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2025/Direct Density Ratio Optimization_ A Statistically Consistent Approach to Aligning Large Language Models/09234321-63a4-438d-bff5-2b190e46e754_origin.pdf

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2025/Direct Density Ratio Optimization_ A Statistically Consistent Approach to Aligning Large Language Models/full.md

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+# Direct Density Ratio Optimization: A Statistically Consistent Approach to Aligning Large Language Models
+
+Rei Higuchi  $^{12}$  Taiji Suzuki  $^{12}$
+
+# Abstract
+
+Aligning large language models (LLMs) with human preferences is crucial for safe deployment, yet existing methods assume specific preference models like Bradley-Terry model. This assumption leads to statistical inconsistency, where more data doesn't guarantee convergence to true human preferences. To address this critical gap, we introduce a novel alignment method Direct Density Ratio Optimization (DDRO). DDRO directly estimates the density ratio between preferred and unpreferred output distributions, circumventing the need for explicit human preference modeling. We theoretically prove that DDRO is statistically consistent, ensuring convergence to the true preferred distribution as the data size grows, regardless of the underlying preference structure. Experiments demonstrate that DDRO achieves superior performance compared to existing methods on many major benchmarks. DDRO unlocks the potential for truly data-driven alignment, paving the way for more reliable and human-aligned LLMs.
+
+# 1. Introduction
+
+Large language models (LLMs) have achieved remarkable performance across various natural language processing tasks, including question answering, text generation, and translation (Achiam et al., 2023; Chowdhery et al., 2023; Touvron et al., 2023). However, they can also generate outputs containing harmful information, biased opinions, and misinformation, posing potential risks to society (Bender et al., 2021; Weidinger et al., 2021; Bai et al., 2022). To develop safe and reliable LLMs, it is essential to align their behavior with human intentions, values, and ethics (Christiano et al., 2017; Gabriel, 2020; Glaese et al., 2022).
+
+<sup>1</sup>University of Tokyo, Tokyo, Japan <sup>2</sup>Center for Advanced Intelligence Project, RIKEN, Tokyo, Japan. Correspondence to: Rei Higuchi <higuchi-rei714@g.ecc.u-tokyo.ac.jp>.
+
+Proceedings of the  $42^{nd}$  International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
+
+Recently, alignment methods for LLMs, such as Reinforcement Learning from Human Feedback (RLHF) (Ouyang et al., 2022; Christiano et al., 2017; Stiannon et al., 2020) and Direct Preference Optimization (DPO) (Rafailov et al., 2024), have yielded significant progress. RLHF learns a reward model based on human feedback and optimizes the LLM's policy via reinforcement learning to maximize this reward. DPO directly optimizes the LLM's policy from human preference data without explicitly learning a reward model, and it has been reported to be more computationally efficient and stable than RLHF.
+
+However, these existing methods rely on the assumption that human preferences follow a specific model (e.g., the Bradley-Terry model (Bradley & Terry, 1952)). This assumption may not accurately capture the complexity of real human preferences (Wu et al., 2022), leading to a fundamental issue of a lack of statistical consistency. Statistical consistency refers to the property that, as the amount of data increases, the learned model converges to the true optimal model (Vapnik, 1998; Mohri et al., 2012). Without statistical consistency, increasing the amount of data does not guarantee performance improvement, raising concerns about the model's reliability and safety. More concretely, even if one perfectly optimizes the loss function based on the Bradley-Terry model, it does not necessarily guarantee the accurate learning of the true underlying human preferences.
+
+In this work, we aim to address the statistical consistency problem in existing methods and develop a more theoretically justified alignment method that benefits from increased data. Specifically, we propose a novel alignment method, "Direct Density Ratio Optimization (DDRO)," which directly estimates the distribution of preferable outputs (Sugiyama et al., 2012a; 2007b; Kanamori et al., 2009) without relying on human preference models. DDRO takes an approach that directly estimates the density ratio between the distributions of preferable and unpreferable outputs. This allows for direct alignment from data without depending on certain human preference models. Our contributions can be summarized as follows:
+
+1. Methodological Contribution: We propose a new preference optimization method called DDRO that does not rely on any human preference model. The idea of our
+
+proposal is to frame the preference optimization problem into a direct density ratio estimation problem between the preferred and unpreferred data distributions, and solve it by matching the true ratio and our estimator's ratio through minimizing their Bregman divergence. DDRO does not require paired data, which contrasts with such a method as DPO that requires paired data. We also discuss how our method is connected to existing work by investigating how the preferred and unpreferred data distributions interact in the loss function.
+
+2. Theoretical Contribution: We theoretically prove that DDRO is statistically consistent, ensuring convergence to the true distribution as the number of data points increases, without dependence on any preference model. This stems from the usage of the Bregman divergence for density ratio estimation because it gives an upper-bound of the  $L^2$ -distance from the true distribution of the preferred data.  
+3. Practical Contribution: We validate DDRO across multiple real-world benchmarks, showing that it achieves performance on par with or surpassing widely-adopted methods such as KTO and BCO. Even when converting paired datasets into an unpaired format, DDRO nearly matches or outperforms DPO, which is a paired method.
+
+The remainder of this paper is organized as follows. Section 2 provides background knowledge on LLMs and alignment, as well as existing alignment methods and their limitations. Section 3 describes the detailed theoretical framework of our proposed method, Direct Density Ratio Optimization (DDRO), including its simplified version, and discusses its statistical consistency. Section 4 presents the experimental setup and results to validate the effectiveness of our proposed method. Section 5 discusses the experimental results, the strengths and weaknesses of our method, and the significance and contributions of this research. Finally, Section 6 concludes this paper and outlines future directions.
+
+# 2. Related Work: Limitations of Existing Alignment Methods
+
+Large language models (LLMs) have achieved remarkable progress in natural language processing. However, they also present risks, notably the generation of harmful or inappropriate content. Aligning LLMs with human values and preferences is therefore a critical challenge (Christiano et al., 2017; Gabriel, 2020; Glaese et al., 2022). Reinforcement Learning from Human Feedback (RLHF) (Ouyang et al., 2022; Christiano et al., 2017; Stiennon et al., 2020) has emerged as a promising paradigm for this alignment, demonstrating effectiveness in enhancing helpfulness, harmlessness, and alignment with complex instructions (Zheng et al., 2024; Dai et al., 2023).
+
+RLHF consists of a two-stage process. The first stage in
+
+volves learning a reward model  $r_{\phi}(x,y)$  that predicts the quality of a response  $y$  to a given prompt  $x$ , based on human preference data. This is achieved using a dataset of human comparisons  $\mathcal{D}_{\mathrm{paired}} = \{(x_i,y_i^+,y_i^-)\}_{i=1}^N$ , where each data point  $(x_i,y_i^+,y_i^-)$  indicates that human annotators prefer response  $y_i^+$  over  $y_i^-$  for prompt  $x_i$ . The objective of reward model learning is to acquire a reward function that is consistent with human preferences.
+
+In RLHF, the learning of the reward model employs a loss function  $\mathcal{L}_{\mathrm{RM}}(\phi)$  based on the Bradley-Terry model. The Bradley-Terry model is used to model human preference probabilities. For responses  $y_{1}, y_{2}$  to a prompt  $x$ , the probability  $\operatorname*{Pr}(y_1\succ y_2|x)$  that response  $y_{1}$  is preferred over  $y_{2}$  is defined as
+
+$$
+\Pr (y _ {1} \succ y _ {2} | x) = \sigma \left(r _ {\phi} (x, y _ {1}) - r _ {\phi} (x, y _ {2})\right)
+$$
+
+where  $\sigma (z) = (1 + e^{-z})^{-1}$  is the sigmoid function.
+
+Based on this model, the loss function  $\mathcal{L}_{\mathrm{RM}}(\phi)$  is formulated to minimize the expected binary cross-entropy loss in predicting human preferences in the paired dataset  $\mathcal{D}_{\mathrm{paired}}$ :
+
+$$
+\mathcal {L} _ {\mathrm {R M}} (\phi) = - \mathbb {E} _ {\mathcal {D} _ {\text {p a i r e d}}} \left[ \log \sigma \left(r _ {\phi} \left(x, y ^ {+}\right) - r _ {\phi} \left(x, y ^ {-}\right)\right) \right]
+$$
+
+In the second stage, the learned reward model is used to optimize the LLM policy  $p_{\theta}(y \mid x)$  using reinforcement learning algorithms. This is typically done using Proximal Policy Optimization (PPO) (Schulman et al., 2017) by maximizing the following objective function:
+
+$$
+\max  _ {\theta} \mathbb {E} _ {x \sim \mathcal {D}, y \sim p _ {\theta} (\cdot | x)} [ r _ {\phi} (x, y) ] - \beta \mathrm {K L} [ p _ {\theta} (y \mid x) \| p _ {\text {r e f}} (y \mid x) ], \tag {1}
+$$
+
+Here,  $p_{\mathrm{ref}}(y \mid x)$  is the reference policy (often the initial LLM policy), and  $\beta$  is a hyperparameter that controls the strength of KL divergence regularization. The KL divergence term contributes to stabilizing training and mitigating reward hacking (Ziegler et al., 2019).
+
+While RLHF has been empirically successful, it has limitations such as high computational costs due to separate reward model and policy learning, and instability inherent in reinforcement learning (Rafailov et al., 2024). To address these issues, Direct Preference Optimization (DPO) (Rafailov et al., 2024) was proposed. DPO aims to streamline the alignment process by directly optimizing the policy from preference data, bypassing explicit reward modeling. DPO leverages the theoretical connection between reward functions and optimal policies, demonstrating improved stability and computational efficiency compared to RLHF. However, DPO inherits a critical assumption from the underlying Bradley-Terry model: that human preferences can be accurately modeled by this specific parametric form. This reliance on the Bradley-Terry model raises concerns about statistical consistency. Specifically, even DPO converges to
+
+the optimal policy under the Bradley-Terry model assumption as paired comparison data increases, convergence to the true optimal policy reflecting actual human preferences is not guaranteed if human preferences deviate from this model.
+
+Proposition 2.1. There exists a class of preferences, none of which can be obtained by minimizing the negative log-likelihood loss under the Bradley-Terry model assumption.
+
+See Appendix A for the proof. Proposition 2.1 suggests that the performance of DPO may be fundamentally limited by the validity of the Bradley-Terry model assumption. Furthermore, both RLHF and DPO rely on paired comparison data. Collecting such data at scale can be expensive and time-consuming, potentially hindering the scalability of these alignment approaches.
+
+To mitigate the reliance on paired comparison data and potentially improve scalability, Kahneman-Tversky Optimization (KTO) (Ethayarajh et al., 2024) introduced the use of unpaired preference data. Unpaired data, where individual responses are labeled with scores or binary feedback (e.g., "good" or "bad"), is often more readily available and scalable as it can be collected from real-world user interactions. KTO aligns LLMs using such unpaired data by leveraging prospect theory (Kahneman & Tversky, 2013; Tversky & Kahneman, 1992), which captures the asymmetry in human value functions, particularly the higher sensitivity to losses compared to gains. While KTO offers a practical approach by utilizing more accessible unpaired data, it also assumes a preference model based on prospect theory. Similar to DPO's reliance on the Bradley-Terry model, the performance and statistical consistency of KTO are contingent on the validity of prospect theory assumption in capturing human preferences.
+
+In contrast to DPO and KTO, which depend on specific parametric models for human preferences, our proposed approach aims to achieve statistical consistency without assuming a particular preference model. While we also utilize unpaired preference data as in KTO, the key novelty of our approach lies in its ability to learn a policy that is guaranteed to converge to the distribution of preferred outputs as the amount of unpaired data increases, regardless of whether human preferences strictly adhere to Bradley-Terry, prospect theory, or other predefined models. Such statistical consistency is crucial for ensuring the reliability and robustness of LLM alignment. Especially in real-world scenarios where human preferences are complex and diverse, and may be poorly modeled by simplistic assumptions, our approach offers a significant advantage.
+
+# 3. Direct Density Ratio Optimization (DDRO)
+
+To avoid relying on a specific preference model such as Bradley-Terry model, we propose a new method called Direct Density Ratio Optimization (DDRO) that estimates the aligned model through density ratio estimation. The basic idea of existing preference optimization methods (DPO, PPO, KTO) is to estimate the density ratio between the aligned model and the given reference model. Indeed, the optimal solution for PPO (1) can be given as  $p^* \propto \exp(-r_\phi / \beta) p_{\mathrm{ref}}$ , that is, PPO obtains the solution  $p^*$  via the density ratio  $p^* / p_{\mathrm{ref}}$  which is proportional to  $\exp(-r_\phi / \beta)$ . The same argument applies to DPO and KTO. Based on this observation, our idea is to directly estimate the density ratio without going through any intermediate step such as estimating an artificial preference model (like Bradley-Terry model) or constructing a reward model. For that purpose, we utilized a technique developed in the field of direct density ratio estimation (Sugiyama et al., 2012b). To introduce our method, we begin by describing the problem setting. We then derive the DDRO objective through Bregman divergence, explain its advantage from a theoretical viewpoint, and conclude by providing a practical, simplified variant of the approach.
+
+# 3.1. Problem Setting
+
+The alignment of large language models can be formulated as an optimization problem aimed at bringing the model's output distribution closer to a desirable human preference distribution. Specifically, given a prompt  $x$  from a prompt set  $\mathcal{X}$ , our goal is to align the conditional distribution of responses  $y$  denoted as  $p_{\theta}(y \mid x)$  with the unknown true preferred response distribution  $p^{+}(y \mid x)$ . This alignment is achieved through density ratio estimation in our method. We also denote  $p_{\mathrm{x}}(x)$  as the probability distribution of a prompt  $x \in \mathcal{X}$ . Here, we consider responses  $y$  to be elements of a response set  $\mathcal{Y}$ .
+
+Assumption 3.1 (Assumption on Response Set). We assume that the prompt set  $\mathcal{X}$  and response set  $\mathcal{Y}$  are finite.
+
+This is a practically motivated assumption. For both prompts and responses, large language models operate on sequences of tokens from a finite vocabulary  $\mathcal{V}$ . Given the context window length  $C$  and generation length limit  $T$ , the size of possible sequences is bounded by  $|\mathcal{V}|^C$  for  $\mathcal{X}$  and  $|\mathcal{V}|^T$  for  $\mathcal{Y}$ , ensuring both sets are finite. Technically, this assumption is particularly required to ensure that we can calculate the density function (or more precisely, the probability mass function). As long as we can obtain a density or mass function, then we may remove this condition.
+
+Our goal is to align this model distribution with a true preferred distribution  $p^{+}(y \mid x)$ , which represents the responses that humans find desirable. We also denote
+
+$p^{-}(y\mid x)$  as the corresponding true unpreferred response distribution. Although  $p^{+}(y\mid x)$  is unknown, we assume access to unpaired preference data  $\mathcal{D}_{+} = (x_{i}^{+},y_{i}^{+})_{i = 1}^{n^{+}}$  and  $\mathcal{D}_{-} = (x_{i}^{-},y_{i}^{-})_{i = 1}^{n^{-}}$  sampled from  $p^+$  and  $p^{-}$  respectively, where  $n^{+} = |\mathcal{D}_{+}|$  and  $n^{-} = |\mathcal{D}_{-}|$ . We remark that our method requires only unpaired data and does not require a human preference modeling that describes the relation between pairs. This property is more like KTO rather than DPO which requires paired data.
+
+Here we consider a situation where the data are generated through the reference model  $p_{\mathrm{ref}}$  and human taggers give labels of "preferred" and "unpreferred." Then, we have the following relation between  $p_{\mathrm{ref}}$ ,  $p^{+}$  and  $p^{-}$ :
+
+$$
+p _ {\text {r e f}} (y \mid x) = p ^ {+} (y \mid x) p (+ \mid x) + p ^ {-} (y \mid x) p (- \mid x), \tag {2}
+$$
+
+where  $p(+|x)$  and  $p(-|x)$  are the probability of preferred and unpreferred labels, respectively. As we have described above, the preference optimization can be reduced to a problem of estimating the density ratio  $r^{*}(y|x) = p_{\mathrm{ref}}(y|x) / p^{+}(y|x)$ . Indeed, if we know  $r^{*}(y|x)$ , then we can recover  $p^{+}(y|x)$  as  $p^{+}(y|x) = p_{\mathrm{ref}}(y|x) / r^{*}(y|x)$ . From Eq. (2), the density ratio can be rewritten as  $r^{*}(y|x) = p(+|x) + p(-|x)\frac{p^{-}(y|x)}{p^{+}(y|x)}$ , which means that estimating  $r^{*}$  is reduced to estimating another density ratio  $\frac{p^{-}(y|x)}{p^{+}(y|x)}$ . Hence, DDRO aims to estimate this density ratio  $g^{*}(y|x) = \frac{p^{-}(y|x)}{p^{+}(y|x)}$  to obtain  $r^{*}$  using  $\mathcal{D}_{+}$  and  $\mathcal{D}_{-}$ . Here, we place the assumption that  $p(+|x)$  is a constant, say  $t\in (0,1)$ , for all input  $x$ .
+
+Assumption 3.2 (Assumption on Reference Distribution). The reference distribution  $p_{\mathrm{ref}}$  can be expressed as a convex combination of  $p^{+}$  and  $p^{-}$ :
+
+$$
+p _ {\text {r e f}} = t p ^ {+} + (1 - t) p ^ {-}, \quad t \in (0, 1).
+$$
+
+In other words, the probability of preferred data is constant:  $p(+|x) = t (\forall x \in \mathcal{X})$ .
+
+Under this condition, we can see  $g^{*}$  and  $r^{*}$  satisfy the following relationship:
+
+$$
+g ^ {*} = \frac {1}{1 - t} r ^ {*} - \frac {t}{1 - t}. \tag {3}
+$$
+
+Therefore, the true density ratio  $r^*$  can be estimated by estimating  $g^*$  from data. In the next section, we derive a Bregman divergence loss to estimate  $g^*$ , which is the key component of our proposal.
+
+# 3.2. DDRO: The Proposed Method
+
+Following the argument described above, DDRO directly matches the ratio between  $p_{\theta}$  and  $p_{\mathrm{ref}}$  to the true ratio  $g^{*}$  by minimizing the Bregman divergence loss: Let  $f:\mathbb{R}\to \mathbb{R}$
+
+be a differentiable strictly convex function with  $f(1) = 0$ , then we define the Bregman divergence loss for the density ratio estimator as
+
+$$
+\mathcal {L} _ {\mathrm {B r e g}} (\theta) = \mathbb {E} _ {x \sim p _ {\mathrm {x}}, y | x \sim p ^ {+}} \left[ \mathrm {B r e g} _ {f} \left(g _ {\theta} \parallel g ^ {*}\right) \right],
+$$
+
+where
+
+$$
+\operatorname {B r e g} _ {f} (g \| \tilde {g}) := f (\tilde {g}) - f (g) - f ^ {\prime} (g) (\tilde {g} - g),
+$$
+
+$$
+g _ {\theta} := \frac {1}{1 - t} \frac {p _ {\mathrm {r e f}}}{p _ {\theta}} - \frac {t}{1 - t}.
+$$
+
+From the convexity of  $f$ ,  $\mathcal{L}_{\mathrm{Breg}}(\theta)$  is always non-negative and equals to 0 if and only if  $g_{\theta} = g^{*}$ ,  $(p^{+} \times p_{\mathrm{x}})$ -almost surely. Moreover, from Eq. (3), we can also notice that  $p_{\theta} = p^{+}$  if and only if  $g_{\theta} = g^{*}$ . This observation ensures that we may estimate the preferred data distribution  $g^{+}$  by minimizing the Bregman divergence loss  $\mathcal{L}_{\mathrm{Breg}}(\theta)$ . Now, we also define  $\tilde{p}_{\theta} = \frac{1}{1 - t} p_{\mathrm{ref}} - \frac{t}{1 - t} p_{\theta}$ , which corresponds to an estimator of the unpreferred data distribution  $p^{-}$ . Then, we have the following statement.
+
+Proposition 3.3. Suppose that  $\hat{\theta}$  minimizes the Bregman divergence loss so that  $\mathcal{L}_{\mathrm{Breg}}(\hat{\theta}) = 0$ , then it holds that
+
+$$
+p _ {\hat {\theta}} (y | x) = p ^ {+} (y | x), \tilde {p} _ {\hat {\theta}} (y | x) = p ^ {-} (y | x) ((p ^ {+} \times p _ {\mathrm {x}}) \cdot a. s.).
+$$
+
+This motivates us to use the Bregman divergence loss for the preference optimization.
+
+Empirical estimate of the Bregman divergence loss. Although we know that minimizing the Bregman divergence loss leads to estimating the preferred data distribution  $p^+$ , it is not obvious how to calculate the loss from the observed data. However, expanding the definition of the Bregman divergence, we see that
+
+$$
+\begin{array}{l} \int f (g ^ {*}) - f (g _ {\theta}) - f ^ {\prime} (g _ {\theta}) (g ^ {*} - g _ {\theta}) \mathrm {d} p _ {+} (y | x) \\ = \int - f (g _ {\theta}) - f ^ {\prime} (g _ {\theta}) \left(\frac {p _ {-}}{p _ {+}} - g _ {\theta}\right) \mathrm {d} p _ {+} (y | x) + (\text {c o n s t .}) \\ = \mathbb {E} _ {p _ {+}} \left[ - f (g _ {\theta}) + f ^ {\prime} (g _ {\theta}) g _ {\theta} \right] - \mathbb {E} _ {p _ {-}} \left[ f ^ {\prime} (g _ {\theta}) \right] + (\text {c o n s t .}), \\ \end{array}
+$$
+
+where  $\mathbb{E}_{p_{\pm}}[\cdot ]$  denotes expectation with respect to  $p_{\pm}(y|x)p_{\mathrm{x}}(\mathrm{x})$ . Therefore, the Bregman divergence is represented by the expectations with respect to the preferred and unpreferred data distributions, which can be well approximated by the empirical average over the fine-tuning training datasets: The empirical risk  $\hat{\mathcal{L}}_{\mathrm{Breg}}(\theta)$  is given by
+
+$$
+\begin{array}{l} \hat {\mathcal {L}} _ {\mathrm {B r e g}} (\theta) = \frac {1}{n ^ {+}} \sum_ {z _ {i} ^ {+} \in \mathcal {D} _ {+}} \left(- f \left(g _ {\theta} \left(z _ {i} ^ {+}\right)\right) + f ^ {\prime} \left(g _ {\theta} \left(z _ {i} ^ {+}\right)\right) g _ {\theta} \left(z _ {i} ^ {+}\right)\right) \\ - \frac {1}{n ^ {-}} \sum_ {z _ {i} ^ {-} \in \mathcal {D} _ {-}} \left(f ^ {\prime} \left(g _ {\theta} \left(z _ {i} ^ {-}\right)\right)\right), \\ \end{array}
+$$
+
+where we abbreviated the notation as  $g_{\theta}(z_i) = g_{\theta}(y_i \mid x_i)$  for  $z_i = (x_i, y_i)$ . This enables us to directly estimate the density ratio with the consistency guarantee (Proposition 3.3).
+
+Since just minimizing the empirical risk  $\hat{\mathcal{L}}_{\mathrm{Breg}}(\theta)$  will lead to catastrophic forgetting, we include a Kullback-Leibler divergence (KL-divergence) regularization in the objective:
+
+$$
+\mathcal {L} _ {\mathrm {D D R O}} (\theta) := \mathcal {L} _ {\mathrm {B r e g}} (\theta) + \gamma \operatorname {K L} \left(p _ {\theta} \| p _ {\text {r e f}}\right), \tag {4}
+$$
+
+where  $\mathrm{KL}(p\parallel q)\coloneqq \mathbb{E}_q[\log (p / q)]$  . This is the population version of our DDRO objective.
+
+Analogously, its empirical version is also defined as
+
+$$
+\begin{array}{l} \hat {\mathcal {L}} _ {\mathrm {D D R O}} (\theta) = \\ \frac {1}{n ^ {+}} \sum_ {z _ {i} ^ {+} \in \mathcal {D} _ {+}} \left(- f \left(g _ {\theta} \left(z _ {i} ^ {+}\right)\right) + f ^ {\prime} \left(g _ {\theta} \left(z _ {i} ^ {+}\right)\right) g _ {\theta} \left(z _ {i} ^ {+}\right) \right. \\ + \gamma t g _ {\theta} \left(z _ {i} ^ {+}\right) ^ {- 1} \log g _ {\theta} \left(z _ {i} ^ {+}\right)) \\ - \frac {1}{n ^ {-}} \sum_ {z _ {i} ^ {-} \in \mathcal {D} _ {-}} \left(f ^ {\prime} \left(g _ {\theta} \left(z _ {i} ^ {-}\right)\right) \right. \\ \left. + \gamma (1 - t) g _ {\theta} ^ {- 1} \left(z _ {i} ^ {-}\right) \log g _ {\theta} \left(z _ {i} ^ {-}\right)\right), \tag {5} \\ \end{array}
+$$
+
+where the KL-divergence is also replaced by its empirical estimate based on Assumption 3.2 to represent  $p_{\mathrm{ref}}$ . We can see that various types of preference optimization methods can be derived from our DDRO formulation (5) by examining various convex functions  $f$  for the Bregman divergence (see Appendix F for an empirical investigation of this choice). This offers modeling flexibility depending on the data property.
+
+Example 3.4. Here, we give one particular example of the Bregman divergence. Suppose that we employ  $f(x) = x\log x - (1 + x)\log (1 + x)$ . Then the Bregman loss is reduced to the logistic loss:
+
+$$
+\mathcal {L} _ {\mathrm {B r e g}} (\theta) = \mathbb {E} _ {p ^ {+}} \left[ \log (1 + g _ {\theta}) \right] + \mathbb {E} _ {p ^ {-}} \left[ \log (1 + g _ {\theta} ^ {- 1}) \right].
+$$
+
+Indeed, if we let a "classifier"  $f_{\theta}$  be  $f_{\theta} = \log (1 / g_{\theta})$ , then the loss becomes  $\mathcal{L}_{\mathrm{Breg}}(\theta) = \mathbb{E}_{p^{+}}[\log (1 + \exp (-f_{\theta})] + \mathbb{E}_{p^{-}}[\log (1 + \exp (f_{\theta}))]$ , which coincides with the loss of the logistic regression. We employ this logistic loss for our implementation. See Sugiyama et al. (2012a) for more choices of density ratio estimators.
+
+Related work on density ratio estimation Direct density ratio estimation has been extensively studied in the literature (Sugiyama et al., 2012b). The idea behind this technique is to directly model the ratio between two densities  $p(x) / q(x)$  is more efficient and robust, especially in high-dimensional spaces. Density ratio estimation has been applied to various machine learning tasks, including covariate shift adaptation (Shimodaira, 2000; Sugiyama et al., 2007a), mutual
+
+information approximation (Suzuki et al., 2008; 2009), and causal inference (Yamada & Sugiyama, 2010).
+
+The Bregman divergence framework that we employed for DDRO was originally established in (Sugiyama et al., 2012b) and is one of the most general frameworks for density ratio estimation. This framework encompasses various density ratio estimation methods depending on the choice of Bregman divergence (i.e., the convex function  $f$ ) and how the density ratio model is parameterized. Notable examples include Least Squares Importance Fitting (LSIF) (Kanamori et al., 2009), Kernel Mean Matching (KMM) (Gretton et al., 2009), and Kullback-Leibler Importance Estimation Procedure (KLIEP) (Sugiyama et al., 2007b). Basically, we may employ any density ratio estimator for the preference optimization. However, we employ the logistic loss for our implementation due to its simplicity.
+
+# 3.3. Practical modification of DDRO
+
+While the DDRO is effective in terms of consistency as shown in Theorem 3.3 and Theorem 4.1 below, we found that its training becomes unstable in preliminary experiments due to the appearance of gradient spikes during training (see Appendix E). To mitigate this instability, we propose a practical variant of the DDRO loss that introduces a monotonically increasing function  $S$  in the loss of each data point:
+
+$$
+\begin{array}{l} \hat {\mathcal {L}} _ {\mathrm {D D R O}} (\theta) = \frac {1}{n ^ {+}} \sum_ {z _ {i} ^ {+} \in \mathcal {D} _ {+}} \left[ S \left(\log \left(1 + g _ {\theta} \left(z _ {i} ^ {+}\right)\right)\right) \right. \\ \left. + \gamma t g _ {\theta} \left(z _ {i} ^ {+}\right) ^ {- 1} \log g _ {\theta} \left(z _ {i} ^ {+}\right) \right] \\ + \frac {1}{n ^ {-}} \sum_ {z _ {i} ^ {-} \in \mathcal {D} _ {-}} \left[ S \left(\log \left(1 + g _ {\theta} ^ {- 1} \left(z _ {i} ^ {+}\right)\right)\right) \right. \\ \left. \left. + \gamma (1 - t) g _ {\theta} ^ {- 1} \left(z _ {i} ^ {-}\right) \log g _ {\theta} \left(z _ {i} ^ {-}\right) \right]. \right. \tag {6} \\ \end{array}
+$$
+
+We employed  $S(x) = \log (\sigma (x))$  for the sigmoid function  $\sigma$  in our experiments, which showed good empirical results. We note that we may choose other possibilities for  $S$ . We provide a comparison of different  $S$  in Appendix E. The rationale behind the choice of log-sigmoid function is as follows: Applying  $S$  clips large values of  $g_{\theta}$  and its gradient, which prevents gradient spikes and also works as regularization, so that  $p_{\theta}$  does not go far away from the original reference model  $p_{\mathrm{ref}}$ . Indeed, we observed that this modification can mitigate gradient spikes (see Appendix E).
+
+# 4. Theoretical Analysis
+
+In this section, we give some theoretical analysis of our proposed method. First, we give the error bound of the estimated distribution via the density ratio estimator, which gives justification to use the Bregman divergence loss. Next, we investigate a connection to existing approaches by study
+
+Table 1. Unified perspective on preference optimization objectives. Existing methods apply (partially) increasing and (partially) convex functions such as sigmoid  $\sigma (\cdot)$ , quadratic,  $-\log \sigma ((\mathrm{const.}) - \cdot)$ . For simplicity, we omit the regularization terms.  
+
+<table><tr><td>Method</td><td>Reference</td><td>Loss Function</td></tr><tr><td>DPO</td><td>Rafailov et al. 2024</td><td>- log σ(a - b)</td></tr><tr><td>IPO</td><td>Azar et al. 2024</td><td>(a - b - 1)2</td></tr><tr><td>SPPO</td><td>Wu et al. 2024</td><td>(a - 1/2)2 + (-b - 1/2)2</td></tr><tr><td>KTO</td><td>Ethayarajh et al. 2024</td><td>σ(-a) + σ(b)</td></tr><tr><td>BCO</td><td>Jung et al. 2024</td><td>- log σ(a - δ) - log σ(-b - δ)</td></tr><tr><td>DDRO</td><td>Ours</td><td>-a - b + (const.)</td></tr></table>
+
+ing how the preferred data distribution and the unpreferred data distribution interact in the loss.
+
+# 4.1. Estimation error bound
+
+First, we give an estimation error bound of our method with respect to the  $L^2$ -norm that includes the model misspecification error and a generalization error. Here, let  $\Theta$  be the set of parameters of our model and its corresponding set of distributions be  $\mathcal{H} = \{p_{\theta} \mid \theta \in \Theta\}$ . Let the Rademacher complexity of the model  $\mathcal{H}$  be denoted by  $\mathcal{R}_n(\mathcal{H})$ :
+
+$$
+\mathcal {R} _ {n} (\mathcal {H}) := \max  _ {p \in p ^ {+}, p ^ {-}} \mathbb {E} _ {\mathcal {D} \sim p ^ {n}} \left[ \mathbb {E} _ {\sigma} \left[ \sup  _ {\theta \in \Theta} \left| \frac {1}{n} \sum_ {z _ {i} \in \mathcal {D}} \sigma_ {i} p _ {\theta} (z ^ {i}) \right| \right] \right],
+$$
+
+where  $(\sigma_{i})_{i = 1}^{n}$  is an i.i.d. Rademacher sequence, that is,  $P(\sigma_{i} = 1) = P(\sigma_{i} = -1) = 1 / 2$  (see Mohri et al. (2012) for a reference). It is known that the Rademacher complexity measures the variance of the empirical risk minimizer that reflects the complexity of the model. Let  $\tilde{p} = \frac{1}{1 - t} p_{\mathrm{ref}} - \frac{t}{1 - t} p$  for  $p\in \mathcal{H}$ , and let  $h^{\pm}:\mathbb{R}\to \mathbb{R}$  be  $h^{+}(p) = -f(\tilde{p} /p) + (\tilde{p} /p)f'(\tilde{p} /p)$  and  $h^{-}(p) = -f'(\tilde{p} /p)$ . Then, we have the following theorem.
