
+<table><tr><td>Method</td><td>Training Iter</td><td>Batch Size</td><td>FID(↓)</td><td>test FID(↓)</td></tr><tr><td>CD</td><td>800k</td><td>512</td><td>3.66</td><td>5.88</td></tr><tr><td>CD</td><td>840k</td><td>512</td><td>3.65</td><td>5.85</td></tr><tr><td>LCSC</td><td>100k</td><td>128</td><td>3.21</td><td>5.48</td></tr><tr><td>LCSC</td><td>250k</td><td>512</td><td>2.66</td><td>4.88</td></tr><tr><td>LCSC</td><td>800k</td><td>512</td><td>2.44</td><td>4.70</td></tr><tr><td>LCSC</td><td>800+40k</td><td>512</td><td>2.50</td><td>4.75</td></tr></table>
+
+Table 19: Results of LCSC on Stable Diffusion Checkpoints.  
+
+<table><tr><td></td><td>NFE</td><td>FID</td><td>PKS</td><td>WR@PKS</td><td>CLIP Score</td></tr><tr><td>LCSC</td><td>15</td><td>16.30</td><td>0.53(v.s. SD)/0.53(v.s. LoRA)</td><td>55%(v.s. SD)/56%(v.s. LoRA)</td><td>26.69</td></tr><tr><td>SDv1-5</td><td>15</td><td>17.55</td><td>0.47</td><td>44%</td><td>26.60</td></tr><tr><td>LoRA tuning</td><td>15</td><td>17.05</td><td>0.47</td><td>45%</td><td>26.61</td></tr><tr><td>LCSC</td><td>10</td><td>16.68</td><td>0.59(v.s. SD)/0.51(v.s. LoRA)</td><td>64%(v.s. SD)/53%(v.s. LoRA)</td><td>26.61</td></tr><tr><td>SDv1-5</td><td>10</td><td>18.16</td><td>0.41</td><td>36%</td><td>26.57</td></tr><tr><td>LoRA tuning</td><td>10</td><td>17.35</td><td>0.49</td><td>47%</td><td>26.56</td></tr></table>
+
+![](images/d520c81903fe2edfb62bb7a6608a0182aa1a201117a73a4b85210c72d2732a27.jpg)  
+Figure 9: Visualization of weight combination coefficients obtained using LCSC compared to those from the default EMA on ImageNet-64.
+
+![](images/30ac2fd4c07176c5862d1d4952f39f61e1796b76e304056d405a8962849cfb09.jpg)
+
+![](images/22b34a2e3cf1a55217d19e7e8d1bb4a63ce786c6bc8917e1f994f6156b3ff632.jpg)
+
+![](images/82e724aefb03d2316fceb3cafa9b5d0813c50e3284d5ccb5911af695679a4535.jpg)  
+Figure 10: FID of checkpoints with varying magnitude of coefficients.
+
+![](images/cb07176d4d003b6272918a8752091334b7895deeb7f5c3e8da5e4fdb2ad9c040.jpg)
+
+basins exist within the weight checkpoint subspace, and LCSC converges to one of them depending on the randomness. Previous study demonstrates that local minimal in networks are connected by simple curves over which losses are nearly constant (Garipov et al., 2018). To investigate whether solutions found by LCSC are also connected we average the search patterns obtained from multiple random seeds and evaluate the performance of the resulting model. As shown in Fig. 12, the averaged search pattern becomes homogeneous and contains smaller coefficients, yet it achieves comparable performance, as detailed in the figure caption. This observation suggests that the search results of LCSC may also be connected by low-loss curve.
+
+![](images/027801b0c15006fe2aab115fafe08a4f71fb698df7d1417d813cd315f1ed5f04.jpg)  
+Figure 11: Visualization of linear combination coefficients obtained by LCSC with different random seeds using CD on CIFAR10. Their FID scores are: 2.76 (seed 1) and 2.69 (seed 2).
+
+![](images/10fc2cb1d9826b995bfb4ffd436ff70249d902e1be9ba80edc9f9a799301979a.jpg)  
+Figure 12: Visualization of averaged coefficients obtained by LCSC across different random seeds using CD on CIFAR10. Their FID scores are: 2.76 (left), 2.70 (middle), 2.83 (right).
+
+![](images/6779ead357de899ab79585813f2b0493b76cd5b85437748b6cd790d43cffa699.jpg)
+
+![](images/b85a6485bb35e73f29f3e9da45644babf501e3550aca7961fab2efb0bad2942c.jpg)
+
+# F.1.3 THE CRITICAL ROLE OF SEARCH SPACE CONFIGURATION
+
+As explored in Sec. 6, restricting averaged weights to the convex hull of candidate checkpoints, where all combination coefficients are non-negative and their sum equals 1, might limit search efficacy. To test this hypothesis, we perform searches under convex combination conditions by clipping coefficients to non-negative values and normalizing their sum to 1. Tab. 20 presents these findings, with "w/o" indicating no restriction on coefficients (allowing values below zero) and "w" representing the convex hull restriction. The results clearly show superior performance without the convex restriction, underscoring its limiting effect on search outcomes. Furthermore, we discover that normalizing the coefficient sum to 1 is vital for effective search exploration, a practice we continue even after lifting the convex condition, as detailed in Eq. (7).
+
+# F.1.4 FORMATION OF THE SEARCH PATTERN
+
+We observe that the search pattern is established during the early stage of the search, while the later stage primarily amplifies it to a certain magnitude, as shown in Fig. 13.
+
+Table 20: Performance of LCSC with or without restriction to the convex hull of all saved checkpoints. "w/o" in the column "Restriction" is our default setting in main experiments, while "w" means the restriction holds.  
+
+<table><tr><td>Model</td><td>Method</td><td>Training Iter</td><td>Batch Size</td><td>NFE</td><td>Restriction</td><td>FID(↓)</td><td>IS(↑)</td></tr><tr><td rowspan="2">CD</td><td rowspan="2">LCSC</td><td rowspan="2">250K</td><td rowspan="2">512</td><td rowspan="2">1</td><td>w/o</td><td>2.76</td><td>9.71</td></tr><tr><td>w</td><td>3.38</td><td>9.36</td></tr><tr><td rowspan="2">DM</td><td rowspan="2">LCSC</td><td rowspan="2">350K</td><td rowspan="2">128</td><td rowspan="2">15</td><td>w/o</td><td>3.56</td><td>9.35</td></tr><tr><td>w</td><td>3.59</td><td>9.29</td></tr></table>
+
+![](images/e3a3f822ce8b17523fc05f8d1750f39fa7c06059785c8bccd1f4b41d26d68792.jpg)  
+Figure 13: Visualization of coefficient pattern at different number of search iterations. The experiment is conducted on ImageNet using CT.
+
+# F.2 REGULARIZED EVOLUTIONARY SEARCH
+
+We conduct additional experiments to evaluate the impact of regularization. Specifically, during the search process, we clip all coefficients to be below 1, thereby constraining the search to a restricted space. Using the same initialization and random seed, Fig. 14 shows that LCSC produces smaller and more homogeneous coefficients with regularization compared to those obtained without it. However, as noted in the figure caption, applying regularization results in worse performance, suggesting that constraining LCSC to smaller and more homogeneous coefficients is not advantageous.
+
+# F.3 CONVERGENCE CURVE OF LCSC
+
+Fig. 15a illustrates the convergence curve of the models searched by LCSC. At each point, corresponding to a specific number of training iterations, we perform LCSC with 2K search iterations using the most recent checkpoints available at that point. The results indicate that as the number of training iterations increases, the models searched by LCSC converge to progressively lower FID values. Notably, the convergence curve of LCSC exhibits a similar trend to that of the EMA model's
+
+![](images/a8d72ae4457ff0332960c89c22ee4d77d66f189f5fb6cea03923c296a28222e2.jpg)  
+(a) Visualization of searched coefficients with regularization.
+
+![](images/061481a9752b745337295b3772d25edb75c884845e050d38b374a67726341d03.jpg)  
+(b) Visualization of searched coefficients without regularization.
+
+![](images/4dce8950caf7cfdaa30ffb3943bd9e43fc9fb304561a163b7e29b809162df104.jpg)  
+(a) Training convergence curve with LCSC.  
+Figure 15: Convergence curves of LCSC on CIFAR-10 with CD.
+
+![](images/ad0c74894fa6aa317777df2f7bbcb0fc1b32cb64a1cbff0327f66f231b48e077.jpg)  
+Figure 14: Visualization of searched coefficients with and without regularization. Respective FID/IS/Prec/Rec are: 13.40/33.4/0.64/0.55 (left), 10.5/36.8/0.66/0.56 (right).  
+(b) Search convergence curve of LCSC.
+
+convergence curve. However, LCSC consistently achieves lower FID values compared to the EMA model, highlighting its enhanced ability to effectively merge historical checkpoints. Additionally, Fig. 15b shows the convergence curve of LCSC itself. This curve consistently reduces as the number of search iteration increases.
+
+# F.4 DISCUSSION ON THE TRAINING VARIANCE OF DM AND CM
+
+In Sec. 3, we mentioned that the training objectives of DM and CM tend to introduce substantial variance in gradient estimations. We further discuss the potential reasons or hypothesis of such phenomenon as follows.
+
+# F.4.1 UNBIASED ESTIMATION AS OBJECTIVE
+
+The objectives used in DM and CT are not accurate for any single batch. Instead, they serve as an unbiased estimation. Specifically, for DM, the neural network learns the score function at any  $(\boldsymbol{x}_t,t)$ , denoted as  $s_\theta (\boldsymbol{x}_t,t)$ . The ground truth score function  $\nabla \log p_t(\boldsymbol{x}_t)$  is given as (Song et al., 2020b):
+
+$$
+\nabla \log p _ {t} \left(\boldsymbol {x} _ {t}\right) = \mathbb {E} _ {\boldsymbol {y} \sim p _ {\text {d a t a}}, \varepsilon \sim \mathcal {N} \left(\boldsymbol {0}, \sigma (t) ^ {2} \boldsymbol {I}\right)} \left(- \frac {\varepsilon}{\sigma_ {t}} \mid \alpha_ {t} \boldsymbol {y} + \sigma_ {t} \varepsilon = \boldsymbol {x} _ {t}\right) \tag {78}
+$$
+
+This formula implies that every sample  $\pmb{y}$  in the dataset contributes to the ground truth score function at any  $(x_{t}, t)$ . However, the training objective Eq. (2) provides only an unbiased estimation of the ground truth score Eq. (78). At every iteration, the current target for the network is not identical to the ground truth score function but is determined by the sample  $\pmb{y}$  randomly drawn from the dataset and the noise level  $\sigma_{t}$ . Thus, even if  $s_{\theta}(\pmb{x}_{t}, t)$  exactly matches  $\nabla p_{t}(\pmb{x}_{t})$  for  $\forall (\pmb{x}_{t}, t)$ , its gradient estimation using a mini-batch of data does not converge to zero. The approximation of ground truth score function can only be obtained through the expectation over many training iterations.
+
+For CT, the situation is very similar. To simulate the current ODE solution step without a teacher diffusion model, Song et al. (Song et al., 2023) utilize the Monte Carol estimation  $-\frac{\boldsymbol{x}_t - \boldsymbol{y}}{t^2}$  to replace the ground truth score function  $\nabla \log p_t(\boldsymbol{x}_t) = \mathbb{E}_{\boldsymbol{y} \sim p_{\mathrm{data}}, \boldsymbol{x}_t \sim \mathcal{N}(\boldsymbol{y}, t^2 \mathbf{I})} (-\frac{\boldsymbol{x}_t - \boldsymbol{y}}{t^2} | \boldsymbol{x}_t)$  under the noise schedule from EDM (Karras et al., 2022) as denoted in Eq. (4). Therefore, the objective is also not accurate for any single batch of data and the model has to learn to fit the target model output after solving the current step with the expectation of score function among many training iterations. These types of objectives, which are not accurate for any single batch, introduce high variance to the gradient estimation.
+
+# F.4.2 MODEL ACROSS DIFFERENT TIME STEPS
+
+For both DM and CM, the neural network has to be trained at various time steps. Since the model input and output follow different distributions, the gradients at different time steps may conflict with each other. This phenomenon of negative transfer between timesteps has been studied in several previous work, where they train different diffusion neural networks at different timesteps and achieve better performance (Hang et al., 2023; Go et al., 2024). The misalignment between the objectives of different time steps may also contribute to the high variance in the gradient estimation.
+
+# F.4.3 ERROR ACCUMULATION OF CM
+
+For CM, the model learns to approximate the output of the target model at the previous step. This could potentially introduce the problem of error accumulation (Berthelot et al., 2023). Consequently, any noise introduced during training at early time steps is likely to lead to inaccuracies in the target model, which may be magnified in subsequent timesteps. This property of CM amplifies the high training variance, particularly for one-step sampling.
+
+# G FUTURE WORK
+
+LCSC represents a novel optimization paradigm, indicating its potential for widespread application. We recommend future investigations focus on three key areas:
+
+- Expanded Search Space: Presently, LCSC applies a uniform coefficient across an entire model. Considering that different model layers might benefit from distinct combination coefficients, partitioning model weights into segments for unique coefficient assignments could enhance the search space. For DMs, adopting variable coefficients across different timesteps may also offer further improvements.  
+- Efficient Optimization Methods: The current reliance on evolutionary methods, characterized by their dependency on randomness, limits efficiency and risks convergence to local optima. Investigating more effective optimization strategies presents a promising avenue for enhancing LCSC.  
+- Broader Application Scope: While the initial motivation for LCSC stems from managing the high training variance observed in DMs and CMs, its utility is not confined to these models. Exploring LCSC's applicability to other domains, such as fine-tuning language models or additional vision models, could unlock new performance gains.
+
+# H VISUALIZATION
+
+In this section, visualizations of images generated by LCM-LoRA (Fig. 16), DM (Figs. 17 and 18), CD (Figs. 19 and 20), and CT (Figs. 21 and 22) are presented. Many images produced by the EMA model are observed to be similar to those produced by the model derived from our LCSC when using the same noise input. This is expected given that both models are based on weights from the same training cycle. To more clearly highlight the distinctions between EMA and LCSC, we generate  $50\mathrm{K}$  images from EMA and LCSC using the same set of noise inputs, order the image pairs according to their Euclidean distances in the inception feature space, and randomly select the images with large distances. In general, images generated by the LCSC model are found to display enhanced sharpness, diminished noise, and more distinct object representation as well as details.
+
+![](images/f5fb5c1506d3435868b33f2272f86f9d2ee83d18b0e35fe94b9c1b0812dc1774.jpg)  
+(a) Vanilla training
+
+![](images/072536ae064691f86065fd4481f105892da35b621a4973072009df1122e4fa00.jpg)  
+(b) LCSC
+
+![](images/c93e8bec7dd6bfe9995d093d857b0316b871b67460985ca38ab0574f89fcf026.jpg)  
+(c) Vanilla training
+
+![](images/e02c2b9a308ccb593cc468935864ed877e8f249458623333c0adb157624fc180.jpg)  
+(d) LCSC
+
+![](images/191b4723820cf6a210ac54cbae6fdc71c83e10aedc6dece42879bfccd95337b8.jpg)  
+(e) Vanilla training
+
+![](images/b4a47a0f16a4c449f34a0bf7c4eb702efda45341649427e49f3c8ecd3aa1c75d.jpg)  
+(f) LCSC
+
+![](images/80cac115b331341301ce5bce5b695d972fffa2b23468e239b7763c9406dbe9b7.jpg)  
+(g) Vanilla training  
+Figure 16: Examples of the images generated by LCSC and vanilla training. The prompts for the images are 'A large passenger airplane flying through the air', 'A brown purse is sitting on a green bench', 'A man getting a drink from a water fountain that is a toilet', 'A picture of a man playing a violin in a kitchen' respectively.
+
+![](images/a6f19b60b653e0ca778e4f63aaac22ed19498b092ba527fd61117da32b0f7d3a.jpg)  
+(h) LCSC
+
+![](images/2bac6dee56288a4afc1fe2593dc34c213560cc4f6d449865433f700e564b5572.jpg)
+
+![](images/b6e73cca0177d107dece86fba38d3b5e96f69e45adad8aef7bd3b1ca2952cbf6.jpg)  
+(a) EMA model (FID: 4.16).  
+(b) Model obtained through LCSC (FID: 3.18).  
+Figure 17: Generated images from a DM trained on CIFAR10 over 800K iterations.
+
+![](images/d7af4f48d733ffc2a01b6d0b58b73846cfef85151fabed97fe43248381ec26ad.jpg)  
+(a) EMA model (FID: 19.8).
+
+![](images/1ce5cc06241f3331ce0ca2c9f32a8c4425cdb56c5920622dd4bea1ed89fecbe7.jpg)  
+(b) Model obtained through LCSC (FID: 15.3).
+
+![](images/3c78b5cf5387e483aab12d601b201728681c74c810210285d3c3cea6e0db07f5.jpg)  
+Figure 18: Generated images from a DM trained on ImageNet-64 over 500K iterations.
+
+![](images/b46fecdcf061c74260fe574e7436cd13a8186fc696f70d8a885e000df4bad97e.jpg)  
+(a) EMA model (FID: 3.66).  
+(b) Model obtained through LCSC (FID: 2.42).  
+Figure 19: Generated images from a CD model trained on CIFAR10 for 800K iterations.
+
+![](images/825b9da2d2222b9d746b22e88962f48003071e3f6b220e41539e86a4639edc54.jpg)  
+(a) EMA model (FID: 7.30).
+
+![](images/720b8364ea4dd51471c70100272a9c673acbfe86842f830a17120fd21bb72cb2.jpg)  
+(b) Model obtained through LCSC (FID: 5.54).
+
+![](images/383848221930e76930a7ddd3a594a5c9f03b494c90266c1b5e35813e68422631.jpg)  
+Figure 20: Images from a CD model trained on ImageNet-64 for 600K iterations.
+
+![](images/85996f7130a08ff7998baf351c033049b901a84f46173d3456d426fbd8ae2543.jpg)  
+(a) EMA model (FID: 12.08).  
+(b) Model obtained through LCSC (FID: 8.60).  
+Figure 21: Generated images from a CT model trained on CIFAR10 for 400K iterations.
+
+![](images/02ee313e29ffcb51edeadbd19602ddc77d13f12b5a07e5a7bf55afa189ff5e0f.jpg)  
+(a) EMA model (FID: 16.6).
+
+![](images/8d76f60e2f4e30dfc7ae4199ca12ea48be1b451887e591361710e7348eaf2c83.jpg)  
+(b) Model obtained through LCSC (FID: 12.1).  
+Figure 22: Images from a CT model trained on ImageNet-64 for 600K iterations.
\ No newline at end of file
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+# LINEAR MULTISTEP SOLVER DISTILLATION FOR FAST SAMPLING OF DIFFUSION MODELS
+
+Yuchen Liang, Xiangzhong Fang  
+School of Mathematical Sciences  
+Peking University  
+ycliang@pku.edu.cn
+
+Hanting Chen, Yunhe Wang  
+Huawei Noah's Ark Lab  
+chenhanting@huawei.com
+
+# ABSTRACT
+
+Sampling from diffusion models can be seen as solving the corresponding probability flow ordinary differential equation (ODE). The solving process requires a significant number of function evaluations (NFE), making it time-consuming. Recently, several solver search frameworks have attempted to find better-performing model-specific solvers. However, predicting the impact of intermediate solving strategies on final sample quality remains challenging, rendering the search process inefficient. In this paper, we propose a novel method for designing solving strategies. We first introduce a unified prediction formula for linear multistep solvers. Subsequently, we present a solver distillation framework, which enables a student solver to mimic the sampling trajectory generated by a teacher solver with more steps. We utilize the mean Euclidean distance between the student and teacher sampling trajectories as a metric, facilitating rapid adjustment and optimization of intermediate solving strategies. The design space of our framework encompasses multiple aspects, including prediction coefficients, time step schedules, and time scaling factors. Our framework has the ability to complete a solver search for Stable-Diffusion in under 12 total GPU hours. Compared to previous reinforcement learning-based search frameworks, our approach achieves over a  $10 \times$  increase in search efficiency. With just 5 NFE, we achieve FID scores of 3.23 on CIFAR10, 7.16 on ImageNet-64, 5.44 on LSUN-Bedroom, and 12.52 on MS-COCO, resulting in a  $2 \times$  sampling acceleration ratio compared to handcrafted solvers.
+
+# 1 INTRODUCTION
+
+Diffusion models (Sohl-Dickstein et al., 2015; Ho et al., 2020; Song et al., 2021b) have gained widespread success in various applications including image generation (Dhariwal & Nichol, 2021; Rombach et al., 2022), audio synthesis (Kong et al., 2021; Chen et al., 2021), video generation (Ho et al., 2022b;a; Blattmann et al., 2023), and text-to-image synthesis (Saharia et al., 2022; Ruiz et al., 2023; Podell et al., 2024; Esser et al., 2024). When generating samples, diffusion models perform reverse solving of a predefined Stochastic Differential Equation (SDE) or its corresponding Probability Flow Ordinary Differential Equation (ODE) (Song et al., 2021b). This solving process often required hundreds of function evaluations (NFE), making it extremely time-consuming compared to classical generative models like Generative Adversarial Networks (GANs) (Goodfellow et al., 2014).
+
+Fortunately, significant advancements have been made in accelerating the sampling process of diffusion models. Existing acceleration methods can be broadly categorized into two classes. The first class of methods involves an additional distillation training phase (Luhman & Luhman, 2021; Salimans & Ho, 2022; Song et al., 2023; Sauer et al., 2023; Luo et al., 2024). Distilled models require only 1-4 NFE to generate high-quality samples. However, the distillation phase typically requires thousands of GPU hours, posing a significant training cost, especially for large scale models. Additionally, many of distilled models lack the ability to perform image processing tasks, such as image editing and restoration (Yu et al., 2023; He et al., 2023; Wang et al., 2022; Chung et al., 2023; Zhu et al., 2023). We will provide a more detailed discussion in Sec. 2.2.
+
+![](images/ca5807bf22f8e27d5fe81d2f8ca640f2ced3a48095cc743a53b98db35d8d76de.jpg)  
+NFE=5  
+NFE=10
+
+![](images/b17aff3ea4b9177446ce86ec9fdf4662e45bf7c04dc18867e10dfc3b533dcf21.jpg)
+
+![](images/d3fc0a5af21a69a021113962217a399c1a315dc567f39e9fbd45fac7d10f816c.jpg)
+
+![](images/324528dccbcc2f765d66ebda1cfe0226a9940a80bb8e5c9e346c6cffd149d1f2.jpg)
+
+![](images/db71e9cbb1b1577c9b15c72c4b2d9adc0f3e20d8abedb0ef13a9354e4bd94681.jpg)  
+DEIS  
+(Zhang & Chen, 2022)
+
+![](images/12ff9907d201e2244ee2bddae1a5d05495e1fc677824f5305f1373640d53e11f.jpg)  
+DPM-Solver++  
+(Lu et al., 2022b)
+
+![](images/bd3bde507ad8009731cff382dac85d0d57a0597a2d0fb2de2bf8883353b7cd4a.jpg)  
+UniPC  
+(Zhao et al., 2023)
+
+![](images/0300c4b5b085b6ca7c8df525c5164965cfb5510ebd0d838d9ca5beee9150234f.jpg)  
+DLMS  
+Ours  
+Figure 1: Synthesized images of Stable-Diffusion (Rombach et al., 2022) with default classifier-free guidance scale 7.5 and text prompt "A large brown bear walking through a forest". Our proposed DLMS can generate more realistic and visually detailed images compared with previous handcrafted samplers (Zhang & Chen, 2022; Lu et al., 2022b; Zhao et al., 2023).
+
+And the second class, which is also widely regarded, focuses on designing efficient solvers without further training the model. Techniques such as parameterization, exponential integrators, and higher-order solvers have successfully reduced the NFE to 15-20 for the sampling process (Lu et al., 2022b; Zhao et al., 2023; Zhang & Chen, 2022; Liu et al., 2022; Lu et al., 2022a; Song et al., 2021a; Karras et al., 2022). However, when the NFE is less than 10, the sample quality deteriorates significantly.
+
+The challenges have given rise to a series of search frameworks seeking more efficient solvers. However, due to the iterative nature of solving processes, it is difficult to predict the impact of intermediate solving strategies on the final sample quality, making the search process difficult and inefficient. These frameworks still often require tens of GPU hours and were limited to specific design space such as time steps (Watson et al., 2022; Li et al., 2023; Liu et al., 2023a; Sabour et al., 2024; Chen et al., 2024), parameterization mode (Zheng et al., 2023), or solver combination (Liu et al., 2023b). Recently, some work (Shaul et al., 2024; Zhang et al., 2024) has attempted to use solver distillation to design solvers, but the performance improvements have been relatively limited.
+
+In this paper, we propose a novel solver distillation framework that greatly accelerates the search efficiency. We begin by presenting a unified prediction formula for linear multistep solvers. Next, we introduce our solver distillation algorithm, which allows a student solver to replicate the sampling trajectory of a teacher solver that uses more steps. We employ the mean Euclidean distance between the student and teacher sampling trajectories as a metric, enabling quick adjustments and optimization of intermediate solving strategies. Compared to previous reinforcement learning-based search frameworks, our approach achieves over a  $10\times$  increase in search efficiency.
+
+Our framework has the ability to complete a solver distillation for Stable-Diffusion (Rombach et al., 2022) in less than  $1.5\mathrm{h}$  on 8 NVIDIA V100 GPUs. The design scope includes the time steps, the time scaling factors and the prediction coefficients. We extensively evaluate our approach on various resolution datasets in both pixel space and latent space. The Distilled Linear Multistep Solver (DLMS) significantly surpasses previous handcrafted and search-based solvers. Compared to handcrafted solvers, DLMS achieves a  $2\times$  sampling acceleration ratio.
+
+# 2 BACKGROUND AND RELATED WORK
+
+# 2.1 DIFFUSION MODELS
+
+Given the data distribution  $p_0$ , diffusion models employ a forward process  $\mathbf{x}_t = \alpha_t\mathbf{x}_0 + \sigma_t\epsilon, t \in [0,T]$  with marginal distribution  $\{p_t\}_{0}^T$  to gradually degenerate the data  $\mathbf{x}_0 \sim p_0$  with Gaussian
+
+noise  $\epsilon \sim \mathcal{N}(\mathbf{0},\mathbf{I})$ . The noise schedule  $\alpha_{t},\sigma_{t} > 0$  is designed to make  $p_T$  approximately a pure Gaussian distribution  $\mathcal{N}(\mathbf{0},\tilde{\sigma}^2\mathbf{I})$ . Notably, there exists a corresponding Probability Flow ordinary differential equation (PF-ODE) (Song et al., 2021b):
+
+$$
+\mathrm {d} \mathbf {x} _ {t} = \left[ f (t) \mathbf {x} _ {t} - \frac {1}{2} g ^ {2} (t) \nabla \log p _ {t} (\mathbf {x} _ {t}) \right] \mathrm {d} t, \tag {1}
+$$
+
+where  $f(t) = \frac{\mathrm{d}\log\alpha_t}{\mathrm{d}t}, g^2(t) = \frac{\mathrm{d}\sigma_t^2}{\mathrm{d}t} - 2\frac{\mathrm{d}\log\alpha_t}{\mathrm{d}t}\sigma_t^2$  (Kingma et al., 2021). The PF-ODE shares the same marginal distribution  $p_t$  as the forward process. And the score function  $\nabla \log p_t(\mathbf{x}_t)$  can be expressed using Tweedie's formula (Robbins, 1992):
+
+$$
+\nabla \log p _ {t} (\mathbf {x} _ {t}) = \frac {\alpha_ {t} \mathbb {E} [ \mathbf {x} _ {0} | \mathbf {x} _ {t} ] - \mathbf {x} _ {t}}{\sigma_ {t} ^ {2}} = - \frac {\mathbb {E} [ \epsilon | \mathbf {x} _ {t} ]}{\sigma_ {t}}. \tag {2}
+$$
+
+To estimate the score function, a neural network  $\epsilon_{\theta}(\mathbf{x}_t,t)$  is trained to predict  $\mathbb{E}[\epsilon |\mathbf{x}_t]$  via the least square estimation by minimizing the  $L_{2}$  loss
+
+$$
+\mathbb {E} _ {\mathbf {x} _ {0} \sim p _ {\text {d a t a}}, \epsilon \sim \mathcal {N} (\mathbf {0}, I)} \| \epsilon_ {\theta} (\mathbf {x} _ {t}, t) - \epsilon \| _ {2} ^ {2} \tag {3}
+$$
+
+for each  $t \in [0, T]$ . Consequently, we can sample from diffusion models by solving the empirical PF-ODE:
+
+$$
+\mathrm {d} \mathbf {x} _ {t} = \left[ f (t) \mathbf {x} _ {t} + \frac {g ^ {2} (t)}{2 \sigma_ {t}} \epsilon_ {\theta} (\mathbf {x} _ {t}, t) \right] \mathrm {d} t \tag {4}
+$$
+
+from time  $T$  to time 0. Additionally, the conditional sampling can be carried out by guided sampling (Dhariwal & Nichol, 2021; Ho & Salimans, 2021). Remarkably, classifier-free guidance (Ho & Salimans, 2021) defines a guided noise predictor:
+
+$$
+\tilde {\epsilon} _ {\theta} \left(\mathbf {x} _ {t}, t, c\right) = s \cdot \epsilon_ {\theta} \left(\mathbf {x} _ {t}, t, c\right) + (1 - s) \cdot \epsilon_ {\theta} \left(\mathbf {x} _ {t}, t, \emptyset\right), \tag {5}
+$$
+
+where  $c$  is the condition,  $\emptyset$  stand for the unconditional sampling, and  $s > 0$  is the guidance scale.
+
+In practice, except for the noise predictor  $\epsilon_{\theta}(\mathbf{x}_t,t)$ , diffusion models can also be parameterized as data predictor  $\pmb{x}_{\theta}(\mathbf{x}_t,t)$  to predict  $\mathbf{x}_0$  or velocity predictor  $\pmb{v}_{\theta}(\mathbf{x}_t,t)$  to predict  $\alpha_{t}\epsilon -\sigma_{t}\mathbf{x}_{0}$ , where the parameterizations are theoretically equivalent, but have impact in practice performance (Karras et al., 2022; Hang et al., 2023).
+
+# 2.2 IMAGE PROCESSING WITH PRE-TRAINED DIFFUSION MODELS
+
+Pre-trained diffusion models are versatile tools for a variety of image processing tasks. When aiming to manipulate or restore a reference image  $y$ , one can simply adjust the sampling score function using Bayes' Rule:
+
+$$
+\underbrace {\nabla \log p _ {t} (\mathbf {x} _ {t} | \mathbf {y})} _ {\text {c o n d i t i o n a l s c o r e}} = \underbrace {\nabla \log p _ {t} (\mathbf {x} _ {t})} _ {\text {s c o r e}} + \underbrace {\nabla \log p _ {t} (\mathbf {y} | \mathbf {x} _ {t})} _ {\text {g u i d a n c e t e r m}} \tag {6}
+$$
+
+The guidance term can be derived from the gradient of a differentiable loss function. For example, FreeDoM (Yu et al., 2023) uses Gram matrices (Johnson et al., 2016) of the intermediate layers of the CLIP image encoder as the loss function.
+
+Another image processing technique is post-processing. These methods achieve the goal of image processing tasks by projecting the estimated value  $\pmb{x}_{\theta}(\mathbf{x}_t,t)$  at each time step onto a predefined manifold  $\mathcal{M}_y$  (He et al., 2023; Wang et al., 2022; Chung et al., 2023; Zhu et al., 2023).
+
+While there are efforts towards distillation work for the guided diffusion model (Meng et al., 2023), the high distillation costs make it impractical to distill for every kind of guidance. In contrast, efficient ODE solvers are much more flexible.
+
+# 2.3 FAST SAMPLING WITH EXPONENTIAL INTEGRATORS
+
+Samplers based on exponential integrator have been found to be more efficient than directly solving the ODE (4). Given an initial value  $\mathbf{x}_s$  at time  $s$ , the ODE solution  $\mathbf{x}_t$  can be analytically computed
+
+![](images/56fa0300763acc830a52d507591d5235e3a617a73af4368aff5e4affb31f3b53.jpg)  
+Figure 2: Designer network architecture. We concatenate the embeddings of the bottleneck feature, time step, and step index, and pass them through a fully connected layer to obtain the parameters required for the prediction formula (8, 9).
+
+with the following exponentially weighted integral by changing the variable from  $t$  to half log-SNR  $\lambda_{t} := \log (\alpha_{t} / \sigma_{t})$  (Lu et al., 2022a;b):
+
+$$
+\mathbf {x} _ {t} = \frac {\alpha_ {t}}{\alpha_ {s}} \mathbf {x} _ {s} - \alpha_ {t} \int_ {\lambda_ {s}} ^ {\lambda_ {t}} e ^ {\lambda} \hat {\epsilon} _ {\theta} (\hat {\mathbf {x}} _ {\lambda}, \lambda) d \lambda , \mathbf {x} _ {t} = \frac {\sigma_ {t}}{\sigma_ {s}} \mathbf {x} _ {s} + \sigma_ {t} \int_ {\lambda_ {s}} ^ {\lambda_ {t}} e ^ {- \lambda} \hat {\boldsymbol {x}} _ {\theta} (\hat {\mathbf {x}} _ {\lambda}, \lambda) d \lambda \tag {7}
+$$
+
+where  $\hat{\epsilon}_{\theta}(\cdot, \lambda) \coloneqq \epsilon_{\theta}(\cdot, t(\lambda)), \hat{\mathbf{x}}_{\theta}(\cdot, \lambda) \coloneqq \pmb{x}_{\theta}(\cdot, t(\lambda))$  and  $\hat{\mathbf{x}}_{\lambda} \coloneqq \mathbf{x}_{t(\lambda)}$ . To predict the integral part, DEIS (Zhang & Chen, 2022) approximates  $\epsilon_{\theta}(\mathbf{x}_t, t)$  with polynomial interpolation w.r.t  $t$ , similarly DPM-Solver (Lu et al., 2022a) approximate  $\hat{\epsilon}_{\theta}(\hat{\mathbf{x}}_{\lambda}, \lambda)$  w.r.t  $\lambda$  (equation 7, left) and DPM-Solver++ (Lu et al., 2022b) approximate  $\hat{\pmb{x}}_{\theta}(\hat{\mathbf{x}}_{\lambda}, \lambda)$  w.r.t  $\lambda$  (equation 7, right). UniPC (Zhao et al., 2023) introduces a correcting strategy and various scale functions. And AMED-Solver (Zhou et al., 2024) adopts an approximate mean value method instead polynomial interpolation.
+
+# 2.4 EFFICIENT SAMPLER SEARCH AND DISTILLATION
+
+Recently, several frameworks have incorporated a search phase to enhance the efficiency of samplers. The time step schedule is a commonly explored aspect (Chen et al., 2024; Sabour et al., 2024; Li et al., 2023; Watson et al., 2022; Liu et al., 2023a). Moreover, DPM-Solver-v3 (Zheng et al., 2023) aims to optimize the parameterization mode beyond noise prediction and data prediction. USF (Liu et al., 2023b) strives to identify the most suitable solver for each time step.
+
+Bespoke solver (Shaul et al., 2024) is also a type of solver distillation method, but it only uses the final output to adjust hyperparameters, with a deep and complex computation graph that hinders effective hyperparameter tuning. Zhang et al. (2024) proposed a method that utilizes local mean squared error (MSE) to refine existing solvers, which incurs minimal cost but yields limited effectiveness. We actively use intermediate values in the trajectory and judiciously stop unnecessary gradient backpropagation, greatly enhancing the effectiveness of solver distillation method.
+
+# 3 METHODOLOGY
+
+In this section, we present our method to solve the PF-ODE (4). We start with the discussion about the optimal linear multistep prediction strategy. And then we introduce how to achieve the optimal solver with an adaptive time schedule and time scaling factors. Finally, we introduce some practical techniques to further improve performance. A comparison with related work is provided in Appendix A.
+
+# 3.1 TOWARDS OPTIMAL LINEAR MULTISTEP SOLVER
+
+Given a time schedule  $\{t_n\}_{n=0}^N$  decreasing from  $t_0 = T$  to  $t_N = 0$ , the solvers are trying to predict the ground truth trajectory  $\{\mathbf{x}_{t_n}^G\}_{n=0}^N$  with a numerical simulation trajectory  $\{\mathbf{x}_{t_n}^S\}_{n=0}^N$  with
+
+$\mathbf{x}_{t_0}^S = \mathbf{x}_{t_0}^G$ . Linear multistep solvers aim to predict each local intermediate  $\mathbf{x}_{t_n}^G$ ,  $n \geq 1$  by leveraging the linear combination of  $p$  previous outputs from the denoising network. Take the data prediction type (equation 7, right) for example, the prediction formula can be expressed as:
+
+$$
+\boldsymbol {D} _ {n} = \sum_ {k = 1} ^ {p} a _ {k} \boldsymbol {x} _ {\theta} \left(\mathbf {x} _ {t _ {n - k}} ^ {S}, s _ {n - k} t _ {n - k}\right), \quad \sum_ {k = 1} ^ {p} a _ {k} = 1 \tag {8}
+$$
+
+$$
+\mathbf {x} _ {t _ {n}} ^ {S} = \frac {\sigma_ {t _ {n}}}{\sigma_ {t _ {n - 1}}} \mathbf {x} _ {t _ {n - 1}} ^ {S} - \alpha_ {t _ {n}} \left(e ^ {\lambda_ {t _ {n - 1}} - \lambda_ {t _ {n}}} - 1\right) \boldsymbol {D} _ {n} \tag {9}
+$$
+
+where  $\{a_k\}_{k=1}^p$  are the prediction coefficients and the time scaling factors  $\{s_n\}_{n=0}^N$  are usually set to 1. This implies that the candidate  $\mathbf{x}_{t_n}^S$  lies in a  $p$ -dimensional hyperplane and hence the theoretically optimal  $\mathbf{x}_{t_n}^S$  is the linear projection of  $\mathbf{x}_{t_n}^G$  onto the hyperplane.
+
+It is worth noting that for each trajectory  $\{\mathbf{x}_{t_n}^G\}_{n = 0}^N$  and for each step  $n$  the optimal coefficients  $\{a_k^*\}_{k = 1}^p$  are different. Nevertheless, some studies have shown that the ODE trajectories from diffusive models have similar properties(Chen et al., 2024), so it is possible to design a unified  $\{a_{k}\}_{k = 1}^{p}$  that works well on all trajectories just as in most of previous works, but we consider this to be a suboptimal choice.
+
+As mentioned in the previous paragraph, we aim to assign distinct coefficients for each trajectory and step. To achieve this, we establish a mapping  $g: (\mathbf{x}_{t_{n-1}}^S, t_{n-1}, n-1) \longmapsto \{a_k\}_{k=1}^p$ . Drawing inspiration from Zhou et al. (2024) we utilize the bottleneck feature  $h_{t_{n-1}}$  obtained from the pretrained denoising model instead of  $\mathbf{x}_{t_{n-1}}^S$  to circumvent additional computational costs. Subsequently, we employ an extremely lightweight designer network with approximately 9k parameters, denoted as  $g_\phi(h_{t_{n-1}}, t_{n-1}, n-1)$ , to generate the coefficients  $\{a_k\}_{k=1}^p$ . The network architecture is shown in Fig. 2.
+
+Given the availability of the ground truth trajectory  $\{\mathbf{x}_{t_n}^G\}_{n=0}^N$ , we can establish the optimal solver by minimizing the square distance  $d(\mathbf{x}_{t_n}^S, \mathbf{x}_{t_n}^G)$  where the prediction  $\mathbf{x}_{t_n}^S$  is computed using equations (8) and (9) with  $\{a_k\}_{k=1}^p = g_\phi(h_{t_{n-1}}, t_{n-1}, n-1)$ .
+
+In practical terms, we utilize a numerical ground truth trajectory  $\mathbf{x}_{t_n}^T$  generated by a teacher solver  $\Phi_t$  such as DPM-Solver++ with  $M$  interpolation time steps between  $t_{n - 1}$  and  $t_n$ , which we denote as  $\mathbf{x}_{t_n}^T = \Phi_t(\mathbf{x}_{t_{n - 1}}^T,t_{n - 1},t_n,M)$ .
+
+# 3.2 ADAPTIVE TIME SCHEDULE
+
+Adaptive time schedule is a class of approaches that can significantly improve the efficiency of various solvers. Some previous works (Xia et al., 2024; Zhou et al., 2024) also show that adjusting the time scaling factors  $\{s_n\}_{k=1}^N$  properly can improve the accuracy of the solution. In Sec. 3.1, we described how to get the distilled solver for a fixed time schedule  $\{t_n\}_{n=0}^N$ . In this section, we show how to use our distillation framework to get adaptive time schedules.
+
+When adjusting the prediction coefficients  $\{a_k\}_{k=1}^p$  in Sec. 3.1, we chose  $\mathbf{x}_{t_{n-1}}^S$  at time  $t_{n-1}$  as the starting point to predict  $\mathbf{x}_{t_n}^G$ . Now, we aim to incorporate  $t_{n-1}$  and  $s_{n-1}$  into the design space. Therefore, we modify the starting point to be  $\mathbf{x}_{t_{n-2}}^S$  at time  $t_{n-2}$ . To ensure each trajectory has an independent adaptive time schedule rather than a unified schedule, we include the time schedules  $\{t_n\}_{n=0}^N$  and  $\{s_n\}_{n=0}^N$  in the output of  $g_\phi$ . This is achieved by setting  $\{a_k\}_{k=1}^p$ ,  $t_{n-1}$ ,  $s_{n-1} = g_\phi(h_{t_{n-2}}, t_{n-2}, n-2)$  for  $n \geq 2$ . Let  $Q$  denote the buffer list that collects outputs  $\mathbf{x}_\theta(\mathbf{x}_{t_n}^S, s_n t_n)$ . We write the prediction formula (8, 9) as:
+
+$$
+\mathbf {x} _ {n} ^ {S} = \Phi \left(\mathbf {x} _ {n - 1} ^ {S}, t _ {n - 1}, t _ {n}, \left\{a _ {k} \right\} _ {k = 1} ^ {p}, Q\right). \tag {10}
+$$
+
+Thus, the distillation phase can be summarized by Algorithm 1. After the distillation phase, Algorithm 2 shows the sampling process with a trained designer network  $g_{\phi}$ .
+
+It is worth noting that we have employed the stop gradient operation multiple times in Algorithm 1, which is crucial for the proper functioning of the algorithm. When using the state  $\mathbf{x}_{t_{n-2}}^S$  at time  $t_{n-2}$  to predict  $\mathbf{x}_{t_n}^T$  at time  $t_n$ , retaining the gradients of  $t_{n-2}, t_n, \mathbf{x}_{t_{n-2}}^S, \mathbf{x}_{t_n}^T$  will lead to two issues:
+
+Algorithm 1 Linear Multistep Solver Distillation  
+Require: Designer network  $g_{\phi}$  , teacher solver  $\Phi_t$  , number of time steps  $N$  , number of interpolation time steps  $M$  , initial timestep  $T$  , max order  $p$    
+1: repeat   
+2: Sample initial value  $\mathbf{x}_{t_0}^S = \mathbf{x}_{t_0}^T\sim \mathcal{N}(\mathbf{0},\tilde{\sigma}^2\mathbf{I})$    
+3:  $t_0\gets T,s_0\gets 1$  . Initialize an empty buffer  $Q$    
+4: Set the parameter gradient of  $\phi$  to 0   
+5:  $Q\stackrel {\mathrm{buffer}}{\leftarrow}\pmb{x}_{\theta}(\pmb{x}_{t_0}^S,s_0t_0)$    
+6: Extract the bottleneck feature  $h_{t_0}$    
+7: for  $n = 2$  to  $N$  do   
+8:  $\{a_k\}_{k = 1}^{\max (n - 1,p)},t_{n - 1},s_{n - 1}\gets g_\phi (h_{t_{n - 2}},t_{n - 2},n - 2)$    
+9:  $\mathbf{x}_{t_{n - 1}}^{S}\gets \Phi (\mathbf{x}_{n - 2}^{S},t_{n - 2},t_{n - 1},\{a_{k}\}_{k = 1}^{\max (n - 1,p)},Q)$ $\triangleright$  Generate prediction  $\mathbf{x}_{t_n - 1}^S$    
+10:  $Q\stackrel {\mathrm{buffer}}{\leftarrow}\pmb{x}_{\theta}(\pmb{x}_{t_{n - 1}}^S,s_{n - 1}t_{n - 1})$    
+11: Extract the bottleneck feature  $h_{t_{n - 1}}$    
+12:  $\{a_k\}_{k = 1}^{\max (n,p)},t_n,s_n\gets g_\phi (h_{t_{n - 1}},t_{n - 1},n - 1)$    
+13:  $t_n\gets \mathrm{sg}(t_n)$ $\triangleright$  Stop the gradient of  $t_n$    
+14:  $\mathbf{x}_{t_n}^S\gets \Phi (\mathbf{x}_{n - 1}^S,t_{n - 1},t_n,\{a_k\}_{k = 1}^{\max (n,p)},Q)$ $\triangleright$  Generate prediction  $\mathbf{x}_{t_n}^S$    
+15: if n=2 then   
+16:  $\mathbf{x}_{t_{n - 1}}^T\gets \mathrm{sg}(\Phi_t(\mathbf{x}_{t_{n - 2}}^T,t_{n - 2},t_{n - 1},\mathrm{M}))$    
+17: end if   
+18:  $\mathbf{x}_{t_n}^T\gets \mathrm{sg}(\Phi_t(\mathbf{x}_{t_{n - 1}}^T,t_{n - 1},t_n,\mathrm{M}))$ $\triangleright$  Solve the numerical ground truth  $\mathbf{x}_{t_n}^T$    
+19:  $\mathcal{L}_n(\phi)\gets d(\mathbf{x}_{t_n}^S,\mathbf{x}_{t_n}^T)$ $\triangleright$  Calculate the prediction error at time  $t_n$    
+20: Perform backpropagation for  $\mathcal{L}_n(\phi)$    
+21:  $Q\gets \mathrm{sg}(Q),t_{n - 1}\gets \mathrm{sg}(t_{n - 1})$ $\triangleright$  Stop the gradient of buffers and  $t_{n - 1}$    
+22: end for   
+23: Update the parameter  $\phi$    
+24: until convergence
+
+Algorithm 2 Distilled Solver Sampling  
+Require: Designer network  $g_{\phi}$  , number of time steps  $N$  , initial timestep  $T$  , max order  $p$  1: Sample initial value  $\mathbf{x}_{t_0}^S\sim \mathcal{N}(\mathbf{0},\tilde{\sigma}^2\mathbf{I})$    
+2:  $t_0\gets T,s_0\gets 1$  . Initialize an empty buffer  $Q$    
+3: for  $n = 1$  to  $N$  do   
+4:  $Q\stackrel {\mathrm{buffer}}{\leftarrow}\pmb{x}_{\theta}(\pmb{x}_{t_{n - 1}}^{S},s_{n - 1}t_{n - 1})$    
+5: Extract the bottleneck feature  $h_{t_n - 1}$    
+6:  $\{a_k\}_{k = 1}^{\max (n,p)},t_n,s_n\gets g_\phi (h_{t_{n - 1}},t_{n - 1},n - 1)$    
+7:  $\mathbf{x}_{t_n}^S\gets \Phi (\mathbf{x}_{n - 1}^S,t_{n - 1},t_n,\{a_k\}_{k = 1}^{\max (n,p)},Q)$    
+8: end for   
+9: return  $\mathbf{x}_{t_N}^S$
+
+- Firstly, the gradients from past time steps will not be removed from the computation graph, resulting in a linear increase in memory usage with each step  $n$ .  
+- Secondly,  $t_n$  will move towards  $t_{n - 2}$  to ease the prediction difficulty, ultimately resulting in the collapse of the entire time schedule. This is not an ideal outcome.
+
+In addition, we find it necessary to adopt data prediction type (equation 7, right). We believe this is because the data prediction type can automatically align the noise intensity with the time step  $t$ , thus stabilizing the prediction error for the solvers.
+
+# 3.3 PRACTICAL TECHNIQUES
+
+In this section, we introduce several practical techniques to further improve the performance of DLMS.
+
+High-order initialization. Since the prediction formula (8, 9) covers almost all linear multi-step solvers with exponential integrators such as DDIM (Song et al., 2021a), PLMS (iPNDM) (Zhang & Chen, 2022; Liu et al., 2022), DPM-Solver++ (Lu et al., 2022b), UniP- $p$  (Zhao et al., 2023) and DEIS (Zhang & Chen, 2022). We can initialize with these pre-designed solvers by setting the output biases of the designer network  $g_{\phi}$ . Therefore, DLMS has a theoretical accuracy that is at least on par with the aforementioned methods. Thus, we still refer to the number of history outputs  $p$  used in each step as the "order". Although our framework still works with DDIM (Song et al., 2021a) initialization, using higher-order solvers allows the distillation phase to be completed more quickly with fewer generated trajectories. In our experiments, initializing with any higher-order methods did not show significant differences; for simplicity, we recommend PLMS (Zhang & Chen, 2022; Liu et al., 2022) as the initial solving strategy.
+
+Analytical First Step (AFS). In Algorithm 1, the initial time step  $t_0 = T$  and time scaling factor  $s_0 = 1$  are excluded from the design space. Instead, we can treat  $t_0 = T$  as a virtual initial step. Specifically, we set  $\boldsymbol{x}_{\theta}(\mathbf{x}_{t_0}^S, s_0 t_0) = \mathbf{0}$  and  $h_{t_0} = \mathbf{0}$ , without employing the denoising network. This approach ensures that  $t_1$  is the actual first time step at which the diffusion model is engaged, while  $t_1$  and  $s_1$  remain in the design space. This strategy is fundamentally equivalent to the analytical first step (AFS) proposed by Dockhorn et al. (2022).
+
+Exponential Moving Average (EMA). Due to NFE limitations, the student solver cannot fully replicate the sampling trajectory generated by the teacher solver. This leads to a non-zero stochastic gradient even at convergence. To reduce the impact of parameter oscillations on solver performance, we introduce EMA updates inspired by diffusion models. During solver distillation, we apply EMA updates with half-lives of 1, 2, and 3 kimg, selecting the best-performing configuration from four parameter sets.
+
+Inception distance at the final step. Some search-based frameworks directly use Fréchet Inception Distance (FID) (Heusel et al., 2017) as the optimization objective (Liu et al., 2023b; Watson et al., 2022; Li et al., 2023). In contrast, our method employs square distance as the optimization target. Since square distance is less sensitive to high-frequency information, we observe that optimizing square distance does not always lead to better FID scores. Therefore, in experiments conducted in pixel space, we replace the final step's objective from pixel square distance to Inception Distance (the square distance of features from the Inception network). We only make this replacement in the final step because we find that Inception Distance does not effectively capture the differences between noisy images. In latent space tasks, the issue of high-frequency information is not present. To avoid unnecessary calls to the network modules, we do not perform this replacement in those cases.
+
+# 4 EXPERIMENTS
+
+In this section, we demonstrate that DLMS exhibits significant advantages in both unconditional and conditional sampling with pixel-space and latent-space diffusion models. We conducted experiments across multiple datasets with resolutions ranging from 32 to 512 and compared our approach with current state-of-the-art both handcrafted and search-based solvers. Then, we showcase the benefits from adaptive time steps and time scales, as well as ablation studies of the practical techniques we provided. Finally, we visualize the adaptive time schedules and the samples generated by DLMS.
+
+# 4.1 SETTINGS
+
+In our experiments, we uniformly use the noise schedule  $\alpha_{t} = 1$ ,  $\sigma_{t} = t$  from Karras et al. (2022). We initialized the prediction coefficients with PLMS (Zhang & Chen, 2022; Liu et al., 2022), using a uniform time schedule (Ho et al., 2020) and time scaling factors of 1. We use DPM-Solver++ (Lu et al., 2022b) to generate ground truth trajectories. The designer network  $g_{\phi}$  consists of a two-layer MLP with a total parameter count of only 9k. We use Adam as the optimizer with a learning rate of  $5\times 10^{-3}$ . To ensure fairness, we use the same random seed to evaluate the FID score. For details, see Appendix B.
+
+# 4.2 MAIN RESULTS
+
+We select the current state-of-the-art artificially designed solvers as baseline methods, including DEIS Zhang & Chen (2022), DPM-Solver++ (Lu et al., 2022b), and UniPC (Zhao et al., 2023). All results are obtained from an open-source toolbox<sup>1</sup>, utilizing the recommended settings from the original papers. Detailed results can be found in the Appendix C.
+
+![](images/aacd6518237e6851de6870526f770d55b42df321605b65c198e681948a92667f.jpg)  
+(a) CIFAR10  $32 \times 32$
+
+![](images/19e6d197fd190bb299c847cfe498c74c77375462939ecef3757e917a35a5142e.jpg)  
+(Pixel space, unconditional)  
+(d) LSUN-Bedroom  $256\times 256$  
+(Latent space, unconditional)  
+Figure 3: Comparison of FID↓ scores between DLMS and handcrafted solvers. DLMS demonstrates a significant advantage over handcrafted methods.
+
+![](images/e9c4e863b44603de9d61063dfd995cbae80505b2be247824e6212ff1ddab4041.jpg)  
+(b) FFHQ  $64 \times 64$
+
+![](images/6790b137d9619e1699b2f97541aa7cdf784729c13d5470a279bcc4c07cc888c6.jpg)  
+(Pixel space, unconditional)  
+(e) ImageNet  $256 \times 256$  
+(Pixel space, guidance scale 2.0)
+
+![](images/8aaba14084c53dbc5d5d38c55601aa0687008942d883bc28fa165a7970760eb4.jpg)  
+(c) ImageNet-64  $64 \times 64$
+
+![](images/a35ece0e3e01a5eeef2cfa4ebb931557ccaa1d3942a5d8cfe6de37a28e7bec54.jpg)  
+(Pixel space, conditional)  
+(f) MS-COCO  $512 \times 512$  
+(Latent space, guidance scale 7.5)
+
+Fig. 3 presents a comparison of FID scores between DLMS and previous handcrafted methods. As shown in Fig. 3, handcrafted solvers based on polynomial interpolation exhibit a sharp decline in performance as NFE decreases. In contrast, our proposed DLMS maintains high-quality sampling, demonstrating a significant advantage over baseline methods. In text-to-image generation with Stable-Diffusion (Rombach et al., 2022), DLMS achieves an FID of 12.52 with only 5 NFEs, while the baseline methods require more than 10 NFEs for comparable performance, resulting in a  $2 \times$  acceleration ratio.
+
+We further compare our method with other search frameworks. The methods include in the comparison are the reinforcement learning-based framework USF (Liu et al., 2023b), the parameterization-focused DPM-Solver-v3 (Zheng et al., 2023). Additionally, we include GITS (Chen et al., 2024), an outstanding adaptive time scheduling method, and AMED-Plugin (Zhou et al., 2024), a closely related work. As shown in Tab. 1, DLMS still stands out among various search frameworks.
+
+Table 1: Comparison of FID  $\downarrow$  on CIFAR10 between DLMS and search-based solvers.  
+
+<table><tr><td rowspan="2">Solver</td><td colspan="7">NFE</td></tr><tr><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td></tr><tr><td>DPM-Solver-v3</td><td>-</td><td>12.21</td><td>8.56</td><td>-</td><td>3.50</td><td>-</td><td>2.51</td></tr><tr><td>USF</td><td>11.50</td><td>6.86</td><td>5.18</td><td>3.81</td><td>3.41</td><td>3.02</td><td>2.69</td></tr><tr><td>AMED-Plugin</td><td>-</td><td>6.61</td><td>-</td><td>3.65</td><td>-</td><td>2.63</td><td>-</td></tr><tr><td>GITS</td><td>10.11</td><td>6.77</td><td>4.29</td><td>3.43</td><td>2.70</td><td>2.42</td><td>2.28</td></tr><tr><td>DLMS</td><td>4.52</td><td>3.23</td><td>2.81</td><td>2.53</td><td>2.43</td><td>2.37</td><td>2.24</td></tr></table>
+
+In Tab. 2, we compare the total GPU hours required for a 7 NFE solver with USF and DPM-Solverv3. Due to limitations in code availability, the reported time costs are sourced from original papers and measured on different devices. Nevertheless, our method demonstrates a significant order-of-magnitude advantage, achieving a  $10\times$  increase in search efficiency.
+
+Table 2: Comparison of total GPU hours for search phase. DLMS demonstrates a significant order-of-magnitude advantage.  
+
+<table><tr><td>Solver</td><td>CIFAR10</td><td>MS-COCO</td><td>Device</td></tr><tr><td>DPM-Solver-v3</td><td>28</td><td>88</td><td>NVIDIA A40</td></tr><tr><td>USF</td><td>12.15</td><td>106.64</td><td>NVIDIA 3090/A100</td></tr><tr><td>DLMS</td><td>0.7</td><td>10</td><td>NVIDIA V100</td></tr></table>
+
+# 4.3 ABLATION STUDY
+
+Tab. 3 demonstrates the ablation effects of each component in the DLMS framework. As shown in the results, the time step-related time schedule and time scaling contribute the most significant improvements.
+
+Table 3: FID results of ablation study on CIFAR10. The time step-related time schedule and time scaling contribute the most significant improvements.  
+
+<table><tr><td>NFE</td><td>4</td><td>6</td><td>8</td><td>10</td></tr><tr><td>DLMS</td><td>4.52</td><td>2.81</td><td>2.43</td><td>2.24</td></tr><tr><td>w/o AFS</td><td>6.48</td><td>3.30</td><td>2.42</td><td>2.30</td></tr><tr><td>w/o bottleneck feature</td><td>4.71</td><td>3.40</td><td>2.46</td><td>2.25</td></tr><tr><td>w/o high-order initialization</td><td>4.92</td><td>3.25</td><td>2.94</td><td>2.44</td></tr><tr><td>w/o Inception distance</td><td>6.67</td><td>3.77</td><td>3.10</td><td>2.80</td></tr><tr><td>w/o time scaling</td><td>7.75</td><td>3.86</td><td>3.07</td><td>2.41</td></tr><tr><td>w/o adaptive time schedule</td><td>10.41</td><td>6.18</td><td>3.17</td><td>3.03</td></tr><tr><td>Handcrafted (best)</td><td>25.66</td><td>9.40</td><td>3.99</td><td>2.89</td></tr></table>
+
+Another finding is that DLMS performance does not always improve as the number of interpolation time steps  $M$  increases. In fact, the fitting capacity of student solvers is limited. When the teacher solver is significantly superior to the student, side effects can arise due to fitting difficulties. Especially for models in the latent space, setting  $M = 1$ , meaning the teacher has about twice NFEs of the student, yields the best results. The table below illustrates the effects of increasing  $M$  on the LSUN-Bedroom dataset.
+
+Table 4: The sensitivity of  $M$  on LSUN-Bedroom. Setting  $M = 1$  yields the best results on latent diffusion models.  
+
+<table><tr><td>NFE</td><td>4</td><td>6</td><td>8</td><td>10</td></tr><tr><td>M=1</td><td>8.20</td><td>4.44</td><td>3.89</td><td>3.50</td></tr><tr><td>M=2</td><td>8.86</td><td>4.56</td><td>4.51</td><td>3.69</td></tr><tr><td>M=3</td><td>11.07</td><td>4.88</td><td>4.54</td><td>3.81</td></tr></table>
+
+# 4.4 VISUALIZATIONS
+
+Visual Quality. We present qualitative comparisons in Fig. 1. Handcrafted solvers struggle to generate vegetation and accurately colored bears with 5 NFEs. In contrast, DLMS effectively learns the generation results of the teacher solver with double NFEs. With 10 NFEs, DLMS is the only method that successfully produces the correct lighting and head pose. Additional samples are provided in Appendix D.
+
+Visualization of adaptive time schedule. Fig. 4 shows the adaptive time schedules (average of different trajectories) obtained from DLMS with 10 NFE using AFS. By comparing these schedules with handcrafted time schedules such as logSNR (Zheng et al., 2023; Lu et al., 2022b), polynomial (Karras et al., 2022), time uniform (Ho et al., 2020) and time square (Zhang & Chen, 2022), we uncover some intriguing findings. The adaptive time schedules exhibit striking differences across various datasets. For the CIFAR10 and FFHQ datasets, the adaptive time schedules are similar to the time square schedule. In contrast, for ImageNet-64 and LSUN-Bedroom, they resemble the time uniform schedule. Notably, the schedule learned for the Stable-Diffusion model aligns closely with the uniform logSNR schedule, except for the final step.
+
+The observed differences may arise from several factors, including the solving strategy, image resolution, solving space, and guidance. For instance, in the EDM context, DPM-Solver++ (Lu et al., 2022b) and UniPC (Zhao et al., 2023) perform better with logSNR, while DEIS (Zhang & Chen, 2022) demonstrates superior performance with the time square schedule. However, on other models, all three solvers tend to prefer the time uniform schedule. This highlights the importance of simultaneously searching for both time schedules and solving strategies within our framework across different datasets.
+
+![](images/b98b6c7f9612e6144471b19d577e3302a06571997bf4f6d2123be679dd693993.jpg)  
+(a) EDM
+
+![](images/dd8013a87075949740bfb343ecd0c7d619a4efec330379910aca8615dc1ada3f.jpg)  
+(b) Latent-Diffusion  
+Figure 4: Visualization of adaptive time schedule. We use dotted lines to represent handcrated time schedules, and use solid lines to represent adaptive time schedules learned on each data set. DLMS learns different adaptive schedules on different data sets.
+
+![](images/6df75ab421a4ceafe55bcac92f2e083db9e44424b15a62f4e056d93535224e64.jpg)  
+(c) Stable-Diffusion
+
+# 5 CONCLUSION
+
+We propose a linear multistep solver distillation framework. Our framework enables the student solver to replicate the sampling trajectory of a teacher solver that utilizes more steps, facilitating rapid adjustments and optimization of prediction coefficients, time step schedules, and time scaling factors. Experiments demonstrate the effectiveness of our framework across various resolution datasets, using both pixel-space and latent-space pre-trained diffusion models, and reveal a significant improvement in sample quality with  $4\sim 10$  NFEs.
+
+Limitations and Future Work. Our framework is currently limited to ODE solvers, while in practice, stochastic samplers (Xue et al., 2024) often outperform deterministic samplers. Therefore, extending our method to stochastic samplers is a promising direction. Additionally, integrating our work with approaches such as Deepcache (Ma et al., 2024) and FreeU (Si et al., 2024) is also worth exploring.
+
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+
+# A COMPARING WITH RELATED METHODS
+
+Comparing with Progressive Distillation. Progressive distillation (Salimans & Ho, 2022) is a distillation method for diffusion models that gradually distills the results of multi-step solvers into fewer steps. In our framework, we also employ a multi-step teacher solver and aim to distill the results into our student solver with fewer steps. However, our approach does not require training the parameters of the diffusion model, while the parameter requirements and training duration for progressive distillation (PD) are thousands of times greater than those of our framework.
+
+Comparing with AMED-Plugin. AMED-Plugin (Zhou et al., 2024) is a method for selecting intermediate time steps for existing solvers and time schedules. The designer network  $g_{\phi}$  used in our work is modified from AMED-Plugin. In contrast, the DLMS in this paper do not rely on existing solvers or time schedules. AMED-Plugin can be seen as adjusting half of the time schedule, while our proposed adaptive time schedule method is for full time schedule adjustment.
+
+Comparing with USF. USF (Liu et al., 2023b) is a search framework based on reinforcement learning. The authors train a predictor network to estimate the FID performance of a solver with specific hyperparameters, using this to guide an evolutionary search process for hyperparameter optimization. The relationship between hyperparameters and performance is complex, making the training of a predictor network both challenging and time-consuming.
+
+In contrast, our framework leverages local prediction error to design an optimal solution strategy, which aligns more closely with the iterative nature of the sampling process and the criteria for manually designing solvers. Furthermore, while USF simply combines existing handcrafted solvers, our approach utilizes prediction formulas (8, 9) to achieve a true unification of multiple solving strategies. This enables us to develop new solving strategies by learning the prediction coefficients  $\{a_k\}_{k=1}^p$ .
+
+Comparing with Bespoke solver. Bespoke solver (Shaul et al., 2024) is also a type of solver distillation method, but it only uses the final output to adjust hyperparameters, with a deep and complex computation graph that hinders effective hyperparameter tuning. We actively use intermediate values in the trajectory and judiciously stop unnecessary gradient backpropagation, greatly enhancing the effectiveness of solver distillation method. For EDM trained on CIFAR10, our method achieved an FID of 2.53 with 7 NFEs, outperforming their result of 2.75 with 20 NFEs.
+
+# B DETAILED SETTINGS
+
+EDM on CIFAR10, FFFQ, ImageNet-64. The sampling on CIFAR10 (Krizhevsky et al., 2009)  $32 \times 32$ , FFFQ (Karras et al., 2019)  $64 \times 64$ , ImageNet-64 (Deng et al., 2009)  $64 \times 64$  is based on the pretrained pixel-space diffusion model provided by EDM (Karras et al., 2022). Among these, ImageNet-64 is for conditional sampling and CIFAR10, FFFQ are for unconditional Sampling. The order  $p$  for student solver DLMS is set to 4. The number of interpolation time steps  $M$  is set to 4. For each dataset, we conduct solver distillation on  $20k$  trajectories. The distillation times are approximately  $40 \times NFE$  seconds,  $80 \times NFE$  seconds, and  $150 \times NFE$  seconds, respectively, on 8 NVIDIA V100 GPUs. We measure sample quality using the FID score calculated on  $50k$  generated images.
+
+Latent-Diffusion on LSUN-Bedroom. The unconditional sampling on LSUN-Bedroom (Yu et al., 2015)  $256 \times 256$ , is based on the pretrained latent-space diffusion model provided by Latent-Diffusion (Rombach et al., 2022). The order  $p$  for student solver DLMS is set to 3. The number of interpolation time steps  $M$  is set to 1. We conduct solver distillation on 10k trajectories. The distillation times are approximately  $3 \times \mathrm{NFE}$  mins on 8 NVIDIA V100 GPUs. We measure sample quality using the FID score calculated on 50k generated images.
+
+Guided-Diffusion on ImageNet. The conditional sampling on ImageNet (Deng et al., 2009)  $256 \times 256$ , is based on the pretrained pixel-space diffusion model provided by Guided-Diffusion (Dhariwal & Nichol, 2021). The order  $p$  for student solver DLMS is set to 3. The number of interpolation time steps  $M$  is set to 3. We conduct solver distillation on 5k trajectories with default guidance scale 2.0. The distillation times are approximately  $6 \times \mathrm{NFE}$  mins on 8 NVIDIA V100 GPUs. We measure sample quality using the FID score calculated on 10k generated images.
+
+Stable-Diffusion on MS-COCO prompts. The text-to-image sampling on MS-COCO (2014) (Lin et al., 2014)  $512 \times 512$ , is based on the pretrained latent-space diffusion model provided by StableDiffusion v1.5 (Rombach et al., 2022). The order  $p$  for student solver DLMS is set to 2. The number of interpolation time steps  $M$  is set to 1. We conduct solver distillation on  $5\mathrm{k}$  trajectories with default guidance scale 7.5. The distillation times are approximately  $9 \times \mathrm{NFE}$  mins on 8 NVIDIA V100 GPUs. We measure sample quality using the FID score calculated on  $30\mathrm{k}$  generated images generated by  $30\mathrm{k}$  prompts from the MS-COCO validation set.
+
+# C FID RESULTS FOR DLMS AND HANDCRAFTED SOLVERS.
+
+Table 5: Comparison of  $\mathrm{{FID}} \downarrow$  scores between DLMS and handcrafted solvers.  
+
+<table><tr><td rowspan="2">Dataset</td><td rowspan="2">Solver</td><td colspan="7">NFE</td></tr><tr><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td></tr><tr><td rowspan="4">CIFAR10</td><td>DEIS</td><td>25.66</td><td>14.39</td><td>9.40</td><td>6.94</td><td>5.55</td><td>4.68</td><td>4.09</td></tr><tr><td>DPM-Solver++</td><td>46.52</td><td>24.97</td><td>11.99</td><td>6.74</td><td>4.54</td><td>3.42</td><td>3.00</td></tr><tr><td>UniPC</td><td>45.20</td><td>23.98</td><td>11.14</td><td>5.83</td><td>3.99</td><td>3.21</td><td>2.89</td></tr><tr><td>DLMS</td><td>4.52</td><td>3.23</td><td>2.81</td><td>2.53</td><td>2.43</td><td>2.37</td><td>2.24</td></tr><tr><td rowspan="4">FFHQ</td><td>DEIS</td><td>28.31</td><td>17.36</td><td>12.25</td><td>9.37</td><td>7.59</td><td>6.39</td><td>5.56</td></tr><tr><td>DPM-Solver++</td><td>45.95</td><td>22.51</td><td>13.74</td><td>8.44</td><td>6.04</td><td>4.77</td><td>4.12</td></tr><tr><td>UniPC</td><td>44.78</td><td>21.40</td><td>12.85</td><td>7.44</td><td>5.50</td><td>4.47</td><td>3.84</td></tr><tr><td>DLMS</td><td>9.63</td><td>6.85</td><td>5.82</td><td>5.16</td><td>4.81</td><td>4.23</td><td>4.12</td></tr><tr><td rowspan="4">ImageNet-64</td><td>DEIS</td><td>23.53</td><td>14.75</td><td>12.57</td><td>8.20</td><td>6.84</td><td>5.97</td><td>5.34</td></tr><tr><td>DPM-Solver++</td><td>56.63</td><td>25.49</td><td>15.06</td><td>10.14</td><td>7.84</td><td>6.48</td><td>5.67</td></tr><tr><td>UniPC</td><td>55.63</td><td>24.36</td><td>14.30</td><td>9.57</td><td>7.52</td><td>6.34</td><td>5.53</td></tr><tr><td>DLMS</td><td>10.07</td><td>7.16</td><td>7.08</td><td>6.31</td><td>5.93</td><td>4.57</td><td>4.30</td></tr><tr><td rowspan="4">LSUN-Bedroom</td><td>DEIS</td><td>12.89</td><td>6.83</td><td>5.35</td><td>4.78</td><td>4.43</td><td>4.25</td><td>4.05</td></tr><tr><td>DPM-Solver++</td><td>48.49</td><td>18.44</td><td>8.39</td><td>5.18</td><td>4.12</td><td>3.77</td><td>3.60</td></tr><tr><td>UniPC</td><td>39.66</td><td>13.76</td><td>6.46</td><td>4.52</td><td>3.96</td><td>3.72</td><td>3.56</td></tr><tr><td>DLMS</td><td>8.20</td><td>5.44</td><td>4.44</td><td>3.99</td><td>3.89</td><td>3.70</td><td>3.50</td></tr><tr><td rowspan="3">ImageNet</td><td>DPM-Solver++</td><td>26.30</td><td>17.08</td><td>13.06</td><td>11.08</td><td>9.95</td><td>9.25</td><td>8.72</td></tr><tr><td>UniPC</td><td>24.99</td><td>15.71</td><td>11.95</td><td>10.24</td><td>9.36</td><td>8.82</td><td>8.45</td></tr><tr><td>DLMS</td><td>19.74</td><td>13.83</td><td>10.90</td><td>9.66</td><td>9.07</td><td>8.89</td><td>8.22</td></tr><tr><td rowspan="4">MS-COCO</td><td>DEIS</td><td>-</td><td>15.16</td><td>13.95</td><td>13.61</td><td>13.58</td><td>13.46</td><td>13.38</td></tr><tr><td>DPM-Solver++</td><td>-</td><td>17.40</td><td>15.78</td><td>15.08</td><td>14.77</td><td>14.43</td><td>14.18</td></tr><tr><td>UniPC</td><td>-</td><td>17.33</td><td>15.70</td><td>15.06</td><td>14.74</td><td>14.44</td><td>14.28</td></tr><tr><td>DLMS</td><td>-</td><td>12.52</td><td>11.87</td><td>11.65</td><td>11.63</td><td>11.52</td><td>11.45</td></tr></table>
+
+# D MORE QUALITATIVE RESULTS
+
+![](images/6d67627f8e126af4716afa0a000c8a7f677038b328eea2f5576350702a820e3c.jpg)  
+(a) DEIS, FID=25.66  
+Figure 5: Uncurated samples on CIFAR10  $32 \times 32$  with 4 NFE.
+
+![](images/880b60eabebb0ff145749d9b4223c87f8b669a9f365db957f295aafbabfda771.jpg)  
+(b) DLMS, FID=4.52
+
+![](images/a796794108853796ef0053bd6187030bfe4217c4d8b37f7b30dafe4a9d03f669.jpg)  
+(a) DEIS, FID=28.31  
+Figure 6: Uncurated samples on FFHQ  $64 \times 64$  with 4 NFE.
+
+![](images/4919ba774d994d53808be0243bf4cf03549b9bf58ad0377e81fe8a7e351eda06.jpg)  
+(b) DLMS, FID=9.63
+
+![](images/e3613fa6f771e0db363de269e0da5d0470ce77660088155098fe01a971041b04.jpg)  
+(a) DEIS,  $\mathrm{FID} = 14.75$  
+Figure 7: Uncurated samples on ImageNet-64  $64 \times 64$  with 5 NFE.
+
+![](images/2e193f69839077b3554b667b9244cff758f034a40b6d26c424b49fcaf2ec6fb8.jpg)  
+(b) DLMS, FID=7.16
+
+![](images/8803c729d559408e12eea429cbfccf3ac591061f1675cb0070386150e2434617.jpg)  
+(a) DEIS, FID=12.89  
+Figure 8: Uncurated samples on LSUN-Bedroom  $256 \times 256$  with 4 NFE.
+
+![](images/780546cbcf97ac13b5537f98d0b8dceef8343ead85e53b6496fd15cf433f5d7a.jpg)  
+(b) DLMS, FID=8.20
+
+Table 6: Additional samples of Stable-Diffusion (Rombach et al., 2022) with a classifier-free guidance scale 7.5, using only 10 NFE and selected text prompts.  
+
+<table><tr><td>Text Prompts</td><td>DPM-Solver++ (FID=14.18)</td><td>DEIS (FID=13.38)</td><td>DLMS (FID=11.45)</td></tr><tr><td>“a sandwich on wheat bread sits on a plate”</td><td></td><td></td><td></td></tr><tr><td>“three big elephants walking across a wide river”</td><td></td><td></td><td></td></tr><tr><td>“Air force jet in a take off position above the tree line.”</td><td></td><td></td><td></td></tr><tr><td>“An empty bench next to a potted tree up against a brick wall.”</td><td></td><td></td><td></td></tr><tr><td>“A star hangs upon a canopied bed in a bedroom.”</td><td></td><td></td><td></td></tr></table>
\ No newline at end of file
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+# LINEAR PARTIAL GROMOV-WASSERSTEIN EMBEDDING
+
+Yikun Bai $^{1}$ , Abihith Kothapalli $^{*,1}$ , Hengrong Du $^{*,2}$
+
+Rocio Diaz Martin\*3, and Soheil Kolouri
+
+$^{1}$ Department of Computer Science, Vanderbilt University
+
+$^{2}$ Department of Mathematics, University of California, Irvine
+
+$^{3}$ Department of Mathematics, Tufts University
+
+$^{1}$ yikun.bai, abi.kothapalli, soheil.kolouri@vanderbilt.edu
+
+2 hengrond@uci.edu
+
+<sup>3</sup>rocio.diaz_martin@tufts.edu
+
+# ABSTRACT
+
+The Gromov-Wasserstein (GW) problem, a variant of the classical optimal transport (OT) problem, has attracted growing interest in the machine learning and data science communities due to its ability to quantify similarity between measures in different metric spaces. However, like the classical OT problem, GW imposes an equal mass constraint between measures, which restricts its application in many machine learning tasks. To address this limitation, the partial Gromov-Wasserstein (PGW) problem has been introduced. It relaxes the equal mass constraint, allowing the comparison of general positive Radon measures. Despite this, both GW and PGW face significant computational challenges due to their non-convex nature. To overcome these challenges, we propose the linear partial Gromov-Wasserstein (LPGW) embedding, a linearized embedding technique for the PGW problem. For  $K$  different metric measure spaces, the pairwise computation of the PGW distance requires solving the PGW problem  $\mathcal{O}(K^2)$  times. In contrast, the proposed linearization technique reduces this to  $\mathcal{O}(K)$  times. Similar to the linearization technique for the classical OT problem, we prove that LPGW defines a valid metric for metric measure spaces. Finally, we demonstrate the effectiveness of LPGW in practical applications such as shape retrieval and learning with transport-based embeddings, showing that LPGW preserves the advantages of PGW in partial matching while significantly enhancing computational efficiency. The code is available at https://github.commint-vu/Linearized_Partial_Gromov_Wasserstein.
+
+# 1 INTRODUCTION
+
+Optimal transport (OT), unbalanced and partial OT, and linearized OT. The core objective of this work, and a general challenge in machine learning, is to match objects and devise practical, computationally efficient notions of similarity between shapes. One of the most well-known techniques for this task arises from optimal transport (OT) theory, which has found extensive applications in generative modeling (Arjovsky et al., 2017; Genevay et al., 2016), domain adaptation (Courty et al., 2016; Balaji et al., 2020), representation learning (Wu et al., 2023), and many other domains. The classical OT problem, as detailed by (Villani, 2009), involves matching two probability distributions to minimize the expected transportation cost, giving rise to the so-called Wasserstein distances. Although the classical OT problem imposes a mass conservation requirement, recent advances have extended the utility of OT beyond probability measures, allowing for comparisons of non-negative measures with different total masses and partial matching of sets through the unbalanced and partial OT frameworks (Chizat et al., 2018a;b; Fatras et al., 2021; Séjourné et al., 2021). Additionally, in large-scale machine learning applications, since the OT approach is computationally expensive, it has motivated numerous approximations that lead to significant speedups, such as those in (Cuturi, 2013; Chizat et al., 2020; Scuton & Cuturi,
+
+2022), and the linearization technique, termed linear optimal transportation (LOT), proposed by (Wang et al., 2013). There have also been advances in reducing the computational complexity of comparing unbalanced measures (Cai et al., 2022; Bai et al., 2023).
+
+The Gromov-Wasserstein (GW) distance, unbalanced and partial GW, and linearized GW. Another line of research has focused on the comparison of probability measures across different metric spaces using Gromov-Wasserstein (GW) distances (Mémoli, 2011). These innovations have applications in areas ranging from quantum chemistry (Peyré et al., 2016) to natural language processing (Alvarez-Melis & Jaakkola, 2018), enhancing fundamental tasks such as clustering (Chowdhury & Needham, 2021), dimensionality reduction (Van Assel et al., 2024), shape correspondence (Kong et al., 2024), shape analysis (Mémoli & Needham, 2022), and inference-based simulation (Hur et al., 2024). The GW distance is a valuable tool across a variety of domains, as it allows for comparisons between distributions while being independent of the ambient spaces. To extend GW to unbalanced settings, GW-like comparisons have been developed for metric measure spaces with differing total masses (Liu et al., 2020; Chapel et al., 2020; Séjourné et al., 2021; Bai et al., 2024; Zhang et al., 2022). Furthermore, to address the computational expense of pairwise GW distance calculations in large-scale applications, a linearized version of GW has been proposed in (Beier et al., 2022). Similar to the LOT framework (Wang et al., 2013), this approach is based on barycentric projection maps of transport plans.
+
+# Contributions.
+
+- We propose the linear Partial Gromov-Wasserstein embedding (LPGW) technique, which extends the classical LOT embedding technique to the partial GW setting.  
+- Based on the embedding, we propose the so-called LPGW distance, which we prove to be a metric under certain assumptions. Given  $K$  different metric measure spaces, computing their pairwise PGW distances requires solving the PGW problem  $\mathcal{O}(K^2)$  times. However, computing the LPGW distance pairwise requires only  $\mathcal{O}(K)$  distance computations.  
+- Numerically, we test our proposed LPGW embedding and LPGW distance in two experiments: shape retrieval and learning with transform-based embeddings. In both experiments, we observe that the LPGW-based approach can preserve the partial matching property of PGW while significantly improving computational efficiency.
+
+# 2 BACKGROUND
+
+# 2.1 THE WASSERSTEIN DISTANCE
+
+Let  $\mathcal{P}(\mathbb{R}^d)$  be the space of probability measures on the Borel  $\sigma$ -algebra on  $\mathbb{R}^d$ , and let  $\mathcal{P}_2(\mathbb{R}^d) := \{\mu \in \mathcal{P}(\mathbb{R}^d) : \int_{\mathbb{R}^d} |x|^2 d\mu(x) < \infty\}$ . Given a (measurable) mapping  $T: \mathbb{R}^d \to \mathbb{R}^d$ , the pushforward measure  $T_\# \mu$  is defined as  $T_\# \mu(B) := \mu(T^{-1}(B))$  for all Borel sets  $B \subseteq \mathbb{R}^d$ , where  $T^{-1}(B) := \{x : T(x) \in B\}$  is the preimage of  $B$  under  $T$ .
+
+Given  $\mu, \nu \in \mathcal{P}_2(\mathbb{R}^d)$ , the 2-Wasserstein distance is defined as
+
+$$
+W _ {2} ^ {2} (\mu , \nu) := \min  _ {\gamma \in \Gamma (\mu , \nu)} \int_ {\mathbb {R} ^ {d} \times \mathbb {R} ^ {d}} \| x - y \| ^ {2} d \gamma (x, y) \tag {1}
+$$
+
+where  $\Gamma(\mu, \nu)$  is a set of joint probability measures on  $\mathbb{R}^d \times \mathbb{R}^d$  whose first and second marginals are  $\mu$  and  $\nu$ , respectively. The classical OT theory (Villani, 2021; 2009) guarantees that a minimizer to (1) always exists, so such a formulation is well-defined. In fact, there might exist multiple minimizers, and we denote by  $\Gamma^{*}(\mu, \nu)$  the set of all optimal transportation plans for problem (1).
+
+When an optimal transportation plan  $\gamma$  is induced by a mapping  $T:\mathbb{R}^d\to \mathbb{R}^d$ , that is,
+
+$$
+\gamma = (\operatorname {i d} \times T) _ {\#} \mu \in \Gamma^ {*} (\mu , \nu), \quad \text {w i t h} \quad T _ {\#} \mu = \nu , \tag {2}
+$$
+
+we say that the Monge mapping assumption holds, and the function  $T$  is called a Monge mapping. In particular, by Brenier's theorem, when  $\mu$  is a continuous measure (i.e.,
+
+absolutely continuous with respect to the Lebesgue measure on  $\mathbb{R}^d$ , a Monge mapping  $T$  always exists and it is unique.
+
+We refer to Appendix A for a detailed introduction to the LOT formulation.
+
+# 2.2 THE GROMOV-WASSERSTEIN DISTANCE
+
+Consider two compact gauged measure spaces (gm-spaces)  $\mathbb{X} = (X,g_X,\mu)$  and  $\mathbb{Y} = (Y,g_Y,\nu)$ , where  $X\subset \mathbb{R}^{d_x},Y\subset \mathbb{R}^{d_y}$  are non-empty (compact) convex sets (with  $d_{x},d_{y}\in \mathbb{N}$ );  $g_{X}:X^{\otimes 2}\to \mathbb{R}$  is a gauge function in  $L_{sys}^{2}(X^{\otimes 2},\mu^{\otimes 2})$ , i.e., the space of symmetric, real-valued functions on  $X^{\otimes 2}\coloneqq X\times X$  that are square-integrable with respect to  $\mu^{\otimes 2}\coloneqq \mu \otimes \mu$  (similarly for  $g_{Y}\in L_{sys}^{2}(Y^{\otimes 2},\nu^{\otimes 2})$ ; and  $\mu ,\nu$  are probability measures defined on  $X,Y$  respectively. In addition, assume that  $\mathbb{X}$  and  $\mathbb{Y}$  are metric measure spaces (mm-spaces); that is, there exist distances  $d_{X}$  and  $d_{Y}$  on  $X$  and  $Y$ , respectively. In general, we will assume that  $g_{X}$  is a continuous function on  $X^{\otimes 2}$  with respect to the metric  $d_{X^{\otimes 2}}((x_1,x_1'),(x_2,x_2'))\coloneqq d_X(x_1,x_2) + d_X(x_1',x_2')$  on  $X^{\otimes 2}$  (respectively, for  $g_{Y}$ ). Thus, since  $X$  is compact,  $g_{X}$  (respectively,  $g_{Y}$ ) is a Lipschitz continuous function, i.e., there exists  $K\in \mathbb{R}_{>0}$  such that
+
+$$
+\left| g _ {X} \left(x _ {1}, x _ {1} ^ {\prime}\right) - g _ {X} \left(x _ {2}, x _ {2} ^ {\prime}\right) \right| \leq K \left(d _ {X} \left(x _ {1}, x _ {2}\right) + d _ {X} \left(x _ {1} ^ {\prime}, x _ {2} ^ {\prime}\right)\right), \quad \forall x _ {1}, x _ {1} ^ {\prime}, x _ {2}, x _ {2} ^ {\prime} \in X. \tag {3}
+$$
+
+For the reader's convenience, unless stated otherwise, we default to  $g_{X}(x_{1},x_{2}) = d_{X}^{2}(x_{1},x_{2})$  (which is the classical setting for the Gromov-Wasserstein problem (Alvarez-Melis & Jaakkola, 2018; Vayer, 2020)), or to the inner product  $g_{X}(x_{1},x_{2}) = x_{1}^{\top}x_{2}$  (which has been studied in (Vayer, 2020; Sturm, 2012)). The same applies to  $g_{Y}$ .
+
+The Gromov-Wasserstein (GW) problem between  $\mathbb{X}$  and  $\mathbb{Y}$  is defined as
+
+$$
+G W ^ {2} (\mathbb {X}, \mathbb {Y}) := \min  _ {\gamma \in \Gamma (\mu , \nu)} \left\langle | g _ {X} - g _ {Y} | ^ {2}, \gamma^ {\otimes 2} \right\rangle \tag {4}
+$$
+
+where
+
+$$
+\langle | g _ {X} - g _ {Y} | ^ {2}, \gamma^ {\otimes 2} \rangle := \int_ {(X \times Y) ^ {\otimes 2}} | g _ {X} (x, x ^ {\prime}) - g _ {Y} (y, y ^ {\prime}) | ^ {2} d \gamma (x, y) d \gamma (x ^ {\prime}, y ^ {\prime}).
+$$
+
+A minimizer for (4) always exists (Mémoli, 2011), and we use  $\Gamma^{*}(\mathbb{X},\mathbb{Y})$  to denote the set of all optimal transportation plans  $\gamma \in \Gamma (\mu ,\nu)$  for the GW problem defined in (4). In addition, when  $g_{X} = d_{X}^{s}$  and  $g_{Y} = d_{Y}^{s}$  for  $s\geq 1$ ,  $GW(\cdot ,\cdot)$  defines a metric on the space of mm-spaces up to isomorphism<sup>1</sup>.
+
+Similar to classical OT theory, we say that the GW-Monge mapping assumption holds if there exists a minimizer  $\gamma$  for (4) of the form  $\gamma = (\mathrm{id} \times T)_{\#} \mu$ , where  $\mathrm{id}: X \to X$  is the identity map on  $X$ , and  $T: X \to Y$  is a measurable function that pushes forward  $\mu$  to  $\nu$ . In recent years, the existence of Monge mappings for the GW problem has been studied in works such as (Vayer, 2020; Maron & Lipman, 2018; Dumont et al., 2024).
+
+Note that the GW minimization problem defined in (4) is quadratic but non-convex. The computational cost of  $GW(\cdot, \cdot)$  for discrete measures  $\mu, \nu$  of size  $n$  is  $\mathcal{O}(n^3 \cdot L)$  to obtain a solution that is a local minimum, where  $n^3$  is the computational cost of one iteration and  $L$  is the required number of iterations $^2$ . Thus, when faced with the task of computing the pairwise GW distances between discrete gm-spaces  $\mathbb{Y}^1, \ldots, \mathbb{Y}^K$  with  $n$  points each, the resulting complexity is  $\mathcal{O}(K^2 \cdot n^3 \cdot L)$ . To reduce the complexity of computing the pairwise distances in this setting, (Beier et al., 2022) introduces a linear embedding technique called linear Gromov-Wasserstein (LGW), which we discuss in the following section.
+
+# 2.3 THE LINEAR GROMOV-WASSERSTEIN EMBEDDING AND DISTANCE
+
+Consider a fixed gm-space  $\mathbb{X} = (X,g_X,\mu)$ , which serves as a reference space. Given a target gm-space  $\mathbb{Y} = (Y,g_Y,\nu)$ , suppose that the GW-Monge mapping assumption holds, i.e., there exists an optimal transportation plan  $\gamma = (\mathrm{id}\times T)_{\#}\mu$  for the GW problem
+
+(4) between  $\mathbb{X}$  and  $\mathbb{Y}$ . Then, the linear Gromov-Wasserstein (LGW) embedding is defined as the following mapping
+
+$$
+\mathbb {Y} \mapsto k _ {\gamma} := g _ {X} (\cdot_ {1}, \cdot_ {2}) - g _ {Y} (T (\cdot_ {1}), T (\cdot_ {2})), \tag {5}
+$$
+
+which assigns each gm-space  $\mathbb{Y}$  to a gauge function  $k_{\gamma}:X\times X\to \mathbb{R}$  defined by (5). This embedding can recover the GW distance  $GW(\mathbb{X},\mathbb{Y})$  by the following:
+
+$$
+G W (\mathbb {X}, \mathbb {Y}) = \int_ {X \times X} | k _ {\gamma} (x, x ^ {\prime}) | ^ {2} d \mu (x) d \mu (x ^ {\prime}).
+$$
+
+Given two gm-spaces  $\mathbb{Y}^1 = \{Y^1,g_{Y^1},\nu^1\} ,\mathbb{Y}^2 = \{Y^2,g_{Y^2},\nu^2\}$ , assume that there exist optimal transportation plans  $\gamma^{1} = (\mathrm{id}\times T^{1})_{\#}\mu \in \Gamma^{*}(\mathbb{X},\mathbb{Y}^{1}),\gamma^{2} = (\mathrm{id}\times T^{2})_{\#}\mu \in \Gamma^{*}(\mathbb{X},\mathbb{Y}^{2})$ . Let  $k_{\gamma^1},k_{\gamma^2}$  denote the corresponding linear GW embeddings as defined by (5). Then, the LGW distance conditioned on  $\gamma^1,\gamma^2$  is defined as
+
+$$
+L G W \left(\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}\right) := \left\| k _ {\gamma^ {1}} - k _ {\gamma^ {2}} \right\| _ {L ^ {2} \left(\mu^ {\otimes 2}\right)}. \tag {6}
+$$
+
+When the GW-Monge mapping assumption does not hold, (Beier et al., 2022) extend the above distance conditioned on a pair of arbitrary plans  $\gamma^1 \in \Gamma^*(\mathbb{X}, \mathbb{Y}^1)$ ,  $\gamma^2 \in \Gamma^*(\mathbb{X}, \mathbb{Y}^2)$ :
+
+$$
+\begin{array}{l} L G W \left(\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}\right) = \inf  _ {\gamma \in \Gamma \left(\gamma^ {1}, \gamma^ {2}; \mu\right)} \left\langle \left| g _ {Y ^ {1}} - g _ {Y ^ {2}} \right| ^ {2}, \gamma^ {\otimes 2} \right\rangle \tag {7} \\ = \inf _ {\gamma \in \Gamma (\gamma^ {1}, \gamma^ {2}; \mu)} \int_ {(X \times Y ^ {1} \times Y ^ {2}) \otimes 2} | g _ {Y ^ {1}} (y ^ {1}, y ^ {1 \prime}) - g _ {Y ^ {2}} (y ^ {2}, y ^ {2 \prime}) | ^ {2} d \gamma (x, y ^ {1}, y ^ {2}) \gamma (x ^ {\prime}, y ^ {1 \prime}, y ^ {2 \prime}), \\ \end{array}
+$$
+
+where
+
+$$
+\Gamma (\gamma^ {1}, \gamma^ {2}; \mu) := \left\{\gamma \in \mathcal {M} _ {+} (X \times Y ^ {1} \times Y ^ {2}): (\pi_ {X, Y ^ {1}}) _ {\#} \gamma = \gamma^ {1}, (\pi_ {X, Y ^ {2}}) _ {\#} \gamma = \gamma^ {2}, (\pi_ {X}) _ {\#} \gamma = \mu \right\},
+$$
+
+for  $\pi_{X,Y^i}(x,y^1,y^2)\coloneqq (x,y^i)$  for  $i = 1,2$ , and  $\pi_X(x,y^1,y^2)\coloneqq x$ . Finally, the LGW distance between  $\mathbb{Y}^1$  and  $\mathbb{Y}^2$  is defined as
+
+$$
+L G W (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}) = \inf  _ {\substack {\gamma^ {1} \in \Gamma^ {*} (\mu , \nu^ {1}) \\ \gamma^ {2} \in \Gamma^ {*} (\mu , \nu^ {2})}} L G W (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}). \tag{8}
+$$
+
+We refer to Appendix B for further details.
+
+To simplify the computation of the above formulation, (Beier et al., 2022) propose the barycentric projection method, which we will briefly review here.
+
+Let  $\gamma \in \Gamma(\mu, \nu)$  be a transportation plan between  $\mu \in \mathcal{P}(X)$  and  $\nu \in \mathcal{P}(Y)$ . By using [Definition 5.4.2, Ambrosio et al. (2005)], the barycentric projection map  $\mathcal{T}_{\gamma}: X \to Y$  is defined as
+
+$$
+\mathcal {T} _ {\gamma} (x) := \int_ {Y} y d \gamma_ {Y | X} (y | x). \tag {9}
+$$
+
+where  $\{\gamma_{Y|X}(\cdot|x)\}_{x\in X}$  is the corresponding disintegration of  $\gamma$  with respect to its first marginal  $\mu$ . That is,  $\{\gamma_{Y|X}(\cdot|x)\}_{x\in X}$  is a family of measures satisfying
+
+$$
+\int_ {X \times Y} \phi (x, y) d \gamma (x, y) = \int_ {X} \int_ {Y} \phi (x, y) d \gamma_ {Y | X} (y | x) d \mu (x), \quad \forall \phi \in C (X \times Y).
+$$
+
+If  $\gamma^{*}\in \Gamma^{*}(\mathbb{X},\mathbb{Y})$ , we define the measure  $\widetilde{\nu}_{\gamma^{*}}:= (\mathcal{T}_{\gamma^{*}})_{\#}\mu \in \mathcal{P}(Y)$  and the gm-space  $\widetilde{\mathbb{Y}}_{\gamma^{*}}:= (Y,g_{Y},\nu_{\gamma^{*}})$ . We provide the following results as a complement to the work in (Beier et al., 2022).
+
+Proposition 2.1. Let  $\gamma^{*}\in \Gamma^{*}(\mathbb{X},\mathbb{Y})$  and  $\gamma^0\in \Gamma^* (\mathbb{X},\mathbb{X})$ . Then, we have the following:
+
+(1) The LGW embedding of  $\mathbb{Y}$  can recover the  $GW$  distance, i.e.,
+
+$$
+L G W (\mathbb {X}, \mathbb {Y}; \mathbb {X}, \gamma^ {0}, \gamma^ {*}) = G W (\mathbb {X}, \mathbb {Y}).
+$$
+
+(2) If  $T$  is a Monge mapping for  $GW(\mathbb{X},\mathbb{Y})$  such that  $\gamma^{*} = (\mathrm{id}\times T)_{\#}\mu$ , then  $\mathcal{T}_{\gamma^*} = T$ ,  $\mu$ -a.e. As a consequence,  $\widetilde{\nu}_{\gamma^{*}} = \nu$ , and thus  $\widetilde{\mathbb{Y}}_{\gamma^{*}} = \mathbb{Y}$ .  
+(3) If  $g_{Y}(\cdot_{1},\cdot_{2}) = \alpha \langle \cdot_{1},\cdot_{2}\rangle_{Y}$ , where  $\langle \cdot_{1},\cdot_{2}\rangle_{Y}$  is an inner product restricted to  $Y$  and  $\alpha \in \mathbb{R}$ , then the barycentric projection map  $\mathcal{T}_{\gamma^{*}}$  defined by (9) is a Monge mapping in the sense that  $\widetilde{\gamma} = (\mathrm{id} \times \mathcal{T}_{\gamma^{*}})_{\#} \mu$  is optimal for the GW problem  $GW(\mathbb{X},\widetilde{\mathbb{Y}}_{\gamma^{*}})$ .
+
+We refer the reader to Appendix E for the proof.
+
+Based on the above proposition, given  $\mathbb{Y}^1 = (Y^1,g_{Y^1},\nu^1)$  and  $\mathbb{Y}^2 = (Y^2,g_{Y^2},\nu^2)$ , let  $\gamma^{1}\in \Gamma^{*}(\mathbb{X},\mathbb{Y}^{1})$  and  $\gamma^2\in \Gamma^* (\mathbb{X},\mathbb{Y}^2)$ . The following approximated LGW (aLGW) distance can be used as a proxy for the LGW and GW distances. Furthermore, in practice, we can fix a single pair  $(\gamma^1,\gamma^2)$  to compute the distance rather than compute the infimum:
+
+$$
+\begin{array}{l} aLGW(\mathbb{Y}^{1},\mathbb{Y}^{2};\mathbb{X}):= \inf_{\substack{\gamma^{1}\in \Gamma^{*}(\mathbb{X},\mathbb{Y}^{1})\\ \gamma^{2}\in \Gamma^{*}(\mathbb{X},\mathbb{Y}^{2})}}LGW(\mathbb{Y}_{\gamma^{1}}^{1},\mathbb{Y}_{\gamma^{2}}^{2};\mathbb{X},\gamma^{1},\gamma^{2}) \\ = \inf  _ {\substack {\gamma^ {1} \in \Gamma^ {*} (\mathbb {X}, \mathbb {Y} ^ {1}) \\ \gamma^ {2} \in \Gamma^ {*} (\mathbb {X}, \mathbb {Y} ^ {2})}} \| g _ {Y ^ {1}} \left(\mathcal {T} _ {\gamma^ {1}} (\cdot_ {1}), \mathcal {T} _ {\gamma^ {1}} (\cdot_ {2})\right) - g _ {Y ^ {2}} \left(\mathcal {T} _ {\gamma^ {2}} (\cdot_ {1}), \mathcal {T} _ {\gamma^ {2}} (\cdot_ {2})\right) \| _ {L ^ {2} (\mu^ {\otimes 2})} ^ {2} (10) \\ \approx \left\| g _ {Y ^ {1}} \left(\mathcal {T} _ {\gamma^ {1}} (\cdot_ {1}), \mathcal {T} _ {\gamma^ {1}} (\cdot_ {2})\right) - g _ {Y ^ {2}} \left(\mathcal {T} _ {\gamma^ {2}} (\cdot_ {1}), \mathcal {T} _ {\gamma^ {2}} (\cdot_ {2})\right) \right\| _ {L ^ {2} (\mu^ {\otimes 2})} ^ {2}. (11) \\ \end{array}
+$$
+
+# 2.4 THE PARTIAL GROMOW-WASSERSTEIN DISTANCE
+
+Partial Gromov-Wasserstein (PGW), or more generally, unbalanced GW, relaxes the assumption that  $\mu, \nu$  are normalized probability measures (Chapel et al., 2020; Séjourné et al., 2021; Bai et al., 2024). In particular, the PGW distance is defined as
+
+$$
+P G W _ {\lambda} ^ {2} (\mathbb {X}, \mathbb {Y}) := \inf  _ {\gamma \in \Gamma_ {\leq} (\mu , \nu)} \left\langle | g _ {X} - g _ {Y} | ^ {2}, \gamma^ {\otimes 2} \right\rangle + \lambda \left(| \mu^ {\otimes 2} - \gamma_ {X} ^ {\otimes 2} | + | \nu^ {\otimes 2} - \gamma_ {Y} ^ {\otimes 2} |\right), \tag {12}
+$$
+
+where  $\lambda > 0$  is a parameter;  $\gamma_{X} \coloneqq (\pi_{X})_{\#}\gamma, \gamma_{Y} \coloneqq (\pi_{Y})_{\#}\gamma$  are the corresponding marginals of  $\gamma$  on  $X$  and  $Y$ , respectively; and  $\Gamma_{\leq}(\mu, \nu) \coloneqq \{\gamma \in \mathcal{M}_{+}(X \times Y) : \gamma_{X} \leq \mu, \gamma_{Y} \leq \nu\}$ , where  $\mathcal{M}_{+}(X \times Y)$  denotes the set of finite non-negative measures on  $X \times Y$ , and  $\gamma_{X} \leq \mu$  denotes that for any Borel set  $A$ ,  $\gamma_{X}(A) \leq \mu(A)$  (similarly for  $\gamma_{Y} \leq \nu$ ).
+
+In the discrete setting, the PGW problem can be solved by variants of the Frank-Wolfe algorithm (see the algorithms in (Bai et al., 2024) for details). From a theoretical perspective, there always exists a minimizer for the optimization problem in (12) (see [Proposition 3.3, (Bai et al., 2024)]). We use the notation  $\Gamma_{\leq,\lambda}^{*}(\mathbb{X},\mathbb{Y})$  to denote the set of all optimal transportation plans for  $PGW_{\lambda}(\mathbb{X},\mathbb{Y})$ . For convenience, when  $\lambda$  is fixed, we may omit the subscript  $\lambda$  and write  $\Gamma_{\leq}^{*}(\mathbb{X},\mathbb{Y})$  instead.
+
+We say that the PGW-Monge mapping assumption holds if there exists an optimal transportation plan  $\gamma$  for the optimization problem in (12) of the form  $\gamma = (\mathrm{id} \times T)_{\#} \gamma_X$  for some  $T: X \to Y$ .
+
+# 3 LINEAR PARTIAL GROMOV-WASSERSTEIN EMBEDDING AND DISTANCE
+
+Inspired by the work in (Beier et al., 2022), we extend the LGW distance to the unbalanced setting. In particular, we present a linearization technique for the PGW problem proposed in (Bai et al., 2024).
+
+Given gm-spaces  $\mathbb{X} = (X,g_X,\mu)$  and  $\mathbb{Y} = (Y,g_Y,\nu)$ , let  $\gamma \in \Gamma_{\leq ,\lambda}^{*}(\mathbb{X},\mathbb{Y})$ . We first suppose that the PGW-Monge mapping assumption holds; that is,  $\gamma = (\mathrm{id}\times T)_{\#}\gamma_{X}$  for some  $T:X\to Y$ . We subsequently define the linear partial Gromov-Wasserstein (LPGW) embedding as
+
+$$
+\mathbb {Y} \mapsto (\gamma_ {X}, k _ {\gamma}, \gamma_ {c}), \tag {13}
+$$
+
+where  $\gamma_{X} := (\pi_{X})_{\#}\gamma$ ,  $k_{\gamma}(\cdot_{1},\cdot_{2}) := g_{X}(\cdot_{1},\cdot_{2}) - g_{Y}(T(\cdot_{1}),T(\cdot_{2}))$ , and  $\gamma_{c} := \nu^{\otimes 2} - (\gamma_{Y})^{\otimes 2}$  for  $\gamma_{Y} = (\pi_{Y})_{\#}\gamma$ . Here, the first component of the embedding,  $\gamma_{X}$ , encodes the mass transportation from the source domain  $X$ . The second component,  $k_{\gamma}$ , describes the transportation cost function. Finally, the third component,  $\gamma_{c}$ , encodes the mass creation in the target domain  $Y$ .
+
+Similarly to the LGW embedding given by (5), we show in Proposition 3.1 that the above embedding can recover  $PGW_{\lambda}(\mathbb{X}, \mathbb{Y})$  in the sense that
+
+$$
+\| k _ {\gamma^ {*}} \| _ {L ^ {2} \left(\left(\gamma_ {X} ^ {*}\right) ^ {\otimes 2}\right)} ^ {2} + \lambda \left(| \mu | ^ {2} - | \gamma_ {X} ^ {*} | ^ {2} + | \gamma_ {c} ^ {*} |\right) = P G W _ {\lambda} (\mathbb {X}, \mathbb {Y}).
+$$
+
+Given two gm-spaces  $\mathbb{Y}^1, \mathbb{Y}^2$ , let  $\gamma^1 \in \Gamma_{\leq,\lambda}^*(\mathbb{X}, \mathbb{Y}^1)$  and  $\gamma^2 \in \Gamma_{\leq,\lambda}^*(\mathbb{X}, \mathbb{Y}^2)$ . Suppose  $\gamma^1, \gamma^2$  are induced by Monge mappings  $T^1, T^2$  and compute the corresponding LPGW embeddings via (13). We define the LPGW discrepancy conditioned on  $\gamma^1, \gamma^2$  by
+
+$$
+L P G W _ {\lambda} \left(\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}\right) := \inf  _ {\mu^ {\prime} \leq \gamma_ {X} ^ {1} \wedge \gamma_ {X} ^ {2}} \| k _ {\gamma^ {1}} - k _ {\gamma^ {2}} \| _ {L ^ {2} \left(\mu^ {\prime} \otimes^ {2}\right)} ^ {2} + \lambda \left(| \nu^ {1} | ^ {2} + | \nu^ {2} | ^ {2} - 2 \left| \mu^ {\prime} \right| ^ {2}\right), \tag {14}
+$$
+
+where  $(\gamma_X^1 \wedge \gamma_X^2)(E) = \min_{F \subset E} \{\gamma_X^1(E) + \gamma_X^2(E \setminus F)\}$ . Note that the above formulation can be written entirely in terms of the LPGW embeddings since  $|\nu^1|^2 = |\gamma_c^1| + |\gamma_X^1|^2$ , and similarly  $|\nu^2|^2 = |\gamma_c^2| + |\gamma_X^2|^2$ .
+
+When the PGW-Monge mapping assumption does not hold, we extend the above LPGW discrepancy conditioned on arbitrary transportation plans  $\gamma^1\in \Gamma_{\leq ,\lambda}^* (\mathbb{X},\mathbb{Y}^1),\gamma^2\in \Gamma_{\leq ,\lambda}^* (\mathbb{X},\mathbb{Y}^2)$  as follows:
+
+$$
+L P G W _ {\lambda} \left(\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}\right) := \inf  _ {\gamma \in \Gamma_ {\leq} \left(\gamma^ {1}, \gamma^ {2}; \mu\right)} \| g _ {Y ^ {1}} - g _ {Y ^ {2}} \| _ {L ^ {2} \left(\gamma^ {\otimes 2}\right)} ^ {2} + \lambda \left(| \nu^ {1} | ^ {2} + | \nu^ {2} | ^ {2} - 2 | \gamma | ^ {2}\right), \tag {15}
+$$
+
+where
+
+$$
+\Gamma_ {\leq} (\gamma^ {1}, \gamma^ {2}; \mu) := \{\gamma \in \mathcal {M} _ {+} (X \times Y ^ {1} \times Y ^ {2}): (\pi_ {X, Y ^ {1}}) _ {\#} \gamma \leq \gamma^ {1}, (\pi_ {X, Y ^ {2}}) _ {\#} \gamma \leq \gamma^ {2} \}.
+$$
+
+Finally, to gain independence from the transportation plans  $\gamma^1,\gamma^2$ , we formally define the LPGW discrepancy as
+
+$$
+L P G W _ {\lambda} \left(\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}\right) := \inf  _ { \begin{array}{c} \gamma^ {1} \in \Gamma_ {\leq , \lambda} ^ {*} (\mathbb {X}, \mathbb {Y} ^ {1}) \\ \gamma^ {2} \in \Gamma_ {\leq , \lambda} ^ {*} (\mathbb {X}, \mathbb {Y} ^ {2}) \end{array} } L P G W _ {\lambda} \left(\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}\right). \tag {16}
+$$
+
+Proposition 3.1. Given compact gm-spaces  $\mathbb{X} = (X,g_X,\mu)$  and  $\mathbb{Y} = (Y,g_Y,\nu)$ , let  $\gamma^0\in \Gamma_{\leq ,\lambda}^{*}(\mathbb{X},\mathbb{X})$  and  $\gamma^{*}\in \Gamma_{\leq ,\lambda}^{*}(\mathbb{X},\mathbb{Y})$ . We have the following:
+
+(1) Problem (15) admits a minimizer.  
+(2) Under the PGW-Monge mapping assumption, the problems (14) and (15) coincide.  
+(3) Under the PGW-Monge mapping assumption, when
+
+$$
+2 \lambda \geq \max  _ { \begin{array}{c} y ^ {1}, y ^ {1 \prime} \in Y ^ {1}, \\ y ^ {2}, y ^ {2 \prime} \in Y ^ {2} \end{array} } | g _ {Y} \left(y ^ {1}, y ^ {1 \prime}\right) - g _ {Y} \left(y ^ {2}, y ^ {2 \prime}\right) | ^ {2} \tag {17}
+$$
+
+problem (14) achieves the value
+
+$$
+L P G W _ {\lambda} \left(\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}\right) := \| k _ {\gamma^ {1}} - k _ {\gamma^ {2}} \| _ {L \left(\left(\gamma_ {X} ^ {1} \wedge \gamma_ {X} ^ {2}\right) ^ {\otimes 2}\right)} ^ {2} + \lambda \left(\left| \nu^ {1} \right| ^ {2} + \left| \nu^ {2} \right| ^ {2} - 2 \left| \gamma_ {X} ^ {1} \wedge \gamma_ {X} ^ {2} \right| ^ {2}\right). \tag {18}
+$$
+
+(4) The LPGW embedding of  $\mathbb{Y}$  can recover  $PGW_{\lambda}(\mathbb{X},\mathbb{Y})$  in the sense that, under the PGW-Monge mapping assumption, we have
+
+$$
+\| k _ {\gamma^ {*}} \| _ {L ^ {2} \left(\left(\gamma_ {X} ^ {*}\right) \otimes 2\right)} ^ {2} + \lambda \left(| \mu | ^ {2} - \left| \gamma_ {X} ^ {*} \right| ^ {2} + \left| \gamma_ {c} ^ {*} \right|\right) = P G W _ {\lambda} (\mathbb {X}, \mathbb {Y}),
+$$
+
+and in general, we have
+
+$$
+L P G W _ {\lambda} (\mathbb {X}, \mathbb {Y}; \gamma^ {0}, \gamma^ {*}) = P G W _ {\lambda} (\mathbb {X}, \mathbb {Y}).
+$$
+
+(5) The LPGW discrepancies defined in (15) and (16) are pseudo-metrics. Indeed, under certain conditions, (16) is a rigorous metric. We refer to Appendix F for detailed discussion.
+
+The proof of the above proposition is included in Appendix F.
+
+Similar to LGW, to accelerate the computation of the above formulation, we propose the following barycentric projection method in the LPGW setting.
+
+First, similar to Proposition 2.1, we present the following theorem, for which we provide the proof in Appendix H.
+
+Theorem 3.2. Given gm-spaces  $\mathbb{X} = (X,g_X,\mu)$  and  $\mathbb{Y} = (Y,g_Y,\nu)$ , let  $\gamma^0\in \Gamma_{\leq ,\lambda}^{*}(\mathbb{X},\mathbb{X})$  and  $\gamma^{*}\in \mathcal{M}_{+}(X\times Y)$ . Further, let  $\widetilde{\nu} := \widetilde{\nu}_{\gamma^{*}}:= (\mathcal{T}_{\gamma^{*}})_{\#}\gamma$  and  $\widetilde{\mathbb{Y}}_{\gamma^{*}} = (Y,d_{Y},\widetilde{\nu})$ . Then,
+
+(1) If  $\gamma^{*}$  is induced by a mapping  $T$ , then  $\mathcal{T}_{\gamma^{*}} = T, \gamma_{X}^{*} - a.s.$  
+(2) If  $\gamma^{*} \in \Gamma_{\leq}^{*}(\mathbb{X}, \mathbb{Y})$ , and the conditions for  $g_{Y}(\cdot_{1}, \cdot_{2})$  in statement (3) of Proposition 2.1 hold, then  $\widetilde{\gamma} := (\mathrm{id} \times \mathcal{T}_{\gamma^{*}})_{\#} \gamma_{X}^{*}$  is optimal for  $PGW_{\lambda}(\mathbb{X}, \widetilde{\mathbb{Y}}_{\gamma^{*}})$ .  
+(3) If  $\gamma^1\in \Gamma_{\leq}^{*}(\mathbb{X},\mathbb{Y}^1)$  and  $\gamma^2\in \Gamma_{\leq}^{*}(\mathbb{X},\mathbb{Y}^2)$ , and the PGW-Monge mapping assumption holds, then
+
+$$
+L P G W _ {\lambda} \left(\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}\right) = L P G W _ {\lambda} \left(\widetilde {\mathbb {Y}} _ {\gamma^ {1}}, \widetilde {\mathbb {Y}} _ {\gamma^ {2}}; \mathbb {X}, \widetilde {\gamma} ^ {1}, \widetilde {\gamma} ^ {2}\right) + \lambda \left(\left| \gamma_ {c} ^ {1} \right| + \left| \gamma_ {c} ^ {2} \right|\right). \tag {19}
+$$
+
+Based on part (3) in the above theorem, we define the approximated LPGW (aLPGW) discrepancy by
+
+$$
+a L P G W _ {\lambda} \left(\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}\right) := \inf  _ {\substack {\gamma^ {1} \in \Gamma_ {\leq , \lambda} ^ {*} (\mathbb {X}, \mathbb {Y} ^ {1}), \\ \gamma^ {2} \in \Gamma_ {\leq , \lambda} ^ {*} (\mathbb {X}, \mathbb {Y} ^ {2})}} L P G W _ {\lambda} \left(\widetilde {\mathbb {Y}} _ {\gamma^ {1}}, \widetilde {\mathbb {Y}} _ {\gamma^ {2}}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}\right) + \lambda \left(\left| \gamma_ {c} ^ {1} \right| + \left| \gamma_ {c} ^ {2} \right|\right). \tag{20}
+$$
+
+Proposition 3.3. If  $X, Y$  are compact sets, when  $\lambda$  is sufficiently large, in particular, (17) is satisfied, the above formulation becomes:
+
+$$
+a L P G W _ {\lambda} \left(\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}\right) = \inf  _ {\substack {\gamma^ {1} \in \Gamma_ {\leq , \lambda} ^ {*} (\mathbb {X}, \mathbb {Y} ^ {1}), \\ \gamma^ {2} \in \Gamma_ {\leq , \lambda} ^ {*} (\mathbb {X}, \mathbb {Y} ^ {2})}} \| k _ {\gamma^ {1}} - k _ {\gamma^ {2}} \| _ {(\gamma_ {X} ^ {1} \wedge \gamma_ {X} ^ {2}) \otimes 2} ^ {2} + \lambda (| \nu^ {1} | ^ {2} + | \nu^ {2} | ^ {2} - 2 | \gamma_ {X} ^ {1} \wedge \gamma_ {X} ^ {2} | ^ {2}). \tag{21}
+$$
+
+In practice, similar to LGW, we only select one pair of optimal transportation plans  $\gamma^1, \gamma^2$  to compute the embedding and the above aLPGW discrepancy. In addition, we apply (21) to approximate the original formula (20). That is, in our experiments, we directly compute
+
+$$
+\left\| k _ {\gamma^ {1}} - k _ {\gamma^ {2}} \right\| _ {\left(\gamma_ {X} ^ {1} \wedge \gamma_ {X} ^ {2}\right) ^ {\otimes 2}} ^ {2} + \lambda \left(| \nu^ {1} | ^ {2} + | \nu^ {2} | ^ {2} - 2 \left| \gamma_ {X} ^ {1} \wedge \gamma_ {X} ^ {2} \right| ^ {2}\right). \tag {22}
+$$
+
+# 3.1 NUMERICAL IMPLEMENTATION OF LPGW IN DISCRETE SETTING
+
+In practice, we consider discrete distributions (or, more generally, discrete Radon measures). Suppose  $\mathbb{X} = \{X,\| \cdot \|_{d_0},\mu = \sum_{i = 1}^n p_i\delta_{x_i}\}$ , where  $X\subset \mathbb{R}^{d_0}$  is a convex compact set containing  $\{0_{d_0},x_1,\ldots x_n\}$ . We similarly let  $\mathbb{Y}^1 = (Y^1,\| \cdot \|_{d_1},\nu = \sum_{j = 1}^{m_1}q_j^1\delta_{y_j^1})$  and  $\mathbb{Y}^2 = (Y^2,\| \cdot \|_{d^2},\nu^2 = \sum_{j = 1}^{m_2}q_j^2\delta_{y_j^2})$ .
+
+Suppose  $\gamma^1\in \mathbb{R}_+^{n\times m_1}$  is an optimal transportation plan for  $PGW_{\lambda}(\mathbb{X},\mathbb{Y}^1)$ . By convexity of  $Y$ , the barycentric projection given by (9) becomes
+
+$$
+\mathcal {T} _ {\gamma^ {1}} \left(x _ {i}\right) = \left\{ \begin{array}{l l} \widetilde {y} _ {i} = \frac {1}{\widetilde {q} _ {i}} \left[ \gamma^ {1} y ^ {1} \right] _ {i} & \text {i f} \widetilde {q} _ {i} > 0 \\ 0 _ {d _ {1}} & \text {i f} \widetilde {q} _ {i} = 0 \end{array} \quad \text {w h e r e} \widetilde {q} = \gamma_ {X} ^ {1} = \gamma^ {1} 1 _ {m _ {1}}. \right. \tag {23}
+$$
+
+Thus, the corresponding projected measure becomes  $\widetilde{\nu}_{\gamma^1} = \sum_i\widetilde{q}_i\delta_{\widetilde{y}_i}$ . In addition, we have  $|\gamma_c| = (\sum_{i = 1}^{n}p_i)^2 -(\sum_{i\in 1}^{n}\widetilde{q}_i^1)^2$ . Let
+
+$$
+\widetilde {K} ^ {1} := k _ {\widetilde {\gamma} ^ {*}} = \left[ \| \widetilde {y} _ {i} - \widetilde {y} _ {i} ^ {\prime} \| ^ {2} - \| x _ {i} - x _ {i} ^ {\prime} \| ^ {2} \right] _ {i, i ^ {\prime} \in [ 1: n ]} \in \mathbb {R} ^ {n \times n}. \tag {24}
+$$
+
+Suppose  $\gamma^2\in \mathbb{R}_+^{n\times m_2}$  is an optimal transportation plan for  $PGW_{\lambda}(\mathbb{X},\mathbb{Y}^2)$ . We define all corresponding terms analogously.
+
+Let  $\widetilde{q}^{1,2} = \widetilde{q}^1\wedge \widetilde{q}^2$ $|\widetilde{q}^1 -\widetilde{q}^2 |_{TV} = \sum_{i = 1}^n |\widetilde{q}_i^1 -\widetilde{q}_i^2 |,|q^1 | = \sum_{i = 1}^{m_1}q_i^1$  . The aLPGW distance given by (20) and (21) becomes
+
+$$
+\begin{array}{l} a L P G W (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}) \\ \approx \sum_ {i = 1} ^ {n} \left(\| \widetilde {y} _ {i} ^ {1} - \widetilde {y} _ {i ^ {\prime}} ^ {1} \| ^ {2} - \| \widetilde {Y} _ {i} ^ {2} - \widetilde {Y} _ {i ^ {\prime}} ^ {2} \| ^ {2}\right) ^ {2} \widetilde {q} _ {i} ^ {1 2} \widetilde {q} _ {i ^ {\prime}} ^ {1 2} + \lambda (| q ^ {1} | ^ {2} + | q ^ {2} | ^ {2} - 2 | \widetilde {q} ^ {1 2} | ^ {2}) \\ = \left(\widetilde {q} ^ {1 2}\right) ^ {\top} \left[ \left(\widetilde {K} _ {1} - \widetilde {K} _ {2}\right) ^ {2} \right] \left(\widetilde {q} ^ {1 2}\right) + \lambda \left(\left| \gamma_ {c} ^ {1} \right| + \left| \gamma_ {c} ^ {2} \right| + \left| \widetilde {q} _ {1} ^ {\otimes 2} - \widetilde {q} _ {2} ^ {\otimes 2} \right| _ {T V}\right), \tag {25} \\ \end{array}
+$$
+
+where  $(\widetilde{K}_1 - \widetilde{K}_2)^2$  denotes the element-wise squared matrix and  $\widetilde{q}_1^{\otimes 2} = \widetilde{q}_1\widetilde{q}_1^\top$  (similarly for  $\widetilde{q}_2^{\otimes 2}$ ). The above quantity (25) can be used to approximate the original PGW distance between  $\mathbb{Y}^1$  and  $\mathbb{Y}^2$ .
+
+In addition, the original LPGW embedding (13) of  $\mathbb{Y}^1$  (in fact, of  $\widetilde{\mathbb{Y}}^1$ ) is  $(\widetilde{K}^1,\widetilde{\nu}_{\gamma^1},\gamma_c^1)$ . We reduce this embedding to  $(\widetilde{K}^1,\widetilde{q}^1,|\gamma_c^1|)$  which is sufficient to compute the above aLPGW discrepancy. Thus,  $(\widetilde{K}^1,\widetilde{q}^1,|\gamma_c^1|)$  can be regarded as numerical implementation of the LPGW embedding (13).
+
+# 4 EXPERIMENTS
+
+# 4.1 ELLIPTICAL DISKS
+
+In this experiment, we apply the LPGW distance and the PGW distance to the 2D dataset presented in (Beier et al., 2022), consisting of 100 elliptical disks. We first compute the pairwise distances between the samples using the PGW distance and then compare them against the LPGW distance using nine different reference spaces. We present the resulting wall-clock time for each method and evaluate the quality of the approximation of PGW by LPGW using the mean relative error (MRE) and the Pearson correlation coefficient (PCC).
+
+Experiment setup. We represent each 2D shape as an mm-space  $\mathbb{X}^i = (\mathbb{R}^2, \| \cdot \|_2, \mu^i)$ , where  $\mu^i = \sum_{i=1}^{n} \frac{1}{n} \delta_{x_i}$ . We normalize each shape so that the largest pairwise distance in each mm-space is 1. Based on [Lemma E.2, Bai et al. (2023)], the largest possible choice of  $\lambda$  is given by  $2\lambda = 1$ . We hence test  $\lambda \in \{0.05, 0.08, 0.1, 0.3, 0.5\}$ , and for each reference space, we compute the pairwise LPGW distances and compute the wall-clock time, MRE, and PCC. We refer to Appendix K for full numerical details, MDS visualizations, and complete results.
+
+Performance analysis. We present the results when  $\lambda = 0.1$  in Table 1. First, we observe that LPGW is significantly faster than PGW as it only requires  $N = 100$  transport plan computations, whereas the PGW methods require  $\binom{N}{2}$  transport plan computations. Second, we observe that when the reference space is one of  $\mathbb{S}_6, \mathbb{S}_7, \mathbb{S}_8, \mathbb{S}_9$ , LPGW admits a relatively smaller MRE and higher PCC, demonstrating the importance of the chosen reference space.
+
+<table><tr><td></td><td>PGW</td><td>S1</td><td>S2</td><td>S3</td><td>S4</td><td>S5</td><td>S6</td><td>S7</td><td>S8</td><td>S9</td></tr><tr><td>points</td><td></td><td>441</td><td>676</td><td>625</td><td>52</td><td>289</td><td>545</td><td>882</td><td>882</td><td>317</td></tr><tr><td>time (min)</td><td>46.97</td><td>0.76</td><td>3.78</td><td>3.13</td><td>0.08</td><td>0.62</td><td>1.56</td><td>1.91</td><td>1.98</td><td>0.71</td></tr><tr><td>MRE ↓</td><td>—</td><td>0.1941</td><td>0.1264</td><td>0.1431</td><td>0.2542</td><td>0.0538</td><td>0.0444</td><td>0.0205</td><td>0.0198</td><td>0.0245</td></tr><tr><td>PCC ↑</td><td>—</td><td>0.5781</td><td>0.5738</td><td>0.5881</td><td>0.8581</td><td>0.9871</td><td>0.9930</td><td>0.9952</td><td>0.9954</td><td>0.9949</td></tr></table>
+
+Table 1: Experimental results when  $\lambda = 0.1$ . The first row shows the total wall-clock time for PGW and LPGW. The second and third rows display the MRE (mean relative error) and PCC (Pearson correlation coefficient), respectively.
+
+# 4.2 SHAPE RETRIEVAL
+
+Experiment setup. We now employ the PGW distance to distinguish between 2D and 3D shapes, similarly to as done in (Beier et al., 2022). We use GW, LGW, and PGW as baselines for comparison against the results of our proposed LPGW. Given a series of shapes, we represent the shapes as mm-spaces  $\mathbb{X}^i = (\mathbb{R}^d,\| \cdot \| _2,\mu^i)$ , where  $\mu^i = \sum_{k = 1}^{n^i}\alpha^i\delta_{x_k^i}$
+
+For the GW and LGW methods, we normalize the mass for the balanced mass constraint setting (i.e.,  $\alpha^i = \frac{1}{n^i}$ ), and for the PGW and LPGW methods, we let  $\alpha^i = \alpha$  for all the shapes, where  $\alpha > 0$  is a fixed constant. In this manner, we compute the pairwise distances between the shapes.
+
+Next, using the approach given by (Beier et al., 2022), we combine each distance matrix with a support vector machine (SVM), applying stratified 10-fold cross-validation. In each iteration of cross-validation, we train an SVM using  $\exp(-\sigma D)$  as the kernel, where  $D$  is the matrix of pairwise distances (w.r.t. one of the considered distances) restricted to 9 folds, and compute the accuracy of the model on the remaining fold. We report the accuracy averaged over all 10 folds for each model.
+
+Dataset setup. We test one 2D shape dataset and one 3D shape dataset. We use the 2D dataset presented in (Bai et al., 2024), consisting of 8 classes, each containing 20 shapes. The 3D dataset is given by (Pan et al., 2021), which provides 100-200 complete shapes in each of 16 different classes, and for each complete shape, provides 26 corresponding "incomplete shapes." We choose four classes of shapes from this dataset, and for each, we sample 15 complete and 45 incomplete shapes, yielding a total of 240 shapes. The datasets are visualized in Figure 4 in the Appendix.
+
+Performance analysis. We refer to Appendix L for full numerical details, parameter settings, and the visualization of the resulting confusion matrices. We visualize the resulting pairwise distance matrices in Figure 7. For the SVM experiments, on the 2D dataset, GW achieves the highest accuracy,  $98.1\%$ , while the second best method is LPGW,  $97.5\%$ . On the 3D dataset, PGW achieves the highest accuracy of  $93.8\%$ , and LPGW achieves the next highest accuracy of  $93.7\%$ . We refer to Table 2 for details.
+
+<table><tr><td></td><td>GW</td><td>PGW</td><td>LGW</td><td>LPGW(ours)</td></tr><tr><td colspan="5">2D Dataset</td></tr><tr><td>Accuracy ↑</td><td>98.1%</td><td>96.2%</td><td>93.7%</td><td>97.5%</td></tr><tr><td>Time (s) ↓</td><td>43s</td><td>39s</td><td>0.4s</td><td>0.5s</td></tr><tr><td colspan="5">3D Dataset</td></tr><tr><td>Accuracy ↑</td><td>92.5%</td><td>93.8%</td><td>92.5%</td><td>93.7%</td></tr><tr><td>Time (m) ↓</td><td>203.0m</td><td>203.6m</td><td>1.3m</td><td>1.8m</td></tr></table>
+
+Table 2: Accuracy and wall-clock time comparison in shape retrieval experiment.
+
+In addition, we report the wall-clock time required to compute all pairwise distances for each distance in Table 2. In both datasets, we observe that LGW/LPGW admit similar computational time and are much faster than GW/PGW. This difference is more obvious in the 3D dataset. We note that when  $\lambda$  is sufficiently large, the performance between the LPGW method and the LGW method is the same (see Appendix L for details).
+
+4.3 LEARNING WITH TRANSFORM-BASED EMBEDDINGS  
+
+<table><tr><td>Data</td><td>Method</td><td>LOT</td><td>LGW</td><td>LPGW (ours)</td></tr><tr><td rowspan="2">no rotation</td><td>Accuracy ↑</td><td>89.0%</td><td>82.5%</td><td>82.5%</td></tr><tr><td>Time ↓</td><td>183s+16s</td><td>405s+77s</td><td>309s+84s</td></tr><tr><td rowspan="2">η = 0</td><td>Accuracy ↑</td><td>51.2%</td><td>82.5%</td><td>82.5%</td></tr><tr><td>Time ↓</td><td>183s+15s</td><td>405s+77s</td><td>309s+84s</td></tr><tr><td rowspan="2">η = 0.1</td><td>Accuracy ↑</td><td>13.4%</td><td>17.0%</td><td>81.8%</td></tr><tr><td>Time ↓</td><td>183s+17s</td><td>405s+88s</td><td>309s+91s</td></tr><tr><td rowspan="2">η = 0.3</td><td>Accuracy ↑</td><td>12.5%</td><td>13.3%</td><td>75.8%</td></tr><tr><td>Time ↓</td><td>183s+23s</td><td>405s+145s</td><td>309s+123s</td></tr><tr><td rowspan="2">η = 0.5</td><td>Accuracy ↑</td><td>12.5%</td><td>12.5%</td><td>72.9%</td></tr><tr><td>Time ↓</td><td>183s+27s</td><td>405s+248s</td><td>309s+168s</td></tr></table>
+
+Table 3: Accuracy and wall-clock time comparison for learning with transform-based embeddings experiments. We report each time as  $t_1 + t_2$ , where  $t_1$  is the time required to compute the training set embeddings and  $t_2$  is the time required to compute the testing set embeddings.
+
+In this experiment, we perform machine learning tasks using transform-based embeddings. Specifically, given training and testing datasets, we apply various transform-based embedding methods to both. Additionally, we assume that the testing data is randomly rotated and has been corrupted with random noise. The objective is to evaluate the robustness of
+
+machine learning models trained using the different embedding techniques. The full numerical details for this experiment can be found in Appendix M.
+
+Baseline methods. In this experiment, we consider several classical embedding methods, namely the LOT embedding (Wang et al., 2013; Kolouri et al., 2020; Nenna & Pass, 2023), LGW embedding (Beier et al., 2022), and our proposed LPGW embedding.
+
+![](images/02708715db407a7a6f51d358f708dbcffe29b33138e6c008de7089f4cf6398ff.jpg)
+
+![](images/a46a3ec46a288ca61d66eeb4538c1b9c0c0962d62bc0c2f6824d472a5adab1de.jpg)  
+(a) MNIST point cloud dataset.  
+(b) Reconstructed digits based on the LOT/LGW/LPGW embeddings.  
+Figure 1: MNIST digits and its reconstruction.
+
+Dataset setup. We adopt the MNIST 2D point cloud dataset for this experiment. Specifically, for each digit, we sample  $N_{1} = 500$  point clouds per class from the training set and  $N_{2} = 100$  point clouds per class from the testing set. Each point cloud is represented as a gm-space in  $\mathbb{R}^2$  with the measure  $\mu = \sum_{i=1}^{n} p_i \delta_{x_i}$ , where  $p_i > 0, \forall i \in \{1, \dots, n\}$  are the pixel intensities provided by the original dataset. We normalize  $p_i$  such that  $\sum_{i=1}^{n} p_i = 1$  for all sampled point clouds.
+
+For each shape in the training set, we randomly rotate or flip the shape horizontally and add the transformed shape to the training set. As a result, each class contains  $500 \times 2 = 1000$  shapes in the training set. For each test shape, we first randomly rotate or flip the shape, and we then "corrupt" the shape by adding uniformly distributed noise. The mass of each added point is  $\frac{1}{n}$ , where  $n$  is the number of points in the original shape, and the total mass of the added points is  $\eta \in \{0, 0.1, 0.3, 0.5\}$ .
+
+Performance analysis. For each method (LOT, LGW, and LPGW), we first compute the embeddings of the training data and optimize a classifica
+
+tion model (logistic regression) using these embeddings. Then, we compute the embeddings for the test dataset and evaluate the accuracy of the model on the test embeddings. The results are presented in Table 3. In this table, we observe that for the original test dataset, LOT achieves the highest accuracy at  $89.0\%$ . However, on the corrupted dataset, LOT's performance drops significantly, classifying only  $51\%$  correctly when  $\eta = 0$  (no noise but with random rotation/flip) and falling dropping to  $12.5\%$  (i.e., random guessing) when  $\eta > 0.1$ . LGW is more robust to the data with random rotations/flips, but its accuracy also reduces to random guessing as  $\eta$  is increased. In contrast, LPGW embeddings exhibit substantially greater performance, with its accuracy ranging from  $70 - 85\%$  when  $\eta \geq 0.1$ .
+
+For an intuitive comparison of the performance of these embeddings, we present the data reconstruction results in Figure 1b and the t-SNE visualization of the embeddings produced by each method in Figure 9. Further details can be found in Appendix N.
+
+# 5 SUMMARY
+
+In this paper, we propose the linear partial Gromov-Wasserstein (LPGW) embedding, a linearization technique for the PGW problem. Theoretically, we prove that LPGW admits a metric with certain assumptions, and numerically, we demonstrate the utility of our proposed LPGW method in shape retrieval and learning with transform-based embedding tasks. We demonstrate that the LPGW-based method can preserve the partial matching property of PGW while significantly improving computational efficiency.
+
+# ACKNOWLEDGMENTS
+
+This research was partially supported by the NSF CAREER Award No. 2339898.
+
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+
+# A BACKGROUND: LINEAR OPTIMAL TRANSPORTATION DISTANCE
+
+The space of measures  $\mathcal{P}_2(\mathbb{R}^d)$  can be endowed with the two-Wasserstein metric  $W_{2}$ , and the resulting space defines a Riemannian manifold (see, e.g., (Villani, 2021)), which is referred to in the literature as the Wasserstein space. Under the Monge mapping assumption, consider  $\mu \in \mathcal{P}_2(\mathbb{R}^d)$  and let  $\mathrm{Tan}_{\mu}(\mathcal{P}_2(\mathbb{R}^d))$  denote the corresponding tangent space at  $\mu$ . Then, each tangent vector  $v \in \mathrm{Tan}_{\mu}(\mathcal{P}_2(\mathbb{R}^d))$  can be regarded as an  $L^2(\mu)$ -function, that is,  $v: \mathbb{R}^d \to \mathbb{R}^d$  such that
+
+$$
+\| v \| _ {L ^ {2} (\mu)} ^ {2} = \int_ {\mathbb {R} ^ {d}} \| v (x) \| ^ {2} d \mu (x) <   \infty .
+$$
+
+Given  $\nu \in \mathcal{P}_2(\mathbb{R}^d)$ , suppose  $\gamma = (\mathrm{id}\times T)_{\#}\mu$  is an optimal transportation plan for  $W_{2}(\mu ,\nu)$ . Then, the logarithm mapping is defined by
+
+$$
+\mathcal {P} _ {2} \left(\mathbb {R} ^ {d}\right) \ni \nu \mapsto v _ {\gamma} := (T - \mathrm {i d}) \in \operatorname {T a n} _ {\mu} \left(\mathcal {P} _ {2} \left(\mathbb {R} ^ {d}\right)\right), \tag {26}
+$$
+
+and the resulting image  $v_{\gamma}$  is the so-called Linear Optimal Transportation (LOT) embedding (Wang et al., 2013).
+
+The LOT embedding  $v_{\gamma}$  is a "representation" of the measure  $\nu$  and encodes the optimal displacement from the fixed measure  $\mu$  to  $\nu$ . Indeed,
+
+$$
+W _ {2} ^ {2} (\mu , \nu) = \| v _ {\gamma} - 0 \| _ {L ^ {2} (\mu)} ^ {2}.
+$$
+
+In general, given two probability measures  $\nu^1,\nu^2\in \mathcal{P}_2(\mathbb{R}^d)$ , if  $\gamma^1,\gamma^2$  are optimal solutions for  $W_{2}(\mu ,\nu^{1})$ ,  $W_{2}(\mu ,\nu^{2})$  of the form (2), one can use the following so-called LOT distance to approximate the original OT distance  $W_{2}(\nu^{1},\nu^{2})$  (Moosmüller & Cloninger, 2023):
+
+$$
+L O T _ {2} ^ {2} \left(\nu^ {1}, \nu^ {2}; \mu , \gamma^ {1}, \gamma^ {2}\right) := \left\| v _ {\gamma^ {1}} - v _ {\gamma^ {2}} \right\| _ {L ^ {2} (\mu)} ^ {2}. \tag {27}
+$$
+
+The above definition relies on the Monge mapping assumption. In (Wang et al., 2013; Moosmüller & Cloninger, 2023), the authors generalize the LOT distance without the Monge mapping assumption; however, in what follows, we will avoid such an assumption.
+
+First, we discuss the notion of a geodesic in the classical OT setting. Suppose  $\gamma$  is an optimal transportation plan for the OT problem between  $\mu, \nu \in \mathcal{P}_2(\mathbb{R}^d)$ . Then, the geodesic from  $\mu$  to  $\nu$  is defined by
+
+$$
+t \mapsto \gamma_ {t} := ((1 - t) \pi_ {X} + t \pi_ {Y}) _ {\#} \gamma \quad (0 \leq t \leq 1). \tag {28}
+$$
+
+Remark A.1. In the particular case where  $\gamma$  is of the form  $\gamma = (\mathrm{id} \times T)_{\#} \mu$ , the above formula reduces to
+
+$$
+t \mapsto ((1 - t) \mathrm {i d} + t T) _ {\#} \mu \quad (0 \leq t \leq 1).
+$$
+
+Remark A.2. In the discrete case,  $\mu = \sum_{i=1}^{n} p_i \delta_{x_i}$  and  $\nu = \sum_{j=1}^{m} q_j \delta_{y_j}$ . By abuse of notation, let  $\gamma \in \Gamma^*(\mu, \nu)$  be interpreted as a matrix. Then, the above geodesic becomes
+
+$$
+t \mapsto \sum_ {i = 1} ^ {n} \sum_ {j = 1} ^ {m} \delta_ {(1 - t) x _ {i} + t y _ {j}} \gamma_ {i j} \qquad (0 \leq t \leq 1).
+$$
+
+Given curves  $\{\gamma_t^1\}_{t\in [0,1]}$ ,  $\{\gamma_t^2\}_{t\in [0,1]}$  both originating from  $\mu$ , we claim they are equivalent if there exists  $\epsilon > 0$  such that for all  $t\in [0,\epsilon]$ ,  $\gamma_t^1 = \gamma_t^2$ . Let  $G_{\mu}$  denote the set of all the equivalence classes. The generalized tangent space at  $\mu$ , denoted by  $\mathrm{Tan}_{\mu}^{g}(\mathcal{P}_2(\mathbb{R}^d))$ , is defined as the closure of  $G_{\mu}$  (see (Beier et al., 2022) for details).
+
+Now, given  $\nu^1,\nu^2\in \mathcal{P}_2(\mathbb{R}^d)$ , let  $\gamma^1,\gamma^2$  be some optimal transportation plan in  $\Gamma^{*}(\mu ,\nu^{1})$ ,  $\Gamma^{*}(\mu ,\nu^{2})$ , respectively. We define the following distance, defined on the space  $\mathrm{Tan}_{\mu}^{g}\mathcal{P}_{2}(\mathbb{R}^{d})$ , as the LOT distance conditioned on  $\gamma^1,\gamma^2$ :
+
+$$
+L O T _ {2} ^ {2} \left(\nu^ {1}, \nu^ {2}; \mu , \gamma^ {1}, \gamma^ {2}\right) := \inf  _ {\gamma \in \Gamma \left(\gamma^ {1}, \gamma^ {2}; \mu\right)} \int_ {\mathbb {R} ^ {3 d}} \| y ^ {1} - y ^ {2} \| ^ {2} d \gamma (x, y ^ {1}, y ^ {2}). \tag {29}
+$$
+
+where  $\Gamma (\gamma^1,\gamma^2;\mu)\coloneqq \{\gamma \in \mathcal{P}(\mathbb{R}^{3d}):\left(\pi_{01}\right)_{\#}\gamma = \gamma^1,\left(\pi_{02}\right)_{\#}\gamma = \gamma^2\}$ , where  $\pi_{01}(x,y^1,y^2) = (x,y^1)$ ,  $\pi_{02}(x,y^1,y^2) = (x,y^2)$ .
+
+Remark A.3. When  $\gamma^1 = (\mathrm{id}\times T^1)_{\#}\mu ,\gamma^2 = (\mathrm{id}\times T^2)_{\#}\mu$  for some mappings  $T^1,T^2$ , there exists only one element in  $\Gamma (\gamma^1,\gamma^2;\mu)$ , which is  $(\mathrm{id}\times T^{1}\times T^{2})_{\#}\mu$ . Thus, the right-hand side of (29) becomes
+
+$$
+\left\| T ^ {1} (x) - T ^ {2} (x) \right\| _ {L ^ {2} (\mu)} ^ {2}. \tag {30}
+$$
+
+That is, (29) and (27) coincide.
+
+Finally, to formulate a distance that is independent of the optimal transportation plans  $\gamma^1,\gamma^2$ , we define the LOT distance between  $\nu^{1},\nu^{2}$  with respect to  $\mu$  by
+
+$$
+L O T _ {2} ^ {2} \left(\nu^ {1}, \nu^ {2}; \mu\right) := \inf  _ {\substack {\gamma^ {1} \in \Gamma^ {*} (\mu , \nu^ {1}) \\ \gamma^ {2} \in \Gamma^ {*} (\mu , \nu^ {2})}} L O T _ {2} ^ {2} \left(\nu^ {1}, \nu^ {2}; \mu , \gamma^ {1}, \gamma^ {2}\right). \tag{31}
+$$
+
+Barycentric projection. The above LOT distance is complicated to compute. To simplify the computation, (Wang et al., 2013) introduces the "barycentric projection" method that we will recall here.
+
+First, we review some basic concepts in measure theory. Given  $\gamma \in \mathcal{P}(\mathbb{R}^{2d})$ , with  $\pi_{1\#}\gamma = \mu$  where  $\pi_1:\mathbb{R}^{2d}\to \mathbb{R}^d$  is given by  $\pi_1(x,y) = x$ , by the disintegration theorem (see, for e.g., (Ambrosio et al., 2005, Thm 5.3.1) and (Dumont et al., 2024, Theorem 6)), there exists a  $\mu$ -a.e. uniquely defined family of probability measures  $\{\gamma (\cdot |x)\}_{x\in \mathbb{R}^d}\subset \mathcal{P}(\mathbb{R}^d)$ , such that
+
+$$
+\int_ {\mathbb {R} ^ {2 d}} \phi (x, y) d \gamma (x, y) = \int_ {\mathbb {R} ^ {d}} \int_ {\mathbb {R} ^ {d}} \phi (x, y) d \gamma (y | x) d \mu (x), \quad \forall \phi \in C _ {0} \left(\mathbb {R} ^ {2 d}\right). \tag {32}
+$$
+
+In this paper, we call  $\gamma (\cdot |x)$  the disintegration of  $\gamma$  with respect to its first marginal  $(\mu)$ , given  $x$  (for each  $x\in \operatorname {supp}(\mu)$ ).
+
+The barycentric projection map  $\mathcal{T}_{\gamma}:\mathbb{R}^{d}\to \mathbb{R}^{d}$  of  $\gamma$  is defined by
+
+$$
+\mathcal {T} _ {\gamma} (x) := \arg \min  _ {y \in Y} \int_ {Y} \| y - y ^ {\prime} \| ^ {2} d \gamma \left(y ^ {\prime} \mid x ^ {\prime}\right) = \int_ {Y} y ^ {\prime} d \gamma \left(y ^ {\prime} \mid x ^ {\prime}\right). \tag {33}
+$$
+
+Note that the second equality holds since the distance between  $y, y'$  is simply the quadratic Euclidean distance. In the third term, we have a vector integration.
+
+Given an optimal transportation plan  $\gamma \in \Gamma^{*}(\mu, \nu)$ , we define  $\widetilde{\nu}_{\gamma} := (\mathcal{T}_{\gamma})_{\#} \mu$ .
+
+Proposition A.4 (Prop. II.4 (Beier et al., 2022)). If  $\gamma \in \Gamma^{*}(\mu, \nu)$ , then the transportation plan induced by the barycentric projection map  $\mathcal{T}_{\gamma}$ , denoted as  $\widetilde{\gamma} \coloneqq (\mathrm{id} \times \mathcal{T}_{\gamma})_{\#} \mu$  is optimal for the OT problem  $W_{2}(\mu, \widetilde{\nu}_{\gamma})$ .
+
+In addition, if  $\gamma = (\mathrm{id} \times T)_{\#} \mu$  satisfies the Monge mapping assumption, then  $\mathcal{T}_{\gamma^*} = T$ $\mu$ -a.e. and is a Monge map.
+
+Thus, if we replace  $\nu$  by  $\widetilde{\nu}_{\gamma}$ , the OT problem  $W_{2}(\mu, \widetilde{\nu}_{\gamma})$  can be solved by the map  $\mathcal{T}_{\gamma}$ . Based on this property, given  $\nu^{1}, \nu^{2} \in \mathcal{P}(\mathbb{R}^{d})$ , consider optimal transportation plans  $\gamma^{1} \in \Gamma^{*}(\mu, \nu^{1}), \gamma^{2} \in \Gamma(\mu, \nu^{2})$ , then define the following approximated LOT distance (aLOT) to estimate  $W_{2}(\nu^{1}, \nu^{2})$ :
+
+$$
+\begin{array}{l} aLOT_{2}^{2}(\nu^{1},\nu^{2};\mu):= \inf_{\substack{\gamma^{1}\in \Gamma^{*}(\mu ,\nu^{1})\\ \gamma^{2}\in \Gamma^{*}(\mu ,\nu^{2})}}LOT_{2}^{2}(\widetilde{\nu}_{\gamma^{1}},\widetilde{\nu}_{\gamma^{2}};\mu ,\widetilde{\gamma}^{1},\widetilde{\gamma}^{2}) \\ = \inf  _ {\substack {\gamma^ {1} \in \Gamma^ {*} (\mu , \nu^ {1}) \\ \gamma^ {2} \in \Gamma^ {*} (\mu , \nu^ {2})}} \| \mathcal {T} _ {\gamma^ {1}} - \mathcal {T} _ {\gamma^ {2}} \| _ {L ^ {2} (\mu)} ^ {2} (34) \\ \approx \left\| \mathcal {T} _ {\gamma^ {1}} - \mathcal {T} _ {\gamma^ {2}} \right\| _ {L ^ {2} (\mu)} ^ {2} (35) \\ \end{array}
+$$
+
+Note, in most of the literature, (35) is referred to as  $LOT_{2}$ , and for simplicity, it is approximated by any pair of optimal transportation plans  $(\gamma^{1},\gamma^{2})$
+
+# B BACKGROUND: LINEAR GROMOV-WASSERSTEIN DISTANCE
+
+Let  $X \subset \mathbb{R}^{d_0}$  be a compact and convex subset, and consider a Borel measure  $\mu$  on  $X$ . A symmetric function  $g_{X}: X^{\otimes 2} \to \mathbb{R}$  with  $\int_{X^{\otimes 2}} g_{X}(x, x') d\mu^{\otimes 2} < \infty$  is called a gauge function. The space of all gauge functions is denoted as  $L_{sym}^{2}(X^{\otimes 2}, \mu^{\otimes 2})$ . In this paper, without specific description, we default set  $g_{X}$  to be continuous (thus, Lipschitz continuous) on  $X$ . For example  $g_{X}(x, x') = \| x - x' \|^{2}$  or  $g_{X}(x, x') = x^{\top} x'$ . The space  $\mathbb{X} = (X, g_{X}, \mu)$  is called a gauged measure space (gm-space), which can be regarded as a generalized version of an mm-space. The GW problem between two gm-spaces  $\mathbb{X}, \mathbb{Y}$  is:
+
+$$
+G W ^ {2} (\mathbb {X}, \mathbb {Y}) := \inf  _ {\gamma \in \Gamma (\mu , \nu)} \int_ {(X \times X) ^ {\otimes 2}} | g _ {X} (x, x ^ {\prime}) - g _ {Y} (y, y ^ {\prime}) | ^ {2} d \gamma^ {\otimes 2}.
+$$
+
+Lemma B.1. There exists a minimizer  $\gamma$  of  $GW(\mathbb{X},\mathbb{Y})$  in  $\Gamma (\mu ,\nu)$
+
+The existence of minimizers of  $GW(\mathbb{X},\mathbb{Y})$  could be shown by a standard compact-SNEss argument, see for instance, Lemma 10.3 in (Mémoli, 2011). Suppose  $\gamma$  is one optimal solution of the above generalized GW problem. The geodesic between  $\mathbb{X}$  and  $\mathbb{Y}$  is defined by
+
+$$
+t \mapsto \gamma_ {t} := (X \times Y, (1 - t) g _ {X} + t g _ {Y}, \gamma), t \in [ 0, 1 ]. \tag {36}
+$$
+
+Note, even if  $g_{X}, g_{Y}$  are metrics,  $\gamma_{t}$  is a gm-space (rather than mm-space), and this is the reason mm-space is generalized to gm-spaces in (Beier et al., 2022) and in this paper.
+
+Two gm-spaces  $\mathbb{X},\mathbb{Y}$  are "equivalent" if  $GW^{2}(\mathbb{X},\mathbb{Y}) = 0$ . Let  $[\mathbb{X}]$  denote the equivalence class of  $\mathbb{X}$ . The tangent space at  $[\mathbb{X}]$  is defined by
+
+$$
+\left. \operatorname {T a n} _ {\mathbb {X}} := \left(\bigcup_ {\left(X ^ {\prime}, g _ {X ^ {\prime}}, \mu^ {\prime}\right) \in [ \mathbb {X} ]} L _ {s y m} ^ {2} \left(X ^ {\prime} \times X ^ {\prime}, \mu^ {\prime} ^ {\otimes 2}\right)\right) / \sim \right. \tag {37}
+$$
+
+where, if  $\mathbb{T}_{k_1} = (X^1,g_{X^1},\mu^1),\mathbb{T}_{k_2} = (X^2,g_{X^2},\mu^2)\in [\mathbb{X}]$  are representatives of  $k_{1}\in L^{2}_{sys}((X^{1})^{\otimes 2},(\mu^{1})^{\otimes 2}),k_{2}\in L^{2}_{sys}((X^{2})^{\otimes 2},(\mu^{2})^{\otimes 2})$  , we say that  $k_{1}\sim k_{2}$  if there exists  $\gamma \in \Gamma (\mathbb{X}^1,\mathbb{X}^2)$  such that  $GW(\mathbb{X}^1,\mathbb{X}^2) = 0$  . (We refer to (Beier et al., 2022) for more details.)
+
+Given spaces  $\mathbb{X}^1 \coloneqq (X \times Y^1, d_X, \gamma^1)$ ,  $\mathbb{X}^2 \coloneqq (X \times Y^2, d_X, \gamma^2)$  where  $\gamma^1, \gamma^2$  are optimal transportation plans for  $GW(\mathbb{X}, \mathbb{Y}^1)$ ,  $GW(\mathbb{X}, \mathbb{Y}^2)$ , respectively, it is straightforward to verify
+
+$$
+G W (\mathbb {X}, \mathbb {X} ^ {1}) = G W (\mathbb {X}, \mathbb {X} ^ {2}) = 0,
+$$
+
+thus  $\mathbb{X}^1, \mathbb{X}^2 \in [\mathbb{X}]$ .
+
+In addition, choose  $k_{1}\in L^{2}_{sys}(X^{1})^{\otimes 2},(\gamma^{1})^{\otimes 2}),k_{2}\in L^{2}_{sys}((X^{2})^{\otimes 2},(\gamma^{2})^{\otimes 2}),$  then we have
+
+$$
+k _ {1}, k _ {2} \in \operatorname {T a n} _ {\mathbb {X}},
+$$
+
+where  $\Gamma^{*}(\mathbb{T}_{k_1},\mathbb{T}_{k_2})$  is same to  $\Gamma^{*}(\gamma^{1},\gamma^{2})$  in  $GW(\mathbb{X}^1,\mathbb{X}^2)$
+
+Their inner product distance is defined by
+
+$$
+D^{2}(k_{1},k_{2}):= \inf_{\gamma \in \Gamma^{*}(\mathbb{T}_{k_{1}},\mathbb{T}_{k_{2}})}\| k_{1} - k_{2}\|_{\gamma^{\otimes 2}}^{2}.
+$$
+
+Now, we set  $k_{1} = k_{\gamma^{1}} = g_{Y^{1}} - g_{X}, k_{2} = k_{\gamma^{2}} = g_{Y^{2}} - g_{X}$ . Thus, the linear GW distance between  $\mathbb{Y}^1$  and  $\mathbb{Y}^2$  given  $\gamma^1, \gamma^2$  is defined by the above inner product distance. That is,
+
+$$
+\begin{array}{l} \operatorname {L G W} \left(\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}\right) := D \left(k _ {1}, k _ {2}\right) \\ = \inf  _ {\gamma \in \Gamma^ {*} (\mathbb {T} _ {k _ {1}}, \mathbb {T} _ {k _ {2}})} \| k _ {1} - k _ {2} \| _ {L ^ {2} (\gamma^ {\otimes 2})} ^ {2} \\ = \inf  _ {\gamma \in \Gamma \left(\gamma^ {1}, \gamma^ {2}; \mu\right)} \int_ {\left(X \times Y ^ {1} \times Y ^ {2}\right) \otimes 2} \| g _ {Y ^ {1}} \left(y ^ {1}, y ^ {1 ^ {\prime}}\right) - g _ {Y ^ {2}} \left(y ^ {2}, y ^ {2 ^ {\prime}}\right) \| ^ {2} d \gamma^ {\otimes 2} \tag {38} \\ \end{array}
+$$
+
+where  $\Gamma(\gamma^1, \gamma^2; \mu) \coloneqq \{\gamma \in \mathcal{P}(X \times Y^1 \times Y^2): (\pi_{X,Y^1})_\# \gamma = \gamma^1, (\pi_{X,Y^2})_\# \gamma = \gamma^2\}$ . The third line follows from Proposition III.1 in (Beier et al., 2022) (or equivalently from the identity (49) in the next section). Note, under Monge mapping assumptions,  $\gamma^1 = (\mathrm{id} \times T^1)_\# \mu$ ,  $\gamma^2 = (\mathrm{id} \times T^2)_\# \mu$ ,  $\Gamma(\gamma^1, \gamma^2; \mu) = \{(\mathrm{id} \times T^1 \times T^2)_\# \mu\}$  and the above distance (38) coincides with (7).
+
+The original Linear Gromov-Wasserstein distance between  $\mathbb{Y}^1$  and  $\mathbb{Y}^2$  with respect to  $\mathbb{X}$  is defined as
+
+$$
+\operatorname {L G W} \left(\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}\right) := \inf  _ {\substack {\gamma^ {1} \in \Gamma (\mathbb {X}, \mathbb {Y} ^ {1}) \\ \gamma^ {2} \in \Gamma (\mathbb {X}, \mathbb {Y} ^ {2})}} L G W \left(\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}\right). \tag{39}
+$$
+
+# C LOT/LGW/LPGW EMBEDDING WITHOUT MONGE MAPPING.
+
+In this section, we briefly discuss the linear transportation embedding under linear optimal transport, linear Gromov Wasserstein, and linear partial Gromov Wasserstein setting. In summary, the formulations of these embeddings do not rely on the Monge mapping assumption.
+
+Linear OT embedding without Monge mapping. Given probability measures  $\nu^1, \nu^2 \in \mathcal{P}(\Omega)$ , and reference measure  $\mu \in \mathcal{P}(\Omega)$ , where  $\Omega \subset \mathbb{R}^d$  is non-empty set. Choose optimal transportation plans  $\gamma^1 \in \Gamma^*(\mu, \nu^1), \gamma^2 \in \Gamma^*(\mu, \nu^2)$ , the linear OT embedding 26 can be generalized to
+
+$$
+\nu^ {1} \mapsto \hat {T} ^ {1} := \{x \mapsto \gamma^ {1} (\cdot | x) \}, \tag {40}
+$$
+
+where the conditional distribution  $\gamma^1 (\cdot |x)$  can be treated as a "random mapping". When  $\gamma^{1} = (\mathrm{id}\times T^{1})_{\#}\mu$ , it is straightforward to verify  $T^1 = \hat{T}^1$ . That is, the embedding (40),(26) coincide.
+
+We define the "distance" between two such two random mappings  $T^1, T^2$  via:
+
+$$
+\left\| \hat {T} ^ {1} - \hat {T} ^ {2} \right\| _ {L (\mu)} ^ {2} := \int_ {\gamma \in \Gamma \left(\gamma^ {1} (\cdot | x), \gamma^ {2} (\cdot | x)\right)} \left\| y ^ {1} - y ^ {2} \right\| ^ {2} d \gamma \left(y ^ {1}, y ^ {2}\right) d \mu (x) \tag {41}
+$$
+
+From (29), we have
+
+$$
+\begin{array}{l} L O T _ {2} ^ {2} \left(\nu^ {1}, \nu^ {2}; \mu\right) := \int_ {\gamma \in \Gamma \left(\nu^ {1}, \nu^ {2}; \mu\right)} \| y ^ {1} - y ^ {2} \| \gamma (x, y ^ {1}, y ^ {2}) \\ = \int_ {\Omega} \inf  _ {\gamma \in \Gamma (\gamma^ {1} (\cdot | x), \gamma^ {2} (\cdot | x)} \| y ^ {1} - y ^ {2} \| ^ {2} d \mu (x) \\ = \| \hat {T} ^ {1} - \hat {T} ^ {2} \| _ {L (\mu)} ^ {2}. \\ \end{array}
+$$
+
+Linear GW embedding without Monge mapping. Similar to above formulation, given two gm spaces  $\mathbb{Y}^1 = (Y^1,g_{Y^1},\nu^1),\mathbb{Y}^2 = (Y^1,g_{Y^2},\nu^2)$  and a reference  $gm$ -space  $\mathbb{X} = (X,g_X,\mu)$ , where  $\mu ,\nu^{1},\nu^{2}$  are probability measures. Suppose  $\gamma^1,\gamma^2$  are optimal transportation plans for  $GW(\mathbb{X},\mathbb{Y}^1),GW(\mathbb{X},\mathbb{Y}^2)$  respectively. Then the embedding (5) can be generalized to:
+
+$$
+\mathbb {Y} ^ {1} \mapsto \hat {k} ^ {1} := \left\{\left(x, x ^ {\prime}\right) \mapsto \left(g _ {Y ^ {1}} \left(\cdot_ {1}, \cdot_ {2}\right)\right) _ {\#} \left(\gamma^ {1} \left(\cdot^ {1} | x\right) \gamma^ {1} \left(\cdot^ {2} | x ^ {\prime}\right)\right). \right. \tag {42}
+$$
+
+Note, similar to LOT, when  $\gamma^1 = (\mathrm{id} \times T^1)_{\#} \mu$ , we have  $\hat{T}^1 = T^1$ . That is (5), (42) coincide.
+
+Similarly, we define the "distance" between two embeddings  $\hat{k}^1,\hat{k}^2$  via:
+
+$$
+\begin{array}{l} \| \hat {k} ^ {1} - \hat {k} ^ {2} \| _ {L (\mu^ {\otimes 2})} \\ := \int_ {X ^ {2}} \inf  _ {\substack {\gamma \in \Gamma (\gamma^ {1} (\cdot | x), \gamma^ {2} (\cdot | x) \\ \gamma^ {\prime} \in \Gamma (\gamma^ {1} (\cdot | x ^ {\prime}), \gamma^ {2} (\cdot | x ^ {\prime}))}} \| g _ {Y ^ {1}} (y ^ {1}, y ^ {1 \prime}) - g _ {Y ^ {2}} (y ^ {2}, y ^ {2 \prime}) \| ^ {2} d \gamma ((y ^ {1}, y ^ {2} | x) d \gamma^ {\prime} (y ^ {1 \prime}, y ^ {2 \prime} | x ^ {\prime}) d \mu^ {\otimes 2} (x, x ^ {\prime}). \tag{43} \\ \end{array}
+$$
+
+From (8), we have
+
+$$
+\begin{array}{l} L G W (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}, \mathbb {X}) \\ = \inf  _ {\gamma \in \Gamma (\gamma^ {1}, \gamma^ {2}; \mu)} | g _ {Y ^ {1}} (y ^ {1}, y ^ {1 ^ {\prime}}) - g _ {Y ^ {2}} (y ^ {2}, y ^ {2 ^ {\prime}}) | ^ {2} \gamma^ {\otimes 2} (x, y ^ {1}, y ^ {2}) \\ = \int_{X^{2}}\inf_{\substack{\gamma \in \Gamma (\gamma^{1}(\cdot |x),\gamma^{2}(\cdot |x)\\ \gamma^{\prime}\in \Gamma (\gamma^{1}(\cdot |x^{\prime}),\gamma^{2}(\cdot |x^{\prime}))}}\| g_{Y^{1}}(y^{1},y^{1^{\prime}}) - g_{Y^{2}}(y^{2},y^{2^{\prime}})\|^{2}d\gamma ((y^{1},y^{2}|x)d\gamma^{\prime}(y^{1^{\prime}},y^{2^{\prime}}|x^{\prime})d\mu^{\otimes 2}(x,x^{\prime}) \\ = \| \hat {k} ^ {1} - \hat {k} ^ {2} \| _ {L (\mu^ {\otimes 2})}. \\ \end{array}
+$$
+
+That is, the distance between two embeddings coincides with the LGW distance between two gm spaces.
+
+LPGW embedding without Monge mapping. Similar to above setting, given gm-spaces  $\mathbb{X},\mathbb{Y}^1,\mathbb{Y}^2$  and suppose  $\gamma^1,\gamma^2$  are optimal transportation plans for  $PGW_{\lambda}(\mathbb{X},\mathbb{Y}^{1}),PGW_{\lambda}(\mathbb{X},\mathbb{Y}^{2})$ , the LPGW embedding in general case is defined by
+
+$$
+\mathbb {Y} ^ {1} \mapsto E ^ {1} := \left(\hat {k} ^ {1}, \gamma_ {X} ^ {1}, \gamma_ {c} ^ {1} := \left(\nu^ {1}\right) ^ {\otimes 2} - \left(\gamma_ {Y ^ {1}} ^ {1}\right) ^ {\otimes 2}\right). \tag {44}
+$$
+
+Note, when the Monge mapping assumption holds, (13),(44) coincide.
+
+The distance between two embeddings is defined as:
+
+$$
+D \left(E ^ {1}, E ^ {2}\right) := \inf  _ {\mu^ {\prime} \leq \gamma_ {X} ^ {1} \wedge \gamma_ {X} ^ {2}} \| \hat {k} ^ {1} - \hat {k} ^ {1} \| _ {(\mu^ {\prime}) ^ {\otimes 2}} + \lambda \left(| \gamma_ {X} ^ {1} | ^ {2} + | \gamma_ {X} ^ {2} | ^ {2} - 2 | \mu^ {\prime} | ^ {2} + | \gamma_ {c} ^ {1} | + | \gamma_ {c} ^ {2} |\right). \tag {45}
+$$
+
+We have
+
+$$
+\begin{array}{l} L P G W _ {\lambda} (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}) \\ = \inf  _ {\gamma \in \Gamma_ {\leq} (\nu^ {1}, \nu^ {2}; \mu)} \int_ {(X \times Y ^ {1} \times Y ^ {2}) ^ {2}} \| g _ {Y ^ {1}} (y ^ {1}, y ^ {1 \prime}) - g _ {Y ^ {2}} (y ^ {2}, y ^ {2 \prime}) \| ^ {2} d \gamma (x, y ^ {1}, y ^ {2}) d \gamma (x ^ {\prime}, y ^ {1 \prime}, y ^ {2 \prime}) \\ + \lambda \left(\left| \nu^ {1} \right| ^ {2} + \left| \nu^ {2} \right| ^ {2} - | \gamma | ^ {2}\right) \\ = \inf  _ {\mu^ {\prime} \leq \gamma^ {1} \wedge \gamma^ {2}} \underbrace {\inf  _ {\substack {\gamma \in \Gamma (\gamma^ {1} (\cdot | x) , \gamma^ {2} (\cdot | x)) \\ \gamma^ {\prime} \in \Gamma (\gamma^ {1} (\cdot | x ^ {\prime}) , \gamma^ {2} (\cdot | x ^ {\prime}))}} \int_ {X ^ {2}} \| g _ {Y ^ {1}} (y ^ {1} , y ^ {1 \prime}) - g _ {Y ^ {2}} (y ^ {2} , y ^ {2 \prime}) \| d (\mu^ {\prime}) ^ {\otimes 2} (x , x ^ {\prime})} _ {\| \hat {k} ^ {1} - \hat {k} ^ {2} \| _ {L ((\mu^ {\prime}) ^ {\otimes 2})}} \\ + \lambda (\underbrace {| \nu^ {1} | ^ {2} - | \gamma_ {X} ^ {1} | ^ {2}} _ {| \gamma_ {c} ^ {1} |} + \underbrace {| \nu^ {2} | ^ {2} - | \gamma_ {X} ^ {2} | ^ {2}} _ {| \gamma_ {X} ^ {2} | ^ {2}} + | \gamma_ {X} ^ {1} | ^ {2} + | \gamma_ {X} ^ {2} | ^ {2} - 2 | \mu^ {\prime} | ^ {2}) \\ = D \left(E ^ {1}, E ^ {2}\right) \\ \end{array}
+$$
+
+That is, the distance between two embeddings coincides with the LPGW distance between the two gm-spaces.
+
+# D COMPUTATIONAL COMPLEXITY OF LPGW
+
+Computational Complexity of GW and PGW. The classical solvers for the GW and PGW problems are variants of the Frank-Wolf algorithms (Chapel et al., 2020; Bai et al., 2024).
+
+Consider metric measure spaces  $\mathbb{X}$  and  $\mathbb{Y}$ , with measures given by  $\mu = \sum_{i=1}^{n} p_i \delta_{x_i}$  and  $\nu = \sum_{j=1}^{m} q_j \delta_{y_j}$ , respectively. To achieve an  $\epsilon$ -accurate solution for the GW problem, the number of required iterations of the FW algorithm is
+
+$$
+\frac {\operatorname* {m a x} \left\{2 L _ {1} , 2 \cdot n ^ {2} m ^ {2} \operatorname* {m a x} (\{2 (C ^ {X}) ^ {2} + 2 (C ^ {Y}) ^ {2} \}) \right\} ^ {2}}{\epsilon^ {2}},
+$$
+
+and for PGW is
+
+$$
+\frac {\left. \max \left\{2 L _ {1} , 2 \min (| \mathrm {p} | , | \mathrm {q} |) \cdot n ^ {2} m ^ {2} \max (\{2 (C ^ {X}) ^ {2} + 2 (C ^ {Y}) ^ {2} , 2 \lambda \}) \right\} \right\rbrace^ {2}}{\epsilon^ {2}}.
+$$
+
+Here,  $L_{1}$  is a value determined by the initial guess  $\gamma^0$  used in the FW algorithm and  $C^{X},C^{Y}$  are the corresponding cost matrices for  $\mathbb{X},\mathbb{Y}$ . Note, since the largest value for  $\lambda$  is  $2\lambda = \max (2(C^{X})^{2},2(C^{Y})^{2})$ , the above two quantities coincide.
+
+In each iteration of FW, we are required to solve an OT/POT problem, whose complexity is  $\mathcal{O}(nm(n + m))$  (Bonneel et al., 2011). Thus, the theoretical time complexity to solve GW/PGW is  $\mathcal{O}\left(\frac{1}{\epsilon^2} nm(n + m)n^2 m^2\right)$ .
+
+Given  $K$  mm-spaces,  $\mathbb{X}_1,\ldots \mathbb{X}_K$ , then, computing their pairwise GW/PGW distances requires  $\binom{K}{2}\cdot\mathcal{O}\left(\frac{1}{\epsilon^2}nm(n+m)n^2m^2\right)$ . Meanwhile, computing the pairwise LPGW distances between them requires only
+
+$$
+K \mathcal {O} \left(\frac {1}{\epsilon^ {2}} n m (n + m) + n ^ {2} m ^ {2}\right) + \binom {K} {2} \mathcal {O} \left(n _ {0} ^ {2}\right), \tag {46}
+$$
+
+where the term  $\mathcal{O}(n_0^2)$  represents the time complexity of computing the distance between two embeddings, and  $n_0$  denotes the size of the reference space. In general, we would like to set  $n_0$  to be the mean/median/maximum of the sizes of  $K$  mm-spaces, and thus, this term can be ignored as compared to the first term.
+
+Remark D.1. We generally impose a fixed limit on the maximum number of iterations to be used in the FW algorithm (e.g., in PythonOT (Flamary et al., 2021), it is set to 1000 by default). In practice, this maximum number of iterations is generally not achieved. Hence, un-rigorously speaking, we can think of the complexity of GW/PGW as being  $\mathcal{O}(nm(n + m))$ .
+
+# E PROOF OF PROPOSITION 2.1
+
+# E.1 CLARIFICATION
+
+Although statements (1) and (2) in this proposition are not explicitly introduced in (Beier et al., 2022), they are implicitly mentioned, for example, in Proposition III.1 of such article. Thus, we do not claim the proofs of statements (1) and (2) as contributions of this paper. Additionally, to the best of our knowledge, the statement (3) has not yet been studied. We present the related proof as a complement to the work in (Beier et al., 2022).
+
+# E.2 CONJECTURE AND UNDERSTANDING
+
+Regarding statement (3), we conjecture that this conclusion can be extended to the case where  $g_{X}$  and  $g_{Y}$  are squared Euclidean distances. However, this statement may not hold for other gauge mappings. In our understanding, for general gauge mappings, achieving a similar result to the statement (3) would require defining a generalized barycentric projection mapping that is dependent on the specific gauge mapping.
+
+The formulation of a new barycentric projection mapping based on the chosen gauge mapping, as well as a generalized version of statement (3), is left for future work.
+
+# E.3 PROOF OF PROPOSITION 2.1: PART (1)
+
+Proof in a particular case. We first show the result in a simplified case. In particular, suppose  $\gamma^{*}$  satisfies the Monge mapping, i.e.  $\gamma^{*} = (\mathrm{id}\times T)_{\#}\mu$ , where  $T:X\to Y$ , and  $\gamma^0 = (\mathrm{id}\times \mathrm{id})_{\#}\mu \in \Gamma^* (\mathbb{X},\mathbb{X})$ . Then  $k_{\gamma^0}\equiv 0$  and thus
+
+$$
+\begin{array}{l} L G W (\mathbb {X}, \mathbb {Y}; \mathbb {X}, \gamma^ {0}, \gamma^ {*}) = \| k _ {\gamma^ {*}} - k _ {\gamma^ {0}} \| _ {\mu^ {\otimes 2}} ^ {2} = \| k _ {\gamma^ {*}} \| _ {\mu^ {\otimes 2}} ^ {2} \\ = \int_ {(X \times Y) ^ {\otimes 2}} \left| g _ {X} \left(x, x ^ {\prime}\right) - g _ {Y} \left(T (x), T \left(x ^ {\prime}\right)\right) \right| ^ {2} d \mu^ {\otimes 2} \\ = G W (\mathbb {X}, \mathbb {Y}). \\ \end{array}
+$$
+
+Proof in the general case. Consider  $\gamma^0\in \Gamma^* (\mathbb{X},\mathbb{X})$  , and  $\gamma \in \Gamma (\gamma^0,\gamma^*)$
+
+We claim the following lemma
+
+Lemma E.1. Choose  $\gamma^0\in \Gamma^* (\mathbb{X},\mathbb{X})$ $\gamma^{*}\in \mathcal{M}_{+}(\mathbb{X},\mathbb{Y})$  , for each  $\gamma \in \Gamma (\gamma^0,\gamma^*;\mu)$  , we have the following identity:
+
+$$
+\begin{array}{l} \int_ {(X \times X \times Y) ^ {\otimes 2}} \left| g _ {X} \left(x ^ {2}, x ^ {2 \prime}\right) - g _ {Y} \left(y, y ^ {\prime}\right) \right| ^ {2} d (\gamma) ^ {\otimes 2} \left(\left(x ^ {1}, x ^ {1 \prime}\right), \left(x ^ {2}, x ^ {2 \prime}\right), \left(y, y ^ {\prime}\right)\right) \\ = \int_ {(X \times Y) ^ {\otimes 2}} \left| g _ {X} \left(x, x ^ {\prime}\right) - g _ {Y} \left(y, y ^ {\prime}\right) \right| ^ {2} d \left(\gamma^ {*}\right) ^ {\otimes 2} \left(\left(x, x ^ {\prime}\right), \left(y, y ^ {\prime}\right)\right) \tag {47} \\ \end{array}
+$$
+
+Proof of Lemma (E.1). Since
+
+$$
+G W (\mathbb {X}, \mathbb {X}) = \int_ {(X \times X) ^ {\otimes 2}} \left| g _ {X} \left(x ^ {1}, x ^ {1 \prime}\right) - g _ {X} \left(x ^ {2}, x ^ {2 \prime}\right) \right| ^ {2} d \gamma^ {0} = 0,
+$$
+
+we have
+
+$$
+g _ {X} \left(x ^ {1}, x ^ {1 /}\right) - g _ {X} \left(x ^ {2}, x ^ {2 /}\right) = 0, \quad \gamma^ {0} - a. s.
+$$
+
+For convenience, let  $X^1 = X^2 = X$  be independent copies of  $X$ . We have:
+
+$$
+\begin{array}{l} \int_ {(X \times X \times Y) ^ {\otimes 2}} | g _ {X} (x ^ {2}, x ^ {2 \prime}) - g _ {Y} (y, y ^ {\prime}) | ^ {2} d \gamma^ {\otimes 2} ((x ^ {1}, x ^ {1 \prime}), (x ^ {2}, x ^ {2 \prime}), (y, y ^ {\prime})) \\ = \int_ {(X \times X ^ {0} \times Y) ^ {\otimes 2}} | g _ {X} (x ^ {2}, x ^ {2 \prime}) - g _ {X} (x ^ {1}, x ^ {1 \prime}) + g _ {X} (x ^ {1}, x ^ {1 \prime}) - g _ {Y} (y, y ^ {\prime}) | ^ {2} d \gamma^ {\otimes 2} ((x ^ {1}, x ^ {1 \prime}) (x ^ {2}, x ^ {2 \prime}), (y, y ^ {\prime})) \\ = \int_ {(X ^ {1} \times X ^ {2}) ^ {\otimes 2}} \left[ \int_ {(Y) ^ {\otimes 2}} \left| g _ {X} \left(x ^ {2}, x ^ {2 \prime}\right) - g _ {X} \left(x ^ {1}, x ^ {1 \prime}\right) + g _ {X} \left(x ^ {1}, x ^ {1 \prime}\right) - g _ {Y} \left(y, y ^ {\prime}\right) \right| ^ {2} \right. \\ \left. d \gamma_ {Y | \left(X ^ {1}, X ^ {2}\right)} ^ {\otimes 2} \left(\left(y, y ^ {\prime}\right) \mid \left(x ^ {1}, x ^ {1 ^ {\prime}}\right), \left(x ^ {2}, x ^ {2 ^ {\prime}}\right)\right) \right] d \left(\gamma^ {0}\right) ^ {\otimes 2} \left(\left(x ^ {1}, x ^ {1 ^ {\prime}}\right), \left(x ^ {2}, x ^ {2 ^ {\prime}}\right)\right) (48) \\ = \int_ {(X ^ {1} \times X ^ {2}) ^ {\otimes 2}} \left[ \int_ {Y ^ {\otimes 2}} | g _ {X} (x ^ {1}, x ^ {1 \prime}) - g _ {Y} (y, y ^ {\prime}) | ^ {2} d \gamma_ {Y | X ^ {1}, X ^ {2}} ^ {\otimes 2} ((y, y ^ {\prime}) | (x ^ {1}, x ^ {1 \prime}), (x ^ {2}, x ^ {2 \prime}) \right] \\ d \left(\gamma^ {0}\right) ^ {\otimes 2} \left(\left(x ^ {1}, x ^ {1 ^ {\prime}}\right), \left(x ^ {2}, x ^ {2 ^ {\prime}}\right)\right) \\ = \int_ {(X ^ {1} \times Y) ^ {\otimes 2}} | g _ {X} (x ^ {1}, x ^ {1 \prime}) - g _ {Y} (y, y ^ {\prime}) | ^ {2} d \gamma^ {\otimes 2} ((x ^ {1}, x ^ {2}, y), (x ^ {1 \prime}, x ^ {2 \prime}, y ^ {\prime})) \\ = \int_ {(X \times Y) ^ {\otimes 2}} \left| g _ {X} \left(x, x ^ {\prime}\right) - g _ {Y} \left(y, y ^ {\prime}\right) \right| ^ {2} d \left(\gamma^ {*}\right) ^ {\otimes 2} \left(\left(x, y\right), \left(x ^ {\prime}, y ^ {\prime}\right)\right) (49) \\ \end{array}
+$$
+
+where in (48)  $\gamma_{Y|X^1,X^2}(\cdot |x^1,x^2)$  is the disintegration of  $\gamma$  with respect to  $(\pi_{X^1,X^2})_\# \gamma = \gamma^0$ .
+
+Based on the identity (47) and the fact  $\gamma^1$  is optimal for problem  $GW(\mathbb{X},\mathbb{Y})$ , we have:
+
+$$
+\begin{array}{l} L G W (\mathbb {X}, \mathbb {Y}; \mathbb {X}, \gamma^ {0}, \gamma^ {*}) = \inf  _ {\gamma \in \Gamma (\gamma^ {0}, \gamma^ {*})} \int_ {(X \times X \times Y) ^ {\otimes 2}} | d _ {X} (x ^ {0}, x ^ {0 \prime}) - d _ {Y} (y, y ^ {\prime}) | ^ {2} d \gamma^ {\otimes 2} ((x, x ^ {\prime}), (x ^ {0}, x ^ {0 \prime}), (y, y ^ {\prime})) \\ = \inf  _ {\gamma \in \Gamma (\gamma^ {0}, \gamma^ {*})} G W (\mathbb {X}, \mathbb {Y}) \\ = G W (\mathbb {X}, \mathbb {Y}), \\ \end{array}
+$$
+
+and we have completed the proof.
+
+# E.4 PROOF OF PROPOSITION 2.1: PART (2)
+
+In this case,  $\gamma^{*} = (\mathrm{id}\times T)_{\#}\mu$ , for each  $x\in \operatorname{Supp}(\mu)$ , we have that the disintegration of  $\gamma^{*}$  with respect to its first marginal  $(\pi_X)_{\#}\gamma^{*} = \gamma_X^*$ , given first component is  $x$ , is
+
+$$
+\gamma_ {Y | X} ^ {*} (\cdot_ {2} | y ^ {\prime}) = \delta (\cdot_ {2} - T (x)),
+$$
+
+where  $\delta$  is the Dirac measure. Thus, for all  $x\in \mathrm{supp}(\mu)$ , we have:
+
+$$
+\mathcal {T} _ {\gamma^ {*}} (x) = \int_ {Y} y d \gamma_ {Y | X} ^ {*} (y | x) = \int_ {Y} y d \delta (y - T (x)) = T (x).
+$$
+
+where the last equality holds from the fact  $T(x) \in \mathrm{supp}(\nu) \subset Y$ . Therefore,  $\mathcal{T}_{\gamma^*} = T \mu$ -a.s. Since  $T_{\#} \mu = \nu$ , we have  $\widetilde{\nu}_{\gamma^*} := (\mathcal{T}_{\gamma^*})_{\#} \mu = T_{\#} \mu = \nu$ .
+
+# E.5 PROOF OF PROPOSITION 2.1: PART (3)
+
+# E.5.1 PROOF IN THE DISCRETE CASE
+
+We first demonstrate the proof in the following simplified discrete case: Suppose  $g_{X}(x,x^{\prime}) = x^{\top}x^{\prime}$  and  $g_{Y}(y,y^{\prime}) = y^{\top}y^{\prime}$ , and consider
+
+$$
+\mu = \sum_ {i = 1} ^ {n} p _ {i} \delta_ {x _ {i}}, \quad \nu = \sum_ {j = 1} ^ {m} q _ {j} \delta_ {y _ {j}} \quad \text {w i t h} \sum_ {i = 1} ^ {n} p _ {i} = 1 = \sum_ {j = 1} ^ {m} q _ {k}.
+$$
+
+In addition, we suppose that  $X$ ,  $Y$  are compact convex sets which contains 0 (besides containing  $\{x_1, \ldots, x_n\}$  and  $\{y_1, \ldots, y_m\}$ , respectively).
+
+Let  $\gamma^{*}\in \Gamma^{*}(\mathbb{X},\mathbb{Y})$  be an optimal transportation plan. Then, the corresponding barycentric projected measure is given by
+
+$$
+\widetilde {\nu} = \widetilde {\nu} _ {\gamma^ {*}} := \sum_ {i = 1} ^ {n} p _ {i} \delta_ {\widetilde {y} _ {i}}, \qquad \mathrm {w h e r e} \widetilde {y} _ {i} := \frac {1}{p _ {i}} \sum_ {j = 1} ^ {m} \gamma_ {i j} ^ {*} y _ {j}.
+$$
+
+By convexity,  $\widetilde{y}_1,\ldots ,\widetilde{y}_n\in Y$  . Thus  $\widetilde{\mathbb{Y}} := \widetilde{\mathbb{Y}}_{\gamma^*} = (X,g_Y,\widetilde{\nu})$
+
+Then, it induces a transportation plan  $\widetilde{\gamma}^{*} \coloneqq \mathrm{diag}(p_{1}^{0}, \dots, p_{n}^{0}) \in \mathbb{R}_{+}^{n \times n}$  with first and second marginals  $\mu$  and  $\widetilde{\nu}$ , respectively. We will show that for any transportation plan  $\widetilde{\gamma} \in \Gamma(\mu, \widetilde{\nu})$ , it holds that
+
+$$
+C (\widetilde {\gamma} ^ {*}; \mathbb {X}, \widetilde {\mathbb {Y}}) \leq C (\widetilde {\gamma}; \mathbb {X}, \widetilde {\mathbb {Y}}),
+$$
+
+where
+
+$$
+C (\gamma ; \mathbb {X}, \mathbb {Y}) = \sum_ {i, i ^ {\prime} = 1} ^ {n} \sum_ {j, j ^ {\prime} = 1} ^ {m} (x _ {i} ^ {\top} x _ {i ^ {\prime}} - y _ {j} ^ {\top} y _ {j ^ {\prime}}) ^ {2} \gamma_ {i j} \gamma_ {i ^ {\prime} j ^ {\prime}},
+$$
+
+in other words, we will show that  $\widetilde{\gamma}^*$  is optimal for  $\mathrm{GW}(\mathbb{X},\widetilde{\mathbb{Y}})$ .
+
+First, we notice that  $(\widetilde{\gamma}^{*})^{-1} = \mathrm{diag}(1 / p_{1}^{0},\dots ,1 / p_{n}^{0})$  . Then, we set
+
+$$
+\gamma := \widetilde {\gamma} (\widetilde {\gamma} ^ {*}) ^ {- 1} \gamma^ {*} \in \mathbb {R} _ {+} ^ {n \times m}.
+$$
+
+Secondly, we check that  $\gamma \in \Gamma(\mu, \nu)$ . In fact, let  $p = (p_1, \ldots, p_n)$ ,  $q = (q_1, \ldots, q_m)$  be the vectors of the weights of finite discrete measures  $\mu$  and  $\nu$ , respectively, then, since  $\gamma^* 1_m = p$  and  $\tilde{\gamma}^T 1_n = q$ , we have
+
+$$
+\gamma 1 _ {m} = \widetilde {\gamma} (\widetilde {\gamma} ^ {*}) ^ {- 1} \gamma^ {*} 1 _ {m} = \widetilde {\gamma} (\widetilde {\gamma} ^ {*}) ^ {- 1} p = \widetilde {\gamma} 1 _ {n} = q,
+$$
+
+$$
+\gamma^ {\top} 1 _ {n} = (\gamma^ {*}) ^ {\top} (\widetilde {\gamma} ^ {*}) ^ {- 1} \widetilde {\gamma} ^ {\top} 1 _ {n} = (\gamma^ {*}) ^ {\top} (\widetilde {\gamma} ^ {*}) ^ {- 1} q = (\gamma^ {*}) ^ {\top} 1 _ {m} = p.
+$$
+
+Lastly, we compute the costs as follows:
+
+$$
+\begin{array}{l} C (\gamma^ {*}; \mathbb {X}, \mathbb {Y}) = \sum_ {i, i ^ {\prime} = 1} ^ {n} (x _ {i} ^ {\top} x _ {i ^ {\prime}}) ^ {2} p _ {i} p _ {i ^ {\prime}} + \sum_ {j, j ^ {\prime} = 1} ^ {m} (y _ {j} ^ {\top} y _ {j ^ {\prime}}) ^ {2} q _ {j} q _ {j ^ {\prime}} - 2 \sum_ {i, i ^ {\prime} = 1} ^ {n} \sum_ {j, j ^ {\prime} = 1} ^ {m} (x _ {i} ^ {\top} x _ {i ^ {\prime}}) (y _ {j} ^ {\top} y _ {j ^ {\prime}}) \gamma_ {i j} ^ {*} \gamma_ {i ^ {\prime} j ^ {\prime}} ^ {*} \\ = J _ {1} - 2 \sum_ {i, i ^ {\prime} = 1} ^ {n} \sum_ {j, j ^ {\prime} = 1} ^ {m} (x _ {i} ^ {\top} x _ {i ^ {\prime}}) (y _ {j} ^ {\top} y _ {j ^ {\prime}}) \gamma_ {i j} ^ {*} \gamma_ {i ^ {\prime} j ^ {\prime}} ^ {*}. \\ \end{array}
+$$
+
+where  $J_{1} = \sum_{i,i^{\prime} = 1}^{n}(x_{i}^{\top}x_{i^{\prime}})^{2}p_{i}p_{i^{\prime}} + \sum_{j,j^{\prime} = 1}^{m}(y_{j}^{\top}y_{j^{\prime}})^{2}q_{j}q_{j^{\prime}}$  is independent of  $\gamma^{*}$ . By the multilinearity of the last term in the identity, we can show
+
+$$
+\begin{array}{l} C (\widetilde {\gamma} ^ {*}; \mathbb {X}, \widetilde {\mathbb {Y}}) = J _ {2} - 2 \sum_ {i, i ^ {\prime} = 1} ^ {n} \sum_ {\ell , \ell^ {\prime} = 1} ^ {n} (x _ {i} ^ {\top} x _ {i ^ {\prime}}) (\widetilde {y} _ {\ell} ^ {\top} \widetilde {y} _ {\ell^ {\prime}}) \widetilde {\gamma} _ {i \ell} ^ {*} \widetilde {\gamma} _ {i ^ {\prime}, \ell^ {\prime}} ^ {*} \\ = J _ {2} - 2 \sum_ {i, i ^ {\prime} = 1} ^ {n} \sum_ {\ell , \ell^ {\prime} = 1} ^ {n} x _ {i} ^ {\top} x _ {i ^ {\prime}} \sum_ {j, j ^ {\prime} = 1} ^ {m} \left(\frac {\gamma_ {\ell j} ^ {*} y _ {j} ^ {\top}}{p _ {\ell}} \frac {\gamma_ {\ell^ {\prime} j ^ {\prime}} ^ {*} y _ {j ^ {\prime}}}{p _ {\ell^ {\prime}}}\right) \widetilde {\gamma} _ {i \ell} \widetilde {\gamma} _ {i ^ {\prime} \ell^ {\prime}} \tag {50} \\ = J _ {2} - J _ {1} + C \left(\gamma^ {*}; \mu , \nu^ {1}\right), \\ \end{array}
+$$
+
+where  $J_{2} = \sum_{i,i^{\prime} = 1}^{n}(x_{i}^{\top}x_{i^{\prime}})^{2}p_{i}p_{i^{\prime}}^{0} + \sum_{\ell ,\ell^{\prime} = 1}^{n}(\widetilde{y}_{\ell}^{\top}\widetilde{y}_{\ell^{\prime}})^{2}p_{\ell}^{0}p_{\ell^{\prime}}^{0}$ . Similarly, we can compute
+
+$$
+C (\widetilde {\gamma}; \mathbb {X}, \widetilde {\mathbb {Y}}) = J _ {2} - J _ {1} + C (\gamma ; \mathbb {X}, \mathbb {Y}) \tag {51}
+$$
+
+The optimality of  $\gamma^{*}$  together with identities (50) and (51) implies that  $C(\widetilde{\gamma}^{*};\mathbb{X},\widetilde{\mathbb{Y}})\leq C(\widetilde{\gamma};\mathbb{X},\widetilde{\mathbb{Y}})$ .
+
+# E.5.2 PROOF IN THE GENERAL CASE
+
+Similar to the discrete case, given  $\gamma^{*}\in \Gamma^{*}(\mathbb{X},\mathbb{Y})$ , the barycentric projection is given by
+
+$$
+\mathcal {T} _ {\gamma^ {*}} (x) = \int_ {Y} y d \gamma_ {Y | X} ^ {*} (y | x).
+$$
+
+We have that  $\widetilde{\gamma}^{*} = (\mathrm{id}\times \mathcal{T}_{\gamma^{*}})_{\#}\mu$  is a joint measure with first and second marginals  $\mu$  and  $\widetilde{\nu} := \widetilde{\nu}_{\gamma^{*}} := (\mathcal{T}_{\gamma^{*}})_{\#}\mu$ , respectively. The goal is to show that  $\widetilde{\gamma}^{*}$  is optimal for  $GW(\mathbb{X},\widetilde{\mathbb{Y}})$  where  $\widetilde{\mathbb{Y}} := \widetilde{\mathbb{Y}}_{\gamma^{*}} := (\mathcal{T}_{\gamma^{*}})_{\#}\mu$ .
+
+Let  $\widetilde{\gamma} \in \Gamma(\mu, \widetilde{\nu}_{\gamma^*})$  be an arbitrary plan. Let  $\widetilde{\gamma}(\cdot_2|x) \coloneqq \widetilde{\gamma}_{Y|X}(\cdot_2|x)$  denote the disintegration of  $\widetilde{\gamma}$  with respect to  $\mu$ , given that the first component is  $x$ , for each  $x \in \operatorname{Supp}(\mu)$ . Similarly, we adopt the notation  $\gamma^*(\cdot_2|x)$  for each  $x \in \operatorname{Supp}(\mu)$ .
+
+We define the joint measure
+
+$$
+\gamma := \gamma_ {Y | X} ^ {*} \widetilde {\gamma} _ {X | Y} ^ {*} \widetilde {\gamma} \tag {52}
+$$
+
+on  $X\times Y$ . In particular, the above notation means that for each test function  $\phi \in C_0(X\times Y)$ , we have
+
+$$
+\begin{array}{l} \int_ {X \times Y} \phi (x, y) d \gamma (x, y) = \int_ {X \times Y \times X \times Y} \phi (x, y) d \gamma^ {*} (y | x ^ {0}) d \widetilde {\gamma} ^ {*} (x ^ {0} | \widetilde {y}) d \widetilde {\gamma} ^ {*} (x, \widetilde {y}) \\ = \int_ {X \times Y} \left(\int_ {X} \left(\int_ {Y} \phi (x, y) d \gamma^ {*} (y | x ^ {0})\right) d \widetilde {\gamma} ^ {*} (x ^ {0} | \widetilde {y})\right) d \widetilde {\gamma} ^ {*} (x, \widetilde {y}) \\ \end{array}
+$$
+
+For any test functions  $\phi_X\in C_0(X),\phi_Y\in C_0(Y)$  we have
+
+$$
+\begin{array}{l} \int_ {X \times Y} \phi_ {X} (x) d \gamma (x, y) = \int_ {X \times Y \times X \times Y} \phi_ {X} (x) d \gamma^ {*} (y | x ^ {0}) d \widetilde {\gamma} ^ {*} (x ^ {0} | \widetilde {y}) d \widetilde {\gamma} (x, \widetilde {y}) \\ = \int_ {X \times Y} \phi_ {X} (x) d \widetilde {\gamma} (x, \widetilde {y}) \\ = \int_ {X} \phi_ {X} (x) d \mu (x) \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \int_ {X \times Y} \phi_ {Y} (y) d \gamma (x, y) = \int_ {X \times Y \times X \times Y} \phi_ {Y} (y) d \gamma^ {*} (y | x ^ {0}) d \widetilde {\gamma} ^ {*} (x ^ {0} | \widetilde {y}) d \widetilde {\gamma} (x, \widetilde {y}) \\ = \int_ {X \times Y \times X} \phi_ {Y} (y) d \gamma^ {*} (y | x ^ {0}) d \widetilde {\gamma} ^ {*} (x ^ {0} | \widetilde {y}) d \widetilde {\nu} (\widetilde {y}) \\ = \int_ {X \times Y} \phi_ {Y} (y) d \gamma^ {*} (y | x ^ {0}) d \mu (x ^ {0}) \\ = \int_ {X \times Y} \phi_ {Y} (y) d \gamma^ {*} (x ^ {0}, y) \\ = \int_ {Y} \phi_ {Y} (y) d \nu (y) \\ \end{array}
+$$
+
+Thus  $\gamma \in \Gamma (\mu ,\nu)$
+
+In addition, we observe that we have the following property that holds for the case of considering inner products:
+
+Lemma E.2. Suppose  $X \subset \mathbb{R}^{d_x}, Y \subset \mathbb{R}^{d_y}$  are finite-dimensional, convex compact sets. Let  $\gamma \in \mathcal{M}_{+}(X \times Y)$  and define  $\mathcal{T}_{\gamma}$  as in (9). In addition, let  $g_{Y}(y, y') = \alpha \langle y, y' \rangle$ , where  $(y, y') \mapsto \langle y, y' \rangle := \sum_{i=1}^{d_1} y_i y_i'$  is an inner product (the standard one, for example). Then for each  $x, x' \in X$ , let  $\widetilde{y} = \mathcal{T}_{\gamma}(x)$ ,  $\widetilde{y}' = \mathcal{T}_{\gamma}(x')$ , we have:
+
+$$
+g _ {Y} (\widetilde {y}, \widetilde {y} ^ {\prime}) = \int_ {Y \times Y} g _ {Y} (y, y ^ {\prime}) d \gamma (y | x) d \gamma \left(y ^ {\prime} \mid x ^ {\prime}\right). \tag {53}
+$$
+
+Proof of Lemma E.2. For each  $i \in [1:d_y]$ , we have:
+
+$$
+\begin{array}{l} \widetilde {y} [ i ] \widetilde {y} ^ {\prime} [ i ] = \int_ {X} y [ i ] d \gamma (y | x) \int_ {X} y ^ {\prime} [ i ] d \gamma \left(y ^ {\prime} \mid x ^ {\prime}\right) \\ = \int_ {X} y [ i ] \int_ {y ^ {\prime}} y ^ {\prime} [ i ] d \gamma (y | x) d \gamma \left(y ^ {\prime} \mid x ^ {\prime}\right) \\ = \int_ {Y \times Y} y [ i ] y ^ {\prime} [ i ] d \gamma (y | x) d \gamma \left(y ^ {\prime} \mid x ^ {\prime}\right) \\ \end{array}
+$$
+
+where the third equality follows from Fubini's theorem.
+
+Thus,
+
+$$
+\begin{array}{l} g _ {Y} (\widetilde {y}, \widetilde {y} ^ {\prime}) = \alpha \sum_ {i = 1} ^ {d _ {y}} \widetilde {y} [ i ] \widetilde {y} ^ {\prime} [ i ] \\ = \alpha \int_ {Y \times Y} y [ i ] y ^ {\prime} [ i ] d \gamma (y | x) d \gamma \left(y ^ {\prime} \mid x ^ {\prime}\right) \\ = \int_ {Y \times Y} \alpha \sum_ {i = 1} ^ {d _ {y}} y [ i ] y ^ {\prime} [ i ] d \gamma (y | x) d \gamma \left(y ^ {\prime} | x ^ {\prime}\right) \\ = \int_ {Y \times Y} g _ {X} (y, y ^ {\prime}) d \gamma (y | x) d \gamma \left(y ^ {\prime} \mid x ^ {\prime}\right) \\ \end{array}
+$$
+
+![](images/42d69c223549f92de6a61caa54c30ee694510c1e47ea58ace44069a4f81a4cf1.jpg)
+
+Now, we continue with the proof of Proposition 2.1, part (3).
+
+We have
+
+$$
+\begin{array}{l} C \left(\gamma^ {*}; \mathbb {X}, \mathbb {Y}\right) = \int_ {\left(X \times Y\right) ^ {\otimes 2}} \left| g _ {X} \left(x, x ^ {\prime}\right) - g _ {Y} \left(y, y ^ {\prime}\right) \right| ^ {2} d \gamma^ {*} (x, y) d \gamma^ {*} \left(x ^ {\prime}, y ^ {\prime}\right) \\ = \| g _ {X} (x, x ^ {\prime}) \| _ {L ^ {2} (\mu^ {\otimes 2})} ^ {2} + \| g _ {Y} (y, y ^ {\prime}) \| _ {L ^ {2} (\nu^ {\otimes 2})} ^ {2} - 2 \int_ {(X \times X) ^ {\otimes 2}} g _ {X} (x, x ^ {\prime}) g _ {Y} (y, y ^ {\prime}) d \gamma^ {*} (x, y) d \gamma^ {*} (x ^ {\prime}, y ^ {\prime}) \\ = J _ {1} - 2 \left\langle g _ {X} g _ {Y}, \left(\gamma^ {*}\right) ^ {\otimes 2} \right\rangle \tag {54} \\ \end{array}
+$$
+
+where  $J_{1} = \| g_{X}\|_{L^{2}(\mu^{\otimes 2})}^{2} + \| g_{Y}\|_{L^{2}(\nu^{\otimes 2})}^{2}$  is independent to  $\gamma^{*}$ . Similarly,
+
+$$
+\begin{array}{l} C (\widetilde {\gamma} ^ {*}; \mathbb {X}, \widetilde {\mathbb {Y}}) \\ = \int_ {(X \times Y) ^ {\otimes 2}} | g _ {X} (x, x ^ {\prime}) - g _ {Y} (\widetilde {y}, \widetilde {y} ^ {\prime}) | ^ {2} d \widetilde {\gamma} ^ {*} (x, y) d \widetilde {\gamma} ^ {*} (x ^ {\prime}, y ^ {\prime}) \\ = \| g _ {X} \| _ {L ^ {2} \left(\mu^ {\otimes 2}\right)} ^ {2} + \| g _ {Y} \| _ {L ^ {2} \left(\widetilde {\nu} ^ {\otimes 2}\right)} ^ {2} - 2 \int_ {(X \times Y) ^ {\otimes 2}} g _ {X} \left(x, x ^ {\prime}\right) g _ {Y} \left(\widetilde {y}, \widetilde {y} ^ {\prime}\right) d \widetilde {\gamma} ^ {*} (x, \widetilde {y}) d \widetilde {\gamma} ^ {*} \left(x ^ {\prime}, \widetilde {y} ^ {\prime}\right) \\ = J _ {2} - 2 \int_ {X ^ {\otimes 2}} g _ {X} \left(x, x ^ {\prime}\right) g _ {Y} \left(\mathcal {T} _ {\gamma^ {*}} (x), \mathcal {T} _ {\gamma^ {*}} \left(x ^ {\prime}\right)\right) d \mu (x) d \mu \left(x ^ {\prime}\right) (55) \\ = J _ {2} - 2 \int_ {X ^ {\otimes 2}} g _ {X} \left(x, x ^ {\prime}\right) \left[ \int_ {Y ^ {\otimes 2}} g _ {Y} \left(y, y ^ {\prime}\right) d \gamma^ {*} (y | x) d \gamma^ {*} \left(y ^ {\prime} \mid y ^ {\prime}\right) \right] d \mu (x) d \mu \left(x ^ {\prime}\right) (56) \\ = J _ {2} - 2 \int_ {(X \times Y) ^ {\otimes 2}} g _ {X} (x, x ^ {\prime}) g _ {Y} (y, y ^ {\prime}) d \gamma^ {*} (x, y) d \gamma^ {*} (x ^ {\prime}, y ^ {\prime}) \\ = J _ {2} - 2 \left\langle g _ {X} g _ {Y}, \left(\gamma^ {*}\right) ^ {\otimes 2} \right\rangle (57) \\ \end{array}
+$$
+
+where in (55),  $J_{2} = \| g_{X}(\cdot_{1},\cdot_{2})\|_{L^{2}(\mu^{\otimes 2})}^{2} + \| g_{Y}(\cdot_{1},\cdot_{2})\|_{L^{2}(\widetilde{\nu}^{\otimes 2})}^{2}$  (which is independent of  $\widetilde{\gamma}^*$ ); and (56) follows from Lemma E.2. Combining (54) and (57), we have
+
+$$
+C \left(\widetilde {\gamma} ^ {*}; \mathbb {X}, \widetilde {\mathbb {Y}}\right) = C \left(\gamma^ {*}; \mathbb {X}, \mathbb {Y}\right) - J _ {1} + J _ {2}. \tag {58}
+$$
+
+Similarly, we can show that
+
+$$
+C (\widetilde {\gamma}; \mathbb {X}, \widetilde {\mathbb {Y}}) = C (\gamma ; \mathbb {X}, \mathbb {Y}) - J _ {1} + J _ {2}. \tag {59}
+$$
+
+Also, similarly to (54) and (57), we have
+
+$$
+C (\gamma ; \mathbb {X}, \mathbb {Y}) = J _ {1} - 2 \langle g _ {X} g _ {Y}, \gamma^ {\otimes 2} \rangle \tag {60}
+$$
+
+$$
+C (\widetilde {\gamma}; \mathbb {X}, \widetilde {\mathbb {Y}}) = J _ {2} - 2 \langle g _ {X} g _ {Y}, \widetilde {\gamma} ^ {\otimes 2} \rangle \tag {61}
+$$
+
+It remains to show
+
+$$
+\langle g _ {X} g _ {Y}, \gamma^ {\otimes 2} \rangle = \langle g _ {X} g _ {Y}, \widetilde {\gamma} ^ {\otimes 2} \rangle . \tag {62}
+$$
+
+Indeed,
+
+$$
+\begin{array}{l} \langle g _ {X} g _ {Y}, \widetilde {\gamma} ^ {\otimes 2} \rangle = \int_ {(X \times Y) ^ {\otimes 2}} g _ {X} (x, x ^ {\prime}) g _ {Y} (\widetilde {y}, \widetilde {y} ^ {\prime}) d \widetilde {\gamma} (x, \widetilde {y}) d \widetilde {\gamma} (x ^ {\prime}, \widetilde {y} ^ {\prime}) \\ = \int_ {(X \times Y) ^ {\otimes 2}} g _ {X} (x, x ^ {\prime}) g _ {Y} (\widetilde {y}, \widetilde {y} ^ {\prime}) d \widetilde {\gamma} (\widetilde {y} | x) d \widetilde {\gamma} (\widetilde {y} ^ {\prime} | x ^ {\prime \prime}) d \mu (x) d \mu \left(x ^ {\prime}\right) \\ = \int_ {(X \times Y) ^ {\otimes 2}} g _ {X} (x, x ^ {\prime}) \left[ \int_ {X ^ {\otimes 2}} g _ {Y} \left(\mathcal {T} _ {\gamma^ {*}} \left(x ^ {0}\right), \mathcal {T} _ {\gamma^ {*}} \left(x ^ {0 ^ {\prime}}\right)\right) d \widetilde {\gamma} ^ {*} \left(x ^ {0} | \widetilde {y}\right) d \widetilde {\gamma} ^ {*} \left(x ^ {0 ^ {\prime}} | \widetilde {y} ^ {\prime}\right) \right] \\ d \widetilde {\gamma} (\widetilde {y} | x) d \widetilde {\gamma} \left(\widetilde {y} ^ {\prime} \mid x ^ {\prime}\right) d \mu (x) d \mu \left(x ^ {\prime}\right) (63) \\ = \int_ {(X \times Y \times X) ^ {\otimes 2}} g _ {X} (x, x ^ {\prime}) g _ {Y} \left(\mathcal {T} _ {\gamma^ {*}} \left(x ^ {0}\right), \mathcal {T} _ {\gamma^ {*}} \left(x ^ {0 ^ {\prime}}\right)\right) \\ d \widetilde {\gamma} ^ {*} \left(x ^ {0} | \widetilde {y}\right) d \widetilde {\gamma} ^ {*} \left(x ^ {0 ^ {\prime}} | \widetilde {y} ^ {\prime}\right) d \widetilde {\gamma} (\widetilde {y} | x) d \widetilde {\gamma} \left(\widetilde {y} ^ {\prime} | x ^ {\prime \prime}\right) d \mu (x) d \mu \left(x ^ {\prime}\right) \\ = \int_ {(X \times Y \times X \times Y) ^ {\otimes 2}} g _ {X} (x, x ^ {\prime}) \left[ \int_ {Y ^ {\otimes 2}} g _ {Y} (y, y ^ {\prime}) d \gamma^ {*} (y | x ^ {0}) d \gamma^ {*} (y ^ {\prime} | x ^ {0 \prime}) \right] \\ d \widetilde {\gamma} ^ {*} \left(x ^ {0} | \widetilde {y}\right) d \widetilde {\gamma} ^ {*} \left(x ^ {0 ^ {\prime}} | \widetilde {y} ^ {\prime}\right) d \widetilde {\gamma} (\widetilde {y} | x) d \widetilde {\gamma} \left(\widetilde {y} ^ {\prime} | x ^ {\prime}\right) d \mu (x) d \mu \left(x ^ {\prime}\right) (64) \\ = \int_ {(X \times Y \times X \times Y) ^ {\otimes 2}} g _ {X} (x, x ^ {\prime}) g _ {Y} (y, y ^ {\prime}) \left(d \gamma^ {*} (y | x ^ {0}) d \widetilde {\gamma} ^ {*} (x ^ {0} | \widetilde {y}) d \widetilde {\gamma} (\widetilde {y} | x) d \mu (x)\right) \\ \left(d \gamma^ {*} \left(y ^ {\prime} \mid x ^ {\prime 0 ^ {\prime}}\right) d \widetilde {\gamma} ^ {*} \left(x ^ {0 ^ {\prime}} \mid \widetilde {y} ^ {\prime}\right) d \widetilde {\gamma} \left(\widetilde {y} ^ {\prime} \mid x ^ {\prime}\right) d \mu \left(x ^ {\prime}\right)\right) \\ = \int_ {(X \times Y) ^ {\otimes 2}} g _ {X} (x, x ^ {\prime}) g _ {Y} (y, y ^ {\prime}) d \gamma (x, x) d \gamma (x ^ {\prime}, y ^ {\prime}) \\ = \langle g _ {X} g _ {Y}, \gamma^ {\otimes 2} \rangle \\ \end{array}
+$$
+
+where (63) holds since given  $\widetilde{y}$ , the probability measure  $\widetilde{\gamma}^{*}(\cdot_{1}|\widetilde{y})$  is only supported on  $\{x^0\in X:\widetilde{y} = \mathcal{T}_{\gamma^*}(x^0)\}$ , similarly to measure  $\gamma (\cdot_1|\widetilde{y}')$ ; (64) follows from Lemma E.2 (that is, from equality (53)).
