Daily Papers

Off-policy actor-critic methods in reinforcement learning train a critic with temporal-difference updates and use it as a learning signal for the policy (actor). This design typically achieves higher sample efficiency than purely on-policy methods. However, critic networks tend to overestimate value estimates systematically. This is often addressed by introducing a pessimistic bias based on uncertainty estimates. Current methods employ ensembling to quantify the critic's epistemic uncertainty-uncertainty due to limited data and model ambiguity-to scale pessimistic updates. In this work, we propose a new algorithm called Stochastic Actor-Critic (STAC) that incorporates temporal (one-step) aleatoric uncertainty-uncertainty arising from stochastic transitions, rewards, and policy-induced variability in Bellman targets-to scale pessimistic bias in temporal-difference updates, rather than relying on epistemic uncertainty. STAC uses a single distributional critic network to model the temporal return uncertainty, and applies dropout to both the critic and actor networks for regularization. Our results show that pessimism based on a distributional critic alone suffices to mitigate overestimation, and naturally leads to risk-averse behavior in stochastic environments. Introducing dropout further improves training stability and performance by means of regularization. With this design, STAC achieves improved computational efficiency using a single distributional critic network.

To improve the sample efficiency of policy-gradient based reinforcement learning algorithms, we propose implicit distributional actor-critic (IDAC) that consists of a distributional critic, built on two deep generator networks (DGNs), and a semi-implicit actor (SIA), powered by a flexible policy distribution. We adopt a distributional perspective on the discounted cumulative return and model it with a state-action-dependent implicit distribution, which is approximated by the DGNs that take state-action pairs and random noises as their input. Moreover, we use the SIA to provide a semi-implicit policy distribution, which mixes the policy parameters with a reparameterizable distribution that is not constrained by an analytic density function. In this way, the policy's marginal distribution is implicit, providing the potential to model complex properties such as covariance structure and skewness, but its parameter and entropy can still be estimated. We incorporate these features with an off-policy algorithm framework to solve problems with continuous action space and compare IDAC with state-of-the-art algorithms on representative OpenAI Gym environments. We observe that IDAC outperforms these baselines in most tasks. Python code is provided.

Reinforcement learning (RL) has driven significant progress in robotics, but its complexity and long training times remain major bottlenecks. In this report, we introduce FastTD3, a simple, fast, and capable RL algorithm that significantly speeds up training for humanoid robots in popular suites such as HumanoidBench, IsaacLab, and MuJoCo Playground. Our recipe is remarkably simple: we train an off-policy TD3 agent with several modifications -- parallel simulation, large-batch updates, a distributional critic, and carefully tuned hyperparameters. FastTD3 solves a range of HumanoidBench tasks in under 3 hours on a single A100 GPU, while remaining stable during training. We also provide a lightweight and easy-to-use implementation of FastTD3 to accelerate RL research in robotics.

The overestimation bias is one of the major impediments to accurate off-policy learning. This paper investigates a novel way to alleviate the overestimation bias in a continuous control setting. Our method---Truncated Quantile Critics, TQC,---blends three ideas: distributional representation of a critic, truncation of critics prediction, and ensembling of multiple critics. Distributional representation and truncation allow for arbitrary granular overestimation control, while ensembling provides additional score improvements. TQC outperforms the current state of the art on all environments from the continuous control benchmark suite, demonstrating 25% improvement on the most challenging Humanoid environment.

Multilingual Retrieval-Augmented Generation (mRAG) leverages cross-lingual evidence to ground Large Language Models (LLMs) in global knowledge. However, we show that current mRAG systems suffer from a language bias during reranking, systematically favoring English and the query's native language. By introducing an estimated oracle evidence analysis, we quantify a substantial performance gap between existing rerankers and the achievable upper bound. Further analysis reveals a critical distributional mismatch: while optimal predictions require evidence scattered across multiple languages, current systems systematically suppress such ``answer-critical'' documents, thereby limiting downstream generation performance. To bridge this gap, we propose \textbf{Language-Agnostic Utility-driven Reranker Alignment (LAURA)}, which aligns multilingual evidence ranking with downstream generative utility. Experiments across diverse languages and generation models show that LAURA effectively mitigates language bias and consistently improves mRAG performance.

Reinforcement learning (RL) has shown remarkable success in solving complex decision-making and control tasks. However, many model-free RL algorithms experience performance degradation due to inaccurate value estimation, particularly the overestimation of Q-values, which can lead to suboptimal policies. To address this issue, we previously proposed the Distributional Soft Actor-Critic (DSAC or DSACv1), an off-policy RL algorithm that enhances value estimation accuracy by learning a continuous Gaussian value distribution. Despite its effectiveness, DSACv1 faces challenges such as training instability and sensitivity to reward scaling, caused by high variance in critic gradients due to return randomness. In this paper, we introduce three key refinements to DSACv1 to overcome these limitations and further improve Q-value estimation accuracy: expected value substitution, twin value distribution learning, and variance-based critic gradient adjustment. The enhanced algorithm, termed DSAC with Three refinements (DSAC-T or DSACv2), is systematically evaluated across a diverse set of benchmark tasks. Without the need for task-specific hyperparameter tuning, DSAC-T consistently matches or outperforms leading model-free RL algorithms, including SAC, TD3, DDPG, TRPO, and PPO, in all tested environments. Additionally, DSAC-T ensures a stable learning process and maintains robust performance across varying reward scales. Its effectiveness is further demonstrated through real-world application in controlling a wheeled robot, highlighting its potential for deployment in practical robotic tasks.

In reinforcement learning (RL), function approximation errors are known to easily lead to the Q-value overestimations, thus greatly reducing policy performance. This paper presents a distributional soft actor-critic (DSAC) algorithm, which is an off-policy RL method for continuous control setting, to improve the policy performance by mitigating Q-value overestimations. We first discover in theory that learning a distribution function of state-action returns can effectively mitigate Q-value overestimations because it is capable of adaptively adjusting the update stepsize of the Q-value function. Then, a distributional soft policy iteration (DSPI) framework is developed by embedding the return distribution function into maximum entropy RL. Finally, we present a deep off-policy actor-critic variant of DSPI, called DSAC, which directly learns a continuous return distribution by keeping the variance of the state-action returns within a reasonable range to address exploding and vanishing gradient problems. We evaluate DSAC on the suite of MuJoCo continuous control tasks, achieving the state-of-the-art performance.

Growth in the penetration of renewable energy sources makes supply more uncertain and leads to an increase in the system imbalance. This trend, together with the single imbalance pricing, opens an opportunity for balance responsible parties (BRPs) to perform energy arbitrage in the imbalance settlement mechanism. To this end, we propose a battery control framework based on distributional reinforcement learning (DRL). Our proposed control framework takes a risk-sensitive perspective, allowing BRPs to adjust their risk preferences: we aim to optimize a weighted sum of the arbitrage profit and a risk measure while constraining the daily number of cycles for the battery. We assess the performance of our proposed control framework using the Belgian imbalance prices of 2022 and compare two state-of-the-art RL methods, deep Q learning and soft actor-critic. Results reveal that the distributional soft actor-critic method can outperform other methods. Moreover, we note that our fully risk-averse agent appropriately learns to hedge against the risk related to the unknown imbalance price by (dis)charging the battery only when the agent is more certain about the price.

Reinforcement learning with verifiable rewards (RLVR) has significantly improved reasoning in large language models (LLMs), yet the token-level mechanisms underlying these improvements remain unclear. We present a systematic empirical study of RLVR's distributional effects organized around three main analyses: (1) token-level characterization of distributional shifts between base and RL models, (2) the impact of token-level distributional shifts on sequence-level reasoning performance through cross-sampling interventions, and (3) fine-grained mechanics of these shifts at the token level. We find that RL fine-tuning induces highly sparse and targeted changes, with only a small fraction of token distributions exhibiting meaningful divergence between the base and RL policies. We further characterize the structure and evolution of these shifts through analyses of token entropy, positional concentration, and reallocation of probability mass. To assess the functional importance of these sparse changes, we conduct cross-sampling experiments that selectively swap token choices between the base and RL models with varying intervention budgets. We show that inserting only a small fraction of RL-sampled tokens into base generations progressively recovers RL performance gains, while injecting a similarly small number of base token choices into otherwise RL-generated sequences collapses performance to base levels, isolating a small set of token-level decisions directly responsible for RLVR's performance gains. Finally, we explore divergence-weighted variants of the advantage signal as a diagnostic intervention, finding that they can yield improvements over baselines. Together, our results shed light on the distributional changes induced by RLVR and provide a fine-grained, token-level lens for understanding RLVR fine-tuning as a targeted refinement process.

Language models (LMs) are increasingly used as simulacra for people, yet their ability to match the distribution of views of a specific demographic group and be distributionally aligned remains uncertain. This notion of distributional alignment is complex, as there is significant variation in the types of attributes that are simulated. Prior works have underexplored the role of three critical variables -- the question domain, steering method, and distribution expression method -- which motivates our contribution of a benchmark explicitly addressing these dimensions. We construct a dataset expanding beyond political values, create human baselines for this task, and evaluate the extent to which an LM can align with a particular group's opinion distribution to inform design choices of such simulation systems. Our analysis reveals open problems regarding if, and how, LMs can be used to simulate humans, and that LLMs can more accurately describe the opinion distribution than simulate such distributions.

Exploration remains a critical issue in deep reinforcement learning for an agent to attain high returns in unknown environments. Although the prevailing exploration Random Network Distillation (RND) algorithm has been demonstrated to be effective in numerous environments, it often needs more discriminative power in bonus allocation. This paper highlights the "bonus inconsistency" issue within RND, pinpointing its primary limitation. To address this issue, we introduce the Distributional RND (DRND), a derivative of the RND. DRND enhances the exploration process by distilling a distribution of random networks and implicitly incorporating pseudo counts to improve the precision of bonus allocation. This refinement encourages agents to engage in more extensive exploration. Our method effectively mitigates the inconsistency issue without introducing significant computational overhead. Both theoretical analysis and experimental results demonstrate the superiority of our approach over the original RND algorithm. Our method excels in challenging online exploration scenarios and effectively serves as an anti-exploration mechanism in D4RL offline tasks. Our code is publicly available at https://github.com/yk7333/DRND.

This paper introduces PG-Rainbow, a novel algorithm that incorporates a distributional reinforcement learning framework with a policy gradient algorithm. Existing policy gradient methods are sample inefficient and rely on the mean of returns when calculating the state-action value function, neglecting the distributional nature of returns in reinforcement learning tasks. To address this issue, we use an Implicit Quantile Network that provides the quantile information of the distribution of rewards to the critic network of the Proximal Policy Optimization algorithm. We show empirical results that through the integration of reward distribution information into the policy network, the policy agent acquires enhanced capabilities to comprehensively evaluate the consequences of potential actions in a given state, facilitating more sophisticated and informed decision-making processes. We evaluate the performance of the proposed algorithm in the Atari-2600 game suite, simulated via the Arcade Learning Environment (ALE).

Autonomous driving involves multiple, often conflicting objectives such as safety, efficiency, and comfort. In reinforcement learning (RL), these objectives are typically combined through weighted summation, which collapses their relative priorities and often yields policies that violate safety-critical constraints. To overcome this limitation, we introduce the Preordered Multi-Objective MDP (Pr-MOMDP), which augments standard MOMDPs with a preorder over reward components. This structure enables reasoning about actions with respect to a hierarchy of objectives rather than a scalar signal. To make this structure actionable, we extend distributional RL with a novel pairwise comparison metric, Quantile Dominance (QD), that evaluates action return distributions without reducing them into a single statistic. Building on QD, we propose an algorithm for extracting optimal subsets, the subset of actions that remain non-dominated under each objective, which allows precedence information to shape both decision-making and training targets. Our framework is instantiated with Implicit Quantile Networks (IQN), establishing a concrete implementation while preserving compatibility with a broad class of distributional RL methods. Experiments in Carla show improved success rates, fewer collisions and off-road events, and deliver statistically more robust policies than IQN and ensemble-IQN baselines. By ensuring policies respect rewards preorder, our work advances safer, more reliable autonomous driving systems.

High-quality pretraining data is the fossil fuel of large language models (LLMs), yet its reserves are running low for frontier models. In this paper, we introduce RePro, a novel web recycling method that trains a relatively small LM with reinforcement learning to generate effective and faithful rephrasings of pretraining data. Specifically, we design one quality reward and three faithfulness rewards, optimizing the LM rephraser to convert organic data into high-quality rephrasings while maintaining its core semantics and structure. In our experiment, we train a 4B rephraser to recycle 72B tokens sampled from DCLM-RefinedWeb. Pretraining results on 400M and 1.4B models demonstrate that RePro delivers 4.7%-14.0% relative accuracy gains over organic-only baseline on 22 downstream tasks. RePro also outperforms ReWire, the state-of-the-art web recycling method that prompts a 70B rephraser, as well as the organic baseline with a 4x larger data pool. Experiments with different amounts of recycled data highlight that RePro improves organic data efficiency by 2-3x. Individual and distributional analyses validate that RePro preserves more critical information and faithfully reflects the characteristics of organic data compared to prompting-based methods. Together, these results show that RePro provides an efficient and controllable path to effectively harness the fossil fuel of LLM pretraining. We open-source our code, rephraser, and recycled data at https://github.com/cxcscmu/RePro.

Trajectory prediction models often fail in real-world automated driving due to distributional shifts between training and test conditions. Such distributional shifts, whether behavioural or environmental, pose a critical risk by causing the model to make incorrect forecasts in unfamiliar situations. We propose a self-supervised method that trains a decoder in a post-hoc fashion on the self-supervised task of forecasting the second half of observed trajectories from the first half. The L2 norm of the gradient of this forecasting loss with respect to the decoder's final layer defines a score to identify distribution shifts. Our approach, first, does not affect the trajectory prediction model, ensuring no interference with original prediction performance and second, demonstrates substantial improvements on distribution shift detection for trajectory prediction on the Shifts and Argoverse datasets. Moreover, we show that this method can also be used to early detect collisions of a deep Q-Network motion planner in the Highway simulator. Source code is available at https://github.com/Michedev/forecasting-the-past.

LLM-based agents are increasingly deployed to autonomously solve complex tasks, raising urgent needs for IP protection and regulatory provenance. While content watermarking effectively attributes LLM-generated outputs, it fails to directly identify the high-level planning behaviors (e.g., tool and subgoal choices) that govern multi-step execution. Critically, watermarking at the planning-behavior layer faces unique challenges: minor distributional deviations in decision-making can compound during long-term agent operation, degrading utility, and many agents operate as black boxes that are difficult to intervene in directly. To bridge this gap, we propose AgentMark, a behavioral watermarking framework that embeds multi-bit identifiers into planning decisions while preserving utility. It operates by eliciting an explicit behavior distribution from the agent and applying distribution-preserving conditional sampling, enabling deployment under black-box APIs while remaining compatible with action-layer content watermarking. Experiments across embodied, tool-use, and social environments demonstrate practical multi-bit capacity, robust recovery from partial logs, and utility preservation. The code is available at https://github.com/Tooooa/AgentMark.