+
+Theorem 4.1. Let  $\hat{\theta} = \arg \min_{\theta \in \Theta}\mathcal{L}_{\mathrm{DDRO}}(\theta)$  be the optimal solution of the DDRO loss function with  $\gamma = 0$ . We assume that there exists some open bounded interval  $I\subset \mathbb{R}$  such that  $\tilde{p} (y|x) / p(y|x)\in I$  for all  $p\in \mathcal{H}$ ,  $x\in \mathcal{X}$ , and  $y\in \mathcal{V}$ . We also assume that  $h^+$  and  $h^{-}$  are Lipschitz continuous on the interval  $I$  with a Lipschitz constant  $\operatorname {Lip}(h)$ . Then, the estimation error of  $p_{\hat{\theta}}$  with respect to the true preferred distribution  $p^+$  satisfies:
+
+$$
+\begin{array}{l} \mathbb {E} _ {\mathcal {D} _ {\pm}} \left[ \left\| p _ {\hat {\theta}} - p ^ {+} \right\| _ {L ^ {2} (p ^ {+})} ^ {2} \right] \leq \frac {2 (1 - t) ^ {2}}{t ^ {2} m _ {+} ^ {2} \mu} \times \\ \left[ 4 \mathrm {L i p} (h) (\mathcal {R} _ {n ^ {+}} (\mathcal {H}) + \mathcal {R} _ {n ^ {-}} (\mathcal {H})) + \inf _ {\theta \in \Theta} \mathcal {L} _ {\mathrm {B r e g}} (g _ {\theta}) \right], \\ \end{array}
+$$
+
+where the expectation is taken over the realization of the supervised data  $\mathcal{D}_{\pm}$ ,  $m_{+} := \min_{x \in \mathcal{X}, y \in \mathrm{supp}(p^{+}(\cdot | x))} p^{+}(y | x) > 0$ , and  $\mu := \inf_{s \in I} f''(s) > 0$ .
+
+The proof is given in Appendix C. Since usual Rademacher complexity bounds yield  $\mathcal{R}_n(\mathcal{H}) = \mathcal{O}(1 / \sqrt{n})$  (Mohri
+
+et al., 2012), the right hand side can be evaluated as  $\mathcal{O}\left(\frac{1}{\sqrt{\min\{n^{+},n^{-}\}} + \inf_{\theta \in \Theta} \mathcal{L}_{\mathrm{Breg}}(g_{\theta})}\right)$ . The term  $\inf_{\theta \in \Theta} \mathcal{L}_{\mathrm{Breg}}(g_{\theta})$  represents how far the model is misspecified from the true distribution. If the true preferred data distribution is included in the model, this term vanishes. Hence, we see that the bound represents the bias and variance trade-off to estimate an appropriately aligned model. As the data size increases, DDRO becomes more accurate even though we are not putting any assumption on the human preference model.
+
+# 4.2. Connection to existing methods
+
+To elucidate the connection between our DDRO framework and existing methods, we first consider a simplified version of our loss by setting  $t = 1/2$  and  $\gamma = 0$ . Under these conditions, the DDRO loss (4) is reduced to
+
+$$
+\mathcal {L} _ {\mathrm {D D R O}} (\theta) = \mathbb {E} _ {p ^ {+}} \left[ \log 2 - \log \frac {p _ {\theta}}{p _ {\mathrm {r e f}}} \right] + \mathbb {E} _ {p ^ {-}} \left[ \log 2 - \log \frac {\tilde {p} _ {\theta}}{p _ {\mathrm {r e f}}} \right].
+$$
+
+To establish a unified comparison, we follow the single-pair setup from Wu et al. (2024) where only one preference pair  $(x,y^{+},y^{-})$  is available. Define the log-density-ratio terms on the data points  $y^{+}$  and  $y^{-}$  as
+
+$$
+a = \log \frac {p _ {\theta} (y ^ {+} \mid x)}{p _ {\mathrm {r e f}} (y ^ {+} \mid x)}, \quad b = \log \frac {p _ {\theta} (y ^ {-} \mid x)}{p _ {\mathrm {r e f}} (y ^ {-} \mid x)}.
+$$
+
+Then, the existing approaches can be summarized in Table 1. We see that existing methods can be expressed as  $S(-a + b)$  or  $S(-a) + S(b)$  using specific (partially) increasing and (partially) convex functions  $S(\cdot)$ .
+
+On the other hand, the simplified DDRO can be expressed as  $-a - \tilde{b} + (\mathrm{const.})$ , where  $\tilde{b}$  is defined by
+
+$$
+\tilde {b} = \log \frac {\tilde {p} _ {\theta} (y ^ {-} \mid x)}{p _ {\mathrm {r e f}} (y ^ {-} \mid x)}.
+$$
+
+The biggest difference from existing methods is that, while existing ones try to reduce fitting of  $p_{\theta}$  to unpreferred data by decreasing  $b$ , DDRO tries to increase fitting of  $\tilde{p}_{\theta}$  to the unpreferred data by increasing  $\tilde{b}$ . Without any assumption
+
+![](images/e93180b8cf9e10952ee61e987643792f0ac9600d1130c276621c36013e73274e.jpg)
+
+![](images/ad5bd215a006bc8ad0a573781607bbabcdb0369ba82aa54e3df08d09c0931142.jpg)
+
+![](images/5d928f72fd3620a71a363f9603d72c489794c192c670271aed1a24faf3a78c50.jpg)
+
+![](images/29d764e21df5a18fde7d8f0b1e61f0c9f55d6054cdad6762e784d6518f4cf767.jpg)  
+Figure 1. Performance comparison of three methods (KTO, BCO, DDRO) on various benchmarks (BBH, GSM8K, MMLU, TruthfulQA, and AlpacaEval LC Winrate). The methods are applied to four different model sizes: Pythia 1.4B, Pythia 2.8B, Pythia 6.9B, and LLaMA 7B, each using the UF-G dataset.
+
+![](images/922e073c7700fc33d76af79c472846bde84b6fa11a53eb5e1504289719bfe58a.jpg)
+
+on the human preference model, decreasing the value of  $b$  does not necessarily lead to good fit on the preferred data. On the other hand, our method gives good fit of both  $p_{\theta}$  and  $\tilde{p}_{\theta}$  to preferred and unpreferred data respectively, leading to its better explanation of the whole data. This property stems from the proper specification of the loss derived from the Bregman divergence that yields statistical consistency as shown in the previous section.
+
+# 5. Experiments
+
+In this section, we evaluate our proposed method on two datasets: UF-G (unpaired) and UF-B (paired). We first demonstrate that DDRO can achieve performance comparable to or better than existing unpaired methods (KTO, BCO) on UF-G. Next, we investigate how well DDRO can approach the performance of a paired method (DPO) by converting the paired UF-B dataset into an unpaired format for DDRO training. Surprisingly, DDRO remains comparable or superior to DPO on all benchmarks, despite losing information when transforming paired data into unpaired data.
+
+# 5.1. Experimental Setup
+
+Datasets. We conduct experiments on two publicly available datasets for LLM alignment: ultrafeedback-gpt-3.5-turbo-helpfulness (UF-G) and ultrafeedback_binarized (UF-B). Both of these datasets are available through datasets library (Lhoest et al., 2021). The UF-G dataset is a filtered version of the UltraFeedback dataset (Cui et al., 2023), specifically filtered by helpfulness. After filtering, the UF-G
+
+dataset is transformed into an unpaired dataset. The UF-B dataset is a preference dataset built from the GPT-4 scored UltraFeedback dataset (64k prompts, each with 4 completions). It is created by binarizing the dataset, where the highest scoring completion is labeled as "chosen" and a random completion is labeled as "rejected." This makes UF-B a paired dataset.
+
+Models. We train the following base pre-trained LLMs: Pythia (1.4B, 2.8B, 6.9B) (Biderman et al., 2023), LLaMA 7B (Touvron et al., 2023).
+
+Benchmarks. We evaluate performance on several downstream benchmarks that assess helpfulness, correctness, or factual consistency. These benchmarks include BBH (Suzgun et al., 2022), which consists of a subset of BIG-Bench (Srivastava et al., 2022) tasks; GSM8K (Cobbe et al., 2021), which features grade-school math problems; MMLU (Hendrycks et al., 2021), a multi-task language understanding benchmark; TruthfulQA (Lin et al., 2022), which evaluates the truthfulness of responses; and AlpacaEval LC Winrate (Dubois et al., 2024), which focuses on length-controlled preference evaluation. We employ Llama3-70B-Instruct (AI@Meta, 2024) as an evaluator to determine preferences between responses.
+
+# 5.2. Results on UF-G (Unpaired Dataset)
+
+We first compare DDRO with existing unpaired methods. Figure 1 shows that DDRO achieves performance on par with or exceeding KTO and BCO across four out of five benchmarks. One exception is observed in the AlpacaEval
+
+![](images/52934ee4ddf1aaba3714083dc388248ba8b9fc0b182ff8ffb372498b4ab98579.jpg)
+
+![](images/b235dbe314e86a4713662123537e2ebe205bde054ec084755effabc4dbc69086.jpg)
+
+![](images/3fabc962337dc101e8280b8be26d03fce6e44543e77851abb12d01e794894585.jpg)
+
+![](images/b6547f51bdee1fe39e781aaa49b8e610db1cc3265272f61b9243f7f90b43764c.jpg)  
+Figure 2. Performance comparison of DDRO and DPO on various benchmarks (BBH, GSM8K, MMLU, TruthfulQA, and AlpacaEval LC Winrate). The methods are applied to four different model sizes: Pythia 1.4B, Pythia 2.8B, Pythia 6.9B, and LLaMA 7B, each using the UF-B dataset.
+
+![](images/78ad839240828eb343af7c3199f37b9aee646d5853530e4e01e97e5b261111f8.jpg)
+
+LC Winrate benchmark, where KTO slightly outperforms DDRO. These results confirm that DDRO is highly competitive in the unpaired setting.
+
+# 5.3. Results on UF-B (Paired Dataset) vs. DPO
+
+Next, we examine how effectively DDRO, an unpaired method, can approach or match the performance of a paired method (DPO). For DDRO, we convert the paired samples into unpaired form by taking each prompt-completion pair separately (and discarding explicit pairwise preference links). This artificially creates an unpaired dataset from originally paired data.
+
+Figure 2 presents an interesting result: despite converting a paired dataset into an unpaired form and thus discarding direct pairwise preference information, DDRO consistently performs comparably to or slightly better than DPO across all evaluated benchmarks (BBH, GSM8K, MMLU, TruthfulQA, AlpacaEval). This outcome is noteworthy because DDRO, by design an unpaired approach, manages to achieve performance on par with or exceeding a paired method even under conditions of inherent information loss, suggesting that the direct density ratio estimation technique effectively leverages the remaining preference signals.
+
+# 6. Conclusion
+
+In this paper, we addressed the critical issue of statistical inconsistency in the alignment of large language models (LLMs) and proposed Direct Density Ratio Optimization (DDRO), a novel framework for alignment based on direct
+
+density ratio estimation from data. This framework overcomes the limitations of existing methods that rely on specific preference model assumptions. By directly optimizing the density ratio between preferred and unpreferred distributions using Bregman divergence loss, DDRO theoretically guarantees statistical consistency (Theorem 4.1).
+
+Experimental results on multiple benchmarks (BBH, GSM8K, MMLU, TruthfulQA, and AlpacaEval LC Win-rate) confirm that DDRO compares favorably with prior methods, achieving on-par or better performance. It is noteworthy that in comparisons using paired datasets, it achieved results comparable to or better than DPO, despite DPO's ability to leverage preference information.
+
+Furthermore, a key feature of our proposed framework is the flexibility to select the convex function  $f$  used in the Bregman divergence. By changing the choice of  $f$ , we can vary the estimated divergence (Sugiyama et al., 2012b). A more comprehensive investigation of its impact on performance could potentially reveal new benefits. Indeed, (Huang et al., 2024) has shown that preference optimization using the  $\chi^2$  divergence as a regularization term instead of the conventional KL divergence offers specific advantages, such as improved robustness.
+
+Overall, our findings highlight the potential of DDRO to address fundamental challenges in LLM alignment. By discarding parametric assumptions on human preferences, DDRO enables truly data-driven alignment, leading to more reliable and human-aligned LLMs.
+
+# Acknowledgements
+
+RH was partially supported by JST CREST (JPMJCR2015). TS was partially supported by JSPS KAKENHI (24K02905) and JST CREST (JPMJCR2115). This research is supported by the National Research Foundation, Singapore, Infocomm Media Development Authority under its Trust Tech Funding Initiative, and the Ministry of Digital Development and Information under the AI Visiting Professorship Programme (award number AIVP-2024-004). Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not reflect the views of National Research Foundation, Singapore, Infocomm Media Development Authority, and the Ministry of Digital Development and Information.
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
+
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+
+# A. Proof of Proposition 2.1
+
+Proof. We aim to demonstrate the existence of a class of preferences for which no member can be obtained through optimization under the Bradley-Terry model assumption. To this end, let us consider a set of prompts  $\mathcal{X} = \{x\}$  and a set of responses  $\mathcal{Y} = \{y_a, y_b, y_c\}$ .
+
+We define a class of preferences  $\mathcal{C}_{\mathrm{pref}}$  parametrized by  $t \in [0,1/2) \cup (1/2,1]$ . Each preference  $P_t \in \mathcal{C}_{\mathrm{pref}}$  is characterized by the specific pairwise comparison probabilities:
+
+$$
+\Pr \left[ y _ {a} \succ y _ {b} | x \right] = \Pr \left[ y _ {b} \succ y _ {c} | x \right] = \Pr \left[ y _ {c} \succ y _ {a} | x \right] = t.
+$$
+
+If a preference  $P_{t}$  were obtainable, there must exist a reward  $r_{\phi}$  such that
+
+$$
+\sigma \left(r _ {\phi} (x, y _ {a}) - r _ {\phi} (x, y _ {b})\right) = \sigma \left(r _ {\phi} (x, y _ {b}) - r _ {\phi} (x, y _ {c})\right) = \sigma \left(r _ {\phi} (x, y _ {c}) - r _ {\phi} (x, y _ {a})\right) = t.
+$$
+
+Here  $\sigma$  is a sigmoid function. Due to the strict monotonicity of  $\sigma$ , it follows that
+
+$$
+r _ {\phi} (x, y _ {a}) - r _ {\phi} (x, y _ {b}) = r _ {\phi} (x, y _ {b}) - r _ {\phi} (x, y _ {c}) = r _ {\phi} (x, y _ {c}) - r _ {\phi} (x, y _ {a}).
+$$
+
+Since these differences add up to 0 by canceling each other out, we have
+
+$$
+r _ {\phi} (x, y _ {a}) - r _ {\phi} (x, y _ {b}) = r _ {\phi} (x, y _ {b}) - r _ {\phi} (x, y _ {c}) = r _ {\phi} (x, y _ {c}) - r _ {\phi} (x, y _ {a}) = 0,
+$$
+
+which implies  $r_{\phi}(x,y_a) = r_{\phi}(x,y_b) = r_{\phi}(x,y_c)$ . Then, the model generates a probability:
+
+$$
+\Pr \left[ y _ {a} \succ y _ {b} | x \right] = \sigma \left(r _ {\phi} (x, y _ {a}) - r _ {\phi} (x, y _ {b})\right) = \sigma (0) = 1 / 2.
+$$
+
+This contradicts  $t \in [0, 1/2) \cup (1/2, 1]$ , which indicates that  $P_t$  cannot be obtained by optimizing under the Bradley-Terry model assumption. This completes the proof.
+
+# B. Proof of Proposition 3.3
+
+Proof. We are given that  $\hat{\theta}$  minimizes the Bregman divergence loss,  $\mathcal{L}_{\mathrm{Breg}}(\theta)$ , such that  $\mathcal{L}_{\mathrm{Breg}}(\hat{\theta}) = 0$ . The Bregman divergence loss is defined as  $\mathcal{L}_{\mathrm{Breg}}(\theta) = \mathbb{E}_{x \sim p_x, y|x \sim p^+}[\mathrm{Breg}_f(g_\theta||g^*)]$ , where  $f$  is a strictly convex function. A fundamental property of the Bregman divergence is that  $\mathrm{Breg}_f(g_\theta||g^*) \geq 0$ , with equality holding if and only if  $g_\theta = g^*$ . Also, since the expected value of a non-negative random variable is zero if and only if the probability of the random variable taking a non-zero value is zero,  $g_{\hat{\theta}}(y|x) = g^*(y|x)$  holds  $(p^+ \times p_x)$ -almost surely. The paper establishes a direct relationship, stating that  $p_\theta = p^+$  if and only if  $g_\theta = g^*$ . Since we have shown  $g_{\hat{\theta}} = g^* (p^+ \times p_x)$ -almost surely, it follows that  $p_{\hat{\theta}}(y|x) = p^+(y|x) (p^+ \times p_x)$ -almost surely. This proves the first assertion of the proposition.
+
+Next, we examine the estimator for the unpreferred distribution,  $\tilde{p}_{\theta}$ , defined as  $\tilde{p}_{\theta} = \frac{1}{1 - t} p_{\mathrm{ref}} - \frac{t}{1 - t} p_{\theta}$ . Substituting  $\hat{\theta}$  for  $\theta$  and using the established result  $p_{\hat{\theta}} = p^{+}$ , we obtain  $\tilde{p}_{\hat{\theta}}(y|x) = \frac{1}{1 - t} p_{\mathrm{ref}}(y|x) - \frac{t}{1 - t} p^{+}(y|x)$ . Assumption 3.2 provides the decomposition of the reference distribution as  $p_{\mathrm{ref}}(y|x) = tp^{+}(y|x) + (1 - t)p^{-}(y|x)$ . Inserting this into the expression for  $\tilde{p}_{\hat{\theta}}$  yields:
+
+$$
+\tilde {p} _ {\hat {\theta}} (y | x) = \frac {1}{1 - t} [ t p ^ {+} (y | x) + (1 - t) p ^ {-} (y | x) ] - \frac {t}{1 - t} p ^ {+} (y | x) = p ^ {-} (y | x).
+$$
+
+This equality holds  $(p^{+} \times p_{x})$ -almost surely, as it relies on the condition  $p_{\hat{\theta}}(y|x) = p^{+}(y|x)$ , which itself holds  $(p^{+} \times p_{x})$ -almost surely. This demonstrates the second assertion of the proposition.
+
+Thus, we have shown that if  $\mathcal{L}_{\mathrm{Breg}}(\hat{\theta}) = 0$ , then both  $p_{\hat{\theta}}(y|x) = p^{+}(y|x)$  and  $\tilde{p}_{\hat{\theta}}(y|x) = p^{-}(y|x)$  hold  $((p^{+}\times p_{x}) - a.s.)$ .
+
+# C. Proof of Theorem 4.1
+
+Proof. First note that, for any  $\theta \in \Theta$ , we have that
+
+$$
+\begin{array}{l} \mathcal {L} _ {\mathrm {B r e g}} (\hat {\theta}) = \mathcal {L} _ {\mathrm {B r e g}} (\hat {\theta}) - \mathcal {L} _ {\mathrm {B r e g}} (\theta) + \mathcal {L} _ {\mathrm {B r e g}} (\theta) + \hat {\mathcal {L}} _ {\mathrm {B r e g}} (\hat {\theta}) - \hat {\mathcal {L}} _ {\mathrm {B r e g}} (\hat {\theta}) + \hat {\mathcal {L}} _ {\mathrm {B r e g}} (\theta) - \hat {\mathcal {L}} _ {\mathrm {B r e g}} (\theta) \\ = \mathcal {L} _ {\mathrm {B r e g}} (\theta) + \underbrace {\left(\mathcal {L} _ {\mathrm {B r e g}} (\hat {\theta}) - \hat {\mathcal {L}} _ {\mathrm {B r e g}} (\hat {\theta})\right)} _ {(i)} - \underbrace {\left(\mathcal {L} _ {\mathrm {B r e g}} (\theta) - \hat {\mathcal {L}} _ {\mathrm {B r e g}} (\theta)\right)} _ {(i i)} + \underbrace {\left(\hat {\mathcal {L}} _ {\mathrm {B r e g}} (\hat {\theta}) - \hat {\mathcal {L}} _ {\mathrm {B r e g}} (\theta)\right)} _ {(i i i)}. \tag {7} \\ \end{array}
+$$
+
+We see that  $(iii) \leq 0$  for any  $\theta \in \Theta$  because  $\hat{\theta}$  minimizes  $\hat{\mathcal{L}}_{\mathrm{Breg}}(\theta)$  in the model. Next, we bound the terms  $(i)$  and  $(ii)$ . Since both terms can be bounded as
+
+$$
+(i), (i i) \leq \sup  _ {\theta \in \Theta} \left| \hat {\mathcal {L}} _ {\mathrm {B r e g}} (\theta) - \mathcal {L} _ {\mathrm {B r e g}} (\theta) \right|,
+$$
+
+we just need to bound the right hand side. Define the functions  $h^{+}(p;(x,y))$  and  $h^{-}(p;(x,y))$  as
+
+$$
+h ^ {+} (p; (x, y)) = - f \left(\frac {\tilde {p} (y \mid x)}{p (y \mid x)}\right) + \frac {\tilde {p} (y \mid x)}{p (y \mid x)} f ^ {\prime} \left(\frac {\tilde {p} (y \mid x)}{p (y \mid x)}\right),
+$$
+
+$$
+h ^ {-} (p; (x, y)) = - f ^ {\prime} \left(\frac {\tilde {p} (y \mid x)}{p (y \mid x)}\right).
+$$
+
+Then, we have that
+
+$$
+\begin{array}{l} \mathbb {E} _ {\mathcal {D} _ {\pm}} \sup  _ {\theta \in \Theta} \left| \hat {\mathcal {L}} _ {\mathrm {B r e g}} (\theta) - \mathcal {L} _ {\mathrm {B r e g}} (\theta) \right| = \mathbb {E} _ {\mathcal {D} _ {\pm}} \left[ \sup  _ {p \in \mathcal {H}} \left| \frac {1}{n ^ {+}} \sum_ {z ^ {+} \in \mathcal {D} _ {+}} h ^ {+} (p; z ^ {+}) - \mathbb {E} _ {p ^ {+}} [ h ^ {+} (p) ] \right. \right. \\ + \left. \frac {1}{n ^ {-}} \sum_ {z ^ {-} \in \mathcal {D} _ {-}} h ^ {-} (p; z ^ {-}) - \mathbb {E} _ {p ^ {-}} \left[ h ^ {-} (p) \right] \right| \\ \leq \mathbb {E} _ {\mathcal {D} _ {+}} \left[ \sup  _ {p \in \mathcal {H}} \left| \frac {1}{n ^ {+}} \sum_ {z _ {i} ^ {+} \in \mathcal {D} _ {+}} h ^ {+} (p; z _ {i} ^ {+}) - \mathbb {E} _ {p ^ {+}} [ h ^ {+} (p) ] \right| \right] \\ + \mathbb {E} _ {\mathcal {D} _ {-}} \left[ \sup  _ {p \in \mathcal {H}} \left| \frac {1}{n ^ {-}} \sum_ {z _ {i} ^ {-} \in \mathcal {D} _ {-}} h ^ {-} (p; z _ {i} ^ {-}) - \mathbb {E} _ {p ^ {-}} \left[ h ^ {-} (p) \right] \right| \right]. \\ \end{array}
+$$
+
+Here, by the standard symmetrization trick (Theorem 3.1 of Mohri et al. (2012)) and the contraction inequality (Theorem 11.6 of Boucheron et al. (2013) or Theorem 4.12 of Ledoux & Talagrand (1991)), we have
+
+$$
+\mathbb {E} _ {\mathcal {D} _ {\pm}} \left[ \sup _ {p \in \mathcal {H}} \left| \frac {1}{n ^ {\pm}} \sum_ {z _ {i} ^ {\pm} \in \mathcal {D} _ {\pm}} h ^ {\pm} (p; z _ {i} ^ {\pm}) - \mathbb {E} _ {p ^ {\pm}} [ h ^ {\pm} (p) ] \right| \right] \leq 2 \mathcal {R} _ {n ^ {\pm}} (h ^ {\pm} \circ \mathcal {H}) \leq 2 \mathrm {L i p} (h) \mathcal {R} _ {n ^ {\pm}} (\mathcal {H}).
+$$
+
+Finally, the left hand side of Eq. (7) can be lower bounded by
+
+$$
+\frac {\| p _ {\hat {\theta}} - p ^ {+} \| _ {L ^ {2} (p ^ {+})} ^ {2}}{\frac {2 (1 - t) ^ {2}}{t ^ {2} m _ {+} ^ {2} \mu}} \leq \mathcal {L} _ {\mathrm {B r e g}} (\hat {\theta}),
+$$
+
+from Lemma C.1. Finally, by taking infimum over  $\theta \in \Theta$  in the right hand side of Eq. (7), we obtain the assertion.
+
+![](images/ff488a08927185792613b084c2e07f96121e5fbb444f4d6ab24dbb5a2592e382.jpg)
+
+# C.1. Auxiliary lemma
+
+Lemma C.1. Suppose that the same assumption as Theorem 4.1 holds. For any  $p \in \mathcal{H}$ , there exists  $\mu > 0$  such that
+
+$$
+\| p - p ^ {+} \| _ {L ^ {2} \left(p ^ {+}\right)} ^ {2} \leq \frac {2 (1 - t) ^ {2}}{t ^ {2} m _ {+} ^ {2} \mu} \mathcal {L} _ {\mathrm {B r e g}} (g)
+$$
+
+where  $m_{+} \coloneqq \min_{x \in \mathcal{X}, y \in \operatorname{supp}(p^{+}(\cdot | x))} p^{+}(y \mid x) > 0$ .
+
+Proof. Lemma 4 in Kato & Teshima 2021 states that for any  $p \in \mathcal{H}$ , there exists  $\mu > 0$  such that
+
+$$
+\left\| g - g ^ {*} \right\| _ {L ^ {2} \left(p ^ {+}\right)} ^ {2} \leq \frac {2}{\mu} \mathcal {L} _ {\mathrm {B r e g}} (g) \tag {8}
+$$
+
+We now demonstrate that when  $g$  and  $g^*$  are close,  $p$  and  $p^+$  are also close. Let  $x \in \mathcal{X}$  be an arbitrary prompt. For notational simplicity, we omit the conditioning on  $x$ .
+
+$$
+\begin{array}{l} \left\| g - g ^ {*} \right\| _ {L ^ {2} (p ^ {+})} ^ {2} = \left\| \frac {\tilde {p} _ {\theta}}{p} - \frac {p ^ {-}}{p ^ {+}} \right\| _ {L ^ {2} (p ^ {+})} ^ {2} \\ = \left\| \frac {p _ {\mathrm {r e f}}}{(1 - t) p p ^ {+}} (p - p ^ {+}) \right\| _ {L ^ {2} (p ^ {+})} ^ {2} \\ \end{array}
+$$
+
+Due to the finiteness of  $\mathcal{V}, p^+$  has a positive minimum value on its support:
+
+$$
+m _ {+} := \min  _ {y \in \mathcal {Y}} p ^ {+} (y) > 0
+$$
+
+Furthermore, by Assumption 3.2, on the support of  $p^+$ , we have  $p_{\mathrm{ref}} \geq tp^+ \geq tm^+$ . Consequently:
+
+$$
+\begin{array}{l} \left\| g - g ^ {*} \right\| _ {L ^ {2} (p ^ {+})} ^ {2} \geq \left\| \frac {p _ {\text {r e f}}}{(1 - t) p p ^ {+}} (p - p ^ {+}) \right\| _ {L ^ {2} (p ^ {+})} ^ {2} \\ \geq \left\| \frac {t m _ {+}}{(1 - t) p p ^ {+}} (p - p ^ {+}) \right\| _ {L ^ {2} (p ^ {+})} ^ {2} \\ \geq \frac {t ^ {2} m _ {+} ^ {2}}{(1 - t) ^ {2}} \left\| p - p ^ {+} \right\| _ {L ^ {2} (p ^ {+})} ^ {2} \\ \end{array}
+$$
+
+By combining this with (8), we obtain that
+
+$$
+\frac {t ^ {2} m _ {+} ^ {2}}{(1 - t) ^ {2}} \left\| p - p ^ {+} \right\| _ {L ^ {2} (p ^ {+})} ^ {2} \leq \frac {2}{\mu} \mathcal {L} _ {\mathrm {B r e g}} (g).
+$$
+
+Rearranging terms, we find that
+
+$$
+\left\| p - p ^ {+} \right\| _ {L ^ {2} (p ^ {+})} ^ {2} \leq \frac {2 (1 - t) ^ {2}}{t ^ {2} m _ {+} ^ {2} \mu} \mathcal {L} _ {\mathrm {B r e g}} (g).
+$$
+
+# D. Training Details
+
+This section provides details regarding the training procedures employed for each model in our experiments.
+
+Training Setup: All models were trained using the TRL library (von Werra et al., 2020). The batch size was set to 64, and training was conducted for 1 epoch across all methods and datasets. The AdamW optimizer (Loshchilov & Hutter, 2017) was used for optimization.
+
+Learning Rates: The learning rates for each method were as follows: 5e-7 for DPO, 5e-6 for KTO, 1e-6 for BCO, and 5e-7 for DDRO. For existing methods, we utilized the learning rate values reported in their respective papers (Rafailov et al., 2024; Ethayarajh et al., 2024; Jung et al., 2024). It is important to note that the DDRO learning rate may not be optimal for every choice of the set  $S$ . Since the scale of the loss function can vary depending on the selection of  $S$ , the learning rate might need to be adjusted accordingly.
+
+Other Details: Following the approach in Ethayarajh et al. 2024, we did not include the KL regularization term in the gradient calculation for training stability.
+
+# E. Impact of the Choice of Smoothing Function  $S(x)$  on Training Stability
+
+To analyze in detail the impact of the choice of the smoothing function  $S(x)$ , which was introduced in the loss function of our proposed method DDRO (Equation (6)) to encourage stable learning, we conducted an ablation study. Theoretically,  $S(x)$  is expected to smooth the Bregman Divergence term in the loss function and stabilize the optimization process. As described in Section 3.3, while  $S(x)$  should ideally satisfy monotonicity and convexity from a theoretical perspective, practically, it is not always necessary to adhere strictly to these constraints, allowing for the exploration of various functional forms.
+
+We compared four functional forms for  $S(x)$ . As a baseline without smoothing, we used the identity function  $(S(x) = x)$ . We also tested  $S(x) = \sigma(x), \log \sigma(x), -\log \sigma(-x)$ .
+
+Figure 3 illustrates the transition of the gradient norm during training for each choice of  $S(x)$ . The figure reveals that with the identity function  $(S(x) = x)$ , gradient spikes occur during training, suggesting unstable learning dynamics. In contrast, when using the other functions, the gradient spikes are suppressed, confirming more stable training.
+
+These results demonstrate the crucial role of smoothing with  $S(x)$  for the stable learning of DDRO. The gradient spikes observed with the identity function  $(S(x) = x)$  indicate that without smoothing, the loss landscape may become complex, potentially leading to convergence to suboptimal solutions or unstable learning. Conversely, we observed stabilization of learning with any choice of  $S$  other than the identity function.
+
+![](images/5c98db440b2c24ba7e04d21471c121f116af16ce5eb4332587e42e7685669ba8.jpg)  
+Figure 3. Gradient norm during training for different smoothing functions  $S(x)$  (identity:  $S(x) = x$ , logsig:  $S(x) = \log \sigma(x)$ , neglogsigneg:  $S(x) = -\log \sigma(-x)$ , sig:  $S(x) = \sigma(x)$ ).