+
+Combining (59), (58), (62) and the fact  $C(\gamma^{*};\mathbb{X},\mathbb{Y}) \leq C(\gamma ;\mathbb{X},\mathbb{Y})$ , we obtain that  $C(\widetilde{\gamma}^{*};\mathbb{X},\mathbb{Y}) \leq C(\widetilde{\gamma};\mathbb{X},\mathbb{Y})$  which completes the proof.
+
+# F PROOF OF PROPOSITION 3.1
+
+Consider gm-spaces  $\mathbb{X},\mathbb{Y}^1,\mathbb{Y}^2$ , and select  $\gamma^1\in \Gamma_{\leq ,\lambda}^{*}(\mathbb{X},\mathbb{Y}^{1}),\gamma^{2}\in \Gamma_{\leq ,\lambda}^{*}(\mathbb{X},\mathbb{Y}^{2})$ . Inspired by the LGW distance (38), we propose the LPGW distance (conditioned on  $\gamma^1,\gamma^2$ ) (15) in the general case:
+
+$$
+\begin{array}{l} L P G W _ {\lambda} (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}) := \inf  _ {\gamma \in \Gamma_ {\leq} (\gamma^ {1}, \gamma^ {2}; \mu)} \int_ {(X \times Y ^ {1} \times Y ^ {2}) ^ {\otimes 2}} | g _ {Y ^ {1}} (y ^ {1}, y ^ {1 \prime}) - g _ {Y ^ {2}} (y ^ {2}, y ^ {2 \prime}) | ^ {2} d \gamma^ {\otimes 2} \\ + \lambda \left(\left| \nu^ {1} \right| ^ {2} + \left| \nu^ {2} \right| ^ {2} - 2 | \gamma | ^ {2}\right). \tag {65} \\ \end{array}
+$$
+
+where  $\Gamma_{\leq}(\gamma^{1},\gamma^{2};\mu)\coloneqq \{\gamma \in \mathcal{M}_{+}(X\times Y^{1}\times Y^{2}):\left(\pi_{X,Y^{1}}\right)_{\#}\gamma \leq \gamma^{1},\left(\pi_{X,Y^{2}}\right)_{\#}\gamma \leq \gamma^{2}\}$ .
+
+Next, we discuss the proof of Proposition 3.1.
+
+# F.1 PROOF OF PROPOSITION 3.1: PART (1)
+
+In this section, we discuss the proof of statement (1) of Proposition 3.1; namely, we prove the existence of a minimizer to the LPGW problem.
+
+First, we introduce a series of lemmas.
+
+Lemma F.1. Given Radon measures  $\mu, \nu^1, \nu^2$ , and  $\gamma^1 \in \Gamma_{\leq}(\mu, \nu^1)$ ,  $\gamma^2 \in \Gamma_{\leq}(\mu, \nu^2)$ , then
+
+$$
+\Gamma_ {\leq} (\gamma^ {1}, \gamma^ {2}; \mu)
+$$
+
+is sequentially compact set.
+
+Proof. The main idea is similar to the proof in, e.g., Lemma B.2 (Liu et al., 2023). In particular, consider a sequence  $(\gamma^n) \subset \Gamma_{\leq, X}(\gamma_X, \gamma_Y)$ . It is straightforward to verify that this sequence is bounded above in total variation (since  $\gamma^1$  and  $\gamma^2$  are finite measures) and that it is also a tight sequence. This verification is even simpler because  $X$ ,  $Y^1$ , and  $Y^2$  are compact. Thus, by Prokhorov's theorem for signed measures, the closure of  $\Gamma_{\leq}(\gamma^1, \gamma^2)$  (in the weak topology) is weakly sequentially compact in  $\mathcal{M}(X \times Y^1 \times Y^2)$ .
+
+It remains to show  $\Gamma_{\leq}(\gamma^{1},\gamma^{2};\mu)$  is closed.
+
+Suppose  $\gamma^n\xrightarrow{w*}\gamma\in\mathcal{M}(X\times Y^1\times Y^2)$ , for each nonnegative test function  $\phi\in C(X\times Y^1)$  we have
+
+$$
+\begin{array}{l} \lim  _ {n \rightarrow \infty} \int_ {X \times Y ^ {1}} \phi (x, y ^ {1}) d \left(\pi_ {X, Y ^ {1}}\right) _ {\#} \gamma^ {n} (x, y ^ {1}) = \lim  _ {n \rightarrow \infty} \int_ {X \times Y ^ {1} \times Y ^ {2}} \phi (x, y ^ {1}) d \gamma^ {n} (x, y ^ {1}, y ^ {2}) \\ = \int_ {X \times Y ^ {1} \times Y ^ {2}} \phi (x, y ^ {1}) d \gamma (x, y ^ {1}, y ^ {2}) \\ = \int_ {X \times Y ^ {1}} \phi (x, y ^ {1}) d \left(\pi_ {X, Y ^ {1}}\right) _ {\#} \gamma (x, y ^ {1}) \\ \end{array}
+$$
+
+That is,  $(\pi_{X,Y^1})_\# \gamma^n \stackrel{w*}{\rightharpoonup} (\pi_{X,Y^1})_\# \gamma$ . By Lemma B.1 in (Liu et al., 2023), we have  $(\pi_{X,Y^1})_\# \gamma \leq \gamma^1$ . Similarly we have  $(\pi_{X,Y^2})_\# \gamma \leq \gamma^2$ .
+
+Thus,  $\gamma \in \Gamma_{\leq}(\gamma^{1},\gamma^{2};\mu)$  and we have completed the proof.
+
+![](images/318b0be044e4b61795e1ce593a5bc0380c468348210a40f34e1e7a688a337027.jpg)
+
+Lemma F.2. Suppose  $g_{X}, g_{Y}$  are continuous functions (see assumption (3)), we claim that the mapping
+
+$$
+(X \times Y) ^ {\otimes 2} \ni ((x, y), (x ^ {\prime}, y ^ {\prime})) \mapsto | g _ {X} (x, x ^ {\prime}) - g _ {Y} (y, y ^ {\prime}) | ^ {2} \in \mathbb {R}
+$$
+
+is a Lipschitz function with respect to metric
+
+$$
+D _ {X \times Y} ((x, y), (x ^ {\prime}, y ^ {\prime})) = d _ {X} (x, x ^ {\prime}) + d _ {Y} (y, y ^ {\prime}).
+$$
+
+Proof of Lemma F.2. Let the mapping in the statement of the lemma be denoted by  $\Phi$ . For  $x_{1}, x_{1}^{\prime}, x_{2}, x_{2}^{\prime} \in X, y_{1}, y_{1}^{\prime}, y_{2}, y_{2}^{\prime} \in Y$  we have
+
+$$
+\begin{array}{l} \left| \Phi \left(\left(x _ {1}, y _ {1}\right), \left(x _ {1} ^ {\prime}, x _ {1} ^ {\prime}\right)\right) - \Phi \left(\left(x _ {2}, y _ {2}\right), \left(x _ {2} ^ {\prime}, y _ {2} ^ {\prime}\right)\right) \right| \\ = \left| \left| g _ {X} \left(x _ {1}, x _ {1} ^ {\prime}\right) - g _ {Y} \left(y _ {1}, y _ {1} ^ {\prime}\right) \right| ^ {2} - \left| g _ {X} \left(x _ {2}, x _ {2} ^ {\prime}\right) - g _ {Y} \left(y _ {2}, y _ {2} ^ {\prime}\right) \right| ^ {2} \right| \\ \leq \left| \left| g _ {X} \left(x _ {1}, x _ {1} ^ {\prime}\right) - g _ {Y} \left(y _ {1}, y _ {1} ^ {\prime}\right) \right| ^ {2} - \left| g _ {X} \left(x _ {2}, x _ {2} ^ {\prime}\right) - g _ {Y} \left(y _ {1}, y _ {1} ^ {\prime}\right) \right| ^ {2} \right| \\ + \left| \left| g _ {X} \left(x _ {2}, x _ {2} ^ {\prime}\right) - g _ {Y} \left(y _ {1}, y _ {1} ^ {\prime}\right) \right| ^ {2} - \left| g _ {X} \left(x _ {2}, x _ {2} ^ {\prime}\right) - g _ {Y} \left(y _ {2}, y _ {2} ^ {\prime}\right) \right| ^ {2} \right| \\ \leq K _ {1} \left(\left| g _ {X} \left(x _ {1}, x _ {1} ^ {\prime}\right) - g _ {X} \left(x _ {2}, x _ {2} ^ {\prime}\right) \right| + \left| g _ {Y} \left(y _ {1}, y _ {1} ^ {\prime}\right) - g _ {Y} \left(y _ {2}, y _ {2} ^ {\prime}\right) \right|\right) (66) \\ = K _ {1} K _ {2} \left| d _ {X} \left(x _ {1}, x _ {2}\right) + d _ {X} \left(x _ {1} ^ {\prime}, x _ {2} ^ {\prime}\right) + d _ {Y} \left(y _ {1}, y _ {2}\right) + d _ {Y} \left(y _ {1} ^ {\prime}, y _ {2} ^ {\prime}\right) \right| (67) \\ \end{array}
+$$
+
+where (66) follows from the fact that the function  $r_1 \mapsto |r_1 - r_2|^2$  is Lipschitz on a compact set (see, e.g., Lemma C.1 in (Bai et al., 2024)), and  $K_1 \geq 0$  is its Lipschitz constant. Then, (67) follows from assumption (3) for  $g_X$  and  $g_Y$ , with  $K_2 \geq 0$  being the maximum of their Lipschitz constants.
+
+Consider  $(\gamma^n)\subset \Gamma_{\leq}(\gamma^1,\gamma^2;\mu)$ , such that
+
+$$
+\langle |g_{Y^{1}} - g_{Y^{2}}|^{2} - 2\lambda ,(\gamma^{n})^{\otimes 2}\rangle \to \inf_{\gamma \in \Gamma_{\leq}(\gamma^{1},\gamma^{2};\mu)}\langle |g_{Y^{1}} - g_{Y^{2}}|^{2} - 2\lambda ,\gamma^{\otimes 2}\rangle .
+$$
+
+By compactness of  $\Gamma_{\leq}(\gamma^{1},\gamma^{2};\mu)$  (see Lemma F.1), we have that there exists  $\gamma^{*}\in$ $\Gamma_{\leq}(\gamma^{1},\gamma^{2};\mu)$  which is sub-sequence limit of  $\gamma^n$  (in weak convergence).
+
+By Lemma F.2 and Lemma C.2.(3) in (Bai et al., 2024), we have
+
+$$
+\langle | g _ {Y ^ {1}} - g _ {Y ^ {2}} | ^ {2} - 2 \lambda , (\gamma^ {*}) ^ {\otimes 2} \rangle = \inf  _ {\gamma \in \Gamma_ {\leq} (\gamma^ {1}, \gamma^ {2}; \mu)} \langle | g _ {Y ^ {1}} - g _ {Y ^ {2}} | ^ {2} - 2 \lambda , \gamma^ {\otimes 2} \rangle .
+$$
+
+Thus,  $\gamma^{*}$  is a minimizer, and we have completed the proof.
+
+# F.2 PROOF OF PROPOSITION 3.1: PART (2)
+
+Proof. Under the Monge mapping assumptions, we have
+
+$$
+\Gamma_ {\leq} (\gamma^ {1}, \gamma^ {2}; \mu) = \left\{\left(\operatorname {i d} \times T ^ {1} \times T ^ {2}\right) _ {\#} \mu^ {\prime}: \mu^ {\prime} \leq \gamma_ {X} ^ {1} \wedge \gamma_ {X} ^ {2} \right\}.
+$$
+
+Thus, (65) becomes
+
+$$
+\begin{array}{l} L P G W (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}) \\ = \inf  _ {\mu^ {\prime} \leq \gamma_ {x} ^ {1} \wedge \gamma_ {X} ^ {2}} \int_ {X ^ {\otimes 2}} | g _ {Y ^ {1}} (T ^ {1} (x), T ^ {1} (x ^ {\prime})) - g _ {Y ^ {2}} (T ^ {2} (x), T ^ {2} (x ^ {\prime})) | ^ {2} + \lambda (| \nu^ {1} | ^ {2} + | \nu^ {2} | ^ {2} - 2 | (\mathrm {i d} \times T ^ {1} \times T ^ {2}) _ {\#} \mu^ {\prime} | ^ {2}) \\ = \inf  _ {\mu^ {\prime} \leq \gamma_ {X} ^ {\prime} \wedge \gamma_ {X} ^ {2}} \int_ {X ^ {\otimes 2}} \left| g _ {Y ^ {1}} \left(T ^ {1} (x), T ^ {1} \left(x ^ {\prime}\right)\right) - g _ {Y ^ {2}} \left(T ^ {2} (x), T ^ {2} \left(x ^ {\prime}\right)\right) \right| ^ {2} + \lambda \left(\left| \nu^ {1} \right| ^ {2} + \left| \nu^ {2} \right| ^ {2} - 2 \left| \mu^ {\prime} \right| ^ {2}\right) \\ = (1 4). \\ \end{array}
+$$
+
+and we have proven the statement.
+
+![](images/72bc40c884d57060db5d8b44069d8c7c5b76d34dd034a50edffcc7eed5307790.jpg)
+
+# F.3 PROOF OF PROPOSITION 3.1: PART (3)
+
+Since  $X, Y^1, Y^2$  are compact, by continuity (see (3)), we have that  $g_{X}, g_{Y^1}, g_{Y^2}$  are bounded functions.
+
+By Lemma E.2 in (Bai et al., 2024), when  $2\lambda > \max(|g_{Y^1}(y^1, y^{1'}) - g_{Y^2}(y^2, y^{2'})|^2)$ , pick any optimal
+
+$$
+\gamma \in \Gamma_ {\leq} (\gamma^ {1}, \gamma^ {2}; \mu) = \{(\mathrm {i d} \times T ^ {1} \times T ^ {2}) _ {\#} \mu^ {\prime}: \mu^ {\prime} \leq \gamma_ {X} ^ {1} \wedge \gamma_ {X} ^ {2} \},
+$$
+
+we have  $|\gamma| = \min(|\gamma^1|, |\gamma^2|) = \min(|\gamma_X^1|, |\gamma_X^2|)$ . Thus  $\gamma = (\mathrm{id} \times T^1 \times T^2)_{\#} \gamma_X^1 \wedge \gamma_X^2$ . Plug in this optimal  $\gamma$  into the LPGW problem (14) (or (15)), we obtain
+
+$$
+\begin{array}{l} L P G W (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}) \\ = \left\langle \left| g _ {Y ^ {1}} - g _ {Y ^ {2}} \right| ^ {2}, \left(\gamma_ {X} ^ {1} \wedge \gamma_ {X} ^ {2}\right) ^ {\otimes 2} \right\rangle + \lambda \left(\left| \nu^ {1} \right| ^ {2} + \left| \nu^ {2} \right| ^ {2} - 2 \left| \gamma_ {X} ^ {1} \wedge \gamma_ {X} ^ {2} \right| ^ {2}\right) \\ = \left\langle \left| k _ {\gamma^ {1}} - k _ {\gamma^ {2}} \right| ^ {2}, \left(\gamma_ {X} ^ {1} \wedge \gamma_ {X} ^ {2}\right) ^ {\otimes 2} \right\rangle + \lambda \left(\left| \nu^ {1} \right| ^ {2} + \left| \nu^ {2} \right| ^ {2} - 2 \left| \gamma_ {X} ^ {1} \wedge \gamma_ {X} ^ {2} \right| ^ {2}\right) \\ \end{array}
+$$
+
+and we complete the proof.
+
+# F.4 PROOF OF PROPOSITION 3.1: PART (4)
+
+First, under the Monge mapping assumption, an optimal PGW plan is of the form  $\gamma^{*} = (\mathrm{id}\times T)_{\#}\gamma_{X}^{*}$  for some  $\gamma_X^*\leq \mu$ , we have
+
+$$
+\begin{array}{l} \left\| k _ {\gamma^ {*}} \right\| _ {L ^ {2} \left(\left(\gamma_ {X} ^ {*}\right) ^ {\otimes 2}\right)} ^ {2} + \lambda \left(\left| \mu \right| ^ {2} - \left| \gamma_ {X} ^ {*} \right| ^ {2} + \left| \gamma_ {c} \right|\right) \\ = \int_ {X ^ {\otimes 2}} | g _ {X} (x, x ^ {\prime}) - g _ {Y} (T (x), T (x ^ {\prime})) | ^ {2} d (\gamma^ {*}) ^ {\otimes 2} + \lambda \left(| \mu^ {\otimes 2} - \left(\gamma_ {X} ^ {*}\right) ^ {\otimes 2} | + | \nu^ {\otimes 2} - \left(\gamma_ {Y} ^ {*}\right) ^ {\otimes 2} |\right) \\ = P G W _ {\lambda} (\mathbb {X}, \mathbb {Y}). \\ \end{array}
+$$
+
+In the general case, we first introduce the following lemma.
+
+Lemma F.3. If  $\gamma^0\in \Gamma_{\leq ,\lambda}^* (\mathbb{X},\mathbb{X})$  then  $(\pi_X)_{\#}\gamma^0 = \mu ,(\pi_Y)_{\#}\gamma^0 = \mu$
+
+Proof of Lemma F.3. Pick any  $\gamma \in \Gamma_{\leq}(\mathbb{X},\mathbb{X})$ , suppose  $|\gamma| < |\mu|$ , we have
+
+$$
+\begin{array}{l} C (\gamma ; \mathbb {X}, \mathbb {X}, \lambda) = \int_ {X ^ {\otimes 2}} | g _ {X} (x, x ^ {\prime}) - g _ {X} \left(x ^ {0}, x ^ {0 \prime}\right) | d \gamma^ {\otimes 2} + \lambda \left(2 | \mu | ^ {2} - 2 | \gamma | ^ {2}\right) \\ \geq \lambda (2 | \mu | ^ {2} - 2 | \gamma | ^ {2}) \\ > 0. \tag {68} \\ \end{array}
+$$
+
+However,  $PGW_{\lambda}(\mathbb{X}, \mathbb{X}) = 0$ . Thus,  $\gamma$  is not optimal.
+
+![](images/0572c830049c721ec8252155b0c88692dfaae5f75db39fe0da2ba26c62d5acf0.jpg)
+
+By the above lemma, we have  $|\gamma^{*}|\leq |\gamma^{0}|$ . For each  $\gamma \in \Gamma_{\leq}(\gamma^0,\gamma^*;\mu)$ , from (Bai et al., 2024), there exists  $\gamma^{\prime}\in \Gamma_{\leq}(\gamma^{0},\gamma^{*})$  such that  $(\pi_{X,Y})_{\#}\gamma^{\prime} = \gamma^{*}$  and  $\gamma \leq \gamma^\prime$ . Thus we have  $|\gamma^{\prime}| = |\gamma^{*}|$ .
+
+Let
+
+$$
+\begin{array}{l} C (\gamma ; \mathbb {X}, \mathbb {X}, \mathbb {Y}, \lambda) := \int_ {(X \times X \times Y) ^ {\otimes 2}} | g _ {X} (x, x ^ {\prime}) - g _ {Y} (y, y ^ {\prime}) | ^ {2} d \gamma^ {\otimes 2} ((x, x ^ {\prime}), (x, x ^ {\prime}), (y, y ^ {\prime})) \\ + \lambda (| \mu | ^ {2} + | \nu | ^ {2} - 2 | \gamma | ^ {2}) \\ \end{array}
+$$
+
+and similarly we define  $C(\gamma'; \mathbb{X}, \mathbb{X}, \mathbb{Y}, \lambda)$ . Then
+
+$$
+\begin{array}{l} C (\gamma^ {\prime}; \mathbb {X}, \mathbb {X}, \mathbb {Y}, \lambda) - C (\gamma ; \mathbb {X}, \mathbb {X}, \mathbb {Y}, \lambda) \\ = \int_ {(X \times X \times Y) ^ {\otimes 2}} \left[ | g _ {X} (x, x ^ {\prime}) - g _ {Y} (y, y ^ {\prime}) | ^ {2} - 2 \lambda \right] d (\gamma^ {\prime \otimes 2} - \gamma^ {\otimes 2}) ((x, x ^ {\prime}), (x, x ^ {\prime}), (y, y ^ {\prime})) \\ \leq 0. \\ \end{array}
+$$
+
+In addition,
+
+$$
+\begin{array}{l} C (\gamma^ {\prime}; \mathbb {X}, \mathbb {X}, \mathbb {Y}, \lambda) \\ = \int_ {(X \times X \times Y) ^ {\otimes 2}} \left| g _ {X} \left(x, x ^ {\prime}\right) - g _ {Y} \left(y, y ^ {\prime}\right) \right| ^ {2} d \gamma^ {\prime \otimes 2} \left(\left(x, x ^ {\prime}\right), \left(x, x ^ {\prime}\right), \left(y, y ^ {\prime}\right)\right) + \lambda \left(\left| \mu \right| ^ {2} + \left| \nu \right| ^ {2} - 2 \left| \gamma^ {\prime} \right| ^ {2}\right) \\ = \int_ {(X \times Y) ^ {\otimes 2}} | g _ {X} (x, x ^ {\prime}) - g _ {Y} (y, y ^ {\prime}) | ^ {2} d (\gamma^ {*}) ^ {\otimes 2} ((x, x ^ {\prime}), (y, y ^ {\prime})) + \lambda (| \mu | ^ {2} + | \nu | ^ {2} - 2 | \gamma^ {*} | ^ {2}) \\ = P G W _ {\lambda} (\mathbb {X}, \mathbb {Y}) \\ \end{array}
+$$
+
+where the second equality holds by (49) and the fact  $|\gamma^{\prime}| = |\gamma^{*}|$ . Thus, we have
+
+$$
+L P G W _ {\lambda} (\mathbb {X}, \mathbb {Y}; \mathbb {X}, \gamma^ {0}, \gamma^ {*}) \leq C (\gamma^ {\prime}; \mathbb {X}, \mathbb {X}, \mathbb {Y}, \lambda) = P G W _ {\lambda} (\mathbb {X}, \mathbb {Y}).
+$$
+
+Another direction is trivial since for each  $\gamma \in \Gamma_{\leq}(\gamma^0,\gamma^*;\mu)$ ,  $(\pi_{2,3})_{\#}\gamma \coloneqq (\pi_{X,Y})_{\#}\gamma \in \Gamma_{\leq}(\mu ,\nu)$ . Thus
+
+$$
+P G W _ {\lambda} (\mathbb {X}, \mathbb {Y}; \gamma) \leq C (\gamma ; \mathbb {X}, \mathbb {X}, \mathbb {Y}, \lambda) \leq L P G W _ {\lambda} (\mathbb {X}, \mathbb {Y}; \mathbb {X}, \gamma^ {0}, \gamma^ {*}).
+$$
+
+and we have completed the proof.
+
+# F.5 PROOF OF PROPOSITION 3.1: PART (5)
+
+This section discusses the proof of statement (5) of Proposition 3.1. That is, we discuss the metric properties of the proposed LPGW distance.
+
+# F.5.1 FORMAL STATEMENT
+
+In this section, we set  $g_{Y} = d_{Y}^{s}$ . Let
+
+$$
+\mathcal {G} := \{\mathbb {Y} = (Y, g _ {Y}, \nu) \}
+$$
+
+where  $Y \subset \mathbb{R}^d$  for some  $d \in \mathbb{N}$  such that  $Y$  is nonempty convex compact set;  $\nu \in \mathcal{M}_{+}(Y)$  with  $\nu^{\otimes 2}(|g_Y|) < \infty$ .
+
+Then  $\forall \mathbb{Y}^1, \mathbb{Y}^2 \in \mathcal{G}$ , we define  $\mathbb{Y}^1 \sim \mathbb{Y}^2$  iff  $GW(\mathbb{Y}^1, \mathbb{Y}^2) = 0$ . By (Mémoli, 2011), in this case, optimal transportation plan  $GW(\mathbb{Y}^1, \mathbb{Y}^2)$  is induced by the Monge mapping  $T$ . Furthermore,  $T$  is bijection  $\nu^1$ -a.s. and is called "isomorphism" since
+
+$$
+d _ {Y ^ {1}} \left(y ^ {1}, y ^ {1 ^ {\prime}}\right) = d _ {Y ^ {2}} \left(T \left(y ^ {1}\right), T \left(y ^ {1 ^ {\prime}}\right)\right), \nu^ {1} - a. s.
+$$
+
+Next, given  $\mathcal{G}' \subset \mathcal{G}$  we introduce the following assumptions:
+
+Assumption F.4. For each  $\mathbb{Y} \in \mathcal{G}'$ , the problem  $PGW_{\lambda}(\mathbb{X}, \mathbb{Y})$  admits a unique solution.
+
+Assumption F.5. For subset  $\mathcal{G}'$ , define  $K_{1}, K_{2}$  by
+
+$$
+K _ {1} = \sup  \left\{\left| \nu \right|, \mathbb {Y} = \left(Y, g _ {Y}, \nu\right) \in \mathcal {G} ^ {\prime} \right\},
+$$
+
+$$
+K _ {2} = \sup  \left\{\max  _ {y, y ^ {\prime} \in \operatorname {s u p p} (\nu)} | g _ {Y} (y, y ^ {\prime}) | ^ {2}: \mathbb {Y} = (X, g _ {Y}, \nu) \in \mathcal {G} ^ {\prime}) \right\}
+$$
+
+We suppose  $K_{1}, K_{2} < \infty$ .
+
+Assumption F.6.  $g_{X}, g_{Y^{1}}, g_{Y^{2}}$  are induced by metrics. In particular,  $g_{X}(\cdot, \cdot) = d_{X}^{q}(\cdot, \cdot)$ ,  $g_{Y^{1}}(\cdot, \cdot) = d_{Y^{1}}^{q}(\cdot, \cdot)$ ,  $g_{Y^{2}}(\cdot, \cdot) = d_{Y^{2}}^{q}(\cdot, \cdot)$ , where  $d_{X}$  is a metric defined in  $X$ ,  $q \geq 1$ , similar to  $d_{Y^{1}}, d_{Y^{2}}$ .
+
+Remark F.7. The uniqueness of optimal transportation plan assumption (F.4) is also introduced by (Nenna & Pass, 2023) to prove the triangle inequality of linear optimal transport. The assumption (F.5) discusses the bounded total mass and transportation cost.
+
+Proposition F.8 (Metric property of LPGW). Fix  $\mathbb{X} \in \mathcal{G}$  and for each  $\mathbb{Y}^1$ , we select  $\gamma^1 \in \Gamma_{\leq, \lambda}^*(\mathbb{X}, \mathbb{Y}^1)$ ,  $\gamma^2 \in \Gamma_{\leq, \lambda}^*(\mathbb{X}, \mathbb{Y}^2)$ . In addition, set  $g_X = d_X^s$ ,  $g_Y = d_Y^s$  where  $s \geq 1$ . Then we have:
+
+(1) The LPGW distances (15), (16) are nonnegative:
+
+$$
+L P G W (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}, \lambda), L P G W _ {\lambda} (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}) \geq 0 \tag {69}
+$$
+
+(2) The LPGW distances (15),(16) are symmetric:
+
+$$
+L P G W (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}, \lambda) = L P G W (\mathbb {Y} ^ {2}, \mathbb {Y} ^ {1} \mathbb {X}, \gamma^ {2}, \gamma^ {1}, \lambda)
+$$
+
+$$
+L P G W \left(\mathbb {Y} ^ {2}, \mathbb {Y} ^ {1}; \mathbb {X}, \lambda\right) = L P G W _ {\lambda} \left(\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}\right). \tag {70}
+$$
+
+(3) The LPGW distances (15) satisfy triangle inequality.
+
+In addition, under assumption (F.4), (16) satisfies the triangle inequality.
+
+(4) Under assumption (F.6), if  $LPGW_{\lambda}(\mathbb{Y}^1,\mathbb{Y}^2;\mathbb{X},\gamma^1,\gamma^2) = 0$ , we have  $\mathbb{Y}^1\sim \mathbb{Y}^2$ . Similarly to  $LPGW_{\lambda}(\mathbb{Y}^1,\mathbb{Y}^2;\mathbb{X})$
+
+(5) Under assumption (F.5), suppose  $|\mu| \geq K_1$ , and  $2\lambda \geq K_2 + \sup_{x,x' \in supp(\mu)} |g_X(x,x')|^2$ , and  $\mathbb{Y}^1 \sim \mathbb{Y}^2$ , where  $K_1, K_2$  are defined in (F.5). Then there exists  $\gamma^1, \gamma^2$  such that
+
+$$
+L P G W (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}, \lambda) = L P G W _ {\lambda} (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}) = 0.
+$$
+
+Therefore, under the assumptions (F.4), (F.5), and (F.6), we have that (16) defines a metric in  $\mathcal{G}^{\prime} / \sim$ .
+
+Note, the above statement implies the metric property of LGW by setting  $\lambda \to \infty$ . We refer to the next section for details.
+
+# F.5.2 PROOF OF PROPOSITION F.8: PARTS (1), (2), (4), (5)
+
+Proof. Choose  $\mathbb{Y}^1, \mathbb{Y}^2, \mathbb{Y}^3 \in \mathcal{G}$ .
+
+(1), (2) It is straightforward to verify the nonnegativity (1) and symmetry (2).
+
+(4) In this setting, we have:
+
+$$
+\begin{array}{l} 0 = L P G W _ {\lambda} \left(\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}\right) \\ = \inf  _ {\gamma \in \Gamma_ {\leq} (\gamma^ {1}, \gamma^ {2}; \mu)} \int_ {(X \times Y ^ {1} \times Y ^ {2}) ^ {\otimes 2}} | g _ {Y ^ {1}} (y ^ {1}, y ^ {1 \prime}) - g _ {Y ^ {2}} (y ^ {2}, y ^ {2 \prime}) | ^ {2} d \gamma^ {\otimes 2} + \lambda (| \nu^ {1} | ^ {2} + | \nu^ {2} | - 2 | \gamma | ^ {2}) \\ \end{array}
+$$
+
+Pick one corresponding minimizer  $\gamma$ , we obtain
+
+$$
+\int_ {(X \times Y ^ {1} \times Y ^ {2}) ^ {\otimes 2}} | g _ {Y ^ {1}} (y ^ {1}, y ^ {1 \prime}) - g _ {Y ^ {2}} (y ^ {2}, y ^ {2 \prime}) | ^ {2} d \gamma^ {\otimes 2} = 0,
+$$
+
+and
+
+$$
+| \nu^ {1} | ^ {2} + | \nu^ {2} | ^ {2} - 2 | \gamma | ^ {2} = 0.
+$$
+
+Thus  $\gamma \in \Gamma (\nu^1,\nu^2)$  and therefore
+
+$$
+G W (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}) = 0.
+$$
+
+That is  $\mathbb{Y}^1\sim \mathbb{Y}^2$
+
+By a similar process, if  $LPGW_{\lambda}(\mathbb{Y}^1,\mathbb{Y}^2;\mathbb{X}) = 0$  , we have  $\mathbb{Y}^1\sim \mathbb{Y}^2$
+
+(5) Since  $\mathbb{Y}^1\sim \mathbb{Y}^2$  , from triangle inequality of  $PGW_{\lambda}(\cdot ,\cdot)$  , we have
+
+$$
+P G W _ {\lambda} (\mathbb {X}, \mathbb {Y} ^ {1}) = P G W _ {\lambda} (\mathbb {X}, \mathbb {Y} ^ {2}).
+$$
+
+In addition, there exists a bijection mapping  $T: \operatorname{supp}(\nu^1) \to \operatorname{supp}(\nu^2)$  such that  $g_{Y^1}(x_1, x_1') = g_{X_2}(T(x_1), T(x_1'))$ ,  $\nu^1$ -a.s. such that  $T_\# \nu^1 = \nu^2$ .
+
+Pick  $\gamma^1\in \Gamma_{\leq ,\lambda}^* (\mathbb{X},\mathbb{Y}^1)$ . Let  $\gamma^2\coloneqq (\mathrm{id}\times T)_{\#}\gamma^1$ , that is, for each test function  $\phi \in C_0(X\times Y^2)$ ,
+
+$$
+\gamma^ {2} (\phi) := \int_ {X \times Y ^ {1}} \phi (x, T (y ^ {1})) d \gamma^ {1} (x, y ^ {1}).
+$$
+
+We claim  $\gamma^2$  is optimal for  $PGW_{\lambda}(\mathbb{X}^1,\mathbb{Y}^2)$
+
+$$
+\begin{array}{l} (\pi_ {1}) _ {\#} \gamma^ {2} = (\pi_ {1}) _ {\#} \gamma^ {1} \leq \mu , \\ \left(\pi_ {2}\right) _ {\#} \gamma^ {2} = T _ {\#} \left(\left(\pi_ {2}\right) _ {\#} \gamma^ {1})\right) \leq \nu^ {2} \quad \text {s i n c e} \left(\pi_ {Y}\right) _ {\#} \gamma^ {1} \leq \nu^ {1}. \\ \end{array}
+$$
+
+Thus,  $\gamma^2\in \Gamma_{\leq}(\mu ,\nu^2)$  . In addition,
+
+$$
+\begin{array}{l} C (\gamma^ {2}; \mathbb {X}, \mathbb {Y} ^ {2}) \\ = \int_ {(X \times Y ^ {1}) ^ {\otimes 2}} | g _ {X} (x, x ^ {\prime}) - g _ {Y ^ {2}} (T (y ^ {1}), T (y ^ {1}) ^ {\prime}) | ^ {2} d (\gamma^ {1}) ^ {\otimes 2} + \lambda \left(| \mu | ^ {2} + | \nu^ {2} | ^ {2} - 2 | \gamma^ {2} | ^ {2}\right) \\ = \int_ {(X \times Y ^ {1}) ^ {\otimes 2}} | g _ {X} (x, x ^ {\prime}) - g _ {Y ^ {2}} (T (y ^ {1}), T (y ^ {1}) ^ {\prime}) | ^ {2} d (\gamma^ {1}) ^ {\otimes 2} (x, x ^ {\prime} | y ^ {1}, y ^ {1 ^ {\prime}})) d (\nu^ {1}) ^ {\otimes 2} (y ^ {1}, y ^ {1 ^ {\prime}}) \\ + \lambda (| \mu | ^ {2} + | \nu^ {1} | - 2 | \gamma^ {1} | ^ {1}) \\ = \int_ {(X \times Y ^ {1}) ^ {\otimes 2}} \left| g _ {X} \left(x, x ^ {\prime}\right) - g _ {Y ^ {1}} \left(y ^ {1}, y ^ {1 \prime}\right) \right| ^ {2} d \left(\gamma^ {1}\right) ^ {\otimes 2} \left(x, x ^ {\prime} \mid y ^ {1}, y ^ {1 \prime}\right)) d \left(\nu^ {1}\right) ^ {\otimes 2} \left(y ^ {1}, y ^ {1 \prime}\right) \\ + \lambda \left(\left| \mu \right| ^ {2} - \left| \gamma^ {1} \right| ^ {2}\right) \tag {71} \\ = P G W _ {\lambda} (\mathbb {X}, \mathbb {Y} ^ {1}) = P G W _ {\lambda} (\mathbb {X}, \mathbb {Y} ^ {2}). \\ \end{array}
+$$
+
+and thus  $\gamma^2$  is optimal in  $PGW_{\lambda}(\mathbb{X},\mathbb{Y}^2)$
+
+Since  $2\lambda$  is sufficiently large, by lemma E.2 in (Bai et al., 2024), there exists optimal  $\gamma^1$  such that:
+
+$$
+| \gamma^ {1} | = \min  (| \mu |, | \nu^ {1} |) = | \nu^ {1} |.
+$$
+
+Thus we have  $|\gamma^2| = |\gamma^1| = |\nu^1| = |\nu^2|$ . Plug  $\gamma^1, \gamma^2$  into (15), we have:
+
+$$
+\begin{array}{l} L P G W (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}) \\ = \int g _ {Y ^ {1}} \left(y ^ {1}, y ^ {1 ^ {\prime}}\right) - g _ {Y ^ {2}} \left(T \left(y ^ {1}\right), T \left(y ^ {1 ^ {\prime}}\right)\right) d \left(\nu^ {1}\right) ^ {\otimes 2} \left(y ^ {1}, y ^ {1 ^ {\prime}}\right) \\ = 0. \tag {72} \\ \end{array}
+$$
+
+![](images/6aa6e13a67e656dc4c01b7575a083cd3d1bc9f9defbc951f9b7fb3b58e06ff40.jpg)
+
+# F.5.3 PROOF OF PROPOSITION F.8: PART (3)
+
+Notation Setup and Summary Choose  $\gamma^1\in \Gamma_{\leq ,\lambda}^* (\mathbb{X},\mathbb{Y}^1),\gamma^2\in \Gamma_{\leq ,\lambda}^* (\mathbb{X},\mathbb{Y}^2),\gamma^3\in$ $\Gamma_{<,\lambda}^{*}(\mathbb{X},\mathbb{Y}^{3})$  , the goal is to show triangle inequality in this section:
+
+$$
+L P G W _ {\lambda} \left(\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}\right) \leq L P G W _ {\lambda} \left(\mathbb {Y} ^ {1}, \mathbb {Y} ^ {3}; \mathbb {X}, \gamma^ {1}, \gamma^ {3}\right) + L P G W _ {\lambda} \left(\mathbb {Y} ^ {2}, \mathbb {Y} ^ {3}; \mathbb {X}, \gamma^ {2}, \gamma^ {3}\right). \tag {73}
+$$
+
+# Related Lemmas
+
+Lemma F.9 (Variant Gluing lemma). Suppose  $\gamma^1 \in \Gamma(\mu, \nu^1)$ ,  $\gamma^2 \in \Gamma(\mu, \nu^2)$ ,  $\gamma^3 \in \Gamma(\mu, \nu^3)$  are probability measures. Pick  $\gamma^{1,2} \in \Gamma(\gamma^1, \gamma^2; \mu)$ ,  $\gamma^{1,3} \in \Gamma(\gamma^1, \gamma^3; \mu)$ ,  $\gamma^{2,3'} \in \Gamma(\gamma^1, \gamma^{3'}; \mu)$ , there exists  $\gamma \in \mathcal{P}(X \times Y^1 \times Y^2 \times Y^3)$  such that
+
+$$
+\left\{ \begin{array}{l} \left(\pi_ {X, Y ^ {1}, Y ^ {3}}\right) _ {\#} \gamma = \gamma^ {1, 3}, \\ \left(\pi_ {X, Y ^ {2}, Y ^ {3}}\right) _ {\#} \gamma = \gamma^ {2, 3}. \end{array} \right. \tag {74}
+$$
+
+Note, if we set  $X = \{0\}$  and  $\mu = \delta_0$ , the above lemma becomes the classical Gluing lemma (see e.g. Lemma 5.5 (Santambrogio, 2015)). For this, we call it the variant gluing lemma.
+
+Proof of Lemma F.9. For convenience, let  $X', Y^{3'}$  be independent copy of  $X, Y^3$ . By gluing lemma, there exists  $\gamma' \in \mathcal{P}((X \times Y^1 \times Y^2) \times X')$  such that
+
+$$
+\left\{ \begin{array}{l} \left(\pi_ {X, Y ^ {1}, Y ^ {2}}\right) _ {\#} \gamma = \gamma^ {1, 2}, \\ \left(\pi_ {X, X ^ {\prime}}\right) _ {\#} \gamma = (\mathrm {i d} \times \mathrm {i d}) _ {\#} \mu . \end{array} \right. \tag {75}
+$$
+
+Apply gluing lemma again between  $\gamma^\prime$  and  $\gamma^{1,3}$ , we can find  $\gamma^{\prime \prime}\in \mathcal{P}((X\times Y^1\times Y^3)\times (X^{\prime}\times$ $Y^{2}\times Y^{3^{\prime}}))$  such that
+
+$$
+\left\{ \begin{array}{l} (\pi_ {X, Y ^ {1}, Y ^ {3}, X ^ {\prime}}) _ {\#} \gamma = \gamma^ {\prime}, \\ (\pi_ {X ^ {\prime}, Y ^ {2}, Y ^ {3}}) _ {\#} \gamma = \gamma^ {2, 3}. \end{array} \right.
+$$
+
+Applying  $\gamma = (\pi_{X,Y^1,Y^2,Y^3})_\# \gamma''$  we complete the proof.
+
+Alternatively, we can set
+
+$$
+\gamma = \gamma_ {Y ^ {1}, Y ^ {3} | X} ^ {1, 3} \gamma_ {Y ^ {2}, Y ^ {3} | X} ^ {2, 3} \mu (x)
+$$
+
+and it is straightforward to verify  $\gamma$  satisfies (74).
+
+![](images/5211f48cb53b408ecb56e7b151bfb170690c8b0b184259a0365d06c4ff9060ac.jpg)
+
+Lemma F.10 (Triangle inequality for LGW). Suppose  $\mathbb{X},\mathbb{Y}^1,\mathbb{Y}^2,\mathbb{Y}^3$  are under the balanced GW setting, choose  $\gamma^{1}\in \Gamma^{*}(\mu ,\nu^{1}),\gamma^{2}\in \Gamma^{*}(\mu ,\nu^{2}),\gamma^{3}\in \Gamma^{*}(\mu ,\nu^{3})$ , we have
+
+$$
+L G W \left(\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}\right) \leq L G W \left(\mathbb {Y} ^ {1}, \mathbb {Y} ^ {3}; \mathbb {X}, \gamma^ {1}, \gamma^ {3}\right) + L G W \left(\mathbb {Y} ^ {2}, \mathbb {Y} ^ {3}; \mathbb {X}, \gamma^ {2}, \gamma^ {3}\right) \tag {76}
+$$
+
+Proof of lemma F.10. Pick the  $\gamma$  from lemma F.9, we have
+
+$$
+\left(\pi_ {X, Y ^ {1}}\right) _ {\#} \gamma_ {X, Y ^ {1}} = \left(\pi_ {X, Y ^ {1}}\right) _ {\#} \gamma^ {1, 3} = \gamma^ {1},
+$$
+
+$$
+\left(\pi_ {X, Y ^ {2}}\right) _ {\#} \gamma_ {X, Y ^ {2}} = \left(\pi_ {X, Y ^ {2}}\right) _ {\#} \gamma^ {2, 3} = \gamma^ {2}.
+$$
+
+Thus  $(\pi_{X,Y^1,Y^2})_\# \gamma \in \Gamma (\gamma^1,\gamma^2;\mu)$ . Then we have
+
+$$
+\begin{array}{l} L G W (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}) \\ = \left\langle \left| g _ {Y ^ {1}} - g _ {Y ^ {2}} \right| ^ {2}, (\gamma^ {1, 2}) ^ {\otimes 2} \right\rangle^ {1 / 2} \\ \leq \left\langle \left| g _ {Y ^ {1}} - g _ {Y ^ {2}} \right| ^ {2}, \left(\left(\pi_ {X, Y ^ {1}, Y ^ {2}}\right) _ {\#} \gamma\right) ^ {\otimes 2} \right\rangle^ {1 / 2} (77) \\ = \left(\int_ {X \times Y \times Y ^ {2} \times Y ^ {3}} | g _ {Y ^ {1}} \left(y ^ {1}, y ^ {1 \prime}\right) - g _ {Y ^ {2}} \left(y ^ {2}, y ^ {2 \prime}\right) | ^ {2} d \gamma^ {\otimes 2} \left(\left(x, x ^ {\prime}\right), \left(y ^ {1}, y ^ {1 \prime}\right), \left(y ^ {2}, y ^ {2 \prime}\right), \left(y ^ {3}, y ^ {3 \prime}\right)\right)\right) ^ {1 / 2} \\ \leq \left(\int_ {X \times Y \times Y ^ {2} \times Y ^ {3}} \left(| g _ {Y ^ {1}} (y ^ {1}, y ^ {1 \prime}) - g _ {Y ^ {3}} (y ^ {3}, y ^ {3 \prime}) | + | g _ {Y ^ {2}} (y ^ {2}, y ^ {2 \prime}) - g _ {Y ^ {3}} (y ^ {3}, y ^ {3 \prime} |\right) ^ {2} d \gamma^ {\otimes 2}\right) ^ {1 / 2} \\ \leq \left(\int_ {X \times Y \times Y ^ {2} \times Y ^ {3}} \left(| g _ {Y ^ {1}} (y ^ {1}, y ^ {1 \prime}) - g _ {Y ^ {3}} (y ^ {3}, y ^ {3 \prime}) |\right) ^ {2} d \gamma^ {\otimes 2}\right) ^ {1 / 2} \\ + \left(\int_ {X \times Y \times Y ^ {2} \times Y ^ {3}} \left(| g _ {Y ^ {2}} \left(y ^ {2}, y ^ {2 \prime}\right) - g _ {Y ^ {3}} \left(y ^ {3}, y ^ {3 \prime} \right|\right) ^ {2} d \gamma^ {\otimes 2}\right) ^ {1 / 2} (78) \\ = L G W (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {3}; \mathbb {X}, \gamma^ {1}, \gamma^ {3}) + L G W (\mathbb {Y} ^ {2}, \mathbb {Y} ^ {3}; \mathbb {X}, \gamma^ {2}, \gamma^ {3}) \\ \end{array}
+$$
+
+where (77) follows from the fact  $\gamma^{1,2}$  is optimal in  $\Gamma (\gamma^1,\gamma^2;\mu)$ ; (78) holds by the Minkowski inequality. In the uniqueness assumption (F.4), we have
+
+$$
+L G W (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}) \leq L G W (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {3}; \mathbb {X}) + L G W (\mathbb {Y} ^ {2}, \mathbb {Y} ^ {3}; \mathbb {X}).
+$$
+
+![](images/8eaa6e00402673ba74598d904cf338c661371178ea0e0c22b863727c33f5fee3.jpg)
+
+Convert LPGW to LGW The main idea of this step is similar to the proof of Proposition 3.3 in (Bai et al., 2024).
+
+# Notations Setup
+
+We can redefine  $\Gamma_{\leq}(\gamma^{1},\gamma^{2};\mu)$  as
+
+$$
+\begin{array}{l} \Gamma_ {\leq} (\gamma^ {1}, \gamma^ {2}; \mu) = \left\{\gamma \in \mathcal {M} _ {+} (X \times Y ^ {1} \times X \times Y ^ {2}): (\pi_ {1, 2}) _ {\#} \gamma \leq \gamma^ {1}, (\pi_ {3, 4}) _ {\#} \gamma \leq \gamma^ {2}, \right. \\ \left(\pi_ {1, 3}\right) _ {\#} \gamma = (\mathrm {i d} \times \mathrm {i d}) _ {\#} \mu^ {\prime}, \mu^ {\prime} \leq \mu \}. \tag {79} \\ \end{array}
+$$
+
+We define auxiliary points  $\hat{\infty}_0, \hat{\infty}_1, \hat{\infty}_2, \hat{\infty}_3$  and then define
+
+$$
+\hat {X} = X \cup \{\hat {\infty} _ {0} \},
+$$
+
+$$
+\hat {Y} ^ {1} = Y ^ {1} \cup \{\hat {\infty} _ {1}, \hat {\infty} _ {2}, \hat {\infty} _ {3} \},
+$$
+
+$$
+\hat {Y} ^ {2} = Y ^ {2} \cup \{\hat {\infty} _ {1}, \hat {\infty} _ {2}, \hat {\infty} _ {3} \},
+$$
+
+$$
+\hat {Y} ^ {3} = Y ^ {3} \cup \{\hat {\infty} _ {1}, \hat {\infty} _ {2}, \hat {\infty} _ {3} \}.
+$$
+
+and
+
+$$
+\hat {\gamma} ^ {1} = \gamma^ {1} + | \gamma^ {2} | \delta_ {\hat {\infty} _ {0}, \hat {\infty} _ {2}} + | \gamma^ {3} | \delta_ {\hat {\infty} _ {0}, \hat {\infty} _ {3}} \in \mathcal {M} _ {+} (\hat {X} \times \hat {Y} ^ {1}),
+$$
+
+$$
+\hat {\gamma} ^ {2} = \gamma^ {2} + | \gamma^ {1} | \delta_ {\hat {\infty} _ {0}, \hat {\infty} _ {1}} + | \gamma^ {3} | \delta_ {\hat {\infty} _ {0}, \hat {\infty} _ {3}} \in \mathcal {M} _ {+} (\hat {X} \times \hat {Y} ^ {2}),
+$$
+
+$$
+\hat {\gamma} ^ {3} = \gamma^ {3} + | \gamma^ {1} | \delta_ {\hat {\infty} _ {0}, \hat {\infty} _ {1}} + | \gamma^ {2} | \delta_ {\hat {\infty} _ {0}, \hat {\infty} _ {2}} \in \mathcal {M} _ {+} (\hat {X} \times \hat {Y} ^ {3}).
+$$
+
+Define
+
+$$
+g _ {\hat {Y} ^ {1}} (y ^ {1}, y ^ {1 \prime}) = \left\{ \begin{array}{l l} g _ {Y ^ {1}} (y ^ {1}, y ^ {1 \prime}) & \text {i f} y ^ {1}, y ^ {1} \in Y ^ {1}, \\ \infty & \text {e l s e w h e r e} \end{array} \right.
+$$
+
+and  $g_{\hat{Y}^2}, g_{\hat{Y}^3}$  are defined similarly.
+
+Finally, define
+
+$$
+D _ {\lambda} (r _ {1}, r _ {2}) = \left\{ \begin{array}{l l} | r _ {1} - r _ {2} | & \text {i f} r _ {1}, r _ {2} \in \mathbb {R}, \\ \sqrt {\lambda} & \text {e l s e w h e r e .} \end{array} \right..
+$$
+
+Let  $\hat{X}^{\prime} = X^{\prime}\cup \{\hat{\infty}_{0}\}$  to be a copy of set  $\hat{X}$ . Then, we can set
+
+$$
+\hat {\Gamma} (\hat {\gamma} ^ {1}, \hat {\gamma} ^ {2}; \mu) := \left\{\hat {\gamma} \in \mathcal {M} _ {+} (\hat {X} \times \hat {Y} ^ {1} \times \hat {X} \times \hat {Y} ^ {2}), (\pi_ {1, 2}) _ {\#} \hat {\gamma} = \hat {\gamma} ^ {1}, (\pi_ {3, 4}) _ {\#} \hat {\gamma} = \hat {\gamma} ^ {2}, \right. \tag {80}
+$$
+
+$$
+\left(\pi_ {1, 3}\right) _ {\#} \hat {\gamma} | _ {X ^ {\otimes 2}} = (\mathrm {i d} \times \mathrm {i d}) _ {\#} \mu^ {\prime}, \mu^ {\prime} \leq \mu \}. \tag {81}
+$$
+
+Consider the mapping:
+
+$$
+\begin{array}{l} \mathcal {F}: \Gamma_ {\leq} (\gamma^ {1}, \gamma^ {2}; \mu) \mapsto \hat {\Gamma} (\hat {\gamma} ^ {1}, \gamma^ {2}; \mu) \\ \gamma \mapsto \hat {\gamma} := \gamma + (\gamma^ {1} - (\pi_ {1, 2}) _ {\#} \gamma) \otimes \delta_ {(\hat {\infty} _ {0}, \hat {\infty} _ {1})} + \delta_ {(\hat {\infty} _ {0}, \hat {\infty} _ {2})} \otimes (\gamma^ {2} - (\pi_ {3, 4}) _ {\#} \gamma) + | \gamma | \delta_ {(\hat {\infty} _ {0}, \hat {\infty} _ {2}), (\infty_ {0}, \hat {\infty} _ {1})} \\ + | \gamma^ {3} | \delta_ {(\hat {\infty} _ {0}, \hat {\infty} _ {3}), (\hat {\infty} _ {0}, \hat {\infty} _ {3})}. \tag {82} \\ \end{array}
+$$
+
+By the following lemma, we show that  $\mathcal{F}$  defines an equivalent relation between the two sets.
+
+Lemma F.11. The mapping  $\mathcal{F}$  defined in (82) is well-defined. In addition, it is a bijection if we set the identity  $\hat{\infty}_0 = \hat{\infty}_1 = \hat{\infty}_2 = \hat{\infty}_3$ .
+
+Proof of Lemma F.11. By (Bai et al., 2022, Proposition B.1) we have for each  $\gamma \in \Gamma(\gamma^1, \gamma^2; \mu)$ ,
+
+$$
+(\pi_ {1, 2}) _ {\#} \hat {\gamma} = \hat {\gamma} ^ {1}, (\pi_ {3, 4}) _ {\#} \hat {\gamma} = \hat {\gamma} ^ {2}.
+$$
+
+In addition,
+
+$$
+(\pi_ {1, 3}) _ {\#} \hat {\gamma} = (\pi_ {1, 3}) _ {\#} \bar {\gamma} + ((\pi_ {1}) _ {\#} \gamma^ {1} - (\pi_ {1}) _ {\#} \gamma) \otimes \delta_ {\infty_ {0}} + \delta_ {\infty_ {0}} \otimes ((\pi_ {1}) _ {\#} \gamma^ {2} - (\pi_ {3}) _ {\#} \gamma) + (| \bar {\gamma} | + | \gamma^ {3} |) \delta_ {\infty_ {0}, \infty_ {0}}.
+$$
+
+Thus,
+
+$$
+\left(\pi_ {1, 3}\right) _ {\#} \hat {\gamma} | _ {X ^ {\otimes 2}} = \left(\pi_ {1, 3}\right) _ {\#} \gamma = (\mathrm {i d} \times \mathrm {i d}) _ {\#} \left(\pi_ {1}\right) _ {\#} \gamma ,
+$$
+
+where  $(\pi_1)_{\#}\gamma \leq (\pi_1)_{\#}\gamma^1\leq \mu$  , and we have  $\mathcal{F}$  is well-defined.
+
+It remains to show  $\mathcal{F}$  is a bijection. For each  $\hat{\gamma} \in \hat{\Gamma}_{\leq}(\hat{\gamma}^1, \hat{\gamma}^2)$  and we define mapping
+
+$$
+\mathcal {F} ^ {- 1}: \widehat {\Gamma} \left(\widehat {\gamma} ^ {1}, \widehat {\gamma} ^ {2}; \mu\right)\rightarrow \Gamma_ {\leq} \left(\gamma^ {1}, \gamma^ {2}; \mu\right) \tag {83}
+$$
+
+$$
+\hat {\gamma} \mapsto \hat {\gamma} \mid_ {X \times Y ^ {1} \times X \times Y ^ {2}}
+$$
+
+It remains to show  $\mathcal{F}^{-1}$  is inverse of  $\mathcal{F}$ .
+
+First, we claim  $\mathcal{F}^{-1}$  is well-defined. Pick  $\hat{\gamma} \in \hat{\Gamma}(\hat{\gamma}^1, \hat{\gamma}^2)$ , by Lemma (Bai et al., 2022, Lemma B.1.), we have  $(\pi_{X,Y^1})_{\#} \hat{\gamma} \leq \gamma^1$ ,  $(\pi_{3,4})_{\#} \hat{\gamma} \leq \gamma^2$ . In addition,
+
+$$
+\left(\pi_ {1, 3}\right) _ {\#} \gamma = \left(\pi_ {1, 3}\right) _ {\#} \hat {\gamma} | _ {X ^ {\otimes 2}} = (\mathrm {i d} \times \mathrm {i d}) _ {\#} \mu^ {\prime},
+$$
+
+where  $\mu^{\prime}\leq \mu$  . Thus,  $\mathcal{F}^{-1}(\hat{\gamma})\in \Gamma_{\leq}(\gamma^{1},\gamma^{2};\mu)$
+
+In addition, it is straightforward to verify  $\mathcal{F}^{-1}(\mathcal{F}(\gamma)) = \gamma ,\forall \bar{\gamma}\in \Gamma_{\leq}(\gamma^{1},\gamma^{2};\mu),$  and then we complete the proof.
+
+It is straightforward to verify:
+
+$$
+\langle D _ {\lambda} ^ {2} \left(g _ {\hat {Y} ^ {1}}, g _ {\hat {Y} ^ {2}}, \hat {\gamma} ^ {\otimes 2}\right) \rangle = \langle \left| g _ {Y ^ {1}} - g _ {Y ^ {2}} \right| ^ {2}, \gamma^ {\otimes 2} \rangle + \lambda \left(\left| \gamma^ {1} \right| ^ {2} + \left| \gamma^ {2} \right| ^ {2} - \left| \gamma \right| ^ {2}\right)
+$$
+
+Combine it with the above lemma, take the infimum over  $\Gamma_{\leq}(\gamma^{1},\gamma^{2};\mu)$  (equivalently, over  $\hat{\Gamma} (\hat{\gamma}^1,\hat{\gamma}^2;\mu)$ , we obtain:
+
+$$
+\begin{array}{l} \inf_{\hat{\gamma}\in \widehat{\Gamma} (\hat{\gamma}^{1},\hat{\gamma}^{2};\mu)}\langle D_{\lambda}^{2}(g_{\hat{Y}^{1}},g_{\hat{Y}^{2}},\hat{\gamma}^{\otimes 2})\rangle = \inf_{\gamma \in \Gamma_{\leq}(\gamma^{1},\gamma^{2};\mu)}\langle |g_{Y^{1}} - g_{Y^{2}}|^{2},\gamma^{\otimes 2}\rangle +\lambda (|\gamma^{1}|^{2} + |\gamma^{2}|^{2} - |\gamma |^{2}) \\ = L P G W (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}) \\ \end{array}
+$$
+
+where the second equality holds from the fact  $|\nu^1| = |\nu^2| = |\nu^3| = |\gamma^1| = |\gamma^2| = |\gamma^3|$ . Similar identity holds for  $LPGW(\mathbb{Y}^1, \mathbb{Y}^3; \mathbb{X}, \gamma^1, \gamma^3)$ ,  $LPGW(\mathbb{Y}^2, \mathbb{Y}^3; \mathbb{X}, \gamma^2, \gamma^3)$ .
+
+Verifying Inequality It remains to show
+
+$$
+\inf_{\hat{\gamma}\in \widehat{\Gamma} (\hat{\gamma}^{1},\hat{\gamma}^{2};\mu)}\langle D_{\lambda}^{2}(g_{\hat{Y}^{1}},g_{\hat{Y}^{2}},\hat{\gamma}^{\otimes 2})\rangle^{1 / 2}\leq \inf_{\hat{\gamma}\in \widehat{\Gamma} (\hat{\gamma}^{1},\hat{\gamma}^{3};\mu)}\langle D_{\lambda}^{2}(g_{\hat{Y}^{1}},g_{\hat{Y}^{3}},\hat{\gamma}^{\otimes 2})\rangle^{1 / 2} + \inf_{\hat{\gamma}\in \widehat{\Gamma} (\hat{\gamma}^{2},\hat{\gamma}^{3};\mu)}\langle D_{\lambda}^{2}(g_{\hat{Y}^{2}},g_{\hat{Y}^{3}},\hat{\gamma}^{\otimes 2})\rangle^{1 / 2}.
+$$
+
+Pick  $\gamma^{1,2} \in \Gamma_{\leq}(\gamma^1, \gamma^2; \mu)$  that is optimal for  $\langle D_{\lambda}^{2}(g_{\hat{Y}^1}, g_{\hat{Y}^2}, \hat{\gamma}^{\otimes 2}) \rangle^{1/2}$ . Similarly, pick optimal  $\gamma^{1,3} \in \Gamma_{\leq}(\gamma^1, \gamma^3), \gamma^{2,3} \in \Gamma_{\leq}(\gamma^2, \gamma^3)$ , we construct the corresponding optimal  $\hat{\gamma}^{1,3} \in \Gamma(\hat{\gamma}^1, \hat{\gamma}^3; \mu), \hat{\gamma}^{2,3} \in \Gamma(\hat{\gamma}^2, \hat{\gamma}^3; \mu)$ .
+
+Thus, by Gluing lemma, there exists  $\gamma \in \mathcal{M}_{+}((X\times Y^{1})\times (X\times Y^{2})\times (X\times Y^{3}))$  such that
+
+$$
+\big (\pi_ {(1, 2), (5, 6)} \big) _ {\#} \gamma = \hat {\gamma} ^ {1, 3},
+$$
+
+$$
+\left(\pi_ {(3, 4), (5, 6)}\right) _ {\#} \gamma = \hat {\gamma} ^ {2, 3}.
+$$
+
+Now by the definition of  $\hat{\Gamma} (\hat{\gamma}^1,\hat{\gamma}^3;\mu),\hat{\Gamma} (\hat{\gamma}^2,\hat{\gamma}^3;\mu)$  , we have
+
+$$
+\begin{array}{l} \left(\pi_ {1, 5}\right) _ {\#} \gamma = (\mathrm {i d} \times \mathrm {i d}) _ {\#} \mu^ {\prime}, (84) \\ \left(\pi_ {3, 5}\right) _ {\#} \gamma = (\mathrm {i d} \times \mathrm {i d}) _ {\#} \mu^ {\prime \prime} (85) \\ \end{array}
+$$
+
+for some radon measures  $\mu^{\prime},\mu^{\prime \prime}\leq \mu$  . Thus, we have  $\mu^{\prime} = \mu^{\prime \prime}\leq \mu$  . Therefore,
+
+$$
+\left(\pi_ {1, 3}\right) _ {\#} \gamma = (\mathrm {i d} \times \mathrm {i d}) _ {\#} \mu^ {\prime}.
+$$
+
+In addition,
+
+$$
+(\pi_ {1, 2}) _ {\#} \gamma = (\pi_ {1, 2}) _ {\#} \hat {\gamma} ^ {1, 3} = \hat {\gamma} ^ {1},
+$$
+
+$$
+(\pi_ {3, 4}) _ {\#} \gamma = (\pi_ {1, 2}) _ {\#} \hat {\gamma} ^ {2, 3} = \hat {\gamma} ^ {2}.
+$$
+
+So  $(\pi_{1,2,3,4})_{\#}\gamma \in \Gamma (\hat{\gamma}^{1},\hat{\gamma}^{2};\mu)$
+
+Therefore, we have
+
+$$
+\begin{array}{l} \left\langle D _ {\lambda} ^ {2} \left(k _ {Y ^ {1}}, k _ {Y ^ {2}}\right), \left(\hat {\gamma} ^ {1, 2}\right) ^ {\otimes 2} \right\rangle^ {1 / 2} \\ \leq \left\langle D _ {\lambda} ^ {2} \left(k _ {Y ^ {1}}, k _ {Y ^ {2}}\right), \gamma^ {\otimes 2} \right\rangle^ {1 / 2} \\ \leq \left\langle D _ {\lambda} ^ {2} \left(k _ {Y ^ {1}}, k _ {Y ^ {3}}\right), \gamma^ {\otimes 2} \right\rangle^ {1 / 2} + \left\langle D _ {\lambda} ^ {2} \left(k _ {Y ^ {2}}, k _ {Y ^ {3}}\right), \gamma^ {\otimes 2} \right\rangle^ {1 / 2} \\ \end{array}
+$$
+
+where the second inequality holds from (Bai et al., 2024, Eq.(57)).
+
+# G RELATION BETWEEN LPGW AND LGW
+
+Theorem G.1. Suppose  $|\mu| = |\nu^1| = |\nu^2| = 1$  and  $x, x$  is bounded,  $g_{Y^1}, g_{Y^2}$  are continuous function.
+
+Choose sequence  $\lambda_{n}\to \infty$  , if  $n$  is sufficiently large, we have  $\lambda_{n}$  , then
+
+$$
+L P G W _ {\lambda_ {n}} (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}) \rightarrow L G W _ {\lambda_ {n}} (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}),
+$$
+
+$$
+a L P G W _ {\lambda_ {n}} (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}) \to a L G W (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}).
+$$
+
+Proof of theorem G.1. By the lemma F.1 in (Bai et al., 2024), when  $n$  is sufficiently large, in particular, when
+
+$$
+\lambda_ {n} > M := \sup  \quad | g _ {Y ^ {1}} \left(y ^ {1}, y ^ {1 ^ {\prime}}\right) - g _ {Y ^ {2}} \left(y ^ {1}, y ^ {1 ^ {\prime}}\right) |,
+$$
+
+$$
+\begin{array}{c} y ^ {1}, y ^ {1 ^ {\prime}} \in Y ^ {1} \\ y ^ {2}, y ^ {2 ^ {\prime}} \in Y ^ {2} \end{array}
+$$
+
+for each  $\gamma^1\in \Gamma_{\leq}^{*}(\mathbb{X},\mathbb{Y}^{1})$ $|\gamma^1 | = \min (|\mu |,|\nu^1 |) = 1$  . That is
+
+$$
+\Gamma_ {\leq , \lambda_ {n}} ^ {*} (\mathbb {X}, \mathbb {Y} ^ {1}) = \Gamma^ {*} (\mathbb {X}, \mathbb {Y} ^ {1}).
+$$
+
+Similarly, when  $n$  is sufficiently large, we have  $\Gamma_{\leq ,\lambda_n}^* (\mathbb{X},\mathbb{Y}^2) = \Gamma^* (\mathbb{X},\mathbb{Y}^2)$ .
+
+Pick  $\gamma^1\in \Gamma_{\leq}^{*}(\mathbb{X},\mathbb{Y}^1) = \Gamma^{*}(\mathbb{X},\mathbb{Y}^{1}),\gamma^{2}\in \Gamma_{\leq}^{*}(\mathbb{X},\mathbb{Y}^{2}) = \Gamma^{*}(\mathbb{X},\mathbb{Y}^{2})$ , since  $|\gamma^1 | = |\mu | = |\nu^1 | = |\nu^2 | = 1$ , the mass penalty term vanishes in (65), thus we have
+
+$$
+L P G W _ {\lambda_ {n}} (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}) = L G W (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}).
+$$
+
+Take the infimum over all  $\gamma^1, \gamma^2$  and the take a limit for  $n \to \infty$ , we prove
+
+$$
+L P G W _ {\lambda_ {n}} (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}) \rightarrow L G W (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}).
+$$
+
+Similarly, in this case,  $\gamma_c^1 = \gamma_c^2 = 0$
+
+$$
+L P G W _ {\lambda_ {n}} \big (\widetilde {\mathbb {Y}} _ {\gamma^ {1}}, \widetilde {\mathbb {Y}} _ {\gamma^ {2}}; \mathbb {X}, \widetilde {\gamma} ^ {1}, \widetilde {\gamma} ^ {2} \big) + \lambda \big (| \gamma_ {c} ^ {1} | + | \gamma_ {c} ^ {2} | \big) = L G W \big (\widetilde {\mathbb {Y}} _ {\gamma^ {1}}, \widetilde {\mathbb {Y}} _ {\gamma^ {2}}; \mathbb {X}, \widetilde {\gamma} ^ {1}, \widetilde {\gamma} ^ {2} \big).
+$$
+
+Take the infimum on both sides over  $\gamma^1, \gamma^2$  and take  $\lambda_n \to \infty$ , we obtain
+
+$$
+a L P G W (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}) \to a L G W (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}).
+$$
+
+# H PROOF OF THEOREM 3.2
+
+# H.1 PROOF OF THEOREM 3.2: PARTS (1), (3)
+
+(1) Similar to proof in Theorem 2.1, in this case, we have  $\mathcal{T}_{\gamma^*} = T$ ,  $\gamma_X^* - a.s.$  
+(3) Based on statement (2) of Theorem 3.2 (see the proof in next section), we have  $\widetilde{\gamma}^1, \widetilde{\gamma}^2$  are optimal solutions for  $PGW_{\lambda}(\mathbb{X}, \widetilde{\mathbb{Y}}_{\gamma^1}), PGW_{\lambda}(\mathbb{X}, \widetilde{\mathbb{Y}}_{\gamma^2})$  respectively.
+
+By (1), we have  $T^{1} = \mathcal{T}_{\gamma^{1}}, T^{2} = \mathcal{T}_{\gamma^{2}} \gamma_{X}^{1} \wedge \gamma_{Y}^{2}$ -a.s. Thus, we have:
+
+$$
+\begin{array}{l} L P G W _ {\lambda} (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}) \\ = \inf  _ {\mu^ {\prime} \leq \gamma_ {X} ^ {1} \wedge \gamma_ {X} ^ {2}} \int_ {X ^ {\otimes 2}} | d _ {Y ^ {1}} (T ^ {1} (\cdot_ {1}), T ^ {1} (\cdot_ {2})) - d _ {Y ^ {2}} (T ^ {1} (\cdot_ {1}), T ^ {2} (\cdot_ {2})) | ^ {2} d (\mu^ {\prime}) ^ {\otimes 2} + \lambda (| \nu^ {1} | ^ {2} + | \nu^ {2} | ^ {2} - 2 | \mu^ {\prime} | ^ {2}) \\ = \inf  _ {\mu^ {\prime} \leq \gamma_ {X} ^ {1} \wedge \gamma_ {X} ^ {2}} \int_ {X ^ {\otimes 2}} | d _ {Y ^ {1}} (\mathcal {T} _ {\gamma^ {1}} (\cdot_ {1}), \mathcal {T} _ {\gamma^ {1}} (\cdot_ {2})) - d _ {Y ^ {2}} (\mathcal {T} _ {\gamma^ {2}} (\cdot_ {1}), \mathcal {T} _ {\gamma^ {2}} (\cdot_ {1}) | ^ {2} d (\mu^ {\prime}) ^ {\otimes 2} + \lambda (| \widetilde {\nu} _ {\gamma^ {1}} | ^ {2} + | \widetilde {\nu} _ {\gamma^ {2}} | ^ {2} - 2 | \mu^ {\prime} | ^ {2}) \\ + \lambda \left(| \nu^ {1} | ^ {2} - | \widetilde {\nu} _ {\gamma 1} | ^ {2} + | \nu^ {2} | ^ {2} - | \widetilde {\nu} _ {\gamma 2} | ^ {2}\right) \\ = \inf  _ {\mu^ {\prime} \leq \gamma_ {X} ^ {1} \wedge \gamma_ {X} ^ {2}} \left[ \right. \int_ {X ^ {\otimes 2}} \left| \right. d _ {Y ^ {1}} \left(\mathcal {T} _ {\gamma^ {1}} (\cdot_ {1}), \mathcal {T} _ {\gamma^ {1}} (\cdot_ {2})\right) - d _ {Y ^ {2}} \left( \right.\mathcal {T} _ {\gamma^ {2}} (\cdot_ {1}), \mathcal {T} _ {\gamma^ {2}} (\cdot_ {1}) \left. \right| ^ {2} d \left(\mu^ {\prime}\right) ^ {\otimes 2} + \lambda \left(\left| \widetilde {\nu} _ {\gamma^ {1}} \right| ^ {2} + \left| \widetilde {\nu} _ {\gamma^ {2}} \right| ^ {2} - 2 \left| \mu^ {\prime} \right| ^ {2}\right)\left. \right] \\ + \lambda \left(\left| \gamma_ {c} ^ {1} \right| + \left| \gamma_ {c} ^ {2} \right|\right) \tag {86} \\ = L P G W _ {\lambda} \left(\widetilde {\mathbb {Y}} _ {\gamma^ {1}}, \widetilde {\mathbb {Y}} _ {\gamma^ {2}}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}\right) + \lambda \left(\left| \gamma_ {c} \right| ^ {1} + \left| \gamma_ {c} \right| ^ {2}\right) \\ \end{array}
+$$
+
+where (86) holds since
+
+$$
+\widetilde {\nu} ^ {1} = \mathcal {T} _ {\gamma^ {1}} \gamma_ {1} ^ {1} = T ^ {1} \gamma_ {1} ^ {1} = (\pi_ {Y}) _ {\#} \gamma^ {1} \leq \nu^ {1}.
+$$
+
+Thus  $|\nu^1|^2 - |\widetilde{\nu}^1|^2 = |(\nu^1)^{\otimes 2} - ((\pi_Y)_\# \gamma^1)^{\otimes 2}| = |\gamma_c^1|$ ; and similarly we have  $|\nu^2|^2 - |\widetilde{\nu}^2|^2 = |\gamma_c^2|$ .
+
+# H.2 PROOF OF THEOREM 3.2: PART (2)
+
+# H.2.1 PROOF IN THE DISCRETE CASE.
+
+# Notation setup
+
+We first demonstrate a simplified proof in the discrete case. Next, we will provide the proof of the statement in the general case.
+
+Suppose
+
+$$
+\mu = \sum_ {i = 1} ^ {n} p _ {i} x _ {i}, \qquad \nu = \sum_ {j = 1} ^ {m} q _ {j} y _ {j},
+$$
+
+$$
+g _ {X} (x, x ^ {\prime}) = x ^ {\top} x ^ {\prime}, \quad \forall x, x ^ {\prime} \in X,
+$$
+
+$$
+g _ {X} (y, y ^ {\prime}) = y ^ {\top} y ^ {\prime}, \quad \forall y, y ^ {\prime} \in Y.
+$$
+
+Choose  $\gamma^{*}\in \Gamma_{\leq}^{*}(\mathbb{X},\mathbb{Y})$  , we obtain the corresponding barycentric projected measure:
+
+$\widetilde{\nu} \coloneqq \widetilde{\nu}_{\gamma^{*}} = \sum_{i=1}^{n} \widetilde{q}_{i} \widetilde{y}_{i}$  with
+
+$$
+\widetilde {q} _ {i} = \sum_ {j = 1} ^ {n} \gamma_ {i, j} ^ {*}, \quad \forall i \in [ 1: n ], \tag {87}
+$$
+
+$$
+\widetilde {y _ {i}} = \left\{ \begin{array}{l l} \frac {1}{\widetilde {q} _ {i} ^ {\Gamma}} \gamma_ {i j} ^ {*} x _ {i} & \text {i f} \widetilde {q} _ {i} > 0, \\ 0 & \text {e l s e w h e r e .} \end{array} \right. \tag {88}
+$$
+
+Then  $\widetilde{\mathbb{Y}}_{\gamma^*} = (Y,g_Y,\widetilde{\nu})$ ,  $\widetilde{\gamma}^{*} = \mathrm{diag}(\widetilde{q}_{1},\ldots \widetilde{q}_{n})$ . Our goal is to show  $\widetilde{\gamma}^{*} = \mathrm{diag}(\widetilde{q}_{1},\ldots ,\widetilde{q}_{n})$  is optimal in  $PGW_{\lambda}(\mathbb{X},\widetilde{\mathbb{Y}}_{\gamma^*})$ .
+
+Pick  $\widetilde{\gamma} \in \Gamma_{\leq}(p, \widetilde{q}) \coloneqq \Gamma_{\leq}(\mu, \widetilde{\nu})$ , similar to section E.5, we set diagonal matrix  $(\widetilde{\gamma}^{*})^{-1} \in \mathbb{R}_{+}^{n \times n}$  as
+
+$$
+(\widetilde {\gamma} ^ {*}) _ {i i} ^ {- 1} = \left\{ \begin{array}{l l} \frac {1}{\widetilde {q} _ {i}} & \text {i f} \widetilde {q} _ {i} > 0, \\ 0 & \text {e l s e w h e r e .} \end{array} \right.
+$$
+
+Let  $\gamma \coloneqq \widetilde{\gamma} (\widetilde{\gamma}^{*})^{-1}\gamma^{*}\in \mathbb{R}_{+}^{n\times m}$ . Let  $\gamma_{X}\coloneqq \gamma 1_{m}$  and  $\gamma_{Y}\coloneqq \gamma^{\top}1_{n}$ .
+
+In addition, we set  $D = \{i : \widetilde{q}_i > 0\}$ , and  $1_D \in \mathbb{R}^n$  with  $1_D[i] = 1, \forall i \in D$  and  $1_D[i] = 0$  elsewhere. Then we have:
+
+$$
+\gamma 1 _ {m} = \widetilde {\gamma} (\widetilde {\gamma} ^ {*}) ^ {- 1} \gamma^ {*} 1 _ {m} = \widetilde {\gamma} (\widetilde {\gamma} ^ {*}) ^ {- 1} \widetilde {q} = \widetilde {\gamma} 1 _ {D} = \widetilde {\gamma} _ {X} \leq p _ {0}, \tag {89}
+$$
+
+$$
+\gamma^ {\top} 1 _ {n} = \left(\gamma^ {*}\right) ^ {\top} \left(\widetilde {\gamma} ^ {*}\right) ^ {- 1} \widetilde {\gamma} ^ {\top} 1 _ {n} \leq \left(\gamma^ {*}\right) ^ {\top} \left(\widetilde {\gamma} ^ {*}\right) ^ {- 1} \widetilde {q} = \left(\gamma^ {*}\right) ^ {\top} 1 _ {D} = \gamma_ {Y} ^ {*} \leq q. \tag {90}
+$$
+
+Therefore,  $\gamma_{X}\leq \widetilde{\gamma}_{X}\leq p$ $\gamma_{Y}\leq \gamma_{Y}^{*}\leq q$  . Thus,  $\gamma \in \Gamma_{\leq}(p,q)$
+
+Relation between  $PGW_{\lambda}(\mathbb{X}, \mathbb{Y})$  and  $PGW_{\lambda}(\mathbb{X}, \widetilde{\mathbb{Y}})$ .
+
+Similar to (54),(57), we obtain
+
+$$
+\begin{array}{l} C (\gamma^ {*}; \mathbb {X}, \mathbb {Y}, \lambda) = \langle (x ^ {\top} x ^ {\prime}) ^ {2}, (\gamma_ {X} ^ {*}) ^ {\otimes 2} \rangle + \langle (y ^ {\top} y ^ {\prime}) ^ {2}, (\gamma_ {Y} ^ {*}) ^ {\otimes 2} \rangle + \lambda (| p | ^ {2} + | q | ^ {2} - 2 | \gamma^ {*} |) \\ - 2 \sum_ {i, i ^ {\prime} = 1} ^ {n} \sum_ {j, j ^ {\prime} = 1} ^ {m} x _ {i} ^ {\top} x _ {i} ^ {\prime} y _ {j} ^ {\top} y _ {j} ^ {\prime} \gamma_ {i, j} ^ {*} \gamma_ {i ^ {\prime}, j ^ {\prime}} ^ {*} \\ \end{array}
+$$
+
+$$
+\begin{array}{l} C \left(\widetilde {\gamma} ^ {*}; \mathbb {X}, \widetilde {\mathbb {Y}}, \lambda\right) = \left\langle \left(x ^ {\top} x ^ {\prime}\right) ^ {2}, \left(\widetilde {\gamma} _ {X} ^ {*}\right) ^ {\otimes 2} \right\rangle + \left\langle \left(y ^ {\top} y ^ {\prime}\right) ^ {2}, \left(\widetilde {\gamma} _ {Y} ^ {*}\right) ^ {\otimes 2} \right\rangle + \lambda \left(| p | ^ {2} + | \widetilde {q} | ^ {2} - 2 \left| \widetilde {\gamma} ^ {*} \right|\right) \\ - 2 \sum_ {i, i ^ {\prime} = 1} ^ {n} \sum_ {j, j ^ {\prime} = 1} ^ {m} x _ {i} ^ {\top} x _ {i} ^ {\prime} y _ {j} ^ {\top} y _ {j} \gamma_ {i, j} ^ {*} \gamma_ {i ^ {\prime}, j ^ {\prime}} ^ {*} \\ \end{array}
+$$
+
+where  $(x^{\top}x^{\prime})^{2} := [(x_{i}^{\top}x_{i^{\prime}}^{\prime})^{2}]_{i,i^{\prime}\in [1:n]}\in \mathbb{R}^{n\times n},(\gamma_{X}^{*})^{\otimes 2} = \gamma_{X}^{*}(\gamma_{X}^{*})^{\top},\langle (x^{\top}x^{\prime})^{2},(\gamma_{X}^{*})^{\otimes 2}\rangle := \sum_{i,i^{\prime} = 1}^{n}(x_{i}^{\top}x_{i^{\prime}}^{\prime})^{2}(\gamma_{X}^{*})_{i}(\gamma_{X}^{*})_{i^{\prime}}$  is the element-wise dot product. All other notations are defined similarly.
+
+Combined the above two equalities with the fact  $|\gamma^{*}| = |\gamma_{Y}^{*}|$  and  $\gamma_X^* = \widetilde{\gamma}_X^*$ , we obtain:
+
+$$
+C \left(\widetilde {\gamma} ^ {*}; \mu , \widetilde {\nu}, \lambda\right) = C \left(\gamma^ {*}; \mathbb {X}, \mathbb {Y}, \lambda\right) + \left\langle \left(\widetilde {y} ^ {\top} \widetilde {y} ^ {\prime}\right) ^ {2}, \left(\widetilde {\gamma} _ {Y} ^ {*}\right) ^ {\otimes 2} \right\rangle - \left\langle \left(y ^ {\top} y ^ {\prime}\right) ^ {2}, \left(\gamma_ {Y} ^ {*}\right) ^ {\otimes 2} \right\rangle + \lambda \left(| q | ^ {2} - | \widetilde {q} | ^ {2}\right). \tag {91}
+$$
+
+Similarly, from the fact  $\gamma_{X} = \widetilde{\gamma}_{X}$  (see (89)) we obtain
+
+$$
+C (\widetilde {\gamma}; \mu , \widetilde {\nu}, \lambda) = C (\gamma ; \mathbb {X}, \mathbb {Y}, \lambda) + \langle (\widetilde {y} ^ {\top} \widetilde {y} ^ {\prime}) ^ {2}, (\widetilde {\gamma} _ {Y}) ^ {\otimes 2} \rangle - \langle (y ^ {\top} y ^ {\prime}) ^ {2}, (\gamma_ {Y}) ^ {\otimes 2} \rangle + \lambda (| q | ^ {2} - | \widetilde {q} | ^ {2}). \tag {92}
+$$
+
+Thus
+
+$$
+\begin{array}{l} C (\widetilde {\gamma} ^ {*}; \mathbb {X}, \widetilde {\mathbb {Y}}, \lambda) - C (\widetilde {\gamma}; \mathbb {X}, \widetilde {\mathbb {Y}}, \lambda) \\ = C \left(\gamma^ {*}; \mathbb {X}, \mathbb {Y}, \lambda\right) - C \left(\gamma ; \mathbb {X}, \mathbb {Y}, \lambda\right) + \left\langle \left(\widetilde {y} ^ {\top} \widetilde {y} ^ {\prime}\right) ^ {2}, \left(\widetilde {\gamma} _ {Y} ^ {*}\right) ^ {\otimes 2} - \left(\widetilde {\gamma} _ {Y}\right) ^ {\otimes 2} \right\rangle - \left\langle \left(y ^ {\top} y ^ {\prime}\right) ^ {2}, \left(\gamma_ {Y} ^ {*}\right) ^ {\otimes 2} - \left(\gamma_ {Y}\right) ^ {\otimes 2} \right\rangle \\ \leq \left\langle \left(\widetilde {y} ^ {\top} \widetilde {y} ^ {\prime}\right) ^ {2}, \left(\widetilde {\gamma} _ {Y} ^ {*}\right) ^ {\otimes 2} - \left(\widetilde {\gamma} _ {Y}\right) ^ {\otimes 2} \right\rangle - \left\langle \left(y ^ {\top} y ^ {\prime}\right) ^ {2}, \left(\gamma_ {Y} ^ {*}\right) ^ {\otimes 2} - \left(\gamma_ {Y}\right) ^ {\otimes 2} \right\rangle (93) \\ = \sum_ {i, i ^ {\prime} \in D} (\widetilde {y} _ {i} ^ {\top} \widetilde {y} _ {i ^ {\prime}}) ^ {2} ((\widetilde {\gamma} _ {Y} ^ {*}) _ {i} (\widetilde {\gamma} _ {Y} ^ {*}) _ {i ^ {\prime}} - (\widetilde {\gamma} _ {Y}) _ {i} (\widetilde {\gamma} _ {Y}) _ {i ^ {\prime}}) - \sum_ {j, j ^ {\prime} = 1} ^ {m} (y _ {j} ^ {\top} y _ {j ^ {\prime}}) ^ {2} ((\gamma_ {Y} ^ {*}) _ {j} (\gamma_ {Y} ^ {*}) _ {j ^ {\prime}} - (\gamma_ {Y}) _ {j} (\gamma_ {Y}) _ {j ^ {\prime}}) \\ \leq \sum_ {i, i ^ {\prime} \in D} \sum_ {j, j ^ {\prime} = 1} ^ {m} \frac {\gamma_ {i , j} ^ {*} \gamma_ {i , j ^ {\prime}} ^ {*}}{\widetilde {q} _ {i} ^ {1} \widetilde {q} _ {i ^ {\prime}} ^ {1}} ((x _ {i} ^ {\top} x _ {i ^ {\prime}}) ^ {2}) ((\widetilde {\gamma} _ {Y} ^ {*}) _ {i} (\widetilde {\gamma} _ {Y} ^ {*}) _ {i ^ {\prime}} - (\widetilde {\gamma} _ {Y}) _ {i} (\widetilde {\gamma} _ {Y}) _ {i ^ {\prime}}) \\ - \sum_ {j, j ^ {\prime} = 1} ^ {m} \left(y _ {j} ^ {\top} y _ {j ^ {\prime}}\right) ^ {2} \left(\left(\gamma_ {Y} ^ {*}\right) _ {j} \left(\gamma_ {Y} ^ {*}\right) _ {j ^ {\prime}} - \left(\gamma_ {Y}\right) _ {j} \left(\gamma_ {Y}\right) _ {j ^ {\prime}}\right) (94) \\ = \sum_ {j, j ^ {\prime} = 1} ^ {m} \left(y _ {j} ^ {\top} y _ {j ^ {\prime}}\right) ^ {2} \left(\sum_ {i, i ^ {\prime} \in D} \frac {\gamma_ {i , j} ^ {*} \gamma_ {i , j ^ {\prime}} ^ {*}}{\widetilde {q} _ {i} ^ {1} \widetilde {q} _ {i ^ {\prime}} ^ {1}} \left(\left(\widetilde {\gamma} _ {Y} ^ {*}\right) _ {i} \left(\widetilde {\gamma} _ {Y} ^ {*}\right) _ {i ^ {\prime}} - \left(\widetilde {\gamma} _ {Y}\right) _ {i} \left(\widetilde {\gamma} _ {Y}\right) _ {i ^ {\prime}}\right) - \left(\left(\gamma_ {Y} ^ {*}\right) _ {j} \left(\gamma_ {Y} ^ {*}\right) _ {j ^ {\prime}} - \left(\gamma_ {Y}\right) _ {j} \left(\gamma_ {Y}\right) _ {j ^ {\prime}}\right)\right) \\ = 0 (95) \\ \end{array}
+$$
+
+where (93) holds since  $\gamma^1$  is optimal in  $PGW_{\lambda}(\mathbb{X},\mathbb{Y})$ ; (94) follows from the facts  $\gamma_Y \leq \gamma_Y^*$  (see (90)), and the fact
+
+$$
+\widetilde {y} _ {i} ^ {\top} \widetilde {y} _ {i} ^ {\prime} = \sum_ {j = 1} ^ {m} \sum_ {j ^ {\prime} = 1} ^ {m} \frac {\gamma_ {i , j} \gamma_ {i ^ {\prime} , j ^ {\prime}} y _ {j} ^ {\top} y _ {j ^ {\prime}} ^ {\prime}}{\widetilde {q} _ {i} \widetilde {q} _ {i} ^ {\prime}}, \forall i, i ^ {\prime} \in D
+$$
+
+and Jensen's inequality:
+
+$$
+(\widetilde {y} _ {i} ^ {\top} \widetilde {y} _ {i} ^ {\prime}) ^ {2} \leq \sum_ {j = 1} ^ {m} \sum_ {j ^ {\prime} = 1} ^ {m} \frac {\gamma_ {i , j} \gamma_ {i ^ {\prime} , j ^ {\prime}} (y _ {j} ^ {\top} y _ {i} ^ {\prime}) ^ {2}}{\widetilde {q} _ {i} \widehat {q} _ {i} ^ {\prime}}.
+$$
+
+The last equality (95) holds from the following:
+
+For each  $(j,j^{\prime}\in [1:m])$ , we have
+
+$$
+\begin{array}{l} \sum_ {i, i ^ {\prime} \in D} \frac {\gamma_ {i , j} ^ {*} \gamma_ {i , j ^ {\prime}} ^ {*}}{\widetilde {q} _ {i} ^ {1} \widetilde {q} _ {i ^ {\prime}} ^ {1}} ((\widetilde {\gamma} _ {Y} ^ {1}) _ {i} (\widetilde {\gamma} _ {Y} ^ {*}) _ {i ^ {\prime}} - (\widetilde {\gamma} _ {Y}) _ {i} (\widetilde {\gamma} _ {Y}) _ {i ^ {\prime}}) - ((\gamma_ {Y} ^ {*}) _ {j} (\gamma_ {Y} ^ {*}) _ {j ^ {\prime}} - (\gamma_ {Y}) _ {j} (\gamma_ {Y}) _ {j ^ {\prime}}) \\ = \underbrace {\left(\sum_ {i , i ^ {\prime} \in D} \frac {\gamma_ {i , j} ^ {*} \gamma_ {i , j ^ {\prime}} ^ {*}}{\widetilde {q} _ {i} ^ {1} \widetilde {q} _ {i ^ {\prime}} ^ {1}} (\widetilde {\gamma} _ {Y} ^ {*}) _ {i} (\widetilde {\gamma} _ {Y} ^ {*}) _ {i ^ {\prime}}\right) - (\gamma_ {Y} ^ {*}) _ {j} (\gamma_ {Y} ^ {*}) _ {j ^ {\prime}}} _ {A} - \underbrace {\left(\sum_ {i , i ^ {\prime} \in D} \frac {\gamma_ {i , j} ^ {*} \gamma_ {i , j ^ {\prime}} ^ {*}}{\widetilde {q} _ {i} \widetilde {q} _ {i ^ {\prime}}} (\widetilde {\gamma} _ {Y}) _ {i} (\widetilde {\gamma} _ {Y}) _ {i ^ {\prime}}\right) - (\gamma_ {Y}) _ {j} (\gamma_ {Y}) _ {j ^ {\prime}}} _ {B}, \\ \end{array}
+$$
+
+where
+
+$$
+\begin{array}{l} A = \sum_ {i, i ^ {\prime} \in D} \frac {\gamma_ {i , j} ^ {*} \gamma_ {i ^ {\prime} , j ^ {\prime}} ^ {*}}{\widetilde {q} _ {i} ^ {1} \widetilde {q} _ {i ^ {\prime}} ^ {1}} \widetilde {q} _ {i} ^ {1} \widetilde {q} _ {i ^ {\prime}} ^ {1} - \widetilde {q} _ {j} ^ {1} \widetilde {q} _ {j ^ {\prime}} ^ {1} \\ = \sum_ {i, i ^ {\prime} \in D} \gamma_ {i, j} ^ {*} \gamma_ {i ^ {\prime}, j ^ {\prime}} ^ {*} - \widetilde {q} _ {j} ^ {1} \widetilde {q} _ {j ^ {\prime}} ^ {1} \\ = \left(\sum_ {i \in D} \gamma_ {i, j} ^ {*}\right) \left(\sum_ {i ^ {\prime} \in D} \gamma_ {i ^ {\prime}, j ^ {\prime}} ^ {*}\right) - \tilde {q} _ {j} ^ {1} \tilde {q} _ {j ^ {\prime}} ^ {1} = 0 \\ \end{array}
+$$
+
+$$
+\begin{array}{l} B = \sum_ {i, i ^ {\prime} \in D} \frac {\gamma_ {i , j} ^ {*} \gamma_ {i , j ^ {\prime}} ^ {*}}{\widetilde {q} _ {i} \widetilde {q} _ {i ^ {\prime}}} \sum_ {k, k ^ {\prime} = 1} ^ {n} \widetilde {\gamma} _ {k, i} \widetilde {\gamma} _ {k ^ {\prime}, i ^ {\prime}} - \sum_ {k, k ^ {\prime} = 1} ^ {n} \gamma_ {k, j} \gamma_ {k ^ {\prime}, j ^ {\prime}} \\ = \sum_ {i, i ^ {\prime} \in D} \sum_ {k, k ^ {\prime} = 1} ^ {n} \frac {\gamma_ {i , j} ^ {*} \gamma_ {i , j ^ {\prime}} ^ {*}}{\widetilde {q} _ {i} ^ {1} \widetilde {q} _ {i ^ {\prime}}} \widetilde {\gamma} _ {k, i} \widetilde {\gamma} _ {k ^ {\prime}, i ^ {\prime}} - \sum_ {k, k ^ {\prime} = 1} ^ {n} \left(\sum_ {i \in D} \frac {\widetilde {\gamma} _ {k , i} \gamma_ {i , j} ^ {*}}{\widetilde {q} _ {i}}\right) \left(\sum_ {i \in D} \frac {\widetilde {\gamma} _ {k ^ {\prime} , i ^ {\prime}} \gamma_ {i ^ {\prime} , j ^ {\prime}} ^ {*}}{\widetilde {q} _ {i ^ {\prime}}}\right) \\ = \sum_ {i, i ^ {\prime} \in D} \sum_ {k, k ^ {\prime} = 1} ^ {n} \frac {\gamma_ {i , j} ^ {*} \gamma_ {i , j ^ {\prime}} ^ {*}}{\widetilde {q} _ {i} ^ {1} \widetilde {q} _ {i ^ {\prime}}} \widetilde {\gamma} _ {k, i} \widetilde {\gamma} _ {k ^ {\prime}, i ^ {\prime}} - \sum_ {i, i ^ {\prime} \in D} \sum_ {k, k ^ {\prime} = 1} ^ {n} \frac {\widetilde {\gamma} _ {k , i} \widetilde {\gamma} _ {k ^ {\prime} , i ^ {\prime}} \gamma_ {i , j} ^ {*} \gamma_ {i ^ {\prime} , j ^ {\prime}} ^ {*}}{\widetilde {q} _ {i} ^ {1} \widetilde {q} _ {i ^ {\prime}} ^ {1}} \\ = 0 \\ \end{array}
+$$
+
+and thus we complete the proof.
+
+# H.3 PROOF FOR THE GENERAL CASE
+
+# H.3.1 NOTATION SETUP AND RELATED LEMMA
+
+Suppose
+
+$$
+g _ {X} (x, x ^ {\prime}) = \alpha_ {0} x ^ {\top} x ^ {\prime}, g _ {Y} = \alpha x ^ {\top} y ^ {\prime}, \forall x, x ^ {\prime} \in X, Y, y ^ {\prime} \in X.
+$$
+
+Choose  $\gamma^{*}\in \Gamma_{\leq ,\lambda}^{*}(\mathbb{X},\mathbb{Y})$  . The barycentric projection mapping is
+
+$$
+\mathcal {T} _ {\gamma^ {*}} (x) := \int y ^ {\prime} d \gamma^ {*} \left(y ^ {\prime} \mid x ^ {\prime}\right), \forall x \in \operatorname {s u p p} (\mu).
+$$
+
+We obtain:  $\widetilde{\nu} := \widetilde{\nu}_{\gamma^{*}} = (\mathcal{T}_{\gamma^{*}})_{\#}\mu$ ,  $\widetilde{\gamma}^{*} = (\mathrm{id} \times \mathcal{T}_{\gamma^{*}})_{\#}\mu$  and we  $\widetilde{\mathbb{Y}}_{\gamma^{*}} := (X, g_{Y}, \widetilde{\nu})$ .
+
+Our goal is to show  $\widetilde{\nu}$  is optimal in  $PGW_{\lambda}(\mathbb{X},\widetilde{\mathbb{Y}})$ . Equivalently, pick  $\widetilde{\gamma}\in \Gamma_{\leq}(\mu ,\widetilde{\nu})$ , we need to show
+
+$$
+C (\widetilde {\gamma} ^ {*}; \mathbb {X}, \mathbb {Y}, \lambda) \leq C (\widetilde {\gamma}; \mathbb {X}, \mathbb {Y}, \lambda).
+$$
+
+Set  $\gamma$  from (52):
+
+$$
+\gamma = \gamma_ {Y | X} ^ {*} \widetilde {\gamma} _ {X | Y} ^ {*} \widetilde {\gamma}.
+$$
+
+We have  $\gamma$  satisfies the following two lemma:
+
+Lemma H.1. Choose  $\gamma^{*}\in \Gamma_{\leq}(\mu ,\nu)$ $\widetilde{\gamma}\in \Gamma_{\leq}(\mu ,\widetilde{\nu})$  . Set  $\widetilde{\nu} = (\mathcal{T}_{\gamma^{*}})_{\#}\mu$  , then  $\gamma$  defined in (52) satisfies the following:
+
+(a)  $\gamma \in \Gamma_{\leq}((\pi_X)_{\#}\widetilde{\gamma},(\pi_Y)_{\#}\gamma^*)\subset \Gamma_{\leq}(\mu ,\nu^{1})$  , furthermore:
+
+$$
+\left(\pi_ {X}\right) _ {\#} \gamma = \left(\pi_ {X}\right) _ {\#} \widetilde {\gamma} \tag {96}
+$$
+
+$$
+\left(\pi_ {Y}\right) _ {\#} \gamma \leq \left(\pi_ {Y}\right) _ {\#} \gamma^ {*} \tag {97}
+$$
+
+(b) If  $\widetilde{\gamma} = \widetilde{\gamma}^*$ , then  $\gamma = \gamma^*$ .
+
+(c) Regarding the second marginal of  $\gamma$ , we have for each test function  $\phi_X \in C_0(X)$ :
+
+$$
+\int_ {X} \phi_ {Y} (y) d \gamma_ {Y} (y) = \int_ {Y \times X \times Y \times X} \phi_ {Y} (y) d \widetilde {\gamma} ^ {*} (y | x ^ {0}) d \widetilde {\gamma} ^ {*} (x ^ {0} | \widetilde {y}) d \widetilde {\gamma} _ {Y} (\widetilde {y}). \tag {98}
+$$
+
+Proof.
+
+(a) Note, in a discrete setting, this statement has been proved in (89),(90). Pick test functions  $\phi_X\in C_0(X),\phi_Y\in C_0(Y)$  with  $\phi_Y\geq 0$  , we have:
+
+$$
+\begin{array}{l} \langle \phi_ {X}, \gamma \rangle = \int_ {X \times Y} \phi_ {X} (x) d \gamma (x, y) \\ = \int_ {X \times Y} \int_ {X} \int_ {Y} \phi_ {X} (x) d \gamma^ {*} (y | x ^ {0}) d \widetilde {\gamma} ^ {*} (x ^ {0} | \widetilde {y}) d \widetilde {\gamma} (x, \widetilde {y}) \\ = \int_ {X \times Y} \phi_ {X} (x) d \widetilde {\gamma} (x, \widetilde {y}) \\ = \int_ {X} \phi_ {X} (x) d \widetilde {\gamma} _ {X} (x) \\ \langle \phi_ {Y}, \gamma_ {Y} \rangle = \int_ {X \times Y} \phi_ {Y} (y) d \gamma (x, y) \\ = \int_ {X \times Y \times X \times Y} \phi_ {Y} (y) d \gamma^ {*} (y | x ^ {0}) d \widetilde {\gamma} ^ {*} (x ^ {0} | \widetilde {y}) d \widetilde {\gamma} (x, \widetilde {y}) \\ = \int_ {X \times X} \int_ {X} \int_ {Y} \phi_ {Y} (y) d \gamma^ {*} (y | x ^ {0}) d \widetilde {\gamma} ^ {*} (x ^ {0} | \widetilde {y}) d \widetilde {\gamma} _ {Y} (\widetilde {y}) \\ \leq \int_ {X \times Y} \int_ {X} \int_ {Y} \phi_ {Y} (y) d \gamma^ {*} (y | x ^ {0}) d \widetilde {\gamma} ^ {*} (x ^ {0} | \widetilde {y}) d \widetilde {\gamma} _ {Y} ^ {*} (\widetilde {y}) \\ = \int_ {X \times Y} \phi_ {Y} (y) d \gamma^ {*} (y | x ^ {0}) d \widetilde {\gamma} _ {X} ^ {*} (x ^ {0}) \\ = \int_ {X \times Y} \phi_ {Y} (y) d \gamma^ {*} (y | x ^ {0}) d \gamma_ {X} ^ {*} (x ^ {0}) \\ = \int_ {Y} \phi_ {Y} (y) d \gamma_ {Y} ^ {*} (y) \\ \end{array}
+$$
+
+where the inequality holds from the fact  $\widetilde{\gamma}_Y\leq \widetilde{\gamma}_Y^*$
+
+(b) If  $\widetilde{\gamma} = \widetilde{\gamma}^*$ , pick  $\phi \in C_0(X \times X)$ , we have:
+
+$$
+\begin{array}{l} \langle \phi , \gamma \rangle = \int_ {X \times Y} \int_ {X} \int_ {Y} \phi (x, y) d \gamma^ {*} (y | x ^ {0}) d \widetilde {\gamma} ^ {*} (x ^ {0} | \widetilde {y}) d \widetilde {\gamma} ^ {*} (\widetilde {y} | x) d \widetilde {\gamma} _ {X} ^ {*} (x) \\ = \int_ {X \times Y} \int_ {X} \int_ {Y} \phi (x, y) d \gamma^ {*} (y | x ^ {0}) d \widetilde {\gamma} _ {X} ^ {*} (x) \\ = \int_ {X \times Y} \int_ {X} \phi (x, y) d \gamma^ {*} (y | x ^ {0}) \delta (x - x ^ {0}) d \gamma_ {X} ^ {*} (x) \\ = \int_ {X \times Y} \phi (x, y) d \gamma^ {*} (y | x) d \gamma_ {X} ^ {*} (x) \\ = \int_ {X \times X} \phi (x, y) d \gamma^ {*} (x, y) \\ \end{array}
+$$
+
+(c) It follows directly from the definition of  $\gamma$
+
+![](images/aa03e1fdb4c8a2943ce9f1a5e85a39cf546a58ad7d0d0a00df49f4e21149c793.jpg)
+
+# H.3.2 RELATIONBETWEEN  $PGW_{\lambda}(\mathbb{X},\mathbb{Y})$  AND  $PGW_{\lambda}(\mathbb{X},\tilde{\mathbb{Y}})$
+
+From (54),(57),(60),(61) and (62), we obtain:
+
+$$
+C \left(\widetilde {\gamma} ^ {*}; \mathbb {X}, \widetilde {\mathbb {Y}}, \lambda\right) = C \left(\gamma^ {*}; \mathbb {X}, \mathbb {Y}, \lambda\right) + \left\langle g _ {Y} ^ {2}, \left(\widetilde {\gamma} _ {Y} ^ {*}\right) ^ {\otimes 2} \right\rangle - \left\langle g _ {Y} ^ {2}, \left(\gamma_ {Y} ^ {*}\right) ^ {\otimes 2} \right\rangle + \lambda \left(| \nu | ^ {2} - | \widetilde {\nu} | ^ {2}\right) \tag {99}
+$$
+
+$$
+C (\widetilde {\gamma}; \mathbb {X}, \widetilde {\mathbb {Y}}, \lambda) = C (\gamma ; \mathbb {X}, \mathbb {Y}, \lambda) + \left\langle g _ {Y} ^ {2}, \left(\widetilde {\gamma} _ {Y}\right) ^ {\otimes 2} \right\rangle - \left\langle g _ {Y} ^ {2}, \left(\gamma_ {Y}\right) ^ {\otimes 2} \right\rangle + \lambda (| \nu | ^ {2} - | \widetilde {\nu} | ^ {2}) \tag {100}
+$$
+
+From (99)-(100), we obtain
+
+$$
+\begin{array}{l} C (\widetilde {\gamma} ^ {*}; \mathbb {X}, \widetilde {\mathbb {Y}}, \lambda) - C (\widetilde {\gamma}; \mathbb {X}, \widetilde {\mathbb {Y}}, \lambda) \\ = (C (\gamma^ {*}; \mathbb {X}, \mathbb {Y}, \lambda) - C (\gamma ; \mathbb {X}, \mathbb {Y}, \lambda)) + ((\widetilde {\gamma} ^ {*}) ^ {\otimes 2} - \widetilde {\gamma} ^ {\otimes 2})) (g _ {Y} ^ {2} (\widetilde {y}, \widetilde {y} ^ {\prime})) - ((\gamma^ {*}) ^ {\otimes 2} - \gamma^ {\otimes 2}) (g _ {Y} ^ {2} (x, y ^ {\prime})) \\ \leq \underbrace {\left\langle g _ {Y} ^ {2} , \left(\widetilde {\gamma} _ {Y} ^ {*}\right) ^ {\otimes 2} - \left(\widetilde {\gamma} _ {Y}\right) ^ {\otimes 2} \right\rangle} _ {A} - \left\langle g _ {Y} ^ {2}, \left(\gamma_ {Y} ^ {*}\right) ^ {\otimes 2} - \gamma_ {Y} ^ {\otimes 2} \right\rangle \tag {101} \\ \end{array}
+$$
+
+where (101) holds from the fact  $\gamma^{*}$  is optimal in  $PGW_{\lambda}(\mathbb{X},\mathbb{Y})$
+
+In addition, pick  $\widetilde{y},\widetilde{y}^{\prime}\in \mathrm{supp}(\widetilde{\gamma}_{Y}^{*})$  , then for each  $x\in \operatorname {supp}(\widetilde{\gamma}^{*}(\cdot_{1}|y))$ $x^{\prime}\in \operatorname {supp}(\widetilde{\gamma}^{*}(\cdot_{1}|y^{\prime}))$  , we have:
+
+$$
+\widetilde {y} = \mathcal {T} _ {\gamma^ {*}} (x), \widetilde {y} ^ {\prime} = \mathcal {T} _ {\gamma^ {*}} (x ^ {\prime}).
+$$
+
+Thus,
+
+$$
+A = \int_ {(X \times Y) ^ {\otimes 2}} \int_ {X ^ {\otimes 2}} g _ {Y} ^ {2} \left(\mathcal {T} _ {\gamma^ {*}} \left(x ^ {0}\right), \mathcal {T} _ {\gamma^ {*}} \left(x ^ {0 ^ {\prime}}\right)\right) d \left(\gamma_ {X | Y} ^ {\otimes 2}\right) \left(\left(x ^ {0}, x ^ {0 ^ {\prime}}\right) \mid \left(\tilde {y}, \tilde {y} ^ {\prime}\right)\right) d \left(\left(\tilde {\gamma} _ {Y} ^ {*}\right) ^ {\otimes 2} - \left(\tilde {\gamma} _ {Y}\right) ^ {\otimes 2}\right) \left(\left(x, x ^ {\prime}\right), \left(\tilde {y}, \tilde {y} ^ {\prime}\right)\right)
+$$
+
+From Lemma E.2 and Jensen's inequality
+
+$$
+\begin{array}{l} g _ {Y} ^ {2} \left(\mathcal {T} _ {\gamma^ {*}} \left(x ^ {0}\right), \mathcal {T} _ {\gamma^ {*}} \left(x ^ {0 ^ {\prime}}\right)\right) \\ = \left(\int_ {Y ^ {\otimes 2}} g _ {Y} \left(y, y ^ {\prime}\right) d \gamma^ {*} \left(y \mid x ^ {0}\right) d \gamma^ {*} \left(y ^ {\prime} \mid x ^ {\prime 0 \prime}\right)\right) ^ {2} \\ \leq \int_ {Y ^ {\otimes 2}} g _ {Y} ^ {2} (y, y ^ {\prime}) d \gamma^ {*} (y | x ^ {0}) d \gamma^ {*} (y ^ {\prime} | x ^ {\prime 0 \prime}). \\ \end{array}
+$$
+
+Combined it with the fact  $\widetilde{\gamma}_Y^* \leq \widetilde{\gamma}_Y$  (see Lemma H.1 (a)), we obtain
+
+$$
+\begin{array}{l} A \leq \int_ {(X \times Y) ^ {\otimes 2}} \int_ {X ^ {\otimes 2}} g _ {Y} ^ {2} (y, y ^ {\prime}) d \gamma^ {*} (y | x ^ {0}) d \gamma^ {*} (y ^ {\prime} | x ^ {\prime 0 \prime}) d \widetilde {\gamma} ^ {*} (x ^ {0} | y) d \widetilde {\gamma} ^ {*} (x ^ {0 \prime} | y ^ {\prime}) d \left(\left(\widetilde {\gamma} _ {Y} ^ {*}\right) ^ {\otimes 2} - \widetilde {\gamma} _ {Y} ^ {\otimes 2}\right) ((x, x ^ {\prime}), (y, y ^ {\prime})) \\ = \left\langle g _ {Y} ^ {2}, \left(\gamma^ {*}\right) _ {Y | X} ^ {\otimes 2} \left(\widetilde {\gamma} ^ {*}\right) _ {X | Y} ^ {\otimes 2} \left(\left(\widetilde {\gamma} _ {Y} ^ {*}\right) ^ {\otimes 2} - \left(\widetilde {\gamma} _ {Y}\right) ^ {\otimes 2}\right) \right\rangle . \tag {102} \\ \end{array}
+$$
+
+Thus, we can continue bound (101):
+
+$$
+\begin{array}{l} (1 0 1) \leq \left\langle g _ {Y} ^ {2}, \left(\gamma^ {*}\right) _ {Y | X} ^ {\otimes 2} \left(\widetilde {\gamma} ^ {*}\right) _ {X | Y} ^ {\otimes 2} \left(\left(\widetilde {\gamma} _ {Y} ^ {*}\right) ^ {\otimes 2} - \left(\widetilde {\gamma} _ {Y}\right) ^ {\otimes 2}\right) \right\rangle - \left\langle g _ {Y} ^ {2}, \left(\gamma_ {Y} ^ {*}\right) ^ {\otimes 2} - \gamma_ {Y} ^ {\otimes 2} \right\rangle \\ = \underbrace {\langle g _ {Y} , (\gamma^ {*}) _ {Y | X} ^ {\otimes 2} (\widetilde {\gamma} ^ {*}) _ {X | Y} ^ {\otimes 2} (\widetilde {\gamma} _ {Y} ^ {*}) ^ {\otimes 2} - (\gamma_ {Y} ^ {*}) ^ {\otimes 2} \rangle} _ {B _ {1}} \\ - \underbrace {\left\langle g _ {Y} , \left(\gamma^ {*}\right) _ {Y | X} ^ {\otimes 2} \left(\widetilde {\gamma} ^ {*}\right) _ {X | Y} ^ {\otimes 2} \left(\widetilde {\gamma} _ {Y}\right) ^ {\otimes 2} - \left(\gamma_ {Y}\right) ^ {\otimes 2} \right\rangle} _ {B _ {2}}, \tag {103} \\ \end{array}
+$$
+
+where  $B_{2} = 0$  from lemma H.1 (c) and  $B_{1} = B_{2}$  from lemma H.1(b). Therefore w, we complete the proof.
+
+# H.4 PROOF OF PROPOSITION 3.3
+
+The proposition is directly implied by definition (20) and Proposition 3.1 (3).
+
+# I SPECIAL CASE: LINEAR MASS-CONSTRAINED PARTIAL GROMOV-WASSERSTEIN DISTANCE
+
+In this section, we introduce the linearization technique for the "mass-constraint" partial Gromov-Wasserstein distance, which can be regarded as a special case of the proposed LPGW distance.
+
+Mass-constraint Partial Gromov-Wasserstein distance is defined as:
+
+$$
+M P G W _ {\rho} (\mathbb {X}, \mathbb {Y}) = \inf  _ {\gamma \in \Gamma_ {\leq} ^ {\eta} (\mu , \nu)} \int_ {(X \times Y) ^ {\otimes 2}} | g _ {X} (x, x ^ {\prime}) - g _ {Y} (y, y ^ {\prime}) | ^ {2} d \gamma^ {\otimes}, \tag {104}
+$$
+
+where  $\eta \in [0,\min (|\mu |,|\nu^1 |)]$  is a fixed number.
+
+$$
+\Gamma_ {\leq} ^ {\eta} (\mu , \nu) = \left\{\gamma \in \mathcal {M} _ {+} (X \times Y): \left(\pi_ {X}\right) _ {\#} \gamma \leq \mu , \left(\pi_ {Y}\right) _ {\#} \gamma \leq \nu , | \gamma | = \eta \right\},
+$$
+
+We use  $\Gamma_{\leq}^{\eta}(\mathbb{X},\mathbb{Y}^{1})$  to denote the set of all optimal transportation plans.
+
+Consider the following gm-spaces  $\mathbb{X} = (X,g_X,\mu),\mathbb{Y}^1 = (Y^1,g_{Y^1},\nu^1),\mathbb{Y}^2 = (Y^2,g_{Y^2},\nu^2)$  given  $\eta \in [0,\min (|\nu^1 |,|\nu^2 |)$  and we suppose  $|\mu | = \eta$  . Choose  $\gamma^1\in \Gamma_\leq^\eta (\mathbb{X},\mathbb{Y}^1),\gamma^2\in \Gamma_\leq^\eta (\mathbb{X},\mathbb{Y}^2)$  Suppose the Monge mapping assumption holds and let  $T^1,T^2$  be the corresponding transportation plan. We define
+
+$$
+k _ {\gamma^ {1}}: (\cdot_ {1}, \cdot_ {2}) \mapsto d _ {X} (\cdot_ {1}, \cdot_ {2}) - d _ {Y ^ {1}} \left(T _ {1} (\cdot_ {1}), T _ {1} (\cdot_ {2})\right)
+$$
+
+as linear MPGW embedding of  $\mathbb{Y}^1$  given  $\gamma^1$ .  $k_{\gamma^2}$  is defined similarly.
+
+Thus, similar to (15) and (7), the linear M-PGW distance between  $\mathbb{Y}^1,\mathbb{Y}^2$ , given  $\gamma^1,\gamma^2$  is defined as
+
+$$
+L M P G W _ {\eta} \left(\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}\right) := \| k _ {\gamma^ {1}} - k _ {\gamma^ {2}} \| _ {\mu^ {\otimes 2}} ^ {2}. \tag {105}
+$$
+
+and similar to (7),(15), the general case without Monge mapping assumption is
+
+$$
+L M P G W _ {\eta} \left(\mathbb {Y} ^ {1}, \mathbb {Y}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}\right) := \inf  _ {\gamma \in \Gamma_ {\leq} ^ {\eta} \left(\gamma^ {1}, \gamma^ {2}; \mu\right)} \int_ {(X \times Y) ^ {\otimes 2}} \left| g _ {X} - g _ {Y} \right| ^ {2} d \gamma^ {\otimes 2}, \tag {106}
+$$
+
+where  $\Gamma_{\leq}^{\rho}(\gamma^{1},\gamma^{2};\mu)\coloneqq \{\gamma \in \mathcal{M}_{+}(X\times Y^{1}\times Y^{2}):\left(\pi_{X,Y^{1}}\right)_{\#}\gamma \leq \gamma^{1},\left(\pi_{X,Y^{2}}\right)_{\#}\gamma \leq \gamma^{2},|\gamma | = \eta \}$ . And similar to (10),(20), by applying barycentric projection  $\mathcal{T}_{\gamma^1},\mathcal{T}_{\gamma^2}$ , we define the approximated linear-MPGW distance as follows:
+
+$$
+\begin{array}{l} a L M P G W _ {\eta} (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}) := \inf  _ { \begin{array}{c} \gamma^ {1} \in \Gamma_ {\leq} ^ {\rho} (\mathbb {X}, \mathbb {Y} ^ {1}) \\ \gamma^ {2} \in \Gamma_ {\leq} ^ {\rho} (\mathbb {X}, \mathbb {Y} ^ {2}) \end{array} } L M P G W (\widehat {\mathbb {Y}} ^ {1}, \widehat {\mathbb {Y}} ^ {2}; \mathbb {X}) \\ = \inf  _ {\substack {\gamma^ {1} \in \Gamma_ {\leq} ^ {\rho} (\mathbb {X}, \mathbb {Y} ^ {1}) \\ \gamma^ {2} \in \Gamma_ {\leq} ^ {\rho} (\mathbb {X}, \mathbb {Y} ^ {2})}} \int_ {X ^ {\otimes 2}} | d _ {Y ^ {1}} \left(\mathcal {T} _ {\gamma^ {1}} (x), \mathcal {T} _ {\gamma^ {1}} \left(x ^ {\prime}\right) - d _ {Y ^ {2}} \left(\mathcal {T} _ {\gamma^ {2}} (x), \mathcal {T} _ {\gamma^ {2}} \left(x ^ {\prime}\right)\right) \right| d \mu^ {\otimes 2}. \tag{107} \\ \end{array}
+$$
+
+Similar to theorem 3.2, we have:
+
+Proposition I.1. Choose  $\gamma^{*}\in \Gamma_{\leq}^{\rho ,*}(\mathbb{X},\mathbb{Y}),\gamma^{1}\in \Gamma_{\leq}^{\rho ,*}(\mathbb{X},\mathbb{Y}^{1}),\gamma^{2}\in \Gamma_{\leq}^{\rho ,*}(\mathbb{X},\mathbb{Y}^{2})$  , we have:
+
+(1) linear MPGW embedding of  $\mathbb{Y}^1$  can recover MPGW discrepancy between  $\mathbb{X}$  and  $\mathbb{Y}^1$ , i.e.
+
+$$
+\left\| k _ {\gamma^ {*}} \right\| _ {\mu^ {\otimes 2}} ^ {2} = M P G W _ {\eta} (x).
+$$
+
+In general, we have:
+
+$$
+L M P G W _ {\eta} (\mathbb {X}, \mathbb {Y}; \gamma^ {0}, \gamma^ {*}) := M P G W _ {\eta} (\mathbb {X}, \mathbb {Y}), \forall \gamma^ {0} \in \Gamma_ {\leq} ^ {\eta , *} (\mathbb {X}, \mathbb {X}).
+$$
+
+(2) If Monge mapping assumption hold for  $\gamma^1, \gamma^2$ , then (106),(105) coincide.  
+(3)  $\hat{\gamma}^{*} = (\mathrm{id}\times \mathcal{T}_{\gamma^{*}})_{\#}\mu$  is optimal for  $MPGW_{\eta}(\mathbb{X},\widetilde{\mathbb{Y}}_{\gamma^{*}}) = GW(\mathbb{X},\widetilde{\mathbb{Y}}_{\gamma^{*}})$ .  
+(4) When  $\lambda$  is sufficiently large, we have:
+
+$$
+L M P G W _ {\eta} (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}) = L P G W _ {\lambda} (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}) - \lambda (| \nu^ {1} | ^ {2} + | \nu^ {2} | ^ {2} - 2 \eta^ {2}).
+$$
+
+$$
+a L M P G W _ {\eta} (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}) = a L P G W _ {\lambda} (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}) - \lambda (| \nu^ {1} | ^ {2} + | \nu^ {2} | ^ {2} - 2 \eta^ {2}).
+$$
+
+(5) If  $|\mu| = |\nu^1| = |\nu^2| = \eta = 1$ ,  $LMPGW_{\eta}, LGW$  coincide and  $aLMPGW_{\eta}, aLGW$  coincide.
+
+Proof.
+
+(1) If Monge mapping assumption holds for  $\gamma^{*}$ , i.e.  $\gamma^{*} = (\mathrm{id} \times T)_{\#} \mu$ , we have:
+
+$$
+\begin{array}{l} M P G W _ {\eta} (\mathbb {X}, \mathbb {Y}) = \int_ {(X \times Y) ^ {\otimes 2}} \left| g _ {X} (x, x ^ {\prime}) - g _ {Y} (y, y ^ {\prime}) \right| ^ {2} d \left(\gamma^ {*}\right) ^ {\otimes 2} \\ = \| k _ {\gamma^ {*}} \| _ {\mu^ {\otimes 2}} ^ {2}. \\ \end{array}
+$$
+
+Without Monge mapping assumption, pick  $\gamma^0\in \Gamma_{\leq}^{\eta ,*}(\mathbb{X},\mathbb{X})$ , since  $|\gamma^0 | = \eta = |\mu |$  we have  $\gamma^0\in \Gamma (\mu ,\mu)$ . Thus
+
+$$
+\int_ {(X \times X) ^ {\otimes 2}} | g _ {X} (x, x ^ {\prime}) - g _ {X} (x ^ {0}, x ^ {0 \prime}) | ^ {2} d ((\gamma^ {0}) ^ {2}) ^ {\otimes 2} = M P G W _ {\eta} (\mathbb {X}, \mathbb {X}) = 0 = G W (\mathbb {X}, \mathbb {X}).
+$$
+
+Therefore  $\gamma^0\in \Gamma^* (\mathbb{X},\mathbb{X})$
+
+Pick  $\gamma \in \Gamma_{\leq}^{\eta}(\gamma^{0},\gamma^{*}) = \Gamma (\gamma^{0},\gamma^{*})$  , by (49), we have
+
+$$
+\begin{array}{l} \int_ {(X \times X \times Y) ^ {\otimes 2}} | g _ {X} (x, x ^ {\prime}) - g _ {Y} (y, y ^ {\prime}) | ^ {2} d \gamma^ {\otimes 2} \\ = \int_ {(X \times Y) ^ {\otimes 2}} \left| g _ {X} (x, x ^ {\prime}) - g _ {Y} (y, y ^ {\prime}) \right| ^ {2} d \left(\gamma^ {*}\right) ^ {\otimes 2} \\ = M P G W _ {\eta} (\mathbb {X}, \mathbb {Y} ^ {1}; \mathbb {X}, \gamma^ {0}, \gamma^ {*}) \\ \end{array}
+$$
+
+It holds for all  $\gamma \in \Gamma_{\leq}^{\eta}(\gamma^{0},\gamma^{*})$  , thus
+
+$$
+L M P G W _ {\eta} (\mathbb {X}, \mathbb {Y}; \mathbb {X}, \gamma^ {0}, \gamma^ {*}) = M P G W _ {\eta} (\mathbb {X}, \mathbb {Y} ^ {*}).
+$$
+
+(2) Under Monge mapping assumption, we have  $\gamma^1 = (\mathrm{id}\times T^1)_{\#}\mu ',\gamma^2 = (\mathrm{id}\times T^2)_{\#}\mu ''$  for some mappings  $T^1,T^2$  , where  $\mu^{\prime},\mu^{\prime \prime}\leq \mu$  . Since  $|\mu^{\prime}| = |\mu^{\prime \prime}| = \eta = |\mu |$  , then we have  $\mu '' = \mu ' = \mu$  . Thus,
+
+$$
+\begin{array}{l} \Gamma_ {\leq} (\gamma^ {1}, \gamma^ {2}; \mu) = \left\{(\mathrm {i d} \times T ^ {1} \times T ^ {2}) _ {\#} \mu^ {\prime}, \mu^ {\prime} \leq \mu , | \mu^ {\prime} | = \eta \right\} \\ = \left\{\left(\mathrm {i d} \times T ^ {1} \times T ^ {2}\right) _ {\#} \mu \right\} \\ \end{array}
+$$
+
+and we have (106),(105) coincide.
+
+(3) Since  $\eta = |\mu|$ , we have  $|\widetilde{\nu}_{\gamma^1}| = |\gamma^*| = \eta = |\mu|$ , thus
+
+$$
+\Gamma_ {\leq} ^ {\eta} (\mu , \widetilde {\nu} _ {\gamma^ {1}}) = \Gamma (\mu , \widetilde {\nu} _ {\gamma^ {1}}).
+$$
+
+Thus,  $MPGW_{\eta}(\mathbb{X},\widetilde{\mathbb{Y}}_{\gamma^{*}}) = GW(\mathbb{X},\widetilde{\mathbb{Y}}_{\gamma^{*}})$ . Since  $\gamma^{*}$  is optimal in  $MPGW_{\eta}(\mathbb{X},\mathbb{Y}) = GW(\mathbb{X},\mathbb{Y})$ , from Proposition 2.1, we have  $\widetilde{\gamma}_{\gamma^1}$  is optimal for  $GW(\mathbb{X},\mathbb{Y}_{\gamma^1}) = MPGW_{\eta}(\mathbb{X},\widetilde{\mathbb{Y}}_{\gamma^1})$  and we complete the proof.
+
+(4) Since  $X, Y^1, Y^2$  are compact, by Lemma E.2 in (Bai et al., 2024) when  $\lambda$  is sufficiently large, for each  $\gamma^1 \in \Gamma_{\leq, \lambda}^*(\mathbb{X}, \mathbb{Y}^1)$ ,  $\gamma^2 \in \Gamma_{\leq, \lambda}^*(\mathbb{X}, \mathbb{Y}^2)$ , we have  $|\gamma^1| = |\gamma^2| = \eta = |\mu|$ . By Proposition M.1. (Bai et al., 2024), we have  $\gamma^1, \gamma^2$  are optimal for  $MPGW_{\eta}(\mathbb{X}, \mathbb{Y}^1)$ ,  $MPGW_{\eta}(\mathbb{X}, \mathbb{Y}^2)$ . That is
+
+$$
+\Gamma_ {\leq , \lambda} ^ {*} (\mathbb {X}, \mathbb {Y} ^ {1}) \subset \Gamma_ {\leq} ^ {\eta , *} (\mathbb {X}, \mathbb {Y} ^ {1}), \Gamma_ {\leq , \lambda} ^ {*} (\mathbb {X}, \mathbb {Y} ^ {2}) \subset \Gamma_ {\leq} ^ {\eta , *} (\mathbb {X}, \mathbb {Y} ^ {2}).
+$$
+
+For the other direction, pick  $\gamma^1\in \Gamma_{\leq}^{\eta ,*}(\mathbb{X},\mathbb{Y}^1)$ ,  $\gamma^\prime \in \Gamma_{\leq ,\lambda}^{*}(\mathbb{X},\mathbb{Y}^{1})$ . Thus  $|\gamma^{\prime}| = \eta$ , and we have
+
+$$
+\begin{array}{l} C (\gamma^ {1}; \mathbb {X}, \mathbb {Y} ^ {1}, \lambda) \\ = \int_ {(X \times Y ^ {1}) ^ {\otimes 2}} | g _ {X} (x, x ^ {\prime}) - g _ {Y ^ {1}} (y ^ {1}, y ^ {1 \prime}) | ^ {2} d (\gamma^ {1}) ^ {\otimes 2} + \gamma (| \mu | ^ {2} + | \nu^ {1} | ^ {2} - 2 \eta^ {2}) \\ \leq \int_ {(X \times Y ^ {1}) ^ {\otimes 2}} | g _ {X} (x, x ^ {\prime}) - g _ {Y ^ {1}} (y ^ {1}, y ^ {1 \prime}) | ^ {2} d (\gamma^ {\prime}) ^ {\otimes 2} + \gamma (| \mu | ^ {2} + | \nu^ {1} | ^ {2} - 2 \eta^ {2}) \\ = C (\gamma^ {\prime}; \mathbb {X}, \mathbb {Y} ^ {1}, \lambda) \\ = P G W _ {\lambda} (\mathbb {X}, \mathbb {Y} ^ {1}) \\ \end{array}
+$$
+
+Thus,  $\gamma^1\in \Gamma_{\leq ,\lambda}^* (\mathbb{X},\mathbb{Y}^1)$
+
+Therefore,
+
+$$
+\Gamma_ {\leq , \lambda} ^ {*} (\mathbb {X}, \mathbb {Y} ^ {1}) = \Gamma_ {\leq} ^ {\eta , *} (\mathbb {X}, \mathbb {Y} ^ {1}), \Gamma_ {\leq , \lambda} ^ {*} (\mathbb {X}, \mathbb {Y} ^ {2}) = \Gamma_ {\leq} ^ {\eta , *} (\mathbb {X}, \mathbb {Y} ^ {2}).
+$$
+
+Pick  $\gamma^1\in \Gamma_{\leq ,\lambda}^* (\mathbb{X},\mathbb{Y}^1) = \Gamma_{\leq}^{\eta ,*}(\mathbb{X},\mathbb{Y}^1),\gamma^2\in \Gamma_{\leq ,\lambda}^* (\mathbb{X},\mathbb{Y}^2) = \Gamma_{\leq}^{\eta ,*}(\mathbb{X},\mathbb{Y}^2)$ , we have
+
+$$
+M L P G W _ {\eta} (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}) = L P G W _ {\lambda} (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}, \gamma^ {1}, \gamma^ {2}) - \lambda (| \nu^ {1} | + | \nu^ {2} | - 2 \eta^ {2})
+$$
+
+Take the infimum over all  $\gamma^1, \gamma^2$ , we obtain
+
+$$
+M L P G W _ {\eta} (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}) = L P G W _ {\lambda} (\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}) - \lambda (| \nu^ {1} | ^ {2} + | \nu^ {2} | ^ {2} - 2 \eta^ {2})
+$$
+
+Similarly, we have
+
+$$
+a M L P G W _ {\eta} \left(\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}\right) = a L P G W _ {\lambda} \left(\mathbb {Y} ^ {1}, \mathbb {Y} ^ {2}; \mathbb {X}\right) - \lambda \left(\left| \nu^ {1} \right| ^ {2} + \left| \nu^ {2} \right| ^ {2} - 2 \eta^ {2}\right) \tag {108}
+$$
+
+(5) In this case, we have  $\Gamma_{\leq}^{\eta}(\mu ,\nu^{1}) = \Gamma (\mu ,\nu^{1}),\Gamma_{\leq}^{\eta}(\mu ,\nu^{2} = \Gamma (\mu ,\nu^{2})$  . Thus
+
+$$
+\Gamma_ {\leq} ^ {\eta , *} (\mathbb {X}, \mathbb {Y} ^ {1}) = \Gamma^ {*} (\mathbb {X}, \mathbb {Y} ^ {1}), \Gamma_ {\leq} ^ {\eta , *} (\mathbb {X}, \mathbb {Y} ^ {2}) = \Gamma^ {*} (\mathbb {X}, \mathbb {Y} ^ {2}).
+$$
+
+Thus  $LMPGW, LGW$  coincide,  $aLMPGW, aLGW$  coincide.
+
+![](images/d2569464a1ab40c4ce184cf991aedab4add72365ddd37ba90311a823e3ab75d3.jpg)
+
+# J NUMERICAL IMPLEMENTATION OF LOT, LGW, LPGW DISTANCE.
+
+LOT distance In previous sections, we introduce LOT distance (31), aLOT distance (34) and its approximation formulation (35). Their relationship can be described as follows:
+
+- LOT distance (31) is proposed to approximate OT distance.  
+- aLOT distance (34) is proposed to approxiamte LOT distance.
+
+![](images/bc88a1454fa2a3ed1e4a4b95de0116f97b0b70c7294384602f68aa67f875d8ee.jpg)  
+(a) Shapes in ellipse dataset.
+
+![](images/548eef17a5afa809a6f3cadea2ece9cc287b19ade97e97770cdc3ba3017b7a4f.jpg)  
+(b) Reference spaces.  
+Figure 2: In the first figure, we visualize the ellipse dataset (Beier et al., 2022). For each ellipse shape  $X \subset \mathbb{R}^{n \times 2}$ , we normalize the scaling of the shape such that  $\max_{i,i' \in [1:n]} \| X[i,:] - X[i',:) \| = 1$ . The sizes of these shapes range from 90 to 650. In the second figure, we visualize the reference spaces. In each shape, the color represents the value of the probability mass at the corresponding location.
+
+- Formulation (35) is proposed to approximate aLOT distance.
+
+However, in practice, it is not a multi-layer approximation. The distance LOT (31) and the distance LOT (34) are only proposed for theoretical completeness and are not computationally feasible. In practice, formulation (35),
+
+$$
+\left\| \mathcal {T} _ {\gamma^ {1}} - \mathcal {T} _ {\gamma^ {2}} \right\| _ {L (\mu)},
+$$
+
+is used to approximate  $OT$  distance between  $\nu^1$  and  $\nu^2$  and in most of the reference which cite (Wang et al., 2013), (34) is referred as LOT distance, the original formulation (31),(34) are not mentioned.
+
+A similar convention is adapted to LGW and LPGW.
+
+LGW distance The relation between LGW distance (39), aLGW distance (10) and its approximation formulation (11) can be described as follows:
+
+- LGW distance (39) is proposed to approximate GW distance.  
+- aLGW distance (10) is proposed to approximate LGW distance.  
+- Formulation (11) is proposed to approximate aLGW distance.
+
+Similarly, it is not a multi-layer approximation. LGW distance (39) and aLGW distance  $(\ref{eq:11})$  are only proposed for theoretical completeness and are not computationally feasible. In practice, formulation (11),
+
+$$
+\left| g _ {Y ^ {1}} \left(\mathcal {T} _ {\gamma^ {1}} (\cdot_ {1}), \mathcal {T} _ {\gamma^ {2}} (\cdot_ {2})\right) - g _ {Y ^ {2}} \left(\mathcal {T} _ {\gamma^ {2}} (\cdot_ {1}), \mathcal {T} _ {\gamma^ {2}} (\cdot_ {2})\right) \right| _ {L (\mu^ {\otimes 2})} ^ {2},
+$$
+
+is used to approximate  $GW$  distance.
+
+LPGW distance The relation between LPGW distance (16), aLGW distance (20) and its approximation formulation (22) can be described as follows:
+
+- LPGW distance (16) is proposed to approximate PGW distance.  
+- aLGW distance (20) is proposed to approximate LPGW distance.  
+- Formulation (22) is proposed to approximate aLPGW distance.
+
+Similarly, it is not a multi-layer approximation. LPGW distance (16) and aLGW distance (20) are only proposed for theoretical completeness and are not computationally feasible. In practice, formulation (22),
+
+$$
+| g _ {Y ^ {1}} (\mathcal {T} _ {\gamma^ {1}} (\cdot_ {1}), \mathcal {T} _ {\gamma^ {2}} (\cdot_ {2})) - g _ {Y ^ {2}} (\mathcal {T} _ {\gamma^ {2}} (\cdot_ {1}), \mathcal {T} _ {\gamma^ {2}} (\cdot_ {2})) | _ {L ((\gamma_ {X} ^ {1} \wedge \gamma_ {X} ^ {2}) ^ {\otimes 2})} ^ {2} + \lambda (| \nu | ^ {1} + | \nu^ {2} | ^ {2} - 2 | \gamma_ {X} ^ {1} \wedge \gamma_ {X} ^ {2} | ^ {2})
+$$
+
+is used to approximate PGW distance.
+
+# K DETAILS OF ELLIPTICAL DISKS EXPERIMENT
+
+In this section, we present the details of the elliptical disks experiment.
+
+Dataset and numerical setting details. The dataset we used is the ellipse dataset given by (Beier et al., 2022), which consists of 100 distinct ellipses. Each ellipse is represented as an  $n \times 2$  matrix, where  $n$  ranges from 90 to 600. For reference shapes, we selected 9 different 2D shapes, including disks, squares, triangles, and others. See Figure 2 for a visualization of the dataset and the reference spaces. In this experiment, each shape is modeled with an empirical measure  $\sum_{i=1}^{n} \frac{1}{n} \delta_{x_i}$ .
+
+# Performance analysis
+
+We present the results in Table 4. As mentioned in the main text, LPGW is significantly faster than PGW, as it requires only  $N = 100$  PGW computations, while PGW requires  $\binom{N}{2}$  computations. Furthermore, we observe that for some reference spaces (e.g.,  $\mathbb{S}_5$ ,  $\mathbb{S}_7$ ,
+
+<table><tr><td></td><td></td><td>PGW</td><td>S1</td><td>S2</td><td>S3</td><td>S4</td><td>S5</td><td>S6</td><td>S7</td><td>S8</td><td>S9</td></tr><tr><td>λ</td><td>points</td><td></td><td>441</td><td>676</td><td>625</td><td>52</td><td>289</td><td>545</td><td>882</td><td>882</td><td>317</td></tr><tr><td rowspan="3">0.05</td><td>time (mins)</td><td>45.34</td><td>0.86</td><td>3.8</td><td>3.08</td><td>0.66</td><td>0.63</td><td>1.43</td><td>1.89</td><td>2.1</td><td>0.69</td></tr><tr><td>MRE</td><td>—</td><td>0.1982</td><td>0.1263</td><td>0.1428</td><td>0.6315</td><td>0.4732</td><td>0.1454</td><td>0.0394</td><td>0.0793</td><td>0.0245</td></tr><tr><td>PCC</td><td>—</td><td>0.5774</td><td>0.5741</td><td>0.5884</td><td>0.5584</td><td>0.6514</td><td>0.7698</td><td>0.9303</td><td>0.8711</td><td>0.9944</td></tr><tr><td rowspan="3">0.08</td><td>time (mins)</td><td>43.86</td><td>1.02</td><td>3.33</td><td>3.55</td><td>0.34</td><td>0.99</td><td>1.25</td><td>1.29</td><td>1.55</td><td>1.22</td></tr><tr><td>MRE</td><td>—</td><td>0.1941</td><td>0.1264</td><td>0.1431</td><td>0.2542</td><td>0.0705</td><td>0.0444</td><td>0.0205</td><td>0.0198</td><td>0.0245</td></tr><tr><td>PCC</td><td>—</td><td>0.5781</td><td>0.5738</td><td>0.5881</td><td>0.8581</td><td>0.8741</td><td>0.993</td><td>0.9952</td><td>0.9954</td><td>0.9949</td></tr><tr><td rowspan="3">0.1</td><td>time (mins)</td><td>46.97</td><td>0.76</td><td>3.78</td><td>3.13</td><td>0.08</td><td>0.62</td><td>1.56</td><td>1.91</td><td>1.98</td><td>0.71</td></tr><tr><td>MRE</td><td>—</td><td>0.1941</td><td>0.1264</td><td>0.1431</td><td>0.2542</td><td>0.0538</td><td>0.0444</td><td>0.0205</td><td>0.0198</td><td>0.0245</td></tr><tr><td>PCC</td><td>—</td><td>0.5781</td><td>0.5738</td><td>0.5881</td><td>0.8581</td><td>0.9871</td><td>0.993</td><td>0.9952</td><td>0.9954</td><td>0.9949</td></tr><tr><td rowspan="3">0.5</td><td>time (mins)</td><td>44.77</td><td>0.74</td><td>3.68</td><td>3.02</td><td>0.08</td><td>0.61</td><td>1.51</td><td>1.89</td><td>1.98</td><td>0.69</td></tr><tr><td>MRE</td><td>—</td><td>0.1932</td><td>0.1262</td><td>0.1428</td><td>0.2542</td><td>0.0538</td><td>0.0443</td><td>0.0205</td><td>0.0198</td><td>0.0246</td></tr><tr><td>PCC</td><td>—</td><td>0.5779</td><td>0.5737</td><td>0.5879</td><td>0.8583</td><td>0.9871</td><td>0.993</td><td>0.9952</td><td>0.9953</td><td>0.9949</td></tr></table>
+
+Table 4: In the first column, the values 0.05, 0.08, 0.1, 0.5, represent the selected  $\lambda$  values. For each  $\lambda$ , the first row shows the wall-clock time for PGW and LPGW. The second and third rows display the MRE (mean relative error) and PCC (Pearson correlation coefficient), respectively.
+
+$\mathbb{S}_9)$ , the MRE is relatively lower. Moreover, for most reference spaces, including  $\mathbb{S}_7$ ,  $\mathbb{S}_8$ , and  $\mathbb{S}_9$ , LPGW admits a PCC greater than 0.85. Finally, when  $\lambda$  is larger, the PCC tends to be higher, and the MRE is lower across all reference spaces, as discussed further in the next section. These results highlight the importance of the choice of reference space, as is commonly the case for linear OT-based methods.
+
+Relative error analysis Given  $\nu^1, \nu^2, \ldots, \nu^K$  and reference measure  $\mu$ , the relative error is defined as:
+
+$$
+\mathrm {M R E} = \frac {1}{\binom {K} {2}} \sum_ {i \neq j} \frac {\left| P G W \left(\nu^ {i} , \nu^ {j}\right) - L P G W \left(\nu^ {i} , \nu^ {j} ; \mu\right) \right|}{P G W \left(\nu^ {i} , \nu^ {j}\right)}. \tag {109}
+$$
+
+Remark K.1. For numerical stability, when  $\lambda$  is small (i.e.,  $\lambda = 0.05, 0.08$ ), we remove the PGW/LPGW distance whenever  $PGW \leq 1 \cdot 10^{-10}$  since  $1 \cdot 10^{-10}$  is the tolerance in the PGW algorithm. In this case,  $PGW \approx 1 \cdot 10^{-10}$  and  $LPGW \approx 1 \cdot 10^{-11}$  which renders the relative error uninformative.
+
+We decompose this error into the following four aspects:
+
+- The transportation plan induced by LPGW may not necessarily be the optimal transportation plan for the PGW problem.  
+- In practice, we use the barycentric projected measure  $\hat{\nu}^i$  to approximate  $\nu^i$  for each  $i$ . These two measures can be distinct, especially when the optimal transportation plan is not generated by the Monge mapping.  
+- The solvers for both PGW and LPGW rely on the Frank-Wolfe algorithm. Due to the non-convexity of these problems, the Frank-Wolfe algorithm may yield a local minimizer instead of a global one. Therefore, the computed PGW transportation plan might not be optimal.  
+- In practice, we approximate the original LPGW distance (16) (or (10)) using the approximation formulation (22). This introduces a gap between the real LPGW distance and the approximation.
+
+It is important to note that the first issue arises from a theoretical perspective, while the remaining aspects are due to the numerical implementation. The first two errors are also present in other linear OT-based methods (Wang et al., 2013; Cai et al., 2022; Bai et al., 2023; Beier et al., 2022). The third issue stems from the non-convexity of GW/PGW, affecting both LGW and LPGW similarly. The final error is specific to LPGW due to the approximation formulation used.
+
+In practice, several methods can be employed to reduce these errors. For example, the dataset can be represented as empirical measures with equal mass at each point. This approach is effective for unbalanced linear OT techniques (e.g., (Bai et al., 2023; Cai et al., 2022)), as these methods do not require mass normalization. With high probability, this will result in a Monge mapping, reducing the second error. Additionally, to minimize the last error, as stated in Theorem 3.2, using a higher  $\lambda$  leads to a lower error, and when  $\lambda$  is sufficiently large, this error becomes zero.
+
+MDS visualization and analysis. We visualize the multi-dimensional scaling (MDS) embeddings for both PGW and LPGW with respect to each reference space in Figure 3. We observe that when  $\lambda$  is large (i.e.  $\lambda = 0.1, 0.5$ ), LPGW with reference space  $\mathbb{S}_5, \mathbb{S}_6, \mathbb{S}_7, \mathbb{S}_8, \mathbb{S}_9$  admit similar patterns. When  $\lambda = 0.05$  or  $\lambda = 0.08$ , we observe LPGW with reference  $\mathbb{S}_7, \mathbb{S}_8, \mathbb{S}_9$  and PGW admit similar patterns. From this figure, we observe that  $\mathbb{S}_7, \mathbb{S}_8, \mathbb{S}_9$  admit better performance than other reference spaces.
+
+Summary. Although LPGW methods are subject to potential errors, the high PCC observed between LPGW and PGW (when, e.g., the reference space is  $\mathbb{S}_7,\mathbb{S}_8,\mathbb{S}_9$ ), suggests that LPGW can serve as a good proxy for PGW, rather than simply as an approximation.
+
+![](images/e93d9b2d5b2483cf9ebb0865398ad6d6c64e706125fd54a1cf2182e3d999dde4.jpg)
+
+![](images/74426b15ee54e48dc7ef26fc0285ad1a28e09ff554342d04d014060f3e8a684c.jpg)  
+(a)  $\lambda = 0.05$
+
+![](images/e905702c0bb7a4e60d5c6bd0747e21b224412beab20cea183e563545e8b925e7.jpg)  
+(b)  $\lambda = 0.08$
+
+![](images/de3420e1b95128dc147be5643357b9096d1f73b69eb8edd2b508bf58d3b0cd16.jpg)  
+(c)  $\lambda = 0.1$  
+(d)  $\lambda = 0.5$  
+Figure 3: MDS visualization for  $\lambda = 0.05, 0.08, 0.10, 0.50$ . Each subfigure shows the MDS visualization for PGW and LPGW based on different reference spaces. In each figure, the first subfigure in the first row is the MDS visualization of PGW.
+
+# L DETAILS OF SHAPE RETRIEVAL EXPERIMENT
+
+Dataset details. We test two datasets in this experiment: a 2D dataset and a 3D dataset. The visualization of the two datasets is presented in Figure 4.
+
+In addition, for each shape  $X = \{x_{1},\ldots x_{n}\}$ , we normalize the scaling such that  $\max_{i\neq j}\| x_i - x_j\| = 1$ . For the 3D dataset, we apply the  $k$ -means clustering to reduce the size of each shape to 500, from its original size 2048.
+
+![](images/3cb145b7c717843e9c1bd9842f5f59df284ff09b0cbc83e9512eb372a72c89fe.jpg)  
+(a) 2D Dataset
+
+Numerical details. We represent the shapes in each dataset as mm-spaces  $\mathbb{X}^i = \left(\mathbb{R}^d, \| \cdot \|_2, \mu^i = \sum_{k=1}^{n^i} \alpha^i \delta_{x_k^i}\right)$ .
+
+We use  $\alpha^i = \frac{1}{n^i}$  to compute the GW/LGW distances for the balanced mass constraint setting.
+
+For the PGW/LPGW distances, we set  $\alpha = \frac{1}{N}$ , where  $N$  is the median number of points across all shapes in the dataset.
+
+![](images/9c9b9a94384f3e6c22abfa6c841a14512db20268bf9d000767930a37e07f092c.jpg)  
+(b) 3D Dataset  
+Figure 4: Datasets for shape retrieval experiment.
+
+For the SVM experiments, we use  $\exp(-\sigma D)$  as the kernel for
+
+the SVM model. Here, we normalize the matrix  $D$  and choose the best  $\sigma \in \{0.001, 0.01, 0.1, 1, 5, 8, 10, 100\}$  for each method used in order to facilitate a fair comparison of the resulting performances. We note that the resulting kernel matrix is not necessarily positive semidefinite.
+
+Parameter selection for PGW and LPGW. The parameter  $\lambda$  for PGW and LPGW is chosen as follows. For the 2D dataset, we set  $\lambda$  such that  $\lambda \leq \lambda_{max} = \frac{1}{2}\max_i(|C^i|^2) = 0.5$ , since all shapes are normalized to a scale of 1. We perform a line search for  $\lambda$  over the set  $\{0.2, 0.3, 0.5\}$  and select  $\lambda = 0.2$  for both PGW and LPGW.
+
+For the 3D dataset, we randomly select 2-3 shapes and compute the PGW distance between these shapes and the reference space. We find that  $\lambda_{max} \approx 0.15$ . Thus, we conduct a line search for  $\lambda$  over the range  $\{0.02, 0.03, 0.05, 0.06, 0.070, 0.08, 0.080, 1, 0.15\}$  and choose  $\lambda = 0.06$  for both PGW and LPGW.
+
+Reference space setting for LGW and LPGW. Similar to the classical OT method (Wang et al., 2013), the ideal reference space should represent the "center" of the tested measures. In practice, we typically use the GW barycenter (Peyre et al., 2016) or the PGW barycenter (Bai et al., 2024).
+
+For each dataset, we randomly select one point cloud from each class. Specifically, for the 2D dataset, we select 8 shapes, and for the 3D dataset, we select 4 shapes. We then compute the GW/PGW barycenter between these selected shapes. The wall-clock time for this computation on the 2D dataset is about 10 seconds, and for the 3D dataset, it is about 5 minutes. For fairness in comparison, we apply the same reference space for both the LGW and LPGW methods in all experiments.
+
+Nearest neighbor classification. Once the pairwise distance matrices have been computed between the shapes using each method, we use the computed distances for a nearest-
+
+![](images/040167d5b8d4bb44accf159d3bb40f2706f1731f29d58d42f48a13228e74b37f.jpg)
+
+![](images/e590a309f0f3f1e646d08ba76bac169dbe079b22fdc4e0120e552863083761f4.jpg)  
+(a) 2D Dataset.  
+(b) 3D Dataset.  
+Figure 5: Confusion matrices computed from nearest neighbor classification experiments.
+
+neighbor classification experiment. We choose a single representative at random from each class in the dataset and classify each shape according to its nearest representative. This is repeated over 10,000 iterations, and we generate a confusion matrix for each distance used.
+
+Performance analysis. The pairwise distance matrices are visualized for each dataset in Figure 7, and the confusion matrices from the nearest neighbor classification experiment on each dataset are shown in Figure 5. Finally, the classification accuracy with the SVM experiments is reported in Table 2. The results indicate that the LPGW distance is able to obtain high performance across both data sets consistently.
+
+From Figure 7, we observe that PGW and LPGW qualitatively admit a more reasonable similarity measure compared to the other considered methods. For example, on the 2D dataset, class "bone" and "rectangle" should have a relatively smaller distance than "bone" and "annulus". Ideally, a reasonable distance should satisfy the following:  $0 < d(\text{bone}, \text{rectangle}) < d(\text{bone}, \text{annulus})$ . However, we do not observe this relation in GW or LGW.
+
+In the 3D dataset, the main challenge is the presence of incomplete shapes, which represent only a part of the corresponding complete shape and can have a large dissimilarity to each other. In this experiment, we observe that PGW/LPGW admits slightly better performance. We suspect due the the unbalanced setting of these two methods, these two methods are more robust to the incomplete shape classification.
+
+![](images/57b4c32ceec0ec57ccf944903c7b26cebb1c79e9c9238d457f02fbe14b26ce24.jpg)  
+Figure 6: We visualize the accuracy of LPGW for different  $\lambda$ . For each  $\lambda$ , we select the best  $\sigma$  to compute the accuracy. We observe that when  $\lambda \geq 0.08$ , LPGW and LGW have the same accuracy.
+
+![](images/87e77f439b9f9b60bd5e65ba53de355ae7da4e602b2f12ae658a89c9e7a72242.jpg)  
+(a) 2D Dataset.
+
+![](images/18af3ef3808b594b4b26aaeb1b2a3ca4205f3dfb15b0d441de32099585dfff93.jpg)  
+(b) 3D Dataset.  
+Figure 7: Visualization of pairwise distance matrices resulting from each of the considered methods: GW, PGW, LGW, and LPGW (ours). In (b), the label "inc. airplane" denotes the incomplete airplane shape class, and similarly for each of the other 3 such classes.
+
+Note that in this dataset all shapes have the same size. Thus, when  $\lambda$  is sufficiently large (i.e.  $\lambda = 0.5$ ), LPGW can recover the performance of LGW. We refer to Figure 6 for visualization.
+
+# M DETAILS OF LEARNING WITH TRANSFORM-BASED EMBEDDINGS EXPERIMENT
+
+Reference space. Similar to the shape retrieval experiment, for each class/digit, we select one shape from the training set and compute the GW barycenter based on the selected shapes. Note, in this step, the reference space we obtain is  $(M_0\in \mathbb{R}^{n_0\times n_0},p^0)$ , where  $n_0\in \mathbb{N}$  is the size of the reference space,  $M_0$  denotes the pairwise distance matrix, and  $p^0$  is the PMF of the reference measure. Here,  $n_0$ ,  $p^0$  are inputs to the GW barycenter algorithm, and in this experiment, we set  $n_0$  to be the mean of the sizes of the selected shapes, and  $p^0$  is set to be  $p^0 = \frac{1}{n_0} 1_{n_0}$ .
+
+However,  $M_0$  cannot be applied as reference space for the LOT method. Thus, we apply the MDS method for  $M_0$  and obtain the supported points  $X^0 = \{x_1^0, \ldots x_{n_0}^0\}$  of the reference space, i.e.,
+
+$$
+X = \arg \min  _ {X} \sum_ {i = 1} ^ {n} \| (M _ {0}) _ {i, j} - \| x _ {i} - x _ {j} \| \| ^ {2}, \text {w h e r e} x _ {i} = X [ i,: ], \forall i.
+$$
+
+We use  $(M_0, p^0, \sum_{i=1}^{n} x_i)$  as the reference space for LOT/LGW/LPGW. The wall-clock time of barycenter computation is 26 seconds.
+
+Numerical details. Note, since the goal of this experiment is to test the performance of learning from embeddings under corruption by random rotation/flipping and noise, we remove digits  $\{5,9\}$  from the data since the rotated and flipped digit "2" is highly similar to "5". Similarly, rotated "6" is nearly identical to "9".
+
+The classification model we selected is the logistic regression model as provided by the scikit-learn package. For the LOT and LGW methods, due to the balanced mass requirement, we
+
+![](images/aeb8c3ff5821b28e3d7999449f5d6ce473b60978eab71d8f38e041cabf1b0868.jpg)  
+Figure 8: We visualize the interpolation between two shapes using GW, PGW, LGW, and LPGW. In the first column, the three shapes shown are the source shape, the target shape, and the reference space used in LGW and LPGW. The target shape, digit "4", is corrupted by the addition of noise points, with a total mass of  $\eta = 0.3$ .
+
+normalize the PMF of all shapes in the testing dataset. For LPGW, this normalization is not applied.
+
+Parameter setting for LPGW. Note that in this case, the training data is not corrupted. Thus, we can set  $\lambda = \lambda_{max} = \frac{1}{2}\max_i C_i^2$  where  $C_i$  is the cost matrix for shape  $i$ . In this experiment, we set  $\lambda = 40$ . In addition, to improve the computational speed of LGW/LPGW, the augmented training data obtained by rotation and flipping is NOT contained in these two methods since the rotated shape is equivalent to the original shape in the setting of GW/PGW.
+
+Performance analysis. The LOT method achieves  $51.3\%$  accuracy when  $\eta = 0$ . This indicates that even with the addition of rotated/flipped data to the training set, the LOT embedding method struggles to classify the rotated digits with high accuracy. In contrast, LGW/LPGW achieve  $82.5\%$  accuracy, indicating that these two embedding techniques are more robust to corruption by rotation/flipping.
+
+When  $\eta \geq 0.1$ , the accuracy of both LOT and LGW drops to  $10 - 20\%$ . However, we observe that the LPGW embedding maintains a strong accuracy between  $70\%$  and  $85\%$ . This demonstrates that the LPGW embedding is more robust to corrupted test data.
+
+# M.1 TOY EXAMPLE: POINT CLOUD INTERPOLATION
+
+In this subsection, we select one shape from the training data and a noise-corrupted shape from the testing data (with rotation and flipping removed). We demonstrate the interpolation between these two shapes using the GW barycenter (Peyre et al., 2016), PGW barycenter (Bai et al., 2024), LGW geodesic (Beier et al., 2022), and our LPGW interpolation. The goal is to intuitively visualize the LGW/LPGW embeddings for the reader's understanding.
+
+Our method and baseline methods. Let  $\mathbb{X}$  be the reference mm-space applied in the classification experiment. Note, numerically, it can be described by  $(C^0,X^0,p^0)$ , where  $p^0\in \mathbb{R}_+^{n_0}$  is the PMF;  $n_0$  is the size of  $p^0$ ;  $X^0\in \mathbb{R}^{n_0\times 2}$  is the set of 2D supported points; and  $C^0 = [\| X_i^0 -X_j^0\|_{i,j\in [1:n_0]}^2 ]$  is the corresponding cost matrix.
+
+That is, we can use  $(C^0,p^0)$  to represent  $\mathbb{X}^3$ . Similarly, we use  $(C^{1},p^{1})$  and  $(C^2,p^2)$  to represent the source and target mm-spaces (shapes).
+
+Now we introduce the  $\mathbf{GW}$  barycenter method. For each time  $t\in \{0,1 / 6,\dots ,6 / 6 = 1\}$ , we solve the barycenter problem
+
+$$
+C ^ {*} = \operatorname * {a r g   m i n} _ {C \in \mathbb {R} ^ {n _ {0} \times n _ {0}}} \big ((1 - t) G W ((C, p ^ {0}), (C ^ {1}, p ^ {1})) + t G W ((C, p ^ {0}), (C ^ {2}, p ^ {2})) \big).
+$$
+
+When  $t = 0$ ,  $(C^*, p^0)$  and  $(C^1, p^1)$  represent similar shapes. In fact, if  $p^0 = p^1$ , the two shapes are identical. Similarly, if  $t = 1$ ,  $(C^*, p^1)$  is similar to the target  $(C^2, p^2)$ . For  $t \in (0, 1)$ ,  $(C^*, p^0)$  represents an interpolation shape between the source shape  $(C^1, p^1)$  and the target shape  $(C^2, p^2)$ .
+
+The PGW barycenter method is defined similarly.
+
+In the LGW geodesic method, let  $\sum_{i=1}^{n} \delta_{\hat{y}_i^1} p_i^0$  denote the barycentric projection obtained by  $\gamma$ , where  $\gamma$  is the optimal transportation plan for the GW problem between  $(C^0, p^0)$  and  $(C^1, p^1)$ . The numerical implementation of the LGW embedding is given by
+
+$$
+E ^ {1} = C ^ {0} - \left[ \| \hat {y} _ {i} ^ {1} - \hat {y} _ {j} ^ {1} \| ^ {2} \right] _ {i, j \in [ 1: n _ {0} ]} \in \mathbb {R} ^ {n _ {0} \times n _ {0}}
+$$
+
+(we refer to (5) for the original formulation). Similarly, we can define the LGW embedding for the target shape  $(C^2,p^2)$ , denoted as  $E_{2}\in \mathbb{R}^{n_{0}\times n_{0}}$ .
+
+Then for each  $t$ , the interpolation/geodesic of LGW is defined by
+
+$$
+C ^ {0} + (1 - t) E _ {1} + t E _ {2}. \tag {110}
+$$
+
+For the LPGW interpolation method, the numerical LPGW embeddings  $E_{1}, E_{2} \in \mathbb{R}^{n_{0} \times n_{0}}$  are defined similarly, and we refer to (24) for details. We use (110) for the interpolation.
+
+To visualize these interpolations, for each  $C_t$ , we adapt the MDS method, and the solution  $X_{t}\in \mathbb{R}^{n_{0}\times 2}$  is a point cloud. We visualize these point clouds in Figure 8.
+
+Performance comparison. In this experiment, the size of these shapes is in the range of 200-250. GW/PGW requires 230-270 seconds, while LGW/LPGW requires 1-2 seconds. In Figure 8, we observe that the shapes generated by GW/LGW contain more noise points. Additionally, at times such as  $t = 3 / 6$  and  $t = 4 / 6$ , the generated shapes are difficult to distinguish due to the noise points.
+
+In contrast, the shapes generated by the PGW/LPGW methods contain significantly fewer noise points.
+
+Note that at  $t = 0$  and  $t = 1$ , the shapes generated by the LGW/LPGW methods can be treated as visualizations of the corresponding embeddings for the source shape  $(C^1, p^1)$  and target shape  $(C^2, p^2)$ , respectively.
+
+# M.2 OTHER BASELINES OT/GW/PGW
+
+Note that OT, GW, and PGW methods cannot be directly applied to this experiment since we use a linear model (logistic regression) as the classifier. However, we can adapt the kernel-SVM methods described in the shape retrieval experiment for this classification task as well. The primary distinction between these three methods and LOT, LGW, and LPGW lies in their computational complexity.
+
+Suppose  $N_{\mathrm{train}}$  and  $N_{\mathrm{test}}$  represent the sizes of the training and testing datasets, respectively. Let  $n$  and  $d$  denote the average size and dimension of all the point clouds or digits. Additionally, let  $\mathcal{N}$  represent the average number of iterations required by the FW algorithms for GW and PGW.
+
+The computational cost of these methods is summarized as follows:
+
+In methods OT and LOT, the term  $(n^3 + n^2 d)$  refers to the complexity of computing one OT distance, while  $nd$  represents the complexity of computing one LOT distance given two embeddings.
+
+<table><tr><td>Method</td><td>Training Process</td><td>Testing Process</td></tr><tr><td>OT</td><td>O(N2train·(n3+n2d))</td><td>O(NtrainNtest(n3+n2d))</td></tr><tr><td>LOT</td><td>O(Ntrain·(n3+n2d))</td><td>O(Ntest·(n3+n2d) + NtestNtrain·(nd))</td></tr><tr><td>GW</td><td>O(N2train·(n3N+ n2d))</td><td>O(NtestNtrain·(n3N+n2d))</td></tr><tr><td>LGW</td><td>O(Ntrain(n3N+ n2d))</td><td>O(Ntest(n3N+n2d) + NtestNtrain·(n2))</td></tr><tr><td>PGW</td><td>O(N2train·(n3N+n2d))</td><td>O(NtestNtrain·(n3N+n2d))</td></tr><tr><td>LPGW</td><td>O(Ntrain(n3N+n2d))</td><td>O(Ntest(n3N+n2d) + NtestNtrain(n2))</td></tr></table>
+
+Table 5: Computational complexity of different methods.
+
+Similarly, in methods GW and LGW, the term  $(n^3\mathcal{N} + n^2 d)$  refers to the computational complexity of computing one GW distance, and the term  $(n^2)$  represents the complexity of computing one LGW distance given two embeddings.
+
+The complexities for PGW and LPGW follow a similar pattern.
+
+In this experiment,  $N_{\mathrm{train}} = 8000$  and  $N_{\mathrm{test}} = 1000$ . OT-based methods require approximately 50 hours (2-3 days) to train the model, while GW and PGW require about 400 hours (over 15 days). In contrast, LOT, LGW, and LPGW take only 3-5 minutes.
+
+# N COMPLEMENTAL RESULTS MNIST CLASSIFICATION EXPERIMENT
+
+Data reconstruction. We visualize the reconstructed digits using LOT, LGW, and LPGW embeddings under two settings:  $\eta = 0$  and  $\eta = 0.2$  (See figure 1b). When the data is not corrupted by noise points ( $\eta = 0$ ), the reconstructed digits from all methods closely resemble the original data. However, when  $20\%$  of noise points are added ( $\eta = 0.2$ ), LPGW's reconstructed digits effectively exclude most of the noise points. In contrast, the embeddings produced by OT and LGW retain information from the noisy data. This demonstrates that the LPGW embedding leverages the partial matching property of PGW, ensuring that most of the noise points are excluded during the embedding process. This explains the robustness of LPGW embeddings to noise corruption.
+
+t-SNE. From the figure 9, we can observe that when  $\eta \geq 0.1$ , LPGW embeddings show greater separability than those of LOT or LGW.
+
+# O COMPUTE RESOURCES
+
+All experiments presented in this paper are conducted on a computational machine with an AMD EPYC 7713 64-Core Processor,  $8 \times 32\mathrm{GB}$  DIMM DDR4,  $3200\mathrm{MHz}$ , and an NVIDIA RTX A6000 GPU.
+
+![](images/09817ab5508f3db02c58c8ee2c079ed8aa592e99c91ef207266cac515a592df4.jpg)  
+Figure 9: t-SNE visualization of embeddings. In the first column, the label "orig" indicates the experimental results for the original testing dataset without random rotations/flips or noise. In the remaining columns, random rotations/flips are applied to the testing data, and  $\eta$  denotes the total mass of noise points.
\ No newline at end of file
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+# LINEAR TRANSFORMER TOPOLOGICAL MASKING WITH GRAPH RANDOM FEATURES
+
+Isaac Reid\*12, Avinava Dubey\*2, Deepali Jain\*3, Will Whitney\*3, Amr Ahmed\*2, Joshua Ainslie\*3, Alex Bewley\*3, Mithun George Jacob\*3, Aranyak Mehta\*2, David Rendleman\*3, Connor Schenck\*3, Richard E. Turner\*1, René Wagner\*3, Adrian Weller\*14, Krzysztof Choromanski\*35
+
+<sup>1</sup>University of Cambridge, <sup>2</sup>Google Research, <sup>3</sup>Google DeepMind, <sup>4</sup>Alan Turing Institute,
+
+5 Columbia University
+
+# ABSTRACT
+
+When training transformers on graph-structured data, incorporating information about the underlying topology is crucial for good performance. Topological masking, a type of relative position encoding, achieves this by upweighting or downweighting attention depending on the relationship between the query and keys in a graph. In this paper, we propose to parameterise topological masks as a learnable function of a weighted adjacency matrix – a novel, flexible approach which incorporates a strong structural inductive bias. By approximating this mask with graph random features (for which we prove the first known concentration bounds), we show how this can be made fully compatible with linear attention, preserving  $\mathcal{O}(N)$  time and space complexity with respect to the number of input tokens. The fastest previous alternative was  $\mathcal{O}(N\log N)$  and only suitable for specific graphs. Our efficient masking algorithms provide strong performance gains for tasks on image and point cloud data, including with  $>30k$  nodes.
+
+# 1 INTRODUCTION
+
+Across data modalities, transformers have emerged as a leading machine learning architecture (Vaswani, 2017; Dosovitskiy et al., 2020; Arnab et al., 2021). They derive their success from modelling complex dependencies between the tokens using an attention mechanism. However, in its vanilla instantiation the transformer is a set function, meaning it is invariant to permutation of its inputs. This is often undesirable; e.g. words tend to be arranged in a particular sequence, and graph nodes are connected to a particular set of neighbours. Therefore, it has become standard to modify the attention matrix depending on any underlying structure, building in inductive bias. For instance, causal attention applies a triangular binary mask to ensure that tokens only attend to preceding elements of the sequence (Yang et al., 2019). Meanwhile, relative position encoding (RPE) captures the spatial relationship between the tokens (Shaw et al., 2018). In some cases, the queries and keys can be arranged in a graph  $\mathcal{G}$  and a function  $\mathbf{M}(\mathcal{G})$  can be used to modulate the transformer attention. This is referred to as topological masking (Choromanski et al., 2022). It provides a powerful avenue for incorporating structural inductive bias from  $\mathcal{G}$  into transformers, in some cases closing the performance gap with the best-customised graph neural networks (Ying et al., 2021).
+
+Topological masking of low-rank attention. Topological masking is simple when the attention is full-rank. In this case, the entire attention matrix  $\mathbf{A} \in \mathbb{R}^{N \times N}$  (with  $N$  the number of input tokens) is materialised in memory, so its individual entries  $\mathbf{A}_{ij}$  can be pointwise multiplied by a (fixed or learnable) mask  $\mathbf{M}(\mathcal{G})_{ij}$ . But full-rank attention famously incurs a quadratic  $\mathcal{O}(N^2)$  space and time complexity cost, which may become prohibitive on large inputs. This has motivated a number of low-rank decompositions of  $\mathbf{A}$ . These improve scalability by avoiding computing  $\mathbf{A}$  explicitly, instead doing so implicitly in feature space. The question of how to perform topological masking in the low-rank setting is challenging and not fully solved. Previously proposed algorithms are limited to inexpressive functions  $\mathbf{M}$  and restricted classes of graphs  $\mathcal{G}$  (e.g. trees and grids), and are often  $\mathcal{O}(N \log N)$  time complexity rather than  $\mathcal{O}(N)$  (App. B; Choromanski et al., 2022; Liutkus et al.,
+
+![](images/78571aa896a3460b294e1b30d8585dd78eaa5f3e1028bac7e18564303bea2d06.jpg)  
+Figure 1: Schematic overview. Regular attention is  $\mathcal{O}(N^2)$ , with  $N$  the number of input tokens. Topological masking modulates  $\mathbf{A}$  by a graph function  $\mathbf{M}(\mathcal{G})$ , improving predictive performance. Linear attention reduces the time complexity to  $\mathcal{O}(N)$  by leveraging a low-rank decomposition. Our contribution (blue) is the first algorithm to achieve both -  $\mathcal{O}(N)$  topological masking of low-rank attention - by approximating  $\mathbf{M}(\mathcal{G})$  with graph random features (GRFs). GRFs are sparse vectors (denoted 'sp.') computed by sampling random walks, constructed so that  $\mathbb{E}(\widehat{\Phi}_{\mathbf{Q},\mathcal{G}}\widehat{\Phi}_{\mathbf{K},\mathcal{G}}^{\top}) = \mathbf{A} \odot \mathbf{M}(\mathcal{G})$  with strong concentration properties (Thm. 3.2).
+
+2021). Addressing these shortcomings is the chief focus of this manuscript. Our central question is: how can we implement efficient yet flexible topological (graph-based) masking of transformer attention, in the low-rank setting where  $\mathbf{A}$  is never explicitly instantiated?
+
+Graph random features. To answer the question posed above, we propose to use graph random features (GRFs) (Choromanski, 2023; Reid et al., 2024b). GRFs are sparse, randomised vectors whose dot product gives an unbiased estimate of arbitrary functions of a weighted adjacency matrix, a class that captures the structure of  $\mathcal{G}$  and is popular in classical graph-based machine learning (Smola and Kondor, 2003; Kondor and Lafferty, 2002). GRFs have been used for approximate kernelised node clustering and in scalable graph-based Gaussian processes (Reid et al., 2023a; 2024c), but never before in transformers. Advancing understanding of the convergence properties of GRFs is another key goal of this work.
+
+Our contributions. We present the first known  $\mathcal{O}(N)$  algorithm for topological masking of linear attention transformers with general  $\mathcal{G}$ , using graph random features (Fig. 1).
+
+1. We propose to modulate the transformer attention mechanism with a learnable function of a weighted adjacency matrix - a simple, general topological masking trick which brings a strong inductive bias and substantial improvements to predictive performance.  
+2. To scale, we replace the exact, deterministic mask with a sparse, stochastic estimate constructed using graph random features (GRFs). We use a novel exponential concentration bound, requiring only a mild technical assumption on  $\mathcal{G}$  as  $N$  grows, to prove that our method is linear time and space complexity with respect to input size: the first known algorithm with this property.  
+3. We demonstrate strong accuracy gains on image data, as well as for modelling the dynamics of massive point clouds ( $>30\mathrm{k}$  particles) in robotics applications where efficiency is essential.
+
+# 2 PRELIMINARIES
+
+Masked attention. Denote by  $N$  the number of input tokens and  $d$  the dimensionality of their latent representations. Consider matrices  $\mathbf{Q},\mathbf{K},\mathbf{V}\in \mathbb{R}^{N\times d}$  whose rows are given by the query  $(\{\pmb {q}_i\}_{i = 1}^N)$ , key  $(\{\pmb {k}_j\}_{i = 1}^N)$  and value vectors  $(\{\pmb {v}_i\}_{i = 1}^N)$  respectively. The attention mechanism, a basic processing unit of the transformer, can be written:
+
+$$
+\operatorname {A t t} (\mathbf {Q}, \mathbf {K}, \mathbf {V}) := \mathbf {D} ^ {- 1} \mathbf {A V}, \quad \mathbf {A} := \mathcal {K} (\mathbf {Q}, \mathbf {K}), \quad \mathbf {D} := \mathbf {A} \mathbf {1} _ {N}. \tag {1}
+$$
+
+Here, the function  $\mathcal{K}$  assigns a similarity score to each query-key pair and  $\mathbf{1}_N\coloneqq [1]_{i = 1}^N\in \mathbb{R}^N$ . For regular softmax attention, one simply takes  $\mathcal{K}(\mathbf{Q},\mathbf{K})_{ij} = \exp (q_i^\top k_j)$ , optionally also normalising by  $d$ . To incorporate masking, we make the modification
+
+$$
+\mathbf {A} _ {\mathbf {M}} := \mathbf {M} \odot \mathcal {K} (\mathbf {Q}, \mathbf {K}), \tag {2}
+$$
+
+where  $\mathbf{M} \in \mathbb{R}^{N \times N}$  is a (hard or soft) masking matrix and  $\odot$  denotes the Hadamard (elementwise) product. For softmax, this is equivalent to adding a bias term  $b_{ij} \coloneqq \log \mathbf{M}_{ij}$  to  $\pmb{q}_i^\top \pmb{k}_j$  before exponentiating. As remarked in Sec. 1, this can be used to enforce causality or as a relative position encoding (RPE) (Shaw et al., 2018). Taking  $\mathbf{M}(\mathcal{G})$  so that attention is modulated by a function of the underlying graph structure is called topological masking. Topological masking is not just a mechanism to sparsify attention for efficiency gains; its goal is to incorporate topological signal from the underlying graph, improving predictive performance via a useful inductive bias.
+
+Improving efficiency with low-rank attention. The time-complexity of Eq. 1 is  $\mathcal{O}(N^2)$ , which famously limits the ability of regular transformers to scale to long inputs. A leading approach to mitigate this and recover  $\mathcal{O}(N)$  complexity is linear or low-rank attention (Katharopoulos et al., 2020; Choromanski et al., 2020). This takes a feature mapping  $\phi : \mathbb{R}^d \to \mathbb{R}^m$  with  $m \ll N$  and constructs the matrices  $\Phi_{\mathbf{Q}} \coloneqq [\phi(\mathbf{q}_i)]_{i=1}^N$  and  $\Phi_{\mathbf{K}} \coloneqq [\phi(\mathbf{k}_i)]_{i=1}^N$ , where  $\Phi_{\mathbf{Q},\mathbf{K}} \in \mathbb{R}^{N \times m}$ . One then defines  $\mathcal{K}_{\mathrm{LR}}(\mathbf{Q},\mathbf{K}) \coloneqq \Phi_{\mathbf{Q}}\Phi_{\mathbf{K}}^\top$  and computes
+
+$$
+\operatorname {A t t} _ {\mathrm {L R}} (\mathbf {Q}, \mathbf {K}, \mathbf {V}) := \mathbf {D} ^ {- 1} \left(\Phi_ {\mathbf {Q}} \left(\Phi_ {\mathbf {K}} ^ {\top} \mathbf {V}\right)\right), \quad \mathbf {D} := \Phi_ {\mathbf {Q}} \left(\Phi_ {\mathbf {K}} ^ {\top} \mathbf {1} _ {N}\right), \tag {3}
+$$
+
+where the parentheses show the order of computation. Exploiting associativity, the time complexity is reduced to  $\mathcal{O}(Nmd)$ , albeit usually with some sacrifice in performance compared to full-rank softmax attention. Since we care about scaling with graph size, we will henceforth write this as  $\mathcal{O}(N)$ . Different choices of feature maps  $\phi$  exist; for example, one can take a random feature map that provides a Monte Carlo estimate of the softmax kernel (Performer; Choromanski et al., 2020), or a deterministic nonlinearity like  $\mathrm{elu}(x) + x$  (linear transformer; Katharopoulos et al., 2020).
+
+Efficient masking of low-rank attention. As Eq. 3 suggests, low-rank attention derives its speed from avoiding computing  $\mathbf{A} = \mathcal{K}_{\mathrm{LR}}(\mathbf{Q},\mathbf{K})$  explicitly, preferring to evaluate faster products like  $\Phi_{\mathbf{K}}^{\top}\mathbf{V}$  and  $\Phi_{\mathbf{K}}^{\top}\mathbf{1}_N$  first and use them to weight the query features  $\Phi_{\mathbf{Q}}$ . This makes the direct Hadamard product implementation of masking in Eq. 2 impossible: if the matrix  $\mathbf{A}$  is never explicitly materialised in memory, one cannot pointwise multiply each of its elements. Algorithms have been proposed to mask implicitly (without instantiating  $\mathbf{A}$  or  $\mathbf{M}$ ) in very simple cases, e.g. causal masking (Choromanski et al., 2020) or one-dimensional RPE (Liutkus et al., 2021; Luo et al., 2021). Subquadratic implicit masking is also possible when  $\mathbf{M}$  is very structured: e.g. functions of shortest path distance on grid graphs and trees (Choromanski et al., 2022), or of the degrees of the pair of input nodes (Chen et al., 2024). See App. B for a detailed review. However, an  $\mathcal{O}(N)$  algorithm capable of masking with flexible functions on general graphs has, until now, remained out of reach.
+
+Remainder of manuscript. In Sec. 3.1 we parameterise topological masks  $\mathbf{M}(\mathcal{G})$  as learnable functions of weighted adjacency matrices, which incorporates strong structural inductive bias into attention and boosts performance. Sec. 3.2 shows how this can be implemented implicitly, without instantiating  $\mathbf{A}$  or  $\mathbf{M}$  in memory. Sec. 3.3 offers a faster stochastic alternative using GRFs, which we prove is  $\mathcal{O}(N)$  time and space complexity. Sec. 4 gives experimental evaluations across data modalities, including images and point clouds. The Appendices present proofs and extra details too long for the main text, and extended discussion about related work.
+
+# 3 TOWARDS LINEAR TOPOLOGICAL MASKING
+
+This section details our main results. Consider an undirected graph  $\mathcal{G}(\mathcal{N},\mathcal{E})$  where  $\mathcal{N}\coloneqq \{v_1,\ldots ,v_N\}$  is the set of nodes and  $\mathcal{E}$  is the set of edges, with  $(v_{i},v_{j})\in \mathcal{E}$  if and only if there exists an edge between  $v_{i}$  and  $v_{j}$  in  $\mathcal{G}$ . The size of the graph  $N$  corresponds to the number of tokens. Denote by  $\mathbf{M}(\mathcal{G})\in \mathbb{R}^{N\times N}$  the topological masking matrix.
+
+# 3.1 GRAPH NODE KERNELS AS TOPOLOGICAL MASKS
+
+As remarked above, a common choice for  $\mathbf{M}(\mathcal{G})_{ij}$  is a function of the shortest path distance between nodes  $v_{i}$  and  $v_{j}$  (Ying et al., 2021). Though effective at improving performance, this requires computation for every pair of nodes so it cannot straightforwardly be applied to low-rank attention in a scalable way – except for special structured  $\mathcal{G}$  where the mask becomes simple (Choromanski et al., 2022). Shortest path distances are also very sensitive to changes like the addition or removal of edges, and need access to the whole graph upfront.
+
+A new topological mask: graph node kernels. Given that we intend  $\mathbf{M}(\mathcal{G})_{ij}$  to quantify some sense of topological similarity between  $v_{i}$  and  $v_{j}$ , a more principled choice may be graph node
+
+kernels: positive definite, symmetric functions  $k:\mathcal{N}\times \mathcal{N}\to \mathbb{R}$  mapping from pairs of graph nodes to real numbers. These are widely used in bioinformatics (Borgwardt et al., 2005), community detection (Kloster and Gleich, 2014) and recommender systems (Yajima, 2006), with more recent applications in manifold learning for deep generative modelling (Zhou et al., 2020) and solving single- and multiple-source shortest path problems (Crane et al., 2017). They are also used to model the covariance function in geometric Gaussian processes (Zhi et al., 2023; Borovitskiy et al., 2021).
+
+Graph node kernels are often parameterised as follows. Let  $\mathbf{W} \in \mathbb{R}^{N \times N}$  be a weighted adjacency matrix, so that  $\mathbf{W} \coloneqq [w_{ij}]_{i,j \in \mathcal{N}}$  with  $w_{ij}$  nonzero if and only if  $(v_i, v_j) \in \mathcal{E}$ . A common choice is  $w_{ij} = 1 / \sqrt{d_i d_j}$ , with  $d_i$  the degree of  $v_i$ . Then consider the power series
+
+$$
+\mathbf {M} _ {\alpha} (\mathcal {G}) := \sum_ {k = 0} ^ {\infty} \alpha_ {k} \mathbf {W} ^ {k} \tag {4}
+$$
+
+where  $(\alpha_{k})_{k = 0}^{\infty}\subset \mathbb{R}$  is a set of real-valued Taylor coefficients. This is positive definite if and only if  $\sum_{k = 0}^{\infty}\alpha_k\lambda_i^k >0\forall \lambda_i\in \Lambda (\mathbf{W})$  , where  $\Lambda (\mathbf{W})$  denotes the set of eigenvalues of  $\mathbf{W}$  . With different choices for  $(\alpha_{k})_{k = 0}^{\infty}$  , Eq. 4 includes many of the most popular graph node kernels in the literature, e.g. heat, diffusion, cosine and  $p$  -step random walk. Kernel learning can be implemented by optimising  $\alpha_{k}$  (Reid et al., 2024b; Zhi et al., 2023).
+
+Eq. 4 provides a very effective parameterisation for topological attention masks, striking the right balance between flexibility and hard-coded structural inductive bias. In Sec. 4 we will show that it boosts performance compared to unmasked attention. However, explicitly computing  $\mathbf{M}_{\alpha}(\mathcal{G}) \in \mathbb{R}^{N \times N}$  and storing it in memory is incompatible with  $\mathcal{O}(N)$  low-rank attention. The next section will show how this can be avoided by masking implicitly.
+
+# 3.2 IMPLICIT MASKING WITH GRAPH FEATURES
+
+Linear attention is fast because it replaces softmax attention  $\exp (\mathbf{q}_i^\top \mathbf{k}_j)$  with a 'featurised' alternative  $\phi (\mathbf{q}_i)^\top \phi (\mathbf{k}_j)$ . Can we do the same for  $\mathbf{M}_{\alpha}(\mathcal{G})$ ? The following is true.
+
+Lemma 3.1 (Explicit  $N$ -dimensional features for graph node kernels (Reid et al., 2024b)). Let  $(f_k)_{k=0}^{\infty} \subset \mathbb{R}$  denote the deconvolution of  $(\alpha_k)_{k=0}^{\infty}$ , i.e.  $\sum_{p=0}^{k} f_p f_{k-p} = \alpha_k \forall k$ . There exists a feature matrix  $\Phi_{\mathcal{G}} \in \mathbb{R}^{N \times N}$  satisfying  $\mathbf{M}_{\alpha}(\mathcal{G}) = \Phi_{\mathcal{G}} \Phi_{\mathcal{G}}^{\top}$ , given by  $\Phi_{\mathcal{G}} := \sum_{k=0}^{\infty} f_k \mathbf{W}^k$ .
+
+This is can be shown in a few lines of algebra; see App. A.1. In general, one requires some extra technical assumptions to ensure that  $\mathbf{M}_{\alpha}(\mathcal{G})$  and  $\Phi_{\mathcal{G}}$  converge, which we also describe in App. A.1. We refer to the rows of the feature matrix  $\Phi_{\mathcal{G}}$  as graph features, letting  $\Phi_{\mathcal{G}} =: [\phi_{\mathcal{G}}(v_i)]_{i=1}^{N}$  with  $\phi_{\mathcal{G}}(v_i) \in \mathbb{R}^N$  the graph feature of node  $v_i$ . For every kernel there trivially exists a feature map (Gretton, 2013), but unusually in this case it is finite-dimensional and admits a simple closed form.
+
+Let  $\otimes : \mathbb{R}^m \times \mathbb{R}^N \to \mathbb{R}^{m \times N}$  denote the outer product which maps a pair of vectors to a matrix, and  $\operatorname{vec} : \mathbb{R}^{m \times N} \to \mathbb{R}^{mN}$  denote the vectorising operation that flattens a matrix into a vector. Then  $\phi(\mathbf{q}_i) \otimes \phi_{\mathcal{G}}(v_i)$  is an  $m \times N$  matrix whose  $jk$ th element is  $\phi(\mathbf{q}_i)_j \phi_{\mathcal{G}}(v_i)_k$ , and  $\operatorname{vec}(\phi(\mathbf{q}_i) \otimes \phi_{\mathcal{G}}(v_i))$  is an  $mN$ -dimensional vector whose  $(N(j - 1) + k)$ th entry is  $\phi(\mathbf{q}_i)_j \phi_{\mathcal{G}}(v_i)_k$ . Observe the following:
+
+$$
+\begin{array}{l} \operatorname {vec}\left(\phi (\mathbf{q}_{i})\otimes \phi_{\mathcal{G}}(v_{i})\right)^{\top}\operatorname {vec}\left(\phi (\mathbf{k}_{j})\otimes \phi_{\mathcal{G}}(v_{j})\right) = \sum_{s = 1}^{mN}\operatorname {vec}\left(\phi (\mathbf{q}_{i})\otimes \phi_{\mathcal{G}}(v_{i})\right)_{s}\operatorname {vec}\left(\phi (\mathbf{k}_{j})\otimes \phi_{\mathcal{G}}(v_{j})\right)_{s} \\ = \sum_ {k = 1} ^ {m} \sum_ {l = 1} ^ {N} \phi (\mathbf {q} _ {i}) _ {k} \phi_ {\mathcal {G}} (v _ {i}) _ {l} \phi (\mathbf {k} _ {j}) _ {k} \phi_ {\mathcal {G}} (v _ {j}) _ {l} = \left(\phi (\mathbf {q} _ {i}) ^ {\top} \phi (\mathbf {k} _ {j})\right) \left(\phi_ {\mathcal {G}} (v _ {i}) ^ {\top} \phi_ {\mathcal {G}} (v _ {j})\right) = \mathcal {K} _ {\mathrm {L R} i j} \mathbf {M} _ {\alpha i j}. \tag {5} \\ \end{array}
+$$
+
+Hence, taking  $\{\mathrm{vec}(\phi (\{\mathbf{q}_i,\mathbf{k}_i\})\otimes \phi_{\mathcal{G}}(v_i))\}_{v_i\in \mathcal{N}}\subset \mathbb{R}^{mN}$  as features, we can compute masked attention implicitly, without materialising  $\mathbf{A}$  and  $\mathbf{M}_{\alpha}$ . Concretely, we have that:
+
+$$
+\Phi_ {\mathbf {Q}, \mathcal {G}} \Phi_ {\mathbf {K}, \mathcal {G}} ^ {\top} = \mathcal {K} _ {\mathrm {L R}} \odot \mathbf {M} _ {\alpha} \quad \text {i f} \quad \Phi_ {\{\mathbf {Q}, \mathbf {K} \}, \mathcal {G}} := \left[ \operatorname {v e c} \left(\phi \left(\left\{\mathbf {q} _ {i}, \mathbf {k} _ {i} \right\}\right) \otimes \phi_ {\mathcal {G}} (v _ {i})\right) \right] _ {i = 1} ^ {N} \in \mathbb {R} ^ {N \times N m}. \tag {6}
+$$
+
+Masked low-rank attention can then be computed using  $\Phi_{\{\mathbf{Q},\mathbf{K}\},\mathcal{G}}$  as the feature matrices:
+
+$$
+\operatorname {A t t} _ {\mathrm {L R}, \mathbf {M}} (\mathbf {Q}, \mathbf {K}, \mathbf {V}, \mathcal {G}) := \mathbf {D} ^ {- 1} \left(\Phi_ {\mathbf {Q}, \mathcal {G}} \left(\Phi_ {\mathbf {K}, \mathcal {G}} ^ {\top} \mathbf {V}\right)\right), \quad \mathbf {D} := \Phi_ {\mathbf {Q}, \mathcal {G}} \left(\Phi_ {\mathbf {K}, \mathcal {G}} ^ {\top} \mathbf {1} _ {N}\right). \tag {7}
+$$
+
+Time complexity. Eqs 6 and 7 give a recipe for computing masked attention implicitly in feature space, without needing to materialise  $\mathbf{A}$  or  $\mathbf{M}$  in memory. However, this approach relies on computing the feature matrix  $\Phi_{\{\mathbf{Q},\mathbf{K}\},\mathcal{G}} \in \mathbb{R}^{N \times Nm}$  and evaluating products like  $\Phi_{\mathbf{K},\mathcal{G}}^{\top}\mathbf{1}_N$ , which still incurs  $\mathcal{O}(N^2)$  space and time complexity. For a linear masking algorithm, we must reduce this to  $\mathcal{O}(N)$ . This can be achieved by approximating the features.
+
+# 3.3 FASTER IMPLICIT MASKING WITH GRAPH RANDOM FEATURES
+
+Our goal is now to replace  $\Phi_{\{\mathbf{Q},\mathbf{K}\} ,\mathcal{G}}$  by Monte Carlo estimates  $\widehat{\Phi}_{\{\mathbf{Q},\mathbf{K}\} ,\mathcal{G}}$  that satisfy
+
+$$
+\mathbb {E} \left(\widehat {\boldsymbol {\Phi}} _ {\mathbf {Q}, \mathcal {G}} \widehat {\boldsymbol {\Phi}} _ {\mathbf {K}, \mathcal {G}} ^ {\top}\right) = \boldsymbol {\Phi} _ {\mathbf {Q}, \mathcal {G}} \boldsymbol {\Phi} _ {\mathbf {K}, \mathcal {G}} ^ {\top} = \mathcal {K} _ {\mathrm {L R}} \odot \mathbf {M} _ {\alpha}, \quad \widehat {\boldsymbol {\Phi}} _ {\mathbf {K}, \mathcal {G}} ^ {\top} \boldsymbol {v} \sim \mathcal {O} (N) \forall \boldsymbol {v} \in \mathbb {R} ^ {N}, \tag {8}
+$$
+
+meaning we apply the desired topological mask in expectation but in linear time. This can be achieved using graph random features (GRFs) (Choromanski, 2023; Reid et al., 2024b).
+
+At a high-level, GRFs can be understood as follows. From Lemma 3.1, the graph feature for node  $v_{i}$  is given by  $\phi_{\mathcal{G}}(v_{i}) \coloneqq [\sum_{k=0}^{\infty} f_{k} \mathbf{W}_{ij}^{k}]_{j=1}^{N}$ . Since  $\mathbf{W}$  is an adjacency matrix,  $\bar{\mathbf{W}}_{ij}^{k}$  counts the number of graph walks of length  $k$  between nodes  $i$  and  $j$ , weighted by the product of traversed edge weights. So  $\phi_{\mathcal{G}}(v_{i})_{j}$  can be interpreted as a weighted sum over all walks of any length between nodes  $v_{i}$  and  $v_{j}$ , with longer contributions suppressed since edge weights are at most 1 (by normalisation of  $\mathbf{W}$ ). GRFs apply importance sampling, approximating this infinite sum by a Monte Carlo estimate  $\hat{\phi}_{\mathcal{G}}(v_{i})$  using random walks.
+
+Using GRFs. Concretely, to build the GRF  $\widehat{\phi}_{\mathcal{G}}(v_i)$  one samples  $n$  random walks  $\{\omega_k^{(i)}\}_{k = 1}^n\subset \Omega$  out of node  $v_{i}$ , where  $\Omega \coloneqq \{(v_{i})_{i = 1}^{l}|v_{i}\in \mathcal{N},(v_{i},v_{i + 1})\in \mathcal{E},l\in \mathbb{N}\}$  is the set of all walks. Denote by  $p(\omega)$  the probability of walk  $\omega$  which is known from the sampling mechanism.  $p(\omega)$  is typically chosen so that walks randomly halt, prioritising sampling lower powers of  $\mathbf{W}$  that contribute more to the kernel. Let  $\widetilde{\omega}:\Omega \to \mathbb{R}$  be a function computing the product of traversed edge weights,
+
+$$
+\widetilde {\omega} (\omega) := \left\{\prod_ {i = 1} ^ {\text {l e n} (\omega)} \mathbf {W} _ {\omega [ i ] \omega [ i + 1 ]} \text {i f l e n} (\omega) > 1, \quad 1 \text {o t h e r w i s e} \right\}, \tag {9}
+$$
+
+where  $\mathrm{len}(\omega)$  is the number of hops and  $\omega [i]$  is the node visited at the  $i$ th timestep. Then:
+
+$$
+\widehat {\phi} _ {\mathcal {G}} \left(v _ {i}\right) _ {q} := \frac {1}{n} \sum_ {k = 1} ^ {n} \sum_ {\omega_ {i q} \in \Omega_ {i q}} \frac {\widetilde {\omega} \left(\omega_ {i q}\right) f _ {\text {l e n} \left(\omega_ {i q}\right)}}{p \left(\omega_ {i q}\right)} \mathbb {I} \left(\omega_ {i q} \text {p r e f i x s u b w a l k} \omega_ {k} ^ {(i)}\right), \quad q = 1, \dots , N, \tag {10}
+$$
+
+where  $\Omega_{iq}$  denotes the set of all walks between nodes  $v_{i}$  and  $v_{q}$ , of which  $\omega_{iq}$  is a member. 'Prefix subwalk' means that the walk  $\omega_{k}^{(i)}$  beginning at node  $v_{i}$  initially follows the sequence of nodes  $\omega_{iq}$ , then optionally continues. This complicated notation belies a very simple algorithm: one simulates  $n$  random walks out of node  $v_{i}$ . At every visited node, one adds a contribution to the corresponding GRF coordinate that depends on 1) the product of traversed edge weights, 2) the probability of the subwalk and 3) the function  $f$ . The result is normalised by dividing by the number of walks  $n$ .
+
+GRFs give unbiased approximation of graph node kernels:  $\mathbb{E}[\widehat{\phi}_{\mathcal{G}}(v_i)^\top \widehat{\phi}_{\mathcal{G}}(v_j)] = \mathbf{M}_{\alpha ij}$ . This follows because  $\mathbb{E}[\mathbb{I}(\omega_{iq}$  prefix subwalk of  $\omega_k^{(i)})] = p(\omega_{iq})$ . Unbiasedness is one of our required properties; we also need  $\widehat{\Phi}_{\{\mathbf{Q},\mathbf{K}\},\mathcal{G}}$  to support  $\mathcal{O}(N)$  matrix-vector multiplication. We will show this by providing the first known proof that GRFs are sparse:  $\widehat{\phi} (v_i)$  has  $\mathcal{O}(1)$  nonzero entries  $\forall v_{i}\in \mathcal{N}$ .
+
+# 3.4 NOVEL THEORETICAL RESULTS FOR GRFS: SHARP ESTIMATORS WITH SPARSE FEATURES
+
+It is intuitive that GRFs are sparse. The walk sampler  $p(\omega)$  is chosen so that walks are short, e.g. taking simple random walks with geometrically distributed lengths. From Eq. 10, this means that  $\widehat{\phi}_{\mathcal{G}}(v_i)$  is only nonzero at nodes 'close' to  $v_i$ . Most of its entries are 0. However, to prove that  $\widehat{\Phi}_{\{\mathbf{Q},\mathbf{K}\} \mathcal{G}}$  supports  $\mathcal{O}(N)$  matrix-vector multiplication we need to rigorously relate this to the quality of approximation of  $\mathbf{M}_{\alpha}(\mathcal{G})$ . This requires concentration inequalities for GRFs, but no such bounds were previously known. The following result is novel.
+
+Theorem 3.2 (GRF exponential concentration bounds). Consider a graph  $\mathcal{G}$  with adjacency matrix  $\mathbf{W}$  and node degrees  $\{d_i\}_{v_i\in \mathcal{N}}$ . Suppose we construct GRFs  $\{\widehat{\phi}_{\mathcal{G}}(v_i)\}_{v_i\in \mathcal{G}}$  by sampling  $n$  random
+
+walks  $\{\omega_k^{(i)}\}_{v_i\in \mathcal{N}, k\in [1,n]}$  with geometrically distributed lengths, terminating with probability  $p_{halt}$  at each timestep. Let  $c\coloneqq \sum_{k = 0}^{\infty}|f_k|(\max_{v_i,v_j\in \mathcal{N}}\frac{|\mathbf{W}_{ij}|d_i}{1 - p_{halt}})^k$ . Suppose  $c$  is finite, which is guaranteed e.g. for bounded  $f$  and suitably normalised  $\mathbf{W}$ , or if there exists some  $i_{max}\in \mathbb{N}$  such that  $f_{i}$  vanishes for all  $i > i_{max}$  (see App. A.2). Then:
+
+$$
+\Pr \left(\left| \widehat {\phi} _ {\mathcal {G}} \left(v _ {i}\right) ^ {\top} \widehat {\phi} _ {\mathcal {G}} \left(v _ {j}\right) - \mathbf {M} _ {\alpha i j} \right| > t\right) \leq 2 \exp \left(- \frac {t ^ {2} n ^ {3}}{2 (2 n - 1) ^ {2} c ^ {4}}\right). \tag {11}
+$$
+
+Proof sketch. Full details are in App. A.2; here, we provide an overview. Treat each pair of random walks  $X_{k} \coloneqq (\omega_{k}^{(i)}, \omega_{k}^{(j)}) \in \Omega \times \Omega$ , simulated out of nodes  $v_{i}$  and  $v_{j}$  respectively, as a random variable. Let  $k \in [1, n]$  so we sample  $n$  such pairs of walks in total. Consider modifying one of the pairs of walks. If  $c$  is finite, the maximum load it is capable of depositing is finite so we can bound the  $L_{1}$ -norm of the contribution made to  $\widehat{\phi}_{\mathcal{G}}(v_i)$  and  $\widehat{\phi}_{\mathcal{G}}(v_j)$ . This allows us to bound the change in the estimator  $\widehat{\phi}_{\mathcal{G}}(v_i)^{\top} \widehat{\phi}_{\mathcal{G}}(v_j)$  if  $X_{k}$  is modified. Applying McDiarmid's inequality, the result follows.
+
+Thm 3.2 provides an exponential bound on the probability of a topological mask estimate deviating from its true value. Moreover, assuming that  $c$  remains constant (i.e. we fix a maximum edge weight and node degree as the graph grows), this probability is independent of the number of nodes  $N$ . Therefore, if we fix  $t$  and the probability  $\operatorname{Pr}$ , we can solve for the minimum number of walkers  $n$  to guarantee a good mask estimate with high probability, independent of graph size. Now observe the following.
+
+Lemma 3.3 (GRF sparsity). Consider a graph random feature  $\widehat{\phi}_{\mathcal{G}}(v_i)$  constructed using  $m$  random walkers each terminating with probability  $p_{halt}$  at every timestep. With probability at least  $1 - \delta$ ,  $\widehat{\phi}_{\mathcal{G}}(v_i)$  will have  $n \log (1 - (1 - \delta)^{1 / n}) \log (1 - p_{halt})^{-1}$  or fewer nonzero entries.
+
+Proof. Since the walk lengths are geometrically distributed, the probability that a single walk is of length  $b$  or shorter is  $1 - (1 - p_{\mathrm{halt}})^b$ .  $n$  independent walkers are all  $b$  or shorter with probability  $\left(1 - (1 - p_{\mathrm{halt}})^b\right)^n$ . Let this be equal to  $1 - \delta$  and solve for  $b$ . At most  $bn$  entries of  $\widehat{\phi}_{\mathcal{G}}(v_i)$  can then be nonzero, so the sparsity guarantee follows.
+
+Lemma 3.3 means that, if we fix the number of walkers  $n$  (e.g. using the bound in Eq. 11), we can also bound GRF sparsity with high probability. Crucially: the number of nonzero GRF entries is independent of the graph size  $N$ . We conclude the following.
+
+Corollary 3.4 (GRFs implement  $\mathcal{O}(N)$  topological masking). Suppose we construct a set of graph random features  $\{\widehat{\phi}_{\mathcal{G}}(v_i)\}_{v_i\in \mathcal{N}}$ , choosing the number of walkers  $n$  as described above with  $c$ ,  $p_{halt}$ ,  $\delta$ ,  $t$  and the deviation probability  $\operatorname*{Pr}\left(|\phi_{\mathrm{GRF}}(v_i)^\top \phi_{\mathrm{GRF}}(v_j) - \mathbf{M}_{\alpha ij}| > t\right)$  fixed as hyperparameters. Then the number of nonzero entries in  $\widehat{\Phi}_{\{\mathbf{Q},\mathbf{K}\}}, \mathcal{G} \in \mathbb{R}^{N\times Nm}$  is linear in  $N$ , which means that topological masking is implemented with  $\mathcal{O}(N)$  time complexity.
+
+Proof. Lemma 3.3 shows that the number of nonzero entries in  $\widehat{\phi}_{\mathcal{G}}(v_i)$  is independent of the number of nodes  $N$ . This must also be true of  $\operatorname{vec}(\phi(\mathbf{k}_i) \otimes \widehat{\phi}_{\mathcal{G}}(v_i))$ , so the number of nonzero entries in  $\widehat{\Phi}_{\mathbf{K},\mathcal{G}} := [\operatorname{vec}(\phi(\mathbf{k}_i) \otimes \widehat{\phi}_{\mathcal{G}}(v_i))]_{i=1}^N$  is proportional to  $N$ . The same holds for the equivalent matrix for the queries,  $\widehat{\Phi}_{\mathbf{Q},\mathcal{G}}$ . The time complexity of the sparse matrix-matrix multiplication  $\widehat{\Phi}_{\mathbf{K},\mathcal{G}}^\top \mathbf{V}$  is  $\mathcal{O}(Nmd)$  and  $\widehat{\Phi}_{\mathbf{K},\mathcal{G}}^\top \mathbf{1}_N$  is  $\mathcal{O}(Nm)$ . Both these time complexities are linear in  $N$ . This is also the case for the subsequent computations involving  $\widehat{\Phi}_{\mathbf{Q},\mathcal{G}}$ .
