Daily Papers

Reinforcement Learning from Human Feedback (RLHF) has emerged as a dominant approach for aligning LLM outputs with human preferences. Inspired by the success of RLHF, we study the performance of multiple algorithms that learn from feedback (Expert Iteration, Proximal Policy Optimization (PPO), Return-Conditioned RL) on improving LLM reasoning capabilities. We investigate both sparse and dense rewards provided to the LLM both heuristically and via a learned reward model. We additionally start from multiple model sizes and initializations both with and without supervised fine-tuning (SFT) data. Overall, we find all algorithms perform comparably, with Expert Iteration performing best in most cases. Surprisingly, we find the sample complexity of Expert Iteration is similar to that of PPO, requiring at most on the order of 10^6 samples to converge from a pretrained checkpoint. We investigate why this is the case, concluding that during RL training models fail to explore significantly beyond solutions already produced by SFT models. Additionally, we discuss a trade off between maj@1 and pass@96 metric performance during SFT training and how conversely RL training improves both simultaneously. We then conclude by discussing the implications of our findings for RLHF and the future role of RL in LLM fine-tuning.

State-of-the-art (SOTA) reinforcement learning (RL) methods enable the vision-language agents to learn from interactions with the environment without human supervision. However, they struggle with learning inefficiencies in tackling real-world complex sequential decision-making tasks, especially with sparse reward signals and long-horizon dependencies. To effectively address the issue, we introduce Variational Subgoal-Conditioned RL (VSC-RL), which reformulates the vision-language sequential decision-making task as a variational goal-conditioned RL problem, allowing us to leverage advanced optimization methods to enhance learning efficiency. Specifically, VSC-RL optimizes the SubGoal Evidence Lower BOund (SGC-ELBO), which consists of (a) maximizing the subgoal-conditioned return via RL and (b) minimizing the subgoal-conditioned difference with the reference policy. We theoretically demonstrate that SGC-ELBO is equivalent to the original optimization objective, ensuring improved learning efficiency without sacrificing performance guarantees. Additionally, for real-world complex decision-making tasks, VSC-RL leverages the vision-language model to autonomously decompose the goal into feasible subgoals, enabling efficient learning. Across various benchmarks, including challenging real-world mobile device control tasks, VSC-RL significantly outperforms the SOTA vision-language agents, achieving superior performance and remarkable improvement in learning efficiency.

Off-policy dynamic programming (DP) techniques such as Q-learning have proven to be important in sequential decision-making problems. In the presence of function approximation, however, these techniques often diverge due to the absence of Bellman completeness in the function classes considered, a crucial condition for the success of DP-based methods. In this paper, we show how off-policy learning techniques based on return-conditioned supervised learning (RCSL) are able to circumvent these challenges of Bellman completeness, converging under significantly more relaxed assumptions inherited from supervised learning. We prove there exists a natural environment in which if one uses two-layer multilayer perceptron as the function approximator, the layer width needs to grow linearly with the state space size to satisfy Bellman completeness while a constant layer width is enough for RCSL. These findings take a step towards explaining the superior empirical performance of RCSL methods compared to DP-based methods in environments with near-optimal datasets. Furthermore, in order to learn from sub-optimal datasets, we propose a simple framework called MBRCSL, granting RCSL methods the ability of dynamic programming to stitch together segments from distinct trajectories. MBRCSL leverages learned dynamics models and forward sampling to accomplish trajectory stitching while avoiding the need for Bellman completeness that plagues all dynamic programming algorithms. We propose both theoretical analysis and experimental evaluation to back these claims, outperforming state-of-the-art model-free and model-based offline RL algorithms across several simulated robotics problems.

Recent research in offline reinforcement learning (RL) has demonstrated that return-conditioned supervised learning is a powerful paradigm for decision-making problems. While promising, return conditioning is limited to training data labeled with rewards and therefore faces challenges in learning from unsupervised data. In this work, we aim to utilize generalized future conditioning to enable efficient unsupervised pretraining from reward-free and sub-optimal offline data. We propose Pretrained Decision Transformer (PDT), a conceptually simple approach for unsupervised RL pretraining. PDT leverages future trajectory information as a privileged context to predict actions during training. The ability to make decisions based on both present and future factors enhances PDT's capability for generalization. Besides, this feature can be easily incorporated into a return-conditioned framework for online finetuning, by assigning return values to possible futures and sampling future embeddings based on their respective values. Empirically, PDT outperforms or performs on par with its supervised pretraining counterpart, especially when dealing with sub-optimal data. Further analysis reveals that PDT can extract diverse behaviors from offline data and controllably sample high-return behaviors by online finetuning. Code is available at here.

Offline reinforcement learning (RL) has progressed with return-conditioned supervised learning (RCSL), but its lack of stitching ability remains a limitation. We introduce Q-Aided Conditional Supervised Learning (QCS), which effectively combines the stability of RCSL with the stitching capability of Q-functions. By analyzing Q-function over-generalization, which impairs stable stitching, QCS adaptively integrates Q-aid into RCSL's loss function based on trajectory return. Empirical results show that QCS significantly outperforms RCSL and value-based methods, consistently achieving or exceeding the maximum trajectory returns across diverse offline RL benchmarks.

Future- or return-conditioned supervised learning is an emerging paradigm for offline reinforcement learning (RL), where the future outcome (i.e., return) associated with an observed action sequence is used as input to a policy trained to imitate those same actions. While return-conditioning is at the heart of popular algorithms such as decision transformer (DT), these methods tend to perform poorly in highly stochastic environments, where an occasional high return can arise from randomness in the environment rather than the actions themselves. Such situations can lead to a learned policy that is inconsistent with its conditioning inputs; i.e., using the policy to act in the environment, when conditioning on a specific desired return, leads to a distribution of real returns that is wildly different than desired. In this work, we propose the dichotomy of control (DoC), a future-conditioned supervised learning framework that separates mechanisms within a policy's control (actions) from those beyond a policy's control (environment stochasticity). We achieve this separation by conditioning the policy on a latent variable representation of the future, and designing a mutual information constraint that removes any information from the latent variable associated with randomness in the environment. Theoretically, we show that DoC yields policies that are consistent with their conditioning inputs, ensuring that conditioning a learned policy on a desired high-return future outcome will correctly induce high-return behavior. Empirically, we show that DoC is able to achieve significantly better performance than DT on environments that have highly stochastic rewards and transition

Upside down reinforcement learning (UDRL) flips the conventional use of the return in the objective function in RL upside down, by taking returns as input and predicting actions. UDRL is based purely on supervised learning, and bypasses some prominent issues in RL: bootstrapping, off-policy corrections, and discount factors. While previous work with UDRL demonstrated it in a traditional online RL setting, here we show that this single algorithm can also work in the imitation learning and offline RL settings, be extended to the goal-conditioned RL setting, and even the meta-RL setting. With a general agent architecture, a single UDRL agent can learn across all paradigms.

Recently, reward-conditioned reinforcement learning (RCRL) has gained popularity due to its simplicity, flexibility, and off-policy nature. However, we will show that current RCRL approaches are fundamentally limited and fail to address two critical challenges of RCRL -- improving generalization on high reward-to-go (RTG) inputs, and avoiding out-of-distribution (OOD) RTG queries during testing time. To address these challenges when training vanilla RCRL architectures, we propose Bayesian Reparameterized RCRL (BR-RCRL), a novel set of inductive biases for RCRL inspired by Bayes' theorem. BR-RCRL removes a core obstacle preventing vanilla RCRL from generalizing on high RTG inputs -- a tendency that the model treats different RTG inputs as independent values, which we term ``RTG Independence". BR-RCRL also allows us to design an accompanying adaptive inference method, which maximizes total returns while avoiding OOD queries that yield unpredictable behaviors in vanilla RCRL methods. We show that BR-RCRL achieves state-of-the-art performance on the Gym-Mujoco and Atari offline RL benchmarks, improving upon vanilla RCRL by up to 11%.

Recently, supervised learning (SL) methodology has emerged as an effective approach for offline reinforcement learning (RL) due to their simplicity, stability, and efficiency. However, recent studies show that SL methods lack the trajectory stitching capability, typically associated with temporal difference (TD)-based approaches. A question naturally surfaces: How can we endow SL methods with stitching capability and bridge its performance gap with TD learning? To answer this question, we introduce Q-conditioned maximization supervised learning for offline goal-conditioned RL, which enhances SL with the stitching capability through Q-conditioned policy and Q-conditioned maximization. Concretely, we propose Goal-Conditioned Reinforced Supervised Learning (GCReinSL), which consists of (1) estimating the Q-function by CVAE from the offline dataset and (2) finding the maximum Q-value within the data support by integrating Q-function maximization with Expectile Regression. In inference time, our policy chooses optimal actions based on such a maximum Q-value. Experimental results from stitching evaluations on offline RL datasets demonstrate that our method outperforms prior SL approaches with stitching capabilities and goal data augmentation techniques.

Offline goal-conditioned reinforcement learning (GCRL) aims at solving goal-reaching tasks with sparse rewards from an offline dataset. While prior work has demonstrated various approaches for agents to learn near-optimal policies, these methods encounter limitations when dealing with diverse constraints in complex environments, such as safety constraints. Some of these approaches prioritize goal attainment without considering safety, while others excessively focus on safety at the expense of training efficiency. In this paper, we study the problem of constrained offline GCRL and propose a new method called Recovery-based Supervised Learning (RbSL) to accomplish safety-critical tasks with various goals. To evaluate the method performance, we build a benchmark based on the robot-fetching environment with a randomly positioned obstacle and use expert or random policies to generate an offline dataset. We compare RbSL with three offline GCRL algorithms and one offline safe RL algorithm. As a result, our method outperforms the existing state-of-the-art methods to a large extent. Furthermore, we validate the practicality and effectiveness of RbSL by deploying it on a real Panda manipulator. Code is available at https://github.com/Sunlighted/RbSL.git.

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

Offline goal-conditioned reinforcement learning (GCRL) learns goal-conditioned policies from static pre-collected datasets. However, accurate value estimation remains a challenge due to the limited coverage of the state-action space. Recent physics-informed approaches have sought to address this by imposing physical and geometric constraints on the value function through regularization defined over first-order partial differential equations (PDEs), such as the Eikonal equation. However, these formulations can often be ill-posed in complex, high-dimensional environments. In this work, we propose a physics-informed regularization derived from the viscosity solution of the Hamilton-Jacobi-Bellman (HJB) equation. By providing a physics-based inductive bias, our approach grounds the learning process in optimal control theory, explicitly regularizing and bounding updates during value iterations. Furthermore, we leverage the Feynman-Kac theorem to recast the PDE solution as an expectation, enabling a tractable Monte Carlo estimation of the objective that avoids numerical instability in higher-order gradients. Experiments demonstrate that our method improves geometric consistency, making it broadly applicable to navigation and high-dimensional, complex manipulation tasks. Open-source codes are available at https://github.com/HrishikeshVish/phys-fk-value-GCRL.

Many classic Reinforcement Learning (RL) algorithms rely on a Bellman operator, which involves an expectation over the next states, leading to the concept of bootstrapping. To introduce a form of pessimism, we propose to replace this expectation with an expectile. In practice, this can be very simply done by replacing the L_2 loss with a more general expectile loss for the critic. Introducing pessimism in RL is desirable for various reasons, such as tackling the overestimation problem (for which classic solutions are double Q-learning or the twin-critic approach of TD3) or robust RL (where transitions are adversarial). We study empirically these two cases. For the overestimation problem, we show that the proposed approach, ExpectRL, provides better results than a classic twin-critic. On robust RL benchmarks, involving changes of the environment, we show that our approach is more robust than classic RL algorithms. We also introduce a variation of ExpectRL combined with domain randomization which is competitive with state-of-the-art robust RL agents. Eventually, we also extend \ExpectRL with a mechanism for choosing automatically the expectile value, that is the degree of pessimism

In recent years, Reinforcement Learning (RL) has been applied to real-world problems with increasing success. Such applications often require to put constraints on the agent's behavior. Existing algorithms for constrained RL (CRL) rely on gradient descent-ascent, but this approach comes with a caveat. While these algorithms are guaranteed to converge on average, they do not guarantee last-iterate convergence, i.e., the current policy of the agent may never converge to the optimal solution. In practice, it is often observed that the policy alternates between satisfying the constraints and maximizing the reward, rarely accomplishing both objectives simultaneously. Here, we address this problem by introducing Reinforcement Learning with Optimistic Ascent-Descent (ReLOAD), a principled CRL method with guaranteed last-iterate convergence. We demonstrate its empirical effectiveness on a wide variety of CRL problems including discrete MDPs and continuous control. In the process we establish a benchmark of challenging CRL problems.

In this work, we take a fresh look at some old and new algorithms for off-policy, return-based reinforcement learning. Expressing these in a common form, we derive a novel algorithm, Retrace(λ), with three desired properties: (1) it has low variance; (2) it safely uses samples collected from any behaviour policy, whatever its degree of "off-policyness"; and (3) it is efficient as it makes the best use of samples collected from near on-policy behaviour policies. We analyze the contractive nature of the related operator under both off-policy policy evaluation and control settings and derive online sample-based algorithms. We believe this is the first return-based off-policy control algorithm converging a.s. to Q^* without the GLIE assumption (Greedy in the Limit with Infinite Exploration). As a corollary, we prove the convergence of Watkins' Q(λ), which was an open problem since 1989. We illustrate the benefits of Retrace(λ) on a standard suite of Atari 2600 games.