+
+# F. Impact of the Choice of Convex Function  $f$  in Bregman divergence on Benchmark Performance
+
+In our proposed framework (5), there remains flexibility in the choice of the convex function  $f$  used for the Bregman divergence. This section investigates how performance on various benchmarks changes when  $f$  is altered.
+
+For the experimental setup, we trained Pythia 1.4B (Biderman et al., 2023) on the UF-G dataset (see Section 5) with different convex functions  $f$ . The candidates for  $f$ , referencing (Sugiyama et al., 2012b), are as follows:
+
+$$
+\begin{array}{l} f (t) = t \log t - (1 + t) \log (1 + t), \quad t \log t - t, \quad (t - 1) ^ {2} / 2, \\ (t ^ {1 + \alpha} - t) / \alpha \quad (\alpha = 0. 1, 0. 5, 1. 0, 2. 0, 5. 0) \\ \end{array}
+$$
+
+The learning rate was individually tuned for each function  $f$  by selecting from 5e-7, 5e-8, or 5e-9 to accommodate the variation in loss scaling.
+
+The results are presented in Table 2. We note that  $f(t) = t\log t - (1 + t)\log (1 + t)$ , listed in the first row of the table, is the function used in the main experiments of this paper, and its corresponding results are identical to those shown in Figure 1 and Table 3. As indicated in Table 2, AlpacaEval was the benchmark where the choice of  $f$  led to the most remarkable differences in performance. For the other benchmarks, the performance variations were generally within a range of  $1 - 2\%$ . Notably, the function employed in the main experiments of this paper demonstrated superior performance over the other functions in AlpacaEval, which supports the validity of our choice.
+
+Table 2. Benchmark performance of Pythia 1.4B with varying convex functions  $f\left( t\right)$  for Bregman divergence. Scores are reported for BBH, GSM8K, MMLU, TruthfulQA, and AlpacaEval, alongside the learning rate (Ir) used for each function. The function  $f\left( t\right)  = t\log t - \left( {1 + t}\right) \log \left( {1 + t}\right)$  achieved the highest AlpacaEval score (bolded), which is adopted in our main experiment.  
+
+<table><tr><td>f(t)</td><td>lr</td><td>BBH</td><td>GSM8K</td><td>MMLU</td><td>TruthfulQA</td><td>AlpacaEval</td></tr><tr><td>t log t - (1 + t) log(1 + t)</td><td>5e-7</td><td>0.2593</td><td>0.0167</td><td>0.2416</td><td>0.2313</td><td>0.5836</td></tr><tr><td>t log t - t</td><td>5e-9</td><td>0.2594</td><td>0.0227</td><td>0.2426</td><td>0.2264</td><td>0.4962</td></tr><tr><td>(t-1)2/2</td><td>5e-9</td><td>0.2603</td><td>0.0235</td><td>0.2423</td><td>0.2264</td><td>0.5000</td></tr><tr><td>(α = 0.1)</td><td>5e-9</td><td>0.2591</td><td>0.0227</td><td>0.2425</td><td>0.2264</td><td>0.4977</td></tr><tr><td>(α = 0.5)</td><td>5e-8</td><td>0.2593</td><td>0.0220</td><td>0.2428</td><td>0.2264</td><td>0.4872</td></tr><tr><td>(t1+α - t)/α</td><td>5e-8</td><td>0.2599</td><td>0.0174</td><td>0.2436</td><td>0.2301</td><td>0.4742</td></tr><tr><td>(α = 2.0)</td><td>5e-9</td><td>0.2589</td><td>0.0220</td><td>0.2425</td><td>0.2264</td><td>0.4955</td></tr><tr><td>(α = 5.0)</td><td>5e-8</td><td>0.2597</td><td>0.0205</td><td>0.2426</td><td>0.2264</td><td>0.4820</td></tr></table>
+
+# G. Benchmark Performance Details
+
+This section provides the detailed numerical results for the experiments presented in Section 5. We report performance metrics on five standard benchmarks: BBH, GSM8K, MMLU, TruthfulQA, and AlpacaEval (LC Winrate). The evaluated methods include DPO, KTO, BCO, and our method DDRO and we applied them to Pythia and LLaMA model families.
+
+In Table 3 and Table 4, for each model and benchmark combination, the highest score among the compared methods is indicated in bold.
+
+Table 3. Comparison of KTO, BCO, and DDRO performance across benchmarks on training models on UF-G dataset (unpaired). DDRO is comparable to or outperforms other methods on all evaluated benchmarks except for AlpacaEval.  
+
+<table><tr><td colspan="2">Models</td><td>BBH</td><td>GSM8K</td><td>MMLU</td><td>TruthfulQA</td><td>AlpacaEval</td></tr><tr><td rowspan="3">Pythia 1.4B</td><td>+ KTO</td><td>0.2634</td><td>0.0190</td><td>0.2432</td><td>0.1958</td><td>0.7364</td></tr><tr><td>+ BCO</td><td>0.2503</td><td>0.0159</td><td>0.2422</td><td>0.2228</td><td>0.4284</td></tr><tr><td>+ DDRO</td><td>0.2593</td><td>0.0167</td><td>0.2416</td><td>0.2313</td><td>0.5836</td></tr><tr><td rowspan="3">Pythia 2.8B</td><td>+ KTO</td><td>0.2797</td><td>0.0235</td><td>0.2539</td><td>0.1946</td><td>0.7214</td></tr><tr><td>+ BCO</td><td>0.2877</td><td>0.0212</td><td>0.2504</td><td>0.2032</td><td>0.4933</td></tr><tr><td>+ DDRO</td><td>0.2854</td><td>0.0265</td><td>0.2532</td><td>0.2240</td><td>0.5740</td></tr><tr><td rowspan="3">Pythia 6.9B</td><td>+ KTO</td><td>0.2758</td><td>0.0212</td><td>0.2526</td><td>0.1885</td><td>0.7341</td></tr><tr><td>+ BCO</td><td>0.2755</td><td>0.0220</td><td>0.2590</td><td>0.2301</td><td>0.6469</td></tr><tr><td>+ DDRO</td><td>0.2794</td><td>0.0265</td><td>0.2531</td><td>0.2338</td><td>0.5986</td></tr><tr><td rowspan="3">LLaMA 7B</td><td>+ KTO</td><td>0.3213</td><td>0.0591</td><td>0.2877</td><td>0.1701</td><td>0.6636</td></tr><tr><td>+ BCO</td><td>0.3345</td><td>0.0819</td><td>0.3123</td><td>0.2166</td><td>0.5108</td></tr><tr><td>+ DDRO</td><td>0.3330</td><td>0.0993</td><td>0.3182</td><td>0.2191</td><td>0.5280</td></tr></table>
+
+Table 4. Comparison of DPO and DDRO performance across benchmarks on training models on UF-B dataset (paired). DDRO is comparable to or outperforms DPO regardless of the limitation that it cannot directly leverage the paired structure of the data.  
+
+<table><tr><td colspan="2">Models</td><td>BBH</td><td>GSM8K</td><td>MMLU</td><td>TruthfulQA</td><td>AlpacaEval</td></tr><tr><td rowspan="2">Pythia 1.4B</td><td>+ DPO</td><td>0.2648</td><td>0.0167</td><td>0.2423</td><td>0.2277</td><td>0.5785</td></tr><tr><td>+ DDRO</td><td>0.2715</td><td>0.0182</td><td>0.2396</td><td>0.2277</td><td>0.6609</td></tr><tr><td rowspan="2">Pythia 2.8B</td><td>+ DPO</td><td>0.2907</td><td>0.0235</td><td>0.2485</td><td>0.2191</td><td>0.5886</td></tr><tr><td>+ DDRO</td><td>0.2910</td><td>0.0205</td><td>0.2582</td><td>0.2326</td><td>0.6695</td></tr><tr><td rowspan="2">Pythia 6.9B</td><td>+ DPO</td><td>0.2766</td><td>0.0303</td><td>0.2582</td><td>0.2191</td><td>0.5804</td></tr><tr><td>+ DDRO</td><td>0.2892</td><td>0.0296</td><td>0.2542</td><td>0.2375</td><td>0.6405</td></tr><tr><td rowspan="2">LLaMA 7B</td><td>+ DPO</td><td>0.3281</td><td>0.1008</td><td>0.3332</td><td>0.2215</td><td>0.5564</td></tr><tr><td>+ DDRO</td><td>0.3350</td><td>0.1001</td><td>0.3220</td><td>0.2240</td><td>0.5973</td></tr></table>

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+# Directly Forecasting Belief for Reinforcement Learning with Delays
+
+Qingyuan Wu  $^{*1}$  Yuhui Wang  $^{*2}$  Simon Sinong Zhan  $^{*3}$
+
+Yixuan Wang  $^{3}$  Chung-Wei Lin  $^{4}$  Chen Lv  $^{5}$  Qi Zhu  $^{3}$  Jürgen Schmidhuber  $^{2,6}$  Chao Huang
+
+# Abstract
+
+Reinforcement learning (RL) with delays is challenging as sensory perceptions lag behind the actual events: the RL agent needs to estimate the real state of its environment based on past observations. State-of-the-art (SOTA) methods typically employ recursive, step-by-step forecasting of states. This can cause the accumulation of compounding errors. To tackle this problem, our novel belief estimation method, named Directly Forecasting Belief Transformer (DFBT), directly forecasts states from observations without incrementally estimating intermediate states step-by-step. We theoretically demonstrate that DFBT greatly reduces compounding errors of existing recursively forecasting methods, yielding stronger performance guarantees. In experiments with D4RL offline datasets, DFBT reduces compounding errors with remarkable prediction accuracy. DFBT's capability to forecast state sequences also facilitates multistep bootstrapping, thus greatly improving learning efficiency. On the MuJoCo benchmark, our DFBT-based method substantially outperforms SOTA baselines. Code is available at https://github.com/QingyuanWuNothing/DFBT.
+
+# 1. Introduction
+
+Reinforcement learning (RL) has achieved impressive success in various scenarios, including board games (Silver et al., 2016; Schrittwieser et al., 2020), video games (Mnih et al., 2013; Berner et al., 2019) and cyber-physical systems (Wei et al., 2017; Wang et al., 2023b;c; Zhan et al., 2024). The timing factor is critical to enable RL in real
+
+*Equal contribution 1University of Southampton 2GenAI, King Abdullah University of Science and Technology 3Northwestern University 4National Taiwan University 5Nanyang Technological University 6The Swiss AI Lab IDSIA/USI/SUPSI. Correspondence to: Chao Huang <chao.huang@soton.ac.uk>.
+
+Proceedings of the  $42^{nd}$  International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
+
+world applications, particularly the delays that occur in the interaction between the agent and the environment due to physical distance, computational processing, and signal transmission. Otherwise, delays fundamentally affect the system's safety (Mahmood et al., 2018), performance (Cao et al., 2020) and efficiency (Hwangbo et al., 2017). Unlike rewards-delays-arising credit assignment issues that have been well studied (Arjona-Medina et al., 2019; Wang et al., 2024), observation-delays and action-delays disrupt the Markovian property of the Markov Decision Process (MDP), posing more challenges in RL. Most of the literature (Kim et al., 2023; Wang et al., 2023a; Wu et al., 2024b) mainly focuses on observation-delays, which has been proved as a superset of action-delays (Katsikopoulos & Engelbrecht, 2003; Nath et al., 2021). Therefore, this work mainly focuses on the RL with delays  $\Delta$  in observation: at time step  $t$ , the agent can not observe the real environment state  $s_t$  but only the history state  $s_{t - \Delta}$ $\Delta$  steps ago.
+
+To enable RL in environments with delayed observations, it is essential to restore the Markovian property (Altman & Nain, 1992; Katsikopoulos & Engelbrecht, 2003). Augmentation-based methods (Bouteiller et al., 2020; Kim et al., 2023) are based on an observation that the information state  $x_{t} = \{s_{t - \Delta}, a_{t - \Delta:t - 1}\}$ , augmented from the last observable state and the sequential actions, carries the equivalent information with  $s_{t}$  (Bertsekas, 2012). Thus, the policy learning over the original state space  $S$  can be transferred to the policy learning over the augmented state space  $\mathcal{X}$ , which is memoryless and can be addressed by nominal RL. However, as delays  $\Delta$  increase, the dimensionality of the augmented state space  $\mathcal{X}$  expands significantly, leading to a drastic deterioration in learning efficiency due to the exponentially increased sample complexity (Wu et al., 2024b).
+
+To improve learning efficiency in augmentation-based methods, belief-based methods (Walsh et al., 2009; Chen et al., 2021a) propose to conduct RL in the original state space  $S$ , by estimating the instant unobservable state  $s_t$  from the information state  $x_t$ , of which the mapping is referred to as belief. The belief function is commonly forecasted recursively (Chen et al., 2021a; Karamzade et al., 2024): for  $i = 1, \dots, \Delta$ ,  $s_{t - \Delta + i}$  is estimated by applying approximate dynamic function with the previous state  $s_{t - \Delta + i - 1}$  and the
+
+previous action  $a_{t - \Delta +i - 1}$ . Obtaining the estimation of the instant state  $s_t$ , the delayed RL problem is effectively reduced to a delay-free RL problem without enlarging the state space, mitigating the curse of dimensionality. However, this recursive process is evidently affected by the error accumulation of the approximate dynamic function across  $\Delta$  steps: the compounding errors grow exponentially with the delays  $\Delta$ . This fundamental limitation of such recursive methodology for belief forecasting leads to significant performance degradation, especially in environments with long-delayed signals.
+
+This work aims to squarely address the compounding errors in existing belief-based techniques with recursive strategy, using a direct strategy in sequence modeling. Specifically, we present the Directly Forecasting Belief Transformer (DFBT), a novel directly forecasting belief method by reformulating belief forecasting as a sequence modeling problem. DFBT first represents the information state  $x_{t}$  and reward signals  $r_{t - \Delta :t - 1}$  to  $\Delta$  tokens in the form  $\{s_{t - \Delta},a_{t - \Delta +i},r_{t - \Delta +i}\}_{i = 0}^{\Delta -1}$ . Unlike recursively forecasting belief methods that invoke the forward prediction process  $\Delta$  times, DFBT simultaneously forecasts the states  $s_{t - \Delta +1:t}$  without introducing accumulated errors, effectively addressing the issue of compounding errors. We then integrate DFBT with the Soft Actor-Critic (SAC) method (Haarnoja et al., 2018) in online learning, incorporating multi-step bootstrapping on accurate state predictions generated from DFBT, which further improves learning efficiency. Theoretically, we demonstrate how the compounding errors in recursively forecasting belief affect the RL performance and how it is alleviated by our direct strategy. Empirically, using the D4RL benchmark (Fu et al., 2020), we show that DFBT achieves significantly higher prediction accuracy compared to other belief methods. On the MuJoCo benchmark (Todorov et al., 2012), across various delays settings, we demonstrate that our DFBT-SAC consistently outperforms SOTA augmentation-based and belief-based methods in both learning efficiency and overall performance.
+
+In Section 3, we introduce the delayed RL problem and the concept of belief representation. Section 4 presents our proposed DFBT, which directly forecasts states from delayed observations, avoiding the need for recursive single-step predictions. Additionally, we develop DFBT-SAC by leveraging multi-step bootstrapping on the states predicted by DFBT. Through theoretical analysis in Section 5, we show that DFBT effectively mitigates the compounding errors associated with recursively forecasting belief methods, ensuring superior performance. Finally, in Section 6, we empirically demonstrate the superior prediction accuracy of DFBT on the D4RL datasets and the remarkable efficacy of DFBT-SAC compared to state-of-the-art approaches on MuJoCo across various delays settings. Overall, our key
+
+contributions are summarized as follows:
+
+- We present Directly Forecasting Belief Transformer (DFBT), a novel directly forecasting belief method that effectively addresses compounding errors in recursively generated belief.  
+- We propose DFBT-SAC, a novel delayed RL method that further improves the learning efficiency via multi-step bootstrapping on the DFBT.  
+- We theoretically demonstrate that our directly forecasting belief significantly reduces compounding errors compared to existing recursively forecasting belief approaches, offering a more robust performance guarantee.  
+- We empirically demonstrate that our DFBT method effectively forecasts state sequences with significantly higher prediction accuracy compared to baseline approaches.  
+- We empirically show that our DFBT-SAC outperforms SOTAs in terms of sample efficiency and performance on the MuJoCo benchmark.
+
+# 2. Related Works
+
+In classical control theory, delays are modelled with delay-differential equations (DDEs) (Cooke, 1963). A rich toolbox is available for analysing their reachability (Fridman & Shaked, 2003; Xue et al., 2020), stability (Feng et al., 2019), and safety (Xue et al., 2021). These techniques, however, assume fully known dynamics and become impractical for high-dimensional systems. Delayed reinforcement learning (RL) is far closer to real deployments than the delay-free setting that dominates the literature. Latencies are intrinsic to robotics (Mahmood et al., 2018; Hwangbo et al., 2017), high-frequency trading (Hasbrouck & Saar, 2013), intelligent transportation (Cao et al., 2020), and many other domains. Depending on the methodology of retrieving the Markovian property, existing delayed RL approaches can be mainly categorized as augmentation-based and belief-based approaches.
+
+Augmentation-based Approaches. Augmentation-based approaches retrieve the Markovian property by augmenting the original state with the sequence of actions within the delays window, optimizing the policy in the resulting augmented MDP to mitigate performance degradation caused by errors in belief representation approximation. Specifically, DIDA (Liotet et al., 2022) learns the delayed policy in the augmented MDP by imitating the behaviours from the delay-free expert policy in the original MDP; DC/AC (Bouteiller et al., 2020) suggest using the delays correction operator
+
+to improve the learning efficiency in the augmented MDP; ADRL (Wu et al., 2024b) introduces the auxiliary delayed task with the smaller augmented MDP for bootstrapping the larger augmented MDP; VDPO (Wu et al., 2024a) proposes to learn the delayed policy through the lens of variational inference, improving the learning efficiency via extensive optimization tools. However, augmentation-based approaches still operate on an increasingly large augmented state space as delays grow, inevitably suffering from the curse of dimensionality and resulting in significant learning inefficiencies.
+
+Belief-based Approaches. The belief-based approach retrieves the Markovian property by forecasting the unobservable state through belief representation, then optimizing policy within the original state space. Thus, correspondingly, the prediction accuracy of approximated belief representation is highly related to the overall performance. DATS (Chen et al., 2021a) is the first to propose approximating the belief representation using a Gaussian distribution. Inspired by the world models (Schmidhuber, 1990b;a; Ha & Schmidhuber, 2018), D-Dreamer (Karamzade et al., 2024) employ a recurrent neural network to capture temporal dependencies in the belief through forward prediction in the latent space, enabling adaptation to high-dimensional observations. D-SAC (Liotet et al., 2021) utilizes the attention mechanism in the causal transformer to integrate the information across the delays. Unfortunately, the approximation errors of recursively forecasting belief accumulated at each step finally lead to compounding errors, which hinder prediction accuracy and performance.
+
+Time Series Forecasting in RL. Time series forecasting plays a critical role in predicting future states in RL, particularly within the context of world model learning (Ha & Schmidhuber, 2018), which simulates the dynamics of the environment, predicting future states and planning effectively without directly interacting with the real environment. World model learning often leverages time series forecasting techniques to capture temporal dependencies, enhancing the agent's understanding of the environment. Applications of world models include robotics, autonomous driving, and gaming scenarios (Hafner et al., 2019). Additionally, the RL problem can be treated as a sequence modeling problem, which can be effectively solved by the transformer (Vaswani et al., 2017). Directly predicting the action from the desired outcomes (Schmidhuber, 2019), decision transformer (Chen et al., 2021b) predict future actions based on historical trajectories and desired outcomes. Similarly, trajectory transformer (Janner et al., 2021) models trajectories as sequences, enabling policy optimization by leveraging autoregressive modeling. Therefore, in this paper, we investigate how to leverage the advanced time series forecasting approaches in addressing the issue of compounding errors existing in the recursively forecasting belief. We also present a com
+
+The prehensive theoretical analysis of how compounding errors seriously degenerate performance as delays are increased.
+
+# 3. Preliminaries
+
+# 3.1. From Delay-free RL to Delayed RL
+
+A conventional delay-free RL problem is usually formalized as a Markov Decision Process (MDP) represented as a tuple  $\langle S, \mathcal{A}, \mathcal{P}, \mathcal{R}, \rho, \gamma \rangle$ , where  $\mathcal{S}$  is the state space,  $\mathcal{A}$  is the action space,  $\mathcal{P}: S \times \mathcal{A} \times S \to [0,1]$  is the dynamic function,  $\mathcal{R}: S \times \mathcal{A} \to \mathbb{R}$  is the reward function,  $\rho$  is the initial state distribution, and  $\gamma \in (0,1)$  is the discount factor. The agent selects an action  $a_t \sim \pi(\cdot | s_t)$  according to the policy  $\pi: S \times \mathcal{A} \to [0,1]$  based on the current state  $s_t$  at time step  $t$ . The agent will then observe the next state  $s_{t+1} \sim \mathcal{P}(\cdot | s_t, a_t)$  and the reward  $r_t = \mathcal{R}(s_t, a_t)$  from the environment. The objective of the agent is to find the optimal policy  $\pi^*$  that maximizes the expected discounted return  $G := \mathbb{E}_{\tau \sim p_{\pi}(\tau)}[\sum_{t=0}^{\infty} \gamma^t \mathcal{R}(s_t, a_t)]$  where  $p_{\pi}(\tau)$  represents the distribution of trajectories induced by policy  $\pi$ .
+
+A delayed RL problem can be formalized as an augmented MDP by applying the augmentation technique (Altman & Nain, 1992; Katsikopoulos & Engelbrecht, 2003) to retrieve the Markovian property. For a delayed RL problem with constant delays  $\Delta$ , the newly formed MDP is represented as  $\langle \mathcal{X}, \mathcal{A}, \mathcal{P}_{\Delta}, \mathcal{R}_{\Delta}, \rho_{\Delta}, \gamma \rangle$ , where  $\mathcal{X} \coloneqq S \times \mathcal{A}^{\Delta}$  is the augmented state space,  $\mathcal{A}$  is the original action space, the delayed dynamic function is  $\mathcal{P}_{\Delta}(x_{t+1}|x_t, a_t) \coloneqq \mathcal{P}(s_{t-\Delta+1}|s_{t-\Delta}, a_{t-\Delta}) \delta_{a_t}(a_t') \prod_{i=1}^{\Delta-1} \delta_{a_{t-i}}(a_{t-i}')$  where  $\delta$  is the Dirac distribution, the delayed reward function is defined as  $\mathcal{R}_{\Delta}(x_t, a_t) \coloneqq \mathbb{E}_{s_t \sim b(|x_t|)} [\mathcal{R}(s_t, a_t)]$  where  $b: \mathcal{X} \times S \to [0,1]$  is the belief representation mapping from the augmented state space  $\mathcal{X}$  to the original state space  $S$ ,  $\rho_{\Delta} = \rho \prod_{i=1}^{\Delta} \delta_{a_{-i}}$  is the initial augmented state distribution.
+
+# 3.2. Belief Representation in Delayed RL
+
+Delayed RL can be viewed as a special form of a partially observable RL problem where the observation is contaminated by noise, instead of being delayed. Therefore, similar to partially observable RL, delayed RL also have the belief representation defined as follows:
+
+$$
+\begin{array}{l} b \left(s _ {t} \mid x _ {t}\right) := \\ \int_ {\mathcal {S} ^ {\Delta}} \prod_ {i = 0} ^ {\Delta - 1} \mathcal {P} \left(s _ {t - \Delta + i + 1} \mid s _ {t - \Delta + i}, a _ {t - \Delta + i}\right) \mathrm {d} s _ {t - \Delta + i + 1}. \tag {1} \\ \end{array}
+$$
+
+The belief representation can retrieve the Markovian property via mapping the augmented state space  $\mathcal{X}$  to  $\mathcal{S}$ , recasting the delayed RL problem in the original MDP without augmenting the state space. The belief representation can be viewed as the recursive forward prediction of the dynamics
+
+$\mathcal{P}$ . With the belief representation, the agent can directly learn in the original state space  $S$ . In this work, we use  $s$  and  $\hat{s}$  to represent the true state of the environment and the predicted state of the approximate belief, respectively.
+
+# 4. Our Approach
+
+In this section, we present the Directly Forecasting Belief Transformer (DFBT), a new directly forecasting belief method. By framing belief forecasting as a sequence modeling problem, DFBT directly predicts the unobservable states. Instead of recursively predicting the next state step-by-step, DFBT can effectively capture the dependency across the delays via the attention mechanism of the transformer model. Therefore, our DFBT can effectively reduce the compounding errors of the recursively forecasting belief, especially in the long delays. Specifically, as summarized in Figure 1, we train the DFBT using a pre-collected trajectory dataset, latter deploying the trained DFBT in the environment with delays to reconstruct the delay-free environment for training. Furthermore, we integrate multi-step bootstrapping with the predicted states of DFBT to improve learning efficiency.
+
+# 4.1. Directly Forecasting Belief
+
+The belief representation learning problem can be viewed as a sequence modeling problem. Given a pre-collected delay-free trajectory  $\{s_i, a_i, r_i\}_{i=0}^T$ , we sample the sub-trajectory with  $\Delta$  timesteps  $\{s_{t-\Delta+i}, a_{t-\Delta+i}, r_{t-\Delta+i}\}_{i=0}^{\Delta}$ . We model the past reward signals  $r_{t-\Delta:t-1}$  into the augmented state  $x_t = \{s_{t-\Delta}, a_{t-\Delta:t-1}\}$  for considering more dynamic information in belief. Then, we have reformed the representation of the augmented state to  $\Delta$  tokens for sequence modeling:  $x_t^{\mathrm{tokens}} = \{s_{t-\Delta}, a_{t-\Delta+i}, r_{t-\Delta+i}\}_{i=0}^{\Delta-1}$ . The Directly Forecasting Belief Transformer (DFBT)  $b_\theta$  leverages the transformer architecture (Vaswani et al., 2017), utilizing the attention mechanism to effectively capture dependencies across long delays. Inputting with  $x_t^{\mathrm{tokens}}$ , DFBT simultaneously predicts the unobserved  $\Delta$  states  $\{s_{t-\Delta+i}\}_{i=1}^{\Delta}$  via autoregressive modeling with loss:
+
+$$
+\nabla_ {\theta} \left[ \sum_ {i = 1} ^ {\Delta} \left[ - \log b _ {\theta} ^ {(i)} \left(s _ {t - \Delta + i} \mid x _ {t} ^ {\text {t o k e n s}}\right) \right] \right], \tag {2}
+$$
+
+where  $b_{\theta}^{(i)}(\cdot |x_t)$  represents the  $i$ -th prediction. In a deterministic environment with a deterministic belief function, we typically replace Equation (2) with the Mean-Square-Error (MSE) loss.
+
+# 4.2. Multi-step Bootstrapping with DFBT
+
+Next, we directly deploy the trained DFBT  $b_{\theta}$  in the online environment with delayed signals to reconstruct the delay-free environment where the agent can directly learn with the original state space  $S$  instead of the augmented state
+
+Algorithm 1 DFBT-SAC  
+Input: offline dataset  $\mathcal{D}$  ,DFBT  $b_{\theta}$  ,critic  $Q_{\psi}$  ,actor #training DFBT on the offline dataset   
+for Epoch  $= 1$  .do Update  $b_{\theta}$  on the  $\mathcal{D}$  via Equation (2)   
+end for   
+# learning with DFBT on the online environment   
+for Epoch  $= 1$  .do Update  $Q_{\psi}$  on via Equation (3) Update  $\pi_{\phi}$  on via Equation (4)   
+end for   
+Output:  $b_{\theta}$  and  $\pi_{\phi}$
+
+space  $\mathcal{X}$ , thus maintaining the superior sample complexity in online learning.
+
+Then, we present our practical delayed RL method, named DFBT-SAC by incorporating the trained DFBT  $b_{\theta}$  with Soft Actor-Critic (Haarnoja et al., 2018). To improve the learning efficiency with the DFBT, the critic of DFBT-SAC is bootstrapped on the states predicted by the DFBT with the multi-step learning (Sutton & Barto, 2018; Hessel et al., 2018) and delay-free training techniques (Wu et al., 2024b; Kim et al., 2023). Specifically, given multi-step data  $(x_{t}^{\mathrm{tokens}}, s_{t - \Delta + 1:t})$ , the critic  $Q_{\psi}$  parameterized by  $\psi$  is updated via:
+
+$$
+\nabla_ {\psi} \left[ \frac {1}{2} \left(Q _ {\psi} \left(s _ {t - \Delta}, a _ {t - \Delta}\right) - \mathbb {Y}\right) ^ {2} \right], \tag {3}
+$$
+
+where  $N$ -step temporal difference target  $\mathbb{Y}$  is defined as:
+
+$$
+\begin{array}{l} \mathbb {Y} := \sum_ {i = 0} ^ {N - 1} \gamma^ {i} r _ {t - \Delta + i} \\ + \gamma^{N}\underset { \begin{array}{c}a\sim \pi (\cdot |\hat{s}_{t - \Delta +N})\\ \hat{s}_{t - \Delta +N}\sim b_{\theta}^{(N)}(\cdot |\hat{s}_{t^{\rm{tokens}}}) \end{array} }{\mathbb{E}}\left[Q(s_{t - \Delta +N},a) + \log \pi (a|\hat{s}_{t - \Delta +N})\right], \\ \end{array}
+$$
+
+and  $N(\leq \Delta)$  is the bootstrapping steps on the DBFT. In this work, we set  $N = 8$  as default, and we also conduct the ablation study (Section 6) to investigate bootstrapping steps settings. The actor  $\pi_{\phi}$  parameterized by  $\phi$  is updated by:
+
+$$
+\nabla_ {\phi} \underset { \begin{array}{c} a \sim \pi (\cdot | \hat {s} _ {t - \Delta + N}) \\ \hat {s} _ {t - \Delta + N} \sim b _ {\theta} ^ {(N)} (\cdot | x _ {t} ^ {\text {t o k e n s}}) \end{array} } {\mathbb {E}} \left[ \log \pi_ {\phi} (a | \hat {s} _ {t - \Delta + N}) - Q _ {\psi} (s _ {t - \Delta + N}, a) \right]. \tag {4}
+$$
+
+The pseudo-code of DFBT-SAC is provided in Algorithm 1.
+
+# 5. Theoretical Analysis
+
+In this section, we theoretically illustrate that performance degeneration is rooted in compounding errors of recursively forecasting belief (Section 5.1), which can be effectively addressed by our DFBT, directly forecasting belief, thus achieving a better performance guarantee (Section 5.2).
+
+![](images/39bf6dc9d148bfe64e3c26d984482272854f0f845e3e100bc7079f7ae424924f.jpg)  
+Figure 1. Pipeline of DFBT-SAC. (a) Training DFBT on the trajectory dataset. (b) The agent can interact and learn with the delay-free environment recovered by the DFBT as highlighted in the dashline box. (c) Multi-step bootstrapping on the forecasted states from DFBT.
+
+Before starting the theoretical analysis, we introduce the definition of the performance degeneration of the groundtruth belief  $b$  as follows.
+
+Definition 5.1 (Performance Degeneration of Ground-truth Belief (Liotet al., 2022)). For policies  $\pi$  and  $\pi_{\Delta}$  with delays  $\Delta$ . Given any  $x_{t} \in \mathcal{X}$ , the performance difference of the ground-truth belief  $b$  is denoted as  $I_{\Delta}^{\mathrm{true}}(x_t)$
+
+$$
+I_{\Delta}^{\text{true}}(x_{t}) = \frac{1}{1 - \gamma}\underset { \begin{array}{c}\hat{s}\sim b(\cdot |\hat{x})\\ \hat{a}\sim \pi_{\Delta}(\cdot |\hat{x})\\ \hat{x}\sim d_{x_{t}}^{\pi} \end{array} }{\mathbb{E}}\left[V^{\pi}(\hat{s}) - Q^{\pi}(\hat{s},\hat{a})\right],
+$$
+
+where  $d_{x_t}^\pi$  is the augmented state distribution induced by policy  $\pi$  with the initial state  $x_t$ .
+
+Furthermore, we also introduce the Lipschitz Continuity of MDP (Definition 5.2) and value function (Definition 5.3), which are common and mild assumptions in the literature.