Modern deep convolutional networks (CNNs) are often criticized for not generalizing under distributional shifts. However, several recent breakthroughs in transfer learning suggest that these networks can cope with severe distribution shifts and successfully adapt to new tasks from a few training examples. In this work we study the interplay between out-of-distribution and transfer performance of modern image classification CNNs for the first time and investigate the impact of the pre-training data size, the model scale, and the data preprocessing pipeline. We find that increasing both the training set and model sizes significantly improve the distributional shift robustness. Furthermore, we show that, perhaps surprisingly, simple changes in the preprocessing such as modifying the image resolution can significantly mitigate robustness issues in some cases. Finally, we outline the shortcomings of existing robustness evaluation datasets and introduce a synthetic dataset SI-Score we use for a systematic analysis across factors of variation common in visual data such as object size and position.

Transformers trained via Reinforcement Learning (RL) with outcome-based supervision can spontaneously develop the ability to generate intermediate reasoning steps (Chain-of-Thought). Yet the mechanism by which sparse rewards drive policy gradient to discover such systematic reasoning remains poorly understood. We address this by analyzing the policy gradient dynamics of single-layer Transformers on a synthetic graph traversal task that cannot be solved without Chain-of-Thought but admits a simple iterative solution. We prove that despite training solely on final-answer correctness, policy gradient drives the Transformer to converge to a structured, interpretable algorithm that iteratively traverses the graph vertex-by-vertex. We characterize the distributional properties required for this emergence, identifying the critical role of "simple examples": instances requiring fewer reasoning steps. When the training distribution places sufficient mass on these simpler examples, the Transformer learns a generalizable traversal strategy that extrapolates to longer chains; when this mass vanishes, policy gradient learning becomes infeasible. We corroborate our theoretical results through experiments on synthetic data and with real-world language models on mathematical reasoning tasks, validating that our theoretical findings carry over to practical settings.

Retrieval-augmented generation (RAG) systems have advanced large language models (LLMs) in complex deep search scenarios requiring multi-step reasoning and iterative information retrieval. However, existing approaches face critical limitations that lack high-quality training trajectories or suffer from the distributional mismatches in simulated environments and prohibitive computational costs for real-world deployment. This paper introduces SimpleDeepSearcher, a lightweight yet effective framework that bridges this gap through strategic data engineering rather than complex training paradigms. Our approach synthesizes high-quality training data by simulating realistic user interactions in live web search environments, coupled with a multi-criteria curation strategy that optimizes the diversity and quality of input and output side. Experiments on five benchmarks across diverse domains demonstrate that SFT on only 871 curated samples yields significant improvements over RL-based baselines. Our work establishes SFT as a viable pathway by systematically addressing the data-scarce bottleneck, offering practical insights for efficient deep search systems. Our code is available at https://github.com/RUCAIBox/SimpleDeepSearcher.

Post-training of large language models involves a fundamental trade-off between supervised fine-tuning (SFT), which efficiently mimics demonstrations but tends to memorize, and reinforcement learning (RL), which achieves better generalization at higher computational cost. Dynamic Fine-Tuning (DFT) recently emerged as a promising middle ground, reweighting SFT objectives with token probabilities and achieving improvements in certain reasoning domains, though it exhibits instability in other tasks. We provide a analysis of DFT through the reward-weighted regression (RWR) framework, revealing that it corresponds to a specific auxiliary distribution choice that yields provably tighter RL bounds than standard SFT. However, our analysis also uncovers a critical limitation: this construction lacks distributional anchoring, leading to progressive drift that undermines training stability. To address this, we propose Anchored Supervised Fine-Tuning (ASFT), which augments DFT's reweighting with lightweight KL regularization to preserve tightness while ensuring stability. Empirically, ASFT consistently outperforms both SFT and DFT across mathematical reasoning, medical knowledge grounding, and code generation, achieving substantial improvements with minimal computational overhead. Our RWR framework provides a systematic lens for understanding post-training methods and demonstrates that principled theoretical analysis leads to both stronger guarantees and practical gains.

Diffusion Transformers are fundamental for video and image generation, but their efficiency is bottlenecked by the quadratic complexity of attention. While block sparse attention accelerates computation by attending only critical key-value blocks, it suffers from degradation at high sparsity by discarding context. In this work, we discover that attention scores of non-critical blocks exhibit distributional stability, allowing them to be approximated accurately and efficiently rather than discarded, which is essentially important for sparse attention design. Motivated by this key insight, we propose PISA, a training-free Piecewise Sparse Attention that covers the full attention span with sub-quadratic complexity. Unlike the conventional keep-or-drop paradigm that directly drop the non-critical block information, PISA introduces a novel exact-or-approximate strategy: it maintains exact computation for critical blocks while efficiently approximating the remainder through block-wise Taylor expansion. This design allows PISA to serve as a faithful proxy to full attention, effectively bridging the gap between speed and quality. Experimental results demonstrate that PISA achieves 1.91 times and 2.57 times speedups on Wan2.1-14B and Hunyuan-Video, respectively, while consistently maintaining the highest quality among sparse attention methods. Notably, even for image generation on FLUX, PISA achieves a 1.2 times acceleration without compromising visual quality. Code is available at: https://github.com/xie-lab-ml/piecewise-sparse-attention.

Open weight models, which are ubiquitous, rarely provide access to their training data or loss function. This makes modifying such models for tasks such as pruning or unlearning constrained by this unavailability an active area of research. Existing techniques typically require gradients or ground-truth labels, rendering them infeasible in settings with limited computational resources. In this work, we investigate the fundamental question of identifying components that are critical to the model's predictive performance, without access to either gradients or the loss function, and with only distributional access such as synthetic data. We theoretically demonstrate that the global reconstruction error is linearly bounded by local reconstruction errors for Lipschitz-continuous networks such as CNNs and well-trained Transformers (which, contrary to existing literature, we find exhibit Lipschitz continuity). This motivates using the locally reconstructive behavior of component subsets to quantify their global importance, via a metric that we term Subset Fidelity. In the uncorrelated features setting, selecting individual components via their Subset Fidelity scores is optimal, which we use to propose ModHiFi, an algorithm for model modification that requires no training data or loss function access. ModHiFi-P, for structured pruning, achieves an 11% speedup over the current state of the art on ImageNet models and competitive performance on language models. ModHiFi-U, for classwise unlearning, achieves complete unlearning on CIFAR-10 without fine-tuning and demonstrates competitive performance on Swin Transformers.

Reinforcement Learning with Verifiable Rewards (RLVR) has emerged as a promising paradigm for post-training reasoning models. However, group-based methods such as Group Relative Policy Optimization (GRPO) face a critical dilemma in sparse-reward settings: pure Reinforcement Learning (RL) suffers from advantage collapse and high-variance gradient estimation, while mixed-policy optimization introduces persistent distributional bias. To resolve this dilemma, we introduce Hindsight-Anchored Policy Optimization (HAPO). HAPO employs the Synthetic Success Injection (SSI) operator, a hindsight mechanism that selectively anchors optimization to teacher demonstrations during failure. This injection is governed by a Thompson sampling-inspired gating mechanism, creating an autonomous, self-paced curriculum. Theoretically, we demonstrate that HAPO achieves asymptotic consistency: by naturally annealing the teacher signal as the policy improves, HAPO recovers the unbiased on-policy gradient. This ensures off-policy guidance acts as a temporary scaffold rather than a persistent ceiling, enabling the model to surpass the limitations of static teacher forcing.

Flow matching has recently emerged as a promising alternative to diffusion-based generative models, particularly for text-to-image generation. Despite its flexibility in allowing arbitrary source distributions, most existing approaches rely on a standard Gaussian distribution, a choice inherited from diffusion models, and rarely consider the source distribution itself as an optimization target in such settings. In this work, we show that principled design of the source distribution is not only feasible but also beneficial at the scale of modern text-to-image systems. Specifically, we propose learning a condition-dependent source distribution under flow matching objective that better exploit rich conditioning signals. We identify key failure modes that arise when directly incorporating conditioning into the source, including distributional collapse and instability, and show that appropriate variance regularization and directional alignment between source and target are critical for stable and effective learning. We further analyze how the choice of target representation space impacts flow matching with structured sources, revealing regimes in which such designs are most effective. Extensive experiments across multiple text-to-image benchmarks demonstrate consistent and robust improvements, including up to a 3x faster convergence in FID, highlighting the practical benefits of a principled source distribution design for conditional flow matching.

Recursive self-improvement is moving from theory to practice: modern systems can critique, revise, and evaluate their own outputs, yet iterative self-modification risks subtle alignment drift. We introduce SAHOO, a practical framework to monitor and control drift through three safeguards: (i) the Goal Drift Index (GDI), a learned multi-signal detector combining semantic, lexical, structural, and distributional measures; (ii) constraint preservation checks that enforce safety-critical invariants such as syntactic correctness and non-hallucination; and (iii) regression-risk quantification to flag improvement cycles that undo prior gains. Across 189 tasks in code generation, mathematical reasoning, and truthfulness, SAHOO produces substantial quality gains, including 18.3 percent improvement in code tasks and 16.8 percent in reasoning, while preserving constraints in two domains and maintaining low violations in truthfulness. Thresholds are calibrated on a small validation set of 18 tasks across three cycles. We further map the capability-alignment frontier, showing efficient early improvement cycles but rising alignment costs later and exposing domain-specific tensions such as fluency versus factuality. SAHOO therefore makes alignment preservation during recursive self-improvement measurable, deployable, and systematically validated at scale.

In reinforcement learning an agent interacts with the environment by taking actions and observing the next state and reward. When sampled probabilistically, these state transitions, rewards, and actions can all induce randomness in the observed long-term return. Traditionally, reinforcement learning algorithms average over this randomness to estimate the value function. In this paper, we build on recent work advocating a distributional approach to reinforcement learning in which the distribution over returns is modeled explicitly instead of only estimating the mean. That is, we examine methods of learning the value distribution instead of the value function. We give results that close a number of gaps between the theoretical and algorithmic results given by Bellemare, Dabney, and Munos (2017). First, we extend existing results to the approximate distribution setting. Second, we present a novel distributional reinforcement learning algorithm consistent with our theoretical formulation. Finally, we evaluate this new algorithm on the Atari 2600 games, observing that it significantly outperforms many of the recent improvements on DQN, including the related distributional algorithm C51.

In contrast to classical reinforcement learning, distributional reinforcement learning algorithms aim to learn the distribution of returns rather than their expected value. Since the nature of the return distribution is generally unknown a priori or arbitrarily complex, a common approach finds approximations within a set of representable, parametric distributions. Typically, this involves a projection of the unconstrained distribution onto the set of simplified distributions. We argue that this projection step entails a strong inductive bias when coupled with neural networks and gradient descent, thereby profoundly impacting the generalization behavior of learned models. In order to facilitate reliable uncertainty estimation through diversity, this work studies the combination of several different projections and representations in a distributional ensemble. We establish theoretical properties of such projection ensembles and derive an algorithm that uses ensemble disagreement, measured by the average 1-Wasserstein distance, as a bonus for deep exploration. We evaluate our algorithm on the behavior suite benchmark and find that diverse projection ensembles lead to significant performance improvements over existing methods on a wide variety of tasks with the most pronounced gains in directed exploration problems.

Reinforcement learning (RL) allows an agent interacting sequentially with an environment to maximize its long-term expected return. In the distributional RL (DistrRL) paradigm, the agent goes beyond the limit of the expected value, to capture the underlying probability distribution of the return across all time steps. The set of DistrRL algorithms has led to improved empirical performance. Nevertheless, the theory of DistrRL is still not fully understood, especially in the control case. In this paper, we present the simpler one-step distributional reinforcement learning (OS-DistrRL) framework encompassing only the randomness induced by the one-step dynamics of the environment. Contrary to DistrRL, we show that our approach comes with a unified theory for both policy evaluation and control. Indeed, we propose two OS-DistrRL algorithms for which we provide an almost sure convergence analysis. The proposed approach compares favorably with categorical DistrRL on various environments.

In dynamic programming (DP) and reinforcement learning (RL), an agent learns to act optimally in terms of expected long-term return by sequentially interacting with its environment modeled by a Markov decision process (MDP). More generally in distributional reinforcement learning (DRL), the focus is on the whole distribution of the return, not just its expectation. Although DRL-based methods produced state-of-the-art performance in RL with function approximation, they involve additional quantities (compared to the non-distributional setting) that are still not well understood. As a first contribution, we introduce a new class of distributional operators, together with a practical DP algorithm for policy evaluation, that come with a robust MDP interpretation. Indeed, our approach reformulates through an augmented state space where each state is split into a worst-case substate and a best-case substate, whose values are maximized by safe and risky policies respectively. Finally, we derive distributional operators and DP algorithms solving a new control task: How to distinguish safe from risky optimal actions in order to break ties in the space of optimal policies?

We propose a Distributional Approach for addressing Controlled Text Generation from pre-trained Language Models (LMs). This approach permits to specify, in a single formal framework, both "pointwise" and "distributional" constraints over the target LM -- to our knowledge, the first model with such generality -- while minimizing KL divergence from the initial LM distribution. The optimal target distribution is then uniquely determined as an explicit EBM (Energy-Based Model) representation. From that optimal representation we then train a target controlled Autoregressive LM through an adaptive distributional variant of Policy Gradient. We conduct a first set of experiments over pointwise constraints showing the advantages of our approach over a set of baselines, in terms of obtaining a controlled LM balancing constraint satisfaction with divergence from the initial LM. We then perform experiments over distributional constraints, a unique feature of our approach, demonstrating its potential as a remedy to the problem of Bias in Language Models. Through an ablation study, we show the effectiveness of our adaptive technique for obtaining faster convergence. (Code available at https://github.com/naver/gdc)

We introduce distributional dynamic programming (DP) methods for optimizing statistical functionals of the return distribution, with standard reinforcement learning as a special case. Previous distributional DP methods could optimize the same class of expected utilities as classic DP. To go beyond expected utilities, we combine distributional DP with stock augmentation, a technique previously introduced for classic DP in the context of risk-sensitive RL, where the MDP state is augmented with a statistic of the rewards obtained so far (since the first time step). We find that a number of recently studied problems can be formulated as stock-augmented return distribution optimization, and we show that we can use distributional DP to solve them. We analyze distributional value and policy iteration, with bounds and a study of what objectives these distributional DP methods can or cannot optimize. We describe a number of applications outlining how to use distributional DP to solve different stock-augmented return distribution optimization problems, for example maximizing conditional value-at-risk, and homeostatic regulation. To highlight the practical potential of stock-augmented return distribution optimization and distributional DP, we combine the core ideas of distributional value iteration with the deep RL agent DQN, and empirically evaluate it for solving instances of the applications discussed.

In this work, we build on recent advances in distributional reinforcement learning to give a generally applicable, flexible, and state-of-the-art distributional variant of DQN. We achieve this by using quantile regression to approximate the full quantile function for the state-action return distribution. By reparameterizing a distribution over the sample space, this yields an implicitly defined return distribution and gives rise to a large class of risk-sensitive policies. We demonstrate improved performance on the 57 Atari 2600 games in the ALE, and use our algorithm's implicitly defined distributions to study the effects of risk-sensitive policies in Atari games.

