Daily Papers

Reinforcement fine-tuning (RFT) is essential for enhancing the reasoning capabilities of large language models (LLM), yet the widely adopted Group Relative Policy Optimization (GRPO) suffers from entropy collapse, where entropy monotonically decreases, exploration vanishes, and policies converge prematurely. Existing entropy-regularized methods only partially alleviate this issue while introducing bias and instability, leaving entropy control unresolved and the connection between entropy, exploration, and performance unclear. We propose Arbitrary Entropy Policy Optimization (AEPO), which eliminates entropy collapse by replacing entropy bonuses with REINFORCE policy gradient on temperature-adjusted distributions and stabilizing entropy through temperature regulation. AEPO integrates three key designs: policy gradient as regularization, distribution as regularization, and REINFORCE as regularization, enabling precise entropy control without distorting optimization. Experiments demonstrate three major contributions: AEPO (1) stabilizes entropy at arbitrary target levels, effectively removing collapse in GRPO; (2) reveals a non-monotonic relation where performance first improves then declines with increasing entropy, clarifying the link between entropy, exploration, and reasoning; and (3) generalizes beyond entropy, providing a broader RFT paradigm where superior target distributions can serve as REINFORCE regularizers.

Training LLM agents in multi-turn environments with sparse rewards, where completing a single task requires 30+ turns of interaction within an episode, presents a fundamental challenge for reinforcement learning. We identify a critical failure mode unique to this setting: the exploration-exploitation cascade failure. This cascade begins with early-stage policy premature convergence, where sparse feedback causes agents to commit to flawed, low-entropy strategies. Subsequently, agents enter late-stage policy collapse, where conventional entropy regularization becomes counterproductive, promoting chaotic exploration that destabilizes training. We propose Entropy-regularized Policy Optimization (EPO), a general framework that breaks this failure cycle through three synergistic mechanisms: (1) adopting entropy regularization in multi-turn settings to enhance exploration, (2) an entropy smoothing regularizer that bounds policy entropy within historical averages to prevent abrupt fluctuations, and (3) adaptive phase-based weighting that balances exploration and exploitation across training. Our analysis justifies that EPO guarantees monotonically decreasing entropy variance while maintaining convergence. EPO achieves up to 152% performance improvement on ScienceWorld and up to 19.8% on ALFWorld. Our work demonstrates that multi-turn sparse-reward settings require fundamentally different entropy control than traditional RL, with broad implications for LLM agent training.

Reinforcement learning from verifiable rewards has significantly advanced the reasoning capabilities of large language models. However, Group Relative Policy Optimization (GRPO) typically assigns a uniform, sequence-level advantage to all tokens, thereby overlooking the intrinsic information heterogeneity along reasoning chains. We show that this coarse-grained credit assignment leads to premature entropy collapse and encourages the model to generate redundant, low-quality reasoning paths. Through systematic empirical analysis, we identify Critical Decision Pivots (CDPs): transient high-entropy states where the policy's trajectory is most sensitive to perturbations. These pivots represent the "forks in the road" where effective multi-path exploration is most crucial yet often suppressed by uniform advantage signals. Building on these insights, we propose Entropy-Regulated Policy Optimization (ERPO), which transitions the optimization focus from coarse sequences to fine-grained token dynamics. ERPO introduces three synergistic components: (i) Entropy-aware Gating, which adaptively amplifies exploration at CDPs to facilitate diverse path discovery; (ii) Bucket-based Implicit Normalization, which mitigates difficulty bias by aligning token progress windows; and (iii) Result-anchored Advantage Synthesis, which re-weights token-level signals via outcome-driven anchors. Extensive experiments on competitive mathematical benchmarks demonstrate that ERPO significantly outperforms GRPO. Notably, ERPO not only boosts reasoning accuracy but also yields significantly more concise and robust derivation paths, while achieving performance comparable to large models with orders of magnitude more parameters.

Reinforcement Learning has emerged as the fundamental technique for enhancing reasoning in LLMs. However, existing algorithms apply uniform optimization to all tokens, ignoring their different roles in reasoning process. To address this limitation, we introduce Heterogeneous Adaptive Policy Optimization (HAPO), a comprehensive token-aware algorithm that dynamically adapts optimization based on token entropy. For rollout sampling, we propose Adaptive Temperature Sampling, which adjusts sampling temperature in real time, promoting exploration at high-entropy tokens while preserving coherence at low-entropy ones. For advantage calculation, we introduce Token Level Group Average that normalizes advantages at token level, jointly accounting for sequence-length as in token-mean loss while preserving non-biased treatment. We then develop Differential Advantage Redistribution that leverages entropy and importance ratios to modulate rewards-adjusting updates for tokens with clear signals. For clipping loss, we design Asymmetric Adaptive Clipping, allowing aggressive probability reduction for noisy low-entropy tokens while enabling exploration for high-entropy tokens. Through systematic investigation between entropy and training dynamics, we embedded token-level treatment into every stages to achieve fine-grained control. Extensive experiments demonstrate that HAPO consistently outperforms DAPO across multiple model scales. Our code can be found in https://github.com/starriver030515/HAPO.

Iterative generative policies, such as diffusion models and flow matching, offer superior expressivity for continuous control but complicate Maximum Entropy Reinforcement Learning because their action log-densities are not directly accessible. To address this, we propose Field Least-Energy Actor-Critic (FLAC), a likelihood-free framework that regulates policy stochasticity by penalizing the kinetic energy of the velocity field. Our key insight is to formulate policy optimization as a Generalized Schrödinger Bridge (GSB) problem relative to a high-entropy reference process (e.g., uniform). Under this view, the maximum-entropy principle emerges naturally as staying close to a high-entropy reference while optimizing return, without requiring explicit action densities. In this framework, kinetic energy serves as a physically grounded proxy for divergence from the reference: minimizing path-space energy bounds the deviation of the induced terminal action distribution. Building on this view, we derive an energy-regularized policy iteration scheme and a practical off-policy algorithm that automatically tunes the kinetic energy via a Lagrangian dual mechanism. Empirically, FLAC achieves superior or comparable performance on high-dimensional benchmarks relative to strong baselines, while avoiding explicit density estimation.

Discrete diffusion models have recently gained significant attention due to their ability to process complex discrete structures for language modeling. However, fine-tuning these models with policy gradient methods, as is commonly done in Reinforcement Learning from Human Feedback (RLHF), remains a challenging task. We propose an efficient, broadly applicable, and theoretically justified policy gradient algorithm, called Score Entropy Policy Optimization (SEPO), for fine-tuning discrete diffusion models over non-differentiable rewards. Our numerical experiments across several discrete generative tasks demonstrate the scalability and efficiency of our method. Our code is available at https://github.com/ozekri/SEPO.

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.

This paper aims to overcome a major obstacle in scaling RL for reasoning with LLMs, namely the collapse of policy entropy. Such phenomenon is consistently observed across vast RL runs without entropy intervention, where the policy entropy dropped sharply at the early training stage, this diminished exploratory ability is always accompanied with the saturation of policy performance. In practice, we establish a transformation equation R=-a*e^H+b between entropy H and downstream performance R. This empirical law strongly indicates that, the policy performance is traded from policy entropy, thus bottlenecked by its exhaustion, and the ceiling is fully predictable H=0, R=-a+b. Our finding necessitates entropy management for continuous exploration toward scaling compute for RL. To this end, we investigate entropy dynamics both theoretically and empirically. Our derivation highlights that, the change in policy entropy is driven by the covariance between action probability and the change in logits, which is proportional to its advantage when using Policy Gradient-like algorithms. Empirical study shows that, the values of covariance term and entropy differences matched exactly, supporting the theoretical conclusion. Moreover, the covariance term stays mostly positive throughout training, further explaining why policy entropy would decrease monotonically. Through understanding the mechanism behind entropy dynamics, we motivate to control entropy by restricting the update of high-covariance tokens. Specifically, we propose two simple yet effective techniques, namely Clip-Cov and KL-Cov, which clip and apply KL penalty to tokens with high covariances respectively. Experiments show that these methods encourage exploration, thus helping policy escape entropy collapse and achieve better downstream performance.

In long-horizon tasks, recent agents based on Large Language Models (LLMs) face a significant challenge that sparse, outcome-based rewards make it difficult to assign credit to intermediate steps. Previous methods mainly focus on creating dense reward signals to guide learning, either through traditional reinforcement learning techniques like inverse reinforcement learning or by using Process Reward Models for step-by-step feedback. In this paper, we identify a fundamental problem in the learning dynamics of LLMs: the magnitude of policy gradients is inherently coupled with the entropy, which leads to inefficient small updates for confident correct actions and potentially destabilizes large updates for uncertain ones. To resolve this, we propose Entropy-Modulated Policy Gradients (EMPG), a framework that re-calibrates the learning signal based on step-wise uncertainty and the final task outcome. EMPG amplifies updates for confident correct actions, penalizes confident errors, and attenuates updates from uncertain steps to stabilize exploration. We further introduce a bonus term for future clarity that encourages agents to find more predictable solution paths. Through comprehensive experiments on three challenging agent tasks, WebShop, ALFWorld, and Deep Search, we demonstrate that EMPG achieves substantial performance gains and significantly outperforms strong policy gradient baselines. Project page is at https://empgseed-seed.github.io/

Recent advances in multimodal large reasoning models (MLRMs) have substantially improved their ability to solve complex textual and visual tasks. However, these models tend to overthink on simple problems, producing unnecessarily lengthy reasoning traces, while under-exploring on challenging ones, leading to missed solutions. To address this imbalance, we propose ARES, a unified open-source framework for adaptive reasoning that dynamically allocates exploration effort based on task difficulty. Our approach is motivated by two key empirical findings: (i) while single-token entropy is noisy, high window-entropy (HWE) tokens (token-level entropies averaged under a sliding window) can reliably capture reasoning-critical moments; and (ii) reducing HWE usage benefits easy problems, while increasing it is essential for solving hard ones. Building on these insights, ARES introduces a two-stage training pipeline. In the Adaptive Cold-Start stage, we curate multimodal and textual data paired with reasoning traces of length proportional to problem difficulty, equipping the model with initial difficulty awareness. In the second stage, we develop Adaptive Entropy Policy Optimization (AEPO), which uses HWE tokens as exploration triggers to decide when to explore, and a hierarchical entropy reward with dynamic KL control to decide how much to explore. Extensive experiments demonstrate that ARES achieves superior performance and reasoning efficiency across diverse mathematical, logical, and multimodal benchmarks, while closing the gap to leading commercial systems under significantly lower inference costs.

Policy optimization methods are popular reinforcement learning algorithms in practice. Recent works have built theoretical foundation for them by proving T regret bounds even when the losses are adversarial. Such bounds are tight in the worst case but often overly pessimistic. In this work, we show that in tabular Markov decision processes (MDPs), by properly designing the regularizer, the exploration bonus and the learning rates, one can achieve a more favorable polylog(T) regret when the losses are stochastic, without sacrificing the worst-case guarantee in the adversarial regime. To our knowledge, this is also the first time a gap-dependent polylog(T) regret bound is shown for policy optimization. Specifically, we achieve this by leveraging a Tsallis entropy or a Shannon entropy regularizer in the policy update. Then we show that under known transitions, we can further obtain a first-order regret bound in the adversarial regime by leveraging the log-barrier regularizer.

Hybrid training methods for large language models combine supervised fine tuning (SFT) on expert demonstrations with reinforcement learning (RL) on model rollouts, typically at the sample level. We propose Entropy Gated Selective Policy Optimization (EGSPO), a three stage framework that extends sample level mixing with token level gradient modulation. Stage 1, SFT expert learning, establishes a reliable warm up policy using expert demonstrations with a pure SFT loss. Stage 2, RL rollout generation, samples trajectories from the current policy and computes per token predictive entropy. Stage 3, the EGSPO mechanism, applies entropy gated gradient allocation: a predictive entropy module routes high entropy tokens to full PPO updates to encourage exploration, and low entropy tokens to attenuated PPO updates to reduce variance and preserve knowledge. Critically, both branches incorporate the advantage function A_t, ensuring that incorrect trajectories receive consistent negative learning signals and preventing reinforcement of confident errors. EGSPO achieves consistent improvements on mathematical reasoning benchmarks, with gains of 3.8 percent on AIME and 2.9 percent on MATH over the CHORD phi baseline, while incurring only 3.4 percent additional computational overhead.

Reinforcement learning (RL) is a pivotal task for enhancing Large Language Model (LLM) reasoning. Conventional algorithms, however, typically adhere to a coarse-grained credit assignment paradigm, applying a uniform reward to all tokens in a sequence, a critical flaw in long-chain reasoning tasks. In this paper, we address this challenge and propose Dynamic Entropy Weighting, a novel mechanism that facilitates fine-grained rewards through two new algorithms: Group Token Policy Optimization (GTPO), which assigns an entropy-weighted reward to each token, and the analogous algorithm Sequence-Level GRPO (GRPO-S). Our approach is founded on the hypothesis that high policy entropy within a reasoning path is a powerful heuristic for cognitive effort at pivotal junctures, which can be repurposed into a learning signal. By repurposing policy entropy for reward shaping, we achieve true per-token credit assignment. Experimental results across challenging reasoning benchmarks validate the superiority of our approach, showing our methods significantly outperform a strong DAPO baseline and confirming our entropy-weighting mechanism as the key driver of this performance boost.

Long-term training of large language models (LLMs) requires maintaining stable exploration to prevent the model from collapsing into sub-optimal behaviors. Entropy is crucial in this context, as it controls exploration and helps avoid premature convergence to sub-optimal solutions. However, existing reinforcement learning methods struggle to maintain an appropriate level of entropy, as the training process involves a mix of positive and negative samples, each affecting entropy in different ways across steps. To address this, we propose Entropy stablilization via Proportional-Integral Control (EntroPIC), a novel method that adaptively adjusts the influence of positive and negative samples by dynamically tuning their loss coefficients. This approach stabilizes entropy throughout training, ensuring efficient exploration and steady progress. We provide a comprehensive theoretical analysis for both on-policy and off-policy learning settings, demonstrating that EntroPIC is effective at controlling entropy in large-scale LLM training. Experimental results show that our method successfully maintains desired entropy levels, enabling stable and optimal RL training for LLMs.

Recent reinforcement learning has enhanced the flow matching models on human preference alignment. While stochastic sampling enables the exploration of denoising directions, existing methods which optimize over multiple denoising steps suffer from sparse and ambiguous reward signals. We observe that the high entropy steps enable more efficient and effective exploration while the low entropy steps result in undistinguished roll-outs. To this end, we propose E-GRPO, an entropy aware Group Relative Policy Optimization to increase the entropy of SDE sampling steps. Since the integration of stochastic differential equations suffer from ambiguous reward signals due to stochasticity from multiple steps, we specifically merge consecutive low entropy steps to formulate one high entropy step for SDE sampling, while applying ODE sampling on other steps. Building upon this, we introduce multi-step group normalized advantage, which computes group-relative advantages within samples sharing the same consolidated SDE denoising step. Experimental results on different reward settings have demonstrated the effectiveness of our methods.