+
+Definition 5.2 (Lipschitz Continuity of MDP (Rachelson & Lagoudakis, 2010)). An MDP is  $L_{\mathcal{P}}$ -LC, if  $\forall (s_1, a_1), (s_2, a_2) \in S \times \mathcal{A}$ , dynamic function  $\mathcal{P}$  satisfies:
+
+$$
+\mathcal {W} \left(\mathcal {P} (\cdot | s _ {1}, a _ {1}) | | \mathcal {P} (\cdot | s _ {2}, a _ {2})\right) \leq L _ {\mathcal {P}} \left(d _ {\mathcal {S}} \left(s _ {1}, s _ {2}\right) + d _ {\mathcal {A}} \left(a _ {1}, a _ {2}\right)\right),
+$$
+
+where  $\mathcal{W}$  is the L1-Wasserstein distance,  $d_S$  and  $d_{\mathcal{A}}$  are distance measure of the  $S$  and  $\mathcal{A}$ , respectively.
+
+Definition 5.3 (Lipschitz Continuity of Value Function (Rachelson & Lagoudakis, 2010)). Consider a  $L_{Q}$ -LC Q-function  $Q^{\pi}$  of the  $L_{\pi}$ -LC policy  $\pi$ , value function  $V^{\pi}$  satisfies that
+
+$$
+\left|\underset { \begin{array}{c} s _ {1} \sim \mu \\ s _ {2} \sim v \end{array} }{\mathbb{E}}\left[ V ^ {\pi} (s _ {1}) - V ^ {\pi} (s _ {2}) \right] \right| \leq L _ {V} \mathcal {W} \left(\mu | | v\right),
+$$
+
+where  $L_{V} = L_{Q}(1 + L_{\pi})$  and  $\mu, \nu$  are two arbitrary distributions over  $\mathcal{S}$ .
+
+# 5.1. Recursively Forecasting Belief: Compounding Errors Analysis
+
+In this section, we show that the performance degeneration of recursively forecasting belief is influenced by com
+
+pounding errors, which consist of the recursive error and are exponentially increased with delays. We assume that the recursively forecasting belief  $\mathcal{P}_{\theta}$  has the approximated error bound as shown in Assumption 5.4.
+
+Assumption 5.4 (Approximated Dynamic Difference Bound). The distance between the approximated dynamic function  $\mathcal{P}_{\theta}$  parameterized by  $\theta$  and the ground-truth dynamic function  $\mathcal{P}$  is bounded, it satisfies that  $\forall (s,a)\in S\times \mathcal{A}$
+
+$$
+\mathcal {W} \left(\mathcal {P} _ {\theta} (\cdot | s, a) | | \mathcal {P} (\cdot | s, a)\right) \leq \epsilon_ {\mathcal {P}}.
+$$
+
+Then, we demonstrate that the performance difference of the recursively forecasting belief is determined by the compounding errors (Theorem 5.5).
+
+Theorem 5.5 (Performance Difference of Recursively Forecasting Belief, Proof in Theorem B.1). For the delay-free policy  $\pi$  and the delayed policy  $\pi_{\Delta}$ . Given any  $x_{t} \in \mathcal{X}$ , the performance difference  $I^{\text{recursive}}(x_{t})$  of the recursively forecasting belief  $b_{\theta}$  can be bounded as follows, respectively.
+
+For deterministic delays  $\Delta$ , we have
+
+$$
+\left| I ^ {\text {r e c u r s i v e}} (x _ {t}) \right| \leq \left| I _ {\Delta} ^ {\text {t r u e}} (x _ {t}) \right| + L _ {V} \underbrace {\frac {1 - L _ {\mathcal {P}} ^ {\Delta}}{1 - L _ {\mathcal {P}}} \epsilon_ {\mathcal {P}}} _ {\text {c o m p o u n d i n g e r r o r s}}.
+$$
+
+And for stochastic delays  $\delta \sim d_{\Delta}(\cdot)$ , we have
+
+$$
+\left| I ^ {r e c u r s i v e} (x _ {t}) \right| \leq \underset {\delta \sim d _ {\Delta} (\cdot)} {\mathbb {E}} \left[ | I _ {\delta} ^ {t r u e} (x _ {t}) | + L _ {V} \underbrace {\frac {1 - L _ {\mathcal {P}} ^ {\delta}}{1 - L _ {\mathcal {P}}} \epsilon_ {\mathcal {P}}} _ {c o m p o u n d i n g e r r o r s} \right].
+$$
+
+Theorem 5.5 tells that the compounding errors are exponentially increased with the delays, seriously degenerating the performance.
+
+# 5.2. Directly Forecasting Belief: Errors Analysis
+
+Next, we theoretically show that our directly forecasting belief can effectively address the compounding errors. The
+
+![](images/a7febd2c35c9fc567286cf514623178a69abc705f5a6fb451be09ca29267f97c.jpg)  
+(a) HalfCheetah-v2
+
+![](images/f72ac49395b26501f0d8d9ceea98bc279a26f03e4ec41477b5e9b7c7e47b8ae4.jpg)  
+(b) Hopper-v2
+
+![](images/e9c58d89e778027f38d90c373f8c218767c47c4e2a16bacf83d35c647b755c1c.jpg)  
+(c) Walker2d-v2
+
+![](images/0b248b14f2e1c3ee26579c110f3ddbda4b36e3481ad01f8c646f76f1e6788bac.jpg)  
+Figure 2. Belief errors comparison on (a) HalfCheetah-v2, (b) Hopper-v2, and (c) Walker2d-v2.
+
+sequence-modeling method estimates the states  $s_{t - \Delta +1:t}$  based on the augmented state  $x_{t}$  in parallel, instead of invoking  $\Delta$  times iteratively, which effectively alleviates the source of compounding errors. Especially, the attention mechanism in the transformer can selectively capture the long-range relationships within the augmented state with long delays. We assume that the belief error between the directly forecasting belief and the ground-truth belief is bounded (Assumption 5.6).
+
+Assumption 5.6 (Directly Forecasting Belief Difference Bound). The distance between the directly forecasting belief belief  $b_{\theta}$  parameterized by  $\theta$  and the ground-truth belief  $b$  is bounded, it satisfies that  $\forall x_{t} \in \mathcal{X}$ , we have  $\mathcal{W}(b(\cdot |x_t)||b_\theta (\cdot |x_t)) \leq \epsilon_{direct}$ , where  $\epsilon_{direct} := \max_{i = 1,\dots,\Delta} \mathcal{W}(b^{(i)}(\cdot |x_t)||b_\theta^{(i)}(\cdot |x_t))$ .
+
+From Theorem 5.5, we can derive the performance degeneration bound of directly forecasting belief as follows.
+
+Proposition 5.7 (Performance Degeneration Bound of Directly Forecasting Belief, Proof in Proposition B.2). For the delay-free policy  $\pi$  and the delayed policy  $\pi_{\Delta}$ . Given any  $x_{t} \in \mathcal{X}$ , the performance degeneration  $I^{\text{direct}}$  of the directly forecasting belief  $b_{\theta}$  can be bounded as follows respectively.
+
+For deterministic delays  $\Delta$ , we have
+
+$$
+\left| I ^ {\text {d i r e c t}} \left(x _ {t}\right) \right| \leq \left| I _ {\Delta} ^ {\text {t r u e}} \left(x _ {t}\right) \right| + L _ {V} \epsilon_ {\text {d i r e c t}}.
+$$
+
+For stochastic delays  $\delta \sim d_{\Delta}(\cdot)$ , we have
+
+$$
+\left| I ^ {\text {d i r e c t}} \left(x _ {t}\right) \right| \leq \mathbb {E} _ {\delta \sim d _ {\Delta} (\cdot)} \left[ \left| I _ {\delta} ^ {\text {t r u e}} \left(x _ {t}\right) \right| \right] + L _ {V} \epsilon_ {\text {d i r e c t}}.
+$$
+
+Then, we have the performance degeneration comparison between directly forecasting belief and recursively forecasting belief.
+
+Proposition 5.8 (Performance Degeneration Comparison, Proof in Proposition B.3). Directly forecasting belief could achieve a better performance guarantee  $\left| I^{\text{direct}}(x_t) \right| \leq \left| I^{\text{recursive}}(x_t) \right|$ , if we have
+
+$$
+\epsilon_ {d i r e c t} \leq \frac {1 - L _ {\mathcal {P}} ^ {\Delta}}{1 - L _ {\mathcal {P}}} \epsilon_ {\mathcal {P}}
+$$
+
+for deterministic delays  $\Delta$ , and
+
+$$
+\epsilon_ {d i r e c t} \leq \underset {\delta \sim d _ {\Delta} (\cdot)} {\mathbb {E}} \left[ \frac {1 - L _ {\mathcal {P}} ^ {\delta}}{1 - L _ {\mathcal {P}}} \right] \epsilon_ {\mathcal {P}}
+$$
+
+for stochastic delays  $\delta \sim d_{\Delta}(\cdot)$ .
+
+Remark 5.9 (Empirical Validation of Proposition 5.8). It is obvious that the belief errors of recursively forecasting grows much faster than the directly forecasting, which is not strictly related with the delay length. We also empirically show that Proposition 5.8 is always held in Section 6.2.1. In the future, we will theoretically investigate the sample complexity of recursive and direct forecasting beliefs.
+
+Remark 5.10 (General Error Distribution Case). In the context of time forecasting, our theoretical results of performance degeneration comparison have the potential to extend to the variance analysis commonly discussed in related literature (Taieb & Atiya, 2015; Clements & Hendry, 1996; Chevillon, 2007).
+
+# 6. Experiments
+
+# 6.1. Experimental Setting
+
+We adopt D4RL (Fu et al., 2020) and MuJoCo (Todorov et al., 2012) as the offline dataset and the benchmark respectively to evaluate our DFBT-SAC. For the baselines, we choose the SOTA augmentation-based methods (A-SAC (Haarnoja et al., 2018), BPQL (Kim et al., 2023), and ADRL (Wu et al., 2024b)) and belief-based methods (DATS (Chen et al., 2021a), D-Dreamer (Karamzade et al., 2024), and D-SAC (Liotet et al., 2021)). All of the belief-based baselines and our DFBT-SAC are trained on the same D4RL dataset. We first investigate the prediction accuracy of beliefs (Section 6.2.1), followed by the performance comparison with deterministic and stochastic delays (Section 6.2.2). Additionally, we conduct ablation studies on the multi-step bootstrapping of our DFBT-SAC (Section 6.2.3).
+
+Table 1. Performance on MuJoCo with Deterministic Delays. The best performance is underlined, the best belief-based method is in red.  
+
+<table><tr><td rowspan="2">Task</td><td rowspan="2">Delays</td><td colspan="3">Augmentation-based</td><td colspan="5">Belief-based</td></tr><tr><td>A-SAC</td><td>BPQL</td><td>ADRL</td><td>DATS</td><td>D-Dreamer</td><td>D-SAC</td><td>DFBT-SAC(1) (ours)</td><td>DFBT-SAC (ours)</td></tr><tr><td rowspan="3">HalfCheetah-v2</td><td>8</td><td>0.10±0.01</td><td>0.40±0.04</td><td>0.44±0.03</td><td>0.08±0.01</td><td>0.08±0.01</td><td>0.12±0.06</td><td>0.38±0.03</td><td>0.35±0.12</td></tr><tr><td>32</td><td>0.02±0.02</td><td>0.40±0.03</td><td>0.26±0.04</td><td>0.11±0.04</td><td>0.08±0.00</td><td>0.08±0.02</td><td>0.40±0.07</td><td>0.42±0.03</td></tr><tr><td>128</td><td>0.04±0.06</td><td>0.08±0.13</td><td>0.14±0.02</td><td>0.10±0.08</td><td>0.15±0.05</td><td>0.09±0.04</td><td>0.40±0.06</td><td>0.41±0.03</td></tr><tr><td rowspan="3">Hopper-v2</td><td>8</td><td>0.61±0.31</td><td>0.87±0.09</td><td>0.95±0.16</td><td>0.41±0.31</td><td>0.11±0.01</td><td>0.16±0.05</td><td>0.92±0.28</td><td>0.77±0.18</td></tr><tr><td>32</td><td>0.11±0.02</td><td>0.89±0.14</td><td>0.73±0.20</td><td>0.07±0.04</td><td>0.11±0.05</td><td>0.11±0.01</td><td>0.60±0.23</td><td>0.68±0.20</td></tr><tr><td>128</td><td>0.04±0.01</td><td>0.08±0.02</td><td>0.07±0.01</td><td>0.08±0.01</td><td>0.09±0.03</td><td>0.06±0.01</td><td>0.16±0.02</td><td>0.20±0.03</td></tr><tr><td rowspan="3">Walker2d-v2</td><td>8</td><td>0.44±0.26</td><td>1.07±0.02</td><td>0.97±0.10</td><td>0.13±0.05</td><td>0.11±0.06</td><td>0.09±0.05</td><td>0.95±0.14</td><td>0.99±0.03</td></tr><tr><td>32</td><td>0.10±0.02</td><td>0.37±0.25</td><td>0.16±0.08</td><td>0.02±0.03</td><td>0.08±0.05</td><td>0.08±0.02</td><td>0.57±0.21</td><td>0.64±0.10</td></tr><tr><td>128</td><td>0.06±0.00</td><td>0.07±0.08</td><td>0.08±0.01</td><td>0.02±0.02</td><td>0.08±0.05</td><td>0.11±0.06</td><td>0.38±0.11</td><td>0.40±0.08</td></tr></table>
+
+Table 2. Performance on MuJoCo with Stochastic Delays. The best performance is underlined, and the best belief-based method is in red.  
+
+<table><tr><td rowspan="2">Task</td><td rowspan="2">Delays</td><td colspan="3">Augmentation-based</td><td colspan="4">Belief-based</td></tr><tr><td>A-SAC</td><td>BPQL</td><td>ADRL</td><td>DATS</td><td>D-Dreamer</td><td>D-SAC</td><td>DFBT-SAC (ours)</td></tr><tr><td rowspan="3">HalfCheetah-v2</td><td>U(1,8)</td><td>0.09±0.01</td><td>0.21±0.07</td><td>0.17±0.07</td><td>0.09±0.03</td><td>0.02±0.01</td><td>0.03±0.01</td><td>0.37±0.12</td></tr><tr><td>U(1,32)</td><td>0.01±0.00</td><td>0.33±0.07</td><td>0.23±0.02</td><td>0.11±0.04</td><td>0.02±0.00</td><td>0.01±0.01</td><td>0.31±0.16</td></tr><tr><td>U(1,128)</td><td>0.01±0.01</td><td>0.03±0.03</td><td>0.15±0.02</td><td>0.16±0.03</td><td>0.16±0.00</td><td>0.02±0.00</td><td>0.39±0.04</td></tr><tr><td rowspan="3">Hopper-v2</td><td>U(1,8)</td><td>0.17±0.05</td><td>0.20±0.04</td><td>0.18±0.04</td><td>0.04±0.01</td><td>0.07±0.05</td><td>0.14±0.04</td><td>0.86±0.18</td></tr><tr><td>U(1,32)</td><td>0.05±0.01</td><td>0.07±0.09</td><td>0.05±0.01</td><td>0.05±0.01</td><td>0.04±0.01</td><td>0.03±0.01</td><td>0.43±0.21</td></tr><tr><td>U(1,128)</td><td>0.03±0.01</td><td>0.04±0.01</td><td>0.04±0.02</td><td>0.05±0.00</td><td>0.03±0.01</td><td>0.03±0.00</td><td>0.14±0.01</td></tr><tr><td rowspan="3">Walker2d-v2</td><td>U(1,8)</td><td>0.36±0.24</td><td>0.40±0.32</td><td>0.41±0.15</td><td>0.07±0.01</td><td>0.07±0.05</td><td>0.12±0.04</td><td>1.11±0.10</td></tr><tr><td>U(1,32)</td><td>0.12±0.03</td><td>0.16±0.04</td><td>0.11±0.05</td><td>0.09±0.04</td><td>0.12±0.04</td><td>0.05±0.02</td><td>0.67±0.15</td></tr><tr><td>U(1,128)</td><td>0.06±0.01</td><td>0.06±0.06</td><td>0.04±0.02</td><td>0.10±0.04</td><td>0.15±0.07</td><td>0.03±0.04</td><td>0.30±0.13</td></tr></table>
+
+# 6.2. Experimental Results
+
+# 6.2.1. BELIEF PREDICTION ACCURACY
+
+We first evaluate the state prediction accuracy of our DFBT in D4RL offline datasets. Using the L1 norm as the belief error, We report the error curves increasing with delays of the belief of DATS, D-Dreamer, D-SAC and our DFBTSAC in Figure 2. All methods are trained on the D4RL mixed dataset including random, medium and expert policy demonstrations. The implementation details are provided in Appendix A. From the results, we can tell that our DFBT can address the compounding errors effectively via directly forecasting delayed observations, thus maintaining the best prediction accuracy with increased delays, which is consistent with our theoretical results. We further provide the belief qualitative comparison in Appendix E.
+
+# 6.2.2. PERFORMANCE COMPARISON
+
+We report the normalized return  $R_{\mathrm{norm}} \coloneqq \frac{R_{\mathrm{alg}} - R_{\mathrm{random}}}{R_{\mathrm{sac}} - R_{\mathrm{random}}}$ , where  $R_{\mathrm{alg}}, R_{\mathrm{sac}}, R_{\mathrm{random}}$  are the return of the algorithm, delay-free SAC, and random returns, respectively. Each method is evaluated across 5 random seeds, and the implementation details are provided in the Appendix A. Additional experiment results and learning curves for different delays are provided in the Appendix C and Appendix D, respectively.
+
+Deterministic Delays. The performance of DFBT-SAC and baselines are evaluated on MuJoCo with deterministic delays (8, 32, and 128) reported in Table 1, showing that our DFBT-SAC overall outperforms other belief-based methods significantly. Specifically, DFBT-SAC shows comparable performance with the SOTA augmentation-based methods (BPQL and ADRL) on tasks with relatively short delays (8). However, our DFBT-SAC yields a leading performance on challenging tasks when the delays increase to long delays (32 and 128). Additionally, we also compare DFBT-SAC(1), the single-step bootstrapping version of DFBT-SAC. The results imply that the multi-step bootstrapping technique can further improve the performance effectively, validating our statement in Section 4.2. We will investigate the bootstrapping steps in the later ablation study.
+
+Stochastic Delays. We also evaluate DFBT-SAC on MuJoCo with stochastic delays which follow the uniform distribution  $U$ . As shown in Table 2, our DFBT-SAC remarkably outperforms all belief-based baselines on all tasks with all delays settings ( $U(1,8)$ ,  $U(1,32)$ , and  $U(1,128)$ ). Especially for  $U(1,128)$  delays, DFBT-SAC also performs approximately  $144\%$ ,  $180\%$ , and  $100\%$  better than the second best baselines on HalfCheetah-v2, Hopper-v2, and Walker2d-v2, respectively.
+
+Learning Curves on MuJoCo with Challenging Delays. As summarized in Figure 3, we report the learning curves on MuJoCo with 128 and  $U(1,128)$  delays. Our DFBT-
+
+![](images/6ac98fbdb7b6d8bd4e281ea8d26bce3862b214ab7e81ded29d1fa11f7668a1a0.jpg)
+
+![](images/ef33657d161149aa0c08862b862d701850c961a54b4c6938926b38245d22c4cf.jpg)
+
+![](images/a82784a65f7691cf30274c7ad1a95c92538f29ddcdc21deca17f4d2c8188de82.jpg)
+
+![](images/0b0dcf6f4a9cbfc29a465765d0b1c899b6497da9dc3899956db5f479706803c3.jpg)  
+(a) HalfCheetah-v2 (128 Delays)
+
+![](images/5edf7dfeed187794336b33144b95028224e6faa8a01f3e18f5ded0bcbfbd0bd2.jpg)  
+(b) Hopper-v2 (128 Delays)  
+(e) Hopper-v2  $(U(1,128)$  Delays)
+
+![](images/e0f8307b8df65920255f04bb7fbf0f82f5facf5f07445c794c35adc455020be5.jpg)  
+(c) Walker2d-v2 (128 Delays)  
+(f) Walker2d-v2  $(U(1,128)$  Delays)
+
+![](images/dcaf9fe489be50c0c4917296336212d024e7d37023bef558838af75106ef6093.jpg)  
+(d) HalfCheetah-v2  $(U(1,128)$  Delays)  
+Figure 3. Learning Curves on MuJoCo with 128 Delays and  $U(1,128)$  Delays.
+
+SAC exhibits a leading learning efficiency across all these challenging delays. Especially in the MuJoCo with 128 delays, DFBT-SAC can learn remarkably faster than all baselines, resulting in  $173.3\%$  (HalfCheetah-v2),  $122.2\%$  (Hopper-v2), and  $263.6\%$  (Walker2d-v2) final performance improvement better than the second-best baselines.
+
+Results Analysis and Discussion. Based on the above experimental results, we can observe a general trend in both deterministic and stochastic delays that DFBT-SAC shows comparable performance with SOTA augmentation-based methods under short delays scenarios, in which augmentation-based methods generally have better performance than other belief-based approaches. Since when delays are short, the dimensionality of the augmented state space is proper for learning, and without the approximation errors from belief representation. However, as delays increase in both scenarios, DFBT-SAC unleashes its advantages. This empirical finding validates our theoretical analysis in Section 5. For augmentation-based methods, as the delays increase, the augmented state in augmentation-based methods grows too rapidly, making efficient policy optimization infeasible. For recursively forecasting belief methods, the compounding errors in belief function approximation grow exponentially with increasing delays, as demonstrated in Figure 2 and Section 6.2.1. This results in inaccurate predictions of delayed observations and undermines subsequent policy training. In contrast, DFBT's prediction error remains largely unaffected by the length of delays, enabling DFBT-SAC to maintain strong performance even in scenarios with long delays.
+
+# 6.2.3. ABLATION STUDY ON BOOTSTRAPPING STEPS
+
+As mentioned in Section 4.2, we conduct the ablation study on the bootstrapping steps in DFBT-SAC. The result presented in Table 3 tells us that DFBT-SAC with 8 bootstrapping steps achieves the averaged best performance compared to other choices. It is also confirmed that bootstrapping with the states predicted by the trained DFBT can effectively improve performance.
+
+Table 3. Final Performance on Walker2d-v2 of DFBT-SAC with different bootstrapping steps. The best performance is underlined.  
+
+<table><tr><td rowspan="2">Delays</td><td colspan="4">Bootstrapping Steps N</td></tr><tr><td>1</td><td>2</td><td>4</td><td>8</td></tr><tr><td>8</td><td>0.86±0.07</td><td>0.96±0.07</td><td>1.00±0.14</td><td>1.11±0.10</td></tr><tr><td>16</td><td>0.84±0.24</td><td>0.76±0.21</td><td>1.02±0.12</td><td>0.99±0.06</td></tr><tr><td>32</td><td>0.63±0.22</td><td>0.53±0.15</td><td>0.67±0.20</td><td>0.67±0.15</td></tr><tr><td>64</td><td>0.34±0.24</td><td>0.28±0.22</td><td>0.29±0.15</td><td>0.41±0.10</td></tr><tr><td>128</td><td>0.24±0.03</td><td>0.29±0.07</td><td>0.27±0.14</td><td>0.30±0.13</td></tr></table>
+
+Table 4. Performance comparison of DFBT-SAC with different belief training methods on MuJoCo tasks with 32 delays. The best performance is underlined.  
+
+<table><tr><td>Task</td><td>Online</td><td>Offline</td><td>Offline + Fine-tuning</td></tr><tr><td>HalfCheetah-v2</td><td>0.11 ± 0.39</td><td>0.42 ± 0.12</td><td>0.39 ± 0.04</td></tr><tr><td>Hopper-v2</td><td>0.10 ± 0.58</td><td>0.68 ± 0.20</td><td>0.84 ± 0.04</td></tr><tr><td>Walker2d-v2</td><td>0.09 ± 0.27</td><td>0.64 ± 0.10</td><td>0.96 ± 0.32</td></tr></table>
+
+# 6.2.4. EVALUATION ON ADDITIONAL MUJOCO TASKS.
+
+we conducted experiments on the Pusher-v2, Reacher-v2, and Swimmer-v2 with deterministic 32 delays. The offline datasets (500k samples) are collected from a SAC policy, with other settings unchanged. The results are shown in Table 5, showing that DFBT-SAC achieves superior performance on these tasks.
+
+Table 5. Performance comparison on additional MuJoCo tasks with 32 deterministic delays. The best performance is underlined.  
+
+<table><tr><td>Task</td><td>A-SAC</td><td>BPQL</td><td>ADRL</td><td>DATS</td><td>D-Dreamer</td><td>D-SAC</td><td>DFBT-SAC (ours)</td></tr><tr><td>Pusher-v2</td><td>0.05 ± 0.00</td><td>0.93 ± 0.58</td><td>0.95 ± 0.24</td><td>0.92 ± 0.10</td><td>0.87 ± 0.18</td><td>0.94 ± 0.04</td><td>1.04 ± 0.24</td></tr><tr><td>Reacher-v2</td><td>0.89 ± 0.08</td><td>0.83 ± 0.06</td><td>0.85 ± 0.01</td><td>0.82 ± 0.13</td><td>0.84 ± 0.02</td><td>0.88 ± 0.07</td><td>0.93 ± 0.06</td></tr><tr><td>Swimmer-v2</td><td>0.27 ± 0.05</td><td>0.80 ± 0.14</td><td>0.60 ± 0.06</td><td>0.25 ± 0.05</td><td>0.21 ± 0.07</td><td>0.30 ± 0.07</td><td>1.01 ± 0.27</td></tr></table>
+
+Table 6. Performance on stochastic MuJoCo with deterministic 128 delays. The best performance is underlined.  
+
+<table><tr><td>Task</td><td>A-SAC</td><td>BPQL</td><td>ADRL</td><td>DATS</td><td>D-Dreamer</td><td>D-SAC</td><td>DFBT-SAC (ours)</td></tr><tr><td>HalfCheetah-v2</td><td>0.00 ± 0.03</td><td>0.01 ± 0.05</td><td>0.13 ± 0.04</td><td>0.13 ± 0.03</td><td>0.07 ± 0.01</td><td>0.00 ± 0.04</td><td>0.35 ± 0.04</td></tr><tr><td>Hopper-v2</td><td>0.03 ± 0.04</td><td>0.08 ± 0.05</td><td>0.06 ± 0.04</td><td>0.05 ± 0.06</td><td>0.04 ± 0.05</td><td>0.04 ± 0.05</td><td>0.13 ± 0.22</td></tr><tr><td>Walker2d-v2</td><td>0.06 ± 0.02</td><td>0.04 ± 0.01</td><td>0.08 ± 0.01</td><td>0.06 ± 0.03</td><td>0.10 ± 0.03</td><td>0.05 ± 0.02</td><td>0.30 ± 0.07</td></tr></table>
+
+# 6.2.5. ABLATION RESULTS OF BELIEF TRAINING.
+
+Some task-specific information may be missing if the belief is frozen in the RL process, leading to limited performance improvement. This issue can be mitigated by fine-tuning the belief within the RL process. The results, shown in Table 4, demonstrate that fine-tuning helps the DFBT capture the task-specific information with better performance. Note that there are many potential methods for capturing task-specific information, not limited to fine-tuning DFBT. Belief learning from scratch in the online RL process always suffers from instability issues. Therefore, in this paper, we separate belief learning from the online RL process, which allows us to investigate the belief component solely, eliminate potential influences from the RL side.
+
+# 6.2.6. STOCHASTIC MUJOCo.
+
+We conducted additional experiments on the stochastic MuJoCo tasks with a probability of 0.001 for the unaware noise and deterministic 128 delays. As shown in Table 6, the results demonstrate that our DFBT-SAC achieves superior performance in these stochastic MuJoCo tasks.
+
+# 6.2.7. INFERENCE SPEED COMPARISON.
+
+We conducted additional experiments on computational efficiency. As shown in Table 7, the results demonstrate that directly forecasting belief maintains a consistent and stable inference speed (around  $4\mathrm{ms}$ ) across different delays. In contrast, the recursively forecasting belief experiences inference speed issues as delays increase. In HalfCheetah-v2 with 128 delays, the training times of DATS and D-Dreamer are around 10 hours and 15 hours, respectively, while those of D-SAC and DFBT-SAC are both around 6 hours.
+
+Table 7. Inference speed (ms) comparison in HalfCheetah-v2.  
+
+<table><tr><td>Delays</td><td>DATS</td><td>D-Dreamer</td><td>D-SAC</td><td>DFBT-SAC</td></tr><tr><td>8</td><td>1.10 ± 0.02</td><td>1.85 ± 0.01</td><td>4.03 ± 0.03</td><td>4.18 ± 0.04</td></tr><tr><td>32</td><td>3.85 ± 0.06</td><td>6.80 ± 0.04</td><td>4.03 ± 0.04</td><td>4.18 ± 0.04</td></tr><tr><td>128</td><td>14.97 ± 0.22</td><td>26.51 ± 0.19</td><td>4.03 ± 0.03</td><td>4.15 ± 0.05</td></tr></table>
+
+# 6.3. Limitations and Challenges
+
+In this work, we empirically validate that our approach can address the compounding errors of recursively forecasting belief with significant performance improvement. However, there remain some limitations and challenges as follows.
+
+Online Belief Learning. In this paper, we mainly consider learning DFBT from the offline dataset, which means the performance of the belief is subject to the quality and quantity of the dataset. However, direct learning belief in the online environment will introduce an auxiliary representation task, destabilizing the learning process.
+
+Sample Complexity of Directly Forecasting Belief. Although we have empirically demonstrated that directly forecasting belief can effectively reduce the compounding errors of recursively forecasting belief. However, it is worth theoretically analysing the sample complexity of belief learning, especially in online scenarios.
+
+# 7. Conclusion
+
+This work investigates the challenges of RL in environments where inherent delays exist between actions and their corresponding outcomes. Existing belief-based approaches usually suffer from the compounding errors issue of the recursively forecasting belief as the delays are increased, seriously hindering the performance. To resolve this issue, we present DFBT, a new directly forecasting belief representation method. Furthermore, we present DFBT-SAC, which facilitates multi-step bootstrapping in learning the value function via the state prediction from the DFBT, effectively improving the sample efficiency. We demonstrate that our DFBT greatly reduces compounding errors, yielding stronger performance guarantees. Our empirical results validate that DFBT has remarkable prediction accuracy in the D4RL datasets. We also empirically show that our DFBT-SAC not only effectively enhance the learning efficiency with superior performance in the MuJoCo benchmark.
+
+# Acknowledgement
+
+We sincerely appreciate the insightful suggestions provided by ICML 2025 reviewers, which help us improve the quality of this paper. We sincerely acknowledge the support by the grant EP/Y002644/1 under the EPSRC ECR International Collaboration Grants program, funded by the International Science Partnerships Fund (ISPF) and the UK Research and Innovation. We also sincerely acknowledge the support by Taiwan NSTC under Grant Number NSTC-112-2221-E-002-168-MY3, and from King Abdullah University of Science and Technology (KAUST) - Center of Excellence for Generative AI, under award number 5940. Simon Sinong Zhan and Qi Zhu are partially supported by the US National Science Foundation grants 2324936 and 2328973.
+
+# Impact Statement
+
+This paper aims to advance the field of Machine Learning. While acknowledging there are many potential societal consequences of our work, we believe that none need to be specially highlighted here.
+
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+
+# A. Implementation Details
+
+The implementation of DFBT and DFBT-SAC is based on CORL (Tarasov et al., 2022) and CleanRL (Huang et al., 2022). The codebase for reproducing our experimental results is also provided in the Supplementary Material. We detail the hyperparameter settings of DFBT and DFBT-SAC in Table 8 and Table 9, respectively.
+
+Table 8. Hyper-parameters table of DFBT.  
+
+<table><tr><td>Hyper-parameter</td><td>Value</td></tr><tr><td>Epoch</td><td>1e3</td></tr><tr><td>Batch Size</td><td>256</td></tr><tr><td>Attention Heads Num</td><td>4</td></tr><tr><td>Layers Num</td><td>10</td></tr><tr><td>Hidden Dim</td><td>256</td></tr><tr><td>Attention Dropout Rate</td><td>0.1</td></tr><tr><td>Residual Dropout Rate</td><td>0.1</td></tr><tr><td>Hidden Dropout Rate</td><td>0.1</td></tr><tr><td>Learning Rate</td><td>1e-4</td></tr><tr><td>Optimizer</td><td>AdamW</td></tr><tr><td>Weight Decay</td><td>1e-4</td></tr><tr><td>Betas</td><td>(0.9, 0.999)</td></tr></table>
+
+Table 9. Hyper-parameters table of DFBT-SAC.  