+
+Further comments on theoretical results. We have provided the first concentration bounds for GRFs and quantification of the tradeoff between sparsity and kernel approximation quality. These results may be of interest independent from topological masking, e.g. for applications in scalable geometric Gaussian processes (Reid et al., 2024c; Borovitskiy et al., 2021). Thm. 3.2 is remarkably general, requiring only the modest assumption about  $\mathcal{G}$  and  $f$  that  $c$  is finite. To guarantee  $\mathcal{O}(N)$  scaling we assume that  $c$  is fixed as  $\mathcal{G}$  changes, which is guaranteed by e.g. fixed maximum node degree and edge weight.
+
+# 3.5 ALGORITHM AND INTUITIVE PICTURE
+
+Alg. 1 presents our method. The results of Secs 3.1-3.4 can be intuitively understood as follows (see Fig. 2 for an overview). Graph node kernels, which compute an infinite weighted sum of walks
+
+![](images/880265d29c05b7bbc1d6c276b9bd8216072c6ede4fa7aa97507528615635682b.jpg)  
+Figure 2: Visual overview. A graph  $\mathcal{G}$  (left) has a weighted adjacency matrix  $\mathbf{W}$  (centre left). A learnable power series  $\mathbf{M}_{\alpha}(\mathcal{G}) \coloneqq \sum_{i=0}^{\infty} \alpha_i \mathbf{W}^i$  is an effective topological mask or graph RPE (centre right).  $\mathbf{M}_{\alpha}(\mathcal{G})$  can be efficiently approximated using graph random features (centre top), which perform importance sampling of halting random walks. The feature  $\widehat{\phi}_{\mathcal{G}}(v_i)$  is only nonzero at entries visited by the ensemble of walks beginning at  $v_i$ . In Thm. 3.2, we prove that the number of such entries is  $\mathcal{O}(1)$  whilst still accurately estimating  $\mathbf{M}_{\alpha}(\mathbf{W})_{ij}$  with high probability, so GRFs are sparse. This unlocks  $\mathcal{O}(N)$  topological masking. Note that  $\phi_{\mathcal{G}}(v_i)^{\top} \widehat{\phi}_{\mathcal{G}}(v_j)$  is only nonzero if the features are nonzero at some of the same coordinates, which happens if their respective ensembles of walks 'hit'. This incorporates a strong structural inductive bias. The equations on the right formalise our method mathematically.
+
+between pairs of input nodes, provide an effective RPE mechanism for transformers. They capture the topological relationship between queries and keys. Using graph features, we can implement this implicitly, without needing to actually materialise the mask matrix. Replacing these dense features by graph random features, a sparse approximation, we can reduce the cost of implicit masking to  $\mathcal{O}(N)$  with low performance loss. Eq. 11 gives a remarkably tight bound on the quality of mask estimation. Physically, masking with GRFs means that nodes  $v_{i}$  and  $v_{j}$  only attend to one another if members of their respective ensembles of random walks hit, such that they visit some of the same nodes. This incorporates a strong structural inductive bias from  $\mathcal{G}$ . It upweights crucial interactions between nearby nodes, whilst also sampling longer-range attention with lower probability.
+
+# Algorithm 1  $\mathcal{O}(N)$  topologically-masked attention for general graphs
+
+Input: query matrix  $\mathbf{Q} \in \mathbb{R}^{N \times d}$ , key matrix  $\mathbf{K} \in \mathbb{R}^{N \times d}$ , value matrix  $\mathbf{V} \in \mathbb{R}^{N \times d}$ , graph  $\mathcal{G}$  with weighted adjacency matrix  $\mathbf{W} \in \mathbb{R}^{N \times N}$ , learnable mask parameters  $(f_i)_{i=0}^{\lambda_{\max}}$ , number of random walks to sample  $n \in \mathbb{N}$ , query/key feature map  $\phi(\cdot): \mathbb{R}^d \to \mathbb{R}^m$ .
+
+Output: masked attention  $\mathrm{Att}_{\mathrm{LR},\hat{\mathbf{M}}}(\mathbf{Q},\mathbf{K},\mathbf{V},\mathcal{G})$  in  $\mathcal{O}(N)$  time.
+
+1: Simulate  $n$  terminating random walks  $\{\omega_k^{(i)}\}_{k = 1}^n$  out of every node  $v_{i}\in \mathcal{N}$  
+2: Compute sparse graph random features  $\{\widehat{\phi}_{\mathcal{G}}(v_i)\}_{v_i\in \mathcal{N}}\subset \mathbb{R}^N$  using the walks  $\{\omega_k^{(i)}\}_{k = 1}^n$  and learnable mask parameters  $(f_{i})_{i = 0}^{i_{\mathrm{max}}}$  
+3: Compute the query and key features  $\{\phi (\mathbf{q}_i)\}_{i = 1}^N,\{\phi (\mathbf{k}_i)\}_{i = 1}^N$ , with  $\phi$  the chosen linear-attention nonlinearity (e.g.  $\mathrm{elu}(x) + x$  (Katharopoulos et al., 2020) or a Monte Carlo estimate of softmax (Choromanski et al., 2020))  
+4: Combine the query/key features and graph features into a sparse topology-enhanced feature matrix,  $\widehat{\Phi}_{\{\mathbf{Q},\mathbf{K}\},\mathcal{G}}\coloneqq [\mathrm{vec}(\phi (\{\mathbf{q}_i,\mathbf{k}_i\})\otimes \widehat{\phi_{\mathcal{G}}} (v_i))]_{i = 1}^N\in \mathbb{R}^{N\times Nm}$  
+5: Compute low-rank attention with topological masking in  $\mathcal{O}(N)$  time by
+
+$$
+\operatorname {A t t} _ {\mathrm {L R}, \widehat {\mathbf {M}}} (\mathbf {Q}, \mathbf {K}, \mathbf {V}, \mathcal {G}) := \mathbf {D} ^ {- 1} \left(\widehat {\boldsymbol {\Phi}} _ {\mathbf {Q}, \mathcal {G}} \left(\widehat {\boldsymbol {\Phi}} _ {\mathbf {K}, \mathcal {G}} ^ {\top} \mathbf {V}\right)\right), \quad \mathbf {D} := \widehat {\boldsymbol {\Phi}} _ {\mathbf {Q}, \mathcal {G}} \left(\widehat {\boldsymbol {\Phi}} _ {\mathbf {K}, \mathcal {G}} ^ {\top} \mathbf {1} _ {N}\right). \tag {12}
+$$
+
+Distributed computation, expressivity and GATs. Alg. 1 is simple to distribute for massive graphs that cannot be stored on a single machine; to simulate random walks one only needs a node's immediate neighbours rather than all of  $\mathcal{G}$  upfront. This desirable property is not shared by masking algorithms that rely on the fast Fourier transform (Luo et al., 2021; Choromanski et al., 2022). Moreover, our method can distinguish graphs identical under the 1-dimensional Weisfeiler Lehman graph isomorphism heuristic (with colours replaced by node features) because their adjacency matrices and hence masks differ. In this sense, GRF transformers are more expressive than standard graph
+
+neural networks and unmasked transformers, which notoriously fail this test (Morris et al., 2019; Xu et al., 2019). Finally, we remark that our algorithm can be interpreted as a stochastic extension of graph attention networks (GATs; Velicković et al., 2018), with a greater focus on linear attention and scalability. Tokens still attend to a subset of other tokens informed by the topology of  $\mathcal{G}$ , but importance sampling random walks permits interactions beyond nearest-neighbour. Incorporating learnable parameters  $(f_{i})_{i = 0}^{i_{\mathrm{max}}}$  lets us estimate a principled function of  $\mathbf{W}$ .
+
+# 4 EXPERIMENTS
+
+In this section, we test our algorithms for topological masking with GRFs. We consider data modalities with different graph topologies: images and point clouds.
+
+# 4.1 TIME COMPLEXITY
+
+For a hardware-agnostic comparison, we first compute the total number of FLOPs for evaluating (i) unmasked softmax, (ii) unmasked linear and (iii) GRF-masked linear attention for graphs of different sizes  $N$ . Fig. 3 shows the results. For concreteness we use 1-dimensional grid graphs, though the theoretical results in Sec. 3.4 are independent of  $\mathcal{G}$  (provided  $c$  is upper bounded). We take  $n = 4$  random walkers per node with termination probability  $p_{\mathrm{halt}} = 0.5$ . The latent dimension is  $d = 8$  and the feature size is  $m = 8$ . In the masked case we show the mean number of FLOPs over 10 seeds since GRF sparsity is itself a random variable. As anticipated, the time complexity of our mechanism is  $\mathcal{O}(N)$ ; there is a constant multiplicative cost for topological masking.
+
+We clearly improve upon the time complexity of  $\mathcal{O}(N^2)$  softmax attention.
+
+![](images/4e27d42f8601f842d6e770f9ca47d6c6b6c30e01f9d92f75b2ed3efff3321edb.jpg)  
+Figure 3: Number of FLOPs vs. number of graph nodes for softmax attention, linear attention, and linear attention with GRF topological masking (ours).
+
+# 4.2 VITs: GRID GRAPHS AND IMAGE DATA
+
+Next, we use our topological masking algorithm to improve the vision transformer (ViT; Dosovitskiy et al., 2020). The underlying graph  $\mathcal{G}$  is a square grid of size num_x_patches  $\times$  num_y_patches, with nodes connected by an edge if they are neighbours in  $x$  or  $y$ . Since the graph is fixed, random walks can be pre-computed and frozen for each attention head and layer before training and inference. Full architectural and training details are in App. C. All our ViT models also endow tokens with learned additive absolute position embeddings.
+
+Table 1 shows the final test accuracies for ImageNet (Deng et al., 2009), iNaturalist2021 (Horn et al., 2018) and Places365 (Zhou et al., 2018). For the slow (superlinear) variants, we include (i) unmasked softmax and (ii) Toeplitz-masked linear (Choromanski et al., 2022) attention. Toeplitz can only be used for this specific, structured graph topology. We also include (iii)  $\mathbf{M}_{\alpha}(\mathcal{G})$ -masked linear, which takes an explicit power series in  $\mathbf{W}$  (Eq. 4) to directly modulate the  $N\times N$  attention matrix. This is equal to our Monte Carlo GRFs algorithm in the  $n\to \infty$  limit, so it indicates the asymptotic performance of GRFs with very many random walkers. For the fast (linear) variants,
+
+<table><tr><td></td><td>Variant</td><td>Time comp.</td><td>ImageNet 1M</td><td>iNaturalist2021 2.7M</td><td>Places365 (Small) 1.8M</td></tr><tr><td>SLOW</td><td>Unmasked softmax Toeplitz-masked linear \( {\mathbf{M}}_{\alpha }\left( \mathcal{G}\right) \)-masked linear</td><td>\( \mathcal{O}\left( {N}^{2}\right) \) \( \mathcal{O}\left( {N\log N}\right) \) \( \mathcal{O}\left( {N}^{2}\right) \)</td><td>0.741 0.733 0.741</td><td>0.699 0.674 0.692</td><td>0.567 0.550 0.551</td></tr><tr><td>FAST</td><td>Unmasked linear GRF-masked linear</td><td>\( \mathcal{O}\left( N\right) \) \( \mathcal{O}\left( N\right) \)</td><td>0.693 0.730</td><td>0.667 0.689</td><td>0.543 0.548</td></tr></table>
+
+Table 1: Final transformer test accuracies across attention masking variants and datasets. Our algorithms are shown in **bold**. Topological masking with GRFs is the **best O(N) mechanism**, matching or beating more expensive alternatives.
+
+we include (iv) unmasked linear and (v) GRF-masked linear attention. Our algorithm gives strong improvements above the unmasked baseline, by  $+3.7\%$ ,  $+2.2\%$  and  $+0.5\%$  respectively. Remarkably, it is competitive with much more expensive variants: GRFs often match or beat Toeplitz. For ImageNet and iNaturalist2021, it even substantially narrows the gap with  $\mathcal{O}(N^2)$  softmax.
+
+# 4.3 ABLATION STUDIES
+
+In App. C.2, we perform ablation studies for the ViT experiments. We find that predictive performance improves with the number of walkers  $n$  because the accuracy of the Monte Carlo estimate  $\widehat{\mathbf{M}}_{\alpha}(\mathcal{G})$  increases. Moreover, importance sampling walks is crucial; if we fail to upweight improbable walks, the gains vanish. Lastly, the inductive bias from parameterising  $\mathbf{M}_{\alpha}(\mathcal{G})$  as a function of  $\mathbf{W}$  is key. Fully learnable graph features of the same dimensionality  $N$ , an impractical and expensive alternative to our method, give much smaller gains.
+
+# 4.4 PCTS: PREDICTING PARTICLE DYNAMICS FOR ROBOTICS
+
+As a final task, we use our algorithm to model high-density visual particle dynamics (HD-VPD; Whitney et al., 2023; 2024), predicting the physical evolution of scenes involving a bi-manual robot using a point cloud transformer (PCT). Images from a pair of cameras are unprojected to a set of 3D particles. Their interactions, conditioned on specific robot actions, are modelled by a dynamics network. We predict updates to the latent point cloud, render the result to an image with a conditional NeRF (Mildenhall et al., 2021), and train end-to-end with video prediction loss. Such learned dynamics models are attracting growing interest in robotics as a more flexible alternative to physical simulators, but to scale to the massive point clouds needed for detailed predictions efficiency and effective structural inductive biases are key.
+
+For the dynamics network, we use an Interlacer: a PCT which alternates unmasked linear attention and message passing-style updates between neighbouring particles (Whitney et al., 2024). This architecture is twice as fast as GNNs with the same prediction quality, and can handle  $4 \times$  bigger point clouds. To test the ability of GRFs to efficiently capture local topological structure, we replace message passing by GRF-masked linear attention, taking query/key features  $\phi(\cdot) = \mathrm{ReLU}(\cdot)$ . Implementation details for HD-VPD, including the robotic arm and camera configurations, are the same as reported by Whitney et al. (2024); see App. C.3 for discussion. To disambiguate, we will refer to our model as the GRF Interlacer and Whitney et al.'s as the message passing (MP) Interlacer.
+
+In contrast to our previous experiments, for HD-VPD the underlying graph  $\mathcal{G}$  is no longer fixed so walks cannot be pre-computed. Instead, we construct them on the fly, using an approximate  $k$ -nearest neighbours solver to get adjacency
+
+![](images/125bf7f90af2a1de1d8e4327b9ba0eb119829200da72233709620b270e59960b.jpg)  
+Figure 4: Accuracy comparison. Structural similarity index measure (SSIM) between ground truth camera frames and predictive NeRF renderings after 100k training steps, plotted against rollout timestep. Higher is better. GRFs improve the accuracy of dynamics predictions.
+
+lists of length  $k = 3$  for each node. We use repelling random walks for variance reduction (Reid et al., 2024a), sampling 3 hops. For easy integration with existing Interlacer code, we use asymmetric GRFs (App. D; Reid et al., 2024b). We consider point clouds with  $N = 32768$  particles, which is too many to explicitly instantiate  $\mathbf{A}$  or  $\mathbf{M}$  in memory.
+
+Fig. 5 shows example NeRF renderings of test set predictive dynamics for four rollouts. Since the model is deterministic, predictions blur with long time horizons. Incorporating a diffusion head (Song et al., 2020) is left to future work. Fig. 4 shows predicted video frame accuracy (image SSIM) vs. rollout timestep for each model. GRFs outperform the vanilla transformer and MP Interlacer baselines, modelling the particle dynamics more faithfully. We include examples of predictive videos in the supplementary material.
+
+In addition to modelling cloud dynamics more accurately, our method is faster. While the MP Interlacer trains at 0.925 steps/second, the GRFs Interlacer trains at 0.982 steps/second. This makes it faster by  $6\%$  – a gain our method enjoys because it avoids expensive GNN-style message passing. The unmasked baseline, which struggles to capture local structure so gives poor accuracy, trains at 1.49 steps/second.
+
+![](images/6511dc7f9b85188d0baf421cd606ba988c2444c19b277c18b55111e3e22ea04b.jpg)  
+Figure 5: Rendered rollouts. NeRF renderings of the predictive dynamics of a bimanual Kuka robot, conditioned on the initial scene and sequence of robot actions. We show four tasks: using a dustpan and brush, lifting a can, moving a green block, and dropping a can. GRF Interlacers model point cloud dynamics more accurately, so the predicted frame rendings are closer to the ground truth.
+
+# 5 CONCLUSION
+
+We have presented the first linear algorithm for transformer topological masking on general graphs, based on sampling random walks. It uses graph random features, for which we have proved the first known concentration inequalities and sparsity guarantees. Our algorithm brings strong performance gains across data modalities, including on massive point clouds with  $>30\mathrm{k}$  particles for which computational efficiency is essential. Future work might include improving hardware and libraries for high-dimensional sparse linear algebra to fully benefit from the guaranteed complexity improvements, and investigating whether dimensionality reduction techniques like the Johnson-Lindenstrauss transform (Dasgupta et al., 2010) can be applied to GRFs for further speedups.
+
+# 6 ETHICS AND REPRODUCIBILITY
+
+Ethics statement: Our work is foundational with no immediate ethical concerns apparent to us. However, increases in scalability provided by improvements to efficient transformers could exacerbate existing and incipient risks of machine learning, from bad actors or as unintended consequences.
+
+Reproducibility statement: We have made every effort to ensure the work's reproducibility. The core algorithm is presented clearly in Alg. 1. Theoretical results are proved with accompanying assumptions in Sec. 3.4 and App. A.2. Proof sketches are also included for clarity. Apart from the robotics dataset, all datasets are standard and freely available online. Exhaustive experimental details about the training and architectures are reported in App. C; see especially Table 2.
+
+# 7 CONTRIBUTIONS AND ACKNOWLEDGEMENTS
+
+Relative contributions. IR and KC conceptualised the project and co-designed the algorithms. IR proved the theoretical results, implemented the GRFs code, ran the Interlacer experiments and wrote the text. KAD was crucially involved throughout, running the ViT and ViViT experiments and guiding the project's direction. DJ and WW helped set up the Interlacer experiments. AA, JA, AB, MJ, AM, DR, CS, RT, RW and AW provided thoughtful feedback on the project and manuscript, helped set up experimental infrastructure, and made Cambridge and Google great places to work.
+
+Acknowledgements and funding. IR acknowledges support from a Trinity College External Studentship and a Google PhD Fellowship. This work was completed whilst employed as a student researcher at Google. IR sincerely thanks Silvio Lattanzi for making this possible. RET is supported by the EPSRC Probabilistic AI Hub (EP/Y028783/1). AW acknowledges support from a Turing AI fellowship under grant EP/V025279/1 and the Leverhulme Trust via CFI.
+
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+William F Whitney, Tatiana Lopez-Guevara, Tobias Pfaff, Yulia Rubanova, Thomas Kipf, Kimberly Stachenfeld, and Kelsey R Allen. Learning 3d particle-based simulators from rgb-d videos. arXiv preprint arXiv:2312.05359, 2023. URL https://doi.org/10.48550/arXiv.2312.05359.  
+William F Whitney, Jacob Varley, Deepali Jain, Krzysztof Choromanski, Sumeet Singh, and Vikas Sindhwani. Modeling the real world with high-density visual particle dynamics. arXiv preprint arXiv:2406.19800, 2024. URL https://doi.org/10.48550/arXiv.2406.19800.  
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+
+# A PROOFS
+
+# A.1 PROOF OF LEMMA 3.1
+
+This result was presented by Reid et al. (2024b); a short demonstration is included below for the reader's convenience. Observe that
+
+$$
+\begin{array}{l} \mathbf {M} _ {\alpha} (\mathcal {G}) _ {i j} := \sum_ {k = 0} ^ {\infty} \alpha_ {k} \mathbf {W} _ {i j} ^ {k} \stackrel {(1)} {=} \sum_ {v = 1} ^ {N} \sum_ {k = 0} ^ {\infty} \alpha_ {k} \mathbf {W} _ {i v} ^ {k - p} \mathbf {W} _ {v j} ^ {p} \stackrel {(2)} {=} \sum_ {v = 1} ^ {N} \sum_ {k = 0} ^ {\infty} \sum_ {p = 0} ^ {k} f _ {k - p} f _ {p} \mathbf {W} _ {i v} ^ {k - p} \mathbf {W} _ {v j} ^ {p} \tag {13} \\ \stackrel {(3)} {=} \sum_ {v = 1} ^ {N} \left(\sum_ {k _ {1} = 0} ^ {\infty} f _ {k _ {1}} \mathbf {W} _ {i v} ^ {k _ {1}}\right) \left(\sum_ {k _ {2} = 0} ^ {\infty} f _ {k _ {2}} \mathbf {W} _ {j v} ^ {k _ {2}}\right) ^ {\top} \stackrel {(4)} {=} \boldsymbol {\Phi} _ {\mathcal {G}} \boldsymbol {\Phi} _ {\mathcal {G}} ^ {\top} \\ \end{array}
+$$
+
+where  $\Phi_{\mathcal{G}} \coloneqq \sum_{k=0}^{\infty} f_k \mathbf{W}^k$ . Here, step (1) splits the  $\mathbf{W}^k$  into a product of  $\mathbf{W}^{k-p}$  and  $\mathbf{W}^p$  with  $p \in \mathbb{N}$ ; step (2) replaces  $\alpha_k$  by the convolution  $\alpha_k \eqqcolon \sum_{p=0}^{k} f_p f_{k-p}$ ; step (3) changes the summation indices; step (4) arrives at the desired form.
+
+Convergence. The brief analysis above has made some implicit assumptions. First, in the definition  $\mathbf{M}_{\alpha}(\mathcal{G}) \coloneqq \sum_{k=0}^{\infty} \alpha_k \mathbf{W}^k$ , we have taken that the Taylor series converges. This occurs if all the eigenvalues of  $\mathbf{W}$  lie within the series' radius of convergence, meaning  $\rho(\mathbf{W}) < (\lim \sup_{n \to \infty} \sqrt[n]{|\alpha_n|})^{-1}$ . Here,  $\rho(\mathbf{W}) \coloneqq \max \{|\lambda_1|, \dots, |\lambda_N|\}$  denotes the spectral radius. This is a common problem in the graph node kernel literature. It is typically ensured by regularising  $\mathbf{W}$  to control its spectral radius (Chung, 1997). We have also taken that  $\mathbf{M}_{\alpha}(\mathcal{G})$  is positive definite, meaning all its eigenvalues are greater than 0. This can be ensured e.g. by taking  $\alpha_0 > 0$ , again regularising  $\mathbf{W}$  to control its spectral radius, and appealing to Weyl's perturbation inequality (Bai et al., 2000).
+
+We have also assumed that  $(f)_{i = 0}^{\infty}$  exists which satisfies the discrete convolution relation, and that the Taylor series  $\Phi_{\mathcal{G}} = \sum_{i = 0}^{\infty}f_i\mathbf{W}^i$  also converges. Recall that
+
+$$
+(1 - z) ^ {\frac {1}{2}} = \sum_ {n = 0} ^ {\infty} (- 1) ^ {n} \binom {\frac {1}{2}} {n} z ^ {n}, \tag {14}
+$$
+
+which is well-known to converge if  $\|z\| \leq 1$ . Letting  $z = 1 - \mathbf{M}_{\alpha}(\mathcal{G})$
+
+$$
+\mathbf {M} _ {\alpha} (\mathcal {G}) ^ {\frac {1}{2}} = \sum_ {n = 0} ^ {\infty} (- 1) ^ {n} \binom {\frac {1}{2}} {n} (1 - \mathbf {M} _ {\alpha} (\mathcal {G})) ^ {n} = \sum_ {n = 0} ^ {\infty} (- 1) ^ {n} \binom {\frac {1}{2}} {n} (1 - \sum_ {i = 0} ^ {\infty} \alpha_ {i} \mathbf {W} ^ {i}) ^ {n}. \tag {15}
+$$
+
+To converge, we require that  $\rho(1 - \mathbf{M}_{\alpha}(\mathcal{G})) \leq 1$ . Since  $\mathbf{M}_{\alpha}(\mathcal{G})$  is positive definite, it is sufficient that  $\rho(\mathbf{M}_{\alpha}(\mathcal{G})) \leq 2$ . Again, this is achieved by regularising  $\mathbf{W}$  to control its spectral radius and choosing suitable  $(\alpha_{i})_{i=0}^{\infty}$ . Inspecting Eq. 15, it is clear that  $\mathbf{M}_{\alpha}(\mathcal{G})^{\frac{1}{2}}$  can be expanded in powers of  $\mathbf{W}$ , so we let
+
+$$
+\mathbf {M} _ {\alpha} (\mathcal {G}) ^ {\frac {1}{2}} = \sum_ {k = 0} ^ {\infty} f _ {k} \mathbf {W} ^ {k}. \tag {16}
+$$
+
+Substituting this back in, we recover the convolution relationship  $\alpha_{i} \coloneqq \sum_{k=0}^{i} f_{k} f_{i-k}$ . One can find the deconvolution  $(f_{k})_{k=0}^{\infty}$  given a sequence  $(\alpha_{i})_{i=0}^{\infty}$  using an iterative formula, given in Eq. 6 by Reid et al. (2024b). We take  $\Phi_{\mathcal{G}} = \sum_{k=0}^{\infty} f_{k} \mathbf{W}^{k}$ , then  $\mathbf{M}_{\alpha}(\mathcal{G}) = \Phi_{\mathcal{G}} \Phi_{\mathcal{G}}^{\top}$  since  $\mathbf{W}$  is symmetric.
+
+The case for implicit kernel learning. We have seen that, to guarantee convergence and positive definiteness of  $\sum_{k=0}^{\infty} \alpha_k \mathbf{W}^k$ , one must in general control the spectrum of  $\mathbf{W}$ . This can be achieved by normalising it by suitably downscaling the weights (Chung, 1997; Reid et al., 2024b), or choosing  $(\alpha_k)_{k=0}^{\infty}$  to stay within the radius of convergence. To avoid a tricky constrained optimisation, we therefore prefer to instead directly optimise the set of coefficients  $(f_i)_{i=0}^{i_{\max}}$ , up to some maximum order  $i_{\max}$  above which  $f$  vanishes. In doing so, we automatically guarantee both convergence and positive definiteness: the learned  $\Phi_G$  must be finite and they implicitly define a kernel in feature space (Gretton, 2013). This is why we choose optimise mask parameters in Alg. 1, elegantly sidestepping any convergence and positive definiteness issues.
+
+# A.2 PROOF OF THM. 3.2 (NOVEL)
+
+The following concentration bounds are new. Referring back to Eq. 10, recall that graph random features are computed by
+
+$$
+\widehat {\phi} _ {\mathcal {G}} \left(v _ {i}\right) _ {q} := \frac {1}{n} \sum_ {k = 1} ^ {n} \psi \left(\omega_ {k} ^ {(i)}\right) _ {q}, \quad q = 1, \dots , N, \tag {17}
+$$
+
+where we define the projection vector  $\psi (\cdot):\Omega \to \mathbb{R}^N$  such that
+
+$$
+\psi \left(\omega_ {k} ^ {(i)}\right) _ {q} := \sum_ {\omega_ {i q} \in \Omega_ {i q}} \frac {\widetilde {\omega} \left(\omega_ {i q}\right) f _ {\mathrm {l e n} \left(\omega_ {i q}\right)}}{p \left(\omega_ {i q}\right)} \mathbb {I} \left(\omega_ {i q} \text {p r e f i x s u b w a l k o f} \omega_ {k} ^ {(i)}\right). \tag {18}
+$$
+
+$\Omega_{iq}$  is the set of walks between nodes  $v_{i}$  and  $v_{q}$ ;  $p(\omega_{iq})$  is the probability of a particular walk  $\omega_{iq}$ , known from the sampling mechanism;  $\widetilde{\omega}(\omega_{iq})$  is a function that returns the product of weights of edges traversed by  $\omega_{iq}$  (Eq. 9);  $\mathrm{len}(\omega_{iq})$  is the length of  $\omega_{iq}$ ; and  $f: \mathbb{N} \to \mathbb{R}$  is a sequence of a real number that determines the graph node kernel to be estimated. Following convention, we will assume that we sample random walks with geometrically distributed lengths, halting with probability  $p_{\mathrm{halt}}$  at each timestep, though in principle different importance sampling schemes are possible.
+
+Suppose we are to estimate a single entry of the graph node kernel,  $\mathbf{M}_{\alpha}(\mathcal{G})_{ij}$ . We do so by taking
+
+$$
+\widehat {\mathbf {M}} _ {\alpha i j} := \widehat {\phi} _ {\mathcal {G}} (v _ {i}) ^ {\top} \widehat {\phi} _ {\mathcal {G}} (v _ {j}) = \frac {1}{n ^ {2}} \sum_ {k _ {1} = 1} ^ {n} \sum_ {k _ {2} = 1} ^ {n} \psi \left(\omega_ {k _ {1}} ^ {(i)}\right) ^ {\top} \psi \left(\omega_ {k _ {2}} ^ {(j)}\right) \tag {19}
+$$
+
+where  $\mathbb{E}(\widehat{\mathbf{M}}_{\alpha ij}) = \mathbf{M}_{\alpha ij}$  since the estimate is unbiased. We sample  $n$  independent random walks starting from every node of the graph.
+
+Let us now consider a single pair of random walks from  $v_{i}$  and  $v_{j}$  respectively. Denote this by the random variable
+
+$$
+X _ {k} := \left(\omega_ {k} ^ {(i)}, \omega_ {k} ^ {(j)}\right) \in \Omega \times \Omega , \tag {20}
+$$
+
+where  $n$  such random variables  $\{X_{k}\}_{k = 1}^{n}$  are used in total. Suppose we modify  $X_{k}$ . Under mild assumptions, we can prove that the change in the estimator  $|\Delta \widehat{\mathbf{M}}_{\alpha ij}|$  is bounded, which will enable us to use derive concentration inequalities.
+
+This can be understood as follows. To begin, we will bound the  $L_{1}$ -norm of  $\psi (\omega_k^{(i)})$ . Observe that
+
+$$
+\| \psi \left(\omega_ {k} ^ {(i)}\right) \| _ {1} \leq \max  _ {\omega_ {k} ^ {(i)} \in \Omega} \sum_ {\omega \text {p . s .} \omega_ {k} ^ {(i)}} \left| \frac {\widetilde {\omega} (\omega) f _ {\mathrm {l e n} (\omega)}}{p (\omega)} \right| \leq \sum_ {k = 0} ^ {\infty} | f _ {k} | \left(\max  _ {v _ {i}, v _ {j} \in \mathcal {N}} \frac {| \mathbf {W} _ {i j} | d _ {i}}{1 - p _ {\mathrm {h a l t}}}\right) ^ {k} =: c, \tag {21}
+$$
+
+which corresponds to the case of an infinite length walk that happens almost never. Here,  $\omega$  p.s.  $\omega_{k}^{(i)}$  means that  $\omega$  is a 'prefix subwalk' of  $\omega_{k}^{(i)}$ , so the first  $\mathrm{len}(\omega)$  nodes of  $\omega_{k}^{(i)}$  are exactly the nodes of  $\omega$ .
+
+Suppose that  $c$  is finite. This can be guaranteed for bounded  $f$  by normalisation of  $\mathbf{W}$ , multiplying by a scalar so that  $|\max_{v_i, v_j \in \mathcal{N}} \frac{\mathbf{W}_{ij} d_i}{1 - p}| < 1$ . More pragmatically, it can be achieved by stipulating that there exists some integer  $i_{\mathrm{max}} \in \mathbb{N}$  such that  $f_i = 0 \forall i > i_{\mathrm{max}}$ , so we only consider walks up to a certain length. This is a natural choice in experiments since we want to keep the set of learnable parameters finite. It also coincides with the intuition that very long walks should not reflect the relationship between the nodes of a graph of finite size.
+
+We have seen that  $0 < \| \psi (\omega_k^{(i)})\| _1\leq c$ . It follows that  $-c^2\leq \psi (\omega_{k_1}^{(i)})^\top \psi (\omega_{k_2}^{(j)})\leq c^2\forall (k_1,k_2,i,j)$ , whereupon the change in  $\widehat{\mathbf{M}}_{\alpha ij}$  if we modify  $X_{k}$  obeys
+
+$$
+\left| \Delta \widehat {\mathbf {M}} _ {\alpha i j} \right| \leq 2 \frac {2 n - 1}{n ^ {2}} c ^ {2}. \tag {22}
+$$
+
+With straightforward application of McDiarmid's inequality (Doob, 1940), it follows that
+
+$$
+\Pr \left(\left| \widehat {\phi} _ {\mathcal {G}} \left(v _ {i}\right) ^ {\top} \widehat {\phi} _ {\mathcal {G}} \left(v _ {j}\right) - \mathbf {M} _ {\alpha i j} \right| > t\right) \leq 2 \exp \left(- \frac {t ^ {2} n ^ {3}}{2 (2 n - 1) ^ {2} c ^ {4}}\right), \tag {23}
+$$
+
+completing the proof.
+
+![](images/e4f6705a5569743668ce702ac252002ddc4bdb71c1224d529330ed2caa0e7a79.jpg)
+
+As noted in the main text, this is a remarkably general and powerful result. It is the first known concentration inequality for GRFs.
+
+# B EXTENDED DISCUSSION ON RELATED WORK
+
+In this appendix, we give a detailed explanation of the relationship to previous work.
+
+1. Stochastic position encoding (SPE) (Liutkus et al., 2021). To our knowledge, this is the earliest RPE strategy compatible with  $\mathcal{O}(N)$  transformers, incorporating information about the 'lag' between tokens in the linear attention setting by drawing samples from random processes with a prescribed covariance structure. The time complexity is  $\mathcal{O}(N)$  but the method is only applicable to sequence data, i.e. the very special case that  $\mathcal{G}$  is a 1-dimensional grid.  
+2. Toeplitz masks and the fast Fourier transform (FFT) (Luo et al., 2021). Again considering sequence data, suppose that RPE between a query  $\mathbf{q}_i$  and a key  $\mathbf{k}_j$  is implemented by adding a bias term  $b_{j - i}$  to the dot product before exponentiating. Then the corresponding mask  $\mathbf{M}_{ij} = e^{b_j - i}$  is Toeplitz (constant on all diagonals). From numerical linear algebra, multiplying a Toeplitz matrix and a vector can be achieved with  $\mathcal{O}(N\log N)$  time complexity rather than  $\mathcal{O}(N^2)$ , using the FFT. This is subquadratic, though not linear, and again only specific  $\mathcal{G}$  are considered.  
+3. Graphorner (Ying et al., 2021). In this paper, positional encodings based on the shortest path distances between the nodes of the graph (spatial encoding) and their respective degrees (centrality encoding) are seen to substantially improve transformer performance, closing the gap with graph neural networks. There is no attempt to incorporate these techniques with linear attention, but the paper provides strong evidence of the effectiveness of graph-based attention modulation (topological masking).  
+4. Block Toeplitz matrices and differential equations on graphs (Choromanski et al., 2022). This paper proves that subquadratic matrix-vector multiplication is sufficient for masking to be applied efficiently, and suggests a number of algorithms to achieve this in special cases.
+
+- Block Toeplitz matrices. This method extends the work of Luo et al. (2021) to show that the analogous mask on a more general  $d$ -dimensional grid is  $d$ -level block Toeplitz. Hence, this mask parameterisation also supports  $\mathcal{O}(N\log N)$  masking, but now for a slightly more general class of graphs.  
+- Affine mappings on forests. If  $\mathcal{G}$  is a forest and  $\mathbf{M}_{ij} = \exp(\tau(\mathrm{dist}(v_i, v_j)))$ , where  $\mathrm{dist}(v_i, v_j)$  is the shortest path distance between nodes  $v_i$  and  $v_j$  and  $\tau$  is an affine mapping, then topological masking can be implemented in  $\mathcal{O}(N)$ . The algorithm uses dynamic programming techniques for rooted trees. Notwithstanding its speed, the flexibility of this parameterisation is limited and only particular topologies can be treated.  
+- Heat kernels on hypercubes. Kondor and Lafferty (2002) derived a closed form for the heat kernel on a hypercube, which also turns out give a Toeplitz mask that can be applied in  $\mathcal{O}(N\log N)$ . This is a special instantiation of our power series masks  $\mathbf{M}_{\alpha}(\mathbf{W})$  for a specific choice of  $\mathcal{G}$ . The authors did not empirically test heat kernels; we found them to perform very poorly compared to our more general, faster method.  
+- Random walk graph node kernels. Here, the authors define an ad-hoc graph kernel by taking the dot product between two 'frequency vectors', recording the nodes visited by the random walks beginning at the two nodes of interest. This method is the closest to our approach, but with some crucial differences: (i) we use importance sampling of random walks, which we find to be crucial for good performance (see App. C); (ii) we use learnable weights that depend on the length of random walk, permitting flexible upweighting or downweighting of longer- and shorter-range interactions; (iii) our method provides a Monte Carlo estimate of a learnable function of a weighted adjacency matrix, which enjoys certain regularisation properties (Reid et al., 2024b). More importantly, we are able to derive strong concentration bounds that prove that our method is  $\mathcal{O}(N)$ , whereas Choromanski et al. (2022) do not explicitly guarantee any time complexities.
+
+5. Centrality degree encoding (Chen et al., 2024). This paper takes the mask  $\mathbf{M}_{ij} = \sin\left(\frac{\pi}{4}[z(d_i) + z(d_j)]\right)$ , where  $d_{\{i,j\}}$  is the degree of node  $v_{\{i,j\}}$  and  $z: \mathbb{N} \to (0,1)$  is a learnable
+
+<table><tr><td></td><td>ImageNet 1M</td><td>iNaturalist2021 2.7M</td><td>Places365 (Small) 1.8M</td></tr><tr><td>Num. layers</td><td>12</td><td>12</td><td>12</td></tr><tr><td>Num. heads</td><td>12</td><td>12</td><td>12</td></tr><tr><td>Num. patches</td><td>16 × 16</td><td>16 × 16</td><td>16 × 16</td></tr><tr><td>Hidden size</td><td>768</td><td>768</td><td>768</td></tr><tr><td>MLP dim.</td><td>3072</td><td>3072</td><td>3072</td></tr><tr><td>Optimiser</td><td>Adam</td><td>Adam</td><td>Adam</td></tr><tr><td>Epochs</td><td>90</td><td>5</td><td>5</td></tr><tr><td>Base learning rate</td><td>3 × 10-3</td><td>3 × 10-3</td><td>3 × 10-3</td></tr><tr><td>Final learning rate</td><td>1 × 10-5</td><td>1 × 10-5</td><td>1 × 10-5</td></tr><tr><td>Learning rate schedule</td><td colspan="3">Linear warmup (104steps), constant, cosine decay</td></tr><tr><td>Batch size</td><td>4096</td><td>64</td><td>64</td></tr><tr><td>φ(·)</td><td>ReLU(·)</td><td>ReLU(·)</td><td>ReLU(·)</td></tr><tr><td>Pretrained?</td><td>×</td><td>✓</td><td>✓</td></tr><tr><td>Num. walks</td><td>100</td><td>20</td><td>20</td></tr><tr><td>phalt</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>Max. walk length (imax)</td><td>100</td><td>10</td><td>10</td></tr></table>
+
+Table 2: Architecture, hyperparameters and training details for ViT experiments.
+
+function thereof. This can be implemented with linear transformers on general graphs in  $\mathcal{O}(N)$  time complexity, but since it is completely insensitive to the relative positions of the nodes in the graph it cannot be considered a general topological masking mechanism.
+
+6. Kernel decomposition linear graph transformer (KDLGT; Wu et al.) This paper also attempts to efficiently incorporate topological information into graph transformers by defining a graph node kernel with a low-rank decomposition. It is based off a 'structure extractor' for a subgraph centred at each respective node or the 'shortest anchor path distance'. However, bias is incorporated as an additive term, rather multiplicatively which is our chief goal (Eq. 8).  
+7. Graph attention networks (Velicković et al., 2018). This seminal work incorporates masked self-attention into graph neural networks, with each node attending only to its neighbours. The focus is on developing a convolution-style neural network for graph-structured data rather than on efficient masking of linear attention transformers. Our mask parameterisation is more general than this hard nearest-neighbours example, but shares the finding that modulating attention by local topological structure boosts performance.
+
+# C FURTHER EXPERIMENTS AND DETAILS
+
+Here, we include extra experiments and details too long for the main text.
+
+# C.1 ARCHITECTURES, HYPERPARAMETERS AND TRAINING DETAILS
+
+Table 2 gives details for the ViT experiments.
+
+# C.2 VIT ABLATIONS
+
+Here, we run extra experiments to further evidence the findings reported in the main text. Unless explicitly stated, architectures and training details are as reported in Table 2.
+
+Number of walkers  $n$ . We construct GRFs using  $\{1, 10, 100, 1000\}$  walkers with a termination probability  $p_{\mathrm{halt}} = 0.5$ , retraining from scratch using the respective topological masks. See Fig. 6. Larger  $n$  reduces the mask variance and improves predictive performance, but also makes the features  $\{\widehat{\phi}_{\mathcal{G}}(v_i)\}_{v_i \in \mathcal{N}}$  less sparse so increases computational cost. Since the estimator is unbiased, the  $n \to \infty$  limit gives dense features  $\{\phi_{\mathcal{G}}(v_i)\}_{v_i \in \mathcal{N}}$  whose dot product is exactly a Taylor mask  $\mathbf{M}_{\alpha}(\mathcal{G})$ .
+
+Importance sampling. The crux of our approach is modulating attention by an unbiased Monte Carlo approximation of a function of a weighted adjacency matrix  $\mathbf{W}$ . Using GRFs, this involves
+
+sampling random walks which we reweight by (i) their probability under the sampling mechanism, (ii) the product of traversed edge weights  $\widetilde{\omega} (\omega)$ , and (iii) a learnable function  $f$  (see Eq. 10). To investigate the importance of this principled sampling mechanism, we ablate (i) and (ii) and compare with an ad-hoc empirical random walk kernel defined by the features
+
+$$
+\phi_ {\text {a d - h o c}} \left(v _ {i}\right) _ {q} := \frac {1}{n} \sum_ {k = 1} ^ {n} \sum_ {\omega_ {i q} \in \Omega_ {i q}} f _ {\text {l e n} \left(\omega_ {i q}\right)} \mathbb {I} \left(\omega_ {i q} \text {p r e f i x s u b w a l k o f} \omega_ {k} ^ {(i)}\right), \quad q = 1, \dots , N. \tag {24}
+$$
+
+Eq. 24 still simulates random walks and records their destinations, giving an empirical measure of the overlap between respective nodes' neighborhoods. However, in absence of dependence on  $\widetilde{\omega} (\omega)$  and  $p(\omega)$  this cannot be interpreted as a MC estimate of a function of  $\mathbf{W}$ . This is similar to the approach employed by Choromanski et al. (2022) but with extra learnability via  $f$ .
+
+Comparing ViT performance with  $n = 100$  random walks and  $p_{\mathrm{halt}} = 0.5$ , the results are striking; see Table 3. In contrast to our method, the mask defined by Eq. 24 is worse than the unmasked baseline. This shows the crucial importance of the theory in Sec. 3.4. Intuitively, without upweighting long, improbable walks by  $p(\omega)^{-1}$  it is difficult to capture long-range dependencies in the topological mask.
+
+Fully learnable features. As a final check, we endow each graph with a fully learnable, dense  $N$ -dimensional feature  $\phi_{\mathrm{dense,learned}}(v_i) \in \mathbb{R}^N$ , relaxing the parameterisation as a function of a weighted adjacency matrix  $\mathbf{M}_{\alpha}(\mathcal{G})$ . This uses many more parameters and is  $\mathcal{O}(N^2)$  rather than  $\mathcal{O}(N)$ , but it technically includes GRFs as a special case. In making this choice we have removed all structural inductive bias from  $\mathcal{G}$ , allowing the model to implicitly learn a masking kernel without any constraints (except feature dimension). The model performs badly, providing only a very slight improvement over the unmasked linear baseline. See Table 3.
+
+![](images/74dc6401774fbea50717c9a0f02e21963831f564521b771dfc909ac1ec832716.jpg)  
+Figure 6: Final test accuracy vs. number of walkers  $n$ . Monte Carlo mask variance drops with the number of walks so predictions improve.
+
+Table 3: Ablation studies. Importance sampling of random walks and mask parameterisation as a function of  $\mathbf{W}$  are crucial for good performance.  
+
+<table><tr><td>Variant</td><td>Test acc.</td></tr><tr><td>GRF-masked</td><td>0.730</td></tr><tr><td>Unmasked</td><td>0.693</td></tr><tr><td>Ad-hoc masked (Eq. 24)</td><td>0.689</td></tr><tr><td>Fully learnable features</td><td>0.696</td></tr></table>
+
+Time complexity scaling with number of walkers. Readers may also be interested in how the total number of FLOPs scales with the number of walkers  $n$ , for a given graph  $\mathcal{G}$  and termination probability  $p_{\mathrm{halt}}$ . There are two main contributions to consider. First, one must sample the random walks, the cost of which is  $\mathcal{O}(n)$ . This is done once at the start of training, so is not a big consideration in practice. Second, the number of walks determines the sparsity of the GRF, and therefore the time complexity of implicit masking via Eq. 7. We have see that this is at most  $\mathcal{O}(n)$ , but the details will vary depending on the graph topology (see Lemma 3.3). Fig. 7 gives an example for a 1D grid with  $n = 64$ , taking  $p_{\mathrm{halt}} = 0.5$  and  $d = m = 8$ . It reflects the behaviour described above.
+
+![](images/706bf98e87221bcc728f2a1f48393afc87af8028a683e43d75ee55445612c9f3.jpg)  
+Figure 7: Time complexity scaling wrt number of walkers  $n$ .
+
+Linear attention variant. As a brief note, our topological masking algorithm is compatible with different choices for the nonlinearity  $\phi(\cdot)$ . In the text we have generally taken  $\phi(\cdot) = \mathrm{ReLU}(\cdot)$  so that  $m = d$ , but other choices are possible. For instance, one could use a randomised mapping that approximates the softmax kernel (Choromanski et al., 2020; Likhosherstov et al., 2024), which is commonly referred to as the Performer.
+
+To demonstrate, we replace the ReLU features by FAVOR+ Monte Carlo attention approximation, considering the same ViT architecture and training as reported in Sec. 4 (see Table 2). For ImageNet, the unmasked Performer achieves a test accuracy of 0.693. Meanwhile, the GRF-masked performer achieves a test accuracy of 0.734 - a  $3.9\%$  absolute improvement. This shows how Alg. 1 can be expected to provide gains across a range of linear attention mechanisms.
+
+# C.3 HD-VPD EXPERIMENTAL DETAILS
+
+Here, we describe the high density visual particle dynamics (HD-VPD) experiment from Sec. 4 in more detail. The setup closely follows that of (Whitney et al., 2024), to which we direct the interested reader for an exhaustive information about hardware and modelling techniques. We will provide a brief overview for convenience.
+
+Dynamics modelling as video prediction. The HD-VPD stack involves three steps: (1) encode the input images as 3D point clouds; (2) predict updates to the point clouds, conditioned on robot actions; (3) render the predicted new state into an image. We train end-to-end using video prediction loss. In more detail:
+
+1. Encode the input images. The model receives two RGB-D (colour channels plus depth) images at times  $t - 1$ ,  $t - 2$ . Each RGB image is encoded into per-pixel features using a U-Net (Ronneberger et al., 2015), and unprojected using known camera intrinsics and extrinsics to a point cloud  $(\pmb{x}_i, \pmb{f}_i)_{i=1}^{N_{\mathrm{pixels}}}$  with  $\pmb{x} \in \mathbb{R}^3$  and  $\pmb{f} \in \mathbb{R}^{16}$ . For every timestep, the particles across different cameras are merged and subsampled uniformly at random to get  $N = 32768$ . The robot actions at timesteps  $\{t - 2, t - 1, t\}$ , representing the robot joints and where they plan to move, are also encoded as kinematic particles.  
+2. Predict the point cloud dynamics. With an Interlacer (see below), use the point clouds and kinematic particles to predict  $(\Delta x_{i},\Delta f_{i})_{i = 1}^{N}$ , and hence the updated state  $(x_{i}^{t - 1} + \Delta x_{i},f_{i}^{t - 1}+$ $\Delta f_i)_{i = 1}^N$  
+3. Render the new state to an image. Use a ray-based renderer similar to Point-NeRF (Xu et al., 2022) to render an image for the updated point cloud.
+
+The dynamics model can be recursively applied to make predictions multiple rollout timesteps into the future. To train, we sample a subset of rays to render at each timestep and compute the  $L_{2}$  loss between the predicted and observed RGB values.
+
+Interlacers as PCTs. The Interlacer is a type of point cloud transformer (PCT) designed to scale to tens of thousands of particles. It alternates linear-attention transformer layers that capture global dependencies between tokens with local neighbour-attender layers that model fine-grained geometric structure (see Fig. 3 by Whitney et al. (2024)). This is reported to outcompete both graph neural
+
+networks and vanilla linear-attention transformers. For the neighbour-attender, the authors use a message passing-type algorithm that updates the features of a subset of anchor particles depending on the  $k$ -nearest neighbours' position and feature vectors, and then updates the remaining particles depending on their closest anchor. In Sec. 4, we show that this can be replaced by GRF-masked linear attention. This also captures local structure and obviates expensive message passing.
+
+Training and architecture details. This exactly follows App. E by Whitney et al. (2024), apart from:
+
+1. We average over 5 seeds, discarding one where training does not converge.  
+2. For our algorithm, we use asymmetric GRFs. (Sec. D). We use  $n = 3^3 = 27$  repelling random walks (Reid et al., 2023b). Instead of terminating, we sample 3 hops deterministically since the computational bottleneck is finding the  $k$ -nearest neighbours list rather than sampling lengths once it has been computed. We take  $\phi(\cdot) = \mathrm{ReLU}(\cdot)$  and compute GRF-masked attention with a single head, projecting the queries and keys with a dense layer.  
+3. To construct  $\mathcal{G}$ , we find  $k = 3$  neighbours for every node. This is done on the fly since the neighbours can change at every timestep.  
+4. For Fig. 4, we train for 100k steps, using a learning rate of  $3 \times 10^{-4}$  until step 1000 then  $1 \times 10^{-4}$  until step 100k.
+
+To enumerate some other important details: all models are trained with a batch size of 16; we use the AdamW optimiser (Loshchilov, 2017) with weight decay  $10^{-3}$ , clipping the gradient norm to 0.01; models are trained with 6 step rollouts, with losses computed on 128 sampled rays;  $\Delta x_{i}$  are constrained to  $[-0.25, 0.25]$  by a scaled tanh nonlinearity so particles do not move more than 25cm per timestep. For the rendering, like Whitney et al. (2024) we use four concentric annular kernels of radii [0, 0.01, 0.02, 0.05] with bandwidths [0.01, 0.01, 0.01, 0.05], approximated using 16 nearest neighbours. The near plane is  $0.1\mathrm{m}$  and the far plane is  $2\mathrm{m}$ , with the background set to solid white.
+
+# C.4 VIVIT: TOPOLOGICAL MASKING FOR VIDEO DATA
+
+Here, we present preliminary results for incorporating topological masking with graph random features into video vision transformers (ViViT; Arnab et al., 2021). Prompted by the original paper, we use a factorised encoder, where the spatial encoder follows the ViT architecture from earlier in the paper (see Sec. 4 and App. C.1) with patch size  $16 \times 16 \times 2$ . The final index mixes pairs of successive frames. The temporal encoder, which subsequently models interactions between the frame-level rep
+
+Table 4: ViViT test accuracies on the Kinetics benchmark. Topological masking with GRFs boosts performance.
+
+<table><tr><td>VARIANT</td><td>TEST ACC.</td></tr><tr><td>SOFTMAX</td><td>0.754</td></tr><tr><td>SOFTMAX + GRFs</td><td>0.758</td></tr></table>
+
+resentations, also follows this architecture but with 4 layers instead of 12. We train for 30 epochs using the Adam optimiser, with base learning rate  $10^{-1}$  and final learning rate  $10^{-4}$ . We take the same schedule as in Table 2. We train and evaluate on the Kinetics 400 benchmark (Kay et al., 2017). Table 4 shows the results for (i) unmasked softmax attention and (ii) softmax attention with GRFs. Topological masking boosts predictive performance by  $+0.4\%$ . To our knowledge, this is the first application of topological masking to the video modality.
+
+# D ASYMMETRIC GRAPH RANDOM FEATURES
+
+In this Appendix, we present a variant of our core topological masking algorithm that relaxes the assumption that the function  $f$  for the query and key ensembles of walkers are identical, instead using asymmetric graph random features. These are believed (but not proved) to generally give higher variance mask estimates (Reid et al., 2024b), but making a judicious choice can also bring computational and implementation benefits. Using asymmetric GRFs for topological masking is also a novel contribution of this paper. In Sec. 4, it is found to be a convenient choice for the Interlacer (Whitney et al., 2023) because it can be straightforwardly swapped into existing code.
+
+Asymmetric GRFs. It is straightforward to see that Eq. 13 of App. A.1 generalises to
+
+$$
+\mathbf {M} _ {\alpha} (\mathcal {G}) _ {i j} = \sum_ {v = 1} ^ {N} \left(\sum_ {k _ {1} = 0} ^ {\infty} f _ {k _ {1}} ^ {(1)} \mathbf {W} _ {i v} ^ {k _ {1}}\right) \left(\sum_ {k _ {2} = 0} ^ {\infty} f _ {k _ {2}} ^ {(2)} \mathbf {W} _ {j v} ^ {k _ {2}}\right) \tag {25}
+$$
+
+if the discrete convolution
+
+$$
+\sum_ {p = 0} ^ {k} f _ {p} ^ {(1)} f _ {k - p} ^ {(2)} = \alpha_ {k} \tag {26}
+$$
+
+holds. We do not necessarily require that  $f^{(1)} = f^{(2)}$ ; this is a special case. Another special case is  $f_{p}^{(2)} = \{1 \text{ if } p = 0, 0 \text{ otherwise}\}$ , whereupon we need  $f_{p}^{(1)} = \alpha_{p} \forall p$ . Sampling GRFs in this manner will still give an unbiased estimate of  $\mathbf{M}_{\alpha}(\mathcal{G})_{ij}$ , but with different concentration properties.
+
+Computational benefits. Supposing that  $f_{p}^{(2)} = \{1 \text{ if } p = 0, 0 \text{ otherwise}\}$ ,  $\phi_{\mathcal{G}}^{(2)}(v_i)_n = \mathbb{I}(i = n)$ , a one-hot vector only nonzero at the coordinate corresponding to  $v_i$ . In other words, we do not need to simulate any random walks to obtain GRFs for the keys;  $\Phi_{\mathcal{G}}^{(2)} = \mathbf{I}_N$  with  $\mathbf{I}_N \in \mathbb{R}^{N \times N}$  the identity matrix. It follows that
+
+$$
+\widehat {\mathbf {M}} _ {\alpha} (\mathcal {G}) = \boldsymbol {\Phi} _ {\mathcal {G}} ^ {(1)} \boldsymbol {\Phi} _ {\mathcal {G}} ^ {(2) ^ {\top}} = \boldsymbol {\Phi} _ {\mathcal {G}} ^ {(1)} =: \left[ \phi_ {\mathcal {G}} ^ {(1)} \left(v _ {i}\right) \right] _ {i = 1} ^ {N}. \tag {27}
+$$
+
+But  $\phi^{(1)}(v_i)$  is only nonzero at nodes visited by the ensemble of walkers beginning at  $v_{i}$ . This makes masked attention straightforward to efficiently compute, obviating the outer products and vectorisation. Simply,
+
+$$
+\left(\mathbf {A} \odot \widehat {\mathbf {M}} _ {\alpha} (\mathcal {G})\right) \mathbf {V} = \left(\mathbf {A} \odot \boldsymbol {\Phi} _ {\mathcal {G}} ^ {(1)}\right) \mathbf {V} = \left[ \sum_ {j = 1} ^ {N} \mathbf {A} _ {i j} \phi_ {\mathcal {G}} ^ {(1)} \left(v _ {i}\right) _ {j} \boldsymbol {v} _ {j} \right] _ {i = 1} ^ {N} \tag {28}
+$$
+
+where  $\mathbf{A}_{ij} = \exp (\pmb {q}_i^\top \pmb {k}_j)$  (softmax) or  $\mathbf{A}_{ij} = \pmb {q}_i^\top \pmb {k}_j$  (linear) and  $\{\pmb {v}_j\}_{j = 1}^N$  are the value vectors. Since  $\phi_{\mathcal{G}}^{(1)}(v_i)$  is sparse, the sum in Eq. 28 is computed in constant time so attention is computed in  $\mathcal{O}(N)$ . Even more explicitly, for an ensemble of walks  $\{\omega_k^{(i)}\}_{v_i\in \mathcal{N}, k\in [1,n]}$ , we can rewrite the above as
+
+$$
+\left[ \sum_ {k = 1} ^ {n} \sum_ {\omega \text {p . s .} \omega_ {k} ^ {(i)}} \mathbf {A} _ {i \omega [ - 1 ]} \frac {\widetilde {\omega} (\omega) f _ {\mathrm {l e n} (\omega)}}{p (\omega)} \boldsymbol {v} _ {\omega [ - 1 ]} \right] _ {i = 1} ^ {N} \tag {29}
+$$
+
+where  $\omega[-1]$  stands for the final node of the walk  $\omega$  and p.s. means 'prefix subwalk' (see App. A.2). The number of FLOPs to compute attention for node  $v_i$  depends on the length of all the geometrically distributed walks beginning at node  $v_i$ , but we have seen that this is independent of graph size (Sec. 3.4). Computing the entire matrix is only  $\mathcal{O}(N)$ . We give pseudocode for this approach in Alg. 2.
+
+Benefits and limitations of asymmetric GRFs. The chief benefit of asymmetric GRFs is that Eq. 29 makes efficient masked attention very straightforward to compute, without relying on sparse matrix operations that may not be well-optimised in machine learning libraries. We use it for the HDVPD experiments because it can be easily integrated with the existing code: instead of finding each node's (approximate)  $k$ -nearest neighbours and implementing the complicated message passing-type algorithm by Whitney et al. (2024), we find the 3-hop neighbourhood and straightforwardly compute weighted attention with a learnable function of walk length. The drawback is that the variance of the mask estimate with asymmetric GRFs tends to be greater, though proving this is an open problem.
+
+# E ADDENDUM: ADDITIONAL GRF RESULTS
+
+Thm. 3.2 provides a novel bound on the GRF estimator concentration, specifying the number of walkers  $n$  and termination probability  $p_{\mathrm{halt}}$  needed to ensure sharp mask estimates with high probability. Under mild assumptions on graph (namely, that  $c$  does not change as  $N$  grows), this can be used to upper bound GRF sparsity and thus the time complexity of the algorithm. The bound derived in Lemma 3.3 implicitly assumes the worst case that walkers never backtrack and do not visit
+
+Algorithm 2  $\mathcal{O}(N)$  topological masking with asymmetric GRFs
+
+Input: query matrix  $\mathbf{Q} \in \mathbb{R}^{N \times d}$ , key matrix  $\mathbf{K} \in \mathbb{R}^{N \times d}$ , value matrix  $\mathbf{V} \in \mathbb{R}^{N \times d}$ , graph  $\mathcal{G}$  with weighted adjacency matrix  $\mathbf{W} \in \mathbb{R}^{N \times N}$ , learnable mask parameters  $\{f_i\}_{i=0}^{i_{\max}}$ , number of random walks to sample  $n \in \mathbb{N}$ , query/key feature map  $\phi(\cdot): \mathbb{R}^d \to \mathbb{R}^m$ .
+
+Output: masked attention  $\mathrm{Att}_{\mathrm{LR},\widehat{\mathbf{M}}}(\mathbf{Q},\mathbf{K},\mathbf{V},\mathcal{G})$  in  $\mathcal{O}(N)$  time, without using sparse linear algebra libraries.
+
+1: Simulate  $n$  terminating random walks  $\{\omega_k^{(i)}\}_{k = 1}^n$  out of every node  $v_{i}\in \mathcal{N}$  
+2: for  $v_{i} \in \mathcal{N}$  do  
+3: Initialise output feature,  $\mathbf{f}_i \gets \mathbf{0}_d$  
+4: Initialise attention normalisation,  $S_{i} \gets 0$  
+5: for  $\omega^{(i)}$  in  $\{\omega_k^{(i)}\}_{k = 1}^n$  do  
+6: for  $(t,v_{j})$  in enumerate  $(\omega^{(i)})$  do  
+7:  $\pmb{f}_i + = \phi (\pmb {q}_i)^\top \phi (\pmb {k}_j)\times \pmb {f}_t\times \frac{\widetilde{\omega}(\omega^{(i)}[:t])}{p(\omega^{(i)}[:t])}\times \pmb {v}_j$  
+8:  $S_{i} + = \phi (\pmb{q}_{i})^{\top}\phi (\pmb{k}_{j})\times \pmb{f}_{t}\times \frac{\widetilde{\omega}(\omega^{(i)}[:t])}{p(\omega^{(i)}[:t])}$  
+9: end for  
+10: end for  
+11: Normalise output feature,  $\pmb{f}_i / = (S_i \times n)$  
+12: end for  
+13: Return low-rank attention with topological masking in  $\mathcal{O}(N)$  time by
+
+$$
+\operatorname {A t t} _ {\mathrm {L R}, \widehat {\mathbf {M}}} (\mathbf {Q}, \mathbf {K}, \mathbf {V}, \mathcal {G}) = [ \boldsymbol {f} _ {i} ] _ {i = 1} ^ {N} \tag {30}
+$$
+
+any of the same nodes, so in practice how tight this bound is may depend on the particular graph being considered. In particular, it will tend to be tighter for denser, bigger graphs. Our algorithm is even more efficient for smaller, sparser graphs. This behaviour, beyond asymptotic time complexity results, may be of interest to practitioners.
+
+To supplement with some empirical results, we consider using GRFs to estimate the heat kernel  $\exp (\sigma^2\mathbf{W})$ , where  $\sigma \in \mathbb{R}$  is a lengthscale parameter and  $\mathbf{W}_{ij} = 1 / \sqrt{d_id_j}$  as usual. We consider Erdős-Rényi graphs, where edges are independent Bernoulli random variables with probability  $p$ .
+
+Scaling with graph sparsity. The left pane of Fig. 8 shows the Frobenius norm error on the mask estimate,  $\| \widehat{\mathbf{M}} - \mathbf{M} \|_2 / \| \mathbf{M} \|_2$  where  $\widehat{\mathbf{M}}$  is the estimate, vs. the edge generation probability  $p$ . The number of nodes is  $N = 100$ , and we take  $n = 4$  walks per node with  $\sigma = 0.5$ . We average the results over 100 trials for standard errors. As expected, the approximation error becomes a little worse for denser graphs, though it remains excellent throughout. The right hand pane of Fig. 8 shows the RF sparsity (proportion of nonzero entries in  $\widehat{\Phi}_{\{\mathbf{Q}, \mathbf{K}\}, \mathcal{G}}$ ) over the same parameters. Again, the number of nonzero entries grows as graphs become denser because visiting the same node multiple times is less likely, but GRFs are sparse in every case. We emphasise once more that the bounds still hold in every case; this is simply a matter of how tight they are.
+
+Scaling with graph size. Fig. 9 performs the same analysis for Erdős-Rényi graphs with fixed edge probability  $p = 0.5$ , now varying the size of the graph  $N$ . The approximation error grows slightly for bigger graphs but soon plateaus, and again remains excellent for all graphs considered. The GRF sparsity drops because the number of visited nodes is  $\mathcal{O}(1)$  in every instance, meaning the proportion on nonzero GRF entries must scale as  $\mathcal{O}(1 / N)$ .
+
+We have included this section for the convenience of the interested reader. For more exhaustive details, we invite them to read the previous works of Reid et al. (2024b) and Choromanski (2023), which first introduced GRFs and explored their properties.
+
+![](images/95fffd0d3d1cc7230212e7a70945fdd2fa47c21d7c2c77c9c70ca3d13375fe63.jpg)  
+Figure 8: GRF approximation quality and sparsity for Erdős-Rényi graphs of different sparsities.
+
+![](images/09e6e035f7db3cd6de16016be3daf80fb98d3308e166c8a7d4aa590683b89a7c.jpg)  
+Figure 9: GRF approximation quality and sparsity for Erdős-Rényi graphs of different sizes.
\ No newline at end of file
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+# LINEAR COMBINATIONS OF LATENTS IN GENERATIVE MODELS: SUBSPACES AND BEYOND
+
+Erik Bodin
+
+Alexandru Stere4
+
+Dragos D. Margineantu<sup>5</sup>
+
+Carl Henrik Ek<sup>1,3</sup>
+
+Henry Moss1,2
+
+<sup>1</sup>University of Cambridge
+
+$^{2}$ Lancaster University
+
+$^{3}$ Karolinska Institutet
+
+4 Boeing Commercial Airplanes
+
+5 Boeing AI
+
+# ABSTRACT
+
+Sampling from generative models has become a crucial tool for applications like data synthesis and augmentation. Diffusion, Flow Matching and Continuous Normalising Flows have shown effectiveness across various modalities, and rely on latent variables for generation. For experimental design or creative applications that require more control over the generation process, it has become common to manipulate the latent variable directly. However, existing approaches for performing such manipulations (e.g. interpolation or forming low-dimensional representations) only work well in special cases or are network or data-modality specific. We propose Latent Optimal Linear combinations (LOL) as a general-purpose method to form linear combinations of latent variables that adhere to the assumptions of the generative model. As LOL is easy to implement and naturally addresses the broader task of forming any linear combinations, e.g. the construction of subspaces of the latent space, LOL dramatically simplifies the creation of expressive low-dimensional representations of high-dimensional objects.
+
+# 1 INTRODUCTION
+
+Generative models are a cornerstone of machine learning, with diverse applications including image synthesis, data augmentation, and creative content generation. Diffusion models (Ho et al., 2020; Song et al., 2020a,b) have emerged as a particularly effective approach to generative modeling for various modalities, such as for images (Ho et al., 2020), audio (Kong et al., 2020), video (Ho et al., 2022), and 3D models (Luo & Hu, 2021). A yet more recent approach to generative modelling is Flow Matching (Lipman et al., 2022), built upon Continuous Normalising Flows (Chen et al., 2018), that generalises diffusion to allow for different probability paths between data and latent distribution, e.g. defined through optimal transport (Gulrajani et al., 2017; Villani et al., 2009).
+
+As well as generation, these models allow inversion where, by running the generative procedure in the opposite direction, data objects can be transformed deterministically into a corresponding realisation in the latent space. Such invertible connections between latent and data space provide a convenient mechanism for controlling generated objects by manipulating their latent vectors. The most common manipulation is to attempt semantically meaningful interpolation of two generated objects (Song et al., 2020a;b; Luo & Hu, 2021) by interpolating their corresponding latent vectors. However, the optimal choice of interpolant remains an open question, with simple approaches like linear interpolation leading to intermediates for which the model fails to generate plausible objects.
+
+White (2016) argues that poor quality generation under linear interpolation is due to a mismatch between the norms of the intermediate vectors and those of the Gaussian vectors that the model has been trained to expect. Indeed, it is well-known that the squared norm of a  $D$ -dimensional unit Gaussian follows the chi-squared distribution. Consequently, likely samples are concentrated in a tight annulus around a radius  $\sqrt{D}$  — a set which is not closed under linear interpolation. Because of this it is common to rely instead on spherical interpolation (Shoemake, 1985) (SLERP) that maintains similar
+
+norms as the endpoints. Motivated by poor performance of interpolation between the latent vectors provided by inverting natural images, alternatives to SLERP have recently been proposed (Samuel et al., 2023; Zheng et al., 2024). However, these procedures are costly, have parameters to tune, and like SLERP — are challenging to generalise to other types of manipulations beyond interpolation, such as constructing the subspaces needed for latent space optimisation methods (Gómez-Bombarelli et al., 2018), or adapting to more diverse latent distributions beyond Gaussian.
+
+In this work, we demonstrate that successful generation from latent vectors is strongly dependent on whether their broader statistical characteristics match those of samples — a stronger condition than just having e.g. likely norms. We show that effective manipulation of latent spaces can be achieved by following the simple guiding principle of adhering to the modelling assumptions of the generation process. Our primary contributions are as follows:
+
+- We show that even if a latent leads to a valid object upon generation, it can fail to provide effective interpolation if it lacks the typical characteristics of samples from the latent distribution — as diagnosed by statistical distribution tests.  
+- We introduce Latent Optimal Linear combinations (LOL) — an easy-to-implement method for ensuring that interpolation intermediates continue to match the latent distribution.  
+- We show that LOL — a closed-form transform that makes no assumptions about the network structure or data modality — constitutes a Monge optimal transport map for linear combinations to a general class of latent distributions.  
+- We demonstrate that LOL can be applied to define general linear combinations beyond interpolations, including centroids (Figure 2) and, in particular, to define meaningful low-dimensional latent representations (Figure 1)—a goal achieved previously with significant expense and only for specific architectures.
+
+Implementation and examples: https://github.com/bodin-e/linear-combinations-of-latents
+
+![](images/7d349c654c3cbe9efb78aead54ce411f3342659deaccbe1312e3dce35fb425d3.jpg)  
+Figure 1: Low-dimensional latent subspaces. A 5-dimensional subspace from the flow matching model Stable Diffusion 3 (Esser et al., 2024) extracted using LOL (left) from the latents corresponding to images  $\boldsymbol{x}_1, \ldots, \boldsymbol{x}_5$ . The left plot show generations from uniform grid points across an axis-aligned slice of the subspace coordinate system, centered around the coordinate for  $\boldsymbol{x}_1$ . Each coordinate in the subspace correspond to a linear combination of latents, which define basis vectors. The right plot shows the corresponding subspace without the proposed LOL transformation. See Figure 6, Figure 7 and Section G in the appendix for additional examples.
+
+# 2 BACKGROUND
+
+# 2.1 GENERATIVE MODELLING WITH LATENT VARIABLES
+
+The methodology in this paper applies to any generative model that transforms samples from a known distribution in order to model another distribution — typically, a complicated distribution of high-
+
+![](images/97e41f6edf00e8cf5f92b26b12ac08906d8d0d0b1e517fcde803e062de1e61cd.jpg)  
+Figure 2: Centroid determination. Generation using Stable Diffusion 2.1 (Rombach et al., 2022) from the centroid of the latents corresponding to images  $\boldsymbol{x}_1, \boldsymbol{x}_2, \boldsymbol{x}_3$  using different methods. Note that our proposed method removes several artifacts, such as unrealistic headlights and chassis texture.
+
+dimensional objects. For clarity of exposition, we focus on popular Diffusion and Flow Matching models using Gaussian latents, and then extend to general latent distributions in the appendix.
+
+Diffusion models learn a process that reverses the effect of gradually adding noise to training data (Ho et al., 2020). The noise level is governed by a schedule designed so that the noised samples at the final time index  $T$  follow the latent distribution  $\pmb{x}^{(T)} \sim p(\pmb{x})$ . To sample from the diffusion model, one starts by drawing a sample from the latent distribution before iteratively evaluating the reverse process with learnt parameters  $\theta$ , denoising step-by-step, generating a sequence  $\{\pmb{x}^{(T)}, \pmb{x}^{(T-1)}, \dots, \pmb{x}^{(0)}\}$
+
+$$
+\boldsymbol {x} ^ {(t - 1)} \sim p _ {\boldsymbol {\theta}} (\boldsymbol {x} ^ {(t - 1)} \mid \boldsymbol {x} ^ {(t)}),
+$$
+
+finally producing a generated object  $\pmb{x}^{(0)}$ . Diffusion has also been extended to the continuous time setting by Song et al. (2020b), where the diffusion is expressed as a stochastic differential equation. Note that, by using the Denoising Diffusion Implicit Model (DDIM) (Song et al., 2020a) or the probability flow formulation in Song et al. (2020b), the generation process can be made deterministic, i.e. the latent representation  $\pmb{x}^{(T)} \sim p(\pmb{x})$  completely specifies the generated object  $\pmb{x}^{(0)}$ .
+
+Flow Matching Lipman et al. (2022) is an efficient approach to train Continuous Normalising Flows (CNF) Chen et al. (2018) — an alternative class of generative models that builds complicated distributions from simple latent distributions using differential equations. CNFs model the evolution of data points over continuous time using an ordinary differential equation (ODE) parameterized by a neural network, providing a deterministic relationship between latent and object space. Mathematically, the transformation of a latent sample  $\boldsymbol{x}^{(T)} \sim p(\boldsymbol{x})$  at time  $t = T$  to  $\boldsymbol{x}^{(t)}$  at time  $t$  is governed by  $f(\boldsymbol{x}^{(t)}, t; \boldsymbol{\theta})$ , where  $f(\cdot; \boldsymbol{\theta})$  is a neural network with parameters  $\boldsymbol{\theta}$ . Flow matching allows the transformation dynamics of the CNF to be learnt without needing to simulate the entire forward process during training, which improves stability and scalability.
+
+Note that both diffusion and flow matching models can generate in a deterministic manner, where the realisation of the latent variable completely specifies the generated data object, e.g. the image. Moreover, by running their deterministic generation formulation in reverse, one can obtain the corresponding latent representation associated with a known object, referred to as "inversion". In this paper we focus on the effect of properties of the latent vectors, and their manipulation, on the generation process. For simplicity of notation, henceforth we drop the time index and refer to the latent variable  $\boldsymbol{x}^{(T)}$  as  $\boldsymbol{x}$ , and use subscript indexing to denote realisations of the latent variable.
+
+# 2.2 INTERPOLATION OF LATENTS
+
+Linear Interpolation. The most common latent space manipulation is interpolation, where we are given two latent vectors  $x_{1}$  and  $x_{2}$ , referred to as seed latents (or seeds), and obtain intermediate latent vectors. The simplest approach is to interpolate linearly between the seeds to get intermediates
+
+$$
+\boldsymbol {y} _ {\ln} ^ {w} = w \boldsymbol {x} _ {1} + (1 - w) \boldsymbol {x} _ {2} \quad \text {f o r} \quad w \in [ 0, 1 ].
+$$
+
+Spherical Interpolation. However, as discussed in (White, 2016) in the context of Variational Autoencoders (Kingma, 2013) and Generative Adversarial Networks (Goodfellow et al., 2014), linear interpolation yields intermediates with unlikely norms for Gaussian samples and results in highly implausible generated objects (e.g. all-green images). Consequently, it is common to instead use
+
+spherical interpolation (SLERP) (Shoemake, 1985), which instead builds interpolants via
+
+$$
+\boldsymbol {y} _ {\mathrm {S L E R P}} ^ {w} = \frac {\sin w \theta}{\sin \theta} \boldsymbol {x} _ {1} + \frac {\sin ((1 - w) \theta)}{\sin \theta} \boldsymbol {x} _ {2} \quad \text {f o r} \quad w \in [ 0, 1 ] \quad \cos \theta = \frac {\langle \boldsymbol {x} _ {1} , \boldsymbol {x} _ {2} \rangle}{| | \boldsymbol {x} _ {1} | | | | \boldsymbol {x} _ {2} | |},
+$$
+
+to maintain similar norms for the intermediates as the endpoints. SLERP has become popular and is the standard method for interpolation also in the more recent models (Song et al., 2020a;b).
+
+Norm-aware Optimisation. Motivated by poor performance of SLERP when interpolating between the latent vectors corresponding to inverted natural images (rather than using those sampled directly from the model's latent distribution), Zheng et al. (2024) propose to add additional Gaussian noise to interpolants via the inversion procedure to control a trade-off between generation quality and adherence to the seed images (i.e. the images corresponding to the latents at the interpolation endpoints). Alternatively, Samuel et al. (2023) advocate for Norm-Aware Optimisation (NAO) which uses back-propagation to identify interpolation paths  $\gamma :[0,1]\to \mathbb{R}^d$  that solve
+
+$$
+\inf _ {\gamma} - \int \log \mathbb {P} (\gamma (s)) d s \quad \mathrm {s . t .} \quad \gamma (0) = \boldsymbol {x} _ {1}, \gamma (1) = \boldsymbol {x} _ {2},
+$$
+
+where  $\log \mathbb{P}:\mathbb{R}^d\to \mathbb{R}$  is the log likelihood of the squared norm under its sampling distribution  $\chi^2 (D)$  for unit Gaussian samples. NAO can also calculate centroids of  $K$  latents by simultaneous optimisation of paths between the centroid and each respective latent.
+
+All the proposed interpolation methods above aim to make the intermediates adhere to statistics of Gaussian samples, however, require significant computation, have hyperparameters to tune, or lack theoretical guarantees. Moreover — in common with spherical interpolation — they are difficult to generalise beyond Gaussian latent variables or to more general expressions of the latents beyond interpolation, such as building subspaces based on their span — as required to build expressive low-dimensional representations.
+
+In this work we will propose a simple technique that guarantees that interpolations follow the latent distribution if the original (seed) latents do, applies to a general class of latent distributions, and enables us go beyond interpolation. Moreover, we will address how to assess the distributional assumption of the seed latents, to simplify the diagnosis of errors at the source.
+
+![](images/141b454ee6939896988e5d29b232e0a65f5307b525b5555fdddc0b676f8f1bfc.jpg)  
+Figure 3: Likelihood and norm insufficient. Column (1) of each panel shows an image generated using a random sample from the associated Gaussian latent distribution for the diffusion model of Rombach et al. (2022) (left side) and the flow matching model of Esser et al. (2024) (right side). Columns (2) and (3) both show images generated from latents with the most likely norm according to their respective latent distribution. Columns (2) use the same Gaussian samples as in columns (1) but rescaled to have this norm, also yielding realistic images. Meanwhile, columns (3) show the failed generation from constant vectors  $sI$  scaled to have the most likely norm according to the latent distribution but lacking other characteristics (e.g. not having all-equal values) that the network was trained to expect, even though its likelihood  $\mathcal{N}(sI; \mu, \Sigma)$  is typical of real samples. Moreover, the distribution mode  $\mu$ , which also lacks needed characteristics, has vastly higher log likelihood than any realistic sample; -33875 and -135503 for the two models, respectively. See Table 2 for an example of failed generation using the mode  $\mu$ .
+
+# 3 ASSESSING VALIDITY OF LATENTS VIA DISTRIBUTION TESTING
+
+Before introducing our proposed method, we first explore evidence for our central hypothesis that tackles a misconception underpinning current interpolation methods for Gaussian latents:
+
+Having a latent vector with a norm that is likely for a sample from the latent distribution  $\mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$  is not a sufficient condition for plausible sample generation. Rather, plausible generation requires
+
+a latent vector with characteristics that match those of a sample from  $\mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$  more generally (as evaluated by normality tests), with a likely norm being only one such characteristic.
+
+The characteristics of the random variable  $\pmb{x}$  may be described by a collection of statistics, e.g. its mean, variance or norm. Above, we hypothesise that if a specified latent  $\pmb{x}_{*}$  has an extremely unlikely value for a characteristic for which the network has come to rely, then implausible generation will follow. A norm is unlikely if it has a low likelihood under its sampling distribution; for unit Gaussians it is  $\chi(D)$ , see Section 1. While the norm has been shown to often be an important statistic (Samuel et al., 2023), our Figure 3 illustrates on a popular diffusion model (Rombach et al., 2022) and a flow matching model (Esser et al., 2024) that a latent with a likely norm — even the most likely — can still induce failure in the generative process if other evidently critical characteristics are unmet.
+
+![](images/3115ad2e2ef312e547690ad3a399ce97607630179b95d62d857fa9e34e7fd7ee.jpg)  
+Figure 4: Normality testing of latent vectors obtained from inversion. We show the LPIPS (Zhang et al., 2018) reconstruction errors (second row, in red) of 200 inverted randomly selected images across 50 random classes from ImageNet1k (Deng et al., 2009), the p-values of their inversions (third row), and rejection rates (bottom row) of the Kolmogorov-Smirnov normality test applied to the corresponding latent obtained from inversion under various step budgets. We use the diffusion model of Rombach et al. (2022), always using its maximum number of steps (999) for generation, and denote the 10th, 50th and 90th percentiles with black lines. The first row shows image reconstructions using its inversion at each budget, highlighted in red when the latent was rejected  $(p < 1e^{-3})$ , with the interpretation that the characteristics of the latent were unlikely for a real sample according to the KS test. We note the strong correlation between inversion budgets providing low reconstruction errors and those for which the p-values of the latents are realistic — taking values likely to occur by chance for real samples. However, as we will see in Figure 5, there are still many latents with low reconstruction error yet extremely low p-value, and this often severely affects the quality of its interpolants.
+
+In general, neural networks have many degrees of freedom, and so it is difficult to determine and list the input characteristics that a specific model relies on. Therefore, rather than trying to determine all necessary statistics for each model, we propose instead to rely on standard methods for distribution testing, a classic area of statistics (Razali et al., 2011; Kolmogorov, 1933; Shapiro & Wilk, 1965) — here exploring the (null) hypothesis that a latent vector  $\boldsymbol{x}_{*} \in \mathbb{R}^{D}$  is drawn from  $\mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ . Popular normality tests consider broad statistics associated with normality. For example, the Kolmogorov-Smirnov (Kolmogorov, 1933) test considers the distribution of the largest absolute difference between the empirical and the theoretical cumulative density function across all values.
+
+![](images/097a7bef42b7daf67634b5cf12186d02475febf45a8a68da6d83052c9dbe4cbd.jpg)  
+Figure 5: Lack of normality is linked to failure of interpolants The left and middle panels show LPIPS (Zhang et al., 2018) reconstruction error (after 999 generation steps) and the Kolmogorov-Smirnov p-value for all inversions presented in Figure 4, split into two plots due to the vast dynamic range of p-values. Although latents with high (realistic) p-values tend to have low reconstruction errors, there are many latents with low reconstruction errors that also have low p-values. The right panel shows Q-Align visual quality scores (Wu et al., 2023) for spherical interpolants between pairs of inversions selected from matching ImageNet1k (Deng et al., 2009) image classes, demonstrating that choosing seed latents with both low reconstruction error  $(< 0.05)$  and high p-values  $(>1e^{-3})$  allows us to avoid low-quality interpolants that would arise when choosing seeds by reconstruction error alone. For reference, we include examples of interpolants at each visual quality level.
+
+# 3.1 THE SAMPLE CHARACTERISTICS OF SEED LATENTS AFFECT THEIR INTERPOLATIONS
+
+An ability to identify latents that lack the necessary characteristics for good quality generation is critical when manipulating latents that are not acquired by explicitly sampling from the latent distribution. Indeed, as we show below, using such deficient latents as seeds for interpolation can severely affect the quality of interpolants, for example when interpolating between inverted natural images (the most common way to obtain a seed latent, see Section 2). As inversion is subject to a finite computation budget (i.e. finite inversion steps), numerical imprecision, and because the particular data instance may not be drawn from the distribution of the model's training data, the resulting latent vector  $\boldsymbol{x}_{*}$  may not correspond to a sample from the latent distribution. We will now show that distribution testing is helpful for identifying such deficient latents, providing an opportunity to apply corrections or review the examples or inversion setup, before performing interpolation.
+
+Figure 4 demonstrates that inversions having realistic p-values are strongly correlated with them reproducing their original image with good quality. The p-value indicates the probability of observing the latent vector by chance given that it is drawn from the latent distribution  $\mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ . If this value is extremely low that is a strong indicator that the latent lacks characteristics expected of samples from  $\mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ . In the figure we see that at low inversion budgets (with high reconstruction errors) most p-values are  $1e^{-50}$  or lower, and then reach values you expect to see by chance for real samples (around  $1e^{-3}$ ) at budgets where the reconstruction errors tend to be small. However, we will show that the p-value provides additional information about the quality of the latents than provided by reconstruction error alone.
+
+We now examine how the p-values of two inversions relate to our ability to interpolate them with good quality. Figure 5 shows that although latents with realistic p-values tend to have low reconstruction errors, there are many latents with low reconstruction errors that have low p-values. Therefore, just because the process of going from object to latent and back reproduces the object, this does not necessarily mean that the resulting latent is well characterised as a sample from  $\mathcal{N}(\boldsymbol{\mu},\boldsymbol{\Sigma})$ . The lack of such characteristics can be inherited when manipulating the latent; in Appendix 8 we show that interpolants typically inherit low p-values from their seed latents. Figure 5 also demonstrates that if
+
+we beyond requiring inversions to reconstruct their original objects also require their p-values to be realistic — at levels expected of real samples — we are able to avoid many low quality interpolants downstream, by not using the rejected latents as seeds. In other words, the normality test helps us discover when the latents acquired from inversion lack characteristics of real samples, as this lack prevents them from being reliably used as seeds. Distribution testing the latents can be helpful for debugging the inversion setup — which typically involves solving an ODE — and assessing the settings used. In Appendix E we investigate the effectiveness of a range of other classic normality tests and tests based on the likelihood  $\mathcal{N}(\boldsymbol{x}_*;\boldsymbol{\mu},\boldsymbol{\Sigma})$  as well as the likelihood of the norm statistic. We assess these methods across a suite of test cases including vectors with unlikely characteristics, and for which we know, through failed generation, violate the assumptions of the generative model.
+
+# 4 LINEAR COMBINATIONS OF LATENTS
+
+Now, equipped with the knowledge that matching broad characteristics of a sample from the latent distribution is critical, we propose a simple scheme for forming linear combinations of seed latents — which we now will assume are sampled from the latent distribution — to maintain their sample characteristics. We will focus on Gaussian latents in this section, and present an extension to general latent distributions in Appendix A. In this section we change the notation slightly, where a seed latent  $\boldsymbol{x}_k \in \mathbb{R}^D$  rather than being a realisation of a random variable — a vector of known values — it is a random variable following the latent distribution (here,  $\mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ ). We assume access to  $K$  such seed latent variables  $\{\boldsymbol{x}_k\}_{k=1}^K$  and attempt to form new variables following the same distribution.
+
+Let  $\pmb{y}$  be a linear combination of the  $K$  Gaussian latent variables  $\pmb{x}_k\sim \mathcal{N}(\pmb {\mu},\pmb {\Sigma})$
+
+$$
+\boldsymbol {y} := \sum_ {k = 1} ^ {K} w _ {k} \boldsymbol {x} _ {k} = \boldsymbol {w} ^ {T} \boldsymbol {X}, \tag {1}
+$$
+
+where  $w_{k}\in \mathbb{R},\pmb {w} = [w_{1},w_{2},\dots ,w_{K}]$  and  $\pmb {X} = [\pmb {x}_1,\pmb {x}_2,\dots ,\pmb {x}_K]$ . Then we have that  $\pmb{y}$  is also a Gaussian random variable, with mean and covariance
+
+$$
+\boldsymbol {y} \sim \mathcal {N} (\alpha \boldsymbol {\mu}, \beta \boldsymbol {\Sigma}) \quad \alpha = \sum_ {k = 1} ^ {K} w _ {k} \quad \beta = \sum_ {k = 1} ^ {K} w _ {k} ^ {2}. \tag {2}
+$$
+
+In other words,  $\pmb{y}$  is only distributed as  $\mathcal{N}(\pmb{\mu}, \pmb{\Sigma})$  in the specific case where (a)  $\alpha \pmb{\mu} = \pmb{\mu}$  and (b)  $\beta \pmb{\Sigma} = \pmb{\Sigma}$  — an observation which we now use to explain the empirically observed behaviour of existing interpolation methods. Firstly, for linear interpolation, where  $\pmb{w} = [v, 1 - v], v \in [0, 1]$ , (b) holds only for the endpoints  $v = \{0, 1\}$ , and so leads to implausible generations for interpolants (as we demonstrate empirically in Figure 11). In contrast, in the popular case of high-dimensional unit Gaussian latent vectors, spherical interpolants have  $\beta \approx 1$ ,  $\forall v \in [0, 1]$ , as proven in Appendix C, and (a) is met as  $\alpha \pmb{0} = \pmb{0}$ , which is consistent with plausible interpolations (see Figure 11).
+
+In this work, we instead propose transforming linear combinations such that  $\alpha = \beta = 1$ , for any  $\boldsymbol{w} \in \mathbb{R}^{K}$ , thus exactly meeting the criteria for any linear combination and any choice of  $\mu$  and  $\Sigma$ . We define a transformed random variable  $\boldsymbol{z}$  to use as the latent instead of the linear combination  $\boldsymbol{y}$
+
+$$
+\boldsymbol {z} := \mathcal {T} _ {\boldsymbol {w}, \boldsymbol {\mu}} (\boldsymbol {y}) = \mathcal {T} _ {\boldsymbol {w}, \boldsymbol {\mu}} \left(\sum_ {k = 1} ^ {K} w _ {k} \boldsymbol {x} _ {k}\right), \tag {3}
+$$
+
+where  $\mathcal{T}_{\boldsymbol{w},\boldsymbol{\mu}}(\boldsymbol{y}) \coloneqq (1 - \frac{\alpha}{\sqrt{\beta}})\boldsymbol{\mu} + \frac{\boldsymbol{y}}{\sqrt{\beta}}$ , for which it holds that  $z \sim \mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$  given latent variables  $\boldsymbol{x}_k \sim \mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ . Here,  $\mathcal{T}_{\boldsymbol{w},\boldsymbol{\mu}}: \mathbb{R}^D \to \mathbb{R}^D$  is the map that transforms samples from the distribution of a linear combination of  $\mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ -variables with weights  $\boldsymbol{w}$  into  $\mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ . In Appendix B we show that this transport map is Monge optimal, and extend it to general distributions in Appendix A. The weights  $\boldsymbol{w}$ , which via  $\alpha$  and  $\beta$  together with the set of  $K$  seed latents specify the transformed linear combination  $\boldsymbol{z}$ , depend on the operation, represented as particular linear combinations. Below are a few examples of popular operations (linear combinations) to form  $\boldsymbol{y}$ ; these are used as above to, for the corresponding weights  $\boldsymbol{w}$ , obtain  $\boldsymbol{z}$  following the distribution expected by the generative model.
+
+- Interpolation:  $\pmb{y} = \pmb{w}^T\pmb{X}$ , where  $\pmb{X} = [x_1, x_2]$ ,  $\pmb{w} = [w_1, 1 - w_1]$  and  $w_1 \in [0,1]$ .
+
+- Centroid Determination:  $\pmb{y} = \pmb{w}^T\pmb{X}$ , where  $\pmb{X} = [x_1, \dots, x_K]$ ,  $\pmb{w} = [\frac{1}{K}]^K$ .  
+- Subspaces: Suppose we wish to build a navigable subspace spanned by linear combinations of  $K$  latent variables. By performing the QR decomposition of  $\mathbf{X} \coloneqq [\pmb{x}_1, \pmb{x}_2, \dots, \pmb{x}_K] \in \mathbb{R}^{D \times K}$  to produce a semi-orthonormal matrix  $\mathbf{U} \in \mathbb{R}^{D \times K}$  (as the Q-matrix), we can then define a subspace projection of any new  $\pmb{x}$  into the desired subspace via  $s(\pmb{x}) \coloneqq U\mathbf{U}^T\mathbf{x} = U\mathbf{h} \in \mathbb{R}^D$ . The weights  $\pmb{w}$  for a given point in the subspace  $s(\pmb{x})$  are given by  $\pmb{w} = \pmb{X}_{\dagger}\pmb{s}(\pmb{x}) = \pmb{X}_{\dagger}\pmb{U}\pmb{h} \in \mathbb{R}^{K}$  where  $\pmb{X}_{\dagger}$  is the Moore-Penrose inverse of  $\pmb{X}$ . See the derivation of the weights and proof in Appendix D. One can directly pick coordinates  $\pmb{h} \in \mathbb{R}^{K}$ , compute the subspace projection  $\pmb{y} = U\pmb{h}$ , and then subsequently use the transport map (defined by Equation 3) to the latent space to obtain  $\pmb{z}$ . In Figure 1 we use grids in  $\mathbb{R}^{K}$  to set  $h$ , used as above together with a basis (defined by  $U$ ) from a set of latents.
+
+# 5 EXPERIMENTS
+
+We now assess our proposed transformation scheme LOL experimentally. To verify that LOL matches or exceeds current methods for Gaussian latents for currently available operations — interpolation and centroid determination — we perform qualitative and quantitative comparisons to their respective baselines. We then demonstrate new capabilities with several examples of low-dimensional subspaces on popular diffusion models and a popular flow matching model.
+
+# 5.1 INTERPOLATION AND CENTROID DETERMINATION
+
+For the application of interpolation, we compare our proposed LOL to linear interpolation (LERP), spherical linear interpolation (SLERP), and Norm-Aware Optimization (NAO) Samuel et al. (2023).
+
+In contrast to the all the other considered interpolation methods (including LOL) which only involve closed-form expressions, NAO requires numerical optimisation. We closely follow the evaluation protocol in Samuel et al. (2023), basing the experiments on Stable Diffusion (SD) 2.1 (Rombach et al., 2022) and inversions of random images from 50 random classes from ImageNet1k (Deng et al., 2009), and assess visual quality and preservation of semantics using Fréchet Inception Distance (FID) (Heusel et al., 2017) and class prediction accuracy, respectively. For the interpolation we (randomly without replacement) pair the 50 images per class into 25 pairs, forming 1250 image pairs in total.
+
+<table><tr><td colspan="4">Interpolation</td></tr><tr><td>Method</td><td>Accuracy</td><td>FID ↓</td><td>Time</td></tr><tr><td>LERP</td><td>3.92%</td><td>199</td><td>6e-3s</td></tr><tr><td>SLERP</td><td>64.6%</td><td>42.6</td><td>9e-3s</td></tr><tr><td>NAO</td><td>62.1%</td><td>46.0</td><td>30s</td></tr><tr><td>LOL (ours)</td><td>67.4%</td><td>38.9</td><td>6e-3s</td></tr><tr><td colspan="4">Centroid determination</td></tr><tr><td>Method</td><td>Accuracy</td><td>FID ↓</td><td>Time</td></tr><tr><td>Euclidean</td><td>0.286%</td><td>310</td><td>4e-4s</td></tr><tr><td>Standardised Euclidean</td><td>44.6%</td><td>88.8</td><td>1e-3s</td></tr><tr><td>Mode norm Euclidean</td><td>44.6%</td><td>88.4</td><td>1e-3s</td></tr><tr><td>NAO</td><td>44.0%</td><td>93.0</td><td>90s</td></tr><tr><td>LOL (ours)</td><td>46.3%</td><td>87.7</td><td>6e-4s</td></tr></table>
+
+Table 1: Quantitative comparisons of baselines.
+
+For the centroid determination we compare to NAO, the Euclidean centroid  $\bar{x} = \frac{1}{K}\sum_{k=1}^{K}x_k$  and two transformations thereof; "standardised Euclidean", where  $\bar{x}$  is subsequently standardised to have mean zero and unit variance (as SD 2.1 assumes), and "mode norm Euclidean", where  $\bar{x}$  is rescaled for its norm to equal the maximum likelihood norm  $\sqrt{D}$ . For each class, we form 10 3-groups, 10 5-groups, 4 10-groups and 1 25-group, sampled without replacement per group, for a total of 1250 centroids per method. For more details on the experiment setup and settings, see Appendix F.
+
+In Table 1 we show that our method outperforms or maintains the performance of the baselines in terms of FID distance and accuracy, as calculated using a pre-trained classifier following the evaluation methodology of Samuel et al. (2023). For an illustration of centroids and interpolations, see Figure 2 and Figure 16, respectively. The evaluation time of an interpolation path and centroid, shown with one digit of precision, illustrate that the closed-form expressions are significantly faster than NAO. Surprisingly, NAO did not perform as well as spherical interpolation and several other baselines, despite using their implementation, which was outperforming these methods in Samuel et al. (2023). We note one discrepancy is that we report FID distances using (2048) high-level features, while in their work they are using (64) low-level features, which in Seitzer (2020) is recommended
+
+against as it does not necessarily correlate with visual quality. In the appendix we include FID distances using all settings of features. We note that, in our setup, the baselines perform substantially better than reported in Samuel et al. (2023) — including NAO in terms of FID distances using the 64 low-level feature setting (1.30 in our setup vs 6.78 in their setup), and class accuracy during interpolation (62% vs 52%). See Section H in the appendix for more details and ablations.
+
+![](images/b806bd39a5391d28e7b55f084c4f2d4fe38f78fe04e2ee01259273c0950e2ad3.jpg)  
+Figure 6: Low-dimensional subspaces. The latents  $x_{1}, \ldots, x_{5}$  (corresponding to images) are converted into basis vectors and used to define a 5-dimensional subspace. The grids show generations from the flow matching model Stable Diffusion 3 (Esser et al., 2024) over uniform grid points in the subspace coordinate system, where the left and right grids are for the dimensions  $\{1, 2\}$  and  $\{3, 4\}$ , respectively, centered around the coordinate for  $x_{1}$ . Each coordinate in the subspace corresponds to a linear combination of the basis vectors, which through LOL all yield high-quality generations.
+
+# 5.2 LOW-DIMENSIONAL SUBSPACES
+
+![](images/6ffe7959084058235a4694216544de10c692aa186e10831ec403401a89fd730e.jpg)  
+Figure 7: Model-agnostic subspace definitions. The latents  $x_{1}, x_{2}$  and  $x_{3}$  — which via the generative model correspond to 3D designs — are converted into basis vectors and used to define a three-dimensional subspace. The left plot shows the generated designs corresponding to a ten-by-ten grid in a two-dimensional slice of the subspace, shown over a region. The right plot shows the wavedrag, as evaluated by simulation in OpenVSP (McDonald & Gloudemans, 2022) for the designs over a 100-by-100 grid in the same region, with the respective design coordinates (from the left plot) shown as black dots. We trained the SLIDE (Lyu et al., 2023) diffusion model on ShapeNet (Chang et al., 2015). LOL allows subspaces to be defined without any model-specific treatment.
+
+In Figure 1 and Figure 6 we illustrate slices of a 5-dimensional subspace of the latent space of a flow matching model, indexing high-dimensional images of sports cars and rocking chairs, respectively.
+
+The subspaces here are defined using five images (one per desired dimension), formed using our LOL method described in Section 4 to transform the linear combinations of the corresponding projections. Our approach is dimensionality and model-agnostic, which we illustrate in Figure 7, where the same procedure is used on a completely different model without adaptations — where we define a three-dimensional subspace based on three designs in a point cloud diffusion model with mesh reconstruction. We evaluate designs in this subspace using a simple computational fluid dynamics simulation to illustrate that our approach allows objective functions to be defined over the space, in turn allowing off-the-shelf optimisation methods to be applied to search for designs. In Figure 9 in the appendix we show corresponding grids to the sports cars in Figure 1 without the proposed LOL transformation, which either leads to implausible images or the same image for each coordinate, depending on the model. In the appendix (Section G) we also include more slices and examples; including subspaces using the diffusion model Stable Diffusion 2.1 (Rombach et al., 2022).
+
+# 6 RELATED WORK
+
+Generative models with non-Gaussian priors. In Fadel et al. (2021) and Davidson et al. (2018), in the context of normalizing flows and VAEs, respectively, Dirichlet and von Mises-Fisher prior distributions are explored with the goal of improving interpolation performance. However, these methods require training the model with the new latent distribution, which is impractical and untested for the large pretrained models we consider.
+
+Conditional Diffusion. An additional way to control the generation process of a diffusion model is to guide the generative process with additional information, such as text and labels, to produce outputs that meet specific requirements, such as where samples are guided towards a desired class (Dhariwal & Nichol, 2021; Ho & Salimans, 2022), or to align outputs with text descriptions (Radford et al., 2021). Conditioning is complementary to latent space manipulation. For example, when making Figure 1 we used conditioning (a prompt) to constrain the generation to sports cars, however, the variation of images fulfilling this constraint is encoded in the latent space.
+
+Low-dimensional representations. We have shown that LOL can provide expressive low-dimensional representations of latent spaces of generative models. To the best of our knowledge, the most closely related line of work was initiated in Kwon et al. (2022), where it was shown that semantic edit directions can recovered from activations of a UNet (Ronneberger et al., 2015) denoiser network during generation. Using a pretrained CLIP (Contrastive Language-Image Pre-Training) (Radford et al., 2021) model to define directions within the inner-most feature map of the UNet architecture, named h-space, they show that some semantic edits, such as adding glasses to an image of a person, correspond to linear edits in the h-space. More recently, Haas et al. (2024) demonstrate that h-space edit directions can also be found through Principle Component Analysis, and Park et al. (2023) propose a pullback metric that transfers edits in the h-space into the original latent space. However, while these three approaches demonstrate impressive editing capabilities on images, they all require a modification to the generative process and are limited to diffusion models built with UNet architectures. In our work low-dimensional representations are instead formed in closed-form as linear subspaces based on a (free) choice of latent basis vectors. These basis vectors could be obtained through various methodologies, including via the pull-back metric in Park et al. (2023), which may be interesting to explore in future work.
+
+# 7 CONCLUSION
+
+In this paper we propose LOL, a general and simple scheme to define linear combinations of latents that maintain a prespecified latent distribution and demonstrate its effectiveness for interpolation and defining subspaces within the latent spaces of generative models. Of course, these linear combinations only follow the desired distribution if the original (seed) latents do. Therefore, we propose the adoption of distribution tests to assess the validity of latents obtained from e.g. inversions. Still, the existing methods for distribution testing may not align perfectly with a given network's expectations — e.g. many tests may be stricter than necessary for the network — and it may be an interesting direction of future work to develop tailored methods, along with more extensive comparisons of existing tests in the context of generative models.
+
+# ACKNOWLEDGMENTS
+
+We thank Samuel Willis and Daattavya Aggarwal (University of Cambridge) for their valuable contributions. Samuel contributed important ideas related to computational fluid dynamics simulation and the SLIDE model, and constructed the setup and plots used in Figure 7. Daattavya assisted in verifying proofs and offered comments that greatly improved the manuscript.
+
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+
+# A OPTIMAL TRANSPORT MAP FOR GENERAL LATENT LINEAR COMBINATIONS WITH INDEPENDENT ELEMENTS
+
+In Section 4 we introduced LOL for Gaussian latent variables, the most commonly used latent distribution in modern generative models. We now extend LOL to general latent distributions with independent real-valued elements, provided the cumulative distribution function (CDF) and its inverse is known for each respective element.
+
+Let  $p(\boldsymbol{x})$  be a distribution with independent components across elements, such that  $p(\boldsymbol{x}) = \prod_{d=1}^{D} p_d(x^{(d)})$ , where  $p_d$  is the distribution of element  $x^{(d)} \in \mathbb{R}$  with CDF  $\Phi_d$  and inverse CDF  $\Phi_d^{-1}$ .
+
+Given linear combination weights  $\{w_k\}_{k=1}^K$ ,  $w_k \in \mathbb{R}$ , and seeds  $\{\pmb{x}_k\}_{k=1}^K$ ,  $\pmb{x}_k \sim p$ , we obtain  $z \sim p$  via the transport map  $\mathcal{T}_{\{w\}} := \{\mathcal{T}_{\{w\}}^{(d)}\}_{d=1}^D$  defined per latent element  $z^{(d)}$  as:
+
+$$
+z ^ {(d)} := \mathcal {T} _ {\{w \}} ^ {(d)} \left(\left\{x _ {k} ^ {(d)} \right\}\right) = \Phi_ {d} ^ {- 1} \left(\Phi_ {\epsilon} \left(\epsilon^ {(d)}\right)\right) \quad \epsilon^ {(d)} = \mathcal {T} _ {\boldsymbol {w}, 0} \left(\sum_ {k = 1} ^ {K} w _ {k} \epsilon_ {k} ^ {(d)}\right) \quad \epsilon_ {k} ^ {(d)} = \Phi_ {\epsilon} ^ {- 1} \left(\Phi_ {d} \left(x _ {k} ^ {(d)}\right)\right), \tag {4}
+$$
+
+where  $\Phi_{\epsilon}$  is the CDF of the standard Gaussian, and  $\mathcal{T}_{w,0}:\mathbb{R}\to \mathbb{R}$  is the transport map for standard Gaussians as defined in Equation 3. We will now show that the proposed map leads to the target distribution  $p$  and then show that this map is Monge optimal, leveraging results from Section B.
+
+Lemma 1. Let  $\mathbf{z}$  be a transformed variable defined through the transport map  $\mathcal{T}_{\{w\}}$  applied to a linear combination of independent latent variables  $\mathbf{x}_k \sim p(\mathbf{x})$ , such that
+
+$$
+z ^ {(d)} = \mathcal {T} _ {\{w \}} ^ {(d)} (\{x _ {k} ^ {(d)} \}) = \Phi_ {d} ^ {- 1} (\Phi_ {\epsilon} (\mathcal {T} _ {\boldsymbol {w}, 0} (\sum_ {k = 1} ^ {K} w _ {k} \epsilon_ {k} ^ {(d)})))
+$$
+
+where
+
+$$
+\epsilon_ {k} ^ {(d)} = \Phi_ {\epsilon} ^ {- 1} (\Phi_ {d} (x _ {k} ^ {(d)})).
+$$
+
+Then,  $z \sim p(x)$ , meaning that the transformed variable follows the target latent distribution.
+
+Proof. We prove that  $z \sim p(\pmb{x})$  by verifying that each element  $z^{(d)}$  follows the corresponding marginal distribution  $p_d$ .
+
+Since  $x_{k}^{(d)} \sim p_{d}$ , its cumulative distribution function (CDF) is given by  $\Phi_d(x_k^{(d)})$ . Defining the standard normal equivalent  $\epsilon_{k}^{(d)} = \Phi_{\epsilon}^{-1}(\Phi_{d}(x_{k}^{(d)}))$ , we know that  $\epsilon_{k}^{(d)} \sim \mathcal{N}(0,1)$  because the transformation via  $\Phi_{\epsilon}^{-1}$  preserves uniformity.
+
+Now, consider the weighted sum:
+
+$$
+\epsilon^ {(d)} = \sum_ {k = 1} ^ {K} w _ {k} \epsilon_ {k} ^ {(d)}.
+$$
+
+Since the  $\epsilon_k^{(d)}$  are independent standard normal variables, the resulting sum follows:
+
+$$
+\epsilon^ {(d)} \sim \mathcal {N} (\alpha \cdot 0, \beta \cdot 1) = \mathcal {N} (0, \beta),
+$$
+
+where  $\alpha = \sum_{k=1}^{K} w_k$  and  $\beta = \sum_{k=1}^{K} w_k^2$ .
+
+Applying the optimal Gaussian transport map  $\mathcal{T}_{\boldsymbol{w},0}$  (as proven in Section B) to  $\epsilon^{(d)}$ , we obtain:
+
+$$
+\mathcal {T} _ {\boldsymbol {w}, 0} (\epsilon^ {(d)}) = \frac {\epsilon^ {(d)}}{\sqrt {\beta}}.
+$$
+
+Since  $\epsilon^{(d)}\sim \mathcal{N}(0,\beta)$ , it follows that:
+
+$$
+\frac {\epsilon^ {(d)}}{\sqrt {\beta}} \sim \mathcal {N} (0, 1).
+$$
+
+Thus, applying the standard normal CDF, we get:
+
+$$
+\Phi_ {\epsilon} (\mathcal {T} _ {\boldsymbol {w}, 0} (\epsilon^ {(d)})) \sim U (0, 1),
+$$
+
+which is a uniform distribution.
+
+Finally, applying the inverse CDF  $\Phi_d^{-1}$  of the target distribution  $p_d$  results in:
+
+$$
+z ^ {(d)} = \Phi_ {d} ^ {- 1} \left(\Phi_ {\epsilon} \left(\mathcal {T} _ {\boldsymbol {w}, 0} \left(\epsilon^ {(d)}\right)\right)\right) \sim p _ {d}.
+$$
+
+Since this holds for each  $d$ , we conclude that  $z \sim p(\pmb{x})$ , proving that the transport map correctly transforms the linear combination into the desired latent distribution.
+
+Lemma 2. The transport map  $\mathcal{T}_{\{w\}}$  defined element-wise as
+
+$$
+z ^ {(d)} = \mathcal {T} _ {\{w \}} ^ {(d)} (\{x _ {k} ^ {(d)} \}) = \Phi_ {d} ^ {- 1} (\Phi_ {\epsilon} (\mathcal {T} _ {\boldsymbol {w}, 0} (\sum_ {k = 1} ^ {K} w _ {k} \epsilon_ {k} ^ {(d)})))
+$$
+
+where
+
+$$
+\epsilon_ {k} ^ {(d)} = \Phi_ {\epsilon} ^ {- 1} (\Phi_ {d} (x _ {k} ^ {(d)})),
+$$
+
+is the Monge optimal transport map, minimizing the quadratic Wasserstein distance  $W_{2}$  between the source distribution of the weighted sum of independent elements and the target latent distribution  $p$ .
+
+Proof. We want to show that  $\mathcal{T}_{\{w\}}$  minimizes the quadratic Wasserstein distance  $W_{2}$  between the source distribution (the weighted sum of independent seeds) and the target distribution.
+
+Step 1: General Theorem for One-Dimensional Optimal Transport For two probability measures  $P$  and  $Q$  on  $\mathbb{R}$  with cumulative distribution functions (CDFs)  $F_{P}$  and  $F_{Q}$ , respectively, the optimal transport map in the  $W_{2}$  sense (which minimizes the expected squared Euclidean distance) is given by the monotone increasing function:
+
+$$
+\mathcal {T} ^ {*} = F _ {Q} ^ {- 1} \circ F _ {P}.
+$$
+
+This result follows from classical optimal transport theory for univariate distributions (see Remark 2.30 in Peyre & Cuturi (2020)).
+
+Step 2: Applying to Our Case For each element  $d$ , the source distribution is given by the weighted sum:
+
+$$
+\epsilon^ {(d)} = \sum_ {k = 1} ^ {K} w _ {k} \epsilon_ {k} ^ {(d)}.
+$$
+
+Each  $\epsilon_k^{(d)}$  is obtained by transforming the original variable  $x_{k}^{(d)}$  using:
+
+$$
+\epsilon_ {k} ^ {(d)} = \Phi_ {\epsilon} ^ {- 1} (\Phi_ {d} (x _ {k} ^ {(d)})).
+$$
+
+Since the  $\epsilon_k^{(d)}$  are independent standard normal variables, their weighted sum follows:
+
+$$
+\epsilon^ {(d)} \sim \mathcal {N} (0, \beta), \quad \text {w h e r e} \quad \beta = \sum_ {k = 1} ^ {K} w _ {k} ^ {2}.
+$$
+
+To map this distribution to the standard normal  $\mathcal{N}(0,1)$ , the optimal transport map for Gaussians (derived in Section B) is:
+
+$$
+\mathcal {T} _ {\boldsymbol {w}, 0} (\epsilon^ {(d)}) = \frac {\epsilon^ {(d)}}{\sqrt {\beta}}.
+$$
+
+Applying the standard normal CDF  $\Phi_{\epsilon}$  to this transformation ensures that:
+
+$$
+\Phi_ {\epsilon} \left(\mathcal {T} _ {\boldsymbol {w}, 0} \left(\epsilon^ {(d)}\right)\right) \sim U (0, 1).
+$$
+
+By the general optimal transport theorem stated in Step 1, the optimal transport map to the target distribution  $p_d$  is then:
+
+$$
+z ^ {(d)} = \Phi_ {d} ^ {- 1} \left(\Phi_ {\epsilon} \left(\mathcal {T} _ {\boldsymbol {w}, 0} \left(\epsilon^ {(d)}\right)\right)\right).
+$$
+
+Since each element  $d$  is mapped independently using the element-wise Monge optimal transport map, the full transformation  $\mathcal{T}_{\{w\}}$  is optimal in the Monge sense, minimizing the quadratic Wasserstein distance  $W_{2}$ .
+
+Thus,  $\mathcal{T}_{\{w\}}$  is the Monge optimal transport map from the distribution of the weighted sum of independent elements to the target latent distribution  $p$ , completing the proof.
+
+# B OPTIMAL TRANSPORT MAP FOR GAUSSIAN LATENT LINEAR COMBINATIONS
+
+In Section 4 we introduced LOL, which we use to transform random variables arising from linear combinations — for example interpolations or subspace projections — to the latent distribution  $\mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$  via a transport map  $\mathcal{T}_{\boldsymbol{w}, \boldsymbol{\mu}}$  defined per set of linear combination weights  $\boldsymbol{w} = [w_1, \dots, w_K]$ ,  $w_k \in \mathbb{R}$ . We will now derive the distribution of a (uncorrected) linear combination  $\boldsymbol{y}$  — which we show does not match the latent distribution — followed by the distribution of the transformed variable  $\boldsymbol{z}$ , which we show does. We will then show that the proposed transport map  $\mathcal{T}_{\boldsymbol{w}, \boldsymbol{\mu}}$  is Monge optimal.
+
+Lemma 3. Let  $\mathbf{y}$  be a linear combination of  $K$  i.i.d. random variables  $\pmb{x}_k\sim \mathcal{N}(\pmb {\mu},\pmb {\Sigma})$ , where  $\pmb{y}$  is defined as
+
+$$
+\boldsymbol {y} := \sum_ {k = 1} ^ {K} w _ {k} \boldsymbol {x} _ {k} = \boldsymbol {w} ^ {T} \boldsymbol {X},
+$$
+
+with  $w_{k}\in \mathbb{R},\pmb {y}\in \mathbb{R}^{D},\pmb{x}_{k}\in \mathbb{R}^{D},\pmb {w} = [w_{1},w_{2},\dots ,w_{K}],K\in \mathbb{N}_{>0}$  and  $\pmb {X} = [\pmb {x}_1,\pmb {x}_2,\dots ,\pmb {x}_K]$  Then  $\pmb{y}$  is a Gaussian random variable with the distribution
+
+$$
+\boldsymbol {y} \sim \mathcal {N} (\alpha \boldsymbol {\mu}, \beta \boldsymbol {\Sigma}),
+$$
+
+where
+
+$$
+\alpha = \sum_ {k = 1} ^ {K} w _ {k} \quad a n d \quad \beta = \sum_ {k = 1} ^ {K} w _ {k} ^ {2}.
+$$
+
+Proof. Given  $\pmb{x}_k \sim \mathcal{N}(\pmb{\mu}, \pmb{\Sigma})$ , we know that each  $\pmb{x}_k$  has mean  $\pmb{\mu}$  and covariance matrix  $\pmb{\Sigma}$ .
+
+Define  $\pmb{y} = \pmb{w}^T\pmb{X} = \sum_{k=1}^{K} w_k \pmb{x}_k$ .
+
+First, we calculate the mean of  $\pmb{y}$ :
+
+$$
+\mathbb {E} [ \boldsymbol {y} ] = \mathbb {E} \left[ \sum_ {k = 1} ^ {K} w _ {k} \boldsymbol {x} _ {k} \right] = \sum_ {k = 1} ^ {K} w _ {k} \mathbb {E} [ \boldsymbol {x} _ {k} ] = \sum_ {k = 1} ^ {K} w _ {k} \boldsymbol {\mu} = \left(\sum_ {k = 1} ^ {K} w _ {k}\right) \boldsymbol {\mu} = \alpha \boldsymbol {\mu}.
+$$
+
+Next, we calculate the covariance of  $\pmb{y}$ :
+
+$$
+\operatorname {C o v} (\boldsymbol {y}) = \operatorname {C o v} \left(\sum_ {k = 1} ^ {K} w _ {k} \boldsymbol {x} _ {k}\right) = \sum_ {k = 1} ^ {K} w _ {k} ^ {2} \operatorname {C o v} (\boldsymbol {x} _ {k}),
+$$
+
+since the  $\pmb{x}_k$  are i.i.d. and thus  $\mathrm{Cov}(\pmb{x}^{(i)},\pmb{x}^{(j)}) = 0$  for  $i\neq j$ .
+
+Given that  $\pmb{x}_k \sim \mathcal{N}(\pmb{\mu}, \pmb{\Sigma})$ , we have  $\mathrm{Cov}(\pmb{x}_k) = \pmb{\Sigma}$ . Therefore,
+
+$$
+\operatorname {C o v} (\boldsymbol {y}) = \sum_ {k = 1} ^ {K} w _ {k} ^ {2} \boldsymbol {\Sigma} = \left(\sum_ {k = 1} ^ {K} w _ {k} ^ {2}\right) \boldsymbol {\Sigma} = \beta \boldsymbol {\Sigma}.
+$$
+
+Hence,  $\pmb{y} \sim \mathcal{N}(\alpha \pmb{\mu}, \beta \pmb{\Sigma})$  with  $\alpha = \sum_{k=1}^{K} w_k$  and  $\beta = \sum_{k=1}^{K} w_k^2$ .
+
+Lemma 4. Let  $z$  be defined as
+
+$$
+\boldsymbol {z} = \mathcal {T} _ {\boldsymbol {w}, \boldsymbol {\mu}} (\boldsymbol {y}) \quad \mathcal {T} _ {\boldsymbol {w}, \boldsymbol {\mu}} (\boldsymbol {y}) = (1 - \frac {\alpha}{\sqrt {\beta}}) \boldsymbol {\mu} + \frac {\boldsymbol {y}}{\sqrt {\beta}} \quad \alpha = \sum_ {k = 1} ^ {K} w _ {k} \quad \beta = \sum_ {k = 1} ^ {K} w _ {k} ^ {2}
+$$
+
+with  $\pmb{y} \sim \mathcal{N}(\alpha \pmb{\mu}, \beta \pmb{\Sigma})$ . Then  $\pmb{z} \sim \mathcal{N}(\pmb{\mu}, \pmb{\Sigma})$ .
+
+Proof. Given  $\pmb{y} \sim \mathcal{N}(\alpha \pmb{\mu}, \beta \pmb{\Sigma})$ , we need to show that  $z \sim \mathcal{N}(\pmb{\mu}, \pmb{\Sigma})$ .
+
+First, we calculate the mean of  $z$ :
+
+$$
+\mathbb {E} [ z ] = \mathbb {E} [ \boldsymbol {a} + \boldsymbol {B} \boldsymbol {y} ] = \boldsymbol {a} + \boldsymbol {B} \mathbb {E} [ \boldsymbol {y} ] = \left(1 - \frac {\alpha}{\sqrt {\beta}}\right) \boldsymbol {\mu} + \frac {1}{\sqrt {\beta}} \alpha \boldsymbol {\mu} = \boldsymbol {\mu},
+$$
+
+where
+
+$$
+\boldsymbol {a} = \left(1 - \frac {\alpha}{\sqrt {\beta}}\right) \boldsymbol {\mu} \quad \text {a n d} \quad \boldsymbol {B} = \frac {1}{\sqrt {\beta}} \boldsymbol {I}.
+$$
+
+Next, we calculate the covariance of  $z$ :
+
+$$
+\operatorname {C o v} (\boldsymbol {z}) = \operatorname {C o v} (\boldsymbol {a} + \boldsymbol {B} \boldsymbol {y}) = \boldsymbol {B} \operatorname {C o v} (\boldsymbol {y}) \boldsymbol {B} ^ {T} = \frac {1}{\sqrt {\beta}} \beta \boldsymbol {\Sigma} \frac {1}{\sqrt {\beta}} = \boldsymbol {\Sigma}.
+$$
+
+Since  $z$  has mean  $\mu$  and covariance  $\Sigma$ , we conclude that
+
+$$
+z \sim \mathcal {N} (\boldsymbol {\mu}, \boldsymbol {\Sigma}).
+$$
+
+![](images/a815497e054927a2cb2d9fe748801b38898d703c7e221bd9079d220671b81bfd.jpg)
+
+Lemma 5. The transport map  $\mathcal{T}_{w,\mu}$  defined by
+
+$$
+\mathcal {T} _ {\boldsymbol {w}, \boldsymbol {\mu}} (\boldsymbol {y}) = (1 - \frac {\alpha}{\sqrt {\beta}}) \boldsymbol {\mu} + \frac {\boldsymbol {y}}{\sqrt {\beta}}
+$$
+
+is the Monge optimal transport map, minimizing the quadratic Wasserstein distance  $W_{2}$  between  $\mathcal{N}(\alpha \pmb{\mu}, \beta \pmb{\Sigma})$  and  $\mathcal{N}(\pmb{\mu}, \pmb{\Sigma})$ .
+
+Proof. In optimal transport theory, the Monge problem seeks a transport map  $\mathcal{T}:\mathbb{R}^D\to \mathbb{R}^D$  that pushes forward the source distribution  $\bar{\mathcal{N}} (\pmb {\mu}_y,\pmb {\Sigma}_y)$  to the target distribution  $\mathcal{N}(\pmb {\mu}_z,\pmb {\Sigma}_z)$  while minimizing the expected squared Euclidean distance:
+
+$$
+W _ {2} ^ {2} \left(\mathcal {N} \left(\boldsymbol {\mu} _ {y}, \boldsymbol {\Sigma} _ {y}\right), \mathcal {N} \left(\boldsymbol {\mu} _ {z}, \boldsymbol {\Sigma} _ {z}\right)\right) = \inf  _ {\mathcal {T} \in \mathcal {M}} \mathbb {E} \left[ \| \mathcal {T} (\boldsymbol {y}) - \boldsymbol {y} \| ^ {2} \right].
+$$
+
+The optimal transport map for Gaussian distributions is given by the affine transformation:
+
+$$
+\mathcal {T} (\boldsymbol {y}) = \boldsymbol {\mu} _ {z} + \boldsymbol {A} (\boldsymbol {y} - \boldsymbol {\mu} _ {y}),
+$$
+
+where  $A$  is defined as
+
+$$
+\boldsymbol {A} := \boldsymbol {\Sigma} _ {y} ^ {- 1 / 2} (\boldsymbol {\Sigma} _ {y} ^ {1 / 2} \boldsymbol {\Sigma} _ {z} \boldsymbol {\Sigma} _ {y} ^ {1 / 2}) ^ {1 / 2} \boldsymbol {\Sigma} _ {y} ^ {- 1 / 2}.
+$$
+
+See Remark 2.31 in Peyre & Cuturi (2020).
+
+Setting the parameters
+
+$$
+\boldsymbol {\mu} _ {y} = \alpha \boldsymbol {\mu}, \quad \boldsymbol {\Sigma} _ {y} = \beta \boldsymbol {\Sigma}, \tag {5}
+$$
+
+$$
+\boldsymbol {\mu} _ {z} = \boldsymbol {\mu}, \quad \boldsymbol {\Sigma} _ {z} = \boldsymbol {\Sigma}, \tag {6}
+$$
+
+we compute  $A$ :
+
+$$
+\boldsymbol {A} = (\beta \boldsymbol {\Sigma}) ^ {- 1 / 2} \left(\left(\beta \boldsymbol {\Sigma}\right) ^ {1 / 2} \boldsymbol {\Sigma} \left(\beta \boldsymbol {\Sigma}\right) ^ {1 / 2}\right) ^ {1 / 2} \left(\beta \boldsymbol {\Sigma}\right) ^ {- 1 / 2} = \frac {1}{\sqrt {\beta}} \boldsymbol {I}. \tag {7}
+$$
+
+Thus, the optimal transport map simplifies to
+
+$$
+z := \mathcal {T} (\boldsymbol {y}) = \boldsymbol {\mu} + \frac {1}{\sqrt {\beta}} (\boldsymbol {y} - \alpha \boldsymbol {\mu}) = (1 - \frac {\alpha}{\sqrt {\beta}}) \boldsymbol {\mu} + \frac {\boldsymbol {y}}{\sqrt {\beta}}. \tag {8}
+$$
+
+Since this matches exactly with  $\mathcal{T}_{w,\mu}$ , we conclude that  $\mathcal{T}_{w,\mu}$  is the Monge optimal transport map minimising the quadratic Wasserstein distance.
+
+![](images/dc1c5f6c8b02eec2652726be8ac7985f431feeef16b06fb422cfd198c530bcee.jpg)
+
+# C SPHERICAL INTERPOLANTS OF HIGH-DIMENSIONAL UNIT GAUSSIAN RANDOM VARIABLES ARE APPROXIMATELY UNIT GAUSSIAN
+
+Spherical linear interpolation (SLERP) (Shoemake, 1985) is defined as
+
+$$
+\boldsymbol {y} = w _ {1} \boldsymbol {x} _ {1} + w _ {2} \boldsymbol {x} _ {2} \tag {9}
+$$
+
+where
+
+$$
+w _ {i} := \frac {\sin v _ {i} \theta}{\sin \theta}, \tag {10}
+$$
+
+$v_{i}\in [0,1]$  and  $v_{2} = 1 - v_{1}$  and  $\cos \theta = \frac{\langle\pmb{x}_1,\pmb{x}_2\rangle}{||\pmb{x}_1||||\pmb{x}_2||}$ , where  $\cos \theta$  is typically referred to as the cosine similarity of  $\pmb{x}_{1}$  and  $\pmb{x}_{2}$ .
+
+As such, using Equation 2, we obtain
+
+$$
+\alpha = \sum_ {k = 1} ^ {K} w _ {k} = \frac {\sin v _ {1} \theta}{\sin \theta} + \frac {\sin v _ {2} \theta}{\sin \theta} \quad \text {a n d} \quad \beta = \frac {\sin^ {2} v _ {1} \theta}{\sin^ {2} \theta} + \frac {\sin^ {2} v _ {2} \theta}{\sin^ {2} \theta} \tag {11}
+$$
+
+As discussed in Section 4, for a linear combination  $\pmb{y}$  to be a random variable following distribution  $\mathcal{N}(\pmb{\mu},\pmb{\Sigma})$ , given that  $\pmb{x}_1$  and  $\pmb{x}_2$  do, it must be true that  $\alpha \pmb{\mu} = \pmb{\mu}$  and  $\beta \pmb{\Sigma} = \pmb{\Sigma}$ .
+
+A common case is using unit Gaussian latents (as in e.g. the models Esser et al. (2024) and Rombach et al. (2022) used in this paper), i.e. where  $\pmb{\mu} = \mathbf{0}$ ,  $\pmb{\Sigma} = \pmb{I}$ . In this case it trivially follows that  $\alpha \pmb{\mu} = \pmb{\mu}$  since  $\alpha \mathbf{0} = \mathbf{0}$ . We will now show that  $\beta \approx 1$  in this special (i.e. unit Gaussian) case.
+
+Lemma 6. Let
+
+$$
+\beta = \frac {\sin^ {2} v \theta}{\sin^ {2} \theta} + \frac {\sin^ {2} (1 - v) \theta}{\sin^ {2} \theta},
+$$
+
+where  $\cos \theta = \frac{\langle\pmb{x}_1,\pmb{x}_2\rangle}{\|\pmb{x}_1\|\|\pmb{x}_2\|}$  and  $\pmb {x}_1,\pmb {x}_2\sim \mathcal{N}(\pmb {0},\pmb {I})$  . Then  $\beta \approx 1$  for large  $D,\forall v\in [0,1]$
+
+Proof. Since  $\pmb{x}_1, \pmb{x}_2 \sim \mathcal{N}(\pmb{0}, \pmb{I})$ , each component  $x_{1j}$  and  $x_{2j}$  for  $j = 1, \dots, D$  are independent standard normal random variables. The inner product  $\langle \pmb{x}_1, \pmb{x}_2 \rangle$  is given by:
+
+$$
+\langle \boldsymbol {x} _ {1}, \boldsymbol {x} _ {2} \rangle = \sum_ {j = 1} ^ {D} x _ {1 j} x _ {2 j}.
+$$
+
+The product  $x_{1j}x_{2j}$  follows a distribution known as the standard normal product distribution. For large  $D$ , the sum of these products is approximately normal due to the Central Limit Theorem (CLT), with:
+
+$$
+\langle \boldsymbol {x} _ {1}, \boldsymbol {x} _ {2} \rangle \sim \mathcal {N} (0, D).
+$$
+
+Next, consider the norms  $\| \pmb{x}_1\|$  and  $\| \pmb{x}_2\|$ . Each  $\| \pmb{x}_i\|^2 = \sum_{j=1}^{D} x_{ij}^2$  is a chi-squared random variable with  $D$  degrees of freedom. For large  $D$ , by the central limit theorem,  $\| \pmb{x}_i\|^2 \sim \mathcal{N}(D, 2D)$ , and therefore  $\| \pmb{x}_i\|$  is approximately  $\sqrt{D}$ .
+
+Thus, for large  $D$
+
+$$
+\cos (\theta) = \frac {\langle \boldsymbol {x} _ {1} , \boldsymbol {x} _ {2} \rangle}{\| \boldsymbol {x} _ {1} \| \| \boldsymbol {x} _ {2} \|} \approx \frac {\mathcal {N} (0 , D)}{\sqrt {D} \cdot \sqrt {D}} = \frac {\mathcal {N} (0 , D)}{D} = \mathcal {N} \left(0, \frac {1}{D}\right).
+$$
+
+Thus,  $\theta \approx \pi /2$ , which implies  $\sin (\theta)\approx 1$ . Therefore:
+
+$$
+\beta = \frac {\sin^ {2} (v \theta)}{\sin^ {2} \theta} + \frac {\sin^ {2} ((1 - v) \theta)}{\sin^ {2} \theta} \approx \sin^ {2} (v \theta) + \sin^ {2} ((1 - v) \theta).
+$$
+
+Using the identity  $\sin^2 (a) + \sin^2 (b) = 1 - \cos^2 (a - b)$ ,
+
+$$
+\beta \approx 1 - \cos^ {2} (v \theta - (1 - v) \theta) = 1 - \cos^ {2} (\theta)
+$$
+
+Given  $v \in [0,1]$  and  $\theta \approx \pi / 2$  for large  $D$ , the argument of  $\cos(\cdot) \approx 0$ . Hence,
+
+$$
+\beta \approx 1.
+$$
+
+Therefore, for large  $D$ , it follows that  $\beta \approx 1$ .
+
+![](images/cfab57935cc3ff5510a25beba97b742127b9fc9984a0d9d9e93f7e2a1f82f9e0.jpg)
+
+Empirical verification To confirm the effectiveness of the approximations used above for values of  $D$  which are typical for popular generative models that use unit Gaussian latents, we estimate the confidence interval of  $\beta$  using 10k samples of  $x_{1}$  and  $x_{2}$ , respectively, where  $x_{i} \sim \mathcal{N}(0, I)$ . For Stable Diffusion 3 (Esser et al., 2024), a flow matching model with  $D = 147456$ , the estimated  $99\%$  confidence interval of  $\beta$  is [0.9934, 1.0067] for  $v = 0.5$  where the error is largest. For Stable Diffusion 2.1 (Rombach et al., 2022), a diffusion model with  $D = 36864$ , the corresponding confidence interval of  $\beta$  is [0.9868, 1.014].
+
+# D LINEAR COMBINATION WEIGHTS OF SUBSPACE PROJECTIONS
+
+In Section 4 we introduced LOL, which we use to transform latent variables arising from linear combinations such that they maintain the the latent distribution. This transformation depends on the weights  $w$  which specify the linear combination. We will now derive the linear combination weights for subspace projections.
+
+Lemma 7. Let  $\mathbf{U} \in \mathbb{R}^{D \times K}$  be a semi-orthonormal matrix. For a given point  $\mathbf{x} \in \mathbb{R}^D$ , the subspace projection is  $s(\mathbf{x}) = \mathbf{U} \mathbf{U}^T \mathbf{x}$ . The weights  $\mathbf{w} \in \mathbb{R}^K$  such that  $s(\mathbf{x})$  is a linear combination of  $\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_K$  (columns of  $\mathbf{A} = [\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_K] \in \mathbb{R}^{D \times K}$ ) can be expressed as  $\mathbf{w} = \mathbf{A}_{\dagger} s(\mathbf{x})$ , where  $\mathbf{A}_{\dagger}$  is the Moore-Penrose inverse of  $\mathbf{A}$ .
+
+Proof. The subspace projection  $s(\pmb{x})$  of  $\pmb{x} \in \mathbb{R}^D$  is defined as:
+
+$$
+s (\boldsymbol {x}) = \boldsymbol {U} \boldsymbol {U} ^ {T} \boldsymbol {x}.
+$$
+
+We aim to express  $s(\pmb{x})$  as a linear combination of the columns of  $\pmb{A} = [\pmb{x}_1, \pmb{x}_2, \dots, \pmb{x}_K] \in \mathbb{R}^{D \times K}$ . That is, we seek  $\pmb{w} \in \mathbb{R}^K$  such that:
+
+$$
+s (\boldsymbol {x}) = \boldsymbol {A} \boldsymbol {w}.
+$$
+
+By definition, the Moore-Penrose inverse  $A_{\dagger}$  of  $\pmb{A}$  satisfies the following properties:
+
+1.  $AA_{\dagger}A = A$  
+2.  $A_{\dagger}AA_{\dagger} = A_{\dagger}$
+
+3.  $(\mathbf{A}\mathbf{A}_{\dagger})^{T} = \mathbf{A}\mathbf{A}_{\dagger}$  
+4.  $(\mathbf{A}_{\dagger}\mathbf{A})^{T} = \mathbf{A}_{\dagger}\mathbf{A}$
+
+Since  $s(\pmb{x})$  is in the subspace spanned by the columns of  $\pmb{A}$ , there exists a  $\pmb{w}$  such that:
+
+$$
+s (\boldsymbol {x}) = \boldsymbol {A} \boldsymbol {w}.
+$$
+
+Consider a  $\pmb{w}^{\prime} \in \mathbb{R}^{K}$  constructed using the Moore-Penrose inverse  $A_{\dagger}$ :
+
+$$
+\boldsymbol {w} ^ {\prime} = \boldsymbol {A} _ {\dagger} s (\boldsymbol {x}).
+$$
+
+We now verify that this  $\boldsymbol{w}'$  satisfies the required equation. Substituting back
+
+$$
+\boldsymbol {A} \boldsymbol {w} ^ {\prime} = \boldsymbol {A} (\boldsymbol {A} _ {\dagger} s (\boldsymbol {x}))
+$$
+
+and using the property of the Moore-Penrose inverse  $AA_{\dagger}A = A$  , we get:
+
+$$
+A A _ {\dagger} s (\pmb {x}) = s (\pmb {x}).
+$$
+
+Thus:
+
+$$
+\boldsymbol {A} \boldsymbol {w} ^ {\prime} = s (\boldsymbol {x}),
+$$
+
+which shows that  $\pmb{w}^{\prime} = \pmb{A}_{\dagger}s(\pmb {x})$  is indeed the correct expression for the weights.
+
+From uniqueness of  $\pmb{w}$  for a given set of columns  $x_{1}, x_{2}, \ldots, x_{K}$  (see Lemma 8), we have proven that the weights  $\pmb{w}$  for a given point in the subspace  $s(\pmb{x})$  are given by:
+
+$$
+\boldsymbol {w} = \boldsymbol {A} _ {\dagger} s (\boldsymbol {x}).
+$$
+
+![](images/42427acec7bd90f4efac9b4889331c7683f94f6a87d69bb7b9749d840fd533ae.jpg)
+
+Lemma 8. The weights  $\pmb{w} \in \mathbb{R}^{K}$  such that  $s(\pmb{x})$  is a linear combination of  $\pmb{x}_1, \pmb{x}_2, \dots, \pmb{x}_K$  (columns of  $\pmb{A} = [\pmb{x}_1, \pmb{x}_2, \dots, \pmb{x}_K] \in \mathbb{R}^{D \times K}$ ) are unique.
+
+Proof. Suppose there exist two different weight vectors  $\pmb{w}_1$  and  $\pmb{w}_2$  such that both satisfy the equation:
+
+$$
+s (\boldsymbol {x}) = \boldsymbol {A} \boldsymbol {w} _ {1} = \boldsymbol {A} \boldsymbol {w} _ {2}.
+$$
+
+Then, subtracting these two equations gives:
+
+$$
+\boldsymbol {A} \boldsymbol {w} _ {1} - \boldsymbol {A} \boldsymbol {w} _ {2} = \boldsymbol {0}.
+$$
+
+This simplifies to:
+
+$$
+\boldsymbol {A} \left(\boldsymbol {w} _ {1} - \boldsymbol {w} _ {2}\right) = \boldsymbol {0}.
+$$
+
+Let  $\pmb{v} = \pmb{w}_1 - \pmb{w}_2$ . Then:
+
+$$
+\boldsymbol {A} \boldsymbol {v} = \boldsymbol {0}.
+$$
+
+Since  $\pmb{v} \in \mathbb{R}^{K}$ , this equation implies that  $\pmb{v}$  lies in the null space of  $\pmb{A}$ . However, the assumption that  $\pmb{A}$  has full column rank (since  $\pmb{A}$  is used to represent a linear combination for  $s(\pmb{x})$ ) implies that  $\pmb{A}$  has no non-zero vector in its null space, i.e.,  $\pmb{A}\pmb{v} = \pmb{0}$  only when  $\pmb{v} = \pmb{0}$ .
+
+Therefore:
+
+$$
+\boldsymbol {v} = \boldsymbol {0} \Rightarrow \boldsymbol {w} _ {1} = \boldsymbol {w} _ {2}.
+$$
+
+This shows that the weights  $\boldsymbol{w}$  are unique, and there cannot be two distinct sets of weights  $\boldsymbol{w}_1$  and  $\boldsymbol{w}_2$  that satisfy the equation  $s(\boldsymbol{x}) = \boldsymbol{A}\boldsymbol{w}$ .
+
+Hence, we conclude that the weights  $\pmb{w}$  such that  $s(\pmb{x})$  is a linear combination of  $\pmb{x}_1, \pmb{x}_2, \dots, \pmb{x}_K$  are unique.
+
+# E NORMALITY TEST COMPARISON
+
+We will now investigate the effectiveness of a range of classic normality tests and tests based on the likelihood  $\mathcal{N}(\pmb{x}_*;\pmb{\mu},\pmb{\Sigma})$  as well as the likelihood of the norm statistic. We assess these methods across a suite of test cases including for vectors with unlikely characteristics and which we know, through failed generation, violate the assumptions of the model. In order to improve statistical power and detect even small deviations from normality of a single instance  $\pmb{x}_* \in \mathbb{R}^D$ , we transform the single high-dimensional latent into a large collection of  $(D)$  i.i.d. unit Gaussian samples  $\{\epsilon_i\}_{i=1}^D$ , and test the hypothesis  $\epsilon_1, \ldots, \epsilon_d \sim \mathcal{N}(0,1)$  where  $\epsilon = \pmb{\Sigma}^{-1}(\pmb{x} - \pmb{\mu}), \epsilon = [\epsilon_1, \dots, \epsilon_d]$ . Note that rejecting the hypothesis  $\epsilon_1, \dots, \epsilon_d \sim \mathcal{N}(0,1)$  corresponds to rejecting the hypothesis that  $\pmb{x}_* \sim \mathcal{N}(\pmb{\mu},\pmb{\Sigma})$ .
+
+We compare popular alternatives for normality testing that assume a known mean and variance and that are applicable to very large sample sizes, including Kolmogorov-Smirnov (Kolmogorov, 1933), Shapiro-Wilk (Shapiro & Wilk, 1965), Chi-square (Pearson, 1900), and Cramér-von Mises (Cramér, 1928). We assess these methods on a suite of test cases of vectors which are extremely unlikely under the model's latent distribution and which we know through failed generation violates the characteristic assumptions of the diffusion model, as well as assess the "positive case" where the sample does follow the latent distribution,  $\boldsymbol{x}_{*} \sim \mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ . We report the [0.1%, 99.9%]-confidence interval of the p-value produced by each respective method on  $1e^{4}$  random samples in the stochastic test cases. We also include the likelihood  $\mathcal{N}(\boldsymbol{x}_{*}; \boldsymbol{\mu}, \boldsymbol{\Sigma})$  and the likelihood of the norm statistic.
+
+The results are reported in Table 2. We find that Kolmogorov-Smirnov and Cramér-von Mises both succeed in reporting a lower 99.9th percentile p-value for each tested failure case than the 0.1th percentile assigned to real samples. Kolmogorov-Smirnov did so with the largest gap, reporting extremely low p-values for all failure cases while reporting calibrated p-values for the positive case. Shapiro-Wilk, which in the literature often is regarded as one of the most powerful tests (Razali et al., 2011), did not work well in our context in contrast to the other normality testing methods, as it produced high p-values also for several of the failure cases. An explanation may be that in our context we have sample sizes of tens of thousands (same as the dimensionality of the latent  $D$ ), while comparisons in the literature typically focuses on smaller sample sizes, such as up to hundreds or a few thousand (Razali et al., 2011). The Chi-square test is a test for discrete distributions. However, it is commonly used on binned data (Larsen & Marx, 2005) to approximate a continuous distribution as discrete. We do this, using histograms of 30 bins. This test successfully produced very low p-values for each failure case, but also did so for many valid samples, yielding a  $0.1\%$  of 0. The likelihood  $\mathcal{N}(\boldsymbol{x}_*; \boldsymbol{\mu}, \boldsymbol{\Sigma})$  assigns high likelihoods to vectors  $\boldsymbol{x}_*$  near the mode irrespective of its characteristics. We note that it fails to distinguish to failure cases from real samples, and in some cases assigns much higher likelihood to the failure cases than real data. The insufficiency of the likelihood to describe the feasibility of samples is a phenomenon present in cases of high-dimensionality and low joint dependence between these dimensions; this is due to the quickly shrinking concentration of points near the centre (or close to  $\boldsymbol{\mu}$ , where the likelihood is highest) as the dimensionality is increased (Talagrand, 1995; Ash, 2012; Nalisnick et al., 2019). The norm statistic we note successfully assigns low likelihoods to some failure cases, where the norm is atypical, but fails to do so in most of the tested cases.
+
+In summary, we find that Kolmogorov-Smirnov and Cramér-von Mises both correctly rejects the failure cases (by assigning low p-values) while reporting calibrated p-values for real samples, with Kolmogorov-Smirnov doing so with the lowest p-values for all failure cases.
+
+# F INTERPOLATION AND CENTROID DETERMINATION SETUP DETAILS
+
+Baselines For the application of interpolation, we compare to linear interpolation (LERP), spherical linear interpolation (SLERP), and Norm-Aware Optimization (NAO) (Samuel et al., 2023), a recent approach which considers the norm of the noise vectors. In contrast to the other approaches which only involve closed-form expressions, NAO involves a numerical optimization scheme based on a discretion of a line integral. For the application of centroid determination we compare to NAO, the Euclidean centroid  $\bar{\pmb{x}} = \frac{1}{K}\sum_{k = 1}^{K}\pmb{x}_k$ , and two transformations of the Euclidean centroid; "standardised Euclidean", where  $\bar{\pmb{x}}$  is subsequently standardised to have mean zero and unit variance, and "mode norm Euclidean", where  $\bar{\pmb{x}}$  is rescaled to have the norm equal to the (square root of the) mode
+
+![](images/5e4d14276e33d4a8376ab21e74fa92d2157785ca53d565e450bd029d1da7722c.jpg)  
+Table 2: Normality testing of latents Reported is the  $[0.1\%, 99.9\%]$ -confidence interval of the p-value produced by each respective normality testing method on  $1e^4$  random samples in the stochastic test cases. We also report the log likelihood of the vector  $\pmb{x}^{(i)}$  under the latent distribution and the likelihood of the norm statistic. The diffusion model in Rombach et al. (2022) is used in this example. Green colour for the respective failure test case indicate the method assigning a lower p-value or likelihood to the failure case to at least  $99.9\%$  of the failure samples than the lowest  $0.1\%$  of real samples, to illustrate that it was successful in distinguishing the failure samples. Chi-square assigns low p-values to many real samples as well, which we indicate in red. The images show the generated image from a random failure sample of the test case. In (failure) Test 1 in column 1 the latent is  $\mathbf{0}$ , the mode of the latent distribution for this model  $\mathcal{N}(\mathbf{0}, \mathbf{I})$ . In Test 2 it is the constant vector with the maximum likelihood norm of the norm sampling distribution. In Test 3 the constant vector is instead normalised to have its first two moments matching the latent's marginal distribution.  $n(\cdot)$  indicate that the vector(s) of the test case is normalised to have zero mean and unit variance. In Test 4 a linspace vector  $[-1, \dots, 1]^D$  have been similarly normalised. In Test 5 and 6 the normal samples (from the left most column) have been transformed with scaling and a bias, respectively. In Test 7 random selections of  $1\%$  of the dimensions  $[1, \dots, D]$  of the normal samples have been repeated (100 times) and subsequently normalised by  $n(\cdot)$ . In Test 8 and 9 a proportion of the dimensions of the normal samples have been multiplied by 5 (with indices selected randomly) and the whole vectors are subsequently normalised by  $n(\cdot)$ , where the proportion is  $5\%$  and  $1\%$ , respectively.
+
+of  $\chi^2(D)$ , the chi-squared distribution with  $D$  degrees of freedom, which is the maximum likelihood norm given that  $\pmb{x}$  has been generated from a unit Gaussian with  $D$  dimensions.
+
+Evaluation sets We closely follow the evaluation protocol in Samuel et al. (2023), where we base the experiments on Stable Diffusion 2 (Rombach et al., 2022) and inversions of random images from ImageNet1k (Deng et al., 2009). We (uniformly) randomly select 50 classes, each from which we randomly select 50 unique images, and find their corresponding latents through DDIM inversion (Song et al., 2020a) using the class name as prompt. We note that the DDIM inversion can be sensitive to the number of steps. Therefore, we made sure to use a sufficient number of steps for the inversion (we used 400 steps), which we then matched for the generation; see Figure 15 for an illustration of the importance of using a sufficient number of steps. We used the guidance scale 1.0 (i.e. no guidance) for the inversion, which was then matched during generation. Note that using no guidance is important both for accurate inversion as well as to not introduce a factor (the dynamics of prompt guidance) which would be specific to the Stable Diffusion 2.1 model.
+
+For the interpolation setup we randomly (without replacement) pair the 50 images per class into 25 pairs, forming 1250 image pairs in total. In between the ends of each respective pair, each method then is to produce three interpolation points (and images). For the NAO method, which needs additional interpolation points to approximate the line integral, we used 11 interpolation points and selected three points from these at uniform (index) distance, similar to in Samuel et al. (2023).
+
+For the centroid determination setup we for each class form 10 3-groups, 10 5-groups, 4 10-groups and 1 25-group, sampled without replacement per group<sup>1</sup>; i.e. each method is to form 25 centroids per class total, for an overall total of 1250 centroids per method. Similarly to the interpolation setup, for NAO we used 11 interpolations points per path, which for their centroid determination method entails  $K$  paths per centroid.
+
+![](images/e9485566739844b8ddc59627084fe755e7607ff88939323f4af3ccf02ed562b6.jpg)  
+Figure 8: Low p-values are inherited by interpolants The x-axis show the lower Kolmogorov-Smirnov p-value of the two seed latents for each corresponding spherical interpolation in Figure 5, and the y-axis show the p-value of the resulting interpolant. Due to the vast dynamic range of the p-values, the scatter plot is shown in full on the left, and zoomed in on the upper-right quadrant on the right. We note that when any of the two seed latents have a small p-value this largely tend to be inherited by the interpolant. The Pearson correlation coefficient is 0.79 for the indicator of acceptance (defined as 1 if the p-value is greater than  $1e^{-3}$  and 0 otherwise) and 0.66 for the  $\log_{10}$  p-values.
+
+![](images/66de5610baac638e7f0acb11289c7c04a3bfc0f372ba89b7d5693425afb6add4.jpg)
+
+Evaluation We, as in Samuel et al. (2023), assess the methods quantitatively based on visual quality and preservation of semantics using FID distances and class prediction accuracy, respectively.
+
+The FID distances are computed using the pytorch-fid library (Seitzer, 2020), using all evaluation images produced per method for the interpolation and centroid determination respectively, to maximize the FID distance estimation accuracy.
+
+For the classification accuracy, we used a pre-trained classifier, the MaxViT image classification model (Tu et al., 2022) as in Samuel et al. (2023), which achieves a top-1 of  $88.53\%$  and  $98.64\%$  top-5 accuracy on the test-set of ImageNet.
+
+See results in Section H.
+
+# G ADDITIONAL QUALITATIVE RESULTS
+
+See Figure 9 for a demonstration of subspaces of latents without the LOL transformation introduced in Equation 3, using a diffusion model (Rombach et al., 2022) and a flow matching model (Esser et al., 2024), respectively. The setup is identical (with the same original latents) as in Figure 12 and Figure 1, respectively, except without applying the proposed (LOL) transformation.
+
+See Figure 10 for additional slices of the sports car subspace shown in Figure 1.
+
+See Figure 12 and Figure 13 for Stable Diffusion 2.1 (SD2.1) versions of the Stable Diffusion 3 (SD3) examples in Figure 1 and Figure 6, with an otherwise identical setup including the prompts. Just like in the SD3 examples LOL defines working subspaces. However, as expected since SD2.1 is an older model than SD3, the visual quality of the generations is better using SD3.
+
+See Figure 11 for an interpolation example.
+
+In the examples above the SD2.1 and SD3 models are provided with a text prompt during the generation. See Figure 14 for an example where the original latents  $\{\pmb{x}_i\}$  were obtained using DDIM inversion (from images) without a prompt (and guidance scale 1.0, i.e. no guidance), allowing generation without a prompt. This allows us to also interpolate without conditioning on a prompt. We note that, as expected without a prompt, the intermediates are not necessarily related to the end points (the original images) but still realistic images are obtained as expected (except for using linear interpolation, see discussion in Section 4) and the interpolations yield smooth gradual changes.
+
+![](images/abde7c7167e51553eb7635feb7ec2464210d85947a123207f648b2f016685360.jpg)  
+Figure 9: Without LOL transformation. The setup here is exactly the same as in Figure 12 and Figure 1, respectively, except without the proposed (LOL) transformation (see Equation 3). The prompt used is "A high-quality photo of a parked, colored sports car taken with a DLSR camera with a 45.7MP sensor. The entire sports car is visible in the centre of the image. The background is simple and uncluttered to keep the focus on the sports car, with natural lighting enhancing its features." We note that the diffusion model does not produce images of a car without the transformation, and neither model produces anything else than visually the same image for all coordinates.
+
+![](images/5e194c6667af42e2dbd24926703a9d0c14dd6f16741bdb6659a831e3b4ecbc96.jpg)  
+Figure 10: Additional slices of a sports car subspace. The latents  $x_{1}, \ldots, x_{5}$  (corresponding to images) are converted into basis vectors and used to define a 5-dimensional subspace. The grids show generations from uniform grid points in the subspace coordinate system, where the left and right grids are for the dimensions  $\{1, 2\}$  and  $\{3, 4\}$ , respectively, centered around the coordinate for  $x_{1}$ . Each coordinate in the subspace corresponds to a linear combination of the basis vectors. The flow matching model Stable Diffusion 3 (Esser et al., 2024) is used in this example. See Figure 1 for another slice of the latent subspace.
+
+# H ADDITIONAL QUANTITATIVE RESULTS AND ANALYSIS
+
+In Table 1 we show that our method outperforms or maintains the performance of the baselines in terms of FID distances and accuracy, as calculated using a pre-trained classifier following the evaluation methodology of Samuel et al. (2023). For an illustration of centroids and interpolations, see Figure 2 and Figure 16, respectively. The evaluation time of an interpolation path and centroid, shown with one digit of precision, illustrate that the closed-form expressions are significantly faster than NAO. Surprisingly, NAO did not perform as well as spherical interpolation and several other baselines, despite using their implementation, which was outperforming these methods in Samuel
+
+![](images/d2c1c2adcc64c4ce6ccd90476a85e2b27100e4df54e78d710c70a729e79ea57f.jpg)  
+Figure 11: Interpolation. Shown are generations from equidistant interpolations of latents  $\boldsymbol{x}_1$  and  $\boldsymbol{x}_2$  (in  $v \in [0,1]$ ) using the respective method. The diffusion model Stable Diffusion 2.1 (Rombach et al., 2022) is used in this example.
+
+![](images/c06090f6a9df9d7817990daffbc2dec0309a78dc1c027a80dee86fe960f91987.jpg)  
+Figure 12: Low-dimensional subspaces. The latents  $x_{1}, \ldots, x_{5}$  (corresponding to images) are converted into basis vectors and used to define a 5-dimensional subspace. The grids show generations from uniform grid points in the subspace coordinate system, where the left and right grids are for the dimensions  $\{1,3\}$  and  $\{2,4\}$ , respectively, centered around the coordinate for  $x_{1}$ . Each coordinate in the subspace corresponds to a linear combination of the basis vectors. The diffusion model Stable Diffusion 2.1 (Rombach et al., 2022) is used in this example.
+
+et al. (2023). We note one discrepancy is that we report FID distances using (2048) high-level features, while in their work they are using (64) low-level features, which in Seitzer (2020) is recommended against as it does not necessarily correlate with visual quality. In Table 3 we report results with all feature settings. We note that, in our setup, the baselines perform substantially better than reported in Samuel et al. (2023) — including NAO in terms of FID distances using the 64 low-level feature setting (1.30 in our setup vs 6.78 in their setup), and class accuracy during interpolation (62% vs 52%), which we study below.
+
+The number of DDIM inversion steps used for the ImageNet images is not reported in the NAO setup (Samuel et al., 2023), but if we use 1 step instead of 400 (400 is what we use in all our experiments) the class preserving accuracy of NAO for centroid determination, shown in Table 4, resemble theirs more, yielding  $58\%$ . As we illustrate in Figure 5 and Figure 15, a sufficient number of inversion steps is critical to acquire latents which not only reconstructs the original image but which can be successfully interpolated. We hypothesise that a much lower setting of DDIM inversions
+
+![](images/d5ebb426636f8ea5781dc35f7176d52516b7048e4dfb527e6391c3fa2f830c6c.jpg)  
+Figure 13: Low-dimensional subspaces. The latents  $x_{1}, \ldots, x_{5}$  (corresponding to images) are converted into basis vectors and used to define a 5-dimensional subspace. The grids show generations from uniform grid points in the subspace coordinate system, where the left and right grids are for the dimensions  $\{1, 3\}$  and  $\{2, 4\}$ , respectively, centered around the coordinate for  $x_{1}$ . Each coordinate in the subspace corresponds to a linear combination of the basis vectors. The diffusion model Stable Diffusion 2.1 (Rombach et al., 2022) is used in this example.
+
+steps were used in their setup than in ours, which would be consistent with their norm-correcting optimization method having a large effect, making otherwise blurry/superimposed images intelligible for the classification model.
+
+<table><tr><td colspan="7">Interpolation</td></tr><tr><td>Method</td><td>Accuracy</td><td>FID 64</td><td>FID 192</td><td>FID 768</td><td>FID 2048</td><td>Time</td></tr><tr><td>LERP</td><td>3.92%</td><td>63.4</td><td>278</td><td>2.18</td><td>199</td><td>6e-3s</td></tr><tr><td>SLERP</td><td>64.6%</td><td>2.28</td><td>4.95</td><td>0.173</td><td>42.6</td><td>9e-3s</td></tr><tr><td>NAO</td><td>62.1%</td><td>1.30</td><td>4.11</td><td>0.195</td><td>46.0</td><td>30s</td></tr><tr><td>LOL (ours)</td><td>67.4%</td><td>1.72</td><td>3.62</td><td>0.156</td><td>38.9</td><td>6e-3s</td></tr><tr><td colspan="7">Centroid determination</td></tr><tr><td>Method</td><td>Accuracy</td><td>FID 64</td><td>FID 192</td><td>FID 768</td><td>FID 2048</td><td>Time</td></tr><tr><td>Euclidean</td><td>0.286%</td><td>67.7</td><td>317</td><td>3.68</td><td>310</td><td>4e-4s</td></tr><tr><td>Standardized Euclidean</td><td>44.6%</td><td>5.92</td><td>21.0</td><td>0.423</td><td>88.8</td><td>1e-3s</td></tr><tr><td>Mode norm Euclidean</td><td>44.6%</td><td>6.91</td><td>21.6</td><td>0.455</td><td>88.4</td><td>1e-3s</td></tr><tr><td>NAO</td><td>44.0%</td><td>4.16</td><td>15.6</td><td>0.466</td><td>93.0</td><td>90s</td></tr><tr><td>LOL (ours)</td><td>46.3%</td><td>9.38</td><td>25.5</td><td>0.455</td><td>87.7</td><td>6e-4s</td></tr></table>
+
+Table 3: Full table with class accuracy and FID distances using each setting in Seitzer (2020). Note that FID 64, 192, and 768 is not recommended by Seitzer (2020), as it does not necessarily correlate with visual quality. FID 2048 is based on the final features of InceptionV3 (as in Heusel et al. (2017)), while FID 64, 192 and 768 are based on earlier layers; FID 64 being the first layer.
+
+# I ADDITIONAL IMAGENET INTERPOLATIONS AND CENTROIDS
+
+![](images/22db95d7522d9b3d2da49f5876c030b584d51b204db94147ed36abaf13f6d6a5.jpg)  
+Figure 14: Interpolation during unconditional generation. Shown are generations from equidistant interpolations of latents  $\pmb{x}_1$  and  $\pmb{x}_2$  using the respective method. The latents  $\pmb{x}_1$  and  $\pmb{x}_2$  were obtained from DDIM inversion (Song et al., 2020a) with an empty prompt, and all generations in this example are then carried out with an empty prompt. The larger images show interpolation index 8 for SLERP, NAO and LOL, respectively. The diffusion model Stable Diffusion 2.1 (Rombach et al., 2022) is used in this example.
+
+<table><tr><td colspan="7">Centroid determination of invalid latents</td></tr><tr><td>Method</td><td>Accuracy</td><td>FID 64</td><td>FID 192</td><td>FID 768</td><td>FID 2048</td><td>Time</td></tr><tr><td>Euclidean</td><td>42.3%</td><td>25.7</td><td>59.3</td><td>1.23</td><td>170</td><td>4e-4s</td></tr><tr><td>Standardized Euclidean</td><td>46.6%</td><td>3.35</td><td>16.6</td><td>1.23</td><td>175</td><td>1e-3s</td></tr><tr><td>Mode norm Euclidean</td><td>50.6%</td><td>2.08</td><td>8.74</td><td>1.19</td><td>173</td><td>1e-3s</td></tr><tr><td>NAO</td><td>57.7%</td><td>3.77</td><td>13.3</td><td>1.05</td><td>150</td><td>90s</td></tr><tr><td>LOL (ours)</td><td>48.6%</td><td>2.96</td><td>12.5</td><td>1.22</td><td>171</td><td>6e-4s</td></tr></table>
+
+Table 4: Centroid determination results if we would use a single step for both the DDIM inversion and generation (not recommended). This results in latents which do not follow the correct distribution  $\mathcal{N}(\boldsymbol{\mu},\boldsymbol{\Sigma})$  and which, despite reconstructing the original image exactly if generating also using a single step, cannot be used successfully for interpolation, see Figure 5. As illustrated in the figure, interpolations of latents obtained through a single step of DDIM inversion merely results in the two images being superimposed. Note that the class accuracy is much higher for all methods on such superimposed images, compared to using DDIM inversion settings which allow for realistic interpolations (see Table 3). Moreover, the FID distances set to use fewer features are lower for these superimposed images, while with recommended setting for FID distances (with 2048 features) they are higher (i.e. worse), as one would expect by visual inspection. The NAO method, initialized by the latent yielding the superimposed image, then optimises the latent yielding a norm close to norms of typical latents, which as indicated in the table leads to better "class preservation accuracy". However, the images produced with this setting are clearly unusable as they do not look like real images.
+
+![](images/d0c8acaee11cc9c7dab2c3f6bd691740f800a74c6c45aecd9185711d35931f85.jpg)  
+Figure 15: Accurate DDIM inversions are critical for interpolation. Shown is the interpolation (center) of Image 1 and Image 2 using each respective method, after a varying number of budgets of DDIM inversion steps (Song et al., 2020a). For each budget setting, the inversion was run from the beginning. We note that although a single step of DDIM inversion yields latents which perfectly reconstructs the original image such latents do not lead to realistic images, but merely visual "superpositions" of the images being interpolated.
+
+![](images/8bdffb810fbbfa940add93806339c65a616728373140b742889a218874a0e153.jpg)  
+Figure 16: Interpolation. Shown are generations from equidistant interpolations of latents  $\boldsymbol{x}_1$  and  $\boldsymbol{x}_2$  using the respective method. The latents  $\boldsymbol{x}_1$  and  $\boldsymbol{x}_2$  were obtained from DDIM inversion (Song et al., 2020a) of two random image examples from one of the randomly selected ImageNet classes ("Chesapeake Bay retriever") described in Section 5. The larger images show interpolation index 6 for SLERP, NAO and LOL, respectively. The diffusion model Stable Diffusion 2.1 (Rombach et al., 2022) is used in this example.
+
+![](images/a94356e791f3267440f77d3f67dd7d7e1a575e9a6cb26900b309c11aabd31e30.jpg)  
+Figure 17: ImageNet centroid determination. The centroid of the latents  $\boldsymbol{x}_1, \boldsymbol{x}_2, \boldsymbol{x}_3$  as determined using the different methods, with the result shown in the respective (right-most) plot. The diffusion model Stable Diffusion 2.1 (Rombach et al., 2022) is used in this example.
+
+![](images/3e74686c6b990b4d37ff083c16000b5047c8b74abc488370b6a0ee456eb698ed.jpg)  
+Figure 18: Centroid determination of many latents. The centroid of the latents as determined using the different methods, with the result shown in the respective (right-most) plot. Ten and 25 latents are used in the first two and second two examples (rows), respectively, and the plots show a subset of these in the left-most plots. We noted during centroid determination of many latents (such as 10 and 25) that the centroids were often unsatisfactory using all methods; blurry, distorted etc. For applications needing to find meaningful centroids of a large number of latents, future work is needed. The diffusion model Stable Diffusion 2.1 (Rombach et al., 2022) is used in this example.
\ No newline at end of file
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+# LINES OF THOUGHT IN LARGE LANGUAGE MODELS
+
+# Raphaël Sarfati
+
+School of Civil and Environmental Engineering
+
+Cornell University, USA
+
+raphael.sarfati@cornell.edu
+
+# Toni J.B. Liu
+
+Department of Physics
+
+Cornell University, USA
+
+toni.liu@cornell.edu
+
+# Nicolas Boulle
+
+Department of Mathematics
+
+Imperial College London, UK
+
+n.boulle@imperial.ac.uk
+
+# Christopher J. Earls
+
+Center for Applied Mathematics
+
+School of Civil and Environmental Engineering
+
+Cornell University, USA
+
+earls@cornell.edu
+
+# ABSTRACT
+
+Large Language Models achieve next-token prediction by transporting a vectorized piece of text (prompt) across an accompanying embedding space under the action of successive transformer layers. The resulting high-dimensional trajectories realize different contextualization, or 'thinking', steps, and fully determine the output probability distribution. We aim to characterize the statistical properties of ensembles of these 'lines of thought.' We observe that independent trajectories cluster along a low-dimensional, non-Euclidean manifold, and that their path can be well approximated by a stochastic equation with few parameters extracted from data. We find it remarkable that the vast complexity of such large models can be reduced to a much simpler form, and we reflect on implications. Code for trajectory generation, visualization, and analysis is available on Github at https://github.com/lapsar/lines-of-thought.
+
+# 1 INTRODUCTION
+
+How does a large language model (LLM) think? In other words, how does it abstract the prompt "Once upon a time, a facetious" to suggest adding, e.g., "chatbot", and, by repeating the operation, continue on to generate a respectable fairy tale à la Perrault? What we know is by design. A piece of text is mapped into a set of high-dimensional vectors, which are then transported across their embedding (latent) space through successive transformer layers (Vaswani et al., 2017), each allegedly distilling different syntactic, semantic, informational, contextual aspects of the input (Valeriani et al., 2023; Song & Zhong, 2024). The final position is then projected onto an embedded vocabulary to create a probability distribution about what the next word should be. Why these vectors land where they do eludes human comprehension due to the concomitant astronomical numbers of arithmetic operations which, taken individually, do nothing, but collectively confer the emergent ability of language.
+
+Our inability to understand the inner workings of LLMs is problematic and, perhaps, worrisome. While LLMs are useful to write college essays or assist with filing tax returns, they are also often capricious, disobedient, and hallucinatory (Sharma et al., 2023; Zhang et al., 2023). That's because, unlike traditional 'if-then' algorithms, instructions have been only loosely, abstractly, encoded in the structure of the LLM through machine learning, that is, without human intervention.
+
+In return, language models, trained primarily on textual data to generate language, have demonstrated curious abilities in many other domains (in-context learning), such as extrapolating time series (Gruver et al., 2024; Liu et al., 2024), writing music (Zhou et al., 2024), or playing chess (Ruoss et al., 2024). Such emergent, but unpredicted, capabilities lead to questions about what other abilities LLMs may possess. For these reasons, current research is attempting to break down internal processes to make LLMs more interpretable. $^{1}$  Recent studies have notably revealed some aspects of
+
+![](images/9300058ce99981fe7413fd6cfb59e0218072210172d2ff536196a4efc40042fd.jpg)  
+Figure 1: (a) Lines of thought (blue to red) for an ensemble of 1000 pseudo-sentences of 50 tokens each, projected along the first 3 singular vectors after the last layer ( $t = 24$ ). They appear to form a tight bundle, with limited variability around a common average path. (b) Representation of the low-dimensional, ribbon-shaped manifold in  $S$  (projected along 3 Cartesian coordinates). Positions are plotted for  $t = 12$  (green) to  $t = 24$  (yellow).
+
+![](images/34c5a3b01373e6504f6a4d1467349860c8d06b9cd62398d217c4e90cef4354e9.jpg)
+
+the self-attention mechanism (Vig, 2019), patterns of neuron activation (Bricken et al., 2023; Templeton et al., 2024), signatures of 'world models' (Gurnee & Tegmark, 2023; Marks & Tegmark, 2023), geometrical relationships between concepts (Jiang et al., 2024), or proposed mathematical models of transformers (Geshkovski et al., 2024).
+
+This work introduces an alternative approach inspired by physics, treating an LLM as a complex dynamical system. We investigate which large-scale, ensemble properties can be inferred experimentally without concern for the 'microscopic' details.<sup>3</sup> Specifically, we are interested in the trajectories, or 'lines of thought' (LoT), that embedded tokens realize in the latent space when passing through successive transformer layers (Aubry et al., 2024). By splitting a large input text into  $N$ -token sequences, we study LoT ensemble properties to shed light on the internal, average processes that characterize transformer transport.
+
+We find that, even though transformer layers perform  $10^{6} - 10^{9}$  individual computations, the resulting trajectories can be described with far fewer parameters. In particular, we first identify a low-dimensional manifold that explains most of LoT transport (see Fig. 1). Then, we demonstrate that trajectories can be well approximated by an average linear transformation, whose parameters are extracted from ensemble properties, along with a random component with well characterized statistics. Eventually, this allows us to describe trajectories as a kind of diffusive process, with a linear drift and a modified stochastic component.
+
+# Main contributions.
+
+1. We provide a framework to discover low-dimensional structures in an LLM's latent space.  
+2. We find that token trajectories cluster on a non-Euclidean, low-dimensional manifold.  
+3. We introduce a stochastic model to describe trajectory ensembles with few parameters and extend them to continuous paths.
+
+# 2 METHODS
+
+This section describes our algorithm for generating and analyzing an ensemble of tokens trajectories in the latent space of LLMs. Our code is provided in the corresponding Github repository (Sarfati et al.).
+
+Language models. We rely primarily on the 355M-parameter ('medium') version of the GPT-2 model (Radford et al., 2019). It presents the core architecture of ancestral (circa 2019) LLMs: transformer-based, decoder-only.4 It consists of  $N_{L} = 24$  transformer layers5 operating in a latent space  $S$  of dimension  $D = 1024$ . The vocabulary  $\mathcal{V}$  contains  $N_{\mathcal{V}} = 50257$  tokens. A layer normalization (Ba et al., 2016) is applied to the last latent space position before projection onto  $\mathcal{V}$  to form the logits. (This final normalization is not included in our trajectories.) We later extend our analysis to the Llama 2 7B (Touvron et al., 2023), Mistral 7B v0.1 (Jiang et al., 2023), and small Llama 3.2 models (1B and 3B) (MetaAI, 2024).
+
+Input ensembles. We study statistical properties of trajectory ensembles obtained by passing a set of input prompts through GPT-2. We generate inputs by tokenizing (Wolf et al., 2020) a large text and then chopping it into 'pseudo-sentences', i.e., chunks of a fixed number of tokens  $N_{k}$  (see Algorithm 1). Unless otherwise noted,  $N_{k} = 50$ . These non-overlapping chunks are consistent in terms of token cardinality, and possess the structure of language, but have various meanings and endings (see Appendix A.1). The main corpus in this study comes from Henry David Thoreau's Walden, obtained from the Gutenberg Project (Project Gutenberg, 2024).<sup>6</sup> We typically use a set of  $N_{s} \simeq 3000 - 14000$  pseudo-sentences.
+
+Trajectory collection. We form trajectories by collecting the successive vector outputs, within the latent space, after each transformer layer (hidden_states). For conciseness, we identify layer number with a notional 'time',  $t$ . Even though all embedded tokens of a prompt voyage across the latent space, only the embedding corresponding to the last token form the logits (by projection onto  $\mathcal{V}$ ) for next-token inference. Hence, here, we only consider the trajectory of this last (or 'pilot') token. The trajectory  $M_{k}$  of sentence  $k$ 's pilot is the sequence of 24 successive time positions  $\{\pmb{x}_k(1),\pmb{x}_k(2),\dots,\pmb{x}_k(24)\}$ , concatenated as a column matrix (Algorithm 1).
+
+Latent space bases. The latent space is spanned by the Cartesian basis  $\mathcal{E} = \{\pmb {e}_i\}_{i = 1\dots D}$  (the orthogonal set of one-hot unit vectors with a 1 in  $i^{\mathrm{th}}$  position, 0 elsewhere). Additionally, we will often refer to the bases  $\mathcal{U}(t) = \{\pmb {u}_i^{(t)}\}_{i = 1\dots D}$  formed by the left-singular vectors of the singular value decomposition (SVD) of the  $D\times N_{\mathrm{s}}$  matrix after layer  $t$ :  $M = U\Sigma V^{\top}$ , with  $M_{\cdot ,k}^{(t)} = x_k(t)$ . Vectors  $\pmb {u}_i$  are organized according to their corresponding singular values,  $\sigma_{i}$ , in descending order. Note that because trajectory clusters evolve over time there are 24 distinct bases.
+
+# 3 RESULTS
+
+We present and characterize results pertaining to ensembles of trajectories as they travel within the latent space  $\mathcal{S}$ .
+
+# 3.1 LINES OF THOUGHT CLUSTER ALONG SIMILAR PATHWAYS
+
+We endeavor to visualize and characterize the trajectories of pilot tokens in the latent space  $\mathcal{S}$ . The high dimensionality makes it non-trivial. Which projections should we consider?
+
+Algorithm 1 Trajectory generation in transformer-based model  
+1: Input: Large text: "It was the best of times, it was the worst of times, it was the age ..."  
+2: Tokenize text into token sequence: [1027, 374, 263, 1267, 287, 1662, 12, ...]  
+3: Split token sequence into  $n$ -token pseudo-sentences:  
+[ s_1 = [1027, 374, 263], \quad s_2 = [1267, 287, 1662], \quad \ldots ]  
+4: for each pseudo-sentence  $s_i$  do  
+5: Semantic embedding:  
+[ E_S = [v(1027), v(374), v(263)] \quad \text{for} \, s_1 ]  
+6:  $E^0 = E_S + E_P$  {add positional embeddings  $P$ }  
+7: for  $t = 1 \rightarrow 23$  do  
+8:  $E^{t+1} = \text{TransformerLayer}_t(E^t)$  {update embeddings through transformer layer}  
+9:  $x(t+1) = E_{:, \text{end}}^{(t+1)}$  {extract last token representation}  
+10:  $M_{:, t+1} = x(t+1)$  {save trajectory array}  
+11: end for  
+12: end for  
+13: Output: Final embeddings  $x(t+1)$  for all pseudo-sentences
+
+There is no reason a priori for the Cartesian axes,  $e_i$ , to align with any meaningful directions, so we seek relevant alternative bases informed by the data. Naturally, we consider the bases  $\mathcal{U}(t)$  formed by the singular vectors of pilots ensemble after each layer  $t$ .
+
+Using these bases aligned with the data's intrinsic directions, we observe in Fig. 1a that trajectories tend to cluster together, instead of producing an isotropic and homogeneous filling of  $S$ . Indeed, LoTs for different, independent pseudo-sentences follow a common path (forming bundles), augmented by individual variability. Specifically, there exist directions with significant displacement relative to the spread (mean over standard deviation); and positions at different times form distinct clusters, as shown in Fig. A1.
+
+# 3.2 LINES OF THOUGHT FOLLOW A LOW-DIMENSIONAL MANIFOLD
+
+We remark in Fig. 2a that the intrinsic bases  $\mathcal{U}(t)$  rotate only slightly across successive timepoints  $t$ . Besides, Fig. 2b shows that the corresponding singular values decay quickly over several orders of magnitude. Both suggest that LoTs may be described by a low-dimensional curved subspace.
+
+But how many dimensions are relevant? Singular values relate to ensemble variance along their corresponding directions. Since the embedding space is high-dimensional, however, the curse of dimensionality looms, hence the significance of Euclidean distances crumbles. To circumvent this limitation, we consider a more practical metric: how close to the original output distribution on the vocabulary does a reduction in dimensionality get us?
+
+To investigate this question, we express token positions  $\pmb{x}(t)$  in the singular vector basis  $\mathcal{U}(t)$ :
+
+$$
+\boldsymbol {x} (t) = \sum_ {i = 1} ^ {K} a _ {i} ^ {(t)} \boldsymbol {u} _ {i} ^ {(t)},
+$$
+
+where the  $\pmb{u}_i^{(t)},$  s are organized by descending order of their corresponding singular values. By default  $K = D$ , and the true output distribution  $\pmb{p}^{\nu}$  is obtained. Now, we examine what happens when, instead of passing the full basis set, we truncate it, after each layer, to keep only the first  $K < D$  principal components. We compare the resulting output distribution,  $\mathbf{p}_K^\nu$  to the true distribution  $\mathbf{p}^\nu$  using KL divergence  $D_{\mathrm{KL}}(\mathbf{p}_K^\nu \| \mathbf{p}^\nu)$ . In Fig. 2c, we see that  $D_{\mathrm{KL}}$  decreases very slowly with decreasing  $K$ , up to about  $K_0 = 256$ . At that point,  $D_{\mathrm{KL}}$  is only about  $10\%$  of its uncorrelated baseline value, implying that most of the true distribution is recovered when keeping only about  $K_0 = 256$ , or  $25\%$ , of the principal components. In other words, for the purpose of next-token prediction, LoTs are quasi-256-dimensional.
+
+(a)  
+![](images/966fed2a1f4436838e38514522b78a7a69759b716bf6ebdf7339f657b201b950.jpg)  
+(baseline for dissimilar distributions).
+
+![](images/2139a5c2c1127bfe95054b065ab0b9a72de93efd379032ec840eddc022af0ca9.jpg)  
+(b)
+
+![](images/923db087dc1d0f2c31905928ed20af57ad6b0a9579d1c69f6b080f34c87f387a.jpg)  
+(c)  
+Figure 2: (a) Angle between the first 4 singular vectors at  $(t_1, t_2)$ ,  $\arccos(\boldsymbol{u}_i^{(t_1)} \cdot \boldsymbol{u}_i^{(t_2)})$ , for  $i = \{1, 2, 3, 4\}$  (top-left, top-right, bottom-left, bottom-right, respectively). (b) Singular values for  $t = 1, \dots, 24$  (blue to red). Clusters stretch more and more after each layer. The leading singular values,  $\sigma_1(t)$ , have been omitted for clarity. (c) Average (over all trajectories) KL divergence between reduced dimensionality trajectories output and true output distributions, as the dimensionality  $K$  is increased. The red dashes line shows the average KL divergence for output distributions from unrelated inputs
+
+If these principal directions remained constant at each layer, this would imply that  $75\%$  of the latent space could be discarded with no consequence. This seems unrealistic. In fact, the principal directions rotate slightly over time, as displayed in Fig. 2a. Eventually, between  $t = 1$  and  $t = 24$ , the full Cartesian basis  $\mathcal{E}$  is necessary to express the first singular directions. Thus, we conclude that lines of thoughts evolve on a low-dimensional curved manifold of about 256 dimensions, that is contained within the full latent space (Fig. 1b).
+
+# 3.3 LINEAR APPROXIMATION OF TRAJECTORIES
+
+Examination of the singular vectors and values at each time step indicates that LoT bundles rotate and stretch smoothly after passing through each layer (Fig. 2). This suggests that token trajectories could be approximated by the linear transformations described by the ensemble, and extrapolated accordingly, from an initial time  $t$  to a later time  $t + \tau$ . Evidently, it is improbable that a transformer layer could be replaced by a mere linear transformation. We rather hypothesize that, in addition to this deterministic average path, a token's location after layer  $t + \tau$  will depart from its linear approximation from  $t$  by an unknown component  $w(t,\tau)$ .<sup>7</sup> We propose the following model:
+
+$$
+\boldsymbol {x} (t + \tau) = \boldsymbol {R} (t + \tau) \boldsymbol {\Lambda} (t, \tau) \boldsymbol {R} (t) ^ {\top} \boldsymbol {x} (t) + \boldsymbol {w} (t, \tau), \tag {1}
+$$
+
+where  $\boldsymbol{x}(t)$  is the pilot token's position in the Cartesian basis, and  $\boldsymbol{R}, \boldsymbol{\Lambda}$  are rotation (orthonormal) and stretch (diagonal) matrices, respectively. Eq. (1) formalizes the idea that, to approximate  $\boldsymbol{x}(t + \tau)$ , given  $\boldsymbol{x}(t)$ , we first project  $\boldsymbol{x}$  in the ensemble intrinsic basis at  $t (\boldsymbol{R}^\top \boldsymbol{x})$ , then stretch the coordinates by the amount given by  $\boldsymbol{\Lambda}$ , and finally rotate according to how much the singular directions have rotated between  $t$  and  $t + \tau$ ,  $\boldsymbol{R}(t + \tau)$  (see also Fig. A2 in Appendix B). Consequently, we can express these matrices as a function of the set of singular vectors  $(\boldsymbol{U})$  and values  $(\boldsymbol{\Sigma})$ :
+
+$$
+\boldsymbol {R} (t) = \boldsymbol {U} (t), \quad \boldsymbol {\Lambda} (t, \tau) = \operatorname {d i a g} (\sigma_ {i} (t + \tau) / \sigma_ {i} (t)) = \boldsymbol {\Sigma} (t + \tau) \boldsymbol {\Sigma} ^ {- 1} (t).
+$$
+
+Fig. 3 shows the close agreement, at the ensemble level, between the true and extrapolated positions.
+
+This is confirmed by the observation that the two sets are not separable with a trained linear classifier, including at large  $\tau$  (see Appendix). Eq. (1) is merely a linear approximation as it is similar to assuming that LoT clusters deform like an elastic solid, where each point maintains the same vicinity, as illustrated in Fig. A2. The actual coordinates ought to include an additional random component  $\mathbf{w}(t,\tau)$ , which a priori depends on both  $t$  and  $\tau$ .
+
+![](images/ae5e7571219510496193d69fee77b8a33516de4a4b0e9d3b1b963afd9a802f99.jpg)  
+Figure 3: Extrapolated token positions  $\tilde{\pmb{x}}^{(k)}$  (blue) from  $t = \{12,14,16,18\}$  to  $t + \tau = \{t + 1,\dots ,21\}$ , compared to their true positions  $\pmb{x}^{(k)}$  (gray), projected in the  $(\pmb{u}_2^{(t)},\pmb{u}_3^{(t)})$  planes.
+
+Is it possible to express  $\pmb{w}$  in probabilistic terms? We consider the empirical residuals
+
+$$
+\delta \boldsymbol {x} (t, \tau) = \boldsymbol {x} (t + \tau) - \tilde {\boldsymbol {x}} (t, \tau)
+$$
+
+between true positions  $\pmb{x}$  and linear approximations  $\tilde{\pmb{x}}(t, \tau) = \pmb{R}(t + \tau)\pmb{\Lambda}(t, \tau)\pmb{R}(t)^{\top}\pmb{x}(t)$ . We investigate the distributions and correlations of  $\delta\pmb{x}(t, \tau)$  across layer combinations  $(t, t + \tau)$ .
+
+From the data, Fig. 4 shows that, for all  $(t,t + \tau)\in \{1,\dots ,23\} \times \{t + 1,\dots ,24\}$ , the ensemble of  $\delta x(t,\tau)$  has the following characteristics: 1) it is Gaussian, 2) with zero mean, 3) and variance scaling as  $\exp (t + \tau)$ . In addition, Fig. A3 shows that the distribution is isotropic, with no evidence of spatial cross-correlations. Hence, we propose:
+
+$$
+\mathrm {w} _ {i} (t, \tau) \sim \mathcal {N} (0, \alpha e ^ {\lambda (t + \tau)}), \tag {2}
+$$
+
+i.e., each coordinate  $\mathbf{w}_i$  of  $\mathbf{w}$  is a Gaussian random variable with mean zero and variance  $\alpha e^{\lambda (t + \tau)}$ . Linear fitting of the logarithm of the variance yields  $\alpha \simeq 0.64$  and  $\lambda \simeq 0.18$ . Even though this formulation ignores some variability across times and dimensions, it is a useful minimal modelling form to describe the ensemble dynamics with as few parameters as possible.
+
+# 3.4 LANGEVIN DYNAMICS FOR CONTINUOUS TIME TRAJECTORIES
+
+Just like the true positions  $\pmb{x}(t)$ , matrices  $\pmb{R}$  and  $\pmb{\Lambda}$  are known (empirically) only for integers values of  $t$ .<sup>8</sup> Can we extend Eq. (1) to a continuous time parameter  $t \in [1,24]$ ? Indeed, it is possible to interpolate  $\pmb{R}$  and  $\pmb{\Lambda}$  between their known values (Absil et al., 2008). Specifically,  $\pmb{R}(t)$  remains orthogonal and rotates from its endpoints; singular values can be interpolated by a spline function.
+
+In return, this allows us to interpolate trajectories between transformer layers. Thus, we extend Eq. (1) to a continuous time variable  $t$ , and write in infinitesimal terms the Langevin equation for the dynamics:
+
+$$
+d \boldsymbol {x} (t) = \left[ \dot {\boldsymbol {R}} (t) \boldsymbol {R} (t) ^ {\top} + \boldsymbol {R} (t) \dot {\boldsymbol {S}} (t) \boldsymbol {R} (t) ^ {\top} \right] \boldsymbol {x} (t) d t + \sqrt {\alpha \lambda \exp (\lambda t)} d \mathbf {w} (t), \tag {3}
+$$
+
+where  $\dot{S} = \mathrm{diag}\left(\dot{\sigma}_i / \sigma_i\right)$  and  $d\mathbf{w}(t)$  is a differential of a Wiener process (Pavliotis, 2014). We defer the mathematical derivation to Appendix A.2. This equation artificially extends LoTs to continuous paths across  $S$ . It provides a stochastic approximation to any token's trajectory, at all times  $t$ .
+
+![](images/3a422ffd9bd0b1fa471674cfe9061c32374277ed5b27c5e270f68e829f956b3f.jpg)  
+Figure 4: Statistics of  $\delta x(t,\tau)$ : mean  $\mu$ , variance  $\sigma^2$ , excess kurtosis  $\kappa$ . Brackets  $\langle \dots \rangle$  denote average over directions  $e_i$  (see Fig. A4 for details). (a) For all  $(t,t + \tau)$ ,  $\mu \simeq 0$  (that is,  $\mu/\sigma \ll 1$ ). (b)  $\log (\sigma^2)$  increases linearly in time, only depends on  $t + \tau$ . (c) The excess kurtosis (kurtosis minus 3) remains close to 0, indicating Gaussianity (except in early layers).
+
+![](images/5d423ab34f4dd27aed214c61c9d2e7db5b540371b12a80222cf155bee9177075.jpg)
+
+![](images/f38cdadac4290718974a8d01d346cfe3d8fef9f7762a69853d3ef68f3aeda18b.jpg)
+
+# 3.5 FOKKER-PLANCK FORMULATION
+
+Eq. (3) is a stochastic differential equation (SDE) describing individual trajectories with a random component. Since the noise distribution is well characterized (see Eq. (2)), we can write an equivalent formulation for the deterministic evolution of the probability density  $P(\pmb{x}, t)$  of tokens  $\pmb{x}$  over time (Pavliotis, 2014). The Fokker-Planck equation<sup>10</sup> associated to Eq. (3) reads:
+
+$$
+\frac {\partial P (\boldsymbol {x} , t)}{\partial t} = - \nabla_ {\boldsymbol {x}} \cdot \left[ \left(\dot {\boldsymbol {R}} \boldsymbol {R} ^ {\top} + \boldsymbol {R} \dot {\boldsymbol {S}} \boldsymbol {R} ^ {\top}\right) \boldsymbol {x} P (\boldsymbol {x}, t) \right] + \frac {1}{2} \alpha \lambda e ^ {\lambda t} \nabla_ {\boldsymbol {x}} ^ {2} P (\boldsymbol {x}, t). \tag {4}
+$$
+
+This equation captures trajectory ensemble dynamics in a much simpler form, and with far fewer parameters, than the computation actually performed by the transformer stack on the fully embedded prompt. The price paid for this simplification is a probabilistic, rather than deterministic, path for LoTs. We now test our model and assess the extent and limitations of our results.
+
+# 4 TESTING AND VALIDATION
+
+# 4.1 SIMULATIONS OF THE STOCHASTIC MODEL
+
+We test our continuous-time model described above. Due to the high dimensionality of the space, numerical integration of the Fokker-Planck equation, Eq. (4), is computationally prohibitive. Instead, we simulate an ensemble of trajectories based on the Langevin formulation, Eq. (3). The technical details are provided in Appendix A.3.
+
+The results presented in Fig. 5 show that the simulated ensembles closely reproduce the ground truth of true trajectory distributions. We must note that Eqs. (3) and (4) are not path-independent;
+
+![](images/1eff6186e4039b36078614d886cb3dd354a88f3a1309606e825d7b1547ae4cd6.jpg)
+
+![](images/be4981789831c2b6ee623b7032006513046c5a9bfd1b2beaa92081bc690c06b3.jpg)
+
+![](images/04ed3e9ffa1824b8dbdf194f87d5c9a4e94161d431c885d1fc2f4698d3afb773.jpg)
+
+![](images/e09987612572c9aef61aaf7dbb2513e183792cd0ea1bee66a78fb2319747809a.jpg)
+
+![](images/a9268496e22c2491240babb446a71a27e31c047f318e76af0c06dec582951bd0.jpg)
+
+![](images/03fc246e8ae522f10e711e2e061cdd66d23081827548a6066e4ad8cb15079cd4.jpg)  
+Figure 5: Simulated distributions for  $t = 12$ ,  $t + \tau = \{12, 13, 14, 15, 16\}$ , projected on the  $(\pmb{u}_1, \pmb{u}_2)$  plane (top row) and the  $(\pmb{u}_3, \pmb{u}_4)$  plane (bottom row). Distributions have been approximated from ensemble trajectories, 10 trajectories for each initial point. Background lines indicate true distributions, thin lines on top indicate simulations.
+
+![](images/5aefb88662b878f3377365a11f9088ab98603788b6ee11b78485a09a255623b4.jpg)
+
+![](images/952fed7ffd853ea47d4b97c64699c95629025a0dcb9fcf2318d79d12e86f6eb3.jpg)
+
+![](images/c3431a609c8fd2277a5ba17d8bb45c3d702ce6a64a736c8ade870c2c2e9d6812.jpg)
+
+![](images/7cedc27593842b1ac2db0da0ada5782da5a0d92093a2a11979ba64f9247f1080.jpg)
+
+therefore, their solution depends on the value of  $R(t)$ ,  $S(t)$  at all time  $t$ . Since there is no 'true' value for the matrices in-between layers, the output of numerical integration naturally depends on the interpolation scheme. Hence, discrepancies are to be expected.
+
+# 4.2 NULL TESTING
+
+We now examine trajectory patterns for non-language inputs and untrained models.
+
+# 4.2.1 GIBBERISH
+
+We generate non-language ('gibberish') pseudo-sentences by assembling  $N$ -token sequences of random tokens in the vocabulary, and pass them as input to GPT-2. The resulting trajectories also cluster around a path similar to that of language. However, the two ensembles, language and gibberish, are linearly separable at all layers (see Fig. A5 in Appendix B.6), indicating that they travel on two distinct, yet adjacent, manifolds.
+
+# 4.2.2 UNTRAINED & ABLATED MODELS
+
+We compare previous observations with the null baseline of an untrained model.
+
+First, we collect trajectories of the Walden ensemble passing through a reinitialized version of GPT-2 (the weights have been reset to a random seed). We observe that while LoTs get transported away from their starting point, the trajectories follow straight, quasi-parallel paths, maintaining their vicinity (see Fig. A5). Furthermore, the model of Eqs. (1) and (2) does not hold; Fig. A6 shows that the variance of  $\delta x$  does not follow the  $\exp(t + \tau)$  scaling, and the distributions are far from Gaussian.
+
+Next, we consider an ablated model, where only layers 13 to 24 have been reinitialized. When reaching the untrained layers, the trajectories stop and merely diffuse about their  $t = 12$  location (Fig. A5).
+
+In conclusion, upon training, the weights evolve to constitute a specific type of transport in the latent space.
+
+# 4.3 RESULTS WITH OTHER MODELS
+
+We repeat the same approach with a set of larger and more recent LLMs. We collect the trajectories of the Walden ensemble in their respective latent spaces.
+
+Llama 2 7B. We first investigate the Llama 2 7B model (Touvron et al., 2023). Remarkably, the pattern of GPT-2 repeats. Token positions at  $t + \tau$  can be extrapolated from  $t$  by rotation and stretch using the singular vectors and values of the ensemble. The residuals are distributed as those of GPT-2, with  $\mathrm{w}_i(t,\tau) \sim \mathcal{N}(0,\alpha e^{\lambda (t + \tau)})$ , see Fig. A7. The values for the parameters  $\alpha$  and  $\lambda$ , however, differ from those of GPT-2 (here,  $\alpha \simeq -5.4$ ,  $\lambda \simeq 0.27$ ).
+
+Mistral 7B. Trajectories across the Mistral 7B (v0.1) model (Jiang et al., 2023)<sup>12</sup> also follow the same pattern (Fig. A8). We note, however, that Eq. (4) only holds up until layer 31. It seems as though the last layer is misaligned with the rest of the trajectories, as linear extrapolation produces an error that is much larger than expected.
+
+Llama 3.2. The last layer anomaly is also apparent for Llama  $3.2\mathrm{1B}^{13}$ , both in the mean and variance of  $\delta x(t,16)$  (see Fig. A9). However, the rest of the trajectories follows Eq. (1). The same pattern is observed for Llama  $3.2\mathrm{3B}^{14}$  in Fig. A10.
+
+It is noteworthy that these three recent models feature the same anomaly at the last layer. The reason is not immediately evident, and perhaps worth investigating further. In addition, we remark that all models also show deviations from predicted statistics across the very first layers (top-left corners). We conjecture that these anomalies might be an effect of re-alignment or fine-tuning, as the first and last layers are the most exposed to perturbations which might not propagate deep into the stack.
+
+# 5 CONCLUSION
+
+Summary. This work began with the prospect of visualizing token trajectories in their embedding space  $S$ . The space is not only high-dimensional, but also isotropic: all coordinates are a priori equivalent.[15] Hence, we sought directions and subspaces of particular significance in shaping token trajectories[16], some kind of 'eigenvectors' of the transformer stack.
+
+Instead of spreading chaotically, lines of thought travel along a low-dimensional manifold. We used this pathway to extrapolate token trajectories from a known position at  $t$  to a later time, based on the geometry of the ensemble. Individual trajectories deviate from this average path by a random amount with well-defined statistics. Consequently, we could interpolate token dynamics to a continuous time in the form of a stochastic differential equation, Eq. (3). The same ensemble behavior holds for various transformer-based pre-trained LLMs, but collapses for untrained (reinitialized) ones.
+
+This approach aims to extract important features of language model internal computation. Unlike much of prior research on interpretability, it is agnostic to the syntactic and semantic aspects of inputs and outputs. We also proposed geometrical interpretations of ensemble properties which avoid relying on euclidean metrics, as they become meaningless in high-dimensional spaces.
+
+Limitations. This method is limited to open-source models, as it requires extracting hidden states; fine-tuned, heavily re-aligned models might exhibit different patterns. In addition, it would be compelling to connect the latent space with the space of output distributions, for example by investigating the relative arrangement of final positions with respect to embedded vocabulary. However, this is complicated by the last layer normalization which typically precedes projection onto the vocabulary. This normalization has computational benefits, but its mathematical handling is cumbersome: it is highly non-linear as it involves the mean and standard deviation of the input vector.
+
+$^{11}$ Decoder-only, 32 layers, 4096 dimensions; released July 2023 by Meta AI.  
+$^{12}$ Decoder-only, 32 layers, 4096 dimensions; released September 2023 by Mistral AI.  
+$^{13}$ Decoder-only, 16 layers, 2048 dimensions; released September 2024 by Meta AI.  
+$^{14}$ Decoder-only, 28 layers, 3072 dimensions; released September 2024 by Meta AI.  
+15Unlike other types of datasets where different dimensions might have well-defined meaning, for example: temperature, pressure, wind speed, etc.  
+And hence defining next-token distribution outputs
+
+Implications. Just like molecules in a gas or birds in a flock, the complex system formed by billions of artificial neurons in interaction exhibits some simple, macroscopic properties. It can be described by ensemble statistics with a well defined random component. Previously, Aubry et al. (2024) had also uncovered specific dynamical features, notably token alignment, in transformer stacks of a wide variety of trained models.
+
+Patterns are explanatory. Our concern here has been primarily to discover some of the mechanisms implicitly encoded in the weights of trained language models. Yet, there are also concrete and potentially practical implications to our findings.
+
+For interpretability, finding low-dimensional structures is consequential. It is one of the most efficient ways to break down the inherent complexity of large models into more elementary constituents. Our dynamical system approach reveals a surprising dimensionality reduction of token embeddings. It suggests, notably, that the true "meaning" of embeddings is contained within individual variability (possibly orthogonal to the average pathway collectively followed by all LoTs). This is also merely a first-order approximation, which could be extended to more complete and precise equations, where the "noise term" becomes smaller and smaller. Eventually, we anticipate the possibility for hybrid architectures where the deterministic part of trajectories is delegated to a small system of equations, while the variable part, where meaning is encoded, is handled by a neural network; potentially with many fewer weights.
+
+Our theoretical model in Eqs. (3) and (4) not only reveals low-dimensionality, but also extends token trajectories to continuous paths. In the past, the Neural Ordinary Differential Equation paradigm by Chen et al. (2019) showed that converting a discrete neural network into a continuous dynamical system had many advantages. Notably, it offers opportunities for compression and stability, while pointing towards efficient hybrid architectures. Our paper demonstrates that transformers can also been seen through the lens of dynamical systems, with a similar continuous extension as seen in neural ODEs.
+
+Finally, the new methodology that we introduced is portable and widely applicable. Incidentally, it can also serve as a diagnostic method to highlight intrinsic differences between transformer layers. Fig. A8 to Fig. A10, for example, show significant deviations in the last layer (and to a lesser extent in the early ones). This suggests that these layers achieve a different kind of processing than intermediate layers, possibly following fine-tuning and/or re-alignment. It's not immediately obvious to us how these "anomalies" could be detected through a different approach.
+
+# ACKNOWLEDGMENTS
+
+This work was supported by the SciAI Center, and funded by the Office of Naval Research (ONR), under Grant Numbers N00014-23-1-2729 and N00014-23-1-2716.
+
+# REFERENCES
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+Jesse Vig. Visualizing attention in transformer-based language representation models, 2019. URL https://arxiv.org/abs/1904.02679.  
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+Ziya Zhou, Yuhang Wu, Zhiyue Wu, Xinyue Zhang, Ruibin Yuan, Yinghao Ma, Lu Wang, Emmanouil Benetos, Wei Xue, and Yike Guo. Can llms "reason" in music? an evaluation of llms' capability of music understanding and generation. 2024. URL https://arxiv.org/abs/2407.21531.
+
+# Appendix
+
+# A ADDITIONAL METHODS AND DERIVATIONS
+
+# A.1 PSEUDO-SENTENCES
+
+Random sample of 10-token pseudo-sentences (non-consecutive) extracted from Walden. Similar chunks, but of 50 tokens, were passed through GPT2 to form trajectories.
+
+```txt
+| not been made by my townsmen concerning my mode |
+| --- | --- |
+| to pardon me if I undertake to answer some of |
+| writer, first or last, a simple and sincere |
+| would fain say something, not so much concerning |
+| Brahmins sitting exposed to four fires and looking |
+| more incredible and astonishing than the scenes which I daily |
+| and farming tools; for these are more easily acquired |
+|. How many a poor immortal soul have I met |
+| into the soil for compost. By a seeming fate |
+| as Raleigh rhymes it in his sonorous way |
+| il, are too clumsy and tremble too much |
+| the bloom on fruits, can be preserved only by
+```
+
+# A.2 LANGEVIN EQUATION DERIVATION
+
+Starting from
+
+$$
+\boldsymbol {x} (t + \tau) = \boldsymbol {R} (t + \tau) \boldsymbol {\Lambda} (t, \tau) \boldsymbol {R} (t) \boldsymbol {x} (t) + \boldsymbol {w} (t, \tau),
+$$
+
+with  $\pmb{\Lambda}(t,\tau) = \pmb{\Sigma}(t + \tau)\pmb{\Sigma}^{-1}(t)$ , and assuming now that  $t,\tau$  are variables in  $\mathbb{R}$ , as  $\tau$  goes to 0 we can approximate:
+
+$$
+\boldsymbol {R} (t + \tau) \approx \boldsymbol {R} (t) + \tau \dot {\boldsymbol {R}} (t)
+$$
+
+and
+
+$$
+\boldsymbol {\Sigma} (t + \tau) \approx \boldsymbol {\Sigma} (t) + \tau \dot {\boldsymbol {\Sigma}} (t),
+$$
+
+leading to:
+
+$$
+\boldsymbol {\Lambda} (t, \tau) \approx (\boldsymbol {\Sigma} (t) + \tau \dot {\boldsymbol {\Sigma}} (t)) \boldsymbol {\Sigma} ^ {- 1} (t) = \boldsymbol {I} + \tau \boldsymbol {\Sigma} ^ {- 1} (t) \dot {\boldsymbol {\Sigma}} (t).
+$$
+
+Hence:
+
+$$
+\begin{array}{l} \boldsymbol {R} (t + \tau) \boldsymbol {\Lambda} (t, \tau) \boldsymbol {R} (t) ^ {\top} \approx \left(\boldsymbol {R} (t) + \tau \dot {\boldsymbol {R}} (t)\right) \left(\boldsymbol {I} + \tau \dot {\boldsymbol {\Sigma}} (t) \boldsymbol {\Sigma} ^ {- 1} (t)\right) \boldsymbol {R} (t) ^ {\top} \\ \approx \boldsymbol {I} + \tau \left(\dot {\boldsymbol {R}} (t) \boldsymbol {R} (t) ^ {\top} + \boldsymbol {R} (t) \dot {\boldsymbol {S}} (t) \boldsymbol {R} (t) ^ {\top}\right), \\ \end{array}
+$$
+
+given that  $\pmb{R}\pmb{R}^{\top} = \pmb{I}$  and with  $\pmb {S}(t) = \mathrm{diag}\left(\ln \sigma_{i}(t)\right)$  and thus  $\dot{S} (t) = \mathrm{diag}(\dot{\sigma}_i / \sigma_i)$
+
+The variance of the noise term is given by:
+
+$$
+\operatorname {v a r} = \alpha \exp (\lambda (t + \tau)) \approx \alpha \exp (\lambda t) (1 + \lambda \tau).
+$$
+
+The increment of variance over time  $\tau$  is:
+
+$$
+\delta [ \mathrm {v a r} ] = \alpha \lambda \exp (\lambda t) \tau .
+$$
+
+This means the noise term can be expressed as:
+
+$$
+\boldsymbol {w} (t, \tau) = \sqrt {\alpha \lambda \exp (\lambda t) \tau} \cdot \vec {\eta},
+$$
+
+where  $\vec{\eta}$  is a vector of standard Gaussian random variables.
+
+Putting everything together:
+
+$$
+\pmb {x} (t + \tau) - \pmb {x} (t) = \tau (\dot {\pmb {R}} (t) \pmb {R} (t) ^ {\top} + \pmb {R} (t) \dot {\pmb {S}} (t) \pmb {R} (t) ^ {\top}) \pmb {x} (t) + \sqrt {\alpha \lambda \exp (\lambda t) \tau} \eta (t).
+$$
+
+And finally:
+
+$$
+d \pmb {x} (t) = \left(\dot {\pmb {R}} (t) \pmb {R} (t) ^ {\top} + \pmb {R} (t) \dot {\pmb {S}} (t) \pmb {R} (t) ^ {\top}\right) \pmb {x} (t) d t + \sqrt {\alpha \lambda \exp (\lambda t)} d \pmb {w} (t),
+$$
+
+with  $d\pmb{w}(t)$  a Wiener process.
+
+# A.3 NUMERICAL INTEGRATION
+
+Numerical integration of Eq. (3) requires to interpolate the singular vectors and values, and their derivatives, at non-integer times.
+
+Interpolation of (scalar) singular values is straightforward. We use a polynomial interpolation scheme for each value, and compute the corresponding polynomial derivative. This yields  $\dot{\sigma}_i(t) / \sigma_i(t)$  for every coordinate  $i$  at any time  $t\in [1,24]$ , and hence  $\dot{S} (t)$ .
+
+Interpolating sets of orthogonal vectors presents significant challenges. A rigorous approach involves performing the interpolation within the compact Stiefel manifold, followed by a reprojection onto the horizontal space Praveen et al. (2023). However, this method is computationally expensive and can introduce discontinuities, which are problematic for numerical integration. To address these issues, we used an approximation based on the matrix logarithm, which simplifies the process while maintaining an acceptable level of accuracy. To interpolate between  $U_{1}$  and  $U_{2}$  at  $t_1, t_2$ , we compute the relative rotation matrix  $R = U_{1}^{\top}U_{2}$  and interpolate using
+
+$$
+\boldsymbol {U} (t) = \boldsymbol {U} _ {1} \exp_ {\mathrm {M}} (\alpha \ln_ {\mathrm {M}} \boldsymbol {R}). \tag {5}
+$$
+
+where  $\alpha = (t - t_1) / (t_2 - t_1)$  and with  $\ln_{\mathrm{M}},\exp_{\mathrm{M}}$  denoting the matrix logarithm and exponential, respectively. This also yields the derivative  $\dot{\pmb{U}} (t) = [\pmb {U}\ln_{M}\pmb {R}] / (t_{2} - t_{1})$  . Indeed:
+
+$$
+\dot {\pmb {U}} = \pmb {U} _ {1} \cdot \frac {d}{d t} \exp (\alpha (t) \ln \pmb {R}) = \pmb {U} _ {1} \dot {\alpha} \ln \pmb {R} \exp \alpha (t) \ln \pmb {R} = \dot {\alpha} \pmb {U} \ln \pmb {R}.