Recent advances in Reinforcement Learning with Verifiable Rewards (RLVR) for Large Language Model (LLM) reasoning have been hindered by a persistent challenge: exploration collapse. The semantic homogeneity of random rollouts often traps models in narrow, over-optimized behaviors. While existing methods leverage policy entropy to encourage exploration, they face inherent limitations. Global entropy regularization is susceptible to reward hacking, which can induce meaningless verbosity, whereas local token-selective updates struggle with the strong inductive bias of pre-trained models. To address this, we propose Latent Policy Optimization via Iterative Information Bottleneck (IIB-LPO), a novel approach that shifts exploration from statistical perturbation of token distributions to topological branching of reasoning trajectories. IIB-LPO triggers latent branching at high-entropy states to diversify reasoning paths and employs the Information Bottleneck principle both as a trajectory filter and a self-reward mechanism, ensuring concise and informative exploration. Empirical results across four mathematical reasoning benchmarks demonstrate that IIB-LPO achieves state-of-the-art performance, surpassing prior methods by margins of up to 5.3% in accuracy and 7.4% in diversity metrics.

While Reinforcement Learning with Verifiable Rewards (RLVR) can enhance LLM reasoning, its training process poses a critical risk: entropy collapse. This phenomenon is a rapid loss of policy diversity, stemming from the exploration-exploitation imbalance and leading to a lack of generalization. Recent entropy-intervention methods aim to prevent entropy collapse, yet their underlying mechanisms remain unclear. In this paper, we conduct a quantitative analysis to reveal token-level entropy changes and how existing entropy intervention methods help avoid entropy collapse. Our findings point out a fundamental limitation of existing methods: they attempt to control entropy dynamics indirectly. By only affecting related factors, such as the advantage signal and generation probability, their effectiveness is inherently limited and could potentially fail. To address this limitation, we introduce an entropy-change-aware reweighting scheme, namely Stabilizing Token-level Entropy-changE via Reweighting (STEER), that adaptively stabilizes entropy dynamics through fine-grained token-level adjustments. Our approach mitigates over-exploitation while fostering robust exploration. Extensive experiments demonstrate that STEER significantly mitigates entropy collapse, stabilizes entropy dynamics, and achieves stronger downstream performance across various mathematical reasoning benchmarks \footnote{Our code is available at https://github.com/zz-haooo/STEER.

Recently, Agentic Reinforcement Learning (Agentic RL) has made significant progress in incentivizing the multi-turn, long-horizon tool-use capabilities of web agents. While mainstream agentic RL algorithms autonomously explore high-uncertainty tool-call steps under the guidance of entropy, excessive reliance on entropy signals can impose further constraints, leading to the training collapse. In this paper, we delve into the challenges caused by entropy and propose the Agentic Entropy-Balanced Policy Optimization (AEPO), an agentic RL algorithm designed to balance entropy in both the rollout and policy update phases. AEPO comprises two core components: (1) a dynamic entropy-balanced rollout mechanism that adaptively allocate global and branch sampling budget through entropy pre-monitoring, while imposing a branch penalty on consecutive high-entropy tool-call steps to prevent over-branching issues; and (2) Entropy-Balanced Policy Optimization that inserts a stop-gradient operation into the high-entropy clipping term to preserve and properly rescale gradients on high-entropy tokens, while incorporating entropy-aware advantage estimation to prioritize learning on high-uncertainty tokens. Results across 14 challenging datasets show that AEPO consistently outperforms 7 mainstream RL algorithms. With just 1K RL samples, Qwen3-14B with AEPO achieves impressive results: 47.6% on GAIA, 11.2% on Humanity's Last Exam, and 43.0% on WebWalker for Pass@1; 65.0% on GAIA, 26.0% on Humanity's Last Exam, and 70.0% on WebWalker for Pass@5. Further analysis reveals that AEPO improves rollout sampling diversity while maintaining stable policy entropy, facilitating scalable web agent training.

Large language models (LLMs) have shown promise in performing complex multi-step reasoning, yet they continue to struggle with mathematical reasoning, often making systematic errors. A promising solution is reinforcement learning (RL) guided by reward models, particularly those focusing on process rewards, which score each intermediate step rather than solely evaluating the final outcome. This approach is more effective at guiding policy models towards correct reasoning trajectories. In this work, we propose an entropy-regularized process reward model (ER-PRM) that integrates KL-regularized Markov Decision Processes (MDP) to balance policy optimization with the need to prevent the policy from shifting too far from its initial distribution. We derive a novel reward construction method based on the theoretical results. Our theoretical analysis shows that we could derive the optimal reward model from the initial policy sampling. Our empirical experiments on the MATH and GSM8K benchmarks demonstrate that ER-PRM consistently outperforms existing process reward models, achieving 1% improvement on GSM8K and 2-3% improvement on MATH under best-of-N evaluation, and more than 1% improvement under RLHF. These results highlight the efficacy of entropy-regularization in enhancing LLMs' reasoning capabilities.

While large language models (LLMs) have demonstrated exceptional capabilities in challenging tasks such as mathematical reasoning, existing methods to enhance reasoning ability predominantly rely on supervised fine-tuning (SFT) followed by reinforcement learning (RL) on reasoning-specific data after pre-training. However, these approaches critically depend on external supervisions--such as human labelled reasoning traces, verified golden answers, or pre-trained reward models--which limits scalability and practical applicability. In this work, we propose Entropy Minimized Policy Optimization (EMPO), which makes an early attempt at fully unsupervised LLM reasoning incentivization. EMPO does not require any supervised information for incentivizing reasoning capabilities (i.e., neither verifiable reasoning traces, problems with golden answers, nor additional pre-trained reward models). By continuously minimizing the predictive entropy of LLMs on unlabeled user queries in a latent semantic space, EMPO enables purely self-supervised evolution of reasoning capabilities with strong flexibility and practicality. Our experiments demonstrate competitive performance of EMPO on both mathematical reasoning and free-form commonsense reasoning tasks. Specifically, without any supervised signals, EMPO boosts the accuracy of Qwen2.5-Math-7B Base from 30.7\% to 48.1\% on mathematical benchmarks and improves truthfulness accuracy of Qwen2.5-7B Instruct from 87.16\% to 97.25\% on TruthfulQA.

Policy optimization methods are powerful algorithms in Reinforcement Learning (RL) for their flexibility to deal with policy parameterization and ability to handle model misspecification. However, these methods usually suffer from slow convergence rates and poor sample complexity. Hence it is important to design provably sample efficient algorithms for policy optimization. Yet, recent advances for this problems have only been successful in tabular and linear setting, whose benign structures cannot be generalized to non-linearly parameterized policies. In this paper, we address this problem by leveraging recent advances in value-based algorithms, including bounded eluder-dimension and online sensitivity sampling, to design a low-switching sample-efficient policy optimization algorithm, LPO, with general non-linear function approximation. We show that, our algorithm obtains an varepsilon-optimal policy with only O(text{poly(d)}{varepsilon^3}) samples, where varepsilon is the suboptimality gap and d is a complexity measure of the function class approximating the policy. This drastically improves previously best-known sample bound for policy optimization algorithms, O(text{poly(d)}{varepsilon^8}). Moreover, we empirically test our theory with deep neural nets to show the benefits of the theoretical inspiration.

For RL algorithms, appropriate entropy control is crucial to their effectiveness. To control the policy entropy, a commonly used method is entropy regularization, which is adopted in various popular RL algorithms including PPO, SAC and A3C. Although entropy regularization proves effective in robotic and games RL conventionally, studies found that it gives weak to no gains in LLM-RL training. In this work, we study the issues of entropy bonus in LLM-RL setting. Specifically, we first argue that the conventional entropy regularization suffers from the LLM's extremely large response space and the sparsity of the optimal outputs. As a remedy, we propose AEnt, an entropy control method that utilizes a new clamped entropy bonus with an automatically adjusted coefficient. The clamped entropy is evaluated with the re-normalized policy defined on certain smaller token space, which encourages exploration within a more compact response set. In addition, the algorithm automatically adjusts entropy coefficient according to the clamped entropy value, effectively controlling the entropy-induced bias while leveraging the entropy's benefits. AEnt is tested in math-reasoning tasks under different base models and datasets, and it is observed that AEnt outperforms the baselines consistently across multiple benchmarks.

When deploying artificial agents in real-world environments where they interact with humans, it is crucial that their behavior is aligned with the values, social norms or other requirements of that environment. However, many environments have implicit constraints that are difficult to specify and transfer to a learning agent. To address this challenge, we propose a novel method that utilizes the principle of maximum causal entropy to learn constraints and an optimal policy that adheres to these constraints, using demonstrations of agents that abide by the constraints. We prove convergence in a tabular setting and provide an approximation which scales to complex environments. We evaluate the effectiveness of the learned policy by assessing the reward received and the number of constraint violations, and we evaluate the learned cost function based on its transferability to other agents. Our method has been shown to outperform state-of-the-art approaches across a variety of tasks and environments, and it is able to handle problems with stochastic dynamics and a continuous state-action space.

Entropy serves as a critical metric for measuring the diversity of outputs generated by large language models (LLMs), providing valuable insights into their exploration capabilities. While recent studies increasingly focus on monitoring and adjusting entropy to better balance exploration and exploitation in reinforcement fine-tuning (RFT), a principled understanding of entropy dynamics during this process is yet to be thoroughly investigated. In this paper, we establish a theoretical framework for analyzing the entropy dynamics during the RFT process, which begins with a discriminant expression that quantifies entropy change under a single logit update. This foundation enables the derivation of a first-order expression for entropy change, which can be further extended to the update formula of Group Relative Policy Optimization (GRPO). The corollaries and insights drawn from the theoretical analysis inspire the design of entropy control methods, and also offer a unified lens for interpreting various entropy-based methods in existing studies. We provide empirical evidence to support the main conclusions of our analysis and demonstrate the effectiveness of the derived entropy-discriminator clipping methods. This study yields novel insights into RFT training dynamics, providing theoretical support and practical strategies for optimizing the exploration-exploitation balance during LLM fine-tuning.

Behavior constrained policy optimization has been demonstrated to be a successful paradigm for tackling Offline Reinforcement Learning. By exploiting historical transitions, a policy is trained to maximize a learned value function while constrained by the behavior policy to avoid a significant distributional shift. In this paper, we propose our closed-form policy improvement operators. We make a novel observation that the behavior constraint naturally motivates the use of first-order Taylor approximation, leading to a linear approximation of the policy objective. Additionally, as practical datasets are usually collected by heterogeneous policies, we model the behavior policies as a Gaussian Mixture and overcome the induced optimization difficulties by leveraging the LogSumExp's lower bound and Jensen's Inequality, giving rise to a closed-form policy improvement operator. We instantiate offline RL algorithms with our novel policy improvement operators and empirically demonstrate their effectiveness over state-of-the-art algorithms on the standard D4RL benchmark. Our code is available at https://cfpi-icml23.github.io/.

Reinforcement Learning (RL) has emerged as a powerful tool for neural combinatorial optimization, enabling models to learn heuristics that solve complex problems without requiring expert knowledge. Despite significant progress, existing RL approaches face challenges such as diminishing reward signals and inefficient exploration in vast combinatorial action spaces, leading to inefficiency. In this paper, we propose Preference Optimization, a novel method that transforms quantitative reward signals into qualitative preference signals via statistical comparison modeling, emphasizing the superiority among sampled solutions. Methodologically, by reparameterizing the reward function in terms of policy and utilizing preference models, we formulate an entropy-regularized RL objective that aligns the policy directly with preferences while avoiding intractable computations. Furthermore, we integrate local search techniques into the fine-tuning rather than post-processing to generate high-quality preference pairs, helping the policy escape local optima. Empirical results on various benchmarks, such as the Traveling Salesman Problem (TSP), the Capacitated Vehicle Routing Problem (CVRP) and the Flexible Flow Shop Problem (FFSP), demonstrate that our method significantly outperforms existing RL algorithms, achieving superior convergence efficiency and solution quality.

Reinforcement learning (RL) has enabled complex reasoning abilities in large language models (LLMs). However, most RL algorithms suffer from performance saturation, preventing continued gains as RL training scales. This problem can be characterized by the collapse of entropy, a key diagnostic for exploration in RL. Existing attempts focus on preventing entropy collapse through regularization or clipping. However, their resulting entropy curves often exhibit instability in the long term, which hinders performance gains. In this paper, we introduce Entrocraft, a simple rejection-sampling approach that realizes user-customized entropy schedule by biasing the advantage distributions. Entrocraft requires no objective regularization and is advantage-estimator-agnostic. Theoretically, we relate per-step entropy change to the advantage distribution under minimal assumptions. This explains the behavior of existing RL and entropy-preserving methods. Entrocraft also enables a systematic study of entropy schedules, which reveals that linear annealing, which starts high and decays to a slightly lower target, performs best. Empirically, Entrocraft addresses performance saturation, significantly improving generalization, output diversity, and long-term training. It enables a 4B model to outperform an 8B baseline, sustains improvement for up to 4x longer before plateauing, and raises pass@K by 50% over the baseline.

We describe an iterative procedure for optimizing policies, with guaranteed monotonic improvement. By making several approximations to the theoretically-justified procedure, we develop a practical algorithm, called Trust Region Policy Optimization (TRPO). This algorithm is similar to natural policy gradient methods and is effective for optimizing large nonlinear policies such as neural networks. Our experiments demonstrate its robust performance on a wide variety of tasks: learning simulated robotic swimming, hopping, and walking gaits; and playing Atari games using images of the screen as input. Despite its approximations that deviate from the theory, TRPO tends to give monotonic improvement, with little tuning of hyperparameters.

Group Relative Policy Optimization (GRPO) has emerged as the de facto Reinforcement Learning (RL) objective driving recent advancements in Multimodal Large Language Models. However, extending this success to open-source multimodal generalist models remains heavily constrained by two primary challenges: the extreme variance in reward topologies across diverse visual tasks, and the inherent difficulty of balancing fine-grained perception with multi-step reasoning capabilities. To address these issues, we introduce Gaussian GRPO (G^2RPO), a novel RL training objective that replaces standard linear scaling with non-linear distributional matching. By mathematically forcing the advantage distribution of any given task to strictly converge to a standard normal distribution, N(0,1), G^2RPO theoretically ensures inter-task gradient equity, mitigates vulnerabilities to heavy-tail outliers, and offers symmetric update for positive and negative rewards. Leveraging the enhanced training stability provided by G^2RPO, we introduce two task-level shaping mechanisms to seamlessly balance perception and reasoning. First, response length shaping dynamically elicits extended reasoning chains for complex queries while enforce direct outputs to bolster visual grounding. Second, entropy shaping tightly bounds the model's exploration zone, effectively preventing both entropy collapse and entropy explosion. Integrating these methodologies, we present OpenVLThinkerV2, a highly robust, general-purpose multimodal model. Extensive evaluations across 18 diverse benchmarks demonstrate its superior performance over strong open-source and leading proprietary frontier models.