The goal of reinforcement learning (RL) is to find a policy that maximizes the expected cumulative return. It has been shown that this objective can be represented as an optimization problem of state-action visitation distribution under linear constraints. The dual problem of this formulation, which we refer to as dual RL, is unconstrained and easier to optimize. In this work, we first cast several state-of-the-art offline RL and offline imitation learning (IL) algorithms as instances of dual RL approaches with shared structures. Such unification allows us to identify the root cause of the shortcomings of prior methods. For offline IL, our analysis shows that prior methods are based on a restrictive coverage assumption that greatly limits their performance in practice. To fix this limitation, we propose a new discriminator-free method ReCOIL that learns to imitate from arbitrary off-policy data to obtain near-expert performance. For offline RL, our analysis frames a recent offline RL method XQL in the dual framework, and we further propose a new method f-DVL that provides alternative choices to the Gumbel regression loss that fixes the known training instability issue of XQL. The performance improvements by both of our proposed methods, ReCOIL and f-DVL, in IL and RL are validated on an extensive suite of simulated robot locomotion and manipulation tasks. Project code and details can be found at this https://hari-sikchi.github.io/dual-rl.

Reinforcement Learning with Verifiable Rewards (RLVR) has substantially improved the reasoning abilities of large language models (LLMs), yet training often stalls as problems become saturated. We identify the core challenge as the poor accessibility of informative failures: learning signals exist but are rarely encountered during standard rollouts. To address this, we propose failure-prefix conditioning, a simple and effective method for learning from saturated problems. Rather than starting from the original question, our approach reallocates exploration by conditioning training on prefixes derived from rare incorrect reasoning trajectories, thereby exposing the model to failure-prone states. We observe that failure-prefix conditioning yields performance gains matching those of training on medium-difficulty problems, while preserving token efficiency. Furthermore, we analyze the model's robustness, finding that our method reduces performance degradation under misleading failure prefixes, albeit with a mild trade-off in adherence to correct early reasoning. Finally, we demonstrate that an iterative approach, which refreshes failure prefixes during training, unlocks additional gains after performance plateaus. Overall, our results suggest that failure-prefix conditioning offers an effective pathway to extend RLVR training on saturated problems.

Real-world reinforcement learning (RL) is often severely limited since typical RL algorithms heavily rely on the reset mechanism to sample proper initial states. In practice, the reset mechanism is expensive to implement due to the need for human intervention or heavily engineered environments. To make learning more practical, we propose a generic no-regret reduction to systematically design reset-free RL algorithms. Our reduction turns reset-free RL into a two-player game. We show that achieving sublinear regret in this two-player game would imply learning a policy that has both sublinear performance regret and sublinear total number of resets in the original RL problem. This means that the agent eventually learns to perform optimally and avoid resets. By this reduction, we design an instantiation for linear Markov decision processes, which is the first provably correct reset-free RL algorithm to our knowledge.

Offline goal-conditioned reinforcement learning (GCRL) is a promising approach for pretraining generalist policies on large datasets of reward-free trajectories, akin to the self-supervised objectives used to train foundation models for computer vision and natural language processing. However, scaling GCRL to longer horizons remains challenging due to the combination of sparse rewards and discounting, which obscures the comparative advantages of primitive actions with respect to distant goals. Hierarchical RL methods achieve strong empirical results on long-horizon goal-reaching tasks, but their reliance on modular, timescale-specific policies and subgoal generation introduces significant additional complexity and hinders scaling to high-dimensional goal spaces. In this work, we introduce an algorithm to train a flat (non-hierarchical) goal-conditioned policy by bootstrapping on subgoal-conditioned policies with advantage-weighted importance sampling. Our approach eliminates the need for a generative model over the (sub)goal space, which we find is key for scaling to high-dimensional control in large state spaces. We further show that existing hierarchical and bootstrapping-based approaches correspond to specific design choices within our derivation. Across a comprehensive suite of state- and pixel-based locomotion and manipulation benchmarks, our method matches or surpasses state-of-the-art offline GCRL algorithms and scales to complex, long-horizon tasks where prior approaches fail. Project page: https://johnlyzhou.github.io/saw/

Offline goal-conditioned RL (GCRL) offers a way to train general-purpose agents from fully offline datasets. In addition to being conservative within the dataset, the generalization ability to achieve unseen goals is another fundamental challenge for offline GCRL. However, to the best of our knowledge, this problem has not been well studied yet. In this paper, we study out-of-distribution (OOD) generalization of offline GCRL both theoretically and empirically to identify factors that are important. In a number of experiments, we observe that weighted imitation learning enjoys better generalization than pessimism-based offline RL method. Based on this insight, we derive a theory for OOD generalization, which characterizes several important design choices. We then propose a new offline GCRL method, Generalizable Offline goAl-condiTioned RL (GOAT), by combining the findings from our theoretical and empirical studies. On a new benchmark containing 9 independent identically distributed (IID) tasks and 17 OOD tasks, GOAT outperforms current state-of-the-art methods by a large margin.

In online reinforcement learning (RL), instead of employing standard structural assumptions on Markov decision processes (MDPs), using a certain coverage condition (original from offline RL) is enough to ensure sample-efficient guarantees (Xie et al. 2023). In this work, we focus on this new direction by digging more possible and general coverage conditions, and study the potential and the utility of them in efficient online RL. We identify more concepts, including the L^p variant of concentrability, the density ratio realizability, and trade-off on the partial/rest coverage condition, that can be also beneficial to sample-efficient online RL, achieving improved regret bound. Furthermore, if exploratory offline data are used, under our coverage conditions, both statistically and computationally efficient guarantees can be achieved for online RL. Besides, even though the MDP structure is given, e.g., linear MDP, we elucidate that, good coverage conditions are still beneficial to obtain faster regret bound beyond O(T) and even a logarithmic order regret. These results provide a good justification for the usage of general coverage conditions in efficient online RL.

In reinforcement learning (RL), there are two major settings for interacting with the environment: online and offline. Online methods explore the environment at significant time cost, and offline methods efficiently obtain reward signals by sacrificing exploration capability. We propose semi-offline RL, a novel paradigm that smoothly transits from offline to online settings, balances exploration capability and training cost, and provides a theoretical foundation for comparing different RL settings. Based on the semi-offline formulation, we present the RL setting that is optimal in terms of optimization cost, asymptotic error, and overfitting error bound. Extensive experiments show that our semi-offline approach is efficient and yields comparable or often better performance compared with state-of-the-art methods.

Reinforcement learning (RL) promises a framework for near-universal problem-solving. In practice however, RL algorithms are often tailored to specific benchmarks, relying on carefully tuned hyperparameters and algorithmic choices. Recently, powerful model-based RL methods have shown impressive general results across benchmarks but come at the cost of increased complexity and slow run times, limiting their broader applicability. In this paper, we attempt to find a unifying model-free deep RL algorithm that can address a diverse class of domains and problem settings. To achieve this, we leverage model-based representations that approximately linearize the value function, taking advantage of the denser task objectives used by model-based RL while avoiding the costs associated with planning or simulated trajectories. We evaluate our algorithm, MR.Q, on a variety of common RL benchmarks with a single set of hyperparameters and show a competitive performance against domain-specific and general baselines, providing a concrete step towards building general-purpose model-free deep RL algorithms.

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/.

We consider the problem of preference based reinforcement learning (PbRL), where, unlike traditional reinforcement learning, an agent receives feedback only in terms of a 1 bit (0/1) preference over a trajectory pair instead of absolute rewards for them. The success of the traditional RL framework crucially relies on the underlying agent-reward model, which, however, depends on how accurately a system designer can express an appropriate reward function and often a non-trivial task. The main novelty of our framework is the ability to learn from preference-based trajectory feedback that eliminates the need to hand-craft numeric reward models. This paper sets up a formal framework for the PbRL problem with non-markovian rewards, where the trajectory preferences are encoded by a generalized linear model of dimension d. Assuming the transition model is known, we then propose an algorithm with almost optimal regret guarantee of mathcal{O}left( SH d log (T / delta) T right). We further, extend the above algorithm to the case of unknown transition dynamics, and provide an algorithm with near optimal regret guarantee mathcal{O}((d + H^2 + |S|)dT +|mathcal{S||A|TH} ). To the best of our knowledge, our work is one of the first to give tight regret guarantees for preference based RL problems with trajectory preferences.

Offline preference-based reinforcement learning (PbRL) typically operates in two phases: first, use human preferences to learn a reward model and annotate rewards for a reward-free offline dataset; second, learn a policy by optimizing the learned reward via offline RL. However, accurately modeling step-wise rewards from trajectory-level preference feedback presents inherent challenges. The reward bias introduced, particularly the overestimation of predicted rewards, leads to optimistic trajectory stitching, which undermines the pessimism mechanism critical to the offline RL phase. To address this challenge, we propose In-Dataset Trajectory Return Regularization (DTR) for offline PbRL, which leverages conditional sequence modeling to mitigate the risk of learning inaccurate trajectory stitching under reward bias. Specifically, DTR employs Decision Transformer and TD-Learning to strike a balance between maintaining fidelity to the behavior policy with high in-dataset trajectory returns and selecting optimal actions based on high reward labels. Additionally, we introduce an ensemble normalization technique that effectively integrates multiple reward models, balancing the tradeoff between reward differentiation and accuracy. Empirical evaluations on various benchmarks demonstrate the superiority of DTR over other state-of-the-art baselines.

Policy gradient algorithms have been successfully applied to enhance the reasoning capabilities of large language models (LLMs). Despite the widespread use of Kullback-Leibler (KL) regularization in policy gradient algorithms to stabilize training, the systematic exploration of how different KL divergence formulations can be estimated and integrated into surrogate loss functions for online reinforcement learning (RL) presents a nuanced and systematically explorable design space. In this paper, we propose regularized policy gradient (RPG), a systematic framework for deriving and analyzing KL-regularized policy gradient methods in the online RL setting. We derive policy gradients and corresponding surrogate loss functions for objectives regularized by both forward and reverse KL divergences, considering both normalized and unnormalized policy distributions. Furthermore, we present derivations for fully differentiable loss functions as well as REINFORCE-style gradient estimators, accommodating diverse algorithmic needs. We conduct extensive experiments on RL for LLM reasoning using these methods, showing improved or competitive results in terms of training stability and performance compared to strong baselines such as GRPO, REINFORCE++, and DAPO. The code is available at https://github.com/complex-reasoning/RPG.

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.

Offline reinforcement learning (RL), where the agent aims to learn the optimal policy based on the data collected by a behavior policy, has attracted increasing attention in recent years. While offline RL with linear function approximation has been extensively studied with optimal results achieved under certain assumptions, many works shift their interest to offline RL with non-linear function approximation. However, limited works on offline RL with non-linear function approximation have instance-dependent regret guarantees. In this paper, we propose an oracle-efficient algorithm, dubbed Pessimistic Nonlinear Least-Square Value Iteration (PNLSVI), for offline RL with non-linear function approximation. Our algorithmic design comprises three innovative components: (1) a variance-based weighted regression scheme that can be applied to a wide range of function classes, (2) a subroutine for variance estimation, and (3) a planning phase that utilizes a pessimistic value iteration approach. Our algorithm enjoys a regret bound that has a tight dependency on the function class complexity and achieves minimax optimal instance-dependent regret when specialized to linear function approximation. Our work extends the previous instance-dependent results within simpler function classes, such as linear and differentiable function to a more general framework.

Generalization in Reinforcement Learning (RL) aims to learn an agent during training that generalizes to the target environment. This paper studies RL generalization from a theoretical aspect: how much can we expect pre-training over training environments to be helpful? When the interaction with the target environment is not allowed, we certify that the best we can obtain is a near-optimal policy in an average sense, and we design an algorithm that achieves this goal. Furthermore, when the agent is allowed to interact with the target environment, we give a surprising result showing that asymptotically, the improvement from pre-training is at most a constant factor. On the other hand, in the non-asymptotic regime, we design an efficient algorithm and prove a distribution-based regret bound in the target environment that is independent of the state-action space.

Reinforcement learning with verifiable rewards (RLVR) is a promising approach for training language models (LMs) on reasoning tasks that elicit emergent long chains of thought (CoTs). Unlike supervised learning, it updates the model using both correct and incorrect samples via policy gradients. To better understand its mechanism, we decompose the learning signal into reinforcing correct responses and penalizing incorrect ones, referred to as Positive and Negative Sample Reinforcement (PSR and NSR), respectively. We train Qwen2.5-Math-7B and Qwen3-4B on a mathematical reasoning dataset and uncover a surprising result: training with only negative samples -- without reinforcing correct responses -- can be highly effective: it consistently improves performance over the base model across the entire Pass@k spectrum (k up to 256), often matching or surpassing PPO and GRPO. In contrast, reinforcing only correct responses improves Pass@1 but degrades performance at higher k, due to reduced diversity. These inference-scaling trends highlight that solely penalizing incorrect responses may contribute more to performance than previously recognized. Through gradient analysis, we show that NSR works by suppressing incorrect generations and redistributing probability mass toward other plausible candidates, guided by the model's prior beliefs. It refines the model's existing knowledge rather than introducing entirely new behaviors. Building on this insight, we propose a simple variant of the RL objective that upweights NSR, and show that it consistently improves overall Pass@k performance on MATH, AIME 2025, and AMC23. Our code is available at https://github.com/TianHongZXY/RLVR-Decomposed.