+$$
+
+# B SUPPLEMENTARY FIGURES AND SCHEMATICS
+
+# B.1 TRAJECTORY CLUSTERING
+
+In Fig. A1, we show evidence of trajectory clustering in the latent space. In particular, all pilot tokens get transported away from the origin (or their starting point) by a comparable amount, resulting in narrow distributions along the first singular direction. Another signature of clustering is the fact that token positions at different times form distinct clusters, as showed by low-dimensional t-SNE representation (van der Maaten & Hinton, 2008).
+
+# B.2 TRAJECTORY EXTRAPOLATION
+
+In Fig. A2, we provide a schematic to explain the reasoning behind Eq. (1). If the cluster rotated and stretched like a solid, the position of a point  $\pmb{x}^{\prime}$  at  $t^{\prime}$  could be inferred exactly from its position  $\pmb{x}$  at  $t$ , using the formula outlined. However, unsurprisingly, the token ensemble does not maintain its topology and the points move around the clusters, requiring the stochastic term  $\mathbf{w}$  injected in Eq. (1).
+
+# B.3 SEPARABILITY OF TRUE AND EXTRAPOLATED POSITIONS
+
+To characterize the similarity between the ensemble of true positions at  $t + \tau$ ,  $\boldsymbol{x}(t + \tau)$ , from the positions extrapolated from  $t$ ,  $\tilde{\boldsymbol{x}}(t, \tau) = \boldsymbol{R}(t + \tau)\boldsymbol{\Lambda}(t, \tau)\boldsymbol{R}(t)^{\top}\boldsymbol{x}(t)$ , we evaluate how much the two sets can be separated with a linear classifier. We train a Support Vector Machine Model with a linear kernel for each set of extrapolations  $\{\tilde{\boldsymbol{x}}(t, \tau)\}$  (70/30 train/test). We then apply the classifier to predict whether points in the test set are true or extrapolated. In Table 1, we report results for the panels corresponding to Fig. 3. The accuracy of the classifier lies in the 50%-60% range, barely above random guessing (50%).
+
+![](images/92649440eb05f7b5fe412b909079dd8d87d99103b00369e11b48bf042ce6b666.jpg)  
+Figure A1: (Left) Distributions along the first singular vector at different times. (Right) Low-dimensional (t-SNE) visualization of the clustering of tokens, notably across different times. Same color legend.
+
+![](images/f011213432dda975ed999e27b87dcbe74bddea4533921e176d8442ca65b059de.jpg)
+
+![](images/697f2e4c631edd828824da4619baa6212e93c6643433f3f6146c57c325df97f0.jpg)  
+Figure A2: Extrapolation between  $t$  and  $t'$ . The extrapolated location  $\pmb{x}'$  corresponds to the rotated and stretched position of  $\pmb{x}$ . Given that  $\vec{u} = \pmb{R}\vec{e}$ ,  $\vec{u}' = \pmb{R}'\vec{e}$  and  $\pmb{R}^{-1} = \pmb{R}^{\top}$ , we have  $\vec{e} = \pmb{R}^{\top}\vec{u} = \pmb{R}'^{\top}\vec{u}'$  and thus  $\vec{u}' = \pmb{R}'\pmb{R}^{\top}\vec{u}$ .
+
+Table 1: Accuracy of linear classifier to separate  $\mathbf{x}\left( {t + \tau }\right)$  and  $\widetilde{\mathbf{x}}\left( {t,\tau }\right)$  ,in percent.  
+
+<table><tr><td></td><td>t=12</td><td>t=14</td><td>t=16</td><td>t=18</td></tr><tr><td>t+τ=13</td><td>48</td><td></td><td></td><td></td></tr><tr><td>t+τ=15</td><td>51</td><td>49</td><td></td><td></td></tr><tr><td>t+τ=17</td><td>56</td><td>53</td><td>47</td><td></td></tr><tr><td>t+τ=19</td><td>54</td><td>56</td><td>55</td><td>46</td></tr><tr><td>t+τ=21</td><td>56</td><td>60</td><td>61</td><td>55</td></tr></table>
+
+# B.4 NOISE STATISTICS
+
+Fig. A3 provides additional details pertaining to the distribution of residuals  $\delta_x$ . Since they are many dimensions and time points, it gives only representative snapshots. It intends to substantiate the results that:
+
+- the  $\delta x$  are Gaussian (Fig. A3A);  
+- the variance is exponential in  $(t + \tau)$ , with no dependency on  $t$  (Fig. A3B);  
+- all components  $\delta x_{i}$  of  $\delta x$  have the same distribution (Fig. A3C), i.e., isotropy;  
+- there are no spatial cross-correlations, i.e.  $\langle \delta x_i\delta x_j\rangle = \delta_{ij}$  (Dirac function) (Fig. A3D).
+
+![](images/913ec1303435d5b930ff4523552fd18c18b5c9f9cb106d47e48cf3b8054059dd.jpg)  
+Figure A3: Statistics of  $\delta \mathbf{x}$ . (A) Empirical PDF of  $\delta x_{42}(10,t + \tau)$ , with  $t + \tau = 12,14,16$ . The curves appear Gaussian. (B) Variance of  $\delta x_{i}$  for  $i = 1\dots 8$ , for  $t = 4,8,12,16$  and  $t + \tau > t$ . (C) Empirical PDF of  $\delta x_{i}(12,14)$  for  $i = 1\dots 1024$ . The curves are similar for almost all coordinates. (D) Cross correlations of  $\delta x_{i}$  and  $\delta x_{j}$ .
+
+![](images/765a0d4a9be48c15b49cb627e4bb0b655963644a1cccd6466d278867db871076.jpg)
+
+![](images/481a96745950a9d792caf2aa44e90c35b4f2777adc07cf5e203dae2f3668dc3e.jpg)
+
+![](images/da9d8f7b0d6ff04db279d16ce66dcbf8a89ff6f62303be8ca7460b27057ee9f7.jpg)
+
+# B.5 DETAILS ON NOISE AGGREGATED STATISTICS (FIG. 4)
+
+Fig. A4 explains how the noise plots such as Fig. 4 are created. We use ensemble averages  $\langle \dots \rangle$  of the absolute values for  $|\mu_i|, |\kappa_i|$  since we are interested in the average distances from 0.
+
+# B.6 NULL TESTING
+
+Fig. A5 shows the trajectories of language vs gibberish, as well as the linear separability of the two ensemble. It also shows trajectories for an untrained GPT-2 shell, and a model with only the last 12 layers reinitialized.
+
+# B.7 RESULTS WITH OTHER MODELS
+
+![](images/a2c0ac8544c26df6ab45835a70434f0711b97edd3aee77c2a15f9b4202a04cea.jpg)  
+Figure A4: Schematic to explain the noise figures such as Fig. 4. Each square represents a summary statistics. Specifically, the square at  $(t,t + \tau)$  represents the distribution of  $\{\delta \pmb{x}^{(k)}(t,t + \tau)\}_{k}$ , with  $k$  indexing individual tokens. The  $\delta \pmb{x}$  along each coordinate  $i$  form a distribution, from which one can extract the corresponding  $\mu_{i},\sigma_{i},\kappa_{i}$  (mean, variance, kurtosis). These 1D moments are then averaged along all coordinates  $i$  ( $\langle \mu_i\rangle_i$ , etc.), forming the value displayed in the square.
+
+![](images/37fdc265ca9010b17bf08efbc86029a01f39579d01b9e3fdc3396b38a5eb3dc4.jpg)
+
+![](images/fd911e102a638f5e5f239aae3dd7e654f04fc406ec669de52c7d6f171dd6eac7.jpg)
+
+![](images/f9da7f000a5be971ad85687b4f4231819ef52fa37b4d396712c26f8c9c7b3648.jpg)  
+Figure A5: (Top-left) Trajectories of non-language (red) vs language (black), plotted in the same axes (10-token pseudo-sentences). (Top-right) Accuracy of linear separability between language and non-language for each layer. Obtained by training a Perceptron (train/test: 0.7/0.3; 14000 trajectories). (Bottom-left) Trajectories in the untrained GPT-2 model. They are transported in straight lines. (Bottom-right) Trajectories in the mixed model. After being transported by trained layers 1-12, the trajectories stop. Layers 13-24 with random weights do not transport tokens any further.
+
+![](images/50373b485fcade7aa6453cce84393698645c6dc19a0cf5e9c39f9b9139438e3a.jpg)
+
+![](images/be16d2693315650dfe501d1d2d1688fb0ee5599baf014c03212891a1b7e42873.jpg)  
+Figure A6: GPT-2 UNTRAINED. The averaged excess kurtoses  $\langle |\kappa |\rangle$  fall in the 1-1.5 range, indicating strong non-gaussianity. The variance does not scale solely with  $t + \tau$ .
+
+![](images/32e485d8719862c32d92094c7ed6ba8a0bfa14cde10f84cdfa02f4cc073a6b90.jpg)
+
+![](images/ae82a6573cc2dc0231545c9dd7db2789457565aa060d5693aaf870686692554f.jpg)
+
+![](images/b746c515937549e7f4d3b4da1d461bce8c2ce5bcfd287a86100d20841e50e28b.jpg)  
+Figure A7: LLAMA 2 7B: noise statistics,  $\delta \pmb{x}(t,t + \tau) = \pmb{x}(t + \tau) - \tilde{\pmb{x}}(t,\tau)$ , averaged  $\langle \dots \rangle$  over all Cartesian dimensions, for 1000 trajectories (50-token chunks). (a) Mean over standard deviation. (b) Logarithm of variance. (c) Excess kurtosis (0 means Gaussian).
+
+![](images/c0c82e09a7412722646afcbc69a56126832a26ec0345fbeee1d46112b7d4de89.jpg)
+
+![](images/f0a7ef0627344e51da83e856c70ebde4ad17ffcf53c11f33808fc16200c56020.jpg)
+
+![](images/c0c3365f6e470292095430d5ac9dc6a547b161633a1d69415aa1d7b04c5572f6.jpg)  
+Figure A8: MISTRAL 7B v0.1. The last layer (32) appears to have an anomalously large variance.
+
+![](images/f6a1b743da961ccbc115b2c10433da46eb2697a9c07437cb76723100c2ab6fe1.jpg)
+
+![](images/7fc5a567c363f9ae40f5f7e8f7b50d500ffca30a085845934b4a62b834efd809.jpg)
+
+![](images/49c55291c149e2e85577f7ea0c9ce8bd0753308aeff84e62702a05f2ca1397cd.jpg)  
+Figure A9: LLAMA 3.2 1B. This small model all present an out-of-distribution last layer.
+
+![](images/871f091027c21948f42c6f638945fea274b663c2eaa3670d2f8bfca32361d0c1.jpg)
+
+![](images/87fbd182ea00dd21232b12942581d46bd103af5d034fa777c3d0db183fef73c6.jpg)
+
+![](images/65ff36b5d7cea60fc21f02a32a5c2d926e3c160793bfc7046ea0f1f9d7c3244a.jpg)  
+Figure A10: LLAMA 3.2 3B. The last layer anomaly is also present.
+
+![](images/b2b143b11d3f8ddf73d53d2f5315e1519624b337e85bcfdaaba2eca160fb4c4b.jpg)
+
+![](images/3af10071ed0fac25927357bb69f7527454dc57043b97c49b656afd00548f79e9.jpg)
\ No newline at end of file
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diff --git a/ICLR/2025/Lipschitz Bandits in Optimal Space/full.md b/ICLR/2025/Lipschitz Bandits in Optimal Space/full.md
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+# LIPSCHITZ BANDITS IN OPTIMAL SPACE
+
+Xiaoyi Zhu
+
+School of Data Science
+
+Fudan University
+
+Shanghai, China
+
+zhuxy22@m.fudan.edu.cn
+
+Zengfeng Huang *
+
+School of Data Science
+
+Fudan University
+
+Shanghai, China
+
+huangzf@fudan.edu.cn
+
+# ABSTRACT
+
+This paper considers the Lipschitz bandit problem, where the set of arms is continuous and the expected reward is a Lipschitz function over the arm space. This problem has been extensively studied. Prior algorithms need to store the reward information of all visited arms, leading to significant memory consumption. We address this issue by introducing an algorithm named Log-space Lipschitz bandits (Log-Li), which achieves an optimal (up to logarithmic factors) regret of  $\widetilde{O}\left(T^{\frac{dz + 1}{dz + 2}}\right)$  while only uses  $O(\log T)$  bits of memory. Additionally, we provide a complexity analysis for this problem, demonstrating that  $\Omega (\log T)$  bits of space are necessary for any algorithm to achieve the optimal regret. We also conduct numerical simulations, and the results show that our new algorithm achieves regret comparable to the state-of-the-art while reducing memory usage by orders of magnitude.
+
+# 1 INTRODUCTION
+
+In a multi-armed bandit (MAB) problem, an online algorithm must select from a set of strategies over a sequence of  $n$  trials, with the objective of maximizing the cumulative payoff of the chosen strategies. These problems form the fundamental theoretical framework for analyzing the trade-offs between exploration and exploitation that are intrinsic to sequential decision-making under conditions of uncertainty. Algorithms and methodologies for these problems find successful applications in areas such as online auctions, adaptive routing, and the theory of learning in games (Silver et al., 2016; Schneider & Zimmert, 2024).
+
+In this paper, we study the Lipschitz bandit problem - a specific type of MAB problem where the expected reward is Lipschitz continuous(Kleinberg et al., 2008). Existing Lipschitz bandit algorithms heavily rely on storing the reward statistics (e.g. the number of pulls and the mean reward of an arm observed so far) for the visited arms in the memory. Due to the intrinsic assumption in Lipschitz bandit problems that arms belong to a large set, these algorithms need to store information for  $poly(T)$  arms, leading to significant space consumption. For instance, the Zooming algorithm (Kleinberg et al., 2008) and Hierarchical Optimistic Optimization (HOO) algorithm (Bubeck et al., 2011) require  $O(T)$  space, while the state-of-the-art A-BLiN (Feng et al., 2022) has a space complexity of  $O\left(T^{\frac{d_z + 1}{d_z + 2}}(\log T)^{-\frac{d_z + 1}{d_z + 2}}\right)(d_z$  is the zooming dimension). These space requirements are impractical in many real-world scenarios with high time horizons  $T$ , leading to a natural question:
+
+What is the memory cost of achieving optimal regret for the Lipschitz bandit problem?
+
+Recently, learning with constrained memory has garnered significant attention due to its myriad practical applications. Extensive research has been conducted in various fields, including kernel methods(Williams & Seeger, 2000), convex optimization(Marsden et al., 2022; Blanchard et al., 2023), and general machine learning algorithms(Mitliagkas et al., 2013). Memory constraints have also been explored in the context of multi-armed bandit (MAB) settings, though most studies have focused on stochastic MAB in a streaming setting (Maiti et al., 2021; Jin et al., 2021; Assadi & Wang, 2024). These approaches, however, do not apply to the Lipschitz bandit problem, which involves an infinite set of arms without a stream.
+
+To solve the problem, we present a novel algorithm, named Log-space Lipschitz(Log-Li) that only uses a memory of  $O(\log T)$  bits space while achieving an optimal regret rate of  $\widetilde{O}\left(T^{\frac{d_z + 1}{d_z + 2}}\right)$ . Log-Li maintains the empirically best arm and strategically narrows the arm set to learn regions of high reward. This suggests the main finding in this paper that, surprisingly, there is almost no tradeoff between regret and space efficiency for the Lipschitz bandit problem. We further complement this result with a lower-bound complexity analysis and show that  $\Omega (\log T)$  is necessary for achieving optimal regret rate. This means that Log-Li is optimal for both space complexity and regret rate.
+
+# 1.1 PROBLEM DEFINITIONS AND PRELIMINARIES
+
+We consider a compact doubling metric space  $(\mathcal{A},d_{\mathcal{A}})$  of arms. Each arm is associated a sub-Gaussian distribution where the expected reward  $\mu :\mathcal{A}\to \mathbb{R}$  satisfies the 1-Lipschitz condition that  $|\mu (x_{1}) - \mu (x_{2})|\leq d_{\mathcal{A}}(x_{1},x_{2})$  for any  $x_{1},x_{2}\in \mathcal{A}$ . For simplicity, we assume the parameter for the subgaussian distribution to be 1, as extending to other value is straightforward.
+
+Let  $\mu^{\star} = \max_{x\in \mathcal{A}}\mu (x)$  denote the maximal expected payoff. The objective of the algorithm is to minimize regret, defined as  $R(T) = \sum_{t = 1}^{T}\left(\mu^{\star} - \mu \left(x_{t}\right)\right)$ . For convenience, we define the optimal gap of  $x$  as  $\Delta_x = \mu^\star -\mu (x)$  for all  $x\in \mathcal{A}$ .
+
+Consistent with other works, we restrict our consideration to the metric space  $\left([0,1]^d,\| \cdot \|_{\infty}\right)$  for simplicity, since we can always embed a doubling space into a Euclidean space with some distortion of metric by the Assouad's embedding theorem (Assouad, 1983). The ideas underlying our algorithms can be generalized to other doubling metric spaces.
+
+In accordance with previous work, we define the zooming dimension as follows.
+
+Definition 1. For a problem instance with arm set  $\mathcal{A}$ , metric  $d_{\mathcal{A}}$  and expected payoff  $\mu$ . Let  $X_r = \{x \in \mathcal{A} : \Delta_x = \mu^* - \mu(x) \leq r\}$ . We define the  $r$ -zooming number as  $N_r = \mathcal{N}\left(X_{16r}, \frac{r}{2}\right)$ , which is the  $\frac{r}{2}$ -packing number of  $X_{16r}$ . The zooming dimension is then defined as
+
+$$
+d _ {z} := \min  \left\{d \geq 0: \exists a > 0, N _ {r} \leq a r ^ {- d}, \forall 0 <   r <   1 \right\}
+$$
+
+Moreover, we define the zooming constant  $C_z$  as
+
+$$
+C _ {z} = \min  \left\{a > 0: N _ {r} \leq a r ^ {- d _ {z}}, \forall 0 <   r <   1 \right\}
+$$
+
+The zooming dimension  $d_{z}$  can be significantly smaller if the set of near-optimal arms is "small" in terms of the packing number.
+
+# 1.2 MAIN RESULTS
+
+This paper investigates the space complexity for the Lipschitz bandit problem and offers both upper and lower bounds.
+
+Space upper bounds. Our main contribution is the introduction of the Log-Li algorithm, which solves Lipschitz bandits using logarithmic space. With the Doubling Edge-length Sequence  $r_m = 2^{-m + 1}$ , Log-Li achieves the optimal regret rate of  $\widetilde{O}\left(T^{\frac{d_z + 1}{d_z + 2}}\right)$  using  $O(\log T)$  bits.
+
+Theorem 1. With probability exceeding  $1 - 2\delta$ , the  $T$ -step total regret  $R(T)$  of Log-Li with Doubling Edge-length Sequence  $r_m = 2^{-m + 1}$  satisfies
+
+$$
+R (T) \leq (5 1 2 C _ {z} + 1 6) \cdot T ^ {\frac {d _ {z} + 1}{d _ {z} + 2}} (\log (T / \delta)) ^ {\frac {1}{d _ {z} + 2}},
+$$
+
+where  $d_{z}$  is the zooming dimension of the problem instance. Moreover, the space complexity of Log-Li is  $O(\log T)$  bits.
+
+The details of the algorithm can be found in Algorithm 2 and Algorithm 1. The proof of Theorem 1 can be found in Section 3. The core of bandit algorithms lies in balancing exploration and exploitation. Prior research addressed this balance by storing the results of all explored arms. This approach allowed their algorithms to deactivate undesirable regions, thereby limiting the number of pulls in these areas and enabling continued exploration in promising directions.
+
+Log-Li, however, adopts a different strategy to reduce space consumption. Instead of deactivating suboptimal regions entirely, it allows revisits to less favorable areas. The frequency of these revisits is controlled based on each area's contribution to regret. With careful analysis, the regret rate of Log-Li is shown to be  $\widetilde{O}\left(T^{\frac{d_x + 1}{d_z + 2}}\right)$ .
+
+In contrast, seminal studies (Kleinberg et al., 2008; Bubeck et al., 2011; Feng et al., 2022) demonstrate that the optimal regret bound for Lipschitz bandits without space limitations is  $R(T) \lesssim T^{\frac{d_z + 1}{d_z + 2}} \cdot (\log T)^{\frac{1}{d_z + 2}}$ . Consequently, Log-Li attains the optimal regret rate for Lipschitz bandits while utilizing minimal space.
+
+Space lower bounds. Our next contribution is the space lower bounds for the Lipschitz bandit.
+
+Theorem 2. Consider Lipschitz bandit problems with time horizon  $T$  larger than a large enough constant and zooming dimension  $d_z \leq d$ . Then for any algorithm, if the regret bound  $\mathbb{E}[R(T)] \leq \frac{1}{12} T^{\frac{d_z + 1}{d_z + 2}}$ , we must have that the space complexity of the algorithm is at least  $\frac{1}{2} \log T$  bits.
+
+Consequently, Log-Li algorithm is optimal in terms of both regret and space. We also conduct numerical simulations in Section 5, and the results show that our Log-Li achieves regret comparable to the state-of-the-art while reducing memory usage by orders of magnitude.
+
+# 1.3 OTHER RELATED WORK
+
+Prior work on MAB. The history of the Multi-Armed Bandit (MAB) problem dates back to Thompson (1933), with a significant surge of activity in recent decades. Several notable algorithms, such as UCB (Agrawal, 1995), the arm elimination strategy(Even-Dar et al., 2006; Perchet & Rigollet, 2013), the  $\varepsilon$ -greedy method(Auer, 2002), the exponential weights and mirror descent frameworkAuer et al. (2002), have been shown to achieve order-optimal cumulative regret. The Lipschitz bandit problem was first introduced in Kleinberg et al. (2008) and holds significant importance on its own. The Zooming algorithm (Kleinberg et al., 2008) and the Hierarchical Optimistic Optimization (HOO) algorithm (Bubeck et al., 2011) were developed for general doubling metric space.
+
+Learning under limited space. Recently, there has been a surge of work on understanding learning under information constraints such as limited memory or communication constraints. One line of research follows the breakthrough paper of Raz (2017), which shows that any learning algorithm for parity problem requires either a memory of quadratic size or an exponential number of samples. Subsequent works have extended these techniques to other learning problems, such as linear regression (Sharan et al., 2019) and noisy version of the parity problem (Garg et al., 2021). There is also significant work on memory lower bounds for random-order streaming models, addressing problems such as entropy estimation (Acharya et al., 2019) and the needle problem (Lovett & Zhang, 2023).
+
+MAB under limited space. Most works focus on the (stochastic) multi-arm bandit problem in the streaming setting where both regret minimization and pure exploration are studied. The streaming pure exploration MAB was first introduced by Assadi & Wang (2020), and algorithms are proposed to find  $\varepsilon$ -best arms with  $O(1)$  memory and  $O\left(\frac{K}{\varepsilon^2}\right)$  pulls. These algorithms were further developed by Assadi & Wang (2022) to achieve instance-optimal sample complexity. For the regret minimization problem, building on earlier algorithms by Liau et al. (2018); Chaudhuri & Kalyanakrishnan (2020), Agarwal et al. (2022) provided upper and lower regret bounds that are tight in  $T$ . The lower bound was further improved by Li et al. (2023). Additionally, there is another line of work focusing on the closely related expert learning problem (Srinivas et al., 2022; Peng & Zhang, 2023).
+
+# 2 ALGORITHM
+
+With space constraints, the agent only has knowledge about a constant number of arms in the environment. To fully learn the landscape of the reward, the agent must keep exploring different regions of the arm set. Since the agent forgets information about most regions, it may revisit areas that have previously been eliminated, leading to sub-optimal regrets. To overcome this problem, we gradually partition the regions of the arm set and perform exploration in an iterative deepening manner.
+
+Log-Li operates based on finite partitions of the arm space. To keep exploring while avoiding consistent visits to suboptimal areas, we control the maximum depth the algorithm can search in each
+
+round. The algorithm must start from depth 1 each round, as it has already forgotten which areas were previously eliminated. This process may cause redundant visits to undesirable regions. To mitigate this problem, the algorithm maintains the empirically best arm in the last round and uses it to perform arm elimination. This strategy prevents us from searching too deeply in suboptimal areas. By carefully designing partitions and determining the number of pulls for each round, we can effectively control the overall regret. The learning process is summarized in Algorithm 1 and 2.
+
+Algorithm 1: Logarithmic Space Lipschitz for Each round (RoundFunc)  
+Input: Time horizon  $T$ ; current time  $t$ ; maximum depth  $m$ ; current depth  $h$ ; current cube  $C$ ; comparison arm reward  $\tilde{\mu}^{m-1}$ ; current round max reward  $\tilde{\mu}^m$ .  
+if  $t + n_h > T$  then  
+ $\sqsubseteq$  return  
+Play arm  $x_{C,1}^m, \dots, x_{C,n_h}^m$  from  $C$ . Collect the rewards of these pulls  $y_{C,1}^m, \dots, y_{C,n_h}^m$  and compute the average payoff  $\hat{\mu}_h^m(C) = \frac{\sum_{i=1}^{n_h} y_{C,i}^m}{n_h}$ .  
+if  $h$  equals to  $m$  then  
+ $\sqsubseteq$  Compute  $\tilde{\mu}^m = \max \{\tilde{\mu}^m, \hat{\mu}_h^m(C)\}$ .  
+else  
+if  $\tilde{\mu}^{m-1} - \hat{\mu}_h^m(C) < 4r_h$  then  
+ $\sqsubseteq$  Equally partition current cube into  $(r_h / r_{h+1})^d$  subcubes and define  $\mathcal{B}$  as the collection of these subcubes.  
+foreach subcube  $B \in \mathcal{B}$  do  
+ $\sqsubseteq$  RoundFunc $(T, t + n_h, m, h+1, B, \tilde{\mu}^{m-1}, \tilde{\mu}^m)$ .  
+else  
+ $\sqsubseteq$  Eliminate the cube  $C$  implicitly by ending the recursion.
+
+Algorithm 2: Logarithmic Space Lipschitz(Log-Li)  
+Input: Arm set  $\mathcal{A} = [0,1]^d$ ; time horizon  $T$ .  
+Initialize: Error probability  $\delta$ ; Number of rounds  $B$ ; Edge-length sequence  $\{r_m\}_{m=1}^{B+1}$ ; current time  $t = 0$ ; comparison arm reward  $\tilde{\mu}^0 = 0$   
+Compute  $n_m = \frac{16\log(T / \delta)}{r_m^2}$  for  $m = 1,2,\dots,B$ .  
+for  $m = 1,2,\dots,B$  do  
+    Equally partition  $\mathcal{A}$  to  $\left(\frac{1}{r_1}\right)^d$  subcubes and define  $\mathcal{B}$  as the collection of these subcubes. Set  $\tilde{\mu}^m = 0$ .  
+    for each subcube  $B \in \mathcal{B}$  do  
+        RoundFunc $(T,t,m,1,B,\tilde{\mu}^{m-1},\tilde{\mu}^m)$ .  
+Cleanup: play the arm corresponding to  $\tilde{\mu}^{m-1}$  until all  $T$  steps are used.
+
+A running example of Log-Li can be found in figure 1. In each round, Log-Li starts from depth 1 and re-examines the rewards of the cubes. By leveraging the empirically best reward from the last round, we can eliminate suboptimal areas (dark cubes) at the same depth as the previous round without significantly impacting our regret. Simultaneously, we collect rewards from the promising regions (white cubes) for further exploration.
+
+It is worth mentioning that the partition process in the algorithm is detailed explicitly for clarity and comprehension. In practice, the algorithm only requires iterating over the newly partitioned subcubes sequentially, without the need to explicitly store information for each individual subcube.
+
+Moreover, The overhead running time of the algorithm is  $O(T)$  since at each time stamp we simply collect a reward and compare it with  $\tilde{\mu}^m$ . In comparison, the time complexity of the Zooming Algorithm (Kleinberg et al., 2008) is  $O(T^2)$ , while the time complexity of the HOO algorithm(Bubeck et al., 2011) is  $O(T\log T)$ .
+
+![](images/c77acfdbb3003d93021475eb27d773a72bb09e146223774d699bfd072704abca.jpg)  
+Figure 1: Runing example of Log-Li. The  $i$ th subfigure reflects the results after the  $i$ th round. The dark cubes are the eliminated areas and the white cubes are regions with decent rewards at the maximum depth of the current round.
+
+![](images/61d853dceaf1ae9ef77411022486b27e03c44513d732b71907264e2a3b368247.jpg)
+
+![](images/d33e1e0f93f1c4e943fbafdee69c6a2cbb31aa599ce0fd0a75a4de9753845a92.jpg)
+
+# 3 SPACE AND REGRET ANALYSIS
+
+In this section, we provide space regret analysis for Log-Li. The highlight of the finding is that  $O(\log T)$  bits are sufficient to achieve optimal regret rate of  $\widetilde{O}\left(T^{\frac{dz + 1}{d_z + 2}}\right)$  summarized in Theorem 1.
+
+# 3.1 SPACE ANALYSIS
+
+We would start with the space analysis which is more straightforward.
+
+Lemma 1. Algorithm 1 and 2 consumes  $O(\log T)$  bits.
+
+Proof. Note that at each timestamp, Algorithm 1 and 2 only needs to store:
+
+- The id and the empirical mean of the best arm in the last round  $\tilde{\mu}^{m - 1}$ .  
+- The id and the empirical mean of the best arm in the current round  $\tilde{\mu}^m$ .  
+- The id and the empirical mean of the current running arm  $\hat{\mu}_h^m (C)$
+
+Since the number of subcubes we may run is bound by  $T$ , we can use  $\log T$  bits to encode the id of each subcube. For the empirical mean, we need to store the number of pulls and the sum of the rewards, which are both bound by  $T$  since reward of each pull is less than 1. Therefore, we can encode the empirical mean in  $2\log T$  bits. The total bits consumed by Algorithm 1 and 2 is bound by  $3\times (\log T + 2\log T) = O(\log T)$  bits.
+
+![](images/2ecf1f20d677c9f954539da61066eb21237266dafea5fb86ee2c6226b19c3bc5.jpg)
+
+# 3.2 REGRET ANALYSIS
+
+In this subsection, we would prove the regret upper bound of Log-Li with Doubling Edge-length Sequence  $r_m = 2^{-m + 1}$ . Compared to previous research, Log-Li permits revisits to less favorable directions, which may result in additional regret. However, we will manage the number of these revisits by controlling the number of rounds to ensure that the overall regret remains bounded.
+
+Consistent with other studies, we first show that the estimator  $\hat{\mu}$  is concentrated to the true expected reward  $\mu$  (Lemma 2). In the following analysis, we let  $B_{\mathrm{stop}}$  be the total number of rounds and  $\mathcal{A}_h$  be the set of cubes we visit when the depth is  $h$  in round  $m$ .
+
+Lemma 2. Define
+
+$$
+\mathcal {E} := \left\{\left| \mu (x) - \hat {\mu} _ {h} ^ {m} (C) \right| \leq r _ {h} + \sqrt {\frac {1 6 \log (T / \delta)}{n _ {h}}}, \forall 1 \leq h \leq m \leq B _ {s t o p} - 1, \forall C \in \mathcal {A} _ {h} ^ {m}, \forall x \in C \right\}.
+$$
+
+It holds that  $\mathbb{P}(\mathcal{E})\geq 1 - 2\delta$
+
+Proof. Fix any cube  $C \in \mathcal{A}_h^m$ , we have
+
+$$
+\hat {\mu} _ {h} ^ {m} (C) = \frac {\sum_ {i = 1} ^ {n _ {h}} y _ {C , i} ^ {m}}{n _ {h}}, \mathbb {E} \left[ \hat {\mu} _ {h} ^ {m} (C) \right] = \frac {\sum_ {i = 1} ^ {n _ {h}} \mu \left(x _ {C , i} ^ {m}\right)}{n _ {h}}.
+$$
+
+Since  $\hat{\mu}_h^m (C) - \mathbb{E}\left[\hat{\mu}_h^m (C)\right]$  obeys subgaussian distribution with parameter  $\frac{1}{n_h}$ . Applying Hoeffding inequality
+
+$$
+\mathbb {P} \left(| \hat {\mu} _ {h} ^ {m} (C) - \mathbb {E} [ \hat {\mu} _ {h} ^ {m} (C) ] | \geq \sqrt {\frac {1 6 \log (T / \delta)}{n _ {h}}}\right) \leq \frac {2 \delta}{T ^ {8}}.
+$$
+
+By the Lipschitz condition of  $\mu$  that  $|\mathbb{E}[\hat{\mu}_h^m (C)] - \mu (x)|\leq r_h,\forall x\in C$  , we have
+
+$$
+\mathbb {P} \left(\sup  _ {x \in C} | \mu (x) - \hat {\mu} _ {h} ^ {m} (C) | \leq r _ {h} + \sqrt {\frac {1 6 \log (T / \delta)}{n _ {h}}}\right) \geq 1 - \frac {2 \delta}{T ^ {8}}.
+$$
+
+Any cube  $C \in \mathcal{A}_h^m$  is played for not less than 1 time, and thus  $|\mathcal{A}_h^m| \leq T$ . On the other hand, there are at most  $B_{stop} \leq T$  rounds. Taking a union bound over  $C \in \mathcal{A}_h^m$  and  $1 \leq h \leq m \leq B_{\mathrm{stop}}$ , we would have that  $\mathcal{E}$  holds with probability at least  $1 - 2\delta$ .
+
+We then show that the optimal arm survives all eliminations with high probability (Lemma 3).
+
+Lemma 3. Under event  $\mathcal{E}$  (defined in Lemma 2), the optimal arm  $x^{*} = \arg \max \mu (x)$  is not eliminated in any round.
+
+Proof. Fix any round  $m \in [B]$ . We use  $(C_h^m)^*$  to denote the cube containing  $x^*$  in  $\mathcal{A}_h^m$ . We will show that  $(C_h^m)^*$  is not eliminated in depth  $h$  for  $1 \leq h \leq m - 1$ . Under event  $\mathcal{E}$ , for any cube  $C \in \mathcal{A}_{m - 1}^{m - 1}$  and  $x \in C$ , we have
+
+$$
+\begin{array}{l} \hat {\mu} _ {m - 1} ^ {m - 1} (C) - \hat {\mu} _ {h} ^ {m} \left(\left(C _ {h} ^ {m}\right) ^ {*}\right) \leq \mu (x) + \sqrt {\frac {1 6 \log (T / \delta)}{n _ {m - 1}}} + r _ {m - 1} - \mu \left(x ^ {*}\right) + \sqrt {\frac {1 6 \log (T / \delta)}{n _ {h}}} + r _ {h} \\ \leq 2 r _ {m - 1} + 2 r _ {h} \leq 4 r _ {h}, \\ \end{array}
+$$
+
+where the second inequality holds for  $n_m = \frac{16\log(T / \delta)}{r_m^2}$  and the last inequality holds for decreasing property of Doubling Edge-length Sequence. By the elimination rule,  $(C_h^m)^*$  is not eliminated.
+
+Based on lemma 2 and 3, we can control the loss of cubes in each depth.
+
+Lemma 4. Under event  $\mathcal{E}$  (defined in Lemma 2), for any  $1\leq h\leq m\leq B_{stop}$ , any  $C\in \mathcal{A}_h^m$  and any  $x\in C,\Delta_x$  satisfies
+
+$$
+\Delta_ {x} \leq 8 r _ {h - 1}
+$$
+
+Proof. We again fix any round  $m \in [B]$ . For  $h = 1$ , this is straightforward from the Lipschitz condition of  $\mu$ . For  $h > 1$ , let  $\left(C_{m-1}^{m-1}\right)^*$  be the cube in  $\mathcal{A}_{m-1}^{m-1}$  such that  $x^* \in \left(C_{m-1}^{m-1}\right)^*$ . This cube  $\left(C_{m-1}^{m-1}\right)^*$  is well-defined under  $\mathcal{E}$  by Lemma 3. For any cube  $C \in \mathcal{A}_h^m$  and  $x \in C$ , it is obvious that  $x$  is also in the parent of  $C$  (the cube in the previous depth that contains  $C$ ), denoted as  $C_{\mathrm{par}}$ . For any  $x \in C$ , we have
+
+$$
+\Delta_ {x} = \mu^ {*} - \mu (x) \leq \hat {\mu} _ {m - 1} ^ {m - 1} \left(\left(C _ {m - 1} ^ {m - 1}\right) ^ {*}\right) + \sqrt {\frac {1 6 \log (T / \delta)}{n _ {m - 1}}} + r _ {m - 1} - \hat {\mu} _ {h - 1} ^ {m} \left(C _ {\mathrm {p a r}}\right) + \sqrt {\frac {1 6 \log (T / \delta)}{n _ {h - 1}}} + r _ {h - 1}.
+$$
+
+Due to our choice of  $n_m = \frac{16\log(T / \delta)}{r_m^2}$ , we have
+
+$$
+\begin{array}{l} \Delta_ {x} \leq \hat {\mu} _ {m - 1} ^ {m - 1} \left(\left(C _ {m - 1} ^ {m - 1}\right) ^ {*}\right) - \hat {\mu} _ {h - 1} ^ {m} \left(C _ {\text {p a r}}\right) + 2 r _ {m - 1} + 2 r _ {h - 1} \\ \leq \hat {\mu} _ {m - 1} ^ {m - 1} \left(\left(C _ {m - 1} ^ {m - 1}\right) ^ {*}\right) - \hat {\mu} _ {h - 1} ^ {m} \left(C _ {\mathrm {p a r}}\right) + 4 r _ {h - 1}, \\ \end{array}
+$$
+
+where the last inequality holds from the decreasing property of Doubling Edge-length Sequence. It is obvious that  $\hat{\mu}_{m - 1}^{m - 1}\left(\left(C_{m - 1}^{m - 1}\right)^{*}\right)\leq \tilde{\mu}^{m - 1}$ . Since the cube  $C_\mathrm{par}$  is not eliminated, we can easily derive the following from the elimination rule
+
+$$
+\Delta_ {x} \leq 4 r _ {h - 1} + 4 r _ {h - 1} \leq 8 r _ {h - 1}.
+$$
+
+![](images/bf30864d87fb9755cd306e97ada1bce5733db209d4c63d80192dea6b69d2b668.jpg)
+
+We are now ready to prove Theorem 1.
+
+Proof of Theorem 1. Let  $R_{m}$  denote regret of the  $m$ -th round. Fixing any positive number  $B$ , the total regret  $R(T)$  can be divided into two parts:  $R(T) = \sum_{m \leq B} R_{m} + \sum_{m > B} R_{m}$ . In the following, we bound two parts separately and then determine  $B$  to obtain the upper bound of the total regret.
+
+We firstly fix any round  $m$ . Recall that  $\mathcal{A}_h^m$  is a set of the survived cubes in depth  $h$ . According to Lemma 4, for any  $x \in \cup_{C \in \mathcal{A}_h^m} C$ , we have  $\Delta_x \leq 8r_{h-1} = 16r_h$ . Each cube in  $\mathcal{A}_h^m$  is a  $\|\cdot\|_{\infty}$ -ball with radius  $\frac{r_h}{2}$ , and is a subset of  $S(16r_h)$ . Therefore,  $\mathcal{A}_h^m$  forms a  $\left(\frac{r_h}{2}\right)$ -packing of  $S(16r_h)$ , and the definition of zooming dimension yields that
+
+$$
+\left| \mathcal {A} _ {h} ^ {m} \right| \leq N _ {r _ {h}} \leq C _ {z} r _ {h} ^ {- d _ {z}} \leq C _ {z} 2 ^ {(h - 1) d _ {z}},
+$$
+
+where the last inequality holds since  $r_h = 2^{-h + 1}$ . The total regret of the  $m$ -th round is
+
+$$
+\begin{array}{l} R _ {m} = \sum_ {h = 1} ^ {m} \sum_ {C \in \mathcal {A} _ {h} ^ {m}} \sum_ {i = 1} ^ {n _ {h}} \Delta_ {x _ {C, i} ^ {m}} \leq \sum_ {h = 1} ^ {m} | \mathcal {A} _ {h} ^ {m} | \cdot \frac {1 6 \log (T / \delta)}{r _ {h} ^ {2}} \cdot 1 6 r _ {h} \\ \leq \sum_ {h = 1} ^ {m} C _ {z} 2 ^ {(h - 1) d _ {z}} \cdot \frac {2 5 6 \log (T / \delta)}{r _ {h}} = \sum_ {h = 1} ^ {m} C _ {z} 2 ^ {(h - 1) (d _ {z} + 1)} \cdot 2 5 6 \log (T / \delta). \\ \end{array}
+$$
+
+Therefore, we have
+
+$$
+\begin{array}{l} \sum_ {m \leq B} R _ {m} = \sum_ {m \leq B} \sum_ {h = 1} ^ {m} C _ {z} 2 ^ {(h - 1) (d _ {z} + 1)} \cdot 2 5 6 \log (T / \delta) \\ = \sum_ {m \leq B} (B - m + 1) C _ {z} 2 ^ {(m - 1) \left(d _ {z} + 1\right)} \cdot 2 5 6 \log (T / \delta). \\ \end{array}
+$$
+
+By standard calculation, we should have that for any real number  $a$ ,
+
+$$
+\begin{array}{l} \sum_ {m \leq B} (B - m + 1) a ^ {m} \\ = \sum_ {m \leq B} (B + 1) a ^ {m} - \sum_ {m \leq B} m a ^ {m} = (B + 1) \sum_ {m \leq B} a ^ {m} - \left(\frac {B}{a - 1} a ^ {B + 1} - \frac {1}{a - 1} \sum_ {m \leq B} a ^ {m}\right) \\ = \left(B + 1 + \frac {1}{a - 1}\right) \cdot \sum_ {m \leq B} a ^ {m} - \frac {B}{a - 1} a ^ {B + 1} \leq \left(B + 1 + \frac {1}{a - 1}\right) \cdot \frac {a ^ {B + 1}}{a - 1} - \frac {B}{a - 1} a ^ {B + 1} \\ = \left(1 + \frac {1}{a - 1}\right) \cdot \frac {a ^ {B + 1}}{a - 1} \leq 2 a ^ {B}, \\ \end{array}
+$$
+
+which gives that
+
+$$
+\begin{array}{l} \sum_ {m \leq B} R _ {m} = 2 5 6 C _ {z} \log (T / \delta) \cdot 2 ^ {- (d _ {z} + 1)} \sum_ {m \leq B} (B - m + 1) \left(2 ^ {(d _ {z} + 1)}\right) ^ {m} \\ \leq 5 1 2 C _ {z} \log (T / \delta) \cdot 2 ^ {(B - 1) (d _ {z} + 1)}. \\ \end{array}
+$$
+
+On the other hand, Lemma 4 implies that the arm corresponding to  $\tilde{\mu}^{B - 1}$  must satisfy that  $\Delta_x\leq$ $8r_{B - 1} = 16r_B = 16\cdot 2^{-B + 1}$ . Therefore, we finally have
+
+$$
+R (T) = \sum_ {m \leq B} R _ {m} + \sum_ {m > B} R _ {m} \leq 5 1 2 C _ {z} \log (T / \delta) \cdot 2 ^ {(B - 1) (d _ {z} + 1)} + 1 6 \cdot 2 ^ {- B + 1} T.
+$$
+
+This inequality holds for any positive  $B$ . By choosing  $B^{*} = 1 + \frac{\log\frac{T}{\log(T / \delta)}}{d_{z} + 2}$ , we have
+
+$$
+R (T) \leq (5 1 2 C _ {z} + 1 6) \cdot T ^ {\frac {d _ {z} + 1}{d _ {z} + 2}} (\log (T / \delta)) ^ {\frac {1}{d _ {z} + 2}}.
+$$
+
+![](images/c190695c64ca6b14db0967f3e092d895e8ee6bf073f8e90a9eb4a516ff2da9a9.jpg)
+
+# 4 SPACE LOWER BOUND
+
+In this section, we present space lower bounds for Lipschitz bandit problem as summarized in Theorem 2. Essentially, Theorem 2 says that if we want to achieve the optimal regret bound, we must consume at least  $\Omega (\log T)$  bits. Similar to most works, we would first construct problem instances that are difficult to differentiate and then prove lower bounds on these instances.
+
+# 4.1 HARD CASE
+
+We would construct a set of problem instances that are difficult to distinguish. Let  $r = \frac{1}{T^{\frac{1}{d_z + 2}}}$  and  $K = \frac{c_z}{r^{d_z}} = c_z T^{\frac{d_z}{d_z + 2}}$ . Here  $c_z$  is a small enough constant satisfying by the definition of zooming dimension, we can find a set of arms  $\mathcal{U} = \{u_1, \dots, u_K\}$  such that  $d(u_i, u_j) \geq r$  for any  $i \neq j$ . Then we consider a set of problem instances  $\mathcal{I} = \{I_1, \dots, I_K\}$ . The expected reward for  $I_i$  is
+
+$$
+\mu_ {i} (x) = \left\{ \begin{array}{l} \frac {1}{2} + r, \text {i f} x = u _ {i}, \\ \max  \left\{\frac {1}{2}, \mu_ {i} (u _ {i}) - d (x, u _ {i}) \right\}, \text {o t h e r w i s e}. \end{array} \right.
+$$
+
+Note that here we would have that  $\log K = \frac{d_z}{d_z + 2}\log c_z\log T = \Theta (\log T)$
+
+# 4.2 PROOF OF THEOREM 2
+
+We consider the "best-cube identification" problem that after  $t$  rounds, the algorithm outputs a cube  $y_{t}$ : a prediction for which cube is optimal (has the highest mean reward). For this problem, we may find that if our space is small, the prediction quality will be very poor.
+
+Lemma 5. Consider "best-cube identification" problem with  $K \geq 10$  and space complexity less than  $\frac{1}{2} \log(K)$ . Then there exists at least  $\left\lceil \frac{K}{3} \right\rceil$  cubes that, for problem instances in Section 4.1,
+
+$$
+\operatorname * {P r} \left[ y _ {t} = i \mid \mathcal {I} _ {i} \right] <   \frac {3}{4}.
+$$
+
+Proof. We consider that summing up all the instance cases and use  $M$  to denote the memory state.
+
+$$
+\begin{array}{l} \sum_ {i = 1} ^ {K} \Pr \left[ y _ {t} = i \mid \mathcal {I} _ {i} \right] = \sum_ {i = 1} ^ {K} \sum_ {m} \Pr \left[ y _ {t} = i \mid M = m, \mathcal {I} _ {i} \right] \Pr \left[ M = m \mid \mathcal {I} _ {i} \right] \\ = \sum_ {i = 1} ^ {K} \sum_ {m} \Pr \left[ y _ {t} = i \mid M = m \right] \Pr \left[ M = m \mid \mathcal {I} _ {i} \right] \leq \sum_ {i = 1} ^ {K} \sum_ {m} \Pr \left[ y _ {t} = i \mid M = m \right] \\ = \sum_ {m} \sum_ {i = 1} ^ {K} \Pr \left[ y _ {t} = i \mid M = m \right] = \sum_ {m} 1 \leq 2 ^ {\frac {1}{2} \log (K)} = K ^ {\frac {1}{2}}, \\ \end{array}
+$$
+
+where the second equation holds since the prediction is determined by the memory state, and the last inequation is due to our bound of space complexity. To prove the lemma, assume for contradiction that we have more than  $\frac{2K}{3}$  cubes with  $\operatorname*{Pr}[y_t = i\mid \mathcal{I}_i]\geq \frac{3}{4}$ . We would have that
+
+$$
+\sum_ {i = 1} ^ {K} \Pr \left[ y _ {t} = i \mid \mathcal {I} _ {i} \right] \geq \frac {2 K}{3} \times \frac {3}{4} = \frac {1}{2} K > K ^ {\frac {1}{2}},
+$$
+
+where the last inequation holds for  $K \geq 10$  and leads to the contradiction.
+
+![](images/8fc6646f7421e499240edf6e6a104f3ddd9dbeca14faab9bb6a13ac1c62c4099.jpg)
+
+Lemma 5 immediately implies this corollary.
+
+Corollary 1. Consider a "best-cube identification" problem with  $K \geq 10$  and space complexity less than  $\frac{1}{2} \log(K)$ . Choose a cube  $i$  uniformly at random, and run the algorithm on instance  $\mathcal{I}_i$ . Then  $\operatorname*{Pr}[y_t \neq i] \geq \frac{1}{12}$ , where the probability is over the choice of cube  $i$ , the randomness in rewards and the algorithm.
+
+Proof. By Lemma 5, we can find a set  $\mathcal{A}$  such that
+
+$$
+\Pr \left[ y _ {t} = i \mid \mathcal {I} _ {i} \right] <   \frac {3}{4}, \forall i \in \mathcal {A}, | \mathcal {A} | \geq \lceil \frac {K}{3} \rceil .
+$$
+
+Since we choose the cube uniformly at random, we would have that
+
+$$
+\Pr \left[ y _ {t} \neq i \right] \geq \frac {1}{K} \sum_ {i = 1} ^ {K} \Pr \left[ y _ {t} \neq i \mid \mathcal {I} _ {i} \right] \geq \frac {1}{K} \sum_ {i \in \mathcal {A}} \Pr \left[ y _ {t} \neq i \mid \mathcal {I} _ {i} \right] \geq \frac {1}{K} \cdot \lceil \frac {K}{3} \rceil \cdot \frac {1}{4} \geq \frac {1}{1 2}
+$$
+
+![](images/affd5e43421c99fa4a6805ad54fad7865d5e97d2a52c619a5e14cfc8aca2d220.jpg)
+
+Finally, we use Corollary 1 to finish our proof of space complexity.
+
+Proof of Theorem 2. For  $T$  larger than a large enough constant, we would have that  $K = c_{z}T^{\frac{d_{z}}{d_{z} + 2}} \geq 10$ . Assume for contradiction that we have space complexity less than  $\frac{1}{2}\log (K)$ . Fix round  $t$ , Let us interpret the algorithm as a "best-cube identification" algorithm, where the prediction is the closest  $u_{i}$  to the arm  $x$  that the algorithm chooses. Note that here choosing the closet  $u_{i}$  will only decrease the regret by our hard instance setting. We can apply Corollary 1 to have that  $\operatorname*{Pr}[y_t \neq i] \geq \frac{1}{12}$ . In words, the algorithm chooses a non-optimal cube with probability at least  $\frac{1}{12}$ , and choosing a non-optimal cube will incur  $\Delta(y_{t}) = \mu^{*} - \mu_{i}(u_{y_{t}}) = r$  regret. Therefore,
+
+$$
+\mathbb {E} [ R (T) ] \geq \sum_ {t = 1} ^ {T} \mathbb {E} [ \Delta (y _ {t}) ] = \sum_ {t = 1} ^ {T} \Pr [ y _ {t} \neq i ] \cdot r \geq \frac {T}{1 2} \cdot \frac {1}{T ^ {\frac {1}{d _ {z} + 2}}} \geq \frac {1}{1 2} T ^ {\frac {d _ {z} + 1}{d _ {z} + 2}},
+$$
+
+which leads to contradiction.
+
+![](images/7e0588f86c7290b7f0c6accf3e6afc6d0fb60720c497d33b7c272285d8d117b4.jpg)
+
+# 5 EXPERIMENTS
+
+![](images/b883134b4dbe659661d363bce1d273c6fa585b16f57b033d8396695e98225c0e.jpg)  
+$x_{1} = (0.3,0.1)$  
+$x_{2} = (0.9,0.9)$
+
+![](images/7c96b868921832bf5b601203e4b78a162c797f433ddb0bd7f7b0df241127fcaf.jpg)  
+$x_{1} = (0.8,0.7)$  
+$x_{2} = (0.7,0.3)$  
+Figure 2: the landscape of  $\mu$ .
+
+![](images/ac3715d11de19a039648450e5669e2db8b6d2a91801caf647960a51e58d38188.jpg)  
+$x_{1} = (0.45, 0.55)$  
+$x_{2} = (0.1,0.9)$
+
+In this section, we evaluate the Log-Li algorithm. In the experiment, the time horizon  $T = 100,000$ , the arm space is  $[0,1]^2$  and the expect reward function is  $\mu (x) = 1 - \| x - x_1\| _2 - 0.5\| x - x_2\| _2$  for different values of  $x_{1}$  and  $x_{2}$ . The landscape of  $\mu$  can be found in figure 2, where the optimal arm is always at  $x^{\star} = x_{1}$ . We select A-BLiN(Feng et al., 2022) and Zooming(Kleinberg et al., 2008) for comparison. Zooming algorithm achieves an optimal regret rate but with a relatively high time complexity of  $O(T^2)$  and a space complexity of  $O(T)$ ; A-BLiN delivers performance comparable to the Zooming algorithm while utilizing space proportional to  $O\left(T^{\frac{d_z + 1}{d_z + 2}}(\log T)^{-\frac{d_z + 1}{d_z + 2}}\right)$  and operating in  $O(T)$  time. We will compare the regret rate and space complexity of three algorithms to illustrate the sublinear trend of regret and space efficiency of our algorithm.
+
+![](images/2877f92c5736df723e76e71f9d70772d3f9a125ede7bc6fa78a99c969dc64773.jpg)  
+$x_{1} = (0.3,0.1)$
+
+![](images/5c2279c9c63e5885afc214fa869a09d62f15e10c409e51739062135176f55089.jpg)  
+$x_{2} = (0.9,0.9)$  
+$x_{1} = (0.8,0.7)$  
+$x_{2} = (0.7,0.3)$
+
+![](images/fc06713de3e27436fd0a6364b8354c98c53c2e25e5a9847de821ddd90eece3cf.jpg)  
+$x_{1} = (0.45, 0.55)$  
+$x_{2} = (0.1,0.9)$
+
+![](images/8725f90957b6092033f197f3c9feae884ccfb3075479491163928c0f16baf83a.jpg)  
+Figure 3: The cumulative regret v.s. time horizon  $t$ .  
+Figure 4: Number of words Saved by Log-Li, A-BLiN and Zooming for Different  $\mu$ .
+
+Figure 3 illustrates the performance of the three algorithms under different expect reward  $\mu$ . When  $t$  is small, all of the three algorithms are in their initial exploration phases, leading to similar performance. As  $t$  increases, the performance of Log-Li tends to be relatively worse compared to the other two algorithms because it restarts the search from depth 1 each time to save space consumption. This could also be a reason why Log-Li experiences increases in regret during the latter stages and exhibits more instability during the exploration process. However, we should also notice that as the search progresses further, Log-Li ultimately achieves a sublinear regret curve, with a growth trend similar to that of A-BLiN. This indicates that the regret rate of Log-Li is asymptotically optimal even with limited space, which is consistent with our analysis in section 3.
+
+We also report the space usage of the three algorithms in figure 4. It is clear that Log-Li significantly reduces the number of arms needed to save and thus consumes much less space. It is noteworthy that this gap will further widen as the time horizon  $T$  increases because Log-Li needs to save information about only 3 arms regardless of  $T$ , whereas the other two algorithms require saving statistics of an increasing number of arms as  $T$  grows.
+
+# 6 CONCLUSIONS
+
+In this study, we explore the space complexity for the Lipschitz bandit problem, presenting both upper and lower bounds. Our algorithm, Log-Li, is proven to achieve the optimal regret rate using only  $O(\log T)$  bits, whereas prior algorithms require  $poly(T)$  bits. This significant improvement in space efficiency greatly reduces memory costs and suggests that there is almost no tradeoff between regret and space efficiency for the Lipschitz bandit problem. Furthermore, the space lower bounds for the Lipschitz bandit indicate that achieving the optimal regret bound necessitates consuming at least  $\Omega (\log T)$  bits. Hence, Log-Li is optimal in terms of both regret and space. We also conduct experiments to visualize the performance of our algorithm Log-Li. Our work provides novel insights into designing memory-limited bandit algorithms. It would be valuable to apply the principles of Log-Li to other related bandit problems.
+
+# 7 ACKNOWLEDGEMENTS
+
+This work is supported by National Natural Science Foundation of China No. 62276066, No. U2241212.
+
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+David Silver, Aja Huang, Chris J Maddison, Arthur Guez, Laurent Sifre, George Van Den Driessche, Julian Schrittwieser, Ioannis Antonoglou, Veda Panneershelvam, Marc Lanctot, et al. Mastering the game of go with deep neural networks and tree search. nature, 529(7587):484-489, 2016.  
+Vaidehi Srinivas, David P Woodruff, Ziyu Xu, and Samson Zhou. Memory bounds for the experts problem. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pp. 1158-1171, 2022.  
+William R Thompson. On the likelihood that one unknown probability exceeds another in view of the evidence of two samples. Biometrika, 25(3-4):285-294, 1933.  
+Christopher Williams and Matthias Seeger. Using the nystrom method to speed up kernel machines. Advances in neural information processing systems, 13, 2000.
\ No newline at end of file
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+# LIVECODEBENCH: HOLISTIC AND CONTAMINATION FREE EVALUATION OF LARGE LANGUAGE MODELS FOR CODE
+
+Naman Jain† King Han† Alex Gu* $ Wen-Ding Li*‡ Fanjia Yan*† Tianjun Zhang*†
+
+Sida I. Wang Armando Solar-Lezama $^{\ddagger}$  Koushik Sen $^{\dagger}$  Ion Stoica $^{\dagger}$
+
+† UC Berkeley $ MIT ‡ Cornell
+
+{naman_jain,kingh0730,fanjiayan,tianjunz,ksen,istoica}@berkeley.edu  
+{gua,asolar}@csail.mit.edu wl678@cornell.edu sidawang88@gmail.com
+
+# ABSTRACT
+
+Large Language Models (LLMs) applied to code-related applications have emerged as a prominent field, attracting significant interest from academia and industry. However, as new and improved LLMs are developed, existing evaluation benchmarks (e.g., HUMANEVAL, MBPP) are no longer sufficient for assessing their capabilities suffering from data contamination, overfitting, saturation, and focus on merely code generation. In this work, we propose LIVECODEBENCH, a comprehensive and contamination-free evaluation of LLMs for code, which collects new problems over time from contests across three competition platforms, LEETCODE, ATCODER, and CODEFORCES. Notably, our benchmark also focuses on a broader range of code-related capabilities, such as self-repair, code execution, and test output prediction, beyond just code generation. Currently, LIVECODEBENCH hosts over six hundred coding problems that were published between May 2023 and Aug 2024. We evaluate over 50 LLMs on LIVECODEBENCH (LCB for brevity) presenting the largest evaluation study of code LLMs on competition problems. Based on the study, we present novel empirical findings on contamination, overfitting, and holistic evaluations. We demonstrate that time-segmented evaluations serve as a robust approach to evade contamination; they are successful at detecting contamination across a wide range of open and closed models including GPT-4-O, CLAUSEDEPSEK, and CODESTRAL. Next, we highlight overfitting and saturation of traditional coding benchmarks like HUMANEVAL and demonstrate LCB allows more reliable evaluations. Finally, our holistic evaluation scenarios allow for measuring the different capabilities of programming agents in isolation.
+
+# 1 INTRODUCTION
+
+Code has emerged as an important application area for LLMs, with a proliferation of code-specific models (Chen et al., 2021; Austin et al., 2021; Nijkamp et al., 2022; Li et al., 2023b; Roziere et al., 2023; Guo et al., 2024; Ridnik et al., 2024; Lozhkov et al., 2024) and their applications across various tasks such as program repair (Zheng et al., 2024; Olausson et al., 2023), optimization (Madaan et al., 2023a), test generation (Steenhoek et al., 2023), documentation (Luo et al., 2024), SQL (Sun et al., 2023). In contrast with these rapid advancements, evaluations have remained relatively stagnant, and current benchmarks like HUMANETVAL and MBPP paint a misleading picture. Firstly, while coding is a multi-faceted skill, these benchmarks only focus on code generation, thus overlooking broader code-related capabilities. Moreover, these benchmarks suffer from contamination or overfitting, as benchmark samples are present in the training datasets.
+
+![](images/cf571a3e5147eb8ebfca08c94a58310262c9377c316e0a82a139bb51edd6a851.jpg)  
+Figure 1: Left. LIVECODEBENCH comprises problems marked with release dates, allowing evaluations over different time windows. The figure depicts performances of DEEPSEEK, GPT-4-O, and CLAUSED3s models over bimonthly time windows ( $\sim$ 40 LEETCODE problems) showcasing a stark drop after their cutoff dates, highlighting contamination. Thus, we can detect and avoid contamination by evaluating models only on time-windows after the model's cutoff date (green region) $^{1}$ . Right. We evaluate LLMs across four scenarios that capture key coding capabilities necessary for building programming agents: code generation, repair, testing, and comprehension. The figure depicts various model performances across the scenarios available in LIVECODEBENCH in a radial plot - highlighting relative differences changing across the scenarios.
+
+![](images/4cb48cf181628b1390735ab96de808b639089638e60aad7d3926e877b193044a.jpg)
+
+Motivated by these shortcomings, we introduce LIVECODEBENCH, built on the following principles:
+
+1. Live updates to prevent contamination. LLMs are trained on massive inscrutable corpora and current benchmarks suffer from the risk of data contamination as they could be included in those training datasets. While previous works have attempted decontamination using both exact and fuzzy matches (Li et al., 2023b;d), it can be a non-trivial task (Team, 2024; Yang et al., 2023). Existing competition programming benchmarks (like APPS and CODESCOPE) are already contaminated and may fail to provide reliable evaluation of code LLM capabilities. In LIVECODEBENCH, to prevent the risk of problem contamination, we evaluate models on new problems tagged with a release date. Next, for newer models, we only consider problems released after the model's cutoff date to ensure that the model has not been trained on it as demonstrated in Figure 1 left.  
+2. Holistic Evaluation. Current code evaluations primarily focus on natural language to code generation. However, programming is a multi-faceted task that requires capabilities beyond those measured by code generation. These broader capabilities are even more relevant for constructing programming agents that can interact with the execution environment. Therefore, we evaluate the execution-feedback-based multi-turn coding using the self-repair scenario (Olausson et al., 2023), assess code comprehension capabilities using the code execution scenario (Gu et al., 2024), and introduce the test output prediction scenario to evaluate the models' test generation capabilities.  
+3. High-quality problems and tests. High-quality problems and tests are crucial for reliable evaluation of LLMs. However, existing benchmarks suffer from multiple deficiencies. First, existing competitive programming benchmarks (APPS, CODE-CONTESTS, XCODEVAL, CODESCOPE) contain problems not amenable to input-output-based auto-grading. For example, CODE-CONTESTS (Li et al., 2022) (page 39, second-to-last paragraph) reports that about twenty-five percent of the problems in the benchmark accept multiple correct outputs for a single input. This incorrectly penalizes correct solutions, adding noise to a considerable fraction of the benchmark. Next, prior benchmarks like HUMANEVAL and APPS contain insufficient tests, further exacerbating noise in evaluations (Liu et al. (2023a) reports  $8\%$  drop in model performance). In LIVECODEBENCH, we source high-quality problems from reputable platforms and implement heuristics to detect and remove problems not amenable to input-output-based auto-grading. Finally, for every problem, we provide a substantial number of tests (over 18 on average) for reliable and efficient evaluations. In contrast to
+
+Table 1: Comparing LCB with existing coding and competition programming benchmarks.  
+
+<table><tr><td>Benchmark</td><td>Contamination Prevention</td><td>Problem Curation</td><td>Robust Test Based Eval.</td><td>Varied Difficulties</td><td>Not Saturated</td><td>Broader Eval. Scenarios</td><td>Comp. Analysis across Models</td></tr><tr><td>HUMANETVAL (Chen et al., 2021)</td><td>X</td><td>✓</td><td>X</td><td>✓</td><td>X</td><td>X</td><td>✓</td></tr><tr><td>HUMANETVAL+ (Liu et al., 2023b)</td><td>X</td><td>✓</td><td>✓</td><td>✓</td><td>X</td><td>X</td><td>✓</td></tr><tr><td>MBPP (Austin et al., 2021)</td><td>X</td><td>✓</td><td>X</td><td>✓</td><td>X</td><td>X</td><td>✓</td></tr><tr><td>APPS (Hendrycks et al., 2021)</td><td>X</td><td>X</td><td>X</td><td>✓</td><td>✓</td><td>X</td><td>X</td></tr><tr><td>CODE-CONTESTS (Li et al., 2022)</td><td>X</td><td>X</td><td>✓</td><td>X</td><td>✓</td><td>X</td><td>X</td></tr><tr><td>XCODEVAL (Khan et al., 2023)</td><td>X</td><td>X</td><td>-</td><td>X</td><td>✓</td><td>✓2</td><td>X</td></tr><tr><td>CODESCOPE (Yan et al., 2023)</td><td>X</td><td>X</td><td>-</td><td>X</td><td>✓</td><td>✓2</td><td>X</td></tr><tr><td>TACO (Li et al., 2023c)</td><td>X</td><td>X</td><td>X</td><td>X</td><td>✓</td><td>X</td><td>X</td></tr><tr><td>USCAOBENCH (Shi et al., 2024)</td><td>X</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>X</td><td>X</td></tr><tr><td>LIVECODEBENCH (Ours)</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr></table>
+
+prior works, we also include several large tests designed for stress-testing solutions ensuring weak or worse complexity solutions do not pass test harnesses.
+
+4. Difficulty Guided Problem Curation. Competitive programming is a challenging domain even for strong LLMs. As a result, these problems can be unsuitable for meaningfully comparing today's open LLMs, because the variance in performance is low, often relying on less than  $1\%$  performance differences (within the margin of error). Therefore, we use problem difficulty ratings (sourced from the competition websites) to curate our problems, avoiding those that are too difficult for current models. In particular, we avoid collecting CODE-FORCES problems used by prior works (CODE-CONTESTS, CODESCOPE, XCODEVAL) since we find they do not sufficiently distinguish models due to model performances  $(\text{PASS} @ 1)$  tending to zero. Indeed, LCB easy and medium problems can separate 7B models that are indistinguishable on the hard subset (see DS-7B vs SC2-7B).
+
+With these principles in mind, we build LIVECODEBENCH, a continuously updated benchmark that avoids data contamination. Particularly, we collect 612 problems from contests occurring between May 2023 and Aug 2024 and use them to construct the different scenarios.
+
+Empirical Findings. We have evaluated over 50 (18 base models and 34 instruction-tuned) models across different LIVECODEBENCH scenarios. Based on this study and following analysis, we present novel empirical findings which have not been revealed in prior benchmarks.
+
+1. Contamination. We observe a stark drop in the performance of DEEPSEEK, GPT-4-O, CODESTRAL and CLAUDE-3S on LEETCODE problems released after Aug 2023, Oct 2023, Jan 2024, and April 2024 (Figure 1 left). These results highlight likely contamination in older problems and time-segmented evaluations prove effective for performing fair comparisons across varied set of models from both open and closed domains.  
+2. Holistic Evaluation. Our evaluations reveal that model performances are correlated across tasks, but the relative differences do vary. For example, in Figure 1 right, the gap between open and closed models further increases on tasks like self-repair or test output prediction. Similarly, CLAUSED-3-OPUS surpasses GPT-4 on the test output prediction scenario.  
+3. HUMANETVAL Overfitting. Upon comparing LIVECODEBENCH with HUMANETVAL, we find that models cluster into two groups, ones that perform well on both benchmarks, and others that perform well on HUMANETVAL but not on LIVECODEBENCH (see Figure 4). The latter group primarily comprises fine-tuned open-access models while the former comprises base and closed models. This indicates some level of overfitting to HUMANETVAL.  
+4. Model Comparisons (Figure 3). We provide model comparisons across different groups like base, instruct, open, and closed models and across groups like open vs closed models.
+
+Concurrent Work. (Huang et al., 2023) also evaluate LLMs in a time-segmented manner. However, they only focus on CODEFORCES problems, while we combine problems across platforms and additionally propose a holistic evaluation across multiple code-related scenarios. Liu et al. (2024) evaluate the code comprehension capabilities of LLMs using execution. (Singhal et al., 2024) also evaluates LLMs on more tasks but focus on non-functional-correctness aspects of programming. Shi et al. (2024) evaluate LLMs on USACO problems while we focus on simple competition scenarios.
+
+![](images/3bcdb6f45e4d9aba5cad8ccdd4bc72b83811ea076845c23ee4288c874ce0d03c.jpg)  
+Figure 2: Overview of the four scenarios present in LIVECODEBENCH.
+
+# 2 HOLISTIC EVALUATION
+
+Code capabilities of LLMs are evaluated and compared using natural language to code generation tasks. However, this only captures one dimension of code-related capabilities. AlphaCodium (Ridnik et al., 2024) developed an intricate LLM pipeline for solving competition coding problems. By combining natural language reasoning, test case generation, code generation, and self-repair, they achieve significant improvements over a naive direct code generation baseline, showcasing the importance of these broader capabilities. Motivated by this, we propose a holistic evaluation of LLMs using a suite of evaluation setups that capture a broader range of code-related capabilities.
+
+Specifically, we evaluate code LLMs in four scenarios, namely code generation, self-repair, code execution, and test output prediction. Our selection criterion was to pick settings that are useful components in code LLM workflows and in addition, have clear and automated evaluation metrics.
+
+Following we describe each of these scenarios in detail.
+
+Code Generation. The code generation scenario follows the standard setup for generating code from natural language. The model is given a problem statement, which includes a natural language description and example tests (input-output pairs), and is tasked with generating a correct solution. The evaluation is performed based on functional correctness, using a set of unseen test cases. We use the PASS@1 metric measured as the fraction of the problems for which the model was able to generate a program passing all tests. Figure 2 (left) provides an example of this scenario.
+
+Self Repair. The self-repair scenario is based on previous works that tested the self-repair capabilities of LLMs (Olausson et al., 2023; Shinn et al., 2023; Chen et al., 2023). Here, the model is given a problem statement from which it generates a candidate program (similar to the single-step code generation scenario above). However, in case of a mistake, the model is additionally provided with error feedback (either the exception message or a failing test case) and is tasked with generating the fixed solution. Similar to the code generation scenario, the evaluation is performed via functional correctness on the final program, i.e. either the single-step correct generation or the attempted repair. We use the PASS@1 metric to measure the combined performance after the repair step. Figure 2 (mid-left) provides an example of this scenario.
+
+Code Execution. The code execution scenario is based on the output prediction setup used in CRUXEVAL (Gu et al., 2024). The model is provided a program snippet consisting of a function (f) along with a test input to the program and is tasked with predicting the output of the program on the input test case. The evaluation is performed via an execution based on an exact match correctness metric. Figure 2 (right) provides an example of the code execution scenario.
+
+Test Case Output Prediction. Finally, we introduce a new task that is designed to study natural language reasoning and test generation. In this task, the model is given the problem statement along with a test case input, and it is tasked with generating the expected output for that input. This task follows a setup similar to the one used in CodeT (Chen et al., 2022), where tests are generated solely
+
+from problem statements, without the need for the function's implementation. A key difference is that we provide a fixed set of test inputs for each problem in our dataset, and the models are then prompted to only predict the expected output for those specific inputs. This approach allows for a straightforward evaluation of the test generation capabilities by avoiding test input prediction, a hard-to-evaluate task. Figure 2 (mid-right) provides an example of this scenario.
+
+# 3 BENCHMARK CURATION
+
+We curate our problems from three coding competition websites: LEETCODE, ATCODER, and CODE-FORCES. These websites periodically host contests containing problems that assess the coding and problem-solving skills of participants. The problems consist of a natural language problem statement along with example input-output examples, and the goal is to write a program that passes a set of hidden tests. Further, thousands of participants participate, solving these problems thus ensuring that the problems are vetted for clarity.
+
+# 3.1 DATA COLLECTION
+
+We have written automated HTML scrapers for each of the above websites to collect problems and the corresponding metadata. To ensure quality and consistency, we parse mathematical formulas and exclude problems with images. We also exclude problems that are not suitable for grading by input-output examples, such as those that accept multiple correct answers or require the construction of data structures. Specifically, we use keyword-based heuristic filters to filter interactive problems and problems accepting multiple correct solutions. Besides parsing the problem descriptions, we also collect associated ground truth solutions and test cases whenever directly available. Thus for each problem, we collect tuples of natural language problem statement  $P$ , test cases  $T$ , and ground truth solution  $S$ . Finally, we associate the contest date  $D$  to mark the release date of each problem and use the collected attributes to construct problems for our four scenarios (detailed in Section C.2 ahead). Note that this process is completely automated and human involvement is only involved in modifying high-level design decisions such as updating problem difficulty settings and improving the keyword-based heuristics for filtering problems over different "live updates".
+
+Scrolling through time. As noted, we associate the contest date  $D$  for each problem. The release date allows us to measure the performance of LLMs over different time windows by filtering problems based on whether the problem release date falls within a time window (referred to as "scrolling" through time). This is crucial for evaluating and comparing models trained at different times. Specifically, for a new model and the corresponding cutoff date (normalized to the release date if the training cutoff date is not published), we can measure the performance of the model on benchmark problems released after the cutoff date. We have developed a UI that allows comparing models on problems released during different time windows (Figure 10).
+
+Test collection. Tests are crucial for assessing the correctness of the generated outputs and are used in all four scenarios. We collect tests available on platform websites whenever possible and use them for the benchmark. Otherwise, following (Liu et al., 2023b), we use a LLM (here GPT-4-TURBO) to generate test inputs for the problems. A key difference between our test generation approach is that instead of generating inputs directly using the LLM, we construct generators that sample inputs based on the problem specifications using in-context learning. Details of this approach can be found in Section A.2. Finally, we collect a small fraction of failing tests from the platforms for robust adversarial tests. Section A.4 describes our design decisions to determine the number of tests.
+
+Problem difficulty. Competition programming has remained a challenge for LLMs, with GPT-4 achieving an average CODEFORCES rating (ELO) of 392, placing it in the bottom 5 percentile (OpenAI, 2023). This makes it difficult to compare LLMs, as the variation in performance across models is low. In LIVECODEBENCH, we collect problems of diverse difficulties as labeled in competition platforms, excluding problems that are rated above a certain threshold that are likely too difficult for even the best models<sup>3</sup>. Further, we use these ratings to classify problems as EASY, MEDIUM, and HARD for more granular model comparisons, estimating difficulty roughly based on HUMANEVAL scores.
+
+We defer the platform and scenario specific curation details to the Appendix (Section C)
+
+# 4 EXPERIMENT SETUP
+
+We describe the experimental setup in this section. First, we provide the common setup across the scenarios, followed by the scenario-specific setups in Section 4.1.
+
+Models. We evaluate 52 models across various sizes, ranging from 1.3B to 70B, including base models, instruction models, and both open and closed models. Our experiments include models from different classes, such as GPTs, CLAUSES, GEMINIS, MISTRAL, LLAMA-3S, DEEPSEEKS, CODELLAMAS, STARCODER2, CODEQWEN and their variants. Appendix D.1 provides the list of models.
+
+Evaluation Metrics. We use the PASS@1 (Kulal et al., 2019; Chen et al., 2021) metric for our evaluations. Specifically, we generate 10 candidate answers for each problem from model providers or using vLLM (Kwon et al., 2023). We use nucleus sampling with temperature 0.2 and top_p 0.95 and calculate the fraction of correct solutions.
+
+# 4.1 SCENARIO-SPECIFIC SETUP
+
+The setup for each scenario is presented below. Note that the base models are only used in the code generation scenario since they do not easily follow the format for the other scenarios.
+
+Code Generation. For the instruction-tuned models, we use a zero-shot prompt and follow the approach of Hendrycks et al. (2021) by adding appropriate instructions to generate solutions in either functional or stdin format (one-shot for base-models). Section D.2 depicts the prompt.
+
+Self Repair. Similar to prior work Olausson et al. (2023), we use the programs generated during the code generation scenario along with the corresponding error feedback to build the zero-shot prompt for the self-repair scenario. The error feedback can be syntax errors, runtime errors, wrong answers, and time-limit errors. Section D.3 provides error feedback and the corresponding prompt.
+
+Code Execution. We use few-shot prompts for the code execution scenario, both with and without chain-of-thought prompting (COT). Particularly, we use a 2-shot prompt without COT and a 1-shot prompt with COT with manually detailed steps. The prompts are detailed in Section D.4.
+
+Test Output Prediction. We use a zero-shot prompt that queries the model to complete assertions, given the problem, function signature, and test input. We provide the prompt in Section D.5.
+
+# 5 RESULTS
+
+We first describe how LIVECODEBENCH helps detect and avoid benchmark contamination in Section 5.1. Next, we present the findings from our evaluations on LIVECODEBENCH in Section 5.2.
+
+# 5.1 AVOIDING CONTAMINATION
+
+Contamination in DEEPSEEK and GPT-4-O. LIVECODEBENCH curates problems released since May 2023. However, DEEPSEEK was released Sep 2023 and GPT-4-O's official cutoff date is Nov 2023. We can measure the performance of these models on the benchmark problems released after these dates, thereby estimating the performance of the model on previously unseen problems. Figure 1 shows the performance of these models on LEETCODE problems released in different months from May 2023 and Aug 2024. First, we notice stark drops in model performance - DS-INS-33B model after Aug. 2023, GPT-4-O model after Nov. 2023, CLAUDE-3s model after Apr. 2023. These drops align with their release or cutoff dates which suggests that the earlier problems might indeed be contaminated. This trend is consistent across other LIVECODEBENCH scenarios like repair and code execution, as depicted in Figure 11. Concurrently, (Guo et al., 2024) (Section 4.1, last paragraph) also acknowledge the possibility of LEETCODE contamination, noting that "models achieved higher scores in the LeetCode Contest held in July and August".
+
+Interestingly, we find that this drop in performance primarily occurs for the LEETCODE problems only and that the model performance is relatively smooth across the months for problems from other platforms. Figure 12 shows a relatively stable performance for all models on ATCODER problems.
+
+Performances of other models. We study performance variations in other models released more recently. Particularly, GPT-4-TURBO, MISTRAL-L, and CLAUSEDs models were released in Nov'2023,
+
+![](images/80fe172012fb83d65b6506f76e2b1948dec567b6c41e5ee1bfdfb150a93c03de.jpg)
+
+![](images/5959cab75daed33ecc05ec0be411f795538ce2116270943924c89b4b46cf383c.jpg)
+
+![](images/d7efae3527da24a95e7e4e047d5483d803cbb9c9208c314e4e067ab551a48263.jpg)  
+Figure 3: Model performances across the four scenarios available in LIVECODEBENCH.
+
+![](images/6886459b93b8246862fb48352373a96183e38199a3cb44f12096997bc135c1db.jpg)
+
+Feb'2024, and Mar'2024 respectively. Empirically, we do not observe significant performance variations across the months, as shown in Figure 13. Interestingly, we find that even the DS-BASE-33B model also suffers from contamination dropping from  $\mathrm{PASS} @ 1 \sim 60$  in May problems to  $\mathrm{PASS} @ 1 \sim 0$  in September LEETCODE problems. Finally, CODESTRAL achieves  $\mathrm{PASS} @ 1$  36.5 on problems released between May'23 and Jan'24 and  $\mathrm{PASS} @ 1$  28.3 on problems post Jan'24.
+
+# 5.2 PERFORMANCE AND MODEL COMPARISONS
+
+We provide our model comparison findings here (and in Appendix E). To overcome contamination issues in DEEPSEEK models, we only consider problems released since Sep 2023 for all evaluations below. Figure 3 shows the performance of a subset of models across the four scenarios.
+
+Holistic Evaluations. We have evaluated the models across the four scenarios currently available in LIVECODEBENCH. Figure 1 displays the performance of models on all scenarios along the axes of the polar chart. First, we observe that the relative order of models remains mostly consistent across the scenarios. This is also supported by high correlations between PASS@1 metric across the scenarios – over 0.88 across all pairs as shown in Figure 14. However, despite the strong correlation, the relative differences in performance do vary across the scenarios. For example, GPT-4-TURBO further gains performance gap over GPT-4 in the self-repair scenario after already leading in the code generation scenario. Similarly, CLAUDE-3-OPUS and MISTRAL-L perform well in tasks involving COT, particularly in the code execution and test output prediction scenarios. For instance, CLAUDE-3-OPUS even outperforms GPT-4-TURBO in the test output prediction scenario. These differences highlight the need for holistic evaluations beyond measuring code generation capabilities.
+
+Comparison to HUMANEVAL. Next, we compare how code generation performance metrics translate between LIVECODEBENCH and HUMANEVAL, the primary benchmark used for evaluating coding capabilities. Note that we use HUMANEVAL+ providing more accurate evaluations. Figure 4 shows a scatter plot of PASS@1 on HUMANEVAL+ versus LCB-Easy code generation scenario. We find only a moderate correlation of 0.72, with much larger performance variations on LCB-Easy.
+
+Additionally, we observe that the models cluster into two groups, shaded in red and green. The models in the green-shaded region lie close to the  $x = y$  line, indicating that they perform similarly on both
+
+![](images/f94038474db0842c68b2c37cdf3247f6b3aaae1860df325d7385f501c15db0b7.jpg)  
+LCB-Easy vs HUMANETVAL+  
+PASS@1 on LCB-Easy
+
+GPT-4-Turbo  
+GPT-3.5-Turbo  
+GPT-4  
+Gemini-Pro-1.5
+
+Gemini-Flash-1.5  
+★Claude-3-O  
+Claude-3-S  
+Claude-Ins-1
+
+Mistral-L  
+Mix-8x22B-Ins  
+Phind-34B  
+MC-6.7B
+
+![](images/f81dd61ecdb44ec1adf78a3418981869a0bb26a6be3a11bac4e419330205784c.jpg)  
+Figure 4: Scatter plot comparing PASS@1 of models on HUMANETVAL+ versus LCB-Easy (time-window Sep 2023 to May 2024). We find that the models are separated into two groups – the green-shaded region where performances on the two datasets are aligned and the red-shaded region where models perform well on HUMANETVAL+ but perform poorly on LIVECODEBENCH. This indicates potential overfitting on HUMANETVAL+ and primarily occurs in the fine-tuned variants of open-access models. For example, DS-INS-1.3B achieves PASS@1 of 60 and 26 on HUMANETVAL+ and LCB-Easy subset. Thus, while it ranks above CMD-R+ ( $\sim$ 100x larger!) on HUMANETVAL+, it performs significantly worse on the LCB. Similarly, DS-INS-6.7B and CODEQWEN outperform SONNET on HUMANETVAL+ but are  $>20$  points behind on LCB-Easy.
+
+LLama3-70b-Ins  
+LLama3-8b-Ins  
+Lamae-8b-Base
+
+![](images/645280522297886580cbf39ad580a9c1c67d09ee4f7890ec00a8bcbebb1e1ef8.jpg)
+
+□DS-Base-6.7B   
+DS-Base-13B  
+SC2-Base-15B
+
+![](images/f2d07ea6a0874e620f130969722701781692c0da97579936a00097c788d0a18e.jpg)
+
+DS-Ins-6.7b  
+DS-Ins-1.3b  
+CodeQwen-Chat
+
+benchmarks. On the other hand, models in the red shaded region perform well on HUMANEVAL+ but not as well on LIVECODEBENCH. Interestingly, the green-shaded cluster contains base models or closed-access models, while the red-shaded cluster primarily comprises fine-tuned variants of open-access models. The well-separated clusters suggest that many models that perform well on HUMANEVAL might be overfitting on the benchmark, and their performances do not translate well to problems from other domains or difficulty levels like those present in LIVECODEBENCH. Indeed, HUMANEVAL has small and isolated programming problems and thus easier to overfit. In contrast, LIVECODEBENCH problems are sourced from reputable coding platforms offering more challenging problems with higher diversity and difficulty levels. We detail instances of this overfitting (DS-INS-1.3B, DS-INS-6.7B, and CODEQWEN) in Figure 4 caption.
+
+Comparing Base Models. We use four families of base models and compare them on the code generation scenario. A one-shot prompt is used for all models to avoid any formatting and answer extraction issues. We find L3-BASE and DS-B models are significantly better than both CODELLAMA and STARCODER2 base models with a DS-BASE-6.7B model even outperforming both CL-BASE-34B and SC2-BASE-15B models. Note that some LCB specific differences can potentially be attributed to data curation approaches. For instance, SC models (and potentially DS as discussed in Section 5.1) use competition problems during pre-training.
+
+Role of Post Training. Post-training improves performance on both HUMANEVAL+ and LIVE-CODEBENCH for the code generation scenario. Particularly, on LCB L3-INS-70B, DS-INS-33B and PHIND-34B improve PASS@1 over their base models by 8.2, 7.3 and 9.5 points respectively. This highlights the importance of good post-training datasets for building strong LLMs. At the same time, we note that the base models have aligned performances on LCB code generation and HUMANEVAL+ benchmarks and lie within or close to the green shaded region in Figure 4. However, the fine-tuned open models exhibit a larger performance gap, with much better performances on HUMANEVAL+. On the other hand, the closed-access models are still aligned across both benchmarks. This suggests the necessity of more diverse post-training data mixtures.
+
+Open-Access vs Closed-Access Models. In general, closed (API) access model families generally outperform the open access models. The gap is only closed by three models, namely L3-INS-70B, MIXTRAL, and DS-INS-33B which reach the performance levels of the closed models. For instance, in
+
+![](images/589226a7fb8368828619ee712d520137643936bd1fb51e9b44cac6c4d1372bb7.jpg)  
+Figure 5: Comparison of open access and (closed) API access models on LIVECODEBENCH code generation scenario. We find that closed-access models consistently outperform the open models with only strong instruction-tuned variants of  $>30\mathrm{B}$  models (specifically L3-INS-70B, MIXTRAL and DS-INS-33B models) crossing the performance gap.
+
+the code generation scenario (Figure 5), these models reach close to or even outperform closed access models like GEMINI-PRO, GPT-3.5-TURBO, and CLAUDE-3-SONNET. The performances vary across scenarios with the closed-access models faring better in test output and code execution scenarios.
+
+Highlighting gap between SoTA One distinct observation from our evaluations is the large gap between SoTA models and open models across all scenarios. Particularly, GPT-4-TURBO, GPT-4, GEMINI-PRO-1.5 and CLAUDE-3-OPUS lead across the benchmarks with wide performance margins over other models. This distinguishes LIVECODEBENCH from prior benchmarks (like HUMANEVAL) where various open models have achieved similar or better performance. For example, DS-INS-33B is merely 4.3 point behind GPT-4-TURBO on HUMANEVAL+ but 16.2 points  $(69\%)$  on LCB code generation scenario. This gap either holds or sometimes even amplifies across other scenarios.
+
+Comparing open-access instruction-tuned models. Here, we compare various fine-tuned variants of the L3-BASE, DEEPSEEK and CODELLAMA base models across different model sizes. We find that fine-tuned L3-BASE and DEEPSEEK models lead in performance, followed by PHIND-34B and CODELLAMA models across most scenarios. Broadly, we find that model performances correlate with model sizes. For example, PHIND-34B model outperforms the 6.7B models across all scenarios.
+
+# 6 RELATED WORK
+
+Language Models for Code Generation. Starting with Codex (Chen et al., 2021), there are over a dozen code LLMs. These include CodeT5 (Wang et al., 2021; 2023), CodeGen (Nijkamp et al., 2022), SantaCoder (Allal et al., 2023), StarCoder (Li et al., 2023b; Lozhkov et al., 2024), InCoder (Fried et al., 2022), CodeGeeX (Zheng et al., 2023), L3-BASE, DEEPSEEK (Bi et al., 2024) and CODELLAMA (Roziere et al., 2023).
+
+Code Generation Benchmarks. Many benchmarks have been proposed to compare and evaluate these models. These primarily focus on natural language to Python code generation: HUMAN EVAL (Chen et al., 2021), HUMAN+ (Liu et al., 2023b), APPS (Hendrycks et al., 2021), CODE-CONTESTS (Li et al., 2022), MBPP (Austin et al., 2021), L2CEval (Ni et al., 2023). Their variants have been proposed to cover more languages, (Wang et al., 2022a; Zheng et al., 2023; Cassano et al., 2022; Athiwaratkun et al., 2022). Many benchmarks have focused on code generation in APIs. Benchmarks like DS-1000 (Lai et al., 2023), ARCADE (Yin et al., 2022), NumpyEval (Zhang et al., 2023b), and PandasEval (Jain et al., 2022) focus on data science APIs. Other benchmarks measure using broader APIs or general software engineering tasks, such as JuICe (Agashe et al., 2019), APIBench (Patil et al., 2023), RepoBench (Liu et al., 2023c), ODEX (Wang et al., 2022b), SWE-Bench (Jimenez et al.,
+
+2023), GoogleCodeRepo (Shrivastava et al., 2023), RepoEval (Zhang et al., 2023a), ClasEval (Du et al., 2023) and Cocomic-Data (Ding et al., 2022).
+
+A few benchmarks specifically measure competitive programming, such as APPS (Hendrycks et al., 2021), CodeContests (Li et al., 2022), CodeScope (Yan et al., 2023), xCodeEval (Khan et al., 2023), and LeetCode-Hard (Shinn et al., 2023), and TACO (Li et al., 2023c). Table 1 highlights differences between them. Methods such as AlphaCode (Li et al., 2022), AlphaCode 2 (Gemini Team et al., 2023), ALGO (Zhang et al., 2023d), Parsel (Zelikman et al., 2022), code cleaning (Jain et al., 2023), code explanations (Li et al., 2023a), analogical reasoning (Yasunaga et al., 2023), and AlphaCodium (Ridnik et al., 2024) have focused on improving LLMs.
+
+# 6.1 HOLISTIC TASKS AND CONTAMINATION
+
+Code Repair. (Chen et al., 2023; Olausson et al., 2023; Madaan et al., 2023b; Peng et al., 2023; Zhang et al., 2023c) have investigated self-repair for existing code LLM benchmarks. These methods use error feedback for models to improve inspiring our code repair scenario.
+
+Code Execution. Code execution was first studied in (Austin et al., 2021; Nye et al., 2021) LIVE-CODEBENCH's execution scenario is particularly inspired by CRUXEval (Gu et al., 2024), a recent benchmark measuring the reasoning and execution abilities of code LLMs. We differ from CRUXEval in that our benchmark is live, and our functions are more complex and human-produced.
+
+Test Generation. Test generation using LLMs has been explored in (Yuan et al., 2023; Schäfer et al., 2024; Tufano et al., 2022; Watson et al., 2020). Here, we decouple the test inputs and outputs for fair evaluations. Finally, some works have additionally studied other tasks and scenarios like type prediction (Mir et al., 2022; Wei et al., 2023; Malik et al., 2019), code summarization (LeClair et al., 2019; Iyer et al., 2016; Barone & Sennrich, 2017; Hasan et al., 2021; Alon et al., 2018), code security (Liguori et al., 2022; Pearce et al., 2022; Tony et al., 2023), etc.
+
+Contamination. Data contamination and test-case leakage have received considerable attention Oren et al. (2024); Golchin & Surdeanu (2023); Weller et al. (2023); Roberts et al. (2024) recently. (Sainz et al., 2023) demonstrated contamination by simply prompting the model to highlight its contamination. Some detection methods have also been built to avoid these cases (Shi et al., 2023; Zhou et al., 2023). For code, (Riddell et al., 2024) use edit distance and AST-similarity to detect contamination. (Ma et al., 2021; Li et al., 2024) have explore "live" benchmark in other domains.
+
+# 7 LIMITATIONS AND CONCLUSION
+
+We describe key limitations here and provide a deeper discussion in Appendix G.
+
+Evaluation Noise. We anticipate noise from benchmark size, prompts, and sampling. For benchmark size, while 612 problems is fairly larger than existing code benchmarks (e.g. 164 problems in HUMANEVAL), this set reduces on "scrolling" over the more recent problems. Next, prompting can cause large variations in model performance and we do not tune prompts across models. For sampling, we use bootstrapped Pass@1 using 10 completions, which should limit this noise considerably.
+
+Problem Domain. LIVECODEBENCH comprises competition programming problems which might not correlate with how LLMs are used in practice. Even then, our results are well correlated with findings from human evaluations like Chatbot-Arena Chiang et al. (2024) thus providing useful signal.
+
+Conclusion. In this work, we propose LIVECODEBENCH, a new benchmark for evaluating LLMs for code. LIVECODEBENCH provides an extensible framework that will keep on updating with new problems. Our benchmark mitigates contamination issues in existing benchmarks by introducing live evaluations and emphasizing scenarios beyond code generation to account for the broader coding abilities of LLMs. Our evaluations reveal novel findings such as contamination detection, holistic evaluations, and potential overfitting on HUMANETVAL. We hope LIVECODEBENCH with serve to advance understanding of current code LLMs and also guide future research through our findings.
+
+# ACKNOWLEDGEMENT
+
+This work was supported in part by NSF grants CCF:1900968, CCF:1908870 and by SKY Lab industrial sponsors and affiliates Astronomer, Google, IBM, Intel, Lacework, Microsoft, Mohamed Bin Zayed University of Artificial Intelligence, Nexla, Samsung SDS, Uber, and VMware. A. Gu is supported by the NSF Graduate Research Fellowship under Grant No. 2141064. A. Solar-Lezama is supported by the NSF and Intel Corporation through NSF Grant CCF:2217064. Any opinions, findings, conclusions, or recommendations in this paper are solely those of the authors and do not necessarily reflect the position of the sponsors.
+
+Finally, we thank Manish Shetty, Wei-Lin Chiang, Jierui Li, Horace He, Federico Cassano, Pengcheng Yin, Kexun Zhang, Rishabh Singh, and Aman Madaan for helpful feedback at various stages of the work. We particularly thank Kexun Zhang for providing results for  $\mathrm{C + + }$  and Nikhil Thakur for helping maintain the benchmark.
+
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+
+# A DATASET
+
+# A.1 LEGAL COMPLIANCE AND LICENSE
+
+Our benchmark does not comprise and personal identifiable information or offensive content.
+
+Similar to (Hendrycks et al., 2021), we scrape only the problem statements, ground-truth solutions, and test cases from competition websites – LEETCODE, ATCODER, and CODEFORCES. Further, we primarily scrape publicly visible portions of websites, avoiding any data collection that might be pay-walled or require login or interaction with the website. Following, (Hendrycks et al., 2021) we abide by Fair Use §107: “the fair use of a copyrighted work, including such use by ... scholarship, or research, is not an infringement of copyright”, where fair use is determined by “the purpose and character of the use, including whether such use is of a commercial nature or is for nonprofit educational purposes”, “the amount and substantiality of the portion used in relation to the copyrighted work as a whole”, and “the effect of the use upon the potential market for or value of the copyrighted work.” Finally, we use the collected problems for academic purposes only and in addition, do not train on the collected problems.
+
+# A.2 GENERATOR BASED TEST GENERATION
+
+We use GPT-4-TURBO to construct input generators. The following prompts (Figures 6 and 7) provide one-shot prompt templates used for synthesizing random and adversarial input generators. These generators define a function returns the arguments sampled in some distribution. These generators are then executed to construct inputs which validated on the collected correct programs. We use separate generators for random and adversarial setting since often times programming problems have corner cases which might not be captured by randomly sampling over the inputs. We build 2 random input generators, 4 adversarial input generators and check if the sampled inputs work for the correct programs. Finally, the number of collected inputs is thresholded to 100 for efficient grading (using random selection). We find that our generators can already function well but future work can study the design space of constructing such generators.
+
+Note that for CODEFORCES, we construct the generators in semi-autonomous manner since only 9 problems were used.
+
+# A.3 CODE EXECUTION
+
+The code execution split of LiveCodeBench consists of 479 samples from 85 distinct problems. To encourage diversity in our benchmark while keeping our benchmark small and usable, we place a limit of six samples for each given problem. These sample programs and corresponding test cases are chosen uniformly at random from all those passing the filter.
+
+Filtering Criteria: The specific filtering criteria are as follows:
+
+- Compile time: length of code is between 100 and 500 characters, no syntax errors, all necessary imports are included  
+- Runtime: no floating point operations, true division, exp, other integer operations must have at least one argument  $\leq 3$ , string and list operations must have at least one argument with length  $\leq 3$ , finish running in 2 seconds, "reasonable" number of steps (roughly, under 1000 Python bytecode operations).
+
+We give two examples of two programs that are filtered out in the Listings below. Our final benchmark consists of 479 samples from 85 problems, but will increase in size due to its live nature.
+
+Dataset Statistics: As mentioned, we filter for codes between 100 and 500 characters, as well as below 1000 steps. The statistics for programs in our dataset are shown in Fig. 8.
+
+# A.4 NUMBER OF TESTS
+
+We can sample large amounts of diverse inputs using our generator-based input sampling approach. However, the number of inputs produces a tradeoff in terms of the usability of the benchmark, slowing
+
+```txt
+You are an expert python competitive programma and your goal is to  $\rightarrow$  construct input-generators for testing programming contest  $\rightarrow$  problems. You will write relevant generators and finally  $\rightarrow$  construct 'construct_entries' function that returns a list of  $\rightarrow$  diverse inputs sampled from the generator. Remember to strictly  $\rightarrow$  follow the instructions and constraints present in the problem  $\rightarrow$  statement   
+<USER> {EXAMPLE_PROBLEM}   
+Construct a random input generator.   
+</USER>   
+<ASSISTANT> import numpy as np   
+def random_input_generator(weight_min, weight_max, size_min, size_max): weights_size = np.random.randint(size_min, size_max+1) weights  $=$  np.random.randint(weight_min, weight_max, size=  $\leftrightarrow$  weights_size).tolist() k  $=$  np.random.randint(1, len(weights)+1) return weights, k   
+def construct_entries(): inputs_list  $=$  [] #small inputs for i in range(15): inputs_list.append(random_input_generator(1, 10**3, 1, 10)) # medium inputs for i in range(15): inputs_list.append(random_input_generator(1, 10**6, 1, 10**3)) # large inputs for i in range(15): inputs_list.append(random_input_generator(1, 10**9, 1, 10**5)) return inputs_list   
+</ASSISTANT>   
+<USER> {PROBLEM}   
+Construct a random input generator. Use the format used in the above  $\leftrightarrow$  example by returning a single function that builds diverse  $\leftrightarrow$  inputs named 'construct_entries'   
+</USER>
+```
+
+Figure 6: Random Input Generator Prompt
+
+down the evaluation process. Hence, we threshold the number of sampled inputs to less than 25 based on problem length, difficulty, and number of inputs.
+
+We base this decision by estimating model performances when using different sets of inputs at random for testing in Figure 9. Particularly, we vary the number of private tests between 0, 1, 5, 10, 20, 40, 60, 80, and 100 and observe an exponentially decaying curve. We find the  $\sim 20$  inputs provides a good threshold which balances evaluation time with evaluation rigor.
+
+```txt
+You are an expert python competitive programmaar and your goal is to  $\nrightarrow$  construct input-generators for testing programming contest  $\nrightarrow$  problems. You will write relevant generators and finally  $\nrightarrow$  construct 'construct_entries' function that returns a list of  $\nrightarrow$  diverse inputs sampled from the generator. Remember to strictly  $\nrightarrow$  follow the instructions and constraints present in the problem  $\nrightarrow$  statement   
+<USER> {EXAMPLE_PROBLEM}   
+Construct an adversarial input generator.   
+</USER>   
+<ASSISTANT> import numpy as np   
+# case 1 - alternating large and small weights def generate_adversarial_entries_1(weight_size, max_weight, k): weights  $=$  [1 if i%2==0 else max_weight for i in range(weight_size)] return weights, k   
+# case 2 - equalweights   
+def adversarial_input_generator_2(weight_size, max_weight, k): weights  $=$  [max_weight for _ in range(weight_size)] return weights, k   
+# Case 3 - Large weights at the ends   
+def adversarial_input_generator_3(weight_size, max_weight, k): pass ## truncatede   
+def construct_entries(): inputs_list = [] weight_sizes  $= [10,1000,100000]$  maxweights  $= [10**3,10**6,10**9]$  for weight_size in weight_sizes: for max_weight in maxweights: ks  $= [1,2,5,$  weight_size//2, weight_size-1, weight_size] for k in ks: inputs_list.append(generate_adversarial_entries_1(  $\nrightarrow$  weight_size, max_weight, k)) # truncated return inputs_list   
+</ASSISTANT>   
+{PROBLEM}   
+Construct an adversarial input generator. Use the format used in the  $\nrightarrow$  above example by returning a single function that builds diverse  $\nrightarrow$  inputs named 'construct_entries'   
+</USER>
+```
+
+Figure 7: Adversarial Input Generator Prompt
+
+Program filtered because of multiplication  
+```python
+def check(x, t):
+    if x == ':':
+        return t == 0
+    if t < 0:
+        return False
+    for i in range(len(x)):
+        if check(x[:i], t - int(x[i:]))
+            return True
+    return False
+@cache
+def punishmentNumber(n: int) -> int:
+    if n == 0:
+        return 0
+    ans = punishmentNumber(n-1)
+    if check(str(n * n), n):
+        ans += n * n
+    return ans
+assert punishmentNumber(n = 37) == 1478
+```
+
+Program filtered because of control flow  
+```python
+dp = [True for _ in range(int(1e6 + 5))]
+MAXN = int(1e6 + 5)
+p = []
+dp[0] = False
+dp[1] = False
+for i in range(2, MAXN):
+    if not dp[i]: continue
+    p.append(i)
+    for j in range(2 * i, MAXN, i):
+        dp[j] = False
+def findPrimePairs(n: int) -> List[List[int]):
+    res = []
+    for i in range(1, n):
+        if n % 2 == 1 and i > n // 2: break
+            if n % 2 == 0 and i > n // 2: break
+                if dp[i] and dp[n - i]:
+                    res.append([i, n - i])
+    return res
+assert findPrimePairs(n = 2) == []
+```
+
+![](images/a44142ccd95c241a0023bcb712934d9616267ce4a9233991f8f8e7481b29b4b6.jpg)  
+Figure 8: Distribution of code lengths and number of execution steps
+
+![](images/d1e0a251c018edf87178ea98c1d7aeb32f8c5008d81f4fd50e948fdd463dd8ce.jpg)
+
+![](images/d41185e66e0b73f8ade419c8e61d528da83b811ed4f88562aa833d3f8036af40.jpg)  
+Pass@1 vs Number of Private Test Cases  
+Figure 9: Model performance with varying counts of private inputs used for evaluation. We find the PASS@1 tends to stabilize when sampling about 20 inputs providing a good tradeoff in evaluation time and rigor.
+
+# B UI
+
+# LiveCodeBench
+
+![](images/6d82691ea21324a5bd7c5ba7deba2c562211d9789a57f06d8fc3fd21a40d684d.jpg)
+
+# LiveCodeBench
+
+![](images/472169d2a9dd522b34bc409e1ac6b59c55e60af388ba96345cd19a774815ac1b.jpg)  
+Figure 10: UI of LIVECODEBENCH showing two views - May-Jan and Sep-Jan. The contaminated models are blurred and the performance difference is visible across the two views. The scroll on the top allows selecting different periods of time highlighting the live nature of the benchmark.
+
+# C CURATION
+
+# C.1 PLATFORM SPECIFIC CURATION
+
+We describe the curation process for each platform.
+
+LEETCODE. We collect problems from all weekly and biweekly contests on LEETCODE that have taken place after April'23. For each problem, we collect the problems, public tests, and user solutions. The platform also provides a difficulty label for each problem which we use to tag the problems as EASY, MEDIUM, and HARD. Since LEETCODE provides a starter code for each problem, we also collect it and provide it to the LLM in the STDIN format. Since the hidden tests are not directly available, we use our generator-based test input generation approach (Section A.2) and also collect the auto grader failing tests for some of the recent problems.
+
+ATCODER. We collect problems from the abc (beginner round) contests on ATCODER that have taken place after April'23. We deliberately avoid the more challenging arc and agc contests which are designed for more advanced Olympiad participants. The problems are assigned numeric difficulty ratings, and we exclude abc problems with a rating of more than 500. We also use these numeric ratings to tag the problems as EASY, MEDIUM, and HARD. Specifically, we use the rating brackets  $[0 - 200)$ ,  $[200 - 400)$ , and  $[400 - 500]$  to perform the classification. ATCODER provides public and hidden tests for each problem which we directly use in the benchmark.
+
+CODEFORCES. We have collected problems from the Division 3 and Division 4 contests on CODEFORCES. Notably, we find that even with this filter, the problems are harder than the other two platforms. CODEFORCES also provides difficulty ratings for the problems which we use to tag the problems as EASY, MEDIUM, and HARD using the rating brackets  $\{800\}$ ,  $(800 - 1000]$ , and  $(1000 - 1300]$  respectively. Due to the higher difficulty, we only consider a small fraction of problems from CODEFORCES and semi-automatically construct test case generators, as they do not provide complete tests on the platform (long tests are truncated).
+
+Table 2 provides various statistics about the problems that we have collected for LIVECODEBENCH.
+
+# C.2 SCENARIO-SPECIFIC BENCHMARK CONSTRUCTION
+
+Code Generation and Self-Repair. We use the natural language problem statement as the problem statement for these scenarios. For LEETCODE, as noted above, an additional starter code is provided for the functional input format. For ATCODER and CODEFORCES problems, we use the standard input format (similar to (Hendrycks et al., 2021)). The collected or generated tests are then used to evaluate the correctness of the generated programs. Our final dataset consists of 511 problem instances across the three platforms.
+
+Code Execution. We draw inspiration from the benchmark creation procedure used in (Gu et al., 2024). First, we collect a large pool of  $\sim 2000$  correct, human-submitted solutions from the
+
+<table><tr><td>Platform</td><td>Total Count</td><td>#Easy</td><td>#Medium</td><td>#Hard</td><td>Average Tests</td></tr><tr><td>LCB (May-end)</td><td>511</td><td>182</td><td>206</td><td>123</td><td>17.0</td></tr><tr><td>LCB (Sep-end)</td><td>349</td><td>125</td><td>136</td><td>88</td><td>18.0</td></tr><tr><td>ATCODER</td><td>267</td><td>99</td><td>91</td><td>77</td><td>15.6</td></tr><tr><td>LEETCODE</td><td>235</td><td>79</td><td>113</td><td>43</td><td>19.0</td></tr><tr><td>CODEFORCES</td><td>9</td><td>4</td><td>2</td><td>3</td><td>11.1</td></tr><tr><td>LCB-Easy</td><td>182</td><td>182</td><td>0</td><td>0</td><td>16.1</td></tr><tr><td>LCB-Medium</td><td>206</td><td>0</td><td>206</td><td>0</td><td>17.4</td></tr><tr><td>LCB-Hard</td><td>123</td><td>0</td><td>0</td><td>123</td><td>18.0</td></tr></table>
+
+Table 2: The statistics of problems collected in LIVECODEBENCH (LCB). We present the number of problems, their difficulty distributions and the average number of tests per problem. We present the results on the following subsets of LIVECODEBENCH (used throughout this manuscript) - (a) problems in the May-Feb and Sep-Feb time windows, (b) problems sourced from the three platforms, and (c) problems in the LCB-Easy, LCB-Medium, and LCB-Hard subsets.
+
+LEETCODE subset. However, many of these programs have multiple nested loops, complex numerical computations, and a large number of execution steps. Therefore, we apply compile-time and run-time filters to ensure samples are reasonable, and we double-check this with a manual inspection. More details on the filtering criteria and statistics of the dataset can be found in Appendix A.3. Our final dataset consists of 479 samples from 85 problems.
+
+Test Case Output Prediction. We use the natural language problem statement from the LEETCODE platform and the example test inputs to construct our test case output prediction dataset. Since the example test inputs in the problems are reasonable test cases for humans to reason about and understand the problems, they also serve as ideal test inputs for LLMs to process. Our final dataset consists of 442 problem instances from a total of 181 LEETCODE problems.
+
+# D EXPERIMENTAL SETUP
+
+# D.1 MODELS
+
+We describe the details of models considered in our study in Table 3.
+
+<table><tr><td>Model ID</td><td>Short Name</td><td>Approximate Cutoff Date</td><td>Link</td></tr><tr><td>deepseek-ai/deepseek-coder-33b-instruct</td><td>DSCoder-33b-Ins</td><td>08/30/2023</td><td>deepseek-coder-33b-instruct</td></tr><tr><td>deepseek-ai/deepseek-coder-6.7b-instruct</td><td>DSCoder-6.7b-Ins</td><td>08/30/2023</td><td>deepseek-coder-6.7b-instruct</td></tr><tr><td>deepseek-ai/deepseek-coder-1.3b-instruct</td><td>DSCoder-1.3b-Ins</td><td>08/30/2023</td><td>deepseek-coder-1.3b-instruct</td></tr><tr><td>codellama/CodeLlama-70b-Instruct-hf</td><td>CodeLlama-70b-Ins</td><td>01/01/2023</td><td>CodeLlama-70b-Instruct-hf</td></tr><tr><td>openbmb/Eurus-70b-sft</td><td>Eurus-70B-SFT (n=1)</td><td>01/01/2023</td><td>Eurus-70b-sft</td></tr><tr><td>openbmb/Eurux-8x22b-nca</td><td>Eurux-8x22b-NCA (n=1)</td><td>04/30/2023</td><td>Eurux-8x22b-nca</td></tr><tr><td>codellama/CodeLlama-34b-Instruct-hf</td><td>CodeLlama-34b-Ins</td><td>01/01/2023</td><td>CodeLlama-34b-Instruct-hf</td></tr><tr><td>codellama/CodeLlama-13b-Instruct-hf</td><td>CodeLlama-13b-Ins</td><td>01/01/2023</td><td>CodeLlama-13b-Instruct-hf</td></tr><tr><td>codellama/CodeLlama-7b-Instruct-hf</td><td>CodeLlama-7b-Ins</td><td>01/01/2023</td><td>CodeLlama-7b-Instruct-hf</td></tr><tr><td>meta-liama/Meta-Llama-3-8B-Instruct</td><td>LLama3-8b-Ins</td><td>01/01/2023</td><td>Meta-Llama-3-8B-Instruct</td></tr><tr><td>meta-liama/Meta-Llama-3-70B-Instruct</td><td>LLama3-70b-Ins</td><td>01/01/2023</td><td>Meta-Llama-3-70B-Instruct</td></tr><tr><td>Phind/Phind-CodeLlama-34B-v2</td><td>Phind-34B-V2</td><td>01/01/2023</td><td>Phind-CodeLlama-34B-v2</td></tr><tr><td>Smaug-2-72B</td><td>Smaug-2-72B</td><td>01/01/2023</td><td>Smaug-2-72B</td></tr><tr><td>Qwen-1.5-72B-Chat</td><td>Qwen-1.5-72B-Chat</td><td>01/01/2023</td><td>Qwen-1.5-72B-Chat</td></tr><tr><td>Qwen/CodeQwen1.5-7B</td><td>CodeQwen15-7B</td><td>08/30/2023</td><td>CodeQwen1.5-7B</td></tr><tr><td>Qwen/CodeQwen1.5-7B Chat</td><td>CodeQwen15-7B-Chat</td><td>08/30/2023</td><td>CodeQwen1.5-7B-Chat</td></tr><tr><td>gpt-3.5-turbo-0301</td><td>GPT-3.5-Turbo-0301</td><td>10/01/2021</td><td>gpt-3.5-turbo-0301</td></tr><tr><td>gpt-3.5-turbo-0125</td><td>GPT-3.5-Turbo-0125</td><td>10/01/2021</td><td>gpt-3.5-turbo-0125</td></tr><tr><td>gpt-4-0613</td><td>GPT-4-0613</td><td>10/01/2021</td><td>gpt-4-0613</td></tr><tr><td>gpt-4-1106-preview</td><td>GPT-4-Turbo-1106</td><td>04/30/2023</td><td>gpt-4-1106-preview</td></tr><tr><td>gpt-4-turbo-2024-04-09</td><td>GPT-4-Turbo-2024-04-09</td><td>04/30/2023</td><td>gpt-4-turbo-2024-04-09</td></tr><tr><td>gpt-4o-2024-05-13</td><td>GPT-4O-2024-05-13</td><td>10/30/2023</td><td>gpt-4o-2024-05-13</td></tr><tr><td>claude-2</td><td>Claude-2</td><td>12/31/2022</td><td>claude-2</td></tr><tr><td>claude-instant-1</td><td>Claude-Instant-1</td><td>12/31/2022</td><td>claude-instant-1</td></tr><tr><td>claude-3-opus-20240229</td><td>Claude-3-Opus</td><td>04/30/2023</td><td>claude-3-opus-20240229</td></tr><tr><td>claude-3-sonnet-20240229</td><td>Claude-3-Sonnet</td><td>04/30/2023</td><td>claude-3-sonnet-20240229</td></tr><tr><td>claude-3-haiku-20240307</td><td>Claude-3-Haiku</td><td>04/30/2023</td><td>claude-3-haiku-20240307</td></tr><tr><td>codestral-latest</td><td>Codestral-Latest</td><td>01/31/2024</td><td>codestral-latest</td></tr><tr><td>gemini-pro</td><td>Gemini-Pro</td><td>04/30/2023</td><td>gemini-pro</td></tr><tr><td>gemini-1.5-pro-latest</td><td>Gemini-Pro-1.5-May</td><td>04/30/2023</td><td>gemini-1.5-pro-latest</td></tr><tr><td>gemini-1.5-flash-latest</td><td>Gemini-Flash-1.5-May</td><td>04/30/2023</td><td>gemini-1.5-flash-latest</td></tr><tr><td>ise-uic/Magicoder-S-DS-6.7B</td><td>MagiCoderS-DS-6.7B</td><td>08/30/2023</td><td>Magicoder-S-DS-6.7B</td></tr><tr><td>ise-uic/Magicoder-S-CL-7B</td><td>MagiCoderS-CL-7B</td><td>01/01/2023</td><td>Magicoder-S-CL-7B</td></tr><tr><td>bigcode/starcoder2-3b</td><td>StarCoder2-3b</td><td>01/01/2023</td><td>starcoder2-3b</td></tr><tr><td>bigcode/starcoder2-7b</td><td>StarCoder2-7b</td><td>01/01/2023</td><td>starcoder2-7b</td></tr><tr><td>bigcode/starcoder2-15b</td><td>StarCoder2-15b</td><td>01/01/2023</td><td>starcoder2-15b</td></tr><tr><td>codellama/CodeLlama-70b-hf</td><td>CodeLlama-70b-Base</td><td>01/01/2023</td><td>CodeLlama-70b-hf</td></tr><tr><td>codellama/CodeLlama-34b-hf</td><td>CodeLlama-34b-Base</td><td>01/01/2023</td><td>CodeLlama-34b-hf</td></tr><tr><td>codellama/CodeLlama-13b-hf</td><td>CodeLlama-13b-Base</td><td>01/01/2023</td><td>CodeLlama-13b-hf</td></tr><tr><td>codellama/CodeLlama-7b-hf</td><td>CodeLlama-7b-Base</td><td>01/01/2023</td><td>CodeLlama-7b-hf</td></tr><tr><td>deepseek-ai/deepseek-coder-33b-base</td><td>DSCoder-33b-Base</td><td>08/30/2023</td><td>deepseek-coder-33b-base</td></tr><tr><td>deepseek-ai/deepseek-coder-6.7b-base</td><td>DSCoder-6.7b-Base</td><td>08/30/2023</td><td>deepseek-coder-6.7b-base</td></tr><tr><td>deepseek-ai/deepseek-coder-1.3b-base</td><td>DSCoder-1.3b-Base</td><td>08/30/2023</td><td>deepseek-coder-1.3b-base</td></tr><tr><td>google/codegemma-7b</td><td>CodeGemma-7b-Base</td><td>01/01/2023</td><td>codegemma-7b</td></tr><tr><td>google/codegemma-2b</td><td>CodeGemma-2b-Base</td><td>01/01/2023</td><td>codegemma-2b</td></tr><tr><td>google/gemma-7b</td><td>Gemma-7b-Base</td><td>01/01/2023</td><td>gemma-7b</td></tr><tr><td>google/gemma-2b</td><td>Gemma-2b-Base</td><td>01/01/2023</td><td>gemma-2b</td></tr><tr><td>meta-llama/Meta-Llama-3-70B</td><td>LLama3-70b-Base</td><td>01/01/2023</td><td>Meta-Llama-3-70B</td></tr><tr><td>meta-llama/Meta-Llama-3-8B</td><td>LLama3-8b-Base</td><td>01/01/2023</td><td>Meta-Llama-3-8B</td></tr><tr><td>mistral-large-latest</td><td>Mistral-Large</td><td>01/01/2023</td><td>mistral-large-latest</td></tr><tr><td>open-mixtral-8x22b</td><td>Mixtral-8x22B-Ins</td><td>01/01/2023</td><td>open-mixtral-8x22b</td></tr><tr><td>open-mixtral-8x7b</td><td>Mixtral-8x7B-Ins</td><td>01/01/2023</td><td>open-mixtral-8x7b</td></tr><tr><td>m-a-p/OpenCodeInterpreter-DS-33B</td><td>OC-DS-33B</td><td>08/30/2023</td><td>OpenCodeInterpreter-DS-33B</td></tr><tr><td>m-a-p/OpenCodeInterpreter-DS-6.7B</td><td>OC-DS-6.7B</td><td>08/30/2023</td><td>OpenCodeInterpreter-DS-6.7B</td></tr><tr><td>m-a-p/OpenCodeInterpreter-DS-1.3B</td><td>OC-DS-1.3B</td><td>08/30/2023</td><td>OpenCodeInterpreter-DS-1.3B</td></tr><tr><td>command-r</td><td>Command-R</td><td>01/01/2023</td><td>command-r</td></tr><tr><td>command-r+</td><td>Command-R+</td><td>01/01/2023</td><td>command-r+</td></tr></table>
+
+Table 3: Language Models Overview
+
+We use variety of GPUs based on availability for running local models (A6000, L4, A100).
+
+# D.2 CODE GENERATION
+
+Below we provide the prompt format (with appropriate variants adding special tokens accommodating each instruct-tuned model) used for this scenario.
+
+# D.3 SELF REPAIR
+
+Below we provide the prompt format (with appropriate variants adding special tokens accommodating each instruct-tuned model) used for this scenario.
+
+# D.4 CODE EXECUTION
+
+Below we provide the prompts for code execution with and without CoT. The prompts are modified versions of those from (Gu et al., 2024) to fit the format of the samples in our benchmark.
+
+# D.5 TEST OUTPUT PREDICTION
+
+Below we provide the prompt format (with appropriate variants adding special tokens accommodating each instruct-tuned model) used for this scenario.
+
+Code Generation Prompt  
+```txt
+You are an expert Python programmer. You will be given a question (  $\leftrightarrow$  problem specification) and will generate a correct Python  $\leftrightarrow$  program that matches the specification and passes all tests. You  $\leftrightarrow$  will NOT return anything except for the program   
+```python
+You are an expert Python programmer. You will be given a question (  $\leftrightarrow$  problem specification) and will generate a correct Python  $\leftrightarrow$  program that matches the specification and passes all tests. You  $\leftrightarrow$  will NOT return anything except for the program   
+```python
+You are an expert Python programmer. You will be given a question (  $\leftrightarrow$  problem.search) and will generate a correct Python  $\leftrightarrow$  program that matches the specification and passes all tests. You  $\leftrightarrow$  will NOT return anything except for the program   
+```python
+You are an expert Python programmer. You will be given a question (  $\leftrightarrow$  problem.search) and will generate a correct Python  $\leftrightarrow$  program that matches the specification and passes all tests. You  $\leftrightarrow$  will NOT return anything except for the program   
+```python
+You are an Expert Python programmer. You will be given a question (  $\leftrightarrow$  problem.search) and will generate a correct Python  $\leftrightarrow$  program that matches the specification and passes all tests. You  $\leftrightarrow$  will NOT return anything except for the program   
+```python
+You are an expert Python programmer. You will be given a question (  $\leftrightarrow$  problem.search) and will generate a correct Python  $\leftrightarrow$  program that matches the specification and passes all tests. You  $\leftrightarrow$ 
+```
+You are an expert Python programmer. You will be given a question (  $\leftrightarrow$  problem.search) and will generate a correct Python  $\leftrightarrow$  program that matches the specification and passes all tests. You  $\leftrightarrow$  will NOT return anything except for the program   
+```python
+You are an expert Python programmer. You will be given a question (  $\leftrightarrow$  problem.search) and will generate a correct Python  $\leftrightarrow$  program that matches the specification and passes all tests. Your answer: (use the provided format with back ticks)
+```
+
+```txt
+{if check_result(result_status is "Wrong Answer")  
+The above code is incorrect and does not pass the testcase.  
+Input: {wrong_testcase_input}  
+Output: {wrong_testcase_output}  
+Expected: {wrong_testcaseEXPECTED}  
+{elif check_result(result_status is "Time Limit Exceeded")  
+The above code is incorrect and exceeds the time limit.  
+Input: {wrong_testcase_input}  
+{elif check_result(result_status is "Runtime Error")  
+The above code is incorrect and has a runtime error.  
+Input: {wrong_testcase_input}  
+Error Message: {wrong_testcase_error_message}  
+{endif}  
+Self Repair Error Feedback Pseudocode
+```
+
+```txt
+You are a helpful programming assistant and an expert Python programmer  $\leftrightarrow$ . You are helping a user write a program to solve a problem. The  $\leftrightarrow$  user has written some code, but it has some errors and is not  $\leftrightarrow$  passing the tests. You will help the user by first giving a  $\leftrightarrow$  concise (at most 2-3 sentences) textual explanation of what is  $\leftrightarrow$  wrong with the code. After you have pointed out what is wrong  $\leftrightarrow$  with the code, you will then generate a fixed version of the  $\leftrightarrow$  program. You must put the entered fixed program within code  $\leftrightarrow$  delimiters only for once.
+```
+
+```markdown
+```java
+```java
+Question: \n{question(question_content)}  
+```java
+Answer: ` ` python
+{code.code_to_be.corrected}  
+```
+## # Format: {PromptConstants.FORMATTING_CHECK_ERROR_MESSAGE}  
+## # Answer: (use the provided format with back ticks)
+```
+
+# Self-Repair Prompt
+
+```txt
+You are given a Python function and an assertion containing an input to  $\leftrightarrow$  the function. Complete the assertion with a literal (no  $\leftrightarrow$  unsimplified expressions, no function calls) containing the  $\leftrightarrow$  output when executing the provided code on the given input, even  $\leftrightarrow$  if the function is incorrect or incomplete. Do NOT output any  $\leftrightarrow$  extra information. Provide the full assertion with the correct  $\leftrightarrow$  output in [ANSWER] and [/ANSWER] tags, following the examples.   
+[PYTHON]   
+def repeatNumber(number : int) -> int: return number   
+assert repeatNumber(number = 17) == ??   
+[/PYTHON]   
+[ANSWER]   
+assert repeatNumber(number = 17) == 17   
+[/ANSWER]   
+[PYTHON]   
+def addCharacterA(string : str) -> str: return string + "a"   
+assert addCharacterA(string  $=$  "x9j")  $= = ??$    
+[/PYTHON]   
+[ANSWER]   
+assert addCharacterA(string  $=$  "x9j")  $= = ?x9ja$    
+[/ANSWER]   
+[PYTHON]   
+{code}   
+assert {input}  $= = ??$    
+[/PYTHON]   
+[ANSWER]
+```
+
+# Code Execution Prompt
+
+```txt
+You are given a Python function and an assertion containing an input to  $\leftrightarrow$  the function. Complete the assertion with a literal (no  $\leftrightarrow$  unsimplified expressions, no function calls) containing the  $\leftrightarrow$  output when executing the provided code on the given input, even  $\leftrightarrow$  if the function is incorrect or incomplete. Do NOT output any  $\leftrightarrow$  extra information. Execute the program step by step before  $\leftrightarrow$  arriving at an answer, and provide the full assertion with the  $\leftrightarrow$  correct output in [ANSWER] and [/ANSWER] tags, following the  $\leftrightarrow$  examples.   
+[PYTHON]   
+def performOperation(s): s  $= \mathrm{s} + \mathrm{s}$  return "b"  $^+$  s  $^+$  "a"   
+assert performOperation(s  $=$  "hi")  $= = ??$    
+[/PYTHON]   
+[THOUGHT]   
+Let's execute the code step by step:   
+1. The function performOperation is defined, which takes a single  $\leftrightarrow$  argument s.   
+2. The function is called with the argument "hi", so within the  $\leftrightarrow$  function, s is initially "hi".   
+3. Inside the function, s is concatenated with itself, so s becomes "  $\leftrightarrow$  hihi".   
+4. The function then returns a new string that starts with "b",  $\leftrightarrow$  followed by the value of s (which is now "hihi"), and ends with  $\leftrightarrow$  "a".   
+5. The return value of the function is therefore "bhihia".   
+[/THOUGHT]   
+[ANSWER]   
+assert performOperation(s  $=$  "hi")  $= =$  "bhihia"   
+[/ANSWER]   
+[PYTHON]   
+{code}   
+assert {input}  $= = ??$    
+[/PYTHON]   
+[THOUGHT]
+```
+
+Code Execution Prompt with CoT
+
+```txt
+```python
+# Instruction: You are a helpful programming assistant and an expert
+     $\leftrightarrow$  Python programmer. You are helping a user to write a test case
+     $\leftrightarrow$  to help to check the correctness of the function. The user has
+     $\leftrightarrow$  written a input for the testcase. You will calculate the output
+     $\leftrightarrow$  of the testcase and write the whole assertion statement in the
+     $\leftrightarrow$  markdown code block with the correct output.
+Problem:
+{problem_statement}
+Function:
+...
+{function_signature}
+...
+Please complete the following test case:
+...
+assert {function_name}({testcase_input}) == #Todo
+...
+## Response:
+```
+
+Test Output Prediction Prompt
+
+# E RESULTS
+
+# E.1 CONTAMINATION
+
+Figure 11 demonstrates contamination in DEEPSEEK in self repair and test output prediction scenarios.
+
+![](images/e35a7649b09fa4a8c3235d410ceea8ff4e4aff8e894aae9c3a434d2b91da221e.jpg)
+
+![](images/658bbc6e870200939f599b5fcc1d0d5e7b029eab73519bc755ca09221818357f.jpg)
+
+![](images/39d83f5bbc2c8ac6514d7433298519f914c09949712927e680ebdadc3bb393e9.jpg)  
+Figure 11: Contamination in DS-B models across self-repair and code execution (without COT) scenarios over time. Note that code execution currently runs between May and November
+
+![](images/53712ccdfecbfe0608f3508fc30c5f592ce038726c036f781612ac873aafc8fb.jpg)
+
+![](images/394e172a8d10e98be2aee7b125e81d1155b6bc5592c24fc1a8538fa19c459369.jpg)  
+Figure 12: Performance on problems released over different months for ATCODER
+
+# E.2 ALL RESULTS
+
+Below we provide the tables comprising of results across different LIVECODEBENCH scenarios.
+
+Comparing Closed Models. We evaluate a range of closed (API access) models ranging from different model families like GPTs, CLAUSES, GEMINI, and MISTRAL. We find the GPT-4-TURBO and
+
+![](images/7166131fc12fe5c175df11b42e839424043d6858f9f4cecc37dadb5f0978c378.jpg)  
+Figure 13: Live evaluation over time for various models on code generation scenario in LIVE-CODEBENCH. We consider many recently released models and do not find significant performance variations across months except for DS-B models.
+
+![](images/eef048c5d150282238702278557e2a7a88a7f78cabc7f7ab551eb0d818ad4ad0.jpg)
+
+![](images/ee9655311274e3483ce555b2b3165ab1fdb520500a771f12a149d002966fb766.jpg)  
+Figure 14: Correlations across different scenarios studied in LIVECODEBENCH
+
+CLAUDE-3-OPUS rank at the top across all scenarios followed by MISTRAL-L and CLAUDE-3-SONNET models. Finally, GEMINI-PRO and GPT-3.5-TURBO lie on the lower end of the models. The relative differences between the models vary across the scenarios. For example, GPT-4-TURBO demonstrates remarkable improvement from self-repair (24.5% to 36.9% on the LCB-Medium problems) while GEMINI-PRO only improves from 8.5% to 9.4%. Similarly, as identified above, CLAUDE-3-OPUS and MISTRAL-L perform considerably better on test output prediction and code execution scenarios.
+
+# E.3 C++ RESULTS
+
+We evaluate model performances with  $\mathrm{C + + }$  in Table 8. As we see, model performances roughly align across the two languages.
+
+![](images/e0410545edd3e7a332e01580dd158374a636dab7ce024e07366de36b6d508bda.jpg)  
+Figure 15: Comparison of open access and (closed) API access models on LIVECODEBENCH-Easy code generation scenario. We find that closed-access models consistently outperform the open models with only strong instruction-tuned variants of  $>30\mathrm{B}$  models (specifically L3-INS-70B, MIXTRAL and DS-INS-33B models) crossing the performance gap.
+
+<table><tr><td>Model Name</td><td>Easy</td><td>Medium</td><td>Hard</td><td>Total</td></tr><tr><td>Claude-2</td><td>61.80</td><td>4.90</td><td>0.20</td><td>22.30</td></tr><tr><td>Claude-3-Haiku</td><td>63.00</td><td>4.30</td><td>1.10</td><td>22.80</td></tr><tr><td>Claude-3-Opus</td><td>78.80</td><td>16.30</td><td>3.20</td><td>32.80</td></tr><tr><td>Claude-3-Sonnet</td><td>67.60</td><td>6.20</td><td>1.10</td><td>25.00</td></tr><tr><td>Claude-Instant-1</td><td>60.70</td><td>4.30</td><td>1.10</td><td>22.10</td></tr><tr><td>CodeGemma-2b-Base</td><td>18.30</td><td>0.40</td><td>0.00</td><td>6.30</td></tr><tr><td>CodeGemma-7b-Base</td><td>35.70</td><td>2.60</td><td>0.10</td><td>12.80</td></tr><tr><td>CodeLlama-13b-Base</td><td>24.60</td><td>0.90</td><td>0.00</td><td>8.50</td></tr><tr><td>CodeLlama-13b-Ins</td><td>36.60</td><td>2.40</td><td>0.00</td><td>13.00</td></tr><tr><td>CodeLlama-34b-Base</td><td>32.20</td><td>1.80</td><td>0.10</td><td>11.40</td></tr><tr><td>CodeLlama-34b-Ins</td><td>33.70</td><td>2.40</td><td>1.10</td><td>12.40</td></tr><tr><td>CodeLlama-70b-Base</td><td>15.80</td><td>1.20</td><td>0.00</td><td>5.70</td></tr><tr><td>CodeLlama-70b-Ins</td><td>7.80</td><td>0.60</td><td>0.00</td><td>2.80</td></tr><tr><td>CodeLlama-7b-Base</td><td>19.00</td><td>0.40</td><td>0.00</td><td>6.50</td></tr><tr><td>CodeLlama-7b-Ins</td><td>28.60</td><td>2.50</td><td>0.00</td><td>10.40</td></tr><tr><td>CodeQwen15-7B</td><td>40.40</td><td>4.80</td><td>0.00</td><td>15.10</td></tr><tr><td>CodeQwen15-7B-Chat</td><td>39.20</td><td>13.10</td><td>0.50</td><td>17.60</td></tr><tr><td>Codestral-Latest</td><td>69.00</td><td>18.70</td><td>0.90</td><td>29.50</td></tr><tr><td>Command-R</td><td>39.00</td><td>3.60</td><td>0.00</td><td>14.20</td></tr><tr><td>Command-R+</td><td>56.60</td><td>6.80</td><td>0.00</td><td>21.10</td></tr><tr><td>DSCoder-1.3b-Base</td><td>17.30</td><td>0.70</td><td>0.00</td><td>6.00</td></tr><tr><td>DSCoder-1.3b-Ins</td><td>22.90</td><td>1.50</td><td>0.00</td><td>8.10</td></tr><tr><td>DSCoder-33b-Base</td><td>39.40</td><td>2.30</td><td>0.00</td><td>13.90</td></tr><tr><td>DSCoder-33b-Ins</td><td>55.60</td><td>9.00</td><td>0.70</td><td>21.80</td></tr><tr><td>DSCoder-6.7b-Base</td><td>34.30</td><td>1.40</td><td>0.10</td><td>11.90</td></tr><tr><td>DSCoder-6.7b-Ins</td><td>46.40</td><td>5.80</td><td>0.70</td><td>17.60</td></tr><tr><td>GPT-3.5-Turbo-0125</td><td>56.80</td><td>10.80</td><td>0.10</td><td>22.60</td></tr><tr><td>GPT-3.5-Turbo-0301</td><td>53.40</td><td>8.80</td><td>0.20</td><td>20.80</td></tr><tr><td>GPT-4-0613</td><td>78.40</td><td>21.20</td><td>2.30</td><td>33.90</td></tr><tr><td>GPT-4-Turbo-1106</td><td>84.40</td><td>24.00</td><td>0.50</td><td>36.30</td></tr><tr><td>GPT-4-Turbo-2024-04-09</td><td>85.30</td><td>33.00</td><td>5.10</td><td>41.10</td></tr><tr><td>GPT-4O-2024-05-13</td><td>88.30</td><td>33.20</td><td>4.20</td><td>41.90</td></tr><tr><td>Gemini-Flash-1.5-May</td><td>68.10</td><td>12.60</td><td>2.70</td><td>27.80</td></tr><tr><td>Gemini-Pro-1.5-April (n=1)</td><td>56.50</td><td>14.30</td><td>3.60</td><td>24.80</td></tr><tr><td>Gemini-Pro-1.5-May</td><td>76.00</td><td>19.40</td><td>3.50</td><td>33.00</td></tr><tr><td>Gemma-2b-Base</td><td>6.10</td><td>0.00</td><td>0.00</td><td>2.00</td></tr><tr><td>Gemma-7b-Base</td><td>27.00</td><td>0.90</td><td>0.00</td><td>9.30</td></tr><tr><td>LLama3-70b-Base</td><td>52.20</td><td>3.20</td><td>0.60</td><td>18.60</td></tr><tr><td>LLama3-70b-Ins</td><td>60.70</td><td>15.80</td><td>1.40</td><td>26.00</td></tr><tr><td>LLama3-8b-Base</td><td>32.90</td><td>1.50</td><td>0.00</td><td>11.50</td></tr><tr><td>LLama3-8b-Ins</td><td>38.60</td><td>3.50</td><td>0.50</td><td>14.20</td></tr><tr><td>MagiCoderS-CL-7B</td><td>32.80</td><td>2.40</td><td>0.00</td><td>11.70</td></tr><tr><td>MagiCoderS-DS-6.7B</td><td>49.20</td><td>7.50</td><td>0.00</td><td>18.90</td></tr><tr><td>Mistral-Large</td><td>60.20</td><td>10.90</td><td>0.90</td><td>24.00</td></tr><tr><td>Mixtral-8x22B-Ins</td><td>59.80</td><td>12.70</td><td>0.00</td><td>24.20</td></tr><tr><td>Mixtral-8x7B-Ins</td><td>31.60</td><td>2.60</td><td>0.00</td><td>11.40</td></tr><tr><td>OC-DS-1.3B</td><td>11.30</td><td>0.10</td><td>0.00</td><td>3.80</td></tr><tr><td>OC-DS-33B</td><td>53.90</td><td>5.10</td><td>0.00</td><td>19.70</td></tr><tr><td>OC-DS-6.7B</td><td>46.30</td><td>4.50</td><td>0.00</td><td>16.90</td></tr><tr><td>Phind-34B-V2</td><td>53.40</td><td>4.70</td><td>0.10</td><td>19.40</td></tr><tr><td>StarCoder2-15b</td><td>37.30</td><td>2.20</td><td>0.00</td><td>13.20</td></tr><tr><td>StarCoder2-3b</td><td>28.20</td><td>0.70</td><td>0.00</td><td>9.60</td></tr><tr><td>StarCoder2-7b</td><td>29.90</td><td>1.20</td><td>0.00</td><td>10.40</td></tr></table>
+
+Table 4: Code Generation Performances
+
+<table><tr><td>Model Name</td><td>Easy</td><td>Medium</td><td>Hard</td><td>Total</td></tr><tr><td>Claude-2</td><td>66.20</td><td>10.30</td><td>0.40</td><td>25.60</td></tr><tr><td>Claude-3-Haiku</td><td>66.50</td><td>8.70</td><td>2.50</td><td>25.90</td></tr><tr><td>Claude-3-Opus</td><td>83.10</td><td>23.70</td><td>6.70</td><td>37.80</td></tr><tr><td>Claude-3-Sonnet</td><td>72.60</td><td>11.80</td><td>2.20</td><td>28.90</td></tr><tr><td>Claude-Instant-1</td><td>64.40</td><td>7.10</td><td>2.20</td><td>24.60</td></tr><tr><td>CodeLlama-13b-Ins</td><td>43.10</td><td>3.00</td><td>0.00</td><td>15.30</td></tr><tr><td>CodeLlama-34b-Ins</td><td>31.50</td><td>3.50</td><td>1.80</td><td>12.30</td></tr><tr><td>CodeLlama-7b-Ins</td><td>31.90</td><td>3.10</td><td>1.50</td><td>12.10</td></tr><tr><td>Codestral-Latest</td><td>72.50</td><td>25.90</td><td>3.30</td><td>33.90</td></tr><tr><td>DSCoder-1.3b-Ins</td><td>29.50</td><td>2.10</td><td>0.00</td><td>10.60</td></tr><tr><td>DSCoder-33b-Ins</td><td>60.70</td><td>8.10</td><td>1.50</td><td>23.40</td></tr><tr><td>DSCoder-6.7b-Ins</td><td>49.90</td><td>5.70</td><td>1.10</td><td>18.90</td></tr><tr><td>GPT-3.5-Turbo-0125</td><td>59.30</td><td>11.90</td><td>0.50</td><td>23.90</td></tr><tr><td>GPT-3.5-Turbo-0301</td><td>58.40</td><td>11.60</td><td>0.70</td><td>23.60</td></tr><tr><td>GPT-4-0613</td><td>79.30</td><td>25.00</td><td>2.40</td><td>35.60</td></tr><tr><td>GPT-4-Turbo-1106</td><td>86.90</td><td>36.90</td><td>4.00</td><td>42.60</td></tr><tr><td>GPT-4-Turbo-2024-04-09</td><td>88.70</td><td>39.70</td><td>8.40</td><td>45.60</td></tr><tr><td>GPT-4O-2024-05-13</td><td>92.60</td><td>46.40</td><td>8.20</td><td>49.10</td></tr><tr><td>Gemini-Flash-1.5-May</td><td>73.40</td><td>16.40</td><td>4.40</td><td>31.40</td></tr><tr><td>Gemini-Pro</td><td>53.80</td><td>9.40</td><td>0.20</td><td>21.10</td></tr><tr><td>Gemini-Pro-1.5-April (n=1)</td><td>71.80</td><td>19.40</td><td>5.50</td><td>32.20</td></tr><tr><td>Gemini-Pro-1.5-May</td><td>84.80</td><td>30.10</td><td>7.30</td><td>40.70</td></tr><tr><td>LLama3-70b-Ins</td><td>69.60</td><td>19.00</td><td>1.80</td><td>30.10</td></tr><tr><td>LLama3-8b-Ins</td><td>47.10</td><td>6.10</td><td>0.00</td><td>17.70</td></tr><tr><td>MagiCoderS-CL-7B</td><td>36.50</td><td>3.10</td><td>0.00</td><td>13.20</td></tr><tr><td>MagiCoderS-DS-6.7B</td><td>50.60</td><td>8.60</td><td>0.00</td><td>19.70</td></tr><tr><td>Mistral-Large</td><td>71.20</td><td>15.60</td><td>3.60</td><td>30.10</td></tr><tr><td>OC-DS-1.3B</td><td>20.00</td><td>0.40</td><td>0.00</td><td>6.80</td></tr><tr><td>OC-DS-33B</td><td>58.90</td><td>7.20</td><td>1.30</td><td>22.50</td></tr><tr><td>OC-DS-6.7B</td><td>50.90</td><td>6.30</td><td>0.20</td><td>19.10</td></tr><tr><td>Phind-34B-V2</td><td>62.00</td><td>6.50</td><td>0.90</td><td>23.10</td></tr></table>
+
+Table 5: Self Repair Performances
+
+<table><tr><td>Model Name</td><td>Pass@1</td></tr><tr><td>Claude-2</td><td>32.70</td></tr><tr><td>Claude-3-Haiku</td><td>32.90</td></tr><tr><td>Claude-3-Opus</td><td>58.70</td></tr><tr><td>Claude-3-Sonnet</td><td>34.10</td></tr><tr><td>Claude-Instant-1</td><td>25.40</td></tr><tr><td>CodeLlama-13b-Ins</td><td>24.40</td></tr><tr><td>CodeLlama-34b-Ins</td><td>23.00</td></tr><tr><td>CodeLlama-70b-Ins</td><td>16.10</td></tr><tr><td>CodeLlama-7b-Ins</td><td>15.30</td></tr><tr><td>Codestral-Latest</td><td>41.80</td></tr><tr><td>DSCoder-1.3b-Ins</td><td>12.50</td></tr><tr><td>DSCoder-33b-Ins</td><td>28.30</td></tr><tr><td>DSCoder-6.7b-Ins</td><td>26.50</td></tr><tr><td>GPT-3.5-Turbo-0125</td><td>35.40</td></tr><tr><td>GPT-3.5-Turbo-0301</td><td>32.50</td></tr><tr><td>GPT-4-0613</td><td>52.90</td></tr><tr><td>GPT-4-Turbo-1106</td><td>55.70</td></tr><tr><td>GPT-4-Turbo-2024-04-09</td><td>66.10</td></tr><tr><td>GPT-4O-2024-05-13</td><td>68.90</td></tr><tr><td>Gemini-Flash-1.5-May</td><td>38.10</td></tr><tr><td>Gemini-Pro</td><td>29.50</td></tr><tr><td>Gemini-Pro-1.5-April (n=1)</td><td>49.60</td></tr><tr><td>Gemini-Pro-1.5-May</td><td>44.80</td></tr><tr><td>LLama3-70b-Ins</td><td>41.40</td></tr><tr><td>LLama3-8b-Ins</td><td>24.40</td></tr><tr><td>MagiCoderS-CL-7B</td><td>21.30</td></tr><tr><td>MagiCoderS-DS-6.7B</td><td>27.10</td></tr><tr><td>Mistral-Large</td><td>46.50</td></tr><tr><td>Mixtral-8x22B-Ins</td><td>44.70</td></tr><tr><td>Mixtral-8x7B-Ins</td><td>31.80</td></tr><tr><td>OC-DS-1.3B</td><td>7.80</td></tr><tr><td>OC-DS-33B</td><td>11.30</td></tr><tr><td>OC-DS-6.7B</td><td>18.30</td></tr><tr><td>Phind-34B-V2</td><td>27.20</td></tr></table>
+
+Table 6: Test Output Prediction Performances
+
+<table><tr><td>Model Name</td><td>Pass@1</td><td>Pass@1 (COT)</td></tr><tr><td>Claude-2</td><td>31.50</td><td>43.80</td></tr><tr><td>Claude-3-Haiku</td><td>0.70</td><td>28.30</td></tr><tr><td>Claude-3-Opus</td><td>36.50</td><td>80.10</td></tr><tr><td>Claude-3-Sonnet</td><td>29.30</td><td>39.40</td></tr><tr><td>Claude-Instant-1</td><td>20.00</td><td>34.80</td></tr><tr><td>Cllama-13b-Ins</td><td>23.50</td><td>14.10</td></tr><tr><td>Cllama-34b-Ins</td><td>28.90</td><td>24.50</td></tr><tr><td>Cllama-7b-Ins</td><td>20.60</td><td>14.20</td></tr><tr><td>CodeLlama-70b-Ins</td><td>31.20</td><td>-1.00</td></tr><tr><td>Codestral-Latest</td><td>37.90</td><td>41.80</td></tr><tr><td>DSCoder-1.3b-Base</td><td>19.00</td><td>13.40</td></tr><tr><td>DSCoder-1.3b-Ins</td><td>18.10</td><td>17.00</td></tr><tr><td>DSCoder-33b-Base</td><td>29.90</td><td>29.10</td></tr><tr><td>DSCoder-33b-Ins</td><td>26.60</td><td>31.70</td></tr><tr><td>DSCoder-6.7b-Base</td><td>23.50</td><td>25.10</td></tr><tr><td>DSCoder-6.7b-Ins</td><td>23.10</td><td>23.80</td></tr><tr><td>GPT-3.5-Turbo-0301</td><td>33.90</td><td>34.80</td></tr><tr><td>GPT-4-0613</td><td>44.30</td><td>64.80</td></tr><tr><td>GPT-4-Turbo-1106</td><td>40.50</td><td>83.60</td></tr><tr><td>GPT-4-Turbo-2024-04-09</td><td>45.90</td><td>83.80</td></tr><tr><td>GPT-4O-2024-05-13</td><td>39.10</td><td>91.00</td></tr><tr><td>Gemini-Flash-1.5-May</td><td>21.40</td><td>57.10</td></tr><tr><td>Gemini-Pro</td><td>27.70</td><td>37.40</td></tr><tr><td>Gemini-Pro-1.5 (April) (n=1)</td><td>30.30</td><td>64.40</td></tr><tr><td>Gemini-Pro-1.5-May</td><td>42.10</td><td>72.10</td></tr><tr><td>LLama3-70b-Ins</td><td>29.60</td><td>55.50</td></tr><tr><td>LLama3-8b-Ins</td><td>18.40</td><td>29.40</td></tr><tr><td>MagiCoderS-CL-7B</td><td>21.20</td><td>-1.00</td></tr><tr><td>MagiCoderS-DS-6.7B</td><td>27.20</td><td>-1.00</td></tr><tr><td>Mistral-Large</td><td>36.60</td><td>54.40</td></tr><tr><td>Phind-34B-V2</td><td>26.90</td><td>-1.00</td></tr><tr><td>StarCoder</td><td>20.30</td><td>-1.00</td></tr><tr><td>WCoder-34B-V1</td><td>28.40</td><td>-1.00</td></tr></table>
+
+Table 7: Code Execution Performances  
+
+<table><tr><td>Model</td><td>PASS@1</td></tr><tr><td>GPT-4o</td><td>49.8</td></tr><tr><td>Claude-3.5-Sonnet</td><td>45.9</td></tr></table>
+
+Table 8: Results for models with  $\mathrm{C} +  +$  evaluations on the entire 713 LIVECODEBENCH problems
+
+# F QUALITATIVE EXAMPLES
+
+# F.1 CODE EXECUTION
+
+We show 5 examples from the code execution task that GPT-4 (gpt-4-1106-preview) still struggles to execute, even with CoT.
+
+```python
+def countWays(nums: List[int]) -> int:
+    nums.sort()
+    n = len(nums)
+    ans = 0
+    for i in range(n + 1):
+        if i and nums[i-1] >= i: continue
+            if i < n and nums[i] <= i: continue
+                ans += 1
+    return ans
+assert countWays(nums = [6, 0, 3, 3, 6, 7, 2, 7]) == 3
+# GPT-4 + CoT Outputs: 1, 2, 4, 5
+```
+
+# Mistake 1
+
+```python
+def minimumCoins(pprices: List[int]) -> int:
+    @cache
+    def dfs(i, free_until):
+        if i >= len(pprices):
+            return 0
+        res = prices[i] + dfs(i + 1, min(len(pprices) - 1, i + i + 1))
+        if free_until >= i:
+            res = min(res, dfs(i + 1, free_until))
+        return res
+    dfs.cache_clear()
+    return dfs(0, -1)
+    assert minimumCoins(pprices = [3, 1, 2]) == 4
+# GPT-4 + CoT Outputs: 1, 3, 5, 6
+```
+
+# Mistake 2
+
+```python
+def sortVowels(s: str) -> str:
+    q = dequesorted((ch for ch in s if vowel(ch)))
+    res = []
+    for ch in s:
+        if vowel(ch):
+            res.append(q.popleft())
+        else:
+            res.append(ch)
+        return False. join(res)
+assert sortVowels(s = 'lEetcOde') == 'lEOtcede'
+# GPT-4 + CoT Outputs: "leetecode", "lEetcOde", "leetcede", "leetcEde
+ $\leftrightarrow$  ", "leetcOde"
+```
+
+# Mistake 3
+
+```python
+def relocateMarbles(nums:List[int],moveFrom:List[int],moveTo:List[  $\leftrightarrow$  int]）->List[int]:   
+nums  $=$  sorted(list(set(nums))) dd  $= \{\}$  for item in nums: dd[item]  $= 1$  fora,binzip(moveFrom，moveTo): del dd[a] dd[b]  $= 1$  11  $=$  dd_keys() return sorted(ll)   
+assert relocateMarbles(nums  $= [1,6,7,8]$  ,moveFrom  $= [1,7,2]$  ，  $\leftrightarrow$  moveTo  $= [2,9,5]) = = [5,6,8,9]$    
+#GPT-4  $^+$  CoT Outputs:[2,6,8,9],[2,5,6,8,9]，KeyError
+```
+
+# Mistake 4
+
+```python
+def minimumSum(nums: List[int]) -> int:
+    left, right, ans = [inf], [inf], inf
+    for num in nums:
+        left.append(min(left[-1], num))
+    for num in nums[::-1]:
+        right.append(min(right[-1], num))
+    right.reverse()
+    for i, num in enumerate(nums):
+        if left[i] < num and right[i + 1] < num:
+            ans = min(ans, num + left[i] + right[i + 1])
+    return ans if ans < inf else -1
+assert minimumSum(nums = [6, 5, 4, 3, 4, 5]) == -1
+# GPT-4 + CoT Outputs: 10, 11, 12
+```
+
+# Mistake 5
+
+# G LIMITATIONS
+
+Benchmark Size. LIVECODEBENCH code generation scenario currently hosts over 500 instances from problems released since May 2023. To account for contamination in DEEPSEEK, we only perform evaluations on problems released after the model cutoff date. This leads to only 349 problems used in our final evaluations which might add noise due to problem set samples. We currently estimate  $1 - 2\%$  performance variations in LCB code generation due to this issue (measured by bootstrapping 349 sized problem sets from the full dataset). Other scenarios, i.e. self-repair, code execution, and test output prediction comprise 238, 188, and 254 problems would have similar performance variations. We thus recommend exercising proper judgement when comparing models with small performance differences. Note that HUMANEVAL has 164 problems and would also struggle with similar issues.
+
+This issue is also exacerbated for newer models, with more recent cutoff dates, as they might only have access to a smaller evaluation set. We propose two solutions addressing this issue as we evolve LIVECODEBENCH. First, we will use other competition platforms for problem collection, allowing larger number of recent problems to be added to the benchmark. In addition, we also hope supplement this with an unreleased private test set constructed specifically for model evaluation. These problems will use a similar flavor to current problems and will be used when models are submitted for evaluation to the LIVECODEBENCH platform. This would reduce the reliance on public accessible problems and provide a more robust evaluation of the models while providing community public access to similar problems, similar to strategies employed by popular platforms like KAGGLE.
+
+Focus on Python. LIVECODEBENCH currently only focuses on Python which might not provide enough signal about model capabilities in other languages. However, since we collected problem statements and serialized tests, adding new programming languages would be straightforward once appropriate evaluation engines are used.
+
+Robustness to Prompts. Recent works have identified huge performance variances that can be caused due to insufficient prompt. Here, we either do not tune prompts across models or make minor adjustments based on the system prompts and delimiter tokens. This can lead to performance variance in our results. Our findings and model comparison orders generalize across LIVECODEBENCH scenarios and mostly match the performance trends observed on HUMANETVAL making this a less prominent issue.
+
+This issue can be particularly observed open models on the code execution scenario with COT prompting. Interestingly, often the open models perform even worse in comparison to the direct code execution baseline. Note that we used same prompts for the closed models all of which show noticeable improvement from COT. While the used prompts might be sub-optimal, this highlights how open-models perform worse against the closed models at performing chain-of-thought.
+
+Problem Domain. Programming is a vast domain and occurs in various forms such as programming puzzles, competition programming, and real-world software development. Different domains might have individual requirements, constraints, challenges, and difficulty levels. LIVECODEBENCH currently focuses on competition problems sourced from three platforms. This might not be representative of the "most general" notion of LLM programming capabilities. Particularly, real-world usage of LLMs is drawn upon open-ended and unconstrained problems raised by users. We therefore recommend using LIVECODEBENCH as a starting point for evaluating LLMs and further using domain-specific evaluations to measure and compare LLMs in specific settings as required.
\ No newline at end of file
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+# LIVEXIV - A MULTI-MODAL LIVE BENCHMARK BASED ON ARXIV PAPERS CONTENT
+
+Nimrod Shabtay $^{1,2}$ , Felipe Maia Polo $^{3}$ , Sivan Doveh $^{2}$ , Wei Lin $^{4}$ , M. Jehanzeb Mirza $^{5}$ , Leshem Choshen $^{6}$ , Mikhail Yurochkin $^{6}$ , Yuekai Sun $^{3}$ , Assaf Arbelle $^{2}$ , Leonid Karlinsky $^{6}$ , Raja Giryes $^{1}$
+
+$^{1}$  Faculty of Engineering Tel-Aviv University,  $^{2}$  IBM Research,  
+<sup>3</sup> Department of Statistics, University of Michigan, USA <sup>4</sup> JKU Linz, Austria, <sup>5</sup> MIT CSAIL. <sup>6</sup> MIT-IBM
+
+# ABSTRACT
+
+The large-scale training of multi-modal models on data scraped from the web has shown outstanding utility in infusing these models with the required world knowledge to perform effectively on multiple downstream tasks. However, one downside of scraping data from the web can be the potential sacrifice of the benchmarks on which the abilities of these models are often evaluated. To safeguard against test data contamination and to truly test the abilities of these foundation models we propose LiveXiv: A scalable evolving live benchmark based on scientific ArXiv papers. LiveXiv accesses domain-specific manuscripts at any given timestamp and proposes to automatically generate visual question-answer pairs (VQA). This is done without any human-in-the-loop, using the multi-modal content in the manuscripts, like graphs, charts, and tables. Moreover, we introduce an efficient evaluation approach that estimates the performance of all models on the evolving benchmark using evaluations of only a subset of models. This significantly reduces the overall evaluation cost. We benchmark multiple open and proprietary Large Multi-modal Models (LMMs) on the first version of our benchmark, showing its challenging nature and exposing the models' true abilities, avoiding contamination. Lastly, in our commitment to high quality, we have collected and evaluated a manually verified subset. By comparing its overall results to our automatic annotations, we have found that the performance deviation is indeed minimal ( $< 2.5\%$ ). Our dataset is available online on HuggingFace and our code is available on GitHub.
+
+# 1 INTRODUCTION
+
+![](images/d42fe4d0bf3ce2c7ee485571348fc581386d85a422cdbca3fff082c5b6c71625.jpg)  
+Figure 1: Static benchmark contamination. As training data increases, the risk for test set contamination grows and static benchmarks become saturated, reflecting falsely improved capabilities.
+
+The internet, with its vast and ever-growing repository of information, serves as a rich data source for training Large Language Models (LLMs) (Brown et al., 2020; OpenAI, 2023; Chiang et al., 2023; Raffel et al., 2019; Touvron et al., 2023a;b; Dubey et al., 2024) and Large Multi-modal Models (LMMs) (OpenAI, 2023; Liu et al., 2023c; Li et al., 2024c; Zhu et al., 2023; Chen et al., 2023a; Alayrac et al., 2022; Radford et al., 2021a). This diverse and continuously updated data fits precisely the need to cover varying knowledge in scale in the training data.
+
+Training on such data enables the models to achieve superhuman performance across a wide range of tasks on multiple common benchmarks (Fu et al., 2023; Yue et al., 2024; Li et al., 2024d; Liu et al., 2023d).
+
+![](images/4d6118e2e016a94ec0d890cda88f7bfc31a524e1b70276e633cf489bebadeaf7.jpg)  
+Figure 2: We propose LiveXiv, a new method for generating Live multi-modal dataset for Visual Question-Answering based on ArXiv content. Our pipeline automatically generates scalable and reliable questions along with an efficient evaluation method to reduce the computational and logistic overheads required for continually evaluating past and present models on new versions of the dataset.
+
+We hypothesize that a portion of LLMs' reported improvements are due to data contamination (Figure 1) and pose the following question: To what extent does the potential for test set contamination during large-scale training affect our perception of the abilities of LMMs?
+
+One possible way to safeguard against the contamination of static benchmarks is to design a live benchmark that can continuously harness data from the web and turn it into an ever-evolving benchmark to test the abilities of these models. A live benchmark may be used in one of the following ways: (a) Expand the dataset over time and evaluate the models' overall knowledge over all collected data, while taking into account that the data might be contaminated. (b) Use only the latest version to assess model capabilities while keeping data contamination risk minimal. While we focus on (b), we share key properties of our efficient evaluation method can be applicable to both cases.
+
+Although, a live benchmark is a promising direction, it still comes with its fair share of challenges. A live benchmark should ideally be updated frequently, consistently, and automatically, i.e., it should be able to scrape the data from the web and formulate it into a benchmark for automated evaluations. Furthermore, as the benchmark is ever-evolving, each time a new version arrives, all the participant models need to be re-evaluated, making the update procedure prohibitively expensive both in time and compute. This requires a methodology for efficient evaluation of these models on a continuously updating benchmark. Such a methodology should ease the computational burden of evaluating all the models on each new version of the dataset and reduce the logistic overhead of maintaining inaccessible old models.
+
+In this work, we take a step in this direction and propose LiveXiv – a novel fully automated multi-modal live benchmark that focuses on scientific domains. LiveXiv starts with scraping category-specific (e.g., cs.CV, eess.SY, q-bio.BM, etc.) manuscripts from ArXiv and generates visual question answers from figures, charts, and tables present in these manuscripts through a capable multi-modal model, namely, GPT-4o. As it is challenging to directly feed information-rich PDF documents to GPT-4o, as a pre-processing step, we extract relevant information from the papers by processing it with a structured document parsing pipeline (Team, 2022) to obtain pertinent information like placements of figures, charts, tables, and the text in the captions or in the tables.
+
+This information is used to extract, e.g., by cropping, relevant information from the manuscripts, which is fed to GPT-4o to generate visual questions and answers. Although very capable, GPT-4o is still prone to errors, e.g., due to hallucinations, and may even generate questions that can be answered without visual information. Thus, to mitigate these issues, we add an extensive filtering stage that automatically filters questions requiring only textual information to answer them, and reduce hallucinations through obtaining agreement about the generated questions with another capable multi-modal model, namely, Claude-Sonnet. After the extensive filtering, we obtain a large corpora of VQA pairs which are incorporated into our LiveXiv live benchmark.
+
+Over time, the benchmark is expected to grow, either in the size of the dataset or the amount of models to be evaluated, which increases the required resources for evaluation. Moreover, comparing a new model to existing models at different times requires re-evaluating the existing models over the latest version of the dataset, which can cause additional overhead for continuous evaluation and comparison to prior works. To make the evaluations on LiveXiv feasible, we take inspiration from Maia Polo et al. (2024a;b) and propose a method to approximate the performance of the existing models in new versions of LiveXiv just by re-evaluating a small portion of them. Figure 2 provides a conceptualized overview of our approach.
+
+We summarize our contributions as follows: (a) We propose a scalable live benchmark without any human in the loop that automatically harnesses data from online scientific manuscripts, generates multiple
+
+VQA pairs, filters these questions to reduce errors, and formulates them in the form of a benchmark to test the evolving landscape of LMMs; (b) We introduce an efficient evaluation pipeline that requires LMMs to be tested only on a fraction of the data to infer its performance on the latest version of the benchmark, reducing the overall needed evaluations by at least  $\approx 75\%$ ; (c) We benchmark multiple open and proprietary LMMs on the first version of our benchmark highlighting its challenging nature and providing interesting insights about the models' behavior when evaluated on less contaminated data.
+
+# 2 RELATED WORKS
+
+Large multi-modal Models (LMMs). LMMs have shown significant advancements in enabling billion-parameter scale LLMs to perform multi-modal tasks such as image captioning, visual reasoning, and visual question answering. Academia and industry have endeavored to develop LMMs targeting the multi-modal competence of advanced proprietary models like GPT-4o (OpenAI, 2023) and Claude (cla, 2024). Instruct-BLIP performs instruction tuning on the pre-trained BLIP-2 (Li et al., 2023) covering 11 vision-language tasks. The LLaVA series models (Liu et al., 2023c;a;b; Li et al., 2024b) develop the pipeline of collection of instruction-following data and visual instruction tuning with enhanced vision capabilities. The internLM-XComposer (IXC) series (Dong et al., 2024a;b) target free-form vision-language composition and multilingual comprehension. Models from Idefics release (Laurençon et al., 2024b;a) benefit from the massive collection of instruction-following data from over 50 vision-language databases, enhancing capabilities of OCR, document understanding, and visual reasoning. In this work, we include 17 top-performing LMMs in our multi-modal live benchmark LiveXiv, covering both open-sourced and proprietary representatives.
+
+Static evaluation benchmarks for LMMs. Most existing LMM benchmarks offer static evaluation with fixed questions and answers (Fu et al., 2023; Yue et al., 2024; Li et al., 2024d; Liu et al., 2023d; Huang et al., 2024; Lin et al., 2024; Zhang et al., 2024b; Onoe et al., 2024; Yanuka et al., 2025). Ben-Kish et al. (2024) use generative models to extend such benchmarks to open settings. MME (Fu et al., 2023) offers evaluation of perception and cognition on 14 tasks and MMMU (Yue et al., 2024) includes 11.5K questions from college exams, quizzes and text books from six major disciplines. Although these benchmarks cover a large variety of multi-modal domain knowledge, evaluation on them is faced with two hazards: the excessive evaluation cost and test data contamination. In this work, we tackle both challenges by proposing a suite that enables efficient evaluation on a contamination-free live benchmark.
+
+Contamination-free benchmarks. As large foundation models like LLMs and LMMs are trained on combined sources of tremendous amount of web data or repurposed version of existing open-sourced datasets, there is a high risk of overlap between training data and samples from evaluation benchmarks. Reported evidence and analysis show impact of data contamination on evaluation benchmarks for LLMs (Wei et al., 2023; Zhang et al., 2024a; Cobbe et al., 2021; Roberts et al., 2023; Jain et al., 2024) and LMMs (Chen et al., 2024), indicating the significance of contamination-free evaluation benchmarks. Recent works targeted at LLMs are discussed in A.1.
+
+For LMMs, Vibe-Eval (Padlewski et al., 2024) and LLaVA-Wilder (Li et al., 2024a) perform contamination check on the collected samples that reflect real-world user requests. Most related to our work, the LMMs-Eval LiveBench (Zhang et al., 2024b) collects images from sources of new websites and online forums and employs proprietary LMMs for design and revision of questions. However, the LMMs-Eval LiveBench requires human manual verification of questions which impedes the scalability. Furthermore, it contains only open-ended questions that require LMM-as-a-judge which is time-consuming, susceptible to judge biases, and difficult to scale. In comparison, our LiveXiv constructs a fully-automated data collection pipeline which generates multiple-choice questions which are challenging to the top-performing LMMs.
+
+Efficient benchmarks. With the increasing amount of tasks and samples in current benchmarks, evaluation of the full suite is time-consuming and cost-intensive. Efforts are underway to develop efficient benchmarks that reduce computation costs without sacrificing reliability. In A.1 at the appendix we cover the works related to efficient benchmarks.
+
+# 3 LIVEXIV
+
+At a higher level, our automated LiveXiv is created by first obtaining the domain-specific scientific manuscripts from ArXiv at any given timestamp. Then, to obtain pertinent information from the manuscripts, we pass them through a structured document parsing pipeline and then generate visual
+
+![](images/ada7d920d155bd4d4e505ed36fa5a9e0065293c8687c17fa57d47ec69cfa2702.jpg)  
+Figure 3: Our live dataset generation consists of several stages. We first extract the images and their corresponding metadata (i.e. captions and table contents), then we classify the figures into categories using meta-prompting. All the extracted data is then fed to GPT4o to generate multiple questions-answer pairs per image. Since generative models are prone to errors, we apply several filtering steps, using an LLM and LMM to ensure that our dataset is truly multi-modal and reliable.
+
+question answers through a capable LMM (Section 3.1). However, the generated questions can contain errors due to hallucinations or might be too straightforward to answer. Thus, to mitigate these issues, we offer an extensive filtering stage (Section 3.2). To evaluate the benchmark, we propose an efficient evaluation framework to infer the overall performance on the benchmark using only a small subset of evaluations, making the evaluations extremely resource-efficient (Section 3.3). The data acquisition and filtering steps are schematically visualized in Figure 3.
+
+# 3.1 DATA ACQUISITION AND VQA GENERATION
+
+We start with the data acquisition phase, then pre-process the data to obtain the required metadata (e.g., placements of figures, captions, etc.), and then generate the first iteration of VQA from the multi-modal data (figures and tables) from the manuscripts.
+
+Data Acquisition: At any given timestamp, we begin by acquiring only ArXiv papers which have non-exclusive license to distribute from predefined domains such as Computer Science (cs.AI, cs.CV), Electrical Engineering (eess.SP, eess.SY), and Quantitative Biology (q-bio.BM, q-bio.GN). However, these manuscripts contain a lot of information that might not be necessary for the task of VQA data generation. Thus, to extract pertinent information we require a pre-processing step.
+
+Pre-processing: The downloaded PDFs undergo a structured document parsing pipeline using the DeepSearch toolkit (Team, 2022), which extracts a comprehensive layout of each document, including the positions of figures, tables, captions, and other elements. This structured layout forms the basis for extracting the multi-modal data required for subsequent tasks. To enrich the dataset with additional metadata not captured by the parsing pipeline, we employ a meta-prompting approach with CLIP (Radford et al., 2021b), similar to the method used by Mirza et al. (2024). Specifically, we classify the figures into three distinct categories: Block Diagram, Chart, and Qualitative visual examples which facilitates a more granular, domain-specific evaluation of LMM performance.
+
+VQA Generation: For Visual Question Answering (VQA), we construct pairs of figures and their corresponding captions, and for generating VQA from the data present in the tables, we obtain (e.g., crop) images of tables accompanied by their corresponding data.
+
+The VQA process involves two steps using GPT-4o. First, we input the figure and its caption to GPT-4o to generate a detailed description of the figure, employing a Chain-of-Thought (CoT) approach (Wei et al., 2022). Next, the detailed description and figure are fed back into GPT-4o, with prompts adapted from 'ConMe' (Huang et al., 2024) to suit our scientific use case, enabling the generation of relevant VQA questions. For questions from the tables, we utilize the table's content directly, presenting both the image of the table and its data in markdown format to GPT-4o to produce questions that require common-sense reasoning and data manipulation. The automated nature of this process ensures a robust and comprehensive evaluation framework for LMMs, tailored to scientific literature specifics. Detailed prompt templates can be found in Appendix A.6.
+
+# 3.2 FILTERING PHASE
+
+Even though GPT-4o is powerful and has been reported to outperform humans on many different benchmarks (OpenAI, 2023), still it is prone to errors and sometimes can even result in VQA pairs that are answerable without requiring the visual information. Thus, to ensure that the benchmark remains competitive and also has minimum errors, we propose an extensive automatic filtering step. At a higher
+
+level, the filtering phase consists of two main parts, each designed to mitigate a separate issue that can arise due to the automatic dataset generation.
+
+Blind test with an LLM: To ensure that the generated VQA pairs are truly multi-modal, we pass them through a Large Language Model (LLM), namely LLama-3.1-70B (Meta, 2024), without providing any associated images or image descriptions. This process, referred to as a blind test, aims to identify questions that the LLM can answer correctly even in the absence of visual context, indicating they are not truly multi-modal. To ensure robustness, this blind evaluation is repeated multiple times to eliminate any potential lucky guesses by the LLM. Questions that are consistently answered correctly by the LLM are filtered out, resulting in the removal of approximately  $30\%$  of the generated questions. This step ensures that the remaining questions in the dataset are inherently multi-modal and cannot be answered solely based on linguistic context. The filtered dataset thus represents a more challenging benchmark for evaluating multi-modal capabilities of vision-language models.
+
+Agreement between disjoint models: Generative models, including LMMs, are prone to hallucination, where the model generates incorrect or not grounded information. In our case, these hallucinations can lead to erroneous VQA pairs. To address this issue, we introduce an additional filtering step. All questions that pass the initial "blind test" are reviewed along with their generated answers by a different LMM, in this case Claude-Sonnet (cla, 2024), which is provided with the image, question, and the ground truth answer which were all generated by GPT-4o. This second model is asked to either agree or disagree with the generated answer, considering the visual context.
+
+We point out that agreement between models is a nuanced process; incorporating more models to validate answers may lead to the exclusion of difficult questions, thereby diluting the difficulty of the dataset. Therefore, we limit this validation step to models with comparable performance to the generation model (i.e. GPT-4o). Our preliminary manual evaluation on a subset of the dataset indicates that this agreement step significantly reduces the proportion of incorrect ground-truth (GT) questions, with a reduction of  $38.5\%$ , while minimally impacting the retention of high-quality question-GT pairs, with only a  $6.15\%$  removal of valid pairs. This refinement ensures that the final dataset is both challenging and accurate for the evaluation of LMMs' multi-modal reasoning capabilities. The generated corpus of data is ready to be incorporated into LiveXiv and can be updated automatically without any human intervention.
+
+# 3.3 EFFICIENT EVALUATION
+
+Since LiveXiv is a dynamic benchmark, evaluation can be costly: ideally, whenever a new version of the benchmark is released, all models must be re-evaluated on the updated data, which can pose an engineering challenge and become computationally expensive when handling dozens of models. In this section, we describe our approach to efficient evaluation, which avoids re-evaluating all models at each step, making LiveXiv's maintenance economically feasible. Our idea is based on Item Response Theory (IRT) (Cai et al., 2016; Van der Linden, 2018; Lord et al., 1968; Maia Polo et al., 2024b;a), a collection of statistical models traditionally used in psychometrics and educational assessment. We briefly give some background on IRT and detail how we use it for our evaluations.
+
+# 3.3.1 ITEM RESPONSE THEORY (IRT)
+
+We use the IRT model to predict the probability of a certain LMM  $i$  answering correctly on a sample (question)  $j$ . In mathematical terms, let  $Y_{ij} \in \{0,1\}$  denote the correctness on sample  $j$  when responded by LMM  $i$ :
+
+$$
+Y _ {i j} \sim \mathrm {B e r n o u l l i} (\mu (\theta_ {i}, \beta_ {j})),
+$$
+
+where  $\theta_{i}$  is an LMM-specific parameter,  $\beta_{j}$  is a sample-specific parameter, and  $\mu$  is a function that maps those parameters to the probability of correctness. In this work, we follow Maia Polo et al. (2024b) and assume the parameters live in the real line while  $\mu$  induces a logistic regression model. In more detail, we assume
+
+$$
+\mathbb {P} \left(Y _ {i j} = 1; \theta_ {i}, \beta_ {j}\right) = \frac {1}{1 + \exp \left[ - \left(\theta_ {i} - \beta_ {j}\right) \right]}. \tag {1}
+$$
+
+Here,  $\theta_{i}$  can be interpreted as the skill level of LMM  $i$  while  $\beta_{j}$  is seen as the hardness of sample  $j$ . By equation 1, if  $\theta_{i}$  is much greater (resp. smaller) than  $\beta_{j}$ , then the probability  $\mathbb{P}(Y_{ij} = 1; \theta_{i}, \beta_{j})$  will be close to one (resp. zero). This version of the IRT model is known as the Rasch model (Georg, 1960; Chen et al.,
+
+2023b), and it is widely used in fields such as recommendation systems (Starke et al., 2017), educational testing (Clements et al., 2008), and evaluation of language models (Maia Polo et al., 2024b). Moreover, it has a similar formulation to the popular Bradley-Terry model (Bradley & Terry, 1952) used in Chatbot Arena (Chiang et al., 2024), a popular and dynamic benchmark for AI-powered chatbots. We fit the Rasch model using maximum likelihood estimation as in Chen et al. (2023b) and Maia Polo et al. (2024b).
+
+# 3.3.2 EFFICIENT EVALUATION WITH IRT
+
+We can estimate old model scores on new data without reevaluating those models. Let  $\mathcal{I}_t$  and  $\mathcal{J}_t$  represent sets of non-negative integers corresponding to LMMs and samples at time  $t \geq 0$ . It is usually the case that  $\mathcal{I}_t \subseteq \mathcal{I}_{t+1}$  since the set of available models does not shrink over time, except in cases of depreciation, and  $\mathcal{J}_{t_1} \cap \mathcal{J}_{t_2} = \emptyset$  for  $t_1 \neq t_2$  because samples are not repeated across different time steps. Let the set of evaluated models at time  $t$  be denoted by  $\hat{\mathcal{I}}_t$ . For  $t > 0$ , we assume that  $\mathcal{I}_t \backslash \mathcal{I}_{t-1}$  is a proper subset of  $\hat{\mathcal{I}}_t$ , meaning that all newly introduced models are evaluated on the new batch of samples along with some previously existing models. At  $t = 0$ , we assume that  $\hat{\mathcal{I}}_t = \mathcal{I}_t$ , meaning all models are evaluated on all samples. Furthermore, we assume that  $|\hat{\mathcal{I}}_t|$  is much smaller than  $|\mathcal{I}_t|$  when  $t > 0$  so computing power and evaluation time can be saved.
+
+Our goal at time  $t > 0$  is to estimate the performance of a model  $i \notin \hat{\mathcal{I}}_t$  on the set of samples  $\mathcal{J}_t$ , using only the correctness scores  $\mathcal{D}_t = \{Y_{ij} : (i,j) \in \Omega_t\}$ , where  $\Omega_t \triangleq \cup_{t' \leq t} \hat{\mathcal{I}}_{t'} \times \mathcal{J}_{t'}$ . Specifically, we aim to approximate  $S_{it} = \frac{1}{|\mathcal{J}_t|} \sum_{j \in \mathcal{J}_t} Y_{ij}$  by estimating its expectation
+
+$$
+\mathbb {E} \left[ S _ {i t} \right] = \frac {1}{| \mathcal {I} _ {t} |} \sum_ {j \in \mathcal {I} _ {t}} \mathbb {P} \left(Y _ {i j} = 1; \theta_ {i}, \beta_ {j}\right). \tag {2}
+$$
+
+For a moment, let us assume that  $\Omega_t$  is known. Using  $\mathcal{D}_t$ , we can estimate the skill parameters  $\theta_i$ 's of all models in  $\mathcal{I}_t$  and the difficulty parameters  $\beta_j$ 's of all samples in  $\cup_{t' \leq t} \mathcal{I}_{t'}$ ; we denote these estimates as  $\hat{\theta}_i$ 's and  $\hat{\beta}_j$ 's. Finally, we obtain an approximation for equation 2,  $\hat{\mathbb{E}}[S_{it}]$ , by substituting  $\theta_i$  and  $\beta_j$ 's by their estimates. The estimator  $\hat{\mathbb{E}}[S_{it}]$  is known as the Performance-IRT estimator (Maia Polo et al., 2024b;a).
+
+Now, we provide a method to obtain  $\Omega_{t}$  assuming  $\Omega_{t - 1}$  is given; in summary, we need to decide which models in  $\mathcal{I}_{t - 1}$  are going to be in  $\hat{\mathcal{I}}_t$ . Our approach to choosing which models are going to be re-evaluated is inspired by the concept of optimal design of tests (Van der Linden, 2017, Chapter 9) but in which we choose LMMs instead of samples. First, we set a budget  $m_t$  representing the maximum number of models to be re-evaluated at time step  $t$ . Second, assuming that the level of difficulty of the new samples  $\mathcal{I}_t$  is not very different from the ones in  $\mathcal{I}_{t - 1}$ , we choose a set of  $m_t$  representative samples in  $\mathcal{I}_{t - 1}$  by ordering  $\hat{\beta}_j$ 's and choosing equally spaced samples, based on their quantiles, from the 5th to the 95th percentiles; this will give us questions with a variety of difficulties, excluding outliers. For example, if  $m_t = 3$  we would choose questions with difficulties in the 5th, 50th, and 95th percentiles. Denote the chosen core set of samples as  $\{j_0,\dots ,j_{m_t - 1}\}$  and, for each one of these samples  $j_{k}$ , we choose a model  $i$  in  $\mathcal{I}_{t - 1}$  such that the following Fisher information criterion
+
+$$
+F _ {j _ {k}} (i) = \mathbb {P} \left(Y _ {i j _ {k}} = 1; \hat {\theta} _ {i}, \hat {\beta} _ {j _ {k}}\right) \left[ 1 - \mathbb {P} \left(Y _ {i j _ {k}} = 1; \hat {\theta} _ {i}, \hat {\beta} _ {j _ {k}}\right) \right]
+$$
+
+is maximized. The model that maximizes  $F_{j_k}$  is maximally informative about the parameter of sample  $j_k$  and, consequently, about all samples with similar difficulty levels in the new version of LiveXiv; this will help us estimate the difficulties of new samples. We note that some models in  $\mathcal{I}_{t-1}$  might not be available at step  $t$ , e.g., due to depreciation; when choosing models, we do not consider them, but note that we can still estimate their performance on the new batches of data. Moreover, the model selection procedure can also take convenience into account; for example, if two models have very similar Fisher information, we opt for the one that is cheaper to evaluate.
+
+Our experiments demonstrate that re-evaluating just 5 (or even 3) models per step provides accurate model evaluation. With 50 total models, this approach can reduce computing costs by at least  $\times 10$ , particularly when selecting less expensive models without significantly impacting  $F_{j_k}$ . In this paper, we use a maximum of 19 models; thus, re-evaluating 3 to 5 models represents a saving of  $\approx 75\%$  to  $85\%$  in the number of evaluated models.
+
+# 4 RESULTS & ANALYSIS
+
+This section presents the results obtained on the first version of LiveXiv. First, we start by describing the experimental settings. Then, we present the results and finally conclude with a detailed analysis of our dataset.
+
+# 4.1 EXPERIMENTAL SETTINGS
+
+Evaluation Protocol: After the generation of the question-answer pairs from our automated pipeline explained in Section 3, we transform the benchmark to multiple-choice questions. We resort to the 'generate' inference employed extensively by previous works, such as Li et al. (2024d); Huang et al. (2024); Liu et al. (2023d). The model is prompted to choose the letter corresponding to the correct choice and answer with the letter directly. The output letter is then compared with the ground truth and the accuracy is measured. For ease of assimilation and to obtain insights into what type of data the models flourish at, we provide the results from data generated on tables and figures separately. The data generated from figures is labeled as part of Visual Question and Answers (VQA) and the data from the tables is labeled as Table Question and Answers (TQA). Examples for the multiple-choice formulation of the question-answer pairs are added to the Appendix Section A.3.
+
+Size of dataset: The first version of our LiveXiv consists of 7328 questions on figures, and 9000 questions on tables, both are generated from 250 papers (25 papers from 10 domains). Overall our first version of the dataset has 16328 questions in total. Thanks to the continual growth in the number of publications in our target domains and the fully automatic nature of our proposed LiveXiv pipeline for benchmark data generation, we will grow LiveXiv by adding an equal-sized large amount of new VQA & TQA data (around 7K VQA and 9K TQA) on a monthly basis.
+
+Models: We extensively evaluate our benchmark by employing a total of 17 LMMs. Specifically, we employ 5 models from the LLaVA family of models including LLaVA 1.5-7B and LLaVA 1.5-13B (Liu et al., 2023c), LLaVA-1.6-7B and LLaVA 1.6-34B (Liu et al., 2023b), and LLaVA One-Vision (Li et al., 2024b). Furthermore, we employ IntstructBLIP (Dai et al., 2023), InternVL2-2B and InternVL2-8B (Chen et al., 2023c), InternLM-Xcomposer2-4KHD (Dong et al., 2024b), InternLM-Xcomposer2.5 (Chen et al., 2023c), Mantis (Jiang et al., 2024), Phi3v (Abdin et al., 2024), Idefics2 (Laurençon et al., 2024b) and Idefics3 (Laurençon et al., 2024a), Qwen2-VL (Wang et al., 2024) and API models Claude-Sonnet (cla, 2024) and GPT-4o (OpenAI, 2023) for our evaluations. These models have been chosen because of their varying characteristics and strong performance on multiple current benchmarks. All the models (except GPT-4o and Cloude-Sonnet) are accessed from the huggingface API, which makes our framework modular for an extension to more models as they are being added to the hub in the future.
+
+Additional LiveXiv versions: While this section mainly focuses on the analysis of the first version of LiveXiv, new versions are continuously being uploaded to the HuggingFace Hub, at the time of writing, four additional versions exist: one of past ArXiv papers (v0) and three of more recent papers (v2 - v4). Version 2 consists of  $18K$  samples, introduces 4 new domains from physics (namely, physics optics, physics.bio-ph, physics.app-ph, physics.data-an), and includes two additional LMMs (Pixtral and Molmo-7B). Version 3 utilizes multiple generation and filtering models (GPT-4o, Claude-Sonnet, Qwen2-VL-72B) to and increase diversity and measure the effect of potential biases. In Version 4, we restrict diversity to two models, GPT-4o and Claude-Sonnet, but maintain variability by having each model take on different roles each time. In addition, to perform a deeper analysis, we created a variant of version 1 with opposite roles (i.e., Claude as the QA generation model and GPT as the filter model). We found that the question and visual content diversity remained roughly the same which kept the model performance and ranking unaffected. Lastly, version 0 is generated from papers from more than 10 years ago. Our experiments reassure our findings about the performance of our efficient evaluation, strengthening its validity even in the presence of a large time gap between the dataset versions. All the details can be found in Appendix A.4.
+
+# 4.2 EXPERIMENTAL RESULTS
+
+Results on LiveXiv v1: We evaluated 17 LMMs across two prominent tasks, VQA and TQA. Table 1 provides a detailed summary of the performance across both tasks. One interesting observation is Claude's superior performance on all tasks. This substantial performance gap suggests that Claude's architecture and underlying methodologies are particularly well-suited for both VQA and TQA tasks. The results align with other relatively close benchmarks, DocVQA (Mathew et al., 2021), ChartQA (Masry et al., 2022) and AI2D (Kembhavi et al., 2016), where we see a similar trend: Claude has significantly higher performance
+
+Table 1: LiveXiv v1 VQA and TQA average accuracy across ArXiv taxonomy. (V is VQA, T-TQA)  
+
+<table><tr><td>VQA&amp;TQA Acc. # Samples</td><td>eess.SP V T 651</td><td>q-bio.BM T 429</td><td>q-bio.CB V T 900</td><td>cs.AI 1624</td><td>cs.AI T 697</td><td>esss.SY 1069</td><td>cs.CV 735</td><td>cs.RO 472</td><td>q-bio.GN 720</td><td>cs.LG 570</td><td>q-bio.TO 647</td><td>Mean 634</td><td>894</td><td>900</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>InstructBLIP-7B</td><td>21.2</td><td>18.1</td><td>25.2</td><td>16.6</td><td>19.5</td><td>20.2</td><td>24.5</td><td>21.8</td><td>23.4</td><td>18.9</td><td>21.3</td><td>20.7</td><td>22.6</td><td>22.8</td><td>24.9</td><td>16.9</td><td>24.1</td><td>18.5</td><td>21.1</td><td>18.2</td><td>23.6</td><td>19.1</td></tr><tr><td>LLVA-1.5-7B</td><td>29.0</td><td>24.9</td><td>27.8</td><td>20.9</td><td>29.5</td><td>25.4</td><td>31.9</td><td>23.5</td><td>30.5</td><td>25.0</td><td>31.0</td><td>21.8</td><td>34.9</td><td>24.9</td><td>29.1</td><td>22.6</td><td>29.3</td><td>24.9</td><td>32.8</td><td>25.4</td><td>30.4</td><td>23.5</td></tr><tr><td>LLVA-1.6-Mistral-7B</td><td>28.1</td><td>31.9</td><td>28.7</td><td>26.8</td><td>28.6</td><td>29.7</td><td>33.9</td><td>29.2</td><td>31.0</td><td>36.4</td><td>31.4</td><td>29.0</td><td>33.3</td><td>30.5</td><td>27.0</td><td>27.6</td><td>27.9</td><td>30.4</td><td>29.5</td><td>29.1</td><td>29.9</td><td>29.3</td></tr><tr><td>Mantis-LLama3-8B</td><td>32.3</td><td>31.2</td><td>28.6</td><td>28.0</td><td>32.7</td><td>30.0</td><td>33.7</td><td>30.0</td><td>30.2</td><td>33.5</td><td>36.9</td><td>29.1</td><td>32.6</td><td>32.5</td><td>29.2</td><td>29.9</td><td>30.8</td><td>30.0</td><td>34.9</td><td>30.3</td><td>32.1</td><td>30.0</td></tr><tr><td>LLVA-1.5-13B</td><td>32.6</td><td>30.5</td><td>29.4</td><td>28.9</td><td>31.5</td><td>33.1</td><td>33.4</td><td>31.5</td><td>33.2</td><td>35.6</td><td>35.9</td><td>31.5</td><td>35.7</td><td>33.0</td><td>30.6</td><td>29.8</td><td>30.0</td><td>29.4</td><td>32.2</td><td>30.8</td><td>32.3</td><td>30.9</td></tr><tr><td>Idefics2-8B</td><td>35.6</td><td>37.1</td><td>38.4</td><td>35.9</td><td>35.9</td><td>43.2</td><td>40.7</td><td>39.2</td><td>40.5</td><td>42.8</td><td>38.6</td><td>35.0</td><td>39.6</td><td>40.0</td><td>30.3</td><td>38.7</td><td>36.9</td><td>37.0</td><td>38.8</td><td>38.9</td><td>37.6</td><td>38.2</td></tr><tr><td>IXC2-KHD-7B</td><td>33.0</td><td>41.1</td><td>36.7</td><td>40.8</td><td>33.0</td><td>46.8</td><td>40.1</td><td>38.4</td><td>35.8</td><td>47.2</td><td>45.7</td><td>37.0</td><td>44.5</td><td>41.9</td><td>37.9</td><td>42.8</td><td>35.8</td><td>41.3</td><td>36.1</td><td>42.6</td><td>37.7</td><td>41.5</td></tr><tr><td>IXC2.5-7B</td><td>46.2</td><td>42.3</td><td>46.1</td><td>40.5</td><td>48.2</td><td>48.4</td><td>53.3</td><td>39.8</td><td>50.5</td><td>50.8</td><td>45.1</td><td>39.2</td><td>47.0</td><td>44.6</td><td>47.9</td><td>47.5</td><td>49.4</td><td>36.7</td><td>46.8</td><td>40.7</td><td>48.1</td><td>42.1</td></tr><tr><td>InternVL2-2B</td><td>48.4</td><td>42.7</td><td>48.1</td><td>44.2</td><td>50.4</td><td>53.7</td><td>53.4</td><td>48.9</td><td>50.5</td><td>58.3</td><td>46.3</td><td>44.7</td><td>54.2</td><td>51.6</td><td>48.4</td><td>52.0</td><td>48.2</td><td>49.2</td><td>50.9</td><td>52.0</td><td>49.8</td><td>49.1</td></tr><tr><td>LLVA-1.6-34B</td><td>48.4</td><td>49.5</td><td>45.6</td><td>49.4</td><td>47.4</td><td>55.7</td><td>55.9</td><td>49.2</td><td>52.5</td><td>59.7</td><td>51.8</td><td>48.9</td><td>54.9</td><td>50.7</td><td>47.9</td><td>49.2</td><td>47.9</td><td>46.6</td><td>50.2</td><td>50.8</td><td>50.0</td><td>50.2</td></tr><tr><td>Idefics3</td><td>54.4</td><td>47.2</td><td>50.6</td><td>48.1</td><td>52.3</td><td>55.5</td><td>57.2</td><td>48.9</td><td>57.0</td><td>57.8</td><td>53.3</td><td>47.2</td><td>54.6</td><td>51.9</td><td>51.5</td><td>51.7</td><td>51.5</td><td>48.0</td><td>56.6</td><td>51.8</td><td>53.7</td><td>50.2</td></tr><tr><td>LLVA-OneVision-7B</td><td>53.1</td><td>46.2</td><td>49.7</td><td>47.8</td><td>51.8</td><td>53.9</td><td>57.2</td><td>50.3</td><td>52.8</td><td>57.8</td><td>57.2</td><td>47.7</td><td>57.6</td><td>53.2</td><td>51.6</td><td>51.2</td><td>51.1</td><td>50.5</td><td>59.1</td><td>52.7</td><td>53.9</td><td>50.6</td></tr><tr><td>Phi3v</td><td>60.1</td><td>51.4</td><td>54.4</td><td>48.7</td><td>59.9</td><td>54.8</td><td>64.5</td><td>52.8</td><td>61.8</td><td>57.8</td><td>56.0</td><td>48.7</td><td>58.5</td><td>51.4</td><td>58.9</td><td>51.0</td><td>56.0</td><td>51.5</td><td>58.2</td><td>56.2</td><td>58.7</td><td>51.8</td></tr><tr><td>GPT-4o</td><td>64.1</td><td>50.7</td><td>55.9</td><td>51.8</td><td>58.8</td><td>56.2</td><td>62.9</td><td>54.3</td><td>64.4</td><td>62.3</td><td>60.1</td><td>50.8</td><td>60.3</td><td>56.1</td><td>55.2</td><td>56.3</td><td>59.0</td><td>55.1</td><td>64.4</td><td>55.0</td><td>60.3</td><td>54.5</td></tr><tr><td>InternVL2-8B</td><td>64.5</td><td>57.5</td><td>56.9</td><td>57.5</td><td>61.4</td><td>65.3</td><td>67.0</td><td>57.5</td><td>65.3</td><td>67.2</td><td>59.9</td><td>60.1</td><td>65.3</td><td>61.8</td><td>58.4</td><td>60.8</td><td>61.4</td><td>59.1</td><td>65.6</td><td>61.4</td><td>62.3</td><td>60.2</td></tr><tr><td>Qwen2-VL</td><td>68.0</td><td>60.3</td><td>62.4</td><td>59.6</td><td>71.8</td><td>67.3</td><td>67.2</td><td>59.7</td><td>69.3</td><td>70.1</td><td>63.3</td><td>62.6</td><td>64.6</td><td>64.6</td><td>64.5</td><td>61.1</td><td>63.7</td><td>59.2</td><td>71.9</td><td>65.0</td><td>66.6</td><td>62.1</td></tr><tr><td>Claude-Sonnet</td><td>78.9</td><td>84.0</td><td>72.3</td><td>81.2</td><td>77.4</td><td>80.3</td><td>77.7</td><td>84.5</td><td>78.4</td><td>85.6</td><td>69.9</td><td>84.0</td><td>74.1</td><td>86.5</td><td>72.9</td><td>82.9</td><td>76.4</td><td>86.4</td><td>75.9</td><td>82.3</td><td>75.4</td><td>83.5</td></tr></table>
+
+Table 2: Performance change between LiveXiv (v1) and a manually verified subset averaged across all evaluated models. LiveXiv is robust, thanks to excessive filtering steps which keep the labeling errors low.  
+
+<table><tr><td></td><td>LiveXiv</td><td>Verified Subset</td><td>Absolute Avg.</td></tr><tr><td>VQA</td><td>46.734</td><td>47.273</td><td>2.336</td></tr><tr><td>TQA</td><td>45.101</td><td>46.028</td><td>2.105</td></tr></table>
+
+over the runner-up models such as Qwen2-VL, GPT-4o and InternVL2-8B. See Table 4 for more details. However, a notable caveat is that Claude plays an integral role in the question-filtering process, which may introduce a potential bias in favor of questions it is predisposed to solve effectively. This implies that while Claude's overall performance remains strong, the evaluation might not fully reflect its robustness to novel or more diverse question types outside the scope of this filtering. Surprisingly, GPT-4o has low performance. To understand the root cause of this observation, we experimented with various hyper-parameters such as temperature, image resolution, and diverse textual prompts. Nevertheless, these all yielded similar results.
+
+We further observe that newer models, such as InternVL2-8B and Qwen2-VL, consistently outperform older models like LLaVA-1.6 and Idefics2, suggesting rapid advancements in LMM development over the past few months. This trend highlights the continual improvement in both architecture and training paradigms, leading to better generalization across multi-modal tasks.
+
+Zooming into the domain-specific performance using an ArXiv-based taxonomy, we evaluate each model's effectiveness in distinct scientific fields such as biology, electrical engineering, and mathematics. Our results show that certain models, particularly the newer architectures, exhibit a higher degree of robustness across diverse domains, highlighting that the models' training data might already have potential contamination issues. Conversely, for VQA, models in the Intern-VL2 and the LLaVA families appear to be more sensitive to domain shifts, performing inconsistently across different scientific areas, as oppose to the more recent models like Qwen2-VL, Claude and GPT-4o, see Figures 6, 7 for more details. For TQA, it's not the case, probably since the questions test more specific skills such a retrieval and arithmetic manipulations, see Figures 8, 9. This domain-specific sensitivity emphasizes the need for further refinements in LMMs, especially when applied to specialized scientific knowledge domains. Overall, this analysis not only underscores the ongoing evolution of LMMs but also highlights areas for further investigation, especially concerning model adaptability to diverse content domains and the potential biases introduced by models.
+
+Contamination free effect: Interestingly, focusing on new data that came after the LMMs were trained, allows LiveXiv to provide a new, contamination-free, perspective on the relative performance ranking between strong LMMs. For example, taking the official results from original publications and computing the average ranking of the evaluated LMMs over the established DocVQA (Mathew et al., 2021), ChartQA (Masry et al., 2022) and AI2D (Kembhavi et al., 2016) benchmarks, and comparing those to the average rankings provided by LiveXiv, we observe some significant ranking changes. e.g., IXC2.5 and IXC2-4KHD drop over 4 points in average ranking. See Table 5 in the appendix for details.
+
+Performance on manually filtered dataset: To further verify our proposed automated question-answer generation and filtering methodology and to obtain a measure of errors in the generated data, we manually verified a subset of 1000 samples (500 for both, VQA and TQA) and evaluated all models on this subset. Table 2 presents the results for VQA and TQA on the filtered subset. We see that on average
+
+![](images/9971588ab8689f18534cc60833957b2724a72317a91b96bb6513d8fb17dc2446.jpg)  
+Figure 4: Average prediction errors and rank correlations for overall performance. We report MAE (± mean absolute deviation) for non-re-evaluated models and Spearman's rank correlation across 19 LMMs on different LiveXiv versions. Re-evaluating just 3-5 models is generally sufficient for accurate performance prediction.
+
+![](images/54796bf7b19bd855aeed547a69efb29ff940df784124411b83925305165b0daa.jpg)
+
+![](images/c4489f77b2bce5287946ca170973cdce2fd2e078c4b477aa7cc05734fec584bc.jpg)
+
+![](images/3c2dd9166bd420761dc389e25b29a182780e970f279e15c5a2923103d548eee6.jpg)
+
+![](images/114bb28df9d5b9222368f8feaed22fec80c61e267bc093bf17c8f0bfac2ad491.jpg)  
+Figure 5: Average prediction errors and rank correlations when we predict performance for each domain separately. We report average MAE (± average mean absolute deviation) for non-re-evaluated models and average Spearman's rank correlation across 19 LMMs on different LiveXiv versions. A worse performance LiveXiv v3 can be explained by a smaller number of evaluations.
+
+![](images/8615b683d33b1bb162b3aa704f640a0347def225c37312df7aa8fcc88c1f18cb.jpg)
+
+![](images/82ca9d99a939e1394dba30396d5bae9a05a7cd4c3e73bbd7c8600a641268ee6c.jpg)
+
+![](images/ee81586fa145f66b1c27d0de2fc648624e2f9e6def908cbb3e3bb2638ac546d9.jpg)
+
+the performance only fluctuates by  $2.3\%$  and  $2.1\%$  for VQA and TQA when comparing the results obtained by all the models on the entire dataset and the manually verified subset. These results hint that our automated question-answer generation pipeline and the filtering methodology is quite robust. Detailed results can be found at the Appendix, Tables 6 and 7.
+
+Efficient evaluations of LMMs: In this section, we empirically validate the effectiveness of our proposed efficient re-evaluation method for LMMs using LiveXiv v1-v4. Dynamic benchmarks like LiveXiv present a challenge in terms of evaluation costs since each time a new version of the benchmark is released, all models should be re-evaluated on the updated data. This process, however, can become computationally prohibitive when dealing with numerous models. Our goal is to demonstrate that by re-evaluating only a small subset of models on a new version of LiveXiv, we can still reliably predict the performance of the remaining models. For this experiment, we evaluate 17 models on the first version of LiveXiv but only re-evaluate  $m \in \{3,5,8,12\}$  models on the subsequent versions of the benchmark using the model selection methodology detailed in Section 3.3; we focus on VQA or TQA (but not both simultaneously) and consider all ArXiv domains. From LiveXiv v2 onwards, two new extra models were added to the analysis; these do not count towards the budget of  $m$  models at v2.
+
+At every step, an IRT model is fitted to the full observed data and we predict the performance of the non-re-evaluated models on the full VQA and TQA versions of LiveXiv and on each ArXiv domain using empirical versions of equation 2. Figure 4 presents the results for the case we aim to predict the performance of the non-re-evaluated models on the full VQA and TQA versions of LiveXiv while Figure 5 presents the averages of errors and correlations across domains. They report both the mean absolute error (MAE) ( $\pm$  mean absolute deviation) for the non-re-evaluated models when predicting their accuracy and Spearman's rank correlation across all 19 LMMs on different LiveXiv versions when comparing real accuracy and predicted accuracy. These results suggest that re-evaluating just 3 or 5 models is likely to be sufficient for accurately predicting the performance of the remaining models and the ranking of all models, especially when our focus is predicting the overall performances. This last observation makes sense since the full benchmarks contain many more data points when compared to individual domains and, in that case, individual prediction errors tend to cancel out in the empirical version of equation 2. For the same reason, we ended up having a poorer performance on LiveXiv v3, which has fewer data points; as detailed in Appendix A.4.4, LiveXiv v3 has questions generated by three models as part of an ablation study, however, we ultimately use the questions generated by GPT-4o and Claude-Sonnet in this study to keep procedures more standardized with v4.
+
+In Appendix A.5, we present additional results to further validate the effectiveness of our method. Specifically, we (i) show a detailed error analysis of the results for LiveXiv v4 (from Figures 4/5), (ii)
+
+Table 3: LiveXiv accuracy (%) on different categories of question and partitions averaged over all evaluated models.  
+
+<table><tr><td></td><td>Data Analysis</td><td>Reasoning</td><td>Attribute</td><td>Localization</td><td>Reading</td><td>Arithmetic</td><td>Charts</td><td>Block Diagram</td><td>Qualitative</td></tr><tr><td>VQA</td><td>46.93</td><td>47.95</td><td>46.18</td><td>41.91</td><td>47.83</td><td>46.87</td><td>44.17</td><td>52.69</td><td>48.60</td></tr><tr><td>TQA</td><td>46.02</td><td>63.61</td><td>68.69</td><td>51.66</td><td>59.35</td><td>35.56</td><td>-</td><td>-</td><td>-</td></tr></table>
+
+test a situation with high distribution shift using a hypothetical version of LiveXiv consisting of ArXiv papers from 2010 (v0), (iii) validate our efficient evaluation method when questions are also generated by Qwen2-VL-72B in v3, and (iv) test our approach on MM-LiveBench (Zhang et al., 2024b).
+
+# 4.3 ANALYSIS AND ABLATIONS
+
+To analyze various aspects of LiveXiv we provide an extensive ablation study. We start by providing an analysis of the results from different models obtained w.r.t the language content partitions, then provide results for different models w.r.t the visual data partitions.
+
+Language analysis - performance according to question type. To discover error slices of models for an analysis of mistakes they commonly make, we classify the questions present in the benchmark into one of the following categories: reasoning, data analysis, reading, localization, attribute, and arithmetic. To achieve this classification, we employ the Llama-3.1 (Meta, 2024) LLM and prompt the model with the question and the list of categories to choose for this question. The prompt is provided in the Appendix Figure 21. Table 3 summarizes the results for all the models. We see that the performance of these models on the arithmetic partition is the lowest on average as compared to other partitions highlighting room for potential improvement. We also provide the detailed results for all models on these partitions for VQA and TQA in Tables 9 and 10 of the Appendix.
+
+Vision analysis - performance according to figure type. For a more fine-grained analysis of LMM performance on different types of visual data present in our benchmark, we first categorize the data through Meta-Prompting for CLIP, proposed by Mirza et al. (2024), in a zero-shot classification setup. Specifically, we classify the image content into three categories of figures: Block diagrams, Qualitative visual results, and Charts. We summarize the results in Table 3. Detailed results for each model's performance can be found in Table 8 in the Appendix. The results reveal a significant variance in performance across figure types for nearly all models. In most cases, block diagrams are the most favorable category for models. However, InternLM-Xcomposer2-4KHD-7B (Dong et al., 2024b) stands out by achieving the highest accuracy on Qualitative figures. Overall, Charts emerge as the most challenging figure type on average, suggesting a lack of sufficient examples in the training data for this category. This kind of analysis can be further expanded to include more categories and discover error slices on which different models struggle so that potential targeted improvements can be designed for these models to mitigate the shortcomings.
+
+Diversity & Difficulty The visual content of scientific papers (figures and tables) evolves over time, and we aim to ensure our questions remain diverse across dataset versions. To achieve this, we use an LLM (Llama3.1-70B Meta (2024)) to classify questions into predefined categories, allowing us to monitor and maintain diversity. Table 18 confirms that LiveXiv maintains question diversity across versions. Additionally, we assess question difficulty by analyzing how many models answer each question correctly. Figure 10 shows that the difficulty distribution for both VQA and TQA tasks remains stable across dataset versions.
+
+# 5 LIMITATIONS AND CONCLUSIONS
+
+Limitations. LiveXiv relies on capable proprietary LMMs in order to be fully automatic, and with high quality. However, relying on proprietary LMMs is a limitation since we do not have full control over the models, they can change through time and might affect LiveXiv. Nevertheless, we commonly expect them to continuously improve leading to a positive impact on LiveXiv effectiveness.
+
+Conclusions. We propose LiveXiv, an ever-evolving, fully automatic, multi-modal benchmark focused on scientific domains to tackle test set contamination issues and consequently allow a new (contamination-free) perspective on relative ranking of advanced LMMs. We utilize ArXiv, as the data source, carefully and extensively crafting a quality dataset to evaluate LMMs. To significantly reduce the computational and logistical overhead of maintaining the dataset throughout time and models, we propose an efficient evaluation method that can save more than  $70\%$  of the evaluated models on each dataset version. Our method can be extended to other archives such as BioRXiv to extend our dataset to new domains. For future work, we propose enhancing LMM evaluation by incorporating free-form questions to assess generative abilities and complex question types to evaluate reasoning capabilities.
+
+# 6 ETHICS STATEMENT
+
+This work introduces LiveXiv, a live multi-modal benchmark for evaluating LMMs using scientific ArXiv papers. By relying solely on publicly available ArXiv manuscripts with proper licenses, we ensure compliance with copyright and distribution policies. The automated generation of Visual Question Answering (VQA) and Table Question Answering (TQA) pairs enables scalable evaluation of LMMs without human involvement, minimizing the risk of human biases in data collection. However, we acknowledge the potential for unintentional biases within the models or dataset itself. Continuous evaluation and refinement are necessary to mitigate these biases and promote the responsible deployment of LMMs in wider applications.
+
+# 7 ACKNOWLEDGMENTS
+
+This research was partially supported by the TAD center at Tel Aviv University. The authors thank Eli Schwartz for his valuable feedback and assistance.