Reinforcement learning (RL) has enabled machine learning models to achieve significant advances in many fields. Most recently, RL has empowered frontier language models to solve challenging math, science, and coding problems. However, central to any RL algorithm is the reward function, and reward engineering is a notoriously difficult problem in any domain. In this paper, we propose RENT: Reinforcement Learning via Entropy Minimization -- a fully unsupervised RL method that requires no external reward or ground-truth answers, and instead uses the model's entropy of its underlying distribution as an intrinsic reward. We find that by reinforcing the chains of thought that yield high model confidence on its generated answers, the model improves its reasoning ability. In our experiments, we showcase these improvements on an extensive suite of commonly-used reasoning benchmarks, including GSM8K, MATH500, AMC, AIME, and GPQA, and models of varying sizes from the Qwen, Mistral, and Llama families. The generality of our unsupervised learning method lends itself to applicability in a wide range of domains where external supervision is unavailable.

Reinforcement learning (RL) has become a powerful paradigm for optimizing large language models (LLMs) to handle complex reasoning tasks. A core challenge in this process lies in managing policy entropy, which reflects the balance between exploration and exploitation during training. Existing methods, such as proximal policy optimization (PPO) and its variants, discard valuable gradient signals from low-probability tokens due to the clipping mechanism. We systematically analyze the entropy dynamics and reveal that these clipped tokens play a critical yet overlooked role in regulating entropy evolution. We propose Controlling Entropy via Gradient-Preserving Policy Optimization (CE-GPPO), a novel algorithm that reintroduces gradients from clipped tokens in native PPO in a gentle and bounded manner. By controlling the magnitude of gradients from tokens outside the clipping interval, CE-GPPO is able to achieve an exploration-exploitation trade-off. We provide theoretical justification and empirical evidence showing that CE-GPPO effectively mitigates entropy instability. Extensive experiments on mathematical reasoning benchmarks show that CE-GPPO consistently outperforms strong baselines across different model scales.

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.

Varying dynamics parameters in simulation is a popular Domain Randomization (DR) approach for overcoming the reality gap in Reinforcement Learning (RL). Nevertheless, DR heavily hinges on the choice of the sampling distribution of the dynamics parameters, since high variability is crucial to regularize the agent's behavior but notoriously leads to overly conservative policies when randomizing excessively. In this paper, we propose a novel approach to address sim-to-real transfer, which automatically shapes dynamics distributions during training in simulation without requiring real-world data. We introduce DOmain RAndomization via Entropy MaximizatiON (DORAEMON), a constrained optimization problem that directly maximizes the entropy of the training distribution while retaining generalization capabilities. In achieving this, DORAEMON gradually increases the diversity of sampled dynamics parameters as long as the probability of success of the current policy is sufficiently high. We empirically validate the consistent benefits of DORAEMON in obtaining highly adaptive and generalizable policies, i.e. solving the task at hand across the widest range of dynamics parameters, as opposed to representative baselines from the DR literature. Notably, we also demonstrate the Sim2Real applicability of DORAEMON through its successful zero-shot transfer in a robotic manipulation setup under unknown real-world parameters.

We consider the reinforcement learning (RL) problem with general utilities which consists in maximizing a function of the state-action occupancy measure. Beyond the standard cumulative reward RL setting, this problem includes as particular cases constrained RL, pure exploration and learning from demonstrations among others. For this problem, we propose a simpler single-loop parameter-free normalized policy gradient algorithm. Implementing a recursive momentum variance reduction mechanism, our algorithm achieves mathcal{O}(epsilon^{-3}) and mathcal{O}(epsilon^{-2}) sample complexities for epsilon-first-order stationarity and epsilon-global optimality respectively, under adequate assumptions. We further address the setting of large finite state action spaces via linear function approximation of the occupancy measure and show a mathcal{O}(epsilon^{-4}) sample complexity for a simple policy gradient method with a linear regression subroutine.

Reinforcement learning (RL) plays a crucial role in shaping the behavior of large language and reasoning models (LLMs/LRMs). However, it often produces brittle and unstable policies, leading to critical failures such as spurious reasoning, deceptive alignment, and instruction disobedience that undermine the trustworthiness and safety of LLMs/LRMs. Currently, these issues lack a unified theoretical explanation and are typically addressed using ad-hoc heuristics. This paper presents a rigorous mathematical framework for analyzing the stability of the mapping from a reward function to the optimal policy. We show that policy brittleness often stems from non-unique optimal actions, a common occurrence when multiple valid traces exist in a reasoning task. This theoretical lens provides a unified explanation for a range of seemingly disparate failures, reframing them as rational outcomes of optimizing rewards that may be incomplete or noisy, especially in the presence of action degeneracy. We extend this analysis from the fundamental single-reward setting to the more realistic multi-reward RL across diverse domains, showing how stability is governed by an "effective reward" aggregation mechanism. We also prove that entropy regularization restores policy stability at the cost of increased stochasticity. Our framework provides a unified explanation for recent empirical findings on deceptive reasoning, instruction-following trade-offs, and RLHF-induced sophistry, and is further validated through perturbation experiments in multi-reward RL. This work advances policy-stability analysis from empirical heuristics towards a principled theory, offering essential insights for designing safer and more trustworthy AI systems.

Large language model (LLM) reinforcement learning has increasingly relied on group-based policy optimization frameworks, such as GRPO and GSPO, to achieve stable fine-tuning at scale. However, a fundamental trade-off persists between optimization granularity and training stability. While GSPO improves robustness via sequence-level optimization, its monolithic treatment of sequences introduces severe inefficiencies: its conservative clipping mechanism indiscriminately discards valid training samples-a phenomenon we term gradient underutilization-and its uniform credit assignment fails to capture the heterogeneous contributions of critical reasoning steps. In this work, we propose Entropy Importance Sampling Policy Optimization (ESPO), a novel framework that reconciles fine-grained control with training stability. ESPO decomposes sequences into groups based on predictive entropy, enabling (1) Entropy-driven Importance Sampling to capture intra-sequence heterogeneity, and (2) Entropy-adaptive Clipping to dynamically allocate trust regions based on model uncertainty. Extensive experiments on mathematical reasoning benchmarks demonstrate that ESPO not only accelerates convergence but also achieves state-of-the-art performance, notably improving accuracy on the challenging HMMT benchmark from 4.4% to 13.13%.

In the post-training of large language models (LLMs), Reinforcement Learning from Human Feedback (RLHF) is an effective approach to achieve generation aligned with human preferences. Direct Preference Optimization (DPO) allows for policy training with a simple binary cross-entropy loss without a reward model. The objective of DPO is regularized by reverse KL divergence that encourages mode-seeking fitting to the reference policy. Nonetheless, we indicate that minimizing reverse KL divergence could fail to capture a mode of the reference distribution, which may hurt the policy's performance. Based on this observation, we propose a simple modification to DPO, H-DPO, which allows for control over the entropy of the resulting policy, enhancing the distribution's sharpness and thereby enabling mode-seeking fitting more effectively. In our experiments, we show that H-DPO outperformed DPO across various tasks, demonstrating superior results in pass@k evaluations for mathematical tasks. Moreover, H-DPO is simple to implement, requiring only minor modifications to the loss calculation of DPO, which makes it highly practical and promising for wide-ranging applications in the training of LLMs.

Reasoning models have advanced rapidly, but the dominant reinforcement learning from verifiable rewards (RLVR) recipe remains surprisingly narrow: sample many responses and reward each with a single bit indicating whether the final answer is correct. Yet many settings provide rich feedback, including execution traces, tool outputs, expert corrections, and model self-evaluations. We study how to use such feedback through a distributional variant of the classic imitation learning algorithm DAgger, where the learner has local access to an expert distribution on states visited by the current policy. This yields a simple forward cross-entropy objective that admits a blackbox expert and whose sequence-level gradient {conduct rich credit assignment by propagating} future expert-student disagreement back to earlier decisions. We show that prior RL with self-distillation objectives based on reverse KL or Jensen-Shannon fail to guarantee monotonic policy improvement: even when the expert has higher reward, their updates may increase probability on worse actions. In contrast, we show that forward cross-entropy admits monotonic policy improvement and enjoys guarantees on regret. We further show that our objective optimizes a lower bound on teacher-weighted likelihood of success, leading to improved Pass@N. Empirically, our approach, DistIL, improves over RLVR and RL with self-distillation baselines across a variety of domains: scientific reasoning, coding, and solving hard mathematical problems.

Reinforcement learning with verifiable rewards (RLVR) has demonstrated superior performance in enhancing the reasoning capability of large language models (LLMs). However, this accuracy-oriented learning paradigm often suffers from entropy collapse, which reduces policy exploration and limits reasoning capabilities. To address this challenge, we propose an efficient reinforcement learning framework that leverages entropy signals at both the semantic and token levels to improve reasoning. From the data perspective, we introduce semantic entropy-guided curriculum learning, organizing training data from low to high semantic entropy to guide progressive optimization from easier to more challenging tasks. For the algorithmic design, we adopt non-uniform token treatment by imposing KL regularization on low-entropy tokens that critically impact policy exploration and applying stronger constraints on high-covariance portions within these tokens. By jointly optimizing data organization and algorithmic design, our method effectively mitigates entropy collapse and enhances LLM reasoning. Experimental results across 6 benchmarks with 3 different parameter-scale base models demonstrate that our method outperforms other entropy-based approaches in improving reasoning.

Reinforcement learning with verifiable rewards (RLVR) has become a core technique for post-training of Large Language Models (LLMs). While policy optimization is driven by all sampled tokens under a globally broadcast scalar reward, the heterogeneous policy behaviors exhibited along trajectories are largely overlooked without differentiation. Existing works address this by credit allocation, including token-level advantage reweighting, and selective token optimization, however, the allocation criterion are principally stagnant throughout training, limiting resilient policy evolution. In this work, we argue that when learning signals are scheduled can be as important as where they are allocated across tokens, and introduce the temporal dimension that scheduling the credit allocation criteria over the course of RLVR optimization. We find that prioritizing targeted tokens emphasized with specific policy behaviors, and gradually attenuating toward general optimization leads to more stable and efficient learning dynamics. Furthermore, we show that simple trajectory percentiles provide a natural perspective for distinguishing policy behaviors, and works effectively with temporal scheduling. Our analysis reveals that standard optimization substantially sacrifices policy entropy when simultaneously accommodating heterogeneous behaviors, whereas temporal scheduling yields healthier policy evolution dynamics. Experiments across mathematical and general reasoning benchmarks demonstrate consistent improvements, suggesting that temporal scheduling constitutes a promising optimization dimension.

Existing reinforcement learning (RL) approaches treat large language models (LLMs) as a single unified policy, overlooking their internal mechanisms. Understanding how policy evolves across layers and modules is therefore crucial for enabling more targeted optimization and raveling out complex reasoning mechanisms. In this paper, we decompose the language model policy by leveraging the intrinsic split of the Transformer residual stream and the equivalence between the composition of hidden states with the unembedding matrix and the resulting samplable policy. This decomposition reveals Internal Layer Policies, corresponding to contributions from individual layers, and Internal Modular Policies, which align with the self-attention and feed-forward network (FFN) components within each layer. By analyzing the entropy of internal policy, we find that: (a) Early layers keep high entropy for exploration, top layers converge to near-zero entropy for refinement, with convergence patterns varying across model series. (b) LLama's prediction space rapidly converges in the final layer, whereas Qwen-series models, especially Qwen3, exhibit a more human-like, progressively structured reasoning pattern. Motivated by these findings, we propose Bottom-up Policy Optimization (BuPO), a novel RL paradigm that directly optimizes the internal layer policy during early training. By aligning training objective at lower layer, BuPO reconstructs foundational reasoning capabilities and achieves superior performance. Extensive experiments on complex reasoning benchmarks demonstrates the effectiveness of our method. Our code is available at https://github.com/Trae1ounG/BuPO.

In this work, we investigate the application of Taylor expansions in reinforcement learning. In particular, we propose Taylor expansion policy optimization, a policy optimization formalism that generalizes prior work (e.g., TRPO) as a first-order special case. We also show that Taylor expansions intimately relate to off-policy evaluation. Finally, we show that this new formulation entails modifications which improve the performance of several state-of-the-art distributed algorithms.

Reinforcement Learning (RL) has demonstrated tremendous empirical success across numerous challenging domains. However, we lack a strong theoretical understanding of the statistical complexity of RL in environments with large state spaces, where function approximation is required for sample-efficient learning. This thesis addresses this gap by rigorously examining the statistical complexity of RL with function approximation from a learning theoretic perspective. Departing from a long history of prior work, we consider the weakest form of function approximation, called agnostic policy learning, in which the learner seeks to find the best policy in a given class Pi, with no guarantee that Pi contains an optimal policy for the underlying task. We systematically explore agnostic policy learning along three key axes: environment access -- how a learner collects data from the environment; coverage conditions -- intrinsic properties of the underlying MDP measuring the expansiveness of state-occupancy measures for policies in the class Pi, and representational conditions -- structural assumptions on the class Pi itself. Within this comprehensive framework, we (1) design new learning algorithms with theoretical guarantees and (2) characterize fundamental performance bounds of any algorithm. Our results reveal significant statistical separations that highlight the power and limitations of agnostic policy learning.

Sparse rewards and heterogeneous task sequences remain persistent challenges in Reinforcement Learning (RL), often resulting in slow convergence, weak generalization, and inefficient exploration. We propose Uncertainty-Aware LLM-Guided Policy Shaping (ULPS), a novel framework that integrates a calibrated Large Language Model (LLM) into the RL training loop to provide structured, uncertainty-modulated behavioral guidance. ULPS employs an A*-based oracle to synthesize optimal symbolic trajectories, which are used to fine-tune a BERT-based language model. During training, this model supplies action suggestions whose influence is conditioned on epistemic uncertainty estimated via Monte Carlo (MC) dropout. An entropy-based blending mechanism adaptively balances LLM guidance and the learned policy (via Proximal Policy Optimization, PPO), allowing the agent to prioritize reliable priors while preserving adaptability. We evaluate ULPS on the MiniGridUnlockPickup benchmark and observe consistent improvements in success rate, reward efficiency, and sample complexity over unguided, uncalibrated, and standard RL baselines. ULPS achieves more than 9% improvement in execution accuracy after fine-tuning, requires fewer environment interactions, and yields higher reward AUC. Our results demonstrate that integrating symbolic A* trajectories, pretrained language priors, and uncertainty-aware control offers a principled and effective approach to multi-task reinforcement learning in sparse-reward domains, with potential extensibility to partially observable and multi-agent settings.

Two main challenges in Reinforcement Learning (RL) are designing appropriate reward functions and ensuring the safety of the learned policy. To address these challenges, we present a theoretical framework for Inverse Reinforcement Learning (IRL) in constrained Markov decision processes. From a convex-analytic perspective, we extend prior results on reward identifiability and generalizability to both the constrained setting and a more general class of regularizations. In particular, we show that identifiability up to potential shaping (Cao et al., 2021) is a consequence of entropy regularization and may generally no longer hold for other regularizations or in the presence of safety constraints. We also show that to ensure generalizability to new transition laws and constraints, the true reward must be identified up to a constant. Additionally, we derive a finite sample guarantee for the suboptimality of the learned rewards, and validate our results in a gridworld environment.