Typical reinforcement learning (RL) methods for LLM reasoning waste compute on hard problems, where correct on-policy traces are rare, policy gradients vanish, and learning stalls. To bootstrap more efficient RL, we consider reusing old sampling FLOPs (from prior inference or RL training) in the form of off-policy traces. Standard off-policy methods supervise against off-policy data, causing instabilities during RL optimization. We introduce PrefixRL, where we condition on the prefix of successful off-policy traces and run on-policy RL to complete them, side-stepping off-policy instabilities. PrefixRL boosts the learning signal on hard problems by modulating the difficulty of the problem through the off-policy prefix length. We prove that the PrefixRL objective is not only consistent with the standard RL objective but also more sample efficient. Empirically, we discover back-generalization: training only on prefixed problems generalizes to out-of-distribution unprefixed performance, with learned strategies often differing from those in the prefix. In our experiments, we source the off-policy traces by rejection sampling with the base model, creating a self-improvement loop. On hard reasoning problems, PrefixRL reaches the same training reward 2x faster than the strongest baseline (SFT on off-policy data then RL), even after accounting for the compute spent on the initial rejection sampling, and increases the final reward by 3x. The gains transfer to held-out benchmarks, and PrefixRL is still effective when off-policy traces are derived from a different model family, validating its flexibility in practical settings.

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

While Reinforcement Learning for Verifiable Rewards (RLVR) is powerful for training large reasoning models, its training dynamics harbor a critical challenge: RL overfitting, where models gain training rewards but lose generalization. Our analysis reveals this is driven by policy over-specialization and catastrophic forgetting of diverse solutions generated during training. Standard optimization discards this valuable inter-step policy diversity. To address this, we introduce RLoop, a self-improving framework built on iterative policy initialization. RLoop transforms the standard training process into a virtuous cycle: it first uses RL to explore the solution space from a given policy, then filters the successful trajectories to create an expert dataset. This dataset is used via Rejection-sampling Fine-Tuning (RFT) to refine the initial policy, creating a superior starting point for the next iteration. This loop of exploration and exploitation via iterative re-initialization effectively converts transient policy variations into robust performance gains. Our experiments show RLoop mitigates forgetting and substantially improves generalization, boosting average accuracy by 9% and pass@32 by over 15% compared to vanilla RL.

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

We study the problem of nonepisodic reinforcement learning (RL) for nonlinear dynamical systems, where the system dynamics are unknown and the RL agent has to learn from a single trajectory, i.e., without resets. We propose Nonepisodic Optimistic RL (NeoRL), an approach based on the principle of optimism in the face of uncertainty. NeoRL uses well-calibrated probabilistic models and plans optimistically w.r.t. the epistemic uncertainty about the unknown dynamics. Under continuity and bounded energy assumptions on the system, we provide a first-of-its-kind regret bound of O(Gamma_T T) for general nonlinear systems with Gaussian process dynamics. We compare NeoRL to other baselines on several deep RL environments and empirically demonstrate that NeoRL achieves the optimal average cost while incurring the least regret.

We study how vision-language-action (VLA) models can improve through real-world deployments via reinforcement learning (RL). We present a general-purpose method, RL with Experience and Corrections via Advantage-conditioned Policies (RECAP), that provides for RL training of VLAs via advantage conditioning. Our method incorporates heterogeneous data into the self-improvement process, including demonstrations, data from on-policy collection, and expert teleoperated interventions provided during autonomous execution. RECAP starts by pre-training a generalist VLA with offline RL, which we call π^{*}_{0.6}, that can then be specialized to attain high performance on downstream tasks through on-robot data collection. We show that the π^{*}_{0.6} model trained with the full RECAP method can fold laundry in real homes, reliably assemble boxes, and make espresso drinks using a professional espresso machine. On some of the hardest tasks, RECAP more than doubles task throughput and roughly halves the task failure rate.

Prevalent reinforcement learning~(RL) methods for fine-tuning LLM reasoners, such as GRPO or Leave-one-out PPO, abandon the learned value function in favor of empirically estimated returns. This hinders test-time compute scaling that relies on using the value-function for verification. In this work, we propose RL^V that augments any ``value-free'' RL method by jointly training the LLM as both a reasoner and a generative verifier using RL-generated data, adding verification capabilities without significant overhead. Empirically, RL^V boosts MATH accuracy by over 20\% with parallel sampling and enables 8-32times efficient test-time compute scaling compared to the base RL method. RL^V also exhibits strong generalization capabilities for both easy-to-hard and out-of-domain tasks. Furthermore, RL^V achieves 1.2-1.6times higher performance when jointly scaling parallel and sequential test-time compute with a long reasoning R1 model.

Risk-sensitive reinforcement learning (RL) has become a popular tool to control the risk of uncertain outcomes and ensure reliable performance in various sequential decision-making problems. While policy gradient methods have been developed for risk-sensitive RL, it remains unclear if these methods enjoy the same global convergence guarantees as in the risk-neutral case. In this paper, we consider a class of dynamic time-consistent risk measures, called Expected Conditional Risk Measures (ECRMs), and derive policy gradient updates for ECRM-based objective functions. Under both constrained direct parameterization and unconstrained softmax parameterization, we provide global convergence and iteration complexities of the corresponding risk-averse policy gradient algorithms. We further test risk-averse variants of REINFORCE and actor-critic algorithms to demonstrate the efficacy of our method and the importance of risk control.

Reinforcement learning (RL) theory has largely focused on proving minimax sample complexity bounds. These require strategic exploration algorithms that use relatively limited function classes for representing the policy or value function. Our goal is to explain why deep RL algorithms often perform well in practice, despite using random exploration and much more expressive function classes like neural networks. Our work arrives at an explanation by showing that many stochastic MDPs can be solved by performing only a few steps of value iteration on the random policy's Q function and then acting greedily. When this is true, we find that it is possible to separate the exploration and learning components of RL, making it much easier to analyze. We introduce a new RL algorithm, SQIRL, that iteratively learns a near-optimal policy by exploring randomly to collect rollouts and then performing a limited number of steps of fitted-Q iteration over those rollouts. Any regression algorithm that satisfies basic in-distribution generalization properties can be used in SQIRL to efficiently solve common MDPs. This can explain why deep RL works, since it is empirically established that neural networks generalize well in-distribution. Furthermore, SQIRL explains why random exploration works well in practice. We leverage SQIRL to derive instance-dependent sample complexity bounds for RL that are exponential only in an "effective horizon" of lookahead and on the complexity of the class used for function approximation. Empirically, we also find that SQIRL performance strongly correlates with PPO and DQN performance in a variety of stochastic environments, supporting that our theoretical analysis is predictive of practical performance. Our code and data are available at https://github.com/cassidylaidlaw/effective-horizon.

In risk-averse reinforcement learning (RL), the goal is to optimize some risk measure of the returns. A risk measure often focuses on the worst returns out of the agent's experience. As a result, standard methods for risk-averse RL often ignore high-return strategies. We prove that under certain conditions this inevitably leads to a local-optimum barrier, and propose a soft risk mechanism to bypass it. We also devise a novel Cross Entropy module for risk sampling, which (1) preserves risk aversion despite the soft risk; (2) independently improves sample efficiency. By separating the risk aversion of the sampler and the optimizer, we can sample episodes with poor conditions, yet optimize with respect to successful strategies. We combine these two concepts in CeSoR - Cross-entropy Soft-Risk optimization algorithm - which can be applied on top of any risk-averse policy gradient (PG) method. We demonstrate improved risk aversion in maze navigation, autonomous driving, and resource allocation benchmarks, including in scenarios where standard risk-averse PG completely fails.

Reinforcement learning (RL) is frequently employed in fine-tuning large language models (LMs), such as GPT-3, to penalize them for undesirable features of generated sequences, such as offensiveness, social bias, harmfulness or falsehood. The RL formulation involves treating the LM as a policy and updating it to maximise the expected value of a reward function which captures human preferences, such as non-offensiveness. In this paper, we analyze challenges associated with treating a language model as an RL policy and show how avoiding those challenges requires moving beyond the RL paradigm. We start by observing that the standard RL approach is flawed as an objective for fine-tuning LMs because it leads to distribution collapse: turning the LM into a degenerate distribution. Then, we analyze KL-regularised RL, a widely used recipe for fine-tuning LMs, which additionally constrains the fine-tuned LM to stay close to its original distribution in terms of Kullback-Leibler (KL) divergence. We show that KL-regularised RL is equivalent to variational inference: approximating a Bayesian posterior which specifies how to update a prior LM to conform with evidence provided by the reward function. We argue that this Bayesian inference view of KL-regularised RL is more insightful than the typically employed RL perspective. The Bayesian inference view explains how KL-regularised RL avoids the distribution collapse problem and offers a first-principles derivation for its objective. While this objective happens to be equivalent to RL (with a particular choice of parametric reward), there exist other objectives for fine-tuning LMs which are no longer equivalent to RL. That observation leads to a more general point: RL is not an adequate formal framework for problems such as fine-tuning language models. These problems are best viewed as Bayesian inference: approximating a pre-defined target distribution.

When language models (LMs) are trained via reinforcement learning (RL) to generate natural language "reasoning chains", their performance improves on a variety of difficult question answering tasks. Today, almost all successful applications of RL for reasoning use binary reward functions that evaluate the correctness of LM outputs. Because such reward functions do not penalize guessing or low-confidence outputs, they often have the unintended side-effect of degrading calibration and increasing the rate at which LMs generate incorrect responses (or "hallucinate") in other problem domains. This paper describes RLCR (Reinforcement Learning with Calibration Rewards), an approach to training reasoning models that jointly improves accuracy and calibrated confidence estimation. During RLCR, LMs generate both predictions and numerical confidence estimates after reasoning. They are trained to optimize a reward function that augments a binary correctness score with a Brier score -- a scoring rule for confidence estimates that incentivizes calibrated prediction. We first prove that this reward function (or any analogous reward function that uses a bounded, proper scoring rule) yields models whose predictions are both accurate and well-calibrated. We next show that across diverse datasets, RLCR substantially improves calibration with no loss in accuracy, on both in-domain and out-of-domain evaluations -- outperforming both ordinary RL training and classifiers trained to assign post-hoc confidence scores. While ordinary RL hurts calibration, RLCR improves it. Finally, we demonstrate that verbalized confidence can be leveraged at test time to improve accuracy and calibration via confidence-weighted scaling methods. Our results show that explicitly optimizing for calibration can produce more generally reliable reasoning models.

Reinforcement Learning with Verifiable Rewards (RLVR) has become the leading paradigm for enhancing reasoning in Large Language Models (LLMs). However, standard RLVR algorithms suffer from a well-documented pathology: while they improve Pass@1 accuracy through sharpened sampling, they simultaneously narrow the model's reasoning boundary and reduce generation diversity. We identify a root cause that existing methods overlook: the uniform penalization of errors. Current approaches -- whether data-filtering methods that select prompts by difficulty, or advantage normalization schemes -- treat all incorrect rollouts within a group identically. We show that this uniformity allows overconfident errors (incorrect reasoning paths that the RL process has spuriously reinforced) to persist and monopolize probability mass, ultimately suppressing valid exploratory trajectories. To address this, we propose the Asymmetric Confidence-aware Error Penalty (ACE). ACE introduces a per-rollout confidence shift metric, c_i = log(pi_theta(y_i|x) / pi_ref(y_i|x)), to dynamically modulate negative advantages. Theoretically, we demonstrate that ACE's gradient can be decomposed into the gradient of a selective regularizer restricted to overconfident errors, plus a well-characterized residual that partially moderates the regularizer's strength. We conduct extensive experiments fine-tuning Qwen2.5-Math-7B, Qwen3-8B-Base, and Llama-3.1-8B-Instruct on the DAPO-Math-17K dataset using GRPO and DAPO within the VERL framework. Evaluated on MATH-500 and AIME 2025, ACE composes seamlessly with existing methods and consistently improves the full Pass@k spectrum across all three model families and benchmarks.

Reinforcement learning from Human Feedback (RLHF) learns from preference signals, while standard Reinforcement Learning (RL) directly learns from reward signals. Preferences arguably contain less information than rewards, which makes preference-based RL seemingly more difficult. This paper theoretically proves that, for a wide range of preference models, we can solve preference-based RL directly using existing algorithms and techniques for reward-based RL, with small or no extra costs. Specifically, (1) for preferences that are drawn from reward-based probabilistic models, we reduce the problem to robust reward-based RL that can tolerate small errors in rewards; (2) for general arbitrary preferences where the objective is to find the von Neumann winner, we reduce the problem to multiagent reward-based RL which finds Nash equilibria for factored Markov games under a restricted set of policies. The latter case can be further reduce to adversarial MDP when preferences only depend on the final state. We instantiate all reward-based RL subroutines by concrete provable algorithms, and apply our theory to a large class of models including tabular MDPs and MDPs with generic function approximation. We further provide guarantees when K-wise comparisons are available.

Decoding-based regression, which reformulates regression as a sequence generation task, has emerged as a promising paradigm of applying large language models for numerical prediction. However, its progress is hindered by the misalignment between discrete token-level objectives (e.g., cross-entropy) and continuous numerical values. Existing approaches relying on token-level constraints often fail to capture the global magnitude of the target value, limiting their precision and generalization. In this paper, we propose to unlock the potential of decoding-based regression via Reinforcement Learning (RL). We formulate the generation process as a Markov Decision Process, utilizing sequence-level rewards to enforce global numerical coherence. Extensive experiments on tabular regression and code metric regression demonstrate that our method (specifically with ReMax and GRPO) consistently outperforms both state-of-the-art token-level baselines and traditional regression heads, showing the superiority of introducing sequence-level signals. Our analysis further reveals that RL significantly enhances sampling efficiency and predictive precision, establishing decoding-based regression as a robust and accurate paradigm for general-purpose numerical prediction.