+
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+
+# A APPENDIX
+
+# A.1 RELATED WORKS
+
+# A.1.1 EFFICIENT BENCHMARKS
+
+With the increasing amount of tasks and samples in current benchmarks, evaluation of the full suite is time-consuming and cost-intensive. Efforts are underway to develop efficient benchmarks that reduce computation costs without sacrificing reliability. For LLMs, Perlitz et al. (2023) proposed the first systematic study of the effects of language model benchmark designs on reliability and efficiency, and applied efficient benchmark practices on the HELM benchmark (Liang et al., 2022), leading to  $\times 100$  computation reduction with minimal loss on reliability. Lifelong benchmarks (Prabhu et al., 2024) has an ever-expanding pool of test samples for the categories in CIFAR10 (Krizhevsky & Hinton, 2009) and ImageNet (Deng et al., 2009); to make this design economically feasible, it reuses past model evaluations on a sample set through dynamic programming to enable efficient evaluation of new incoming models, drastically reducing the evaluation cost. Most related to our work, tinyBenchmarks (Maia Polo et al., 2024a) and PromptEval (Maia Polo et al., 2024b) propose using Item Response Theory (IRT) (Lord et al., 1968) to estimate the performance of LLMs on unseen samples, making efficient evaluation possible by only conducting a small fraction of the total number of evaluations. Inspired by the last two works, we leverage IRT to estimate the performance of older models in new batches of data. More specifically, at each version of LiveXiv, we choose a small core set of models ( $\leq 5$ ) previously added to the leaderboard and re-evaluate them on the new data. Depending on their responses to the new samples, we estimate the performance of the remaining old models on the new benchmark version.
+
+# A.1.2 LLM CONTAMINATION-FREE BENCHMARKS
+
+For LLMs, LMSys Chatbot Arena (Chiang et al., 2024) and AI2 WildVision (Lu et al., 2024) create a user-focused platform that provides contamination-free environment for proper evaluations. However, it is expensive to collect tens of thousands of human preferences on the compared language models. Furthermore, Seal Benchmark (AI, 2024) proposes private questions paired with human evaluations. Srivastava et al. (2024) update the questions in the MATH dataset (Hendrycks et al., 2021) by changing numbers in the math questions. LiveBench White et al. (2024) collects frequently updated questions from diverse information sources e.g. math competitions, arXiv papers and news articles and more challenging versions of existing benchmark tasks. Concurrently, CodeBench (Jain et al., 2024) contributes a live benchmark on broader code-related capabilities. Note that these datasets focus on language data only.
+
+# A.2 ANALYSIS & ABLATIONS
+
+Table 4: Average results for relatively close benchmarks (DocVQA, ChartQA and AI2D). We can see that Claude, GPT4o, Qwen2-VL and InternVL2 are the top models. The overall ordering is align with our benchmark.  
+
+<table><tr><td></td><td>ChartQA</td><td>DocVQA</td><td>AI2D</td></tr><tr><td>InstructBLIP-7B</td><td>10.9</td><td>74</td><td>40.6</td></tr><tr><td>LLaVA-1.6-Mistral-7B</td><td>51.8</td><td>72.2</td><td>69</td></tr><tr><td>Mantis</td><td>42.9</td><td>-</td><td>60.4</td></tr><tr><td>LLaVA-1.5-7B</td><td>17.8</td><td>74.4</td><td>55.5</td></tr><tr><td>LLaVA-1.5-13B</td><td>18.2</td><td>77.5</td><td>70</td></tr><tr><td>Idefics2</td><td>-</td><td>74</td><td>72.3</td></tr><tr><td>InternVL2-2B</td><td>76.2</td><td>86.9</td><td>74.1</td></tr><tr><td>IXC2-4KHD-7B</td><td>81</td><td>90</td><td>81</td></tr><tr><td>IXC2.5-7B</td><td>82.2</td><td>90.9</td><td>81.6</td></tr><tr><td>LLaVA-OneVision-7B</td><td>80</td><td>83.7</td><td>81.4</td></tr><tr><td>Phi3v</td><td>72</td><td>84.9</td><td>77.8</td></tr><tr><td>Idefics3</td><td>-</td><td>87.7</td><td>76.5</td></tr><tr><td>LLaVA-1.6-34B</td><td>67.6</td><td>84</td><td>78.9</td></tr><tr><td>GPT-4o</td><td>85.7</td><td>92.8</td><td>94.2</td></tr><tr><td>Qwen2-VL</td><td>83</td><td>94.5</td><td>83</td></tr><tr><td>InternVL2-8B</td><td>83.3</td><td>91.6</td><td>83.8</td></tr><tr><td>Claude-Sonnet</td><td>90.8</td><td>95.2</td><td>94.7</td></tr></table>
+
+![](images/b63e422de7a0f468efb5d16dd7f858a6a57cc64434ca11927e9c149dfd20e283.jpg)  
+Figure 6: Domain sensitivity according to domains. We visualize the performance of each model across all domains. Clear trends revealed where old models or models with a small LLM are "under-fitting" and perform worse across all domains. In the middle we have the mid-level models that are sensitive to the domain, indicating their lack of generalization across domain without any additional training. Lastly the newest models (open-source and proprietary) are robust to domain shifts and present a stable performance across the domains.
+
+Table 5: Average ranking on static benchmarks (ChartQA, DocVQA and AI2D) and LiveXiv (v1). We can see from the ranking difference column that some models have a significant drop (negative difference) in the relative ranking in LiveXiv compared to the static datasets. The gap is highlighting a potential risk of test data contamination when using static (frozen in time) benchmark datasets.  
+
+<table><tr><td>Model</td><td>Static datasets</td><td>LiveXiv</td><td>Difference (static - livexiv)</td></tr><tr><td>InstructBLIP-7B</td><td>15.33</td><td>17.00</td><td>-1.67</td></tr><tr><td>LLaVA-1.6-Mistral-7B</td><td>13.67</td><td>16.00</td><td>-2.33</td></tr><tr><td>Mantis</td><td>14.50</td><td>14.50</td><td>0.00</td></tr><tr><td>LLaVA-1.5-7B</td><td>14.33</td><td>14.50</td><td>-0.17</td></tr><tr><td>LLaVA-1.5-13B</td><td>12.67</td><td>13.00</td><td>-0.33</td></tr><tr><td>Idefics2</td><td>13.00</td><td>12.00</td><td>1.00</td></tr><tr><td>InternVL2-2B</td><td>9.00</td><td>10.00</td><td>-1.00</td></tr><tr><td>IXC2-4KHD-7B</td><td>6.33</td><td>10.50</td><td>-4.17</td></tr><tr><td>IXC2.5-7B</td><td>5.00</td><td>9.50</td><td>-4.50</td></tr><tr><td>LLaVA-OneVision-7B</td><td>8.00</td><td>6.50</td><td>1.50</td></tr><tr><td>Phi3v</td><td>9.00</td><td>6.00</td><td>3.00</td></tr><tr><td>Idefics3</td><td>8.50</td><td>6.50</td><td>2.00</td></tr><tr><td>LLaVA-1.6-34B</td><td>9.33</td><td>6.50</td><td>2.83</td></tr><tr><td>GPT-4o</td><td>2.33</td><td>4.00</td><td>-1.67</td></tr><tr><td>Qwen2-VL</td><td>3.33</td><td>3.00</td><td>0.33</td></tr><tr><td>InternVL2-8B</td><td>3.33</td><td>2.00</td><td>1.33</td></tr><tr><td>Claude-Sonnet</td><td>1.00</td><td>1.00</td><td>0.00</td></tr></table>
+
+![](images/829c1b1ded99c7044204e5ebba9062e59603f01707e8513c78a89fc693bae5c7.jpg)  
+Figure 7: LMMs performance based on domain. To complement our analysis form Figure 6 we visualize the statistical properties of each domain. One clear trend is that across all modesl, the performance on cs.CV and cs.AI is the most concentrated, hinting lower variance between models.
+
+![](images/1f2eede8a5308b86590d69cea366c061c92e80b5946a111592644c68fcfbed4e.jpg)  
+Figure 8: Domain sensitivity according to domains. As opposed to the high variance some models demonstrated in Figure 6, in TQA the tasks and he visual content are more limited thus shrunken the performance variance greatly.
+
+![](images/38b97162aa9029b5dfd02ca288d41e2884ff9cf15e6459a26aa3344a798b8489.jpg)  
+Figure 9: LMMs performance based on domain. The domains are very similar in their statistical properties showing high variance in performance. This is probably due to wide range of models that differ significantly in their performance.
+
+# A.2.1 PERFORMANCE CHANGE COMPARED TO MANUALLY CURATED SUBSET
+
+Table 6: VQA Performance change between LiveXiv (v1) and a manually curated subset (500 examples).  
+
+<table><tr><td>Model</td><td>LiveXiv (%)</td><td>Manual (%)</td><td>Performance Change</td><td>LiveXiv Rank</td><td>Manual Rank</td><td>Change</td></tr><tr><td>InstructBLIP-7B</td><td>23.216</td><td>21.346</td><td>-1.87</td><td>17</td><td>17</td><td>0</td></tr><tr><td>InternLM-Xcomposer2.5-7B</td><td>47.839</td><td>50.769</td><td>2.93</td><td>10</td><td>9</td><td>1</td></tr><tr><td>InternVL2-8B</td><td>61.558</td><td>66.154</td><td>4.596</td><td>3</td><td>3</td><td>0</td></tr><tr><td>LLaVA-1.6-Mistral-7B</td><td>29.163</td><td>26.346</td><td>-2.816</td><td>16</td><td>16</td><td>0</td></tr><tr><td>LLaVA-OneVision-Qwen2-7B</td><td>52.864</td><td>56.154</td><td>3.29</td><td>7</td><td>6</td><td>1</td></tr><tr><td>LLaVA-1.5-13B</td><td>31.859</td><td>30.385</td><td>-1.475</td><td>14</td><td>13</td><td>1</td></tr><tr><td>LLaVA-1.5-7B</td><td>29.983</td><td>28.654</td><td>-1.329</td><td>15</td><td>14</td><td>1</td></tr><tr><td>LLaVA-1.6-34B</td><td>49.196</td><td>53.269</td><td>4.073</td><td>9</td><td>7</td><td>2</td></tr><tr><td>Mantis-LLama3-8B</td><td>32.094</td><td>28.654</td><td>-3.44</td><td>13</td><td>14</td><td>-1</td></tr><tr><td>Phi3v</td><td>58.141</td><td>58.654</td><td>0.513</td><td>5</td><td>5</td><td>0</td></tr><tr><td>InternLM-Xcomposer2-4KHD-7B</td><td>36.801</td><td>33.654</td><td>-3.147</td><td>12</td><td>12</td><td>0</td></tr><tr><td>Idefics2-8B</td><td>36.851</td><td>36.731</td><td>-0.12</td><td>11</td><td>11</td><td>0</td></tr><tr><td>InternVL2-2B</td><td>49.548</td><td>48.654</td><td>-0.894</td><td>8</td><td>10</td><td>-2</td></tr><tr><td>Claude-Sonnet</td><td>75.942</td><td>79.615</td><td>3.673</td><td>1</td><td>1</td><td>0</td></tr><tr><td>Qwen2-VL</td><td>66.248</td><td>71.346</td><td>5.098</td><td>2</td><td>2</td><td>0</td></tr><tr><td>GPT-4o</td><td>60.303</td><td>60.577</td><td>0.274</td><td>4</td><td>4</td><td>0</td></tr><tr><td>Idefics3</td><td>52.881</td><td>52.692</td><td>-0.189</td><td>6</td><td>8</td><td>-2</td></tr><tr><td>Average (absolute) change</td><td></td><td></td><td>2.336</td><td></td><td></td><td></td></tr></table>
+
+Table 7: TQA Performance change between LiveXiv (v1) and a manually curated subset (500 examples).  
+
+<table><tr><td>Model</td><td>LiveXiv (%)</td><td>Manual (%)</td><td>Performance Change</td><td>LiveXiv Rank</td><td>Manual Rank</td><td>Change</td></tr><tr><td>InstructBLIP-7B</td><td>19.1</td><td>18.5</td><td>-0.6</td><td>17</td><td>17</td><td>0</td></tr><tr><td>InternLM-Xcomposer2.5-7B</td><td>49.1</td><td>45.9</td><td>-3.2</td><td>9</td><td>9</td><td>0</td></tr><tr><td>InternVL2-8B</td><td>62.1</td><td>65.3</td><td>3.2</td><td>2</td><td>2</td><td>0</td></tr><tr><td>LLaVA-1.6-Mistral-7B</td><td>23.5</td><td>23.2</td><td>-0.3</td><td>16</td><td>16</td><td>0</td></tr><tr><td>LLaVA-OneVision-Qwen2-7B</td><td>50.2</td><td>51.6</td><td>1.4</td><td>7</td><td>8</td><td>-1</td></tr><tr><td>LLaVA-1.5-13B</td><td>30.9</td><td>31.2</td><td>0.3</td><td>13</td><td>13</td><td>0</td></tr><tr><td>LLaVA-1.5-7B</td><td>30.0</td><td>29.6</td><td>-0.3</td><td>14</td><td>14</td><td>0</td></tr><tr><td>LLaVA-1.6-34B</td><td>51.8</td><td>52.2</td><td>0.4</td><td>5</td><td>7</td><td>-2</td></tr><tr><td>Mantis-LLama3-8B</td><td>29.3</td><td>28.0</td><td>-1.3</td><td>15</td><td>15</td><td>0</td></tr><tr><td>Phi3v</td><td>50.2</td><td>54.1</td><td>4.0</td><td>7</td><td>6</td><td>1</td></tr><tr><td>InternLM-Xcomposer2-4KHD-7B</td><td>42.1</td><td>41.7</td><td>-0.4</td><td>10</td><td>11</td><td>-1</td></tr><tr><td>Idefics2-8B</td><td>38.2</td><td>42.0</td><td>3.8</td><td>12</td><td>10</td><td>-2</td></tr><tr><td>InternVL2-2B</td><td>41.5</td><td>39.5</td><td>-2.0</td><td>11</td><td>12</td><td>-1</td></tr><tr><td>Claude-Sonnet</td><td>83.5</td><td>89.2</td><td>5.6</td><td>1</td><td>1</td><td>0</td></tr><tr><td>Qwen2-VL</td><td>60.2</td><td>58.3</td><td>-1.9</td><td>3</td><td>3</td><td>0</td></tr><tr><td>GPT-4o</td><td>54.5</td><td>55.7</td><td>1.3</td><td>4</td><td>5</td><td>-1</td></tr><tr><td>Idefics3</td><td>50.6</td><td>56.4</td><td>5.7</td><td>6</td><td>4</td><td>2</td></tr><tr><td>Average (absolute) change</td><td></td><td></td><td>2.105</td><td colspan="3"></td></tr></table>
+
+# A.2.2 FIGURE TYPE
+
+We provide all the details for VQA performance according to figure type content in Table 8. We divide the performance to the following figure types: "Chart", "Block Diagram" and "Qualitative".
+
+# A.2.3 QUESTION CATEGORY
+
+In Tables 9 and 10 we provide detailed results for VQA and TQA performance according to the category of the questions as classified by an LLM. We divide the performance to the following categories: "Data Analysis", "Attribute", "Reasoning", "Reading", "Localization" and "Arithmetic"
+
+# A.2.4 QUESTIONS'DIFFICULTY
+
+We provide additional analyses regarding the diverse difficulties of LiveXiv's generated questions. Figure 10 demonstrates that for all LiveXiv tasks and for both the first and second versions, a wide range of difficulties is present, where most of the questions concentrate in the middle, suggesting our dataset is indeed challenging.
+
+Table 8: Performance of LMMs over different figure types from the LiveXiv v1 VQA set (the amount of samples for each figure type is in brackets).  
+
+<table><tr><td>Model</td><td>Chart (4354)</td><td>block_diagram (2110)</td><td>Qualitative (864)</td></tr><tr><td>InstructBLIP-7B</td><td>22.9</td><td>22.8</td><td>22.5</td></tr><tr><td>InternLM-Xcomposer2.5-7B</td><td>46.7</td><td>53.5</td><td>41.7</td></tr><tr><td>InternVL2-8B</td><td>59.1</td><td>68.4</td><td>63.8</td></tr><tr><td>LLaVA-1.6-Mistral-7B</td><td>27.3</td><td>33.6</td><td>33.4</td></tr><tr><td>LLaVA-OneVision-Qwen2-7B</td><td>48.7</td><td>62.8</td><td>57.9</td></tr><tr><td>LLaVA-1.5-13B</td><td>29.3</td><td>35.4</td><td>40.2</td></tr><tr><td>LLaVA-1.5-7B</td><td>28.1</td><td>33.1</td><td>36.0</td></tr><tr><td>LLaVA-1.6-34B</td><td>44.6</td><td>60.0</td><td>52.7</td></tr><tr><td>Mantis-LLama3-8B</td><td>29.3</td><td>36.2</td><td>36.1</td></tr><tr><td>Phi3v</td><td>56.1</td><td>65.5</td><td>55.2</td></tr><tr><td>InternLM-Xcomposer2-4KHD-7B</td><td>32.9</td><td>43.7</td><td>47.1</td></tr><tr><td>Idefics2-8B</td><td>34.3</td><td>43.0</td><td>41.1</td></tr><tr><td>InternVL2-2B</td><td>47.2</td><td>54.9</td><td>50.0</td></tr><tr><td>Claude-Sonnet</td><td>73.7</td><td>81.3</td><td>69.1</td></tr><tr><td>Qwen2-VL</td><td>63.1</td><td>73.9</td><td>66.0</td></tr><tr><td>GPT-4o</td><td>56.5</td><td>68.6</td><td>59.5</td></tr><tr><td>idefics3</td><td>51.1</td><td>59.0</td><td>54.1</td></tr></table>
+
+Table 9: LiveXiv v1 VQA Performance by Question Categories (the amount of samples for each category is in brackets).  
+
+<table><tr><td>Model</td><td>Data Analysis (2291)</td><td>Reasoning (872)</td><td>Attribute (903)</td><td>Localization (1596)</td><td>Reading (1470)</td><td>Arithmetic (154)</td></tr><tr><td>InstrcutBLIP</td><td>21.16</td><td>29.29</td><td>22.77</td><td>31.25</td><td>23.87</td><td>23.01</td></tr><tr><td>InterLM-XC-2.5</td><td>47.52</td><td>43.43</td><td>48.51</td><td>31.25</td><td>50.46</td><td>47.28</td></tr><tr><td>InternVL2-8B</td><td>61.74</td><td>63.64</td><td>65.35</td><td>56.25</td><td>62.54</td><td>62.41</td></tr><tr><td>LLaVA1.6-7B</td><td>28.91</td><td>31.31</td><td>30.69</td><td>12.50</td><td>30.00</td><td>30.43</td></tr><tr><td>LLaVA-OneVision</td><td>52.57</td><td>54.55</td><td>60.40</td><td>56.25</td><td>56.76</td><td>52.85</td></tr><tr><td>LLaVA1.5-13B</td><td>32.92</td><td>25.25</td><td>25.74</td><td>43.75</td><td>31.85</td><td>32.58</td></tr><tr><td>LLaVA1.5-7B</td><td>30.56</td><td>28.28</td><td>29.70</td><td>25.00</td><td>29.71</td><td>30.89</td></tr><tr><td>LLaVA1.6-34B</td><td>50.87</td><td>43.43</td><td>45.54</td><td>37.50</td><td>49.60</td><td>50.00</td></tr><tr><td>Mantis</td><td>32.50</td><td>33.33</td><td>30.69</td><td>43.75</td><td>31.97</td><td>31.80</td></tr><tr><td>Phi3v</td><td>58.05</td><td>63.64</td><td>59.41</td><td>43.75</td><td>58.15</td><td>59.07</td></tr><tr><td>InterLM-XC-4Khd</td><td>37.51</td><td>43.43</td><td>39.60</td><td>18.75</td><td>38.67</td><td>37.09</td></tr><tr><td>Idefics2</td><td>37.79</td><td>39.39</td><td>35.64</td><td>31.25</td><td>39.54</td><td>36.34</td></tr><tr><td>InternVL2-2B</td><td>50.26</td><td>46.46</td><td>44.55</td><td>43.75</td><td>51.45</td><td>48.62</td></tr><tr><td>Claude-Sonnet</td><td>74.78</td><td>78.79</td><td>67.33</td><td>81.25</td><td>75.49</td><td>75.64</td></tr><tr><td>Qwen2-VL</td><td>66.93</td><td>66.67</td><td>68.32</td><td>43.75</td><td>68.73</td><td>65.01</td></tr><tr><td>GPT4o</td><td>61.41</td><td>60.61</td><td>59.41</td><td>62.50</td><td>59.83</td><td>59.76</td></tr><tr><td>Idefics3</td><td>52.34</td><td>63.64</td><td>51.49</td><td>50.00</td><td>54.51</td><td>54.00</td></tr></table>
+
+Table 10: LiveXiv v1 TQA Performance by Question Categories (the amount of samples for each category is in brackets).  
+
+<table><tr><td>Model</td><td>Data Analysis (2582)</td><td>Reasoning (123)</td><td>Attribute (121)</td><td>Localization (23)</td><td>Reading (2127)</td><td>Arithmetic (3934)</td></tr><tr><td>InstructBLIP-7B</td><td>24.7</td><td>27.6</td><td>34.7</td><td>43.5</td><td>20.4</td><td>13.6</td></tr><tr><td>InternLM-Xcomposer2.5-7B</td><td>54.9</td><td>75.6</td><td>79.3</td><td>60.9</td><td>74.6</td><td>29.7</td></tr><tr><td>InternVL2-8B</td><td>58.6</td><td>73.2</td><td>78.5</td><td>65.2</td><td>79.8</td><td>54.1</td></tr><tr><td>LLaVA-1.6-Mistral-7B</td><td>30.4</td><td>39.0</td><td>52.1</td><td>30.4</td><td>32.0</td><td>13.0</td></tr><tr><td>LLaVA-OneVision-Qwen2-7B</td><td>47.5</td><td>72.4</td><td>80.2</td><td>60.9</td><td>69.1</td><td>40.3</td></tr><tr><td>LLaVA-1.5-13B</td><td>31.6</td><td>49.6</td><td>47.9</td><td>30.4</td><td>32.4</td><td>28.5</td></tr><tr><td>LLaVA-1.5-7B</td><td>30.8</td><td>39.0</td><td>42.1</td><td>43.5</td><td>31.6</td><td>27.9</td></tr><tr><td>LLaVA-1.6-34B</td><td>46.4</td><td>69.1</td><td>76.9</td><td>47.8</td><td>66.8</td><td>46.1</td></tr><tr><td>Mantis-LLama3-8B</td><td>28.7</td><td>39.0</td><td>46.3</td><td>21.7</td><td>32.4</td><td>27.3</td></tr><tr><td>Phi3v</td><td>52.2</td><td>76.4</td><td>77.7</td><td>60.9</td><td>72.4</td><td>35.2</td></tr><tr><td>InternLM-Xcomposer2-4KHD-7B</td><td>45.4</td><td>69.1</td><td>77.7</td><td>60.9</td><td>66.2</td><td>25.0</td></tr><tr><td>Idefics2-8B</td><td>35.0</td><td>57.7</td><td>57.9</td><td>43.5</td><td>51.1</td><td>32.4</td></tr><tr><td>InternVL2-2B</td><td>36.9</td><td>57.7</td><td>70.2</td><td>30.4</td><td>62.2</td><td>32.1</td></tr><tr><td>Claude-Sonnet</td><td>85.1</td><td>90.2</td><td>91.7</td><td>87.0</td><td>91.0</td><td>78.1</td></tr><tr><td>Qwen2-VL</td><td>64.6</td><td>86.2</td><td>90.1</td><td>65.2</td><td>82.0</td><td>44.0</td></tr><tr><td>GPT-4o</td><td>57.5</td><td>79.7</td><td>86.8</td><td>69.6</td><td>73.0</td><td>40.6</td></tr><tr><td>Idefics3</td><td>52.0</td><td>79.7</td><td>77.7</td><td>56.5</td><td>72.0</td><td>36.6</td></tr></table>
+
+![](images/b606bcd302613bd15d077d22d9b815c87708ba981dddd9f6970b9d32adeadc84.jpg)
+
+![](images/e6719ae8e0679156472d4081c0d63df8ab33ef1d5f4ee3792340ce6c173d1ff0.jpg)
+
+![](images/be13cc1c33bcd163fffd52bad11a67bf2b9f3ba7f2cf20fdcef79678e5992bbf.jpg)  
+Figure 10: LiveXiv questions' difficulty diversity. LiveXiv is composed of questions at varying levels of difficulty. Each row shows a version of the dataset on both of its task (VQA and TQA). LiveXiv demonstrates questions in a wide range of questions' difficulties, from questions that no model was able to answer correctly up to questions that all models were able to easily answer. The center mass consternates in the middle, hinting the dataset is indeed challenging and the fact that the difficulty distribution remains the same for both the first version and second version suggests the stability of our diverse and challenging dataset.
+
+![](images/62548ab54a84272a4873253271250b4d5e9f1bc2fc9ed4cc16f005bc16cbcc4b.jpg)
+
+# A.3 DETAILED EXAMPLES FOR VQA AND TQA GENERATION
+
+Here we present full and detailed examples of our flow from ArXiv papers until constructing verified multi-choice Q&A. Figure 11 shows the full example for generating questions from figures (VQA). Figure 12 shows the full examples for TQA.
+
+![](images/32fdfb022277cf72ce96eca144d139879aa8b3da8c8ab36f80a4eca54d87f6b7.jpg)  
+Figure 11: A detailed example for VQA questions generation.
+
+![](images/d652667fccbfaaaea58fe97245521c48cb75c2d3cc1a89109dec0876a9fa5615.jpg)  
+Figure 12: A detailed example for TQA question generation.
+
+Table 11: Version 0 - VQA and TQA average accuracy across ArXiv taxonomy (the number of samples is in brackets).  
+
+<table><tr><td>VQA Accuracy</td><td>cs.AI(188)</td><td>cs.CV(200)</td><td>cs.LG(196)</td><td>cs.RO(321)</td><td>eess.SP(269)</td><td>eess.SY(286)</td><td>q-bio.BM(154)</td><td>q-bio.CB(206)</td><td>q-bio.GN(80)</td><td>q-bio.TO(167)</td><td>Mean(2067)</td></tr><tr><td>InstrcutBLIP</td><td>0.29</td><td>0.24</td><td>0.20</td><td>0.22</td><td>0.24</td><td>0.25</td><td>0.27</td><td>0.22</td><td>0.19</td><td>0.25</td><td>0.24</td></tr><tr><td>InterLM-XC-2.5</td><td>0.42</td><td>0.42</td><td>0.49</td><td>0.48</td><td>0.56</td><td>0.48</td><td>0.51</td><td>0.49</td><td>0.57</td><td>0.46</td><td>0.49</td></tr><tr><td>InternVL2-2B</td><td>0.30</td><td>0.28</td><td>0.30</td><td>0.34</td><td>0.32</td><td>0.36</td><td>0.26</td><td>0.25</td><td>0.22</td><td>0.30</td><td>0.29</td></tr><tr><td>InternVL2-8B</td><td>0.61</td><td>0.66</td><td>0.64</td><td>0.65</td><td>0.69</td><td>0.68</td><td>0.58</td><td>0.55</td><td>0.59</td><td>0.57</td><td>0.62</td></tr><tr><td>LLaVA1.6-7B</td><td>0.54</td><td>0.56</td><td>0.52</td><td>0.62</td><td>0.55</td><td>0.60</td><td>0.50</td><td>0.50</td><td>0.60</td><td>0.43</td><td>0.54</td></tr><tr><td>LLaVA-OneVision</td><td>0.32</td><td>0.36</td><td>0.33</td><td>0.38</td><td>0.35</td><td>0.35</td><td>0.31</td><td>0.27</td><td>0.32</td><td>0.22</td><td>0.32</td></tr><tr><td>LLaVA1.5-13B</td><td>0.29</td><td>0.35</td><td>0.31</td><td>0.35</td><td>0.33</td><td>0.33</td><td>0.33</td><td>0.28</td><td>0.29</td><td>0.25</td><td>0.31</td></tr><tr><td>LLaVA1.5-7B</td><td>0.44</td><td>0.50</td><td>0.63</td><td>0.52</td><td>0.52</td><td>0.53</td><td>0.46</td><td>0.40</td><td>0.44</td><td>0.48</td><td>0.49</td></tr><tr><td>LLaVA1.6-34B</td><td>0.30</td><td>0.39</td><td>0.33</td><td>0.36</td><td>0.29</td><td>0.36</td><td>0.33</td><td>0.32</td><td>0.32</td><td>0.31</td><td>0.33</td></tr><tr><td>Mantis</td><td>0.55</td><td>0.60</td><td>0.65</td><td>0.57</td><td>0.67</td><td>0.61</td><td>0.54</td><td>0.58</td><td>0.57</td><td>0.42</td><td>0.58</td></tr><tr><td>Phi3v</td><td>0.42</td><td>0.42</td><td>0.41</td><td>0.41</td><td>0.29</td><td>0.35</td><td>0.39</td><td>0.36</td><td>0.39</td><td>0.26</td><td>0.37</td></tr><tr><td>InterLM-XC-4Khd</td><td>0.39</td><td>0.44</td><td>0.42</td><td>0.41</td><td>0.43</td><td>0.45</td><td>0.38</td><td>0.34</td><td>0.32</td><td>0.40</td><td>0.40</td></tr><tr><td>Idefics2</td><td>0.47</td><td>0.48</td><td>0.48</td><td>0.52</td><td>0.50</td><td>0.49</td><td>0.49</td><td>0.43</td><td>0.49</td><td>0.48</td><td>0.48</td></tr><tr><td>Claude-Sonnet</td><td>0.77</td><td>0.74</td><td>0.81</td><td>0.77</td><td>0.89</td><td>0.79</td><td>0.72</td><td>0.74</td><td>0.55</td><td>0.78</td><td>0.75</td></tr><tr><td>Qwen2-VL-7B</td><td>0.62</td><td>0.66</td><td>0.71</td><td>0.72</td><td>0.77</td><td>0.70</td><td>0.69</td><td>0.59</td><td>0.66</td><td>0.61</td><td>0.67</td></tr><tr><td>GPT4o</td><td>0.49</td><td>0.52</td><td>0.55</td><td>0.55</td><td>0.58</td><td>0.52</td><td>0.47</td><td>0.49</td><td>0.45</td><td>0.49</td><td>0.51</td></tr><tr><td>Idefics3</td><td>0.52</td><td>0.63</td><td>0.54</td><td>0.58</td><td>0.54</td><td>0.54</td><td>0.55</td><td>0.47</td><td>0.55</td><td>0.51</td><td>0.54</td></tr><tr><td>TQA Accuracy</td><td>cs.AI(167)</td><td>cs.CV(473)</td><td>cs.LG(516)</td><td>cs.RO(247)</td><td>eess.SP(45)</td><td>eess.SY(188)</td><td>q-bio.BM(118)</td><td>q-bio.CB(108)</td><td>q-bio.GN(398)</td><td>q-bio.TO(115)</td><td>Mean(2375)</td></tr><tr><td>InstrcutBLIP</td><td>0.20</td><td>0.19</td><td>0.23</td><td>0.19</td><td>0.20</td><td>0.21</td><td>0.19</td><td>0.14</td><td>0.24</td><td>0.21</td><td>0.20</td></tr><tr><td>InterLM-XC-2.5</td><td>0.46</td><td>0.38</td><td>0.36</td><td>0.51</td><td>0.38</td><td>0.56</td><td>0.45</td><td>0.44</td><td>0.47</td><td>0.44</td><td>0.45</td></tr><tr><td>InternVL2-2B</td><td>0.41</td><td>0.36</td><td>0.36</td><td>0.51</td><td>0.27</td><td>0.49</td><td>0.42</td><td>0.43</td><td>0.40</td><td>0.42</td><td>0.41</td></tr><tr><td>InternVL2-8B</td><td>0.50</td><td>0.56</td><td>0.55</td><td>0.68</td><td>0.64</td><td>0.75</td><td>0.62</td><td>0.64</td><td>0.54</td><td>0.73</td><td>0.62</td></tr><tr><td>LLaVA1.6-7B</td><td>0.26</td><td>0.19</td><td>0.22</td><td>0.29</td><td>0.24</td><td>0.26</td><td>0.20</td><td>0.28</td><td>0.22</td><td>0.30</td><td>0.25</td></tr><tr><td>LLaVA-OneVision</td><td>0.48</td><td>0.41</td><td>0.43</td><td>0.57</td><td>0.33</td><td>0.64</td><td>0.49</td><td>0.57</td><td>0.43</td><td>0.55</td><td>0.49</td></tr><tr><td>LLaVA1.5-13B</td><td>0.37</td><td>0.29</td><td>0.28</td><td>0.36</td><td>0.31</td><td>0.32</td><td>0.29</td><td>0.31</td><td>0.30</td><td>0.31</td><td>0.31</td></tr><tr><td>LLaVA1.5-7B</td><td>0.32</td><td>0.26</td><td>0.29</td><td>0.32</td><td>0.33</td><td>0.30</td><td>0.28</td><td>0.36</td><td>0.30</td><td>0.30</td><td>0.31</td></tr><tr><td>LLaVA1.6-34B</td><td>0.54</td><td>0.50</td><td>0.46</td><td>0.55</td><td>0.47</td><td>0.63</td><td>0.58</td><td>0.57</td><td>0.47</td><td>0.54</td><td>0.53</td></tr><tr><td>Mantis</td><td>0.31</td><td>0.24</td><td>0.29</td><td>0.35</td><td>0.33</td><td>0.32</td><td>0.19</td><td>0.32</td><td>0.32</td><td>0.32</td><td>0.30</td></tr><tr><td>Phi3v</td><td>0.47</td><td>0.41</td><td>0.43</td><td>0.51</td><td>0.49</td><td>0.60</td><td>0.48</td><td>0.56</td><td>0.46</td><td>0.48</td><td>0.49</td></tr><tr><td>InterLM-XC-4Khd</td><td>0.43</td><td>0.31</td><td>0.36</td><td>0.50</td><td>0.24</td><td>0.53</td><td>0.48</td><td>0.44</td><td>0.38</td><td>0.43</td><td>0.41</td></tr><tr><td>Idefics2</td><td>0.36</td><td>0.30</td><td>0.34</td><td>0.38</td><td>0.33</td><td>0.43</td><td>0.34</td><td>0.41</td><td>0.34</td><td>0.30</td><td>0.35</td></tr><tr><td>Claude-Sonnet</td><td>0.77</td><td>0.79</td><td>0.84</td><td>0.85</td><td>0.80</td><td>0.85</td><td>0.84</td><td>0.79</td><td>0.78</td><td>0.83</td><td>0.81</td></tr><tr><td>Qwen2-VL-7B</td><td>0.59</td><td>0.53</td><td>0.52</td><td>0.63</td><td>0.53</td><td>0.64</td><td>0.53</td><td>0.66</td><td>0.53</td><td>0.63</td><td>0.58</td></tr><tr><td>GPT4o</td><td>0.46</td><td>0.41</td><td>0.40</td><td>0.55</td><td>0.47</td><td>0.57</td><td>0.49</td><td>0.50</td><td>0.45</td><td>0.48</td><td>0.48</td></tr><tr><td>Idefics3</td><td>0.49</td><td>0.43</td><td>0.44</td><td>0.57</td><td>0.36</td><td>0.57</td><td>0.45</td><td>0.58</td><td>0.46</td><td>0.52</td><td>0.49</td></tr></table>
+
+# A.4 ADDITIONAL LIVEXIV DATASET VERSIONS
+
+In this section we describe in detail the additional LiveXiv versions, v0 (generated form papers from 2010), v1-opposite which is the same raw data as v1 but this time Claude is the QA generation model and GPT-4o does the filtering, and v2 and v3 which consist of multiple participating models. We start by describing each version and the performance of the evaluated models. Lastly, we summarize the diversity of our generated dataset, both in terms of visual content and in terms of the different categories of questions.
+
+# A.4.1 LIVEXIV V0
+
+We analyzed 100 scientific papers from 2010 and developed v0, which contains 4,500 questions. Due to the dynamic nature of scientific data, v0 represents a significant distribution shift from v1. We used our efficient evaluation method to predict v1 from v0, and our high-accuracy predictions demonstrate the method's robustness across a large temporal gap. Refer to Table 11 for performance details.
+
+# A.4.2 LIVEXIV V1-OPPOSITE
+
+To show the impact of the role of each model in the dataset creation, we created an opposite version of v1, where Claude is the model that generates the questions and GPT is the model that does the filtering. Table 12 demonstrates the changes in the absolute performance and in the ranking between the models. Overall, switching roles between Claude and GPT-4o has minimal impact on most models. However, GPT-4o shows slightly more variation, dropping slightly in the rankings when it takes on the role of filtering questions. This variation occurs because generating the questions provides some advantage.
+
+# A.4.3 LIVEXIV V2
+
+LiveXiv second version (v2) consists of 10.4K questions for VQA and 7.7K questions for TQA. In addition, we introduced 4 new domains from physics, namely, physics.app-ph, physics optics,
+
+Table 12: We compared LiveXiv first version with different configurations, The original configuration where GPT4o generates the questions and Claude filters and an opposite version where Claude generates the questions and GPT4o performs the filtering. We see in it that overall, the ranking remains similar and the changes are minor. Indeed, for the generating model there is some advantage as we see that GPT goes down a bit in the ranking (Claude remains first in both cases).  
+
+<table><tr><td></td><td colspan="2">VQA</td><td colspan="2">TQA</td></tr><tr><td></td><td>Ranking change</td><td>Average change (%)</td><td>Ranking change</td><td>Average change (%)</td></tr><tr><td>InstructBLIP-7B</td><td>0</td><td>-1.78</td><td>0</td><td>9.66</td></tr><tr><td>IXC2.5-7B</td><td>0</td><td>-0.83</td><td>0</td><td>13.73</td></tr><tr><td>InternVL2-8B</td><td>0</td><td>-1.18</td><td>0</td><td>2.37</td></tr><tr><td>LLaVA-1.6-Mistral-7B</td><td>2</td><td>-8.64</td><td>0</td><td>10.72</td></tr><tr><td>LLaVA-OneVision-7B</td><td>1</td><td>-3.66</td><td>-2</td><td>13.42</td></tr><tr><td>LLaVA-v1.5-13B</td><td>0</td><td>-6.49</td><td>0</td><td>8.48</td></tr><tr><td>LLaVA-v1.5-7B</td><td>0</td><td>-8.00</td><td>-1</td><td>12.72</td></tr><tr><td>LLaVA-v1.6-34B</td><td>2</td><td>-5.85</td><td>0</td><td>-0.35</td></tr><tr><td>Mantis-LLama3-8B</td><td>2</td><td>-7.13</td><td>0</td><td>10.24</td></tr><tr><td>Phi3v</td><td>1</td><td>-0.49</td><td>0</td><td>13.856</td></tr><tr><td>IXC2-4KHD-7B</td><td>-5</td><td>-0.47</td><td>3</td><td>8.60</td></tr><tr><td>Idefics2-8B</td><td>1</td><td>-6.00</td><td>0</td><td>3.21</td></tr><tr><td>InternVL2-2B</td><td>0</td><td>-1.42</td><td>0</td><td>2.61</td></tr><tr><td>Claude-Sonnet</td><td>0</td><td>-3.31</td><td>0</td><td>10.11</td></tr><tr><td>Qwen2-VL</td><td>0</td><td>-0.31</td><td>0</td><td>2.81</td></tr><tr><td>GPT-4o</td><td>-4</td><td>4.74</td><td>0</td><td>13.46</td></tr><tr><td>Idefics3</td><td>0</td><td>-1.89</td><td>0</td><td>12.23</td></tr><tr><td>Average</td><td>0</td><td>-3.10</td><td>0</td><td>8.70</td></tr></table>
+
+Table 13: Version 2 - VQA and TQA average accuracy across ArXiv taxonomy (the number of samples is in brackets).  
+
+<table><tr><td>VQA Accuracy</td><td>cs.AI (790)</td><td>cs.CV (1071)</td><td>cs.LG (853)</td><td>cs.RG (834)</td><td>eess.SP (889)</td><td>eess.SY (772)</td><td>physics.app-ph (944)</td><td>physics.bio-ph (868)</td><td>physics.data-an (841)</td><td>physics optics (819)</td><td>q-bio.BM (890)</td><td>q-bio.CB (184)</td><td>q-bio.GN (321)</td><td>q-bio.TO (299)</td><td>Mean (10375)</td></tr><tr><td>InstrunctBLIP</td><td>0.25</td><td>0.24</td><td>0.23</td><td>0.27</td><td>0.28</td><td>0.22</td><td>0.22</td><td>0.25</td><td>0.27</td><td>0.22</td><td>0.20</td><td>0.25</td><td>0.22</td><td>0.17</td><td>0.24</td></tr><tr><td>InterLM-XC-2.5</td><td>0.51</td><td>0.49</td><td>0.43</td><td>0.49</td><td>0.54</td><td>0.52</td><td>0.46</td><td>0.52</td><td>0.49</td><td>0.51</td><td>0.47</td><td>0.42</td><td>0.54</td><td>0.51</td><td>0.49</td></tr><tr><td>InterLM-2.8B</td><td>0.66</td><td>0.62</td><td>0.62</td><td>0.65</td><td>0.65</td><td>0.65</td><td>0.63</td><td>0.68</td><td>0.64</td><td>0.66</td><td>0.62</td><td>0.59</td><td>0.64</td><td>0.64</td><td>0.64</td></tr><tr><td>InterMLV2.2B</td><td>0.34</td><td>0.39</td><td>0.34</td><td>0.36</td><td>0.33</td><td>0.32</td><td>0.30</td><td>0.34</td><td>0.32</td><td>0.31</td><td>0.34</td><td>0.29</td><td>0.30</td><td>0.34</td><td>0.33</td></tr><tr><td>LLaVAL6.7B</td><td>0.55</td><td>0.60</td><td>0.47</td><td>0.59</td><td>0.54</td><td>0.59</td><td>0.57</td><td>0.59</td><td>0.55</td><td>0.59</td><td>0.55</td><td>0.47</td><td>0.58</td><td>0.56</td><td>0.56</td></tr><tr><td>LLaVA-OneVision</td><td>0.35</td><td>0.36</td><td>0.33</td><td>0.39</td><td>0.35</td><td>0.36</td><td>0.34</td><td>0.35</td><td>0.33</td><td>0.34</td><td>0.36</td><td>0.33</td><td>0.35</td><td>0.35</td><td>0.35</td></tr><tr><td>LLaVAL5.1-13B</td><td>0.32</td><td>0.35</td><td>0.33</td><td>0.35</td><td>0.31</td><td>0.34</td><td>0.31</td><td>0.34</td><td>0.35</td><td>0.35</td><td>0.33</td><td>0.35</td><td>0.32</td><td>0.35</td><td>0.33</td></tr><tr><td>LLaVAL5.7B</td><td>0.53</td><td>0.56</td><td>0.42</td><td>0.57</td><td>0.54</td><td>0.54</td><td>0.48</td><td>0.58</td><td>0.45</td><td>0.55</td><td>0.51</td><td>0.47</td><td>0.45</td><td>0.56</td><td>0.52</td></tr><tr><td>LLaVAL6.3-4B</td><td>0.35</td><td>0.35</td><td>0.29</td><td>0.35</td><td>0.30</td><td>0.33</td><td>0.31</td><td>0.35</td><td>0.31</td><td>0.36</td><td>0.33</td><td>0.35</td><td>0.33</td><td>0.35</td><td>0.33</td></tr><tr><td>Mantis</td><td>0.59</td><td>0.64</td><td>0.51</td><td>0.63</td><td>0.64</td><td>0.60</td><td>0.60</td><td>0.66</td><td>0.60</td><td>0.62</td><td>0.60</td><td>0.49</td><td>0.58</td><td>0.62</td><td>0.60</td></tr><tr><td>Phi3v</td><td>0.43</td><td>0.45</td><td>0.30</td><td>0.42</td><td>0.32</td><td>0.35</td><td>0.32</td><td>0.33</td><td>0.36</td><td>0.32</td><td>0.36</td><td>0.24</td><td>0.38</td><td>0.32</td><td>0.35</td></tr><tr><td>InterLM-XC-Khld</td><td>0.48</td><td>0.44</td><td>0.36</td><td>0.43</td><td>0.36</td><td>0.40</td><td>0.38</td><td>0.43</td><td>0.39</td><td>0.39</td><td>0.39</td><td>0.33</td><td>0.34</td><td>0.39</td><td>0.39</td></tr><tr><td>Idlefs2</td><td>0.50</td><td>0.54</td><td>0.45</td><td>0.47</td><td>0.47</td><td>0.52</td><td>0.52</td><td>0.55</td><td>0.51</td><td>0.53</td><td>0.48</td><td>0.43</td><td>0.48</td><td>0.51</td><td>0.50</td></tr><tr><td>Claude-Sonnet</td><td>0.81</td><td>0.78</td><td>0.75</td><td>0.76</td><td>0.83</td><td>0.80</td><td>0.78</td><td>0.80</td><td>0.76</td><td>0.80</td><td>0.77</td><td>0.69</td><td>0.76</td><td>0.78</td><td>0.78</td></tr><tr><td>Owen2-VL-7B</td><td>0.68</td><td>0.71</td><td>0.61</td><td>0.68</td><td>0.72</td><td>0.66</td><td>0.70</td><td>0.75</td><td>0.69</td><td>0.73</td><td>0.67</td><td>0.58</td><td>0.69</td><td>0.71</td><td>0.68</td></tr><tr><td>GPT4o</td><td>0.58</td><td>0.61</td><td>0.50</td><td>0.60</td><td>0.62</td><td>0.57</td><td>0.61</td><td>0.63</td><td>0.59</td><td>0.61</td><td>0.59</td><td>0.53</td><td>0.54</td><td>0.67</td><td>0.59</td></tr><tr><td>Idlefs3</td><td>0.57</td><td>0.60</td><td>0.50</td><td>0.55</td><td>0.58</td><td>0.58</td><td>0.52</td><td>0.59</td><td>0.53</td><td>0.54</td><td>0.55</td><td>0.52</td><td>0.60</td><td>0.58</td><td>0.56</td></tr><tr><td>Pixtral</td><td>0.73</td><td>0.71</td><td>0.66</td><td>0.73</td><td>0.78</td><td>0.70</td><td>0.76</td><td>0.77</td><td>0.73</td><td>0.73</td><td>0.70</td><td>0.67</td><td>0.72</td><td>0.77</td><td>0.73</td></tr><tr><td>Molmo</td><td>0.62</td><td>0.67</td><td>0.52</td><td>0.62</td><td>0.62</td><td>0.62</td><td>0.59</td><td>0.65</td><td>0.61</td><td>0.63</td><td>0.61</td><td>0.49</td><td>0.55</td><td>0.67</td><td>0.60</td></tr><tr><td>TQA Accuracy</td><td>cs.AI (1208)</td><td>cs.AI (901)</td><td>cs.LG (785)</td><td>cs.RG (1315)</td><td>eess.SP (164)</td><td>eess.SY (458)</td><td>physics.app-ph (998)</td><td>physics.bio-ph (482)</td><td>physics.data-an (340)</td><td>physics optics (198)</td><td>q-bio.BM (252)</td><td>q-bio.CB (203)</td><td>q-bio.TO (141)</td><td>q-bio.TO (267)</td><td>Mean (7712)</td></tr><tr><td>InstrunctBLIP</td><td>0.20</td><td>0.22</td><td>0.21</td><td>0.19</td><td>0.19</td><td>0.19</td><td>0.15</td><td>0.13</td><td>0.18</td><td>0.18</td><td>0.18</td><td>0.21</td><td>0.20</td><td>0.18</td><td>0.19</td></tr><tr><td>InterLM-XC-2.5</td><td>0.41</td><td>0.43</td><td>0.45</td><td>0.49</td><td>0.42</td><td>0.50</td><td>0.46</td><td>0.56</td><td>0.52</td><td>0.43</td><td>0.45</td><td>0.38</td><td>0.46</td><td>0.41</td><td>0.46</td></tr><tr><td>InterMLV2.2B</td><td>0.37</td><td>0.34</td><td>0.39</td><td>0.37</td><td>0.36</td><td>0.41</td><td>0.41</td><td>0.47</td><td>0.39</td><td>0.37</td><td>0.38</td><td>0.34</td><td>0.41</td><td>0.30</td><td>0.38</td></tr><tr><td>InterLM-2.8B</td><td>0.56</td><td>0.55</td><td>0.59</td><td>0.64</td><td>0.63</td><td>0.65</td><td>0.65</td><td>0.69</td><td>0.56</td><td>0.60</td><td>0.61</td><td>0.51</td><td>0.57</td><td>0.57</td><td>0.60</td></tr><tr><td>LLaVAL6.7B</td><td>0.23</td><td>0.21</td><td>0.22</td><td>0.23</td><td>0.21</td><td>0.24</td><td>0.22</td><td>0.21</td><td>0.26</td><td>0.25</td><td>0.24</td><td>0.15</td><td>0.24</td><td>0.21</td><td>0.22</td></tr><tr><td>LLaVA-OneVision</td><td>0.42</td><td>0.44</td><td>0.50</td><td>0.48</td><td>0.45</td><td>0.53</td><td>0.49</td><td>0.56</td><td>0.52</td><td>0.50</td><td>0.48</td><td>0.46</td><td>0.49</td><td>0.47</td><td>0.49</td></tr><tr><td>LLaVAL5.1-13B</td><td>0.30</td><td>0.28</td><td>0.33</td><td>0.32</td><td>0.34</td><td>0.28</td><td>0.29</td><td>0.35</td><td>0.31</td><td>0.31</td><td>0.29</td><td>0.25</td><td>0.28</td><td>0.27</td><td>0.30</td></tr><tr><td>LLaVAL5.7B</td><td>0.28</td><td>0.26</td><td>0.31</td><td>0.31</td><td>0.28</td><td>0.30</td><td>0.27</td><td>0.33</td><td>0.28</td><td>0.29</td><td>0.31</td><td>0.33</td><td>0.25</td><td>0.27</td><td>0.29</td></tr><tr><td>LLaVAL6.3-4B</td><td>0.45</td><td>0.45</td><td>0.48</td><td>0.54</td><td>0.54</td><td>0.59</td><td>0.55</td><td>0.55</td><td>0.55</td><td>0.52</td><td>0.54</td><td>0.48</td><td>0.49</td><td>0.51</td><td>0.52</td></tr><tr><td>Mantis</td><td>0.28</td><td>0.28</td><td>0.29</td><td>0.32</td><td>0.33</td><td>0.28</td><td>0.28</td><td>0.33</td><td>0.34</td><td>0.25</td><td>0.28</td><td>0.22</td><td>0.27</td><td>0.24</td><td>0.29</td></tr><tr><td>Phi3v</td><td>0.43</td><td>0.45</td><td>0.48</td><td>0.42</td><td>0.42</td><td>0.53</td><td>0.47</td><td>0.60</td><td>0.54</td><td>0.48</td><td>0.47</td><td>0.39</td><td>0.49</td><td>0.47</td><td>0.49</td></tr><tr><td>InterLM-XC-Khld</td><td>0.37</td><td>0.37</td><td>0.40</td><td>0.43</td><td>0.34</td><td>0.42</td><td>0.40</td><td>0.46</td><td>0.44</td><td>0.47</td><td>0.47</td><td>0.39</td><td>0.24</td><td>0.41</td><td>0.30</td></tr><tr><td>Idlefs2</td><td>0.32</td><td>0.33</td><td>0.35</td><td>0.37</td><td>0.34</td><td>0.44</td><td>0.35</td><td>0.44</td><td>0.39</td><td>0.36</td><td>0.36</td><td>0.29</td><td>0.37</td><td>0.32</td><td>0.36</td></tr><tr><td>Claude-Sonnet</td><td>0.79</td><td>0.82</td><td>0.81</td><td>0.87</td><td>0.81</td><td>0.81</td><td>0.83</td><td>0.85</td><td>0.78</td><td>0.80</td><td>0.83</td><td>0.77</td><td>0.81</td><td>0.84</td><td>0.82</td></tr><tr><td>Owen2-VL-7B</td><td>0.51</td><td>0.54</td><td>0.56</td><td>0.59</td><td>0.60</td><td>0.63</td><td>0.57</td><td>0.73</td><td>0.59</td><td>0.53</td><td>0.56</td><td>0.51</td><td>0.61</td><td>0.52</td><td>0.58</td></tr><tr><td>GPT4o</td><td>0.47</td><td>0.45</td><td>0.45</td><td>0.50</td><td>0.50</td><td>0.50</td><td>0.52</td><td>0.55</td><td>0.46</td><td>0.46</td><td>0.47</td><td>0.43</td><td>0.49</td><td>0.54</td><td>0.49</td></tr><tr><td>Idlefs3</td><td>0.43</td><td>0.48</td><td>0.47</td><td>0.47</td><td>0.48</td><td>0.57</td><td>0.47</td><td>0.55</td><td>0.49</td><td>0.48</td><td>0.47</td><td>0.39</td><td>0.50</td><td>0.49</td><td>0.48</td></tr><tr><td>Pixtral</td><td>0.57</td><td>0.56</td><td>0.58</td><td>0.59</td><td>0.57</td><td>0.63</td><td>0.60</td><td>0.66</td><td>0.58</td><td>0.57</td><td>0.58</td><td>0.54</td><td>0.61</td><td>0.57</td><td>0.59</td></tr><tr><td>Molmo</td><td>0.49</td><td>0.48</td><td>0.51</td><td>0.53</td><td>0.49</td><td>0.55</td><td>0.52</td><td>0.64</td><td>0.53</td><td>0.50</td><td>0.53</td><td>0.44</td><td>0.51</td><td>0.49</td><td>0.51</td></tr></table>
+
+physics.bio-ph, physics.data-an to a total of 14 domains. Lastly, we also added Two new models, Pixtral (Agrawal et al., 2024) and Molmo-7B (Deitke et al., 2024). Due to our high frequency updates, Quantitative Biology related domains have overleaping papers between versions. Thus, we filtered these overleaping papers out. V2 shows similar diversity to v1 (see Table 18), similar performance of the evaluated models, resulting an accurate and reliable efficient evaluation (see Figure ??) and lastly, demonstrating LiveXiv to be evolving scalable live dataset. Table 13 reports the performance summary.
+
+# A.4.4 LIVEXIV V3
+
+To further mitigate the potential bias due to single generation and single filtering model, v3 is a combination of 3 participant models, GPT-4o, Claude-Sonnet and Qwen2-VL-72B. v3 is actually 3 subsets, where each time a different model takes a different role. Table 14 presents the results for all 3 subsets. Table 15 shows the performance per subset. Overall we can see that the model ranking remains similar to previous version. This hints that the bias of using a single model is small. We extended our manual verification process with an additional check of v3. Tables 16 and 17 present these results in detail. The manual check revealed more variations compared to the manual check on v1, which can be attributed to the participation of three models in v3. As shown in Table 15, Qwen2-VL-72B demonstrated weaker performance compared to GPT and Claude. Consequently, its role as the filtering and generation model likely introduced additional noise into the dataset. However, the average performance change for both VQA and TQA remains below  $5\%$ .
+
+Table 14: Version 3 - VQA and TQA average accuracy across ArXiv taxonomy (the number of samples is in brackets).  
+
+<table><tr><td>VQA Accuracy</td><td>cs.AI(609)</td><td>cs.CV(892)</td><td>cs.LG(792)</td><td>cs.RO(798)</td><td>eess.SP(835)</td><td>eess.SY(597)</td><td>physics.app-ph(1035)</td><td>physics.bio-ph(908)</td><td>physics.data-an(425)</td><td>physics.optics(628)</td><td>q-bio.BM(803)</td><td>q-bio.CB(640)</td><td>q-bio.GN(425)</td><td>q-bio.TO(555)</td><td>Mean(9942)</td></tr><tr><td>InstrunctBLIP</td><td>0.24</td><td>0.26</td><td>0.23</td><td>0.26</td><td>0.21</td><td>0.21</td><td>0.21</td><td>0.22</td><td>0.22</td><td>0.20</td><td>0.22</td><td>0.23</td><td>0.26</td><td>0.24</td><td>0.23</td></tr><tr><td>InterLM-XC-2.5</td><td>0.50</td><td>0.50</td><td>0.46</td><td>0.49</td><td>0.48</td><td>0.47</td><td>0.53</td><td>0.50</td><td>0.50</td><td>0.52</td><td>0.49</td><td>0.48</td><td>0.51</td><td>0.52</td><td>0.50</td></tr><tr><td>InterML-VL-2B</td><td>0.37</td><td>0.38</td><td>0.31</td><td>0.42</td><td>0.33</td><td>0.35</td><td>0.34</td><td>0.31</td><td>0.28</td><td>0.34</td><td>0.33</td><td>0.35</td><td>0.32</td><td>0.35</td><td>0.34</td></tr><tr><td>InterML-VL-8B</td><td>0.65</td><td>0.66</td><td>0.60</td><td>0.71</td><td>0.67</td><td>0.65</td><td>0.69</td><td>0.64</td><td>0.62</td><td>0.68</td><td>0.62</td><td>0.64</td><td>0.61</td><td>0.69</td><td>0.65</td></tr><tr><td>LLaVA1.6-7B</td><td>0.58</td><td>0.59</td><td>0.51</td><td>0.65</td><td>0.57</td><td>0.55</td><td>0.61</td><td>0.57</td><td>0.52</td><td>0.58</td><td>0.51</td><td>0.56</td><td>0.52</td><td>0.57</td><td>0.56</td></tr><tr><td>LLaVA-OneVision</td><td>0.38</td><td>0.40</td><td>0.33</td><td>0.45</td><td>0.33</td><td>0.39</td><td>0.36</td><td>0.34</td><td>0.31</td><td>0.37</td><td>0.36</td><td>0.38</td><td>0.33</td><td>0.34</td><td>0.36</td></tr><tr><td>LLaVA1.5-3B</td><td>0.37</td><td>0.38</td><td>0.32</td><td>0.44</td><td>0.31</td><td>0.36</td><td>0.35</td><td>0.33</td><td>0.29</td><td>0.36</td><td>0.32</td><td>0.36</td><td>0.34</td><td>0.36</td><td>0.35</td></tr><tr><td>LLaVA1.5-7B</td><td>0.54</td><td>0.54</td><td>0.47</td><td>0.64</td><td>0.56</td><td>0.56</td><td>0.57</td><td>0.52</td><td>0.47</td><td>0.59</td><td>0.59</td><td>0.59</td><td>0.59</td><td>0.57</td><td>0.54</td></tr><tr><td>LLaVA1.6-34B</td><td>0.37</td><td>0.40</td><td>0.31</td><td>0.42</td><td>0.33</td><td>0.37</td><td>0.37</td><td>0.35</td><td>0.30</td><td>0.36</td><td>0.34</td><td>0.37</td><td>0.34</td><td>0.37</td><td>0.36</td></tr><tr><td>Mantis</td><td>0.63</td><td>0.64</td><td>0.59</td><td>0.65</td><td>0.62</td><td>0.59</td><td>0.67</td><td>0.63</td><td>0.60</td><td>0.63</td><td>0.56</td><td>0.60</td><td>0.60</td><td>0.59</td><td>0.61</td></tr><tr><td>Phi3v</td><td>0.38</td><td>0.42</td><td>0.34</td><td>0.46</td><td>0.33</td><td>0.33</td><td>0.38</td><td>0.32</td><td>0.31</td><td>0.37</td><td>0.31</td><td>0.31</td><td>0.37</td><td>0.38</td><td>0.36</td></tr><tr><td>InterLM-XC-4Khd</td><td>0.40</td><td>0.42</td><td>0.35</td><td>0.47</td><td>0.36</td><td>0.41</td><td>0.42</td><td>0.39</td><td>0.35</td><td>0.38</td><td>0.36</td><td>0.39</td><td>0.39</td><td>0.45</td><td>0.40</td></tr><tr><td>Idefics2</td><td>0.54</td><td>0.55</td><td>0.47</td><td>0.58</td><td>0.49</td><td>0.51</td><td>0.55</td><td>0.50</td><td>0.49</td><td>0.50</td><td>0.48</td><td>0.53</td><td>0.49</td><td>0.50</td><td>0.51</td></tr><tr><td>Claude-Sonnet</td><td>0.75</td><td>0.77</td><td>0.76</td><td>0.78</td><td>0.80</td><td>0.80</td><td>0.81</td><td>0.80</td><td>0.73</td><td>0.81</td><td>0.76</td><td>0.75</td><td>0.77</td><td>0.78</td><td>0.78</td></tr><tr><td>Qwen2-VL-7B</td><td>0.68</td><td>0.69</td><td>0.66</td><td>0.74</td><td>0.68</td><td>0.66</td><td>0.75</td><td>0.72</td><td>0.69</td><td>0.72</td><td>0.62</td><td>0.70</td><td>0.68</td><td>0.73</td><td>0.69</td></tr><tr><td>GPT4o</td><td>0.50</td><td>0.54</td><td>0.50</td><td>0.56</td><td>0.51</td><td>0.54</td><td>0.52</td><td>0.48</td><td>0.48</td><td>0.48</td><td>0.46</td><td>0.51</td><td>0.51</td><td>0.52</td><td>0.51</td></tr><tr><td>Idefics3</td><td>0.59</td><td>0.58</td><td>0.50</td><td>0.60</td><td>0.54</td><td>0.57</td><td>0.58</td><td>0.53</td><td>0.52</td><td>0.56</td><td>0.49</td><td>0.52</td><td>0.57</td><td>0.54</td><td>0.55</td></tr><tr><td>Pixtral</td><td>0.69</td><td>0.71</td><td>0.69</td><td>0.74</td><td>0.72</td><td>0.70</td><td>0.76</td><td>0.70</td><td>0.72</td><td>0.74</td><td>0.69</td><td>0.71</td><td>0.70</td><td>0.74</td><td>0.71</td></tr><tr><td>Molmo</td><td>0.62</td><td>0.64</td><td>0.58</td><td>0.66</td><td>0.64</td><td>0.60</td><td>0.66</td><td>0.60</td><td>0.58</td><td>0.62</td><td>0.57</td><td>0.62</td><td>0.60</td><td>0.65</td><td>0.62</td></tr><tr><td>TQA Accuracy</td><td>cs.AI(802)</td><td>cs.CV(939)</td><td>cs.LG(890)</td><td>cs.RO(502)</td><td>eess.SP(386)</td><td>eess.SY(273)</td><td>physics.app-ph(270)</td><td>physics.bio-ph(199)</td><td>physics.data-an(149)</td><td>physics.optics(241)</td><td>q-bio.BM(821)</td><td>q-bio.CB(283)</td><td>q-bio.GN(650)</td><td>q-bio.TO(248)</td><td>Mean(6653)</td></tr><tr><td>InstrunctBLIP</td><td>0.18</td><td>0.25</td><td>0.16</td><td>0.18</td><td>0.17</td><td>0.22</td><td>0.24</td><td>0.18</td><td>0.18</td><td>0.19</td><td>0.22</td><td>0.18</td><td>0.20</td><td>0.23</td><td>0.20</td></tr><tr><td>InterLM-XC-2.5</td><td>0.49</td><td>0.47</td><td>0.51</td><td>0.49</td><td>0.49</td><td>0.59</td><td>0.66</td><td>0.65</td><td>0.52</td><td>0.53</td><td>0.50</td><td>0.56</td><td>0.56</td><td>0.59</td><td>0.54</td></tr><tr><td>InterML-VL-2B</td><td>0.44</td><td>0.38</td><td>0.44</td><td>0.46</td><td>0.46</td><td>0.56</td><td>0.63</td><td>0.58</td><td>0.46</td><td>0.46</td><td>0.42</td><td>0.49</td><td>0.47</td><td>0.47</td><td>0.48</td></tr><tr><td>InterML-VL-8B</td><td>0.69</td><td>0.60</td><td>0.68</td><td>0.65</td><td>0.67</td><td>0.77</td><td>0.79</td><td>0.77</td><td>0.72</td><td>0.72</td><td>0.62</td><td>0.75</td><td>0.66</td><td>0.67</td><td>0.69</td></tr><tr><td>LLaVA1.6-7B</td><td>0.26</td><td>0.24</td><td>0.24</td><td>0.21</td><td>0.23</td><td>0.32</td><td>0.35</td><td>0.29</td><td>0.29</td><td>0.26</td><td>0.24</td><td>0.27</td><td>0.23</td><td>0.26</td><td>0.26</td></tr><tr><td>LLaVA-OneVision</td><td>0.52</td><td>0.52</td><td>0.50</td><td>0.52</td><td>0.52</td><td>0.66</td><td>0.73</td><td>0.66</td><td>0.60</td><td>0.57</td><td>0.48</td><td>0.62</td><td>0.53</td><td>0.56</td><td>0.57</td></tr><tr><td>LLaVA1.5-3B</td><td>0.35</td><td>0.32</td><td>0.33</td><td>0.29</td><td>0.34</td><td>0.38</td><td>0.33</td><td>0.32</td><td>0.17</td><td>0.35</td><td>0.33</td><td>0.29</td><td>0.33</td><td>0.24</td><td>0.31</td></tr><tr><td>LLaVA1.5-7B</td><td>0.32</td><td>0.32</td><td>0.32</td><td>0.30</td><td>0.34</td><td>0.37</td><td>0.33</td><td>0.30</td><td>0.24</td><td>0.30</td><td>0.32</td><td>0.31</td><td>0.30</td><td>0.29</td><td>0.31</td></tr><tr><td>LLaVA1.6-34B</td><td>0.58</td><td>0.51</td><td>0.47</td><td>0.56</td><td>0.60</td><td>0.66</td><td>0.72</td><td>0.72</td><td>0.60</td><td>0.65</td><td>0.55</td><td>0.67</td><td>0.56</td><td>0.54</td><td>0.60</td></tr><tr><td>Mantis</td><td>0.32</td><td>0.27</td><td>0.33</td><td>0.31</td><td>0.29</td><td>0.40</td><td>0.33</td><td>0.32</td><td>0.33</td><td>0.35</td><td>0.29</td><td>0.30</td><td>0.30</td><td>0.26</td><td>0.31</td></tr><tr><td>Phi3v</td><td>0.52</td><td>0.51</td><td>0.53</td><td>0.51</td><td>0.55</td><td>0.62</td><td>0.74</td><td>0.68</td><td>0.59</td><td>0.61</td><td>0.53</td><td>0.63</td><td>0.58</td><td>0.51</td><td>0.58</td></tr><tr><td>InterLM-XC-4Khd</td><td>0.45</td><td>0.39</td><td>0.46</td><td>0.44</td><td>0.44</td><td>0.51</td><td>0.57</td><td>0.55</td><td>0.43</td><td>0.48</td><td>0.46</td><td>0.46</td><td>0.44</td><td>0.47</td><td>0.47</td></tr><tr><td>Idefics2</td><td>0.40</td><td>0.31</td><td>0.39</td><td>0.33</td><td>0.40</td><td>0.46</td><td>0.51</td><td>0.45</td><td>0.42</td><td>0.47</td><td>0.35</td><td>0.42</td><td>0.42</td><td>0.44</td><td>0.41</td></tr><tr><td>Claude-Sonnet</td><td>0.82</td><td>0.78</td><td>0.78</td><td>0.85</td><td>0.85</td><td>0.87</td><td>0.85</td><td>0.86</td><td>0.86</td><td>0.88</td><td>0.78</td><td>0.88</td><td>0.83</td><td>0.83</td><td>0.84</td></tr><tr><td>Qwen2-VL-7B</td><td>0.62</td><td>0.60</td><td>0.63</td><td>0.63</td><td>0.62</td><td>0.68</td><td>0.77</td><td>0.72</td><td>0.72</td><td>0.73</td><td>0.62</td><td>0.70</td><td>0.66</td><td>0.64</td><td>0.67</td></tr><tr><td>GPT4o</td><td>0.45</td><td>0.44</td><td>0.45</td><td>0.44</td><td>0.51</td><td>0.59</td><td>0.62</td><td>0.56</td><td>0.54</td><td>0.53</td><td>0.48</td><td>0.60</td><td>0.54</td><td>0.51</td><td>0.52</td></tr><tr><td>Idefics3</td><td>0.55</td><td>0.49</td><td>0.57</td><td>0.47</td><td>0.51</td><td>0.59</td><td>0.68</td><td>0.66</td><td>0.54</td><td>0.59</td><td>0.52</td><td>0.58</td><td>0.56</td><td>0.53</td><td>0.56</td></tr><tr><td>Pixtral</td><td>0.40</td><td>0.41</td><td>0.50</td><td>0.42</td><td>0.37</td><td>0.48</td><td>0.16</td><td>0.14</td><td>0.49</td><td>0.54</td><td>0.36</td><td>0.42</td><td>0.39</td><td>0.37</td><td>0.39</td></tr><tr><td>Molmo</td><td>0.35</td><td>0.35</td><td>0.42</td><td>0.38</td><td>0.34</td><td>0.47</td><td>0.14</td><td>0.12</td><td>0.14</td><td>0.52</td><td>0.31</td><td>0.33</td><td>0.34</td><td>0.31</td><td>0.34</td></tr></table>
+
+Table 15: Average performance (\%) on different versions. We clearly can see that version subset 3 (denote v3-s3) has a large shift in the average performance for TQA, suggests that Qwen2-VL-72B might generate easier questions.  
+
+<table><tr><td></td><td>VQA</td><td>TQA</td></tr><tr><td>v1</td><td>47.1</td><td>45.1</td></tr><tr><td>v2</td><td>50.7</td><td>44.0</td></tr><tr><td>v3 - s1</td><td>51.6</td><td>44.5</td></tr><tr><td>v3 - s2</td><td>49.1</td><td>41.7</td></tr><tr><td>v3 - s3</td><td>52.2</td><td>52.5</td></tr><tr><td>v3</td><td>50.8</td><td>46.3</td></tr></table>
+
+Table 16: VQA Manual verification. We re-verified our dataset by curating a subset of 300 questions (100 from each subset). We can see that version 3 even though composed of 3 different subsets, remains accurate with a bounded performance change overall and minimal ranking changes.  
+
+<table><tr><td></td><td>Overall</td><td>Manual</td><td>Performance change</td><td>Rank Overall</td><td>Rank Manual</td><td>Rank change</td></tr><tr><td>instructclip-vicuna-7b</td><td>22.90</td><td>26.75</td><td>-3.84</td><td>19</td><td>19</td><td>0</td></tr><tr><td>internlm-xcomposer2d5-7b</td><td>49.62</td><td>51.37</td><td>-1.75</td><td>12</td><td>12</td><td>0</td></tr><tr><td>InternVL2-8B</td><td>65.43</td><td>68.09</td><td>-2.66</td><td>4</td><td>5</td><td>-1</td></tr><tr><td>llava-v1.6-mistral-7b</td><td>34.21</td><td>40.12</td><td>-5.91</td><td>18</td><td>17</td><td>1</td></tr><tr><td>llava-onevision-qwen2-7b-ov</td><td>56.69</td><td>59.88</td><td>-3.19</td><td>7</td><td>9</td><td>-2</td></tr><tr><td>llava-v1.5-13b</td><td>36.48</td><td>43.16</td><td>-6.68</td><td>14</td><td>15</td><td>-1</td></tr><tr><td>llava-v1.5-7b</td><td>34.92</td><td>39.51</td><td>-4.59</td><td>17</td><td>18</td><td>-1</td></tr><tr><td>llava-v1.6-34b</td><td>54.59</td><td>61.40</td><td>-6.81</td><td>9</td><td>8</td><td>1</td></tr><tr><td>Mantis-8B-siglip-llama3</td><td>35.87</td><td>41.34</td><td>-5.47</td><td>16</td><td>16</td><td>0</td></tr><tr><td>phi3v</td><td>61.79</td><td>64.74</td><td>-2.95</td><td>6</td><td>7</td><td>-1</td></tr><tr><td>internlm-xcomposer2-4khd-7b</td><td>36.07</td><td>46.20</td><td>-10.13</td><td>15</td><td>13</td><td>2</td></tr><tr><td>idefics2-8b</td><td>39.64</td><td>45.59</td><td>-5.95</td><td>13</td><td>14</td><td>-1</td></tr><tr><td>InternVL2-2B</td><td>51.67</td><td>54.10</td><td>-2.43</td><td>10</td><td>10</td><td>0</td></tr><tr><td>claude</td><td>77.95</td><td>81.46</td><td>-3.51</td><td>1</td><td>1</td><td>0</td></tr><tr><td>qwen2vl</td><td>69.56</td><td>73.56</td><td>-3.99</td><td>3</td><td>2</td><td>1</td></tr><tr><td>gpt</td><td>50.82</td><td>51.67</td><td>-0.85</td><td>11</td><td>11</td><td>0</td></tr><tr><td>idefics3</td><td>54.94</td><td>64.74</td><td>-9.80</td><td>8</td><td>7</td><td>1</td></tr><tr><td>pixtral</td><td>71.49</td><td>72.64</td><td>-1.15</td><td>2</td><td>3</td><td>-1</td></tr><tr><td>molmo</td><td>61.93</td><td>69.91</td><td>-7.98</td><td>5</td><td>4</td><td>1</td></tr><tr><td>mean</td><td>50.87</td><td>55.59</td><td>4.72</td><td></td><td></td><td></td></tr></table>
+
+Table 17: TQA Manual verification. We re-verified our dataset by curating a subset of 300 questions (100 from each subset). We can see that version 3 even though composed of 3 different subsets, remains accurate with small performance change overall and minimal ranking changes.  
+
+<table><tr><td></td><td>Overall</td><td>Manual</td><td>Performance change</td><td>Rank Overall</td><td>Rank Manual</td><td>Rank change</td></tr><tr><td>instructclip-vicuna-7b</td><td>19.93</td><td>17.96</td><td>1.97</td><td>19</td><td>19</td><td>0</td></tr><tr><td>internlm-xcomposer2d5-7b</td><td>52.28</td><td>49.40</td><td>2.88</td><td>9</td><td>11</td><td>-2</td></tr><tr><td>InternVL2-2B</td><td>45.69</td><td>43.71</td><td>1.98</td><td>12</td><td>13</td><td>-1</td></tr><tr><td>InternVL2-8B</td><td>65.93</td><td>73.65</td><td>-7.73</td><td>2</td><td>2</td><td>0</td></tr><tr><td>llava-v1.6-mistral-7b</td><td>25.22</td><td>26.35</td><td>-1.13</td><td>18</td><td>18</td><td>0</td></tr><tr><td>llava-onevision-qwen2-7b-ov</td><td>53.99</td><td>56.29</td><td>-2.30</td><td>8</td><td>6</td><td>2</td></tr><tr><td>llava-v1.5-13b</td><td>32.24</td><td>31.74</td><td>0.50</td><td>15</td><td>17</td><td>-2</td></tr><tr><td>llava-v1.5-7b</td><td>31.50</td><td>33.23</td><td>-1.73</td><td>16</td><td>15</td><td>1</td></tr><tr><td>llava-v1.6-34b</td><td>56.74</td><td>62.57</td><td>-5.83</td><td>5</td><td>3</td><td>2</td></tr><tr><td>Mantis-8B-siglip-llama3</td><td>30.74</td><td>32.34</td><td>-1.60</td><td>17</td><td>16</td><td>1</td></tr><tr><td>phi3v</td><td>55.49</td><td>52.40</td><td>3.10</td><td>6</td><td>7</td><td>-1</td></tr><tr><td>internlm-xcomposer2-4khd-7b</td><td>45.44</td><td>40.12</td><td>5.32</td><td>13</td><td>14</td><td>-1</td></tr><tr><td>idefics2-8b</td><td>39.13</td><td>46.71</td><td>-7.58</td><td>14</td><td>12</td><td>2</td></tr><tr><td>claude</td><td>81.92</td><td>88.62</td><td>-6.70</td><td>1</td><td>1</td><td>0</td></tr><tr><td>qwen2vl</td><td>64.36</td><td>62.28</td><td>2.09</td><td>3</td><td>4</td><td>-1</td></tr><tr><td>gpt</td><td>49.23</td><td>50.30</td><td>-1.07</td><td>11</td><td>10</td><td>1</td></tr><tr><td>idefics3</td><td>54.41</td><td>52.10</td><td>2.32</td><td>7</td><td>8</td><td>-1</td></tr><tr><td>pixtral</td><td>57.45</td><td>56.36</td><td>1.09</td><td>4</td><td>5</td><td>-1</td></tr><tr><td>molmo</td><td>50.12</td><td>51.36</td><td>-1.25</td><td>10</td><td>9</td><td>1</td></tr><tr><td>mean</td><td>47.99</td><td>48.82</td><td>3.06</td><td></td><td></td><td></td></tr></table>
+
+# A.4.5 DIVERSITY THROUGH TIME
+
+We evaluate LiveXiv diversity though time in Table 18. It shows that each version has a steady distribution of visual content and a diverse set of questions for both figures and tables.
+
+Table 18: Questions and visual content diversity through LiveXiv versions, we can see that the diversity is persistent and LiveXiv has a rich set of visual content as well diverse questions throughout the new versions.  
+
+<table><tr><td>VQA</td><td>Data Analysis</td><td>Reasoning</td><td>Attribute</td><td>Localization</td><td>Reading</td><td>Arithmetic</td><td>Charts</td><td>Block Diagram</td><td>Qualitative</td></tr><tr><td>v0</td><td>768</td><td>280</td><td>256</td><td>365</td><td>320</td><td>42</td><td>1239</td><td>578</td><td>214</td></tr><tr><td>v1</td><td>2291</td><td>872</td><td>903</td><td>1596</td><td>1470</td><td>154</td><td>4354</td><td>2110</td><td>864</td></tr><tr><td>v2</td><td>3005</td><td>1471</td><td>1610</td><td>1797</td><td>1732</td><td>112</td><td>5480</td><td>2805</td><td>1442</td></tr><tr><td>v3</td><td>3265</td><td>1196</td><td>1459</td><td>1202</td><td>1916</td><td>321</td><td>5113</td><td>2141</td><td>1573</td></tr><tr><td>TQA</td><td>Data Analysis</td><td>Reasoning</td><td>Attribute</td><td>Localization</td><td>Reading</td><td>Arithmetic</td><td colspan="3"></td></tr><tr><td>v0</td><td>637</td><td>45</td><td>40</td><td>14</td><td>425</td><td>1168</td><td colspan="3"></td></tr><tr><td>v1</td><td>2582</td><td>123</td><td>121</td><td>23</td><td>2127</td><td>3934</td><td colspan="3"></td></tr><tr><td>v2</td><td>2289</td><td>228</td><td>74</td><td>23</td><td>1125</td><td>3436</td><td colspan="3"></td></tr><tr><td>v3</td><td>1808</td><td>130</td><td>97</td><td>16</td><td>1590</td><td>2572</td><td colspan="3"></td></tr></table>
+
+# A.5 EXPLORING MORE DETAILS ON EFFICIENT EVALUATION
+
+# A.5.1 DETAILED RESULTS FOR LIVEXIV V4
+
+In Figure 13, we present a more detailed set of results extracted from the experiment presented in Figures 4/5 for LiveXiv v4. Interestingly, we see that the models selected for re-evaluation cover the possible values of performances and that our method can make reasonable predictions for domains that were not included in LiveXiv v1.
+
+# A.5.2 LIVEXIV V0 (PAPERS FROM 2010) EFFICIENT EVALUATION RESULTS
+
+In Figure 15 we show the performance prediction results of our efficient re-evaluation method on v1 from v0, when re-evaluating 5 models. Despite the (temporal) big distribution shift, our efficient evaluation method does still work well.
+
+# A.5.3 LIVEXIV V3 EFFICIENT EVALUATION
+
+RESULTS CONSIDERING ALL QUESTIONS (GENERATED BY DIFFERENT MODELS)
+
+The third version of LiveXiv (denoted as v3) introduced multiple participating models for QA generation and filtering. Unlike the default pipeline for the two first versions, in which we have GPT4o as the generation model and Claude-Sonnet for the filtering, we augment LiveXiv v3 with more questions but (i) using Claude-Sonnet as the generation model and Qwen2-VL-72B as the filtering model, and using (ii) Qwen2-VL-72B as the generation model and GPT4o as the filtering model. The results can be seen in Figure 14; in summary, we have a greater distribution shift when it comes to TQA and when Qwen2-VL-72B is included as the generation model, making the prediction task for the IRT model harder, thus increasing the prediction errors for v3 when compared to Figures 4/5. In v4 we narrow the participants only to GPT-4o and Claude-Sonnet playing both roles, and reducing both potential bias from a single model for generation or filtering and maintaining accurate predictions.
+
+# A.5.4 EFFICIENT EVALUATION ON MM-LIVEBENCH
+
+In this section, we challenge our efficient evaluation method, by examining its performance over another type of multi-modal live dataset Zhang et al. (2024b). The dataset has 3 versions (May 2024, June 2024, and July 2024), and each version has roughly 250-300 samples of open-ended questions scraped from newspapers. To evaluate our method we use GPT-4o to convert the open-ended questions into closed-form of questions where the true answer is rephrased and 3 more negative answers are proposed. Then we evaluate 13 LMMs over all the dataset versions. We use the first version as a training set and we predict the performance over the new concatenated sets using our IRT-based method. Figure 16 shows that our method still performs well on a different benchmark when re-evaluating only 5 models.
+
+<table><tr><td></td><td colspan="2">VQA</td><td colspan="2">TQA</td></tr><tr><td></td><td>MAE</td><td>Rank Corr.</td><td>MAE</td><td>Rank Corr.</td></tr><tr><td>cs.AI</td><td>1.7±1.3</td><td>99.8</td><td>2.3±1.4</td><td>98.6</td></tr><tr><td>cs.CV</td><td>3.8±2.9</td><td>92.6</td><td>3.6±1.8</td><td>97.0</td></tr><tr><td>cs.LG</td><td>1.6±1.0</td><td>97.4</td><td>2.6±1.4</td><td>97.9</td></tr><tr><td>cs.RO</td><td>3.3±1.9</td><td>98.6</td><td>3.5±2.3</td><td>98.5</td></tr><tr><td>eess.S</td><td>1.7±0.9</td><td>99.5</td><td>2.6±1.5</td><td>96.8</td></tr><tr><td>eess.SY</td><td>2.6±1.4</td><td>98.9</td><td>4.8±2.2</td><td>97.0</td></tr><tr><td>physics.app-ph</td><td>2.7±1.3</td><td>98.6</td><td>7.6±3.9</td><td>94.3</td></tr><tr><td>physics.bio-ph</td><td>2.1±1.6</td><td>96.3</td><td>3.3±2.1</td><td>96.0</td></tr><tr><td>physics.data-an</td><td>2.4±1.8</td><td>96.8</td><td>3.0±1.8</td><td>96.0</td></tr><tr><td>physics.optics</td><td>2.9±2.1</td><td>98.9</td><td>3.8±2.0</td><td>95.3</td></tr><tr><td>q-bio.BM</td><td>2.0±1.1</td><td>96.8</td><td>2.8±1.6</td><td>97.7</td></tr><tr><td>q-bio.CB</td><td>3.2±2.2</td><td>96.9</td><td>2.2±1.3</td><td>98.6</td></tr><tr><td>q-bio.GN</td><td>2.4±1.5</td><td>97.4</td><td>2.3±1.5</td><td>96.6</td></tr><tr><td>q-bio.TO</td><td>2.9±1.6</td><td>96.8</td><td>3.0±2.5</td><td>96.6</td></tr></table>
+
+![](images/3b30eb40e6c7744b1523952b3f9d964d23618eda7cf76c2384daa6f506fc5349.jpg)  
+Figure 13: Performance prediction results of our efficient re-evaluation method on v4 when re-evaluating 5 models.
+
+![](images/25913ad2fff211c7fd1b33c4902b5dd3cec2dda7ab7feea918f0c35e203c3f51.jpg)
+
+![](images/1742f23594752a69b4c9ac3c6f9dbfd21c8d0bd3ad2249b5aa43b614483573dc.jpg)
+
+![](images/c044dcf03d80886389956e197709a7d6af953dc92db741a84c3321a819970097.jpg)
+
+![](images/b40ac9abcc151bae326db8eebbfd7ac971f2bd009d1b7a1e1a853ddda19927f0.jpg)
+
+![](images/fb717a7244a5a396ff03ee8e4ed2e113ce6b9b86efe8eb7d9bee3012eca4b5ed.jpg)
+
+![](images/12d5743de4db898052d8f5490c2d201af4109664050515706d948d96905e5c9a.jpg)  
+Figure 14: This figure is a reproduction of Figures 4 (first row) and 5 (second row) considering the full LiveXiv v3 and not only questions generated by GPT4o and Claude-Sonnet. In this case, we can see that the prediction performance of IRT can be deteriorated due to distribution shifts in the data.
+
+![](images/1fe598143d7cdf00040902b6e981c5ae2562ca4fc96a604c3c44f41e4fa4c92b.jpg)
+
+![](images/b018303aa1db650c1e0ed9d3467be9a2a56cd1b932565b93eb79dd7af286e888.jpg)
+
+![](images/97c5caa5e66df8c4fed9dab307623bbf10b388f88c98aa2ff633a52f36468f5c.jpg)
+
+![](images/769f04e33aa4b3e5839167f7e67b4e1a088b7a74aca0558eb3baccc47d28214c.jpg)  
+Figure 15: Performance prediction results of our efficient re-evaluation method on v1 from v0 when re-evaluating 5 models. We can see that even in a large temporal gap, our method managed to predict accurately.
+
+![](images/e08410fa06adc5418a2124052df1e3035f7d32be30b97956a10c238dac03a257.jpg)  
+Figure 16: Our efficient evaluation method applied to MM-LiveBench. We check that our method could be successfully applied in this other context, here we convert MM-LiveBench to multiple choice questions and apply our efficient evaluation method between MM-LiveBench versions. The results for MM-LiveBench are optimistic, showing our method predict the results between its version accurately.
+
+# A.6 PROMPT TEMPLATE FOR QA GENERATIONS
+
+Figure 17: Prompt template for general detailed caption.  
+```txt
+This is a figure from a scientific paper with the following caption: {text_desc}. Please describe the image in as much details as possible. For all the details you are confident about include everything you see, and be as specific as possible, such as existing numbers, describing objects, attributes ...
+```
+
+Figure 18: Prompt template for visual question-answering.  
+```txt
+Compositional reasoning defines the understanding of attributes, relations and word order significance. A good vision -language model should be able to accurately answer composition reasoning questions about an image. Your task is to fool a vision -language model by generating challenging compositional reasoning questions about the figure. Given the image and the description you generated: {detailed_description}, generate {n_questions} diverse and challenging compositional reasoning questions which a vision-language model would incorrectly answer. For each question include the following: - A compositional reasoning question - A correct answer - 3 hard negative options. Each negative option should differ only subtly from the correct answer but still be clearly incorrect given the image, and the question. The goal is for a vision-language model to choose the negative option over the positive option when you asked to answer the question in binary multiple choice format. Only include questions you are confident in your answer and make sure there is indeed only a single correct answer and the others are false answers. Format your response as a string in the format {{"Q":<question>, "a":<correct answer >, "n1":<negative option 1>, "n2":<negative option 2>, ...}}.
+```
+
+```txt
+Document and table understanding defines the understanding of values, metrics and perform arithmetic operations over numerical values and commonsense reasoning. A good language model should be able to accurately answer {commonsense_reasoning / arithmetic manipulation} questions from a given table. Your task is to fool a language model by generating challenging table {commonsense_reasoning / arithmetic manipulation} questions about the table. Given the table: {table_content} Generate {n_questions} diverse and challenging {commonsense_reasoning / arithmetic manipulation} questions on the table questions which a language model would incorrectly answer .For each question include the following: - A question - A correct answer - 3 hard negative options. Each negative option should differ only subtly from the correct answer but still be clearly incorrect given the figure, caption and the question. The goal is for a language model to choose the negative option over the positive option when you asked to answer the question in binary multiple choice format. Only include questions you are confident in your answer and make sure there is indeed only a single correct answer and the others are false answers. Format your response as a string in the format {{"Q":<question>, "a":<correct answer >, "n1":<negative option 1>, "n2":<negative option 2>, ...}}.
+```
+
+Figure 19: Prompt template for table question-answering.
+
+```txt
+Think step by step before answering. For the given image and question: {question} write only the words yes or no if think the option {correct_answer} is indeed the correct answer out of {options} for this question?
+```
+
+Figure 20: Prompt template for agreement filtering.
+
+```txt
+You are an insightful assistant, for the question/options pair provided by the user, pick a question category from the list below: Question category: - attribute: the question asks about the presence or visibility of an attribute of an object (e.g. "What is the color of circles in plot (a)?" ["A. Blue, B. White, C. Green, D. Red']) - reasoning: the question asks about understanding the figure (e.g "What is the object inside the red box?" ["A. Bottle, B. Table, C. Tree, D. Nothing']) - localization: the question asks about the presence or visibility at a specific location in the image (e.g "On which subplot does the scatter is the most spread?" ["A. Top-Left, B. Bottom-Right, C. 'Middle-Left', D. 'Top-Right']) - reading: the question asks about reading some text from the figure (e.g "What is name of the method presneted as a green line?" ["A. GPSK, B. FDAH, C. TQWA, D.Ours]") - arithmetic: the questions asks about mathematical arithmetic of numbers (e.g if the maximum accuracy of SIFT would be doubled? what would be the value?" ["A. 2, B. 4 , C. 100, D. 50]) - data analysis: the question asks about understanding of a graph (e.g., "Which values intersect at  $T = 2$  ?""[A. N1, B. N2, C. N3, D. N4]") Respond with a JSON object with the following format: {"Question category": "category"}
+```
+
+Figure 21: Prompt template for question categories analysis.
\ No newline at end of file
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+# LOCA: LOCATION-AWARE COSINE ADAPTATION FOR PARAMETER-EFFICIENT FINE-TUNING
+
+Zhekai  $\mathbf{Du}^{\dagger,\ddagger,*}$ , Yinjie  $\mathbf{Min}^\diamond$ , Jingjing  $\mathbf{Li}^{\dagger\boxtimes}$ , Ke  $\mathbf{Lu}^\dagger$ , Changliang  $\mathbf{Zou}^\diamond$ , Liuhua Peng‡ Tingjin  $\mathbf{Chu}^\ddagger$ , Mingming Gong‡,
+
+† University of Electronic Science and Technology of China ‡ The University of Melbourne
+
+$\diamond$  Nankai University  $\star$  Mohamed bin Zayed University of Artificial Intelligence
+
+{zhekaid, jj1, kel}@uestc.edu.cn, {nk.yjmin, nk.chlzou}@gmail.com
+
+{liuhua.peng, tingjin.chu, mingming.gong}@unimelb.edu.au
+
+# ABSTRACT
+
+Low-rank adaptation (LoRA) has become a prevalent method for adapting pretrained large language models to downstream tasks. However, the simple low-rank decomposition form may constrain the hypothesis space. To address this limitation, we introduce Location-aware Cosine Adaptation (LoCA), a novel frequency-domain parameter-efficient fine-tuning method based on inverse Discrete Cosine Transform (iDCT) with selective locations of learnable components. We begin with a comprehensive theoretical comparison between frequency-domain and low-rank decompositions for fine-tuning pre-trained large models. Our analysis reveals that frequency-domain decomposition with carefully selected frequency components can surpass the expressivity of traditional low-rank-based methods. Furthermore, we demonstrate that iDCT offers a more efficient implementation compared to inverse Discrete Fourier Transform (iDFT), allowing for better selection and tuning of frequency components while maintaining equivalent expressivity to the optimal iDFT-based adaptation. By employing finite-difference approximation to estimate gradients for discrete locations of learnable coefficients on the DCT spectrum, LoCA dynamically selects the most informative frequency components during training. Experiments on diverse language and vision fine-tuning tasks demonstrate that LoCA offers enhanced parameter efficiency while maintains computational feasibility comparable to low-rank-based methods.
+
+# 1 INTRODUCTION
+
+Pre-trained large language models (LLMs) (Radford et al., 2019; Liu et al., 2019; Brown et al., 2020) have shown strong capabilities in learning language knowledge and adapting to various natural language processing (NLP) tasks through fine-tuning (FT). This FT paradigm has extended to vision (Dosovitskiy et al., 2020; Liu et al., 2021) and multi-modal domains (Radford et al., 2021; Li et al., 2022), leveraging the Transformer architecture (Vaswani et al., 2017). However, as models grow larger, fine-tuning the entire model becomes too costly for practical use.
+
+To address this challenge, various Parameter-Efficient Fine-Tuning (PEFT) methods (Houlsby et al., 2019) have been developed. Adapter-based methods (Hu et al., 2023; He et al., 2021) insert small trainable modules into Transformer layers. Prompt-based approaches (Lester et al., 2021; Wang et al., 2023) prepend learnable vectors to input or hidden states. However, these methods often introduce non-negligible inference overhead. Partial  $FT$  (Zaken et al., 2021; Xu et al., 2021) selectively updates a subset of existing model parameters, but they still suffer from suboptimal performance compared to full FT. To address these limitations, Low-Rank Adaptation (LoRA) (Hu et al., 2021) offers an alternative by reparameterizing incremental updates of pre-trained weights using low-rank decomposition. For a pre-trained weight matrix  $W_0 \in \mathbb{R}^{p \times q}$  in an attention layer or a feed-forward layer, LoRA reparameterizes fine-tuned weights as  $W' = W_0 + \Delta W = W_0 + BA$ , where  $B \in \mathbb{R}^{p \times r}$ ,  $A \in \mathbb{R}^{r \times q}$ , and  $r \ll \min(p, q)$ . During FT, only  $A$  and  $B$  are updated. This allows LoRA to significantly reduce the number of trainable parameters while still achieving impressive performance.
+
+The success of LoRA has inspired a series of subsequent work. These LoRA variants typically aim to better utilize the parameter budget (Zhang et al., 2023b; Valipour et al., 2022; Kopiczko et al., 2023), improve computational efficiency (Dettmers et al., 2024; Zhang et al., 2023a; Hedegaard et al., 2024), enable diverse learning patterns (Liu et al., 2024), or achieve a higher rank (Hyeon-Woo et al., 2021; Edalati et al., 2022; Hao et al., 2024). However, they still reparameterize weight update with the low-rank decomposition form, which may limit the hypothesis space and prevent further parameter reduction. To address this issue, FourierFT (Gao et al., 2024) proposes to reparameterize  $\Delta W$  with a randomly selected set of frequency-domain components by inverse Discrete Fourier Transform (iDFT). This implicitly allows for enhanced expressivity and flexible parameter budget.
+
+While FourierFT has shown empirical success, its advantages over low-rank methods have not been theoretical analyzed. To fill this gap, we aim to provide a comprehensive understanding of frequency-domain PEFT. We begin with a systematic analysis of weight updates during FT, and identify the asymptotic normality of weight incremental matrices through both empirical observations and theoretical justification. This foundation enables a rigorous mathematical comparison of the expressivity between frequency-domain and low-rank methods. Interestingly, our analysis reveals that iDFT-based methods with randomly selected locations of learnable frequency components exhibit lower expressivity than low-rank methods. In response, we design iDFT-based variants with carefully selected components, which consequently surpass the expressivity of low-rank-based methods. We further demonstrate that the best choice of iDFT-based variants can be equivalently and more efficiently implemented using inverse Discrete Cosine Transform (iDCT).
+
+Building on these insights, we introduce Location-aware Cosine Adaptation (LoCA), an iDCT-based PEFT method that optimizes both the coefficients and locations of frequency components. By employing finite-difference approximation to estimate gradients for discrete location variables, LoCA dynamically selects the most informative frequency components for each weight update matrix. We demonstrate that LoCA offers enhanced parameter efficiency while maintaining computational feasibility comparable to low-rank methods. Experiments across various language and vision tasks show that LoCA matches state-of-the-art PEFT performance using significantly fewer parameters.
+
+# 2 PRELIMINARY ANALYSIS OF FINE-TUNING MODERN LLMS
+
+Modern LLMs are predominantly built upon the Transformer architecture (Vaswani et al., 2017), where each Transformer block has a multi-head self-attention (MHSA) and a feed-forward network (FFN). For input  $x \in \mathbb{R}^{n \times d}$ , MHSA projects  $x$  into query, key, and value matrices per head  $h$  using  $W_{q}^{h}, W_{k}^{h}, W_{v}^{h} \in \mathbb{R}^{d \times d / H}$ , where  $H$  is the number of heads. The FFN then processes the attention output using  $W_{f1} \in \mathbb{R}^{d \times d_m}$  and  $W_{f2} \in \mathbb{R}^{d_m \times d}$ , where  $d_m$  is the hidden dimension.
+
+To systematically analyze the behavior of fine-tuning LLMs, we fine-tune a pretrained LLaMA-7b model (Touvron et al., 2023a) on the Alpaca-52K dataset (Taori et al., 2023). For each fine-tuned weight matrix  $W' \in \mathbb{R}^{p \times q}$  ( $p \geq q$ ), we get the incremental matrix  $\Delta W = W' - W_0$  and examine its properties from various perspectives. Our empirical observations reveal that the weights in each  $\Delta W$  closely approximate a Gaussian distribution (Fig. 1a). We claim that this normality can be theoretically justified. Consider a pre-trained model  $f$  with a pre-trained weight matrix  $W_0$ . Assume the fine-tuning dataset is sampled from  $P(X, Y; \overline{W})$ , where  $\overline{W}$  can be considered as the distribution parameter as well as the oracle solution of fine-tuning,  $X$  and  $Y$  denote the input data and corresponding labels, respectively. During the FT process, we obtain the parameter  $W'$  by minimizing the empirical loss. Consequently,  $W'$  can be regarded as an M-estimator of  $\overline{W}$ , which satisfies  $\mathbb{P}_n \psi(W') \stackrel{\text{def.}}{=} \mathbb{P}_n \nabla \ell[Y - f(X; W')]^2 = 0$ , where  $\mathbb{P}_n$  is the empirical average over  $n$  samples drawn from  $P(X, Y; \overline{W})$ ,  $\psi$  is the score function, and  $\ell$  is an objective function. Under fairly general conditions,  $W' - \overline{W}$  is known to be asymptotically normal (Yohai & Maronna, 1979):  $\sqrt{n} (W' - \overline{W})^V \stackrel{d}{\to} \mathcal{N}_{pq}(0, \Sigma_{\overline{W}})$ , where  $^V$  denotes vectorization. We further assert that, under some mild assumptions, the incremental matrix  $\Delta W$  also exhibits asymptotic normality.
+
+Proposition 1. Let  $W_0 \in \mathbb{R}^{K \times K}$  and  $W' \in \mathbb{R}^{K \times K}$  be the pre-trained weight matrix and fine-tuned weight trained on datasets with  $N$  and  $n'$  data samples, respectively. Assume that (A1) The pre-training dataset follows  $P(X, Y; \overline{W}_0)$ . For real-world fine-tuning datasets, the vectorized  $\overline{W}^V$  follows a prior distribution  $\mathcal{N}_{K^2}(\overline{W}_0^V, \overline{\sigma}^2 I_{K^2})$ , where  $\overline{\sigma}$  is a constant. (A2) For any given  $\overline{W}$ , let  $W'$
+
+![](images/9795264fa77a7ab2f4a4c49b04dc26097412f835514d81e87514b15bf5792963.jpg)  
+(a) Empirical Distribution of  $\Delta W$
+
+![](images/9ce04aa38ebfde6902d235270e2c59a4e3bfac2873f57012edeac02328514930.jpg)  
+(b) Hypothesis Testing  
+Figure 1: Analysis of the weight incremental matrices. (a) Empirical distribution of the incremental query  $(\Delta W_{q})$  and value  $(\Delta W_{v})$  projection matrices for a representative middle layer. (b) p-values of the hypothesis test for  $\Delta W_{q}$  and  $\Delta W_{v}$  across different layers. (c) Empirical spectral density (ESD) of  $\Delta W_{q}$  and  $\Delta W_{v}$  for layer 4. Same phenomena are observed in other weight matrices.
+
+![](images/3a9d0ae0c9b65a27553a13809d1049d1a30b1409eed96dcff5d54dfcbc99436a.jpg)  
+(c) Empirical Spectral Density
+
+be an  $M$ -estimator that satisfies asymptotic normality. The elements on  $W' - \overline{W}$  are asymptotically independent and identically distributed, and the estimation error  $W' - \overline{W}$  is independent of  $\overline{W}$ . Under these assumptions, there exists  $\sigma_0 > 0$ , the weight update matrix  $\Delta W = W' - W_0$  satisfies:
+
+$$
+\Delta W ^ {V} \sim \mathcal {N} _ {K ^ {2}} \left(0, \left(\frac {\sigma_ {0} ^ {2}}{n ^ {\prime}} + \bar {\sigma} ^ {2}\right) I _ {K ^ {2}}\right) + o _ {P} \left(\frac {1}{\sqrt {n ^ {\prime}}}\right) + O _ {P} \left(\frac {1}{\sqrt {N}}\right).
+$$
+
+We justify the reasonability of these assumptions in Appendix A. For ease of representation, we use square matrices for theoretical analysis without loss of generality. Proposition 1 shows that during FT, the weight update follows an isotropic Gaussian, plus two error terms. In practice, the second term can be assumed to be zero due to the vast amount of pre-training data. However, the last term, which is related to the size of the FT dataset, causes the final distribution to deviate slightly from a Gaussian distribution. To examine the impact of this error term, we design a hypothesis test, where the null hypothesis posits that the total variation (TV) between the distribution of parameters  $w \in \Delta W$  and the normal distribution is less than a constant  $\varepsilon$ , i.e.,  $H_0: d_{TV}(P(w), \mathcal{N}(w; \hat{\mu}, \hat{\sigma}^2)) \leq \epsilon$ , where  $d_{TV}(\cdot, \cdot)$  denotes the total variation,  $P(w)$  is the true distribution of  $w$ ,  $\hat{\mu}$  and  $\hat{\sigma}$  are the empirical mean and standard deviation of  $w$  respectively. We use the TV between the empirical distribution of  $w$  and  $\mathcal{N}(w; \hat{\mu}, \hat{\sigma}^2)$  as the test statistic and employ a bootstrap-like method to estimate its distribution (the details are described in Appendix B). Fig. 1b illustrates the results for  $\Delta W_q$  and  $\Delta W_v$  across different layers. We choose  $\epsilon = 0.001$  and significance level 0.05 for this test. The large p-values across all tests in Fig. 1b mean that the null hypothesis  $H_0$  cannot be rejected, i.e., the parameter updates indeed asymptotically follow a Gaussian distribution.
+
+Another observation from Proposition 1 is that the parameters in  $\Delta W$  are asymptotically i.i.d. To examine this, we analyze the empirical spectral density (ESD) of each  $\Delta W$ , which is defined as the probability density of the eigenvalues  $\{\lambda_i\}_{i=1}^q$  of the correlation matrix  $\Delta C = \frac{1}{p}\Delta W^T\Delta W \in \mathbb{R}^{q \times q}$ . ESD is extensively studied in random matrix theory and helps understand the asymptotic behavior of the eigenvalues of large random matrices with i.i.d. elements. According to the Marchenko-Pastur (MP) law (Yang et al., 2012), as  $p, q \to \infty$  with a fixed aspect ratio  $Q = p/q$ , the ESD for a random matrix converges to the MP distribution determined by the element-wise variance  $\sigma_{mp}^2$ . The agreement between the ESD and the MP distribution in Fig. 1c suggests that  $\Delta W$  behaves like an i.i.d. random matrix. This property will help us to better analyze various PEFT methods.
+
+# 3 COMPARISON BETWEEN FREQUENCY-SPACE AND LOW-RANK ADAPTATION
+
+Given the asymptotic Gaussian nature of  $\Delta W$ , we can now analytically compare the expressivities of low-rank-based and frequency-space-based adaptation methods. We regard expressivity as the ability to approximate a fully fine-tuned weight incremental matrix using the same parameter budget.
+
+Given any  $\Delta W\in \mathbb{R}^{p\times q}$  obtained through full fine-tuning, low-rank-based methods approximate it as  $\hat{W}_R = BA$  with  $N_0 = (p + q)r$  parameters, where  $r$  is the chosen rank. In contrast, FourierFT (Gao et al., 2024) adopts a frequency-domain approach by randomly selecting  $N_{1}$  components on the Fourier spectrum  $F = \mathcal{F}(\Delta W)$  to learn, setting others to zero, and approximates  $\Delta W$  as
+
+$\hat{W}_F^{(1)} = \mathcal{F}^{-1}(\hat{F}^{(1)})$ , where  $\mathcal{F}, \mathcal{F}^{-1}$  denote the FFT and inverse FFT respectively, and  $\hat{F}^{(1)} \in \mathbb{C}^{p \times q}$  is the learned spectrum, which has non-zero values at randomly selected locations  $Id^{(1)} = \{id_i^{(1)} = (x_i^{(1)}, y_i^{(1)})\}_{i=1}^{N_1}$ . However, FourierFT only considers learning the real part on  $\hat{F}^{(1)}$ , and simply discards the imaginary part after the inverse FFT. Besides, it fails to exploit the conjugate symmetry property inherent in the Fourier spectra for real-valued matrices. We argue that this could lead to information loss and inefficient utilization of the parameter budget. To address these concerns, we consider a more comprehensive approach that leverages both the real and imaginary parts of the Fourier spectrum while exploiting the conjugate symmetry property. Specifically, we select learnable locations only on the non-redundant half (i.e., the left half) of  $F$ , and learn both real and imaginary coefficients at these locations. We still denote the result of the improved version as  $\hat{W}_F^{(1)}$ .
+
+Intuitively, when approximating a matrix through low-rank decomposition, the learned low-rank matrices are effectively the left and right singular matrices corresponding to the largest  $r$  singular values of  $\Delta W$ . However, for frequency-domain methods, this order statistic is not inherently involved. To incorporate this information, we consider an oracle variant that selects  $N_{2}$  locations in the nonredundant half of  $\mathcal{F}(\Delta W)$  with the largest amplitude values (the search space is  $\Omega_1 = [p] \times [q/2]$ ), and sets other locations to 0. We denote the resulting sparse Fourier spectrum with optimal locations as  $\hat{F}^{(2)}$ , yielding  $\hat{W}_F^{(2)} = \mathcal{F}^{-1}(\hat{F}^{(2)})$ . Furthermore, we explore an additional variant leveraging the fact that each location in the Fourier spectrum has a real and an imaginary coefficient, which need not be bound together for selection. We propose selecting  $N_{3}$  learnable coefficients individually with a search space  $\Omega_2 = [p] \times [q/2] \times [2]$ . In this case, the optimal strategy is to choose the top  $N_{3}$  coefficients with the largest absolute values in the non-redundant half of  $\mathcal{F}(\Delta W)$  for learning. Denoting the spectrum with these optimal coefficients as  $\hat{F}^{(3)}$ , we obtain  $\hat{W}_F^{(3)} = \mathcal{F}^{-1}(\hat{F}^{(3)})$ . We show that, given the asymptotic Gaussian nature of  $\Delta W$ , we can mathematically compare these PEFT methods. In our theoretical analysis, we account for location indexing within the parameter budget. For a fair comparison with rank  $r$  decomposition, we set  $N_1 = N_3 = 1/2N_0$  and  $N_2 = 2/3N_0^{-1}$ .
+
+Theorem 1. Let  $W \in \mathbb{R}^{K \times K} \sim G$  be a weight matrix where each element independently follows a standard normal distribution  $\mathcal{N}(0,1)$ . Define the reconstruction error  $L(W, \hat{\mathcal{W}}) = ||W - \hat{\mathcal{W}}||_F^2$ , where  $\hat{\mathcal{W}}$  can be  $\hat{W}_R, \hat{W}_F^{(1)}, \hat{W}_F^{(2)}$ , or  $\hat{W}_F^{(3)}$  stated above. Then, for  $r < K/3$ , we have
+
+$$
+\mathbb {E} _ {I d ^ {(1)}} \mathbb {E} _ {W \sim G} [ L (W, \hat {W} _ {F} ^ {(1)}) ] > \mathbb {E} _ {W \sim G} [ L (W, \hat {W} _ {R}) ] > \mathbb {E} _ {W \sim G} [ L (W, \hat {W} _ {F} ^ {(2)}) ] > \mathbb {E} _ {W \sim G} [ L (W, \hat {W} _ {F} ^ {(3)}) ].
+$$
+
+Note that we use  $\mathcal{N}(0,1)$  in Theorem 1 without loss of generality, as any matrix can be rescaled to have zero mean and unit variance. Importantly, Theorem 1 shows that randomly selecting learnable coefficients in the frequency domain, i.e.,  $\hat{W}_F^{(1)}$ , has worse expressivity than all other method, highlighting the importance of strategic selection of frequency components. On the other hand, the superior performance of  $\hat{W}_F^{(3)}$ , which allows for individual selection of (real or imaginary) coefficients, indicates that this increased flexibility in frequency component selection can lead to better expressivity. These findings have significant implications for the design of PEFT methods.
+
+# 4 LOCATION-AWARE COSINE ADAPTATION
+
+# 4.1 PROBLEM FORMULATION
+
+In this work, we regard the goal of PEFT as effectively reparameterizing a weight incremental matrix. Building on our previous analysis, we aim to propose a frequency-domain PEFT method that considers both the coefficients and locations of frequency components. Formally, given a pre-trained weight matrix  $W_0 \in \mathbb{R}^{p \times q}$ , our objective is to fine-tune it on a specific dataset to obtain the finetuned weight matrix  $W' = W_0 + \Delta W = W_0 + \alpha \mathcal{F}^{-1}(\mathcal{S}(\boldsymbol{a},\boldsymbol{l},\boldsymbol{k}))$ , where  $\alpha$  is a scaling coefficient,  $\boldsymbol{a} = \{a_i\}_{i=1}^{\mathcal{B}}$  represents the learnable coefficients,  $l = \{(l_i^1,l_i^2)\}_{i=1}^{\mathcal{B}}$  stores the component locations,  $\boldsymbol{k} = \{0,1\}^{\mathcal{B}}$  indicates real (1) or imaginary (0) coefficients,  $\mathcal{B}$  is the component budget, and  $S(\cdot)$  is an operator that scatters  $\boldsymbol{a}$  onto a zero matrix according to  $l$  and  $\boldsymbol{k}$ .
+
+However, its practical implementation presents significant challenges, primarily due to the requirement for extensive discrete optimization of  $l$  and  $k$ . This motivates our exploration of alternative formulations that balance the benefits of frequency-space adaptation with computational feasibility.
+
+# 4.2 INVERSE DISCRETE COSINE TRANSFORM-BASED REPARAMETERIZATION
+
+Individually selecting learnable coefficients requires deciding whether to learn the real or imaginary part on each location in  $l$ , which involves extensive discrete optimization of  $k$  in practical implementation. To address this issue, we introduce the discrete cosine transform (DCT). We prove that in this problem, individually selecting learnable coefficients on the Fourier spectrum is equivalent to selecting locations on the DCT spectrum, which involves only real-valued coefficients.
+
+Theorem 2. Let  $W \in \mathbb{R}^{K \times K} \sim G$  be a weight matrix where each element independently follows a standard normal distribution  $\mathcal{N}(0,1)$ . Let  $\mathcal{D}(\cdot)$  and  $\mathcal{D}^{-1}(\cdot)$  denote the discrete cosine transform (DCT) and inverse DCT, respectively, and  $\mathcal{F}(\cdot)$  denote the discrete Fourier transform. Define  $F_{D}$  as the sparse matrix that preserves the  $N_{D}$  coefficients with the largest absolute values on  $\mathcal{D}(W)$  and sets others to 0. With  $\hat{W}_{D} = \mathcal{D}^{-1}(F_{D})$ , and  $L(\cdot,\cdot), N_{3}, \hat{W}_{F}^{(3)}$  stated above, if  $N_{D} = N_{3}$ , then:
+
+$$
+E _ {W \sim G} [ L (W, \hat {W} _ {F} ^ {(3)}) ] = E _ {W \sim G} [ L (W, \hat {W} _ {D}) ].
+$$
+
+Theorem 2 guides us towards a more efficient alternative by utilizing the iDCT instead of the iDFT. By reparameterizing  $\Delta W$  using iDCT, We can maintain the equivalent expressivity while avoiding the optimization of  $k$ . This is because DCT operates in the real domain, which simplifies computations and reduces the complexity of parameter selection. It is known that iDCT is essentially a linear transformation (Ahmed et al., 1974). We can express the reparameterization based on 2D iDCT by
+
+$$
+W ^ {\prime} = W _ {0} + \Delta W = W _ {0} + \alpha \left[ C ^ {T} \mathcal {S} (\boldsymbol {a}, \boldsymbol {l}, \boldsymbol {1}) D \right], \tag {1}
+$$
+
+where  $C \in \mathbb{R}^{p \times p}$ ,  $D \in \mathbb{R}^{q \times q}$  are the DCT matrices. The elements of  $C$  are defined as:
+
+$$
+C _ {i j} = \sqrt {\frac {2}{p}} \cdot k _ {i} \cdot \cos \left(\frac {\pi (2 j + 1) i}{2 p}\right), \text {w h e r e} k _ {i} = \left\{ \begin{array}{l l} \frac {1}{\sqrt {2}}, & \text {i f} i = 0 \\ 1, & \text {i f} i > 0. \end{array} \right. \tag {2}
+$$
+
+The formulation is similar for  $D$ . In practice, when  $\mathcal{S}(\pmb{a},\pmb{l},\pmb{1})$  is highly sparse, we can further simplify the computation by  $\Delta W = \alpha [C^T S(\pmb{a},\pmb{l},\pmb{1})D] = \alpha \sum_{i=1}^{\mathcal{B}} a_i C_{l_i^1}^T D_{l_i^2}$ , where  $C_{l_i^1}$  is the  $l_i^1$ -th row of  $C$ , and  $D_{l_i^2}$  is the  $l_i^2$ -th row of  $D$ . This simplification reduces the computation complexity of iDCT from  $O(p^2 q^2)$  to  $O(\mathcal{B}pq)$ . In contrast, when more frequency components are needed, it is recommended to use the fast DCT algorithm with an asymptotic complexity of  $O(log(pq)pq)$ . A detailed discussion of computation complexity can be found in Appendix J. Noting that we can pre-generate  $C$  and  $D$  with only one global copy, which does not consume additional memory usage.
+
+# 4.3 ESTIMATING LOCATION GRADIENT USING FINITE-DIFFERENCE APPROXIMATION
+
+While the coefficients  $\pmb{a}$  can be directly optimized through backpropagation, the operation  $S(\cdot)$  does not produce gradients with respect to the locations  $l$ . Furthermore,  $l$  needs to be treated as a discrete variable, which prevents us from directly learning the locations through backpropagation.
+
+To address this issue, we draw inspiration from the straight-through estimator (STE) (Bengio et al., 2013), a technique that allows gradient-based optimization of neural networks with discrete variables by using a surrogate gradient. However, unlike traditional STE that simply bypasses the gradient computation for discrete variables, e.g., the STE used in VQ-VAE (Van Den Oord et al., 2017), we estimate their gradients using the central difference approximation, as we elaborate below.
+
+Forward Pass. To enable gradient-based learning of location variables, we first redefine the locations  $\pmb{l}$  as continuous variables. During the forward pass, we discretize  $\pmb{l}$  by  $\hat{l} = \text{round}(\pmb{l}) = \{ (\hat{l}_i^1, \hat{l}_i^2) \}_{i=1}^{\mathcal{B}}$ , where  $\text{round}(\cdot)$  maps each element of  $\pmb{l}$  to its nearest integer.
+
+Backward Pass. During the backward propagation, we estimate the gradient of the loss function  $\mathcal{L}$  to each element in  $l$ . For clarity, we take  $l_n^1$  and  $a_{n}$  as an example. The location gradient is
+
+$$
+\frac {\partial \mathcal {L}}{\partial l _ {n} ^ {1}} = \sum_ {i = 1} ^ {p} \sum_ {j = 1} ^ {q} \frac {\partial \mathcal {L}}{\partial \Delta W _ {i j}} \frac {\partial \Delta W _ {i j}}{\partial l _ {n} ^ {1}} = t r \left[ \left(\frac {\partial \mathcal {L}}{\partial \Delta W}\right) ^ {T} \left(\frac {\partial \Delta W}{\partial l _ {n} ^ {1}}\right) \right]. \tag {3}
+$$
+
+Here,  $\partial \mathcal{L} / \partial \Delta W$  can be obtained directly through backpropagation. The tricky part is how to estimate  $\partial \Delta W / \partial l_n^1$ . In this work, we choose to use central difference approximation, i.e.,
+
+$$
+\frac {\partial \Delta W}{\partial l _ {n} ^ {1}} = \frac {\alpha C ^ {T} [ \mathcal {S} (a _ {n} , (\hat {l _ {n} ^ {1}} + 1 , \hat {l _ {n} ^ {2}}) , 1) - \mathcal {S} (a _ {n} , (\hat {l _ {n} ^ {1}} - 1 , \hat {l _ {n} ^ {2}}) , 1) ] D}{2}. \tag {4}
+$$
+
+For simplicity, we denote  $S(a_{n},(\hat{l}_{n}^{1} + 1,\hat{l}_{n}^{2}),1) - S(a_{n},(\hat{l}_{n}^{1} - 1,\hat{l}_{n}^{2}),1)$  as  $\Delta S$ , then Eq. (3) becomes
+
+$$
+\frac {\partial \mathcal {L}}{\partial l _ {n} ^ {1}} = \frac {\alpha}{2} t r \left[ \left(\frac {\partial \mathcal {L}}{\partial \Delta W}\right) ^ {T} C ^ {T} \Delta \mathcal {S} D \right] = \frac {\alpha}{2} t r \underbrace {\left[ D \left(\frac {\partial \mathcal {L}}{\partial \Delta W}\right) ^ {T} C ^ {T} \right.} _ {D C T} \Delta \mathcal {S} ]. \tag {5}
+$$
+
+Eq. (5) demonstrates that the gradient estimate for  $l_{n}^{1}$  can be obtained by first applying a DCT to  $(\partial \mathcal{L} / \partial \Delta W)^T$  (we denote the resulting matrix as  $Z$ ), and then multiplying it with  $\Delta S$ . Note that  $\Delta S$  is a matrix with non-zero elements only at locations  $(l_{n}^{\hat{1}} - 1, l_{n}^{\hat{2}})$  and  $(l_{n}^{\hat{1}} + 1, l_{n}^{\hat{2}})$ . Therefore, the result of Eq. (5) can be simplified as  $\alpha a_{n}(Z_{l_{n}^{\hat{2}}, l_{n}^{\hat{1}} + 1} - Z_{l_{n}^{\hat{2}}, l_{n}^{\hat{1}} - 1}) / 2$ . Since  $Z$  can be reused for computing gradients for all locations  $l$  and coefficients  $\pmb{a}$  (the gradient to  $\pmb{a}$  can also be obtained from  $Z$ ), Eq. (5) introduces almost no additional computational burden (see Appendix I).
+
+# 4.4 ALTERNATING OPTIMIZATION STRATEGY
+
+To effectively optimize both the coefficients  $\pmb{a}$  and locations  $l$ , we implement an alternating optimization scheme inspired by coordinate ascent methods (Wright, 2015), which have shown remarkable efficacy in tackling multi-variable optimization problems. Specifically, we initially train the coefficients  $\pmb{a}$  for  $B_{a}$  steps while maintaining fixed locations  $l$ . Subsequently, we fix  $\pmb{a}$  and optimize the locations  $l$  for  $B_{l}$  steps. This alternating process continues for totally  $\mathcal{B}_s$  iterations. After that, we only optimize the coefficients  $\pmb{a}$  until convergence. This strategy facilitates an efficient exploration of the frequency domain while progressively refining the selected components in the early training state, while focusing on the coefficients of the identified important frequency components in the remaining stage. A detailed training procedure can be found in Appendix E.
+
+# 5 EXPERIMENTS
+
+We mainly evaluate LoCA across four domains: natural language understanding (NLU), natural language generation (NLG), instruction tuning, and computer vision. For NLU tasks, we fine-tune RoBERTa models on the GLUE benchmark (Wang et al., 2018). For NLG, we fine-tune GPT-2 (medium/large) on E2E NLG Challenge. For instruction tuning, we fine-tune LLaMA-family models on the Alpaca-52K dataset (Taori et al., 2023) and evaluate them on the MT-Bench (Zheng et al., 2024) and Vicuna (Chiang et al., 2023) datasets. For vision tasks, we fine-tune Vision Transformer (ViT) models on 8 classification datasets. More experiments can be found in Appendix.
+
+Implementation Details. We implement our method using the PyTorch framework. Our code is built on the PEFT library (Mangrulkar et al., 2022) from Huggingface, and all pre-trained models are sourced from Huggingface's Transformers library (Wolf et al., 2020). For the alternating optimization, we used  $\mathcal{B}_a = 10$  and  $\mathcal{B}_l = 20$ . The coefficients  $a$  are initialized to be zeros and the locations  $l$  are randomly initialized with a uniform distribution. We scale  $l$  to the range [0, 1] for optimization. All PEFT experiments are conducted on a single NVIDIA Tesla H100 GPU. Noting that while LoCA initially optimizes both  $a$  and  $l$ , the locations are fixed after  $\mathcal{B}_s$  iterations. Therefore, the reported number of trainable parameters only includes the final coefficient parameters.
+
+Baseline Methods. We compare our LoCA with Full fine-tuning (FF), BitFit (Zaken et al., 2021), Adapter-based methods (Houlsby et al., 2019), LoRA (Hu et al., 2021), AdaLoRA (Zhang et al., 2023b), VeRA (Kopiczko et al., 2023), DoRA (Liu et al., 2024) and FourierFT (Gao et al., 2024).
+
+# 5.1 NATURAL LANGUAGE UNDERSTANDING
+
+We evaluate our method on NLU tasks using the GLUE benchmark (Wang et al., 2018), which consists of diverse tasks that cover various aspects of language understanding, including single-sentence classification, similarity and paraphrase, and inference task. For our experiments, we fine-tune RoBERTa-base and RoBERTa-large models (Liu et al., 2019) on 8 GLUE tasks using different adaptation methods. Following Zhang et al. (2023b); Gao et al. (2024), we report the best results on the validation set for each task. Mean results are reported after 3 runs with different random seeds.
+
+Implementation Details. For LoRA and its variants, we use a rank  $r = 8$  and a scaling value  $\alpha = 8$ . To maintain consistency with FourierFT, we set the number of frequency components  $\mathcal{B}$
+
+Table 1: Fine-tuning results with RoBERTa-base/large on the GLUE benchmark. We report the overall accuracy (matched and mismatched) for MNLI, Matthew's correlation coefficient (MCC) for CoLA and use the Pearson correlation coefficient (PCC) for STS-B. Accuracy (Acc.) is reported for all other tasks.  $\dagger$ ,  $\ddagger$ , * denote values from prior works. Best results are shown in bold.  
+
+<table><tr><td>Model</td><td>FT Method</td><td>Param.</td><td>CoLA MCC</td><td>MNLI Acc</td><td>MRPC Acc</td><td>QNLI Acc</td><td>QQP Acc</td><td>RTE Acc</td><td>SST-2 Acc</td><td>STS-B PCC</td><td>All Avg.</td></tr><tr><td rowspan="9">RoBERTa-base</td><td>FT‡</td><td>125M</td><td>63.6</td><td>87.6</td><td>90.2</td><td>92.8</td><td>91.9</td><td>78.7</td><td>94.8</td><td>91.2</td><td>86.4</td></tr><tr><td>BitFit‡</td><td>0.1M</td><td>62.0</td><td>84.7</td><td>92.7</td><td>91.8</td><td>84.0</td><td>81.5</td><td>93.7</td><td>90.8</td><td>85.2</td></tr><tr><td>AdapterD‡</td><td>0.9M</td><td>62.6</td><td>87.3</td><td>88.4</td><td>93.0</td><td>90.6</td><td>75.9</td><td>94.7</td><td>90.3</td><td>85.4</td></tr><tr><td>LoRA</td><td>0.3M</td><td>62.8</td><td>86.6</td><td>89.7</td><td>93.3</td><td>90.8</td><td>79.3</td><td>94.9</td><td>91.4</td><td>86.1</td></tr><tr><td>AdaLoRA</td><td>0.3M</td><td>63.0</td><td>86.8</td><td>90.2</td><td>93.4</td><td>90.9</td><td>80.4</td><td>94.6</td><td>90.9</td><td>86.3</td></tr><tr><td>DoRA</td><td>0.31M</td><td>63.5</td><td>87.0</td><td>90.2</td><td>93.1</td><td>91.4</td><td>78.6</td><td>95.2</td><td>91.5</td><td>86.3</td></tr><tr><td>VeRA†</td><td>0.043M</td><td>65.6</td><td>85.1</td><td>89.5</td><td>91.8</td><td>89.6</td><td>78.7</td><td>94.6</td><td>90.7</td><td>85.7</td></tr><tr><td>FourierFT*</td><td>0.024M</td><td>63.8</td><td>84.9</td><td>90.0</td><td>92.2</td><td>88.2</td><td>79.1</td><td>94.2</td><td>90.8</td><td>85.4</td></tr><tr><td>LoCA</td><td>0.024M</td><td>64.5</td><td>85.2</td><td>90.5</td><td>92.0</td><td>88.7</td><td>81.5</td><td>94.6</td><td>90.9</td><td>86.0</td></tr><tr><td rowspan="8">RoBERTa-large</td><td>FT‡</td><td>355M</td><td>68.0</td><td>90.2</td><td>90.9</td><td>94.7</td><td>92.2</td><td>86.6</td><td>96.4</td><td>92.4</td><td>88.9</td></tr><tr><td>AdapterH‡</td><td>6M</td><td>66.5</td><td>89.9</td><td>88.7</td><td>94.7</td><td>92.1</td><td>83.4</td><td>96.2</td><td>91.0</td><td>87.8</td></tr><tr><td>LoRA</td><td>0.8M</td><td>68.4</td><td>90.5</td><td>90.2</td><td>94.4</td><td>91.6</td><td>85.7</td><td>96.2</td><td>92.4</td><td>88.7</td></tr><tr><td>AdaLoRA</td><td>0.8M</td><td>67.9</td><td>90.6</td><td>90.6</td><td>94.2</td><td>91.6</td><td>86.4</td><td>95.9</td><td>92.7</td><td>88.7</td></tr><tr><td>DoRA</td><td>0.83M</td><td>68.3</td><td>90.5</td><td>90.7</td><td>94.8</td><td>91.8</td><td>85.4</td><td>96.3</td><td>92.4</td><td>88.8</td></tr><tr><td>VeRA†</td><td>0.061M</td><td>68.0</td><td>90.2</td><td>90.9</td><td>94.4</td><td>90.3</td><td>85.9</td><td>96.1</td><td>91.7</td><td>88.4</td></tr><tr><td>FourierFT*</td><td>0.048M</td><td>67.1</td><td>88.9</td><td>90.9</td><td>94.4</td><td>89.2</td><td>87.4</td><td>96.0</td><td>91.9</td><td>88.2</td></tr><tr><td>LoCA</td><td>0.048M</td><td>68.8</td><td>89.4</td><td>91.0</td><td>94.4</td><td>90.0</td><td>87.9</td><td>96.4</td><td>92.0</td><td>88.7</td></tr></table>
+
+to 1000 for both frequency-domain methods, resulting in significantly less parameters compared to low-rank decomposition methods. Since FourierFT does not report results for the MNLI and QQP tasks, we obtained these results by our own runs with tuned hyperparameters. Following the settings in Hu et al. (2021); Gao et al. (2024), all low-rank decomposition methods and frequency-domain decomposition methods are applied only to the query and value matrices, and the best performance on the validation set for each run is recorded. Detailed hyperparameters can be found in Table 6.
+
+Experimental Results. Table 1 presents the results for RoBERTa-base and RoBERTa-large models. Our LoCA achieves competitive average scores of 86.0 and 88.7 respectively, approaching cutting-edge performance while using significantly fewer parameters. LoCA consistently outperforms FourierFT across most tasks despite the same parameter budget, and shows comparable or superior results to LoRA-family methods on several tasks. Notably, LoCA achieves the highest scores on CoLA for both model sizes, surpassing even FF. For challenging tasks (e.g., QQP), we will show in Section 5.5 that if we appropriately increase the parameter budget, the performance of LoCA will improve significantly, which eventually surpasses LoRA with the same parameter budget.
+
+# 5.2 NATURAL LANGUAGE GENERATION
+
+We evaluate LoCA on the E2E NLG Challenge dataset (Novikova et al., 2017), a widely-used benchmark for data-to-text generation. The dataset consists of over 50K samples in the restaurant domain, with each input being a set of slot-value pairs and the corresponding output being a natural language description. We conduct experiments on both GPT-2 medium and GPT-2 large.
+
+Implementation Details. Following Hu et al. (2021), we train our models using AdamW optimizer with a linear learning rate decay schedule for 5 epochs. We set the batch size to 32 and use a label
+
+smoothing factor of 0.1. We only adapt the query and value matrices, with 1000 frequency components for both LoCA and FourierFT. See Table 7 for more details.
+
+Experimental Results. Table 2 shows that LoCA achieves superior performance compared to previous PEFT methods including FourierFT and LoRA across multiple metrics. Specifically, when using GPT-2 large as the base model, LoCA outperforms others on BLEU, METEOR and ROUGE-L scores.
+
+Table 2: Results of tuning GPT-2 Medium/Large models on the E2E benchmark. Higher values indicate better performance for all metrics.  $\dagger ,\ddagger ,*$  denote values from prior works.  
+
+<table><tr><td>Model</td><td>FT Method</td><td>Param.</td><td>BLEU</td><td>NIST</td><td>METEOR</td><td>ROUGE-L</td><td>CIDEr</td></tr><tr><td rowspan="7">GPT-2 Medium</td><td>FF*</td><td>354.92M</td><td>68.2</td><td>8.62</td><td>46.2</td><td>71.0</td><td>2.47</td></tr><tr><td>AdptL*</td><td>11.09M</td><td>68.9</td><td>8.71</td><td>46.1</td><td>71.3</td><td>2.47</td></tr><tr><td>AdptH*</td><td>11.09M</td><td>67.3±6</td><td>8.5±07</td><td>46.0±2</td><td>70.7±2</td><td>2.44±01</td></tr><tr><td>LoRA ‡</td><td>0.35M</td><td>68.9±3</td><td>8.76±06</td><td>46.6±1</td><td>71.5±1</td><td>2.53±03</td></tr><tr><td>VeRA †</td><td>0.098M</td><td>70.1</td><td>8.81</td><td>46.6</td><td>71.5</td><td>2.50</td></tr><tr><td>FourierFT ‡</td><td>0.048M</td><td>69.1±1</td><td>8.82±05</td><td>47.0±3</td><td>71.8±1</td><td>2.51±02</td></tr><tr><td>LoCA</td><td>0.048M</td><td>69.7±2</td><td>8.85±04</td><td>46.6±2</td><td>72.1±3</td><td>2.52±06</td></tr><tr><td rowspan="6">GPT-2 Large</td><td>FF*</td><td>774.03M</td><td>68.5</td><td>8.78</td><td>46.0</td><td>69.9</td><td>2.45</td></tr><tr><td>AdptL*</td><td>23.00M</td><td>68.9±3</td><td>8.70±04</td><td>46.1±1</td><td>71.3±2</td><td>2.45±02</td></tr><tr><td>LoRA ‡</td><td>0.77M</td><td>70.1±3</td><td>8.83±02</td><td>46.8±2</td><td>72.0±3</td><td>2.47±02</td></tr><tr><td>VeRA †</td><td>0.17M</td><td>70.3</td><td>8.85</td><td>46.9</td><td>71.6</td><td>2.54</td></tr><tr><td>FourierFT ‡</td><td>0.072M</td><td>70.2±2</td><td>8.90±02</td><td>47.0±2</td><td>71.8±1</td><td>2.50±02</td></tr><tr><td>LoCA</td><td>0.072M</td><td>70.4±2</td><td>8.88±05</td><td>47.2±02</td><td>72.1±2</td><td>2.54±02</td></tr></table>
+
+# 5.3 INSTRUCTION TUNING
+
+We fine-tune various LLaMA-family models (Touvron et al., 2023a;b) using the Alpaca-52K dataset (Taori et al., 2023). The Alpaca-52K dataset, derived from the self-instruct technique, provides a diverse set of instruction-following examples. In this experiment, we mainly compare our method with FF, LoRA and FourierFT. After fine-tuning, we evaluate the model on the MT-Bench (Zheng et al., 2024) and Vicuna (Chiang et al., 2023) datasets, which offer challenging multi-turn and open-ended scenarios for LLM evaluation. We employed GPT-4 to assign scores on a scale of 1-10 based on the quality, relevance, and coherence of the responses.
+
+Implementation Details. We apply all PEFT LoRA, we set the rank  $r$  to 64 and the scaling value  $\alpha$  to 16. For FourierFT, we use 150K frequency components and tune other hyperparameters to ensure the optimal performance, since we cannot reproduce the results in Gao et al. (2024). For LoCA, we also use 150K frequency components, and set the scaling value  $\alpha$  to 1. We utilize the LLM-as-a-Judge repository (Zheng et al., 2024) for fair evaluation. We train LLaMA-1-7b/LLaMA-2-7b for 3 epochs and LLaMA-1-13b/LLaMA-2-13b for 1 epoch. Quantization (Dettmers et al., 2024) is used for LLaMA-1-13b/LLaMA-2-13b to ensure feasible FT on a single GPU. Detailed hyperparameters can be found in Table 8.
+
+Experimental Results. The results in Table 3 demonstrate the competitive performance of our method across various LLaMA model sizes and architectures. Notably, LoCA consistently outperforms FourierFT and, in many scenarios, either approaches or surpasses the performance
+
+of LoRA, despite the latter utilizing a larger parameter budget. This underscores the superior efficiency of LoCA in parameter utilization and its effectiveness in acquiring task-specific knowledge.
+
+methods to the query and value matrices. For
+
+Table 3: Evaluation results for fine-tuned LLaMA-family models on MT-Bench and Vicuna datasets, using GPT-4 as the judge with a 1-10 scoring scale. Bold and underlined values indicate the best and second best results, respectively.  
+
+<table><tr><td>Model</td><td>FT Method</td><td>Param.</td><td>MT-Bench</td><td>Vicuna</td></tr><tr><td rowspan="4">LLaMA1-7b</td><td>FF</td><td>6.8B</td><td>4.46</td><td>7.24</td></tr><tr><td>LoRA</td><td>33.5M</td><td>4.52</td><td>7.52</td></tr><tr><td>FourierFT</td><td>9.6M</td><td>4.33</td><td>6.97</td></tr><tr><td>LoCA</td><td>9.6M</td><td>4.47</td><td>7.18</td></tr><tr><td rowspan="4">LLaMA1-13b</td><td>FF</td><td>13B</td><td>4.78</td><td>7.68</td></tr><tr><td>LoRA</td><td>52.4M</td><td>4.87</td><td>7.82</td></tr><tr><td>FourierFT</td><td>12M</td><td>4.70</td><td>7.61</td></tr><tr><td>LoCA</td><td>12M</td><td>4.83</td><td>7.85</td></tr><tr><td rowspan="4">LLaMA2-7b</td><td>FF</td><td>6.8B</td><td>4.94</td><td>7.81</td></tr><tr><td>LoRA</td><td>33.5M</td><td>4.67</td><td>7.68</td></tr><tr><td>FourierFT</td><td>9.6M</td><td>4.65</td><td>7.62</td></tr><tr><td>LoCA</td><td>9.6M</td><td>4.82</td><td>7.78</td></tr><tr><td rowspan="4">LLaMA2-13b</td><td>FF</td><td>13B</td><td>5.55</td><td>8.13</td></tr><tr><td>LoRA</td><td>52.4M</td><td>5.48</td><td>8.03</td></tr><tr><td>FourierFT</td><td>12M</td><td>5.37</td><td>7.95</td></tr><tr><td>LoCA</td><td>12M</td><td>5.52</td><td>8.11</td></tr></table>
+
+# 5.4 IMAGE CLASSIFICATION
+
+We evaluate our method on computer vision tasks by conducting experiments on 8 image classification datasets, including OxfordPets (Parkhi et al., 2012), StanfordCars (Krause et al., 2013), CIFAR10 (Krizhevsky et al., 2009), DTD (Cimpoi et al., 2014), EuroSAT (Helber et al., 2019), FGVC (Maji et al., 2013), RESISC45 (Cheng et al., 2017) and CIFAR100 (Krizhevsky et al., 2009). We fine-tune ViT/16-base and ViT/16-large models (Dosovitskiy et al., 2020), both pre-trained on ImageNet-21k (Ridnik et al., 2021). In this experiment, we compares LoCA against several baselines: Linear Probing (LP), FF, LoRA, and FourierFT. Noting that we encountered significant discrepancies when attempting to reproduce the results reported in Gao et al. (2024), possibly due to the lack of detailed hyperparameter setup. To ensure a fair comparison, we re-run all methods using our own hyperparameter settings. All results are obtained after 5 random trials.
+
+Implementation Details. To ensure a fair comparison across all methods, the classification head is configured identically for all approaches. For LoRA, we a rank of 16 and a scaling factor  $\alpha$  of 16. Following Gao et al. (2024), FourierFT is implemented with 3000 and 10,000 frequency components and a scaling factor of 300. For our LoCA, we also evaluate 3000 and 10,000 frequency components for both base and large models. The learning rates for all methods are carefully tuned to ensure good performance across different tasks and model sizes. We report the number of trainable parameters excluding the classification head to provide a clear comparison of parameter efficiency. Detailed hyperparameter configurations for all methods can be found in Table 9.
+
+Experimental Results. The results are presented in Table 4. Notably, LoCA achieves superior performance compared to FourierFT while using the same number of parameters. For instance, with ViT-Base, LoCA using 72K parameters outperforms FourierFT on most datasets, with obvious
+
+Table 4: Fine-tuning results on 8 image classification datasets with ViT-base and ViT-large models. For fair comparison, we report the accuracy  $(\%)$  and standard deviation after 10 epochs of training for all methods. Best results are shown in bold.  
+
+<table><tr><td>Model</td><td>FT Method</td><td>Param.</td><td>OxfordPets</td><td>StanfordCars</td><td>CIFAR10</td><td>DTD</td><td>EuroSAT</td><td>FGVC</td><td>RESISC45</td><td>CIFAR100</td><td>Avg.</td></tr><tr><td rowspan="7">VIT-base</td><td>LP</td><td>-</td><td>92.94±0.12</td><td>47.02±0.23</td><td>96.82±0.01</td><td>76.47±0.22</td><td>94.78±0.02</td><td>29.21±1.33</td><td>86.13±0.10</td><td>86.05±0.08</td><td>76.18</td></tr><tr><td>FF</td><td>85.8M</td><td>93.09±0.11</td><td>84.71±0.03</td><td>98.89±0.00</td><td>77.37±0.30</td><td>98.91±0.09</td><td>63.83±1.13</td><td>95.72±0.21</td><td>90.72±0.23</td><td>87.91</td></tr><tr><td>LoRA</td><td>581K</td><td>93.26±0.28</td><td>82.12±0.22</td><td>98.51±0.07</td><td>79.54±0.72</td><td>98.65±0.06</td><td>55.67±1.24</td><td>94.82±0.45</td><td>91.51±0.12</td><td>86.76</td></tr><tr><td>FourierFT</td><td>72K</td><td>93.07±0.34</td><td>73.74±0.13</td><td>98.64±0.02</td><td>77.72±0.74</td><td>98.32±0.05</td><td>48.24±1.09</td><td>92.89±0.07</td><td>91.23±0.04</td><td>84.23</td></tr><tr><td>LoCA</td><td>72K</td><td>93.36±0.03</td><td>77.78±0.14</td><td>98.66±0.21</td><td>78.44±0.31</td><td>98.94±0.06</td><td>53.23±0.96</td><td>93.88±0.20</td><td>91.40±0.11</td><td>85.71</td></tr><tr><td>FourierFT</td><td>239K</td><td>93.44±0.31</td><td>79.34±0.14</td><td>98.70±0.08</td><td>79.43±1.15</td><td>98.81±0.05</td><td>52.26±1.50</td><td>94.19±0.06</td><td>91.60±0.15</td><td>86.02</td></tr><tr><td>LoCA</td><td>239K</td><td>94.10±0.21</td><td>80.11±0.58</td><td>98.62±0.21</td><td>80.15±0.61</td><td>99.04±0.08</td><td>54.86±0.65</td><td>94.73±0.18</td><td>91.68±0.43</td><td>86.66</td></tr><tr><td rowspan="7">VITLarge</td><td>LP</td><td>-</td><td>91.93±0.21</td><td>43.24±0.30</td><td>97.78±0.23</td><td>72.52±0.35</td><td>93.76±0.18</td><td>26.55±0.86</td><td>83.52±0.38</td><td>88.73±0.34</td><td>74.75</td></tr><tr><td>FF</td><td>303.3M</td><td>94.13±0.12</td><td>85.84±0.17</td><td>99.22±0.15</td><td>81.64±0.29</td><td>99.13±0.07</td><td>63.33±0.37</td><td>96.21±0.11</td><td>94.67±0.09</td><td>89.27</td></tr><tr><td>LoRA</td><td>1.57M</td><td>94.34±0.36</td><td>85.92±0.24</td><td>98.93±0.02</td><td>79.90±0.88</td><td>98.91±0.07</td><td>64.47±0.63</td><td>95.63±0.13</td><td>92.37±0.02</td><td>88.81</td></tr><tr><td>FourierFT</td><td>144K</td><td>94.52±0.53</td><td>75.35±0.32</td><td>99.12±0.42</td><td>79.78±0.76</td><td>98.79±0.35</td><td>48.32±0.89</td><td>94.18±0.41</td><td>93.01±0.14</td><td>85.38</td></tr><tr><td>LoCA</td><td>144K</td><td>94.60±0.03</td><td>82.04±0.25</td><td>98.92±0.03</td><td>79.02±0.18</td><td>98.97±0.05</td><td>57.62±0.02</td><td>94.41±91.76</td><td>91.76±0.09</td><td>87.17</td></tr><tr><td>FourierFT</td><td>480K</td><td>94.78±0.09</td><td>82.27±0.30</td><td>99.00±0.08</td><td>79.03±0.04</td><td>98.95±0.10</td><td>56.96±1.09</td><td>95.53±0.03</td><td>92.56±0.04</td><td>87.39</td></tr><tr><td>LoCA</td><td>480K</td><td>94.47±0.82</td><td>83.47±0.32</td><td>99.02±0.03</td><td>80.21±0.66</td><td>99.03±0.18</td><td>63.02±0.61</td><td>95.49±0.15</td><td>92.65±0.22</td><td>88.42</td></tr></table>
+
+improvements on StanfordCars and FGVC. Furthermore, when increasing the parameter budget to 10,000 for LoCA, we observe performance comparable to LoRA across most tasks. These results demonstrate that LoCA achieves a favorable balance between parameter efficiency and performance.
+
+# 5.5 ANALYTICAL EXPERIMENTS
+
+Effectiveness of Gradient Estimation. To validate the reliability of our estimated location gradients, we present the training process on 4 selected datasets in Fig. 2. The left figure shows
+
+that during the alternating optimization phase, the validation loss generally decreases in most steps, particularly for StanfordCars and CI-FAR10. The right figure demonstrates corresponding improvements in validation accuracy (or Pearson correlation). These trends indicate that our central difference approximation method effectively guides the optimization process, enabling successful updates to frequency component locations. We also conduct a toy experiment to show the convergence of the alternating optimization strategy in Appendix M.
+
+Performance under Different Parameter Budgets. Fig. 3 compares various methods under same parameter budgets. Here we focus on QQP and FHVC, which present significant
+
+challenges for LoRA. The parameter budget is standardized using LoRA's rank  $r$  as the base unit. Our results reveal that FourierFT often underperforms LoRA when using fewer parameters. This observation aligns with expectations, as the locations of frequency components become increasingly critical under constrained parameter budgets. Notably, LoCA consistently outperforms LoRA and FourierFT across the tested scenarios. It is worth noting that our theoretical analysis centers on expected performance. While specific task structures may allow FourierFT to surpass LoRA in certain instances, these exceptions do not undermine our overall conclusions and analytical framework.
+
+Choice of Scaling value  $\alpha$  and Alternating Optimization Steps  $\mathcal{B}_s$ . Fig. 4 demonstrates the impact of different choices of  $\alpha$  and  $\mathcal{B}_s$  on the MRPC task. We empirically find that a scaling value between 1-2 can achieve better results. Additionally, setting  $\mathcal{B}_s$  to between  $10\% - 20\%$  of the total training steps is more appropriate (with a total of 5750 steps for the MRPC task).
+
+Ablation Study of the Alternating Optimization Strategy. Table 5 compares several variants of our method: V1 only optimizes coefficients with randomly initialized locations. V2 alternately optimizes coefficients and locations throughout the training. V3 jointly optimizes locations and coefficients in each step for  $B_{s}$  steps. V4 and V5 use forward and backward difference approximation for gradient estimation, respectively. Hyperparameters are identical
+
+![](images/4c2c7410f1e37d520fbcd1a1a1b38dad979100ba70567cadaf245c38e60477bd.jpg)  
+Figure 2: Evaluation loss (left) and performance (right) of our method with RoBERTa-base and ViT-base models. We record every 10 steps. The solid lines represent alternating optimization of coefficients and locations, while the dashed lines represent optimizing coefficients only.
+
+![](images/26248a44534a6dc3c8bd186198e6ed087a04d748d899cd8f913339fb01de43c0.jpg)
+
+standardized using LoRA's rank  $r$  as the base unit. In forms LoRA when using fewer parameters. This obvi- 7 ans of frequency components becomes increasingly notable, LoCA consistently outperforms LoRA and is in noting that our theoretical analysis centers on ex- 8 tmes may allow FourierFT to surpass LoRA in certain or overall conclusions and analytical framework.
+
+![](images/d506c37c1807403af565cf818f5fc2198753c4ec0554597a148e518205ec7007.jpg)  
+Figure 3: Performance comparison under different parameter budgets on QQP (RoBERTa-base) and FGVC (ViT-base).
+
+![](images/d0ba52432351268cd9724ac36198a586b0e8f94c91384abc8f04cef9d53eb614.jpg)
+
+![](images/774961e606a24fc5ef97cbe5b7b7bee29d795b1abf720f80fd8fe91ea345c3da.jpg)  
+Figure 4: Influence of  $\alpha$  and  $\mathcal{B}_s$  on MRPC (RoBERTa-base).
+
+to the ones in Section 5.1 and 5.4. It can be observed that alternating optimization throughout the entire process leads to instability, resulting in a suboptimal performance. Simultaneously optimizing coefficients makes convergence not guaranteed, thus being less effective than alternating optimization. Both one-side (forward and backward) difference approximations show effectiveness, but it is challenging to theoretically analyze which is superior. Therefore, we choose using the central difference approximation as the default implementation.
+
+Table 5: Comparison between different optimization strategies on 4 datasets. We use RoBERTa-base and ViT-base models for this experiment. Best results are shown in bold.  
+
+<table><tr><td rowspan="2">Variants</td><td colspan="2">Vision Tasks (B =5000)</td><td colspan="2">Language Tasks (B =1000)</td></tr><tr><td>OxfordPets</td><td>DTD</td><td>QQP</td><td>CoLA</td></tr><tr><td>V1</td><td>92.8</td><td>76.8</td><td>87.7</td><td>63.2</td></tr><tr><td>V2</td><td>91.9</td><td>76.3</td><td>86.5</td><td>61.6</td></tr><tr><td>V3</td><td>93.4</td><td>79.1</td><td>88.0</td><td>64.1</td></tr><tr><td>V4</td><td>93.8</td><td>79.5</td><td>88.6</td><td>64.3</td></tr><tr><td>V5</td><td>93.8</td><td>79.7</td><td>88.4</td><td>64.4</td></tr><tr><td>LoCA</td><td>93.8</td><td>79.7</td><td>88.7</td><td>64.5</td></tr></table>
+
+# 6 RELATED WORK
+
+The recent surge in LLM research has reignited interest in PEFT research. To pursue favorable task performance while using only a small number of trainable parameters, current PEFT methods primarily lie in four categories: adding extra trainable modules (Houlsby et al., 2019; Rücklé et al., 2020), selectively training a small subset of key parameters (Zaken et al., 2021; Lawton et al., 2023), employing reparameterization techniques like low-rank decomposition to the incremental matrices (Hu et al., 2021; Zhang et al., 2023b; Liu et al., 2024; Hao et al., 2024), or combining multiple strategies (Chen et al., 2023). Among them, low-rank methods have garnered significant attention due to their mergeable nature and parameter efficiency. These low-rank methods, which aim to approximate large weight matrices using a few principal components, is highly analogous to techniques employed in data compression. In fact, low-rank decomposition (or singular value decomposition) and frequency-domain decomposition (e.g., JPEG compression) represents two fundamental tools in image compression and signal processing.
+
+For image compression, frequency-domain reconstruction (e.g., DCT) are preferred due to the inherent smoothness prior of image data (Wallace, 1991). However, when dealing with the complex data structures of neural network parameter matrices, the relative efficacy of these approaches remains unexplored. To the best of our knowledge, although FourierFT (Gao et al., 2024) has made an empirical study of frequency-domain PEFT by employing Fourier Transform, no prior work has conducted a rigorous comparison between low-rank and frequency-domain decomposition methods in the context of PEFT. Our work aims to bridge this gap by providing a comprehensive theoretical analysis and designing a more efficient frequency-domain PEFT method.
+
+# 7 CONCLUSION
+
+This paper provides a theoretical foundation for frequency-domain PEFT methods. We prove that carefully selected frequency components can outperform low-rank approaches, leading to the development of location-aware frequency-domain PEFT method. Our method optimizes both coefficients and locations of frequency components using iDCT and difference approximation. We show that our method enhances expressiveness while maintaining computational efficiency. Extensive experiments across NLP and computer vision tasks demonstrate the superior performance and parameter efficiency compared to existing PEFT methods.
+
+# ACKNOWLEDGEMENT
+
+This work was supported in part by the National Natural Science Foundation of China under Grant 62273071, 62176042, 11925106, 12231011 & 12326325, and in part by TCL Technology Innovation Funding SS2024105, and in part by the Fundamental Research Funds for the Central Universities (UESTC) under Grant ZYGX2024Z008, and in part by the National Key R&D Program of China (Grant Nos. 2022YFA1003703, 2022YFA1003800), and in part by the Natural Science Foundation of Xinjiang Uyghur Autonomous Region, China (No. 2024D01B07), and in part by China Scholarship Council (CSC). MG was supported by ARC DE210101624, ARC DP240102088, and WIS-MBZUAI 142571.
+
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+
+# A JUSTIFICATION OF ASSUMPTIONS
+
+In the pre-training and fine-tuning paradigm, deep neural networks are initially trained on a large dataset with distribution  $P(X,Y;\overline{W}_0)$  and subsequently fine-tuned on a specific down-stream dataset with distribution  $P(X,Y;\overline{W})$ . In this context,  $\overline{W}$  becomes a random variable associated with a specific data distribution.
+
+First for assumption (A1), the large dataset used for pre-training represents an aggregation of numerous sub-datasets. Each sub-dataset contributes to the overall distribution  $P(X, Y; \overline{W}_0)$ . The parameter  $\overline{W}_0$  can be seen as the central tendency (mean) of the parameters for all sub-datasets. This aggregation naturally leads to a central limit theorem effect, where the mixture of multiple sub-datasets can be approximated by a normal distribution around  $\overline{W}_0$ , which also reflects the idea of symmetry in the distribution of sub-datasets. In the absence of strong directional biases, it is reasonable to consider that the parameters for different sub-datasets are symmetrically distributed. Note that our proposition is based on all sub-datasets, which also follows the philosophy of the No Free Lunch (NFL) theorem in machine learning. By modeling  $\overline{W}$  as a distribution centered on  $\overline{W}_0$ , we account for the variability across different sub-datasets.
+
+![](images/47075e2f959e91fbfe7d0868c5b65699c8ead505744b4baa11f6270c6b1b2242.jpg)  
+(a) Layer 10
+
+![](images/57843d02de7c7427699912facaedbe990b962966892636a8b439e32d1a3b1fcd.jpg)  
+(b) Layer 20
+
+![](images/465375bd03c70a9294122effdd340fb2c869ce5645cfd8613b293236ff7776e8.jpg)  
+(c) Layer 30  
+Figure 5: Empirical spectral density of the fine-tuned  $W'$  across multiple layers. The experimental settings are the same as those in Section 2.
+
+Regarding assumption (A2), the asymptotic normality of M-estimators is a commonly used assumption in statistics and machine learning. The strongest assumption here should be that the elements of  $W' - \overline{W}$  are asymptotically independent and identically distributed given  $\overline{W}$ . To demonstrate the reasonability of this assumption. We first consider the asymptotically i.i.d. property of  $W'$ . While the strict i.i.d. property of parameters in trained neural networks remains a subject of ongoing research, several studies have shwon that certain statistical properties of these parameters resemble those of random i.i.d. matrices (Thamm et al., 2022; Martin & Mahoney, 2021; Lee et al., 2017). Our work extends this line by examining the spectral properties of the trained weight during LLM fine-tuning. Specifically, we use the Marchenko-Pastur (MP) law to test the fit between the empirical spectral densities of  $W'$  and that of random matrices. The MP law is a fundamental result in random matrix theory. It describes the asymptotic behavior of the eigenvalue distribution of large random matrices. The law can be formally stated as follows: Consider a  $p \times q$  random matrix  $W$ , where each element is an independent and identically distributed random variable with mean 0 and variance  $\sigma^2$ . Let  $C = (1/p)W'^T W'$  be the covariance matrix. As  $p, q \to \infty$  with a fixed aspect ratio, the empirical spectral distribution of the eigenvalues of  $C$  converges almost surely to a deterministic probability distribution known as the Marchenko-Pastur distribution. Here we are dealing with large Transformer weight matrices. If they are asymptotically i.i.d. matrices, the ESD of them should closely approximate the MP distribution corresponding to their current aspect ratios. We visualize the ESD of the fine-tuned  $W'$  across multiple layers, as shown in Fig. 5. And the results show that  $W'$  behaves like an i.i.d random matrix. As each element on  $\overline{W}$  is permutable due to the equal role of different positions, we can summarize that  $\overline{W}$  has a zero expectation influence on  $W' - \overline{W}$ . Therefore, the asymptotically i.i.d property of  $W' - \overline{W}$  does not violate our observations. The assumption that  $W' - \overline{W}$  and  $\overline{W}$  are independent is analogous to treating  $W' - \overline{W}$  as noise, while  $\overline{W}$  is the true signal. This is a common assumption in the context of asymptotic analysis, where the estimation error (or noise) is considered to be independent of the true parameter.
+
+# B DETAILS OF THE HYPOTHESIS TESTING
+
+We now describe the detailed procedure of the hypothesis testing adopted in Section 2. Recall that our goal is to test whether the elements  $w$  from the weight incremental matrix  $\Delta W$  follows a distribution that is close to a Gaussian. Formally, we have the following hypothesis setup and test statistic.
+
+# Hypothesis Setup:
+
+$$
+H _ {0}: d _ {T V} (P (w), \mathcal {N} (w; \hat {\mu}, \hat {\sigma} ^ {2})) \leq \epsilon , H _ {1}: d _ {T V} (P (w), \mathcal {N} (w; \hat {\mu}, \hat {\sigma} ^ {2})) > \epsilon
+$$
+
+Where  $d_{TV}(\cdot ,\cdot)$  denotes the total variation distance,  $P(w)$  is the true distribution of elements in  $\Delta W$ , and  $\mathcal{N}(\hat{\mu},\hat{\sigma}^2)$  is the normal distribution with sample mean and variance as parameters.
+
+# Test Statistic:
+
+$$
+T = d _ {T V} \left(\hat {P} _ {n} (w), \mathcal {N} \left(w; \hat {\mu}, \hat {\sigma} ^ {2}\right)\right)
+$$
+
+Where  $\hat{P}_n(w)$  is the empirical distribution of  $w$ .
+
+# Testing Procedure:
+
+Given a  $\Delta W\in \mathbb{R}^{p\times q}$  yielded by full fine-tuning, our test procedure consists of the following steps.
+
+1. From the observed  $\Delta W$ , compute the empirical mean  $\hat{\mu}$  and variance  $\hat{\sigma}^2$ .  
+2. Generate  $1e^{5}$  samples from  $\mathcal{N}(w; \hat{\mu}, \hat{\sigma}^2)$ , denoted this set of samples by  $\mathcal{G}$ .
+
+3. Generate  $B$  perturbed distributions:
+
+- Add small random perturbations  $e \sim \mathcal{N}(e; 0, \sigma_e^2)$  to the  $M$  samples, where  $\sigma_e = 1e^{-5}$ .  
+- Calculate the empirical distribution of the perturbed samples.  
+- Compute the total variation distance between the obtained empirical distribution and  $\mathcal{G}$ .  
+- If the total variation distance is less than  $\epsilon$ , keep this distribution.  
+- Repeat until 100 valid perturbed distributions are obtained.
+
+4. For each of the 100 perturbed distributions:
+
+- Sample 10 sets of  $p \times q$  points.  
+- For each set, calculate the total variation distance between the empirical distribution of this set and  $\mathcal{G}$ . This results in  $M \times P$  total variation distances, forming the distribution of the test statistic under  $H_0$ .
+
+5. Calculate the total variation distance between the empirical distribution of  $\Delta W$  and  $\mathcal{G}$ , denoted by  $T$ .  
+6. The p-value is the percentile of  $T$  in the  $M \times P$  total variation distances.  
+7. Reject  $H_0$  if the p-value is less than the chosen significance level (e.g., 0.05). Otherwise, accept  $H_0$ .
+
+Note that although this process is not strictly a bootstrap (as it does not directly resample from the original data), it does use the idea of repeated sampling to generate the distribution of the test statistic. Traditional bootstrap typically resamples with replacement directly from the original data, whereas our method first generates a series of perturbed distributions and then samples from these distributions. The advantage of this approach is that it allows us to explore the behavior of distributions that are close to a Gaussian distribution, while allowing for small variations. This method is more akin to a Monte Carlo simulation, used to estimate the distribution of total variation under the null hypothesis.
+
+# C DETAILS ABOUT BASELINE METHODS
+
+- Full fine-tuning (FF) updates all parameters of the pre-trained model during the fine-tuning process, allowing for comprehensive adaptation at the cost of significant computational resources.
+
+- BitFit (Zaken et al., 2021) solely fine-tunes the bias weights while keeping other parameters frozen.  
+- Adapter-based methods inject extra trainable modules into pre-trained models and keep the original model parameters frozen. In our comparison, we primarily focused on three types of Adapters:  $\text{Adapter}^{\text{H}}$  (Houlsby et al., 2019), which inserts a two-layer adapter between the self-attention module (or the FFN module) and the subsequent residual connections,  $\text{Adapter}^{\text{L}}$  (Lin et al., 2020) that inserts a lightweight adapter layer with a bottleneck architecture after the MLP module and a LayerNorm layer in each Transformer block, and  $\text{Adapter}^{\text{D}}$  (Rücklé et al., 2020) that further enhances efficiency by strategically dropping inactive adapter layers.  
+- LoRA (Hu et al., 2021) reparameterizes  $\Delta W$  using two trainable low-rank matrices. Therefore, the number of trainable parameters is controlled by the chosen rank and the shape of weight matrices.  
+- AdaLoRA (Zhang et al., 2023b) extends LoRA by introducing an adaptive mechanism to dynamically allocate the rank budget across different parameter matrices.  
+- VeRA (Kopiczko et al., 2023) extends LoRA by introducing trainable scaling vectors  $(d$  and  $b)$  to adaptively adjust the contribution of each dimension in the low-rank matrices, achieving comparable performance with significantly fewer parameters.  
+- DoRA (Liu et al., 2024) is a LoRA variant that decomposes pre-trained weights into magnitude and direction components for fine-tuning. It demonstrates learning patterns closer to full fine-tuning.  
+- FourierFT (Gao et al., 2024) treats weight changes as spatial-domain matrices and reparameterizes them with a set of learnable frequency components. The number of trainable parameters is controlled by the number of frequency components, allowing for more flexible scaling of parameter budgets.
+
+# D HYPERPARAMETERS
+
+Table 6, 8, 7 and 9 summarize the hyperparameters we used in each experiment. It is worth noting that for LoCA, the weight decay is not applied to the optimization of the location variables. Regarding the total number of alternating learning steps  $\mathcal{B}_s$ , we set it to approximately  $10\%$  of the total training steps, based on the size of different datasets.
+
+It is worth noting that our method has very stable hyperparameters (especially the scaling value) across different tasks on GLUE, while FourierFT requires extensive parameter tuning to achieve satisfactory results, as can be seen from Gao et al. (2024).
+
+Table 6: Hyperparameters for our method on the GLUE benchmark.  
+
+<table><tr><td>Model</td><td>Datasets</td><td>CoLA</td><td>MNLI</td><td>MRPC</td><td>QNLI</td><td>QQP</td><td>RTE</td><td>SST2</td><td>STS-B</td></tr><tr><td rowspan="9">Common</td><td>Optimizer</td><td></td><td></td><td></td><td>AdamW</td><td></td><td></td><td></td><td></td></tr><tr><td>LR Schedule</td><td></td><td></td><td></td><td>Linear</td><td></td><td></td><td></td><td></td></tr><tr><td>Batch Size</td><td></td><td></td><td></td><td>32</td><td></td><td></td><td></td><td></td></tr><tr><td>Where</td><td></td><td></td><td></td><td>Query, Value</td><td></td><td></td><td></td><td></td></tr><tr><td>Warmup Ratio</td><td></td><td></td><td></td><td>0.06</td><td></td><td></td><td></td><td></td></tr><tr><td>B</td><td></td><td></td><td></td><td>1000</td><td></td><td></td><td></td><td></td></tr><tr><td>Learning Rate (Positions)</td><td></td><td></td><td></td><td>1e-4</td><td></td><td></td><td></td><td></td></tr><tr><td>Scaling Value α</td><td></td><td></td><td></td><td>1</td><td></td><td></td><td></td><td></td></tr><tr><td>Random Seeds</td><td></td><td></td><td></td><td>{6,66,666}</td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="6">RoBERTa-base</td><td>Learning Rate (Head)</td><td>5e-3</td><td>5e-4</td><td>6e-3</td><td>1e-3</td><td>5e-4</td><td>6e-3</td><td>1e-3</td><td>1e-3</td></tr><tr><td>Learning Rate (Coefficients)</td><td>5e-3</td><td>5e-4</td><td>1e-2</td><td>5e-3</td><td>5e-4</td><td>5e-3</td><td>5e-3</td><td>5e-3</td></tr><tr><td>Max Seq. Len</td><td></td><td></td><td></td><td>512</td><td></td><td></td><td></td><td></td></tr><tr><td>Weight Decay</td><td>1e-4</td><td>1e-4</td><td>1e-4</td><td>5e-4</td><td>1e-4</td><td>0</td><td>5e-4</td><td>5e-4</td></tr><tr><td>Epochs</td><td>80</td><td>30</td><td>50</td><td>40</td><td>35</td><td>80</td><td>30</td><td>50</td></tr><tr><td>Bs</td><td>2100</td><td>3000</td><td>600</td><td>3000</td><td>3000</td><td>600</td><td>3000</td><td>600</td></tr><tr><td rowspan="6">RoBERTa-large</td><td>Learning Rate (Head)</td><td>5e-3</td><td>5e-4</td><td>5e-3</td><td>1e-3</td><td>5e-4</td><td>5e-3</td><td>1e-3</td><td>1e-3</td></tr><tr><td>Learning Rate (Coefficients)</td><td>5e-3</td><td>5e-4</td><td>1e-2</td><td>5e-3</td><td>5e-4</td><td>5e-3</td><td>5e-3</td><td>5e-3</td></tr><tr><td>Max Seq. Len</td><td></td><td></td><td></td><td>512</td><td></td><td></td><td></td><td></td></tr><tr><td>Weight Decay</td><td>1e-4</td><td>1e-4</td><td>25</td><td>20</td><td>50</td><td>20</td><td>50</td><td>5e-4</td></tr><tr><td>Epochs</td><td>40</td><td>15</td><td>30</td><td>3000</td><td>3000</td><td>300</td><td>3000</td><td>600</td></tr><tr><td>Bs</td><td>1000</td><td>3000</td><td>400</td><td>3000</td><td>3000</td><td>300</td><td>3000</td><td>600</td></tr></table>
+
+Table 7: Hyperparameter configuration of LoCA on the E2E benchmark.  
+
+<table><tr><td>Hyperparameter</td><td>GPT-2 Medium/Large</td></tr><tr><td>Optimizer</td><td>AdamW</td></tr><tr><td>Dropout</td><td>0</td></tr><tr><td>Warmup Steps</td><td>100</td></tr><tr><td>Epochs</td><td>5</td></tr><tr><td>Where</td><td>Query, Value</td></tr><tr><td>Label Smooth</td><td>0.1</td></tr><tr><td>LR Schedule</td><td>Linear</td></tr><tr><td>Learning Rate (Coefficients)</td><td>5e-3</td></tr><tr><td>Learning Rate (Positions)</td><td>1e-4</td></tr><tr><td>Learning Rate (Head)</td><td>2e-4</td></tr><tr><td>Batch Size</td><td>32</td></tr><tr><td>Weight Decay</td><td>0.01</td></tr><tr><td>B</td><td>1000</td></tr><tr><td>Learning iterations Bs</td><td>1200</td></tr><tr><td>Scaling Value α</td><td>1</td></tr></table>
+
+Table 8: Hyperparameter configuration for all methods on the instruction tuning task.  
+
+<table><tr><td>Method</td><td>Hyperparameter</td><td>LLaMA-7B</td><td>LLaMA-13B</td></tr><tr><td rowspan="7">Common</td><td>Optimizer</td><td colspan="2">AdamW</td></tr><tr><td>LR schedule</td><td colspan="2">Linear</td></tr><tr><td>Batch Size</td><td colspan="2">16</td></tr><tr><td>Where</td><td colspan="2">Query, Value</td></tr><tr><td>Weight Decay</td><td colspan="2">0</td></tr><tr><td>Epochs</td><td>3</td><td>1</td></tr><tr><td>Accumulation Steps</td><td></td><td>4</td></tr><tr><td rowspan="3">LoRA</td><td>Rank</td><td colspan="2">64</td></tr><tr><td>Scaling Value</td><td colspan="2">16</td></tr><tr><td>Learning Rate</td><td colspan="2">3e-4</td></tr><tr><td>FF</td><td>Learning Rate</td><td>2e-5</td><td>1e-5</td></tr><tr><td rowspan="3">FourierFT</td><td>Frequency Components</td><td colspan="2">150000</td></tr><tr><td>Scaling Value</td><td colspan="2">64</td></tr><tr><td>Learning Rate</td><td colspan="2">1e-3</td></tr><tr><td rowspan="5">LoCA</td><td>Frequency Components</td><td colspan="2">150000</td></tr><tr><td>Learning Rate (coefficient)</td><td colspan="2">5e-4</td></tr><tr><td>Scaling Value</td><td colspan="2">1</td></tr><tr><td>Learning iterations (B_s)</td><td>600</td><td>300</td></tr><tr><td>Learning Rate (locations)</td><td colspan="2">1e-4</td></tr></table>
+
+Table 9: Hyperparameter configuration for all methods on eight image classification datasets.  
+
+<table><tr><td>Method</td><td>Hyperparameter</td><td>ViT-Base</td><td>ViT-Large</td></tr><tr><td rowspan="7">Common</td><td>Optimizer</td><td></td><td>AdamW</td></tr><tr><td>LR schedule</td><td></td><td>Linear</td></tr><tr><td>Batch Size</td><td></td><td>128</td></tr><tr><td>Where</td><td></td><td>Query, Value</td></tr><tr><td>Learning Rate (Head)</td><td>1e-2</td><td>1e-3</td></tr><tr><td>Weight Decay</td><td></td><td>5e-5</td></tr><tr><td>Random Seeds</td><td></td><td>{2020, 2021, 2022, 2023, 2024}</td></tr><tr><td rowspan="3">LoRA</td><td>Rank</td><td></td><td>16</td></tr><tr><td>Scaling Value</td><td></td><td>16</td></tr><tr><td>Learning Rate (ViT)</td><td></td><td>5e-3</td></tr><tr><td rowspan="3">FourierFT</td><td>Frequency Components</td><td></td><td>3000 and 10,000</td></tr><tr><td>Scaling Value</td><td></td><td>300</td></tr><tr><td>Learning Rate (ViT)</td><td></td><td>5e-2</td></tr><tr><td rowspan="5">LoCA</td><td>Frequency Components</td><td></td><td>3000 and 10,000</td></tr><tr><td>Learning Rate (ViT)</td><td></td><td>5e-2</td></tr><tr><td>Scaling Value</td><td></td><td>1 and 0.5</td></tr><tr><td>Learning iterations (B_s)</td><td></td><td>120</td></tr><tr><td>Learning Rate (locations)</td><td></td><td>1e-4</td></tr></table>
+
+# E TRAINING PROCEDURE
+
+We provide a pseudo code of our LoCA fine-tuning method in Algorithm 1.
+
+Algorithm 1 LoCA Fine-tuning  
+Require: Pre-trained weight  $W_{0}$  dataset  $\mathcal{D}$  , learning rates  $\eta_{a},\eta_{l}$  number of alternating iterations   
+ $\mathcal{B}_s$  , number of coefficient update steps  $\mathcal{B}_a$  , number of location update steps  $\mathcal{B}_l$  , total iterations   
+ $T$  scaling factor  $\alpha$    
+Ensure: Fine-tuned weight  $W^{\prime}$    
+1: Initialize  $a\gets 0,l$  randomly   
+2: for  $t = 1$  to  $T$  do   
+3: Sample a mini-batch  $\mathcal{D}$  and compute the training loss  $\mathcal{L}$    
+4: if  $t\leq B_s$  then   
+5: if  $t$  mod  $(B_a + B_l) <   B_a$  then   
+6: Update  $a$  by  $a\gets a - \eta_a\nabla_a\mathcal{L}$    
+7: else   
+8: Update  $l$  by  $l\gets l - \eta_l\frac{\partial\mathcal{L}}{\partial l}$  using Eq. (5)   
+9: end if   
+10: else   
+11: Update  $a$  by  $a\gets a - \eta_a\nabla_a\mathcal{L}$    
+12: end if   
+13: end for   
+14: return  $W^{\prime} = W_{0} + \alpha [C^{T}\mathcal{S}(a,l,1)D]$
+
+# F DERIVATION OF PROPOSITION 1
+
+Given any parameter  $\overline{W}$  for a down-stream dataset, we assume that the M-estimator  $W^{\prime}$  has asymptotic normality, the estimation error  $W^{\prime} - \overline{W}$  is independent of  $\overline{W}$  and are asymptotically independent and identically distributed, which can be specified as
+
+$$
+\sqrt {n ^ {\prime}} \left(W ^ {\prime} - \bar {W}\right) ^ {V} \mid \bar {W} \xrightarrow {d _ {i}} \mathcal {N} _ {K ^ {2}} \left(0, \sigma_ {0} ^ {2} I _ {K ^ {2}}\right), \tag {6}
+$$
+
+where  $n'$  is the number of samples in the dataset,  $K$  is the width (length) of the weight matrix and  $\sigma_0 > 0$  is a constant independent of  $\overline{W}$ .
+
+Lemma 1. Let  $X_1, X_2, \ldots$  be a sequence of  $k$ -dimensional random variables, and let  $g(X_n, s)$  be a parameterized function with parameter space  $S$ , such that for all  $s \in S$ ,  $\sqrt{n} g(X_n, s) \xrightarrow{d} \mathcal{N}_k(0, I_k)$ . Then, for any random variable  $S$  taking values in  $S$  and independent of  $X_n$ , we have  $\sqrt{n} g(X_n, S) \xrightarrow{d} \mathcal{N}_k(0, I_k)$ .
+
+Proof. Fix any point  $t \in \mathbb{R}^k$ , denote all coordinates of  $X_{n}$  not larger than  $t$  by  $X_{n} \leq t$ . Assume the distribution of  $S$  and  $X_{n}$  are  $P_{S}, P_{n}$  respectively. Thus
+
+$$
+\begin{array}{l} \mathbb {P} \left(\sqrt {n} g (X _ {n}, S) \leq t\right) = \int_ {\sqrt {n} g (x, s) \leq t} d P _ {S} (s) d P _ {n} (x) \\ = \int_ {s \in \mathcal {S}} \mathbb {P} \left(\sqrt {n} g (X _ {n}, s) \leq t\right) d P _ {S} (s). \\ \end{array}
+$$
+
+As  $\sqrt{n} g(X_n,s)\stackrel {d}{\to}\mathcal{N}_k(0,I_k),\forall s\in S$  implies  $\mathbb{P}(\sqrt{n} g(X_n,s)\leq t)\rightarrow \Phi_k(t),\forall s\in S,$  where  $\Phi_k(\cdot)$  is the C.D.F of standard multivariate normal distribution. Based on dominate convergence theorem and  $\mathbb{P}(\sqrt{n} g(X_n,s)\leq t)\leq 1$  , we have
+
+$$
+\mathbb {P} \left(\sqrt {n} g (X _ {n}, S) \leq t\right)\rightarrow \Phi_ {k} (t),
+$$
+
+which is  $\sqrt{n} g(X_n, S) \stackrel{d}{\to} \mathcal{N}_k(0, I_k)$ .
+
+![](images/c72be8cba6f6d9dcad93f28c7954ee60f3b688fdff0536a578caa11f2077152b.jpg)
+
+Note that we can replace  $\mathcal{N}_k(0,I_k)$  with any continuous distribution in  $\mathbb{R}^k$  and the result still holds. Based on our assumption and Eq. (6), we consider  $\sqrt{n'}\left(W' - \overline{W}\right)^V\mid \overline{W}$  as a random variable parameterized by  $\overline{W}$ . Therefore, there exists a constant  $\sigma_0$  such that we have:
+
+$$
+\sqrt {n ^ {\prime}} \left(W ^ {\prime} - \overline {{W}}\right) ^ {V} \xrightarrow {d _ {i}} \mathcal {N} _ {K ^ {2}} \left(0, \sigma_ {0} ^ {2} I _ {K ^ {2}}\right),
+$$
+
+in other words,
+
+$$
+\left(W ^ {\prime} - \bar {W}\right) ^ {V} = \mathcal {N} _ {K ^ {2}} \left(0, \frac {\sigma_ {0} ^ {2}}{n ^ {\prime}} I _ {K ^ {2}}\right) + o _ {P} \left(\frac {1}{\sqrt {n ^ {\prime}}}\right). \tag {7}
+$$
+
+Besides, the assumption gives
+
+$$
+\left(\overline {{W}} - \overline {{W}} _ {0}\right) ^ {V} = \mathcal {N} _ {K ^ {2}} \left(0, \overline {{\sigma}} ^ {2} I _ {K ^ {2}}\right).
+$$
+
+Adding it to Eq. (7), we have
+
+$$
+\left(W ^ {\prime} - \bar {W} _ {0}\right) ^ {V} = \mathcal {N} _ {K ^ {2}} \left(0, \left(\frac {\sigma_ {0} ^ {2}}{n ^ {\prime}} + \bar {\sigma} ^ {2}\right) I _ {K ^ {2}}\right) + o _ {P} \left(\frac {1}{\sqrt {n ^ {\prime}}}\right). \tag {8}
+$$
+
+On the other hand,  $W_{0}$  is the M-estimator of  $\overline{W}_{0}$  using  $N$  samples, we have
+
+$$
+W _ {0} - \overline {{W}} _ {0} = O _ {P} \left(\frac {1}{\sqrt {N}}\right).
+$$
+
+Combining it with Eq. (8) we have
+
+$$
+\Delta W ^ {V} = \left(W ^ {\prime} - W _ {0}\right) ^ {V} = \mathcal {N} _ {K ^ {2}} \left(0, \left(\frac {\sigma_ {0} ^ {2}}{n ^ {\prime}} + \overline {{\sigma}} ^ {2}\right) I _ {K ^ {2}}\right) + o _ {P} \left(\frac {1}{\sqrt {n ^ {\prime}}}\right) + O _ {P} \left(\frac {1}{\sqrt {N}}\right).
+$$
+
+# G PROOF OF THEOREM 1
+
+Before proving the proposed theorem, we first give a proposition.
+
+For any matrix  $W \in \mathbb{R}^{K \times K}$ , let its singular values be  $|\lambda_1| \geq \ldots \geq |\lambda_K|$ . Define the discrete Fourier transform of  $W$  as  $\mathcal{F}(W) = HWH$ , where  $H \in \mathbb{C}^{K \times K}$  is the DFT matrix. More specifically, we can express  $H$  as  $H = Re(H) + iIm(H)$ , where  $i$  is the imaginary unit, and  $Re(H), Im(H) \in$
+
+$\mathbb{R}^{K\times K}$  are the real and imaginary coefficients, respectively. Let  $F = (F_{ij})_{1\leq i,j\leq K} = \mathcal{F}(W)$ . For each location  $(i,j)$ , we define a reference matrix  $R = (R_{ij})_{1\leq i,j\leq K}$  as follows:
+
+$$
+R _ {i j} = \left\{ \begin{array}{l l} - 1, & \text {i f F _ {i j} h a s a s y m m e t r i c c o u p e r t a r p a n d (i , j) s a t i s f i e s c o n d i t i o n U} \\ 1, & \text {i f F _ {i j} h a s a s y m m e t r i c c o u p e r t a r p a n t b u t (i , j) d o e s n o t s a t i s f y c o n d i t i o n U} \\ 0, & \text {o t h e r w i s e} \end{array} \right.
+$$
+
+Here the condition  $U$  is a set of conjugate that
+
+$$
+\begin{array}{l} [ (i = 0) \wedge (j > K - j) ] \vee [ (j = 0) \wedge (i > K - i) ] \vee \\ [ (j > 0) \wedge (j > K - j) ] \vee [ (j = n - j) \wedge (i > K - i). ] \\ \end{array}
+$$
+
+We then define the half matrix of  $F$  by  $F^{H} = (F_{ij}^{H})_{1\leq i,j\leq K}$ , where
+
+$$
+F _ {i j} ^ {H} = \left\{2 \mathbb {1} \left(R _ {i j} = 1\right) + \mathbb {1} \left(R _ {i j} = 0\right) \right\} \left| F _ {i j} \right| ^ {2}.
+$$
+
+Similarly, we define the real and imaginary part half matrix of  $F$  by  $F^R$  and  $F^I$ , where
+
+$$
+F _ {i j} ^ {R} = \left\{2 \mathbb {1} \left(R _ {i j} = 1\right) + \mathbb {1} \left(R _ {i j} = 0\right) \right\} R e \left(F _ {i j}\right) ^ {2},
+$$
+
+$$
+F _ {i j} ^ {I} = \left\{2 \mathbb {1} \left(R _ {i j} = 1\right) + \mathbb {1} \left(R _ {i j} = 0\right) \right\} I m \left(F _ {i j}\right) ^ {2}.
+$$
+
+Based on the definition, we have  $F^H = F^R + F^I$ . We then sort  $F^H$  in descending order, denoting it as  $F_{(1)}^H \geq \ldots \geq F_{(K^2)}^H = 0$ . It can be inferred that approximately half of these elements are equal to 0. Consider the separate matrix  $F^S = (F^R, F^I) \in \mathbb{R}^{K \times 2K}$ , and also sort it in descending order, denoted as  $F_{(1)}^S \geq \ldots \geq F_{(2K^2)}^S = 0$ . There are also about half of these elements equal to 0. For the simplicity of notations, we define  $L_R = \mathbb{E}_{W \sim_G} L(W, \hat{W}_R)$ ,  $L_F^{(i)} = \mathbb{E}_{W \sim_G} L(W, \hat{W}_F^{(i)})$  for  $i = 1, 2, 3$ . Denote  $\widetilde{Id}^{(1)}$  be the set of locations that are symmetric counterparts of  $Id^{(1)}$ .
+
+Proposition 2. With the notations defined above, for  $r < K$ , we have
+
+$$
+L _ {R} = \sum_ {i = K - r + 1} ^ {K} | \lambda_ {i} | ^ {2},
+$$
+
+$$
+L _ {F} ^ {(1)} = \sum_ {(i, j) \notin I d ^ {(1)} \cup \widetilde {I d} ^ {(1)}} | F _ {i j} | ^ {2}, L _ {F} ^ {(2)} = \sum_ {i = N _ {2} + 1} ^ {K ^ {2}} F _ {(i)} ^ {H}, L _ {F} ^ {(3)} = \sum_ {i = N _ {3} + 1} ^ {2 K ^ {2}} F _ {(i)} ^ {S},
+$$
+
+$$
+s. t. | | W | | _ {2} ^ {2} = | | F | | _ {2} ^ {2} = \sum_ {i = 1} ^ {K} | \lambda_ {i} | ^ {2} = \sum_ {i = 1} ^ {K} \sum_ {j = 1} ^ {K} | F _ {i j} | ^ {2} = \sum_ {i = 1} ^ {K ^ {2}} F _ {(i)} ^ {H} = \sum_ {i = 1} ^ {2 K ^ {2}} F _ {(i)} ^ {S},
+$$
+
+Proof. First let us explore the reconstruction loss of low rank approximation. For any  $W \in \mathbb{R}^{K \times K}$ , its SVD decomposition is given by
+
+$$
+W = U \Lambda V ^ {T}, \Lambda = d i a g (\lambda_ {1}, \dots , \lambda_ {K}),
+$$
+
+$$
+U ^ {T} U = V ^ {T} V = I _ {K}, \left| \lambda_ {1} \right| \geq \dots \geq \left| \lambda_ {K} \right|.
+$$
+
+The best  $\hat{W}_R$  that minimize the reconstruction loss in terms of Frobenius norm is
+
+$$
+\hat {W} _ {R} = \hat {U} \hat {V} ^ {T}, \hat {U} = U \Lambda_ {r} ^ {1 / 2}, \hat {V} = V \Lambda_ {r} ^ {1 / 2},
+$$
+
+$$
+\Lambda_ {r} = \left(d i a g (\lambda_ {1}, \dots , \lambda_ {r}), 0 _ {r \times (K - r)}\right) ^ {T}.
+$$
+
+Thus we can easily calculate the reconstruction loss
+
+$$
+\begin{array}{l} L _ {R} = | | W - \hat {W} _ {R} | | _ {2} ^ {2} = | | U (\Lambda - \Lambda_ {r}) V ^ {T} | | _ {2} ^ {2} \\ = t r \left(\left\{U (\Lambda - \Lambda_ {r}) V ^ {T} \right\} ^ {T} \left\{U (\Lambda - \Lambda_ {r}) V ^ {T} \right\}\right) \\ = t r \left(\left(\Lambda - \Lambda_ {r}\right) ^ {T} (\Lambda - \Lambda_ {r})\right) \\ = \sum_ {i = K - r + 1} ^ {K} | \lambda_ {i} | ^ {2}. \\ \end{array}
+$$
+
+Before moving on to  $L_F^{(i)}, i = 1,2,3$ , we introduce discrete Parseval theorem first.
+
+Lemma 2 (Discrete Parseval Theorem). For a matrix  $X$  of size  $K \times K$ , with its Discrete Fourier Transform (DFT) denoted by  $F$ , the sum of the squares of the elements in the original matrix is equal to the sum of the squares of the elements in the DFT matrix, scaled by  $1 / K$ . Formally, if  $X$  is the original matrix and  $F$  is its DFT, then:
+
+$$
+| | X | | _ {2} ^ {2} = \sum_ {i = 0} ^ {K - 1} \sum_ {j = 0} ^ {K - 1} | X _ {i j} | ^ {2} = \frac {1}{K} \sum_ {i = 0} ^ {K - 1} \sum_ {j = 0} ^ {K - 1} | F _ {i j} | ^ {2} = \frac {1}{K} | | F | | _ {2} ^ {2}.
+$$
+
+Since  $F = \mathcal{F}(W), W = \mathcal{F}^{-1}(F)$ , and Fourier transform is linear transform, we have
+
+$$
+\begin{array}{l} L _ {F} ^ {(i)} = \left\| W - \hat {W} _ {F} ^ {(i)} \right\| _ {2} ^ {2} = \left\| W - \mathcal {F} ^ {- 1} (\hat {F} ^ {(i)}) \right\| _ {2} ^ {2} \\ = \left\| \mathcal {F} ^ {- 1} (F) - \mathcal {F} ^ {- 1} \left(\hat {F} ^ {(i)}\right) \right\| _ {2} ^ {2} \\ \end{array}
+$$
+
+linearity of Fourier Transformation  $= ||\mathcal{F}^{-1}(F - \hat{F}^{(i)})||_2^2$
+
+Parseval Theorem  $= ||F - \hat{F}^{(i)}||_2^2$
+
+Check  $i = 1,2,3$  separately and we have
+
+$$
+L _ {F} ^ {(1)} = \sum_ {(i, j) \notin I d ^ {(1)} \cup \widetilde {I d} ^ {(1)}} | F _ {i j} | ^ {2}, L _ {F} ^ {(2)} = \sum_ {i = N _ {2} + 1} ^ {K ^ {2}} F _ {(i)} ^ {H}, L _ {F} ^ {(3)} = \sum_ {i = N _ {3} + 1} ^ {2 K ^ {2}} F _ {(i)} ^ {S}.
+$$
+
+![](images/b5b0703a48048a9affd2c40f5b6f518c59371b95a1a827fa8697d357cd75b340.jpg)
+
+As we assume  $W \sim \mathcal{N}_{K,K}(0, I_K, I_K)$ , we then define  $A = W^T W \sim W_K(K, I_K, 0)$ , which follows a central Wishart distribution. Recall the SVD of  $W$ , i.e.,  $W = U\Lambda V^T$ , and
+
+$$
+A = W ^ {T} W = V \Lambda^ {2} V ^ {T}, \Lambda^ {2} = d i a g (\lambda_ {1} ^ {2}, \dots , \lambda_ {K} ^ {2}),
+$$
+
+we can conclude that  $\lambda_i' = \lambda_i^2$  is the eigenvalue of the matrix that follows  $W_K(K, I_K, 0)$  distribution.
+
+Next we present a commonly used result about the Wishart distribution in random matrix theory.
+
+Lemma 3. The joint density of  $\Lambda^2 = \text{diag}(\lambda_1', \dots, \lambda_K') = \text{diag}(\lambda_1^2, \dots, \lambda_K^2)$  is
+
+$$
+g _ {L} (\Lambda^ {2}) = C \left[ \prod_ {i = 1} ^ {K} \lambda_ {i} ^ {\prime - 1 / 2} e ^ {- \lambda_ {i} ^ {\prime} / 2} \right] \left[ \prod_ {i <   j} | \lambda_ {i} ^ {\prime} - \lambda_ {j} ^ {\prime} | \right].
+$$
+
+Noting that Lemma 3 is a direct corollary of Weyl's Integration Formula in Lemma 4 and 5.
+
+Lemma 4. (Bröcker & Tom Dieck, 2013). If  $X \in \mathbb{R}^{K \times K}$  is a real symmetric random matrix with density  $g(\lambda_1', \ldots, \lambda_K')$ , where  $g$  is exchangeable, and  $\lambda_1', \ldots, \lambda_K'$  are eigenvalues. Thus the joint density of  $(\lambda_1', \ldots, \lambda_K')$  is
+
+$$
+f ^ {\prime} \left(\lambda_ {1} ^ {\prime}, \dots , \lambda_ {K} ^ {\prime}\right) = C g \left(\lambda_ {1} ^ {\prime}, \dots , \lambda_ {K} ^ {\prime}\right) \prod_ {i <   j} \left| \lambda_ {i} ^ {\prime} - \lambda_ {j} ^ {\prime} \right|,
+$$
+
+where  $C$  is some constant such that
+
+$$
+\int C g (\lambda_ {1} ^ {\prime}, \ldots , \lambda_ {K} ^ {\prime}) \prod_ {i <   j} | \lambda_ {i} ^ {\prime} - \lambda_ {j} ^ {\prime} | d \lambda_ {1} ^ {\prime} \ldots d \lambda_ {K} ^ {\prime} = 1.
+$$
+
+Remark. Exchangeable function  $g$  means for any permutation  $\pi : [K] \to [K]$  and  $\lambda_1', \ldots, \lambda_K'$ ,
+
+$$
+g (\lambda_ {1} ^ {\prime}, \ldots , \lambda_ {K} ^ {\prime}) = g (\lambda_ {\pi (1)} ^ {\prime}, \ldots , \lambda_ {\pi (K)} ^ {\prime}).
+$$
+
+Wishart distribution  $W_{K}(K, I_{K}, 0)$  has density
+
+$$
+g (A) = \frac {| A | ^ {- 1 / 2} \exp \left\{- t r (A) / 2 \right\}}{2 ^ {K ^ {2} / 2} \pi^ {K (K - 1) / 4} \prod_ {i = 1} ^ {K} \Gamma ((K - i + 1) / 2)},
+$$
+
+where
+
+$$
+\left| A \right| ^ {- 1 / 2} = \left(\prod_ {i = 1} ^ {K} \lambda_ {i} ^ {\prime}\right) ^ {- 1 / 2} = \prod_ {i = 1} ^ {K} \lambda_ {i} ^ {- 1},
+$$
+
+$$
+t r (A) = \sum_ {i = 1} ^ {K} \lambda_ {i} ^ {\prime} = \sum_ {i = 1} ^ {K} \lambda_ {i} ^ {2}.
+$$
+
+This directly yields an unordered version of the result in Lemma 3. Specifically, let  $\lambda_1', \ldots, \lambda_K'$  be the unordered eigenvalues of  $A$ . To avoid confusion, we denote these unordered eigenvalues as  $\tilde{\Lambda}^2 = (\tilde{\lambda}_1', \ldots, \tilde{\lambda}_K')$ . Their joint density function is given by:
+
+$$
+\tilde {g} _ {L} \left(\tilde {\Lambda} ^ {2}\right) = \tilde {C} \left[ \prod_ {i = 1} ^ {K} \tilde {\lambda} _ {i} ^ {\prime - 1 / 2} e ^ {- \tilde {\lambda} _ {i} ^ {\prime} / 2} \right] \left[ \prod_ {i <   j} \left| \tilde {\lambda} _ {i} ^ {\prime} - \tilde {\lambda} _ {j} ^ {\prime} \right| \right]. \tag {9}
+$$
+
+Note that in the density function of  $\Lambda^2$ , all  $\lambda_1', \ldots, \lambda_K'$  are exchangeable, and for exchangeable random variables we have Lemma 5.
+
+Lemma 5. For any  $K$  exchangeable variables  $X_{1},\ldots ,X_{K}$ , which means for any permutation  $\pi :[K]\to [K]$ , the following equation holds,
+
+$$
+\left(X _ {1}, \ldots , X _ {K}\right) \stackrel {{d.}} {{=}} \left(X _ {\pi (1)}, \ldots , X _ {\pi (K)}\right).
+$$
+
+Let  $g$  be the density function of  $X_{1},\ldots ,X_{K}$ . Denote their order statistics as  $X_{(1)}\geq \dots \geq X_{(K)}$ . If we use  $\overline{g}$  to represent the joint distribution of these order statistics, then we have:
+
+$$
+\bar {g} (x _ {(1)}, \dots , x _ {(K)}) = K! g (x _ {1}, \dots , x _ {K}).
+$$
+
+Based on Lemma 5 and Eq. (9), let  $g_{L}$  denote the density function of the random variables with joint density  $\tilde{g}_{L}$ , and we finally have
+
+$$
+g _ {L} (\Lambda^ {2}) = C \left[ \prod_ {i = 1} ^ {K} \lambda_ {i} ^ {\prime - 1 / 2} e ^ {- \lambda_ {i} ^ {\prime} / 2} \right] \left[ \prod_ {i <   j} | \lambda_ {i} ^ {\prime} - \lambda_ {j} ^ {\prime} | \right],
+$$
+
+where the constant  $C$  has following representation (Muirhead, 2009):
+
+$$
+C = \left(\frac {\pi}{2}\right) ^ {K ^ {2} / 2} \frac {1}{\Gamma_ {K} ^ {2} (K / 2)},
+$$
+
+here  $\Gamma_p(a)$  is the multivariate gamma function. To summarize, we can calculate  $L_{R}$  by taking expectation over distribution  $g_{L}$ ,
+
+$$
+L _ {R} = \int \sum_ {i = K - r + 1} ^ {K} \lambda_ {i} ^ {\prime} g _ {L} (\Lambda^ {2}) d \lambda_ {1} ^ {\prime} \dots d \lambda_ {K} ^ {\prime}.
+$$
+
+Note that if  $K / 2\in \mathbb{N}$ , there are in total  $C_{K^2 /2 + 2}^{Kr}$  possible choice of  $Id^{(1)}$  with equal probability.
+
+$$
+\begin{array}{l} \mathbb {E} _ {I d ^ {(1)}} \left[ \mathbb {E} _ {W \sim G} \left(K ^ {2} - L _ {F} ^ {(1)}\right) \right] \\ = \frac {1}{C _ {K ^ {2} / 2 + 2} ^ {K r}} \sum_ {I d ^ {(1)}} \mathbb {E} _ {W \sim G} \left(K ^ {2} - L _ {F} ^ {(1)}\right) \\ = \frac {1}{C _ {K ^ {2} / 2 + 2} ^ {K r}} \sum_ {I d ^ {(1)}} \mathbb {E} _ {W \sim G} \left\{\sum_ {i d \in I d ^ {(1)} \cup \widetilde {I d} ^ {(1)}} | F _ {i d} | ^ {2} \right\} \\ = \frac {1}{C _ {K ^ {2} / 2 + 2} ^ {K r}} \sum_ {I d ^ {(1)}} \mathbb {E} _ {W \sim G} \left[ \sum_ {i = 1} ^ {K} \sum_ {j = 1} ^ {K} | F _ {i j} | ^ {2} \mathbb {1} \left\{(i, j) \in I d ^ {(1)} \cup \widetilde {I d} ^ {(1)} \right\} \right] \\ = \frac {1}{C _ {K ^ {2} / 2 + 2} ^ {K r}} \mathbb {E} _ {W \sim G} \left[ \sum_ {i = 1} ^ {K} \sum_ {j = 1} ^ {K} | F _ {i j} | ^ {2} \sum_ {I d ^ {(1)}} \mathbb {1} \left\{(i, j) \in I d ^ {(1)} \cup \widetilde {I d} ^ {(1)} \right\} \right] \\ = \frac {C _ {K ^ {2} / 2 + 1} ^ {K r - 1}}{C _ {K ^ {2} / 2 + 2} ^ {K r}} \mathbb {E} _ {W \sim G} \left(\sum_ {i = 1} ^ {K} \sum_ {j = 1} ^ {K} | F _ {i j} | ^ {2}\right) \\ = \frac {K ^ {3} r}{K ^ {2} / 2 + 2} <   2 K r, \\ \end{array}
+$$
+
+which aligns with intuition that random choice gives average performance. Similarly, if  $(K + 1) / 2\in \mathbb{N}$ , there are in total  $C_{(K^2 +1) / 2}^{Kr}$  possible choices of  $Id^{(1)}$  with equal probability. And
+
+$$
+\mathbb {E} _ {I d ^ {(1)}} \mathbb {E} _ {W \sim G} \left(K ^ {2} - L _ {F} ^ {(1)}\right) = \frac {K ^ {3} r}{(K ^ {2} + 1) / 2} <   2 K r.
+$$
+
+On the other hand,
+
+$$
+\begin{array}{l} \mathbb {E} _ {W \sim G} \left(K ^ {2} - L _ {R}\right) = \mathbb {E} _ {W \sim G} \left(\sum_ {i = 1} ^ {r} | \lambda_ {i} | ^ {2}\right) \\ = \int g _ {L} \left(\Lambda^ {2}\right) \sum_ {i = 1} ^ {r} \lambda_ {i} ^ {\prime} d \lambda_ {K} ^ {\prime} \dots d \lambda_ {1} ^ {\prime}. \\ \end{array}
+$$
+
+This calculation is complicated and does not have a closed-form expression. Next, we demonstrate
+
+$$
+\mathbb {E} _ {W \sim G} \left(K ^ {2} - L _ {R}\right) > \mathbb {E} _ {I d ^ {(1)}} \mathbb {E} _ {W \sim G} \left(K ^ {2} - L _ {F} ^ {(1)}\right).
+$$
+
+We begin by proving that this inequality holds for the case where  $r = 1$  and  $K$  is sufficiently large. Following this, we extend our analysis by numerically approximating the exact values of the integrals for various combinations of  $r$  and  $K$ . We first prove that for  $r = 1$  and sufficiently large  $K$ , the inequality  $\mathbb{E}_{W\sim G}|\lambda_1|^2 = \mathbb{E}_{W\sim G}\lambda_1' > 2Kr$  holds.  $\lambda_1',\ldots ,\lambda_K'$  has density
+
+$$
+g _ {L} \left(\Lambda^ {2}\right) = \left(\frac {\pi}{2}\right) ^ {K ^ {2} / 2} \frac {1}{\Gamma_ {K} ^ {2} (K / 2)} \left[ \prod_ {i = 1} ^ {K} \lambda_ {i} ^ {\prime - 1 / 2} e ^ {- \lambda_ {i} ^ {\prime} / 2} \right] \left[ \prod_ {i <   j} \left| \lambda_ {i} ^ {\prime} - \lambda_ {j} ^ {\prime} \right| \right], \tag {10}
+$$
+
+and  $\lambda_1^\prime$  is the largest eigenvalue of a standard Wishart ensemble. We refer to the large deviation result under this circumstance that for large  $K$  there exists  $c\leq 1$  and
+
+$$
+\lambda_ {1} ^ {\prime} = \left(\frac {1}{\sqrt {c}} + 1\right) ^ {2} K + c ^ {1 / 6} \left(\frac {1}{\sqrt {c}} + 1\right) ^ {4 / 3} K ^ {1 / 3} \chi , \tag {11}
+$$
+
+where the random variable  $\chi$  has an  $K$ -independent limiting distribution, which is Tracy-Widom distribution (Vivo et al., 2007; Johnstone, 2001; Johansson, 2000). Take expectation on both sides of Eq. (11) and
+
+$$
+\mathbb {E} _ {W \sim G} \left(K ^ {2} - L _ {R}\right) = \mathbb {E} \lambda_ {1} ^ {\prime} = \left(\frac {1}{\sqrt {c}} + 1\right) ^ {2} K + O (K ^ {1 / 3}).
+$$
+
+Thus  $\frac{\mathbb{E}_{W\sim G}(1 - L_R)}{K}\to \left(\frac{1}{\sqrt{c}} +1\right)^2\geq 4 > 2$ , which concludes the first inequality in Theorem 1. For  $r = 1$  but not sufficiently large  $K$ , we directly calculate the  $\mathbb{E}\lambda_1^{\prime}$  and compare it with  $2K$ . For  $r > 1$  we can apply similar analysis but that will be much more complex. We demonstrate the result in later numerical approximation (Fig. 6 and 7).
+
+Now we turn to  $L_F^{(i)}, i = 1,2,3$ . Remember we have
+
+$$
+\begin{array}{l} \mathcal {F} (W) = \left\{R e (H) + i I m (H) \right\} W \left\{R e (H) + i I m (H) \right\} \\ = \left\{R e (H) W R e (H) - I m (H) W I m (H) \right\} + i \left\{I m (H) W R e (H) + R e (H) W I m (H) \right\} \\ = R e (\mathcal {F} (W)) + i I m (\mathcal {F} (W)). \\ \end{array}
+$$
+
+After vectorization,
+
+$$
+R e (\mathcal {F} (W)) ^ {V} = \left\{R e (H) \otimes R e (H) - I m (H) \otimes I m (H) \right\} W ^ {V},
+$$
+
+$$
+I m (\mathcal {F} (W)) ^ {V} = \{R e (H) \otimes I m (H) + I m (H) \otimes R e (H) \} W ^ {V}.
+$$
+
+As  $W^{V} \sim N_{K^{2}}(0, I_{K^{2}})$ , and the linear transform of multivariate normal is still normal, we have
+
+$$
+R e (\mathcal {F} (W)) ^ {V} \sim N _ {n ^ {2}} (0, \Sigma_ {R}), I m (\mathcal {F} (W)) ^ {V} \sim N _ {n ^ {2}} (0, \Sigma_ {I}), \text {w h e r e}
+$$
+
+$$
+\Sigma_ {R} = \left\{R e (H) \otimes R e (H) - I m (H) \otimes I m (H) \right\} \left\{R e (H) \otimes R e (H) - I m (H) \otimes I m (H) \right\} ^ {T},
+$$
+
+$$
+\Sigma_ {I} = \left\{R e (H) \otimes I m (H) + I m (H) \otimes R e (H) \right\} \left\{R e (H) \otimes I m (H) + I m (H) \otimes R e (H) \right\} ^ {T}.
+$$
+
+Next we propose that  $Re(H) \operatorname{Im}(H) = 0$ .
+
+Lemma 6. For any  $K$ ,  $H$  is the 2d DFT  $K \times K$  matrix defined by
+
+$$
+H _ {u, v} = \frac {1}{\sqrt {K}} \left\{\cos (2 \pi u v / K) - i \sin (2 \pi u v / K) \right\},
+$$
+
+we have  $Re(H) \operatorname{Im}(H) = 0$ .
+
+Proof. First, let us denote the real part  $R$  and the imaginary part  $I$  of  $H$  as follows:
+
+$$
+R _ {u, v} = \frac {1}{\sqrt {K}} \cos \left(\frac {2 \pi u v}{K}\right), I _ {u, v} = - \frac {1}{\sqrt {K}} \sin \left(\frac {2 \pi u v}{K}\right)
+$$
+
+We calculate the matrix product  $R \cdot I$ , where  $R$  and  $I$  are  $K \times K$  matrices. The element of the resulting matrix at location  $(u, w)$  is given by:
+
+$$
+(R I) _ {u, w} = \sum_ {v = 0} ^ {K - 1} R _ {u, v} I _ {v, w}.
+$$
+
+Substituting the expressions for  $R$  and  $I$ :
+
+$$
+\begin{array}{l} (R I) _ {u, w} = \sum_ {v = 0} ^ {K - 1} \left\{\frac {1}{\sqrt {K}} \cos \left(\frac {2 \pi u v}{K}\right) \right\} \left\{- \frac {1}{\sqrt {K}} \sin \left(\frac {2 \pi v w}{K}\right) \right\} \\ = - \frac {1}{K} \sum_ {v = 0} ^ {K - 1} \cos \left(\frac {2 \pi u v}{K}\right) \sin \left(\frac {2 \pi v w}{K}\right). \\ \end{array}
+$$
+
+Next, we use the trigonometric identity that  $\cos (x)\sin (y) = [\sin (x + y) - \sin (x - y)] / 2$ . Applying this identity, we have
+
+$$
+\cos \left(\frac {2 \pi u v}{K}\right) \sin \left(\frac {2 \pi v w}{K}\right) = \frac {1}{2} \left\{\sin \left(\frac {2 \pi u v}{K} + \frac {2 \pi v w}{K}\right) - \sin \left(\frac {2 \pi u v}{K} - \frac {2 \pi v w}{K}\right) \right\}.
+$$
+
+Substituting back, we get
+
+$$
+(R I) _ {u, w} = - \frac {1}{2 K} \sum_ {v = 0} ^ {K - 1} \left\{\sin \left(\frac {2 \pi (u + w) v}{K}\right) - \sin \left(\frac {2 \pi (u - w) v}{K}\right) \right\} = 0.