In real-world applications of reinforcement learning, it is often challenging to obtain a state representation that is parsimonious and satisfies the Markov property without prior knowledge. Consequently, it is common practice to construct a state larger than necessary, e.g., by concatenating measurements over contiguous time points. However, needlessly increasing the dimension of the state may slow learning and obfuscate the learned policy. We introduce the notion of a minimal sufficient state in a Markov decision process (MDP) as the subvector of the original state under which the process remains an MDP and shares the same reward function as the original process. We propose a novel SEquEntial Knockoffs (SEEK) algorithm that estimates the minimal sufficient state in a system with high-dimensional complex nonlinear dynamics. In large samples, the proposed method achieves selection consistency. As the method is agnostic to the reinforcement learning algorithm being applied, it benefits downstream tasks such as policy learning. Empirical experiments verify theoretical results and show the proposed approach outperforms several competing methods regarding variable selection accuracy and regret.

Reinforcement Learning with Verifiable Rewards (RLVR) has recently demonstrated remarkable potential in enhancing the reasoning capability of Large Reasoning Models (LRMs). However, RLVR often leads to entropy collapse, resulting in premature policy convergence and performance saturation. While manipulating token-level entropy has proven effective for promoting policy exploration, we argue that the latent dynamics underlying token generation encode a far richer computational structure for steering policy optimization toward a more effective exploration-exploitation tradeoff. To enable tractable analysis and intervention of the latent dynamics of LRMs, we leverage Koopman operator theory to obtain a linearized representation of their hidden-state dynamics. This enables us to introduce Dynamic Spectral Dispersion (DSD), a new metric to quantify the heterogeneity of the model's latent dynamics, serving as a direct indicator of policy exploration. Building upon these foundations, we propose Reasoning with Latent eXploration (ReLaX), a paradigm that explicitly incorporates latent dynamics to regulate exploration and exploitation during policy optimization. Comprehensive experiments across a wide range of multimodal and text-only reasoning benchmarks show that ReLaX significantly mitigates premature convergence and consistently achieves state-of-the-art performance.

Recent advances in fine-tuning multimodal large language models (MLLMs) using reinforcement learning have achieved remarkable progress, particularly with the introduction of various entropy control techniques. However, the role and characteristics of entropy in perception-oriented tasks like visual grounding, as well as effective strategies for controlling it, remain largely unexplored. To address this issue, we focus on the visual grounding task and analyze the role and characteristics of entropy in comparison to reasoning tasks. Building on these findings, we introduce ECVGPO (Entropy Control Visual Grounding Policy Optimization), an interpretable algorithm designed for effective entropy regulation. Through entropy control, the trade-off between exploration and exploitation is better balanced. Experiments show that ECVGPO achieves broad improvements across various benchmarks and models.

Reinforcement learning (RL) has recently become the core paradigm for aligning and strengthening large language models (LLMs). Yet, applying RL in off-policy settings--where stale data from past policies are used for training--improves sample efficiency, but remains challenging: policy entropy declines sharply, optimization often becomes unstable and may even collapse. Through theoretical and empirical analysis, we identify two key insights: (i) an imbalance in optimization, where negative-advantage samples dominate the policy gradient, suppressing useful behaviors and risking gradient explosions; and (ii) the derived Entropy-Clip Rule, which reveals that the fixed clipping mechanism in PPO-like objectives systematically blocks entropy-increasing updates, thereby driving the policy toward over-exploitation at the expense of exploration. Building on these insights, we propose BAlanced Policy Optimization with Adaptive Clipping (BAPO), a simple yet effective method that dynamically adjusts clipping bounds to adaptively re-balance positive and negative contributions, preserve entropy, and stabilize RL optimization. Across diverse off-policy scenarios--including sample replay and partial rollout--BAPO achieves fast, stable, and data-efficient training. On AIME 2024 and AIME 2025 benchmarks, our 7B BAPO model surpasses open-source counterparts such as SkyWork-OR1-7B, while our 32B BAPO model not only achieves state-of-the-art results among models of the same scale but also outperforms leading proprietary systems like o3-mini and Gemini-2.5-Flash-Thinking.

In recent years, various powerful policy gradient algorithms have been proposed in deep reinforcement learning. While all these algorithms build on the Policy Gradient Theorem, the specific design choices differ significantly across algorithms. We provide a holistic overview of on-policy policy gradient algorithms to facilitate the understanding of both their theoretical foundations and their practical implementations. In this overview, we include a detailed proof of the continuous version of the Policy Gradient Theorem, convergence results and a comprehensive discussion of practical algorithms. We compare the most prominent algorithms on continuous control environments and provide insights on the benefits of regularization. All code is available at https://github.com/Matt00n/PolicyGradientsJax.

We propose SmoothCruiser, a new planning algorithm for estimating the value function in entropy-regularized Markov decision processes and two-player games, given a generative model of the environment. SmoothCruiser makes use of the smoothness of the Bellman operator promoted by the regularization to achieve problem-independent sample complexity of order O~(1/epsilon^4) for a desired accuracy epsilon, whereas for non-regularized settings there are no known algorithms with guaranteed polynomial sample complexity in the worst case.

Wild Test-Time Adaptation (WTTA) is proposed to adapt a source model to unseen domains under extreme data scarcity and multiple shifts. Previous approaches mainly focused on sample selection strategies, while overlooking the fundamental problem on underlying optimization. Initially, we critically analyze the widely-adopted entropy minimization framework in WTTA and uncover its significant limitations in noisy optimization dynamics that substantially hinder adaptation efficiency. Through our analysis, we identify region confidence as a superior alternative to traditional entropy, however, its direct optimization remains computationally prohibitive for real-time applications. In this paper, we introduce a novel region-integrated method ReCAP that bypasses the lengthy process. Specifically, we propose a probabilistic region modeling scheme that flexibly captures semantic changes in embedding space. Subsequently, we develop a finite-to-infinite asymptotic approximation that transforms the intractable region confidence into a tractable and upper-bounded proxy. These innovations significantly unlock the overlooked potential dynamics in local region in a concise solution. Our extensive experiments demonstrate the consistent superiority of ReCAP over existing methods across various datasets and wild scenarios.

Model-free reinforcement learning algorithms have seen remarkable progress, but key challenges remain. Trust Region Policy Optimization (TRPO) is known for ensuring monotonic policy improvement through conservative updates within a trust region, backed by strong theoretical guarantees. However, its reliance on complex second-order optimization limits its practical efficiency. Proximal Policy Optimization (PPO) addresses this by simplifying TRPO's approach using ratio clipping, improving efficiency but sacrificing some theoretical robustness. This raises a natural question: Can we combine the strengths of both methods? In this paper, we introduce Simple Policy Optimization (SPO), a novel unconstrained first-order algorithm. By slightly modifying the policy loss used in PPO, SPO can achieve the best of both worlds. Our new objective improves upon ratio clipping, offering stronger theoretical properties and better constraining the probability ratio within the trust region. Empirical results demonstrate that SPO outperforms PPO with a simple implementation, particularly for training large, complex network architectures end-to-end.

Supervised Fine-Tuning (SFT) is the standard paradigm for domain adaptation, yet it frequently incurs the cost of catastrophic forgetting. In sharp contrast, on-policy Reinforcement Learning (RL) effectively preserves general capabilities. We investigate this discrepancy and identify a fundamental distributional gap: while RL aligns with the model's internal belief, SFT forces the model to fit external supervision. This mismatch often manifests as "Confident Conflicts" tokens characterized by low probability but low entropy. In these instances, the model is highly confident in its own prediction but is forced to learn a divergent ground truth, triggering destructive gradient updates. To address this, we propose Entropy-Adaptive Fine-Tuning (EAFT). Unlike methods relying solely on prediction probability, EAFT utilizes token-level entropy as a gating mechanism to distinguish between epistemic uncertainty and knowledge conflict. This allows the model to learn from uncertain samples while suppressing gradients on conflicting data. Extensive experiments on Qwen and GLM series (ranging from 4B to 32B parameters) across mathematical, medical, and agentic domains confirm our hypothesis. EAFT consistently matches the downstream performance of standard SFT while significantly mitigating the degradation of general capabilities.

Reinforcement Learning with Verifiable Rewards (RLVR) has emerged as an indispensable paradigm for enhancing reasoning in Large Language Models (LLMs). However, standard policy optimization methods, such as Group Relative Policy Optimization (GRPO), often converge to low-entropy policies, leading to severe mode collapse and limited output diversity. We analyze this issue from the perspective of sampling probability dynamics, identifying that the standard objective disproportionately reinforces the highest-likelihood paths, thereby suppressing valid alternative reasoning chains. To address this, we propose a novel Advantage Re-weighting Mechanism (ARM) designed to equilibrate the confidence levels across all correct responses. By incorporating Prompt Perplexity and Answer Confidence into the advantage estimation, our method dynamically reshapes the reward signal to attenuate the gradient updates of over-confident reasoning paths, while redistributing probability mass toward under-explored correct solutions. Empirical results demonstrate that our approach significantly enhances generative diversity and response entropy while maintaining competitive accuracy, effectively achieving a superior trade-off between exploration and exploitation in reasoning tasks. Empirical results on Qwen2.5 and DeepSeek models across mathematical and coding benchmarks show that ProGRPO significantly mitigates entropy collapse. Specifically, on Qwen2.5-7B, our method outperforms GRPO by 5.7% in Pass@1 and, notably, by 13.9% in Pass@32, highlighting its superior capability in generating diverse correct reasoning paths.

Expressive generative policies such as diffusion and flow models are appealing for MaxEnt online reinforcement learning because of their ability to model multimodal and highly non-Gaussian action distributions. However, training effective soft generative policies faces two obstacles that often arise together. First, marginal action densities are often unavailable, so existing methods typically rely on entropy bounds, heuristic proxies or approximations. Second, iterative shared-parameter samplers raise inference cost and require backpropagation through time over repeated network evaluations, increasing memory cost and destabilizing policy optimization. These obstacles motivate us to seek a generative policy that exposes a tractable MaxEnt objective while requiring only a single sampled actor forward pass for action generation. To this end, we propose soft generative actor-critic (SoftGAC), whose actor defines a stochastic bridge from a fixed base latent to a terminal action latent in pre-tanh space. This structured bridge allows us to lift the MaxEnt objective as an analytically tractable path-wise relative-entropy objective against a high-entropy reference process. In practical finite-step implementation, this relative entropy reduces exactly to sampled transition control energy and thus provides principled soft regularization. Moreover, we keep the single-pass actor lightweight by using small step-specific bridge transitions, each evaluated only once per sampled action, while maintaining a parameter budget comparable to strong actor baselines. Extensive experiments on challenging continuous-control benchmarks show that SoftGAC attains higher or competitive returns than strong generative policy baselines, including diffusion and flow-matching policies, while staying in the low-latency regime of one-pass actors and showing considerable improvements in the compute-return tradeoff.

Reasoning ability has become a defining capability of Large Language Models (LLMs), with Reinforcement Learning with Verifiable Rewards (RLVR) emerging as a key paradigm to enhance it. However, RLVR training often suffers from policy entropy collapse, where the policy becomes overly deterministic, hindering exploration and limiting reasoning performance. While entropy regularization is a common remedy, its effectiveness is highly sensitive to the fixed coefficient, making it unstable across tasks and models. In this work, we revisit entropy regularization in RLVR and argue that its potential has been largely underestimated. Our analysis shows that (i) tasks of varying difficulty demand distinct exploration intensities, and (ii) balanced exploration may require the policy entropy to be maintained within a moderate range below its initial level. Therefore, we propose Adaptive Entropy Regularization (AER)--a framework that dynamically balances exploration and exploitation via three components: difficulty-aware coefficient allocation, initial-anchored target entropy, and dynamic global coefficient adjustment. Experiments on multiple mathematical reasoning benchmarks show that AER consistently outperforms baselines, improving both reasoning accuracy and exploration capability.

This paper proposes a new optimization algorithm called Entropy-SGD for training deep neural networks that is motivated by the local geometry of the energy landscape. Local extrema with low generalization error have a large proportion of almost-zero eigenvalues in the Hessian with very few positive or negative eigenvalues. We leverage upon this observation to construct a local-entropy-based objective function that favors well-generalizable solutions lying in large flat regions of the energy landscape, while avoiding poorly-generalizable solutions located in the sharp valleys. Conceptually, our algorithm resembles two nested loops of SGD where we use Langevin dynamics in the inner loop to compute the gradient of the local entropy before each update of the weights. We show that the new objective has a smoother energy landscape and show improved generalization over SGD using uniform stability, under certain assumptions. Our experiments on convolutional and recurrent networks demonstrate that Entropy-SGD compares favorably to state-of-the-art techniques in terms of generalization error and training time.

We address the challenge of exploration in reinforcement learning (RL) when the agent operates in an unknown environment with sparse or no rewards. In this work, we study the maximum entropy exploration problem of two different types. The first type is visitation entropy maximization previously considered by Hazan et al.(2019) in the discounted setting. For this type of exploration, we propose a game-theoretic algorithm that has mathcal{O}(H^3S^2A/varepsilon^2) sample complexity thus improving the varepsilon-dependence upon existing results, where S is a number of states, A is a number of actions, H is an episode length, and varepsilon is a desired accuracy. The second type of entropy we study is the trajectory entropy. This objective function is closely related to the entropy-regularized MDPs, and we propose a simple algorithm that has a sample complexity of order mathcal{O}(poly(S,A,H)/varepsilon). Interestingly, it is the first theoretical result in RL literature that establishes the potential statistical advantage of regularized MDPs for exploration. Finally, we apply developed regularization techniques to reduce sample complexity of visitation entropy maximization to mathcal{O}(H^2SA/varepsilon^2), yielding a statistical separation between maximum entropy exploration and reward-free exploration.

While Large Language Model (LLM) agents excel at general tasks, they inherently struggle with continual adaptation due to the frozen weights after deployment. Conventional reinforcement learning (RL) offers a solution but incurs prohibitive computational costs and the risk of catastrophic forgetting. We introduce Just-In-Time Reinforcement Learning (JitRL), a training-free framework that enables test-time policy optimization without any gradient updates. JitRL maintains a dynamic, non-parametric memory of experiences and retrieves relevant trajectories to estimate action advantages on-the-fly. These estimates are then used to directly modulate the LLM's output logits. We theoretically prove that this additive update rule is the exact closed-form solution to the KL-constrained policy optimization objective. Extensive experiments on WebArena and Jericho demonstrate that JitRL establishes a new state-of-the-art among training-free methods. Crucially, JitRL outperforms the performance of computationally expensive fine-tuning methods (e.g., WebRL) while reducing monetary costs by over 30 times, offering a scalable path for continual learning agents. The code is available at https://github.com/liushiliushi/JitRL.

We provide a theoretical framework for Reinforcement Learning with Human Feedback (RLHF). Our analysis shows that when the true reward function is linear, the widely used maximum likelihood estimator (MLE) converges under both the Bradley-Terry-Luce (BTL) model and the Plackett-Luce (PL) model. However, we show that when training a policy based on the learned reward model, MLE fails while a pessimistic MLE provides policies with improved performance under certain coverage assumptions. Additionally, we demonstrate that under the PL model, the true MLE and an alternative MLE that splits the K-wise comparison into pairwise comparisons both converge. Moreover, the true MLE is asymptotically more efficient. Our results validate the empirical success of existing RLHF algorithms in InstructGPT and provide new insights for algorithm design. Furthermore, our results unify the problem of RLHF and max-entropy Inverse Reinforcement Learning (IRL), and provide the first sample complexity bound for max-entropy IRL.