Aligning language models with preferences can be posed as approximating a target distribution representing some desired behavior. Existing approaches differ both in the functional form of the target distribution and the algorithm used to approximate it. For instance, Reinforcement Learning from Human Feedback (RLHF) corresponds to minimizing a reverse KL from an implicit target distribution arising from a KL penalty in the objective. On the other hand, Generative Distributional Control (GDC) has an explicit target distribution and minimizes a forward KL from it using the Distributional Policy Gradient (DPG) algorithm. In this paper, we propose a new approach, f-DPG, which allows the use of any f-divergence to approximate any target distribution that can be evaluated. f-DPG unifies both frameworks (RLHF, GDC) and the approximation methods (DPG, RL with KL penalties). We show the practical benefits of various choices of divergence objectives and demonstrate that there is no universally optimal objective but that different divergences present different alignment and diversity trade-offs. We show that Jensen-Shannon divergence strikes a good balance between these objectives, and frequently outperforms forward KL divergence by a wide margin, leading to significant improvements over prior work. These distinguishing characteristics between divergences persist as the model size increases, highlighting the importance of selecting appropriate divergence objectives.

Reinforcement learning from human feedback (RLHF) has become a key method for aligning large language models (LLMs) with human preferences through the use of reward models. However, traditional reward models typically generate point estimates, which oversimplify the diversity and complexity of human values and preferences. In this paper, we introduce Quantile Reward Models (QRMs), a novel approach to reward modeling that learns a distribution over rewards instead of a single scalar value. Our method uses quantile regression to estimate a full, potentially multimodal distribution over preferences, providing a more powerful and nuanced representation of preferences. This distributional approach can better capture the diversity of human values, addresses label noise, and accommodates conflicting preferences by modeling them as distinct modes in the distribution. Our experimental results show that QRM outperforms comparable traditional point-estimate models on RewardBench. Furthermore, we demonstrate that the additional information provided by the distributional estimates can be utilized in downstream applications, such as risk-aware reinforcement learning, resulting in LLM policies that generate fewer extremely negative responses. Our code and model are released at https://github.com/Nicolinho/QRM.

Large language models (LLMs) have revolutionized the role of AI, yet also pose potential risks of propagating unethical content. Alignment technologies have been introduced to steer LLMs towards human preference, gaining increasing attention. Despite notable breakthroughs in this direction, existing methods heavily rely on high-quality positive-negative training pairs, suffering from noisy labels and the marginal distinction between preferred and dispreferred response data. Given recent LLMs' proficiency in generating helpful responses, this work pivots towards a new research focus: achieving alignment using solely human-annotated negative samples, preserving helpfulness while reducing harmfulness. For this purpose, we propose Distributional Dispreference Optimization (D^2O), which maximizes the discrepancy between the generated responses and the dispreferred ones to effectively eschew harmful information. We theoretically demonstrate that D^2O is equivalent to learning a distributional instead of instance-level preference model reflecting human dispreference against the distribution of negative responses. Besides, D^2O integrates an implicit Jeffrey Divergence regularization to balance the exploitation and exploration of reference policies and converges to a non-negative one during training. Extensive experiments demonstrate that our method achieves comparable generation quality and surpasses the latest baselines in producing less harmful and more informative responses with better training stability and faster convergence.

Teaching large language models (LLMs) to critique and refine their outputs is crucial for building systems that can iteratively improve, yet it is fundamentally limited by the ability to provide accurate judgments and actionable suggestions. In this work, we study LLM critics for code generation and propose CTRL, a framework for Critic Training via Reinforcement Learning, which trains a critic model to generate feedback that maximizes correction performance for a fixed generator model without human supervision. Our results demonstrate that critics trained with CTRL significantly enhance pass rates and mitigate compounding errors across both base and stronger generator models. Furthermore, we show that these critic models act as accurate generative reward models and enable test-time scaling through iterative critique-revision, achieving up to 106.1% relative improvements across challenging code generation benchmarks.

A growing trend for value-based reinforcement learning (RL) algorithms is to capture more information than scalar value functions in the value network. One of the most well-known methods in this branch is distributional RL, which models return distribution instead of scalar value. In another line of work, hybrid reward architectures (HRA) in RL have studied to model source-specific value functions for each source of reward, which is also shown to be beneficial in performance. To fully inherit the benefits of distributional RL and hybrid reward architectures, we introduce Multi-Dimensional Distributional DQN (MD3QN), which extends distributional RL to model the joint return distribution from multiple reward sources. As a by-product of joint distribution modeling, MD3QN can capture not only the randomness in returns for each source of reward, but also the rich reward correlation between the randomness of different sources. We prove the convergence for the joint distributional Bellman operator and build our empirical algorithm by minimizing the Maximum Mean Discrepancy between joint return distribution and its Bellman target. In experiments, our method accurately models the joint return distribution in environments with richly correlated reward functions, and outperforms previous RL methods utilizing multi-dimensional reward functions in the control setting.

It is well known that ensemble methods often provide enhanced performance in reinforcement learning. In this paper, we explore this concept further by using group-aided training within the distributional reinforcement learning paradigm. Specifically, we propose an extension to categorical reinforcement learning, where distributional learning targets are implicitly based on the total information gathered by an ensemble. We empirically show that this may lead to much more robust initial learning, a stronger individual performance level, and good efficiency on a per-sample basis.

Critique-guided reinforcement learning (RL) has emerged as a powerful paradigm for training LLM agents by augmenting sparse outcome rewards with natural-language feedback. However, current methods often rely on static or offline critic models, which fail to adapt as the policy evolves. In on-policy RL, the agent's error patterns shift over time, causing stationary critics to become stale and providing feedback of diminishing utility. To address this, we introduce ECHO (Evolving Critic for Hindsight-Guided Optimization)}, a framework that jointly optimizes the policy and critic through a synchronized co-evolutionary loop. ECHO utilizes a cascaded rollout mechanism where the critic generates multiple diagnoses for an initial trajectory, followed by policy refinement to enable group-structured advantage estimation. We address the challenge of learning plateaus via a saturation-aware gain shaping objective, which rewards the critic for inducing incremental improvements in high-performing trajectories. By employing dual-track GRPO updates, ECHO ensures the critic's feedback stays synchronized with the evolving policy. Experimental results show that ECHO yields more stable training and higher long-horizon task success across open-world environments.

We study the problem of temporal-difference-based policy evaluation in reinforcement learning. In particular, we analyse the use of a distributional reinforcement learning algorithm, quantile temporal-difference learning (QTD), for this task. We reach the surprising conclusion that even if a practitioner has no interest in the return distribution beyond the mean, QTD (which learns predictions about the full distribution of returns) may offer performance superior to approaches such as classical TD learning, which predict only the mean return, even in the tabular setting.

In this paper we argue for the fundamental importance of the value distribution: the distribution of the random return received by a reinforcement learning agent. This is in contrast to the common approach to reinforcement learning which models the expectation of this return, or value. Although there is an established body of literature studying the value distribution, thus far it has always been used for a specific purpose such as implementing risk-aware behaviour. We begin with theoretical results in both the policy evaluation and control settings, exposing a significant distributional instability in the latter. We then use the distributional perspective to design a new algorithm which applies Bellman's equation to the learning of approximate value distributions. We evaluate our algorithm using the suite of games from the Arcade Learning Environment. We obtain both state-of-the-art results and anecdotal evidence demonstrating the importance of the value distribution in approximate reinforcement learning. Finally, we combine theoretical and empirical evidence to highlight the ways in which the value distribution impacts learning in the approximate setting.

In vision-language modeling, critic models are typically trained to evaluate outputs -- assigning scalar scores or pairwise preferences -- rather than to generate responses. This separation from policy models, which produce the responses, is so entrenched that critics are rarely considered for direct policy use. In this work, we challenge this convention. We propose to reorganize preference-labeled critic datasets into verifiable training signals and perform reinforcement learning directly on a base generative model, producing LLaVA-Critic-R1, a multimodal critic trained to optimize preference judgments while retaining full generation ability. Surprisingly, LLaVA-Critic-R1 emerges not only as a top-performing critic but also as a competitive policy model -- matching or surpassing specialized reasoning VLMs trained with in-domain data across 26 visual reasoning and understanding benchmarks, with an average gain of +5.7% over its base model (Qwen-2.5-VL-7B). Extending this approach to existing strong reasoning VLMs yields LLaVA-Critic-R1+, which further advances policy performance without sacrificing critic quality, achieving a SoTA performance of 71.9 on MMMU at the 7B scale. Finally, we show that the enhanced critic ability benefits inference: applying self-critique at test time yields an average +13.8% improvement on five representative reasoning tasks without additional training. Our results reveal that RL training on critic data can produce a unified model excelling at both evaluation and generation, offering a simple path toward scalable, self-improving multimodal systems.

Training reinforcement learning (RL) systems in real-world environments remains challenging due to noisy supervision and poor out-of-domain (OOD) generalization, especially in LLM post-training. Recent distributional RL methods improve robustness by modeling values with multiple quantile points, but they still learn each quantile independently as a scalar. This results in rough-grained value representations that lack fine-grained conditioning on state information, struggling under complex and OOD conditions. We propose DFPO (Distributional Value Flow Policy Optimization with Conditional Risk and Consistency Control), a robust distributional RL framework that models values as continuous flows across time steps. By scaling value modeling through learning of a value flow field instead of isolated quantile predictions, DFPO captures richer state information for more accurate advantage estimation. To stabilize training under noisy feedback, DFPO further integrates conditional risk control and consistency constraints along value flow trajectories. Experiments on dialogue, math reasoning, and scientific tasks show that DFPO outperforms PPO, FlowRL, and other robust baselines under noisy supervision, achieving improved training stability and generalization.

Reinforcement learning (RL) can align language models with non-differentiable reward signals, such as human preferences. However, a major challenge arises from the sparsity of these reward signals - typically, there is only one reward for the entire generation. This sparsity of rewards can lead to inefficient and unstable learning. In this paper, we introduce a novel framework leveraging the critique ability of LLMs to produce dense rewards throughout the learning process. Our approach incorporates a critic language model alongside the policy model. This critic is prompted with the task description, question, policy model's output, and environment's reward signal as input, and provides token or span-level dense rewards that reflect the quality of each segment of the output. We assess our approach on three text generation tasks: sentiment control, language model detoxification, and summarization. Experimental results show that incorporating artificial dense rewards in training yields consistent performance gains over the PPO baseline with holistic rewards. Furthermore, in a setting where the same model serves as both policy and critic, we demonstrate that "self-critique" rewards also boost learning efficiency.

Instruction following is a fundamental ability of Large Language Models (LLMs), requiring their generated outputs to follow multiple constraints imposed in input instructions. Numerous studies have attempted to enhance this ability through preference optimization or reinforcement learning based on reward signals from LLM-as-a-Judge. However, existing evaluation models for instruction following still possess many deficiencies, such as substantial costs and unreliable assessments. To this end, we propose IF-CRITIC, an LLM critic that can provide efficient and reliable assessments of constraint following in the instructions. We first develop a checklist generator to decompose instructions and generate constraint checklists. With the assistance of the checklists, we collect high-quality critique training data through a multi-stage critique filtering mechanism and employ a constraint-level preference optimization method to train IF-CRITIC. Extensive experiments demonstrate that the evaluation performance of IF-CRITIC can beat strong LLM-as-a-Judge baselines, including Deepseek-R1 and o4-mini. With the scalable reward signals provided by IF-CRITIC, LLMs can achieve substantial performance gains in instruction-following optimization under lower computational overhead compared to strong LLM critic baselines.

Recent developments in large language models (LLMs) have been impressive. However, these models sometimes show inconsistencies and problematic behavior, such as hallucinating facts, generating flawed code, or creating offensive and toxic content. Unlike these models, humans typically utilize external tools to cross-check and refine their initial content, like using a search engine for fact-checking, or a code interpreter for debugging. Inspired by this observation, we introduce a framework called CRITIC that allows LLMs, which are essentially "black boxes" to validate and progressively amend their own outputs in a manner similar to human interaction with tools. More specifically, starting with an initial output, CRITIC interacts with appropriate tools to evaluate certain aspects of the text, and then revises the output based on the feedback obtained during this validation process. Comprehensive evaluations involving free-form question answering, mathematical program synthesis, and toxicity reduction demonstrate that CRITIC consistently enhances the performance of LLMs. Meanwhile, our research highlights the crucial importance of external feedback in promoting the ongoing self-improvement of LLMs.

Translating natural language mathematical statements into formal, executable code is a fundamental challenge in automated theorem proving. While prior work has focused on generation and compilation success, little attention has been paid to the critic phase-the evaluation of whether generated formalizations truly capture the semantic intent of the original problem. In this paper, we introduce CriticLean, a novel critic-guided reinforcement learning framework that elevates the role of the critic from a passive validator to an active learning component. Specifically, first, we propose the CriticLeanGPT, trained via supervised fine-tuning and reinforcement learning, to rigorously assess the semantic fidelity of Lean 4 formalizations. Then, we introduce CriticLeanBench, a benchmark designed to measure models' ability to distinguish semantically correct from incorrect formalizations, and demonstrate that our trained CriticLeanGPT models can significantly outperform strong open- and closed-source baselines. Building on the CriticLean framework, we construct FineLeanCorpus, a dataset comprising over 285K problems that exhibits rich domain diversity, broad difficulty coverage, and high correctness based on human evaluation. Overall, our findings highlight that optimizing the critic phase is essential for producing reliable formalizations, and we hope our CriticLean will provide valuable insights for future advances in formal mathematical reasoning.

We introduce LLaVA-Critic, the first open-source large multimodal model (LMM) designed as a generalist evaluator to assess performance across a wide range of multimodal tasks. LLaVA-Critic is trained using a high-quality critic instruction-following dataset that incorporates diverse evaluation criteria and scenarios. Our experiments demonstrate the model's effectiveness in two key areas: (1) LMM-as-a-Judge, where LLaVA-Critic provides reliable evaluation scores, performing on par with or surpassing GPT models on multiple evaluation benchmarks; and (2) Preference Learning, where it generates reward signals for preference learning, enhancing model alignment capabilities. This work underscores the potential of open-source LMMs in self-critique and evaluation, setting the stage for future research into scalable, superhuman alignment feedback mechanisms for LMMs.

Adversarial training for LLMs is one of the most promising methods to reliably improve robustness against adversaries. However, despite significant progress, models remain vulnerable to simple in-distribution exploits, such as rewriting prompts in the past tense or translating them into other languages. We argue that this persistent fragility stems from a fundamental limitation in current adversarial training algorithms: they minimize adversarial loss on their training set but inadequately cover the data distribution, resulting in vulnerability to seemingly simple attacks. To bridge this gap, we propose Distributional Adversarial Training, DAT. We leverage Diffusion LLMs to approximate the true joint distribution of prompts and responses, enabling generation of diverse, high-likelihood samples that address generalization failures. By combining optimization over the data distribution provided by the diffusion model with continuous adversarial training, DAT achieves substantially higher adversarial robustness than previous methods.