On-policy reinforcement learning (RL) has become a popular framework for solving sequential decision problems due to its computational efficiency and theoretical simplicity. Some on-policy methods guarantee every policy update is constrained to a trust region relative to the prior policy to ensure training stability. These methods often require computationally intensive non-linear optimization or require a particular form of action distribution. In this work, we show that applying KL penalization alone is nearly sufficient to enforce such trust regions. Then, we show that introducing a "fixup" phase is sufficient to guarantee a trust region is enforced on every policy update while adding fewer than 5% additional gradient steps in practice. The resulting algorithm, which we call FixPO, is able to train a variety of policy architectures and action spaces, is easy to implement, and produces results competitive with other trust region methods.

Offline Goal-Conditioned Reinforcement Learning (GCRL) is tasked with learning to achieve multiple goals in an environment purely from offline datasets using sparse reward functions. Offline GCRL is pivotal for developing generalist agents capable of leveraging pre-existing datasets to learn diverse and reusable skills without hand-engineering reward functions. However, contemporary approaches to GCRL based on supervised learning and contrastive learning are often suboptimal in the offline setting. An alternative perspective on GCRL optimizes for occupancy matching, but necessitates learning a discriminator, which subsequently serves as a pseudo-reward for downstream RL. Inaccuracies in the learned discriminator can cascade, negatively influencing the resulting policy. We present a novel approach to GCRL under a new lens of mixture-distribution matching, leading to our discriminator-free method: SMORe. The key insight is combining the occupancy matching perspective of GCRL with a convex dual formulation to derive a learning objective that can better leverage suboptimal offline data. SMORe learns scores or unnormalized densities representing the importance of taking an action at a state for reaching a particular goal. SMORe is principled and our extensive experiments on the fully offline GCRL benchmark composed of robot manipulation and locomotion tasks, including high-dimensional observations, show that SMORe can outperform state-of-the-art baselines by a significant margin.

Reinforcement learning has become the standard for improving reasoning in large language models, yet evidence increasingly suggests that RL does not teach new strategies; it redistributes probability mass over solutions the base model already contains. In this work, we ask: if RL merely steers the model toward paths it already knows, is the RL optimization loop itself necessary? Through token-level analysis across multiple model families and RL algorithms, we find that RL's beneficial footprint is a sparse, predictable correction concentrated at high-entropy decision points where the model is uncertain which branch to take. Only 1--3\% of token positions are affected, the promoted token always lies within the base model's top-5 alternatives, and targeted corrections at those few positions causally recover a large fraction of RL's accuracy gain, while random corrections fail. The base model's own entropy identifies these positions without any RL-trained model, and the entire correction is low-dimensional, representable in a tiny fraction of model parameters. These findings reframe reasoning improvement as sparse policy selection, not capability acquisition. We translate this insight into ReasonMaxxer, a minimal RL-free method that applies contrastive loss only at entropy-gated decision points, using a few hundred base-model rollouts and no online generation. Across three model families, six scales, and six math reasoning benchmarks, ReasonMaxxer matches or exceeds full RL performance while requiring only tens of problems and minutes of single-GPU training, a reduction in training cost of roughly three orders of magnitude.

Off-policy, value-based reinforcement learning methods such as Q-learning are appealing because they can learn from arbitrary experience, including data collected by older policies or other agents. In practice, however, bootstrapping makes long-horizon learning brittle: estimation errors at later states propagate backward through temporal-difference (TD) updates and can compound over time. We propose long-horizon Q-learning (LQL), which introduces a principled backstop against compounding error when learning the optimal action-value function. LQL builds on a prior optimality tightening observation: any realized action sequence lower-bounds what the optimal policy can achieve in expectation, so acting optimally earlier should not be worse than following the observed actions for several steps before switching to optimal behavior. Our contribution is to turn this inequality into a practical stabilization mechanism for Q-learning by using a hinge loss to penalize violations of these bounds. Importantly, LQL computes these penalties using network outputs already produced for the TD error, requiring no auxiliary networks and no additional forward passes relative to Q-learning. When combined with multiple state-of-the-art methods on a range of online and offline-to-online benchmarks, LQL consistently outperforms both 1-step TD and n-step TD learning at similar runtime.

Reinforcement learning (RL) for continuous control often requires large amounts of online interaction data. Value-based RL methods can mitigate this burden by offering relatively high sample efficiency. Some studies further enhance sample efficiency by incorporating offline demonstration data to "kick-start" training, achieving promising results in continuous control. However, they typically compute the Q-function independently for each action dimension, neglecting interdependencies and making it harder to identify optimal actions when learning from suboptimal data, such as non-expert demonstration and online-collected data during the training process. To address these issues, we propose Auto-Regressive Soft Q-learning (ARSQ), a value-based RL algorithm that models Q-values in a coarse-to-fine, auto-regressive manner. First, ARSQ decomposes the continuous action space into discrete spaces in a coarse-to-fine hierarchy, enhancing sample efficiency for fine-grained continuous control tasks. Next, it auto-regressively predicts dimensional action advantages within each decision step, enabling more effective decision-making in continuous control tasks. We evaluate ARSQ on two continuous control benchmarks, RLBench and D4RL, integrating demonstration data into online training. On D4RL, which includes non-expert demonstrations, ARSQ achieves an average 1.62times performance improvement over SOTA value-based baseline. On RLBench, which incorporates expert demonstrations, ARSQ surpasses various baselines, demonstrating its effectiveness in learning from suboptimal online-collected data. Project page is at https://sites.google.com/view/ar-soft-q

Reinforcement learning (RL) for mathematical reasoning can suffer from reward sparsity: for challenging problems, LLM fails to sample any correct trajectories, preventing RL from receiving meaningful positive feedback. At the same time, there often exist human-written reference solutions along with the problem (e.g., problems from AoPS), but directly fine-tuning on these solutions offers no benefit because models often cannot imitate human proofs that lie outside their own reasoning distribution. We introduce Reference-Guided Fine-Tuning (ReGFT), a simple and effective method that utilizes human-written reference solutions to synthesize positive trajectories on hard problems and train on them before RL. For each problem, we provide the model with a partial reference solution and let it generate its own reasoning trace, ensuring the resulting trajectories remain in the model's reasoning space while still benefiting from reference guidance. Fine-tuning on these reference-guided trajectories increases the number of solvable problems and produces a checkpoint that receives more positive rewards during RL. Across three benchmarks (AIME24, AIME25, BeyondAIME), ReGFT consistently improves supervised accuracy, accelerates DAPO training, and raises the final performance plateau of RL. Our results show that ReGFT effectively overcomes reward sparsity and unlocks stronger RL-based mathematical reasoning.

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

Offline goal-conditioned reinforcement learning (GCRL) offers a practical learning paradigm where goal-reaching policies are trained from abundant unlabeled (reward-free) datasets without additional environment interaction. However, offline GCRL still struggles with long-horizon tasks, even with recent advances that employ hierarchical policy structures, such as HIQL. By identifying the root cause of this challenge, we observe the following insights: First, performance bottlenecks mainly stem from the high-level policy's inability to generate appropriate subgoals. Second, when learning the high-level policy in the long-horizon regime, the sign of the advantage signal frequently becomes incorrect. Thus, we argue that improving the value function to produce a clear advantage signal for learning the high-level policy is essential. In this paper, we propose a simple yet effective solution: Option-aware Temporally Abstracted value learning, dubbed OTA, which incorporates temporal abstraction into the temporal-difference learning process. By modifying the value update to be option-aware, the proposed learning scheme contracts the effective horizon length, enabling better advantage estimates even in long-horizon regimes. We experimentally show that the high-level policy extracted using the OTA value function achieves strong performance on complex tasks from OGBench, a recently proposed offline GCRL benchmark, including maze navigation and visual robotic manipulation environments.

Reinforcement Learning with Verifiable Rewards (RLVR) strengthens LLM reasoning, but training often oscillates between {entropy collapse} and {entropy explosion}. We trace both hazards to the mean baseline used in value-free RL (e.g., GRPO and DAPO), which improperly penalizes negative-advantage samples under reward outliers. We propose {Quantile Advantage Estimation} (QAE), replacing the mean with a group-wise K-quantile baseline. QAE induces a response-level, two-regime gate: on hard queries (p <= 1 - K) it reinforces rare successes, while on easy queries (p > 1 - K) it targets remaining failures. Under first-order softmax updates, we prove {two-sided entropy safety}, giving lower and upper bounds on one-step entropy change that curb explosion and prevent collapse. Empirically, this minimal modification stabilizes entropy, sparsifies credit assignment (with tuned K, roughly 80% of responses receive zero advantage), and yields sustained pass@1 gains on Qwen3-8B/14B-Base across AIME 2024/2025 and AMC 2023. These results identify {baseline design} -- rather than token-level heuristics -- as the primary mechanism for scaling RLVR.

Policy gradient is a widely utilized and foundational algorithm in the field of reinforcement learning (RL). Renowned for its convergence guarantees and stability compared to other RL algorithms, its practical application is often hindered by sensitivity to hyper-parameters, particularly the step-size. In this paper, we introduce the integration of the Polyak step-size in RL, which automatically adjusts the step-size without prior knowledge. To adapt this method to RL settings, we address several issues, including unknown f* in the Polyak step-size. Additionally, we showcase the performance of the Polyak step-size in RL through experiments, demonstrating faster convergence and the attainment of more stable policies.

Recent advances in Reinforcement Learning with Verifiable Rewards (RLVR) have empowered large language models (LLMs) to tackle challenging reasoning tasks such as mathematics and programming. RLVR leverages verifiable outcome rewards to guide policy optimization, enabling LLMs to progressively improve output quality in a grounded and reliable manner. Despite its promise, the RLVR paradigm poses significant challenges, as existing methods often suffer from sparse reward signals and unstable policy gradient updates, particularly in RL-based approaches. To address the challenges, we propose PACS, a novel RLVR framework that achieves imPlicit Actor Critic coupling via a Supervised learning framework. By treating the outcome reward as a predictable label, we reformulate the RLVR problem into a supervised learning task over a score function parameterized by the policy model and optimized using cross-entropy loss. A detailed gradient analysis shows that this supervised formulation inherently recovers the classical policy gradient update while implicitly coupling actor and critic roles, yielding more stable and efficient training. Benchmarking on challenging mathematical reasoning tasks, PACS outperforms strong RLVR baselines, such as PPO and GRPO, achieving superior reasoning performance. For instance, PACS achieves 59.78\% at pass@256 on AIME 2025, representing improvements of 13.32 and 14.36 points over PPO and GRPO. This simple yet powerful framework offers a promising avenue for LLMs post-training with verifiable rewards. Our code and data are available as open source at https://github.com/ritzz-ai/PACS.

Reinforcement Learning with Verifiable Rewards (RLVR) has emerged as a key method for improving Large Language Models' reasoning capabilities, yet recent evidence suggests it may paradoxically shrink the reasoning boundary rather than expand it. This paper investigates the shrinkage issue of RLVR by analyzing its learning dynamics and reveals two critical phenomena that explain this failure. First, we expose negative interference in RLVR, where learning to solve certain training problems actively reduces the likelihood of correct solutions for others, leading to the decline of Pass@k performance, or the probability of generating a correct solution within k attempts. Second, we uncover the winner-take-all phenomenon: RLVR disproportionately reinforces problems with high likelihood, correct solutions, under the base model, while suppressing other initially low-likelihood ones. Through extensive theoretical and empirical analysis on multiple mathematical reasoning benchmarks, we show that this effect arises from the inherent on-policy sampling in standard RL objectives, causing the model to converge toward narrow solution strategies. Based on these insights, we propose a simple yet effective data curation algorithm that focuses RLVR learning on low-likelihood problems, achieving notable improvement in Pass@k performance. Our code is available at https://github.com/mail-research/SELF-llm-interference.

A widely used technique for improving policies is success conditioning, in which one collects trajectories, identifies those that achieve a desired outcome, and updates the policy to imitate the actions taken along successful trajectories. This principle appears under many names -- rejection sampling with SFT, goal-conditioned RL, Decision Transformers -- yet what optimization problem it solves, if any, has remained unclear. We prove that success conditioning exactly solves a trust-region optimization problem, maximizing policy improvement subject to a χ^2 divergence constraint whose radius is determined automatically by the data. This yields an identity: relative policy improvement, the magnitude of policy change, and a quantity we call action-influence -- measuring how random variation in action choices affects success rates -- are exactly equal at every state. Success conditioning thus emerges as a conservative improvement operator. Exact success conditioning cannot degrade performance or induce dangerous distribution shift, but when it fails, it does so observably, by hardly changing the policy at all. We apply our theory to the common practice of return thresholding, showing this can amplify improvement, but at the cost of potential misalignment with the true objective.

Reinforcement learning algorithms are commonly analyzed (and designed) under the Markov assumption. This is unrealistic, as most environments encountered in practice are either partially observable, or require function approximation that restricts the agent to access non-Markovian state features. We consider the problem of learning an optimal reactive policy in a finite environment with deterministic observations (or equivalently, hard state aggregation). We introduce a new algorithm, Committed Q-learning, and prove almost-sure convergence to the optimal reactive policy under an intuitive assumption we call rewire-robustness. This assumption is strictly weaker than the q_star-realizability condition used in prior work. Our algorithm is a variant of classical Q-learning in which the behavior policy commits to a single action upon entering a feature, and only resamples actions when the observed feature changes. A crucial part of our analysis is the introduction of quasi-Markov environments.