+
+<table><tr><td>Hyper-parameter</td><td>Value</td></tr><tr><td>Bootstrapping Steps N</td><td>8</td></tr><tr><td>Learning Rate (Actor)</td><td>3e-4</td></tr><tr><td>Learning Rate (Critic)</td><td>1e-3</td></tr><tr><td>Learning Rate (Entropy)</td><td>1e-3</td></tr><tr><td>Train Frequency (Actor)</td><td>2</td></tr><tr><td>Train Frequency (Critic)</td><td>1</td></tr><tr><td>Soft Update Factor (Critic)</td><td>5e-3</td></tr><tr><td>Batch Size</td><td>256</td></tr><tr><td>Neurons</td><td>[256, 256]</td></tr><tr><td>Layers</td><td>3</td></tr><tr><td>Hidden Dim</td><td>256</td></tr><tr><td>Activation</td><td>ReLU</td></tr><tr><td>Optimizer</td><td>Adam</td></tr></table>
+
+# B. Theoretical Analysis
+
+Theorem B.1 (Performance Difference of Recursively Forecasting Belief). For the delay-free policy  $\pi$  and the delayed policy  $\pi_{\Delta}$ . Given any  $x_{t} \in \mathcal{X}$ , the performance difference  $I^{\text{recursive}}(x_{t})$  of the recursively forecasting belief  $b_{\theta}$  can be bounded as follows, respectively.
+
+For deterministic delays  $\Delta$ , we have
+
+$$
+\left| I ^ {\text {r e c u r s i v e}} (x _ {t}) \right| \leq \left| I _ {\Delta} ^ {\text {t r u e}} (x _ {t}) \right| + L _ {V} \underbrace {\frac {1 - L _ {\mathcal {P}} ^ {\Delta}}{1 - L _ {\mathcal {P}}} \epsilon_ {\mathcal {P}}} _ {\text {c o m p o u n d i n g e r r o r s}}.
+$$
+
+And for stochastic delays  $\delta \sim d_{\Delta}(\cdot)$ , we have
+
+$$
+\left| I ^ {\text {r e c u r s i v e}} (x _ {t}) \right| \leq \underset {\delta \sim d _ {\Delta} (\cdot)} {\mathbb {E}} \left[ | I _ {\delta} ^ {\text {t r u e}} (x _ {t}) | + L _ {V} \underbrace {\frac {1 - L _ {\mathcal {P}} ^ {\delta}}{1 - L _ {\mathcal {P}}} \epsilon_ {\mathcal {P}}} _ {\text {c o m p o u n d i n g e r r o r s}} \right].
+$$
+
+Proof. For deterministic delays  $\Delta$ , the performance difference  $I^{\mathrm{recursive}}$  can be written as:
+
+$$
+\begin{array}{l} I ^ {\text {r e c u r s i v e}} \left(x _ {t}\right) = \underset {s _ {t} \sim b _ {\theta} (\cdot | x _ {t})} {\mathbb {E}} \left[ V ^ {\pi} \left(s _ {t}\right) \right] - V ^ {\pi_ {\Delta}} \left(x _ {t}\right) \\ = I _ {\Delta} ^ {\mathrm {t r u e}} (x _ {t}) + \underset {s _ {t} \sim b _ {\theta} (\cdot | x _ {t})} {\mathbb {E}} \left[ V ^ {\pi} (s _ {t}) \right] - \underset {s _ {t} \sim b (\cdot | x _ {t})} {\mathbb {E}} \left[ V ^ {\pi} (s _ {t}) \right]. \\ \end{array}
+$$
+
+And recall that we have the Lipschitz continuity of the value function  $V^{\pi}$ :
+
+$$
+\left| \underset {s _ {t} \sim b _ {\theta} (\cdot | x _ {t})} {\mathbb {E}} \left[ V ^ {\pi} (s _ {t}) \right] - \underset {s _ {t} \sim b (\cdot | x _ {t})} {\mathbb {E}} \left[ V ^ {\pi} (s _ {t}) \right] \right| \leq L _ {V} \mathcal {W} \left(b _ {\theta} (\cdot | x _ {t}) | | b (\cdot | x _ {t})\right).
+$$
+
+For  $\mathcal{W}(b_{\theta}(\cdot |x_t)||b(\cdot |x_t))$  , we follow the proof sketch of (Asadi et al., 2018).
+
+We use  $b^{(i)}(\cdot |x_t)(i = 1,\dots ,\Delta)$  to note that the belief function with the specific delays  $i$ . For instance,  $b^{(\Delta)}(\cdot |x_{t}) = b(\cdot |x_{t})$  and  $b^{(1)}(\cdot |x_t) = \mathcal{P}(\cdot |s_t,a_t)$ .
+
+Then, we have
+
+$$
+\begin{array}{l} \mathcal {W} \left(b _ {\theta} ^ {(\Delta)} (\cdot | x _ {t}) | | b ^ {(\Delta)} (\cdot | x _ {t})\right) \\ = \mathcal {W} \left(\mathcal {P} _ {\theta} \left(\cdot | b _ {\theta} ^ {(\Delta - 1)} (\cdot | x _ {t}), a _ {t - 1}\right) \mid \mid \mathcal {P} \left(\cdot | b ^ {(\Delta - 1)} (\cdot | x _ {t}), a _ {t - 1}\right)\right) \\ \leq \mathcal {W} \left(\mathcal {P} _ {\theta} \left(\cdot | b _ {\theta} ^ {(\Delta - 1)} (\cdot | x _ {t}), a _ {t - 1}\right) | | \mathcal {P} \left(\cdot | b _ {\theta} ^ {(\Delta - 1)} (\cdot | x _ {t}), a _ {t - 1}\right)\right) + \mathcal {W} \left(\mathcal {P} \left(\cdot | b _ {\theta} ^ {(\Delta - 1)} (\cdot | x _ {t}), a _ {t - 1}\right) | | \mathcal {P} \left(\cdot | b ^ {(\Delta - 1)} (\cdot | x _ {t}), a _ {t - 1}\right)\right) \\ \leq \epsilon_ {\mathcal {P}} + L _ {\mathcal {P}} \mathcal {W} \left(b _ {\theta} ^ {\Delta - 1} (\cdot | x _ {t}) \mid \mid b ^ {\Delta - 1} (\cdot | x _ {t})\right) \\ \leq \left(1 + L _ {\mathcal {P}}\right) \epsilon_ {\mathcal {P}} + L _ {\mathcal {P}} ^ {2} \mathcal {W} \left(b _ {\theta} ^ {(\Delta - 2)} (\cdot | x _ {t}) | | b ^ {(\Delta - 2)} (\cdot | x _ {t})\right) \\ \leq \dots \\ \leq \left(1 + \dots + L _ {\mathcal {P}} ^ {\Delta - 2}\right) \epsilon_ {\mathcal {P}} + L _ {\mathcal {P}} ^ {\Delta - 1} \mathcal {W} \left(b _ {\theta} ^ {(1)} (\cdot | x _ {t}) \right\lVert b ^ {(1)} (\cdot | x _ {t})) \\ \leq \left(1 + \dots + L _ {\mathcal {P}} ^ {\Delta - 1}\right) \epsilon_ {\mathcal {P}} \\ = \frac {1 - L _ {\mathcal {P}} ^ {\Delta}}{1 - L _ {\mathcal {P}}} \epsilon_ {\mathcal {P}}. \\ \end{array}
+$$
+
+Therefore, we have
+
+$$
+\left| I ^ {\text {r e c u r s i v e}} \left(x _ {t}\right) \right| \leq \left| I _ {\Delta} ^ {\text {t r u e}} \left(x _ {t}\right) \right| + L _ {V} \frac {1 - L _ {\mathcal {P}} ^ {\Delta}}{1 - L _ {\mathcal {P}}} \epsilon_ {\mathcal {P}}.
+$$
+
+The above theoretical results can be extended to the stochastic delays  $\delta \sim d_{\Delta}(\cdot)$  easily. The performance difference of the ground-truth belief  $I_{\delta}^{\mathrm{true}}$  is defined as:
+
+$$
+I_{\delta}^{\text{true}}(x_{t}) = \frac{1}{1 - \gamma}\underset { \begin{array}{c}\hat{s}\sim b_{\delta}(\cdot |\hat{x})\\ \hat{a}\sim \pi_{\delta}(\cdot |\hat{x})\\ \hat{x}\sim d_{\delta}^{\pi}(\cdot |x_{t}) \end{array} }{\mathbb{E}}\left[V^{\pi}(\hat{s}) - Q^{\pi}(\hat{s},\hat{a})\right].
+$$
+
+Finally, we have
+
+$$
+\left| I ^ {\text {r e c u r s i v e}} \left(x _ {t}\right) \right| \leq \sum_ {\delta = 1} ^ {\Delta} d _ {\Delta} (\delta) \left[ \left| I _ {\delta} ^ {\text {t r u e}} \left(x _ {t}\right) \right| + L _ {V} \frac {1 - L _ {\mathcal {P}} ^ {\delta}}{1 - L _ {\mathcal {P}}} \epsilon_ {\mathcal {P}} \right].
+$$
+
+![](images/83e9fd083bb9b4a8a086e9cd12ddf52928cc1ede2677c11cb2ec4cc17d28dbc7.jpg)
+
+Proposition B.2 (Performance Degeneration Bound of Directly Forecasting Belief). For the delay-free policy  $\pi$  and the delayed policy  $\pi_{\Delta}$ . Given any  $x_{t} \in \mathcal{X}$ , the performance degeneration  $I^{direct}$  of the directly forecasting belief  $b_{\theta}$  can bounded as follows respectively.
+
+For deterministic delays  $\Delta$ , we have
+
+$$
+\left| I ^ {\text {d i r e c t}} \left(x _ {t}\right) \right| \leq \left| I _ {\Delta} ^ {\text {t r u e}} \left(x _ {t}\right) \right| + L _ {V} \epsilon_ {\text {d i r e c t}}.
+$$
+
+For stochastic delays  $\delta \sim d_{\Delta}(\cdot)$ , we have
+
+$$
+\left| I ^ {d i r e c t} (x _ {t}) \right| \leq \underset {\delta \sim d _ {\Delta} (\cdot)} {\mathbb {E}} [ | I _ {\delta} ^ {t r u e} (x _ {t}) | ] + L _ {V} \epsilon_ {d i r e c t}.
+$$
+
+Proof. Applying Assumption 5.6 and the proof of Theorem B.1.
+
+![](images/16002f0c0ae3e4d8894f1ea084a3846e58f5c803753a3229cf9cd47dde7de96e.jpg)
+
+Proposition B.3 (Performance Degeneration Comparison). Directly forecasting belief could achieve a better performance guarantee  $\left| I^{\text{direct}}(x_t) \right| \leq \left| I^{\text{recursive}}(x_t) \right|$ , if we have
+
+$$
+\epsilon_ {d i r e c t} \leq \frac {1 - L _ {\mathcal {P}} ^ {\Delta}}{1 - L _ {\mathcal {P}}} \epsilon_ {\mathcal {P}}
+$$
+
+for deterministic delays  $\Delta$ , and
+
+$$
+\epsilon_ {d i r e c t} \leq \underset {\delta \sim d _ {\Delta} (\cdot)} {\mathbb {E}} \left[ \frac {1 - L _ {\mathcal {P}} ^ {\delta}}{1 - L _ {\mathcal {P}}} \right] \epsilon_ {\mathcal {P}}
+$$
+
+for stochastic delays  $\delta \sim d_{\Delta}(\cdot)$ .
+
+Proof. For deterministic delays  $\Delta$ , if we have
+
+$$
+\epsilon_ {d i r e c t} \leq \frac {1 - L _ {\mathcal {P}} ^ {\Delta}}{1 - L _ {\mathcal {P}}} \epsilon_ {\mathcal {P}},
+$$
+
+then it is obvious that we have
+
+$$
+\underbrace {\left| I _ {\Delta} ^ {\text {t r u e}} \left(x _ {t}\right) \right| + L _ {V} \epsilon_ {\text {d i r e c t}}} _ {\left| I ^ {\text {d i r e c t}} \left(x _ {t}\right) \right|} \leq \underbrace {\left| I _ {\Delta} ^ {\text {t r u e}} \left(x _ {t}\right) \right| + L _ {V} \frac {1 - L _ {\mathcal {P}} ^ {\Delta}}{1 - L _ {\mathcal {P}}} \epsilon_ {\mathcal {P}}} _ {\left| I ^ {\text {r e c u r s i v e}} \left(x _ {t}\right) \right|}.
+$$
+
+For stochastic delays  $\delta \sim d_{\Delta}(\cdot)$ , if we have
+
+$$
+\epsilon_ {d i r e c t} \leq \underset {\delta \sim d _ {\Delta} (\cdot)} {\mathbb {E}} \left[ \frac {1 - L _ {\mathcal {P}} ^ {\delta}}{1 - L _ {\mathcal {P}}} \epsilon_ {\mathcal {P}} \right],
+$$
+
+then it is obvious that we have
+
+$$
+\underbrace {\left| I _ {\Delta} ^ {\text {t r u e}} (x _ {t}) \right| + L _ {V} \epsilon_ {\text {d i r e c t}}} _ {| I ^ {\text {d i r e c t}} (x _ {t}) |} \leq \underbrace {\left| I _ {\Delta} ^ {\text {t r u e}} (x _ {t}) \right| + L _ {V} \underset {\delta \sim d _ {\Delta} (\cdot)} {\mathbb {E}} \left[ \frac {1 - L _ {\mathcal {P}} ^ {\delta}}{1 - L _ {\mathcal {P}}} \epsilon_ {\mathcal {P}} \right]} _ {| I ^ {\text {r e c u r s i v e}} (x _ {t}) |}.
+$$
+
+![](images/280e0a19dc93fd35b41195c468961195e94081ad4ef7e73f18d4a309a18c63aa.jpg)
+
+# C. Additional Results on MuJoCo.
+
+We report additional experimental results on MuJoCo, including more different patterns of deterministic and stochastic delays in Table 10 and Table 11, respectively.
+
+Table 10. Performance on MuJoCo with Deterministic Delays. The best performance is underlined, the best belief-based method is in red.  
+
+<table><tr><td rowspan="2">Task</td><td rowspan="2">Delays</td><td colspan="3">Augmentation-based</td><td colspan="4">Belief-based</td></tr><tr><td>A-SAC</td><td>BPQL</td><td>ADRL</td><td>DATS</td><td>D-Dreamer</td><td>D-SAC</td><td>DFBT-SAC (ours)</td></tr><tr><td rowspan="5">HalfCheetah-v2</td><td>8</td><td>0.10±0.01</td><td>0.40±0.04</td><td>0.44±0.03</td><td>0.08±0.01</td><td>0.08±0.01</td><td>0.12±0.06</td><td>0.35±0.12</td></tr><tr><td>16</td><td>0.06±0.03</td><td>0.42±0.02</td><td>0.29±0.10</td><td>0.08±0.01</td><td>0.08±0.01</td><td>0.11±0.10</td><td>0.40±0.05</td></tr><tr><td>32</td><td>0.02±0.02</td><td>0.40±0.03</td><td>0.26±0.04</td><td>0.11±0.04</td><td>0.08±0.00</td><td>0.08±0.02</td><td>0.42±0.03</td></tr><tr><td>64</td><td>0.01±0.01</td><td>0.15±0.12</td><td>0.16±0.02</td><td>0.12±0.05</td><td>0.11±0.05</td><td>0.12±0.07</td><td>0.39±0.06</td></tr><tr><td>128</td><td>0.04±0.06</td><td>0.08±0.13</td><td>0.14±0.02</td><td>0.10±0.08</td><td>0.15±0.05</td><td>0.09±0.04</td><td>0.41±0.03</td></tr><tr><td rowspan="5">Hopper-v2</td><td>8</td><td>0.61±0.31</td><td>0.87±0.09</td><td>0.95±0.16</td><td>0.41±0.31</td><td>0.11±0.01</td><td>0.16±0.05</td><td>0.77±0.18</td></tr><tr><td>16</td><td>0.17±0.06</td><td>0.92±0.16</td><td>0.94±0.17</td><td>0.24±0.31</td><td>0.19±0.13</td><td>0.11±0.01</td><td>0.89±0.13</td></tr><tr><td>32</td><td>0.11±0.02</td><td>0.89±0.14</td><td>0.73±0.20</td><td>0.07±0.04</td><td>0.11±0.05</td><td>0.11±0.01</td><td>0.68±0.20</td></tr><tr><td>64</td><td>0.05±0.00</td><td>0.23±0.30</td><td>0.11±0.03</td><td>0.13±0.00</td><td>0.09±0.05</td><td>0.08±0.02</td><td>0.19±0.02</td></tr><tr><td>128</td><td>0.04±0.01</td><td>0.08±0.02</td><td>0.07±0.01</td><td>0.08±0.01</td><td>0.09±0.03</td><td>0.06±0.01</td><td>0.20±0.03</td></tr><tr><td rowspan="5">Walker2d-v2</td><td>8</td><td>0.44±0.26</td><td>1.07±0.02</td><td>0.97±0.10</td><td>0.13±0.05</td><td>0.11±0.06</td><td>0.09±0.05</td><td>0.99±0.03</td></tr><tr><td>16</td><td>0.13±0.02</td><td>0.96±0.05</td><td>0.67±0.21</td><td>0.06±0.10</td><td>0.12±0.03</td><td>0.08±0.04</td><td>0.95±0.11</td></tr><tr><td>32</td><td>0.10±0.02</td><td>0.37±0.25</td><td>0.16±0.08</td><td>0.02±0.03</td><td>0.08±0.05</td><td>0.08±0.02</td><td>0.64±0.10</td></tr><tr><td>64</td><td>0.07±0.01</td><td>0.14±0.03</td><td>0.10±0.01</td><td>0.01±0.02</td><td>0.08±0.03</td><td>0.08±0.04</td><td>0.41±0.14</td></tr><tr><td>128</td><td>0.06±0.00</td><td>0.07±0.03</td><td>0.08±0.01</td><td>0.02±0.02</td><td>0.08±0.05</td><td>0.11±0.06</td><td>0.40±0.08</td></tr></table>
+
+Table 11. Performance on MuJoCo with Stochastic Delays. The best performance is underlined, and the best belief-based method is in red.  
+
+<table><tr><td rowspan="2">Task</td><td rowspan="2">Delays</td><td colspan="3">Augmentation-based</td><td colspan="4">Belief-based</td></tr><tr><td>A-SAC</td><td>BPQL</td><td>ADRL</td><td>DATS</td><td>D-Dreamer</td><td>D-SAC</td><td>DFBT-SAC (ours)</td></tr><tr><td rowspan="5">HalfCheetah-v2</td><td>U(1,8)</td><td>0.09±0.01</td><td>0.21±0.07</td><td>0.17±0.07</td><td>0.09±0.03</td><td>0.02±0.01</td><td>0.03±0.01</td><td>0.37±0.12</td></tr><tr><td>U(1,16)</td><td>0.04±0.04</td><td>0.31±0.08</td><td>0.24±0.04</td><td>0.13±0.03</td><td>0.03±0.02</td><td>0.01±0.01</td><td>0.37±0.06</td></tr><tr><td>U(1,32)</td><td>0.01±0.00</td><td>0.33±0.07</td><td>0.23±0.02</td><td>0.11±0.04</td><td>0.02±0.00</td><td>0.01±0.01</td><td>0.31±0.16</td></tr><tr><td>U(1,64)</td><td>0.06±0.11</td><td>0.23±0.06</td><td>0.17±0.02</td><td>0.16±0.03</td><td>0.04±0.03</td><td>0.01±0.00</td><td>0.40±0.06</td></tr><tr><td>U(1,128)</td><td>0.01±0.01</td><td>0.03±0.03</td><td>0.15±0.02</td><td>0.16±0.03</td><td>0.16±0.00</td><td>0.02±0.00</td><td>0.39±0.04</td></tr><tr><td rowspan="5">Hopper-v2</td><td>U(1,8)</td><td>0.17±0.05</td><td>0.20±0.04</td><td>0.18±0.04</td><td>0.04±0.01</td><td>0.07±0.05</td><td>0.14±0.04</td><td>0.86±0.18</td></tr><tr><td>U(1,16)</td><td>0.08±0.02</td><td>0.11±0.11</td><td>0.07±0.04</td><td>0.04±0.01</td><td>0.03±0.01</td><td>0.04±0.02</td><td>0.89±0.17</td></tr><tr><td>U(1,32)</td><td>0.05±0.01</td><td>0.07±0.09</td><td>0.05±0.01</td><td>0.05±0.01</td><td>0.04±0.01</td><td>0.03±0.01</td><td>0.43±0.21</td></tr><tr><td>U(1,64)</td><td>0.03±0.01</td><td>0.03±0.01</td><td>0.03±0.01</td><td>0.05±0.01</td><td>0.03±0.01</td><td>0.03±0.01</td><td>0.17±0.05</td></tr><tr><td>U(1,128)</td><td>0.03±0.01</td><td>0.04±0.01</td><td>0.04±0.02</td><td>0.05±0.00</td><td>0.03±0.01</td><td>0.03±0.00</td><td>0.14±0.01</td></tr><tr><td rowspan="5">Walker2d-v2</td><td>U(1,8)</td><td>0.36±0.24</td><td>0.40±0.32</td><td>0.41±0.15</td><td>0.07±0.01</td><td>0.07±0.05</td><td>0.12±0.04</td><td>1.11±0.10</td></tr><tr><td>U(1,16)</td><td>0.19±0.10</td><td>0.27±0.17</td><td>0.24±0.10</td><td>0.08±0.02</td><td>0.13±0.08</td><td>0.07±0.02</td><td>0.99±0.06</td></tr><tr><td>U(1,32)</td><td>0.12±0.03</td><td>0.16±0.04</td><td>0.11±0.05</td><td>0.09±0.04</td><td>0.12±0.04</td><td>0.05±0.02</td><td>0.67±0.15</td></tr><tr><td>U(1,64)</td><td>0.08±0.02</td><td>0.09±0.08</td><td>0.06±0.01</td><td>0.08±0.04</td><td>0.15±0.05</td><td>0.06±0.03</td><td>0.41±0.10</td></tr><tr><td>U(1,128)</td><td>0.06±0.01</td><td>0.06±0.06</td><td>0.04±0.02</td><td>0.10±0.04</td><td>0.15±0.07</td><td>0.03±0.04</td><td>0.30±0.13</td></tr></table>
+
+# D. Learning Curves on MuJoCo.
+
+We report learning curves on MuJoCo with different patterns of deterministic and stochastic delays in Figure 4 and Figure 5, respectively.
+
+![](images/75c4e3fa587b59388150f0a8ec21bc7538352119e917df3c6cfd30088c7ce0ee.jpg)  
+(a) HalfCheetah-v2 (8 Delays)
+
+![](images/6aef442e94889248ac82e60d3e89911734351f17ea9f81bb3a1ef0a5dc8f1bac.jpg)  
+(b) Hopper-v2 (8 Delays)
+
+![](images/cf588416fddf5affaede8657cda7e8b85faac390e46db71f7222f80db085ea79.jpg)  
+(c) Walker2d-v2 (8 Delays)
+
+![](images/c257a59a6196b1c180ba050c3bc2faec9f4efa934dbd8d28e1193b8d3656a75c.jpg)  
+(d) HalfCheetah-v2 (16 Delays)
+
+![](images/7702c33f859185b010e5417ee9819298b8c1be6b442b68438fe6c0af602c6f13.jpg)  
+(e) Hopper-v2 (16 Delays)
+
+![](images/24eee9bf7062c29443d72b90696222f0807c3d2949f85e890eaeaad13bb24c6e.jpg)  
+(f) Walker2d-v2 (16 Delays)
+
+![](images/c19b103e0a2be638a704b9fae92c9585c0c486da8c1e8e5f297175ec1ccb6cc6.jpg)  
+(g) HalfCheetah-v2 (32 Delays)
+
+![](images/88a257db4e84ff5817e92d90c1e47691346e68a42ed26adba3763e7f8d35f243.jpg)  
+(h) Hopper-v2 (32 Delays)
+
+![](images/b8be688aea575ff6b1f28d093cbfa90ccfc48991c6ed99272c99da5ad6ead567.jpg)  
+(i) Walker2d-v2 (32 Delays)
+
+![](images/ee670e4fb2ae4b3927801faebb84faccedbf06bff1051afc0f327566aba80a1a.jpg)  
+(j) HalfCheetah-v2 (64 Delays)
+
+![](images/2edc3625d5fc577a28c01f93c0a5b38833e3a44ef5948dc9133604a2a272a624.jpg)  
+(k) Hopper-v2 (64 Delays)
+
+![](images/2a588068a32dcd711d145b720c5c7e70805c0859e4777012c33411ea1c04e006.jpg)  
+(1) Walker2d-v2 (64 Delays)
+
+![](images/c29e65969ac0075df461cf9f0b8d8674abf548986a0d279d657aacd164702c20.jpg)  
+(m) HalfCheetah-v2 (128 Delays)
+
+![](images/35d6b802ea2a09b5052b4fdff3da7e3631123b90354c63aae6f986d8674a322f.jpg)  
+(n) Hopper-v2 (128 Delays)
+
+![](images/81ef6ae0043d3cb89861cea31422652f386d7ee4a24f4dea546c739e1abf7b33.jpg)  
+Figure 4. Learning Curves on MuJoCo with Deterministic Delays.
+
+![](images/01dd371d78ab3db2d7ce6dc6b049800c97efff140cdf50ec4c8d0307ef7a3a91.jpg)  
+(o) Walker2d-v2 (128 Delays)
+
+![](images/90a273cccadad2b2495628369804ca4bc527e48943e4f91581f86d4ac68d266e.jpg)  
+(a) HalfCheetah-v2  $(U(1,8)$  Delays)
+
+![](images/2b2b017ef9446813bf613304cdd6bf14c6e5c9906fa506dac7268127d787eaf3.jpg)  
+(b) Hopper-v2  $(U(1,8)$  Delays)
+
+![](images/e58bbafeda9b4bf81d27a99440a31617ce86b734238132bef007efeade2c23af.jpg)  
+(c) Walker2d-v2  $(U(1,8)$  Delays)
+
+![](images/975110b45776e57f36abd755ddb155f438598d4488b9aa81c3dd31e00d18b7dc.jpg)  
+(d) HalfCheetah-v2  $(U(1,16)$  Delays)
+
+![](images/d57f30ee595830b07aa7fbed65c3c0a53817c57fd014ed2f34c852f6509d6128.jpg)  
+(e) Hopper-v2  $(U(1,16)$  Delays)
+
+![](images/dc007014463cb667cd39f1b3a249875bc9a27b2b6de361e40c1c04e1007a5b94.jpg)  
+(f) Walker2d-v2  $(U(1,16)$  Delays)
+
+![](images/1b324130a987f56316cf336f093c165ca51781f814a94502053d95fbf72fc32f.jpg)  
+(g) HalfCheetah-v2  $(U(1,32)$  Delays)
+
+![](images/c5aea3c8cca1c8d47072da0ae2ec63562fd8f3bac3103d63af4861a6e0b54deb.jpg)  
+(h) Hopper-v2  $(U(1,32)$  Delays)
+
+![](images/aedc6849f5ae1c99536dc2f32a01f549f3be9b8d36cd6920f38b10ab89dce888.jpg)  
+(i) Walker2d-v2  $(U(1,32)$  Delays)
+
+![](images/fca8f7da751e24d448d1ec3979c50e32d22e6dfae8f42427284fb98f09cd69d0.jpg)  
+(j) HalfCheetah-v2  $(U(64)$  Delays)
+
+![](images/1d3bb7b9b8b51eb943d1109de013886a601438edb55be1f935b2505fd1cf47e1.jpg)  
+(k) Hopper-v2  $(U(1,64)$  Delays)
+
+![](images/1b26a23b03ef52cf3874df89a6ba9f1ac934299cabc45560532a298ee896a90a.jpg)  
+(1) Walker2d-v2  $(U(1,64)$  Delays)
+
+![](images/0bead70f19954898e15fbb145129fe4aed26b0c06518296ac9034d017917d551.jpg)  
+(m) HalfCheetah-v2  $(U(1,128)$  Delays)
+
+![](images/b5356e7082a866a125c22b719e6b9c9d22953c2b95c784a0ce15d84c1f85df41.jpg)  
+(n) Hopper-v2  $(U(1,128)$  Delays)
+
+![](images/cfcf46995a58a686e75c76d13e9501bd9dda6ffc1d73d5680e204f7143dcacde.jpg)  
+(o) Walker2d-v2  $(U(1,128)$  Delays)  
+Figure 5. Learning Curves on MuJoCo with Stochastic Delays.
+
+![](images/808046c0c9ae4b9d88679787b959cee276553630c395e326aceb40c78e6c366b.jpg)
+
+# E. Belief Qualitative Comparison
+
+We report the qualitative comparison of the beliefs on HalfCheetah-v2, Hopper-v2, and Walker2d-v2 in Figure 6, Figure 7, and Figure 8, respectively.
+
+![](images/fee2398569edc228bf342341f931fd0989418d7b86b61580294ed7242c9337b9.jpg)  
+(a) Truth (8 Delays)
+
+![](images/2d5324b157bb4e85d0ee7c50d592ff757e92d8fbf1cdc916e29cf279adbe2dc6.jpg)  
+(b) DATS (8 Delays)
+
+![](images/eb243f32c3d8b3bf68b7905f62154fe0e087885f27fc8eab3932708fd3af9505.jpg)  
+(c) D-Dreamer (8 Delays)
+
+![](images/56cf8d3c4164f3cb0f88856d35a630eb6a38fb3bcf5b58c15028f71560a55d53.jpg)  
+(d) D-SAC (8 Delays)
+
+![](images/3988a2216fd88c528b59a38a8e3840deb3a5db41c6181b298074dcf9c109d068.jpg)  
+(e) DFBT (8 Delays)
+
+![](images/36e9827c751e74175c1fa70d03665c8e66534039f17354d214350a06cb44b794.jpg)  
+(f) Truth (16 Delays)
+
+![](images/9405281ea2c7fce877388c685103f3fefb5a438cdbdfae9d2e6e20ac256347a3.jpg)  
+(g) DATS (16 Delays)
+
+![](images/e1435caa3a5f89362d743eb6864aa9773af6f2cc4333e893b886782969d34848.jpg)  
+(h) D-Dreamer (16 Delays)
+
+![](images/8d0928d320c60359f1aa36ec62f4c67d6e76ad59e049857b6ad77f4a133c94f3.jpg)  
+(i) D-SAC (16 Delays)
+
+![](images/41cb53a319ebbee7051193e182fecb034dd0eb643d0b274c43ff254b17e50e89.jpg)  
+(j) DFBT (16 Delays)
+
+![](images/e86be15189fe923e95ff50428a45d24a0a288ed67966ae48e51fc8834f2907bd.jpg)  
+(k) Truth (32 Delays)
+
+![](images/81e7243414dbc31aae6fd1f7ad51830d991c618820a088c963ed4231f2ae18c2.jpg)  
+(1) DATS (32 Delays)
+
+![](images/0324ff34d6b83c8e088aca04e8d9b7cdf23fe3e118f05b5d21b5e48954eaea9b.jpg)  
+(m) D-Dreamer (32 Delays)
+
+![](images/9e6064d073dce26770b8b2ec22166095269bb3ccf91ef076ba8439617bad40b4.jpg)  
+(n) D-SAC (32 Delays)
+
+![](images/b7bd0474f8231a78292683eeb18e38bbdf52105bdc71af2db191b011f5cbc648.jpg)  
+(o) DFBT (32 Delays)
+
+![](images/abab4cf3945e1140146c51c574e7cfb1a75ae9377503eef697eba259719b49ef.jpg)  
+(p) Truth (64 Delays)
+
+![](images/08b839afa3b85a00b444c9ec93aa8d20cba00a344b7bdd6f7ef4277c05361a45.jpg)  
+(q) DATS (64 Delays)
+
+![](images/6f6b7700b5f4a38b265e109a8771e48a6c3d93672927cb00fd2e2475665b7232.jpg)  
+(r) D-Dreamer (64 Delays)
+
+![](images/0109f2dde0a1ec98a842c6ef303002a6a23910316af7607e1f979e9eca6a0f8a.jpg)  
+(S) D-SAC (64 Delays)
+
+![](images/9de1ad897d907a0d0ebcdd97beef311fd616711851e872a8f4283610aba91884.jpg)  
+(t) DFBT (64 Delays)
+
+![](images/05d1cd0ac3481b201c200df42975374a154fffe4aec3385f79571a25bebccc3b.jpg)  
+(u) Truth (128 Delays)
+
+![](images/7f635bff6b24451158cf47e515b56c93e9a45bf2ab4394651c6e4da8879ff0b9.jpg)  
+(V) DATS (128 Delays)
+
+![](images/adba03ac11c26cc0a0f736e3602ec87f9a2d6c8262d7acb8fe21a16ce1a03900.jpg)  
+(W) D-Dreamer (128 Delays)
+
+![](images/93253ac2abdd8b61d98b1605c9f71b0dac7d580194eac61c8b2c881aa1004e48.jpg)  
+(X) D-SAC (128 Delays)
+
+![](images/9cfd8f58adc59a65d0b9c93e1f63c4dbfdb618d8526e1b164898f768669eb59a.jpg)  
+(y) DFBT (128 Delays)  
+Figure 6. Belief qualitative comparison on HalfCheetah-v2 with different delays.