As black-box models increasingly power high-stakes applications, a variety of data-driven explanation methods have been introduced. Meanwhile, machine learning models are constantly challenged by distributional shifts. A question naturally arises: Are data-driven explanations robust against out-of-distribution data? Our empirical results show that even though predict correctly, the model might still yield unreliable explanations under distributional shifts. How to develop robust explanations against out-of-distribution data? To address this problem, we propose an end-to-end model-agnostic learning framework Distributionally Robust Explanations (DRE). The key idea is, inspired by self-supervised learning, to fully utilizes the inter-distribution information to provide supervisory signals for the learning of explanations without human annotation. Can robust explanations benefit the model's generalization capability? We conduct extensive experiments on a wide range of tasks and data types, including classification and regression on image and scientific tabular data. Our results demonstrate that the proposed method significantly improves the model's performance in terms of explanation and prediction robustness against distributional shifts.

Recent advances in reinforcement learning (RL)-based post-training have led to notable improvements in large language models (LLMs), particularly in enhancing their reasoning capabilities to handle complex tasks. However, most existing methods treat the training data as a unified whole, overlooking the fact that modern LLM training often involves a mixture of data from diverse distributions-varying in both source and difficulty. This heterogeneity introduces a key challenge: how to adaptively schedule training across distributions to optimize learning efficiency. In this paper, we present a principled curriculum learning framework grounded in the notion of distribution-level learnability. Our core insight is that the magnitude of policy advantages reflects how much a model can still benefit from further training on a given distribution. Based on this, we propose a distribution-level curriculum learning framework for RL-based LLM post-training, which leverages the Upper Confidence Bound (UCB) principle to dynamically adjust sampling probabilities for different distrubutions. This approach prioritizes distributions with either high average advantage (exploitation) or low sample count (exploration), yielding an adaptive and theoretically grounded training schedule. We instantiate our curriculum learning framework with GRPO as the underlying RL algorithm and demonstrate its effectiveness on logic reasoning datasets with multiple difficulties and sources. Our experiments show that our framework significantly improves convergence speed and final performance, highlighting the value of distribution-aware curriculum strategies in LLM post-training. Code: https://github.com/ZhentingWang/DUMP.

Recent work shows that flow matching can be effective for scalar Q-value function estimation in reinforcement learning (RL), but it remains unclear why or how this approach differs from standard critics. Contrary to conventional belief, we show that their success is not explained by distributional RL, as explicitly modeling return distributions can reduce performance. Instead, we argue that the use of integration for reading out values and dense velocity supervision at each step of this integration process for training improves TD learning via two mechanisms. First, it enables robust value prediction through test-time recovery, whereby iterative computation through integration dampens errors in early value estimates as more integration steps are performed. This recovery mechanism is absent in monolithic critics. Second, supervising the velocity field at multiple interpolant values induces more plastic feature learning within the network, allowing critics to represent non-stationary TD targets without discarding previously learned features or overfitting to individual TD targets encountered during training. We formalize these effects and validate them empirically, showing that flow-matching critics substantially outperform monolithic critics (2times in final performance and around 5times in sample efficiency) in settings where loss of plasticity poses a challenge e.g., in high-UTD online RL problems, while remaining stable during learning.

Reward modeling is crucial for aligning large language models (LLMs) with human preferences, especially in reinforcement learning from human feedback (RLHF). However, current reward models mainly produce scalar scores and struggle to incorporate critiques in a natural language format. We hypothesize that predicting both critiques and the scalar reward would improve reward modeling ability. Motivated by this, we propose Critic-RM, a framework that improves reward models using self-generated critiques without extra supervision. Critic-RM employs a two-stage process: generating and filtering high-quality critiques, followed by joint fine-tuning on reward prediction and critique generation. Experiments across benchmarks show that Critic-RM improves reward modeling accuracy by 3.7%-7.3% compared to standard reward models and LLM judges, demonstrating strong performance and data efficiency. Additional studies further validate the effectiveness of generated critiques in rectifying flawed reasoning steps with 2.5%-3.2% gains in improving reasoning accuracy.

State-of-the-art large language model (LLM) pipelines rely on bootstrapped reasoning loops: sampling diverse chains of thought and reinforcing the highest-scoring ones, mainly optimizing correctness. We analyze how this design choice is sensitive to the collapse of the model's distribution over reasoning paths, slashing semantic entropy and undermining creative problem-solving. To analyze this failure, we introduce Distributional Creative Reasoning (DCR), a unified variational objective that casts training as gradient flow through probability measures on solution traces. STaR, GRPO, and DPO, as well as entropy bonuses, and other methods, all constitute special cases of the same loss. The framework delivers three core results: (i) the diversity decay theorem, describing how correctness-based objectives lead to distinct modes of diversity decay for STaR, GRPO, and DPO; (ii) designs that ensure convergence to a stable and diverse policy, effectively preventing collapse; and (iii) simple, actionable recipes to achieve this in practice. DCR thus offers the first principled recipe for LLMs that remain both correct and creative.

Policy gradient methods are reinforcement learning algorithms that adapt a parameterized policy by following a performance gradient estimate. Conventional policy gradient methods use Monte-Carlo techniques to estimate the gradient, which tend to have high variance, requiring many samples and resulting in slow convergence. We first propose a Bayesian framework for policy gradient, based on modeling the policy gradient as a Gaussian process. This reduces the number of samples needed to obtain accurate gradient estimates. Moreover, estimates of the natural gradient and a measure of the uncertainty in the gradient estimates, namely, the gradient covariance, are provided at little extra cost. Since the proposed framework considers system trajectories as its basic observable unit, it does not require the dynamics within trajectories to be of any particular form, and can be extended to partially observable problems. On the downside, it cannot exploit the Markov property when the system is Markovian. To address this, we supplement our Bayesian policy gradient framework with a new actor-critic learning model in which a Bayesian class of non-parametric critics, based on Gaussian process temporal difference learning, is used. Such critics model the action-value function as a Gaussian process, allowing Bayes rule to be used to compute the posterior distribution over action-value functions, conditioned on the observed data. Appropriate choices of the policy parameterization and of the prior covariance (kernel) between action-values yield closed-form expressions for the posterior of the gradient of the expected return with respect to the policy parameters. We perform detailed experimental comparisons of the proposed Bayesian policy gradient and actor-critic algorithms with classic Monte-Carlo based policy gradient methods, on a number of reinforcement learning problems.

Existing Large Language Model (LLM) agents struggle in interactive environments requiring long-horizon planning, primarily due to compounding errors when simulating future states. To address this, we propose ProAct, a framework that enables agents to internalize accurate lookahead reasoning through a two-stage training paradigm. First, we introduce Grounded LookAhead Distillation (GLAD), where the agent undergoes supervised fine-tuning on trajectories derived from environment-based search. By compressing complex search trees into concise, causal reasoning chains, the agent learns the logic of foresight without the computational overhead of inference-time search. Second, to further refine decision accuracy, we propose the Monte-Carlo Critic (MC-Critic), a plug-and-play auxiliary value estimator designed to enhance policy-gradient algorithms like PPO and GRPO. By leveraging lightweight environment rollouts to calibrate value estimates, MC-Critic provides a low-variance signal that facilitates stable policy optimization without relying on expensive model-based value approximation. Experiments on both stochastic (e.g., 2048) and deterministic (e.g., Sokoban) environments demonstrate that ProAct significantly improves planning accuracy. Notably, a 4B parameter model trained with ProAct outperforms all open-source baselines and rivals state-of-the-art closed-source models, while demonstrating robust generalization to unseen environments. The codes and models are available at https://github.com/GreatX3/ProAct

Despite the remarkable capabilities of large language models (LLMs) in various reasoning tasks, they still struggle with table reasoning tasks, particularly in maintaining consistency throughout multi-step reasoning processes. While existing approaches have explored various decomposition strategies, they often lack effective mechanisms to identify and correct errors in intermediate reasoning steps, leading to cascading error propagation. To address these issues, we propose Table-Critic, a novel multi-agent framework that facilitates collaborative criticism and iterative refinement of the reasoning process until convergence to correct solutions. Our framework consists of four specialized agents: a Judge for error identification, a Critic for comprehensive critiques, a Refiner for process improvement, and a Curator for pattern distillation. To effectively deal with diverse and unpredictable error types, we introduce a self-evolving template tree that systematically accumulates critique knowledge through experience-driven learning and guides future reflections. Extensive experiments have demonstrated that Table-Critic achieves substantial improvements over existing methods, achieving superior accuracy and error correction rates while maintaining computational efficiency and lower solution degradation rate.

Tool-Integrated Reasoning (TIR) with search engines enables large language models to iteratively retrieve up-to-date external knowledge, enhancing adaptability and generalization in complex question-answering tasks. However, existing search agent pipelines typically depend on reinforcement learning based optimization, which often suffers from sparse outcome rewards, leading to inefficient exploration and unstable training. We introduce CriticSearch, a fine-grained credit-assignment framework that supplies dense, turn-level feedback via a retrospective critic mechanism. During training, a frozen, asymmetric critique LLM retrospectively evaluates each turn using privileged information from the full trajectory and gold answers, converting these assessments into stable, dense rewards that guide policy improvement. Experimental results across diverse multi-hop reasoning benchmarks demonstrate that CriticSearch consistently outperforms existing baselines, achieving faster convergence, improved training stability, and higher performance.

Self-critic has become an important mechanism for enhancing the reasoning performance of LLMs. However, current approaches mainly involve basic prompts without further training, which tend to be over-simplified, leading to limited accuracy.Moreover, there is a lack of in-depth investigation of the relationship between LLM's ability to criticism and its task-solving performance.To address these issues, we propose Critic-CoT, a novel framework that pushes LLMs toward System-2-like critic capability, via step-wise CoT reasoning format and distant-supervision data construction, without the need for human annotation. Experiments on GSM8K and MATH show that via filtering out invalid solutions or iterative refinement, our enhanced model boosts task-solving performance, which demonstrates the effectiveness of our method. Further, we find that training on critique and refinement alone improves the generation. We hope our work could shed light on future research on improving the reasoning and critic ability of LLMs.

Steering language generation towards objectives or away from undesired content has been a long-standing goal in utilizing language models (LM). Recent work has demonstrated reinforcement learning and weighted decoding as effective approaches to achieve a higher level of language control and quality with pros and cons. In this work, we propose a novel critic decoding method for controlled language generation (CriticControl) that combines the strengths of reinforcement learning and weighted decoding. Specifically, we adopt the actor-critic framework to train an LM-steering critic from non-differentiable reward models. And similar to weighted decoding, our method freezes the language model and manipulates the output token distribution using called critic, improving training efficiency and stability. Evaluation of our method on three controlled generation tasks, namely topic control, sentiment control, and detoxification, shows that our approach generates more coherent and well-controlled texts than previous methods. In addition, CriticControl demonstrates superior generalization ability in zero-shot settings. Human evaluation studies also corroborate our findings.

The ability of large language models (LLMs) to utilize external tools has enabled them to tackle an increasingly diverse range of tasks. However, as the tasks become more complex and long-horizon, the intricate tool utilization process may trigger various unexpected errors. Therefore, how to effectively handle such errors, including identifying, diagnosing, and recovering from them, has emerged as a key research direction for advancing tool learning. In this work, we first extensively analyze the types of errors encountered during the function-calling process on several competitive tool evaluation benchmarks. Based on it, we introduce CRITICTOOL, a comprehensive critique evaluation benchmark specialized for tool learning. Building upon a novel evolutionary strategy for dataset construction, CRITICTOOL holds diverse tool-use errors with varying complexities, which better reflects real-world scenarios. We conduct extensive experiments on CRITICTOOL, and validate the generalization and effectiveness of our constructed benchmark strategy. We also provide an in-depth analysis of the tool reflection ability on various LLMs, offering a new perspective on the field of tool learning in LLMs. The code is available at https://github.com/Shellorley0513/CriticTool{https://github.com/Shellorley0513/CriticTool}.

We introduce Distribution Contractive Reinforcement Learning (DICE-RL), a framework that uses reinforcement learning (RL) as a "distribution contraction" operator to refine pretrained generative robot policies. DICE-RL turns a pretrained behavior prior into a high-performing "pro" policy by amplifying high-success behaviors from online feedback. We pretrain a diffusion- or flow-based policy for broad behavioral coverage, then finetune it with a stable, sample-efficient residual off-policy RL framework that combines selective behavior regularization with value-guided action selection. Extensive experiments and analyses show that DICE-RL reliably improves performance with strong stability and sample efficiency. It enables mastery of complex long-horizon manipulation skills directly from high-dimensional pixel inputs, both in simulation and on a real robot. Project website: https://zhanyisun.github.io/dice.rl.2026/.

Multimodal large language models (MLLMs) struggle with numerical regression under long-tailed target distributions. Token-level supervised fine-tuning (SFT) and point-wise regression rewards bias learning toward high-density regions, leading to regression-to-the-mean behavior and poor tail performance. We identify the lack of cross-sample relational supervision as a key limitation of existing MLLM training paradigms. To address it, we propose a distribution-aware reinforcement learning framework based on Group Relative Policy Optimization, which introduces batch-level comparison-based supervision via the Concordance Correlation Coefficient-based reward to align predicted and ground-truth distributions in terms of correlation, scale, and mean. The framework is plug-and-play, requiring no architectural modification. Experiments on a unified suite of long-tailed regression benchmarks show consistent improvements over SFT and existing MLLM regression methods, with particularly strong gains in medium- and few-shot regimes.

As Large Language Models (LLMs) are rapidly evolving, providing accurate feedback and scalable oversight on their outputs becomes an urgent and critical problem. Leveraging LLMs as critique models to achieve automated supervision is a promising solution. In this work, we focus on studying and enhancing the math critique ability of LLMs. Current LLM critics provide critiques that are too shallow and superficial on each step, leading to low judgment accuracy and struggling to offer sufficient feedback for the LLM generator to correct mistakes. To tackle this issue, we propose a novel and effective two-stage framework to develop LLM critics that are capable of deliberately critiquing on each reasoning step of math solutions. In the first stage, we utilize Qwen2.5-72B-Instruct to generate 4.5K long-form critiques as seed data for supervised fine-tuning. Each seed critique consists of deliberate step-wise critiques that includes multi-perspective verifications as well as in-depth critiques of initial critiques for each reasoning step. Then, we perform reinforcement learning on the fine-tuned model with either existing human-labeled data from PRM800K or our automatically annotated data obtained via Monte Carlo sampling-based correctness estimation, to further incentivize its critique ability. Our developed critique model built on Qwen2.5-7B-Instruct not only significantly outperforms existing LLM critics (including the same-sized DeepSeek-R1-distill models and GPT-4o) on various error identification benchmarks, but also more effectively helps the LLM generator refine erroneous steps through more detailed feedback.