Function calling capabilities are crucial for deploying Large Language Models in real-world applications, yet current training approaches fail to develop robust reasoning strategies. Supervised fine-tuning produces models that rely on superficial pattern matching, while standard reinforcement learning methods struggle with the complex action space of structured function calls. We present a novel reinforcement learning framework designed to enhance group relative policy optimization through strategic entropy based exploration specifically tailored for function calling tasks. Our approach addresses three critical challenges in function calling: insufficient exploration during policy learning, lack of structured reasoning in chain-of-thought generation, and inadequate verification of parameter extraction. Our two-stage data preparation pipeline ensures high-quality training samples through iterative LLM evaluation and abstract syntax tree validation. Extensive experiments on the Berkeley Function Calling Leaderboard demonstrate that this framework achieves state-of-the-art performance among open-source models with 86.02\% overall accuracy, outperforming standard GRPO by up to 6\% on complex multi-function scenarios. Notably, our method shows particularly strong improvements on code-pretrained models, suggesting that structured language generation capabilities provide an advantageous starting point for reinforcement learning in function calling tasks. We will release all the code, models and dataset to benefit the community.

In multi-agent tasks, the central challenge lies in the dynamic adaptation of strategies. However, directly conditioning on opponents' strategies is intractable in the prevalent deep reinforcement learning paradigm due to a fundamental ``representational bottleneck'': neural policies are opaque, high-dimensional parameter vectors that are incomprehensible to other agents. In this work, we propose a paradigm shift that bridges this gap by representing policies as human-interpretable source code and utilizing Large Language Models (LLMs) as approximate interpreters. This programmatic representation allows us to operationalize the game-theoretic concept of Program Equilibrium. We reformulate the learning problem by utilizing LLMs to perform optimization directly in the space of programmatic policies. The LLM functions as a point-wise best-response operator that iteratively synthesizes and refines the ego agent's policy code to respond to the opponent's strategy. We formalize this process as Programmatic Iterated Best Response (PIBR), an algorithm where the policy code is optimized by textual gradients, using structured feedback derived from game utility and runtime unit tests. We demonstrate that this approach effectively solves several standard coordination matrix games and a cooperative Level-Based Foraging environment.

Learning to optimize has emerged as a powerful framework for various optimization and machine learning tasks. Current such "meta-optimizers" often learn in the space of continuous optimization algorithms that are point-based and uncertainty-unaware. To overcome the limitations, we propose a meta-optimizer that learns in the algorithmic space of both point-based and population-based optimization algorithms. The meta-optimizer targets at a meta-loss function consisting of both cumulative regret and entropy. Specifically, we learn and interpret the update formula through a population of LSTMs embedded with sample- and feature-level attentions. Meanwhile, we estimate the posterior directly over the global optimum and use an uncertainty measure to help guide the learning process. Empirical results over non-convex test functions and the protein-docking application demonstrate that this new meta-optimizer outperforms existing competitors.

Large language models (LLMs) have achieved remarkable progress in reasoning tasks, yet the optimal integration of Supervised Fine-Tuning (SFT) and Reinforcement Learning (RL) remains a fundamental challenge. Through comprehensive analysis of token distributions, learning dynamics, and integration mechanisms from entropy-based perspectives, we reveal key differences between these paradigms: SFT induces coarse-grained global changes to LLM policy distributions, while RL performs fine-grained selective optimizations, with entropy serving as a critical indicator of training effectiveness. Building on these observations, we propose Supervised Reinforcement Fine-Tuning (SRFT), a single-stage method that unifies both fine-tuning paradigms through entropy-aware weighting mechanisms. Our approach simultaneously applies SFT and RL to directly optimize the LLM using demonstrations and self-exploration rollouts rather than through two-stage sequential methods. Extensive experiments show that SRFT achieves 59.1% average accuracy, outperforming zero-RL methods by 9.0% on five mathematical reasoning benchmarks and 10.9% on three out-of-distribution benchmarks.

While advancements in the reasoning abilities of LLMs have significantly enhanced their performance in solving mathematical problems, coding tasks, and general puzzles, their effectiveness in accurately adhering to instructions remains inconsistent, particularly with more complex directives. Our investigation identifies lazy reasoning during the thinking stage as the primary factor contributing to poor instruction adherence. To mitigate this issue, we propose a comprehensive framework designed to enable rigorous reasoning processes involving preview and self-checking, essential for satisfying strict instruction constraints. Specifically, we first generate instructions with complex constraints and apply a filtering process to obtain valid prompts, resulting in three distinct prompt datasets categorized as hard, easy, and pass. Then, we employ rejection sampling on the pass prompts to curate a small yet high-quality dataset, enabling a cold-start initialization of the model and facilitating its adaptation to effective reasoning patterns. Subsequently, we employ an entropy-preserving supervised fine-tuning (Entropy-SFT) strategy coupled with token-wise entropy-adaptive (TEA-RL) reinforcement learning guided by rule-based dense rewards. This approach encourages the model to transform its reasoning mechanism, ultimately fostering generalizable reasoning abilities that encompass preview and self-checking. Extensive experiments conducted on instruction-following benchmarks demonstrate remarkable performance improvements across various model scales. Notably, our Light-IF-32B model surpasses both larger open-source models such as DeepSeek-R1 and closed-source models like Doubao-1.6.

We develop a Multi-Agent Reinforcement Learning (MARL) method to learn scalable control policies for target tracking. Our method can handle an arbitrary number of pursuers and targets; we show results for tasks consisting up to 1000 pursuers tracking 1000 targets. We use a decentralized, partially-observable Markov Decision Process framework to model pursuers as agents receiving partial observations (range and bearing) about targets which move using fixed, unknown policies. An attention mechanism is used to parameterize the value function of the agents; this mechanism allows us to handle an arbitrary number of targets. Entropy-regularized off-policy RL methods are used to train a stochastic policy, and we discuss how it enables a hedging behavior between pursuers that leads to a weak form of cooperation in spite of completely decentralized control execution. We further develop a masking heuristic that allows training on smaller problems with few pursuers-targets and execution on much larger problems. Thorough simulation experiments, ablation studies, and comparisons to state of the art algorithms are performed to study the scalability of the approach and robustness of performance to varying numbers of agents and targets.

Masked Diffusion Models (MDMs) offer flexible, non-autoregressive generation, but this freedom introduces a challenge: final output quality is highly sensitive to the decoding order. We are the first to formalize this issue, attributing the variability in output quality to the cumulative predictive uncertainty along a generative path. To quantify this uncertainty, we introduce Denoising Entropy, a computable metric that serves as an internal signal for evaluating generative process. Leveraging this metric, we propose two algorithms designed to optimize the decoding path: a post-hoc selection method and a real-time guidance strategy. Experiments demonstrate that our entropy-guided methods significantly improve generation quality, consistently boosting accuracy on challenging reasoning, planning, and code benchmarks. Our work establishes Denoising Entropy as a principled tool for understanding and controlling generation, effectively turning the uncertainty in MDMs from a liability into a key advantage for discovering high-quality solutions.

Flow-based generative models, including diffusion models, excel at modeling continuous distributions in high-dimensional spaces. In this work, we introduce Flow Policy Optimization (FPO), a simple on-policy reinforcement learning algorithm that brings flow matching into the policy gradient framework. FPO casts policy optimization as maximizing an advantage-weighted ratio computed from the conditional flow matching loss, in a manner compatible with the popular PPO-clip framework. It sidesteps the need for exact likelihood computation while preserving the generative capabilities of flow-based models. Unlike prior approaches for diffusion-based reinforcement learning that bind training to a specific sampling method, FPO is agnostic to the choice of diffusion or flow integration at both training and inference time. We show that FPO can train diffusion-style policies from scratch in a variety of continuous control tasks. We find that flow-based models can capture multimodal action distributions and achieve higher performance than Gaussian policies, particularly in under-conditioned settings.

On-policy distillation is a promising approach for transferring knowledge between language models, where a student learns from dense token-level signals along its own trajectories. This framework typically uses reverse KL divergence, encouraging the student to match the teacher's high-confidence predictions. However, we show that the mode-seeking property of reverse KL reduces generation diversity and yields unstable learning signals when the teacher distribution has high entropy. To address this, we introduce Entropy-Aware On-Policy Distillation. Our key idea is augmenting the standard reverse KL objective with forward KL when teacher entropy is high, capturing the full range of plausible outputs while retaining precise imitation elsewhere. It balances mode-seeking precision with mode-covering robustness without sacrificing on-policy training efficiency. Experiments show that our method maintains generation diversity (sustained token-level entropy) and improves student-teacher alignment (lower forward KL on high-entropy tokens). Across six math reasoning benchmarks, this yields Pass@8 accuracy gains of +1.37 for Qwen3-0.6B-Base, +2.39 for Qwen3-1.7B-Base, and +5.05 for Qwen3-4B-Base compared to baseline on-policy distillation methods. These results demonstrate that accounting for teacher uncertainty is essential for maintaining diversity and achieving effective knowledge transfer.

Imitation learning uses data for training policies to solve complex tasks. However, when the training data is collected from human demonstrators, it often leads to multimodal distributions because of the variability in human actions. Most imitation learning methods rely on a maximum likelihood (ML) objective to learn a parameterized policy, but this can result in suboptimal or unsafe behavior due to the mode-averaging property of the ML objective. In this work, we propose Information Maximizing Curriculum, a curriculum-based approach that assigns a weight to each data point and encourages the model to specialize in the data it can represent, effectively mitigating the mode-averaging problem by allowing the model to ignore data from modes it cannot represent. To cover all modes and thus, enable diverse behavior, we extend our approach to a mixture of experts (MoE) policy, where each mixture component selects its own subset of the training data for learning. A novel, maximum entropy-based objective is proposed to achieve full coverage of the dataset, thereby enabling the policy to encompass all modes within the data distribution. We demonstrate the effectiveness of our approach on complex simulated control tasks using diverse human demonstrations, achieving superior performance compared to state-of-the-art methods.

While Experience Replay - the practice of storing rollouts and reusing them multiple times during training - is a foundational technique in general RL, it remains largely unexplored in LLM post-training due to the prevailing belief that fresh, on-policy data is essential for high performance. In this work, we challenge this assumption. We present a systematic study of replay buffers for LLM post-training, formalizing the optimal design as a trade-off between staleness-induced variance, sample diversity and the high computational cost of generation. We show that strict on-policy sampling is suboptimal when generation is expensive. Empirically, we show that a well-designed replay buffer can drastically reduce inference compute without degrading - and in some cases even improving - final model performance, while preserving policy entropy.

We study the computational complexity of approximating general constrained Markov decision processes. Our primary contribution is the design of a polynomial time (0,ε)-additive bicriteria approximation algorithm for finding optimal constrained policies across a broad class of recursively computable constraints, including almost-sure, chance, expectation, and their anytime variants. Matching lower bounds imply our approximation guarantees are optimal so long as P neq NP. The generality of our approach results in answers to several long-standing open complexity questions in the constrained reinforcement learning literature. Specifically, we are the first to prove polynomial-time approximability for the following settings: policies under chance constraints, deterministic policies under multiple expectation constraints, policies under non-homogeneous constraints (i.e., constraints of different types), and policies under constraints for continuous-state processes.

Unsupervised reinforcement learning aims at learning a generalist policy in a reward-free manner for fast adaptation to downstream tasks. Most of the existing methods propose to provide an intrinsic reward based on surprise. Maximizing or minimizing surprise drives the agent to either explore or gain control over its environment. However, both strategies rely on a strong assumption: the entropy of the environment's dynamics is either high or low. This assumption may not always hold in real-world scenarios, where the entropy of the environment's dynamics may be unknown. Hence, choosing between the two objectives is a dilemma. We propose a novel yet simple mixture of policies to address this concern, allowing us to optimize an objective that simultaneously maximizes and minimizes the surprise. Concretely, we train one mixture component whose objective is to maximize the surprise and another whose objective is to minimize the surprise. Hence, our method does not make assumptions about the entropy of the environment's dynamics. We call our method a Mixture Of SurpriseS (MOSS) for unsupervised reinforcement learning. Experimental results show that our simple method achieves state-of-the-art performance on the URLB benchmark, outperforming previous pure surprise maximization-based objectives. Our code is available at: https://github.com/LeapLabTHU/MOSS.

Reinforcement learning (RL) is the dominant paradigm for sharpening strategic tool use capabilities of LLMs on long-horizon, sparsely-rewarded agent tasks, yet it faces a fundamental challenge of exploration-exploitation trade-off. Existing studies stimulate exploration through the lens of policy entropy, but such mechanical entropy maximization is prone to RL training instability due to the multi-turn distribution shifting. In this paper, we target the progressive exploration-exploitation balance under the guidance of the agent own experiences without succumbing to either entropy collapsing or runaway divergence. We propose SPEAR, a curriculum-based self-imitation learning (SIL) recipe for training agentic LLMs. It extends the vanilla SIL framework, where a replay buffer stores self-generated promising trajectories for off-policy update, by gradually steering the policy evolution within a well-balanced range of entropy across stages. Specifically, our approach incorporates a curriculum to manage the exploration process, utilizing intrinsic rewards to foster skill-level exploration and facilitating action-level exploration through SIL. At first, the auxiliary tool call reward plays a critical role in the accumulation of tool-use skills, enabling broad exposure to the unfamiliar distributions of the environment feedback with an upward entropy trend. As training progresses, self-imitation gets strengthened to exploit existing successful patterns from replayed experiences for comparative action-level exploration, accelerating solution iteration without unbounded entropy growth. To further stabilize training, we recalibrate the advantages of experiences in the replay buffer to address the potential policy drift. Reugularizations such as the clipping of tokens with high covariance between probability and advantage are introduced to the trajectory-level entropy control to curb over-confidence.

Direct Preference Optimization (DPO) has become a standard technique for aligning language models with human preferences in a supervised manner. Despite its empirical success, the theoretical justification behind its log-ratio reward parameterization remains incomplete. In this work, we address this gap by utilizing the Differential Information Distribution (DID): a distribution over token sequences that captures the information gained during policy updates. First, we show that when preference labels encode the differential information required to transform a reference policy into a target policy, the log-ratio reward in DPO emerges as the uniquely optimal form for learning the target policy via preference optimization. This result naturally yields a closed-form expression for the optimal sampling distribution over rejected responses. Second, we find that the condition for preferences to encode differential information is fundamentally linked to an implicit assumption regarding log-margin ordered policies-an inductive bias widely used in preference optimization yet previously unrecognized. Finally, by analyzing the entropy of the DID, we characterize how learning low-entropy differential information reinforces the policy distribution, while high-entropy differential information induces a smoothing effect, which explains the log-likelihood displacement phenomenon. We validate our theoretical findings in synthetic experiments and extend them to real-world instruction-following datasets. Our results suggest that learning high-entropy differential information is crucial for general instruction-following, while learning low-entropy differential information benefits knowledge-intensive question answering. Overall, our work presents a unifying perspective on the DPO objective, the structure of preference data, and resulting policy behaviors through the lens of differential information.