Most offline reinforcement learning (RL) algorithms return a target policy maximizing a trade-off between (1) the expected performance gain over the behavior policy that collected the dataset, and (2) the risk stemming from the out-of-distribution-ness of the induced state-action occupancy. It follows that the performance of the target policy is strongly related to the performance of the behavior policy and, thus, the trajectory return distribution of the dataset. We show that in mixed datasets consisting of mostly low-return trajectories and minor high-return trajectories, state-of-the-art offline RL algorithms are overly restrained by low-return trajectories and fail to exploit high-performing trajectories to the fullest. To overcome this issue, we show that, in deterministic MDPs with stochastic initial states, the dataset sampling can be re-weighted to induce an artificial dataset whose behavior policy has a higher return. This re-weighted sampling strategy may be combined with any offline RL algorithm. We further analyze that the opportunity for performance improvement over the behavior policy correlates with the positive-sided variance of the returns of the trajectories in the dataset. We empirically show that while CQL, IQL, and TD3+BC achieve only a part of this potential policy improvement, these same algorithms combined with our reweighted sampling strategy fully exploit the dataset. Furthermore, we empirically demonstrate that, despite its theoretical limitation, the approach may still be efficient in stochastic environments. The code is available at https://github.com/Improbable-AI/harness-offline-rl.

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.

Model-based and model-free reinforcement learning are traditionally viewed as separate paradigms: instead of learning a model of the transition kernel P, model-free agents typically estimate value functions tied to a specific policy and reward. In this paper, we challenge this dichotomy by proving that value-based agents trained on a sufficiently rich set of reward functions, e.g. using goal-conditioned RL, implicitly encode a unique and accurate world model. To extract this model in practice, we introduce P-learning, an inverse analogue to Q-learning that samples from an agent's Q-values, policies and rewards to decode its internal model of the environment. We then provide sufficient conditions on the type and number of goals for which agents encode the true kernel P, covering both stochastic and deterministic MDPs over finite or continuous state spaces. Even when our assumptions are violated, we empirically demonstrate that agents trained on a handful of reward functions encode accurate dynamics in Reacher, MountainCar and stochastic variants of FourRooms. Surprisingly, we find that policies trained exclusively on a Reacher agent's implicit world model are quasi-optimal on out-of-distribution, velocity-based goals despite position-only training -- suggesting that agents contain hidden generalisation capabilities and providing a new lens into the connection between model-based, model-free, and goal-conditioned RL.

Offline (or batch) reinforcement learning (RL) algorithms seek to learn an optimal policy from a fixed dataset without active data collection. Based on the composition of the offline dataset, two main categories of methods are used: imitation learning which is suitable for expert datasets and vanilla offline RL which often requires uniform coverage datasets. From a practical standpoint, datasets often deviate from these two extremes and the exact data composition is usually unknown a priori. To bridge this gap, we present a new offline RL framework that smoothly interpolates between the two extremes of data composition, hence unifying imitation learning and vanilla offline RL. The new framework is centered around a weak version of the concentrability coefficient that measures the deviation from the behavior policy to the expert policy alone. Under this new framework, we further investigate the question on algorithm design: can one develop an algorithm that achieves a minimax optimal rate and also adapts to unknown data composition? To address this question, we consider a lower confidence bound (LCB) algorithm developed based on pessimism in the face of uncertainty in offline RL. We study finite-sample properties of LCB as well as information-theoretic limits in multi-armed bandits, contextual bandits, and Markov decision processes (MDPs). Our analysis reveals surprising facts about optimality rates. In particular, in all three settings, LCB achieves a faster rate of 1/N for nearly-expert datasets compared to the usual rate of 1/N in offline RL, where N is the number of samples in the batch dataset. In the case of contextual bandits with at least two contexts, we prove that LCB is adaptively optimal for the entire data composition range, achieving a smooth transition from imitation learning to offline RL. We further show that LCB is almost adaptively optimal in MDPs.

Large Language Models (LLMs) trained via Reinforcement Learning (RL) have exhibited strong reasoning capabilities and emergent reflective behaviors, such as backtracking and error correction. However, conventional Markovian RL confines exploration to the training phase to learn an optimal deterministic policy and depends on the history contexts only through the current state. Therefore, it remains unclear whether reflective reasoning will emerge during Markovian RL training, or why they are beneficial at test time. To remedy this, we recast reflective exploration within the Bayes-Adaptive RL framework, which explicitly optimizes the expected return under a posterior distribution over Markov decision processes. This Bayesian formulation inherently incentivizes both reward-maximizing exploitation and information-gathering exploration via belief updates. Our resulting algorithm, BARL, instructs the LLM to stitch and switch strategies based on the observed outcomes, offering principled guidance on when and how the model should reflectively explore. Empirical results on both synthetic and mathematical reasoning tasks demonstrate that BARL outperforms standard Markovian RL approaches at test time, achieving superior token efficiency with improved exploration effectiveness. Our code is available at https://github.com/shenao-zhang/BARL.

As Reinforcement Learning (RL) agents are increasingly employed in diverse decision-making problems using reward preferences, it becomes important to ensure that policies learned by these frameworks in mapping observations to a probability distribution of the possible actions are explainable. However, there is little to no work in the systematic understanding of these complex policies in a contrastive manner, i.e., what minimal changes to the policy would improve/worsen its performance to a desired level. In this work, we present COUNTERPOL, the first framework to analyze RL policies using counterfactual explanations in the form of minimal changes to the policy that lead to the desired outcome. We do so by incorporating counterfactuals in supervised learning in RL with the target outcome regulated using desired return. We establish a theoretical connection between Counterpol and widely used trust region-based policy optimization methods in RL. Extensive empirical analysis shows the efficacy of COUNTERPOL in generating explanations for (un)learning skills while keeping close to the original policy. Our results on five different RL environments with diverse state and action spaces demonstrate the utility of counterfactual explanations, paving the way for new frontiers in designing and developing counterfactual policies.

We study risk-sensitive Reinforcement Learning (RL), where we aim to maximize the Conditional Value at Risk (CVaR) with a fixed risk tolerance tau. Prior theoretical work studying risk-sensitive RL focuses on the tabular Markov Decision Processes (MDPs) setting. To extend CVaR RL to settings where state space is large, function approximation must be deployed. We study CVaR RL in low-rank MDPs with nonlinear function approximation. Low-rank MDPs assume the underlying transition kernel admits a low-rank decomposition, but unlike prior linear models, low-rank MDPs do not assume the feature or state-action representation is known. We propose a novel Upper Confidence Bound (UCB) bonus-driven algorithm to carefully balance the interplay between exploration, exploitation, and representation learning in CVaR RL. We prove that our algorithm achieves a sample complexity of Oleft(H^7 A^2 d^4{tau^2 epsilon^2}right) to yield an epsilon-optimal CVaR, where H is the length of each episode, A is the capacity of action space, and d is the dimension of representations. Computational-wise, we design a novel discretized Least-Squares Value Iteration (LSVI) algorithm for the CVaR objective as the planning oracle and show that we can find the near-optimal policy in a polynomial running time with a Maximum Likelihood Estimation oracle. To our knowledge, this is the first provably efficient CVaR RL algorithm in low-rank MDPs.

Risk-sensitive reinforcement learning (RL) aims to optimize policies that balance the expected reward and risk. In this paper, we present a novel risk-sensitive RL framework that employs an Iterated Conditional Value-at-Risk (CVaR) objective under both linear and general function approximations, enriched by human feedback. These new formulations provide a principled way to guarantee safety in each decision making step throughout the control process. Moreover, integrating human feedback into risk-sensitive RL framework bridges the gap between algorithmic decision-making and human participation, allowing us to also guarantee safety for human-in-the-loop systems. We propose provably sample-efficient algorithms for this Iterated CVaR RL and provide rigorous theoretical analysis. Furthermore, we establish a matching lower bound to corroborate the optimality of our algorithms in a linear context.

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) provides a theoretical framework for continuously improving an agent's behavior via trial and error. However, efficiently learning policies from scratch can be very difficult, particularly for tasks with exploration challenges. In such settings, it might be desirable to initialize RL with an existing policy, offline data, or demonstrations. However, naively performing such initialization in RL often works poorly, especially for value-based methods. In this paper, we present a meta algorithm that can use offline data, demonstrations, or a pre-existing policy to initialize an RL policy, and is compatible with any RL approach. In particular, we propose Jump-Start Reinforcement Learning (JSRL), an algorithm that employs two policies to solve tasks: a guide-policy, and an exploration-policy. By using the guide-policy to form a curriculum of starting states for the exploration-policy, we are able to efficiently improve performance on a set of simulated robotic tasks. We show via experiments that JSRL is able to significantly outperform existing imitation and reinforcement learning algorithms, particularly in the small-data regime. In addition, we provide an upper bound on the sample complexity of JSRL and show that with the help of a guide-policy, one can improve the sample complexity for non-optimism exploration methods from exponential in horizon to polynomial.

Reinforcement learning (RL) has become a standard paradigm for refining large language models (LLMs) beyond pre-training and instruction tuning. A prominent line of work is RL with verifiable rewards (RLVR), which leverages automatically verifiable outcomes (e.g., correctness or executability) to generate reward signals. While efficient, this framework faces two key limitations: First, its binary feedback is too sparse to capture the quality of the reasoning process. Second, its coarse-grained rewards potentially lead to vanishing gradients. Inspired by observations from human learning, we introduce a RL technique that integrates verifiable outcomes with the model's own confidence estimates. This joint design enriches the reward signal, providing finer-grained feedback and implicitly supervising the reasoning process. Experimental results demonstrate that our proposed method enhances RL performance across multiple datasets and reduces token consumption during inference, while incurring negligible additional training cost. Moreover, it can be used as a plug-in module to enhance other state-of-the-art RL methods.

Reinforcement learning with verifiable rewards (RLVR) has become a standard recipe for improving large language models (LLMs) on reasoning tasks, with Group Relative Policy Optimization (GRPO) widely used in practice. Yet GRPO wastes substantial compute on negative groups: groups in which no sampled response is correct yield zero advantage and thus no gradient. We ask whether negative groups can be leveraged without extra supervision. Starting from a maximum-likelihood (MLE) objective in reward modeling, we show that the MLE gradient is equivalent to a policy gradient for a modified value function. This value function adds a confidence-weighted penalty on incorrect responses, imposing larger penalties on more confident mistakes. We refer to this as Likelihood Estimation with Negative Samples (LENS). LENS modifies GRPO to assign non-zero, confidence-dependent rewards to incorrect generations, making negative groups informative and converting previously wasted samples into useful gradient updates. On the MATH benchmark with Llama-3.1-8B and Qwen-2.5-3B, the proposed variant consistently outperforms GRPO baseline, with significant gains on harder items. These results demonstrate a principled and practical way to "rescue" negative groups, improving efficiency and performance in RLVR.

Reinforcement learning with verifiable rewards (RLVR) has proven effective in eliciting complex reasoning in large language models (LLMs). However, standard RLVR training often leads to excessively verbose processes (in reasoning tasks) and inefficient exploration trajectories (in agentic settings), as outcome-only rewards provide no incentive for efficiency and the high variance in response length within relatively small rollout groups results in noisy optimization signals. To address this, we propose Rollout Response Recomposition (RoRecomp), a plug-and-play method that guides models toward concise reasoning by strategically recomposing the training data. RoRecomp separates responses into two distinct batch types: 1) priority batches, which combine short-correct and long-incorrect responses selected from online batches to provide a clear gradient signal for brevity, and 2) compensation batches, which utilize remaining responses from a replay buffer to maintain stability and prevent model collapse. To comprehensively evaluate effectiveness, we test RoRecomp across three settings where results demonstrate substantial efficiency gains: reducing reasoning length by 27.7% in zero RL training, reducing unnecessary tool calls by 46.8% while improving accuracy in agentic RL, and achieving up to 52.5% length reduction in thinking compression, all with minimal performance impact.

Some reinforcement learning (RL) algorithms can stitch pieces of experience to solve a task never seen before during training. This oft-sought property is one of the few ways in which RL methods based on dynamic-programming differ from RL methods based on supervised-learning (SL). Yet, certain RL methods based on off-the-shelf SL algorithms achieve excellent results without an explicit mechanism for stitching; it remains unclear whether those methods forgo this important stitching property. This paper studies this question for the problems of achieving a target goal state and achieving a target return value. Our main result is to show that the stitching property corresponds to a form of combinatorial generalization: after training on a distribution of (state, goal) pairs, one would like to evaluate on (state, goal) pairs not seen together in the training data. Our analysis shows that this sort of generalization is different from i.i.d. generalization. This connection between stitching and generalisation reveals why we should not expect SL-based RL methods to perform stitching, even in the limit of large datasets and models. Based on this analysis, we construct new datasets to explicitly test for this property, revealing that SL-based methods lack this stitching property and hence fail to perform combinatorial generalization. Nonetheless, the connection between stitching and combinatorial generalisation also suggests a simple remedy for improving generalisation in SL: data augmentation. We propose a temporal data augmentation and demonstrate that adding it to SL-based methods enables them to successfully complete tasks not seen together during training. On a high level, this connection illustrates the importance of combinatorial generalization for data efficiency in time-series data beyond tasks beyond RL, like audio, video, or text.

Mixed integer linear programs are commonly solved by Branch and Bound algorithms. A key factor of the efficiency of the most successful commercial solvers is their fine-tuned heuristics. In this paper, we leverage patterns in real-world instances to learn from scratch a new branching strategy optimised for a given problem and compare it with a commercial solver. We propose FMSTS, a novel Reinforcement Learning approach specifically designed for this task. The strength of our method lies in the consistency between a local value function and a global metric of interest. In addition, we provide insights for adapting known RL techniques to the Branch and Bound setting, and present a new neural network architecture inspired from the literature. To our knowledge, it is the first time Reinforcement Learning has been used to fully optimise the branching strategy. Computational experiments show that our method is appropriate and able to generalise well to new instances.