Vision-language models~(VLMs) have shown remarkable advancements in multimodal reasoning tasks. However, they still often generate inaccurate or irrelevant responses due to issues like hallucinated image understandings or unrefined reasoning paths. To address these challenges, we introduce Critic-V, a novel framework inspired by the Actor-Critic paradigm to boost the reasoning capability of VLMs. This framework decouples the reasoning process and critic process by integrating two independent components: the Reasoner, which generates reasoning paths based on visual and textual inputs, and the Critic, which provides constructive critique to refine these paths. In this approach, the Reasoner generates reasoning responses according to text prompts, which can evolve iteratively as a policy based on feedback from the Critic. This interaction process was theoretically driven by a reinforcement learning framework where the Critic offers natural language critiques instead of scalar rewards, enabling more nuanced feedback to boost the Reasoner's capability on complex reasoning tasks. The Critic model is trained using Direct Preference Optimization (DPO), leveraging a preference dataset of critiques ranked by Rule-based Reward(RBR) to enhance its critic capabilities. Evaluation results show that the Critic-V framework significantly outperforms existing methods, including GPT-4V, on 5 out of 8 benchmarks, especially regarding reasoning accuracy and efficiency. Combining a dynamic text-based policy for the Reasoner and constructive feedback from the preference-optimized Critic enables a more reliable and context-sensitive multimodal reasoning process. Our approach provides a promising solution to enhance the reliability of VLMs, improving their performance in real-world reasoning-heavy multimodal applications such as autonomous driving and embodied intelligence.

Open-ended generation tasks require outputs to satisfy diverse and often implicit task-specific evaluation rubrics. The sheer number of relevant rubrics leads to prohibitively high verification costs and incomplete assessments of a response, making reinforcement learning (RL) post-training with rubric-based rewards difficult to scale. This problem is exacerbated by the fact that often the best way to combine these rubrics into one single reward is also highly prompt-specific. We propose Reinforcement Learning with Adversarial Critic (RLAC), a post-training approach that addresses these challenges via dynamic rubric verification. Our approach employs a large language model (LLM) as a critic that dynamically identifies only the most likely failure modes (e.g., a factual error or unhandled edge case), which are then verified by an external validator to optimize both generator and critic jointly. By training both the generator and the critic, this game enhances the critic's error detection and the generator's output quality while reducing required verifications. Our experiments demonstrate that RLAC improves factual accuracy in text generation and correctness in code generation, while also outperforming exhaustive verification and reward model methods. We show that dynamic critics are more effective than fixed critics, showcasing the potential of RLAC for scaling RL post-training to free-form generation tasks.

Discrete-action reinforcement learning algorithms often falter in tasks with high-dimensional discrete action spaces due to the vast number of possible actions. A recent advancement leverages value-decomposition, a concept from multi-agent reinforcement learning, to tackle this challenge. This study delves deep into the effects of this value-decomposition, revealing that whilst it curtails the over-estimation bias inherent to Q-learning algorithms, it amplifies target variance. To counteract this, we present an ensemble of critics to mitigate target variance. Moreover, we introduce a regularisation loss that helps to mitigate the effects that exploratory actions in one dimension can have on the value of optimal actions in other dimensions. Our novel algorithm, REValueD, tested on discretised versions of the DeepMind Control Suite tasks, showcases superior performance, especially in the challenging humanoid and dog tasks. We further dissect the factors influencing REValueD's performance, evaluating the significance of the regularisation loss and the scalability of REValueD with increasing sub-actions per dimension.

Training critiquing language models to assess and provide feedback on model outputs is a promising way to improve LLMs for complex reasoning tasks. However, existing approaches typically rely on stronger supervisors for annotating critique data. To address this, we propose Critique-RL, an online RL approach for developing critiquing language models without stronger supervision. Our approach operates on a two-player paradigm: the actor generates a response, the critic provides feedback, and the actor refines the response accordingly. We first reveal that relying solely on indirect reward signals from the actor's outputs for RL optimization often leads to unsatisfactory critics: while their helpfulness (i.e., providing constructive feedback) improves, the discriminability (i.e., determining whether a response is high-quality or not) remains poor, resulting in marginal performance gains. To overcome this, Critique-RL adopts a two-stage optimization strategy. In stage I, it reinforces the discriminability of the critic with direct rule-based reward signals; in stage II, it introduces indirect rewards based on actor refinement to improve the critic's helpfulness, while maintaining its discriminability via appropriate regularization. Extensive experiments across various tasks and models show that Critique-RL delivers substantial performance improvements. For example, it achieves a 9.02% gain on in-domain tasks and a 5.70% gain on out-of-domain tasks for Qwen2.5-7B, highlighting its potential.

Policy gradient methods rely on a baseline to measure the relative advantage of an action, ensuring the model reinforces behaviors that outperform its current average capability. In the training of Large Language Models (LLMs) using Actor-Critic methods (e.g., PPO), this baseline is typically estimated by a Value Model (Critic) often as large as the policy model itself. However, as the policy continuously evolves, the value model requires expensive, synchronous incremental training to accurately track the shifting capabilities of the policy. To avoid this overhead, Group Relative Policy Optimization (GRPO) eliminates the coupled value model by using the average reward of a group of rollouts as the baseline; yet, this approach necessitates extensive sampling to maintain estimation stability. In this paper, we propose V_0, a Generalist Value Model capable of estimating the expected performance of any model on unseen prompts without requiring parameter updates. We reframe value estimation by treating the policy's dynamic capability as an explicit context input; specifically, we leverage a history of instruction-performance pairs to dynamically profile the model, departing from the traditional paradigm that relies on parameter fitting to perceive capability shifts. Focusing on value estimation at State Zero (i.e., the initial prompt, hence V_0), our model serves as a critical resource scheduler. During GRPO training, V_0 predicts success rates prior to rollout, allowing for efficient sampling budget allocation; during deployment, it functions as a router, dispatching instructions to the most cost-effective and suitable model. Empirical results demonstrate that V_0 significantly outperforms heuristic budget allocation and achieves a Pareto-optimal trade-off between performance and cost in LLM routing tasks.

Large language models (LLMs) have recently transformed from text-based assistants to autonomous agents capable of planning, reasoning, and iteratively improving their actions. While numerical reward signals and verifiers can effectively rank candidate actions, they often provide limited contextual guidance. In contrast, natural language feedback better aligns with the generative capabilities of LLMs, providing richer and more actionable suggestions. However, parsing and implementing this feedback effectively can be challenging for LLM-based agents. In this work, we introduce Critique-Guided Improvement (CGI), a novel two-player framework, comprising an actor model that explores an environment and a critic model that generates detailed nature language feedback. By training the critic to produce fine-grained assessments and actionable revisions, and the actor to utilize these critiques, our approach promotes more robust exploration of alternative strategies while avoiding local optima. Experiments in three interactive environments show that CGI outperforms existing baselines by a substantial margin. Notably, even a small critic model surpasses GPT-4 in feedback quality. The resulting actor achieves state-of-the-art performance, demonstrating the power of explicit iterative guidance to enhance decision-making in LLM-based agents.

This paper studies distributional model risk in marginal problems, where each marginal measure is assumed to lie in a Wasserstein ball centered at a fixed reference measure with a given radius. Theoretically, we establish several fundamental results including strong duality, finiteness of the proposed Wasserstein distributional model risk, and the existence of an optimizer at each radius. In addition, we show continuity of the Wasserstein distributional model risk as a function of the radius. Using strong duality, we extend the well-known Makarov bounds for the distribution function of the sum of two random variables with given marginals to Wasserstein distributionally robust Markarov bounds. Practically, we illustrate our results on four distinct applications when the sample information comes from multiple data sources and only some marginal reference measures are identified. They are: partial identification of treatment effects; externally valid treatment choice via robust welfare functions; Wasserstein distributionally robust estimation under data combination; and evaluation of the worst aggregate risk measures.

We study the problem of predicting numeric labels that are constrained to the integers or to a subrange of the integers. For example, the number of up-votes on social media posts, or the number of bicycles available at a public rental station. While it is possible to model these as continuous values, and to apply traditional regression, this approach changes the underlying distribution on the labels from discrete to continuous. Discrete distributions have certain benefits, which leads us to the question whether such integer labels can be modeled directly by a discrete distribution, whose parameters are predicted from the features of a given instance. Moreover, we focus on the use case of output distributions of neural networks, which adds the requirement that the parameters of the distribution be continuous so that backpropagation and gradient descent may be used to learn the weights of the network. We investigate several options for such distributions, some existing and some novel, and test them on a range of tasks, including tabular learning, sequential prediction and image generation. We find that overall the best performance comes from two distributions: Bitwise, which represents the target integer in bits and places a Bernoulli distribution on each, and a discrete analogue of the Laplace distribution, which uses a distribution with exponentially decaying tails around a continuous mean.

Deep reinforcement learning (RL) has achieved significant success, yet its application in real-world scenarios is often hindered by a lack of robustness to environmental uncertainties. To solve this challenge, some robust RL algorithms have been proposed, but most are limited to tabular settings. In this work, we propose Distributionally Robust Soft Actor-Critic (DR-SAC), a novel algorithm designed to enhance the robustness of the state-of-the-art Soft Actor-Critic (SAC) algorithm. DR-SAC aims to maximize the expected value with entropy against the worst possible transition model lying in an uncertainty set. A distributionally robust version of the soft policy iteration is derived with a convergence guarantee. For settings where nominal distributions are unknown, such as offline RL, a generative modeling approach is proposed to estimate the required nominal distributions from data. Furthermore, experimental results on a range of continuous control benchmark tasks demonstrate our algorithm achieves up to 9.8 times the average reward of the SAC baseline under common perturbations. Additionally, compared with existing robust reinforcement learning algorithms, DR-SAC significantly improves computing efficiency and applicability to large-scale problems.

Learning from human feedback (LHF) -- and in particular learning from pairwise preferences -- has recently become a crucial ingredient in training large language models (LLMs), and has been the subject of much research. Most recent works frame it as a reinforcement learning problem, where a reward function is learned from pairwise preference data and the LLM is treated as a policy which is adapted to maximize the rewards, often under additional regularization constraints. We propose an alternative interpretation which centers on the generative process for pairwise preferences and treats LHF as a density estimation problem. We provide theoretical and empirical results showing that for a family of generative processes defined via preference behavior distribution equations, training a reward function on pairwise preferences effectively models an annotator's implicit preference distribution. Finally, we discuss and present findings on "annotator misspecification" -- failure cases where wrong modeling assumptions are made about annotator behavior, resulting in poorly-adapted models -- suggesting that approaches that learn from pairwise human preferences could have trouble learning from a population of annotators with diverse viewpoints.

Flow-based generative models have shown remarkable success in text-to-image generation, yet fine-tuning them with intermediate feedback remains challenging, especially for continuous-time flow matching models. Most existing approaches solely learn from outcome rewards, struggling with the credit assignment problem. Alternative methods that attempt to learn a critic via direct regression on cumulative rewards often face training instabilities and model collapse in online settings. We present AC-Flow, a robust actor-critic framework that addresses these challenges through three key innovations: (1) reward shaping that provides well-normalized learning signals to enable stable intermediate value learning and gradient control, (2) a novel dual-stability mechanism that combines advantage clipping to prevent destructive policy updates with a warm-up phase that allows the critic to mature before influencing the actor, and (3) a scalable generalized critic weighting scheme that extends traditional reward-weighted methods while preserving model diversity through Wasserstein regularization. Through extensive experiments on Stable Diffusion 3, we demonstrate that AC-Flow achieves state-of-the-art performance in text-to-image alignment tasks and generalization to unseen human preference models. Our results demonstrate that even with a computationally efficient critic model, we can robustly finetune flow models without compromising generative quality, diversity, or stability.

Critical thinking is essential for rational decision-making and problem-solving. This skill hinges on the ability to provide precise and reasoned critiques and is a hallmark of human intelligence. In the era of large language models (LLMs), this study explores the ability of LLMs to deliver accurate critiques across various tasks. We are interested in this topic as a capable critic model could not only serve as a reliable evaluator, but also as a source of supervised signals for model tuning. Particularly, if a model can self-critique, it has the potential for autonomous self-improvement. To examine this, we introduce a unified evaluation framework for assessing the critique abilities of LLMs. We develop a benchmark called CriticBench, which comprises 3K high-quality natural language queries and corresponding model responses; and annotate the correctness of these responses. The benchmark cover tasks such as math problem-solving, code completion, and question answering. We evaluate multiple LLMs on the collected dataset and our analysis reveals several noteworthy insights: (1) Critique is generally challenging for most LLMs, and this capability often emerges only when models are sufficiently large. (2) In particular, self-critique is especially difficult. Even top-performing LLMs struggle to achieve satisfactory performance. (3) Models tend to have lower critique accuracy on problems where they are most uncertain. To this end, we introduce a simple yet effective baseline named self-check, which leverages self-critique to improve task performance for various models. We hope this study serves as an initial exploration into understanding the critique abilities of LLMs, and aims to inform future research, including the development of more proficient critic models and the application of critiques across diverse tasks.

Reinforcement Learning (RL) has emerged as a popular training paradigm, particularly when paired with reasoning models. While effective, it primarily focuses on generating responses and lacks mechanisms to explicitly foster critique or reflection. Several recent studies, like Critique-Fine-Tuning (CFT) and Critique-Guided-Distillation (CGD) have shown the benefits of explicitly teaching LLMs how to critique. Motivated by them, we propose Critique Reinforcement Learning (CRL), where the model is tasked with generating a critique for a given (question, solution) pair. The reward is determined solely by whether the final judgment label c in {True, False} of the generated critique aligns with the ground-truth judgment c^*. Building on this point, we introduce Critique-Coder, which is trained on a hybrid of RL and CRL by substituting 20\% of the standard RL data with CRL data. We fine-tune multiple models (Critique-Coder) and evaluate them on different benchmarks to show their advantages over RL-only models. We show that Critique-Coder consistently outperforms RL-only baselines on all the evaluated benchmarks. Notably, our Critique-Coder-8B can reach over 60\% on LiveCodeBench (v5), outperforming other reasoning models like DeepCoder-14B and GPT-o1. Beyond code generation, Critique-Coder also demonstrates enhanced general reasoning abilities, as evidenced by its better performance on logic reasoning tasks from the BBEH dataset. This indicates that the application of CRL on coding datasets enhances general reasoning and critique abilities, which are transferable across a broad range of tasks. Hence, we believe that CRL works as a great complement to standard RL for LLM reasoning.

The exploration-exploitation (EE) trade-off is a central challenge in reinforcement learning (RL) for large language models (LLMs). With Group Relative Policy Optimization (GRPO), training tends to be exploitation driven: entropy decreases monotonically, samples convergence, and exploration fades. Most existing fixes are sample-centric: they seek or bonus rare samples, assuming exploration comes from novel trajectories and tokens. These heuristics depend on the "luck" of informative samples, lack principled control of the policy, and often yield limited or inconsistent gains. In this work, we are the first to introduce a distribution-centric perspective for RL, in which exploration is always guided by a "better" target distribution, and reveal that a policy's ability to resist entropy collapse is governed by the distribution itself rather than individual samples. Building on this insight, we propose Distribution-Centric Policy Optimization (DCPO), which reformulates entropy regulation as distribution-level regularization. DCPO achieves controllable entropy fully on-policy without sampling from external distributions, enabling efficient exploration while maintaining training stability. Across multiple models and seven benchmarks, DCPO improves over GRPO by about 20\% on average. Overall, DCPO replaces sample-level heuristics with distribution-level principles, offering a theoretically grounded and flexible framework for controllable exploration and a stronger EE trade-off. The code is available in https://github.com/597358816/DCPO.