+$$
+
+This lemma gives  $Re(H) \operatorname{Im}(H) = \operatorname{Im}(H) \operatorname{Re}(H) = 0$ . Therefore
+
+$$
+\begin{array}{l} \{R e (H) \otimes R e (H) - I m (H) \otimes I m (H) \} \{R e (H) \otimes I m (H) + I m (H) \otimes R e (H) \} \\ = \left\{R e (H) \right\} ^ {2} \otimes R e (H) I m (H) + R e (H) I m (H) \otimes \left\{R e (H) \right\} ^ {2} - \\ I m (H) R e (H) \otimes \left\{I m (H) \right\} ^ {2} - \left\{I m (H) \right\} ^ {2} \otimes I m (H) R e (H) \\ = 0, \\ \end{array}
+$$
+
+which indicates  $\Sigma_R\Sigma_I = 0$ , due to the normality,  $Re(\mathcal{F}(W))$  is independent of  $Im(\mathcal{F}(W))$ .  $H$  has slightly different property when  $K$  is an odd or even number. For the simplicity of proof, we assume  $K / 2 \in \mathbb{N}$ , the odd case can be proved similarly.
+
+Lemma 7. When  $K / 2 \in \mathbb{N}$ ,  $Re(H)Re(H)^T$  and  $Im(H)Im(H)^T$  have the following property:
+
+$$
+\left\{R e (H) R e (H) ^ {T} \right\} _ {u, w} = \left\{ \begin{array}{l} 1, u = w = 0, K / 2, \\ \frac {1}{2}, u = w \neq 0, K / 2, \\ \frac {1}{2}, u \neq w, u + w = K, \\ 0, o t h e r w i s e. \end{array} \right.
+$$
+
+$$
+\bigl\{I m(H)I m(H)^{T}\bigr \}_{u,w} = \left\{ \begin{array}{l}0,u = w = 0,K / 2,\\ \frac{1}{2},u = w\neq 0,K / 2,\\ -\frac{1}{2},u\neq w,u + w = K,\\ 0,otherwise. \end{array} \right.
+$$
+
+Proof. Follow previous notations,
+
+$$
+\left(R R ^ {T}\right) _ {u, w} = \frac {1}{2 K} \sum_ {v = 0} ^ {K - 1} \left\{\cos \left(\frac {2 \pi (u + w) v}{K}\right) + \cos \left(\frac {2 \pi (u - w) v}{K}\right) \right\}.
+$$
+
+First we get  $\left(RR^{T}\right)_{0,0} = \left(RR^{T}\right)_{K / 2,K / 2} = 1$ . When  $u = w \neq 0, K / 2$ ,
+
+$$
+\left(R R ^ {T}\right) _ {u, w} = \frac {1}{2 K} \sum_ {v = 0} ^ {K - 1} \cos \left(\frac {2 \pi (u + w) v}{K}\right) + \frac {1}{2} = \frac {1}{2},
+$$
+
+since  $K \nmid (u + w)$ . When  $u \neq w$  but  $u + w = K$ ,
+
+$$
+\left(R R ^ {T}\right) _ {u, w} = \frac {1}{2 K} \sum_ {v = 0} ^ {K - 1} \cos \left(\frac {2 \pi (u - w) v}{K}\right) + \frac {1}{2} = \frac {1}{2},
+$$
+
+since  $K \nmid (u - w)$ . For other  $u, w$ , it is easy to derive  $\left(RR^T\right)_{u,w} = 0$ .
+
+Moreover,  $H\overline{H}^T = I_K$ , where  $\bar{\cdot}$  means conjugation, indicating that  $R R^T + I I^T = I_K$ , and we get the result for  $II^T$ .
+
+As  $Re(H)Im(H) = Im(H)Re(H) = 0$ , we can calculate
+
+$$
+\Sigma_ {R} = \left\{R e (H) R e (H) ^ {T} \right\} \otimes \left\{R e (H) R e (H) ^ {T} \right\} + \left\{I m (H) I m (H) ^ {T} \right\} \otimes \left\{I m (H) I m (H) ^ {T} \right\},
+$$
+
+$$
+\Sigma_ {I} = \left\{R e (H) R e (H) ^ {T} \right\} \otimes \left\{I m (H) I m (H) ^ {T} \right\} + \left\{I m (H) I m (H) ^ {T} \right\} \otimes \left\{R e (H) R e (H) ^ {T} \right\}.
+$$
+
+Based on Lemma (7), we can assert that there are 4 locations in  $\Sigma_R$  containing the element 1. These locations are  $(0,0)$ ,  $(K/2, K/2)$ ,  $(K^2/2, K^2/2)$ , and  $((K^2 + K)/2, (K^2 + K)/2)$ . Excluding rows and columns  $0$ ,  $K/2$ ,  $K^2/2$ , and  $(K^2 + K)/2$ , each of the remaining rows and columns contains 2 locations with the value 0.5. There exists a row permutation matrix  $U \in \mathbb{R}^{K \times K}$ , such that
+
+$$
+U \Sigma_ {R} U ^ {T} = \left( \begin{array}{c c c c c} I _ {4} & & & & \\ & \Delta_ {2} & & & \\ & & \Delta_ {2} & & \\ & & & \ddots & \\ & & & & \Delta_ {2} \end{array} \right), \quad \Delta_ {2} = \left( \begin{array}{c c} 0. 5 & 0. 5 \\ 0. 5 & 0. 5 \end{array} \right). \tag {12}
+$$
+
+Since
+
+$$
+\begin{array}{l} \Sigma_ {R} + \Sigma_ {I} = \left\{R e (H) R e (H) ^ {T} + I m (H) I m (H) ^ {T} \right\} \otimes \left\{R e (H) R e (H) ^ {T} + I m (H) I m (H) ^ {T} \right\} \\ = \left(H \bar {H} ^ {T}\right) \otimes \left(H \bar {H} ^ {T}\right) = I _ {K} \otimes I _ {K} = I _ {K ^ {2}}, \\ \end{array}
+$$
+
+we have similar results on  $\Sigma_I$  that
+
+$$
+U \Sigma_ {I} U ^ {T} = \left( \begin{array}{c c c c c} \mathbf {0} _ {4} & & & & \\ & \Delta_ {2} ^ {-} & & & \\ & & \Delta_ {2} ^ {-} & & \\ & & & \ddots & \\ & & & & \Delta_ {2} ^ {-} \end{array} \right), \quad \Delta_ {2} ^ {-} = \left( \begin{array}{c c} 0. 5 & - 0. 5 \\ - 0. 5 & 0. 5 \end{array} \right). \tag {13}
+$$
+
+This analysis aligns with the definitions of  $F^R$  and  $F^I$ . Given that  $W^V$  follows a standard normal distribution and  $\Sigma_R \Sigma_I = 0$ , we can represent  $\Sigma_R$  and  $\Sigma_I$  as shown in Eq. (12) and Eq. (13), respectively. Let  $R$  be the reference matrix, for  $i, j$  with  $R_{ij} = 0$ , the  $i, j$ -th element corresponds to the element with variance 1, and  $F_{ij}^R \sim \chi_1^2$ ,  $F_{ij}^I = 0$ ; for  $i, j$  with  $R_{ij} = 1$ ,  $F_{ij}^R$ ,  $F_{ij}^I \sim \chi_1^2$ ; for  $i, j$  such that  $R_{ij} = -1$ ,  $F_{ij}^R = F_{ij}^I = 0$ . And for all  $i, j$  with  $R_{ij} \neq -1$ ,  $F_{ij}^R$  and  $F_{ij}^I$  are independent.
+
+When  $R_{ij} = 0$ ,  $\left|F_{ij}^{H}\right|^{2} = Re(F_{ij})^{2} \sim \chi_{1}^{2}$ ; when  $R_{ij} = 1$ ,  $\left|F_{ij}^{H}\right|^{2} = 2Re(F_{ij})^{2} + 2Im(F_{ij})^{2} \sim \chi_{2}^{2}$ . Thus we can reformulate  $L_{F}^{(2)}$  and  $L_{F}^{(3)}$  in a more clear way. Define  $\psi_{1}, \ldots, \psi_{K^{2}} \stackrel{i.i.d.}{\sim} \chi_{1}^{2}$ ,  $\phi_{1}, \ldots, \phi_{(K^{2}-4)/2} \stackrel{i.i.d.}{\sim} \chi_{2}^{2}, \phi_{(K^{2}-2)/2}, \ldots, \phi_{(K^{2}+4)/2} \stackrel{i.i.d.}{\sim} \chi_{1}^{2}$ . Denote the order statistics of  $\psi_{i}, \phi_{i}$  as  $\psi_{(1)} \geq \ldots \geq \psi_{(K^{2})}$  and  $\phi_{(1)} \geq \ldots \geq \phi_{((K^{2}+4)/2)}$ , we then have
+
+$$
+L _ {F} ^ {(2)} \stackrel {{d.}} {{=}} \sum_ {i = N _ {2} + 1} ^ {\left(K ^ {2} + 4\right) / 2} \phi_ {(i)}, L _ {F} ^ {(3)} \stackrel {{d.}} {{=}} \sum_ {i = N _ {3} + 1} ^ {K ^ {2}} \psi_ {(i)}, \tag {14}
+$$
+
+where  $\stackrel{d}{=}$  means equality in distribution. In other words,
+
+$$
+\mathbb {E} _ {W \sim G} \left(K ^ {2} - L _ {F} ^ {(2)}\right) = \sum_ {i = 1} ^ {N _ {2}} \mathbb {E} \phi_ {(i)}, \mathbb {E} _ {W \sim G} \left(K ^ {2} - L _ {F} ^ {(3)}\right) = \sum_ {i = 1} ^ {N _ {3}} \mathbb {E} \psi_ {(i)},
+$$
+
+i.e.,  $\mathbb{E}_{W\sim G}\left(K^2 -L_F^{(3)}\right)$  is the summation of i.i.d. chi square order statistics' expectation. Similarly, we can bound  $\mathbb{E}_{W\sim G}\left(K^2 -L_F^{(2)}\right)$ , by defining
+
+$$
+\xi_ {1} ^ {(1)}, \ldots , \xi_ {(K ^ {2} - 4) / 2} ^ {(1)}, \xi_ {1} ^ {(2)}, \ldots , \xi_ {(K ^ {2} + 4) / 2} \sim \chi_ {2} ^ {2},
+$$
+
+and corresponding order statistics
+
+$$
+\xi_ {(1)} ^ {(1)} \geq \dots \geq \xi_ {((K ^ {2} - 4) / 2)} ^ {(1)}, \xi_ {(1)} ^ {(2)} \geq \dots \geq \xi_ {((K ^ {2} + 4) / 2)} ^ {(2)}.
+$$
+
+Define  $M_1 = \sum_{i=1}^{N_2} \mathbb{E}\xi_{(i)}^{(1)}$  and  $M_2 = \sum_{i=1}^{N_2} \mathbb{E}\xi_{(i)}^{(2)}$ , we have  $M_1 \leq \mathbb{E}_{W \sim G}\left(K^2 - L_F^{(2)}\right) \leq M_2$ .
+
+Lemma 8. For any  $n$  i.i.d. random variables with pdf  $f(x)$  and cdf  $H(x)$ , the  $l$ -th largest order statistic has density  $h_l(x) = nC_{n-1}^{l-1}h(x)H(x)^{l-1}\left\{1 - H(x)\right\}^{n-l}$ .
+
+We claim that for given  $r < K / 3$
+
+$$
+\sum_ {i = 1} ^ {N _ {3}} \mathbb {E} \psi_ {(i)} \geq M _ {2} \geq M _ {1} \geq \int g _ {L} \left(\Lambda^ {2}\right) \sum_ {i = 1} ^ {r} \lambda_ {i} ^ {\prime} d \lambda_ {K} ^ {\prime} \dots d \lambda_ {1} ^ {\prime}, \tag {15}
+$$
+
+where  $g_{L}(\Lambda^{2})$  is given in Eq. (10). We verify this inequality by numerical calculation, since each item in Eq. (15) is already a closed form integration. Specifically, we compare the ratios  $\frac{L}{K^2}$  for various combinations of  $K$  and  $r$ , where  $L$  represents  $L_{R}, L_{F}^{(1)}, L_{F}^{(2)}$ , and  $L_{F}^{(3)}$ . For commonly used
+
+$r$  values, we examined  $K$  from 100 to 300, while for larger matrices with  $K = 768$  and  $K = 4096$ , we tested  $r$  values from 8 to 32. Throughout these tests, we employ specific definitions for the different  $L$  values:  $L_F^{(1)} = 2Kr$ ,  $K^2 - M_2 \leq L_F^{(2)} \leq K^2 - M_1$ , and  $L_F^{(3)} = L_D$ , with the last definition verified by Theorem 2.
+
+Remark. Given that the four integrals in Eq. (15) are not easily expressed in a straightforward manner, directly proving the inequality is impractical. Beyond numerical approximation for commonly used  $r$  and  $K$  in Fig. 6 and 7, we offer an intuitive illustration to show why the inequality holds.
+
+A tight bound on order statistics is given by Arnold & Groeneveld (1979); Bertsimas et al. (2006): for  $X_{1}, \dots, X_{n}$  i.i.d. with expectation  $\mu$  and variance  $\sigma^2$ , the expectation of  $l$ -th order statistic is bounded by  $\mu + \sigma \sqrt{\frac{n - l}{l}}$ . Consider using this bound to approximate  $\sum_{i=1}^{N_3} \mathbb{E}\psi_{(i)}$  and  $M_{1}, M_{2}$ :
+
+$$
+n _ {1} = K ^ {2}, \mu_ {1} = \mathbb {E} \psi_ {i} = 1, \sigma_ {1} = \sqrt {\operatorname {V a r} (\psi_ {i})} = \sqrt {2},
+$$
+
+$$
+n _ {2} = K ^ {2} / 2 + 2, \mu_ {2} = \mathbb {E} \xi_ {i} ^ {(2)} = 2, \sigma_ {2} = \sqrt {V a r (\xi_ {i} ^ {(2)})} = 2.
+$$
+
+Thus
+
+$$
+\begin{array}{l} 2 \sum_ {i = 1} ^ {2 K r / 3} \sqrt {\frac {K ^ {2} / 2 + 2 - i}{i}} = \sqrt {2} \sum_ {i = 1} ^ {2 K r / 3} \sqrt {\frac {K ^ {2}}{i} + \frac {4}{i} - 2} \\ \approx \sqrt {2} \sum_ {i = 1} ^ {2 K r / 3} \sqrt {\frac {K ^ {2}}{i} - 1} \\ <   \sqrt {2} \sum_ {i = 1} ^ {K r} \sqrt {\frac {K ^ {2}}{i} - 1}, \\ \end{array}
+$$
+
+which gives the upper bound of  $M_2$  is smaller than that of  $\sum_{i=1}^{N_3} \mathbb{E}\psi_{(i)}$ .
+
+![](images/6366d63bf7bb208c8ee53641a29486dac6535a56caa1a1759bfe24c61a2a9b7c.jpg)  
+Figure 6: Reconstruction errors of different  $r$ ,  $K$  and methods. Specify  $r = 8,16,24,32$  and  $K \in [100,300]$ . R denotes the low rank method, the curve is  $L_{R} / K^{2}$ ; M1 and M2 denotes  $1 - M_1 / K^2, 1 - M_2 / K^2$  respectively; D denotes  $L_F^{(3)} / K^2$  or  $L_D / K^2$ ; U denotes  $1 - 2r / K$ .
+
+![](images/0002be8b356eebe0dc43b826f386c945ab067f6863e2c00e5e1c642ef33bdd5e.jpg)  
+Figure 7: Reconstruction errors of different  $r$ ,  $K$  and methods. Specify  $K = 768, 4096$  and  $r \in [8,32]$ . R denotes the low rank method, the curve is  $L_{R} / K^{2}$ ; M1 and M2 denotes  $1 - M_{1} / K^{2}$ ,  $1 - M_{2} / K^{2}$  respectively; D denotes  $L_{F}^{(3)} / K^{2}$  or  $L_{D} / K^{2}$ ; U denotes  $1 - 2r / K$ .
+
+# H PROOF OF THEOREM 2
+
+Proof. Theorem 2 is a corollary of Eq. (14). For notation simplicity, denote the expectation of reconstruction loss of DCT method as  $L_{D} = \mathbb{E}_{W\sim G}\left\{L(W,\hat{W}_{D})\right\}$
+
+Denote discrete cosine transform as  $D = \mathcal{D}(W) = QWQ^T$ , where  $Q \in \mathbb{R}^{K \times K}$  is the DCT matrix satisfies  $QQ^T = I_K$ . Vectorize  $D$  we get
+
+$$
+D ^ {V} = (Q \otimes Q) W ^ {V} \sim \mathcal {N} _ {K ^ {2}} (0, \Sigma_ {D}),
+$$
+
+where  $\Sigma_D = (Q\otimes Q)(Q\otimes Q)^T = (QQ^T)\otimes (QQ^T) = I_{K^2}$ .
+
+Denote the order statistics of  $D$ 's elements as  $D_{(1)} \geq \ldots \geq D_{(K^2)}$ . The Parseval theorem also holds for DCT, thus
+
+$$
+L _ {D} = \mathbb {E} _ {W \sim G} \left\{\sum_ {i = N _ {D} + 1} ^ {K ^ {2}} | D _ {(i)} | ^ {2} \right\} = K ^ {2} - \mathbb {E} _ {W \sim G} \left\{\sum_ {i = 1} ^ {N _ {D}} | D _ {(i)} | ^ {2} \right\}.
+$$
+
+Under the situation of  $W \sim G$ ,  $|D_{ij}|^2 \sim \chi_1^2$  and  $K^2 - L_D$  is the expectation of the largest  $N_D$  out of  $K^2$  random  $\chi_1^2$  variables, which exactly equals to the  $K^2 - L_F^{(3)}$  in Eq. (14) when  $N_D = N_3$ .
+
+# I COMPUTATIONAL EFFICIENCY OF GRADIENT ESTIMATION
+
+Recall that the 2D iDCT can be represented as  $\Delta W = \alpha [C^T S(\pmb {a},\pmb {l},\mathbf{1})D]$ , here  $C^T$  and  $D$  are iDCT transformation matrices for rows and columns respectively. We show that the gradient of location  $\pmb{l}$  is computed alongside with the gradient of  $\pmb{a}$ , introducing only negligible additional computations.
+
+For ease of representation, we denote the sparse matrix  $S(\pmb{a}, \pmb{l}, \mathbf{1})$  as  $W_{s}$ . We first show how a change at location  $(i,j)$  in  $W_{s}$  affects location  $(m,n)$  in  $\Delta W^{2}$ :
+
+$$
+\frac {\partial \Delta W [ m , n ]}{\partial W _ {s} [ i , j ]} = \alpha C ^ {T} [ m, i ] D [ j, n ]. \tag {16}
+$$
+
+Now, consider  $\partial \mathcal{L} / \partial \Delta W$  that we get during backpropagation, if we want to get the gradient of an element in  $\pmb{a}$  (indexed by  $i,j$ ), we need to compute:
+
+$$
+\frac {\partial \mathcal {L}}{\partial W _ {s} [ i , j ]} = \sum_ {m, n} \left(\frac {\partial \mathcal {L}}{\partial \Delta W [ m , n ]} \frac {\partial \Delta W [ m , n ]}{\partial W _ {s} [ i , j ]}\right). \tag {17}
+$$
+
+Expanding Eq. (17), we have
+
+$$
+\frac {\partial \mathcal {L}}{\partial W _ {s} [ i , j ]} = \alpha \sum_ {m, n} \left(\frac {\partial \mathcal {L}}{\partial \Delta W [ m , n ]} C ^ {T} [ m, i ] D [ j, n ]\right) = \alpha \underbrace {\left(D \left(\frac {\partial \mathcal {L}}{\partial \Delta W}\right) ^ {T} C ^ {T}\right) ^ {T}} _ {D C T, \text {m a t r i x} Z} [ i, j ]. \tag {18}
+$$
+
+Therefore, to get the gradient of  $\mathbf{a}$ , we also need to compute the matrix  $Z$  in Eq. (5). The gradient of each element in  $\mathbf{a}$  can be directly indexed by locations, while the gradient of each element in  $l$  can be estimated according to Section 4.3, which will introduce only negligible additional computation.
+
+# J COMPUTATIONAL COMPLEXITY AND MEMORY COST COMPARISON
+
+As discussed in Section 4.2, the original implementation of DCT, i.e., Eq. (1) can take two enhanced forms depending on the sparsity of the DCT spectrum: a sparse matrix-based implementation and a fast algorithm-based implementation. The computational complexity of using the sparse matrix implementation is  $O(\mathcal{B}pq)$ , where  $\mathcal{B}$  is the number of frequency components, and  $p$  and  $q$  are the dimensions of the weight matrix. The fast algorithm-based implementation has a complexity of  $O(pq\log(pq))$ . It is worth noting that PyTorch currently lacks a specialized fast algorithm for DCT. To address this, we implemented a fast DCT based on FFT. In comparison, LoRA has a complexity of  $O(rpq)$ , where  $r$  is the rank. FourierFT, which utilizes iFFT algorithm, has an asymptotic complexity of  $O(pq\log(pq))$ .
+
+From an asymptotic analysis perspective, the fast implementations of LoCA and FourierFT have the same complexity, while the complexity of LoRA is lower when  $r < \log(pq)$ . However, noting that the practical performance can differ significantly from theoretical asymptotic analysis due to various factors such as implementation details, hardware-specific optimizations, the constant coefficient in computation complexity and the actual values of  $\mathcal{B}$ ,  $r$ , and  $pq$ . In our experimental comparisons, we observed that the actual running times of these methods are much closer than the asymptotic analysis might suggest. Table 10 presents a comparison of the three methods.
+
+Table 10: Comparison of actual training speed and memory costs on a single Tesla H100 GPU.  $\mathsf{LoCA}^1$  represents the sparse matrix-based iDCT implementation, and  $\mathsf{LoCA}^2$  refers to the fast iDCT implementation based on iFFT.  $\mathsf{LoCA}^3$  is the DCT implementation in the original matrix multiplication form (default implementation). All experimental configurations are consistent with the ones in main experiments.
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Asymptotic Complexity</td><td colspan="2">MRPC RoBERTa-base</td><td colspan="2">Alpaca-52K LLaMA-1-7b</td><td colspan="2">StanfordCars ViT-base</td></tr><tr><td>Training Speed (iterations/s)</td><td>Memory Cost (MB)</td><td>Training Speed (iterations/s)</td><td>Memory Cost (MB)</td><td>Training Speed (iterations/s)</td><td>Memory Cost (MB)</td></tr><tr><td>LoCA1</td><td>O(B log(pq))</td><td>28.56</td><td>3,828</td><td>-</td><td>-</td><td>2.28</td><td>4,214</td></tr><tr><td>LoCA2</td><td>O(pq log(pq))</td><td>25.12</td><td>3,834</td><td>0.63</td><td>57,152</td><td>1.01</td><td>3,782</td></tr><tr><td>LoCA3</td><td>O(p2q2)</td><td>27.77</td><td>3,793</td><td>0.87</td><td>57,888</td><td>2.33</td><td>3,754</td></tr><tr><td>FourierFT</td><td>O(pq log(pq))</td><td>28.82</td><td>4,050</td><td>0.89</td><td>58,868</td><td>2.35</td><td>3,760</td></tr><tr><td>LoRA</td><td>O(rpq)</td><td>31.14</td><td>3,758</td><td>1.18</td><td>53,154</td><td>2.78</td><td>3,708</td></tr></table>
+
+As shown in Table 10, despite the differences in asymptotic complexities, the actual running speeds of LoCA and FourierFT are very close, with LoRA being slightly faster (since the matrix multiplication operation is highly optimized on the GPU). This suggests that the implementation efficiency and hardware utilization play significant roles in practical performance. For the memory consumption, both LoCA and FourierFT exhibit marginally higher memory usage compared to LoRA. However, LoCA consistently maintains a lower memory footprint than FourierFT across all test scenarios.
+
+Notably, there is still potential for further optimization for our method. Since the current fast DCT implementation is based on FFT, a lot of redundant computation is introduced to construct a DCT into the form of a DFT. A specialized fast DCT algorithm could potentially improve the speed of LoCA even further, potentially bringing its speed closer to or even surpassing that of LoRA while maintaining its superior adaptation capability. For FourierFT, while FFT can handle complex data, it introduces computational redundancies when processing real-valued data, making DCT theoretically superior in terms of both computational speed and memory usage for real-number operations. We leave the development and implementation of such optimizations as future work.
+
+# K EXAMPLES OF INSTRUCTION TUNING EXPERIMENTS
+
+Fig. 8 and Fig. 9 show multi-turn conversations on two MT-Bench questions of a fine-tuned LLaMA model using LoCA and a pre-trained LLaMA model. From the dialogue, we can see that the instruction-following ability of the pre-trained LLaMA is quite limited. The first-turn answer does not perfectly address the given question, and in the second round, it barely responds according to the user request at all. However, after LoCA fine-tuning, there is a noticeable improvement in the quality of the answers, which can be seen from the evaluation of GPT-4.
+
+# Turn 1 Please describe the concept of machine learning. Could you elaborate on the differences between supervised, unsupervised, and reinforcement learning? Provide real-world examples of each.
+
+![](images/a9de9c30fc5f164037e2ae092b71ab41eb32e00e7a66ef46891fef106ea3d34e.jpg)
+
+Machine learning is a subfield of computer science that focuses on the development of algorithms that can learn from data. It is a branch of artificial intelligence that deals with the design and development of algorithms that can learn from data and use the information to make predictions. Machine learning algorithms can be divided into three categories: supervised, unsupervised, and reinforcement learning. Supervised learning algorithms are trained using labeled data. The algorithms are given a set of labeled data, and they learn to make predictions based on the data. Unsupervised learning algorithms are trained using unlabeled data. The algorithms are given a set of unlabeled data, and they learn to make predictions based on the data. Reinforcement learning algorithms are trained using a reward system. The algorithms are given a set of unlabeled data, and they learn to make predictions based on the data. The algorithms are also given a reward system, and they learn to make predictions based on the data and the reward system. Supervised learning algorithms are used in applications like spam filtering, speech recognition, and computer vision. Unsupervised learning algorithms are used in applications like clustering, anomaly detection, and recommender systems. Reinforcement learning algorithms are used in applications like autonomous driving, robotics, and game playing.
+
+Machine learning is a subfield of artificial intelligence that involves the development of algorithms that can learn from and make predictions based on data. These algorithms are trained on large datasets and can improve their performance over time as they receive more data. There are three main types of machine learning: supervised learning, unsupervised learning, and reinforcement learning. Supervised Learning: In supervised learning, the algorithm is trained on labeled data, meaning that the correct output is provided for each input. The algorithm learns to predict the output for new data by finding patterns in the training data. For example, a supervised learning algorithm could be trained on a dataset of images of cats and dogs, and then be able to accurately classify new images as either cats or dogs. Unsupervised Learning: In unsupervised learning, the algorithm is trained on unlabeled data, meaning that the correct output is not provided for each input. The algorithm learns to find patterns and structure in the data on its own. For example, an unsupervised learning algorithm could be trained on a dataset of customer transactions, and then be able to identify clusters of similar customers based on their purchasing habits. Reinforcement Learning: In reinforcement learning, the algorithm learns by interacting with its environment and receiving feedback in the form of rewards or penalties. The algorithm learns to make decisions that maximize the rewards and minimize the penalties. For example, a reinforcement learning algorithm could be trained to play a game like chess or Go, where it learns to make moves that lead to victory by receiving rewards for winning games and penalties for losing games. In summary, machine learning is a powerful tool that can be used to make predictions and decisions based on data. The three main types of machine learning are supervised learning, unsupervised learning, and reinforcement learning, each with its own strengths and applications.
+
+![](images/d7a81f53a3bcc55c3cc54cf7cbafc402508afccc8eada90c53f3d4cdf80f2860.jpg)
+
+![](images/5c84e0ec6a79fed2ae3c186681144be42eabd3f6bcc6aef76a453ece1883fb10.jpg)
+
+The assistant's response provides a basic explanation of machine learning and the three primary categories: supervised, unsupervised, and reinforcement learning. However, there are several repetition and clarity issues. The response lacks depth and fails to sufficiently elaborate on how these learning methods differ. The examples provided are appropriate but could be expanded upon to provide more real-world context. There is also a factual inaccuracy in the description of reinforcement learning. Overall, the response is somewhat helpful and relevant but lacks accuracy, depth, and clarity.
+
+The assistant's response is accurate, relevant, and detailed. It provides a clear and concise definition of machine learning and elaborates on the differences between supervised, unsupervised, and reinforcement learning. The assistant also provides real-world examples for each type of machine learning, which helps to illustrate the concepts in a practical context. The response is comprehensive and informative, providing a good understanding of the topic. Therefore, I would rate this response highly.
+
+![](images/11a1d831430a9aa64395b9de10cc202044add06b22ba9d173064566f75f910df.jpg)
+
+Rating: [[9]]
+
+# Rating: [[5]]
+
+# Turn 2 In your last example of reinforcement learning, can we use supervised learning to solve it?
+
+![](images/93e5542b57e096993ab57817564d5c596776c58cd3849598a34be5790c7ad214.jpg)
+
+Reinforcement learning is a subfield of machine learning that deals with the design and development of algorithms that can learn from data and use the information to make predictions. Reinforcement learning algorithms are trained using a reward system. The algorithms are given a set of unlabeled data, and they learn to make predictions based on the data and the reward system.
+
+No, reinforcement learning is a type of machine learning that is distinct from supervised learning. Reinforcement learning involves learning from interactions with an environment and receiving feedback in the form of rewards or penalties. Supervised learning, on the other hand, involves training an algorithm on labeled data, where the correct output is provided for each input. While both methods can be used to solve problems, they are different approaches that are better suited to different types of problems.
+
+![](images/c07b1ce61168dac87cd3a236cd3e7b4ee6f2c767e447a84ef1fe09ee31941cb9.jpg)
+
+![](images/0e4e90ee2e8cba89788b9c5f64f09367577d8b09ef187d68565c03775921af96.jpg)
+
+The assistant's response is problematic. The assistant repeats the basic definition of reinforcement learning, which was already provided in the previous answer. This repetition does not address the question. The response does not answer the user's question about whether supervised learning can be used in place of reinforcement learning. It completely misses the opportunity to explain the fundamental differences between these approaches and when one might be preferable over the other. Overall, the response fails to address the user's question, repeats irrelevant information, and perpetuates an earlier inaccuracy.
+
+The assistant's response is accurate and relevant to the user's question. It correctly explains why supervised learning cannot be used in place of reinforcement learning, highlighting the differences between the two. The assistant also provides a clear and concise explanation of the unique characteristics of reinforcement learning and supervised learning. However, the assistant could have provided a more detailed explanation or example to illustrate why supervised learning would not be suitable for a problem typically solved by reinforcement learning, such as a game like chess or Go. Overall, the response is helpful and informative, but could benefit from additional depth and detail.
+
+![](images/1db4a5a2d25c0276f6073fbc368ffa1b3218d1b7c0ada00884c8020ef510a9d7.jpg)
+
+Rating: [[2]]
+
+Rating: [[8]]
+
+After Fine-Tuning
+
+Figure 8: Question 149 from MT-Bench with the multi-turn interactions of pre-trained LLaMA-7b (left) and fine-tuned LLaMA-7b using LoCA (right). We also attach the single-answer grading GPT-4 judgment for reference.
+
+# Turn 1
+
+![](images/02b68dcbf2568daf9272cd8d9d3dfed73b9d562b24a87f5fa85010751f6bce70.jpg)
+
+Draft a professional email seeking your supervisor's feedback on the 'Quarterly Financial Report' you prepared. Ask specifically about the data analysis, presentation style, and the clarity of conclusions drawn. Keep the email short and to the point.
+
+Here is a draft email you can send:\nSubject:
+
+Review and feedback on Quarterly Financial Report
+
+Dear [Supervisor],
+
+I have attached the Quarterly Financial Report for your review and feedback.
+
+As requested, I would like to specifically seek your feedback on the following areas:
+
+- Data analysis: Was the data accurate, relevant, and presented in a clear and concise manner?  
+- Presentation style: Was the layout of the report visually appealing and easy to follow?  
+- Clarity of conclusions drawn: Were the conclusions drawn from the data analysis clear and supported by the data?
+
+Thank you in advance for your valuable feedback. I look forward to discussing the report with you further
+
+Dear [Supervisor],  
+I hope this email finds you well. I am writing to seek your feedback on the 'Quarterly Financial Report' that I have prepared.  
+Firstly, I would like to express my gratitude for the opportunity to present the report to you. I have analyzed the data and presented the findings in a clear and concise manner. I believe that the report accurately reflects the financial performance of the company during the quarter.  
+Secondly, I would like to know if you find the presentation style of the report effective. I have used graphs and charts to highlight the key points and make the data easily understandable.  
+Lastly, I would appreciate your feedback on the conclusions drawn from the report. I have presented the findings in a logical manner and have provided recommendations for future action.  
+Thank you for taking the time to review the report. Please let me know if you have any further comments or questions. I look forward to your feedback.
+
+![](images/90fe0d893e0c4c203842c8be6b9a7e36558e70cb619d0a5940efa0bab847ad83.jpg)
+
+The assistant's response is relevant, accurate, and helpful. It provides a professional and concise draft email that addresses the user's request. The assistant specifically asks for feedback on the data analysis, presentation style, and clarity of conclusions drawn, as requested by the user. The assistant's response is also creative and detailed, providing a clear structure for the email and suggesting a subject line. Therefore, the assistant's response is of high quality.
+
+Rating: [[8]]
+
+The assistant's response is professional, concise, and to the point. It covers all the points requested by the user, including seeking feedback on the data analysis, presentation style, and clarity of conclusions drawn in the 'Quarterly Financial Report'. The assistant also expresses gratitude for the opportunity to present the report, which adds a nice touch of politeness and professionalism. The assistant's response is well-structured and easy to understand, making it an effective communication. Therefore, I would rate it highly.
+
+Rating: [[9]]
+
+![](images/93fd75f49ed890f6219ae971436ce69f74920a0d7a5a43ac26f1b0da700b1f82.jpg)
+
+# Turn 2
+
+![](images/5b4ce883a5824d1b43f8cae1ddd47e4e01ccf7002e9771ad5b6dbdf591d9beba.jpg)
+
+Take a moment to evaluate and critique your own response.
+
+![](images/265793f33d1b6189f3891e872a77ab201a38340e8118bba9c3e337d628601043.jpg)  
+Figure 9: Question 82 from MT-Bench with the multi-turn interactions of pre-trained LLaMA2-13b (left) and fine-tuned LLaMA2-13b using LoCA (right). We also attach the single-answer grading GPT-4 judgment for reference.
+
+Here are some things to consider when critiquing your own work:
+
+-Did you follow all the instructions and meet the stated requirements?  
+- Was the email concise and to the point?  
+- Was the subject line clear and appropriate?
+- Did you use professional and polite language in the email?  
+- Did you include any relevant attachments?  
+Was the tone appropriate for the situation? - Did you leave any typos or grammatical error
+
+By critiquing your own work, you can identify areas for improvement and refine your writing skills.
+
+My response was well-structured and professional. I addressed the specific points mentioned in the email and provided clear and concise information. The language used was appropriate and polite. Overall, the response was effective in conveying the message.
+
+![](images/c12d56f9cdeb3bef3a94c9def08232acfc0842acf1000c27fba60305727de118.jpg)
+
+![](images/b96365a5c17717aa97630c5d71a5ebde998b5fbcbcaa8ddab2663d692aec5b8f.jpg)  
+Fig. 10 shows the responses of LLaMA-2-7b models fine-tuned with LoCA and FourierFT on two open-ended questions from the Vicuna dataset. It can be observed that while both fine-tuned models answered the questions well, the LoCA fine-tuned model provided more detailed and insightful responses, as evidenced by GPT-4's evaluation.
+
+The assistant's response is relevant and accurate, as it provides a list of factors to consider when evaluating the quality of the email draft. However, it doesn't directly critique the assistant's own response to the user's first question. The assistant should have provided a self-evaluation of the email draft, pointing out its strengths and weaknesses based on the factors listed. The response lacks depth and detail because it doesn't provide a specific critique of the email draft. Therefore, the assistant's response is not as helpful as it could be.
+
+Rating: [[5]]
+
+The assistant's self-evaluation is accurate. The assistant's response to the user's request was indeed well-structured, professional, and addressed the specific points mentioned. The language used was appropriate and polite, and the message was effectively conveyed. The assistant demonstrated a good understanding of the task and executed it well. However, the assistant could have been more critical in its self-evaluation, perhaps pointing out areas where it could improve or change its approach. Nonetheless, the assistant's self-evaluation was largely accurate and fair.
+
+Rating: [[8]]
+
+![](images/bbf3a018bcbf20bacea9c0dd1c951d381827789f54669b857885f2e4907fcf9d.jpg)
+
+# Question 6
+
+![](images/1557241812f3b225fe4cf2aa3508a2d62a693640c57d7f17ca21dd2a6927948e.jpg)
+
+What are the differences between plant-based and animal-based protein sources?
+
+# Assistant A (Ours)
+
+![](images/62a49352c3e1f370b39686a39ef475380eef05c01f6afedb6a0fbfafe7e42cfc.jpg)
+
+Plant-based and animal-based protein sources differ in several ways:\n1. Nutritional Composition: Plant-based protein sources like beans, lentils, and nuts are generally richer in fiber, vitamins, and minerals, while animal-based protein sources like meat and eggs are richer in essential amino acids and other nutrients.\n2. Protein Quality: Plant-based protein sources are generally considered to have lower protein quality than animal-based sources. This is because they generally have a lower biological value, meaning that the body may not absorb as much of the protein from plant sources as it would from animal sources.\n3. Environmental Impact: Plant-based protein sources have a lower environmental impact than animal-based sources. Raising animals for food requires a lot of land, water, and feed, and produces greenhouse gases. Plant-based protein sources require less resources to produce and have a smaller environmental footprint.\n4. Processing and Cooking: Plant-based protein sources are generally easier to process and cook than animal-based sources. For example, it is easier to boil, steam or grill beans or lentils than to grill or fry meat.\n5. Cost: Plant-based protein sources are generally cheaper than animal-based sources. However, the cost may vary depending on the specific foods and the location.\nIn conclusion, both plant-based and animal-based protein sources have their advantages and disadvantages. It's important to choose sources that align with your dietary preferences, nutritional needs, and environmental values.
+
+# Assistant B (FourierFT)
+
+Plant-based and animal-based protein sources differ in several ways.  $\backslash$ First, the source of the protein. Animal-based protein comes from meat, fish, eggs, and dairy products, while plant-based protein comes from sources like legumes, nuts, seeds, and grains.  $\backslash$ Second, the nutritional value. Plant-based proteins tend to be richer in fiber, vitamins, and minerals, while animal-based proteins are generally higher in essential fatty acids and vitamin B12.  $\backslash$ Third, the impact on the environment. Animal-based protein production requires a lot of land, water, and resources, while plant-based protein production is generally more sustainable.  $\backslash$ Finally, the taste and texture. Animal-based proteins tend to be more flavorful and have a more satisfying texture, while plant-based proteins can vary in taste and texture depending on the source.  $\backslash$ In summary, plant-based and animal-based protein sources differ in their source, nutritional value, environmental impact, and taste and texture. Ultimately, it's up to the individual to choose the protein source that best meets their needs and preferences.
+
+![](images/b392101c170e121860af7795260db8295e4caa5f535f344ce102c13e515b6850.jpg)
+
+![](images/120c64808df5ce898317c8064cf5c09b72bdc809f67cb917e571dd0148219c95.jpg)
+
+Both assistants provide comprehensive and accurate responses to the user's question, detailing the differences between plant-based and animal-based protein sources. They both discuss the source of the protein, nutritional value, environmental impact, and other factors.  $\backslash \backslash$ However, Assistant A goes a step further by discussing the protein quality, processing and cooking, and cost, which Assistant B does not mention. Assistant A also explains the concept of biological value, which is an important aspect when discussing protein quality. Therefore, Assistant A provides a more detailed and informative response. Final Verdict: [[A]]
+
+# Question 7
+
+![](images/38f286b00e7db51c21a59b90d15f57fd6708eb956e3a4f3c2792899794f79345.jpg)
+
+How can I develop my critical thinking skills?
+
+# Assistant A (Ours)
+
+![](images/3ac54a3633149fc5b82c1b5ff1e86f7bfd840a1625c96345fc135153a7c33d9d.jpg)
+
+Here are some tips to help you develop your critical thinking skills:\n1. Read widely: Reading is an excellent way to develop critical thinking. Read widely, including nonfiction books, essays, and articles, to broaden your perspective and expose yourself to different ideas and arguments.\n2. Ask questions: Always ask questions, whether they be in your head or out loud. This will help you to challenge assumptions, explore different perspectives, and dig deeper into issues.\n3. Think for yourself: Don't accept everything you hear at face value. Use your own judgment and reasoning to evaluate information and arguments.\n4. Be skeptical: Always be skeptical of claims, especially those that seem too good to be true. Don't believe everything you read or hear, but instead, use your critical thinking skills to evaluate the evidence and arguments.\n5. Consider multiple perspectives: Try to consider multiple perspectives when evaluating an issue. Don't just see things from one side, but try to understand the arguments and perspectives of others.\n6. Be open-minded: Don't be afraid to change your mind if new evidence or arguments present themselves. Be open to new ideas and perspectives, and don't be afraid to challenge your own beliefs.\n7. Practice: Like any skill, critical thinking takes practice. Try to apply these tips to your daily life and see how they can help you think more critically.\nRemember, critical thinking is a skill that takes time and effort to develop, but with practice and dedication, you can improve your abilities.
+
+# Assistant B (FourierFT)
+
+Here are some tips to help you develop your critical thinking skills:\n1. Read widely: Reading is an excellent way to develop critical thinking skills. When you read, try to analyze the author's arguments, evaluate the evidence, and identify logical fallacies.\n2. Practice problem-solving: Solving problems is a great way to exercise your critical thinking skills. Try to approach problems from different angles, and think creatively to find solutions.\n3. Ask questions: Asking questions is a great way to develop critical thinking skills. When you're presented with new information, try to analyze it and ask questions to understand it better.\n4. Engage in debate: Debate is a great way to exercise your critical thinking skills. Try to evaluate the arguments of others and form your own opinions.\n5. Learn to think logically: Logical thinking is the foundation of critical thinking. Try to learn the basics of logic, such as how to identify and evaluate arguments, and how to spot logical fallacies.\n6. Be open-minded: Being open-minded is essential for critical thinking. Try to keep an open mind when you're presented with new information, and don't be afraid to challenge your own beliefs.\n7. Practice active listening: Active listening is a great way to develop critical thinking skills. When you're listening to someone, try to analyze their arguments and evaluate their evidence.\n8. Keep learning: Finally, never stop learning. The more you learn, the better your critical thinking skills will become. So, keep an open mind, and keep exploring new topics and ideas.
+
+![](images/12eb79bac417f1a4bed0e13b70230c3b77ff1470a85943cb4aaf6826864a77d1.jpg)
+
+![](images/2d59a785b562f936f2d3b6808a625fb7e56f187a39c14577392592e05881f436.jpg)
+
+Both Assistant A and Assistant B provide helpful and relevant responses to the user's question about developing critical thinking skills. They both suggest reading widely, asking questions, being open-minded, and practicing regularly. However, Assistant A goes a step further by suggesting additional strategies such as engaging in debate, learning to think logically, practicing active listening, and continuous learning. This makes Assistant A's response more detailed and comprehensive, providing the user with more strategies to develop their critical thinking skills. Therefore, Assistant A's response is superior in this case. Final Verdict: [[A]]
+
+Figure 10: Question 6 and 7 from Vicuna-Bench. We present the pairwise-version judgment by GPT-4 to differentiate the performance of our fine-tuned LLaMA2-7b (left) and FourierFT fine-tuned LLaMA2-7b (right).
+
+# L INSTRUCTION TUNING STABLE DIFFUSION
+
+We show how our PEFT method can be used to fine-tune Stable Diffusion (Rombach et al., 2022) so that it can perform specific image editing tasks according to instructions. Our experiment is based on InstructPix2Pix (Brooks et al., 2023), which performs instruction fine-tuning on numerous generated
+
+image pairs and prompts using pretrained Stable Diffusion checkpoints. The public InstructPix2Pix model is good at executing general instructions, but may not be skilled at specific instructions.
+
+Following Paul (2023), we choose cartoonization as the target task for fine-tuning. The fine-tuning dataset includes 5000 paired image-cartoon images as well as the corresponding prompting texts. The original images are randomly sampled from the training set of ImageNette (Howard & Gugger, 2020), and the corresponding edited images are obtained with the Whitebox Cartoonizer model (Wang & Yu, 2020). The prompts are generated using ChatGPT  $^{3}$ . All pretrained models are from the Huggingface Diffusers  $^{4}$  library. We apply PEFT methods to the Key, Query, Value and Out matrix in the Unet of Stable Diffusion for fine-tuning. After fine-tuning, we randomly choose some images from the photo domain of the PACS dataset (Li et al., 2017) for evaluation, using the prompt Change the natural image to a cartoon-style image. We provide the hyperparameters for our LoCA and FourierFT in Table 11.
+
+![](images/aeca83666b88a0157448d0ac9d98d36845da2d25f0a8c2e9f9f50c94d69fce60.jpg)  
+Figure 11: Comparison of the instruction-following abilities of InstructPix2Pix, FourierFT and Our LoCA on the cartoonization task.
+
+From Fig. 11, we can see that the pre-trained InstructPix2Pix model does not perform perfectly on this specific cartoonization task, especially in terms of preserving the original content. After fine-tuning, there is a noticeable improvement in the quality of the edits. However, the images produced by our fine-tuning method show better detail preservation compared to those generated by FourierFT.
+
+Table 11: Hyperparameters of FourierFT and LoCA for the Stable Diffusion fine-tuning experiment.  
+
+<table><tr><td>Hyperparameter</td><td>FourierFT</td><td>LoCA</td></tr><tr><td>Optimizer</td><td colspan="2">AdamW</td></tr><tr><td>Weight Decay</td><td colspan="2">1e-2</td></tr><tr><td>Learning Rate</td><td>1e-3</td><td>1e-4</td></tr><tr><td>Scaling Value</td><td>64</td><td>1</td></tr><tr><td>Where</td><td colspan="2">Key, Query, Value, Out</td></tr><tr><td>Accumulation Steps</td><td colspan="2">4</td></tr><tr><td>Batch Size</td><td colspan="2">2</td></tr><tr><td>Training Steps</td><td colspan="2">10000</td></tr><tr><td>Learning iterations (B_s)</td><td>-</td><td>1200</td></tr></table>
+
+# M TOY EXPERIMENT OF THE CONVERGENCE
+
+To visually demonstrate the convergence process of our method, we designed a toy experiment based on a regression task.
+
+Data Generation. We generated 5000 6-dimensional samples  $X \in \mathbb{R}^{5000 \times 6}$ , where each dimension of each sample was independently sampled from a Gaussian distribution  $\mathcal{N}(0,20)$ .
+
+Network and Ground-truth Labels Preparation. We design a simple three-layer neural network with parameter matrices  $W_{1}, W_{2}$ , and  $W_{3}$ , each with a shape of  $6 \times 6$ . We reparameterized  $W_{2}$  as  $W_{2} = \mathsf{iDCT}(F_{2})$ , where  $F_{2}$  is a sparse frequency domain matrix with only 3 non-zero coefficients. Then, we randomly initialize  $W_{1}$ , the coefficients of  $F_{2}$ , and  $W_{3}$  using  $\mathcal{N}(0,0.2)$ , and initialize the locations of  $F_{2}$ 's non-zero coefficients using a uniform distribution. We denote these initialized network weights as the ground-truth weights  $W_{1}^{*}, F_{2}^{*}, W_{3}^{*}$ , and use them to generate ground-truth labels, i.e.,  $Y = W_{3}^{*}\mathsf{iDCT}(F_{2}^{*})W_{1}^{*}X$ .
+
+**Optimization Details.** We now fix  $W1^*$  and  $W3^*$ , and re-initialize the coefficient locations of  $F_2$ , and set its coefficients to zero (the same as that in our method design). We aim to explore whether, through our alternating optimization method, the zero matrix  $F_2$  could converge to  $F_2^{*5}$ . The entire optimization process uses an SGD optimizer and mean queue error loss function. We set the learning rate of coefficients and locations to 0.02 and 0.05, respectively, and alternately optimize the coefficients and locations of  $F_2$  in a period of 10 steps.
+
+Experimental Results. From Fig. 12, we can see that after re-initialization, the locations of the learnable coefficients in  $F_{2}$  have changed. If we only learn the coefficients without changing their locations, it would be impossible to converge to  $F_{2}^{*}$ . Through our alternating optimization strategy, the locations of the learnable coefficients begin to change gradually after 200 steps and eventually converge successfully to the ground-truth locations. At that, if we fix the locations and only learn the coefficients, we can perfectly converge to  $F_{2}^{*}$ , which can be observed in Fig. 13. This is the rationale behind the design of our optimization method.
+
+Remark. It is worth noting that our location gradient is estimated through difference approximation and is not entirely precise. The most accurate calculation method would be to compute the onesided gradients in 8 directions separately and then choose the direction with the largest gradient for movement. However, this approach would introduce implementation difficulties and additional computational costs. In our experiments, we find that the difference approximation generally works well. Given the large scale of the weight matrix in Transformer, using an approximate method is a more reasonable approach. Although in practical applications, it may be too demanding to require
+
+every coefficient to converge to its optimal locations, we show that even if one parameter moves in a better direction, it will improve the training effect. This can be observed from the loss descent curve in Fig. 13.
+
+![](images/9caa888cc4ec67b3df5f5663dcc75a7d794f8e462d358c7d928fb3a912e9e24a.jpg)
+
+![](images/49f02432354db079128ce3d4990231d696f69ef8f53e3608632db90ecc6f9bc1.jpg)
+
+![](images/c6ed81563da3b9fc8d1550ebd4365b2b76443054092ad0d9cc35b95f10205050.jpg)
+
+![](images/1b00b22ea2f454116508b4959a856c091597e13204dc1b00ba9bcd6988b9b4f0.jpg)
+
+![](images/d6791a5a27f2a384c54cef6c53dd6d3cca35b16b912ffc0c0ae41e721e516550.jpg)
+
+![](images/051d8bb08a1e3a043067c092197220500ae0f204330b8b3a99f03e6f7a7d2343.jpg)
+
+![](images/d83093db3a51e59764d27989491096179668c1f1acf0044963eaa5ff9bab7489.jpg)
+
+![](images/22c0ac1da86934b4103cec215c5d3f735a614749a702670f00f499809f3b63fb.jpg)
+
+![](images/249e648d7f401cc297ce33e45b6b578c7195f75d5b319a053ee65e06b3f6c23f.jpg)
+
+![](images/731a033eb936cc9ad3507db49a6a1b34a469b5c2ec67db2fb69b13aaade51bae.jpg)  
+Figure 12: Optimization process of  $F_{2}$  for the toy experiment.
+
+![](images/02cd5378d9002d8c79eeda9fda2171bbe02df75b53815b22332883f878691b10.jpg)
+
+![](images/ed5b801a1ecf95c3ad90a48a1e0319e3e639e877417504b869fa57035b78d88e.jpg)
+
+![](images/dd7ab2fa623d7984fb444e78804b81be34a9b89efbbc013aeeee5ec840550e84.jpg)  
+Figure 13: Comparison of the training loss of our method with and without alternating optimization strategy on the toy experiment.
+
+# N COMPARISON OF LEARNING PATTERNS IN DIFFERENT FINE-TUNING METHODS
+
+![](images/0fdc83be9320c311765e49b7e9fc8ecb312745c3abe897e412ee52cf71781610.jpg)  
+Figure 14: Visualization of learned  $\Delta W_{q}$  and  $\Delta W_{v}$  in different fine-tuning methods with RoBERTabase. We choose layer 6 and layer 8 tuned on MNLI task as an example. For a clearer presentation, we use average pooling to downsample to  $1/8$  of the original size.
+
+To visually compare the differences in learning patterns between frequency domain methods and low-rank decomposition methods, we present in Fig. 14 the incremental matrices learned through FF, LoRA, and our LoCA. The hyperparameter settings for the experiment are the same as in Section 5.1. It can be observed that the  $\Delta W$  obtained from full fine-tuning shows more degrees of freedom across the entire matrix, exhibiting a Gaussian-like distribution. This aligns with the asymptotic normality we proposed in Proposition 1. In contrast, the incremental weights learned by LoRA display a structured absence of many elements on the matrix, likely due to its low-rank approximation. This suggests that the optimization of LoRA may be constrained and it may not effectively capture the information present in the weight updates. LoCA circumvents the constraints of low-rank decomposition through frequency domain decomposition. As can be seen from Fig. 14, the pattern of LoCA is more diverse compared to LoRA, thus enabling it to better capture the learning pattern of full fine-tuning.
+
+# O EXTENDED ANALYSIS ON OTHER LORA VARIANTS
+
+Our theoretical analysis in Theorem 1 focuses specifically on the classical low-rank reconstruction method LoRA (Hu et al., 2021), which potentially constrains our comparative analysis with various LoRA variants. While it may not be feasible to encompass all low-rank methods within a single theorem, as some methods like VeRA (Kopiczko et al., 2023) are not explicitly designed for reconstruction, we can conduct case-by-case analyses since all low-rank-based methods are inherently bounded in their reconstruction capabilities.
+
+For a given  $\Delta W \in \mathbb{R}^{n \times n}$ , VeRA decomposes it to  $\Lambda_b B \Lambda_d A$  where  $B, A$  are drawn i.i.d. from a certain distribution and frozen and shared over all training steps and layers,  $\Lambda_b, \Lambda_d$  are learnable diagonal matrix. From a reconstruction perspective, the  $i$ -th element of  $\Lambda_b$  is the ordinary least squares (OLS) coefficient while setting the response as  $i$ -th row of  $\Delta W$  and covariate as  $i$ -th row of  $B \Lambda_d A$ . This idea enables us to find  $\Lambda_d$  that maximize the correlation between  $i$ -th row of  $\Delta W$  and  $i$ -th row of  $B \Lambda_d A$ . However  $A$  and  $B$  are chosen randomly independent of  $\Delta W$ , the reconstruction error is approximately the error we learn from white noise.
+
+We can conduct a detailed theoretical analysis of DoRA (Liu et al., 2024), here we only give the outline. For a given  $\Delta W$ , DoRA first decomposes it as  $\Delta W = A\Lambda$  where  $\Lambda$  is diagonal and each column of  $A$  has magnitude 1. The  $r$ -rank approximation is  $A_{r}\Lambda$ , where  $A_{r} = U_{r}\Lambda_{r}V_{r}^{T}$ , and  $U_{r}, V_{r} \in \mathbb{R}^{n \times r}$  and  $\Lambda_{r}$  contains  $r$  largest eigenvalues of  $A$ . If each element in  $\Delta W$  follows i.i.d. standard normal, we can derive the independency of  $A$  and  $\Lambda$ . Using total expectation, we have the following reconstruction loss
+
+$$
+\mathbb {E} (\| A \Lambda - A _ {r} \Lambda \| ^ {2}) = \mathbb {E} \{\mathbb {E} (\| A \Lambda - A _ {r} \Lambda \| ^ {2} | A) \} = \sqrt {2} \frac {\Gamma ((n + 1) / 2)}{\Gamma (n / 2)} \mathbb {E} (\| A - A _ {r} \| ^ {2})
+$$
+
+As each non-zero element in  $\Lambda$  follows i.i.d.  $\chi(n)$  distribution. Subsequent calculations only require computing the reconstruction loss based on the distribution of  $A$ . At this point, the reconstruction loss is consistent with the LoRA method, except that the distributions are different. This requires complex calculations, but since each column of  $A$  is the direction of a random normal vector, the difference should not be significant. The loss corresponding to DoRA should therefore be approximately the same as that of LoRA.
+
+# P ANALYSIS OF NON-I.I.D. EFFECTS
+
+While our main theoretical analysis assumes independence of weight updates for analytical tractability, practical neural network training through gradient-based optimization introduces dependencies between parameters. In this section, we provide a detailed analysis of how deviations from the i.i.d. assumption affect our theoretical results.
+
+Correlation Structure. To systematically study the impact of parameter dependencies, we consider a controlled correlation setting where the vectorized weight updates follow a multivariate normal distribution:
+
+$$
+W ^ {T} \sim N _ {K ^ {2}} (0, \Sigma) \tag {19}
+$$
+
+where the covariance matrix  $\Sigma$  takes the form:
+
+$$
+\Sigma = \rho \mathbb {1} \mathbb {1} ^ {T} + I _ {K ^ {2}} \tag {20}
+$$
+
+Here,  $\mathbb{1} = (1,\dots ,1)^T\in \mathbb{R}^{K^2}$  is the all-ones vector,  $I_{K^2}$  is the identity matrix, and  $\rho$  controls the strength of uniform correlation between all pairs of parameters. This structure allows us to precisely control the degree of dependency while maintaining the marginal distributions of individual parameters.
+
+Critical Correlation Analysis. We conduct extensive numerical experiments to identify the critical correlation levels where the relative performance of different adaptation methods changes significantly. For these experiments, we fix the matrix size to  $300 \times 300$  and vary the rank  $r$  used in low-rank approximations. For each rank setting, we identified the critical correlation value  $\rho_{c}$  where LoRA's reconstruction ability begins to outperform LoCA. The results are shown in Fig. 15.
+
+![](images/027458f89fd61571b3c3d363919f58b2d63b15012a2f5403e9dc91c43e8b18e6.jpg)
+
+![](images/56129c190c64b29699d9fd3b80128685684ae39a811fca67e7b953984391b964.jpg)
+
+![](images/2595367e56cc8785e497d4ca5b118ec1dd2d58e9195ff4c5345bdc372fa7240e.jpg)  
+Correlation value
+
+![](images/d2425cf82949a71065d129aabdccdc7f1c00d3dee7ef8a317e705dc54d02436e.jpg)  
+Figure 15: Reconstruction errors of different  $r$  under different correlation values  $\rho$ . R, M1, M2, D, U denote the same meaning in Fig. 6.
+
+The results show that when  $r = 8,16,24$ , and 32, the critical values  $\rho_{c}$  are 0.09, 0.14, 0.17, and 0.19, respectively, which are quite high and indicate our method remains effective under substantial dependencies.
+
+Statistical Detection of Correlation. To validate that these critical correlation levels represent statistically significant departures from independence, we developed a test based on the Marchenko-Pastur (MP) law. According to the MP law, under independence, the eigenvalues of the sample correlation matrix should fall within a specific interval  $[\lambda_{-},\lambda_{+}]$ . We define a test statistic:
+
+$$
+T = \frac {\sum_ {\lambda \notin [ \lambda_ {-} , \lambda_ {+} ]} \lambda}{\sum \lambda}. \tag {21}
+$$
+
+This statistic measures the proportion of eigenvalue mass that falls outside the MP bounds. Through Monte Carlo simulation, we determined that the critical value at the 0.95 significance level is 0.005. For our identified critical correlation values  $\rho_{c} = 0.09$ , 0.14, 0.17, 0.19, the corresponding test statistics are:
+
+-  $\rho_{c} = 0.09$ :  $T = 0.086$  
+-  $\rho_{c} = 0.14$ :  $T = 0.134$
+
+-  $\rho_{c} = 0.17$ :  $T = 0.143$  
+-  $\rho_{c} = 0.19$ :  $T = 0.146$
+
+All these test statistics substantially exceed the critical value, confirming that these levels of correlation are readily detectable and represent significant departures from independence.
+
+Implications for Theory. These findings have several important implications:
+
+1. The critical correlation values where method performance characteristics change are statistically significant and detectable using standard random matrix theory diagnostics.  
+2. The monotonic increase in critical correlation with rank suggests that higher-dimensional representations are more robust to dependencies.  
+3. Even under substantial and detectable correlations, the performance advantages of frequency-domain methods persist, supporting the practical validity of our theoretical framework.
+
+These results demonstrate that while strict independence is violated in practice, our theoretical insights remain valid under realistic levels of parameter dependency. The robustness of our results to substantial correlations, as quantified by both performance analysis and statistical tests, supports the practical applicability of frequency-domain adaptation methods.
\ No newline at end of file
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+# LOLCATS: ON LOW-RANK LINEARIZING OF LARGE LANGUAGE MODELS
+
+Michael Zhang\*, Simran Arora†, Rahul Chalamala‡§, Alan Wu§, Benjamin Spector†, Aaryan Singhal†, Krithik Ramesh‡§§, and Christopher Ré†
+
+†Stanford University, ‡Together AI, §California Institute of Technology, §MIT mzhang@cs.stanford.edu
+
+# ABSTRACT
+
+Recent works show we can linearize large language models (LLMs)—swapping the quadratic attentions of popular Transformer-based LLMs with subquadratic analogs, such as linear attention—avoiding the expensive pretraining costs. However, linearizing LLMs often significantly degrades model quality, still requires training over billions of tokens, and remains limited to smaller 1.3B to 7B LLMs. We thus propose Low-rank Linear Conversion via Attention Transfer (LoLCATs), a simple two-step method that improves LLM linearizing quality with orders of magnitudes less memory and compute. We base these steps on two findings. First, we can replace an LLM's softmax attentions with closelyapproximating linear attentions, simply by training the linear attentions to match their softmax counterparts with an output MSE loss ("attention transfer"). Then, this enables adjusting for approximation errors and recovering LLM quality simply with low-rank adaptation (LoRA). LoLCATs significantly improves linearizing quality, training efficiency, and scalability. We significantly reduce the linearizing quality gap and produce state-of-the-art subquadratic LLMs from Llama 3 8B and Mistral 7B v0.1, leading to  $20+$  points of improvement on 5-shot MMLU. Furthermore, LoLCATs does so with only  $0.2\%$  of past methods' model parameters and  $0.04 - 0.2\%$  of their training tokens. Finally, we apply LoLCATs to create the first linearized 70B and 405B LLMs ( $50 \times$  that of prior work). When compared with prior approaches under the same compute budgets, LoLCATs significantly improves linearizing quality, closing the gap between linearized and original Llama 3.1 70B and 405B LLMs by  $77.8\%$  and  $78.1\%$  on 5-shot MMLU.
+
+# 1 INTRODUCTION
+
+"Linearizing" large language models (LLMs)—or converting existing Transformer-based LLMs into attention-free or subquadratic alternatives—has shown promise for scaling up efficient architectures. While many such architectures offer complexity-level efficiency gains, like linear-time and constant-memory generation, they are often limited to smaller models pretrained on academic budgets (Gu & Dao, 2023; Peng et al., 2023a; Yang et al., 2023; Arora et al., 2024; Beck et al., 2024). In a complementary direction, linearizing aims to start with openly available LLMs—e.g., those with  $7\mathrm{B}+$  parameters pretrained on trillions of tokens (AI, 2024; Jiang et al., 2023)—and (i) swap their softmax attentions with subquadratic analogs, before (ii) further finetuning to recover quality. This holds exciting promise for quickly scaling up subquadratic capabilities.
+
+However, to better realize this promise and allow anyone to convert LLMs into subquadratic models, we desire methods that are (1) quality-preserving, e.g., to recover the zero-shot abilities of modern LLMs; (2) parameter and token efficient, to linearize LLMs on widely accessible compute; and (3) highly scalable, to support linearizing  $70\mathrm{B}+$  LLMs available today (Touvron et al., 2023a;b).
+
+Existing methods present opportunities to improve all three criteria. On quality, despite using motivated subquadratic analogs such as RetNet-inspired linear attentions (Sun et al., 2023; Mercat et al., 2024) or state-space model (SSM)-based Mamba layers (Gu & Dao, 2023; Bick et al., 2024; Wang et al., 2024), prior works significantly reduce performance on popular LM Evaluation Harness tasks (LM Eval) (Gao et al., 2023) (up to 23.4-28.2 pts on 5-shot MMLU (Hendrycks et al., 2020)). On parameter and token efficiency, to adjust for architectural differences, prior methods update all model
+
+![](images/9dc7019333f1bd54fa53296dbe391a700626d9fe87ee763e88995b00f4516b5a.jpg)  
+Figure 1: LoLCATs framework. We linearize LLMs by (1) training attention analogs to approximate softmax attentions (attention transfer), before swapping attentions and (2) minimally adjusting (with LoRA).
+
+![](images/0dccd1aef7ec46cc49806136411e4f16e3fb72c32256b011554166e8a74d5879.jpg)
+
+![](images/972f4193a6ca221b42cf293a670f45ee709f303eed14ec2ca1441eee9ea7baad.jpg)
+
+parameters in at least one stage of training (Mercat et al., 2024; Wang et al., 2024; Bick et al., 2024), and use 20 - 100B tokens to linearize 7B LLMs. On scalability, these training costs make linearizing larger models on academic compute more difficult; existing works only linearize up to 8B LLMs. This makes it unclear how to support linearizing 70B to 405B LLMs (Dubey et al., 2024).
+
+In this work, we thus propose LoLCATs (Low-rank Linear Conversion with Attention Transfer), a simple approach to improve the quality, efficiency, and scalability of linearizing LLMs. As guiding motivation, we ask if we can linearize LLMs by simply reducing architectural differences, i.e.,
+
+1. Starting with simple softmax attention analogs such as linear attention (Eq. 2), and training their parameterizations explicitly to approximate softmax attention ("attention transfer").  
+2. Subsequently only training with low-cost finetuning to adjust for any approximation errors, e.g., with low-rank adaptation (LoRA) (Hu et al., 2021) ("low-rank linearizing").
+
+In evaluating this hypothesis, we make several contributions. First, to better understand linearizing feasibility, we empirically study attention transfer and low-rank linearizing with existing linear attentions. While intuitive—by swapping in perfect subquadratic softmax attention approximators, we could get subquadratic LLMs with no additional training—prior works suggest linear attentions struggle to match softmax expressivity (Keles et al., 2023; Qin et al., 2022) or need full-model updates to recover linearizing quality (Kasai et al., 2021; Mercat et al., 2024). In contrast, we find that while either attention transfer or LoRA alone is insufficient, we can rapidly recover quality by simply doing both (Figure 3, Table 2). At the same time, we do uncover quality issues related to attention-matching architecture and training. With prior linear attentions, the best low-rank linearized LLMs still significantly degrade in quality vs. original Transformers (up to 42.4 pts on 5-shot MMLU). With prior approaches that train all attentions jointly (Zhang et al., 2024), we also find that later layers can result in  $200 \times$  the MSE of earlier ones (Figure 7). We later find this issue aggravated by larger LLMs; jointly training all Llama 3.1 405B's 126 attention layers fails to viably linearize.
+
+Next, to resolve these issues and improve upon our original criteria, we detail LoLCATs' method components. For quality, we generalize prior notions of learnable linear attentions to sliding window + linear attention variants. These remain subquadratic to compute yet consistently yield better attention transfer via lower mean-squared error (MSE) on attention outputs. For parameter and token efficiency, we maintain our simple 2-step framework of (1) training subquadratic attentions to match softmax attentions, before (2) adjusting for any errors via only LoRA. For scalability, we use finer-grained "block-by-block" training. We split LLMs into blocks of  $k$  layers before jointly training attentions only within each block to improve layer-wise attention matching. We pick  $k$  to balance the speed of training blocks in parallel with the memory of saving hidden state outputs of prior blocks (as inputs for later ones). We provide a simple cost model to navigate these tradeoffs.
+
+Finally, in experiments, we validate that LoLCATs improves on each of our desired criteria.
+
+- On quality, when linearizing popular LLMs such as Mistral-7B and Llama 3 8B, LoLCATs substantially improves past linearizing methods (by 1.1–8.6 points (pts) on zero-shot LM Eval tasks; +17.2 pts on 5-shot MMLU)). With Llama 3 8B, LoLCATs for the first time closes the zero-shot LM Eval gap between linearized and Transformer models (73.1 vs 74.2 pts), while supporting  $3 \times$  the throughput and  $64 \times$  the batch sizes vs. popular FlashAttention-2 (Dao, 2023)
+
+<table><tr><td>Name</td><td>Architecture</td><td>Quality Preserving</td><td>Parameter Efficient</td><td>Token Efficient</td><td>Validated at Scale</td></tr><tr><td>Pretrained</td><td>Attention</td><td>✓✓</td><td>XX</td><td>XX</td><td>✓✓</td></tr><tr><td>SUPRA</td><td>Linear Attention</td><td>X</td><td>X</td><td>✓</td><td>✓</td></tr><tr><td>Mohawk</td><td>Mamba (2)</td><td>X</td><td>X</td><td>✓</td><td>X</td></tr><tr><td>Mamba in Llama</td><td>Mamba (2)</td><td>X</td><td>X</td><td>✓</td><td>✓</td></tr><tr><td>LoLCATs</td><td>Softmax-Approx. Linear Attention</td><td>✓</td><td>✓</td><td>✓✓</td><td>✓✓</td></tr></table>
+
+![](images/0b7d045f21241e5fddaa82d4481f201947ea77f41e38628fc336749f58970ed5.jpg)  
+Figure 2: Linearizing comparison. LoLCATs significantly improves LLM linearizing quality and training efficiency. No. of  $\checkmark$  or  $x$  indicate relatively better or worse support.
+
+models (generating 4096 tokens on an 80GB H100). As an alternative to pretraining, LoLCATs outperforms strong pretrained 7B subquadratic LLMs and hybrids by 1.2–9.9 pts (LM Eval avg.).
+
+- On parameter and token-efficiency, by only training linear attention feature maps in Stage 1, while only using LoRA on linear attention projections in Stage 2, LoLCATs enables these gains while updating only  $< 0.2\%$  of past linearizing methods' model parameters, making a single 40GB GPU sufficient for 7B LLM linearizing. This also only takes 40M tokens, i.e.,  $0.003\%$  and  $0.04\%$  of prior pretraining and linearizing methods' token counts.  
+- On scalability, with LoLCATs we scale up linearizing to support Llama 3.1 70B and 405B LLMs (Dubey et al., 2024). LoLCATs presents the first viable approach to linearizing larger LLMs. We create the first linearized 70B LLM, taking only 18 hours on one  $8 \times 80\mathrm{GB}$  H100 node, and the first linearized 405B LLM with a combination of 5 hours on 14 80GB H100 GPUs (attention transfer) + 16 hours on three  $8 \times 80\mathrm{GB}$  H100 nodes (LoRA finetuning) for Llama 3.1 405B. For both models, this amount to under half the total GPU hours than prior methods reported to linearize 8B models (5 days on  $8 \times 80\mathrm{GB}$  A100s) (Wang et al., 2024). Furthermore, under these computational constraints, LoLCATs significantly improves quality versus prior linearizing approaches without attention transfer. With Llama 3.1 70B and 405B, we close  $77.8\%$  and  $78.1\%$  of the 5-shot MMLU gap between Transformers and linearized variants respectively.
+
+# 2 PRELIMINARIES
+
+To motivate LoLCATs, we first go over Transformers, attention, and linear attention. We then briefly discuss related works on linearizing Transformers and Transformer-based LLMs.
+
+Transformers and Attention. Popular LLMs such as Llama 3 8B (AI@Meta, 2024a) and Mistral 7B (Jiang et al., 2023) are decoder-only Transformers, with repeated blocks of multi-head softmax attention followed by MLPs (Vaswani et al., 2017). For one head, attention computes outputs  $\pmb{y} \in \mathbb{R}^{l \times d}$  from inputs  $\pmb{x} \in \mathbb{R}^{l \times d}$  (where  $l$  is sequence length,  $d$  is head dimension) with query, key, and value weights  $\pmb{W}_q, \pmb{W}_k, \pmb{W}_v \in \mathbb{R}^{d \times d}$ . In causal language modeling, we compute  $\pmb{q} = \pmb{x}\pmb{W}_q$ ,  $\pmb{k} = \pmb{x}\pmb{W}_k$ ,  $\pmb{v} = \pmb{x}\pmb{W}_v$ , before getting attention weights  $\pmb{a}$  and outputs  $\pmb{y}$  via
+
+$$
+a _ {n, i} = \frac {\exp \left(\boldsymbol {q} _ {n} ^ {\top} \boldsymbol {k} _ {i} / \sqrt {d}\right)}{\sum_ {i = 1} ^ {n} \exp \left(\boldsymbol {q} _ {n} ^ {\top} \boldsymbol {k} _ {i} / \sqrt {d}\right)}, \quad \boldsymbol {y} _ {n} = \sum_ {i = 1} ^ {n} a _ {n, i} \boldsymbol {v} _ {i}, \quad \text {f o r} n \text {i n} [ 1, \dots , l ] \tag {1}
+$$
+
+Multi-head attention maintains inputs, outputs, and weights for each head, e.g.,  $\pmb{x} \in \mathbb{R}^{h \times l \times d}$  or  $W_{q} \in \mathbb{R}^{h \times d \times d}$  ( $h$  being number of heads), and computes Eq. 1 for each head. In both cases, we compute final outputs by concatenating  $y_{n}$  across heads, before using output weights  $W_{o} \in \mathbb{R}^{hd \times hd}$  to compute  $y_{n}W_{o} \in \mathbb{R}^{l \times hd}$ . While expressive, causal softmax attention requires all  $\{k_{i}, v_{i}\}_{i \leq n}$  to compute  $y_{n}$ . For long context or large batch settings, this growing KV cache can incur prohibitive memory costs even with state-of-the-art implementations such as FlashAttention (Dao, 2023).
+
+Linear Attention. To get around this, Katharopoulos et al. (2020) show a similar attention operation, but with linear time and constant memory over generation length (linear time and space when processing inputs). To see how, note that softmax attention's exponential is a kernel function  $\mathcal{K}(\pmb{q}_n,\pmb{k}_i)$ , which in general can be expressed as the dot product of feature maps  $\phi :\mathbb{R}^d\mapsto \mathbb{R}^{d'}$ . Swapping  $\exp (\pmb{q}_n^\top \pmb{k}_i / \sqrt{d})$  with  $\phi (\pmb {q}_n)^\top \phi (\pmb {k}_i)$  in Eq. 1 gives us linear attention weights and outputs:
+
+$$
+\hat {\boldsymbol {y}} _ {n} = \sum_ {i = 1} ^ {n} \frac {\phi \left(\boldsymbol {q} _ {n}\right) ^ {\top} \phi \left(\boldsymbol {k} _ {i}\right) \boldsymbol {v} _ {i}}{\sum_ {i = 1} ^ {n} \phi \left(\boldsymbol {q} _ {n}\right) ^ {\top} \phi \left(\boldsymbol {k} _ {i}\right)} = \frac {\phi \left(\boldsymbol {q} _ {n}\right) ^ {\top} \left(\sum_ {i = 1} ^ {n} \phi \left(\boldsymbol {k} _ {i}\right) \boldsymbol {v} _ {i} ^ {\top}\right)}{\phi \left(\boldsymbol {q} _ {n}\right) ^ {\top} \sum_ {i = 1} ^ {n} \phi \left(\boldsymbol {k} _ {i}\right)} \tag {2}
+$$
+
+This lets us compute both the numerator  $s_n = \sum_{i=1}^n \phi(\pmb{k}_i) \pmb{v}_i^\top$  and denominator  $z_n = \sum_{i=1}^n \phi(\pmb{k}_i)$  as recurrent "KV states". With  $s_0 = 0$ ,  $z_0 = 0$ , we recurrently compute linear attention outputs as
+
+$$
+\hat {\boldsymbol {y}} _ {n} = \frac {\phi (\boldsymbol {q} _ {n}) ^ {\top} \boldsymbol {s} _ {n}}{\phi (\boldsymbol {q} _ {n}) ^ {\top} \boldsymbol {z} _ {n}} \text {f o r} \boldsymbol {s} _ {n} = \boldsymbol {s} _ {n - 1} + \phi (\boldsymbol {k} _ {n}) \boldsymbol {v} _ {n} ^ {\top} \text {a n d} \boldsymbol {z} _ {n} = \boldsymbol {z} _ {n - 1} + \phi (\boldsymbol {k} _ {n}) \tag {3}
+$$
+
+Eq. 2 lets us compute attention over an input sequence of length  $n$  in  $\mathcal{O}(ndd')$  time and space, while Eq. 3 lets us compute  $n$  new tokens in  $\mathcal{O}(ndd')$  time and  $\mathcal{O}(dd')$  memory. Especially during generation, when softmax attention has to compute new tokens sequentially anyway, Eq. 3 enables time and memory savings if  $d' < (\text{prompt length} + \text{prior generated tokens})$ .
+
+Linearizing Transformers. To combine efficiency with quality, various works propose different  $\phi$ , (e.g.,  $\phi(x) = 1 + \mathrm{ELU}(x)$  as in Katharopoulos et al. (2020)). However, they typically train linear attention Transformers from scratch. We build upon recent works that swap the softmax attentions of existing Transformers with linear attention before finetuning the modified models with next-token prediction to recover language modeling quality. These include methods proposed for LLMs (Mercat et al., 2024), and those for smaller Transformers—e.g., 110M BERTs (Devlin et al., 2018)—reasonably adaptable to modern LLMs (Kasai et al., 2021; Mao, 2022; Zhang et al., 2024).
+
+# 3 METHOD: LINEARIZING LLMS WITH LOLCATS
+
+We now study how to build a high-quality and highly efficient linearizing method. In Section 3.1, we present our motivating framework, which aims to (1) learn good softmax attention approximators with linear attentions and (2) enable low-rank adaptation for recovering linearized quality. In Section 3.2, we find that while this attention transfer works surprisingly well for low-rank linearizing with existing linear attentions, on certain tasks, it still results in sizable quality gaps compared to prior methods. We also find that attention-transfer quality strongly corresponds with the final linearized model's performance. In Section 3.3, we use our learned findings to overcome prior issues, improving attention transfer to subsequently improve low-rank linearizing quality.
+
+# 3.1 LOLCATS PART 1: A FRAMEWORK FOR LOW-COST LINEARIZING
+
+In this section, we present our initial LoLCATs framework for linearizing LLMs in an effective yet efficient manner. Our main hypothesis is that by first learning linear attentions that approximate softmax, we can then swap these attentions in as drop-in subquadratic replacements. We would then only need a minimal amount of subsequent training—e.g., that is supported by low-rank updates—to recover LLM quality in a cost-effective manner effectively. We thus proceed in two steps.
+
+1. Parameter-Efficient Attention Transfer. For each softmax attention in an LLM, we aim to learn a closely-approximating linear attention, i.e., one that computes attention outputs  $\hat{\pmb{y}}\approx \pmb{y}$  for all natural inputs  $\pmb{x}$ . We call this "attention transfer", as we aim to transfer the attention modeling of existing softmax attentions into target linear attentions. Due to architectural similarity, we can treat this as a feature map learning problem, learning  $\phi$  to approximate softmax. For each head and layer, let  $\phi_q$  and  $\phi_{k}$  be query and key feature maps. Per head, we compute:
+
+$$
+\boldsymbol {y} _ {n} = \underbrace {\sum_ {i = 1} ^ {n} \frac {\exp \left(\boldsymbol {q} _ {n} ^ {\top} \boldsymbol {k} _ {i} / \sqrt {d}\right)}{\sum_ {i = 1} ^ {n} \exp \left(\boldsymbol {q} _ {n} ^ {\top} \boldsymbol {k} _ {i} / \sqrt {d}\right)} \boldsymbol {v} _ {i}} _ {\text {S o f t m a x A t t e n t i o n}}, \quad \hat {\boldsymbol {y}} _ {n} = \underbrace {\sum_ {i = 1} ^ {n} \frac {\phi_ {q} \left(\boldsymbol {q} _ {n}\right) ^ {\top} \phi_ {k} \left(\boldsymbol {k} _ {i}\right)}{\sum_ {i = 1} ^ {n} \phi_ {q} \left(\boldsymbol {q} _ {n}\right) ^ {\top} \phi_ {k} \left(\boldsymbol {k} _ {i}\right)} \boldsymbol {v} _ {i}} _ {\text {L i n e a r A t t e n t i o n}} \tag {4}
+$$
+
+for all  $n \in [l]$  with input  $\in \mathbb{R}^{l \times d}$ , and train  $\phi_q, \phi_k$  to minimize sample mean squared error (MSE)
+
+$$
+\ell_ {\mathrm {M S E}} = \frac {1}{M H} \sum_ {m = 1} ^ {M} \sum_ {h = 1} ^ {H} \ell_ {\mathrm {M S E}} ^ {h, m}, \quad \ell_ {\mathrm {M S E}} ^ {h, m} = \frac {1}{d} \sum_ {n = 1} ^ {d} \left(\boldsymbol {y} _ {n} - \hat {\boldsymbol {y}} _ {n}\right) ^ {2} \tag {5}
+$$
+
+i.e., jointly for each head  $h$  in layer  $m$ . Similar to past work (Kasai et al., 2021; Zhang et al., 2024), rather than manually design  $\phi$ , we parameterize each  $\phi : \mathbb{R}^d \mapsto \mathbb{R}^{d'}$  as a learnable layer:
+
+$$
+\phi_ {q} (\boldsymbol {q} _ {n}) := f \left(\boldsymbol {q} _ {n} \tilde {\boldsymbol {W}} _ {(q)} + \tilde {\boldsymbol {b}} _ {(q)}\right), \quad \phi_ {k} (\boldsymbol {k} _ {i}) := f \left(\boldsymbol {k} _ {i} \tilde {\boldsymbol {W}} _ {(k)} + \tilde {\boldsymbol {b}} _ {(k)}\right)
+$$
+
+Here  $\tilde{W} \in \mathbb{R}^{d \times d'}$  and  $\tilde{b} \in \mathbb{R}^{d'}$  are trainable weights and optional biases,  $f(\cdot)$  is a nonlinear activation, and  $d'$  is an arbitrary feature dimension (set to equal head dimension  $d$  in practice).
+
+2. Low-rank Adjusting. After training the linearizing layers, we replace the full-parameter training of prior work with low-rank adaptation (LoRA) (Hu et al., 2021). Like prior work, to adjust for the modifying layers and recover language modeling quality, we now train the modified LLM end-to-end over tokens to minimize a sample next-token prediction loss  $\ell_{\mathrm{xent}} = -\sum \log P_{\Theta}(\boldsymbol{u}_{t+1} \mid \boldsymbol{u}_{1:t})$ . Here  $P_{\Theta}$  is the modified LLM,  $\Theta$  is the set of LLM parameters, and we aim to maximize the probability of true  $\boldsymbol{u}_{t+1}$  given past tokens  $\boldsymbol{u}_{1:t}$  (Fig. 1 right). However, rather than train all LLM parameters, we only train the swapped linear attention  $\boldsymbol{W}_q, \boldsymbol{W}_k, \boldsymbol{W}_v, \boldsymbol{W}_o$  with LoRA, updating  $\boldsymbol{W}' \gets \boldsymbol{W} + \Delta \boldsymbol{W}$  with  $\Delta \boldsymbol{W}$  as the product of two low-rank matrices  $\boldsymbol{BA}, \boldsymbol{B} \in \mathbb{R}^{d \times r}, \boldsymbol{A} \in \mathbb{R}^{r \times d}$ . This enables parameter efficiency for  $r \ll d$ .
+
+Training footprint and efficiency. Both steps remain parameter-efficient. For Step 1, optimizing Eq. 5 is similar to a layer-by-layer cross-architecture distillation. We compute layer-wise  $(\boldsymbol{x}, \boldsymbol{y})$  as pretrained attention inputs and outputs, using an LLM forward pass over natural language samples (Fig. 1 middle). However, to keep our training footprint low, we freeze the original pretrained attention layer's parameters and simply insert new  $\phi_q, \phi_k$  after  $W_q, W_k$  in each softmax attention (Fig. 1 left). We compute outputs  $\boldsymbol{y}, \hat{\boldsymbol{y}}$  with the same attention weights in separate passes (choosing either Eq. 1 or Eq. 2; "teacher-forcing" by only sending softmax outputs  $\boldsymbol{y}$  to future layers and preventing error propagation, Fig. 1 middle). For Llama 3 8B or Mistral 7B, training  $\phi_q, \phi_k$  with  $d' = 64$  then only takes 32 layers  $\times 32$  heads  $\times 2$  feature maps  $\times (128 \times 64)$  weights  $\approx 16.8\mathrm{M}$  trainable weights (0.2% of LLM sizes). Furthermore, Eq. 5 keeps attention transfer memory-efficient. While Zhang et al. (2024) also train layer-wise  $\phi$  to approximate softmax attention, they supervise by matching on the  $n^2$  attention weights computed for  $n$ -token inputs (App. D, Eq. 12). This scales poorly for large  $n$ . Instead, by only needing attention outputs, we can reduce memory from  $\mathcal{O}(n^2)$  to  $\mathcal{O}(n)$ , computing softmax attention  $\boldsymbol{y}$  with FlashAttention (Dao et al., 2022; Dao, 2023) and linear attention  $\hat{\boldsymbol{y}}$  with Eq. 2 (or 6) both in  $\mathcal{O}(n)$  memory by not materializing attention weights. Despite these savings, we show similar attention weight recovery in App. E.2. In Step 2, LoRA with  $r = 8$  on all attention projections suffices for state-of-the-art quality. This updates just  $< 0.09\%$  of 7B parameters.
+
+# 3.2 BASELINE STUDY: ATTENTION TRANSFER AND LOW-RANK LINEARIZING
+
+As a first step, we aim to understand if attention transfer and low-rank adjusting as proposed are sufficient for linearizing LLMs. While simple, it is unclear whether these steps can lead to high-quality linearizing, as all prior works default to more involved approaches (Mercat et al., 2024; Bick et al., 2024; Wang et al., 2024). They use linearizing layers featuring GroupNorms (Wu & He, 2018) and decay factors (Sun et al., 2023), or alternate SSM-based architectures (Gu & Dao, 2023; Dao & Gu, 2024). They also all use full-LLM training after swapping in the subquadratic layers. In contrast, as a first contribution we find that simple linear attentions can lead to viable linearizing, with attention transfer + LoRA obtaining competitive quality on  $4 / 6$  popular LM Eval tasks.
+
+Experimental Setup. We test the LoLCATs framework by linearizing two popular base LLMs, Llama 3 8B (AI, 2024) and Mistral 7B v0.1 (Jiang et al., 2023). For linearizing layers, we study two feature maps used in prior work (Table 1). To support the rotary positional embeddings (RoPE) (Su et al., 2024) in these LLMs, we apply the feature maps  $\phi$  after
+
+<table><tr><td>Feature Map</td><td>φ(q) (same for k)</td><td>Weight Shapes</td></tr><tr><td>T2R</td><td>ReLU(qW + b)</td><td>W: (128, 128), b: (128,)</td></tr><tr><td>Hedgehog</td><td>[SMd(qW) ⊕ SMd(-qW)]</td><td>W: (128, 64)</td></tr></table>
+
+Table 1: Learnable feature maps. Transformer to RNN (T2R) from Kasai et al. (2021), Hedgehog from Zhang et al. (2024), both  $\oplus$  (concat) and  $\mathbf{SM}_d$  (softmax) apply over feature dimension.
+
+RoPE, $^{1}$  i.e., computing query features  $\phi_q(\pmb{q}) = f(\mathrm{RoPE}(\pmb{q})\tilde{\pmb{W}}_q + \tilde{\pmb{b}})$ . For linearizing data, we wish to see if LoLCATs with a small amount of data can recover general zero-shot and instruction-following LLM abilities. We use the 50K samples of a cleaned Alpaca dataset $^{2}$ , due to its ability to improve general instruction-following in 7B LLMs despite its relatively small size (Taori et al., 2023). We train all feature maps jointly. Training code and implementation details are in App. C.
+
+To study the effects of attention transfer and low-rank linearizing across LLMs and linear attention architectures, we evaluate their validation set perplexity (Table 2, Fig. 3) and downstream LM Eval
+
+![](images/2d87c6446956e382b34a3d2b8e33a994afd8deb28965cd00436017099156b2fc.jpg)  
+Figure 3: Attention transfer training efficiency. Even accounting for initial training steps, low-rank linearizing with attention transfer still consistently achieves lower perplexity faster across feature maps and LLMs.
+
+![](images/14cdffe7a84d718b1e4b2bf2c97cf438fa68c010e979c12e604be38188a9b61a.jpg)
+
+![](images/359de366d4843a59d001017082a4d83f62f17d9210e8da9f93030d363f79581b.jpg)
+
+![](images/3594d8772a1d0928ab6d2b716733b91ae445a69751832ce06c40ae1b6121a7c9.jpg)
+
+zero-shot quality (Table 4). We use the same data for both stages, early stopping, and either 2 epochs for attention transfer and LoRA adjusting or 4 epochs for either alone ( $\approx$ 40M total training tokens). We use LoRA  $r = 8$  by popular default (Hu et al., 2021), training  $0.2\%$  of LLM parameter counts.
+
+<table><tr><td rowspan="3">Attention Transfer?</td><td colspan="4">Llama 3 8B</td><td colspan="4">Mistral 7B</td></tr><tr><td colspan="2">T2R</td><td colspan="2">Hedgehog</td><td colspan="2">T2R</td><td colspan="2">Hedgehog</td></tr><tr><td>PPL@0</td><td>PPL@2/4</td><td>PPL@0</td><td>PPL@2/4</td><td>PPL@0</td><td>PPL@2/4</td><td>PPL@0</td><td>PPL@2/4</td></tr><tr><td>No ✘</td><td>1539.39</td><td>16.05</td><td>2448.01</td><td>9.02</td><td>2497.13</td><td>8.85</td><td>561.47</td><td>4.87</td></tr><tr><td>Yes ✓</td><td>79.33</td><td>4.11</td><td>60.86</td><td>3.90</td><td>32.78</td><td>3.29</td><td>18.94</td><td>3.04</td></tr></table>
+
+Attention Transfer + LoRA Enables Fast LLM Linearizing. In Table 2 and Fig. 3, we report the validation PPL of linearized LLMs, abating attention transfer and LoRA adjusting. We find that while attention transfer alone is often insufficient (c.f., PPL@0, Table 2), a single low-rank update rapidly recovers performance by 15-75 PPL (Fig. 3), where training to approximate softmax leads to up to 11.9 lower PPL than no attention transfer. Somewhat surprisingly, this translates to performing competitively with prior linearizing methods that train all model parameters (Mercat et al., 2024; Wang et al., 2024) (within 5 accuracy points on  $4/6$  popular LM Eval tasks; Table 4), while only training with  $0.04\%$  of their token counts and  $0.2\%$  of their parameter counts. The results suggest we can linearize 7B LLMs at orders-of-magnitude less training costs than previously shown.
+
+LoL SAD: Limitations of Low-Rank Linearizing. At the same time, we note quality limitations with the present framework. While sometimes close, low-rank linearized LLMs perform worse than full-parameter alternatives and original Transformers on  $5/6$  LM Eval tasks (up to 42.4 points on 5-shot MMLU; Table 4). To understand the issue, we study if the attention transfer stage can produce close linear attention approximations of LLM softmax attentions. We note three observations:
+
+1. Attention transfer quality (via output MSE) strongly ties to final linearized LLM quality (via PPL) (Fig. 5), suggesting we can improve quality by reducing MSE with softmax attentions.  
+2. However, larger MSEs coincide with lower softmax attention weight entropies (Fig. 6a). Zhang et al. (2024) find linear attentions struggle to approximate such "spikier" distributions, suggesting we may need better attention-matching layers to reduce MSE and improve final linearized quality.  
+3. When training layers jointly like in prior work (Zhang et al., 2024), larger MSEs also heavily concentrate in later layers (Fig. 6b). To bring the MSE in these layers down, we may thus need more fine-grained layer-wise supervision, rather than the objective over all layers in Eq. 5.
+
+Table 2: Alpaca validation set perplexity (PPL) of linearized LLMs, comparing attention transfer, no LoRA adjusting (PPL@0) and PPL after training (PPL@2/4; 2 with attention transfer, 4 without, for equal total steps).  
+
+<table><tr><td>Model</td><td>Tokens (B)</td><td>PiQA</td><td>ARC-E</td><td>ARC-C</td><td>HS</td><td>WG</td><td>MMLU</td></tr><tr><td>Llama 3 8B</td><td>-</td><td>79.9</td><td>80.1</td><td>53.3</td><td>79.1</td><td>73.1</td><td>66.6</td></tr><tr><td>→ Mamba2</td><td>100</td><td>76.8</td><td>74.1</td><td>48.0</td><td>70.8</td><td>58.6</td><td>43.2</td></tr><tr><td>→ LoRA Hedgehog</td><td>0.04</td><td>77.4</td><td>71.1</td><td>40.6</td><td>66.5</td><td>54.3</td><td>24.2</td></tr><tr><td>Mistral 7B</td><td>-</td><td>82.1</td><td>80.9</td><td>53.8</td><td>81.0</td><td>74.0</td><td>62.4</td></tr><tr><td>→ SUPRA</td><td>100</td><td>80.4</td><td>75.9</td><td>45.8</td><td>77.1</td><td>70.3</td><td>34.2</td></tr><tr><td>→ LoRA Hedgehog</td><td>0.04</td><td>79.3</td><td>76.4</td><td>45.1</td><td>73.1</td><td>57.5</td><td>28.2</td></tr></table>
+
+Figure 4: Linearizing comparison on LM Eval. Task names in Table 4. Acc. norm: ARC-C, HS. Acc. otherwise. 5-shot MMLU. 0-shot otherwise.
+
+![](images/8a17a9819bf02fa07bd8f85c97e45d1cdef6f563645f0659e09bd607d8f3984e.jpg)  
+Figure 5: Attention MSE vs. PPL. Across feature maps, LLMs; lower MSE coincides with better linearized quality.
+
+![](images/bf946a1a9aa3a4d837dab30c8cff27a2acf74e34b5fca00c18d5c53f3fad65c8.jpg)
+
+![](images/5e37bbad70fa4a591c50068c5f13a7e2dc916e8aef39f507a456864a1fee4c6e.jpg)  
+(a) Attention Entropy
+
+![](images/6d0cccb303fb8dc88e8a5349d1a8e6c5bfa3ceef30ad8f59398cbf70c364241c.jpg)  
+(b) Model Layer  
+Figure 6: Sources of Attention Transfer Error with Llama 3 8B. We find two potential sources of attention transfer difficulty: (a) low softmax attention entropy and (b) attentions in later layers.
+
+# 3.3 LOLCATS PART 2: COMPONENTS TO IMPROVE LOW-RANK LINEARIZING
+
+Following our motivating hypotheses, framework, and observations, we now introduce two simple improvements to improve linearized LLM quality by reducing MSE: (1) better attention-matching architectures (Section 3.3.1), and (2) finer-grained layer-wise attention transfer (Section 3.3.2).
+
+# 3.3.1 ARCHITECTURE: GENERALIZING LEARNABLE LINEAR ATTENTIONS
+
+As described, we can apply our framework with any linear attentions with learnable  $\phi$  (e.g., T2R and Hedgehog, Figure 3). However, to improve attention-matching quality, we introduce a hybrid  $\phi$  parameterization combining linear attention and sliding window attention. Motivated by prior works that show quality improvements when combining attention layers with linear attentions (Arora et al., 2024; Munkhdalai et al., 2024), we combine short sliding windows of softmax attention (Beltagy et al., 2020; Zhu et al., 2021) (size 64 in experiments) followed by linear attention in a single layer. This allows attending to all prior tokens for each layer while keeping the entire LLM subquadratic. For window size  $w$  and token indices  $[1,\dots ,n - w,\dots ,n]$ , we apply the softmax attention over the  $w$  most recent tokens, and compute attention outputs  $\hat{\pmb{y}}_n$  as
+
+$$
+\hat {\mathbf {y}} _ {n} = \frac {\sum_ {i = n - w + 1} ^ {n} \gamma \exp \left(\boldsymbol {q} _ {n} ^ {\top} \boldsymbol {k} _ {i} / \sqrt {d} - \boldsymbol {c} _ {n}\right) \boldsymbol {v} _ {i} + \phi_ {q} \left(\boldsymbol {q} _ {n}\right) ^ {\top} \left(\sum_ {j = 1} ^ {n - w} \phi_ {k} \left(\boldsymbol {k} _ {j}\right) \boldsymbol {v} _ {j} ^ {\top}\right)}{\sum_ {i = n - w + 1} ^ {n} \gamma \exp \left(\boldsymbol {q} _ {n} ^ {\top} \boldsymbol {k} _ {i} / \sqrt {d} - \boldsymbol {c} _ {n}\right) + \phi_ {q} \left(\boldsymbol {q} _ {n}\right) ^ {\top} \left(\sum_ {j = 1} ^ {n - w} \phi_ {k} \left(\boldsymbol {k} _ {j}\right) ^ {\top}\right)} \tag {6}
+$$
+
+$\gamma$  is a learnable mixing term, and  $\pmb{c}_n$  is a stabilizing constant as in log-sum-exp calculations ( $\pmb{c}_n = \max_i \left\{\pmb{q}_n^\top \pmb{k}_i / \sqrt{d} : i \in [n - w + 1, \dots, n]\right\}$ ). Like before, we can pick any learnable  $\phi$ .
+
+Subquadratic efficiency. The hybrid layer retains linear time and constant memory generation. For  $n$ -token prompts, we initially require  $\mathcal{O}(w^2 d)$  and  $\mathcal{O}((n - w) dd')$  time and space for window and linear attention respectively, attending over a  $w$ -sized KV-cache and computing KV and K-states (Eq. 3). For generation, we only need  $\mathcal{O}(w^2 d + dd')$  time and space for every token. We evict the KV-cache's first  $k$ ,  $v$ , compute  $\phi_k(k)$ , and add  $\phi_k(k)v^\top$  and  $\phi_k(k)$  to KV and K-states respectively.
+
+Hardware-aware implementation. To make Eq. 6 competitive with modern softmax attentions like FlashAttention-2 (Dao, 2023), we provide a "hardware-aware" Eq. 6 with the Hedgehog feature map. For space, we defer implementation details to App. A, C.2. We evaluate this version by default.
+
+![](images/9cd47576c926bc9e0a78162ebe2659b1c29bb0f3f8befff7d902d1efa1c619bf.jpg)  
+Figure 7: Improving Attention matching MSE. Linearizing with linear + sliding window attention better matches LLM softmax attentions (lower MSE) over attention entropy values and LLM layers.
+
+![](images/8d9b85a162e5f42103129b83a6a8da31dfaf2af1e5ccfdf8daf5dfb5f147e7e0.jpg)
+
+# 3.3.2 TRAINING: LAYER (OR BLOCK)-WISE ATTENTION TRANSFER
+
+We describe our training approach and provide a simplified model to show its cost-quality tradeoffs. Based on the layer-wise MSE differences in Sec. 3.2 from training all layers jointly (Eq. 5), we instead generalize to training over finer-grained  $b$ -layer blocks, and train each block independently:
+
+$$
+\ell_ {\mathrm {M S E}} ^ {\text {b l o c k}} = \frac {1}{b H} \sum_ {m = i} ^ {i + b} \sum_ {h = 1} ^ {H} \ell_ {\mathrm {M S E}} ^ {h, m} \quad (\text {f o r b l o c k s s t a r t i n g a t l a y e r s} i = 0, b, 2 b, \dots) \tag {7}
+$$
+
+We choose  $b$  to balance quality and cost in both memory and training time, which we find particularly helpful for linearizing larger LLMs. Several block-wise training approaches exist, including joint training with separate optimizer groups per block, sequentially training separate blocks, or precomputing hidden states and training blocks in parallel across GPUs. For space, we report results on LLM quality trade-offs in App. B.7.3, where we find smaller  $b$  improves LoLCATs Llama 3.1 405B PPL by 1.02 points (Table 24). Below we discuss primary cost trade-offs:
+
+- Compute: While the joint training of Llama 3.1 405B in 16-bit precision uses multiple nodes (e.g., NVIDIA H100 8×80GB nodes), an individual block of  $b = 9$  or fewer layers can be trained on a single GPU (e.g., H100 80GB GPU) at sequence length 1024.  
+- Time: However, doing so on a single GPU multiples total training time by  $M / b$ . With multiple GPUs, we can distribute training different blocks on difference devices in parallel. To train blocks at layers  $b, 2b, \ldots$ , we need the outputs from each prior block. As we teacher-force with the "true" softmax attention outputs, we can simply precompute these with the original Transformer.  
+- **Memory:** Lastly, we need to save each precomputed block's outputs to disk. The total disk space required is  $2 \times T \times d \times \frac{L}{k}$  for total training tokens  $T$ , model dimension  $d$ , number of layers  $L$  and 2-byte (16-bit) precision. For Llama 3.1 405B, saving states per-layer ( $b = 1$ ) for just 50M tokens requires over 200TB of disk space. Larger  $b$  divides this storage, potentially at the cost of quality.
+
+For each target LLM, we thus aim to make low-rank linearizing feasible by first doing block-wise attention transfer, adjusting parameters based on linearizing quality and what is feasible in compute, time, and memory. We summarize LOLCATs with Alg. 1, 2, providing pseudocode in App. C.1.
+
+<table><tr><td colspan="2">Algorithm 1 LoLCATs Step 1: Attn. Transfer</td></tr><tr><td colspan="2">Input: Pretrained Transformer with M attn. layers; input tokens u</td></tr><tr><td colspan="2">Input: Linear attn. feature map params. {φmq, φmkb, γm : m ∈ [M]}, window size w; layer block-size b, learning rate α</td></tr><tr><td colspan="2">1: Freeze all Transformer parameters</td></tr><tr><td colspan="2">2: Initialize block-wise losses {ℓi← 0 : i ∈ [M//b]}</td></tr><tr><td colspan="2">3: Compute initial attn. input x1 ← embed(u)</td></tr><tr><td colspan="2">4: for attn. layer m ∈ [M] do ▷ Computation</td></tr><tr><td colspan="2">5: q, k, v = xmWq^m, xmWk^m, xmWv^m</td></tr><tr><td colspan="2">6: y^m = softmax_attn(q, k, v) (Eq. 1) ▷ (No grad)</td></tr><tr><td colspan="2">7: y^m = linear_attn(q, k, v, w, φq^m, φk^m, γ^m) (Eq. 6)</td></tr><tr><td colspan="2">8: ℓm/b ← ℓm/b + ℓMSE(y^m, y^m) (Eq. 7)</td></tr><tr><td colspan="2">9: x^m+1 = mlp^m(y^mW_o^m) ▷ (Teacher-force next layer</td></tr><tr><td colspan="2">10: for attn. layer m ∈ [M] do ▷ Update feature map</td></tr><tr><td colspan="2">11: for weights θ ∈ {φq^m, φk^m, γ^m} do</td></tr><tr><td colspan="2">12: Update θ ← θ - α ∂/∂θ ℓm/b</td></tr></table>
+
+# Algorithm 2 LoLCATs Step 2: LoRA Adjust
+
+<table><tr><td colspan="2">Input: Linearized Transformer with M attn. layers; input tokens u</td></tr><tr><td colspan="2">Input: Trained feature map params. {φq^m, φk^m, γ^m : m ∈ [M]}, window size w; learning rate α</td></tr><tr><td colspan="2">1: for attn. layer m ∈ [M] do ▷ (Add LoRA weights) Init. LoRA A^m, B^m for each W_q^m, W_k^m, W_v^m, W_o^m</td></tr><tr><td colspan="2">2: Compute initial attn. input x^1 ← embed(u)</td></tr><tr><td colspan="2">3: for attn. layer m ∈ [M] do ▷ Compute attentions</td></tr><tr><td colspan="2">4: q = x^m(W_q^m + B_q^m A_q^m) ▷ (LoRA forward pass)</td></tr><tr><td colspan="2">5: k = x^m(W_k^m + B_k^m A_k^m)</td></tr><tr><td colspan="2">6: v = x^m(W_v^m + B_v^m A_v^m)</td></tr><tr><td colspan="2">7: y^m = linear_attn(q, k, v, w, φq^m, φk^m, γ^m) (Eq. 6)</td></tr><tr><td colspan="2">8: x^{m+1} = mlp^m(y^m(W_o^m + B_o^m A_o^m))</td></tr><tr><td colspan="2">9: Compute next-token pred u = l_mhead(x^M) ▷ Train LoRA</td></tr><tr><td colspan="2">10: Compute sample loss ℓ = ℓCrossEnt(ˆu_{0:n-1}, u_{1:n})</td></tr><tr><td colspan="2">11: Update ℓ← ℓ - α ∂ℓ/∂θ for all ℓ in all A, B</td></tr></table>
+
+# 4 EXPERIMENTS
+
+Through experiments, we study: (1) if LoLCATs linearizes LLMs with higher quality than existing subquadratic alternatives and linearizations, and higher generation efficiency than original Transformers (Sec. 4.1); (2) how ablations on attention transfer loss, subquadratic architecture, and parameter and token counts impact LLM quality (Sec. 4.2); and (3) how LoLCATs' quality and efficiency holds up to 70B and 405B LLMs by linearizing the complete Llama 3.1 family (Sec. 4.3).
+
+# 4.1 MAIN RESULTS: LOLCATS EFFICIENTLY RECOVERS QUALITY IN LINEARIZED LLMS
+
+In our main evaluation, we linearize the popular base Llama 3 8B (AI, 2024) and Mistral 7B (Jiang et al., 2023) LLMs. We first test if LoLCATs can efficiently create high-quality subquadratic LLMs from strong base Transformers, comparing to existing linearized LLMs from prior methods. We also
+
+<table><tr><td>Model</td><td>Training Tokens (B)</td><td>PiQA</td><td>ARC-e</td><td>ARC-c (norm)</td><td>HellaSwag (norm)</td><td>Wino-grande</td><td>MMLU (5-shot)</td><td>Avg.</td><td>Avg. (no MMLU)</td></tr><tr><td>Mistral 7B</td><td>-</td><td>82.1</td><td>80.9</td><td>53.8</td><td>81.0</td><td>74.0</td><td>62.4</td><td>72.4</td><td>74.4</td></tr><tr><td>Mistral 7B SUPRA</td><td>100</td><td>80.4</td><td>75.9</td><td>45.8</td><td>77.1</td><td>70.3</td><td>34.2</td><td>64.0</td><td>69.9</td></tr><tr><td>Mistral 7B LoLCATs (Ours)</td><td>0.04</td><td>81.5</td><td>81.7</td><td>54.9</td><td>80.7</td><td>74.0</td><td>51.4</td><td>70.7</td><td>74.5</td></tr><tr><td>Llama 3 8B</td><td>-</td><td>79.9</td><td>80.1</td><td>53.3</td><td>79.1</td><td>73.1</td><td>66.6</td><td>72.0</td><td>73.1</td></tr><tr><td>Mamba2-Llama 3</td><td>20</td><td>76.8</td><td>74.1</td><td>48.0</td><td>70.8</td><td>58.6</td><td>43.2</td><td>61.9</td><td>65.6</td></tr><tr><td>Mamba2-Llama 3, 50% Attn.</td><td>20</td><td>81.5</td><td>78.8</td><td>58.2</td><td>79.5</td><td>71.5</td><td>56.7</td><td>71.0</td><td>73.9</td></tr><tr><td>Llama 3 8B Hedgehog</td><td>0.04</td><td>77.4</td><td>71.1</td><td>40.6</td><td>66.5</td><td>54.3</td><td>24.2</td><td>55.7</td><td>62.0</td></tr><tr><td>Llama 3 8B LoLCATs (Ours)</td><td>0.04</td><td>80.9</td><td>81.7</td><td>54.9</td><td>79.7</td><td>74.1</td><td>52.8</td><td>70.7</td><td>74.2</td></tr></table>
+
+Table 3: LoLCATs comparison among linearized 7B+ LLMs. Among linearized 7B+ LLMs, LoLCATs-linearized Mistral 7B and Llama 3 8B consistently achieve best or 2nd-best performance on LM Eval tasks. LoLCATs closes the Transformer quality gap by  $79.8\%$  (Mistral 7B) and  $86.6\%$  (Llama 3 8B) (average over all tasks; numbers except Hedgehog cited from original works), despite only using 40M tokens to linearize.  
+
+<table><tr><td>Model</td><td>Tokens (B)</td><td>PiQA</td><td>ARC-e</td><td>ARC-c(acc. norm)</td><td>HellaSwag(acc. norm)</td><td>Winogrande</td><td>MMLU(5-shot)</td><td>Avg.(w MMLU)</td><td>Avg.(no MMLU)</td></tr><tr><td>Transformer</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Gemma 7B</td><td>6000</td><td>81.9</td><td>81.1</td><td>53.2</td><td>80.7</td><td>73.7</td><td>62.9</td><td>72.3</td><td>74.1</td></tr><tr><td>Mistral 7B</td><td>8000*</td><td>82.1</td><td>80.9</td><td>53.8</td><td>81.0</td><td>74.0</td><td>62.4</td><td>72.4</td><td>74.4</td></tr><tr><td>Llama 3 8B</td><td>15000</td><td>79.9</td><td>80.1</td><td>53.3</td><td>79.1</td><td>73.1</td><td>66.6</td><td>72.0</td><td>73.1</td></tr><tr><td>Subquadratic</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Mamba 7B</td><td>1200</td><td>81.0</td><td>77.5</td><td>46.7</td><td>77.9</td><td>71.8</td><td>33.3</td><td>64.7</td><td>71.0</td></tr><tr><td>RWKV-6 World v2.1 7B</td><td>1420</td><td>78.7</td><td>76.8</td><td>46.3</td><td>75.1</td><td>70.0</td><td>-</td><td>69.4</td><td>69.4</td></tr><tr><td>TransNormerLLM 7B</td><td>1400</td><td>80.1</td><td>75.4</td><td>44.4</td><td>75.2</td><td>66.1</td><td>43.1</td><td>64.1</td><td>68.2</td></tr><tr><td>Hawk 7B</td><td>300</td><td>80.0</td><td>74.4</td><td>45.9</td><td>77.6</td><td>69.9</td><td>35.0</td><td>63.8</td><td>69.6</td></tr><tr><td>Griffin 7B</td><td>300</td><td>81.0</td><td>75.4</td><td>47.9</td><td>78.6</td><td>72.6</td><td>39.3</td><td>65.8</td><td>71.1</td></tr><tr><td>Hybrid Softmax</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>StripedHyena-Nous-7B</td><td>-</td><td>78.8</td><td>77.2</td><td>40.0</td><td>76.4</td><td>66.4</td><td>26.0</td><td>60.8</td><td>67.8</td></tr><tr><td>Zamba 7B</td><td>1000</td><td>81.4</td><td>74.5</td><td>46.6</td><td>80.2</td><td>76.4</td><td>57.7</td><td>69.5</td><td>71.8</td></tr><tr><td>Linearized</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Mistral 7B LoLCATs (Ours)</td><td>0.04</td><td>81.5</td><td>81.7</td><td>54.9</td><td>80.7</td><td>74.0</td><td>51.4</td><td>70.7</td><td>74.5</td></tr><tr><td>Llama 3 8B LoLCATs (Ours)</td><td>0.04</td><td>80.9</td><td>81.7</td><td>54.9</td><td>79.7</td><td>74.1</td><td>52.8</td><td>70.7</td><td>74.2</td></tr></table>
+
+Table 4: LoLCATs comparison to pretrained subquadratic LLMs. LoLCATs-linearized Mistral 7B and Llama 3 8B outperform pretrained Transformer alternatives by 1.2 to 9.9 points (Avg.), only training  $0.2\%$  of their parameter counts on 0.013 to  $0.003\%$  of their training token counts. \*Reported in Mercat et al. (2024).
+
+test if LOLCATs can create subquadratic LLMs that outperform modern Transformer alternatives pretrained from scratch. For space, we defer linearizing training details to Appendix A.
+
+In Table 4, we report results on six popular LM Evaluation Harness (LM Eval) tasks (Gao et al., 2023). Compared to recent linearizing methods, LoLCATs significantly improves quality and training efficiency across tasks and LLMs. On quality, LoLCATs closes  $79.8\%$  and  $86.6\%$  of the Transformer-linearizing gap for Mistral 7B and Llama 3 8B respectively, notably improving 5-shot MMLU by  $60.9\%$  and  $40.9\%$  over next best fully subquadratic models (17.2 and 9.6 points). On efficiency, we achieve these results while only training  $< 0.2\%$  of model parameters via LoRA versus prior full-parameter training. We also only use 40M tokens versus the prior  $20 - 100\mathrm{B}$  (a  $500 - 2500\times$  improvement in "tokens-to-model" efficiency). Among all 7B LLMs, LoLCATs-linearized LLMs further outperform strong subquadratic Transformer alternatives, representing RNNs or linear attentions (RWKV-v6 (Peng et al., 2024), Hawk, Griffin (De et al., 2024), TransNormer (Qin et al., 2023)), state-space models (Mamba (Gu & Dao, 2023)), and hybrid architectures with some full attention (StripedHyena (Poli et al., 2023b), Zamba (Glorioso et al., 2024)).
+
+# 4.2 LOLCATS COMPONENT PROPERTIES AND ABLATIONS
+
+We next validate that LoLCATs enables subquadratic efficiency, and study how each of LoLCATs' components contribute to these quality gains. We include additional ablations in Appendix B.
+
+Subquadratic Generation Throughput and Memory. We measure generation throughput and memory of LoLCATs LLMs, validating that linearizing LLMs can significantly improve their generation efficiency. We use the popular Llama 3 8B HuggingFace model $^3$ , and compare LoL-
+
+![](images/9313b0bda3a5b78c4a4f3504d0840d1f80e8fd7714b300073a25a598dea88c13.jpg)  
+Figure 8: Generation Efficiency, Llama 3 8B.
+
+![](images/38edc804b2cfbaa58a825036d5c95b68c07905314fc13064a54db6e356724a03.jpg)
+
+<table><tr><td>Feature Map</td><td>LM Eval Metric</td><td>Swap &amp; Finetune</td><td>+Attention Transfer</td><td>+Sliding Window, +Attention Transfer</td><td>+ Sliding Window, No Attention Transfer</td></tr><tr><td rowspan="2">Hedgehog</td><td>Average</td><td>44.20</td><td>55.32</td><td>70.66</td><td>68.78</td></tr><tr><td>MMLU (5-shot)</td><td>23.80</td><td>23.80</td><td>52.77</td><td>45.80</td></tr><tr><td rowspan="2">T2R</td><td>Average</td><td>38.84</td><td>54.83</td><td>68.28</td><td>39.52</td></tr><tr><td>MMLU (5-shot)</td><td>23.20</td><td>23.10</td><td>40.70</td><td>23.80</td></tr></table>
+
+Table 5: LoLCATs component ablations, linearizing Llama 3 8B over 1024-token sequences. LoLCATs default shaded. Across Hedgehog and T2R feature maps, LoLCATs' attention transfer and sliding window increasingly improve linearized LLM quality. Full task results in App. B.1.1.  
+
+<table><tr><td></td><td>PiQA acc</td><td>ARC Easy acc</td><td>ARC Challenge (acc norm)</td><td>HellaSwag (acc norm)</td><td>WinoGrande acc</td><td>MMLU (5-shot) acc</td></tr><tr><td>Llama 3.1 8B</td><td>79.87</td><td>81.52</td><td>53.58</td><td>79.01</td><td>73.48</td><td>66.11</td></tr><tr><td>Linearized, no attn. transfer</td><td>78.67</td><td>78.11</td><td>49.83</td><td>77.83</td><td>68.51</td><td>51.44</td></tr><tr><td>LoLCATs (Ours)</td><td>80.96</td><td>82.37</td><td>54.44</td><td>79.07</td><td>69.69</td><td>54.88</td></tr><tr><td>Llama 3.1 70B</td><td>83.10</td><td>87.30</td><td>60.60</td><td>85.00</td><td>79.60</td><td>78.80</td></tr><tr><td>Linearized, no attn. transfer</td><td>81.99</td><td>80.89</td><td>54.44</td><td>82.29</td><td>71.19</td><td>28.74</td></tr><tr><td>LoLCATs (Ours)</td><td>82.10</td><td>84.98</td><td>60.50</td><td>84.62</td><td>73.72</td><td>67.70</td></tr><tr><td>Llama 3.1 405B</td><td>85.58</td><td>87.58</td><td>66.21</td><td>87.13</td><td>79.40</td><td>82.98</td></tr><tr><td>Linearized, no attn. transfer</td><td>84.44</td><td>86.62</td><td>64.33</td><td>86.19</td><td>79.87</td><td>33.86</td></tr><tr><td>LoLCATs (Ours)</td><td>85.58</td><td>88.80</td><td>67.75</td><td>87.41</td><td>80.35</td><td>72.20</td></tr></table>
+
+Table 6: Linearizing Llama 3.1 8B, 70B, and 405B. Among the first linearized 70B and 405B LLMs (via low-rank linearizing), LOLCATS significantly improves zero- and few-shot quality.
+
+CATs implemented in HuggingFace Transformers with the supported FlashAttention-2 (FA2) (Dao, 2023). We benchmark LoLCATs with Hedgehog feature map and linear + sliding window attention in FP32 and BF16 on one 80GB H100. Fig. 8 reports scaling batch size on throughput (left) and memory (right). We measure throughput as (generated tokens × batch size / total time), with 128-token prompts and 4096-token generations. With larger batch size, LoLCATs-linearized LLMs achieve higher throughput than FA2. This corresponds with lower memory, where FA2 exceeds memory at batch size 64. Meanwhile, LoLCATs supports 3000 tokens / second at batch size 2048 (Fig. 8 left), with fixed "KV state" vs. softmax attention's growing KV cache (Fig. 8 right).
+
+Ablations. We study how attention transfer and linear + sliding window attention in LoLCATs contribute to downstream linearized Llama 3 8B performance (Table 5). We start with prior linear attentions (Hedgehog, Zhang et al. (2024); T2R, Kasai et al. (2021)), using the prior linearizing approach that swaps attentions and finetunes the model to predict next tokens (Mercat et al., 2024). We then add (i) attention transfer, (ii) sliding window attentions, or (iii) both. On average LM Eval score and 5-shot MMLU accuracy, LoLCATs' default performs best across feature maps.
+
+# 4.3 SCALING UP LINEARIZING TO 70B AND 405B LLMS
+
+We finally use LoLCATs to scale up linearizing to Llama 3.1 70B and 405B models. In Table 6, LoLCATs provides the first practical solution for linearizing larger LLMs, achieving significant quality improvements over prior linearizing approaches of swapping in attentions and finetuning (Mercat et al., 2024). With the same linear + sliding window layer, LoLCATs gets +39.0 points in 5-shot MMLU accuracy on Llama 3.1 70B, and +38.3 on Llama 3.1 405B. These results highlight LoLCATs' ability to linearize large-scale models with greater efficiency and improved performance, showing for the first time that we can scale up linearizing to 70B+ LLMs.
+
+# 5 CONCLUSION
+
+We propose LoLCATs, an efficient LLM linearizing method that (1) trains linear attentions to match an LLM's self-attention, before (2) swapping the attentions and only finetuning the replacing attentions with LoRA. This reduces linearizing to learning good softmax attention approximations, and we study how to do so. On popular LM Eval tasks, LoLCATs enables LLMs with high quality and inference efficiency, outperforming prior Transformers alternatives while only updating  $0.2\%$  of model parameters and requiring  $0.003\%$  of LLM pretraining tokens. Our findings substantially improve linearizing quality and accessibility, enabling the first linearized 70B and 405B LLMs.
+
+# ETHICS STATEMENT
+
+Our work deals with improving the efficiency of open-weight models. While promising for beneficial applications, increasing their accessibility also raises concerns about potential misuse. Bad actors could leverage our technique to develop LLMs capable of generating harmful content, spreading misinformation, or enabling other malicious activities. We focus primarily on base models, but acknowledge that linearizing could also be used on instruction-tuned LLMs; research on whether linearizing preserves guardrails is still an open question. We acknowledge the risks and believe in the responsible development and deployment of efficient and widely accessible models.
+
+# REPRODUCIBILITY
+
+We include experimental details in Appendix A, and further implementation details with sample code for linearizing architectures and training in Appendix C.1. Our code is also available at https://github.com/HazyResearch/lolcats
+
+# ACKNOWLEDGEMENTS
+
+We thank Mayee Chen, Ben Viggiano, Gautam Machiraju, Dan Fu, Sabri Eyuboglu, Tri Dao, and anonymous reviewers for helpful discussions on linear attention and paper feedback. We gratefully acknowledge the support of NIH under No. U54EB020405 (Mobilize), NSF under Nos. CCF2247015 (Hardware-Aware), CCF1763315 (Beyond Sparsity), CCF1563078 (Volume to Velocity), and 1937301 (RTML); US DEVCOM ARL under Nos. W911NF-23-2-0184 (Long-context) and W911NF-21-2-0251 (Interactive Human-AI Teaming); ONR under Nos. N000142312633 (Deep Signal Processing); Stanford HAI under No. 247183; NXP, Xilinx, LETI-CEA, Intel, IBM, Microsoft, NEC, Toshiba, TSMC, ARM, Hitachi, BASF, Accenture, Ericsson, Qualcomm, Analog Devices, Google Cloud, Salesforce, Total, the HAI-GCP Cloud Credits for Research program, the Stanford Data Science Initiative (SDSI), and members of the Stanford DAWN project: Meta, Google, and VMWare. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views, policies, or endorsements, either expressed or implied, of NIH, ONR, or the U.S. Government.
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+
+# A EXPERIMENTAL DETAILS
+
+# A.1 MAIN RESULTS, LINEARIZING 7B AND 8B LLMS
+
+Setup. We describe our setup for linearizing Mistral 7B (v0.1) (Jiang et al., 2023), Llama 3 8B (AI@Meta, 2024a), and Llama 3.1 8B (Dubey et al., 2024).
+
+For linearizing layers, we replace softmax attentions with hybrid linear + sliding window analogs (Section 3.3.1), using Hedgehog's feature map for its prior quality (Zhang et al., 2024).
+
+For the sliding window implementation, we considered two options: a standard sliding window where  $w$  is the same for all tokens, and a "terraced" window where  $w$  changes based on token index (Figure 9). While we found both comparable in quality (Table 18), the latter lets us exploit the new ThunderKittens (TK) DSL's (Spector et al., 2024) primitives for implementing fast CUDA kernels. Here we prefer contiguous blocks of size  $w = 64$ , which can quickly be computed in parallel on modern GPUs. We use this "terrace" implementation in our main results, and include further implementation details in Appendix C.2.
+
+![](images/f1bce7a0da46db61d6ad403d2948626374d4e8c0fc0bc0aa2c33088179d5468a.jpg)  
+(a) Sliding window  
+Figure 9: We apply softmax attention locally and attend to all past tokens with linear attention.
+
+![](images/88ccda2a4371226e4652bd6446f9795376535545872a6e5a7cf8795ac82460c2.jpg)  
+(b) TK "terrace"
+
+For linearizing data, we use the Alpaca linearizing data setup in Section 3.2 unless otherwise noted. We also tried a more typical pretraining corpus (a subset $^4$  of RedPajama (Computer, 2023)), but found comparable performance when controlling for number of token updates (Appendix B.5.1). To linearize, we simply train all feature maps in parallel for two epochs with learning rate 1e-2, before applying LoRA on the attention projection layers for two epochs with learning rate 1e-4. By default, we use LoRA rank  $r = 8$ , and scale LoRA updates by 2 ( $\alpha = 16$  in HuggingFace PEFT $^5$ ), amounting to training  $< 0.09\%$  of all model parameters. For both stages, we train with early stopping, AdamW optimizer (Loshchilov & Hutter, 2017), and packing into 1024-token sequences with batch size 8. We evaluate the best checkpoints based on validation set perplexity.
+
+Hyperparameters. We list all model and training hyperparameters in Table 7. For learning rates, we did an initial sweep over  $\{1\mathrm{e} - 2,1\mathrm{e} - 3,1\mathrm{e} - 4\}$ , choosing the best based on final validation set perplexity during step 2: low-rank adjusting, and checkpointing with early stopping. We did not tune batch size or choice of optimizer, and used default values informed by prior work for other design parameters such as sliding window size (Arora et al., 2024), LoRA rank, and LoRA projection layers (Hu et al., 2021). In Appendix B, we study the effect of sweeping various values such as window sizes, ranks, and LoRA modules as ablations.
+
+Compute Resources. For each linearizing run we use one NVIDIA 40GB A100 GPU. With batch size 1 and gradient accumulation over 8 batches, attention transfer takes  $\approx 2$  hours and post-swap finetuning takes  $\approx 4.5$  hours, i.e., 6.5 total GPU hours to linearize an 8B LLM.
+
+# A.2 LINEARIZING LLAMA 3.1 70B
+
+We provide experimental details corresponding to the 70B parameter results reported in Table 6.
+
+Setup. We compare the quality of two linearization approaches to the quality of the original Llama 3.1 70B model, including (1) the baseline linearization without attention transfer, which is representative of the approach used in prior work (Mercat et al., 2024; Yang et al., 2024; Wang et al., 2024) and (2) our approach, LoLCATs. For both the baseline and LoLCATs, we start with Llama 3.1
+
+<table><tr><td></td><td>Hedgehog</td><td>LoLCATs</td></tr><tr><td colspan="3">Model</td></tr><tr><td>Precision</td><td colspan="2">16-bit (float16)</td></tr><tr><td>Sequence length</td><td colspan="2">1024</td></tr><tr><td>Linearizing attention</td><td>(Linear)</td><td>(Linear + Sliding Window)</td></tr><tr><td>Linear attn feature map</td><td></td><td>Hedgehog</td></tr><tr><td>Linear attn feature dimension</td><td colspan="2">64 (effectively 128, see Table 1)</td></tr><tr><td>Linear attn feature activation</td><td colspan="2">Softmax (across feature dim)</td></tr><tr><td>Sliding window implementation</td><td>N/A</td><td>Terrace</td></tr><tr><td>Sliding window attn size</td><td>N/A</td><td>64</td></tr><tr><td colspan="3">Optimizer and LR Schedule</td></tr><tr><td>Optimizer</td><td colspan="2">AdamW</td></tr><tr><td>Global batch size</td><td colspan="2">8</td></tr><tr><td>Gradient accumulation</td><td colspan="2">8</td></tr><tr><td>Gradient clipping threshold</td><td colspan="2">1.0</td></tr><tr><td>Learning rate schedule</td><td colspan="2">Reduce LR on Plateau</td></tr><tr><td colspan="3">Step 1: Attention Transfer</td></tr><tr><td>Number of epochs</td><td colspan="2">2</td></tr><tr><td>Tokens per epoch</td><td colspan="2">10M</td></tr><tr><td>Learning rate</td><td colspan="2">0.01</td></tr><tr><td colspan="3">Step 2: Low-rank Adjusting</td></tr><tr><td>Number of epochs</td><td colspan="2">2</td></tr><tr><td>Tokens per epoch</td><td colspan="2">20M</td></tr><tr><td>Learning rate</td><td colspan="2">1e-4</td></tr><tr><td>LoRA rank and alpha</td><td colspan="2">r=8, α=16</td></tr><tr><td>LoRA dropout</td><td colspan="2">0.0</td></tr><tr><td>LoRA projections</td><td colspan="2">Wq, Wk, Wv, Wo</td></tr></table>
+
+Table 7: Hyperparameters for Mistral 7B, Llama 3 8B, and Llama 3.1 8B experiments.
+
+70B and replace the softmax attentions with the linear attention architecture defined in Section 3.3.1, Equation (6). The training procedure involves:
+
+- Baseline: We introduce LoRA parameters to the attention  $W_{q}$ ,  $W_{k}$ ,  $W_{v}$ ,  $W_{o}$  projection matrices. We train the linear attention feature maps, learnable mixing term  $\gamma$ , and the LoRA parameters during the fine-tuning adjustment stage of the linearization process.  
+- LoLCATs: We first perform layer-wise attention transfer following Equation (7), with  $k = 80$  (i.e., we optimize over all layers together). We then introduce LoRA parameters to the attention  $W_{q}, W_{k}, W_{v}, W_{o}$  projection matrices. During fine-tuning we only train the LoRA parameters, freezing the linear attention map and  $\gamma$  weights.
+
+We use an MSE loss for the layer-wise attention transfer stage and cross-entropy loss for the next-token prediction fine-tuning (Low-rank Finetuning) stage.
+
+Hyperparameters. We include the hyperparameters for the baseline and LoLCATs approaches in Table 8, following the same sweep as the 8B models. Since the baseline does not use attention transfer, we mark these values with "N/A". We linearize each model using the same randomly sampled 20M tokens of the RedPajama pre-training corpus (Computer, 2023). We pack the sequences to fill full context length, and evaluate the best checkpoints based on validation set perplexity.
+
+Compute Resources. We linearize using a single NVIDIA  $8 \times 80\mathrm{GB}$  H100 node. Attention transfer takes 4 hours and fine-tuning takes 14 hours. We use PyTorch FSDP with activation checkpointing for distributed training.
+
+<table><tr><td></td><td>Baseline</td><td>LoLCATS</td></tr><tr><td colspan="3">Model</td></tr><tr><td>Precision</td><td colspan="2">16-bit (bfloat16)</td></tr><tr><td>Sequence length</td><td colspan="2">1024</td></tr><tr><td>Linearizing attention</td><td colspan="2">Linear + Sliding Window</td></tr><tr><td>Linear attn feature map</td><td colspan="2">Hedgehog</td></tr><tr><td>Linear attn feature dimension</td><td colspan="2">64 (effectively 128, see Table 1)</td></tr><tr><td>Linear attn feature activation</td><td colspan="2">Softmax (across feature dim)</td></tr><tr><td>Sliding window implementation</td><td colspan="2">Terrace</td></tr><tr><td>Sliding window attn size</td><td colspan="2">64</td></tr><tr><td colspan="3">Optimizer and LR Schedule</td></tr><tr><td>Optimizer</td><td colspan="2">AdamW</td></tr><tr><td>Global batch size</td><td colspan="2">8</td></tr><tr><td>Gradient accumulation</td><td colspan="2">8</td></tr><tr><td>Gradient clipping threshold</td><td colspan="2">1.0</td></tr><tr><td>Learning rate schedule</td><td colspan="2">Reduce LR on Plateau</td></tr><tr><td colspan="3">Stage 1: Attention Transfer</td></tr><tr><td>Number of epochs</td><td>N/A</td><td>1</td></tr><tr><td>Tokens per epoch</td><td>N/A</td><td>20M</td></tr><tr><td>Learning rate</td><td>N/A</td><td>0.01</td></tr><tr><td colspan="3">Stage 2: Low-rank Adjusting</td></tr><tr><td>Number of epochs</td><td colspan="2">1</td></tr><tr><td>Tokens per epoch</td><td>20M</td><td>20M</td></tr><tr><td>Learning rate</td><td colspan="2">1e-4</td></tr><tr><td>LoRA rank and alpha</td><td colspan="2">r=8, α=16</td></tr><tr><td>LoRA dropout</td><td colspan="2">0.0</td></tr><tr><td>LoRA projections</td><td colspan="2">Wq, Wk, Wv, Wo</td></tr></table>
+
+Table 8: Hyperparameters for Llama 3.1 70B experiments.
+
+# A.3 LINEARIZING LLAMA 3.1 405B
+
+Setup. We compare the quality of two linearization approaches to the quality of the original Llama 3.1 405B model, including (1) the baseline linearization without attention transfer, which is representative of the approach used in prior work (Mercat et al., 2024) and (2) our approach, LoLCATs. For both the baseline and LoLCATs, we start with Llama 3.1 405B and replace the softmax attentions with the linear attention architecture defined in Section 3.3.1, Equation (6). The training procedure involves:
+
+- Baseline: We introduce LoRA parameters to the attention  $W_{q}, W_{k}, W_{v}, W_{o}$  projection matrices. We train the linear attention feature maps, learnable mixing term  $\gamma$ , and the LoRA parameters during the fine-tuning adjustment stage of the linearization process.  
+- LoLCATs: We first perform block-wise attention transfer following Equation (7), with  $k = 9$  as the block size. To perform attention transfer for block  $i$ , we save the hidden states outputted by block  $i - 1$  to disk and then use this as training data for block  $i$ . We then introduce LoRA parameters to the attention  $W_{q}, W_{k}, W_{v}, W_{o}$  projection matrices. During fine-tuning we only train the LoRA parameters, freezing the linear attention map and  $\gamma$  weights.
+
+For the reported checkpoints, we train the layer-wise attention transfer stage using use a weighted combination of the MSE loss on attention outputs plus a cross-entropy loss between the softmax and linear attention maps. We use cross-entropy loss for Stage 2 Low-rank Linearizing.
+
+Hyperparameters. We include the hyperparameters for the baseline and LoLCATs approaches in Table 9, following the same sweep as the 8B models. Since the baseline does not use attention transfer, we mark it with "N/A". We linearize each model using the same randomly sampled 20M
+
+tokens of the RedPajama pre-training corpus (Computer, 2023). We pack the sequences to fill the full context length, and evaluate the best checkpoints based on validation set perplexity.
+
+<table><tr><td></td><td>Baseline</td><td>LoLCATS</td></tr><tr><td colspan="3">Model</td></tr><tr><td>Precision</td><td colspan="2">16-bit (FP16)</td></tr><tr><td>Sequence length</td><td colspan="2">1024</td></tr><tr><td>Linearizing attention</td><td colspan="2">Linear + Sliding Window</td></tr><tr><td>Linear attn feature map</td><td colspan="2">Hedgehog</td></tr><tr><td>Linear attn feature dimension</td><td colspan="2">64 (effectively 128, see Table 1)</td></tr><tr><td>Linear attn feature activation</td><td colspan="2">Softmax (across feature dim)</td></tr><tr><td>Sliding window implementation</td><td colspan="2">Terrace</td></tr><tr><td>Sliding window attn size</td><td colspan="2">64</td></tr><tr><td colspan="3">Optimizer and LR Schedule</td></tr><tr><td>Optimizer</td><td colspan="2">AdamW</td></tr><tr><td>Global batch size</td><td colspan="2">8</td></tr><tr><td>Gradient accumulation</td><td colspan="2">8</td></tr><tr><td>Gradient clipping threshold</td><td colspan="2">1.0</td></tr><tr><td>Learning rate schedule</td><td colspan="2">Reduce LR on Plateau</td></tr><tr><td colspan="3">Stage 1: Attention Transfer</td></tr><tr><td>Number of epochs</td><td>N/A</td><td>1</td></tr><tr><td>Tokens per epoch</td><td>N/A</td><td>20M</td></tr><tr><td>Learning rate</td><td>N/A</td><td>0.01</td></tr><tr><td>MSE and X-ent weights</td><td>N/A</td><td>1000, 1</td></tr><tr><td colspan="3">Stage 2: Low-rank Adjusting</td></tr><tr><td>Number of epochs</td><td colspan="2">1</td></tr><tr><td>Tokens per epoch</td><td>20M</td><td>20M</td></tr><tr><td>Learning rate</td><td colspan="2">1e-4</td></tr><tr><td>LoRA rank and alpha</td><td colspan="2">r = 4, α = 8</td></tr><tr><td>LoRA dropout</td><td colspan="2">0.5</td></tr><tr><td>LoRA projections</td><td colspan="2">Wq, Wk, Wv, Wo</td></tr></table>
+
+Table 9: Hyperparameters for Llama 3.1 405B experiments.
+
+Compute Resources. We linearize the baseline using three NVIDIA  $8 \times 80$ GB H100 nodes, evaluating the best validation checkpoint after 19.5 hours. For LoLCATs, we perform attention transfer with 180GB H100 GPU for 5 hours per block, and we finetune with 3 NVIDIA  $8 \times 80$ GB H100 nodes for 16 hours. We use PyTorch FSDP with activation checkpointing for distributed training.
+
+# B ADDITIONAL EXPERIMENTS AND RESULTS
+
+To better understand LoLCATs' properties and performance, we now report extended results on LoLCATs. We first report expanded results of our main paper, including task-specific ablation numbers, multiple seeds, and additional comparison at the 1B LLM scale. We then extend our ablations by studying how different amounts of parameter updates, different data sources, and different amounts of training data affect LoLCATs quality for various tasks such as zero-shot LM Eval, 5-shot MMLU, and passkey retrieval. We finally expand on the layer-wise training dynamics of LoLCATs: how layer-wise MSE between learned and softmax attentions changes during LoRA adjusting (App. B.7.1), how LoRA weights update over time depending on attention transfer and softmax attention-matching quality (App. B.7.2), and how block-wise attention transfer both is motivated by layer-wise MSE and improves linearized LLM quality for larger LLMs (App. B.7.3).
+
+# B.1 EXPANDED RESULTS
+
+# B.1.1 TASK-SPECIFIC RESULTS FOR LOLCATS COMPONENT ABLATIONS
+
+We report the task-specific LM Eval results when ablating the attention transfer and linear + sliding window attention in LoLCATs, expanding on Table 5 in Table 10. Across all but one task (ARC-easy), we validate that the LoLCATs proposed combination leads to best performance.
+
+<table><tr><td>Feature Map</td><td>+Attention Transfer</td><td>+Sliding Window</td><td>PiQA</td><td>ARC-e</td><td>ARC-c (acc. norm)</td><td>HellaSwag (acc. norm)</td><td>Winogrande</td><td>MMLU (5-shot)</td><td>Average</td></tr><tr><td rowspan="4">Hedgehog</td><td>X</td><td>X</td><td>67.8</td><td>58.1</td><td>28.9</td><td>35.8</td><td>50.8</td><td>23.8</td><td>44.2</td></tr><tr><td>✓</td><td>X</td><td>76.5</td><td>72.6</td><td>40.1</td><td>65.6</td><td>53.3</td><td>23.8</td><td>55.3</td></tr><tr><td>X</td><td>✓</td><td>80.5</td><td>80.6</td><td>53.4</td><td>78.8</td><td>73.6</td><td>45.8</td><td>68.8</td></tr><tr><td>✓</td><td>✓</td><td>80.9</td><td>81.7</td><td>54.9</td><td>79.7</td><td>74.1</td><td>52.8</td><td>70.7</td></tr><tr><td rowspan="4">T2R</td><td>X</td><td>X</td><td>62.0</td><td>42.1</td><td>24.7</td><td>32.7</td><td>48.3</td><td>23.2</td><td>38.8</td></tr><tr><td>✓</td><td>X</td><td>76.1</td><td>72.8</td><td>40.8</td><td>63.6</td><td>52.6</td><td>23.1</td><td>54.8</td></tr><tr><td>X</td><td>✓</td><td>54.8</td><td>26.3</td><td>26.2</td><td>56.4</td><td>49.6</td><td>23.8</td><td>39.5</td></tr><tr><td>✓</td><td>✓</td><td>80.7</td><td>82.0</td><td>54.6</td><td>79.5</td><td>72.2</td><td>40.7</td><td>68.3</td></tr></table>
+
+# B.1.2 LLAMA 38B LOLCATS RESULTS ACROSS MULTIPLE RUNS
+
+In Table 11, we report the LM Eval results for Llama 3 8B models linearized with two linear attention approaches, Hedgehog (Zhang et al., 2024) and LoLCATs (ours), after doing linearizing across three seeds  $(0,1,2)$ . We report means and standard deviations (in parentheses). Across all tasks, variation across seeds is low (under 1 point, other than Winogrande), with the difference in task performance across methods being much higher (greater than 10 points).
+
+Table 10: LoLCATs component ablations on individual LM Eval tasks, linearizing Llama 3 8B, expanded view of Table 5. LoLCATs default shaded. Across Hedgehog and T2R feature maps, LoLCATs' attention transfer and sliding window increasingly improve linearized LLM quality on popular LM Eval tasks both on average and specifically for 5-shot MMLU scores.  
+
+<table><tr><td>Llama 3 8B</td><td>PiQA</td><td>ARC-e</td><td>ARC-c (norm)</td><td>HellaSwag (norm)</td><td>Winogrande</td><td>MMLU (5-shot)</td><td>Average</td><td>Average (no MMLU)</td></tr><tr><td>Hedgehog</td><td>76.86 (0.32)</td><td>73.27 (0.67)</td><td>40.76 (0.69)</td><td>65.77 (0.38)</td><td>53.42 (0.22)</td><td>24.22 (0.62)</td><td>55.72 (0.35)</td><td>62.02 (0.35)</td></tr><tr><td>LoLCATs (Ours)</td><td>80.79 (0.11)</td><td>81.62 (0.41)</td><td>54.73 (0.41)</td><td>79.48 (0.07)</td><td>72.92 (1.02)</td><td>52.74 (0.64)</td><td>70.38 (0.33)</td><td>73.91 (0.29)</td></tr></table>
+
+Table 11: LoLCATs comparison with prior linearizing methods, multiple runs; expanded view of linear attention Llama 3 8B results in Table 3. We report mean and (standard deviation) across three seeds, comparing LoLCATs and prior linear attention Hedgehog linearizing method.
+
+# B.2 CLOSING PERFORMANCE GAPS WITH HYBRIDIZING LOLCATS ACROSS LAYERS
+
+We now study how LoLCATs models perform while keeping some layers as full softmax attention. For Llama 3 8B, we keep the first 16 layers (half of all layers) as full softmax attention and only linearize the last half with the sliding window + linear attention layer for Llama 3 8B. We then just do attention transfer over Alpaca samples, instead of the two-stage attention transfer and subsequent LoRA finetuning as default in Table 7 (keeping Stage 1 details the same). In Table 12, we find that keeping just  $50\%$  softmax attention, while only doing Stage 1 transfer on 20 million tokens, substantially closes the MMLU performance gap by  $94.2\%$  (65.8% hybrid versus  $52.8\%$  default LoLCATs, compared to Transformer  $66.6\%$  accuracy). Furthermore, while just training on 20 million tokens (half that of the default protocol), we recover performance on all other tasks.
+
+# B.3 LOLCATS EVALUATION FOR LINEARIZING 1B+ LLMS
+
+We now study LoLCATs' performance when linearizing  $\sim 1$ B parameter LLMs. We choose two popular models, Llama 3.2 1B (AI@Meta, 2024b) and Phi 1.5 1.3B (Li et al., 2023), and linearize with LoLCATs using the same hyperparameters, architectures, and training details (e.g., 40M tokens over Alpaca) as Llama 3 8B (Table 7) except for linear attention feature dimension, instead adjusting this to the 1B model head dimensions (i.e.,  $0.5 \times$  head dimension  $64 = 32$ ). To evaluate
+
+<table><tr><td>Softmax Attn. Layers</td><td>Tokens (B)</td><td>PiQA</td><td>ARC-E</td><td>ARC-C</td><td>HellaSwag</td><td>WinoGrande</td><td>MMLU</td></tr><tr><td>All (Llama 3 8B baseline)</td><td>0</td><td>79.9</td><td>80.1</td><td>53.3</td><td>79.1</td><td>73.1</td><td>66.6</td></tr><tr><td>0-15 (LoLCATS 50%, Just Attn Transfer)</td><td>0.02</td><td>79.5</td><td>80.2</td><td>53.4</td><td>79.2</td><td>73.6</td><td>65.8</td></tr><tr><td>None (LoLCATs, Attn Transfer + LoRA)</td><td>0.04</td><td>80.9</td><td>81.7</td><td>54.9</td><td>79.0</td><td>74.1</td><td>52.8</td></tr></table>
+
+Table 12: Hybrid LoLCATs, Llama 3 8B, LM Eval. With just  $50\%$  softmax attention, training on 20 million tokens (half that of the default protocol) results in substantially closing the gap to full softmax attention Transformers on MMLU (+13 points absolute), and maintaining performance on other LM Eval tasks.  
+
+<table><tr><td>Model</td><td>PiQA</td><td>ARC-e</td><td>ARC-c(acc. norm)</td><td>HellaSwag(acc. norm)</td><td>Winogrande</td><td>MMLU(5-shot)</td><td>Avg.</td><td>Avg.(No MMLU)</td><td>Avg. % ofTransformer</td></tr><tr><td>Llama 3.2 1B</td><td>74.4</td><td>65.5</td><td>35.8</td><td>63.7</td><td>60.5</td><td>31.9</td><td>55.3</td><td>60.0</td><td>-</td></tr><tr><td>→T2R</td><td>69.2</td><td>58.2</td><td>29.9</td><td>42.6</td><td>54.1</td><td>23.3</td><td>46.2</td><td>50.8</td><td>84.7</td></tr><tr><td>→Hedgehog</td><td>70.1</td><td>55.8</td><td>29.8</td><td>47.7</td><td>50.7</td><td>23.0</td><td>46.2</td><td>50.8</td><td>84.7</td></tr><tr><td>→LoLCATs (Ours)</td><td>74.6</td><td>63.0</td><td>35.1</td><td>63.7</td><td>61.5</td><td>27.3</td><td>54.2</td><td>59.6</td><td>99.3</td></tr></table>
+
+Table 13: LoLCATs comparison with linearizing methods, Llama 3.2 1B, LM Eval. Following the two-step linearizing procedure in LoLCATs, we compare the LoLCATs architecture against the prior Transformer-to-RNN (T2R) and Hedgehog linear attentions. At the 1B scale, LoLCATs substantially closes the performance gap with original Llama 3.2 1B Transformer.
+
+LoLCATs quality, we compare the LM Eval Harness performance of LoLCATs with other available linearizing methods on each Transformer LLM (Table 13, Table 14), and against competitive 1B subquadratic LLMs pretrained from scratch (Table 15), reporting results from Bick et al. (2024).
+
+In all evaluation settings, LoLCATs demonstrates competitive or state-of-the-art linearizing quality. When first controlling for the original pretrained Transformer, LoLCATs is able to outperform both prior linear attentions, as well as pure and hybrid Phi-Mamba models created with MOHAWK, another method to distill Transformers into SSM-based Mamba architectures (Bick et al., 2024). Notably, LoLCATs makes this possible while only using  $1.33\%$  of the training tokens in MOHAWK (Table 14), and again only using parameter-efficient updates. Furthermore, by linearizing Llama 3.2 1B and Phi 1.5 1.3B, LoLCATs efficiently creates subquadratic 1B+ LLMs that outperform various LLMs pretrained from scratch (Table 15). These results suggest LoLCATs remains a competitive option for linearizing LLMs at multiple parameter counts.
+
+# B.4 STUDY ON PARAMETER-EFFICIENT TRAINING
+
+In our main results, we found that simple default initializations (e.g., rank 8, applied to all attention projection layers) could recover high quality linearizing while only updating  $< 0.2\%$  of model parameters. In this section, we study how changing different aspects of low-rank adaptation and sliding window size impact linearizing performance.
+
+# B.4.1 EFFECT OF LORA RANK
+
+We study the effect of LoRA rank in post-swap finetuning for zero-shot linearized LLM performance. Following standard implementations (Hu et al., 2021), we consider two factors. First: rank  $r$ , which determines the rank of the low-rank matrices  $A, B$  we decompose the weight deltas into. Second: alpha  $\alpha$ , where  $\alpha / r$  is a scaling factor that controls the degree to which  $BA$  affect the output (i.e., LoRA output  $y = Wx + \frac{\alpha}{r}BAx$ ).
+
+Setup. We sweep over ranks  $\{4, 8, 16, 32, 64, 128, 256\}$ , and adjust  $\alpha$  such that  $\alpha / r = 2$  as a default scaling factor (Raschka, 2023). For comparison, we also do a full finetuning run, where after attention transfer we do the stage 2 adjustment but training all Llama 3 8B parameters. For all runs, we linearize Llama 3 8B. For LoRA runs, we use the same default experimental setup as our main results, while for the full finetuning we decreased learning rate to 1e-5 due to training instability. We start with the linear + "terrace" window attentions, training the feature maps via attention transfer over 20M tokens over Alpaca (2 full epochs). We then freeze feature maps and apply LoRA with the above ranks on all attention weight projections (freezing all other parameters such as those in MLPs or GroupNorms), finetuning for two more epochs over Alpaca using the hyperparameters in Table 7. We evaluate with LM Eval tasks.
+
+<table><tr><td>Model</td><td>Training Tokens (B)</td><td>PiQA</td><td>ARC-e</td><td>ARC-c (acc. norm)</td><td>HellaSwag (acc. norm)</td><td>Winogrande</td><td>MMLU (5-shot)</td><td>Avg. (No MMLU)</td><td>Avg. % of Transformer</td></tr><tr><td>Phi 1.5 1.3B (MOHAWK)</td><td>150</td><td>76.6</td><td>75.6</td><td>48.0</td><td>62.6</td><td>73.4</td><td>-</td><td>67.2</td><td>-</td></tr><tr><td>Phi 1.5 1.3B (Our run)</td><td>150</td><td>76.6</td><td>76.1</td><td>47.6</td><td>62.6</td><td>72.8</td><td>43.6</td><td>67.1</td><td>-</td></tr><tr><td>Phi-Mamba 1.5</td><td>3</td><td>75.5</td><td>74.0</td><td>44.1</td><td>60.2</td><td>71.7</td><td>-</td><td>65.1</td><td>96.8</td></tr><tr><td>Hybrid Phi-Mamba 1.5</td><td>3</td><td>76.5</td><td>75.3</td><td>45.8</td><td>60.6</td><td>72.0</td><td>-</td><td>66.0</td><td>98.2</td></tr><tr><td>Phi 1.5 1.3B T2R</td><td>0.04</td><td>71.0</td><td>69.1</td><td>36.6</td><td>46.2</td><td>53.6</td><td>24.3</td><td>55.3</td><td>82.4</td></tr><tr><td>Phi 1.5 1.3B Hedgehog</td><td>0.04</td><td>72.7</td><td>70.9</td><td>38.0</td><td>49.4</td><td>54.1</td><td>23.5</td><td>57.0</td><td>85.0</td></tr><tr><td>Phi 1.5 1.3B LoLCATs (Ours)</td><td>0.04</td><td>76.9</td><td>77.0</td><td>46.9</td><td>62.3</td><td>72.7</td><td>39.2</td><td>67.2</td><td>100.1</td></tr></table>
+
+Table 14: LoLCATs comparison with linearizing methods, Phi 1.5 1.3B, LM Eval. We compare LoLCATs with linearizing with prior linear attentions, and available results from Bick et al. (2024), who distill Phi 1.5B into Mamba and hybrid Mamba-Transformer architectures with their MOHAWK method (Phi-Mamba 1.5, Hybrid Phi-Mamba 1.5). LoLCATs similarly outperforms prior linearizing methods, closing the gap to the Transformer Phi 1.5 1.3B on PiQA, ARC-Easy and average zero-shot LM Eval (no MMLU).  
+
+<table><tr><td>Model</td><td>Training 
+Tokens (B)</td><td>PiQA</td><td>ARC-e</td><td>ARC-c 
+(acc. norm)</td><td>HellaSwag 
+(acc. norm)</td><td>Winogrande</td><td>MMLU 
+(5-shot)</td><td>Avg. 
+(No MMLU)</td></tr><tr><td colspan="9">Transformer</td></tr><tr><td>Pythia 1.4B</td><td>300</td><td>71.1</td><td>60.6</td><td>26.0</td><td>52.1</td><td>57.3</td><td>26.6</td><td>53.4</td></tr><tr><td>Llama 3.2 1B</td><td>9000</td><td>74.4</td><td>65.5</td><td>35.8</td><td>63.7</td><td>60.5</td><td>31.9</td><td>60.0</td></tr><tr><td>Phi 1.5 1B</td><td>150</td><td>76.6</td><td>76.1</td><td>47.6</td><td>62.6</td><td>72.8</td><td>43.6</td><td>67.1</td></tr><tr><td colspan="9">Subquadratic</td></tr><tr><td>xLSTM 1.4B</td><td>300</td><td>74.6</td><td>64.3</td><td>32.6</td><td>60.9</td><td>60.6</td><td>-</td><td>58.6</td></tr><tr><td>Finch 1.6B (RWKV-v6)</td><td>1100</td><td>72.6</td><td>64.2</td><td>34.1</td><td>57.3</td><td>59.4</td><td>-</td><td>57.5</td></tr><tr><td>DeltaNet 1.3B</td><td>100</td><td>71.2</td><td>57.2</td><td>28.3</td><td>50.2</td><td>53.6</td><td>-</td><td>52.1</td></tr><tr><td>GLA 1.3B</td><td>100</td><td>71.8</td><td>57.2</td><td>26.6</td><td>49.8</td><td>53.9</td><td>-</td><td>51.9</td></tr><tr><td>Mamba 1.4B</td><td>315</td><td>74.2</td><td>65.5</td><td>32.8</td><td>59.1</td><td>61.5</td><td>-</td><td>58.6</td></tr><tr><td>Mamba 2 1.3B</td><td>315</td><td>73.2</td><td>64.3</td><td>33.3</td><td>59.9</td><td>60.9</td><td>-</td><td>58.3</td></tr><tr><td>Llama 3.2 1B LoLCATs (Ours)</td><td>0.04</td><td>74.6</td><td>63.0</td><td>35.1</td><td>63.7</td><td>61.5</td><td>27.3</td><td>59.6</td></tr><tr><td>Phi 1.5 1.3B LoLCATs (Ours)</td><td>0.04</td><td>76.9</td><td>77.0</td><td>46.9</td><td>62.3</td><td>72.7</td><td>39.2</td><td>67.2</td></tr></table>
+
+Table 15: LoLCATs comparison to pretrained 1B LLMs. LoLCATs-linearized Llama 3.2 1B and Phi 1.5 1.3B consistently outperform strong subquadratic  $1\mathrm{B}+$  LLMs pretrained from scratch, achieving best or second-best accuracy on all tasks other than ARC-easy. Subquadratic results reported from Bick et al. (2024).
+
+Results. We report results in Table 16. We find that when applying LoRA to adjust in linearizing after attention transfer, larger rank updates, e.g.,  $r = 128$  or 256, do not necessarily lead to improved zero or few-shot downstream performance. Full finetuning also does not improve performance. Somewhat surprisingly, using just  $r = 4$  leads to overall best performance, while  $r = 128$  and  $r = 8$  achieve best and second-best 5-shot MMLU accuracy. While we leave further exploration for future work, we hypothesize that low-rank updates may improve quality by preventing large and potentially harmful updates with the linearizing data to pretrained weights. During the second step of adjusting, if the linearizing data is not particularly diverse or large (e.g., for efficient linearizing), LoRA can then reduce the risk of overfitting to the linearizing data and losing LLM pretrained generalization.
+
+# B.4.2 EFFECT OF LORA PROJECTION LAYER
+
+We next compare performance when applying LoRA to different weight matrices of the linear attention layers. With the same training and evaluation setup as Appendix B.4.1, but fixing  $r = 8, \alpha = 16$ , we now apply LoRA to different combinations of  $W_{q}, W_{k}, W_{v}, W_{o}$  weights after swapping in attention-transferred linear attentions. We use the same combination for each layer.
+
+We report results in Table 17. Interestingly, when isolating for projections updated, LoRA on projections not involved in computing layer-wise attention weights (value  $W_{v}$  and output  $W_{o}$  projections) improves quality compared to query  $W_{q}$  or key projections  $W_{k}$ . Somewhat surprisingly, updating just  $W_{v}$  or  $W_{o}$  achieves comparable performance to updating all projections (c.f., average zero-shot accuracy of  $74.19\%$  when updating just  $\{W_{v}\}$ ,  $74.09\%$  for  $\{W_{o}\}$  versus  $74.24\%$  updating  $\{W_{q}, W_{k}, W_{v}, W_{o}\}$ ). Meanwhile, updating just  $W_{q}$  or  $W_{k}$  performs significantly worse (72.68%, 72.29%; versus 74.24%). This suggests much of the quality recovery in Stage-2 low-rank fine-tuning comes from adjusting values and outputs to the learned attention weights—as opposed to further refining attention weight computation. While best results do come from a combination of adapting either  $W_{q}$  or  $W_{k}$  with value and output projections, we may be able to achieve even more parameter-efficient linearizing—with comparable quality—by focusing on subsets with the latter.
+
+<table><tr><td>LoRA Rank</td><td>PiQA</td><td>ARC-e</td><td>ARC-c (norm)</td><td>HellaSwag (norm)</td><td>Wino-grande</td><td>MMLU (5-shot)</td><td>Avg.</td><td>Avg. (no MMLU)</td></tr><tr><td>4</td><td>80.7</td><td>82.5</td><td>56.0</td><td>79.6</td><td>73.6</td><td>50.1</td><td>71.3</td><td>74.5</td></tr><tr><td>8</td><td>80.9</td><td>81.7</td><td>54.9</td><td>79.7</td><td>74.1</td><td>52.8</td><td>70.7</td><td>74.2</td></tr><tr><td>16</td><td>80.7</td><td>81.9</td><td>54.5</td><td>79.7</td><td>73.8</td><td>48.9</td><td>69.9</td><td>74.1</td></tr><tr><td>32</td><td>81.1</td><td>81.5</td><td>54.5</td><td>79.7</td><td>72.8</td><td>51.0</td><td>70.1</td><td>73.9</td></tr><tr><td>64</td><td>80.9</td><td>81.9</td><td>54.5</td><td>79.3</td><td>72.1</td><td>51.7</td><td>71.1</td><td>73.8</td></tr><tr><td>128</td><td>80.5</td><td>80.9</td><td>52.8</td><td>78.4</td><td>72.2</td><td>53.4</td><td>69.7</td><td>73.0</td></tr><tr><td>256</td><td>80.7</td><td>80.2</td><td>52.1</td><td>78.8</td><td>71.4</td><td>52.1</td><td>69.2</td><td>72.7</td></tr><tr><td>Full Finetune</td><td>80.6</td><td>81.9</td><td>54.3</td><td>79.4</td><td>72.4</td><td>52.1</td><td>70.1</td><td>73.7</td></tr></table>
+
+Table 16: LoRA rank  $r$  comparison. Evaluation on LM Evaluation Harness tasks. When adapting all projections  ${\mathbf{W}}_{q},{\mathbf{W}}_{k},{\mathbf{W}}_{v},{\mathbf{W}}_{o}$  after attention transfer, we find smaller ranks  $r = 4,8$  are surprisingly sufficient. Larger ranks with more parameter-heavy updates do not necessarily improve downstream performance.  
+
+<table><tr><td>LoRA Projection</td><td>PiQA</td><td>ARC-e</td><td>ARC-c (norm)</td><td>HellaSwag (norm)</td><td>Winogrande</td><td>Average</td></tr><tr><td>Wq</td><td>79.49</td><td>81.06</td><td>51.45</td><td>79.18</td><td>72.22</td><td>72.68</td></tr><tr><td>Wk</td><td>79.82</td><td>79.80</td><td>50.68</td><td>78.54</td><td>72.61</td><td>72.29</td></tr><tr><td>Wv</td><td>80.69</td><td>82.49</td><td>56.66</td><td>79.28</td><td>71.82</td><td>74.19</td></tr><tr><td>Wo</td><td>80.09</td><td>81.65</td><td>55.63</td><td>79.35</td><td>73.72</td><td>74.09</td></tr><tr><td>Wq, Wk</td><td>79.76</td><td>81.19</td><td>51.02</td><td>79.29</td><td>72.22</td><td>72.70</td></tr><tr><td>Wq, Wv</td><td>80.96</td><td>81.90</td><td>54.35</td><td>79.01</td><td>72.30</td><td>73.70</td></tr><tr><td>Wq, Wo</td><td>81.28</td><td>81.86</td><td>54.78</td><td>79.20</td><td>73.72</td><td>74.17</td></tr><tr><td>Wk, Wv</td><td>80.74</td><td>82.62</td><td>56.83</td><td>79.26</td><td>71.98</td><td>74.28</td></tr><tr><td>Wk, Wo</td><td>80.69</td><td>82.15</td><td>55.46</td><td>79.53</td><td>73.01</td><td>69.33</td></tr><tr><td>Wv, Wo</td><td>80.69</td><td>82.79</td><td>55.89</td><td>79.57</td><td>71.59</td><td>74.10</td></tr><tr><td>Wq, Wk, Wv</td><td>80.90</td><td>82.20</td><td>56.48</td><td>79.13</td><td>73.95</td><td>74.53</td></tr><tr><td>Wq, Wk, Wo</td><td>80.41</td><td>80.89</td><td>54.18</td><td>79.18</td><td>74.35</td><td>73.80</td></tr><tr><td>Wk, Wv, Wo</td><td>80.36</td><td>82.32</td><td>54.61</td><td>79.13</td><td>73.56</td><td>74.00</td></tr><tr><td>Wq, Wk, Wv, Wo</td><td>80.85</td><td>81.73</td><td>54.86</td><td>79.65</td><td>74.11</td><td>74.24</td></tr></table>
+
+Table 17: LoRA projection comparison. Evaluation on zero-shot LM Evaluation Harness tasks. We apply LoRA with the same rank  $r = 8$  to different combinations of the attention projections, shading scores by increasing quality (darker is better). When isolating for projections updated, LoRA on projections not involved in layer-wise attention weight computations (value  $W_v$  and output  $W_o$  projections) achieves higher quality over query  $W_q$  or key projections  $W_k$ . This suggests  $W_v$ ,  $W_o$  may be more important to adapt after attention transfer, although best results involve a combination across attention weight and output projections.
+
+# B.4.3 EFFECT OF WINDOW SIZE
+
+We now compare model performance using different window sizes in the LoLCATs linear + sliding window attention layer. With the standard sliding window implementation (Fig 9), we compare LM Eval performance after linearizing with window sizes  $w \in \{4,16,64,256\}$ . We also compare against the ThunderKittens-motivated "terraced" implementation used in our main experiments with window size 64. In Table 18, we find that in each of these settings, having more softmax attention generally improves performance ( $c,f$ ,  $w = 16,64$  versus  $w = 4$ ). However, more softmax attention does not always lead to better quality. Window size 256 results in up to an 8.8 point drop in 5-shot MMLU accuracy, suggesting we may not necessarily trade-off more softmax attention for higher quality. Comparing the standard sliding window and terracing implementations, we find similar performance (70.3 versus 70.7 average accuracy across tasks).
+
+# B.5 STUDY ON LINEARIZING DATA
+
+While most of our work focuses on architecture and training procedure for improving LLM linearizing quality, we now study how data selection affects LoLCATs performance.
+
+<table><tr><td>Window Size</td><td>PiQA</td><td>ARC-e</td><td>ARC-c (norm)</td><td>HellaSwag (norm)</td><td>Winogrande</td><td>MMLU (5-shot)</td><td>Avg.</td><td>Avg. (no MMLU)</td></tr><tr><td>4</td><td>80.7</td><td>81.4</td><td>55.8</td><td>76.6</td><td>72.1</td><td>40.8</td><td>67.9</td><td>73.3</td></tr><tr><td>16</td><td>80.5</td><td>82.2</td><td>56.0</td><td>78.2</td><td>73.9</td><td>50.3</td><td>70.2</td><td>74.1</td></tr><tr><td>64</td><td>80.7</td><td>81.7</td><td>54.7</td><td>79.1</td><td>75.3</td><td>50.3</td><td>70.3</td><td>74.3</td></tr><tr><td>256</td><td>80.6</td><td>81.8</td><td>55.0</td><td>75.6</td><td>74.9</td><td>41.5</td><td>68.3</td><td>73.6</td></tr><tr><td>TK 64</td><td>80.9</td><td>81.7</td><td>54.9</td><td>79.7</td><td>74.1</td><td>52.8</td><td>70.7</td><td>74.2</td></tr></table>
+
+Table 18: Window size comparison. We ablate the window size in LoLCATs linear + sliding window attention. For each window size  $w$ , the layer applies softmax attention to the  $w$ -most recent positions, combined with Hedgehog linear attention applied for all prior positions (Eq. 6, Figure 9). We compare  $w \in \{4, 16, 64, 256\}$  using the standard sliding window implementation with our default  $w = 64$  terraced window setup motivated by ThunderKittens (TK 64). Window size 64 performs best, where both implementations perform comparably.  
+
+<table><tr><td>Model</td><td>Linearizing Dataset</td><td>PiQA</td><td>ARC-e</td><td>ARC-c (acc. norm)</td><td>HellaSwag (acc. norm)</td><td>Wino-grande</td><td>MMLU (5-shot)</td><td>Avg.</td><td>Avg. (no MMLU)</td></tr><tr><td>Mistral 7B (v0.1)</td><td>-</td><td>82.1</td><td>80.9</td><td>53.8</td><td>81.0</td><td>74.0</td><td>62.4</td><td>72.4</td><td>74.4</td></tr><tr><td>→LoLCATs</td><td>Alpaca Clean</td><td>81.5</td><td>81.7</td><td>54.9</td><td>80.7</td><td>74.0</td><td>51.4</td><td>70.7</td><td>74.5</td></tr><tr><td>→LoLCATs</td><td>RedPajama</td><td>80.1</td><td>77.6</td><td>49.0</td><td>80.3</td><td>71.7</td><td>53.2</td><td>68.6</td><td>71.7</td></tr><tr><td>Llama 3 8B</td><td>-</td><td>79.9</td><td>80.1</td><td>53.3</td><td>79.1</td><td>73.1</td><td>66.6</td><td>72.0</td><td>73.1</td></tr><tr><td>→LoLCATs</td><td>Alpaca Clean</td><td>80.9</td><td>81.7</td><td>54.9</td><td>79.7</td><td>74.1</td><td>52.8</td><td>70.7</td><td>74.2</td></tr><tr><td>→LoLCATs</td><td>RedPajama</td><td>78.9</td><td>79.0</td><td>52.0</td><td>78.1</td><td>72.6</td><td>55.2</td><td>69.3</td><td>72.1</td></tr></table>
+
+Table 19: Linearizing data comparison. For linearizing Mistral 7B (v0.1) and Llama 3 8B, LoLCATs with Alpaca and RedPajama subsets perform comparably (c.f. prior methods, Table 3), though we find that Alpaca actually leads to higher accuracy for most tasks other than 5-shot MMLU.
+
+# B.5.1 DATA SOURCE: ALPACA VERSUS REDPAJAMA
+
+We study the effect of linearizing data for downstream LLM performance. While we initially found that just using the  $\sim 50\mathrm{K}$  samples of a cleaned Alpaca dataset (Taori et al., 2023) could lead to surprisingly high performance on popular zero-shot LM Eval tasks, prior linearizing works (Mercat et al., 2024) use more typical pretraining datasets to linearize such as RefinedWeb (Penedo et al., 2023). We thus also try linearizing with a random subset of RedPajama (Computer, 2023) to evaluate how LoLCATs works with pretraining data, albeit without any special curation. For both setups, we pack samples into 1024 token sequences and randomly subsample the RedPajama data so that we use the same number of training tokens (20M) for both attention transfer and finetune stages (40M tokens overall). We use the setup as described in Appendix A.1 for all other hyperparameters.
+
+In Table 19, we find that across Mistral 7B (v0.1) and Llama 3 8B, using the Alpaca cleaned dataset actually leads to better downstream task quality for all tasks except for 5-shot MMLU, where linearizing with RedPajama consistently leads to  $\sim 2$  percentage point improvements. LoLCATs with both of these datasets leads to comparable or higher performance than prior methods trained on  $2500\times$  the data (c.f., Table 3; SUPRA trained on 100B tokens gets Avg. accuracy of  $64.0\%$ ), suggesting that LoLCATs can robustly improve linearizing quality over different data sources.
+
+# B.5.2 MATCHING LINEARIZING DATA TO DOWNSTREAM (RETRIEVAL) TASK
+
+Among LM Eval tasks, we note a sizable gap between linearized and Transformer-based LLMs on MMLU. We hypothesize one source of this gap is due to MMLU's evaluation setup, which not only tests for knowledge recall in pretrained weights, but also retrieval over the input context (Hendrycks et al., 2020). In the default 5-shot multiple choice setup, models must be able to retrieve and produce the letter associated with the right answer choice in context. However, prior works have shown that linear attentions and non-softmax attention models perform worse on retrieval, both explicitly on MMLU (Waleffe et al., 2024) and in retrieval tasks at large (Waleffe et al., 2024; Shen et al., 2024).
+
+To counteract these effects, we study if linearizing LLMs with data that explicitly reflects the target downstream task can improve performance. We report two such settings next.
+
+Improving MMLU. First, to test improving performance on MMLU, we linearize with additional data from CommonsenseQA (Talmor et al., 2019) (CQA), another multiple-choice dataset. We construct a linearizing dataset by using the same 5-shot in-context template as in MMLU, i.e., <bos><question 1><answer choices 1><answer 1>, <question 2><answer choices 2><answer 2>, ..., <question 5><answer choices 5>---<eos>, where the next token to predict --- corresponds to the correct answer among <answer choices 5>, using the  $\sim 10\mathrm{k}$  samples of the CQA training set. For both stages, we linearize Llama 3 8B over the combined set of 5-shot CQA samples and our default  $\sim 50\mathrm{k}$  Alpaca samples, comparing against just Alpaca or CQA samples. We pack all samples to context length 1024.
+
+In Table 20, we report the MMLU scores following each linearizing data choice. We find that adding a small amount of multiple-choice samples can substantially improve MMLU accuracy ( $\sim$ 2 points). However, CQA alone performs  $\sim$ 10 points worse than Alpaca. This suggests quantity and diversity of samples may still be necessary for linearizing; we break this down further and study the amount of linearizing needed during both attention transfer and low-rank adjusting in Appendix B.6.
+
+<table><tr><td rowspan="2">Model Linearizing Data</td><td colspan="3">Llama 3 8B LoLCATs</td><td rowspan="2">Llama 3 8B N/A</td></tr><tr><td>Alpaca</td><td>Alpaca + CQA</td><td>CQA</td></tr><tr><td>MMLU (5-shot) Acc.</td><td>52.8</td><td>54.5</td><td>43.9</td><td>66.6</td></tr></table>
+
+Evaluating and improving needle-in-a-haystack retrieval. Next, to further test linearizing performance on downstream tasks, we evaluate the LoLCATS-linearized Llama 3 8B LLM on "needle-in-a-haystack" tasks. We use the passkey-retrieval task setup introduced in Mohtashami & Jaggi (2023) and evaluated in various prior works (Chen et al., 2023a;b; Tworkowski et al., 2024), where a model must retrieve a hidden passkey uniform randomly placed inside a text span. We use randomly generated prompts and 5-digit passkeys (Listing 1), and test correctness by whether the model outputs the passkey exactly (exact-match). We evaluate over various text lengths (2048 to 10240 tokens), and report accuracy binned by which decile the passkey occurs in.
+
+Like in our MMLU study, we evaluate LoLCATs with two linearizing data setups. First, we use the default Alpaca data, but this time test linearizing Llama 3 8B at its max context length by packing the 50K Alpaca samples into 8192-token chunks. Second, to see if linearizing explicitly with retrieval data helps, we generate 10K passkey retrieval samples with  $\sim 8192$  tokens, and use these samples for both stages of LoLCATs linearizing. For both, we use the same hyperparameters as Table 7.
+
+In Table 21, we first report the results for Llama models evaluated on 8192-token prompts. We compare the LoLCATs Llama models against the base Llama 3 8B, the instruction-tuned version (Llama 3 8B Instruct), and a Transformer Llama 3 8B LoRA-finetuned on Alpaca data (Llama 3 8B (Alpaca); using the same LoRA parameters and second stage training procedure as in Table 7). We find that with LoLCATs models, linearizing data plays a particularly strong role in downstream performance. As a potential drawback of LoLCATs, when just using packed Alpaca samples, the linearized Llama 3 8B fails to get even a single passkey retrieval sample correct (LoLCATs Llama 3 8B (Alpaca)). Meanwhile, LoRA-finetuning the non-linearized Llama 3 8B maintains high passkey retrieval accuracy. However, linearizing with passkey samples (LoLCATs Llama 3 8B (Passkey)) recovers  $100\%$  accuracy. This suggests that with LoLCATs linearizing, the linear + sliding window attention is able to do passkey retrieval with similar performance to full softmax attention models.
+
+Table 20: MMLU comparison with task-specific linearizing data. Adding multiple-choice samples (CommonsenseQA; CQA) to linearizing training data modestly improves downstream MMLU performance. However, CQA alone is insufficient to achieve competitive quality.  
+
+<table><tr><td>Passkey Placement</td><td>0-10%</td><td>10-20%</td><td>20-30%</td><td>30-40%</td><td>40-50%</td><td>50-60%</td><td>60-70%</td><td>70-80%</td><td>80-90%</td><td>90-100%</td></tr><tr><td>Llama 3 8B</td><td>100.00</td><td>100.00</td><td>100.00</td><td>100.00</td><td>92.86</td><td>100.00</td><td>94.44</td><td>94.12</td><td>92.31</td><td>100.00</td></tr><tr><td>Llama 3 8B Instruct</td><td>100.00</td><td>100.00</td><td>100.00</td><td>100.00</td><td>100.00</td><td>100.00</td><td>100.00</td><td>100.00</td><td>100.00</td><td>100.00</td></tr><tr><td>Llama 3 8B (Alpaca)</td><td>100.00</td><td>100.00</td><td>100.00</td><td>100.00</td><td>100.00</td><td>100.00</td><td>100.00</td><td>100.00</td><td>100.00</td><td>100.00</td></tr><tr><td>LoLCATs Llama 3 8B (Alpaca)</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr><tr><td>LoLCATs Llama 3 8B (Passkey)</td><td>100.00</td><td>100.00</td><td>100.00</td><td>100.00</td><td>100.00</td><td>100.00</td><td>100.00</td><td>100.00</td><td>100.00</td><td>100.00</td></tr></table>
+
+Table 21: 8K Context Passkey Retrieval. We report passkey retrieval accuracy for various Llama 3 8B models, using 8192-token samples and binning by passkey placement decile. Linearizing with just packed Alpaca samples fails, but using passkey samples as the linearizing data recovers  $100\%$  retrieval performance.
+
+```txt
+There is an important piece of info hidden inside a lot of irrelevant text. Find it and memorize it. I will quiz you about the important information there. The grass is green. The sky is blue. The sun is yellow. Here we go. There and back again. The grass is green. The sky is blue. The sun is yellow. Here we go. There and back again. The grass is green. The sky is blue. The sun is yellow. Here we go. There and back again. The grass is green. The sky is blue.. ... <EVEN MORE FILLER>   
+The pass key is <PASSKEY>. Remember it. <PASSKEY> is the pass key.   
+The grass is green. The sky is blue. The sun is yellow. Here we go. There and back again. The grass is green. The sky is blue. The sun is yellow. Here we go. There and back again... .. <MORE FILLER>   
+What is the pass key? The pass key is
+```
+
+![](images/8cf8634b47597cd5d75e8eea07205ac93ebd9d25a749f2e057129c790241f7b4.jpg)  
+Listing 1: Passkey Retrieval Prompt Template  
+Figure 10: Needle-in-a-haystack with passkey retrieval. LoLCATs-linearized Llama 3 8B with passkey retrieval linearizing data results in comparable retrieval to original Llama 3 8B and instruction-tuned variants.
+
+We further test the robustness of this retrieval across various context lengths (2048 to 10240 tokens) in Figure 10. We report accuracies across input context lengths and passkey placements, finding similar strong retrieval performance for samples under Llama 3 8B's 8192 context length. Interestingly, only the Transformer Llama 3 8B LoRA-finetuned on Alpaca is able to do retrieval over longer context samples (10240 tokens).
+
+# B.5.3 SAMPLE LENGTHS: EFFECT OF EFFECTIVE SEQUENCE LENGTHS
+
+We further study the impact of sample sequence length for linearizing quality. By default, for linearizing data we pack original data samples into sequences of consistent length, e.g., 1024 tokens. As done in prior work (Raffel et al., 2020), this allows us to pack multiple short data samples together into longer training sequences, improving training efficiency and removing any padding tokens. However, it may also introduce situations where our linearizing sequences only carry short-context dependencies, i.e., because we pack together many samples with few tokens, or split longer samples into multiple sequences. Especially with attention transfer, linearized LLMs may model longer samples less well (e.g., the 5-example in-context samples in 5-shot MMLU) because we never learn to approximate attentions over "long enough" sequence lengths.
+
+Effective sequence length. To study this data effect, we define an "effective sequence length" (ESL) metric. This roughly captures for each query how far back a layer needs to attend to capture all non-zero softmax attention weights. For query at position  $i$ , we define the ESL per query as
+
+$$
+\operatorname {E S L} \left(\boldsymbol {q} _ {i}\right) := \sum_ {j = 1} ^ {i} (i - j) \frac {\exp \left(\boldsymbol {q} _ {i} ^ {\top} \boldsymbol {k} _ {j} / \sqrt {d}\right)}{\sum_ {m \leq j} \exp \left(\boldsymbol {q} _ {i} ^ {\top} \boldsymbol {k} _ {m} / \sqrt {d}\right)} \tag {8}
+$$
+
+We compute a sample's ESL per head as the sum over all query ESLs, i.e.,  $\sum_{i=1}^{n} \operatorname{ESL}(q_i)$  for a sample with  $n$  tokens. We average this over all heads and layers to measure a sample's overall ESL.
+
+We hypothesize that if our linearizing data only has samples with shorter ESL than those encountered at test time, then we would poorly model these test samples. Conversely, we may be able to improve linearizing quality by specifically filtering for samples with longer ESL. We report two findings next.
+
+![](images/3520719aedd424cfd80f362ae81910e924ad74ff6d6f586dfa194f1190fea14f.jpg)  
+Figure 11: Effective sequence length distributions. Although we pack data samples into 1024-token training sequences, different data sources can vary in effective sequence lengths (left). Furthermore, we can selectively filter for longer ESL samples (right). We find linearizing with longer ESLs coincides with improved MMLU scores, albeit at the cost of other LM Eval tasks (Table 22).
+
+![](images/b639569bb7b35bea7c85f6fc5af4f0a07e4d31b4dc2be449d2b7db3936e7f508.jpg)
+
+Finding 1: ESL corresponds with RedPajama versus Alpaca performance. In Figure 11, we first plot the distribution of sample ESLs computed with Llama 3 8B on Alpaca and RedPajama linearizing data subsets. We find RedPajama samples on average display longer ESLs, which coincides with improved MMLU score (c.f., Table B.5.1).
+
+Finding 2: Filtering for higher ESL improves MMLU. Furthermore, we can increase the ESLs in linearizing data by actively filtering for high ESL samples (Figure 11 right). Here we actively filter for the top 20,000 packed RedPajama samples with the highest ESLs, amounting to 20M tokens. When doing one epoch of attention transfer and low-rank linearizing with this subset, we further improve MMLU accuracy by 1.3 points (Table 22). However, this comes at a cost for all other LM Eval tasks, dropping quality compared to random RedPajama packing by 0.2 to 2.1 points.
+
+# B.6 STUDY ON LINEARIZINGTOKEN BUDGET
+
+We further study how varying the number of tokens used for both attention transfer and low-rank adaptation impacts LoLCATs linearizing quality.
+
+Impact of minimal tokens. To first test how efficient we can be with attention transfer, we linearize Llama 3 8B with varying numbers of attention transfer steps (0 - 1800), before low-rank adjusting for up to 2000 steps. We use the Alpaca dataset and the same packed random sampling as our main experiments, and measure evaluation perplexity on validation samples both in-distribution (held-out
+
+<table><tr><td>Linearizing Data</td><td>PiQA</td><td>ARC-easy</td><td>ARC-challenge</td><td>HellaSwag (acc. norm)</td><td>Winogrande (acc. norm)</td><td>MMLU (5-shot)</td><td>Avg. (no MMLU)</td></tr><tr><td>Alpaca</td><td>80.9</td><td>81.7</td><td>54.9</td><td>79.7</td><td>74.1</td><td>52.8</td><td>74.2</td></tr><tr><td>RedPajama</td><td>78.9</td><td>79.0</td><td>52.0</td><td>78.1</td><td>72.6</td><td>55.2</td><td>72.1</td></tr><tr><td>RedPajama (Sample Top ESL)</td><td>78.4</td><td>77.0</td><td>49.8</td><td>78.0</td><td>71.4</td><td>56.5</td><td>70.9</td></tr></table>
+
+Table 22: Effect of ESL on linearized LLM quality, Llama 3 8B. While linearizing with longer ESLs—e.g., RedPajama samples or specifically filtering for top ESLs in RedPajama (Sample Top ESL)—improves 5-shot MMLU accuracy up to 3.7 points, it reduces quality on all other evaluated LM Eval tasks by 1.7 to 5.0 points.
+
+Alpaca samples) and out-of-distribution (RedPajama validation samples) over different combinations of steps (Figure 13). Without attention transfer, low-rank adaptation converges significantly higher on in-distribution samples (Figure 12a), suggesting poorer quality linearizing. However, we find similar held-out perplexities after relatively few attention transfer steps (c.f., 1000 - 1800 updates, the former amounting to just 8 million tokens for attention transfer), where all runs improve in-distribution PPL by  $\sim 0.23$  points after LoRA finetuning for 2000 steps.
+
+![](images/5f7b21546f6caa51ba9d4198a6eb0c86f779fc2ff1faaa40f86d166c73a1d983.jpg)  
+(a) Alpaca PPL
+
+![](images/a8c2714305727502e67ddb4b567949089aa86278bb965d77a8c5e96476ddf0d6.jpg)  
+Figure 12: Evaluation curves over number of training steps. Llama 3 8B linearized with Alpaca. We report the impact of steps allotted to attention transfer versus LoRA linearizing, using validation set perplexity (PPL) over both in-distribution (held-out Alpaca samples) and out-of-distribution (RedPajama validation samples). We first run linearizing with 0 - 1800 attention transfer steps, before LoRA-finetuning for up to 2000 steps. Without any attention transfer (0 steps), linearized LLMs get much higher perplexity (Fig 12a). On the other hand, we observe similar convergence after attention transfer over only 1000 steps.
+
+![](images/1fab9e9d9e926e419d8c2931fc8f4c6031fb6174b24493b57991ce5739977134.jpg)  
+(b) Alpaca Val. PPL (Zoomed)  
+(c) RedPajama Val. PPL (Zoomed)
+
+In Table 23, we report the numerical values for held-out perplexities at the end of linearizing (1800 attention transfer steps + 2000 low-rank adaptation steps), as well as the average LM Eval score over zero-shot tasks. We similarly find competitive generalized zero-shot quality with relatively few attention transfer steps (200 steps), all achieving 7.70–8.16 higher points than the next best Mamba-Llama model (0.16–0.62 higher points than the  $50\%$  softmax attention variant, c.f., Table 3). Without any attention transfer, linearized LLMs perform drastically worse on out-of-distribution samples (Table 23, RedPajama and LM Eval metrics).
+
+Impact of more pretraining data. We next study how linearizing over larger amounts of pretraining data impacts quality. We randomly sample a larger set of unique RedPajama training sequences (1024-token packed; 72,000 such samples overall), allowing us to linearize Llama 3 8B with different combinations of up to 9,000 attention transfer updates and 2,000 low-rank linearizing updates. To test language modeling recovery, we report both held-out validation sample perplexity (Table 13a) and general zero-shot LM Evaluation Harness quality (Table 13b). Increasing both Stage 1 attention transfer steps and Stage 2 low-rank adjusting steps notably improves validation perplexity. However, we similarly find competitive zero-shot LM scores across all evaluated attention transfer steps. Across checkpoints at different numbers of low-rank updates, attention transfer with up to  $9 \times$  more unique tokens does not seem to monotonically improve downstream quality (Table 13b). Meanwhile, we do find that across various amounts of attention transfer steps, subsequent low-rank adaptation consistently improves average zero-shot LM score by  $>1$  points.
+
+# B.7 LAYER-WISE TRAINING DYNAMICS AND ANALYSIS
+
+Finally, we further study LoLCATs layer-wise training dynamics, such as the resulting linear attention approximation quality to softmax attention at every layer (App. B.7.1), how this corresponds
+
+<table><tr><td>Attn. Transfer Steps</td><td>0</td><td>200</td><td>400</td><td>600</td><td>800</td><td>1000</td><td>1200</td><td>1400</td><td>1600</td><td>1800</td></tr><tr><td>Alpaca Eval PPL</td><td>3.211</td><td>2.802</td><td>2.792</td><td>2.782</td><td>2.782</td><td>2.776</td><td>2.778</td><td>2.775</td><td>2.782</td><td>2.776</td></tr><tr><td>RedPajama Eval PPL</td><td>61.305</td><td>10.459</td><td>9.799</td><td>9.699</td><td>9.628</td><td>9.679</td><td>9.420</td><td>9.328</td><td>9.413</td><td>9.358</td></tr><tr><td>Avg. Zero-shot LM Eval Acc.</td><td>56.86</td><td>73.68</td><td>73.34</td><td>73.70</td><td>73.66</td><td>73.74</td><td>73.47</td><td>73.70</td><td>73.80</td><td>73.66</td></tr></table>
+
+Table 23: Effect of attention transfer steps. With Llama 3 8B linearized on Alpaca data, we report the final evaluation perplexities after 2000 LoRA steps in Fig 13, as well as downstream LM Eval performance averaged over zero-shot tasks. We again find competitive quality with relatively few attention transfer steps.
+
+![](images/09635a1d137b13a4ed3d01ab8190ddfb423073ad18e5ce604b8c8eca1180e5e1.jpg)  
+(a) RedPajama Validation Perplexity
+
+![](images/566b24249075ecde02164871073a157070a83bc4610a08ef94cf56c2e2e0a207.jpg)  
+(b) LM Eval Score
+
+![](images/01993f9b66fb724293e1ab300c3ab6ebdd760ba38579474c822177b4286ca188.jpg)  
+Figure 13: Evaluation quality from ablating RedPajama linearizing updates.  
+Figure 14: Layer-wise MSE and deltas. Layer-wise MSE and change in MSE (absolute) before and after LoRA adjusting with the Hedgehog linear attention and LoLCATs linear + sliding window attention on Llama 3 8B. Both use the Hedgehog feature map. With larger initial MSEs after attention transfer (Fig. 14a), LoRA adjusting with Hedgehog results in larger MSE improvements in MSE across layers (Fig. 14b).
+
+![](images/7ee4ca7f38f34ee82f2bc8850a87428a53aca35aecf3fdd8c351c4274c42d68c.jpg)  
+(a) Layer-wise MSE before and after LoRA adjusting  
+(b) Layer-wise MSE change after LoRA adjusting
+
+with LoRA updates after attention transfer (App. B.7.2), and how these layer-wise attention MSEs (a) relate to model size and (b) further motivate block-wise training (App. B.7.3).
+
+# B.7.1 LAYER-WISE SOFTMAX ATTENTION RECOVERY WITH LORA ADJUSTING
+
+We now study how LoRA can explicitly improve softmax attention approximation. Given learned feature maps, can we recover softmax attention better by adjusting the attention projections?
+
+In Figure 14a, we report the layer-wise MSE between learned linear attentions and softmax attention using either the pure linear attention (Hedgehog) or linear + sliding window attention (LoLCATs). We plot the mean MSE computed over all samples in our Alpaca validation set, averaging over all heads, token positions, and heads per layer. In addition to plotting this metric after attention transfer (as in Figure 7), we also plot the MSE after an additional round of LoRA adjusting (+ LoRA). Like before, we freeze the linear attention weights and add LoRA weights to query, key, value, and output projections. However, rather than train these weights end-to-end for next-token prediction, we update LoRAs to explicitly minimize the MSE between our trainable linear attention and original softmax attention outputs like in Stage 1. We use the same hyperparameters as in Table 7.
+
+We report both the absolute MSEs (Figure 14a) and the change in MSE (Figure 14b). LoRA reduces MSEs with both linear attentions—frequently reducing MSE more when starting with worse approximations after attention transfer—which may suggest some type of compensatory role. This occurs both across layers for each linear attention, where we see greater MSE deltas for the later layers  $(c.f., \text{LoLCATs MSE change, Figure 14b})$ , and between linear attentions  $(c.f., \text{Hedgehog MSE change, Figure 14b})$ , where LoRA generally improves MSE more with the Hedgehog linear attention versus LoLCATs linear + sliding window. Despite these greater improvements, LoRA alone does not close the MSE gap (Figure 14a), suggesting that linearizing architecture still plays an important role in learning to match softmax attention.
+
+# B.7.2 LAYER-SPECIFIC LORA TRAINING DYNAMICS
+
+Next, we further study how LoLCATs layers behave during LoRA adjusting, and plot the cumulative weight updates to LoRA low-rank  $A$  and  $B$  weight matrices while training LLMs end-to-end for next-token prediction. As reference points, we compare against LoRA finetuning (1) the original
+
+Transformer LLMs with softmax attention, (2) linearized LLMs without attention transfer (using the LoLCATs linear + sliding window attention, Hedgehog feature map, but not trained to match softmax first), (3) linearized LLMs with attention transfer and the Hedgehog pure linear attention, and (4) linearized LLMs with Hedgehog pure linear attention without attention transfer. We finetune Llama 3 8B with two epochs on the Alpaca dataset, following Step 2 hyperparameters in Table 7.
+
+We plot these updates per LoRA weight and projection in Figure 15, with layer-specific plots in Figure 16 and 17. We specifically show cumulative sums of the update magnitudes over 2500 steps.
+
+![](images/f0a30a1bb18c7210b45a2c7f4e789774787a3b560513f46a0d5d26bdd948c1b1.jpg)
+
+![](images/8f4f0fa759a8494782a51eeb1c7eeddf1f44cdcd530b7489a28e11a0a8234877.jpg)
+
+![](images/4ed5b09fa9216a5acde546fab0f3b912f22910110d446fcccc4584370fa4efe0.jpg)
+
+![](images/bdc59a20276c800181d3655b4a8cb7a622a9a74b8643dd50808e26f632d93f9a.jpg)
+
+![](images/6c956a0f7d16b63d37c41996501e3ef83691bc07347e7f89bd7bbdcf51c9141d.jpg)
+
+![](images/4949eec020f8095b769fd986118fbd9f6d99985e954bd6a3832c9ded6f332a1d.jpg)  
+Figure 15: LoRA  $A$ ,  $B$  weight updates by attention projection over training, averaged over all layers.
+
+![](images/c3ae4deadca3568ebd777f98e1437b310991691206eacb75af4e1a05a33faeb2.jpg)
+
+![](images/e8194d9e8570ebce44a2698353e6d2108ac4c10765fb6bd0ca346eeb3da8542d.jpg)
+
+![](images/814114f6c8e7562afb11163d51e2eaadbb94f332f04c3da93f73c93f710ad5c2.jpg)
+
+![](images/2361cc3897e1c4f4413c47cfc4865b965c5f4b3ad2bfa2034a0f4fdb57891dc7.jpg)
+
+Across plots, we find several interesting findings. First, LoLCATs with attention transfer results in much more similar cumulative updates to softmax attention for value and output projection LoRAs, where both result in noticeably larger updates than any other configuration. This may suggest the LoLCATs attention transfer linear attentions are sufficiently similar to softmax attention, such that when LoRA finetuning end-to-end, the weights also behave similarly. Furthermore, for query and key projections—i.e., those involved in the attention weight computation—the situation is reversed. The untrained or no attention transfer linearized LLMs display greater cumulative updates earlier during training for  $A$  weights and consistently for  $B$  weights than either trained linear attention or softmax attention. As we train all layers jointly, we find that with the untrained linear attentions, relatively more "weight" is diverted to learning to compute attention weights via query and key projection updates. How this impacts the downstream model—i.e., by updating the value and output projections less than a softmax attention Transformer—is an interesting question for future work.
+
+When plotting these updates per layer (Figure 16, 17) we find similar dynamics. Interestingly, despite LoLCATs with attention transfer resulting in the worst softmax attention approximations for the layer 31 (c.f., attention MSE, Figure 7, 14a), the resulting LoRA weight deltas do not seem to track the softmax attention LoRA deltas noticeably worse than other layers.
+
+# B.7.3 BLOCK-WISE ATTENTION TRANSFER
+
+Finally, we study how attention transfer properties change with model scale, motivating LoLCATs's block-wise training approach (Section 3.3.2). In particular, we note that attention output MSEs can vary quite a bit across layers, and this can be aggravated by model size ( $c.f$ , Llama 3.1 70B, Table 25; and Llama 3.1 405B, Table 26).
+
+Recall that attention transfer involves training the LoLCATs linear attentions to match the outputs of softmax attention at each layer by minimizing the MSE between the softmax and linear attention outputs. Since the MSE loss is scale-sensitive and MSE already varies across layers after attention transfer for Llama 3 8B (Figure 6b), we hypothesize that jointly training all 126 Transformer layers in Llama 3.1 405B – by summing the MSE losses across all layers – may be difficult. Correspondingly, in Table 24, we find that block-wise attention transfer leads to lower language modeling perplexity for 405B linearized LLMs compared to joint training. However, we find that joint training is sufficient and performs similarly to block-wise training at the smaller scales (8B, 70B).
+
+![](images/c544e884f7109dbd720014d394c08ff2147f7245085ad80a39421109fe073691.jpg)  
+Figure 16: LoRA  $A$  weight updates by attention projection and layer over training.
+
+![](images/9d030360f741656e3d91b19abdc95a43ea238acee2a44b702a40d31e6eebd82f.jpg)  
+Figure 17: LoRA  $B$  weight updates by attention projection and layer over training.
+
+Tradeoffs between quality and efficiency. We show that the block-wise approach improves linearization quality at large model scales and improves the user's ability to flexibly balance compute and memory efficiency tradeoffs. We compare (1) joint  $(k = 126)$  training, where we load the full model once and compute the loss as the sum of layer-wise MSEs, and (2) block-wise  $(k = 9)$  training, where we break the model into blocks and train each independently. Per our discussion in Sec. 3.3.2, this increases storage costs to save
+
+<table><tr><td>Method</td><td>b × M//b</td><td>PPL</td></tr><tr><td>Joint</td><td>126×1</td><td>4.23</td></tr><tr><td>Block-wise</td><td>9×14</td><td>3.21</td></tr></table>
+
+Table 24: Validation perplexity after LoRA adjusting for Llama 3.1 405B, comparing attention transfer with  $M//b$  blocks of  $b$  layers.  $M = {126}$  is number of total layers.
+
+precomputed hidden states. However, we hypothesize that optimizing a block-wise attention transfer loss can improve quality.
+
+For quality, we compare the evaluation perplexity of both the attention transfer approaches, after LoRA adjusting on the same subset of RedPajama data. The block-wise and joint approaches perform similarly at the 8B and 70B scales, however, as shown in Table 24, the block-wise approach performs 1.02 points better than joint attention transfer at the 405B scale. These results support our study in Section 3.2, which shows the variation in MSE magnitudes grows large at the 405B scale.
+
+Layer-wise MSE. We report the attention MSEs across model scales, finding that variation in MSE magnitudes corresponds with model size. In Tables 25 and 26, we report magnitude of the MSE loss as the depth of the layers in the block increases. MSE variation increases with model size: while the largest MSEs at the 70B scale are between  $2.5 - 3 \times$  the MSE of the final block at the 405B scale is 48.66 (over  $13 \times$  that of the next largest MSE by block, 3.62).
+
+<table><tr><td>Block (Layer range)</td><td>Eval MSE</td></tr><tr><td>0-4</td><td>8e-4</td></tr><tr><td>5-9</td><td>0.03</td></tr><tr><td>10-14</td><td>0.06</td></tr><tr><td>15-19</td><td>0.10</td></tr><tr><td>20-24</td><td>0.28</td></tr><tr><td>25-29</td><td>0.73</td></tr><tr><td>30-34</td><td>0.28</td></tr><tr><td>35-39</td><td>0.28</td></tr><tr><td>40-44</td><td>0.25</td></tr><tr><td>45-49</td><td>1.08</td></tr><tr><td>50-54</td><td>0.26</td></tr><tr><td>55-59</td><td>0.18</td></tr><tr><td>60-64</td><td>0.45</td></tr><tr><td>65-69</td><td>0.50</td></tr><tr><td>70-74</td><td>2.91</td></tr><tr><td>75-79</td><td>2.56</td></tr></table>
+
+Table 25: Attention transfer block-wise MSE We report the eval MSE by 5-layer block for each of the 16 blocks in the 80 Transformer layer Llama 3.1 70B model. Each block is trained on the exact same set of RedPajama data at sequence length 1024.  
+
+<table><tr><td>Block (Layer range)</td><td>Eval MSE</td></tr><tr><td>0-8</td><td>0.03</td></tr><tr><td>9-17</td><td>0.02</td></tr><tr><td>18-26</td><td>0.09</td></tr><tr><td>27-35</td><td>0.04</td></tr><tr><td>36-44</td><td>0.36</td></tr><tr><td>45-53</td><td>0.42</td></tr><tr><td>54-62</td><td>1.59</td></tr><tr><td>63-71</td><td>2.75</td></tr><tr><td>72-80</td><td>3.62</td></tr><tr><td>81-89</td><td>2.49</td></tr><tr><td>90-98</td><td>0.29</td></tr><tr><td>99-107</td><td>1.11</td></tr><tr><td>108-116</td><td>3.42</td></tr><tr><td>117-126</td><td>48.66</td></tr></table>
+
+Table 26: Attention transfer block-wise MSE We report the eval MSE by 9-layers block for the 14 blocks in the 126 Transformer layer Llama 3.1 405B model. Each block is trained on the exact same set of RedPajama data at sequence length 1024.
+
+# C IMPLEMENTATION DETAILS
+
+# C.1 PSEUDOCODE IMPLEMENTATION
+
+Below we provide further details on implementing LoLCATs with PyTorch-like code and example demonstrations from the HuggingFace Transformers library.
+
+Learnable Linear Attention. To start, we simply replace the softmax attentions in an LLM with a linear attention. We define such a class below.
+
+Listing 2: LoLCATs Linear Attention Class  
+```python
+import copy   
+import torch.nn as nn   
+from einops import rearrange   
+class LolcatsLlamaAttention(nnModule): def __init__(self, feature_dim: int, base attn: nnModule, # original Transformer attn. ) -> None: super().__init_(   ) # Inherit pretrained weights self.q_proj = base attn.q_proj self.k_proj = base attn.k_proj self.v_proj = base attn.v_proj self.o_proj = base attn.o_proj # Inherit other attention things self.rotary_emb = base attn.rotary_emb self.base_attn = base_attn # keep for attention transfer self.num_heads = base_attn(num_heads self.head_dim = base_attn.head_dim # Initialize feature maps, see Hedgehog definition below self.feature_map_q = HedgehogFeatureMap( self.num_heads, self.head_dim, feature_dim ) self.feature_map_k = copy.deepcopy(self.feature_map_q) def forward(self, x: torch.Tensor) -> torch.Tensor: "" Compute linear attention (assume no GQA) (b: batch_size, h: num_heads, l: seq_len, d: head_dim) "" q = self.q_proj(x) # assume all are (b, h, l, d) k = self.k_proj(x) v = self.v_proj(x) # Apply rotary embeddings q = self.rotary_emb(q) k = self.rotary_emb(k) # Apply feature maps q = self.feature_map_q(q) # (b, h, l, feature_dim) k = self.feature_map_k(k) # (b, h, l, feature_dim) # Compute linear attention kv = torch.einsum('bhlf,bhld->bhfd', k, v) y = torch.einsum('bhlf,bhfd->bhld', q, kv) y /= torch.einsum('bhlf,bhlf->bh1', q, k.cumsum(dim=2)) [... , None] # Apply output projection return self.o_proj(rearrange(y,'b h 1d -> b 1 (h d)'))
+```
+
+Linear + Sliding Window Attention. We can augment this linear attention with the linear attention and sliding window formulation described in Eq. 6. We first define standalone functions for linear attention and sliding window softmax, before defining such as hybrid class below.
+
+Listing 3: Linear Attention in Linear + Sliding Window Attention  
+```python
+def sliding_window softmaxattention(q:torch,Tensor, k:torch,Tensor, v:torch,Tensor, window_size:int, window_factor:float):   
+""Compute sliding window softmax attention in O(n) time and space   
+by not materializing O(n^2) attention weights   
+""d=q.shape[-1] # Compute windows for keys and values, shifting by window size window_kwarges  $=$  {'dimension':2,'size':window_size,'step':1} k=F_pad(k,(0,0, window_size-1,0),value=0).unfold(\*\* window_kwarges)  $\mathbf{v} = \mathbf{F}$  .pad(v,(0,0, window_size-1,0),value=0).unfold(\*\* window_kwarges) #Compute windowed softmax(qk); causal in its construction a_sm  $=$  torch.einsum('bhld,bhldw->bhlw',q,k)* (d\*\*-0.5) #heuristic for zeroing out padding above a_sm[a_sm  $= = 0]$ $=$  -torch.finfo(q.dtype).max   
+#Compute softmax terms for combining attentions (attn and sum) a_sm_max  $=$  torch.amax(a_sm,dim=-1,keepdim=True) a_sm  $=$  window_factor  $\star$  torch.exp(a_sm-a_sm_max) sum_sm  $=$  a_sm-sum(dim=-1,keepdim=True) return torch.einsum('bhlw,bhldw->bhld',a_sm,v),sum_sm
+```
+
+Listing 4: Linear Attention in Linear + Sliding Window Attention  
+```python
+def under_windowlinearattention(f_q:torch,Tensor, # phi(q) f_k:torch,Tensor, # phi(k) v:torch,Tensor, window_size:int, linear_factor:float):   
+"   
+Compute hybrid window attention dot product with linear complexity in q_len   
+""   
+dtype  $=$  f_q.dtype   
+#Shift keys and valuues for window   
+w  $=$  window_size   
+f_k  $=$  F_pad(f_k,(0,0,w,0),value=0)[(:, :, :-w, :]   
+v  $=$  F_pad(v,(0,0,w,0),value=0)[(:, :, :-w, :]   
+# Compute linear terms for combining attentions   
+kv  $=$  torch.einsum('bhlf,bhld->bhfd',k,v)   
+qkv  $=$  linear_factor  $\star$  torch.einsum('bhlf,bhfd->bhld',q,kv)   
+sum_f_k  $=$  f_k.float().cumsum(dim=2).todtype=dtype)   
+sum_qk  $=$  linear_factor  $\star$  torch.einsum("bhld,bhld->bh1", f_q,sum_f_k)[...None]   
+return qkv, sum_qk
+```
+
+```python
+class LolcatsSlidingWindowLlamaAttention(LolcatsLlamaAttention):
+def __init__(self, window_size: int = 64, **kwargs: any,):
+super().__init__(**kwargs)
+selfwindow_size = window_size # sliding window size
+selfwindowFactors = nn_PARAMETER( # gamma mixing term
+torch.ones(1, self.num_heads, 1, 1))
+)
+```
+
+Listing 5: LoLCATs Linear + Sliding Window Attention Class
+
+Hedgehog Feature Map. We implement the Hedgehog feature map following Zhang et al. (2024).  
+```python
+import torch.nn as nn   
+class HedgehogFeatureMap(nnModule): def __init__(self, num_heads  $= 32$  : int, # defaults for 8B LLMs head_dim  $= 128$  : int, feature_dim  $= 64$  : int, ) -> None: super().__init_() self.num_heads  $=$  num_heads self.head_dim  $=$  head_dim self.feature_dim  $=$  feature_dim # Initialize trainable feature map weights selfweights  $=$  nn.Parameters( torch.zeros(self.num_heads,self.head_dim,self.feature_dim) def self.activation(self:torch.Tensor) ->torch.Tensor: ""Softmax across feature dims activation"" return torch.cat([ torch softmax(x，dim=-1)，torch softmax(-x，dim=-1) ],dim=-1) def forward(self,x:torch.Tensor) ->torch.Tensor: "" Assume x.shape is (b,h,l,d) (b: batch_size,h: num_heads,l: seq_len,d: head_dim) ""  $\mathbf{x} =$  torch.einsum('hdf,bhld->bhlf'，selfweights,x) return selfactivation(x)
+```
+
+# Listing 6: Hedgehog Feature Map
+
+Linearizing LLM Setup. To initialize an LLM for linearizing, we simply replace each softmax attention in the Transformer's layers with our LoLCATs linear attention class. We illustrate this with a Huggingface Transformer's class below.