The Information Bottleneck (IB) method frequently suffers from unstable optimization, characterized by abrupt representation shifts near critical points of the IB trade-off parameter, beta. In this paper, I introduce a novel approach to achieve stable and convex IB optimization through symbolic continuation and entropy-regularized trajectories. I analytically prove convexity and uniqueness of the IB solution path when an entropy regularization term is included, and demonstrate how this stabilizes representation learning across a wide range of eta values. Additionally, I provide extensive sensitivity analyses around critical points (beta) with statistically robust uncertainty quantification (95% confidence intervals). The open-source implementation, experimental results, and reproducibility framework included in this work offer a clear path for practical deployment and future extension of my proposed method.

Test-time reinforcement learning generates multiple candidate answers via repeated rollouts and performs online updates using pseudo-labels constructed by majority voting. To reduce overhead and improve exploration, prior work introduces tree structured rollouts, which share reasoning prefixes and branch at key nodes to improve sampling efficiency. However, this paradigm still faces two challenges: (1) high entropy branching can trigger rollout collapse, where the branching budget concentrates on a few trajectories with consecutive high-entropy segments, rapidly reducing the number of effective branches; (2) early pseudo-labels are noisy and biased, which can induce self-reinforcing overfitting, causing the policy to sharpen prematurely and suppress exploration. To address these issues, we propose Entropy Confidence Hybrid Group Relative Policy Optimization (ECHO). During rollout, ECHO jointly leverages local entropy and group level confidence to adaptively control branch width, and further introduces online confidence-based pruning to terminate persistently low confidence branches, avoiding high entropy traps and mitigating collapse. During policy updates, ECHO employs confidence adaptive clipping and an entropy confidence hybrid advantage shaping approach to enhance training robustness and mitigate early stage bias. Experiments demonstrate that ECHO achieves consistent gains on multiple mathematical and visual reasoning benchmarks, and generalizes more effectively under a limited rollout budget.

Reinforcement learning has significantly enhanced the reasoning capabilities of Large Language Models (LLMs) in complex problem-solving tasks. Recently, the introduction of DeepSeek R1 has inspired a surge of interest in leveraging rule-based rewards as a low-cost alternative for computing advantage functions and guiding policy optimization. However, a common challenge observed across many replication and extension efforts is that when multiple sampled responses under a single prompt converge to identical outcomes, whether correct or incorrect, the group-based advantage degenerates to zero. This leads to vanishing gradients and renders the corresponding samples ineffective for learning, ultimately limiting training efficiency and downstream performance. To address this issue, we propose a consistency-aware policy optimization framework that introduces a structured global reward based on outcome consistency, the global loss based on it ensures that, even when model outputs show high intra-group consistency, the training process still receives meaningful learning signals, which encourages the generation of correct and self-consistent reasoning paths from a global perspective. Furthermore, we incorporate an entropy-based soft blending mechanism that adaptively balances local advantage estimation with global optimization, enabling dynamic transitions between exploration and convergence throughout training. Our method introduces several key innovations in both reward design and optimization strategy. We validate its effectiveness through substantial performance gains on multiple mathematical reasoning benchmarks, highlighting the proposed framework's robustness and general applicability. Code of this work has been released at https://github.com/hijih/copo-code.git.

Reinforcement learning is crucial for aligning large language models to perform complex reasoning tasks. However, current algorithms such as Group Relative Policy Optimization suffer from coarse grained, sequence level credit assignment, which severely struggles to isolate pivotal reasoning steps within long Chain of Thought generations. Furthermore, the standard unbounded Kullback Leibler divergence penalty induces severe gradient instability and mode seeking conservatism, ultimately stifling the discovery of novel reasoning trajectories. To overcome these limitations, we introduce Distribution Guided Policy Optimization, a novel critic free reinforcement learning framework that reinterprets distribution deviation as a guiding signal rather than a rigid penalty. DGPO replaces the volatile KL divergence with the bounded Hellinger distance to safely quantify token level exploration without the risk of gradient explosion. To effectively distinguish genuine reasoning breakthroughs from hallucinatory noise, we propose an entropy gating mechanism that scales this deviation by the policy`s epistemic uncertainty. By dynamically redistributing the coarse sequence-level advantage to individual tokens based on these gated scores, DGPO heavily incentivizes critical exploratory steps while suppressing unwarranted, low-entropy deviations. Consequently, DGPO completely eliminates the traditional token-level KL penalty and achieves fine-grained credit reallocation without the computational overhead of an additional value network. Extensive empirical evaluations demonstrate that DGPO sets a new state-of-the-art for critic free alignment. Notably, on the Qwen2.5-32B architecture, DGPO achieves 60.0% Avg@32 accuracy and 46.0% Avg@32 accuracy on the challenging AIME2024 and AIME2025 benchmarks respectively, substantially outperforming competitive baselines like DAPO.

The varying significance of distinct primitive behaviors during the policy learning process has been overlooked by prior model-free RL algorithms. Leveraging this insight, we explore the causal relationship between different action dimensions and rewards to evaluate the significance of various primitive behaviors during training. We introduce a causality-aware entropy term that effectively identifies and prioritizes actions with high potential impacts for efficient exploration. Furthermore, to prevent excessive focus on specific primitive behaviors, we analyze the gradient dormancy phenomenon and introduce a dormancy-guided reset mechanism to further enhance the efficacy of our method. Our proposed algorithm, ACE: Off-policy Actor-critic with Causality-aware Entropy regularization, demonstrates a substantial performance advantage across 29 diverse continuous control tasks spanning 7 domains compared to model-free RL baselines, which underscores the effectiveness, versatility, and efficient sample efficiency of our approach. Benchmark results and videos are available at https://ace-rl.github.io/.

Recent advancements in Large Language Models have yielded significant improvements in complex reasoning tasks such as mathematics and programming. However, these models remain heavily dependent on annotated data and exhibit limited adaptability in unsupervised scenarios. To address these limitations, test-time reinforcement learning (TTRL) has been proposed, which enables self-optimization by leveraging model-generated pseudo-labels. Despite its promise, TTRL faces several key challenges, including high inference costs due to parallel rollouts and early-stage estimation bias that fosters overconfidence, reducing output diversity and causing performance plateaus. To address these challenges, we introduce an entropy-based mechanism to enhance the exploration-exploitation balance in test-time reinforcement learning through two strategies: Entropy-fork Tree Majority Rollout (ETMR) and Entropy-based Advantage Reshaping (EAR). Compared with the baseline, our approach enables Llama3.1-8B to achieve a 68 percent relative improvement in Pass at 1 metric on the AIME 2024 benchmark, while consuming only 60 percent of the rollout tokens budget. This highlights our method's ability to effectively optimize the trade-off between inference efficiency, diversity, and estimation robustness, thereby advancing unsupervised reinforcement learning for open-domain reasoning tasks.

We argue that inventory management presents unique opportunities for the reliable application of deep reinforcement learning (DRL). To enable this, we emphasize and test two complementary techniques. The first is Hindsight Differentiable Policy Optimization (HDPO), which uses pathwise gradients from offline counterfactual simulations to directly and efficiently optimize policy performance. Unlike standard policy gradient methods that rely on high-variance score-function estimators, HDPO computes gradients by differentiating through the known system dynamics. Via extensive benchmarking, we show that HDPO recovers near-optimal policies in settings with known or bounded optima, is more robust than variants of the REINFORCE algorithm, and significantly outperforms generalized newsvendor heuristics on problems using real time series data. Our second technique aligns neural policy architectures with the topology of the inventory network. We exploit Graph Neural Networks (GNNs) as a natural inductive bias for encoding supply chain structure, demonstrate that they can represent optimal and near-optimal policies in two theoretical settings, and empirically show that they reduce data requirements across six diverse inventory problems. A key obstacle to progress in this area is the lack of standardized benchmark problems. To address this gap, we open-source a suite of benchmark environments, along with our full codebase, to promote transparency and reproducibility. All resources are available at github.com/MatiasAlvo/Neural_inventory_control.

Entropy minimization (EM) trains the model to concentrate even more probability mass on its most confident outputs. We show that this simple objective alone, without any labeled data, can substantially improve large language models' (LLMs) performance on challenging math, physics, and coding tasks. We explore three approaches: (1) EM-FT minimizes token-level entropy similarly to instruction finetuning, but on unlabeled outputs drawn from the model; (2) EM-RL: reinforcement learning with negative entropy as the only reward to maximize; (3) EM-INF: inference-time logit adjustment to reduce entropy without any training data or parameter updates. On Qwen-7B, EM-RL, without any labeled data, achieves comparable or better performance than strong RL baselines such as GRPO and RLOO that are trained on 60K labeled examples. Furthermore, EM-INF enables Qwen-32B to match or exceed the performance of proprietary models like GPT-4o, Claude 3 Opus, and Gemini 1.5 Pro on the challenging SciCode benchmark, while being 3x more efficient than self-consistency and sequential refinement. Our findings reveal that many pretrained LLMs possess previously underappreciated reasoning capabilities that can be effectively elicited through entropy minimization alone, without any labeled data or even any parameter updates.

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.

Reinforcement learning for large language models faces a fundamental trade-off between sample efficiency and asymptotic performance: strictly on-policy methods discard trajectories after a single update, while off-policy reuse introduces distribution mismatch that existing trust-region techniques mitigate primarily by enforcing conservative optimization, often leaving rich training signals underutilized. To investigate this, we perform extensive off-policy updates on fixed data. Our experiments reveal that aggressive multi-step optimization brings rapid initial gains, but excessive updates cause trajectory probabilities to deviate and entropy to collapse, with performance plateauing early. Tightening KL constraints merely lowers the ceiling without resolving the degradation. This motivates Extreme Region Policy Distillation (ERPD), a two-stage framework that decouples sample efficiency from KL efficiency. The first stage performs weakly constrained off-policy optimization on fixed data to maximally extract training signals. The resulting policy provides token-level supervision. In the second stage, we distill these signals into the base policy under trust-region constraints, filtering harmful drift while preserving useful signals. The distilled policy achieves comparable or better performance with substantially smaller KL divergence, indicating that much of the first-stage divergence was spent on unnecessary drift rather than genuine improvement. Crucially, ERPD accommodates both strong and weak teachers: when aggressive optimization yields no stronger policy, even degenerate teachers provide effective supervision via alternative signal construction strategies. We validate ERPD on mathematical reasoning, showing gains for strong base models where on-policy training plateaus, and reliable improvements with weak teachers.

The alignment of language models with human preferences is vital for their application in real-world tasks. The problem is formulated as optimizing the model's policy to maximize the expected reward that reflects human preferences with minimal deviation from the initial policy. While considered as a straightforward solution, reinforcement learning (RL) suffers from high variance in policy updates, which impedes efficient policy improvement. Recently, direct preference optimization (DPO) was proposed to directly optimize the policy from preference data. Though simple to implement, DPO is derived based on the optimal policy that is not assured to be achieved in practice, which undermines its convergence to the intended solution. In this paper, we propose efficient exact optimization (EXO) of the alignment objective. We prove that EXO is guaranteed to optimize in the same direction as the RL algorithms asymptotically for arbitary parametrization of the policy, while enables efficient optimization by circumventing the complexities associated with RL algorithms. We compare our method to DPO with both theoretical and empirical analyses, and further demonstrate the advantages of our method over existing approaches on realistic human preference data.

The pervasiveness of proprietary language models has raised critical privacy concerns, necessitating advancements in private inference (PI), where computations are performed directly on encrypted data without revealing users' sensitive information. While PI offers a promising solution, its practical deployment is hindered by substantial communication and latency overheads, primarily stemming from nonlinear operations. To address this, we introduce an information-theoretic framework to characterize the role of nonlinearities in decoder-only language models, laying a principled foundation for optimizing transformer-architectures tailored to the demands of PI. By leveraging Shannon's entropy as a quantitative measure, we uncover the previously unexplored dual significance of nonlinearities: beyond ensuring training stability, they are crucial for maintaining attention head diversity. Specifically, we find that their removal triggers two critical failure modes: {\em entropy collapse} in deeper layers that destabilizes training, and {\em entropic overload} in earlier layers that leads to under-utilization of Multi-Head Attention's (MHA) representational capacity. We propose an entropy-guided attention mechanism paired with a novel entropy regularization technique to mitigate entropic overload. Additionally, we explore PI-friendly alternatives to layer normalization for preventing entropy collapse and stabilizing the training of LLMs with reduced-nonlinearities. Our study bridges the gap between information theory and architectural design, establishing entropy dynamics as a principled guide for developing efficient PI architectures. The code and implementation are available at https://github.com/Nandan91/entropy-guided-attention-llm{entropy-guided-llm}.

Entropy regularization is a standard technique in reinforcement learning (RL) to enhance exploration, yet it yields negligible effects or even degrades performance in Large Language Models (LLMs). We attribute this failure to the cumulative tail risk inherent to LLMs with massive vocabularies and long generation horizons. In such environments, standard global entropy maximization indiscriminately dilutes probability mass into the vast tail of invalid tokens rather than focusing on plausible candidates, thereby disrupting coherent reasoning. To address this, we propose Trust Region Entropy (TRE), a method that encourages exploration strictly within the model's trust region. Extensive experiments across mathematical reasoning (MATH), combinatorial search (Countdown), and preference alignment (HH) tasks demonstrate that TRE consistently outperforms vanilla PPO, standard entropy regularization, and other exploration baselines. Our code is available at https://github.com/WhyChaos/TRE-Encouraging-Exploration-in-the-Trust-Region.

Reinforcement learning with verifiable rewards (RLVR) has shown great promise in enhancing the reasoning abilities of large reasoning models (LRMs). However, it suffers from a critical issue: entropy collapse and premature convergence. Naive entropy regularization, a common approach for encouraging exploration in the traditional RL literature, fails to address this problem in the context of LRM. Our analysis reveals that this failure stems from the vast action space and long trajectories in LRMs, which easily trigger a global entropy explosion as the model indiscriminately explores all possible actions and states. To address this, we propose SIREN (SelectIve entRopy rEgularizatioN), a method that confines exploration to a meaningful subset of actions and states. SIREN achieves this through a two-step entropy masking mechanism, consisting of a top-p mask and a peak-entropy mask. In addition, regularization is transformed into a self-anchored form to stabilize training. Across five mathematical benchmarks, SIREN attains superior average performance over previous entropy-related RLVR approaches, exemplified by a +6.6 maj@k improvement on AIME24/25 with Qwen2.5-Math-7B. Further analysis confirms that SIREN promotes greater response diversity and maintains entropy at an appropriate level, which helps to preserve the validation pass@k throughout training. This effectively mitigates the premature convergence problem common in RLVR for LRM.

Multi-agent systems (MAS) have emerged as a prominent paradigm for leveraging large language models (LLMs) to tackle complex tasks. However, the mechanisms governing the effectiveness of MAS built upon publicly available LLMs, specifically the underlying rationales for their success or failure, remain largely unexplored. In this paper, we revisit MAS through the perspective of entropy, considering both intra- and inter-agent dynamics by investigating entropy transitions during problem-solving across various topologies, six reasoning benchmarks, and two agentic tasks. By analyzing 245 features spanning token-, agent-, and round-level entropy, we counterintuitively find that a single agent outperforms MAS in approximately 43.3\% of cases, and that entropy dynamics are largely determined during the first round of interaction. Furthermore, we provide three key observations: 1) Certainty Preference: peak entropy directly harms and stable entropy directly benefits MAS correctness; 2) Base Entropy: base models with lower entropy during problem-solving causally drive MAS performance; and 3) Task Awareness: entropy dynamics of MAS play varying roles across different tasks. Building on these insights, we introduce a simple yet effective algorithm, the Entropy Judger, to select solutions from MAS's pass@k results, leading to consistent accuracy improvements across all MAS configurations and tasks. Our source code is available at https://github.com/AgenticFinLab/multiagent-entropy{this https URL}.