The reasoning performance of large language models (LLMs) can be substantially improved by training them with reinforcement learning (RL). The RL objective for LLM training involves a regularization term, which is the reverse Kullback-Leibler (KL) divergence between the trained policy and the reference policy. Since computing the KL divergence exactly is intractable, various estimators are used in practice to estimate it from on-policy samples. Despite its wide adoption, including in several open-source libraries, there is no systematic study analyzing the numerous ways of incorporating KL estimators in the objective and their effect on the downstream performance of RL-trained models. Recent works show that prevailing practices for incorporating KL regularization do not provide correct gradients for stated objectives, creating a discrepancy between the objective and its implementation. In this paper, we further analyze these practices and study the gradients of several estimators configurations, revealing how design choices shape gradient bias. We substantiate these findings with empirical observations by RL fine-tuning Qwen2.5-7B, Llama-3.1-8B-Instruct and Qwen3-4B-Instruct-2507 with different configurations and evaluating their performance on both in- and out-of-distribution tasks. Through our analysis, we observe that, in on-policy settings: (1) estimator configurations with biased gradients can result in training instabilities; and (2) using estimator configurations resulting in unbiased gradients leads to better performance on in-domain as well as out-of-domain tasks. We also investigate the performance resulting from different KL configurations in off-policy settings and observe that KL regularization can help stabilize off-policy RL training resulting from asynchronous setups.

Specifying rewards for reinforcement learned (RL) agents is challenging. Preference-based RL (PbRL) mitigates these challenges by inferring a reward from feedback over sets of trajectories. However, the effectiveness of PbRL is limited by the amount of feedback needed to reliably recover the structure of the target reward. We present the PRIor Over Rewards (PRIOR) framework, which incorporates priors about the structure of the reward function and the preference feedback into the reward learning process. Imposing these priors as soft constraints on the reward learning objective reduces the amount of feedback required by half and improves overall reward recovery. Additionally, we demonstrate that using an abstract state space for the computation of the priors further improves the reward learning and the agent's performance.

We consider the exploration-exploitation dilemma in finite-horizon reinforcement learning problems whose state-action space is endowed with a metric. We introduce Kernel-UCBVI, a model-based optimistic algorithm that leverages the smoothness of the MDP and a non-parametric kernel estimator of the rewards and transitions to efficiently balance exploration and exploitation. For problems with K episodes and horizon H, we provide a regret bound of Oleft( H^3 K^{2d{2d+1}}right), where d is the covering dimension of the joint state-action space. This is the first regret bound for kernel-based RL using smoothing kernels, which requires very weak assumptions on the MDP and has been previously applied to a wide range of tasks. We empirically validate our approach in continuous MDPs with sparse rewards.

Reinforcement Learning with Verifiable Rewards (RLVR) is widely used to improve reasoning in multiple domains, yet outcome-only scalar rewards are often sparse and uninformative, especially on failed samples, where they merely indicate failure and provide no insight into why the reasoning fails. In this paper, we investigate how to leverage richer verbal feedback to guide RLVR training on failed samples, and how to convert such feedback into a trainable learning signal. Specifically, we propose a multi-turn feedback-guided reinforcement learning framework. It builds on three mechanisms: (1) dynamic multi-turn regeneration guided by feedback, triggered only on failed samples, (2) two complementary learning signals for within-turn and cross-turn optimization, and (3) structured feedback injection into the model's reasoning process. Trained on sampled OpenR1-Math, the approach outperforms supervised fine-tuning and RLVR baselines in-domain and generalizes well out-of-domain.

Reinforcement learning is the method of choice to train models in sampling-based setups with binary outcome feedback, such as navigation, code generation, and mathematical problem solving. In such settings, models implicitly induce a likelihood over correct rollouts. However, we observe that reinforcement learning does not maximize this likelihood, and instead optimizes only a lower-order approximation. Inspired by this observation, we introduce Maximum Likelihood Reinforcement Learning (MaxRL), a sampling-based framework to approximate maximum likelihood using reinforcement learning techniques. MaxRL addresses the challenges of non-differentiable sampling by defining a compute-indexed family of sample-based objectives that interpolate between standard reinforcement learning and exact maximum likelihood as additional sampling compute is allocated. The resulting objectives admit a simple, unbiased policy-gradient estimator and converge to maximum likelihood optimization in the infinite-compute limit. Empirically, we show that MaxRL Pareto-dominates existing methods in all models and tasks we tested, achieving up to 20x test-time scaling efficiency gains compared to its GRPO-trained counterpart. We also observe MaxRL to scale better with additional data and compute. Our results suggest MaxRL is a promising framework for scaling RL training in correctness based settings.

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.

Applying reinforcement learning (RL) to real-world problems is often made challenging by the inability to interact with the environment and the difficulty of designing reward functions. Offline RL addresses the first challenge by considering access to an offline dataset of environment interactions labeled by the reward function. In contrast, Preference-based RL does not assume access to the reward function and learns it from preferences, but typically requires an online interaction with the environment. We bridge the gap between these frameworks by exploring efficient methods for acquiring preference feedback in a fully offline setup. We propose Sim-OPRL, an offline preference-based reinforcement learning algorithm, which leverages a learned environment model to elicit preference feedback on simulated rollouts. Drawing on insights from both the offline RL and the preference-based RL literature, our algorithm employs a pessimistic approach for out-of-distribution data, and an optimistic approach for acquiring informative preferences about the optimal policy. We provide theoretical guarantees regarding the sample complexity of our approach, dependent on how well the offline data covers the optimal policy. Finally, we demonstrate the empirical performance of Sim-OPRL in various environments.

Developing an agent capable of adapting to unseen environments remains a difficult challenge in imitation learning. This work presents Adaptive Return-conditioned Policy (ARP), an efficient framework designed to enhance the agent's generalization ability using natural language task descriptions and pre-trained multimodal encoders. Our key idea is to calculate a similarity between visual observations and natural language instructions in the pre-trained multimodal embedding space (such as CLIP) and use it as a reward signal. We then train a return-conditioned policy using expert demonstrations labeled with multimodal rewards. Because the multimodal rewards provide adaptive signals at each timestep, our ARP effectively mitigates the goal misgeneralization. This results in superior generalization performances even when faced with unseen text instructions, compared to existing text-conditioned policies. To improve the quality of rewards, we also introduce a fine-tuning method for pre-trained multimodal encoders, further enhancing the performance. Video demonstrations and source code are available on the project website: https://sites.google.com/view/2023arp.

Reinforcement Learning with Verifiable Rewards (RLVR) can elicit strong multi-step reasoning, yet it often encourages overly verbose traces. Moreover, naive length penalties in group-relative optimization can severely hurt accuracy. We attribute this failure to two structural issues: (i) Dilution of Length Baseline, where incorrect responses (with zero length reward) depress the group baseline and over-penalize correct solutions; and (ii) Difficulty-Penalty Mismatch, where a static penalty cannot adapt to problem difficulty, suppressing necessary reasoning on hard instances while leaving redundancy on easy ones. We propose Dynamic Decoupled Conditional Advantage (DDCA) to decouple efficiency optimization from correctness. DDCA computes length advantages conditionally within the correct-response cluster to eliminate baseline dilution, and dynamically scales the penalty strength using the group pass rate as a proxy for difficulty. Experiments on GSM8K, MATH500, AMC23, and AIME25 show that DDCA consistently improves the efficiency--accuracy trade-off relative to adaptive baselines, reducing generated tokens by approximately 60% on simpler tasks (e.g., GSM8K) versus over 20% on harder benchmarks (e.g., AIME25), thereby maintaining or improving accuracy. Code is available at https://github.com/alphadl/DDCA.

Offline reinforcement learning requires reconciling two conflicting aims: learning a policy that improves over the behavior policy that collected the dataset, while at the same time minimizing the deviation from the behavior policy so as to avoid errors due to distributional shift. This trade-off is critical, because most current offline reinforcement learning methods need to query the value of unseen actions during training to improve the policy, and therefore need to either constrain these actions to be in-distribution, or else regularize their values. We propose an offline RL method that never needs to evaluate actions outside of the dataset, but still enables the learned policy to improve substantially over the best behavior in the data through generalization. The main insight in our work is that, instead of evaluating unseen actions from the latest policy, we can approximate the policy improvement step implicitly by treating the state value function as a random variable, with randomness determined by the action (while still integrating over the dynamics to avoid excessive optimism), and then taking a state conditional upper expectile of this random variable to estimate the value of the best actions in that state. This leverages the generalization capacity of the function approximator to estimate the value of the best available action at a given state without ever directly querying a Q-function with this unseen action. Our algorithm alternates between fitting this upper expectile value function and backing it up into a Q-function. Then, we extract the policy via advantage-weighted behavioral cloning. We dub our method implicit Q-learning (IQL). IQL demonstrates the state-of-the-art performance on D4RL, a standard benchmark for offline reinforcement learning. We also demonstrate that IQL achieves strong performance fine-tuning using online interaction after offline initialization.

Predicting and reasoning about the future lie at the heart of many time-series questions. For example, goal-conditioned reinforcement learning can be viewed as learning representations to predict which states are likely to be visited in the future. While prior methods have used contrastive predictive coding to model time series data, learning representations that encode long-term dependencies usually requires large amounts of data. In this paper, we introduce a temporal difference version of contrastive predictive coding that stitches together pieces of different time series data to decrease the amount of data required to learn predictions of future events. We apply this representation learning method to derive an off-policy algorithm for goal-conditioned RL. Experiments demonstrate that, compared with prior RL methods, ours achieves 2 times median improvement in success rates and can better cope with stochastic environments. In tabular settings, we show that our method is about 20 times more sample efficient than the successor representation and 1500 times more sample efficient than the standard (Monte Carlo) version of contrastive predictive coding.

Reinforcement Learning with Verifiable Rewards (RLVR) is increasingly viewed as a tree pruning mechanism. However, we identify a systemic pathology termed Recursive Space Contraction (RSC), an irreversible collapse driven by the combined dynamics of positive sharpening and negative squeezing, where the sampling probability of valid alternatives vanishes. While Kullback-Leibler (KL) regularization aims to mitigate this, it imposes a rigid Shape Matching constraint that forces the policy to mimic the reference model's full density, creating a gradient conflict with the sharpening required for correctness. We propose Anchored Policy Optimization (APO), shifting the paradigm from global Shape Matching to Support Coverage. By defining a Safe Manifold based on the reference model's high-confidence support, APO permits aggressive sharpening for efficiency while selectively invoking a restorative force during error correction to prevent collapse. We theoretically derive that APO serves as a gradient-aligned mechanism to maximize support coverage, enabling an Elastic Recovery that re-inflates valid branches. Empirical evaluations on mathematical benchmarks demonstrate that APO breaks the accuracy-diversity trade-off, significantly improving Pass@1 while restoring the Pass@K diversity typically lost by standard policy gradient methods.

Past analyses of reinforcement learning from human feedback (RLHF) assume that the human fully observes the environment. What happens when human feedback is based only on partial observations? We formally define two failure cases: deception and overjustification. Modeling the human as Boltzmann-rational w.r.t. a belief over trajectories, we prove conditions under which RLHF is guaranteed to result in policies that deceptively inflate their performance, overjustify their behavior to make an impression, or both. To help address these issues, we mathematically characterize how partial observability of the environment translates into (lack of) ambiguity in the learned return function. In some cases, accounting for partial observability makes it theoretically possible to recover the return function and thus the optimal policy, while in other cases, there is irreducible ambiguity. We caution against blindly applying RLHF in partially observable settings and propose research directions to help tackle these challenges.

A central paradox in fine-tuning Large Language Models (LLMs) with Reinforcement Learning with Verifiable Reward (RLVR) is the frequent degradation of multi-attempt performance (Pass@k) despite improvements in single-attempt accuracy (Pass@1). This is often accompanied by catastrophic forgetting, where models lose previously acquired skills. While various methods have been proposed, the choice and function of the divergence term have been surprisingly unexamined as a proactive solution. We argue that standard RLVR objectives -- both those using the mode-seeking reverse KL-divergence and those forgoing a divergence term entirely -- lack a crucial mechanism for knowledge retention. The reverse-KL actively accelerates this decay by narrowing the policy, while its absence provides no safeguard against the model drifting from its diverse knowledge base. We propose a fundamental shift in perspective: using the divergence term itself as the solution. Our framework, Diversity-Preserving Hybrid RL (DPH-RL), leverages mass-covering f-divergences (like forward-KL and JS-divergence) to function as a rehearsal mechanism. By continuously referencing the initial policy, this approach forces the model to maintain broad solution coverage. Extensive experiments on math and SQL generation demonstrate that DPH-RL not only resolves the Pass@k degradation but improves both Pass@1 and Pass@k in- and out-of-domain. Additionally, DPH-RL is more training-efficient because it computes f-divergence using generator functions, requiring only sampling from the initial policy and no online reference model. Our work highlights a crucial, overlooked axis for improving RLVR, demonstrating that the proper selection of a divergence measure is a powerful tool for building more general and diverse reasoning models.

Reinforcement learning (RL) has significantly advanced code generation for large language models (LLMs). However, current paradigms rely on outcome-based rewards from test cases, neglecting the quality of the intermediate reasoning process. While supervising the reasoning process directly is a promising direction, it is highly susceptible to reward hacking, where the policy model learns to exploit the reasoning reward signal without improving final outcomes. To address this, we introduce a unified framework that can effectively incorporate the quality of the reasoning process during RL. First, to enable reasoning evaluation, we develop LCB-RB, a benchmark comprising preference pairs of superior and inferior reasoning processes. Second, to accurately score reasoning quality, we introduce an Optimized-Degraded based (OD-based) method for reward model training. This method generates high-quality preference pairs by systematically optimizing and degrading initial reasoning paths along curated dimensions of reasoning quality, such as factual accuracy, logical rigor, and coherence. A 7B parameter reward model with this method achieves state-of-the-art (SOTA) performance on LCB-RB and generalizes well to other benchmarks. Finally, we introduce Posterior-GRPO (P-GRPO), a novel RL method that conditions process-based rewards on task success. By selectively applying rewards to the reasoning processes of only successful outcomes, P-GRPO effectively mitigates reward hacking and aligns the model's internal reasoning with final code correctness. A 7B parameter model with P-GRPO achieves superior performance across diverse code generation tasks, outperforming outcome-only baselines by 4.5%, achieving comparable performance to GPT-4-Turbo. We further demonstrate the generalizability of our approach by extending it to mathematical tasks. Our models, dataset, and code are publicly available.