+
+```python
+from transformers import AutoModelForCausalLM   
+def convert_model(model:AutoModelForCausalLM, window_size:int  $= 64$  feature_dim:int  $= 64$  ：   
+"""Setup linearizing attentions""" for layer in model.model.layers: ifwindow_size  $\equiv = 0$  layer.self_attn  $=$  LolcatsLlamaAttention( feature_dim  $\equiv$  feature_dim, base_attn  $\equiv$  layer.self_attn, ） else: layer.self_attn  $=$  LolcatsSlidingWindowLlamaAttention( window_size  $\equiv$  window_size, feature_dim  $\equiv$  feature_dim, base_attn  $\equiv$  layer.self_attn, ） return model
+```
+
+# Listing 7: Linearizing LLM Setup
+
+Attention Transfer Training. We can then train LoLCATs layers in a simple end-to-end loop. Although doing this attention transfer is akin to a layer-by-layer cross-architecture distillation, due to architectural similarities we implement linearizing with the same footprint as finetuning a single model. Furthermore, as we freeze all parameters except for the newly introduced feature map weights, this amounts to parameter-efficient finetuning, training  $< 0.2\%$  of a 7B+ LLM's parameters.
+
+```python
+```
+Example attention transfer training loop for Llama 3.1 8B
+```
+import torch.nn as nn
+from transformers import AutoModelForCausalLM
+# Load Llama 3.1 8B
+model_config = {
+pretrained_model_name_or_path': 'meta-llama/Meta-Llama-3.1-8B'
+}
+model = AutoModelForCausalLM.from_pretrained(**model_config)
+# Freeze all pretrained weights
+for p in model.params():
+    prequires_grad = False
+# Prepare LoLCATs linearizing layers
+model = convert_model(model)
+# Setup MSE loss criterion
+mse_loss = nn.MSELoss()
+block_size = 32 # default end-to-end for 7B+ LLMs
+num_blocks = len(model.layers) // block_size
+# Get some linearizing data
+trainloader = load_data(**data_kwargs)
+# Train LoLCATs layers via attention transfer
+for ix, input_ids in enumerate(trainloader):
+    losses = [0] * range(num_blocks) # Attention transfer loss here
+    x = model_embedding_tokens(input_ids) # Input embeddings from tokens
+# Forward pass thru model
+for lix, layer in enumerate(model.layers):
+    # *** Start Attention *** 
+    _x = layer-input_layernorm(x) # Just Llama things
+    ## Attention Transfer part
+    with torch.no_grad():
+        y_true = layer.self_attn.base_ATtn(_x)
+        y_pred = layer.self_ATtn(_x)
+        _idx = lix // block_size # Add layer or block-wise MSE
+        losses[_idx] += mse_loss(y_pred, y_true)
+        _x = y_true # Pass true attention outputs
+        # thru to rest of model
+        x = _x + x
+        # *** End Attention *** 
+# start MLP *** 
+_x = layer.postattention_layernorm(x)
+_x = self.mlp(_x)
+x = _x + x
+# *** End MLP *** 
+for loss in losses: # End-to-end attention transfer
+loss.backup()
+```
+
+Listing 8: End-to-end attention transfer pseudocode (Stage 1).
+
+Low-rank Adjusting. Finally, after attention transfer, we train the model end-to-end with next-token prediction. This allows the linearized LLM to adjust to the learned linear attentions, which may still not perfect approximations of the softmax attentions. However, with LoLCATs we hope to make these errors small enough such that we can adjust and recover pretrained LLM capabilities with parameter-efficient low-rank updates (e.g., LoRA finetuning).
+
+Listing 9: Defining a LoRA layer  
+```python
+class LoRALayer(torch.nnModule): def __init__(self, base_layer: nnModule, rank: int = 8, alpha: float = 16): super().__init_(in_low-rank_parameters) in_dim = base_layer.weight.shape[1] out_dim = base_layer.weight.shape[0] self.A = nn_PARAMETER(torch.random(in_dim, rank)) self.B = nn_PARAMETER(torch.zeros(rank, out_dim)) self.alpha = alpha self.base_layer = base_layer def low_rank_forward(self, x: torch.Tensor) -> torch.Tensor: ""Compute LoRA pass"" x = torch.einsum('...d,dr->...r', x, self.A) x = torch.einsum('...r,rd->...d', x, self.B) return self.alpha * x def forward(self, x: torch.Tensor) -> torch.Tensor: ""Actual forward"" x = self.base_layer(x) + self.low_rank_forward(x) return x
+```
+
+```python
+model  $=$  attention_transfer(model）#Do Step 1 of LoLCATs   
+#0.Prepare model for LoRA   
+lora_kwarg  $\equiv$  {'rank':8,'alpha':16} #examples   
+#1.Freeze all pretrained weights   
+for p in model.params():prequires_grad  $=$  False   
+#2.Add LoRA weights to Q,K,V,O projections   
+for layer in model.layers: for proj in ['q_proj', 'k_proj', 'v_proj', 'o_proj']: _layer  $=$  getattr(layer.self_attn,proj) _layer  $=$  LoRALayer(_layer,\*\*lora_kwargseattr(layer.self_attn,proj,lora_layer)   
+#3.Finetune LLM with LoRA   
+xent_loss  $=$  nn.CrossEntropyLoss()   
+trainloader  $=$  load_data(\*\*data_kwargse)   
+for ix，input_ids in enumerate(trainloader): #Process input tokens next_token_ids  $=$  model(input_ids) #Train model to predict next token y_pred  $=$  next_token_ids[.,:1] y_true  $=$  input_ids[.,1:] loss  $=$  xent_loss(y_pred,y_true) loss.backup()
+```
+
+Listing 10: End-to-end low-rank adjusting pseudocode (Stage 2).
+
+# C.2 HARDWARE-AWARE IMPLEMENTATION OF LOLCATS SLIDING WINDOW
+
+Despite the theoretical efficiency of linear attention, existing implementations have long underperformed well-optimized attention implementations (e.g., FlashAttention) in wall clock speed (Dao et al., 2022). To translate the benefits of LoLCATs to wall clock speedups, we develop a custom hardware-aware algorithm for LoLCATs prefetch using the ThunderKittens CUDA framework. We first briefly review the GPU execution model and then detail our algorithm.
+
+# C.2.1 GPU EXECUTION MODEL
+
+GPUs workloads are executed by independent streaming multiprocessors (SMs), which contain warps, groups of 32 threads, that operate in parallel.
+
+Memory hierarchy. ML workloads involve moving large tensors (weights, activations) in and out of memory to perform computation. GPUs have a memory hierarchy, which includes global memory (HBM), shared memory (SRAM), and registers. Reading from and writing data to memory, referred to as I/O operations, takes time. There is a large amount of HBM, which has high I/O costs, and a small amount of SRAM and registers have much costs. All SMs access global memory, warps within an SM threadblock can access shared memory, and threads within a threadblock have independent register memory. To reduce the I/O costs, locality is key - kernels should perform as many operations as possible on data that has already been loaded into fast memory (i.e., thread registers) before writing the results back to slower memory.
+
+Compute units. GPUs have increasingly heterogeneous compute units on newer generations of hardware. Tensor cores—specialized compute units for matrix-matrix multiplications—are the fastest units, operating at 1.0 PetaFLOPS on Nvidia H100 GPUs in contrast to 67 TeraFLOPS for the general non Tensor core units. ML workloads should thus ideally exploit the tensor cores.
+
+Cost model. Overall, workloads may either be compute or memory bound, depending on whether they are bottlenecked by the compute speed or I/O costs. To hide latencies from either expensive compute or I/O, a classic principle in systems is to pipeline computation among parallel workers.
+
+# C.2.2 THUNDERKITTENS CUDA KERNEL FOR PREFILL
+
+We describe our overall approach below and provide pseudocode in Algorithm 3, designed around the three principles above: memory locality, tensor core utilization, and pipelined execution.
+
+The kernel fuses the entire LoLCATs layer, taking as input the attention queries, keys, and values, for  $\mathbf{q}, \mathbf{k}, \mathbf{v} \in \mathbb{R}^{N \times d}$  with sequence length  $N$  and head dimension  $d$  and outputting the result of the  $\mathbf{y} \in \mathbb{R}^{N \times d}$ . Following Llama 3 (AI@Meta, 2024a), we let  $d = 128$  in the discussion below.
+
+Pipeline execution overview. Each thread block handles a single batch and head element of size  $N \times d$ . The kernel loops over chunks of length 64 along the sequence dimension  $N$ , loading  $64 \times 128$  tiles of  $q, k, v$ , which we'll refer to as  $q_t, k_t, v_t$ , in each iteration  $t$ . We use 8 warps (workers) per thread block, splitting them into two groups of 4 workers that pipeline the computation. One "warpgroup" is in charge of launching memory loads and stores and computing the relatively cheap terraced window attention, while the other focuses on computing the more expensive linear attention computation and recurrent state updates.
+
+Warpgroup 1 (Window attention). For the diagonal  $64 \times 128$  sized tiles, recall that the output is simply the window attention result. At iteration  $t$ , the warpgroup loads  $\pmb{q}_t, \pmb{k}_t, \pmb{v}_t$  into thread registers, use the tensor cores to multiply queries and keys, apply a causal mask, apply the Softmax, and use the tensor cores to multiply the attention scores with the values. Note that we can use Nvidia's new warpgroup operations (e.g., WGMMA) introduced in the H100 architecture to perform these operations. We refer to the terraced window output as  $\text{terrace}_0$ .
+
+Because the window attention is relatively cheap, warpgroup 1 also helps handle loads and stores between HBM and SRAM for the entire kernel. We use tensor memory acceleration (TMA), a new H100 capability for asynchronous memory movement, to perform these loads and stores.
+
+Warpgroup 2 (Linear attention). We briefly review the linear attention equation. The formulation on the left shows a quadratic view, wherein  $\phi (\pmb {q}_n)^\top$  and  $\phi (\pmb {k}_i)$  are multiplied first, while the right formulation shows a linear view, wherein  $\pmb{k}_i$  and  $\pmb{v}_i^\top$  are multiplied first.
+
+$$
+\hat {\boldsymbol {y}} _ {n} = \sum_ {i = 1} ^ {n} \frac {\left(\phi \left(\boldsymbol {q} _ {n}\right) ^ {\top} \phi \left(\boldsymbol {k} _ {i}\right)\right) \boldsymbol {v} _ {i}}{\sum_ {i = 1} ^ {n} \phi \left(\boldsymbol {q} _ {n}\right) ^ {\top} \phi \left(\boldsymbol {k} _ {i}\right)} = \frac {\phi \left(\boldsymbol {q} _ {n}\right) ^ {\top} \left(\sum_ {i = 1} ^ {n} \phi \left(\boldsymbol {k} _ {i}\right) \boldsymbol {v} _ {i} ^ {\top}\right)}{\phi \left(\boldsymbol {q} _ {n}\right) ^ {\top} \sum_ {i = 1} ^ {n} \phi \left(\boldsymbol {k} _ {i}\right)} \tag {9}
+$$
+
+Since LoLCATs uses no linear attention on the diagonal tiles, the linear attention contribution for tile  $t$  is as follows, where queries are multiplied by the cumulative KV state from the prior iterations up to  $t - 1$ :
+
+$$
+\hat {\boldsymbol {y}} _ {t} \frac {\phi \left(\boldsymbol {q} _ {t}\right) ^ {\top} \left(\sum_ {i = 1} ^ {t - 1} \phi \left(\boldsymbol {k} _ {i}\right) \boldsymbol {v} _ {i} ^ {\top}\right)}{\phi \left(\boldsymbol {q} _ {n}\right) ^ {\top} \sum_ {i = 1} ^ {t - 1} \phi \left(\boldsymbol {k} _ {i}\right)} \tag {10}
+$$
+
+At  $t = 0$ , the KV state and K state are initialized to 0, maintained in warpgroup 2's registers.
+
+At iteration  $t$ , warpgroup 2 loads in the learned feature maps into register and computes  $q\pmb{f}_t$ . This result gets multiplied by the running KV state so far up until  $t - 1$  (again 0 at iteration 0), and the result, linear<sub>O</sub> gets written to SMEM.
+
+The warps then update the KV state to prepare for the next iteration by featurizing  $k_{t}$  to  $k f_{t}$  using the learned feature map, and multiplying by  $v_{t}$  with WGMMA operations. Note that because the KV state in linear attention is somewhat large  $(d \times d)$ , we leave the state in register throughout the kernel execution to avoid I/O costs.
+
+Combining the results. Warpgroup 1 loads the linearo contribution from SMEM to its registers, adds the terracedo component, normalizes the overall result, and stores it back to HBM using TMA asynchronous store operations. We provide pseudocode in Algorithm 3.
+
+To recap, our overall algorithm uses three classical systems ideas to run efficiently: (1) pipelining the different attention and I/O operations, (2) keeping the fastest compute—the tensor cores—occupied, and (3) keeping the recurrent (KV) state local in fast memory (thread registers).
+
+Algorithm 3 LoLCATs ThunderKittens prefetch kernel  
+Output: LoLCATs attention output  $\pmb{o} \in \mathbb{R}^{N \times d}$  
+```txt
+Input: Attention queries, keys, and values  $\mathbf{q}, \mathbf{k}, \mathbf{v} \in \mathbb{R}^{N \times d}$  for head dimension  $d$  and sequence length  $N$
+```
+
+```txt
+Let  $\mathrm{local}_{KV}$  be the cumulative recurrent state ("KV-state") initialized to 0 in warpgroup 2's registers.
+```
+
+```javascript
+1: for  $t \gets 0$  to  $\frac{N}{64}$  do
+```
+
+```txt
+$\triangleright$  Warpgroup 1, Terraced attention
+```
+
+2: Load  $\pmb{q}_t, \pmb{k}_t, \pmb{v}_t \in \mathbb{R}^{64 \times d}$  from HBM to SMEM.  
+```latex
+3: Compute the terraced attention output tile  $\mathbf{terrace}_{\mathbf{o}_t} \in \mathbb{R}^{64 \times d}$  in register using WGMMA operations.
+```
+
+```txt
+$\triangleright$  Warpgroup 2, Linear attention
+```
+
+4: Featurize  $\pmb{q}_t$  by multiplying with the learned feature map to obtain  $\pmb{q}\pmb{f}_t$  
+6: Write linear  $\mathbf{O}_t$  from register to SMEM  
+```latex
+5: Compute the linear attention output tile  $\mathbf{linear}_{\mathbf{ot}} \in \mathbb{R}^{64 \times d}$  in register, using  $\mathbf{q} \mathbf{f}_t$  and  $\mathbf{local}_{KV}$ .
+```
+
+8: Update  $\mathbf{local}_{KV}$  by multiplying  $\pmb{k}\pmb{f}_t$  and  $\pmb{v}_t$ , and adding the result to  $\mathbf{local}_{KV}$  in place, all in register  
+```txt
+7: Featurize  $k_{t}$  by multiplying with the learned feature map to obtain  $k f_{t}$
+```
+
+```txt
+$\triangleright$  Warpgroup 1, Combine results
+```
+
+9: Load linear  $\mathbf{a}_{0t}$  from SMEM to register  
+```txt
+10: Add  $\pmb{o}_t = \mathbf{linear}_{\pmb{o}t} + \mathbf{t e r r a c e}_{\mathbf{o}}$  in register
+```
+
+```txt
+11: Write  $o_t$  to HBM
+```
+
+# D EXTENDED RELATED WORK
+
+# D.1 LINEARIZING TRANSFORMERS
+
+In this work, we build upon both approaches explicitly proposed to linearize LLMs (Mercat et al., 2024), as well as prior methods focusing on smaller Transformers reasonably adaptable to modern LLMs (Kasai et al., 2021; Mao, 2022; Zhang et al., 2024). We highlight two approaches most related to LOLCATS and their extant limitations next.
+
+Scalable UPtraining for Recurrent Attention (SUPRA). Mercat et al. (2024) linearize LLMs by swapping softmax attentions with linear attentions similar to Retentive Network (RetNet) layers (Sun et al., 2023), before jointly training all model parameters on the RefinedWeb pretraining dataset (Penedo et al., 2023). In particular, they suggest that linearizing LLMs with the vanilla linear attention in Eq. 2 is unstable, and swap attentions with
+
+$$
+\hat {\boldsymbol {y}} _ {n} = \operatorname {G r o u p N o r m} \left(\sum_ {i = 1} ^ {n} \gamma^ {n - i} \phi \left(\boldsymbol {q} _ {n}\right) ^ {\top} \phi \left(\boldsymbol {k} _ {i}\right) \boldsymbol {v} _ {i}\right) \tag {11}
+$$
+
+GroupNorm (Wu & He, 2018) is used as the normalization in place of the  $\sum_{i=1}^{n} \phi(\pmb{q}_n)^\top \phi(\pmb{k}_i)$  denominator in Eq. 2,  $\gamma$  is a decay factor as in RetNet, and  $\phi$  is a modified learnable feature map from Transformer-to-RNN (T2R) (Kasai et al., 2021) with rotary embeddings (Su et al., 2024). In other words,  $\phi(\pmb{x}) = \mathrm{RoPE}(\mathrm{ReLU}(\pmb{x}\pmb{W} + \pmb{b}))$  with  $\pmb{W} \in \mathbb{R}^{d \times d}$  and  $\pmb{b} \in \mathbb{R}^d$  as trainable weights and biases. With this approach, they recover zero-shot capabilities in linearized Llama 2 7B (Touvron et al., 2023b) and Mistral 7B (Jiang et al., 2023) models on popular LM Evaluation Harness (Gao et al., 2023) and SCROLLS (Shaham et al., 2022) tasks.
+
+Hedgehog. Zhang et al. (2024) show we can train linear attentions to approximate softmax attentions, improving linearized model quality by swapping in the linear attentions as learned drop-in replacements. They use the standard linear attention (Eq. 2), where query, key, value, and output projections (the latter combining outputs in multi-head attention (Vaswani et al., 2017)) are first copied from an existing softmax attention. They then specify learnable feature maps  $\phi (\pmb {x}) = [\mathrm{softmax}(\pmb {x}\pmb {W} + \pmb {b})\oplus \mathrm{softmax}(-\pmb {x}\pmb {W} - \pmb {b})]$  (where  $\oplus$  denotes concatenation, and both  $\oplus$  and the softmax are applied over the feature dimension) for  $q$  and  $\pmb{k}$  in each head and layer, and train  $\phi$  such that linear attention weights  $\hat{\pmb{a}}$  match a Transformer's original softmax weights  $\pmb{a}$ . Given some sample data, they update  $\phi$  with a cross-entropy-based distillation to minimize:
+
+$$
+\mathcal {L} _ {n} = - \sum_ {i = 1} ^ {n} \frac {\exp \left(\boldsymbol {q} _ {n} ^ {\top} \boldsymbol {k} _ {i} / \sqrt {d}\right)}{\sum_ {i = 1} ^ {n} \exp \left(\boldsymbol {q} _ {n} ^ {\top} \boldsymbol {k} _ {i} / \sqrt {d}\right)} \log \frac {\phi \left(\boldsymbol {q} _ {n}\right) ^ {\top} \phi \left(\boldsymbol {k} _ {i}\right)}{\sum_ {i = 1} ^ {n} \phi \left(\boldsymbol {q} _ {n}\right) ^ {\top} \phi \left(\boldsymbol {k} _ {i}\right)} \tag {12}
+$$
+
+as the softmax and linear attention weights are both positive and sum to 1. As they focus on task-specific linearization (e.g., GLUE classification (Wang et al., 2018) or WikiText-103 language modeling (Merit et al., 2017)), for both attention and model training they use task-specific training data. By doing this "attention distillation", they show significant linearized quality improvements over T2R mainly on smaller Transformers (e.g., 110M parameter BERTs (Devlin et al., 2018) and 125M GPT-2s (Radford et al., 2019)). They further show LLM linearizing by linearizing Llama 2 7B for a specific SAMSum summarization task (Gliwa et al., 2019).
+
+# D.2 EFFICIENT ARCHITECTURES
+
+Subquadratic Attention Alternatives. Many prior works study more efficient sequence modeling modules compared to Transformer self-attention, commonly training proposed architectures from scratch. While our work is most similar to and compatible to linear attentions (Katharopoulos et al., 2020; Choromanski et al., 2020; Chen et al., 2021b; Xiong et al., 2021; Zhang et al., 2024; Qin et al., 2022; Schlag et al., 2021; Yang et al., 2023) and sparse attentions (Beltagy et al., 2020; Chen et al., 2021a; Zaheer et al., 2020), other efficient architectures include convolutional models (Poli et al., 2023a; Gu et al., 2021), recurrent neural nets (RNNs) (not strictly following the attention inductive bias) (Peng et al., 2023a; De et al., 2024; Beck et al., 2024), and state-space model (SSM)-inspired neural net layers (Gu et al., 2021; Gu & Dao, 2023; Dao & Gu, 2024).
+
+Improving LLMs for Long Contexts. We share motivation various works focus on improving context length quality (Chen et al., 2023a; kai; Peng et al., 2023b). They that the rotary embeddings (Su et al., 2024) of popular LLMs (Biderman et al., 2023; Touvron et al., 2023b; AI@Meta, 2024a; Jiang et al., 2023) struggle to handle contexts longer than those used in training. Chen et al. (2023b) notably study the training efficiency of this via LoRA. Our work is complementary, where we reduce the efficiency bottleneck by replacing self-attention. Further work lies in how to improve long context quality of linearized LLMs, e.g., by integrating these methods. Other works also improve efficiency by optimizing the KV Cache from a systems standpoint (Kwon et al., 2023). These may present complementary optimizations to linear attention's algorithmic improvements.
+
+# E ADDITIONAL ARTIFACTS
+
+# E.1 SAMPLE GENERATIONS
+
+We include sample generations for various prompts, using Llama 3 8B models linearized with different LoLCATs-related linearizing setups on the Alpaca dataset. For reference, we also include responses from a standard (softmax attention) Llama 3 8B finetuned with LoRA (rank = 8, alpha = 16, applied to all attention projections) on the Alpaca training set. LoLCATs as presented often leads to greatest coherence among linearized LLMs.
+
+# Prompt
+
+<|begin_of_text|> Below is an instruction that describes a task. Write a response that appropriately completes the request.
+
+Instruction:
+
+What is the difference between H&M and Balenciaga?
+
+Response:
+
+# Hedgehog (Linear Only) (No LoRA)
+
+Hindi: the difference is that in the first one is a little bit more complicated, but the other one is that is the same. <|end_of_text|>
+
+# Hedgehog (Linear Only) (Attention Transfer + LoRA)
+
+H&M and Balenciaga are two of the most popular fashion brands in the world, but there are some differences between them.
+
+H&M is a fast fashion retailer that sells trendy and affordable clothing. It was founded in Sweden in 1947 and is now available in over 60 countries worldwide. The brand is known for its fast fashion and affordable prices, making it a popular choice for those on a budget.
+
+Balenciaga, on the other hand, is a luxury fashion house that was founded in 1977. It is known for its high-end ready-to-wear pieces, including leather jackets, dresses, and shoes. The brand is known for its use of luxurious fabrics and intricate designs, as well as its collaborations with other fashion houses such as Versace and Versace.
+
+In summary, H&M and Balenci are two popular fashion brands that offer different styles and price points. While H&M is known for its fast fashion and affordable fashion, Balenciaga is known for its luxury and high-end ready-to-wear pieces.<|end_of_text|>
+
+# LoLCATs (Linear + Sliding Window) (No LoRA)
+
+The difference between H&M and Balenciaga is that H&M is a high street brand while Balenciaga is a luxury brand. H&M is more affordable than Balenciaga, which is a luxury brand.<|end_of_text|>
+
+# LoLCATs (Linear + Sliding Window) (No Attention Transfer)
+
+H&M and Balenciaga are both clothing brands, but there are some key differences between the two.
+
+H&M is a fast fashion brand known for its affordable and trendy clothing. The company was founded in Sweden and has stores in over 60 countries. It is known for its seasonal collections, which include clothing, accessories, and home decor. H&M has a reputation for its ethical and sustainable practices, and the company works with designers and brands to create its collections.
+
+On the other hand, Burberry is a British luxury fashion house founded in 1856. It is known for its high-end fashion, including clothing, accessories, and fragrances. The company has a strong heritage, and its designs are inspired by British culture and history. Burberry is known for its classic trench coats, and its signature check pattern is instantly recognizable.
+
+Overall, both brands are high-end fashion companies, but they have different target audiences and design styles. H&M is more accessible and affordable, while Burberry is more expensive and known for its classic, timeless designs. <|end_of_text|>
+
+# LOLCATs (Linear + Sliding Window) (Attention Transfer + LoRA)
+
+H&M and Balenciaga are both fashion brands, but they have significant differences.
+
+H&M is a fast fashion brand that offers a wide range of clothing, accessories, and home decor at affordable prices. The brand is known for its trendy and stylish designs, and it often collaborates with high-profile designers to create limited-edition collections. H&M is a mass-market retailer, and its products are available in its own stores, as well as in department stores and online.
+
+Balenciaga, on the other hand, is a luxury fashion house founded in 1917 by Spanish designer Cristobal Balenciaga. The brand is known for its avant-garde designs, high-quality materials, and attention to detail. Balenciaga's collections are often bold, experimental, and daring, and the brand has a strong following among fashion enthusiasts and celebrities. H&M, on the other hand, is a fast-fashion retailer that offers affordable clothing and accessories inspired by the latest trends. While H&M also produces its own clothing and accessories, the brand is known for its collaborations with high-end designers and its ability to produce fashion at a much lower price point than other luxury brands.<|end_of_text|>
+
+# Softmax Attention
+
+H&M and Balenciaga are two fashion brands that are quite different from each other.
+
+H&M is a fast fashion brand that offers affordable clothing and accessories for both men and women. The brand is known for its trendy and on-trend designs, and its collections often feature collaborations with high-profile designers and celebrities. H&M is a mass-market brand that aims to make fashion accessible to everyone.
+
+Balenciaga, on the other hand, is a luxury fashion brand that is known for its avant-garde designs and high-end materials. The brand is associated with couture and haute couture, and its collections often feature bold and experimental designs. Balenciaga is a high-end brand that caters to a more exclusive clientele.
+
+In summary, the main difference between H&M and Balenciaga is their target audience, price point, and design philosophy. H&M is a fast fashion brand that offers affordable and trendy designs, while Balenciaga is a luxury brand that focuses on high-end and avant-garde designs.<|end_of_text|>
+
+# Prompt
+
+<|begin_of_text|> Below is an instruction that describes a task. Write a response that appropriately completes the request.
+
+Instruction:
+
+Explain why the following fraction is equivalent to 1/4. 4/16
+
+\*\*\* Response:
+
+# Hedgehog (Linear Only) (No LoRA)
+
+The number 4/4 is a fraction. This is a fraction that is equivalent to 4/4. The number 4/4 is the number  $\frac{4}{4} / \frac{4}{4} / \frac{4}{4} < \frac{1}{\text{end_of_text}}$ .
+
+# Hedgehog (Linear Only) (Attention Transfer + LoRA)
+
+The fraction 1/4 is equivalent to 4/16. To see why, we can multiply both the numerator and denominator by 4 to obtain the equivalent fraction 16/16. This is because the multiplication of two equal numbers will not change the value of the fraction.
+
+The number 4/16 can be rewritten as 16/4, which is equivalent to 4/4. Similarly, the numerator 4/4 can be rewritten as 4/4. Therefore, the fraction 4/4 is equivalent to 4/16, which is the same as 1/4. <|end_of_text|>
+
+# LoLCATs (Linear + Sliding Window) (No LoRA)
+
+The fraction 4/16 is equivalent to  $\frac{1}{4}$  because the numerator and denominator are both divisible by 2. This is because 4/16 is equal to 4/16. This is because the numerator and denominator are both divisible by 2. This means that 4/16 is equal to  $\frac{1}{4}$ . Therefore,  $\frac{1}{4}$  is equal to  $\frac{1}{4}$ , which is  $\frac{1}{4}$ . Therefore, 4/16 is equal to  $\frac{1}{4}$ , and  $\frac{1}{4}$  is equal to  $\frac{1}{4} < \left| \text{end_of_text} \right| >$
+
+# LoLCATS (Linear + Sliding Window) (No Attention Transfer)
+
+$$
+1 / 4 = 4 / 1 6
+$$
+
+To explain why the fraction 1/4 is equivalent to 4/16, we first need to understand what a fraction is. A fraction is a way to represent a part of a whole, where the whole is divided into equal parts. In this case, the whole is the whole number 4, which is divided into 4 equal parts, each of which is represented by the fraction 1/4. The symbol used to represent a fraction is a fraction bar, which is a horizontal line that goes between the numerator and the denominator. So, the fraction 1/4 can be written as 1/4 or  $\frac{1}{4}$ .
+
+# LoLCATs (Linear + Sliding Window) (Attention Transfer + LoRA)
+
+The fraction 4/16 is equivalent to 1/4 because both fractions represent the same value. When simplifying 4/16, we can divide both the numerator and denominator by 4 to get 1/4. This is because 4 divided by 4 is equal to 1, and 16 divided by 4 is also equal to 4. So, we can cancel out the 4s to get  $\frac{1}{4} < |\text{end-of-text}| >$
+
+# Softmax Attention
+
+The fraction 4/16 is equivalent to  $\frac{1}{4}$  because both fractions represent the same value. To see why this is the case, we can simplify the fraction 4/16 by dividing both the numerator and the denominator by their greatest common factor.
+
+The greatest common factor (GCF) of 4 and 16 is 4. We can simplify the fraction by dividing both the numerator and the denominator by 4:
+
+$$
+4 \div 4 = 1
+$$
+
+$$
+1 6 \div 4 = 4
+$$
+
+Thus, the simplified fraction is 1/4. Since the simplified fraction is equivalent to the original fraction, 4/16 is equivalent to  $\frac{1}{4} < |\text{end_of_text}| >$
+
+# Prompt
+
+<|begin_of_text|> Below is an instruction that describes a task. Write a response that appropriately completes the request.
+
+Instruction:
+
+Why did we linearize the chungus among us?
+
+\*\*\* Response:
+
+# Hedgehog (Linear Only) (No LoRA)
+
+- The reason that the request is to request the request is to use the request to request.<|end_of_text|>
+
+# Hedgehog (Linear Only) (Attention Transfer + LoRA)
+
+The reason we linearized the chunus among us is to make it easier for us to understand and compare their behavior. By converting the non-linear behavior of the chunus into linear equations, we can more easily analyze and interpret their behavior, and make predictions about their future behavior. Additionally, it allows us to compare the behavior of the chunus among themselves, and to compare their behavior over time.<|end_of_text|>
+
+# LoLCATs (Linear + Sliding Window) (No LoRA)
+
+Because we have no other choice.<|end_of_text|>
+
+# LoLCATS (Linear + Sliding Window) (No Attention Transfer)
+
+The word "chungus" is not in the English language, so it cannot be linearized.<|end_of_text|>
+
+# LoLCATS (Linear + Sliding Window) (Attention Transfer + LoRA)
+
+Linearizing the chungus among us is a process of organizing and arranging the information in a logical and coherent manner. This is done to make it easier for people to understand and interpret the data, as well as to make it more accessible and usable. By linearizing the data, we are able to extract meaningful insights and patterns from the information, which can then be used to make informed decisions and take action.<|end-of-text|>
+
+# Softmax Attention
+
+I'm sorry, but I cannot respond to this question as I do not have the necessary information to provide a meaningful response. Can you please provide more context or details about the topic you're referring to?<|end_of_text|>
+
+# E.2 SAMPLE ATTENTION WEIGHTS
+
+We finally visualize sample attention weights from LoLCATs Mistral 7b and Llama 3 8B layers, computed over held-out 1024-token packed Alpaca samples. Despite only training to match attention outputs via an MSE loss, we find LoLCATs often recovers softmax attention weights qualitatively, with comparable quality to explicitly matching attention weights via a cross-entropy loss (XENT). We validate that this is not just due to upweighting softmax attention terms in our sliding window setup, plotting query-key interactions for 1024-long samples between the last 32 queries and the first and last 32 keys (spanning distances well over the 64-token window sizes). LoLCATs linear attentions frequently recover attention weights over these distances, suggesting that we learn both linear attention feature maps and weighting factors to match softmax attention weights.
+
+Finally, we note two observations on LoLCATs training and architectures. Newly initialized LoL-CATs linear + sliding window attentions do not capture softmax attention weights (init.), supporting earlier results that attention transfer is necessary to reduce architectural differences and improve linearizing efficiency. Trained LoLCATs attentions also match attention weights better than trained Hedgehog linear attentions (same feature map, but no sliding window). These results suggest LoL-CATs attention transfer and linear + sliding window layers allow us to learn better approximations of softmax attention weights, coinciding with improved lienarizing quality.
+
+![](images/d301d2868414792c9a8a68a348dbcbf0b6625ac8f99e9e628620f3c7e0c24c9e.jpg)
+
+![](images/4ab5ebc6356ab7474cc41c1a3d7cb9222af9254436d036bf5b837c6e36e469bd.jpg)
+
+![](images/20ec6e691628fba49fcc5a9c71d703b7d6eb62f7ebd5e1dbaaa92db198e5e4e0.jpg)
+
+![](images/6788d52463cdea51b7462defb093edb4eb75d5940212ce203af34f196df3321a.jpg)
+
+![](images/51754df6ccacaf176c1ebdd781a88673c3c27e4f37e3af9f1e2a02d6af59d387.jpg)
+
+![](images/1d38ca8de62837d6773f24f0cdf51780624c0f4294a925c0568833377663e5fb.jpg)  
+Figure 18: Llama 3 8B attention weights; head 0; layers 0, 8, 16, 24, 31.
+
+![](images/1b9b3754d6190399870dfd0f9c3e72f61b9ce64bb7be979bfe1624d35436c1b6.jpg)  
+Figure 19: Llama 3 8B attention weights; head 1; layers 0, 8, 16, 24, 31.
+
+![](images/1b82f117ae5b1ca830d01d08604b94a951034d859aa2b8f5faa1bcea4e5fa872.jpg)  
+Figure 20: Llama 3 8B attention weights; head 2.; layers 0, 8, 16, 24, 31.
+
+![](images/4b8e41aa553dd4c29cf8a15066e4881cf27c1ad92ce036747b9bee14c7a05a12.jpg)  
+Figure 21: Llama 3 8B attention weights; head 3; layers 0, 8, 16, 24, 31.
+
+![](images/2d4d5024da71a94336699406f247ac6a82cb2a1eedb115495a121070958064c7.jpg)  
+Figure 22: Mistral 7B v0.1 attention weights; head 0; layers 0, 8, 16, 24, 31.
+
+![](images/406a9e1fe3e305aa7b66b74f5594a4ccfd81b98fd5b1ef3cae60a4d3aae8ec1d.jpg)  
+Figure 23: Mistral 7B v0.1 attention weights; head 1; layers 0, 8, 16, 24, 31.
+
+![](images/322fa5917953011e9506b743930cefc5574b153b7edf8fdb16d51d56a22f8e57.jpg)  
+Figure 24: Mistral 7B v0.1 attention weights; head 2; layers 0, 8, 16, 24, 31.
+
+![](images/6b2c936905b4a334b5e36bb07270330b9b714982bcf580990474e42b196ed103.jpg)  
+Figure 25: Mistral 7B v0.1 attention weights; head 3; layers 0, 8, 16, 24, 31.
\ No newline at end of file
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+# LOR-VP: LOW-RANK VISUAL PROMPTING FOR EFFICIENT VISION MODEL ADAPTATION
+
+Can Jin $^{1}$ , Ying Li $^{2}$ , Mingyu Zhao $^{1}$ , Shiyu Zhao $^{1}$ , Zhenting Wang $^{1}$ , Xiaoxiao He $^{1}$ , Ligong Han $^{3,4}$ , Tong Che $^{5}$ , Dimitris N. Metaxas $^{1}$
+
+$^{1}$ Rutgers University,  $^{2}$ Zhejiang University,  $^{3}$ Red Hat AI Innovation,
+
+$^{4}$ MIT-IBM Watson AI Lab,  $^{5}$ NVIDIA Research
+
+# ABSTRACT
+
+Visual prompting has gained popularity as a method for adapting pre-trained models to specific tasks, particularly in the realm of parameter-efficient tuning. However, existing visual prompting techniques often pad the prompt parameters around the image, limiting the interaction between the visual prompts and the original image to a small set of patches while neglecting the inductive bias present in shared information across different patches. In this study, we conduct a thorough preliminary investigation to identify and address these limitations. We propose a novel visual prompt design, introducing Low-Rank matrix multiplication for Visual Prompting (LoR-VP), which enables shared and patch-specific information across rows and columns of image pixels. Extensive experiments across seven network architectures and four datasets demonstrate significant improvements in both performance and efficiency compared to state-of-the-art visual prompting methods, achieving up to  $6 \times$  faster training times, utilizing  $18 \times$  fewer visual prompt parameters, and delivering a  $3.1\%$  improvement in performance. Code is available at https://github.com/jincan333/LoR-VP.
+
+# 1 INTRODUCTION
+
+Many applications in computer vision (CV) and natural language processing (NLP) rely on adapting large-scale, pre-trained models to multiple downstream tasks (Liu et al., 2021; Dosovitskiy et al., 2020; Ridnik et al., 2021; Brown et al., 2020; Peng et al., 2024). Recent advances in large language models (LLMs) have highlighted data-centric techniques such as in-context learning (Brown et al., 2020; Shin et al., 2020; Liu et al., 2022) and prompting (Li & Liang, 2021; Liu et al., 2023). These techniques well-designed prompts or input templates to significantly enhance the performance of LLMs across a wide range of tasks. Inspired by these methods, visual prompting has gained substantial attention as a means of adapting pre-trained vision models by modifying input pixels or output transformations (Bahng et al., 2022; Chen et al., 2023; Tsao et al., 2024).
+
+Existing visual prompting methods, such as those proposed by CLIP-VP (Bahng et al., 2022), ILM-VP (Chen et al., 2023), and AutoVP (Tsao et al., 2024), have demonstrated the capability to enhance the performance of pre-trained vision models across various downstream tasks. A natural question arises: why does the addition or padding of tunable parameters to the original image pixels improves adaptation performance? A plausible explanation is that the introduced visual prompts (VPs) provide task-specific information that not only alters the representation of the original images but also influences the attention distribution across image patches. This is particularly significant in pretrained models such as Vision Transformers (ViTs) (Dosovitskiy et al., 2020), where VPs interact with patch tokens and guide the model's attention to different parts of the image.
+
+However, current VP designs primarily focus on adding or padding tunable parameters at the periphery of the image (see Part 1 of Figure 1 for existing VP designs), thus only allowing boundary patches to be modified, while the central patches remain unchanged. These designs present two notable limitations: (1) The VP parameters are restricted to interacting with the original image in a limited set of patches, leaving a substantial portion of the image unmodified. As a result, VPs
+
+can only influence the model's interpretation of specific regions of the image, while other regions-potentially containing critical information-remain unaffected. (2) The VPs applied to each patch operate independently, disregarding the inductive biases present in the shared information and positional encoding across different patches. For instance, one patch may represent part of an object, while an adjacent patch represents another part of the same object. By ignoring these inter-patch relationships, visual prompting may limit the model's ability to capture the global context of the image effectively.
+
+To address these limitations, we first conduct a thorough investigation to understand the shortcomings of existing methods. Building on this analysis, we propose a novel visual prompting method, termed LOR-VP, aimed at more efficient and effective adaptation of vision models. Our approach not only influences every patch of the image but also introduces inductive biases between the rows and columns of image patches. By leveraging LOR-VP, we achieve superior performance compared to state-of-the-art (SOTA) methods while significantly reducing the number of parameters required. In summary, our contributions are organized around the following three thrusts:
+
+$\star$  (Preliminary Study) We conduct an in-depth preliminary study to identify and illustrate the limitations present in current visual prompting methods and explore potential solutions to overcome these challenges.  
+$\star$  (Novel Approach) To mitigate these limitations, we propose a novel visual prompting technique, named LoR-VP, which optimizes the visual prompts uniformly across all patches and introduces inductive biases between the rows and columns of the image patches.  
+$\star$  (Experiments) We perform extensive experiments across a wide range of large-scale models and datasets. The empirical results consistently demonstrate the significant improvements in both performance and efficiency achieved by LoR-VP, validating its practical effectiveness. For instance, LoR-VP surpasses the previous SOTA method, AutoVP (Tsao et al., 2024), by an average of  $3.1\%$  on seven network architectures and four datasets using  $6\times$  less training time.
+
+# 2 RELATED WORKS
+
+# 2.1 VISUAL PROMPTING
+
+The concept of prompting originated in the field of NLP as a technique for adapting pre-trained models to downstream tasks (Shin et al., 2020; Liu et al., 2022; Li & Liang, 2021; Liu et al., 2023; Han et al., 2023; 2024; Jin et al., 2023). This design philosophy was later extended to CV by Bahng et al. (2022), who introduced tunable parameters directly into input images to create what is known as a Visual Prompt (VP). A typical VP framework consists of two primary modules: input design and output transformation (Bahng et al., 2022; Tsai et al., 2020; Tsao et al., 2024; Cai et al., 2024). Various strategies have been proposed for constructing VPs. For instance, Bahng et al. (2022) modify input images by adding a frame of visual prompting parameters, whereas Chen et al. (2023) incorporate the visual prompting parameters around resized images. Wu et al. (2022) explore efficient methods for generating visual prompts that enhance performance across different tasks, and Oh et al. (2023) develop visual prompts designed for adapting models to black-box, inaccessible models. Since the output logits of pre-trained models remain in the source domain, an additional output transformation (e.g., label mapping) is required to accurately predict the targets. A simple approach is to randomly map source labels (RLM) onto target labels. Tsai et al. (2020) propose a frequency-based label mapping (FLM) technique, which derives the mapping based on frequency statistics. Chen et al. (2023) further introduces iterative label mapping (ILM), which improves the performance of visual prompting. Yang et al. (2023) proposes a semantics-based label mapping approach that aligns source and target classes based on semantic similarity. Additionally, Tsao et al. (2024) introduces full mapping (FM), utilizing an automated system to select the most appropriate label mapping (LM) method to optimize performance across diverse downstream tasks.
+
+# 2.2 LOW-RANK STRUCTURES IN DEEP LEARNING
+
+Low-rank structures are widely observed in machine learning, as many problems inherently exhibit low-rank properties (Li et al., 2016; Cai et al., 2010; Li et al., 2018; Grasedyck et al., 2013).
+
+It has been found that for numerous deep learning tasks, especially those involving heavily overparameterized neural networks, the resulting models tend to exhibit low-rank characteristics after training (Oymak et al., 2019; Khodak et al., 2021). Some prior work has explicitly integrated low-rank constraints during the training process of neural networks (Sainath et al., 2013; Zhang et al., 2014; Zhao et al., 2016). From a theoretical perspective, neural networks have been shown to outperform classical learning methods, including finite-width neural tangent kernels, when the underlying concept class has a low-rank structure (Allen-Zhu et al., 2019; Li & Liang, 2018; Ghorbani et al., 2020; Allen-Zhu & Li, 2019; 2020). Additionally, Allen-Zhu et al. (2020) highlight that low-rank adaptations can be beneficial in adversarial training scenarios. The LoRA method, introduced by Hu et al. (2021), along with its variants (Zhang, 2023; Yeh, 2023), is particularly noteworthy for not introducing additional inference burdens, thus improving the parameter efficiency of adapting large pre-trained models. These methods employ low-rank matrices to approximate weight updates during fine-tuning, enabling a seamless integration with pre-trained weights prior to inference.
+
+# 3 PRELIMINARY STUDY
+
+# 3.1 ANALYSIS OF PAD PROMPTING
+
+Visual prompting is proposed to address the problem of adapting a pre-trained source model to downstream tasks without fine-tuning the network weights. Consider a downstream target image dataset  $\mathcal{D} = \{(\mathbf{x}_1, y_1), \dots, (\mathbf{x}_n, y_n)\}$  with typical color channels  $c$  (usually 3) and a pre-trained vision model  $f$  with a resolution of  $L \times L$  (the default value of  $L$  is 224 for simplicity). Visual prompting modifies the images by adding tunable parameters to the image pixels. Among various methods, Pad Prompting(Tsao et al., 2024; Chen et al., 2023; Bahng et al., 2022) is the most prevalent, which involves resizing the initial images to a specific size  $s$  (typically smaller than  $L$ , such as 128), and then surrounding the resized image with a tunable parameter border of size  $p$  such that  $s + 2p = L$ , resulting in a prompted image of dimensions  $L \times L$ . An illustration of Pad Prompting is depicted in Part 1 of Figure 1. The optimal values for  $s$  and  $p$  generally vary across different models and tasks. AutoVP, currently the SOTA in visual prompting, automates the selection of  $s$  and  $p$  to enhance performance across various models and tasks (Tsao et al., 2024).
+
+![](images/07427c5700a3271d6e52821154e36280d1aea7a7129f77e1fd190412e3466b0a.jpg)  
+Figure 1: Illustration of various visual prompting methods applied to target domain data:  $①$  AutoVP(Pad): Focuses on optimizing the balance between image scaling and tunable parameter integration to enhance model responsiveness.  $②$  Patch-Pad: Aims to enhance localized learning by surrounding each image patch with tunable visual prompts.  $③$  Patch-Free: Provides maximum adaptability by allowing independent tuning of visual prompts for each patch, catering to diverse feature requirements across the image.  $④$  Patch-Same: Promotes consistency in model training by applying uniform visual prompts across all patches, ensuring coherent feature learning across input.
+
+Despite AutoVP's automated process for optimizing prompt size, Pad Prompting inherently only adjusts the prompts in the peripheral patches of the resized images, leaving the central part unchanged.
+
+Furthermore, the parameters in different patches are optimized independently, disregarding the inductive biases that arise from shared information and positional encoding across patches. We hypothesize that a more effective visual prompting strategy would allow interaction with each patch while also considering the inductive biases among them. This hypothesis has led us to explore new VP designs:
+
+- Patch-Pad: As shown in Part 2 of Figure 1, after resizing the image, we evenly split it into different patches and then pad the tunable parameters around each patch to form a resolution  $L \times L$  image. For instance, when using an ImageNet-21K (Deng et al., 2009) pre-trained ViT-B/32 (Dosovitskiy et al., 2020) and fine-tuned on ImageNet-1K model, the resize image is split into  $7 \times 7$  patches, and each patch is padded to a size of  $32 \times 32$  using tunable parameters. The Patch-Pad method can influence each patch of the image but split the image into discontinuous parts and potentially deviate the information in the images.  
+- Patch-Free: As shown in Part 3 of Figure 1, to avoid dropping information in the images, we directly resize the image to resolution  $L \times L$ , then evenly add independent tunable parameters (initialized as 0) to each patch of the resized image. The Patch-Free design can influence each patch of the image without splitting the resized image thus maintaining the information of the image, but this design updates the patch VPs independently and doesn't consider shared information on different patches.  
+- Patch-Same: As shown in Part 4 of Figure 1, to enable shared prompting among each patch, we initialize a patch of tunable parameters and repeatedly add it to all patches of the image. Patch-Same enables shared visual prompting among different patches but constrains the shared information as the same for all patches.
+
+# 3.2 PERFORMANCE INVESTIGATION OF DIFFERENT VP DESIGNS
+
+To investigate the performance of the aforementioned four VP designs, we conducted experiments using ImageNet-21K pre-trained ViT-B/32 and ViT-B/16 (both fine-tuned on ImageNet-1K) on CIFAR10/100 (Krizhevsky et al., 2009). The performance of AutoVP (Pad Prompting) was used as a benchmark, with all methods employing a fully connected label (FM) (Tsao et al., 2024) for fair comparison. The results, depicted in Figure 2, indicate that  $①$  Patch-Pad underperforms across all models and datasets, the most likely reason is it split the image patches thus might damage the image information.  $②$  Patch-Free outperforms Patch-Pad, confirming that maintaining image continuity is beneficial. However, Patch-Free is less effective than Patch-Same, which suggests that shared visual prompting can enhance performance.  $③$  Patch-Same outperforms AutoVP, underscoring the importance of shared prompting information across patches.  $④$  The performance gap between Patch-Same and Pad Prompting shrinks in models that have smaller patch sizes, suggesting that when using the ViT-B/16 model, Patch-Same constrains more patches to learn a smaller visual prompt, which may bring too strong constraints to the visual prompts, and a better way is utilizing visual prompts that not only introduce inductive bias in different patches but also allow for patch-specific visual prompting.
+
+![](images/0ac52899bd873c09ed155e361ce0f43b21cbe14f8af7d6fcb44ce562cc7734f9.jpg)  
+Figure 2: Preliminary Investigation Results. Performance comparison of various VP designs. Our VP method demonstrates competitive or superior performance in several configurations. The final performance of each method is marked by  $\star$  or  $\bullet$ , with all results averaged over three runs.
+
+# 4 METHODOLOGY
+
+Inspired by the observations in Section 3, we propose a novel visual prompt design that facilitates prompting across all patches while enabling both shared and patch-specific information. This approach leverages low-rank matrix multiplication to efficiently manage visual prompts.
+
+# 4.1 LOW-RANK VISUAL PROMPTING
+
+In order to facilitate comprehensive pixel information and interaction across all patches, we resize the image to a uniform size of  $L \times L$ . This resizing strategy is designed to minimize information loss, which is a common issue with the traditional Pad Prompting method that often resize images to dimensions smaller than  $L$ . To enable the sharing of visual prompt information across different patches, we introduce two low-rank matrix parameters,  $\mathbf{B} \in \mathbb{R}^{c \times L \times r}$  and  $\mathbf{A} \in \mathbb{R}^{c \times r \times L}$ , where  $r \ll L$ . The product of these matrices,  $\mathbf{B} \cdot \mathbf{A}$ , serves as the visual prompt.
+
+This configuration allows the visual prompt to act as a linear combination of the row vectors in  $\mathbf{A}$  and the column vectors in  $\mathbf{B}$ , facilitating shared information across the rows and columns of the image. Additionally, this design supports patch-specific information, as the coefficients of each row and column are independently adjustable. Based on our observations in Section 3, this approach to visual prompting is likely to yield superior performance. The visual prompt is directly added to the resized image, resulting in the prompted image being expressed as:
+
+$$
+\mathcal {P} (\mathbf {x}) = \operatorname {R e s i z e} _ {L} (\mathbf {x}) + \mathbf {B} \cdot \mathbf {A}, \quad \mathbf {x} \in \mathcal {D}, \tag {1}
+$$
+
+where  $\mathrm{Resize}_L(\cdot)$  resizes the image  $\mathbf{x}$  into a size of  $L\times L$ , and matrices  $\mathbf{B}$  and  $\mathbf{A}$  are the initialized visual prompt parameters, we utilize zero initialization of  $\mathbf{B}$  and a random Gaussian initialization of  $\mathbf{A}$  so  $\mathbf{B}\cdot \mathbf{A}$  is zero at the beginning of training.
+
+Utilizing a rank  $r = 4$  in  $\mathbf{B}$  and  $\mathbf{A}$ , we conduct experiments using the same configurations as Section 3. From the experimental results shown in Figure 2, we can observe that our Low-Rank matrices multiplication Visual Prompting (LoR-VP) achieve the best performance among all designs, further validate our hypothesis in Section 3.
+
+This method simplifies the visual prompting process compared to Pad Prompting, which involves complex resizing, padding, and mask manipulations. By employing low-rank matrices, we reduce the number of tunable parameters from  $cL^2$  to  $crL$ , enhancing parameter efficiency significantly.
+
+# 4.2 OUTPUT TRANSFORMATION
+
+The output of the pre-trained model  $f$  on the prompted image  $\mathcal{P}(\mathbf{x})$  remains in the source domain. To align these predictions with target labels in downstream tasks, we apply an output transformation, denoted as  $\mathcal{M}$ :
+
+$$
+\underset {\delta , \mathcal {M}} {\text {m i n i m i z e}} \quad \mathbb {E} _ {(\mathbf {x}, y) \in \mathcal {D}} \mathcal {L} (\mathcal {M} (f (\mathcal {P} (\mathbf {x})), y), \tag {2}
+$$
+
+We utilize Linear Probing (LP) as our output transformation method, which adjusts the output features of the classifier head to suit downstream classes. This method proves more efficient than existing methods such as iterative label mapping (ILM) (Chen et al., 2023) and fully connected layer mapping (FM) (Tsao et al., 2024) on large models and datasets. For instance, when using ImageNet-21K pre-trained Swin-B (Liu et al., 2021) and tuning on ImageNet-1K, ILM costs too much time and GPU storage to calculate and store the mapping sequences (a  $21,841 \times 1,000$  matrix) and AutoVP also achieves inferior performance due to the ineffectiveness to learn a  $21,841 \times 1,000$  full connected layer (see Figure 5 for performance details).
+
+# 5 EXPERIMENTS
+
+To assess the effectiveness and efficiency of our novel visual prompting method, we adopt the most widely used evaluation protocol for VPs, i.e., evaluating models pre-trained on large datasets across
+
+![](images/6c3d31eb0902943c5d780bb1c65709d6649736ac45a434e72fe21ef4e3e6da56.jpg)  
+Figure 3: Our VP Design. We resize the image to a resolution of  $L \times L$  and initialize two low-rank matrices B and A as tunable parameters. B·A serves as the visual prompt and is directly added to the resized images. This design allows shared information in rows and columns and also allows patch-specific information in different patches.
+
+various visual domains. Unlike previous works such as ILM-VP (Chen et al., 2023) and AutoVP (Tsao et al., 2024), which primarily utilize ImageNet-1K (Deng et al., 2009) pre-trained models and fine-tune on smaller downstream datasets, we extend our exploration to larger pre-training datasets, such as ImageNet-21K (Deng et al., 2009), as well as larger downstream datasets, including ImageNet-1K, to examine the scalability of existing VP methods. Furthermore, we conduct extensive empirical evaluations, focusing on the following aspects: (1) Demonstrating the superior performance and faster convergence of LoR-VP across different datasets and architectures; (2) Investigating the out-of-distribution robustness of LoR-VP; (3) Showcasing the efficiency of LoR-VP in terms of training epochs, runtime, and parameter usage, etc; (4) Performing ablation studies to evaluate the effectiveness of our VP approach under various label mapping methods, the optimal rank configuration in LoR-VP, and the contribution of different components within LoR-VP.
+
+# 5.1 IMPLEMENTATION DETAILS
+
+Datasets. For pre-training, we utilize the ImageNet-1K dataset (Deng et al., 2009), which contains 1K classes and 1.3M images, the ImageNet-21K-P dataset (Ridnik et al., 2021), comprising 11K classes and 12M images, and the ImageNet-21K dataset (Deng et al., 2009), which includes 21K classes and 14M images. We evaluate the effectiveness and efficiency of LoR-VP across four downstream datasets: ImageNet-1K, Tiny-ImageNet (Le & Yang, 2015), and CIFAR-10/100 (Krizhevsky et al., 2009). To assess the out-of-distribution robustness of LoR-VP, we conduct experiments on ImageNet-R (Hendrycks et al., 2021a), ImageNet-Sketch (Wang et al., 2019), ImageNet-A (Hendrycks et al., 2021b), and ImageNet-V2 (Recht et al., 2019). Additional details about the datasets are in Table 6.
+
+Networks. We employ six architectures for our experiments, all of which operate at a resolution of  $224 \times 224$ . (1) ResNet-18 and ResNet-50 (He et al., 2016) pre-trained on ImageNet-1K, and ViT-B/32 (Dosovitskiy et al., 2020) pre-trained on ImageNet-21K and fine-tuned on ImageNet-1K, each with a classifier head of 1000 classes; (2) ResNet-50-P and ViT-B/16-P (Dosovitskiy et al., 2020), pre-trained on ImageNet-21K-P, with classifier heads for 11,221 classes; (3) Swin-B (Liu et al., 2021), pre-trained on ImageNet-21K, with a classifier head for 21,841 classes; (4) CLIP (Radford et al., 2021), a vision-language model that uses a ViT-B/32 architecture as its vision encoder. The
+
+weights for these models are all publicly available through the official PyTorch Model Zoo or the Hugging Face Timm Library. Further details of the network architectures can be found in Table 7.
+
+Baselines. We select four representative SOTA methods as our baselines: (1)  $CLIP - VP$  (Bahng et al., 2022), which extends prompt tuning to the computer vision domain by incorporating prompt parameters directly into input images using CLIP models; (2)  $ILM - VP$  (Chen et al., 2023), which explores the impact of frequency-based label mapping (FLM) in visual prompting and introduces iterative label mapping (ILM) for improved performance; (3) AutoVP (Tsao et al., 2024), a SOTA method in visual prompting that automates the selection of VP configurations, including prompt sizes and label mapping (LM) strategies—our experiments use the optimal configuration provided by AutoVP; (4)  $LP$ , which modifies the classifier head of the pre-trained model to adapt to downstream tasks, a commonly used technique in transfer learning, serving as a baseline akin to LoR-VP without the novel VPs introduced in our approach.
+
+Training and Evaluation. The results for the baseline methods, including CLIP-VP, ILM-VP, and AutoVP, are reproduced using the same configurations as described in their respective original papers. For LoR-VP, we resize all input images to  $224 \times 224$  and use a rank of 4 in our VP design. As a result, the two sets of parameters in LoR-VP have dimensions of  $3 \times 224 \times 4$  and  $3 \times 4 \times 224$ , respectively, meaning that the total number of parameters in the visual prompts is only 5K. The optimal hyperparameters for LoR-VP are determined through grid search. All experiments are conducted on NVIDIA Quadro RTX8000 GPUs with 48GB of memory. Additional implementation details for LoR-VP are provided in Table 8.
+
+# 5.2 MAIN RESULTS
+
+Performance of ImageNet-1K and CLIP Pre-trained Models. To demonstrate the effectiveness of LoR-VP on widely used ImageNet-1K pre-trained models, we evaluate its performance across several downstream datasets using ImageNet-21K pre-trained ViT-B/32 (fine-tuned on ImageNet-1K), ImageNet-1K pre-trained ResNet-18 and ResNet-50, as well as CLIP models. As shown in Figure 4, we can observe that: ① LoR-VP consistently outperforms all baselines across all network and dataset combinations, achieving an average improvement of  $3.2\%$  and  $2.1\%$  over AutoVP and LP, respectively. ② LoR-VP converges significantly faster than the baselines, reaching optimal performance with  $5\times$  fewer training epochs than AutoVP and  $10\times$  fewer than ILM-VP.
+
+![](images/bbb47825bb69e2b03492aa12a40887122f051b5329deb443021e3a87a4c930f6.jpg)
+
+![](images/455ec58213ac3b9b74e2c543db64212fe90a2397a5893d31d227a8192bf12fdc.jpg)
+
+![](images/5809e49fcd3fcd83ab1f54fc16871ac1c4011a52318f72f04d059dfc23f6b4a0.jpg)
+
+![](images/0631d01d176a059866363bf536ece6a544db2722249d1f33bd5c760ed6a942ef.jpg)
+
+![](images/d6f21fcc135e94850ce3fe010cafe184a1159acd216ad49d92b6b158bc6c0024.jpg)  
+Figure 4: Performance of ImageNet-1K and CLIP Pre-trained Models on Downstream Datasets. Overview of the performance of LoR-VP compared to four baseline methods. The final performance of each method is indicated by  $\star$  or  $\bullet$ , and all results are averaged over three runs. LoR-VP consistently outperforms all baselines across various models and datasets.
+
+![](images/7f849c3c02052af916947c50ca3e16ce423e7d94927ca37d2f079a10f131e4e0.jpg)
+
+![](images/f933411b41c0d0372fd747177a574cb7bf2c0094c8d7f698823522e3e612f3db.jpg)
+
+![](images/b96b22f5d44a2b133450aedb692771df02ca99b1528e59cec984d0a8fe96e156.jpg)
+
+![](images/875781e0a4b9c99c173ce25a8728120042d891b9c14202f69d74cfb604a63fec.jpg)
+
+![](images/2f521eaec5dd3e9185cf2a203fb51cb732df22b116392fcf66c4618db05fd42b.jpg)
+
+![](images/e052eda939cb733fe05a008a2bd9697cbf9dee57c9a391cbe16c58ee0664aa89.jpg)
+
+![](images/2a9b7343adca50b76c3bafb9d09decbf0593b0e8b52f167619487e1af099f0ba.jpg)
+
+![](images/9ec324513633c9781c75c80800c9901a05aff4a6e4c64c65da58a19914941361.jpg)
+
+Performance of ImageNet-21K Pre-trained Models. To further evaluate the effectiveness of LoR-VP and existing visual prompting methods on larger models and datasets with a greater number of classifier classes, we conduct experiments using ImageNet-21K-P pre-trained ResNet-50-P and ViT-B/16-P, as well as ImageNet-21K pre-trained Swin-B models, tuning them on ImageNet-1K and Tiny-ImageNet. These models have significantly more classifier output features compared to ImageNet-1K pre-trained models, providing additional evidence of the effectiveness of LoR-VP and other VP methods on large-scale models and datasets. For the ImageNet-1K experiments, we focus on the strongest baselines, such as AutoVP and LP, running them for 30 epochs in line with the implementation in Liu et al. (2021), due to resource constraints. We found it challenging to run ILM-VP on our GPUs, as the ILM process is computationally expensive in terms of both training time and GPU memory. The results of the experiments are presented in Figure 5, where we observe the following: ① LoR-VP consistently achieves the best performance across large models and datasets, outperforming all baselines. ② The performance gap between LoR-VP and AutoVP increases to 5.06 on ImageNet-1K. A likely explanation is that the full mapping (FM) method used in AutoVP is less effective in this scenario, as it struggles to efficiently train a fully connected layer with 21,841 input features and 1,000 output features.
+
+![](images/e10c2210dbddb13de539b91b1de125e8c3a0b71cc3a81dc8f27a81552351280c.jpg)
+
+![](images/066e80c714ea02e896f4ac13be57c3906e23369184b50efda2ec1b9e624fd562.jpg)
+
+![](images/744a7a5942f51cef8cd8c2bd8824e2fa8cd6cdf0f5b688cb673d77dc10cc3a6a.jpg)
+
+![](images/b7309ceec86308f649599d5cd3e4cc136a6fde2c4b50899353551df916dd779e.jpg)
+
+![](images/ac0a2a590288265aa3f75dbf1eca917157851114b16642d8323a8038e9300db5.jpg)  
+AutoVP  
+Figure 5: Performance of ImageNet-21K Pre-trained Models on ImageNet-1K and TinyImageNet. Performance comparison of LoR-VP and four baseline methods. The models are pretrained on either ImageNet-21K-P or ImageNet-21K and then tuned on the respective downstream datasets. The final performance results are denoted by  $\star$  or  $\bullet$ . All results are averaged over three runs. LoR-VP consistently outperforms all baselines across different models and datasets.
+
+![](images/764f65dc58a7cc5b271c14f89d581ca5db57199b9364f78b212e73b9320f22f2.jpg)  
+ILM-VP
+
+![](images/597f09b53fbf37c81761bfec157b7b45a085ed8c26f78bac709b13e2f7efe0f0.jpg)
+
+![](images/98c8fa4a71269cf4edb744c9eb9568dd41a6bb972e53ed5ea7fb15293e226bb0.jpg)  
+P
+
+# 5.3 ROBUSTNESS OF LOR-VP
+
+To investigate the out-of-distribution robustness of LoR-VP and explore its potential for enhancing real-world applications, we conduct experiments using ImageNet-21K pre-trained Swin-B. We apply LoR-VP on ImageNet-1K and then evaluate the performance of the resulting model and visual prompts on four out-of-distribution datasets. The performance of LoR-VP and the two strongest baselines are presented in Table 1. We find that LoR-VP consistently demonstrates the best out-of-distribution robustness and generalization performance across all baselines, achieving an average improvement of 10.6 over AutoVP across the four datasets. These results further highlight the superior out-of-distribution robustness of LoR-VP, confirming its advantage over SOTA prompting methods in terms of generalization.
+
+Table 1: Out-of-Distribution Generalization Performance. Evaluation of the out-of-distribution generalization performance using the ImageNet-21K pre-trained Swin-B, with visual prompting applied on ImageNet-1K, and tested across four out-of-distribution datasets.  
+
+<table><tr><td rowspan="2">Method</td><td>Source</td><td colspan="4">Target</td></tr><tr><td>ImageNet-1K</td><td>ImageNet-R</td><td>ImageNet-Sketch</td><td>ImageNet-A</td><td>ImageNet-V2</td></tr><tr><td>AutoVP [ICLR24]</td><td>78.98</td><td>38.14</td><td>28.89</td><td>17.91</td><td>67.38</td></tr><tr><td>LP</td><td>83.52</td><td>51.48</td><td>40.07</td><td>27.54</td><td>71.86</td></tr><tr><td>LoR-VP</td><td>84.04</td><td>52.27</td><td>41.13</td><td>27.89</td><td>72.38</td></tr></table>
+
+# 5.4 EFFICIENCY OF LOR-VP
+
+As shown in Figures 4 and 5, LoR-VP achieves superior performance compared to SOTA methods with fewer training epochs. To further examine the efficiency of LoR-VP in contrast to the baselines, we assess its performance using several criteria: training epochs, training time, tunable parameters (including visual prompt parameters), GPU memory usage during training, and inference latency. Evaluations are conducted using an ImageNet-21K-P pre-trained ViT-B/16-P with visual prompting applied on Tiny-ImageNet, and an ImageNet-1K pre-trained ResNet-18 with visual prompting applied on CIFAR-10. The results are presented in Table 2, we can observe that: ① LoR-VP converges the fastest among all methods, requiring  $5 \times$  fewer epochs and  $6 \times$  less training time compared to AutoVP, and  $10 \times$  fewer epochs and  $15 \times$  less training time compared to ILM-VP. ILM-VP, in particular, converges the slowest and incurs the highest time cost, as it requires an additional epoch for every training epoch to compute the LM sequences. ② LoR-VP is highly parameter-efficient. For small models and datasets, such as ResNet-18 and CIFAR-10, LoR-VP only requires 10K parameters to achieve optimal performance, which is  $15 \times$  and  $11 \times$  fewer than ILM-VP and AutoVP, respectively. Notably, LoR-VP requires just 5K visual prompt parameters, which is  $18 \times$  and  $30 \times$  fewer than AutoVP and ILM-VP, on average. ③ GPU usage and inference speed for LoR-VP are comparable to AutoVP, whereas ILM-VP consumes the most GPU memory on larger models due to the additional computation and storage required for LM sequences. ④ LoR-VP achieves the best performance with the fewest visual prompt parameters and the shortest training time, making it an ideal choice for adapting pre-trained vision models to downstream tasks, particularly for resource-constrained environments such as mobile devices.
+
+Table 2: Training and Inference Efficiency. Comparison of the training and inference efficiency of LOR-VP, AutoVP, and ILM-VP, evaluated using ImageNet-21K-P pre-trained ViT-B/16-P on Tiny-ImageNet and ImageNet-1K pre-trained ResNet-18 on CIFAR-10.  
+
+<table><tr><td>Network</td><td>Dataset</td><td>Method</td><td>Epochs</td><td>Time</td><td># VP Params</td><td># Tunable Params</td><td>GPU Usage</td><td>Latency</td><td>Accuracy</td></tr><tr><td rowspan="3">ResNet-18</td><td rowspan="3">CIFAR10</td><td>ILM-VP[CVPR23]</td><td>200</td><td>5.76h</td><td>147K</td><td>147K</td><td>4.51GB</td><td>4.61ms</td><td>66.39</td></tr><tr><td>AutoVP[ICLR24]</td><td>100</td><td>2.61h</td><td>101K</td><td>111K</td><td>4.51GB</td><td>4.59ms</td><td>87.55</td></tr><tr><td>LoR-VP</td><td>20</td><td>0.50h</td><td>5K</td><td>10K</td><td>4.49GB</td><td>4.53ms</td><td>88.64</td></tr><tr><td rowspan="3">ViT-B/16-P</td><td rowspan="3">Tiny-ImageNet</td><td>ILM-VP[CVPR23]</td><td>200</td><td>25.55h</td><td>147K</td><td>147K</td><td>17.24GB</td><td>14.55ms</td><td>26.17</td></tr><tr><td>AutoVP[ICLR24]</td><td>100</td><td>8.08h</td><td>74K</td><td>2,318K</td><td>13.59GB</td><td>14.40ms</td><td>81.42</td></tr><tr><td>LoR-VP</td><td>20</td><td>1.32h</td><td>5K</td><td>159K</td><td>13.34GB</td><td>14.29ms</td><td>84.40</td></tr></table>
+
+# 5.5 ALBATION STUDIES
+
+How does Output Transformation Impact LoR-VP's Performance? To further explore how different output transformations affect the performance of LoR-VP, we conduct experiments by combining LoR-VP with FLM, ILM, and FM, referred to as LoR-VP w. FLM, LoR-VP w. ILM, and LoR-VP w. FM, respectively. These experiments are performed using ImageNet-21K pre-trained Swin-B, ImageNet-21K-P pre-trained ViT-B/16-P, ImageNet-21K pre-trained ViT-B/32 (fine-tuned on ImageNet-1K), and ImageNet-1K pre-trained ResNet-18 on CIFAR-100 and TinyImageNet. The results are presented in Table 3, where we observe the following:  $\bullet$  LoR-VP with LP as the output transformation achieves the overall best performance across all methods, networks, and datasets.  $\bullet$  Even when using the same output transformations as ILM-VP and AutoVP, LoR-VP consistently outperforms these methods, further demonstrating the superiority of our visual prompt design.
+
+What is the optimal rank in LoR-VP? To provide deeper insights into the optimal rank selection in LoR-VP, we conduct experiments with various configurations: LoR-VP, LoR-VP combined with ILM as the output transformation named as LoR-VPw. ILM, and LoR-VP combined with FM as the output transformation named as LoR-VPw. FM. These experiments are performed using the ImageNet-21K-P pre-trained ViT-B/16-P on Tiny-ImageNet and the ImageNet-21K pre-trained ViT-B/32 (fine-tuned on ImageNet-1K) on CIFAR-100. The results are presented in Figure 6, from which we can derive the following observations: ① For output transformations such as LP and FM, the optimal rank is 4; increasing the rank beyond 4 does not yield any further performance improvements. ② When LoR-VP is combined with ILM, the optimal rank is around 16. A plausible explanation is that ILM lacks tunable parameters, so a higher rank is needed to enhance the expressive power of the visual prompts and achieve optimal performance.
+
+Table 3: The Impact of Output Transformation. The performance comparison of utilizing FLM, ILM, and FM as the output transformation of LoR-VP and the baselines. LoR-VP achieves the overall best performance among all output transformation methods, networks, and datasets.  
+
+<table><tr><td rowspan="2">Dataset</td><td rowspan="2">Method</td><td rowspan="2">Output Transformation</td><td colspan="4">Network</td></tr><tr><td>Swin-B</td><td>ViT-B/16-P</td><td>ViT-B/32</td><td>ResNet-18</td></tr><tr><td rowspan="7">Tiny-ImageNet</td><td>ILM-VP[CVPR23]</td><td>ILM</td><td>56.28</td><td>26.17</td><td>32.58</td><td>14.13</td></tr><tr><td>AutoVP[ICLR24]</td><td>FM</td><td>84.81</td><td>81.42</td><td>82.43</td><td>59.68</td></tr><tr><td>LP</td><td>LP</td><td>86.54</td><td>82.75</td><td>83.95</td><td>65.17</td></tr><tr><td>LoR-VP w. FLM</td><td>FLM</td><td>82.15</td><td>41.76</td><td>82.89</td><td>57.45</td></tr><tr><td>LoR-VP w. ILM</td><td>ILM</td><td>84.85</td><td>43.50</td><td>84.86</td><td>62.20</td></tr><tr><td>LoR-VP w. FM</td><td>FM</td><td>85.59</td><td>83.15</td><td>86.03</td><td>65.63</td></tr><tr><td>LoR-VP</td><td>LP</td><td>88.28</td><td>84.40</td><td>85.85</td><td>68.28</td></tr><tr><td rowspan="7">CIFAR100</td><td>ILM-VP[CVPR23]</td><td>ILM</td><td>65.78</td><td>41.49</td><td>40.10</td><td>25.36</td></tr><tr><td>AutoVP[ICLR24]</td><td>FM</td><td>86.83</td><td>88.58</td><td>85.96</td><td>63.77</td></tr><tr><td>LP</td><td>LP</td><td>87.37</td><td>88.90</td><td>86.21</td><td>67.06</td></tr><tr><td>LoR-VP w. FLM</td><td>FLM</td><td>74.08</td><td>46.35</td><td>72.81</td><td>34.58</td></tr><tr><td>LoR-VP w. ILM</td><td>ILM</td><td>77.22</td><td>48.53</td><td>78.23</td><td>39.07</td></tr><tr><td>LoR-VP w. FM</td><td>FM</td><td>86.25</td><td>89.10</td><td>88.48</td><td>68.64</td></tr><tr><td>LoR-VP</td><td>LP</td><td>90.42</td><td>89.69</td><td>88.65</td><td>69.88</td></tr></table>
+
+Ablation of Components in LoR-VP. We perform ablation studies on the two key components of LoR-VP: the low-rank VP design and the linear probing output transformation. For experiments without label mapping, we apply FLM prior to training and keep this mapping sequence fixed during visual prompt training to ensure valid results. We conduct these experiments using the ImageNet-21K-P pre-trained ViT-B/16-P, ImageNet-21K pre-trained Swin-B, and ImageNet-1K pre-trained ResNet-18 on Tiny-ImageNet. The results, shown in Table 5, reveal the following: ① LoR-VP achieves the highest performance when both the low-rank VP and the output transformation are employed, demonstrating the effectiveness of these components in LoR-VP. ② Our VP design improves model performance, regardless of whether a fixed mapping sequence or linear probing is used.
+
+![](images/dfc924d0999928e75cb17de8b509a122967bbdef12f6e179c6b29af83a4e81de.jpg)  
+Figure 6: The Impact of Rank in LOR-VP.
+
+Table 5: Ablation of Components in LoRVP. The results of using ImageNet-21K-P pretrained ViT-B/16-P, ImageNet-21K pre-trained Swin-B, and ImageNet-1K pre-train ResNet18 on Tiny-ImageNet.  
+
+<table><tr><td colspan="2">Components</td><td colspan="3">Network</td></tr><tr><td>VP</td><td>Output Trans.</td><td>ViT-B/16-P</td><td>Swin-B</td><td>ResNet-18</td></tr><tr><td>X</td><td>X</td><td>36.09</td><td>80.61</td><td>40.79</td></tr><tr><td>✓</td><td>X</td><td>41.76</td><td>82.15</td><td>57.45</td></tr><tr><td>X</td><td>✓</td><td>82.75</td><td>86.54</td><td>65.17</td></tr><tr><td>✓</td><td>✓</td><td>84.40</td><td>88.28</td><td>68.28</td></tr></table>
+
+# 6 CONCLUSION
+
+Visual prompting has emerged as a powerful technique for adapting pre-trained models to specific tasks through parameter-efficient tuning. Traditional methods, however, often restrict the interaction between visual prompts and the original image to a limited number of patches, overlooking the potential benefits of shared information across different patches. Addressing these shortcomings, our study introduces a novel approach, termed Low-Rank Matrix Multiplication for Visual Prompting (LoR-VP), which facilitates both shared and patch-specific information dissemination throughout the image. Extensive experiments over seven networks and eight datasets consistently demonstrate the effectiveness and efficiency of our method.
+
+# 7 ACKNOWLEDGEMENTS
+
+This research is partially funded by research grants to Metaxas from NSF: 2310966, 2235405, 2212301, 2003874, 1951890, AFOSR 23RT0630, and NIH 2R01HL127661.
+
+# 8 REPRODUCIBILITY STATEMENT
+
+The authors have made an extensive effort to ensure the reproducibility of the results presented in the paper. First, the details of the experimental settings are provided Section 5.1 and in the Appendix A. This paper investigates eight datasets, and the details about each dataset are described in Table 6. The evaluation metrics are also clearly introduced in Section 5.1. Second, the baseline methods' implementation particulars are elucidated in Section 5.1. Simultaneously, the implementation details of our method, LoR-VP, are included in Section 5.1 and Appendix A. Third, the codes are included in the supplementary material for further reference.
+
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+
+# A IMPLEMENTATION DETAILS
+
+Table 6: Dataset Information.  
+
+<table><tr><td>Dataset</td><td>Original Resolution</td><td># Training Set Images</td><td># Test Set Images</td><td># Classes</td></tr><tr><td>ImageNet-21K (Deng et al., 2009)</td><td>Varies</td><td>14M</td><td>-</td><td>21,843</td></tr><tr><td>ImageNet-21K-P (Ridnik et al., 2021)</td><td>224 × 224</td><td>12M</td><td>0.6M</td><td>11,221</td></tr><tr><td>ImageNet-1K (Deng et al., 2009)</td><td>Varies</td><td>1.3M</td><td>50K</td><td>1,000</td></tr><tr><td>ImageNet-R (Hendrycks et al., 2021a)</td><td>Varies</td><td>-</td><td>30K</td><td>200</td></tr><tr><td>ImageNet-Sketch (Wang et al., 2019)</td><td>Varies</td><td>-</td><td>50K</td><td>1,000</td></tr><tr><td>ImageNet-A (Hendrycks et al., 2021a)</td><td>Varies</td><td>-</td><td>7.5K</td><td>1,000</td></tr><tr><td>ImageNet-V2 (Recht et al., 2019)</td><td>Varies</td><td>-</td><td>10K</td><td>1,000</td></tr><tr><td>Tiny-ImageNet (Le &amp; Yang, 2015)</td><td>64 × 64</td><td>100K</td><td>10K</td><td>200</td></tr><tr><td>CIFAR100 (Krizhevsky et al., 2009)</td><td>32 × 32</td><td>50K</td><td>10K</td><td>100</td></tr><tr><td>CIFAR10 (Krizhevsky et al., 2009)</td><td>32 × 32</td><td>50K</td><td>10K</td><td>10</td></tr></table>
+
+Table 7: Network Information.  
+
+<table><tr><td>Network</td><td>Pre-trained Dataset</td><td># Model Params</td><td>Resolution</td><td>Classifier Head Input Features</td><td>Classifier Head Output Features</td></tr><tr><td>ResNet-18 (He et al., 2016)</td><td>ImageNet-1K</td><td>12M</td><td>224 × 224</td><td>512</td><td>1,000</td></tr><tr><td>ResNet-50 (He et al., 2016)</td><td>ImageNet-1K</td><td>26M</td><td>224 × 224</td><td>2,048</td><td>1,000</td></tr><tr><td>ResNet-50-P (He et al., 2016)</td><td>ImageNet-21K-P</td><td>46M</td><td>224 × 224</td><td>2,048</td><td>11,221</td></tr><tr><td>ViT-B/16-P (Dosovitskiy et al., 2020)</td><td>ImageNet-21K-P</td><td>94M</td><td>224 × 224</td><td>768</td><td>11,221</td></tr><tr><td>ViT-B/16 (Dosovitskiy et al., 2020)</td><td>ImageNet-21K, ImageNet-1K</td><td>87M</td><td>224 × 224</td><td>768</td><td>1,000</td></tr><tr><td>ViT-B/32 (Dosovitskiy et al., 2020)</td><td>ImageNet-21K, ImageNet-1K</td><td>88M</td><td>224 × 224</td><td>768</td><td>1,000</td></tr><tr><td>Swin-B (Liu et al., 2021)</td><td>ImageNet-21K</td><td>109M</td><td>224 × 224</td><td>1,024</td><td>21,841</td></tr><tr><td>CLIP (Radford et al., 2021)</td><td>WebImageText</td><td>86M</td><td>224 × 224</td><td>512</td><td>-</td></tr></table>
+
+Table 8: Implementation Details.  
+
+<table><tr><td>Network</td><td>Pre-trained Data</td><td>Downstream Data</td><td>Resolution</td><td>Optimizer</td><td>LR</td><td>Label Mapping</td><td>LOR-VP Rank</td><td>Epochs</td><td>Batch Size</td></tr><tr><td>ResNet-18</td><td>ImageNet-1K</td><td>CIFAR100</td><td>224 × 224</td><td>SGD</td><td>0.02</td><td>Linear Probing</td><td>4</td><td>20</td><td>256</td></tr><tr><td>ResNet-50</td><td>ImageNet-1K</td><td>CIFAR100</td><td>224 × 224</td><td>SGD</td><td>0.02</td><td>Linear Probing</td><td>4</td><td>20</td><td>256</td></tr><tr><td>ViT-B/32</td><td>ImageNet-21K, ImageNet-1K</td><td>CIFAR100</td><td>224 × 224</td><td>SGD</td><td>0.02</td><td>Linear Probing</td><td>4</td><td>20</td><td>256</td></tr><tr><td>ResNet-50-P</td><td>ImageNet-21K-P</td><td>Tiny-ImageNet</td><td>224 × 224</td><td>SGD</td><td>0.02</td><td>Linear Probing</td><td>4</td><td>20</td><td>256</td></tr><tr><td>ViT-B/16-P</td><td>ImageNet-21K-P</td><td>Tiny-ImageNet</td><td>224 × 224</td><td>SGD</td><td>0.02</td><td>Linear Probing</td><td>4</td><td>20</td><td>256</td></tr><tr><td>Swin-B</td><td>ImageNet-21K</td><td>Tiny-ImageNet</td><td>224 × 224</td><td>SGD</td><td>0.02</td><td>Linear Probing</td><td>4</td><td>20</td><td>256</td></tr><tr><td>CLIP</td><td>WebImageText</td><td>Tiny-ImageNet</td><td>224 × 224</td><td>SGD</td><td>40</td><td>Linear Probing</td><td>4</td><td>20</td><td>256</td></tr><tr><td>Swin-B</td><td>ImageNet-21K</td><td>ImageNet-1K</td><td>224 × 224</td><td>SGD</td><td>0.01</td><td>Linear Probing</td><td>4</td><td>10</td><td>256</td></tr></table>
+
+# B ADDITIONAL INVESTIGATION
+
+Visual Prompting in Object Detection and Semantic Segmentation. In this paper, we primarily focus on image classification tasks, following previous works such as AutoVP and ILM-VP. To further explore the applicability of LoR-VP to object detection and semantic segmentation tasks, we conduct experiments using YOLOv4 (Bochkovskiy et al., 2020) for detection and DeepLabv3+ (Chen et al., 2018) for segmentation. Both models use ImageNet-1K pre-trained ResNet-50 as the backbone. Hyperparameters such as the number of epochs and the rank in LoR-VP are kept consistent with those used in classification tasks.
+
+For object detection, we train on the Pascal VOC 2012 and 2007 training sets and evaluate on the Pascal VOC 2007 test set, following the setup in He et al. (2020). The bounding box head is modified for output transformation, and a learning rate of 0.0001 is applied. For semantic segmentation, we train on the Pascal VOC 2012 training set and evaluate on its validation set, with the DeepLabv3+ head adapted for downstream segmentation and a learning rate of 0.01. The experimental results for detection are presented in Table 9, while the segmentation results are shown in Table 10. Our method, LoR-VP, outperforms AutoVP by nearly  $4\%$  in  $\mathrm{AP}_{50}$  on VOC 2007 detection and by  $1.1\%$  on VOC 2012 segmentation, demonstrating the effectiveness of LoR-VP for object detection and semantic segmentation tasks.
+
+Additional Investigation on Diverse Downstream Tasks. To assess the performance of LOR-VP across a broader range of classification tasks, including those involving natural and artificial
+
+Table 9: Performance (AP,  $\mathrm{AP}_{50}$ , and  $\mathrm{AP}_{75}$ ) for object detection using YOLOv4 with an ImageNet-1K pre-trained ResNet-50 backbone, evaluated on the Pascal VOC 2007 test set.  
+
+<table><tr><td>Method</td><td>AP</td><td>AP50</td><td>AP75</td></tr><tr><td>LP</td><td>42.87</td><td>75.25</td><td>47.74</td></tr><tr><td>AutoVP</td><td>41.72</td><td>73.07</td><td>44.85</td></tr><tr><td>LoR-VP</td><td>43.21</td><td>77.02</td><td>48.07</td></tr></table>
+
+Table 10: Performance (mIOU) for semantic segmentation using DeepLabv3+ with an ImageNet-1K pre-trained ResNet-50 backbone, evaluated on Pascal VOC 2012 validation set.  
+
+<table><tr><td>Method</td><td>mIOU</td></tr><tr><td>LP</td><td>67.82</td></tr><tr><td>AutoVP</td><td>67.42</td></tr><tr><td>LoR-VP</td><td>68.55</td></tr></table>
+
+objects, scenes, and textures, we conduct experiments on ten downstream datasets. These experiments use ViT-B/32 pre-trained on ImageNet-21K and fine-tuned on ImageNet-1K, following the methodologies of AutoVP and ILM-VP, to further evaluate the generalization and robustness of our approach. Although LoR-VP primarily focuses on pixel-level visual prompt designs, we extend our comparison to include VPT-DEEP, as described by Jia et al. (2022), which modifies the transformer layers. This allows for a more comprehensive evaluation against additional baselines. The experimental results, presented in Table 11, show that LoR-VP achieves superior average performance across the ten datasets compared to VPT and AutoVP. Specifically, LoR-VP improves performance by  $1.4\%$  over AutoVP and  $1.3\%$  over VPT, further demonstrating its effectiveness in diverse scenarios and against a wider range of baselines.
+
+Table 11: Comparison of accuracy between LoR-VP and four baseline methods using ViT-B/32 pre-trained on ImageNet-21K and fine-tuned on ImageNet-1K across ten datasets.  
+
+<table><tr><td>Method</td><td>Tiny-ImageNet</td><td>EuroSAT</td><td>OxfordPets</td><td>Food101</td><td>DTD</td><td>Flowers102</td><td>CIFAR10</td><td>CIFAR100</td><td>SVHN</td><td>GTSRB</td><td>Average</td></tr><tr><td>LP</td><td>83.95</td><td>95.67</td><td>91.90</td><td>82.18</td><td>69.83</td><td>97.98</td><td>96.51</td><td>86.21</td><td>83.09</td><td>83.21</td><td>87.05</td></tr><tr><td>VPT [ECCV22]</td><td>83.54</td><td>95.90</td><td>92.27</td><td>82.29</td><td>72.11</td><td>98.47</td><td>96.02</td><td>86.22</td><td>81.48</td><td>88.29</td><td>87.66</td></tr><tr><td>ILM-VP [CVPR23]</td><td>32.58</td><td>88.12</td><td>78.92</td><td>48.24</td><td>42.65</td><td>64.27</td><td>85.27</td><td>40.10</td><td>80.81</td><td>67.88</td><td>62.88</td></tr><tr><td>AutoVP [ICLR24]</td><td>82.43</td><td>96.25</td><td>92.12</td><td>82.86</td><td>70.81</td><td>98.42</td><td>95.45</td><td>85.96</td><td>85.24</td><td>86.39</td><td>87.59</td></tr><tr><td>LoR-VP</td><td>85.85</td><td>96.25</td><td>92.18</td><td>83.51</td><td>72.49</td><td>98.58</td><td>97.52</td><td>88.65</td><td>86.31</td><td>88.07</td><td>88.94</td></tr></table>
+
+Table 12: Additional comparison of LOR-VP and AutoVP using LP and FM as output transformations, respectively.  
+
+<table><tr><td rowspan="2">Dataset</td><td rowspan="2">Method</td><td rowspan="2">Output Transformation</td><td colspan="4">Network</td></tr><tr><td>Swin-B</td><td>ViT-B/16-P</td><td>ViT-B/32</td><td>ResNet-18</td></tr><tr><td rowspan="5">Tiny-ImageNet</td><td>LP</td><td>LP</td><td>86.54</td><td>82.75</td><td>83.95</td><td>65.17</td></tr><tr><td>AutoVP[ICLR24]</td><td>FM</td><td>84.81</td><td>81.42</td><td>82.43</td><td>59.68</td></tr><tr><td>AutoVP w. LP[ICLR24]</td><td>LP</td><td>86.45</td><td>82.92</td><td>83.31</td><td>65.58</td></tr><tr><td>LoR-VP w. FM</td><td>FM</td><td>85.59</td><td>83.15</td><td>86.03</td><td>65.63</td></tr><tr><td>LoR-VP</td><td>LP</td><td>88.28</td><td>84.40</td><td>85.85</td><td>68.28</td></tr><tr><td rowspan="5">CIFAR100</td><td>LP</td><td>LP</td><td>87.37</td><td>88.90</td><td>86.21</td><td>67.06</td></tr><tr><td>AutoVP[ICLR24]</td><td>FM</td><td>86.83</td><td>88.58</td><td>85.96</td><td>63.77</td></tr><tr><td>AutoVP w. LP[ICLR24]</td><td>LP</td><td>88.70</td><td>89.34</td><td>87.00</td><td>68.10</td></tr><tr><td>LoR-VP w. FM</td><td>FM</td><td>86.25</td><td>89.10</td><td>88.48</td><td>68.64</td></tr><tr><td>LoR-VP</td><td>LP</td><td>90.42</td><td>89.69</td><td>88.65</td><td>69.88</td></tr></table>
+
+Additional Investigation of Output Transformation. Table 3 presents the impact of output transformations on LoR-VP, demonstrating that LoR-VP outperforms baseline methods, such as AutoVP and ILM-VP, when using the same output transformations. These results validate the effectiveness of our visual prompt designs. To further examine the influence of different output transformations and reinforce the superiority of our approach, we conduct additional ablation studies comparing LoR-VP and AutoVP using LP and FM as output transformations. The experiments are performed on the same architectures and datasets as those in Table 3. The results, shown in Table 12, indicate that LoR-VP consistently outperforms AutoVP with LP as the output transformation
+
+across all models and datasets. Interestingly, AutoVP with LP achieves higher performance than LP alone on CIFAR100 but performs comparably on Tiny-ImageNet. This variation may stem from the scaling factors employed in AutoVP, which likely affect visual prompting performance differently across datasets. Notably, LoR-VP adopts a fixed visual prompt size of  $224 \times 224$ , simplifying its design by avoiding the need to account for scaling size, further underscoring its simplicity and adaptability.
\ No newline at end of file
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+# LORA-X: BRIDGING FOUNDATION MODELS WITH TRAINING-FREE CROSS-MODEL ADAPTATION
+
+Farzad Farhadzadeh*, Debasmit Das, Shubhankar Borse, Fatih Porikli
+
+Qualcomm AI Research†, San Diego, CA 92121, USA
+
+{ffarhadz,debadas,sborse,fporikli}@qti.QUALCOMM.COM
+
+# ABSTRACT
+
+The rising popularity of large foundation models has led to a heightened demand for parameter-efficient fine-tuning methods, such as Low-Rank Adaptation (LoRA), which offer performance comparable to full model fine-tuning while requiring only a few additional parameters tailored to the specific base model. When such base models are deprecated and replaced, all associated LoRA modules must be retrained, requiring access to either the original training data or a substantial amount of synthetic data that mirrors the original distribution. However, the original data is often inaccessible due to privacy or licensing issues, and generating synthetic data may be impractical and insufficiently representative. These factors complicate the fine-tuning process considerably. To address this challenge, we introduce a new adapter, Cross-Model Low-Rank Adaptation (LoRA-X), which enables the training-free transfer of LoRA parameters across source and target models, eliminating the need for original or synthetic training data. Our approach imposes the adapter to operate within the subspace of the source base model. This constraint is necessary because our prior knowledge of the target model is limited to its weights, and the criteria for ensuring the adapter's transferability are restricted to the target base model's weights and subspace. To facilitate the transfer of LoRA parameters of the source model to a target model, we employ the adapter only in the layers of the target model that exhibit an acceptable level of subspace similarity. Our extensive experiments demonstrate the effectiveness of LoRA-X for text-to-image generation, including Stable Diffusion v1.5 and Stable Diffusion XL.
+
+# 1 INTRODUCTION
+
+Large foundation models (LFMs) have demonstrated outstanding performance across various domains, including natural language processing (OpenAI et al., 2023; Gemini Team et al., 2023; Anthropic, 2024; AI@Meta, 2024) and computer vision (Ho et al., 2020; Rombach et al., 2022). Due to their remarkable capabilities, fine-tuning LFMs for a wide array of downstream tasks has become common practice. In the full fine-tuning approach, each new model tailored to a specific task generally retains the same number of parameters as the original model. As models increase in size and customization requirements grow, the need to store such fully fine-tuned checkpoints also rises, leading to substantial storage and memory costs.
+
+To address this challenge, Parameter-Efficient Fine-Tuning (PEFT) (Xu et al., 2023) methods, such as Low-Rank Adaptation (LoRA) (Hu et al., 2022), offer a promising solution. PEFT methods aim to reduce the number of parameters that need to be updated during fine-tuning, making the process more computationally efficient and less resource-intensive.
+
+LoRA, in particular, addresses this by representing the weight changes in the model using two low-rank matrices,  $\mathbf{A}$  and  $\mathbf{B}$ . Specifically, the weight update is expressed as  $\mathbf{W}_0 +$
+
+![](images/e200a1e79be4fe10f7979fd2bed8fd13d0849518fa6a3e6b00b846c7235cd7fe.jpg)  
+Figure 1: LoRA-X training-free transfer (a) from source SD-v1.5 to targets SD Eff-v1.0 and RV-v3.0, (b) from source SDXL to targets SSD-1B and RVXL-v3.0.
+
+$\Delta W = W_0 + BA$ , where  $W_0$  is the original weight matrix, and  $\Delta W$  is the change applied during fine-tuning. By decomposing the weight changes into these low-rank matrices, LoRA significantly reduces the number of parameters that need to be adjusted, thus making the fine-tuning process more efficient.
+
+Nevertheless, a consequential limitation of the LoRA approach is its dependency on the base model. A LoRA adapter fine-tuned for a specific task is intrinsically tied to its base model and cannot function independently of it. Moreover, it cannot be directly transferred to another base model without additional training. This dependency poses a profound challenge when the base model is deprecated and replaced by a newer version. In such cases, applications utilizing LoRA adapters from the deprecated model must transition their adapters to the newer base model. This process can be cumbersome, resource-intensive, and, crucially, infeasible if the original fine-tuning data is no longer available.
+
+This naturally leads us to an important question: Can we design a LoRA adapter that can be transferred between different base models without requiring additional training or access to the original data? A solution addressing this problem allows the longevity of LoRA adapters and eliminates the need for repeated fine-tunings as base models evolve.
+
+In this work, we introduce LoRA-X, a compact adapter that can be transferred between different versions of a base model without the need for fine-tuning with the original data. The core idea behind LoRA-X is to maintain the adapter within the column-row subspace of the base model weights. This innovative strategy ensures that the adapter remains compatible with alternative versions of the base model, simplifying the transition process and enhancing the adaptability of fine-tuned models.
+
+To the best of our knowledge, LoRA-X is the first adapter designed for transferability without additional training. Our main contributions are (See Figure 1):
+
+- Concept of LoRA-X: We introduce LoRA-X, a versatile cross-model adapter that can be seamlessly transferred across various base models without the need for additional training. This innovation simplifies the process of adapting models to new tasks or domains.  
+- Subspace Similarity and Transferrability Metrics: We propose a subspace similarity metric to identify pairs of modules between source and target base models that share a common subspace. Additionally, we compute a scalar transferability cost metric to quantify the difficulty of transferring adapters. This metric is derived from the optimal transport solution to pair-wise subspace similarities between different modules of the source and target models.
+
+- Transfer Method: We present a practical yet highly effective method to transfer LoRA-X from a source model to a target model. This approach ensures that the target model inherits the capabilities of the source model, enabling efficient knowledge transfer and improved performance.
+
+Throughout this paper, we refer to "Source" as the case where the LoRA-X adapter is trained from scratch on a specific base model (pretrained weights) using a training dataset. Conversely, "Target" denotes the case where the LoRA-X adapter, transferred from a different base model without any additional training, is applied to a new base model.
+
+# 2 RELATED WORK
+
+Parameter Efficient Finetuning (PEFT) (Xu et al., 2023) has emerged as a pivotal research area, particularly in transfer learning, where the challenge lies in adapting large pretrained models to specific tasks without extensive retraining. The literature on PEFT includes various strategies, each aiming to modify a minimal number of parameters while maintaining competitive performance. Several PEFT methods have been introduced, such as Adapter Modules (Sung et al., 2022), Prompt Tuning (Lester et al., 2021), and popular Low-Rank Adaptation techniques like LoRA (Hu et al., 2022) and VeRA (Kopiczko et al., 2023). Among these, recent methods such as SVDiff (Han et al., 2023), PiSSA (Meng et al., 2024), SVFT (Lingam et al., 2024) and LoRA-XS (Balazy et al., 2024) fine-tune the singular values of base model weight matrices. However, SVDiff and its follow-up works primarily aim to reduce the number of parameters during fine-tuning rather than the transferability of the adapter, which is our main objective. A drawback of existing PEFT techniques is their lack of transferability across base models. Our approach addresses this challenge, providing a novel solution for the first time.
+
+Knowledge Distillation (Hinton, 2015; Gou et al., 2021; Kim & Rush, 2016; Park et al., 2019; Bui Thi Mai & Lampert, 2019) is a technique where knowledge from a larger, typically more complex model (teacher) is transferred to a smaller, more efficient model (student). Variants of knowledge distillation include Self-Distillation (Zhang et al., 2019; 2021; Zhang & Sabuncu, 2020), where the same model acts as both teacher and student, and Weak-to-Strong Distillation (Bang et al., 2021; Kaplun et al., 2022; Wang et al., 2022), which can help the stronger model avoid overfitting under certain circumstances. While these approaches have shown promise in transferring knowledge between models, they still rely on a training dataset for the distillation process, making them problematic to apply in data-free scenarios.
+
+Adapter Transfer: Both (Wang et al., 2024) and (Ran et al., 2023) aim to transfer adapters across different base models. The primary difference between (Wang et al., 2024) and our method lies in their use of synthetic data generated by the source model, along with a small subset of the original dataset, for the transfer process. (Ran et al., 2023) proposed a universal mapper capable of transferring adapters from a source diffusion model to a target model. However, this mapper requires training for each target model using a dataset subset common to both the source and target models. In contrast, we develop an adapter transfer methodology using a closed-form solution, which eliminates the need for additional training. Additionally, we introduce adapters that are inherently transferable.
+
+# 3 MOTIVATION
+
+A LoRA adapter is fine-tuned for a specific task using a designated base model and dataset, making it dependent on that model. This dependency becomes problematic if the base model is updated or if a user wants to use a distilled version, like SDXL to SSD-1B. Additionally, users might lack access to the original training dataset, complicating transfers. To address this, we propose LoRA-X, an adapter that can be easily transferred between different base model versions. This approach leverages the strong correlation between layers of different base model versions (Samragh et al., 2023), especially in deeper layers, which significantly impact fine-tuned task performance (Frenkel et al., 2024).
+
+![](images/978f4e6e8b63a7db3a0cfe52c831cf26acef6547a881e740a37f4d21519eab7d.jpg)  
+Figure 2: (a) Source: only  $\Delta \Sigma_{s}$  is finetuned for a down stream task, Target: for a given  $\Delta \Sigma_{s}$  from a source, first computes  $\Delta \Sigma_{s\leftarrow t}$  and then reconstruct the weight change using its own left and right singular matrices. (b) Samples produced by diffusion target models SD Eff-v1.0 and SSD-1B, utilizing training-free transferred adapters from sources SD-v1.5 and SDXL, respectively.
+
+![](images/3d80d56d75bf0361275e53499a29fced957cd88a5dccb5c7ab328354fd5bd991.jpg)
+
+LoRA-X is designed to stay within the same subspace as the base model, focusing on amplifying or diminishing features relevant to specific tasks without introducing new feature extractors. This method enhances flexibility and adaptability, ensuring fine-tuned tasks benefit from base model updates without losing specific adaptations. Figure2 (b) shows samples generated using a diffusion base model (e.g., SD Eff-v1.0) with various training-free transferred style adapters. Unlike Transferred LoRA, the samples generated using Transferred LoRA-X—transferred from a LoRA-X adapter trained on a different base model (e.g., SD-v1.5) without additional training—successfully capture the BlueFire style.
+
+# 4 METHOD
+
+This section is organized as follows: LoRA-X: We introduce LoRA-X and define it based on the Source base model. Methods for Transferring LoRA-X: We then explain how to transfer LoRA-X from the Source to the Target model without additional training.
+
+# 4.1 LoRA-X
+
+We design LoRA-X based on the following two criteria: First, (Hu et al., 2022) demonstrated that LoRA can amplify key features for specific downstream tasks that were learned but not highlighted in the general pre-training model. Second, we only have access to the target pre-trained model. Therefore, any similarity measurement between different modules of the source and target models can only be based on their pre-trained model weights and their corresponding subspace. Building on these crucial insights, we constrain the adapter to remain within the subspace of the (source) base model. This allows for straightforward transfer to another (target) base model by projecting it into the target's subspace.
+
+To ensure that the adapter  $\Delta W \in \mathbb{R}^{m \times n}$  remains in the same subspace as the base model weight  $W_0 \in \mathbb{R}^{m \times n}$ ,  $\Delta W$  should adhere to a specific structure. This typically involves ensuring that  $\Delta W$  is aligned with the principal components or directions of  $W_0$ . One common approach is to express  $\Delta W$  in terms of the singular value decomposition (SVD) of  $W_0$ . If  $\boldsymbol{W}_0 = \boldsymbol{U} \boldsymbol{\Sigma} \boldsymbol{V}^\top$ , where  $\boldsymbol{U}$  and  $\boldsymbol{V}$  are left and right singular matrices, respectively, and  $\boldsymbol{\Sigma} = \mathrm{diag}(\boldsymbol{\sigma})$  and  $\boldsymbol{\sigma} = [\sigma_1, \dots, \sigma_n]$  is a diagonal matrix of singular values in descending order<sup>1</sup>, then  $\Delta W$  can be structured as:
+
+$$
+\Delta \boldsymbol {W} = \tilde {\boldsymbol {U}} \Delta \boldsymbol {\Sigma} \tilde {\boldsymbol {V}} ^ {\top} \tag {1}
+$$
+
+Here,  $\tilde{U} \in \mathbb{R}^{m \times r}$  and  $\tilde{V} \in \mathbb{R}^{n \times r}$  are truncated left and right singular matrices obtained by zeroing out the  $n - r$  smallest singular values from  $\pmb{U}$  and  $\pmb{V}$  respectively. The matrix
+
+$\Delta \Sigma = \mathrm{diag}([\delta \sigma_1, \dots, \delta \sigma_r]) \in \mathbb{R}^{r \times r}$  is a diagonal matrix satisfying  $\sigma_i + \delta \sigma_i \geq 0, \forall 1 \leq i \leq r$ . The parameter  $r$  represents the rank of  $\Delta W$  identifying the rank of the adapter. During fine-tuning task,  $\tilde{U}$  and  $\tilde{V}$  are frozen and only  $\Delta \Sigma$  learns the change of the singular values of the pre-trained model.
+
+By projecting  $\Delta W$  onto  $W$  subspace, we have
+
+$$
+\boldsymbol {U} \boldsymbol {U} ^ {\top} \Delta \boldsymbol {W} \boldsymbol {V} \boldsymbol {V} ^ {\top} = \boldsymbol {U} \boldsymbol {U} ^ {\top} \tilde {\boldsymbol {U}} \Delta \boldsymbol {\Sigma} \tilde {\boldsymbol {V}} ^ {\top} \boldsymbol {V} \boldsymbol {V} ^ {\top} = \tilde {\boldsymbol {U}} \Delta \boldsymbol {\Sigma} \tilde {\boldsymbol {V}} ^ {\top} = \Delta \boldsymbol {W} \tag {2}
+$$
+
+where  $UU^{\top}\Delta W\mathbf{V}\mathbf{V}^{\top}$  gives the "projection" of  $\Delta W$  onto the subspace spanned by  $\mathbf{W}$ . As shown in equation 2 by projection we get  $\Delta W$  back, indicating  $\Delta W$  was already within the  $\mathbf{W}$  subspace. Figure 2 (a) illustrates LoRA-X training on a source base model, while (b) shows the transfer of LoRA-X into a target model.
+
+Generally, the matrix  $\Delta \Sigma$  can be any arbitrary square matrix and does not need to be diagonal. This flexibility allows the adapter to capture a wide range of transformations. However, if the general pre-trained model has already learned the key features necessary for specific downstream tasks, imposing a diagonal constraint on  $\Delta \Sigma$  can be sufficient. This is because the diagonal elements can scale the learned features appropriately without needing to mix them. In cases where the pre-trained model's features are not perfectly aligned with the downstream task, the adapter must learn a linear combination of these features. This is achieved by allowing  $\Delta \Sigma$  to be a full matrix, enabling the adapter to reweight and combine the pre-trained features in a more complex manner.
+
+(Han et al., 2023) proposed a similar adapter structure called SVDiff, targeting the fine-tuning of singular values within weight matrices. Our method differs in two key ways: we apply truncated SVD, modifying only the  $r$  largest singular values, and we apply LoRA-X only to attention modules, while SVDiff applies the adapter to all modules, including convolutional ones. SVDiff aims to minimize the number of parameters adjusted during fine-tuning, which is beneficial in resource-limited scenarios.
+
+In contrast, our work emphasizes the transferability of the adapter, enhancing adaptability across different base models. By focusing on transferability, we aim to create an adapter that can be effectively reused in various LFMs, improving both efficiency and effectiveness. This distinction highlights the different priorities: SVDiff focuses on parameter efficiency, while our approach emphasizes transferability and adaptability.
+
+# 4.2 METHODS FOR TRANSFERRING LORA-X
+
+Assume a source model with the base model weight  $\pmb{W}_{s,0} = \pmb{U}_s\pmb{\Sigma}_s\pmb{V}_s^\top \in \mathbb{R}^{m\times n}$  and the LoRA-X weight  $\Delta \pmb{W}_s = \tilde{\pmb{U}}_s\Delta \pmb{\Sigma}_s\tilde{\pmb{V}}_s^\top \in \mathbb{R}^{m\times n}$ . Our goal is to transfer  $\Delta \pmb{W}_s \in \mathbb{R}^{m\times n}$  to another target model with the base weight  $\pmb{W}_{t,0} = \pmb{U}_t\pmb{\Sigma}_t\pmb{V}_t^\top \in \mathbb{R}^{m' \times n'}$  without training.
+
+# 4.2.1 SAME DIMENSION
+
+In the case,  $m = m'$  and  $n = n'$ , by projecting the source adapter weight  $\Delta \pmb{W}_s$  into the target pre-trained based weight  $\pmb{W}_t = \pmb{U}_t \pmb{\Sigma}_t \pmb{V}_t^\top$  we have
+
+$$
+\Delta \boldsymbol {W} _ {t \leftarrow s} = \boldsymbol {U} _ {t} \boldsymbol {U} _ {t} ^ {\top} \Delta \boldsymbol {W} _ {s} \boldsymbol {V} _ {t} \boldsymbol {V} _ {t} ^ {\top} = \boldsymbol {U} _ {t} \boldsymbol {U} _ {t} ^ {\top} \tilde {\boldsymbol {U}} _ {s} \Delta \boldsymbol {\Sigma} _ {s} \tilde {\boldsymbol {V}} _ {s} ^ {\top} \boldsymbol {V} _ {t} \boldsymbol {V} _ {t} ^ {\top} = \boldsymbol {U} _ {t} \Delta \boldsymbol {\Sigma} _ {t \leftarrow s} \boldsymbol {V} _ {t} ^ {\top} \tag {3}
+$$
+
+where  $\Delta \pmb{\Sigma}_{t\leftarrow s} = \pmb{U}_t^\top \tilde{\pmb{U}}_s\Delta \pmb{\Sigma}_s\tilde{\pmb{V}}_s^\top \pmb{V}_t$  rotates  $\Delta \pmb{\Sigma}_s$  in a direction that the source and target have highest subspace similarity. Consequently,  $\Delta \pmb{\Sigma}_{t\leftarrow s}$  can influence the target's modules that have pre-trained weight subspaces highly similar to those in the source model. Moreover,  $\Delta \pmb{\Sigma}_{t\leftarrow s}$  is no longer a diagonal matrix unless the source and target subspaces are perfectly aligned, i.e.,  $\pmb{U}_t\pmb{U}_t^\top \tilde{\pmb{U}}_s = \tilde{\pmb{U}}_t$  and  $\tilde{\pmb{V}}_s^\top \pmb{V}_t\pmb{V}_t^\top = \tilde{\pmb{V}}_t^\top$ .
+
+# 4.2.2 DIFFERENT DIMENSIONS
+
+If the dimensions of the source and target base model weights do not match, with either  $m \neq m'$  or  $n \neq n'$ , we cannot compute  $U_t^\top \tilde{U}_s$  or  $\tilde{V}_s^\top V_t$  in equation 3. Instead, we need to find a subspace of the same dimension that captures the highest correlation. One possible
+
+solution is to find a linear transformation that minimizes the Frobenius norm difference. Assuming  $m \neq m'$ , the linear transformation can be evaluated as  $\hat{\pmb{P}} = \arg \min_{\pmb{P}} \| \pmb{P} \pmb{U}_s - \pmb{U}_t \|_F^2$ . Consequently, we have  $\tilde{\pmb{U}}_s = \pmb{U}_t \pmb{U}_s^\top (\pmb{U}_s \pmb{U}_s^\top)^{-1} \pmb{U}_s$ . Similarly, if  $n \neq n'$ , an approximate of right singular matrix is given by  $\tilde{\pmb{V}}_s = \pmb{V}_s (\pmb{V}_s^\top \pmb{V}_s)^{-1} \pmb{V}_s^\top \pmb{V}_t$ .
+
+# 4.2.3 SUBSPACE SIMILARITY
+
+To capture subspace similarity, we use unweighted similarity,
+
+$$
+\Phi_ {l} (A, B) = \Psi (\boldsymbol {U} _ {A}, \boldsymbol {U} _ {B}) = \frac {\left\| \boldsymbol {U} _ {A} ^ {\top} \boldsymbol {U} _ {B} \right\| _ {F} ^ {2}}{n} = \frac {\sum_ {i} \sum_ {j} \left\langle \boldsymbol {u} _ {A} ^ {i} , \boldsymbol {u} _ {B} ^ {j} \right\rangle^ {2}}{n} \tag {4}
+$$
+
+to measure the subspace similarity between two column orthonormal matrices  $\mathbf{U}_A \in \mathbb{R}^{m \times n}$  and  $\mathbf{U}_B \in \mathbb{R}^{m \times n}$ , obtained by taking columns of the left singular matrices of  $A \in \mathbb{R}^{m \times n}$  and  $B \in \mathbb{R}^{m \times n}$ . Similarly, we use  $\Phi_r(A, B) = \Psi(\mathbf{V}_A, \mathbf{V}_B) = \frac{\|\mathbf{V}_A^\top \mathbf{V}_B\|_F^2}{n}$  by taking columns of  $\mathbf{V}_A \in \mathbb{R}^{n \times n}$  and  $\mathbf{V}_B \in \mathbb{R}^{n \times n}$  the right singular matrices of  $A$  and  $B$ , respectively. Refer to Appendix C for additional variants of the subspace similarity metric.
+
+# 4.2.4 TRANSFERABILITY METRIC ACROSS MODELS
+
+To determine the transferability of adapters across models, we use subspace similarity between modules of a source and target base model. We propose a transferability metric to guide training-free transfers between architectures, such as from SDXL to SD1.5. This involves developing a cost function for transferring adapters between architectures.
+
+Assume that the source model has  $S$  locations for attaching adapters, with weight matrices  $\mathbf{W}_s^i$  for  $i\in \{1,2,\ldots S\}$ . Similarly, the target model has  $T$  locations with weight matrices  $\mathbf{W}_t^j$  for  $j\in \{1,2,\dots T\}$ . The cost of transferring adapter  $i$  from the source to  $j$  in the target is computed using subspace similarity  $\Phi (\mathbf{W}_s^i,\mathbf{W}_t^j)$ . This similarity is then used to construct a cost matrix. The cost matrix is employed to compute an optimal transport map, resulting in an optimal transport cost, which we refer to as the Adapter Transferability Cost (ATC). The ATC, normalized between 0 and 1, indicates the cost of transferring adapters, with higher values representing greater difficulty. Details on obtaining the ATC are in Appendix D
+
+# 5 EXPERIMENT
+
+This section describes our experiments evaluating the effectiveness of LoRA-X, detailing the setup and presenting our evaluation. We analyze and quantify LoRA-X through text-to-image generation experiments in Section 5.1 and text-generation experiments in Appendix E.3.
+
+# 5.1 EXPERIMENTAL SETUP FOR TEXT-TO-IMAGE GENERATION
+
+To evaluate the quality of images generated by LoRA-X and its training-free transferred version, we consider two scenarios. In the first scenario, "Trained", we train the LoRA-X on a specific base model (e.g., SD Eff-v1.0) using a training dataset and generate samples with this trained LoRA-X. In the second scenario, "Transferred", we transfer a LoRA-X trained on a different base model (e.g., SD-v1.5) to the same base model used in the "Trained" scenario (SD Eff-v1.0) and generate samples with the transferred LoRA-X.
+
+Datasets: For style transfer, we evaluate LoRA-X trained from scratch on base source models and training-free transferred LoRA-X using public datasets like BlueFire, Origami Styles, and Paintings. We follow the setup described in (Borse et al., 2024). Additional details are in AppendixB.
+
+Models: For text-to-image generation tasks, we employ Stable Diffusion v1.5 (SDv1.5) (Rombach et al., 2022) and Stable Diffusion XL (SDXL) (Podell et al., 2024) as the source models. The target models include Stable Diffusion Efficient v1.0 (SD Eff-v1.0) (see Appendix A for details), Realistic Vision v3.0 (RealVis-v3.0), Segmind Stable Diffusion 1B (SSD-1B) (Gupta et al., 2024), and Realistic Vision XL v3.0 (RealVisXL-v3.0).
+
+![](images/db69c006d8cb6de26fb7638c83ea2bb5b0c89b8593f6260886fbaacee727e126.jpg)  
+Figure 3: Generated samples using LoRA-X style adapter on the SDXL and SD-v1.5 as the source base models and corresponding training-free transferred samples using SSD-1B and SD Eff-v1.0 as the target based models.
+
+Table 1: Evaluation of LoRA-X trained from scratch on base models versus training-free transferred LoRA-X from a source model into a target model. LoRA-X modifies the 320 largest singular values of the pre-trained weights. Results are averaged over 30 seeds.  
+
+<table><tr><td>Datasets</td><td>Base Model</td><td>Adapter</td><td>Training-Free</td><td>HPSv2 (↑)</td><td>LPIPS diversity (↑)</td><td>DINOv2 (↑)</td></tr><tr><td rowspan="8">BlueFire (900 images)</td><td rowspan="2">RealVis-v3.0</td><td>Trained</td><td></td><td>0.331</td><td>0.524</td><td rowspan="2">0.882</td></tr><tr><td>Transferred</td><td>✓</td><td>0.332 (+0.3%)</td><td>0.540 (+2.9%)</td></tr><tr><td rowspan="2">SD Eff-v1.0</td><td>Trained</td><td></td><td>0.296</td><td>0.534</td><td rowspan="2">0.851</td></tr><tr><td>Transferred</td><td>✓</td><td>0.307 (+3.6%)</td><td>0.538 (+0.7%)</td></tr><tr><td rowspan="2">RealVisXL-v3.0</td><td>Trained</td><td></td><td>0.319</td><td>0.484</td><td rowspan="2">0.947</td></tr><tr><td>Transferred</td><td>✓</td><td>0.319 (0.0%)</td><td>0.456 (-6.1%)</td></tr><tr><td rowspan="2">SSD-1B</td><td>Trained</td><td></td><td>0.316</td><td>0.428</td><td rowspan="2">0.969</td></tr><tr><td>Transferred</td><td>✓</td><td>0.300 (-5.3%)</td><td>0.392 (-8.4%)</td></tr><tr><td rowspan="8">Paintings (630 images)</td><td rowspan="2">RealVis-v3.0</td><td>Trained</td><td></td><td>0.319</td><td>0.502</td><td rowspan="2">0.928</td></tr><tr><td>Transferred</td><td>✓</td><td>0.329 (+3.0%)</td><td>0.441 (-11.8%)</td></tr><tr><td rowspan="2">SD Eff-v1.0</td><td>Trained</td><td></td><td>0.298</td><td>0.485</td><td rowspan="2">0.820</td></tr><tr><td>Transferred</td><td>✓</td><td>0.292 (-2.0%)</td><td>0.476 (-2.0%)</td></tr><tr><td rowspan="2">RealVisXL-v3.0</td><td>Trained</td><td></td><td>0.333</td><td>0.467</td><td rowspan="2">0.945</td></tr><tr><td>Transferred</td><td>✓</td><td>0.325 (-2.5%)</td><td>0.421 (-9.6%)</td></tr><tr><td rowspan="2">SSD-1B</td><td>Trained</td><td></td><td>0.319</td><td>0.409</td><td rowspan="2">0.961</td></tr><tr><td>Transferred</td><td>✓</td><td>0.320 (+0.3%)</td><td>0.355 (-13.2%)</td></tr></table>
+
+Metrics: To evaluate image quality in the "Trained" and "Transferred" scenarios, we report DINOv2 (Oquab et al., 2024), HPSv2.1 (Wu et al., 2023), and LPIPS (Zhang et al., 2018) diversity scores. DINOv2 measures similarity based on embedded representations, HPSv2 assesses image quality and prompt/style alignment, and LPIPS captures diversity among generated images across different seeds.
+
+# 5.2 LORA-X PERFORMANCE
+
+Table 1 compares the performance of LoRA-X in the "Trained" scenario, where it is trained on various base models using the BlueFire and Painting datasets, with its performance in the "Transferred" scenario, where it is moved from a source model with a different base model to the target model using the same base model as in the "Trained" scenario. Appendix E presents LoRA-X's performance on the Origami dataset.
+
+To enable training-free transfer from a source model to a target model, we begin by identifying the correlated modules between the source and target using Equation equation 4. For additional details, please refer to Appendix C. Subsequently, we project the source's LoRA-X onto the corresponding module in the target model using Equation equation 2. Appendix F provides a PyTorch pseudocode for LoRA-X transfer.
+
+The HPSv2 and LPIPS scores in both scenarios are very similar, indicating that the Transferred LoRA-X performs comparably to the one Trained with the datasets, demonstrating the effective transferability of LoRA-X. Additionally, the high DINOV2 scores suggest that the generated samples in both scenarios are highly correlated.
+
+Figure 3 shows samples generated by LoRA-X based on the BlueFire style dataset. The first row displays samples from the source models SDXL and SD-v1.5 using LoRA-X trained on BlueFire. The second and third rows show samples from training-free transferred LoRA-X to the target models SSD-1B, SD Eff-v1.0, RVXL-v3.0, and RV-v3.0.
+
+# 5.3 EFFECT OF SUBSPACE CONSTRAINT
+
+# 5.3.1 COMPARISON WITH LORA
+
+Table 2 demonstrates the impact of the subspace constraint imposed in the LoRA-X structure (equation 1) by comparing its transferability with LoRA (Hu et al., 2022). For this comparison, we used the BlueFire dataset and fine-tuned both LoRA and LoRA-X on SD-v1.5 as the source model, with SD Eff-v1.0 as the target. This comparison highlights how the subspace constraint affects the adapter's ability to transfer across different base models. In this experiment, we trained the LoRA  $\Delta W = BA$  of rank  $r = 32$  on SD-v1.5 as the base model. Subsequently, we transferred LoRA to SD Eff-v1.0 by projecting into its pre-trained weights subspace, i.e.,  $BA_{t\leftarrow s(\mathrm{SD - v1.5})} = U_tU_t^\top BA V_tV_t^\top$ . The evaluation demonstrates that subspace constraint is essential for maintaining quality and diversity of transferred style. We repeated the experiment for different LoRA ranks to show how LoRA's transferability drops as rank is reduced, though its total size remains much higher than LoRA-X. Appendix E.1 presents the effect of constraint based on the Origami dataset.
+
+Table 2: LoRA-X subspace constraint effect on transferability of style adapter. BlueFire dataset, SD-v1.5 as the source model and SD Eff-v1.0 as the target.  
+
+<table><tr><td>Method</td><td>Adapter</td><td>Rank</td><td>HPSv2 (↑)</td><td>LPIPS diversity (↑)</td><td>DINOv2 (↑)</td><td>Total size (MB)</td></tr><tr><td rowspan="2">LoRA-X</td><td>Trained</td><td rowspan="2">320</td><td>0.2958</td><td>0.5340</td><td rowspan="2">0.8513</td><td rowspan="2">0.16</td></tr><tr><td>Transferred</td><td>0.3073 (+3.7%)</td><td>0.5376 (+0.6%)</td></tr><tr><td rowspan="6">LoRA</td><td>Trained</td><td rowspan="2">32</td><td>0.3153</td><td>0.5049</td><td rowspan="2">0.8471</td><td rowspan="2">34.07</td></tr><tr><td>Transferred</td><td>0.2466 (-27.8%)</td><td>0.4834 (-4.4%)</td></tr><tr><td>Trained</td><td rowspan="2">16</td><td>0.2652</td><td>0.5248</td><td rowspan="2">0.8266</td><td rowspan="2">17.08</td></tr><tr><td>Transferred</td><td>0.2408 (-10.1%)</td><td>0.5224 (-0.5%)</td></tr><tr><td>Trained</td><td rowspan="2">1</td><td>0.2650</td><td>0.5312</td><td rowspan="2">0.8228</td><td rowspan="2">1.15</td></tr><tr><td>Transferred</td><td>0.2355 (-12.5%)</td><td>0.5274 (-0.7%)</td></tr></table>
+
+# 5.3.2 COMPARISON WITH DORA AND FOURA
+
+Additionally, we showed results for DoRA (Liu et al., 2024) and FouRA (Borse et al., 2024) adapters in Table 3. From the results, we see that the projection idea works well on both these type of adapters. However, the DINO score of the transfer is relatively small compared to that of LoRA-X transfer. Moreover, the percentage change of transferred and trained adapters are higher suggesting that LoRA-X transfers better.
+
+Table 3: Transferability of style adapters DoRA & FouRA. For DoRA, SDXL is the source model and SSD-1B is the target model. For FouRA, SD-v1.5 is the source model and SD Eff-v1.0 is the target model.  
+
+<table><tr><td>Method</td><td>Adapter</td><td>Rank</td><td>Dataset</td><td>HPSv2 (↑)</td><td>LPIPS diversity (↑)</td><td>DINOv2 (↑)</td></tr><tr><td rowspan="2">DoRA</td><td>Trained</td><td rowspan="2">8</td><td rowspan="2">Paintings</td><td>0.3042</td><td>0.4624</td><td rowspan="2">0.9138</td></tr><tr><td>Transferred</td><td>0.2764 (-9.1%)</td><td>0.4526 (-2.1%)</td></tr><tr><td rowspan="2">DoRA</td><td>Trained</td><td rowspan="2">8</td><td rowspan="2">Origami</td><td>0.2491</td><td>0.3408</td><td rowspan="2">0.9441</td></tr><tr><td>Transferred</td><td>0.2224 (-10.7%)</td><td>0.3073 (-9.8%)</td></tr><tr><td rowspan="2">FouRA</td><td>Trained</td><td rowspan="2">64</td><td rowspan="2">Paintings</td><td>0.3034</td><td>0.4686</td><td rowspan="2">0.9153</td></tr><tr><td>Transferred</td><td>0.2891 (-4.7%)</td><td>0.4446 (-5.1%)</td></tr></table>
+
+# 5.4 COMPARISON WITH X-ADAPTER
+
+We compare the performance of transferred LoRA-X using our training-free method based on equation 3 with X-Adapter (Ran et al., 2023), which uses plug-and-play modules trained on the target model. Table 4 shows the comparison: the "Transferred" row for LoRA-X indicates our training-free transfer from SSD-1B to SDXL, while for X-Adapter, it refers to the transfer method using X-adapter modules trained for SD-v1.5 to SDXL. The "Trained" row for both methods refers to trained LoRA-X adapter from scratch using BlueFire dataset.
+
+Table 4: Evaluation of training-free transferred LoRA-X from SSD-1B to SDXL versus LoRA-X trained on SDXL from scratch using BlueFire dataset using our training-free transfer method and training-based X-adapter. Wall clock time is measured on A100 GPU  
+
+<table><tr><td>Method</td><td>Adapter</td><td>HPSv2 (↑)</td><td>LPIPS diversity (↑)</td><td>DINOv2 (↑)</td><td>Time (↓)</td></tr><tr><td rowspan="2">LoRA-X</td><td>Trained</td><td>0.306</td><td>0.422</td><td rowspan="2">0.953</td><td rowspan="2">3.7s</td></tr><tr><td>Transferred</td><td>0.279 (-9.5%)</td><td>0.433 (+2.6%)</td></tr><tr><td rowspan="2">X-Adapter</td><td>Trained</td><td>0.306</td><td>0.422</td><td rowspan="2">0.892</td><td rowspan="2">17.1 s</td></tr><tr><td>Transferred</td><td>0.282 (-7.8%)</td><td>0.406 (-3.7%)</td></tr></table>
+
+Results show change in performance for HPSv2 & LPIPS from the trained baseline is in similar. However, our LoRA-X transfer produces higher DINO score mainly because it is transferred from a source in the similar family i.e SSD-1B. Also inference time for X-adapter is higher due to processing through base model, transferred model and the adapter.
+
+# 5.5 ABLATION STUDIES
+
+# 5.5.1 EFFECT OF SUBSPACE PROJECTION
+
+Table 5 illustrates the impact of subspace projection in LoRA-X by comparing the training-free transfer of LoRA-X from a source model using subspace projection equation 2 with the method of directly copying  $\Delta \Sigma_{s}$  from the source to the target, i.e.,  $\Delta \widehat{W}_{t\leftarrow s} = U_t\Delta \Sigma_s V_t^\top$ . This analysis focuses on the effect of the alignment of left and right singular matrices between
+
+Table 5: Evaluation of training-free transformed LoRA-X by copying singular value modifications from the source to the target versus subspace projected one.
+
+<table><tr><td>Subspace Proj.</td><td>HPSv2 (↑)</td><td>LPIPS diversity (↑)</td><td>DINOv2 (↑)</td></tr><tr><td></td><td>0.1235</td><td>0.4804</td><td>0.7046</td></tr><tr><td>✓</td><td>0.3073</td><td>0.5376</td><td>0.8513</td></tr></table>
+
+the source and target models. As shown in the table, without subspace projection, the performance of transferred LoRA-X significantly drops, indicating that subspace projection is crucial. Appendix E shows the effect of subspace projection in the SDXL family.
+
+# 5.5.2 LORA-X RANK
+
+We analyze LoRA-X performance at various ranks, indicating the number of modified singular values. Table 6 shows that trained LoRA-X (trained on SD Eff-v1.0) performance declines as rank decreases. In contrast, transferred LoRA-X (from SD-v1.5) maintains performance close to the Trained version.
+
+Table 6: LoRA-X subspace constraint effect on transferability of style adapter. BlueFire dataset, SD-v1.5 as the source model and SD Eff-v1.0 as the target.  
+
+<table><tr><td>Method</td><td>Adapter</td><td>Rank</td><td>HPSv2 (↑)</td><td>LPIPS diversity (↑)</td><td>DINOv2 (↑)</td><td>Total size (MB)</td></tr><tr><td rowspan="6">LoRA-X</td><td>Trained</td><td rowspan="2">320</td><td>0.2958</td><td>0.5340</td><td rowspan="2">0.8513</td><td rowspan="2">0.16</td></tr><tr><td>Transferred</td><td>0.3073 (+3.7%)</td><td>0.5376 (+0.6%)</td></tr><tr><td>Trained</td><td rowspan="2">160</td><td>0.2850</td><td>0.5310</td><td rowspan="2">0.8352</td><td rowspan="2">0.1</td></tr><tr><td>Transferred</td><td>0.2849 (-0.03%)</td><td>0.5263 (-0.8%)</td></tr><tr><td>Trained</td><td rowspan="2">80</td><td>0.2782</td><td>0.5294</td><td rowspan="2">0.8300</td><td rowspan="2">0.05</td></tr><tr><td>Transferred</td><td>0.2723 (-2.1%)</td><td>0.5224 (-1.3%)</td></tr></table>
+
+# 5.5.3 SMALLER SOURCE
+
+In previous sections, we examined the transferability of LoRA-X from larger or similarly sized source models to target models. Here, we demonstrate the performance of transfer
+
+from a smaller source model. Table 7 presents the performance of LoRA-X transfer from SD EFF-v1.0 to SD-v1.5. The Trained row indicates LoRA-X performance when trained on SD-v1.5 from scratch using the BlueFire dataset. The Transferred row indicates LoRA-X training-free transfer from the source model SD Eff-v1.0 to the target model SD-v1.5. See Appendix E for evaluation of LoRA-X transfer from smaller SSD-1B to larger SDXL.
+
+Table 7: Evaluation of Transferred LoRA-X from the smaller source (SD Eff-v1.0) to the larger target (SD-v1.5) versus LoRA-X Trained on SD-v1.5 using BlueFire dataset.  
+
+<table><tr><td>Adapter</td><td>HPSv2 score (↑)</td><td>LPIPS diversity (↑)</td><td>DINOv2 (↑)</td></tr><tr><td>Trained</td><td>0.2959</td><td>0.5386</td><td rowspan="2">0.8312</td></tr><tr><td>Transferred</td><td>0.2834 (-4.4%)</td><td>0.5322 (-1.2%)</td></tr></table>
+
+# 5.6 CROSS MODEL TRANSFERABILITY METRIC
+
+We measure ATC for different pairs of source and target among following models: SD-v1.5, SD Eff-v1.0, RV-v3.0, SDXL, SSD-1B and RVXL-v3.0. The cost is computed for both the left and the right subspace similarities. The results are shown in Figure 4 (a) and (b), respectively. ATC is normalized to be between 0 and 1, with higher values indicating a larger cost of transferring adapters. From the plot, we observe that for both the left and right subspace similarities, ATC among Family 1 (SD-v1.5, SD Eff-v1.0, and RV-v3.0) is less than 0.5. Similarly, ATC among Family 2 (SDXL, SSD
+
+1B, and RVXL-v3.0) is also less than 0.5. However, ATC across different families shows higher cost, suggesting difficulty in training-free adapter transferability. As expected, ATC is lower between same architectures, such as between SD-v1.5 and RV-v3.0 or SDXL and RVXL-v3.0.
+
+![](images/47a1223e20e94213027ff064270e3b19fb929bfaf721e182b1be6c351e5beb0f.jpg)  
+Figure 4: Adapter transferability cost (ATC) between source and target models using (a) left subspace similarity and (b) right subspace similarity. Lower cost implies easier transfer.
+
+# 5.7 DISCUSSION AND FUTURE WORK
+
+In this paper, we primarily focus on the training-free transferability of LoRA-X in text-to-image generation tasks. To determine transferability, we use subspace similarity between different modules of a source base model and a target base model to assess if LoRA-X is transferable from source to target. We mainly considered architectures from the same family when transferring LoRA. Training-free transfer across different architecture families is a challenge, as demonstrated by our transferability cost. Future research in this area is expected to address these challenges. While our primary focus is on transferring style LoRA-X, future research could extend this to acceleration adapters such as LCM-LoRA (Luo et al., 2023). Additionally, another promising area for future work could involve training-free transfer across different architectures of Large Language Models (LLMs).
+
+# 6 CONCLUSION
+
+The increasing reliance on LFMs has amplified the need for PEFT methods like LoRA, which offer comparable performance to full model fine-tuning with minimal additional parameters. However, the necessity to retrain LoRA modules when base models are replaced poses significant challenges, especially when the original training data is inaccessible. To address this, we proposed the LoRA-X method, a compact adapter enabling training-free transfer of LoRAs across different base models. This is achieved by maintaining the adapter within the base model's subspace and integrating it into layers with similar subspace characteristics. This innovative approach has been validated with text-to-image generation models, demonstrating its potential to streamline the adaptation process in scenarios where data privacy or availability is a concern.
+
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+
+# A EFFICIENT UNET ARCHITECTURE
+
+The SD-v1.5 UNet architecture has an attention block in the first three downsampling and the last three upsampling stages. The highest input dimension feature maps to these stages are  $64 \times 64$ , which are prevalent in the first upsampling stage and the last downsampling stage. Hence, each cross attention block contains linear layers comprising of  $4096 \times 4096$  matrices. From an on-device latency standpoint, these blocks incur majority of the computational bottleneck. Furthermore, we observe that much of text and spatial-semantics interaction is captured in low-resolution stages of UNet. These consist of  $32 \times 32$  or lesser dimensional feature vectors, along with a higher number of channels and capacity, critical for Diffusion based Generative Models. Hence, we distill our pruned UNet, "SD Efficient-v1", presented in Figure 5, without these  $4096 \times 4096$  dimensional cross-attention blocks.
+
+![](images/231eb44ff6f6e9a5d9d5162adfb55b2ae7057c68268898b956cdf950ce454e9c.jpg)  
+Figure 5: Our Efficient UNet architecture.
+
+# B DATASETS
+
+In this section, we provide more details on the style transfer datasets we use for vision adaptation experiments. We followed the licensing terms for every dataset which was curated.
+
+BlueFire (Training): The BlueFire dataset is created by collecting images from open public domain and consist of 6 concepts - car, dragon, bird, fox, man and castle. The dataset has a total of 54 images covering all the concepts.
+
+BlueFire (Validation): The Bluefire validation set consists of 30 curated text prompts, of which 9 prompts contain one of 6 categories on which the model was trained, and the remaining 21 prompts correspond to categories which the low-rank adapter has not been fine-tuned on. These contain categories such as: (football, monster, sword, chess rook, lion, tiger, dog, cat, koala, panda).
+
+For all training experiments validating on this dataset, we produce 30 images per prompt, varying the input seed. Hence, the HPS analysis is over 900 image and LPIPS-diversity analysis is over 14500 image pairs.
+
+Paintings: On similar lines, the Paintings dataset is also a collection of images from public domain (CC0 license). The dataset has a total of 90 images cover 9 concepts - fire, bird, elephants, ship, horse, flower, woman, man and tiger.
+
+Paintings (Validation): The Paintings validation set consists of 21 curated text prompts, of which 9 prompts contain one of 9 categories on which the model was trained, and the remaining 12 prompts correspond to categories which the low-rank adapter has not been fine-tuned on. These contain categories such as: (lion, tiger, dog, cat, koala, panda, and other landscapes)
+
+Origami: The Origami dataset is also a collection of origami images from public domains. The dataset has a total of 52 images covering 7 concepts - bird, boat, flower, cat, dog, fox and house.
+
+**Pokemon:** The **Pokemon** dataset is a collect of Pokemon images, initially introduced in (Liu et al., 2020). It consists of 833 images and captions, the captions have been labeled using BLIP (Li et al., 2022).
+
+# B.1 LORA-X ON SDXL FAMILY
+
+To fine-tune SDXL for a specific downstream tasks, we applied LoRA-X exclusively to the UNet attention processors and the  $\left[\text{'to_q', 'to_v', 'to_k', 'to_out'}\right]$  modules. The rank of the  $\Delta \Sigma_s$  is  $r = 320$ , i.e.,  $\Delta W_s = \tilde{U}_s \Delta \Sigma_s \tilde{V}_s^\top$ ,  $\tilde{U}_s \in \mathbb{R}^{m \times r}$ ,  $\tilde{V}_s \in \mathbb{R}^{n \times r}$ , meaning the total number of parameters we fine-tune per module is only 320, which is significantly lower than LoRA (Hu et al., 2022). For the adaptation, we use mixed precision training of FP16, a batch size of 8, gradient accumulation steps of 1, use gradient checkpointing and learning rate of 1e-3 with a constant scheduler. We use SNR Gamma  $= 5.0$  and train for 5000 iterations. For training LoRA-X on SSD-1B and RealVisXL-V3.0, as well as for all the datasets: BlueFire, Paintings and Origami, we follow the same set of hyper-parameters. The implementation is derived from the codebase<sup>2</sup>, where the hyper-parameters are described as above.
+
+# B.1.1 LORA-X TRANSFER TO SSD-1B
+
+To transfer LoRA-X trained on a source model to a target model, we begin by identifying correlated modules between the source and target models through subspace similarity across their various modules.
+
+Figure 6 (a) illustrates the subspace similarity between the common attention blocks of down-blocks in SDXL and SSD-1B. It shows that certain attention blocks in down- and up-blocks exhibit low subspace similarity. For instance, transformer block 3 (tb.3) in downblock 2 (db.2) has a similarity of less than 0.4, which is significantly lower than other blocks.
+
+To address this, we seek another transformer block within the same module that may have higher similarity. As shown in Figure 6 (b), db.2 attentions.0.tb.3 of SSD-1B exhibits higher similarity with db.2 attentions.0.tb.6 of SDXL. Therefore, we apply LoRA-X of that SDXL block on SSD-1B. We observed the similar behavior in db.2 attentions.1.tb.3 and proceeded accordingly.
+
+Finally, as shown in Figure 7, up.0 attentions.2.tb.4 to tb.7 do not exhibit this behavior, and hence we do not apply LoRA-X of SDXL on those blocks of SSD-1B. For up.0 attentions.0.tb.3 and up.0 attentions.1.tb.3, we proceeded to seek another transformer block using the process described in Figure 6 (b).
+
+After identifying correlated modules, we transfer the LoRA-X of a specific module from the source model to the target model by projecting the LoRA-X onto its pre-trained weight using equation 2.
+
+# B.1.2 LORA-X TRANSFER TO REALVISXL-v3.0
+
+To transfer LoRA-X trained on SDXL into RealVisXL-v3.0, we followed the same steps as outlined in section B.1.1. First, we identified the correlated attention blocks, and then we projected LoRA-X into the RealVis-v3.0 subspace.
+
+# B.2 LORA-X ON SD-v1.5 FAMILY
+
+To train LoRA-X on SD-v1.5 for a specific downstream tasks, we applied LoRA-X to the UNet and the text-encoder attention processors and the ['to_q', 'to_v', 'to_k', 'to_out'] modules. The rank of the  $\Delta \Sigma_{s}$  is  $r = 320$ .
+
+![](images/177182af2422bf55d63a904cd1d989c4d69eb0512ac29a283b3000a6907468fa.jpg)
+
+![](images/32a8e4135f8678c295734866ea1e14e38dc52375a5f69d29f39e0f85e0bf431c.jpg)  
+(a)  
+(b)  
+Figure 6: (a) Subspace similarity of common attention blocks of down-blocks (db) in SDXL and SSD-1B, (b) Subspace similarity between db.2 attentions.0.tb.3 of SSD-1B and db.2 attentions.0.tb.3 to db.2 attentions.0.tb.9 of SDXL.
+
+For the adaptation, we use mixed precision training of FP16, a batch size of 8, gradient accumulation steps of 1 and learning rate of 1e-4 with a cosine scheduler. We train for 5000 steps. For training LoRA-X on SD Eff-v1.0 and RealVis-V3.0, as well as for all the datasets: BlueFire, Paintings and Origami, we follow the same set of hyper-parameters.
+
+Table 8 presents the ablation study on hyperparameters, including Steps and Batch size, which differ from the default values in the kohya-ss repository<sup>3</sup> for LoRA finetuing.
+
+Table 8: Ablation on different hyper parameters on training LoRA-X using base model SDv1.5 and BlueFire dataset.  
+
+<table><tr><td>Steps</td><td>Batch size</td><td>HPSv2 (↑)</td><td>LPIPS (↑)</td></tr><tr><td>5000</td><td>4</td><td>0.284</td><td>0.528</td></tr><tr><td>2000</td><td>4</td><td>0.260</td><td>0.518</td></tr><tr><td>2000</td><td>8</td><td>0.266</td><td>0.517</td></tr><tr><td>5000</td><td>8</td><td>0.296</td><td>0.539</td></tr></table>
+
+# B.2.1 LoRA-X TRANSFER
+
+We followed the procedures outlined in section B.1.1 to transfer LoRA-X trained on SD-v1.5 into both SD Eff-v1.0 and RealVis-v3.0.
+
+To analyze subspace similarity, we followed the same as in section B.1.1. The detailed analysis has been shown in Figure 8 and Figure 9. In fact, the correlation between different subspaces are quite high i.e. greater than 0.8 suggesting direct transfer of the LoRA-X's from the source model to the corresponding target model.
+
+# C SUBSPACE SIMILARITY
+
+Weighted similarity, one can use
+
+$$
+\Phi_ {l} (A, B) = \frac {\| \boldsymbol {A} ^ {\top} \boldsymbol {B} \| _ {F} ^ {2}}{\| \boldsymbol {A} ^ {\top} \boldsymbol {A} \| _ {F} \| \boldsymbol {B} ^ {\top} \boldsymbol {B} \| _ {F}} = \frac {\sum_ {i} \sum_ {j} \sigma_ {A} ^ {i} \sigma_ {B} ^ {j} \left\langle \boldsymbol {u} _ {A} ^ {i} , \boldsymbol {u} _ {B} ^ {j} \right\rangle^ {2}}{\sqrt {\sum_ {i} \left(\sigma_ {A} ^ {i}\right) ^ {2}} \sqrt {\sum_ {i} \left(\sigma_ {B} ^ {i}\right) ^ {2}}} \tag {5}
+$$
+
+to measure the subspace similarity between two subspaces spanned by the columns of  $\mathbf{A}$  and  $\mathbf{B}$ . In this scenario, similarity is influenced by singular values, with larger singular values contributing more significantly to subspace similarity. However, because the adapter can enhance features associated with very small singular values, this similarity measure might not be particularly effective for transferring LoRA-X from a source model to a target model.
+
+![](images/4960f1d5c1016864909622aa6e1181da8b8b1eae99a5073f4fd6a40bb52af089.jpg)  
+Figure 7: Subspace similarity of common attention blocks in up-blocks (ub) of SDXL vs. SSD-1B.
+
+# D OPTIMAL TRANSPORT SOLUTION
+
+In this section, we describe the method to compute the optimal transport cost. We calculate  $\Phi(\pmb{W}_s^i, \pmb{W}_t^j)$  for all  $i \in 1,2,\ldots,S$  and  $j \in 1,2,\ldots,T$ , creating a cost matrix  $C \in \mathbb{R}^{S \times T}$ , where  $C_{ij} = 1 - \Phi(\pmb{W}_s^i, \pmb{W}_t^j)$ . Subspace similarity is invalid if: (a)  $\pmb{W}_s^i$  and  $\pmb{W}_t^j$  have different row and column counts, making it impossible to minimize the Frobenius Norm difference. (b)  $\pmb{W}_s^i$  and  $\pmb{W}_t^j$  belong to different network parts (e.g., up, down, mid block of UNet) or different attention operations (e.g., query, key, value, output). In such invalid cases, subspace similarity  $\Phi(\cdot)$  is considered 0.
+
+This allows us to frame the adapter transfer problem as an optimal transport problem. We solve for the transport map  $X \in \mathbb{R}^{S \times T}$  by minimizing:
+
+$$
+\underset {X} {\operatorname {m i n i m i z e}} \operatorname {T r} \left(C ^ {\top} X\right) \quad \text {s u c h t h a t} \quad X a = \frac {1}{S} b, \quad X ^ {\top} b = \frac {1}{T} a, \quad X \geq 0
+$$
+
+where  $a$  is a  $T \times 1$ -dimensional vector of ones and  $b$  is a  $S \times 1$ -dimensional vector of ones. The standard solution to this problem is using a simplex algorithm, which is inherently
+
+![](images/ff8e6141752ea441ab9d034165cb173a927778a8fab800f66fb5038879cb803d.jpg)  
+Figure 8: Subspace similarity of common attention blocks in SD-v1.5 vs. SD Eff-v1.
+
+![](images/e21b5784d824265feabd8034058bc8227d8610f20bef7730a19358e3a2caf1fe.jpg)  
+Figure 9: Subspace similarity of common attention blocks in SD-v1.5 and RV-v3.0.
+
+slow. Alternately, we use the network simplex approach which is a graph-theoretic version of the simplex approach. In this version, the linear program is converted into a min-cost flow problem with a bipartite directed graph:
+
+- There are two sets of nodes: source and target. The source has  $S$  nodes while the target has  $T$  nodes, corresponding to the number of source and target adapter locations.  
+- The direction of flow is from the source node to the target node, where the supply at each source node is  $\frac{1}{S}$  while demand at target node is  $\frac{1}{T}$ .  
+- There is a cost associated with flow transfer from  $i^{th}$  source node to the  $j^{th}$  target node, which is denoted as  $C_{ij}$ .
+
+Using this setup, we vectorize  $X$  into  $x$  and  $C$  into  $c$  and convert the optimization problem to that of
+
+$$
+\underset {x} {\text {m i n i m i z e}} \quad c ^ {\top} x \quad \text {s u c h t h a t} \quad I x = d, \quad x \geq 0, \tag {6}
+$$
+
+where  $I \in \mathbb{R}^{(S + T) \times ST}$  is an incidence matrix of the bipartite graph. The rows of the incidence matrix represent the nodes in the graph while the columns represent the edges in the graph.  $d \in \mathbb{R}^{(S + T) \times 1}$  is the demand vector where  $d_k = -\frac{1}{S}$  for the source nodes and  $d_k = \frac{1}{T}$  for the target nodes for  $k = \{1,2,3,\dots,S + T\}$ . The optimal solution is a basic feasible solution to the problem and can be obtained as a spanning tree of the bipartite graph. The initial solution starts with a spanning tree followed by pivoting to another spanning tree until the one with minimal cost is obtained.
+
+# E MORE RESULTS
+
+We also evaluate our proposed LoRA-X on the the Origami dataset. On this dataset, we show results of transferring from SDv1.5 to RealVis-v3.0 or SD Eff-v1.0 and from SDXL to RealVisXL-3.0 or SSD-1B in Table 9. The HPSv2 score for both the transferred LoRA-X and that trained from scratch is similar in score. In fact, transferred LoRA-X produces higher performance in most of the cases. This can be attributed to better text-image alignment of the source model from which LoRA-X is transferred. In terms of diversity, the transferred LoRA-X mostly produces poorer performance. This can be due to the fact that all modes of the dataset are not transferred during the subspace projection. However, DINO scores among the dataset generated using transferred LoRA-X and that trained from scratch is high. This suggests that the generated data using both the methods are well correlated.
+
+Table 9: Evaluation of LoRA-X trained from scratch on base source models  $(\Delta \Sigma_{s})$  versus training-free transferred LoRA-X from a source base model into a target model  $(\Delta \Sigma_{t\leftarrow s})$  in text-to-image tasks. LoRA-X modifies the 320 largest singular values of the pre-trained weights. Results are averaged over 30 seeds.  
+
+<table><tr><td>Datasets</td><td>Base Model</td><td>Adapter</td><td>Training-Free</td><td>HPSv2 (↑)</td><td>LPIPS diversity (↑)</td><td>DINOv2 (↑)</td></tr><tr><td rowspan="8">Origami (900 images)</td><td rowspan="2">RealVis-v3.0</td><td>Trained</td><td></td><td>0.3344</td><td>0.4476</td><td rowspan="2">0.9193</td></tr><tr><td>Transferred</td><td>✓</td><td>0.3294 (+1.5%)</td><td>0.4497 (+0.5%)</td></tr><tr><td rowspan="2">SD Eff-v1.0</td><td>Trained</td><td></td><td>0.2645</td><td>0.5210</td><td rowspan="2">0.8184</td></tr><tr><td>Transferred</td><td>✓</td><td>0.2703 (+2.1%)</td><td>0.4838 (-7.7%)</td></tr><tr><td rowspan="2">RealVisXL-v3.0</td><td>Trained</td><td></td><td>0.2450</td><td>0.4522</td><td rowspan="2">0.8847</td></tr><tr><td>Transferred</td><td>✓</td><td>0.2724 (+9.7%)</td><td>0.4180 (-8.1%)</td></tr><tr><td rowspan="2">SSD-1B</td><td>Trained</td><td></td><td>0.2437</td><td>0.4124</td><td rowspan="2">0.9413</td></tr><tr><td>Transferred</td><td>✓</td><td>0.2695 (+9.5%)</td><td>0.3880 (-6.3%)</td></tr></table>
+
+Furthermore, we show visual results on the Painting and Origami datasets in Figure 10 and Figure 11, Figure 12 and Figure 13, respectively, with the SDXL and SD-v1.5 families. Results are shown for the source models as well as cases of transferring LoRA-X to the target model. Visually, we do not see much difference between the generated image styles and contents of both the source or the target models.
+
+# E.1 EFFECT OF SUBSPACE CONSTRAINT
+
+Here we show the effect of subspace constraint using Origami dataset. Table 10 demonstrates the impact of the subspace constraint imposed in the LoRA-X structure (equation 1) by comparing its transferability with LoRA (Hu et al., 2022). For this comparison, we used the Oriami dataset and fine-tuned both LoRA and LoRA-X on SD-v1.5 as the source model, with SD Eff-v1.0 as the target.
+
+# E.2 ABLATION STUDIES ON SDXL FAMILY
+
+In Table 11, we show quantitative results when transferring LoRA-X from SSD-1B to SDXL i.e. from a smaller source to a larger target. The HPSv2 scores and LPIPS diversity scores are quite similar with that of SDXL LoRA-X trained form scratch. This suggests that our LoRA-X is effective even for smaller sources in the SDXL family. Furthermore, high DINO score suggests that image fidelity is quite high between the images generated from both the models.
+
+Figure 10: Generated samples using LoRA-X style adapter for painting style on the SDXL as source model and our proposed training-free transfer to SSD-1B and RVXL v3.0. Results are also shown when SSD-1B and RVXL v3.0 are trained from scratch.  
+![](images/15cb56774ca184a0290c40400de509e8a0e9f87796e6e8ba224b5ba459396443.jpg)  
+Adapter: "Painting", Prompt: 1) "astronaut floating in space" 2) "bear riding bike, traffic light" 3) "bird flying in the sky" 4) "elephant in a grassland" 5) "car on a winding road, mean headlights, thunderstorms, blue flames" 6) "horses eating grass, wooden hut".
+
+Table 10: LoRA-X subspace constraint effect on transferability of style adapter. SD-v1.5 as the source model and SD Eff-v1.0 as the target.  
+
+<table><tr><td>Dataset</td><td>Method</td><td>Adapter</td><td>Rank</td><td>HPSv2 (↑)</td><td>LPIPS diversity (↑)</td><td>DINOv2 (↑)</td><td>Total size (MB)</td></tr><tr><td rowspan="8">Origami</td><td rowspan="2">LoRA-X</td><td>Trained</td><td rowspan="2">320</td><td>0.265</td><td>0.521</td><td rowspan="2">0.819</td><td rowspan="2">0.16</td></tr><tr><td>Transferred</td><td>0.330 (+1.5%)</td><td>0.484 (-7.7%)</td></tr><tr><td rowspan="6">LoRA</td><td>Trained</td><td rowspan="2">32</td><td>0.253</td><td>0.414</td><td rowspan="2">0.812</td><td rowspan="2">34.07</td></tr><tr><td>Transferred</td><td>0.226 (-10.6%)</td><td>0.482 (+16.4%)</td></tr><tr><td>Trained</td><td rowspan="2">16</td><td>0.261</td><td>0.460</td><td rowspan="2">0.781</td><td rowspan="2">17.08</td></tr><tr><td>Transferred</td><td>0.229 (-12.2%)</td><td>0.475 (+3.2%)</td></tr><tr><td>Trained</td><td rowspan="2">1</td><td>0.255</td><td>0.480</td><td rowspan="2">0.798</td><td rowspan="2">1.15</td></tr><tr><td>Transferred</td><td>0.230 (-9.4%)</td><td>0.492 (+2.5%)</td></tr></table>
+
+Table 11: Evaluation of training-free transferred LoRA-X from the smaller source (SSD-1B) to the larger target (SDXL) versus LoRA-X trained on SDXL from scratch using BlueFire dataset.  
+
+<table><tr><td>Adapter</td><td>HPSv2 (↑)</td><td>LPIPS diversity (↑)</td><td>DINOv2 (↑)</td></tr><tr><td>Trained</td><td>0.3060</td><td>0.4216</td><td rowspan="2">0.9528</td></tr><tr><td>Transferred</td><td>0.2793 (9.5%)</td><td>0.4329 (2.6%)</td></tr></table>
+
+In Table 12, we show that copying the singular values from source model (SDXL) to target model (SSD-1B) produces slightly poorer performance compared to LoRA-X.
+
+Transferred Trained  
+Figure 11: Generated samples using LoRA-X style adapter for painting style on the SD-v1.5 as source model and our proposed training-free transfer to SD Eff-v1.0 and RV-v3.0. Results are also shown when SD Eff-v1.0 and RV-v3.0 are trained from scratch.  
+Table 12: Evaluation of training-free transformed LoRA-X by copying singular value modifications from the source (SDXL) to the target (SSD-1B) versus subspace projected one on the BlueFire Dataset.  
+![](images/522d5678cd7d7e6368ebf2dd30675efc82a0a8471bf4eb904df0b564b2ccb992.jpg)  
+Adapter: "Painting", Prompt: 1) "astronaut floating in space" 2) "bear riding bike, traffic light" 3) "bird flying in the sky" 4) "elephant in a grassland" 5) "horses eating grass, wooden hut" 6) "wild dolphins swimming".
+
+<table><tr><td>Subspace Proj.</td><td>HPSv2 (↑)</td><td>LPIPS diversity (↑)</td><td>DINOv2 (↑)</td></tr><tr><td></td><td>0.296</td><td>0.351</td><td>0.966</td></tr><tr><td>✓</td><td>0.300</td><td>0.392</td><td>0.969</td></tr></table>
+
+# E.3 EXPERIMENTAL SETUP FOR TEXT GENERATION
+
+We have implemented a LoRA-X application to fine-tune TinyLlama, a large language model Zhang et al. (2024), and successfully transferred it from TinyLlama 3T to TinyLlama 2.5T for a prompt generation task using the 'awesome chatgpt prompts' dataset. In this experiment, the rank of LoRA-X is set to  $r = 32$ . We used the default hyperparameters from the PEFT repository (Mangrulkar et al., 2022) for LoRA finetuning on the CAUSAL task over 3 epochs. Table 13 shows the results for the Trained case versus the Transferred case, indicating that Transferred LoRA-X outperforms in terms of BLEU and ROUGE metrics.
+
+To further assess the generalizability of our approach, we evaluated LoRA-X transferability on two standard text generation benchmarks used in the original LoRA paper (Hu et al., 2022): text-to-text generation on the restaurant review dataset (E2E NLG) (Novikova et al., 2017) (Table 14) and text summarization on the SamSum dataset (Gliwa et al., 2019) (Table 15). In both tasks, we observed only minor differences in BLEU and ROUGE scores between LoRA-X adaptations trained from scratch on the target model versus those trans
+
+Transferred Trained  
+Figure 12: Generated samples using LoRA-X style adapter for origami style on the SDXL as the source model and our proposed training-free transfer to SSD-1B and RVXL v3.0. Results are also shown when adapters on SSD-1B and RVXL v3.0 are trained from scratch.  
+![](images/958b2c7d73216f01bb39a555df8de954a18b854214d66820e558bf9e13f56e66.jpg)  
+Adapter: "Origami", Prompt: 1) "elephant" 2) "bird with spread wings" 3) "doberman dog" 4) "dragon" 5) "flower" 6) "truck".
+
+ferred from the source model. These results demonstrate the potential of LoRA-X for efficient knowledge transfer across different language tasks and model variants.
+
+Table 13: Evaluation of LoRA-X trained from scratch on the base model TinyLlama 2.5T versus training-free transferred LoRA-X from another base model TinyLlama 3T to the target model TinyLlama 2.5T in a text-generation task using the "awesome chatgpt prompts" dataset.  
+
+<table><tr><td>Method</td><td>Adapter</td><td>Bleu (↑)</td><td>ROUGE-1 (↑)</td><td>ROUGE-2 (↑)</td><td>ROUGE-L (↑)</td><td>ROUGE-LSum (↑)</td></tr><tr><td rowspan="2">LoRA-X</td><td>Trained</td><td>0.8612</td><td>0.9349</td><td>0.9346</td><td>0.9349</td><td>0.9349</td></tr><tr><td>Transferred</td><td>0.8819</td><td>0.9874</td><td>0.9873</td><td>0.9874</td><td>0.9874</td></tr></table>
+
+Table 14: Evaluation of LoRA-X trained from scratch on the base model TinyLlama 2.5T versus training-free transferred LoRA-X from another base model TinyLlama 3T to the target model TinyLlama 2.5T in a text-generation task using the E2E-NLG dataset.  
+
+<table><tr><td>Method</td><td>Adapter</td><td>Bleu (↑)</td><td>ROUGE-1 (↑)</td><td>ROUGE-2 (↑)</td><td>ROUGE-L (↑)</td><td>ROUGE-LSum (↑)</td></tr><tr><td rowspan="2">LoRA-X</td><td>Trained</td><td>0.6503</td><td>0.7689</td><td>0.6267</td><td>0.7533</td><td>0.7533</td></tr><tr><td>Transferred</td><td>0.6603</td><td>0.7661</td><td>0.6423</td><td>0.7624</td><td>0.7621</td></tr></table>
+
+# F IMPLEMENTATION
+
+We also share the implementation of our LoRA-X transfer technique. It takes the LoRA-X from the source model, the target model and filter blocks. The filter blocks are modules
+
+Transferred Trained  
+Figure 13: Generated samples using LoRA-X style adapter for origami style on the SD-v1.5 as the source model and our proposed training-free transfer to SD Eff-v1.0 and RV-v3.0. Results are also shown when adapters on SD Eff-v1.0 and RV-v3.0 are trained from scratch.  
+![](images/53bf023083cf5c6943cf50c03fdf64c46c43c93279d9b3240cf437e2f5bcd0d2.jpg)  
+Adapter: "Origami", Prompt: 1) "elephant" 2) "bird with spread wings" 3) "doberman dog" 4) "dragon" 5) "flower" 6) "truck".
+
+Table 15: Evaluation of LoRA-X trained from scratch on the base model TinyLlama 2.5T versus training-free transferred LoRA-X from another base model TinyLlama 3T to the target model TinyLlama 2.5T in a text-generation task using the SamSum dataset.  
+
+<table><tr><td>Method</td><td>Adapter</td><td>ROUGE-1 (↑)</td><td>ROUGE-2 (↑)</td><td>ROUGE-L (↑)</td><td>ROUGE-LSum (↑)</td></tr><tr><td rowspan="2">LoRA-X</td><td>Trained</td><td>0.3394</td><td>0.1394</td><td>0.2731</td><td>0.2731</td></tr><tr><td>Transferred</td><td>0.3568</td><td>0.1526</td><td>0.2884</td><td>0.2882</td></tr></table>
+
+where we do not apply the transfer due to low subspace similarity between the source and target model. The output is the LoRA-X for the target model.
+
+Algorithm 1 Simplistic Pytorch style pseudocode for LoRA-X transfer  
+```python
+def forward(ref_lora, tar_model_tensors, filter_blocks):  
+    ref_lora: LoRA weights of the reference/source model. For LoRA-X case, singular matrix is absorbed into down matrix tar_model_tensors: weights of the target model filter_blocks: model weights which have low cross similarity  
+    tar_lora = {}  
+for key, model_key in zip(ref_lora_keys(), tar_model_tensors.keys(): # Consider tensors not in the filter_blocks if not key.startswith(tuple(filters_blocks)): if key.endsWith("down.weight"): continue  
+    tar_model_w = tar_model_tensors[model_key]  
+    lora_up_w = ref_lora[key]  
+    lora_down_key = key.replace('/_lora.up', '/_lora.down')  
+    lora_down_w = ref_lora[lora_down_key]  
+    lora_rank = lora_down_weight.shape[0]  
+    u_model_w, s_model_w, vh_model_w = \torch.linalg.svd(tar_model_w, full_matrices=False)  
+    # to project LoRA weights on base model weight  
+    proj_lora_up_w = u_model_w @ u_model_w.T @ lora_up_w  
+    proj_lora_down_w = lora_down_w @ vh_model_w.T @ vh_model_w  
+    tar_lora[key] = proj_lora_up_w  
+    tar_lora[lora_down_key] = proj_lora_down_w  
+return tar_lora
+```
\ No newline at end of file
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+# LORANPAC: LOW-RANK RANDOM FEATURES AND PRE-TRAINED MODELS FOR BRIDGING THEORY AND PRACTICE IN CONTINUAL LEARNING
+
+Liangzu Peng
+
+University of Pennsylvania
+
+lpenn@seas.upenn.edu
+
+Juan Elenter Litwin
+
+Spotify
+
+juane@spotify.com
+
+Joshua Agterberg
+
+University of Illinois Urbana-Champaign
+
+jagt@illinois.edu
+
+Alejandro Ribeiro
+
+University of Pennsylvania
+
+aribeiro@seas.upenn.edu
+
+René Vidal
+
+University of Pennsylvania
+
+vidalr@seas.upenn.edu
+
+# ABSTRACT
+
+The goal of continual learning (CL) is to train a model that can solve multiple tasks presented sequentially. Recent CL approaches have achieved strong performance by leveraging large pre-trained models that generalize well to downstream tasks. However, such methods lack theoretical guarantees, making them prone to unexpected failures. Conversely, principled CL approaches often fail to achieve competitive performance. In this work, we aim to bridge this gap between theory and practice by designing a simple CL method that is theoretically sound and highly performant. Specifically, we lift pre-trained features into a higher dimensional space and formulate an over-parametrized minimum-norm least-squares problem. We find that the lifted features are highly ill-conditioned, potentially leading to large training errors (numerical instability) and increased generalization errors. We address these challenges by continually truncating the singular value decomposition of the lifted features. Our approach, termed LoRanPAC, is stable with respect to the choice of hyperparameters, can handle hundreds of tasks, and outperforms state-of-the-art CL methods on multiple datasets. Importantly, our method satisfies a recurrence relation throughout its continual learning process, which allows us to prove it maintains small training and test errors by appropriately truncating a fraction of SVD factors. This results in a stable continual learning method with strong empirical performance and theoretical guarantees. Code available:
+
+https://github.com/liangzu/loranpac.
+
+# 1 INTRODUCTION
+
+Continual learning (CL) requires training a model that performs well on multiple tasks presented sequentially. A primary challenge in CL is acquiring new knowledge without causing catastrophic forgetting (i.e., substantial performance degradation on previously learned tasks). However, the gap that prevails in the CL literature, as we review in Section 5 and Appendix H, is that theoretically grounded CL methods tend to be impractical (Evron et al., 2022; Peng & Risteski, 2022; Peng et al., 2023; Cai & Diakonikolas, 2024), while highly performant methods involve solving intricate, non-convex training problems, for which deriving informative theoretical guarantees is challenging (Wang et al., 2022b;c;a; Smith et al., 2023; Wang et al., 2023; Jung et al., 2023; Tang et al., 2023; Gao et al., 2024b; Roy et al., 2024; Kim et al., 2024). In this paper, we aim to bridge the gap by designing a simple CL method that is both theoretically grounded and highly performant.
+
+Towards this goal, we consider learning downstream image classification tasks continually with large pre-trained models, which are widely available nowadays. As their weights are typically frozen, they provide highly generalizable features that significantly boost performance with little computational overhead (Wang et al., 2022d;a; McDonnell et al., 2023; Zhou et al., 2024a). Crucially, pre-trained models simplify network design, as concatenating a pre-trained model with a shallow trainable network often attains competitive performance (Zhou et al., 2023; McDonnell et al., 2023).
+
+Motivated by the RanPAC method of McDonnell et al. (2023), we use a shallow trainable network that is now commonly known as the random feature model. With this network, we lift pre-trained features into a higher dimensional space and then train a linear classifier on the lifted features simply by least-squares fitting. Yet, the lifted features are double-edged: while they tend to boost performance by increasing feature separability (Telgarsky, 2022; Min et al., 2024), they are also highly ill-conditioned, making it computationally difficult to train the linear classifier. For example, the ill-conditioned features would decelerate the convergence of gradient-based methods such as Orthogonal Gradient Descent (OGD) and PCA-OGD (Farajtabar et al., 2020; Doan et al., 2021). Also due to ill-conditioning, the implementation of the Ideal Continual Learner (ICL) (Peng et al., 2023) based on incremental Singular Value Decomposition (SVD) will be numerically unstable. While RanPAC alleviates the numerical instability by using ridge regression, its performance is sensitive to the choice of the regularization parameter, which can make it ill-suited for long task sequences.
+
+We identify that the ill-conditioning and instability arise as more tasks are observed and then the smallest singular values of the lifted features plummet, while the largest singular values remain almost constant. This finding motivates our method, termed LoRanPAC, which truncates these smallest singular values prior to least-squares fitting. LoRanPAC bridges the gap between theory and practice by delivering stable and strong performance with theoretical guarantees. Concretely:
+
+- We provide a continual implementation of LoRanPAC to train an over-parameterized linear classifier with highly ill-conditioned features in a numerically stable fashion (Section 3). We show it is more stable, more scalable, and more efficient than RanPAC.  
+- We derive theoretical guarantees for LoRanPAC, proving that it has small training and test errors when a suitable fraction of SVD factors are truncated (Theorems 1 and 2, Section 4). These results stem from a non-trivial recurrence relation that allows us to capture the continual learning dynamics of LoRanPAC (Lemma 1, Appendix D).  
+- We conduct extensive experiments on multiple datasets, showing that LoRanPAC uniformly outperforms prior works and specifically RanPAC (Section 6). Thanks to our stable implementation, LoRanPAC outmatches RanPAC by a significant margin in the CIL setting with one class given at a time (Inc-1), where hundreds of tasks (classes) are sequentially presented (Table 2).
+
+# 2 TECHNICAL BACKGROUND
+
+Problem Setting. We consider classification tasks in the class-incremental learning (CIL) setting, where each incoming task contains only unseen classes. Following conventions (Yan et al., 2021; Zhou et al., 2023), we write  $\mathrm{B - q_1}$ ,  $\mathrm{Inc - q_2}$  to mean that the model is given  $q_{1}$  classes in the first task and then  $q_{2}$  classes in each of the subsequent tasks ( $q_{1} = 0$  means all tasks have  $q_{2}$  distinct classes). We use vision transformers (ViTs) of Dosovitskiy et al. (2021) as pre-trained models.
+
+Pretrained Features and Labels. Given  $m_t$  images of task  $t$ , we feed them to pre-trained ViTs, obtaining the output features  $\mathbf{X}_t \in \mathbb{R}^{d \times m_t}$ . Here,  $d$  is the feature dimension ( $d = 768$  in the ViTs used). Corresponding to  $\mathbf{X}_t$  is the label matrix  $\mathbf{Y}_t \in \mathbb{R}^{c_t \times m_t}$ . Every column of  $\mathbf{Y}_t$  is a one-hot vector, that is some standard basis vector in  $\mathbb{R}^{c_t}$ , where  $c_t$  is the total number of classes observed so far. We thus have  $c_1 \leq \dots \leq c_i \leq \dots \leq c_t$ . Let  $M_t := m_1 + \dots + m_t$ . While  $\mathbf{Y}_i \in \mathbb{R}^{c_i \times m_t}$  might have a different number of rows as  $c_i$  varies, one can pad  $c_t - c_i$  zero rows to  $\mathbf{Y}_i$  when new class information is revealed; so, with a slight abuse of notation,  $\mathbf{Y}_i$  is viewed as having  $c_t$  rows. We denote by  $\mathbf{Y}_{1:t}$  the label matrix of the first  $t$  tasks:  $\mathbf{Y}_{1:t} = [\mathbf{Y}_1, \ldots, \mathbf{Y}_t] \in \mathbb{R}^{c_t \times M_t}$ .
+
+Random ReLU Features. Let  $\mathrm{relu}:\xi \mapsto \max \{0,\xi \}$  be a ReLU layer and  $P\in \mathbb{R}^{E\times d}$  denote a random Gaussian matrix with i.i.d.  $\mathcal{N}(0,1)$  entries; here we assume  $E > d$  . These allow us to embed  $\pmb{X}_{t}$  into a higher dimensional space and get random ReLU features  $\pmb {H}_t\in \mathbb{R}^{E\times m_t}$  via
+
+$$
+\boldsymbol {H} _ {t} := \operatorname {r e l u} \left(\boldsymbol {P} \boldsymbol {X} _ {t}\right), \quad \boldsymbol {H} _ {1: t} := \left[ \boldsymbol {H} _ {1}, \dots , \boldsymbol {H} _ {t} \right] \in \mathbb {R} ^ {E \times M _ {t}}. \tag {1}
+$$
+
+Note that relu is a pointwise non-linearity, applied to  $\mathbf{P}\mathbf{X}_t$  entry-wise. The goal is to learn a linear classifier  $\mathbf{W} \in \mathbb{R}^{c_t \times E}$  continually, using features  $\mathbf{H}_t$  and labels  $\mathbf{Y}_t$  of task  $t$ .
+
+An alternative way to view these setups is through a two-layer neural network
+
+$$
+\boldsymbol {X} \mapsto \boldsymbol {W} \cdot \operatorname {r e l u} (\boldsymbol {P X}),
+$$
+
+where  $P$  are randomly generated, then fixed, and  $W$  are trainable weights. Networks of this form predate the early work of Schmidt et al. (1992); Pao & Takefuji (1992) and are later known as extreme learning machines and random feature models. In Appendix H.2 we review these lines of research. In principle, the randomness of  $P$  would have some effects on learning  $W$ , but algorithmically, this randomness is irrelevant as  $P$  is fixed after random generation. In the paper we adopt this algorithmic viewpoint, treating  $H_{1:t} = \mathrm{relu}(PX_{1:t})$  as fixed, and largely ignore the randomness of  $P$ .
+
+Table 16 of our appendix shows random ReLU features  $H_{1:t}$  boosts the performance (compared to the original pre-trained features  $X_{1:t}$ , ReLU features  $\mathrm{relu}(X_t)$ , or randomly embedded features  $P X_{1:t}$ ). Fig. 6 in the appendix furthermore shows that the performance improves as the embedding dimension  $E$  increases. These experiments justify the use of random ReLU features in high embedding dimensions (we use  $E = 10^5$  unless otherwise specified).
+
+Minimum Norm Solution and Its Instability. If  $E \gg M_t$ , there are infinitely many  $W$  satisfying  $WH_i = Y_i (\forall i)$ . Among them, a common choice is the minimum-norm solution:
+
+$$
+\min  _ {\boldsymbol {W} \in \mathbb {R} ^ {c _ {t} \times E}} \| \boldsymbol {W} \| _ {\mathrm {F}} ^ {2} \quad \text {s . t .} \quad \boldsymbol {W} \boldsymbol {H} _ {i} = \boldsymbol {Y} _ {i}, \quad i = 1, \dots , t, \tag {Min-Norm}
+$$
+
+where  $\| \cdot \|_{\mathrm{F}}$  denotes the Frobenius norm. While Min-Norm is an offline formulation that assumes the availability of all seen data  $\{(H_i, Y_i)\}_{i=1}^t$ , it can be implemented in a CL fashion, e.g., via incremental SVD (Remark 2). But such an intuitive implementation fails. Fig. 1a plots the eigenvalues of  $H_{1:t}^{\top} H_{1:t}$ , showing that  $H_{1:t}^{\top} H_{1:t}$  is highly ill-conditioned: it has just a few largest and smallest eigenvalues, respectively of order  $10^{11}$  and  $10^{-5}$ , outnumbered by the eigenvalues in between that decay more slowly. Fig. 1b plots extreme eigenvalues of  $H_{1:t}^{\top} H_{1:t}$ , revealing that the minimum eigenvalue drastically drops after a certain number of tasks. Comparing Fig. 1b, c, d, we see that the training MSE loss  $\frac{1}{M_t} \| \boldsymbol{W} \boldsymbol{H}_{1:t} - \boldsymbol{Y}_{1:t} \|_{\mathrm{F}}^2$  explodes up, and the test accuracy plummets, exactly when the smallest eigenvalues (of order  $10^{-5}$ ) emerge and begin to invade the spectrum. In summary, the incremental SVD solution to Min-Norm is unable to handle the highly ill-conditioned features  $H_{1:t}$ , resulting in numerical errors.
+
+# 3 LORANPAC: STABLE CONTINUAL LEARNING VIA LOW-RANK RANDOM FEATURES
+
+Offline Formulation. The numerical evidence collected in Fig. 1 suggests that the instability of Min-Norm relates to the emergence of very small eigenvalues that make  $\pmb{H}_{1:t} \in \mathbb{R}^{E \times M_t}$  ill-conditioned. This motivates a simple remedy, called LoRanPAC (offline formulation), which consists of truncating the smallest singular values (vectors) of  $\pmb{H}_{1:t}$  and then solving Min-Norm with its truncated version. More concretely, write the SVD of  $\pmb{H}_{1:t}$  as  $\sigma_1 \pmb{u}_1 \pmb{v}_1^\top + \dots + \sigma_{M_t} \pmb{u}_{M_t} \pmb{v}_{M_t}^\top$  with ordered singular values  $\sigma_1 \geq \dots \geq \sigma_{M_t}$ . The truncation can then be described with some integer  $k_t \in [0, M_t]$  by a function  $\tau_{k_t}$  that maps  $\pmb{H}_{1:t}$  to  $\sigma_1 \pmb{u}_1 \pmb{v}_1^\top + \dots + \sigma_{k_t} \pmb{u}_{k_t} \pmb{v}_{k_t}^\top$ , where  $k_t$  is the number of top SVD factors preserved. Since  $\tau_{k_t}(\cdot)$  preserves the shape of its input,  $\tau_{k_t}(\pmb{H}_{1:t})$  is of the same size  $E \times M_t$  as  $\pmb{H}_{1:t}$ . Applying this idea of truncation to Min-Norm means solving the following program:
+
+$$
+\overline {{\boldsymbol {W}}} _ {t} \in \underset {\boldsymbol {W} \in \mathbb {R} ^ {c _ {t} \times E}} {\operatorname {a r g m i n}} \| \boldsymbol {W} \| _ {\mathrm {F}} ^ {2} \quad \text {s . t .} \quad \boldsymbol {W} \tau_ {k _ {t}} (\boldsymbol {H} _ {1: t}) = \boldsymbol {Y} _ {1: t}. \qquad \qquad (\text {L o R a n P A C})
+$$
+
+LoRanPAC is thus named, as it draws inspiration from RanPAC (McDonnell et al., 2023) and leverages low-rank random features. It is safe to assume  $k_{t} \leq \mathrm{rank}(H_{1:t})$ , for otherwise the truncation has no effects. For simplicity one might assume  $H_{1:t}$  has full rank, that is  $\mathrm{rank}(H_{1:t}) = \min \{E, M_t\}$ .
+
+Continual Implementation. To solve LoRanPAC continually, we first write down the closed-form expression of  $\overline{\boldsymbol{W}}_t$ . Let  $\overline{\boldsymbol{U}}_{1:t}\overline{\boldsymbol{\Sigma}}_{1:t}\overline{\boldsymbol{V}}_{1:t}^{\top}$  be a compact SVD of  $\tau_{k_t}(H_{1:t})$ ; here,  $\overline{\boldsymbol{U}}_{1:t}$  is of size  $E\times k_t$  and  $\overline{\boldsymbol{\Sigma}}_{1:t}$  is invertible of size  $k_{t}\times k_{t}$ . Similarly, let  $\boldsymbol{U}_{1:t}\boldsymbol{\Sigma}_{1:t}\boldsymbol{V}_{1:t}^{\top}$  be a compact SVD of  $\boldsymbol{H}_{1:t}$ , where
+
+![](images/b33151d3ba036796f4fd0c6806240deff519d4807c5b69bbe08fbfaa4a27ba37.jpg)  
+(1a) The eigenvalues of  $\pmb{H}_{1:t}^{\top}\pmb{H}_{1:t}$  arranged in descending order (  $t$  fixed).
+
+![](images/da0b9be8bf3d8c430cf5eb70b64aa747560554698e541ef91f30c9ccb15feed6.jpg)
+
+![](images/3c490a80e1158f0302b9ac462f99735bbdb2131e0c7aa02aa91abbc6f3d77f11.jpg)
+
+![](images/a4400e437e0039376948c7eebb612caedbac706eb9713e880e7c563955a01a94.jpg)
+
+![](images/8cc17bf5d9a01dc3752a9b4229520597273ccd98ccee17fc2c089acca9279169.jpg)
+
+![](images/6e8a9304f9a9ec98d48ae63c37b4397c13ed0ef0044cf8ab2198ddd6a6783d0c.jpg)
+
+![](images/7d177758754e50b055b5a3ef9fc33922db2daa97455ecb792d35c95b62c116fb.jpg)
+
+![](images/2a151cc38650af469b3627d4ed73aebf5026f0969196bc2940a3be0d5efbe677.jpg)
+
+![](images/8632e0c430946154e59296ffd91a03ac545aad5d707c66a1210e9feed44e4c89.jpg)  
+(1b) Maximum and minimum eigenvalues of  $\pmb{H}_{1:t}^{\top}\pmb{H}_{1:t}$  as the number  $t$  of seen tasks increases.
+
+![](images/592543d30f0ae1f21031373fbac01fdda1e1836df7f44d966814c2d44eeeb3ec.jpg)
+
+![](images/d4a59c220fcf1eb08d02f415c78c85727faa64097ced2ebd2fc1ab8778c88609.jpg)
+
+![](images/b40344cbbd8b5f81c2d34283eaa720ab2d336d06a1fc5f5259871dae61125d39.jpg)
+
+![](images/640412e3fdffa624567ead907d4ab6e64df1443360f51af7703080e06f91455b.jpg)  
+(1c) Training MSE loss  $\frac{1}{M_t}\| \pmb {W}\pmb {H}_{1:t} - \pmb {Y}_{1:t}\| _F^2$  of the incremental SVD solution to Min-Norm.
+
+![](images/ee78f306f5350169b3981e78614d6a466892107290647a0f75a2f70054ed734a.jpg)  
+(1d) Final test accuracy of the incremental SVD solution to Min-Norm.
+
+![](images/e7be48a6382ab437a8f8099cc22aae198da70b8087696f3832c4f398721bd66e.jpg)  
+Figure 1: Spectrum of  $\pmb{H}_{1:t}^{\top}\pmb{H}_{1:t}$  and its impact on training losses & test accuracy  $(E = 10^{5})$ ; see also Appendix K.8. The matrix  $H_{1:t}^{\top}H_{1:t}$  is ill-conditioned (1a); training loss increases (1c) and test accuracy drops (1d), drastically, when small eigenvalues (of order  $10^{-5}$ ) invade the spectrum (1b).
+
+![](images/7bfc2a5bf6a79f030932bbcd2dbc2513f7cf2a9b76ac1ff8630e44411a79bccf.jpg)
+
+Algorithm 1: Continual Solver of LoRanPAC (detailed version in Algorithm 4, Appendix C)  
+Input (Task t): Features  $\pmb{H}_t\in \mathbb{R}^{E\times m_t}$  (1), labels  $\pmb {Y}_t\in \mathbb{R}^{c_t\times m_t}$  , truncation percentage  $\zeta \in [0,1]$  For  $t\gets 1,2,\ldots$  .  $k_{t}\gets \lceil (1 - \zeta)\min \{E,M_{t}\} \rceil ;$  //  $M_{t}\coloneqq m_{1} + \dots +m_{t}$  can be updated online  $J_{t}\leftarrow Y_{1:t}H_{1:t}^{\top};$  // online update of  $J_{t}$  detailed in Algorithm 5, Appendix C Form  $B_{t}$  as per (3);   
+ $(\widetilde{U}_{1:t},\widetilde{\Sigma}_{1:t})\gets$  Top-  $k_{t}$  SVD factors of  $B_{t}$  // Algorithm 3 if  $t = 1$  , or Algorithm 2 if  $t > 1$    
+Compute linear classifier  $\widetilde{\boldsymbol{W}}_t\coloneqq J_t\widetilde{U}_{1:t}\widetilde{\Sigma}_{1:t}^{-2}\widetilde{U}_{1:t}^{\top};$  // cf. (2) and (4)
+
+$\pmb{U}_{1:t}$  has  $\mathrm{rank}(\pmb{H}_{1:t})$  columns and contains  $\overline{\pmb{U}}_{1:t}$  as a submatrix. We can then write  $\overline{\pmb{W}}_t$  as
+
+$$
+\begin{array}{l} \overline {{\boldsymbol {W}}} _ {t} = \boldsymbol {Y} _ {1: t} \overline {{\boldsymbol {V}}} _ {1: t} \overline {{\boldsymbol {\Sigma}}} _ {1: t} ^ {- 1} \overline {{\boldsymbol {U}}} _ {1: t} ^ {\top} = \boldsymbol {Y} _ {1: t} \overline {{\boldsymbol {V}}} _ {1: t} \overline {{\boldsymbol {\Sigma}}} _ {1: t} \overline {{\boldsymbol {U}}} _ {1: t} ^ {\top} \left(\overline {{\boldsymbol {U}}} _ {1: t} \overline {{\boldsymbol {\Sigma}}} _ {1: t} ^ {- 2} \overline {{\boldsymbol {U}}} _ {1: t} ^ {\top}\right) \tag {2} \\ \stackrel {{\mathrm {(i)}}} {{=}} \mathbf {Y} _ {1: t} \mathbf {V} _ {1: t} \boldsymbol {\Sigma} _ {1: t} \mathbf {U} _ {1: t} ^ {\top} \left(\overline {{\mathbf {U}}} _ {1: t} \overline {{\boldsymbol {\Sigma}}} _ {1: t} ^ {- 2} \overline {{\mathbf {U}}} _ {1: t} ^ {\top}\right) = \mathbf {Y} _ {1: t} \mathbf {H} _ {1: t} ^ {\top} \left(\overline {{\mathbf {U}}} _ {1: t} \overline {{\boldsymbol {\Sigma}}} _ {1: t} ^ {- 2} \overline {{\mathbf {U}}} _ {1: t} ^ {\top}\right), \\ \end{array}
+$$
+
+where (i) holds as the column vectors of  $\mathbf{U}_{1:t}$  not shown in  $\overline{\mathbf{U}}_{1:t}$  are orthogonal to  $\overline{\mathbf{U}}_{1:t}$ . Given (2), it now suffices to update  $\mathbf{J}_t \coloneqq \mathbf{Y}_{1:t}\mathbf{H}_{1:t}^\top \in \mathbb{R}^{c_t\times E}$ ,  $\overline{\mathbf{U}}_{1:t} \in \mathbb{R}^{E\times k_t}$ , and  $\overline{\mathbf{\Sigma}}_{1:t} \in \mathbb{R}^{k_t\times k_t}$  in an online fashion. This procedure is described in Algorithm 1, where the following points are considered:
+
+- Since the columns of  $\mathbf{Y}_{1:t}$  are one-hot vectors, we can compute  $\mathbf{Y}_{1:t}\mathbf{H}_{1:t}^{\top}$  incrementally by matrix addition rather than (sparse) matrix multiplication.  
+- An exact update of  $\overline{U}_{1:t}$  and  $\overline{\Sigma}_{1:t}$  would require computing the SVD factors of the full data  $H_{1:t}$ . However, past data  $H_{1:t-1}$  is not available when observing task  $t$ . Thus, we consider approximating
+
+![](images/eefe5832a735c756900869fd2fcd5fb93efaeef77cf7b23aa082026ba33269dd.jpg)  
+Figure 2: Final test accuracy as the truncation percentage  $\zeta$  varies.
+
+![](images/2bbc8c096aa4af56f6f15d2eb1fd37c8b421cc2aca3770a592a2b2664b264e5b.jpg)
+
+![](images/cd5b97e08552eb5b5bbb1853fcc5624fd43248f498a5bc4045f9515ad828cbb8.jpg)
+
+![](images/9a8168b9032675bbeb46b8440f8a80f2478dd91f05ebfe1a1b8a40ae9e9ca014.jpg)
+
+$\overline{U}_{1:t}, \overline{\Sigma}_{1:t}$  by two respective matrices  $\widetilde{U}_{1:t}, \widetilde{\Sigma}_{1:t}$  of the same sizes. Specifically, as shown in Line 6, we set  $\widetilde{U}_{1:t}, \widetilde{\Sigma}_{1:t}$  to be the top  $k_t$  SVD factors of  $B_t$ , where  $B_t$  is defined as
+
+$$
+\boldsymbol {B} _ {t} := \left\{ \begin{array}{l l} \boldsymbol {H} _ {1} & \text {i f} t = 1; \\ \left[ \tilde {\boldsymbol {U}} _ {1: t - 1} \tilde {\boldsymbol {\Sigma}} _ {1: t - 1}, \boldsymbol {H} _ {t} \right] & \text {o t h e r w i s e .} \end{array} \right. \tag {3}
+$$
+
+Note that  $B_{1}$  is of size  $E \times m_{1}$ , while for  $t > 1$  we have  $B_{t}$  of size  $E \times (k_{t - 1} + m_{t})$ . The top-  $k_{t}$  singular values of  $B_{t}$  and  $H_{1:t}$  are close to each other (cf. Fig. 1a, Fig. 8, Fig. 9, and Theorem 6), which indicates the effectiveness of our continual updating strategy. Then, as shown in Line 7 and recalling (2) and  $J_{t} = Y_{1:t}H_{1:t}^{\top}$ , we construct a linear classifier via
+
+$$
+\widetilde {\boldsymbol {W}} _ {t} \leftarrow \boldsymbol {J} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}. \tag {4}
+$$
+
+- In Algorithm 1,  $\zeta$  denotes the truncation percentage. Given  $\zeta$ , we set the number  $k_{t}$  of top SVD factors preserved at task  $t$  to  $k_{t} = \lceil (1 - \zeta) \min \{E, M_{t}\} \rceil$ , where  $\lceil \cdot \rceil$  converts an input number to the closest integer no smaller than it. Fig. 2 visualizes the effect of  $\zeta$  on the final test accuracy, highlighting nearly zero accuracy with  $\zeta = 0$  and stable performance with  $\zeta \geq 5\%$ .
+
+Remark 1. At test time, given a sample, we make a forward pass and obtain its random ReLU feature  $\pmb{h}$ . The predicted class is then set according to the maximum entry of  $\widetilde{\pmb{W}}_t\pmb{h}$ .
+
+Remark 2. Algorithm 1 with  $\zeta = 0$  is the incremental SVD method that we use to solve Min-Norm. Remark 3 (Time and Space Complexity). The major cost of Algorithm 1 is to compute the top- $k_{t}$  SVD factors of the  $E \times (k_{t-1} + m_{t})$  matrix  $\pmb{B}_{t}$ , which takes  $O(E(k_{t-1} + m_{t})^{2})$  time. Furthermore, Algorithm 1 uses  $O(2Ec_{t} + Ek_{t} + k_{t}^{2})$  memory to store  $\pmb{J}_{t}$ ,  $\widetilde{\pmb{U}}_{1:t}$ , and  $\widetilde{\Sigma}_{1:t}$ , and  $\widetilde{\pmb{W}}_{t}$ , and  $O(Ek_{t-1} + Em_{t})$  memory to construct  $\pmb{B}_{t}$ , and some extra working memory to compute the top- $k_{t}$  SVD factors of  $\pmb{B}_{t}$ . We refer the reader to Appendix C for more details.
+
+# 4 PROVABLY CONTROLLED TRAINING AND TEST ERRORS
+
+In this section, we present Theorems 1 and 2, which bound the training and test error of the output (4) of our approach (Algorithm 1).
+
+Notations. Denote by  $\mu_k(\cdot)$  the  $k$ -th largest eigenvalue of a symmetric matrix. Define
+
+$$
+\gamma_ {1} := 1, \quad \gamma_ {t} := \frac {\mu_ {k _ {t}} \left(\boldsymbol {B} _ {t} \boldsymbol {B} _ {t} ^ {\top}\right)}{\max  _ {i = 1 , \dots , t - 1} \left\{\mu_ {k _ {i} + 1} \left(\boldsymbol {B} _ {i} \boldsymbol {B} _ {i} ^ {\top}\right) \right\}}, \quad \forall t > 1. \tag {5}
+$$
+
+The quantity  $\gamma_{t}$  relates to the stability-plasticity tradeoff, as it is the ratio between the minimum preserved eigenvalue  $\mu_{k_t}\left(B_tB_t^\top\right)$  at task  $t$  and the maximum eigenvalues being truncated in the past,  $\mu_{k_i + 1}\bigl (B_iB_i^\top\bigr)$ . Clearly  $\gamma_{t} > 0$ , as we truncate only non-zero eigenvalues. Furthermore, instead of determining the number of preserved SVD factors  $k_{t}$  based on the truncation percentage  $\zeta$ , we can take a threshold hyperparameter  $\delta$  and truncate eigenvalues smaller than  $\delta$ ; this  $\delta$  implicitly determines  $k_{t}$ 's, and we have  $\mu_{k_t}\bigl (B_tB_t^\top\bigr)\geq \delta >\mu_{k_i + 1}\bigl (B_iB_i^\top\bigr)$ , which implies  $\gamma_{t}\geq 1$ . Finally, as suggested by Fig. 1a,  $\gamma_{t}$  can be as large as  $10^{10}$ : If we set  $\delta = 10^{-2}$  in the case of Fig. 1a, then the maximum truncated eigenvalue is of order  $10^{-5}$  and the minimum preserved is of order  $10^5$ .
+
+Then, the accumulative error  $a_{t}$  is defined as
+
+$$
+a _ {0} := 0, \quad a _ {t} := \sum_ {i = 1} ^ {t} \mu_ {k _ {i} + 1} \left(\boldsymbol {B} _ {i} \boldsymbol {B} _ {i} ^ {\top}\right), \quad \forall t \geq 1. \tag {6}
+$$
+
+The term  $a_{t}$  reflects the information ignored by our algorithm, as  $\mu_{k_i + 1}\left(\boldsymbol {B}_i\boldsymbol {B}_i^\top\right)$  is the maximum eigenvalue truncated at task  $i$ . Note that, even when observing thousands of tasks (e.g.,  $t \approx 10^{3}$ ), if we truncate the smallest eigenvalues (of order  $10^{-5}$ ),  $a_{t}$  is in the order of  $10^{-2}$ .
+
+Model Assumption. We consider a noisy linear regression model. Specifically, we assume there is some ground-truth weight matrix  $W_{t}^{*}\in \mathbb{R}^{c_{t}\times E}$  and noise  $\mathcal{E}_{1:t}\in \mathbb{R}^{c_t\times M_t}$  satisfying
+
+$$
+\boldsymbol {Y} _ {1: t} = \boldsymbol {W} _ {t} ^ {*} \boldsymbol {H} _ {1: t} + \mathcal {E} _ {1: t}. \tag {7}
+$$
+
+The quantities  $W_{t}^{*}$  and  $\mathcal{E}_{1:t}$  are colored to reflect the fact that they are unknown and not computable. The model in (7) is related to probabilistic principal component analysis (PPCA); cf. Tipping & Bishop (1999) and Chapter 2.2 of Vidal et al. (2016). The two main differences with PPCA are that we make no probabilistic assumptions on  $H_{1:t}$  or  $\mathcal{E}_{1:t}$  (except in Appendix F); and we consider the over-parameterized case with large  $E$ , while PPCA assumes  $W_{t}^{*}$  is a tall matrix (i.e.,  $E < c_{t}$ ).
+
+Bound The Training MSE. In the over-parametrized regime  $E \gg M_t$ , a solution to Min-Norm should, in principle, perfectly fit the data and achieve zero training MSE. However, solving Min-Norm is numerically unstable and empirically entails huge losses (Fig. 4). As a remedy, our approach truncates the data spectrum continually, trading off between perfectly fitting training data and increasing numerical stability. The following theorem, whose proof can be found in Appendix E, connects the eigenvalue ratio  $\gamma_t$  and the accumulative error  $a_t$  with our method's training loss, showing that the training MSE is provably under control:
+
+Theorem 1. Let  $B_{t}, \gamma_{t}, a_{t}$  be defined as in (3), (5), and (6) respectively. If  $\mathbf{Y}_{1:t} = \mathbf{W}_t^* \mathbf{H}_{1:t} + \mathcal{E}_{1:t}$  (7), then the output  $\widetilde{\mathbf{W}}_t = \mathbf{Y}_{1:t} \mathbf{H}_{1:t}^\top \widetilde{\mathbf{U}}_{1:t} \widetilde{\boldsymbol{\Sigma}}_{1:t}^{-2} \widetilde{\mathbf{U}}_{1:t}^\top$  of our method (4) satisfies
+
+$$
+\begin{array}{l} \frac {1}{M _ {t}} \left\| \widetilde {\boldsymbol {W}} _ {t} \boldsymbol {H} _ {1: t} - \boldsymbol {Y} _ {1: t} \right\| _ {\mathrm {F}} ^ {2} \leq 4 \cdot \left\| \boldsymbol {W} _ {t} ^ {*} \right\| _ {\mathrm {F}} ^ {2} \left(\frac {a _ {t}}{M _ {t}} + \frac {a _ {t - 1} (t - 1)}{\gamma_ {t} M _ {t}} + \frac {a _ {t - 1} (t - 1) ^ {2}}{\gamma_ {t} ^ {2} M _ {t}}\right) \tag {8} \\ + 2 \cdot \| \mathcal {E} _ {1: t} \| ^ {2} \left(\frac {(M _ {t} - k _ {t})}{M _ {t}} + \frac {(t - 1) \min \left\{M _ {t - 1} - k _ {t - 1} , (t - 1) k _ {t} \right\}}{\gamma_ {t} ^ {2} M _ {t}}\right). \\ \end{array}
+$$
+
+In (8),  $\| \cdot \|$  is an overloaded notation, denoting the spectrum norm of a matrix and also the Euclidean norm of a vector. One of the main quantities governing the bound in Theorem 1 is  $a_{t} / M_{t}$ , which reflects the truncation process for the current task. When truncating the smallest eigenvalues (of order  $10^{-5}$ ) and observing hundreds of tasks,  $a_{t}$  is in the order of  $10^{-3}$ , which makes  $a_{t} / M_{t}$  insignificant. Then, the terms  $(t - 1) / \gamma_{t}$  and  $a_{t - 1} / M_{t}$  capture the continual past truncations and are equal to zero for  $t = 1$ . Similarly to  $a_{t}$ , when truncating only the smallest eigenvalues, we have  $a_{t - 1} \approx 10^{-5}(t - 1)$  and  $(t - 1) / \gamma_{t} \approx 10^{-10}(t - 1)$ . Hence, all terms involving  $(t - 1) / \gamma_{t}$  and  $a_{t - 1} / M_{t}$  are under control for hundreds- even thousands- of tasks. Finally, although the ground-truth  $W_{t}^{*}$  and noise  $\mathcal{E}_{1:t}$  are unknown, we empirically verify that the minimum-norm solution to LoRanPAC achieves high accuracy (Section 6). This suggests the linear model assumption is adequate, and that  $\| \mathcal{E}_{1:t} \|^{2}$  and  $\| W_{t}^{*} \|_{\mathrm{F}}^{2}$  are reasonably small. In summary, the upper bound (8) shown in Theorem 1 behaves well and is quite small if we truncate the eigenvalues suitably (which makes  $\gamma_{t}$  large and  $a_{t}$  small).
+
+Bound The Test MSE. Consider a test sample  $(\pmb{h},\pmb{y})$  satisfying  $\pmb{y} = \pmb{W}_t^*\pmb{h} + \pmb{\epsilon}$  for some noise vector  $\pmb{\epsilon}$ . To derive a bound on the test MSE, we assume that  $\pmb{h}$  is randomly sampled from some distribution with a finite second-order moment ( $\Lambda := \mathbb{E}[hh^\top] < \infty$ ), and that  $\pmb{\epsilon}$  is random, independent of  $\pmb{h}$ . Given the output  $\widetilde{\pmb{W}}_t$  of our method (4), we bound its test error  $\mathbb{E}_{h,\epsilon}\left[\|\widetilde{\pmb{W}}_t\pmb{h} - \pmb{y}\|^2\right]$  over the randomness of  $\pmb{h},\pmb{\epsilon}$  as follows:
+
+Theorem 2. Let  $B_t, \gamma_t, a_t$  be defined as in (3), (5), and (6) respectively. Assume  $\mathbf{Y}_{1:t} = \mathbf{W}_t^* \mathbf{H}_{1:t} + \mathcal{E}_{1:t}(7)$  and  $\mathbf{y} = \mathbf{W}_t^* \mathbf{h} + \epsilon$  with  $\Lambda := \mathbb{E}[h \mathbf{h}^\top]$ . The output  $\widetilde{\mathbf{W}}_t$  of Algorithm 1 satisfies
+
+$$
+\mathbb {E} _ {\boldsymbol {h}, \epsilon} \left\| \widetilde {\boldsymbol {W}} _ {t} \boldsymbol {h} - \boldsymbol {y} \right\| ^ {2} \leq 4 \cdot \left\| \boldsymbol {W} _ {t} ^ {*} \right\| _ {\mathrm {F}} ^ {2} \cdot \mathbb {B} _ {t} + 4 \cdot \left\| \mathcal {E} _ {1: t} \right\| ^ {2} \cdot \mathbb {V} _ {t} + 2 \cdot \mathbb {E} _ {\epsilon} \left[ \| \epsilon \| ^ {2} \right], \tag {9}
+$$
+
+where  $\mathbb{B}_t$  and  $\mathbb{V}_t$  are defined as follows:
+
+$$
+\begin{array}{l} \mathbb {B} _ {t} = \left\| \boldsymbol {\Lambda} - \frac {1}{M _ {t}} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \right\| \left(1 + \frac {(t - 1) ^ {2}}{\gamma_ {t} ^ {2}}\right) + \left(\frac {a _ {t}}{M _ {t}} + \frac {a _ {t - 1} (t - 1)}{\gamma_ {t} M _ {t}} + \frac {a _ {t - 1} (t - 1) ^ {2}}{\gamma_ {t} ^ {2} M _ {t}}\right) \\ \mathbb {V} _ {t} = \left\| \boldsymbol {\Lambda} - \frac {1}{M _ {t}} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \right\| \cdot \frac {\left(\frac {1}{\gamma_ {t}} \min  \left\{M _ {t - 1} - k _ {t - 1} , (t - 1) k _ {t} \right\} + k _ {t}\right)}{\mu_ {k _ {t}} \left(\boldsymbol {B} _ {t} \boldsymbol {B} _ {t} ^ {\top}\right)} \tag {10} \\ + \frac {k _ {t}}{M _ {t}} + \left(\frac {t - 1}{\gamma_ {t} ^ {2} M _ {t}} + \frac {2}{\gamma_ {t} M _ {t}}\right) \cdot \min \left\{M _ {t - 1} - k _ {t - 1}, (t - 1) k _ {t} \right\}. \\ \end{array}
+$$
+
+There are two major terms in  $\mathbb{B}_t$  (10). The term in the right-most large parenthesis also appears in the error bound of Theorem 1; and reflects the fact that training losses impact test errors. Then, the term  $\left\| \Lambda - \frac{1}{M_t} H_{1:t} H_{1:t}^\top \right\|$  is commonly seen in covariance estimation (Wainwright, 2019), where  $h$  and the columns of  $H_{1:t}$  are assumed to be independent i.i.d. Gaussian vectors. In this case, if  $\Lambda$  furthermore satisfies some boundedness condition, we can show  $\left\| \Lambda - \frac{1}{M_t} H_{1:t} H_{1:t}^\top \right\|$  converges to 0 as  $M_t \to \infty$ ; cf. Theorem 9 of Koltchinskii & Lounici (2017). On the other hand, the Gaussian assumption is sufficient but not necessary, and a similar conclusion is reached if we take a much weaker assumption called hypercontractivity (Jirak et al., 2024). Note that  $\mathbb{V}_t$  is independent of noise, so the rest of the terms in (9), which are weighted by noise magnitudes  $\| \mathcal{E}_{1:t} \|^2$ ,  $\mathbb{E}_{\epsilon}[\| \epsilon \|^2]$ , are negligible if the noise is sufficiently small.
+
+# 5 RELATED WORK
+
+We now discuss related works that are the most relevant to our method and theory. A more extensive review of the literature and context is in Appendix H.
+
+RanPAC. The RanPAC method of McDonnell et al. (2023) motivates our use of random ReLU features  $H_{1}, \ldots, H_{t}$ . It amounts to solving the ridge regression problem (with some  $\lambda > 0$ )
+
+$$
+\min  _ {\boldsymbol {W} \in \mathbb {R} ^ {c _ {t} \times E}} \lambda \cdot \| \boldsymbol {W} \| _ {\mathrm {F}} ^ {2} + \| \boldsymbol {W} \boldsymbol {H} _ {1: t} - \boldsymbol {Y} _ {1: t} \| _ {\mathrm {F}} ^ {2}. \quad (\text {R a n P A C})
+$$
+
+The choice of hyperparameter  $\lambda$  is crucial; RanPAC with a small regularization  $\lambda$  fails to achieve competitive performance, while it might work well with a large enough  $\lambda$  (e.g.,  $\lambda = 10^4$ ); cf. Fig. 12. In constraint, Prabhu et al. (2024) finds that small  $\lambda$  (of order  $10^{-5}$ ) works better if  $H_{1:t}$  is replaced with random Fourier features. This implies the optimal choice of  $\lambda$  depends, among other factors, on the scale of the features and the noise level. Our method also has a hyperparameter, the truncation percentage  $\zeta$ , while the choice of  $\zeta$  is less sensitive to these factors (Fig. 2). In the implementation of McDonnell et al. (2023), RanPAC selects  $\lambda$  from the predefined set  $\{10^{-8}, 10^{-7}, \dots, 10^{8}\}$  via cross-validation on a small fraction of training data. Although this stabilizes RanPAC in some cases, cross-validation can fail when the validation (or training) set is small and not representative of test data. Unfortunately, this failure occurs often in CIL with small increments (cf. Table 2, Section 6).
+
+In more detail, for every task  $t$  and every each candidate choice of  $\lambda$ , RanPAC maintains the covariances  $\pmb{H}_{1:t}\pmb{H}_{1:t}^{\top}, \pmb{Y}_{1:t}\pmb{H}_{1:t}^{\top}$ , to solve the normal equations  $\pmb{W}(\pmb{H}_{1:t}\pmb{H}_{1:t}^{\top} + \lambda \pmb{I}_E) = \pmb{Y}_{1:t}\pmb{H}_{1:t}^{\top}$  in variable  $\pmb{W}$  using off-the-shelf solvers implemented in PyTorch, which in general takes  $O(E^3)$  time. In contrast, LoRanPAC has  $O(E(k_{t-1} + m_t)^2)$  time complexity, and this is why it is slower than LoRanPAC for the same  $E$ , particularly when  $E$  is large (cf. Fig. 3, Section 6). Certainly, both RanPAC and LoRanPAC can potentially be implemented more efficiently. For example, RanPAC involves inverting the regularized covariance  $\pmb{H}_{1:t}\pmb{H}_{1:t}^{\top} + \lambda \pmb{I}_E$ , and this inverse can be updated continually via the Sherman-Morrison-Woodbury formula. This formula is at the heart of the classic recursive least-squares method (Sayed, 2008), and its philosophy is also found in recent continual learning papers (Min et al., 2022; Zhuang et al., 2022; 2023). However, it is known that such a scheme can be numerically unstable, brittle for ill-conditioned data. Indeed, in our setting, We find the implementation based on the Sherman-Morrison-Woodbury formula suffers from numerical errors and is unable to maintain good accuracy. Moreover, numerical errors accumulate over time, leading to worse performance for longer sequences of tasks. Finally, even if we know the numerical errors might arise in these methods, there is no obvious way to remedy them. This is different from our implementation based on robust truncated SVD, which has the advantage that we could (empirically) reduce numerical errors by re-orthogonalizing  $\widetilde{\pmb{U}}_{1:t}$  (see Remark 4 and Algorithm 2).