+
+![](images/bbd4e3f63ee87107f779b7d1bd0a53cdd93437edd09162e6288ee11864ac2f26.jpg)  
+(a) Truth (8 Delays)
+
+![](images/4fb72a4b7493a743b5fe5367adcf4da6117769f9870e4576ac33bf9d8b0f8440.jpg)  
+(b) DATS (8 Delays)
+
+![](images/13e53c214aa1e17f15bc276dc39fdf693840501e6661e02639aa23ca729f3e8e.jpg)  
+(c) D-Dreamer (8 Delays)
+
+![](images/affa9cc4473c0171f6f99a756dd6ca36300119de031dc4413d8faa3b858359d6.jpg)  
+(d) D-SAC (8 Delays)
+
+![](images/a9f1b639ee06577741eb37c77904f5eb5fb077b730808c8bab18060b2860f655.jpg)  
+(e) DFBT (8 Delays)
+
+![](images/4655f263b2c66e5fb072b4fc824444d1db35abb79f4379913df6faecf30e4c42.jpg)  
+(f) Truth (16 Delays)
+
+![](images/62da7ad1e398c51855f17b332221c9f90d14b44080a480c6f46857b388d3bb0c.jpg)  
+(g) DATS (16 Delays)
+
+![](images/5f556f3a6737024f2d7b38f0b518affbbf5530d54bb348143321956e9f38848b.jpg)  
+(h) D-Dreamer (16 Delays)
+
+![](images/fcb0eb6c25d2eb0709d30e1a02cc174a0c75c143904ed30ec2881baf885b7545.jpg)  
+(i) D-SAC (16 Delays)
+
+![](images/fd6461602e8a9eb72725602238eb58020c587d08964aa9f8474f0d4338b7dc87.jpg)  
+(j) DFBT (16 Delays)
+
+![](images/9c1e907cc75018bccfa1b07b38aa9b6cfce2d837d2ebcb5647654194c27e6d02.jpg)  
+(k) Truth (32 Delays)
+
+![](images/079f9d3cc9877062035ebc8349ff7adde45a5ce68026098417740eab3bf42ff2.jpg)  
+(1) DATS (32 Delays)
+
+![](images/c9ea57eff1e38efb474d3e904ca01614152d3d41853ac57ac65d977475310aee.jpg)  
+(m) D-Dreamer (32 Delays)
+
+![](images/0098124bea67b6e0fc1509bedda964ffc7c70912e8219ed4e60e1dca0e432ec6.jpg)  
+(n) D-SAC (32 Delays)
+
+![](images/73149b59a499204f75059e8418a611b02e528be10ff589dd866b766f6a784c4c.jpg)  
+(0) DFBT (32 Delays)
+
+![](images/74067af2bbe5c506ea0f631e2abe3b031a642460073c38ffbdfe217e27ad8b3c.jpg)  
+(p) Truth (64 Delays)
+
+![](images/dc66172961066a56cabee5321600a003bb35034c176d387c4eb4b1ef8a83cfd4.jpg)  
+(q) DATS (64 Delays)
+
+![](images/92873404f9083d4b8899d727a6c7a1885c09d82fd2364eb6706d281107577ed9.jpg)  
+(r) D-Dreamer (64 Delays)
+
+![](images/18a79cea70d96500afe08adf84485b5f0f9b46a851ebe2d854e14faab0e58052.jpg)  
+(S) D-SAC (64 Delays)
+
+![](images/6d3d2fe4b505b88e68de0f85d92038049297a237c3712aae3b02d4412c909eba.jpg)  
+(t) DFBT (64 Delays)
+
+![](images/8f479320fba4e9f5137e89e0f9c8395aadcfce13e6967cbe1cadaa485babb108.jpg)  
+(u) Truth (128 Delays)
+
+![](images/b68d0588dccf45cf351721ae69087508b3ccbce56bc70dc38d053c7faa7fb819.jpg)  
+(V) DATS (128 Delays)
+
+![](images/f1345cfa0e19a62c7e592b4fdbe1b044f8b758b8da52001e49e1885d28054cae.jpg)  
+(W) D-Dreamer (128 Delays)
+
+![](images/2b46b87a8bf3dce974071b59dbc6d2f7f8a81aba28702a10a2fe2c33c6671fbf.jpg)  
+(X) D-SAC (128 Delays)
+
+![](images/e4dab5352b9f14db868734e9bb3f8d8850700332aec4a22b7a071a64e475e74f.jpg)  
+(y) DFBT (128 Delays)  
+Figure 7. Belief qualitative comparison on Hopper-v2 with different delays.
+
+![](images/15b8856b657f02341c3a29b71af1159b329d006a837d24a363325f8615029534.jpg)  
+(a) Truth (8 Delays)
+
+![](images/ec52cfc226c696eb11f7a84ecfec3baf471b7f85045b5bdd4870f8e725bbe6ea.jpg)  
+(b) DATS (8 Delays)
+
+![](images/8ca2bd1a0a0a0da8bb17cb9a04a30f3017cfac700cc9df25fc1e58937b65529f.jpg)  
+(c) D-Dreamer (8 Delays)
+
+![](images/83b392f9a2efc28f728e7496dadfb60720944a5d079131b466f9b68f1f07e914.jpg)  
+(d) D-SAC (8 Delays)
+
+![](images/51179739fa887b22b5def045cb34a1bfa9ff8593d43692ba2b99651c376f7cbe.jpg)  
+(e) DFBT (8 Delays)
+
+![](images/a8e203e0a6389dcc757269740833940559298d08b6f9875c8feb17793eefe571.jpg)  
+(f) Truth (16 Delays)
+
+![](images/d507024f37fa4571a2f2296e162fae313979e2f8d5c1a757f91450cf3b7d762c.jpg)  
+(g) DATS (16 Delays)
+
+![](images/35388b06b9c0f61f166f143da403ab0a362b23f8080431b62736c9971c35a1ff.jpg)  
+(h) D-Dreamer (16 Delays)
+
+![](images/007d5ba63763dffe395827fac5cb47fb1f01c1ecb8903f003431305cd7452836.jpg)  
+(i) D-SAC (16 Delays)
+
+![](images/56815446ae94ecc7eb6c941fcf21524ac2fa8cdbdd0cad3c7405395eb71361ed.jpg)  
+(j) DFBT (16 Delays)
+
+![](images/4849dda0d4e0f2e42b30701ab5f2058d29dc5856f0d56fe72f3aaa0aab020f1e.jpg)  
+(k) Truth (32 Delays)
+
+![](images/fb2c1fc2cee34ad1543ccf5d6fad2fd46b4b1edcff93d7608d4199a80c26f661.jpg)  
+(1) DATS (32 Delays)
+
+![](images/bd61756f6d51859524d196f162c7e4dc9ba4424b4fac2f668a734b04f37b3783.jpg)  
+(m) D-Dreamer (32 Delays)
+
+![](images/cbebc6dcb3d3d9a63f00e5ced1c0cd4a883f0d2817dce50b0491dc22fb1aa2d5.jpg)  
+(n) D-SAC (32 Delays)
+
+![](images/dcb7c4e9d085bfcfe50ef0c299fa32d6b1b84d53d28c2815933d96ee5cebbbc2.jpg)  
+(0) DFBT (32 Delays)
+
+![](images/762480dda715f4e73d8a8f285a607e6b5f957d4048faa4c7766b0c82d100e7be.jpg)  
+(p) Truth (64 Delays)
+
+![](images/4a1ef574dfc95dbe14fef267e4309a378b480e96301f4ed192ec232eb6d2eda3.jpg)  
+(q) DATS (64 Delays)
+
+![](images/f37e96978342e3b5d1663cdb888d68c8d499855014966643097f898879bcbcba.jpg)  
+(r) D-Dreamer (64 Delays)
+
+![](images/59a7f1f675548812edca6ab3c48b71be18fee9b074ee78b0561d62ac2f698bbb.jpg)  
+(S) D-SAC (64 Delays)
+
+![](images/7b39e3bd26dfc766e0bb6d849f36b0fa090d211e2a328140926cc213dc20f76f.jpg)  
+(t) DFBT (64 Delays)
+
+![](images/593d0632b2369315907e5038eb15ff7649d92855f560c65e12c556b3ef6711e8.jpg)  
+(u) Truth (128 Delays)
+
+![](images/efbe7cca54de43047388ffeed80f1872bfb3e90e663078f304154282e1229636.jpg)  
+(V) DATS (128 Delays)
+
+![](images/d739f1c8c66ccefb6d0fa521d5e052e1c8962c6361d52567f736c11cba19b0b2.jpg)  
+(W) D-Dreamer (128 Delays)
+
+![](images/2371c9a118495211cb92f6ec2c50f708d6fb74eaf9c778f5f3ccca4d29e3e4a5.jpg)  
+(X) D-SAC (128 Delays)
+
+![](images/410ac67835388d3cc92aa83260bfec1ece07330725fe88bc7d61b27757a61462.jpg)  
+(y) DFBT (128 Delays)  
+Figure 8. Belief qualitative comparison on Walker2d-v2 with different delays.

2025/Directly Forecasting Belief for Reinforcement Learning with Delays/images.zip

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2025/Discovering Global False Negatives On the Fly for Self-supervised Contrastive Learning/44963d82-7cf7-40b3-a145-20250e157f0f_origin.pdf

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2025/Discovering Global False Negatives On the Fly for Self-supervised Contrastive Learning/full.md

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+# Discovering Global False Negatives On the Fly for Self-supervised Contrastive Learning
+
+Vicente Balmaseda<sup>1</sup> Bokun Wang<sup>1</sup> Ching-Long Lin<sup>2</sup> Tianbao Yang<sup>1</sup>
+
+# Abstract
+
+In self-supervised contrastive learning, negative pairs are typically constructed using an anchor image and a sample drawn from the entire dataset, excluding the anchor. However, this approach can result in the creation of negative pairs with similar semantics, referred to as "false negatives", leading to their embeddings being falsely pushed apart. To address this issue, we introduce GLOFND, an optimization-based approach that automatically learns on the fly the threshold for each anchor data to identify its false negatives during training. In contrast to previous methods for false negative discovery, our approach globally detects false negatives across the entire dataset rather than locally within the mini-batch. Moreover, its per-iteration computation cost remains independent of the dataset size. Experimental results on image and image-text data demonstrate the effectiveness of the proposed method. Our implementation is available at "https://github.com/vibalcam/GloFND".
+
+# 1. Introduction
+
+Representation learning is a fundamental problem in machine learning that aims to learn a good representation of the data for downstream tasks. Conventional supervised approaches rely on large quantities of high-quality labeled data, which is hard to collect. Recently, self-supervised learning has achieved promising performance for image representation learning (Chen et al., 2020a; Grill et al., 2020). Its success extends to bimodal learning (Radford et al., 2021) and semi-supervised learning (Chen et al., 2020b). These methods exploit unlabeled data to acquire general-purpose representations that exhibit robust performance and transfer
+
+$^{1}$ Department of Computer Science and Engineering, Texas A&M University, College Station, USA  $^{2}$ . Correspondence to: Vicente Balmaseda <vibalcam@tamu.edu>.
+
+Proceedings of the  $42^{nd}$  International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
+
+Anchor False negatives Anchor False negatives
+
+![](images/aa90d0b3e763a4709a51f3d03f4e4282a2f677941f36c3262e5ac0e2319c541c.jpg)  
+Figure 1: Examples of false negative images seen during training on ImageNet. The left column is the anchor image and the rest are observed false negative samples.
+
+ability across diverse downstream tasks.
+
+Notably, many self-supervised learning approaches center around contrastive learning. Contrastive learning operates on a straightforward principle: it seeks to bring together the embeddings of positive (similar) pairs while simultaneously pushing apart those of negative (dissimilar) pairs. This principle is combined with well-chosen data augmentations to improve the model's invariance to non-semantic variations.
+
+In the absence of reliable labels to determine whether a pair of data is positive or negative, many methods resort to instance discrimination. To this end, positive pairs are defined as distinct augmented views of the anchor data, while negative pairs are generated by sampling from the whole dataset excluding the anchor data, irrespective of their semantics (Chen et al., 2020a; Yuan et al., 2022). However, negative pairs produced through this method lack reliability. Specifically, augmented views from images sharing similar semantic meanings are incorrectly deemed negative, leading to their embeddings being pushed apart. This inadvertently encourages the model to discard crucial semantic information. We term these undesirable negative pairs as false negatives (FN). Figure 1 illustrates examples of false negatives encountered during training on ImageNet, where the anchor images are different from their negative samples yet semantically similar.
+
+The presence of false negatives detrimentally impacts the representations learned through contrastive learning (Saunshi et al., 2019), with this effect becoming more pronounced in large-scale datasets featuring numerous semantic concepts (Chen et al., 2022). For instance, during Sog-CLR (Yuan et al., 2022) pretraining on ImageNet100 (100 classes), approximately  $1\%$  of all negative pairs are false
+
+![](images/ac9df205a70d9cc62797b37e6ba8948fb0a89b5e9f895895b91ca3262e083414.jpg)  
+(a) SogCLR
+
+![](images/88062f622db15be2018a7e46c1fe0e6c8553d112f89ed5e0de468d8caf8b2e9d.jpg)  
+(b) GLOFND (Ours)  
+Figure 2: ImageNet100 features (t-SNE projected) learned by SogCLR (Yuan et al., 2022) and GLOFND (this work).
+
+negatives. This translates to around 20,000 false negatives per batch with a batch size of 1024, and about 325 with a batch size of 128. The presence of such false negatives during training can significantly degrade the quality of learned representations: a linear classifier trained on top of such representations achieves up to  $10\%$  lower accuracy in a semi-supervised setting compared to representations learned without the false negatives.
+
+Given an approach to identify false negatives, we can take corrective actions such as filtering them from the training set (i.e., false negative elimination) or incorporating them as additional positive pairs (i.e., false negative attraction) (Huynh et al., 2022). However, confidently identifying the potential false negatives in the absence of labels poses a challenging problem. The desire to eliminate false negatives is motivated by the goal of improving representation learning, yet the identification of these instances may necessitate some level of semantic knowledge to determine whether two pairs are indeed negative. Looking at Figure 1, we can observe some of the false negatives are not straightforward to identify, especially after data augmentation.
+
+Previous works addressing this problem fall into two categories: local (batch-wise) and global (dataset-wise) approaches. Local methods (Zheng et al., 2021; Huynh et al., 2022) identify false negatives for an anchor by assessing its similarities or adjacency to other data within the same mini-batch. However, the most similar or adjacent item to the anchor in the mini-batch may not necessarily be semantically similar to the anchor in the entire data space, particularly when the mini-batch size is small. The global approach IFND (Chen et al., 2022) aims to discover false negatives for each anchor in the whole dataset. However, their method involves clustering the entire dataset at specific epochs, which could be computationally expensive for large-scale datasets.
+
+This paper addresses existing limitations in false negative detection by introducing a novel algorithm, named Global False Negative Discovery (GLOFND), which learns global and dynamic thresholds for each anchor in the dataset. This enables the selection of the top- $\alpha\%$  most similar negative
+
+data points from the entire dataset on the fly, with top- $\alpha\%$  being the set above the  $(1 - \alpha)$ -quantile ( $\alpha \in [0,1]$ ). The GLOFND algorithm alternates between two key steps: i) Updating the per-anchor thresholds by SGD to solve a convex optimization problem of finding a threshold that can filter out the top- $\alpha\%$  of a set of scores. ii) Updating the parameters of the encoder network by using a stochastic gradient estimator of the modified contrastive loss that takes care of the false negatives identified via the learned thresholds (e.g., excluding them).
+
+GLOFND can be integrated with various CL techniques with minimal computational overhead. It effectively identifies false negatives for each sample, offering flexibility in how these are addressed. We demonstrate the empirical success of our method in unimodal, bimodal, and semi-supervised contrastive learning on several CL techniques without using a large batch size. Figure 2(a) and 2(b) qualitatively showcases that identifying and removing false negatives using GLOFND achieves better separation between the learned representations of different classes. One example of this observation is that clusters close to the periphery appear more tightly packed and distinct.
+
+# 2. Related Work
+
+Self-supervised learning (SSL). SSL has garnered substantial attention for its capacity to generate general-purpose representations from unlabeled data, facilitating scalability to large-scale datasets (Gui et al., 2023). SSL learns a data encoder network by leveraging intrinsic relationships within the data. The encoder network is then used to learn predictive models in downstream tasks through transfer learning. Noteworthy applications span computer vision (Kolesnikov et al., 2019), natural language processing (Lan et al., 2019), and healthcare (Sowrirajan et al., 2021), among others.
+
+Early efforts in SSL formulated pretext tasks to enable models to learn representations from unlabeled data. Examples include predicting the relative offset between two patches within the same image (Doersch et al., 2016), solving jigsaw puzzles (Noroozi & Favaro, 2017), colorizing grayscale images (Zhang et al., 2016), and unsupervised deep clustering (Caron et al., 2019). However, these methods necessitate carefully crafted pretext tasks, which may not always apply to diverse domains, leading to a lack of generality.
+
+Contrastive learning (CL). CL has emerged as a prevalent paradigm in SSL, primarily grounded in instance discrimination (Wu et al., 2018; Zhao et al., 2021). This approach employs contrastive losses, compelling the model to bring the embedding vectors of positive pairs closer while simultaneously pushing those of negative pairs apart. In essence, it promotes the learning of representations with high similarity among positive pairs and low similarity among negative
+
+pairs. Positive pairs can be easily generated from different views of the same image. However, generating quality negative pairs is more challenging. MoCo (He et al., 2020) addresses this through (i) a momentum encoder network that generates representations of images for contrast with the anchor, and (ii) a long queue to provide a large number of negative samples. SimCLR (Chen et al., 2020a) instead uses a large batch size and data augmentations, generating negative pairs with augmented views of images other than the anchor image. However, its performance degrades a lot as the batch size decreases. To address this issue, Yuan et al. (2022) propose a stochastic algorithm called SogCLR that does not rely on large batch size. Ge et al. (2024) generates negatives that preserve superfluous instead of semantic features. Shah et al. (2022) introduced a max-margin criterion inspired by support vector machines (SVMs). While these methods have shown promising results, they overlook the semantic relationship when generating negative pairs. Despite two images being semantically similar, their augmented views are treated as negative pairs, a phenomenon referred to as "false negatives". The false negatives result in the loss of crucial semantic information, consequently impacting representation learning (Saunshi et al., 2019).
+
+Semantic-aware CL. Recently, several studies have enhanced instance-discrimination-based CL by leveraging the underlying semantics. Qiu et al. (2023) introduce the iSog-CLR algorithm that learns individualized temperatures for each sample depending on the frequency of its semantics to increase the tolerance of false negatives. In contrast, GLOFND tackles the more challenging task of detecting each sample's false negatives, thus providing more freedom on how to deal with them. Supervised CL (SupCon) (Khosla et al., 2021) has demonstrated that employing the CL objective with labels to define positive and negative pairs (i.e., avoiding false negatives) can be more effective than the conventional supervised cross-entropy loss. Weakly supervised CL (WCL) (Zheng et al., 2021) constructs an undirected graph based on auxiliary embeddings of minibatch data, whose connected components are used to define weak labels for the SupCon. Huynh et al. (2022) adopt a different approach called FNC, addressing false negatives for each anchor by selecting the top  $k$  similar samples in the batch. The limitation of the last two works is that the top similar negative samples in the batch may not be reliable false negatives when the mini-batch size is small.
+
+Instead of detecting false negatives within the mini-batch, Chen et al. (2022) introduce an incremental dataset-wide clustering-based approach. At specific epochs, embeddings are computed for all samples in the dataset, followed by clustering using  $k$ -means. Pairs of samples within the same cluster are designated as false negatives. Nevertheless, this approach entails high computational costs, particularly for large-scale datasets, due to the necessity of computing em
+
+beddings for the entire dataset and subsequently applying  $k$ -means clustering. Hence, there remains a need for a global (dataset-wise) false-negative discovery approach that is agnostic to batch size and scalable for large-scale datasets.
+
+# 3. CL with Global False Negative Identification
+
+Let  $\mathcal{D} = \{\mathbf{x}_1,\dots ,\mathbf{x}_n\}$  denote a dataset of size  $n$  and let  $\mathcal{P}$  be a collection of data augmentation operators.  $E_{\mathbf{w}}(\cdot)$  represents an encoder network parametrized by  $\mathbf{w}\in \mathbb{R}^d$
+
+The global contrastive objective (Yuan et al., 2022)  $\mathcal{L}_{\mathrm{GCL}}(\mathbf{w})$  contrasts each  $\mathbf{x}_i \in \mathcal{D}$  with negative data  $S_i^- = \{A(\mathbf{x}) \mid \forall A \in \mathcal{P}, \forall \mathbf{x} \in \mathcal{D} \backslash \{\mathbf{x}_i\}\}$  in the whole dataset. Let  $\mathbf{z}_i = E_{\mathbf{w}}(A(\mathbf{x}_i))$ ,  $\mathbf{z}_i' = E_{\mathbf{w}}(A'(x_i))$  denote the embeddings and  $\sin(\cdot, \cdot)$  the cosine similarity. Then,
+
+$$
+\mathcal {L} _ {\mathrm {G C L}} (\mathbf {w}) = \mathbf {E} _ {\mathbf {x} _ {i} \in \mathcal {D}, A, A ^ {\prime} \in \mathcal {P}} \left[ \ell (\mathbf {w}; \mathbf {x} _ {i}, A, A ^ {\prime}) \right], \tag {1}
+$$
+
+$$
+\ell (\mathbf {w}; \mathbf {x} _ {i}, A, A ^ {\prime}) = - \tau \log \frac {\exp \left(\sin \left(\mathbf {z} _ {i} , \mathbf {z} _ {i} ^ {\prime}\right) / \tau\right)}{\sum_ {\mathbf {x} \in S _ {i} ^ {-}} \exp \left(\sin \left(\mathbf {z} _ {i} , E _ {\mathbf {w}} (\mathbf {x})\right) / \tau\right)},
+$$
+
+While GLOFND is not restricted to any CL technique, the following sections present how GLOFND can be integrated with the global contrastive loss. GLOFND's application to other CL techniques can be done in a similar manner and we empirically show its effectiveness in Section 4. Since GLOFND's focus is on detecting false negatives, it does not restrict what actions to take with the detected false negatives. This paper will consider the straightforward approach of filtering them from the loss, leaving how to make the best use of false negatives for future work.
+
+# 3.1. Learning Dynamic Per-Anchors Thresholds
+
+The challenge in the false negative discovery problem appears akin to a chicken-and-egg dilemma: the reliable identification of false negatives demands good representations, yet achieving quality representation learning necessitates detecting and dealing with false negatives. Thus, our approach starts with a sufficiently pre-trained encoder network  $E_{\mathbf{w}}$  (this can be done with a warm-up stage using existing CL methods) and further refines it by systematically and dynamically eliminating the identified false negatives. Moreover, we assume that the top  $\alpha \%$  most similar negative data share similar semantics with the anchor data based on their current representations, where  $\alpha$  is a hyper-parameter to be tuned that allows adapting to different settings. We will identify as false negatives of the anchor  $\mathbf{x}_i$  the negative data with similarity scores above the  $(1 - \alpha)$ -quantile  $(\alpha \in [0,1])$ .
+
+The hyperparameter  $\alpha$  allows GLOFND to adapt to different definitions of false negatives, which are inherently dependent on the desired level of granularity. For example, in a dataset like ImageNet, consider two classification settings: (1) a coarse-grained task of classifying between cars and
+
+animals, and (2) fine-grained task of classifying dog breeds. Two images of a dog from different breeds might be considered a false negative in the coarse-grained case (1), but not in the fine-grained one (2). Consequently, the optimal percentage of false negatives, controlled by  $\alpha$ , varies with the chosen granularity. By adjusting  $\alpha$ , GLOFND can flexibly align with different levels of semantic resolution, i.e., granularity. Its value can be set based on prior knowledge (e.g., expected rate of false negatives or desired granularity of learned representations) or tuned like other hyperparameters, such as the temperature in contrastive learning.
+
+Previous work (Huynh et al., 2022) either selects the top  $k$  most similar negatives or sets a threshold on similarity scores. However, the former involves the expensive computation and ranking of cosine similarities across the entire dataset, while the latter presents challenges due to the need for manually crafted scheduling of the threshold for each anchor. Moreover, both approaches are problematic as similarity scores change when we update the parameters of the encoding network.
+
+Instead, we choose an optimization-based approach to automatically learn the per-anchor threshold  $\lambda_{i}$  to select the top  $\alpha \%$  most similar negative data for the  $i$ -th anchor. To achieve this, we cast the problem of finding the  $(1 - \alpha)$ -quantile of all similarity scores between  $\mathbf{x}_i$  and all other samples, i.e.,  $R_{i} = \{\mathrm{sim}(E_{\mathbf{w}}(\mathbf{x}_{i}),E_{\mathbf{w}}(\mathbf{x}))|\mathbf{x}\in S_{i}^{-}\}$  as the following optimization problem (Ogryczak & Tamir, 2003):
+
+$$
+\lambda_ {i} = \arg \min  _ {\nu \in [ - 1, 1 ]} \nu \alpha + \frac {1}{| R |} \sum_ {r \in R _ {i}} (r - \nu) _ {+} \tag {2}
+$$
+
+Then, we will have a set of threshold  $\{\lambda_i\}_{i\in [|\mathcal{D}|]}$ . The following lemma shows that the solution  $\lambda_{i}$  to (2) can be used to select the top- $\alpha \%$  most similar negative data.
+
+Lemma 3.1. Let  $k = \lceil \alpha |S_i^- |\rceil$ . The solution  $\lambda_{i}$  to (2) is either the  $k$ -th largest value or between  $k$ -th and  $(k + 1)$ -th largest value in the set  $R_{i}$ .
+
+We modify the contrastive loss in (1) to eliminate the false negatives, yielding the following optimization problem:
+
+$$
+\min  _ {\mathbf {w}} \mathcal {L} _ {\mathrm {G C L}} (\mathbf {w}, \boldsymbol {\lambda}) = \frac {1}{n} \sum_ {\mathbf {x} _ {i} \in \mathcal {D}} \mathbf {E} _ {A, A ^ {\prime}} [ \ell (\mathbf {w}, \lambda_ {i}; \mathbf {x} _ {i}) ], \tag {3}
+$$
+
+$$
+\ell (\mathbf {w}, \lambda_ {i}; \mathbf {x} _ {i}) = - \operatorname {s i m} \left(\mathbf {z} _ {i}, \mathbf {z} _ {i} ^ {\prime}\right) + \tau \log \left(\left| \widetilde {\mathcal {S}} _ {i} ^ {-} \right| g \left(\mathbf {w}, \lambda_ {i}; \mathbf {x} _ {i}, \widetilde {\mathcal {S}} _ {i} ^ {-}\right)\right),
+$$
+
+$$
+g (\mathbf {w}, \lambda_ {i}; \mathbf {x} _ {i}, \widetilde {\mathcal {S}} _ {i} ^ {-}) = \frac {1}{| \widetilde {\mathcal {S}} _ {i} ^ {-} |} \sum_ {\mathbf {x} \in \widetilde {\mathcal {S}} _ {i} ^ {-}} \exp (\mathrm {s i m} (\mathbf {z} _ {i}, E _ {\mathbf {w}} (\mathbf {x})) / \tau),
+$$
+
+where  $\widetilde{S}_i^- = \{\mathbf{x} \mid \mathbf{x} \in S_i^-, \mathrm{sim}(\mathbf{z}_i, E_{\mathbf{w}}(\mathbf{x})) \leq \lambda_i\}$  is obtained by removing the false negatives (identified via the threshold  $\lambda_i$ ) in the negative dataset  $S_i^-$  for anchor  $\mathbf{x}_i$ .
+
+Note that  $\min_{\mathbf{w}}\mathcal{L}_{\mathrm{GCL}}(\mathbf{w},\lambda)$  can be viewed as a stochastic bilevel optimization problem (Ghadimi & Wang, 2018)
+
+since the minimization of  $\mathcal{L}_{\mathrm{GCL}}(\mathbf{w},\boldsymbol {\lambda})$  involves the solution  $\pmb{\lambda}$  to a lower level problem in (2). However, the problem in (3) is more challenging than most bilevel problems in the literature (Ghadimi & Wang, 2018; Ji et al., 2021) because the lower-level problem in (2) is non-smooth and non-strongly convex while the upper-level function  $\mathcal{L}_{\mathrm{GCL}}(\mathbf{w},\boldsymbol {\lambda})$  is nondifferentiable to  $\pmb{\lambda}$ . To tackle this challenge, we just ignore the hypergradient of  $\pmb{\lambda}$  in terms of  $\mathbf{w}$ , which has been used in model-agnostic meta-learning (Finn et al., 2017).
+
+# 3.2. GLOFND for Unimodal CL
+
+We propose an efficient algorithm called GLOFND for dynamically discovering and eliminating the false negatives in contrastive learning. GLOFND can be combined with previous contrastive learning algorithms, e.g., SogCLR (Yuan et al., 2022). In each iteration,  $\mathrm{SogCLR} + \mathrm{GLOFND}$  first randomly samples a batch of data  $\mathcal{B} \subset \mathcal{D}$  and data augmentations  $A$ ,  $A'$ . Then, it alternatively executes two steps: (i) updating the thresholds  $\lambda_{i}, i \in \mathcal{B}$ ; (ii) removing the identified false negatives from the loss function and updating the parameters  $\mathbf{w}$  of the encoding network. Notably, step (i), GLOFND's computation of  $\lambda_{i}$ , is independent of the specific contrastive loss used, as it relies solely on the embedding similarity of negative pairs. The contrastive loss only influences how the filtered false negatives are handled during training.