Sequential decision making algorithms often struggle to leverage different sources of unstructured offline interaction data. Imitation learning (IL) methods based on supervised learning are robust, but require optimal demonstrations, which are hard to collect. Offline goal-conditioned reinforcement learning (RL) algorithms promise to learn from sub-optimal data, but face optimization challenges especially with high-dimensional data. To bridge the gap between IL and RL, we introduce Distance Weighted Supervised Learning or DWSL, a supervised method for learning goal-conditioned policies from offline data. DWSL models the entire distribution of time-steps between states in offline data with only supervised learning, and uses this distribution to approximate shortest path distances. To extract a policy, we weight actions by their reduction in distance estimates. Theoretically, DWSL converges to an optimal policy constrained to the data distribution, an attractive property for offline learning, without any bootstrapping. Across all datasets we test, DWSL empirically maintains behavior cloning as a lower bound while still exhibiting policy improvement. In high-dimensional image domains, DWSL surpasses the performance of both prior goal-conditioned IL and RL algorithms. Visualizations and code can be found at https://sites.google.com/view/dwsl/home .

Reinforcement Learning with Verifiable Rewards (RLVR) enhances reasoning of Large Language Models (LLMs) but usually exhibits limited generation diversity due to the over-incentivization of positive rewards. Although methods like Negative Sample Reinforcement (NSR) mitigate this issue by upweighting penalty from negative samples, they may suppress the semantic distributions shared between positive and negative responses. To boost reasoning ability without losing diversity, this paper proposes negative sample projection Residual Reinforcement Learning (ResRL) that decouples similar semantic distributions among positive and negative responses. We theoretically link Lazy Likelihood Displacement (LLD) to negative-positive head-gradient interference and derive a single-forward proxy that upper-bounds representation alignment to guide conservative advantage reweighting. ResRL then projects negative-token hidden representations onto an SVD-based low-rank positive subspace and uses projection residuals to modulate negative gradients, improving reasoning while preserving diversity and outperforming strong baselines on average across twelve benchmarks spanning Mathematics, Code, Agent Tasks, and Function Calling. Notably, ResRL surpasses NSR on mathematical reasoning by 9.4\% in Avg@16 and 7.0\% in Pass@128. Code is available at https://github.com/1229095296/ResRL.git.

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

Reinforcement Learning with Verifiable Rewards (RLVR) has become a standard paradigm for reasoning in Large Language Models. However, optimizing solely for final-answer correctness often drives models into aimless, verbose exploration, where they rely on exhaustive trial-and-error tactics rather than structured planning to reach solutions. While heuristic constraints like length penalties can reduce verbosity, they often truncate essential reasoning steps, creating a difficult trade-off between efficiency and verification. In this paper, we argue that discriminative capability is a prerequisite for efficient generation: by learning to distinguish valid solutions, a model can internalize a guidance signal that prunes the search space. We propose JudgeRLVR, a two-stage judge-then-generate paradigm. In the first stage, we train the model to judge solution responses with verifiable answers. In the second stage, we fine-tune the same model with vanilla generating RLVR initialized from the judge. Compared to Vanilla RLVR using the same math-domain training data, JudgeRLVR achieves a better quality--efficiency trade-off for Qwen3-30B-A3B: on in-domain math, it delivers about +3.7 points average accuracy gain with -42\% average generation length; on out-of-domain benchmarks, it delivers about +4.5 points average accuracy improvement, demonstrating enhanced generalization.

We propose a stronger formulation of RL on top of RWKV-style recurrent sequence models, in which the fixed-size recurrent state is explicitly interpreted as a belief state rather than an opaque hidden vector. Instead of conditioning policy and value on a single summary h_t, we maintain a compact uncertainty-aware state b_t = (μ_t, Σ_t) derived from RWKV-style recurrent statistics and let control depend on both memory and uncertainty. This design targets a key weakness of plain fixed-state policies in partially observed settings: they may store evidence, but not necessarily confidence. We present the method, a theoretical program, and a pilot RL experiment with hidden episode-level observation noise together with a test-time noise sweep. The pilot shows that belief-state policies nearly match the best recurrent baseline overall while slightly improving return on the hardest in-distribution regime and under a held-out noise shift. Additional ablations show that this simple belief readout is currently stronger than two more structured extensions, namely gated memory control and privileged belief targets, underscoring the need for richer benchmarks.

Large language models trained with reinforcement learning (RL) for mathematical reasoning face a fundamental challenge: on problems the model cannot solve at all - "cliff" prompts - the RL gradient vanishes entirely, preventing any learning signal from reaching these failure modes. We introduce Hybrid Distillation Policy Optimization (HDPO), which augments standard RL with privileged self-distillation targeting cliff prompts. On each training step, HDPO identifies prompts where all rollouts fail, generates privileged rollouts by providing the model with ground-truth information, filters for correct solutions, and distills the teacher's token-level distribution into the student. Because teacher and student share the same weights - differing only in their input - the realizability gap is provably bounded, unlike cross-model distillation. We prove that R=1 filtered privileged generation recovers the optimal KL-regularized RL policy in the hard-threshold limit. Experiments on OpenMathInstruct-2 with Qwen2.5-Math-1.5B-Instruct show that HDPO consistently improves coverage metrics (pass@4 by +0.8-1.1%, pass@8 by +0.4-1.7%) while maintaining greedy accuracy, with the distillation weight lambda providing direct control over the exploration-exploitation tradeoff.

While many real-world problems that might benefit from reinforcement learning, these problems rarely fit into the MDP mold: interacting with the environment is often expensive and specifying reward functions is challenging. Motivated by these challenges, prior work has developed data-driven approaches that learn entirely from samples from the transition dynamics and examples of high-return states. These methods typically learn a reward function from high-return states, use that reward function to label the transitions, and then apply an offline RL algorithm to these transitions. While these methods can achieve good results on many tasks, they can be complex, often requiring regularization and temporal difference updates. In this paper, we propose a method for offline, example-based control that learns an implicit model of multi-step transitions, rather than a reward function. We show that this implicit model can represent the Q-values for the example-based control problem. Across a range of state-based and image-based offline control tasks, our method outperforms baselines that use learned reward functions; additional experiments demonstrate improved robustness and scaling with dataset size.

Reinforcement learning with verifiable rewards (RLVR) has recently enhanced the reasoning capabilities of large language models (LLMs), particularly for mathematical problem solving. However, a fundamental limitation remains: as the sampling budget increases, the advantage of RLVR-trained models over their pretrained bases often diminishes or even vanishes, revealing a strong dependence on the base model's restricted search space. We attribute this phenomenon to the widespread use of the reverse Kullback-Leibler (KL) divergence regularizer, whose mode-seeking behavior keeps the policy trapped inside the base model's support region and hampers wider exploration. To address this issue, we propose RAPO (Rewards-Aware Policy Optimization), an algorithm to promote broader yet focused exploration. Our method (i) utilizes the forward KL penalty to replace the reverse KL penalty for out-of-distribution exploration, and (ii) reweights the reference policy to facilitate adaptive in-distribution exploration. We train Qwen2.5-3B and 7B models with RAPO on the 8K SimpleRL-Zero dataset, without supervised fine-tuning, and evaluate them on AIME2024 and AIME2025. Results show that RAPO consistently improves problem-solving performance. Notably, RAPO enables models to surpass the base model's performance ceiling and solves previously intractable problems, advancing the frontier of RLVR for challenging reasoning tasks.

Offline reinforcement learning faces a significant challenge of value over-estimation due to the distributional drift between the dataset and the current learned policy, leading to learning failure in practice. The common approach is to incorporate a penalty term to reward or value estimation in the Bellman iterations. Meanwhile, to avoid extrapolation on out-of-distribution (OOD) states and actions, existing methods focus on conservative Q-function estimation. In this paper, we propose Conservative State Value Estimation (CSVE), a new approach that learns conservative V-function via directly imposing penalty on OOD states. Compared to prior work, CSVE allows more effective in-data policy optimization with conservative value guarantees. Further, we apply CSVE and develop a practical actor-critic algorithm in which the critic does the conservative value estimation by additionally sampling and penalizing the states around the dataset, and the actor applies advantage weighted updates extended with state exploration to improve the policy. We evaluate in classic continual control tasks of D4RL, showing that our method performs better than the conservative Q-function learning methods and is strongly competitive among recent SOTA methods.

Reinforcement learning (RL) has emerged as a promising strategy for finetuning small language models (SLMs) to solve targeted tasks such as math and coding. However, RL algorithms tend to be resource-intensive, taking a significant amount of time to train. We propose RAPID, a novel RL algorithm that can substantially reduce the running time of RL. Our key insight is that RL tends to be costly due to the need to perform both inference and backpropagation during training. To maximize use of computational resources, our algorithm performs inference in large batches, and then performs off-policy policy gradient updates in mini-batches. For off-policy updates, we incorporate group advantage estimation into the policy gradient algorithm, and derive an importance weighted estimator to correct for the bias arising from off-policy learning. Our experiments demonstrate that our algorithm can reduce running time by 11%-34% on three benchmarks compared to state-of-the-art RL algorithms while maintaining similar or better accuracy.

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 (RL) from Human Preference-based feedback is a popular paradigm for fine-tuning generative models, which has produced impressive models such as GPT-4 and Claude3 Opus. This framework often consists of two steps: learning a reward model from an offline preference dataset followed by running online RL to optimize the learned reward model. In this work, leveraging the idea of reset, we propose a new RLHF algorithm with provable guarantees. Motivated by the fact that offline preference dataset provides informative states (i.e., data that is preferred by the labelers), our new algorithm, Dataset Reset Policy Optimization (DR-PO), integrates the existing offline preference dataset into the online policy training procedure via dataset reset: it directly resets the policy optimizer to the states in the offline dataset, instead of always starting from the initial state distribution. In theory, we show that DR-PO learns to perform at least as good as any policy that is covered by the offline dataset under general function approximation with finite sample complexity. In experiments, we demonstrate that on both the TL;DR summarization and the Anthropic Helpful Harmful (HH) dataset, the generation from DR-PO is better than that from Proximal Policy Optimization (PPO) and Direction Preference Optimization (DPO), under the metric of GPT4 win-rate. Code for this work can be found at https://github.com/Cornell-RL/drpo.

Reinforcement Learning with Verifiable Rewards (RLVR) is an effective paradigm for improving the reasoning capabilities of large language models. However, existing RLVR methods utilize rollouts in an indiscriminate and short-horizon manner: responses of heterogeneous quality within each prompt are treated uniformly, and historical rollouts are discarded after a single use. This leads to noisy supervision, poor sample efficiency, and suboptimal policy updates. We address these issues by formulating rollout scheduling in RLVR as a contextual bandit problem and proposing a unified neural scheduling framework that adaptively selects high-value rollouts throughout training. Each rollout is treated as an arm whose reward is defined by the induced performance gain between consecutive optimization steps. The resulting scheduler supports both noise-aware intra-group selection and adaptive global reuse of historical rollouts within a single principled framework. We provide theoretical justification by deriving sublinear regret bounds and showing that enlarging the rollout buffer improves the achievable performance upper bound. Experiments on six mathematical reasoning benchmarks demonstrate consistent gains in performance and training efficiency across multiple RLVR optimization methods.

Reinforcement learning (RL) has emerged as the de-facto paradigm for improving the reasoning capabilities of large language models (LLMs). We have developed RLAX, a scalable RL framework on TPUs. RLAX employs a parameter-server architecture. A master trainer periodically pushes updated model weights to the parameter server while a fleet of inference workers pull the latest weights and generates new rollouts. We introduce a suite of system techniques to enable scalable and preemptible RL for a diverse set of state-of-art RL algorithms. To accelerate convergence and improve model quality, we have devised new dataset curation and alignment techniques. Large-scale evaluations show that RLAX improves QwQ-32B's pass@8 accuracy by 12.8% in just 12 hours 48 minutes on 1024 v5p TPUs, while remaining robust to preemptions during training.

Automated chest X-ray report generation has recently benefited from reinforcement learning (RL) and large language models. However, RL training often suffers from instability or limited exploration due to fixed Kullback-Leibler (KL) regularization and a static reference policy that accumulates KL pressure over time. We propose Response-Weighted and Validation-Anchored Policy Optimization (REVA-PO), a RL framework that stabilizes long-term training via Response-Weighted Regularization (RER) and Validation-Anchored Policy Reset (VAPR). RER dynamically adjusts per-response KL weights based on advantage and reference-policy entropy, relaxing constraints for high-quality responses while tightening them for low-quality ones. Complementarily, VAPR periodically synchronizes the reference and current policies to the best validation checkpoint, resetting accumulated regularization pressure to expand the viable exploration space. To ensure a robust starting point, we employ a three-stage pipeline consisting of warm-up training, classifier-guided supervised fine-tuning, and RL. Extensive evaluations on MIMIC-CXR and IU-Xray demonstrate that REVA-PO sets new state-of-the-art benchmarks in both linguistic quality and clinical accuracy. Notably, BLEU-4 improves by 5.1% on MIMIC-CXR and 3.6% on IU-Xray, while CheXpert F1 and RadGraph F1 scores increase by 4.5% and 12.8%, respectively, over prior leading methods. The code is publicly available at https://github.com/LiGuo12/REVA_PO/.