We investigate a narrow but common failure mode of GRPO-style reinforcement learning in the context of sparse verifiable rewards: early updates contain more responses with negative advantages than those with positive advantages, while response-level length normalization ties the magnitude of the update to the length of the output. We propose Hysteretic Policy Optimization (HPO), a minimal modification of GRPO that reduces the weight of negative-advantage updates and replaces per-response length normalization with mean-length normalization. We further introduce Adaptive HPO (A-HPO), which sets the hysteretic weight based on batch-level advantage-sign statistics, thereby removing the need for tuning a fixed hysteretic weight. In our TeleLogs and Countdown experiments, A-HPO improves the reward per update compared to GRPO, with the largest gains in early sparse reward regimes. On TeleLogs, A-HPO achieves a final reward of 0.84, outperforming SAPO by 5%, GSPO by 11%, and GRPO by 15%, while maintaining a comparable response-length. On Countdown, A-HPO achieves the largest gains in initial and most difficult configurations across 1.5B-7B models. Ablation studies on the hysteretic weight show that the gains of A-HPO come from better balancing the contributions of positive and negative advantages compared to positive-only or fully symmetric updates.

Optimization (PPO) has been positioned by recent literature as the canonical method for the RL part of RLHF. PPO performs well empirically but has a heuristic motivation and handles the KL-divergence constraint used in LM-RLHF in an ad-hoc manner and suffers form reward oscillations, entropy collapse, value function drift, and sudden policy divergence that require frequent restarts and extensive hyperparameter tuning. In this paper, we develop a new pure on policy actor-critic RL method for the LM-RLHF setting. We present SAFE (Stable Alignment Finetuning with Entropy-aware control),a novel RLHF algorithm that combines a Double Soft-Min Critic for pessimistic value estimation with a new multi-layer stabilization framework combining entropy-gated KL regulation, and PID-controlled adaptive thresholds. Unlike standard PPO's symmetric KL penalties, SAFE distinguishes high-entropy exploration from low-entropy mode collapse and adjusts penalties dynamically based on reward velocity. Experiments on a 3B parameter model show SAFE achieves +5.15\% training-average reward than PPO (0.725 vs 0.689), negligible reward crashes, and superior KL control than ppo . Our method adds minimal computational overhead and provides an interpretable, crash-resistant RLHF framework that maintains aggressive learning speed while ensuring stable long-horizon optimization suitable for production deployment. Code is available at https://github.com/ryyzn9/SAFE

We introduce Entropy-Guided Sequence Weighting (EGSW), a novel approach that enhances the exploration-exploitation tradeoff by dynamically assigning weights to generated outputs based on their advantage and entropy for Reinforcement Learning-based Large Language Model fine-tuning. EGSW integrates entropy regularization with advantage-based weighting to balance policy updates, enabling efficient exploration in high-dimensional state spaces. By employing temperature-scaled softmax weighting over sequences, EGSW prioritizing high-reward, high-uncertainty steps while maintaining training stability. Although originally developed to improve Group Relative Policy Optimization (GRPO) during large language model (LLM) fine-tuning, EGSW is generalizable to other reinforcement learning (RL) algorithms and can be implemented in both step-wise and trajectory-wise settings. Empirical evaluations demonstrate that EGSW enhances GRPO reasoning ability, yielding improvements in sample efficiency. Future work will explore the application of EGSW to advanced RL methodologies.

Reinforcement learning is structurally harder than supervised learning because the policy changes the data distribution it learns from. The resulting fragility is especially visible in large-model training, where the training and rollout systems differ in numerical precision, sampling, and other implementation details. Existing methods manage this fragility by adding hyper-parameters to the training objective, which makes the algorithm more sensitive to its configuration and requires retuning whenever the task, model scale, or distribution mismatch changes. This fragility traces to two concerns that current objectives entangle through hyper-parameters set before training begins: a trust-region concern, that updates should not move the policy too far from its current value, and an off-policy concern, that data from older or different behavior policies should influence the update only to the extent that it remains reliable. Neither concern is a constant to set in advance, and their severity is reflected in the policy-ratio distribution of the current batch. We present a simple yet effective batch-adaptive objective that replaces fixed clipping with the normalized effective sample size of the policy ratios. The same statistic caps the score-function weight and sets the strength of an off-policy regularizer, so the update stays close to the usual on-policy score-function update when ratios are nearly uniform, and tightens automatically when stale or mismatched data cause ratio concentration, while retaining a nonzero learning signal on high-ratio tokens. Experiments across a wide range of settings show that our method matches or exceeds tuned baselines, introducing no new objective hyper-parameters and removing several existing ones. The code is available at https://github.com/FeynRL-project/FeynRL.

Among on-policy reinforcement learning algorithms, Proximal Policy Optimization (PPO) demonstrates is widely favored for its simplicity, numerical stability, and strong empirical performance. Standard PPO relies on surrogate objectives defined via importance ratios, which require evaluating policy likelihood that is typically straightforward when the policy is modeled as a Gaussian distribution. However, extending PPO to more expressive, high-capacity policy models such as continuous normalizing flows (CNFs), also known as flow-matching models, is challenging because likelihood evaluation along the full flow trajectory is computationally expensive and often numerically unstable. To resolve this issue, we propose PolicyFlow, a novel on-policy CNF-based reinforcement learning algorithm that integrates expressive CNF policies with PPO-style objectives without requiring likelihood evaluation along the full flow path. PolicyFlow approximates importance ratios using velocity field variations along a simple interpolation path, reducing computational overhead without compromising training stability. To further prevent mode collapse and further encourage diverse behaviors, we propose the Brownian Regularizer, an implicit policy entropy regularizer inspired by Brownian motion, which is conceptually elegant and computationally lightweight. Experiments on diverse tasks across various environments including MultiGoal, PointMaze, IsaacLab and MuJoCo Playground show that PolicyFlow achieves competitive or superior performance compared to PPO using Gaussian policies and flow-based baselines including FPO and DPPO. Notably, results on MultiGoal highlight PolicyFlow's ability to capture richer multimodal action distributions.

Reinforcement learning with verifiable reward has recently emerged as a central paradigm for post-training large language models (LLMs); however, prevailing mean-based methods, such as Group Relative Policy Optimization (GRPO), suffer from entropy collapse and limited reasoning gains. We argue that these issues stem from overemphasizing high-probability output sequences while neglecting rare but informative reasoning paths. To address these challenges, we propose Risk-based Policy Optimization (RiskPO), which substitutes classical mean-based objectives with principled risk measures. Specifically, we introduce a Mixed Value-at-Risk objective that integrates weighted attention over multiple regions of the reward distribution, thereby amplifying gradient signals on challenging instances and preventing overconfident convergence. We further design a bundling scheme that aggregates multiple questions into bundles, thus enriching the feedback signal and yielding more stable and informative training dynamics. Theoretically, we prove that the risk-averse update alleviates entropy collapse and promotes exploration. Numerically, RiskPO achieves consistent and significant improvements in mathematical reasoning, multi-modal reasoning, and code generation benchmarks, surpassing GRPO and its variants on both Pass@1 and Pass@k metrics. Our results demonstrate that risk-based optimization provides a rigorous and effective paradigm for enhancing LLM reasoning capabilities.

Reinforcement Learning with Verifiable Rewards (RLVR) has propelled Large Language Models in complex reasoning, yet its scalability is often hindered by a training bottleneck where performance plateaus as policy entropy collapses, signaling a loss of exploration. Previous methods typically address this by maintaining high policy entropy, yet the precise mechanisms that govern meaningful exploration have remained underexplored. Our analysis suggests that an unselective focus on entropy risks amplifying irrelevant tokens and destabilizing training. This paper investigates the exploration dynamics within RLVR and identifies a key issue: the gradual elimination of valuable low-probability exploratory tokens, which we term \textit{reasoning sparks}. We find that while abundant in pre-trained models, these sparks are systematically extinguished during RLVR due to over-penalization, leading to a degeneracy in exploration. To address this, we introduce Low-probability Regularization (Lp-Reg). Its core mechanism regularizes the policy towards a heuristic proxy distribution. This proxy is constructed by filtering out presumed noise tokens and re-normalizing the distribution over the remaining candidates. The result is a less-noisy proxy where the probability of reasoning sparks is amplified, which then serves as a soft regularization target to shield these valuable tokens from elimination via KL divergence. Experiments show that Lp-Reg enables stable on-policy training for around 1,000 steps, a regime where baseline entropy-control methods collapse. This sustained exploration leads to state-of-the-art performance, achieving a 60.17% average accuracy on five math benchmarks, an improvement of 2.66% over prior methods. Code is available at https://github.com/CarlanLark/Lp-Reg.

We trained 13,440 large language models and found that entropy minimization requires only a single unlabeled data and 10 steps optimization to achieve performance improvements comparable to or even greater than those obtained using thousands of data and carefully designed rewards in rule-based reinforcement learning. This striking result may prompt a rethinking of post-training paradigms for large language models. Our code is avaliable at https://github.com/zitian-gao/one-shot-em.

Stateful policies play an important role in reinforcement learning, such as handling partially observable environments, enhancing robustness, or imposing an inductive bias directly into the policy structure. The conventional method for training stateful policies is Backpropagation Through Time (BPTT), which comes with significant drawbacks, such as slow training due to sequential gradient propagation and the occurrence of vanishing or exploding gradients. The gradient is often truncated to address these issues, resulting in a biased policy update. We present a novel approach for training stateful policies by decomposing the latter into a stochastic internal state kernel and a stateless policy, jointly optimized by following the stateful policy gradient. We introduce different versions of the stateful policy gradient theorem, enabling us to easily instantiate stateful variants of popular reinforcement learning and imitation learning algorithms. Furthermore, we provide a theoretical analysis of our new gradient estimator and compare it with BPTT. We evaluate our approach on complex continuous control tasks, e.g., humanoid locomotion, and demonstrate that our gradient estimator scales effectively with task complexity while offering a faster and simpler alternative to BPTT.

Recent Reinforcement Learning (RL) algorithms making use of Kullback-Leibler (KL) regularization as a core component have shown outstanding performance. Yet, only little is understood theoretically about why KL regularization helps, so far. We study KL regularization within an approximate value iteration scheme and show that it implicitly averages q-values. Leveraging this insight, we provide a very strong performance bound, the very first to combine two desirable aspects: a linear dependency to the horizon (instead of quadratic) and an error propagation term involving an averaging effect of the estimation errors (instead of an accumulation effect). We also study the more general case of an additional entropy regularizer. The resulting abstract scheme encompasses many existing RL algorithms. Some of our assumptions do not hold with neural networks, so we complement this theoretical analysis with an extensive empirical study.

Reinforcement Learning, particularly through policy gradient methods, has played a central role in enabling reasoning capabilities of Large Language Models. However, the optimization stability of policy gradients in this setting remains understudied. As a result, existing implementations often resort to conservative hyperparameter choices to ensure stability, which requires more training samples and increases computational costs. Hence, developing models for reliably tracking the underlying optimization dynamics and leveraging them into training enables more sample-efficient regimes and further unleashes scalable post-training. We address this gap by formalizing the stochastic optimization problem of policy gradients with explicit consideration of second-order geometry. We propose a tractable computational framework that tracks and leverages curvature information during policy updates. We further employ this framework to design interventions in the optimization process through data selection. The resultant algorithm, Curvature-Aware Policy Optimization (CAPO), identifies samples that contribute to unstable updates and masks them out. Theoretically, we establish monotonic improvement guarantees under realistic assumptions. On standard math reasoning benchmarks, we empirically show that CAPO ensures stable updates under aggressive learning regimes where baselines catastrophically fail. With minimal intervention (rejecting fewer than 8% of tokens), CAPO achieves up to 30x improvement in sample efficiency over standard GRPO for LLM reasoning.

The increasing capabilities of large language models (LLMs) raise opportunities for artificial general intelligence but concurrently amplify safety concerns, such as potential misuse of AI systems, necessitating effective AI alignment. Reinforcement Learning from Human Feedback (RLHF) has emerged as a promising pathway towards AI alignment but brings forth challenges due to its complexity and dependence on a separate reward model. Direct Preference Optimization (DPO) has been proposed as an alternative, and it remains equivalent to RLHF under the reverse KL regularization constraint. This paper presents f-DPO, a generalized approach to DPO by incorporating diverse divergence constraints. We show that under certain f-divergences, including Jensen-Shannon divergence, forward KL divergences and alpha-divergences, the complex relationship between the reward and optimal policy can also be simplified by addressing the Karush-Kuhn-Tucker conditions. This eliminates the need for estimating the normalizing constant in the Bradley-Terry model and enables a tractable mapping between the reward function and the optimal policy. Our approach optimizes LLMs to align with human preferences in a more efficient and supervised manner under a broad set of divergence constraints. Empirically, adopting these divergences ensures a balance between alignment performance and generation diversity. Importantly, f-DPO outperforms PPO-based methods in divergence efficiency, and divergence constraints directly influence expected calibration error (ECE).

Proximal Policy Optimization (PPO) has become the predominant algorithm for on-policy reinforcement learning due to its scalability and empirical robustness across domains. However, there is a significant disconnect between the underlying foundations of trust region methods and the heuristic clipped objective used in PPO. In this paper, we bridge this gap by introducing the Bounded Ratio Reinforcement Learning (BRRL) framework. We formulate a novel regularized and constrained policy optimization problem and derive its analytical optimal solution. We prove that this solution ensures monotonic performance improvement. To handle parameterized policy classes, we develop a policy optimization algorithm called Bounded Policy Optimization (BPO) that minimizes an advantage-weighted divergence between the policy and the analytic optimal solution from BRRL. We further establish a lower bound on the expected performance of the resulting policy in terms of the BPO loss function. Notably, our framework also provides a new theoretical lens to interpret the success of the PPO loss, and connects trust region policy optimization and the Cross-Entropy Method (CEM). We additionally extend BPO to Group-relative BPO (GBPO) for LLM fine-tuning. Empirical evaluations of BPO across MuJoCo, Atari, and complex IsaacLab environments (e.g., Humanoid locomotion), and of GBPO for LLM fine-tuning tasks, demonstrate that BPO and GBPO generally match or outperform PPO and GRPO in stability and final performance.