+
+# 3.2.1. UPDATING THE THRESHOLD  $\lambda$
+
+First, the threshold  $\lambda_{i}$  can be updated by calculating the stochastic subgradient of (2) and employing the regular SGD update. Given the predetermined sampled negative data of  $\mathbf{x}_i$  in the mini-batch, i.e.,  $\mathcal{B}_i^- = \{A(\mathbf{x}), A'(\mathbf{x}) \mid \mathbf{x} \in \mathcal{B} \backslash \{\mathbf{x}_i\}\} \subset S_i^-$ , we can compute an stochastic estimator  $\widehat{\nabla}_{\lambda_i}$  of the subgradient of (2) w.r.t.  $\lambda_{i}$ . Then, we update those  $\lambda_{i}$ 's that correspond to those sampled anchors  $\mathbf{x}_i \in \mathcal{B}$  while keeping others unchanged, i.e.,
+
+$$
+\widehat {\nabla} _ {\lambda_ {i}} = \alpha - \frac {1}{| \mathcal {B} _ {i} ^ {-} |} \sum_ {\mathbf {x} \in \mathcal {B} _ {i} ^ {-}} \mathbb {I} (\operatorname {s i m} (\mathbf {z} _ {i}, E _ {\mathbf {w}} (\mathbf {x})) > \lambda_ {i})
+$$
+
+$$
+\lambda_ {i} \leftarrow \left\{ \begin{array}{l l} \Pi_ {[ - 1, 1 ]} \left[ \lambda_ {i} - \theta \widehat {\nabla} _ {\lambda_ {i}} \right], & \mathbf {x} _ {i} \in \mathcal {B}, \\ \lambda_ {i}, & \mathbf {x} _ {i} \notin \mathcal {B}, \end{array} \right. \tag {4}
+$$
+
+where  $\mathbb{I}(\cdot)$  is the indicator function,  $\Pi_{[-1,1]}\cdot$  denotes the projection onto the interval  $[-1,1]$  due to that are similarity scores are in  $[-1,1]$ , and  $\theta$  is the learning rate of  $\lambda_{i}$ . In this way, we keep track of a threshold  $\lambda_{i}$  for detecting global false negatives across the whole dataset for each anchor  $\mathbf{x}_i\in \mathcal{D}$ , while ensuring that computation remains minibatch-wise.
+
+# 3.2.2. UPDATING ENCODER NETWORK  $E_{\mathbf{w}}$
+
+For each anchor  $\mathbf{x}_i\in \mathcal{B}$ , we eliminate the false negatives identified through the threshold  $\lambda_{i}$  from its negative data batch  $\mathcal{B}_i^-$ , resulting in  $\tilde{\mathcal{B}}_i^- = \{\mathbf{x}|\mathbf{x}\in \mathcal{B}_i^-, \mathrm{sim}(\mathbf{z}_i,E_{\mathbf{w}}(\mathbf{x}))\leq \lambda_i\}$ . Following the SogCLR algorithm (Yuan et al., 2022), we use a moving average estimator of  $g(\mathbf{w},\lambda_i;\mathbf{x}_i,\widetilde{S}_i^-)$  for each anchor  $\mathbf{x}_i\in \mathcal{D}$  to alleviate the requirement of a large batch size. For each  $\mathbf{x}_i$ , we maintain a scalar  $u_{i}$  to estimate  $g(\mathbf{w},\lambda_i;\mathbf{x}_i,S_i^-)$  as
+
+$$
+u _ {i} \leftarrow \left\{ \begin{array}{l l} (1 - \gamma) u _ {i} + \gamma \widehat {g} (\mathbf {w}, \lambda_ {i}; \mathbf {x} _ {i}, \widetilde {\mathcal {B}} _ {i} ^ {-}), & \mathbf {x} _ {i} \in \mathcal {B}, \\ u _ {i}, & \mathbf {x} _ {i} \notin \mathcal {B}, \end{array} \right. \tag {5}
+$$
+
+$$
+\widehat {g} (\mathbf {w}, \lambda_ {i}; \mathbf {x} _ {i}, \widetilde {\mathcal {B}} _ {i} ^ {-}) = \frac {1}{| \widetilde {\mathcal {B}} _ {i} ^ {-} |} \sum_ {\mathbf {x} \in \widetilde {\mathcal {B}} _ {i} ^ {-}} \exp (\mathrm {s i m} (\mathbf {z} _ {i}, E _ {\mathbf {w}} (\mathbf {x})) / \tau),
+$$
+
+where  $\gamma \in [0,1]$  is the parameter and  $\widehat{g} (\mathbf{w},\lambda_i;\mathbf{x}_i,\widetilde{\mathcal{B}}_i^{-})$  is an stochastic estimator of  $g(\mathbf{w},\lambda_i;\mathbf{x}_i,\widetilde{\mathcal{S}}_i^{-})$ . Finally, we can update  $\mathbf{w}$  by computing a stochastic estimator of the gradient of  $\mathcal{L}_{\mathrm{GCL}}(\mathbf{w},\boldsymbol {\lambda})$  in (3) w.r.t. the parameters of encoding network  $\mathbf{w}$  as:
+
+$$
+\widehat {\nabla} _ {\mathbf {w}} = \frac {1}{| \mathcal {B} |} \sum_ {\mathbf {x} _ {i} \in \mathcal {B}} - \nabla_ {\mathbf {w}} \mathrm {s i m} (\mathbf {z} _ {i}, \mathbf {z} _ {i} ^ {\prime}) + \frac {\tau \nabla_ {\mathbf {w}} \widehat {g} (\mathbf {w} , \lambda_ {i} ; \mathbf {x} _ {i} , \widetilde {\mathcal {B}} _ {i} ^ {-})}{u _ {i}}.
+$$
+
+The whole algorithm, incorporating GLOFND to address the false negative issue in SogCLR through filtering, is outlined in Algorithm 1. Noteworthy differences compared to the vanilla SogCLR algorithm are highlighted in blue. Note that GLOFND adds little overhead, since it just requires basic matrix computations and runs in  $O(B^{2})$ , which is relatively negligible compared to the cosine similarity and forward/backward computations.
+
+Algorithm 1 SogCLR + GLOFND
+
+1: Initialize:  $\mathbf{w} \in \mathbb{R}^d$ , initialize  $\mathbf{u} \in \mathbb{R}^n$  and  $\lambda \in \mathbb{R}^n$  
+2: for  $t = 1, \dots, T$  do  
+3: Draw a batch of  $B$  samples  $\mathcal{B} \subset \mathcal{D}$  and data augmentations  $A, A'$ , and construct  $\mathcal{B}_i^- = \{A(\mathbf{x}), A'(\mathbf{x})\mid \mathbf{x} \in \mathcal{B} \setminus \{\mathbf{x}_i\}\}$  for each  $\mathbf{x}_i \in \mathcal{B}$  
+4: for  $\mathbf{x}_i\in \mathcal{B}$  do  
+5: Update  $\lambda_{i}$  according to (4)  
+6: Construct  $\mathcal{B}_i^-$  by excluding the false negatives identified via  $\lambda_i$  and compute  $\widehat{g}(\mathbf{w}; \mathbf{x}_i, A, \widetilde{\mathcal{B}}_i^-)$  
+7: Update  $u_{i,t}$  according to (5)  
+8: end for  
+9: Compute the gradient estimator  $\widehat{\nabla}_{\mathbf{w}}$  
+10: Update  $\mathbf{w}$  by the momentum or Adam method  
+11: end for
+
+# 3.3. Extension to Bimodal CL
+
+Our approach GLOFND can be extended to resolve the global false negative discovery in bimodal CL, e.g.,
+
+CLIP (Radford et al., 2021). This can be achieved by learning a threshold for each instance for two modalities and following the same general procedure as for unimodal CL. In this section, we provide a brief overview of GLOFND's extension to bimodal CL.
+
+Let  $\mathcal{D} = \{(\mathbf{x}_1,\mathbf{t}_1),\dots ,(\mathbf{x}_n,\mathbf{t}_n)\}$  be a set of image-text pairs, and denote the encoder network by  $E_{I}$  for images and the encoder network by  $E_{T}$  for text parametrized by w. For each  $(\mathbf{x}_i,\mathbf{t}_i)\in \mathcal{D}$  the negative dataset for each anchor image  $\mathbf{x}_i$  is  $S_{I,i}^{-} = \{\mathbf{t}\mid \forall (\mathbf{x},\mathbf{t})\in \mathcal{D}\setminus \{\mathbf{x}_i,t_i\} \}$  while the negative dataset for each anchor text  $\mathbf{t}_i$  is  $S_{T,i}^{-} = \{\mathbf{x}\mid \forall (\mathbf{x},\mathbf{t})\in \mathcal{D}\setminus \{\mathbf{x}_i,t_i\}\}$ . Let  $\mathbf{z}_{I,i}$  be the representation of the  $i$ -th anchor image  $\mathbf{x}_i$  and  $\mathbf{z}_{T,i}$  be the representation of the  $i$ -th anchor text  $\mathbf{t}_i$ , and  $\lambda_{I,i},\lambda_{T,i}\in [-1,1]$  their respective associated thresholds defined through (2). Then, the problem is formulated as:
+
+$$
+\min  _ {\mathbf {w}} \frac {1}{n} \sum_ {(\mathbf {x} _ {i}, \mathbf {t} _ {i}) \in \mathcal {D}} \mathbf {E} [ \ell (\mathbf {w}, \lambda_ {I, i}, \lambda_ {T, i}; \mathbf {x} _ {i}, \mathbf {t} _ {i}) ]
+$$
+
+$$
+\begin{array}{l} \ell (\mathbf {w}, \lambda_ {I, i}, \lambda_ {T, i}; \mathbf {x} _ {i}, \mathbf {t} _ {i}) = - 2 \operatorname {s i m} (\mathbf {z} _ {I, i}, \mathbf {z} _ {T, i}) \\ + \tau \log | \widetilde {\mathcal {S}} _ {I, i} ^ {-} | g _ {I} (\mathbf {w}, \lambda_ {I, i}; \mathbf {x} _ {i}, \widetilde {\mathcal {S}} _ {I, i} ^ {-}) \\ + \tau \log | \widetilde {\mathcal {S}} _ {T, i} ^ {-} | g _ {T} (\mathbf {w}, \lambda_ {T, i}; \mathbf {t} _ {i}, \widetilde {\mathcal {S}} _ {T, i} ^ {-}), \\ \end{array}
+$$
+
+$$
+g _ {I} (\mathbf {w}, \lambda_ {I, i}; \mathbf {x} _ {i}, \widetilde {\mathcal {S}} _ {I, i} ^ {-}) = \frac {1}{| \widetilde {\mathcal {S}} _ {I , i} ^ {-} |} \sum_ {\mathbf {t} \in \widetilde {\mathcal {S}} _ {I, i} ^ {-}} \exp (\mathrm {s i m} (\mathbf {z} _ {I, i}, E _ {T} (\mathbf {t})) / \tau),
+$$
+
+$$
+g _ {T} (\mathbf {w}, \lambda_ {T, i}; \mathbf {t} _ {i}, \widetilde {\mathcal {S}} _ {T, i} ^ {-}) = \frac {1}{| \widetilde {\mathcal {S}} _ {T , i} ^ {-} |} \sum_ {\mathbf {x} \in \widetilde {\mathcal {S}} _ {T, i} ^ {-}} \exp (\sin (\left. E _ {I} (\mathbf {x}), \mathbf {z} _ {T, i}\right) / \tau),
+$$
+
+where  $\widetilde{\mathcal{S}}_{I,i}^{-} = \{\mathbf{t}\mid \mathbf{t}\in \mathcal{S}_{I,i}^{-},\mathrm{sim}(\mathbf{z}_{I,i},E_{T}(\mathbf{t}))\leq \lambda_{I,i}\}$  and  $\widetilde{S}_{T,i}^{-} = \{\mathbf{x}\mid \mathbf{x}\in S_{T,i}^{-},\mathrm{sim}(E_{I}(\mathbf{x}),\mathbf{z}_{T,i})\leq \lambda_{T,i}\}$
+
+Then, we can easily extend the GLOFND to the bimodal setting similarly as in the unimodal setting. We refer readers to the Appendix B for more details.
+
+# 4. Experiments
+
+In this section, we evaluate GLOFND in unimodal, semi-supervised unimodal, and bimodal scenarios. It is not our focus to leverage multiple techniques for achieving state-of-the-art performance, but to showcase GLOFND's improvements in identifying false negatives across different settings while being scalable to large-scale datasets (with negligible overhead) and compatible with small batch sizes. Additionally, we perform an ablation study to analyze the effect of the different components of GLOFND. We report the score average and standard deviation in parenthesis over 3 runs with different random seeds.
+
+The unimodal experiments are run on a single NVIDIA A30 with 24GB memory size, while the bimodal experiments make use of a multi-node setup with 2 nodes, each with 2
+
+Table 1: Linear evaluation results in unimodal semi-supervised scenario. We train the linear classifiers with different percentages of randomly sampled labeled training data and present their top-1 accuracies  $(\%)$  on the validation set. We include the overall average and, in parentheses, its improvement WRT SogCLR baseline. We also report the average of recall, precision, and f1-score of the identified false negatives over the final epoch of pretraining.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="5">Top-1 Accuracy</td><td colspan="3">False Negatives Identification</td></tr><tr><td>100.0%</td><td>10.0%</td><td>1.0%</td><td>0.1%</td><td>Average</td><td>Precision</td><td>Recall</td><td>F1-Score</td></tr><tr><td>SogCLR</td><td>76.55 (0.09)</td><td>72.24 (0.14)</td><td>62.92 (0.34)</td><td>34.94 (0.60)</td><td>61.66</td><td>—</td><td>—</td><td>—</td></tr><tr><td>+FNC</td><td>77.12 (0.14)</td><td>72.89 (0.25)</td><td>64.29 (0.34)</td><td>36.11 (0.70)</td><td>62.60 (+0.94)</td><td>27.57 (0.03)</td><td>53.67 (0.27)</td><td>36.42 (0.07)</td></tr><tr><td>+GLOFND</td><td>77.59 (0.03)</td><td>73.36 (0.15)</td><td>65.09 (0.49)</td><td>37.38 (0.97)</td><td>63.36 (+1.70)</td><td>48.40 (0.65)</td><td>58.81 (0.60)</td><td>53.10 (0.31)</td></tr></table>
+
+Table 2: Unimodal transfer learning results. We report the overall average and its improvement WRT SogCLR baseline.  
+
+<table><tr><td>Method</td><td>CIFAR10</td><td>CIFAR100</td><td>Food101</td><td>Caltech101</td><td>Cars</td><td>DTD</td><td>Pets</td><td>Flowers</td><td>Average</td></tr><tr><td>SogCLR</td><td>82.46 (0.36)</td><td>60.2 (0.22)</td><td>59.66 (0.2)</td><td>77.73 (0.17)</td><td>25.99 (0.64)</td><td>57.80 (0.30)</td><td>60.63 (0.34)</td><td>76.91 (0.33)</td><td>62.67</td></tr><tr><td>+FNC</td><td>82.77 (0.36)</td><td>60.96 (0.31)</td><td>59.82 (0.14)</td><td>78.34 (0.92)</td><td>26.28 (0.50)</td><td>58.60 (0.29)</td><td>61.67 (0.66)</td><td>78.77 (0.51)</td><td>63.40 (+0.73)</td></tr><tr><td>+GloFND</td><td>82.81 (0.23)</td><td>61.94 (0.24)</td><td>59.87 (0.16)</td><td>79.18 (0.52)</td><td>27.88 (0.44)</td><td>58.97 (0.96)</td><td>63.91 (0.22)</td><td>79.89 (0.09)</td><td>64.31 (+1.64)</td></tr></table>
+
+# NVIDIA A100 GPUs with 40GB each.
+
+For unimodal and semi-supervised experiments, we use Sog-CLR (Yuan et al., 2022) and compare with FNC (Huynh et al., 2022). FNC computes the top  $k$  for negative data within a mini-batch by utilizing a support set. The support set includes additional views for each image, and the similarity scores are averaged across these views. For a fair comparison, we set  $k = \alpha |\mathcal{D}|$  and use a support set of size 1. For bimodal experiments, we compare GLOFND with SogCLR and FastCLIP (Wei et al., 2024).
+
+# 4.1. Unimodal and Semi-supervised Experiments
+
+Dataset. We run our experiments on ImageNet100 (Wu et al., 2019), a subset of ImageNet with 100 randomly selected classes (about 128k images), and report scores on its official validation split. Additionally, we examine the transfer learning performance on Food-101 (Bossard et al., 2014), CIFAR-10 and CIFAR-100 (Krizhevsky, 2009), Stanford Cars (Krause et al., 2013), Describable Textures Dataset (DTD) (Cimpoi et al., 2014), Oxford-IIIT Pets (Parkhi et al., 2012), Caltech-101 (Li et al., 2022), and Oxford 102 Flowers (Nilsback & Zisserman, 2008).
+
+Experiment Setup. Following previous work (Yuan et al., 2022), we pretrain ResNet-50 (He et al., 2015) with a 2-layer  $128 \times 128$  projection head on top of the backbone encoder. We pretrain for 200 epochs with a batch size of 128 and the same set of augmentations as in SogCLR. We use LARS optimizer (You et al., 2017) with square root learning rate scaling  $(0.075 \times \text{sqrt}(\text{BatchSize}))$  and cosine decay schedule without restart. For SogCLR, we set the temperature  $(\tau)$  to 0.1 and  $\gamma = 0.9$ . We start using GLOFND when we reach 70 epochs. We use  $\alpha = 0.01$ , initialize  $\lambda_i = 1$ , and learn it with Adam with a learning rate of 0.05 ( $\beta_1 = 0.9$ ,  $\beta_2 = 0.98$ ) during the remaining
+
+epochs. For FNC, we set  $\alpha = 0.01$  and tune the starting epoch in  $\{10,30,50,70,90,110,130\}$ , choosing the value that achieves the best semi-supervised average performance. More details on hyperparameters can be found in Appendix D.1.
+
+Evaluation. We evaluate our model in three ways: false negative identification, semi-supervised linear evaluation, and transfer learning. First, given that GLOFND's main objective is false negative identification, we assess its effectiveness to correctly detect false negatives. To construct the ground truth, we compare the labels of each sample pair, if both samples have the same label they are considered a false negative. We report precision, recall, and F1-scores for the final epoch of pretraining. Second, we evaluate GLOFND's ability to achieve better representations through linear evaluation. That is, we freeze the weights of the encoder at the last iteration of pretraining, remove its projection head, and train a linear classifier on top of the encoder's output. We follow a semi-supervised learning setup, where we use different fractions of labeled training data during linear evaluation, i.e., we train on random subsets of  $100\%$  (full dataset),  $10\%$ ,  $1\%$ , and  $0.1\%$  of the training data. We report each top-1 accuracy on the validation set and average the performance across percentages obtaining the overall semi-supervised score. Lastly, we evaluate the transfer learning performance of the learned representations. We train an  $\ell_2$ -regularized logistic regression classifier on features extracted from the frozen pretrained network after removing the projector head. For each method, we report linear evaluation and transfer learning results for the model that achieves the highest semi-supervised average performance.
+
+Results. We report false negative identification performance and top-1 accuracies by linear evaluation in Table 1. GLOFND achieves significant improvements in false negative identification over FNC, with a  $20.83\%$  and  $5.14\%$  in
+
+Table 3: Results in bimodal zero-shot downstream tasks. Datacomp provides the average across 38 tasks, Retrieval averages the performance on 3 image-text retrieval datasets, and IN & Variants averages 7 ImageNet datasets.  
+
+<table><tr><td>Method</td><td>Datacomp</td><td>Retrieval</td><td>IN &amp; Variants</td></tr><tr><td>SogCLR (Wei et al., 2024)</td><td>24.87 (0.13)</td><td>29.28 (0.30)</td><td>18.86 (0.09)</td></tr><tr><td>+ FNC</td><td>24.55 (0.20)</td><td>29.69 (0.19)</td><td>18.69 (0.62)</td></tr><tr><td>+ GLOFND</td><td>25.37 (0.16)</td><td>29.92 (0.35)</td><td>19.44 (0.16)</td></tr><tr><td>FastCLIP (Wei et al., 2024)</td><td>24.76 (0.26)</td><td>30.36 (0.18)</td><td>19.08 (0.16)</td></tr><tr><td>+ FNC</td><td>24.63 (0.72)</td><td>28.87 (0.93)</td><td>18.51 (0.19)</td></tr><tr><td>+ GLOFND</td><td>25.37 (0.13)</td><td>30.22 (0.32)</td><td>19.38 (0.15)</td></tr></table>
+
+crease in mean precision and recall, leading to an F1-score of  $53.10\%$ , which is  $16.68\%$  higher than FNC. Observe that simply removing the false negatives identified by FNC or GLOFND improves both the semi-supervised and transfer learning performance of SogCLR. GLOFND achieves greater improvements in both scenarios, achieving  $1.04\% - 2.44\%$  improvement in the semi-supervised scenario and an average  $1.64\%$  improvement in transfer learning, while increasing the per-epoch computation by only  $2\%$  (from 427 s to 435 s). More details can be found in Appendix D.3. Note these improvements are achieved by simply removing the false negatives identified by GLOFND from the loss, while a more careful treatment can potentially improve the performance even further.
+
+# 4.2. Bimodal Experiments
+
+Datasets. For bimodal learning, we use the Conceptual Captions 3M (CC3M) (Sharma et al., 2018) dataset. We evaluate the performance by leveraging the Datacomp Benchmark (Gadre et al., 2023), which includes 38 zero-shot downstream tasks. We report the average performance, named Datacomp. For each scenario, we select the model with the best Datacomp average and also report its average performance on two subsets of the tasks: zero-shot image classification on ImageNet-1k (Russakovsky et al., 2015) and 6 ImageNet distribution shift datasets (Wang et al., 2019; Recht et al., 2019; Hendrycks et al., 2021b;a; Barbu et al., 2019) (IN & Variants), and zero-shot cross-modal image-text retrieval on Flickr30K (Plummer et al., 2017), MSCOCO (Lin et al., 2015), and WinoGAViL (Bitton et al., 2022).
+
+Experiment Setup. Following previous work (Wei et al., 2024), we use a 12-layer transformer (Vaswani et al., 2017) as the text encoder, and ResNet50 as the vision encoder. All experiments are conducted in a multi-node setting with 2 nodes, each with two A100 40GB GPUs. We pretrain for 37 epochs with a global batch size of 1024. For SogCLR, we start using FNC/GLOFND after 15 epochs, setting  $\alpha = 5e - 4$ , while for FastCLIP we start after 20 epochs with  $\alpha = 1e - 3$ . For both losses, we initialize  $\lambda_{i} = 1$  and use Adam updates with a learning rate of 0.05. More details on hyperparameters can be found in Appendix D.2.
+
+![](images/7eef94ae2cc9b72347ca167c8d975ee074d3b9b05bd7f87a228bd564fabcb88d.jpg)  
+(a) Threshold distributions
+
+![](images/263aa4e387662d0f10ba61e67becc94b0b8e84b22362de5378e7bc2f860d66ab.jpg)  
+(b) % of predicted FN
+
+![](images/875d205282d33fb620a3c9efd71b9063f25cb42d791207bbce8c44ce8d93170b.jpg)  
+Figure 3: Analysis of learned  $\lambda_{i}$  for GLOFND with  $\alpha = 0.01$ . (a) Kernel density estimation of the distributions of GLOFND's  $\lambda_{i}$ , the approximated optimal  $\lambda_{i}^{a}$ , and 20 randomly sampled FNC thresholds. (b) Average percentage of negative pairs predicted to be false negatives during training (i.e.,  $1 - |\widetilde{S}_i^-| / |S_i^-|$ ).
+
+![](images/137941698db4148e623c7006cedc9d1251803d1e83f141d7274c5dd1c96c033e.jpg)  
+(b)  $10\%$  labeled
+
+![](images/5dbc47eebf84fdf3e436b7ea64e8fd6b41eb2be41d9ad538478b646ea90957e3.jpg)  
+(a)  $100\%$  labeled  
+(c)  $1\%$  labeled  
+Figure 4: Linear evaluation (top-1 accuracy %) on ImageNet100 for SogCLR without FN identification, with FNC (top  $1\%$ ), with a single learned threshold ( $\alpha = 0.1$ ), and GLOFND ( $\alpha = 0.01$ ).
+
+![](images/4e43bac6ab4c846ac138d6d9d61e29aeeb2a2ebabd5694474028b03e7d1a8cd0.jpg)  
+(d)  $0.1\%$  labeled
+
+Results. We present the bimodal results in Table 3. Despite using a larger batch compared to the unimodal case, the bimodal scenario proves more challenging for FNC, which generally underperforms compared to the baseline models. In contrast, GLOFND enhances both SogCLR and Fast-CLIP across most metrics. Notably, it improves the overall Datacomp score for both models. These results highlight the benefit of integrating false negative detection into bimodal contrastive losses, demonstrating that GLOFND is an effective approach for this task.
+
+# 4.3. Ablation Study
+
+Verification of Algorithm Design. We empirically validate three aspects of GLOFND's design: (i) the necessity to have a global threshold (as opposed to batch-wise), (ii) the necessity to have a different  $\lambda_{i}$  for each anchor  $\mathbf{x}_i\in \mathcal{D}$ , and (iii) the quality of the learned  $\lambda_{i}$  threshold. We will cover (ii) and (iii) in this section and (i) in the next section.
+
+(ii) Do we need a different threshold for each anchor?
+
+We first examine the distribution of  $\lambda_{i}$  learned by GLOFND after pretraining. Rather than being concentrated around a single value, we expect it to span a range, indicating that different anchors adopt different thresholds when computing their top  $(1 - \alpha)$ -quantile. Figure 3(a) illustrates this distribution for ImageNet100 (red line). As expected,  $\lambda_{i}$  varies within the range [0.1, 0.6], highlighting the necessity of a per-anchor threshold.
+
+To empirically validate this, we compare GLOFND, which assigns a distinct  $\lambda_{i}$  per anchor, against a variant that uses a single  $\lambda$  for all anchors, referred to as "Single  $\lambda$ ." Figure 4 reports the semi-supervised performance on ImageNet100, demonstrating that GLOFND with per-anchor  $\lambda_{i}$  consistently outperforms the single-threshold variant.
+
+(iii) How good are the learned  $\lambda_{i}$  thresholds?
+
+We train SogCLR on ImageNet100 and apply GLOFND  $(\alpha = 0.01)$  with SGD updates starting at epoch 20. We monitor the percentage of negative pairs predicted to be false negatives, computed as  $1 - |\tilde{S}_i^-| / |S_i^-|$ , throughout training. We initialize  $\lambda_{i} = 1$  (indicating no false negatives) and expect the percentage to converge to the target  $\alpha$ . As shown in Figure 3(b), GLOFND successfully reaches and maintains the desired  $\alpha$  after a few epochs.
+
+Next, we evaluate the error of the learned  $\lambda_{i}$  relative to its optimal value. Since computing the exact optimal  $\lambda_{i}$  is intractable, requiring similarity calculations for every anchor against all other samples under different augmentations, we approximate it instead. We freeze the network and estimate the optimal threshold  $\lambda_{i}^{a}$  by randomly selecting 100,000 samples per anchor. Empirically, GLOFND approximates the desired threshold significantly better than FNC with a batch size of 128, achieving a Mean Absolute Error (MAE) of 0.1 and a Root Mean Squared Error (RMSE) of 0.13 ( $\lambda_{i} \in [-1,1]$ ), whereas FNC obtains MAE and RMSE of 0.21 and 0.28, respectively. This means GLOFND has less than half the error of FNC. Qualitatively, Figure 3(a) compares the distribution of the learned  $\lambda_{i}$ , the approximated  $\lambda_{i}^{a}$ , and the thresholds used by FNC. Since FNC computes thresholds at the mini-batch level, we sample 20 random batches per anchor and plot their respective distributions. The results show that GLOFND learns a  $\lambda_{i}$  distribution that more closely aligns with the desired threshold than FNC.
+
+Impact of Starting Epoch. GLOFND requires a sufficiently pre-trained network to ensure that the similarity between embeddings reflects semantic similarity. This is achieved by applying GLOFND after a certain number of training epochs. However, the optimal start time involves a trade-off. If GLOFND is applied too early, the embedding space may not be well-formed, leading to incorrect classification of pairs as false negatives. Conversely, if applied too late, the
+
+![](images/4ee26cd2223a4ff0e34d913481bd70ff2f3160932ec55ca80617f5bfd1a513dd.jpg)  
+(a) Precision
+
+![](images/cbd56a1a5f0e4cdd92d820b0b4c106402cf30d16b636d7f90eb2605d35bbfab4.jpg)  
+(b) Recall
+
+![](images/8adab5050e7cdc30aabcd15ee51946bbf908063c3d6616e6643d061a2c9b861f.jpg)  
+(c) F1-Score
+
+![](images/f2667bfbcc8ce6b8e11314be7be81c1259332725c6f9112f302f10fffff1016a.jpg)  
+(d)  
+Figure 5: FNC and GLOFND comparison. (a): False negative prediction performance scores for ImageNet100 using the labels as ground truth. We report the per-epoch mean precision, recall, and f1-score for SogCLR + FNC and SogCLR + GLOFND ( $\alpha = 0.01$ ). (b): Starting epoch comparison on linear evaluation performance.
+
+potential benefits of GLOFND may be limited due to insufficient training time. To assess the necessity and impact of this "wait" period, we evaluate GLOFND's semi-supervised performance on ImageNet100 as we vary the start epoch in  $\{10, 30, 50, 70, 90, 110\}$ , keeping the total number of training epochs fixed at 200. The linear evaluation results, presented in Figure 5(d), show that GLOFND's performance improves as the wait time increases, peaking at 70 epochs. Beyond this point, performance begins to decline. FNC shows a similar trend, with its performance degrading when applied after 110 epochs.
+
+Computational Efficiency. When using GloFND with a contrastive method based on embedding similarity (e.g., SimCLR, SogCLR, and CLIP), pairwise similarities are already computed as part of the loss function. Thus, the only additional computations required are: (1) updating the  $\lambda_{i}$  values for the mini-batch samples, which involves a simple gradient computation (Equation 4), and (2) filtering the false negatives, which can be done through simple matrix operations. Both operations involve basic matrix computations and run in linear time with respect to the number of pairs in a batch ( $O(B^{2})$ , where  $B$  is the batch size). The computational overhead of GLOFND is minimal compared to the cost of cosine similarity computation and forward/backpropagation. Our experiments on ImageNet100 with a batch size of 128 show that GLOFND introduces only a  $2\%$  increase in per-epoch training time for SogCLR (435.19 s for SogCLR + GLOFND vs. 426.67 s for SogCLR). This overhead is comparable to that of the batch-wise method FNC (434.06 s).
+
+# 4.4. Comparison with Mini-batch Top- $k$  Method
+
+In this section, we assess the necessity of a global threshold by comparing GLOFND's global thresholds to FNC, which computes a threshold for each mini-batch.
+
+Semi-supervised Linear Evaluation. We present the semi-supervised linear evaluation performance for different percentages of labeled training data in Figure 4. Both GLOFND and FNC outperform not addressing false negatives, highlighting the importance of handling them. Furthermore, GLOFND consistently outperforms FNC across all settings, with improvements ranging from  $0.47\%$  to  $1.27\%$ . This demonstrates the advantage of using a global threshold, as opposed to a threshold specific to each mini-batch.
+
+Quality of Found False Negatives. We analyze the quality of the false negatives identified by GLOFND and FNC. In Section 4.3 (iii), we discussed how GLOFND matches more closely the optimal dataset-wide threshold than FNC, with half the approximation error. Here, we examine how this improved threshold alignment affects the quality of the false negatives identified. To do so, we calculate the per-epoch mean precision, recall, and F1-score for each method, using the class labels as ground truth (i.e., a pair is considered a false negative if both samples share the same label). As training progresses and the embedding space improves, we expect these metrics to increase, reflecting better alignment between embedding and semantic similarity. Furthermore, for GLOFND, we expect an increase in recall as  $\lambda_{i}$  reaches the desired quantile, capturing more false negatives. This should lead to a decrease in precision due to early representations not being sufficiently pretrained. After some oscillation, GLOFND should follow a steady upward trend.
+
+The results are presented in Figure 5. We observe that GLOFND behaves as expected, with the oscillations diminishing and becoming minimal around epoch 120. Regardless, GLOFND shows a  $14.89\%$  average improvement in F1-score, surpassing FNC for all but 4 epochs (indicated by the red area in Figure 5). Moreover, after epoch 120, GLOFND consistently maintains a mean F1-score between  $14.64\%$  and  $18.05\%$  higher than FNC. This underscores GLOFND's superiority in identifying false negatives.
+
+This is quantitatively illustrated in Figure 6, which shows examples of false negatives identified in a mini-batch by GLOFND and FNC. While the number of false negatives identified by FNC remains constant across mini-batches, GLOFND's dynamic threshold allows this number to vary, adapting to each mini-batch more effectively. For instance, in the second and third rows, FNC's fixed top- $k$  approach results in the selection of negative samples that are not sufficiently similar, leading to errors. In contrast, GLOFND is not constrained to a fixed number and instead selects only the most similar samples according to  $\lambda_{i}$ . The opposite
+
+![](images/c4404ea8d0a459024b90977454c3e3f9193a5d5cce78cfff2e2174274876cf52.jpg)  
+Figure 6: Examples of false negatives identified for ImageNet100 by GLOFND and FNC. The left column shows the anchor images, while the middle and right columns present the false negatives identified by GLOFND and FNC respectively.