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.

In Reinforcement Learning (RL), regularization has emerged as a popular tool both in theory and practice, typically based either on an entropy bonus or a Kullback-Leibler divergence that constrains successive policies. In practice, these approaches have been shown to improve exploration, robustness and stability, giving rise to popular Deep RL algorithms such as SAC and TRPO. Policy Mirror Descent (PMD) is a theoretical framework that solves this general regularized policy optimization problem, however the closed-form solution involves the sum of all past Q-functions, which is intractable in practice. We propose and analyze PMD-like algorithms that only keep the last M Q-functions in memory, and show that for finite and large enough M, a convergent algorithm can be derived, introducing no error in the policy update, unlike prior deep RL PMD implementations. StaQ, the resulting algorithm, enjoys strong theoretical guarantees and is competitive with deep RL baselines, while exhibiting less performance oscillation, paving the way for fully stable deep RL algorithms and providing a testbed for experimentation with Policy Mirror Descent.

Self-distillation has emerged as a powerful framework for post-training LLMs, where a teacher conditioned on extra information guides a student without it, both from the same model. While this guidance is useful when the student has failed, on successful rollouts, the same mechanism instead overwrites the student's choices and suppresses it's own reasoning. Therefore, we propose reading the original self-distillation signal in reverse: when the student succeeds along a path the teacher would not have predicted, these tokens reflect its self-driven reasoning. Building on this, we propose RLRT (RLVR with Reversed Teacher), which augments GRPO by reinforcing these tokens on correct rollouts. We interpret this as a new form of exploration in RLVR: not uniform diversity, but valuable exploration grounded in the student's own success. Across base, instruction-tuned, and thinking-tuned Qwen3 checkpoints, RLRT substantially outperforms self-distillation and exploration-based baselines, establishing information asymmetry as a new, principled design axis for RLVR.

R1-style Reinforcement Learning (RL) significantly enhances Large Language Models' reasoning capabilities, yet the mechanism behind rule-based RL remains unclear. We found that small-scale SFT has significant influence on RL but shows poor efficiency. To explain our observations, we propose an analytical framework and compare the efficiency of SFT and RL by measuring sample effect. Hypothetical analysis show that SFT efficiency is limited by training data. Guided by our analysis, we propose Re-distillation, a technique that fine-tunes pretrain model through small-scale distillation from the RL-trained policy. Experiments on Knight & Knave and MATH datasets demonstrate re-distillation's surprising efficiency: re-distilled models match RL performance with far fewer samples and less computation. Empirical verification shows that sample effect is a good indicator of performance improvements. As a result, on K&K dataset, our re-distilled Qwen2.5-1.5B model surpasses DeepSeek-V3-0324 with only 1K SFT samples. On MATH, Qwen2.5-1.5B fine-tuned with re-distilled 500 samples matches its instruct-tuned variant without RL. Our work explains several interesting phenomena in R1-style RL, shedding light on the mechanisms behind its empirical success. Code is available at: https://github.com/on1262/deep-reasoning

Iterative generative modeling techniques, such as flow matching, provide powerful tools to model complex behaviors for effective offline reinforcement learning (RL). In this work, we propose a new off-policy RL algorithm that trains a flow policy based on prior data. Our idea starts from the "expanded" Markov decision process (MDP) framework, which treats individual flow refinement steps as separate actions in an MDP. To enable off-policy RL within this framework, we apply two techniques: we generate virtual on-policy trajectories (by "reversing" flows) to make this framework compatible with prior data, and we apply a bias-and-variance reduction technique to mitigate the curse of horizon in off-policy RL. We call the resulting algorithm Reversal Q-learning (RQL). RQL has several advantages over previous flow-based RL methods: it does not suffer from backpropagation through time, makes better use of the learned value function, and directly trains the full, expressive flow policy. Through our experiments on 50 challenging simulated robotic tasks, we show that RQL leads to the best average offline RL performance compared to state-of-the-art flow-based offline RL algorithms.

Probabilistic dynamics model ensemble is widely used in existing model-based reinforcement learning methods as it outperforms a single dynamics model in both asymptotic performance and sample efficiency. In this paper, we provide both practical and theoretical insights on the empirical success of the probabilistic dynamics model ensemble through the lens of Lipschitz continuity. We find that, for a value function, the stronger the Lipschitz condition is, the smaller the gap between the true dynamics- and learned dynamics-induced Bellman operators is, thus enabling the converged value function to be closer to the optimal value function. Hence, we hypothesize that the key functionality of the probabilistic dynamics model ensemble is to regularize the Lipschitz condition of the value function using generated samples. To test this hypothesis, we devise two practical robust training mechanisms through computing the adversarial noise and regularizing the value network's spectral norm to directly regularize the Lipschitz condition of the value functions. Empirical results show that combined with our mechanisms, model-based RL algorithms with a single dynamics model outperform those with an ensemble of probabilistic dynamics models. These findings not only support the theoretical insight, but also provide a practical solution for developing computationally efficient model-based RL algorithms.

Despite recent progress in large-scale reinforcement learning (RL) for reasoning, the training recipe for building high-performing reasoning models remains elusive. Key implementation details of frontier models, such as DeepSeek-R1, including data curation strategies and RL training recipe, are often omitted. Moreover, recent research indicates distillation remains more effective than RL for smaller models. In this work, we demonstrate that large-scale RL can significantly enhance the reasoning capabilities of strong, small- and mid-sized models, achieving results that surpass those of state-of-the-art distillation-based models. We systematically study the RL training process through extensive ablations and propose a simple yet effective approach: first training on math-only prompts, then on code-only prompts. Notably, we find that math-only RL not only significantly enhances the performance of strong distilled models on math benchmarks (e.g., +14.6% / +17.2% on AIME 2025 for the 7B / 14B models), but also code reasoning tasks (e.g., +6.8% / +5.8% on LiveCodeBench for the 7B / 14B models). In addition, extended code-only RL iterations further improve performance on code benchmarks with minimal or no degradation in math results. We develop a robust data curation pipeline to collect challenging prompts with high-quality, verifiable answers and test cases to enable verification-based RL across both domains. Finally, we identify key experimental insights, including curriculum learning with progressively increasing response lengths and the stabilizing effect of on-policy parameter updates. We find that RL not only elicits the foundational reasoning capabilities acquired during pretraining and supervised fine-tuning (e.g., distillation), but also pushes the limits of the model's reasoning ability, enabling it to solve problems that were previously unsolvable.

We propose a method for meta-learning reinforcement learning algorithms by searching over the space of computational graphs which compute the loss function for a value-based model-free RL agent to optimize. The learned algorithms are domain-agnostic and can generalize to new environments not seen during training. Our method can both learn from scratch and bootstrap off known existing algorithms, like DQN, enabling interpretable modifications which improve performance. Learning from scratch on simple classical control and gridworld tasks, our method rediscovers the temporal-difference (TD) algorithm. Bootstrapped from DQN, we highlight two learned algorithms which obtain good generalization performance over other classical control tasks, gridworld type tasks, and Atari games. The analysis of the learned algorithm behavior shows resemblance to recently proposed RL algorithms that address overestimation in value-based methods.

Outcome-reward reinforcement learning (RL) is a common and increasingly significant way to refine the step-by-step reasoning of multimodal large language models (MLLMs). In the multiple-choice setting - a dominant format for multimodal reasoning benchmarks - the paradigm faces a significant yet often overlooked obstacle: unfaithful trajectories that guess the correct option after a faulty chain of thought receive the same reward as genuine reasoning, which is a flaw that cannot be ignored. We propose Self-Consistency Sampling (SCS) to correct this issue. For each question, SCS (i) introduces small visual perturbations and (ii) performs repeated truncation and resampling of an initial trajectory; agreement among the resulting trajectories yields a differentiable consistency score that down-weights unreliable traces during policy updates. Based on Qwen2.5-VL-7B-Instruct, plugging SCS into RLOO, GRPO, and REINFORCE++ series improves accuracy by up to 7.7 percentage points on six multimodal benchmarks with negligible extra computation. SCS also yields notable gains on both Qwen2.5-VL-3B-Instruct and InternVL3-8B, offering a simple, general remedy for outcome-reward RL in MLLMs.

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.

Reinforcement learning with verifiable rewards (RLVR) is a powerful recipe for improving language-model reasoning, but it is expensive to repeat on every new strong model because the target model must generate many rollouts during training. As models scale, post-training itself becomes a bottleneck. We study a weak-to-strong alternative: run RL on a smaller model where rollouts are cheaper, then reuse what that RL run learned to improve a stronger target model. Directly distilling the post-RL weak teacher is not enough, because the teacher's final policy mixes useful RL gains with the limitations of the smaller model. We propose Direct On-Policy Distillation (Direct-OPD), which transfers the teacher's RL-induced policy shift instead. Direct-OPD compares the post-RL teacher with its own pre-RL reference and treats their log-ratio as a dense implicit reward for the student. In plain terms, the checkpoint pair tells us which actions RL made the weak model more or less likely to take, and Direct-OPD applies that signal on the stronger student's own on-policy states. This directly reuses the weak model's RL supervision signal without running sparse-reward RL on the target model. Empirically, Direct-OPD consistently leverages weaker teachers to improve stronger target models; notably, it boosts Qwen3-1.7B from 48.3% to 58.3% on AIME 2024 in just 4 hours on 8 A100 GPUs. It outperforms step-matched direct RL and enables the sequential composition of multiple policy shifts. Our results show that RL outcomes can be reused across model scales as implicit reward signals, not merely as final models to imitate.

Large language models (LLMs) excel at reasoning, yet post-training remains critical for aligning their behavior with task goals. Existing reinforcement learning (RL) methods often depend on costly human annotations or external reward models. We propose Reinforcement Learning via Self-Confidence (RLSC), which uses the model's own confidence as reward signals-eliminating the need for labels, preference models, or reward engineering. Applied to Qwen2.5-Math-7B with only 16 samples per question and 10 or 20 training steps, RLSC improves accuracy by +13.4% on AIME2024, +21.2% on MATH500, +21.7% on Minerva Math, +20.8% on Olympiadbench, and +9.7% on AMC23. RLSC provides a simple, scalable post-training method for inference models, requiring only a small number of samples and unlabelled supervision.

Offline reinforcement learning (RL) provides a promising direction to exploit massive amount of offline data for complex decision-making tasks. Due to the distribution shift issue, current offline RL algorithms are generally designed to be conservative in value estimation and action selection. However, such conservatism can impair the robustness of learned policies when encountering observation deviation under realistic conditions, such as sensor errors and adversarial attacks. To trade off robustness and conservatism, we propose Robust Offline Reinforcement Learning (RORL) with a novel conservative smoothing technique. In RORL, we explicitly introduce regularization on the policy and the value function for states near the dataset, as well as additional conservative value estimation on these states. Theoretically, we show RORL enjoys a tighter suboptimality bound than recent theoretical results in linear MDPs. We demonstrate that RORL can achieve state-of-the-art performance on the general offline RL benchmark and is considerably robust to adversarial observation perturbations.

Recent improvements in conditional generative modeling have made it possible to generate high-quality images from language descriptions alone. We investigate whether these methods can directly address the problem of sequential decision-making. We view decision-making not through the lens of reinforcement learning (RL), but rather through conditional generative modeling. To our surprise, we find that our formulation leads to policies that can outperform existing offline RL approaches across standard benchmarks. By modeling a policy as a return-conditional diffusion model, we illustrate how we may circumvent the need for dynamic programming and subsequently eliminate many of the complexities that come with traditional offline RL. We further demonstrate the advantages of modeling policies as conditional diffusion models by considering two other conditioning variables: constraints and skills. Conditioning on a single constraint or skill during training leads to behaviors at test-time that can satisfy several constraints together or demonstrate a composition of skills. Our results illustrate that conditional generative modeling is a powerful tool for decision-making.

With respect to improving the reasoning accuracy of LLMs, the representative reinforcement learning (RL) method GRPO faces failure due to insignificant reward variance, while verification methods based on process reward models (PRMs) suffer from difficulties with training data acquisition and verification effectiveness. To tackle these problems, this paper introduces ReST-RL, a unified LLM RL paradigm that significantly improves LLM's code reasoning ability by combining an improved GRPO algorithm with a meticulously designed test time decoding method assisted by a value model (VM). As the first stage of policy reinforcement, ReST-GRPO adopts an optimized ReST algorithm to filter and assemble high-value training data, increasing the reward variance of GRPO sampling, thus improving the effectiveness and efficiency of training. After the basic reasoning ability of LLM policy has been improved, we further propose a test time decoding optimization method called VM-MCTS. Through Monte-Carlo Tree Search (MCTS), we collect accurate value targets with no annotation required, on which VM training is based. When decoding, the VM is deployed by an adapted MCTS algorithm to provide precise process signals as well as verification scores, assisting the LLM policy to achieve high reasoning accuracy. We validate the effectiveness of the proposed RL paradigm through extensive experiments on coding problems. Upon comparison, our approach significantly outperforms other reinforcement training baselines (e.g., naive GRPO and ReST-DPO), as well as decoding and verification baselines (e.g., PRM-BoN and ORM-MCTS) on well-known coding benchmarks of various levels (e.g., APPS, BigCodeBench, and HumanEval), indicating its power to strengthen the reasoning ability of LLM policies. Codes for our project can be found at https://github.com/THUDM/ReST-RL.