+
+One more component in RanPAC is a preprocessing step called first-session adaptation. That is, before using the pre-trained model for continual learning, one fine-tunes it with data from the first task in a parameter-efficient way (Panos et al., 2023). This needs extra hyperparameters and yields different features than  $H_{1:t}$ . We study the impact of this step in Table 1, Section 6.
+
+The final point that relates LoRanPAC to RanPAC is this:  $\overline{W}_t$  in (2) is a global minimizer of
+
+$$
+\min  _ {\boldsymbol {W} \in \mathbb {R} ^ {c _ {t} \times E}} \| \boldsymbol {W} \tau_ {k _ {t}} (\boldsymbol {H} _ {1: t}) - \boldsymbol {Y} _ {1: t} \| _ {\mathrm {F}} ^ {2}. \tag {11}
+$$
+
+Both LoRanPAC and RanPAC aim to minimize the MSE loss; the former uses truncation and the latter uses regularization to make the problem better conditioned. The MSE loss typically yields
+
+similar performance to the cross-entropy loss in many settings (Janocha & Czarnecki, 2017; Hui & Belkin, 2021), and the MSE loss is preferred here as it allows for a closed-form least-squares solution to be rapidly computed and continually updated.
+
+ICL. LoRanPAC is also related to the Ideal Continual Learner (ICL) of Peng et al. (2023), which in the linear, over-parameterized case is the following linearly constrained quadratic problem:
+
+$$
+\min  _ {\boldsymbol {W} \in \mathbb {R} ^ {c _ {t} \times E}} \| \boldsymbol {W} \boldsymbol {H} _ {t} - \boldsymbol {Y} _ {t} \| _ {\mathrm {F}} ^ {2} \quad \text {s . t .} \quad \boldsymbol {W} \boldsymbol {H} _ {i} = \boldsymbol {Y} _ {i}, \quad i = 1, \dots , t - 1. \tag {ICL}
+$$
+
+Proposition 5 of Peng et al. (2023) gives a method based on SVD to solve ICL; it is proved in Peng & Vidal (2025) that this method implicitly finds the solution to Min-Norm. But we have seen in Fig. 1 that solving Min-Norm is numerically challenging due to highly ill-conditioned features  $H_{1:t}$ . Proposition 6 of Peng et al. (2023) further suggests that solving ICL by a gradient-based method gives the approach of Farajtabar et al. (2020), known as Orthogonal Gradient Descent (OGD). Subsequently, OGD is combined with the idea of SVD truncation in the PCA-OGD method (Doan et al., 2021). As gradient-based methods, OGD and PCA-OGD converge slowly for ill-conditioned data and would be less efficient than our LoRanPAC implementation; the differences between PCA-OGD and our method are thoroughly discussed in our rebuttal. The OR-Fit method of Min et al. (2022) improves PCA-OGD by devising carefully chosen step sizes that facilitate solving the current task. Their proposed step sizes are related to ICL and recursive least-squares in an intriguing manner; we refer the reader to Peng & Vidal (2025) for the precise mathematical connections and a unifying perspective on the aforementioned methods.
+
+Principal Component Regression. LoRanPAC combines principal component analysis and ordinary least-squares, which is analogous to principal component regression (PCR) (Xu & Hsu, 2019; Huang et al., 2022; Tucker & Wahl, 2023; Bach, 2024; Green & Romanov, 2024). These papers consider the offline setting, where truncation is performed only once. In contrast, we analyze the effect of continual truncation, which is most pertinent for CL. Indeed, for  $t = 1$ ,  $\mathbb{B}_1$  of Theorem 2 is equal to the corresponding term in Theorem 1 of Huang et al. (2022) up to a constant. More importantly, these papers have statistical assumptions on  $H_{1:t}$ , which are potentially violated by generating  $H_{1:t}$  via  $H_t := \mathrm{relu}(PX_t)$ , with  $X_t$  consisting of features from pre-trained models. In contrast, Theorems 1 and 2 have few assumptions, and so they apply, at least in principle, to the full architecture (i.e., a pre-trained model and random ReLU feature model in cascade).
+
+# 6 NUMERICAL VALIDATION
+
+This section highlights the performance and efficiency of LoRanPAC in the CIL setting across a diverse range of datasets and increments. For additional results, see Appendix K, particularly Appendix K.4 for experimental outcomes in the DIL (domain-incremental learning) setting.
+
+# 6.1 SETUP
+
+Baselines. The most relevant baseline to compare is RanPAC (McDonnell et al., 2023). Additional competitive baselines include L2P (Wang et al., 2022d), DualPrompt (Wang et al., 2022c), Codaprompt (Smith et al., 2023), SimpleCIL, ADAM (Zhou et al., 2023) and EASE (Zhou et al., 2024a). We also compare LoRanPAC with a joint linear classifier, that is, a linear model trained using either the pre-trained features  $X_{1:T}$  of all  $T$  tasks, or the random ReLU features  $H_{1:T}$ . We denote these two methods by LC  $(X_{1:T})$  and LC  $(H_{1:T})$ . To ensure a fair comparison, all experiments are conducted based on the PILOT GitHub repository of Sun et al. (2023). Additional experimental details, as well as a comprehensive review of relevant baselines is given in Appendix J and Appendix H.
+
+Pre-trained Models. We use ViT models pre-trained on ImageNet-1K; specifically the model vit_base_batch16_224 from the timm repository (Wightman, 2019). Experiments using ViTs pre-trained on ImageNet-21K are presented in Appendix K.2.
+
+Datasets. Following prior works (Zhou et al., 2023; McDonnell et al., 2023), we run CIL experiments with B- $q_{1}$ , Inc- $q_{2}$  on continual learning versions of the following datasets: CIFAR100 (Krizhevsky et al., 2009), ImageNet-R (Hendrycks et al., 2021a), ImageNet-A (Hendrycks et al., 2021b), CUB-200 (Wah et al., 2011), ObjectNet (Barbu et al., 2019), OmniBenchmark (Zhang et al., 2022), VTAB (Zhai et al., 2019), and StanfordCars (Krause et al., 2013). We set  $q_{1} = 0$  for most cases, but since
+
+Table 1: Final accuracy with pre-trained ViTs. Large accuracy gaps between RanPAC and LoRanPAC (ours) are shown in bold.  $\dagger$ : Methods using first-session adaptation with the hyperparameters set as per RanPAC†.  $*$ : Methods using first-session adaptation with the hyperparameters set as per EASE* (Zhou et al., 2024a). Table 14 reports standard deviation. Appendix J reports experimental details.  
+
+<table><tr><td>(Part 1)</td><td colspan="3">CIFAR100 (B-0)</td><td colspan="3">ImageNet-R (B-0)</td><td colspan="3">ImageNet-A (B-0)</td><td colspan="3">CUB-200 (B-0)</td><td>Avg.</td></tr><tr><td>\( LC(X_{1:T}) \)</td><td colspan="3">87.56</td><td colspan="3">72.42</td><td colspan="3">58.85</td><td colspan="3">88.76</td><td>76.90</td></tr><tr><td>\( LC(H_{1:T}) \)</td><td colspan="3">87.76</td><td colspan="3">73.00</td><td colspan="3">59.25</td><td colspan="3">88.72</td><td>77.18</td></tr><tr><td></td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td></td></tr><tr><td>L2P</td><td>80.25</td><td>83.53</td><td>83.57</td><td>67.92</td><td>71.78</td><td>73.42</td><td>44.50</td><td>48.52</td><td>51.28</td><td>53.60</td><td>59.20</td><td>67.81</td><td>65.45</td></tr><tr><td>DualPrompt</td><td>80.85</td><td>83.86</td><td>84.59</td><td>67.12</td><td>71.57</td><td>72.87</td><td>49.70</td><td>53.72</td><td>56.75</td><td>54.79</td><td>63.99</td><td>69.93</td><td>67.48</td></tr><tr><td>CodaPrompt</td><td>82.93</td><td>86.31</td><td>87.87</td><td>67.80</td><td>72.73</td><td>74.85</td><td>34.43</td><td>49.57</td><td>59.51</td><td>36.39</td><td>60.18</td><td>71.29</td><td>65.32</td></tr><tr><td>SimpleCIL</td><td>80.48</td><td>80.48</td><td>80.48</td><td>63.47</td><td>63.47</td><td>63.47</td><td>58.72</td><td>58.72</td><td>58.72</td><td>80.45</td><td>80.45</td><td>80.45</td><td>70.78</td></tr><tr><td>RanPAC</td><td>86.71</td><td>87.02</td><td>87.10</td><td>71.90</td><td>71.97</td><td>72.50</td><td>56.48</td><td>62.34</td><td>61.75</td><td>88.08</td><td>87.15</td><td>88.13</td><td>76.76</td></tr><tr><td>LoRanPAC</td><td>88.18</td><td>88.18</td><td>88.21</td><td>73.67</td><td>73.72</td><td>73.63</td><td>62.74</td><td>63.20</td><td>63.20</td><td>89.36</td><td>89.27</td><td>89.23</td><td>78.55</td></tr><tr><td>\( ADAM^† \)</td><td>83.55</td><td>85.13</td><td>85.86</td><td>63.73</td><td>65.03</td><td>71.40</td><td>58.72</td><td>58.66</td><td>58.99</td><td>80.49</td><td>80.66</td><td>81.00</td><td>72.77</td></tr><tr><td>\( RanPAC^† \)</td><td>88.73</td><td>90.04</td><td>90.74</td><td>70.80</td><td>73.37</td><td>78.80</td><td>62.34</td><td>62.08</td><td>62.28</td><td>88.42</td><td>87.57</td><td>88.68</td><td>78.65</td></tr><tr><td>\( LoRanPAC^† \)</td><td>89.73</td><td>90.82</td><td>91.44</td><td>73.58</td><td>74.55</td><td>79.13</td><td>62.74</td><td>62.80</td><td>62.94</td><td>89.14</td><td>89.19</td><td>89.27</td><td>79.61</td></tr><tr><td>\( EASE^* \)</td><td>84.43</td><td>86.48</td><td>88.16</td><td>73.53</td><td>77.02</td><td>77.55</td><td>58.26</td><td>61.69</td><td>62.28</td><td>80.66</td><td>81.68</td><td>81.13</td><td>76.07</td></tr><tr><td>\( LoRanPAC^* \)</td><td>89.46</td><td>90.90</td><td>91.67</td><td>78.73</td><td>80.43</td><td>81.45</td><td>63.40</td><td>64.45</td><td>65.64</td><td>89.14</td><td>89.19</td><td>89.44</td><td>81.16</td></tr><tr><td>(Part 2)</td><td colspan="3">ObjectNet (B-0)</td><td colspan="3">OmniBenchmark (B-0)</td><td colspan="3">VTAB (B-10)</td><td colspan="3">StanfordCars (B-16)</td><td>Avg.</td></tr><tr><td>\( LC(X_{1:T}) \)</td><td colspan="3">59.70</td><td colspan="3">79.55</td><td colspan="3">91.32</td><td colspan="3">74.12</td><td>76.17</td></tr><tr><td>\( LC(H_{1:T}) \)</td><td colspan="3">59.96</td><td colspan="3">80.02</td><td colspan="3">91.17</td><td colspan="3">73.65</td><td>76.20</td></tr><tr><td></td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td></td></tr><tr><td>L2P</td><td>45.53</td><td>52.05</td><td>55.49</td><td>54.50</td><td>57.29</td><td>60.50</td><td>59.32</td><td>73.25</td><td>78.91</td><td>13.70</td><td>27.46</td><td>43.68</td><td>51.81</td></tr><tr><td>DualPrompt</td><td>47.56</td><td>53.68</td><td>55.64</td><td>56.14</td><td>59.18</td><td>62.39</td><td>64.10</td><td>77.78</td><td>83.75</td><td>11.38</td><td>18.84</td><td>27.89</td><td>51.53</td></tr><tr><td>CodaPrompt</td><td>46.61</td><td>54.44</td><td>59.17</td><td>60.00</td><td>64.98</td><td>68.25</td><td>68.77</td><td>76.81</td><td>86.32</td><td>7.96</td><td>11.29</td><td>30.74</td><td>52.95</td></tr><tr><td>SimpleCIL</td><td>51.66</td><td>51.66</td><td>51.66</td><td>70.19</td><td>70.19</td><td>70.19</td><td>82.53</td><td>82.53</td><td>82.53</td><td>35.46</td><td>35.46</td><td>35.46</td><td>59.96</td></tr><tr><td>RanPAC</td><td>58.77</td><td>57.66</td><td>57.69</td><td>77.63</td><td>77.63</td><td>77.46</td><td>91.15</td><td>91.58</td><td>91.58</td><td>58.03</td><td>71.40</td><td>71.40</td><td>73.50</td></tr><tr><td>LoRanPAC</td><td>60.83</td><td>60.86</td><td>60.77</td><td>79.50</td><td>79.60</td><td>79.70</td><td>92.46</td><td>92.55</td><td>92.56</td><td>74.21</td><td>74.39</td><td>74.39</td><td>76.82</td></tr><tr><td>\( ADAM^† \)</td><td>52.16</td><td>53.94</td><td>55.97</td><td>70.54</td><td>70.53</td><td>70.38</td><td>82.55</td><td>82.55</td><td>82.55</td><td>35.61</td><td>35.61</td><td>35.61</td><td>60.67</td></tr><tr><td>\( RanPAC^† \)</td><td>59.14</td><td>61.54</td><td>64.59</td><td>78.10</td><td>78.46</td><td>78.86</td><td>91.48</td><td>91.86</td><td>91.86</td><td>58.65</td><td>72.24</td><td>72.24</td><td>74.56</td></tr><tr><td>\( LoRanPAC^† \)</td><td>61.78</td><td>63.56</td><td>66.48</td><td>80.07</td><td>80.28</td><td>80.45</td><td>92.55</td><td>92.53</td><td>92.60</td><td>74.87</td><td>74.89</td><td>75.13</td><td>77.93</td></tr><tr><td>\( EASE^* \)</td><td>49.28</td><td>53.88</td><td>57.05</td><td>70.33</td><td>70.68</td><td>70.84</td><td>89.85</td><td>93.48</td><td>93.49</td><td>32.43</td><td>31.77</td><td>29.00</td><td>61.84</td></tr><tr><td>\( LoRanPAC^* \)</td><td>61.57</td><td>63.40</td><td>66.29</td><td>80.02</td><td>80.42</td><td>80.82</td><td>92.68</td><td>92.71</td><td>92.67</td><td>75.91</td><td>75.71</td><td>75.96</td><td>78.18</td></tr></table>
+
+StanfordCars and VTAB have 196 and 50 classes, respectively, we take  $q_{1} = 16$  and  $q_{1} = 10$  for them. We let  $q_{2}$  vary in  $\{5, 10, 20\}$ , and also consider the more challenging case  $q_{2} = 1$ .
+
+Metrics. After learning task  $t$  we evaluate the top-1 classification accuracy  $\mathcal{A}_{i,t}$  for every  $i = 1, \dots, t$ . For a total of  $T$  tasks, the accuracy matrix  $\mathcal{A}$  is defined as a  $T \times T$  upper triangular matrix with its  $(i,t)$ -th entry being  $\mathcal{A}_{i,t}$ . Final accuracy is defined as the average  $\frac{1}{T}\sum_{i=1}^{T}\mathcal{A}_{i,T}$  of the last column of  $\mathcal{A}$ . Total accuracy is defined as the average  $\frac{1}{T(T-1)}\sum_{1 \leq i \leq t \leq T}\mathcal{A}_{i,t}$  of all upper triangular entries. Following common practices, we use total accuracy and final accuracy as our evaluation metrics.
+
+# 6.2 EXPERIMENTAL RESULTS AND ANALYSIS
+
+Table 1 contains the main results for  $q_{2} = 5, 10, 20$  on 8 different CIL datasets. First observe that L2P, DualPrompt, and CodaPrompt are unstable as their accuracy varies significantly in different datasets for different values of  $q_{2}$ . Second, SimpleCIL, ADAM, and EASE are unstable as their performance is largely compromised on StanfordCars. Then, RanPAC is unstable with respect to  $q_{2}$  as it exhibits a large performance gap compared to LoRanPAC for  $q_{2} = 5$  on ImageNet-A and StanfordCars. Finally, we see LoRanPAC has more stable performance across datasets and for varying  $q_{2}$ .
+
+Why Does LoRanPAC Uniformly Outperform RanPAC? The first reason is that LoRanPAC's high efficiency and scalability enable the use of a larger embedding dimension. Indeed, LoRanPAC uses  $E = 10^{5}$ , taking advantage of the scaling law (Fig. 6, Appendix K.5), while RanPAC uses its default choice  $E = 10^{4}$ . Note that this is a fair comparison since LoRanPAC's implementation is more scalable and more efficient than RanPAC's. Specifically, LoRanPAC has  $O(E(k_{t-1} + m_t)^2)$
+
+![](images/0cc31386a0a7a46ebf9fe8df2f1347478f29e39eadaf6efa6d3f7fd60d21ebfa.jpg)  
+Figure 3: Training times (in minutes) for varying embedding dimensions  $E$ .
+
+![](images/a1119b9b4891184e8e6f551b3a77119b82aaf53f3efe9a5b31eab200eeb625ce.jpg)
+
+![](images/f712a79197f6bf2aea337ab62e4ea32c991befade4dc69e536a1555717cc4c87.jpg)
+
+![](images/e5f1a3eb371a19a8cfe6c441e217fccd22f681dda4f6f7e939d947ca68f878c5.jpg)
+
+time complexity while RanPAC takes  $O(E^3)$  time for each task  $t$ . An alternative way to make a fair comparison is to set the same embedding dimension  $E$  for both methods, in which case LoRanPAC can be up to 1000 times faster than RanPAC (e.g., see  $E = 25000$  in Fig. 3).
+
+The second reason, as mentioned earlier, is that the cross-validation strategy of McDonnell et al. (2023) might fail to choose a suitable regularization  $\lambda$  for RanPAC when the validation set is small. This is the case in ImageNet-A (B-0, Inc-5) and StanfordCars (B-16, Inc-5) of Table 1, where the validation sets are small and RanPAC's performance is severely degraded. A more careful analysis of these two failure cases shows that the accuracy matrices of RanPAC have multiple columns with nearly zero entries (cf. Fig. 15a and Fig. 17c), exposing RanPAC's instability.
+
+Table 2: Final and total accuracy in CIL datasets with  ${q}_{2} = 1$  (Inc-1).  
+
+<table><tr><td></td><td>CIFAR100</td><td>ImageNet-R</td><td>ImageNet-A</td><td>CUB</td><td>ObjectNet</td><td>OmniBenchmark</td><td>VTAB</td><td>StanfordCars</td></tr><tr><td colspan="9">Final Accuracy</td></tr><tr><td>RanPAC</td><td>86.99±0.06</td><td>70.12±0.39</td><td>36.6±25.35</td><td>55.15±37.14</td><td>57.14±0.24</td><td>77.9±0.04</td><td>91.47±0.3</td><td>35.56±24.75</td></tr><tr><td>LoRanPAC</td><td>88.19±0.05</td><td>73.66±0.07</td><td>62.76±0.16</td><td>89.19±0.06</td><td>60.82±0.15</td><td>79.3±0.06</td><td>92.51±0.05</td><td>74.32±0.11</td></tr><tr><td colspan="9">Total Accuracy</td></tr><tr><td>RanPAC</td><td>90.46±0.73</td><td>69.1±0.37</td><td>44.23±0.46</td><td>74.67±2.87</td><td>62.37±2.1</td><td>85.23±0.56</td><td>74.67±3.07</td><td>56.27±0.78</td></tr><tr><td>LoRanPAC</td><td>92.18±0.56</td><td>78.87±0.34</td><td>70.08±0.86</td><td>92.89±0.59</td><td>70.54±1.94</td><td>86.51±0.59</td><td>96.41±0.31</td><td>81.18±0.68</td></tr></table>
+
+Inc-1: One Class at A Time. In light of the above analysis, we consider the CIL setting, with one class given at each iteration (Inc-1). In this setting, a new task has much fewer training samples and CL methods need to cope with hundreds of tasks (classes) on certain datasets. Note that adapter-based methods such as EASE are infeasible for CIL with Inc-1 (cf. Appendix H.1). In this setting, the fragility of RanPAC with respect to the choice of  $\lambda$  is amplified (see Table 2), and the method exhibits a significant performance drop compared to Table 1. In contrast, LoRanPAC's performance is stable, exhibiting high accuracy comparable to the cases of Inc-{5, 10, 20} in Table 1. Accuracy matrices associated with Table 2 are plotted in Figs. 14 to 18 of Appendix K.10.3, where we present similar results for Inc-{1, 2, 4, 5}.
+
+# 7 CONCLUSION
+
+This work puts forward a simple method that bridges the gap between empirical performance and theoretical guarantees in continual learning with pre-trained models. By addressing the ill-conditioning of lifted features through continual SVD truncation, our approach achieves both stability and strong performance. Extensive experiments demonstrated that our method outperforms state-of-the-art methods across multiple datasets and can handle sequences with hundreds of tasks. Theoretically, we proved that our method maintains small training and test errors by appropriately truncating SVD factors. This work underscores the potential of combining empirical techniques with principled frameworks to develop robust and scalable continual learning systems, and will encourage follow-up works to achieve so as well.
+
+# ACKNOWLEDGMENTS
+
+This work is supported by the project ULEARN "Unsupervised Lifelong Learning", funded by the Research Council of Norway (grant number 316080), and by NSF-Simons Research Collaborations on the Mathematical and Scientific Foundations of Deep Learning (NSF grant 2031985).
+
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+
+# A OVERVIEW OF THE APPENDIX
+
+- In Appendix B, we complie all the mathematical notations used throughout the paper.  
+- In Appendix C we describe the implementation details of LoRanPAC. There, we also discuss other potential implementations and our design choice.  
+- In Appendix D, we present auxiliary lemmas that are useful for proving our main theorems.  
+- In Appendix E, we prove the theorems displayed in the main paper (Theorems 1 and 2).  
+- In Appendix F we present results similar to Appendix E, with the difference that we now assume the noise is Gaussian, which gives slightly tighter error bounds.  
+- In Appendix G, we prove some extra theoretical results such as perturbation bounds on eigenvalues and eigenvectors (Theorem 6).  
+- In Appendix H we review related works on continual learning, focusing on CL methods with pretrained models and existing theoretical developments.  
+- In Appendix I we report the statistics of the datasets we use for experiments.  
+- In Appendix J we specify the experimental setup.  
+- In Appendix K we report extra experimental results, figures, and tables.
+
+# B NOTATIONS
+
+Here in Table 3 we compile all the notations used in the paper.
+
+Table 3: Notations  
+
+<table><tr><td>d</td><td>dimension of pre-trained features</td></tr><tr><td>E</td><td>embedding dimension</td></tr><tr><td>mt</td><td>number of training samples for task t</td></tr><tr><td>Mt</td><td>m1+···+mt</td></tr><tr><td>ct</td><td>total number of classes seen in the first t tasks</td></tr><tr><td>T</td><td>total number of tasks</td></tr><tr><td>N(0,1)</td><td>Gaussian distribution with mean 0 and variance 1</td></tr><tr><td>[·]</td><td>the ceiling operator, which maps a number to the closest integer no smaller than it</td></tr><tr><td>IE</td><td>E×E identity matrix</td></tr><tr><td>Xt</td><td>d×mtmatrix, whose columns are output features of pre-trained models</td></tr><tr><td>P</td><td>E×d random embedding matrix with N(0,1) entries</td></tr><tr><td>Ht</td><td>RandomReLU features relu(PXt) as defined in (1)</td></tr><tr><td>λ</td><td>ridge regularization parameter in RanPAC</td></tr><tr><td>Bt</td><td>the matrix whose SVDs are truncated by Algorithm 4, defined in (3)</td></tr><tr><td>kt</td><td>the number of singular values and vectors preserved for the first t tasks</td></tr><tr><td>τkt(·)</td><td>function that computes the best rank-kt approximation of a matrix</td></tr><tr><td>μk(·)</td><td>the k-th largest eigenvalue of a symmetric matrix</td></tr><tr><td>U1:tΣ1:tV1:t</td><td>SVD of H1:t</td></tr><tr><td>U1:tΣ1:tV1:t</td><td>SVD of τkt(H1:t)</td></tr><tr><td>U1:t,Σ1:t</td><td>SVD factors of Bt</td></tr><tr><td>at</td><td>accumulative error defined in (6)</td></tr><tr><td>γt</td><td>the eigengap between the present and past, defined in (5)</td></tr></table>
+
+# C IMPLEMENTATION DETAILS FOR LORANPAC
+
+In this section, we give full details of our algorithm for LoRanPAC. Note that Algorithm 1 of the main paper is a concise version of our approach, used to illustrate the methodology at high level.
+
+In Appendix C.1, we introduce Algorithm 2, our implementation of the incremental SVD approach. Note that Algorithm 2 dates back at least to Bunch & Nielsen (1978) and has been applied to image processing, computer vision, and latent semantic indexing (Zha & Simon, 1999; Levey & Lindenbaum, 2000; Brand, 2002; Artac et al., 2002; Ross et al., 2008). Recently a link between continual learning and incremental SVD was built (Peng et al., 2023). However, it has not been applied to the context we consider here to the best of our knowledge, and suitable modifications are needed to incorporate incremental SVD for solving LoRanPAC satisfactorily. For example, we truncate the SVD factors in each continual update as shown Algorithm 2, and the outputs  $(\widetilde{U}_{1:t}, \widetilde{\Sigma}_{1:t})$  of Algorithm 2 are not necessarily equal to the top- $k_t$  SVD factors of  $H_{1:t}$ . It is then our contribution to arm Algorithm 2 with theoretical guarantees (cf. Lemma 1 and Theorem 6).
+
+In Appendix C.2, we introduce Algorithm 4, a continual learning method that stably solves LoRanPAC.
+
+# C.1 INIncrementAL TRUNCATED SVD
+
+We first explain the design choice as suggested by (2): Should we maintain all SVD factors  $\widetilde{U}_{1:t}, \widetilde{\Sigma}_{1:t}$ , and  $\widetilde{V}_{1:t}$ , or should we just maintain the singular values  $\widetilde{\Sigma}_{1:t}$  and and left singular vectors  $\widetilde{U}_{1:t}$ ? In the main paper, we suggested taking the latter choice, as we empirically found continually updating all SVD factors  $\widetilde{U}_{1:t}, \widetilde{\Sigma}_{1:t}$ , and  $\widetilde{V}_{1:t}$  lead to large test errors.
+
+We now describe how to update the top  $k_{t}$  SVD factors  $\widetilde{U}_{1:t},\widetilde{\Sigma}_{1:t}$  from the previous estimates  $\widetilde{U}_{1:t - 1},\widetilde{\Sigma}_{1:t - 1}$  and new data  $\pmb{H}_{t}$ . Let  $\pmb{Q}_t\pmb{R}_t$  be the QR decomposition of  $(I_E - \widetilde{U}_{1:t - 1}\widetilde{U}_{1:t - 1}^{\top})\pmb{H}_t$ . Then we have
+
+$$
+\left[ \widetilde {\boldsymbol {U}} _ {1: t - 1} \widetilde {\boldsymbol {\Sigma}} _ {1: t - 1}, \boldsymbol {H} _ {t} \right] = \left[ \widetilde {\boldsymbol {U}} _ {1: t - 1}, \boldsymbol {Q} _ {t} \right] \left[ \begin{array}{c c} \widetilde {\boldsymbol {\Sigma}} _ {1: t - 1} & \widetilde {\boldsymbol {U}} _ {t - 1} ^ {\top} \boldsymbol {H} _ {t} \\ 0 & \boldsymbol {R} _ {t} \end{array} \right].
+$$
+
+Note that  $[\widetilde{U}_{1:t-1}, Q_t]$  is already orthogonal, we can do a truncated SVD on the smaller  $(k_{t-1} + m_t) \times (k_{t-1} + m_t)$  matrix of the right-hand side. The full procedure is summarized below:
+
+Algorithm 2: Incremental Truncated Singular Value Decomposition  
+Input: data matrix  $\pmb{H}_{t}\in \mathbb{R}^{E\times m_{t}}$  of task  $t$ , desired output rank  $k_{t}\leq \min \{E,M_{t}\}$ $(M_t\coloneqq m_1 + \dots +m_t)$ , truncated SVD factors  $\widetilde{\boldsymbol{U}}_{1:t - 1}\in \mathbb{R}^{E\times k_{t - 1}}$  and  $\widetilde{\Sigma}_{1:t - 1}\in \mathbb{R}^{k_{t - 1}\times k_{t - 1}}$  of the previous  $t - 1$  tasks;   
+2 Compute the QR decomposition  $Q_{t}R_{t}$  of  $(I_E - \widetilde{U}_{1:t - 1}\widetilde{U}_{1:t - 1}^{\top})H_{t}$    
+3 Set  $(\Sigma_{\mathrm{tmp}},\pmb{U}_{\mathrm{tmp}})$  to the top-  $k_{t}$  SVD components of // Algorithm 3   
+4  $\begin{bmatrix} \widetilde{\Sigma}_{1:t - 1} & \widetilde{U}_{t - 1}^{\top}\pmb {H}_t\\ 0 & \pmb {R}_t \end{bmatrix} \in \mathbb{R}^{(k_{t - 1} + m_t)\times (k_{t - 1} + m_t)};$    
+5 Set  $\widetilde{\Sigma}_{1:t}\gets \Sigma_{\mathrm{tmp}}$  and  $\widetilde{U}_{1:t}\gets [\widetilde{U}_{t - 1}Q_t]U_{\mathrm{tmp}}$    
+6  $\widetilde{U}_{1:t}\gets$  The orthogonal factor of QR decomposition of  $\widetilde{U}_{1:t}$  // improve numerical stability   
+7 Output:  $(\widetilde{U}_{1:t},\widetilde{\Sigma}_{1:t})$
+
+Remark 4. Since  $[\widetilde{U}_{1:t-1}Q_t]$  and  $U_{\mathrm{tmp}}$  are orthogonal,  $\widetilde{U}_{1:t}$  is expected to be orthogonal as well. However, the multiplication  $\widetilde{U}_{1:t} = [\widetilde{U}_{t-1}Q_t]U_{\mathrm{tmp}}$  might lose orthogonality due to numerical errors, especially when  $t$  gets large. This is fixed by an extra post-processing step that orthogonalizes  $\widetilde{U}_{1:t}$ .
+
+Memory Complexity Analysis. The extra working memory of this approach is roughly:
+
+-  $O(Em_{t} + m_{t}^{2})$ , for the QR factors  $Q_{t}R_{t}$  
+-  $O((k_{t - 1} + m_t)^2)$ , for the matrix in (12) and its SVD factors;
+
+Algorithm 3: Truncated Singular Value Decomposition (TSVD)  
+1 Input: matrix  $\pmb{H} \in \mathbb{R}^{E \times m}$  and desired output rank  $r \leq \min\{E, m\}$ ;  
+2 tmp  $\leftarrow$  min\{E, m\};  
+3 Compute the SVD  $\sigma_{1} \pmb{u}_{1} \pmb{v}_{1}^{\top} + \dots + \sigma_{\mathrm{tmp}} \pmb{u}_{\mathrm{tmp}} \pmb{v}_{\mathrm{tmp}}^{\top}$  of  $\pmb{H}$ ;  
+4 Set  $\widetilde{\Sigma} \gets \mathrm{diag}(\sigma_{1}, \ldots, \sigma_{r}), \widetilde{\pmb{U}} \gets [\pmb{u}_{1}, \ldots, \pmb{u}_{r}]$ ;  
+5 Output:  $(\widetilde{\pmb{U}}, \widetilde{\Sigma})$ ;
+
+Hence, for large  $E$ , this is less than the  $O(E(k_{t-1} + m_t))$  memory used by the direct SVD method.
+
+Time Complexity Analysis. The major cost is the SVD of (12), which takes  $O((k_{t-1} + m_t)^3)$  time. While in principle the QR orthogonalization for the post-processing of  $\widetilde{U}_{1:t}$  takes  $O(E(k_{t-1} + m_t)^2)$  time, it is significantly faster than SVD as the constants behind its  $O(\cdot)$  is very small. Therefore, one would expect the SVD on the matrix of (12) in  $O((k_{t-1} + m_t)^3)$  time should be much faster than the SVD on the matrix  $[\widetilde{U}_{1:t-1}\widetilde{\Sigma}_{1:t-1}, H_t]$ , which needs  $O(E(k_{t-1} + m_t)^2)$  time, where  $E$  is far larger than  $k_{t-1} + m_t$  (e.g.,  $E = 10^5$  and  $k_{t-1} + m_t = 10^4$ ). This is true on a sequential machine, but their running time difference is not significant for highly parallel GPU implementations in our experience (e.g., computing the inner product between two  $E$ -dimensional vectors has similar running times to computing the inner product between  $(k_{t-1} + m_t)$ -dimensional vectors, due to parallelism). Hence, for a parallel implementation, the main advantage of doing SVD on the matrix in (12) is that it takes less working memory than SVD on  $[\widetilde{U}_{1:t-1}\widetilde{\Sigma}_{1:t-1}, H_t]$ .
+
+Remark 5. Algorithm 3 can, in fact, be implemented by randomized linear algebra techniques (Halko et al., 2011). Some of these techniques compute by design only the top  $k$  SVD factors. Intuitively this could save time and memory if  $k$  is very small. One such method is conveniently implemented in PyTorch as well (torch.svd_lowrank). However, in our rudimentary attempts at using randomized approaches, we found this PyTorch routine does not seem to be as efficient or accurate as our present implementation (we consistently set truncation percentage  $\zeta$  to  $25\%$ ). This observation aligns with the PyTorch document of torch.svd_lowrank: In general, use the full-rank SVD implementation torch.linalg.svd() for dense matrices due to its 10-fold higher performance characteristics. The low-rank SVD will be useful for huge sparse matrices that torch.linalg.svd() cannot handle. For the moment, we conclude that it needs deeper investigations to see whether randomized techniques are suitable for the continual learning contexts.
+
+# C.2 CONTINUAL SOLVER FOR LORANPAC
+
+The proposed algorithm is shown in Algorithm 4. Here are a few details that we have not yet mentioned in the main paper. First, note that Algorithm 4 formally updates  $M_t$  ad  $J_t$  continually. At Line 8 of Algorithm 4 we compute the label-feature covariance matrix  $J_1 := Y_1H_1^\top \in \mathbb{R}^{c_1 \times E}$ , and then at lines 10 and 11 we update  $J_{t-1}$  into  $J_t$  via  $J_t \gets J_{t-1} + J_{\mathrm{tmp}}$ . The attentive reader might find that  $J_{t-1}$  is of size  $c_{t-1} \times E$  while  $J_{\mathrm{tmp}}$  is of size  $c_t \times E$ . But it could be that  $c_{t-1} < c_t$ , so it might not make sense to add  $J_{t-1}$  and  $J_{\mathrm{tmp}}$  as in Line 11. Note that we wrote Line 11 just for simplicity. The implementation would pad  $c_t - c_{t-1}$  zero rows to  $J_{t-1}$  in a similar fashion to how we extend  $Y_{t-1}$  into  $Y_t$  when more classes are given, and this is what Line 11 should mean.
+
+Second, we add an extra parameter  $r_{\mathrm{max}}$ , to control the maximum allowable rank, that is the maximum number of columns  $\widetilde{U}_{1:t}$  is allowed to have. The purpose is to control the time complexity of Algorithm 4 and allow it to run more efficiently on large datasets such as DomainNet (cf. Tables 7 and 8). We argue both the truncation percentage  $\zeta$  and maximum allowable rank  $r_{\mathrm{max}}$  are needed: With  $\zeta$  alone, the method might run slowly or even exceed the memory for large datasets such as DomainNet (cf. Tables 7 and 8); with  $r_{\mathrm{max}}$  alone, truncation is not activated before receiving  $r_{\mathrm{max}}$  samples, and numerical instability if it arises, can not be prevented before truncation is in effect. Table 4 gives the values of  $\zeta$  and  $r_{\mathrm{max}}$  we use for each dataset.
+
+<table><tr><td>Dataset</td><td>Truncation Percentage ζ</td><td>Embedding Dimension E</td><td>Maximum Allowable Rank rmax</td></tr><tr><td>CIFAR100</td><td>25%</td><td>10^5</td><td>10000</td></tr><tr><td>ImageNet-R</td><td>25%</td><td>10^5</td><td>10000</td></tr><tr><td>ImageNet-A</td><td>25%</td><td>10^5</td><td>10000</td></tr><tr><td>CUB-200</td><td>25%</td><td>10^5</td><td>10000</td></tr><tr><td>ObjectNet</td><td>25%</td><td>10^5</td><td>20000</td></tr><tr><td>OmniBenchmark</td><td>25%</td><td>10^5</td><td>20000</td></tr><tr><td>VTAB†</td><td>25%</td><td>10^5</td><td>10000</td></tr><tr><td>StanfordCars</td><td>25%</td><td>10^5</td><td>10000</td></tr></table>
+
+Table 4: Hyperparameters we use for each dataset.
+
+Algorithm 4: Continual Solver of LoRanPAC (concise version in Algorithm 1)  
+1 Input of Task t: Random ReLU features  $\pmb{H}_{t}\in \mathbb{R}^{E\times m_{t}}$  , label matrix  $\pmb {Y}_t\in \mathbb{R}^{c_t\times m_t}$  truncation percentage  $\zeta \in [0,1]$  , maximum allowable rank  $r_{\mathrm{max}}$  .   
+2 For  $t\gets 1,2,\ldots$  ..   
+3  $M_{t}\leftarrow M_{t - 1} + m_{t};$  // update the total number of samples  $M_{t}$    
+4  $k_{t}\leftarrow \min \{r_{\mathrm{max}},(1 - \zeta)M_{t})\} ;$  // perserve  $k_{t}$  SVD factors for the first  $t$  tasks   
+5 Form  $\pmb{B}_{t}$  as per (3);   
+6  $(\widetilde{\pmb{U}}_{1:t},\widetilde{\pmb{\Sigma}}_{1:t})\gets$  Top-  $k_{t}$  SVD factors of  $\pmb{B}_{t}$  // use Algorithm 3 if  $t = 1$  , or Algorithm 2 if  $t > 1$    
+7 If  $t = 1$  // Continual update of  $J_{t}\coloneqq Y_{1:t}H_{1:t}^{\top}$    
+8  $\pmb{J}_{1}\gets$  Output of Algorithm 5 run with inputs  $\pmb {H}_1,\pmb {Y}_1;$  // label-feature covariance of task 1   
+9 Else:   
+10  $\pmb{J}_{\mathrm{tmp}}\gets$  Output of Algorithm 5 run with inputs  $\pmb {H}_t,\pmb {Y}_t;$  // label-feature covariance of task t   
+11  $\pmb{J}_{t}\gets \pmb{J}_{t - 1} + \pmb{J}_{\mathrm{tmp}};$    
+12 Form the linear classifier  $\widetilde{\pmb{W}}_t\coloneqq J_t\left(\widetilde{\pmb{U}}_{1:t}\widetilde{\pmb{\Sigma}}_{1:t}^{-2}\widetilde{\pmb{U}}_{1:t}^{\top}\right);$  // cf. (2) and (4)
+
+# D AUXILIARY LEMMAS
+
+The following lemma provides an explicit expression for the difference between the continually truncated factors  $\widetilde{U}_{1:t}\widetilde{\Sigma}_{1:t}^{2}\widetilde{U}_{1:t}^{\top}$  and covariance  $H_{1:t}H_{1:t}^{\top}$ .
+
+Lemma 1. Let  $B_{t}$  be the matrix whose SVDs are truncated by Algorithm 4, as defined in (3). We have  $\overline{U}_1 = \tilde{U}_1$  and  $\overline{\Sigma}_{1} = \tilde{\Sigma}_{1}$ . Moreover, we have
+
+$$
+\boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} - \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} = \sum_ {i = 1} ^ {t} \left(\boldsymbol {B} _ {i} \boldsymbol {B} _ {i} ^ {\top} - \tau_ {k _ {i}} \left(\boldsymbol {B} _ {i} \boldsymbol {B} _ {i} ^ {\top}\right)\right).
+$$
+
+Proof of Lemma 1. It should be clear that  $\overline{U}_1 = \widetilde{U}_1$  and  $\overline{\Sigma}_1 = \widetilde{\Sigma}_1$ . For every  $i = 1, \dots, t$  we have
+
+$$
+\widetilde {U} _ {1: i} \widetilde {\Sigma} _ {1: i} ^ {2} \widetilde {U} _ {1: i} ^ {\top} = \tau_ {k _ {i}} \left(\boldsymbol {B} _ {i} \boldsymbol {B} _ {i} ^ {\top}\right).
+$$
+
+and therefore
+
+$$
+\widetilde {\boldsymbol {U}} _ {1: i} \widetilde {\boldsymbol {\Sigma}} _ {1: i} ^ {2} \widetilde {\boldsymbol {U}} _ {1: i} ^ {\top} - \boldsymbol {B} _ {i} \boldsymbol {B} _ {i} ^ {\top} = \tau_ {k _ {i}} \left(\boldsymbol {B} _ {i} \boldsymbol {B} _ {i} ^ {\top}\right) - \boldsymbol {B} _ {i} \boldsymbol {B} _ {i} ^ {\top}.
+$$
+
+Algorithm 5: Compute The Label-Feature Covariance Matrix  
+1 Input: matrix  $\pmb{H} = [h_1,\dots,h_m]\in \mathbb{R}^{E\times m}$  and label matrix  $\pmb {Y}\in \mathbb{R}^{c\times m}$    
+2 Convert  $\pmb{Y}$  into a vector of indices  $\pmb {y} = [y_{1},\dots,y_{m}]$  such that the  $i$  -th column of  $\pmb{Y}$  is the  $y_{i}$  -th standard basis vector (i.e., one-hot vector with 1 at position  $y_{i}$  .   
+3 Initialize  $\pmb {S} = [\pmb {s}_1,\dots,\pmb {s}_c]$  to be the  $E\times c$  zero matrix;   
+4 For  $i = 1,\ldots ,m$  // parallel implementation via torch.Tensor.index_add_   
+5  $S_{y_i}\gets S_{y_i} + h_i;$    
+6 Output:  $\pmb{S}^{\top}$  //  $S$  is equal to  $\pmb{H}\pmb{Y}^{\top}$
+
+A key observation is that summing the above equality over  $i = 2, \ldots, t$  yields
+
+$$
+\begin{array}{l} \sum_ {i = 2} ^ {t} \left(\widetilde {\boldsymbol {U}} _ {1: i} \widetilde {\boldsymbol {\Sigma}} _ {1: i} ^ {2} \widetilde {\boldsymbol {U}} _ {1: i} ^ {\top} - \boldsymbol {B} _ {i} \boldsymbol {B} _ {i} ^ {\top}\right) = \sum_ {i = 2} ^ {t} \left(\tau_ {k _ {i}} \left(\boldsymbol {B} _ {i} \boldsymbol {B} _ {i} ^ {\top}\right) - \boldsymbol {B} _ {i} \boldsymbol {B} _ {i} ^ {\top}\right) \\ \Leftrightarrow \sum_ {i = 2} ^ {t} \left(\widetilde {\boldsymbol {U}} _ {1: i} \widetilde {\boldsymbol {\Sigma}} _ {1: i} ^ {2} \widetilde {\boldsymbol {U}} _ {1: i} ^ {\top} - \widetilde {\boldsymbol {U}} _ {1: i - 1} \widetilde {\boldsymbol {\Sigma}} _ {1: i - 1} ^ {2} \widetilde {\boldsymbol {U}} _ {1: i - 1} ^ {\top} - \boldsymbol {H} _ {i} \boldsymbol {H} _ {i} ^ {\top}\right) = \sum_ {i = 2} ^ {t} \left(\tau_ {k _ {i}} \left(\boldsymbol {B} _ {i} \boldsymbol {B} _ {i} ^ {\top}\right) - \boldsymbol {B} _ {i} \boldsymbol {B} _ {i} ^ {\top}\right) \\ \Leftrightarrow \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} - \widetilde {\boldsymbol {U}} _ {1} \widetilde {\boldsymbol {\Sigma}} _ {1} ^ {2} \widetilde {\boldsymbol {U}} _ {1} ^ {\top} - \boldsymbol {H} _ {2: t} \boldsymbol {H} _ {2: t} ^ {\top} = \sum_ {i = 2} ^ {t} \left(\tau_ {k _ {i}} \left(\boldsymbol {B} _ {i} \boldsymbol {B} _ {i} ^ {\top}\right) - \boldsymbol {B} _ {i} \boldsymbol {B} _ {i} ^ {\top}\right) \tag {13} \\ \Leftrightarrow \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} - \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} = \overline {{\boldsymbol {U}}} _ {1} \overline {{\boldsymbol {\Sigma}}} _ {1} ^ {2} \overline {{\boldsymbol {U}}} _ {1} ^ {\top} - \boldsymbol {H} _ {1} \boldsymbol {H} _ {1} ^ {\top} + \sum_ {i = 2} ^ {t} \left(\tau_ {k _ {i}} \left(\boldsymbol {B} _ {i} \boldsymbol {B} _ {i} ^ {\top}\right) - \boldsymbol {B} _ {i} \boldsymbol {B} _ {i} ^ {\top}\right) \\ \Leftrightarrow \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} - \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} = \sum_ {i = 1} ^ {t} \left(\tau_ {k _ {i}} \left(\boldsymbol {B} _ {i} \boldsymbol {B} _ {i} ^ {\top}\right) - \boldsymbol {B} _ {i} \boldsymbol {B} _ {i} ^ {\top}\right). \\ \end{array}
+$$
+
+The last equality also holds for  $t = 1$ . This finishes the proof.
+
+![](images/6dec44786d1122ce78ac09afe5230dcbd9bb08a3f6d8ef203f5f3dca3d2bc030.jpg)
+
+The lemma below is a direct consequence of Von Neumann's trace inequality, and its proof is omitted.
+
+Lemma 2. Given two square matrices  $A, B$  with  $A$  positive semidefinite, we have
+
+$$
+\operatorname {t r} (A B) \leq \operatorname {t r} (A) \cdot \| B \|.
+$$
+
+Lemma 3 presented below is elementary.
+
+Lemma 3. Assume  $C$  is a positive semidefinite matrix. Then we have
+
+$$
+\operatorname {t r} \left(D A C B D ^ {\top}\right) + \operatorname {t r} \left(D B ^ {\top} C A ^ {\top} D ^ {\top}\right) \leq \operatorname {t r} \left(D A C A ^ {\top} D ^ {\top}\right) + \operatorname {t r} \left(D B C B ^ {\top} D ^ {\top}\right),
+$$
+
+where  $A, B, C, D$  are matrices of compatible sizes. Therefore, it holds that
+
+$$
+\operatorname {t r} \left(D (A + B) C (A + B) D ^ {\top}\right) \leq 2 \operatorname {t r} \left(D A C A ^ {\top} D ^ {\top}\right) + 2 \operatorname {t r} \left(D B C B ^ {\top} D ^ {\top}\right).
+$$
+
+The following two lemmas provide upper bounds on several terms appearing naturally in our main results.
+
+Lemma 4. Let  $B_{t}$  be defined in (3),  $\gamma_{t}$  in (5), and  $a_{t}$  in (6). Define
+
+$$
+\boldsymbol {D} _ {t} := \sum_ {i = 1} ^ {t} \left(\boldsymbol {B} _ {i} \boldsymbol {B} _ {i} ^ {\top} - \tau_ {k _ {i}} \left(\boldsymbol {B} _ {i} \boldsymbol {B} _ {i} ^ {\top}\right)\right). \tag {14}
+$$
+
+We have
+
+$$
+\left\| \boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \right\| \leq \frac {t - 1}{\gamma_ {t}},
+$$
+
+$$
+\left\| \tilde {\boldsymbol {U}} _ {1: t} \tilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \tilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {D} _ {t} \right\| \leq \frac {t - 1}{\gamma_ {t}},
+$$
+
+$$
+\operatorname {t r} \left(\boldsymbol {D} _ {t} \tilde {\boldsymbol {U}} _ {1: t} \tilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \tilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \leq \frac {1}{\gamma_ {t}} \min  \left\{M _ {t - 1} - k _ {t - 1}, (t - 1) k _ {t} \right\},
+$$
+
+$$
+\mathrm {t r} \left(\boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 4} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \leq \frac {1}{\mu_ {k _ {t}} (\boldsymbol {B} _ {t} \boldsymbol {B} _ {t} ^ {\top})} \cdot \frac {1}{\gamma_ {t}} \min \left\{M _ {t - 1} - k _ {t - 1}, (t - 1) k _ {t} \right\}.
+$$
+
+Proof of Lemma 4. It follows from definition that
+
+$$
+\left(\boldsymbol {B} _ {i} \boldsymbol {B} _ {i} ^ {\top} - \tau_ {k _ {i}} \left(\boldsymbol {B} _ {i} \boldsymbol {B} _ {i} ^ {\top}\right)\right) \widetilde {\boldsymbol {U}} _ {1: t} = 0,
+$$
+
+hence  $D_{t}\widetilde{U}_{1:t} = D_{t - 1}\widetilde{U}_{1:t}$ . This means
+
+$$
+\begin{array}{l} \left\| \boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \right\| = \left\| \boldsymbol {D} _ {t - 1} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \right\| \\ \leq \| \boldsymbol {D} _ {t - 1} \| \cdot \left\| \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \right\| \\ = \| \boldsymbol {D} _ {t - 1} \| \cdot \frac {1}{\mu_ {k _ {t}} \left(\boldsymbol {B} _ {t} \boldsymbol {B} _ {t} ^ {\top}\right)}, \\ \end{array}
+$$
+
+where the last equality follows by definition. On the other hand, we have
+
+$$
+\left\| \boldsymbol {D} _ {t - 1} \right\| \leq \sum_ {i = 1} ^ {t - 1} \mu_ {k _ {i} + 1} \left(\boldsymbol {B} _ {i} \boldsymbol {B} _ {i} ^ {\top}\right) \leq \frac {(t - 1)}{\gamma_ {t}} \mu_ {k _ {t}} \left(\boldsymbol {B} _ {t} \boldsymbol {B} _ {t} ^ {\top}\right).
+$$
+
+Combining the above proves the first required equality. The second inequality follows similarly.
+
+For the final trace inequality, we have  $(k_0\coloneqq 0)$
+
+$$
+\begin{array}{l} \operatorname {t r} \left(\boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) = \operatorname {t r} \left(\boldsymbol {D} _ {t - 1} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \\ \stackrel {\mathrm {(i)}} {\leq} \operatorname {t r} \left(\boldsymbol {D} _ {t - 1}\right) \cdot \left\| \tilde {\boldsymbol {U}} _ {1: t} \tilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \tilde {\boldsymbol {U}} _ {1: t} ^ {\top} \right\| \\ = \left(\sum_ {i = 1} ^ {t - 1} \sum_ {j = k _ {i} + 1} ^ {m _ {i} + k _ {i - 1}} \mu_ {j} \left(\boldsymbol {B} _ {i} \boldsymbol {B} _ {i} ^ {\top}\right)\right) \cdot \frac {1}{\mu_ {k _ {t}} \left(\boldsymbol {B} _ {t} \boldsymbol {B} _ {t} ^ {\top}\right)} \\ \leq \frac {1}{\gamma_ {t}} \sum_ {i = 1} ^ {t - 1} \left(m _ {i} + k _ {i - 1} - k _ {i}\right) \\ = \frac {1}{\gamma_ {t}} \left(M _ {t - 1} - k _ {t - 1}\right), \\ \end{array}
+$$
+
+where (i) holds as  $D_{t - 1}$  is positive semidefinite (cf. Lemma 2).
+
+We can also bound  $\operatorname{tr}\left(\boldsymbol{D}_t\widetilde{\boldsymbol{U}}_{1:t}\widetilde{\boldsymbol{\Sigma}}_{1:t}^{-2}\widetilde{\boldsymbol{U}}_{1:t}^{\top}\right)$  alternatively as follows:
+
+$$
+\begin{array}{l} \operatorname {t r} \left(\boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) = \operatorname {t r} \left(\boldsymbol {D} _ {t - 1} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \\ \leq \| \boldsymbol {D} _ {t - 1} \| \cdot \operatorname {t r} \left(\widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \\ \leq \frac {(t - 1) k _ {t}}{\gamma_ {t}} \\ \end{array}
+$$
+
+Combining the two bounds on  $\operatorname{tr}\left(\boldsymbol{D}_t\widetilde{\boldsymbol{U}}_{1:t}\widetilde{\boldsymbol{\Sigma}}_{1:t}^{-2}\widetilde{\boldsymbol{U}}_{1:t}^{\top}\right)$  proves the third inequality. The fourth inequality follows similarly.
+
+Lemma 5. Using the notations in Lemma 4, we have
+
+$$
+\left. \right. \operatorname {t r} \left(\boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \leq \frac {t - 1}{\gamma_ {t} ^ {2}} \min  \left\{M _ {t - 1} - k _ {t - 1}, (t - 1) k _ {t} \right\},
+$$
+
+$$
+\begin{array}{l} \left\| \boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {D} _ {t} \right\| \leq a _ {t - 1} \cdot \left(\frac {(t - 1) ^ {2}}{\gamma_ {t} ^ {2}} + \frac {t - 1}{\gamma_ {t}}\right), \\ \left\| \left(\boldsymbol {I} _ {E} - \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \left(\boldsymbol {I} _ {E} - \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \right\| \leq a _ {t}, \\ \left\| \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} \right\| _ {\mathrm {F}} ^ {2} \leq \left(\frac {t - 1}{\gamma_ {t} ^ {2}} + \frac {2}{\gamma_ {t}}\right) \min  \left\{M _ {t - 1} - k _ {t - 1}, (t - 1) k _ {t} \right\} + k _ {t}, \\ \operatorname {t r} \left(\widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \leq \frac {1}{\mu_ {k _ {t}} \left(\boldsymbol {B} _ {t} \boldsymbol {B} _ {t} ^ {\top}\right)} \cdot \left(\frac {\min \left\{M _ {t - 1} - k _ {t - 1} , (t - 1) k _ {t} \right\}}{\gamma_ {t}} + k _ {t}\right). \\ \end{array}
+$$
+
+Proof of Lemma 5. Since  $D_{t-1}$  is positive semidefinite, let  $L_{t-1}L_{t-1}^{\top}$  be its Cholesky decomposition. Then we have
+
+$$
+\begin{array}{l} \operatorname {t r} \left(\boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \\ = \operatorname {t r} \left(\boldsymbol {D} _ {t - 1} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {D} _ {t - 1} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \\ = \operatorname {t r} \left(\boldsymbol {L} _ {t - 1} \boldsymbol {L} _ {t - 1} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {L} _ {t - 1} \boldsymbol {L} _ {t - 1} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \\ = \operatorname {t r} \left(\boldsymbol {L} _ {t - 1} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {L} _ {t - 1} \boldsymbol {L} _ {t - 1} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {L} _ {t - 1}\right) \\ \stackrel {(i)} {\leq} \operatorname {t r} \left(\boldsymbol {L} _ {t - 1} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {L} _ {t - 1}\right) \cdot \left\| \boldsymbol {L} _ {t - 1} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {L} _ {t - 1} \right\| \\ = \operatorname {t r} \left(\boldsymbol {D} _ {t - 1} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \cdot \left\| \boldsymbol {L} _ {t - 1} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {L} _ {t - 1} \right\| \\ \stackrel {\mathrm {(i i)}} {\leq} \frac {1}{\gamma_ {t}} \min  \left\{M _ {t - 1} - k _ {t - 1}, (t - 1) k _ {t} \right\} \cdot \left\| \boldsymbol {L} _ {t - 1} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {L} _ {t - 1} \right\|. \\ \end{array}
+$$
+
+In the above, (i) follows from Lemma 2, and (ii) follows from Lemma 4. Continuing with the above inequality, we have
+
+$$
+\begin{array}{l} \left\| \boldsymbol {L} _ {t - 1} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {L} _ {t - 1} \right\| \leq \| \boldsymbol {L} _ {t - 1} \| ^ {2} \cdot \frac {1}{\mu_ {k _ {t}} \left(\boldsymbol {B} _ {t} \boldsymbol {B} _ {t} ^ {\top}\right)} \\ = \| \boldsymbol {D} _ {t - 1} \| \cdot \frac {1}{\mu_ {k _ {t}} \left(\boldsymbol {B} _ {t} \boldsymbol {B} _ {t} ^ {\top}\right)} \\ \leq \frac {t - 1}{\gamma_ {t}}. \\ \end{array}
+$$
+
+Recall the fact  $D_{t}\widetilde{U}_{1:t} = D_{t - 1}\widetilde{U}_{1:t}$ . The second inequality in Lemma 5 can be proved as follows:
+
+$$
+\begin{array}{l} \left\| \boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {D} _ {t} \right\| \\ \stackrel {\text {(i)}} {=} \left\| \boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \left(\boldsymbol {D} _ {t} + \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {D} _ {t} \right\| \\ = \left\| \boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {D} _ {t} + \boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {D} _ {t} \right\| \\ \leq \| \boldsymbol {D} _ {t - 1} \| \cdot \left(\left\| \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {D} _ {t} \right\| + \left\| \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {D} _ {t} \right\|\right) \\ \leq a _ {t - 1} \cdot \left(\frac {(t - 1) ^ {2}}{\gamma_ {t} ^ {2}} + \frac {t - 1}{\gamma_ {t}}\right). \\ \end{array}
+$$
+
+In the above, (i) follows from Lemma 1.
+
+The third inequality is proved as follows:
+
+$$
+\begin{array}{l} \left\| \left(\boldsymbol {I} _ {E} - \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \left(\boldsymbol {I} _ {E} - \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \right\| \\ = \left\| \left(\boldsymbol {I} _ {E} - \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \left(\boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} - \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} ^ {2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \left(\boldsymbol {I} _ {E} - \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \right\| \\ = \left\| \left(\boldsymbol {I} _ {E} - \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \boldsymbol {D} _ {t} \left(\boldsymbol {I} _ {E} - \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \right\| \\ \leq \left\| D _ {t} \right\| = a _ {t}. \\ \end{array}
+$$
+
+We now prove the fourth inequality:
+
+$$
+\begin{array}{l} \left\| \boldsymbol {H} _ {1: t} ^ {\top} \tilde {\boldsymbol {U}} _ {1: t} \tilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \tilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} \right\| _ {\mathrm {F}} ^ {2} \\ = \operatorname {t r} \left(\boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \\ \stackrel {\mathrm {(i)}} {=} \operatorname {t r} \left(\left(\boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} + \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \left(\boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} + \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {U}} _ {1: \cdot t} ^ {\top}\right)\right) \\ = \operatorname {t r} \left(\boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} + 2 \boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) + k _ {t} \\ \stackrel {\mathrm {(i i)}} {\leq} \left(\frac {t - 1}{\gamma_ {t} ^ {2}} + \frac {2}{\gamma_ {t}}\right) \min  \left\{M _ {t - 1} - k _ {t - 1}, (t - 1) k _ {t} \right\} + k _ {t}. \\ \end{array}
+$$
+
+In the above, (i) follows from Lemma 1, and (ii) follows from Lemma 4 and the first inequality we just proved for Lemma 5.
+
+The fifth inequality can be proved as follows:
+
+$$
+\begin{array}{l} \operatorname {t r} \left(\widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \\ \stackrel {\text {(i)}} {=} \operatorname {t r} \left(\widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \left(\boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} + \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right)\right) \\ = \operatorname {t r} \left(\boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 4} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) + \operatorname {t r} \left(\widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2}\right) \\ \stackrel {\mathrm {(i i)}} {\leq} \frac {1}{\mu_ {k _ {t}} (\boldsymbol {B} _ {t} \boldsymbol {B} _ {t} ^ {\top})} \cdot \frac {1}{\gamma_ {t}} \min  \left\{M _ {t - 1} - k _ {t - 1}, (t - 1) k _ {t} \right\} + \frac {k _ {t}}{\mu_ {k _ {t}} (\boldsymbol {B} _ {t} \boldsymbol {B} _ {t} ^ {\top})}. \\ \end{array}
+$$
+
+Here, (i) holds as a result of Lemma 1 and (ii) follows from Lemma 4.
+
+![](images/713006c6fd1c20fdc902b0fdc90ccdedbe76ab3d850d49266d2735615b5921c2.jpg)
+
+Lemma 6. Using the notations in Lemma 4, we have for any  $W$  that
+
+$$
+\left\| \boldsymbol {W} (\boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} - \boldsymbol {I} _ {M _ {t}}) \boldsymbol {H} _ {1: t} \right\| _ {\mathrm {F}} ^ {2} \leq 2 \cdot \| \boldsymbol {W} \| _ {\mathrm {F}} ^ {2} \left(a _ {t} + \frac {a _ {t - 1} (t - 1)}{\gamma_ {t}} + \frac {a _ {t - 1} (t - 1) ^ {2}}{\gamma_ {t} ^ {2}}\right).
+$$
+
+Proof. We have
+
+$$
+\left\| \boldsymbol {W} (\boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \tilde {\boldsymbol {U}} _ {1: t} \tilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \tilde {\boldsymbol {U}} _ {1: t} ^ {\top} - \boldsymbol {I} _ {M _ {t}}) \boldsymbol {H} _ {1: t} \right\| _ {\mathrm {F}} ^ {2}
+$$
+
+$$
+= \operatorname {t r} \left(\boldsymbol {W} \left(\boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} - \boldsymbol {I} _ {M _ {t}}\right) \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \left(\widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} - \boldsymbol {I} _ {M _ {t}}\right) (\boldsymbol {W}) ^ {\top}\right)
+$$
+
+$$
+\leq \operatorname {t r} \left(\boldsymbol {W} \left(\boldsymbol {D} _ {t} \tilde {\boldsymbol {U}} _ {1: t} \tilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \tilde {\boldsymbol {U}} _ {1: t} ^ {\top} + \tilde {\boldsymbol {U}} _ {1: t} \tilde {\boldsymbol {U}} _ {1: t} ^ {\top} - \boldsymbol {I} _ {E}\right) \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \left(\tilde {\boldsymbol {U}} _ {1: t} \tilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \tilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {D} _ {t} + \tilde {\boldsymbol {U}} _ {1: t} \tilde {\boldsymbol {U}} _ {1: t} ^ {\top} - \boldsymbol {I} _ {E}\right) (\boldsymbol {W}) ^ {\top}\right)
+$$
+
+$$
+\begin{array}{l} \stackrel {\text {(i i)}} {\leq} 2 \operatorname {t r} \left(\boldsymbol {W} \left(\widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} - \boldsymbol {I} _ {E}\right) \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \left(\widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} - \boldsymbol {I} _ {E}\right) \left(\boldsymbol {W}\right) ^ {\top}\right) \\ + 2 \operatorname {t r} \left(\boldsymbol {W} \boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {D} _ {t} (\boldsymbol {W}) ^ {\top}\right) \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \stackrel {(i i i)} {\leq} 2 \cdot \| \boldsymbol {W} \| _ {\mathrm {F}} ^ {2} \cdot \left\| \left(\boldsymbol {I} _ {E} - \tilde {\boldsymbol {U}} _ {1: t} \tilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \left(\boldsymbol {I} _ {E} - \tilde {\boldsymbol {U}} _ {1: t} \tilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \right\| \\ + 2 \cdot \| \boldsymbol {W} \| _ {\mathrm {F}} ^ {2} \cdot \left\| \boldsymbol {D} _ {t} \tilde {\boldsymbol {U}} _ {1: t} \tilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \tilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \tilde {\boldsymbol {U}} _ {1: t} \tilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \tilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {D} _ {t} \right\| \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \stackrel {\text {(i v)}} {\leq} 2 \cdot \| \boldsymbol {W} \| _ {\mathrm {F}} ^ {2} \cdot a _ {t} + 2 \cdot \| \boldsymbol {W} \| _ {\mathrm {F}} ^ {2} \cdot a _ {t - 1} \cdot \left(\frac {(t - 1) ^ {2}}{\gamma_ {t} ^ {2}} + \frac {t - 1}{\gamma_ {t}}\right) \\ = 2 \cdot \| \boldsymbol {W} \| _ {\mathrm {F}} ^ {2} \left(a _ {t} + \frac {a _ {t - 1} (t - 1)}{\gamma_ {t}} + \frac {a _ {t - 1} (t - 1) ^ {2}}{\gamma_ {t} ^ {2}}\right). \\ \end{array}
+$$
+
+In the above, (i) follows from Lemma 1, (ii) from Lemma 3, (iii) from Lemma 2, and (iv) from Lemma 5. The proof is complete.
+
+# E PROOFS OF THEOREM 1 AND THEOREM 2
+
+Proof of Theorem 1. Let  $\mathbf{I}_{M_t}$  be the  $M_t \times M_t$  identity matrix. The training loss can be written as
+
+$$
+\begin{array}{l} \left\| \widetilde {\boldsymbol {W}} _ {t} \boldsymbol {H} _ {1: t} - \boldsymbol {Y} _ {1: t} \right\| _ {\mathrm {F}} ^ {2} \\ = \left\| Y _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} - \boldsymbol {Y} _ {1: t} \right\| _ {\mathrm {F}} ^ {2} \\ = \left\| Y _ {1: t} \left(\boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} - \boldsymbol {I} _ {M _ {t}}\right) \right\| _ {\mathrm {F}} ^ {2} \\ = \left\| \left(\boldsymbol {W} _ {t} ^ {*} \boldsymbol {H} _ {1: t} + \mathcal {E} _ {1: t}\right) \left(\boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} - \boldsymbol {I} _ {M _ {t}}\right) \right\| _ {\mathrm {F}} ^ {2} \\ \leq 2 \cdot \left\| \boldsymbol {W} _ {t} ^ {*} (\boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} - \boldsymbol {I} _ {M _ {t}}) \boldsymbol {H} _ {1: t} \right\| _ {\mathrm {F}} ^ {2} + 2 \cdot \left\| \mathcal {E} _ {1: t} (\boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} - \boldsymbol {I} _ {M _ {t}}) \right\| _ {\mathrm {F}} ^ {2}. \\ \end{array}
+$$
+
+We can now bound the first term by Lemma 6 as follows:
+
+$$
+\left\| \boldsymbol {W} _ {t} ^ {*} (\boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} - \boldsymbol {I} _ {M _ {t}}) \boldsymbol {H} _ {1: t} \right\| _ {\mathrm {F}} ^ {2} \leq 2 \cdot \| \boldsymbol {W} _ {t} ^ {*} \| _ {\mathrm {F}} ^ {2} \left(a _ {t} + \frac {a _ {t - 1} (t - 1)}{\gamma_ {t}} + \frac {a _ {t - 1} (t - 1) ^ {2}}{\gamma_ {t} ^ {2}}\right).
+$$
+
+The second term is bounded above as follows:
+
+$$
+\begin{array}{l} 2 \cdot \left\| \mathcal {E} _ {1: t} \left(\boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} - \boldsymbol {I} _ {M _ {t}}\right) \right\| _ {\mathrm {F}} ^ {2} \\ \leq 2 \cdot \| \mathcal {E} _ {1: t} \| ^ {2} \cdot \left\| \left(\boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} - \boldsymbol {I} _ {M _ {t}}\right) \right\| _ {\mathrm {F}} ^ {2} \\ \leq 2 \cdot \| \mathcal {E} _ {1: t} \| ^ {2} \cdot \left(M _ {t} - k _ {t} + \frac {t - 1}{\gamma_ {t} ^ {2}} \min  \left\{M _ {t - 1} - k _ {t - 1}, (t - 1) k _ {t} \right\}\right), \\ \end{array}
+$$
+
+where the first inequality follows from Lemma 2 and the last inequality from Proposition 1.
+
+![](images/f7b51b308566ca7ff211cf0f6bc7a918ee5b349e4ad877ff3d1698acfea18872.jpg)
+
+Proof of Theorem 2. Define  $\pmb{D}_t \coloneqq \sum_{i=1}^t \left( \pmb{B}_i \pmb{B}_i^\top - \tau_{k_i} \left( \pmb{B}_i \pmb{B}_i^\top \right) \right)$ . Note that  $\pmb{D}_t$  is a symmetric and positive semi-definite matrix.
+
+Note that we have
+
+$$
+\begin{array}{l} \mathbb {E} _ {\boldsymbol {h}} \left[ \left\| \widetilde {\boldsymbol {W}} _ {t} \boldsymbol {h} - \boldsymbol {y} \right\| ^ {2} \right] = \mathbb {E} _ {\boldsymbol {h}} \left[ \left\| \widetilde {\boldsymbol {W}} _ {t} \boldsymbol {h} - \boldsymbol {W} _ {t} ^ {*} \boldsymbol {h} - \epsilon \right\| ^ {2} \right] \\ = 2 \cdot \mathbb {E} _ {\boldsymbol {h}} \left[ \left\| \widetilde {\boldsymbol {W}} _ {t} \boldsymbol {h} - \boldsymbol {W} _ {t} ^ {*} \boldsymbol {h} \right\| ^ {2} \right] + 2 \cdot \| \epsilon \| ^ {2}, \\ \end{array}
+$$
+
+so we next focus on bounding  $\mathbb{E}_h\left[\| \widetilde{\boldsymbol{W}}_t\boldsymbol {h} - \boldsymbol {W}_t^*\boldsymbol {h}\| ^2\right]$ . With the  $E\times E$  identity matrix  $\pmb{I}_{E}$  and  $\Lambda \coloneqq \mathbb{E}[hh^{\top}]$ , we have
+
+$$
+\begin{array}{l} \mathbb {E} _ {\boldsymbol {h}} \left[ \| \widetilde {\boldsymbol {W}} _ {t} \boldsymbol {h} - \boldsymbol {W} _ {t} ^ {*} \boldsymbol {h} \| ^ {2} \right] \\ = \mathbb {E} _ {\boldsymbol {h}} \left[ \| \boldsymbol {Y} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {h} - \boldsymbol {W} _ {t} ^ {*} \boldsymbol {h} \| ^ {2} \right] \\ = \mathbb {E} _ {\boldsymbol {h}} \left[ \left\| \left(\boldsymbol {W} _ {t} ^ {*} \boldsymbol {H} _ {1: t} + \mathcal {E} _ {1: t}\right) \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {h} - \boldsymbol {W} _ {t} ^ {*} \boldsymbol {h} \right\| ^ {2} \right] \\ = \mathbb {E} _ {\boldsymbol {h}} \left[ \| \left(\boldsymbol {W} _ {t} ^ {*} \boldsymbol {H} _ {1: t} + \mathcal {E} _ {1: t}\right) \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {h} - \boldsymbol {W} _ {t} ^ {*} \boldsymbol {h} \| ^ {2} \right] \\ \leq 2 \cdot \mathbb {E} _ {\boldsymbol {h}} \left[ \| \boldsymbol {W} _ {t} ^ {*} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {h} - \boldsymbol {W} _ {t} ^ {*} \boldsymbol {h} \| ^ {2} \right] + 2 \cdot \mathbb {E} _ {\boldsymbol {h}} \left[ \| \mathcal {E} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {h} \| ^ {2} \right] \\ \leq 2 \cdot \mathbb {E} _ {\boldsymbol {h}} \left[ \| \boldsymbol {W} _ {t} ^ {*} (\boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} - \boldsymbol {I} _ {E}) \boldsymbol {h} \| ^ {2} \right] + 2 \cdot \| \mathcal {E} _ {1: t} \| ^ {2} \cdot \mathbb {E} _ {\boldsymbol {h}} \left[ \| \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {h} \| ^ {2} \right] \\ \end{array}
+$$
+
+The term  $\mathbb{E}_h\| H_{1:t}^\top \widetilde{U}_{1:t}\widetilde{\Sigma}_{1:t}^{-2}\widetilde{U}_{1:t}^\top h\|^2$  can be bounded above as follows:
+
+$$
+\begin{array}{l} \mathbb {E} _ {\boldsymbol {h}} \left\| \boldsymbol {H} _ {1: t} ^ {\top} \tilde {\boldsymbol {U}} _ {1: t} \tilde {\Sigma} _ {1: t} ^ {- 2} \tilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {h} \right\| ^ {2} \\ = \operatorname {t r} \left(\boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \Lambda \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t}\right) \\ = \operatorname {t r} \left(\left(\boldsymbol {\Lambda} - \frac {1}{M _ {t}} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} + \frac {1}{M _ {t}} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top}\right) \tilde {\boldsymbol {U}} _ {1: t} \tilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \tilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \tilde {\boldsymbol {U}} _ {1: t} \tilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \tilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \\ \stackrel {(i)} {\leq} \left\| \boldsymbol {\Lambda} - \frac {1}{M _ {t}} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \right\| \cdot \operatorname {t r} \left(\widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \\ + \frac {1}{M _ {t}} \operatorname {t r} \left(\boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \\ \stackrel {\text {(i i)}} {\leq} \left\| \boldsymbol {\Lambda} - \frac {1}{M _ {t}} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \right\| \cdot \frac {\left(\frac {1}{\gamma_ {t}} \min  \left\{M _ {t - 1} - k _ {t - 1} , (t - 1) k _ {t} \right\} + k _ {t}\right)}{\mu_ {k _ {t}} \left(\boldsymbol {B} _ {t} \boldsymbol {B} _ {t} ^ {\top}\right)} \\ + \frac {1}{M _ {t} \gamma_ {t}} \left(\frac {t - 1}{\gamma_ {t} ^ {2} M _ {t}} + \frac {2}{\gamma_ {t} M _ {t}}\right) \cdot \min  \left\{M _ {t - 1} - k _ {t - 1}, (t - 1) k _ {t} \right\} + \frac {k _ {t}}{M _ {t}} \\ =: \mathbb {V} _ {t} \\ \end{array}
+$$
+
+Here, (i) follows from Lemma 2 and (ii) follows from Lemma 5.
+
+The term  $\mathbb{E}_h\left[\| W_t^* (\pmb {H}_{1:t}\pmb{H}_{1:t}^\top \widetilde{\pmb{U}}_{1:t}\widetilde{\pmb{\Sigma}}_{1:t}^{-2}\widetilde{\pmb{U}}_{1:t}^\top -\pmb {I}_E)\pmb {h}\| ^2\right]$  satisfies:
+
+$$
+\mathbb {E} _ {\boldsymbol {h}} \left[ \| W _ {t} ^ {*} \left(\boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} - \boldsymbol {I} _ {E}\right) \boldsymbol {h} \| ^ {2} \right]
+$$
+
+$$
+= \operatorname {t r} \left(\boldsymbol {W} _ {t} ^ {*} \left(\boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} - \boldsymbol {I} _ {E}\right) \boldsymbol {\Lambda} \left(\widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} - \boldsymbol {I} _ {E}\right) \left(\boldsymbol {W} _ {t} ^ {*}\right) ^ {\top}\right)
+$$
+
+$$
+\stackrel {\mathrm {(i)}} {=} \operatorname {t r} \left(\boldsymbol {W} _ {t} ^ {*} (\boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} + \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} - \boldsymbol {I} _ {E}) \boldsymbol {\Lambda} (\widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {D} _ {t} + \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} - \boldsymbol {I} _ {E}) (\boldsymbol {W} _ {t} ^ {*}) ^ {\top}\right)
+$$
+
+$$
+\begin{array}{l} \stackrel {\text {(i i)}} {\leq} 2 \operatorname {t r} \left(\boldsymbol {W} _ {t} ^ {*} (\widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} - \boldsymbol {I} _ {E}) \boldsymbol {\Lambda} (\widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} - \boldsymbol {I} _ {E}) (\boldsymbol {W} _ {t} ^ {*}) ^ {\top}\right) \\ + 2 \operatorname {t r} \left(\boldsymbol {W} _ {t} ^ {*} \boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {\Lambda} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {D} _ {t} \left(\boldsymbol {W} _ {t} ^ {*}\right) ^ {\top}\right) \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \stackrel {(i i i)} {\leq} 2 \cdot \left\| W _ {t} ^ {*} \left(\boldsymbol {I} _ {E} - \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \right\| _ {\mathrm {F}} ^ {2} \cdot \left\| \left(\boldsymbol {I} _ {E} - \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \Lambda \left(\boldsymbol {I} _ {E} - \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \right\| \\ + 2 \cdot \| W _ {t} ^ {*} \| _ {\mathrm {F}} ^ {2} \cdot \left\| D _ {t} \tilde {U} _ {1: t} \tilde {\Sigma} _ {1: t} ^ {- 2} \tilde {U} _ {1: t} ^ {\top} \Lambda \tilde {U} _ {1: t} \tilde {\Sigma} _ {1: t} ^ {- 2} \tilde {U} _ {1: t} ^ {\top} D _ {t} \right\| \\ \end{array}
+$$
+
+where the above three steps, (i), (ii), and (iii), follow from Lemma 1, Lemma 3, and Lemma 2 respectively. To bound  $\mathbb{B}_{t1} \coloneqq \left\| (\boldsymbol{I}_E - \widetilde{\boldsymbol{U}}_{1:t}\widetilde{\boldsymbol{U}}_{1:t}^\top)\boldsymbol{\Lambda}(\boldsymbol{I}_E - \widetilde{\boldsymbol{U}}_{1:t}\widetilde{\boldsymbol{U}}_{1:t}^\top) \right\|$ , we have
+
+$$
+\begin{array}{l} \mathbb {B} _ {t 1} = \left\| \left(\boldsymbol {I} _ {E} - \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \left(\boldsymbol {\Lambda} - \frac {1}{M _ {t}} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \left(\boldsymbol {I} _ {E} - \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) \right\| \\ \leq \left\| \Lambda - \frac {1}{M _ {t}} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \right\| \\ \leq \left\| \boldsymbol {\Lambda} - \frac {1}{M _ {t}} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \right\| + \frac {1}{M _ {t}} \cdot \left\| \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} - \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \right\| \\ = \left\| \boldsymbol {\Lambda} - \frac {1}{M _ {t}} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \right\| + \frac {1}{M _ {t}} \cdot \| \boldsymbol {D} _ {t} \| \\ = \left\| \boldsymbol {\Lambda} - \frac {1}{M _ {t}} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \right\| + \frac {a _ {t}}{M _ {t}}. \\ \end{array}
+$$
+
+To bound  $\mathbb{B}_{t2} := \left\| D_t \widetilde{U}_{1:t} \widetilde{\Sigma}_{1:t}^{-2} \widetilde{U}_{1:t}^\top \Lambda \widetilde{U}_{1:t} \widetilde{\Sigma}_{1:t}^{-2} \widetilde{U}_{1:t}^\top D_t \right\|$ , we have
+
+$$
+\begin{array}{l} \mathbb {B} _ {t 2} = \left\| \boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \Lambda \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {D} _ {t} \right\| \\ = \left\| \boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \left(\boldsymbol {\Lambda} - \frac {1}{M _ {t}} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} + \frac {1}{M _ {t}} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top}\right) \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {D} _ {t} \right\| \\ \leq \left\| \boldsymbol {\Lambda} - \frac {1}{M _ {t}} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \right\| \cdot \left\| \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {D} _ {t} \right\| ^ {2} \\ + \frac {1}{M _ {t}} \left\| \boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {D} _ {t} \right\| \\ \leq \left\| \boldsymbol {\Lambda} - \frac {1}{M _ {t}} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \right\| \cdot \frac {(t - 1) ^ {2}}{\gamma_ {t} ^ {2}} + a _ {t - 1} \cdot \left(\frac {(t - 1) ^ {2}}{\gamma_ {t} ^ {2}} + \frac {t - 1}{\gamma_ {t}}\right), \\ \end{array}
+$$
+
+where the last step follows from Lemma 5. Putting together, we have obtained
+
+$$
+\begin{array}{l} \frac {\mathbb {E} _ {\boldsymbol {h}} \left[ \| \boldsymbol {W} _ {t} ^ {*} (\boldsymbol {H} _ {1 : t} \boldsymbol {H} _ {1 : t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1 : t} \widetilde {\boldsymbol {\Sigma}} _ {1 : t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1 : t} ^ {\top} - \boldsymbol {I} _ {E}) \boldsymbol {h} \| ^ {2} \right]}{2 \cdot \| \boldsymbol {W} _ {t} ^ {*} \| _ {\mathrm {F}} ^ {2}} \\ \leq \mathbb {B} _ {t 1} + \mathbb {B} _ {t 2} \\ \leq \left\| \Lambda - \frac {1}{M _ {t}} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \right\| \cdot \left(1 + \frac {(t - 1) ^ {2}}{\gamma_ {t} ^ {2}}\right) + \frac {a _ {t - 1}}{M _ {t}} \cdot \left(\frac {(t - 1) ^ {2}}{\gamma_ {t} ^ {2}} + \frac {t - 1}{\gamma_ {t}}\right) + \frac {a _ {t}}{M _ {t}} =: \mathbb {B} _ {t}. \\ \end{array}
+$$
+
+Combining the above finishes the proof.
+
+![](images/a989b03bbcefd84c0ea971e4ac7c5073ce0223ba42927cbd33d7c4697907c289.jpg)
+
+# F THEORETICAL GUARANTEES UNDER GAUSSIAN ASSUMPTIONS
+
+In this section, we prove slightly tighter results than Theorems 1 and 2 presented in the main paper. The key idea is to make certain Gaussian assumptions on noise. Specifically, we assume both the
+
+training noise  $\mathcal{E}_{1:t}$  and test noise  $\epsilon$  have i.i.d.  $\mathcal{N}(0,\nu^2)$  entries. With these, we present and prove Theorems 3 to 5 below.
+
+Theorem 3. On top of the settings of Theorem 1, furthermore assume  $\mathcal{E}_{1:t}$  consists of i.i.d.  $\mathcal{N}(0,\nu^2)$  entries. Then the output  $\widetilde{\boldsymbol{W}}_t = \boldsymbol{Y}_{1:t}\boldsymbol{H}_{1:t}^\top\widetilde{\boldsymbol{U}}_{1:t}\widetilde{\boldsymbol{\Sigma}}_{1:t}^{-2}\widetilde{\boldsymbol{U}}_{1:t}^\top$  of our method (4) satisfies
+
+$$
+\begin{array}{l} \frac {1}{M _ {t}} \mathbb {E} _ {\varepsilon_ {1: t}} \left\| \widetilde {\boldsymbol {W}} _ {t} \boldsymbol {H} _ {1: t} - \boldsymbol {Y} _ {1: t} \right\| _ {\mathrm {F}} ^ {2} \leq 2 \cdot \left\| \boldsymbol {W} _ {t} ^ {*} \right\| _ {\mathrm {F}} ^ {2} \left(\frac {a _ {t}}{M _ {t}} + \frac {a _ {t - 1} (t - 1)}{\gamma_ {t} M _ {t}} + \frac {a _ {t - 1} (t - 1) ^ {2}}{\gamma_ {t} ^ {2} M _ {t}}\right) \tag {15} \\ + c _ {t} \nu^ {2} \left(\frac {(M _ {t} - k _ {t})}{M _ {t}} + \frac {(t - 1) \min \{M _ {t - 1} - k _ {t - 1} , (t - 1) k _ {t} \}}{\gamma_ {t} ^ {2} M _ {t}}\right). \\ \end{array}
+$$
+
+Proof of Theorem 3. Let  $\mathbf{I}_{M_t}$  be the  $M_t \times M_t$  identity matrix. The training loss can be written as
+
+$$
+\begin{array}{l} \mathbb {E} _ {\mathcal {E} _ {1: t}} \left\| \widetilde {\boldsymbol {W}} _ {t} \boldsymbol {H} _ {1: t} - \boldsymbol {Y} _ {1: t} \right\| _ {\mathrm {F}} ^ {2} \\ = \mathbb {E} _ {\mathcal {E} _ {1: t}} \left\| \boldsymbol {Y} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} - \boldsymbol {Y} _ {1: t} \right\| _ {\mathrm {F}} ^ {2} \\ = \mathbb {E} _ {\mathcal {E} _ {1: t}} \left\| \boldsymbol {Y} _ {1: t} \left(\boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} - \boldsymbol {I} _ {M _ {t}}\right) \right\| _ {\mathrm {F}} ^ {2} \\ = \mathbb {E} _ {\mathcal {E} _ {1: t}} \left\| \left(\boldsymbol {W} _ {t} ^ {*} \boldsymbol {H} _ {1: t} + \mathcal {E} _ {1: t}\right) \left(\boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} - \boldsymbol {I} _ {M _ {t}}\right) \right\| _ {\mathrm {F}} ^ {2} \\ = \left\| W _ {t} ^ {*} \left(\boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} - \boldsymbol {I} _ {M _ {t}}\right) \boldsymbol {H} _ {1: t} \right\| _ {\mathrm {F}} ^ {2} + \mathbb {E} _ {\mathcal {E} _ {1: t}} \left\| \mathcal {E} _ {1: t} \left(\boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} - \boldsymbol {I} _ {M _ {t}}\right) \right\| _ {\mathrm {F}} ^ {2}. \\ \end{array}
+$$
+
+We can now bound the first term by Lemma 6 as follows:
+
+$$
+\left\| \boldsymbol {W} _ {t} ^ {*} (\boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} - \boldsymbol {I} _ {M _ {t}}) \boldsymbol {H} _ {1: t} \right\| _ {\mathrm {F}} ^ {2} \leq 2 \cdot \| \boldsymbol {W} _ {t} ^ {*} \| _ {\mathrm {F}} ^ {2} \left(a _ {t} + \frac {a _ {t - 1} (t - 1)}{\gamma_ {t}} + \frac {a _ {t - 1} (t - 1) ^ {2}}{\gamma_ {t} ^ {2}}\right).
+$$
+
+The second term can be bounded above as follows:
+
+$$
+\begin{array}{l} \left. \mathbb {E} _ {\mathcal {E}} \right\| \mathcal {E} _ {1: t} \left(\boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} - \boldsymbol {I} _ {M _ {t}}\right) \| _ {\mathrm {F}} ^ {2} \\ = \mathbb {E} _ {\mathcal {E}} \operatorname {t r} \left(\left(\boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} - \boldsymbol {I} _ {M _ {t}}\right) \mathcal {E} _ {1: t} ^ {\top} \mathcal {E} _ {1: t} \left(\boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} - \boldsymbol {I} _ {M _ {t}\right)\right) \\ = c _ {t} \nu^ {2} \cdot \left\| \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} - \boldsymbol {I} _ {M _ {t}} \right\| _ {\mathrm {F}} ^ {2} \\ \leq c _ {t} \nu^ {2} \cdot (M _ {t} - k _ {t}) + c _ {t} \nu^ {2} \cdot \frac {t - 1}{\gamma_ {t} ^ {2}} \min  \left\{M _ {t - 1} - k _ {t - 1}, (t - 1) k _ {t} \right\}. \\ \end{array}
+$$
+
+The last inequality follows from Proposition 1. Combining the above finishes the proof.
+
+While in Theorem 3 bounds the average training MSE loss  $\frac{1}{M_t}\mathbb{E}_{\mathcal{E}_{1:t}}\left\|\widetilde{\boldsymbol{W}}_t\boldsymbol{H}_{1:t}-\boldsymbol{Y}_{1:t}\right\|_{\mathrm{F}}^2$ , an alternative is to give a bound on  $\frac{1}{M_t}\mathbb{E}_{\mathcal{E}_{1:t}}\left\|\widetilde{\boldsymbol{W}}_t\boldsymbol{H}_{1:t}-\boldsymbol{W}_t^*\boldsymbol{H}_{1:t}\right\|_{\mathrm{F}}^2$ . The latter term evaluates the difference between the prediction of  $\widetilde{\boldsymbol{W}}_t$  and the ground-truth  $\boldsymbol{W}_t^*$  on training data  $\boldsymbol{H}_{1:t}$ . The difference between the two terms is that  $\boldsymbol{Y}_{1:t} = \boldsymbol{W}_t^*\boldsymbol{H}_{1:t} + \mathcal{E}_{1:t}$  is contaminated by noise. We bound  $\frac{1}{M_t}\mathbb{E}_{\mathcal{E}_{1:t}}\left\|\widetilde{\boldsymbol{W}}_t\boldsymbol{H}_{1:t}-\boldsymbol{W}_t^*\boldsymbol{H}_{1:t}\right\|_{\mathrm{F}}^2$  in the next result.
+
+Theorem 4. On top of the settings of Theorem 1, furthermore assume  $\mathcal{E}_{1:t}$  consists of i.i.d.  $\mathcal{N}(0,\nu^2)$  entries. Then the output  $\widetilde{\boldsymbol{W}}_t = \boldsymbol{Y}_{1:t}\boldsymbol{H}_{1:t}^\top \widetilde{\boldsymbol{U}}_{1:t}\widetilde{\boldsymbol{\Sigma}}_{1:t}^{-2}\widetilde{\boldsymbol{U}}_{1:t}^\top$  of our method (4) satisfies
+
+$$
+\begin{array}{l} \frac {1}{M _ {t}} \mathbb {E} _ {\mathcal {E} _ {1: t}} \left\| \widetilde {\boldsymbol {W}} _ {t} \boldsymbol {H} _ {1: t} - \boldsymbol {W} _ {t} ^ {*} \boldsymbol {H} _ {1: t} \right\| _ {\mathrm {F}} ^ {2} \leq 2 \cdot \left\| \boldsymbol {W} _ {t} ^ {*} \right\| _ {\mathrm {F}} ^ {2} \left(\frac {a _ {t}}{M _ {t}} + \frac {a _ {t - 1} (t - 1)}{\gamma_ {t} M _ {t}} + \frac {a _ {t - 1} (t - 1) ^ {2}}{\gamma_ {t} ^ {2} M _ {t}}\right) \\ + c _ {t} \nu^ {2} \cdot \left(\frac {k _ {t}}{M _ {t}} + \left(\frac {t - 1}{\gamma_ {t} ^ {2} M _ {t}} + \frac {2}{\gamma_ {t} M _ {t}}\right) \min \left\{M _ {t - 1} - k _ {t - 1}, (t - 1) k _ {t} \right\}\right). \\ \end{array}
+$$
+
+Proof of Theorem 4. We have
+
+$$
+\begin{array}{l} \mathbb {E} _ {\mathcal {E} _ {1: t}} \left\| \widetilde {\boldsymbol {W}} _ {t} \boldsymbol {H} _ {1: t} - \boldsymbol {W} _ {t} ^ {*} \boldsymbol {H} _ {1: t} \right\| _ {\mathrm {F}} ^ {2} \\ = \mathbb {E} _ {\mathcal {E} _ {1: t}} \left\| \boldsymbol {Y} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} - \boldsymbol {W} _ {t} ^ {*} \boldsymbol {H} _ {1: t} \right\| _ {\mathrm {F}} ^ {2} \\ = \mathbb {E} _ {\mathcal {E} _ {1: t}} \left\| W _ {t} ^ {*} \boldsymbol {H} _ {1: t} \left(\boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} - \boldsymbol {I} _ {M _ {t}}\right) + \mathcal {E} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} \right\| _ {\mathrm {F}} ^ {2} \\ = \left\| W _ {t} ^ {*} \left(\boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} - \boldsymbol {I} _ {M _ {t}}\right) \boldsymbol {H} _ {1: t} \right\| _ {\mathrm {F}} ^ {2} + \mathbb {E} _ {\mathcal {E} _ {1: t}} \left\| \mathcal {E} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} \right\| _ {\mathrm {F}} ^ {2}. \\ \end{array}
+$$
+
+The first term is identical to that of Theorem 3, and it remains to bound the second term:
+
+$$
+\begin{array}{l} \mathbb {E} _ {\mathcal {E} _ {1: t}} \left\| \mathcal {E} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} \right\| _ {\mathrm {F}} ^ {2} \\ = \mathbb {E} _ {\mathcal {E}} \operatorname {t r} \left(\boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} \mathcal {E} _ {1: t} ^ {\top} \mathcal {E} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t}\right) \\ = c _ {t} \nu^ {2} \cdot \left\| \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} \right\| _ {\mathrm {F}} ^ {2} \\ \leq c _ {t} \nu^ {2} \cdot \left(\frac {t - 1}{\gamma_ {t} ^ {2}} + \frac {2}{\gamma_ {t}}\right) \min  \left\{M _ {t - 1} - k _ {t - 1}, (t - 1) k _ {t} \right\} + c _ {t} \nu^ {2} \cdot k _ {t}. \\ \end{array}
+$$
+
+The last inequality follows from Lemma 5.
+
+![](images/61e9c8e17b714091ef4043f2a549c8ae547cf515f9b7b8088211e42f08276990.jpg)
+
+Theorem 5. On top of the settings of Theorem 2, furthermore assume both  $\mathcal{E}_{1:t}$  and  $\epsilon$  consists of i.i.d.  $\mathcal{N}(0,\nu^2)$  entries. Then the output  $\widetilde{\boldsymbol{W}}_t = \boldsymbol{Y}_{1:t}\boldsymbol{H}_{1:t}^\top \widetilde{\boldsymbol{U}}_{1:t}\widetilde{\boldsymbol{\Sigma}}_{1:t}^{-2}\widetilde{\boldsymbol{U}}_{1:t}^\top$  of our method (4) satisfies
+
+$$
+\mathbb {E} _ {\mathcal {E} _ {1: t}, \boldsymbol {h}, \epsilon} \| \widetilde {\boldsymbol {W}} _ {t} \boldsymbol {h} - \boldsymbol {y} \| ^ {2} \leq 2 \cdot \| \boldsymbol {W} _ {t} ^ {*} \| _ {\mathrm {F}} ^ {2} \cdot \mathbb {B} _ {t} + c _ {t} \nu^ {2} \cdot \mathbb {V} _ {t} + c _ {t} \nu^ {2}. \tag {16}
+$$
+
+where  $\mathbb{B}_t$  and  $\mathbb{V}_t$  are defined in (10) and also shown below:
+
+$$
+\mathbb {B} _ {t} = \left\| \boldsymbol {\Lambda} - \frac {1}{M _ {t}} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \right\| \left(1 + \frac {(t - 1) ^ {2}}{\gamma_ {t} ^ {2}}\right) + \left(\frac {a _ {t}}{M _ {t}} + \frac {a _ {t - 1} (t - 1)}{\gamma_ {t} M _ {t}} + \frac {a _ {t - 1} (t - 1) ^ {2}}{\gamma_ {t} ^ {2} M _ {t}}\right)
+$$
+
+$$
+\begin{array}{l} \mathbb {V} _ {t} = \left\| \boldsymbol {\Lambda} - \frac {1}{M _ {t}} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \right\| \cdot \frac {\left(\frac {1}{\gamma_ {t}} \min  \left\{M _ {t - 1} - k _ {t - 1} , (t - 1) k _ {t} \right\} + k _ {t}\right)}{\mu_ {k _ {t}} (\boldsymbol {B} _ {t} \boldsymbol {B} _ {t} ^ {\top})} \\ + \frac {k _ {t}}{M _ {t}} + \left(\frac {t - 1}{\gamma_ {t} ^ {2} M _ {t}} + \frac {2}{\gamma_ {t} M _ {t}}\right) \cdot \min \left\{M _ {t - 1} - k _ {t - 1}, (t - 1) k _ {t} \right\}. \\ \end{array}
+$$
+
+Proof of Theorem 5. Recall the definition of  $D_{t}$  in (14). Note that  $D_{t}$  is a symmetric and positive semi-definite matrix.
+
+Note that for any  $\mathbf{W} \in \mathbb{R}^{c_t \times E}$  we have
+
+$$
+\begin{array}{l} \mathbb {E} _ {\mathcal {E} _ {1: t}, \boldsymbol {h}, \epsilon} \left[ \| \boldsymbol {W h} - \boldsymbol {y} \| ^ {2} \right] = \mathbb {E} _ {\mathcal {E} _ {1: t}, \boldsymbol {h}, \epsilon} \left[ \| \boldsymbol {W h} - \boldsymbol {W} _ {t} ^ {*} \boldsymbol {h} - \epsilon \| ^ {2} \right] \\ = \mathbb {E} _ {\mathcal {E} _ {1: t}, \boldsymbol {h}} \left[ \| \boldsymbol {W} \boldsymbol {h} - \boldsymbol {W} _ {t} ^ {*} \boldsymbol {h} \| ^ {2} \right] + c _ {t} \nu^ {2}. \\ \end{array}
+$$
+
+Denote by  $\mathbf{I}_E$  the  $E\times E$  identity matrix. With  $\widetilde{\boldsymbol{W}}_t = \boldsymbol {Y}_{1:t}\boldsymbol{H}_{1:t}^\top \widetilde{\boldsymbol{U}}_{1:t}\widetilde{\boldsymbol{\Sigma}}_{1:t}^{-2}\widetilde{\boldsymbol{U}}_{1:t}^\top$  and  $\mathbf{Y}_{1:t} = \mathbf{W}_t^*\mathbf{H}_{1:t} + \mathcal{E}_{1:t}$  we obtain
+
+$$
+\begin{array}{l} \mathbb {E} _ {\mathcal {E} _ {1: t}, \boldsymbol {h}} \left[ \| \widetilde {\boldsymbol {W}} _ {t} \boldsymbol {h} - \boldsymbol {W} _ {t} ^ {*} \boldsymbol {h} \| ^ {2} \right] \\ = \mathbb {E} _ {\mathcal {E} _ {1: t}, \boldsymbol {h}} \left[ \| \boldsymbol {Y} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {h} - \boldsymbol {W} _ {t} ^ {*} \boldsymbol {h} \| ^ {2} \right] \\ = \mathbb {E} _ {\mathcal {E} _ {1: t}, \boldsymbol {h}} \left[ \| \left(\boldsymbol {W} _ {t} ^ {*} \boldsymbol {H} _ {1: t} + \mathcal {E} _ {1: t}\right) \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {h} - \boldsymbol {W} _ {t} ^ {*} \boldsymbol {h} \| ^ {2} \right] \\ = \mathbb {E} _ {\mathcal {E} _ {1: t}, \boldsymbol {h}} \left[ \| \left(\boldsymbol {W} _ {t} ^ {*} \boldsymbol {H} _ {1: t} + \mathcal {E} _ {1: t}\right) \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {h} - \boldsymbol {W} _ {t} ^ {*} \boldsymbol {h} \| ^ {2} \right] \\ = \mathbb {E} _ {\boldsymbol {h}} \left[ \| \boldsymbol {W} _ {t} ^ {*} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {h} - \boldsymbol {W} _ {t} ^ {*} \boldsymbol {h} \| ^ {2} \right] + \mathbb {E} _ {\mathcal {E} _ {1: t}, \boldsymbol {h}} \left[ \| \mathcal {E} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {h} \| ^ {2} \right] \\ = \mathbb {E} _ {\boldsymbol {h}} \left[ \| \boldsymbol {W} _ {t} ^ {*} (\boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} - \boldsymbol {I} _ {E}) \boldsymbol {h} \| ^ {2} \right] + c _ {t} \nu^ {2} \mathbb {E} _ {\boldsymbol {h}} \left[ \| \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {h} \| ^ {2} \right] \\ \end{array}
+$$
+
+The rest of the proof is identical to that of Theorem 2.
+
+![](images/136a63d42b555315886194f1ad3c12a717764a50ae16a5c55971b7c2a131eb93.jpg)
+
+# G ADDITIONAL THEORETICAL RESULTS
+
+Given the weight  $\widetilde{\boldsymbol{W}}_t\coloneqq \boldsymbol{Y}_{1:t}\boldsymbol{H}_{1:t}^\top \widetilde{\boldsymbol{U}}_{1:t}\widetilde{\boldsymbol{\Sigma}}_{1:t}^{-2}\widetilde{\boldsymbol{U}}_{1:t}^\top$  computed by our continual implementation in Section 3, here we aim to derive upper bounds on the training MSE losses  $\frac{1}{M_t}\left\| \widetilde{\boldsymbol{W}}_t\boldsymbol {H}_{1:t} - \boldsymbol {Y}_{1:t}\right\| _F^2$  without the linear model assumption  $Y_{1:t} = W_{t}^{*}H_{1:t} + \mathcal{E}_{1:t}$  as used in the main paper.
+
+First observe that
+
+$$
+\left\| \widetilde {\boldsymbol {W}} _ {t} \boldsymbol {H} _ {1: t} - \boldsymbol {Y} _ {1: t} \right\| _ {\mathrm {F}} ^ {2} = \left\| \boldsymbol {Y} _ {1: t} \left(\boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} - \boldsymbol {I} _ {M _ {t}}\right) \right\| _ {\mathrm {F}} ^ {2},
+$$
+
+where we recall  $\mathbf{I}_{M_t}$  is the  $M_t \times M_t$  identity matrix. This motivates us to give a bound on  $\| (\mathbf{H}_{1:t}^\top \widetilde{\mathbf{U}}_{1:t} \widetilde{\boldsymbol{\Sigma}}_{1:t}^{-2} \widetilde{\mathbf{U}}_{1:t}^\top \mathbf{H}_{1:t} - \mathbf{I}_{M_t}) \|_F^2$ :
+
+Proposition 1. It holds for every  $t \geq 1$  that  $(M_0 := 0, k_0 := 0)$ .
+
+$$
+\left\| \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} - \boldsymbol {I} _ {M _ {t}} \right\| _ {\mathrm {F}} ^ {2} \leq M _ {t} - k _ {t} + \frac {t - 1}{\gamma_ {t} ^ {2}} \min  \left\{M _ {t - 1} - k _ {t - 1}, (t - 1) k _ {t} \right\}.
+$$
+
+Remark 6. The term  $M_{t} - k_{t}$  is inevitable as we truncate  $M_{t} - k_{t}$  eigenvalues. Indeed,  $M_{t} - k_{t}$  is precisely equal to  $\| H_{1:t}^{\top}\overline{U}_{1:t}\overline{\Sigma}_{1:t}^{-2}\overline{U}_{1:t}^{\top}H_{1:t} - I_{M_t}\| _F^2$ , and it is the minimum of a rank-  $k_{t}$  approximation problem:
+
+$$
+M _ {t} - k _ {t} = \min  _ {\boldsymbol {L} \in \mathbb {R} ^ {M _ {t} \times k _ {t}}} \| \boldsymbol {L L} ^ {\top} - \boldsymbol {I} _ {M _ {t}} \| _ {\mathrm {F}} ^ {2}.
+$$
+
+The term  $\frac{t - 1}{\gamma_t^2} \min \left\{ M_{t - 1} - k_{t - 1}, (t - 1)k_t \right\}$  arises as we solve LoRanPAC continually rather than offline. With  $\gamma_t = 1$ , this term is upper bounded by  $(t - 1)(M_{t - 1} - k_{t - 1})$ . With  $\gamma_t = 10^{10}$  (as discussed in the main paper), this term is negligible for even hundreds of tasks.
+
+Proof of Proposition 1. From Lemma 1 it follows that
+
+$$
+\boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} = \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} + \boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}, \tag {17}
+$$
+
+where we recall  $D_{t}$  is defined as  $\pmb{D}_{t} = \sum_{i = 1}^{t}\left(\pmb{B}_{i}\pmb{B}_{i}^{\top} - \tau_{k_{i}}\left(\pmb{B}_{i}\pmb{B}_{i}^{\top}\right)\right)$  in (14). Then we have
+
+$$
+\begin{array}{l} \left\| \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} - \boldsymbol {I} _ {M _ {t}} \right\| _ {\mathrm {F}} ^ {2} \\ = \operatorname {t r} \left(\boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} - 2 \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} + \boldsymbol {I} _ {M _ {t}}\right) \\ = \operatorname {t r} \left(\boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} - 2 \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) + M _ {t} \\ \stackrel {(1 7)} {=} \operatorname {t r} \left( \right.\left(\tilde {\boldsymbol {U}} _ {1: t} \tilde {\boldsymbol {U}} _ {1: t} ^ {\top} + \boldsymbol {D} _ {t} \tilde {\boldsymbol {U}} _ {1: t} \tilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \tilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right)\left(\tilde {\boldsymbol {U}} _ {1: t} \tilde {\boldsymbol {U}} _ {1: t} ^ {\top} + \boldsymbol {D} _ {t} \tilde {\boldsymbol {U}} _ {1: t} \tilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \tilde {\boldsymbol {U}} _ {1: t} ^ {\top})\right) + M _ {t} \\ - 2 \left(\widetilde {U} _ {1: t} \widetilde {U} _ {1: t} ^ {\top} + D _ {t} \widetilde {U} _ {1: t} \widetilde {\Sigma} _ {1: t} ^ {- 2} \widetilde {U} _ {1: t} ^ {\top}\right) \\ = \operatorname {t r} \left(\boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} - \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) + M _ {t} \\ = \operatorname {t r} \left(\boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}\right) + M _ {t} - k _ {t} \\ \stackrel {\mathrm {(i)}} {\leq} \frac {t - 1}{\gamma_ {t} ^ {2}} \min  \left\{M _ {t - 1} - k _ {t - 1}, (t - 1) k _ {t} \right\} + M _ {t} - k _ {t} \\ \end{array}
+$$
+
+where (i) is due to Lemma 5.
+
+![](images/e86109011e6dbc50d74768c1d54f6018d682b8df0fe55ae295ca56464945a414.jpg)
+
+A simple corollary of Proposition 1 now follows:
+
+Corollary 1. The output  $\widetilde{W}_t$  of Algorithm 4 satisfies
+
+$$
+\frac {1}{M _ {t}} \left\| \widetilde {\boldsymbol {W}} _ {t} \boldsymbol {H} _ {1: t} - \boldsymbol {Y} _ {1: t} \right\| _ {\mathrm {F}} ^ {2} \leq \frac {\left\| \boldsymbol {Y} _ {1 : t} ^ {\top} \boldsymbol {Y} _ {1 : t} \right\|}{M _ {t}} \left(M _ {t} - k _ {t} + \frac {t - 1}{\gamma_ {t} ^ {2}} \min  \left\{M _ {t - 1} - k _ {t - 1}, (t - 1) k _ {t} \right\}\right).
+$$
+
+Remark 7. In classification, the columns of  $\mathbf{Y}_{1:t}$  are one-hot vectors. Hence, up to permutation,  $\mathbf{Y}_{1:t}^{\top} \mathbf{Y}_{1:t} \in \mathbb{R}^{M_t \times M_t}$  is a block diagonal matrix with  $c_t$  block, where the  $i$ -th diagonal block is a  $n_i \times n_i$  matrix of all ones  $\mathbf{1}_{n_i}$ ; here  $n_i$  is the number of labels in class  $i$ . In other words, there exists a permutation matrix  $\Pi$  such that
+
+$$
+\mathbf {Y} _ {1: t} ^ {\top} \mathbf {Y} _ {1: t} = \boldsymbol {\Pi} \operatorname {d i a g} (\mathbf {1} _ {n _ {1}}, \mathbf {1} _ {n _ {2}}, \dots , \mathbf {1} _ {n _ {c _ {t}}}) \boldsymbol {\Pi} ^ {\top}.
+$$
+
+Since the maximum eigenvalue of  $\mathbf{1}_{m_i}$  is  $m_{i}$ , we know
+
+$$
+\left\| \boldsymbol {Y} _ {1: t} ^ {\top} \boldsymbol {Y} _ {1: t} \right\| = \max  _ {i = 1, \dots , c _ {t}} \left\{n _ {i} \right\}.
+$$
+
+Substitute this into Corollary 1 and we obtain
+
+$$
+\frac {1}{M _ {t}} \big \| \widetilde {\boldsymbol {W}} _ {t} \boldsymbol {H} _ {1: t} - \boldsymbol {Y} _ {1: t} \big \| _ {\mathrm {F}} ^ {2} \leq \frac {\max _ {i = 1 , . . . , c _ {t}} \{n _ {i} \}}{M _ {t}} \left(M _ {t} - k _ {t} + \frac {t - 1}{\gamma_ {t} ^ {2}} \min \left\{M _ {t - 1} - k _ {t - 1}, (t - 1) k _ {t} \right\}\right).
+$$
+
+It is also of interest to bound the distances between the SVD factors computed online and offline, namely the distances between  $\widetilde{\Sigma}_{1:t},\overline{\Sigma}_{1:t}$  and between  $\widetilde{U}_{1:t},\overline{U}_{1:t}$ . We do so in the next result.
+
+Theorem 6. Let  $a_{t}$  be defined as in (6). For  $t \geq 1$  define
+
+$$
+\operatorname {g a p} _ {t} := \mu_ {k _ {t}} \left(\boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top}\right) - \mu_ {k _ {t} + 1} \left(\boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top}\right). \tag {18}
+$$
+
+Then it always holds that
+
+$$
+\left\| \overline {{\boldsymbol {\Sigma}}} _ {1: t} ^ {2} - \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {2} \right\| _ {\infty} \leq a _ {t - 1}. \tag {19}
+$$
+
+Moreover, if  $a_{t-1} < \left(1 - \sqrt{2}\right) \mathrm{gap}_t$ , then for any  $t \geq 1$  we have
+
+$$
+\min  _ {\boldsymbol {O} \in \mathcal {O} (k)} \left\| \overline {{\boldsymbol {U}}} _ {1: t} - \widetilde {\boldsymbol {U}} _ {1: t} \boldsymbol {O} \right\| _ {\mathrm {F}} \leq \left\| \overline {{\boldsymbol {U}}} _ {1: t} \overline {{\boldsymbol {U}}} _ {1: t} ^ {\top} - \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \right\| \leq \frac {\sqrt {2} a _ {t - 1}}{\operatorname {g a p} _ {t}}, \tag {20}
+$$
+
+where  $\mathcal{O}(k)$  be the set of  $k\times k$  orthogonal matrices, defined as
+
+$$
+\mathcal {O} (k) := \left\{\boldsymbol {O} \in \mathbb {R} ^ {k \times k}: \boldsymbol {O} ^ {\top} \boldsymbol {O} = \boldsymbol {O O} ^ {\top} = \boldsymbol {I} _ {k} \right\}.
+$$
+
+Proof of Theorem 6. It is clear that  $\overline{U}_1 = \widetilde{U}_1$  and  $\overline{\Sigma}_1 = \widetilde{\Sigma}_1$ . We now consider the case  $t \geq 2$ . Note that  $\widetilde{U}_{1:t}\widetilde{\Sigma}_{1:t}\widetilde{U}_{1:t}^{\top}$  is the eigen decomposition of  $\tau_{k_t}(\widetilde{U}_{1:t-1}\widetilde{\Sigma}_{1:t-1}^2\widetilde{U}_{1:t-1}^{\top} + H_tH_t^{\top})$ , and  $\overline{U}_{1:t}\overline{\Sigma}_{1:t}\overline{U}_{1:t}^{\top}$  is the eigen decomposition of  $\tau_{k_t}(H_{1:t}H_{1:t}^{\top})$ . We can compute
+
+$$
+\begin{array}{l} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} - \left(\widetilde {\boldsymbol {U}} _ {1: t - 1} \widetilde {\boldsymbol {\Sigma}} _ {1: t - 1} ^ {2} \widetilde {\boldsymbol {U}} _ {1: t - 1} ^ {\top} + \boldsymbol {H} _ {t} \boldsymbol {H} _ {t} ^ {\top}\right) = \boldsymbol {H} _ {1: t - 1} \boldsymbol {H} _ {1: t - 1} ^ {\top} - \widetilde {\boldsymbol {U}} _ {1: t - 1} \widetilde {\boldsymbol {\Sigma}} _ {1: t - 1} ^ {2} \widetilde {\boldsymbol {U}} _ {1: t - 1} ^ {\top} \\ \stackrel {\text {(i)}} {=} \sum_ {i = 1} ^ {t - 1} \left(\boldsymbol {B} _ {i} \boldsymbol {B} _ {i} ^ {\top} - \tau_ {k _ {i}} \left(\boldsymbol {B} _ {i} \boldsymbol {B} _ {i} ^ {\top}\right)\right) =: \boldsymbol {D} _ {t}, \\ \end{array}
+$$
+
+where (i) follows from Lemma 1. We can therefore apply Weyl's inequality to obtain
+
+$$
+\left\| \overline {{\boldsymbol {\Sigma}}} _ {1: t} ^ {2} - \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {2} \right\| _ {\infty} \leq \| \boldsymbol {D} _ {t} \| = a _ {t - 1}.
+$$
+
+This proves (19). On the other hand, (20) follows from the Davis-Kahan theorem (Davis & Kahan, 1970), or more precisely, from Corollary 2.8 of Chen et al. (2021).  $\square$
+
+# G.1 RELATION BETWEEN THE OFFLINE AND ONLINE SOLUTIONS
+
+Recall the definitions of the offline solution  $\overline{\mathbf{W}}_t$  in (2) and the output  $\widetilde{\mathbf{W}}_t$  of LoRanPAC in (4):
+
+$$
+\overline {{\boldsymbol {W}}} _ {t} = \boldsymbol {Y} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \overline {{\boldsymbol {U}}} _ {1: t} \overline {{\boldsymbol {\Sigma}}} _ {1: t} ^ {- 2} \overline {{\boldsymbol {U}}} _ {1: t} ^ {\top}, \quad \widetilde {\boldsymbol {W}} _ {t} = \boldsymbol {Y} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top}.
+$$
+
+Here we aim to bound the distance  $\| \overline{W}_t - \widetilde{W}_t\| _F$ . We consider the model  $Y_{1:t} = W_t^* H_{1:t}$ ; this is the  $Y_{1:t} = W_t^* H_{1:t} + \mathcal{E}_{1:t}$  with  $\mathcal{E}_{1:t}$ . Here we make this assumption for simplicity, and the result here can be extended to the case with noise.
+
+Recall the definition of  $\mathrm{gap}_t$  in (18):
+
+$$
+\operatorname {g a p} _ {t} := \mu_ {k _ {t}} \left(\boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top}\right) - \mu_ {k _ {t} + 1} \left(\boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top}\right).
+$$
+
+Based on Theorem 6, we prove the following result.
+
+Theorem 7. Let  $a_{t}$  be defined as in (6) and  $\mathrm{gap}_t$  as in (18). Assume  $a_{t-1} < \left(1 - \frac{1}{\sqrt{2}}\right)\mathrm{gap}_t$ . Suppose  $\mathbf{Y}_{1:t} = \mathbf{W}_t^*\mathbf{H}_{1:t}$ . Then we have
+
+$$
+\left\| \overline {{\boldsymbol {W}}} _ {t} - \widetilde {\boldsymbol {W}} _ {t} \right\| _ {\mathrm {F}} \leq \left\| \boldsymbol {W} _ {t} ^ {*} \right\| _ {\mathrm {F}} \cdot \left(\frac {\sqrt {2} a _ {t - 1}}{\operatorname {g a p} _ {t}} + \frac {t - 1}{\gamma_ {t}}\right). \tag {21}
+$$
+
+Proof. Note that
+
+$$
+\begin{array}{l} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \overline {{\boldsymbol {U}}} _ {1: t} \overline {{\boldsymbol {\Sigma}}} _ {1: t} ^ {- 2} \overline {{\boldsymbol {U}}} _ {1: t} ^ {\top} = \overline {{\boldsymbol {U}}} _ {1: t} \overline {{\boldsymbol {U}}} _ {1: t} ^ {\top} \\ \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \tilde {\boldsymbol {U}} _ {1: t} \tilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \tilde {\boldsymbol {U}} _ {1: t} ^ {\top} = \tilde {\boldsymbol {U}} _ {1: t} \tilde {\boldsymbol {U}} _ {1: t} ^ {\top} + \boldsymbol {D} _ {t} \tilde {\boldsymbol {U}} _ {1: t} \tilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \tilde {\boldsymbol {U}} _ {1: t} ^ {\top}, \\ \end{array}
+$$
+
+where  $D_{t}$  is defined in (14) and the second equality follows from Lemma 1. So we have
+
+$$
+\begin{array}{l} \left\| \overline {{\boldsymbol {W}}} _ {t} - \widetilde {\boldsymbol {W}} _ {t} \right\| _ {\mathrm {F}} = \left\| \boldsymbol {W} _ {t} ^ {*} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \overline {{\boldsymbol {U}}} _ {1: t} \overline {{\boldsymbol {\Sigma}}} _ {1: t} ^ {- 2} \overline {{\boldsymbol {U}}} _ {1: t} ^ {\top} - \boldsymbol {W} _ {t} ^ {*} \boldsymbol {H} _ {1: t} \boldsymbol {H} _ {1: t} ^ {\top} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \right\| _ {\mathrm {F}} (22) \\ \leq \left\| W _ {t} ^ {*} \right\| _ {\mathrm {F}} \cdot \left\| H _ {1: t} H _ {1: t} ^ {\top} \bar {U} _ {1: t} \bar {\Sigma} _ {1: t} ^ {- 2} \bar {U} _ {1: t} ^ {\top} - H _ {1: t} H _ {1: t} ^ {\top} \tilde {U} _ {1: t} \tilde {\Sigma} _ {1: t} ^ {- 2} \tilde {U} _ {1: t} ^ {\top} \right\| (23) \\ = \left\| W _ {t} ^ {*} \right\| _ {\mathrm {F}} \cdot \left\| \overline {{\boldsymbol {U}}} _ {1: t} \overline {{\boldsymbol {U}}} _ {1: t} ^ {\top} - \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} - \boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \right\| (24) \\ \leq \left\| W _ {t} ^ {*} \right\| _ {\mathrm {F}} \cdot \left(\left\| \overline {{\boldsymbol {U}}} _ {1: t} \overline {{\boldsymbol {U}}} _ {1: t} ^ {\top} - \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \right\| + \left\| \boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \right\|\right) (25) \\ \leq \left\| \boldsymbol {W} _ {t} ^ {*} \right\| _ {\mathrm {F}} \cdot \left(\left\| \overline {{\boldsymbol {U}}} _ {1: t} \overline {{\boldsymbol {U}}} _ {1: t} ^ {\top} - \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \right\| + \left\| \boldsymbol {D} _ {t} \widetilde {\boldsymbol {U}} _ {1: t} \widetilde {\boldsymbol {\Sigma}} _ {1: t} ^ {- 2} \widetilde {\boldsymbol {U}} _ {1: t} ^ {\top} \right\|\right) (26) \\ \leq \left\| W _ {t} ^ {*} \right\| _ {\mathrm {F}} \cdot \left(\frac {\sqrt {2} a _ {t - 1}}{\operatorname {g a p} _ {t}} + \frac {t - 1}{\gamma_ {t}}\right) (27) \\ \end{array}
+$$
+
+where the last inequality is due to Theorem 6 and Lemma 4. The proof is complete.
+
+![](images/c475f14fc44a02304669d9ec2ca512c858b701bdadbd6aaa97629afe35a9706d.jpg)
+
+# H REVIEW OF RELATED WORKS
+
+In Appendix H.1 we review related work on CL. Recent surveys on CL include Parisi et al. (2019); van de Ven et al. (2022); Shaheen et al. (2022); Zhou et al. (2024b); Wang et al. (2024); Shi et al. (2024). See also the GitHub repo of Liu (2024) for an extensive list of CL papers.
+
+In Appendix H.2 we review related work on random feature models.
+
+# H.1 MORE RELATED WORK ON CONTINUAL LEARNING
+
+Many CL methods have been proposed without explicitizing the use of pre-trained models (Ruvolo & Eaton, 2013; Kirkpatrick et al., 2017; Rebuffi et al., 2017; Lopez-Paz & Ranzato, 2017; Zeng et al., 2019; Chaudhry et al., 2019; Yan et al., 2021; Saha et al., 2021; Douillard et al., 2022; Elenter et al., 2023). An easy way to boost their performance is to adapt them for the context of pre-trained models. There are two natural approaches to do so. One approach is to use the pre-trained model as initialization and run these CL algorithms to fine-tune the pre-trained model; see, e.g., Li et al. (2024). The other approach is to train a shallow network with the output features of the pre-trained model and either of these CL algorithms. We do not explore these directions here. In what follows, we review existing CL methods explicitly designed for leveraging pre-trained models, and we review theoretical developments for CL as well.
+
+Prior Work on CL with Pre-trained Models. The availability of pre-trained models has motivated new insights into designing CL methods. CL methods such designed can be roughly divided into two categories. In one category, the pre-trained model is completely frozen, and their output features are used as inputs for a tailored CL method. A straightforward method in this category, known as SimpleCIL (Zhou et al., 2023) or Nearest Mean Classifier (NMC) (Mensink et al., 2013; Rebuffi et al., 2017; Panos et al., 2023; Janson et al., 2022), is to classify a test image based on the (cosine) distances of its feature to class means of the training features. While this method is stable, hyperparameter-free, and can handle long task sequences, to the best of our knowledge, it does not have theoretical guarantees. Of course, RanPAC and LoRanPAC also fall into this category. Other methods in the category include Ahrens et al. (2024); Prabhu et al. (2024). Both methods make certain modifications on top of RanPAC:
+
+- The method of Ahrens et al. (2024) replaces the random ReLU features with the concatenation of the output features of intermediate layers. We identify that this generalizes the idea of Pao & Takefuji  $(1992)^{2}$ .
+
+- The method of Prabhu et al. (2024) replaces the random ReLU features with random Fourier features (which were used by Rahimi & Recht (2007) for learning kernel machines), and the ridge regression solver of RanPAC with linear discriminant analysis (LDA) (Hayes & Kanan, 2020). Note that LDA optimizes an objective that is in general different from the MSE training loss, for which Prabhu et al. (2024) have not provided theoretical guarantees.
+
+Similarly to RanPAC, the methods of Ahrens et al. (2024); Prabhu et al. (2024) need  $O(E^3)$  time per task to invert the (regularized)  $E \times E$  covariance matrix. Prabhu et al. (2024) uses  $E = 25000$  in their experiments, which might constitute the current computational limit of performing the inversion (cf. Figs. 3 and 5). Also, both methods lack theoretical guarantees. In contrast, the running time of our LoRanPAC method depends only linearly on  $E$  and can handle  $E \geq 10^5$  with stable performance and theoretical guarantees.
+
+In the other category of methods, the weights of the pre-trained models remain fixed, but the output features of the pre-trained models are changed. The catch is that these methods either change the input or change the network architecture. Such change could be applied layer-wise, therefore, in order to describe the idea, it is the simplest to assume the pre-trained model  $f$  is a single-layer network.
+
+- If we keep both the input and architecture fixed, then the network would take an input  $X$  and output  $f(X)$ .  
+- A popular way to change the input is to stack some trainable parameters  $Z$  with input  $X$ , where  $Z$  and  $X$  have the same number of columns. The network outputs  $f([X;Z])$ . Here, it is implicitly assumed that  $f$  can take input matrices with different number of rows (i.e., different number of tokens). For instance,  $f$  can be a single-layer vision transformer. In this case  $Z$  is often called (visual) prompts (Jia et al., 2022), and CL methods using this strategy are often called prompt-based methods; see, e.g., (Wang et al., 2022b;c;a; Smith et al., 2023; Wang et al., 2023; Jung et al., 2023; Tang et al., 2023; Gao et al., 2024b; Roy et al., 2024; Kim et al., 2024).  
+- A popular way to change the architecture is to replace the input-output map  $X \mapsto f(X)$  with  $X \mapsto f(\pmb{X}) + g(\pmb{X})$ , where  $g$  is some simple shallow network parametrized by the extra trainable parameters. For instance,  $g$  could be a simple two-layer linear network of the form  $g(\pmb{X}) = AB\pmb{X}$  or  $g(\pmb{X}) = A\mathrm{relu}(B\pmb{X})$ , where  $A, B$  are trainable. In these case,  $A, B$  are called adapters (Houlsby et al., 2019; Hu et al., 2022; Chen et al., 2022), and CL methods using this strategy are often called adapter-based methods (Zhou et al., 2023; Liang & Li, 2024; Tan et al., 2024; Gao et al., 2024a).
+
+Clearly, prompt-based and adapter-based methods can both be viewed as expansion-based methods that enlarge the capacity of a network in order to learn new tasks (Rusu et al., 2016; Yoon et al., 2018; Li et al., 2019; Ramesh & Chaudhari, 2022).
+
+Despite their popularity, both prompt-based and adapter-based methods need to solve highly nonconvex training problems, for which deriving informative theoretical guarantees is a significant challenge. Their lack of theoretical guarantees makes them prone to unexpected failures. For example, prompt-based methods such as  $L2P$  (Wang et al., 2022b), DualPrompt (Wang et al., 2022c), CodaPrompt (Smith et al., 2023), have their performance highly sensitive to the choice of hyperparameters and therefore to the pre-trained model in use (Wang et al., 2023), dataset, and problem setting (cf. Table 1); indeed, a small perturbation in learning rates might change the accuracy drastically (Zhang et al., 2023). While they are often equipped with dataset-specific hyperparameters released by authors (cf. Appendix J), their instability still emerges when applied to a long sequence of tasks. This is because new prompts or adapters are often needed to maintain high accuracy on new tasks (Zhou et al., 2024a), but doing so eventually becomes infeasible. Indeed, to train on the 100 tasks of the CIFAR100 dataset in the CIL setting with one class given at a time (B-0, Inc-1), running the adapter-based method of Zhou et al. (2024a), called EASE, with default hyperparameters, would create more than 117M parameters for its growing number of adapters, while the pre-trained ViTs in use have less than 87M parameters.
+
+In Table 5 we summarize the conceptual differences of our approach from prior works.
+
+Prior Work on CL Theory. Theoretical developments on CL have been chasing the current CL practice, with a majority of the theory CL papers limiting themselves to the linear, two-layer, or kernel setting (Doan et al., 2021; Heckel, 2022; Evron et al., 2022; Peng & Risteski, 2022; Lin et al., 2023;
+
+Table 5: Conceptual comparison to prior work.  
+
+<table><tr><td></td><td>Theoretical Guarantees?</td><td>Stable?</td><td>Can Handle Long Task Sequences?</td></tr><tr><td>L2P, Dual Prompt, CodaPrompt</td><td>None</td><td>No</td><td>No</td></tr><tr><td>EASE</td><td>None</td><td>No</td><td>No</td></tr><tr><td>SimpleCIL</td><td>None</td><td>Yes</td><td>Yes</td></tr><tr><td>RanPAC</td><td>None</td><td>No</td><td>No</td></tr><tr><td>LoRanPAC (Ours)</td><td>Theorems 1 and 2</td><td>Yes</td><td>Yes</td></tr></table>
+
+Swartworth et al., 2023; Goldfarb et al., 2024; Zhao et al., 2024; Ding et al., 2024). While these works cover various theoretical aspects (e.g., generalization bounds, sample complexity, and convergence rates), there has arguably been a huge gap between their simplified settings and deep networks that state-of-the-art CL methods use. On the other hand, we have seen that deep pre-trained models in cascade with shallow trainable networks can provide competitive performance, thus it now makes sense to revise and extend these theoretical contributions within this cascaded architecture, thereby providing meaningful guarantees for learning the shallow networks. We believe this viewpoint would greatly reduce the gap between the theory and practice in the current CL literature.
+
+# H.2 RANDOM VECTOR FUNCTIONAL LINK NETWORK AND RANDOM FEATURE MODELS
+
+Here we review related works on random feature models. Recall our model is a two-layer network of the form
+
+$$
+\boldsymbol {X} \mapsto \boldsymbol {W} \cdot \operatorname {r e l u} (\boldsymbol {P X}) \tag {28}
+$$
+
+where  $P$  is randomly generated and fixed, and  $W$  consists of trainable parameters.
+
+Random Vector Functional Link Network. In independent efforts, Schmidt et al. (1992) and Pao & Takefuji (1992) considered models of form (1). Schmidt et al. (1992) used the sigmoid activation function  $\xi \mapsto \frac{1}{1 + \exp(-\xi)}$ , while Pao & Takefuji (1992) specified an arbitrary activation function as inspired by Hornik et al. (1989) and stack the features  $X$  and  $H$  together (see, e.g., Section 2.1 of Malik et al. (2023)). The model Pao & Takefuji (1992) proposed has been known as random vector functional link (RVFL), and the model of Schmidt et al. (1992) is referred to, according to a recent review (Malik et al., 2023), as Schmidt neural network (SNN).
+
+Models of form (28) are best combined with MSE losses, as training  $\mathbf{W}$  with  $\mathbf{P}$  fixed amounts to solving a least-squares problem, which admits a closed-form solution as shown by Schmidt et al. (1992) and even earlier by Webb & Lowe (1990).
+
+The model Schmidt et al. (1992) and Pao & Takefuji (1992) advocated was proposed, again, by Huang et al. (2004; 2006) under the name extreme learning machine (ELM). While Huang et al. (2004) claimed ELMs to be a new learning scheme in the paper title, it was criticized by (Wang & Wan, 2008; Authors) that ELMs are ideas stolen from the last century (Schmidt et al., 1992; Pao & Takefuji, 1992), which Huang et al. (2004) were aware of yet did not cite. Despite the criticism, and perhaps because of its "fancy" name, ELMs had once been popular and attracted many follow-up variants. We shall not review these variants here.
+
+The model we considered, therefore, follows in spirit the framework put forth by Schmidt et al. (1992); Pao & Takefuji (1992). Crucially, our approach is a modern instantiation of their framework (cf. Table 6), where we consider larger-scale problems with ill-conditioned data, online solvers with GPU implementations. But does it make sense to use the random ReLU features  $\pmb{H}_t \coloneqq \mathrm{relu}(\pmb{P}\pmb{X}_t)$  rather than the pre-trained features  $\pmb{X}_t$  for regression? Would the transformation  $\pmb{P} \in \mathbb{R}^{E \times d}$  even harm the pre-trained knowledge? McDonnell et al. (2023) empirically verified that having the first layer  $\pmb{P}$  is beneficial to performance as long as  $E \geq d$ , and the accuracy tends to be higher for larger  $E$  (see Table A5 of Appendix F.6 of McDonnell et al. (2023)). While RanPAC is limited to  $E \approx 10^4$ , our approach is inherently more scalable, allowing us to take  $E = 10^5$ . The pursuit in higher embedding dimension  $E$  brings us into the over-parameterized territory, where the corresponding MSE objective has infinitely many solutions, and this is different from the classic works on RVFLs or SNNs that largely focus on the case where there are just a few hundred neurons (e.g.,  $E \approx 100$ ).
+
+Table 6: Comparison between our approach and classic methods for extreme learning machines.  
+
+<table><tr><td></td><td>Problem Scale</td><td>Solver</td><td>Data</td><td>Compute Platform</td></tr><tr><td>Schmidt et al. (1992); Pao &amp; Takefuji (1992)</td><td>small</td><td>offline</td><td>well-conditioned</td><td>CPU</td></tr><tr><td>LoRanPAC (Ours)</td><td>large</td><td>online</td><td>ill-conditioned</td><td>GPU</td></tr></table>
+
+Random Feature Models. Model (28) is also studied under the name random feature model (RFM) with the origin of RFMs often attributed to Rahimi & Recht (2007). The RFM is considered to be a simple proxy model for understanding deep networks, and hence it has recently been popular (Belkin et al., 2018; 2019; Bartlett et al., 2020; Hastie et al., 2022; Mei & Montanari, 2022; Tsigler & Bartlett, 2023). Some of these works make statistical assumptions on  $X_{t}$  and address technical challenges in analyzing the nonlinear map  $W \cdot \mathrm{relu}(PX_{t})$ .
+
+Alternatively, one could conduct analysis conditioned on  $H_{t} \coloneqq \mathrm{relu}(PX_{t})$ , which would be more manageable as it reduces to linear models. For example, the work of Xu & Hsu (2019); Huang et al. (2022); Bach (2024); Green & Romanov (2024) truncates the SVD factors of the features before applying least-squares. This is similar to ours, with an important difference that they apply LoRanPAC only once, while we apply it continually. Also, their results make statistical (e.g., Gaussian) assumptions on  $H_{1:t}$ , which could violate our context that  $H_{1:t}$  is generated via  $H_{t} \coloneqq \mathrm{relu}(PX_{t})$ . In contrast, our results in the main paper, namely Theorems 1 and 2, make no assumptions on  $H_{1:t}$  and are therefore applicable to random feature models and to the pre-trained models bridged with a random feature model.
+
+# I DATASET DETAILS
+
+For convenience and completeness, we collect some details about the datasets in Tables 7 and 8.
+
+Table 7: Datasets used for class-incremental learning experiments.†: ObjectNet, OmniBenchmark, and VTAB contain a large number of classes, and we use a subset of these datasets delivered by Zhou et al. (2023); see their Table 4 and also Table A2 of McDonnell et al. (2023).  
+
+<table><tr><td>Dataset Name</td><td>Origin</td><td>Training Set Size</td><td>Test Set Size</td><td># of Classes</td><td>Link</td></tr><tr><td>CIFAR100</td><td>(Krizhevsky et al., 2009)</td><td>50,000</td><td>10,000</td><td>100</td><td>Here</td></tr><tr><td>ImageNet-R</td><td>(Hendrycks et al., 2021a)</td><td>24,000</td><td>6,000</td><td>200</td><td>Here</td></tr><tr><td>ImageNet-A</td><td>(Hendrycks et al., 2021b)</td><td>5,981</td><td>5,985</td><td>200</td><td>Here</td></tr><tr><td>CUB-200</td><td>(Wah et al., 2011)</td><td>9,430</td><td>2,358</td><td>200</td><td>Here</td></tr><tr><td>\( ObjectNet^{\dagger} \)</td><td>(Barbu et al., 2019)</td><td>26,509</td><td>6,628</td><td>200</td><td>Here</td></tr><tr><td>\( OmniBenchmark^{\dagger} \)</td><td>(Zhang et al., 2022)</td><td>89,697</td><td>5,985</td><td>300</td><td>Here</td></tr><tr><td>\( VTAB^{\dagger} \)</td><td>(Zhai et al., 2019)</td><td>1,796</td><td>8,619</td><td>50</td><td>Here</td></tr><tr><td>StanfordCars</td><td>(Krause et al., 2013)</td><td>8,144</td><td>8,041</td><td>196</td><td>Here</td></tr></table>
+
+Table 8: Datasets used for domain-incremental learning. See Table A3 of McDonnell et al. (2023) for even more details.  
+
+<table><tr><td>Dataset Name</td><td>Origin</td><td>Training Set Size</td><td>Test Set Size</td><td># of Classes</td><td>Link</td></tr><tr><td>CORe50</td><td>(Lomonaco &amp; Maltoni, 2017)</td><td>119,894</td><td>44,972</td><td>50</td><td>Here</td></tr><tr><td>CDDB-Hard</td><td>(Li et al., 2023)</td><td>16,068</td><td>5,353</td><td>2</td><td>Here</td></tr><tr><td>DomainNet</td><td>(Peng et al., 2019)</td><td>409,832</td><td>176,743</td><td>345</td><td>Here</td></tr></table>
+
+# J EXPERIMENTAL SETUP DETAILS
+
+The details of how we run each of the methods are specified as follows. For L2P, DualPrompt, CodaPrompt, we use the hyperparameters available in the PILOT repo (Sun et al., 2023) for the CIFAR100 and ImageNet-R datasets. Since no official hyperparameters are released for other datasets, we simply use their respective hyperparameters of CIFAR100 for other datasets. One might notice that these methods have large accuracy drops on other datasets, suggesting that they are sensitive to hyperparameters and the good and dataset-specific hyperparameters, if they exist, are to be found for these methods to perform well. This is an inherent drawback as they involve minimizing a highly non-convex training objective. On the other hand, many other methods, including SimpleCIL, RanPAC, and ours, are almost parameter-free. Specifically, the only hyperparameter of our approach is the truncation threshold and its role is clearly explained in the main paper.
+
+For joint linear classifiers, that is LC  $(X_{1:T})$  or LC  $(H_{1:T})$ , we train for 20 epochs using the cross-entropy loss, batch size 48, weight decay 0.0005, and SGD with the cosine annealing schedule. We run LC  $(X_{1:T})$  and LC  $(H_{1:T})$  with different initial learning rates  $\{0.001, 0.005, 0.01, 0.02, 0.03\}$  and take report the maximum accuracy (Table 9). Note that LC  $(X_{1:T})$  and LC  $(H_{1:T})$  are trained using the cross-entropy loss, not the MSE loss. The reason is that the features  $H_{1:t}$  are highly ill-conditioned (Fig. 1), which makes SGD converge very slowly with the MSE loss. Comparing this to (11), we conclude that the MSE loss in our setting is useful when the objective is minimized via robust numerical computation techniques (e.g., our LoRanPAC implementation in Appendix C) instead of SGD.
+
+We also consider the idea of first-session adaptation. This idea introduces a few hyperparameters such as the learning rate and schedule. We run experiments with two sets of hyperparameters, given respectively by RanPAC and EASE. We attach the symbol  $\dagger$  to the method name when we use the hyperparameters of RanPAC (e.g., ADAM $\dagger$ , LoRanPAC $\dagger$ , RanPAC $\dagger$ ). We attach the symbol * when we use the hyperparameters of EASE (e.g., LoRanPAC $*$ , EASE\*).
+
+For RanPAC, the official hyperparameters given by McDonnell et al. (2023) vary for different datasets. We try to unify the setup by keeping using the hyperparameters most frequently used by McDonnell
+
+Table 9: Accuracy of training joint linear classifiers given all data with different initial learning rates  $\{0.001,0.005,0.01,0.02,0.03\}$  and the corresponding maximum accuracy. LC  $(X_{1:T})$  trains a linear classifier using all pre-trained features  $X_{1:T}$  and LC  $(H_{1:T})$  uses all embedded features.  
+
+<table><tr><td rowspan="2"></td><td colspan="6">LC (X1:T)</td><td colspan="6">LC (H1:T)</td></tr><tr><td>0.001</td><td>0.005</td><td>0.01</td><td>0.02</td><td>0.03</td><td>Max</td><td>0.001</td><td>0.005</td><td>0.01</td><td>0.02</td><td>0.03</td><td>Max</td></tr><tr><td colspan="13">ViTs pre-trained on ImageNet-1K (vit_base_batch16_224):</td></tr><tr><td>CIFAR100</td><td>86.39</td><td>87.56</td><td>87.47</td><td>87.09</td><td>86.99</td><td>87.56</td><td>87.76</td><td>86.68</td><td>86.36</td><td>86.75</td><td>86.46</td><td>87.76</td></tr><tr><td>ImageNet-R</td><td>70.25</td><td>72.42</td><td>72.22</td><td>72.02</td><td>71.52</td><td>72.42</td><td>73.00</td><td>71.08</td><td>71.18</td><td>70.70</td><td>71.15</td><td>73.00</td></tr><tr><td>ImageNet-A</td><td>54.64</td><td>58.85</td><td>58.46</td><td>57.67</td><td>56.55</td><td>58.85</td><td>59.25</td><td>56.16</td><td>56.09</td><td>56.48</td><td>56.42</td><td>59.25</td></tr><tr><td>CUB-200</td><td>82.32</td><td>87.62</td><td>88.59</td><td>88.63</td><td>88.76</td><td>88.76</td><td>88.72</td><td>88.13</td><td>88.08</td><td>88.17</td><td>87.83</td><td>88.72</td></tr><tr><td>ObjectNet</td><td>57.53</td><td>59.70</td><td>59.22</td><td>58.68</td><td>58.43</td><td>59.70</td><td>59.96</td><td>56.14</td><td>55.63</td><td>55.48</td><td>55.34</td><td>59.96</td></tr><tr><td>Omnibenchmark</td><td>76.11</td><td>79.00</td><td>79.55</td><td>79.43</td><td>79.50</td><td>79.55</td><td>80.02</td><td>79.62</td><td>79.57</td><td>79.62</td><td>79.43</td><td>80.02</td></tr><tr><td>VTAB</td><td>86.40</td><td>90.89</td><td>91.32</td><td>91.23</td><td>90.89</td><td>91.32</td><td>91.17</td><td>89.86</td><td>89.99</td><td>90.16</td><td>89.99</td><td>91.17</td></tr><tr><td>StanfordCars</td><td>39.56</td><td>62.54</td><td>69.29</td><td>72.86</td><td>74.12</td><td>74.12</td><td>72.43</td><td>73.65</td><td>73.54</td><td>72.81</td><td>72.49</td><td>73.65</td></tr><tr><td colspan="13">ViTs pre-trained on ImageNet-21K (vit_base_batch16_224_in21k):</td></tr><tr><td>CIFAR100</td><td>86.15</td><td>86.78</td><td>86.33</td><td>85.86</td><td>85.17</td><td>86.78</td><td>85.80</td><td>85.16</td><td>85.05</td><td>85.31</td><td>85.4</td><td>85.80</td></tr><tr><td>ImageNet-R</td><td>67.22</td><td>68.63</td><td>67.12</td><td>65.90</td><td>65.28</td><td>68.63</td><td>68.65</td><td>68.00</td><td>68.17</td><td>68.23</td><td>67.78</td><td>68.65</td></tr><tr><td>ImageNet-A</td><td>46.68</td><td>51.42</td><td>50.03</td><td>49.24</td><td>48.58</td><td>51.42</td><td>51.15</td><td>50.16</td><td>49.44</td><td>48.98</td><td>49.64</td><td>51.15</td></tr><tr><td>CUB-200</td><td>85.58</td><td>88.89</td><td>89.06</td><td>88.72</td><td>88.21</td><td>89.06</td><td>89.31</td><td>88.17</td><td>88.63</td><td>88.46</td><td>88.51</td><td>89.31</td></tr><tr><td>ObjectNet</td><td>58.01</td><td>58.39</td><td>57.50</td><td>55.87</td><td>54.42</td><td>58.39</td><td>56.93</td><td>53.33</td><td>54.44</td><td>54.47</td><td>54.54</td><td>56.93</td></tr><tr><td>Omnibenchmark</td><td>78.95</td><td>79.67</td><td>79.73</td><td>79.45</td><td>79.33</td><td>79.73</td><td>79.62</td><td>79.26</td><td>79.11</td><td>78.91</td><td>79.05</td><td>79.62</td></tr><tr><td>VTAB</td><td>87.71</td><td>90.78</td><td>90.97</td><td>90.71</td><td>90.11</td><td>90.97</td><td>90.78</td><td>91.03</td><td>90.42</td><td>90.44</td><td>90.27</td><td>91.03</td></tr><tr><td>StanfordCars</td><td>44.80</td><td>64.22</td><td>68.06</td><td>68.92</td><td>68.71</td><td>68.92</td><td>69.38</td><td>67.64</td><td>67.65</td><td>67.79</td><td>67.39</td><td>69.38</td></tr></table>
+
+et al. (2023). Specifically, we set the embedding dimension  $E$  to 10000 for RanPAC; note the exception that McDonnell et al. (2023) run the CDDB experiments with  $E = 5000$ , even though their Table A5 showed that larger  $E$  in general leads to higher accuracy on CIFAR100. The main hyperparameters of RanPAC used for first-session adaptation are as follows:
+
+```json
+{"tuned_epoch":20, "init_lr":0.01, "batch_size":48, "weight Decay":0.0005}
+```
+
+We run ADAM†, LoRanPAC†, RanPAC† where first-session adaptation uses these hyperparameters consistently for all datasets.
+
+The EASE approach of Zhou et al. (2024a) performs fine-tuning, not just for the first session, but for every session, in an interesting way. The hyperparameters in their released code vary for different sessions and different datasets, and we refer the reader to the official GitHub repo of EASE for details. We also run LoRanPAC* with the hyperparameters of EASE for first-session adaptation. Note that since EASE does not show experiments on StanfordCars, or does not release hyperparameters on this dataset, we run EASE with its CIFAR100 hyperparameters for StanfordCars; see also Table 10 of Appendix K.1 where we tune the initial learning rates of EASE on StanfordCars, showing that the accuracy is still low.
+
+Finally, we note that in all tables, some approaches are marked in gray; they are not directly comparable to our approach as the methodology can be very different and it is in fact possible to combine one with another for even better performance. On the other hand, RanPAC is the most related to our method, hence we highlight the comparison with the purple background.
+
+# K EXTRA EXPERIMENTS, FIGURES, AND TABLES
+
+# K.1 PERFORMANCE ON STANFORDCARS
+
+It is observed in Table 1 that many methods exhibit significant performance drops on StanfordCars. Are these methods inherently unable to handle this dataset, or is it our taking inappropriate hyperparameters that lead to poor performance? Note that the authors of these works did not test their
+
+Table 10: Accuracy of EASE with different initial learning rates  $\{0.001,0.005,0.01,0.02,0.03\}$  on StanfordCars.  
+
+<table><tr><td colspan="3">0.001</td><td colspan="3">0.005</td><td colspan="3">0.01</td><td colspan="3">0.02</td><td colspan="3">0.03</td></tr><tr><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td></tr><tr><td>33.42</td><td>32.27</td><td>19.38</td><td>33.48</td><td>32.33</td><td>19.46</td><td>33.39</td><td>32.65</td><td>20.06</td><td>32.58</td><td>32.11</td><td>24.96</td><td>32.41</td><td>31.77</td><td>29.76</td></tr></table>
+
+Table 11: Final accuracy of different methods using ViTs pre-trained on ImageNet-21K (vit_base_batch16_224_in21k). Compare this with Table 1 of the main paper.  
+
+<table><tr><td colspan="15">(Part 1) CIFAR100 (B-0) ImageNet-R (B-0) ImageNet-A (B-0) CUB-200 (B-0) Avg.</td></tr><tr><td>\( LC\left( {{X}_{1 : T}}\right) \)</td><td>86.78</td><td></td><td></td><td>68.63</td><td></td><td></td><td>51.42</td><td></td><td></td><td>89.06</td><td></td><td></td><td>73.97</td><td></td></tr><tr><td>\( LC\left( {{H}_{1 : T}}\right) \)</td><td>85.80</td><td></td><td></td><td>68.65</td><td></td><td></td><td>51.15</td><td></td><td></td><td>89.31</td><td></td><td></td><td>73.73</td><td></td></tr><tr><td></td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td></td><td></td></tr><tr><td>RanPAC</td><td>87.58</td><td>87.65</td><td>87.75</td><td>68.18</td><td>70.03</td><td>70.13</td><td>37.59</td><td>52.01</td><td>52.73</td><td>89.48</td><td>89.65</td><td>89.44</td><td>73.52</td><td></td></tr><tr><td>LoRanPAC</td><td>88.62</td><td>88.63</td><td>88.67</td><td>70.85</td><td>70.93</td><td>70.72</td><td>54.71</td><td>54.71</td><td>55.10</td><td>90.33</td><td>90.33</td><td>90.46</td><td>76.17</td><td></td></tr><tr><td>EASE*</td><td>85.85</td><td>87.67</td><td>89.47</td><td>70.27</td><td>74.53</td><td>75.88</td><td>43.05</td><td>47.53</td><td>54.51</td><td>86.77</td><td>86.81</td><td>85.50</td><td>73.99</td><td></td></tr><tr><td>RanPAC*</td><td>90.26</td><td>91.39</td><td>91.97</td><td>75.47</td><td>76.10</td><td>77.33</td><td>47.60</td><td>58.00</td><td>62.74</td><td>83.21</td><td>89.57</td><td>89.69</td><td>77.78</td><td></td></tr><tr><td>LoRanPAC*</td><td>90.55</td><td>91.88</td><td>92.39</td><td>76.48</td><td>76.82</td><td>77.25</td><td>56.95</td><td>58.85</td><td>62.74</td><td>90.63</td><td>90.71</td><td>90.67</td><td>79.66</td><td></td></tr><tr><td colspan="14">(Part 2) ObjectNet (B-0) OmniBenchmark (B-0) VTAB (B-10) StanfordCars (B-16) Avg.</td><td></td></tr><tr><td>\( LC\left( {{X}_{1 : T}}\right) \)</td><td>58.39</td><td></td><td></td><td>79.73</td><td></td><td></td><td>90.97</td><td></td><td></td><td>68.92</td><td></td><td></td><td>74.50</td><td></td></tr><tr><td>\( LC\left( {{H}_{1 : T}}\right) \)</td><td>56.93</td><td></td><td></td><td>79.62</td><td></td><td></td><td>91.03</td><td></td><td></td><td>69.38</td><td></td><td></td><td>74.24</td><td></td></tr><tr><td></td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td></td><td></td></tr><tr><td>RanPAC</td><td>59.14</td><td>59.23</td><td>59.29</td><td>78.38</td><td>78.40</td><td>78.06</td><td>92.37</td><td>92.84</td><td>92.84</td><td>43.60</td><td>65.78</td><td>65.78</td><td>72.14</td><td></td></tr><tr><td>LoRanPAC</td><td>61.35</td><td>61.36</td><td>61.33</td><td>80.12</td><td>80.03</td><td>80.13</td><td>92.90</td><td>92.76</td><td>92.92</td><td>68.96</td><td>68.76</td><td>68.95</td><td>75.80</td><td></td></tr><tr><td>EASE*</td><td>54.98</td><td>57.50</td><td>60.35</td><td>72.88</td><td>73.50</td><td>73.87</td><td>88.24</td><td>93.46</td><td>93.43</td><td>35.43</td><td>34.62</td><td>37.32</td><td>64.63</td><td></td></tr><tr><td>RanPAC*</td><td>64.56</td><td>66.55</td><td>66.05</td><td>78.55</td><td>79.15</td><td>79.23</td><td>92.77</td><td>92.84</td><td>93.72</td><td>2.31</td><td>69.11</td><td>69.11</td><td>71.16</td><td></td></tr><tr><td>LoRanPAC*</td><td>66.76</td><td>66.67</td><td>66.93</td><td>80.55</td><td>80.82</td><td>81.55</td><td>93.85</td><td>93.79</td><td>93.76</td><td>71.46</td><td>71.58</td><td>71.71</td><td>78.29</td><td></td></tr></table>
+
+methods on StanfordCars, nor they released the corresponding hyperparameters. To rule out the case of hyperparameter misspecification, we take the EASE* method of Zhou et al. (2024a) for example, and we run it with different initial learning rates  $\{0.001, 0.005, 0.01, 0.02, 0.03\}$  for the first task (this hyperparameter is called init_lr in the JSON file of the code repo of Sun et al. (2023); all other hyperparameters are set to the corresponding hyperparameters Zhou et al. (2024a) gave for CIFAR100. The results are in Table 10. It shows that different initial learning rates for EASE do not improve the performance on StanfordCars too much, suggesting that StanfordCars is perhaps inherently difficult for these types of methods.
+
+# K.2 EXPERIMENTS WITH VIT FEATURES PRE-TRAINED ON IMAGENET-21K
+
+Note that by default we use ViTs pre-trained on ImageNet-1K (vit_base_batch16_224). Here in Table 11 we show experiments with ViTs pre-trained on ImageNet-21K. Comparing this with Table 1, we obtain a similar conclusion that LoRanPAC is more stable than and outperforms RanPAC.
+
+# K.3 EXPERIMENTS THAT SHOW LORANPAC STABILIZES THE TRAINING LOSSES
+
+In Fig. 2 of the main paper we showed that LoRanPAC has stable test accuracy by truncating the SVD factors, compared to the nearly zero test accuracy of Min-Norm. In Fig. 4, we show that the truncation also stabilizes the average training MSE losses  $\frac{1}{M_t} \| \boldsymbol{W} \boldsymbol{H}_{1:t} - \boldsymbol{Y}_{1:t} \|_F^2$ , which eliminates the numerical issues and furthermore enables generalization in test scenarios.
+
+# K.4 EXPERIMENTS ON DOMAIN-INCREMENTAL LEARNING (DIL)
+
+Here we consider domain-incremental learning (DIL), where each task has images of all objects collected from different sources or domains, e.g., objects in the images of task 1 could be hand-written sketches of cars, and images of task 2 could be colored cars.
+
+![](images/de104410d33736e81163c5fd24cbb04d493c2dac82e053709a8e6dfcf041395d.jpg)  
+Figure 4: The average training MSE loss  $\frac{1}{M_t}\| W H_{1:t} - Y_{1:t}\| _F^2$  of the incremental SVD solution to Min-Norm explodes when eigenvalues of order  $10^{-5}$  emerge (Fig. 1b). LoRanPAC (25%) truncates 25% minimum singular values and implements LoRanPAC online, stabilizing Min-Norm.
+
+![](images/d16f744e7e8415ac1af6c95d6ccbfb898bbf1150c22a801688df09b857384207.jpg)
+
+![](images/58cfa267d10cc0dc94f64ef9bb38a9d8fd5980fe88392746af51d5914fbd2dde.jpg)
+
+![](images/0d83a0950a6fca251648b1306bc8f2b006437b4c5ec914085879122defe1559d.jpg)
+
+Table 12: Final accuracies of RanPAC and LoRanPAC with pre-trained ViTs for domain incremental learning.  
+
+<table><tr><td></td><td>CORe50</td><td>CDDB-Hard</td><td>DomainNet</td><td>Avg.</td></tr><tr><td>RanPAC</td><td>94.98</td><td>75.14</td><td>64.20</td><td>78.11</td></tr><tr><td>LoRanPAC</td><td>96.06</td><td>79.21</td><td>67.18</td><td>80.82</td></tr></table>
+
+We follow the work of McDonnell et al. (2023) to run domain-incremental learning experiments on 3 datasets, CORe50 (Lomonaco & Maltoni, 2017), CDDB-Hard (Li et al., 2023), and DomainNet (Peng et al., 2019). The corresponding experimental results are shown in Table 12, from which we observe a similar phenomenon: LoRanPAC is more stable and has higher accuracy.
+
+# K.5 SCALING LAWS OF THE EMBEDDING DIMENSION
+
+![](images/d1c89c44e5cc7c6c8de6a9b4b1ddf0516e9c0264593a3c3b50a19efc3c25cb8b.jpg)  
+Figure 5: Training times (in minutes) of LoRanPAC and RanPAC for different embedding dimensions  $E$ . See also Fig. 3 in the main paper for similar results on the other four datasets.
+
+![](images/f017d9549fd4b4c68c72c4b5cd22ad82570a872742b1b21016aea18a8983386f.jpg)
+
+![](images/5503458bea183659b81ca02c55ef9149c4a8a83ba6e4d385e90955860334faf4.jpg)
+
+![](images/e62b7ca2462a8bc518a224495491316cce31bf76584abd0fdc5e876a6704d85a.jpg)
+
+Fig. 6 exhibits a scaling law where the accuracy of LoRanPAC grows with the embedding dimension  $E$ . A similar phenomenon is also shown for RanPAC in Table A5 of Appendix F.6 of McDonnell et al. (2023). However, the experiments of McDonnell et al. (2023) are limited to  $E = 15000$  as RanPAC is not scalable (cf. Figs. 3 and 5), and also limited to only the CIFAR100 dataset. Here, since our LoRanPAC implementation is more stable and scalable, we can run it with  $E$  as large as  $10^5$ , therefore visualizing the scaling phenomenon for eight different datasets in Fig. 6.
+
+It is clearly tempting to scale the embedding dimension even more, but doing so brings up two issues. First, a large  $E$  entails a large memory use, and will therefore eventually be infeasible. Second, we have not found any significant performance gain with even larger  $E$  (e.g.,  $E = 150000$  or  $E = 200000$ ), which is why we stopped at dimension  $E = 10^5$ . Note that enlarging  $E$  amounts to increasing the width of the corresponding layer. It was suggested that increasing the width is beneficial, theoretically (Peng et al., 2023) and empirically (Mirzadeh et al., 2022). On the other hand, Fig. 6 suggests the benefit of increasing the width empirically diminishes, e.g., the accuracies for  $E = 50\mathrm{K}$  and  $E = 100\mathrm{K}$  are comparable on ImageNet-A, CUB-200, VTAB, and StanfordCars. This is empirically corroborated by Guha & Lakshman (2024) and theoretically confirmed by Hu et al. (2024).
+
+# K.6 RUNNING TIMES
+
+In Fig. 3 we compare the running times of several methods in addition to RanPAC (see Figs. 3 and 5). We see that LoRanPAC is faster than prompt-based methods (e.g., L2P, CodaPrompt) and adapter-based methods (e.g., EASE) on CIFAR100, ImageNet-R, ImageNet-A, and CUB.
+
+![](images/e4f4c73a1a2dc02d33e9097c949738ada7a1bb8d01d327dcf0ae0479ca536183.jpg)
+
+![](images/9b5a86632671c5b0a93873c740dbc1c6f38a1e6feb43b693851fb4f177c0da1e.jpg)
+
+![](images/d716584e00bfae114f1d571913a94176de93925d8ee27bd33b3271adb698bfe2.jpg)
+
+![](images/1e0422656b272bf5638abad1cfb03235b49f440d7f3e02b94792d2b240c64449.jpg)
+
+![](images/92543663f723b56e0efafeee91cf75ca96a598495d84df05388a76c8f068560c.jpg)  
+Figure 6: Final accuracy of LoRanPAC for varying embedding dimensions  $E$ . See also Table 15 in comparison to RanPAC.
+
+![](images/10d5da4c9b40bd9cd4eff7be1edc5e3eb54d51073da5f285e73dd779341b9946.jpg)
+
+![](images/20dc468cb78895a25a7d5f71e6c62721e904f964a1dc945c44e7777dba7b3e4e.jpg)
+
+![](images/c223d76e30aed67bb8aa18ee516e64b012b412998e08cfc1f4b315c1ea0d2d56.jpg)
+
+Table 13: Running times (in minutes) of various methods on CIL datasets with  $q_{2} = 5$  (Inc-5). This is the same setting as Fig. 3.  
+
+<table><tr><td></td><td>CIFAR100</td><td>ImageNet-R</td><td>ImageNet-A</td><td>CUB</td></tr><tr><td>L2P</td><td>52.09</td><td>72.31</td><td>26.96</td><td>17.62</td></tr><tr><td>CodaPrompt</td><td>174.9</td><td>246.94</td><td>27.28</td><td>39.91</td></tr><tr><td>EASE</td><td>139.74</td><td>128.35</td><td>54.0</td><td>66.23</td></tr><tr><td>LoRanPAC (E=100K)</td><td>31.24</td><td>27.0</td><td>1.29</td><td>3.73</td></tr></table>
+
+# K.7 MULTIPLE RUNS OF THE EXPERIMENTS
+
+In Table 14 we report results under the same setting of Table 1. We report the mean and standard deviations over 3 random seeds.
+
+Table 14: Final accuracy with pre-trained ViTs under the same setting of Table 1. Reported results are mean and standard deviations over 3 random seeds. See also Table 1.  
+
+<table><tr><td colspan="4">(Part 1) CIFAR100 (B-0)</td><td colspan="3">ImageNet-R (B-0)</td><td colspan="3">ImageNet-A (B-0)</td><td colspan="3">CUB-200 (B-0)</td></tr><tr><td></td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td></tr><tr><td>L2P</td><td>78.22±1.49</td><td>82.33±1.14</td><td>83.61±0.24</td><td>68.36±0.31</td><td>72.01±0.37</td><td>73.74±0.16</td><td>44.5±0.18</td><td>49.31±0.65</td><td>50.87±0.35</td><td>52.6±0.76</td><td>59.46±0.55</td><td>67.47±1.03</td></tr><tr><td>DualPrompt</td><td>80.39±0.59</td><td>83.92±0.12</td><td>85.58±0.7</td><td>67.67±0.4</td><td>71.58±0.25</td><td>72.6±0.16</td><td>49.24±1.1</td><td>53.59±0.19</td><td>56.46±0.28</td><td>53.04±1.32</td><td>61.96±2.04</td><td>68.39±1.25</td></tr><tr><td>CodaPrompt</td><td>82.53±0.51</td><td>85.85±0.43</td><td>87.62±0.2</td><td>67.74±0.33</td><td>72.31±0.37</td><td>74.67±0.33</td><td>33.42±0.56</td><td>48.56±0.76</td><td>58.29±0.93</td><td>38.48±0.29</td><td>58.61±0.87</td><td>70.67±1.38</td></tr><tr><td>SimpleCIL</td><td>80.48±0.0</td><td>80.48±0.0</td><td>80.48±0.0</td><td>63.47±0.0</td><td>63.47±0.0</td><td>63.47±0.0</td><td>58.72±0.0</td><td>58.72±0.0</td><td>58.72±0.0</td><td>80.45±0.0</td><td>80.45±0.0</td><td>80.45±0.0</td></tr><tr><td>RanPAC</td><td>86.93±0.15</td><td>87.06±0.04</td><td>87.02±0.06</td><td>71.95±0.13</td><td>71.85±0.06</td><td>72.29±0.23</td><td>60.11±3.03</td><td>61.07±1.4</td><td>61.71±0.32</td><td>88.07±0.33</td><td>87.53±0.33</td><td>87.83±0.54</td></tr><tr><td>LoRanPAC</td><td>88.21±0.02</td><td>88.23±0.03</td><td>88.24±0.04</td><td>73.62±0.1</td><td>73.6±0.13</td><td>73.67±0.02</td><td>62.63±0.08</td><td>62.96±0.19</td><td>62.89±0.23</td><td>89.2±0.11</td><td>89.16±0.16</td><td>89.14±0.18</td></tr><tr><td>ADAM†</td><td>83.44±0.22</td><td>84.81±0.2</td><td>86.03±0.13</td><td>63.8±0.11</td><td>64.94±0.06</td><td>70.82±0.51</td><td>58.72±0.0</td><td>58.73±0.03</td><td>58.86±0.19</td><td>80.48±0.02</td><td>80.69±0.02</td><td>80.99±0.12</td></tr><tr><td>RanPAC†</td><td>88.76±0.12</td><td>90.14±0.11</td><td>90.62±0.16</td><td>71.81±0.83</td><td>73.43±0.06</td><td>78.07±0.59</td><td>21.26±29.0</td><td>61.71±0.32</td><td>62.06±0.17</td><td>87.91±0.43</td><td>87.56±0.19</td><td>88.31±0.33</td></tr><tr><td>LoRanPAC†</td><td>89.62±0.08</td><td>90.85±0.12</td><td>91.54±0.07</td><td>73.76±0.19</td><td>74.58±0.04</td><td>78.79±0.58</td><td>62.56±0.4</td><td>62.76±0.38</td><td>62.98±0.32</td><td>89.17±0.02</td><td>89.24±0.04</td><td>89.36±0.07</td></tr><tr><td>EASE*</td><td>84.68±0.4</td><td>86.87±0.25</td><td>88.46±0.24</td><td>73.44±0.26</td><td>76.2±0.69</td><td>77.54±0.18</td><td>59.67±1.05</td><td>59.56±1.7</td><td>62.41±0.24</td><td>80.56±0.09</td><td>81.07±0.45</td><td>80.92±0.56</td></tr><tr><td>LoRanPAC*</td><td>89.97±0.44</td><td>90.89±0.11</td><td>91.58±0.06</td><td>78.42±0.26</td><td>80.07±0.35</td><td>81.48±0.13</td><td>63.18±0.17</td><td>64.12±0.34</td><td>65.81±0.2</td><td>89.1±0.07</td><td>89.24±0.11</td><td>89.38±0.05</td></tr><tr><td colspan="4">(Part 2) ObjectNet (B-0)</td><td colspan="3">OmniBenchmark (B-0)</td><td colspan="3">VTAB (B-10)</td><td colspan="3">StanfordCars (B-16)</td></tr><tr><td></td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td></tr><tr><td>L2P</td><td>46.63±0.73</td><td>52.3±0.3</td><td>55.51±0.65</td><td>53.74±0.59</td><td>57.32±0.2</td><td>60.9±1.13</td><td>59.03±5.12</td><td>72.32±2.15</td><td>76.66±1.67</td><td>14.49±0.77</td><td>27.14±0.79</td><td>41.35±2.04</td></tr><tr><td>DualPrompt</td><td>47.71±0.4</td><td>52.83±0.67</td><td>56.03±0.54</td><td>56.56±0.48</td><td>59.45±0.24</td><td>62.4±0.3</td><td>64.67±4.34</td><td>75.98±3.22</td><td>79.58±3.09</td><td>11.91±0.46</td><td>18.92±0.76</td><td>29.04±3.23</td></tr><tr><td>CodaPrompt</td><td>46.7±0.37</td><td>54.32±0.79</td><td>59.26±0.39</td><td>59.33±0.67</td><td>64.87±0.65</td><td>68.37±0.45</td><td>62.39±0.58</td><td>74.93±0.6</td><td>85.24±0.52</td><td>7.89±0.17</td><td>11.63±0.35</td><td>29.46±1.43</td></tr><tr><td>SimpleCIL</td><td>51.66±0.0</td><td>51.66±0.0</td><td>51.66±0.0</td><td>70.19±0.0</td><td>70.19±0.0</td><td>70.19±0.0</td><td>82.53±0.0</td><td>82.53±0.0</td><td>82.53±0.0</td><td>35.46±0.0</td><td>35.46±0.0</td><td>35.46±0.0</td></tr><tr><td>RanPAC</td><td>58.15±0.17</td><td>58.17±0.33</td><td>58.21±0.43</td><td>77.74±0.09</td><td>77.48±0.1</td><td>77.45±0.2</td><td>91.53±0.14</td><td>91.77±0.08</td><td>91.77±0.13</td><td>55.53±3.65</td><td>71.55±0.11</td><td>71.54±0.12</td></tr><tr><td>LoRanPAC</td><td>60.71±0.09</td><td>60.67±0.16</td><td>60.61±0.12</td><td>79.63±0.09</td><td>79.65±0.04</td><td>79.76±0.04</td><td>92.47±0.01</td><td>92.56±0.06</td><td>92.53±0.04</td><td>74.23±0.11</td><td>74.35±0.13</td><td>74.28±0.09</td></tr><tr><td>ADAM†</td><td>52.2±0.15</td><td>53.78±0.59</td><td>55.9±0.09</td><td>70.53±0.04</td><td>70.43±0.14</td><td>70.25±0.18</td><td>82.58±0.05</td><td>82.58±0.05</td><td>82.58±0.05</td><td>35.56±0.04</td><td>35.56±0.04</td><td>35.56±0.04</td></tr><tr><td>RanPAC†</td><td>59.33±0.42</td><td>61.3±0.6</td><td>64.37±0.18</td><td>78.29±0.08</td><td>78.2±0.23</td><td>78.39±0.37</td><td>91.76±0.11</td><td>91.88±0.07</td><td>91.98±0.1</td><td>44.02±31.09</td><td>72.32±0.09</td><td>72.32±0.09</td></tr><tr><td>LoRanPAC†</td><td>61.43±0.17</td><td>63.29±0.74</td><td>66.24±0.38</td><td>79.85±0.16</td><td>80.19±0.24</td><td>80.08±0.2</td><td>92.57±0.03</td><td>92.54±0.02</td><td>92.59±0.09</td><td>74.67±0.18</td><td>74.66±0.21</td><td>74.67±0.33</td></tr><tr><td>EASE*</td><td>50.2±0.4</td><td>53.92±0.53</td><td>56.78±0.31</td><td>70.4±0.05</td><td>70.67±0.22</td><td>70.81±0.18</td><td>89.86±0.24</td><td>93.56±0.12</td><td>93.67±0.15</td><td>32.46±0.73</td><td>31.64±1.03</td><td>29.79±1.91</td></tr><tr><td>LoRanPAC*</td><td>63.67±0.6</td><td>66.52±0.79</td><td>66.96±0.27</td><td>80.07±0.37</td><td>80.34±0.17</td><td>80.41±0.37</td><td>93.16±0.67</td><td>93.17±0.67</td><td>93.16±0.7</td><td>75.63±0.22</td><td>75.56±0.14</td><td>75.67±0.24</td></tr></table>
+
+Table 15: Final accuracies of RanPAC and LoRanPAC with pre-trained ViTs. RanPAC takes its default choice  $E = 10\mathrm{K}$ , while for LoRanPAC we set three different values for  $E$ :  $E = 10\mathrm{K}$ ,  $E = 50\mathrm{K}$ , and  $E = 100\mathrm{K}$ .  
+
+<table><tr><td>(Part 1)</td><td colspan="3">CIFAR100 (B-0)</td><td colspan="3">ImageNet-R (B-0)</td><td colspan="3">ImageNet-A (B-0)</td><td colspan="3">CUB-200 (B-0)</td><td>Avg.</td></tr><tr><td></td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td></td></tr><tr><td>RanPAC</td><td>86.71</td><td>87.02</td><td>87.10</td><td>71.90</td><td>71.97</td><td>72.50</td><td>56.48</td><td>62.34</td><td>61.75</td><td>88.08</td><td>87.15</td><td>88.13</td><td>76.76</td></tr><tr><td>LoRanPAC (E=10K)</td><td>87.48</td><td>87.49</td><td>87.42</td><td>70.77</td><td>70.85</td><td>70.48</td><td>58.85</td><td>58.46</td><td>59.38</td><td>86.73</td><td>86.85</td><td>87.19</td><td>76.00</td></tr><tr><td>LoRanPAC (E=50K)</td><td>88.13</td><td>88.05</td><td>88.04</td><td>73.05</td><td>73.07</td><td>73.05</td><td>62.80</td><td>62.80</td><td>62.48</td><td>89.23</td><td>89.23</td><td>89.19</td><td>78.26</td></tr><tr><td>LoRanPAC (E=100K)</td><td>88.18</td><td>88.18</td><td>88.21</td><td>73.67</td><td>73.72</td><td>73.63</td><td>62.74</td><td>63.20</td><td>63.20</td><td>89.36</td><td>89.27</td><td>89.23</td><td>78.55</td></tr><tr><td>(Part 2)</td><td colspan="3">ObjectNet (B-0)</td><td colspan="3">OmniBenchmark (B-0)</td><td colspan="3">VTAB (B-10)</td><td colspan="3">StanfordCars (B-16)</td><td>Avg.</td></tr><tr><td></td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td>Inc-5</td><td>Inc-10</td><td>Inc-20</td><td></td></tr><tr><td>RanPAC</td><td>58.77</td><td>57.66</td><td>57.69</td><td>77.63</td><td>77.63</td><td>77.46</td><td>91.15</td><td>91.58</td><td>91.58</td><td>58.03</td><td>71.40</td><td>71.40</td><td>73.50</td></tr><tr><td>LoRanPAC (E=10K)</td><td>57.97</td><td>57.82</td><td>58.06</td><td>76.79</td><td>76.96</td><td>76.91</td><td>91.89</td><td>91.81</td><td>91.91</td><td>64.03</td><td>64.43</td><td>64.72</td><td>72.78</td></tr><tr><td>LoRanPAC (E=50K)</td><td>59.41</td><td>59.4</td><td>59.54</td><td>79.33</td><td>79.35</td><td>79.43</td><td>92.48</td><td>92.47</td><td>92.54</td><td>73.32</td><td>73.66</td><td>73.47</td><td>76.20</td></tr><tr><td>LoRanPAC (E=100K)</td><td>60.83</td><td>60.86</td><td>60.77</td><td>79.50</td><td>79.60</td><td>79.70</td><td>92.46</td><td>92.55</td><td>92.56</td><td>74.21</td><td>74.39</td><td>74.39</td><td>76.82</td></tr></table>
+
+![](images/a3f89be783888d27305436353abb4c6639571c516ee9c3e772d6a34106b43a49.jpg)  
+(7a) The effective rank  $r_1(\pmb{H}_{1:t}^\top \pmb{H}_{1:t})$  as the number  $t$  of seen tasks increases.
+
+![](images/1b902f1dec0fbc480781a2169ce444df71c67a7e3d26d830a51c5e5ba66625d8.jpg)
+
+![](images/43bcaaa0bcf75b4e19d7cc02c336e945ff248451743a1d91a32d92160502bf26.jpg)
+
+![](images/8f15fe0a7f2c3f651dab84f62df18a7c71086c27f57678c6cfd8a6f30825632f.jpg)
+
+![](images/2fd301ac93f61beaac1d760085e5ffc788fb47d8dcb99a6d128f2fa7e2f09713.jpg)  
+(7b) The eigenvalues of  $\pmb{H}_{1:t}^{\top}\pmb{H}_{1:t}$  arranged in descending order (  $t$  fixed).
+
+![](images/b5640baa5247e5883f5a02aeefa6af1f570ae68a8f73215cb60cc90642e33356.jpg)  
+Figure 7: The effective ranks of  $\pmb{H}_{1:t}^{\top}\pmb{H}_{1:t}$ .
+
+![](images/d2a237b69c0fd9458cd95c2081e2e815609e880f5b0b0c4d03e4336ac1e21646.jpg)
+
+![](images/eba1999bcee1f933e234b148a6764a6e3d28e0c2babc04fcbc87b9d7686277e4.jpg)
+
+# K.8 MORE FIGURES FOR DATA ANALYSIS
+
+Recall  $\mu_{k}(\cdot)$  denotes the  $k$ -th largest eigenvalue of a matrix and  $M_{t} \coloneqq m_{1} + \dotsm + m_{t}$ . Define the notion of effective rank, as by Tsigler & Bartlett (2023):
+
+$$
+r _ {k} \left(\boldsymbol {H} _ {1: t} ^ {\top} \boldsymbol {H} _ {1: t}\right) := \frac {\sum_ {j \geq k} \mu_ {j} \left(\boldsymbol {H} _ {1 : t} ^ {\top} \boldsymbol {H} _ {1 : t}\right)}{\mu_ {k} \left(\boldsymbol {H} _ {1 : t} ^ {\top} \boldsymbol {H} _ {1 : t}\right)}, \quad \forall k = 1, \dots , M _ {t} \tag {29}
+$$
+
+Note that  $r_1(\cdot)$  is the standard definition of effective rank (sometimes called stable rank), while  $r_k(\cdot)$  generalizes it by only considering the eigenvalues starting from the  $k$  largest.
+
+Fig. 7a plots the effective rank  $r_1(\pmb{H}_{1:t}^\top \pmb{H}_{1:t})$  as the number of tasks increases and Fig. 7b plots  $r_k(\pmb{H}_{1:t}^\top \pmb{H}_{1:t})$  for different values of  $k$ . We observe similar curves on different datasets:  $r_1(\pmb{H}_{1:t}^\top \pmb{H}_{1:t})$  is smaller than 3, and  $r_k(\pmb{H}_{1:t}^\top \pmb{H}_{1:t})$  first increases and then decreases as a function of  $k$ .
+
+Fig. 8 plots the top  $k_{t}$  eigenvalues of  $B_{t}B_{t}^{\top}$  (recall that we only preserve top  $k_{t}$  singular values of  $B_{t}$ ). It shows that the condition number is now of order  $10^{5}(10^{11} / 10^{6})$ , which is much smaller than the condition number of  $H_{1:t}^{\top}H_{1:t}$ .
+
+Fig. 9 plots the normalized differences  $\| \overline{\Sigma}_{1:t} - \widetilde{\Sigma}_{1:t}\|_{\infty} / \| \overline{\Sigma}_{1:t}\|_{\infty}$  between the eigenvalues given by LoRanPAC and its continual implementation (Algorithm 4). This empirically verifies that the differences between the two are insignificant compared to the largest eigenvalue  $\| \overline{\Sigma}_{1:t}\|_{\infty}$  (just of order  $10^{-3}$ ). This experiment assists understanding Theorem 6.
+
+![](images/b7396c687e2320945bd218745ac733dafed4625c1bebc12f90e2bae35474d8a5.jpg)  
+Figure 8: The eigenvalues of  $\widetilde{\Sigma}_{1:t}$  for  $t = 1, \ldots, 9$ . These are by definition the top  $k_{t}$  eigenvalues of  $\pmb{B}_{t}\pmb{B}_{t}^{\top}$ . It shows that our continual implementation (Algorithm 4) prunes the smallest eigenvalues (of order  $10^{-5}$ ) so that the condition number is now of order  $10^{5}(10^{11} / 10^{6})$ ; compare this figure with Fig. 1a. In this experiment we truncate  $25\%$  of the eigenvalues; that is, given  $M_{t}$ , we select  $k_{t}$  such that  $k_{t} / M_{t} = 75\%$ .
+
+![](images/fe788e1359dbb1712577550402ddfe04ec6c47407f44f7ad2d9969b425e3a3f9.jpg)
+
+![](images/e348a1a9da4eaa7ebe428a34c537b64f38352da7ecb715c9830cd1e70a010d0e.jpg)
+
+![](images/fe5ca7399114ba0d279caab4cbf95ff78b0c6c6cca6249a07fb0511002a50972.jpg)
+
+![](images/c3d22ca0b58c55ba704aaa5762b80ae5336f8773448fee1cda99a3c60519eabd.jpg)  
+Figure 9: Normalized differences  $\| \overline{\mathbf{\Sigma}}_{1:t} - \widetilde{\mathbf{\Sigma}}_{1:t}\|_{\infty} / \| \overline{\mathbf{\Sigma}}_{1:t}\|_{\infty}$  between the eigenvalues given by LoRanPAC and its continual implementation (Algorithm 4). This empirically verifies that the differences between the two are insignificant compared to the largest eigenvalue  $\| \overline{\mathbf{\Sigma}}_{1:t}\|_{\infty}$  (just of order  $10^{-3}$ ). See Fig. 8 and Fig. 1a. See also Theorem 6 where we formally bound the distances  $\| \overline{\mathbf{\Sigma}}_{1:t} - \widetilde{\mathbf{\Sigma}}_{1:t}\|_{\infty}$  for every  $t$ .
+
+![](images/a0a8a8ab8dd92280f047142798341d365665d34e30b35af1688e2901a692ce30.jpg)
+
+![](images/aaa24d6449a12253a046a79bc19f0767edf07660c2fd1b331da2c3b64bac6bc5.jpg)
+
+![](images/ac52976ea91107d4382cb6dfc5547d6b4c94f89aeeb96c0f1e01e325d1bed8fd.jpg)
+
+Fig. 10 plots the eigenvalues of  $\pmb{H}_{1:t}^{\top}\pmb{H}_{1:t} \in \mathbb{R}^{M_t \times M_t} (M_t \leq 10^4)$  with the embedding dimension  $E$  varying in  $\{10000, 25000, 50000, 75000\}$ . It shows that the "shape" of the spectrum is similar for different values of  $E$  and on different datasets (see also Fig. 1a for the case  $E = 10^5$ ).
+
+Fig. 11 depicts how the random embedding  $\pmb{P} \in \mathbb{R}^{E \times d}$  and ReLU layer affects the spectrum of the features. In Fig. 11a we plot the output features  $\pmb{X} \in \mathbb{R}^{d \times M}$  of the ImageNet-A dataset from pre-trained ViTs ( $d = 768, M = 5981, E = 10^5$ ). We see  $\pmb{X}\pmb{X}^\top$  is relatively well-conditioned: Its maximum eigenvalue is of order  $10^5$  and minimum eigenvalue of order 10. Fig. 11b shows that  $\pmb{X}^\top\pmb{P}^\top\pmb{P}\pmb{X} \in \mathbb{R}^{E \times M}$  is ill conditioned. This is because  $\pmb{P}\pmb{X}$  has rank at most  $d$ , and the smallest  $M - d$  eigenvalues of  $\pmb{X}^\top\pmb{P}^\top\pmb{P}\pmb{X}$  should be zero, while we get these small and non-zero eigenvalues in Fig. 11b due to numerical errors in (incremental) SVD; these eigenvalues should be truncated (set to zero), in order to solve Min-Norm accurately. Fig. 11c shows that  $\mathrm{relu}(P\pmb{X})$  also has these small and non-zero eigenvalues. While the rank of  $\mathrm{relu}(P\pmb{X})$  is unclear, its smallest yet non-zero eigenvalues are likely inherent from  $P\pmb{X}$ , and we suggest truncating them as well. See also Table 16.
+
+# K.9 ABLATION STUDY ON RANDOM RELU FEATURES
+
+In Table 16 we study the effects of the random ReLU model. Recall that, given the output features  $\mathbf{X}_t$  of the data of task  $t$  from a pre-trained model, we use the random ReLU features  $H_t \coloneqq \mathrm{relu}(P\mathbf{X}_t)$  and labels  $\mathbf{Y}_t$  to train a linear classifier via continually solving Min-Norm or LoRanPAC. We could
+
+![](images/aaae235f841bd22848031d7e7f0750739fb28c630c9ea560d92c052da7fd7423.jpg)  
+Figure 10: Eigenvalues of  $\pmb{H}_{1:t}^{\top}\pmb{H}_{1:t} \in \mathbb{R}^{M_t\times M_t}$  ( $M_t \leq 10^4$ ) with the embedding dimension  $E$  varying in {10000, 25000, 50000, 75000}. We find that the "shape" of the spectrum is similar for different  $E$  and on different datasets (see also Fig. 1a for the case  $E = 10^5$ ).
+
+![](images/af96364a183ef57ddeee1e15511e4634b4e390e8bc0a1ac04c304d1a95c41b83.jpg)
+
+![](images/944a3d05751760b83f90d93917d3fb99e328de9d783772164eca42332b6e0345.jpg)
+
+![](images/fa39900759006349e16ee9c97bebd9e320d3165437ba02a2658efa7486d78454.jpg)
+
+![](images/211b1d3b93f3677816a6497d9287585c0c4def144a73fb7892da3b2d166fb291.jpg)  
+Figure 11: Feeding  $M$  data samples to a pre-trained model gives its  $d$ -dimensional output features  $\mathbf{X} \in \mathbb{R}^{d \times M}$ . Passing it through a random embedding layer  $\mathbf{P}$  and a ReLU layer yields the random ReLU features  $\mathbf{H} \in \mathbb{R}^{E \times M}$ . Plotted in (a), (b), (c), respectively, are eigenvalues of  $\mathbf{X}\mathbf{X}^{\top}$ ,  $\mathbf{X}^{\top}\mathbf{P}^{\top}\mathbf{P}\mathbf{X}$ , and  $\mathbf{H}^{\top}\mathbf{H}$  in descending order, where  $\mathbf{X}$  consists of pre-trained ViT features of the ImageNet-A dataset ( $d = 768$ ,  $M = 5981$ ,  $E = 10^{5}$ ). It is seen that  $\mathbf{P}\mathbf{X}$  and  $\mathbf{H}$  are more ill-conditioned than  $\mathbf{X}$ . See also Table 16.
+
+instead use  $X_{t}$  or  $P X_{t}$  or  $\mathrm{relu}(X_{t})$  as the features to train the linear classifier. To see the effects of these alternative choices, we make Table 16 from which we have the following observations:
+
+- The random ReLU features  $\mathbf{H}_t$  gives the highest accuracy, while using the ReLU layer alone or random embedding alone does not make improvements over the original pre-trained features  $\mathbf{X}_t$ .  
+- Solving Min-Norm via Incremental SVD exhibits numerical failures as soon as we use random embedding  $\mathbf{P}$ . This is because  $\mathbf{P}\mathbf{X} \in \mathbb{R}^{E \times M}$  has rank at most  $d$  and we would get some  $M - d$  small yet non-zero eigenvalues accounting for the numerical errors of (incremental) SVD solvers (see, e.g., Fig. 11). While they damage the accuracy, truncating these singular values and the corresponding singular vectors restores the performance.
+
+Table 16: Final accuracy of Min-Norm and LoRanPAC when using different features,  $\mathbf{X}_t \in \mathbb{R}^{d \times m_t}$ ,  $\mathrm{relu}(X_t)$ ,  $\mathbf{PX}_t$ , and  $\mathbf{H}_t := \mathrm{relu}(\mathbf{PX}_t)$ . Here  $\mathbf{P} \in \mathbb{R}^{E \times d}$  is a random Gaussian matrix with  $\mathcal{N}(0,1)$  entries and  $E = 10^5$ , and  $\mathbf{H}_t$  consists of random ReLU features we use by default. Note that both Min-Norm and LoRanPAC are solved by the incremental SVD method, with a difference that the latter truncates the SVDs. Incremental SVD without truncation is not scalable enough to handle all 50000 data samples of CIFAR100, so we mark "N.A." in the table for Min-Norm.  
+
+<table><tr><td></td><td>CIFAR100 (B-0, Inc-10)</td><td>ImageNet-R (B-0, Inc-20)</td><td>ImageNet-A (B-0, Inc-20)</td><td>CUB (B-0, Inc-20)</td><td>Avg.</td></tr><tr><td colspan="6">Final Accuracy of Min-Norm</td></tr><tr><td>Xt</td><td>85.11</td><td>69.22</td><td>58.92</td><td>84.90</td><td>74.54</td></tr><tr><td>relu(Xt)</td><td>84.07</td><td>66.43</td><td>55.23</td><td>84.14</td><td>72.47</td></tr><tr><td>Pxt</td><td>N.A.</td><td>1.35</td><td>2.37</td><td>0.55</td><td>N.A.</td></tr><tr><td>Ht</td><td>N.A.</td><td>0.42</td><td>0.92</td><td>0.72</td><td>N.A.</td></tr><tr><td colspan="6">Final Accuracy of LoRanPAC</td></tr><tr><td>Xt</td><td>84.61</td><td>68.23</td><td>59.45</td><td>84.14</td><td>74.11</td></tr><tr><td>relu(Xt)</td><td>83.51</td><td>66.20</td><td>56.62</td><td>84.01</td><td>72.59</td></tr><tr><td>Pxt</td><td>78.96</td><td>48.50</td><td>42.73</td><td>63.15</td><td>58.33</td></tr><tr><td>Ht (default)</td><td>88.18</td><td>73.65</td><td>63.20</td><td>89.23</td><td>78.57</td></tr></table>
+
+# K.10 EXTRA EMPIRICAL STUDY OF RANPAC
+
+In Appendix K.10.2, we analyze the training losses of RanPAC. In Appendix K.10.3 we show RanPAC is unstable with respect to small increments, while LoRanPAC is more stable.
+
+# K.10.1 RANPAC IS UNSTABLE WITH RESPECT TO REGULARIZATION PARAMETERS
+
+In Fig. 12a, we show that RanPAC is unstable as it fails if the regularization  $\lambda$  is small. In Fig. 12b, we extend the solution of (4) into  $J_{t}\widetilde{U}_{1:t}(\widetilde{\Sigma}_{1:t}^{2} + \lambda I_{E})^{-1}\widetilde{U}_{1:t}^{\top}$  for ridge regression; this amounts to RanPAC with truncation. This extension stabilizes RanPAC and is practically immune to changes in the regularization parameter  $\lambda$ .
+
+![](images/3b55a5480ca4c29b3d7808af5752a2c0b8a403a108ed69ede5bdc0f0debd69f2.jpg)
+
+![](images/5ca0bda2a90e2d1d58389020b52fd815453d6ba914339466dba3b1bfb9ecec35.jpg)
+
+![](images/7fa6888bbf5ed5340fc07e56867019fe0f077d1e6c6d0022287c2815d1407c89.jpg)
+
+![](images/95bbb0151dc06a0f89ae88d2fa0ad17922b20cbb76e3f36b088c0e784e40843c.jpg)
+
+![](images/bcf5ae8b935f4c18267d312897462161024670e91a408df809c0902a178063cd.jpg)  
+(12a) RanPAC breaks down for the small regularization  $\lambda$ .
+
+![](images/a0d42b62129cd3dde3df9766a277756b31d308c42de4ee01c30de918f056ba7e.jpg)  
+(12b) RanPAC with truncation (25%) has stable performance for varying regularization parameter  $\lambda$ .
+
+![](images/0fa5242237069e03bfc2e32cd2d9b8b38b5ae8085c97fadb7cf301dc75403827.jpg)
+
+![](images/ffae89c614d0af97b04cb33410a5ca7168e1ce16018ac31cd2bab401bf66e2df.jpg)
+
+![](images/062f6f0b2c4bb9e1208e94222c461b6933720dcd896cc3958fd95e80a2027494.jpg)  
+Figure 12: We extend the solution of (4) into  $J_{t}\widetilde{U}_{1:t}(\widetilde{\Sigma}_{1:t}^{2} + \lambda I_{E})^{-1}\widetilde{U}_{1:t}^{\top}$  for ridge regression; this amounts to RanPAC with truncation. This extension stabilizes RanPAC.
+
+![](images/d1ace505b9a9074fc2a8e8f31180f9a4b7e4fb9d39a5cdf046de47fae1aafbce.jpg)  
+Figure 13: The average training MSE loss  $\frac{1}{M_t}\| \pmb{W}\pmb{H}_{1:t} - \pmb{Y}_{1:t}\|_{\mathrm{F}}^2$  of RanPAC for different ridge regularization parameters  $\lambda \in \{0,1,10,100,1000\}$ . Compare this with Fig. 12 and Fig. 4.
+
+![](images/85eb01dfe20b3351ca15e408e800c2fa55439e92e90076fb1c432133dac4efd9.jpg)
+
+![](images/1cef77d8bddf679edffe1382cbc5cd8eebbb908a0116833432fdd573bf24e53d.jpg)
+
+# K.10.2 TRAINING LOSSES OF RANPAC
+
+Fig. 13 plots the training loss of RanPAC for different ridge regularization parameters  $\lambda \in \{0,1,10,100,1000\}$ . We observe that, in the case of  $\lambda = 0$ , the training loss of RanPAC is smaller than 1. Fig. 4 shows that the incremental SVD implementation of Min-Norm could have its training loss larger than  $10^{10}$ . This difference is because the incremental SVD implementation (without truncation) can be unstable and accumulates errors over time, while RanPAC is implemented by solving the normal equations  $\boldsymbol{W}(\boldsymbol{H}_{1:t}\boldsymbol{H}_{1:t}^{\top} + \lambda \boldsymbol{I}_E) = \boldsymbol{Y}_{1:t}\boldsymbol{H}_{1:t}^{\top}$  in variable  $\boldsymbol{W}$  (the covariances  $\boldsymbol{H}_{1:t}\boldsymbol{H}_{1:t}^{\top}$  and  $\boldsymbol{Y}_{1:t}\boldsymbol{H}_{1:t}^{\top}$  are updated continually). The advantage is that maintaining  $\boldsymbol{H}_{1:t}\boldsymbol{H}_{1:t}^{\top}$  and  $\boldsymbol{Y}_{1:t}\boldsymbol{H}_{1:t}^{\top}$  is easy and does not entail numerical errors, so solving the normal equations  $\boldsymbol{W}(\boldsymbol{H}_{1:t}\boldsymbol{H}_{1:t}^{\top} + \lambda \boldsymbol{I}_E) = \boldsymbol{Y}_{1:t}\boldsymbol{H}_{1:t}^{\top}$  directly is expected to be stable, as long as the solver invoked is numerically stable (the built-in PyTorch solver is used). This appears to be the case, as RanPAC maintains small training errors. On the other hand, the corresponding test accuracy can be nearly zero with small  $\lambda$  (e.g., when  $\lambda = 0,1$  as shown in Fig. 12).
+
+# K.10.3 RANPAC IS UNSTABLE FOR CIL WITH THE SMALLEST INCREMENTS
+
+In the experiments of the main paper, we see that RanPAC is unstable for small increments (e.g., Inc-5). Moreover, its instability is exacerbated in the extreme case Inc-1 (Table 2).
+
+Here, we show similar experimental results in Figs. 14 to 17, suggesting that RanPAC is significantly more unstable for Inc-1, Inc-2, and Inc-4. In particular, in the extreme case of Inc-1, RanPAC presents failures on all datasets except CIFAR100 (as indicated by the vehicle blue line in Figs. 14 to 17. On the contrary, these figures, including Fig. 18, show that our LoRanPAC method is stable for different small increments (1, 2, 4, 5).
+
+![](images/5da435eaade1cc7921b5e9c5164c3d0d745f4bbc5bf50e66fd4e081b65ca7d25.jpg)
+
+![](images/80b295268e162484c6cb7bbf2e2a637cdd78ab45c679a999e057ac900700d1f8.jpg)
+
+![](images/ed6da499162570d190eeffd4cedf15272b0df00bcaec0a38e0f2ba4353c6ead7.jpg)
+
+![](images/533f68aa583633f39df1df8804769ac8378be8e66ab4c0d3d1bf358dd4f661be.jpg)
+
+![](images/111750163164bdd226c74b56fecc4209c1a74a61589f873aec48aeb983c82a37.jpg)
+
+![](images/34cbf1b72d35e8624a6f1103e06bceeedd3874573889f55cde499cef2887011e.jpg)
+
+![](images/990190620b6d29052b4b8575653fa4664681a166eb8e3c013f6dd08de3163d6b.jpg)
+
+![](images/8e4e522324d3892b9053eadb66056b3590583442fed1c3a58a2ec90e49c5c5f4.jpg)
+
+![](images/404c21ce27ab9725f9edcaea2cddf57c056124ddf351347cf8cb2bb95126a570.jpg)
+
+![](images/2fb3d70077406b0bf59ae963bbb09b1bfcfb5736ef3e94938eae62270adccc20.jpg)
+
+![](images/896eafe431aa0572a700f3d8c145e22b9e5f26d1e863d15e135265e53ee021a1.jpg)
+
+![](images/24e623237eb00e86b8ead63083ab079f3acd71d131c07e517b44ff5ad7c1096a.jpg)
+
+![](images/51897bfbf0731650c9ea998fdd15d82ac53d7d5a11626890735dad3cf4a0d992.jpg)  
+(14a) RanPAC
+
+![](images/da38fa853ddcff309349923df66a2f96c5c7a63be8096307255cb4087a75ed34.jpg)  
+(14b) LoRanPAC
+
+![](images/814ce7562f6e3e2f00b1e534d277cd5677793705ba1a0c68730d303efbca7b34.jpg)  
+(14c) RanPAC
+
+![](images/67d9e54c82487be4fc7d62338a1dfc62d417494a79d9f80199b655be2980284d.jpg)  
+(14d) LoRanPAC  
+Figure 14: Upper triangular accuracy matrices on CIFAR100 and ImageNet-R (Inc-1, 2, 4, 5).
+
+![](images/09bd8f55fae68cc1d74236f0e2c8dceea073e278ea60f4f597c31a814d8b7a13.jpg)
+
+![](images/c12cfd54b84d0a808f0185b9e8fb0b482c97ec70617e0bf52df7eb280c8846a6.jpg)
+
+![](images/818a70f55ccf80c7c14dc5834a0e7492ff405843a86d4cf4d136f87b08161555.jpg)
+
+![](images/8164cbb2042822750eaf4cd0d1f16359b8ab6032886ed2d17006cc55b3771997.jpg)
+
+![](images/c7f3202c1ba5ae22c623366a867e5847964e6e498013fb793a4d6f678b424819.jpg)
+
+![](images/84b395c4a77895d7dc198716e7f8a50b6b0567e3ffe827e574c7ed3a3af86744.jpg)
+
+![](images/62d04ae1c40d1f570a94fdcce3ba6e1f9a0b9d77edcef9a52fe4574ae71aa714.jpg)
+
+![](images/6e0716fd7a47852bd91a2d606f59804b98954e084183088c3592db2ef9c0bc61.jpg)
+
+![](images/36f7be6ec11e852d74ee730e41a8e217b4746415f1f4f0a343a3ef6360f585bc.jpg)
+
+![](images/54167278119acd884de2d11f018b9f32404cc4bfdf94077ca2403d43bb8d207b.jpg)
+
+![](images/d6e46cef322aace1735b46c8be6400860e4cf7f20226f54f548e8952b1a1604b.jpg)
+
+![](images/ef8814f8b861480de0703122d09f4fadb18ecb729a0869ec5444cf87dd22fe65.jpg)
+
+![](images/5f10f537cde6d7588ce40d146e9f89201dc8918436486ec20f896017705dd8de.jpg)  
+(15a) RanPAC  
+Figure 15: Upper triangular accuracy matrices on ImageNet-A and CUB-200 (Inc-1, 2, 4, 5).
+
+![](images/6e724cbfb2544dcb005bb89a1689828a2a1324bee4d18df3c559618ce2268176.jpg)  
+(15b) LoRanPAC
+
+![](images/c9602b5958f3ae9736cec5fbd5fa1d8c0d5351afaf497cbce7118fb2dfc0314c.jpg)  
+(15c) RanPAC
+
+![](images/0ec02665425e18b852ceb7b85e19af12dd790f2567f441da9a4f0ef3b1646344.jpg)  
+(15d) LoRanPAC
+
+![](images/370560497b42da854972ad307d23c26f96ecbec16c05bcfe9d49b06b0d00fb3f.jpg)
+
+![](images/ce67bfd22269a6627fc4c8d35244952ac278b2d2e8075722c48cd178eb3e0235.jpg)
+
+![](images/e617f4d60a7344e903c480130212d85f03cc045b8a89089750aa838144260ad7.jpg)
+
+![](images/48ea7df6fd84283573d538525177ea222b9e62ea41b8a4bf9a5840b59cc3d187.jpg)
+
+![](images/c737dfc58c9ae1406b793a347da60d483822bbde860b05721c9e4bf36eeb6f89.jpg)
+
+![](images/fc31431223908561d3750496298b97bdbc92851d96119b9a98093993510d0860.jpg)
+
+![](images/34430b53ec2d58fdb0fcdcf76d27baeb106d864b647fe73a5bef2a0eb300397d.jpg)
+
+![](images/0c41a6495df5f71a18ef059c07af80c05b978692127b27475e1fdda724d992d9.jpg)
+
+![](images/1504ee5b87d675679143fd6f39a043ba19088f6ee56d1f4810bfdc8eed8263b3.jpg)
+
+![](images/e9f07fde6ab5de3fbac09b515cc6e6bd4f52a347a8fc0d4acf0d9d939ef7404e.jpg)
+
+![](images/4b271398fc4bfc3963c7f821d2b334a8bde8bb5355eab3db3d593b5281d4c7b0.jpg)
+
+![](images/c48cd2ce353f2423cc867aa16fc18a751d2863bfa5af90c2e58428a0c8505e86.jpg)
+
+![](images/1936021a8b4bc694afdc1ee3264bb28ced1ce2411fee129998f9ded40b92ded2.jpg)  
+(16a) RanPAC
+
+![](images/0aede3a453ed0f85219a0e8988b42cf873daf0984e010582f8c72f5f86ef0358.jpg)  
+(16b) LoRanPAC
+
+![](images/d2086112fcb5481347108d17ca367fc72d6d1b7c374a8ad7e63ebdbde12ecb59.jpg)  
+(16c) RanPAC
+
+![](images/e97a1314c5e84bae4ba778f1c2d2584383629582f6ec6543a274f31f958d2f9f.jpg)  
+(16d) LoRanPAC  
+Figure 16: Upper triangular accuracy matrices on ObjectNet and OmniBenchmark (Inc-1, 2, 4, 5).
+
+![](images/564baeaa2a46d02ebc835fb0010112e5fa3c49f3dae14cebcd70ab2dca0a41be.jpg)
+
+![](images/30dbc58911f5ce3d43dac220e995087628a61094f42685e5001aa7cf3a9ce6d8.jpg)
+
+![](images/de7a64bbc88e12fc9ea90ed2f40f30069a8aa65b86a19d0435606aff7e56cbbd.jpg)
+
+![](images/18a64a7acc001a81bb86808c0b245dc34e67764502969086d0dc27b20bb0c0e2.jpg)
+
+![](images/ae91baf879bb8c4379bc2f924b2c420ca6e51d10284a3fdfacc259e41ce3d5d4.jpg)
+
+![](images/7724d7713f02c0352d2494c3f2567a8d3a8ad33ac41982848473cd01dbbb93c6.jpg)
+
+![](images/d4c8b2981e30737253825cfcd900e6752afe6eda09bf38c97083a4dffbf71e7a.jpg)
+
+![](images/144d37ae8abaf46f0be4d3590714514ffcd336f0470a8551309d73ec46b4365c.jpg)
+
+![](images/1573119f00778d5b7e880c641ee1490ced2465acdb51fd4a19c8aca71509ae15.jpg)
+
+![](images/c03cd1a8fb616c03e8e4d215a111a7819d40b262fc9abf340f9a8adaaba5ea72.jpg)
+
+![](images/2c12cbd10a748d626a00acaf969a9c43db712e85b8a8080eb1ff7293bf6870be.jpg)
+
+![](images/15d465d69300f3617b0e75aaa56d45fc962fb908dab759b88962b575776a3b32.jpg)
+
+![](images/fd8385b1e6541c4db374192068ca18335de6e7f1cd07429fdb5fbb6ffd7042a1.jpg)  
+(17a) RanPAC  
+Figure 17: Upper triangular accuracy matrices on VTAB and StanfordCars (Inc-1, 2, 4, 5).
+
+![](images/39f21b758b692b91396d7b01ac5feb0a600e1b1bb791e844f9fa16525a9a16a3.jpg)  
+(17b) LoRanPAC
+
+![](images/9f65a006af59fb00473383295835d3526a4c62aaab5b9f8e100a9996e3a8ddb9.jpg)  
+(17c) RanPAC
+
+![](images/2bc24db510667b19a0c65b8493b33119d6effcd76a7dedb400d7e439ac9f54f5.jpg)  
+(17d) LoRanPAC
+
+![](images/ab2ca7579de555cb0fac5aaad031a863bd89d683c41610b477acbc8dd3984e47.jpg)
+
+![](images/d5dbaf459f4d4c07ddf4ef54a7c5635116d79b30c1aab58c91b1c68111d7bc81.jpg)
+
+![](images/c08a74d127bc4627d345d5dac5eacf69ab02b9eaf63e82043906138377a8e591.jpg)
+
+![](images/7fc8c574392419763ba535c23f4caceea59d5493a4697603deaa1881a0fd1cf6.jpg)
+
+![](images/91f2d8a9b77a11ba0496a7a6d5f63767bebb871ba707daf9f4baabff8eef2a84.jpg)
+
+![](images/1226692ac89d579943b26e32b66aa494aca53496fae52a74614046a505b47a33.jpg)
+
+![](images/cd903605f798287d3262910870a6b7c009fe74c2e9e822eaf751e3f65f49e85a.jpg)
+
+![](images/239af9f842aa45de5ac8e2bab85d4a15772575c9ae6f46bafce9167ec7ce6e6e.jpg)
+
+![](images/8fc1dc0bc231ff663390223d713e432500cdc222e8c3cacdc25e17baf521dca1.jpg)
+
+![](images/938881b3c663f8812e501f4171a3cfe18857ecc47bed816bdf947c49685c5995.jpg)
+
+![](images/ddd0720d586b5496ea93dd7f049d1939b4bff2f4425837f95d9aba2575f01353.jpg)
+
+![](images/146a034422f9b8c4405dceca4cc42eaed175cdbdffaa1b6f16fd7fcbf3340b46.jpg)
+
+![](images/d071818772f786bfc9e0744285edff333060c3c948ca233bfbc8fb3babaef743.jpg)  
+Figure 18: Our LoRanPAC method is stable with respect to class increments of each task (Inc-1, 2, 4, 5), while the accuracy of RanPAC drops for smaller increments. The first two rows plot the total accuracy. The last two rows plot the final accuracy. The figures here essentially plot the averages of the upper triangular accuracy matrices (total accuracy) or its last column (final accuracy) of Figs. 14 to 17. The numerical values of the total and final accuracy for Inc-1 are shown in Table 2.
+
+![](images/8d6234cc96f6ad3283686c9e2c0ab6968232c8c9cb53d67d2849f31c3acc27e8.jpg)
+
+![](images/6e0c457159d8485ee8eab026b528083e9ba39f9e6a47696659daf7d54cbed0f1.jpg)
+
+![](images/1798306d21f4b65627678bab22f6f20b08e00219922a2fe34e5b713b86a951bf.jpg)
\ No newline at end of file
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