Large Language Models (LLMs) offer a promising basis for creating agents that can tackle complex tasks through iterative environmental interaction. Existing methods either require these agents to mimic expert-provided trajectories or rely on definitive environmental feedback for reinforcement learning which limits their application to specific scenarios like gaming or code generation. This paper introduces a novel training method for LLM-based agents using weakly supervised signals from a critic LLM, bypassing the need for expert trajectories or definitive feedback. Our agents are trained in iterative manner, where they initially generate trajectories through environmental interaction. Subsequently, a critic LLM selects a subset of good trajectories, which are then used to update the agents, enabling them to generate improved trajectories in the next iteration. Extensive tests on the API-bank dataset show consistent improvement in our agents' capabilities and comparable performance to GPT-4, despite using open-source models with much fewer parameters.

Reinforcement learning (RL) post-training is crucial for LLM alignment and reasoning, but existing policy-based methods, such as PPO and DPO, can fall short of fixing shortcuts inherited from pre-training. In this work, we introduce Qsharp, a value-based algorithm for KL-regularized RL that guides the reference policy using the optimal regularized Q function. We propose to learn the optimal Q function using distributional RL on an aggregated online dataset. Unlike prior value-based baselines that guide the model using unregularized Q-values, our method is theoretically principled and provably learns the optimal policy for the KL-regularized RL problem. Empirically, Qsharp outperforms prior baselines in math reasoning benchmarks while maintaining a smaller KL divergence to the reference policy. Theoretically, we establish a reduction from KL-regularized RL to no-regret online learning, providing the first bounds for deterministic MDPs under only realizability. Thanks to distributional RL, our bounds are also variance-dependent and converge faster when the reference policy has small variance. In sum, our results highlight Qsharp as an effective approach for post-training LLMs, offering both improved performance and theoretical guarantees. The code can be found at https://github.com/jinpz/q_sharp.

In recent years, Multimodal Large Language Models (MLLMs) have been extensively utilized for multimodal reasoning tasks, including Graphical User Interface (GUI) automation. Unlike general offline multimodal tasks, GUI automation is executed in online interactive environments, necessitating step-by-step decision-making based on real-time status of the environment. This task has a lower tolerance for decision-making errors at each step, as any mistakes may cumulatively disrupt the process and potentially lead to irreversible outcomes like deletions or payments. To address these issues, we introduce a pre-operative critic mechanism that provides effective feedback prior to the actual execution, by reasoning about the potential outcome and correctness of actions. Specifically, we propose a Suggestion-aware Gradient Relative Policy Optimization (S-GRPO) strategy to construct our pre-operative critic model GUI-Critic-R1, incorporating a novel suggestion reward to enhance the reliability of the model's feedback. Furthermore, we develop a reasoning-bootstrapping based data collection pipeline to create a GUI-Critic-Train and a GUI-Critic-Test, filling existing gaps in GUI critic data. Static experiments on the GUI-Critic-Test across both mobile and web domains reveal that our GUI-Critic-R1 offers significant advantages in critic accuracy compared to current MLLMs. Dynamic evaluation on GUI automation benchmark further highlights the effectiveness and superiority of our model, as evidenced by improved success rates and operational efficiency.

Actor-critic algorithms have become a cornerstone in reinforcement learning (RL), leveraging the strengths of both policy-based and value-based methods. Despite recent progress in understanding their statistical efficiency, no existing work has successfully learned an epsilon-optimal policy with a sample complexity of O(1/epsilon^2) trajectories with general function approximation when strategic exploration is necessary. We address this open problem by introducing a novel actor-critic algorithm that attains a sample-complexity of O(dH^5 log|A|/epsilon^2 + d H^4 log|F|/ epsilon^2) trajectories, and accompanying T regret when the Bellman eluder dimension d does not increase with T at more than a log T rate. Here, F is the critic function class, A is the action space, and H is the horizon in the finite horizon MDP setting. Our algorithm integrates optimism, off-policy critic estimation targeting the optimal Q-function, and rare-switching policy resets. We extend this to the setting of Hybrid RL, showing that initializing the critic with offline data yields sample efficiency gains compared to purely offline or online RL. Further, utilizing access to offline data, we provide a non-optimistic provably efficient actor-critic algorithm that only additionally requires N_{off} geq c_{off}^*dH^4/epsilon^2 in exchange for omitting optimism, where c_{off}^* is the single-policy concentrability coefficient and N_{off} is the number of offline samples. This addresses another open problem in the literature. We further provide numerical experiments to support our theoretical findings.

As large language models (LLMs) are increasingly used across various applications, there is a growing need to control text generation to satisfy specific constraints or requirements. This raises a crucial question: Is it possible to guarantee strict constraint satisfaction in generated outputs while preserving the distribution of the original model as much as possible? We first define the ideal distribution - the one closest to the original model, which also always satisfies the expressed constraint - as the ultimate goal of guaranteed generation. We then state a fundamental limitation, namely that it is impossible to reach that goal through autoregressive training alone. This motivates the necessity of combining training-time and inference-time methods to enforce such guarantees. Based on this insight, we propose GUARD, a simple yet effective approach that combines an autoregressive proposal distribution with rejection sampling. Through GUARD's theoretical properties, we show how controlling the KL divergence between a specific proposal and the target ideal distribution simultaneously optimizes inference speed and distributional closeness. To validate these theoretical concepts, we conduct extensive experiments on two text generation settings with hard-to-satisfy constraints: a lexical constraint scenario and a sentiment reversal scenario. These experiments show that GUARD achieves perfect constraint satisfaction while almost preserving the ideal distribution with highly improved inference efficiency. GUARD provides a principled approach to enforcing strict guarantees for LLMs without compromising their generative capabilities.

Neural architecture search (NAS) has achieved remarkable results in deep neural network design. Differentiable architecture search converts the search over discrete architectures into a hyperparameter optimization problem which can be solved by gradient descent. However, questions have been raised regarding the effectiveness and generalizability of gradient methods for solving non-convex architecture hyperparameter optimization problems. In this paper, we propose L^{2}NAS, which learns to intelligently optimize and update architecture hyperparameters via an actor neural network based on the distribution of high-performing architectures in the search history. We introduce a quantile-driven training procedure which efficiently trains L^{2}NAS in an actor-critic framework via continuous-action reinforcement learning. Experiments show that L^{2}NAS achieves state-of-the-art results on NAS-Bench-201 benchmark as well as DARTS search space and Once-for-All MobileNetV3 search space. We also show that search policies generated by L^{2}NAS are generalizable and transferable across different training datasets with minimal fine-tuning.

Despite their unprecedented success, even the largest language models make mistakes. Similar to how humans learn and improve using feedback, previous work proposed providing language models with natural language feedback to guide them in repairing their outputs. Because human-generated critiques are expensive to obtain, researchers have devised learned critique generators in lieu of human critics while assuming one can train downstream models to utilize generated feedback. However, this approach does not apply to black-box or limited access models such as ChatGPT, as they cannot be fine-tuned. Moreover, in the era of large general-purpose language agents, fine-tuning is neither computationally nor spatially efficient as it results in multiple copies of the network. In this work, we introduce RL4F (Reinforcement Learning for Feedback), a multi-agent collaborative framework where the critique generator is trained to maximize end-task performance of GPT-3, a fixed model more than 200 times its size. RL4F produces critiques that help GPT-3 revise its outputs. We study three datasets for action planning, summarization and alphabetization and show improvements (~5% on average) in multiple text similarity metrics over strong baselines across all three tasks.

We give the first efficient algorithm for learning halfspaces in the testable learning model recently defined by Rubinfeld and Vasilyan (2023). In this model, a learner certifies that the accuracy of its output hypothesis is near optimal whenever the training set passes an associated test, and training sets drawn from some target distribution -- e.g., the Gaussian -- must pass the test. This model is more challenging than distribution-specific agnostic or Massart noise models where the learner is allowed to fail arbitrarily if the distributional assumption does not hold. We consider the setting where the target distribution is Gaussian (or more generally any strongly log-concave distribution) in d dimensions and the noise model is either Massart or adversarial (agnostic). For Massart noise, our tester-learner runs in polynomial time and outputs a hypothesis with (information-theoretically optimal) error opt + epsilon for any strongly log-concave target distribution. For adversarial noise, our tester-learner obtains error O(opt) + epsilon in polynomial time when the target distribution is Gaussian; for strongly log-concave distributions, we obtain O(opt) + epsilon in quasipolynomial time. Prior work on testable learning ignores the labels in the training set and checks that the empirical moments of the covariates are close to the moments of the base distribution. Here we develop new tests of independent interest that make critical use of the labels and combine them with the moment-matching approach of Gollakota et al. (2023). This enables us to simulate a variant of the algorithm of Diakonikolas et al. (2020) for learning noisy halfspaces using nonconvex SGD but in the testable learning setting.

The rapid deployment of AI systems in high-stakes domains, including those classified as high-risk under the The EU AI Act (Regulation (EU) 2024/1689), has intensified the need for reliable compliance auditing. For binary classifiers, regulatory risk assessment often relies on global fairness metrics such as the Disparate Impact ratio, widely used to evaluate potential discrimination. In typical auditing settings, the auditee provides a subset of its dataset to an auditor, while a supervisory authority may verify whether this subset is representative of the full underlying distribution. In this work, we investigate to what extent a malicious auditee can construct a fairness-compliant yet representative-looking sample from a non-compliant original distribution, thereby creating an illusion of fairness. We formalize this problem as a constrained distributional projection task and introduce mathematically grounded manipulation strategies based on entropic and optimal transport projections. These constructions characterize the minimal distributional shift required to satisfy fairness constraints. To counter such attacks, we formalize representativeness through distributional distance based statistical tests and systematically evaluate their ability to detect manipulated samples. Our analysis highlights the conditions under which fairness manipulation can remain statistically undetected and provides practical guidelines for strengthening supervisory verification. We validate our theoretical findings through experiments on standard tabular datasets for bias detection. Code is publicly available at https://github.com/ValentinLafargue/Inspection.

We propose a novel framework for incorporating unlabeled data into semi-supervised classification problems, where scenarios involving the minimization of either i) adversarially robust or ii) non-robust loss functions have been considered. Notably, we allow the unlabeled samples to deviate slightly (in total variation sense) from the in-domain distribution. The core idea behind our framework is to combine Distributionally Robust Optimization (DRO) with self-supervised training. As a result, we also leverage efficient polynomial-time algorithms for the training stage. From a theoretical standpoint, we apply our framework on the classification problem of a mixture of two Gaussians in R^d, where in addition to the m independent and labeled samples from the true distribution, a set of n (usually with ngg m) out of domain and unlabeled samples are given as well. Using only the labeled data, it is known that the generalization error can be bounded by proptoleft(d/mright)^{1/2}. However, using our method on both isotropic and non-isotropic Gaussian mixture models, one can derive a new set of analytically explicit and non-asymptotic bounds which show substantial improvement on the generalization error compared to ERM. Our results underscore two significant insights: 1) out-of-domain samples, even when unlabeled, can be harnessed to narrow the generalization gap, provided that the true data distribution adheres to a form of the ``cluster assumption", and 2) the semi-supervised learning paradigm can be regarded as a special case of our framework when there are no distributional shifts. We validate our claims through experiments conducted on a variety of synthetic and real-world datasets.

We present Distributional RewArds for Generative OptimizatioN (DRAGON), a versatile framework for fine-tuning media generation models towards a desired outcome. Compared with traditional reinforcement learning with human feedback (RLHF) or pairwise preference approaches such as direct preference optimization (DPO), DRAGON is more flexible. It can optimize reward functions that evaluate either individual examples or distributions of them, making it compatible with a broad spectrum of instance-wise, instance-to-distribution, and distribution-to-distribution rewards. Leveraging this versatility, we construct novel reward functions by selecting an encoder and a set of reference examples to create an exemplar distribution. When cross-modality encoders such as CLAP are used, the reference examples may be of a different modality (e.g., text versus audio). Then, DRAGON gathers online and on-policy generations, scores them to construct a positive demonstration set and a negative set, and leverages the contrast between the two sets to maximize the reward. For evaluation, we fine-tune an audio-domain text-to-music diffusion model with 20 different reward functions, including a custom music aesthetics model, CLAP score, Vendi diversity, and Frechet audio distance (FAD). We further compare instance-wise (per-song) and full-dataset FAD settings while ablating multiple FAD encoders and reference sets. Over all 20 target rewards, DRAGON achieves an 81.45% average win rate. Moreover, reward functions based on exemplar sets indeed enhance generations and are comparable to model-based rewards. With an appropriate exemplar set, DRAGON achieves a 60.95% human-voted music quality win rate without training on human preference annotations. As such, DRAGON exhibits a new approach to designing and optimizing reward functions for improving human-perceived quality. Sound examples at https://ml-dragon.github.io/web.

Proximal Policy Optimization (PPO)-based Reinforcement Learning from Human Feedback (RLHF) is essential for aligning large language models (LLMs) with human preferences. It requires joint training of an actor and critic with a pretrained, fixed reward model for guidance. This approach increases computational complexity and instability due to actor-critic interdependence. Additionally, PPO lacks access to true environment rewards in LLM tasks, limiting its adaptability. Under such conditions, pretraining a value model or a reward model becomes equivalent, as both provide fixed supervisory signals without new ground-truth feedback. To address these issues, we propose Decoupled Value Policy Optimization (DVPO), a lean framework that replaces traditional reward modeling with a pretrained global value model (GVM). The GVM is conditioned on policy trajectories and predicts token-level return-to-go estimates. By decoupling value model from policy training (via frozen GVM-driven RL objectives), DVPO eliminates actor-critic interdependence, reducing GPU memory usage by 40\% and training time by 35\% compared to conventional RLHF. Experiments across benchmarks show DVPO outperforms efficient RLHF methods (e.g., DPO) while matching state-of-the-art PPO in performance.

In value-based reinforcement learning methods such as deep Q-learning, function approximation errors are known to lead to overestimated value estimates and suboptimal policies. We show that this problem persists in an actor-critic setting and propose novel mechanisms to minimize its effects on both the actor and the critic. Our algorithm builds on Double Q-learning, by taking the minimum value between a pair of critics to limit overestimation. We draw the connection between target networks and overestimation bias, and suggest delaying policy updates to reduce per-update error and further improve performance. We evaluate our method on the suite of OpenAI gym tasks, outperforming the state of the art in every environment tested.