+
+occurs in the first and last rows, where GLOFND identifies more false negatives than FNC.
+
+# 5. Conclusions
+
+In this work, we have addressed the problem of identifying global false negatives in self-supervised contrastive learning through an optimization-based approach. We propose identifying as false negatives for a given anchor those negative samples whose similarity exceeds the desired quantile across the entire dataset. We then introduce GLOFND, an optimization-based method that automatically learns a threshold for each anchor, enabling the identification of its false negatives on the fly. Experimental results demonstrate that GLOFND improves existing contrastive learning methods, both for unimodal and bimodal tasks, with minimal computational overhead. An open question is whether GLOFND could be extended to non-CL methods and whether the parameter  $\alpha$  could be individualized.
+
+Limitations. Since the focus of this paper is on false negative detection for contrastive learning, we address false negatives through filtering. While this straightforward approach has proven effective in our settings, future work could explore more advanced methods that may further enhance downstream performance. Additionally, the benefits of GLOFND and similar false-negative techniques on downstream tasks depend on the proportion of false negatives in the pretraining dataset and how false negatives are defined within the downstream task.
+
+# Acknowledgments
+
+VB, BW and TY were partially supported by National Science Foundation Award #2306572 and #2147253, National Institutes of Health Award #R01HL168116. CL was partially supported by National Institutes of Health Award #R01HL168116.
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
+
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+
+# A. Proof of Lemma 3.1
+
+Proof. For simplicity, we denote that  $n_i^- = |S_i^-|$  and  $s_j$  is the  $j$ -th largest value in  $\{\mathrm{sim}(\mathbf{z}_i, E_{\mathbf{w}}(\mathbf{x})) \mid \mathbf{x} \in S_i^-\}$ , i.e.,  $s_1 \geq s_2 \geq \ldots \geq s_{n_i^-}$ . A subgradient  $\phi_i'(\lambda_i)$  of the objective at  $\lambda_i$  in (2) is
+
+$$
+\phi_ {i} ^ {\prime} (\lambda_ {i}) = \alpha n _ {i} ^ {-} - \sum_ {j = 1} ^ {n _ {i} ^ {-}} \psi (s _ {j} - \lambda_ {i}), \quad \psi (s _ {j} - \lambda_ {i}) = \left\{ \begin{array}{l l} 1, & s _ {j} > \lambda_ {i} \\ \epsilon , & s _ {j} = \lambda_ {i} \\ 0, & s _ {j} <   \lambda_ {i}, \end{array} \right.
+$$
+
+where  $\epsilon \in [0,1]$ . We define  $k = \lceil \alpha n_{i}^{-}\rceil$ .  $0\in \partial \phi_{i}(\lambda_{i})$  only happens when  $\lambda_{i}\in [s_{k^{\prime \prime}},s_{k^{\prime}})$  since  $k - 1 < \alpha n_{i}^{-}\leq k$ , where  $k^{\prime} = \max \{j\mid s_{j} > s_{k},j > k\}$ ,  $k'' = \min \{k'''',k + 1\}$ ,  $k'''' = \max \{j\mid s_j < s_k,j > k\}$ . Thus,  $s_k$  is a solution to (2). Besides, when  $\alpha n_i^{-}$  is an integer (i.e.  $k = \alpha n_i^{-} = \lceil \alpha n_i^{-}\rceil$ ), any value between  $[s_k,s_{k + 1}]$  is also a solution to (2), which could be different from  $s_k$ .
+
+![](images/a7f3a67d7c85a046307e16fec044f0716b52fb3674979316adbe1174c7f03d51.jpg)
+
+# B. More Details on Extension to Bimodal CL
+
+Our approach GLOFND can be extended to resolve the global false negative discovery in bimodal CL, e.g., CLIP (Radford et al., 2021). Consider a dataset of image-text pairs  $\mathcal{D} = \{(\mathbf{x}_1,\mathbf{t}_1),\dots ,(\mathbf{x}_n,\mathbf{t}_n)\}$ , a collection of image augmentation operators  $\mathcal{P}_I$ , and a collection of text augmentation operators  $\mathcal{P}_T$ . Suppose that the encoder network  $E_{I}$  for images and the encoder network  $E_{T}$  for text are parametrized by  $\mathbf{w}$ . For each  $(\mathbf{x}_i,\mathbf{t}_i)\in \mathcal{D}$ , the negative dataset for each anchor image  $\mathbf{x}_i$  is  $\mathcal{S}_{I,i}^{-} = \{A_{T}(\mathbf{t})\mid \forall A_{T}\in \mathcal{P}_{T},\forall (\mathbf{x},\mathbf{t})\in \mathcal{D}\}$  while the negative dataset for each anchor text  $\mathbf{t}_i$  is  $\mathcal{S}_{T,i}^{-} = \{A_{I}(\mathbf{x})\mid \forall A_{I}\in \mathcal{P}_{I},\forall (\mathbf{x},\mathbf{t})\in \mathcal{D}\}$ . For the  $i$ -th anchor image  $\mathbf{x}_i$  with representation  $\mathbf{z}_{I,i}$ , the threshold  $\lambda_{I,i}\in [-1,1]$  for finding the top  $\alpha \%$  text neighbors among all negatives  $\mathcal{S}_{I,i}^{-}$  can be solved by
+
+$$
+\min_{\substack{\nu \in [-1,1]}}\phi_{I,i}(\nu),
+$$
+
+$$
+\phi_ {I, i} (\nu) = \nu \alpha + \frac {1}{| \mathcal {S} _ {I , i} ^ {-} |} \sum_ {\mathbf {t} \in \mathcal {S} _ {I, i} ^ {-}} \left(\mathrm {s i m} (\mathbf {z} _ {I, i}, E _ {T} (\mathbf {t})) - \nu\right) _ {+}.
+$$
+
+Similarly, we can obtain the threshold  $\lambda_{T,i} \in [-1,1]$  for the  $i$ -th anchor text  $\mathbf{t}_i$ . Given the thresholds  $\lambda_I, \lambda_T$ , the bimodal contrastive loss can be written as
+
+$$
+\mathcal {L} _ {\mathrm {B G C L}} (\mathbf {w}, \boldsymbol {\lambda} _ {I}, \boldsymbol {\lambda} _ {T}) = \frac {1}{n} \sum_ {(\mathbf {x} _ {i}, \mathbf {t} _ {i}) \in \mathcal {D}} \mathbf {E} [ \ell (\mathbf {w}, \boldsymbol {\lambda} _ {I, i}, \boldsymbol {\lambda} _ {T, i}; \mathbf {x} _ {i}, \mathbf {t} _ {i}) ],
+$$
+
+$$
+\begin{array}{l} \ell (\mathbf {w}, \lambda_ {I, i}, \lambda_ {T, i}; \mathbf {x} _ {i}, \mathbf {t} _ {i}) = - 2 \operatorname {s i m} (\mathbf {z} _ {I, i}, \mathbf {z} _ {T, i}) \\ + \tau \log | \mathcal {S} _ {I, i} ^ {-} | g _ {I} (\mathbf {w}, \lambda_ {I, i}; \mathbf {x} _ {i}, \widetilde {\mathcal {S}} _ {I, i} ^ {-}) \\ + \tau \log | \mathcal {S} _ {T, i} ^ {-} | g _ {T} (\mathbf {w}, \lambda_ {T, i}; \mathbf {t} _ {i}, \tilde {\mathcal {S}} _ {T, i} ^ {-}), \\ \end{array}
+$$
+
+$$
+g _ {I} (\mathbf {w}, \lambda_ {I, i}; \mathbf {x} _ {i}, \widetilde {\mathcal {S}} _ {I, i} ^ {-}) = \frac {1}{| \widetilde {\mathcal {S}} _ {I , i} ^ {-} |} \sum_ {\mathbf {t} \in \widetilde {\mathcal {S}} _ {I, i} ^ {-}} \exp (\operatorname {s i m} (\mathbf {z} _ {I, i}, E _ {T} (\mathbf {t})) / \tau),
+$$
+
+$$
+g _ {T} (\mathbf {w}, \lambda_ {T, i}; \mathbf {t} _ {i}, \widetilde {\mathcal {S}} _ {T, i} ^ {-}) = \frac {1}{| \widetilde {\mathcal {S}} _ {T , i} ^ {-} |} \sum_ {\mathbf {x} \in \widetilde {\mathcal {S}} _ {T, i} ^ {-}} \exp (\mathrm {s i m} (E _ {I} (\mathbf {x}), \mathbf {z} _ {T, i}) / \tau),
+$$
+
+where  $\widetilde{S}_{I,i}^{-} = \{\mathbf{t}\mid \mathbf{t}\in S_{I,i}^{-},\mathrm{sim}(\mathbf{z}_{I,i},E_{T}(\mathbf{t}))\leq \lambda_{I,i}\}$ ,  $\widetilde{S}_{T,i}^{-} = \{\mathbf{x}\mid \mathbf{x}\in S_{T,i}^{-},\mathrm{sim}(E_{I}(\mathbf{x}),\mathbf{z}_{T,i})\leq \lambda_{T,i}\}$  are the negative datasets for anchor  $(\mathbf{x}_i,\mathbf{t}_i)$  excluding the false negatives identified through the learned thresholds  $\lambda_{I,i},\lambda_{T,i}$ .
+
+We extend the GLOFND to the bimodal setting as follows. First, we sample a mini-batch of image-text pairs  $\mathcal{B} \subset \mathcal{D}$ , sampled image augmentations  $A_{I}$ , and text augmentations  $A_{T}$ , we construct the sampled negative sets  $\mathcal{B}_{I,i}^{-} = \{A_{T}(\mathbf{t}) \mid (\mathbf{x},\mathbf{t}) \in \mathcal{B} \setminus \{(\mathbf{x}_{i},\mathbf{t}_{i})\}\}$ ,  $\mathcal{B}_{T,i}^{-} = \{A_{I}(\mathbf{x}) \mid (\mathbf{x},\mathbf{t}) \in \mathcal{B} \setminus \{(\mathbf{x}_{i},\mathbf{t}_{i})\}\}$  for each  $(\mathbf{x}_{i},\mathbf{t}_{i}) \in \mathcal{B}$ . Given the image embedding  $\mathbf{z}_{I,i} = E_{I}(A_{I}(\mathbf{x}_{i}))$  and text embedding  $\mathbf{z}_{T,i} = E_{T}(A_{T}(\mathbf{t}_{i}))$  for anchor  $(\mathbf{x}_{i},\mathbf{t}_{i})$ , the thresholds  $\lambda_{I}$  for images can be
+
+updated by
+
+$$
+\widehat {\nabla} _ {\lambda_ {I, i}} = \alpha - \frac {1}{| \mathcal {B} _ {I , i} ^ {-} |} \sum_ {\mathbf {t} \in \mathcal {B} _ {I, i} ^ {-}} \mathbb {I} (\mathrm {s i m} (\mathbf {z} _ {I, i}, E _ {T} (\mathbf {t})) > \lambda_ {I, i}),
+$$
+
+$$
+\lambda_ {I, i} \leftarrow \left\{ \begin{array}{l l} \Pi_ {[ - 1, 1 ]} \left[ \lambda_ {I, i} - \theta \widehat {\nabla} _ {\lambda_ {I, i}} \right], & (\mathbf {x} _ {i}, \mathbf {t} _ {i}) \in \mathcal {B}, \\ \lambda_ {I, i}, & (\mathbf {x} _ {i}, \mathbf {t} _ {i}) \notin \mathcal {B}, \end{array} \right.
+$$
+
+Similarly, we can update the thresholds  $\lambda_{T}$  for texts. Given the thresholds  $\lambda_{I}$  for images and thresholds  $\lambda_{T}$  for texts, we can construct  $\widetilde{\mathcal{B}}_{I,i}^{-} = \{\mathbf{t} \mid \mathbf{t} \in \mathcal{B}_{I,i}^{-}, \mathrm{sim}(\mathbf{z}_{I,i}, E_{T}(\mathbf{t})) \leq \lambda_{I,i}\}$  and  $\widetilde{\mathcal{B}}_{T,i}^{-} = \{\mathbf{x} \mid \mathbf{x} \in \mathcal{B}_{T,i}^{-}, \mathrm{sim}(E_{I}(\mathbf{x}), \mathbf{z}_{T,i}) \leq \lambda_{T,i}\}$  by excluding the false negative images and texts identified via the thresholds  $\lambda_{I,i}$  and  $\lambda_{T,i}$ .
+
+Then, we employ the moving-average estimators  $u_{I,i}$ ,  $u_{T,i}$  for  $g_{I}(\mathbf{w},\lambda_{I,i};\mathbf{x}_{i},\widetilde{\mathcal{S}}_{I,i}^{-})$ ,  $g_{T}(\mathbf{w},\lambda_{T,i};\mathbf{t}_{i},\widetilde{\mathcal{S}}_{T,i}^{-})$ , respectively.
+
+$$
+\widehat {g} _ {I} (\mathbf {w}, \lambda_ {I, i}; \mathbf {x} _ {i}, \widetilde {\mathcal {B}} _ {I, i} ^ {-}) = \frac {1}{| \widetilde {\mathcal {B}} _ {I , i} ^ {-} |} \sum_ {\mathbf {t} \in \widetilde {\mathcal {B}} _ {I, i} ^ {-}} \exp (\sin (\mathbf {z} _ {I, i}, E _ {T} (\mathbf {t})) / \tau),
+$$
+
+$$
+\widehat {g} _ {T} (\mathbf {w}, \lambda_ {T, i}; \mathbf {t} _ {i}, \widetilde {\mathcal {B}} _ {T, i} ^ {-}) = \frac {1}{| \widetilde {\mathcal {B}} _ {T , i} ^ {-} |} \sum_ {\mathbf {x} \in \widetilde {\mathcal {B}} _ {T, i} ^ {-}} \exp (\mathrm {s i m} (E _ {I} (\mathbf {x}), \mathbf {z} _ {T, i}) / \tau),
+$$
+
+$$
+u _ {I, i} \leftarrow \left\{ \begin{array}{l l} (1 - \gamma) u _ {I, i} + \gamma \widehat {g} (\mathbf {w}, \lambda_ {I, i}; \mathbf {x} _ {i}, \widetilde {\mathcal {B}} _ {I, i} ^ {-}), & (\mathbf {x} _ {i}, \mathbf {t} _ {i}) \in \mathcal {B}, \\ u _ {I, i}, & (\mathbf {x} _ {i}, \mathbf {t} _ {i}) \notin \mathcal {B}, \end{array} \right.
+$$
+
+$$
+u _ {T, i} \leftarrow \left\{ \begin{array}{l l} (1 - \gamma) u _ {T, i} + \gamma \widehat {g} (\mathbf {w}, \lambda_ {T, i}; \mathbf {t} _ {i}, \widetilde {\mathcal {B}} _ {T, i} ^ {-}), & (\mathbf {x} _ {i}, \mathbf {t} _ {i}) \in \mathcal {B}, \\ u _ {T, i}, & (\mathbf {x} _ {i}, \mathbf {t} _ {i}) \notin \mathcal {B}. \end{array} \right.
+$$
+
+Finally, we can update the parameters  $\mathbf{w}$  for image-text encoder networks by the stochastic gradient estimator.
+
+$$
+\begin{array}{l} \widehat {\nabla} _ {\mathbf {w}} = \frac {1}{| \mathcal {B} |} \sum_ {(\mathbf {x} _ {i}, \mathbf {t} _ {i}) \in \mathcal {B}} \left[ - 2 \nabla_ {\mathbf {w}} \operatorname {s i m} (\mathbf {z} _ {I, i}, \mathbf {z} _ {T, i}) \right. \\ + \frac {\tau}{u _ {I , i}} \nabla_ {1} \widehat {g} _ {I} (\mathbf {w}, \lambda_ {I, i}; \mathbf {x} _ {i}, \widetilde {\mathcal {B}} _ {I, i} ^ {-}) \\ \left. + \frac {\tau}{u _ {T , i}} \nabla_ {1} \widehat {g} _ {T} (\mathbf {w}, \lambda_ {T, i}; \mathbf {t} _ {i}, \widetilde {\mathcal {B}} _ {T, i} ^ {-}) \right]. \\ \end{array}
+$$
+
+# C. High-level Intuition for  $\alpha$  Hyperparameter
+
+The hyperparameter  $\alpha$  allows GLOFND to adapt to different definitions of false negatives, which are inherently dependent on the desired level of granularity. For example, in a dataset like ImageNet, consider two classification settings: (1) a coarse-grained task of classifying between cars and animals, and (2) fine-grained task of classifying dog breeds. Two images of a dog from different breeds might be considered a false negative in the coarse-grained case (1), but not in the fine-grained one (2). Consequently, the optimal percentage of false negatives, controlled by  $\alpha$ , varies with the chosen granularity. By adjusting  $\alpha$ , GLOFND can flexibly align with different levels of semantic resolution, i.e., granularity.
+
+The value of  $\alpha$  can be set based on prior knowledge (e.g., expected rate of false negatives or desired granularity of learned representations) or tuned like other hyperparameters such as the temperature. If  $\alpha$  is too low, GLOFND may fail to identify sufficient false negatives, leading to minimal impact on the learned representations, though not degrading performance, as setting  $\alpha = 0$  is equivalent to disabling GLOFND. Conversely, if  $\alpha$  is too high, GLOFND may identify too many false negatives. If these are filtered out during training, the reduced number of negative pairs can limit contrastive learning, potentially harming performance. For tuning, it is recommended to start with a low  $\alpha$  and gradually increase it until no further performance gains are observed.
+
+# D. More Details on Experiments
+
+All the experiments are implemented using the PyTorch (Paszke et al., 2019) library. The unimodal experiments are run on a single NVIDIA A30 with 24GB memory size, while the bimodal experiments make use of a multi-node setup with 2 nodes, each with 2 NVIDIA A100 GPUs with 40GB each. The estimated amount of time to run a single experiment is 1.5 days for ImageNet100, and 14 hours for CC3M.
+
+# D.1. Additional Details for Unimodal Experiments
+
+Experiment Setup. Following prior work (Yuan et al., 2022), we pretrain a ResNet-50 (He et al., 2015) with a 2-layer  $128 \times 128$  projection head on top of the backbone encoder. We use square root learning rate scaling  $(0.075 \times \sqrt{\text{BatchSize}})$  with a cosine decay schedule without restart. Additionally, we apply a linear learning rate warm-up for 10 epochs, where the learning rate linearly increases to its maximum value.
+
+We adopt the same augmentation pipeline as in SogCLR (Yuan et al., 2022), utilizing the torchvision implementation. This includes RandomResizedCrop (resizing to  $224 \times 224$ ), random ColorJitter, RandomGrayscale, random GaussianBlur, RandomHorizontalFlip, and normalization using ImageNet statistics.
+
+The network is pretrained for 200 epochs with a batch size of 128. We use the LARS optimizer (You et al., 2017) with a weight decay of  $1e - 6$  and momentum of 0.9. The temperature  $(\tau)$  is set to 0.1, and for SogCLR,  $\gamma = 0.9$ . For GLOFND, we set  $\alpha = 0.01$  and tune the starting epoch from  $\{70,90,110\}$ , after which we begin updating  $\lambda_{i}$  with Adam updates using a learning rate of 0.05,  $\beta_{1} = 0.9$ , and  $\beta_{2} = 0.98$ . For FNC,  $\alpha = 0.01$  and the starting epoch is tuned from  $\{10,30,50,70,90,110,130\}$ , selecting the value that yields the best semi-supervised average performance.
+
+Linear Evaluation. We evaluate GLOFND's ability to produce better representations through linear evaluation. Specifically, we freeze the encoder's weights at the last iteration of pretraining, remove its projection head, and train a linear classifier (a single fully connected layer) on top of the encoder's output.
+
+Additionally, we employ a semi-supervised learning setup, using different fractions of labeled training data during linear evaluation. We train on random subsets of  $100\%$  (full dataset),  $10\%$ ,  $1\%$ , and  $0.1\%$  of the training data. For each fraction, we report the top-1 accuracy on the validation set and average the performance across the different percentages to obtain the overall semi-supervised performance.
+
+We train for 90, 285, 900, and 900 epochs corresponding to  $100\%$ ,  $10\%$ ,  $1\%$ , and  $0.1\%$  labeled data, respectively, with a batch size of 1024 and early stopping if the validation accuracy does not improve for 100 epochs. We use AdamW (Loshchilov & Hutter, 2019) with a weight decay of 0, momentum of 0.9, and a learning rate of 0.1. The same augmentation pipeline used in SogCLR is applied for linear evaluation. For training, we use RandomResizedCrop (resizing to  $224 \times 224$ ), RandomHorizontalFlip, and normalization. For testing, we resize the images to  $256 \times 256$ , apply CenterCrop to  $224 \times 224$ , and normalize.
+
+Transfer Learning Datasets. We additionally examine the transfer learning performance on Food-101 (Bossard et al., 2014), CIFAR-10 and CIFAR-100 (Krizhevsky, 2009), Stanford Cars (Krause et al., 2013), Describable Textures Dataset (DTD) (Cimpoi et al., 2014), Oxford-IIIT Pets (Parkhi et al., 2012), Caltech-101 (Li et al., 2022), and Oxford 102 Flowers (Nilsback & Zisserman, 2008). We follow the evaluation protocols in the papers introducing these datasets, i.e., we report top-1 accuracy for Food-101, CIFAR-10, CIFAR-100, Stanford Cars, and DTD; and mean per-class accuracy for Oxford-IIIT Pets, Caltech-101, and Oxford 102 Flowers. We report results on the test set and, for DTD, we report results only for the first split. Caltech-101 defines no train/test split, so we randomly select  $20\%$  of images per class to create the test set.
+
+Transfer Learning Evaluation. We train an  $\ell_2$ -regularized multinomial logistic regression classifier on features extracted from the frozen pretrained network after removing the projector head. For each method, we select the pretrained network that achieved the highest semi-supervised average performance, as used in the semi-supervised results.
+
+We employ L-BFGS and apply the same preprocessing as during validation in the linear evaluation setting: resizing to 256, center-cropping to 224, and normalizing. We report the best test performance across different  $\ell_2$  regularization parameters, selecting from a range of 10 logarithmically spaced values between  $10^{-6}$  and  $10^{5}$ .
+
+# D.2. Additional Details for Bimodal Experiments
+
+Datasets. For bimodal learning, we use the Conceptual Captions 3M (CC3M) (Sharma et al., 2018) dataset. Because some links have expired, our downloaded training set of CC3M contains 2, 723, 840 image-text pairs. We evaluate the performance by leveraging the Datacomp Benchmark (Gadre et al., 2023), which includes 38 zero-shot downstream tasks. We report the average performance, named Datacomp. For each scenario, we select the model with the best Datacomp average and also report its average performance on two subsets of the tasks: zero-shot image classification on ImageNet and its different variants (IN & Variants), and zero-shot cross-modal image-text retrieval. IN & Variants includes ImageNet-1k (Russakovsky et al., 2015) and 6 ImageNet distribution shift datasets (i.e., ImageNet-Sketch (Wang et al., 2019), ImageNet-V2 (Recht et al., 2019), ImageNet-A (Hendrycks et al., 2021b), ImageNet-O (Hendrycks et al., 2021b), ImageNet-R (Hendrycks et al., 2021a), and ObjectNet (Barbu et al., 2019)). Retrieval tasks consist of Flickr30K (Plummer et al., 2017), MSCOCO (Lin et al., 2015), and WinoGAViL (Bitton et al., 2022).
+
+Experiment Setup. Following previous work (Wei et al., 2024), we use a 12-layer transformer (Vaswani et al., 2017) as the text encoder and ResNet50 as the vision encoder. All experiments are conducted in a multi-node setting with 2 nodes, each equipped with two A100 40GB GPUs. We pretrain for 37 epochs with a global batch size of 1024. The AdamW (Loshchilov & Hutter, 2019) optimizer is used with  $(\beta_{1},\beta_{2}) = (0.9,0.999)$ ,  $\epsilon = 1e - 8$ , and a learning rate of  $1e - 3$ . A weight decay of 0.1 is applied, with a warm-up period of 10k steps. The learning rate follows a cosine schedule, initially increasing linearly during the warm-up phase and then decreasing according to a cosine function. A cosine  $\gamma$  schedule is employed, with a minimum  $\gamma$  of 0.2 and decay epochs set to 18.
+
+For SogCLR, the temperature parameter is set to 0.03. In FastCLIP, we set the initial temperature parameter to 0.07,  $\rho$  to 6.5, and the learning rate for  $\tau$  to  $2e - 4$ . Additionally, the learning rate of  $\tau$  decays to one-third of its original value when  $\tau$  falls below 0.03. The complete set of hyperparameters is summarized in Table 4. We tune  $\alpha \in \{5e - 4,1e - 3\}$  and the starting epoch in  $\{15,20\}$ . For SogCLR, we start using FNC/GLOFND after 15 epochs with  $\alpha = 5e - 4$ , while for FastCLIP we start after 20 epochs with  $\alpha = 1e - 3$ . For both losses, we initialize  $\lambda_{i} = 1$  and use Adam updates with a learning rate of 0.05.
+
+Table 4: Hyperparameters for FastCLIP Training  
+
+<table><tr><td>Hyperparameter</td><td>CC3M</td></tr><tr><td>Optimizer</td><td>AdamW</td></tr><tr><td>β1, β2</td><td>(0.9, 0.999)</td></tr><tr><td>ε</td><td>1e-8</td></tr><tr><td>Learning rate</td><td>1e-3</td></tr><tr><td>Weight decay</td><td>0.1</td></tr><tr><td>Warm-up steps</td><td>10k</td></tr><tr><td>Cosine γ min</td><td>0.2</td></tr><tr><td>Decay epochs</td><td>18</td></tr><tr><td>Temperature (SogCLR)</td><td>0.03</td></tr><tr><td>Initial temperature (FastCLIP)</td><td>0.07</td></tr><tr><td>ρ (FastCLIP)</td><td>6.5</td></tr><tr><td>Learning rate of τ (FastCLIP)</td><td>2e-4</td></tr></table>
+
+# D.3. Additional Experimental Results
+
+# Statistical Significance.
+
+We check for statistical significance between using the false negative approaches against not using them, which we consider the baseline. Thus, we compute p-values via a paired t-test between SogCLR + FNC/GLOFND and SogCLR baseline across multiple runs, with the alternative hypothesis testing for performance greater than the baseline.
+
+We report in Tables 5 and 6 the p-values with respect to the baseline for the unimodal semi-supervised and transfer learning experiments respectively, and consider a standard significance level of  $5\%$ . For the semi-supervised scenario, we can observe GLOFND achieves a p-value below 0.018 on all scenarios, thus achieving statistically significant improvements in all scenarios. Moreover, GLOFND achieves statistical significance below the  $1\%$  level on both the  $100\%$  and  $1\%$  scenarios.
+
+For the transfer learning case, GLOFND achieves statistically significant improvements on 6 out of 8 datasets, while FNC only achieves it on 2 of 8.
+
+Table 5: Unimodal semi-supervised linear evaluation p-values WRT the baseline (SogCLR). We color red those above the 0.05 threshold. The p-values are calculated via a paired t-test across multiple runs with the alternative hypothesis testing for performance greater than the baseline.
+
+<table><tr><td>Method</td><td>100.0%</td><td>10.0%</td><td>1.0%</td><td>0.1%</td></tr><tr><td>SogCLR + FNC</td><td>0.011</td><td>0.035</td><td>0.014</td><td>0.024</td></tr><tr><td>SogCLR + GLOFND</td><td>0.002</td><td>0.011</td><td>&lt;0.001</td><td>0.018</td></tr></table>
+
+Table 6: Unimodal transfer learning evaluation p-values WRT the baseline (SogCLR). We color red those above the 0.05 threshold. The p-values are calculated via a paired t-test across multiple runs with the alternative hypothesis testing for performance greater than the baseline.  
+
+<table><tr><td>Method</td><td>CIFAR10</td><td>CIFAR100</td><td>Food101</td><td>Caltech101</td><td>Cars</td><td>DTD</td><td>Pets</td><td>Flowers</td></tr><tr><td>SogCLR + FNC</td><td>0.158</td><td>0.056</td><td>0.219</td><td>0.249</td><td>0.235</td><td>0.080</td><td>0.032</td><td>0.036</td></tr><tr><td>SogCLR + GLOFND</td><td>0.143</td><td>0.005</td><td>0.021</td><td>0.024</td><td>0.006</td><td>0.123</td><td>0.002</td><td>0.005</td></tr></table>
+
+# SimCLR Experiment.
+
+We evaluate the performance of GLOFND in conjunction with SimCLR. Unlike SogCLR, SimCLR identifies false negatives only within each mini-batch, necessitating the use of a larger batch size. Table 7 presents the linear evaluation results for SimCLR under the experimental setup described in Appendix D.1, using a batch size of 512.
+
+Table 7: Linear evaluation results in unimodal semi-supervised scenario. We train the linear classifiers with different percentages of randomly sampled labeled training data and present their top-1 accuracies  $(\%)$  on the validation set. We include the overall average and, in parentheses, its improvement WRT SimCLR baseline.  
+
+<table><tr><td>Method</td><td>100.0%</td><td>10.0%</td><td>1.0%</td><td>0.1%</td><td>Average</td></tr><tr><td>SimCLR</td><td>76.88</td><td>73.38</td><td>66.40</td><td>33.56</td><td>62.56</td></tr><tr><td>+FNC</td><td>76.90</td><td>73.10</td><td>64.88</td><td>34.20</td><td>62.27</td></tr><tr><td>+GloFND</td><td>77.14</td><td>73.66</td><td>66.50</td><td>35.58</td><td>63.22</td></tr></table>
+
+# CLIP fine-tuning.
+
+To evaluate the effectiveness of GLOFND in fine-tuning large pretrained models to mitigate the impact of false negatives, we fine-tune OpenAI's ResNet-50 CLIP model on CC3M. The experimental setup closely follows Appendix D.2, with the key difference being that we initialize from OpenAI's pretrained weights and train for 15 epochs, using FNC/GLOFND after the first epoch  $(\alpha = 10^{-4})$ . Validation image-text retrieval results are reported in Table 8.
+
+Table 8: We fine-tune OpenAI's ResNet-50 CLIP model on CC3M and report image-text retrieval results on its validation set.  
+
+<table><tr><td>Method</td><td>IR@1</td><td>IR@5</td><td>IR@10</td><td>IR Average</td><td>TR@1</td><td>TR@5</td><td>TR@10</td><td>TR Average</td></tr><tr><td>Base</td><td>27.58</td><td>48.17</td><td>57.53</td><td>44.43</td><td>26.30</td><td>48.14</td><td>57.69</td><td>44.04</td></tr><tr><td>GloFND</td><td>36.07</td><td>58.94</td><td>67.22</td><td>54.08</td><td>35.76</td><td>59.19</td><td>67.44</td><td>54.13</td></tr><tr><td>+ FNC</td><td>33.69</td><td>58.01</td><td>67.51</td><td>53.07</td><td>33.61</td><td>57.95</td><td>67.53</td><td>53.03</td></tr><tr><td>+ GloFND</td><td>36.52</td><td>59.44</td><td>68.02</td><td>54.66</td><td>35.71</td><td>59.27</td><td>67.97</td><td>54.32</td></tr></table>
+
+# More examples of false negatives identified by GLOFND.
+
+Figure 7 shows examples of false negatives identified by GLOFND for ImageNet100 with  $\alpha = 0.01$  during training. We can observe that the number of false negatives identified is not constant for all anchors since we are using a dynamic threshold for each anchor, as opposed to a mini-batch top k approach. Moreover, the false negatives identified by GLOFND are
+
+![](images/f043d5d9310c72f7babc4c63b7d3583c4564a575e956f4fd3747a9999f775c94.jpg)  
+Figure 7: Examples of false negatives identified for ImageNet100 by GLOFND with  $\alpha = 1\%$ . The left column is the anchor image and the rest are identified false negative samples.
+
+semantically similar to the anchor image, which is what we aim to achieve.

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