Recently, the impressive empirical success of policy gradient (PG) methods has catalyzed the development of their theoretical foundations. Despite the huge efforts directed at the design of efficient stochastic PG-type algorithms, the understanding of their convergence to a globally optimal policy is still limited. In this work, we develop improved global convergence guarantees for a general class of Fisher-non-degenerate parameterized policies which allows to address the case of continuous state action spaces. First, we propose a Normalized Policy Gradient method with Implicit Gradient Transport (N-PG-IGT) and derive a mathcal{O}(varepsilon^{-2.5}) sample complexity of this method for finding a global varepsilon-optimal policy. Improving over the previously known mathcal{O}(varepsilon^{-3}) complexity, this algorithm does not require the use of importance sampling or second-order information and samples only one trajectory per iteration. Second, we further improve this complexity to mathcal{mathcal{O} }(varepsilon^{-2}) by considering a Hessian-Aided Recursive Policy Gradient ((N)-HARPG) algorithm enhanced with a correction based on a Hessian-vector product. Interestingly, both algorithms are (i) simple and easy to implement: single-loop, do not require large batches of trajectories and sample at most two trajectories per iteration; (ii) computationally and memory efficient: they do not require expensive subroutines at each iteration and can be implemented with memory linear in the dimension of parameters.

Reward guidance has been applied to great success in the test-time adaptation of continuous diffusion models; it updates each denoising step using the gradients from a downstream reward model. We study reward guidance for discrete diffusion language models, where one cannot differentiate through the natural outputs of the model because they are discrete tokens. Existing approaches either replace these discrete tokens with continuous relaxations, or employ techniques like the straight-through estimator. In this work, we show the downsides of both these methods. The former degrades gradient feedback because the reward model has never been trained with continuous inputs. The latter involves incorrect optimization because the gradient evaluated at discrete tokens is used to update continuous logits. Our key innovation is to go beyond this tradeoff by introducing a novel mechanism called EntRGi: Entropy aware Reward Guidance that dynamically regulates the gradients from the reward model. By modulating the continuous relaxation using the model's confidence, our approach substantially improves reward guidance while providing reliable inputs to the reward model. We empirically validate our approach on a 7B-parameter diffusion language model across 3 diverse reward models and 3 multi-skill benchmarks, showing consistent improvements over state-of-the-art methods.

Group relative policy optimization (GRPO), a core methodological component of DeepSeekMath and DeepSeek-R1, has emerged as a cornerstone for scaling reasoning capabilities of large language models. Despite its widespread adoption and the proliferation of follow-up works, the theoretical properties of GRPO remain less studied. This paper provides a unified framework to understand GRPO through the lens of classical U-statistics. We demonstrate that the GRPO policy gradient is inherently a U-statistic, allowing us to characterize its mean squared error (MSE), derive the finite-sample error bound and asymptotic distribution of the suboptimality gap for its learned policy. Our findings reveal that GRPO is asymptotically equivalent to an oracle policy gradient algorithm -- one with access to a value function that quantifies the goodness of its learning policy at each training iteration -- and achieves asymptotically optimal performance within a broad class of policy gradient algorithms. Furthermore, we establish a universal scaling law that offers principled guidance for selecting the optimal group size. Empirical experiments further validate our theoretical findings, demonstrating that the optimal group size is universal, and verify the oracle property of GRPO.

Balancing exploration and exploitation is a central goal in reinforcement learning (RL). Despite recent advances in enhancing language model (LM) reasoning, most methods lean toward exploitation, and increasingly encounter performance plateaus. In this work, we revisit entropy -- a signal of exploration in RL -- and examine its relationship to exploratory reasoning in LMs. Through empirical analysis, we uncover strong positive correlations between high-entropy regions and three types of exploratory reasoning actions: (1) pivotal tokens that determine or connect logical steps, (2) reflective actions such as self-verification and correction, and (3) rare behaviors under-explored by the base LMs. Motivated by this, we introduce a minimal modification to standard RL with only one line of code: augmenting the advantage function with an entropy-based term. Unlike traditional maximum-entropy methods which encourage exploration by promoting uncertainty, we encourage exploration by promoting longer and deeper reasoning chains. Notably, our method achieves significant gains on the Pass@K metric -- an upper-bound estimator of LM reasoning capabilities -- even when evaluated with extremely large K values, pushing the boundaries of LM reasoning.

The entropic fictitious play (EFP) is a recently proposed algorithm that minimizes the sum of a convex functional and entropy in the space of measures -- such an objective naturally arises in the optimization of a two-layer neural network in the mean-field regime. In this work, we provide a concise primal-dual analysis of EFP in the setting where the learning problem exhibits a finite-sum structure. We establish quantitative global convergence guarantees for both the continuous-time and discrete-time dynamics based on properties of a proximal Gibbs measure introduced in Nitanda et al. (2022). Furthermore, our primal-dual framework entails a memory-efficient particle-based implementation of the EFP update, and also suggests a connection to gradient boosting methods. We illustrate the efficiency of our novel implementation in experiments including neural network optimization and image synthesis.

We study LLM policy synthesis: using a large language model to iteratively generate programmatic agent policies for multi-agent environments. Rather than training neural policies via reinforcement learning, our framework prompts an LLM to produce Python policy functions, evaluates them in self-play, and refines them using performance feedback across iterations. We investigate feedback engineering (the design of what evaluation information is shown to the LLM during refinement) comparing sparse feedback (scalar reward only) against dense feedback (reward plus social metrics: efficiency, equality, sustainability, peace). Across two canonical Sequential Social Dilemmas (Gathering and Cleanup) and two frontier LLMs (Claude Sonnet 4.6, Gemini 3.1 Pro), dense feedback consistently matches or exceeds sparse feedback on all metrics. The advantage is largest in the Cleanup public goods game, where providing social metrics helps the LLM calibrate the costly cleaning-harvesting tradeoff. Rather than triggering over-optimization of fairness, social metrics serve as a coordination signal that guides the LLM toward more effective cooperative strategies, including territory partitioning, adaptive role assignment, and the avoidance of wasteful aggression. We further perform an adversarial experiment to determine whether LLMs can reward hack these environments. We characterize five attack classes and discuss mitigations, highlighting an inherent tension in LLM policy synthesis between expressiveness and safety. Code at https://github.com/vicgalle/llm-policies-social-dilemmas.

Recent reasoning Large Language Models (LLMs) demonstrate remarkable problem-solving abilities but often generate long thinking traces whose utility is unclear. Our work aims to improve their efficiency, enabling them to reach high performance without overthinking. First, we analyze the entropy of token probabilities in reasoning traces. Across three models, we observe a consistent U-shaped entropy pattern: high entropy on easy problems despite high accuracy, low entropy on problems with medium difficulty, and high entropy on hard problems reflecting uncertainty. Specifically, we notice 22--25\% entropy reduction from easy to medium difficulty regions, suggesting an {overthinking} phenomenon on easy instances. Building on these insights, we introduce DiffAdapt, a lightweight framework that selects Easy/Normal/Hard inference strategies per question based on their difficulty and reasoning trace entropy. Each inference strategy consists of a fixed prompt, temperature and maximum token length. In contrast to existing efficiency optimization methods, our approach does not fine-tune base LLM but a small probe that classifies LLM's final hidden state, allowing inexpensive adaptation. We comprehensively evaluate our method on five models and eight benchmarks. Our method achieves comparable or improved accuracy while reducing token usage by up to 22.4\%, establishing a practical path toward compute-efficient reasoning.

In this work, we consider and analyze the sample complexity of model-free reinforcement learning with a generative model. Particularly, we analyze mirror descent value iteration (MDVI) by Geist et al. (2019) and Vieillard et al. (2020a), which uses the Kullback-Leibler divergence and entropy regularization in its value and policy updates. Our analysis shows that it is nearly minimax-optimal for finding an varepsilon-optimal policy when varepsilon is sufficiently small. This is the first theoretical result that demonstrates that a simple model-free algorithm without variance-reduction can be nearly minimax-optimal under the considered setting.

Inverse reinforcement learning (IRL) is typically formulated as maximizing entropy subject to matching the distribution of expert trajectories. Classical (dual-ascent) IRL guarantees monotonic performance improvement but requires fully solving an RL problem each iteration to compute dual gradients. More recent adversarial methods avoid this cost at the expense of stability and monotonic dual improvement, by directly optimizing the primal problem and using a discriminator to provide rewards. In this work, we bridge the gap between these approaches by enabling monotonic improvement of the reward function and policy without having to fully solve an RL problem at every iteration. Our key theoretical insight is that a trust-region-optimal policy for a reward function update can be globally optimal for a smaller update in the same direction. This smaller update allows us to explicitly optimize the dual objective while only relying on a local search around the current policy. In doing so, our approach avoids the training instabilities of adversarial methods, offers monotonic performance improvement, and learns a reward function in the traditional sense of IRL--one that can be globally optimized to match expert demonstrations. Our proposed algorithm, Trust Region Inverse Reinforcement Learning (TRIRL), outperforms state-of-the-art imitation learning methods across multiple challenging tasks by a factor of 2.4x in terms of aggregate inter-quartile mean, while recovering reward functions that generalize to system dynamics shifts.

Policy Optimization (PO) is one of the most popular methods in Reinforcement Learning (RL). Thus, theoretical guarantees for PO algorithms have become especially important to the RL community. In this paper, we study PO in adversarial MDPs with a challenge that arises in almost every real-world application -- delayed bandit feedback. We give the first near-optimal regret bounds for PO in tabular MDPs, and may even surpass state-of-the-art (which uses less efficient methods). Our novel Delay-Adapted PO (DAPO) is easy to implement and to generalize, allowing us to extend our algorithm to: (i) infinite state space under the assumption of linear Q-function, proving the first regret bounds for delayed feedback with function approximation. (ii) deep RL, demonstrating its effectiveness in experiments on MuJoCo domains.

Model-free algorithms for reinforcement learning typically require a condition called Bellman completeness in order to successfully operate off-policy with function approximation, unless additional conditions are met. However, Bellman completeness is a requirement that is much stronger than realizability and that is deemed to be too strong to hold in practice. In this work, we relax this structural assumption and analyze the statistical complexity of off-policy reinforcement learning when only realizability holds for the prescribed function class. We establish finite-sample guarantees for off-policy reinforcement learning that are free of the approximation error term known as inherent Bellman error, and that depend on the interplay of three factors. The first two are well known: they are the metric entropy of the function class and the concentrability coefficient that represents the cost of learning off-policy. The third factor is new, and it measures the violation of Bellman completeness, namely the mis-alignment between the chosen function class and its image through the Bellman operator. In essence, these error bounds establish that off-policy reinforcement learning remains statistically viable even in absence of Bellman completeness, and characterize the intermediate situation between the favorable Bellman complete setting and the worst-case scenario where exponential lower bounds are in force. Our analysis directly applies to the solution found by temporal difference algorithms when they converge.

We introduce Diffusion Policy Policy Optimization, DPPO, an algorithmic framework including best practices for fine-tuning diffusion-based policies (e.g. Diffusion Policy) in continuous control and robot learning tasks using the policy gradient (PG) method from reinforcement learning (RL). PG methods are ubiquitous in training RL policies with other policy parameterizations; nevertheless, they had been conjectured to be less efficient for diffusion-based policies. Surprisingly, we show that DPPO achieves the strongest overall performance and efficiency for fine-tuning in common benchmarks compared to other RL methods for diffusion-based policies and also compared to PG fine-tuning of other policy parameterizations. Through experimental investigation, we find that DPPO takes advantage of unique synergies between RL fine-tuning and the diffusion parameterization, leading to structured and on-manifold exploration, stable training, and strong policy robustness. We further demonstrate the strengths of DPPO in a range of realistic settings, including simulated robotic tasks with pixel observations, and via zero-shot deployment of simulation-trained policies on robot hardware in a long-horizon, multi-stage manipulation task. Website with code: diffusion-ppo.github.io

As trajectories sampled by policies used by reinforcement learning (RL) and generative flow networks (GFlowNets) grow longer, credit assignment and exploration become more challenging, and the long planning horizon hinders mode discovery and generalization. The challenge is particularly pronounced in entropy-seeking RL methods, such as generative flow networks, where the agent must learn to sample from a structured distribution and discover multiple high-reward states, each of which take many steps to reach. To tackle this challenge, we propose an approach to incorporate the discovery of action abstractions, or high-level actions, into the policy optimization process. Our approach involves iteratively extracting action subsequences commonly used across many high-reward trajectories and `chunking' them into a single action that is added to the action space. In empirical evaluation on synthetic and real-world environments, our approach demonstrates improved sample efficiency performance in discovering diverse high-reward objects, especially on harder exploration problems. We also observe that the abstracted high-order actions are interpretable, capturing the latent structure of the reward landscape of the action space. This work provides a cognitively motivated approach to action abstraction in RL and is the first demonstration of hierarchical planning in amortized sequential sampling.

Exploration is essential for solving complex Reinforcement Learning (RL) tasks. Maximum State-Visitation Entropy (MSVE) formulates the exploration problem as a well-defined policy optimization problem whose solution aims at visiting all states as uniformly as possible. This is in contrast to standard uncertainty-based approaches where exploration is transient and eventually vanishes. However, existing approaches to MSVE are theoretically justified only for discrete state-spaces as they are oblivious to the geometry of continuous domains. We address this challenge by introducing Geometric Entropy Maximisation (GEM), a new algorithm that maximises the geometry-aware Shannon entropy of state-visits in both discrete and continuous domains. Our key theoretical contribution is casting geometry-aware MSVE exploration as a tractable problem of optimising a simple and novel noise-contrastive objective function. In our experiments, we show the efficiency of GEM in solving several RL problems with sparse rewards, compared against other deep RL exploration approaches.

Reinforcement learning (RL) has emerged as a powerful tool for fine-tuning large language models (LLMs) to improve complex reasoning abilities. However, state-of-the-art policy optimization methods often suffer from high computational overhead and memory consumption, primarily due to the need for multiple generations per prompt and the reliance on critic networks or advantage estimates of the current policy. In this paper, we propose A*-PO, a novel two-stage policy optimization framework that directly approximates the optimal advantage function and enables efficient training of LLMs for reasoning tasks. In the first stage, we leverage offline sampling from a reference policy to estimate the optimal value function V*, eliminating the need for costly online value estimation. In the second stage, we perform on-policy updates using a simple least-squares regression loss with only a single generation per prompt. Theoretically, we establish performance guarantees and prove that the KL-regularized RL objective can be optimized without requiring complex exploration strategies. Empirically, A*-PO achieves competitive performance across a wide range of mathematical reasoning benchmarks, while reducing training time by up to 2times and peak memory usage by over 30% compared to PPO, GRPO, and REBEL. Implementation of A*-PO can be found at https://github.com/ZhaolinGao/A-PO.

Reinforcement learning (RL) typically models the interaction between the agent and environment as a Markov decision process (MDP), where the rewards that guide the agent's behavior are always observable. However, in many real-world scenarios, rewards are not always observable, which can be modeled as a monitored Markov decision process (Mon-MDP). Prior work on Mon-MDPs have been limited to simple, tabular cases, restricting their applicability to real-world problems. This work explores Mon-MDPs using function approximation (FA) and investigates the challenges involved. We show that combining function approximation with a learned reward model enables agents to generalize from monitored states with observable rewards, to unmonitored environment states with unobservable rewards. Therefore, we demonstrate that such generalization with a reward model achieves near-optimal policies in environments formally defined as unsolvable. However, we identify a critical limitation of such function approximation, where agents incorrectly extrapolate rewards due to overgeneralization, resulting in undesirable behaviors. To mitigate overgeneralization, we propose a cautious police optimization method leveraging reward uncertainty. This work serves as a step towards bridging this gap between Mon-MDP theory and real-world applications.