Reinforcement learning from human feedback (RLHF) is a promising solution to align large language models (LLMs) more closely with human values. Off-policy preference optimization, where the preference data is obtained from other models, is widely adopted due to its cost efficiency and scalability. However, off-policy preference optimization often suffers from a distributional gap between the policy used for data collection and the target policy, leading to suboptimal optimization. In this paper, we propose a novel strategy to mitigate this problem by simulating on-policy learning with off-policy preference data. Our Weighted Preference Optimization (WPO) method adapts off-policy data to resemble on-policy data more closely by reweighting preference pairs according to their probability under the current policy. This method not only addresses the distributional gap problem but also enhances the optimization process without incurring additional costs. We validate our method on instruction following benchmarks including Alpaca Eval 2 and MT-bench. WPO not only outperforms Direct Preference Optimization (DPO) by up to 5.6% on Alpaca Eval 2 but also establishes a remarkable length-controlled winning rate against GPT-4-turbo of 48.6% based on Llama-3-8B-Instruct, making it the strongest 8B model on the leaderboard. We will release the code and models at https://github.com/wzhouad/WPO.

We study the problem of learning a single neuron with respect to the L_2^2-loss in the presence of adversarial label noise. We give an efficient algorithm that, for a broad family of activations including ReLUs, approximates the optimal L_2^2-error within a constant factor. Our algorithm applies under much milder distributional assumptions compared to prior work. The key ingredient enabling our results is a novel connection to local error bounds from optimization theory.

It is well known that accurate probabilistic predictors can be trained through empirical risk minimisation with proper scoring rules as loss functions. While such learners capture so-called aleatoric uncertainty of predictions, various machine learning methods have recently been developed with the goal to let the learner also represent its epistemic uncertainty, i.e., the uncertainty caused by a lack of knowledge and data. An emerging branch of the literature proposes the use of a second-order learner that provides predictions in terms of distributions on probability distributions. However, recent work has revealed serious theoretical shortcomings for second-order predictors based on loss minimisation. In this paper, we generalise these findings and prove a more fundamental result: There seems to be no loss function that provides an incentive for a second-order learner to faithfully represent its epistemic uncertainty in the same manner as proper scoring rules do for standard (first-order) learners. As a main mathematical tool to prove this result, we introduce the generalised notion of second-order scoring rules.

Reinforcement learning for LLM reasoning has rapidly emerged as a prominent research area, marked by a significant surge in related studies on both algorithmic innovations and practical applications. Despite this progress, several critical challenges remain, including the absence of standardized guidelines for employing RL techniques and a fragmented understanding of their underlying mechanisms. Additionally, inconsistent experimental settings, variations in training data, and differences in model initialization have led to conflicting conclusions, obscuring the key characteristics of these techniques and creating confusion among practitioners when selecting appropriate techniques. This paper systematically reviews widely adopted RL techniques through rigorous reproductions and isolated evaluations within a unified open-source framework. We analyze the internal mechanisms, applicable scenarios, and core principles of each technique through fine-grained experiments, including datasets of varying difficulty, model sizes, and architectures. Based on these insights, we present clear guidelines for selecting RL techniques tailored to specific setups, and provide a reliable roadmap for practitioners navigating the RL for the LLM domain. Finally, we reveal that a minimalist combination of two techniques can unlock the learning capability of critic-free policies using vanilla PPO loss. The results demonstrate that our simple combination consistently improves performance, surpassing strategies like GRPO and DAPO.

With the rapid advancement of Large Language Models (LLMs), developing effective critic modules for precise guidance has become crucial yet challenging. In this paper, we initially demonstrate that supervised fine-tuning for building critic modules (which is widely adopted in current solutions) fails to genuinely enhance models' critique abilities, producing superficial critiques with insufficient reflections and verifications. To unlock the unprecedented critique capabilities, we propose RefCritic, a long-chain-of-thought critic module based on reinforcement learning with dual rule-based rewards: (1) instance-level correctness of solution judgments and (2) refinement accuracies of the policy model based on critiques, aiming to generate high-quality evaluations with actionable feedback that effectively guides model refinement. We evaluate RefCritic on Qwen2.5-14B-Instruct and DeepSeek-R1-Distill-Qwen-14B across five benchmarks. On critique and refinement settings, RefCritic demonstrates consistent advantages across all benchmarks, e.g., 6.8\% and 7.2\% gains on AIME25 for the respective base models. Notably, under majority voting, policy models filtered by RefCritic show superior scaling with increased voting numbers. Moreover, despite training on solution-level supervision, RefCritic outperforms step-level supervised approaches on ProcessBench, a benchmark to identify erroneous steps in mathematical reasoning.

State-of-the-art large language models (LLMs) exhibit impressive problem-solving capabilities but may struggle with complex reasoning and factual correctness. Existing methods harness the strengths of chain-of-thought and retrieval-augmented generation (RAG) to decompose a complex problem into simpler steps and apply retrieval to improve factual correctness. These methods work well on straightforward reasoning tasks but often falter on challenging tasks such as competitive programming and mathematics, due to frequent reasoning errors and irrelevant knowledge retrieval. To address this, we introduce Critic-guided planning with Retrieval-augmentation, CR-Planner, a novel framework that leverages fine-tuned critic models to guide both reasoning and retrieval processes through planning. CR-Planner solves a problem by iteratively selecting and executing sub-goals. Initially, it identifies the most promising sub-goal from reasoning, query generation, and retrieval, guided by rewards given by a critic model named sub-goal critic. It then executes this sub-goal through sampling and selecting the optimal output based on evaluations from another critic model named execution critic. This iterative process, informed by retrieved information and critic models, enables CR-Planner to effectively navigate the solution space towards the final answer. We employ Monte Carlo Tree Search to collect the data for training the critic models, allowing for a systematic exploration of action sequences and their long-term impacts. We validate CR-Planner on challenging domain-knowledge-intensive and reasoning-heavy tasks, including competitive programming, theorem-driven math reasoning, and complex domain retrieval problems. Our experiments demonstrate that CR-Planner significantly outperforms baselines, highlighting its effectiveness in addressing challenging problems by improving both reasoning and retrieval.

Post-training processes are essential phases in grounding pre-trained language models to real-world tasks, with learning from demonstrations or preference signals playing a crucial role in this adaptation. We present a unified theoretical framework bridging Supervised Fine-Tuning (SFT) and preference learning in Large Language Model (LLM) post-training. Through rigorous mathematical derivation, we demonstrate that both SFT and preference learning methods like Direct Preference Optimization (DPO) operate within the same optimal policy-reward subspace, with SFT representing a special case of implicit reward learning. Our analysis reveals a critical limitation in conventional SFT: the KL divergence term in distribution matching becomes constant with respect to the policy during optimization, failing to constrain model updates. To address this, we propose a simple yet effective learning rate reduction approach that yields significant performance improvements (up to 25\% relative gain and 6\% absolute win rate increase in instruction following tasks. Additionally, we derive alternative SFT objectives from various f-divergence functions that preserve the KL term during optimization, further enhancing post-DPO model performance. Finally, we extend the theoretical relationship between LLM logits and Q-functions from preference learning to the SFT context, providing mathematical derivations and experimental validation.

Generative models form the backbone of modern machine learning, underpinning state-of-the-art systems in text, vision, and multimodal applications. While Maximum Likelihood Estimation has traditionally served as the dominant training paradigm, recent work have highlighted its limitations, particularly in generalization and susceptibility to catastrophic forgetting compared to Reinforcement Learning techniques, such as Policy Gradient methods. However, these approaches depend on explicit reward signals, which are often unavailable in practice, leaving open the fundamental problem of how to align generative models when only high-quality datasets are accessible. In this work, we address this challenge via a Bilevel Optimization framework, where the reward function is treated as the optimization variable of an outer-level problem, while a policy gradient objective defines the inner-level. We then conduct a theoretical analysis of this optimization problem in a tractable setting and extract insights that, as we demonstrate, generalize to applications such as tabular classification and model-based reinforcement learning. We release the code at https://github.com/abenechehab/nll_to_po .

When the performance of a machine learning model varies over groups defined by sensitive attributes (e.g., gender or ethnicity), the performance disparity can be expressed in terms of the probability distributions of the input and output variables over each group. In this paper, we exploit this fact to reduce the disparate impact of a fixed classification model over a population of interest. Given a black-box classifier, we aim to eliminate the performance gap by perturbing the distribution of input variables for the disadvantaged group. We refer to the perturbed distribution as a counterfactual distribution, and characterize its properties for common fairness criteria. We introduce a descent algorithm to learn a counterfactual distribution from data. We then discuss how the estimated distribution can be used to build a data preprocessor that can reduce disparate impact without training a new model. We validate our approach through experiments on real-world datasets, showing that it can repair different forms of disparity without a significant drop in accuracy.

Critique ability, a meta-cognitive capability of humans, presents significant challenges for LLMs to improve. Recent works primarily rely on supervised fine-tuning (SFT) using critiques generated by a single LLM like GPT-4. However, these model-generated critiques often exhibit flaws due to the inherent complexity of the critique. Consequently, fine-tuning LLMs on such flawed critiques typically limits the model's performance and propagates these flaws into the learned model. To overcome these challenges, this paper proposes a novel data generation pipeline, named MultiCritique, that improves the critique ability of LLMs by utilizing multi-agent feedback in both the SFT and reinforcement learning (RL) stages. First, our data generation pipeline aggregates high-quality critiques from multiple agents instead of a single model, with crucial information as input for simplifying the critique. Furthermore, our pipeline improves the preference accuracy of critique quality through multi-agent feedback, facilitating the effectiveness of RL in improving the critique ability of LLMs. Based on our proposed MultiCritique data generation pipeline, we construct the MultiCritiqueDataset for the SFT and RL fine-tuning stages. Extensive experimental results on two benchmarks demonstrate: 1) the superior quality of our constructed SFT dataset compared to existing critique datasets; 2) additional improvements to the critique ability of LLMs brought by the RL stage. Notably, our fine-tuned 7B model significantly surpasses other advanced 7B-13B open-source models, approaching the performance of advanced 70B LLMs and GPT-4. Codes, datasets and model weights will be publicly available.

The modeling of probability distributions, specifically generative modeling and density estimation, has become an immensely popular subject in recent years by virtue of its outstanding performance on sophisticated data such as images and texts. Nevertheless, a theoretical understanding of its success is still incomplete. One mystery is the paradox between memorization and generalization: In theory, the model is trained to be exactly the same as the empirical distribution of the finite samples, whereas in practice, the trained model can generate new samples or estimate the likelihood of unseen samples. Likewise, the overwhelming diversity of distribution learning models calls for a unified perspective on this subject. This paper provides a mathematical framework such that all the well-known models can be derived based on simple principles. To demonstrate its efficacy, we present a survey of our results on the approximation error, training error and generalization error of these models, which can all be established based on this framework. In particular, the aforementioned paradox is resolved by proving that these models enjoy implicit regularization during training, so that the generalization error at early-stopping avoids the curse of dimensionality. Furthermore, we provide some new results on landscape analysis and the mode collapse phenomenon.

While efficient distribution learning is no doubt behind the groundbreaking success of diffusion modeling, its theoretical guarantees are quite limited. In this paper, we provide the first rigorous analysis on approximation and generalization abilities of diffusion modeling for well-known function spaces. The highlight of this paper is that when the true density function belongs to the Besov space and the empirical score matching loss is properly minimized, the generated data distribution achieves the nearly minimax optimal estimation rates in the total variation distance and in the Wasserstein distance of order one. Furthermore, we extend our theory to demonstrate how diffusion models adapt to low-dimensional data distributions. We expect these results advance theoretical understandings of diffusion modeling and its ability to generate verisimilar outputs.

We consider the distributionally robust optimization (DRO) problem with spectral risk-based uncertainty set and f-divergence penalty. This formulation includes common risk-sensitive learning objectives such as regularized condition value-at-risk (CVaR) and average top-k loss. We present Prospect, a stochastic gradient-based algorithm that only requires tuning a single learning rate hyperparameter, and prove that it enjoys linear convergence for smooth regularized losses. This contrasts with previous algorithms that either require tuning multiple hyperparameters or potentially fail to converge due to biased gradient estimates or inadequate regularization. Empirically, we show that Prospect can converge 2-3times faster than baselines such as stochastic gradient and stochastic saddle-point methods on distribution shift and fairness benchmarks spanning tabular, vision, and language domains.

Academic benchmarks for coding agents tend to reward autonomous task completion, measured by verifiable rewards such as unit-test success. In contrast, real-world coding agents operate with humans in the loop, where success signals are typically noisy, delayed, and sparse. How can we bridge this gap? In this paper, we propose a process to learn a "critic" model from sparse and noisy interaction data, which can then be used both as a reward model for either RL-based training or inference-time scaling. Specifically, we introduce Critic Rubrics, a rubric-based supervision framework with 24 behavioral features that can be derived from human-agent interaction traces alone. Using a semi-supervised objective, we can then jointly predict these rubrics and sparse human feedback (when present). In experiments, we demonstrate that, despite being trained primarily from trace-observable rubrics and sparse real-world outcome proxies, these critics improve best-of-N reranking on SWE-bench (Best@8 +15.9 over Random@8 over the rerankable subset of trajectories), enable early stopping (+17.7 with 83% fewer attempts), and support training-time data curation via critic-selected trajectories.

In reinforcement learning and imitation learning, an object of central importance is the state distribution induced by the policy. It plays a crucial role in the policy gradient theorem, and references to it--along with the related state-action distribution--can be found all across the literature. Despite its importance, the state distribution is mostly discussed indirectly and theoretically, rather than being modeled explicitly. The reason being an absence of appropriate density estimation tools. In this work, we investigate applications of a normalizing flow-based model for the aforementioned distributions. In particular, we use a pair of flows coupled through the optimality point of the Donsker-Varadhan representation of the Kullback-Leibler (KL) divergence, for distribution matching based imitation learning. Our algorithm, Coupled Flow Imitation Learning (CFIL), achieves state-of-the-art performance on benchmark tasks with a single expert trajectory and extends naturally to a variety of other settings, including the subsampled and state-only regimes.

We study a family of loss functions named label-distributionally robust (LDR) losses for multi-class classification that are formulated from distributionally robust optimization (DRO) perspective, where the uncertainty in the given label information are modeled and captured by taking the worse case of distributional weights. The benefits of this perspective are several fold: (i) it provides a unified framework to explain the classical cross-entropy (CE) loss and SVM loss and their variants, (ii) it includes a special family corresponding to the temperature-scaled CE loss, which is widely adopted but poorly understood; (iii) it allows us to achieve adaptivity to the uncertainty degree of label information at an instance level. Our contributions include: (1) we study both consistency and robustness by establishing top-k (forall kgeq 1) consistency of LDR losses for multi-class classification, and a negative result that a top-1 consistent and symmetric robust loss cannot achieve top-k consistency simultaneously for all kgeq 2; (2) we propose a new adaptive LDR loss that automatically adapts the individualized temperature parameter to the noise degree of class label of each instance; (3) we demonstrate stable and competitive performance for the proposed adaptive LDR loss on 7 benchmark datasets under 6 noisy label and 1 clean settings against 13 loss functions, and on one real-world noisy dataset. The code is open-sourced at https://github.com/Optimization-AI/ICML2023_LDR.

Machine learning models frequently experience performance drops under distribution shifts. The underlying cause of such shifts may be multiple simultaneous factors such as changes in data quality, differences in specific covariate distributions, or changes in the relationship between label and features. When a model does fail during deployment, attributing performance change to these factors is critical for the model developer to identify the root cause and take mitigating actions. In this work, we introduce the problem of attributing performance differences between environments to distribution shifts in the underlying data generating mechanisms. We formulate the problem as a cooperative game where the players are distributions. We define the value of a set of distributions to be the change in model performance when only this set of distributions has changed between environments, and derive an importance weighting method for computing the value of an arbitrary set of distributions. The contribution of each distribution to the total performance change is then quantified as its Shapley value. We demonstrate the correctness and utility of our method on synthetic, semi-synthetic, and real-world case studies, showing its effectiveness in attributing performance changes to a wide range of distribution shifts.

Reinforcement Learning has significantly advanced the reasoning capabilities of Multimodal Large Language Models (MLLMs), yet the resulting policies remain brittle against real-world visual degradations such as blur, compression artifacts, and low-resolution scans. Prior robustness techniques from vision and deep RL rely on static data augmentation or value-based regularization, neither of which transfers cleanly to critic-free RL fine-tuning of autoregressive MLLMs. Reinforcing reasoning against such corruptions is non-trivial: naively injecting degraded views during rollout induces reward poisoning, where perceptual occlusions trigger hallucinated trajectories and destabilize optimization. We propose ROMA, an RL fine-tuning framework that modifies the optimization dynamics to reinforce reasoning against visual degradation while preserving clean-input performance. A dual-forward-pass strategy uses teacher forcing to evaluate corrupted views against clean-image trajectories, avoiding new rollouts on degraded inputs. For distributional consistency, we apply a token-level surrogate KL penalty against the worst-case augmentation; to prevent policy collapse under regularization, an auxiliary policy gradient loss anchored to clean-image advantages preserves a reliable reward signal; and to avoid systematically incorrect invariance, correctness-conditioned regularization restricts enforcement to successful trajectories. On Qwen3-VL 4B/8B across seven multimodal reasoning benchmarks, our method improves robustness by +2.4% on seen and +2.3% on unseen corruptions over GRPO while matching clean accuracy.

Critic-free methods like GRPO reduce memory demands by estimating advantages from multiple rollouts but tend to converge slowly, as critical learning signals are diluted by an abundance of uninformative samples and tokens. To tackle this challenge, we propose the Dynamic Dual-Level Down-Sampling (D^3S) framework that prioritizes the most informative samples and tokens across groups to improve the efficient of policy optimization. D^3S operates along two levels: (1) the sample-level, which selects a subset of rollouts to maximize advantage variance (Var(A)). We theoretically proven that this selection is positively correlated with the upper bound of the policy gradient norms, yielding higher policy gradients. (2) the token-level, which prioritizes tokens with a high product of advantage magnitude and policy entropy (|A_{i,t}|times H_{i,t}), focusing updates on tokens where the policy is both uncertain and impactful. Moreover, to prevent overfitting to high-signal data, D^3S employs a dynamic down-sampling schedule inspired by curriculum learning. This schedule starts with aggressive down-sampling to accelerate early learning and gradually relaxes to promote robust generalization. Extensive experiments on Qwen2.5 and Llama3.1 demonstrate that integrating D^3S into advanced RL algorithms achieves state-of-the-art performance and generalization while requiring fewer samples and tokens across diverse reasoning benchmarks. Our code is added in the supplementary materials and will be made publicly available.

Current critic-free RL methods for large reasoning models suffer from severe inefficiency when training on positive homogeneous prompts (where all rollouts are correct), resulting in waste of rollouts due to zero advantage estimates. We introduce a radically simple yet powerful solution to Mine intrinsic mastery (Miner), that repurposes the policy's intrinsic uncertainty as a self-supervised reward signal, with no external supervision, auxiliary models, or additional inference cost. Our method pioneers two key innovations: (1) a token-level focal credit assignment mechanism that dynamically amplifies gradients on critical uncertain tokens while suppressing overconfident ones, and (2) adaptive advantage calibration to seamlessly integrate intrinsic and verifiable rewards. Evaluated across six reasoning benchmarks on Qwen3-4B and Qwen3-8B base models, Miner achieves state-of-the-art performance among the other four algorithms, yielding up to 4.58 absolute gains in Pass@1 and 6.66 gains in Pass@K compared to GRPO. Comparison with other methods targeted at exploration enhancement further discloses the superiority of the two newly proposed innovations. This demonstrates that latent uncertainty exploitation is both necessary and sufficient for efficient and scalable RL training of reasoning models.

Recent advances in reinforcement learning (RL) with numerical feedback, such as scalar rewards, have significantly enhanced the complex reasoning capabilities of large language models (LLMs). Despite this success, we identify three key challenges encountered by RL with solely numerical feedback: performance plateaus, limited effectiveness of self-reflection, and persistent failures. We then demonstrate that RL-finetuned models, even after exhibiting performance plateaus, can generate correct refinements on persistently failed problems by leveraging natural language feedback in the form of critiques. Building on this insight, we propose Critique-GRPO, an online RL framework that integrates both natural language and numerical feedback for effective policy optimization. Critique-GRPO enables LLMs to learn from initial responses and critique-guided refinements simultaneously while maintaining exploration. Extensive experiments using Qwen2.5-7B-Base and Qwen3-8B-Base show that Critique-GRPO consistently outperforms supervised learning-based and RL-based fine-tuning approaches across eight challenging mathematical, STEM, and general reasoning tasks, improving average pass@1 scores by approximately 4.5% and 5%, respectively. Notably, Critique-GRPO surpasses a strong baseline that incorporates expert demonstrations within online RL. Further analysis reveals two critical insights about policy exploration: (1) higher entropy does not always guarantee efficient learning from exploration, and (2) longer responses do not necessarily lead to more effective exploration.

This article introduces Regression Discontinuity Design (RDD) with Distribution-Valued Outcomes (R3D), extending the standard RDD framework to settings where the outcome is a distribution rather than a scalar. Such settings arise when treatment is assigned at a higher level of aggregation than the outcome-for example, when a subsidy is allocated based on a firm-level revenue cutoff while the outcome of interest is the distribution of employee wages within the firm. Since standard RDD methods cannot accommodate such two-level randomness, I propose a novel approach based on random distributions. The target estimand is a "local average quantile treatment effect", which averages across random quantiles. To estimate this target, I introduce two related approaches: one that extends local polynomial regression to random quantiles and another based on local Fr\'echet regression, a form of functional regression. For both estimators, I establish asymptotic normality and develop uniform, debiased confidence bands together with a data-driven bandwidth selection procedure. Simulations validate these theoretical properties and show existing methods to be biased and inconsistent in this setting. I then apply the proposed methods to study the effects of gubernatorial party control on within-state income distributions in the US, using a close-election design. The results suggest a classic equality-efficiency tradeoff under Democratic governorship, driven by reductions in income at the top of the distribution.

This paper considers the Pointer Value Retrieval (PVR) benchmark introduced in [ZRKB21], where a 'reasoning' function acts on a string of digits to produce the label. More generally, the paper considers the learning of logical functions with gradient descent (GD) on neural networks. It is first shown that in order to learn logical functions with gradient descent on symmetric neural networks, the generalization error can be lower-bounded in terms of the noise-stability of the target function, supporting a conjecture made in [ZRKB21]. It is then shown that in the distribution shift setting, when the data withholding corresponds to freezing a single feature (referred to as canonical holdout), the generalization error of gradient descent admits a tight characterization in terms of the Boolean influence for several relevant architectures. This is shown on linear models and supported experimentally on other models such as MLPs and Transformers. In particular, this puts forward the hypothesis that for such architectures and for learning logical functions such as PVR functions, GD tends to have an implicit bias towards low-degree representations, which in turn gives the Boolean influence for the generalization error under quadratic loss.

Scalable Vector Graphics (SVG) are central to digital design due to their inherent scalability and editability. Despite significant advancements in content generation enabled by Visual Language Models (VLMs), existing text-to-SVG generation methods are limited by a core challenge: the autoregressive training process does not incorporate visual perception of the final rendered image, which fundamentally constrains generation quality. To address this limitation, we propose an Introspective SVG Generation Framework (IntroSVG). At its core, the framework instantiates a unified VLM that operates in a closed loop, assuming dual roles of both generator and critic. Specifically, through Supervised Fine-Tuning (SFT), the model learns to draft SVGs and to provide feedback on their rendered outputs; moreover, we systematically convert early-stage failures into high-quality error-correction training data, thereby enhancing model robustness. Subsequently, we leverage a high-capacity teacher VLM to construct a preference dataset and further align the generator's policy through Direct Preference Optimization (DPO). During inference, the optimized generator and critic operate collaboratively in an iterative "generate-review-refine" cycle, starting from imperfect intermediate drafts to autonomously improve output quality. Experimental results demonstrate that our method achieves state-of-the-art performance across several key evaluation metrics, generating SVGs with more complex structures, stronger semantic alignment, and greater editability. These results corroborate the effectiveness of incorporating explicit visual feedback into the generation loop.

Tool-based Agentic Reinforcement Learning (TARL) has emerged as a promising paradigm for training search agents to interact with external tools for a multi-turn information-seeking process autonomously. However, we identify a critical training instability that leads to catastrophic model collapse: Importance Sampling Distribution Drift(ISDD). In Group Relative Policy Optimization(GRPO), a widely adopted TARL algorithm, ISDD manifests as a precipitous decline in the importance sampling ratios, which nullifies gradient updates and triggers irreversible training failure. To address this, we propose Search Agent Policy Optimization (SAPO), which stabilizes training via a conditional token-level KL constraint. Unlike hard clipping, which ignores distributional divergence, SAPO selectively penalizes the KL divergence between the current and old policies. Crucially, this penalty is applied only to positive tokens with low probabilities where the policy has shifted excessively, thereby preventing distribution drift while preserving gradient flow. Remarkably, SAPO requires only one-line code modification to standard GRPO, ensuring immediate deployability. Extensive experiments across seven QA benchmarks demonstrate that SAPO achieves +10.6\% absolute improvement (+31.5\% relative) over Search-R1, yielding consistent gains across varying model scales (1.5B, 14B) and families (Qwen, LLaMA).

Aligning generative models with human preference via RLHF typically suffers from overoptimization, where an imperfectly learned reward model can misguide the generative model to output undesired responses. We investigate this problem in a principled manner by identifying the source of the misalignment as a form of distributional shift and uncertainty in learning human preferences. To mitigate overoptimization, we first propose a theoretical algorithm that chooses the best policy for an adversarially chosen reward model; one that simultaneously minimizes the maximum likelihood estimation of the loss and a reward penalty term. Here, the reward penalty term is introduced to prevent the policy from choosing actions with spurious high proxy rewards, resulting in provable sample efficiency of the algorithm under a partial coverage style condition. Moving from theory to practice, the proposed algorithm further enjoys an equivalent but surprisingly easy-to-implement reformulation. Using the equivalence between reward models and the corresponding optimal policy, the algorithm features a simple objective that combines: (i) a preference optimization loss that directly aligns the policy with human preference, and (ii) a supervised learning loss that explicitly imitates the policy with a (suitable) baseline distribution. In the context of aligning large language models (LLM), this objective fuses the direct preference optimization (DPO) loss with the supervised fune-tuning (SFT) loss to help mitigate the overoptimization towards undesired responses, for which we name the algorithm Regularized Preference Optimization (RPO). Experiments of aligning LLMs demonstrate the improved performance of RPO compared with DPO baselines. Our work sheds light on the interplay between preference optimization and SFT in tuning LLMs with both theoretical guarantees and empirical evidence.

Reward models (RMs) play a critical role in aligning large language models (LLMs) with human preferences. Yet in the domain of tool learning, the lack of RMs specifically designed for function-calling tasks has limited progress toward more capable agentic AI. We introduce ToolRM, a family of lightweight generative RMs tailored for general tool-use scenarios. To build these models, we propose a novel pipeline that constructs pairwise preference data using rule-based scoring and multidimensional sampling. This yields ToolPref-Pairwise-30K, a diverse, balanced, and challenging dataset of critique tasks that supports reinforcement learning with verifiable feedback. To evaluate tool-use RMs, we also introduce TRBench_{BFCL}, a benchmark built on the agentic evaluation suite BFCL. Trained on our constructed data, models from the Qwen3-4B/8B series achieve up to 14.28% higher accuracy, substantially outperforming frontier models such as Claude 4 and OpenAI o3 in pairwise reward judgments. Beyond training objectives, ToolRM generalizes to broader critique tasks, including Best-of-N sampling and self-correction. Experiments on ACEBench highlight its effectiveness and efficiency, enabling inference-time scaling and reducing output token usage by over 66%. We release data and model checkpoints to facilitate future research.

Distribution inference, sometimes called property inference, infers statistical properties about a training set from access to a model trained on that data. Distribution inference attacks can pose serious risks when models are trained on private data, but are difficult to distinguish from the intrinsic purpose of statistical machine learning -- namely, to produce models that capture statistical properties about a distribution. Motivated by Yeom et al.'s membership inference framework, we propose a formal definition of distribution inference attacks that is general enough to describe a broad class of attacks distinguishing between possible training distributions. We show how our definition captures previous ratio-based property inference attacks as well as new kinds of attack including revealing the average node degree or clustering coefficient of a training graph. To understand distribution inference risks, we introduce a metric that quantifies observed leakage by relating it to the leakage that would occur if samples from the training distribution were provided directly to the adversary. We report on a series of experiments across a range of different distributions using both novel black-box attacks and improved versions of the state-of-the-art white-box attacks. Our results show that inexpensive attacks are often as effective as expensive meta-classifier attacks, and that there are surprising asymmetries in the effectiveness of attacks. Code is available at https://github.com/iamgroot42/FormEstDistRisks

Diffusion-based models for robotic control, including vision-language-action (VLA) and vision-action (VA) policies, have demonstrated significant capabilities. Yet their advancement is constrained by the high cost of acquiring large-scale interaction datasets. This work introduces an alternative paradigm for enhancing policy performance without additional model training. Perhaps surprisingly, we demonstrate that the composed policies can exceed the performance of either parent policy. Our contribution is threefold. First, we establish a theoretical foundation showing that the convex composition of distributional scores from multiple diffusion models can yield a superior one-step functional objective compared to any individual score. A Gr\"onwall-type bound is then used to show that this single-step improvement propagates through entire generation trajectories, leading to systemic performance gains. Second, motivated by these results, we propose General Policy Composition (GPC), a training-free method that enhances performance by combining the distributional scores of multiple pre-trained policies via a convex combination and test-time search. GPC is versatile, allowing for the plug-and-play composition of heterogeneous policies, including VA and VLA models, as well as those based on diffusion or flow-matching, irrespective of their input visual modalities. Third, we provide extensive empirical validation. Experiments on Robomimic, PushT, and RoboTwin benchmarks, alongside real-world robotic evaluations, confirm that GPC consistently improves performance and adaptability across a diverse set of tasks. Further analysis of alternative composition operators and weighting strategies offers insights into the mechanisms underlying the success of GPC. These results establish GPC as a simple yet effective method for improving control performance by leveraging existing policies.